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--- abstract: 'In frequency-selective channels linear receivers enjoy significantly-reduced complexity compared with maximum likelihood receivers at the cost of performance degradation which can be in the form of a loss of the inherent frequency diversity order or reduced coding gain. This paper demonstrates that the minimum mean-square error symbol-by-symbol linear equalizer incurs no diversity loss compared to the maximum likelihood receivers. In particular, for a channel with memory $\nu$, it achieves the full diversity order of ($\nu+1$) while the zero-forcing symbol-by-symbol linear equalizer always achieves a diversity order of one.' author: - 'Ali Tajer[^1]Aria Nosratinia [^2] Naofal Al-Dhahir' bibliography: - 'IEEEabrv.bib' - 'IIR\_LE.bib' title: 'Diversity Analysis of Symbol-by-Symbol Linear Equalizers' --- Introduction {#sec:intro} ============ In broadband wireless communication systems, the coherence bandwidth of the fading channel is significantly less than the transmission bandwidth. This results in inter-symbol interference (ISI) and at the same time provides frequency diversity that can be exploited at the receiver to enhance transmission reliability [@Proakis:book]. It is well-known that for Rayleigh [*flat*]{}-fading channels, the error rate decays only linearly with signal-to-noise ratio ($\snr$) [@Proakis:book]. For frequency-selective channels, however, proper exploitation of the available frequency diversity forces the error probability to decay at a possibly higher rate and, therefore, can potentially achieve higher diversity gains, depending on the detection scheme employed at the receiver. While maximum likelihood sequence detection (MLSD) [@forney:ML] achieves optimum performance over ISI channels, its complexity (as measured by the number of MLSD trellis states) grows *exponentially* with the spectral efficiency and the channel memory. As a low-complexity alternative, filtering-based symbol-by-symbol equalizers (both linear and decision feedback) have been widely used over the past four decades (see [@qureshi:adaptive] and [@vitetta] for excellent tutorials). Despite their long history and successful commercial deployment, the performance of symbol-by-symbol linear equalizers over wireless fading channels is not fully characterized. More specifically, it is not known whether their observed sub-optimum performance is due to their inability to fully exploit the channel’s frequency diversity or due to a degraded performance in combating the residual inter-symbol interference. Therefore, it is of paramount importance to investigate the frequency diversity order achieved by linear equalizers, which is the subject of this paper. Our analysis shows that while single-carrier infinite-length symbol-by-symbol minimum mean-square error (MMSE) linear equalization achieves full frequency diversity, zero-forcing (ZF) linear equalizers cannot exploit the frequency diversity provided by frequency-selective channels. A preliminary version of the results of this paper on the MMSE linear equalization has partially appeared in \[5\] and the proofs available in [@ali:ISIT07_1] are skipped and referred to wherever necessary. The current paper provides two key contributions beyond \[5\]. First, the diversity analysis of ZF equalizers is added. Second, the MMSE analysis in \[5\] lacked a critical step that was not rigorously complete; the missing parts that have key role in analyzing the diversity order are provided in this paper. System Descriptions {#sec:descriptions} =================== Transmission Model {#sec:transmission} ------------------ Consider a quasi-static ISI wireless fading channel with memory length $\nu$ and channel impulse response (CIR) denoted by $\bh=[h_0,\dots,h_{\nu}]$. Without loss of generality, we restrict our analyses to CIR realizations with $h_0\neq 0$. The output of the channel at time $k$ is given by $$\label{eq:model_time} y_k=\sum_{i=0}^{\nu}h_ix_{k-i}+n_k\ ,$$ where $x_k$ is the input to the channel at time $k$ satisfying the power constraint $\mathbb{E}[|x_k|^2]\leq P_0$ and $n_k$ is the additive white Gaussian noise term distributed as $\mathcal{N}_\mathbb{C}(0,N_0)$[^3]. The CIR coefficients $\{h_i\}_{i=0}^\nu$ are distributed independently with $h_i$ being distributed as $\mathcal{N}_\mathbb{C}(0,\lambda_i)$. Defining the $D$-transform of the input sequence $\{x_k\}$ as $X(D)=\sum_k x_kD^k$, and similarly defining $Y(D), H(D)$, and $Z(D)$, the baseband input-output model can be cast in the $D$-domain as $Y(D)=H(D)\cdot X(D)+Z(D)$. The superscript $*$ denotes complex conjugate and we use the shorthand $D^{-*}$ for $(D^{-1})^*$. We define $\snr\dff\frac{P_0}{N_0}$ and say that the functions $f(\snr)$ and $g(\snr)$ are *exponentially equal*, indicated by $f(\snr)\doteq g(\snr)$, when $$\label{eq:exp} \lim_{\snr\rightarrow\infty}\frac{\log f(\snr)}{\log \snr}=\lim_{\snr\rightarrow\infty}\frac{\log g(\snr)}{\log \snr}\ .$$ The operators $\dotlt$ and $\dotgt$ are defined in a similar fashion. Furthermore, we say that the *exponential order* of $f(\snr)$ is $d$ if $f(\snr)\doteq \snr^d$. Linear Equalization {#sec:equalization} ------------------- The zero-forcing (ZF) linear equalizers are designed to produce an ISI-free sequence of symbols and ignore the resulting noise enhancement. By taking into account the [*combined*]{} effects of the ISI channel and its corresponding matched-filter, the ZF linear equalizer in the $D$-domain is given by [@cioffi Equation (3.87)] $$\label{eq:zf_eq} W_{\rm zf}(D)= \frac{\|\bh\|}{H(D)H^*(D^{-*})}\ ,$$ where the $\|\bh\|$ is the $\ell_2$-norm of $\bh$, i.e., $\|\bh\|^2=\sum_{i=0}^{\nu}|h_i|^2$. The variance of the noise seen at the output of the ZF equalizer is the key factor in the performance of the equalizer and is given by $$\label{eq:zf_var} \sigma^2_{\rm zf}\dff\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{N_0}{|H(e^{-ju})|^2}\;du\ .$$ Therefore, the decision-point signal-to-noise ratio for any CIR realization $\bh$ and $\snr=\frac{P_0}{N_0}$ is $$\label{eq:zf_snr} {\gamma_{\rm zf}(\snr,\bh)}\dff\snr \bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{|H(e^{-ju})|^2}\;du\bigg]^{-1}.$$ MMSE linear equalizers are designed to strike a balance between ISI reduction and noise enhancement through minimizing the combined residual ISI and noise level. Given the combined effect of the ISI channel and its corresponding matched-filter, the MMSE linear equalizer in the $D$-domain is [@cioffi Equation (3.148)] $$\begin{aligned} \label{eq:mmse_eq} W_{\rm mmse}(D)= \frac{\|\bh\|}{H(D)H^*(D^{-*})+\snr^{-1}}\ .\end{aligned}$$ The variance of the residual ISI and the noise variance as seen at the output of the equalizer is $$\label{eq:mmse_var} \sigma^2_{\rm mmse}\dff\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{N_0}{|H(e^{-ju})|^2+\snr^{-1}}\;du\ .$$ Hence, the *unbiased*[^4] decision-point signal-to-noise ratio at for any CIR realization $\bh$ and $\snr$ is $$\begin{aligned} \label{eq:mmse_snr}{\gamma_{\rm mmse}(\snr,\bh)}\dff\bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\snr|H(e^{-ju})|^2+1}\; du\bigg]^{-1}-1\ .\end{aligned}$$ Diversity Gain {#sec:diversity} -------------- For a transmitter sending information bits at spectral efficiency $R$ bits/sec/Hz, the system is said to be in *outage* if the ISI channel is faded such that it cannot sustain an arbitrarily reliable communication at the intended communication spectral efficiency $R$, or equivalently, the mutual information $I(x_k,\tilde y_k)$ falls below the target spectral efficiency $R$, where $\tilde y_k$ denotes the equalizer output. The probability of such outage for the signal-to-noise ratio $\gamma(\snr,\bh)$ is $$\label{eq:out}{P_{\rm out}(R,\snr)}\dff P_{\bh}\bigg(\log\Big[1+\gamma(\snr,\bh)\Big]<R\bigg)\ ,$$ where the probability is taken over the ensemble of all CIR realizations $\bh$. The outage probability at high transmission powers ($\snr\rightarrow\infty$) is closely related to the *average pairwise error probability*, denoted by ${P_{\rm err}(R,\snr)}$, which is the probability that a transmitted codeword $\bc_i$ is erroneously detected in favor of another codeword $\bc_j$, $j\neq i$, i.e., $$\label{eq:perr} {P_{\rm err}(R,\snr)}\dff \bbe_{\bh}\bigg[ P\Big(\bc_i\rightarrow\bc_j\med {\bh}\Big)\bigg]\ .$$ When deploying channel coding with arbitrarily long code-length, the outage and error probabilities decay at the same rate with increasing $\snr$ and have the same exponential order [@zheng:IT03] and therefore $$\label{eq:equality} {P_{\rm out}(R,\snr)}\doteq{P_{\rm err}(R,\snr)}\ .$$ This is intuitively justified by noting that in high $\snr$ regimes, the effect of channel noise is diminishing and the dominant source of erroneous detection is channel fading which, as mentioned above, is also the source of outage events. As a result, in our setup, diversity order which is the negative of the exponential order of the average pairwise error probability ${P_{\rm err}(R,\snr)}$ is computed as $$\label{eq:diversity} d=-\lim_{\snr\rightarrow\infty}\frac{\log {P_{\rm out}(R,\snr)}}{\log\snr}\ .$$ Diversity Order of MMSE Linear Equalization {#sec:mmse} =========================================== The main result of this paper for the MMSE linear equalizers is given in the following theorem. \[th:mmse\] For an ISI channel with channel memory length $\nu\geq 1$, and symbol-by-symbol MMSE linear equalization we have $$P_{\rm err}^{\rm mmse}(R,\snr)\doteq\snr^{-(\nu+1)}.$$ The sketch of the proof is as follows. First, we find a lower bound on the unbiased decision-point signal-to-noise ratio ($\snr$) and use this lower bound to show that for small enough spectral efficiencies a full diversity order of $(\nu+1)$ is achievable. The proof of the diversity gain for low spectral efficiencies is offered in Section \[sec:low\]. In the second step, we show that increasing the spectral efficiency to any arbitrary level does not incur a diversity loss, concluding that MMSE linear equalization is capable of collecting the full frequency diversity order of ISI channels. Such generalization of the results presented in Section \[sec:low\] to arbitrary spectral efficiencies is analyzed in Section \[sec:full\]. Full Diversity for Low Spectral Efficiencies {#sec:low} -------------------------------------------- We start by showing that for arbitrarily small data transmission spectral efficiencies, $R$, full diversity is achievable. Corresponding to each CIR realization $\bh$, we define the function $f(\bh,u)\dff|H(e^{-ju})|^2-\|\bh\|^2$ for which after some simple manipulations we have $$\begin{aligned} \label{eq:f} f(\bh,u)&= \sum_{k=-\nu}^\nu c_k\; e^{jku}\ ,\;\;\mbox{where}\;\; c_0=0,\;\; c_{-k}=c^*_k, \;\; c_k=\sum_{m=0}^{\nu-k}h_mh^*_{m+k}\; \;\; \mbox{for} \;\;\;\; k\in\{1,\dots, \nu\}\ .\end{aligned}$$ Therefore, $f(\bh,u)$ is a trigonometric polynomial of degree $\nu$ that is periodic with period $2\pi$ and in the open interval $[-\pi,\pi]$ has at most $2\nu$ roots [@powell:book]. Corresponding to the CIR realization $\bh$ we define the set $${\cal D}(\bh)\dff\{u\in[-\pi,\pi]\;:\; f(\bh,u)>0\}\ ,$$ and use the convention $|{\cal D}(\bh)|$ to denote the measure of ${\cal D}(\bh)$, i.e., the aggregate lengths of the intervals over which $f(\bh,u)$ is strictly positive. In the following lemma, we obtain a lower bound on $|{\cal D}(\bh)|$ which is instrumental in finding a lower bound on ${\gamma_{\rm mmse}(\snr,\bh)}$. \[lemma:interval\] There exists a real number $C>0$ such that for all non-zero CIR realizations $\bh$, i.e. $\forall\bh\neq\boldsymbol 0$, we have that $|{\cal D}(\bh)|\geq C \left(2(2\nu+1)^3\right)^{-\frac{1}{2}}$. According to we immediately have $\int_{-\pi}^{\pi}f(\bh,u)\;du=0$. By invoking the definition of ${\cal D}(\bh)$ and noting that $[-\pi,\pi]\backslash{\cal D}(\bh)$ includes the values of $u$ for which $f(\bh,u)$ is negative, we have $$\label{eq:f_int} \int_{{\cal D}(\bh)}f(\bh,u)\; du=-\int_{[-\pi,\pi]\backslash{\cal D}(\bh)}f(\bh,u)\; du\quad\Rightarrow\quad \int_{-\pi}^{\pi}|f(\bh,u)|\;du=2\int_{{\cal D}(\bh)}f(\bh,u)\; du\ .$$ Also by noting that $f(\bh,u)=|H(e^{-ju})|^2-\|\bh\|^2$, $f(\bh,u)$ has clearly a real value for any $u$. Moreover, by invoking from the Cauchy-Schwartz inequality we obtain $$\label{eq:f_CS} f(\bh,u)\leq |f(\bh,u)|\leq\bigg(\sum_{k=-\nu}^\nu |c_k|^2\bigg)^{\frac{1}{2}}\bigg(\sum_{k=-\nu}^\nu |e^{jku}|^2\bigg)^{\frac{1}{2}} = \bigg(2(2\nu+1)\sum_{k=1}^\nu |c_k|^2\bigg)^{\frac{1}{2}}\ .$$ Equations and together establish that $$\label{eq:f_int_bound} |{\cal D}(\bh)|\geq \frac{1}{2}\bigg(2(2\nu+1)\sum_{k=1}^\nu |c_k|^2\bigg)^{-\frac{1}{2}}\; \int_{-\pi}^{\pi}|f(\bh,u)|\;du\ .$$ Next we strive to find a lower bound on $\int_{-\pi}^{\pi}|f(\bh,u)|\;du$, which according to is equivalent to finding a lower bound on the $\ell_1$ norm of a sum of exponential terms. Obtaining lower bounds on the $\ell_1$ norm of exponential sums has a rich literature in the mathematical analysis and we use a relevant result in this literature that is related to Hardy’s inequality [@mcgehse Theorem 2]. *[@mcgehse Theorem 2]* There is a real number $C>0$ such that for any given sequence of increasing integers $\{n_k\}$, and complex numbers $\{d_k\}$, and for any $N\in\mathbb{N}$ we have $$\label{eq:Hardy} \int_{-\pi}^{\pi}\bigg|\sum_{k=1}^Nd_k\;e^{jn_ku}\bigg|\;du\geq C\sum_{k=1}^N\frac{|d_k|}{k}\ .$$ By setting $N=2\nu+1$ and $d_k=c_{k-(\nu+1)}$ and $n_k=k-(\nu+1)$ for $k\in\{1,\dots,2\nu+1\}$ from it is concluded that there exists $C>0$ that for each set of $\{c_{-\nu},\dots, c_\nu\}$ we have $$\begin{aligned} \label{eq:Hardy2} \int_{-\pi}^{\pi}|f(\bh,u)|\;du \geq C\sum_{k=1}^{2\nu+1}\frac{|c_{k-(\nu+1)}|}{k}\geq \frac{C}{2\nu+1}\sum_{k=1}^{2\nu+1}|c_{k-(\nu+1)}|= \frac{2C}{2\nu+1}\sum_{k=1}^{\nu}|c_{k}|\ ,\end{aligned}$$ where the last equality holds by noting that $c_{-k}=c^*_k$ and $c_0=0$. Combining and provides $$\label{eq:f_int_bound2} |{\cal D}(\bh)|\geq C\left(2(2\nu+1)^3\right)^{-\frac{1}{2}} \underset{\geq 1}{\underbrace{\frac{\sum_{k=1}^\nu |c_k|}{\sqrt{\sum_{k=1}^\nu |c_k|^2}}}}\geq C\left(2(2\nu+1)^3\right)^{-\frac{1}{2}}\ ,$$ which concludes the proof. Now by using Lemma \[lemma:interval\] for any CIR realization $\bh$ and $\snr$ we find a lower bound on ${\gamma_{\rm mmse}(\snr,\bh)}$ that depends on $\bh$ through $\|\bh\|$ only. By defining ${\cal D}^c(\bh)=[-\pi,\pi]\backslash{\cal D}(\bh)$ we have $$\begin{aligned} \nonumber 1+{\gamma_{\rm mmse}(\snr,\bh)}&\overset{\eqref{eq:mmse_snr}}{=} \bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\snr|H(e^{ju})|^2+1}\;du\bigg]^{-1} = \bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\snr(f(\bh,u)+\|\bh\|^2)+1}\;du\bigg]^{-1} \\ \nonumber &= \bigg[\frac{1}{2\pi}\int_{{\cal D}(\bh)}\frac{1}{\snr(\underset{> 0}{\underbrace{f(\bh,u)}}+\|\bh\|^2)+1}\;du+ \frac{1}{2\pi} \int_{{\cal D}^c(\bh)}\frac{1}{\snr\underset{\geq 0}{\underbrace{|H(e^{ju})|^2}}+1}\;du\bigg]^{-1}\\ \nonumber &\geq \bigg[\frac{1}{2\pi}\int_{{\cal D}(\bh)}\frac{1}{\snr\|\bh\|^2+1}\;du+ \frac{1}{2\pi} \int_{{\cal D}^c(\bh)}1\;du\bigg]^{-1}\\ \nonumber &= \bigg[\frac{|{\cal D}(\bh)|}{2\pi}\cdot\frac{1}{ \snr\|\bh\|^2+1} + \bigg(1-\frac{|{\cal D}(\bh)|}{2\pi}\bigg)\bigg]^{-1}\\ \nonumber & = \bigg[1-\frac{|{\cal D}(\bh)|}{2\pi}\bigg(1-\frac{1}{\snr\|\bh\|^2+1}\bigg)\bigg]^{-1}\\ \label{eq:mmse_snr_lb} & \overset{\eqref{eq:f_int_bound2}}{\geq} \bigg[1-\frac{C\left(2(2\nu+1)^3\right)^{-\frac{1}{2}}}{2\pi}\bigg(1-\frac{1}{\snr\|\bh\|^2+1}\bigg)\bigg]^{-1} \ .\end{aligned}$$ By defining $C'\dff \frac{C\left(2(2\nu+1)^3\right)^{-\frac{1}{2}}}{2\pi}$, for the outage probability corresponding to the target spectral efficiency $R$ we have $$\begin{aligned} \nonumber {P^{\rm mmse}_{\rm out}(R,\snr)}&\overset{\eqref{eq:out}}{=} P_{\bh} \bigg(1+{\gamma_{\rm mmse}(\snr,\bh)}<2^R\bigg)\overset{\eqref{eq:mmse_snr_lb}}{\leq} P_{\bh}\bigg\{1-C'\bigg(1-\frac{1}{\snr\|\bh\|^2+1}\bigg)>2^{-R}\bigg\}\\ \label{eq:mmse_out_lb1} &= P_{\bh}\bigg\{1-\frac{1-2^{-R}}{C'}<\frac{1}{\snr\|\bh\|^2+1}\bigg\}\ .\end{aligned}$$ If $$\label{eq:rate1} 1-\frac{1-2^{-R}}{C'}> 0\qquad\mbox{or equivalently}\qquad R<R_{\max}\dff\log_2\left(\frac{1}{1-C'}\right)\ ,$$ then the probability term in can be restated as $$\label{eq:mmse_outlb2} P_{\bh}\bigg\{\snr\|\bh\|^2 < \frac{1-2^{-R}} {C'-(1-2^{-R})}\bigg\} = P_{\bh}\bigg\{\snr\|\bh\|^2 < \frac{2^{R}-1} {1-2^{R-R_{\max}}}\bigg\}\ .$$ Therefore, based on (\[eq:mmse\_out\_lb1\])-(\[eq:mmse\_outlb2\]) for all $0<R<R_{\max}$ we have $$\begin{aligned} \nonumber {P^{\rm mmse}_{\rm out}(R,\snr)}&\leq P_{\bh}\bigg\{\snr\|\bh\|^2<\frac{2^{R}-1} {1-2^{R-R_{\max}}}\bigg\} = P_{\bh}\bigg\{\snr\sum_{m=0}^{\nu}|h_m|^2<\frac{2^{R}-1} {1-2^{R-R_{\max}}}\bigg\}\\ \label{eq:mmse_out_lb3} &\leq \prod_{m=0}^{\nu}P_{\bh}\bigg\{|h_m|^2<\frac{2^{R}-1} {\snr(1-2^{R-R_{\max}})}\bigg\}\doteq \snr^{-(\nu+1)}\ .\end{aligned}$$ Therefore, for the spectral efficiencies $R\in(0, R_{\max})$ we have ${P^{\rm mmse}_{\rm out}(R,\snr)}\;\dotlt\;\snr^{-(\nu+1)}$, which in conjunction with (\[eq:equality\]) proves that $P_{\rm err}^{\rm mmse}\;\dotlt\;\snr^{-(\nu+1)}$, indicating that a diversity order of at least $(\nu+1)$ is achievable. On the other hand, since the diversity order cannot exceed the number of the CIR taps, the achievable diversity order is exactly $(\nu+1)$. Also note that the real number $C>0$ given in is a constant independent of the CIR realization $\bh$ and, therefore, $C'$ and, consequently, $R_{\max}$ are also independent of the CIR realization. This establishes the proof of Theorem \[th:mmse\] for the range of the spectral efficiencies $R\in(0, R_{\max})$, where $R_{\max}$ is fixed and defined in (\[eq:rate1\]). Full Diversity for All Rates {#sec:full} ---------------------------- We now extend the results previously found for $R<R_{\max}$ to all spectral efficiencies. \[lemma:linear\] For asymptotically large values of $\snr$, ${\gamma_{\rm mmse}(\snr,\bh)}$ varies linearly with $\snr$, i.e., $$\lim_{\snr \rightarrow \infty} \frac{\partial\;{\gamma_{\rm mmse}(\snr,\bh)}}{\partial\;\snr}=s(\bh),\;\;\; \mbox{where}\;\;\; s(\bh):\mathbb{R}^{\nu+1}\rightarrow\mathbb{R}\ .$$ See Appendix \[app:lemma:linear\]. \[lemma:limit\] For the continuous random variable $X$, variable $y\in\mathbb{R}$, constants $c_1, c_2\in\mathbb{R}$ and function $G(X,y)$ continuous in $y$, we have $$\lim_{y\rightarrow y_0}P_X\Big(c_1 \leq G(X,y) \leq c_2\Big)=P_X\Big(c_1 \leq \lim_{y\rightarrow y_0}G(X,y) \leq c_2\Big)\ .$$ Follows from Lebesgue’s Dominated Convergence theorem [@bartle:B1] and the same line of argument as in [@ali:ISIT07_1 Appendix C] Now, we show that if for some spectral efficiency $R^{\dag}$ the achievable diversity order is $d$, then for all spectral efficiencies [*up*]{} to $R^{\dag}+1$, the same diversity order is achievable. By induction, we conclude that the diversity order remains unchanged by changing the data spectral efficiency $R$. If for the spectral efficiency $R^{\dag}$, the negative of the exponential order of the outage probability is $d$, i.e., $$\label{eq:induction} P_{\bh}\bigg(\log\Big[1+{\gamma_{\rm mmse}(\snr,\bh)}\Big]<R^{\dag}\bigg)\doteq\snr^{-d},$$ then by applying the results of Lemmas \[lemma:linear\] and \[lemma:limit\] for the target spectral efficiency $R^{\dag}+1$ we get $$\begin{aligned} \nonumber {P^{\rm mmse}_{\rm out}(R,\snr)}&=P_{\bh}\bigg(\log\Big[1+{\gamma_{\rm mmse}(\snr,\bh)}\Big]<R^{\dag}+1\bigg)= P_{\bh}\left(1+{\gamma_{\rm mmse}(\snr,\bh)}<2^{R^\dag+1}\right)\\ \label{eq:induction_1} &\doteq P_{\bh}\left(\snr\;s(\bh)<2^{R^\dag+1}\right)= P_{\bh}\left(\Big(\frac{\snr}{2}\Big) s(\bh)<2^{R^\dag}\right)\\ \label{eq:induction_2} &\doteq P_{\bh}\left(1+\gamma_{\rm mmse}\Big(\frac{\snr}{2},\bh\Big)<2^{R^\dag}\right)\doteq P_{\bh}\bigg(\log\Big[1+\gamma_{\rm mmse}\Big(\frac{\snr}{2},\bh\Big)\Big]<R^{\dag}\bigg)\\ \label{eq:induction_3} &\doteq\Big(\frac{\snr}{2}\Big)^{-d}за\doteq\snr^{-d}\ .\end{aligned}$$ Equations (\[eq:induction\_1\]) and (\[eq:induction\_2\]) are derived as the immediate results of Lemmas \[lemma:linear\] and \[lemma:limit\] that enable interchanging the probability and the limit and also show that ${\gamma_{\rm mmse}(\snr,\bh)}\doteq \snr\cdot s(\bh)$. Equations (\[eq:induction\])-(\[eq:induction\_3\]) imply that the diversity orders achieved for the spectral efficiencies up to $R^\dag$ and the spectral efficiencies up to $R^\dag+1$ are the same. As a result, any arbitrary spectral efficiency exceeding $R_{\max}$ achieves the same spectral efficiency as the spectral efficiencies $R\in(0,R_{\max})$ and, therefore, for any arbitrary spectral efficiency $R$, full diversity is achievable via MMSE linear equalization which completes the proof. Figure \[fig:1\] depicts our simulation results for the pairwise error probabilities for two ISI channels with memory lengths $\nu=1$ and 2 and MMSE equalization. For each of these channels we consider signal transmission with spectral efficiencies $R=(1,2,3,4)$ bits/sec/Hz. The simulation results confirm that for a channel with two taps the achievable diversity order is two irrespective of the data spectral efficiency. Similarly, it is observed that for a three-tap channel the achievable diversity order is three. Diversity Order of ZF linear Equalization {#sec:zf} ========================================= In this section, we show that the diversity order achieved by zero-forcing linear equalization, unlike that achievable with MMSE equalization, is independent of the channel memory length and is always 1. \[lemma:zf\] For any arbitrary set of normal complex Gaussian random variables $\bmu\dff(\mu_1,\dots,\mu_m)$ (possibly correlated) and for any $B\in\mathbb{R}^+$ we have $$\label{eq:lemma:zf1} P_{\bmu}\bigg(\sum_{k=1}^m\frac{1}{\snr|\mu_k|^2}>B\bigg)\; \dotgt\; \snr^{-1}\ .$$ Define $W_k \dff -\frac{\log|\mu_k|^2}{\log\snr}$. Since $|\mu_k|^2$ has exponential distribution, it can be shown that for any $k$ the cumulative density function (CDF) at the asymptote of high values of $\snr$ satisfies [@azarian:IT05] $$\label{eq:W} 1-F_{W_k}(w)\doteq\snr^{-w}\ .$$ Thus, by substituting $|\mu_k|^2\doteq\snr^{1-W_k}$ based on (\[eq:W\]) we find that $$\begin{aligned} \label{eq:lemma:zf2} P_{\bmu}\bigg(\sum_{k=1}^m\frac{1}{\snr|\mu_k|^2}>B\bigg)&\doteq P_{\bW}\bigg(\sum_{k=1}^m\snr^{W_k-1}>B \bigg) \doteq P_{\bW}(\max_k W_k>1)\\ \label{eq:lemma:zf3} &\geq P_{W_k}(W_k>1)=1-F_{W_k}(1)\doteq \snr^{-1}\ .\end{aligned}$$ Equation (\[eq:lemma:zf2\]) holds as the term $\snr^{\max W_k-1}$ is the dominant term in the summation $\sum_{k=1}^m\snr^{W_k-1}$. Also, the transition from (\[eq:lemma:zf2\]) to (\[eq:lemma:zf3\]) is justified by noting that $\max_kW_k\geq W_k$ and the last step is derived by taking into account . \[th:zf\] The diversity order achieved by symbol-by-symbol ZF linear equalization is one, i.e., $$P_{\rm err}^{\rm zf}(R,\snr)\doteq \snr^{-1}$$ By recalling the decision-point signal-to-noise ratio of ZF equalization given in (\[eq:zf\_snr\]) we have $$\begin{aligned} \label{eq:zf1} {P^{\rm zf}_{\rm out}(R,\snr)}&=P_{\bh}\Big({\gamma_{\rm zf}(\snr,\bh)}<2^R-1\Big)=P_{\bh}\bigg\{\Big[\frac{1}{2\pi}\int_{\pi}^{\pi}\frac{1}{\snr|H(e^{-ju})|^2}\;du\Big]^{-1}<2^R-1\bigg\}\\ \label{eq:zf2} &=P_{\bh}\bigg\{\lim_{\Delta\rightarrow 0}\bigg[\sum_{k=0}^{\lfloor 2\pi/\Delta\rfloor}\frac{\Delta}{\snr|H(e^{-j(-\pi+k\Delta)})|^2}\bigg]^{-1}<\frac{2^R-1}{2\pi}\bigg\}\\ \label{eq:zf3} &=\lim_{\Delta\rightarrow 0}P_{\bh}\bigg\{\bigg[\sum_{k=0}^{\lfloor 2\pi/\Delta\rfloor}\frac{\Delta}{\snr |H(e^{-j(-\pi+k\Delta)})|^2}\bigg]^{-1}<\frac{2^R-1}{2\pi}\bigg\}\\ \label{eq:zf4}&=\lim_{\Delta\rightarrow 0}P_{\bh}\bigg\{\sum_{k=0}^{\lfloor 2\pi/\Delta\rfloor}\frac{\Delta}{\snr|H(e^{-j(-\pi+k\Delta)})|^2}>\frac{2\pi}{2^R-1}\bigg\} \; \dotgt \; \snr^{-1}\ .\end{aligned}$$ Equation (\[eq:zf2\]) is derived by using Riemann integration, and (\[eq:zf3\]) holds by using Lemma \[lemma:limit\] which allows for interchanging the limit and the probability. Equation (\[eq:zf4\]) holds by applying Lemma \[lemma:zf\] on $\mu_k=H(e^{-j(-\pi+k\Delta)})$ which can be readily verified to have Gaussian distribution. Therefore, the achievable diversity order is 1. Figure \[fig:2\] illustrates the pairwise error probability of two ISI channels with memory lengths $\nu=1$ and 2. The simulation results corroborate our analysis showing that the achievable diversity order is one, irrespective of the channel memory length or communication spectral efficiency. Conclusion {#sec:conclusion} ========== We showed that infinite-length symbol-by-symbol MMSE linear equalization can fully capture the underlying frequency diversity of the ISI channel. Specifically, the diversity order achieved is equal to that of MLSD and in the high-$\snr$ regime, the performance of MMSE linear equalization and MLSD do not differ in diversity gain and the origin of their performance discrepancy is their ability to control the residual inter-symbol interference. We also show that the diversity order achieved by symbol-by-symbol ZF linear equalizers is always one, regardless of channel memory length. Proof of Lemma \[lemma:linear\] {#app:lemma:linear} =============================== We define $g(\bh,u)\dff|H(e^{-ju})|^2$ which has a finite number of zero by following the same line as for $f(\bh,u)$ in the proof of Lemma [\[lemma:interval\]]{}. By using (\[eq:mmse\_snr\]) we get $$\begin{aligned} \nonumber\frac{\partial\;{\gamma_{\rm mmse}(\snr,\bh)}}{\partial\;\snr}&= \frac{\partial}{\partial\;\snr} \left(\bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\snr g(\bh,u)+1}\;du\bigg]^{-1}-1\right)\\ \label{eq:lemma:linear1} &=\bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{g(\bh,u)}{\left(\snr g(\bh,u)+1\right)^2}\;du\bigg]\cdot \bigg[\frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{1}{\snr g(\bh,u)+1}\;du\bigg]^{-2}\\ \label{eq:lemma:linear2} &=\bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{g(\bh,u)}{\left(\snr g(\bh,u)+1\right)^2}\;du\bigg]\cdot \bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{1}{\snr g(\bh,u)+1}\;du\bigg]^{-2},\end{aligned}$$ where (\[eq:lemma:linear2\]) was obtained by removing a finite-number of points from the integral in (\[eq:lemma:linear1\]). \[th:monotone\] *Monotone Convergence* [@bartle:B1 Theorem. 4.6]: if a function $F(u,v)$ defined on $U\times[a,b]\rightarrow \mathbb{R}$, is positive and monotonically increasing in $v$, and there exists an integrable function $\hat F(u)$, such that $\lim_{v\rightarrow\infty}F(u,v)=\hat F(u)$, then $$\label{eq:lemma:linear3} \underset{v\rightarrow\infty}\lim\int_U F(u,v)\;du=\int_U\underset{v\rightarrow\infty}\lim F(u,v )\;du=\int_U \hat{F}(u)\;du.$$ For further simplifying (\[eq:lemma:linear2\]), we define $F_1(u,\snr)$ and $F_2(u,\snr)$ over $\Big\{u\med u\in[-\pi,\pi], g(\bh,u)\neq 0\Big\}\times[1,+\infty]$ as $$\begin{aligned} F_1(u,\snr)&\dff&\frac{1}{g(\bh,u)}-\frac{1}{\snr^2 g(\bh,u)}+\frac{g(\bh,u)}{(\snr g(\bh,u)+1)^2},\\ \mbox{and}\;\;\;F_2(u,\snr)&\dff&\frac{1}{g(\bh,u)}-\frac{1}{\snr g(\bh,u)}+\frac{1}{\snr g(\bh,u)+1}.\end{aligned}$$ It can be readily verified that $F_i(u,\snr) > 0$ and $F_i(u,\snr)$ is increasing in $\snr$. Moreover, there exist $\hat F(u)$ such that $$\hat F(u)=\lim_{\snr\rightarrow\infty}F_1(u,\snr)=\lim_{\snr\rightarrow\infty}F_2(u,\snr)=\frac{1}{g(\bh,u)}.$$ Therefore, by exploiting the result of Theorem \[th:monotone\] we find $$\begin{aligned} &&\lim_{\snr\rightarrow\infty}\int\bigg[\frac{1}{g(\bh,u)}-\frac{1}{\snr^2 g(\bh,u)}+\frac{g(\bh,u)}{(\snr g(\bh,u)+1)^2}\bigg]du=\int\frac{1}{g(\bh,u)}\;du,\\ \mbox{and}&&\lim_{\snr\rightarrow\infty}\int\bigg[\frac{1}{g(\bh,u)}-\frac{1}{\snr g(\bh,u)}+\frac{1}{\snr g(\bh,u)+1}\bigg]du=\int\frac{1}{g(\bh,u)}\;du,\end{aligned}$$ or equivalently, $$\begin{aligned} \label{eq:lemma:linear4} &&\lim_{\snr\rightarrow\infty}\frac{1}{2\pi}\int\frac{g(\bh,u)\;du}{(\snr g(\bh,u)+1)^2}=\lim_{\snr\rightarrow\infty}\frac{1}{2\pi}\int\frac{\;du}{\snr^2 g(\bh,u)},\\ \label{eq:lemma:linear5} \mbox{and}&&\lim_{\snr\rightarrow\infty}\frac{1}{2\pi}\int\frac{\;du}{\snr g(\bh,u)+1}=\lim_{\snr\rightarrow\infty}\frac{1}{2\pi}\int\frac{\;du}{\snr g(\bh,u)}.\end{aligned}$$ By using the equalities in (\[eq:lemma:linear4\])-(\[eq:lemma:linear5\]) and proper replacement in (\[eq:lemma:linear2\]) we get $$\begin{aligned} \nonumber\lim_{\snr\to\infty} & \frac{\partial\;{\gamma_{\rm mmse}(\snr,\bh)}}{\partial\;\snr}\\ &=\lim_{\snr\to\infty}\bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{g(\bh,u)}{\left(\snr g(\bh,u)+1\right)^2}\;du\bigg]\cdot \bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{1}{\snr g(\bh,u)+1}\;du\bigg]^{-2}\\ \nonumber &=\lim_{\snr\to\infty}\bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{1}{\snr^2 g(\bh,u)}\;du\bigg]\cdot \bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{1}{\snr g(\bh,u)}\;du\bigg]^{-2}\\ \nonumber &= \bigg[\frac{1}{2\pi}\int_{g(\bh,u)\neq 0}\frac{1}{ g(\bh,u)}\;du\bigg]^{-1}=s(\bh),\end{aligned}$$ where $s(\bh)$ is independent of $\snr$ and thus the proof is completed. ![Average detection error probability in two-tap and three-tap ISI channels with MMSE linear equalization.[]{data-label="fig:1"}](mmse.eps "fig:"){width="4.5"}\ ![Average detection error probability in two-tap and three-tap ISI channels with ZF linear equalization[]{data-label="fig:2"}](zf.eps "fig:"){width="4.5"}\ [^1]: Electrical Engineering Department, Princeton University, Princeton, NJ 08544. [^2]: Electrical Engineering Department, University of Texas at Dallas, Richardson, TX 75083. [^3]: $\mathcal{N}_\mathbb{C}(a,b)$ denotes a complex Gaussian distribution with mean $a$ and variance $b$. [^4]: All MMSE equalizers are biased. Removing the bias decreases the decision-point signal-to-noise ratio by $1$ (in linear scale) but improves the error probability [@CDEF]. All the results provided in this paper are valid for biased receivers as well.

ArXiv

--- author: - | The IceCube Collaboration[^1]\ [*<http://icecube.wisc.edu/collaboration/authors/icrc19_icecube>*]{}\ E-mail: bibliography: - 'references.bib' title: 'Measurement of the high-energy all-flavor neutrino-nucleon cross section with IceCube' --- Introduction {#sec:intro} ============ Neutrinos above TeV energies that traverse through the Earth may interact before exiting [@Gandhi:1995tf]. At these energies neutrino-nucleon interactions proceed via deep-inelastic scattering (DIS), whereby the neutrino interacts with the constituent quarks within the nucleon. The DIS cross sections can be derived from parton distribution functions (PDF) which are in turn constrained experimentally [@CooperSarkar:2011pa] or by using a color dipole model of the nucleon and assuming that cross-sections increase at high energies as $\ln^2 s$ [@Arguelles:2015wba]. At energies above a PeV, more exotic effects beyond the Standard Model have been proposed that predict a neutrino cross section of up to at $E_\nu > \SI{e19}{eV}$ [@Jain:2000pu]. Thus far, measurements of the high-energy neutrino cross section have been performed using data from the IceCube Neutrino Observatory. One proposed experiment, the ForwArd Search ExpeRiment at the LHC (FASER), plans to measure the neutrino cross section at TeV energies [@Ariga:2019ufm]. The IceCube Neutrino Observatory is a cubic-kilometer neutrino detector installed in the ice at the geographic South Pole [@Aartsen:2016nxy], between depths of and , completed in 2010. Reconstruction of the direction, energy and flavor of the neutrinos relies on the optical detection of Cherenkov radiation emitted by charged particles produced in the interactions of neutrinos in the surrounding ice or the nearby bedrock. As the transmission probability through the Earth is dependent on the neutrino cross section, a change in the cross section affects the arrival flux of neutrinos at IceCube as a function of the energy and zenith angle. Recently, IceCube performed the first measurement of the high-energy neutrino-nucleon cross section using a sample of upgoing muon neutrinos [@Aartsen:2017kpd]. In this paper, we present a measurement of the neutrino-nucleon cross section using the high-energy starting events (HESE) sample with 7.5 years of data [@Schneider:2019icrc_hese]. By using events that start in the detector, the measurement is sensitive to both the northern and southern skies, as well as all three flavors of neutrinos, unlike [@Bustamante:2017xuy] which used only a single class of events in the six-year HESE sample. Analysis method {#sec:method} =============== Several improvements have been incorporated into the HESE-7.5 analysis chain, and are used in this measurement. These include better detector modeling, a three-topology classifier that corresponds to the three neutrino flavors [@Usner:2018qry], improved atmospheric neutrino background calculation [@Arguelles:2018awr], and a new likelihood treatment that accounts for statistical uncertainties [@Arguelles:2019izp]. The selection cuts have remained unchanged and require the total charge associated with the event to be above with the charge in the outer layer of the detector (veto region) to be below . This rejects almost all of the atmospheric muon background, as well as a fraction of atmospheric neutrinos from the southern sky that are accompanied by muons, as shown in the left panel of \[fig:qtotveto\]. There are a total of 102 events that pass the charge cuts. A histogram of their deposited energy and reconstructed cosine zenith angle is shown in the right panel of \[fig:qtotveto\]. For this analysis, only the 60 events with reconstructed energy above are used. A forward-folded likelihood is constructed using deposited energy and $\cos(\theta_z)$ distributions for tracks and cascades separately. For the two double cascades above the likelihood is constructed using a distribution of the cascade-length separation and deposited energy. Neutrino (top left) and antineutrino (top right) transmission probabilities are shown in \[fig:attenuation1d\] for three different variations of the DIS cross section given in [@CooperSarkar:2011pa] (CSMS). They are plotted for each flavor individually as a function of the neutrino energy, $E_\nu$, at $\theta_\nu=180^\circ$, assuming an initial surface flux with spectral index of $\gamma=2$. As the cross section is decreased, the transmission probability increases since neutrinos are less likely to interact on their way through the Earth. On the other hand, a higher cross section implies a higher chance of interaction and the transmission probability decreases. The reason there is a slight flavor-dependence is due to the fact that charged current (CC) ${\overset{\scriptscriptstyle(-)}{\nu}}_e$ and ${\overset{\scriptscriptstyle(-)}{\nu}}_\mu$ interactions produce charged particles that rapidly lose energy in matter, while a CC ${\overset{\scriptscriptstyle(-)}{\nu}}_\tau$ interaction produces a tau lepton, which can immediately decay to a slightly lower energy ${\overset{\scriptscriptstyle(-)}{\nu}}_\tau$. Neutral current (NC) interactions are also non-destructive, producing a secondary neutrino at a slightly lower energy than the parent [@Vincent:2017svp]. Furthermore, there is a dip in the $\bar{\nu}_e$ transmission probability due to the Glashow resonance (GR) [@Glashow:1960zz]. \ In this analysis, the neutrino-nucleon cross section is measured as a function of energy by dividing the CSMS cross section into four bins: , , , and . The overall normalization of the cross section in each bin is allowed to float with four scaling parameters $\bm{x}=(x_0, x_1, x_2, x_3)$, where the index goes from the lowest energy bin to the highest energy bin. We further assume that the ratio of the CC to NC cross section is fixed and that there is no additional flavor dependence. Thus, $\bm{x}$ is applied identically across all flavors and interaction channels on the CSMS prediction. In order to model the effect of varying the cross section on the arrival flux, we used [`nuSQuIDS`]{} [@Delgado:2014kpa]. This allows us to account properly for destructive CC interactions as well as for secondaries from NC interactions and tau-regeneration. The Earth density is set to the preliminary reference Earth model (PREM) [@Dziewonski:1981xy] and the GR cross section is kept fixed to the Standard Model prediction. We also include the nuisance parameters given in \[tab:nuisances\], for a single-power-law astrophysical flux, pion and kaon induced atmospheric neutrino flux by Honda et al., and BERSS prompt atmospheric neutrino flux [@Honda:2006qj; @Bhattacharya:2015jpa]. Parameter Constraint/Prior Range --------------------------------------------------- ------------------ --------------------- $\Phi_\texttt{astro}$ - $[0,\infty)$ $\gamma_\texttt{astro}$ $2.0\pm1.0$ $(-\infty,\infty)$ $\Phi_\texttt{conv}$ $1.0\pm0.4$ $[0, \infty)$ $\Phi_\texttt{prompt}$ $1.0\pm3.0$ $[0, \infty)$ $\pi/K$ $1.0\pm0.1$ $(-\infty, \infty)$ ${2\nu/\left(\nu+\bar{\nu}\right)}_\texttt{atmo}$ $1.0\pm0.1$ $[0,2]$ $\Delta\gamma_\texttt{CR}$ $-0.05\pm 0.05$ $(-\infty,\infty)$ $\Phi_\mu$ $1.0\pm 0.5$ $[0,\infty)$ : Central values and uncertainties on the nuisance parameters included in the fit. Truncated Gaussians are set to zero for all negative parameter values.[]{data-label="tab:nuisances"} As $\bm{x}$ is varied, Monte-Carlo (MC) events are reweighted by the ratio of $x_i \Phi(E_\nu, \theta_\nu, \bm{x})/\Phi(E_\nu, \theta_\nu, \bm{1})$, where $\Phi$ is the arrival flux as calculated by [`nuSQuIDS`]{}, $E_\nu$ is the true neutrino energy, $\theta_\nu$ the true neutrino zenith angle, and $x_i$ the scaling factor for the bin that covers $E_\nu$. The arrival flux is dependent on $\bm{x}$, while the linear factor $x_i$ is due to the increased probability of interaction at the detector. The MC provides a mapping from the true physics space to reconstructed quantities, and allows us to construct a likelihood using the reconstructed zenith and energy distributions for tracks and cascades, and reconstructed energy and cascade length separation distribution for double-cascades [@Schneider:2019icrc_hese]. This likelihood can then be maximized (frequentist) or marginalized (Bayesian) to obtain the set of scalings that best describe the data, $\bm{\hat{x}}$. A likelihood scan over four dimensions was performed to obtain the frequentist confidence regions assuming Wilks’ theorem. A MCMC sampler, [`emcee`]{} [@ForemanMackey:2012ig], was used to obtain the Bayesian credible regions. The effect of changing the overall cross section on the expected event rate in $\cos(\theta_z)$ is shown in the bottom panel of \[fig:attenuation1d\]. Predictions from two alternative cross sections are shown along with the nominal CSMS expectations (orange), all assuming the best-fit, single-power-law flux from [@Schneider:2019icrc_hese]. In the southern sky, $\cos(\theta_z) > 0$, the Earth absorption is negligible so the effect of rescaling the cross section is linear. In the northern sky, $\cos(\theta_z) < 0$, the strength of Earth absorption is dependent on the cross section and the expected number of events is seen to fall off towards $\cos(\theta_z)=-1$ for the nominal and $5\times \sigma_{\rm CSMS}$ (green) cases. This decreased expectation in the northern sky can also be seen in the right panel of \[fig:qtotveto\], from which a relatively fewer number of events arrive as compared to the southern sky. Results {#sec:results} ======= shows the frequentist 68.3% confidence interval (left panel) and Bayesian 68.3% credible interval (right panel) on the CC cross section obtained using the HESE 7.5 sample. For comparison of frequentist results, the measurement from [@Aartsen:2017kpd] (gray region) is included in the left panel. For comparison of Bayesian results, the measurement from from [@Bustamante:2017xuy] (orange error bars) is included in the right panel. The prediction from [@Arguelles:2015wba] is shown as the solid, blue line and the CSMS cross section as the dashed, black line. As the ratio of CC to NC cross section is assumed to be fixed, the NC cross section is identical relative to the CSMS predictions and so is not shown here. In both the frequentist and Bayesian results, the lowest-energy bin prefers a lower cross section while the highest-energy bin prefers a higher cross section than the Standard Model predictions. However, as the uncertainties are large, none of the bins are in significant tension with the models. Summary {#sec:summary} ======= In this proceeding, we have presented a measurement of the neutrino-nucleon cross section above 60 TeV using a sample of high-energy, starting events detected by IceCube. The measurement relies on the Earth as a neutrino attenuator, one that is dependent on the neutrino interaction rate and hence its cross section. The HESE sample used for this analysis spans 7.5 years of livetime with many improvements incorporated into the analysis chain. As the sample is of starting events, events from both the norther and southern sky are included, and with the new three-topology classifier, the likelihood utilizes flavor information as much as possible. The results are obtained in both frequentist and Bayesian statistical frameworks. This allows for direct comparison to the two previously published measurements, with which the results here are consistent. Both frequentist and Bayesian results are consistent with Standard Model calculations. Though the current uncertainties are large, it will be possible to constrain the cross section to better precision with future, planned detector upgrades. [^1]: For collaboration list, see PoS(ICRC2019) 1177.

ArXiv

--- abstract: 'Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and $E_1$, $E_2$ are vector bundles over $C$.In this paper we compute the pseudo effective cones of higher codimension cycles on $X$.' address: | Institute of Mathematical Sciences\ CIT Campus, Taramani, Chennai 600113, India and Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India author: - Rupam Karmakar title: Effective cones of cycles on products of projective bundles over curves --- Introduction ============ The cones of divisors and curves on projective varieties have been extensively studied over the years and by now are quite well understood. However, more recently the theory of cones of cycles of higher dimension has been the subject of increasing interest(see [@F], [@DELV], [@DJV], [@CC] etc). Lately, there has been significant progress in the theoretical understanding of such cycles, due to [@FL1], [@FL2] and others. But the the number of examples where the cone of effective cycles have been explicitly computed is relatively small till date [@F], [@CLO], [@PP] etc. Let $E_1$ and $E_2$ be two vector bundles over a smooth curve $C$ and consider the fibre product $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$. Motivated by the results in [@F], in this paper, we compute the cones of effective cycles on $X$ in the following cases. Case I: When both $E_1$ and $E_2$ are semistable vector bundles of rank $r_1$ and $r_2$ respectively, the cone of effective codimension k-cycles are described in theorem 3.2. Case II: When Neither $E_1$ nor $E_2$ is semistable, the cone of low dimension effective cycles are computed in theorem 3.3 and the remaining cases in therem 3.5. Preliminaries ============= Let $X$ be a smooth projective varity of dimension $n$. $N_k(X)$ is the real vector space of k-cycles on $X$ modulo numerical equivalence. For each $k$, $N_k(X)$ is a real vector space of finite dimension. Since $X$ is smooth, we can identify $N_k(X)$ with the abstract dual $N^{n - k}(X) := N_{n - k}(X)^\vee$ via the intersection pairing $N_k(X) \times N_{n - k}(X) \longrightarrow \mathbb{R}$. For any k-dimensional subvariety $Y$ of $X$, let $[Y]$ be its class in $N_k(X)$. A class $\alpha \in N_k(X)$ is said to be effective if there exists subvarities $Y_1, Y_2, ... , Y_m$ and non-negetive real numbers $n_1, n _2, ..., n_m$ such that $\alpha$ can be written as $ \alpha = \sum n_i Y_i$. The *pseudo-effective cone* $\overline{\operatorname{{Eff}}}_k(X) \subset N_k(X)$ is the closure of the cone generated by classes of effective cycles. It is full-dimensional and does not contain any nonzero linear subspaces. The pseudo-effective dual classes form a closed cone in $N^k(X)$ which we denote by $\overline{\operatorname{{Eff}}}^k(X)$. For smooth varities $Y$ and $Z$, a map $f: N^k(Y) \longrightarrow N^K(Z)$ is called pseudo-effective if $f(\overline{\operatorname{{Eff}}}^k(Y)) \subset \overline{\operatorname{{Eff}}}^k(Z)$. The *nef cone* $\operatorname{{Nef}}^k(X) \subset N^k(X)$ is the dual of $\overline{\operatorname{{Eff}}}^k(X) \subset N^k(X)$ via the pairing $N^k(X) \times N_k(X) \longrightarrow \mathbb{R}$, i.e, $$\begin{aligned} \operatorname{{Nef}}^k(X) := \Big\{ \alpha \in N_k(X) | \alpha \cdot \beta \geq 0 \forall \beta \in \overline{\operatorname{{Eff}}}_k(X) \Big\} \end{aligned}$$ Cone of effective cycles ======================== Let $E_1$ and $E_2$ be two vector bundles over a smooth curve $C$ of rank $r_1$, $r_2$ and degrees $d_1$, $d_2$ respectively. Let $\mathbb{P}(E_1) = \bf Proj $ $(\oplus_{d \geq 0}Sym^d(E_1))$ and $\mathbb{P}(E_2) = \bf Proj $ $(\oplus_{d \geq 0}Sym^d(E_2))$ be the associated projective bundle together with the projection morphisms $\pi_1 : \mathbb{P}(E_1) \longrightarrow C$ and $\pi_2 : \mathbb{P}(E_2) \longrightarrow C$ respectively. Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ be the fibre product over $C$. Consider the following commutative diagram: $$\begin{tikzcd} X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2) \arrow[r, "p_2"] \arrow[d, "p_1"] & \mathbb{P}(E_2)\arrow[d,"\pi_2"]\\ \mathbb{P}(E_1) \arrow[r, "\pi_1" ] & C \end{tikzcd}$$ Let $f_1,f_2$ and $F$ denote the numerical equivalence classes of the fibres of the maps $\pi_1,\pi_2$ and $\pi_1 \circ p_1 = \pi_2 \circ p_2$ respectively. Note that, $X \cong \mathbb{P}(\pi_1^*(E_2)) \cong \mathbb{P}(\pi_2^*(E_1))$. We first fix the following notations for the numerical equivalence classes, $\eta_1 = [\mathcal{O}_{\mathbb{P}(E_1)}(1)] \in N^1(\mathbb{P}(E_1))$ , $\eta_2 = [\mathcal{O}_{\mathbb{P}(E_2)}(1)] \in N^1(\mathbb{P}(E_2)),$ $\xi_1 = [\mathcal{O}_{\mathbb{P}(\pi_1^*(E_2))}(1)]$ , $\xi_2 = [\mathcal{O}_{\mathbb{P}(\pi_2^*(E_1))}(1)]$ , $\zeta_1 = p_1^*(\eta_1)$ , $\zeta_2 = p_2^*(\eta_2) $ $ \zeta_1 = \xi_2$, $\zeta_2 = \xi_1$ , $F= p_1^\ast(f_1) = p_2^\ast(f_2)$ We here summarise some results that has been discussed in [@KMR] ( See section 3 in [@KMR] for more details) : $N^1(X)_\mathbb{R} = \mathbb{R}\zeta_1 \oplus \mathbb{R}\zeta_2 \oplus \mathbb{R}F,$ $\zeta_1^{r_1}\cdot F = 0\hspace{1.5mm},\hspace{1.5mm} \zeta_1^{r_1 + 1} = 0 \hspace{1.5mm}, \hspace{1.5mm} \zeta_2^{r_2}\cdot F = 0 \hspace{1.5mm}, \hspace{1.5mm} \zeta_2^{r_2 + 1} = 0 \hspace{1.5mm}, \hspace{1.5mm} F^2 = 0$ , $\zeta_1^{r_1} = (\deg(E_1))F\cdot\zeta_1^{r_1-1}\hspace{1.5mm}, \hspace{1.5mm} \zeta_2^{r_2} = (\deg(E_2))F\cdot\zeta_2^{r_2-1}\hspace{3.5mm}, \hspace{3.5mm}$ $\zeta_1^{r_1}\cdot\zeta_2^{r_2-1} = \deg(E_1)\hspace{3.5mm}, \hspace{3.5mm} \zeta_2^{r_2}\cdot\zeta_1^{r_1-1} = \deg(E_2)$. Also, The dual basis of $N_1(X)_\mathbb{R}$ is given by $\{\delta_1, \delta_2, \delta_3\}$, where $\delta_1 = F\cdot\zeta_1^{r_1-2}\cdot\zeta_2^{r_2-1}, $ $\delta_2 = F\cdot\zeta_1^{r_1-1}\cdot\zeta_2^{r_2-2},$ $\delta_3 = \zeta_1^{r_1-1}\cdot\zeta_2^{r_2-1} - \deg(E_1)F\cdot\zeta_1^{r_1-2}\cdot\zeta_2^{r_2-1} - \deg(E_2)F\cdot\zeta_1^{r_1-1}\cdot\zeta_2^{r_2-2}.$ Let $r_1 = \operatorname{{rank}}(E_1)$ and $r_2 = \operatorname{{rank}}(E_2)$ and without loss of generality assume that $ r_1 \leq r_2$. Then the bases of $N^k(X)$ are given by $$N^k(X) = \begin{cases} \Big( \{ \zeta_1^i \cdot \zeta_2^{k - i}\}_{i = 0}^k, \{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \}_{j = 0}^ {k - 1} \Big) & if \quad k < r_1\\ \\ \Big( \{ \zeta_1^i \cdot \zeta_2^{k - i} \}_{i = 0} ^{r_1 - 1}, \{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \}_{j = 0}^ {r_1 - 1} \Big) & if \quad r_1 \leq k < r_2 \\ \\ \Big( \{ \zeta_1^i \cdot \zeta_2^{k - i} \}_{i = t+1} ^{r_1 - 1} , \{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \}_{j = t}^{r_1 - 1} \Big) & if \quad k = r_2 + t \quad where \quad t \in \{0, 1, 2, ..., r_1 - 2 \}. \end{cases}$$ To begin with consider the case where $ k < r_1$. We know that $X \cong \mathbb{P}(\pi_2^* E_1)$ and the natural morphism $ \mathbb{P}(\pi_2^*E_1) \longrightarrow \mathbb{P}(E_2)$ can be identified with $p_2$. With the above identifications in place the chow group of $X$ has the following isomorphism \[see theorem 3.3 , page 64 [@Ful]\] $$\begin{aligned} A(X) \cong \bigoplus_{i = 0}^ {r_1 - 1} \zeta_1^i A(\mathbb{P}(E_2)) \end{aligned}$$ Choose $i_1, i_2$ suct that $ 0\leq i_1 < i_2 \leq k$. Consider the $k$- cycle $\alpha := F \cdot \zeta_1^{r_1 - i_1 -1} \cdot \zeta_2^{r_2 + i_1 -k - 1}$. Then $\zeta_1^{i_1} \cdot \zeta_2^{k - i_1} \cdot \alpha = 1$ but $ \zeta_1^{i_2} \cdot \zeta_2^{k - i_2} \cdot \alpha = 0$. So, $\{ \zeta_1^{i_1} \cdot \zeta_2^{k - i_1} \}$ and $\{\zeta_1^{i_2} \cdot \zeta_2^{k - i_2} \}$ can not be numerically equivalent. Similarly, take $j_1, j_2$ such that $ 0 \leq j_1 < j_2 \leq k$ and consider the $k$-cycle\ $\beta := \zeta_1^{r_1 - j_1 - 1} \cdot \zeta_2^{r_2 + j_1 - k}$. Then as before it happens that $F \cdot \zeta_1^{j_1} \cdot \zeta_2^{k - j_1 - 1} \cdot \beta = 1$ but $F\cdot \zeta_1^{j_2} \cdot \zeta_2^{k - j_2 - 1} \cdot \beta = 0$. So $\{ F \cdot \zeta_1^{j_1} \cdot \zeta_2^{k - j_1 - 1}\}$ and $\{ F \cdot \zeta_1^{j_2} \cdot \zeta_2^{k - j_2 - 1} \}$ can not be numerically equivalent. For the remaining case lets assume $0 \leq i \leq j \leq k$ and consider the k-cycle $\gamma := F \cdot \zeta_1^{r_1 - i -1} \cdot \zeta_2^{r_2 + i - 1 - k}$. Then $\{ \zeta_1^{i} \cdot \zeta_2^{k - i} \} \cdot \gamma = 1$ and $ \{F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \} \cdot \gamma = 0$. So, they can not be numerically equivalent. From these observations and $(2)$ we obtain a basis of $N^k(X)$ which is given by $$\begin{aligned} N^k(X) = \Big( \{ \zeta_1^i \cdot \zeta_2^{k - i} \}_{i = 0}^ k, \{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \}_{j = 0}^{k - 1} \Big) \end{aligned}$$ For the case $ r_1 \leq k < r_2$ observe that $ \zeta_1^{r_1 + 1} = 0$, $ F \cdot \zeta_1^ {r_1} = 0$ and $ \zeta_1^ {r_2} = deg(E_1)F \cdot \zeta_1^{ r_1 - 1}$. When $k \geq r_2$ we write as $k = r_2 + t$ where $t$ ranges from $ 0$ to $r_1 - 1$. In that case the observations like $ \zeta_2^{r_2 + 1} = 0$, $F\cdot\zeta_2^{r_2} = 0$ and $\zeta_2^{r_2} = \deg(E_2)F \cdot \zeta_2^{r_2 - 1}$ proves our case. Now we are ready to treat the case where both $E_1$ and $E_2$ are semistable vector bundles over $C$. Let $E_1$ and $E_2$ be two semistable vector bundles over $C$ of rank $r_1$ and $r_2$ respectively with $r_1 \leq r_2$ and $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$. Then for all $k \in \{1, 2, ..., r_1 + r_2 - 1 \}$ $$\overline{\operatorname{{Eff}}}^k(X) = \begin{cases} \Bigg\langle \Big\{ (\zeta_1 - \mu_1F)^i (\zeta_2 - \mu_2F)^{k - i} \Big\}_{i = 0}^k, \Big\{ F \cdot \zeta_1^j \cdot\zeta_2^{k - j - 1} \Big\}_{j = 0}^{k - 1} \Bigg\rangle & if \quad k< r_1 \\ \\ \Bigg\langle \Big\{ (\zeta_1 - \mu_1F)^i (\zeta_2 - \mu_2F)^{k - i} \Big\}_{i = 0}^{r_1 - 1}, \Big\{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \Big\}_{j = 0}^ {r_1 - 1} \Bigg\rangle & if \quad r_1 \leq k < r_2 \\ \\ \Bigg\langle \Big\{ (\zeta_1 - \mu_1F)^i (\zeta_2 - \mu_2F)^{k - i} \Big\}_{i = t +1}^ {r_1 - 1}, \Big\{ F \cdot \zeta_1^j \cdot \zeta_2^{k - j -1} \Big\}_{j = t}^ {r_1 - 1} \Bigg\rangle & if \quad k = r_2 + t, \quad t = 0,..., r_1-1 . \end{cases}$$ where $\mu_1 = \mu(E_1)$ and $\mu_2 = \mu(E_2)$. Firstly, $(\zeta_1 - \mu_1F)^i \cdot(\zeta_2 - \mu_2F)^{k - i}$ and $ F\cdot \zeta_1^i \cdot \zeta_2^{k - j - 1} [ = F \cdot (\zeta_1 - \mu_1F)^i \cdot(\zeta_2 - \mu_2F)^{k - j -1} ]$ are intersections of nef divisors. So, they are pseudo-effective for all $i \in\{0, 1, 2, ..., k \}$. conversely, when $ k < r_1$ notice that we can write any element $C$ of $\overline{\operatorname{{Eff}}}^k(X)$ as $$\begin{aligned} C = \sum_{i = 0}^k a_i(\zeta_1 - \mu_1F)^i \cdot (\zeta_2 - \mu_2F)^{k - i} + \sum_{j = 0}^k b_j F\cdot\zeta_1^j \cdot\zeta_2^{k - j -1}\end{aligned}$$ where $a_i, b_i \in \mathbb{R}$. For a fixed $i_1$ intersect $C$ with $D_{i_1} :=F \cdot(\zeta_1 - \mu_1F)^{r_1 - i_1 - 1} \cdot(\zeta_2 - \mu_2F)^{r_2 - k + i_1 -1}$ and for a fixed $j_1$ intersect $C$ with $D_{j_1}:= (\zeta_1 - \mu_1F)^{r_1 - j_1 - 1} \cdot(\zeta_2 - \mu_2F)^{r_2 +j_1 - k}$. These intersections lead us to $$\begin{aligned} C \cdot D_{i_1} = a_{i_1}\quad and \quad C\cdot D_{j_1} = b_{j_1}\end{aligned}$$ Since $C \in \overline{\operatorname{{Eff}}}^k(X)$ and $D_{i_1}, D_{j_1}$ are intersection of nef divisors, $a_{i_1}$ and $b_{j_1}$ are non-negetive. Now running $i_1$ and$j_1$ through $\{ 0, 1, 2, ..., k \}$ we get all the $a_i$’s and $b_i$’s are non-negetive and that proves our result for $k < r_1$. The cases where $r_1 \leq k < r_2$ and $k \geq r_2$ can be proved very similarly after the intersection products involving $\zeta_1$ and $\zeta_2$ in page $2$ are taken into count. Next we study the more interesting case where $E_1$ and $E_2$ are two unstable vector bundles of rank $r_1$ and $r_2$ and degree $d_1$ and $d_2$ respectively over a smooth curve $C$. Let $E_1$ be the unique Harder-Narasimhan filtration $$\begin{aligned} E_1 = E_{10} \supset E_{11} \supset ... \supset E_{1l_1} = 0\end{aligned}$$ with $Q_{1i} := E_{1(i-1)}/ E_{1i}$ being semistable for all $i \in [1,l_1-1]$. Denote $ n_{1i} = \operatorname{{rank}}(Q_{1i}), \\ d_{1i} = \deg(Q_{1i})$ and $\mu_{1i} = \mu(Q_{1i}) := \frac{d_{1i}}{n_{1i}}$ for all $i$. Similarly, $E_2$ also admits the unique Harder-Narasimhan filtration $$\begin{aligned} E_2 = E_{20} \supset E_{21} \supset ... \supset E_{2l_2} = 0\end{aligned}$$ with $ Q_{2i} := E_{2(i-1)} / E_{2i}$ being semistable for $i \in [1,l_2-1]$. Denote $n_{2i} = \operatorname{{rank}}(Q_{2i}), \\ d_{2i} = \deg(Q_{2i})$ and $\mu_{2i} = \mu(Q_{2i}) := \frac{d_{2i}}{n_{2i}}$ for all $i$. Consider the natural inclusion $ \overline{i} = i_1 \times i_2 : \mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21}) \longrightarrow \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$, which is induced by natural inclusions $i_1 : \mathbb{P}(Q_{11}) \longrightarrow \mathbb{P}(E_1)$ and $i_2 : \mathbb{P}(Q_{21}) \longrightarrow \mathbb{P}(E_2)$. In the next theorem we will see that the cycles of $ \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ of dimension at most $n_{11} + n_{21} - 1$ can be tied down to cycles of $\mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21})$ via $\overline{i}$. Let $E_1$ and $E_2$ be two unstable bundle of rank $r_1$ and $r_2$ and degree $d_1$ and $d_2$ respectively over a smooth curve $C$ and $r_1 \leq r_2$ without loss of generality and $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$. Then for all $k \in \{1, 2, ..., \mathbf{n} \} \, \, (\mathbf{n} := n_{11} + n_{21} - 1)$ $Case(1)$: $n_{11} \leq n_{21}$ $$\overline{\operatorname{{Eff}}}_k(X) = \begin{cases} \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_1 - \mu_{11}F)^i (\zeta_2 - \mu_{21}F)^{\mathbf{n} - k - i} \Big\}_{i = t + 1}^{n_{11} - 1}, \Big\{ F \cdot \zeta_1^{r_1 - n_{11} +j} \cdot \zeta_2^{r_2 + n_{11} - k - j - 2} \Big\}_{j = t}^{n_{11} - 1} \Bigg\rangle & \\ \qquad \qquad if \quad k < n_{11} \quad and \quad t = 0, 1, 2, ..., n_{11} - 2 \\ \\ \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_1 - \mu_{11}F)^i (\zeta_2 - \mu_{21}F)^{\mathbf{n} - k - i}\Big\}_{i = 0}^{n_{11} - 1}, \Big\{ F\cdot \zeta_1^{r_1 - n_{11} + j} \cdot \zeta_2^{r_2 + n_{11} - k - j - 2} \Big\}_{j = 0}^{n_{11} - 1} \Bigg\rangle & \\ \qquad \qquad if \quad n_{11} \leq k < n_{21}. \\ \\ \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_1 - \mu_{11}F)^i (\zeta_2 - \mu_{21}F)^{ \mathbf{n} - k - i}\Big\}_{i = 0}^{\mathbf{n} - k}, \Big\{ F \cdot \zeta_1^{r_1 - n_{11} + j} \cdot \zeta_2^{r_2 + n_{11} - k - j - 2} \Big\}_{j = 0}^{ \mathbf{n} - k} \Bigg\rangle & \\ \qquad \qquad if \quad k \geq n_{21}. \end{cases}$$ $Case(2)$: $n_{21} \leq n_{11}$ $$\overline{\operatorname{{Eff}}}_k(X) = \begin{cases} \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_2 - \mu_{21}F)^i (\zeta_1 - \mu_{11}F)^{\mathbf{n} - k - i} \Big\}_{i = t + 1}^{n_{21} - 1}, \Big\{ F \cdot \zeta_2^{r_2 - n_{21} +j} \cdot \zeta_1^{r_1 + n_{21} - k - j - 2} \Big\}_{j = t}^{n_{21} - 1} \Bigg\rangle & \\ \qquad \qquad if \quad k < n_{21} \quad and \quad t = 0, 1, 2, ..., n_{21} - 2 \\ \\ \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_2 - \mu_{21}F)^i (\zeta_1 - \mu_{11}F)^{\mathbf{n} - k - i}\Big\}_{i = 0}^{n_{21} - 1}, \Big\{ F\cdot \zeta_2^{r_2 - n_{21} + j} \cdot \zeta_1^{r_1 + n_{21} - k - j - 2} \Big\}_{j = 0}^{n_{21} - 1} \Bigg\rangle & \\ \qquad \qquad if \quad n_{21} \leq k < n_{11}. \\ \\ \Bigg\langle\Big\{ [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_2 - \mu_{21}F)^i (\zeta_1 - \mu_{11}F)^{ \mathbf{n} - k - i}\Big\}_{i = 0}^{\mathbf{n} - k}, \Big\{ F \cdot \zeta_2^{r_2 - n_{21} + j} \cdot \zeta_1^{r_1 + n_{21} - k - j - 2} \Big\}_{j = 0}^{ \mathbf{n} - k} \Bigg\rangle & \\ \qquad \qquad if \quad k \geq n_{11}. \end{cases}$$ Thus in both cases $ \overline{i}_\ast$ induces an isomorphism between $ \overline{\operatorname{{Eff}}}_k([\mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21})])$ and $\overline{\operatorname{{Eff}}}_k(X)$ for $ k \leq \mathbf{n}$. to begin with consider $Case(1)$ and then take $ k \geq n_{21}$. Since $ (\zeta_1 - \mu_{11}F)$ and $ (\zeta_2 - \mu_{21}F)$ are nef $$\begin{aligned} \phi_i := [\mathbb{P}(Q_{11}) \times \mathbb{P}(Q_{21})] (\zeta_1 - \mu_{11}F)^i (\zeta_2 - \mu_{21}F)^{ \mathbf{n} - k - i} \in \overline{\operatorname{{Eff}}}_k(X).\end{aligned}$$ for all $i \in \{ 0, 1, 2, ..., \mathbf{n} -k \}$. Now The result in \[Example 3.2.17, [@Ful]\] adjusted to bundles of quotients over curves shows that $$\begin{aligned} [\mathbb{P}(Q_{11})] = \eta_1^{r_1 - n_{11}} + (d_{11} - d_1)\eta_1^{r_1 - n_{11} - 1}f_1\end{aligned}$$ and $$\begin{aligned} [\mathbb{P}(Q_{21})] = \eta_2^{r_2 - n_{21}} + (d_{21} - d_2)\eta_2^{r_2 - n_{21} - 1}f_2\end{aligned}$$ Also, $p_1^\ast[\mathbb{P}(Q_{11})] \cdot p_2^\ast[\mathbb{P}(Q_{21})] = [\mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21})]$ . With little calculations it can be shown that $\phi_i \cdot (\zeta_1 - \mu_{11}F)^{n_{11} - i} \cdot (\zeta_2 - \mu_{21}F)^{k + i + 1 - n_{11}}$\ $ = (\zeta_1^{r_1 - n_{11}} + (d_{11} - d_1)F \cdot \zeta_1^{r_1 - n_{11} - 1})(\zeta_2^{r_2 - n_{21}} + (d_{21} - d_2)F \cdot \zeta_2^{r_2 - n_{21} - 1})(\zeta_1 - \mu_{11}F)^{n_11 - i} \cdot (\zeta_2 - \mu_{21}F)^{k + i + 1 - n_{11}}$ $= (\zeta_1^{r_1} - d_1F \cdot \zeta_1^{r_1 - 1})(\zeta_2^{r_2} - d_2F\cdot \zeta_2^{r_2 - 1})$ $= 0$. So, $\phi_i$ ’s are in the boundary of $\overline{\operatorname{{Eff}}}_k(X)$ for all $ i \in \{0, 1, ..., \mathbf{n} - k \}$. The fact that $F \cdot \zeta_1^{r_1 - n_{11} + j} \cdot \zeta_2^{r_2 + n_{11} - k - j - 2}$ ’s are in the boundary of $\overline{\operatorname{{Eff}}}_k(X)$ for all $ i \in \{0, 1, ..., \mathbf{n} - k \}$ can be deduced from the proof of Theorem 2.2. The other cases can be proved similarly. The proof of $Case(2)$ is similar to the proof of $Case(1)$. Now, to show the isomorphism between pseudo-effective cones induced by $\overline{i}_\ast$ observe that $Q_{11}$ and $Q_{21}$ are semi-stable bundles over $C$. So, Theorem 2.2 gives the expressions for $\overline{\operatorname{{Eff}}}_k([\mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21})])$. Let $\zeta_{11} = \mathcal{O}_{\mathbb{P}(\tilde{\pi}_2^\ast(Q_{11}))}(1) =\tilde{p}_1^\ast(\mathcal{O}_{\mathbb{P}(Q_{11})}(1))$ and $\zeta_{21} = \mathcal{O}_{\mathbb{P}(\tilde{\pi}_1^\ast(Q_{21})}(1) = \tilde{p}_2^\ast(\mathcal{O}_{\mathbb{P}(Q_{21})}(1))$, where $\tilde{\pi_2} = \pi_2|_{\mathbb{P}(Q_{21})}$, $ \tilde{\pi_1} = \pi_1|_{\mathbb{P}(Q_{11})}$ and $\tilde{p}_1 : \mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21}) \longrightarrow \mathbb{P}(Q_{11})$, $\tilde{p}_2 : \mathbb{P}(Q_{11}) \times_C\mathbb{P}(Q_{21}) \longrightarrow \mathbb{P}(Q_{21})$ are the projection maps. Also notice that $ \overline{i}^\ast \zeta_1 = \zeta_{11}$ and $\overline{i}^\ast \zeta_1 = \zeta_{21}$. Using the above relations and projection formula the isomorphism between $ \overline{\operatorname{{Eff}}}_k([\mathbb{P}(Q_{11}) \times_C \mathbb{P}(Q_{21})])$ and $\overline{\operatorname{{Eff}}}_k(X)$ for $ k \leq \mathbf{n}$ can be proved easily. Next we want to show that higher dimension pseudo effective cycles on $X$ can be related to the pseudo effective cycles on $ \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_{21})$. More precisely there is a isomorphism between $\overline{\operatorname{{Eff}}}^k(X)$ and $\overline{\operatorname{{Eff}}}^k([\mathbb{P}(E_{11}) \times_C \mathbb{P}(E_{21})])$ for $ k < r_1 + r_2 - 1 - \mathbf{n}$. Useing the coning construction as in \[ful\] we show this in two steps, first we establish an isomorphism between $\overline{\operatorname{{Eff}}}^k([\mathbb{P}(E_1) \times_C \mathbb{P}(E_2)])$ and $\overline{\operatorname{{Eff}}}^k([\mathbb{P}(E_{11}) \times_C \mathbb{P}(E_2)])$ and then an isomorphism between $\overline{\operatorname{{Eff}}}^k([\mathbb{P}(E_{11}) \times_C \mathbb{P}(E_2)])$ and $\overline{\operatorname{{Eff}}}^k([\mathbb{P}(E_{11}) \times_C \mathbb{P}(E_{21})])$ in similar fashion. But before proceeding any further we need to explore some more facts. Let $E$ be an unstable vector bundle over a non-singular projective variety $V$. There is a unique filtration $$\begin{aligned} E = E^0 \supset E^1 \supset E^2 \supset ... \supset E^l = 0\end{aligned}$$ which is called the harder-Narasimhan filtration of $E$ with $Q^i := E^{i - 1}/E^i$ being semistable for $ i \in [1, l - 1]$. Now the following short-exact sequence $$\begin{aligned} 0 \longrightarrow E^1 \longrightarrow E \longrightarrow Q^1 \longrightarrow 0\end{aligned}$$ induced by the harder-narasimhan filtration of $E$ gives us the natural inclusion $j : \mathbb{P}(Q^1) \hookrightarrow \mathbb{P}(E)$. Considering $\mathbb{P}(Q^1)$ as a subscheme of $\mathbb{P}(E)$ we obtain the commutative diagram below by blowing up $\mathbb{P}(Q^1)$. $$\begin{tikzcd} \tilde{Y} = Bl_{\mathbb{P}(Q^1)}{\mathbb{P}(E)} \arrow[r, "\Phi"] \arrow[d, "\Psi"] & \mathbb{P}(E^1) = Z \arrow [d, "q"] \\ Y = \mathbb{P}(E) \arrow [r, "p"] & V \end{tikzcd}$$ where $\Psi$ is blow-down map. With the above notation,there exists a locally free sheaf $G$ on $Z$ such that $\tilde{Y} \simeq \mathbb{P}_Z(G)$ and $\nu : \mathbb{P}_Z(G) \longrightarrow Z$ it’s corresponding bundle map. In particular if we place $V = \mathbb{P}(E_2)$, $ E = \pi_2^\ast E_1$, $E^1 = \pi_2^\ast E_{11}$ and $ Q^1 = \pi_2^\ast Q_{11}$ then the above commutative diagram becomes $$\begin{tikzcd} \tilde{Y'} = Bl_{\mathbb{P}(\pi_2^\ast Q_{11})}{\mathbb{P}(\pi_2^\ast E_1)} \arrow[r, "\Phi'"] \arrow[d, "\Psi'"] & \mathbb{P}(\pi_2^\ast E_{11}) = Z' \arrow[d, "\overline{p}_2"] \\ Y' = \mathbb{P}(\pi_2^\ast E_1) \arrow[r, "p_2"] & \mathbb{P}(E_2) \end{tikzcd}$$ where $p_2 : \mathbb{P}(\pi_2^\ast E_1) \longrightarrow \mathbb{P}(E_2)$ and $\overline{p}_2 : \mathbb{P}(\pi_2^\ast E_{11}) \longrightarrow \mathbb{P}(E_2)$ are projection maps. and there exists a locally free sheaf $G'$ on $Z'$ such that $\tilde{Y'} \simeq \mathbb{P}_{Z'}(G')$ and $\nu' : \mathbb{P}_{Z'}(G') \longrightarrow Z'$ it’s bundle map. Now let $\zeta_{Z'} = \mathcal{O}_{Z'}(1)$, $\gamma = \mathcal{O}_{\mathbb{P}_{Z'}(G')}(1)$, $F$ the numerical equivalence class of a fibre of $\pi_2 \circ p_2$, $F_1$ the numerical equivalence class of a fibre of $\pi_2 \circ \overline{p}_2$, $\tilde{E}$ the class of the exceptional divisor of $\Psi'$ and $\zeta_1 = p_1^\ast(\eta_1) = \mathcal{O}_{\mathbb{P}(\pi_2^\ast E_1)}(1)$. Then we have the following relations: $$\begin{aligned} \gamma = (\Psi')^\ast \, \zeta_1, \quad (\Phi')^\ast \, \zeta_{Z'} = (\Psi')^\ast \, \zeta_1 - \tilde{E}, \quad (\Phi')^\ast F_1 = (\Psi')^\ast F \end{aligned}$$ $$\begin{aligned} \tilde{E} \cdot (\Psi')^\ast \, (\zeta_1 - \mu_{11}F)^{n_{11}} = 0 \end{aligned}$$ Additionaly, if we also denote the support of the exceptional divisor of $\tilde{Y'}$ by $\tilde{E}$ , then $\tilde{E} \cdot N(\tilde{Y'}) = (j_{\tilde{E}})_\ast N(\tilde{E})$, where $j_{\tilde{E}}: \tilde{E} \longrightarrow \tilde{Y'}$ is the canonical inclusion. With the above hypothesis the following commutative diagram is formed: 0 & q\^E\^1 & q\^E & q\^Q\^1 & 0\ 0 & \_[(E\^1)]{}(1) & G & q\^Q\^1 & 0 where $G$ is the push-out of morphisms $ q^\ast E^1 \longrightarrow q^\ast E$ and $ q^\ast E^1 \longrightarrow \mathcal{O}_{\mathbb{P}(E^1)}(1)$ and the first vertical map is the natural surjection. Now let $W = \mathbb{P}_Z(G)$ and $\nu : W \longrightarrow Z$ be it’s bundle map. So there is a cannonical surjection $\nu^\ast G \longrightarrow \mathcal{O}_{\mathbb{P}_Z(G)}(1)$. Also note that $q^\ast E \longrightarrow G$ is surjective by snake lemma. Combining these two we obtain a surjective morphism $\nu^\ast q^\ast E \longrightarrow \mathcal{O}_{\mathbb{P}_Z(G)}(1)$ which determines $\omega : W \longrightarrow Y$. We claim that we can identify $(\tilde{Y}, \Phi, \Psi)$ and $(W, \nu, \omega)$. Now Consider the following commutative diagram: $$\begin{tikzcd} W = \mathbb{P}_Z(G) \arrow[drr, bend left, "\nu"] \arrow[ddr, bend right, "\omega"] \arrow[dr, "\mathbf{i}"] & & \\ & Y \times_V Z = \mathbb{P}_Z(q^\ast E) \arrow[r, "pr_2"] \arrow[d, "pr_1"] & \mathbb{P}(E^1) = Z \arrow[d, "q"] \\ & Y = \mathbb{P}(E) \arrow[r, "p"] & V \end{tikzcd}$$ where $\mathbf{i}$ is induced by the universal property of the fiber product. Since $\mathbf{i}$ can also be obtained from the surjective morphism $q^\ast E \longrightarrow G$ it is a closed immersion. Let $\mathcal{T}$ be the $\mathcal{O}_Y$ algebra $\mathcal{O}_Y \oplus \mathcal{I} \oplus \mathcal{I}^2 \oplus ...$, where $\mathcal{I}$ is the ideal sheaf of $\mathbb{P}(Q^1)$ in $Y$. We have an induced map of $\mathcal{O}_Y$- algebras $Sym(p^\ast E^1) \longrightarrow \mathcal{T} \ast \mathcal{O}_Y(1)$ which is onto because the image of the composition $ p^\ast E^1 \longrightarrow p^\ast E \longrightarrow \mathcal{O}_Y(1)$ is $ \mathcal{T} \otimes \mathcal{O}_Y(1)$. This induces a closed immersion $\mathbf{i}' : \tilde{Y} = Proj(\mathcal{T} \ast \mathcal{O}_Y(1)) \longrightarrow Proj(Sym(p^\ast E^1) = Y \times_V Z$. $\mathbf{i}'$ fits to a similar commutative diagram as $(5)$ and as a result $\Phi$ and $\Psi$ factor through $pr_2$ and $pr_1$. Both $W$ and $\tilde{Y}$ lie inside $Y \times_V Z$ and $\omega$ and $\Psi$ factor through $pr_1$ and $\nu$ and $\Phi$ factor through $pr_2$. So to prove the identification between $(\tilde{Y}, \Phi, \Psi )$ and $(W, \nu, \omega)$ , it is enough to show that $ \tilde{Y} \cong W$. This can be checked locally. So, after choosing a suitable open cover for $V$ it is enough to prove $\tilde{Y} \cong W$ restricted to each of these open sets. Also we know that $p^{-1}(U) \cong \mathbb{P}_U ^{rk(E) - 1}$ when $E_{|U}$ is trivial and $\mathbb{P}_U^n = \mathbb{P}_{\mathbb{C}}^n \times U$. Now the the isomorphism follows from \[proposition 9.11, [@EH]\] after adjusting the the definition of projectivization in terms of [@H]. We now turn our attention to the diagram $(3)$. observe that if we fix the notations $W' = \mathbb{P}_Z'(G')$ with $\omega' : W' \longrightarrow Y'$ as discussed above then we have an identification between $(\tilde{Y}', \Phi', \Psi')$ and $(W', \nu', \omega')$. $\omega' : W' \longrightarrow Y'$ comes with $(\omega')^\ast \mathcal{O}_ {Y'}(1) = \mathcal{O}_{\mathbb{P}_{Z'}(G')}(1)$. So, $\gamma = (\Psi')^\ast \, \zeta_1$ is achieved. $(\Phi')^\ast F_1 = (\Psi')^\ast F$ follows from the commutativity of the diagram $(3)$. The closed immersion $\mathbf{i}'$ induces a relation between the $\mathcal{O}(1)$ sheaves of $Y \times_V Z$ and $\tilde{Y}$. For $Y \times_V Z$ the $\mathcal{O}(1)$ sheaf is $pr_2^ \ast \mathcal{O}_{Z}(1)$ and for $ Proj(\mathcal{T} \ast \mathcal{O}_Y(1)$ the $\mathcal{O}(1)$ sheaf is $\mathcal{O}_{\tilde{Y}}( - \tilde{E}) \otimes (\Psi)^ \ast \mathcal{O}_Y(1)$. Since $\Phi$ factors through $pr_2$, $(\Phi)^ \ast \mathcal{O}_Z(1) = \mathcal{O}_{\tilde{Y}}( - \tilde{E}) \otimes (\Psi)^ \ast \mathcal{O}_Y(1)$. In the particular case (see diagram $(3)$) $(\Phi')^ \ast \mathcal{O}_{Z'}(1) = \mathcal{O}_{\tilde{Y}'}( - \tilde{E}) \otimes (\Psi')^ \ast \mathcal{O}_{Y'}(1)$ i. e. $(\Phi')^\ast \, \zeta_{Z'} = (\Psi')^\ast \, \zeta_1 - \tilde{E} $. Next consider the short exact sequence: $ 0 \longrightarrow \mathcal{O}_{Z'}(1) \longrightarrow G' \longrightarrow \overline{p}_2^\ast \pi_2^ \ast Q_{11} \longrightarrow 0$ We wish to calculate below the total chern class of $G'$ through the chern class relation obtained from the above short exact sequence. $c(G') = c(\mathcal{O}_{Z'}(1)) \cdot c(\overline{p}_2^\ast \pi_2^ \ast Q_{11}) = (1 + \zeta_{Z'}) \cdot \overline{p}_2^\ast \pi_2^\ast(1 + d_{11}[pt]) = (1 + \xi_{Z'})(1 + d_{11}F_1)$ From the grothendieck relation for $G'$ we have $\gamma^{n_{11} + 1} - {\Phi'} ^ \ast(\zeta_{Z'} + d_{11}F_1) \cdot \gamma^{n_{11}} + {\Phi'} ^ \ast (d_{11}F_1 \cdot \zeta_{Z'}) \cdot \gamma^ {n_{11} - 1} = 0$\ $\Rightarrow \gamma^{n_{11} + 1} - ({\Psi'} ^ \ast \zeta_1 - \tilde{E}) + d_{11}{\Psi'}^ \ast F) \cdot \gamma^{n_{11}} + d_{11}({\Psi'} ^ \ast \zeta_1 - \tilde{E}) \cdot {\Psi'}^ \ast F) \cdot \gamma^{n_{11} - 1} = 0$\ $\Rightarrow \tilde{E} \cdot \gamma^{n_{11}} - d_{11}\tilde{E} \cdot {\Psi'} ^ \ast F \cdot \gamma^{n_{11} - 1} = 0$\ $\Rightarrow \tilde{E} \cdot {\Psi'}^ \ast (\zeta_1 - \mu_{11}F)^{n_{11}} = 0$ For the last part note that $\tilde{E} = \mathbb{P}(\pi_2^\ast Q_{11}) \times_{\mathbb{P}(E_2)} Z'$. Also $N(\tilde{Y}')$ and $N(\tilde{E})$ are free $N(Z')$-module. Using these informations and projection formula, the identity $\tilde{E} \cdot N(\tilde{Y'}) = (j_{\tilde{E}})_\ast N(\tilde{E})$ is obtained easily. Now we are in a position to prove the next theorem. $\overline{\operatorname{{Eff}}}^k(X) \cong \overline{\operatorname{{Eff}}}^k(Y') \cong \overline{\operatorname{{Eff}}}^k(Z')$ and $\overline{\operatorname{{Eff}}}^k(Z') \cong \overline{\operatorname{{Eff}}}^k(Z'')$. So, $\overline{\operatorname{{Eff}}}^k(X) \cong \overline{\operatorname{{Eff}}}^k(Z'')$ for $k < r_1 + r_2 - 1 - \mathbf{n}$ where $Z'= \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_2)$ and $ Z'' = \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_{21})$ Since $Y' = \mathbb{P}(\pi_2^ \ast E_1) \cong \mathbb{P}(E_1) \times_C \mathbb{P}(E_2) = X$, $\overline{\operatorname{{Eff}}}^k(X) \cong \overline{\operatorname{{Eff}}}^k(Y')$ is followed at once. To prove that $\overline{\operatorname{{Eff}}}^k(X) \cong \overline{\operatorname{{Eff}}}^k(Z')$ we first define the the map: $\theta_k: N^k(X) \longrightarrow N^k(Z')$ by $$\begin{aligned} \zeta_1^ i \cdot \zeta_2^ {k - i} \mapsto \bar{\zeta_1}^ i \cdot \bar{\zeta_2}^{k - i}, \quad F\ \cdot \zeta_1^j \cdot \zeta_2^{k - j - 1} \mapsto F_1 \cdot \bar{\zeta}_1^ j \cdot \bar{\zeta}_2^ {k - j - 1}\end{aligned}$$ where $\bar{\zeta_1} = \overline{p}_1 ^\ast(\mathcal{O}_{\mathbb{P}(E_{11})}(1))$ and $\bar{\zeta_2} = \overline{p}_2 ^\ast(\mathcal{O}_{\mathbb{P}(E_2)}(1))$. $ \overline{p}_1 : \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_2) \longrightarrow \mathbb{P}(E_{11})$ and $ \overline{p}_2 : \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_2) \longrightarrow \mathbb{P}(E_2)$ are respective projection maps. It is evident that the above map is in isomorphism of abstract groups. We claim that this induces an isomorphism between $ \overline{\operatorname{{Eff}}}^k(X)$ and $\overline{\operatorname{{Eff}}}(Z')$. First we construct an inverse for $\theta_k$. Define $\Omega_k : N^k(Z') \longrightarrow N^k(X)$ by $\Omega_k (l) = {\Psi'}_\ast {\Phi'}^\ast (l)$ $\Omega_k$ is well defined since $\Phi'$ is flat and $\Psi'$ is birational. $\Omega_k$ is also pseudo-effective. Now we need to show that $\Omega_k$ is the inverse of $\theta_k$. $$\begin{aligned} \Omega_k(\bar{\zeta_1}^ i \cdot \bar{\zeta_2}^{k - i}) & = {\Psi'}_\ast (({\Phi'}^ \ast \bar{\zeta_1})^i \cdot ({\Phi'}^\ast \bar{\zeta_2})^{k - i}) \\ & = {\Psi'}_\ast (({\Phi'}^\ast \zeta_{Z'})^i \cdot ({\Phi'}^\ast \bar{\zeta_2})^{k - i}) \\ & = {\Psi'}_\ast (({\Psi'}^\ast \zeta_1 - \tilde{E})^i \cdot ({\Psi'}^\ast \zeta_2)^{k - i}) \\ & = {\Psi'}_\ast((\sum_{0 \leq c \leq i} (-1)^ i \tilde{E}^c ({\Psi'}^\ast \zeta_1)^{i - c}) \cdot ({\Psi'}^\ast \zeta_2)^{k - i}) \\\end{aligned}$$ Similarly, $$\begin{aligned} \Omega_k(F_1 \cdot \bar{\zeta_1}^j \cdot \bar{\zeta_2}^{k - j -1}) = {\Psi'}_\ast (( \sum_{0 \leq d \leq j} (-1)^j \tilde{E}^d ({\Psi'}^\ast \zeta_1)^{j - d}) \cdot ({\Psi'}^\ast \zeta_2)^{k - j - 1})\end{aligned}$$ So, $\Omega_k\Big(\sum_i a_i\, \bar{\zeta_1}^ i \cdot \bar{\zeta_2}^{k - i} + \sum_j b_j \, F_1 \cdot \bar{\zeta_1}^j \cdot \bar{\zeta_2}^{k - j -1} \Big)$ $$\begin{aligned} = \Big(\sum_i a_i\,{\zeta_1}^ i \cdot{\zeta_2}^{k - i} + \sum_j b_j \, F \cdot{\zeta_1}^j \cdot{\zeta_2}^{k - j -1} \Big) + {\Psi'}_\ast \Big( \sum_i \sum_{1 \leq c \leq i} \tilde{E}^c {\Psi'}^\ast(\alpha_{i, c}) + \sum_j \sum_{1 \leq d \leq j} \tilde{E}^d {\Psi'}^\ast (\beta_{j, d})\Big)\end{aligned}$$ for some cycles $\alpha_{i, c}, \beta_{j, d} \in N(X)$. But, ${\Psi'}^\ast(\tilde{E}^t) = 0$ for all $1 \leq t \leq i \leq r_1 + r_2 - 1 - \mathbf{n}$ for dimensional reasons. Hence, the second part in the right hand side of the above equation vanishes and we make the conclusion that $\Omega_k = \theta_k ^{-1}$. Next we seek an inverse of $\Omega_k$ which is pseudo-effective and meet our demand of being equal to $\theta_k$. Define $\eta_k : N^k(X) \longrightarrow N^k(Z')$ by $$\begin{aligned} \eta_k(s) = {\Phi'}_\ast(\delta \cdot {\Psi'}^\ast s)\end{aligned}$$ where $ \delta = {\Psi'}^\ast (\xi_2 - \mu_{11}F)^{n_{11}}$. By the relations $(5)$ and $(6)$ , ${\Psi'}^ \ast( (\zeta_1^i \cdot \zeta_2^{k - i})$ is ${\Phi'}^\ast(\bar{\zeta_1}^i \cdot \bar{\zeta_2}^{k - i})$ modulo $\tilde{E}$ and $\delta \cdot \tilde{E} = 0$. Also ${\phi'}_\ast \delta = [Z']$ which is derived from the fact that ${\Phi'}_\ast \gamma^{n_{11}} = [Z']$ and the same relations $(5)$ and $(6)$. Therefore $$\begin{aligned} \eta_k(\zeta_1^i \cdot \zeta_2^{k - i}) = {\Phi'}_\ast (\delta \cdot {\Phi'}^\ast(\bar{\zeta_1}^i \cdot \bar{\zeta_2}^{k - i})) = (\bar{\zeta_1}^i \cdot \bar{\zeta_2}^{k - i}) \cdot [Z'] = \bar{\zeta_1}^i \cdot \bar{\zeta_2}^{k - i}\end{aligned}$$ In a similar way, ${\Psi'}^ \ast(F \cdot \zeta_1^ j \cdot \zeta_2^{k - j - 1})$ is ${\Phi'}^\ast (F_1 \cdot \bar{\zeta_1}^j \cdot \bar{\zeta_2}^{k - j -1})$ modulo $\tilde{\mathbf{E}}$ and as a result of this $$\begin{aligned} \eta_k(F \cdot \zeta_1^ j \cdot \zeta_2^{k - j - 1}) = F_1 \cdot \bar{\zeta_1}^j \cdot \bar{\zeta_2}^{k - j -1}\end{aligned}$$ So, $\eta_k = \theta_k$. Next we need to show that $\eta_k$ is a pseudo- effective map. Notice that ${\Psi'}^\ast s = \bar{s} + \mathbf{j}_\ast s'$ for any effective cycle $s$ on $X$, where $\bar{s}$ is the strict transform under $\Psi'$ and hence effective. Now $\delta$ is intersection of nef classes. So, $\delta \cdot \bar{s}$ is pseudo-effective. Also $\delta \cdot \mathbf{j}_\ast s' = 0$ from theorem 2.4 and ${\Phi'}_\ast$ is pseudo-effective. Therefore $\eta_k$ is pseudo-effective and first part of the theorem is proved. We will sketch the prove for the second part i.e. $\overline{\operatorname{{Eff}}}^k(Z') \cong \overline{\operatorname{{Eff}}}^k(Z'')$ which is similar to the proof of the first part. Consider the following diagram: $$\begin{tikzcd} Z'' = \mathbb{P}(E_{11}) \times_C \mathbb{P}(E_{21}) \arrow[r, "\hat{p}_2"] \arrow[d, "\hat{p}_1"] & \mathbb{P}(E_{21})\arrow[d,"\hat{\pi}_2"]\\ \mathbb{P}(E_{11}) \arrow[r, "\hat{\pi}_1" ] & C \end{tikzcd}$$ Define $\hat{\theta}_k : N^k(Z') \longrightarrow N^k(Z'')$ by $$\begin{aligned} \bar{\zeta_1}^ i \cdot \bar{\zeta_2}^ {k - i} \mapsto \hat{\zeta_1}^ i \cdot \hat{\zeta_2}^{k - i}, \quad F \cdot \bar{\zeta_1}^j \cdot \bar{\zeta_2}^ {k - j - 1} \mapsto F_2 \cdot \hat{\zeta_1}^j \cdot \hat{\zeta_2}^{k - j - 1}\end{aligned}$$ where $\hat{\zeta_1} = \hat{p_1}^\ast (\mathcal{O}_{\mathbb{P}(E_{11})}(1)), \hat{\zeta_2} = \hat{p_2}^\ast (\mathcal{O}_{\mathbb{P}(E_{21})}(1))$ and $F_2$ is the class of a fibre of $\hat{\pi_1} \circ \hat{p_1}$. This is a isomorphism of abstract groups and behaves exactly the same as $\theta_k$. The methods applied to get the result for $\theta_k$ can also be applied successfully here. Acknowledgement {#acknowledgement .unnumbered} --------------- The author would like to thank Prof. D.S. Nagaraj, IISER Tirupati for suggestions and discussions at every stage of this work. This work is supported financially by a fellowship from IMSc,Chennai (HBNI), DAE, Government of India. [\*\*\*\*\*]{} D. Chen, I. Coskun *Extremal higher codimension cycles on modulii spaces of curves*, Proc. Lond. Math. Soc. 111 (2015) 181- 204. I. Coskun, J. Lesieutre, J. C. Ottem *Effective cones of cycles on blow-ups of projective space*, Algebra Number Theory, 10(2016) 1983-2014. O. Debarre, L. Ein, R. Lazarsfeld, C. Voisin *Pseudoeffective and nef classes on abelian varieties*, Compos. Math. 147 (2011) 1793-1818. O. Debarre, Z. Jiang, C. Voisin *Pseudo-effective classes and pushforwards*, Pure Appl. Math. Q. 9 (2013) 643-664. D. Eisenbud, J. Harris *3264 and all that- a second course in algebraic geometry*, Cambridge University Press, Cambridge, 2016. Mihai Fulger *The cones of effective cycles on projective bundles over curves*, Math. Z. 269, (2011) 449-459. M. Fulger, B. Lehmann *Positive cones of dual cycle classes*, Algebr. Geom. 4(2017) 1- 28. M. Fulger, B. Lehmann *Kernels of numerical pushforwards*, Adv. Geom. 17(2017) 373-378. William Fulton *Intersection Theory, 2nd ed.*, Ergebnisse der Math. und ihrer Grenzgebiete(3), vol 2, Springer, Berlin(1998) Robin Hartshorne *Algebraic Geometry. Graduate Texts in Mathematics*, Springer- Verlag, New York Heidelberg, (1977) R. Karmakar, S. Misra, N. Ray *Nef and Pseudoeffective cones of product of projective bundles over a curve* Bull. Sci. math. (2018), https://doi.org/10.1016/j.bulsci.2018.12.002 N. Pintye, A. Prendergast-Smith *Effective cycles on some linear blow ups of projective spaces*, 2018, https://arxiv.org/abs/1812.08476.

ArXiv

--- abstract: 'We classify periodically driven quantum systems on a one-dimensional lattice, where the driving process is local and subject to a chiral symmetry condition. The analysis is in terms of the unitary operator at a half-period and also covers systems in which this operator is implemented directly, and does not necessarily arise from a continuous time evolution. The full-period evolution operator is called a quantum walk, and starting the period at half time, which is called choosing another timeframe, leads to a second quantum walk. We assume that these walks have gaps at the spectral points $\pm1$, up to at most finite dimensional eigenspaces. Walks with these gap properties have been completely classified by triples of integer indices (arXiv:1611.04439). These indices, taken for both timeframes, thus become classifying for half-step operators. In addition a further index quantity is required to classify the half step operators, which decides whether a continuous local driving process exists. In total, this amounts to a classification by five independent indices. We show how to compute these as Fredholm indices of certain chiral block operators, show the completeness of the classification, and clarify the relations to the two sets of walk indices. Within this theory we prove bulk-edge correspondence, where second timeframe allows to distinguish between symmetry protected edge states at $+1$ and $-1$ which is not possible with only one timeframe. We thus resolve an apparent discrepancy between our above mentioned index classification for walks, and indices defined (arXiv:1208.2143). The discrepancy turns out to be one of different definitions of the term ‘quantum walk’.' author: - 'C. Cedzich' - 'T. Geib' - 'A. H. Werner' - 'R. F. Werner' bibliography: - 'F2Wbib.bib' title: Chiral Floquet systems and quantum walks at half period ---

ArXiv

--- abstract: 'Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity may cause a current flow in the negative direction.' author: - 'M. Palmero' - 'E. Torrontegui' - 'J. G. Muga' - 'M. Modugno' title: 'Detecting quantum backflow by the density of a Bose-Einstein condensate' --- introduction ============ Quantum backflow is a fascinating quantum interference effect consisting in a negative current density for quantum wave packets without negative momentum components [@allcock]. It reflects a fundamental point about quantum measurements of velocity: usual measurements of momentum (velocity) distributions are performed globally - with no resolution in position - whereas the detection of the velocity field (or of the flux) implies a local measurement, that may provide values outside the domain of the global velocities, due to non commutativity of momentum and position [@local]. Despite its intriguing nature - obviously counterintuitive from a classical viewpoint - quantum backflow has not yet received as much attention as other quantum effects. Firstly discovered by Allcock in 1969 [@allcock], it only started to be studied in the mid 90’s. Bracken and Melloy [@BM] provided a bound for the maximal fraction of probability that can undergo backflow. Then, additional bounds and analytic examples, and its implications in the definition of arrival times of quantum particles where discussed by Muga *et al.* [@muga; @Leav; @Jus]. Recently, Berry [@BerryBack] analyzed the statistics of backflow for random wavefunctions, and Yearsley *et al.* [@year] studied some specific cases, clarifying the maximal backflow limit. However, so far no experiments have been performed, and a clear program to carry out one is also missing. Two important challenges are the measurement of the current density (the existing proposals for local and direct measurements are rather idealized schemes [@Jus]), and the preparation of states with a detectable amount of backflow. In this paper we derive a general relation that connects the current and the particle density, allowing for the detection of backflow by a density measurement, and propose a scheme for its observation with Bose-Einstein condensates in harmonic traps, that could be easily implemented with current experimental technologies. In particular, we show that preparing a condensate with positive-momentum components, and then further transferring a positive momentum kick to part of the atoms, causes under certain conditions, remarkably, a current flow in the negative direction. Bose-Einstein condensates are particularly promising for this aim because, besides their high level of control and manipulation, they are quantum matter waves where the probability density and flux are in fact a density and flux of particles –in contrast to a statistical ensemble of single particles sent one by one–, and in principle allow for the measurements of local properties in a single shot experiment. ![(Color online) a) A condensate is created in the ground state of a harmonic trap with frequency $\omega_{x}$; at $t=0$ we apply a magnetic gradient that shifts the trap by a distance $d$; b) the condensate starts to perform dipole oscillations in the trap; c) when it reaches a desired momentum $\hbar k_{1}$ the trap is switched off, and d) then the condensate is let expand for a time $t$; e) finally, a Bragg pulse is applied in order to transfer part of the atoms to a state of momentum $\hbar k_{2}$.[]{data-label="fig:scheme"}](fig1.eps){width="0.9\columnwidth"} general scheme ============== Let us start by considering a one dimensional Bose-Einstein condensate with a narrow momentum distribution centered around $\hbar k_{1}>0$, with negligible negative components. Then, we apply a Bragg pulse that transfers a momentum $\hbar q>0$ to part of the atoms [@bragg], populating a state of momentum $\hbar k_{2}=\hbar k_{1} + \hbar q$ (see Fig. \[fig:scheme\]). By indicating with $A_{1}$ and $A_{2}$ the amplitudes of the two momentum states, the total wave function is $$\Psi(x,t)=\psi(x,t)\left(A_{1} + A_{2}\exp\left[iq x + i\varphi\right]\right), \label{eq:bragg}$$ where we can assume $A_{1}, A_{2}\in \mathbb{R}^{+}$ without loss of generality (with $A_{1}^{2}+A_{2}^{2}=1$), $\varphi$ being an arbitrary phase. All these parameters (except the phase, that will be irrelevant in our scheme), can be controlled and measured in the experiment. Then, by writing the wave function of the initial wave packet as $$\psi(x,t)=\phi(x,t)\exp[i\theta(x,t)]$$ with $\phi$ and $\theta$ being real valued functions, the expression for the total current density, $J_{\Psi}(x,t) =(\hbar/m)\textrm{Im}\left[\Psi^{*}\nabla\Psi\right]$ can be easily put in the form $$\frac{m}{\hbar}J_{\Psi}(x,t) =(\nabla\theta)\rho_{\Psi}+\frac12q \left[\rho_{\Psi}+|\phi|^{2} (A_{2}^{2} -A_{1}^{2})\right], \label{eq:current}$$ with $\rho_{\Psi}(x,t)=|\phi(x,t)|^{2}\left(A_{1}^{2} + A_{2}^{2}+ 2A_{1}A_{2}\cos\left(qx + \varphi\right)\right)$ being the total density. Therefore, a negative flux, $J_{\Psi}(x,t)<0$, corresponds to the following inequality for the density $$\textrm{sign}[\eta(x,t)]\rho_{\Psi}(x,t)< \frac{1}{|\eta(x,t)| }|\phi(x,t)|^{2} (A_{1}^{2} -A_{2}^{2}),$$ where we have defined $\eta(x,t)= 1 +2\nabla\theta(x,t)/q$. Later on we will show that $\eta(x,t)<0$ corresponds to a *classical regime*, whereas for $\eta(x,t)>0$ the backflow is a purely quantum effect, without any classical counterpart. Therefore, in the *quantum regime*, backflow takes place when the density is below the following critical threshold: $$\rho_{\Psi}^{crit}(x,t)= \frac{q}{q +2\nabla\theta(x,t) }|\phi(x,t)|^{2} (A_{1}^{2} -A_{2}^{2}). \label{eq:denscrit}$$ This is a fundamental relation that allows to detect backflow by a density measurement. It applies to any class of wavepackets of the form (\[eq:bragg\]), including the superposition of two plane waves discussed in [@Jus; @year]. We remark that, while in the ideal case of plane waves backflow repeats periodically at specific time intervals, for the present case it is limited to the transient when the two wave packets with momenta $\hbar k_{1}$ and $\hbar k_{2}$ are superimposed. An implementation with BEC ========================== State preparation ----------------- In order to propose a specific experimental implementation, we consider a condensate in a three dimensional harmonic trap, with axial frequency $\omega_{x}$. We assume a tight radial confinement, $\omega_{\perp}\gg\omega_{x}$, so that the wave function can be factorized in a radial and axial components (in the non interacting case this factorization is exact)[@note]. In the following we will focus on the 1D axial dynamics, taking place in the waveguide provided by the transverse confinement, that is assumed to be always on. We define $a_{x}=\sqrt{\hbar/m\omega_{x}}$, $a_{\perp}=\sqrt{\hbar/m\omega_{\perp}}$. The scheme proceeds as highlighted in Fig.\[fig:scheme\]. The condensate is initially prepared in the ground state $\psi_{0}$ of the harmonic trap. Then, at $t=0$ the trap is suddenly shifted spatially by $d$ (Fig.\[fig:scheme\]a) and the condensate starts to perform dipole oscillations (Fig.\[fig:scheme\]b). Next, at $t=t_{1}$, when the condensate has reached a desired momentum $m v_{1}=\hbar k_{1}=\hbar k(t_{1})$, the trap is switched off (Fig.\[fig:scheme\]c). At this point we let the condensate expand freely for a time $t$ (Fig.\[fig:scheme\]d). Hereinafter we will consider explicitly two cases, namely a noninteracting condensate and the Thomas-Fermi (TF) limit [@dalfovo], that can both be treated analytically. In fact, in both cases the expansion can be expressed by a scaling transformation $$\begin{aligned} \label{eq:scaling} \psi(x,t)&=&\frac{1}{\sqrt{b(t)}}\psi_{0}\left(\frac{x-v_{1}t}{b(t)}\right) \\ &\times&\exp\left[i\frac{m}{2\hbar }x^{2} \frac{\dot{b}(t)}{b(t)} +i k_{1} x\left(1 -\frac{\dot{b}}{b}t\right) + i\beta(t)\right], \nonumber\end{aligned}$$ where $b(t)$ represents the scaling parameter and $\beta(t)$ is an irrelevant global phase (for convenience we have redefined time and spatial coordinates, so that at $t=0$ - when the trap is switched off - the condensate is centered at the origin). This expression can be easily obtained by generalizing the scaling in [@castin; @kagan] to the case of an initial velocity field. In the *non interacting case*, the initial wave function is a minimum uncertainty Gaussian, $\psi_{0}(x)=({1}/{\pi^{\frac14}\sqrt{a_{x}}})\exp \left[-{x^2}/({2 a_{x}^2})\right]$, and the scaling parameter evolves as $b(t)=\sqrt{1+\omega_{x}^{2}t^{2}}$ [@dalfovo; @merzbacher]. For a TF distribution we have $\psi_{0}(x)=\left[\left(\mu- \frac12 m\omega_{x}^{2}x^{2}\right)/g_{1D}\right]^{1/2}$ for $|x|<R_{TF}\equiv\sqrt{2\mu/m\omega_{x}^{2}}$ and vanishing elsewhere, with $g_{1D}=g_{3D}/(2\pi a_{\perp}^{2})$ [@salasnich], and the chemical potential $\mu$ fixed by the normalization condition $\int\!dx|\psi|^{2}=N$, the latter being the number of atoms in the condensate [@dalfovo]. In this case $b(t)$ satisfies $\ddot{b}(t)=\omega_{x}^{2}/b^{2}(t)$, whose asymptotic solution, for $t\gg1/\omega_{x}$, is $b(t)\simeq \sqrt{2} t \omega_{x}$ [@sp]. Finally, we apply a Bragg pulse as discussed previously (Fig.\[fig:scheme\]e). We may safely assume the duration of the pulse to be very short with respect to the other timescales of the problem [@bragg2]. Then, the resulting wave function is that in Eq. (\[eq:bragg\]), with the corresponding critical density for backflow in Eq. (\[eq:denscrit\]). We have $$\phi(x)=\frac{1}{\sqrt{b(t)}}\psi_{0}\left(\frac{x-(\hbar k_{1}/m)t}{b(t)}\right),$$ while the expression for the phase gradient is $$\nabla\theta=\frac{m}{\hbar }x \frac{\dot{b}(t)}{b(t)} + k_{1}\left(1 -\frac{\dot{b}(t)}{b(t)}t\right) \label{eq:grad-theta}$$ that, in the asymptotic limit $t\gg1/\omega_{x}$, yields the same result $\nabla\theta={mx}/{\hbar t}$ for both the non interacting and TF wave packets. Eventually, backflow can be probed by taking a snapshot of the interference pattern just after the Bragg pulse, measuring precisely its minimum, and comparing it to the critical density. Classical effects ----------------- Before proceeding to the quantum backflow, let us discuss the occurrence of a classical backflow. To this end it is sufficient to consider the flux of a single wave packet (before the Bragg pulse), namely $J_{\psi}(x,t) = (\hbar/m)|\phi|^{2}\nabla\theta$. In this case the flux is negative for $x<v_{1}(t-b/\dot{b})= :x_-(t)$ (see Eq. (\[eq:grad-theta\])). Then, by indicating with $R_{0}$ the initial half width of the wave packet and with $R_{L}(t)=-b(t) R_{0} +v_{1}t$ its left border at time $t$, a negative flux occurs when $R_{L}<x_-$, that is for $R_{0}>v_{1}/\dot{b}(t)$. In the asymptotic limit, the latter relation reads $R_{0}>f v_{1}/\omega_{x}$ ($f=1,1/\sqrt{2}$ for the non interacting and TF cases, respectively). On the other hand, the momentum width of the wave packet is $\Delta_{p}\approx\hbar/R_{0}$ [@momentumwidth], so that the negative momentum components can be safely neglected only when $mv_{1}\gg \hbar/R_{0}$. From these two conditions, we get that there is a negative flux (even in the absence of initial negative momenta) when $R_{0}\gg a_{x}$. This can be easily satisfied in the TF regime. In fact, in that case the backflow has a classical counterpart due to the force $F = -\partial_{x}(g_{1D}|\rho_{\psi}(x,t)|^{2})$ implied by the repulsive interparticle interactions [@castin]. These interactions are responsible for the appearance of negative momenta and backflow. Then, sufficient conditions for avoiding these classical effects are $k_{1}\gg 1/a_{x}$ and $R_{0}<f v_{1}/\omega_{x}$. Quantum backflow ---------------- Let us now turn to the quantum backflow. In order to discuss the optimal setup for having backflow, it is convenient to consider the following expression for the current density, $$\begin{aligned} \label{eq:current2} J_{\Psi}(x,t) &=& \frac{\hbar}{m}|\phi|^{2}\left[q\left(A_{2}^{2}+ A_{1}A_{2}\cos\left(q x + \varphi\right)\right) \right. \\ &&+ \left.\nabla\theta\left(A_{1}^{2} + A_{2}^{2}+ 2A_{1}A_{2}\cos\left(q x + \varphi\right)\right)\right], \nonumber\end{aligned}$$ that follows directly from Eq. (\[eq:current\]) and the expression for the total density. Let us focus on its behavior around the center of wave packet, namely at $x\approx(\hbar k_{1}/m)t=d\omega_{x}t$ (the following analysis extends to the whole packet if $d\omega_{x}t$ is much larger than the condensate width). In the asymptotic limit, the phase gradient at the center is $\nabla\theta|_{c}\approx k_{1}$, and the flux in Eq. (\[eq:current2\]) turns out to be proportional to that of the superposition of two plane waves of momenta $k_{1}$ and $k_{2}=k_{1}+q$. This limit is particularly useful because for two plane waves the probability density is a sinusoidal function and the critical density becomes a constant, which in practice makes irrelevant the value of the arbitrary phase $\varphi$ we cannot control. Then, the condition for having backflow at the wave packet center is $$k_{1}A_{1}^{2} + k_{2}A_{2}^{2}+ (k_{1}+k_{2})A_{1}A_{2}\cos\left(q x + \varphi\right)<0.$$ Since all the parameters $k_{i}$ and $A_{i}$ are positive, the minimal condition for having a negative flux is $k_{1}A_{1}^{2} + k_{2}A_{2}^{2} < (k_{1}+k_{2})A_{1}A_{2}$ (for $\cos(\cdot)=-1$), that can be written as $$F(\alpha,A_{2})\equiv 1 + \alpha A_{2}^{2} - (2 + \alpha)A_{2}\sqrt{1-A_{2}^{2}}<0, \label{eq:fmin0}$$ where we have defined $\alpha=q/k_{1}$. The behavior of the function $F(\alpha,A_{2})$ in the region where $F<0$ is depicted in Fig. \[fig:function\]. ![(Color online) Plot of the function $F(\alpha,A_{2})$ defined in Eq. (\[eq:fmin0\]). For a given value of $\alpha$, the maximal backflow is obtained for the value $A_{2}$ that minimizes $F$.[]{data-label="fig:function"}](fig2.eps){width="0.7\columnwidth"} In particular, for a given value of the relative momentum kick $\alpha$, the minimal value of $F$ is obtained for $A_{2}$ that solves $\partial F/\partial A_{2}|_{\alpha}=0$, that is for $$2\alpha A_{2}\sqrt{1-A_{2}^{2}} + (2 + \alpha)(2 A_{2}^{2} -1)=0. \label{eq:minF}$$ In order to maximize the effect of backflow and its detection, one has to satisfy a number of constraints. In principle, Fig. \[fig:function\] shows that the larger the value of $\alpha=q/k_{1}$, the larger the effect of backflow is. However, $q$ cannot be arbitrarily large as it fixes the wavelength $\lambda=2\pi/q$ of the density modulations, which must be above the current experimental spatial resolution $\sigma_{r}$, $\lambda\gg\sigma_{r}$, for allowing a clean experimental detection of backflow (see later on). In addition, as discussed before, $k_{1}$ should be sufficiently large for considering negligible the negative momentum components of the initial wave packet, $k_{1}\gg 1/a_{x}$. Therefore, since the maximal momentum that the condensate may acquire after a shift $d$ of the trap is $\hbar k_{1}=m\omega_{x}d$, the latter condition reads $d\gg a_{x}$. By combining the two conditions above, we get the hierarchy $1\ll d/a_{x}\ll(2\pi/\alpha)(a_{x}/\sigma_{r})$. Furthermore, we recall that in the interacting case we must have $R_{0}<f v_{1}/\omega_{x}=f d$ in order to avoid classical effects. Therefore, given the value of the current imaging resolution, the non interacting case ($R_{0}\approx a_{x}$) appears more favorable than the TF one (where typically $R_{TF}\gg a_{x}$). Nevertheless, the latter condition can be substantially softened if the measurement is performed at the wave packet center, away from the left tail where classical effects take place. ![(Color online) Plot of the flux $J_{\Psi}(x)$. Backflow corresponds to $J_{\Psi}<0$. Positions are measured with respect to the wave packet center.[]{data-label="fig:flux"}](fig3.eps){width="0.7\columnwidth"} An example with $^{7}$Li ------------------------ As a specific example, here we consider the case of an almost noninteracting $^{7}$Li condensate [@salomon] prepared in the ground state of a trap with frequency $\omega_{x}=2\pi\times1~$Hz (yielding $a_{x}\simeq 38~\mu$m). Then, we shift the trap by $d=80~\mu$m, so that after a time $t_{1}=\pi/(2\omega_{x})=250$ ms the condensate has reached its maximal velocity $\hbar k_{1}/m=\omega_{x}d\simeq 0.5$ mm/s. At this point the axial trap is switched off, and the condensate is let expand for a time $t\gg a_{x}/\omega_{x}d$ until it enters the asymptotic plane wave regime (here we use $t=1$ s). Finally we apply a Bragg pulse of momentum $\hbar q=\alpha \hbar k_{1}$, with $\alpha=3$, that transfers $24\%$ of the population to the state of momentum $\hbar k_{2}$, according to Eq. (\[eq:minF\]) ($A_{2}=0.49$, $A_{1}\simeq0.87$) [@bragg]. The resulting flux is shown in Fig. \[fig:flux\], where the backflow is evident, and more pronounced at the wave packet center. The corresponding density is displayed in Fig. \[fig:dens\], where it is compared with the critical value of Eq. (\[eq:denscrit\]). The values obtained around the center are $\rho_{\Psi}^{min}\simeq 8\%$ and $\rho_{\Psi}^{crit}\simeq 17\%$. ![(Color online) Density (solid) and critical density (dashed) for the case discussed in the text. Backflow occurs in the regions where the density is below the critical value, see Fig. \[fig:flux\]. Positions are measured with respect to the wave packet center.[]{data-label="fig:dens"}](fig4.eps){width="0.7\columnwidth"} Effects of imaging resolution ----------------------------- Let us now discuss more thoroughly the implication of a finite imaging resolution $\sigma_r$. First we note that experimentally it is difficult to obtain a precise measurement of the absolute density, because of uncertainties in the calibration of the imaging setup. Instead, measurements in which the densities at two different points are compared are free from calibration errors and therefore are more precise. Owing to this, it is useful to normalize the total density $\rho_{\Psi}(x,t)=|\phi(x,t)|^{2}\left(A_{1}^{2} + A_{2}^{2}+ 2A_{1}A_{2}\cos\left(qx + \varphi\right)\right)$ to its maximal value $\rho_{\Psi}^{max}\simeq|\phi^{max}|^{2}\left(A_{1}^{2} + A_{2}^{2}+ 2A_{1}A_{2}\right)$. In addition, we have to take into account that, due to the finite resolution $\sigma_{r}$ [@resolution], the sinusoidal term $\cos\left(qx + \varphi\right)$ is reduced by a factor $\zeta=\exp[-q^{2}\sigma_{r}^{2}/2]$ after the imaging. Then, by indicating with $x_{min}(t)$ the position of the density minima, we have $$\left.\frac{\rho_{\Psi}(x_{min},t)}{\rho_{\Psi}^{max}}\right|_{exp}=\frac{|\phi(x_{min},t)|^{2}}{|\phi^{max}|^{2}} \frac{A_{1}^{2}+ A_{2}^{2} -2\zeta A_{1}A_{2}}{A_{1}^{2}+ A_{2}^{2}+2\zeta A_{1}A_{2}},$$ where “exp” refers to the experimental conditions. Instead, the normalized critical density is (close the wave packet center, where $\nabla\theta\approx k_{1}$) $$\frac{\rho_{\Psi}^{crit}(x,t)}{\rho_{\Psi}^{max}}=\frac{q}{q +2k_{1}}\frac{|\phi(x,t)|^{2}}{|\phi^{max}|^{2}} \frac{A_{1}^{2} -A_{2}^{2}}{\left(A_{1} + A_{2}\right)^{2}},$$ so that, assuming that $|\phi(x,t)|^{2}$ varies on a scale much larger than $\sigma_{r}$ to be unaffected by the finite imaging resolution, the condition for *observing* a density drop below the critical value reads $$\frac{A_{1}^{2}+ A_{2}^{2} -2\zeta A_{1}A_{2}}{A_{1}^{2}+ A_{2}^{2}+2\zeta A_{1}A_{2}} =\frac{\alpha}{\alpha +2} \frac{A_{1} -A_{2}}{A_{1} + A_{2}}.$$ In particular, in the example case we have discussed, backflow could be clearly detected with an imaging resolution of about $3~\mu$m, which is within reach of current experimental setups. conclusions and outlooks ======================== In conclusion, we have presented a feasible experimental scheme that could lead to the first observation of quantum backflow, namely the presence of a negative flux for states with negligible negative momentum components. By using current technologies for ultracold atoms, we have discussed how to imprint backflow on a Bose-Einstein condensate and how to detect it by a usual density measurement. Remarkably, the presence of backflow is signalled by the density dropping below a critical threshold. Other possible detection schemes could for example make use of *local* velocity-selective internal state transitions in order to spatially separate atoms travelling in opposite directions. Finally, we remark that a comprehensive understanding of backflow is not only important for its fundamental relation with the meaning of quantum velocity, but as also because of its implications on the use of arrival times as information carriers [@muga; @Leav]. It may as well lead to interesting applications such as the development of a matter wave version of an optical tractor beam [@tractor], namely, particles sent from a source that could attract towards the source region other distant particles, in some times and locations. M. M. is grateful to L. Fallani and C. Fort for useful discussions and valuable suggestions. We acknowledge funding by Grants FIS2012-36673-C03-01, No. IT472-10 and the UPV/EHU program UFI 11/55. M. P. and E. T. acknowledge fellowships by UPV/EHU. [10]{} G. R. Allcock, Annals of Physics **53**, 253 (1969); **53**, 286 (1969); **53**, 311 (1969). J. G. Muga, J. P. Palao, and R. Sala, Phys. Lett. A **238**, 90 (1998). A. J. Bracken and G. F. Melloy, J. Phys. A: Math. Gen. **27**, 2197 (1994). J. G. Muga, J. P. Palao and C. R. Leavens, Phys. Lett. **A253**, 21 (1999). J. G. Muga and C. R. Leavens, Phys. Rep. **338**, 353 (2000). J. A. Damborenea, I. L. Egusquiza, G. C. Hegerfeldt, and J. G. Muga, Phys. Rev. A **66**, 052104 (2002). M. V. Berry, J. Phys. A: Math. Theor. **43**, 415302 (2010). J. M. Yearsley, J. J. Halliwell, R. Hartshorn and A. Whitby, Phys. Rev. A **86**, 042116 (2012). M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. **82**, 871 (1999). This factorization hypothesis allows for a full analytic treatment. Nevertheless, we expect backflow to survive even in case of a generic elongated trap. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, S. Stringari, Rev. Mod. Phys. **71**, 463 (1999). Y. Castin and R. Dum, Phys. Rev. Lett. **77**, 5315 (1996). Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A **54**, 1753 (1996). It is easy to check that Eq. (\[eq:scaling\]) can be put in the form of Eq. (15.50) in E. Merzbacher, *Quantum Mechanics*, 3rd ed., (John Wiley and Sons, New York, 1998). L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A **65**, 043614 (2002). L. Sanchez-Palencia, D. Clèment, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Phys. Rev. Lett. **98**, 210401 (2007). The duration of the Bragg pulse can be of the order of few hundreds of $\mu$s, therefore very short with respect to timescale of the condensate dynamics. J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. **82**, 4569 (1999); *ibid.* **84**, 2283 (2000); G. Baym and C. J. Pethick, Phys. Rev. Lett. **76**, 6 (1996). $\Delta_{p}=\hbar/a_{x}$ for a gaussian wave packet [@dalfovo], and $\Delta_{p}=(\sqrt{21/8})\hbar/R_{TF}$ in the TF limit [@stenger]. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. **87**, 080403 (2001). In general, the point-spread-function of the imaging system can be safely approximated with a gaussian function of width $\sigma_r$. S. Sukhov and A. Dogariu, Opt. Lett. **35**, 3847 (2010).

ArXiv

--- abstract: 'In this paper we address the problem of efficient estimation of Sobol sensitivy indices. First, we focus on general functional integrals of conditional moments of the form ${\mathbb{E}}(\psi({\mathbb{E}}(\varphi(Y)|X)))$ where $(X,Y)$ is a random vector with joint density $f$ and $\psi$ and $\varphi$ are functions that are differentiable enough. In particular, we show that asymptotical efficient estimation of this functional boils down to the estimation of crossed quadratic functionals. An efficient estimate of first-order sensitivity indices is then derived as a special case. We investigate its properties on several analytical functions and illustrate its interest on a reservoir engineering case.' author: - 'Sébastien Da Veiga[^1] and Fabrice Gamboa[^2]' title: Efficient Estimation of Sensitivity Indices --- density estimation, semiparametric Cramér-Rao bound, global sensitivity analysis. 2G20, 62G06, 62G07, 62P30 Introduction ============ In the past decade, the increasing interest in the design and analysis of computer experiments motivated the development of dedicated and sharp statistical tools [@santner03]. Design of experiments, sensitivity analysis and proxy models are examples of research fields where numerous contributions have been proposed. More specifically, global Sensitivity Analysis (SA) is a key method for investigating complex computer codes which model physical phenomena. It involves a set of techniques used to quantify the influence of uncertain input parameters on the variability in numerical model responses. Recently, sensitivity studies have been applied in a large variety of fields, ranging from chemistry [@CUK73; @T90] or oil recovery [@IMDR01] to space science [@Carra07] and nuclear safety [@IVD06].\ In general, global SA refers to the probabilistic framework, meaning that the uncertain input parameters are modelled as a random vector. By propagation, every computer code output is itself a random variable. Global SA techniques then consists in comparing the probability distribution of the output with the conditional probability distribution of the output when some of the inputs are fixed. This yields in particular useful information on the impact of some parameters. Such comparisons can be performed by considering various criteria, each one of them providing a different insight on the input-output relationship. For example, some criteria are based on distances between the probability density functions (e.g. $L^1$ and $L^2$ norms ([@borgo07]) or Kullback-Leibler distance ([@LCS06]), while others rely on functionals of conditional moments. Among those, variance-based methods are the most widely used [@salcha00]. They evaluate how the inputs contribute to the output variance through the so-called Sobol sensitivity indices [@SOB93], which naturally emerge from a functional ANOVA decomposition of the output [@hoeff48; @owen94; @anto84]. Interpretation of the indices in this setting makes it possible to exhibit which input or interaction of inputs most influences the variability of the computer code output. This can be typically relevant for model calibration [@kenoha01] or model validation [@bayber07].\ Consequently, in order to conduct a sensitivity study, estimation of such sensitivity indices is of great interest. Initially, Monte-Carlo estimates have been proposed [@SOB93; @MCK95]. Recent work also focused on their asymptotic properties [@janon12]. However, in many applications, calls to the computer code are very expensive, from several minutes to hours. In addition, the number of inputs can be large, making Monte-Carlo approaches untractable in practice. To overcome this problem, recent work focused on the use of metamodeling techniques. The complex computer code is approximated by a mathematical model, referred to as a “metamodel”, which should be as representative as possible of the computer code, with good prediction capability. Once the metamodel is built and validated, it is used in the extensive Monte-Carlo sampling instead of the complex numerical model. Several metamodels can be used: polynomials, Gaussian process metamodels ([@OOH04], [@IMDR01]) or local polynomials ([@SDV09]). However, in these papers, the approach is generally empirical in the sense that no convergence study is performed and do not provide any insight about the asymptotic behavior of the sensitivity indices estimates. The only exception is the work of @SDV09, where the authors investigate the convergence of a local-polynomial based estimate using the work of @FG96 and @WJ94. In particular, this plug-in estimate achieves a nonparametric convergence rate.\ In this paper, we go one step further and propose the first asymptotically efficient estimate for sensitivity indices. More precisely, we investigate the problem of efficient estimation of some general nonlinear functional based on the density of a pair of random variables. Our approach follows the work of @BL96 [@BL05], and we also refer to @LEV78 and @KIKI96 for general results on nonlinear functionals estimation. Such functionals of a density appear in many statistical applications and their efficient estimation remains an active research field [@gine2008a; @gine2008b; @chacon11]. However we consider functionals involving conditional densities, which necessitate a specific treatment. The estimate obtained here can be used for global SA involving general conditional moments, but it includes as a special case Sobol sensitivity indices. Note also that an extension of the approach developed in our work is simultaneously proposed in the context of sliced inverse regression [@loubes11].\ The paper is organized as follows. Section \[sa\] first recaps variance-based methods for global SA. In particular, we point out which type of nonlinear functional appears in sensitivity indices. Section \[model\] then describes the theoretical framework and the proposed methodology for building an asymptotically efficient estimator. In Section \[examples\], we focus on Sobol sensitivity indices and study numerical examples showing the good behavior of the proposed estimate. We also illustrate its interest on a reservoir engineering example, where uncertainties on the geology propagate to the potential oil recovery of a reservoir. Finally, all proofs are postponed to the appendix. Global sensitivity analysis {#sa} =========================== In many applied fields, physicists and engineers are faced with the problem of estimating some sensitivity indices. These indices quantify the impact of some input variables on an output. The general situation may be formalized as follows.\ The output $Y\in{\mathbb{R}}$ is a nonlinear regression of input variables $\boldsymbol{\tau}=(\tau_1,\ldots,\tau_l)$ ($l\geq 1$ is generally large). This means that $Y$ and $\boldsymbol{\tau}$ satisfy the input-output relationship $$Y=\Phi(\boldsymbol{\tau}) \label{sobol}$$ where $\Phi$ is a known nonlinear function. Usually, $\Phi$ is complicated and has not a closed form, but it may be computed through a computer code [@OOH04]. In general, the input $\boldsymbol{\tau}$ is modelled by a random vector, so that $Y$ is also a random variable. A common way to quantify the impact of input variables is to use the so-called Sobol sensitivity indices [@SOB93]. Assuming that all the random variables are square integrable, the Sobol index for the input $\tau_j$ ($j=1,\ldots,l$) is $$\Sigma_j=\frac{{\textrm{Var}}({\mathbb{E}}(Y|\tau_j))}{{\textrm{Var}}(Y)}. \label{si}$$ Observing an i.i.d. sample $(Y_1,\boldsymbol{\tau}^{(1)}),\ldots,(Y_n,\boldsymbol{\tau}^{(n)})$ (with $Y_i=\Phi(\boldsymbol{\tau}^{(i)})$, $i=1,\ldots,n$), the goal is is then to estimate $\Sigma_j$ ($j=1,\ldots,l$). Obviously, (\[si\]) may be rewritten as $$\Sigma_j=\frac{{\mathbb{E}}({\mathbb{E}}(Y|\tau_j)^2)-{\mathbb{E}}(Y)^2}{{\textrm{Var}}(Y)}.$$ Thus, in order to estimate $\Sigma_j$, the hard part is ${\mathbb{E}}({\mathbb{E}}(Y|\tau_j)^2)$. In this paper we will provide an asymptotically efficient estimate for this kind of quantity. More precisely we will tackle the problem of asymptotically efficient estimation of some general nonlinear functional.\ Let us specify the functionals we are interested in. Let $(Y_1,X_1),\ldots,(Y_n,X_n)$ be a sample of i.i.d. random vectors of ${\mathbb{R}}^2$ having a [*regular*]{} density $f$ (see Section \[model\] for the precise frame). We will study the estimation of the nonlinear functional $$\begin{aligned} T(f)&=& {\mathbb{E}}\Big(\psi\big({\mathbb{E}}(\varphi(Y)|X)\big)\Big)\\ &=& \iint \psi\left(\frac{\int \varphi(y)f(x,y)dy}{\int f(x,y)dy}\right)f(x,y)dxdy\end{aligned}$$ where $\psi$ and $\varphi$ are regular functions. Hence, the Sobol indices are the particular case obtained with $\psi(\xi)=\xi^2$ and $\varphi(\xi)=\xi$.\ The method developed in order to obtain an asymptotically efficient estimate for $T(f)$ follows the one developed by @BL96. Roughly speaking, it involves a preliminary estimate $\hat{f}$ of $f$ built on a small part of the sample. This preliminary estimate is used in a Taylor expansion of $T(f)$ up to the second order in a neighbourhood of $\hat{f}$. This expansion allows to remove the bias that occurs when using a direct plug-in method. Hence, the bias correction involves a quadratic functional of $f$. Due to the form of $T$, this quadratic functional of $f$ may be written as $$\theta(f)=\iiint \eta(x,y_1,y_2)f(x,y_1)f(x,y_2)dxdy_1dy_2.$$ This kind of functional does not fall in the frame treated in @BL96 or @gine2008a and have not been studied to the best of our knowledge. We study this problem in Section \[quad\] where we build an asymptotically efficient estimate for $\theta$. Efficient estimation of $T(f)$ is then investigated in Section \[main\]. Model frame and method {#model} ====================== Let $a<b$ and $c<d$, $L^2(dxdy)$ will denote the set of square integrable functions on $[a,b]\times [c,d]$. Further, $L^2(dx)$ (resp. $L^2(dy)$) will denote the set of square integrable functions on $[a,b]$ (resp. $[c,d]$). For sake of simplicity, we work in the whole paper with the Lebesgue measure as reference measure. Nevertheless, most of the results presented can be obtained for a general reference measure on $[a,b]\times [c,d]$. Let $(\alpha_{i_{\alpha}}(x))_{i_{\alpha}\in D_1}$ (resp. $(\beta_{i_{\beta}}(y))_{i_{\beta}\in D_2}$) be a countable orthonormal basis of $L^2(dx)$ (resp. of $L^2(dy)$). We set $p_i(x,y)=\alpha_{i_{\alpha}}(x) \beta_{i_{\beta}}(y)$ with $i=(i_{\alpha},i_{\beta})\in D:=D_1\times D_2$. Obviously $(p_i(x,y))_{i\in D}$ is a countable orthonormal (tensor) basis of $L^2(dxdy)$. We will also use the following subset of $L^2(dxdy)$ : $$\mathcal{E}=\left\{ \sum_{i\in D} e_ip_i : (e_i)_{i\in D}\ \textrm{is a sequence with} \sum_{i\in D} \left|\frac{e_i}{c_i} \right|^2 \leq 1\right\},$$ here $(c_i)_{i\in D}$ is a given fixed positive sequence.\ Let $(X,Y)$ having a bounded joint density $f$ on $[a,b]\times [c,d]$ from which we have a sample $(X_i,Y_i)_{i=1,\ldots,n}$. We will also assume that $f$ lies in the ellipsoid $\mathcal{E}$. Recall that we wish to estimate a conditional functional $${\mathbb{E}}\Big(\psi\big({\mathbb{E}}(\varphi(Y)|X)\big)\Big)$$ where $\varphi$ is a measurable bounded function with $\chi_1\leq \varphi\leq\chi_2$ and $\psi\in C^3([\chi_1,\chi_2])$ the set of thrice continuously differentiable functions on $[\chi_1,\chi_2]$. This last quantity can be expressed in terms of an integral depending on the joint density $f$: $$\begin{aligned} T(f) & = &\iint \psi\left(\frac{\int \varphi(y)f(x,y)dy}{\int f(x,y)dy}\right)f(x,y)dxdy.\\ & =& \iint \psi(m(x))f(x,y)dxdy\end{aligned}$$ where $m(x)=\int \varphi(y)f(x,y)dy/\int f(x,y)dy$ is the conditional expectation of $\varphi(Y)$ given $(X=x)$. We suggest as a first step to consider a preliminary estimator $\hat{f}$ of $f$, and to expand $T(f)$ in a neighborhood of $\hat{f}$. To achieve this goal we first define $F:[0,1]\rightarrow{\mathbb{R}}$ : $$F(u)=T(uf+(1-u)\hat{f}) \quad (u\in[0,1]).$$ The Taylor expansion of $F$ between $0$ and $1$ up to the third order is $$F(1)=F(0)+F'(0)+\frac{1}{2}F''(0)+\frac{1}{6}F'''(\xi)(1-\xi)^3 \label{taylorF}$$ for some $\xi\in]0,1[$. Here, we have $$F(1)=T(f)$$ and $$\begin{aligned} F(0)=T(\hat{f})&=&\iint \psi\left(\frac{\int \varphi(y)\hat{f}(x,y)dy}{\int \hat{f}(x,y)dy}\right)\hat{f}(x,y)dxdy\\ &=&\iint \psi(\hat{m}(x))\hat{f}(x,y)dxdy\end{aligned}$$ where $\hat{m}(x)=\int \varphi(y)\hat{f}(x,y)dy/\int \hat{f}(x,y)dy$. Straightforward calculations also give higher-order derivatives of $F$ : $$F'(0)=\iint \left(\big[\varphi(y)-\hat{m}(x)\big]\dot\psi(\hat{m}(x))+\psi(\hat{m}(x))\right)\Big(f(x,y)-\hat{f}(x,y)\Big)dxdy\\$$ $$\begin{aligned} F''(0)&=&\iiint \frac{\ddot\psi(\hat{m}(x))}{\left(\int\hat{f}(x,y)dy\right)} \big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\\ &&\Big(f(x,y)-\hat{f}(x,y)\Big)\Big(f(x,z)-\hat{f}(x,z)\Big)dxdydz\\\end{aligned}$$ $$\begin{aligned} F'''(\xi)&=&\iiiint \frac{\left(\int\hat{f}(x,y)dy\right)^2}{\left(\int \xi f(x,y)+(1-\xi)\hat{f}(x,y)dy\right)^{5}}\\ &&\left[\big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\big(\hat{m}(x)-\varphi(t)\big)\right.\\ &&\left(\int\hat{f}(x,y)dy\right)\dddot\psi\left(\hat{r}(\xi,x)\right)- 3\big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\\ &&\left.\left(\int [\xi f(x,y)+(1-\xi)\hat{f}(x,y)]dy\right) \ddot\psi\left(\hat{r}(\xi,x)\right)\right]\\ &&\Big(f(x,y)-\hat{f}(x,y)\Big)\Big(f(x,z)-\hat{f}(x,z)\Big)\\ &&\Big(f(x,t)-\hat{f}(x,t)\Big)dxdydzdt\end{aligned}$$ where ${\displaystyle}{\hat{r}(\xi,x)=\frac{\int\varphi(y)[\xi f(x,y)+(1-\xi)\hat{f}(x,y)]dy}{\int [\xi f(x,y)+(1-\xi)\hat{f}(x,y)]dy}}$ and $\dot\psi$, $\ddot\psi$ and $\dddot\psi$ denote the three first derivatives of $\psi$.\ Plugging these expressions into (\[taylorF\]) yields the following expansion for $T(f)$: where $$\begin{aligned} H(\hat{f},x,y)&=& \big[\varphi(y)-\hat{m}(x)\big]\dot\psi(\hat{m}(x))+\psi(\hat{m}(x)),\\ K(\hat{f},x,y,z)&=& \frac{1}{2}\frac{\ddot\psi(\hat{m}(x))}{\left(\int\hat{f}(x,y)dy\right)} \big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big),\\ \Gamma_n&=&\frac{1}{6}F'''(\xi)(1-\xi)^3\end{aligned}$$ for some $\xi\in]0,1[$. Notice that the first term is a linear functional of the density $f$, it will be estimated with $$\frac{1}{n_2}\sum_{j=1}^{n_2} H(\hat{f},X_j,Y_j).$$ The second one involves a crossed term integral which can be written as $$\iiint \eta(x,y_1,y_2)f(x,y_1)f(x,y_2)dxdy_1dy_2 \label{fq}$$ where $\eta:{\mathbb{R}}^3\rightarrow{\mathbb{R}}$ is a bounded function verifying $\eta(x,y_1,y_2)=\eta(x,y_2,y_1)$ for all $(x,y_1,y_2)\in{\mathbb{R}}^3$. In summary, the first term can be easily estimated, unlike the second one which deserves a specific study. In the next section we then focus on the asymptotically efficient estimation of such crossed quadratic functionals. In Section \[main\], these results are finally used to propose an asymptotically efficient estimator for $T(f)$. Efficient estimation of quadratic functionals {#quad} --------------------------------------------- In this section, our aim is to build an asymptotically efficient estimate for $$\theta=\iiint \eta(x,y_1,y_2)f(x,y_1)f(x,y_2)dxdy_1dy_2.$$ We denote $a_i=\int fp_i$ the scalar product of $f$ with $p_i$ as defined at the beginning of Section \[model\]. We will first build a projection estimator achieving a bias equal to $$-\iiint \left[S_Mf(x,y_1)-f(x,y_1)\right]\left[S_Mf(x,y_2)-f(x,y_2)\right]\eta(x,y_1,y_2)dxdy_1dy_2$$ where $S_Mf=\sum_{i\in M} a_ip_i$ and $M$ is a subset of $D$. Thus, the bias would only be due to projection. Developing the previous expression leads to a goal bias equal to $$\begin{aligned} &&2\iiint S_Mf(x,y_1)f(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\notag\\ &&-\iiint S_Mf(x,y_1)S_Mf(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\notag\\ &&-\iiint f(x,y_1)f(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2. \label{biais2}\end{aligned}$$ Consider now the estimator $\hat{\theta}_n$ defined by $$\begin{aligned} \hat{\theta}_{n}&=&\frac{2}{n(n-1)}\sum_{i\in M}\sum_{j\neq k=1}^{n}p_{i}(X_{j},Y_{j})\int p_{i}(X_{k},u)\eta(X_{k},u,Y_{k})du\notag \\ &&-\frac{1}{n(n-1)}\sum_{i,i'\in M}\sum_{j\neq k=1}^{n}p_{i}(X_{j},Y_{j})p_{i'}(X_{k},Y_{k})\notag \\ &&\int p_{i}(x,y_{1})p_{i'}(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}. \label{est}\end{aligned}$$ This estimator achieves the desired bias : \[biais0\] The estimator $\hat{\theta}_{n}$ defined in (\[est\]) estimates $\theta$ with bias equal to $$-\iiint [S_{M}f(x,y_{1})-f(x,y_{1})][S_{M}f(x,y_{2})-f(x,y_{2})]\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}.$$ Since we will carry out an asymptotic analysis, we will work with a sequence $(M_n)_{n\geq 1}$ of subsets of $D$. We will need an extra assumption concerning this sequence:\ - For all $n\geq 1$, we can find a subset $M_n\subset D$ such that $\left(\sup_{i\notin M_n}|c_{i}|^{2}\right)^{2}\approx \frac{|M_n|}{n^2}$ ($A_n\approx B_n$ means $\lambda_1\leq A_n/B_n\leq \lambda_2$ for some positive constants $\lambda_1$ and $\lambda_2$). Furthermore, $\forall t\in L^2(dxdy)$, ${\displaystyle}{\int (S_{M_n}t-t)^2dxdy\rightarrow 0}$ when $n\rightarrow\infty.$ The following theorem gives the most important properties of our estimate $\hat{\theta}_n$ : \[tfq\] Assume A1 hold. Then $\hat{\theta}_{n}$ has the following properties: - If $|M_n|/n\rightarrow 0$ when $n\rightarrow \infty$, then $$\sqrt{n}\left(\hat{\theta}_n-\theta\right)\rightarrow {\mathcal{N}}\left(0,\Lambda(f,\eta)\right), \label{na}$$ $$\left| {\mathbb{E}}\left(\hat{\theta}_n-\theta\right)^2 - \Lambda(f,\eta)\right| \leq \gamma_1\left[ \frac{|M_n|}{n}+\|S_{M_n}f-f\|_2+\|S_{M_n}g-g\|_2\right], \label{ea}$$ where ${\displaystyle}{g(x,y):=\int f(x,u)\eta(x,y,u)du}$ and $$\Lambda(f,\eta)=4 \left[ \iint g(x,y)^2f(x,y)dxdy -\left( \iint g(x,y)f(x,y)dxdy\right)^2\right].$$ - Otherwise $${\mathbb{E}}\left(\hat{\theta}_n-\theta\right)^2 \leq \gamma_2\frac{|M_n|}{n},$$ where $\gamma_1$ and $\gamma_2$ are constants depending only on $\|f\|_{\infty}$, $\|\eta\|_{\infty}$ and $\Delta_Y$ (with $\Delta_Y=d-c$). Moreover, these constants are increasing functions of these quantities. Since in our main result (to be given in the next section) $\eta$ will depend on $n$ through the preliminary estimator $\hat{f}$, we need in (\[ea\]) a bound that depends explicitly on $n$. Note however that (\[ea\]) implies $$\lim_{n\rightarrow\infty} n{\mathbb{E}}\left(\hat{\theta}_n-\theta\right)^2 =\Lambda(f,\eta).$$ The asymptotic properties of $\hat{\theta}_n$ are of particular importance, in the sense that they are optimal as stated in the following theorem. \[cramerrao1\] Consider the estimation of $$\theta=\theta(f)=\iiint \eta(x,y_1,y_2)f(x,y_1)f(x,y_2)dxdy_1dy_2.$$ Let $f_0\in\mathcal{E}$. Then, for all estimator $\hat{\theta}_n$ of $\theta(f)$ and every family $\mathcal{V}(f_0)$ of vicinities of $f_0$, we have $$\inf_{\{\mathcal{V}(f_0)\}} \liminf_{n\rightarrow \infty} \sup_{f\in\mathcal{V}(f_0)} n{\mathbb{E}}(\hat{\theta}_n-\theta(f_0))^2\geq \Lambda(f_0,\eta).$$ In other words, the optimal asymptotic variance for the estimation of $\theta$ is $\Lambda(f_0,\eta)$. As our estimator defined in (\[est\]) achieves this variance, it is therefore asymptotically efficient. We are now ready to use this result to propose an efficient estimator of $T(f)$. Main Theorem {#main} ------------ In this section we come back to our main problem of the asymptotically efficient estimation of $$T(f)=\iint \psi\left(\frac{\int \varphi(y)f(x,y)dy}{\int f(x,y)dy}\right)f(x,y)dxdy.$$ Recall that we have derived in (\[taylorT\]) an expansion for $T(f)$. The key idea is to use here the previous results on the estimation of crossed quadratic functionals. Indeed we have provided an asymptotically efficient estimator for the second term of this expansion, conditionally on $\hat{f}$. A natural and straightforward estimator for $T(f)$ is then $$\begin{aligned} \widehat{T}_n&=& \frac{1}{n_2} \sum_{j=1}^{n_2} H(\hat{f},X_j,Y_j)\\ &&+ \frac{2}{n_2(n_2-1)}\sum_{i\in M}\sum_{j\neq k=1}^{n_2}p_{i}(X_{j},Y_{j})\int p_{i}(X_{k},u)K(\hat{f},X_{k},u,Y_{k})du\\ &&-\frac{1}{n_2(n_2-1)}\sum_{i,i'\in M}\sum_{j\neq k=1}^{n_2}p_{i}(X_{j},Y_{j})p_{i'}(X_{k},Y_{k})\\ &&\int p_{i}(x,y_{1})p_{i'}(x,y_{2})K(\hat{f},x,y_{1},y_{2})dxdy_{1}dy_{2}.\end{aligned}$$ In the above expression, one can note that the remainder $\Gamma_n$ does not appear : we will see in the proof of the following theorem that it is negligible comparing to the two first terms.\ In order to study the asymptotic properties of $\widehat{T}_n$, some assumptions are required concerning the behavior of the joint density $f$ and its preliminary estimator $\hat{f}$ : - $\textrm{supp} f \subset [a,b]\times [c,d]$ and $\forall (x,y)\in \textrm{supp} f$, $0<\alpha\leq f(x,y)\leq\beta$ with $\alpha,\beta\in{\mathbb{R}}$\ - One can find an estimator $\hat{f}$ of $f$ built with $n_1\approx n/\log(n)$ observations, such that $$\forall (x,y)\in \textrm{supp} f,\; 0<\alpha-\epsilon\leq \hat{f}(x,y)\leq\beta+\epsilon.$$ Moreover, $$\forall 2\leq q<+\infty,\; \forall l\in\mathbb{N}^*,\; {\mathbb{E}}_f\|\hat{f}-f\|_q^l\leq C(q,l)n_1^{-l\lambda}$$ for some $\lambda > 1/6$ and some constant $C(q,l)$ not depending on $f$ belonging to the ellipsoid $\mathcal{E}$.\ Here $\textrm{supp} f$ denotes the set where $f$ is different from $0$. Assumption A2 is restrictive in the sense that only densities with compact support can be considered, excluding for example a Gaussian joint distribution.\ Assumption A3 imposes to the estimator $\hat{f}$ a convergence fast enough towards $f$. We will use this result to control the remainder term $\Gamma_n$.\ We can now state the main theorem of the paper. It investigates the asymptotic properties of $\widehat{T}_n$ under assumptions A1, A2 and A3. \[tfec\] Assume that A1, A2 and A3 hold. Then $\widehat{T}_n$ has the following properties if ${\displaystyle}{\frac{|M_n|}{n}\rightarrow 0}$: $${\displaystyle}{\sqrt{n}\left(\widehat{T}_n-T(f)\right)\rightarrow {\mathcal{N}}\left(0,C(f)\right)},\\ \label{na2}$$ $$\lim_{n\rightarrow\infty} n{\mathbb{E}}\left(\widehat{T}_n-T(f)\right)^2 = C(f), \label{ea2}$$ where $C(f)={\mathbb{E}}\bigg({\textrm{Var}}(\varphi(Y)|X)\Big[\dot\psi\big({\mathbb{E}}(Y|X)\big)\Big]^2\bigg)+{\textrm{Var}}\Big(\psi\big({\mathbb{E}}(\varphi(Y)|X)\big)\Big)$.\ We can also compute as in the previous section the semiparametric Cramér-Rao bound for this problem. \[cramerrao2\] Consider the estimation of $$T(f)=\iint\psi\left(\frac{\int \varphi(y) f(x,y)dy}{\int f(x,y)dy}\right) f(x,y)dxdy={\mathbb{E}}\Big(\psi\big({\mathbb{E}}(\varphi(Y)|X)\big)\Big)$$ for a random vector $(X,Y)$ with joint density $f\in\mathcal{E}$. Let $f_0\in\mathcal{E}$ be a density verifying the assumptions of Theorem \[tfec\]. Then, for all estimator $\widehat{T}_n$ of $T(f)$ and every family $\mathcal{V}(f_0)$ of vicinities of $f_0$, we have $$\inf_{\{\mathcal{V}(f_0)\}} \liminf_{n\rightarrow \infty} \sup_{f\in\mathcal{V}(f_0)} n{\mathbb{E}}(\widehat{T}_n-T(f_0))^2\geq C(f_0).$$ Combination of theorems \[tfec\] and \[cramerrao2\] finally proves that $\widehat{T}_n$ is asymptotically efficient. Application to the estimation of sensitivity indices {#examples} ==================================================== Now that we have built an asymptotically efficient estimate for $T(f)$, we can apply it to the particular case we were initially interested it: the estimation of Sobol sensitivity indices. Let us then come back to model (\[sobol\]) : $$Y=\Phi(\boldsymbol{\tau})$$ where we wish to estimate (\[si\]): $$\Sigma_j=\frac{{\textrm{Var}}({\mathbb{E}}(Y|\tau_j))}{{\textrm{Var}}(Y)}=\frac{{\mathbb{E}}({\mathbb{E}}(Y|\tau_j)^2)-{\mathbb{E}}(Y)^2}{{\textrm{Var}}(Y)} \quad j=1,\ldots,l.$$ To do so, we have an i.i.d. sample $(Y_1,\boldsymbol{\tau}^{(1)}),\ldots,(Y_n,\boldsymbol{\tau}^{(n)})$. We will only give here the procedure for the estimation of $\Sigma_1$ since it will be the same for the other sensitivity indices. Denoting $X:=\tau_1$, this problem is equivalent to estimating ${\mathbb{E}}({\mathbb{E}}(Y|X)^2)$ with an i.i.d. sample $(Y_1,X_1),\ldots,(Y_n,X_n)$ with joint density $f$. We can hence apply the estimate we developed previously by letting $\psi(\xi)=\xi^2$ and $\varphi(\xi)=\xi$: $$\begin{aligned} T(f)&=& {\mathbb{E}}({\mathbb{E}}(Y|X)^2)\\ &=& \iint \left(\frac{\int yf(x,y)dy}{\int f(x,y)dy}\right)^2f(x,y)dxdy.\end{aligned}$$ The Taylor expansion in this case becomes $$\begin{aligned} T(f)&=&\iint H(\hat{f},x,y)f(x,y)dxdy\\ &&+\iiint K(\hat{f},x,y,z)f(x,y)f(x,z)dxdydz+\Gamma_n \end{aligned}$$ where $$\begin{aligned} H(\hat{f},x,y)&=& 2y\hat{m}(x)-\hat{m}(x)^2,\\ K(\hat{f},x,y,z)&=&\frac{1}{\left(\int\hat{f}(x,y)dy\right)} \big(\hat{m}(x)-y\big)\big(\hat{m}(x)-z\big)\end{aligned}$$ and the corresponding estimator is $$\begin{aligned} \widehat{T}_n&=& \frac{1}{n_2} \sum_{j=1}^{n_2} H(\hat{f},X_j,Y_j)\\ &&+ \frac{2}{n_2(n_2-1)}\sum_{i\in M}\sum_{j\neq k=1}^{n_2}p_{i}(X_{j},Y_{j})\int p_{i}(X_{k},u)K(\hat{f},X_{k},u,Y_{k})du\\ &&-\frac{1}{n_2(n_2-1)}\sum_{i,i'\in M}\sum_{j\neq k=1}^{n_2}p_{i}(X_{j},Y_{j})p_{i'}(X_{k},Y_{k})\\ &&\int p_{i}(x,y_{1})p_{i'}(x,y_{2})K(\hat{f},x,y_{1},y_{2})dxdy_{1}dy_{2}.\end{aligned}$$ for some preliminary estimator $\hat{f}$ of $f$, an orthonormal basis $(p_i)_{i\in D}$ of $L^2(dxdy)$ and a subset $M\subset D$ verifying the hypotheses of Theorem \[tfec\].\ We propose now to investigate the practical behavior of this estimator on two analytical models and on a reservoir engineering test case. In all subsequent simulation studies, the preliminary estimator $\hat{f}$ will be a kernel density estimator with bounded support built on $n_1=[\log(n)/n]$ observations. Moreover, we choose the Legendre polynomials on $[a,b]$ and $[c,d]$ to build the orthonormal basis $(p_i)_{i\in D}$ and we will take $|M|=\sqrt{n}$. Finally, the integrals in $\widehat{T}_n$ are computed with an adaptive Simpson quadrature.\ Simulation study on analytical functions ---------------------------------------- The first model we investigate is $$Y=\tau_1 + \tau_2^4 \label{model1}$$ where three configurations are considered ($\tau_1$ and $\tau_2$ being independent): - $\tau_j\sim \mathcal{U}(0,1)$, $j=1,2$; - $\tau_j\sim \mathcal{U}(0,3)$, $j=1,2$; - $\tau_j\sim \mathcal{U}(0,5)$, $j=1,2$. For each configuration, we report the results obtained with $n=100$ and $n=10000$ in Table \[tab\_modele31\]. Note that we repeat the estimation 100 times with different different random samples of $\left(\tau_1,\tau_2\right)$. \[tab\_modele31\] The asymptotically efficient estimator $\widehat{T}_n$ gives a very accurate approximation of sensitivity indices when $n=10000$. But surprisingly, it also gives a reasonably accurate estimate when $n$ only equals $100$, whereas it has been built to achieve the best symptotic rate of convergence.\ It is then interesting to compare it with other estimators, more precisely two nonparametric estimators that have been specifically built to give an accurate approximation of sensitivity indices when $n$ is not large. The first one is based on a Gaussian process metamodel [@OOH04], while the other one involves local polynomial estimators [@SDV09]. The comparison is performed on the following model : $$\begin{aligned} Y&=&0.2\exp(\tau_1-3)+2.2|\tau_2|+1.3\tau_2^6-2\tau_2^2-0.5\tau_2^4-0.5\tau_1^4 \notag \\ &&+2.5\tau_1^2+0.7\tau_1^3+\frac{3}{(8\tau_1-2)^2+(5\tau_2-3)^2+1}+\sin(5\tau_1)\cos(3\tau_1^2) \label{model2}\end{aligned}$$ where $\tau_1$ and $\tau_2$ are independent and uniformly distributed on $[-1,1]$. This nonlinear function is interesting since it presents a peak and valleys. We estimate the sensitivity indices with a sample of size $n=100$, the results are given in Table \[comp1\].\ \[comp1\] Globally, the best estimates are given by the local polynomials technique. However, the accuracy of the asymptotically efficient estimator $\widehat{T}_n$ is comparable to that of the nonparametric ones. These results confirm that $\widehat{T}_n$ is a valuable estimator even with a rather complex model and a small sample size (recall that here $n=100$).\ Reservoir engineering example ----------------------------- The PUNQ test case (Production forecasting with UNcertainty Quantification) is an oil reservoir model derived from real field data [@manmez01]. The considered reservoir is surrounded by an aquifer in the north and the west, and delimited by a fault in the south and the east. The geological model is composed of five independent layers, three of good quality and two of poorer quality. Six producer wells (PRO-1, PRO-4, PRO-5, PRO-11, PRO-12 and PRO-15) have been drilled, and production is supported by four additional wells injecting water (X1, X2, X3 and X4). The geometry of the reservoir and the well locations are given in Figure \[punq\], left. In this setting, 7 variables which are characteristic of media, rocks, fluids or aquifer activity, are considered as uncertain: the coefficient of aquifer strength (AQUI), horizontal and vertical permeability multipliers in good layers (MPV1 and MPH1, respectively), horizontal and vertical permeability multipliers in poor layers (MPV2 and MPH2, respectively), residual oil saturation after waterflood and after gas flood (SORW and SORG, respectively). We focus here on the cumulative production of oil of this field during 12 years. In practice, a fluid flow simulator is used to forecast this oil production for every value of the uncertain parameters we might want to investigate. The uncertain parameters are assumed to be uniformly distributed, with ranges given in Table \[tabpunq\]. We draw a random sample of size $n = 200$ of these 7 parameters, and perform the corresponding fluid-flow simulations to compute the cumulative oil production after 12 years. The histogram of the production obtained with this sampling is depicted in Figure \[punq\], right. Clearly, the impact of the uncertain parameters on oil production is large, since different values yield forecats varying by tens of thousands of oil barrels. In this context, reservoir engineers aim at identifying which parameters affect the most the production. This help them design strategies in order to reduce the most influential uncertainties, which will reduce, by propagation, the uncertainty on production forecasts.\ In this context, computation of sensivity indices is of great interest. Starting from the random sample of size $n=200$, we then estimate the first-order sensitivity index of each parameter with the estimator $\widehat{T}_n$. Results are given in Table \[tabpunq\]. \[tabpunq\] As expected, the most influential parameters are the horizontal permeability multiplier in the good reservoir units MPH1 and the residual oil saturation after waterflood SORW. Indeed, fluid dispacement towards the producer wells is mainly driven by the permeability in units with good petrophysical properties and by water injection. More interestingly, vertical permeability multipliers do not seem to impact oil production in this case. This means that fluid displacements are mainly horizontal in this reservoir. Discussion and conclusions ========================== In this paper, we developed a framework to build an asymptotically efficient estimate for nonlinear conditional functionals. This estimator is both practically computable and has optimal asymptotic properties. In particular, we show how Sobol sensitivty indices appear as a special case of our estimator. We investigate its practical behavior on two analytical functions, and illustrate that it can compete with metamodel-based estimators. A reservoir engineering application case is also studied, where geological and petrophysical uncertain parameters affect the forecasts on oil production. The methodology developed here will be extended to other problems in forthcoming work. A very attractive extension is the construction of an adaptive procedure to calibrate the size of $M_n$ as done in @BL05 for the $L^2$ norm. However, this problem is non obvious since it would involve treating refined inequalities on U-statistics such as presented in @HR02. From a sensitivity analysis perspective, we will also investigate efficient estimation of other indices based on entropy or other norms. Ideally, this would give a general framework for building estimates in global sensitivity analysis. Acknowledgements {#acknowledgements .unnumbered} ================ Many thanks are due to A. Antoniadis, B. Laurent and F. Wahl for helpful discussion. This work has been partially supported by the French National Research Agency (ANR) through COSINUS program (project COSTA-BRAVA ANR-09-COSI-015). [9]{} Antoniadis, A. (1984). Analysis of variance on function spaces. *Math. Oper. 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Proofs of Theorems ================== Proof of Lemma \[biais0\] ------------------------- Let $\hat{\theta}_{n}=\hat{\theta}_{n}^1-\hat{\theta}_{n}^2$ where $$\hat{\theta}_{n}^1=\frac{2}{n(n-1)}\sum_{i\in M}\sum_{j\neq k=1}^{n}p_{i}(X_{j},Y_{j})\int p_{i}(X_{k},u)\eta(X_{k},u,Y_{k})du$$ and $$\begin{aligned} \hat{\theta}_{n}^2&=&\frac{1}{n(n-1)}\sum_{i,i'\in M}\sum_{j\neq k=1}^{n}p_{i}(X_{j},Y_{j})p_{i'}(X_{k},Y_{k})\\ &&\int p_{i}(x,y_{1})p_{i'}(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}.\end{aligned}$$ Let us first compute ${\mathbb{E}}(\hat{\theta}_{n}^1)$ : $$\begin{aligned} {\mathbb{E}}(\hat{\theta}_{n}^1)&=&2\sum_{i\in M} \iint p_i(x,y)f(x,y)dxdy \iiint p_i(x,y)\eta(x,u,y)f(x,y)dxdydu\\ &=& 2\sum_{i\in M} a_i \iiint p_i(x,y)\eta(x,u,y)f(x,y)dxdydu\\ &=& 2\iiint \left(\sum_{i\in M} a_i p_i(x,y)\right) \eta(x,u,y)f(x,y)dxdydu\\ &=& 2\iiint S_Mf(x,y)\eta(x,u,y)f(x,y)dxdydu.\end{aligned}$$ Furthermore, $$\begin{aligned} {\mathbb{E}}(\hat{\theta}_{n}^2)&=&\sum_{i,i'\in M} \iint p_i(x,y)f(x,y)dxdy\iint p_{i'}(x,y)f(x,y)dxdy\\ &&\int p_{i}(x,y_{1})p_{i'}(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}\\ &=&\sum_{i,i'\in M} a_ia_{i'}\int p_{i}(x,y_{1})p_{i'}(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}\\ &=& \int \left(\sum_{i\in M}a_ip_{i}(x,y_{1})\right)\left(\sum_{i'\in M}a_{i'}p_{i'}(x,y_{2})\right)\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}\\ &=&\int S_Mf(x,y_1)S_Mf(x,y_2)\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}.\end{aligned}$$ Finally, ${\mathbb{E}}(\hat{\theta}_{n})-\theta={\mathbb{E}}(\hat{\theta}_{n}^1)-{\mathbb{E}}(\hat{\theta}_{n}^2)-\theta$ and we get the desired bias with (\[biais2\]). Proof of Theorem \[tfq\] ------------------------ We will write $M$ instead of $M_n$ for readability and denote $m=|M|$. We want to bound the precision of $\hat{\theta}_n$. We first write $${\mathbb{E}}\left(\hat{\theta}_{n}-\iiint \eta(x,y_{1},y_{2})f(x,y_{1})f(x,y_{2})dxdy_{1}dy_{2}\right)^{2}={\textrm{Bias}}^{2}(\hat{\theta}_{n})+{\textrm{Var}}(\hat{\theta}_{n}).$$ The first term of this decomposition can be easily bounded, since $\hat{\theta}_{n}$ has been built to achieve a bias equal to $$\begin{aligned} {\textrm{Bias}}(\hat{\theta}_{n})&=&-\iiint [S_{M}f(x,y_{1})-f(x,y_{1})][S_{M}f(x,y_{2})-f(x,y_{2})]\\ &&\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}.\end{aligned}$$ We then get the following lemma : \[biais1\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $$|{\textrm{Bias}}(\hat{\theta}_{n})|\leq\Delta_{Y}\|\eta\|_{\infty}\sup_{i\notin M} |c_{i}|^{2}.$$ $$\begin{aligned} |{\textrm{Bias}}(\hat{\theta}_{n})|&\leq&\|\eta\|_{\infty}\int \left(\int |S_{M}f(x,y_{1})-f(x,y_{1})|dy_{1}\right)\\ &&\left(\int |S_{M}f(x,y_{2})-f(x,y_{2})|dy_{2}\right)dx\\ &\leq& \|\eta\|_{\infty}\int\left(\int|S_{M}f(x,y)-f(x,y)|dy\right)^{2}dx\\ &&\leq\Delta_{Y}\|\eta\|_{\infty}\iint (S_{M}f(x,y)-f(x,y))^{2}dxdy\\ &\leq&\Delta_{Y}\|\eta\|_{\infty}\sum_{i\notin M} |a_{i}|^{2}\leq\Delta_{Y}\|\eta\|_{\infty}\sup_{i\notin M} |c_{i}|^{2}.\end{aligned}$$ Indeed, $f\in \mathcal{E}$ and the last inequality follows from Hölder inequality. Bounding the variance of $\hat{\theta}_{n}$ is however less straightforward. Let $A$ and $B$ be the $m\times 1$ vectors with components $$\begin{aligned} a_{i}&:=&\iint f(x,y)p_{i}(x,y)dxdy\quad i=1,\ldots,m\\ b_{i}&:=&\iiint p_{i}(x,y_{1})f(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}\\ &=&\iint g(x,y) p_{i}(x,y)dxdy\quad i=1,\ldots,m\end{aligned}$$ where ${\displaystyle}{g(x,y)=\int f(x,u)\eta(x,y,u)du}$ for each $i\in M$. $a_{i}$ et $b_{i}$ are the components of $f$ and $g$ onto the $i$th component of the basis. Let $Q$ and $R$ be the $m\times 1$ vectors of the centered functions $q_{i}(x,y)=p_{i}(x,y)-a_{i}$ and ${\displaystyle}{r_{i}(x,y)=\int p_{i}(x,u)\eta(x,u,y)du-b_{i}}$ for $i=1,\ldots,m$. Let $C$ be the $m\times m$ matrix of constants ${\displaystyle}{c_{ii'}=\iiint p_{i}(x,y_{1})p_{i'}(x,y_{2})\eta(x,y_{1},y_{2})dxdy_{1}dy_{2}}$ for $i,i'=1,\ldots,m$. Take care that here $c_{ii'}$ is double subscript unlike in the $(c_i)$ sequence appearing in the definition of the ellipsoid $\mathcal{E}$. We denote by $U_{n}$ the process ${\displaystyle}{U_{n}h=\frac{1}{n(n-1)}\sum_{j\neq k=1}^{n}h(X_{j},Y_{j},X_{k},Y_{k})}$ and by $P_{n}$ the empirical measure ${\displaystyle}{P_{n}f=\frac{1}{n}\sum_{j=1}^{n}f(X_{j},Y_{j})}$. With the previous notation, $\hat{\theta}_{n}$ has the following Hoeffding’s decomposition (see chapter 11 of @VV98): $$\hat{\theta}_{n}=U_{n}K+P_{n}L+2{{\vphantom{A}}^{\mathit t}{A}}B-{{\vphantom{A}}^{\mathit t}{A}}CA \label{thetaH}$$ where $$\begin{aligned} K(x_1,y_1,x_2,y_2)&=&2{{\vphantom{Q}}^{\mathit t}{Q}}(x_1,y_1)R(x_2,y_2)-{{\vphantom{Q}}^{\mathit t}{Q}}(x_1,y_1)CQ(x_2,y_2),\\ L(x_1,y_1)&=&2{{\vphantom{A}}^{\mathit t}{A}}R(x_1,y_1)+2{{\vphantom{B}}^{\mathit t}{B}}Q(x_1,y_1)-2{{\vphantom{A}}^{\mathit t}{A}}CQ(x_1,y_1).\end{aligned}$$ Then ${\textrm{Var}}(\hat{\theta}_{n})={\textrm{Var}}(U_{n}K)+{\textrm{Var}}(P_{n}L)+2\;{\textrm{Cov}}(U_{n}K,P_{n}L)$. We have to get bounds for each of these terms : they are given in the three following lemmas. \[var1\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $${\textrm{Var}}(U_{n}K)\leq \frac{20}{n(n-1)}\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}(m+1).$$ Since $U_{n}K$ is centered, ${\textrm{Var}}(U_{n}K)$ equals $$\begin{aligned} {\mathbb{E}}\left(\frac{1}{(n(n-1))^{2}}\sum_{j\neq k=1}^{n}\sum_{j'\neq k'=1}^{n}K(X_{j},Y_{j},X_{k},Y_{k})K(X_{j'},Y_{j'},X_{k'},Y_{k'})\right)\\ =\frac{1}{n(n-1)}{\mathbb{E}}(K^{2}(X_{1},Y_{1},X_{2},Y_{2})+K(X_{1},Y_{1},X_{2},Y_{2})K(X_{2},Y_{2},X_{1},Y_{1})).\\\end{aligned}$$ By the Cauchy-Schwarz inequality, $${\textrm{Var}}(U_{n}K) \leq \frac{2}{n(n-1)}{\mathbb{E}}(K^{2}(X_{1},Y_{1},X_{2},Y_{2})).$$ Moreover, the inequality $2|{\mathbb{E}}(XY)|\leq {\mathbb{E}}(X^{2})+{\mathbb{E}}(Y^{2})$ leads to $$\begin{aligned} {\mathbb{E}}(K^{2}(X_{1},Y_{1},X_{2},Y_{2}))&\leq& 2\left[{\mathbb{E}}\left((2Q'(X_{1},Y_{1})R(X_{2},Y_{2}))^{2}\right)\right.\\ &&\left.+{\mathbb{E}}\left((Q'(X_{1},Y_{1})CQ(X_{2},Y_{2}))^{2}\right) \right].\end{aligned}$$ We have to bound these two terms. The first one is $${\mathbb{E}}\left((2Q'(X_{1},Y_{1})R(X_{2},Y_{2}))^{2}\right)=4(W_{1}-W_{2}-W_{3}+W_{4})$$ where $$\begin{aligned} W_{1}&=&\int \!\!\!\int \!\!\!\int \!\!\!\int \!\!\!\int \!\!\!\int \sum_{i,i'}p_{i}(x,y)p_{i'}(x,y)p_{i}(x',u)p_{i'}(x',v\eta(x',u,y')\eta(x',v,y')\\ &&f(x,y)f(x',y')dudvdxdydx'dy'\\ W_{2}&=&\iint\sum_{i,i'}b_{i}b_{i'}p_{i}(x,y)p_{i'}(x,y)f(x,y)dxdy\\ W_{3}&=&\iiiint \sum_{i,i'}a_{i}a_{i'}p_{i}(x,u)p_{i'}(x,v)\eta(x,u,y)\eta(x,v,y)f(x,y)dxdy\\ W_{4}&=&\sum_{i,i'}a_{i}a_{i'}b_{i}b_{i'}.\end{aligned}$$ Straightforward manipulations show that $W_2\geq 0$ and $W_3\geq 0$. This implies that $${\mathbb{E}}\left((2Q'(X_{1},Y_{1})R(X_{2},Y_{2}))^{2}\right)\leq 4(W_{1}+W_{4}).$$ On the one hand, $$\begin{aligned} W_{1}&=&\iiiint \sum_{i,i'} p_i(x,y)p_{i'}(x,y) \int p_i(x',u)\eta(x',u,y')du\int p_{i'}(x',v)\eta(x',v,y')dvf(x,y)f(x',y')dxdydx'dy'\\ &\leq&\iiiint \left( \sum_ip_i(x,y)\int p_i(x',u)\eta(x',u,y')du\right)^2f(x,y)f(x',y')dxdydx'dy'\\ &\leq&\|f\|_{\infty}^{2} \iiiint \left( \sum_ip_i(x,y)\int p_i(x',u)\eta(x',u,y')du\right)^2dxdydx'dy' \\ &\leq&\|f\|_{\infty}^{2}\iiiint \sum_{i,i'} p_i(x,y)p_{i'}(x,y) \int p_i(x',u)\eta(x',u,y')du\int p_{i'}(x',v)\eta(x',v,y')dvdxdydx'dy'\\ &\leq&\|f\|_{\infty}^{2}\sum_{i,i'} \iint p_i(x,y)p_{i'}(x,y)dxdy\iint \left(\int p_i(x',u)\eta(x',u,y')du\right)\left(\int p_{i'}(x',v)\eta(x',v,y')dv\right) dx'dy'\\ &\leq&\|f\|_{\infty}^{2}\sum_i \iint \left(\int p_i(x',u)\eta(x',u,y')du\right)^2dx'dy'\end{aligned}$$ since the $p_{i}$ are orthonormal. Moreover, $$\begin{aligned} \left(\int p_i(x',u)\eta(x',u,y')du\right)^2 &\leq& \left(\int p_i(x',u)^2du\right)\left(\int \eta(x',u,y')^2du\right)\\ &&\leq \|\eta\|_{\infty}^{2}\Delta_{Y}\int p_i(x',u)^2du,\end{aligned}$$ and then $$\begin{aligned} \iint \left(\int p_i(x',u)\eta(x',u,y')du\right)^2dx'dy'&\leq & \|\eta\|_{\infty}^{2}\Delta_{Y}^2\iint p_i(x',u)^2dudx'\\ && \|\eta\|_{\infty}^{2}\Delta_{Y}^2.\end{aligned}$$ Finally, $$W_{1}\leq\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}m.$$ On the other hand, $$W_{4}=\left(\sum_{i}a_{i}b_{i}\right)^{2}\leq \sum_{i}a_{i}^{2}\sum_{i}b_{i}^{2}\leq\|f\|_{2}^{2}\|g\|_{2}^{2}\leq\|f\|_{\infty}\|g\|_{2}^{2}.$$ By the Cauchy-Scharwz inequality we have $\|g\|_{2}^{2}\leq \|\eta\|_{\infty}^{2}\|f\|_{\infty}\Delta_{Y}^{2}$ and then $$W_{4}\leq\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}$$ which leads to $${\mathbb{E}}\left((2Q'(X_{1},Y_{1})R(X_{2},Y_{2}))^{2}\right)\leq 4\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}(m+1).$$ Let us bound now the second term ${\mathbb{E}}\left((Q'(X_{1},Y_{1})CQ(X_{2},Y_{2}))^{2}\right)=W_{5}-2W_{6}+W_{7}$ where $$\begin{aligned} W_{5}&=&\iiiint\sum_{i,i'}\sum_{i_{1},i'_{1}}c_{ii'}c_{i_{1}i'_{1}}p_{i}(x,y)p_{i_{1}}(x,y)p_{i'}(x',y')p_{i'_{1}}(x',y')f(x,y)f(x',y')dxdydx'dy'\\ W_{6}&=&\sum_{i,i'}\sum_{i_{1},i'_{1}}\iint c_{ii'}c_{i_{1}i'_{1}}a_{i}a_{i_{1}}p_{i'}(x,y)p_{i'_{1}}(x,y)f(x,y)dxdy\\ W_{7}&=&\sum_{i,i'}\sum_{i_{1},i'_{1}}c_{ii'}c_{i_{1}i'_{1}}a_{i}a_{i_{1}}a_{i'}a_{i'_{1}}.\end{aligned}$$ Following the previous manipulations, we show that $W_6\geq 0$. Thus, $${\mathbb{E}}\left((Q'(X_{1},Y_{1})CQ(X_{2},Y_{2}))^{2}\right)\leq W_{5}+W_{7}.$$First, observe that $$\begin{aligned} W_{5}&=&\iiiint\left(\sum_{i,i'}c_{ii'}p_{i}(x,y)p_{i'}(x',y')\right)^{2}f(x,y)f(x',y')dxdydx'dy'\\ &\leq&\|f\|_{\infty}^{2}\iiiint\left(\sum_{i,i'}c_{ii'}p_{i}(x,y)p_{i'}(x',y')\right)^{2}dxdydx'dy'\\ &\leq&\|f\|_{\infty}^{2}\sum_{i,i'}\sum_{i_{1},i'_{1}}c_{ii'}c_{i_{1}i'_{1}}\iiiint p_{i}(x,y)p_{i_{1}}(x,y)\\ &&p_{i'}(x',y')p_{i'_{1}}(x',y')dxdydx'dy'\\ &\leq&\|f\|_{\infty}^{2}\sum_{i,i'}c_{ii'}^{2}\end{aligned}$$ since the $p_{i}$ are orthonormal. Besides, $$\begin{aligned} \sum_{i,i'}c_{ii'}^{2}&=&\iint \sum_{i_{\alpha},i'_{\alpha}} \alpha_{i_{\alpha}}(x)\alpha_{i'_{\alpha}}(x) \alpha_{i_{\alpha}}(x')\alpha_{i'_{\alpha}}(x')\sum_{i_{\beta},i'_{\beta}} \left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x,y_1,y_2)dy_1dy_2\right)\\ &&\left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x',y_1,y_2)dy_1dy_2\right)dxdx'\\ &=& \iint \left( \sum_{i_{\alpha}} \alpha_{i_{\alpha}}(x)\alpha_{i_{\alpha}}(x')\right)^2\sum_{i_{\beta},i'_{\beta}} \left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x,y_1,y_2)dy_1dy_2\right)\\ &&\left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x',y_1,y_2)dy_1dy_2\right)dxdx'.\end{aligned}$$ But $$\begin{aligned} &&\sum_{i_{\beta},i'_{\beta}} \left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x,y_1,y_2)dy_1dy_2\right)\\ &&\left(\iint\beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x',y_1,y_2)dy_1dy_2\right)\\ &=& \sum_{i_{\beta},i'_{\beta}} \iiiint \beta_{i_{\beta}}(y_1)\beta_{i'_{\beta}}(y_2)\eta(x,y_1,y_2) \beta_{i_{\beta}}(y'_1)\beta_{i'_{\beta}}(y'_2)\eta(x',y'_1,y'_2)dy_1dy_2dy'_1dy'_2\\ &=& \iint \sum_{i_{\beta}} \left(\int\beta_{i_{\beta}}(y_1)\eta(x,y_1,y_2)dy_1\right)\beta_{i_{\beta}}(y'_1)\sum_{i'_{\beta}} \left(\int\beta_{i'_{\beta}}(y'_2)\eta(x',y'_1,y'_2)dy'_2\right)\beta_{i'_{\beta}}(y_2) dy'_1dy_2\\ &=& \iint \eta(x,y'_1,y_2) \eta(x',y'_1,y_2) dy'_1dy_2\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2}\end{aligned}$$ using the fact that $(\beta_i)$ is an orthonormal basis. We then get $$\begin{aligned} \sum_{i,i'}c_{ii'}^{2}&\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2}\iint \left( \sum_{i_{\alpha}} \alpha_{i_{\alpha}}(x)\alpha_{i_{\alpha}}(x')\right)^2dxdx'\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2} \iint \sum_{i_{\alpha},i'_{\alpha}} \alpha_{i_{\alpha}}(x)\alpha_{i'_{\alpha}}(x) \alpha_{i_{\alpha}}(x')\alpha_{i'_{\alpha}}(x') dxdx'\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2} \sum_{i_{\alpha},i'_{\alpha}} \left(\int \alpha_{i_{\alpha}}(x)\alpha_{i'_{\alpha}}(x)dx\right)^2\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2} \sum_{i_{\alpha}} \left(\int \alpha_{i_{\alpha}}(x)^2dx\right)^2\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2}m\end{aligned}$$ since the $\alpha_{i}$ are orthonormal. Finally, $$W_{5}\leq\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}m.$$ Besides, $$W_{7}=\left(\sum_{i,i'}c_{ii'}a_{i}a_{i'}\right)^{2}$$ with $$\begin{aligned} \left|\sum_{i,i'}c_{ii'}a_{i}a_{i'}\right|&\leq&\|\eta\|_{\infty}\iiint |S_{M}f(x,y_{1})S_{M}f(x,y_{2})|dxdy_{1}dy_{2}\\ &\leq&\|\eta\|_{\infty}\iint \left(\int|S_{M}f(x,y_{1})S_{M}f(x,y_{2})|dx\right)dy_{1}dy_{2}.\end{aligned}$$ By using the Cauchy-Schwarz inequality twice, we get $$\begin{aligned} \left(\sum_{i,i'}c_{ii'}a_{i}a_{i'}\right)^{2}&\leq&\Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\iint \left(\int|S_{M}f(x,y_{1})S_{M}f(x,y_{2})|dx\right)^{2}dy_{1}dy_{2}\\ &\leq&\Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\iint \left(\int S_{M}f(u,y_{1})^{2}du\right)\left(\int S_{M}f(v,y_{2})^{2}dv\right)dy_{1}dy_{2}\\ &\leq&\Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\iiiint S_{M}f(u,y_{1})^{2}S_{M}f(v,y_{2})^{2}dudvdy_{1}dy_{2}\\ &\leq&\Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\left(\iint S_{M}f(x,y)^{2}dxdy\right)^{2}\\ &\leq&\Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}.\\\end{aligned}$$ Finally, $${\mathbb{E}}\left((Q'(X_{1},Y_{1})CQ(X_{2},Y_{2}))^{2}\right)\leq \|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}(m+1).$$\ Collecting this inequalities, we obtain $${\textrm{Var}}(U_{n}K)\leq \frac{20}{n(n-1)}\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}(m+1)$$ which concludes the proof of Lemma \[var1\]. Let us now deal with the second term of the Hoeffding’s decomposition of $\hat{\theta}_n$ : \[var2\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $${\textrm{Var}}(P_nL)\leq \frac{36}{n}\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}.$$ First note that $${\textrm{Var}}(P_{n}L)=\frac{1}{n}{\textrm{Var}}(L(X_{1},Y_{1})).$$ We can write $L(X_{1},Y_{1})$ as $$\begin{aligned} L(X_{1},Y_{1})&=&2A'R(X_{1},Y_{1})+2B'Q(X_{1},Y_{1})-2A'CQ(X_{1},Y_{1})\\ &=& 2\sum_{i}a_{i}\left(\int p_{i}(X_{1},u)\eta(X_{1},u,Y_{1})du-b_{i}\right)\\ &&+2\sum_{i}b_{i}(p_{i}(X_{1},Y_{1})-a_{i})-2\sum_{i,i'}c_{ii'}a_{i'}(p_{i}(X_{1},Y_{1})-a_{i})\\ &=& 2\int\sum_{i}a_{i}p_{i}(X_{1},u)\eta(X_{1},u,Y_{1})du + 2\sum_{i}b_{i}p_{i}(X_{1},Y_{1})\\ &&-2\sum_{i,i'}c_{ii'}a_{i'}p_{i}(X_{1},Y_{1})-4A'B+2A'CA\\ &=&2\int S_{M}f(X_{1},u)\eta(X_{1},u,Y_{1})du+2S_{M}g(X_{1},Y_{1})\\ &&-2\sum_{i,i'}c_{ii'}a_{i'}p_{i}(X_{1},Y_{1})-4A'B+2A'CA.\\\end{aligned}$$ Let ${\displaystyle}{h(x,y)=\int S_{M}f(x,u)\eta(x,u,y)du}$, we have $$\begin{aligned} S_{M}h(z,t)&=&\sum_{i}\left(\iint h(x,y)p_{i}(x,y)dxdy\right)p_{i}(z,t)\\ &=&\sum_{i}\left(\iiint S_{M}f(x,u)\eta(x,u,y)p_{i}(x,y)dudxdy\right)p_{i}(z,t)\\ &=&\sum_{i,i'}\left(\iiint a_{i'}p_{i'}(x,u)\eta(x,u,y)p_{i}(x,y)dudxdy\right)p_{i}(z,t)\\ &=&\sum_{i,i'}c_{ii'}a_{i'}p_{i}(z,t)\end{aligned}$$ and we can write $$L(X_{1},Y_{1})=2h(X_{1},Y_{1})+2S_{M}g(X_{1},Y_{1})-2S_{M}h(X_{1},Y_{1})-4A'B+2A'CA.$$ Thus, $$\begin{aligned} {\textrm{Var}}(L(X_{1},Y_{1}))&=&4{\textrm{Var}}[h(X_{1},Y_{1})+S_{M}g(X_{1},Y_{1})-S_{M}h(X_{1},Y_{1})]\\ &\leq&4{\mathbb{E}}[(h(X_{1},Y_{1})+S_{M}g(X_{1},Y_{1})-S_{M}h(X_{1},Y_{1}))^{2}]\\ &\leq&12{\mathbb{E}}[(h(X_{1},Y_{1}))^{2}+(S_{M}g(X_{1},Y_{1}))^{2}+(S_{M}h(X_{1},Y_{1}))^{2}].\end{aligned}$$ Each of these three terms has to be bounded : $$\begin{aligned} {\mathbb{E}}((h(X_{1},Y_{1}))^{2})&=&\iint \left(\int S_{M}f(x,u)\eta(x,u,y)du\right)^{2}f(x,y)dxdy\\ &\leq&\Delta_{Y}\iiint S_{M}f(x,u)^{2}\eta(x,u,y)^{2}f(x,y)dxdydu\\ &\leq&\Delta_{Y}^{2}\|f\|_{\infty}\|\eta\|_{\infty}^{2}\iint S_{M}f(x,u)^{2}dxdu\\ &\leq&\Delta_{Y}^{2}\|f\|_{\infty}\|\eta\|_{\infty}^{2}\|S_{M}f\|_{2}^{2}\\ &\leq&\Delta_{Y}^{2}\|f\|_{\infty}\|\eta\|_{\infty}^{2}\|f\|_{2}^{2}\\ &\leq&\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}\\\end{aligned}$$ $${\mathbb{E}}((S_{M}g(X_{1},Y_{1}))^{2})\leq \|f\|_{\infty}\|S_{M}g\|_{2}^{2}\leq \|f\|_{\infty}\|g\|_{2}^{2}\leq \Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}$$ $${\mathbb{E}}((S_{M}h(X_{1},Y_{1}))^{2})\leq \|f\|_{\infty}\|S_{M}h\|_{2}^{2}\leq \|f\|_{\infty}\|h\|_{2}^{2}\leq\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}$$ from previous calculations. Finally, $${\textrm{Var}}(L(X_{1},Y_{1}))\leq 36\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}.$$ The last term of the Hoeffding’s decomposition can also be controled : \[var3\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $${\textrm{Cov}}(U_{n}K,P_{n}L)=0.$$ Since $U_{n}K$ et $P_{n}L$ are centered, we have $$\begin{aligned} {\textrm{Cov}}(U_{n}K,P_{n}L)&=&{\mathbb{E}}(U_{n}KP_{n}L)\\ &=&{\mathbb{E}}\left[\frac{1}{n^{2}(n-1)}\sum_{j\neq k=1}^{n}K(X_{j},Y_{j},X_{k},Y_{k})\sum_{i=1}^{n}L(X_{i},Y_{i})\right]\\ &=&\frac{1}{n}{\mathbb{E}}(K(X_{1},Y_{1},X_{2},Y_{2})(L(X_{1},Y_{1})+L(X_{2},Y_{2})))\\ &=& 0\end{aligned}$$ since $K$, $L$, $Q$ and $R$ are centered. The four previous lemmas give the expected result on the precision of $\hat{\theta}_n$ : \[precision\] Assuming the hypotheses of Theorem \[tfq\] hold, we have : - If $m/n\rightarrow 0$, $${\mathbb{E}}(\hat{\theta}_{n}-\theta)^{2}=O\left(\frac{1}{n}\right),$$ - Otherwise, $${\mathbb{E}}(\hat{\theta}_{n}-\theta)^{2}\leq \gamma_2(m/n^2)$$ where $\gamma_2$ only depends on $\|f\|_{\infty}$, $\|\eta\|_{\infty}$ and $\Delta_Y$. Lemmas \[var1\], \[var2\] and \[var3\] imply $${\textrm{Var}}(\hat{\theta}_{n})\leq \frac{20}{n(n-1)}\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}(m+1)+\frac{36}{n}\Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}.$$ Finally, for $n$ large enough and a constant $\gamma\in {\mathbb{R}}$, $${\textrm{Var}}(\hat{\theta}_{n})\leq \gamma \Delta_{Y}^{2}\|f\|_{\infty}^{2}\|\eta\|_{\infty}^{2}\left(\frac{m}{n^{2}}+\frac{1}{n}\right).$$ Lemma \[biais1\] gives $${\textrm{Bias}}^{2}(\hat{\theta}_{n})\leq \Delta_{Y}^{2}\|\eta\|_{\infty}^{2}\left(\sup_{i\notin M}|c_{i}|^{2}\right)^{2}$$ and by assumption $\left(\sup_{i\notin M}|c_{i}|^{2}\right)^{2}\approx m/n^{2}$. If $m/n\rightarrow 0$, then ${\mathbb{E}}(\hat{\theta}_{n}-\theta)^{2}=O(\frac{1}{n})$. Otherwise ${\mathbb{E}}(\hat{\theta}_{n}-\theta)^{2}\leq \gamma_2(m/n^2)$ where $\gamma_2$ only depends on $\|f\|_{\infty}$, $\|\eta\|_{\infty}$ and $\Delta_Y$. The lemma we just proved gives the result of Theorem \[tfq\] when $m/n$ does not converge to $0$. Let us now study more precisely the semiparametric case, that is when ${\mathbb{E}}(\hat{\theta}_{n}-\theta)^{2}=O(\frac{1}{n})$, to prove the asymptotic normality (\[na\]) and the bound in (\[ea\]). We have $$\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)=\sqrt{n}(U_{n}K)+\sqrt{n}(P_{n}L)+\sqrt{n}(2A'B-A'CA).$$ We will study the asymptotic behavior of each of these three terms. The first one is easily treated : \[var4\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $$\sqrt{n}U_nK\rightarrow 0$$ in probability when $n\rightarrow \infty$ if $m/n\rightarrow 0$. Since ${\displaystyle}{{\textrm{Var}}(\sqrt{n}U_{n}K)\leq \frac{20}{(n-1)}\|\eta\|_{\infty}^{2}\|f\|_{\infty}^{2}\Delta_{Y}^{2}(m+1)}$, $\sqrt{n}U_nK$ converges to $0$ in probability when $n\rightarrow \infty$ if $m/n\rightarrow 0$. The random variable $P_nL$ will be the most important term for the central limit theorem. Before studying its asymptotic normality, we need the following lemma concerning the asymptotic variance of $\sqrt{n}(P_{n}L)$ : \[var5\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $$n{\textrm{Var}}(P_nL)\rightarrow \Lambda(f,\eta)$$ where $$\Lambda(f,\eta)=4 \left[ \iint g(x,y)^2f(x,y)dxdy -\left( \iint g(x,y)f(x,y)dxdy\right)^2\right].$$ We proved in Lemma \[var2\] that $$\begin{aligned} {\textrm{Var}}(L(X_{1},Y_{1}))&=&4{\textrm{Var}}[h(X_{1},Y_{1})+S_{M}g(X_{1},Y_{1})-S_{M}h(X_{1},Y_{1})]\\ &=&4{\textrm{Var}}[A_1+A_2+A_3]\\ &=&4\sum_{i,j=1}^3{\textrm{Cov}}(A_i,A_j).\end{aligned}$$ We will show that $\forall i,j\in \{1,2,3\}^2$, we have $$\begin{aligned} \left| {\textrm{Cov}}(A_i,A_j)-\epsilon_{ij} \left[ \iint g(x,y)^2f(x,y)dxdy -\left( \iint g(x,y)f(x,y)dxdy\right)^2\right]\right| \notag\\ \leq \gamma\left[ \| S_Mf-f\|_2 + \| S_Mg-g\|_2\right] \label{variances}\end{aligned}$$ where $\epsilon_{ij}=-1$ if $i=3$ or $j=3$ and $i\neq j$ and $\epsilon_{ij}=1$ otherwise, and where $\gamma$ depends only on $\|f\|_{\infty}$, $\|\eta\|_{\infty}$ and $\Delta_Y$.\ We shall give the details only for the case $i=j=3$ since the calculations are similar for the other configurations. We have $${\textrm{Var}}(A_3)=\iint S_M^2[h(x,y)]f(x,y)dxdy-\left(\iint S_M[h(x,y)]f(x,y)dxdy\right)^2$$ We first study the quantity $$\left|\iint S_M^2[h(x,y)]f(x,y)dxdy-\iint g(x,y)^2f(x,y)dxdy\right|.$$ It is bounded by prout prout prout prout prout prout prout prout prout prout $$\begin{aligned} &&\iint \left|S_M^2[h(x,y)]f(x,y)-S_M^2[g(x,y)]f(x,y)\right|dxdy\\ &&+\iint \left|S_M^2[g(x,y)]f(x,y)-g(x,y)^2f(x,y)\right|dxdy\\ &&\leq \|f\|_{\infty} \| S_Mh+S_Mg\|_2 \|S_Mh-S_Mg\|_2 + \|f\|_{\infty} \| S_Mg+g\|_2 \|S_Mg-g\|_2.\end{aligned}$$ Using the fact that $S_M$ is a projection, this sum is bounded by $$\begin{aligned} ~&&\|f\|_{\infty} \| h+g\|_2 \| h-g\|_2 + 2\|f\|_{\infty} \| g\|_2 \|S_Mg-g\|_2\\ ~&&\leq \|f\|_{\infty} (\| h\|_2+\|g\|_2) \| h-g\|_2 + 2\|f\|_{\infty} \| g\|_2 \|S_Mg-g\|_2.\end{aligned}$$ We saw previously that $\|g\|_2\leq \Delta_Y \|f\|_{\infty}^{1/2} \|\eta\|_{\infty}$ and $\| h\|_2 \leq \Delta_Y \|f\|_{\infty}^{1/2} \|\eta\|_{\infty}$. The sum is then bounded by $$2\Delta_Y\|f\|_{\infty}^{3/2} \|\eta\|_{\infty} \| h-g\|_2 + 2\Delta_Y\|f\|_{\infty}^{3/2}\|\eta\|_{\infty} \|S_Mg-g\|_2$$ We now have to deal with $ \| h-g\|_2$: $$\begin{aligned} \| h-g\|_2^2&=& \iint \left( \int \left(S_Mf(x,u)-f(x,u)\right)\eta(x,u,y)du\right)^2dxdy\\ &\leq& \iint \left(\int (S_Mf(x,u)-f(x,u))^2du\right)\left(\int\eta(x,u,y)^2du\right)dxdy\\ &\leq& \Delta_Y^2 \|\eta\|_{\infty}^{2} \|S_Mf-f\|_2^2.\end{aligned}$$ Finally, the sum is bounded by $$2\Delta_Y\|f\|_{\infty}^{3/2} \|\eta\|_{\infty} \left( \Delta_Y\|\eta\|_{\infty}\|S_Mf-f\|_2+ \|S_Mg-g\|_2\right).$$ Let us now study the second quantity $$\left|\left(\iint S_M[h(x,y)]f(x,y)dxdy\right)^2-\left(\iint g(x,y)f(x,y)dxdy\right)^2\right|.$$ It is equal to $$\begin{aligned} \left|\left( \iint (S_M[h(x,y)]+g(x,y))f(x,y)dxdy\right)\right.\\ \left.\left( \iint (S_M[h(x,y)]-g(x,y))f(x,y)dxdy\right)\right|.\end{aligned}$$ By using the Cauchy-Schwarz inequality, it is bounded by $$\begin{aligned} ~&&\|f\|_2 \| S_Mh+g\|_2\|f\|_2 \| S_Mh-g\|_2\\ ~&&\leq \|f\|_2^2 (\|h\|_2+\|g\|_2) (\| S_Mh-S_Mg\|_2+\| S_Mg-g\|_2)\\ ~&&\leq 2\Delta_Y\|f\|_{\infty} ^{3/2} \|\eta\|_{\infty} (\| h-g\|_2+\| S_Mg-g\|_2)\\ ~&&\leq 2\Delta_Y\|f\|_{\infty} ^{3/2} \|\eta\|_{\infty} \left(\Delta_Y\|\eta\|_{\infty}\|S_Mf-f\|_2 + \|S_Mg-g\|_2\right)\end{aligned}$$ by using the previous calculations. Collecting the two inequalities gives (\[variances\]) for $i=j=3$.\ Finally, since by assumption $\forall t\in L^2(d\mu)$, $\|S_Mt-t\|_2 \rightarrow 0$ when $n\rightarrow\infty$, a direct consequence of (\[variances\]) is that $$\begin{aligned} &&\lim_{n\rightarrow\infty} {\textrm{Var}}(L(X_1,Y_1))\\ && = 4 \left[ \iint g(x,y)^2f(x,y)dxdy -\left( \iint g(x,y)f(x,y)dxdy\right)^2\right]\\ &&= \Lambda(f,\eta).\end{aligned}$$ We then conclude by noting that ${\textrm{Var}}(\sqrt{n}(P_nL))={\textrm{Var}}(L(X_1,Y_1))$. We can now study the convergence of $\sqrt{n}(P_nL)$, which is given in the following lemma: Assuming the hypotheses of Theorem \[tfq\] hold, we have $$\sqrt{n}P_nL \overset{\mathcal{L}}{\rightarrow} {\mathcal{N}}(0,\Lambda(f,\eta)).$$ We first note that $$\sqrt{n}\left(P_n(2g)-2\iint g(x,y)f(x,y)dxdy\right)\rightarrow {\mathcal{N}}(0,\Lambda(f,\eta))$$ where ${\displaystyle}{g(x,y)=\int \eta(x,y,u)f(x,u)du}$.\ It is then sufficient to show that the expectation of the square of $${\displaystyle}{R=\sqrt{n}\left[P_nL-\left(P_n(2g)-2 \iint g(x,y)f(x,y)dxdy\right)\right]}$$ converges to $0$. We have $$\begin{aligned} {\mathbb{E}}(R^2)&=& {\textrm{Var}}(R)\\ &=& n{\textrm{Var}}(P_nL)+n{\textrm{Var}}(P_n(2g))-2n{\textrm{Cov}}(P_nL,P_n(2g))\end{aligned}$$ We know that $n{\textrm{Var}}(P_n(2g))\rightarrow \Lambda(f,\eta)$ and Lemma \[var5\] shows that $n{\textrm{Var}}(P_nL)\rightarrow \Lambda(f,\eta)$. Then, we just have to prove that $$\lim_{n\rightarrow\infty} n{\textrm{Cov}}(P_nL,P_n(2g)) = \Lambda(f,\eta).$$ We have $$n{\textrm{Cov}}(P_nL,P_n(2g)) = {\mathbb{E}}(2L(X_1,Y_1)g(X_1,Y_1))$$ because $L$ is centered. Since $$L(X_1,Y_1)=2h(X_1,Y_1)+2S_Mg(X_1,Y_1)-2S_Mh(X_1,Y_1)-4A'B+2A'CA,$$ we get $$\begin{aligned} &&n{\textrm{Cov}}(P_nL,P_n(2g)) = 4\iint h(x,y)g(x,y)f(x,y)dxdy\\ && + 4\iint S_Mg(x,y)g(x,y)f(x,y)dxdy\\ && -4\iint S_Mh(x,y)g(x,y)f(x,y)dxdy -8 \sum_i a_ib_i \iint g(x,y)f(x,y)dxdy\\ &&+ 4 A'CA \iint g(x,y)f(x,y)dxdy \end{aligned}$$ which converges to ${\displaystyle}{4\left[ \iint g(x,y)^2f(x,y)dxdy -\left( \iint g(x,y)f(x,y)dudxdy\right)^2\right]}$ which is equal to $\Lambda(f,\eta)$. We finally deduce that $$\sqrt{n}P_nL\rightarrow {\mathcal{N}}(0,\Lambda(f,\eta))$$ in distribution. In order to prove the asymptotic normality of $\hat{\theta}_n$, the last step is to control the remainder term in the Hoeffding’s decomposition: \[var6\] Assuming the hypotheses of Theorem \[tfq\] hold, we have $$\sqrt{n}(2A'B-A'CA-\theta)\rightarrow 0.$$ $\sqrt{n}(2A'B-A'CA-\theta)\rightarrow 0$ is equal to $$\begin{aligned} &&\sqrt{n}\left[2\iint g(x,y)S_Mf(x,y)dxdy\right.\\ && - \iiint S_Mf(x,y_1)S_Mf(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\\ && \left.-\iiint f(x,y_1)f(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\right].\\\end{aligned}$$ By replacing $g$ we get $$\begin{aligned} &&\sqrt{n}\left[2\iiint S_Mf(x,y_1)f(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\right.\\ && - \iiint S_Mf(x,y_1)S_Mf(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\\ &&\left.-\iiint f(x,y_1)f(x,y_2)\eta(x,y_1,y_2)dxdy_1dy_2\right]\\\end{aligned}$$ With integral manipulation, we show it is also equal to $$\begin{aligned} &&\sqrt{n}\left[\iiint S_Mf(x,y_1)(f(x,y_2)-S_Mf(x,y_2))\eta(x,y_1,y_2)dxdy_1dy_2\right.\\ &&\left.- \iiint f(x,y_2)(S_Mf(x,y_1)-f(x,y_1))\eta(x,y_1,y_2)dxdy_1dy_2\right]\\ &&\leq \sqrt{n} \Delta_Y \|\eta\|_{\infty}\left( \|S_Mf\|_2\|S_Mf-f\|_2+\| f\|_2\|S_Mf-f\|_2\right)\\ &&\leq 2\sqrt{n} \Delta_Y \| f\|_2\|\|\eta\|_{\infty}\|S_Mf-f\|_2\\ &&\leq 2\sqrt{n} \Delta_Y \| f\|_2\|\|\eta\|_{\infty} \left(\sup_{i\notin M}|c_{i}|^{2}\right)^{1/2}\\ &&\approx 2\Delta_Y \| f\|_2\|\|\eta\|_{\infty} \sqrt{\frac{m}{n}},\end{aligned}$$ which converges to $0$ when $n\rightarrow \infty$ since $m/n\rightarrow 0$. Collecting now the results of Lemmas \[var4\], \[var5\] and \[var6\] we get (\[na\]) since $$\sqrt{n}\left(\hat{\theta}_{n}-\theta\right)\rightarrow {\mathcal{N}}(0,\Lambda(f,\eta))$$ in distribution. We finally have to prove (\[ea\]). Remark that $$\begin{aligned} n{\mathbb{E}}\left(\hat{\theta}_n-\theta\right)^2&=&n{\textrm{Bias}}^2(\hat{\theta}_n)+n{\textrm{Var}}(\hat{\theta}_n)\\ &=& n{\textrm{Bias}}^2(\hat{\theta}_n)+n{\textrm{Var}}(U_nK)+n{\textrm{Var}}(P_nL)\end{aligned}$$ We previously proved that $$\begin{aligned} n{\textrm{Bias}}^2(\hat{\theta}_n)&\leq& \lambda \Delta_Y^2\|\eta\|_{\infty}^2 \frac{m}{n} ~~\textrm{for some } \lambda\in {\mathbb{R}},\\ n{\textrm{Var}}(U_nK)&\leq& \mu \Delta_Y^2\|f\|_{\infty}^2\|\eta\|_{\infty}^2 \frac{m}{n} ~~\textrm{for some }\mu\in {\mathbb{R}}.\end{aligned}$$ Moreover, (\[variances\]) imply $$\left| n{\textrm{Var}}(P_nL)-\Lambda(f,\eta)\right| \leq \gamma\left[ \|S_Mf-f\|_2+\|S_Mg-g\|_2\right],$$ where $\gamma$ is a increasing function of $\|f\|_{\infty}$,$\|\eta\|_{\infty}$ and $\Delta_Y$. We then deduce (\[ea\]) which ends the proof of Theorem \[tfq\]. Proof of Theorem \[cramerrao1\] ------------------------------- To prove the inequality we will use the work of @IK91 (see also chapter 25 of @VV98) on efficient estimation. The first step is the computation of the Fréchet derivative of $\theta(f)$ at a point $f_0$. Straightforward calculations show that $$\begin{aligned} \theta(f)-\theta(f_0)&=&\iint \left[2\int \psi(x,y,z)f_0(x,z)dz\right]\left(f(x,y)-f_0(x,y)\right)dxdy\\ && + \; O\left(\iint (f(x,y)-f_0(x,y))^2dxdy\right)\end{aligned}$$ from which we deduce that the Fréchet derivative of $\theta(f)$ at $f_0$ is$$\theta'(f_0)\cdot u=\left< 2\int \psi(x,y,z)f_0(x,z)dz, u\right>\quad (u\in L^2(dxdy)),$$ where $\left<\cdot,\cdot\right>$ is the scalar product in $L^2(dxdy)$. We can now use the results of @IK91. Denote $H(f_0)=H(f_0)=\left\{ u\in L^2(dxdy), \iint u(x,y)\sqrt{f_0(x,y)}dxdy=0\right\}$ the set of functions in $L^2(dxdy)$ orthogonal to $\sqrt{f_0}$, $\textrm{Proj}_{H(f_0)}$ the projection on $H(f_0)$, $A_n(t)=(\sqrt{f_0})t/\sqrt{n}$ and $P_{f_0}^{(n)}$ the joint distribution of $(X_1,\ldots,X_n)$ under $f_0$. Since here $X_1,\ldots,X_n$ are i.i.d., $\left\{P_f^{(n)},f\in\mathcal{E}\right\}$ is locally asymptotically normal at all points $f_0\in\mathcal{E}$ in the direction $H(f_0)$ with normalizing factor $A_n(f_0)$. Ibragimov and Khas’minskii result say that under these conditions, denoting $K_n=B_n\theta'(f_0)A_n\textrm{Proj}_{H(f_0)}$ with $B_n(u)=\sqrt{n}u$, if $K_n\rightarrow K$ weakly and if $K(u)=\left<t,u\right>$, then for every estimator $\hat{\theta}_n$ of $\theta(f)$ and every family $\mathcal{V}(f_0)$ of vicinities of $f_0$, we have $$\inf_{\{\mathcal{V}(f_0)\}} \liminf_{n\rightarrow \infty} \sup_{f\in\mathcal{V}(f_0)} n{\mathbb{E}}(\hat{\theta}_n-\theta(f_0))^2\geq \|t\|_{L^2(dxdy)}^2.$$ Here, $$K_n(u)=\sqrt{n}\theta'(f_0)\cdot\frac{1}{\sqrt{n}}\sqrt{f_0} \textrm{Proj}_{H(f_0)}(u)=\theta'(f_0)\cdot \left(\sqrt{f_0}\left(u-\sqrt{f_0}\int u\sqrt{f}_0\right)\right)$$ does not depend on $n$ and $$\begin{aligned} K(u)&=& \iint \left[2\int \psi(x,y,z)f_0(x,z)dz\right] \sqrt{f_0(x,y)}\\ && \left(u(x,y)-\sqrt{f_0(x,y)}\int u\sqrt{f}_0\right) dxdy\\ &=& \iint \left[2\int \psi(x,y,z)f_0(x,z)dz\right] \sqrt{f_0(x,y)}u(x,y)dxdy\\ &&-\iint \left[2\int \psi(x,y,z)f_0(x,z)dz\right]f_0(x,y)dxdy\int u\sqrt{f}_0\\ &=&\left<t,u\right>\end{aligned}$$ where $$\begin{aligned} t(x,y)&=&\left[2\int \psi(x,y,z)f_0(x,z)dz\right] \sqrt{f_0(x,y)}\\ && - \left(\iint \left[2\int \psi(x,y,z)f_0(x,z)dz\right]f_0(x,y)dxdy\right)\sqrt{f_0(x,y)}.\end{aligned}$$ The semiparametric Cramér-Rao bound for our problem is $\|t\|_{L^2(dxdy)}^2$ : $$\begin{aligned} \|t\|_{L^2(dxdy)}^2&=&4 \iint \left[\int \psi(x,y,z)f_0(x,z)dz\right]^2f_0(x,y)dxdy\\ &&-4\left(\iint \left[\int \psi(x,y,z)f_0(x,z)dz\right]f_0(x,y)dxdy\right)^2\\ &=& 4 \iint g_0(x,y)^2f_0(x,y)dxdy-4\left(\iint g_0(x,y)f_0(x,y)\right)^2\\\end{aligned}$$ where ${\displaystyle}{g_0(x,y)=\int \psi(x,y,z)f_0(x,z)dz}$. Finally, we recognize the expression of $\Lambda(f_0,\psi)$ given in Theorem \[tfq\]. Proof of Theorem \[tfec\] ------------------------- We will first control the remainder term $\Gamma_n$ : $$\Gamma_{n}=\frac{1}{6}F'''(\xi)(1-\xi)^{3}.$$ Let us recall that $$\begin{aligned} F'''(\xi)&=&\iiiint \frac{\left(\int\hat{f}(x,y)dy\right)^2}{\left(\int \xi f(x,y)+(1-\xi)\hat{f}(x,y)dy\right)^{5}}\\ &&\left[\big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\big(\hat{m}(x)-\varphi(t)\big)\right.\\ &&\left(\int\hat{f}(x,y)dy\right)\dddot\psi\left(\hat{r}(\xi,x)\right)- 3\big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\\ &&\left.\left(\int [\xi f(x,y)+(1-\xi)\hat{f}(x,y)]dy\right) \ddot\psi\left(\hat{r}(\xi,x)\right)\right]\\ &&\Big(f(x,y)-\hat{f}(x,y)\Big)\Big(f(x,z)-\hat{f}(x,z)\Big)\\ &&\Big(f(x,t)-\hat{f}(x,t)\Big)dxdydzdt\end{aligned}$$ Assumptions A2 and A3 ensure that the first part of the integrand is bounded by a constant $\mu$ : $$\begin{aligned} \Gamma_{n}&\leq&\frac{1}{6}\mu\iiiint |f(x,y)-\hat{f}(x,y)||f(x,z)-\hat{f}(x,z)|\\ &&|f(x,t)-\hat{f}(x,t)|dxdydzdt\\ &\leq&\frac{1}{6}\mu\int \left(\int |f(x,y)-\hat{f}(x,y)|dy\right)^{3}dx\\ &\leq& \frac{1}{6}\mu\Delta_{Y}^{2}\iint |f(x,y)-\hat{f}(x,y)|^{3}dxdy\end{aligned}$$ by the Hölder inequality. Then ${\mathbb{E}}(\Gamma_{n}^{2})=O({\mathbb{E}}[(\int|f-\hat{f}|^{3})^{2}])=O({\mathbb{E}}[\|f-\hat{f}\|_{3}^{6}])$. Since $\hat{f}$ verifies assumption A2, this quantity has order $O(n_{1}^{-6\lambda})$. If we further assume that $n_{1}\approx n/\log(n)$ and $\lambda > 1/6$, we get $E(\Gamma_{n}^{2})=o(\frac{1}{n})$, which proves that the remainder term $\Gamma_n$ is negligible. We will now show that $\sqrt{n}\left(\hat{T}_n-T(f)\right)$ and $Z_n=\frac{1}{n_2}\sum_{j=1}^{n_2} H(f,X_j,Y_j) - \iint H(f,x,y)f(x,y)dxdy$ have the same asymptotic behavior. The idea is that we can easily get a central limit theorem for $Z_n$ with asymptotic variance $$C(f)=\iint H(f,x,y)^2 f(x,y)dxdy-\left( \iint H(f,x,y) f(x,y)dxdy\right)^2,$$ which imply both (\[na2\]) and (\[ea2\]) (we will show at the end of the proof that $C(f)$ can be expressed such as in the theorem). In order to show that $\sqrt{n}\left(\hat{T}_n-T(f)\right)$ and $Z_n$ have the same asymptotic behavior, we will prove that $$R=\sqrt{n}\left[\hat{T}_n-T(f)-\left(\frac{1}{n_2}\sum_{j=1}^{n_2} H(f,X_j,Y_j) - \iint H(f,x,y)f(x,y)dxdy\right)\right]$$ has a second-order moment converging to $0$. Let us note that $R=R_1+R_2$ where $$\begin{aligned} R_1&=& \sqrt{n}\left[\hat{T}_n-T(f)\right.\\ &&\left.-\left(\frac{1}{n_2}\sum_{j=1}^{n_2} H(\hat{f},X_j,Y_j) - \iint H(\hat{f},x,y)f(x,y)dxdy\right)\right],\\ R_2&=&\sqrt{n}\left[\frac{1}{n_2} \sum_{j=1}^{n_2} \left(H(\hat{f},X_j,Y_j) - \iint H(\hat{f},x,y)f(x,y)dxdy\right) \right]\\ &&- \sqrt{n}\left[\frac{1}{n_2} \sum_{j=1}^{n_2} \left(H(f,X_j,Y_j) - \iint H(f,x,y)f(x,y)dxdy\right) \right].\end{aligned}$$ We propose to show that both ${\mathbb{E}}(R_1^2)$ and ${\mathbb{E}}(R_2^2)$ converge to $0$. We can write $R_1$ as follows : $$R_1= -\sqrt{n}\left[ \hat{Q}'-Q'+ \Gamma_n\right]$$ where $$\begin{aligned} Q'&=&\iiint K(\hat{f},x,y,z)f(x,y)f(x,z),\\ K(\hat{f},x,y,z)&=& \frac{1}{2}\frac{\ddot\psi(\hat{m}(x))}{\left(\int\hat{f}(x,y)dy\right)} \big(\hat{m}(x)-\varphi(y)\big)\big(\hat{m}(x)-\varphi(z)\big)\end{aligned}$$ and $\hat{Q}'$ is the corresponding estimator. Since ${\mathbb{E}}\left(\Gamma_n^2\right)=o(1/n)$, we just have to control the expectation of the square of $\sqrt{n}\left[ \hat{Q}'-Q'\right]$ : \[qq\] Assuming the hypotheses of Theorem \[tfec\] hold, we have $$\lim_{n\rightarrow\infty} n{\mathbb{E}}\left(\hat{Q}'-Q'\right)^2 = 0.$$ The bound given in (\[ea\]) states that if $|M_n|/n\rightarrow 0$ we have $$\begin{aligned} &&\left| n{\mathbb{E}}\left[\left(\hat{Q}'-Q'\right)^2|\hat{f}\right]\right.\\ &&\left. - 4 \left[ \iint \hat{g}(x,y)^2f(x,y)dxdy -\left( \iint \hat{g}(x,y)f(x,y)dxdy\right)^2\right]\right|\\ &&\leq \gamma_1(\|f\|_{\infty},\|\psi\|_{\infty},\Delta_Y) \left[ \frac{|M_n|}{n}+\|S_{M}f-f\|_2+\|S_{M}\hat{g}-\hat{g}\|_2\right]\end{aligned}$$ where ${\displaystyle}{\hat{g}(x,y)=\int K(\hat{f},x,y,z)f(x,z)dz}$. By deconditioning, we get $$\begin{aligned} &&\left| n{\mathbb{E}}\left[\left(\hat{Q}'-Q'\right)^2\right]\right.\\ &&\left. - 4 {\mathbb{E}}\left[ \iint \hat{g}(x,y)^2f(x,y)dxdy -\left( \iint \hat{g}(x,y)f(x,y)dxdy\right)^2\right]\right|\\ &&\leq \gamma_1(\|f\|_{\infty},\|\psi\|_{\infty},\Delta_Y) \left[ \frac{|M_n|}{n}+\|S_{M}f-f\|_2+{\mathbb{E}}\left(\|S_{M}\hat{g}-\hat{g}\|_2\right)\right].\end{aligned}$$ Note that $$\begin{aligned} {\mathbb{E}}\left(\|S_{M}\hat{g}-\hat{g}\|_2\right)&\leq&{\mathbb{E}}\left(\|S_{M}\hat{g}-S_Mg\|_2\right) + {\mathbb{E}}\left(\|S_{M}g-g\|_2\right)\\ &\leq& {\mathbb{E}}\left(\|\hat{g}-g\|_2\right) + {\mathbb{E}}\left(\|S_{M}g-g\|_2\right)\end{aligned}$$ where ${\displaystyle}{g(x,y)=\int K(f,x,y,z)f(x,z)dz}$. The second term converges to $0$ since $g\in L^2(dxdy)$ and $\forall t\in L^2(dxdy)$, $\int (S_{M}t-t)^2d\mu\rightarrow 0$. Moreover $$\begin{aligned} \|\hat{g}-g\|_2^2&=& \iint \left[\hat{g}(x,y)-g(x,y)\right]^2f(x,y)dxdy\\ &=& \iint \left[ \int \left(K(\hat{f},x,y,z)-K(f,x,y,z)\right)f(x,z)dz\right]^2f(x,y)dxdy\\ &\leq& \iint \left[ \int \left(K(\hat{f},x,y,z)-K(f,x,y,z)\right)^2dz\right]\\ &&\left[\int f(x,z)^2dz\right]f(x,y)dxdy\\ &\leq& \Delta_Y^2\|f\|_{\infty}^3 \iiint \left(K(\hat{f},x,y,z)-K(f,x,y,z)\right)^2dxdz\\ &\leq& \delta \Delta_Y^3\|f\|_{\infty}^3 \iint (f(x,y)-\hat{f}(x,y))^2dxdy\end{aligned}$$ for some constant $\delta$ by applying the mean value theorem to $K(f,x,y,z)-K(\hat{f},x,y,z)$. Of course, the bound $\delta$ is obtained here by considering assumptions A1, A2 and A3. Since ${\mathbb{E}}(\|f-\hat{f}\|_2)\rightarrow 0$, we get ${\mathbb{E}}\left(\|\hat{g}-g\|_2\right)\rightarrow 0$. Let us now show that the expectation of $$\iint \hat{g}(x,y)^2f(x,y)dxdy-\left( \iint \hat{g}(x,y)f(x,y)dxdy\right)^2$$ converges to 0. We will only develop the proof for the first term : $$\begin{aligned} &&\left|\iint \hat{g}(x,y)^2f(x,y)dxdy- \iint g(x,y)^2f(x,y)dxdy\right|\\ &&\leq \iint \left|\hat{g}(x,y)^2-g(x,y)^2\right|f(x,y)dxdy\\ &&\leq \lambda \iint \left(\hat{g}(x,y)-g(x,y)\right)^2dxdy\\ &&\leq \lambda \|\hat{g}-g\|_2^2\end{aligned}$$ for some constant $\lambda$. By taking the expectation of both sides, we see it is enough to show that ${\mathbb{E}}\left(\|\hat{g}-g\|_2^2\right)\rightarrow 0$, which is done exactly as above. Besides, we can verify that $$\begin{aligned} g(x,y)&=& \int K(f,x,y,z)f(x,z)dz\\ &=& \frac{1}{2}\frac{\ddot\psi(m(x))}{\left(\int f(x,y)dy\right)} \big(m(x)-\varphi(y)\big)\\ &&\left(m(x)\int f(x,z)dz-\int \varphi(z)f(x,z)dz\right)\\ &=& 0,\end{aligned}$$ which proves that the expectation of${\displaystyle}{\iint \hat{g}(x,y)^2f(x,y)dxdy}$ converges to $0$. Similar considerations show that the expectation of the second term ${\displaystyle}{\left( \iint \hat{g}(x,y)f(x,y)dxdy\right)^2}$ also converges to $0$. We finally have $$\lim_{n\rightarrow\infty} n{\mathbb{E}}\left(\hat{Q}'-Q'\right)^2 = 0.$$ Lemma \[qq\] imply that ${\mathbb{E}}(R_1^2)\rightarrow 0$. We will now prove that ${\mathbb{E}}(R_2^2)\rightarrow 0$ : $$\begin{aligned} {\mathbb{E}}(R_2^2)&=&\frac{n}{n_2} {\mathbb{E}}\left[ \iint \left(H(f,x,y)-H(\hat{f},x,y)\right)^2 f(x,y)dxdy\right]\\ &&- \frac{n}{n_2} {\mathbb{E}}\left[ \iint H(f,x,y)f(x,y)dxdy - \iint H(\hat{f},x,y)f(x,y)dxdy\right]^2.\end{aligned}$$ The same arguments as before (mean value theorem and assumptions A2 and A3) show that ${\mathbb{E}}(R_2^2)\rightarrow 0$. At last, we can give another expression for the asymptotic variance : $$C(f)=\iint H(f,x,y)^2 f(x,y)dxdy-\left( \iint H(f,x,y) f(x,y)dxdy\right)^2.$$ We will prove that $$C(f)={\mathbb{E}}\left({\textrm{Var}}(\varphi(Y)|X)\left[\dot\psi\left({\mathbb{E}}(Y|X)\right)\right]^2\right)+{\textrm{Var}}\left(\psi\left({\mathbb{E}}(\varphi(Y)|X)\right)\right).$$ Remark that $$\begin{aligned} \iint H(f,x,y) f(x,y)dxdy&=& \iint \left( \left[ \varphi(y)-m(x)\right] \dot\psi(m(x)) + \psi(m(x))\right)f(x,y)dxdy\notag \\ &=& \iint m(x) \dot\psi(m(x)) f(x,y)dxdy- \iint m(x) \dot\psi(m(x))f(x,y)dxdy\notag\\ &&+ \iint \psi(m(x)) f(x,y)dxdy\notag \\ &=& {\mathbb{E}}\left(\psi\left({\mathbb{E}}(\varphi(Y)|X)\right)\right). \label{hec}\end{aligned}$$ Moreover, $$\begin{aligned} H(f,x,y)^2&=&\left[ \varphi(y)-m(x)\right]^2 \dot\psi(m(x))^2 + \psi(m(x))^2+2\left[ \varphi(y)-m(x)\right] \dot\psi(m(x))\psi(m(x))\\ &=& \varphi(y)^2\dot\psi(m(x))^2 + m(x)^2\dot\psi(m(x))^2 -2\varphi(y)m(x)\dot\psi(m(x))^2\\ && + \psi(m(x))^2+2\left[ \varphi(y)-m(x)\right] \dot\psi(m(x))\psi(m(x)).\end{aligned}$$ We can then rewrite ${\displaystyle}{\iint H(f,x,y)^2f(x,y)dxdy}$ as: $$\begin{aligned} && \iint \varphi(y)^2\dot\psi(m(x))^2f(x,y)dxdy+ \iint m(x)^2\dot\psi(m(x))^2f(x,y)dxdy\\ &&-2\iint \varphi(y)m(x)\dot\psi(m(x))^2f(x,y)dxdy + \iint \psi(m(x))^2f(x,y)dxdy\\ &&+2\iint \varphi(y)\dot\psi(m(x))\psi(m(x))f(x,y)dxdy-2\iint m(x)\dot\psi(m(x))\psi(m(x))f(x,y)dxdy\\ &=& \iint v(x)\dot\psi(m(x))^2f(x,y)dxdy - \iint m(x)^2\dot\psi(m(x))f(x,y)dxdy + \iint \psi(m(x))^2f(x,y)dxdy\\ &=& \iint \left(\left[v(x) -m(x)^2\right]\dot\psi(m(x))^2 + \psi(m(x))^2\right)f(x,y)dxdy\\ &=& {\mathbb{E}}\left(\left[v(X) -m(X)^2\right]\dot\psi(m(X))^2\right) + {\mathbb{E}}\left(\psi(m(X))^2\right)\\ &=& {\mathbb{E}}\left(\left[{\mathbb{E}}(\varphi(Y)^2|X) -{\mathbb{E}}(\varphi(Y)|X)^2\right]\left[\dot\psi({\mathbb{E}}(\varphi(Y)|X))\right]^2\right)+ {\mathbb{E}}\left(\psi({\mathbb{E}}(\varphi(Y)|X))^2\right)\\ &=& {\mathbb{E}}\left({\textrm{Var}}(\varphi(Y)|X)\left[\dot\psi\left({\mathbb{E}}(Y|X)\right)\right]^2\right) + {\mathbb{E}}\left(\psi({\mathbb{E}}(\varphi(Y)|X))^2\right)\end{aligned}$$ where we have set $v(x)=\int \varphi(y)^2f(x,y)dy/\int f(x,y)dy$. This result and (\[hec\]) give the desired form for $C(f)$ which ends the proof of Theorem \[tfec\]. Proof of Theorem \[cramerrao2\] ------------------------------- We follow the proof of Theorem \[cramerrao1\]. Assumptions A2 and A3 imply that $$\begin{aligned} T(f)-T(f_0)&=&\iint \left(\big[\varphi(y)-m_0(x)\big]\dot\psi(m_0(x))+\psi(m_0(x))\right)\\ &&\Big(f(x,y)-f_0(x,y)\Big)dxdy+O\left(\int (f-f_0)^2\right)\end{aligned}$$ where $m_0(x)=\int \varphi(y)f_0(x,y)dy/\int f_0(x,y)dy$. This result shows that the Fréchet derivative of $T(f)$ at $f_0$ is $T'(f_0)\cdot h =\left< H(f_0,\cdot),h\right>$ where $$H(f_0,x,y)=\left(\big[\varphi(y)-m_0(x)\big]\dot\psi(m_0(x))+\psi(m_0(x))\right).$$ We then deduce that $$\begin{aligned} K(h)&=& T'(f_0)\cdot \left(\sqrt{f_0}\left(h-\sqrt{f_0}\int h\sqrt{f}_0\right)\right)\\ &=& \int H(f_0,\cdot) \sqrt{f_0}h- \int H(f_0,\cdot) \sqrt{f_0} \int h\sqrt{f_0}\\ &=& \left<t,h\right>\end{aligned}$$ with $$t=H(f_0,\cdot)\sqrt{f_0}-\left(\int H(f_0,\cdot)f_0\right)\sqrt{f_0}.\\$$ The semiparametric Cramér-Rao bound for this problem is thus $$\|t\|_{L^2(dxdy)}^2=\int H(f_0,\cdot)^2 f_0 - \left(\int H(f_0,\cdot)f_0\right)^2= C(f_0)$$ where we recognize the expression of $C(f_0)$ in Theorem \[cramerrao2\]. [^1]: IFP Energies nouvelles 1 & 4, avenue de Bois-Préau F-92852 Rueil-Malmaison Cedex [[email protected]]{} [^2]: Institut de Mathématiques Université Paul Sabatier F-31062 Toulouse Cedex 9 [http://www.lsp.ups-tlse.fr/Fp/Gamboa.]{} [[email protected]]{}.

ArXiv

--- abstract: | When independent Bose-Einstein condensates (BEC), described quantum mechanically by Fock (number) states, are sent into interferometers, the measurement of the output port at which the particles are detected provides a binary measurement, with two possible results $\pm1$. With two interferometers and two BEC’s, the parity (product of all results obtained at each interferometer) has all the features of an Einstein-Podolsky-Rosen quantity, with perfect correlations predicted by quantum mechanics when the settings (phase shifts of the interferometers) are the same. When they are different, significant violations of Bell inequalities are obtained. These violations do not tend to zero when the number $N$ of particles increases, and can therefore be obtained with arbitrarily large systems, but a condition is that all particles should be detected. We discuss the general experimental requirements for observing such effects, the necessary detection of all particles in correlation, the role of the pixels of the CCD detectors, and that of the alignments of the interferometers in terms of matching of the wave fronts of the sources in the detection regions. Another scheme involving three interferometers and three BEC’s is discussed; it leads to Greenberger-Horne-Zeilinger (GHZ) sign contradictions, as in the usual GHZ case with three particles, but for an arbitrarily large number of them. Finally, generalizations of the Hardy impossibilities to an arbitrarily large number of particles are introduced. BEC’s provide a large versality for observing violations of local realism in a variety of experimental arrangements. author: - | F. Laloë$^{a}$ and W. J. Mullin$^{b}$\ $^{a}$Laboratoire Kastler Brossel, ENS, UPMC, CNRS ; 24 rue Lhomond, 75005 Paris, France\ $^{b}$Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 USA title: 'Interferometry with independent Bose-Einstein condensates: parity as an EPR/Bell quantum variable' --- PACS numbers: 03.65.Ud, 03.75.Gg, 42.50.Xa The original Einstein-Podolsky-Rosen (EPR) argument [@EPR] considers a system of two microscopic particles that are correlated; assuming that various types of measurements are performed on this system in remote locations, and using local realism, it shows that the system contains more elements of reality than those contained in quantum mechanics. Bohr gave a refutation of the argument [@Bohr] by pointing out that intrinsic physical properties should not be attributed to microscopic systems, independently of their measurement apparatuses; in his view of quantum mechanics (often called orthodox), the notion of reality introduced by EPR is inappropriate. Later, Bell extended the EPR argument and used inequalities to show that local realism and quantum mechanics may sometimes lead to contradictory predictions [@Bell]. Using pairs of correlated photons emitted in a cascade, Clauser et al. [@FC] checked that, even in this case, the results of quantum mechanics are correct; other experiments leading to the same conclusion were performed by Fry et al. [@Fry], Aspect et al. [@Aspect], and many others. The body of all results is now such that it is generally agreed that violations of the Bell inequalities do occur in Nature, even if experiments are never perfect and if loopholes (such as sample bias [@Pearle; @CH; @CS]) can still be invoked in principle. All these experiments were made with a small number of particles, generally a pair of photons, so that Bohr’s point of view directly applies to them. In this article, as in [@FL] we consider systems made of an arbitrarily large number of particles, and study some of their variables that can lead to an EPR argument and Bell inequalities. Mermin [@Mermin] has also considered a physical system made of many particles with spins, assuming that the initial state is a so called GHZ state [@GHZ-1; @GHZ-2]; another many-particle quantum state has been studied by Drummond [@Drummond]. Nevertheless, it turns out that considering a double Fock state (DFS) with spins, instead of these states, sheds interesting new light on the Einstein-Bohr debate. The reason is that, in this case, the EPR elements of reality can be macroscopic, for instance the total angular momentum (or magnetization) contained in a large region of space; even if not measured, such macroscopic quantities presumably possess physical reality, which gives even more strength to the EPR argument. Moreover, one can no longer invoke the huge difference of scales between the measured properties and the measurement apparatuses, and Bohr’s refutation becomes less plausible. Double Fock states with spins also lead to violations of the Bell inequalities [@PRL; @LM], so that they are appropriate for experimental tests of quantum violations of local realism. A difficulty, nevertheless, is that the violations require that all $N$ spins be measured in $N$ different regions of space, which may be very difficult experimentally if $N$ exceeds $2$ or $3$; with present experimental techniques, the schemes discussed in [@PRL; @LM] are therefore probably more thought experiments than realistic possibilities. Here we come closer to experiments by studying schemes involving only individual position measurement of the particles, without any necessity of accurate localization. With Bose condensed gases of metastable helium atoms, micro-channel plates indeed allow one to detect atoms one by one [@Saubamea; @Robert]. The first idea that then comes to mind is to consider the interference pattern created by a DFS, a situation that has been investigated theoretically by several authors [@Java; @WCW; @CGNZ; @PS; @Dragan], and observed experimentally [@WK]. The quantum effects occurring in the detection of the fringes have been studied in [@Polkovnikov-1; @Polkovnikov-2], in particular the quantum fluctuations of the fringe amplitude; see also [@DBRP] for a discussion of fringes observed with three condensates, in correlation with populations oscillations. But, for obtaining quantum violations of Bell type inequalities, continuous position measurements are not necessarily optimal; it is more natural to consider measurement apparatuses with a dichotomic result, such as interferometers with two outputs, as in [@CD; @SC]. Experimentally, laser atomic fluorescence may be used to determine at which output of an interferometer atoms are found, without requiring a very accurate localization of their position; in fact, since this measurement process has a small effect on the measured quantity (the position of the atom in one of the arms), one obtains in this way a quantum non-demolition scheme where many fluorescence cycles can be used to ensure good efficiency. Quantum effects taking place in measurements performed with interferometers with 2 input and 2 output ports (Mach-Zhender interferometers) have been studied by several authors; refs [@HB; @Dowling; @DBB] discuss the effect of quantum noise on an accurate measurement of phase, and compare the feeding of interferometers with various quantum states; refs [@PS-1; @PS-2; @PS-3] give a detailed treatment of the Heisenberg limit as well as of the role of the Fisher information and of the Cramer-Rao lower bound in this problem. But, to our knowledge, none of these studies leads to violations of Bell inequalities and local realism. Here, we will consider interferometers with 4 input ports and 4 output ports, in which a DFS is used to feed two of the inputs (the others receive vacuum), and 4 detectors count the particles at the 4 output ports - see Fig. \[fig1\]; we will also consider a similar 6 input-6 output case. This can be seen as a generalization of the work described by Yurke and Stoler in refs. [@YS-1; @YS-2], and also to some extent of the Rarity-Tapster experiment [@Franson; @RT] (even if, in that case, the two photons were not emitted by independent sources). Another aspect of the present work is to address the question raised long ago by Anderson [@Anderson] in the context of a thought experiment and, more recently, by Leggett and Sols [@LS; @Leggett]: Do superfluids that have never seen each other have a well-defined relative phase?.  A positive answer occurs in the usual view: when spontaneous symmetry breaking takes place at the Bose- Einstein transition, each condensate acquires a well-defined phase, though with a completely random value. However, in quantum mechanics, the Bose-Einstein condensates of superfluids are naturally described by Fock states, for which the phase of the system is completely undetermined, in contradiction with this view. Nevertheless, the authors of refs. [@Java; @WCW; @CGNZ; @PS] and [@CD; @MKL] have shown how repeated quantum measurements of the relative phase of two Fock states make a well-defined value emerge spontaneously, with a random value. This seems to remove the contradiction; considering that the relative phase appears under the effect of spontaneous symmetry breaking, as soon as the BEC’s are formed, or later, under the effect of measurements, then appears as a matter of personal preference. But a closer examination of the problem shows that this is not always true [@PRL; @Polkovnikov-2; @LS; @Leggett]: situations do exist where the two points of view are not equivalent, and where the predictions of quantum mechanics for an ensemble of measurements are at variance with those obtained from a classical average over a phase. This is not so surprising after all: the idea of a pre-existing phase is very similar to the notion of  an EPR element of reality - for a double Fock state, the relative phase is nothing but what is often called a hidden variable. The tools offered by the Bell theorem are therefore appropriate to exhibit contradictions between the notion of a pre–existing phase and the predictions of quantum mechanics. Indeed, we will obtain violations of the BCHSH inequalities [@BCHSH], new GHZ contradictions [@GHZ-1; @GHZ-2] as well as Hardy impossibilities [@H-1; @H-2]. Fock-state condensates appear as remarkably versatile, able to create violations that usually require elaborate entangled wave functions, and produce new $N$-body violations. A preliminary short version of this work has been published in [@PRL-2]. The present article gives more details and focusses on some issues that will inevitably appear in the planning of an experiment, such as the effect of non-perfect detection efficiency, losses, or the geometry of the wavefronts in the region of the detectors. In § \[quantum\] we basically use the same method as in [@PRL-2] (unitary transformations of creation operators), following refs [@YS-1] and [@YS-2], but include the treatment of losses; in § \[more elaborate\], we develop a more elaborate theory of many-particle detection and high order correlation signals, performing a calculation in $3N$ configuration space, and including a treatment of the geometrical effects of wavefronts in the detection regions (this section may be skipped by the reader who is not interested in experimental considerations); finally, § \[EPR\] applies these calculations to three situations: BCHSH inequality violations with two sources, GHZ contradictions with three sources, and Hardy contradictions. Appendix I summarizes some useful technical calculations; appendix II extends the calculations to initial states other than the double Fock state (\[1\]), in particular coherent and phase states. Quantum calculation {#quantum} =================== We first calculate the prediction of quantum mechanics for the experiment that is shown schematically in Fig. \[fig1\].  Each of two Bose-Einstein condensates, described by Fock states with populations $N_{\alpha}$ and $N_{\beta}$, crosses a beam splitter; both are then made to interfere at two other beam splitters, sitting in remote regions of space $D_{A}$ and $D_{B}$. There, two operators, Alice and Bob, count the number of particles that emerge from outputs 1 and 2 for Alice, outputs 3 and 4 for Bob. By convention, channels 1 and 3 are ascribed a result $\eta=+1$, channels 2 and 4 a result $\eta=-1$. We call $m_{j}$ the number of particles that are detected at output $j$ (with $j=1$, $2$, $3$, $4$), $m_{A}=m_{1}+m_{2}$ the total number of particles detected by Alice, $m_{B}=m_{3}+m_{4}$ the number of particles detected by Bob, and $M=m_{A}+m_{B}$ the total number of detected particles. From the series of results that they obtain in each run of the experiment, both operators can calculate various functions $\mathcal{A(}\eta_{1},..\eta_{m_{A}})$ and $\mathcal{B(}\eta _{m_{A}+1},..\eta_{M})$ of their results; we will focus on the case where they choose the parity, given by the product of all their $\eta$’s: $\mathcal{A}=(-1)^{m_{2}}$ for Alice, $\mathcal{B}=(-1)^{m_{4}}$ for Bob; for a discussion of other possible choices, see [@LM]. ![ Two Fock states, with populations $N_{\alpha}$ and $N_{\beta}$, enter beam splitters, and are then made to interfere in two different regions of space $D_{A}$ and $D_{B}$, with detectors 1 and 2 in the former, 3 and 4 in the latter. The number of particles $m_{j}$ in each of the channels $j=1,2,3,4$ are counted.[]{data-label="fig1"}](deux-condensats-symetriques.eps){width="3in"} We now calculate the probability of any sequence of results with the same approach as in [@PRL-2]. This provides correct results if one assumes that the experiment is perfect; a more elaborate approach is necessary to study the effects of experimental imperfections, and will be given in § \[more elaborate\]. Probabilities of the various results {#PVR} ------------------------------------ We consider spinless particles and we assume that the initial state is: $$\left\vert \Phi_{0}\right\rangle =\left\vert N_{\alpha},N_{\beta}\right\rangle \equiv~\frac{1}{\sqrt{N_{\alpha}!N_{\beta}!}}~\left[ \left( a_{\alpha }\right) ^{\dagger}\right] ^{N_{\alpha}}\left[ \left( a_{\beta}\right) ^{\dagger}\right] ^{N_{\beta}}\left\vert 0\right\rangle \label{1}$$ where $\left\vert 0\right\rangle $ is the vacuum state; single particle state $\alpha$ corresponds to that populated by the first source, $\beta$ to that populated by the second source. The destruction operators $a_{1}\cdots a_{4}$ of the output modes can be written in terms of those of the modes at the sources $a_{\alpha},$ $a_{\beta},$ $a_{\alpha^{\prime}}$ and $a_{\beta ^{\prime}}$ (including the vacuum input modes, $a_{\alpha^{\prime}}$ and $a_{\beta^{\prime}}$, which are included to maintain unitarity) by tracing back from the detectors to the sources, providing a phase shift of $\pi/2$ at each reflection and $\zeta$ or $\theta$ at the shifters, and a $1/\sqrt{2}$ for normalization at each beam splitter. Thus we find: $$\left( \begin{array} [c]{l}a_{1}\\ a_{2}\\ a_{3}\\ a_{4}\end{array} \right) =\frac{1}{2}\left( \begin{array} [c]{llll}ie^{i\zeta} & e^{i\zeta} & i & -1\\ -e^{i\zeta} & ie^{i\zeta} & 1 & i\\ i & -1 & ie^{i\theta} & e^{i\theta}\\ 1 & i & -e^{i\theta} & e^{i\theta}\end{array} \right) \left( \begin{array} [c]{l}a_{\alpha}\\ a_{\alpha^{\prime}}\\ a_{\beta}\\ a_{\beta^{\prime}}\end{array} \right)$$ Since $a_{\alpha^{\prime}}$ and $a_{\beta^{\prime}}$ do not contribute we can write simply: $$\begin{array} [c]{l}a_{1}=\frac{1}{2}\left[ ie^{i\zeta}a_{\alpha}+ia_{\beta}\right] \\ a_{2}=\frac{1}{2}\left[ -e^{i\zeta}a_{\alpha}+a_{\beta}\right] \\ a_{3}=\frac{1}{2}\left[ ia_{\alpha}+ie^{i\theta}a_{\beta}\right] \\ a_{4}=\frac{1}{2}\left[ a_{\alpha}-e^{i\theta}a_{\beta}\right] \end{array} \label{awm}$$ In short, we write these expressions as: $$a_{j}=v_{j\alpha}a_{\alpha}+v_{j\beta}a_{\beta}. \label{ai}$$ We suppose that Alice finds $m_{1}$ positive results and $m_{2}$ negative results for a total of $m_{A}$ measurements; Bob finds $m_{3}$ positive and $m_{4}$ negative results in his $m_{B}$ total measurements. The quantum probability of this series of results is the squared modulus of the scalar product of state $\left\vert \Phi_{0}\right\rangle $ by the state associated with the measurement:$$\mathcal{P(}m_{1},m_{2},m_{3},m_{4})=\frac{1}{m_{1}!m_{2}!m_{3}!m_{4}!}~\left\vert \left\langle 0\right\vert \left( a_{1}\right) ^{m_{1}}\cdots\left( a_{4}\right) ^{m_{4}}\left\vert N_{\alpha},N_{\beta }\right\rangle \right\vert ^{2} \label{proba}$$ where the matrix element is non-zero only if:$$m_{1}+m_{2}+m_{3}+m_{4}=N_{\alpha}+N_{\beta}=N \label{sum-m}$$ We can calculate this matrix element by substituting (\[1\]) and (\[ai\]) and expanding in binomial series:$$\begin{array} [c]{l}\displaystyle\left\langle 0\right\vert \left( a_{1}\right) ^{m_{1}}\cdots\left( a_{1}\right) ^{m_{4}}\left\vert N_{\alpha},N_{\beta }\right\rangle =\frac{1}{\sqrt{N_{\alpha}!N_{\beta}!}}\left\langle 0\right\vert \prod_{j=1}^{4}\left( v_{j\alpha}a_{\alpha}+v_{j\beta}a_{\beta }\right) ^{m_{j}}\left( a_{\alpha}^{\dagger}\right) ^{N_{\alpha}}\left( a_{\beta}^{\dagger}\right) ^{N_{\beta}}\left\vert 0\right\rangle \\ \multicolumn{1}{c}{\displaystyle=\frac{1}{\sqrt{N_{\alpha}!N_{\beta}!}}\sum_{p_{\alpha1}=0}^{m_{1}}\frac{m_{1}!}{p_{\alpha1}!p_{\beta1}!}\left( v_{1\alpha}\right) ^{p_{\alpha1}}\left( v_{1\beta}\right) ^{p_{\beta1}}...}\\ \multicolumn{1}{r}{\displaystyle..\sum_{p_{\alpha4}=0}^{m_{4}}\frac{m_{4}!}{p_{\alpha4}!p_{\beta4}!}\left( v_{4\alpha}\right) ^{p_{\alpha4}}\left( v_{4\beta}\right) ^{p_{\beta4}}~\left\langle 0\right\vert \left( a_{\alpha }\right) ^{p_{\alpha1}+\cdots+p_{\alpha4}}\left( a_{\beta}\right) ^{p_{\beta1}+\cdots+p_{\beta4}m_{i}}\left( a_{\alpha}^{\dagger}\right) ^{N_{\alpha}}\left( a_{\beta}^{\dagger}\right) ^{N_{\beta}}\left\vert 0\right\rangle }\end{array} \label{CWM}$$ where $p_{\beta j}=m_{j}-p_{\alpha j}$ for any $j$. The matrix element at the end of this expression is:$$N_{\alpha}!N_{\beta}!~\delta_{N_{\alpha},~p_{\alpha1}+\cdots+~p_{\alpha4}}~\delta_{N_{\beta},~p_{\beta1}+\cdots~+p_{\beta4}} \label{matrix element}$$ But by definition the sum of all $p$’s is equal to $m_{1}+m_{2}+m_{3}+m_{4}$ which, according to (\[sum-m\]), is $N_{\alpha}+N_{\beta}$; the two Kronecker delta’s in (\[matrix element\]) are therefore redundant. For the matrix element to be non zero and equal to $N_{\alpha}!N_{\beta}!$, it is sufficient that the difference between the sums $p_{\alpha1}+\cdots +~p_{\alpha4}$ and $p_{\beta1}+\cdots~+p_{\beta4}$ be equal to $N_{\alpha }-N_{\beta}$, a condition which we can express through the integral: $$\int_{-\pi}^{\pi}\frac{d\mu}{2\pi}e^{i(N_{\beta}-N_{\alpha}+p_{\alpha1}+\cdots+~p_{\alpha4}-p_{\beta1}+\cdots~-p_{\beta4})\mu}=\delta_{N_{\alpha }-N_{\beta},~p_{\alpha1}+\cdots+~p_{\alpha4}-p_{\beta1}-\cdots~-p_{\beta4}} \label{delta}$$ When this is inserted into (\[CWM\]), in the second line every $v_{j\alpha}$ becomes $v_{j\alpha}e^{i\mu}$, every $v_{j\beta}$ becomes $v_{j\beta}e^{-i\mu }$, and we can redo the sums and write the probability amplitude as: $$\sqrt{N_{\alpha}!N_{\beta}!}\int_{-\pi}^{\pi}\frac{d\mu}{2\pi}e^{^{i\left( N_{\beta}-N_{\alpha}\right) \mu}}\prod_{j=1}^{4}\left( v_{j\alpha}e^{i\mu }+v_{j\beta}e^{-i\mu}\right) ^{m_{j}} \label{ampl}$$ Thus the probability is:$$\mathcal{P(}m_{1},m_{2},m_{3},m_{4})=\frac{N_{\alpha}!N_{\beta}!}{m_{1}!m_{2}!m_{3}!m_{4}!}\int_{-\pi}^{\pi}\frac{d\mu}{2\pi}\int_{\pi}^{\pi}\frac{d\mu^{\prime}}{2\pi}~~e^{^{i\left( N_{\beta}-N_{\alpha}\right) \left( \mu-\mu^{\prime}\right) }}\prod_{j=1}^{4}\left[ \Omega_{j}^{\ast}(\mu^{\prime})\Omega_{j}(\mu)\right] ^{m_{i}} \label{prob-2}$$ with:$$\Omega_{j}(\mu)=v_{j\alpha}e^{i\mu}+v_{j\beta}e^{-i\mu} \label{omega}$$ Each of the factors $\Omega_{j}^{\ast}(\mu^{\prime})\Omega_{j}(\mu)$ can now be simplified according to:$$\Omega_{j}^{\ast}(\mu^{\prime})\Omega_{j}(\mu)=\left\vert v_{j\alpha }\right\vert ^{2}e^{i\left( \mu-\mu^{\prime}\right) }+\left\vert v_{j\beta }\right\vert ^{2}e^{i\left( \mu^{\prime}-\mu\right) }+v_{j\alpha}^{\ast }v_{j\beta}~e^{-i\left( \mu+\mu^{\prime}\right) }+v_{j\alpha}v_{j\beta }^{\ast}~e^{i\left( \mu+\mu^{\prime}\right) } \label{produit-omega}$$ which, when (\[awm\]) is inserted, gives:$$\frac{1}{2}\left[ \cos\left( \mu-\mu^{\prime}\right) \pm\cos\left( \zeta+\mu+\mu^{\prime}\right) \right] ~~~~~\text{or}~~~~~\frac{1}{2}\left[ \cos\left( \mu-\mu^{\prime}\right) \pm\cos\left( -\theta+\mu+\mu^{\prime }\right) \right]$$ depending on the value of $j$. Now, if we define:$$\begin{array} [c]{l}\lambda=\mu+\mu^{\prime}\\ \Lambda=\mu-\mu^{\prime}\end{array} \label{Lambda}$$ we finally obtain:$$\begin{array} [c]{l}\displaystyle\mathcal{P(}m_{1},m_{2},m_{3},m_{4})=\frac{N_{\alpha}!N_{\beta}!}{m_{1}!m_{2}!m_{3}!m_{4}!}~2^{-N}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi}\cos\left[ (N_{\beta}-N_{\alpha })\Lambda\right] \\ \multicolumn{1}{r}{\displaystyle\times\left[ \cos\Lambda+\cos\left( \zeta+\lambda\right) \right] ^{m_{1}}\left[ \cos\Lambda-\cos\left( \zeta+\lambda\right) \right] ^{m_{2}}\left[ \cos\Lambda+\cos\left( \theta-\lambda\right) \right] ^{m_{3}}\left[ \cos\Lambda-\cos\left( \theta-\lambda\right) \right] ^{m_{4}}}\end{array} \label{19}$$ where we have used $\Lambda$ parity to reduce $e^{i\left( N_{\beta}-N_{\alpha}\right) \Lambda}$ to a cosine. Effects of particle losses -------------------------- We now study cases where losses occur in the experiment; some particles emitted by the sources are missed by the detectors sitting at the four output ports. The total number of particles they detect is $M$, with $M\leq N$; an analogous situation was already considered in the context of spin measurements [@PRL; @LM]. We first focus on losses taking place near the sources of particles, then on those in the detection regions. ### Losses at the sources {#losses-sources} As a first simple model for treating losses, we consider the experimental configuration shown in Fig. \[fig2\], where additional beam splitters divert some particles before they reach the input of the interferometer. If $T$ and $R$ are the transmission and the reflection coefficients of the additional beam splitters, with:$$R+T=1 \label{6a-bis}$$ the unitary transformations become:$$\begin{array} [c]{l}a_{1}=\frac{i\sqrt{T}}{2}\left[ e^{i\zeta}a_{\alpha}+\alpha_{\beta}\right] \\ a_{2}=\frac{\sqrt{T}}{2}\left[ -e^{i\zeta}a_{\alpha}+\alpha_{\beta}\right] \\ a_{3}=\frac{i\sqrt{T}}{2}\left[ a_{\alpha}+e^{i\theta}a_{\beta}\right] \\ a_{4}=\frac{\sqrt{T}}{2}\left[ a_{\alpha}-e^{i\theta}a_{\beta}\right] \\ a_{5}=i\sqrt{R}\left[ a_{\alpha}\right] \\ a_{6}=i\sqrt{R}\left[ a_{\beta}\right] \end{array} \label{6a}$$ ![The experiment is the same as in figure \[fig1\], but now we assume that two beam splitters are inserted between the two sources and the inputs of the interferometer. Then, the total number of particles $M$ measured at the output of the interferometer may be less than $N$.[]{data-label="fig2"}](particules-deviees.eps){width="3in"} The probability amplitude associated with a series of results $m_{1}$, ...$m_{5}$, $m_{6}$ is now:$$\frac{\left\langle 0\right\vert \left( a_{1}\right) ^{m_{1}}\cdots\left( a_{4}\right) ^{m_{4}}\left( a_{5}\right) ^{m_{5}}\left( a_{6}\right) ^{m_{6}}\left\vert N_{\alpha},N_{\beta}\right\rangle }{\sqrt{m_{1}!....m_{5}!~m_{6}!}} \label{s1}$$ or, taking into account the last two equations (\[6a\]):$$R^{^{\left( m_{5}+m_{6}\right) /2}}\sqrt{\frac{N_{\alpha}!}{\left( N_{\alpha}-m_{5}\right) !~m_{5}!}\times\frac{N_{\beta}!}{\left( N_{\beta }-m_{6}\right) !~m_{6}!}}\frac{\left\langle 0\right\vert \left( a_{1}\right) ^{m_{1}}\cdots\left( a_{4}\right) ^{m_{4}}\left\vert N_{\alpha}-m_{5},N_{\beta}-m_{6}\right\rangle }{\sqrt{m_{1}!...m_{4}!}} \label{t1}$$ The fraction on the right of this expression can be obtained from the calculations of § \[PVR\], by just replacing $N_{\alpha}$ by $N_{\alpha }-m_{5}$, $N_{\beta}$ by $N_{\beta}-m_{6}$ in (\[ampl\]). With this substitution, the numerical factor in front of that expression combines with that of (\[t1\]) to give a prefactor:$$R^{^{\left( m_{5}+m_{6}\right) /2}}\sqrt{\frac{N_{\alpha}!N_{\beta}!}{m_{5}!m_{6}!}} \label{pref}$$ The next step is to sum the probabilities over $m_{5}$ and $m_{6}$, keeping the four $m_{1}$, ..$m_{4}$ constant; we vary $m_{5}$ and $m_{6}$ with a constant sum $N-M$, where $M$ is defined as:$$M=m_{1}+m_{2}+m_{3}+m_{4}\leq N \label{def-M}$$ The $\cos\left[ \left( N_{\beta}-N_{\alpha}\right) \Lambda\right] $ inside the integral of (\[19\]) arose from an exponential $e^{i\left[ \left( N_{\beta}-N_{\alpha}\right) \Lambda\right] }$, which now becomes:$$e^{i\left( N_{\beta}-N_{\alpha}+m_{5}-m_{6}\right) \Lambda} \label{expo}$$ so that the summation over $m_{5}$ and $m_{6}$ reconstructs a power of a binomial:$$\frac{1}{\left( N-M\right) !}\left[ e^{i\Lambda}+e^{-i\Lambda}\right] ^{N-M}=\frac{2^{(N-M)}}{\left( N-M\right) !}\left[ \cos\Lambda\right] ^{N-M} \label{binom}$$ When the powers of $R$ and $T$ are included as well as the factors $2^{-M}$ and $2^{(N-M)}$, equation (\[19\]) is now replaced by:$$\begin{array} [c]{l}\displaystyle\mathcal{P}_{M}\mathcal{(}m_{1},m_{2},m_{3},m_{4})=\frac {N_{\alpha}!N_{\beta}!}{m_{1}!m_{2}!m_{3}!m_{4}!}2^{N-2M}\frac{T^{M}~R^{N-M}~}{\left( N-M\right) !}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}\int_{-\pi }^{\pi}\frac{d\Lambda}{2\pi}\cos\left[ (N_{\beta}-N_{\alpha})\Lambda\right] ~\left[ \cos\Lambda\right] ^{N-M}\\ \multicolumn{1}{r}{\displaystyle\times\left[ \cos\Lambda+\cos\left( \zeta+\lambda\right) \right] ^{m_{1}}\left[ \cos\Lambda-\cos\left( \zeta+\lambda\right) \right] ^{m_{2}}}\\ \multicolumn{1}{r}{\displaystyle\times\left[ \cos\Lambda+\cos\left( \theta-\lambda\right) \right] ^{m_{3}}\left[ \cos\Lambda-\cos\left( \theta-\lambda\right) \right] ^{m_{4}}}\end{array} \label{Proba}$$ where $M$ is defined in (\[def-M\]). This result is similar to (\[19\]), but includes a power of $\cos\Lambda$ inside the integral, which we will discuss in § \[discussion\]. We note that this power of $\cos\Lambda$ introduces exactly the same factor as that already obtained in [@PRL], in the context of spin condensates and particles missed in transverse spin measurements. If $T=1$ and $R=0$, expression (\[Proba\]) vanishes unless $M$ has its maximal value $M=N$; then expression (\[19\]) is recovered, as expected. If $R$ and $T$ have intermediate values, $M$ has a probability distribution including any value less than $N$, with of course smaller values favored when $T$ is small and $R$ large. ### Losses at the detectors {#losses-detectors} Instead of inserting additional beam splitters just after the sources, we can put them just before the detectors, as in Fig. \[fig2-2\]; this provides a model for losses corresponding to imperfect detectors with quantum efficiencies less than $100\%$. ![The experiment is the same as that in figure 2, but now 4 beam splitters are inserted just before the 4 particle detectors, which sit in channels $1$,$2$,$3$,$4$; the other channels, $1^{\prime}$, $2^{\prime}$, $3^{\prime}$, $4^{\prime}$ contain no detector. This provides a model for calculating the effect of limited quantum efficiencies of the detectors.[]{data-label="fig2-2"}](particules-deviees-2.eps){width="3in"}  Instead of 6 destruction operators as in (\[6a\]), we now have 8; each operator $a_{j}$ ($j=1,2,3,4)$ corresponding to one of the 4 the detectors is now associated with a second operator $a_{j}^{\prime}$ corresponding to the other output port with no detector. For instance, for $j=1$, one has:$$a_{1}=\frac{i\sqrt{T}}{2}\left[ e^{i\zeta}a_{\alpha}+\alpha_{\beta}\right] \text{ \ \ \ \ \ \ ; \ \ \ \ \ \ \ \ }a_{1}^{\prime}=\frac{-\sqrt{R}}{2}\left[ e^{i\zeta}a_{\alpha}+\alpha_{\beta}\right] \label{a-prime}$$ and similar results for $j=2$,$3$,$4$. The calculation of the probability associated with results $m_{1}$,..$m_{4}$ (with their sum equal to $M$) and $m_{1}^{\prime}$, ..$m_{4}^{\prime}$ (with their sum equal to $N-M$) is very similar to that of § \[PVR\]; formula (\[proba\]) becomes:$$\mathcal{P(}m_{1},..m_{4};m_{1}^{\prime},..m_{4}^{\prime})=\frac{1}{m_{1}!..!m_{4}!\times m_{1}^{\prime}!..!m_{4}^{\prime}!}~\left\vert \left\langle 0\right\vert \left( a_{1}\right) ^{m_{1}}\cdots\left( a_{4}\right) ^{m_{4}}\times\left( a_{1}^{\prime}\right) ^{m_{1}^{\prime}}\cdots\left( a_{4}^{\prime}\right) ^{m_{4}^{\prime}}\left\vert N_{\alpha },N_{\beta}\right\rangle \right\vert ^{2}$$ Since $a_{j}$ and $a_{j}^{\prime}$ are almost the same operator (they just differ by a coefficient), the result is still given by the right hand side of (\[19\]), with the following changes: \(i) each $m_{j}$ is now replaced by the sum $m_{j}+m_{j}^{\prime}$ \(ii) $m_{1}!..!m_{4}!$ in the denominator is multiplied by $m_{1}^{\prime }!..!m_{4}^{\prime}!$ \(ii) a factor $T^{M}\times R^{N-M}$ appears in front of the expression. Now, we consider the observed results $m_{1}$,..$m_{4}$ as fixed, and add the probabilities associated with all possible non-observed values $m_{1}^{\prime }$, ..$m_{4}^{\prime}$; this amounts to distributing $N-M$ unobserved particles in any possible way among all output channels without detectors. The summation is made in two steps: \(i) summations over $m_{1}^{\prime}$ and $m_{2}^{\prime}$ at constant sum $m_{1}^{\prime}+m_{2}^{\prime}=m_{A}^{\prime}$, and over $m_{3}^{\prime}$ and $m_{4}^{\prime}$ at constant sum $m_{3}^{\prime}+m_{4}^{\prime}=m_{B}^{\prime }$ \(ii) summation over $m_{A}^{\prime}$ and $m_{B}^{\prime}$ at constant sum $m_{A}^{\prime}+m_{B}^{\prime}=N-M$ The first summation provides:$$\sum_{m_{1}^{\prime}+m_{2}^{\prime}=m_{A}^{\prime}}\frac{1}{m_{1}^{\prime }!m_{2}^{\prime}!}\left[ \cos\Lambda+\cos\left( \zeta+\lambda\right) \right] ^{m_{1}^{\prime}}\left[ \cos\Lambda-\cos\left( \zeta+\lambda \right) \right] ^{m_{2}^{\prime}}=\frac{1}{m_{A}^{\prime}!}\left[ 2\cos\Lambda\right] ^{m_{A}^{\prime}} \label{sum1}$$ and similarly for the summation over $m_{3}^{\prime}$ and $m_{4}^{\prime}$. Then the last summation provides:$$\sum_{m_{A}^{\prime}+m_{B}^{\prime}=N-M}\frac{1}{m_{A}^{\prime}!m_{B}^{\prime }!}\left[ 2\cos\Lambda\right] ^{m_{A}^{\prime}}\left[ 2\cos\Lambda\right] ^{m_{B}^{\prime}}=\frac{1}{N-M!}\left[ 4\cos\Lambda\right] ^{N-M} \label{sum2}$$ At the end, as in § \[losses-sources\], we see that each unobserved particle introduces a factor $\cos\Lambda$ so that, in the integral giving the probability, a factor $\left[ \cos\Lambda\right] ^{N-M}$ appears; actually, we end up with an expression that is again exactly (\[Proba\]). The presence of this factor inside the integral seems to be a robust property of the effects of imperfect measurements (for brevity, we do not prove the generality of this statement, for instance by studying the effect of additional beam splitters that are inserted elsewhere, for instance in other parts of the interferometer). Discussion ---------- The discussion of the physical content of equation (\[19\]) and (\[Proba\]) is somewhat similar to that for spin measurements [@PRL]. One difference is that, here, we consider that the total number of measurements $m_{A}=m_{1}+m_{2}$ made by Alice, as well as the total number of measurements $m_{B}=m_{3}+m_{4}$ made by Bob, are left to fluctuate with a constant sum $M$; the particles emitted by the sources may localize in any of the four detection regions. With spins, the numbers of detections depends on the number of spin apparatuses used by Alice and Bob, so that it was more natural to assume that $m_{A}$ and $m_{B}$ are fixed. This changes the normalization and the probabilities, but not the dependence on the experimental parameters, which is given by (\[19\]) and (\[Proba\]). A discussion of the normalization integrals is given in the Appendix. ### Effects of $N_{\alpha}$, $N_{\beta}$ and of the number of measurements {#par} If the numbers of particles in the sources are different$\ $($N_{\alpha}\neq N_{\beta}$), a term in $\cos\left[ (N_{\beta}-N_{\alpha })\Lambda\right] $ appears in both equations; let us assume for simplicity that all $N$ particles are measured $(M=N)$, so that equation (\[19\]) applies, and for instance that $N_{\alpha}>N_{\beta}$. Then, in the product of factors inside the integral, only some terms can provide a non-zero contribution after integration over $\Lambda$; we must choose at least $N_{\alpha}-N_{\beta}$ factors contributing through $\cos\left( \Lambda\right) $, and thus at most $N-(N_{\alpha}-N_{\beta})=2N_{\beta}$ factors contributing through the $\theta$ dependent terms. Therefore $2N_{\beta}$ is the maximum number of particles providing results that depends on the settings of the interferometer; all the others have equal probabilities $1/2$, whatever the phase shift is. This is physically understandable, since $(N_{\alpha}-N_{\beta})$ particles from the first source unmatched particles from the other, and can thus not contribute to an interference effect. All particles can contribute coherently to the interference only if $N_{\alpha }=N_{\beta}$. If the numbers of particles in the sources are equal ($N_{\alpha}=N_{\beta}$), the sources are optimal; equation (\[Proba\]) contains the effect of missing some particles in the measurements. If the number of experiments $M$ is much less than a very large $N$, because $\cos\Lambda^{N-M}$ peaks up sharply[^1] at $\Lambda=0$, the result simplifies into: $$\begin{aligned} & \mathcal{P}_{M}\mathcal{(}m_{1},m_{2},m_{3},m_{4})\sim\frac{1}{m_{1}!m_{2}!m_{3}!m_{4}!}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\left[ 1+\cos\left( \zeta+\lambda\right) \right] ^{m_{1}}\left[ 1-\cos\left( \zeta+\lambda\right) \right] ^{m_{2}}\nonumber\\ & \times\left[ 1+\cos\left( \theta-\lambda\right) \right] ^{m_{3}}\left[ 1-\cos\left( \theta-\lambda\right) \right] ^{m_{4}} \label{22}$$ We then recover classical results, similar to those of refs. [@CD] or [@FL-2]. Suppose that we introduce a classical phase $\lambda$ and calculate classically the interference effects at both beam splitters. This leads to intensities proportional to $\left[ 1+\cos\left( \zeta+\lambda\right) \right] $ and $\left[ 1-\cos\left( \zeta+\lambda\right) \right] $ on both sides of the interferometer in $D_{A}$, and similar results for $D_{B}$. Now, if we assume that each particle reaching the beam splitter has crossing and reflecting probabilities that are proportional to these intensities (we treat each of these individual processes as independent), and if we consider that this classical phase is completely unknown, an average over $2\pi$ then reconstructs exactly (\[22\]). In this case, the classical image of a pre-existing phase leads to predictions that are the same as those of quantum mechanics; this phase will take a completely random value for each realization of the experiment, with for instance no way to force it to take related values in two successive runs. All this fits well within the concept of the Anderson phase, originating from spontaneous symmetry breaking at the phase transition (Bose-Einstein condensation): at this transition point, the quantum system chooses a phase, which takes a completely random value, and then plays the role of a classical variable (in the limit of very large systems). On the other hand, if $N-M$ vanishes, the peaking effect of $\cos\Lambda ^{N-M}$ does not occur anymore, $\Lambda$ can take values close to $\pi/2$ , so that the terms in the product inside the integral are no longer necessarily positive; an interpretation in terms of classical probabilities then becomes impossible. In these cases, the phase does not behave as a semi-classical variable, but retains a strong quantum character; the variable $\Lambda$ controls the amount of quantum effects. It is therefore natural to call $\Lambda$ the quantum angle and $\lambda$ the classical phase. One could object that, if expression (\[19\]) contains negative factors, this does not prove that the same probabilities $\mathcal{P}$ can not be obtained with another mathematical expression without negative probabilities. To show that this is indeed impossible, we have to resort to a more general theorem, the Bell/BCHSH theorem, which proves it in a completely general way; this is what we do in § \[EPR\]. ### Perfect correlations {#perfect} We now show that, if $N_{\alpha}$ and $N_{\beta}$ are equal and if the number of measurements is maximal ($M=N$), when Alice and Bob choose opposite[^2] phase shifts ($\theta=-\zeta$) they always measure the same parity. In this case, the integrand of (\[19\]) becomes:$$\left[ \cos\Lambda+\cos\left( \lambda+\zeta\right) \right] ^{m_{1}+m_{3}}\left[ \cos\Lambda-\cos\left( \lambda+\zeta\right) \right] ^{m_{2+}m_{4}} \label{integrand}$$ which can also be written as:$$2^{N}\left[ \sin(\lambda^{\prime}+\frac{\zeta}{2})\right] ^{m_{1}+m_{3}}\left[ \sin(\lambda^{\prime\prime}-\frac{\zeta}{2})\right] ^{m_{1}+m_{3}}\left[ \cos(\lambda^{\prime}+\frac{\zeta}{2})\right] ^{m_{2}+m_{4}}\left[ \cos(\lambda^{\prime\prime}-\frac{\zeta}{2})\right] ^{m_{2}+m_{4}} \label{integrand-2}$$ with the following change of integration variables[^3]:$$\begin{array} [c]{cc}\lambda^{\prime}=\frac{\lambda+\Lambda}{2} & \lambda^{\prime\prime}=\frac{\Lambda-\lambda}{2}\end{array} \label{lambdas}$$ If, for instance, $m_{1}+m_{3}$ is odd, instead of $\lambda^{\prime}$ one can take $(\lambda^{\prime}+\zeta/2)$ as an integration variable, and one can see that the integral vanishes because its periodicity - the same is true of course for the $\lambda^{\prime\prime}$ integration, which also vanishes. Similarly, if $m_{2}+m_{4}$ is odd, one can take $(\lambda^{\prime }-\frac{\zeta}{2})$ and $(\lambda^{\prime\prime}-\frac{\zeta}{2})$ as integration variables, and the result vanishes again. Finally, the probability is non-zero only if both $m_{1}+m_{3}$ and $m_{2}+m_{4}$ are even; the conclusion is that Alice and Bob always observe the same parity for their results. This perfect correlation is useful for applying the EPR reasoning to parities. A more elaborate calculation {#more elaborate} ============================ The advantage of measuring the positions of particles after a beam splitter, that is interference effects providing dichotomic results, is that one has a device that is close to quantum non-demolition experiments. With a resonant laser, one can make an atom fluoresce and emit many photons, without transferring the atom from one arm of the interferometer to the other. This is clearly important in experiments where, as we have seen, all the atoms must be detected to obtain quantum non-local effects. On the other hand, it is well know experimentally that a difficulty with interferometry is the alignment of the devices in order to obtain an almost perfect matching of the wave front. We discuss this problem now. A general assumption behind the calculation of § \[quantum\] is that, in each region of space (for instance at the inputs, or at the 4 outputs), only one mode of the field is populated (only one $a^{\dagger}$ operator is introduced per region). The advantage of this approach is its simplicity, but it nevertheless eludes some interesting questions. For instance, suppose that the energies of the particles emerging from each source differ by some arbitrarily small quantity; after crossing all beam splitters, they would reach the detection regions in two orthogonal modes, so that the probability of presence would be the sum of the corresponding probabilities, without any interference term. On the other hand, all interesting effects obtained in [@PRL-2] are precisely interference effects arising because, in each detection region, there is no way to tell which source emitted the detected particles. Does this mean that these effects disappear as soon as the sources are not strictly identical, so that the quantum interferences will never be observable in practice? To answer this kind of question, here we will develop a more detailed theory of the detection of many particles in coincidence, somewhat similar to Glauber’s theory of photon coincidences [@Glauber]; nevertheless, while in that theory only the initial value of the n-th time derivative was calculated, here we study the whole time dependence of the correlation function. Our result will be that, provided the interferometers and detectors are properly aligned, it is the detection process that restores the interesting quantum effects, even if the sources do not emit perfectly identical wavefronts in the detection regions, as was assumed in the calculations of § \[quantum\]. As a consequence, the reader who is not interested in experimental limitations may skip this section and proceed directly to § \[EPR\]. In this section we change the definition of the single particle states: $\alpha$ now corresponds to a state for which the wave function originates from the first source, is split into two beams when reaching the first beam splitter, and into two beams again when it reaches the beam splitters associated with the regions of measurement $D_{A}$ and $D_{B}$;  the same is true for state $\beta$. In this point of view, all the propagation in the interferometers is already included in the states. We note in passing that the evolution associated with the beam splitters is unitary; the states $\alpha$ and $\beta$ therefore remain perfectly orthogonal, even if they overlap in some regions of space. Having changed the definition of the single particle states, we keep (\[1\]) to define the N particle state of the system. Pixels as independent detectors ------------------------------- We model the detectors sitting after the beam splitters by assuming that they are the juxtaposition of a large number $\mathcal{Q}$ of independent pixels, which we treat as independent detectors. This does not mean that the positions of the impact of all particles are necessarily registered in the experiment; our calculations still apply if, for instance, only the total number of impact in each channel is recorded. The only thing we assume is that the detection of particles in different points leads to orthogonal states of some part of the apparatus (or the environment), so that we can add the probabilities of the events corresponding to different orthogonal states; whether or not the information differentiating these states is recorded in practice does not matter. If the number of pixels $\mathcal{Q}$ is very large, the probability of detection of two bosons at the same pixel is negligible. If we note $N=N_{\alpha}+N_{\beta}$, this probability is bounded[^4] by:$$\frac{1}{\mathcal{Q}}+\frac{2}{\mathcal{Q}}+....+\frac{N}{\mathcal{Q}}=\frac{N(N+1)}{2\mathcal{Q}} \label{2}$$ so that we will assume that:$$\mathcal{Q}\gg N^{2} \label{3}$$ Moreover, we consider events where all $N$ particles are detected (the probabilities of events where some particles are missed can be obtained from the probabilities of these events in a second step, as in [@PRL]). Flux of probability at the pixels in a stationary state {#flux} ------------------------------------------------------- Each pixel $j$ is considered as defining a region of space $\Delta_{j}$ in which the particles are converted into a macroscopic electric current, as in a photomultiplier. The particles enter this region through the front surface $S_{j}$ of the pixel; all particles crossing $S_{j}$ disappear in a conversion process that is assumed to have 100% efficiency. What we need, then, is to calculate the flux of particles entering the front surfaces of the pixels. It is convenient to reason in the $3N$ dimension configuration space, in which the hyper-volume associated with the $N$ different pixels is:$$V_{N}=\Delta_{1}\otimes\Delta_{2}\otimes\Delta_{3}....\otimes\Delta_{N} \label{a-1}$$ which has an front surface given by:$$S_{N}=S_{1}\otimes\Delta_{2}\otimes\Delta_{3}....\otimes\Delta_{N}~+~\Delta_{1}\otimes S_{2}\otimes\Delta_{3}....\otimes\Delta_{N}+~..+~~\Delta_{1}\otimes\Delta_{2}\otimes\Delta_{3}....\otimes S_{N} \label{a-1-s}$$ The density of probability in this space is defined in terms of the boson field operator $\Psi(\mathbf{r})$ as:$$\rho_{N}(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N})=\Psi^{\dagger }(\mathbf{r}_{1})\Psi(\mathbf{r}_{1})~\Psi^{\dagger}(\mathbf{r}_{2})\Psi(\mathbf{r}_{2})~...~\Psi^{\dagger}(\mathbf{r}_{N})\Psi(\mathbf{r}_{N}) \label{a-2}$$ where we assume that all $\mathbf{r}_{j}$’s are different (all pixels are disjoint). The components of the $3N$ dimension current operator $\mathbf{J}_{N}$ are:$$\mathbf{J}_{N}=\left\{ \begin{array} [c]{l}\frac{\hbar}{2mi}\left[ \Psi^{\dagger}(\mathbf{r}_{1})\mathbf{\nabla}\Psi(\mathbf{r}_{1})-\mathbf{\nabla}\Psi^{\dagger}(\mathbf{r}_{1})\Psi(\mathbf{r}_{1})\right] ~~\Psi^{\dagger}(\mathbf{r}_{2})\Psi (\mathbf{r}_{2})~...\Psi^{\dagger}(\mathbf{r}_{N})\Psi(\mathbf{r}_{N})+\\ +\frac{\hbar}{2mi}\Psi^{\dagger}(\mathbf{r}_{1})\Psi(\mathbf{r}_{1})~\left[ \Psi^{\dagger}(\mathbf{r}_{2})\mathbf{\nabla}\Psi(\mathbf{r}_{2})-\mathbf{\nabla}\Psi^{\dagger}(\mathbf{r}_{2})\Psi(\mathbf{r}_{2})\right] ...\Psi^{\dagger}(\mathbf{r}_{N})\Psi(\mathbf{r}_{N})+\\ +...\\ +\frac{\hbar}{2mi}\Psi^{\dagger}(\mathbf{r}_{1})\Psi(\mathbf{r}_{1})~\Psi^{\dagger}(\mathbf{r}_{2})\Psi(\mathbf{r}_{2})~...\left[ \Psi^{\dagger }(\mathbf{r}_{N})\mathbf{\nabla}\Psi(\mathbf{r}_{N})-\mathbf{\nabla}\Psi^{\dagger}(\mathbf{r}_{N})\Psi(\mathbf{r}_{N})\right] \end{array} \right. \label{a-4}$$ In the Heisenberg picture, the quantum operator $\rho_{N}$ obeys the evolution equation:$$\frac{d}{dt}\rho_{N}(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N};t)+\mathbf{\nabla}_{N}\cdot\mathbf{J}_{N}=0 \label{a-3}$$ where $\mathbf{\nabla}_{N}$ is the $N$ dimensional divergence. The flux of probability entering the $3N$ dimension volume $S_{N}$ it then: $$\begin{array} [c]{l}\mathcal{F}(\Delta_{1},\Delta_{2},...\Delta_{N})~=~<\Phi_{0}\mid F\left( \Delta_{1}\right) \times G\left( \Delta_{2}\right) \times....G\left( \Delta_{N}\right) \mid\Phi_{0}>\\ \multicolumn{1}{r}{+~<\Phi_{0}\mid G\left( \Delta_{1}\right) \times F\left( \Delta_{2}\right) \times....G\left( \Delta_{N}\right) \mid\Phi_{0}>+...}\\ \multicolumn{1}{r}{+~<\Phi_{0}\mid G\left( \Delta_{1}\right) \times G\left( \Delta_{2}\right) \times....F\left( \Delta_{N}\right) \mid\Phi_{0}>}\end{array} \label{4}$$ where $F(\Delta_{j})$ is the operator defined as a flux surface integral associated to pixel $j$: $$F\left( \Delta_{j}\right) =\frac{\hbar}{2mi}~\int_{S_{j}}d^{2}\mathbf{s\cdot}~\left[ \Psi^{\dagger}(\mathbf{r}^{^{\prime}})\mathbf{\nabla }\Psi(\mathbf{r}^{^{\prime}})-\mathbf{\nabla}\Psi^{\dagger}(\mathbf{r}^{^{\prime}})\Psi(\mathbf{r}^{^{\prime}})\right] \label{5}$$ ($d^{2}\mathbf{s}$ the differential vector perpendicular to the surface) and where $G(\Delta_{j})$ is a volume integral associated to the same pixel:$$G\left( \Delta_{j}\right) =\int_{\Delta_{j}}d^{3}r^{\prime}~\Psi^{\dagger }(\mathbf{r}^{^{\prime}})\Psi(\mathbf{r}^{^{\prime}}) \label{5-bis}$$ The value of $\mathcal{F}(\Delta_{1},\Delta_{2},...\Delta_{N})$ provides the time derivative of the probability of detection at all selected pixels, which we calculate in § \[probab\]. Now, because the various pixels do not overlap, the field operators commute and we can push all $\Psi^{\dagger}$’s to the left, all $\Psi$’s to the right; then we expand these operators on a basis that has $u_{\alpha}$ and $u_{\beta }$ as its two first vectors:$$\Psi(\mathbf{r})=u_{\alpha}(\mathbf{r})~a_{\alpha}~+u_{\beta}(\mathbf{r})~a_{\beta}+...~ \label{6}$$ The end of the expansion, noted $....$, corresponds to the components of $\Psi(\mathbf{r})$ on other modes that must be added to modes $\alpha$ and $\beta$ to form a complete orthogonal basis in the space of states of one single particle; it is easy to see that they give vanishing contributions to the average in state $\mid\Phi_{0}>$. The structure of any term in (\[4\]) then becomes (for the sake of simplicity, we just write the first term):$$\begin{array} [c]{l}<\Phi_{0}\mid\mathcal{O}_{a,a\dagger}~~\frac{\hbar}{2mi}\left\{ \int _{\Delta_{1}}\left[ u_{\alpha}^{\ast}(\mathbf{r}_{1}^{\prime})a_{\alpha }^{\dagger}+u_{\beta}^{\ast}(\mathbf{r}_{1}^{\prime})a_{\beta}^{\dagger }\right] ~d^{2}\mathbf{s}_{1}\cdot\left[ \mathbf{\nabla}u_{\alpha }(\mathbf{r}_{1}^{\prime})a_{\alpha}+\mathbf{\nabla}u_{\beta}(\mathbf{r}_{1}^{\prime})a_{\beta}\right] -\text{c.c.}\right\} \times\\ \multicolumn{1}{r}{\times{\displaystyle\prod\limits_{j=2}^{N}} \left[ u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})a_{\alpha}^{\dagger }+u_{\beta}^{\ast}(\mathbf{r}_{j}^{\prime})a_{\beta}^{\dagger}\right] \left[ u_{\alpha}(\mathbf{r}_{j}^{\prime})a_{\alpha}+u_{\beta}(\mathbf{r}_{j}^{\prime})a_{\beta}\right] \mid\Phi_{0}>}\end{array} \label{7}$$ where c.c. means complex conjugate and where $\mathcal{O}_{a,a\dagger}$ is the normal ordering operator that puts all the creation operators $a_{\alpha ,\beta}^{\dagger}$ to the left of all annihilation operators $a_{\alpha,\beta }$. Each term of the product inside this matrix element contains a product of operators that give, either zero, or always the same matrix element $N_{\alpha}!N_{\beta}!$. For obtaining a non-zero value, two conditions are necessary: \(i) the number of $a_{\alpha}^{\dagger}$’s should be equal to that of $a_{\alpha}$’s \(ii) the number of $a_{\alpha}$’s, minus that of $a_{\beta}$’s, should be equal to $N_{\alpha}-N_{\beta}$. These conditions are fulfilled with the help of two integrals:$$\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi }~~e^{i(N_{\beta}-N_{\alpha})\Lambda} \label{8}$$ and by multiplying: \(i) every $u_{\alpha}(\mathbf{r}_{j}^{\prime})$ by $e^{i\lambda}$, and every $u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})$ by $e^{-i\lambda}$ (without changing the wave functions related to $\beta$) \(ii) then every $u_{\alpha}(\mathbf{r}_{j}^{\prime})$ by $e^{i\Lambda}$, and every $u_{\beta}(\mathbf{r}_{j}^{\prime})$ by $e^{-i\Lambda}$ (without touching the complex conjugate wave functions). This provides: $$\begin{array} [c]{l}\mathcal{F\sim}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi}~~e^{i(N_{\beta}-N_{\alpha})\Lambda}\\ \frac{\hbar}{2mi}\int_{\Delta_{1}}d^{2}\mathbf{s}_{1}\cdot\left[ u_{\alpha }^{\ast}(\mathbf{r}_{1}^{\prime})\mathbf{\nabla}u_{\alpha}(\mathbf{r}_{1}^{\prime})e^{i\Lambda}+u_{\beta}^{\ast}(\mathbf{r}_{1}^{\prime })\mathbf{\nabla}u_{\beta}(\mathbf{r}_{1}^{\prime})e^{-i\Lambda}+u_{\alpha }^{\ast}(\mathbf{r}_{1}^{\prime})\mathbf{\nabla}u_{\beta}(\mathbf{r}_{1}^{\prime})e^{-i(\lambda+\Lambda)}+u_{\beta}^{\ast}(\mathbf{r}_{1}^{\prime })\mathbf{\nabla}u_{\alpha}(\mathbf{r}_{1}^{\prime})e^{i(\lambda+\Lambda )}\right. \\ \multicolumn{1}{r}{\left. -u_{\alpha}(\mathbf{r}_{1}^{\prime})\mathbf{\nabla }u_{\alpha}^{\ast}(\mathbf{r}_{1}^{\prime})e^{i\Lambda}-u_{\beta}(\mathbf{r}_{1}^{\prime})\mathbf{\nabla}u_{\beta}^{\ast}(\mathbf{r}_{1}^{\prime})e^{-i\Lambda}-u_{\alpha}(\mathbf{r}_{1}^{\prime})\mathbf{\nabla }u_{\beta}^{\ast}(\mathbf{r}_{1}^{\prime})e^{i(\lambda+\Lambda)}-u_{\beta }(\mathbf{r}_{1}^{\prime})\mathbf{\nabla}u_{\alpha}^{\ast}(\mathbf{r}_{1}^{\prime})e^{-i(\lambda+\Lambda)}\right] }\\ \multicolumn{1}{r}{\times{\displaystyle\prod\limits_{j=2}^{N}} \left[ u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\alpha}(\mathbf{r}_{j}^{\prime})e^{i\Lambda}+u_{\beta}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\beta }(\mathbf{r}_{j}^{\prime})e^{-i\Lambda}+u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\beta}(\mathbf{r}_{j}^{\prime})e^{-i(\lambda+\Lambda )}+\text{c.c.}\right] ~+\text{sim.}}\end{array} \label{9}$$ where sim. is for the $N-1$ similar terms where the gradients occur for $j=2,3,..N$, instead of $j=1$. Now we assume that the experiment is properly aligned so that the wavefronts of the wave functions $\alpha$ and $\beta$ coincide in all detection regions; then, in the gradients:$$\mathbf{\nabla}u_{\alpha}(\mathbf{r})=iu_{\alpha}(\mathbf{r})~\mathbf{k}_{\alpha}~\text{ \ \ \ \ \ \ ; \ \ \ \ \ \ \ }\mathbf{\nabla}u_{\beta }(\mathbf{r})=iu_{\beta}(\mathbf{r})~\mathbf{k}_{\beta}~ \label{10-c}$$ the vectors $\mathbf{k}_{\alpha}$ and $\mathbf{k}_{\beta}$ are parallel; actually, on each pixel $j$ we take these two vectors as equal to the same constant value $\mathbf{k}_{\Delta_{j}}$, assuming that the pixels are small and that the wavelengths of the two wave functions are almost equal. Then (\[9\]) simplifies into: $$\begin{array} [c]{l}\mathcal{F\sim}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi}~~e^{i(N_{\beta}-N_{\alpha})\Lambda}\frac{\hbar}{2m}\int_{S_{1}}\mathbf{k}_{\Delta_{1}}\cdot d^{2}\mathbf{s}_{1}...\int _{\Delta_{2}}d^{3}r_{2}^{\prime}...\int_{\Delta_{i}}d^{3}r_{i}^{\prime}~...\\ \multicolumn{1}{r}{{\displaystyle\prod\limits_{j=1}^{N}} \left[ u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\alpha}(\mathbf{r}_{j}^{\prime})e^{i\Lambda}+u_{\beta}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\beta }(\mathbf{r}_{j}^{\prime})e^{-i\Lambda}+u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\beta}(\mathbf{r}_{j}^{\prime})e^{-i(\lambda+\Lambda )}+u_{\beta}^{\ast}(\mathbf{r}_{j}^{\prime})u_{\alpha}(\mathbf{r}_{j}^{\prime })e^{i(\lambda+\Lambda)}\right] ~+\text{ sim.}}\end{array} \label{10-d}$$ In this expression, the integrand in the $\lambda$ and $\Lambda$ integrals is a product of $N$ factors corresponding to the individual pixels, but this does not imply the absence of correlations (the integral of a product is not the product of integrals). The probability flux $\mathcal{F}$ contains surface integrals through the front surfaces of the pixels, as expected, but also volume integrals in other pixels, which is less intuitive[^5]. As a consequence, for obtaining a non-zero probability flux $\mathcal{F}$ in $3N$ dimensions, it is not sufficient to have a non-zero three dimension probability flux through one (or several) pixels; it is also necessary that some probability has already accumulated in the other pixels. In other words, at the very moment where the wave functions reach the front surface of the pixels, the first time derivative of the probability density in $3N$ dimension remains zero, while only the $N$-th order time derivative is non-zero (this will be seen more explicitly in § \[time\]). This is analogous to the photon detection process with $N$ atoms in quantum optics, see for instance Glauber [@Glauber]. Time dependence {#time} --------------- Consider an experiment where each source emits a wave packet in a finite time. We assume that, in each wave packet, all the particles still remain in the same quantum state, but that this state is now time dependent; the state of the system is then still given by (\[1\]), but with time dependent states $\alpha$ and $\beta$, so that the creation operators are now $\left( a_{\alpha}\right) ^{\dagger}(t)$ and $\left( a_{\beta}\right) ^{\dagger}(t)$. All the calculation of the previous section remains valid, the main difference being that the wave functions are time dependent: $u_{\alpha }(\mathbf{r},t)$ and $u_{\beta}(\mathbf{r},t)$. We must therefore take into account possible time dependences of the wave fronts of the two wave functions, as well as those of their amplitudes and phases. If, for instance, the interferometer is perfectly symmetric, and if the two wave packets are emitted at the same time, they will reach the beam splitters of the detection regions at the same time with wave fronts that will perfectly overlap at the detectors; the amplitudes of the two wave functions will always be the same. See for instance ref. [@Hagley; @Simsarian] for a discussion of the time evolution of the phase of Bose-Einstein condensates, including the effects of the interactions within the condensate. If we consider separately each factor inside the $\lambda$ and $\Lambda$ integral of (\[10-d\]), we come back to the usual three dimension space; two different kinds of integrals then occur:$$\begin{array} [c]{l}f_{j}(t)=\frac{\hbar}{2m}\int_{S_{j}}\mathbf{k}_{\Delta_{j}}\cdot d^{2}\mathbf{s}_{j}\left[ u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime },t)u_{\alpha}(\mathbf{r}_{j}^{\prime},t)e^{i\Lambda}+u_{\beta}^{\ast }(\mathbf{r}_{j}^{\prime},t)u_{\beta}(\mathbf{r}_{j}^{\prime},t)e^{-i\Lambda }\right. \\ \multicolumn{1}{r}{\left. +u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime },t)u_{\beta}(\mathbf{r}_{j}^{\prime},t)e^{-i(\lambda+\Lambda)}+u_{\beta }^{\ast}(\mathbf{r}_{j}^{\prime},t)u_{\alpha}(\mathbf{r}_{j}^{\prime },t)e^{i(\lambda+\Lambda)}\right] }\end{array} \label{10-e}$$ and:$$\begin{array} [c]{l}g_{j}(t)=\int_{\Delta_{i}}d^{3}r_{i}^{\prime}\left[ u_{\alpha}^{\ast }(\mathbf{r}_{j}^{\prime},t)u_{\alpha}(\mathbf{r}_{j}^{\prime},t)e^{i\Lambda }+u_{\beta}^{\ast}(\mathbf{r}_{j}^{\prime},t)u_{\beta}(\mathbf{r}_{j}^{\prime },t)e^{-i\Lambda}+\right. \\ \multicolumn{1}{r}{\left. u_{\alpha}^{\ast}(\mathbf{r}_{j}^{\prime },t)u_{\beta}(\mathbf{r}_{j}^{\prime},t)e^{-i(\lambda+\Lambda)}+u_{\beta }^{\ast}(\mathbf{r}_{j}^{\prime},t)u_{\alpha}(\mathbf{r}_{j}^{\prime },t)e^{i(\lambda+\Lambda)}\right] }\end{array} \label{10-f}$$ The conservation law in ordinary space implies that $f_{j}(t)$ is related to the time derivative of $g_{j}(t)$: it gives the contribution of the front surface of the pixel to the time variation of the accumulated probability $g_{j}(t)$ in volume $\Delta_{i}$. The total time derivative of $g_{j}(t)$ is given by:$$\frac{d}{dt}g(t)=f_{j}(t)-f_{j}^{-}(t) \label{10-f-bis}$$ where $f_{j}^{-}(t)$ is the flux of the three dimensional probability current through the lateral and rear surfaces of volume $\Delta_{j}$; the first term in the right hand side is the entering flux, the second term the out-going flux, with a leak through the rear surface that begins to be non-zero as soon as the wave functions have crossed the entire detection volume $\Delta_{j}$. But we do not have to take into account this out-going flux of probability: we assume that the detection process absorbs all bosons. [For instance, ]{}once they enter volume $\Delta_{j}$, [the atoms are ionized and the emitted electron is amplified into an cascade process, as in a photomultiplier; the detection probability accumulated over time does not decrease under the effect of $f_{j}^{-}(t)$. Therefore we must ignore the second term in the rhs of (\[10-f-bis\]), and replace (\[10-f\]) by the more appropriate definition of $g_{j}(t)$:$$g_{j}(t)=\int_{0}^{t}dt^{\prime}~f_{j}(t^{\prime}) \label{10-g}$$ (we assume that time $t=0$ occurs just before the wave packets reach the detectors). With this relation, we no longer have to manipulate two independent functions $f$ and $g$; moreover, the value of $g_{j}(t)$ now depends only of the values of wave functions on the front surface of the detector, which is physically satisfying (while (\[10-f\]) contains contributions of the wave functions in all volume $\Delta_{j}$, an unphysical result if this volume has a large depth). ]{} We have already assumed in (\[10-c\]) that the wavefronts of the two waves are parallel on every pixel; we moreover assume that $\mathbf{k}_{\Delta_{j}}$ is perpendicular to the surface of the pixel, and then call $\varphi (\Delta_{j})$ their relative phase over this pixel, taking it as a constant over the pixel and over time, during the propagation of the wave functions (which is the case if the interferometer is symmetrical, as in the figure). Moreover, we assume that the two wave functions $u_{\alpha}$ and $u_{\beta}$ have the same square modulus $\left\vert u_{\Delta_{j}}(t)\right\vert ^{2}$ at this pixel, so that the interference contrast is optimal (again, this is related to a proper alignment of the interferometer). Then (\[10-e\]) becomes:$$f_{j}(t)\simeq\frac{\hbar}{2m}S_{j}\left\vert \mathbf{k}_{\Delta_{j}}\right\vert \left\vert u_{\Delta_{j}}(t)\right\vert ^{2}\left\{ \cos \Lambda+\cos\left[ \varphi(\Delta_{j})-\Lambda-\lambda\right] \right\} =\frac{d}{dt}p_{j}(t)\times\left\{ \cos\Lambda+\cos\left[ \varphi(\Delta _{j})-\Lambda-\lambda\right] \right\} \label{10-k}$$ with:$$p_{j}(t)=\frac{\hbar}{2m}S_{j}\left\vert \mathbf{k}_{\Delta_{j}}\right\vert \int_{0}^{t}dt^{\prime}\left\vert u_{\Delta_{j}}(t^{\prime})\right\vert ^{2} \label{10-l}$$ where $S_{j}$ is the area of pixel $j$. Finally, we assume that all the pixels are identical so that their detection areas have the same value $S$. We then obtain the simplified expression:$$\mathcal{F(}t\mathcal{)\sim~}S^{N}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi}~~\cos\left[ (N_{\beta}-N_{\alpha })\Lambda\right] ~~\frac{\text{d}}{\text{dt}}{\displaystyle\prod\limits_{j=1}^{N}} p_{j}(t)\left\{ \cos\Lambda+\cos\left[ \varphi(\Delta_{j})-\Lambda -\lambda\right] \right\} ~ \label{11}$$ (we have used $\Lambda$ parity to replace the exponential in $(N_{\beta }-N_{\alpha})\Lambda$ by a cosine, so that the reality of the expression is more obvious) and the accumulated probability at time $t$ is:$$\mathcal{P(}t\mathcal{)\sim}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int _{-\pi}^{\pi}\frac{d\Lambda}{2\pi}~~\cos\left[ (N_{\beta}-N_{\alpha})\Lambda\right] ~\times{\displaystyle\prod\limits_{j=1}^{N}} p_{j}(t)~\left\{ \cos\Lambda+\cos\left[ \varphi(\Delta_{j})-\Lambda -\lambda\right] \right\} \label{11-bis}$$ We finally consider a situation where $m_{1}$ pixels belong to the first detector, $m_{2}$ to the second, etc., each sitting in one detection region after the last beam splitters. We assume that the front surface of the detectors are parallel to the wave fronts, so that all phases differences $\varphi(\Delta_{j})$ collapse into 4 values only, two (in region $D_{A}$) containing the phase shift $\zeta$, two (in region $D_{B}$) containing the phase shift $\theta$:$$\begin{array} [c]{l}\varphi_{A}-\zeta\text{ for the }m_{1}\text{ first measurements}\\ \varphi_{A}-\zeta+\pi\text{ for the next }m_{2}\text{ measurements}\\ \varphi_{B}+\theta\text{ for the next }m_{3}\text{ measurements}\\ \varphi_{B}+\theta+\pi\text{ for the last }m_{4}\text{ measurements}\end{array} \label{12}$$ We note that unitarity (particle conservation) requires that the third and fourth angle are obtained by adding $\pi$ to the first and third angles). So the time derivative of the probability of obtaining a particular sequence $(m_{1},m_{2},m_{3},m_{4})$ with given pixels is (with the new variable $\lambda^{\prime}=\Lambda-\lambda$, $\zeta^{\prime}=\zeta-\varphi_{A}$, $\theta^{\prime}=\theta-\varphi_{B}$):$$\begin{array} [c]{l}\displaystyle\mathcal{P(}t\mathcal{)\sim~}{\displaystyle\prod\limits_{j=1}^{N}} p_{j}(t)\int_{-\pi}^{\pi}\frac{d\lambda^{\prime}}{2\pi}~~\int_{-\pi}^{\pi }\frac{d\Lambda}{2\pi}~~\cos\left[ (N_{\beta}-N_{\alpha})\Lambda\right] ~~\left[ \cos\Lambda+\cos\left( \zeta^{\prime}-\lambda^{\prime}\right) \right] ^{m_{1}}\left[ \cos\Lambda-\cos\left( \zeta^{\prime}-\lambda ^{\prime}\right) \right] ^{m_{2}}\\ \multicolumn{1}{r}{\times\left[ \cos\Lambda+\cos\left( \theta^{\prime }-\lambda^{\prime}\right) \right] ^{m_{3}}\left[ \cos\Lambda-\cos\left( \theta^{\prime}-\lambda^{\prime}\right) \right] ^{m_{4}}}\end{array} \label{13}$$ For short times, when the wave packets begin to reach the detectors, the probabilities $p_{j}(t)$ grow linearly in time from zero, as usual in a 3 dimensional problem. The coincidence probability $\mathcal{P(}t\mathcal{)}$ contains a product of $N$ values of $p_{j}(t)$, so that it will initially grow much more slowly, with only a $N$-th order non-zero time derivative. For longer times, when the $p_{j}(t)$’s have grown to larger values, any derivative of $\mathcal{P(}t\mathcal{)}$ may be non-zero. At the end of the experiment, when the wave packets have entirely crossed the detectors and all the particles are absorbed, the $p_{j}(t)$’s reach their limiting value $\overline{p}_{j}$, and the probability is (from now on, we drop the primes, which just introduce a redefinition of the origin of the angles) :$$\begin{array} [c]{l}\mathcal{P(}m_{1},m_{2},m_{3},m_{4})\mathcal{\sim}{\displaystyle\prod\limits_{j=1}^{N}} \overline{p}_{j}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}~~\int_{-\pi}^{\pi}\frac{d\Lambda}{2\pi}~~\cos\left[ (N_{\beta}-N_{\alpha})\Lambda\right] ~~\left[ \cos\Lambda+\cos\left( \zeta-\lambda\right) \right] ^{m_{1}}\left[ \cos\Lambda-\cos\left( \zeta-\lambda\right) \right] ^{m_{2}}\\ \multicolumn{1}{r}{\times\left[ \cos\Lambda+\cos\left( \theta-\lambda \right) \right] ^{m_{3}}\left[ \cos\Lambda-\cos\left( \theta -\lambda\right) \right] ^{m_{4}}}\end{array} \label{13-bis}$$ Counting factors and probabilities {#probab} ---------------------------------- At this point, we must take counting factors into account. There are:$$\frac{\mathcal{Q}!}{m_{1}!(\mathcal{Q}-m_{1})!} \label{14}$$ different configurations of the pixels in the first detector that lead to the same number of detections $m_{1}$. For the two detectors in $D_{A}$, this number becomes:$$\frac{\mathcal{Q}!}{m_{1}!(\mathcal{Q}-m_{1})!}\frac{\mathcal{Q}!}{m_{2}!(\mathcal{Q}-m_{2})!} \label{15}$$ But, if we note $m_{A}=m_{1}+m_{2}$ and use the Stirling formula, we can approximate:$$\begin{array} [c]{l}\log(\mathcal{Q}-m_{1})!+\log(\mathcal{Q}-m_{2})!\\ \multicolumn{1}{r}{\sim\left( \mathcal{Q}-m_{1}+\frac{1}{2}\right) \left[ \log\mathcal{Q}+\log\left( 1-\frac{m_{1}}{\mathcal{Q}}\right) \right] -\left( \mathcal{Q}-m_{1}\right) +\left( \mathcal{Q}-m_{2}+\frac{1}{2}\right) \left[ \log\mathcal{Q}+\log\left( 1-\frac{m_{2}}{\mathcal{Q}}\right) \right] -\left( \mathcal{Q}-m_{2}\right) }\end{array} \label{16}$$ or, if we expand the logarithms of $(1-m_{1,2}/\mathcal{Q})$:$$\left( 2\mathcal{Q}-m_{A}+1\right) \log\mathcal{Q}-\mathcal{Q}\frac{m_{A}}{\mathcal{Q}}+\frac{m_{1}^{2}+m_{2}^{2}}{\mathcal{Q}}-2\mathcal{Q}+m_{A}+... \label{17}$$ the second and the fifth term cancel each other, the third can be ignored because of (\[3\]); an exponentiation then provides the following term in the denominator of the counting factor:$$\frac{\left( \mathcal{Q}!\right) ^{2}}{\mathcal{Q}^{m_{A}}} \label{18}$$ The $\mathcal{Q}!$ disappear, and the number of different configurations in region $D_{A}$ is:$$\frac{1}{m_{1}!m_{2}!}Q^{m_{A}} \label{18-b}$$ Finally, we also have to take into account the factors $\overline{p}_{j}$ in (\[13-bis\]). These factors fluctuate among all the pixel configurations we have counted, since some pixels near the center of the modes are better coupled to the boson field and have larger $\overline{p}_{j}$’s than those that are on the sides. If we assume that the number of pixels of each detector is much larger than $m_{1}$ and $m_{2}$, in the summation over all possible configurations of the pixels, we can replace each $\overline{p}_{j}$ by its average $<\overline{p}>$ over the detector[^6]. If we assume that all detectors are identical, this introduces a factor $<\overline{p}>^{m_{A}}$ in the counting factor. When we take into account the other detection region $D_{B}$, the factor $Q^{m_{A}}$, together with the factor $\mathcal{Q}^{m_{B}}$, can be grouped with the prefactor $S^{N}$ in (\[13\]) to provide $\left( \mathcal{QS}\right) ^{N}$, which contains the total detection volume to the power $N$, as natural[^7]; on the other hand, the factor $<\overline{p}>^{N}$ is irrelevant, since it does not affect the relative values. At the end, we recover expression (\[19\]) for the probability of obtaining the series of results $\mathcal{(}m_{1},m_{2},m_{3},m_{4})$. This calculation shows precisely what are the experimental parameters that are important to preserve the interesting interference effects, and expresses them in geometrical terms. The main physical idea is that the detection process should not give any indication, even in principle, of the source from which the particles have originated: on the detection surface, the two sources produce indistinguishable wave functions. Therefore, in practice, what is relevant is not the coherence length of the wave functions over the entire detection regions, as the calculation of § \[quantum\] could suggest, since in these regions the modes are defined mathematically in a half-infinite space; what really matters is the parallelism of the wave fronts of the two wave functions with the input surface of the detectors. Moreover, if necessary, formulas such as (\[9\]) and (\[10-e\]) allow us to calculate the corrections introduced by wave front mismatch, and therefore to have a more realistic idea of the experimental requirements; for instance, if the $\mathbf{k}_{\alpha}$ and $\mathbf{k}_{\beta}$ are not strictly parallel and perpendicular to the surface of the photodetectors, one can write write $\mathbf{k}_{\alpha,\beta}=\mathbf{k}_{\Delta}\pm\delta\mathbf{k}(\mathbf{r})$ and calculate the correction to (\[9\]) to first order in $\delta\mathbf{k}$, etc. EPR argument and Bell theorem for parity {#EPR} ======================================== EPR variables are pairs of variables for which the result of a measurement made by Alice can be used to predict the result of a measurement made by Bob with certainty. For instance, the numbers of particles detected by Alice and by Bob are such a pair, provided we assume that the experiment has 100% efficiency (no particle is missed by the detectors): Alice knows that, if she has measured $m_{A}$ particles, Bob will detect $N-m_{A}$ particles. It is therefore possible to use the EPR argument to show that $m_{A}$ and $m_{B}$ correspond to elements of reality that were determined before any measurement took place. Moreover, this also allows us to define an ensemble of events for which $m_{A}$ and $m_{B}$ are fixed as an ensemble that is independent of the settings used by Alice and Bob; this independence is essential for the derivation of the Bell inequalities within local realism [@CS]. So we may either study situations where $m_{A}$ and $m_{B}$ are left to fluctuate freely, or where they are fixed (as with spin condensates [@PRL]). When $N_{\alpha}=N_{\beta}$, we have seen in § \[perfect\] that another pair of EPR variables is provided by the parities $\mathcal{A}=(-1)^{m_{2}}$ and $\mathcal{B}=(-1)^{m_{4}}$ of the results observed by Alice and Bob: if they choose opposite values $\zeta=-\theta$ for their settings, perfect correlations occur, even if Alice and Bob are at an arbitrarily large distance from each other. We now study quantum violations of local realism with these variables. Parity and BCHSH inequalities ----------------------------- We suppose that in the experiment of Fig. 1, Alice and Bob each use two different angle settings, $\zeta$ and $\zeta^{\prime}$ for Alice and $\theta$ and $\theta^{\prime}$ for Bob. Within local realism (EPR argument), for each realization of the experiment the observed results depend only on the local settings. We can then define $\mathcal{A}$ as the parity observed by Alice if she chooses setting $\zeta$, and $\mathcal{A}^{\prime}$ the parity if she chooses $\zeta^{\prime}$; similarly, Bob obtains results $\mathcal{B}$ or $\mathcal{B}^{\prime}$ depending on his choice $\theta$ or $\theta^{\prime}$; all these results are parities equal to $\pm1.$ Then, since either $\mathcal{B}+\mathcal{B}^{\prime}$ or $\mathcal{B}-\mathcal{B}^{\prime}$ vanishes, within local realism we have the relation: $$-2\geq\mathcal{AB+AB}^{\prime}\mathcal{+A}^{\prime}\mathcal{B-A}^{\prime }\mathcal{B}^{\prime}\leq2 \label{CHSHForm}$$ For an ensemble of events, the average of this quantity over many realizations must then also have a value between $-2$ and $+2$ (BCHSH theorem). In quantum mechanics unperformed experiments have no results [@Peres]: any given realization of the experiment necessarily corresponds to one single whole experimental arrangement, and it is never possible to define simultaneously all 4 numbers in Eq. (\[CHSHForm\]). One can nevertheless calculate the quantum average of the product of the results for given settings, and derive the expression: $$Q=\left\langle \mathcal{AB}\right\rangle +\left\langle \mathcal{AB}^{\prime }\right\rangle +\left\langle \mathcal{A}^{\prime}\mathcal{B}\right\rangle -\left\langle \mathcal{A}^{\prime}\mathcal{B}^{\prime}\right\rangle \label{Q}$$ but there is no reason to expect $Q$ to be between $-2$ and $+2$. Since (\[Proba\]) reduces to (\[19\]) (with $M=N$) when $R=0$ and $T=1\,$, we can proceed from the more general formula (\[Proba\]). The calculation of the average $\left\langle \mathcal{AB}\right\rangle $ is very similar to that of section (iv) of the Appendix, but with a factor $(-1)^{m_{2}+m_{4}}$ included in the sum on $m_{1},\cdots,m_{4}.$ The equivalent of (\[app-12\]), obtained after summations over $m_{1}$ and $m_{2}$ (with constant sum $m_{A}$) and over $m_{3}$ and $m_{4}$ (with constant sum $m_{B}$) is:$$\begin{array} [c]{l}\frac{N_{\alpha}!N_{\beta}!}{m_{A}!m_{B}!}\frac{2^{N-2M}}{\left( N-M\right) !}T^{M}R^{N-M}~\int_{-\pi}^{+\pi}\frac{d\Lambda}{2\pi}\cos\left[ \left( N_{\alpha}-N_{\beta}\right) \Lambda\right] \left[ \cos\Lambda\right] ^{N-M}\\ \multicolumn{1}{r}{\times\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}\left[ 2\cos\left( \zeta+\lambda\right) \right] ^{m_{A}}\left[ 2\cos\left( \theta-\lambda\right) \right] ^{m_{B}}}\end{array} \label{P-1}$$ Formula (\[app-8\]) of the Appendix can then be used, with $M$ replaced by $N-M$. Therefore we see that the average $\left\langle \mathcal{AB}\right\rangle $ of the product vanishes, unless the two following conditions are met:$$\left\{ \begin{array} [c]{l}M\text{ is even}\\ M\leq2N_{\alpha}\text{ and }M\leq2N_{\beta}\end{array} \right. \label{P-2}$$ It these two conditions are met, the first line of (\[P-1\]) becomes:$$\frac{N_{\alpha}!N_{\beta}!}{m_{A}!m_{B}!}~2^{-M}\frac{T^{M}R^{N-M}}{\left( N_{\alpha}-\frac{M}{2}\right) !\left( N_{\beta}-\frac{M}{2}\right) !} \label{P-3}$$ while the second line provides, with the help of formula (\[app-3\]) of the Appendix:$$2^{M}\left[ \cos\left( \frac{\zeta+\theta}{2}\right) \right] ^{M}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}\left[ \cos\left( \lambda+\frac {\zeta-\theta}{2}\right) \right] ^{M}=~M!\left[ \left( \frac{M}{2}\right) !\right] ^{-2}\left[ \cos\left( \frac{\zeta+\theta}{2}\right) \right] ^{M} \label{P-4}$$ Finally, the sum over $m_{A}$ and $m_{B}~$(with constant sum $M$) gives the result:$$\left\langle \mathcal{AB}\right\rangle =\frac{N_{\alpha}!N_{\beta}!}{\left( N_{\alpha}-\frac{M}{2}\right) !\left( N_{\beta}-\frac{M}{2}\right) !\left[ \left( \frac{M}{2}\right) !\right] ^{2}}T^{M}R^{N-M}\left[ \cos\left( \frac{\zeta+\theta}{2}\right) \right] ^{M} \label{P-5}$$ If $M$ is left to fluctuate, a summation of this expression over $M$ should be done. But another point of view is to decide to count only the events where $M$ is fixed[^8]. Then this average should be compared with the probability that $M$ particles will be detected, given by formula (\[app-14\]) in the Appendix; dividing (\[P-5\]) by (\[app-14\]) now provides:$$\left\langle \mathcal{AB}\right\rangle =\frac{N_{\alpha}!N_{\beta}!M!(N-M)!}{N!\left( N_{\alpha}-\frac{M}{2}\right) !\left( N_{\beta}-\frac{M}{2}\right) !\left[ \left( \frac{M}{2}\right) !\right] ^{2}}\left[ \cos\left( \frac{\zeta+\theta}{2}\right) \right] ^{M} \label{P-5-bis}$$ In the case $M=N,$ the second condition (\[P-2\]) requires than $N_{\alpha }=N_{\beta}=M/2$, in which case we get: $$\left\langle \mathcal{AB}\right\rangle =\cos\left( \frac{\zeta+\theta}{2}\right) ^{N}\text{ \ \ for~}~\text{~}~M=N~\text{~}~\text{~}~ \label{P-6}$$ One can put this into $Q$ of Eq. (\[Q\]): Alice’s measurement angle is taken for convenience as $\phi_{a}=2\zeta$ and Bob’s as $\phi_{b}=-2\theta.$ Then defining $E(\phi_{a}-\phi_{b})=\cos^{N}\left( \phi_{a}-\phi_{b}\right) $ and setting $\phi_{a}-\phi_{b}=\phi_{b}-\phi_{a^{\prime}}=\phi_{b^{\prime}}-\phi_{a}=\omega$ and $\phi_{b^{\prime}}-\phi_{a^{\prime}}=3\omega$ we can maximize $Q=3E(\omega)-E(3\omega)$ to find the greatest violation of the inequality for each $N$. For $N=2$ we find $Q_{\max}=2.41$ in agreement with Ref. [@YS-1]; for $N=4,$ $Q_{\max}=2.36;$ and for $N\rightarrow\infty,$ $Q_{\max}\rightarrow2.33$. These values are obtained for a value of the angles corresponding to $\omega=\sqrt{\ln3/N}$, which decreases relatively slowly with $N$. The conclusion is that the system continues to violate local realism for *arbitrarily large condensates*. As already noted in § \[par\], this is a direct consequence of the effects of the quantum angle $\Lambda$, since no such violation could occur if this angle was zero. Suppose now we measure $M=N-1$ particles with $N_{\beta}=M/2,$ $N_{\alpha }=M/2+1.$ Then the coefficient of the cosine in Eq. (\[P-5-bis\]) is $(M/2+1)/(M+1)$, which is 2/3 at $M=2$ and smaller for larger $M,$ so that this case never violates the BCHSH inequality since $2/3\times2.41=1.61<2$. If (\[P-5\]) had been used instead of (\[P-5-bis\]), we would be even even further from any violation, since the first average value is smaller than the second. The conclusion is that *even one single missed particle ruins the quantum violation*. Three Fock states and three interferometers; GHZ contradictions {#Three Fock} --------------------------------------------------------------- With a triple-Fock state source (TFS) as shown in Fig.\[fig3\] we can demonstrate GHZ contradictions [@GHZ-1; @GHZ-2]. Such a contradiction occurs when local realism predicts a quantity to be, say, $+1$ while quantum mechanics predicts the opposite, $-1.$ Previous such contradictions were carried out with states known variously as GHZ states, NOON states, or maximally entangled states. These wave functions are of the form $u\left\vert +++\cdots\right\rangle +v\left\vert ---\cdots\right\rangle $ with particular values of the phases $u$ and $v.$ The original GHZ calculations [@GHZ-1] were done with three- and four-body NOON states, and this was generalized to $N$ particles by Mermin [@Mermin]. Yurke and Stoler [@YS-2] showed how an interferometer with three one-particle sources also could give a GHZ contradiction. We will replace their sources with Bose condensates to show how new $N$-body contradictions can be developed. ![Interferometer with three Fock-state condensate sources and three detectors. The particles from each source can reach two detectors. Each detector has two subdetectors, which will register a $+1$ for the odd-numbered subdetector and $-1$ for the even-numbered. We average the quantity $\mathcal{ABC}$ where $\mathcal{A}=\pm1$ for Alice’s detector, and $\mathcal{B}=\pm1$ for Bob’s, and $\mathcal{C}=\pm1$ for Carole’s.[]{data-label="fig3"}](fig-3.eps){width="14pc"} The initial TFS is: $$\left\vert \Phi\right\rangle ~=\left\vert N_{\alpha},N_{\beta},N_{\gamma }\right\rangle =\frac{1}{\sqrt{N_{\alpha}!N_{\beta}!N_{\gamma}!}}a_{\alpha }^{\dagger N_{\alpha}}a_{\beta}^{\dagger N_{\beta}}a_{\gamma}^{\dagger N_{\gamma}}\left\vert 0\right\rangle \label{TFS}$$ As in § \[quantum\], the output modes (destruction operators) $a_{1}\cdots a_{6}$ can be written in terms of the modes at the sources $a_{\alpha},a_{\beta}$ and $a_{\gamma}$ with three phase shifts of $\zeta,$ $\chi,$ or $\theta$. We find: $$\begin{aligned} a_{1} & =\frac{1}{2}\left[ e^{i\zeta}a_{\alpha}-ia_{\beta}\right] ,~\text{~}~\text{~}~\text{~}~a_{2}=\frac{1}{2}\left[ ie^{i\zeta}a_{\alpha }-a_{\beta}\right] ,\nonumber\\ a_{3} & =\frac{1}{2}\left[ e^{i\theta}a_{\beta}-ia_{\gamma}\right] ,~\text{~}~\text{~}~\text{~}~a_{4}=\frac{1}{2}\left[ ie^{i\theta}a_{\beta }-a_{\gamma}\right] ,\nonumber\\ a_{5} & =\frac{1}{2}\left[ -a_{\alpha}+e^{i\chi}a_{\gamma}\right] ,\text{~~}~\text{~}~\text{~}a_{6}=\frac{1}{2}\left[ ia_{\alpha}+ie^{i\chi }a_{\beta}\right] .\end{aligned}$$ We write generally $a_{i}=v_{i\alpha}a_{\alpha}+v_{i\beta}a_{\beta}+v_{i\gamma}a_{\gamma}$. We consider only the case where every particle in the source is detected, so the probability that we find $m_{i}$ particles in detector $i=1\cdots6,$ is: $$\mathcal{P(}m_{1},\cdots,m_{6})\sim\frac{1}{m_{1}!\cdots m_{6}!}\left\vert \left\langle 0\right\vert a_{1}^{m_{1}}\cdots a_{6}^{m_{6}}\left\vert N_{\alpha},N_{\beta},N_{\gamma}\right\rangle \right\vert ^{2}$$ (this relation is actually an equality, but we write it only as a proportionality relation since we will change the normalization below). We can develop the matrix element just as we did in § \[PVR\]:$$\begin{array} [c]{l}\displaystyle\left\langle 0\right\vert \prod_{i=1}^{6}\left( v_{i\alpha }a_{\alpha}+v_{i\beta}a_{\beta}+v_{i\gamma}a_{\gamma}\right) ^{m_{i}}a_{\alpha}^{\dagger N_{\alpha}}a_{\beta}^{\dagger N_{\beta}}a_{\gamma }^{\dagger N_{\gamma}}\left\vert 0\right\rangle =N_{\alpha}!N_{\beta }!N_{\gamma}!~\times\\ \multicolumn{1}{r}{\displaystyle\times\sum_{p_{1=0}}^{m_{1}}...\sum_{p_{6=0}}^{m_{6}}\left[ \prod_{i=1}^{6}\left( \frac{m_{i}!}{p_{i\alpha}!p_{i\beta }!p_{i\gamma}!}u_{i\alpha}^{p_{i\alpha}}u_{i\beta}^{p_{i\beta}}u_{i\gamma }^{p_{i\gamma}}\right) ~\delta_{p_{1\alpha}+\cdots+p_{6\alpha},N_{\alpha}}~\delta_{p_{1\beta}+\cdots+p_{6\beta},N_{\beta}}~\delta_{p_{1\gamma}+\cdots+p_{6\gamma},N_{\gamma}}\right] }\end{array} \label{number}$$ where the sums are over all $p_{i\alpha}$, $p_{i\beta}$ and $p_{i\gamma}$ such that $p_{i\alpha}+p_{i\beta}+p_{i\gamma}=m_{i}$. We now replace the $\delta $-functions by integrals: $$\delta_{p_{1\alpha}+\cdots+p_{6\alpha},N_{\alpha}}=\int_{-\pi}^{\pi}\frac{d\lambda_{\alpha}}{(2\pi)^{3}}e^{i(p_{1\alpha}+\cdots+p_{6\alpha }-N_{\alpha})\lambda_{\alpha}}$$ with similar integrals over $\lambda_{\beta}$ and $\lambda_{\gamma}.$ In the sum above then, we have every $v_{i\alpha}^{p_{i\alpha}}$ replaced by $\left( v_{i\alpha}e^{i\lambda_{\alpha}}\right) ^{p_{i\alpha}}$ etc. so that we can redo the sums over the $p_{i\alpha},$ etc. to find the probability for the $m_{i}$ arrangement under the condition that all the source particles are detected: $$\mathcal{P(}m_{1},\cdots,m_{6})\sim\frac{1}{m_{1}!\cdots m_{6}!}\int d\tau^{\prime}\int d\tau e^{-i[N_{\alpha}(\lambda_{\alpha}-\lambda_{\alpha }^{\prime})+N_{\beta}(\lambda_{\beta}-\lambda_{\beta}^{\prime})+N_{\gamma }(\lambda_{\gamma}-\lambda_{\gamma}^{\prime})]}\prod_{i=1}^{6}\left( \Omega_{i}^{\prime\ast}\Omega_{i}\right) ^{m_{i}} \label{Proba-1-6}$$ where $\Omega_{i}$ $=v_{i\alpha}e^{i\lambda_{\alpha}}+v_{i\beta}e^{i\lambda_{\beta}}+v_{i\gamma}e^{i\lambda_{\gamma}}$ and $\Omega_{i}^{\prime}$ has the same expression with primed $\lambda$’s; $d\tau$ represents the integrals over $\lambda_{\alpha},$ $\lambda_{\beta},$ and $\lambda _{\gamma},$ and $d\tau^{\prime}$ over the $\lambda_{\alpha}^{\prime},$ $\lambda_{\beta}^{\prime},$ and $\lambda_{\gamma}^{\prime}.$ In a ideal experiment with 100% detection efficiency, the numbers of particles detected in each region are EPR variables, since the value of two of these variables determines the value of the third with certainty; these perfect correlations are independent of the settings (phase shifts of $\zeta,$ $\chi,$ or $\theta$), so that choosing the number of detections in each region defines a class of events that is independent of the settings. Here, assuming that each source emits $N/3$ particles (otherwise, we find zero average values, see below):$$N_{\alpha}=N_{\beta}=N_{\gamma}=N/3$$ we will also assume that each detector registers exactly $N/3$ particles[^9]. We can put in this restriction, when we sum on $m_{1}\cdots m_{6}$ to get averages, by including three $\delta$-functions of the form: $$\delta_{m_{1}+m_{2},N/3}=\int_{-\pi}^{\pi}\frac{d\rho_{A}}{2\pi}~e^{i\rho _{A}(m_{1}+m_{2}-N/3)}$$ with similar ones specifying $m_{3}+m_{4}=N/3$ and $m_{5}+m_{6}=N/3$. The $m_{i}$ sums are then done independently of one another giving a normalization sum of: $$\begin{aligned} \mathcal{N} & =\int d\tau_{\rho}\int d\tau^{\prime}\int d\tau e^{-i[N_{\alpha}(\lambda_{\alpha}-\lambda_{\alpha}^{\prime})+N_{\beta}(\lambda_{\beta}-\lambda_{\beta}^{\prime})+N_{\gamma}(\lambda_{\gamma}-\lambda_{\gamma}^{\prime})]}\nonumber\\ & \times e^{-iN/3[\rho_{A}+\rho_{B}+\rho_{C}]}\exp\left[ \sum_{i=1}^{6}\left( \Omega_{i}^{\prime\ast}\Omega_{i}e^{i\rho_{i}}\right) \right]\end{aligned}$$ where $\rho_{1}=\rho_{2}=\rho_{A},$ $\rho_{3}=\rho_{4}=\rho_{B},$ and $\rho_{5}=\rho_{6}=\rho_{C}$ and $\int d\tau_{\rho}$ represents the new three-fold integration. The sum in the exponential is easily done: $$\begin{aligned} \sum_{i=1}^{6}\left( \Omega_{i}^{\prime\ast}\Omega_{i}e^{i\rho_{i}}\right) & =\frac{1}{2}\left[ e^{-i(\lambda_{\alpha}-\lambda_{\alpha}^{\prime})}\left( e^{i\rho_{A}}+e^{i\rho_{C}}\right) \right. \nonumber\\ & \left. +e^{-i(\lambda_{\beta}-\lambda_{\beta}^{\prime})}\left( e^{i\rho_{B}}+e^{i\rho A}\right) +e^{i(\lambda_{\gamma}-\lambda_{\gamma }^{\prime})]}\left( e^{i\rho_{C}}+e^{i\rho_{B}}\right) \right]\end{aligned}$$ We expand the exponential of this quantity in series in $e^{-i(\lambda _{\alpha}-\lambda_{\alpha}^{\prime})},$ $e^{-i(\lambda_{\beta}-\lambda_{\beta }^{\prime})},$ and $e^{i(\lambda_{\gamma}-\lambda_{\gamma}^{\prime})}$ and do the integrals. When each source emits exactly $N/3$ particles, we obtain: $$\mathcal{N}=\frac{1}{2^{N}}\sum_{l=0}^{\infty}\left( \frac{1}{l!(\frac{N}{3}-l)!}\right) ^{3}$$ Similarly we average the quantities $\mathcal{A},$ $\mathcal{B},$ and $\mathcal{C}$ each $\pm1$ measured by Alice, Bob, and Carole according to: $$\left\langle \mathcal{ABC}\right\rangle =\sum_{m_{1}\cdots m_{6}}{}^{\prime }(-1)^{m_{2}+m_{4}+m_{6}}\mathcal{P(}m_{1},\cdots,m_{6})$$ where the prime on the sum means we again restrict the sums to the case of $N/3$ particles reaching each detector. With this requirement the average vanishes unless each source emits exactly $N/3$ particles, which is why above we considered just that case. We have then: $$\left\langle \mathcal{ABC}\right\rangle =\frac{\sum_{q}\left( \frac {N/3!}{(N/3-q)!q!}~\right) ^{3}e^{i(\zeta+\theta+\chi)(N/3-2q)}}{\sum _{q}\left( \frac{N/3!}{(N/3-q)!q!}~\right) ^{3}} \label{GenGHZResult}$$ Note that if $\zeta+\theta+\chi=0$ we find $\left\langle \mathcal{ABC}\right\rangle =1$: perfect correlations exist between the results since their product is fixed. Thus, if we know the parity of the results of two of the experimenters, we immediately know that of the third, even if that person is very far away. Thus an EPR argument applies to these variables. For the case $N=3,$ we find $\left\langle \mathcal{ABC}\right\rangle =\cos(\zeta+\theta+\chi),$ which is the same form of the original GHZ case found from a three-body NOON state. This result agrees with the interferometer result of Ref. [@YS-2] as expected. Local realism predicts that, for each realization of the experiment, the product of the results is given by a product $A(\zeta)B(\theta)C(\chi)$. To get agreement with quantum mechanics in situations of perfect correlations we must have: $$\begin{aligned} A(\pi/2)~B(\pi/2)~C(0) & =-1\nonumber\\ A(\pi/2)~B(0)~C(\pi/2) & =-1\nonumber\\ A(0)~B(\pi/2)~C(\pi/2) & =-1\end{aligned}$$ But then we obtain by product $A(0)B(0)C(0)=-1,$ while quantum mechanics gives $+1$, in complete contradiction. In our case we get new contradictions for larger $N$; consider for instance $N=9,$ in which case: $$\left\langle \mathcal{ABC}\right\rangle =\frac{1}{28}\left[ 27\cos (\zeta+\theta+\chi)+\cos3(\zeta+\theta+\chi)\right]$$ The above argument goes through exactly in the same way. More generally, any time $N/3$ is odd we get a similar result for arbitrary $N.$ Thus the TFS provides new GHZ-type contradictions for $N$ particles *without* having to prepare NOON states. Hardy impossibilities --------------------- Hardy impossibilities are treated by use of the interferometer shown in Fig. \[fig4\], based on the one discussed in Ref. [@H-1] for $N=2$. ![An interferometer with particle sources $\alpha$ and $\beta$, with beam splitters designated by BS and mirrors by M. In both detection regions, the detectors at D$_{i}$ may be replaced by the D$_{i}^{\prime}$, placed before the beam splitters$.$ []{data-label="fig4"}](fig-4.eps){width="2.5in"} The heart of the system is the beam splitter at the center; due to Bose interference it has the property that, if an equal number of particles approaches each side, then an even number must emerge from each side. The detection beam splitters BSA and BSB are each set to have a transmission probability of 1/3, and the path differences are such that, by destructive interference, no particle reaches $D_{2}$ if only source $N_{\alpha}$ is used; similarly, no particle reaches $D_{3}$ if $N_{\beta}$ alone is used. Alice can use either the detectors $D_{1,2}$ after her beam splitter, or $D_{1,2}^{\prime}$ before; Bob can choose either D$_{3,4}$, or D$_{3,4}^{\prime}$. This gives $4$ arrangements of experiments: $DD$, $DD^{\prime}$, $D^{\prime}D$, or $D^{\prime}D^{\prime}$, with probability amplitudes $C_{XY}(m_{1},m_{2};m_{3},m_{4})$, where $XY$ is any of these $4$ arrangements and the $m$ values are the numbers of particles detected at each counter. We find the destruction operators for the detector modes as we have done in previous sections. For the primed detectors we find:$$\begin{array} [c]{l}a_{D_{1}^{\prime}}=\frac{i}{\sqrt{2}}a_{\alpha},~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}a_{D_{2}^{\prime}}=\frac{1}{2}\left( -a_{\alpha}+ia_{\beta}\right) \\ a_{D_{3}^{\prime}}=\frac{1}{2}\left( ia_{\alpha}-a_{\beta}\right) ,~\text{~}~a_{D_{4}^{\prime}}=\frac{i}{\sqrt{2}}a_{\beta}\end{array} \label{eqn}$$ and for the unprimed detectors:$$\begin{array} [c]{l}a_{D_{1}}=-\frac{\sqrt{3}}{2}a_{\alpha}+\frac{i}{2\sqrt{3}}a_{\beta},~\text{~}~\text{~}~\text{~}~a_{D_{2}}=-\frac{1}{\sqrt{6}}a_{\beta}\\ a_{D_{3}}=-\frac{1}{\sqrt{6}}a_{\alpha},~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~\text{~}~a_{D_{4}}=\frac{i}{2\sqrt{3}}a_{\alpha}-\frac{\sqrt{3}}{2}a_{\beta}\end{array} \label{equat}$$ In general we write these results as: $$a_{i}=v_{i\alpha}a_{\alpha}+v_{i\beta}a_{\beta}$$ Note that, because of the $1/3$ transmission probability at BSA and BSB, Bose interference causes $a_{\alpha}$ and $a_{\beta}$ to drop out of the second and third of Eqs (\[equat\]), respectively. The amplitude is given by: $$C_{\mathrm{XY}}(m_{1},m_{2};m_{3},m_{4})\sim\left\langle 0\right\vert \prod_{i=1}^{6}\left( v_{i\alpha}a_{\alpha}+v_{i\beta}a_{\beta}\right) ^{m_{i}}a_{\alpha}^{\dagger N_{\alpha}}a_{\beta}^{\dagger N_{\beta}}\left\vert 0\right\rangle$$ As we have done in previous sections, we expand the binomials, evaluate the operator matrix element, replace the resulting $\delta$-functions by integrals and resum the series to find: $$C_{\mathrm{XY}}(m_{1},m_{2};m_{3},m_{4})\sim\int_{-\pi}^{\pi}\frac {d\lambda_{\alpha}}{2\pi}\int_{-\pi}^{\pi}\frac{d\lambda_{\beta}}{2\pi }e^{^{-iN_{\alpha}\lambda_{\alpha}}}e^{^{-iN_{\beta}\lambda_{\beta}}}\prod_{i=1}^{4}\left( v_{i\alpha}e^{i\lambda_{\alpha}}+v_{i\beta}e^{i\lambda_{\beta}}\right) ^{m_{i}}$$ In all the following we assume that each source emits $N/2$ particles, where $N/2$ is *odd*, and that detector A and detector B each receive exactly $N/2$ particles; this is possible since, as above, the number of particles detected in each region can define a sample of realizations that is independent of the settings (we have to make this assumption since it turns out that the argument works only in this case). First consider both Alice and Bob using primed detectors. The amplitude for receiving $N/2$ particles in each of D$_{2}^{\prime}$ and D$_{3}^{\prime}$ is: $$\begin{aligned} C_{\mathrm{D}^{\prime}\mathrm{D}^{\prime}}(0,\frac{N}{2};0,\frac{N}{2}) & \sim\int_{-\pi}^{\pi}\frac{d\lambda_{\alpha}}{2\pi}\int_{-\pi}^{\pi}\frac{d\lambda_{\beta}}{2\pi}e^{^{-i\frac{N}{2}(\lambda_{\alpha}+\lambda_{\beta})}}\left( -e^{i\lambda_{\alpha}}+ie^{i\lambda_{\beta}}\right) ^{N/2}\left( ie^{i\lambda_{\alpha}}-e^{i\lambda_{\beta}}\right) ^{N/2}\nonumber\\ & \sim\int_{-\pi}^{\pi}\frac{d\lambda_{\alpha}}{2\pi}\int_{-\pi}^{\pi}\frac{d\lambda_{\beta}}{2\pi}e^{^{-i\frac{N}{2}(\lambda_{\alpha}+\lambda_{\beta})}}\left( e^{i2\lambda_{\alpha}}+e^{i2\lambda_{\beta}}\right) ^{N/2}=0\end{aligned}$$ This quantity must vanish because $N/2$ is odd. This situation is an example of the beam splitter rule mentioned above. The result is that D$_{2}^{\prime}$ and D$_{3}^{\prime}$ cannot collect all the particles if $N/2$ are detected on each side. Consider next the case where one experimenter uses a primed set of detectors and the other the unprimed: $$\begin{aligned} C_{\mathrm{DD}^{\prime}}(0,\frac{N}{2};m_{3}^{\prime},m_{4}^{\prime}) & \sim\int_{-\pi}^{\pi}\frac{d\lambda_{\alpha}}{2\pi}\int_{-\pi}^{\pi}\frac{d\lambda_{\beta}}{2\pi}e^{^{-i\frac{N}{2}(\lambda_{\alpha}+\lambda_{\beta})}}\left( e^{i\lambda_{\beta}}\right) ^{N/2}\left( ie^{i\lambda_{\alpha}}-e^{i\lambda_{\beta}}\right) ^{m_{3}^{\prime}}\left( e^{i\lambda_{\beta}}\right) ^{m_{4}^{\prime}}\nonumber\\ & \sim\delta_{m_{4}^{\prime},0} \label{DD'A}$$ The factor $\left( e^{i\lambda_{\beta}}\right) ^{N/2}$ combines with the first exponential so that $m_{4}^{\prime}$ must vanish. This quantity vanishes because of the destructive interference effects at BSA and BSB caused by the 1/3 transmission probability of the beam splitters; but $C_{\mathrm{DD}^{\prime}}(0,\frac{N}{2};\frac{N}{2},0)\neq0$. Thus, if Alice observes $N/2$ particles at D$_{2}$, when Bob uses the primed detectors he observes with certainty $N/2$ particles at D$_{3}^{\prime}$; similarly, if Bob has seen $N/2$ particles in D$_{3}$, in the D$^{\prime}$D configuration Alice must see $N/2$ in D$_{2}^{\prime}$ . We now consider events where both experimenters do unprimed experiments and each of them finds $N/2$ particles in D$_{2}$ and D$_{3}$; the corresponding probability is:$$C_{\mathrm{DD}}(0,\frac{N}{2};\frac{N}{2},0)\sim\int_{-\pi}^{\pi}\frac{d\lambda_{\alpha}}{2\pi}\int_{-\pi}^{\pi}\frac{d\lambda_{\beta}}{2\pi }e^{^{-i\frac{N}{2}(\lambda_{\alpha}+\lambda_{\beta})}}\left( e^{i\lambda _{\beta}}\right) ^{N/2}\left( e^{i\lambda_{\alpha}}\right) ^{N/2}\neq0,$$ (for $N=6$, the normalized value is $1/216$), which means that events exist where $N/2$ particles are detected at both detectors D$_{2}$ and D$_{3}$. However, in any of these events, if Bob had at the last instant changed to the primed detectors, he would surely have obtained $N/2$ particles in D$_{3}^{\prime}$, because of the certainty mentioned above (while Alice still has $N/2$ particles in D$_{2}$). Similarly, if it is Alice who chooses primed detectors at the last moment, she always obtains $N/2$ particles in D$_{2}^{\prime}$ (while Bob continues to have $N/2$ particles in D$_{3}$). Now, had both changed their minds after the emission and chosen the primed arrangement, local realism implies that they would have found $N/2$ particles each in D$_{2}^{\prime}$ and D$_{3}^{\prime}$: such events must exist. But the corresponding quantum probability is zero, in complete contradiction. The result is the Hardy impossibility of Ref. [@H-1] generalized to $N$ particles. Conclusion ========== Fock-state condensates appear as remarkably versatile, able to create violations that usually require elaborate entangled wave functions, and produce new $N$-body violations. Compared to GHZ states or other elaborate quantum states, they have the advantage of being accessible through the phenomenon of Bose-Einstein condensation, with no limitation in principle concerning the number of particles involved. By contrast, the production of GHZ states requires elaborate measurement procedures, so that it seems difficult to produce them with more than a few particles (to our knowledge, the present world record is 5, see [@GHZ-5]); moreover, they are much more sensitive to decoherence, which destroys their quantum coherence properties [@DBB]. From an experimental point of view, the major requirement is that all particles present in the initial double Fock state should be detected, which will of course put a practical limit on the number of particles involved. Using Bose condensed gases of metastable He atoms seems to be an attractive possibility, since the detection of individual atoms is possible with micro-channel plates [@Saubamea; @Robert]. With alkali atoms, one could also measure the position of the particles at the outputs of interferometers by laser fluorescence, obtaining a non-destructive quantum measurement of $m_{1}$, ..$m_{4}$. The realization of interferometers seems also possible, since interferometry with Bose-Einstein condensates has already been performed [@BS-1] with the help of Bragg scattering optical beam splitters [@BS-2; @BS-3]. Another possibility may be to use cavity quantum electrodynamics and quantum non-demolition photon counting methods [@Guerlin] to prepare multiple Fock states. Experiments therefore do not seem to be out of reach. Laboratoire Kastler Brossel is UMR 8552 du CNRS, de l’ENS, et de l’Université Pierre et Marie Curie. APPENDIX I In this appendix, we give some formulas that are useful for the calculations of this article, in particular to check the normalization of (\[19\]) and (\[Proba\]). \(i) Wallis integral. We consider the integral:$$K=\int_{-\pi}^{+\pi}\frac{d\lambda}{2\pi}~\left[ \cos\lambda\right] ^{N} \label{app-1}$$ which, if the limits are changed to $0$ and $\pi/2$, becomes a Wallis integral (divided by $2\pi$ if one takes the usual definition of these integrals). Expanding the integrand provides:$$2^{-N}\left[ e^{i\lambda}+e^{-i\lambda}\right] ^{N}=2^{-N}\sum_{q=0}^{N}\frac{N!}{q!(N-q)!}~e^{i\left( N-2q\right) \lambda} \label{app-2}$$ The only exponentials that survive the $\lambda$ integration are those with vanishing exponent ($N-2q=0$). Therefore:$$\begin{array} [c]{l}\text{if }N\text{ is odd, }K=0\\ \text{if }N\text{ is even, }K=2^{-N}\frac{N!}{\left[ (N/2)!\right] ^{2}}\end{array} \label{app-3}$$ \(ii) Normalization integral. We define:$$J=\int_{-\pi}^{+\pi}\frac{d\Lambda}{2\pi}\cos\left[ \left( N_{\alpha }-N_{\beta}\right) \Lambda\right] \left[ \cos\Lambda\right] ^{M}=\operatorname{Re}\left\{ \int_{-\pi}^{+\pi}\frac{d\Lambda}{2\pi}e^{i\left( N_{\alpha}-N_{\beta}\right) \Lambda}\left[ \cos\Lambda\right] ^{M}\right\} \label{app-4}$$ and expand:$$\left[ \cos\Lambda\right] ^{M}=\left[ \frac{e^{i\Lambda}+e^{-i\Lambda}}{2}\right] ^{M}=2^{-M}\sum_{q=0}^{M}\frac{M!}{q!(M-q)!}~e^{i\left( M-2q\right) \lambda} \label{app-5}$$ Only a term with $M-2q=N_{\alpha}-N_{\beta}$ can survive the integration, so that:$$q=\frac{N_{\beta}-N_{\alpha}-M}{2} \label{app-6}$$ Therefore, $J$ is non-zero only if:$$M\text{ has the same parity as }N_{\alpha}-N_{\beta}\text{ \ ; \ }-M\leq N_{\alpha}-N_{\beta}\leq+M \label{app-7}$$ and then:$$J=2^{-M}\frac{M!}{\left( \frac{M+N_{\alpha}-N_{\beta}}{2}\right) !\left( \frac{M-N_{\alpha}+N_{\beta}}{2}\right) !} \label{app-8}$$ \(iii) Normalization of (\[19\]). We now consider the probabilities $\mathcal{P(}m_{1},m_{2},m_{3},m_{4})$ given by (\[19\]) and calculate their sum over $m_{1},m_{2},m_{3},m_{4}$, when these variables have a constant sum $N$. We do these sums in three steps: a summation over $m_{1}$ and $m_{2}$ (with constant sum $m_{A}$), a summation over $m_{3}$ and $m_{4}$ (with constant sum $m_{A}$), and a summation over $m_{A}$ and $m_{B}$ (with constant sum $M$). The first two sums reconstruct powers of a binomial, the $\lambda$ integral disappears, and we obtain:$$N_{\alpha}!N_{\beta}!~2^{-N}\int_{-\pi}^{+\pi}\frac{d\Lambda}{2\pi}\cos\left[ \left( N_{\alpha}-N_{\beta}\right) \Lambda\right] \times\frac{1}{m_{A}!m_{B}!}\left[ 2\cos\Lambda\right] ^{N} \label{app-9}$$ which, with (\[app-8\]) for $M=N=N_{\alpha}+N_{\beta}$, gives:$$2^{-N}\text{~}\times\frac{M!}{m_{A}!m_{B}!} \label{app-10}$$ Then the summation over $m_{A}$ and $m_{B}$ gives:$$2^{-N}\left( 1+1\right) ^{N}=1 \label{app-11}$$ as expected. \(iv) Normalization of (\[Proba\]). We now consider the probabilities $\mathcal{P(}m_{1},m_{2},m_{3},m_{4})$ given by (\[Proba\]) and calculate their sum over any $m_{1},m_{2},m_{3},m_{4}$. We do this by the same three summation as above (with $m_{A}+m_{B}=M$, instead of $N$), plus a summation over $M$ ranging from $0$ to $N$. The first two summations give:$$\frac{N_{\alpha}!N_{\beta}!}{m_{A}!m_{B}!}\frac{2^{N-2M}}{\left( N-M\right) !}T^{M}R^{N-M}\int_{-\pi}^{+\pi}\frac{d\Lambda}{2\pi}\cos\left[ \left( N_{\alpha}-N_{\beta}\right) \Lambda\right] \left[ \cos\Lambda\right] ^{N-M}\left[ 2\cos\Lambda\right] ^{M} \label{app-12}$$ or, when (\[app-8\]) is inserted:$$\frac{N!}{m_{A}!m_{B}!}\frac{2^{-M}}{\left( N-M\right) !}T^{M}R^{N-M} \label{app-13}$$ The summation over $m_{A}$ and $m_{B}$ with constant sum $M$ then gives:$$\frac{N!}{M!\left( N-M\right) !}T^{M}R^{N-M} \label{app-14}$$ which provides the probability of detecting $M$ particles, independently of which of the 4 detectors is activated. This probability is maximal when:$$\frac{N-M}{M}\frac{T}{R}\sim1~\ \ \ \ \ \ \ \text{\ or \ \ \ \ \ \ \ }\frac {M}{N}\sim T \label{app-14-bis}$$ The larger the transmission coefficient $T$, the larger the most likely value of $M$, as one could expect physically. Finally, a summation of (\[app-14\]) over $M$ between $0$ and $N$ gives:$$\left( R+T\right) ^{N}=1 \label{app-15}$$ and the total probability is $1$, as expected. APPENDIX II In this appendix, we investigate how Eq. (\[19\]) is changed when the initial state $\left\vert \Phi_{0}\right\rangle $ is different from the double Fock state considered in (\[1\]). \(a) Coherent states We first assume that each of the modes $\alpha$, $\beta$ is in a coherent state with phases $\phi_{\alpha}$, $\phi_{\beta}$ and the same amplitude $E$:$$\left\vert \Phi_{0}\right\rangle =\left\vert \phi_{\alpha}\right\rangle \otimes\left\vert \phi_{\beta}\right\rangle \label{aa-1}$$ with the usual expression of the coherent states:$$\left\vert \phi_{\alpha,\beta}\right\rangle \sim\sum_{r=0}^{\infty}\frac{\left[ E~e^{i\phi_{\alpha,\beta}}\right] ^{r}}{\sqrt{r!}}\left\vert N_{\alpha,\beta}=r\right\rangle \label{aa-2}$$ The calculation of § \[PVR\] is then simplified since this state is a common eigenvector of both annihilation operators $a_{\alpha}$ and $a_{\beta}$. There is no need to introduce conservation rules, and neither $\lambda$ nor $\Lambda$ enter the expressions. Eq. (\[19\]) becomes:$$\begin{array} [c]{l}\mathcal{P(}m_{1},m_{2},m_{3},m_{4})\sim\left[ 1+\cos\left( \zeta +\phi_{\alpha}-\phi_{\beta}\right) \right] ^{m_{1}}\left[ 1-\cos\left( \zeta+\phi_{\alpha}-\phi_{\beta}\right) \right] ^{m_{2}}\\ \multicolumn{1}{r}{\times\left[ 1+\cos\left( \theta+\phi_{\beta}-\phi_{\alpha}\right) \right] ^{m_{3}}\left[ 1-\cos\left( -\theta +\phi_{\beta}-\phi_{\alpha}\right) \right] ^{m_{4}}}\end{array} \label{aa-3}$$ Here, no $\lambda$ distribution occurs, in contrast with (\[19\]): the relative phase of the two states is perfectly defined and takes the exact value $\phi_{\alpha}-\phi_{\beta}$. Now, we can also assume that the initial phases of the two coherent states completely random. Then, an average over all possible values of $\phi_{\alpha}$and $\phi_{\beta}$ leads to:$$\begin{array} [c]{l}\mathcal{P(}m_{1},m_{2},m_{3},m_{4})\sim\int\frac{d\phi}{2\pi}~\left[ 1+\cos\left( \zeta+\phi\right) \right] ^{m_{1}}\left[ 1-\cos\left( \zeta+\phi\right) \right] ^{m_{2}}\\ \multicolumn{1}{r}{\left[ 1+\cos\left( -\theta+\phi\right) \right] ^{m_{3}}\left[ 1-\cos\left( -\theta+\phi\right) \right] ^{m_{4}}}\end{array} \label{aa-4}$$ We now obtain an expression that is similar to Eq. (\[19\]), but with a difference: the terms of the product in the integral are always positive, as if the quantum angle $\Lambda$ had been set equal to zero; no violation of Bell inequalities is therefore possible. This was expected: with coherent states, the phase pre-exists the measurement and is not created under the effect of quantum measurement, as was the case with Fock states; an unknown classical variable does not lead to violations local realism. Moreover, the requirement of measuring all particles does not apply in this case, since the initial state does not have an upper bound for the populations. \(b) Phase state We now assume choose a state that has a fixed number of particles, but a well defined phase $\phi_{0}$ between the two modes:$$\left\vert \Phi_{0},N\right\rangle =\frac{1}{\sqrt{N!}}\left[ e^{i\phi_{0}}a_{\alpha}^{\dagger}+a_{\beta}^{\dagger}\right] ^{N}\left\vert 0\right\rangle =\sqrt{N!}\sum_{q=0}^{N}~\frac{e^{iq\phi_{0}}}{q!\left( N-q\right) !}~\left( a_{\alpha}^{\dagger}\right) ^{q}\left( a_{\beta }^{\dagger}\right) ^{N-q}\left\vert 0\right\rangle \label{aa-5}$$ with $N$ even. We assume that all particles are measured: $\sum_{i}m_{i}=N$. The probability amplitude we wish to calculate is: $$C_{m_{1}\cdots m_{4}}=\frac{1}{\sqrt{\prod_{j}m_{j}!}}\left\langle 0\right\vert \prod_{j=1}^{4}a_{j}^{m_{j}}\left\vert \Phi_{0},N\right\rangle \label{aaa-1}$$ where the $a_{i}$ are defined in (\[awm\]) and written more generically in (\[ai\]). We will use two different methods to do the calculation, first a method based on the specific properties of phase states, and then a more generic method extending the results of § \[PVR\]. \(i) The phase state $\left\vert \Phi_{0},N\right\rangle $ is by definition a state where all bosons are created in one single state $\left[ e^{i\phi_{0}}\left\vert \alpha\right\rangle +\left\vert \beta\right\rangle \right] /\sqrt{2}$, none in the orthogonal state $\left[ -e^{i\phi_{0}}\left\vert \alpha\right\rangle +\left\vert \beta\right\rangle \right] /\sqrt{2}$. Therefore the action of the two annihilation operators:$$\begin{array} [c]{l}a_{\phi_{0}}=\frac{e^{-i\phi_{0}}a_{\alpha}+a_{\beta}}{\sqrt{2}}\\ a_{\phi_{0+\pi}}=\frac{-e^{-i\phi_{0}}a_{\alpha}+a_{\beta}}{\sqrt{2}}\end{array} \label{aaa-2}$$ is straightforward: the former transforms $\left\vert \Phi_{0},N\right\rangle $ into $\left\vert \Phi_{0},N-1\right\rangle $, the latter gives zero. Now, we can use (\[ai\]) and (\[aaa-2\]) to expand each $a_{i}$ as:$$a_{j}=v_{j\alpha}e^{i\phi_{0}}\frac{a_{\phi_{0}}-a_{\phi_{0+\pi}}}{\sqrt{2}}+v_{j\beta}\frac{a_{\phi_{0}}+a_{\phi_{0+\pi}}}{\sqrt{2}} \label{aaa-3}$$ where the action of $a_{\phi_{0+\pi}}$ gives zero. We conclude that:$$a_{j}\left\vert \Phi_{0},N\right\rangle =\sqrt{\frac{N}{2}}(e^{i\phi_{0}}v_{j\alpha}+v_{j\beta})\left\vert \Phi_{0},N-1\right\rangle \label{aaa-4}$$ in which case:$$C_{m_{1}\cdots m_{4}}=\sqrt{\frac{N!}{2^{N}\prod_{j}m_{j}!}}\prod_{j=1}^{4}(e^{i\phi_{0}}v_{j\alpha}+v_{j\beta})^{m_{j}} \label{PSProb}$$ The probability is then:$$\begin{array} [c]{l}\displaystyle\mathcal{P}(m_{1},m_{2},m_{3},m_{4})=\frac{N!}{\prod_{j}2^{N}m_{j}!}\prod_{j=1}^{4}\left[ (e^{-i\phi_{0}}v_{j\alpha}^{\ast}+v_{j\beta}^{\ast})(e^{i\phi_{0}}v_{j\alpha}+v_{j\beta})\right] ^{m_{j}}\\ \multicolumn{1}{c}{\displaystyle=\frac{N!}{4^{N}~m_{1}!...m_{4}!}\left[ (1+\cos(\zeta+\phi_{0})\right] ^{m_{1}}\left[ (1-\cos(\zeta+\phi _{0})\right] ^{m_{2}}}\\ \multicolumn{1}{r}{\displaystyle\times\left[ (1+\cos(\theta-\phi_{0})\right] ^{m_{3}}\left[ (1-\cos(\theta-\phi_{0})\right] ^{m_{4}}}\end{array} \label{aaa-5}$$ As in case (a), we have a state for which the quantum angle vanishes so that no violation of the BCHSH inequalities can take place. \(ii) We can also do the calculation by a method that is similar to that of § \[PVR\]. From (\[aa-5\]) and (\[aaa-1\]), we obtain:$$C_{m_{1}\cdots m_{4}}=\frac{\sqrt{N!}}{\sqrt{\prod_{j}m_{j}!}}\sum_{q=0}^{N}~\frac{e^{iq\phi_{0}}}{q!\left( N-q\right) !}~\prod_{j=1}^{4}~\left\langle 0\right\vert a_{j}^{m_{j}}\left( a_{\alpha}^{\dagger}\right) ^{q}\left( a_{\beta}^{\dagger}\right) ^{N-q}\left\vert 0\right\rangle$$ The calculation is the same as in § \[PVR\], with $N_{\alpha}$ replaced by $q$ and $N_{\beta}$ by $N-q$; the prefactors of (\[1\]) and (\[aa-5\]) combine to introduce a factor $\sqrt{N!/q!\left( N-q\right) !}$, and the equivalent of (\[ampl\]) is now:$$\sqrt{N!}\sum_{q=0}^{N}e^{iq\phi_{0}}\int_{-\pi}^{\pi}\frac{d\mu}{2\pi }e^{^{i\left( N-2q\right) \mu}}\prod_{j=1}^{4}\left( v_{j\alpha}e^{i\mu }+v_{j\beta}e^{-i\mu}\right) ^{m_{j}} \label{aa-ampl}$$ In the probability, a sum over $q$ and $q^{\prime}$ appears, including term $q\neq q^{\prime}$ corresponding to non-diagonal probability terms between two different states $\left\langle N_{\alpha}=q^{\prime}~;N_{\beta}=N-q^{\prime }\right\vert $ and $\left\vert N_{\alpha}=q~;N_{\beta}=N-q\right\rangle $. One finally obtains:$$\begin{array} [c]{l}\displaystyle\mathcal{P(}m_{1},m_{2},m_{3},m_{4})=\frac{N!}{m_{1}!m_{2}!m_{3}!m_{4}!}2^{-N}\int_{-\pi}^{\pi}\frac{d\lambda}{2\pi}\int_{-\pi}^{\pi }\frac{d\Lambda}{2\pi}~G\left( \lambda,\Lambda\right) ~\left[ \cos \Lambda+\cos\left( \zeta+\lambda\right) \right] ^{m_{1}}\\ \displaystyle\times\left[ \cos\Lambda-\cos\left( \zeta+\lambda\right) \right] ^{m_{2}}\left[ \cos\Lambda+\cos\left( \theta-\lambda\right) \right] ^{m_{3}}\left[ \cos\Lambda-\cos\left( \theta-\lambda\right) \right] ^{m_{4}}\end{array} \label{aa-6}$$ with:$$G\left( \lambda,\Lambda\right) =\sum_{q,q^{\prime}=0}^{N}e^{i\left( q-q^{\prime}\right) \left( \phi_{0}-\lambda\right) }~e^{i\left( N-q-q^{\prime}\right) \Lambda} \label{aa-7}$$ The result is therefore similar to (\[19\]), except for the presence of the function $G\left( \lambda,\Lambda\right) $, which introduces a distribution of the phase $\lambda$ and of the quantum angle $\Lambda$. Equations (\[aa-6\]) and (\[aa-7\]) are equivalent to (\[aaa-5\]), although they do not contain the same distribution $G\left( \lambda ,\Lambda\right) $. Equation (\[aaa-5\]) corresponds to an infinitely narrow distribution, since it can be obtained by replacing in (\[aa-6\]) $G\left( \lambda,\Lambda\right) $ by the product $\delta(\lambda-\phi_{0})$ $\delta(\Lambda)$; by contrast, (\[aa-7\]) defines a distribution with finite width. This illustrates the fact that, in (\[aa-6\]), different distributions $G\left( \lambda,\Lambda\right) $ may lead to the same set of probabilities. \(c) General state Consider finally the more general state $\left\vert \Phi_{0}\right\rangle $ combining two modes with a fixed total number of particles can be written as:$$\left\vert \Phi_{0}\right\rangle =\sum_{q=0}^{N}x_{q}~\left\vert N_{\alpha }=q~;~N_{\beta}=N-q\right\rangle \label{aa-8}$$ where the complex coefficients $x_{q}$ are arbitrary. The calculation is similar to that of §(ii) above, but now one obtains (\[aa-6\]) with a different expression of $G(\lambda,\Lambda)$:$$G(\lambda,\Lambda)=\sum_{q,q^{\prime}}x_{q}x_{q^{\prime}}^{\ast}~e^{i(N-q-q^{\prime})}e^{-i\lambda(q-q^{\prime})}\sqrt{q^{\prime}!(N-q^{\prime})!q!(N-q)!} \label{aa-9}$$ Depending on the choice of the coefficients $x_{q}$, one can build states in which the initial phase is well determined, as in § (ii), or completely indetermined as for Fock states; a similar conclusion holds for the quantum angle $\Lambda$. 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[^2]: Usually, perfect correlations are obtained when the two settings are the same, not opposite. But, with the geometry shown in figure \[fig1\], $\zeta$ introduces a phase delay of source $\alpha$ with respect to source $\beta$, while $\theta$ does the opposite by delaying source $\beta$ with respect to source $\alpha$. Therefore, the dephasing effects of the two delays are the same in both regions $D_{A}$ and $D_{B}$ when $\theta=-\zeta$. [^3]: Using the periodicity of the integrand, one can give to both integration variables $\lambda^{\prime}$ and $\lambda^{\prime\prime}$ a range $\left[ -\pi ,+\pi\right] $; this doubles the integration domain, but this doubling is cancelled by a factor $1/2$ introduced by the Jacobian. [^4]: We add the probabilities of non-exclusive events, whith provides an upper bound of the real probability of double detection. [^5]: Equation (\[a-1-s\]) shows that, in the definition of surface in $3N$ dimension space, the $2$ dimension front surface of any pixel is associated with all three dimensions of any other pixel, including its depth. These dimensions play the role of transverse dimensions over which an integration has to be performed to obtain the flux (similarly, in $3$ dimensions, the flux through a surface perpendicular to $Oz$ contains an integration over the transverse directions $Ox$ and $Oy$). [^6]: If $m_{1}=1$, the summation provides exactly $<\overline{p}>$ the average by definition. If $m_{1}=2$, the second pixel can not coincide with the first, so that the average of the product $p_{1}p_{2}$ is not exactly $<\overline{p}>^{2}$; nevertheless, if the number of pixels $\mathcal{Q}$ is much larger than $2$, the average is indeed $<\overline{p}>^{2}$ to a very good approximation. By recurrence, as long as the number of detections $m$ remains much smaller than the number of pixels, one can safely replace the average of the product by the product of averages. [^7]: The probability remains invariant if, at constant detection area, the value of the number of pixels $\mathcal{Q}$ is increased. [^8]: With the experimental setup of fig. \[fig2\], one can include the results of measurements given by the detectors in channels 5 and 6 in the preparation procedure; only the events in which $m_{5}+m_{6}=N-M$ are retained in the sample considered. Since this procedure remains independent of the settings $\theta$ and $\zeta$, this does not open a sample bias loophole. [^9]: We have also performed more general calculations—not given here—in which this restriction does not apply, but then we have found no GHZ contradictions.

ArXiv

--- abstract: 'We examine Dirac’s early algebraic approach which introduces the [*standard*]{} ket and show that it emerges more clearly from a unitary transformation of the operators based on the action. This establishes a new picture that is unitarily equivalent to both the Schrödinger and Heisenberg pictures. We will call this the Dirac-Bohm picture for the reasons we discuss in the paper. This picture forms the basis of the Feynman path theory and allows us to show that the so-called ‘Bohm trajectories’ are averages of an ensemble of Feynman paths.' author: - 'B. J. Hiley[^1] and G. Dennis.' bibliography: - 'myfile.bib' date: | TPRU, Birkbeck, University of London, Malet Street,\ London WC1E 7HX.\ Physics Department, University College London, Gower Street, London WC1E 6BT. title: 'The Dirac-Bohm Picture' --- Representations and Pictures ============================ The Stone-von Neumann theorem [@jn31; @jn32; @ms30] proves that the Schrödinger representation is unique up to a unitary transformation. This means that there could be many equivalent representations or ‘pictures’ as they are often called in this context. For example the Schrödinger picture, where all operators are independent of time, while the wave function or ket carries the time dependence, is well known. In this picture the Hamiltonian is written in the form $$\begin{aligned} H_S=H(\hat q,\hat p)\hspace{0.5cm} \mbox{with the wave function }\hspace{0.1cm} \psi(q,t).\end{aligned}$$ Here the momentum operator is written in the form $\hat p=-i\hbar\partial/\partial q$. When considering quantum field theory, it is the Heisenberg picture that comes to the fore. In this picture all the time dependence is taken into the operators by introducing a unitary transform $U(t)$ so that $$\begin{aligned} H_H(t)=U^\dag H_SU\hspace{0.3cm}\mbox{with}\hspace{0.3cm} U(t)=e^{iHt/\hbar}\end{aligned}$$ where $H(q,p)$ is the Hamiltonian. Then we have the following relations $$\begin{aligned} \hat q_H(t)=U^\dag \hat q_SU\hspace{0.3cm} \mbox{and}\hspace{0.3cm} \hat p_H(t)= U^\dag \hat p_S U.\end{aligned}$$ Apart from the interaction picture and the Fock picture, a little-known picture was implicitly introduced by Dirac [@pd47] in which $$\begin{aligned} H_D(\hat q_D, \hat p_D)=V^\dag H_S(\hat q_S, \hat p_S)V\hspace{0.3cm}\hspace{0.3cm}\mbox{with}\hspace{0.3cm}V(t)=e^{iS(q,t)/\hbar} \label{eq:SUTrans}\end{aligned}$$ where $S(q,t)$ is the classical action. In this case we can then write $$\begin{aligned} \hat q_D=V^\dag \hat q_S V \hspace{0.5cm}\Rightarrow\hspace{0.5cm} \hat q_D=\hat q_S,\hspace{1.5cm}\\ \hat p_D=V^\dag \hat p_S V\hspace{0.5cm}\Rightarrow \hspace{0.5cm}\hat p_D=\hat p_S+\left(\frac{\partial S}{\partial q}\right).\end{aligned}$$ To determine the wave function evolution, we use the Schrödinger equation written in the form $$\begin{aligned} i\hbar\frac{\partial \psi(q,t)}{\partial t}=H_D(\hat q_D, \hat p_D)\psi(q,t).\end{aligned}$$ Utilising $\hat p_D=\hat p_S+\left(\frac{\partial S}{\partial q}\right)$, the quadratic term in the Hamiltonian becomes $$\begin{aligned} \hat p^2_D=\left[\hat p_S+\left(\frac{\partial S}{\partial q}\right)\right]^2=\hat p_S^2+\hat p_S\left(\frac{\partial S}{\partial q}\right)+\left(\frac{\partial S}{\partial q}\right)\hat p_S+\left(\frac{\partial S}{\partial q}\right)^2.\end{aligned}$$ If we use $\hat p_S=-i\hbar\partial/\partial q$, we see that $\hat p_D^2$ has real and imaginary parts. The real part can be written as $$\begin{aligned} \Re(\hat p_D^2)=\hat p_S^2+\left(\frac{\partial S}{\partial q}\right)^2,\end{aligned}$$ while the imaginary part becomes $$\begin{aligned} \Im(\hat p_D^2)=-i\hbar\frac{\partial^2S}{\partial q^2}+2\left(\frac{\partial S}{\partial q}\right)\hat p_S.\end{aligned}$$ The Schrödinger equation can now be expressed in the form $$\begin{aligned} i\hbar R(q,t)^{-1}\frac{\partial R(q,t)}{\partial t}-\hbar\frac{\partial S(q,t)}{\partial t}=R^{-1}(q,t)H_D(\hat q_D, \hat p_D)R(q,t) \label{eq:RSS}\end{aligned}$$ where we have written the wave function in its polar form $$\begin{aligned} \psi(q,t)=R(q,t)\exp[iS(q,t)/\hbar]. \label{eq:polarpsi}\end{aligned}$$ In order to see exactly what is going on, choose a Hamiltonian $H=\hat p^2/2m+V(q)$ for simplicity. The Schrödinger equation can then be split into its real and imaginary parts, the real part being $$\begin{aligned} \frac{\partial S(q,t)}{\partial t}+\frac{1}{2m}\left(\frac{\partial S(q,t)}{\partial q}\right)^2-\frac{\hbar^2}{2mR(q,t)}\left(\frac{\partial^2R(q,t)}{\partial q^2}\right)+V(q)=0 \label{eq:QHJ}\end{aligned}$$ which is identical to the quantum Hamilton-Jacobi equation of the Bohm approach [@db52]. The imaginary part of equation (\[eq:RSS\]) becomes $$\begin{aligned} i\hbar \frac{\partial R(q,t)}{\partial t}=\left[-i\hbar\left(\frac{\partial^2S(q,t)}{\partial q^2}\right)+2\left(\frac{\partial S(q,t)}{\partial q}\right)\hat p_S\right]R(q,t)\end{aligned}$$ which can be written in the form $$\begin{aligned} \frac{\partial\rho(q,t)}{\partial t}+\frac{1}{m}\frac{\partial}{\partial q}\left(\rho(q,t)\frac{\partial S(q,t)}{\partial q}\right)=0. \label{eq:ConP}\end{aligned}$$ Here $\rho(q,t)=R^2(q,t)$ is the probability density and the equation is an expression of the conservation of probability. Thus both the Dirac [@pd47] approach and the Bohm approach [@db52] lead to the same equations. For this reason we will call the representation the Dirac-Bohm picture[^2]. Development of the Dirac-Bohm Picture ===================================== In order to derive equations (\[eq:QHJ\]) and (\[eq:ConP\]), Dirac [@pd47] used a new notation which, he claims, “provides a neat and concise way of writing, in a single scheme, both the abstract \[sic, non-commuting\] quantities themselves and their coordinates, and thus leads to a unification" of the wave picture and the non-commutative algebra of operators [@pd39]. It is this structure that we call the ‘algebraic approach’. A key element in this approach was the introduction of a new symbol into the algebra, namely the [*standard* ]{} ket, $\rangle$ \[NB without the vertical line $|$ \][^3]. With this new element, the wave function can be written as a function of [*operators*]{}. Consequently all the information usually encoded in the wave function may then be abstracted from the algebra itself, viz, $\psi(\hat q, \hat p)\rangle$ without the need for a Hilbert space. In order to express the wave function in configuration space, we choose the standard ket to satisfy the relation $\hat p\;\rangle_q=0$. This is a natural result since $\hat p \psi\rangle_q=-i\hbar \frac{\partial\psi}{\partial \hat q}\rangle_q$ if we anticipate the Schrödinger representation. Then if $\psi=\mbox{constant}$, we have $\hat p\;\rangle_q=0$. If we write a general function of the set $\{\hat q,\hat p\}$ in [*normal order*]{}, we obtain $$\begin{aligned} \psi(\hat q,\hat p, t)\rangle_q=\psi(\hat q, t)\rangle_q\rightarrow \psi (q,t).\end{aligned}$$ By introducing a dual symbol, the [*standard*]{} bra, $_q\langle\;$, we can formally write $$\begin{aligned} _q\langle q|\psi(\hat q,\hat p,t)\rangle_q=\psi(q,t).\end{aligned}$$ In this way we arrive at the usual wave function in a configuration space. To choose the $p$-representation, we introduce a standard ket satisfying $\hat q\rangle_p=0$ since in this representation $\hat q=i\hbar \frac{\partial}{\partial p}$. In this case we write the operators in [*anti-normal order*]{} to obtain any element in the $p$-representation. Thus one can derive the two equations (\[eq:ConP\]) and (\[eq:QHJ\]) directly from the Schrödinger equation written in the [*algebraic*]{} form $$\begin{aligned} i\hbar\frac{\partial}{\partial t}\left(Re^{iS/\hbar}\right)\rangle_q=H(\hat q, \hat p)Re^{iS/\hbar}\rangle_q. \label{eq:ASeqn}\end{aligned}$$ Dirac remarked that equation (\[eq:ConP\]) was similar to the continuity equation used in fluid dynamics but did not mention the earlier work of Madelung [@em26], who introduced the hydrodynamical model. He further noted that equation (\[eq:QHJ\]) had the form of the classical Hamilton-Jacobi equation if one neglected terms of order $\hbar^2$ and above. Dirac did not pursue this line of reasoning because he thought it would come into conflict with the uncertainty principle. Full details of this approach can be found in Sections 31 and 32 of his book “The Principles of Quantum Mechanics", together with comments as to why he did not continue the investigation[@pd47]. The Bohm Approach ----------------- Initially Bohm wrote “Quantum Theory" [@db51] in order to provide an explanation of the standard approach to quantum mechanics–a book in which he argued that hidden variables could [*not*]{} be used to provide a deeper understanding of the formalism. However Bohm told one of us \[BJH\] that after completing the book he felt dissatisfied with his account. While investigating the first order WKB approximation, Bohm noticed, just as Dirac had done, that classical ideas still held. Unlike Dirac, Bohm saw no reason for abandoning the conceptual structure when higher order terms were retained. Furthermore Bohm found that he could derive the two equations (\[eq:QHJ\]) and (\[eq:ConP\]) in a much simpler way by taking the real and imaginary parts of the Schrödinger equation under polar decomposition of the wave function, a process that automatically retains all higher powers of $\hbar$. He went on to show that, contrary to Dirac’s conclusion, the uncertainty principle was not violated when retaining higher powers of $\hbar$. Bohm’s interpretation was based on the assumption that a particle had both a position and a momentum at all times. He further assumed that the momentum was specified by retaining the canonical relation $p(q,t)=\partial S(q,t)/\partial q$, when the classical action $S(q,t)$ was the phase of the wave function, an assumption that was not justified in Bohm’s original papers [@db52]. With that assumption Bohm showed that the uncertainty principle would not be violated. The approach provides a consistent interpretation of a Schrödinger particle, providing a clear, intuitive account of all quantum phenomena by removing many of the paradoxes of the standard interpretation [@ph95]. Although it appears to be a return to classical determinism, a closer examination shows it contains properties that are radically different from classical physics as detailed in Bohm and Hiley [@dbbh93]. Quantum Particle Trajectories? =============================== Perhaps the most contentious aspect of the Bohm approach was the unambiguous appearance of what was regarded as ‘particle trajectories’. This follows from the assumption that a particle has a well-defined position and momentum even though it is not possible to measure them simultaneously. From the fact that position and momentum cannot be measured simultaneously, one can draw two opposite conclusions. Namely, quantum particles do not have a simultaneous position and momentum, or if they have, we cannot say anything meaningful about them together. This is the position taken by the majority of physicists. The opposite assumption, that quantum particles do have a well-defined position and momentum, may have experimental consequences that can be fully explored as they arise. Hopefully one of the consequences will lead to new experiments. This is the position adopted by Bohm in his 1952 paper [@db52]. The fact that one could obtain smooth ‘trajectories’ from a basically non-commutative structure was always a concern. Nevertheless the appearance of the paper by Kocsis [*et al.*]{} [@skbbsr11] changed all that. This paper provided a method for measuring and constructing such ‘trajectories’. These pioneering experiments used single photons but the claim that the momentum flow lines were ‘photon trajectories’ was controversial. The further claim that their results vindicated the Bohm approach because the flow lines bore a remarkable similarity to those calculated by Philippidis, Dewdney and Hiley [@cpcd79] unfortunately cannot be sustained [@dmlr15]. Photons are zero rest mass particles travelling at the speed of light and therefore cannot be described by the Schrödinger equation. Flack and Hiley [@rfbh16] have shown that the Kocsis [*et al.*]{} experiments [@skbbsr11] constructed momentum flow lines, rather than photon trajectories, and that these flow lines are determined by the weak Poynting vector. Nevertheless the same techniques can be used on non-relativistic atoms [@bh11; @rfbh18] and such experiments are under construction [@jmpe16]. An analysis of the method shows that the experimentally determined curves are not the trajectories of a single atom, but are actually momentum flow lines averaged over many individual single atom events, which is in agreement with the results of Kocsis [*et al.*]{} [@skbbsr11]. These new factors call into question the validity of the original assumption that particles follow well-defined trajectories and demand a radical re-evaluation of what underlies the Bohm approach. Do we return to the common assumption that it is not possible to talk about trajectories of quantum particles [@llel13] or can we probe more deeply? Dirac Trajectories ================== What seems to have been forgotten is that Dirac [@pd45] attempted to construct what he called ‘quantum trajectories’ within the context of the non-commutative algebra of quantum operators. We will bring this out using the Dirac-Bohm picture which avoids the need to introduce the standard ket symbol $\rangle$ and work directly in Hilbert space. Following Dirac, we note the general ket evolves in time via $$\begin{aligned} |t\rangle=V(t,t_0)|t_0\rangle. \label{eq:VT} \end{aligned}$$ We can write $$\begin{aligned} \frac{d|t_0\rangle}{dt_0}=\lim_{t\rightarrow t_0}\frac{|t\rangle-|t_0\rangle}{t-t_0}=\lim_{t\rightarrow t_0}\frac{V-1}{t-t_0}|t_0\rangle. \end{aligned}$$ If we put $$\begin{aligned} i\hbar \lim_{t\rightarrow t_0}\frac{V-1}{t-t_0}= H(t_0), \end{aligned}$$ we have for general $t$ $$\begin{aligned} i\hbar \frac{d|t\rangle}{dt}=H(t)|t\rangle. \end{aligned}$$ Using equation (\[eq:VT\]) we obtain $$\begin{aligned} i\hbar\frac {dV}{dt}|t_0\rangle=H(t)V|t_0\rangle. \end{aligned}$$ Since this holds for any $|t_0\rangle$, we have $$\begin{aligned} i\hbar\frac{dV}{dt}=H(t)V. \end{aligned}$$ Thus for any operator $\hat r$ we have $$\begin{aligned} \hat r_t=V^{-1}\hat r V \end{aligned}$$ so that $$\begin{aligned} i\hbar\frac{d\hat r_t}{dt}=V^{-1}\hat rHV-V^{-1}HV\hat r_{t}=[\hat r_t, H_t]\quad\\\mbox{with}\quad H_t=V^{-1}H V.\hspace{3cm} \end{aligned}$$ This is a Heisenberg-type equation of motion only now we use a unitary transformation generated by the classical action (\[eq:SUTrans\]). Introducing the bra $\langle q_t|=\langle q_{t_0}|V$, we have $$\begin{aligned} i\hbar\frac{d}{dt}\langle q_t|=i\hbar \langle q_{t_0}|\frac{dV}{dt}=\langle q_{t_0}|HV=\langle q_t|H_t. \end{aligned}$$ Thus we can write $$\begin{aligned} i\hbar\frac{d}{dt}\langle q_t|q'\rangle=\int \langle q_t|H_t|q''_t\rangle dq''_t\langle q''_t|q'\rangle \end{aligned}$$ showing that the transition amplitude \[TA\] $\langle q_t|q'\rangle $ satisfies the Schrödinger wave equation. We can therefore in general write $$\begin{aligned} \langle q_t|q'\rangle=e^{iS'(q_t,q';t,t')/\hbar}. \end{aligned}$$ This then links with equation (\[eq:ASeqn\]) where we have replaced $Re^{iS/\hbar} $ by $e^{iS'(q_t,q'; t,t')/\hbar}$. Here $S'$ is now a complex number. (For more details see Flack and Hiley [@rfbh18].) However for simplicity, let us continue with the special case assuming $S$ real. In the limit $\hbar\rightarrow 0$, when $S$ is real we have at the initial point $q$ $$\begin{aligned} \frac{\partial S}{\partial t_0}=H_c(q, p)\hspace{0.2cm}\mbox{and}\hspace{0.2cm}p=-\frac{\partial S}{\partial q}, \label{eq:ContA} \end{aligned}$$ where $H_c(q,p)$ is the classical Hamiltonian. At the final point $q'$ we have $$\begin{aligned} -\frac{\partial S}{\partial t}=H_c(q', p')\hspace{0.2cm}\mbox{and}\hspace{0.2cm}p'=\frac{\partial S}{\partial q'}. \label{eq:ContB} \end{aligned}$$ These equations define the connection between the variables $(q,p)$ and the variables $(q', p')$ by a contact transformation. The unitary transformation given by equation (\[eq:lift\]) below is the quantum generalisation of the contact transformation, a point that Bohm [@db51] emphasises. See Brown and Hiley [@mbbh00] for a different discussion of this point. The Role of the Transition Amplitudes {#sec:DandF} -------------------------------------- In the Dirac-Bohm picture, the operators are functions of time. Strict attention must therefore be paid to the time order of the appearance of elements in a sequence of operators. In the non-relativistic limit, operators at different times always commute[^4]. Hence a time-ordered sequence of position operators can be uniquely written in the form, $$\langle q_t|q_{t_0}\rangle=\int\dots\int\langle q_t|q_{t_j}\rangle dq_j\langle q_{t_j}|q_{t_{j-1}}\rangle\dots \langle q_{t_2}|q_{t_1}\rangle dq_1\langle q_{t_1}|q_{t_0}\rangle. \label{eq:QTraj}$$ This divides the TA, $\langle q_t|q_{t_0}\rangle$, into a sequence of closely adjacent points, each pair of which is connected by an infinitesimal TA. Dirac writes “…one can regard this as a trajectory…and thus makes quantum mechanics more closely resemble classical mechanics" [@pd45]. In order to analyse the sequence (\[eq:QTraj\]) further, Dirac assumed that for a small time interval $\Delta t=\epsilon$, we can write $$\langle q'|q\rangle_\epsilon= \exp[iS_\epsilon(q',q)] \label{eq:lift}$$ where we will again take $S_\epsilon(q',q)$ to be a real function. This, of course, became the source of Feynman’s propagators [@rpf48]. However Dirac [@pd47] had previously shown that $$p_\epsilon(q',q)=\frac{\langle q'|\hat P|q\rangle_\epsilon}{\langle q'|q\rangle_\epsilon}=\frac{i\hbar\nabla_{q}\langle q'|q\rangle_\epsilon}{\langle q'|q\rangle_\epsilon}= -\nabla_{q} S_\epsilon(q',q) \label{eq:conjM1}$$ and $$p'_\epsilon(q',q)=\frac{\langle q'|\hat P'|q\rangle_\epsilon}{\langle q'|q\rangle_\epsilon}=\frac{-i\hbar\nabla_{q'}\langle q'|q\rangle_\epsilon}{\langle q'|q\rangle_\epsilon}= \nabla_{q'} S_\epsilon(q',q). \label{eq:conjM}$$ Here $\hat P$ is the momentum operator. These equations replace the classical equations (\[eq:ContA\]) and (\[eq:ContB\]) and generate the quantum relation between the points $(q, p)$ and $(q', p')$. It is this analogy that prompted Dirac to introduce the notion of a particle trajectory. The details of this relationship are discussed by de Gosson and Hiley [@mdgbh11]. In that paper it is shown that the quantum trajectories can be formed by lifting the classical trajectories generated by symplectic transformations onto the covering space, the covering group being the metaplectic group and its generalisation. Quantum Trajectories and Feynman Propagators ============================================ The Momentum Propagator {#sec:MomProp} ----------------------- Let us continue to discuss the notion of ‘quantum trajectories’ based on equation (\[eq:QTraj\]). Each TA between two points can be subdivided into a series of infinitesimal TAs $\langle q'| q\rangle_\epsilon$. Furthermore each infinitesimal TA may be regarded as an unfolding of the immediate past into the adjacent present. Formally we can regard this process as a movement from one set of commuting variables into another commuting set of the [*same*]{} variables. What about the other non-commuting sets of variables? In section \[sec:DandF\], we saw that each infinitesimal TA was accompanied by a pair of momentum TAs, $p_\epsilon(q',q)$ and $p_\epsilon'(q',q)$. How do we relate these TAs to a particle moving between $q$ and $q'$? To avoid problems with the uncertainty principle, consider a small but finite volume $\Delta V$ surrounding the point $q$. Imagine a sequence of particles emanating from a point in $\Delta V$, each with a different momentum, so that over time we have a spray of possible momenta emerging from the volume $\Delta V$. Similarly there is a spray of momenta over time arriving at the small volume $\Delta V'$ surrounding the point $q'$. Better still, let us consider a small volume surrounding the midpoint $Q$. At this point there is a spray of momenta arriving and a spray leaving a volume $\Delta V(Q)$ as shown in Figure \[fig:spray\]. ![Behaviour of the momenta sprays at the midpoint of $\langle q', t'| q, t\rangle_\epsilon$[]{data-label="fig:spray"}](Midpoint){width="4in"} To see how the local momenta behave at the midpoint $Q$, recall that for small time differences $t'-t=\epsilon$, we have for the propagator of a free particle (\[eq:lift\]), $$\begin{aligned} S_\epsilon(q',q)=\frac{m}{2}\frac{(q'-q)^2}{\epsilon} \label{eq:act} \end{aligned}$$ which is obtained from the classical Lagrangian. Then we have the momentum TA $$\begin{aligned} P_Q(q',q)=\frac{\partial S_\epsilon(q', q)}{\partial Q}=\frac{\partial S_\epsilon(Q,q)}{\partial Q}+\frac{\partial S_\epsilon(q', Q)}{\partial Q}. \label{eq:momXa} \end{aligned}$$ Using (\[eq:act\]), we find $$P_Q(q',q)=m\left[\frac{(q'-Q)}{\epsilon}-\frac{(Q-q)}{\epsilon}\right]=p'_Q(q',q)+p_Q(q',q). \label{eq:momX}$$ The RHS of this equation comprises exactly the relations that Dirac [@pd45] obtained in equations (\[eq:conjM1\]) and (\[eq:conjM\]) above. Not surprisingly, it is also exactly the momentum TA that Feynman [@rpf48] obtains in his equation (48) at the point $Q$ which lies between the two neighbouring points separated in time by $\Delta t=\epsilon$. Notice that in the limit of $\epsilon\rightarrow 0$, equation (\[eq:momX\]) comprises two ‘derivatives’ at $Q$, namely $$\begin{aligned} D_Q(\mbox{Backward})=\lim_{\epsilon\rightarrow 0}\frac{(Q-q)}{\epsilon}\hspace{1cm} D_Q(\mbox{Forward})=\lim_{\epsilon\rightarrow 0}\frac{(q'-Q)}{\epsilon}. \end{aligned}$$ Such derivatives are associated with a general stochastic process where the ‘trajectory’ joining the two points $q$ to $q'$ is continuous, but the derivatives are not. This situation is known to arise in Brownian motion [@nw66]. Indeed these very derivatives were used by Nelson [@en66] in his derivation of the Schrödinger equation from an underlying stochastic process. (See also the discussion in Bohm and Hiley [@dbbh89] and Prugovecki [@ep82] for alternative views.) The meaning of the non-continuous derivatives here is clear; the basic underlying quantum process connecting infinitesimally neighbouring points is an [*intrinsically*]{} random process, but at this stage the precise form of this stochastic process is unclear. However the spray of possible momenta emanating from a region cannot be completely random since, as Feynman has shown, the transition amplitudes satisfy the Schrödinger equation under certain assumptions. Some clues as to the precise nature of this distribution have already been supplied by Takabayasi [@tt54] and Moyal [@jm49], clues which we will now exploit. We are interested in finding the average behaviour of the momentum, $P_Q$, at the point $Q$. This means we must determine the spray of momenta that is consistent with the wave function $\psi(Q)$ at $Q$. But we have two contributions, one coming from the point $q$ and one leaving for the point $q'$. Feynman’s proposal [@rpf48] that we can think of $\psi(Q)$ as ‘information coming from the past’ and $\psi^*(Q)$ as ‘information coming from the future’, will be used here as this suggests that we can write $$\begin{aligned} \lim_{q\rightarrow Q}\psi(q)=\int\phi(p)e^{ipQ} dp\hspace{0.5cm}\mbox{and}\hspace{0.5cm} \lim_{Q\rightarrow q'}\psi^*(q')=\int \phi^*(p')e^{-ip'Q}dp'.\end{aligned}$$ The $\phi(p)$ contains information regarding the probability distribution of the incoming momentum spray, while $\phi^*(p')$ contains information about the probability distribution of the outgoing momentum spray. These wave functions must be such that in the limit $\epsilon\rightarrow 0$ they are consistent with the wave function $\psi(Q)$. Thus we can define the mean momentum, $\overline {P(Q)}$, at the point $Q$ as $$\begin{aligned} \rho(Q)\overline {P(Q)}=\int\int P\phi^*(p')e^{-ip'Q}\phi(p)e^{ipQ}\delta(P-(p'+p)/2)dPdpdp' \label{eq:MMPP}\end{aligned}$$ where $\rho(Q)$ is the probability density at $Q$. We have added the restriction $\delta(P-(p'+p)/2)$ because we are using the diffeomorphism $(p,p')\rightarrow [(p'+p)/2,(p'-p)]$. It is immediately seen that equation (\[eq:MMPP\]) can be put in the form $$\begin{aligned} \rho(Q)\overline {P(Q)}=\left(\frac{1}{2i}\right)[(\partial_{q_1}-\partial_{q_2})\psi(q_1)\psi(q_2)]_{q_1=q_2=Q}, \label{eq:MMXX} \end{aligned}$$ a form that appears in Moyal [@jm49]. If we write the wave function in polar form, we find that $\overline {P(Q)}$ is just the local momentum $P_B=\nabla S$ that appears in the Bohm interpretation. Since $P_B$ is used to calculate the Bohm trajectories, there must be a close relationship between these trajectories and Feynman paths. If we assume each evolving quantum process, which we will call a particle, actually follows a Feynman stochastic path then a Bohm trajectory can be regarded as an ensemble average of many such paths. Notice however, this gives a very different picture of the Bohm momentum from the usual one used in Bohmian mechanics [@ddst09]. It is not the momentum of a single ‘particle’ passing the point $Q$, but the mean [*momentum flow*]{} at the point in question. This conclusion is supported by the treatment of the electromagnetic field by Flack and Hiley [@rfbh16] where the notion of a photon replaces that of the particle. In their paper it was shown that the ‘photon’ flow lines constructed in the experiments of Kocsis [*et al.*]{} [@skbbsr11] were actually statistical averages producing momentum flow lines which corresponded to those determined by the [*weak*]{} value of the Poynting vector. This agrees with standard quantum electrodynamics, where the notion of a ‘photon trajectory’ has no meaning. Conclusion ========== The algebraic approach to quantum phenomena proposed by Dirac [@pd45] and the apparently very different approach of Bohm [@db52], are two aspects of a deeper structure which we have called the Dirac-Bohm picture. Just as in the Heisenberg picture, the operators become time dependent through a unitary transformation. However in the Dirac-Bohm picture, the unitary transformation involves the classical action rather than the total energy. This has the effect of producing a quantum formalism that is closer to the classical description, enabling one to see the essential differences between classical and quantum phenomena and providing a basis for deformation quantum mechanics [@ahph02] as we will show elsewhere. Acknowledgments =============== We would like to thank Rob Flack, Bob Callaghan and Lindon Neil for many stimulating discussions. 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ArXiv

--- abstract: 'In various interaction tasks using Underwater Vehicle Manipulator Systems (UVMSs) (e.g. sampling of the sea organisms, underwater welding), important factors such as: i) uncertainties and complexity of UVMS dynamic model ii) external disturbances (e.g. sea currents and waves) iii) imperfection and noises of measuring sensors iv) steady state performance as well as v) inferior overshoot of interaction force error, should be addressed during the force control design. Motivated by the above factors, this paper presents a model-free control protocol for force controlling of an Underwater Vehicle Manipulator System which is in contact with a compliant environment, without incorporating any knowledge of the UVMS’s dynamic model, exogenous disturbances and sensor’s noise model. Moreover, the transient and steady state response as well as reduction of overshooting force error are solely determined by certain designer-specified performance functions and are fully decoupled by the UVMS’s dynamic model, the control gain selection, as well as the initial conditions. Finally, a simulation study clarifies the proposed method and verifies its efficiency.' address: - | Control Systems Lab, Department of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou 15780.\ E-mail: {shahab, kkyria}@mail.ntua.gr - | ACCESS Linnaeus Center, School of Electrical Engineering and KTH Center for Autonomous Systems, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden.\ E-mail: {anikou, dimos}@kth.se author: - 'Shahab Heshmati-Alamdari, Alexandros Nikou and Kostas J. Kyriakopoulos[^1]' - 'Shahab Heshmati-alamdari' - Alexandros Nikou - 'Kostas J. Kyriakopoulos' - 'Dimos V. Dimarogonas' bibliography: - 'mybibfilealina.bib' title: A Robust Force Control Approach for Underwater Vehicle Manipulator Systems --- Underwater Vehicle Manipulator System, Nonlinear Control, Autonomous Underwater Vehicle, Marine Robotics, Force Control, Robust Control. Introduction ============ In view of the development of autonomous underwater vehicles, the capability of such vehicles to interact with the environment by the use of a robot manipulator, had gained attention in the literature. Most of the underwater manipulation tasks, such as maintenance of ships, underwater pipeline or weld inspection, surveying, oil and gas searching, cable burial and mating of underwater connector, require the manipulator mounted on the vehicle to be in contact with the underwater object or environment. The aforementioned systems are complex and they are characterized by several strong constraints, namely the complexity in the mathematical model and the difficulty to control the vehicle. These constraints should be taken into consideration when designing a force control scheme. In order to increase the adaptability of UVMS, force control must be included into the control system of the UVMS. Although many force control schemes have been developed for earth-fixed manipulators and space robots, these control schemes cannot be used directly on UVMS because of the unstructured nature of the underwater environment. From the control perspective, achieving these type of tasks requires specific approaches[@Siciliano_Sciavicco]. However, speaking about underwater robotics, only few publications deal with the interaction control using UVMS. One of the first underwater robotic setups for interaction with the environment was presented in [@Casalino20013220]. Hybrid position/force control schemes for UVMS were developed and tested in [@lane1; @lane2]. However, dynamic coupling between the manipulator and the underwater vehicle was not considered in the system model. In order to compensate the contact force, the authors in [@kajita] proposed a method that utilizes the restoring force generated by the thrusters. In the same context, position/force [@lapierre], impedance control [@cui1; @cui2; @cui3] and external force control schemes [@antonelli_tro; @antonelli_cdc; @Antonelli2002251] can be found in the literature. Over the last years, the interaction control of UVMS is gaining significant attention again. Several control issues for an UVMS in view of intervention tasks has been presented in [@Marani2010175]. In [@Cataldi2015524] based on the interaction schemes presented in [@antonelli_tro] and [@Antonelli2002251], the authors proposed a control protocol for turning valve scenarios. Recent study [@moosavian] proposed a multiple impedance control scheme for a dual manipulator mounted on AUV. Moreover, the two recent European projects TRIDENT(see, e.g. [@Fernández2013121],[@Prats201219],[@Simetti2014364]) and PANDORA (see, e.g. [@Carrera2014], [@Carrera2015]) have given boost to underwater interaction with relevant results. In real applications, the UVMS needs to interact with the environment via its end-effector in order to achieve a desired task. During the manipulation process the following issues occur: the environment is potentially unknown, the system is in the presence of unknown (but bounded) external disturbances (sea currents and sea waves) and the sensor measurements are not always accurate (we have noise in the measurements). These issues can cause unpredicted instabilities to the system and need to be tackled during the control design. From the control design perspective, the UVMS dynamical model is highly nonlinear, complicated and has significant uncertainties. Owing to the aforementioned issues, underwater manipulation becomes a challenging task in order to achieve low overshoot, transient and steady state performance. Motivated by the above, in this work we propose a force - position control scheme which does not require any knowledge of the UVMS dynamic parameters, environment model as well as the disturbances. More specifically, it tackles all the aforementioned issues and guarantees a predefined behavior of the system in terms of desired overshoot and prescribed transient/steady state performance. Moreover, noise measurements, UVMS model uncertainties (a challenging issue in underwater robotics) and external disturbance are considered during control design. In addition, the complexity of the proposed control law is significantly low. It is actually a static scheme involving only a few calculations to output the control signal, which enables its implementation on most of current UVMS. The rest of this paper is organized as follows: in Section 2 the mathematical model of UVMS and preliminary background are given. Section 3 provides the problem statement that we aim to solve in this paper. The control methodology is presented in Section 4. Section 5 validates our approach via a simulation study. Finally, conclusions and future work directions are discussed in Section 6. Preliminaries ============= Mathematical model of the UVMS ------------------------------ In this work, the vectors are denoted with lower bold letters whereas the matrices by capital bold letters. The end effector coordinates with respect to (w.r.t) the inertial frame $\{I\}$ are denoted by ${\boldsymbol}{x}_e\in \mathbb{R}^6$. Let ${\boldsymbol}{q}=[{\boldsymbol}{q}^\top_a,~{\boldsymbol}{q}^\top_m]^\top\in \mathbb{R}^n$ be the state variables of the UVMS, where ${\boldsymbol}{q}_a=[{\boldsymbol}{\eta}_1^\top,{\boldsymbol}{\eta_2}^\top]^\top\in \mathbb{R}^6$ is the vector that involves the position vector ${\boldsymbol}{\eta}_{1}=[x,y,z]^\top$ and Euler-angles orientation ${\boldsymbol}{\eta}_{2}=[\phi,\theta,\psi]^\top$ of the vehicle w.r.t to the inertial frame $\{I\}$ and ${\boldsymbol}{q}_m\in \mathbb{R}^{n-6}$ is the vector of angular position of the manipulator’s joints. Thus, we have [@Fossen2; @antonelli]: $$\begin{gathered} \dot{{\boldsymbol}{q}}_a= {\boldsymbol}{J}^a({\boldsymbol}{q}_a){\boldsymbol}{v} \label{eq1}\end{gathered}$$ where $$\begin{aligned} {\boldsymbol}{J}^a({\boldsymbol}{q}_a)= \begin{bmatrix} {\boldsymbol}{J}_t(\eta_2) & {\boldsymbol}{0}_{3 \times 3} \\ {\boldsymbol}{0}_{3 \times 3} & {\boldsymbol}{J}_{r}(\eta_2) \\ \end{bmatrix}\in \mathbb{R}^{6\times 6}\end{aligned}$$ is the Jacobian matrix transforming the velocities from the body-fixed to the inertial frame and where, ${\boldsymbol}{0}_{3\times 3}$ is the zero matrix of the respective dimensions, ${\boldsymbol}{v}$ is the vector of body velocities of the vehicle and ${\boldsymbol}{J}_t(\eta_2)$ and ${\boldsymbol}{J}_r(\eta_2)$ are the corresponding parts of the Jacobian related to position and orientation respectively. Let also $\dot{{\boldsymbol}{\chi}}=[\dot{{\boldsymbol}{\eta}}_{1}^\top,{\boldsymbol}{\omega}^\top]^\top$ denotes the velocity of UVMS’s End-Effector, where $\dot{{\boldsymbol}{\eta}}_{1}$, ${\boldsymbol}{\omega}$ are the linear and angular velocity of the UVMS’s End-Effector, respectively. Without loss of generality, for the augmented UVMS system we have [@antonelli]: $$\dot{{\boldsymbol}{\chi}}= {\boldsymbol}{J}^g({{\boldsymbol}{q}})\boldsymbol{\zeta}\label{eq222}$$ where $\boldsymbol{\zeta}=[{\boldsymbol}{v}^\top,\dot{{\boldsymbol}{q}}_{m,i}^\top]^\top \in \mathbb{R}^{n}$ is the velocity vector including the body velocities of the vehicle as well as the joint velocities of the manipulator and $ {\boldsymbol}{J}^g({\boldsymbol}{q})$ is the geometric Jacobian Matrix [@antonelli]. In this way, the task space velocity vector of UVMS’s End-Effector can be given by: $$\dot{{\boldsymbol}{x}}_e= {\boldsymbol}{J}({{\boldsymbol}{q}})\boldsymbol{\zeta}\label{eq122}$$ where: ${\boldsymbol}{J}({{\boldsymbol}{q}})$ is analytical Jacobian matrix given by: $$\begin{gathered} {\boldsymbol}{J}({{\boldsymbol}{q}})={{\boldsymbol}{J}'({{\boldsymbol}{q}})}^{-1}{\boldsymbol}{J}^g({{\boldsymbol}{q}})\end{gathered}$$ with ${\boldsymbol}{J}'({{\boldsymbol}{q}})$ to be a Jacobian matrix that maps the Euler angle rates to angular velocities ${\boldsymbol}{\omega}$ and is given by: $$\begin{gathered} {\boldsymbol}{J}'({{\boldsymbol}{q}})=\begin{bmatrix} {\boldsymbol}{I}_{3\times 3} & {\boldsymbol}{0}_{3\times 3}\\ {\boldsymbol}{0}_{3\times 3} & {\boldsymbol}{J}''({{\boldsymbol}{q}}) \end{bmatrix},\\ {\boldsymbol}{J}''({{\boldsymbol}{q}})=\begin{bmatrix} 1 & 0&-\sin(\theta)\\ 0&\cos(\phi)&\cos(\theta)\sin(\phi)\\ 0&-\sin(\phi)&\cos(\theta)\cos(\phi) \end{bmatrix}.\end{gathered}$$ Dynamics -------- Without loss of generality, the dynamics of the UVMS can be given as [@antonelli]: $$\begin{gathered} {\boldsymbol}{M}({\boldsymbol}{q})\dot{{\boldsymbol}{\zeta}}\!+\!{\boldsymbol}{C}({{\boldsymbol}{q}},{\boldsymbol}{\zeta}){{\boldsymbol}{\zeta}}\!+\!{\boldsymbol}{D}({{\boldsymbol}{q}},{\boldsymbol}{\zeta}){{\boldsymbol}{\zeta}}\!+{\boldsymbol}{g}( {\boldsymbol}{q})\!+\!{{\boldsymbol}{J}^g}^\top\boldsymbol{\lambda}+\boldsymbol{\delta}(t)=\!{\boldsymbol}{\tau}\!\label{eq4}\end{gathered}$$ where $\boldsymbol{\delta}(t)$ are bounded disturbances including system’s uncertainties as well as the external disturbances affecting on the system from the environment (sea waves and currents), $\boldsymbol{\lambda}=[{\boldsymbol}{f}^\top_e,\boldsymbol{\nu}^\top_e]^\top$ the generalized vector including force ${\boldsymbol}{f}_e$ and torque $\boldsymbol{\nu}_e$ that the UVMS exerts on the environment at its end-effector frame. Moreover, ${\boldsymbol}{\tau} \in \mathbb{R}^n$ denotes the control input at the joint level, ${\boldsymbol}{{M}}({\boldsymbol}{q})$ is the positive definite inertial matrix, ${\boldsymbol}{{C}}({{\boldsymbol}{q}},{\boldsymbol}{\zeta})$ represents coriolis and centrifugal terms, ${\boldsymbol}{{D}}({{\boldsymbol}{q}},{\boldsymbol}{\zeta})$ models dissipative effects, ${\boldsymbol}{{g}}({\boldsymbol}{q})$ encapsulates the gravity and buoyancy effects. Dynamical Systems ----------------- Consider the initial value problem: $$\dot{\xi} = H(t,\xi), \xi(0)=\xi^0\in\Omega_{\xi}, \label{eq:initial_value_problem}$$ with $H:\mathbb{R}_{\geq 0}\times\Omega_{\xi} \to \mathbb{R}^n$, where $\Omega_{\xi}\subseteq\mathbb{R}^n$ is a non-empty open set. [@Sontag] A solution $\xi(t)$ of the initial value problem is maximal if it has no proper right extension that is also a solution of . [@Sontag] \[thm:dynamical systems\] Consider the initial value problem . Assume that $H(t,\xi)$ is: a) locally Lipschitz in $\xi$ for almost all $t\in\mathbb{R}_{\geq 0}$, b) piecewise continuous in $t$ for each fixed $\xi\in\Omega_{\xi}$ and c) locally integrable in $t$ for each fixed $\xi\in\Omega_{\xi}$. Then, there exists a maximal solution $\xi(t)$ of on the time interval $[0,\tau_{\max})$, with $\tau_{\max}\in\mathbb{R}_{> 0}$ such that $\xi(t)\in\Omega_{\xi},\forall t\in[0,\tau_{\max})$. [@Sontag] \[prop:dynamical systems\] Assume that the hypotheses of Theorem \[thm:dynamical systems\] hold. For a maximal solution $\xi(t)$ on the time interval $[0,\tau_{\max})$ with $\tau_{\max}<\infty$ and for any compact set $\Omega'_{\xi}\subseteq\Omega_{\xi}$, there exists a time instant $t'\in[0,\tau_{\max})$ such that $\xi(t')\notin\Omega'_{\xi}$. Problem Statement ================= We define here the problem that we aim to solve in this paper: Given a UVMS system as well as a desired force profile that should be applied by the UVMS on an entirely unknown model compliant environment, assuming the uncertainties on the UVMS dynamic parameters, design a feedback control law such that the following are guaranteed: 1. a predefined behavior of the system in terms of desired overshoot and prescribed transient and steady state performance. 2. robustness with respect to the external disturbances and noise on measurement devises. Control Methodology =================== In this work we assume that the UVMS is equipped with a force/torque sensor at its end-effector frame. However, we assume that its accuracy is not perfect and the system suffers from noise in the force/torque measurements. In order to combine the features of stiffness and force control, a parallel force/position regulator is designed. This can be achieved by closing a force feedback loop around a position/velocity feedback loop, since the output of the force controller becomes the reference input to the dynamic controller of the UVMS. Control Design -------------- Let ${\boldsymbol}{f}_e^d(t)$ be the desired force profile which should be exerted on the environment by the UVMS. Hence, let us define the force error: $$\begin{aligned} {\boldsymbol}{e}_f(t)={\boldsymbol}{f}_e(t)+\Delta{\boldsymbol}{f}_e(t)-{\boldsymbol}{f}_e^d(t)\in \mathbb{R}^3, \label{eq8} \end{aligned}$$ where $\Delta{\boldsymbol}{f}_e(t)$ denotes the bounded noise on the force’s measurement. Also we define the end-effector orientation error as: $$\begin{aligned} {\boldsymbol}{e}_o(t)= {^o{\boldsymbol}{x}}_e(t)- {^o{\boldsymbol}{x}}^d_e(t) \in \mathbb{R}^3, \label{eq_or} \end{aligned}$$ where ${^o{\boldsymbol}{x}}^d_e(t)\in \mathbb{R}^3$ is predefined desired orientation of the end-effector (e.g. ${^o{\boldsymbol}{x}}^d_e(t)=[0,~0,~0]^\top$). Now we can set the vector of desired end-effector configuration as ${\boldsymbol}{x}^d_e(t)= [{\boldsymbol}{f}_e^d(t)^\top,({^o{\boldsymbol}{x}^d_e(t)})^\top]^\top$. In addition the overall error vector is given as: $$\begin{aligned} {\boldsymbol}{e}_x(t)=[e_{x_1}(t),\ldots,e_{x_6}(t)]=[{\boldsymbol}{e}^\top_f(t),{\boldsymbol}{e}^\top_o(t)]^\top\label{eq:ov:er} \end{aligned}$$ A suitable methodology for the control design in hand is that of prescribed performance control, recently proposed in [@Bechlioulis20141217; @C-2011], which is adapted here in order to achieve predefined transient and steady state response bounds for the errors. Prescribed performance characterizes the behavior where the aforementioned errors evolve strictly within a predefined region that is bounded by absolutely decaying functions of time, called performance functions. The mathematical expressions of prescribed performance are given by the inequalities: $-\rho_{x_j}(t)<e_{x_j}(t)<\rho_{x_j}(t),~ j=1,\ldots,6$, where $\rho_{x_j}:[t_0,\infty)\rightarrow\mathbb{R}_{>0}$ with $\rho_{x_j}(t)=(\rho^0_{x_j}-\rho_{x_j}^\infty)e^{-l_{x_j}t}+\rho_{x_j}^\infty$ and $l_{x_j}>0,\rho^0_{x_j}>\rho^\infty_{x_j}>0$, are designer specified, smooth, bounded and decreasing positive functions of time with positive parameters $l_{x_j},\rho^\infty_{x_j}$, incorporating the desired transient and steady state performance respectively. In particular, the decreasing rate of $\rho_{x_j}$, which is affected by the constant $l_{x_j}$ introduces a lower bound on the speed of convergence of $e_{x_j}$. Furthermore, the constants $\rho^\infty_{x_j}$ can be set arbitrarily small, achieving thus practical convergence of the errors to zeros. Now, we propose a state feedback control protocol $\boldsymbol{\tau}(t)$, that does not incorporate any information regarding the UVMS dynamic model and model of complaint environment, and achieves tracking of the smooth and bounded desired force trajectory ${\boldsymbol}{f}_e^d(t)\in \mathbb{R}^3$ as well as ${^o{\boldsymbol}{x}}^d_e(t)$ with an priori specified convergence rate and steady state error. Thus, given the errors : **Step I-a**: Select the corresponding functions $\rho_{x_j}(t)=(\rho^0_{x_j}-\rho_{x_j}^\infty)e^{-l_{x_j}t}+\rho_{x_j}^\infty$ with $\rho^0_{x_j}>|e_{x_j}(t_0)|, \forall j\in\{1\ldots,6\}$ $\rho^0_{x_j}>\rho^\infty_{x_j}>0$, $ l_{x_j}>0,\forall j\in\{1,\ldots 6\}$, in order to incorporate the desired transient and steady state performance specification and define the normalized errors: $$\begin{aligned} \xi_{x_j}(t)=\frac{e_{x_j}(t)}{\rho_{x_j}(t)},~j=\{1,\ldots,6\}\label{eq11}\end{aligned}$$ **Step I-b**: Define the transformed errors $\varepsilon_{x_j}$ as: $$\begin{aligned} \varepsilon_{x_j}(\xi_{x_j})=\ln \Big(\frac{1+\xi_{x_j}}{1-\xi_{x_j}}\Big),~j=\{1,\ldots,6\}\label{eq12}\end{aligned}$$ Now, the reference velocity as $\dot{{\boldsymbol}{x}}^r_e=[\dot{x}^r_{e_1},\ldots,\dot{x}^r_{e_6}]^\top$ is designed as: $$\begin{aligned} \dot{x}^r_{e_j}(t)=-k_{x_j}\varepsilon_{x_j}(\xi_{x_j}),~k_j>0,~j=\{1,\ldots,6\}\label{eq13}\end{aligned}$$ The task-space desired motion profile $\dot{{\boldsymbol}{x}}^r_e$ can be extended to the joint level using the kinematic equation : $${{\boldsymbol}{\zeta}}^r(t)={\boldsymbol}{J}({\boldsymbol}{q})^{+}\dot{{\boldsymbol}{x}}^r_e \label{eq9}$$ where ${\boldsymbol}{J}({\boldsymbol}{q})^{+}$ denotes the Moore-Penrose pseudo-inverse of Jacobian ${\boldsymbol}{J}({\boldsymbol}{q})$. It is worth mentioning that the $\dot{{\boldsymbol}{x}}^r_e$ can also be extended to the joint level via: $$\begin{aligned} {{\boldsymbol}{\zeta}}^r(t)={\boldsymbol}{J}({\boldsymbol}{q})^{\#}\dot{{\boldsymbol}{x}}^r_e+\big({\boldsymbol}{I}_{n\times n}\!-\!{\boldsymbol}{J}({\boldsymbol}{q})^{\#}{\boldsymbol}{J}\big({\boldsymbol}{q}\big)\big)\dot{{\boldsymbol}{x}}^0 \end{aligned}$$ where ${\boldsymbol}{J}({\boldsymbol}{q})^{\#}$ denotes the generalized pseudo-inverse [@citeulike:6536020] of Jacobian ${\boldsymbol}{J}({\boldsymbol}{q})$ and $\dot{{\boldsymbol}{x}}^0$ denotes secondary tasks which can be regulated independently to achieve secondary goals (e.g., maintaining manipulator’s joint limits, increasing of manipulability) and does not contribute to the end effector’s velocity [@Simetti2016877]. **Step II-a**: Define the velocity error vector at the end-effector frame as: $$\begin{aligned} {\boldsymbol}{e}_\zeta(t)=[{e}_{\zeta_1}(t),\ldots,{e}_{\zeta_n}(t)]^\top= {{{\boldsymbol}{\zeta}}}(t)- {{{\boldsymbol}{\zeta}}}^r(t) \label{eq14}\end{aligned}$$ and select the corresponding functions $\rho_{\zeta_j}(t)=(\rho^0_{\zeta_j}-\rho_{\zeta_j}^\infty)e^{-l_{\zeta_j}t}+\rho_{\zeta_j}^\infty$ with $\rho^0_{\zeta_j}>|e_{\zeta_j}(t_0)|,\forall j\in\{1\ldots,n\}$, $\rho^0_{\zeta_j}>\rho^\infty_{\zeta_j}>0$, $ l_{\zeta_j}>0,\forall j\in\{1,\ldots n\}$, and define the normalized velocity errors $\boldsymbol{\xi}_\zeta$ as: $$\begin{aligned} \boldsymbol{\xi}_{\zeta}(t)=[\xi_{\zeta_1},\ldots,\xi_{\zeta_n}]^\top={\boldsymbol}{P}^{-1}_\zeta(t){\boldsymbol}{e}_\zeta(t)\label{eq15}\end{aligned}$$ where ${\boldsymbol}{P}_\zeta(t)=\text{diag}\{\rho_{\zeta_j}\},j\in\{1,\ldots,n\}$.\ **Step II-b**: Define the transformed errors $\boldsymbol{\varepsilon}_{\zeta}(\boldsymbol{\xi}_{\zeta})=[\varepsilon_{\zeta_1}(\xi_{\zeta_1}),\ldots,\varepsilon_{\zeta_n}(\xi_{\zeta_n})]^\top$ and the signal ${\boldsymbol}{R}_{\zeta}(\boldsymbol{\xi}_{\zeta})=\text{diag}\{r_{\zeta_j}\}$, $~j\in\{1,\ldots,n\}$ as: $$\begin{aligned} \label{eq16} &\boldsymbol{\varepsilon}_{\zeta}(\boldsymbol{\xi}_{\zeta})=\Big[\ln \Big(\frac{1+\xi_{\zeta_1}}{1-\xi_{\zeta_1}}\Big),\ldots,\ln \Big(\frac{1+\xi_{\zeta_n}}{1-\xi_{\zeta_n}}\Big)\Big]^\top\\ & {\boldsymbol}{R}_{\zeta}(\boldsymbol{\xi}_{\zeta})\!=\!\text{diag}\{r_{\zeta_j}\!(\xi_{\zeta_j}\!)\}\!=\!\text{diag}\!\Big\{\!\frac{2}{1-\xi_{\zeta_j}^2\!}\Big\},j\!=\!\{1,\ldots,n\}\label{eq17}\end{aligned}$$ and finally design the state feedback control law $\tau_j,~j\in\{1,\ldots,n\}$ as: $$\begin{aligned} \tau_j\!(\xi_{x_j}\!,\xi_{\zeta_j}\!,t)=-k_{\zeta_j}\frac{r_{\zeta_j}(\xi_{\zeta_j})\varepsilon_{\zeta_j}(\xi_{\zeta_j})}{\rho_{\zeta_j}(t)},~j=\{1,\ldots,n\}\label{eq18}\end{aligned}$$ where $k_{\zeta_j}$ to be a positive gain. The control law can be written in vector form as: $$\begin{aligned} {\boldsymbol}{\tau}({\boldsymbol}{e}_x(t),{\boldsymbol}{e}_\zeta(t),t)&=[ \tau_1(\xi_{x_1},\xi_{\zeta_1},t),\ldots, \tau_n(\xi_{x_n},\xi_{\zeta_n},t)]^\top\nonumber\\ &-{\boldsymbol}{K}_\zeta{\boldsymbol}{P}^{-1}(t){\boldsymbol}{R}_{\zeta}(\boldsymbol{\xi}_{\zeta})\boldsymbol{\varepsilon}_{\zeta}(\boldsymbol{\xi}_{\zeta})\label{eq19}\end{aligned}$$ with ${\boldsymbol}{K}_\zeta$ to be the diagonal matrix containing $k_{\zeta_j}$. Now we are ready to state the main theorem of the paper: Given the error defined in and the required transient and steady state performance specifications, select the exponentially decaying performance function $\rho_{x_j}(t)$, $\rho_{\zeta_j}(t)$ such that the desired performance specifications are met. Then the state feedback control law of guarantees tracking of the trajectory ${\boldsymbol}{f}_e^d(t)\in \mathbb{R}^3$ as well as ${^o{\boldsymbol}{x}}^d_e(t)$: $$\begin{aligned} \lim_{t\rightarrow\infty}{\boldsymbol}{f}_e(t)={\boldsymbol}{f}^d_e(t)~\text{and}\lim_{t\rightarrow\infty}{^o{\boldsymbol}{x}_e(t)}={^o{\boldsymbol}{x}^d_e(t)}\end{aligned}$$ with the desired transient and steady state performance specifications. For the proof we follow parts of the approach in [@Bechlioulis20141217]. We start by differentiating and with respect to the time and substituting the system dynamics as well as and and employing and , obtaining: $$\begin{aligned} \dot{\xi}_{x_j}(\xi_{x_j},t)&=h_{x_j}(\xi_{x_j},t)\nonumber\\ &=\rho^{-1}_{x_j}(t)(\dot{e}_{x_j}(t)-\dot{\rho}_{x_j}\!(t)\xi_{x_j} )\nonumber\\ &=\rho^{-1}_{x_j}(t)(-k_{x_j}\varepsilon_{x_j}(\xi_{x_j})+{\boldsymbol}{J}_{(j,:)}{\boldsymbol}{P}_\zeta{\boldsymbol}{\xi}_\zeta-\dot{x}^d_{e_j}(t))\nonumber\\ &-\rho^{-1}_{x_j}(t)(\dot{\rho}_{x_j}\!(t)\xi_{x_j}), \forall j\in\{1,\ldots,6\}\label{eq21}\end{aligned}$$ $$\begin{aligned} \dot{\boldsymbol{{\xi}}}_{\zeta}(\xi_{\zeta},t&)=h_{\zeta}({\boldsymbol}{\xi}_{\zeta},t)\nonumber\\ & = {\boldsymbol}{P}_\zeta^{-1}(\dot{{\boldsymbol}{\zeta}}-\dot{{\boldsymbol}{\zeta}}^r)-\dot{{\boldsymbol}{P}}_\zeta^{-1}{\boldsymbol{\xi}}_\zeta ) \nonumber\\ & = -{\boldsymbol}{K}_\zeta{\boldsymbol}{P}_\zeta^{-1}{\boldsymbol}{M}^{-1}{\boldsymbol}{P}_\zeta^{-1}{\boldsymbol}{R}_\zeta\boldsymbol{\varepsilon}_{\zeta}-\nonumber\\ &-{\boldsymbol}{P}_\zeta^{-1}\Big[{\boldsymbol}{M}^{-1}\Big({\boldsymbol}{C}\cdot({\boldsymbol}{P}_\zeta\boldsymbol{\xi}_\zeta+{{\boldsymbol}{\zeta}}^r)+{\boldsymbol}{D}\cdot({\boldsymbol}{P}_\zeta\boldsymbol{\xi}_\zeta+{{\boldsymbol}{\zeta}}^r)\nonumber\\ &+{\boldsymbol}{g}+{{\boldsymbol}{J}^g}^\top\boldsymbol{\lambda}+\!\boldsymbol{\delta}(t)\Big)+\dot{{\boldsymbol}{P}}_\zeta\boldsymbol{\xi}_\zeta+\frac{\partial}{\partial t}{{\boldsymbol}{\zeta}}^r\Big]\label{eq22}\end{aligned}$$ where ${\boldsymbol}{J}_{(j,:)}$ denotes all elements of jacobian ${\boldsymbol}{J}$ at its $j$ row. Now let us to define the vectors of normalized state error and the generalized normalized error as $\boldsymbol{\xi}_{x}=[{\xi}_{x_1},\ldots,{\xi}_{x_6}]^\top\!$, and $\boldsymbol{\xi}=[\boldsymbol{\xi}_x^\top,\boldsymbol{\xi}_\zeta^\top]^\top$, respectively. Moreover, let us define $\dot{{\boldsymbol}{\xi}}_x=h_x({\boldsymbol}{\xi_x},t)=[h_{x_1}(\xi_{x_1},t),\ldots,h_{x_6}(\xi_{x_6},t)]^\top$. The equations of and now can be written in compact form as: $$\begin{aligned} \dot{\boldsymbol{\xi}}=h(\boldsymbol{\xi},t)= [h_x^\top({\boldsymbol}{\xi_x},t),h_\zeta^\top({\boldsymbol}{\xi_\zeta},t)]\top\label{eq23}\end{aligned}$$ Let us define the open set $\Omega_\xi=\Omega_{\xi_x}\times\Omega_{\xi_\zeta}$ with $\Omega_{\xi_x}=(-1,1)^6$ and $\Omega_{\xi_\zeta}=(-1,1)^n$. In what follows, we proceed in two phases. First we ensure the existence of a unique maximal solution $\boldsymbol{\xi}(t)$ of over the set $\Omega_\xi$ for a time interval $[0,t_{\text{max}}]$ (i.e., $\boldsymbol{\xi}(t)\in\Omega_\xi, \forall t\in[0,t_{\text{max}}]$). Then, we prove that the proposed controller guarantees, for all $t\in[0,t_{\text{max}}]$ the boundedness of all closed loop signal of as well as that $\boldsymbol{\xi}(t)$ remains strictly within the set $\Omega_\xi$, which leads that $t_{\text{max}}=\infty$ completes the proof. **Phase A**: The set $\Omega_\xi$ is nonempty and open, thus by selecting $\rho^0_{x_j}>|e_{x_j}(0)|$, $\forall j\in\{1,\ldots 6\}$ and $\rho^0_{v_j}>|e_{v_j}(0)|$, $\forall j\in\{1,\ldots n\}$ we guarantee that $\boldsymbol{\xi}_x(0)\in\Omega_{\xi_x}$ and $\boldsymbol{\xi}_\zeta(0)\in\Omega_{\xi_\zeta}$. Additionally, $h$ is continuous on $t$ and locally Lipschitz on $\boldsymbol{\xi}$ over $\Omega_\xi$. Therefore, the hypotheses of Theorem\[thm:dynamical systems\] hold and the existence of a maximal solution $\boldsymbol{\xi}(t)$ of on a time interval $[0,t_{\text{max}}]$ such that $\boldsymbol{\xi}(t) \in \Omega_\xi,~\forall t\in[0,t_{\text{max}}]$ is ensured. at (-13.0, 2.87) [$\begin{bmatrix}f_e^d(t)\\ ^o{x}_e^d(t) \end{bmatrix}$]{}; (-10.80, 0.5) rectangle +(7.2, 3.2); at (-7.2, 0.9) [$\text{Proposed Control Algorithm}$]{}; (-12.0,2.20) circle (0.25cm); at (-11.25,2.50) [$e_x(t)$]{}; (-11.75, 2.2) to (-10.5, 2.2); (-13.1, 2.2) to (-12.25, 2.2); (-10.5, 1.58) rectangle +(2.2, 1.3); at (-9.4,2.60) [$\text{first level}$]{}; at (-9.4,1.95) [${{\boldsymbol}{\zeta}}^r\!(\xi_{x_j}\!,t)$]{}; (-10.5, 2.30) – (-8.3,2.30); (-8.3, 2.2) to (-6.0, 2.2); at (-7.2,2.50) [${{\boldsymbol}{\zeta}}^r\!(\xi_{x_j}\!,t)$]{}; (-6.0, 1.58) rectangle +(2.2, 1.3); at (-4.9,2.60) [$\text{second level}$]{}; at (-4.9,1.95) [${\boldsymbol}{\tau}({\boldsymbol}{e}_x,{\boldsymbol}{e}_\zeta,t)$]{}; (-6.00, 2.30) – (-3.8,2.30); (-3.8, 2.2) to (-1.8, 2.2); (-1.8, 1.58) rectangle +(1.80, 1.3); at (-0.9,2.25) [$\text{UVMS}$]{}; (-1.0, 3.7) to (-1.0, 2.90); at (-1.0,4.38) [$\text{external disturbance}$]{}; at (-1.0,3.97) [$\delta(t)$]{}; (0.0, 2.2) to (2.0, 2.2); (2.0, 1.58) rectangle +(2.2, 1.3); at (3.1,2.25) [$\text{Environment}$]{}; (-5.0, -2.00) rectangle +(2.20, 1.3); at (-3.9,-1.33) [$\text{force sensor}$]{}; (-4.0, -2.8) to (-4.0, -2.0); at (-4.0,-3.00) [$\text{noise}$]{}; (-12.03,-1.35) – (-5.0,-1.350); (-12.0, -1.35) to (-12.0, 1.95); (3.335,-1.350) to (-2.8,-1.35); (3.3,-1.35) – (3.3,1.6); at (-7.8,-1.70) [$f_e(t)+\Delta f_e(t)$]{}; **Phase B**: In the Phase A we have proven that $\boldsymbol{\xi}(t) \in \Omega_\xi,~\forall t\in[0,t_{\text{max}}]$, thus it can be concluded that: \[eq24\] $$\begin{gathered} \xi_{x_j}(t)=\frac{e_{x_j}}{\rho_{x_j}}\in(-1,1),~ \forall j\{1,\ldots,6\} \\ \xi_{\zeta_j}(t)=\frac{e_{\zeta_j}}{\rho_{\zeta_j}}\in(-1,1),~ \forall j\{1,\ldots,n\}\end{gathered}$$ for all $t\in[0,t_{\text{max}}]$, from which we obtain that $e_{x_j}(t)$ and $e_{\zeta_j}(t)$ are absolutely bounded by $\rho_{x_j}$ and $\rho_{\zeta_j}$, respectively. Therefore, the error vectors $\varepsilon_{x_j}(\xi_{x_j}),\forall j\in\{1,\ldots,6\}$ and $\varepsilon_{\zeta_j}(\xi_{\zeta_j}),\forall j\in\{1,\ldots,n\}$ defined in and , respectively, are well defined for all $t\in [0,t_{\text{max}}]$. Hence, consider the positive definite and radially unbounded functions $V_{x_j}(\varepsilon_{x_j})=\varepsilon^2_{x_j},~\forall j\{1,\ldots,6\}$. Differentiating of $V_{x_j}$ w.r.t time and substituting , results in: $$\begin{aligned} \dot{V}_{x_j}\!=-\frac{4\varepsilon_{x_j}\rho^{-1}_{x_j}}{(1-\xi^2_{x_j})}\!\Big(k_{x_j}\varepsilon_{x_j}(\xi_{x_j})\!+\!\dot{x}^d_{e_j}\!+\!\dot{\rho}_{x_j}\!(t)\xi_{x_j}\!-\!{\boldsymbol}{J}_{(j,:)}{\boldsymbol}{P}_\zeta{\boldsymbol}{\xi}_\zeta\! \Big)\label{eq25}\end{aligned}$$ It is well known that the Jacobian ${\boldsymbol}{J}$ is depended only on bounded vehicle’s orientation and angular position of manipulator’s joint. Moreover, since, $\dot{x}^d_{e_j}$, $\rho_{x_j}$ and $\rho_{v_j}$ are bounded by construction and $\xi_{x_j}$,$\xi_{v_j}$ are also bounded in $(-1,1)$, owing to , $\dot{V}_{x_j}$ becomes: $$\begin{aligned} \dot{V}_{x_j} \leq\frac{4\rho^{-1}_{x_j}}{(1-\xi^2_{x_j})}\!\Big(B_x |\varepsilon_{x_j}| -k_{x_j}|\varepsilon_{x_j}|^2\Big)\label{eq26}\end{aligned}$$ $\forall t\in [0,t_{\text{max}}]$, where $B_x$ is an unknown positive constant independent of $t_{\text{max}}$ satisfying $B_x>|\dot{x}^d_{e_j}\!+\!\dot{\rho}_{x_j}\!(t)\xi_{x_j}\!-\!{\boldsymbol}{J}_{(j,:)}{\boldsymbol}{P}_\zeta{\boldsymbol}{\xi}_\zeta|$. Therefore, we conclude that $\dot{V}_{x_j}$ is negative when $\varepsilon_{x_j}>\frac{B_x}{k_{j_x}}$ and subsequently that $$\begin{aligned} |\varepsilon_{x_j}(\xi_{x_j}(t)) |\leq \bar{\varepsilon}_{x_j}=\max\{\varepsilon_{x_j}(\xi_{x_j}(0)),\frac{B_x}{k_{j_x}}\}\label{eq27}\end{aligned}$$ $\forall t\in [0,t_{\text{max}}], \forall j\{1,\ldots,6\}$. Furthermore, from , taking the inverse logarithm, we obtain: $$\begin{aligned} -1<\frac{e^{-\bar{\varepsilon}_{x_j}}-1 }{e^{-\bar{\varepsilon}_{x_j}}+1}=\underline{\xi}_{x_j} \leq \xi_{x_j}(t)\leq\bar{\xi}_{x_j} =\frac{e^{\bar{\varepsilon}_{x_j}}-1 }{e^{\bar{\varepsilon}_{x_j}}+1}<1 \label{eq28}\end{aligned}$$ $\forall t\in [0,t_{\text{max}}],~j\in\{1,\ldots,6\}$. Due to , the reference velocity vector $\dot{{\boldsymbol}{x}}^r_e$ as defined in , remains bounded for all $t\in[0,t_{\text{max}}]$. Moreover, invoking $\dot{{\boldsymbol}{x}}_e={\dot{{\boldsymbol}{x}}}^r_e(t)+{\boldsymbol}{P}_v(t){\boldsymbol}{\xi}_v$ from , and , we also conclude the boundedness of $\dot{{\boldsymbol}{x}}_e$ for all $t\in [0,t_{\text{max}}]$. Finally, differentiating ${\dot{{\boldsymbol}{x}}}^r_e(t)$ w.r.t time and employing , and , we conclude the boundedness of $\frac{\partial}{\partial t}{\dot{{\boldsymbol}{x}}}^r_e(t)$, $\forall t\in [0,t_{\text{max}}]$. Applying the aforementioned line of proof, we consider the positive definite and radially unbounded function $V_\zeta({\boldsymbol}{\varepsilon}_\zeta)=\frac{1}{2}||{\boldsymbol}{\varepsilon}_\zeta||^2$. By differentiating $V_\zeta$ with respect to time, substituting and by employing continuity of ${\boldsymbol}{M}$, ${\boldsymbol}{C}$, ${\boldsymbol}{D}$, ${\boldsymbol}{g}$, $\boldsymbol{\lambda}$, $\boldsymbol{\delta}$, $\boldsymbol{\xi}_x$, $\boldsymbol{\xi}_\zeta$,$\dot{{\boldsymbol}{P}}_\zeta$, $\frac{\partial}{\partial t}{{{\boldsymbol}{\zeta}}}^r$, $\forall t\in [0,t_{\text{max}}]$, we obtain: $$\begin{aligned} \dot{V}_\zeta\leq ||{\boldsymbol}{P}^{-1}_\zeta{\boldsymbol}{R}_\zeta(\boldsymbol{\xi}_\zeta)\boldsymbol{\varepsilon}_\zeta||\Big(B_\zeta-{\boldsymbol}{K}_\zeta\lambda_M||{\boldsymbol}{P}^{-1}_\zeta{\boldsymbol}{R}_\zeta(\boldsymbol{\xi}_\zeta) \boldsymbol{\varepsilon}_\zeta|| \Big)\end{aligned}$$ $\forall t\in [0,t_{\text{max}}]$, where $\lambda_M$ is the minimum singular value of the positive definite matrix ${\boldsymbol}{M}^{-1}$ and $B_v$ is a positive constant independent of $t_{\text{max}}$, satisfying $$\begin{aligned} B_\zeta\geq &|| {\boldsymbol}{M}^{-1}\Big( {\boldsymbol}{C}\cdot({\boldsymbol}{P}_\zeta\boldsymbol{\xi}_\zeta +{{\boldsymbol}{\zeta}}^r(t)) + {\boldsymbol}{D}\cdot({\boldsymbol}{P}_\zeta\boldsymbol{\xi}_\zeta +{{\boldsymbol}{\zeta}}^r(t)) \\ &+{\boldsymbol}{g}+{{\boldsymbol}{J}^g}^\top\boldsymbol{\lambda}+\boldsymbol{\delta}(t)+ \dot{{\boldsymbol}{P}}_\zeta\boldsymbol{\xi}_\zeta+\frac{\partial}{\partial t}{{{\boldsymbol}{\zeta}}}^r \Big) ||\end{aligned}$$ Thus, $\dot{V}_\zeta$ is negative when $||{\boldsymbol}{P}^{-1}_\zeta{\boldsymbol}{R}_\zeta(\boldsymbol{\xi}_\zeta) \boldsymbol{\varepsilon}_\zeta|| >B_\zeta({\boldsymbol}{K}_\zeta\lambda_M)^{-1}$, which by employing the definitions of ${\boldsymbol}{P}_\zeta$ and ${\boldsymbol}{R}_\zeta$, becomes $||\boldsymbol{\varepsilon}_\zeta ||> B_\zeta({\boldsymbol}{K}_\zeta\lambda_M)^{-1}\max\{\rho^0_{\zeta_1},\ldots,\rho^0_{\zeta_n}\} $. Therefore, we conclude that: $$\begin{aligned} ||\boldsymbol{\varepsilon}_\zeta (\boldsymbol{\xi}_\zeta\!(\!t\!))\!\leq\boldsymbol{\bar{\varepsilon}}_\zeta=\!\max\!\Big\{\!\boldsymbol{\varepsilon}_\zeta (\boldsymbol{\xi}_\zeta(0)),\! B_\zeta(\!{\boldsymbol}{K}_\zeta\lambda_M)^{-1}\cdot\!\max\{\rho^0_{\zeta_1},\ldots,\rho^0_{\zeta_n}\}\! \Big\}\end{aligned}$$ $\forall t\in [0,t_{\text{max}}]$. Furthermore, from , invoking that $|\varepsilon_{\zeta_j}|\leq || \boldsymbol{\varepsilon}_\zeta||$, we obtain: $$\begin{aligned} -1<\frac{e^{-\bar{\varepsilon}_{\zeta_j}}-1 }{e^{-\bar{\varepsilon}_{\zeta_j}}+1}=\underline{\xi}_{\zeta_j} \leq \xi_{\zeta_j}(t)\leq\bar{\xi}_{\zeta_j} =\frac{e^{\bar{\varepsilon}_{\zeta_j}}-1 }{e^{\bar{\varepsilon}_{\zeta_j}}+1}<1 \label{eq29}\end{aligned}$$ $\forall t\in [0,t_{\text{max}}],~j\in\{1,\ldots,n\}$ which also leads to the boundedness of the control law . Now, we will show that the $t_{\text{max}}$ can be extended to $\infty$. Obviously, notice by and that $\boldsymbol{\xi}(t)\in\Omega^{'}_\xi=\Omega^{'}_{\xi_x}\times \Omega^{'}_{\xi_\zeta},\forall t\in [0,t_{\text{max}}]$, where: $$\begin{aligned} \Omega^{'}_{\xi_x}=[\underline{\xi}_{x_1},\bar{\xi}_{x_1}]\times\ldots,\times[\underline{\xi}_{x_6},\bar{\xi}_{x_6} ],\\ \Omega^{'}_{\xi_\zeta}=[\underline{\xi}_{\zeta_1},\bar{\xi}_{\zeta_1}]\times\ldots,\times[\underline{\xi}_{\zeta_n},\bar{\xi}_{\zeta_n} ],\end{aligned}$$ are nonempty and compact subsets of $\Omega_{\xi_x}$ and $\Omega_{\xi_\zeta}$, respectively. Hence, assuming that $t_{\text{max}}<\infty$ and since $\Omega_\xi\subset \Omega^{'}_\xi$, Proposition 1, dictates the existence of a time instant $t{^{'}}\in \forall t\in [0,t_{\text{max}}]$ such that $\boldsymbol{\xi}(t^{'})\notin \Omega^{'}_\xi$, which is a clear contradiction. Therefore, $t_{\text{max}}=\infty$. Thus, all closed loop signals remain bounded and moreover $\boldsymbol{\xi}(t)\in \Omega^{'}_\xi,\forall t\geq0$. Finally, from and we conclude that: $$\begin{aligned} -\rho_{x_j}<\frac{e^{-\bar{\varepsilon}_{x_j}}-1 }{e^{-\bar{\varepsilon}_{x_j}}+1}\rho_{x_j} \leq e_{x_j}(t)\leq \rho_{x_j}\frac{e^{\bar{\varepsilon}_{x_j}}-1 }{e^{\bar{\varepsilon}_{x_j}}+1}<\rho_{x_j} \label{eq30}\end{aligned}$$ for $j\in\{1,\ldots,6\}$ and for all $t\geq 0$ and consequently, completes the proof. From the aforementioned proof, it is worth noticing that the proposed control scheme is model free with respect to the matrices ${\boldsymbol}{M}$, ${\boldsymbol}{C}$, ${\boldsymbol}{D}$, ${\boldsymbol}{g}$ as well as the external disturbances $\boldsymbol{\delta}$ that affect only the size of $\bar{\varepsilon}_{x_j}$ and of $\bar{\varepsilon}_{\zeta_j}$, but leave unaltered the achieved convergence properties as dictates. In fact, the actual transient and steady state performance is determined by the selection of the performance function $\rho_{c_j},c\in \{x,\zeta\}$. Finally the closed loop block diagram of the proposed control scheme is indicated in Fig.\[fig:closed\_loop\_control\_scheme\]. Simulation Results ================== Simulation studies were conducted employing a hydrodynamic simulator built in MATLAB. The dynamic equation of UVMS used in this simulator is derived following [@Schjølberg94modellingand]. The UVMS model considered in the simulations is the SeaBotix LBV150 equipped with a small 4 DoF manipulator. We consider a scenario involving 3D motion in workspace, where the end-effector of the UVMS is in interaction on a compliant environment with stiffens matrix ${\boldsymbol}{K}_f=\text{diag}\{2\}$ which is unknown for the controller. The workspace at the initial time, including UVMS and the compliant environment are depicted in Fig.\[fig:workspace\]. More specifically, we adopt: ${\boldsymbol}{f}_e(0)=[0,0,0]^\top$ and $^o{\boldsymbol}{x}_e=[0.2,0.2,-0.2]^\top$. It means that at the initial time of the simulation study we assume that the uvms has attached the compliant environment with a rotation at its end-effector frame. The control gains for the two set of the simulation studies were selected as follows: $k_{x_j}=-0.2 j\in\{1,\ldots,6\}$, $k_{v_j}=-5 j\in\{1,\ldots,n\}$. Moreover, the dynamic parameters of UVMSs as well as the stiffens matrix ${\boldsymbol}{K}_f$ were considered unknown for the controller. The parameters of the performance functions in sequel stimulation studies were chosen as follows:$\rho^0_{x_j}=1,~j\in\{1,2,3\}$, $\rho^0_{x_j}=0.9,~j\in\{4,5,6\}$, $\rho^0_{v_j}=1,~j\in\{1,\ldots,n\}$, $\rho^\infty_{x_j}=0.2~j\in\{1,\ldots,6\}$, $\rho^\infty_{v_j}=0.2~j\in\{1,\ldots,6\}$, $\rho^\infty_{v_j}=0.4~j\in\{7,\ldots,n\}$, $l_{x_j}=3~j\in\{1,2,3\}$, $l_{v_j}=2.2~j\in\{1,\ldots,n\}$. Finally, the whole system was running under the influence of external disturbances (e.g., sea current) acting along $x$, $y$ and $z$ axes (on the vehicle body) given by $0.15\sin(\frac{2\pi}{7}t)$, in order to test the robustness of the proposed scheme. Moreover, bounded noise on measurement devices were considered during the simulation study. In the the presented simulation scenario, a desired force trajectory should be exerted to the environment while predefined orientation $^o{\boldsymbol}{x}^d_e=[0.0,0.0,0.0]^\top$ must be kept. One should bear in mind that this is a challenging task owing to dynamic nature of the underwater environment. The UVMS’s model uncertainties, noise of measurement devices as well as external disturbances in this case can easily cause unpredicted instabilities to the system. The desired force trajectory which should be exerted by UVMS is defined as $f^d_{e_1}=0.4\sin(\frac{2\pi}{2}t)+0.4$. The results are depicted in Figs 3-5. Fig\[fig:forc\] show the evolution of the force trajectory. Obviously, the actual force exerted by the UVMS (indicated by red color) converges to the desired one (indicated by green color) without overshooting and follows the desired force profile. The evolution of the errors at the first and second level of the proposed controller are indicated in Fig.\[fig:ppx\_force\] and Fig.\[fig:ppv\_force\], respectively. It can be concluded that even with the influence of external disturbances as well as noise in measurements, the errors in all directions converge close to zero and remain bounded by the performance functions. Conclusions and Future Work =========================== This work presents a robust force/position control scheme for a UVMS in interaction with a compliant environment, which could have direct applications in the underwater robotics (e.g. sampling of the sea organisms, underwater welding, pushing a button). Our proposed control scheme does not required any priori knowledge of the UVMS dynamical parameters as well as environment model. It guarantees a predefined behavior of the system in terms of desired overshot and transient and steady state performance. Moreover, the proposed control scheme is robust with respect to the external disturbances and measurement noises. The proposed controller of this work exhibits the following important characteristics: i) it is of low complexity and thus it can be used effectively in most of today UVMS. ii) The performance of the proposed scheme (e.g. desired overshot, steady state performance of the systems) is a priori and explicitly imposed by certain designer-specified performance functions, and is fully decoupled by the control gains selection, thus simplifying the control design. The simulations results demonstrated the efficiency of the proposed control scheme. Finally, future research efforts will be devoted towards addressing the torque controller as well as conducting experiments with a real UVMS system. [^1]: Shahab Heshmati-Alamdari and Kostas J. Kyriakopoulos are with the Control Systems Lab, Department of Mechanical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou 15780, Greece. Email: [{shahab, [email protected]}]{}. Alexandros Nikou is with the ACCESS Linnaeus Center, School of Electrical Engineering, KTH Royal Institute of Technology, SE-100 44, Stockholm, Sweden and with the KTH Centre for Autonomous Systems. Email: [{anikou}@kth.se]{}.

ArXiv

--- abstract: | The portfolio are a critical factor not only in risk analysis, but also in insurance and financial applications. In this paper, we consider a special class of risk statistics from the perspective of regulator. This new risk statistic can be uesd for the quantification of portfolio risk. By further developing the properties related to regulator-based risk statistics, we are able to derive dual representation for them. Finally, examples are also given to demonstrate the application of this risk statistic.\ author: - Xiaochuan Deng - Fei Sun title: 'Regulator-based risk statistics for portfolios' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore **Introduction** ================ Research on risk is a hot topic in both quantitative and theoretical research, and risk models have attracted considerable attention. The quantitative calculation of risk involves two problems: choosing an appropriate risk model and allocating the risk to individual institutions. This has led to further research on risk statistics.\ In their seminal paper, Burgert and Rüschendorf (2006) firstly introduced the concepts of the scalar multivariate coherent and convex risk measures, see also Rüschendorf (2013). However, the traditional risk statistics failed to capture sufficiently of the regulator-based risk. Namely, the regulators almost only focus on the loss of investment rather than revenue. Especially, the axiom of translative invariance in coherent and convex risk statistics will definitely fail when we only deal with the regulator-based risk. Thus, the study of regulator-based risk statistics is particularly interesting.\ Evaluating the risk of a portfolio consisting of several financial positions, Jouini et al. (2004) pointed out that a set-valued risk measure is more appropriate than a scalar risk measure, especially in the case where several different kinds of currencies are involved when one is determining capital requirements for the portfolio. They first studied the class of set-valued coherent risk measures by proposing some axioms. Hamel (2009) introduced set-valued convex risk measures by an axiomatic approach. For more studies on set-valued risk measures, see Hamel and Heyde (2010), Hamel et al. (2011), Hamel et al. (2013), Labuschagne and Offwood-Le Roux (2014), Ararat et al. (2014), Farkas et al. (2015), Molchanov and Cascos (2016), and the references therein. A natural set-valued risk statistic can be considered as an empirical (or a data-based) version of a set-valued risk measure.\ From the statistical point of view, the behaviour of a random variable can be characterized by its observations, the samples of the random variable. Heyde, Kou and Peng (2007) and Kou, Peng and Heyde (2013) first introduced the class of natural risk statistics, the corresponding representation results are also derived. An alternative proof of the representation result of the natural risk statistics was also derived by Ahmed, Filipovic and Svindland (2008). Later, Tian and Suo (2012) obtained representation results for convex risk statistics, and the corresponding results for quasiconvex risk statistics were obtained by Tian and Jiang (2015). However, all of these risk statistics are designed to quantify the risk of a single financial position (i.e. a random variable) by its samples. A natural question is how to quantify the risk of a portfolio by its samples, especially in the situation where different kinds of currencies are possibly involved in the portfolio.\ The main focus of this paper is a new class of risk statistics for portfolios, named regulator-based risk statistics by an axiomatic approach. By further developing the properties related to regulator-based risk statistics, we are able to derive dual representation for them. This new class of risk statistics can be considered as an extension of those introduced by Cont et al. (2013), Chen et al. (2018) and Sun et al.(2018) from scalar and multivariate risk setting to set-valued multivariate risks setting. Finally, examples are also given to illustrate this new class of risk statistics.\ The remainder of this paper is organized as follows. In Section 2, we briefly introduce the preliminaries. In Section 3, we state the main result of regulator-based risk statistic, including the representation result. In Section 4, we investigate the alternative regulator-based risk statistics. Section 5 discusses the main proof in this paper. Finally, in Section 6, examples of regulator-based risk statistic are also given. ***Preliminaries*** =================== In this section, we briefly introduce the preliminaries that are used throughout this paper. Let $d \geq 1$ be a fixed positive integer. The space $\mathbb{R}^{d\times n}$ represents the set of financial risk positions. With positive values of $X\in \mathbb{R}^{d\times n}$ we denote the gains while the negative denote the losses. The behavior of the $d$-dimensional random vector $ D =(X_1, \cdots, X_d)$ under different scenarios is represented by different sets of data observed or generated under those scenarios because specifying accurate models for $D$ is usually very difficult. Here, we suppose that there always exist $l$ scenarios. Let $n_j$ be the sample size of $D$ in the $j^{th}$ scenario, $j=1, \cdots, l.$ Let $n:= n_1 +\cdots+n_l$. More precisely, suppose that the behavior of $D$ is represented by a collection of data $X=(X_1, \cdots, X_d) \in \mathbb{R}^n \times \cdots \times \mathbb{R}^n$, where $X_i=(X^{i, 1}, \cdots, X^{i, l}) \in \mathbb{R}^n$, $X^{i, j} =(x^{i, j}_1, \cdots, x^{i, j}_{n_j}) \in \mathbb{R}^{n_j}$ is the data subset that corresponds to the $j^{th}$ scenario with respect to $X_i$. For each $j=1, \cdots, l$, $h=1, \cdots, n_j$, $X^j_h:=\left(x^{1, j}_h, x^{2, j}_h, \cdots, x^{d, j}_h\right)$ is the data subset that corresponds to the $h^{th}$ observation of $D$ in the $j^{th}$ scenario, and can be based on historical observations, hypothetical samples simulated according to a model, or a mixture of observations and simulated samples.\ In this paper, an element $z$ of $\mathbb{R}^d$ is denoted by $ z:=(z^1, \cdots, z^d).$ An element $X$ of $\mathbb{R}^{d \times n}$ is denoted by $$\begin{aligned} X:=(X_1, \cdots, X_d):=\Big(x^{1, 1}_1, \cdots, x^{1, 1}_{n_1}, \cdots, x^{1, l}_1, \cdots, x^{1, l}_{n_l}, \cdots, x^{d, 1}_1, \cdots x^{d, 1}_{n_1}, \cdots, x^{d, l}_1, \cdots, x^{d, l}_{n_l}\Big).\end{aligned}$$ The $d\times n$ dimensional financial positions in $\mathbb{R}^{d\times n}$ have a strong realistic interpretation. This is indeed the case if we consider realistic situations where investors have access to different markets and form multi-asset portfolios in the presence of frictions such as transaction costs, liquidity problems, irreversible transfers, etc.\ Let $K$ be a closed convex polyhedral cone of $\mathbb{R}^{d}$ with $K\supseteq \mathbb{R}^{d}_{++}:=\{(x_{1},\ldots,x_{d})\in \mathbb{R}^{d}; x_{i}>0, 1\leq i\leq d\}$ and $K\cap \mathbb{R}^{d}_{-} = \emptyset$ where $\mathbb{R}^{d}_{-}:=\{(x_{1},\ldots,x_{d})\in \mathbb{R}^{d}; x_{i}\leq 0, 1\leq i\leq d\}$. Let $K^{+}$ be the positive dual cone of $K$, that is $K^{+}:=\{u\in \mathbb{R}^{d}:u^{tr} v\geq0 \textrm{ for any } v\in K\}$, where $u^{tr}$ means the transpose of $u$. For any $X = (X_{1}, \ldots, X_{d}), Y = (Y_{1}, \ldots, Y_{d})\in \mathbb{R}^{d\times n}$, $X + Y$ stands for $(X_{1}+Y_{1}, \ldots, X_{d}+Y_{d})$ and $aX$ stands for $(aX_{1}, \ldots, aX_{d})$ for $a\in \mathbb{R}$. For any $z:=(z^1, \cdots, z^d) \in \mathbb{R}^d$, denote $K1_n:=\{(z^{1} 1_n, z^{2} 1_n, \cdots, z^{d} 1_n): z \in K\}$ where $1_n := (1, \cdots, 1) \in \mathbb{R}^n$. We denote the positive dual cone of $(K1_n)^+$ in $\mathbb{R}^{d \times n}$ by $(K1_n)^+$, i.e. $(K1_n)^+:=\{w \in \mathbb{R}^{d \times n}: w Z^{\mathrm{T}} \geq 0 \text{ for any } z \in K\}$. The partial order respect to $K$ is defined as $X\leq_{K1_n} Y$, which means $Y-X\in K1_n$.\ Let $M:=\mathbb{R}^{m}\times \{0\}^{d-m}$ the linear subspace of $\mathbb{R}^{d}$ for $1\leq m\leq d$. The introduction of $M$ was considered by Jouini et al. (2004) and Hamel (2009). Denote $M_{+}:=M\cap \mathbb{R}^{d}_{+}$ where $\mathbb{R}^{d}_{+}:=\{(x^{1},\ldots,x^{d})\in \mathbb{R}^{d}; x^{i}\geq0, 1\leq i\leq d\}$ and $M^{\bot}:=\{0\}^{m}\times {\mathbb{R}}^{d-m}$. Thus, a regulator can only accept security deposits in the first $m$ reference instruments. Denote $K_{M}:=K\cap M$ by the closed convex polyhedral cone in $M$, $K^{+}_{M}:=\{u\in M:u^{tr}z\geq0 \textrm{ for any } z\in K_{M}\}$ the positive dual cone of $K_{M}$ in $M$, $intK_{M}$ the interior of $K_{M}$ in $M$. We denote $Q^{t}_{M}:=\{A\subset M:A=clco(A+K_{M})\}$ and $Q^{t}_{M^{+}}:=\{A\subset K_{M}:A=clco(A+K_{M})\}$, where the $clco(A)$ represents the closed convex hull of $A$.\ The cone $K$ models proportional frictions between the markets and contains those reference vectors which can be transferred (with paying transaction costs) into positions in $\mathbb{R}^{d}_{+}$, see Hamel (2009). The cone $K$ is also introduced to play the role of the solvency set of all positions that can be liquidated without any debt, or equivalently, it allows to define a liquidation value function as we need it to take into account the interdependencies between currencies, e.g. with respect to transaction costs. In this paper, any financial position belongs to $K$ should be regarded as those who need not to pay the capital requirements.\ By Chen and Hu (2017), a set-valued risk statistic is any map $\rho$ $$\rho: \mathbb{R}^{d\times n}\rightarrow 2^{M}$$ which can be considered as an empirical (or a data-based) version of a set-valued risk measure. The axioms related to this set-valued risk statistic are organized as follows, A0 : Normalized: $K_{M}\subseteq \rho(0)$ and $\rho(0)\cap -intK_{M}=\phi$; A1 : Monotonicity: for any $X$,$Y\in \mathbb{R}^{d\times n}$, $X-Y\in \mathbb{R}^{d\times n}\cap K1_{n}$ implies that $\rho(X)\supseteq \rho(Y)$; A3 : M-translative invariance: for any $X\in \mathbb{R}^{d\times n}$ and $z\in \mathbb{R}^{d}$, $\rho(X-z1_{n})=\rho(X)+z$; A3 : Convex: for any $X,Y\in \mathbb{R}^{d\times n}$ and $\lambda\in[0,1]$, $\rho(\lambda(X)+(1-\lambda)Y)\supseteq \lambda\rho(X)+(1-\lambda)\rho(Y)$; A4 : Positive homogeneity: $\rho(\lambda X)=\lambda\rho(X)$ for any $X\in \mathbb{R}^{d\times n}$ and $\lambda> 0$; A5 : Subadditivity: $\rho(X+Y)\supseteq \rho(X)+\rho(Y)$ for any $X,Y\in \mathbb{R}^{d\times n}$. We end this section with more notations. A function $\rho: \mathbb{R}^{d\times n}\rightarrow 2^{M}$ is called proper if $\textrm{dom}\rho:=\{X\in \mathbb{R}^{d\times n}:\rho(X)\neq \emptyset \}\neq \emptyset $ and $\rho(X)\neq M$ for all $X\in \textrm{dom}\rho$. $\rho$ is said to be closed if $\textrm{graph}\rho$ is a closed set with respect to the product topology on $\mathbb{R}^{d\times n}\times M$. ***Regulator-based risk statistics*** ===================================== In this section, we state the main results of this paper, the representation results of regulator-based risk statistics. However, our viewpoint is not the same as Chen and Hu (2017). Instead, we start from the viewpoint of regulators who only care the positions which need to pay capital requirements. Thus, for any $X\in \mathbb{R}^{d\times n}$, we define $X\wedge_{K1_{n}}0$ as $$\label{e21} X\wedge_{K1_{n}}0:=\left\{ \begin{array}{ll} X, & \textrm{$X\notin K1_{n}$},\\ 0, & \textrm{$X\in K1_{n}$}. \end{array} \right.$$ Thus, the positions which belongs to $K$ regarded as $0$ position. Firstly, we show the axioms related to regulator-based risk statistics. \[D1\] [The property of $\mathbf{R1}$ means any fixed negative risk position $-z$ can be canceled by its positive quality $z$; $\mathbf{R2}$ says that if $X_{1}$ is bigger than $X_{2}$ under the partial order under $K$, then the $X_{1}$ need less capital requirement than $X_{2}$, so $\varrho(X_{1})$ contain $\varrho(X_{2})$; $\mathbf{R3}$ means the regulator-based risk statistics start only from the viewpoint of regulators who only care the positions which need to pay capital requirements, while the positions that belong to $K$ regarded as $0$ position.]{.nodecor} [Let $Y\in \mathbb{R}^{d\times n}$, $u\in M$. Define a function $S_{(Y,u)}(X):\mathbb{R}^{d\times n}$ $\rightarrow$ $2^{M}$ as $$S_{(Y,u)}(X):=\{z\in M:X^{tr} Y\leq u^{tr}z\}.$$]{.nodecor} In order to derive the representation result for regulator-based risk statistics, we recall the Legendre-Fenchel conjugate theory for set-valued introduced by Hamel (2009). \[L1\] [(Hamel (2009) Theorem 2) Let $R:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{M}$ be a set-valued closed convex function. Then the Legendre-Fenchel conjugate and the biconjugate of $R$ can be defined, respectively, as $$-R^{\ast}(Y,u):=cl\bigcup_{X\in \mathbb{R}^{d\times n}}\Big(R(X)+S_{(Y,u)}(-X)\Big), \qquad Y\in \mathbb{R}^{d\times n}, u\in \mathbb{R}^{d};$$ and $$R(X)=R^{\ast\ast}(X):=\bigcap_{(Y,u)\in \mathbb{R}^{d\times n}\times K^{+}_{M}\backslash\{0\}}\Big[-R^{\ast}(Y,u)+S_{(Y,u)}(X)\Big],\qquad X\in \mathbb{R}^{d\times n}.$$]{.nodecor} [(Indicator function) For any $Z\subseteq \mathbb{R}^{d\times n}$, the $Q^{t}_{M}$-valued indicator function $I_{Z}: \mathbb{R}^{d\times n}\rightarrow Q^{t}_{M}$ is defined as $$I_{Z}(X):=\left\{ \begin{array}{ll} clK_{M}, & \textrm{$X\in Z$},\\ \phi, & \textrm{$X\notin Z$}. \end{array} \right.$$]{.nodecor} [The conjugate of $Q^{t}_{M}$-valued indicator function $I_{Z}$ is $$-(I_{Z})^{\ast}(Y,u):= cl\bigcup_{X\in Z}S_{(Y,u)}(-X).$$]{.nodecor} \[R1\] [The regulator-based risk statistics $\varrho$ do not have the property of cash additive, which is also said to be translation invariance, see Hamel (2009). However, they have the property of cash sub-additive studied by El Karoui and Ravanelli (2009) and Sun and Hu (2019). Indeed, from the Theorem 6.2 of Hamel and Heyde (2010), $\varrho$ satisfies the Fatou property. Then, consider any $X\in \mathbb{R}^{d\times n}$ and $z\in K_{M}$, for any $\varepsilon\in (0,1)$, we have $$\begin{aligned} \varrho\Big((1-\varepsilon)X-z1_{n}\Big)&=&\varrho\Big((1-\varepsilon)X+\varepsilon(-\frac{z}{\varepsilon})1_{n}\Big)\\ &\supseteq&(1-\varepsilon)\varrho(X)+\varepsilon\varrho(-\frac{z}{\varepsilon}1_{n})\\ &\supseteq&(1-\varepsilon)\varrho(X)+z\end{aligned}$$ where the last inclusion is due to the property $\mathbf{R1}$. Using the arbitrariness of $\varepsilon$, we have the following lemma.]{.nodecor} \[L2\] [Assume that $\varrho$ is a regulator-based risk statistic. For any $z\in \mathbb{R}^{d}_{+}$, $X\in \mathbb{R}^{d\times n}$, $$\varrho(X-z1_{n})\supseteq\varrho(X)+z$$ which also implies $$\varrho(X+z1_{n})\subseteq\varrho(X)-z.$$]{.nodecor} **Proof.** From the Remark \[R1\], one can steadily show Lemma \[L2\].\ [From Lemma \[L2\], the regulator-based risk statistics have the property of cash sub-additive. Even though regulator-based risk statistics are special case of cash sub-additive risk statistics, it still needed to be highlighted and studied separately because that the property of loss-dependence have its unique meaning in the finance practice. In addition, the property $\mathbf{R1}$ was not implied from cash sub-additive and have its rationality to define risk statistics. So it is a great meaning to study this special class of cash sub-additive risk statistics: regulator-based risk statistics.]{.nodecor} \[P1\] [Let $\varrho:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{M^{+}}$ be a proper closed convex regulator-based risk statistic with $u \in \Big\{\Big(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h\Big) +M^{\perp}\Big\}\bigcap K^{+}_{M}\backslash\{0\}$. Then $$-\varrho^{\ast}(Y,u)=\left\{ \begin{array}{ll} cl\bigcup\limits_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X), & \textrm{$Y \in -{\mathbb{R}_{+}^{d\times n}}\cap (K^{+}1_{n})$},\\ M, & \textrm{elsewhere}. \end{array} \right.\\$$]{.nodecor} Next, we state the main result of this paper, the representation result of regulator-based risk statistics.\ \[T1\] [If $\varrho:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{M^{+}}$ is a proper closed convex regulator-based risk statistic, then there is a $-\alpha: (-{\mathbb{R}_{+}^{d\times n}}\cap K^{+}1_{n}) \times K^{+}_{M}\backslash\{0\}$ $\rightarrow$ $Q^{t}_{M^{+}}$, that is not identically $M$ on the set $$\mathcal{W}=\bigg\{(Y,u)\in (-{\mathbb{R}_{+}^{d\times n}}\cap K^{+}1_{n}) \times K^{+}_{M}\backslash\{0\}:u \in \Big(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h\Big) +M^{\perp}\bigg\},$$ such that for any $X\in \mathbb{R}^{d\times n}$, $$\varrho(X)=\bigcap_{(Y,u)\in \mathcal{W}}\Big\{-\alpha(Y,u)+S_{(Y,u)}\big(X\wedge_{K1_{n}}0\big)\Big\}.$$]{.nodecor} ***Alternative versions of regulator-based risk statistics*** ============================================================= In this section, we develop another framework of regulator-based risk statistics. This framework is a little different from the previous one. However, almost all the arguments are the same as those in the previous section. Thus, we only state the corresponding notations and results, and omit all the proofs and relevant explanations.\ We replace $M $ by $\widetilde{M}\in \mathbb{R}^{d\times n}$ which is a linear subspace of $\mathbb{R}^{d\times n}$. We also replace $K $ by $\widetilde{K}\in \mathbb{R}^{d\times n}$ which is a is a closed convex polyhedral cone with $\widetilde{K}\supseteq \mathbb{R}^{d\times n}_{++}$. The partial order respect to $\widetilde{K}$ is defined as $X\leq_{\widetilde{K}} Y$, which means $Y-X\in \widetilde{K}$. Let $\widetilde{M}_{+}:=\widetilde{M}\cap \mathbb{R}^{d\times n}_{+}$. Denote $\widetilde{K}_{\widetilde{M}}:=\widetilde{K}\cap \widetilde{M}$ by the closed convex polyhedral cone in $\widetilde{M}$, $\widetilde{K}^{+}_{\widetilde{M}}:=\{\widetilde{u}\in M:\widetilde{u}^{tr}\widetilde{z}\geq0 \textrm{ for any } \widetilde{z}\in \widetilde{K}_{\widetilde{M}}\}$ the positive dual cone of $\widetilde{K}_{\widetilde{M}}$ in $\widetilde{M}$, $int\widetilde{K}_{\widetilde{M}}$ the interior of $\widetilde{K}_{\widetilde{M}}$ in $\widetilde{M}$. We denote $Q^{t}_{\widetilde{M}}:=\{\widetilde{A}\subset \widetilde{M}:\widetilde{A}=clco(\widetilde{A}+\widetilde{K}_{\widetilde{M}})\}$ and $Q^{t}_{\widetilde{M}^{+}}:=\{\widetilde{A}\subset \widetilde{K}_{\widetilde{M}}:\widetilde{A}=clco(\widetilde{A}+\widetilde{K}_{\widetilde{M}})\}$. We still start from the viewpoint of regulators who only care the positions which need to pay capital requirements. Thus, for any $X\in \mathbb{R}^{d\times n}$, we define $X\wedge_{\widetilde{K}}0$ as $$\label{e41} X\wedge_{\widetilde{K}}0:=\left\{ \begin{array}{ll} X, & \textrm{$X\notin \widetilde{K}$},\\ 0, & \textrm{$X\in \widetilde{K}$}. \end{array} \right.$$ Then, the axioms related to regulator-based risk statistics become as follows. \[D41\] We need more notations. Let $Y\in \mathbb{R}^{d\times n}$, $\widetilde{u}\in \widetilde{M}$. Define a function $S_{(Y,\widetilde{u})}(X):\mathbb{R}^{d\times n}$ $\rightarrow$ $2^{\widetilde{M}}$ as $$S_{(Y,\widetilde{u})}(X):=\{\widetilde{z}\in \widetilde{M}:X^{tr} Y\leq \widetilde{u}^{tr}\widetilde{z}\}.$$ Let $\widetilde{R}:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{\widetilde{M}}$ be a set-valued closed convex function. Then the Legendre-Fenchel conjugate and the biconjugate of $\widetilde{R}$ can be defined, respectively, as $$-\widetilde{R}^{\ast}(Y,u):=cl\bigcup_{X\in \mathbb{R}^{d\times n}}\Big(\widetilde{R}(X)+S_{(Y,\widetilde{u})}(-X)\Big), \qquad Y\in \mathbb{R}^{d\times n}, \widetilde{u}\in \mathbb{R}^{d\times n};$$ and $$\widetilde{R}(X)=\widetilde{R}^{\ast\ast}(X):=\bigcap_{(Y,\widetilde{u})\in \mathbb{R}^{d\times n}\times \widetilde{K}^{+}_{\widetilde{M}}\backslash\{0\}}\Big[-\widetilde{R}^{\ast}(Y,\widetilde{u})+S_{(Y,\widetilde{u})}(X)\Big],\qquad X\in \mathbb{R}^{d\times n}.$$ For any $\widetilde{Z}\subseteq \mathbb{R}^{d\times n}$, the $Q^{t}_{\widetilde{M}}$-valued indicator function $I_{\widetilde{Z}}: \mathbb{R}^{d\times n}\rightarrow Q^{t}_{\widetilde{M}}$ is defined as $$I_{\widetilde{Z}}(X):=\left\{ \begin{array}{ll} cl\widetilde{K}_{\widetilde{M}}, & \textrm{$X\in \widetilde{Z}$},\\ \phi, & \textrm{$X\notin \widetilde{Z}$}. \end{array} \right.$$ The conjugate of $Q^{t}_{\widetilde{M}}$-valued indicator function $I_{\widetilde{Z}}$ is $$-(I_{\widetilde{Z}})^{\ast}(Y,\widetilde{u}):= cl\bigcup_{X\in \widetilde{Z}}S_{(Y,\widetilde{u})}(-X).$$ Assume that $\widetilde{\varrho}$ is a regulator-based risk statistic. For any $\widetilde{z}\in \mathbb{R}^{d\times n}_{+}$, $X\in \mathbb{R}^{d\times n}$, $$\widetilde{\varrho}(X-\widetilde{z})\supseteq\widetilde{\varrho}(X)+\widetilde{z}$$ which also implies $$\widetilde{\varrho}(X+\widetilde{z})\subseteq\widetilde{\varrho}(X)-\widetilde{z}.$$ Next, we state the main results of this section. [Let $\widetilde{\varrho}:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{\widetilde{M}^{+}}$ be a proper closed convex regulator-based risk statistic with $\widetilde{u} \in \Big\{\Big(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h\Big) +\widetilde{M}^{\perp}\Big\}\bigcap \widetilde{K}^{+}_{\widetilde{M}}\backslash\{0\}$. Then $$-\widetilde{\varrho}^{\ast}(Y,\widetilde{u})=\left\{ \begin{array}{ll} cl\bigcup\limits_{X\in \mathbb{R}^{d\times n}}S_{(Y,\widetilde{u})}(-X), & \textrm{$Y \in -{\mathbb{R}_{+}^{d\times n}}\cap (\widetilde{K}^{+})$},\\ \widetilde{M}, & \textrm{elsewhere}. \end{array} \right.\\$$]{.nodecor} [If $\widetilde{\varrho}:\mathbb{R}^{d\times n}$ $\rightarrow$ $Q^{t}_{\widetilde{M}^{+}}$ is a proper closed convex regulator-based risk statistic, then there is a $-\alpha: (-{\mathbb{R}_{+}^{d\times n}}\cap \widetilde{K}^{+}) \times \widetilde{K}^{+}_{\widetilde{M}}\backslash\{0\}$ $\rightarrow$ $Q^{t}_{\widetilde{M}^{+}}$, that is not identically $\widetilde{M}$ on the set $$\widetilde{\mathcal{W}}=\bigg\{(Y,\widetilde{u})\in (-{\mathbb{R}_{+}^{d\times n}}\cap \widetilde{K}^{+}) \times \widetilde{K}^{+}_{\widetilde{M}}\backslash\{0\}:\widetilde{u} \in \Big(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h\Big) +\widetilde{M}^{\perp}\bigg\},$$ such that for any $X\in \mathbb{R}^{d\times n}$, $$\widetilde{\varrho}(X)=\bigcap_{(Y,\widetilde{u})\in \mathcal{W}}\Big\{-\alpha(Y,\widetilde{u})+S_{(Y,\widetilde{u})}\big(X\wedge_{\widetilde{K}}0\big)\Big\}.$$]{.nodecor} ***Proofs of main results*** ============================ **Proof of Proposition \[P1\].** If $Y\notin -\mathbb{R}_{+}^{d\times n}\cap (K^{+}1_{n})$, there exit an $\bar{X}\in \mathbb{R}^{d\times n}\cap (K1_{n})$ such that $\bar{X}^{tr} Y>0$. Using the definition of $S_{(Y,u)}$, we have $S_{(Y,u)}(-t\bar{X})=\{z\in M:-t\bar{X}^{tr} Y\leq u^{tr}z\}$ for $t>0$. Thus, $$cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)\supseteq \bigcup_{t>0}S_{(Y,u)}(-t\bar{X})=M.$$ The last equality is due to $-\bar{X}^{tr} Y\to -\infty$ when $t\to +\infty$. Using the definition of $S_{(Y,u)}$, we conclude that $cl\bigcup\limits_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)\subseteq M$. Hence $$cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)=M \qquad \textrm{whenever}\qquad Y\notin -\mathbb{R}^{d\times n}\cap (K^{+}1_{n}).$$ It is easy to check that for any $X\in \mathbb{R}^{d\times n}$ and $v\in M$, $$\begin{aligned} S_{(Y,u)}(-X-v1_{n})&=&\{z\in M :-X^{tr} Y\leq u^{tr} z+Y^{tr} (v1_{n})\} \\&=&\{z-v\in M:-X^{tr} Y\leq u^{tr} (z-v)+(Y+u1_{n})^{tr} (v1_{n})\}+v\\ &=&\{z\in M :-X^{tr} Y\leq u^{tr} z+(Y+u1_{n})^{tr} (v1_{n})\}+v.\end{aligned}$$ When $(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h)+u\in M^\perp$, we have $S_{(Y,u)}(-X-v1_{n})=S_{(Y,u)}(-X)+v$. However, when $u\notin \big((-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h, \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h)+M^{\perp}\big)$. Thus, $(-\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{1, j}_h,\\ \cdots, -\sum\limits_{j=1}^l\sum\limits_{h=1}^{n_j} Y^{d, j}_h)+u\notin M^\perp$, we can find $v\in M$ , such that for any $z\in M$, $$-X^{tr} Y\leq u^{tr} z+(Y+u1_{n})^{tr} (v1_{n}).$$ Thus, we have $$z+v\in S_{(Y,u)}(-X-v1_{n}).$$ Therefore $$\bigcup_{z,v\in M}(z+v)\subset \bigcup_{v\in M}S_{(Y,u)}(-X-v1_{n}).$$ Thus, $$M\subset \bigcup_{v\in M}S_{(Y,u)}(-X-v1_{n}).$$ From the definition of $S_{(Y,u)}$, the inverse inclusion is always true. So we conclude that $$M=\bigcup_{v\in M}S_{(Y,u)}(-X-v1_{n}).$$ It is also easy to check that $$\begin{aligned} -\varrho^{\ast}(Y,u)&=&cl\bigcup_{X\in \mathbb{R}^{d\times n},v\in M}\Big(\varrho(X+v1_{n})+S_{(Y,u)}(-X-v1_{n})\Big)\\ &=&cl\bigcup_{X\in \mathbb{R}^{d\times n},v\in M}\Big(\varrho(X+v1_{n})+M\Big)\\ &=&M\end{aligned}$$ where the last equality comes from that the $M$ is a linear space and $\varrho(X)\subseteq M$. We now show that $-\varrho^{\ast}(Y,u)=cl\bigcup\limits_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)$. To this end, from $-\varrho^{\ast}(Y,u)=cl\bigcup\limits_{X\in \mathbb{R}^{d\times n}}\Big(\varrho(X)+S_{(Y,u)}(-X)\Big)$, we derive it in two cases:\ Case  1.  When $X\wedge_{K1_{n}}0=0$, using the definition, we have $\varrho(X)=\varrho(0)\ni0$. Hence $$cl\bigcup_{X\in \mathbb{R}^{d\times n}}\Big(\varrho(X)+S_{(Y,u)}(-X)\Big)\supset cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X).$$ Case 2.  When $X\wedge_{K1_{n}}0= X$, we can always find an $\alpha\in K_{M}$ such that $\alpha\in \varrho(X)$. Then $$\varrho(X)+S_{(Y,u)}(-X)\supseteq \alpha+S_{(Y,u)}(-X)=S_{(Y,u)}(-X-\alpha1_{n})=S_{(Y,u)}(-\beta)$$ where $\beta=X+\alpha1_{n}$. It is relatively simple to check that $\beta\in \mathbb{R}^{d\times n}$. Thus $$cl\bigcup_{X\in \mathbb{R}^{d\times n}}\Big(\varrho(X)+S_{(Y,u)}(-X)\Big)\supseteq cl\bigcup_{z\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-z),$$ that is $$-\varrho^{\ast}(Y,u)\supseteq cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X).$$ Consequently, we have $$-\varrho^{\ast}(Y,u)\supseteq cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X).$$ Now, we need only to show that $-\varrho^{\ast}(Y,u)\subseteq cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)$. In fact, for any $z\in \varrho(X)$ and $X\in \mathbb{R}^{d\times n}$, $X+z1_{n}\in \mathbb{R}^{d\times n}$. Thus $$cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)=cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X)\supseteq S_{(Y,u)}(-X-z1_{n})=S_{(Y,u)}(-X)+z.$$ Using the arbitrariness of $z$, we have $$\varrho(X)+S_{(Y,u)}(-X)\subseteq cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X).$$ Thus, $$-\varrho^{\ast}(Y,u)\subseteq cl\bigcup_{X\in \mathbb{R}^{d\times n}}S_{(Y,u)}(-X).$$                                                                                                                           \ **Proof of Theorem \[T1\].** The proof is straightforward from Lemma \[L1\] and Proposition \[P1\].\ ***Examples*** =============== In this section, we construct several examples for regulator-based risk statistics. [The coherent risk measure AV@R was studied by F$\ddot{o}$llmer and Schied (2011) in detail. They have given several representations and many properties like law invariance and the Fatou property. Hamel et al. (2013) first introduced set-valued AV@R, where the representation result is derived. Moveover, they also proved that it is a set-valued coherent risk measure. We now define the set-valued loss average value at risk. For any $X\in \mathbb{R}^{d\times n}$ and $0<\alpha<1$, we define $\varrho(X)$ as $$\begin{aligned} \varrho(X)&:=&AV@R^{loss}_{\alpha}(X)\\ &:=&\inf_{z\in \mathbb{R}^{d}}\Big\{\frac{1}{\alpha}(-(X\wedge_{K1_{n}}0)|_{M}+z)^{+}-z\Big\}+\mathbb{R}^{m}_{+}.\end{aligned}$$ It is clear that $\varrho$ satisfies the cash-loss, monotonicity, loss-dependence properties and convexity, so $\varrho$ is a regulator-based risk statistic. We call such a risk statistic a set-valued loss average value at risk.]{.nodecor} [The shortfall risk measures were first introduced by F$\ddot{o}$llmer and Schied (2002) with the help of loss function, and they proved that it is a special case of convex risk measures. Ararat and Hamel (2014) introduced the set-valued shortfall risk measures with vector loss function $\ell$.We now define the loss loss shortfall risk statistics. For any $X\in \mathbb{R}^{d\times n}$, we define $$\varrho_{\ell}(X):=\big\{z\in \mathbb{R}^{d}\Big|\ell\Big(-(X\wedge_{K1_{n}}0)-z1_{n}\Big)\in x^{0}-C\big\},$$ where $x^{0}$ is the threshold level and $C$ is the threshold set such that $0\in \mathbb{R}^{d\times n}$ is the boundary point of $C$. When we take $x^{0}=\ell(0)$, $\varrho_{\ell}$ satisfies the cash-loss, monotonicity, loss-dependence properties and convexity, so $\varrho_{\ell}$ is a regulator-based risk statistic. We call such a risk statistic a loss shortfall risk statistic.]{.nodecor} [The notion of $\varphi$-divergence was first introduced by Ben-Tal and Teboulle (1986), and further it was extended by Ben-Tal and Teboulle (2007) with the name optimized certainty equivalent as a convex risk measure. Ararat and Hamel (2014) introduced set-valued divergence risk measures with vector loss function $\ell$, and proved it is a set-valued convex risk measure. We now define the loss divergence risk statistics. For any $X\in \mathbb{R}^{d\times n}$, we define $$\varrho_{g,r}(X):=\bigcap_{r,\omega\in \mathbb{R}^{d}_{+}\backslash\{0\}}\{z\in \mathbb{R}^{d}\Big| \omega^\top z\geq \omega^\top \delta_{g,r}(X\wedge_{K1_{n}} 0)+\inf_{x\in C}\omega^\top \textrm{diag}(r)x\}$$ where $$\delta_{g,r}(X)=\big(\delta_{g_{1},r_{1}}(X_{1}),\ldots \delta_{g_{d},r_{d}}(X_{d})\big)$$ with $$\delta_{g_{i},r_{i}}(X_{i}):=\inf_{z_{i}\in \mathbb{R}}\big(z_{i}+r_{i}\mathbb{E}[\ell_{i}(-X_{i}-z_{i}1_{n})]\big)-r_{i}x_{i}^{0},$$ where $x^{0}$ is the threshold level and $C$ is the threshold set such that $0\in \mathbb{R}^{d\times n}$ is the boundary point of $C$. 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--- abstract: 'We have measured the contact angle of the interface of phase-separated $^{3}$He-$^{4}$He mixtures against a sapphire window. We have found that this angle is finite and does not tend to zero when the temperature approaches $T_t$, the temperature of the tri-critical point. On the contrary, it increases with temperature. This behavior is a remarkable exception to what is generally observed near critical points, i.e. “critical point wetting”. We propose that it is a consequence of the “critical Casimir effect” which leads to an effective attraction of the $^{3}$He-$^{4}$He interface by the sapphire near $T_{t}$.' address: | Laboratoire de Physique Statistique de l’Ecole Normale Supérieure\ associé aux Universités Paris 6 et Paris 7 et au CNRS\ 24 rue Lhomond 75231 Paris Cedex 05, France\ author: - 'T. Ueno[^1], S. Balibar, T. Mizusaki[^2], F. Caupin and E. Rolley' title: Critical Casimir effect and wetting by helium mixtures --- [2]{} In 1977, J. W. Cahn predicted that “in any two-phase mixture of fluids near their critical point, contact angles against any third phase become zero in that one of the critical phases completely wets the third phase and excludes contact with the other critical phase” [@cahn]. This “critical point wetting” is a very general phenomenon [@cahn; @heady; @indekeu; @bonn]. We found an exception to it by studying helium mixtures in contact with a sapphire window [@exceptions]. In fact, de Gennes [@pgg] had noticed that long range forces may prevent complete wetting. Nightingale and Indekeu [@night] further explained that if a long range attraction is exerted by the third phase on the interface between the two critical phases, partial wetting may be observed up to the critical point. We propose that, in $^{3}$He-$^{4}$He mixtures near their tri-critical point, this attraction is provided by the confinement of the fluctuations of superfluidity, i.e. a critical Casimir effect [@pgg2; @night; @krech; @garcia; @garcia2] in the $^{4}$He-rich film between the sapphire and the $^{3}$He-rich bulk phase (Fig. \[fig:contactangle\]). For a solid substrate in contact with a phase-separated $^{3}$He-$^{4}$He mixture, complete wetting by the $^{4}$He-rich “d-phase” was generally expected, due to the van der Waals attraction by the substrate [@romagnan; @sornette]. However, we measured the contact angle $\theta$ of the $^{3}$He-$^{4}$He interface on sapphire, and we found that it is finite. Furthermore, it increases between 0.81 and 0.86 K, close to the tri-critical point at $T_{t}$ = 0.87 K [@jltp]. This behavior is opposite to the usual “critical point wetting” where $\theta$ decreases to zero at a wetting temperature $T_{w}$ below the critical point. In this letter, we briefly recall our experimental results before explaining why the “critical Casimir effect” provides a reasonable interpretation of our observations. ![The phase diagram of $^{3}$He-$^{4}$He mixtures (left graph). On the right, a schematic view of the contact angle $\theta$. There is a superfluid film of $^{4}$He rich “d-phase” between the substrate and the c-phase. Its thickness being finite, $\theta$ is non-zero.[]{data-label="fig:contactangle"}](fig1.eps){width="1\linewidth"} We use a dilution refrigerator with optical access [@jltp]. Our liquid sample is at saturated vapor pressure, and confined between two sapphire windows which form an interferometric cavity. The inside of the windows is treated to have a 15$\%$ reflectivity. The cell is made of pure copper and neither the windows nor the helium absorb any light, so that a very good thermal homogeneity is achieved. From fringe patterns, we analyze the profile of the c-d interface near its contact line with one of the windows [@jltp; @etienne]. A fit with a solution of Laplace’s equation gives the interfacial tension $\sigma_{i}$ and the contact angle $\theta$. As $T$ approaches $T_{t}$, the capillary length vanishes so that the region to be analyzed becomes very small. However, our typical resolution is 5 $\rm \mu m$, significantly smaller than the capillary length (from 84 $\rm \mu m$ at 0.81 K to 33 $\rm \mu m$ at 0.86 K). Here, we present results only in this temperature range because, closer to the tri-critical point, we would need a better resolution and, below 0.80 K, refraction effects distort the fringe patterns [@jltp]. ![Our measurements of the interfacial tension agree with Leiderer’s results (solid line). Different symbols correspond to three different positions along the contact line.[]{data-label="fig:tension"}](fig2.eps){width="1\linewidth"} ![Temperature dependence of the contact angle $\theta$. Black symbols correspond to the present calculation.[]{data-label="fig:angle"}](fig3.eps){width="1\linewidth"} For each temperature, we analyzed three pictures at different positions along the contact line. As shown in Fig. \[fig:tension\], our measurements of $\sigma_{i}$ agree well with Leiderer’s result $\sigma_{i}$ = 0.076 t$^{2}$ erg/cm$^{2}$ ($t=(1-T/T_{t})$ is the reduced temperature) [@leiderer]. As for the contact angle $\theta$ (Fig. \[fig:angle\]), we found that it is non-zero and that it increases with $T$. On these measurements, the typical error bar is $\pm$ 15 $^{\circ}$. It originates in several experimental difficulties such as the precise location of the contact line and a slight bending of the windows which are under stress [@jltp]. When cooling down a homogeneous mixture with concentration higher than the tri-critical value $X_{t}$, J.P. Romagnan et al. [@romagnan] found that a superfluid film formed between the bulk mixture and a metallic substrate. As they approached $T_{eq}$ where separation into “c- ” and “d-” phases occurred, they observed a film thickness diverging as $(T-T_{eq})^{-1/3}$. This behaviour is characteristic of the van der Waals attraction by the substrate, which is stronger on the densest phase [@sornette]. One used to believe that van der Waals forces were the only long range forces in this problem, so that the film thickness should diverge to infinity, and complete wetting by the superfluid d-phase should occur. However, Romagnan et al. only measured this thickness up to about 20 atomic layers (60 Å). If other forces act on the film near the tri-critical point, its thickness can saturate at a value larger than 60 Å. R. Garcia and M. Chan[@garcia] have shown that superfluid films of pure $^{4}$He get thinner near $T_{\lambda}$, due to the critical Casimir effect. Our situation is similar: our d-phase film is just below its superfluid transition, and we have calculated that Casimir forces limit the film thickness to a few hundred Å . An increasing variation of $\theta (T)$ is also surprising. Young’s relation writes: $$\cos(\theta) = \frac{\delta \sigma}{\sigma_{i}} = \frac{\sigma_{sc}-\sigma_{sd}}{\sigma_{i}} \label{eq:young}$$ As the critical point is approached, both $\sigma_{i}$ and $\delta \sigma$ tend to zero. It is often assumed that $\delta \sigma$ is proportional to the difference in concentration between the two phases. If this was always true, the critical exponent of $\sigma_{i}$ would always be larger than that of $\delta \sigma$ [@indekeu]. Consequently, $\theta$ would always decrease to zero at a wetting temperature $T_{w}$ below the critical point. Our observations show that this reasoning does not apply to helium mixtures. Let us now follow D. Ross et al. [@ross] to calculate $\theta$. We first calculate the “disjoining pressure” $\Pi(l)$ as a function of the thickness $l$ of the d-phase film (Fig. \[fig:contactangle\]). For this we consider three long range forces: the van der Waals force, the Casimir force and the “Helfrich” force [@helfrich]. At the equilibrium film thickness $l \: = \: l_{eq}$, $\Pi(l)$ has to cross zero with a negative slope. If $l_{eq}$ was macroscopic, the substrate (s) to c-phase interface would be made of an s-d interface plus a c-d interface. Its energy per unit area would thus be $\sigma_{sc} \:=\: (\sigma_{sd}\: +\: \sigma_{i})$. If $l_{eq}$ is small, a correction to the above formula has to be added, which is the integral of the disjoining pressure from infinity to $l_{eq}$. Finally, Young’s relation imply $$\cos(\theta) = \frac{\sigma_{sc}-\sigma_{sd}}{\sigma_{i}} = 1 + \frac{ \int_{l_{eq}}^{\infty} \Pi (l) dl}{\sigma_{i}} \label{eq:young2}$$ Let us start with the van der Waal contribution $\Pi_{vdW}(l)$ to $\Pi (l)$. The net effective force on the interface is the difference between the respective van der Waals attractions on the d- and c-phase. For helium on copper, Garcia found that this attraction is $A_{0}/Vl^{3}$, with $A_{0}$ = 2600 K.Å$^3$ ($V$ is the atomic volume) [@garcia]. For our window with its insulating coating, we expect a smaller value. Sabisky [@sabisky] found $A_{0}$ = 980 K.Å$^{3}$ for liquid $^{4}$He on CaF$_{2}$. We thus estimate $A_{0}* \approx$ 1000 K.Å$^{3}$ in our case. As for the coefficient of the differential force, it is now $$A = A_{0}* \left (\frac{1}{V_{d}*}-\frac{1}{V_{c}*}\right )$$ where $V_{c,d}$ are the respective atomic volumes in the two phases [@kierstead]. We included the retarded term in the van der Waals potential [@garcia] and finally found $$\Pi_{vdW}(l) = \frac{A}{l^{3}(1 + l/193)} = \frac{14.72 t - 2.82 t^{2} + 2.29 t^{3}}{l^{3}(1 + l/193)} \label{eq:vdW}$$ in K/Å$^{3}$ with $l$ in Å. Let us now consider the Casimir force. Following Garcia [@garcia], the confinement of superfluid fluctuations inside a film of thickness $l$ gives a contribution to the disjoining pressure $$\Pi_{Cas}*(l) = \frac{\vartheta (x)\:T_{t}}{l^{3}} \label{eq:casimir}$$ where $x = tl$ and the “scaling function” $\vartheta (x)$ is negative, with a minimum of about -1.5 at $x \approx 10$. The sign of $\vartheta (x)$ depends on the symmetry of the boundary conditions on the two sides of the film [@garcia2; @krech]. In Ref. [@garcia] as in our case, the whole film is superfluid except near both interfaces where the order parameter vanishes on a distance $\xi$, the correlation length. Consequently, the boundary conditions are symmetric for this order parameter and $\vartheta (x)$ is negative, meaning an attractive force. Note that in Garcia’s second experiment [@garcia2] on mixtures, the film was separated into a superfluid subfilm near the wall and a normal one near the liquid-gas interface; Garcia considered this as an anti-symmetric situation leading to a repulsive force between the wall and the liquid-gas interface. This second experiment is different from ours because it measures a Casimir force on a liquid-gas surface while ours has to do with the c-d interface. As a result our experiment is paradoxically more similar to Garcia’s first experiment [@garcia] with pure $^{4}$He which has also symmetric boundary conditions than with Garcia’s second experiment on mixtures [@garcia2]. In order to evaluate $\vartheta (x)$, and in the absence of any other determination, we have taken Garcia’s curve labelled “Cap. 1” in Ref. [@garcia]. It corresponds to a film thickness of about 400 Å , as found below for $t$ = 10$^{-2} $. Fig. \[fig:casimir\] shows that, at this temperature, the resulting Casimir contribution dominates the van der Waals one above about 100 Å. This is because the coefficient $A$ in Eq. 4 vanishes with $t$, so that, for $t$ = $10^{-2}$, it is about 0.15, ten times less than the maximum amplitude of $\vartheta (x)$. We still need to discuss our approximations further. We are dealing with a tri-critical point instead of the lambda transition in Garcia’s case [@garcia]. According to Krech and Dietrich [@krech] the Casimir amplitude is twice as large for tri-critical points compared to ordinary critical points. Doubling Garcia’s scaling function enlarges $\theta$ and improves the agreement with our experiment (see below). Furthermore, in our system, concentration and superfluidity fluctuations are coupled together. Both should be considered in a rigorous calculation which has not yet been done. The boundary conditions are symmetric for superfluidity but they are anti-symmetric for concentration fluctuations since the film is richer in $^{4}$He near the substrate than near the c-phase. We thus believe that a rigorous calculation should include two contributions with opposite sign. We assume that the confinement of superfluidity dominates because the Casimir amplitude is roughly proportional to the dimension $N$ of the order parameter [@krech] ($N$ = 2 for superfluidity and $N$ = 1 for concentration). We hope that our intuition can be confirmed by further theoretical work. ![The three contributions to the disjoining pressure for $t$ = 10$^{-2}$. The total pressure is zero at $l_{eq}$ = 400 Å.[]{data-label="fig:casimir"}](fig4.eps){width="1.1\linewidth"} The third contribution $\Pi_{H}* (l)$ originates in the limitation of the amplitude $z$ of the c-d interface fluctuations to a fraction of the thickness $l$. According to Helfrich[@helfrich], $<(z)^{2}> \approx l^{2}/6$ and, following Ross et al [@ross] $$\Pi_{H} (l) = \frac{T}{2L^{2}*l}\; ,$$ where $L$ is a long wavelength cutoff. L can be calculated from the equipartition theorem as $$L*=*\xi \exp{\left(\frac{2\pi\sigma_{i}* l^{2}*}{6k_{B}*T}\right )}$$ The bulk correlation length $\xi$ is related to the surface tension $\sigma_{i}*$ by $\xi^{2}*\approx k_{B}*T/(3\pi\sigma_{i})$, where the factor 3 is consistent with both Refs. [@leiderer] and [@ross]. Finally $$\Pi_{H}(l) = \frac{3\pi \sigma_{i}*}{2l} \exp{\left ( \frac{-2\pi\sigma_{i}* l^{2}*}{3k_{B}*T}\right )}$$ The disjoining pressure and the equilibrium film thickness are now obtained by adding the three above contributions and by looking for $l_{eq}$ such that $\Pi (l_{eq})$ = 0. Fig. \[fig:casimir\] shows the results of a calculation for $t$ = 10$^{-2}$. If we had the van der Waals contribution only, the disjoining pressure would be positive everywhere and it would repell the film surface to infinity (complete wetting). The Casimir contribution is negative and large enough to induce partial wetting. As for the Helfrich repulsion, it is very large at small thickness but it decreases exponentially so that its effect is to shift the equilibrium thickness by a few hundred Å. Fig. \[fig:casimir\] shows that, for $t$ = 10$^{-2}$, $l_{eq}$ = 400 Å. This is larger than $\xi\approx $ 100 Å, so that the superfluidity is well established in the middle of the d-phase film. At this temperature we finally calculated the contact angle with Eq. \[eq:young2\], and found $\theta$ = 45 degrees, in good agreement with experimental results (Fig. \[fig:angle\]). In order to account for tri-criticality, one could double the Casimir amplitude; this would roughly double (1- $\cos \theta$) and change 45 into 66 degrees, in even better agreement with our data. The most important result is that $\theta$ is finite. Its exact magnitude depends on the many approximations made above, especially on the value of $\vartheta (x)$ which is only known through Garcia’s measurement in a slightly different situation. We repeated the same calculation for $t$ = 5.10$^{-2}$, i.e. $T$ = 0.83 K, and we found $\theta$ = 30 degrees. However, at this temperature, we found a thinner film for which the value of $\vartheta (x)$ is less accurately known. It is reasonable to find that the contact angle vanishes away from $T_{t}$ because the Casimir force vanishes while the van der Waals force increases. Clearly, there is a temperature region where $\theta$ increases with $T$, as found experimentally. As for very close to $T_{t}$, a crossover to a different regime should occur when $l_{eq} \approx \xi$ so that short range forces should dominate; whether the contact angle keeps increasing, or reaches a finite value, or starts decreasing to zero is an additional question to be solved. Let us finally remark that, if we had an ordinary critical point with van der Waals forces and concentration fluctuations only, the Casimir force would be repulsive [@Muk] and favor critical point wetting. In the case of our helium mixtures, it is the symmetric boundary conditions for superfluidity which lead to a Casimir force acting against critical point wetting. One obviously needs more measurements for a more precise determination of $\theta$ and a calculation of the scaling function for a more accurate theoretical prediction. We are grateful to C. Guthmann, S. Moulinet, M. Poujade, D. Bonn, J. Meunier, D. Chatenay and E. Brezin for very helpful discussions. T. Ueno aknowledges support from the JSPS and the Kyoto University Foundation during his stay at ENS. J. W. Cahn, J. Chem. Phys. [**66**]{}, 3667 (1977). R. B. Heady and J. W. Cahn, J. Chem. Phys. [**58**]{}, 896 (1973). See, for instance, the review articles by J. O. Indekeu, Acta Phys. Pol. B [**26**]{}, 1065 (1995) and by D. Bonn and D. Ross, Rep. Prog. Phys. [**64**]{}, 1085 (2001). D. Bonn, H. Kellay, and G. H. Wegdam, Phys. Rev. Lett. [**69**]{}, 1975 (1992). We believe that this is the first experimental evidence for an exception to critical point wetting. S. Ross and R.E. Kornbrekke ([*J. Coll. Interf. Sc.*]{} [**99**]{} 446 (1984)) claimed that they had found such an exception with binary liquid solutions against glass, but M.R. Moldover and J.W. Schmidt ([*Physica*]{} [**12D**]{}, 351, 1984) explained that this was an artefact due to their observation method. Our results are consistent with preliminary indications obtained with the same helium mixtures against an epoxy resin and with a magnetic resonance imaging technique by T. Ueno, M. Fujisawa, K. Fukuda, Y. Sasaki, and T. Mizusaki, (Physica B [**284-288**]{}, 2057 (2000)). However, in the latter work, the evidence was weaker since the error bar on the contact angle $\theta$ was as large as the value of $\theta$ itself near the critical point. P.G. de Gennes, J. Physique Lettres [**42**]{} L-377 (1981) M.P. Nightingale and J.O. Indekeu, Phys. Rev. Lett. [ **54**]{}, 1824 (1985) M.E. Fisher and P.G. de Gennes, C.R. Acad. Sci. Paris [**B 287**]{}, 209 (1978) M. Krech and S. Dietrich, Phys. Rev. Lett. [**66**]{}, 345 (1991); [**67**]{}, 1055 (1991); J. Low Temp. Phys. [**89**]{}, 145 (1992). R. Garcia and M.H.W. Chan, Phys. Rev. Lett. [**83**]{}, 1187 (1999) R. Garcia and M. H. W. Chan, Phys. Rev. Lett. [**88**]{}, 086101 (2002). J.P. Romagan, J.P. Laheurte, J.C. Noiray and W.F. Saam, J. Low Temp. Phys. [**30**]{}, 425 (1978). D. Sornette and J.P. Laheurte, J. Physique (Paris) [**47**]{}, 1951 (1986). T. Ueno, S. Balibar, F. Caupin, T. Mizusaki and E. Rolley, J. Low Temp. Phys. [*130*]{}, 543 (2003). E. Rolley and C. Guthmann, J. Low Temp. Phys. [**108**]{}, 1 (1997); A. Prevost, Ph.D. Thesis, Université de Paris-sud, 1999 (unpublished). P. Leiderer, H. Poisel, and M. Wanner, J. Low Temp. Phys. [**28**]{}, 167 (1977); P. Leiderer, D. R. Watts, and W. W. Webb, Phys. Rev. Lett. [**33**]{}, 483 (1974). D. Ross, D. Bonn and J. Meunier, Nature [**400**]{}, 737 (1999) E.S. Sabisky and C.H. Anderson, Phys. Rev. [ **A7**]{}, 790 (1973). H. A. Kierstead, J. Low Temp. Phys. [**24**]{}, 497 (1976). W. Helfrich and R.M. Servuss, Nuovo Cimento [**3D**]{}, 137 (1984). A. Mukhopadhyay and B.M. Law, Phys. Rev. Lett. [**83**]{}, 772 (1999). [^1]: Present address: Research Laboratory of Electronics and MIT-Harvard Center for Ultracold Atoms, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA [^2]: Permanent address: Department of Physics, Graduate School of Science, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan

ArXiv

--- abstract: 'We perform 1D/2D/3D relativistic hydrodynamical simulations of accretion flows with low angular momentum, filling the gap between spherically symmetric Bondi accretion and disc-like accretion flows. Scenarios with different directional distributions of angular momentum of falling matter and varying values of key parameters such as spin of central black hole, energy and angular momentum of matter are considered. In some of the scenarios the shock front is formed. We identify ranges of parameters for which the shock after formation moves towards or outwards the central black hole or the long lasting oscillating shock is observed. The frequencies of oscillations of shock positions which can cause flaring in mass accretion rate are extracted. The results are scalable with mass of central black hole and can be compared to the quasi-periodic oscillations of selected microquasars (such as GRS 1915+105, XTE J1550-564 or IGR J17091-3624), as well as to the supermassive black holes in the centres of weakly active galaxies, such as Sgr $A^{*}$.' bibliography: - 'Sukova.bib' title: Shocks in the relativistic transonic accretion with low angular momentum --- \[firstpage\] accretion, accretion discs – hydrodynamics – shock waves – X-rays: binaries – stars: black holes – Galaxy: centre Introduction {#s:Introduction} ============ The weakly active galaxies, where the central nucleus emits the radiation at a moderate level with respect to the most luminous quasars, are frequenlty described by the so called hot accretion flows model. In such flows, the plasma is virially hot and optically thin, and due to the advection of energy onto black hole, the flow is radiativelly inefficient. The prototype of an object which can be well described by such model, is the low luminosity black hole in the centre of our Galaxy, the source Sgr $A^{*}$ . Also, the black hole X-ray binaries in their hard and quiescent states can be good representatives for the hot mode of accretion. These states are frequenlty associated with the ejection of relativistic streams of plasma, which form the jets, responsible for the radio emission of the sources [@2004MNRAS.355.1105F]. The matter essentially flows into black hole with the speed of light, while the sound speed at maximum can reach $c/\sqrt{3}$. Therefore, the accretion flow must have transonic nature. The viscous accretion with transonic solution based on the alpha-disk model was first studied by [@1981AcA....31..283P] and [@1982AcA....32....1M]. After that, e.g. [@1989ApJ...336..304A] examined the stability and structure of transonic disks. The possibility of collimation of jets by thick accretion tori was proposed by, e.g., [@1981MNRAS.197..529S]. In respect of the value of angular momentum there are two main regimes of accretion, the Bondi accretion, which refers to spherical accretion of gas without any angular momentum, and the disk-like accretion with Keplerian distribution of angular momentum. In the case of the former, the sonic point is located farther away from the compact object (depending on the energy of the flow) and the flow is supersonic downstream of it. In the latter case, the flow becomes supersonic quite close to the compact object. For gas with a low value of constant angular momentum, hence belonging in between these two regimes, the equations allow for the existence of two sonic points of both types. The possible existence of shocks in low angular momentum flows connected with the presence of multiple critical points in the phase space has been studied from different points of view during the last thirty years. Quasi-spherical distribution of the gas endowed by constant specific angular momentum $\lambda$ and the arisen bistability was studied already by [@1981ApJ...246..314A]. [@1987PASJ...39..309F] studied the existence of critical points for realistic equation of state of the gas and showed the corresponding Rankine-Hugoniot conditions for standing shocks. Later, the significance of this phenomenon related to the variability of some X-ray sources has been pointed out by and soon after, the possibility of the shock existence together with shock conditions in different types of geometries was discussed also by [@1990ApJ...350..281A]. The transonic solution required by the aforementioned boundary conditions is the solution having low subsonic velocity far away from the compact object ($\mathfrak{M}<1$, where $\mathfrak{M}=v/c_{\rm s}$ is the Mach number, $v=-u^r_{\rm BL}$ is the inward radial velocity in Boyer-Lindquist coordinates, and $c_{\rm s}$ is the local sound speed of the gas), and supersonic velocity very near to the horizon ($\mathfrak{M}>1$). Thus the flow can locally pass only through the outer or also through the inner sonic point. The latter is globally achieved due to the shock formation between the two critical points. More recently, the shock existence was found also in the disc-like structure with low angular momentum in hydrostatic equilibrium both in pseudo-Newtonian potential [@2002ApJ...577..880D] and in full relativistic approach [@2012NewA...17..254D]. Regarding the sequence of steady solutions with different values of specific angular momentum, the hysteresis-like behavior of the shock front was proposed in the latter work. Different geometrical configurations with polytropic or isothermal equation of state were studied in the post Newtonian approach with pseudo-Kerr potential [@SAHA201610]. In the general relativistic description the dependence of the flow properties (Mach number, density, temperature and pressure) in the close vicinity of the horizon was studied by [@Das201581], and the asymmetry of prograde versus retrograde accretion was shown. The complete picture of the accreting microquasar consisting of the Keplerian cold disc and the low angular momentum hot corona, the so called two component advective flow (TCAF), was described by [@1995ApJ...455..623C]. This model combined with the propagating oscillatory shock (POS) model was later used to explain the evolution of the low frequency QPO during the outburst in several microquasars and it was also implemented into the `XSPEC` software package [@doi:10.1093/mnrasl/slu024]. The presence of the low angular momentum component in the accretion flow during the outbursts of microquasars seems to be essential for explaining different timing in the hard and soft bands, especially during the rising phase [@0004-637X-565-2-1161]. [@DEBNATH20132143] showed with the spectral fitting, that the flux of the power-law component is higher than the flux of the disc black body in the time period, when the QPOs are seen in H 1743-322. [@doi:10.1093/mnrasl/slu024] showed similar results with the TCAF model, which gives the mass accretion rate of the disc component comparable to the accretion rate of the sub-Keplerian component, when the QPO is seen in three different sources. Further development in this topic includes numerical simulations of low angular momentum flows in different kinds of geometrical setup. Hydrodynamical models of the low angular momentum accretion flows have been studied already in two and three dimensions, e.g. by [@2003ApJ...582...69P], [@2008ApJ...681...58J] and [@2009ApJ...705.1503J]. In those simulations, a single, constant value of the specific angular momentum was assumed, while the variability of the flows occurred due to e.g. non-spherical or non-axisymmetric distribution of the matter. The level of this variability was also dependent on the adiabatic index. However, these studies have not concentrated on the existence of the standing shocks as predicted by the theoretical works mentioned above. The consideration was also put on the problem of viscosity in such flows, especially on the influence of the viscosity on the position of the shock and the shape of the solution [@1990MNRAS.243..610C; @2004MNRAS.349..649C; @2016ApJ...816....7N]. The possible consequences of viscous mechanisms in the shocked accretion flow for the QPOs’ evolution was studied by [@0004-637X-798-1-57]. Later on, [@2016MNRAS.462..850N] added the phenomenon of outflows to the picture. [@doi:10.1093/mnras/stw1327] showed the joint role of viscosity and magnetic field considered in heating of the accretion flow combined with the cooling by Comptonization on the dynamical structure of the global accretion flow. Such models were also studied by hydrodynamical numerical simulations in the pseudo-Newtonian description of gravity e.g. by [@2010MNRAS.403..516G; @2013MNRAS.430.2836G; @2016MNRAS.462.3502D]. Our aim is to provide numerical simulations of low angular momentum flows, which would support or correct the semi-analytical findings about the shock existence and behavior mentioned above. In our previous work, we performed 1D pseudo-Newtonian computations [@our_paper], where we studied the dependence of the shock solution of the parameters and the response of the shock front to the change of angular momentum in the incoming matter. The hysteresis behavior was observed in our simulations and we have seen the repeated creation and disappearance of the shock front due to the oscillations of angular momentum in the flow. Here we aim to provide more advanced numerical study of the flow using the full relativistic treatment of the gravity with the fixed background metric given by the Schwarzschild/Kerr solution, which is performed in one, two and three dimensions. The organization of the paper is as follows. In Section \[s:Shocks\] we briefly recall the semi-analytical treatment of the shock existence in the pseudo-Newtonian approach, which is described in details in [@our_paper]. The numerical setup of our simulations is given in Section \[s:Numerics\], the different initial conditions are described in Subsections \[Ini\_Bondi\], \[Ini\_shock\] and \[Ini\_spin\]. In Section \[s:Results\] we present our results, in particular in \[results\_1D\] we show the 1D simulations with standing or oscillating shock location and we run models with time-dependent outer boundary condition corresponding to periodic change of angular momentum in the incoming matter. The major part of the results is presented in \[results\_2D\], where the outcomes of the 2D simulations with different kind of initial conditions are presented. In \[3D\] we confirm the reliability of the 2D results with two full three dimensional tests. The findings of our study are discussed in Section \[s:Conclusions\]. Appearance of shocks in 1D low angular momentum flows {#s:Shocks} ===================================================== In this paper we follow up our previous study of the flow with constant low angular momentum $\lambda$, which was held in the pseudo-Newtonian framework. Here we briefly recall the semi-analytical results in such setup, which we use as an initial setting for our GR computations (for further details see @our_paper). For the analytical study we consider a non-viscous quasi-spherical polytropic flow with the equation of state $p=K \rho^\gamma$, where $p$ is the pressure, and $\rho$ is the density in the gas. Our EOS holds for the isentropic flow, hence the specific entropy $K$ is constant. Using the continuity equation and energy conservation, we can find the position of the critical point $r_c$ as the root of the equation $$\begin{gathered} \mathcal{E} - \frac{\lambda^2}{2r_c^2} + \frac{1}{2(r_c-1)}-\\ \frac{\gamma+1}{2(\gamma-1)} \left( \frac{r_c}{4(r_c-1)^2} - \frac{\lambda^2}{2r_c^2} \right) = 0, \label{r_c}\end{gathered}$$ where $\mathcal{E}$ stands for energy and where we assume the Paczynski-Wiita gravitational potential in the form of $\Phi(r)=-\frac{1}{2(r-1)}$, so that $r$ is given in the units of $r_g=2GM/c^2$, and where $\lambda$ is the value of specific angular momentum. For a subset of the parameter space ($\mathcal{E}, \lambda, \gamma$) there exists more than one solution of this equation (three actually), hence there are more critical points located at different positions. It can be shown, that only two of the critical points are of a saddle type, so that the solution can pass through it. We will call them the inner, $r_c^{\tt in}$, and the outer, $r_c^{\tt out}>r_c^{\tt in}$, critical points, respectively. We will call this subset as “multicritical region”. For changing $\lambda$ with other parameters kept constant, this region is projected onto one interval of $[\lambda_{\tt min}^{cr},\lambda_{\tt max}^{cr}]$. For decreasing $\mathcal{E}$, the interval is shifting up to a higher angular momentum. Together with equation (\[r\_c\]) determining the values of all variables at the two critical points, the relation for the derivative ${\rm d}v/{\rm d}r$ can be obtained from the continuity equation and the energy conservation, so that the solution can be found by integrating the equations from the critical point downwards and upwards. The resulting two branches of solutions of course have the same parameters ($\mathcal{E}, \lambda, \gamma$), but they differ in value of the constant specific entropy $K$, which is given by $$K = \left( v r^2 \frac{c_s^{\frac{2}{\gamma - 1}}}{\gamma^{\frac{1}{\gamma-1}} \dot{M}} \right)^{\gamma - 1}. \label{konst_K}$$ This is evaluated at the critical point position ($K^{\rm in/out}=K(r^{\rm in/out}, v^{\rm in/out},c_s^{\rm in/out})$, where $\dot{M}$ is the adopted constant accretion rate. Because in our model we study the motion of test matter, which does not contribute to the gravitational field, and we use a simple polytropic equation of state, the accretion rate can be given in arbitrary units and it does not affect the solution. The only possible production of entropy is at the shock front, where jumps in radial velocity, density, and other quantities in the flow occur. Because we are interested in the solution describing the accretion flow, and not winds, the physical requirement for the shock to occur is that the specific entropy at the inner branch is higher than at the outer one ($K^{\tt in} > K^{\tt out}$). Moreover, the shock will appear only if the Rankine-Hugoniot conditions, expressing the conservation of mass, energy and angular momentum at the shock position are satisfied, that means that the equation $$\frac{ \left(\frac{1}{\mathfrak{M}_{\tt in}} + \gamma \mathfrak{M}_{\tt in} \right)^2 }{ \mathfrak{M}_{\tt in}^2(\gamma - 1) + 2 } = \frac{ \left(\frac{1}{\mathfrak{M}_{\tt out}} + \gamma \mathfrak{M}_{\tt out}\right)^2 }{ \mathfrak{M}_{\tt out}^2(\gamma - 1) + 2 } \label{shock}$$ holds at some radius $r_s$. Numerical setup {#s:Numerics} =============== We performed 1D to 3D hydrodynamical simulations of the non-magnetized accreting gas on the fixed background using the `HARMPI` [^1] computational code [@2015MNRAS.454.1848R; @2017MNRAS.467.3604R; @2012MNRAS.423.3083M; @2007MNRAS.379..469T; @2007MNRAS.378.1118M; @2013ApJ...776..105J; @2017ApJ...837...39J] based on the original HARM code [@0004-637X-589-1-444; @0004-637X-611-2-977]. The background spacetime is given by the stationary Kerr solution. The initial conditions are set using Boyer-Lindquist coordinates, and they are transformed into the code coordinates, which are the Kerr-Shild ones. In order to cover the whole accretion structure with sufficiently fine resolution near the black hole we use logarithmic grid in radius with superexponential grid spacing in the outermost region, so that the outer region is covered with low resolution grid and provide the reservoir of gas for accretion. In the innermost region, the grid spans below the horizon thanks to the regularity of Kerr-Shild coordinates, having several zones inside the black hole and the free outflow boundary. The outer boundary condition is mostly given as a free boundary, because the outer boundary is placed far enough from the central region. In the 1D case, when long-term evolution is studied ( $t_f$ up to $5\cdot 10^7 {\rm M}$)), we prescribe the inflow of matter through the outer boundary according to the PN solution. Because the outer boundary is very far away from the centre (typically at $\sim 10^5$ M), the radial inflowing speed is very small and the deviations between GR and PN solution are negligible. The prescribed properties of the inflowing matter can be also time dependent, when the temporal change of the angular momentum of the matter is considered. For 2D computations, the resolution is in the range between 256 x 128, up to 384 x 256 and 576 x 192, while in 3D case we use 256 x 128 x 96 cells. ![a) Profiles of Mach number for four values of energy in the converged stationary state for $\gamma=4/3$ and $\lambda = 3.6M$ at the end of the simulation, $t_f = 10^6$M. Sonic points and shock fronts are located at the points, where the curve crosses the $\mathfrak{M}=1$ line (purple horizontal line). b) The corresponding profiles of density in arbitrary units. c) Radial velocity profiles of the flow $v=- u_{\rm BL}^{\tt r}$ in the units of the speed of light, $c$. \[1D\_sol\]](pic/all_4_Mach_uu_1D_100.png){width="48.00000%"} In our previous work, [@our_paper], we studied the behavior of the shock solution and also its evolution with 1D simulations using the code `ZEUS-MP` [@1992ApJS...80..753S; @2003ApJS..147..197H] supplied by the pseudo-Newtonian Paczynski-Wiita potential , which mimics the strong gravity effects near the black hole. We will refer to the results presented in that paper as the PW simulations. The parameters which we used in that work, corresponded either to the evolving frequency of quasi-periodic oscillations seen in some microquasars (e.g. GRS1915+105 [@1999ApJ...513L..37M], XTE J1550-564 [@1999ApJ...512L..43C], GRO J1655-40 [@1999ApJ...522..397R], or GX 339-4 ), or were estimated from the values holding for Sgr A\* [@2006MNRAS.370..219M; @2012NewA...17..254D]. However, such parameters led to an extended accretion structure, meaning especially that $r_c^{\rm out} \sim 10^4M$ or more. Here, we consider different values of parameters, and we perform higher dimensional simulations. We require that the outer critical point is located inside the computational domain in order to retain the flow subsonic at the outer boundary. However, the total number of grid cells together with the sufficient resolution near the horizon restrict the maximal radius of the outer boundary, even though we use the logarithmic grid in radial direction. Because of a sufficiently large value of the energy in the flow, $\mathcal{E}$, the radius $r_c^{\rm out}$ is located quite close to the centre. The typical values of parameters used in this study are $\mathcal{E} = 0.0005, \gamma = 4/3, \lambda = 3.6M$. The shape of the corresponding solution is given in Fig. \[1D\_sol\]. The evolution of initially non-magnetized gas is simulated with the `HARMPI` package supplied with our own modifications. The code conserves the vanishing magnetic field and there is no spurious magnetic field generated during the evolution. We evolve two different types of initial conditions. In the first case we prescribe $\rho$, $\epsilon$ and $u^r_{\tt BL}$ (radial component of the four-velocity) according to the Bondi solution, and we modify this solution by adding non-zero $u^\phi_{\tt BL}$ component of the four-velocity, where the $u^\alpha_{\tt BL}$ is the four-velocity in the Boyer-Lindquist coordinates. The second option is to prescribe $\rho$, $\epsilon$ and $u^\alpha_{\tt BL}$ in accordance to the 1D shock solution (computed with Paczynski-Wiita potential in section \[s:Shocks\]), and then follow the evolution. Such solution obtained with the simplifying assumptions is of course not the true stationary solution. However, because of the fact, that for the given initial conditions and for chosen global parameters of the system, the gas evolves towards the true solution, the discrepancies between the GR and PN are not expected to be substantial. Within the range of parameter space which allows for a shock solution, this 1D approximation can be used for prescribing the initial conditions. We are interested in the evolution of such initial state towards the correct stationary state. The EOS of the form $p = (\gamma - 1) \rho \epsilon$ is used to close the system of equations, so that the isentropy assumption is not imposed. Hence, the specific entropy $K=p/\rho^\gamma$ is not constant and can evolve during the simulation. The fiducial value of the polytropic index used in our computations is $\gamma=4/3$. Initial conditions – Bondi solution equipped with angular momentum {#Ini_Bondi} ------------------------------------------------------------------ ![Model [B1]{}: Bondi initial conditions with $\lambda^{\rm eq}=3.6$M. The set of four panels shows the initial conditions at $t=0$, in particular the slices of $\mathfrak{M}$ with streamlines of the flow plotted (top panel), its equatorial profile (middle panel), and distributions of the flow density $\rho$ and angular momentum $\lambda$ (bottom panel). \[BondiRotInit\] ](pic/B300_NMRAS_000.png){width="50.00000%"} In our first set of computations we adopted initially $\rho$, $\epsilon$ and $u^r_{\tt BL} = -v$ according to the Bondi solution with $ \epsilon = K \rho^{\gamma-1} /(\gamma-1)$, where $K$ is given by Eq. (\[konst\_K\]) evaluated for the Bondi critical point (solution of Eq. (\[r\_c\]) with $\lambda = 0$). The solution is parametrized by the value of the polytropic exponent $\gamma$ and the energy $\mathcal{E}$, which fix the position of the critical point $r_c^{\rm out}$. We modify the initial conditions by prescribing the rotation according to relations $$\begin{aligned} \lambda = \lambda^{\rm eq} \sin^2{\theta}, \qquad r&>&r_{\rm b}, \label{uphi}\\ \lambda = 0 ,\qquad r&<&r_{\rm a}, \label{uphi_null}\end{aligned}$$ in the Boyer-Lindquist coordinates. Between $r_{\rm a}$ and $r_{\rm b}$ the values are smoothened by a cubic spline. The time component of four-velocity is set from the normalization condition $g_{\mu\nu}u^\mu u^\nu = -1$ assuming $u^\theta_{\tt BL}=0$. The factor $\sin^2 \theta$ in Eq. (\[uphi\]) ensures that angular momentum vanishes smoothly at the axis, hence the maximal value of angular momentum $\lambda^{\rm eq}$ is achieved only in the equatorial plane. One example of such initial conditions is plotted in Fig. \[BondiRotInit\]. Initial conditions – Shock solution {#Ini_shock} ----------------------------------- In the second type of simulations we modify the initial data procedure such that we find the solution with the shock in the same way as in [@our_paper]. The values of $\rho$, $\epsilon$ and $u^r_{\tt BL}$ are set accordingly, with $ \epsilon = K^{\rm in/out} \rho^{\gamma-1} /(\gamma-1)$, where $K^{\rm in/out}$ is given by Eq. (\[konst\_K\]) evaluated at the corresponding critical point $r_c^{\rm in/out}$ for the two branches of solution. ![ Model [C4]{}: Initial data with a shock with $\mathcal{E}=0.0005, \lambda^{\rm eq}=3.72$M. \[K112\_Mach\_Ini\]](pic/K112_NMRAS_000.png){width="50.00000%"} However, the 1D analysis provided in that paper and here in Section \[s:Shocks\] was based on pseudo-Newtonian approximation, while now we use GR MHD code in order to examine the differences between the two approaches. Moreover, the 1D analysis was held under the assumption of the quasi-spherical shape of the flow and the constant value of angular momentum. That in particular means, that the dependence of the mass accretion rate, which is constant along the flow, scales with $r^2$. We cannot straightforwardly extend this model into 3D, in other words we can not set the spherical distribution of matter with the angular momentum which would be constant everywhere, because we need to avoid a non-zero angular momentum near the vertical axis. However, the choice of the angular momentum profile in $\theta$ direction is arbitrary, if it drops to zero near the axis. Therefore, we choose two different profiles of angular momentum for the initial and boundary conditions, to see how much the results depend on this distribution. ### Angular momentum scaled by $\sin^2 \theta$ {#Ini_shock_sph} The first choice is the same as in Section \[Ini\_Bondi\], given by Equations (\[uphi\]) and (\[uphi\_null\]). In this case, the maximal value of angular momentum $\lambda^{\rm eq}$ is obtained only in the equatorial plane and it is lower elsewhere. This type of initial conditions is shown on Fig. \[K112\_Mach\_Ini\]. ### Constant angular momentum in a cone {#Ini_shock_cone} ![ Initial data for the shock and constant angular momentum in a cone, $\mathcal{E}=0.0005, \lambda^{\rm eq}=3.65$M, $\theta_c=\pi/4$ (model [H3]{}). \[K430\_Mach\_Ini\] ](pic/K430_NMRAS_000.png){width="50.00000%"} In the second case we prescribe constant angular momentum in a cone with the half-angle $\theta_c$ centered along the equatorial plane. The values of $\lambda$ are smoothed down from the cone towards the axis. For this smoothening we choose a cubic spline given by the relations: $$\begin{aligned} \lambda = \lambda^{\rm eq} f \qquad \qquad \quad \qquad && \\ f = \frac{\theta^2(3\theta_c - 2\theta)}{\theta_c^3}, \quad \theta<\frac{\pi}{2}-\theta_c \\ f = 1, \quad \theta\in[\frac{\pi}{2}-\theta_c,\frac{\pi}{2}+\theta_c] \label{lambda-cone}\\ f= \frac{(\theta-\pi)^2(\frac{\pi}{2} + 3\theta_c - 2\theta)}{(\theta_c-\frac{\pi}{2})^3}, \quad \theta>\frac{\pi}{2}+\theta_c\end{aligned}$$ Thus, all gas within the cone has the maximum angular momentum of $\lambda^{\rm eq}$, which resembles more the assumptions of the 1D model. We show the initial conditions for this case in Fig. \[K430\_Mach\_Ini\].   Because of these modifications of angular momentum distribution, such initial conditions are not expected to be the stationary solution in higher dimensions. However, our experience with 1D simulations shows, that only the presence of the inner sonic point is essential for the creation of the shock in the flow, and the exact stationary solution is not needed. If the inner sonic point is present, then the shock bubble shape adjusts itself after a short transient time into the appropriate form. ![ Top panel: Position of the shock front depending on angular momentum for different values of energy. In case of oscillations, the minimal and maximal shock position is shown. Comparison with the shock front position obtained in the semi-analytical approach using Paczynski-Wiita potential is shown with small dots. Bottom panel: The amplitude $\mathcal{A}$ of the oscillations of the mass accretion rate and its frequency for the oscillating models. \[1D\_shock\] ](pic/obr_1D_rs-eps-converted-to.pdf "fig:"){width="48.00000%"} ![ Top panel: Position of the shock front depending on angular momentum for different values of energy. In case of oscillations, the minimal and maximal shock position is shown. Comparison with the shock front position obtained in the semi-analytical approach using Paczynski-Wiita potential is shown with small dots. Bottom panel: The amplitude $\mathcal{A}$ of the oscillations of the mass accretion rate and its frequency for the oscillating models. \[1D\_shock\] ](pic/obr_1D-ampl-frek-eps-converted-to.pdf "fig:"){width="48.00000%"} Initial conditions for spinning black hole {#Ini_spin} ------------------------------------------ For a spinning black hole we use the initial data described in \[Ini\_shock\_sph\] (angular momentum is scaled by $\sin^2\theta$). Our semi-analytical 1D shock solution is obtained for the Schwarzschild black hole only. In the case of spinning black hole, the relevant values of angular momentum for the shocks can vary significantly and can be outside the possible existence of shocks in the Schwarzschild space time. Hence, we use two different values of angular momentum: (i) $\lambda^s$, to find the semi-analytic shock solution in non-spinning space time, according to which $\rho$, $\epsilon$, $u^r_{\tt BL}$ and $K$ are set, and (ii) $\lambda^{\rm eq}_g$, which is a different value, according to which the rotation is prescribed using Equations (\[uphi\]) and (\[uphi\_null\]), and the normalization condition for the four-velocity. The value of $\lambda^s$ is not important for the long term evolution; it only enables us to find a configuration with shock which is used for the initial conditions, so that there exists an inner sonic point at the initial time. However, the gas is rotating with $\lambda^{\rm eq}_g$, which determines the evolution of the flow. After a short transition time, the other variables distribution adjusts to the actual angular momentum. Results {#s:Results} ======= 1D computations {#results_1D} --------------- In the general relativistic framework, the local sound speed relative to the fluid is given by $$c_s ^{2} = \gamma p \left(\rho +\frac{ \gamma p}{\gamma -1}\right)^{-1},$$ and the radial Mach number is obtained as $\mathfrak{M} = - u_{\rm BL}^{\tt r}/c_{\rm s}$. The sonic points are the points, where the flow smoothly passes from subsonic to supersonic motion, that is $\mathfrak{M} = 1$, and its value increases inwards. The shock front is at the place, where $\mathfrak{M}$ discontinuously changes from $\mathfrak{M}>1$ to $\mathfrak{M}<1$ along the flow (with decreasing $r$). ### Properties of the flow for a constant angular momentum ![ Position of the shock front and the sonic points for three different models with changing angular momentum at the outer boundary as a function of time. Model [A1]{}: $\lambda^{\rm eq} (0)=3.67$M, $A=0.16$M, $P=2\cdot10^6$M, model [A2]{}: $\lambda^{\rm eq} (0)=3.655$M, $A=0.2$M, $P=2\cdot10^6$M, model [A3]{}: $\lambda^{\rm eq} (0)=3.68$M, $A=0.18$M, $P=2\cdot10^6$M. All models have $\mathcal{E}=0.005, \gamma=4/3$.\[1D\_loop\] ](pic/obr_oscillations_rs-eps-converted-to.pdf){width="48.00000%"} The resulting profile of the Mach number $\mathfrak{M}$ for the converged stationary state is shown in Fig. \[1D\_sol\] for four different values of $\mathcal{E}$. The inferred shock and outer sonic point locations are given in Table \[t:1D-shock\]. $\mathcal{E}$ 0.00002 0.0001 0.0005 0.0025 ------------------------- ------------------ ------------------ ------------------ ------------------ $r_s$ 31 32 36 64 $r_{\rm son}^{\rm out}$ $3.7 \cdot 10^4$ $7.5 \cdot 10^3$ $1.5 \cdot 10^3$ $2.8 \cdot 10^2$ $\mathcal{R}$ 10 9.4 7.6 4 : Location of shock front and the outer sonic point of stationary solutions for different values of $\mathcal{E}$. The values in geometrized units (\[M\]) are inferred from the solution at the end of the simulation, at $t_f=10^6$M. \[t:1D-shock\] For higher energy, the outer sonic point is closer to the black hole unlike the shock position, which is located farther, and the outer supersonic region of the flow is thus shrinking. The strength of the shock is anti-correlated with the energy (and with the location of the shock), so that for increasing energy the ratio of the post-shock density to the preshock density ($\mathcal{R}$) is decreasing (see Table \[t:1D-shock\] and panel b) in Figure \[1D\_sol\]). In comparison with the pseudo-Newtonian analytical estimate, which put the shock position for $\mathcal{E}=0.0001$ at $r_s^{\rm PW}(0.0001) = 21M$, the GR computation tends to put the shock farther from the black hole. On the other hand, the minimal stable shock position is very similar, because in GR computations the shock exists for slightly lower values of $\lambda$, which can be seen on Fig. \[1D\_shock\]. Here the dependence of the shock front position on angular momentum is shown for both the pseudo-Newtonian and GR computations. The radial extend of possible shock existence agrees very well between PW and GR results, however for the same value of angular momentum the GR shock is located farther from the black hole. Similarly like in the case of PW simulations, which we presented in [@our_paper], also in the GR case computed with `HARMPI` we have found oscillations of the shock front for higher angular momentum, which causes also the oscillation of the mass accretion rate through the inner boundary. In Fig. \[1D\_shock\] in the top panel we show the minimal and maximal shock positions during the simulation for the oscillating cases. In the bottom panel, we show the amplitude of the oscillations of the mass accretion rate, which is computed as the ratio of the difference between the maximal and minimal accretion rate to its mean value $\mathcal{A} = ({\rm max}(\dot{M}) - {\rm min}(\dot{M}))/\bar{\dot{M}}$. This ampplitude and the corresponding frequency is presented for the oscillating models. -- ----------- ------- ------ ------ ------ ------ ------- ------ $\lambda$ 3.20 3.21 3.22 3.25 3.30 3.42 3.43 $r_s$ – 11.9 14.0 19.8 31.6 172.9 – $\lambda$ 3.245 3.25 3.26 3.30 $r_s$ – 10.7 13.3 20.7 -- ----------- ------- ------ ------ ------ ------ ------- ------ : Location of shock front, if existent, with $a=0.3,\gamma=4/3$ for different values of $\lambda$ for $\mathcal{E}_1=0.002$ and $\mathcal{E}_2=0.0000033$. The values in geometrized units (\[M\]) are inferred from the solution at the end of the simulation. \[t:1D-Das\] The general relativistic semi-analytical study of the shock solutions in Kerr metric was given in [@2012NewA...17..254D], however those authors considered the disc in hydrostatic equilibrium with vertically averaged quantities. The shape of the different regions in the parameter space is given in [@2012NewA...17..254D], Fig. 1 for $a=0.3$. On Fig. 3 in that paper, the authors show the parameter space of possible shock existence for $\tilde{\mathcal{E}}=1.0000033, \gamma=4/3, a=0.3$, where $\tilde{\mathcal{E}}$ is the total specific energy and corresponds to $\tilde{\mathcal{E}} = \mathcal{E}+1$. To compare their solutions with our results, we performed a set of simulations with $a=0.3$ for $\mathcal{E}_1=0.002$ and $\mathcal{E}_2=0.0000033$. The results are summarised in Table \[t:1D-Das\]. For $\mathcal{E}_1$ [@2012NewA...17..254D] predicts that the shock exists in the subset of the region A (in particular in the region A$_{\rm S}$, which is not shown in the figure), which gives approximately the range 2.8 M $< \lambda <$ 3.08 M. Our simulations show the shock existence for higher values of angular momentum, in particular the shock front is accreted up to $\lambda=3.2$ M and the stationary shock exists for $\lambda \in [3.21 \rm{M},3.42 \rm{M}]$. For $\mathcal{E}_2$, those authors predict the shock existence for $\lambda > 3.239$M. We have found the stable shock solution for $\lambda=3.25$ M, while for $\lambda=3.245$ M the shock is accreted. When the angular momentum is increased, the oscillations develop, for $\lambda=3.4$ M their frequency $f\sim 2.9\cdot 10^{-4}$ M$^{-1}$ with the amplitude $\mathcal{A} = 1.45$. Because a different configuration is considered in the two papers (disc in vertical hydrostatic equilibrium versus quasi-spherical flow), some differences are expected and they are more prominent for higher energy of the flow. Quite recently, [@2016arXiv161206882T] studied the low angular momentum flows with shocks in the Kerr spacetime in several different geometries with different physical conditions at the shock front. They included also the case of the conical flow with energy-preserving shock, which is close to our scenario. Figure 2 of [@2016arXiv161206882T] shows, that indeed the multicritical region for the quasi-spherical flow exists for higher values of angular momentum than for the flow in vertical hydrostatic equilibrium. The quantitative comparison of our results with this paper will be given in the future work. ### Properties of the flow for a time dependent angular momentum ![Model [B1]{}: The snapshot at the end of the simulation shows the evolved gas at $t=10^4$M, which settles into the outer branch, no shock appears during the evolution. \[BondiRot\] ](pic/B300_NMRAS_205.png){width="50.00000%"} Further, we repeated the PW computations done previously with the code `ZEUS` [@our_paper], regarding the hysteresis loop connected with the shock existence with changing angular momentum of the incoming matter. In this case we prescribe the angular momentum of the matter coming through the outer boundary according to the time dependent relation: $$\lambda^{\rm eq} (t) = \lambda^{\rm eq} (0) - A\sin(t/P),$$ where $A$ is the amplitude and $P$ is the period of the perturbation. If we choose $A$ and $P$ such that the angular momentum crosses the boundary of the multicritical region from below and from above, we observe the creation and disappearance of the shock front, similarly like it was in the PW case. The comparison of the shock front movement for three different models is given in Figure \[1D\_loop\]. Model [A1]{} with $\lambda^{\rm eq} (0)=3.67$ M, $A=0.16$ M, $P=2\cdot10^6$ M does not exceed the boundary of the shock existence interval from neither side. The shock is moving in accordance with the changing angular momentum to and from the black hole, spanning the region (16.6 M, 672.5 M). Both sonic points exist persistently during the evolution. Model [A2]{} with $\lambda^{\rm eq} (0)=3.655$M, $A=0.2$M, $P=2\cdot10^6$M crosses the shock existence boundary on both sides. As a consequence, the shock front is being accreted very quickly after it reaches the minimal stable shock position. The time scale of this event is given by the advection time from the minimal stable shock position and does not depend on the perturbation parameters $A$ and $P$. From that moment the inner sonic point vanishes and the flow follows the outer Bondi-like branch of solution (for which only the outer sonic point exists) until the time, when the angular momentum increases such that the outer solution no longer exists. At that point, the shock is formed at the position of the inner sonic point and it is moving very fast outward, where it merges with the outer sonic point, so that the type of accretion flow with the inner sonic point only is established. Later, when the angular momentum of the flow is decreasing again, the shock forms at the outer sonic point and it moves slowly towards the black hole. The third example (model [A3]{} with $\lambda^{\rm eq} (0)=3.68$M, $A=0.18$M, $P=2\cdot10^6$M) shows the case, where only the upper boundary is crossed. Here we have the shock moving slowly towards and away from the centre, where it merges with the outer sonic point. For some time only the accretion with the inner sonic point exists and then the shock appears again. In this case, the timescale of such changes and the velocity of the shock front is uniquely given by the parameters $A$ and $P$. ![ Model [B2]{}: Bondi initial data with $\mathcal{E}=0.0025, \lambda^{\rm eq}=3.8$M. The shock is expanding (the first snapshot at $t=10^3$M) until it merges with the outer sonic surface (second snapshot at $t=2 \cdot 10^4$M). Note the different spatial range of the second snapshot. \[B10\_II\] ](pic/B10_II_NMRAS_050.png "fig:"){width="50.00000%"} ![ Model [B2]{}: Bondi initial data with $\mathcal{E}=0.0025, \lambda^{\rm eq}=3.8$M. The shock is expanding (the first snapshot at $t=10^3$M) until it merges with the outer sonic surface (second snapshot at $t=2 \cdot 10^4$M). Note the different spatial range of the second snapshot. \[B10\_II\] ](pic/B10_II_NMRAS_1000.png "fig:"){width="50.00000%"} The case, in which only the lower boundary is crossed, is not shown, because in this case the shock front is accreted during the first cycle and never appears again. Hence, under certain circumstances, the standing shock can appear in the low angular momentum flow during accretion. Then it can either (i) stay at certain position, (ii) move slowly downwards or upwards in the accretion flow with velocity given by the rate of change of angular momentum given by parameters $A$ and $P$ in our model, (iii) be accreted quickly from the the minimal stable shock position (close to the black hole), or (iv) be formed close to the black hole and move quickly towards the outer sonic point. In the last two cases, the velocity of the shock front does not depend on the perturbation parameters $A$ and $P$, but it is given by the properties of the medium. We will study this issue in the next section with 2D simulations; in general, the velocity of the shock front is a few times lower than the sound speed in the preshock medium. ![ Model [B2]{}: expanding shock with Bondi initial data. We track the position of the transonic points (i.e. $(\mathfrak{M}[i] -1)(\mathfrak{M}[i+1] -1) <1$) in the equatorial plane ($i$ is the index of the radial coordinate on the grid) and we labeled the sonic points ( $\mathfrak{M}[i]>\mathfrak{M}[i+1]$) by yellow points and the shock position ( $\mathfrak{M}[i]<\mathfrak{M}[i+1]$) by brown points. \[B10\_II\_rs\] ](pic/obr_B10_II-eps-converted-to.pdf){width="50.00000%"} ![ Behaviour of the shock front for different values of $\lambda^{\rm eq}$ with $\mathcal{E}=0.0025$. Panel a) shows the position of the shock front, panel b) shows its velocity $v_s$ and panel c) displays the ratio of the preshock medium sound speed $c^{\rm out}_s$ to the shock front velocity $v_s$ during the evolution. \[2D\_Rs\_position\] ](pic/obr_Rychlost-rs-lin-lambda-eps-converted-to.pdf "fig:"){width="48.00000%"} ![ Behaviour of the shock front for different values of $\lambda^{\rm eq}$ with $\mathcal{E}=0.0025$. Panel a) shows the position of the shock front, panel b) shows its velocity $v_s$ and panel c) displays the ratio of the preshock medium sound speed $c^{\rm out}_s$ to the shock front velocity $v_s$ during the evolution. \[2D\_Rs\_position\] ](pic/obr_rychlost-soku-lin-lin-zoom-eps-converted-to.pdf "fig:"){width="48.00000%"} ![ Behaviour of the shock front for different values of $\lambda^{\rm eq}$ with $\mathcal{E}=0.0025$. Panel a) shows the position of the shock front, panel b) shows its velocity $v_s$ and panel c) displays the ratio of the preshock medium sound speed $c^{\rm out}_s$ to the shock front velocity $v_s$ during the evolution. \[2D\_Rs\_position\] ](pic/obr_Rychlost-pomer-csout-to-vs-lambda-eps-converted-to.pdf "fig:"){width="48.00000%"} 2D computations {#results_2D} --------------- After confirming the general outcomes of our previous PW study with full GR 1D simulations, we now continue with simulations in higher dimensions. Because our initial data are rotationally symmetric (does not depend on $\phi$), is it possible to evolve only a slice spanning ($r,\theta$), and with constant $\phi$, thus assuming that the system will remain rotationally symmetric during the evolution. In order to justify to what extend and under which conditions this assumption is valid the comparison with 3D computation was needed. We will comment on that in Section \[3D\]. However, within the 2.5-D approach, the $\phi$ component of the four-velocity is considered. We prescribe this component in Boyer-Lindquist coordinates according to relations given by Eqs. (\[uphi\]) and (\[uphi\_null\]). We chose several exemplary simulations and we show their results in the form of snapshots. Every snapshot contains four panels with the slices of $\mathfrak{M}$ and its equatorial profile, $\rho$ in arbitrary units, and $\lambda$ in geometrized units, and it is labelled by the time $t$ in geometrized units, where $M$ is the mass of central black hole. The axes show the position in geometrized units. The red colour in the slice of radial Mach number corresponds to the supersonic motion, blue regions indicate the subsonic accretion. The shock front is located at the place, where the abrupt change from supersonic to subsonic motion occurs, hence it is represented with the white curve separating red region farther from the centre and blue region closer to the centre. The sonic curves are also white, but they lie between the blue region farther and red region closer to the centre. On top of the radial Mach number map the velocity streamlines are plotted in blue. These streamlines are computed from the velocity field at the given instant of time, hence they do not represent the actual history of a certain fluid parcel. ![ The same as in Fig. \[2D\_Rs\_position\] for different values of $\mathcal{E}$ with $\lambda^{\rm eq}=3.85$ M. \[2D\_Rs\_position\_eps\] ](pic/obr_Rychlost-rs-lin-eps-eps-converted-to.pdf "fig:"){width="48.00000%"} ![ The same as in Fig. \[2D\_Rs\_position\] for different values of $\mathcal{E}$ with $\lambda^{\rm eq}=3.85$ M. \[2D\_Rs\_position\_eps\] ](pic/obr_Rychlost-lin-eps-eps-converted-to.pdf "fig:"){width="48.00000%"} ![ The same as in Fig. \[2D\_Rs\_position\] for different values of $\mathcal{E}$ with $\lambda^{\rm eq}=3.85$ M. \[2D\_Rs\_position\_eps\] ](pic/obr_Rychlost-pomer-csout-to-vs-eps-eps-converted-to.pdf "fig:"){width="48.00000%"} ### Bondi solution equipped with angular momentum At the beginning we check, that for pure Bondi solution with $\lambda=0$ we get the stationary solution of the flow with the properties consistent with the analytical solution. After that we start to study the slowly rotating flows. For the first simulation, we pick the parameters such that $\lambda^{\rm eq}$ belongs into the multicritical region (the region in the parameter space, where both the inner and outer sonic points exist in the 1D model), but it is very close to its upper bound, in particular $\mathcal{E}=0.0025~M$, $\lambda^{\rm eq} = 3.6~M$, and $\gamma=4/3$. In that case, even when the shock solution is possible, it is not expected to appear, because the initial configuration is closer to the outer branch (there is no inner sonic point in the Bondi initial data). In other words, we expect, that the rotation of the gas affects the profile of the Mach number in the innermost region in such a way, that it gets very close to 1, but does not touch it. In Fig. \[BondiRotInit\], the initial conditions at $t=0M$ are shown and the final state at $t_f=10^4M$ of the gas is depicted in Fig \[BondiRot\]. At the later time, the simulation is already relaxed to the stationary state, which resembles the outer branch. It is interesting to note, that for $t=t_f$ the Mach number actually crosses the $\mathfrak{M}=1$ line, however only on a very short radial range. We would expect, that at the moment, when the inner sonic point appears, the shock creates and expands. The reason, why it is not so in this simulation, is that the angular momentum is very close to the upper bound of the multicritical region, which is a boundary between the two distinct types of evolution. In such case, the numerical evolution depends also on the resolution of the grid and other numerical settings. In other words, if the parameters are close to such boundaries, then the simulations with different resolution can lead to a different type of evolution (i.e., the shock creates or not), hence it is difficult to find the exact critical value of the parameter. This confirms our previous observation [@our_paper], that in the multicritical region the evolution of the flow can tend to either the outer “Bondi-like” branch or to the shock solution, and the choice between these two possibilities is given by the initial conditions. In particular, the profile of the Mach number in the innermost region and the presence of the inner sonic point is crucial for the shock development. ![ Model [C4]{}: Initial data with a shock with $\mathcal{E}=0.0005, \lambda^{\rm eq}=3.72$M. The shock bubble develops oscillations and changes size quasiperiodically, but does not acretes, nor does it expand outside $r_{\rm out}$. \[K112\_Mach\] ](pic/K112_NMRAS_1255.png "fig:"){width="50.00000%"} ![ Model [C4]{}: Initial data with a shock with $\mathcal{E}=0.0005, \lambda^{\rm eq}=3.72$M. The shock bubble develops oscillations and changes size quasiperiodically, but does not acretes, nor does it expand outside $r_{\rm out}$. \[K112\_Mach\] ](pic/K112_NMRAS_2000.png "fig:"){width="50.00000%"} In [@PTAproc] we showed a similar computation with $\lambda^{\rm eq}=3.79M, \mathcal{E}=0.001$, which is also close to the upper bound of the multicritical region. Those computations were done in 3D with a different GR code, `Einstein toolkit`[^2] [@ETtoolkit]. The qualitative and quantitative agreement with the 1D solution is discussed there. However the fact, that two very different codes working on different kind of grids (spherical grid logarithmic in radius versus Cartesian-like grid with the mesh refinement) show similar results is a good support for our results. Last example is the case, when $\lambda^{\rm eq}$ is above the multicritical region, hence the outer type of solution is not possible. Therefore, the prescribed initial conditions are not close to a physical solution and the shock bubble has to appear. However, the position of the shock front is not stable and the shock is expanding until it meets the outer sonic point and the distant supersonic region dissolves, yielding only the inner type of accretion. Two snapshots of the evolution in times $10^3M$ and $2\cdot 10^4M$ are given in Fig. \[B10\_II\]. On the first set of panels, the emergent shock bubble is seen, which is the expanding subsonic and more dense region. Even though the shock front is thought to be very thin, in the numerical simulation it spans across several zones, which can be seen on the equatorial profile of $\mathfrak{M}$. In the last snapshot, the outer supersonic region is already dissolved and only a small supersonic region very close to the black hole can be found. The time dependence of the shock and sonic point positions in the equatorial plane is shown in Fig. \[B10\_II\_rs\], where it can be seen, that the shock meets the outer sonic point at about $t\sim15000~M$, which corresponds to about 0.75 s in case of a typical microquasar with $M=10~M_\odot$. This is way too short in comparison with the observational data concerning the quasiperiodic oscillations, which change frequency during few weeks. Hence, if we want to use an oscillatory shock front model to explain the observed low-frequency QPOs, we need to obtain a solution with long lasting shock front oscillating around the slowly changing mean position, and not the fast expanding shock bubble like was observed in this case. The first step is to find an existing stationary solution with a standing shock. As we already mentioned, such solution is not expected to occur, if we start the evolution with initial conditions close to the Bondi solution. We can be, however, interested in the dependence of the shock propagation on the properties of the gas. On that account, we performed two sets of simulations, where we changed the angular momentum and the energy of the flow such that they are above the multicritical solution, and we followed the expansion of the shock front until it met the outer sonic point. In Fig. \[2D\_Rs\_position\] we show the behaviour of the shock front during the evolution for several different values of $\lambda^{\rm eq}$. For $\lambda^{\rm eq}=3.65$ M, just above the multicritical region, the growth of the shock bubble is slow with a long transient period, during which it is unclear, if the bubble converges to the stationary state, or it rather expands outwards. With increasing $\lambda^{\rm eq}$ the bubble expands faster. The velocity of the shock front ranges between $0.005c$ up to $0.4c$ for the highest angular momentum. Panel c) shows the ratio of the preshock medium sound speed $c^{\rm out}_s$ to the shock front velocity $v_s$. Except for the highest $\lambda^{\rm eq}=10$ M, the shock front expands by a velocity, which is a few times lower than the corresponding preshock sound speed, for the lowest $\lambda^{\rm eq}=3.65$ M the ratio $c^{\rm out}_s/v_s$ reaches the value up to 6. In Fig. \[2D\_Rs\_position\_eps\] we present similar results obtained for different values of $\mathcal{E}$. Here, the position of the outer sonic point differs significantly for different $\mathcal{E}$, hence also the time for the shock front to reach the outer sonic point varies considerably. The plots are therefore given in logarithmic scale of time. ![ Model [C4]{}: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary. \[K112\_rs\] ](pic/obr_112-rs-eps-converted-to.pdf){width="50.00000%"} ![ Model [H3]{}: Initial data with a shock and constant angular momentum distribution ($\mathcal{E}=0.0005, \lambda=3.65$M) in the cone with the half-angle $\theta_c=\pi/4$. \[K430\_Mach\] ](pic/K430_NMRAS_200.png "fig:"){width="50.00000%"} ![ Model [H3]{}: Initial data with a shock and constant angular momentum distribution ($\mathcal{E}=0.0005, \lambda=3.65$M) in the cone with the half-angle $\theta_c=\pi/4$. \[K430\_Mach\] ](pic/K430_NMRAS_1800.png "fig:"){width="50.00000%"} The velocity of the shock front decreases with the decreasing energy, and the same holds for its ratio to the sound speed of the preshock medium. The trend is similar as for decreasing angular momentum. The reason is, that for decreasing energy, the multicritical region exists for higher value of angular momentum, as we have seen earlier. Hence, when we decrease the energy and keep the angular momentum constant, we are approaching the multicritical region in the parameter space. That is well documented on the case with the lowest energy $\mathcal{E}=0.0005$ (the purple lines in Fig. \[2D\_Rs\_position\_eps\]), which lies very close the multicritical region boundary and so the shock bubble waits for a very long time before it starts to expand. Therefore, we conclude, that the shock front velocity depends on the distance of our chosen parameters ($\lambda, \mathcal{E}$) from the multicritical region in the parameter space and that it is typically a few times lower than the sound speed in the preshock medium. ### Shock solution Because we want to find out, if there are any stationary or oscillating solutions with shocks also in 2D, similarly as in 1D, we have to prescribe the initial conditions, which are closer to the shock solution branch than the Bondi-like solution. These initial conditions are described in Section \[Ini\_shock\]. Inspired by the range of shock solution in the PW case, we performed a set of simulations with changing angular momentum $\lambda^{\rm eq} \in [3.52M,3.7M]$, while keeping $\mathcal{E}=0.0025$ and $\gamma=4/3$. (Note that the range in 1D paper was defined for $\mathcal{E}=0.0001$, not $0.0025$. Here, the range is such, that for the lowest $\lambda$ the shock bubble accretes and for the higher values it expands, so it covers the whole interesting interval.) These simulations showed, that for lower angular momentum the stationary state is obtained, while for higher angular momentum the shock bubble again expands and merges with the outer sonic surface, which is located around 280M. The details of these computations are given in [@IAUS_proc]. When we choose lower value of energy, the outer sonic surface is located further, so that there is more space, where the shock existence is possible. We performed several simulations with $\mathcal{E}=0.0005$ and $\mathcal{E}=0.0001$, for which $r_{\rm out}=1475~M$ and $r_{\rm out}=7486~M$, respectively. ![Model [H3]{}: The shock and sonic point position and the corresponding mass accretion rate through the inner boundary. \[K430\_rs\] ](pic/obr_K430_rs_mdot-eps-converted-to.pdf){width="50.00000%"} For higher values of angular momentum we found the shock bubble unstable - the eddies emerge in the flow, the bubble is growing for some time, after which a quick accretion occurs accompanied by shrinking of the bubble, which is also oscillating in vertical direction. However, it does not expand or accrete completely. Several snapshots of the evolution are given in Fig. \[K112\_Mach\], where the shock bubble has different shape and size at different times and the asymmetry with respect to equator can be seen. The time dependence of the shock front position in the equatorial plane is given in Fig. \[K112\_rs\]. $\mathcal{E}$ $\lambda^{\rm eq}$ \[M\] $a$ IC res shock $r_s$ \[M\] $\bar{r}_s$ \[M\] f \[M$^{-1}$\] Fig -------- --------------- -------------------------- ------ --------- ---------------- ------- ------------- ------------------- ------------------------- --------------------------------- -- [B1]{} 0.0025 3.6 0 Bondi 256 x 128 no – – \[BondiRot\] [B2]{} 0.0025 3.8 0 Bondi 256 x 128 EX 5 – 282 – \[B10\_II\], \[B10\_II\_rs\] [C1]{} 0.0005 3.58 0 1D+sph 384 x 192 AC – – [C2]{} 0.0005 3.59 0 1D+sph 384 x 192 OS 28 – 41 35 $\sim 3\cdot 10^{-3}$ [C3]{} 0.0005 3.6 0 1D+sph 384 x 192 OS 31 – 45 39 $\sim 5\cdot 10^{-3}$ [C4]{} 0.0005 3.72 0 1D+sph 384 x 192 OS 29 – 177 106 $\sim 1\cdot 10^{-5}$ \[K112\_Mach\], \[K112\_rs\] [C5]{} 0.0005 3.78 0 1D+sph 384 x 192 OS 10 – 521 216 $\sim 4\cdot 10^{-6}$ [D1]{} 0.0001 3.6 0 1D+sph 384 x 192 AC – – [D2]{} 0.0001 3.72 0 1D+sph 384 x 192 OS 26 – 62 43 $\sim 6\cdot 10^{-5}$ [D3]{} 0.0001 3.78 0 1D+sph 384 x 192 OS 21 – 146 70 $\sim 2.2\cdot 10^{-5}$ [D4]{} 0.0001 3.82 0 1D+sph 384 x 192 OS 10 – 226 86 $\sim 1.7\cdot 10^{-5}$ [D5]{} 0.0001 3.86 0 1D+sph 384 x 192 OS 10 – 359 124 $\sim 1.9\cdot 10^{-5}$ [E1]{} 0.0005 3.34 0.3 1D+spin 384 x 192 AC – – [E2]{} 0.0005 3.35 0.3 1D+spin 384 x 192 OS 18 – 36 30 $\sim 7\cdot 10^{-3}$ [E3]{} 0.0005 3.40 0.3 1D+spin 384 x 192 OS 32 – 81 59 $\sim 2.3\cdot 10^{-3}$ [E4]{} 0.0005 3.48 0.3 1D+spin 384 x 192 OS 26 – 200 100 $\sim 2\cdot 10^{-5}$ [E5]{} 0.0005 3.49 0.3 1D+spin 384 x 192 EX – – [F1]{} 0.0005 2.7 0.8 1D+spin 384 x 256 AC – – [F2]{} 0.0005 2.8 0.8 1D+spin 384 x 256 OS 13 – 80 39 $\sim 5.5\cdot 10^{-5}$ \[K126\_UHR\],\[K126\_UHR\_rs\] [F3]{} 0.0005 2.9 0.8 1D+spin 384 x 256 EX 100 – 1450 [G1]{} 0.0005 2.4 0.95 1D+spin 576 x 192 OS 10 – 58 29 $\sim 7\cdot 10^{-5}$ [G2]{} 0.0005 2.42 0.95 1D+spin 576 x 192 OS 11 – 74 31 $\sim 8\cdot 10^{-5}$ \[K680\_rs\] [H1]{} 0.0005 3.5 0.0 1D+cone 384 x 192 AC – – [H2]{} 0.0005 3.6 0.0 1D+cone 384 x 256 OS 110 – 134 121 $\sim 2.8\cdot 10^{-5}$ [H3]{} 0.0005 3.65 0.0 1D+cone 384 x 256 OS 98 – 366 235 $\sim 8\cdot 10^{-6}$ \[K430\_Mach\],\[K430\_rs\] [H4]{} 0.0005 3.72 0.0 1D+cone 384 x 192 EX 90 – 1477 – [I]{} 0.0005 3.65 0.0 1D+sph 256 x 128 x 96 OS 68 – 81 76 \[K500\],\[K500\_rs\] ![ Model [F2]{}: moderately spining black hole ($\mathcal{E}=0.0005, a=0.8, \lambda^{\rm eq}=2.8$M) with oscillating shock. \[K126\_UHR\] ](pic/K126_UHR_NMRAS_1400.png){width="50.00000%"} This repetitive process causes also flaring in the mass accretion rate through the inner boundary, see Fig \[K112\_rs\]. For $10~M_\odot$ black hole, $t=10^6~M$ corresponds to approx. 50 s, hence the flares occur on a similar time scale as in some of the flaring states of microquasars, e.g. the heartbeat state of GRS 1915+105 or IGR J17091-3624 [@2012ApJ...747L...4A]. Moreover, we have shown, that the sources in this state show the evidence of nonlinear mechanism behind the emission of the hard component, which corresponds to the low angular momentum flow [@nas_nonlin; @nas_chaos_small]. However, this fact is only an indirect evidence for the presence of the shock, because there are other possible explanations of the flares, e.g. they can be a result of the radiation pressure instability in the disc . Another type of simulations are those with different distribution of angular momentum (see Section \[Ini\_shock\_cone\]). Three snapshots of such evolution are given in Fig. \[K430\_Mach\] and the corresponding shock position and mass accretion rate is given in Fig. \[K430\_rs\]. We found qualitatively similar behavior for different values of parameters. The results are summarized in Table \[t:2D-models\]. For $\mathcal{E}=0.0005$ we identified the interval of angular momentum values for which there is oscillating shock to be $[3.59{\rm M},\,3.78{\rm M}]$ (models [C2]{}-[C5]{}). For $\mathcal{E}=0.0001$ the interval is $\lambda\in[3.72{\rm M},\,3.86{\rm M}]$ (models [D2]{}-[D5]{}). In both series of simulations it is clearly visible, that increasing the value of angular momentum within the interval causes increase of the amplitude of shock oscillations. So oscillations appear sharpest just before the upper limit of the angular momentum range and the oscillation frequency peaks in the spectrum are clearly visible in those cases. We can compare the series of simulations [C1]{}-[C5]{} with [H1]{}-[H4]{} which have the same values of parameters, but differ in distribution of angular momentum (modulation by $\sin^2\theta$ versus constant angular momentum in a cone). It is clear that the results are very similar: in both cases there is a range of angular momentum in which the shock oscillating with comparable frequencies is present, with amplitude of oscillations increasing with angular momentum value. This shows, that the main conclusions from our simulations are resistant to changes in spatial distribution of rotation profile. ### Rotating black hole We chose three values of the spin $a$, which can represent quite well the estimated range of spins of the know microquasars, in particular $a=0.3$ (which could be a representative value for XTE J1550-564, H1743-322, LMC X-3, A0620-00), $a=0.8$ (M33 X-7, 4U 1543-47, GRO J1655-40) and $a=0.95$ (Cyg X-1, LMC X-1, GRS 1915+105). The estimated values of the spin are taken from [@McClintock2014]. For $a=0.3$ and $\lambda^{\rm eq}_g \in[3.35~\mathrm{M},3.48~\mathrm{M}]$ we observed a long lasting oscillating shock front. For $\lambda^{\rm eq}_g=3.34~\mathrm{M}$ (and lower values) the shock is accreted, and for $\lambda^{\rm eq}_g=3.49~\mathrm{M}$ (and higher values) the shock surface expands. The Table \[t:2D-models\] contains details about five selected simulations of $a=0.3$ scenarios: values on both sides of left and right boundary of interval and one in the middle of the interval of $\lambda^{\rm eq}_g$ values corresponding to stable shock existence. ![Model [F2]{}: moderately spining black hole ($\mathcal{E}=0.0005, a=0.8, \lambda^{\rm eq}=2.8$M) with oscillating shock. \[K126\_UHR\_rs\] ](pic/obr_K126_UHR-eps-converted-to.pdf){width="50.00000%"} Next in the Table \[t:2D-models\] the results of several simulations with $a=0.8$ are shown. In this case the interval of $\lambda^{\rm eq}_g$ for which there are solutions with oscillating shock appears in the range of lower values than for $a=0.3$. For $\lambda^{\rm eq}_g = 2.7$ M (model [F1]{}), the shock is accreted, for $\lambda^{\rm eq}_g = 2.8$ M (model [F2]{}) the accretion flow contains a long lasting oscillating shock front, and for $\lambda^{\rm eq}_g = 2.9$ M (model [F3]{}) the shock bubble is expanding. The qualitative behavior of the shock front is very similar like in the Schwarzschild case, only the exact values of parameters (mainly the angular momentum of the flow) differs. Again increase of the value of angular momentum within the oscillating shock interval corresponds to increase of the amplitude of oscillations. Series of models [E2]{}-[E4]{} is very similar in this regard to series [C2]{}-[C5]{} and [D2]{}-[D5]{}. For higher values of spin we found the computations to be more sensitive on the resolution, mainly in the radial direction close to the black hole. If the resolution is not sufficient, then the the shock bubble can accrete even after a long oscillatory evolution. However when we increase the resolution for the initial conditions, the shock persists in the evolution, even when it comes very close to the black hole. Therefore we increased the radial resolution for $a=0.95$ to $N_{\rm r}=576$. Even with this high resolution, the shock bubbles in simulations with $\lambda^{\rm eq}=2.42$ M and $\lambda^{\rm eq}=2.45$ M are accreted after long evolution with repeated shock front oscillations. We conclude that the qualitative results obtained in the non-spinning case can be generalized for spinning black holes. What can be clearly seen from the simulations, is that the behavior similar to the case of nonrotating BH is observed for much lower values of the angular momentum, so the spin of the black hole “adds” to the rotation of the gas. Increasing the value of the spin of black hole decreases both limiting values of angular momentum of accreted gas and leads to a narrower interval in which the oscillating shock solutions are observed. ![Model [G2]{}: rapidly spinning black hole ($\mathcal{E}=0.0005, a=0.95, \lambda^{\rm eq}=2.42$M) with oscillating shock. \[K680\_rs\] ](pic/obr_K680_rs_mdot-eps-converted-to.pdf){width="50.00000%"} 3D computations {#3D} --------------- We performed two test runs with the initial conditions described in Section \[Ini\_shock\_sph\] with the parameters $\epsilon=0.0005, \lambda^{\rm eq}=3.65, a=0$ in full 3D, and with the resolution $N_{\rm r}$ x $N_{\theta}$ x $N_{\phi}$ equal to 384 x 192 x 64, and 256 x 128 x 96. The simulations were performed on the *Cray* supercomupter cluster, typically using 16 nodes, and the message-passing interface supplemented with the hyperthreading technique was used, similarly as in [@2017ApJ...837...39J]. The code is supposed to conserve the axi-symmetry of the initial state, which we confirm. Because no non-axisymmetric modes appeared, the solution is the same in each $\phi$-slice during the evolution (see Fig. \[fig:3d\]). ![Model [I]{}: Density distribution in the three-dimensional simulation at time 28000M, within the innermost 50 gravitational radii from the black hole. Resolution of this run is 384 x 192 x 64. The colour scale and threshold is adopted to show only the densest parts of the flow. \[fig:3d\] ](pic/torus_3d_t280_rout50.png){width="50.00000%"} We were able evolved the system only up to $t_f = 34100$M, and $t_f=24400$M, for these two runs respectively, which is significantly shorter then in the 2D case. Hence, we cannot directly compare the long term evolution of the flow. However, the main features of the shock bubble evolution, which we observed in 2D case, appears also in 3D simulation. Mainly, the shock bubble has similar shape, as can be seen in Fig \[K500\], and we observe similar oscillations of the bubble and of the mass accretion rate (Fig. \[K500\_rs\]). Conclusions {#s:Conclusions} =========== In this paper we presented an extensive numerical study of the pseudo-spherical accretion flows with low angular momentum. The simulations were performed in the general relativistic framework on the Kerr background metric with the GR MHD code `HARMPI` in one, two and three dimensions. As a first step, we provided set of 1D computations, which we compared to our earlier results obtained within the pseudo-Newtonian approach with the code ZEUS and published in @our_paper. We confirm the qualitative properties of those simulations, which are especially the oscillation of the shock front for higher values of angular momentum and the possibility of the hysteresis effect, when the parameters of the flow are changing in time. The oscillations of the shock front are connected with small oscillations of the value of angular momentum downward from the shock front, hence, they may be triggered by numerical errors at the shock front. The amplitude of the oscillations reaches the maximum for intermediate shock positions and ceases again for very high shock position/angular momentum. ![ Model [I]{}: 3D simulation ($\epsilon=0.0005,\lambda^{\rm eq}=3.65,a=0.0$) with oscillating shock. \[K500\] ](pic/K500_NMRAS_122.png){width="50.00000%"} If the oscillations are physical, their frequency is in good agreement with the observed low frequency quasi-periodic oscillations seen in several black hole binary systems. Those are shifting in the range from hundreds of mHz up to few tens of Hz on the time scale of weeks (e.g. GRS1915+105 [@1999ApJ...513L..37M], XTE J1550-564 [@1999ApJ...512L..43C], GRO J1655-40 [@1999ApJ...522..397R] or GX 339-4 ). For a fiducial mass of the microquasar equal to 10M$_\odot$ this corresponds to the frequencies between $10^{-6}$M$^{-1}$ up to $10^{-3}$M$^{-1}$ in geometrized units[^3], which is the same range as our observed values (see Fig. \[1D\_shock\], bottom panel for 1D results and Table \[t:2D-models\] for 2D simulations). Hence, the change of the observed frequency during the onset and decline of the outburst is possibly connected with the change of angular momentum of the incoming matter (alternative explanation gives e.g. [@0004-637X-798-1-57], who consider rather the change of viscosity parameter $\alpha$). Such change can be either periodic and connected with the orbital motion of the companion, or caused by different conditions during the release of the matter. [@0004-637X-565-2-1161] proposed the scenario of the outburst of XTE J1550-564, in which the low angular momentum component of the accretion flow is released from the magnetic trap inside the Roche lobe of the black hole, kept by the magnetic field of the active companion. Depending on the distribution of angular momentum inside the trap and on the mechanism of breaking the magnetic confinement the angular momentum of the incoming matter from this region can slightly vary with time. Later the angular momentum of the low angular component could be affected by the interaction with the slowly inward propagating Keplerian disc. ![ Model [I]{}: 3D simulation ($\epsilon=0.0005,\lambda^{\rm eq}=3.65,a=0.0$) with oscillating shock. \[K500\_rs\] ](pic/obr_K500-eps-converted-to.pdf){width="50.00000%"} To simulate such scenario, we have studied the case, when the angular momentum of the incoming matter changes periodically with time. As was shown on Fig. \[1D\_loop\], the behaviour of the shock front depends on whether the value of angular momentum crosses one or both of the values $\lambda_{\tt min}^{cr},\lambda_{\tt max}^{cr}$. When the respective boundary is not crossed, the shock is moving slowly through the permitted region and its speed is determined by the parameters of the perturbation of the angular momentum. When the angular momentum is in the corresponding range, the oscillations of the shock front develops. When one of the the critical values is reached, the abrupt change of the flow geometry happens, while the flow transforms from one type of solution into another. This transformation is achieved via the shock propagation, which is either accreted or expanding. The velocity of the shock front agrees within the order of magnitude with the sound speed of the preshock medium, when the angular momentum is just slightly higher than $\lambda_{\tt max}^{cr}$. If the angular momentum is considerably higher, the formation and propagation of the shock can be even higher than the sound speed in the postshock medium (see Figs. \[2D\_Rs\_position\] and \[2D\_Rs\_position\_eps\]). However, in nature we expect the first case to realize, because such situation can appear only if the angular momentum is increasing in the flow, which is in the Bondi-like configuration, hence it crosses smoothly through $\lambda_{\tt max}^{cr}$. When extending our results to higher dimensions, the freedom of the dependence of angular momentum on the angle $\theta$ arises. We have chosen two different configurations, described in Section \[Ini\_shock\]. The comparison of the corresponding models 1D+sph and 1D+cone in Table \[t:2D-models\] shows, that in the latter case, the shock is placed at larger distances. That can be understood as the consequence of the fact, that thanks to the relations given by Eqs. (\[uphi\]) and (\[lambda-cone\]) larger amount of matter posses the maximal value of angular momentum. However, the main features of the solution, including the existence of the range of angular momentum enabling the long term shock presence, the shape of the resulting shock bubble and its oscillations, are similar in both cases. The biggest difference between the two configurations is that in case of constant angular momentum in a cone, the peaks in mass accretion rate are not so prominent as in the case of angular momentum scaled by $\sin^2\theta$. Hence, we conclude, that the physical processes in the low angular momentum accretion flows do not qualitatively depend on the exact distribution of angular momentum, but the observable consequences (e.g. the presence of prominent peaks and their amplitude) may be influenced by the geometry. Similar conclusion holds also for the change of spin of the black hole. For all three considered values of spin of the black hole ($0.3$, $0.8$ and $0.95$) we have found a range of values of angular momentum of falling matter in which the long lasting oscillations of shock surface was observed. We have also observed, that the shock existence interval of angular momentum depends on the spin of the black hole: the higher the spin value, the lower both limiting angular momentum values between which the oscillating shock exists and the narrower is the corresponding interval. Therefore, for rapidly spinning black holes, even a small change of the angular momentum of the incoming matter leads to significant changes in the flow itself (the existence, position and oscillations of the shock front) and also in the timing properties of the outgoing radiation (which we assume to be related to the accretion rate). Abrupt emergence, expansion or accretion of the shock which is connected with the crossing of the boundary of the shock existence region in parameter space and which leads to significant changes in the accretion rate, are more likely in the accretion flows around rapidly spinning black holes. Simulations which we performed cover large variety of configurations: we considered different values of the spin of the central black hole, energy and angular momentum of the accreted gas, and even the distribution of the angular momentum of the matter. Keeping all the other variables constant we have found the range of angular momentum in which there exist oscillating shock solutions in all scenarios under consideration. Hence the oscillating regime seems to be intrinsic to the low angular momentum accretion flows. This finding is supported by [@2010MNRAS.403..516G] who also found an oscillating shock front in their hydrodynamical simulations in the pseudo-Newtonian framework. Our simulations in two and three dimensions show, that for such parameters the oscillating shock front is long-lasting. That is an important ingredient for the POS model to be able to explain the QPO frequency change during outbursts of microquasars. However, the duration of our simulations corresponds to several tens of second for typical microquasar with $M=10 M_\odot$, which is still short in comparison with the time scale of the QPOs frequency change (weeks). Moreover, we did not address this question from the point of view of an analytical stability analysis. Such analysis was provided by [@1980ApJ...235.1038M] for the spherical accretion onto non-rotating black hole and quite recently by [@BOLLIMPALLI2017153] for low angular momentum flow with standing shocks, who also report the stability of the solution. The dependence of the shock existence interval and consequently the position of the shock on the rotation of the black hole could be the probe of the black hole spin. However, there is a degeneracy between the spin and the angular momentum of the accreting matter, which itself is mostly unknown and hard to measure. Hence, the constraints on the spin from the oscillating shock front model explaining QPOs can be posed only when there will be available better observations of the innermost accretion region or better models predicting the angular momentum of the LAF component. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge support from Interdisciplinary Center for Computational Modeling of the Warsaw University (grant GB66-3) and Polish National Science Center (2012/05/E/ST9/03914). PS is supported from Grant No. GACR-17-06962Y. \[lastpage\] [^1]: e.g., see `https://github.com/atchekho/harmpi` [^2]: `https://einsteintoolkit.org/` [^3]: The time unit $1M$ equals to the time, which light needs to travel the half of the Schwarzschild radius of a black hole of mass M, hence for one solar mass $[t]=1M_\odot=\frac{GM_\odot}{c^3}=4.9255\cdot 10^{-6}s$.

ArXiv

--- abstract: 'This paper proves the asymptotic stability of the multidimensional wave equation posed on a bounded open Lipschitz set, coupled with various classes of positive-real impedance boundary conditions, chosen for their physical relevance: time-delayed, standard diffusive (which includes the Riemann-Liouville fractional integral) and extended diffusive (which includes the Caputo fractional derivative). The method of proof consists in formulating an abstract Cauchy problem on an extended state space using a dissipative realization of the impedance operator, be it finite or infinite-dimensional. The asymptotic stability of the corresponding strongly continuous semigroup is then obtained by verifying the sufficient spectral conditions derived by Arendt and Batty (Trans. Amer. Math. Soc., 306 (1988)) as well as Lyubich and V[ũ]{} (Studia Math., 88 (1988)).' author: - Florian Monteghetti and Ghislain Haine and Denis Matignon title: 'Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions' --- <span style="font-variant:small-caps;">Florian Monteghetti$^*$</span> <span style="font-variant:small-caps;">Ghislain Haine and Denis Matignon</span> \#1 Introduction ============ The broad focus of this paper is the asymptotic stability of the wave equation with so-called impedance boundary conditions (IBCs), also known as acoustic boundary conditions. Herein, the impedance operator, related to the Neumann-to-Dirichlet map, is assumed to be continuous linear time-invariant, so that it reduces to a time-domain convolution. *Passive* convolution operators [@beltrami2014distributions § 3.5], the kernels of which have a positive-real Laplace transform, find applications in physics in the modeling of locally-reacting energy absorbing material, such as non perfect conductors in electromagnetism [@yuferev2010sibc] and liners in acoustics [@monteghetti2017tdibc]. As a result, IBCs are commonly used with Maxwell’s equations [@hiptmair2014FastQuadratureIBC], the linearized Euler equations [@monteghetti2017tdibc], or the wave equation [@sauter2017waveCQ]. Two classes of convolution operators are well-known due to the ubiquity of the physical phenomena they model. Slowly decaying kernels, which yield so-called *long-memory* operators, arise from losses without propagation (due to e.g. viscosity or electrical/thermal resistance); they include fractional kernels. On the other hand, lossless propagation, encountered in acoustical cavity for instance, can be represented as a *time delay*. Both effects can be combined, so that time-delayed long-memory operators model a propagation with losses. Stabilization of the wave equation by a boundary damping, as opposed to an internal damping, has been investigated in a wealth of works, most of which employing the equivalent admittance formulation (\[eq:=00005BMOD=00005D\_IBC-Admittance\]), see Remark \[rem:=00005BMOD=00005D\_Terminology\] for the terminology. Unless otherwise specified, the works quoted below deal with the multidimensional wave equation. Early studies established exponential stability with a proportional admittance [@chen1981note; @lagnese1983decay; @komornik1990direct]. A delay admittance is considered in [@nicaise2006stability], where exponential stability is proven under a sufficient delay-independent stability condition that can be interpreted as a passivity condition of the admittance operator. The proof of well-posedness relies on the formulation of an evolution problem using an infinite-dimensional realization of the delay through a transport equation (see [@engel2000semigroup § VI.6] [@curtainewart1995infinitedim § 2.4] and references therein) and stability is obtained using observability inequalities. The addition of a $2$-dimensional realization to a delay admittance has been considered in [@peralta2018delayibc], where both exponential and asymptotic stability results are shown under a passivity condition using the energy multiplier method. See also [@wang2011stabilitydelay] for a monodimensional wave equation with a non-passive delay admittance, where it is shown that exponential stability can be achieved provided that the delay is a multiple of the domain back-and-forth traveling time. A class of space-varying admittance with finite-dimensional realizations have received thorough scrutiny in [@abbas2013polynomial] for the monodimensional case and [@abbas2015stability] for the multidimensional case. In particular, asymptotic stability is shown using the [Arendt-Batty-Lyubich-Vũ]{} (ABLV) theorem in an extended state space. Admittance kernels defined by a Borel measure on $(0,\infty)$ have been considered in [@cornilleau2009ibc], where exponential stability is shown under an integrability condition on the measure [@cornilleau2009ibc Eq. (7)]. This result covers both distributed and discrete time delays, as well as a class of integrable kernels. Other classes of integrable kernels have been studied in [@desch2010stabilization; @peralta2016delayibc; @li2018memoryibc]. Integrable kernels coupled with a $2$-dimensional realization are considered in [@li2018memoryibc] using energy estimates. Kernels that are both completely monotone and integrable are considered in [@desch2010stabilization], which uses the ABLV theorem on an extended state space, and in [@peralta2016delayibc] with an added time delay, which uses the energy method to prove exponential stability. The energy multiplier method is also used in [@alabauboussouira2009ibc] to prove exponential stability for a class of non-integrable singular kernels. The works quoted so far do not cover fractional kernels, which are non-integrable, singular, and completely monotone. As shown in [@matignon2005asymptotic], asymptotic stability results with fractional kernels can be obtained with the ABLV theorem by using their realization; two works that follow this methodology are [@matignon2014asymptotic], which covers the monodimensional Webster-Lokshin equation with a rational IBC, and [@grabowski2013fracIBC], which covers a monodimensional wave equation with a fractional admittance. The objective of this paper is to prove the asymptotic stability of the multidimensional wave equation (\[eq:=00005BMOD=00005D\_Wave-Equation\]) coupled with a wide range of IBCs (\[eq:=00005BMOD=00005D\_IBC\]) chosen for their physical relevance. All the considered IBCs share a common property: the Laplace transform of their kernel is a positive-real function. A common method of proof, inspired by [@matignon2014asymptotic], is employed that consists in formulating an abstract Cauchy problem on an extended state space (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) using a realization of each impedance operator, be it finite or infinite-dimensional; asymptotic stability is then obtained with the ABLV theorem, although a less general alternative based on the invariance principle is also discussed. In spite of the apparent unity of the approach, we are not able to provide a single, unified proof: this leads us to formulate a conjecture at the end of this work, which we hope will motivate further works. This paper is organized as follows. Section \[sec:=00005BMOD=00005D\_Model-and-preliminary-results\] introduces the model considered, recalls some known facts about positive-real functions, formulates the ABLV theorem as Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\], and establishes a preliminary well-posedness result in the Laplace domain that is the cornerstone of the stability proofs. The remaining sections demonstrate the applicability of Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] to IBCs with infinite-dimensional realizations that arise in physical applications. Delay IBCs are covered in Section \[sec:=00005BDELAY=00005D\_Delay-impedance\], standard diffusive IBCs (e.g. fractional integral) are covered in Section \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\], while extended diffusive IBCs (e.g. fractional derivative) are covered in Section \[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\]. The extension of the obtained asymptotic stability results to IBCs that contain a first-order derivative term is carried out in Section \[sec:=00005BDER=00005D\_Addition-of-Derivative\]. Notation {#notation .unnumbered} -------- Vector-valued quantities are denoted in bold, e.g. $\vector f$. The canonical scalar product in $\spaceC^{d}$, $d\in\llbracket1,\infty\llbracket$, is denoted by $(\vector f,\vector g)_{\spaceC^{d}}\coloneqq\sum_{i=1}^{d}f_{i}\overline{g_{i}}$, where $\overline{g_{i}}$ is the complex conjugate. Throughout the paper, scalar products are antilinear with respect to the second argument. Gradient and divergence are denoted by $$\nabla f\coloneqq\left[\partial_{i}f\right]_{i\in\llbracket1,d\rrbracket},\;\opDiv\vector f\coloneqq\sum_{i=1}^{d}\partial_{i}f_{i},$$ where $\partial_{i}$ is the weak derivative with respect to the $i$-th coordinate. The scalar product (resp. norm) on a Hilbert space $\spaceState$ is denoted by $(\cdot,\cdot)_{\spaceState}$ (resp. $\Vert\cdot\Vert_{\spaceState}$). The only exception is the space of square integrable functions $(L^{2}(\Omega))^{d}$, with $\Omega\subset\spaceR^{d}$ open set, for which the space is omitted, i.e. $$(\vector f,\vector g)\coloneqq\int_{\Omega}(\vector f(\coordx),\vector g(\coordx))_{\spaceC^{d}}\,\dinf\coordx,\;\Vert\vector f\Vert\coloneqq\sqrt{(\vector f,\vector f)}.$$ The scalar product on $(H^{1}(\Omega))^{d}$ is $$(\vector f,\vector g)_{H^{1}(\Omega)}\coloneqq(\vector f,\vector g)+(\nabla\vector f,\nabla\vector g).$$ The topological dual of a Hilbert space $\spaceState$ is denoted by $\spaceState^{'}$, and $L^{2}$ is used as a pivot space so that for instance $$H^{\frac{1}{2}}\subset L^{2}\simeq(L^{2})^{'}\subset H^{-\frac{1}{2}},$$ which leads to the following repeatedly used identity, for $\pac\in L^{2}$ and $\test\in H^{\frac{1}{2}}$, $$\langle\pac,\test\rangle_{H^{-\frac{1}{2}},H^{\frac{1}{2}}}=\langle\pac,\test\rangle_{(L^{2})^{'},L^{2}}=(\pac,\overline{\test})_{L^{2}},\label{eq:=00005BPRE=00005D_L2-Pivot-Space}$$ where $\langle\cdot,\cdot\rangle$ denotes the duality bracket (linear in both arguments). All the Hilbert spaces considered in this paper are over $\spaceC$. Other commonly used notations are $\spaceR^{*}\coloneqq\spaceR\backslash\{0\}$, $\Re(s)$ (resp. $\Im(s)$) for the real (resp. imaginary) part of $s\in\spaceC$, $A^{\transpose}$ for the transpose of a matrix $A$, $R(A)$ (resp. $\opKer(A)$) for the range (resp. kernel) of $A$, $\spaceContinuous(\Omega)$ for the space of continuous functions, $\spaceContinuous_{0}^{\infty}(\Omega)$ for the space of infinitely smooth and compactly supported functions, $\spaceD^{'}(\Omega)$ for the space of distributions (dual of $\spaceContinuous_{0}^{\infty}(\Omega)$), $\mathcal{E}^{'}(\Omega)$ for the space of compactly supported distributions, $\spaceBounded(\spaceState)$ for the space of continuous linear operators over $\spaceState$, $\overline{\Omega}$ for the closure of $\Omega$, $Y_{1}:\,\spaceR\rightarrow\{0,1\}$ for the Heaviside function ($1$ over $(0,\infty)$, null elsewhere), and $\delta$ for the Dirac distribution. Model, strategy, and preliminary results\[sec:=00005BMOD=00005D\_Model-and-preliminary-results\] ================================================================================================ Let $\Omega\subset\spaceR^{d}$ be a bounded open set. The Cauchy problem considered in this paper is the wave equation under one of its first-order form, namely $$\partial_{t}\left(\begin{array}{c} \uac\\ \pac \end{array}\right)+\left(\begin{array}{c} \nabla\pac\\ \opDiv\uac \end{array}\right)=\vector 0\quad\text{on }\Omega,\label{eq:=00005BMOD=00005D_Wave-Equation}$$ where $\uac(t,\coordx)\in\spaceC^{d}$ and $\pac(t,\coordx)\in\spaceC$. To (\[eq:=00005BMOD=00005D\_Wave-Equation\]) is associated the so-called *impedance boundary condition* (IBC), formally defined as a time-domain convolution between $\pac$ and $\uac\cdot\normal$, $$\pac=z\star\uac\cdot\normal\quad\text{a.e. on }\partial\Omega,\label{eq:=00005BMOD=00005D_IBC}$$ where $\normal$ is the unit outward normal and $z$ is the impedance kernel. In general, $z$ is a causal distribution, i.e. $z\in\spaceD_{+}^{'}(\spaceR)$, so that the convolution is to be understood in the sense of distributions [@schwartz1966mathphys Chap. III] [@hormander1990pdevol1 Chap. IV]. This paper proves the asymptotic stability of strong solutions of the evolution problem (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]) with an impedance kernel $z$ whose positive-real Laplace transform is given by $$\hat{z}(s)=\left(z_{0}+z_{\tau}e^{-\tau s}\right)+z_{1}s+\hat{z}_{\text{diff},1}(s)+s\,\hat{z}_{\text{diff},2}(s)\quad(\Re(s)>0),\label{eq:=00005BMOD=00005D_Target-Impedance}$$ where $\tau>0$, $z_{\tau}\in\spaceR$, $z_{0}\geq\vert z_{\tau}\vert$, $z_{1}>0$, and $z_{\text{diff},1}$ as well as $z_{\text{diff},2}$ are both locally integrable completely monotone kernels. The motivation behind the definition of this kernel is physical as it models passive systems that arise in e.g. electromagnetics [@garrappa2016models], viscoelasticity [@desch1988exponential; @mainardi1997frac], and acoustics [@helie2006diffusive; @lombard2016fractional; @monteghetti2016diffusive]. By assumption, the right-hand side of (\[eq:=00005BMOD=00005D\_Target-Impedance\]) is a sum of positive-real kernels that each admit a dissipative realization. This property enables to prove asymptotic stability with (\[eq:=00005BMOD=00005D\_Target-Impedance\]) by treating each of the four positive-real kernel separately: this is carried out in Sections \[sec:=00005BDELAY=00005D\_Delay-impedance\]–\[sec:=00005BDER=00005D\_Addition-of-Derivative\]. This modularity property enables to keep concise notation by dealing with the difficulty of each term one by one; it is illustrated in Section \[sec:=00005BDER=00005D\_Addition-of-Derivative\]. As already mentioned in the introduction, the similarity between the four proofs leads us to formulate a conjecture at the end of the paper. The purpose of the remainder of this section is to present the strategy employed to establish asymptotic stability as well as to prove preliminary results. Section \[sub:=00005BMOD=00005D\_Positive-real-facts\] justifies why, in order to obtain a well-posed problem in $L^{2}$, the Laplace transform of the impedance kernel must be a *positive-real* function. Section \[sub:=00005BSTAB=00005D\_Strategy\] details the strategy used to establish asymptotic stability. Section \[sub:=00005BMOD=00005D\_Lemma-Rellich\] proves a consequence of the Rellich identity that is then used in Section \[sub:=00005BMOD=00005D\_Well-posedness-Result\] to obtain a well-posedness result on the Laplace-transformed wave equation, which will be used repeatedly. \[rem:=00005BMOD=00005D\_Terminology\] The boundary condition (\[eq:=00005BMOD=00005D\_IBC\]) can equivalently be written as $$\uac\cdot\normal=y\star\pac\quad\text{a.e. on }\partial\Omega,\label{eq:=00005BMOD=00005D_IBC-Admittance}$$ where $y$ is known as the *admittance* kernel ($y\star z=\delta$, where $\delta$ is the Dirac distribution). This terminology can be justified, for example, by the acoustical application: an acoustic impedance is homogeneous to a pressure divided by a velocity. The asymptotic stability results obtained in this paper still hold by replacing the impedance by the admittance (in particular, the statement “$z\neq0$” becomes “$y\neq0$”). The third way of formulating (\[eq:=00005BMOD=00005D\_IBC\]), not considered in this paper, is the so-called *scattering* formulation [@beltrami2014distributions p. 89] [@lozano2013dissipative § 2.8] $$\pac-\uac\cdot\normal=\beta\star(\pac+\uac\cdot\normal)\quad\text{a.e. on }\partial\Omega,$$ where $\beta$ is known as the *reflection coefficient*. A Dirichlet boundary condition is recovered for $z=0$ ($\beta=-\delta$) while a Neumann boundary condition is recovered for $y=0$ ($\beta=+\delta$), so that the proportional IBC, obtained for $z=z_{0}\delta$ ($\beta=\frac{z_{0}-1}{z_{0}+1}\,\delta$), $z_{0}\geq0$, can be seen as an intermediate between the two. The use of a convolution in (\[eq:=00005BMOD=00005D\_IBC\]) can be justified with the following classical result [@schwartz1966mathphys § III.3] [@beltrami2014distributions Thm. 1.18]: if $\opZ$ is a linear time-invariant and continuous mapping from $\mathcal{E}^{'}(\spaceR)$ into $\spaceD^{'}(\spaceR)$, then $\opZ(u)=\opZ(\delta)\star u$ for all $u\in\mathcal{E}^{'}(\spaceR)$. Why positive-real kernels?\[sub:=00005BMOD=00005D\_Positive-real-facts\] ------------------------------------------------------------------------ Assume that $(\uac,\pac)$ is a strong solution, i.e. that it belongs to $\spaceContinuous([0,T];(H^{1}(\Omega))^{d+1})$. The elementary a priori estimate $$\Vert(\uac,\pac)(T)\Vert^{2}=\Vert(\uac,\pac)(0)\Vert^{2}-2\,\Re\left[\int_{0}^{T}(\pac(\tau),\uac(\tau)\cdot\normal)_{L^{2}(\partial\Omega)}\,\dinf\tau\right]\label{eq:=00005BMOD=00005D_Energy-Estimate}$$ suggests that to obtain a contraction semigroup, the impedance kernel must satisfy a passivity condition, well-known in system theory. This justifies why we restrict ourselves to impedance kernels that are *admissible* in the sense of the next definition, adapted from [@beltrami2014distributions Def. 3.3]. \[def:=00005BMOD=00005D\_Admissible-Impedance\] A distribution $z\in\spaceD^{'}(\spaceR)$ is said to be an *admissible impedance kernel* if the operator $u\mapsto z\star u$ that maps $\mathcal{E}^{'}(\spaceR)$ into $\spaceD^{'}(\spaceR)$ enjoys the following properties: causality, i.e. $z\in\spaceD_{+}^{'}(\spaceR)$; reality, i.e. real-valued inputs are mapped to real-valued outputs; passivity, i.e. $$\forall u\in\spaceContinuous_{0}^{\infty}(\spaceR),\;\forall T>0,\;\Re\left[\int_{-\infty}^{T}(z\star u(\tau),u(\tau))_{\spaceC}\,\dinf\tau\right]\geq0.\label{eq:=00005BMOD=00005D_Passivity-Cond_Z}$$ An important feature of admissible impedance kernels $z$ is that their Laplace transforms $\hat{z}$ are *positive-real* functions, see Definition \[def:=00005BMOD=00005D\_PR\] and Proposition \[prop:=00005BMOD=00005D\_Characterization-LTI-Z\]. Herein, the Laplace transform $\hat{z}$ is an analytic function on an open *right* half-plane, i.e. $$\hat{z}(s)\coloneqq\int_{0}^{\infty}z(t)e^{-st}\,\dinf t\quad\left(s\in\spaceC_{c}^{+}\right),$$ for some $c\geq0$ with $$\spaceC_{c}^{+}\coloneqq\{s\in\spaceC\,\vert\,\Re(s)>c\}.$$ See [@schwartz1966mathphys Chap. 6] and [@beltrami2014distributions Chap. 2] for the definition when $z\in\spaceD_{+}^{'}(\spaceR)$. \[def:=00005BMOD=00005D\_PR\]A function $f:\,\spaceC_{0}^{+}\rightarrow\spaceC$ is *positive-real* if $f$ is analytic in $\spaceC_{0}^{+}$, $f(s)\in\spaceR$ for $s\in(0,\infty)$, and $\Re[f(s)]\geq0$ for $s\in\spaceC_{0}^{+}$. \[prop:=00005BMOD=00005D\_Characterization-LTI-Z\]A causal distribution $z\in\spaceD_{+}^{'}(\spaceR)$ is an admissible impedance kernel if and only if $\hat{z}$ is a positive-real function. See [@lozano2013dissipative § 2.11] for the case where the kernel $z\in L^{1}(\spaceR)$ is a function and [@beltrami2014distributions § 3.5] for the general case where $z\in\spaceD_{+}^{'}(\spaceR)$ is a causal distribution. (Note that, if $z$ is an admissible impedance kernel, then $z$ is also tempered.) \[rem:=00005BMOD=00005D\_PR-growth-at-infinity\]The growth at infinity of positive-real functions is at most polynomial. More specifically, from the integral representation of positive-real functions [@beltrami2014distributions Eq. (3.21)], it follows that for $\Re(s)\geq c>0$, $\vert\hat{z}(s)\vert\leq C(c)P(\vert s\vert)$ where $P$ is a second degree polynomial. Abstract framework for asymptotic stability\[sub:=00005BSTAB=00005D\_Strategy\] ------------------------------------------------------------------------------- Let the causal distribution $z\in\spaceD_{+}^{'}(\spaceR)$ be an admissible impedance kernel. In order to prove the asymptotic stability of (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]), we will use the following strategy in Sections \[sec:=00005BDELAY=00005D\_Delay-impedance\]–\[sec:=00005BDER=00005D\_Addition-of-Derivative\]. We first rely on the knowledge of a realization of the impedance operator $u\mapsto z\star u$ to formulate an abstract Cauchy problem on a Hilbert space $\spaceState$, $$\dot{\state}(t)=\opA\state,\;\state(0)=\state_{0}\in\spaceState,\label{eq:=00005BSTAB=00005D_Abstract-Cauchy-Problem}$$ where the extended state $\state$ accounts for the memory of the IBC. The scalar product $(\cdot,\cdot)_{\spaceState}$ is defined using a Lyapunov functional associated with the realization. Since, by design, the problem has the energy estimate $\Vert\state(t)\Vert_{\spaceState}\leq\Vert\state_{0}\Vert_{\spaceState}$, it is natural to use the Lumer-Phillips theorem to show that the unbounded operator $$\opA:\,\spaceDomain\subset\spaceState\rightarrow\spaceState\label{eq:=00005BSTAB=00005D_Definition-A}$$ generates a strongly continuous semigroup of contractions on $\spaceState$, denoted by $\opT(t)$. For initial data in $\spaceDomain$, the function $$t\mapsto\opT(t)\state_{0}\label{eq:=00005BSTAB=00005D_Solution}$$ provides the unique strong solution in $\spaceContinuous([0,\infty);\spaceDomain)\cap\spaceContinuous^{1}([0,\infty);\spaceState)$ of the evolution problem (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) [@pazy1983stability Thm. 1.3]. For (less regular) initial data in $\spaceState$, the solution is milder, namely $\spaceContinuous([0,\infty);\spaceState)$. To prove the asymptotic stability of this solution, we rely upon the following result, where we denote by $\sigma(\opA)$ (resp. $\sigma_{p}(\opA)$) the spectrum (resp. point spectrum) of $\opA$ [@yosida1980funana § VIII.1]. \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\]Let $\spaceState$ be a complex Hilbert space and $\opA$ be defined as (\[eq:=00005BSTAB=00005D\_Definition-A\]). If - $\opA$ is dissipative, i.e. $\Re(\opA\state,\state)_{H}\leq0$ for every $\state\in\spaceDomain$, - $\opA$ is injective, - $s\opId-\opA$ is bijective for $s\in(0,\infty)\cup i\spaceR^{*}$, then $\opA$ is the infinitesimal generator of a strongly continuous semigroup of contractions $\opT(t)\in\spaceBounded(\spaceState)$ that is asymptotically stable, i.e. $$\forall\state_{0}\in\spaceState,\;\Vert\opT(t)\state_{0}\Vert_{\spaceState}\underset{t\rightarrow\infty}{\longrightarrow}0.\label{eq:=00005BSTAB=00005D_T-asymptotic-stability}$$ The Lumer-Phillips theorem, recalled in Theorem \[thm:=00005BSTAB=00005D\_Lumer-Phillips\], shows that $\opA$ generates a strongly continuous semigroup of contractions $\opT(t)\in\spaceBounded(\spaceState)$. In particular $\opA$ is closed, from the Hille-Yosida theorem [@pazy1983stability Thm. 3.1], so that the resolvent operator $(s\opId-\opA)^{-1}$ is closed whenever it is defined. A direct application of the closed graph theorem [@yosida1980funana § II.6] then yields $$\left\{ s\in\spaceC\;\vert\;s\opId-\opA\;\text{is bijective}\right\} \subset\rho(\opA),$$ where $\rho(\opA)$ denotes the resolvent set of $\opA$ [@yosida1980funana § VIII.1]. Hence $i\spaceR^{*}\subset\rho(\opA)$ and Theorem \[thm:=00005BSTAB=00005D\_Arendt-Batty\] applies since $0\notin\sigma_{p}(\opA)$. Condition (iii) of Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] could be loosened by only requiring that $s\opId-\opA$ be surjective for $s\in(0,\infty)$ and bijective for $s\in i\spaceR^{*}$. However, in the proofs presented in this paper we always prove bijectivity for $s\in(0,\infty)\cup i\spaceR^{*}$. A consequence of the Rellich identity\[sub:=00005BMOD=00005D\_Lemma-Rellich\] ----------------------------------------------------------------------------- Using the Rellich identity, we prove below that the Dirichlet and Neumann Laplacians do not have an eigenfunction in common. \[prop:=00005BMOD=00005D\_Rellich-Lemma\]Let $\Omega\subset\spaceR^{d}$ be a bounded open set. If $\pac\in H_{0}^{1}(\Omega)$ satisfies $$\forall\test\in H^{1}(\Omega),\;(\nabla\pac,\nabla\test)=\lambda(\pac,\test)\label{eq:=00005BMOD=00005D_Rellich-WeakForm}$$ for some $\lambda\in\spaceC$, then $\pac=0$ a.e. in $\Omega$. Let $\pac\in H_{0}^{1}(\Omega)$ be such that (\[eq:=00005BMOD=00005D\_Rellich-WeakForm\]) holds for some $\lambda\in\spaceC$. The proof is divided in two steps. \(a) Let us first assume that $\partial\Omega$ is $\spaceContinuous^{\infty}$. In particular, $$\forall\test\in H_{0}^{1}(\Omega),\;(\nabla\pac,\nabla\test)=\lambda(\pac,\test),$$ so that $\pac$ is either null a.e. in $\Omega$ or an eigenfunction of the Dirichlet Laplacian. In the latter case, since the boundary $\partial\Omega$ is of class $\spaceContinuous^{\infty}$, we have the regularity result $\pac\in\spaceContinuous^{\infty}(\overline{\Omega})$ [@gilbarg2001elliptic Thm. 8.13]. An integration by parts then shows that, for $\test\in H^{1}(\Omega)$, $$(\partial_{n}\pac,\test)_{L^{2}(\partial\Omega)}=(\Delta\pac+\lambda\pac,\test)=0,$$ so that $\partial_{n}\pac=0$ in $\partial\Omega$. However since $\pac$ is $\spaceContinuous^{2}(\overline{\Omega})$ and $\partial\Omega$ is smooth we have [@rellich1940eigenfunctions] $$\Vert\pac\Vert^{2}=\frac{\int_{\partial\Omega}(\partial_{n}\pac)^{2}\partial_{n}(\vert\coordx\vert^{2})\,\dinf\coordx}{4\lambda},\label{eq:=00005BMOD=00005D_Rellich-Formula}$$ which shows that $\pac=0$ in $\Omega$. (The spectrum of the Dirichlet Laplacian does not include $0$ [@gilbarg2001elliptic § 8.12].) \(b) Let us now assume that $\partial\Omega$ is not $\spaceContinuous^{\infty}$. The strategy, suggested to us by Prof. Patrick Ciarlet, is to get back to (a) by extending $\pac$ by zero. Let $\mathcal{B}$ be an open ball such that $\overline{\Omega}\subset\mathcal{B}$. We denote $\tilde{\pac}$ the extension of $\pac$ by zero, i.e. $\tilde{\pac}=\pac$ on $\Omega$ with $\tilde{\pac}$ null on $\mathcal{B}\backslash\Omega$. From Proposition \[prop:Extension-by-zero\], we have $\tilde{\pac}\in H_{0}^{1}\left(\mathcal{B}\right)$. Using the definition of $\tilde{\pac}$, we can write $$\begin{aligned} {1} \forall\test\in H^{1}(\mathcal{B}),\;(\nabla\tilde{\pac},\nabla\test)_{L^{2}\left(\mathcal{B}\right)} & =\left(\nabla\pac,\nabla\left[\test_{\vert\Omega}\right]\right)_{L^{2}\left(\Omega\right)}=\lambda\left(\pac,\test_{\vert\Omega}\right)_{L^{2}\left(\Omega\right)}=\lambda(\tilde{\pac},\test)_{L^{2}\left(\mathcal{B}\right)},\end{aligned}$$ so that applying (a) to $\tilde{\pac}\in H_{0}^{1}\left(\mathcal{B}\right)$ gives $\tilde{\pac}=0$ a.e. in $\mathcal{B}$. A well-posedness result in the Laplace domain\[sub:=00005BMOD=00005D\_Well-posedness-Result\] --------------------------------------------------------------------------------------------- The following result will be used repeatedly. We define $$\overline{\spaceC_{0}^{+}}\coloneqq\{s\in\spaceC\,\vert\,\Re(s)\geq0\}.$$ \[thm:=00005BMOD=00005D\_Weak-Form-p\]Let $\Omega\subset\spaceR^{d}$ be a bounded open set with a Lipschitz boundary. Let $z:\,\overline{\spaceC_{0}^{+}}\backslash\{0\}\rightarrow\spaceC_{0}^{+}$ be such that $z(s)\in\spaceR$ for $s\in(0,\infty)$. For every $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$ and $l\in H^{-1}(\Omega)$ there exists a unique $\pac\in H^{1}(\Omega)$ such that $$\forall\test\in H^{1}(\Omega),\;(\nabla\pac,\nabla\test)+s^{2}(\pac,\test)+\frac{s}{z(s)}(\pac,\test)_{L^{2}(\partial\Omega)}=\overline{l(\test)}.\label{eq:=00005BMOD=00005D_Weak-Form-p}$$ Moreover, there is $C(s)>0$, independent of $\pac$, such that $$\Vert\pac\Vert_{H^{1}(\Omega)}\leq C(s)\,\Vert l\Vert_{H^{-1}(\Omega)}.$$ Note that $s\mapsto z(s)$ need not be continuous, so that Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\] can be used pointwise, i.e. for only some $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$. Although the need for Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\] will appear in the proofs of the next sections, let us give a *formal* motivation for the formulation (\[eq:=00005BMOD=00005D\_Weak-Form-p\]). Assume that $(\uac,\pac)$ is a smooth solution of (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]). Then $\pac$ solves the wave equation $$\partial_{t}^{2}\pac-\Delta\pac=0\quad\text{on }\Omega,$$ with the impedance boundary condition $$\partial_{t}\pac=z\star\partial_{t}\uac\cdot\normal=z\star(-\nabla\pac)\cdot\normal=-z\star\partial_{n}\pac\quad\text{on }\partial\Omega,$$ where $\partial_{n}\pac$ denotes the normal derivative of $\pac$ and the causal kernel $z$ is, say, tempered and locally integrable. An integration by parts with $\test\in H^{1}(\Omega)$ reads $$(\nabla\pac,\nabla\psi)+(\partial_{t}^{2}\pac,\psi)-(\partial_{n}\pac,\psi)_{L^{2}(\partial\Omega)}=0.$$ The formulation (\[eq:=00005BMOD=00005D\_Weak-Form-p\]) then follows from the application of the Laplace transform in time, which gives $\widehat{z\star\partial_{n}\pac}(s)=\hat{z}(s)\partial_{n}\hat{\pac}(s)$ and $\widehat{\partial_{t}\pac}(s)=s\hat{\pac}(s)$ assuming that $\pac(t=0)=0$ on $\partial\Omega$. If $s\in(0,\infty)$ this is an immediate consequence of the Lax-Milgram lemma [@lax2002funana Thm. 6.6]. Define the following bilinear form over $H^{1}(\Omega)\times H^{1}(\Omega)$: $$\overline{a(\pac,\test)}\coloneqq(\nabla\pac,\nabla\test)+s^{2}(\pac,\test)+\frac{s}{z(s)}(\pac,\test)_{L^{2}(\partial\Omega)}.$$ Its boundedness follows from the continuity of the trace $H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ (see Section \[sub:=00005BMISC=00005D\_Embedding-Trace\]). The fact that $z(s)>0$ gives $$a(\test,\test) \geq\min(1,s^{2})\Vert\test\Vert_{H^{1}(\Omega)}^{2},$$ which establishes the coercivity of $a$. Let $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$. The Lax-Milgram lemma does not apply since the sign of $\Re(\overline{s}z(s))$ is indefinite in general, but the Fredholm alternative is applicable. Using the Riesz-Fréchet representation theorem [@lax2002funana Thm. 6.4], (\[eq:=00005BMOD=00005D\_Weak-Form-p\]) can be rewritten uniquely as $$(\opId-\opK(s))\pac=L\quad\text{in }H^{1}(\Omega),\label{eq:=00005BMOD=00005D_Weak-Form-p_Compact}$$ where $L\in H^{1}(\Omega)$ satisfies $\overline{l(\test)}=(L,\test)_{H^{1}(\Omega)}$ and the operator $\opK(s)\in\spaceBounded(H^{1}(\Omega))$ is given by $$(\opK(s)\pac,\text{\ensuremath{\test}})_{H^{1}(\Omega)}\coloneqq(1-s^{2})(\pac,\test)-\frac{s}{z(s)}(\pac,\test)_{L^{2}(\partial\Omega)}.$$ The interest of (\[eq:=00005BMOD=00005D\_Weak-Form-p\_Compact\]) lies in the fact that $\opK(s)$ turns out to be a compact operator, see Lemma \[lem:=00005BMOD=00005D\_Weak-Form-p\_K-compact\]. The Fredholm alternative states that $\opId-\opK(s)$ is injective if and only if it is surjective [@brezis2011fun Thm. 6.6]. Using Lemma \[lem:=00005BMOD=00005D\_Weak-Form-p\_K-eigenvalue\] and the open mapping theorem [@yosida1980funana § II.5], we conclude that $\opId-\opK(s)$ is a bijection with continuous inverse, which yields the claimed well-posedness result. \[lem:=00005BMOD=00005D\_Weak-Form-p\_K-compact\]Let $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$. The operator $\opK(s)\in\spaceBounded(H^{1}(\Omega))$ is compact. Let $\pac,\test\in H^{1}(\Omega)$. The CauchySchwarz inequality and the continuity of the trace $H^{1}(\Omega)\rightarrow L^{2}(\partial\Omega)$ yield the existence of a constant $C>0$ such that $$\left|(\opK(s)\pac,\text{\ensuremath{\test}})_{H^{1}(\Omega)}\right|\leq\left(\left|1-s^{2}\right|\Vert\pac\Vert+C\left|\frac{s}{z(s)}\right|\Vert\pac\Vert_{L^{2}(\partial\Omega)}\right)\Vert\test\Vert_{H^{1}(\Omega)},$$ from which we deduce $$\Vert\opK(s)\pac\Vert_{H^{1}(\Omega)}\leq\left|1-s^{2}\right|\Vert\pac\Vert+C\left|\frac{s}{z(s)}\right|\Vert\pac\Vert_{L^{2}(\partial\Omega)}.$$ Let $\epsilon\in(0,\frac{1}{2})$. The continuous embedding $H^{\frac{1}{2}+\epsilon}(\Omega)\subset L^{2}(\Omega)$ and the continuity of the trace $H^{\frac{1}{2}+\epsilon}(\Omega)\rightarrow L^{2}(\partial\Omega)$, see Section \[sub:=00005BMISC=00005D\_Embedding-Trace\], yield $$\Vert\opK(s)\pac\Vert_{H^{1}(\Omega)}\leq\left(\left|1-s^{2}\right|+C^{'}\left|\frac{s}{z(s)}\right|\right)\Vert\pac\Vert_{H^{\epsilon+\frac{1}{2}}(\Omega)}.$$ The compactness of the embedding $H^{1}(\Omega)\subset H^{\frac{1}{2}+\epsilon}(\Omega)$, see Section \[sub:=00005BMISC=00005D\_Embedding-Trace\], enables to conclude. \[lem:=00005BMOD=00005D\_Weak-Form-p\_K-eigenvalue\]Let $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$. The operator $\opId-\opK(s)$ is injective. Assume that $\opId-\opK(s)$ is not injective. Then there exists $\pac\in H^{1}(\Omega)\backslash\{0\}$ such that $\opK(s)\pac=\pac$, i.e. for any $\test\in H^{1}(\Omega)$, $$(\nabla\pac,\nabla\text{\ensuremath{\test}})+s^{2}(\pac,\test)+\frac{s}{z(s)}(\pac,\test)_{L^{2}(\partial\Omega)}=0.\label{eq:=00005BIBC=00005D_Weak-Form-p_K-eigenvalue_1}$$ In particular, for $\test=\pac$, $$z(s)\Vert\nabla\pac\Vert^{2}+s^{2}z(s)\Vert\pac\Vert^{2}+s\Vert\pac\Vert_{L^{2}(\partial\Omega)}^{2}=0.\label{eq:=00005BIBC=00005D_Weak-Form-p_K-eigenvalue_2}$$ To derive a contradiction, we distinguish between $s\in\spaceC_{0}^{+}$ and $s\in i\spaceR^{*}$.\ ($s\in\spaceC_{0}^{+}$) This is a direct consequence of Lemma \[lem:=00005BMOD=00005D\_Polynomial-Root\].\ ($s\in i\spaceR^{*}$) Let $s=i\omega$ with $\omega\in\spaceR^{*}$. Then (\[eq:=00005BIBC=00005D\_Weak-Form-p\_K-eigenvalue\_2\]) reads $$\begin{cases} \Re(z(i\omega))\left(\Vert\nabla\pac\Vert^{2}-\omega^{2}\Vert\pac\Vert^{2}\right)=0\\ \Im(z(i\omega))\left(\Vert\nabla\pac\Vert^{2}-\omega^{2}\Vert\pac\Vert^{2}\right)+\omega\Vert\pac\Vert_{L^{2}(\partial\Omega)}^{2}=0, \end{cases}$$ so that $\pac\in H_{0}^{1}(\Omega)$. Going back to the first identity (\[eq:=00005BIBC=00005D\_Weak-Form-p\_K-eigenvalue\_1\]), we therefore have $$\forall\test\in H^{1}(\Omega),\;(\nabla\pac,\nabla\text{\ensuremath{\test}})=\omega^{2}(\pac,\test).$$ The contradiction then follows from Proposition \[prop:=00005BMOD=00005D\_Rellich-Lemma\]. \[lem:=00005BMOD=00005D\_Polynomial-Root\]Let $(a_{0},a_{1},a_{2})\in[0,\infty)^{3}$ and $z\in\spaceC_{0}^{+}$. The polynomial $s\mapsto za_{2}\,s^{2}+a_{1}\,s+za_{0}$ has no roots in $\spaceC_{0}^{+}$. The only case that needs investigating is $a_{i}>0$ for $i\in\llbracket0,2\rrbracket$. Let us denote by $\sqrt{\cdot}$ the branch of the square root that has a nonnegative real part, with a cut on $(-\infty,0]$ (i.e. $\sqrt{\cdot}$ is analytic over $\spaceC\backslash(-\infty,0]$). The roots are given by $$s_{\pm}\coloneqq\frac{a_{1}\overline{z}}{2a_{2}\vert z\vert^{2}}\left(-1\pm\sqrt{1-\gamma z^{2}}\right)$$ with $\gamma\coloneqq4\frac{a_{0}a_{2}}{a_{1}^{2}}>0$ so that $$\Re(s_{\pm})=\frac{a_{1}}{2a_{2}\vert z\vert^{2}}f_{\pm}(z)\quad\text{with}\quad f_{\pm}(z)\coloneqq\Re\left[\overline{z}\left(-1\pm\sqrt{1-\gamma z^{2}}\right)\right].$$ The function $f_{\pm}$ is continuous on $\spaceC_{0}^{+}\backslash[\gamma^{-\nicefrac{1}{2}},\infty)$ (but not analytic) and vanishes only on $i\spaceR$ (if $f_{\pm}(z)=0$, then there is $\omega\in\spaceR$ such that $2\omega z=i\left(\omega^{2}-\gamma\vert z\vert^{4}\right)$). The claim therefore follows from $$f_{\pm}\left(\frac{1}{\sqrt{2\gamma}}\right)=\frac{-\sqrt{2}\pm1}{2\sqrt{\gamma}}<0.$$ In view of Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\], in the remainder of this paper, we make the following assumption on the set $\Omega$. The set $\Omega\subset\spaceR^{d}$, $d\in\llbracket1,\infty\llbracket$, is a bounded open set with a Lipschitz boundary. Delay impedance\[sec:=00005BDELAY=00005D\_Delay-impedance\] =========================================================== This section, as well as Sections \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\] and \[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\], deals with IBCs that have an *infinite*-dimensional realization, which arise naturally in physical modeling [@monteghetti2016diffusive]. Let us first consider the time-delayed impedance $$\hat{z}(s)\coloneqq z_{0}+z_{\tau}e^{-\tau s},\label{eq:=00005BDELAY=00005D_Laplace}$$ where $z_{0},z_{\tau},\tau\in\spaceR$, so that the corresponding IBC (\[eq:=00005BMOD=00005D\_IBC\]) reads $$\pac(t)=z_{0}\uac(t)\cdot\normal+z_{\tau}\uac(t-\tau)\cdot\normal\quad\text{a.e. on \ensuremath{\partial\Omega}},\;t>0.\label{eq:=00005BDELAY=00005D_IBC-Temp}$$ The function (\[eq:=00005BDELAY=00005D\_Laplace\]) is positive-real if and only if $$z_{0}\geq\vert z_{\tau}\vert,\;\tau\geq0,\label{eq:=00005BDELAY=00005D_PR-Condition}$$ which is assumed in the following. From now on, in addition to (\[eq:=00005BDELAY=00005D\_PR-Condition\]), we further assume $$\hat{z}(0)\neq0,\;\tau\neq0.$$ This section is organized as follows: a realization of $\hat{z}$ is recalled in Section \[sub:=00005BDELAY=00005D\_Time-delay-realization\] and the stability of the coupled system is shown in Section \[sub:=00005BDELAY=00005D\_Asymptotic-stability\]. In [@nicaise2006stability], exponential (resp. asymptotic) stability is shown under the condition $z_{0}>z_{\tau}>0$ (resp. $z_{0}\geq z_{\tau}>0$) and $\tau>0$. The case of a (memoryless) proportional impedance $\hat{z}(s)\coloneqq z_{0}$ with $z_{0}>0$ is elementary (it is known that exponential stability is achieved [@chen1981note; @lagnese1983decay; @komornik1990direct]) and can be covered by the strategy detailed in Section \[sub:=00005BSTAB=00005D\_Strategy\] without using an extended state space [@monteghetti2018dissertation §4.2.2]. Time-delay realization\[sub:=00005BDELAY=00005D\_Time-delay-realization\] ------------------------------------------------------------------------- Following a well-known device, time-delays can be realized using a transport equation on a bounded interval [@curtainewart1995infinitedim § 2.4] [@engel2000semigroup § VI.6]. Let $u$ be a causal input. The linear time-invariant operator $u\mapsto z\star u$ can be realized as $$z\star u(t)=z_{0}u(t)+z_{\tau}\delay(t,-\tau)\quad(t>0),$$ where the state $\delay\in H^{1}(-\tau,0)$ with $t\geq0$ follows the transport equation $$\left\{ \begin{aligned} & \partial_{t}\delay(t,\theta)=\partial_{\theta}\delay(t,\theta), & \quad & \left(\theta\in(-\tau,0),\;t>0\right), & & \text{(a)}\\ & \delay(0,\theta)=0, & & \left(\theta\in(-\tau,0)\right), & & \text{(b)}\\ & \delay(t,0)=u(t), & & \left(t>0\right). & & \text{(c)} \end{aligned} \right.\label{eq:=00005BDELAY=00005D_Transport-Equation}$$ For $\delay\in\spaceContinuous^{1}([0,T];H^{1}(-\tau,0))$ solution of (\[eq:=00005BDELAY=00005D\_Transport-Equation\]a), we have the following energy balance $$\begin{aligned} {1} \frac{1}{2}\frac{\dinf}{\dinf t}\Vert\delay(t,\cdot)\Vert_{L^{2}(-\tau,0)}^{2} & =\Re(\partial_{\theta}\delay(t,\cdot),\delay(t,\cdot))_{L^{2}(-\tau,0)}\\ & =\frac{1}{2}\left[\vert\delay(t,0)\vert^{2}-\vert\delay(t,-\tau)\vert^{2}\right],\end{aligned}$$ which we shall use in the proof of Lemma \[lem:=00005BDELAY=00005D\_Dissipativity\]. Note that a finite number of time-delays $\tau_{i}>0$ can be accounted for by setting $\tau\coloneqq\max_{i}\tau_{i}$ and writing $$z\star u(t)=z_{0}u(t)+\sum_{i}z_{\tau_{i}}\delay(t,-\tau_{i}).$$ The corresponding impedance $\hat{z}(s)=z_{0}+\sum_{i}z_{\tau_{i}}e^{-\tau_{i}s}$ is positive-real if $z_{0}\geq\sum_{i}\vert z_{\tau_{i}}\vert$. No substantial change to the proofs of Section \[sub:=00005BDELAY=00005D\_Asymptotic-stability\] is required to handle this case. In [@nicaise2006stability], asymptotic stability is proven under the condition $z_{0}\geq\sum_{i}z_{i}>0$ and $z_i>0$. Asymptotic stability\[sub:=00005BDELAY=00005D\_Asymptotic-stability\] --------------------------------------------------------------------- Let $$H_{\opDiv}(\Omega)\coloneqq\left\{ \uac\in L^{2}(\Omega)^{d}\;\vert\;\opDiv\uac\in L^{2}(\Omega)\right\} .$$ The state space is defined as $$\begin{gathered}\spaceState\coloneqq\nabla H^{1}(\Omega)\times L^{2}(\Omega)\times L^{2}(\partial\Omega;L^{2}(-\tau,0)),\\ ((\uac,\pac,\delay),(\srcuac,\srcpac,\srcdelay))_{\spaceState}\coloneqq(\uac,\srcuac)+(\pac,\srcpac)+k(\delay,\srcdelay)_{L^{2}(\partial\Omega;L^{2}(-\tau,0))}, \end{gathered} \label{eq:=00005BDELAY=00005D_Scalar-Prod}$$ where $k\in\spaceR$ is a constant to be tuned to achieve dissipativity, see Lemma \[lem:=00005BDELAY=00005D\_Dissipativity\]. The evolution operator is defined as $$\begin{gathered}\spaceDomain\ni\state\coloneqq\left(\begin{array}{c} \uac\\ \pac\\ \delay \end{array}\right)\longmapsto\opA\state\coloneqq\left(\begin{array}{c} -\nabla\pac\\ -\opDiv\uac\\ \partial_{\theta}\delay \end{array}\right),\\ \spaceDomain\coloneqq\left\{ (\uac,\pac,\delay)\in\spaceState\;\left|\;\begin{alignedat}{1} & (\uac,\pac)\in H_{\opDiv}(\Omega)\times H^{1}(\Omega)\\ & \delay\in L^{2}(\partial\Omega;H^{1}(-\tau,0))\\ & \pac=z_{0}\uac\cdot\normal+z_{\tau}\delay(\cdot,-\tau)\;\text{in }H^{-\frac{1}{2}}(\partial\Omega)\\ & \delay(\cdot,0)=\uac\cdot\normal\;\text{in }H^{-\frac{1}{2}}(\partial\Omega) \end{alignedat} \right.\right\} . \end{gathered} \label{eq:=00005BDELAY=00005D_Definition-A}$$ In this formulation, the IBC (\[eq:=00005BDELAY=00005D\_IBC-Temp\]) is the third equation in $\spaceDomain$. We apply Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\], see the Lemmas \[lem:=00005BDELAY=00005D\_Dissipativity\], \[lem:=00005BDELAY=00005D\_Injectivity\], and \[lem:=00005BDELAY=00005D\_Bijectivity\] below. Lemma \[lem:=00005BDELAY=00005D\_Dissipativity\] shows that the seemingly free parameter $k$ must be restricted for $\Vert\cdot\Vert_{\spaceState}$ to be a Lyapunov functional, as formally highlighted in [@monteghetti2017delay]. \[rem:=00005BDELAY=00005D\_Bochner-Integration\] For the integrability of vector-valued functions, we follow the definitions and results presented in [@yosida1980funana § V.5]. Let $\mathcal{B}$ be a Banach space. We have [@yosida1980funana Thm. V.5.1] $$L^{2}(\partial\Omega;\mathcal{B})=\left\{ f:\,\partial\Omega\rightarrow\mathcal{B}\;\text{strongly measurable}\,\left|\,\Vert f\Vert_{\mathcal{B}}\in L^{2}(\partial\Omega)\right.\right\} .$$ In Sections \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\] and \[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\], we repeatedly use the following result: if $A\in\spaceBounded(\mathcal{B}_{1},\mathcal{B}_{2})$ and $u\in L^{2}(\partial\Omega;\mathcal{B}_{1})$, then $Au\in L^{2}(\partial\Omega;\mathcal{B}_{2})$. \[rem:|Delay=00005D\_Exclusion-Solenoidal-Fields\]Since $\nabla H^{1}(\Omega)$ is a closed subspace of $L^{2}(\Omega)^{d}$, $\spaceState$ is a Hilbert space, see Section \[sub:=00005BMISC=00005D\_Hodge-decomposition\] for some background. In view of the orthogonal decomposition (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]), working with $\nabla H^{1}\left(\Omega\right)$ instead of $L^{2}\left(\Omega\right)^{d}$ enables to get an injective evolution operator $\opA$. The exclusion of the solenoidal fields $\uac$ that belong to $H_{\opDiv0,0}(\Omega)$ from the domain of $\opA$ can be physically justified by the fact that these fields are non-propagating and are not affected by the IBC. \[lem:=00005BDELAY=00005D\_Dissipativity\]The operator $\opA$ given by (\[eq:=00005BDELAY=00005D\_Definition-A\]) is dissipative if and only if $$k\in\left[z_{0}-\sqrt{z_{0}^{2}-z_{\tau}^{2}},z_{0}+\sqrt{z_{0}^{2}-z_{\tau}^{2}}\right].$$ Let $\state\in\spaceDomain$. In particular, $\uac\cdot\normal\in L^{2}(\partial\Omega)$ since $\delay(\cdot,0)\in L^{2}(\partial\Omega)$. Using Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) $$\begin{aligned} {1} \Re(\opA\state,\state)_{\spaceState}= & -\Re\left[\langle\uac\cdot\normal,\overline{\pac}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}\right]+k\,\Re\left(\partial_{\theta}\delay,\delay\right)_{L^{2}(\partial\Omega;L^{2}(-\tau,0))}\\ = & -\Re\left[(\uac\cdot\normal,\pac)_{L^{2}(\partial\Omega)}\right]+\frac{k}{2}\Vert\delay(\cdot,0)\Vert_{L^{2}(\partial\Omega)}^{2}-\frac{k}{2}\Vert\delay(\cdot,-\tau)\Vert_{L^{2}(\partial\Omega)}^{2},\\ = & \left(\frac{k}{2}-z_{0}\right)\Vert\delay(\cdot,0)\Vert_{L^{2}(\partial\Omega)}^{2}-\frac{k}{2}\Vert\delay(\cdot,-\tau)\Vert_{L^{2}(\partial\Omega)}^{2}\\ & -z_{\tau}\Re\left[(\delay(\cdot,0),\delay(\cdot,-\tau))_{L^{2}(\partial\Omega)}\right],\end{aligned}$$ from which we deduce that $\opA$ is dissipative if and only if the matrix $$\left[\begin{array}{cc} z_{0}-\frac{k}{2} & \frac{z_{\tau}}{2}\\ \frac{z_{\tau}}{2} & \frac{k}{2} \end{array}\right]$$ is positive semidefinite, i.e. if and only if its determinant and trace are nonnegative: $$(2z_{0}-k)k\geq z_{\tau}^{2}\quad\text{and}\quad z_{0}\geq0.$$ The conclusion follows the expressions of the roots of $k\mapsto-k^{2}+2z_{0}k-z_{\tau}^{2}$. \[lem:=00005BDELAY=00005D\_Injectivity\]The operator $\opA$ given by (\[eq:=00005BDELAY=00005D\_Definition-A\]) is injective. Assume $\state\in\spaceDomain$ satisfies $\opA\state=0$, i.e. $\nabla\pac=\vector 0$, $\opDiv\uac=0$, and $$\partial_{\theta}\delay(\coordx,\theta)=0\quad\text{a.e. in }\partial\Omega\times(-\tau,0).\label{eq:=00005BDELAY=00005D_Injectivity-1}$$ Hence $\delay(\coordx,\cdot)$ is constant with $$\delay(\cdot,0)=\delay(\cdot,-\tau)=\uac\cdot\normal\quad\text{a.e. in }\mbox{\ensuremath{\partial\Omega}}.\label{eq:=00005BDELAY=00005D_Injectivity-2}$$ Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) yields $$\langle\uac\cdot\normal,\overline{\pac}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}=0,$$ and by combining with the IBC (i.e. the third equation in $\spaceDomain$) and (\[eq:=00005BDELAY=00005D\_Injectivity-2\]) $$\hat{z}(0)\Vert\uac\cdot\normal\Vert_{L^{2}(\partial\Omega)}^{2}=0,$$ where we have used that $\uac\cdot\normal\in L^{2}(\partial\Omega)$ since $\delay(\cdot,0)\in L^{2}(\partial\Omega)$. Since $\hat{z}(0)\neq0$ we deduce that $\uac\in H_{\opDiv0,0}(\Omega)$, hence $\uac=\vector 0$ from (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]) and $\delay=0$. The IBC gives $\pac=0$ a.e. on $\partial\Omega$, hence $\pac=0$ a.e. on $\Omega$. \[lem:=00005BDELAY=00005D\_Bijectivity\]Let $\opA$ be given by (\[eq:=00005BDELAY=00005D\_Definition-A\]). Then, $s\opId-\opA$ is bijective for $s\in(0,\infty)\cup i\spaceR^{*}$. Let $\srcstate\in\spaceState$ and $s\in(0,\infty)\cup i\spaceR^{*}$. We seek a *unique* $\state\in\spaceDomain$ such that $(s\opId-\opA)\state=\srcstate$, i.e. $$\begin{cases} s\uac+\nabla\pac=\srcuac & \text{(a)}\\ s\pac+\opDiv\uac=\srcpac & \text{(b)}\\ s\delay-\partial_{\theta}\delay=\srcdelay. & \text{(c)} \end{cases}\label{eq:=00005BDELAY=00005D_Bijectivity-1}$$ The proof, as well as the similar ones found in the next sections, proceeds in three steps. \(a) As a preliminary step, let us *assume* that (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]) holds with $\state\in\spaceDomain$. Equation (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]c) can be uniquely solved as $$\delay(\cdot,\theta)=e^{s\theta}\uac\cdot\normal+R(s,\partial_{\theta})\srcdelay(\cdot,\theta),\label{eq:=00005BDELAY=00005D_Bijectivity-Delay}$$ where we denote $$R(s,\partial_{\theta})\srcdelay(\coordx,\theta)\coloneqq\left[Y_{1}e^{s\cdot}\star\srcdelay(\coordx,\cdot)\right](\theta)=\int_{0}^{\theta}e^{s(\theta-\tilde{\theta})}\srcdelay(\coordx,\tilde{\theta})\,\dinf\tilde{\theta}.$$ We emphasize that, in the remaining of the proof, $R(s,\partial_{\theta})$ is merely a convenient notation: the operator “$\partial_{\theta}$” cannot be defined independently from $\opA$ (see Remark \[rem:=00005BDelay=00005D\_Formal-Resolvent-Operator\] for a detailed explanation). The IBC (i.e. the third equation in $\spaceDomain$) can then be written as $$\pac=\hat{z}(s)\uac\cdot\normal+z_{\tau}R(s,\partial_{\theta})\srcdelay(\cdot,-\tau)\quad\text{in }H^{-\frac{1}{2}}(\partial\Omega),\label{eq:=00005BDELAY=00005D_Bijectivity-IBC}$$ and this identity actually takes place in $L^{2}(\partial\Omega)$ since $$\coordx\mapsto R(s,\partial_{\theta})\srcdelay(\coordx,-\tau)\in L^{2}(\partial\Omega).$$ Let $\test\in H^{1}(\Omega)$. Combining $(\srcuac,\nabla\test)+s(\srcpac,\test)$ with (\[eq:=00005BDELAY=00005D\_Bijectivity-IBC\]) yields $$\begin{alignedat}{1}(\nabla\pac,\nabla\test)+s^{2}(\pac,\test)+\frac{s}{\hat{z}(s)}(\pac,\test)_{L^{2}(\partial\Omega)}= & (\srcuac,\nabla\test)+s(\srcpac,\test)\\ & +\frac{sz_{\tau}}{\hat{z}(s)}(R(s,\partial_{\theta})\srcdelay(\cdot,-\tau),\test)_{L^{2}(\partial\Omega)}. \end{alignedat} \label{eq:=00005BDELAY=00005D_Bijectivity-Weakp}$$ In summary, $(s\opId-\opA)\state=\srcstate$ with $\state\in\spaceDomain$ implies (\[eq:=00005BDELAY=00005D\_Bijectivity-Weakp\]). \(b) We now construct a state $\state\in\spaceDomain$ such that $(s\opId-\opA)\state=\srcstate$. To do so, we use the conclusion from the preliminary step (a). Let $\pac\in H^{1}(\Omega)$ be the unique solution of (\[eq:=00005BDELAY=00005D\_Bijectivity-Weakp\]) obtained with Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\]. It remains to find suitable $\uac$ and $\delay$ so that $(\uac,\pac,\delay)\in\spaceDomain$. Let us *define* $\uac\in\nabla H^{1}(\Omega)$ by (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]a). Taking $\test\in\spaceContinuous_{0}^{\infty}(\Omega)$ in (\[eq:=00005BDELAY=00005D\_Bijectivity-Weakp\]) shows that $\uac\in H_{\opDiv}(\Omega)$ with (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]b). Using the expressions of $\uac$ and $\opDiv\uac$, and Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]), the weak formulation (\[eq:=00005BDELAY=00005D\_Bijectivity-Weakp\]) can be rewritten as $$\langle\uac\cdot\normal,\overline{\test}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}=\frac{1}{\hat{z}(s)}(\pac,\test)_{L^{2}(\partial\Omega)}-\frac{z_{\tau}}{\hat{z}(s)}(R(s,\partial_{\theta})\srcdelay(\cdot,-\tau),\test)_{L^{2}(\partial\Omega)}.$$ which shows that $\pac$ and $\uac$ satisfy (\[eq:=00005BDELAY=00005D\_Bijectivity-IBC\]). Let us now *define* $\delay$ in $L^{2}(\partial\Omega;H^{1}(-\tau,0))$ by (\[eq:=00005BDELAY=00005D\_Bijectivity-Delay\]). By rewriting (\[eq:=00005BDELAY=00005D\_Bijectivity-IBC\]) as $$\pac=(\hat{z}(s)-z_{\tau}e^{-s\tau})\uac\cdot\normal+z_{\tau}\left(e^{-s\tau}\uac\cdot\normal+R(s,\partial_{\theta})\srcdelay(\cdot,-\tau)\right)\quad\text{in }H^{-\frac{1}{2}}(\partial\Omega),$$ we deduce thanks to (\[eq:=00005BDELAY=00005D\_Laplace\]) and (\[eq:=00005BDELAY=00005D\_Bijectivity-Delay\]) that the IBC holds, i.e. that $(\uac,\pac,\delay)\in\spaceDomain$. \(c) We now show the uniqueness in $\spaceDomain$ of a solution of (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]). The uniqueness of $\pac$ in $H^{1}\left(\Omega\right)$ follows from Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\]. Although $\uac$ is not unique in $H_{\opDiv}(\Omega)$, it is unique in $H_{\opDiv}(\Omega)\cap\nabla H^{1}\left(\Omega\right)$ following (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]). The uniqueness of $\delay$ follows from the fact that (\[eq:=00005BDELAY=00005D\_Bijectivity-1\]c) is uniquely solvable in $\spaceDomain$. \[rem:=00005BDelay=00005D\_Formal-Resolvent-Operator\]In the proof, $R(s,\partial_{\theta})$ is only a notation since $\partial_{\theta}$ (hence also its resolvent operator) cannot be defined separately from $\opA$. Indeed, the definition of $\partial_{\theta}$ would be $$\left|\begin{alignedat}{1}\partial_{\theta}:\, & \spaceD(\partial_{\theta})\subset L^{2}(\partial\Omega;L^{2}(-\tau,0))\rightarrow L^{2}(\partial\Omega;L^{2}(-\tau,0))\\ & \delay\mapsto\partial_{\theta}\delay, \end{alignedat} \right.$$ with domain $$\spaceD(\partial_{\theta})=\left\{ \delay\in L^{2}(\partial\Omega;H^{1}(-\tau,0))\;\vert\;\delay(\cdot,0)=\uac\cdot\normal\right\}$$ that depends upon $\uac$. Standard diffusive impedance\[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\] ==================================================================================== This section focuses on the class of so-called *standard diffusive* kernels [@montseny1998diffusive], defined as $$z(t)\coloneqq\int_{0}^{\infty}e^{-\xi t}Y_{1}(t)\,\dinf\mu(\xi),\label{eq:=00005BDIFF=00005D_Time-Standard-Diffusive}$$ where $t\in\spaceR$ and $\mu$ is a positive Radon measure on $[0,\infty)$ that satisfies the following well-posedness condition $$\int_{0}^{\infty}\frac{\dinf\mu(\xi)}{1+\xi}<\infty,\label{eq:=00005BDIFF=00005D_Well-posedness-Condition}$$ which guarantees that $z\in L_{\textup{loc}}^{1}([0,\infty))$ with Laplace transform $$\hat{z}(s)=\int_{0}^{\infty}\frac{1}{s+\xi}\dinf\mu(\xi).\label{eq:=00005BDIFF=00005D_Laplace-Standard-Diffusive}$$ The estimate $$\forall s\in\overline{\spaceC_{0}^{+}}\backslash\{0\},\quad\frac{1}{\vert s+\xi\vert}\leq\sqrt{2}\max\left[1,\frac{1}{\vert s\vert}\right]\frac{1}{1+\xi},\label{eq:=00005BDIFF=00005D_FirstOrder-Estimate}$$ which is used below, shows that $\hat{z}$ is defined on $\overline{\spaceC_{0}^{+}}\backslash\{0\}$. This class of (positive-real) kernels is physically linked to non-propagating lossy phenomena and arise in electromagnetics [@garrappa2016models], viscoelasticity [@desch1988exponential; @mainardi1997frac], and acoustics [@helie2006diffusive; @lombard2016fractional; @monteghetti2016diffusive]. Formally, $\hat{z}$ admits the following realization $$\left\{ \begin{alignedat}{1} & \vphantom{\int}\partial_{t}\diff(t,\xi)=-\xi\diff(t,\xi)+u(t),\;\diff(0,\xi)=0\quad\left(\xi\in(0,\infty)\right),\\ & z\star u(t)=\int_{0}^{\infty}\diff(t,\xi)\,\dinf\mu(\xi). \end{alignedat} \right.\label{eq:=00005BDIFF=00005D_Diffusive-Realization}$$ The realization (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]) can be given a meaning using the theory of well-posed linear systems [@weiss2001wellposed; @staffans2005well; @matignon2010diffusivewp; @tucsnak2014wellposed]. However, in order to prove asymptotic stability, we need a framework to give a meaning to the *coupled* system (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\],\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]), which, it turns out, can be done without defining a well-posed linear system out of (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]). Similarly to the previous sections, this section is divided into two parts. Section \[sub:=00005BDIFF=00005D\_Abstract-realization\] defines the realization of (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]) and establishes some of its properties. These properties are then used in Section \[sub:=00005BDIFF=00005D\_Asymptotic-stability\] to prove asymptotic stability of the coupled system. The typical standard diffusive operator is the Riemann-Liouville fractional integral [@samko1993fractional § 2.3] [@matignon2008introduction] $$\hat{z}(s)=\frac{1}{s^{\alpha}},\;\dinf\mu(\xi)=\frac{\sin(\alpha\pi)}{\pi}\frac{1}{\xi^{\alpha}}\dinf\xi,\label{eq:=00005BDIFF=00005D_Diffusive-Weight-Fractional}$$ where $\alpha\in(0,1)$. \[rem:=00005BDIFF=00005D\_Completely-Monotonic\]The expression (\[eq:=00005BDIFF=00005D\_Time-Standard-Diffusive\]) arises naturally when inverting multivalued Laplace transforms, see [@duffy2004transform Chap. 4] for applications in partial differential equations. However, a standard diffusive kernel can also be defined as follows: a causal kernel $z$ is said to be *standard diffusive* if it belongs to $L_{\text{loc}}^{1}([0,\infty))$ and is completely monotone on $(0,\infty)$. By Bernstein’s representation theorem [@gripenberg1990volterra Thm. 5.2.5], $z$ is standard diffusive iff (\[eq:=00005BDIFF=00005D\_Time-Standard-Diffusive\],\[eq:=00005BDIFF=00005D\_Well-posedness-Condition\]) hold. Additionally, a standard diffusive kernel $z$ is integrable on $(0,\infty)$ iff $$\mu(\{0\})=0\quad\text{and}\quad\int_{0}^{\infty}\frac{1}{\xi}\dinf\mu(\xi)<\infty,$$ a property which will be referred to in Section \[sub:=00005BDIFF=00005D\_Abstract-realization\]. State spaces for the realization of classes of completely monotone kernels have been studied in [@desch1988exponential; @staffans1994well]. Abstract realization\[sub:=00005BDIFF=00005D\_Abstract-realization\] -------------------------------------------------------------------- To give a meaning to (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]) suited for our purpose, we define, for any $s\in\spaceR$, the following Hilbert space $$V_{s}\coloneqq\left\{ \diff:\,(0,\infty)\rightarrow\spaceC\text{ measurable}\;\left|\;\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}(1+\xi)^{s}\,\dinf\mu(\xi)<\infty\right.\right\} ,$$ with scalar product $$(\diff,\test)_{V_{s}}\coloneqq\int_{0}^{\infty}(\diff(\xi),\test(\xi))_{\spaceC}(1+\xi)^{s}\,\dinf\mu(\xi),$$ so that the triplet $(\spaceDiff_{-1},\spaceDiff_{0},\spaceDiff_{1})$ satisfies the continuous embeddings $$\spaceDiff_{1}\subset\spaceDiff_{0}\subset\spaceDiff_{-1}.\label{eq:=00005BDIFF=00005D_Triplet-Embedding}$$ The space $\spaceDiff_{0}$ will be the energy space of the realization, see (\[eq:=00005BDIFF=00005D\_State-Space\]). Note that the spaces $\spaceDiff_{-1}$ and $\spaceDiff_{1}$ defined above are different from those encountered when defining a well-posed linear system out of (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]), see [@matignon2010diffusivewp]. When $\dinf\mu$ is given by (\[eq:=00005BDIFF=00005D\_Diffusive-Weight-Fractional\]), the spaces $\spaceDiff_{0}$ and $\spaceDiff_{1}$ reduce to the spaces “$H_{\alpha}$” and “$V_{\alpha}$” defined in [@matignon2014asymptotic § 3.2]. On these spaces, we wish to define the unbounded state operator $A$, the control operator $B$, and the observation operator $C$ so that $$A:\,\spaceD(A)\coloneqq\spaceDiff_{1}\subset\spaceDiff_{-1}\rightarrow\spaceDiff_{-1},\;B\in\spaceBounded(\spaceC,\spaceDiff_{-1}),\;C\in\spaceBounded(\spaceDiff_{1},\spaceC).\label{eq:=00005BDIFF=00005D_ABC-Definition}$$ The state operator is defined as the following multiplication operator $$\begin{alignedat}{1} & A:\\ \\ \end{alignedat} \left|\begin{alignedat}{1} & \spaceD\left(A\right)\coloneqq\spaceDiff_{1}\subset\spaceDiff_{-1}\rightarrow\spaceDiff_{-1}\\ & \diff\mapsto(\xi\mapsto-\xi\diff(\xi)). \end{alignedat} \right.\label{eq:=00005BDIFF=00005D_Multiplication-Operator-A}$$ The control operator is simply $$Bu\coloneqq\xi\mapsto u,\label{eq:=00005BDIFF=00005D_Application-Definition-B}$$ and belongs to $\spaceBounded(\spaceC,\spaceDiff_{-1})$ thanks to the condition (\[eq:=00005BDIFF=00005D\_Well-posedness-Condition\]) since, for $u\in\spaceC$, $$\Vert Bu\Vert_{\spaceDiff_{-1}}=\left[\int_{0}^{\infty}\frac{1}{1+\xi}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\vert u\vert.$$ The observation operator is $$C\diff\coloneqq\int_{0}^{\infty}\diff(\xi)\,\dinf\mu(\xi),$$ and $C\in\spaceBounded(\spaceDiff_{1},\spaceC)$ thanks to (\[eq:=00005BDIFF=00005D\_Well-posedness-Condition\]) as, for $\diff\in\spaceDiff_{1}$, $$\vert C\diff\vert\leq\left[\int_{0}^{\infty}\frac{1}{1+\xi}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\Vert\diff\Vert_{\spaceDiff_{1}}.$$ The next lemma gathers properties of the triplet $(A,B,C)$ that are used in Section \[sub:=00005BDIFF=00005D\_Asymptotic-stability\] to obtain asymptotic stability. Recall that if $A$ is closed and $s\in\rho(A)$, then the resolvent operator $R(s,A)$ defined by (\[eq:=00005BSTAB=00005D\_Resolvent-Operator\]) belongs to $\spaceBounded(\spaceDiff_{-1},\spaceDiff_{1})$ [@kato1995ope § III.6.1]. \[lem:=00005BDIFF=00005D\_Multiplication-Operator-A\]The operator $A$ defined by (\[eq:=00005BDIFF=00005D\_Multiplication-Operator-A\]) is injective, generates a strongly continuous semigroup of contractions on $\spaceDiff_{-1}$, and satisfies $\overline{\spaceC_{0}^{+}}\backslash\{0\}\subset\rho(A)$. The proof is split into three steps, (a), (b), and (c). (a) The injectivity of $A$ follows directly from its definition. (b) Let us show that $(0,\infty)\cup i\spaceR^{*}\subset\rho(A)$. Let $\srcdiff\in\spaceDiff_{-1}$, $s\in(0,\infty)\cup i\spaceR^{*}$, and define $$\diff(\xi)\coloneqq\frac{1}{s+\xi}\srcdiff(\xi)\quad\text{a.e. on }(0,\infty).\label{eq:=00005BDIFF=00005D_Multiplication-Operator-Resolvent}$$ Using the estimate (\[eq:=00005BDIFF=00005D\_FirstOrder-Estimate\]), we have $$\begin{aligned} {1} \Vert\diff\Vert_{\spaceDiff_{1}} & \leq\sqrt{2}\max\left[1,\frac{1}{\vert s\vert}\right]\Vert\srcdiff\Vert_{\spaceDiff_{-1}},\end{aligned}$$ so that $\diff$ belongs to $\spaceDiff_{1}$ and $(s\opId-A)\diff=\srcdiff$ is well-posed. (c) For any $\diff\in\spaceDiff_{1}$, we have $\Re\left[(A\diff,\diff)_{\spaceDiff_{-1}}\right]\leq-\Vert\diff\Vert_{\spaceDiff_{0}}^{2}$, so $A$ is dissipative. By the Lumer-Phillips theorem, $A$ generates a strongly continuous semigroup of contractions on $\spaceDiff_{-1}$, so that $\spaceC_{0}^{+}\subset\rho(A)$ [@pazy1983stability Cor. 3.6]. \[lem:=00005BDIFF=00005D\_Admissible\]The triplet of operators $(A,B,C)$ defined above satisfies (\[eq:=00005BDIFF=00005D\_ABC-Definition\]) as well as the following properties. 1. (Stability) $A$ is closed and injective with $\overline{\spaceC_{0}^{+}}\backslash\{0\}\subset\rho(A)$. 2. (Regularity) 1. $A\in\spaceBounded(\spaceDiff_{1},\spaceDiff_{-1})$. 2. For any $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$, $$AR(s,A)_{\vert\spaceDiff_{0}}\in\spaceBounded(\spaceDiff_{0},\spaceDiff_{0}), \label{eq:=00005BDIFF=00005D_Definition-Regularity}$$ where the vertical line denotes the restriction. 3. (Reality) For any $s\in(0,\infty)$, $$CR(s,A)B_{\vert\spaceR}\in\spaceR,\label{eq:=00005BDIFF=00005D_Definition-Reality}$$ 4. (Passivity) For any $(\diff,u)\in\spaceD(A\&B)$, $$\Re\left[(A\diff+Bu,\diff)_{\spaceDiff_{0}}-(u,C\diff)_{\spaceC}\right]\leq0,\label{eq:=00005BDIFF=00005D_Definition-Dissipativity}$$ where we define $$\spaceD(A\&B)\coloneqq\left\{ (\diff,u)\in\spaceDiff_{1}\times\spaceC\;\vert\;A\diff+Bu\in\spaceDiff_{0}\right\} .$$ Let $A,$ $B$, and $C$ be defined as above. Each of the properties is proven below. - This condition is satisfied from Lemma \[lem:=00005BDIFF=00005D\_Multiplication-Operator-A\]. - Let $\diff\in\spaceDiff_{1}$. We have $$\begin{aligned} {1} \Vert A\diff\Vert_{\spaceDiff_{-1}}^{2}= & \int_{0}^{\infty}\vert\diff(\xi)\vert^{2}\frac{\xi^{2}}{1+\xi}\,\dinf\mu(\xi)\\ \leq & \int_{0}^{\infty}\vert\diff(\xi)\vert^{2}(1+\xi)\,\dinf\mu(\xi)=\Vert\diff\Vert_{\spaceDiff_{1}}^{2},\end{aligned}$$ using the inequality $\xi^{2}\leq(1+\xi)^{2}.$ - Let $\srcdiff\in\spaceDiff_{0}$ and $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$, $$\Vert AR(s,A)\srcdiff\Vert_{\spaceDiff_{0}}=\left[\int_{0}^{\infty}\left|\frac{\xi}{s+\xi}\srcdiff\right|^{2}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\leq\Vert\srcdiff\Vert_{\spaceDiff_{0}},$$ where we have used $\left|\frac{\xi}{s+\xi}\right|\leq\frac{\xi}{\Re(s)+\xi}\leq1.$ - Let $s\in(0,\infty)$ and $u\in\spaceR$. The reality condition is fulfilled since $$CR(s,A)Bu=u\int_{0}^{\infty}\frac{\dinf\mu(\xi)}{s+\xi}.$$ - Let $(\diff,u)\in\spaceD(A\&B)$. We have $$\Re\left[(A\diff+Bu,\diff)_{\spaceDiff_{0}}-(u,C\diff)_{\spaceC}\right]=-\Re\left[\int_{0}^{\infty}\xi\vert\diff(\xi)\vert^{2}\,\dinf\mu(\xi)\right]\leq0,\label{eq:=00005BDIFF=00005D_Application-Passivity}$$ so that the passivity condition is satisfied. The space $\spaceD(A\&B)$ is nonempty. Indeed, it contains at least the following one dimensional subspace $$\left\{ (\diff,u)\in\spaceDiff_{1}\times\spaceC\;\vert\;\diff=R(s,A)Bu\right\}$$ for any $s\in\rho(A)$ (which is nonempty from Lemma \[lem:=00005BDIFF=00005D\_Admissible\](i)); this follows from $$\begin{aligned} {1} A\diff+Bu= & AR(s,A)Bu+Bu\\ = & sR(s,A)Bu\in\spaceDiff_{1}.\end{aligned}$$ It also contains $\left\{ (R(s,A)\diff,0)\;\vert\;\diff\in\spaceDiff_{0}\right\} $. For any $s\in\rho(A)$, we define $$\hfun\coloneqq s\mapsto CR(s,A)B,\label{eq:=00005BDIFF=00005D_Transfer-Function-ABCD}$$ which is analytic, from the analyticity of $R(\cdot,A)$ [@kato1995ope Thm. III.6.7]. Additionally, we have $\hfun(s)\in\spaceR$ for $s\in(0,\infty)$ from (\[eq:=00005BDIFF=00005D\_Definition-Reality\]), and $\Re(\hfun(s))\geq0$ from the passivity condition (\[eq:=00005BDIFF=00005D\_Definition-Dissipativity\]) with $\diff\coloneqq R(s,A)Bu\in\spaceD(A\&B)$: $$\Re(s)\Vert R(s,A)Bu\Vert_{\spaceDiff_{0}}^{2}\leq\Re\left[(u,\hfun(s)u)_{\spaceC}\right].$$ Since $\spaceC_{0}^{+}\subset\rho(A)$, the function $\hfun$ defined by (\[eq:=00005BDIFF=00005D\_Transfer-Function-ABCD\]) is positive-real. Asymptotic stability\[sub:=00005BDIFF=00005D\_Asymptotic-stability\] -------------------------------------------------------------------- Let $(A,B,C)$ be defined as in Section \[sub:=00005BDIFF=00005D\_Abstract-realization\]. We further assume that $A$, $B$, and $C$ are non-null operators. The coupling between the wave equation (\[eq:=00005BMOD=00005D\_Wave-Equation\]) and the infinite-dimensional realization $(A,B,C)$ can be formulated as the abstract Cauchy problem (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) using the following definitions. The extended state space is $$\begin{gathered}\spaceState\coloneqq\nabla H^{1}(\Omega)\times L^{2}(\Omega)\times L^{2}(\partial\Omega;\spaceDiff_{0}),\\ ((\uac,\pac,\diff),(\srcuac,\srcpac,\srcdiff))_{\spaceState}\coloneqq(\uac,\srcuac)+(\pac,\srcpac)+(\diff,\srcdiff)_{L^{2}(\partial\Omega;\spaceDiff_{0})}, \end{gathered} \label{eq:=00005BDIFF=00005D_State-Space}$$ and the evolution operator $\opA$ is $$\begin{gathered}\spaceDomain\ni\state\coloneqq\left(\begin{array}{c} \uac\\ \pac\\ \diff \end{array}\right)\longmapsto\opA\state\coloneqq\left(\begin{array}{c} -\nabla\pac\\ -\opDiv\uac\\ A\diff+B\uac\cdot\normal \end{array}\right),\\ \spaceDomain\coloneqq\left\{ (\uac,\pac,\diff)\in\spaceState\;\left|\;\begin{alignedat}{1} & (\uac,\pac,\diff)\in H_{\opDiv}(\Omega)\times H^{1}(\Omega)\times L^{2}(\partial\Omega;\spaceDiff_{1})\\ & (A\diff+B\uac\cdot\normal)\in L^{2}(\partial\Omega;\spaceDiff_{0})\\ & \pac=C\diff\;\text{in }H^{\frac{1}{2}}(\partial\Omega) \end{alignedat} \right.\right\} , \end{gathered} \label{eq:=00005BDIFF=00005D_Definition-A}$$ where the IBC (\[eq:=00005BMOD=00005D\_IBC\],\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]) is the third equation in $\spaceDomain$. In the definition of $\opA$, there is an abuse of notation. Indeed, we still denote by $A$ the following operator $$\left|\begin{alignedat}{1} & L^{2}(\partial\Omega;\spaceDiff_{1})\rightarrow L^{2}(\partial\Omega;\spaceDiff_{-1})\\ & \diff\mapsto(\coordx\mapsto A\diff(\coordx,\cdot)), \end{alignedat} \right.$$ which is well-defined from Lemma \[lem:=00005BDIFF=00005D\_Admissible\](iia) and Remark \[rem:=00005BDELAY=00005D\_Bochner-Integration\]. A similar abuse of notation is employed for $B$ and $C$. Asymptotic stability is proven by applying Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] through Lemmas \[lem:=00005BDIFF=00005D\_Dissipativity\], \[lem:=00005BDIFF=00005D\_Injectivity\], and \[lem:=00005BDIFF=00005D\_Bijectivity\] below. In order to clarify the proofs presented in Lemmas \[lem:=00005BDIFF=00005D\_Dissipativity\] and \[lem:=00005BDIFF=00005D\_Injectivity\], we first prove a regularity property on $\uac$ that follows from the definition of $\spaceDomain$. \[lem:=00005BDIFF=00005D\_Boundary-Regularity-u\]If $\state=(\uac,\pac,\diff)\in\spaceDomain$, then $\uac\cdot\normal\in L^{2}(\partial\Omega)$. Let $\state\in\spaceDomain$. By definition of $\spaceDomain$, we have $\diff\in L^{2}(\partial\Omega;\spaceDiff_{1})$ so that $A\diff\in L^{2}(\partial\Omega;\spaceDiff_{-1})$ from Lemma \[lem:=00005BDIFF=00005D\_Admissible\](iia) and Remark \[rem:=00005BDELAY=00005D\_Bochner-Integration\]. From $$B\uac\cdot\normal=\underbrace{A\diff+B\uac\cdot\normal}_{\mathclap{\in L^{2}(\partial\Omega;\spaceDiff_{0})}}-\overbrace{A\diff}^{\mathclap{\in L^{2}(\partial\Omega;\spaceDiff_{-1})}},$$ we deduce that $B\uac\cdot\normal\in L^{2}(\partial\Omega;\spaceDiff_{-1})$. The conclusion then follows from the definition of $B$ and the condition (\[eq:=00005BDIFF=00005D\_Well-posedness-Condition\]). \[lem:=00005BDIFF=00005D\_Dissipativity\]The operator $\opA$ given by (\[eq:=00005BDIFF=00005D\_Definition-A\]) is dissipative. Let $\state\in\spaceDomain$. In particular, $\uac\cdot\normal\in L^{2}(\partial\Omega)$ from Lemma \[lem:=00005BDIFF=00005D\_Boundary-Regularity-u\]. Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) and the inequality (\[eq:=00005BDIFF=00005D\_Definition-Dissipativity\]) yield $$\begin{aligned} {1} \Re(\opA\state,\state)_{\spaceState} & =\Re\left[(A\diff+B\uac\cdot\normal,\diff)_{L^{2}(\partial\Omega;\spaceDiff_{0})}-\langle\uac\cdot\normal,\overline{\pac}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}\right]\\ & =\Re\left[(A\diff+B\uac\cdot\normal,\diff)_{L^{2}(\partial\Omega;\spaceDiff_{0})}-(\uac\cdot\normal,C\diff){}_{L^{2}(\partial\Omega)}\right]\leq0,\end{aligned}$$ where we have used that $\uac\cdot\normal\in L^{2}(\partial\Omega)$. \[lem:=00005BDIFF=00005D\_Injectivity\]The operator $\opA$ given by (\[eq:=00005BDIFF=00005D\_Definition-A\]) is injective. Assume $\state\in\spaceDomain$ satisfies $\opA\state=0$. In particular $\nabla\pac=\vector 0$ and $\opDiv\uac=0$, so that Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) yields $$\langle\uac\cdot\normal,\overline{\pac}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}=0,$$ and by combining with the IBC (i.e. the third equation in $\spaceDomain$) $$(\uac\cdot\normal,C\diff)_{L^{2}(\partial\Omega)}=0,\label{eq:=00005BDIFF=00005D_Injectivity-1}$$ where we have used that $\uac\cdot\normal\in L^{2}(\partial\Omega)$ from Lemma \[lem:=00005BDIFF=00005D\_Boundary-Regularity-u\]. The third equation that comes from $\opA\state=0$ is $$A\diff(\coordx,\cdot)+B\uac(\coordx)\cdot\normal(\coordx)=0\quad\text{in }\spaceDiff_{0}\text{ for a.e. }\coordx\in\partial\Omega.\label{eq:=00005BDIFF=00005D_Injectivity-2}$$ We now prove that $\state=0$, the key step being solving (\[eq:=00005BDIFF=00005D\_Injectivity-2\]). Since $A$ is injective, (\[eq:=00005BDIFF=00005D\_Injectivity-2\]) has at most one solution $\diff\in L^{2}(\partial\Omega;\spaceDiff_{1})$. Let us distinguish the possible cases. - If $0\in\rho(A)$, then $\diff=R(0,A)B\uac\cdot\normal\in L^{2}(\partial\Omega;\spaceDiff_{1})$ is the unique solution. Inserting in (\[eq:=00005BDIFF=00005D\_Injectivity-1\]) and using (\[eq:=00005BDIFF=00005D\_Transfer-Function-ABCD\]) yields $$(\uac\cdot\normal,\zfun(0)\uac\cdot\normal)_{L^{2}(\partial\Omega)}=0,$$ from which we deduce that $\uac\cdot\normal=0$ since $\zfun(0)$ is non-null. - If $0\in\sigma_{r}(A)\cup\sigma_{c}(A)$, then either $\overline{R(A)}\neq\spaceDiff_{-1}$ (definition of the residual spectrum) or $\overline{R(A)}=\spaceDiff_{-1}$ but $R(A)\neq\spaceDiff_{-1}$ (definition of the continuous spectrum combined with the closed graph theorem, since $A$ is closed). $R\left(A\right)$ is equipped with the norm from $\spaceDiff_{-1}$. If $B\uac\cdot\normal\notin L^{2}(\partial\Omega;R(A))$, then the only solution is $\diff=0$ and $\uac\cdot\normal=0$. If $B\uac\cdot\normal\in L^{2}(\partial\Omega;R(A))$, then $\diff=-A^{-1}B\uac\cdot\normal$ is the unique solution, where $A^{-1}:\,R(A)\rightarrow\spaceDiff_{1}$ is an unbounded closed bijection. Inserting in (\[eq:=00005BDIFF=00005D\_Injectivity-1\]) yields $$(\uac\cdot\normal,(-CA^{-1}B)\uac\cdot\normal)_{L^{2}(\partial\Omega)}=0.$$ Since $(-CA^{-1}B)\in\spaceC$ is non-null, we deduce that $\uac\cdot\normal=0$. In summary, $\uac\in H_{\opDiv0,0}(\Omega)$, $\diff=0$ in $L^{2}(\partial\Omega;\spaceDiff_{1})$, and $\pac=0$ in $L^{2}(\partial\Omega)$. The nullity of $\pac$ follows from $\nabla\pac=0$. The nullity of $\uac$ follows from $H_{\opDiv0,0}(\Omega)\cap\nabla H^{1}(\Omega)=\{0\}$, see (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]). \[lem:=00005BDIFF=00005D\_Bijectivity\]Let $\opA$ be given by (\[eq:=00005BDIFF=00005D\_Definition-A\]). Then, $s\opId-\opA$ is bijective for $s\in(0,\infty)\cup i\spaceR^{*}$. Let $\srcstate\in\spaceState$ and $s\in(0,\infty)\cup i\spaceR^{*}$. We seek a *unique* $\state\in\spaceDomain$ such that $(s\opId-\opA)\state=\srcstate$, i.e. $$\begin{cases} s\uac+\nabla\pac=\srcuac & \text{(a)}\\ s\pac+\opDiv\uac=\srcpac & \text{(b)}\\ s\diff-A\diff-B\uac\cdot\normal=\srcdiff. & \text{(c)} \end{cases}\label{eq:=00005BDIFF=00005D_Bijectivity-Eq}$$ For later use, let us note that Equation (\[eq:=00005BDIFF=00005D\_Bijectivity-Eq\]c) and the IBC (i.e. the third equation in $\spaceDomain$) imply $$\begin{aligned} {2} \diff & =R(s,A)(B\uac\cdot\normal+\srcdiff) & & \quad\text{in }L^{2}(\partial\Omega;\spaceDiff_{1})\label{eq:=00005BDIFF=00005D_Bijectivity-DiffState}\\ \pac & =\zfun(s)\uac\cdot\normal+CR(s,A)\srcdiff & & \quad\text{in }L^{2}(\partial\Omega).\label{eq:=00005BDIFF=00005D_Bijectivity-IBC}\end{aligned}$$ Let $\test\in H^{1}(\Omega)$. Combining $(\srcuac,\nabla\test)+s(\srcpac,\test)$ with (\[eq:=00005BDIFF=00005D\_Bijectivity-IBC\]) yields $$\begin{alignedat}{1}(\nabla\pac,\nabla\test)+s^{2}(\pac,\test)+\frac{s}{\zfun(s)}(\pac,\test)_{L^{2}(\partial\Omega)}=\; & (\srcuac,\nabla\test)+s(\srcpac,\test)\\ & +\frac{s}{\zfun(s)}(CR(s,A)\srcdiff,\test)_{L^{2}(\partial\Omega)}. \end{alignedat} \label{eq:=00005BDIFF=00005D_Bijectivity-WeakForm-p}$$ Note that since $CR(s,A)\in\spaceBounded(\spaceDiff_{-1},\spaceC),$ we have $$\coordx\mapsto CR(s,A)\srcdiff(\coordx)\in L^{2}(\partial\Omega),$$ so that (\[eq:=00005BDIFF=00005D\_Bijectivity-WeakForm-p\]) is meaningful. Moreover, we have $\Re(\zfun(s))\geq0$, and $\zfun(s)\in(0,\infty)$ for $s\in(0,\infty$). Therefore, we can apply Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\], pointwise, for $s\in(0,\infty)\cup i\spaceR^{*}$. Let us denote by $\pac$ the unique solution of (\[eq:=00005BDIFF=00005D\_Bijectivity-WeakForm-p\]) in $H^{1}(\Omega)$, obtained from Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\]. It remains to find suitable $\uac$ and $\diff$. Let us *define* $\uac\in\nabla H^{1}(\Omega)$ by (\[eq:=00005BDIFF=00005D\_Bijectivity-Eq\]a). Taking $\test\in\spaceContinuous_{0}^{\infty}(\Omega)$ in (\[eq:=00005BDIFF=00005D\_Bijectivity-WeakForm-p\]) shows that $\uac\in H_{\opDiv}(\Omega)$ and (\[eq:=00005BDIFF=00005D\_Bijectivity-Eq\]b) holds. Using the expressions of $\uac$ and $\opDiv\uac$, and Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]), the weak formulation (\[eq:=00005BDIFF=00005D\_Bijectivity-WeakForm-p\]) can be rewritten as $$\langle\uac\cdot\normal,\overline{\test}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}=\zfun(s)^{-1}(\pac,\test)_{L^{2}(\partial\Omega)}-\zfun(s)^{-1}(CR(s,A)\srcdiff,\test)_{L^{2}(\partial\Omega)},$$ which shows that $\pac$ and $\uac$ satisfy (\[eq:=00005BDIFF=00005D\_Bijectivity-IBC\]). Let us now *define* $\diff$ with (\[eq:=00005BDIFF=00005D\_Bijectivity-DiffState\]), which belongs to $L^{2}(\partial\Omega;\spaceDiff_{1})$. By rewriting (\[eq:=00005BDIFF=00005D\_Bijectivity-IBC\]) as $$\pac=(\zfun(s)-CR(s,A)B)\uac\cdot\normal+CR(s,A)(B\uac\cdot\normal+\srcdiff),$$ we obtain from (\[eq:=00005BDIFF=00005D\_Transfer-Function-ABCD\]) and (\[eq:=00005BDIFF=00005D\_Bijectivity-DiffState\]) that the IBC holds. To obtain $(\uac,\pac,\diff)\in\spaceDomain$ it remains to show that $A\diff+B\uac\cdot\normal$ belongs to $L^{2}(\partial\Omega;\spaceDiff_{0})$. Using the definition (\[eq:=00005BDIFF=00005D\_Bijectivity-DiffState\]) of $\diff$, we have $$\begin{aligned} {1} A\diff+B\uac\cdot\normal & =AR(s,A)(B\uac\cdot\normal+\srcdiff)+B\uac\cdot\normal\\ & =(AR(s,A)+\opId)B\uac\cdot\normal+AR(s,A)\srcdiff\\ & =sR(s,A)B\uac\cdot\normal+AR(s,A)\srcdiff.\end{aligned}$$ Since $\uac\cdot\normal\in L^{2}(\partial\Omega)$ and $R(s,A)B\in\spaceBounded(\spaceC,\spaceDiff_{1})$, we have $$sR(s,A)B\uac\cdot\normal\in L^{2}(\partial\Omega;\spaceDiff_{1}).$$ The property (\[eq:=00005BDIFF=00005D\_Definition-Regularity\]) implies that $$AR(s,A)\srcdiff\in L^{2}(\partial\Omega;\spaceDiff_{0}),$$ hence that $(\uac,\pac,\diff)\in\spaceDomain$. The uniqueness of $\pac$ follows from Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\], that of $\uac$ from (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]), and that of $\diff$ from the bijectivity of $s\idMat-A$. The time-delay case does not fit into the framework proposed in Section \[sub:=00005BDIFF=00005D\_Abstract-realization\], see Remark \[rem:=00005BDelay=00005D\_Formal-Resolvent-Operator\]. This justifies why delay and standard diffusive IBCs are covered separately. Extended diffusive impedance\[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\] ======================================================================================= In this section, we focus on a variant of the standard diffusive kernel, namely the so-called *extended diffusive* kernel given by $$\hat{z}(s)\coloneqq\int_{0}^{\infty}\frac{s}{s+\xi}\dinf\mu(\xi),\label{eq:=00005BEXTDIFF=00005D_Laplace-Extended-Diffusive}$$ where $\mu$ is a Radon measure that satisfies the condition (\[eq:=00005BDIFF=00005D\_Well-posedness-Condition\]), already encountered in the standard case, and $$\int_{0}^{\infty}\frac{1}{\xi}\dinf\mu(\xi)=\infty.\label{eq:=00005BEXTDIFF=00005D_Mu-Condition}$$ The additional condition (\[eq:=00005BEXTDIFF=00005D\_Mu-Condition\]) implies that $t\mapsto\int_{0}^{\infty}e^{-\xi t}\,\dinf\mu(\xi)$ is not integrable on $(0,\infty)$, see Remark \[rem:=00005BDIFF=00005D\_Completely-Monotonic\]. From (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]), we directly deduce that $\hat{z}$ *formally* admits the realization $$\left\{ \begin{alignedat}{1} & \vphantom{\int}\partial_{t}\diff(t,\xi)=-\xi\diff(t,\xi)+u(t),\;\diff(0,\xi)=0\quad\left(\xi\in(0,\infty)\right),\\ & z\star u(t)=\int_{0}^{\infty}(-\xi\diff(t,\xi)+u(t))\,\dinf\mu(\xi), \end{alignedat} \right.\label{eq:=00005BEXTDIFF=00005D_Extended-Diffusive-Realization}$$ where $u$ is a causal input. The separate treatment of the standard (\[eq:=00005BDIFF=00005D\_Laplace-Standard-Diffusive\]) and extended (\[eq:=00005BEXTDIFF=00005D\_Laplace-Extended-Diffusive\]) cases is justified by the fact that physical models typically yield non-integrable kernels, i.e. $$\int_{0}^{\infty}\dinf\mu(\xi)=+\infty,\label{eq:=00005BEXTDIFF=00005D_Infinite-Integral}$$ which prevents from splitting the observation integral in (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]): the observation and feedthrough operators must be combined into $C\&D$. This justifies why (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]) is only formal. Although a functional setting for (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]) has been obtained in [@monteghetti2017delay § B.3], we shall again follow the philosophy laid out in Section \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\]. Namely, Section \[sub:=00005BEXTDIFF=00005D\_Abstract-realization\] presents an abstract realization framework whose properties are given in Lemma \[lem:=00005BEXTDIFF=00005D\_Admissible\], which slightly differs from the standard case, and Section \[sub:=00005BEXTDIFF=00005D\_Asymptotic-stability\] shows asymptotic stability of the coupled system (\[eq:=00005BEXTDIFF=00005D\_Definition-A\]). \[rem:=00005BEXTDIFF=00005D\_Fractional-Derivative\] Let $\alpha\in(0,1)$. The typical extended diffusive operator is the Riemman-Liouville fractional derivative [@podlubny1999fractional § 2.3] [@matignon2008introduction], obtained for $\hat{z}(s)=s^{1-\alpha}$ and $\dinf\mu$ given by (\[eq:=00005BDIFF=00005D\_Diffusive-Weight-Fractional\]), which satisfies the condition (\[eq:=00005BEXTDIFF=00005D\_Mu-Condition\]). For this measure $\dinf\mu$, choosing the initialization $\varphi(0,\xi)=\nicefrac{u(0)}{\xi}$ in (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]) yields the Caputo derivative [@lombard2016fractional]. Abstract realization\[sub:=00005BEXTDIFF=00005D\_Abstract-realization\] ----------------------------------------------------------------------- To give meaning to the realization (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]) we follow a similar philosophy to the standard case, namely the definition of a triplet of Hilbert spaces $(\spaceDiff_{-1},\spaceDiff_{0},\spaceDiff_{1})$ that satisfies the continuous embeddings (\[eq:=00005BDIFF=00005D\_Triplet-Embedding\]) as well as a suitable triplet of operators $(A,B,C)$. The Hilbert spaces $\spaceDiff_{-1},$ $\spaceDiff_{0}$, and $\spaceDiff_{1}$ are defined as $$\begin{aligned} {1} \spaceDiff_{1} & \coloneqq\left\{ \diff:\,(0,\infty)\rightarrow\spaceC\text{ measurable}\;\left|\;\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}(1+\xi)\,\dinf\mu(\xi)<\infty\right.\right\} \\ \spaceDiff_{0} & \coloneqq\left\{ \diff:\,(0,\infty)\rightarrow\spaceC\text{ measurable}\;\left|\;\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}\xi\,\dinf\mu(\xi)<\infty\right.\right\} \\ \spaceDiff_{-1} & \coloneqq\left\{ \diff:\,(0,\infty)\rightarrow\spaceC\text{ measurable}\;\left|\;\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}\frac{\xi}{1+\xi^{2}}\,\dinf\mu(\xi)<\infty\right.\right\} ,\end{aligned}$$ with scalar products $$\begin{aligned} {1} (\diff,\test)_{\spaceDiff_{1}} & \coloneqq\int_{0}^{\infty}(\diff(\xi),\test(\xi))_{\spaceC}(1+\xi)\,\dinf\mu(\xi)\\ (\diff,\test)_{\spaceDiff_{0}} & \coloneqq\int_{0}^{\infty}(\diff(\xi),\test(\xi))_{\spaceC}\,\xi\,\dinf\mu(\xi)\\ (\diff,\test)_{\spaceDiff_{-1}} & \coloneqq\int_{0}^{\infty}(\diff(\xi),\test(\xi))_{\spaceC}\frac{\xi}{1+\xi^{2}}\,\dinf\mu(\xi),\end{aligned}$$ so that the continuous embeddings (\[eq:=00005BDIFF=00005D\_Triplet-Embedding\]) are satisfied. Note the change of definition of the energy space $\spaceDiff_{0}$, which reflects the fact that the Lyapunov functional of (\[eq:=00005BDIFF=00005D\_Diffusive-Realization\]) is different from that of (\[eq:=00005BEXTDIFF=00005D\_Extended-Diffusive-Realization\]): compare the energy balance (\[eq:=00005BDIFF=00005D\_Application-Passivity\]) with (\[eq:=00005BEXTDIFF=00005D\_Application-Passivity\]). The change in the definition of $\spaceDiff_{-1}$ is a consequence of this new definition of $\spaceDiff_{0}$. When $\dinf\mu$ is given by (\[eq:=00005BDIFF=00005D\_Diffusive-Weight-Fractional\]), the spaces $\spaceDiff_{0}$ and $\spaceDiff_{1}$ reduce to the spaces “$\tilde{H}_{\alpha}$” and “$V_{\alpha}$” defined in [@matignon2014asymptotic § 3.2]. The operators $A$, $B$, and $C$ satisfy (contrast with (\[eq:=00005BDIFF=00005D\_ABC-Definition\])) $$A:\,\spaceD(A)\coloneqq\spaceDiff_{0}\subset\spaceDiff_{-1}\rightarrow\spaceDiff_{-1},\;B\in\spaceBounded(\spaceC,\spaceDiff_{-1}),\;C\in\spaceBounded(\spaceDiff_{1},\spaceC).\label{eq:=00005BEXTDIFF=00005D_ABC-Definition}$$ The state operator $A$ is still the multiplication operator (\[eq:=00005BDIFF=00005D\_Multiplication-Operator-A\]), but with domain $\spaceDiff_{0}$ instead of $\spaceDiff_{1}$. Let us check that this definition makes sense. For any $\diff\in\spaceDiff_{0}$, we have $$\begin{aligned} {1} \Vert A\diff\Vert_{\spaceDiff_{-1}} & =\left[\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}\frac{\xi^{3}}{1+\xi^{2}}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\leq\Vert\diff\Vert_{\spaceDiff_{0}}.\label{eq:=00005BEXTDIFF=00005D_Estimate-Aphi}\end{aligned}$$ The control operator $B$ is defined as (\[eq:=00005BDIFF=00005D\_Application-Definition-B\]) and we have for any $u\in\spaceC$ $$\begin{aligned} {1} \Vert Bu\Vert_{\spaceDiff_{-1}} & =\left[\int_{0}^{\infty}\vert u\vert^{2}\frac{\xi}{1+\xi^{2}}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\leq\tilde{C}\left[\int_{0}^{\infty}\frac{1}{1+\xi}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\vert u\vert,\end{aligned}$$ where the constant $\tilde{C}>0$ is $$\tilde{C}\coloneqq\left\Vert \frac{\xi(1+\xi)}{1+\xi^{2}}\right\Vert _{L^{\infty}(0,\infty)}.$$ The observation operator $C$ is identical to the standard case. For use in Section \[sub:=00005BEXTDIFF=00005D\_Asymptotic-stability\], properties of $(A,B,C)$ are gathered in Lemma \[lem:=00005BEXTDIFF=00005D\_Admissible\] below. \[lem:=00005BEXTDIFF=00005D\_Multiplication-Operator-1\]The operator $A$ generates a strongly continuous semigroup of contractions on $\spaceDiff_{-1}$ and satisfies $\overline{\spaceC_{0}^{+}}\backslash\{0\}\subset\rho(A)$. The proof is similar to that of Lemma \[lem:=00005BDIFF=00005D\_Multiplication-Operator-A\]. Let $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$ and $\srcdiff\in\spaceDiff_{-1}$. Let us define $\diff$ by (\[eq:=00005BDIFF=00005D\_Multiplication-Operator-Resolvent\]). (a) We have $$\begin{aligned} {1} \Vert\diff\Vert_{\spaceDiff_{0}} & =\left[\int_{0}^{\infty}\left|\frac{1}{s+\xi}\srcdiff\right|^{2}\xi\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\\ & \leq\sqrt{2}\max\left[1,\frac{1}{\vert s\vert}\right]\left\Vert \frac{1+\xi^{2}}{(1+\xi)^{2}}\right\Vert _{L^{\infty}(0,\infty)}\Vert\srcdiff\Vert_{\spaceDiff_{-1}},\end{aligned}$$ so that $\diff$ solves $(s\opId-A)\diff=\srcdiff$ in $\spaceDiff_{0}$. Since $s\opId-A$ is injective, we deduce that $s\in\rho(A)$. (b) Let $\diff\in\spaceDiff_{0}$. We have $$(A\diff,\diff)_{\spaceDiff_{-1}}=-\int_{0}^{\infty}\vert\diff(\xi)\vert^{2}\frac{\xi^{2}}{1+\xi^{2}}\,\dinf\mu(\xi)\leq-\Vert\diff\Vert_{\spaceDiff_{0}}^{2},$$ so that $A$ is dissipative. The conclusion follows from the Lumer-Phillips theorem. \[lem:=00005BEXTDIFF=00005D\_Multiplication-Operator-2\]The operators $A$ and $B$ are injective. Moreover, if (\[eq:=00005BEXTDIFF=00005D\_Mu-Condition\]) holds, then $R(A)\cap R(B)=\{0\}$. The injectivity of $A$ and $B$ is immediate. Let $\srcdiff\in R(A)\cap R(B)$, so that there is $\diff\in\spaceDiff_{0}$ and $u\in\spaceC$ such that $A\diff=Bu$, i.e. $-\xi\diff(\xi)=u$ a.e. on $(0,\infty)$. The function $\diff$ belongs to $\spaceDiff_{0}$ if and only if $$\vert u\vert^{2}\int_{0}^{\infty}\frac{1}{\xi}\dinf\mu(\xi)<\infty.$$ So that, assuming (\[eq:=00005BEXTDIFF=00005D\_Mu-Condition\]), $\diff$ belongs to $\spaceDiff_{0}$ if and only if $u=0$ a.e on $(0,\infty)$. \[lem:=00005BEXTDIFF=00005D\_Admissible\]The triplet of operators $(A,B,C)$ defined above satisfies (\[eq:=00005BEXTDIFF=00005D\_ABC-Definition\]) as well as the following properties. 1. (Stability) $A$ is closed with $\overline{\spaceC_{0}^{+}}\backslash\{0\}\subset\rho(A)$ and satisfies $$\forall(\diff,u)\in\spaceD(C\&D),\;A\diff=Bu\Rightarrow(\diff,u)=(0,0),\label{eq:=00005BEXTDIFF=00005D_Definition-Injectivity}$$ where we define $$\spaceD(C\&D)\coloneqq\left\{ (\diff,u)\in\spaceDiff_{0}\times\spaceC\;\vert\;A\diff+Bu\in\spaceDiff_{1}\right\} .$$ 2. (Regularity) 1. $A\in\spaceBounded(\spaceDiff_{0},\spaceDiff_{-1})$. 2. For any $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$, $$AR(s,A)_{\vert\spaceDiff_{0}}\in\spaceBounded(\spaceDiff_{0},\spaceDiff_{1}),\;R(s,A)B\in\spaceBounded(\spaceC,\spaceDiff_{1}).\label{eq:=00005BEXTDIFF=00005D_Definition-Regularity}$$ 3. (Reality) Identical to Lemma \[lem:=00005BDIFF=00005D\_Admissible\](iii). 4. (Passivity) For any $(\diff,u)\in\spaceD(C\&D)$, $$\Re\left[(A\diff+Bu,\diff)_{\spaceDiff_{0}}-(u,C(A\diff+Bu))_{\spaceC}\right]\leq0.\label{eq:=00005BEXTDIFF=00005D_Definition-Dissipativity}$$ Let $(A,B,C)$ be as defined above. Each of the properties is proven below. 1. Follows from Lemmas \[lem:=00005BEXTDIFF=00005D\_Multiplication-Operator-1\] and \[lem:=00005BEXTDIFF=00005D\_Multiplication-Operator-2\]. 2. Follows from (\[eq:=00005BEXTDIFF=00005D\_Estimate-Aphi\]). 3. Let $s\in\overline{\spaceC_{0}^{+}}\backslash\{0\}$, $\srcdiff\in\spaceDiff_{0}$, and $u\in\spaceC$. We have $$\begin{aligned} {1} \Vert AR(s,A)\srcdiff\Vert_{\spaceDiff_{1}} & =\left[\int_{0}^{\infty}\vert\srcdiff(\xi)\vert^{2}\frac{\xi^{2}(1+\xi)}{\vert s+\xi\vert^{2}}\,\dinf\mu(\xi)\right]^{\nicefrac{1}{2}}\\ & \leq\sqrt{2}\max\left[1,\frac{1}{\vert s\vert}\right]\Vert\srcdiff\Vert_{\spaceDiff_{0}},\end{aligned}$$ and $$\begin{aligned} {1} \Vert R(s,A)Bu\Vert_{\spaceDiff_{1}} & =\left(\int_{0}^{\infty}\frac{1+\xi}{\vert s+\xi\vert^{2}}\,\dinf\mu(\xi)\right)^{\nicefrac{1}{2}}\vert u\vert\\ & \leq\sqrt{2}\max\left[1,\frac{1}{\vert s\vert}\right]\left(\int_{0}^{\infty}\frac{1}{1+\xi}\,\dinf\mu(\xi)\right)^{\nicefrac{1}{2}}\vert u\vert.\end{aligned}$$ 4. Immediate. 5. Let $(\diff,u)\in\spaceD(C\&D)$. We have $$\begin{aligned} {1} \Re\Bigl[(A\diff+ & Bu,\diff)_{\spaceDiff_{0}}-(u,C(A\diff+Bu))_{\spaceC}\Bigr]\nonumber \\ & =\Re\left[\int_{0}^{\infty}(-\xi\diff(\xi)+u,\diff(\xi))_{\spaceC}\,\xi\,\dinf\mu(\xi)-\left(u,\int_{0}^{\infty}(-\xi\diff(\xi)+u)\,\dinf\mu(\xi)\right)_{\spaceC}\right]\nonumber \\ & =\Re\left[\int_{0}^{\infty}(-\xi\diff(\xi)+u,\xi\diff(\xi)-u)_{\spaceC}\,\dinf\mu(\xi)\right]\nonumber \\ & =-\Re\left[\int_{0}^{\infty}\vert-\xi\diff(\xi)+u\vert^{2}\,\dinf\mu(\xi)\right]\leq0.\label{eq:=00005BEXTDIFF=00005D_Application-Passivity}\end{aligned}$$ The remarks made for the standard case hold identically (in particular, $\spaceD(C\&D)$ is nonempty). For $s\in\rho(A)$ we define $$\hfun(s)\coloneqq s\,CR(s,A)B.\label{eq:=00005BEXTDIFF=00005D_Laplace}$$ Asymptotic stability\[sub:=00005BEXTDIFF=00005D\_Asymptotic-stability\] ----------------------------------------------------------------------- Let $(A,B,C)$ be the triplet of operators defined in Section \[sub:=00005BEXTDIFF=00005D\_Abstract-realization\], further assumed to be non-null. The abstract Cauchy problem (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) considered herein is the following. The state space is $$\begin{gathered}\spaceState\coloneqq\nabla H^{1}(\Omega)\times L^{2}(\Omega)\times L^{2}(\partial\Omega;\spaceDiff_{0}),\\ ((\uac,\pac,\diff),(\srcuac,\srcpac,\srcdiff))_{\spaceState}\coloneqq(\uac,\srcuac)+(\pac,\srcpac)+(\diff,\srcdiff)_{L^{2}(\partial\Omega;\spaceDiff_{0})}, \end{gathered} \label{eq:=00005BEXTDIFF=00005D_State-Space}$$ and $\opA$ is defined as $$\begin{gathered}\spaceDomain\ni\state\coloneqq\left(\begin{array}{c} \uac\\ \pac\\ \diff \end{array}\right)\longmapsto\opA\state\coloneqq\left(\begin{array}{c} -\nabla\pac\\ -\opDiv\uac\\ A\diff+B\uac\cdot\normal \end{array}\right),\\ \spaceDomain\coloneqq\left\{ (\uac,\pac,\diff)\in\spaceState\;\left|\;\begin{alignedat}{1} & (\uac,\pac)\in H_{\opDiv}(\Omega)\times H^{1}(\Omega)\\ & (A\diff+B\uac\cdot\normal)\in L^{2}(\partial\Omega;\spaceDiff_{1})\\ & \pac=C(A\diff+B\uac\cdot\normal)\;\text{in }H^{\frac{1}{2}}(\partial\Omega) \end{alignedat} \right.\right\} . \end{gathered} \label{eq:=00005BEXTDIFF=00005D_Definition-A}$$ The technicality here is that the operator $(\diff,u)\mapsto C(A\diff+Bu)$ is defined over $\spaceD(C\&D)$, but $CB$ is not defined in general: this is the abstract counterpart of (\[eq:=00005BEXTDIFF=00005D\_Infinite-Integral\]). An immediate consequence of the definition of $\spaceDomain$ is given in the following lemma. \[lem:=00005BEXTDIFF=00005D\_Boundary-Regularity-u\]If $\state=(\uac,\pac,\diff)\in\spaceDomain$, then $\uac\cdot\normal\in L^{2}(\partial\Omega)$. Let $\state\in\spaceDomain$. By definition of $\spaceDomain$, we have $\diff\in L^{2}(\partial\Omega;\spaceDiff_{0})$ so that $A\diff\in L^{2}(\partial\Omega;\spaceDiff_{-1})$ from Lemma \[lem:=00005BEXTDIFF=00005D\_Admissible\](iia) and Remark \[rem:=00005BDELAY=00005D\_Bochner-Integration\]. The proof is then identical to that of Lemma \[lem:=00005BDIFF=00005D\_Boundary-Regularity-u\]. The application of Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] is summarized in the lemmas below, namely Lemmas \[lem:=00005BEXTDIFF=00005D\_Dissipativity\], \[lem:=00005BEXTDIFF=00005D\_Injectivity\], and \[lem:=00005BEXTDIFF=00005D\_Bijectivity\]. Due to the similarities with the standard case, the proofs are more concise and focus on the differences. \[lem:=00005BEXTDIFF=00005D\_Dissipativity\]The operator $\opA$ defined by (\[eq:=00005BEXTDIFF=00005D\_Definition-A\]) is dissipative. Let $\state\in\spaceDomain$. In particular, $\uac\cdot\normal\in L^{2}(\partial\Omega)$ from Lemma \[lem:=00005BEXTDIFF=00005D\_Boundary-Regularity-u\]. Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) and (\[eq:=00005BEXTDIFF=00005D\_Definition-Dissipativity\]) yield $$\begin{aligned} {1} \Re(\opA\state,\state)_{\spaceState}=\Re\Bigl[(A\diff & +B\uac\cdot\normal,\diff)_{L^{2}(\partial\Omega;\spaceDiff_{0})}\\ & -(\uac\cdot\normal,C(A\diff+B\uac\cdot\normal)){}_{L^{2}(\partial\Omega)}\Bigr]\leq0,\end{aligned}$$ using Lemma \[lem:=00005BEXTDIFF=00005D\_Admissible\]. The next proof is much simpler than in the standard case. \[lem:=00005BEXTDIFF=00005D\_Injectivity\]$\opA$, given by (\[eq:=00005BEXTDIFF=00005D\_Definition-A\]), is injective. Assume $\state\in\spaceDomain$ satisfies $\opA\state=0$. In particular $\nabla\pac=\vector 0$, $\opDiv\uac=0$, and $A\diff+B\uac\cdot\normal=0$ in $L^{2}(\partial\Omega;\spaceDiff_{1})$. The IBC (i.e. the third equation in $\spaceDomain$) gives $\pac=0$ in $L^{2}(\partial\Omega)$ hence $p=0$ in $L^{2}(\Omega)$. From Lemma \[lem:=00005BEXTDIFF=00005D\_Boundary-Regularity-u\], $\uac\cdot\normal\in L^{2}(\partial\Omega)$ so we have at least $B\uac\cdot\normal\in L^{2}(\partial\Omega;\spaceDiff_{-1})$. Using (\[eq:=00005BEXTDIFF=00005D\_Definition-Injectivity\]), we deduce $\diff=0$ and $\uac\cdot\normal=0$, hence $\uac=0$ from (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]). \[lem:=00005BEXTDIFF=00005D\_Bijectivity\]$s\opId-\opA$, with $\opA$ given by (\[eq:=00005BEXTDIFF=00005D\_Definition-A\]), is bijective for $s\in(0,\infty)\cup i\spaceR^{*}$. Let $\srcstate\in\spaceState$, $s\in(0,\infty)\cup i\spaceR^{*}$, and $\test\in H^{1}(\Omega)$. We seek a *unique* $\state\in\spaceDomain$ such that $(s\opId-\opA)\state=\srcstate$, i.e. (\[eq:=00005BDIFF=00005D\_Bijectivity-Eq\]), which implies $$\begin{alignedat}{1}(\nabla\pac,\nabla\test)+s^{2}(\pac,\test)+\frac{s}{\zfun(s)}(\pac,\test)_{L^{2}(\partial\Omega)}= & (\srcuac,\nabla\test)+s(\srcpac,\test)\\ & +\frac{s}{\zfun(s)}(CAR(s,A)\srcdiff,\test)_{L^{2}(\partial\Omega)}. \end{alignedat} \label{eq:=00005BEXTDIFF=00005D_Bijectivity-WeakForm-p}$$ Note that, from (\[eq:=00005BEXTDIFF=00005D\_Definition-Regularity\]), the right-hand side defines an anti-linear form on $H^{1}(\Omega)$. Let us denote by $\pac$ the unique solution of (\[eq:=00005BEXTDIFF=00005D\_Bijectivity-WeakForm-p\]) obtained from a pointwise application of Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\] (we rely here on (\[eq:=00005BDIFF=00005D\_Definition-Reality\])). It remains to find suitable $\uac$ and $\diff$, in a manner identical to the standard diffusive case. Taking $\test\in\spaceContinuous_{0}^{\infty}(\Omega)$ in (\[eq:=00005BEXTDIFF=00005D\_Bijectivity-WeakForm-p\]) shows that $\uac\in H_{\opDiv}(\Omega)$ with (\[eq:=00005BDIFF=00005D\_Bijectivity-Eq\]b). Using the expressions of $\uac\in\nabla H^{1}(\Omega)$ and $\opDiv\uac$, and Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]), the weak formulation (\[eq:=00005BEXTDIFF=00005D\_Bijectivity-WeakForm-p\]) shows that $\pac$ and $\uac$ satisfy, in $L^{2}(\partial\Omega)$, $$\pac=\zfun(s)\uac\cdot\normal+CAR(s,A)\srcdiff.\label{eq:=00005BEXTDIFF=00005D_Bijectivity-IBC}$$ Let us now *define* $\diff$ as $$\diff\coloneqq R(s,A)\left(B\uac\cdot\normal+\srcdiff\right)\in L^{2}(\partial\Omega;\spaceDiff_{0}).$$ Using the property (\[eq:=00005BEXTDIFF=00005D\_Definition-Regularity\]), we obtain that $$\begin{aligned} {1} A\diff+B\uac\cdot\normal & =AR(s,A)\left(B\uac\cdot\normal+\srcdiff\right)+B\uac\cdot\normal\\ & =sR(s,A)B\uac\cdot\normal+AR(s,A)\srcdiff\end{aligned}$$ belongs to $L^{2}(\partial\Omega;\spaceDiff_{1})$. We show that the IBC holds by rewriting (\[eq:=00005BEXTDIFF=00005D\_Bijectivity-IBC\]) as $$\begin{aligned} {1} \pac & =C(sR(s,A)B\uac\cdot\normal+AR(s,A)\srcdiff)\\ & =C(AR(s,A)B\uac\cdot\normal+B\uac\cdot\normal+AR(s,A)\srcdiff)\\ & =C(A\diff+B\uac\cdot\normal),\end{aligned}$$ using (\[eq:=00005BEXTDIFF=00005D\_Laplace\]). Thus $(\uac,\pac,\diff)\in\spaceDomain$. The uniqueness of $\pac$ follows from Theorem \[thm:=00005BMOD=00005D\_Weak-Form-p\], that of $\uac$ from (\[eq:=00005BPRE=00005D\_Hodge-Decomposition\]), and that of $\diff$ from $s\in\rho(A)$. Addition of a derivative term\[sec:=00005BDER=00005D\_Addition-of-Derivative\] ============================================================================== By *derivative impedance* we mean $$\hat{z}(s)=z_{1}s,\;z_{1}>0,$$ for which the IBC (\[eq:=00005BMOD=00005D\_IBC\]) reduces to $\pac=z_{1}\partial_{t}\uac\cdot\normal.$ The purpose of this section is to illustrate, on two examples, that the addition of such a derivative term to the IBCs covered so far (\[eq:=00005BDELAY=00005D\_Laplace\],\[eq:=00005BDIFF=00005D\_Laplace-Standard-Diffusive\],\[eq:=00005BEXTDIFF=00005D\_Laplace-Extended-Diffusive\]) leaves unchanged the asymptotic stability results obtained with Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\]: it only makes the proofs more cumbersome as the state space becomes lengthier. This is why this term has not been included in Sections \[sec:=00005BDELAY=00005D\_Delay-impedance\]–\[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\]. The examples will also illustrate why establishing the asymptotic stability of (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]) with (\[eq:=00005BMOD=00005D\_Target-Impedance\]) can be done by treating each positive-real term in (\[eq:=00005BMOD=00005D\_Target-Impedance\]) separately (i.e. by building the realization of each of the four positive-real term separately and then aggregating them), thus justifying a posteriori the structure of the article. Consider the following positive-real impedance kernel $$\hat{z}(s)=z_{0}+z_{1}s,\label{eq:=00005BDER=00005D_Prop-Der-Impedance}$$ where $z_{0},z_{1}>0$. The energy space is $$\begin{gathered}\spaceState\coloneqq\nabla H^{1}(\Omega)\times L^{2}(\Omega)\times L^{2}(\partial\Omega),\\ \begin{alignedat}{1}((\uac,\pac,\deriv),(\srcuac,\srcpac,\srcderiv))_{\spaceState}\coloneqq(\uac,\srcuac)+(\pac,\srcpac)+z_{1}(\deriv,\srcderiv)_{L^{2}(\partial\Omega)}\end{alignedat} \end{gathered}$$ and the corresponding evolution operator is $$\begin{gathered}\spaceDomain\ni\state\coloneqq\left(\begin{array}{c} \uac\\ \pac\\ \deriv \end{array}\right)\longmapsto\opA\state\coloneqq\left(\begin{array}{c} -\nabla\pac\\ -\opDiv\uac\\ \frac{1}{z_{1}}\left[\pac-z_{0}\uac\cdot\normal\right] \end{array}\right),\\ \spaceDomain\coloneqq\left\{ (\uac,\pac,\deriv)\in\spaceState\;\left|\;\begin{alignedat}{1} & (\uac,\pac)\in H_{\opDiv}(\Omega)\times H^{1}(\Omega)\\ & \deriv=\uac\cdot\normal\;\text{in }L^{2}(\partial\Omega) \end{alignedat} \right.\right\} . \end{gathered}$$ Note how the derivative term in (\[eq:=00005BDER=00005D\_Prop-Der-Impedance\]) is accounted for by adding the state variable $\eta\in L^{2}\left(\partial\Omega\right)$. The application of Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] is straightforward. For instance, for $\state\in\spaceDomain$, we have $$\begin{aligned} {1} \Re(\opA\state,\state)_{\spaceState}= & -\Re\left[(\uac\cdot\normal,\pac)_{L^{2}(\partial\Omega)}\right]+\Re\left[(\pac-z_{0}\uac\cdot\normal,\deriv)_{L^{2}(\partial\Omega)}\right]\\ = & -z_{0}\Vert\uac\cdot\normal\Vert_{L^{2}(\partial\Omega)}^{2},\end{aligned}$$ so that $\opA$ is dissipative. The injectivity of $\opA$ and the bijectivity of $s\opId-\opA$ for $s\in(0,\infty)\cup i\spaceR^{*}$ can be proven similarly to what has been done in the previous sections. Let us revisit the delay impedance (\[eq:=00005BDELAY=00005D\_Laplace\]), covered in Section \[sec:=00005BDELAY=00005D\_Delay-impedance\], by adding a derivative term to it: $$\hat{z}(s)\coloneqq z_{1}s+z_{0}+z_{\tau}e^{-\tau s},\label{eq:=00005BDER=00005D_Impedance-derivative}$$ where $z_{1}>0$ and $(z_{0},z_{\tau})$ are defined as in Section \[sec:=00005BDELAY=00005D\_Delay-impedance\], so that $\hat{z}$ is positive-real. The inclusion of the derivative implies the presence of an additional variable in the extended state, i.e. the state space is (compare with (\[eq:=00005BDELAY=00005D\_Scalar-Prod\])) $$\begin{gathered}\spaceState\coloneqq\nabla H^{1}(\Omega)\times L^{2}(\Omega)\times L^{2}(\partial\Omega;L^{2}(-\tau,0))\times L^{2}(\partial\Omega),\\ \begin{alignedat}{1}((\uac,\pac,\delay,\deriv),(\srcuac,\srcpac,\srcdelay,\srcderiv))_{\spaceState}\coloneqq(\uac,\srcuac)+(\pac,\srcpac)+k(\delay,\srcdelay & )_{L^{2}(\partial\Omega;L^{2}(-\tau,0))}\\ & +z_{1}(\deriv,\srcderiv)_{L^{2}(\partial\Omega)}. \end{alignedat} \\ \\ \end{gathered}$$ The operator $\opA$ becomes (compare with (\[eq:=00005BDELAY=00005D\_Definition-A\])) $$\begin{gathered}\spaceDomain\ni\state\coloneqq\left(\begin{array}{c} \uac\\ \pac\\ \delay\\ \deriv \end{array}\right)\longmapsto\opA\state\coloneqq\left(\begin{array}{c} -\nabla\pac\\ -\opDiv\uac\\ \partial_{\theta}\delay\\ \frac{1}{z_{1}}\left[\pac-z_{0}\uac\cdot\normal-z_{\tau}\delay(\cdot,-\tau)\right] \end{array}\right),\\ \spaceDomain\coloneqq\left\{ (\uac,\pac,\delay,\deriv)\in\spaceState\;\left|\;\begin{alignedat}{1} & (\uac,\pac,\delay)\in H_{\opDiv}(\Omega)\times H^{1}(\Omega)\times L^{2}(\partial\Omega;H^{1}(-\tau,0))\\ & \delay(\cdot,0)=\uac\cdot\normal\;\text{in }L^{2}(\partial\Omega)\\ & \deriv=\uac\cdot\normal\;\text{in }L^{2}(\partial\Omega) \end{alignedat} \right.\right\} , \end{gathered}$$ where the IBC (\[eq:=00005BMOD=00005D\_IBC\],\[eq:=00005BDER=00005D\_Impedance-derivative\]) is the third equation in $\spaceDomain$. The application of Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] is identical to Section \[sub:=00005BDELAY=00005D\_Asymptotic-stability\]. For instance, for $\state\in\spaceDomain$, we have $$\begin{aligned} {1} \Re(\opA\state,\state)_{\spaceState}= & -\Re\left[(\uac\cdot\normal,\pac)_{L^{2}(\partial\Omega)}\right]+\Re\left[k(\partial_{\theta}\delay,\delay)_{L^{2}(\partial\Omega;L^{2}(-\tau,0))}\right]\\ & +\Re\left[(\pac-z_{0}\uac\cdot\normal-z_{\tau}\delay(\cdot,-\tau),\deriv)_{L^{2}(\partial\Omega)}\right]\\ = & -\Re\left[(\uac\cdot\normal,\pac)_{L^{2}(\partial\Omega)}\right]+\frac{k}{2}\Re\left[\Vert\uac\cdot\normal\Vert_{L^{2}(\partial\Omega)}^{2}-\Vert\delay(\cdot,-\tau)\Vert_{L^{2}(\partial\Omega)}^{2}\right]\\ & +\Re\left[(\pac-z_{0}\uac\cdot\normal-z_{\tau}\delay(\cdot,-\tau),\uac\cdot\normal)_{L^{2}(\partial\Omega)}\right]\\ = & \left(\frac{k}{2}-z_{0}\right)\Vert\uac\cdot\normal\Vert_{L^{2}(\partial\Omega)}^{2}-\frac{k}{2}\Vert\delay(\cdot,-\tau)\Vert_{L^{2}(\partial\Omega)}^{2}\\ & -z_{\tau}\Re\left[(\delay(\cdot,-\tau),\uac\cdot\normal)_{L^{2}(\partial\Omega)}\right],\end{aligned}$$ so that the expression of $\Re(\opA\state,\state)_{\spaceState}$ is identical to that without a derivative term, see the proof of Lemma \[lem:=00005BDELAY=00005D\_Dissipativity\]. The proof of the injectivity of $\opA$ is also identical to that carried out in Lemma \[lem:=00005BDELAY=00005D\_Injectivity\]: the condition $\opA\state=0$ yields $\delay(\cdot,0)=\delay(\cdot,-\tau)=\uac\cdot\normal=\deriv$ a.e. on $\partial\Omega$. Finally, the proof of Lemma \[lem:=00005BDELAY=00005D\_Bijectivity\] can also be followed almost identically to solve $(s\opId-\opA)\state=\srcstate$ with $\srcstate=(\srcuac,\srcpac,\srcdelay,\srcderiv)$, the additional steps being straightforward; after defining uniquely $\pac$, $\uac$, and $\delay$, the only possibility for $\deriv$ is $\deriv\coloneqq\uac\cdot\normal$, which belongs to $L^{2}(\partial\Omega)$, and $\deriv=\delay(\cdot,0)$ is deduced from (\[eq:=00005BDELAY=00005D\_Bijectivity-Delay\]). Conclusions and perspectives ============================ This paper has focused on the asymptotic stability of the wave equation coupled with positive-real IBCs drawn from physical applications, namely time-delayed impedance in Section \[sec:=00005BDELAY=00005D\_Delay-impedance\], standard diffusive impedance (e.g. fractional integral) in Section \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\], and extended diffusive impedance (e.g. fractional derivative) in Section \[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\]. Finally, the invariance of the derived asymptotic stability results under the addition of a derivative term in the impedance has been discussed in Section \[sec:=00005BDER=00005D\_Addition-of-Derivative\]. The proofs crucially hinge upon the knowledge of a dissipative realization of the IBC, since it employs the semigroup asymptotic stability result given in [@arendt1988tauberian; @lyubich1988asymptotic]. By combining these results, asymptotic stability is obtained for the impedance $\hat{z}$ introduced in Section \[sec:=00005BMOD=00005D\_Model-and-preliminary-results\] and given by (\[eq:=00005BMOD=00005D\_Target-Impedance\]). This suggests the first perspective of this work, formulated as a conjecture. Assume $\hat{z}$ is positive-real, without isolated singularities on $i\spaceR$. Then the Cauchy problem (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]) is asymptotically stable in a suitable energy space. Establishing this conjecture using the method of proof used in this paper first requires building a dissipative realization of the impedance operator $u\mapsto z\star u$. If $\hat{z}$ is assumed rational and proper (i.e. $\hat{z}(\infty)$ is finite), a dissipative realization can be obtained using the celebrated positive-real lemma, also known as the KalmanYakubovichPopov lemma [@anderson1967prmatrices Thm. 3]; the proof of asymptotic stability is then a simpler version of that carried out in Section \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\], see [@monteghetti2018dissertation §4.3] for the details. If $\hat{z}$ is not proper, it can be written as $\hat{z}=a_{1}s+\hat{z}_{\text{p}}$ where $a_{1}>0$ and $\hat{z}_{\text{p}}$ is proper (see Remark \[rem:=00005BMOD=00005D\_PR-growth-at-infinity\]); each term can be covered separately, see Section \[sec:=00005BDER=00005D\_Addition-of-Derivative\]. If $\hat{z}$ is not rational, then a suitable infinite-dimensional variant of the positive-real lemma is required. For instance, [@staffans2002passive Thm. 5.3] gives a realization using system nodes; a difficulty in using this result is that the properties needed for the method of proof presented here do not seem to be naturally obtained with system nodes. This result would be sharp, in the sense that it is known that exponential stability is not achieved in general (consider for instance $\hat{z}(s)=\nicefrac{1}{\sqrt{s}}$ that induces an essential spectrum with accumulation point at $0$). If this conjecture proves true, then the rate of decay of the solution could also be studied and linked to properties of the impedance $\hat{z}$; this could be done by adapting the techniques used in [@stahn2018waveequation]. To illustrate this conjecture, let us give two examples of positive-real impedance kernels that are *not* covered by the results of this paper. Both examples arise in physical applications [@monteghetti2016diffusive] and have been used in numerical simulations [@monteghetti2017tdibc]. The first example is a kernel similar to (\[eq:=00005BMOD=00005D\_Target-Impedance\]), namely $$\hat{z}(s)=z_{0}+z_{\tau}e^{-\tau s}+z_{1}s+\int_{0}^{\infty}\frac{\mu(\xi)}{s+\xi}\,\dinf\xi\quad\left(\Re(s)>0\right),$$ where $\tau>0$, $z_{\tau}\in\spaceR$, $z_{0}\geq\vert z_{\tau}\vert$, $z_{1}>0$, and the weight $\mu\in\spaceContinuous^{\infty}((0,\infty))$ satisfies the condition $\int_{0}^{\infty}\frac{\vert\mu(\xi)\vert}{1+\xi}\dinf\xi<\infty$ and is such that $\hat{z}$ is positive-real. When the sign of $\mu$ is indefinite the passivity condition (\[eq:=00005BDIFF=00005D\_Application-Passivity\]) does not hold, so that this impedance is not covered by the presented results despite the fact that, overall, $\hat{z}$ is positive-real with a realization formally identical to that of the impedance (\[eq:=00005BMOD=00005D\_Target-Impedance\]) defined in Section \[sec:=00005BMOD=00005D\_Model-and-preliminary-results\]. The second and last example is $$\hat{z}(s)=z_{0}+z_{\tau}\frac{e^{-\tau s}}{\sqrt{s}},$$ with $z_{\tau}\geq0$, $\tau>0$, and $z_{0}\geq0$ sufficiently large for $\hat{z}$ to be positive-real (the precise condition is $z_{0}\geq-z_{\tau}\cos(\tilde{x}+\frac{\pi}{4})\sqrt{\nicefrac{\tau}{\tilde{x}}}$ where $\tilde{x}\simeq2.13$ is the smallest positive root of $x\mapsto\tan(x+\nicefrac{\pi}{4})+\nicefrac{1}{2x}$). A simple realization can be obtained by combining Sections \[sec:=00005BDELAY=00005D\_Delay-impedance\] and \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\], i.e. by delaying the diffusive representation using a transport equation: the convolution then reads, for a causal input $u$, $$z\star u=z_{0}u+z_{\tau}\int_{0}^{\infty}\delay(t,-\tau,\xi)\,\dinf\mu(\xi),$$ where $\diff$ and $\mu$ are defined as in Section \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\], and for a.e. $\xi\in(0,\infty)$ the function $\delay(\cdot,\cdot,\xi)$ obeys the transport equation (\[eq:=00005BDELAY=00005D\_Transport-Equation\]ab) but with $\delay(t,0,\xi)=\diff(t,\xi)$. So far, the authors have not been able to find a suitable Lyapunov functional (i.e. a suitable definition of $\Vert\cdot\Vert_{\spaceState}$) for this realization. The second open problem we wish to point out is the extension of the stability result to discontinuous IBCs. A typical case is a split of the boundary $\partial\Omega$ into three disjoint parts: a Neumann part $\partial\Omega_{N}$, a Dirichlet part $\partial\Omega_{D}$, and an impedance part $\partial\Omega_{z}$ where one of the IBCs covered in the paper is applied. Dealing with such discontinuities may involve the redefinition of both the energy space $\spaceState$ and domain $\spaceDomain$, as well as the derivation of compatibility constraints. The proofs may benefit from considering the scattering formulation, recalled in Remark \[rem:=00005BMOD=00005D\_Terminology\], which enables to write the three boundary conditions in a unified fashion. Acknowledgments {#acknowledgments .unnumbered} =============== This research has been financially supported by the French ministry of defense (Direction Générale de l’Armement) and ONERA (the French Aerospace Lab). We thank the two referees for their helpful comments. The authors are grateful to Prof. Patrick Ciarlet for suggesting the use of the extension by zero in the proof of Proposition \[prop:=00005BMOD=00005D\_Rellich-Lemma\]. Miscellaneous results ===================== For the sake of completeness, the key technical results upon which the paper depends are briefly gathered here. Extension by zero\[sub:=00005BMISC=00005D\_Extension-by-zero\] -------------------------------------------------------------- Let us define the zero extension operator as $$E:\,L^{2}(\Omega_{1})\rightarrow L^{2}(\Omega_{2}),\;Eu\coloneqq\left\{ u\;\text{on}\;\Omega_{1},\;0\;\text{on}\;\Omega_{2}\backslash\Omega_{1},\right.$$ where $\Omega_{1}$ and $\Omega_{2}$ are two open subsets of $\spaceR^{d}$ such that $\overline{\Omega_{1}}\subset\Omega_{2}$. \[prop:Extension-by-zero\]Let $\Omega_{1}$ and $\Omega_{2}$ be two bounded open subsets of $\spaceR^{d}$ such that $\overline{\Omega_{1}}\subset\Omega_{2}$. For any $\pac\in H_{0}^{1}\left(\Omega_{1}\right)$, $E\pac\in H_{0}^{1}\left(\Omega_{2}\right)$ with $$\forall i\in\left\llbracket 1,d\right\rrbracket ,\;\partial_{i}\left[E\pac\right]=E\left[\partial_{i}\pac\right]\;\text{a.e. in }\;\Omega_{2}.\label{eq:Extension-by-zero_Derivatives}$$ In particular, $\Vert\pac\Vert_{H^{1}\left(\Omega_{1}\right)}=\Vert E\pac\Vert_{H^{1}\left(\Omega_{2}\right)}$. Note that we do not require any regularity on the boundary of $\Omega_{i}$. This is due to the fact that the proof only relies on the definition of $H_{0}^{1}$ by density. The first part of the proof is adapted from [@adams1975Sobolev Lem.3.22]. By definition of $H_{0}^{1}\left(\Omega_{1}\right)$, there is a sequence $\phi_{n}\in\spaceContinuous_{0}^{\infty}\left(\Omega_{1}\right)$ converging to $\pac$ in the $\Vert\cdot\Vert_{H^{1}\left(\Omega_{1}\right)}$ norm. Since $E\pac\in L^{2}\left(\Omega_{2}\right)$, $E\pac$ is locally integrable and thus belongs to $\spaceD^{'}\left(\Omega_{2}\right)$. For any $\varphi\in\spaceContinuous_{0}^{\infty}\left(\Omega_{2}\right)$, we have $$\begin{aligned} {2} \left\langle \partial_{i}\left[E\pac\right],\varphi\right\rangle _{\spaceD^{'}\left(\Omega_{2}\right),\spaceContinuous_{0}^{\infty}\left(\Omega_{2}\right)} & \coloneqq-\left\langle E\pac,\partial_{i}\varphi\right\rangle _{\spaceD^{'}\left(\Omega_{2}\right),\spaceContinuous_{0}^{\infty}\left(\Omega_{2}\right)}\\ & =-\int_{\Omega_{1}}\pac\partial_{i}\varphi & \qquad\ensuremath{\left(E\pac\in L^{2}\left(\Omega_{2}\right)\right)}\\ & =-\lim_{n\rightarrow\infty}\int_{\Omega_{1}}\phi_{n}\partial_{i}\varphi & \qquad\ensuremath{\left(\pac\in H_{0}^{1}\left(\Omega_{1}\right)\right)}\\ & =\lim_{n\rightarrow\infty}\int_{\Omega_{1}}\partial_{i}\phi_{n}\varphi & \qquad\ensuremath{\left(\phi_{n}\in\spaceContinuous_{0}^{\infty}\left(\Omega_{1}\right)\right)}\\ & =\int_{\Omega_{1}}\partial_{i}\pac\varphi & \qquad\left(\partial_{i}\phi_{n}\xrightarrow[n\rightarrow\infty]{L^{2}\left(\Omega_{1}\right)}\partial_{i}\pac\right)\\ & =\int_{\Omega_{2}}E\left[\partial_{i}\pac\right]\varphi,\end{aligned}$$ hence $E\left[\partial_{i}\pac\right]=\partial_{i}\left[E\pac\right]$ in $\spaceD^{'}\left(\Omega_{2}\right)$. Since $\partial_{i}\pac\in L^{2}\left(\Omega_{1}\right)$ by assumption, we deduce from this identity that $E\left[\partial_{i}\pac\right]\in L^{2}\left(\Omega_{2}\right)$. Hence $E\pac\in H^{1}\left(\Omega_{2}\right)$. Using the fact that $E$ is an isometry from $H_{0}^{1}\left(\Omega_{1}\right)$ to $H^{1}\left(\Omega_{2}\right)$ we deduce $$\Vert E\phi_{n}-E\pac\Vert_{H^{1}\left(\Omega_{2}\right)}=\Vert E\left(\phi_{n}-\pac\right)\Vert_{H^{1}\left(\Omega_{2}\right)}=\Vert\phi_{n}-\pac\Vert_{H^{1}\left(\Omega_{1}\right)}\xrightarrow[n\rightarrow\infty]{}0.$$ Since $E\phi_{n}\in\spaceContinuous_{0}^{\infty}\left(\Omega_{2}\right)$, this shows that $E\pac\in H_{0}^{1}\left(\Omega_{2}\right)$. Compact embedding and trace operator\[sub:=00005BMISC=00005D\_Embedding-Trace\] ------------------------------------------------------------------------------- Let $\Omega\subset\spaceR^{d}$, $d\in\llbracket1,\infty\llbracket$, be a bounded open set with a Lipschitz boundary. The embedding $H^{1}(\Omega)\subset H^{s}(\Omega)$ with $s\in[0,1)$ is compact [@grisvard2011elliptic Thm. 1.4.3.2]. (See [@lionsMagenes1972BVP1 Thm. 16.17] for smooth domains.) The trace operator $H^{s}(\Omega)\rightarrow H^{s-\nicefrac{1}{2}}(\partial\Omega)$ with $s\in(\nicefrac{1}{2},1]$ is continuous and surjective [@grisvard2011elliptic Thm. 1.5.1.2]. (See [@ding1996trace Thm. 1] if $\Omega$ is also simply connected and [@lionsMagenes1972BVP1 Thm. 9.4] for smooth domains.) The trace operator $H_{\opDiv}(\Omega)\rightarrow H^{-\frac{1}{2}}(\partial\Omega)$, $\uac\mapsto\uac\cdot\normal$ is continuous [@giraultraviart1986femNavierStokes Thm. 2.5], and the following Green’s formula holds for $\test\in H^{1}(\Omega)$ [@giraultraviart1986femNavierStokes Eq. (2.17)] $$(\uac,\nabla\test)+(\opDiv\uac,\test)=\langle\uac\cdot\normal,\overline{\test}\rangle_{H^{-\frac{1}{2}}(\partial\Omega),H^{\frac{1}{2}}(\partial\Omega)}.\label{eq:=00005BPRE=00005D_Green-Formula}$$ Hodge decomposition\[sub:=00005BMISC=00005D\_Hodge-decomposition\] ------------------------------------------------------------------ Let $\Omega\subset\spaceR^{d}$, $d\in\llbracket1,\infty\llbracket$, be a connected open set with a Lipschitz boundary. The following orthogonal decomposition holds [@dautraylions1980vol3spectral Prop. IX.1] $$(L^{2}(\Omega))^{d}=\nabla H^{1}(\Omega)\varoplus H_{\opDiv0,0}(\Omega),\label{eq:=00005BPRE=00005D_Hodge-Decomposition}$$ where $$\nabla H^{1}(\Omega)\coloneqq\left\{ \vector f\in(L^{2}(\Omega))^{d}\;\vert\;\exists g\in H^{1}(\Omega):\;\vector f=\nabla g\right\}$$ is a closed subspace of $(L^{2}(\Omega))^{d}$ and $$H_{\opDiv0,0}(\Omega)\coloneqq\left\{ \vector f\in H_{\opDiv}(\Omega)\;\vert\;\opDiv\vector f=0,\;\vector f\cdot\normal=0\;\text{in }H^{-\frac{1}{2}}(\partial\Omega)\right\} .$$ This result still holds true when $\Omega$ is disconnected (the proof of [@dautraylions1980vol3spectral Prop. IX.1] relies on Green’s formula (\[eq:=00005BPRE=00005D\_Green-Formula\]) as well as the compactness of the embedding $H^{1}\left(\Omega\right)\subset L^{2}\left(\Omega\right)$, needed to apply Peetre’s lemma). The space $H_{\opDiv0,0}(\Omega)$ is studied in [@dautraylions1980vol3spectral Chap. IX] for $n=2$ or $3$. For instance, $$\mathbb{H}_{1}\coloneqq H_{\opDiv0,0}(\Omega)\cap\left\{ \vector f\in(L^{2}(\Omega))^{d}\;\vert\;\nabla\times\vector f=\vector 0\right\}$$ has a finite dimension under suitable assumptions on the set $\Omega$ [@dautraylions1980vol3spectral Prop. IX.2]. Semigroups of linear operators\[sub:=00005BMISC=00005D\_Asymptotic-stability-of-Semigroup\] ------------------------------------------------------------------------------------------- \[thm:=00005BSTAB=00005D\_Lumer-Phillips\]Let $\spaceState$ be a complex Hilbert space and $\opA:\,\spaceDomain\subset\spaceState\rightarrow\spaceState$ an unbounded operator. If $\Re(\opA\state,\state)_{H}\leq0$ for every $\state\in\spaceDomain$ and $\opId-\opA$ is surjective, then $\opA$ is the infinitesimal generator of a strongly continuous semigroup of contractions $\opT(t)\in\spaceBounded(\spaceState)$. The result follows from [@pazy1983stability Thms. 4.3 & 4.6] since Hilbert spaces are reflexive [@lax2002funana Thm. 8.9]. \[thm:=00005BSTAB=00005D\_Arendt-Batty\] Let $\spaceState$ be a complex Hilbert space and $\opA:\,\spaceDomain\subset\spaceState\rightarrow\spaceState$ be the infinitesimal generator of a strongly continuous semigroup $\opT(t)\in\spaceBounded(\spaceState)$ of contractions. If $\sigma_{p}(\opA)\cap i\spaceR=\varnothing$ and $\sigma(\opA)\cap i\spaceR$ is countable, then $\opT$ is asymptotically stable, i.e. $\opT(t)\state_{0}\rightarrow0$ in $\spaceState$ as $t\rightarrow\infty$ for any $\state_{0}\in\spaceState$. Application of the invariance principle\[sub:=00005BSTAB=00005D\_Invariance-Principle\] ======================================================================================= The purpose of this appendix is to justify why, in this paper, we rely on Corollary \[cor:=00005BSTAB=00005D\_Asymptotic-Stability\] rather than the invariance principle, commonly used with dynamical systems on Banach spaces. Theorem \[thm:=00005BSTAB=00005D\_Invariance-Principle\] below states the invariance principle for the case of interest herein, i.e. a linear Cauchy problem (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) for which the Lyapunov functional is $\frac{1}{2}\Vert\cdot\Vert_{\spaceState}^{2}$. (For further background, see [@luoGuoMorgul2012stability § 3.7] and [@cazenave1998evolution Chap. 9].) \[thm:=00005BSTAB=00005D\_Invariance-Principle\] Let $\opA$ be the infinitesimal generator of a strongly continuous semigroup of contractions $\opT(t)\in\spaceBounded(\spaceState)$ and $\state_{0}\in\spaceState$. If the orbit $\gamma(\state_{0})\coloneqq\bigcup_{t\geq0}\opT(t)\state_{0}$ lies in a compact set of $\spaceState$, then $\opT(t)\state_{0}\rightarrow M$ as $t\rightarrow\infty$, where $M$ is the largest $\opT$-invariant set in $$\left\{ \state\in\spaceDomain\;\left|\;\Re\left[(\opA\state,\state)_{\spaceState}\right]=0\right.\right\} .\label{eq:=00005BSTAB=00005D_IVP-Stationary-Set}$$ The function $\Lyapunov\coloneqq\frac{1}{2}\Vert\cdot\Vert_{\spaceState}^{2}$ is continuous on $\spaceState$ and satisfies $\Lyapunov(\opT(t)\state)\leq\Lyapunov(\state)$ for any $\state\in\spaceState$ so that it is a Lyapunov functional. The invariance principle [@hale1969invarianceprinciple Thm. 1] then shows that $\opT(t)\state_{0}$ is attracted to the largest invariant set of $$\left\{ \state\in\spaceState\;\left|\;\lim_{t\rightarrow0^{+}}t^{-1}(\Lyapunov(\opT(t)\state)-\Lyapunov(\state))=0\right.\right\} .$$ Let us now discuss the application of Theorem \[thm:=00005BSTAB=00005D\_Invariance-Principle\] to (\[eq:=00005BMOD=00005D\_Wave-Equation\],\[eq:=00005BMOD=00005D\_IBC\]), assuming we know a dissipative realization of the impedance operator $u\mapsto z\star u$ in a state space $\spaceDiff_{0}$. The first step is to establish that the largest invariant subset of (\[eq:=00005BSTAB=00005D\_IVP-Stationary-Set\]) reduces to $\{0\}$, i.e. that the only solution of (\[eq:=00005BSTAB=00005D\_Abstract-Cauchy-Problem\]) in (\[eq:=00005BSTAB=00005D\_IVP-Stationary-Set\]) is null, which is verified by the evolution operators defined in Sections \[sec:=00005BDELAY=00005D\_Delay-impedance\]–\[sec:=00005BDER=00005D\_Addition-of-Derivative\]. This requires to exclude solenoidal fields from $\state_{0}$, see Remark \[rem:|Delay=00005D\_Exclusion-Solenoidal-Fields\]. The second step is to prove the precompactness of the orbit $\gamma(\state_{0})$ for any $\state_{0}$ in $\spaceState$. The following criterion can be used, where for $s\in\rho(\opA)$ we denote the resolvent operator by $$R(s,\opA)\coloneqq(s\opId-\opA)^{-1}.\label{eq:=00005BSTAB=00005D_Resolvent-Operator}$$ \[thm:=00005BIVP=00005D\_Precompactness-Criterion\] Let $\opA$ be the infinitesimal generator of a strongly continuous semigroup of contractions on $\spaceState$. If $R(s,\opA)$ is compact for some $s>0$, then $\gamma(\state_{0})$ is precompact for any $\state_{0}\in\spaceState$. Using Theorem \[thm:=00005BIVP=00005D\_Precompactness-Criterion\] reduces to proving that the embedding $\spaceDomain\subset\spaceState$ is compact, which based on the examples covered in this paper boils down to proving that the embeddings $$H_{\opDiv}(\Omega)\times H^{1}(\Omega)\subset\nabla H^{1}(\Omega)\times L^{2}(\Omega)\quad\left(\text{a}\right),\;L^{2}(\partial\Omega;\spaceDiff_{1})\subset L^{2}(\partial\Omega;\spaceDiff_{0})\quad\left(\text{b}\right)\label{eq:=00005BIVP=00005D_Compact-Embedding}$$ are compact, where $\spaceDiff_{0}$ is the energy space of the extended variables and $\spaceDiff_{1}\subset\spaceDiff_{0}$. The compactness of the embedding (\[eq:=00005BIVP=00005D\_Compact-Embedding\]a) is obvious if $d=1$. If $d=3$, it can be proven using the following regularity result: if $\Omega$ is a bounded simply connected open set with Lipschitz boundary, [@costabel1990regularity Thm. 2] $$H_{\text{curl}}(\Omega)\cap\left\{ \uac\in H_{\opDiv}(\Omega)\;\left|\;\uac\cdot\normal\in L^{2}(\partial\Omega)\right.\right\} \subset H^{\frac{1}{2}}(\Omega)^{d}$$ and $\nabla H^{1}(\Omega)\subset H_{\text{curl}}(\Omega)$ [@giraultraviart1986femNavierStokes Thm. 2.9]. (Note the stringent requirement that $\Omega$ be simply connected.) The compactness of (\[eq:=00005BIVP=00005D\_Compact-Embedding\]b) depends upon both $d$ and the impedance kernel $z$. If $d=1$, then it holds true if $\spaceDiff_{1}\subset\spaceDiff_{0}$ is compact (which is satisfied by the delay impedance covered in Section \[sec:=00005BDELAY=00005D\_Delay-impedance\], where $\spaceDiff_{1}=H^{1}\left(-\tau,0\right)$ and $\spaceDiff_{0}=L^{2}\left(-\tau,0\right)$, but not by the diffusive impedances covered in Sections \[sec:=00005BDIFF=00005D\_Standard-diffusive-impedance\]–\[sec:=00005BEXTDIFF=00005D\_Extended-diffusive-impedance\] ) or if both $\spaceDiff_{1}$ and $\spaceDiff_{0}$ are finite-dimensional (which is verified for a rational impedance). If $d>1$, then it is not obvious. 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ArXiv

--- abstract: 'The proposal of the possibility of change of signature in quantum cosmology has led to the study of this phenomenon in classical general relativity theory, where there has been some controversy about what is and is not possible. We here present a new analysis of such a change of signature, based on previous studies of the initial value problem in general relativity. We emphasize that there are various continuity suppositions one can make at a classical change of signature, and consider more general assumptions than made up to now. We confirm that in general such a change can take place even when the second fundamental form of the surface of change does not vanish.' author: - | [**Mauro Carfora**]{}$^1$ and [**George Ellis**]{}$^{1,2}$\  \ [*$^1$SISSA, Via Beirut 2-4,*]{}\ [*34013 Trieste, Italy*]{}\  \ [*$^2$Department of Applied Mathematics,*]{}\ [*University of Cape Town, 7700 Rondebosch,*]{}\ [*Republic of South Africa*]{} title: The Geometry of classical change of signature --- Introduction ============ Following on recent developments in quantum cosmology \[1-3\], a subject of some interest is the possibility of a change of signature in a classical space-time \[4-12\]. We discuss here in depth the geometry associated with such a classical change of signature. The results obtained differ depending on what smoothness assumptions one makes. We look at the most general case, resulting from concentrating on the 3-dimensional surface where the change of signature occurs, rather than on either the Lorentzian (hyperbolic) or Riemannian (positive definite) enveloping space (the latter is often referred to as Euclidean; however we prefer Riemannian, as ‘Euclidean’ suggests that the space is flat) .\ In our approach we emphasize the initial value problem associated with signature change and the dynamical content of the theory, rather than regarding the problem as just a generalisation of the well-known Israel junction conditions \[13\]. There are more than junction conditions involved. In the case of the surface of a star, junction conditions are rather separated from the role of the initial value problem (because the surface is timelike). In the case of a change of signature, this must take place on a spacelike surface and so is essentially tied in to the nature of the initial value problem. Junction conditions play here a kinematical role, while the real dynamics of the change of signature are captured by the constraints associated with the field equations. This understanding underlies the approach we adopt.\ The first fundamental form must be continuous. The continuity of the the second fundamental form as seen from both sides, is only assumed up to the action of an infinitesimal diffeomorphism corresponding to a Lie derivative. This allows a kink in the geometry - not allowed in the more restrictive assumptions considered up to now. We insist that the constraints are valid for both enveloping metrics. Further junction conditions only arise if the matter is assumed to be smoothly behaved - which may not be required.\ These conditions thus generalise those considered by Ellis et al \[5,6,11\], which in turn are more general than those considered by Hayward et al \[7,8,10\] on the basis of their more restricted approach (placing more stringent restrictions on what is allowed). Our stand- point is that one can adopt any of these views - they are based on different philosophies of how one should approach junction conditions - or indeed one can question whether there should be any conditions other than a gluing condition, such as is adopted here.\ We avoid use of specific coordinate systems, as well as use of abstract notation such as is employed by Hayward \[7\]. Rather we follow the notation of Hawking and Ellis \[14\] and of Fisher and Marsden \[15\]. Approach Taken ============== We let ${\cal S}$ denote a compact oriented three-manifold, and let $$\Theta: {\cal S} \rightarrow (M^{(4)},{\bf g}) \equiv M$$ be an embedding of ${\cal S}$ in a Lorentzian manifold $(M^{(4)},{\bf g})$ such that the imbedded manifold $\Theta({\cal S})$ is space-like, that is the pull-back $$\Theta^*({\bf g}) \equiv {\bf h}$$ is a Riemannian metric on ${\cal S}$.\ Similarly we define $$\hat{\Theta}: {\cal S} \rightarrow (M^{(4)},\hat{{\bf g}}) \equiv \hat{M}$$ as an embedding of ${\cal S}$ in the [*same*]{} 4-dimensional manifold, $M^{(4)}$, but now endowed with a Riemannian metric $\hat{{\bf g}}$, [*viz.*]{}, $(M^{(4)},\hat{{\bf g}})$.\ Our strategy is to think of the metrics ${\bf g}$ and ${\bf \hat{g}}$ as living on the same portion of manifold, and in order to avoid misunderstandings, we wish to stress that $M$ and $\hat{M}$ are just a shorthand notation for the same underlying four-manifold $M^{(4)}$ with different metrics, with ${\bf g}$ Lorentzian, whereas ${\hat{\bf g}}$ is Riemannian. As we are not concerned with global problems we may restrict ourselves to a a tubular neighborhood of $\Theta({\cal S})$ (containing also $\hat{\Theta}({\cal S})$). For the moment ${\bf g}$ and ${\bf \hat{g}}$ are arbitrary. This coexistence of both Riemannian and Lorentzian metrics on the same region of the manifold will in our opinion avoid a lot of problems when thinking of the geometry involved.\ We are going to identify - modulo the action of the diffeomorphisms- the Lorentzian and Riemannian geometry along a common imbedded space-like hypersurface, determined by the constraints associated with the Einstein equations. Geometry ======== In order to define the variables of interest, we need to characterise the foliations employed and the related lapse and shift in both the Riemannian and Lorentzian cases.\ Let $E^\infty({\cal S},\hat{M})$ and $E^\infty({\cal S},M)$ denote the sets of all spacelike imbeddings of ${\cal S}$ in $\hat{M}$ and $M$ respectively.\ Suppose we have a curve in each of these imbedding spaces: namely a one-parameter $(\lambda)$ family of spacelike imbeddings of ${\cal S}$ into $M$, and a similar one-parameter $(\lambda)$ family of imbeddings of ${\cal S}$ into $\hat{M}$. Explicitly, ${\Theta}_{\lambda}\colon{\cal S}\times{I}\to{M}$ and $\hat{\Theta}_{\lambda}\colon{\cal S}\times{I}\to\hat{M}$, where $I\equiv{(-\epsilon,\epsilon)}$ for a suitably small $\epsilon>0$. This family of imbeddings defines a corresponding one-parameter family of vector fields $X^{(4)}_\lambda\colon{\cal S}\to{TM^{(4)}}$ and $\hat{X}^{(4)}_\lambda\colon{\cal S}\to{T\hat{M}^{(4)}} $ by $${d\Theta_\lambda \over d\lambda}(p) = X^{(4)}_\lambda(\Theta_\lambda (p))$$ and $${d\hat{\Theta}_\lambda \over d\lambda}(p) = \hat{X}^{(4)}_\lambda(\hat{\Theta}_\lambda (p))$$ as $p$ varies over ${\cal S}$.\ In order to simplify the notation a bit, we shall denote them simply by $X_\lambda$ and $\hat{X}_\lambda$. Roughly speaking, either in $M$ or in $\hat{M}$ these vectors connect the point $\Theta_\lambda(p)$ with $\Theta_{\lambda+d\lambda}(p)$ (and similarly for $\hat{\Theta}$); namely the images of a given point $p$ in ${\cal S}$ under two infinitesimally near imbeddings.\ If $n$ and $\hat{n}$ respectively denote the forward-pointing unit normals to $\Theta({\cal S})$ and $\hat{\Theta}({\cal S})$ (so $n^a n^b g_{ab} = -1$; $\hat{n^a} \hat{n}^b \hat{g}_{ab} = +1$), we can as usual decompose the vector fields $X$ and $\hat{X}$ into their normal and tangential components: $$X_\lambda = N_\lambda \hat{n} + \beta_\lambda$$ $$\hat{X}_\lambda = \hat{N}_\lambda \hat{n} + \hat{\beta}_\lambda$$ which define the corresponding family of lapse functions on ${\cal S}$, [*i.e.*]{}, $N_{\lambda}\colon{\cal S}\to{\bf R}$ and a corresponding family of shift vector fields again on ${\cal S}$, namely ${\beta}_{\lambda}\colon{\cal S}\to{T{\cal S}}$. We wish to stress the fact (slightly obscured by our simplified nota- tion) that the family of lapse functions $N_\lambda$ are defined on the abstract manifold ${\cal S}$, and similarly the family of shift vector fields $\beta_\lambda$ are defined over ${\cal S}$; similarly for the lapse $\hat{N}_\lambda$ and shift $\hat{\beta}_\lambda$. Here “the lapse and the shift are seen in their proper geometric roles - describ- ing the hypersurface deformations in the enveloping geometries - rather than as pieces of the metric” (Isenberg and Nester \[16\]).\ The metric interpretation comes about for instance if we use the maps $$F: I \times {\cal S} \rightarrow M$$ defined by $$(\lambda,p) \mapsto \Theta_\lambda(p)$$ as a diffeomorphism of $I \times {\cal S}$ onto a tubular neighbourhood of $\Theta_0({\cal S})$. We can then pull back the metric $g$ onto $I \times {\cal S}$ and get the usual expression $$(F^* g)_{\alpha\beta} dx^\alpha dx^\beta = - (N^2_\lambda - \beta_i \beta^i)d\lambda^2 + 2 \beta_i dx^i d\lambda + h_{ij} dx^i dx^j$$ where indices $\alpha$ and $\beta$ run from 1 to 4, $i$ and $j$ run from 1 to 3, $\{x^i\}$ are local coordinates on ${\cal S}$, and $h_{ij}$ is the $\lambda-$dependent one-parameter family of metrics on ${\cal S}$. A similar analysis holds for $\hat{F}$, leading to $$(\hat{F}^* \hat{g})_{\alpha\beta} dy^\alpha dy^\beta = + (\hat{N}^2_\lambda + \hat{\beta}_i \hat{\beta}^i)d\lambda^2 + 2 \hat{\beta}_i dy^i d\lambda + \hat{h}_{ij} dy^i dy^j$$ with the obvious meaning of the symbols.\ There are a number of general comments that should be made at this stage. In particular, we wish to caution the reader to not confuse the abstract manifold ${\cal S}\times{I}$ with its images ${\Theta}_{\lambda}({\cal S})$ and $\hat{\Theta}_{\lambda}({\cal S})$ in $M=(M^{(4)},{\bf g})$ and $\hat{M}=(M^{(4)},\hat{\bf g})$, respectively. Typically, when dealing with the initial value problem, one is accustomed to do so for obvious reasons, and this identification is usually harmless. However making clear the distinction is more than a technical convenience here. By identifying ${\cal S}\times{I}$ with ${\Theta}_{\lambda}({\cal S})$ and $\hat{\Theta}_{\lambda}({\cal S})$ one is lead to an incorrect interpretation of the vector field $\partial/\partial{\lambda}$, which is defined on ${\cal S}\times{I}$, in terms of which the initial value formalism is phrased. Observe that the parameter $\lambda$ is the natural label for all the fields ${\bf h}_{\lambda}$, ${\hat {\bf h}}_{\lambda}$, $N_{\lambda}$, ${\hat N}_{\lambda}$, ${\beta}_{\lambda}$, ${\hat{\beta}}_{\lambda}$, and the extrinsic curvatures (defined below), if they are referred either to the Lorent- zian or to the Riemannian case. This is a rather obvious statement when things are correctly seen, as they should be, on ${\cal S}\times{I}$. It is not an obvious statement at all if we identify ${\cal S}\times{I}$ with its images under ${\Theta}_{\lambda}$ and $\hat{\Theta}_{\lambda}$. In this case, since the foliations ${\Theta}_{\lambda}({\cal S)}$ and $\hat{\Theta}_{\lambda}{\cal S})$ are different, and with different deformation vectors $X_{\lambda}$ and ${\hat {X}}_{\lambda}$, one is incorrectly led to believe that these deformation vectors must be tangent to different deformation coordinates, namely $X_{\lambda} =\partial/\partial{\lambda}$ and ${\hat {X}}_{\lambda}=\partial/\partial{\omega}$, for some other defor- mation parameter ${\omega}$. As stressed before, this is usually harm- less in standard situations where one has just one enveloping space- time, but it is fatal here where the enveloping geometries are two and quite distinct. The source of the error, in proceeding as above, lies in the fact that one is identifying vectors living on different spaces, since the family of vector fields $\partial/\partial{\lambda}$ is defined on ${\cal S}$, while the deformation vectors $X_{\lambda}$ and $\hat{X}_{\lambda}$ are defined on $M$ and $\hat{M}$, respectively. If these two latter are different, their intuitive identification with vectors tangent to a deformation coordinate, ([*i.e.*]{}, with $\partial/\partial{\lambda}$), is problematic and very confusing. One must clearly separate the role of the vector tangent to the deformation coordinate, which is $\partial/\partial{\lambda}$, and which is defined on ${\cal S}$, from the vectors $X_{\lambda}$ and $\hat{X}_{\lambda}$ which are respectively associated to the imbeddings ${\Theta}_{\lambda}$ and $\hat{\Theta}_{\lambda}$, (these vector fields can be thought of as the vector fields covering the two distinct family of imbeddings ${\Theta}_{\lambda}$ and $\hat{\Theta}_{\lambda}$ of ${\cal S}$).\ It is our strategy to address the geometry of signature change exclu- sively in terms of quantities defined on ${\cal S}$ and this should be clearly kept in mind when deciding which quantities should be continuous through a surface of signature change. For instance it would be very unnatural from our viewpoint to assume the continuity of the unit normals, for these quantities live in the embedding spacetimes $M$ and $\hat{M}$, and this is something that an observer living in ${\cal S}$ does not know [*a priori*]{}. It is much more natural for him to assume the continuity of the vector $\partial/\partial{\lambda}$ and of the lapse function and of the shift vector fields, since all such quantities are well defined on ${\cal S}$ and they provide him the complete kinematical framework for describing – from his standpoint – the deformations of ${\cal S}$ which may be compatible with a Riemannian geometry on one side and with a Lorentzian geometry on the other.\ With these general remarks out of the way, we recall that in order to describe the imbeddings $\Theta$ and $\hat{\Theta}$, besides introducing the 3-metrics $h$ and $\hat{h}$ we must also introduce, on ${\cal S}$, two symmetric tensor fields $K$ and $\hat{K}$ to be interpreted as the second fundamental forms of $\Theta_\lambda({\cal S})$ and $\hat{\Theta}_\lambda({\cal S})$ respectively. In our notation, they are defined, at the generic point $x\in{\cal S}$, and for any pair of vectors ${\bf u}$ and ${\bf v}$ in $T_x{\cal S}$ by $$K_x({\bf u},{\bf v}) =<T_x{\Theta}\circ{\bf u}| {\nabla}^{(4)}(T_x{\Theta}\circ{\bf v}){\bf n}>_{g}({\Theta}(x))$$ where ${\nabla}^{(4)}$ denotes the covariant derivative operator in $M$, the brackets $<\cdot|\cdot>_{g}({\Theta}(x))$ stand for the inner product in the Lorentzian metric ${\bf g}$ evaluated at the point ${\Theta}(x)\in{M}$, and $T_x{\Theta}$ stands for the tangential mapping, at $x\in {\cal S}$, associated to the embedding ${\Theta}$. Similarly, and with an obvious meaning of the symbols, $$\hat{K}_x({\bf u},{\bf v}) = <T_x{\hat{\Theta}}\circ{\bf u}| {\hat{\nabla}}^{(4)} (T_x{\hat{\Theta}}\circ{\bf v}){\bf n}>_{\hat{g}} ({\hat{\Theta}}(x))$$ For each given value of $\lambda$ the fields $(h,K)$ and $(\hat{h}, \hat{K})$ cannot be arbitrarily prescribed. From the Gauss-Codazzi relation, one gets that such fields must satisfy four compatibility conditions, namely in the Riemannian case $$R(\hat{h}) - (\hat{K}^{dc} \hat{h}_{dc})^2 + \hat{K}^{ab} \hat{K}^{cd} \hat{h}_{ac} \hat{h}_{bd} = - 2 \Theta^*(G_{\mu\nu}\hat{n}^{\mu}\hat{n}^{\nu})$$ where $\hat{G}_{\mu\nu}$ is the Einstein tensor of $\hat{g}$ and $$\hat{D}_a \hat{K}^{ac} \hat{h}_{cb} - \hat{D}_b \hat{K}^{cd} \hat{h}_{cd} = \hat{\Theta}^* [R_{\mu\nu}(\hat{g}) \hat{n}^{\nu}{\perp}^{\mu}_b]$$ where $\hat{D}$ is the covariant derivative in $({\cal S},\hat{h})$ and $R_{\mu\nu}$ is the Ricci tensor of the metric $\hat{g}$.\ In the Lorentzian case, we obtain $$R(h) + ( K^{dc} h_{dc})^2 - K^{ab} K^{cd} h_{ac} h_{bd} = 2 \Theta^*(G_{\mu\nu}n^{\mu}n^{\nu})$$ where $ G_{\mu\nu}$ is the Einstein tensor of $g$ and $$D_a K^{ac} h_{cb} - D_b K^{cd} h_{cd} = \Theta^* [R_{\mu\nu}(g)n^{\nu}{\perp}^{\mu}_b] \, .$$ Change of Signature =================== Now we are ready to discuss the possibility of change of signature through a regular hypersurface. Till now the embedded hypersurfaces $\Theta_\lambda({\cal S})$ and $\hat{\Theta}_\lambda({\cal S})$ were kept distinct. The basic condition we need in order to be able to speak of a signature change is to choose one of the $\Theta_\lambda ({\cal S})$ to ‘coincide’ with one of the manifolds of the family $\hat{\Theta}_\lambda({\cal S})$.\ Asking directly, as often is implicitly done, that for a given range of $\lambda$, say $-\epsilon<\lambda<\epsilon$, ${\Theta}_\lambda({\cal S})\equiv \hat{\Theta}_\lambda({\cal S})$, is too restrictive. And this is partially the reason for having unnecessary stringent constraints on the second fundamental form on the hypersurface of signature change. It is more natural to assume, at leas t a priori, that the identification between ${\Theta}_\lambda({\cal S})$ and $\hat{\Theta}_\lambda({\cal S})$, $-\epsilon<\lambda<\epsilon$, occurs modulo the action of diffeomorphisms of the manifold ${\cal S}$. More particularly, we consider a $\lambda$ dependent family of diffeomorphisms ${\phi}_{\lambda}\colon{\cal S}\to{\cal S}$, smoothly varying as $-\epsilon<\lambda<\epsilon$, and such that for a given value of $\lambda$, say $\lambda=0$, $$\hat{\Theta}_0(p)={\Theta}_0(p), \forall p\in{\cal S}$$ namely, it is only required that ${\phi}_{\lambda}=id_{\cal S}$ for $\lambda=0$. The strategy will be to use these diffeomorphisms to glue the bottom (Riemannian) region with the top (Lorentzian) region. This will mean - remembering that there are two metrics on ${\cal S}\times{I}$ - we designate the metric $\hat{g}$ as the physical metric in the lower region ${\cal S}\times(0,-\beta)$ and the metric $g$ as the physical metric in the upper region ${\cal S}\times(\beta,0)$. On the zero section, ${\cal S}\times\{0\}$, of ${\cal S}\times{I}$, the constraints associated to the Lorentzian and to the Riemannian imbedding must be simultaneously satisfied.\ It is clear that as far as the three-metrics $h$ and $\hat{h}$ are concerned, the action of the one-parameter group of diffeomorphisms ${\phi}_{\lambda}$ is simply that of having $$\hat{h} = {\phi}_{\lambda}^* h$$ for $-\epsilon<\lambda<\epsilon$, and in particular, $\hat{h} = h$ for $\lambda=0$.\ The situation is less dull as far as concerns the tensor fields $K$ and $\hat{K}$ yielding the second fundamental forms. In order to see how the action of ${\phi}_{\lambda}$ relates $K$ and $\hat{K}$ on ${\cal S}$ let us write the explicit expressions of $K$ and $\hat{K}$ in terms of the three-metrics $h$, $\hat{h}$, and of the vector field (defined over ${\cal S} \times I$), $\frac{\partial}{\partial\lambda}$. We get $$K_{ij}=N_{\lambda}^{-1}[\frac{\partial}{\partial{\lambda}}h_{ij}- L_{{\beta}_{\lambda}}h_{ij}]$$ and similarly $$\hat{K}_{ij}=\hat{N}_{\lambda}^{-1} [\frac{\partial}{\partial{\lambda}}\hat{h}_{ij}- L_{\hat{\beta}_{\lambda}}\hat{h}_{ij}]$$ where $L_{\cdot}$ denotes Lie differentiation along the vector field indicated.\ For $-\epsilon<\lambda<\epsilon$, we have $\hat{h}_{ij}=({\phi}^*_{\lambda}h)_{ij}$ thus $$\hat{K}_{ij}=\hat{N}_{\lambda}^{-1} [\frac{\partial}{\partial{\lambda}}({\phi}^*_{\lambda}h)_{ij}- L_{\hat{\beta}_{\lambda}}({\phi}^*_{\lambda}h)_{ij}]$$ A direct computation shows (see [*e.g.*]{}, DeTurck \[17\]), $$\frac{\partial}{\partial{\lambda}}[({\phi}^*_{\lambda}h)_{ij}(p)]= {\phi}^*_{\lambda} [\frac{\partial}{\partial{\lambda}}h_{ij}({\phi}_{\lambda}(p))]+ {\phi}^*_{\lambda} [L_{{v}_{\lambda}}h_{ij}({\phi}_{\lambda}(p))]$$ where the vector field $v_{\lambda}$ is the generator of the one-parameter group of diffeomorphisms ${\phi}_{\lambda}$ according to $$\frac{\partial}{\partial{\lambda}}{\phi}_{\lambda}(p)= v_{\lambda}(\lambda,{\phi}_{\lambda}(p))$$ with the initial condition ${\phi}_{\lambda}|_{\lambda=0}=id_{\cal S}$.\ Thus $$\hat{K}_{ij}=\hat{N}_{\lambda}^{-1} {\phi}^*_{\lambda}[ \frac{\partial}{\partial{\lambda}}h_{ij}({\phi}_{\lambda}(p))+ L_{{v}_{\lambda}}h_{ij}({\phi}_{\lambda}(p))- L_{\hat{\beta}_{\lambda}}h_{ij}({\phi}_{\lambda}(p))]$$ In particular, for $\lambda=0$, we get $$\hat{K}_{ij}=\hat{N}_{\lambda}^{-1}[ \frac{\partial}{\partial{\lambda}}h_{ij}+ L_{{v}_{\lambda}}h_{ij}- L_{\hat{\beta}_{\lambda}}h_{ij}]$$ which shows that if, as argued in the previous paragraph, we assume continuity of the lapse and the shift for $\lambda=0$: $$\hat{N}_{\lambda}=N_{\lambda}, ~~~\hat{\beta}_{\lambda}={\beta}_{\lambda},$$ and assuming also continuity of $\frac{\partial}{\partial{\lambda}}h_{ij}$, then $$\hat{K}_{ij}=K_{ij}+ N_{\lambda}^{-1}L_{v_{\lambda}}h_{ij}$$ (a similar relation holds for any $-\epsilon<\lambda<\epsilon$ provided that we act by ${\phi}^*_{\lambda}$). Thus, on the hypersurface ${\Theta}_0({\cal S})=\hat{\Theta}_0({\cal S})$ where we seek a change of signature, we may assume that the corresponding second fundamental forms coincide only up the Lie derivative term $N_{\lambda}^{-1}L_{v_{\lambda}}h^{ij}$.\ We wish to stress that by forcing ${\phi}_{\lambda}$ to be the identity for all $\lambda$, one may obviously achieve equality between the second fundamental forms on the transition hypersurface. But fixing [*a priori*]{} the three degrees of freedom (per space point) associated with ${\phi}_{\lambda}$ will be a very bad investment when dealing with the constraints.\ One may also argue that equation (26) is equally compatible with having continuity of the second fundamental form, provided that one allows for a discontinuous shift vector field, namely ${\beta}_{\lambda}=\hat{\beta}_{\lambda}-v_{\lambda}$. Further impositio n of the continuity of the shift would then yield $v_{\lambda}=0$, and the former case of freezing the diffeomorphism group is then recovered. All this is actually related to what one considers standard junction conditions in the setting of signature changes. In ordinary situations, such conditions require the continuity of the four-metric and of the second fundamental form. But whether or not a such conditions can be extended at face value to the case of surfaces of signature change is a very delicate issue. Continuity of the four-metric leads to vanishing of the lapse function, which is quite disturbing. Moreover, the tensor fields $K$ and $\hat{K}$, when interpreted as second fundamental forms, are to be thought of as defined in terms of a unit normal (to the surface of signature change) whose norm changes sign at the junction. Thus it is not obvious at all that the continuity of the second fundamental form is a [*natural*]{} requirement in the case of surfaces of signature change.\ In this respect, it is often argued that the correct answer must come from the field equations. More precisely, one should impose the validity of the field equations everywhere, in particular on the surface of signature change. This point of view is apparently reasonable and interesting, but implies very severe constraints on the resulting solutions. We offer here an alternative point of view, namely we do not force the validity of the full four dimensional field equations on the surface of signature change, but rather we concentrate on the validity of that part of the field equations which is really [*intrinsic*]{} to the surface of signature change, namely we impose the consistency among the four constraints associated with the field equations. In our view, this is a minimal necessary requirement, the basic one. Further restrictions can come only if one has some input from the matter fields present, in particular on how they behave on the surface of signature change; and that is a matter for debate.\ We wish also to stress the following point. From the point of view of analysis and physics, partial differential equations of mixed type, where the type (elliptic, hyperbolic, or parabolic) of the equation is a function of position, are rather familiar. The added difficulty here, in considering surfaces of signature change, lies exactly in the diffeomorphism invariance of the theory. By considering the full field equations at once everywhere, one is behaving as if there exists a general theory of boundary value problems independent of the type of the equation, which is very bold, to say the least. Even in the simplest cases in hydrodynamics, such a theory is very delicate, and general results exist only for equations of special types. The situation becomes hopeless in a general relativity setting. Indeed, Einstein’s equations in the Riemannian regime are a strongly overdetermined elliptic system (owing to diffeomorphism invariance), and the problem of finding a metric with a preassigned Einstein or Ricci tensor is often obstructed even at an infinitesimal level, ([ *i.e.*]{}, there are even obstructions to finding a metric, around a given point, with prescribed Ricci tensor, see \[17\]). The situation changes drastically in the Lorentzian regime. Thus it is fair to say that the study of mixed type Einstein equations is a completely open problem. It follows that forcing the validity of the field equations everywhere, in the case of a surface of signature change, is a formal procedure not really justified from an existing theory, and to which one should give the same [*interlocutory*]{} status as other proposals. In our approach, restricting attention to the constraints forced on the surface of signature change, one is considering what kind of initial data is compatible with a signature change in terms of partial differential equations which do [*not*]{} change type on the surface of signature change. Furthermore, these contain the essential dynamical equations of the theory (for example in the Robertson-Walker case, they include the Friedmann equation), which lead to the Wheeler-de Witt equation which underlies quan- tum cosmology.\ As a final remark, notice that at first reading one may think there is a surface layer present in our formalism because of the allowed discontinuity of the second fundamental form. However, there is no variance with the essence of the junction conditions of Israel \[13\], since we are assuming the continuity of the proper dynamical variables, which are $\frac{\partial} {\partial {\lambda}} h_{ij}$. These conditions are usually written down in terms of adapted coordinates such that the second fundamental form is the time derivative, and so do not allow for the action of a diffeomorphism which is responsible for the Lie derivative terms. Actually, in the geometrical setting discussed here, as stressed above, they should not be taken at face value, since the general remarks discussed in the previous paragraph apply also here. In our setting, the proper variables to match are the lapse $N_{\lambda}$, the shift ${\beta}_{\lambda}$, the three-metric ${\bf h}_{\lambda}$ and its derivative $\frac{\partial} {\partial {\lambda}}{\bf h}$ – as we have done, and no surface layer is present as is clearly shown by imposing the constraints. Constraints ----------- The constraints, both in their Lorentzian and Riemannian version, must hold for $\lambda=0$.\ Let us start from the momentum (or divergence) constraint. We assume that on $M$ both $\hat{g}$ and $g$ satisfy the corresponding form of Einstein field equations, the Riemannian form for the former, the standard Lorentzian form for the latter. Thus in the Riemannian case $$\hat{R}_{\alpha\beta}=\hat{T}_{\alpha\beta}- \frac{1}{2}\hat{g}_{\alpha\beta} \hat{g}^{\gamma\delta}\hat{T}_{\gamma\delta}$$ where $\hat{T}_{\alpha\beta}$ are the components of the Riemannian energy-momentum tensor. Relative to the slicing $\hat{\Theta}_{\lambda}({\cal S})$ we shall write $$\hat{T}_{\alpha\beta}=\hat{\mu}\hat{n}_{\alpha}\hat{n}_{\beta}+ \hat{j}_{\alpha}\hat{n}_{\beta}+ \hat{j}_{\beta}\hat{n}_{\alpha}+\hat{s}_{\alpha\beta}$$ where $\hat{\mu}$, $\hat{j}_{\alpha}$, and $\hat{s}_{\alpha\beta}$ respectively are the normal-normal, normal-tangential, and tangential-tangential projections of $\hat{T}_{\alpha\beta}$ with respect to $\hat{\Theta}_{\lambda}({\cal S})$. In the Lorentzian case, we shall similarly write $${R}_{\alpha\beta}={T}_{\alpha\beta}- \frac{1}{2}{g}_{\alpha\beta} {g}^{\gamma\delta}{T}_{\gamma\delta}$$ where ${T}_{\alpha\beta}$ are the components of the energy-momentum tensor which, relative to the slicing ${\Theta}_{\lambda}({\cal S})$, can be decomposed according to $${T}_{\alpha\beta}={\mu}{n}_{\alpha}{n}_{\beta}+ {j}_{\alpha}{n}_{\beta}+ {j}_{\beta}{n}_{\alpha}+{s}_{\alpha\beta}$$ where ${\mu}$, ${j}_{\alpha}$, and ${s}_{\alpha\beta}$ respectively are the relative density of mass-energy, the relative density of momentum, and the relative spatial stress tensor with respect to ${\Theta}_{\lambda}({\cal S})$.\ In general, there is no a priori need to assume that for $\lambda=0$ the [*matter*]{} variables are continuous. From a phenomenological point of view, there is no obvious evidence that one should assume continuity of the stress tensor components at the change of signature, although one might make that assumption if given no further information. On the other hand, if one has a more fundamental description of the stress tensor, for example as arising from a scalar field, one can work out the continuity properties of the stress tensor components from that description. This was done in \[5,6\] for the case of a classical scalar field. Then the obvious continuity conditions are that the fundamental variables associated with the more fundamental description are continuous, and satisfy whatever requirements there may be to give a good set of initial data for the matter field equations on either side of the signature change surface.\ In general this will result in discontinuous stress tensor components. This is not unreasonable in view of the fact that the usual conservation laws for energy and momentum break down at a change of signature surface \[12\]. The fundamental underlying point is that it is difficult to understand physics in the positive definite region, indeed classical physics in the usual sense will not exist there (although quantum physics will be fine!). Thus one must be open-minded as to what conditions should be imposed on ‘matter’ in the positive definite regime, in a classical discussion of signature change.\ Without making specific assumptions, the momentum constraint forced on ${\cal S}\times\{0\}$ by the Riemannian side is $$\hat{D}^a\hat{K}_{ab}-\hat{D}_b\hat{k}=\hat{j}^b$$ where $\hat{k}\equiv \hat{h}^{cd}\hat{K}_{cd}$ is the rate of volume expansion (the trace of the second fundamental form). Since, for $\lambda=0$, $\hat{D}^a=D^a$ and $\hat{K}_{ab}=K_{ab} +N^{-1}L_vh_{ab}$ we get $$D^aK_{ab}+D^a(N^{-1}L_vh_{ab})-D_bk-D_b(h^{cd}N^{-1}L_vh_{cd})=\hat{j}_b$$ (where as above $k\equiv h^{cd}K_{cd}$). But the momentum constraint forced on ${\cal S}\times\{0\}$ by the Lorentzian side implies that $${D}^a{K}_{ab}-{D}_b{k}=-{j}^b$$ which, when introduced in the previous expression, yields $$D^a(N^{-1}L_vh_{ab})- D_b(h^{cd}N^{-1}L_vh_{cd})=j_b+\hat{j}_b$$ Given $h_{ab}$, the lapse function $N$, and the momentum densities $j_b$, $\hat{j}_b$ the above is a system of partial differential equations determining the vector field $v$ which generates the gluing one-parameter group of diffeomorphisms ${\phi}_{\lambda}$ in the neighbourhood of $\lambda=0$. Notice however that this system is elliptic ([*i.e.*]{}, (36) can be actually inverted) only if the vector field $v$ is divergence-free, $D^av_a=0$. This further requirement implies that $k$, the trace of the second fundamental form, is continuous through the surface of signature change $S$, namely $$k=\hat{k}$$ This result is quite satisfactory since in an initial value approach, the rate of volume expansion is to be considered as a kinematical variable selecting the family of hypersurfaces along which we are following the dynamics of the gravitational field. 0.5 cm Next we can impose that both the Riemannian and the Lorentzian version of the Hamiltonian constraint hold for ${\cal S}\times\{0\}$. This yields $$2{\mu}+2\hat{\mu}+\frac{2}{3}k^2+\\ h^{ac}h^{bd}(\tilde{K}_{ab}+N^{-1}L_vh_{ab})(\tilde{K}_{cd}+N^{-1}L_vh_{ cd})+ \tilde{K}^{ab}\tilde{K}_{ab}=0$$ where $\tilde{K}_{ab}$ denotes the trace-free part of $K_{ab}$. If we assume that $\hat{\mu}\geq 0$, then the above condition, being the sum of algebraically independent non-negative terms, is only compatible with the vanishing of each summand. Thus, in this cas e from $\hat{\mu}\geq 0$ we actually get $\hat{\mu}=0$, $\mu=0$, $k=0$, $\tilde{K}_{ab}=0$, and $N^{-1}L_vh_{ab}=0$. Thus, if we require continuity of the matter variables through a surface of signature change, we found, as expected, that the second fundamental form must vanish correspondingly. Notice that this result follows without requiring the a priori continuity of the 4-metric or the continuity of the second fundamental form. Actually, it is precisely the continuity of the matter variables which forces such a result. It is not in the geometry, and as argued in the previous paragraph, there is no a priori need to assume that for $\lambda=0$ the matter variables are continuous, or satisfy energy conditions reminiscent of the Lorentzian regime.\ In general, without imposing any continuity or sign restriction on $\hat{\mu}$, equation (37) must be considered as a constraint on the Lorentzian rate of volume expansion $k$. In other words, the above compatibility condition between the Hamiltonian constraints sets an origin for the [*extrinsic time*]{} $k$ which parametrizes the time evolution in the Lorentzian region. Geometrically speaking, this condi- tion is simply selecting the hypersurface where the signature change can occur \[5,6\].\ One could use an approach even more closely tuned to the spirit of the initial value problem by using as dynamical variables the conformal part of the 3-metric, and the scaled divergence-free trace-free part of the second fundamental form. However this would complicate the equations without throwing much light on the basic issues we are addressing. We have therefore avoided these complications here, although this more detailed analysis shows signs of raising interesting questions. Relation to other approaches ============================ It is essential to our approach that the 3-metric is continuous through the change of signature. Others have emphasized \[7,8,10\] their belief in the importance of using coordinate systems where [*all*]{} the covariant components of the metric are continuous at a change of signature surface. We have not adopted this view, [*inter alia*]{} because then some of the contravariant metric components will diverge at the surface of change, leading [*inter alia*]{} to the divergence of various Christoffel terms; so the appearance of continuity is somewhat misleading.\ What we do believe is important is that the kinematics should be well-behaved there; this means we demand a well behaved shift and lapse, which determine the 4-dimensional metric structure. In particular the lapse should not go to zero because if it does then one halts the evolution in the coordinate system thereby defined. This means in turn that while the 3-metric components and their first ‘time’ derivative can always be chosen continuous up to a diffeomorphism, if the lapse is regular then the 4-dimensional metric tensor components associated will have a discontinuous component (the time-time component, which is not dynamical).\ In our geometric approach, there is no need to assume [*a priori*]{} that the 4-dimensional metric is continuous, because we have shown that one can match the Lorentzian and Riemannian spacetimes without making such an assumption, by having a perfectly well behaved kinematical description (the lapse and shift are well-behaved in our approach). The kinematics through a signature change surface should as far as possible be free from particular coordinate choices, and one should be free to choose the kinematical data (the lapse and shift) as desired, not forced to make them go to zero.\ The approach of \[7\] is based on a different view: emphasizing more the role of the full space-time metric than the view used here. It also assumes additional differentiability for the solutions, and is therefore more restrictive than the view adopted here; it is not surprising that the results obtained are more restrictive than if one does not impose these extra conditions. However that view also implies the lapse function goes to zero as one approaches the change surface. This ‘collapse of the lapse’ may be expected to cause problems for the dynamics \[18\].\ It will be clear from the above that the generic situation does not require a vanishing of the second fundamental form at the surface of change, which is required for example in both the distributional \[7\] and the Hartle-Hawking approach \[19\] (which uses a complex time variable). Our hope is that the present geometrical analysis of the classical case will be of help in understanding the full generality of what may be possible in the quantum case, through first clarifying the full generality of the analogous classical situation.\ \ We thank the MURST (Italy) for support. We would like to thank Charles Hellaby for many useful criticisms and comments that greatly improved the presentation of the paper.\ \ [**References**]{}\ \[1\] S W Hawking, in [*Astrophysical Cosmology*]{}, Ed. H A Brück, G V Coyne and M S Longair (Pontifica Academia Scientarium, Vatican City, 1982), pp. 563-574.\ \[2\] J B Hartle and S W Hawking, [*Phys Rev*]{} [**D28**]{}, 2960 (1983).\ \[3\] S W Hawking, [*Nucl Phys*]{} [**B239**]{}, 257 (1984).\ \[4\] T Dray, C Manogue and R Tucker, [*Gen Rel Grav*]{} [**23**]{}, 967 (1991).\ \[5\] G F R Ellis, A Sumeruk, D Coule and C Hellaby, [*Class Qu Grav*]{} [**9**]{}, 1535 (1992).\ \[6\] G F R Ellis, [*Gen Rel Grav*]{} [**24**]{}, 1047 (1992).\ \[7\] S A Hayward, [*Class Qu Grav*]{} [**9**]{}, 1851 (1992).\ \[8\] S A Hayward, [*Class Qu Grav*]{} [**10**]{}, L7 (1993).\ \[9\] T Dray, C Manogue and R Tucker, [*Phys Rev*]{} [**D48**]{}, 2587 (1993).\ \[10\] M Kossowski and M Kriele, [*Class Qu Grav*]{} [**10**]{}, 2363 (1993).\ \[11\] G F R Ellis and K Piotrkowska. To appear, [*Proc Les Journées Relativistes*]{}, Brussels, 1993.\ \[12\] C Hellaby and T Dray, ‘Failure of Standard Conservation Laws at a Classical Change of Signature’. 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ArXiv

--- abstract: 'We adapt a method of Voisin to powers of abelian varieties in order to study orbits for rational equivalence of zero-cycles on very general abelian varieties. We deduce that a very general abelian variety of dimension at least $2k-2$ has gonality at least $k+1$. This settles a conjecture of Voisin. We also discuss how upper bounds for the dimension of orbits for rational equivalence can be used to provide new lower bounds on other measures of irrationality. In particular, we obtain a strengthening of the Sommese bound on the degree of irrationality of abelian varieties. In the appendix we present some new identities in the Chow group of zero-cycles of abelian varieties.' address: 'Department of Mathematics, University of Chicago, IL, 60637' author: - Olivier Martin title: On a Conjecture of Voisin on the Gonality of Very General Abelian Varieties --- Introduction ============ In his seminal 1969 paper [@M] Mumford shows that the Chow group of zero-cycles of a smooth projective surface over $\mathbb{C}$ with $p_g>0$ is very large. Building on the work of Mumford, in [@R1] and [@R2] Roĭtman studies the map $$\text{Sym}^k (X)\to CH_0(X)$$ for $X$ a smooth complex projective variety. He shows that fibers of this map, which we call orbits of degree $k$ for rational equivalence[^1], are countable unions of Zariski closed subsets. Moreover, he defines birational invariants $d(X)$ and $j(X)\in \mathbb{Z}_{\geq 0}$ such that for $k\gg 0$ the minimal dimension of orbits of degree $k$ for rational equivalence is $k(\dim X-d(X))-j(X)$. Roĭtman’s generalization of Mumford’s theorem is the following statement: $$\text{If }H^0(X,\Omega^q)\neq 0 \text{ then }d(X)\geq q.$$ In particular, if $X$ has a global holomorphic top form then a very general $x_1+\ldots+x_k\in \text{Sym}^kX$ is contained in a zero-dimensional orbit.\ Abelian varieties are among the simplest examples of varieties admitting a global holomorphic top form. In this article we will focus our attention on this example and take a point of view different from the one mentioned above. Instead of fixing an abelian variety $A$ and considering the minimal dimension of orbits of degree $k$ for rational equivalence, we will be interested in the maximal dimension of such orbits for a very general abelian variety $A$ of dimension $g$ with a given polarization $\theta$.\ This perspective has already been studied by Pirola, Alzati-Pirola, and Voisin (see respectively [@P],[@AP],[@V]) with a view towards the gonality of very general abelian varieties. The story begins with [@P] in which Pirola shows that, given a very general abelian variety $A$, curves of geometric genus less than $\dim A-1$ are rigid in the Kummer variety $K(A)=A/\{\pm 1\}$. This allows him to show: \[P\] A very general abelian variety of dimension $\geq 3$ does not have positive dimensional orbits of degree $2$. In particular it does not admit a non-constant morphism from a hyperelliptic curve. There are several ways in which one might hope to generalize this result. For instance, one can ask for the gonality of a very general abelian variety of dimension $g$. We define the gonality of a smooth projective variety $X$ as the minimal gonality of the normalization of an irreducible curve $C\subset X$. Note that an irreducible curve $C\subset X$ whose normalization $\widetilde{C}$ has gonality $k$ provides us with a positive dimensional orbit in $\text{Sym}^k \widetilde{C}$, and thus with a positive dimensional orbit in $\text{Sym}^k X$. Hence, one can give a lower bound on the gonality of $X$ by giving a lower bound on the degree of positive dimensional orbits. This suggest to consider the function $$\mathscr{G}(k):=\text{min}\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g\\ \text{ does not have a positive dimensional orbit of degree } k\end{rcases},\end{cases}$$ and attempt to find an upper bound[^2] on $\mathscr{G}(k)$. Indeed, a very general abelian variety of dimension at least $\mathscr{G}(k)$ must have gonality at least $k+1$.\ In this direction, a few years after the publication of [@P], Alzati and Pirola improved on Pirola’s results in [@AP], showing that a very general abelian variety $A$ of dimension $\geq 4$ does not have positive dimensional orbits of degree $3$, i.e.that $\mathscr{G}(3)\leq 4$. This suggest that for any $k\in \mathbb{N}$ a very general abelian variety of sufficiently large dimension should not admit a positive dimensional orbit of degree $k$, i.e.that $\mathscr{G}(k)$ is finite for any $k\in \mathbb{Z}_{>0}$. This problem was posed in [@BPEL] and answered positively by Voisin in [@V]. \[Vmainthm\] A very general abelian variety of dimension at least $2^{k}(2k-1)+(2^{k}-1)(k-2)$ does not have a positive dimensional orbit of degree $k$, i.e. $\mathscr{G}(k)\leq 2^{k}(2k-1)+(2^{k}-1)(k-2).$ Voisin provides some evidence suggesting that this bound can be improved significantly. Her main conjecture from [@V] is the following linear bound on the gonality of very general abelian varieties: A very general abelian variety of dimension at least $2k-1$ has gonality at least $k+1$.\[Vweakconj\] The central result of this paper is the proof of this conjecture. It is obtained by generalizing Voisin’s method to powers of abelian varieties. This allows us to rule out the existence of positive dimensional $\text{CCS}_k$ in very general abelian varieties of dimension $g$ for $g$ large compared to $k$. For $k\geq 3$, a very general abelian variety of dimension at least $2k-2$ has no positive dimensional orbits of degree $k$, i.e. $\mathscr{G}(k)\leq 2k-2$. This gives the following lower bound on the gonality of very general abelian varieties: For $k\geq 3$, a very general abelian variety of dimension $\geq 2k-2$ has gonality at least $k+1$. In particular Conjecture [\[Vweakconj\]](Vweakconj) holds. Another approach to generalizing Theorem [\[P\]](P) is the following: Observe that a nonconstant morphism from a hyperelliptic curve $C$ to an abelian surface $A$ gives rise to a positive dimensional orbit of the form $2\{0_A\}$. Indeed, translating $C$ if needed, we can assume that a Weierstrass point of $C$ maps to $0_A$. This suggests to consider the maximal $g$ for which a very general abelian variety $A$ of dimension $g$ contains an irreducible curve $C\subset A$ whose normalization $\pi: \widetilde{C}\to C$ admits a degree $k$ morphism to $\mathbb{P}^1$ with a point of ramification index at least $l$. Here we say that $p\in \widetilde{C}$ has ramification index at least $l$ if the sum of the ramification indices of $\widetilde{C}\to \mathbb{P}^1$ at all points in $\pi^{-1}(\pi(p))$ is at least $l$. This maximal $g$ is less than $$\mathscr{G}_l(k):=\text{min}\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g \text{ does not }\\\text{ admit a positive dimensional orbit of the form } |\sum_{i=1}^{k-l} \{a_i\}+l\{a_0\}|\end{rcases},\end{cases}$$ where, given a smooth projective variety $X$ and $\{x_1\}+\ldots+\{x_k\}\in \text{Sym}^k X$, we denote by $|\{x_1\}+\ldots+\{x_k\}|$ the orbit of $\{x_1\}+\ldots+\{x_k\}$, namely the orbit containing this point. Clearly, we have $$\mathscr{G}_1(k)=\mathscr{G}_0(k)=\mathscr{G}(k),$$ and $$\mathscr{G}_l(k)\leq \mathscr{G}(k).$$ In [@V] the author shows the following: The following inequality holds: $$\mathscr{G}_l(k)\leq 2^{k-l}(2k-1)+(2^{k-l}-1)(k-2).$$ In particular $$\mathscr{G}_k(k)\leq 2k-1.$$ We can improve on Voisin’s result to show the following: A very general abelian variety $A$ of dimension at least $2k+2-l$ does not have a positive dimensional orbit of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$, i.e. $\mathscr{G}_{l}(k)\leq 2k+2-l.$\ Moreover, if $A$ is a very general abelian variety of dimension at least $k+1$ the orbit $|k\{0_A\}|$ is countable, i.e. $\mathscr{G}_{k}(k)\leq k+1.$ Taking a slightly different perpective, one can consider the maximal dimension of orbits of degree $k$ of abelian varieties. One of the main contributions of [@V] is the following extension of results of Alzati-Pirola (the cases $k=2,3$): \[k-1\] Orbits of degree $k$ on an abelian variety have dimension at most $k-1$. As observed by Voisin, one sees easily[^3] that this bound cannot be improved if one considers all abelian varieties. Moreover, it cannot be improved for very general abelian surfaces as shown in Example . Nevertheless Theorem [\[Vmainthm\]](Vmainthm) shows that it can be improved for very general abelian varieties of large dimension. In this direction we have: Orbits of degree $k$ on a very general abelian variety of dimension $\geq k-1$ have dimension at most $k-2$. Recall that the degree of irrationality $\text{Irr}(X)$ of a variety $X$ is the minimal degree of a dominant morphism $X\to \mathbb{P}^{\dim X}$. The previous theorem provides the following improvement of the Sommese bound $\text{Irr}(A)\geq \dim A+1$ (see the appendix to [@BE]) on the degree of irrationality of abelian varieties. If $A$ is a very general abelian variety of dimension $g\geq 3$, then $$\textup{Irr}(A)\geq g+2.$$ It is very likely that one can do better by studying the Gauss map of $\text{CCS}_k$[^4] of very general abelian varieties.\ Finally, in the appendix we present the following generalization of Proposition 0.9 of [@V] along with similar results: Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$ such that $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$\prod_{j=1}^k(\{x_i\}-\{y_j\})=0\in CH_0(A),$$ where the product is the Pontryagin product. In the last part of this introduction we will sketch Voisin’s proof of Theorem [\[Vmainthm\]](Vmainthm) and give an idea of the methods we will use to show Theorem [\[2k-2\]](2k-2). Voisin considers what she calls *naturally defined subsets of abelian varieties*. Given a universal abelian variety $\mathcal{A}/S$ of dimension $g$ with a fixed polarization $\theta$ there are subvarieties $S_\lambda\subset S$ along which $\mathcal{A}_s\sim \mathcal{B}_s^{\lambda}\times \mathcal{E}_s^{\lambda}$, where $\mathcal{B}^{\lambda}/S_\lambda$ is a family of abelian varieties of dimension $(g-1)$ and $\mathcal{E}^\lambda/S_\lambda$ is a family of elliptic curves. Let $S_\lambda(B)=\{s\in S_\lambda: B^\lambda_s=B\}$.\ Voisin shows that, given a naturally defined subset $\Sigma_{\mathcal{A}}\subsetneq \mathcal{A}$, there is an $S_\lambda$ such that the image of the restriction of $p_\lambda: (\mathcal{B}^\lambda_s\times \mathcal{E}_{s}^{\lambda})^k \to (\mathcal{B}^\lambda_s)^k=B^k$ to $\Sigma_{\mathcal{A}_{s}}$ varies with $s\in S_\lambda(B)$, for a generic $B$ in the family $\mathcal{B}^{\lambda}$. This shows that if $\Sigma_{B}$ is $d$-dimensional for a very-general $(g-1)$-dimensional abelian variety $B$ with $0<d<g-1$, then $\Sigma_A$ is at most $(d-1)$-dimensional for a very general abelian variety $A$ of dimension $g$.\ She proceeds to show that the set $$\bigcup_{i=1}^k \text{pr}_i(|k\{0_A\}|)=\left\{a_1\in A: \exists a_2,\ldots, a_k\in A\text{ such that }\sum_{i=1}^k\{a_i\}= k\{0_A\}\in CH_0(A)\right\}$$ is contained in a naturally defined subset $A_k\subset A$ and that $\dim A_k\leq k-1$. In particular, $A_k\neq A$ if $\dim A\geq k$. It follows from the discussion above that $$\dim\bigcup_{i=1}^k \text{pr}_i(|k\{0_A\}|)=0$$ if $\dim A\geq 2k-1$ and $A$ is very general. An induction argument finishes the proof. Voisin’s results are very similar in spirit to those of [@AP] but a key difference is that the latter is concerned with subvarieties of $A^k$ and not of $A$. We will see that this has important technical consequences. While Lemma 1.5 in [@V] shows that the restriction of the projection $B\times E\to B$ to $\Sigma_{B\times E}$ is generically finite on its image, this becomes a serious sticking point in [@AP] (see lemmas 6.8 to 6.10 of [@AP]). One of the innovations of this article is a way to bypass this generic finiteness assumptions by using the inductive nature of the argument. Aknowledgements {#aknowledgements .unnumbered} --------------- This paper owes a lot to the work of Alzati, Pirola, and Voisin. I thank Madhav Nori and Alexander Beilinson for countless useful and insightful conversations as well as for their support. I would also like to extend my gratitude to Claire Voisin for bringing my attention to this circle of ideas by writing [@V], and for kindly answering some questions about the proof of Theorem 0.1 from that article. Preliminaries ============= In this section we fix some notation and establish some facts about positive dimensional orbits for rational equivalence on abelian varieties.\ A variety will mean a quasi-projective reduced scheme of finite type over $\mathbb{C}$. In what follows, $X$ is a smooth projective variety, $\mathcal{X}/S$ is a family of such varieties, $A$ is an abelian variety of dimension $g$ with polarization type $\theta$, and $\mathcal{A}/S$ is a family of such abelian varieties. Mostly we will be concerned with locally complete families of abelian varieties, namely families $\mathcal{A}/S$ such that the corresponding morphism from $S$ to the moduli stack of $g$-dimensional abelian varieties with polarization type $\theta$ is dominant. In an effort to simplify notation, we will often write $\mathcal{X}^k$ instead of $\mathcal{X}_S^k$ to denote the $k$-fold fiber product of $\mathcal{X}$ with itself over $S$. $\mathcal{Z}\subset \mathcal{A}^{k}$ will be a subvariety such that $\mathcal{Z}\to S$ is flat with irreducible fibers $\mathcal{Z}_{s}$. Finally, $CH_0(X)$ will denote the Chow group of zero-cycles of a smooth projective variety $X$ with $\mathbb{Q}$-coefficients. \[convention\] In many of our arguments, we will have a family of varieties $\mathcal{X}\to S$ and a subvariety $\mathcal{Z}\subset \mathcal{X}$, such that $\mathcal{Z}\to S$ is flat with irreducible fibers. We will often need to base change by a generically finite morphism $S'\to S$. To avoid the growth of notation we will denote the base changed family by $\mathcal{X}\to S$ again. Moreover, if $S_\lambda\subset S$ is a Zariski closed subset, the base change of $S_\lambda$ under $S'\to S$ will also be denoted $S_\lambda$. Note that this applies also to the statement of theorems. For instance if we say a statement holds for a family $\mathcal{X}/S$ we mean that it holds for some $\mathcal{X}_{S'}/S'$, where $S'\to S$ is generically finite. Instead of considering orbits for rational equivalence, one can consider subvarieties of orbits. This makes talking about families of orbits somewhat simpler. \[CCS\] A constant cycle subvariety of degree $k$ $\text{CCS}_k$ of $X$ is a Zariski closed subset of $X^k$ contained in a fiber of $X^k\to CH_0(X)$. A Zariski closed subset $\mathcal{Z}\subset \mathcal{X}^k$ is a $\text{CCS}_k/S$ if $\mathcal{Z}_s$ is a $\text{CCS}_k$ of $\mathcal{X}_s^k$ for every $s\in S$. The notion of $\text{CCS}_k$ above is closely related but not to be confused with constant cycle subvarieties in the sense of Huybrechts (see [@H]). Indeed a $CCS_1$ is exactly the analogue of a constant cycle subvariety as defined in [@H] for K3 surfaces. Nonetheless a $\text{CCS}_k$ of $X$ need not be a $CCS_1$ of $X^k$; in the first case we consider rational equivalence of cycles in $X$, and in the other rational equivalence of cycles in $X^k$. We will not only be interested in $\text{CCS}_k$ but in families of $\text{CCS}_k$ and subvarieties of $X^k$ foliated by $\text{CCS}_k$. $\;$ 1. An $(r+d)$-dimensional Zariski closed subset $Z\subset X^k$ is foliated by $d$-dimensional $\text{CCS}_k$ if for all $z\in Z$ we have $$\dim |z|\cap Z\geq d.$$ 2. Similarly $\mathcal{Z}\subset \mathcal{X}^k$ is foliated by $d$-dimensional $\text{CCS}_k$ if $\mathcal{Z}_s$ is foliated by $d$-dimensional $\text{CCS}_k$ for all $s\in S$.\ 3. An $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of $X$ is an $r$-dimensional locally closed subset $D$ of a Chow variety of $X^k$ parametrizing dimension $d$ cycles with a fixed cycle class, such that for each $t\in D$ the corresponding cycle is a $\text{CCS}_k$.\ 4. Similarly, an $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of $\mathcal{X}/S$ is an $r$-dimensional locally closed subset $\mathcal{D}$ of the relative Chow variety of $\mathcal{X}^k$ parametrizing dimension $d$ cycles with a fixed cycle class, such that for each $t\in \mathcal{D}$ the fiber of the corresponding cycle over any $s\in S$ is a $\text{CCS}_k$. Note that given $D$, an $r$-parameter family of $d$-dimensional $\text{CCS}_k$ of a variety $X$, and $\mathcal{Y}\to D$, the corresponding family of cycles, the set $\bigcup_{t\in D}\mathcal{Y}_t\subset X^k$ is foliated by $d$-dimensional $\text{CCS}_k$. Yet its dimension might be less than $(r+d)$. Conversely, any subvariety of $X^k$ foliated by positive dimensional $\text{CCS}_k$ arises in such a fashion from a family of $\text{CCS}_k$ after possibly passing to a Zariski open subset. Indeed, by work of Roĭtman (see [@R1]) the set $$\Delta_{rat}=\left\{((x_1,\ldots, x_k),(x_1',\ldots, x_k'))\in X^k\times X^k: \sum_{i=1}^k x_i=\sum_{i=1}^k x_i'\in CH_0(X)\right\}\subset X^k\times X^k$$ is a countable union of Zariski closed subsets. Consider $$\pi: \Delta_{rat}\cap Z\times Z\to Z.$$ Given an irreducible component $\Delta'$ of $\Delta_{rat}\cap Z\times Z$ that dominates $Z$ and has relative dimension $d$ over $Z$, we can consider the map from an open set in $Z$ to an appropriate Chow variety of $X^k$ taking $z\in Z$ to the cycle $[\Delta_z']$. Letting $D$ be the image of this morphism, we get a family of $\text{CCS}_k$ with the desired property.\ Given an abelian variety $A$, we denote by $A^r_{M}$ (or $A_M$, when $r=1$) the image of $A^r$ in $A^k$ under the embedding $i_M: A^r\to A^k$ given by $$(a_1,\ldots, a_r)\mapsto \left(\sum_{j=1}^r m_{1j}a_j,\sum_{j=1}^r m_{2j}a_j,\ldots, \sum_{j=1}^r m_{kj}a_j\right),$$ where $0\leq r\leq k$, and $M=(m_{ij})\in M_{k\times r}(\mathbb{Z})$ has rank $r$. We will use the same notation $V_M^r$, with $M\in M_{k\times r}({\mathbb{C}})$, for $V$ a vector space over ${\mathbb{C}}$. If $A$ is simple then all abelian subvarieties of $A^k$ are of this form. Let $\text{pr}_i: A^k\to A$ be the projections to the $i^{\text{th}}$ factor and, given a form $\omega\in H^0(A,\Omega^q)$, let $\omega_k:=\sum_{i=1}^k \text{pr}_i^*\omega$.\ For the sake of simplicity we mostly deal with $X^k$ rather than $\text{Sym}^kX$ and we take the liberty to call points of $X^k$ effective zero-cycles of degree $k$. Given an abelian variety $A$, a zero-cycle $z=(z_1,\ldots, z_k)\in A^k$ is called normalized if $z_1+\ldots+z_k=0_A$. We write $A^{k,\,0}$ for the kernel of the summation map, i.e. the set of normalized effective zero-cycles of degree $k$. In Corollary 3.5 of [@AP] the authors show the following generalization of Mumford’s result: \[vanish\] Let $D$ be an an $r$-parameter family of $\text{CCS}_k$ of a variety $X$ with corresponding family of cycles $\mathcal{Z}\to D$. Denote by $g: \mathcal{Z}\to X^k$ the natural map. If $\omega\in H^0(A,\Omega^q)$ and $q>r$, then $$g^*(\omega_k)=0.$$ In proposition 3.2 of [@V], Voisin shows that if $A$ is an abelian variety, and $Z\subset A^k$ is such that $\omega_k|_{Z}=0$ for all $\omega\in H^0(A,\Omega^q)$ and all $q\geq 1$, then $\dim Z\leq k-1$. Along with the above proposition this gives: \[1dimfam\] An abelian variety does not admit a one-parameter family of $(k-1)$-dimensional $\text{CCS}_k$. This non-existence result along with our degeneration and projection argument will provide the proof of Theorem [\[no(k-1)\]](no(k-1)). For most other applications we will use non-existence results for large families of $\text{CCS}_k$ on surfaces $X$ with $p_g\neq 0$. In particular we have: \[vanish2\] Let $X$ be a smooth projective surface with $p_g\neq 0$ and $\omega\neq 0\in H^0(X,\Omega^2_X)$. If $Z\subset X^k$ is a Zariski closed subset of dimension $m$ foliated by $d$-dimensional $\text{CCS}_k$ then $\omega_k^{\lceil(m-d+1)/2\rceil}$ restricts to zero on $Z$. The set of points $z\in Z$ such that $z\in Z_{sm} \cap(|z|\cap Z)_{sm}$ is clearly Zariski dense. Thus it suffices to show that $\omega_k^{\lceil(m-d+1)/2\rceil}$ restricts to zero on $T_{Z,z}$ for such a $z$. Suppose that $m-d$ is odd (the even case is treated in the same way). Given any $(m-d+1)$-dimensional subspace of $T_{Z,z}$, it must meet the tangent space to $T_{|z|\cap Z,z}$ and so we can assume it admits a basis $v_1,\ldots, v_{m-d+1}$ with $v_1\in T_{|z|\cap Z,z}$. Hence $$\iota_{v_1,\ldots, v_{m-d+1}}{\omega_k^{(m-d+1)/2}}$$ will consist of a product of terms of the form $\iota_{v_i,v_j}{\omega_k}$. But $\iota_{v_1,v_j}{\omega_k}=0$ for any $j$ by Proposition [\[vanish\]](vanish). Indeed for any $j$ the space spanned by $v_1$ and $v_j$ is contained in the tangent space to a $(d+1)$-fold foliated by at least $d$-dimensional $\text{CCS}_k$. \[surfbound\] Let $X$ and $\omega$ be as in the previous lemma. If $Z\subset X^k$ is such that $\omega_k^l$ restricts to zero on $Z$, then $\dim Z< k+l$. Pick $z\in Z_{sm}$ and let $v_1,\ldots, v_m$ be a basis of $T_{Z,z}$. Complete it with $v_{m+1},\ldots, v_{2k}$ to a basis of $T_{X^k,z}$. Let $\omega$ be a symplectic form on $X$. Since $\omega_k^k$ is a volume form on $X^k$ we have $\iota_{v_1,\ldots, v_{2k}}\omega^k\neq 0$. If $m\geq k+l$ then $\iota_{v_{m+1},\ldots, v_{2k}}\omega_k^k\in (\omega_k^l)\subset H^0(X^k, \Omega_X^{\bullet})$. The non-vanishing stated above then implies $\omega_k^l$ cannot vanish on $Z$. If $A$ is an abelian variety and $\omega\neq 0\in H^0(A,\Omega_A^2)$, then $$(i_M^*\omega_k)^{\wedge l}\neq 0 \in H^0(A_M^l,\Omega_{A_M^l}^{2l}).$$ In particular, if $\dim A=2$, the form $\omega_k$ restricts to a symplectic form on $A_M^l$. Let $z,w$ be two coordinates on $A$ and $\omega=dz\wedge dw$. Let $z_{i},w_i$ be the corresponding coordinates on the $i^{\text{th}}$ factor of $A^l$. We have $$i_M^*\omega_k=\sum_{i=1}^k\left(\sum_{j=1}^lm_{ij}dz_{j}\wedge \sum_{j'=1}^lm_{ij'}dw_{j'}\right).$$ We claim that $$(i_M^*\omega_k)^{\wedge l}=l!\det \mathbf{G}\; dz_{1}\wedge dw_1\wedge\ldots \wedge dz_{l}\wedge dw_{l},$$ where $\mathbf{G}=(\langle M_i,M_j\rangle)_{1\leq i,j\leq l}$ is the Gram matrix of the columns $M_i$ of the matrix $M$. Since $M$ has maximal rank its Gram matrix has positive determinant. It follows that $(i_M^*\omega_k)^{\wedge l}\neq 0$. To prove the above claim we observe that $$i_M^*\omega_k=\sum_{i=1}^k\sum_{j,j'=1}^lm_{ij}m_{ij'}dz_j\wedge dw_{j'}=\sum_{j,j'=1}^l\langle M_j,M_{j'}\rangle dz_j\wedge dw_{j'}.$$ It follows that $$\begin{aligned} (i_M^*\omega_k)^{\wedge l}&=\sum_{\sigma,\sigma'\in S_l}\prod_{j=1}^l\langle M_{\sigma(j)},M_{\sigma'(j)}\rangle dz_{\sigma(1)}\wedge dw_{\sigma'(1)}\wedge\ldots\wedge dz_{\sigma(l)}\wedge dw_{\sigma'(l)}\\ &=\sum_{\sigma,\sigma'\in S_l}\text{sgn}(\sigma)\text{sgn}(\sigma')\prod_{j=1}^l\langle M_{\sigma(j)},M_{\sigma'(j)}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\ &=\sum_{\sigma'\in S_l}\text{sgn}(\sigma')\sum_{\sigma'\sigma\in S_l}\text{sgn}(\sigma'\sigma)\prod_{j=1}^l\langle M_{\sigma'\sigma(j)},M_{\sigma'(j)}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\ &=\sum_{\sigma'\in S_l}\sum_{\sigma'\sigma\in S_l}\text{sgn}(\sigma)\prod_{j=1}^l\langle M_{\sigma(j)},M_{j}\rangle dz_1\wedge dw_1\wedge\ldots \wedge dz_l\wedge dw_l\\ &=l!\det\mathbf{G}\end{aligned}$$ In particular Lemma [\[vanish2\]](vanish2) implies: \[notfoliated2\] If $Z\subset A^k$ is foliated by positive dimensional $\text{CCS}_k$ and $\dim A=2$ then it cannot be an abelian subvariety of the form $A^l_M$. This corollary will play a crucial technical role in our argument. Using Lemma [\[surfbound\]](surfbound), we see that the proof of Lemma 2.5 also shows the following: If $A$ is an abelian surface and $Z\subset A^{k,\,0}$ is such that $\omega_k^l$ vanishes on $Z$, then $$\dim Z< k+l-1.$$ \[normabsurfbound\] If $A$ is an abelian surface and $Z\subset A^{k,\,0}$ is foliated by $d$-dimensional $\text{CCS}_k$, then $$\dim Z\leq 2(k-1)-d.$$\[absurfbound\] By Lemma [\[vanish2\]](vanish2) $\omega^{\lceil (\dim Z-d+1)/2\rceil}$ restricts to zero on $Z$. Then by Lemma [\[surfbound\]](surfbound) $$\dim Z<k+\lceil(\dim Z-d+1)/2\rceil-1.$$ This gives the stated bound for parity reasons. One could instead seek existence results for subvarieties of $A^k$ foliated by $d$-dimensional $\text{CCS}_k$. Alzati and Pirola show in examples 5.2 and 5.3 of [@AP] that any abelian surface has a $2$-dimensional $\text{CCS}_3$ and a $2$-parameter family of normalized $\text{CCS}_1$. In particular, using the argument from Remark [\[trick\]](trick) we see that Corollary [\[normabsurfbound\]](normabsurfbound) is sharp for $d=0,1,2$. \[HSL\] In [@HSL] Lin shows that Corollary [\[normabsurfbound\]](normabsurfbound) is sharp for every $d$. The methods of [@HSL] can be used to show the following: \[curvecase\] If an abelian variety $A$ of dimension $g$ is the quotient of the Jacobian of a smooth genus $g'$ curve $C$, then $A^k$ contains a $(g(k+1-g'-d)+d)$-dimensional subvariety foliated by $d$-dimensional normalized $\text{CCS}_k$ for all $k\geq g'+d-1$. To simplify notation we identify $C$ with its image in $J(C)$. We can assume that $0_A\in C$. Recall that the summation map $\text{Sym}^lC\to J(C)$ has fibers $\mathbb{P}^{l-g'}$ for all $l\geq g'$. Moreover, if $(c_1,\ldots, c_l)$ and $(c_1',\ldots, c_l')$ are such that $\sum c_i=\sum c_i'\in J(C)$, then the zero cycles $\sum \{c_i\}$ and $\sum \{c_i\}'$ are equal in $CH_0(C)$ and thus in $CH_0(A)$.\ In light of Remark [\[trick\]](trick), it suffices to show that $A^{g'+d-1,\, 0}$ contains a $d$-dimensional $\text{CCS}_{g'+d-1}$. Consider the map $$\psi: C\times C^{g'+d-1}\to A^{g'+d-1}$$ given by $$(c_0,(c_1,\ldots, c_{g'+d-1}))\mapsto (c_1-c_0,\ldots, c_{g'+d-1}-c_0).$$ This morphism is generically finite on its image since the restriction of the summation map $A^2\to A$ to $C^2\subset A^2$ is. The intersection of the image of $\psi$ with $A^{g'+d-1,\, 0}$ is $d$-dimensional and we claim it is a $\text{CCS}_{g'-d+1}$. Indeed, given $$(c_1-c_0,\ldots, c_{g'+d-1}-c_0)\in \text{Im}(\psi)\cap A^{g'+d-1,\, 0},$$we have $$\sum_{i=1}^{g'+d-1} c_i=(g'+d-1)c_0$$ so that $$\sum_{i=1}^{g'+d-1} \{c_i\}=(g'+d-1)\{c_0\}\in CH_0(C).$$ It follows that $$\sum_{i=1}^{g'+d-1}\{c_i-c_0\}= (g'+d-1)\{0_A\}\in CH_0(A).$$ Since the Torelli morphism $\mathcal{M}_{3}\to \mathcal{A}_3$ is dominant the previous proposition provides a one-dimensional orbit of degree $3$ in a very general abelian $3$-fold. Our methods do not seem to provide any way to rule out the existence of a one-parameter family of one-dimensional $\text{CCS}_3$ on a very general abelian $3$-fold. Yet, the study of zero cycles on Jacobians does not seem to readily provide an example of such a family. This motivates the following: Does a very general abelian $3$-fold admit a one-parameter family of normalized one-dimensional orbits of degree $3$?\ Degeneration and Projection =========================== In this section we generalize Voisin’s method from Section 1 of [@V] to powers of abelian varieties. The key difference is that our generalization requires technical assumptions which are automatically satisfied in Voisin’s setting.\ Given $\mathcal{A}/S$ a locally complete family of abelian varieties of dimension $g$, and a positive integer $l< g$, let $S_{\lambda}\subset S$ denote loci along which $$\mathcal{A}_s\sim\mathcal{B}^\lambda_{s}\times \mathcal{E}^\lambda_{s},$$ where $\mathcal{B}^\lambda/S_\lambda$ and $\mathcal{E}^\lambda/S_\lambda$ are locally complete families of abelian varieties of dimension $l$ and $g-l$ respectively, and the index $\lambda\in \Lambda_l$ encodes the structure of the isogeny. Given a positive integer $l'< l$ we will also be concerned, inside each $S_{\lambda}$, with loci $S_{\lambda,\mu}$ along which $$\mathcal{B}_s^\lambda\sim \mathcal{D}_s^{\lambda,\mu}\times\mathcal{F}_s^{\lambda,\mu},$$ where $\mathcal{D}^{\lambda,\mu}/S_{\lambda,\mu}$ and $\mathcal{F}^{\lambda,\mu}/S_{\lambda,\mu}$ are locally complete families of abelian varieties of dimension $l'$ and $l-l'$ respectively, and the index $\mu\in \Lambda_{l'}^{\lambda}$ encodes the structure of the isogeny. For our applications we will mostly be concerned with $(l,l')=(g-1,2)$. Upon passing to a generically finite cover of $S_\lambda$ and $S_{\lambda,\mu}$ we can assume that we have projections $$\begin{aligned} p_{\lambda}: &\mathcal{A}_{S_{\lambda}}^k\to ({\mathcal{B}^{\lambda}})^k\\ p_{\mu}: &(\mathcal{B}^{\lambda}_{S_{\lambda,\mu}})^k\to (\mathcal{D}^{\lambda,\mu})^k,\\\end{aligned}$$ and we let $$p_{\lambda,\mu}:=p_{\mu}\circ p_{\lambda}$$ for $\mu\in \Lambda_{l'}^{\lambda}$. Note that, to keep an already unruly notation in check, we suppress the power $k$ from the notation of the projections. Given a subvariety $\mathcal{Z}\subset \mathcal{A}^k/S$ we consider the following subsets of $S$ and conditions on $\mathcal{Z}$ $$\begin{aligned} R_{gf}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}|_{\mathcal{Z}_s}: \mathcal{Z}_s\to \mathcal{B}^{\lambda}_s\text{ is generically finite on its image}\},\\ R_{ab}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}(\mathcal{Z}_s)\text{ is not an abelian subvariety of }\mathcal{B}^{\lambda}_s\},\\ R_{st}&=\bigcup_{\lambda}\{s\in S_\lambda : p_{\lambda}(\mathcal{Z}_s) \text{ is not stabilized by an abelian subvariety of } \mathcal{B}^{\lambda}_s\},\end{aligned}$$ $$R_{gf}\subset S \text{ is dense} \tag{$*$},$$ $$R_{ab}\cap R_{gf}\subset S \text{ is dense} \tag{$**$},$$ $$R_{st}\cap R_{gf}\subset S \text{ is dense} \tag{$***$}.$$ Note that these sets and conditions depend on $\mathcal{Z}$ and $l$ and, while $\mathcal{Z}$ should ususally be clear from the context, we will say $(*)$ holds for a specified value of $l$. Moreover, we will always assume that $\mathcal{Z}\to S$ is irreducible and has irreducible fibers. Given an abelian variety $A$ we will denote by $T_A:=T_{A,0_A}$ the tangent space to $A$ at $0_A\in A$. We let $$\begin{aligned} \mathscr{T}/S&:=T_{\mathcal{A}}/S,\\ G/S&:=\text{Gr}(g-l,{\mathscr{T}})/S,\\ G'/S&:=\text{Gr}(g-l',{\mathscr{T}})/S,\\ $$ and we consider the following sections $$\begin{aligned} &\sigma_{\lambda}: S_\lambda\to G_{{S_\lambda}}=\text{Gr}(g-l,{\mathscr{T}}_{S_\lambda}),\;\qquad \qquad \qquad \sigma_{\lambda}(s):=T_{\ker(p_{\lambda,s})},\\ &\sigma_{\lambda,\mu}: S_{\lambda,\mu}\to G'_{{S_{\lambda,\mu}}}= \text{Gr}(g-l',{\mathscr{T}}_{S_{\lambda,\mu}}),\qquad \;\;\sigma_{\lambda,\mu}(s):=T_{\ker(p_{\lambda,\mu,s})}. \end{aligned}$$ Let $\mathcal{A}/S$ be a family of abelian varieties, and $\mathcal{Z}\subset \mathcal{A}$ a subvariety which is flat over the base. Then, the set of $s\in S$ such that $\mathcal{Z}_s$ is stabilized by a positive dimensional abelian subvariety of $\mathcal{A}_s$ is closed in $S$. Consider the morphism $\mathcal{Z}\times_S\mathcal{A}\to \mathcal{A}$ given by $(z,a)\mapsto (z+a)$ and let $\mathcal{R}$ be the preimage of $\mathcal{Z}\subset \mathcal{A}$. Since $\mathcal{Z}\to S$ is flat, so is $\mathcal{Z}\times_S\mathcal{A}\to \mathcal{A}$. Since flat morphisms are open, the image of $\mathcal{Z}\times_S\mathcal{A}\setminus \mathcal{R}$ in $\mathcal{A}$ is open. The complement of this image is the closed subset $\mathcal{B}\subset \mathcal{A}$ which is the maximal abelian subscheme stabilizing $\mathcal{Z}$. Finally, the subset of $S$ over which $\mathcal{B}$ has positive dimensional fibers is closed by upper semi-continuity of fiber dimension. $\bigcup_{\lambda\in \Lambda_l}\sigma_\lambda(S_{\lambda})\subset G$ is dense. It suffices to consider the locus of abelian varieties isogenous to $E^g$ for some elliptic curve $E$. This locus is dense in $S$ and, given $s$ in this locus and any $M\in M_{k\times (g-l)}(\mathbb{Z})$ of rank $(g-l)$, the tangent space $T_{E^{g-l}_{M}}\in G_s$ is contained in $\sigma_\lambda(S_{\lambda})$ for some $\lambda\in \Lambda_l$. Since $$\{T_{E^{g-l}_{M}}: M\in M_{k\times (g-l)}(\mathbb{Z}),\; \text{rank} (M)=g-l\}\subset G_s$$ is dense in $G_s$, the result follows. In the following $\mathcal{A}/S$ will be an almost complete family of abelian varieties. If $\mathcal{Z}\subset \mathcal{A}^k$ is foliated by positive dimensional $\text{CCS}_k$ and $\dim A\geq 2$, then, for very general $s\in S$, the subset $\mathcal{Z}_s$ is not an abelian subvariety of the form $A^r_M$. Consider the Zariski closed sets $$\{s\in S: \mathcal{Z}_s=(\mathcal{A}_{s})^r_M\}\subset S.$$ There are countably many such sets so it suffices to show that none of them is all of $S$. Suppose that $\mathcal{A}^r_M$ is foliated by positive dimensional $\text{CCS}_k$. By the previous lemma, there is a $\lambda\in \Lambda_2$ such that $p_\lambda((\mathcal{A}_{s})^r_M)=(\mathcal{B}^\lambda_s)^r_M$ is also foliated by positive dimensional orbits for generic $s\in S_\lambda$. This contradicts Corollary [\[notfoliated2\]](notfoliated2). \[\*implies\*\*\] If $\mathcal{Z}\subset \mathcal{A}^k/S$ is foliated by positive dimensional orbits and $l\geq 2$, then $$(*)\implies (**).$$ First, observe that if $p_{\lambda}|_{\mathcal{Z}_s}: \mathcal{Z}_s\to \mathcal{B}_s^k$ is generically finite on its image, then its image is foliated by positive dimensional orbits. Moreover, if $R_{gf}\cap S_\lambda\neq \emptyset$, then $R_{gf}\cap S_\lambda$ is open in $S_\lambda$. Let $\lambda$ be such that $R_{gf}\cap S_\lambda\neq \emptyset$. For very general $s\in R_{gf}\cap S_\lambda$ the abelian variety $\mathcal{B}_s$ is simple. Thus, if the Zariski closed subset $p_\lambda(\mathcal{Z}_s)$ is an abelian subvariety of $\mathcal{B}_s^k$, it must be abelian subvariety of the form $\mathcal{B}_{M,t}^r$, contradicting Lemma [\[notfoliated2\]](notfoliated2). Hence if $R_{gf}\cap S_\lambda$ is non-empty then $R_{gf}\cap R_{ab}\cap S_\lambda$ is dense in $S_\lambda$. We will also need the following Zariski closed subsets $$\begin{aligned} S_{\lambda}(B)=&\{s\in S_{\lambda}: \mathcal{B}^{\lambda}_s\cong B\}\subset S_{\lambda}\\ S_{\lambda,\mu}(D,F)=&\{s\in S_{\lambda,\mu}: \mathcal{D}^{\lambda,\mu}_s\cong D,\mathcal{F}^{\lambda,\mu}_s\cong F\}\subset S_{\lambda,\mu}.\end{aligned}$$ The main result we will prove in this section is a generalization of Voisin’s method from [@V]. Recall that, given varieties $X,S$, a variety $\mathcal{Z}\subset X_S$ dominant over $S$ gives rise to a morphism from (an open in) $S$ to the Chow variety parametrizing cycles of class $[\mathcal{Z}_{s}]$ on $X$, where $s\in S$ is generic. We remind the reader of the notational convention of Remark [\[convention\]](convention), which allows us to remove the words in parenthesis from the previous sentence. \[prop1\] Let $\mathcal{Z}\subset \mathcal{A}^k$ be a $d$-dimensional variety dominant over $S$, and satisfying $(*)$ and $(**)$. Then there exists a $\lambda\in \Lambda_l$ such that $$p_\lambda(\mathcal{A}_{S_{\lambda}(B)})\subset B_{S_{\lambda}(B)}=\mathcal{B}^\lambda_{S_\lambda(B)}$$ gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety. From Lemma [\[\*implies\*\*\]](*implies**), if $\mathcal{Z}$ is foliated by $\text{CCS}_k$ then it satisfies $(**)$. Hence, the key assumption for our applications will be $(*)$. We propose to use these methods to prove Conjecture [\[Vweakconj\]](Vweakconj) in the following way: Assume that a very general abelian variety of dimension $2k-1$ has a one-dimensional $\text{CCS}_k$. This gives $\mathcal{Z}\subset \mathcal{A}^k/S$ of relative dimension $1$. It is easy to show that $(*)$ holds in this setting so we can use the previous proposition to get a one-parameter family of one-dimensional $\text{CCS}_k$ $$p_{\lambda}(\mathcal{Z}_{S_\lambda}(B))\subset (B_{S_\lambda(B)})^k$$ in a generic abelian variety $B$ of dimension $(g-1)$ with some polarization $\theta^\lambda$. This gives $$\mathcal{Z}'\subset ({\mathcal{B}}^{\lambda})^k/S_{\lambda}$$ which has relative dimension $2$ and such that $\mathcal{Z}'_s$ is foliated by positive dimensional orbits for generic $s\in S_\lambda$.\ We can hope to inductively apply Proposition [\[prop1\]](prop1) to $\mathcal{Z}'/S_\lambda$, eventually getting a large-dimensional subvariety of $B^k$ foliated by positive dimensional orbits, for an abelian surface $B$. Corollary [\[absurfbound\]](absurfbound) would then provide the desired contradiction. The key issue here is to ensure that condition $(*)$ is satisfied. At each step the dimension of the variety $\mathcal{Z}$ to which we apply Proposition [\[prop1\]](prop1) grows. Hence, it gets harder and harder to ensure generic finiteness of the projection. We have found a way around this using the fact that the variety $\mathcal{Z}$ to which we apply the proposition is obtained by successive degenerations and projections. We will introduce this argument in the following section. We first reduce to the case where the condition $(***)$ holds. Consider $s_0\in R_{gf}\cap R_{st}$ such that $\mathcal{B}_{s_0}^{\lambda_0}$ is simple, where $s_0\in S_{\lambda_0}$ and $p_{\lambda_0}|_{\mathcal{Z}_{s_0}}$ is generically finite on its image. Suppose that $p_{\lambda_0}(\mathcal{Z}_0)$ is stabilized by $(\mathcal{B}_{s_0}^{\lambda_0})_M^r$ but not by any larger abelian subvariety of $\mathcal{B}_{s_0}^{\lambda_0}$. Then $$p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}_{s_0}^{\lambda_0})_M^r\subset(\mathcal{B}_{s_0}^{\lambda_0})^k/ \mathcal(\mathcal{B}_{s_0}^{\lambda_0})_M^r$$ is not stabilized by any abelian subvariety of $(\mathcal{B}_{s_0}^{\lambda_0})^k/ \mathcal(\mathcal{B}_{s_0}^{\lambda_0})_M^r$. We have a diagram $$\begin{tikzcd}\label{lambdadiag1} \mathcal{Z}_{S_{\lambda_0}}/(\mathcal{A}_{S_{\lambda_0}})_{M}^r \ar[r,dashed,"g"] \arrow{d}[swap]{p_{\lambda_0}}& \text{Gr}(d,{\mathscr{T}}_{S_{\lambda_0}}^k/T_{(\mathcal{A}_{S_{\lambda_0}})^r_M}) \ar[d,dashed,"\pi_{\sigma_{\lambda_0}(S_{\lambda_0})}"]\\ p_{\lambda_0}(\mathcal{Z}_{S_{\lambda_0}})/(\mathcal{B}_{S_{\lambda_0}})_M^r \ar[r,dashed,"g"]& \text{Gr}\left(d,\text{Gr}(d,{\mathscr{T}}_{S_{\lambda_0}}^k/[\sigma_{\lambda_0}(S_{\lambda_0})^k+T_{(\mathcal{B}^{\lambda_0})^r_M}]\right), \end{tikzcd}$$ where $g$ denotes the Gauss map. Here $\pi_{\sigma_{\lambda_0}(S_{\lambda_0})}$ is the map induced by the quotient $${\mathscr{T}}_{S_{\lambda_0}}^k/T_{(\mathcal{A}_{S_{\lambda_0}})^r_M}\to {\mathscr{T}}_{S_{\lambda_0}}^k/[\sigma_{\lambda_0}(S_{\lambda_0})^k+T_{(\mathcal{B}^{\lambda_0})^r_{M}}].$$ We also denote by $p_{\lambda_0}$ the map induced by $p_{\lambda_0}$ on $ \mathcal{Z}_{S_{\lambda_0}}/(\mathcal{A}_{S_{\lambda_0}})^r_M$. Now, consider the base change by $G\to S$ $$\mathcal{Z}_{G}\subset \mathcal{A}^k_{G}.$$ The section $\sigma_{\lambda_0}$ is a closed immersion $S_{\lambda_0} \to G$ and $\mathcal{Z}_{S_{\lambda_0}}$ is the base change of $\mathcal{Z}_{G}$ under this immersion. The upper right corner of diagram ([\[lambdadiag1\]](lambdadiag1)) is the base-change by $\sigma_{\lambda_0}:S_{\lambda_0}\to G$ of the diagram $$\label{diag1} \begin{tikzcd}\mathcal{Z}_{G}/(\mathcal{A}_{M}^r)_{G}\ar[r,dashed, "g"] & \text{Gr}(d,{\mathscr{T}}_G^k/T_{(\mathcal{A}^r_{M})_G}) \ar[d,dashed, "\pi"]\\ & \text{Gr}(d, [{\mathscr{T}}_{G}/\mathcal{U}]^k/T_{(\mathcal{B}^r_{M})_G}),\\ \end{tikzcd}$$ where $\mathcal{U}\to G:=\text{Gr}(g-l,T)$ is the universal bundle and $\pi$ is the rational map induced by the quotient map. The composition $\pi\circ g$ is defined on a Zariski open in $\mathcal{Z}_G$ which meets $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/\mathcal{A}_{M,\sigma_{\lambda_0}(s_0)}^r$ non-trivially. Indeed $g$ is defined on the smooth locus of $$\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{\sigma_{\lambda_0}(s_0)})^r_M),$$ and the restriction of $p_{\lambda_0}$ to $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{\sigma_{\lambda_0}(s_0)})^r_M)$ is generically finite on its image by the following: Consider an abelian variety $A\sim B\times E$, where $B$ and $E$ are abelian varieties of smaller dimension, and let $p$ be the projection $A^k\to {B}^k$. If $Z\subset A^k$ is such that $p|_{Z}: Z\to p(Z)$ is generically finite, then the projection $p: A^k/A_{M}^r\to {B}^k/{B}_{M}^r$ is such that $p|_{Z/A_M^r}$ is generically finite on its image. Since $p|_{Z/A_M^r}$ is proper and locally of finite presentation, it suffice to show that it is quasi-finite. The fiber of $A^k\to {B}^k/{B}^r_M$ over $p(z)\in p(Z)/{B}^r_M$ for $z\in Z$ is the set of all $A^r_M$-translates of the fiber of $p$ over $p(z)$. Hence the fiber of $p|_{Z/A_M^r}$ over $p(z)$ is finite. We deduce that $q:=g\circ \pi$ is defined in an open in the smooth locus of $\mathcal{Z}_{s_0}/(\mathcal{A}_{M,s_0}^r)$, so that $q$ is defined in an open in $\mathcal{Z}_G$ meeting $\mathcal{Z}_{\sigma_{\lambda_0}(s_0)}/(\mathcal{A}_{s_0})_M^r$ non-trivially.\ Since $p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}_{s_0}^{\lambda_0})_M^r$ is not stabilized by any abelian subvariety of $(\mathcal{B}^{\lambda_0}_{s_0})^k$, the Gauss map $g$ at the bottom of diagram ([\[lambdadiag1\]](lambdadiag1)) is defined on the smooth locus of $p_{\lambda_0}(\mathcal{Z}_{s_0})/(\mathcal{B}^{\lambda_0}_{s_0})_M^r$ and generically finite by results of Griffiths and Harris (see (4.14) in [@GH]). The fact that the restriction of $p_{\lambda_0}$ to $\mathcal{Z}_{s_0}/(\mathcal{A}_{s_0})^r_M$ is generically finite on its image implies that the following composition is generically finite on its image $$\begin{tikzcd}g\circ p_{\lambda_0}: \mathcal{Z}_{s_0}/(\mathcal{A}_{s_0})_M^r\ar[r,dashed] & \text{Gr}\Big(d,{\mathscr{T}}^k/[\sigma_{\lambda_0}(s_0)^k+T_{(\mathcal{B}^{\lambda_0}_{s_0})^r_M}]\Big). \end{tikzcd}$$ It follows that, restricting to an open of $G$ if needed, the map $q_t:=(\pi\circ g)_t$ is defined and is generically finite on its image for all $t\in G$. This generic finiteness statement implies that, for $s\in S_\lambda$ (such that $\sigma_\lambda(s)$ lies in an appropriate open in $G$), the Gauss map of $p_\lambda(\mathcal{Z}_{s})/(\mathcal{B}^\lambda_{s})_M^r$ is generically finite. Namely, for such an $s$, the variety $p_\lambda(\mathcal{Z}_s/(\mathcal{B}^\lambda_{s})_M^r)$ is not stabilized by an abelian subvariety. Moreover, if $\lambda\in \Lambda_l$ and $B$ in the family $\mathcal{B}^\lambda$ are such that the family $$p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)}/\mathcal{A}^r_{M})\subset ({B}^k/{B}^r_{M})_{S_\lambda(B)}$$ gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety of ${B}^k/{B}^{r}_M$, then the family $$p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)})\subset {B}^k_{S_\lambda(B)}$$ also gives rise to a finite morphism from $S_{\lambda}(B)$ to the appropriate Chow variety of $B^k$. Hence, replacing $\mathcal{Z}$ by $\mathcal{Z}/\mathcal{A}_{M}^r$ and $\mathcal{A}^k$ by $\mathcal{A}^k/\mathcal{A}^r_{M}$ we are reduced to the case where $(***)$ holds.\ Now consider the analogue of diagram ([\[lambdadiag1\]](lambdadiag1)) $$\begin{tikzcd}\label{diaglambda} \mathcal{Z}_{S_{\lambda}} \ar[r,dashed,"g"] \arrow{d}[swap]{p_{\lambda}}& \text{Gr}(d,T^k) \ar[d,dashed,"\pi_{\sigma_{\lambda}(S_\lambda)}"]\\ p_{\lambda}(\mathcal{Z}_{S_{\lambda}})\ar[r,dashed,"g"]& \text{Gr}(d,[T/\mathcal{U}_{\sigma_{\lambda}(S_\lambda)}]^k) \end{tikzcd}$$ as well as the analogue of diagram ([\[diag1\]](diag1)) $$\label{diag} \begin{tikzcd}\mathcal{Z}_{G}\ar[r,dashed, "g"] & \text{Gr}(d,T_G^k) \ar[d,dashed, "\pi"]\\ & \text{Gr}(d, [T_{G}/\mathcal{U}]^k).\\ \end{tikzcd}$$ By the discussion above there is an open in $\mathcal{Z}_{G}$ on which the composition $q:=\pi\circ g$ is defined and generically finite on its image. The diagram ([\[diaglambda\]](diaglambda)) provides a factorization of $q$ along each $\sigma_\lambda(S_\lambda)$ $$\begin{tikzcd}\label{diagfact} \mathcal{Z}_{\sigma_\lambda(S_\lambda)}\cong \mathcal{Z}_{S_\lambda} \ar[rr,dashed, "q:=\pi\circ g"] \ar[dr,swap,"p_\lambda"] &\; & \text{Gr}(d, [T_G/\mathcal{U}]^k)\\ & p_{\lambda}(\mathcal{Z}_{S_{\lambda}}).\ar[ur,swap,dashed,"g"]& \end{tikzcd}$$ Hence (3.5) gives a factorization of $q$ on a Zariski dense subset of the base $G$. \[cov1\] Let $\mathcal{Z}/S$ be a family with irreducible fibers and base, and $q: \mathcal{Z}/S\to \mathcal{X}/S$ be such that $q_s: \mathcal{Z}_s\to \mathcal{X}_s$ is generically finite for each $s\in S$. Consider $S'\subset S$, a Zariski dense subset, and suppose that for each $s'\in S'$ we have a factorization of $q_s$ $$\mathcal{Z}_s\xrightarrow{f_{s'}} f_{s'}(\mathcal{Z}_s)\xrightarrow{g_{s'}} \mathcal{X}_s.$$ Then there is a family $\mathcal{Z}'/S$, morphisms $p: \mathcal{Z}\to \mathcal{Z}'$ and $p': \mathcal{Z}'\to \mathcal{X}$, and a Zariski dense subset $S''\subset S'$ such that, for any $s''\in S''$, the morphisms $p_{s''}(\mathcal{Z}_{s''})$ and $f(\mathcal{Z}_{s''})$ are birational, and $p_{s''}$ and $p'_{s''}$ induce the same morphism on function fields as $f_{s''}$ and $g_{s''}$ respectively. Restrict to a Zariski open subset of $\mathcal{X}$ (which we call $\mathcal{X}$ in keeping with remark [\[convention\]](convention)) over which $q$ is finite étale and such that $\mathcal{X}\to S$ is smooth. By work of Hironaka, we can find a compactification $\overline{\mathcal{X}}$ of $\mathcal{X}$ with simple normal crossing divisors at infinity. Restricting to an open in the base, we can assume that we have $\mathcal{X}/S\subset \overline{\mathcal{X}}/S$, such that $\overline{\mathcal{X}}_s\setminus \mathcal{X}_s$ is an snc divisor for any $s\in S$. One can use a version of Ehresmann’s lemma allowing for an snc divisor at infinity to see that $\mathcal{X}\to S$ is a locally-trivial fibration in the category of smooth manifolds.\ It follows that we get a diagram $$\begin{tikzcd} \mathcal{Z}\arrow{r}{q} &\mathcal{X} \arrow{d} \\ &S, \end{tikzcd}$$ where $q$ is a covering. Note that we renamed as $\mathcal{Z}$ an open subset of $\mathcal{Z}$ over which $q$ is étale. To complete the proof of Proposition [\[prop1\]](prop1) we will need the following Lemma. \[cov2\] Given a diagram as above with $\mathcal{Z}_s$ connected for every $s$, and a factorization $$\begin{tikzcd} \mathcal{Z}_{s_0}\ar[rr,"q"] \ar{dr}[swap]{f_{s_0}} & &\mathcal{X}_{s_0} \\ &f_{s_0}(\mathcal{Z}_{s_0}), \ar{ur} & \end{tikzcd}$$ there is a factorization $$\begin{tikzcd} \mathcal{Z}\ar[rr,"q"] \ar{dr}[swap]{f} & &\mathcal{X} \\ &f(\mathcal{Z}), \ar{ur} & \end{tikzcd}$$ which identifies with the original factorization over $s_0\in S$. Consider the Galois closure $\mathcal{Z}'\to \mathcal{X}$ of the covering $q: \mathcal{Z}\to \mathcal{X}$. Note that $\mathcal{Z}'_{s_0}$ is connected. Indeed there is a normal subgroup of the Galois group of $\mathcal{Z}'/\mathcal{X}$ corresponding to a deck transformation inducing the trivial permutation of the connected components of $\mathcal{Z}'_{s_0}$. This subgroup corresponds to a cover above $\mathcal{Z}$ since $\mathcal{Z}_{s_0}$ is connected. It follows that the map $\text{Gal}(\mathcal{Z}'/\mathcal{X})\to \text{Gal}(\mathcal{Z}'_{s_0}/\mathcal{X}_{s_0})$ is injective because a deck transformation which is the identity on the base and on fibers must be the identity. Since $\text{Gal}(\mathcal{Z}'/\mathcal{X})$ has order $$d:=\deg(\mathcal{Z}'/\mathcal{X})=\deg(\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}),$$ and $\text{Gal}(\mathcal{Z}'_{s_0}/\mathcal{X}_{s_0})$ has order at most $d$, this restriction morphism must be an isomorphism and $\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}$ is thus also Galois. One then has an equivalence of categories between the poset of intermediate coverings of $\mathcal{Z}'/\mathcal{X}$ and that of $\mathcal{Z}_{s_0}'/\mathcal{X}_{s_0}$, and hence between the poset of intermediate coverings of $\mathcal{Z}/\mathcal{X}$ and that of $\mathcal{Z}_{s_0}/\mathcal{X}_{s_0}$. By the previous lemma, to each factorization $f_{s'}$ we can uniquely associate an intermediate cover $\mathcal{Z}\to \mathcal{Z}^{s'}$ of $\mathcal{Z}\to \mathcal{X}$, which agrees with $f_{s'}$ at $s'$. Since there are only finitely many intermediate covers of $\mathcal{Z}\to \mathcal{X}$, we get a partition of $S'$ according to the isomorphism type of the cover $\mathcal{Z}\to \mathcal{Z}^{s'}$. One subset $S''\subset S'$ of this partition must be dense in $S$. Let $f: \mathcal{Z}\to f(\mathcal{Z})$ be the corresponding intermediate cover. For the rest of the proof of Proposition [\[prop1\]](prop1) we are back in the situation of diagrams ([\[diaglambda\]](diaglambda)), ([\[diag\]](diag)), and ([\[diagfact\]](diagfact)). There is a variety $\mathcal{Z}'/G$, a morphism $p: \mathcal{Z}\to \mathcal{Z}'$, and a subset $\Lambda_{l,0}\subset \Lambda_{l}$ such that $$\bigcup_{\lambda\in \Lambda_{l,0}}\sigma_{\lambda}(S_{\lambda})\subset G$$ is dense, and such that $p_{\lambda}(\mathcal{Z}_t)$ and $p(\mathcal{Z}_t)$ are birational, and $p_t: \mathcal{Z}_t\to p(\mathcal{Z}_t)$ and $p_{\lambda,t}: \mathcal{Z}_t\to p_\lambda(\mathcal{Z}_t)$ induce the same map on function fields, for any $\lambda\in \Lambda_{l,0}$, $t\in \sigma_{\lambda}(S_\lambda)$. This follows from the previous lemma and its proof once we observe that the intermediate covering of $q$ (or rather of an appropriate finite étale restriction of $q$ as above) associated to the factorization $p_{\lambda,t}: \mathcal{Z}_t\to p_{\lambda}(\mathcal{Z}_t)$ is independent of $t\in \sigma_{\lambda}(S_\lambda)$. A technical point is that the maps $p_\lambda$ are a priori only defined after passing to a generically finite cover of $S_\lambda$. This does not cause problem as $p_{\lambda,s}$ is defined without passing to a generically finite cover and, given a generically finite cover $S_\lambda'\to S_\lambda$, the isomorphism type of the covering $p_{\lambda,s}: \mathcal{Z}_{s}\to p_{\lambda,s}(\mathcal{Z}_s)$ is the same as that of $p_{\lambda,s}: \mathcal{Z}^{\sharp} _{s}\to p_{\lambda,s}(\mathcal{Z}^\sharp_s)$, where $\mathcal{Z}^\sharp$ is obtained from the cover as prescribed in Remark [\[convention\]](convention). To finish the proof of Proposition [\[prop1\]](prop1), consider desingularizations $$\widetilde{p}: \widetilde{\mathcal{Z}}/G\to \widetilde{\mathcal{Z}'}/G$$ with smooth fibers over $G$. We have the inclusion $$j: \mathcal{Z}/G\to \mathcal{A}^k/G$$ as well as $$\widetilde{j}: \widetilde{\mathcal{Z}}/G\to \mathcal{A}^k/G.$$ The morphism $\widetilde{j}$ gives rise to a pullback map $$\widetilde{j}^*: \text{Pic}^0(\mathcal{A}^k/G)\to \text{Pic}^0(\mathcal{\widetilde{Z}}/G).$$ Since $\widetilde{p}$ is generically finite on fibers we can consider the composition $$\widetilde{p}_*\circ \widetilde{j}^* : \text{Pic}^0(\mathcal{A}^k/G)\to \text{Pic}^0(\widetilde{\mathcal{Z}}/G)\to \text{Pic}^0(\widetilde{\mathcal{Z'}}/G).$$ This is a morphism of abelian schemes and we will show that it is non-zero along $\sigma_{\lambda}(S_\lambda)$, $\lambda\in \Lambda_{l,0}$. Consider $t\in \sigma_{\lambda}(S_\lambda)$. Then $\mathcal{A}_t$ is isogenous to $\mathcal{B}_t^{\lambda}\times \mathcal{E}_t^{\lambda}$. We have the following commutative diagram $$\begin{tikzcd} \mathcal{Z}_t \arrow{r}{j} \arrow{d}[swap]{{p_\lambda}|_{\mathcal{Z}_t}} & \mathcal{A}_t^k \arrow{d}{p_{\lambda}}\\ p_{\lambda}(\mathcal{Z}_t) \arrow{r}{j'}& (\mathcal{B}_t^{\lambda})^k . \end{tikzcd}$$ Consider a desingularization of $p_\lambda(\mathcal{Z}_t)$ and the induced map $$\widetilde{j}': \widetilde{p_\lambda(\mathcal{Z}_t)}\to (\mathcal{B}_t^{\lambda})^k.$$ Using the fact that $\widetilde{p}_t$ identifies birationally to $p_{\lambda,t}$, we get the following commutative diagram $$\begin{tikzcd} \text{Pic}^0(\widetilde{\mathcal{Z}}_t) & \text{Pic}^0(\mathcal{A}_t^k) \arrow{l}[swap]{\widetilde{j}^*}\\ \text{Pic}^0(\widetilde{\mathcal{Z}}'_t)\cong \text{Pic}^0(\widetilde{p_\lambda(\mathcal{Z}_t)}) \arrow{u}{\widetilde{p}^*} & \text{Pic}^0({\mathcal{B}_t^{\lambda}}^k) \arrow{l}[swap]{\widetilde{j}'^*} \arrow{u}[swap]{p_{\lambda}^*} . \end{tikzcd}$$ It follows that $$\begin{aligned} \widetilde{p}_*\circ (\widetilde{j}^*\circ {p_{\lambda}}^*)=\widetilde{p}_*\circ (\widetilde{p}^*\circ \widetilde{j}'^*)=(\widetilde{p}_*\circ \widetilde{p}^*)\circ \widetilde{j}'^*=[\deg(\widetilde{p})]\circ \widetilde{j}'.\end{aligned}$$ Since $p_\lambda(\mathcal{\mathcal{Z}}_t)$ is positive dimensional, the morphism $\widetilde{j}'$ is non-zero and so $\widetilde{p}_*\circ \widetilde{j}^*$ is non-zero.\ Hence the kernel of $\widetilde{p}_*\circ \widetilde{j}^*$ is an abelian subscheme of $\mathcal{A}^k$ which is not all of $\mathcal{A}^k$. For very general $t\in G$ the abelian variety $\mathcal{A}_t$ is simple. Therefore, for such a $t$ the abelian subvariety $\ker(\widetilde{p}_*\circ \widetilde{j}^*)_t$ is of the form $(\mathcal{A}_{t})_M^r$, with $M\in M_{k\times r}(\mathbb{Z})$ of rank $r$, and $r\leq k-1$. Choosing $M$ and $r$ such that $$\{t\in G: \ker(\widetilde{p}_*\circ \widetilde{j}^*)_t=(\mathcal{A}_{t})_M^r\}\subset G$$ is dense, and observing that this set is closed, we see that $\ker(\widetilde{p}_*\circ \widetilde{j}^*)_t=(\mathcal{A}_{t})_M^r$ for all $t\in G$.\ Note that, for $t\in \sigma_{\lambda}(S_\lambda)$, we have $$\text{Pic}^0(\ker(p_{\lambda,t}))/\ker (\widetilde{p}_*\circ \widetilde{j}^*)_t\cap\ker(p_{\lambda,t})=\text{Pic}^0(\ker(p_{\lambda,t}))/(\mathcal{A}_{t})_M^r\cap\ker(p_{\lambda,t})\neq 0.$$ Now consider $\lambda\in \Lambda_l$ such that $\sigma_\lambda(S_\lambda)\neq \emptyset$, namely such that some point in $\sigma_{\lambda}(S_\lambda)$ has survived our various restriction to Zariski open subsets, and $B\in \mathcal{B}^\lambda$ such that $\sigma_\lambda(S_{\lambda}(B))\neq \emptyset$. Suppose that there is a curve $C\subset \sigma_\lambda(S_{\lambda}(B))\cong S_\lambda(B)$ such that $p_\lambda(\widetilde{Z}_t)=p_\lambda(\widetilde{Z}_{t'})$ for any $t,t'\in C$, namely such that $C$ is contracted by the morphism from $S_{\lambda}(B)$ to the Chow variety associated to the family $p_{\lambda}(\mathcal{Z}_{S_{\lambda}(B)})\subset B^k_{S_\lambda(B)}$. Now, since $\text{Pic}^0(\widetilde{Z}_t')\cong \text{Pic}^0(\widetilde{p_\lambda(\mathcal{Z}_t)})$ does not depend on $t\in C$, it must contain a variable abelian variety. This provides the desired contradiction and completes the proof of Proposition [\[prop1\]](prop1). Salvaging generic finiteness and a proof of Voisin’s Conjecture =============================================================== In this section we refine the results from the previous section in order to bypass assumption $(*)$ in the inductive application of Proposition [\[prop1\]](prop1). The idea is quite simple: In the last section we saw that we can degenerate to abelian varieties $A$ isogenous to $B\times E$ in such a way that, if we consider the restriction of the projection $A^k\to B^k$ to $Z\subset A^k$, the image of this projection varies with $E$. Here we want to degenerate to abelian varieties isogenous to $D\times F\times E$, where $E$ is an elliptic curve, and consider the restriction of the projections $A^k\to D^k$ and $A^k\to (D\times F)^k$ to $Z\subset A^k$. We can do this in such a way that the images of both of these projections vary with $E$. Hence, if we consider in $(D\times F)^k$ and $D^k$ the union of the image of these projections for every $E$, we get varieties $Z_1\subset (D\times F)^k$ and $Z_2\subset D^k$ of dimension $\dim Z+1$, and the restriction of the projection $(D\times F)^k\to D^k$ to $Z_1$ has image $Z_2$. It follows at once that this restriction is generically finite on its image. We spend this section making this simple idea rigorous and deducing a proof of Theorem [\[2k-2\]](2k-2).\ The families $\mathcal{B}^\lambda/S_\lambda$, $\lambda\in \Lambda_{(g-1)}$ introduced in the last section are families of $(g-1)$-dimensional abelian varieties with some polarization type $\theta^\lambda$. Hence, they give rise to a diagram $$\begin{tikzcd} (\mathcal{B}^\lambda)^k \arrow{r}{\varphi_\lambda} \arrow{d}& \mathcal{A'}^k \arrow{d}\\ S_\lambda \arrow{r}{\psi_\lambda} & S', \end{tikzcd}$$ where $\mathcal{A'}/S'$ is the universal family over the moduli stack of abelian varieties of dimension $(g-1)$ with polarization $\theta^\lambda$. Note that $\mathcal{A'}/S'$ depends of course on $\lambda$ but we suppress $\lambda$ from the notation. We will think of $\mathcal{A}'/S'$ as another locally complete family of abelian varieties. This family comes with its own set $\Lambda_{l'}'$ indexing loci $S_{\eta'}$ along which $\mathcal{A}'_s\sim \mathcal{B'}_s^\eta\times\mathcal{E'}^\eta_s$. Let $\varphi_{\lambda,\mu}$ be the composition of $\varphi_\lambda|_{(\mathcal{B}^\lambda_{S_{\lambda,\mu}})^k}$ with $$(\mathcal{D}^{\lambda,\mu})^k\to (\mathcal{D}^{\lambda,\mu}\times \mathcal{F}^{\lambda,\mu})^k\to (\mathcal{B}^{\lambda}_{S_{\lambda,\mu}})^k,$$ where the last map is the isogeny encoded by $\mu$. We get a diagram $$\begin{tikzcd} (\mathcal{D}^{\lambda,\mu})^k \arrow{r}{\varphi_{\lambda,\mu}} \arrow{d}& (\mathcal{B'}^\eta)^k \arrow{d}\\ S_{\lambda,\mu} \arrow{r} & S'_{\eta}, \end{tikzcd}$$ where $\eta$ is some index in $\Lambda'_{l'}$, and $\psi_{\lambda}(S_{\lambda,\mu})=S_\eta'$. The main result of this section is: \[mainprop\] Suppose that $\mathcal{Z}\subset \mathcal{A}^k$ satisfies $(*)$ and $(**)$ for $l'\geq 2$. Then there exists a $\lambda\in \Lambda_{(g-1)}$ such that $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'$ satisfies $(*)$ for $l'$ and has relative dimension $\dim_S \mathcal{Z}+1$. By Proposition 3.4, there is a $\lambda\in \Lambda_{(g-1)}$ such that $p_{\lambda}(\mathcal{Z}_{t})$ varies with $t\in S_\lambda(B)$, for generic $B$ in the family $\mathcal{B}^\lambda$ (alternatively such that $\dim_{S'} \varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))=\dim_S\mathcal{Z}+1$). The idea is to show that there is a subset $\Lambda_{l',0}^\lambda\subset \Lambda_{l'}^\lambda$ such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda \text{ is dense}$$ and such that $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$, for generic $D,F$ in the families $\mathcal{D}^{\lambda,\mu}, \mathcal{F}^{\lambda,\mu},$ and $\mu\in \Lambda_{l',0}^\lambda$. Indeed, we have a commutative diagram $$\begin{tikzcd} & (\mathcal{B}_{S_{\lambda,\mu}}^{\lambda})^k \ar{rr}{\varphi_\lambda} \ar{dd} \ar{dl}{p_{\lambda,\mu}} & & \mathcal{A'}^k \ar{dd} \ar{dl}{p_{\eta}} \\ ( \mathcal{D}^{\lambda,\mu})^k \ar[crossing over]{rr}{\;\;\;\;\;\;\;\;\;\varphi_{\lambda,\mu}} \ar{dd} & & (\mathcal{B'}^{\eta})^k & \\ & S_{\lambda,\mu} \ar{rr} \ar[dl,equal] & & S'_{\eta} \ar[dl,equal] \\ S_{\lambda,\mu} \ar{rr} && S'_{\eta} \ar[from=uu,crossing over]\;, & \end{tikzcd}$$ where $S'_{\eta}\subset S'$ is the image of $S_{\lambda,\mu}\to S_{\lambda}\to S'$. Consider the restriction of this diagram to $S_{\lambda,\mu}(D,F)$, for $D$ and $F$ in the families $\mathcal{D}^{\lambda,\mu}$ and $\mathcal{F}^{\lambda,\mu}$, namely $$\begin{tikzcd} & (\mathcal{B}_{S_{\lambda,\mu}(D,F)}^{\lambda})^k \ar{rr}{\varphi_\lambda} \ar{dd} \ar{dl}{p_{\lambda,\mu}} & & \mathcal{A'}^k \ar{dd} \ar{dl}{p_{\eta}} \\ ( \mathcal{D}^{\lambda,\mu}_{S_{\lambda,\mu}(D,F)})^k \ar[crossing over]{rr}{\;\;\;\;\;\;\;\;\;\varphi_{\lambda,\mu}} \ar{dd} & & (\mathcal{B'}^{\eta})^k & \\ & S_{\lambda,\mu}(D,F) \ar{rr} \ar[dl,equal] & & S'_{\eta} \ar[dl,equal] \\ S_{\lambda,\mu}(D,F) \ar{rr} && S'_{\eta} \ar[from=uu,crossing over]\; . & \end{tikzcd}$$ Now, if $p_{\lambda,\mu}(\mathcal{Z}_t)$ varies with $t\in S_{\lambda,\mu}(D,F)$, we have $$\dim \varphi_{\lambda}(p_{\lambda}(\mathcal{Z}_{S_{\lambda,\mu}(D,F)}))=\dim \varphi_{\lambda,\mu}(p_{\lambda,\mu}(\mathcal{Z}_{S_{\lambda,\mu}(D,F)}))=\dim_S\mathcal{Z}+1,$$ so that $p_{\eta}$ is generically finite. Hence, if there is a subset $\Lambda_{l',0}^\lambda$ such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda$$ is dense, and $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$ for $\mu\in \Lambda_{l',0}^\lambda$, then $\varphi_\lambda(\mathcal{Z}_{S_{\lambda}})\subset \mathcal{A'}^k$ satisfies condition $(*)$ for $l'$.\ Consider $$R_{st}':=\bigcup_{\lambda,\mu}\{s\in S_{\lambda,\mu} : p_{\lambda,\mu}(\mathcal{Z}_s) \text{ is not stabilized by an abelian subvariety}\}.$$ Following the same argument as in the proof of Proposition [\[prop1\]](prop1), we can assume that $R_{st}'$ is dense in $S$ (this is the analogue of condition $(***)$). Let $\mathcal{U}/G$ and $\mathcal{U'}/G'$ be the universal families over $G:=\mathbb{P}(T)$ and $G':=\text{Gr}(g-l',T)$. Consider the following diagrams analogous to ([\[diaglambda\]](diaglambda)) and ([\[diag\]](diag)): $$\begin{tikzcd} \mathcal{Z}_{S_{\lambda,\mu}} \ar[r,dashed,"g"] \arrow{d}{p_{\lambda}}& \text{Gr}(d,{\mathscr{T}}_{S_{\lambda,\mu}}^k) \ar[d,dashed,"\pi_{\sigma_{\lambda}(S_{\lambda,\mu})}"]\\ p_{\lambda}(\mathcal{Z}_{S_{\lambda,\mu}})\ar[r,dashed,"g"] \arrow{d}{p_{\mu}}& \text{Gr}(d,[{\mathscr{T}}_{S_{\lambda,\mu}}/\mathcal{U}_{\sigma_{\lambda}(S_{\lambda,\mu})}]^k) \ar[d,dashed, "\pi_{\sigma_{\lambda,\mu}(S_{\lambda,\mu})}"]\\ p_{\lambda,\mu}(\mathcal{Z}_{S_{\lambda,\mu}}) \arrow[r,dashed,"g"]& \text{Gr}(d,[{\mathscr{T}}_{S_{\lambda,\mu}}/\mathcal{U'}_{\sigma_{\lambda,\mu}(S_{\lambda,\mu})}]^k), \end{tikzcd}$$ $$\begin{tikzcd}\mathcal{Z}_{G'}\ar[r,dashed, "g"] & \text{Gr}(d,T_{G'}^k) \ar[d,dashed, "\pi"]\\ & \text{Gr}(d, [T_{G'}/\mathcal{U'}]^k).\\ \end{tikzcd}$$ Just as in the proof of Proposition [\[prop1\]](prop1), since $\mathcal{Z}$ satisfies $(*)$ and $(**)$ for $l'$, we can assume that $q_t:=\pi_t\circ g_t$ is generically finite on its image for any $t\in G'$, restricting to a Zariski open in $G'$ if necessary. Note that, for $t\in S_{\lambda,\mu}$, we have a factorization $$\begin{tikzcd} \mathcal{Z}_t \arrow[rrr, dashed,"q"] \arrow{dr}{p_{\lambda}}& & & \text{Gr}(d,[T_{G'}/\mathcal{U}']^k)\\ &p_{\lambda}(\mathcal{Z}_t) \arrow{r}{p_\mu} & p_{\lambda,\mu}(\mathcal{Z}_t) \arrow[ur,dashed,"g"]. & \end{tikzcd}$$ By Proposition [\[prop1\]](prop1), there is a $\lambda$ such that $\dim_{S'} \varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'=\dim_S\mathcal{Z}+1$. One can consider analogues of Lemma [\[cov1\]](cov1) and [\[cov2\]](cov2) and see that there is a partition of $\Lambda_{l'}^{\lambda}$ in finitely many sets according to the isomorphism type of the covering $p_{\mu}$. Hence, since $$\bigcup_{\mu\in \Lambda_{l'}^{\lambda}}S_{\lambda,\mu}\subset S_{\lambda}$$ is dense, there is a subset $\Lambda_{l',0}^\lambda\subset \Lambda_{l'}^\lambda$, such that $$\bigcup_{\mu\in \Lambda_{l',0}^\lambda} S_{\lambda,\mu}\subset S_\lambda$$ is dense, and such that $p_{\lambda,\mu}(\mathcal{Z}_{t})$ varies with $t\in S_{\lambda,\mu}(D,F)$ for generic $D,F$ in the families $\mathcal{B}^{\lambda,\mu}, \mathcal{D}^{\lambda,\mu}$, and $\mu\in \Lambda_{l',0}^\lambda$. Suppose that a very general abelian variety of dimension $g$ has a positive dimensional $\text{CCS}_k$. Then, for a very general abelian variety $A$ of dimension $(g-l)$ there is an $(l+1)$-dimensional subvariety of $A^k$ foliated by positive dimensional $\text{CCS}_k$. Under the assumption of this corollary we get $\mathcal{Z}\subset \mathcal{A}^k/S$, a $\text{CCS}_k/S$, for $\mathcal{A}\to S$ a locally complete family of $g$-dimensional abelian varieties. Apply the previous proposition inductively. By Lemma 3.3 the condition $(**)$ follows from $(*)$. \[conjpf\] Conjecture 1.3 holds: a very general abelian variety of dimension $\geq 2k-1$ has no positive dimensional $\text{CCS}_k$. Note that any $\mathcal{Z}\subset \mathcal{A}^k$ of relative dimension $d$ satisfies $(*)$ for $l\geq d$ since $$\bigcup_{\lambda\in \Lambda_l}\sigma_\lambda(S_\lambda)\subset G$$ is dense. Indeed, if $V$ is a vector space and $W\subset V^k$ has dimension $d<\dim V$, then the restriction of the projection $V^k\to (V/H)^k$ to $W$ is an isomorphism onto its image for a generic $H\in \text{Gr}(\dim V-d,V)$. In particular if $\mathcal{Z}\subset \mathcal{A}^k$ has relative dimension $1$ then it satisfies $(*)$ for any $1\leq l\leq g-1$. Hence, if a very general abelian variety of dimension $2k-1$ has a positive dimensional $\text{CCS}_k$, then a very general abelian surface $B$ will be such that $B^{k,\, 0}$ contains a $(2k-2)$-dimensional subvariety foliated by positive dimensional $\text{CCS}_k$. This does not hold for any abelian surface, let alone generically, by Corollary [\[absurfbound\]](absurfbound). \[2k-2\] For $k\geq 3$, a very general abelian variety of dimension at least $2k-2$ has no positive dimensional orbits of degree $k$, i.e. $\mathscr{G}(k)\leq 2k-2$. Let $\mathcal{Z}\subset \mathcal{A}^{k,\, 0}$ be a one-dimensional normalized $\text{CCS}_k$, where $\dim_S\mathcal{A}=2k-2$. By the previous corollary, there is a $\lambda\in \Lambda_2$ such that $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))/S'$ has relative dimension $2k-3$. This was obtained by successive degenerations and projections. But the morphism from $S_\lambda(B)$ to an appropriate Chow variety of $B^{k,\, 0}$ given by $$s\mapsto [\varphi_\lambda(p_\lambda(\mathcal{Z}_s))]$$ is a $\binom{2k-3}{2}$-parameter family of $\text{CCS}_k$ on $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))$. Hence, $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda}))$ must be foliated by $\text{CCS}_k$ of dimension at least $2$. This contradicts Corollary [\[absurfbound\]](absurfbound). \[gonalitybound\] For $k\geq 3$, a very general abelian variety of dimension $\geq 2k-2$ has gonality at least $k+1$. In particular Conjecture [\[Vweakconj\]](Vweakconj) holds. \[no2dim\] A very general abelian variety of dimension $\geq 2k-4$ does not have a $2$-dimensional $\text{CCS}_k$ for $k\geq 4$. Suppose $\mathcal{A}/S$ is a locally complete family of $(2k-4)$-dimensional abelian varieties with some polarization $\theta$, and that $\mathcal{Z}\subset \mathcal{A}^k$ is a $2$-dimensional $\text{CCS}_k/S$. Using the same argument as in the proof of Corollary [\[conjpf\]](conjpf), we see that $\mathcal{Z}$ satisfies $(*)$, and thus $(**)$. We now follow the proof of Theorem [\[2k-2\]](2k-2). \[k+1\] A very general abelian variety $A$ of dimension at least $2k+2-l$ does not have a positive dimensional orbit of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$, i.e. $\mathscr{G}_{l}(k)\leq 2k+2-l.$\ Moreover, if $A$ is a very general abelian variety of dimension at least $k+1$ the orbit $|k\{0_A\}|$ is countable, i.e. $\mathscr{G}_{k}(k)\leq k+1.$ By the results of [@V], it suffices to show that a very general abelian variety of dimension $2k+2-l$ has no positive dimensional orbits of the form $|\sum_{i=1}^{k-l}\{a_i\}+l\{0_A\}|$. If this were not the case, we could find $\mathcal{Z}\subset \mathcal{A}^k$, a one-dimensional $\text{CCS}_k$, where $\mathcal{A}/S$ is a locally complete family of $(2k+2-l)$-dimensional abelian varieties, and $$\{0_{\mathcal{A}_s}\}\times\ldots\times \{0_{\mathcal{A}_s}\}\times \mathcal{A}_s^{k-l}\cap \mathcal{Z}_s\neq \emptyset$$ for every $s\in S$. By Proposition [\[mainprop\]](mainprop) there is a $\lambda\in \Lambda_2$ such that $\varphi_\lambda( p_{\lambda}(\mathcal{Z}_{S_\lambda}))$ has relative dimension $2k+1-l$. Given a generic $B$ in the family $\mathcal{B}^\lambda$ and $\underline{b}=(a_1,\ldots, a_{k-l}) \in B^{k-l}$, consider $$S_{\lambda}(B,\underline{b}):=\{s\in S_\lambda(B): \underline{b}\in \phi_\lambda(p_\lambda(\mathcal{Z}_s))\}.$$ Clearly $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda(B,\underline{b})}))$ is a $\text{CCS}_k$. In particular, $\varphi_\lambda(p_\lambda(\mathcal{Z}_{S_\lambda(B)}))$ is foliated by $\text{CCS}_k$ of codimension at most $2(k-l)$. This contradicts Corollary [\[absurfbound\]](absurfbound). A similar argument shows $\mathscr{G}_k(k)\leq k+1$. Applications to Other Measures of Irrationality =============================================== We have seen how the minimal degree of a positive dimensional orbit gives a lower bound on the gonality of a smooth projective variety and used this to provide a new lower bound on the gonality of very general abelian varieties. In this section we show how one can use results about the maximal dimension of $\text{CCS}_k$ in order to give lower bounds on other measures of irrationality for very general abelian varieties. We finish by discussing another conjecture of Voisin from [@V] and its implication for the gonality of very general abelian varieties.\ Recall the definitions of some of the measures of irrationality of irreducible $n$-dimensional projective varieties: $$\begin{aligned} \text{irr}(X)&:=\min \left\{\delta>0: \exists \text{ degree } \delta\text{ rational covering } X\to \mathbb{P}^n\right\}\\ \text{gon}(X)&:=\min \left\{c>0: \exists \text{ a non-constant morphism } C\to X, \text{ where } C \text{ has gonality c}\right\}.\end{aligned}$$ Additionally, we will consider the following measure of irrationality which interpolates between the *degree of irrationality* $\text{irr}(X)=\text{irr}_n(X)$ and the *gonality* $\text{gon}(X)=\text{irr}_1(X)$. $$\text{irr}_d(X):=\min \left\{\delta: \exists\text{ a } d\text{-dimensional irreducible subvariety } Z\subset X \text{ with }irr(Z)=\delta\right\}.$$ The methods of the previous section can be applied to get: If $A$ is a very general abelian variety of dimension $\geq 2k-4$ and $k\geq 4$, then $$\text{\textup{irr}}_2(A)\geq k+1.$$ A surface with degree of irrationality $k$ in a smooth projective variety $X$ provides a $2$-dimensional $\text{CCS}_{k}$. The result then follows from Theorem [\[no2dim\]](no2dim). Similarly, we can use bounds on the dimension of a $\text{CCS}_k$ to obtain bounds on the degree of irrationality of abelian varieties. To our knowledge, the best bound currently in the literature is the Sommese bound $\text{irr}(A)\geq \dim A+1$ (see [@BE] Section 4), for any abelian variety $A$. It is an interesting fact that this bound follows easily from Voisin’s Theorem [\[k-1\]](k-1). Indeed, a dominant morphism from $A$ to $\mathbb{P}^{\dim A}$ of degree at most $\dim A$ would provide a $(\dim A)$-dimensional $\text{CCS}_{\dim A}$. Note that Yoshihara and Tokunaga-Yoshihara ([@Y],[@TY]) provide examples of abelian surfaces $A$ with $\text{irr}(A)=3$, so that the Sommese bound is tight for $\dim A=2$. In fact, we do not know of a single example of an abelian surface $A$ with $\dim A>3$.\ Ou results allow us to show that Sommese’s bound is not optimal, at least for very general abelian varieties. \[no(k-1)\] Orbits of degree $k$ on a very general abelian variety of dimension at least $k-1$ have dimension at most $k-2$, for $k\geq 4$. \[sommese\] If $A$ is a very general abelian variety of dimension $g\geq 3$, then $$\text{\textup{irr}}(A)\geq g+2.$$ Suppose that we have $\mathcal{A}/S$, a locally complete family of $(k-1)$-dimensional abelian varieties, and $\mathcal{Z}\subset \mathcal{A}^{k,\, 0}/S$, a $(k-1)$-dimensional $\text{CCS}_k/S$. We claim that $\mathcal{Z}$ satisfies $(*)$ for $l=k-2$. Assuming this, for appropriate $\lambda\in \Lambda_{(k-2)}$ and $B$ in the family $\mathcal{B}^\lambda$, the subvariety $\phi_\lambda(p_{\lambda}(\mathcal{Z}_{S_\lambda(B)}))\subset B^{k,\, 0}$ is $k$-dimensional and foliated by $(k-1)$-dimensional $\text{CCS}_{k}$. This contradicts Corollary [\[1dimfam\]](1dimfam).\ To show that $\mathcal{Z}$ satisfies $(*)$ for $l=k-2$, we will need the following easy lemma which we give without proof: Given $V$ is a $g$-dimensional vector space, and $W\subset V^r$ a $g$-dimensional subspace such that the restriction of $\pi_L: V^r\to (V/L)^r$ to $W$ is not an isomorphism for any $L\in \mathbb{P}(V)$, then $W=V^1_M$ for some $M\in \mathbb{P}({\mathbb{C}}^r)$. Hence, if $\mathcal{Z}$ fails to satisfy $(*)$ for $l=k-2$, for any $s\in S$ and $z\in (\mathcal{Z}_s)_{sm}$ the tangent space $T_{\mathcal{Z}_s,z}$ must be of the form $(T_{\mathcal{A}_s})^1_M\subset T_{\mathcal{A}_s}^{k,\, 0}$ for $M\in \mathbb{P}({\mathbb{C}}^{k,\, 0})$. Here, given a vector space $V$ we use the notation $V^{r\, 0}$ for the kernel of the summation map $V^r\to V$. It follows that for each $s\in S$ we get a morphism $(\mathcal{Z}_s)_{sm}\to \mathbb{P}({\mathbb{C}}^{k,\, 0})$. For very general $s$, the abelian variety $\mathcal{A}_s$ is simple and so $\mathcal{Z}_s$ cannot be stabilized by an abelian subvariety of $\mathcal{A}_s^k$. Thus, the Gauss map of $\mathcal{Z}_s$ is generically finite on its image and so is the morphism $(\mathcal{Z}_s)_{sm}\to \mathbb{P}({\mathbb{C}}^{k,\, 0})$. It follows that the image of this morphism must contain an open in $\mathbb{P}({\mathbb{C}}^{k,\, 0})$. Any open in $\mathbb{P}({\mathbb{C}}^{k,\, 0})$ contains real points and if $M\in \mathbb{P}(\mathbb{R}^{k,\, 0})$, then $(T_{\mathcal{A}_s})^1_M$ cannot be totally isotropic for $\omega_k$ for any non-zero $\omega\in H^0(\mathcal{A}_s,\Omega^2)$. This provides the desired contradiction. The previous corollary motivates the following: Exhibit an abelian threefold $A$ with $d_r(A)\geq 4$. We have reasons to believe that Corollary [\[sommese\]](sommese) is also not optimal. Indeed, the key obstacle to proving a stronger lower bound is the need for $(*)$ to be satisfied. A careful study of the Gauss map of cycles on very general abelian variety is likely to provide stronger results.\ Similarly, we believe that Theorem [\[2k-2\]](2k-2) can be improved. In fact, though Conjecture [\[Vweakconj\]](Vweakconj) is the main conjecture of [@V], it is not the most ambitious. Voisin proposes to attack Conjecture [\[Vweakconj\]](Vweakconj) by studying what she calls the locus $Z_A$ of positive dimensional normalized orbits of degree $k$ $$Z_A:=\left\{a_1\in A: \exists\; a_2,\ldots, a_{k-1}: \dim\left|\{a_1\}+\ldots+\{a_{k-1}\}+\{-\sum_{i=1}^k a_i\}\right|>0\right\}.$$ In particular she suggests to deduce Conjecture [\[Vweakconj\]](Vweakconj) from the following conjecture: \[norm\] If $A$ is a very general abelian variety $$\dim Z_A\leq k-1.$$ Voisin shows that this conjecture implies Conjecture [\[Vweakconj\]](Vweakconj) but it in fact implies the following stronger conjecture: A very general abelian variety of dimension at least $k+1$ does not have a positive dimensional orbit. i.e. $\mathscr{G}(k)\leq k+1$.\[Vstrongconj\] \[trick\]The previous conjecture follows from Conjecture [\[norm\]](norm). Indeed, if a very general abelian variety of dimension $k$ has a positive dimensional orbit $$|\{a_1\}+\ldots+\{a_{k-1}\}|$$ of degree $k-1$, then for any $a\in A$ the orbit $$|\{(k-1)a\}+\{a_1-a\}+\ldots+\{a_{k-1}-a\}| \label{trick}$$ is positive dimensional. This was noticed by Voisin in Example 5.3 of [@V]. It follows that $Z_A=A$ and so $\dim Z_A=k>k-1$. As mentioned above, the results of Pirola and Alzati-Pirola give $\mathscr{G}(2)\leq 3$ and $\mathscr{G}(3)\leq 4$. Our main theorem provides us with the bound $\mathscr{G}(4)\leq 6$. An interesting question is to determine if $\mathscr{G}(4)\leq 5$. This would provide additional evidence in favor of Conjecture [\[Vstrongconj\]](Vstrongconj). Support of zero-cycles on abelian varieties =========================================== In [@V] the author shows the following surprising proposition: \[propV\] Consider an abelian variety $A$ and an effective zero-cycle $\sum_{i=1}^k \{x_i\}$ on A such that $$\sum_{i=1}^k \{x_i\}= k\{0_A\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$(\{x_i\}-\{0_A\})^{*k}=0 \in CH_0(A),$$ where $*$ denotes the Pontryagin product. Voisin defines a subset $A_k:=\{a\in A: (\{a\}-\{0\})^{*k}=0\}\subset A$ and shows that $\dim A_k\leq k-1$. Given a smooth projective variety $X$ and a zero-cycle of the form $z=\sum_{i=1}^k\{x_i\}\in Z_0(X)$ the support of $z$ is $$\text{supp}(z)=\{x_i: i=1,\ldots, k\}\subset X.$$ Similarly we will call the $k$-support of $z$ the following subset of $X$ $$\text{supp}_k(z)=\bigcup_{z'=\sum_{i=1}^k\{x_i'\}: z'\sim z}\text{supp}(z').$$ The previous proposition can then be rephrased as: $\text{supp}_k(k\{0_A\})\subset A_k$. Here we present a generalization of this result. Given $\underline{x}\in X^k$ we let $$A_{k,\underline{x}}:=\{a\in A: (\{a\}-\{x_1\})*\cdots*(\{a\}-\{x_k\})=0\in CH_0(A)\}.$$ One shows easily, using the same argument as Voisin, that $\dim A_{k,\underline{x}}\leq k-1$. \[propA\] Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$ such that $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A).$$ Then for $i=1,\ldots, k$ $$\prod_{j=1}^k(\{x_i\}-\{y_j\})=0\in CH_0(A),$$ where the product is the Pontryagin product. Upon presenting this result to Nori he recognized it as a more effective reformulation of results of his from around 2005. They had been obtained in an attempt to understand work of Colombo-Van Geeman [@CVG] but left unpublished for lack of an application. We present here Nori’s proof as it is more elegant than our original proof. This proof was also suggested to Voisin by Beauville in the context of Proposition [\[propV\]](propV).\ Let $X$ be a smooth projective variety and consider the graded algebra $$\bigoplus_{n=1}^\infty CH_0(X^n).$$ The multiplication is given by extending by linearity $$\begin{aligned} X^{m}\times X^n&\to X^{m+n}\\ ((x_1,\ldots, x_m),(x_1',\ldots, x_n'))&\mapsto (x_1,\ldots, x_m,x_1',\ldots, x_n')\end{aligned}$$ to a product $$Z_0(X^m)\times Z_0(X^n)\to Z_0(X^{n+m}).$$ It is easy to see that the resulting product descends to rational equivalence on the components. Let $$R:=\bigoplus_{n=1}^\infty CH_0(X^n)/(ab-ba)=\bigoplus_{i=1}^\infty R_n$$ be the abelianization of this algebra. Recall that by $CH_0(X^n)$ we mean the Chow group of zero-cycles with rational coefficients. \[Norilemma\] If $z=\sum_{i=1}^k\{x_i\}\in Z_0(X)$ and $y\in \text{supp}_k(z)$, then $$(\{y\}-\{x_1\})(\{y\}-\{x_2\})\cdots(\{y\}-\{x_k\})=0\in R,$$ where the product is taken in $R$ and we consider $\{y\}-\{x_i\}$ as elements of $R_1\subset R$. Since $y\in \text{supp}_k(z)$, there is $\underline{y}=(y_1=y,y_2,\ldots, y_k)\in X^k$ such that $\sum_{i=1}^k \{y_i\}=\sum_{i=1}^k \{x_i\}$. Consider the diagonal embeddings $$\Delta_l: X\to X^l.$$ These give linear maps $${\Delta_l}_*: R_1\to R_l$$ such that $${\Delta_l}_*\left(\sum_{i=1}^k \{y_i\}\right)=\sum_{i=1}^k \{y_i\}^l\in R_l.$$ Since $$\sum_{i=1}^k \{y_i\}=\sum_{i=1}^k \{x_i\}$$ we get $$p_l(\underline{y})=\sum_{i=1}^k \{y_i\}^l=\sum_{i=1}^k \{x_i\}^l=p_l(\{\underline{x}\})\in R_l,$$ where $p_l$ is the $l^{th}$ Newton polynomial and $\{\underline{x}\}=(\{x_1\},\ldots, \{x_{k}\})$. On the other hand we have $$(\{y\}-\{x_1\})(\{y\}-\{x_2\})\ldots(\{y\}-\{x_k\})=\{y\}^k-e_1(\underline{x})\{y\}^{k-1}+\ldots+(-1)^ke_k(\underline{x})\in R_k,$$ where $e_l$ is the $l^{\text{th}}$ elementary symmetric polynomial. Since the elementary symmetric polynomials can be written as polynomials in the Newton polynomials and since $p_l(\underline{y})=p_l(\underline{x})$ for all $l\in \mathbb{N}$, we get $$\prod_{i=1}^k(\{y\}-\{x_i\})=\sum_{i=0}^k(-1)^i\{y\}^{k-i}e_i(\underline{x})=\sum_{i=0}^k(-1)^i\{y\}^{k-i}e_i(\underline{y})=\prod_{i=1}^k(\{y\}-\{y_i\})=0\in R_k.$$ If $X=A$ is an abelian variety we have a summation morphism $A^l\to A$ inducing maps $$CH_0(A^l)\to CH_0(A),$$ and so a map $$\sigma: R\to CH_0(A)$$ such that $$\sigma\left(\prod_{i=1}^k(\{y\}-\{x_i\})\right)=(\{y\}-\{x_1\})*\ldots* (\{y\}-\{x_k\})\in CH_0(A).$$ Lemma [\[Norilemma\]](Norilemma) in fact has many more interesting corollaries. Consider $p(t_1,\ldots, t_k)\in {\mathbb{C}}[t_1,\ldots, t_k]$ and the $S_k$ action on ${\mathbb{C}}[t_1,\ldots, t_k]$ given by permutation of the variables. Let $H_p\subset S_k$ be the subgroup stabilizing $p$. Consider an abelian variety $A$ and effective zero-cycles $\sum_{i=1}^k\{x_i\}$, $\sum_{i=1}^k \{y_i\}$ on $A$. Then $$\sum_{i=1}^k \{x_i\}= \sum_{i=1}^k \{y_i\}\in CH_0(A)$$ if and only if $$\prod_{\sigma\in S_k/H_p}\big(p(\{y_1\},\ldots, \{y_k\})-(\sigma\cdot p)(\{x_1\},\ldots, \{x_k\})\big)=0\in CH_0(A)$$ for every $p\in {\mathbb{C}}[t_1,\ldots, t_k]$. Here the product is the Pontryagin product. The if direction follows trivially from considering $p(t_1,\ldots, t_k)=t_1+\ldots+t_k$. The proof of Lemma [\[Norilemma\]](Norilemma) completes the argument. The special case $p=t_1$ is Proposition [\[propA\]](propA). Another corollary of Lemma [\[Norilemma\]](Norilemma) is the following: Given an effective zero-cycle $z=\sum_{i=1}^k\{x_i\}$ on an abelian variety $A$, and $y_1,\ldots, y_{k+1}\in \text{supp}_k(z)$, the following identity is satisfied $$\prod_{i<j}(\{y_i\}-\{y_j\})=0\in CH_0(A).$$ Let $e_l$ be the $l^{\text{th}}$ elementary symmetric polynomial. By Lemma [\[Norilemma\]](Norilemma) we have $$\{y_i\}^{k}-e_1(\{\underline{x}\})\{y_i\}^{k-1}+\ldots +(-1)^ke_k(\{\underline{x}\})=0\in R_k$$ for $i=1,\ldots, k+1$, where $\{\underline{x}\}=(\{x_1\},\ldots,\{x_k\})$. This gives a non-trivial linear relation between the rows of the Vandermonde matrix $(\{y_{i}\}^{j-1})_{1\leq i,j\leq k+1}$. It follows that the Vandermonde determinant vanishes. Using the morphism $\sigma: R\to CH_0(A)$ from the proof of [\[propA\]](propA) finishes the argument. [9]{} A. Alzati, G. P. Pirola. Rational orbits on three-symmetric products of abelian varieties, Trans. Amer. Math. Soc. 337 (1993), no. 2, 965-980. F. Bastianelli, P. De Poi, L. Ein, R. Lazarsfeld, B. Ullery. Measures of irrationality for hypersurfaces of large degree, Compos. Math. 153 (2017), no. 11, 2368-2393. I. Bernstein, A. L. Edmonds, The degree and branch set of a branched covering, Invent. Math. 45, no. 3, (1978), 213-220. E. Colombo, B. Van Geeman. Note on curves in a Jacobian, Compos. Math. 88 (1993), no. 3, 333-353. P. Griffiths, J. Harris. Algebraic geometry and local differential geometry, Ann. Sci. Éc. Norm. Supér. 12 (1979), no. 3, 355-452. H.-Y. Lin. On the chow group of zero-cycles of a generalized Kummer variety, Advances in Mathematics 298 (2016), 448-472. D. Huybrechts. Curves and cycles on K3 surfaces, Algebraic Geometry, 1 (2014), 69-106. D. Mumford. Rational equivalence of zero cycles on surfaces, J. Math. Kyoto Univ. 9 (1969), 195, 204. G. P. Pirola. Curves on generic Kummer varieties, Duke Math. J. 59 (1989), 701-708. A. A. Roĭtman. $\Gamma$-equivalence of zero-dimensional cycles (in Russian), Math. Sb. (N.S.) 86 (128) (1971), 557-570. \[Translation: Math. USSR-Sb., 15 (1971), 555-567.\] A. A. Roĭtman. Rational equivalence of zero-cycles (in Russian), Math. (N.S.) Sb. 89 (131) (1972), 569-585. \[Translation: Math. USSR-Sb., 18 (1974), 571-588.\] H. Tokunaga, H. Yoshihara. Degree of irrationality of abelian surfaces, J. Algebra 174 (1995), 1111-1121. C. Vial. On the motive of some hyperKähler varieties, J. Reine Angew. Math. 725 (2017), 235-247. C. Voisin. Chow ring and the gonality of general abelian varieties, Preprint ArXiv: 1802.07153v1 (2018). C. Voisin. Remarks and questions on coisotropic subvarieties and 0-cycles of hyper-Kähler varieties, K3 Surfaces and Their Moduli, Proceedings of the Schiermonikoog conference 2014, C. Faber, G. Farkas, G. van der Geer, Editors, Progress in Math 315, Birkhäuser (2016), 365-399. H. Yoshihara. Degree of irrationality of a product of two elliptic curves, Proc. Am. Math. Soc. 124, no. 5, (1996), 1371-1375. [^1]: For simplicity we will call both fibers of $X^k\to CH_0(X)$ and $\text{Sym}^k X\to CH_0(X)$ orbits of degree $k$ for rational equivalence. [^2]: Note that $${\mathbb{Z}}_{\geq \mathscr{G}(k)}=\begin{cases}\begin{rcases}g\in \mathbb{Z}_{>0}: \text{a very general abelian variety of dimension } g\\ \text{ does not have a positive dimensional orbit of degree } k\end{rcases}\end{cases}.$$ To see this, observe that if a very general abelian variety $A$ of dimension $g$ has a positive dimensional orbit $Z\subset A^k$ of degree $k$, we can degenerate $A$ to an abelian variety isogenous to a product $B\times E$ in such a way that the restriction of the projection $p: (B\times E)^k\to B^k$ to $Z$ has a positive dimensional image. [^3]: Consider abelian varieties of dimension $g$ isogenous to $B\times E$, where $B$ is a $(g-1)$-dimensional abelian variety and $E$ is an elliptic curve. [^4]: See Definition [\[CCS\]](CCS).

ArXiv

--- abstract: 'The electronic spin of the nitrogen vacancy (NV) center in diamond forms an atomically sized, highly sensitive sensor for magnetic fields. To harness the full potential of individual NV centers for sensing with high sensitivity and nanoscale spatial resolution, NV centers have to be incorporated into scanning probe structures enabling controlled scanning in close proximity to the sample surface. Here, we present an optimized procedure to fabricate single-crystal, all-diamond scanning probes starting from commercially available diamond and show a highly efficient and robust approach for integrating these devices in a generic atomic force microscope. Our scanning probes consisting of a scanning nanopillar (200 nm diameter, $1-2\,\mu$m length) on a thin ($< 1\mu$m) cantilever structure, enable efficient light extraction from diamond in combination with a high magnetic field sensitivity ($\mathrm{\eta_{AC}}\approx50\pm20\,\mathrm{nT}/\sqrt{\mathrm{Hz}}$). As a first application of our scanning probes, we image the magnetic stray field of a single Ni nanorod. We show that this stray field can be approximated by a single dipole and estimate the NV-to-sample distance to a few tens of nanometer, which sets the achievable resolution of our scanning probes.' author: - Patrick Appel - Elke Neu - Marc Ganzhorn - Arne Barfuss - Marietta Batzer - Micha Gratz - Andreas Tschöpe - Patrick Maletinsky title: Fabrication of all diamond scanning probes for nanoscale magnetometry --- Introduction \[sec:Int\] ======================== The negatively charged nitrogen vacancy center (NV center) in diamond forms a highly promising sensor: On the one hand, its unique combination of long spin coherence times and efficient optical spin readout enables the detection of magnetic [@Maze2008] and electric fields [@Dolde2011] as well as local temperature.[@Toyli2013a; @Acosta2010b] On the other hand, the NV center is a highly photostable single photon source and therefore an ideal emitter for scanning near field [@Tisler2013a] and single photon microscopy.[@Sekatskii1996] Moreover, all properties relevant for sensing are sustained from cryogenic temperatures [@Thiel2015; @Pelliccione2014] up to $550\,$K,[@Toyli2012] rendering NV centers highly promising not only for applications in material sciences and physics but also for applications in the life sciences.[@LeSage2013] As a point defect in the diamond lattice, the NV center can be considered as an ’artificial atom’ with sub-nanometer size. As such, it promises not only highest sensitivity and versatility but in principle also unprecedented nanoscale spatial resolution. Triggered by this multitude of possible applications, various approaches to bring a scanable NV center in close proximity to a sample were recently developed. The first experiments in scanning NV magnetometry employed nanodiamonds (NDs) grafted to atomic force microscope (AFM) tips.[@Balasubramanian2008; @Rondin2012; @Rondin2014; @Tetienne2014] However, NVs in NDs suffer from short coherence times limiting their sensitivity as a magnetic sensor. Secondly, efficient light collection from NDs on scanning probe tips is difficult and limits the resulting sensitivities. Lastly, it has proven challenging to ensure close NV-to-sample separations in this approach. Most published work reported on NDs scanning within $\gtrsim100~$nm from the sample surface, limiting the spatial resolution of the scanning probe imaging. Additionally, the emission of NV centers in single digit NDs is typically unstable without further treatment.[@Bradac2010] Motivated by these drawbacks, a novel approach using all-diamond, single crystalline AFM tips has recently been demonstrated.[@Maletinsky2012] This approach relies on fabricating scanning probes with the NV center placed close to the apex of a scanning diamond nanopillar. Beside close proximity of the NV center to the sample, the pillar’s light guiding properties enhance collection efficiency for the NV fluorescence and the devices can be sculpted out of high purity diamond, which enables long coherence times. Thus, color centers with optimal properties (regarding photo-stability and spin-coherence) in high purity material and efficient light collection can be used as sensors. In this paper, we describe an optimized procedure to fabricate such single-crystal, all-diamond scanning probes. In particular, we present in detail the nanofabrication of diamond nanopillars for scanning probe microscopy and describe a highly efficient and robust approach for integrating these devices in an atomic force microscope (AFM). We discuss the magnetometry performance of the probes and demonstrate high resolution imaging of the stray field of single magnetic Ni nanorods using the all-diamond scanning probes. Fabrication of all diamond scanning probes ========================================== The fabrication procedure that we describe here consists of 6 steps: We start with commercially available, high purity diamond plates ($50~\mathrm{\mu}$m thick, Section \[sec:initialstep\]) in which we create shallow NV centers using ion implantation (Section \[sec:NVcreation\]). Our all diamond scanning probes consist of a cylindrical nanopillar ($200~$nm diameter, $1.5~\mathrm{\mu}$m height) on a $<1~\mathrm{\mu}$m thick cantilever. Thus, it is essential to thin down the commercially available plates to a suitable thickness (Section \[sec:deepetch\]). The thinned membranes are subjected to two consecutive lithography and plasma etching steps to form the pillars and the cantilever (Section \[struct\_scan\]). In the subsequent step, we identify the scanning probes that contain single NV centers (Section \[sec:precharacterization\]). Finally, we mount the selected scanning probes to a tuning fork based AFM head (Section \[sec:transfer\]). Diamond material and initial sample preparation \[sec:initialstep\] ------------------------------------------------------------------- Our nano-fabrication procedure for the all-diamond scanning probe devices is based on commercially available, high purity, synthetic diamond grown by chemical vapor deposition (Element Six, electronic grade, \[N\]${}^s$$<$$5~$ppb, B$<$$1~$ppb).[@e6url] The $500~\mu$m thick diamonds are processed into 30-$100~\mu$m thick diamond plates by laser cutting and subsequent polishing (Delaware Diamond Knives, USA or Almax Easy Lab, Belgium [@Almaxurl]). While our process can be applied to a large range of thicknesses, we found $50~\mu$m thick plates to form the best compromise between mechanical stability, ease of handling and reasonable processing times (see Section \[sec:deepetch\]). The surface roughness of the starting diamond plates is typically $0.7~$nm, as evidenced by AFM imaging \[Fig. \[Fig:membranes\](d)\], and the plates have a wedge of typically several micrometers across the lateral sample dimensions of $4~$mm. We note that such a high quality polish is mandatory for the subsequent processing steps. Initially, we clean the plates using a boiling tri-acid mixture (1:1:1 sulfuric acid, perchloric acid, nitric acid, boil acid mixture until reverts to clear appearance) to remove any surface contamination which might have resulted from polishing.[@Hird2004; @Schuelke2013] Lastly, the sample is cleaned in solvents (deionized water, acetone, ethanol, isopropanol) to remove possible contaminants present in the acids. Mechanical polishing of diamond is known to introduce crystal damage below the polished surface into a depth of up to several micrometers.[@Volpe2009; @Friel2009; @Naamoun2012] The lattice in this highly damaged layer can be strongly deformed and defective: cathodoluminescence (CL) measurements indicate a high concentration of defects[@Volpe2009] and etching away 3-$4~\mu$m of diamond almost recovers the CL of pristine diamond. NVs in this damaged layer might therefore suffer from a unstable charge state or spin decoherence due to trapped paramagnetic defects or fluctuating charges. Furthermore, the highly strained layer might render the NV spins insensitive to magnetic fields in first order and therefore useless for magnetometry.[@Rondin2014] To circumvent these potential obstacles, we remove $\approx 3~\mu$m or more of the damaged surface layer using inductively coupled reactive ion etching (ICP-RIE) as described in the following. For all etch steps, the diamond plates are mounted on Si chips ($1~$cm squares) as carriers; we perform plasma etching using a Sentech SI 500 ICP-RIE apparatus. We initiate the etching by removing roughly the first micrometer of diamond using an ArCl$_2$ plasma step. This plasma chemistry has been reported to remove damaged diamond layers without roughening the surface.[@Friel2009] Note that even slight surface roughening would be detrimental for all subsequent processes. We summarize the plasma parameters used as well as the resulting etch rates \[as determined by an in-situ laser interferometer (SenTech SLI 670)\] in table \[tab:plasma\]. While enabling optimal etching of defective diamond, the ArCl$_2$ plasma also strongly erodes Si carrier wafers routinely used in ICP-RIE processes. The resulting high level of Si contamination introduces a roughening of the diamond surface. To avoid this, we employ a ceramics based carrier system which we find to be more resistant to etching in the ArCl$_2$ plasma consequently avoiding contamination. Diamond surfaces prepared by ArCl$_2$ plasma have been suspected to contain Cl$_2$,[@Tao2014] which might deteriorate the NV spin properties. As a consequence, we terminate etching using an O$_2$ plasma to remove any such potential Cl$_2$ contamination (see table \[tab:plasma\]). ---------- ----------- --------------- ----------------- ---------- ------------ plasma ICP power RF power/bias flux Pressure Etch rate \[W\] \[W\]/\[V\] \[sccm\] \[Pa\] \[nm/min\] ArCl$_2$ 400 100/220 Ar 25 Cl$_2$ 40 1 60 O$_2$ 700 50/120 O$_2$ 60 1.3 150 ArO$_2$ 500 200/120 Ar 50 O$_2$ 50 0.5 150 ---------- ----------- --------------- ----------------- ---------- ------------ : \[tab:plasma\]Plasma parameters for the nano-fabrication procedure. Note that the ArO$_2$ plasma is used to etch the nanopillar structures, while the other plasma types are used for the ’deep etches’ to remove polishing damage and form the thin membrane. The nanopillar etching is carried out using a ($6~$Inch) silicon carrier inside the reactor, while all other etches are performed using a ceramics carrier (96% Al$_2$O$_3$) to avoid silicon contamination. The plasma bias voltage was stable within roughly 10% for runs performed within a time-span of several weeks. Creation of NV color centers \[sec:NVcreation\] ----------------------------------------------- To realize high resolution imaging, it is mandatory to achieve close proximity between NV spin and sample, which implies the creation of NV centers close to the diamond surface. To create such a shallow layer of NV centers, we implant the etched diamond surface with ${}^{14}$N ions at an energy of 6 keV and a dose of $3\times 10^{11}~$cm$^{-2}$ (Ion beam services, France). The estimated resulting stopping depth of the ${}^{14}$N ions in diamond is $9\pm4$ nm.[@SRIM] We anneal the sample in vacuum (chamber base pressure: 3-$4\times10^{-7}$ mbar) partly following the recipe from Ref. . The heating device is a boron nitride plate, directly, electrically heated via buried graphite strips (Tectra, Boralectric HTR-1001). The temperature of the oven is calibrated using a comparison between pyrometer measurements and a thermocouple (tungsten/rhenium) inserted into a bore hole in the heater plate. We use the following sequence of annealing steps: ramp in $1~$h from room temperature to 400$^\circ$C, hold $4~$h at 400$^\circ$C, ramp in $1~$h to 800$^\circ$C, hold at 800$^\circ$C for $2~$h, cool down. We also investigated the effect of a high temperature annealing step at 1200$^\circ$C (ramp in $1~$h 800$^\circ$C to 1200$^\circ$C, hold at 1200$^\circ$C for $2~$h) according to Ref. . However, we did not find any significant effect on the NV yield or the NV spin coherence properties. With the previously described procedure, we create a layer of NV centers with a density of $2.6\times 10^9\, \mathrm{cm}^{-2}$ (see Section \[sec:precharacterization\]). From this, we estimate the yield of the NV creation to be $0.9\,\%$ which is comparable to previously reported values.[@Pezzagna2010] Deep Etching to form diamond membranes \[sec:deepetch\] ------------------------------------------------------- We now introduce an etching process leading to a thinned membrane of several micron thickness and of around $400\times400~\mu$m size supported by the surrounding $50~\mu$m thick diamond plate. Typical etch masks with sub-micron thickness would not withstand the long etching process necessary to thin a $50~\mu$m thick diamond plate down to a few microns. Thus, we employ thin quartz cover slips (SPI supplies, 75-$125~\mu$m thick) as etch masks. Using water jet cutting (Microwater Jet, Switzerland) a slot ($\leq 500~\mu$m width) is cut into the cover slip. The sample is then sandwiched between a Si carrier chip and the mask; the latter is fixed onto the 6 inch carrier wafer using vacuum grease \[see Fig. \[Fig:membranes\](a)\]. The etch resistance of the quartz material allows for a high quality etching, whereas using standard glass cover slips leads to micro-masking and roughening of the etched diamond as a results of low etch resistance. The masks can be reused several times. For the membrane ’deep etch’, we use an ArCl$_2$ and an O$_2$ based plasma, with plasma parameters as summarized in table \[tab:plasma\]. The etching process starts with $5~$mins of ArCl$_2$ plasma, then the following sequence is cycled until the desired etch depth is reached: 5 mins ArCl$_2$, $5~$mins O$_2$, $5~$mins O$_2$. Consecutive etch steps were separated by 5 mins of cooling under Ar ($100~$sccm, $13.2~$Pa). In the ICP-RIE plasma, a trench forms close to the edge of the quartz mask and the sidewalls of the pit etched into the diamond plate, see yellow marker in \[Fig:nanofab\](b). As the depth of this trench can exceed $1~\mu$m during our deep etch, the thinned membrane becomes mechanically unstable as its connection to the thick diamond plate is compromised. The formation of the trench can be explained as follows: the reflection of high energy ions impinging under grazing incidence onto the sidewalls of the mask and the already etched pit leads to a focusing of the ions close to the sidewalls of the pit and a locally enhanced etch rate induces the trench.[@Hoekstra1998] To ensure membrane stability, we exchange the initial etch mask (mostly 400-$500~\mu$m etched area) for a narrower mask (300-$400~\mu$m) when the membrane has reached a thickness of about 8-$10~\mu$m. Due to the shifted mask edge, the trench formation restarts at the new mask edge location \[see e.g. Fig. \[Fig:nanofab\](b), right side\]. The trench formed during the residual etching does not destabilize the membrane. Due to the thick etch mask, we observe a significantly non uniform thickness of the final membrane, which is much thicker close to the mask than in the center. We measure the membrane’s thickness at its free-standing edge using an SEM and estimate the overall thickness variation using a laser scanning confocal microscope \[see Fig. \[Fig:membranes\](c)\]. Our membranes for scanning probe fabrication finally have a thickness of around $1.5~\mu$m in the center and 2.5-$3~\mu$m close to the mask. AFM measurements show that the etching process improves the surface quality of the membrane: Polishing marks observed before the etching \[Fig. \[Fig:membranes\](d), RMS roughness $0.7~$nm\] are not observed anymore after the deep etch \[see Fig. \[Fig:membranes\](e)\] and we find an RMS roughness of $0.3~$nm for the thinned membrane. We note that the trenching at the rim of the membranes as well as the non-uniformity might be reduced or even avoided using quartz masks with angled sidewalls. Such angled sidewalls could reduce the effective thickness of the mask and thus lead to a more uniform etch rate and less trenching. Deep etches using this novel mask geometry engineered using laser cutting (Photonikzentrum Kaiserslautern, Germany) are currently being investigated. Structuring Scanning Probes \[struct\_scan\] -------------------------------------------- Our scanning probes consist of a $20~\mu$m long, $3~\mu$m wide cantilever, which holds a nanopillar for scanning and sensing \[see Fig. \[Fig:nanofab\](c) and Fig. \[Fig:characterization\](a)\]. Following Ref. , we aim for pillars with $\approx200~$nm diameter and a straight, cylindrical shape to enable efficient collection of the NV fluorescence. The cantilevers are connected to a holding bar in the membrane by $500~$nm wide bridges. These bridges are strong enough to reliably fix the cantilever to the membrane, but still allow for easy breaking off of the cantilever for subsequent mounting onto an AFM head. To form these scanning probes, we use two mutually aligned electron beam lithography steps each followed by structuring via ICP-RIE. In the first step, the holding bar pattern together with the cantilevers are formed. Subsequently, pillars are structured on top of the cantilevers, as sketched in Fig. \[Fig:nanofab\](a) For lithography, we use hydrogen silsesquioxane (HSQ) negative electron beam resist (FOX-16, Dow Corning) as an etch mask. To create a thick mask with a high aspect ratio, we evaporate $2~$nm Ti as an adhesion layer before spin coating a $600~$nm thick layer of HSQ, which we bake on a hotplate at $90^{\circ}$C for $10~$min. Note that the Ti layer only efficiently enhances the adhesion when not allowed to oxidize before applying the resist. We use electron beam lithography with $30~$keV to pattern the HSQ layer. To prevent charging of the diamond sample, we expose the mask with currents below $50~$pA and structure our $200~$nm diameter pillar with a dose of $1500~\mu$As/cm$^2$ and the cantilever with a dose of $150~\mu$As/cm$^2$. Finally, we develop the samples for $20~$s in $25~$wt% TMAH and remove the Ti in $70^{\circ}$C hot 37% HCl. Both steps are followed by rinsing in de-ionized water and cleaning in isopropanol. We transfer the HSQ masks into the diamond via an ArO$_2$ plasma (parameters see table \[tab:plasma\]). Our ArO$_2$ plasma enables a highly anisotropic etch while simultaneously creating a smooth surface in-between the etch masks. After each etch step, we remove residual HSQ and Ti using 20:1 buffered oxide etch (10:10:1 deionized water, ammonium fluoride, 40% HF) and clean the sample in a boiling tri-acid mixture and a solvent clean (see Section \[sec:initialstep\]). Fabricating the scanning probes requires multiple steps as illustrated in Fig. \[Fig:nanofab\](a): In the first step, we structure the pattern consisting of the transverse holding bars and the cantilevers. Additionally, markers (crosses) located adjacent to the thin membrane are defined in the HSQ mask and transferred into the surrounding diamond plate simultaneously to the pattern \[markers not shown in Fig. \[Fig:nanofab\](a)\]. In the second step, we spin coat HSQ on top of the etched pattern which on top of the structures forms a homogeneous film. To ease marker identification, we mechanically remove the HSQ film on top of the markers. This allows us to clearly identify the markers during electron beam lithography and use them to align the pillars with respect to the cantilevers. In the last step, we transfer the pillar pattern into the diamond. As only the pillar is protected by an HSQ mask, the previously defined pattern including the membrane is thinned down during this etching. We continue etching, until the membrane is thinned to a point where all diamond material in-between the cantilevers has been etched away and the cantilevers remain free-standing. Note that the length of the pillars is limited by mask erosion and faceting, as well as the formation of a trench around the pillar (see also Section \[sec:deepetch\]) leading to detachment of the pillar from the cantilever. In general, we are able to etch $2\,\mu$m long wires with a $600\,$nm thick HSQ mask. As a consequence, we start with a membrane of $2-3\,\mu$m and etch $\sim 1\,\mu$m deep when we transfer the holding bars and cantilevers into the membrane. In the second step, we are thus able to etch $\sim 2\,\mu$m long pillars while removing all diamond material in-between the cantilevers. It should also be noted, that we have observed micromasking effects forming needles at the edge of the cantilever during this final etch step. While the magnetometry performance remains unaffected, we have explored an alternative approach to eliminate such micromasking effects: based on the work of Ref. , we have also structured the cantilevers and pillars from different sides of the membrane \[examples shown in Fig. \[Fig:nanofab\](b) and (c)\]. Although this approach fully eliminates the above mentioned micromasking problem, the alignment of the pillar with respect to the cantilever becomes challenging. Despite these drawbacks, both techniques allow to produce hundreds of scanning probes on a single membrane \[see Fig. \[Fig:nanofab\](b)\] Furthermore the nano-fabrication results we present have been obtained using (100) oriented diamond material, however first results clearly indicate that our fabrication process is not restricted to this crystal orientation and can be extended to orientations more favorable for NV sensing applications e.g. (111).[@Neu2014] Device characterization \[sec:precharacterization\] --------------------------------------------------- We characterize the scanning probes to identify the most suitable devices to be transferred and integrated into our AFM setup. For this, we employ a homebuilt confocal microscope equipped with microwave control electronics to perform electron spin resonance (ESR) and Hahn echo measurements to determine the NV spin coherence time $T_{2}$. Additionally, the setup is equipped with correlation electronics to perform second order autocorrelation ($g^{(2)}$) measurements to identify single NV centers. Figure\[Fig:characterization\](a) shows a confocal fluorescence map of our structured scanning probe array obtained by redording the photoluminescence (PL) in a spectral window above $550\,$nm. To identify the scanning probes with single and multiple NV centers, we measure the ESR spectra and $g^{(2)}$. Using a resonant microwave driving field, the NV center can be promoted from the ${\left|0\right>}$ state to the less fluorescent ${\left|\pm 1\right>}$ state, which allows for an efficient optical detection of NV ESR, as depicted for a single NV center in Fig. \[Fig:characterization\](b). A static magnetic field leads to a splitting $2 \gamma_{\rm NV} B_{\rm NV}$ of the two NV ESR resonances (${\left|0\right>}$ to ${\left|1\right>}$ and ${\left|0\right>}$ to ${\left|-1\right>}$), where $\gamma_{\rm NV}=2.8\,\rm MHz/G$ is the gyromagnetic ratio and $B_{\rm NV}$ the magnetic field along the NV symmetry axis. Thus, scanning probes with multiple NV centers aligned along more than one of the four equivalent $<111>$ crystal-directions show multiple resonances. While multiple pairs of ESR dips quickly identify multiple NVs, no ESR signal identifies pillars without NV$^-$. Scanning probes with single NV centers are reliably identified by a significant antibunching dip below $0.5$ in the $g^{(2)}$ measurement \[see Fig.\[Fig:characterization\](c)\]. Using these measurements, we classify the scanning probes into devices with no, single and multiple NV. Figure \[Fig:characterization\](d) shows the statistics of the number of NVs found in 79 scanning probes and reveals that approx. $30\,\%$ of them yield single NV centers. As expected, the number of NV centers per scanning probe follows a Poisson distribution. Using the probability for 0 and 1 NV center per pillar, we deduce an average number of NV centers of $0.82\pm 0.13\,$NV centers/scanning probe \[see Fig. \[Fig:characterization\] (d)\] corresponding to a NV density of $2.6\times 10^9\, \mathrm{cm}^{-2}$ and a creation yield of $0.9\,\%$. We note that we observed a high variation of this value between different samples, which we attribute to variations of pillar diameters, uncertainty in the implanted nitrogen dose and possible variations in material properties (e.g. strain or vacancy concentrations). The magnetometry performance of scanning probes with single NV centers is typically characterized by their sensitivity $\eta$ to magnetic fields. The sensitivity set by the spin coherence properties of the NV center and the detected fluorescence rate in the $ {\left|0\right>} $ and $ {\left|1\right>} $ state can be derived from a Hahn-Echo measurement as depicted in Fig.\[Fig:characterization\](e). The data are fitted using the formula [@Childress2006a] $$\begin{aligned} F(\tau) &=& \frac{\alpha_\mathrm{0}+\alpha_\mathrm{1}}{2} \\\nonumber &+&\frac{\alpha_\mathrm{0}-\alpha_\mathrm{1}}{2} \mathrm{exp}[-\left(\frac{\tau}{T_\mathrm{2}}\right)^n\sum_{j} \mathrm{exp}[-\left(\frac{\tau-j\tau_\mathrm{rev}}{T_\mathrm{dec}}\right)^2],\end{aligned}$$ where $\alpha_\mathrm{0}$ and $\alpha_\mathrm{1}$ are the detected fluorescence rates of the NV in the $ {\left|0\right>} $ and $ {\left|1\right>} $ state respectively \[see Fig.\[Fig:characterization\](b)\] and $T_\mathrm{2}$ is the spin coherence time. The exponent $n$ depends on details of the decoherence process,[@Medford2012] whereas $\mathrm{\tau_{rev}}$ indicates the revival period associated with the Larmor precession of the $^{\rm 13}$C nuclear spins and $T_\mathrm{dec}$ the correlation time of the $^{\rm 13}$C nuclear spin bath.[@Childress2006a] For the depicted Hahn echo measurement, we derive $\alpha_1=98\pm1\,\mathrm{kcps}$, $\alpha_0=146\pm1\,\mathrm{kcps}$, $T_{2}=94\pm 4\,\mathrm{\mu s}$, $n=2.1 \pm 0.2$, $\tau_{\mathrm{rev}}=19.8 \pm 0.1\,\mathrm{ns}$ and $T_{\mathrm{dec}}=5.9 \pm 0.2\,\mathrm{ns}$. Note that the detected fluorescence rates are a factor of $\sim 3$ higher compared to shallow implanted NV centers in unstructured samples due to fluorescence waveguiding in the pillar. Finally, the figure of merit of our scanning probes, the shot noise limited sensitivity to AC magnetic fields $\eta_{AC}$ can be calculated via:[@Taylor2008] $$\eta_\mathrm{AC} \approx \frac{\pi }{2 \gamma_{\rm NV} \mathrm{C} \sqrt{\mathrm{T_2}} } \label{eq:sensitivity} ,$$ with $1/\mathrm{C}=\sqrt{1+2\left(\alpha_0+\alpha_1\right)/\left(\alpha_0-\alpha_1\right)^2}$. For the scanning probe measured in Fig.\[Fig:characterization\] (e), we derive a sensitivity of $\mathrm{\eta_{AC}} \approx\,14\,\pm\,1 \,\mathrm{nT}/\sqrt{\mathrm{Hz} } $. For $13$ scanning probes, we determined the magnetic field sensitivities as summarized in Fig. \[Fig:characterization\](f) and find an average sensitivity of $\mathrm{\eta_{AC}} \approx\,50\,\pm\,20\,\,\mathrm{nT}/\sqrt{\mathrm{Hz}} $. The shot noise limited sensitivity to DC magnetic fields can be equivalently determined by using the relation $\mathrm{\eta_{DC}}=2/\pi \sqrt{\mathrm{T_2}/\mathrm{T_2^*}}\,\mathrm{\eta_{AC}}$.[@Taylor2008] Typical values for $T_2^*$ are few $\mu s$ and the resulting average DC sensitivity is therefore $\mathrm{\eta_{DC}}\approx\,200\,\mathrm{nT}/\sqrt{\mathrm{Hz}}$. Transfer to scanning probe setup \[sec:transfer\] ------------------------------------------------- In order to employ the scanning probes for imaging, the individually characterized cantilevers have to be transferred to an AFM head. Previous work employed ion beam assisted metal deposition to attach scanning probes to a quartz rod and subsequent focused ion beam (FIB) milling to detach the diamond scanning probe from the substrate.[@Maletinsky2012] This approach suffers from low yield, high complexity and significant contamination of the scanning probe by the Gallium ions used for FIB. Here we present an alternative method we developed to transfer the scanning probes using micromanipulators (Sutter Instruments, MPC-385) under ambient conditions. Using quartz micropipettes with an end diameter of $\sim3~\mu$m, we apply $\sim3~\mu$m sized droplets of UV curable glue (Thorlabs, NO81) to the device to be transferred \[see Fig. \[Fig:transfer\](b)\]. After curing the glue, we remove the device from the substrate by mechanically breaking the holding bar \[0.5 $\mu$m wide, see e.g. Fig. \[Fig:characterization\](a)\] with the quartz pipette. In a second step, we glue the quartz tip with the scanning probe to a tuning fork attached to an AFM head \[see Fig. \[Fig:transfer\] (c)\]. To that end, we employ a stereo microscope setup which allows precise alignment of the scanning probe with respect to the AFM head and subsequent gluing of the quartz tip to the tuning fork using UV curable optical glue. As a last step, we carefully break the the quartz pipette above its connection (gluing point) to the tuning fork using a diamond scribe \[see Fig. \[Fig:transfer\] (c)\]. With this procedure, we are able to produce tuning fork based AFM heads with the scanning probes aligned within a few degrees to the AFM holder in a robust and fast way. The UV glue forms a strong connecting link that can be used even in cryogenic environment[@Thiel2015] and enables long-term use of the device. Nanoscale scanning probe magnetometry\[sec:mag\] ================================================ We now demonstrate the performance of scanning quantum sensor by showing our device’s capability for quantitatively imaging magnetic fields with nanoscale resolution. Our setup, consisting of a combined AFM and confocal microscope, has been described elsewhere.[@Appel2015] We applied NV magnetometry to study single Ni nanorods. These nanorods have various potential applications such as magneto-optical switches [@Klein2009] or as probe particles in homogeneous immunoassays for the detection of proteins[@Schrittwieser2013] and in microrheology.[@Tschoepe2014] NV center based magnetometry allows us to study the magnetic properties (spin densities, spin textures etc.[@Hingant2015a; @Tetienne2015a]) of individual particles. Here, we present two different approaches for imaging the stray field of single Ni nanorods which have typical diameters $\sim 24\,\mathrm{nm}$ and lengths $\sim 230\,\mathrm{nm}$ and which are deposited from a solution onto a quartz substrate \[see inset of Fig. \[Fig:magnet\](b)\]. Our first imaging method is based on measuring isomagnetic field lines.[@Maletinsky2012] For this purpose, we fix the MW frequency to the NV spin transition frequency as determined in the absence of the sample. In the presence of a magnetic field, e.g. the stray field of the Ni nanorod, the frequency of NV spin transition gets detuned from the MW frequency which results in an increase of NV fluorescence \[see Fig. \[Fig:characterization\](b)\]. While scanning the NV spin at a distance $d$ over the sample, the iso-field line at zero magnetic field is therefore mapped onto decreased NV fluorescence \[Fig. \[Fig:magnet\](b)\]. Such isomagnetic field imaging is a fast method for probing nanomagnetic structures and their dynamics.[@Tetienne2014] For a complete analysis of the magnetic stray field of the nanorod, it is necessary to perform full, quantitative magnetic stray field mapping. To that end, the Zeeman shift induced by the magnetic field needs to be detected. Various methods to measure the Zeeman shift have been discussed.[@Tetienne2015a; @Schoenfeld2011; @Haeberle2013] We pursue the approach presented in Ref.. A feedback loop is used to lock the MW frequency to the NV spin transition frequency. Using such a frequency lock, the magnetic field can be measured while scanning the NV sensor over the sample. Figure\[Fig:magnet\] (c) depicts the full stray field of the Ni nanorod obtained via such a frequency feedback loop. The measured stray field matches the stray field expected for a single dipole. Assuming a point dipole with a magnetic moment of $m=3.75\times10^{-17}$A/m$^2$, as measured for similar rods with different methods,[@Schrittwieser2013] we calculated the magnetic field projected onto the NV axis. With this method we find agreement between measurement and model and estimate a distance of $\sim70\,$nm between the sample surface and NV center. This distance sets the spatial resolution of the presented scanning magnetometer. The NV center can in principle detect changes of magnetic fields on length scales of $\sim1\,$nm, set by the spatial extent of its electronic wavefunction. Consequently, the imaging resolution of our NV magnetometer is not limited by the detector size but solely by the NV-to-sample distance. We emphazise that the distance of $\sim70\,$nm we determined is a rough estimate and a more precise model has to be used to explain in detail the magnetic field profile. Factors that contribute to this larger-than-expected distance include a polymer layer of unknown thickness surrounding the nanorods,[@Guenther2011] a potential water-layer that typically covers samples under ambient conditions, or dirt sticking to the tip and acting as an additional spacing layer. In the absence of such factors we observed NV-to-sample distances between $10$ and $25\,$nm (see Ref. and Ref.) certifying the nanoscale resolution our scanning probes offers. Discussion and Perspectives =========================== The all-diamond scanning probes we fabricated have proven their potential for detecting magnetic fields with high sensitivity and nanoscale resolution. We conclude by highlighting improvements, which are currently investigated to increase the performance of the presented scanning probe technique. To increase the sensitivity of the scanning probes with single NV centers, a long coherence time $T_2$ and high fluorescence rates are required as can be seen in Eq.\[eq:sensitivity\]. Thus, efficiently collecting the NV’s fluorescence is crucial for highly sensitive scanning probes. Using our $200\,$nm diameter, cylindrical pillar, we increase the typical fluorescence count rates by a factor of $\sim 3$ compared to bulk diamond. More complex photonic geometries such as tapered pillars [@Momenzadeh2015] are currently investigated to further enhance the collection efficiency and might be useful for scanning probes. Further improvements are also expected by optimizing the crystal orientation of the employed diamond samples. Here we employ (100) oriented diamond which is the standard orientation of commercially available high purity diamond. However, in (111) oriented diamond, the NV axis can be oriented perpendicularly to the diamond surface, which yields improved photonic properties as compared to (100) oriented nanopillars.[@Neu2014] A central advantage of our scanning probes is the use of high purity diamond which in principle allows long T$_2$ times to be reached. Unfortunately high resolution imaging requires NV centers in close proximity to the surface, which typically comes at the expense of shorter coherence times due to proximal surface spins.[@Romach2015] For the presented scanning probes, we have chosen an implantation depth of $9\pm4$nm which yields coherence times of $T_2=76\pm19\,\mu$s in the diamond plate before nanofabrication of the scanning probes. In our scanning nanopillars however, we find an average $T_2=44\pm26\,\mu$s. We ascribe this reduction of coherence to unwanted and currently unknown surface defects which are created on the diamond surface during etching. Recent work[@Oliveira2015] suggests that a low bias, ’soft’ oxygen plasma can remove such plasma induced surface damage and could thereby provide a remedy for this problem. This and similar methods[@Cui2013; @Osterkamp2013; @Lovchinsky2016] still remain to be tested on diamond scanning probes and their influence on NV spin coherence times remains an open question. Another challenge is the creation of NV centers with a controlled distance to the diamond surface in the nanometer range. The ion implantation employed here partly suffers from a low yield ($<$ 1%) and a significant uncertainty in the resulting NV depth ($9\pm4$ nm). Recent work suggests ’$\delta$-doping’ as an alternative: in this technique down to 2 nm thin, nitrogen enriched layers are engineered during the growth of diamond[@Ohno2012; @Ohno2014]. However, creating the necessary density of NV centers sufficient to yield one NV per pillar still remains an outstanding challenge.[@Ohno2014]. The presented fabrication process is suited for structuring arrays with hundreds of scanning probes. We so far used 50 $\mu$m thin diamond plates and handled them without any permanent bonding to a carrier system. However, permanent bonding to Si carriers as e.g. described in [@RiedrichMoeller2015; @Tao2013] using HSQ e-beam resist might potentially enable the use of thinner diamond plates and structuring of even more device arrays in a single step. Bonding to carriers might potentially facilitate sample handling, enhance the device yield and pave the way towards further scaling of the presented fabrication processes. Conclusion ========== In this paper, we described in detail our advanced fabrication process for all-diamond scanning probes starting from commercially available diamond material. We demonstrated the efficient integration of our tips into a generic AFM setup and imaged the dipolar magnetic field of Ni nanorods with two different measurement techniques. Our state of the art scanning probes, with the NV-center placed $\sim 10\,$nm below surface of the scanning pillar, have sensitivities of $\mathrm{\eta_{AC}}\approx50\pm20\,\mathrm{nT}/\sqrt{\mathrm{Hz}} $. Finally, we highlight future avenues to push NV center based magnetometry to its ultimate limit to yield scanning NV magnetometers capable of detecting weak magnetic signal down to small ensembles of nuclear spins.[@Mamin2013; @Ajoy2015]\ Acknowledgments {#acknowledgments .unnumbered} =============== We thank B. Shields and D. Rohner for fruitful discussions, J. Teissier for assistance with nanofabrication, L. Thiel for support with the experiment control software and A. Kretschmer for creating the illustrations. We gratefully acknowledge financial support through the NCCR QSIT, a competence center funded by the Swiss NSF, and through SNF Grant No. 142697 and 155845. This research has been partially funded by the European Commission’s 7. Framework Program (FP7/2007-2013) under grant agreement number 611143 (DIADEMS). EN acknowledges funding via the NanoMatFutur program of the german ministry of education and research. [54]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1038/nature07279) [****, ()](\doibase 10.1038/nphys1969) [****,  ()](\doibase 10.1073/pnas.1306825110) [**** ()](http://dx.doi.org/10.1103/PhysRevLett.104.070801) [****,  ()](\doibase 10.1021/nl401129m) [****, ()](\doibase 10.1134/1.567024) [ ()](http://arxiv.org/abs/1511.02873) [**** ()](http://journals.aps.org/prapplied/abstract/10.1103/PhysRevApplied.2.054014) [****,  ()](\doibase 10.1103/PhysRevX.2.031001) [****,  ()](\doibase 10.1038/nature12072) [****,  ()](\doibase 10.1038/nature07278) [****,  ()](\doibase 10.1063/1.3703128) [****, ()](http://stacks.iop.org/0034-4885/77/i=5/a=056503) [****,  ()](\doibase 10.1038/ncomms7733) [****,  ()](\doibase 10.1038/NNANO.2010.56) [****,  ()](\doibase 10.1038/NNANO.2012.50) @noop [“,” ]{} @noop [“,” ]{} [****, ()](\doibase 10.1098/rspa.2004.1339) [****,  ()](\doibase 10.1016/j.diamond.2012.11.007) [****,  ()](\doibase 10.1016/j.diamond.2009.04.008) [****,  ()](\doibase doi:10.1016/j.diamond.2009.01.013) [****,  ()](\doibase 10.1002/pssa.201200069) [****,  ()](\doibase doi:10.1038/ncomms4638) @noop [“,” ]{} [****, ()](\doibase 10.1021/nl404836p) [****, ()](http://stacks.iop.org/1367-2630/12/i=6/a=065017) [****,  ()](\doibase 10.1116/1.590135) [****, ()](\doibase 10.1038/NNANO.2010.6) [****,  ()](\doibase 10.1063/1.4871580) [****,  ()](\doibase 10.1126/science.1131871) [**** ()](http://dx.doi.org/10.1103/PhysRevLett.108.086802) [****,  ()](\doibase 10.1038/nphys1075) [****, ()](http://stacks.iop.org/1367-2630/17/i=11/a=112001) [****,  ()](\doibase 10.1063/1.3259365) [****,  ()](\doibase 10.1002/smll.201300023) [****,  ()](\doibase 10.1063/1.4901575) [****,  ()](\doibase 10.1103/PhysRevApplied.4.014003) [****,  ()](\doibase 10.1126/science.1250113) [****,  ()](\doibase 10.1103/PhysRevLett.106.030802) [**** ()](http://dx.doi.org/10.1103/PhysRevLett.111.170801) [****,  ()](http://stacks.iop.org/0953-8984/23/i=32/a=325103) [****,  ()](\doibase 10.1021/nl503326t) [****,  ()](\doibase 10.1103/PhysRevLett.114.017601) [****,  ()](\doibase 10.1063/1.4929356) [****,  ()](\doibase 10.1063/1.4817651) [****,  ()](\doibase 10.1063/1.4829875) [****,  ()](http://dx.doi.org/10.1126/science.aad8022) [****,  ()](\doibase 10.1063/1.4748280) [****,  ()](\doibase 10.1063/1.4890613) [****,  ()](\doibase 10.1063/1.4922117) [****,  ()](\doibase 10.1002/adma.201301343) [****,  ()](\doibase 10.1126/science.1231540) [**** ()](http://journals.aps.org/prx/abstract/10.1103/PhysRevX.5.011001)

ArXiv

[**Duality-invariant bimetric formulation of linearized gravity**]{} Claudio Bunster$^{1,2}$, Marc Henneaux$^{1,3}$ and Sergio Hörtner$^3$ ${}^1$[*Centro de Estudios Científicos (CECs), Casilla 1469, Valdivia, Chile*]{} ${}^2$[*Universidad Andrés Bello, Av. República 440, Santiago, Chile*]{} ${}^3$[*Université Libre de Bruxelles and International Solvay Institutes, ULB-Campus Plaine CP231, B-1050 Brussels, Belgium*]{}\ **Abstract** A formulation of linearized gravity which is manifestly invariant under electric-magnetic duality rotations in the internal space of the metric and its dual, and which contains both metrics as basic variables (rather than the corresponding prepotentials), is derived. In this bimetric formulation, the variables have a more immediate geometrical significance, but the action is non-local in space, contrary to what occurs in the prepotential formulation. More specifically, one finds that: (i) the kinetic term is non-local in space (but local in time); (ii) the Hamiltonian is local in space and in time; (iii) the variables are subject to two Hamiltonian constraints, one for each metric. Based in part on the talk “Gravitational electric-magnetic duality" given by one of us (MH) at the 8-th Workshop “Quantum Field Theory and Hamiltonian Systems" (QFTHS), 19-22 September 2012, Craiova, Romania. To appear in the Proceedings of the Conference (special issue of the Romanian Journal of Physics). Introduction ============ Understanding gravitational duality is one of the important challenges for exhibiting the conjectured infinite-dimensional Kac-Moody algebras (or generalizations thereof) of hidden symmetries of supergravities and M-theory [@Julia:1982gx; @West:2001as; @Damour:2002cu; @Henneaux:2010ys]. Independently of the problem of uncovering these conjectured hidden symmetries, gravitational duality is important in itself as it illuminates the structure of Einstein gravity. In [@Henneaux:2004jw], two of the present authors presented a formulation of linearized gravity in four space-time dimensions that was manifestly invariant under “duality rotations" in the internal space spanned by the graviton and its dual. This was followed by further developments covering higher spins [@Deser:2004xt], the inclusion of a cosmological constant [@Julia:2005ze] and supersymmetry [@Bunster:2012jp]. One crucial aspect of the manifestly duality-invariant formulation of [@Henneaux:2004jw] was the introduction of prepotentials. Technically, these prepotentials arise through the solutions of the constraints present in the Hamiltonian formalism. The prepotential for the metric (spin-2 Pauli-Fierz field in the linearized theory) appears through the solution of the Hamiltonian constraint, while the prepotential for its conjugate momentum appears through the solution of the momentum constraint. Explicitly, if $h_{ij}$ are the spatial components of the metric deviations from flat space and $\pi^{ij}$ the corresponding conjugate momenta, one has $$h_{ij} = {\epsilon}_{irs} \partial^r \Phi^{s}_{\; \; j} + {\epsilon}_{jrs} \partial^r \Phi^{s}_{\; \; i} + \partial_i u_j + \partial_j u_i \label{hPhi0}$$ and $$\pi^{ij} = {\epsilon}^{ipq} {\epsilon}^{jrs} \partial_p \partial_r P_{qs} \label{piP}$$ where $\Phi_{rs} = \Phi_{sr}$ and $P_{rs} = P_{sr}$ are the two prepotentials (the vector $u_i$ can also be thought of as a prepotential but it drops out from the theory so that we shall not put emphasis on it). The second metric $f_{ij}$ dual to $h_{ij}$ is defined in terms of the second prepotential $P_{ij}$ exactly as $h_{ij}$ is defined in terms of $\Phi_{ij}$, $$f_{ij} = {\epsilon}_{irs} \partial^r P^{s}_{\; \; j} + {\epsilon}_{jrs} \partial^r P^{s}_{\; \; i} + \partial_i v_j + \partial_j v_i ,\label{fP0}$$ the vector $v_i$ being another prepotential, which, just as $u_i$, drops from the theory. The expressions (\[hPhi0\]) and (\[fP0\]) satisfy identically the Hamiltonian constraints, $$R[h] = 0 \label{H1constraint}$$ and $$R[f] = 0 \label{H2constraint}$$ where $R[h]$ and $R[f]$ are respectively the three-dimensional spatial curvatures of $h_{ij}$ and $f_{ij}$. Explicitly, $$R[h] = \partial^m \partial^n h_{mn} - \triangle h, \; \; \; \; R[f] = \partial^m \partial^n f_{mn} - \triangle f,$$ where $h$ and $f$ are the traces of $h_{mn}$ and $f_{mn}$, [i.e., ]{}$h = h_i^{\ i}$, $f = f_{i}^{\ i}$ and where $\triangle$ is the Laplacian. When reformulated in terms of the prepotentials, duality symmetry simply amounts to $\textrm{SO}(2)$ rotations in the internal plane of the prepotentials and consequently, also to $\textrm{SO}(2)$ rotations in the internal plane of the metrics. The temporal components of the metrics arise through the integration of the equations of motion, as arbitrary “integration functions". The equations of motion can furthermore be interpreted as twisted self-duality conditions on the curvature tensors of the graviton and its dual [@Bunster:2012km]. \[For some background information on twisted self-duality, see [@Cremmer:1998px; @Hull:2001iu; @Bunster:2011qp].\] The prepotentials are necessary for locality of the action principle but do not have (yet?) an immediate geometrical interpretation. The metrics appear in this formulation as secondary. The purpose of this note is to provide a manifestly duality-invariant formulation of the theory in which the metrics $h_{ij}$ and $f_{ij}$ are the basic variables. As we explicitly show in the next section, this is possible, but the price paid is non-locality of the action principle. Bimetric formulation ==================== The key to the bimetric formulation of the variational principle relies on the fact that one can invert the relations (\[hPhi0\]) and (\[fP0\]) to express, up to gauge transformation terms for $\Phi_{ij}$ and $P_{ij}$ that drop from the action, the prepotentials in terms of the metrics when these latters satisfy the Hamiltonian constraints (\[H1constraint\]) and (\[H2constraint\]). Remarkably enough, the expression takes almost the same form and read $$\Phi_{ij} =- \frac{1}{4} \left[ {\epsilon}_{irs} \triangle^{-1} \left(\partial^r h^{s}_{\; \; j} \right)+ {\epsilon}_{jrs} \triangle^{-1} \left( \partial^r h^{s}_{\; \; i} \right) \right] \label{Phih1}$$ and $$P_{ij} = - \frac{1}{4} \left[ {\epsilon}_{irs} \triangle^{-1} \left(\partial^r f^{s}_{\; \; j} \right)+ {\epsilon}_{jrs} \triangle^{-1} \left( \partial^r f^{s}_{\; \; i} \right) \right]. \label{Pf1}$$ One may easily verify that if one replaces (\[Phih1\]) and (\[Pf1\]) in (\[hPhi0\]) and (\[fP0\]) and uses the constraints (\[H1constraint\]) and (\[H2constraint\]), one recovers indeed $h_{ij}$ and $f_{ij}$, with some definite $u_i$ and $v_i$ that are not needed here and will not be written explicitly. The expressions for the prepotentials are of course not unique since these are determined by the metrics up to prepotential gauge transformations, which have been analyzed in [@Henneaux:2004jw] and which do not matter for our purposes since the theory is gauge invariant. The expressions (\[Phih1\]) and (\[Pf1\]) correspond to a specific choice of gauge. We can now substitute (\[Phih1\]) and (\[Pf1\]) in the manifestly duality action of [@Henneaux:2004jw]. The “$p$-$\dot{q}$"- term $K[Z_a^{\; mn}]$, $$K[Z_a^{\; mn}] = \int dt \int d^3x \, {\epsilon}^{ab} {\epsilon}^{mrs} \left(\partial^p \partial^q \partial_r Z_{aps} - \triangle \partial_r Z_{a\; s}^{\;q}\right) \dot{Z}_{bqm} \label{Kinetic0}$$ ($a, b = 1,2$) where $(Z^a_{ij}) \equiv (P_{ij},\Phi_{ij})$ becomes $$K[h^a_{\; mn}] = \frac{1}{4}\int dt \int d^3x \, {\epsilon}_{ab} {\epsilon}^{mij} \left(\partial_i h^{an}_{\ \ \ j} - \partial^n \partial^r \partial_i \triangle^{-1}(h^{a}_{\ \ r j})\right) \dot{h}^b_{\; mn} \label{Kinetic1}$$ with $(h^a_{ij}) \equiv (f_{ij},h_{ij})$. It is evidently non-local in space (but local in time). By contrast, the Hamiltonian is local and reads $$\begin{aligned} H[h^a_{\; mn}] &=& \int d^3x \delta_{ab} \left[ \frac{1}{4} \partial^r h^{a mn} \partial_r h^{b}_{\ mn} - \frac{1}{2} \partial_m h^{a m}_{\ \ \ \ n} \partial_r h^{brn} \right] \nonumber \\ && + \int d^3x \delta_{ab} \left[\frac{1}{2} \partial^m h^a \partial^n h^{b}_{\ mn} - \frac{1}{4} \partial^m h^{a} \partial_m h^{b} \right] .\end{aligned}$$ Both the kinetic term and the Hamiltonian are invariant under linearized diffeomorphisms, $$\delta h^a_{mn} = \partial_m \xi^a_n + \partial_n \xi^a_m$$ up to surface terms. Since the metrics are subject to the constraints (\[H1constraint\]) and (\[H2constraint\]), one must add these constraints to the action with Lagrange multipliers $n^a$ ($a=1,2$) when trading the prepotentials for the metrics. The Lagrange multipliers turn out to be the linearized lapses of each metric. The bimetric action is therefore $$S[h^a_{\; mn}, n^a] = K[h^a_{\; mn}] - \int dx^0 H[h^a_{\; mn}] - \int dx^0 d^3x \delta_{ab} n^a R^b \label{action2}$$ and is clearly invariant under $\textrm{SO}(2)$-duality rotations in the internal plane of the metrics, $$f'_{ij} = \cos \alpha f_{ij} - \sin \alpha h_{ij}$$ $$h'_{ij} = \sin \alpha f_{ij} + \cos \alpha h_{ij},$$ accompanied with a similar rotation for the lapses $n^1$ and $n^2$. This is because both $\epsilon_{ab}$ and $\delta_{ab}$ are invariant tensors. In (\[action2\]), $R^a$ stands for $R[h^a]$. Note finally that the prepotentials are each worth two independent physical functions since they are unconstrained but possess four independent gauge symmetries (3 spatial diffeomorphisms and 1 Weyl rescaling). This gives $6$ (number of prepotential components) $- 4$ (number of prepotential gauge symmetries) $=2$ physical functions. The metrics each contain the same number of independent physical components, as they should. They are subject to one constraint (Hamiltonian constraint) but possess only three independent gauge symmetries (spatial diffeomorphisms). This gives again $6$ (number of metric components) $-1$ (number of constraints) $-3$ (number of metric gauge symmetries) $=2$ physical functions. The conjugate momentum also contains two independent physical functions, but the counting is this time $6$ (number of momentum components) $-3$ (number of momentum constraints) $-1$ (number of momentum gauge symmetries) $=2$ physical functions. Conclusions and comments ======================== The manifestly duality-invariant metric action (\[action2\]) is our main result. Although it involves two metrics, we stress that this action is strictly equivalent to the linearized Einstein-Hilbert, or Pauli-Fierz, action. In particular, it contains no additional massive or massless spin-2 degree of freedom. The action clearly exhibits that the two metrics are not only duality conjugate, but also canonically conjugate (in a generalized sense taking into account the non trivial, c-number, operator present in the kinetic term that takes the schematic form $\int dx^0 a_{AB} z^A \dot{z}^B$ where $A,B$ run over all discrete ($a$, $(m,n)$) and continuous ($\vec{x}$) indices, and where $a_{AB}$ is a c-number infinite square matrix which is not the standard symplectic matrix $\begin{pmatrix} 0 & -I \\I & 0 \end{pmatrix}$ - it is actually degenerate -, so that the Poisson brackets are not the canonical ones). The detailed Hamiltonian structure will be worked out and studied elsewhere [@BHH]. The bimetric formulation is nonlocal in space (but local in time). This might not really be a drawback since it is expected that the manifestly duality invariant formulation of the interacting theory (if it exists) will be non-local anyway. The formulation underplays the role of the prepotentials, which are absent in (\[action2\]). The prepotentials are nevertheless useful technical devices which enable one to go from the conjugate momenta $\pi^{mn}$ to the second metric $f_{mn}$, and to control also the non-locality of the action. Although the theory is Poincaré invariant, the formulation lacks manifest space-time covariance. This seems to be an unavoidable feature whenever one deals with manifest duality invariance [@Deser:1976iy; @Henneaux:1988gg; @Schwarz:1993vs]. This might indicate that space-time covariance is somewhat secondary. This point of view would seem in any event to be inevitable if spacetime itself is a derived, emergent concept. There exist in fact models in which Poincaré invariance can actually be derived from duality invariance [@Bunster:2012hm]. In higher dimensions, the dual to the Pauli-Fierz spin-2 field is a mixed Young-symmetry tensor field described by the Curtright action [@Curtright:1980yk]. The two-field action analogous to (\[action2\]) involves simultaneously in that case the standard graviton $h_{mn}$ and the Curtright field $T_{m_1 \cdots m_{D-3} n}$. The details will be given elsewhere [@BHH]. Acknowledgments {#acknowledgments .unnumbered} =============== M.H. is grateful to the organizers of the 8-th Workshop “Quantum Field Theory and Hamiltonian Systems" (QFTHS, Craiova, Romania) for their kind invitation. C.B. and M.H. thank the Alexander von Humboldt Foundation for Humboldt Research Awards. The work of M.H.and S.H. is partially supported by the ERC through the “SyDuGraM" Advanced Grant, by IISN - Belgium (conventions 4.4511.06 and 4.4514.08) and by the “Communauté Française de Belgique" through the ARC program. The Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. [99]{} B. Julia, “Kac-moody Symmetry Of Gravitation And Supergravity Theories,” Proc. AMS-SIAM Summer Seminar on Applications of Group Theory in Physics and Mathematical Physics, Chicago 1982, LPTENS preprint 82/22, eds. M. Flato, P. Sally and G. Zuckerman, Lectures in Applied Mathematics, [**21**]{} (1985) 335;\ “Dualities in the classical supergravity limits: Dualizations, dualities and a detour via (4k+2)-dimensions,” In \*Cargese 1997, Strings, branes and dualities\* 121-139 \[hep-th/9805083\]. P. C. West, “E(11) and M theory,” Class. Quant. Grav.  [**18**]{}, 4443 (2001) \[hep-th/0104081\]. T. Damour, M. Henneaux and H. Nicolai, “E(10) and a ’small tension expansion’ of M theory,” Phys. Rev. Lett.  [**89**]{}, 221601 (2002) \[hep-th/0207267\]. M. Henneaux, B. L. Julia and J. Levie, “$E_{11}$, Borcherds algebras and maximal supergravity,” JHEP [**1204**]{}, 078 (2012) \[arXiv:1007.5241 \[hep-th\]\]. M. Henneaux and C. Teitelboim, “Duality in linearized gravity,” Phys. Rev.  D [**71**]{}, 024018 (2005) \[arXiv:gr-qc/0408101\]. S. Deser and D. Seminara, “Duality invariance of all free bosonic and fermionic gauge fields,” Phys. Lett. B [**607**]{}, 317 (2005) \[hep-th/0411169\]. B. Julia, J. Levie and S. Ray, “Gravitational duality near de Sitter space,” JHEP [**0511**]{}, 025 (2005) \[hep-th/0507262\];\ B. L. Julia, “Electric-magnetic duality beyond four dimensions and in general relativity,” hep-th/0512320. C. Bunster and M. Henneaux, “Supersymmetric electric-magnetic duality as a manifest symmetry of the action for super-Maxwell theory and linearized supergravity,” Phys. Rev. D [**86**]{}, 065018 (2012) \[arXiv:1207.1761 \[hep-th\]\]. C. Bunster, M. Henneaux and S. Hörtner, “Gravitational Electric-Magnetic Duality, Gauge Invariance and Twisted Self-Duality,” arXiv:1207.1840 \[hep-th\], to appear in the J. Phys. A special volume on “Higher Spin Theories and AdS/CFT" edited by Matthias Gaberdiel and Misha Vasiliev. E. Cremmer, B. Julia, H. Lu and C. N. Pope, “Dualisation of dualities. II: Twisted self-duality of doubled fields and superdualities,” Nucl. Phys.  B [**535**]{}, 242 (1998) \[arXiv:hep-th/9806106\]. C. M. Hull, “Duality in gravity and higher spin gauge fields,” JHEP [**0109**]{}, 027 (2001) \[hep-th/0107149\]. C. Bunster and M. Henneaux, “The Action for Twisted Self-Duality,” Phys. Rev. D [**83**]{}, 125015 (2011) \[arXiv:1103.3621 \[hep-th\]\]. C. Bunster, M. Henneaux and S. Hörtner, “Twisted Self-Duality for Linearized Gravity in $D$ Dimensions", in preparation. S. Deser and C. Teitelboim, “Duality Transformations Of Abelian And Nonabelian Gauge Fields,” Phys. Rev.  D [**13**]{}, 1592 (1976). M. Henneaux and C. Teitelboim, “Dynamics Of Chiral (selfdual) p-Forms,” Phys. Lett. B [**206**]{}, 650 (1988). J. H. Schwarz and A. Sen, “Duality symmetric actions,” Nucl. Phys. B [**411**]{}, 35 (1994) \[hep-th/9304154\]. 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ArXiv

--- abstract: 'Control of the Néel vector in antiferromagnetic materials is one of the challenges preventing their use as active device components. Several methods have been investigated such as exchange bias, electric current, and spin injection, but little is known about strain-mediated anisotropy. This study of the antiferromagnetic [*L*]{}1$_0$-type MnX alloys MnIr, MnRh, MnNi, MnPd, and MnPt shows that a small amount of strain effectively rotates the direction of the N'' eel vector by 90$^{\circ}$ for all of the materials. For MnIr, MnRh, MnNi, and MnPd, the Néel vector rotates within the basal plane. For MnPt, the Néel vector rotates from out-of-plane to in-plane under tensile strain. The effectiveness of strain control is quantified by a metric of efficiency and by direct calculation of the magnetostriction coefficients. The values of the magnetostriction coefficients are comparable with those from ferromagnetic materials. These results indicate that strain is a mechanism that can be exploited for control of the Néel vectors in this family of antiferromagnets.' author: - In Jun Park - Taehwan Lee - Protik Das - Bishwajit Debnath - 'Greg P. Carman' - 'Roger K. Lake' title: 'Strain control of the Néel vector in Mn-based antiferromagnets' --- There has been a rapidly increasing interest in the use of antiferromagnetic (AFM) materials for use as active device elements [@2018_Tserkovnyak_RMP; @2017_AFM_spintronics_Jungwirth_PSSR; @AFM_spintronics_Jungwirth_NNano16]. AFMs are insensitive to parasitic electromagnetic and magnetic interference. The dipolar coupling is minimal, since there is no net magnetic moment. Their lack of macroscopic magnetic fields allows AFM devices and interconnects to be highly scaled with reduced cross talk and insensitivity to geometrical anisotropy effects. AFM resonant frequencies and magnon velocities are several orders of magnitude higher than those in ferromagnetic materials, and these velocities correlate with similarly higher switching speeds [@gomonay2014spintronics; @AFM_spintronics_Jungwirth_NNano16; @KWang_ULowSwitchingAFM_APL16]. AFM metals and insulators are plentiful, and many have Néel temperatures well above room temperature, a requirement for compatibility with on-chip temperatures in current Si integrated circuits. The high Néel temperatures of the Mn-based equiatomic alloys such as MnIr, MnRh, MnNi, MnPd, and MnPt make them suitable candidates for on-chip applications [@2018_Tserkovnyak_RMP]. Extensive research has been conducted on the electronic [@sakuma1998electronic; @umetsu2002electrical; @umetsu2004pseudogap; @umetsu2006electrical; @umetsu2007electronic], magnetic [@pal1968magnetic; @sakuma1998electronic; @umetsu2006electrical; @umetsu2007electronic], and elastic properties [@wang2013first; @wang2013structural] of these materials. The spins on the Mn atoms are antiferromagnetically coupled with each other in the basal plane, and each plane is coupled ferromagnetically as shown in Fig. \[fig:structure\]. ![\[fig:structure\] Antiferromagnetic [*L*]{}1$_0$-type Mn alloy structures. Mn atoms are the purple spheres with the spin vectors, and the gold spheres indicate the Ir, Rh, Ni, Pd, or Pt atoms. (a) In-plane equilibrium spin texture of MnIr, MnRh, MnNi, and MnPd. (b) Out-of-plane equilibrium spin texture of MnPt. ](Structure2.eps){width="1.0\linewidth"} The positive attributes of speed, scaling, and robustness to stray fields are accompanied by the challenges of manipulating and detecting the antiferromagnetic states. There are several methods to control the magnetic properties of AFM materials such as with exchange bias [@2018_Tserkovnyak_RMP], the use of electric current [@wadley2016electrical], and strain induced by a piezoelectric material [@barra2018voltage; @yan2019piezoelectric]. The recent experimental demonstration of strain control of the Néel vector in MnPt [@yan2019piezoelectric], provides timely motivation for a theoretical study of strain-meditated magnetic anisotropy in the MnX AFM materials. Density functional theory (DFT) is used to analyze the effect of strain on the magnetic anisotropy. The effectiveness of strain control is quantified by a metric of efficiency and by calculation of the magnetostriction coefficients. [c c c c c]{} ------------------------------------------------------------------------ & a ([Å]{}) & b ([Å]{}) & c ([Å]{}) & $\mu_{Mn}$ ([$\mu_{B}$]{})\ MnIr&3.84&3.84&3.64&2.8\ MnRh&3.85&3.85&3.62&3.1\ MnNi&3.62&3.62&3.58&3.2\ MnPd&3.99&3.99&3.69&3.8\ MnPt&3.98&3.98&3.71&3.7\ \[tab1\] First principles calculations are performed as implemented in the Vienna Ab initio Simulation Package (VASP) [@kresse1993ab] to investigate the effect of strain on the magnetic anisotropy of [*L*]{}1$_0$-ordered bulk MnIr, MnRh, MnNi, MnPd, and MnPt. Projector augmented-wave (PAW) potentials [@blochl1994projector] and the generalized gradient approximation (GGA) parameterized by Perdew-Burke-Ernzerhof (PEB) were employed [@perdew1996generalized]. Depending on the materials, different cut-off energies (typically ranging from 420 eV to 450 eV) and k-points grids were used in order to ensure the total energy converged within 10$^{-7}$ eV per unit cell. The initial equilibrium structure consists of a tetragonal unit cell where the fractional coordinates of Mn atoms are (0, 0, 0) and (0.5, 0.5, 0), and those of the X atoms are (0.5, 0, 0.5) and (0, 0.5, 0.5). Compressive or tensile stress along the $a$ axis is applied to each structure, and the structure is fully relaxed along the $b$ and $c$ axes (biaxially) until all forces on each atom are less than 10$^{-4}$ eVÅ$^{-1}$. The relaxed lattice constants for each applied strain are shown in supplementary Fig. S1. The strain is defined as, ${\rm strain} = (a - a_0) / a_0 \times 100 \% $, where $a$ and $a_0$ are the lattice constants with and without strain, respectively. With the relaxed structure, the spin-polarized self-consistent calculation is performed to obtain the charge density. Finally, the magnetic anisotropy energies are determined by calculating the total energies for different Néel vector directions including spin orbit coupling. Table \[tab1\] shows the lattice constants and the magnetic moments of the Mn site in MnX without strain. All of the values are very close to those from previous results [@wang2013first; @wang2013structural]. The local magnetic moments of the X site are zero for all materials. Figures \[fig:Ir\]–\[fig:Pt\] show the differences in the total energies as a function of the strain for MnIr, MnRh, MnNi, MnPd, and MnPt, respectively, where $E_{abc}$ is the ground state energy with the Néel vector along the $[abc]$ direction. The reference energy levels from each figure, which are indicated by the solid black lines, are $E_{001}$ for MnPt and $E_{110}$ for the other materials. The reference energies are the lowest energy state, which means MnIr, MnRh, MnNi, and MnPd have in-plane anisotropy and MnPt has out-of-plane anisotropy without strain. This is consistent with experimental results [@pal1968magnetic]. To show the energy differences more clearly as the strain changes, the reference level is taken at each value of the applied strain. At zero strain, there is no energy difference between $E_{100}$ and $E_{010}$ because of the symmetry of all of the materials. ![\[fig:Ir\] MnIr energy differences $E_{abc} - E_{110}$ for the 3 different orientations of the Néel vector as indicated by the labels. ](MnIr.eps){width=".8\linewidth"} ![\[fig:Rh\] MnRh energy differences $E_{abc} - E_{110}$ for the 3 different orientations of the Néel vector as indicated by the labels. ](MnRh.eps){width=".8\linewidth"} ![\[fig:Ni\] MnNi energy differences $E_{abc} - E_{110}$ for the 3 different orientations of the Néel vector as indicated by the labels. ](MnNi.eps){width=".8\linewidth"} ![\[fig:Pd\] MnPd energy differences $E_{abc} - E_{110}$ for the 3 different orientations of the Néel vector as indicated by the labels. ](MnPd.eps){width=".8\linewidth"} ![\[fig:Pt\] MnPt energy differences $E_{abc} - E_{110}$ for the 3 different orientations of the Néel vector as indicated by the labels. ](MnPt.eps){width=".8\linewidth"} Figures \[fig:Ir\]–\[fig:Pd\] show that sweeping the strain from negative (compressive) to positive (tensile) causes a 90$^{\circ}$ rotation of the Néel vector in the $ab$-plane for the four materials MnIr, MnRh, MnNi, and MnPd. However, the alignment of the Néel vector with compressive or tensile strain depends on the specific material. MnIr and MnRh behave like magnets with a positive magnetostriction coefficient, since tensile strain along \[100\] causes the Néel vector to align in the \[100\] direction. On the other hand, MnNi and MnPd behave like magnets with a negative magnetostriction coefficient, since tensile strain along \[100\] causes the Néel vector to align in the \[010\] direction [@biswas2014energy]. MnPt is unique among the 5 materials. In equilibrium, in the absence of strain, the Néel vector has perpendicular anisotropy. Under compressive (negative) strain along the $[100]$ axis, the Néel vector remains out-of-plane. Under tensile strain along $[100]$, the Néel vector switches from out-of-plane $[001]$ to in-plane aligning in the $[010]$ direction. ![\[fig:Ireff\] MnIr strain energies and efficiency versus strain. (a) The energy difference between two different N' eel vector orientations (black) as shown by the left axis, and the change in total energy (red) as shown by the right axis. (b) The efficiency as a function of the strain. ](MnIr_eff.eps){width="1.0\linewidth"} ![\[fig:Rheff\] MnRh strain energies and efficiency versus strain. (a) The energy difference between two different Néel vector orientations (black) as shown by the left axis, and the change in total energy (red) as shown by the right axis. (b) The efficiency as a function of the strain. ](MnRh_eff.eps){width="1.0\linewidth"} ![\[fig:Nieff\] MnNi strain energies and efficiency versus strain. (a) The energy difference between two different Néel vector orientations (black) as shown by the left axis, and the change in total energy (red) as shown by the right axis. (b) The efficiency as a function of the strain. ](MnNi_eff.eps){width="1.0\linewidth"} ![\[fig:Pdeff\] MnPd strain energies and efficiency versus strain. (a) The energy difference between two different Néel vector orientations (black) as shown by the left axis, and the change in total energy (red) as shown by the right axis. (b) The efficiency as a function of the strain. ](MnPd_eff.eps){width="1.0\linewidth"} ![\[fig:Pteff\] MnPt strain energies and efficiency versus strain. (a) The energy difference between two different Néel vector orientations (black) as shown by the left axis, and the change in total energy (red) as shown by the right axis. (b) The efficiency as a function of the strain. ](MnPt_eff.eps){width="1.0\linewidth"} For applications, it is useful to quantify the efficiency with which strain rotates the Néel vector and to determine the magnetostriction coefficient from the ab initio calculations. The internal efficiency is defined as $${\rm Efficiency} (\%) = \left| \frac{E_{abc} - E_{a'b'c'}}{E_{total} - E_{total}(0)} \right| \times 100 , \label{eq:efficiency}$$ where the total energies $E_{abc}$ and $E_{a'b'c'}$ are defined in the same way as above, i.e. the total energies in the presence of strain with the Néel vector oriented along $[abc]$ or $[a'b'c']$, respectively. The denominator in the Eq. (\[eq:efficiency\]) is the total energy change induced by the strain. For MnIr, MnRh, MnNi, and MnPd, $E_{abc}$ and $E_{a'b'c'}$ are $E_{100}$ and $E_{010}$, respectively. For MnPt, $E_{abc}$ and $E_{a'b'c'}$ are $E_{010}$ and $E_{001}$, respectively. The numerator and denominator of Eq. (\[eq:efficiency\]) are plotted as a function of strain in Figs. \[fig:Ireff\]-\[fig:Pteff\](a), and the resulting efficiencies are plotted as a function of strain in Figs. \[fig:Ireff\]-\[fig:Pteff\](b). The changes in the total energies, shown as red curves in Figs. \[fig:Ireff\]-\[fig:Pteff\](a), are parabolic so that they can be considered as the strain energy proportional to the square of the applied strain. On the other hand, the differences between two energies (the black curves in Figs. \[fig:Ireff\]-\[fig:Pteff\](a)) are approximately linear under small strain ($< 1\%$). Therefore, the efficiency decreases sharply as the amount of strain increases. At $0.5\%$ strain, the highest efficiency for $90^\circ$ in-plane rotation of the Néel vector is $20\%$ for MnIr. For MnRh, MnNi, and MnPd, the efficiencies are smaller and equal to $3.5\%$, $1.5\%$, and $1.4\%$, respectively. To rotate the Néel vector from out-of-plane to in-plane in MnPt, a positive, tensile strain must be applied. The efficiency of this process at $+0.5\%$ strain is $6\%$. Using the data above, the magnetostriction coefficients ($\lambda_s$), which are widely used in ferromagnets, are calculated. The magnetostriction coefficient is defined as $$\lambda_s (ppm) = \frac{2K_{me}}{3Y(\varepsilon_{bb} - \varepsilon_{aa})}, \label{eq:coefficient}$$ where $Y$ and $(\varepsilon_{bb} - \varepsilon_{aa})$ are Young’s modulus and strain, respectively [@bur2011strain]. $K_{me}$ is the magnetoelastic anisotropy constant, which is defined as the difference of the magnetic anisotropy energies with and without strain, and the magnetic anisotropy energy is defined as $E_{100} - E_{010}$. Plots of $E_{100} - E_{010}$ as a function of strain are shown in Supplementary Fig. S2. Young’s moduli for all MnX alloys except MnIr were adopted from previous calculation results [@wang2013first; @wang2013structural], and the value for MnIr was determined as described in the Supplementary Information. For simplicity, we disregard $\varepsilon_{bb}$ which represents a negligible change in the lattice constant along the $b$-axis caused by the applied strain along $a$. The results for $\lambda_s$ are summarized in the Table \[tab2\]. As expected, MnIr and MnRh have positive values of $\lambda_s$, and MnNi, MnPd, and MnPt have negative values. Also, the magnitudes of the magnetostriction coefficients follow the magnitudes of the efficiencies. The magnetostriction coefficients of the MnX alloys are comparable with the ones from ferromagnets [@clark2000magnetostrictive; @panduranga2018polycrystalline; @hall1959single; @huang1995giant; @fritsch2012first], which suggests that strain can be used to control the magnetic anisotropy of these antiferromagnetic materials. MnIr MnRh MnNi MnPd MnPt ------------------- ------ ------ ------ ------ ------ $\lambda_s$ (ppm) 241 43 -15 -17 -196 : Calculated magnetrostriction coefficients of the $L1_0$-type MnX alloys.[]{data-label="table:coef"} \[tab2\] In summary, the Néel vectors of MnIr, MnRh, MnNi, and MnPd can be rotated $90^\circ$ in the basal plane by applying in-plane strain. MnIr and MnRh behave like magnets with positive magnetostriction coefficients, since their Néel vectors align with tensile strain. MnNi and MnRh behave like magnets with negative magnetostriction coefficients, since their Néel vectors align with compressive strain. The internal efficiency of this process is highest for MnIr and it is equal to $20\%$ at $0.5\%$ strain. MnPt is unique among the 5 alloys in that its Néel vector aligns out-of-plane along the \[001\] axis in equilibrium. Applying a tensile strain along \[100\] rotates the Néel vector from out-of-plane \[001\] to in-plane \[010\]. The efficiency of this process at $0.5\%$ tensile strain is $6\%$. Under compressive strain along \[100\], the Néel vector of MnPt remains out-of-plane \[001\]. The magnitudes of the calculated magnetostriction coefficients are comparable with those of ferromagnets, and they follow the same trends as the calculated efficiencies. For in-plane rotation of the Néel vector, MnIr has the highest magnetostriction coefficient of 241 ppm. The magnetostriction coefficient for out-of-plane rotation in MnPt is -196 ppm. These results suggest that strain can be an effective mechanism to control the Néel vectors in this family of antiferromagnets. This work was supported as part of Spins and Heat in Nanoscale Electronic Systems (SHINES) an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award \#DE-SC0012670. This material is based upon work supported by or in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant No. W911NF-17-0364. This work used the Extreme Science and Engineering Discovery Environment (XSEDE) [@towns2014xsede], which is supported by National Science Foundation Grant No. ACI-1548562 and allocation ID TG-DMR130081. [27]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [****,  ()](\doibase 10.1103/RevModPhys.90.015005) [****,  ()](\doibase 10.1002/pssr.201700022) [****,  ()](dx.doi.org/10.1038/nnano.2016.18) @noop [****,  ()]{} [****,  ()](\doibase 10.1063/1.4939446) @noop [****,  ()]{} @noop [****, ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](\doibase 10.1038/s41565-018-0339-0) @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{} @noop [****,  ()]{}

ArXiv

--- abstract: | We discuss several results in electrostatics: Onsager’s inequality, an extension of Earnshaw’s theorem, and a result stemming from the celebrated conjecture of Maxwell on the number of points of electrostatic equilibrium. Whenever possible, we try to provide a brief historical context and references. [**Keywords:**]{}[*[ Electrostatics, potential theory, Onsager’s inequality, Maxwell’s problem, energy equilibria.]{}*]{} address: - 'MS 4242,Texas A$\&$M University, College Station, TX 77843-4242' - '4202 E. Fowler Ave., CMC342, Tampa, FL 33620' - '4202 E. Fowler Ave., CMC342, Tampa, FL 33620' - '4202 E. Fowler Ave., CMC342, Tampa, FL 33620' author: - Artem Abanov - Nathan Hayford - 'Dima Khavinson$^\sharp$' - Razvan Teodorescu date: | December 2019.\ $\quad \quad ^\sharp$The author’s research is supported by the Simons Foundation, under the grant 513381. title: 'Around a theorem of F. Dyson and A. Lenard: Energy Equilibria for Point Charge Distributions in Classical Electrostatics' --- Introduction. ============= Electrostatics is an ancient subject, as far as most mathematicians and physicists are concerned. After several centuries of meticulous study by people like Gauss, Faraday, and Maxwell (to name a few), one wonders if it is still possible to find surprising or new results in the field. Throughout this text, we address several seemingly classical electrostatics problems that have not been fully addressed in the literature, to the best of our knowledge. Let us begin by establishing some notations. For vectors $x,y \in \mathbb{R}^d, \, d \geq 3$, we define the function $K_{d}(x,y)$ by the formula $$\label{Riesz} K_{d}(x, y) = \frac{1}{(2-d) \omega_{d-1}}\frac{1}{|x-y|^{d-2}}.$$ Here, $\omega_{d-1}$ is the surface area of the unit sphere in ${\mathbb{R}}^d$. $K_{d}$ is the fundamental solution for the Laplace operator in $\mathbb{R}^d$ (i.e., $\Delta_y K_{d}(x, y) = \delta_x$). Furthermore, given a locally finite (signed) Borel measure $\mu$ with support $\Sigma$, we define the *Newtonian (or Coulomb) potential* of $\mu$ with respect to the kernel $K_{d}(x,y)$ by $$\left(U^{\mu}_{\Sigma}\right)(x) = \int_{\Sigma} K_{d}(x,y) d\mu(y).$$ If the support of the measure $\mu$ is either clear from context or, otherwise, irrelevant to the problem, we drop the subscript $\Sigma$, and write $U^{\mu}(x)$. We define the *Newtonian (or Coulomb) energy* of a measure $\mu$ with respect to the kernel $K_{d}(x,y)$ by $$W_{\Sigma}[\mu] = \int_{\Sigma} \left (U^{\mu}_{\Sigma}\right)(x) d\mu(x).$$ We also refer to this functional as the *electrostatic energy*, and sometimes use the simpler notation $W_{\Sigma}[\mu] = W[\mu]$ - cf. [@Kellogg; @Landkof] for the basics of Potential Theory. We say a charge distribution (measure) $\mu$ is in *constrained equilibrium*, when its electrostatic potential $${\label}{1} U^{\mu}_{\Sigma}(x) = \int_{\Sigma} K_d(x,y) d\mu(y),$$ is constant (possibly taking different values) on each connected component $\Sigma_j$ of the support of $\mu$, subject to the constraints $${\label}{2} \mu_j = \mu(\Sigma_j) = Q_j, \, j = 1, 2, 3, \ldots, m, \quad \sum_{j=1}^m Q_j = Q.$$ In the case when $\Sigma_j$ consists only of one point, i.e. $\Sigma_j = \{ x_j \}$, the condition that the potential $U^{\mu}_{\Sigma}$ be constant is replaced by the gradient condition $${\label}{3} \left ( \nabla U^{\mu}_{\Sigma \setminus \Sigma_j} \right ) (x_j)= 0.$$ Now, we are ready to state the first problem discussed in this paper. [p1]{} Given a total charge $Q \in \mathbb{R}$ and a locally finite Borel measure $\mu$, such that $\mu(\mathbb{R}^n) = Q$, is there a collection of disjoint compact sets $\{ \Sigma_j \}_{j = 1}^m$, such that $\mu(\Sigma_j) = Q_j$, $ \sum_{j=1}^m Q_j = Q$, and $\mu$ has a constrained equilibrium configuration on $\Sigma := \cup_j \Sigma_j$? It should be noted that, while the associated energy $${\label}{4} W_{\Sigma}[\mu] \equiv \int_{\Sigma} U^{\mu}_{\Sigma}(x) d\mu(x)$$ does solve a variational problem over the set $\mathcal{M}$ of measures constrained by , it is by no means automatically also the solution to the free optimization problem $${\label}{5} W[\mu] = \inf_{\sigma \in \mathcal{M}} W[\sigma].$$ In other words, a solution for Problem  merely gives an equilibrium charge configuration, which need not be also a *stable equilibrium*, i.e. a local minimum of the energy functional, as opposed to a saddle point. The (stronger) stable equilibrium problem may not have a solution, for a generic choice of the support $\Sigma$ and set of constraints . It should also be noted that the choice of total charge $Q$ is not important, except to distinguish between neutral configurations ($Q = 0$) and non-neutral ($Q \ne 0$). This is due to the fact that, under a simple rescaling $$Q = \lambda q, \, \, Q_j = \lambda q_j, \,\, \lambda \in \mathbb{R}\setminus \{0\},$$ the potential scales by a factor of $\lambda$, and the energy by a factor of $\lambda^2$, leaving the variational problems (and their solutions) unchanged. Therefore, the only two distinct cases that need to be considered are $Q = 0$ and $Q = 1$. The second (classical) problem we discuss is the characterization of critical manifolds $C$ (specifically, curves) on which the gradient of the potential of a given charge configuration vanishes. More precisely, the problem in ${\mathbb{R}}^2$ is: [p2]{} Let $\mu$ be a locally finite charge distribution with planar support $\Sigma \subset {\mathbb{R}}^2$. Can the critical manifold $C \subset {\mathbb{R}}^3$ of $\mu$, defined by $${\label}{8} C = \{ x \in \mathbb{R}^3 : \nabla U^{\mu}_{\Sigma}(x) = 0 \} $$ contain a curve in ${\mathbb{R}}^2$? This problem has numerous applications, some of which are discussed in this paper. One of its most obvious implications is that a collection of charges, placed in the plane supporting the distribution $\mu$, cannot have a curve in the same plane as an equilibrium configuration. Problem \[p2\] has a distinguished history, and can be regarded as a special case of Maxwell’s Problem (cf. [@Max] for further references). In short, Maxwell asserted (without proof) that if $n$ point charges $\{x_j\}_{j=1}^n$ are placed in ${\mathbb{R}}^3$, then there are at most $(n-1)^2$ points in ${\mathbb{R}}^3$ at which the electrostatic field vanishes. More precisely, if each $x_j$ has charge $q_j \in {\mathbb{R}}$, then $$\label{Maxwell-bound} \# \bigg\{ x : \nabla \bigg( \sum_{j=1}^n \frac{q_j}{|x-x_j|}\bigg) = 0 \bigg\} \leq (n-1)^2,$$ or is, otherwise, infinite (e.g. contains a curve, the ‘degenerate case’). Thompson, while preparing Maxwell’s works for publication, couldn’t prove it, and the problem has since become known as Maxwell’s conjecture - cf. [@Max] for more details. So far, the only real progress on this problem has been achieved in [@Max], where it was verified that the cardinality of the set of isolated points in equilibrium in is finite. But even in the case of $n = 3$, the best known estimate is $12$, not $4$! No counterexamples to the conjecture have been found. The first examples of curves of degeneracy where holds go back to [@Y]. Futher, partial results, for a particular case when the charges are coplanar can be found in [@Killian; @Peretz]. Killian, in particular, conjectured that in the latter case the degeneracy curves are all transversal to the plane of the charges. This is proven in §4. Finally, note that in the plane, if we use the logarithmic potential, the estimate improves to $(n-1)$ and is an obvious corollary of the Fundamental Theorem of Algebra. This paper is organized as follows: in §2, we discuss Problem \[p1\], first in its classical form (for the Newtonian potential in $\mathbb{R}^3$), and the proof of F. Dyson and A. Lenard for an inequality first discovered by L. Onsager [@Onsager], along with extensions of the same energy inequality to the case of potentials in $\mathbb{R}^n$. In §3 we focus on necessary conditions for the existence of an equilibrium configuration, in particular for the case of Coulomb potentials (in $\mathbb{R}^3$). A necessary condition independent of the support, and which can be expressed as a constraint on the measure density moments, is also discussed in §3 (Intersection Theorem). Section §4 is dedicated to the precise formulation and solution of degeneracy in Maxwell’s problem, for charge configurations constrained to two-dimensional subspaces of ${\mathbb{R}}^3$. In section §5 we pose a fascinating question, originating in approximation theory, that we frivolously label ‘Faraday’s problem’, believing that Sir Michael Faraday would have never hesitated to answer it based on empirical evidence. Variations on Onsager’s Inequality. =================================== The Onsager inequality was originally discussed by Lars Onsager [^1] in a relatively little known paper [@Onsager]. Onsager himself did not provide a proof of this inequality, and it was not until 30 years later that a full proof was given by Freeman Dyson and Andrew Lenard [@Dyson-Lenard]. Here, we shall present their original proof of the inequality, and consider subsequent generalizations to $\mathbb{R}^n$. We remark that this inequality was brought to the attention of the authors by Eero Saksman et. al., who found far-reaching extensions of it in a probablistic context [@Saksman]. F. Dyson and A. Lenard’s Original Proof. ---------------------------------------- Let $\{x_j\}_{j=1}^n$ be a collection of point charges in ${\mathbb{R}}^3$, with charges $\{q_j\}_{j=1}^n$, each of which takes one of the values $\pm 1$. The electrostatic energy due to this collection of point charges is given by $$-\frac{1}{4\pi} \sum_{j < k} \frac{q_j q_k}{|x_j - x_k|}.$$ Furthermore, for each $x_j$, denote the shortest distance to the next point charge by $\delta_j$: $$\delta_j := \min_{k \neq j} |x_j - x_k|.$$ The Onsager inequality states that the electrostatic energy of the point charges is bounded by the sum of inverses of $\delta_j$’s: \[original\] Let $\{x_j\}_{j=1}^n$, $\{q_j\}_{j=1}^n$, and $\{\delta_j\}_{j=1}^n$ be defined as above. Then $$\label{Onsager} \frac{1}{4\pi} \sum_{j < k} -\frac{q_j q_k}{|x_j - x_k|} < \frac{1}{4\pi} \sum_{j=1}^n \frac{1}{\delta_j}.$$ We remark that the inequality agrees with the intuition that spreading the charges out over a larger distance will decrease their electrostatic energy. We now replicate Dyson and Lenard’s original proof of the inequality. It is based on the intuition that replacing point charges with uniform distributions on spheres of the same total charge will not change the electrostatic energy of the configuration, along with the fact that the total energy is always positive. Consider a distribution of charges $\mu$ supported on spheres of radii $\{\rho_j\}_{j=1}^n$ centered at $\{x_j\}_{j=1}^n$, each carrying the total charge $\{q_j\}_{j=1}^n$. In other words, the sphere $B_j$ centered at $x_j$ with radius $\rho_j$ carries uniform surface charge density $\frac{d\mu_j}{dA} = \frac{q_j}{4\pi \rho_j^2}$ (here, $dA$ denotes the surface area measure). At any point $x$ outside of these spheres, this distribution of charge generates the potential $$U(x) = \sum_{j=1}^n \frac{q_j}{4\pi |x - x_j|}.$$ The desired inequality will follow from the positivity of the total energy $W$ of this distribution of charge: $$W = \int U(x) d \mu (x) = \int_{{\mathbb{R}}^3} |\nabla U(x)|^2 dx \geq 0.$$ The latter calculation is well known in potential theory (see [@Landkof], for example), and is a straighforward corollary of Green’s formula. We can decompose the total energy into two parts, the self-energy of the spheres and the mutual pairwise energy: $$W = \sum_{j=1}^n \int_{B_j}U_{\{q_j\}}d\mu_j + \sum_{k\neq j, k = 1}^n \sum_{j=1}^n q_j U_{\{q_k\}}(x_j) = \sum_{j=1}^n\frac{q_j^2}{4\pi \rho_j} + 2 \sum_{1 \leq j < k \leq n} \frac{q_j q_k}{4\pi|x_j - x_k|}.$$ Therefore, the positivity of the total energy can be rewritten as $$\label{inequality1} \sum_{j=1}^n \frac{q_j^2}{4\pi \rho_j} > 2 \sum_{1 \leq j < k \leq n} -\frac{q_j q_k}{4\pi|x_j - x_k|}.$$ Notice that the right hand side is just the expression for the electrostatic energy of the $n$ point charges. We have some freedom in the above inequality in picking the radii of the spheres; one particularly good choice will be to pick the radii to be as large as possible without having any of the spheres intersect; thus, we choose $$\rho_j = \frac{1}{2}\min_{k \neq j}|x_j - x_k| = \frac{1}{2}\delta_j$$ Then the inequality becomes $$\sum_{j=1}^n \frac{q_j^2}{\delta_j} > \sum_{1 \leq j < k \leq n} -\frac{q_j q_k}{|x_j - x_k|}.$$ Finally, in the case where $q_j \in \{\pm 1\}$, we obtain the original Onsager’s inequality: $$\sum_{j=1}^n \frac{1}{\delta_j} > \sum_{1 \leq j < k \leq n} -\frac{q_j q_k}{|x_j - x_k|}.$$ The reader may have noticed that the inequality relies on redistribution of the point charges over spheres, and the object of interest is really the self-energy of the spheres (i.e. the expression on the left hand side of equation ). One may wonder if this inequality can be improved by redistributing the total charge of each point charge differently, for example by replacing the point charge with a uniform volume distribution over the ball of radius $\rho$ (of equal total charge). In fact, the uniform surface distribution yields the optimal estimate from this perspective, since this distribution of charge yields the smallest self-energy. This follows readily from the fact that the equilibrium measure of the ball in ${\mathbb{R}}^3$ is the uniform distribution on its surface. Generalizations of the Onsager Inequality. ------------------------------------------ We can easily generalize the Onsager inequality to ${\mathbb{R}}^d$, by considering the appropriate electrostatic (or, Coulomb) potential in ${\mathbb{R}}^d$: \[Or1\] For $d > 2$, and with all notations adopted from Section 1, the Onsager inequality becomes $$\label{q2} 2^{d-3} \sum_{j=1}^n \frac{q_j^2}{\delta_j^{d-2}} > -\sum_{1\le j<k \le n} \frac{q_j q_k}{|x_j - x_k |^{d-2}}.$$ The proof is virtually identical to the one in ${\mathbb{R}}^3$; we provide a sketch here. Consider a distribution of charges $\mu$ supported on spheres of radii $\{\rho_j\}_{j=1}^n$ centered at $\{x_j\}_{j=1}^n$, each carrying total charge $\{q_j\}_{j=1}^n$. In other words, the sphere centered at $x_j$ with radius $\rho_j$ carries uniform surface charge density $\frac{q_j}{\omega_{d-1} \rho_j^{d-1}}$. (Here, as in §1, $\omega_{d-1}$ denotes the surface area of the unit sphere in ${\mathbb{R}}^d$.) Again, the pivotal fact is the positivity of the total energy: $$W = \sum_{j=1}^n \frac{q_j^2}{\omega_{d-1} \rho_j^{d-2}} + 2\sum_{1\leq j<k \leq n}\frac{q_j q_k}{\omega_{d-1} |x_j - x_k |^{d-2}} > 0.$$ As before, a sharper upper bound is provided by letting each sphere become tangent to its nearest neighbor, i.e., by choosing: $$\rho_j = \frac{1}{2} \min_{k \ne j} |x_j - x_k| = \frac{1}{2} \delta_j.$$ Thus, we obtain that $$2^{d-3} \sum_{j=1}^n \frac{q_j^2}{\delta_j^{d-2}} > \sum_{1\le j<k \le n} -\frac{q_j q_k}{|x_j - x_k |^{d-2}}.$$ Finally, letting $q_j = \pm 1$ yields an inequality equivalent to : $$\label{ineq} 2^{d-3} \sum_{j=1}^n \frac{1}{\delta_j^{d-2}} > \sum_{1\le j<k \le n} -\frac{q_j q_k}{|x_j - x_k |^{d-2}}.$$ We remark that can hold for any number of point charges; both sides of the inequality can be made arbitrarily small by choosing a configuration of charges with the nearest distance $\delta = \min_j{\delta_j}$ to be sufficiently large. Thus, Onsager’s inequality is strict, and only becomes an equality when the charges are moved away to infinity. In $d = 2$ dimensions, the Coulomb interaction is no longer a power law; moreover, the total energy of a distribution of charge is no longer necessarily positive. The positivity of the total energy is indispensible in the proof above; thus no version of the Onsager inequality as general as the one for ${\mathbb{R}}^d$ ($d > 2$) exists. However, if we impose additional conditions [^2] to guarantee the energy postivity constraint, an Onsager-like inequality can be written down. However, the result is rather artificial, since it is only valid for very specific charge configurations; thus, we omit it. Intersection Theorem. ===================== As we have already seen, two-dimensional electrostatics is rather special; the following theorem is no exception to this rule. The theorem may have been known earlier, and the authors would be interested to see if one could find the earliest instance of it in the literature. We remark that it is a theorem about point charges in *unstable* equilibrium, in comparison to the celebrated Earnshaw’s theorem, which ensures that unstable equilibria are the only nontrivial equilibrium configurations. In other words, the potential of an electrostatic field cannot have local minima (or maxima) in space free of charges, only saddle points (the proof, of course, is obvious, since the electrostatic potential is a harmonic function away from the support of the charges – cf. [@Earnshaw]). Intersection Theorem: A Necessary Condition for Equilibrium. ------------------------------------------------------------ Let $\{z_i\}_{i=1}^n$ be point charges in the complex plane, with corresponding charges $\{q_i\}_{i=1}^n$. Assume that the charges are in *equilibrium*, i.e. the electrostatic force acting on each charge is zero: $$\label{equilibrium} \sum_{j \neq i} \frac{q_j}{z_i - z_j} = 0,$$ for each $i = 1, ... , n$ (The field intensity is $-\frac{1}{\pi z}$, as a consequence of the fact that the Coulomb kernel is $\frac{1}{2\pi}\log\frac{1}{|z|}$). The Intersection Theorem provides necessary conditions for these charges to be in equilibrium: \[prop3.1\] Let $\{z_i\}_{i=1}^n$ be point charges (with corresponding charges $\{q_i\}_{i=1}^n$) in equilibrium in the complex plane. Then, necessarily, $$\label{Abanov} \sum_{i=1}^n q_i^2 = \bigg( \sum_{i=1}^n q_i \bigg)^2.$$ Furthermore, for each $k \geq 0$, we must also have that $$\label{eq-relations} (k+1)\sum_{i=1}^n q_i^2 z_i^k = \sum_{\ell = 0}^k \bigg(\sum_{i=1}^n q_i z_i^\ell \bigg)\bigg(\sum_{j=1}^n q_j z_j^{k-\ell}\bigg).$$ \[sphere\] The name chosen for this result follows from a geometric interpretation of Eq. : consider the point in $\mathbb{R}^n$, with coordinates $\{q_k\}_{k=1}^n$. Then a simple way of interpreting is to say that it describes the intersection between the hyperplanes $\sum_k q_k = \pm |Q|$ and the sphere of radius $|Q|$, centered at the origin. For example, this shows that for $n = 2$ the only solution is trivial, i.e. only one charge can be non-zero. For $n > 1$, implies that the charges $\{q_j\}_{j=1}^n$, if we think of them as vectors in ${\mathbb{C}}^n$ with real coordinates, must satisfy infinitely many quadratic equations (i.e., lie in the intersection of infinitely many quadrics in ${\mathbb{C}}^n$, with coefficients depending on the positions $z_j \in {\mathbb{C}}$ where the charges sit. The special case implies that, if $\sum q_j = Q = 0$, equilibrium never occurs. Obviously, the configurations that are in equilibrium (and hence satisfy the infinitely many equations ) are very special. But they do exist! For example, if we equidistribute $n-1$ equal charges $q$ at the vertices of a regular $(n-1)$-gon on the unit circle $\{|z| = 1\}$, and then place a charge $q_n := -\frac{q (n-2)}{2}$ at the center of the circle, the total force acting on each charge will be zero. Hence, for this configuration, (and also ) hold. Define functions $g(z) = \prod_{i=1}^n (z-z_i)^{q_i}$, and $G(z) = (\partial_z \log g(z) )^2$ ($G(z)$ can be interpreted as the square of the complex electric field). Consider the expansion of $G(z)$ near $\infty$ in two different ways: first, write $$\begin{aligned} G(z) &= \sum_{i = 1}^n \frac{q_i^2}{(z-z_i)^2} + \sum_{i \neq j} \frac{q_i q_j}{z_i - z_j}\bigg( \frac{1}{z-z_i} - \frac{1}{z-z_j}\bigg) \label{E-squared}\\ &= \sum_{i = 1}^n \frac{q_i^2}{(z-z_i)^2} \nonumber, \end{aligned}$$ using the equilibrium condition by first summing over $i$ to get rid of the second term in . Expanding $G(z)$ at infinity, we find that $$G(z) = \sum_{k=0}^\infty \frac{1}{z^{k+2}} \bigg((k+1)\sum_{i=1}^n q_i^2 z_i^k\bigg).$$ Now, expand $G(z)$ without taking into account the condition for equilibrium: $$G(z) = \sum_{k=0}^\infty \frac{1}{z^{k+2}} \bigg\{\sum_{\ell = 0}^k \bigg(\sum_{i=1}^n q_i z_i^\ell\bigg) \bigg(\sum_{j=1}^n q_j z_j^{k-\ell}\bigg)\bigg\}.$$ Since these expansions must be the same, we can equate their coefficients termwise and obtain $$(k+1)\sum_{i=1}^n q_i^2 z_i^k = \sum_{\ell = 0}^k \bigg(\sum_{i=1}^n q_i z_i^\ell \bigg)\bigg(\sum_{j=1}^n q_j z_j^{k-\ell}\bigg),$$ for each $k \geq 0$. Every one of the above conditions must necessarily hold for the charges to be in equilibrium. Generalization to Compactly Supported Charge Distributions. ----------------------------------------------------------- The Intersection Theorem may be generalized to compactly supported charge distributions, as well. \[prop3.2\] Let $\rho(z)$ be a continuous density of charge compactly supported on a bounded domain $\Omega \subset {\mathbb{C}}$. Suppose again the charges are in equilibrium. Then, necessarily, $$(k+1)\int_\Omega \rho^2(\zeta) \zeta^k dA(\zeta) = \sum_{\ell = 0}^k \bigg(\int_\Omega \rho(\zeta)\zeta^\ell dA(\zeta) \int_\Omega \rho(\zeta)\zeta^{k -\ell} dA(\zeta)\bigg), \, \forall k \in \mathbb{N},$$ where $dA$ denotes the Lebesgue area measure. The analog of the function $G(z)$ becomes: $$\widetilde{G}(z) = \int_\Omega \int_\Omega \frac{\rho(\zeta) \rho(w)}{(z-\zeta)(z-w)} dA(\zeta) dA(w).$$ Again, let us compute $\widetilde{G}(z)$ in two different ways: first, we rewrite $\widetilde{G}(z)$ as $$\widetilde{G}(z) = \int \int_{\zeta \neq w} \frac{\rho(\zeta) \rho(w)}{(z-\zeta)(z-w)} dA(\zeta) dA(w) +\int_\Omega \frac{\rho^2(\zeta)}{(z-\zeta)^2} dA(\zeta).$$ The second integral is understood in the Cauchy Principal Value sense and is known as the Hilbert transform, or Beurling transform [^3] in 2D – cf. [@Ahlfors; @Dra]. The first integral can be rewritten as $$\begin{aligned} \int \int_{\zeta \neq w} \frac{\rho(\zeta) \rho(w)}{(z-\zeta)(z-w)} dA(\zeta) dA(w) &= \int \int_{\zeta \neq w} \frac{\rho(\zeta) \rho(w)}{\zeta-w} \bigg(\frac{1}{z-\zeta} - \frac{1}{z-w}\bigg) dA(\zeta) dA(w) \\ &= \int_\Omega \frac{\rho(\zeta)}{z-\zeta} \bigg[\int_{w\neq \zeta} \frac{\rho(w)}{\zeta - w}dA(w)\bigg]dA(\zeta) \\ &- \int_\Omega \frac{\rho(w)}{z-w} \bigg[\int_{\zeta\neq w} \frac{\rho(\zeta)}{\zeta - w}dA(\zeta)\bigg]dA(w) \\ &= 0, \end{aligned}$$ as $\int_{\zeta\neq w} \frac{\rho(\zeta)}{\zeta - w}dA(\zeta) = 0$ is the condition for equilibrium. Therefore, we have that $$\widetilde{G}(z) = \int_\Omega \frac{\rho^2(\zeta)}{(z-\zeta)^2} dA(\zeta).$$ Expanding this expression, we find that $$\widetilde{G}(z) = \sum_{k=0}^{\infty} \frac{1}{z^{k+2}} \bigg((k+1)\int_\Omega \rho^2(\zeta) \zeta^k dA(\zeta)\bigg).$$ On the other hand, expanding $\widetilde{G}(z)$ without taking into account the condition for equilibrium, we obtain $$\widetilde{G}(z) = \sum_{k=0}^{\infty} \frac{1}{z^{k+2}} \sum_{\ell = 0}^k \bigg(\int_\Omega \rho(\zeta)\zeta^\ell dA(\zeta) \int_\Omega \rho(\zeta)\zeta^{k -\ell} dA(\zeta)\bigg).$$ Equating the coefficients, we find the following sequence of conditions for equilibrium: $$(k+1)\int_\Omega \rho^2(\zeta) \zeta^k dA(\zeta) = \sum_{\ell = 0}^k \bigg(\int_\Omega \rho(\zeta)\zeta^\ell dA(\zeta) \int_\Omega \rho(\zeta)\zeta^{k -\ell} dA(\zeta)\bigg).$$ In particular, for $k = 0$ we obtain the expected continuous analog of Intersection Theorem : $$\int_\Omega \rho^2(\zeta)dA(\zeta) = \bigg( \int_\Omega \rho(\zeta) dA(\zeta)\bigg)^2.$$ It is interesting to note that the necessary condition for $k=0$ does not involve the actual configuration $\{ z_j \}$, but only the values of the charges $\{ q_j\}$. This peculiar fact can be traced back to the different scaling behavior of equilibrium configurations in $\mathbb{R}^2$ versus $\mathbb{R}^d, \, d \ne 2$: upon scaling an equilibrium configuration $\{ x_j \} \to \{ \lambda x_j \}, \lambda > 0$, the total energy scales as $\lambda^{1-d}$ for $d \ne 2$, but for $d = 2$ it only acquires an additive term proportional to the difference between the two sides in : $$W[\{ \lambda z_j \}] - W[\{z_j\}] = \frac{1}{2\pi} \left ( \left (\sum_j q_j \right )^2 - \sum_j q_j^2\right ) \ln \lambda$$ Therefore, for $d = 2$, the necessary condition follows from the fact that, if it were not satisfied, moving all the charges according to an infinitesimal dilation would lead to a lower energy ($\lambda < 1$ if the right-hand side in is larger than the left, and $\lambda > 1$ otherwise), which would mean the initial configuration was not in equilibrium. For all $d \ne 2$, this reasoning fails, and any necessary condition seems to require the explicit dependence on the configuration itself (positions $\{ x_j\}$). It would be interesting to pursue this theme further. Furthermore, Propositions \[prop3.1\], \[prop3.2\] provide *necessary* conditions for equilibrium; one wonders if, taken collectively, they are indeed also *sufficient*. We think it is a compelling and possibly challenging problem to determine either (a.) if these equations are sufficent for equilibrium, or, if they are not, (b.) find a corresponding collection of sufficient conditions. The above theorem admits an easy generalization to any number of dimensions. Consider a pairwise interaction between a collection of particles $\{x_j\}_{j=1}^n \subset {\mathbb{R}}^d$. Assume that the energy of the interaction depends only on the charges of the particles $\{q_j\}$ and the pairwise distances between them $|x_i - x_j|$; these are natural assumptions. The total pairwise energy of the $n$-particle configuration is then given by $$W := \sum_{i,j, i \neq j} q_i q_j \Phi(|x_i - x_j|),$$ where $\Phi$ is some given function characterizing the interaction. Then, the force acting on particle $i$ is $$F_i = -\nabla_{x_i} \bigg( \sum_{j \neq i} q_i q_j \Phi(|x_i - x_j|) \bigg),$$ and the equilibrium condition then is that $F_i = 0$, $i = 1, ..., n$, i.e. $$\label{cnd} \sum_{j \neq i} q_i q_j \frac{x_i - x_j}{|x_i - x_j|}\Phi'(|x_i - x_j|) = 0,$$ $i = 1, ..., n$. We can rewrite through the logarithmic derivative of $\Phi$, as $$ \sum_{j \neq i} q_i q_j \frac{x_i - x_j}{|x_i - x_j|^2} \left [r \Phi'(r) \right ]_{r = |x_i - x_j|} = 0.$$ Denote $$\label{mass} M_{ij} := \frac{q_i q_j }{|x_i - x_j|^2} \left [r \Phi'(r) \right ]_{r = |x_i - x_j|},$$ then obviously $M_{ij} = M_{ji}, i \ne j$. Setting $M_{jj} : = 0, j = 1, 2, \ldots, n$, becomes $$\sum_{j } M_{ij} (x_i -x_j) = 0, \,\, \forall \, i = 1, 2, \ldots, n, $$ so we can multiply each equation by $x_i$ and sum over $i$, to obtain (since $M_{ij} = M_{ji}$) $$\sum_{i,j } M_{ij} (x_i -x_j)\cdot x_i = 0 \Rightarrow \sum_{i,j } M_{ij} (x_j -x_i)\cdot x_j = 0$$ Adding these equations and using $M_{jj} = 0$ leads to the general form of $$\label{ct} \sum_{i \ne j } M_{ij} |x_i -x_j|^2 = 0 \Rightarrow \sum_{i \neq j} q_i q_j \left [r \Phi'(r) \right ]_{r =|x_i - x_j|} = 0.$$ If $\Phi(r) = -\log(r)$, then $r\Phi' = -1$, and is equivalent to . If $ \Phi(r) = r^{-k}$, $k>0$, then $r\Phi' = -k \Phi$, so becomes $$W = \sum_{i,j, i \neq j} q_i q_j \Phi(|x_i - x_j|) = 0,$$ or, equivalently, if the equilibrium exists, the total energy of the system is zero. Degeneracy in Maxwell’s Problem with Planar Charge Distributions. ================================================================= We now address Problem \[p2\], which stems from Maxwell’s conjecture. The following question was first discussed in [@Killian], also cf. [@Peretz]: Consider a distribution of point charges $\mu$ with support contained in a plane $\mathcal{H} \simeq \mathbb{R}^2$ (without loss of generality, we take $\mathcal{H}$ to be the $xy$-plane). Then the critical manifold $C \subset \mathbb{R}{^3}$ of $\mu$, defined by $$C = \{ x \in \mathbb{R}^3 : \nabla U^{\mu}(x) = 0 \}$$ cannot contain a curve in $\mathcal{H}$. In other words, if $C$ contains a curve on which $\nabla U^{\mu}(x) = 0$, then the latter is necessarily transversal to the plane $\mathcal{H}$. To see this, let us assume $C$ contains a curve in $\mathcal{H}$. By a slight abuse of notation, we shall still denote it by $C$. Since the support of $\mu$ is in the plane, we have that $\frac{\partial U^{\mu}}{\partial z} = 0$ in $\mathcal{H}$. Now, consider the analytic hypersurface $\Gamma_\mu := \{(x,y)\in{\mathbb{C}}^2 : U^{\mu}(x,y,0) = const.\}$ in ${\mathbb{C}}^2$. On the curve $C = \Gamma_\mu \cap \mathcal{H}$, we have that $$\bigg(\frac{\partial U^{\mu}}{\partial x}\bigg)^2 + \bigg(\frac{\partial U^{\mu}}{\partial y}\bigg)^2 = 0,$$ since each term vanishes individually on $C$. This implies that $u := U^\mu(x,y,0)$ (considered as an analytic function defined in ${\mathbb{C}}^2 \setminus {\rm{supp}}\, \mu$) satisfies one of the two equations $$\frac{\partial u}{\partial x} + i \frac{\partial u}{\partial y} = 0,\hspace{3.5mm} \text{or}, \hspace{3.5mm} \frac{\partial u}{\partial x} - i \frac{\partial u}{\partial y} = 0.$$ By analytic continuation, the same equation holds on $\Gamma_\mu$. In other words, on $\Gamma_\mu := \{u = const\}$, we have $\frac{\partial u}{\partial x} / \frac{\partial u}{\partial y} = \pm i$. Therefore, $\Gamma_\mu$ must be a complex line (incidentally, a characteristic line for the two-dimensional Laplacian), i.e., $\Gamma_\mu = \{(x,y) \in {\mathbb{C}}^2 : x + iy = const., \text{ or } x - iy = const.\}$. In either case, the intersection $\Gamma_\mu \cap {\mathbb{R}}^2$ is a point, not a curve $C$. This gives the desired contradiction. ### Further observations - We remark that $\mu$ need not consist only of point charges. The argument extends to arbitrary charge distributions $\mu$ as long as the curve $C$ doesn’t ‘cut’ through the support of $\mu$ - Extending this line of argument to ${\mathbb{R}}^3$, assume that $C \subset {\mathbb{R}}^3$ is a 1-dimensional degenerate curve where $\nabla U^\mu = 0$, where $\mu$ is a point charge distribution in ${\mathbb{R}}^3$. Then $C \subset \Gamma_\mu \subset {\mathbb{C}}^3$, where $\Gamma_\mu$ is an analytic hypersurface. It is clear that we cannot claim that $\nabla U^{\mu} = 0$ on $\Gamma_\mu$, but we do have that $(\nabla U^{\mu})^2 = \big(\frac{{\partial}U^\mu}{{\partial}x}\big)^2 + \big(\frac{{\partial}U^\mu}{{\partial}y}\big)^2 + \big(\frac{{\partial}U^\mu}{{\partial}z}\big)^2 = 0$ on $\Gamma_\mu$, i.e., $\Gamma_\mu$ is characteristic with respect to the Laplacian. Expanding the equation for $\Gamma_\mu$ in Taylor series, we find that the lowest nonzero homogeneous terms $v$ in the expansion still satisfy the eikonal equation $(\nabla v)^2 = 0$. As is shown on p. 178 of [@KL], due to the homogeneity of $v$, we see that either $v$ is linear (i.e., $v(x,y,z) = \alpha x + \beta y + \gamma z$, with $\alpha^2 + \beta^2 + \gamma^2 = 0$), or the level set $\{ v = const.\}$ must be the isotropic cone $\Gamma_0 := \{(x,y,z) \in {\mathbb{C}}^3 : (x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 = 0\}$. The intersection of $\{ v = const.\}$ with ${\mathbb{R}}^3$ then must be either a line, or a circle. Thus, up to terms of order $3$ or higher, the curve $C$ must be either a circle or a line. Note that all known examples of degeneracy support this statement; see, for example, A.I. Janušauskas’ examples of a degenerate line through the center of a square with alternating charges $\pm 1$ at the vertices, or the circle of radius 1, centered at the origin, contained in the $y-z$ plane, perpendicular to the $x$-axis with charges $q$ at $x = \pm 1$, and $-q/\sqrt{2}$ at the origin- cf. [@Y]. - In general, the geometry of an equipotential surface in a neighborhood of a degenerate point where the gradient vanishes is rather mysterious in dimensions higher than 2. Maxwell, for one, conjectured that, similarly to the plane, the equipotential surface splits, in the neighborhood of a critical (degenerate) point, into several equidistributed hypersurfaces intersecting each other at equal angles. This is well known to be false – cf. [@Kellogg], the footnote 1 on p. 276, end of Ch. X. Moreover, A. Szulkin, and later J.C. Wood have constructed harmonic polynomials in $\mathbb{R}^3$ whose level set near a critical point (where the gradient vanishes) is homeomorphic to a plane – a shocking surprise, cf. the discussion in Sect. 2.5 of [@Duren] (also, see [@Szulkin] and [@Wood]). Faraday’s Problem. ================== We conclude this exposition wtih the following question, which came up in connection with the seemingly unrelated problem of uniqueness of the best uniform approximation by harmonic functions from approximation theory [@KS]. However, the problem is, in spirit, very close to the subject of this paper. Let $B = \{ x \in {\mathbb{R}}^d : |x| < 1\}$ be the unit ball in ${\mathbb{R}}^d$, $d > 2$, and $\mu$ be a (signed) charge distribution supported on the closure of $B$ which produces the same electrostatic potential outside of ${\overline}{B}$ as the point charge $\delta_0$ centered at the origin. In other words, $$U^{\mu}(x) = \int_{{\overline}{B}} \frac{d\mu(y)}{|x-y|^{d-2}} = \frac{1}{|x|}, \quad |x| > 1.$$ In essence, this condition says that the effect of the *signed* charge density $\mu$ is the same outside the ball as that of a *positive* point charge placed at the origin. \[Faraday-Problem\] There is a positive charge distribution $m$, absolutely continuous with respect to $|\mu|$, which produces the same potential $U^\mu$ as does $\mu$ outside $B$. The requirement of absolute continuity here implies, in particular, that $m$ cannot contain any charge outside of the support of $\mu$. On physical grounds, this conjecture says that, since $U^\mu$ is the same as the potential of a *postive* charge $\delta_0$, one can expect that it is possible to ‘clean out’ the support of $\mu$, getting rid of all negative charges and redistributing the positive charge in such a way that the new charge produces the same effect outside of its support. Conjecture \[Faraday-Problem\] is true in dimension 2, as shown in [@KS]. Yet, the techniques applied there relied heavily on analytic functions and are not available in higher dimensions. However, the result seems reasonable (on physical grounds, at the very least) in all dimensions. Moreover, if this conjecture holds, then (as explained in [@KS]) it has deep and important consequences for the difficult problems of uniqueness of best harmonic approximation in the uniform norm in dimensions larger than 2. [9]{} L. Ahlfors, [*Lectures on Quasiconformal Mappings*]{}, $2^{nd}$ ed., University Lecture Series 38, AMS (2006) O. Dragičevic, [*Analysis of the Ahlfors-Beurling Transform*]{}, Lecture notes for the summer school at the University of Seville, September 9-13, 2013, p. 1-77 P. Duren, [*[Harmonic Mappings in the Plane]{}*]{}, Cambridge University Press, (2004) F. J. Dyson and A. Lenard, Stability of Matter. I, [*J. Math. Phys.*]{} [**8**]{}, 423 (1967) S. Earnshaw, On the nature of the molecular forces which regulate the constitution of the luminiferous ether, [*Trans. Camb. Phil. Soc.*]{} [**7**]{} (1847), 97-112 A. Gabrielov, D. Novikov, and B. Shapiro, Mystery of Point Charges, [*Proc. London Math. Soc.*]{} [**3**]{}, 95 (2007) A. Janušauskas, Critical points of electrostatic potentials. [*Diff. Uravneniya i Primenen-Trudy Sem. Processov Optimal. Upravleniya. I Sekciya* ]{}[**1**]{} (1971), 84-90 J. Junnila, E. Saksman, and C. Webb, Decompositions of log-correlated fields with applications, [*Ann. Appl. Probab.*]{} [**29**]{}, 6 (2019) O. Kellogg, [*[Foundations of Potential Theory]{}*]{}, Ungar, NY, 4th printing (1970) D. Khavinson and E. Lundberg, [*[Linear Holomorphic Partial Differential Equations and Classical Potential Theory]{}*]{}, AMS Math. Surveys and Monographs, v. 232, (2018) D. Khavinson and H. S. Shapiro, Best approximation in the supremum norm by analytic and harmonic functions, [*Ark. Mat.* ]{}[**39**]{}, 2 (2001) K.  Killian, A remark on Maxwell’s conjecture for planar charges, [*Complex Variables and Elliptic Equations*]{} [**54**]{}, 12 (2009) N. S. Landkof. , volume 180 of [*Grundlehren der Mathematischen Wissenschaften*]{}. Springer-Verlag, Berlin, 1972. Elsevier Publishing Company, Amsterdam, (1972) L.  Onsager, Electrostatic Interaction of Molecules, [*J. Chem. Phys.*]{} [**43**]{}, 189 (1939) R. Peretz, Application of the argument principle to Maxwell’s conjecture for three point charges, [*Complex Var. Elliptic Equ.*]{} [**58**]{}, 5 (2013) E. B. Saff and V. Totik. , volume 316 of [*Grundlehren der Mathematischen Wissenschaften*]{}. Springer-Verlag, Berlin, 1997. A. Szulkin, An example concerning topological character of the zero set of a harmonic function, [*Math. Scand.* ]{} [**43**]{}, 60-62 (1978) J.C. Wood, Lewy’s theorem fails in higher dimensions, [*Math. Scand.*]{} [**69**]{}, 166 (1991) [^1]: Lars Onsager was a theoretical physicist and chemist. He was best known for his work in statistical mechanics, in particular his eponymous relations, which won him the Nobel Prize in Chemistry in 1968, and for his exact solution of the 2D Ising model. For further details about the life and work of Onsager, see [@LO]. [^2]: For example, one could consider only charge configurations with total charge $\sum_j q_j = 0$; such configurations are guaranteed to have positive energy. Alternatively, if one imposes that the charges are all confined to the unit disc, positivity of total energy is again ensured. The proof of these facts can be found in [@Landkof]. [^3]: The Beurling transform is the most studied example of the class of Calderón-Zygmund operators. In short, define $$T\rho(z) := \int_\Omega \frac{\rho^2(\zeta)}{(z-\zeta)^2} dA(\zeta) = \lim_{\epsilon\to 0} \int_{\Omega\cap\{|z-\zeta| > \epsilon\}} \frac{\rho^2(\zeta)}{(z-\zeta)^2} dA(\zeta).$$ Then $T$ is a bounded operator from $L^2$ to $L^2$ (with respect to the area measure) and preserves smooth functions.

ArXiv

--- abstract: 'Spectral variability in hyperspectral images can result from factors including environmental, illumination, atmospheric and temporal changes. Its occurrence may lead to the propagation of significant estimation errors in the unmixing process. To address this issue, extended linear mixing models have been proposed which lead to large scale nonsmooth ill-posed inverse problems. Furthermore, the regularization strategies used to obtain meaningful results have introduced interdependencies among abundance solutions that further increase the complexity of the resulting optimization problem. In this paper we present a novel data dependent multiscale model for hyperspectral unmixing accounting for spectral variability. The new method incorporates spatial contextual information to the abundances in extended linear mixing models by using a multiscale transform based on superpixels. The proposed method results in a fast algorithm that solves the abundance estimation problem only once in each scale during each iteration. Simulation results using synthetic and real images compare the performances, both in accuracy and execution time, of the proposed algorithm and other state-of-the-art solutions.' author: - 'Ricardo Augusto Borsoi, , Tales Imbiriba, , José Carlos Moreira Bermudez,  [^1] [^2] [^3][^4] [^5]' bibliography: - 'references.bib' - 'references\_revpaper.bib' title: A Data Dependent Multiscale Model for Hyperspectral Unmixing With Spectral Variability --- Hyperspectral data, spectral variability, spatial regularization, multiscale, superpixels. Introduction ============ Hyperspectral devices acquire hundreds of contiguous reflectance samples from the observed electromagnetic spectra. This observed reflectance is often mixed at the pixel level and requires unmixing strategies to correctly unveil important information about the materials and their proportion in a target scene [@Keshava:2002p5667]. Hyperspectral unmixing (HU) aims at decomposing the observed reflectance in pure spectral components, *i.e.*, *endmembers*, and their proportions [@Keshava:2002p5667], commonly referred as fractional *abundances*. Different models and strategies have been proposed to solve this problem [@Bioucas-Dias-2013-ID307; @Dobigeon-2014-ID322; @Zare-2014-ID324]. The vast majority of methods considers the Linear Mixing Model (LMM) [@Keshava:2002p5667], which assumes that the observed reflectance vector (*i.e.* a hyperspectral image pixel) can be modeled as a convex combination of the endmembers present in the scene. Although this assumption naturally leads to fast and reliable unmixing strategies, the intrinsic limitation of the LMM cannot cope with relevant nonideal effects intrinsic to practical applications [@Dobigeon-2014-ID322; @Imbiriba2016_tip; @Imbiriba2017_bs_tip]. One such important nonideal effect is endmember variability [@Zare-2014-ID324; @drumetz2016variabilityReviewRecent]. Endmember variability can be caused by a myriad of factors including environmental conditions, illumination, atmospheric and temporal changes [@Zare-2014-ID324]. Its occurrence may result in significant estimation errors being propagated throughout the unmixing process [@thouvenin2016hyperspectralPLMM]. The most common approaches to deal with spectral variability can be divided into three basic classes. 1) to group endmembers in variational sets, 2) to model endmembers as statistical distributions, and 3) to incorporate the variability in the mixing model, often using physically motivated concepts [@drumetz2016variabilityReviewRecent]. This work follows the third approach. Recently, [@thouvenin2016hyperspectralPLMM], [@drumetz2016blindUnmixingELMMvariability] and [@imbiriba2018GLMM] introduced variations of the LMM to cope with spectral variability. The Perturbed LMM model (PLMM) [@thouvenin2016hyperspectralPLMM] introduces an additive perturbation to the endmember matrix. Such perturbation matrix then needs to be estimated jointly with the abundances. Though the perturbation matrix can model arbitrary endmember variations, it lacks physical motivation. The Extended Linear Mixing Model (ELMM) proposed in [@drumetz2016blindUnmixingELMMvariability] increased the flexibility of the LMM model by associating a pixel-dependent multiplicative term to each endmember. This generalization can efficiently model changes in the observed reflectances due to illumination, an important effect [@drumetz2016blindUnmixingELMMvariability]. This model addresses a physically motivated problem, with the advantage of estimating a variability parameter vector of much lower dimension when compared with the additive perturbation matrix in PLMM. Although the ELMM performs well in situations where spectral variability is mainly caused by illumination variations, it lacks flexibility when the endmembers are subject to more complex spectral distortions. This motivated the use of additive low-rank terms to the ELMM to deal with more complex types of spectral variability [@hong2019augmentedLMMvariability]. However, this approach does not provide an explicit representation for the endmembers. In [@imbiriba2018GLMM] a physically-motivated generalization to the ELMM was proposed resulting in the Generalized Linear Mixing Model (GLMM). The GLMM accounted for arbitrary spectral variations in each endmember by allowing the multiplicative scaling factors to vary according to the spectral bands, leading to an increased amount of freedom when compared Though the above described models were shown to be capable to model endmember variability effects with good accuracy, their use in HU leads to severely ill-posed inverse problems, which require sound regularization strategies to yield meaningful solutions. One way to mitigate this ill-posedness is to explore spatial correlations found in typical abundance [@eches2011enhancingHSspatialCorrelations] and variability [@drumetz2016blindUnmixingELMMvariability] maps. For instance, spatial information has been employed both for endmember extraction [@zortea2009spatialPreProcessingEMextraction; @torres2014SpatialMultiscaleEMextraction] and for regularization in linear [@zymnis2007HSlinearUnmixingTV], nonlinear [@Jchen2014nonlinearSpatialTVreg], Bayesian [@eches2011enhancingHSspatialCorrelations; @chen2017sparseBayesianMRFunmixingHS; @altmann2015robustUnmixingAnomaly] and sparse [@iordache2012sparseUnmixingTV] unmixing strategies. Total variation (TV) deserves special mention as a spatial regularization approach that promotes spatially piecewise homogeneous solutions without compromising sharp discontinuities between neighboring pixels. This property is important to handle the type of spatial correlation found in many hyperspectral unmixing applications [@shi2014surveySpatialRegUnmixing; @afonso2011augmented]. Although important to mitigate the ill-posedness of the inverse problem, the use of spatial regularization in spectral-variability-aware HU introduces interdependencies among abundance solutions for different image pixels. This in turn leads to intricate, large scale and computationally demanding optimization problems. Even though some approaches have been investigated to accelerate the minimization of convex TV-regularized functionals [@chambolle2017acceleratedAlternatingDescentDykstra; @chambolle2011primalDualTV], this is still a computationally demanding operation which, in the context of HU, have been primarily addressed using variable splitting (e.g. ADMM) techniques [@thouvenin2016hyperspectralPLMM; @drumetz2016blindUnmixingELMMvariability; @imbiriba2018GLMM]. Such complexity is usually incompatible with recent demands for timely processing of vast amounts of remotely sensed data required by many modern real world applications [@ma2015BigDataRemoteSensing; @chi2016BigDataRemoteSensing]. Thus, it is desirable to search for faster and lower complexity strategies that yield comparable unmixing performances. Two recent works have proposed new regularization techniques for ill-posed HU problems aimed at avoiding the interdependency between pixels introduced by standard regularization methods. In [@imbiriba2018_ULTRA] a low-rank tensor regularization strategy named ULTRA was proposed for regularizing ill-posed HU problems. Although ULTRA avoids the pixel interdependency, it requires a canonical polyadic decomposition at every algorithm iteration, which may negatively impact the complexity of the problem for large datasets. In [@Borsoi2017_multiscale] a multiscale spatial regularization approach was proposed for sparse unmixing. The method uses a signal-adaptive spatial multiscale decomposition to break the unmixing problem down into two simpler problems, one in an approximation domain and another in the original domain. The spatial contextual information is obtained by solving an unregularized unmixing problem in the approximation domain. This information is then mapped back to the original image domain and used to regularize the original unmixing problem. The multiscale approach resulted in a fast algorithm that outperformed competing methods, both in accuracy and in execution time, and promoted piecewise homogeneity in the estimated abundances without compromising sharp discontinuities among neighboring pixels. Motivated by the excellent results in [@Borsoi2017_multiscale], we propose in this paper a novel data dependent multiscale mixture model for use in hyperspectral unmixing accounting for spectral variability of the endmembers. The new model uses a multiscale transform to incorporate spatial contextual information into the abundances of a generic mixing model considering spectral variability. The unmixing problem is then formulated as the minimization of a cost function in which a parametric endmember model (e.g. ELMM, PLMM or GLMM) is used to constraint and reduce the ill-posedness of the endmember estimation problem. However, the dimensionality of this problem is still very high since the spatial regularization ties the abundance solutions of all pixels together. Nevertheless, under a few mild assumptions we are able to devise a computationally efficient solution to the abundance estimation problem that can also be computed separately in the two domains. The contributions of this paper include: 1. The proposal of a new regularization strategy based on a multiscale representation of the hyperspectral images and abundance maps. This regularization is significantly different from and improves our previous work [@Borsoi2017_multiscale]. While in [@Borsoi2017_multiscale] the static/fixed endmember matrix for all pixels allowed the easy separation of the abundance estimation process in different domains, the same approach is not applicable to the present case since the variability of the endmember matrix ties the abundances in the approximation and original image domains. 2. A new approximate multiscale decomposition of the generic mixing model considering spectral variability. The new decomposition leads to a separable abundance estimation problem that allows a simple and efficient solution without significantly sacrificing accuracy. Moreover, the solution can be determined in parallel for all image pixels. When compared with approaches that rely on standard spatial regularization strategies and on variable splitting techniques such as ADMM, the proposed strategy leads to a faster iterative algorithm that at each iteration solves the abundance problem only once in each domain. The new algorithm is named *Multiscale Unmixing Algorithm Accounting for Spectral Variability* (MUA-SV). Simulation results clearly show the advantage of the proposed algorithm, both in accuracy and in execution time, over the competing methods. In this paper we represent scalars by lowercase italic letters, vectors by lowercase bold letters and matrices by uppercase bold letters. We denote the number of pixels in an image by $N$, the number of bands by $L$, and the number of endmembers in the scene by $P$. We denote the hyperspectral image and the abundance maps by ${\boldsymbol{Y}}\in\mathbb{R}^{L\times N}$ and ${\boldsymbol{A}}\in\mathbb{R}^{P\times N}$, respectively, whose $n$-th columns are given by ${\boldsymbol{y}}_n$ and ${\boldsymbol{a}}_n$. The endmember matrix for each pixel is denoted by ${\boldsymbol{M}}_n\in\mathbb{R}^{L\times P}$, and ${\boldsymbol{M}}_0\in\mathbb{R}^{L\times P}$ represents a reference endmember matrix extracted from the image ${\boldsymbol{Y}}$. Matrix ${\boldsymbol{W}}\in\mathbb{R}^{N\times S}$ denotes the multiscale transformation and ${\boldsymbol{W}}^*\in\mathbb{R}^{S\times N}$ its conjugate, and subscripts ${\mathcal{C}}$ and ${\mathcal{D}}$ in variables denote their representation in the coarse and detail spatial scales, respectively. The paper is organized as follows. Section \[sec:LMM\] briefly reviews the linear mixing models and its variants accounting for spectral variability. In Section \[sec:multi\_scale\], we present the proposed multiscale formulation for the mixture model. In Section \[sec:unmixing\_prob\_general\] we formulate the unmixing problem using the multiscale approach. The optimization of the resulting cost function is presented in Section \[sec:algorithm\]. In Section \[sec:probA\_reformulation\], we propose an approximate formulation of the abundance estimation problem that leads to a simple and efficient solution. The resulting MUA-SV algorithm is detailed in Section \[sec:mua\_sv\_alg\]. Simulation results with synthetic and real data are presented in Section \[sec:results\]. Section \[sec:conclusions\] presents the conclusions. Linear mixing models considering spectral variability {#sec:LMM} ===================================================== The classical Linear Mixing Model (LMM) [@Keshava:2002p5667] assumes that an $L$-band pixel ${\boldsymbol{y}}_n = [y_{n,1},\,\ldots, \,y_{n,L} ]^\top$ is represented as $$\begin{aligned} \label{eq:LMM} &{\boldsymbol{y}}_n = {\boldsymbol{M}}{\boldsymbol{a}}_n + {\boldsymbol{e}}_n, \quad \text{subject to }\,{\boldsymbol{1}}^\top{\boldsymbol{a}}_n = 1 \text{ and } {\boldsymbol{a}}_n \geq {\boldsymbol{0}} $$ where ${\boldsymbol{M}}\in\mathbb{R}^{L\times P} $ is the endmember matrix whose columns ${\boldsymbol{m}}_i = [m_{i,1},\,\ldots,\,m_{i,L}]^\top$ are the spectral signatures of pure materials[^6], ${\boldsymbol{a}}_n = [a_{n,1},\,\ldots,\,a_{n,P}]^\top$ is the abundance vector, ${\boldsymbol{e}}_n\sim\mathcal{N}(0, \sigma_n^2{\boldsymbol{I}})$ is an additive white Gaussian noise (WGN), and ${\boldsymbol{I}}$ is the identity matrix. The LMM assumes that the endmember spectra are fixed for all pixels ${\boldsymbol{y}}_n$, $n=1,\ldots,N$, in the hyperspectral image. This assumption can compromise the accuracy of estimated abundances in many circumstances due to the spectral variability existing in a typical scene. Recently, variations of the LMM have been proposed to cope with the variability phenomenon [@thouvenin2016hyperspectralPLMM; @drumetz2016blindUnmixingELMMvariability; @imbiriba2018GLMM; @borsoi2019deepGun; @hong2019augmentedLMMvariability]. This work considers models that are linear on the abundances. The most general form of the LMM considering spectral variability generalizes  to allow a different endmember matrix for each pixel. This results in the following observation model for the $n$-th pixel $$\begin{aligned} \label{eq:model_variab_general} {\boldsymbol{y}}_n = {\boldsymbol{M}}_n {\boldsymbol{a}}_n + {\boldsymbol{e}}_n, \qquad n = 1, \dots, N\end{aligned}$$ where ${\boldsymbol{M}}_n\in\mathbb{R}^{L\times P}$ is the $n$-th pixel endmember matrix. This model can also be written for all pixels as $$\begin{aligned} \label{eq:model_variab_general_allpx} {\boldsymbol{Y}}= \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1, \ldots, {\boldsymbol{M}}_N{\boldsymbol{a}}_N\big] + {\boldsymbol{E}}\,,\end{aligned}$$ where ${\boldsymbol{Y}}=[{\boldsymbol{y}}_1,\ldots,{\boldsymbol{y}}_N]$ is the matrix with all observed pixels and ${\boldsymbol{E}}=[{\boldsymbol{e}}_1,\ldots,{\boldsymbol{e}}_N]$ is the noise. Different models have been recently proposed to represent endmember variability as a parametric function of some reference endmember spectral signatures [@thouvenin2016hyperspectralPLMM; @drumetz2016blindUnmixingELMMvariability; @imbiriba2018GLMM; @borsoi2019deepGun]. These models are generically denoted by $$\begin{aligned} \label{eq:param_mdl_generic} {\boldsymbol{M}}_n = f({\boldsymbol{M}}_0,{\boldsymbol{\theta}}_n)\end{aligned}$$ where $f$ is a parametric function, ${\boldsymbol{M}}_0\in\mathbb{R}^{L\times P}$ is a fixed reference endmember matrix and ${\boldsymbol{\theta}}_n$ is a vector of parameters used to describe the endmember signatures for the $n$-th pixel. Although different forms have been proposed for $f$, two notable examples are given by the Perturbed Linear Mixing Model (PLMM) [@thouvenin2016hyperspectralPLMM] and the Extended Linear Mixing The PLMM proposed in [@thouvenin2016hyperspectralPLMM] models ${\boldsymbol{M}}_n$ as a fixed matrix ${\boldsymbol{M}}_0$ plus a pixel-dependent variation matrix that can accommodate generic spatial variations. Mathematically, $$\begin{aligned} \label{eq:plmm_model_i} {\boldsymbol{y}}_n {}={} \big({\boldsymbol{M}}_0 + {\boldsymbol{d}}{\boldsymbol{M}}_n\big) {\boldsymbol{a}}_n + {\boldsymbol{e}}_n \,,\end{aligned}$$ where the parameters of this model are related to those of  by ${\boldsymbol{\theta}}_n\equiv{\mathrm{vec}}({\boldsymbol{d}}{\boldsymbol{M}}_n)$, where ${\mathrm{vec}}(\cdot)$ is the vectorization operator. This model is not physically motivated. Hence, in most cases all elements of ${\boldsymbol{d}}{\boldsymbol{M}}_n$ must be included as independent variables in the solution of the ill-posed unmixing problem, making the inverse problem very hard to solve. This limitation motivated the development of simpler, physically motivated variability models. The ELMM is a simpler model proposed in [@drumetz2016blindUnmixingELMMvariability]. It incorporates a multiplicative diagonal matrix to LMM, which maintains the directional information of the reference endmembers, but allows them to be independently scaled. The ELMM is expressed as $$\begin{aligned} \label{eq:model_variab_elmm} {\boldsymbol{y}}_n = {\boldsymbol{M}}_0 \,{\mathrm{diag}}({\boldsymbol{\psi}}_n) {\boldsymbol{a}}_n + {\boldsymbol{e}}_n \,,\end{aligned}$$ where ${\boldsymbol{\psi}}_n\in\mathbb{R}^{P}$ is a vector containing a (positive) scaling factor for each endmember, which is related to the parameters of  by ${\boldsymbol{\theta}}_n\equiv{\boldsymbol{\psi}}_n$, and ${\mathrm{diag}}({\boldsymbol{x}})$ denotes a diagonal matrix whose diagonal elements are given by the elements of vector ${\boldsymbol{x}}$. This model is a particular case of  that can model typical endmember variations, such as those caused by illumination variability due to the topography of the scene [@drumetz2016blindUnmixingELMMvariability]. The optimization problem that needs to be solved using model  is much less ill-posed than that generated using model  due to the reduced number of parameters to be estimated. This simplicity is obtained at the price of For both the PLMM and ELMM, the problem of estimating the fractional abundances and the spectral signatures of the endmembers was cast as a large scale, non-convex inverse problem, which was solved using variable splitting procedures [@thouvenin2016hyperspectralPLMM; @drumetz2016blindUnmixingELMMvariability]. The computational cost of these solutions is very high, making them unsuited for processing large amounts of data. Furthermore, the introduction of *a priori* information about the spatial regularity of the abundance maps, which is essential to reduce the ill-posedness of the inverse problem, results in a optimization problem that is not separable per pixel. This significantly increases the computational cost of the solution. Considering this limitation of the models described above, it is of interest to develop new mixture models that combine the generality of the endmember variability patterns that can be considered with the possibility of an efficient solution of the associated inverse problem. In the next section, we introduce a new mixture model that represents separately the image components at different scales using a data-dependent transformation learned from the observed hyperspectral image ${\boldsymbol{Y}}$. This new multiscale representation can be employed to solve the unmixing problem with any parametric model to represent spectral variability that fits the form . The use of this new model results in a method that is able to provide more accurate solutions at a much lower computational cost than the existing methods. A Multiscale Spatial Mixture Model {#sec:multi_scale} ================================== To constrain the set of possible solutions, we propose to separately represent the mixture process in two distinct image scales, namely, the coarse scale containing rough spatial structures, and the fine spatial scale containing small structures and details. By doing so, the conditions for spatial smoothness can be imposed on the relevant parameters in the much simpler coarse scale, and then be translated into the fine scale for further processing. Denote the fractional abundance maps for all pixels by ${\boldsymbol{A}}=[{\boldsymbol{a}}_1,\ldots,{\boldsymbol{a}}_N]$. We consider a transformation ${\boldsymbol{W}}\in\mathbb{R}^{N\times S}$ based on relevant contextual inter-pixel information present in the observed image ${\boldsymbol{Y}}$ to be applied to both ${\boldsymbol{Y}}$ and ${\boldsymbol{A}}$ to unveil the coarse image structures. The transformed matrices are given by $$\begin{aligned} \label{eq:decomposition_calC_i} {\boldsymbol{Y}}_{\!{\mathcal{C}}} = {\boldsymbol{Y}}{\boldsymbol{W}}\,; \quad {\boldsymbol{A}}_{\!{\mathcal{C}}} = {\boldsymbol{A}}{\boldsymbol{W}}\,,\end{aligned}$$ where ${\boldsymbol{Y}}_{\!{\mathcal{C}}}=[{\boldsymbol{y}}_{{\mathcal{C}}_1},\ldots,{\boldsymbol{y}}_{{\mathcal{C}}_S}] \in \mathbb{R}^{L\times S}$ and ${\boldsymbol{A}}_{\!{\mathcal{C}}}=[{\boldsymbol{a}}_{{\mathcal{C}}_1},\ldots,{\boldsymbol{a}}_{{\mathcal{C}}_S}] \in \mathbb{R}^{P\times S}$ with $S \ll N$ are, respectively, the hyperspectral image and the abundance matrix in the coarse approximation scale, denoted by ${\mathcal{C}}$. Being signal dependent, the transformation ${\boldsymbol{W}}$ is nonlinear, and does not bear a direct relationship with the frequency components of the image, though some general relationship exists. The spatial details of the image are represented in the detail scale, denoted by ${\mathcal{D}}$, which is obtained by computing the complement to the transformation ${\boldsymbol{W}}$. Mathematically, $$\begin{aligned} \label{eq:decomposition_calD_i} {\boldsymbol{Y}}_{\!{\mathcal{D}}} = {\boldsymbol{Y}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast) \,; \quad {\boldsymbol{A}}_{\!{\mathcal{D}}} = {\boldsymbol{A}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast) \,,\end{aligned}$$ where ${\boldsymbol{Y}}_{\!{\mathcal{D}}}=[{\boldsymbol{y}}_{{\mathcal{D}}_1},\ldots,{\boldsymbol{y}}_{{\mathcal{D}}_N}]\in \mathbb{R}^{L\times N}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}=[{\boldsymbol{a}}_{{\mathcal{D}}_1},\ldots,{\boldsymbol{a}}_{{\mathcal{D}}_N}] \in \mathbb{R}^{P\times N}$ are the input image and the abundance matrix in the detail scale. Matrix ${\boldsymbol{W}}^\ast\in\mathbb{R}^{S\times N}$ is a conjugate transformation to ${\boldsymbol{W}}$, and takes the images from the coarse domain ${\mathcal{C}}$ back to the original image domain. ${\boldsymbol{Y}}_{\!{\mathcal{D}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ contain the fine scale details of ${\boldsymbol{Y}}$ and ${\boldsymbol{A}}$ in the original image domain. The transformation ${\boldsymbol{W}}$ captures the spatial correlation of the input image, whereas its complement $({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast)$ captures existing fine spatial variabilities. This way it is possible to introduce spatial correlation into the abundance map solutions by separately controlling the regularization strength in each of the scales ${\mathcal{C}}$ and ${\mathcal{D}}$. This is computationally much simpler than to use more complex penalties. By imposing a smaller penalty in the coarse scale ${\mathcal{C}}$ and a larger penalty in the details scale ${\mathcal{D}}$, we effectively favor smooth solutions to the optimization problem. We can define a composite transformation as $$\begin{aligned} \label{eq:TransfcalW} {\mathcal{W}}= [{\boldsymbol{W}}\quad {\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast] \,,\end{aligned}$$ which decomposes the input image into the coarse approximation ${\mathcal{C}}$ and its complement ${\mathcal{D}}$. Note that the transformation is invertible, with a right inverse given by $$\begin{aligned} \label{eq:transf_pinverse} {\mathcal{W}}^\dagger = [{\boldsymbol{W}}^\ast \quad {\boldsymbol{I}}]^\top \,.\end{aligned}$$ Multiplying ${\boldsymbol{Y}}$ from the right by ${\mathcal{W}}$ and considering the generic mixing model for all pixels given in  yields ${\boldsymbol{Y}}{\mathcal{W}}= \big[ {\boldsymbol{Y}}_{\!{\mathcal{C}}} \,\,{\boldsymbol{Y}}_{\!{\mathcal{D}}} \big]$, with $$\begin{aligned} \label{eq:model_decomposed_i} \begin{split} {\boldsymbol{Y}}_{\!{\mathcal{C}}} & = \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N \big] {\boldsymbol{W}}+ {\boldsymbol{E}}_{\!{\mathcal{C}}} \\ {\boldsymbol{Y}}_{\!{\mathcal{D}}} & = \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N \big] ({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast) + {\boldsymbol{E}}_{\!{\mathcal{D}}} \end{split}\end{aligned}$$ where ${\boldsymbol{E}}_{\!{\mathcal{C}}}={\boldsymbol{E}}{\boldsymbol{W}}$ and ${\boldsymbol{E}}_{\!{\mathcal{D}}}={\boldsymbol{E}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast)$ represent the additive noise in the coarse and detail scales, respectively. Image (bands 50, 80 and 100) Superpixels The choice of the multiscale transformation ${\boldsymbol{W}}$ is important for the proposed methodology to achieve a good reconstruction accuracy. Desirable features for this transform are 1) to group image pixels that are spectrally similar and belong to spatially homogeneous regions, and 2) to respect image borders by not grouping pixels that belong to different image structures or features. Additionally, it must be fast to compute. In [@Borsoi2017_multiscale], a superpixel decomposition of the image was considered for the transformation ${\boldsymbol{W}}$. Superpixels satisfy the aforementioned criteria, and have recently been widely applied to hyperspectral imaging tasks, including classification [@jia2017superpixelHIclassification], segmentation [@saranathan2016superpixelsHIsegmentation], endmember detection [@thompson2010superpixelEMdetection], and multiscale regularization [@Borsoi2017_multiscale]. Superpixel algorithms group image pixels into regions with contextually similar spatial information [@achanta2012slicPAMI], decomposing the image into a set of contiguous homogeneous regions. The sizes and regularity of the regions are controlled by adjusting a set of parameters. A particularly fast and efficient algorithm is the Simple Linear Iterative Clustering (SLIC) algorithm [@achanta2012slicPAMI], also considered in [@Borsoi2017_multiscale]. The SLIC algorithm is an adaptation of the k-means algorithm that considers a reduced search space to lower the computational complexity, and a properly defined metric that balances spectral and spatial contributions. The superpixels (clusters) are initialized almost uniformly at low-gradient image neighborhoods to reduce the influence of noise. The number of clusters $S$ is determined as a function of the average superpixel size defined by the user. The clustering employs a normalized distance function that considers both spatial and spectral (color) similarities among pixels. The relative contributions of spatial and spectral components are controlled by a regularity parameter $\gamma$. The parameter $\gamma$ can be increased to emphasize the spatial distance and obtain more compact (lower area to perimeter ratio) superpixels, or decreased to emphasize spectral distances and yield a tighter adherence to image borders (with more irregular shapes). See the supplemental material in [@Borsoi_multiscaleVar_2018] for more details. The decomposition ${\boldsymbol{Y}}{\boldsymbol{W}}$ of the image ${\boldsymbol{Y}}$ returns a set of superpixels. The value of each superpixel is equal to the average of all original pixel values inside that superpixel region. The conjugate transform, ${\boldsymbol{Y}}_{\!\mathcal{C}}{\boldsymbol{W}}^{\ast}$, takes each superpixel in ${\boldsymbol{Y}}_{\!\mathcal{C}}$ and attributes its value to all pixels of the uniform image sampling grid that lie inside its corresponding superpixel region. The successive application of both transforms, ${\boldsymbol{W}}{\boldsymbol{W}}^{\ast}$ effectively consists in averaging all pixels inside each superpixel of the input image. The superpixel decomposition of the Cuprite image using the SLIC algorithm is illustrated The unmixing problem {#sec:unmixing_prob_general} ==================== The spectral unmixing problem with spectral variability can be formulated as the minimization of the cost function $$\begin{aligned} \label{eq:opt_prob_i} \mathcal{J}(\mathbb{M},& {\boldsymbol{\Theta}},{\boldsymbol{A}}) {}={} \frac{1}{2} \big\| {\boldsymbol{Y}}- \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ] \big\|_F^2 \nonumber\\ & + \lambda_A {\mathcal{R}}({\boldsymbol{A}}) + \frac{\lambda_M}{2} \sum_{n=1}^N \|{\boldsymbol{M}}_n-f({\boldsymbol{M}}_0,{\boldsymbol{\theta}}_n)\|_F^2 \nonumber\\ & + \lambda_{{\boldsymbol{\Theta}}} \mathcal{R}({\boldsymbol{\Theta}}) \\ \nonumber & \text{subject to } {\boldsymbol{A}}\geq{\boldsymbol{0}}, \, {\boldsymbol{1}}^\top {\boldsymbol{A}}= {\boldsymbol{1}}^\top, \nonumber \\ \nonumber & \hspace{10ex} {\boldsymbol{M}}_n\geq{\boldsymbol{0}},\,\, n=1,\ldots,N.\end{aligned}$$ where $\mathbb{M}$ is an ${L\times P\times N}$ tensor containing the endmember matrices, with entries given by $[\mathbb{M}]_{:,:,n}={\boldsymbol{M}}_n$ and ${\boldsymbol{\Theta}}=[{\boldsymbol{\theta}}_1,\ldots,{\boldsymbol{\theta}}_N]$ is a matrix containing the parameter vectors of the variability model for all pixels. Note that the generic parametric endmember model ${\boldsymbol{M}}_n=f({\boldsymbol{M}}_0,{\boldsymbol{\theta}}_n)$ of  was included in the cost function  in the form of an additive constraint. This decouples the problem of estimating the abundances from that of estimating the parametric endmember model, allowing the application of the multiscale formulation to other endmember models without loss of generality. Furthermore, this also gives more flexibility to the unmixing solution since the parameter $\lambda_M$ can be adjusted to either allow matrices ${\boldsymbol{M}}_n$ to vary more freely or to strictly enforce the endmember variability model . The regularization functionals $\mathcal{R}({\boldsymbol{A}})$ and $\mathcal{R}({\boldsymbol{\Theta}})$ incorporate prior information about the spatial smoothness of the abundance and about the parameters of the spectral variability model. The abundance maps constraint introduces spatial regularity indirectly through the transformation ${\boldsymbol{W}}$. The constraint is given by $$\begin{aligned} {\mathcal{R}}({\boldsymbol{A}}) & {}={} \frac{\rho}{2}\|{\boldsymbol{A}}{\boldsymbol{W}}\|^2_F + \frac{1}{2} \|{\boldsymbol{A}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast)\|_F^2 \nonumber\\ & {}={} \frac{\rho}{2}\|{\boldsymbol{A}}_{{\mathcal{C}}}\|^2_F + \frac{1}{2} \|{\boldsymbol{A}}_{{\mathcal{D}}}\|^2_F \,\end{aligned}$$ and consists of a quadratic penalization of the multiscale representation of the abundance maps, applied separately to the coarse and detail scales ${\mathcal{C}}$ and ${\mathcal{D}}$. Parameter $\rho$ allows the control of the relative weights of each scale in the abundance penalty. For instance, piecewise smooth abundance solutions to the optimization problem can be promoted by imposing a smaller penalty in the coarse scale ${\mathcal{C}}$ and a larger penalty in the details scale ${\mathcal{D}}$. The constraint $\mathcal{R}({\boldsymbol{\Theta}})$ is selected according to the endmember variability model that is used, and might encode information such as the amount of spectral variability in a scene or spatial correlation in the variables ${\boldsymbol{\theta}}_n$. The parameters $\lambda_{A}$ and $\lambda_{{\boldsymbol{\Theta}}}$ control the balance between the different terms in the cost function. In the following, we employ the ELMM model due to its parsimony and underlying physical motivation [@drumetz2016blindUnmixingELMMvariability]. This results in the following concrete forms for $f$ and ${\boldsymbol{\Theta}}$: $$\begin{aligned} \label{eq:var_mdl_elmm_sel} \begin{split} f({\boldsymbol{M}}_0,{\boldsymbol{\theta}}_n) & {}\equiv{} {\boldsymbol{M}}_0 \,{\mathrm{diag}}({\boldsymbol{\psi}}_n ) \\ {\boldsymbol{\Theta}}& {}\equiv{} {\boldsymbol{\Psi}}\end{split}\end{aligned}$$ where ${\boldsymbol{\Psi}}=[{\boldsymbol{\psi}}_1,\ldots,{\boldsymbol{\psi}}_N]$ is a matrix whose $n$-th column contains scaling factors ${\boldsymbol{\psi}}_n$ of the ELMM model . The scaling maps constraint $\mathcal{R}({\boldsymbol{\Theta}})$ is selected to introduce spatial smoothness to the endmember scaling factors, and is given by $$\begin{aligned} \label{eq:var_reg_elmm_sel} \mathcal{R}({\boldsymbol{\Theta}}) & {}\equiv{} \mathcal{R}({\boldsymbol{\Psi}}) \nonumber \\ & {}={} \|\mathcal{H}_h({\boldsymbol{\Psi}})\|_{F}^2 + \|\mathcal{H}_v({\boldsymbol{\Psi}})\|_{F}^2 \,,\end{aligned}$$ where $\mathcal{H}_h$ and $\mathcal{H}_v$ are linear operators that compute the vertical and horizontal gradients of a bi-dimensional signal, acting separately for each material. In the following, we make the variable substitutions outlined in  and , which turns the cost function in  into $\mathcal{J}(\mathbb{M},{\boldsymbol{\Psi}},{\boldsymbol{A}})$. The estimated abundance maps, endmember matrices and scaling factors can be obtained by minimizing  with respect to (w.r.t.) these variables, resulting in the following optimization problem $$\begin{aligned} \label{eq:opt_prob_i_b} \widehat{\mathbb{M}}, \widehat{{\boldsymbol{\Psi}}}, \widehat{\!{\boldsymbol{A}}} = \mathop{\arg\min}_{\mathbb{M},{\boldsymbol{\Psi}},{\boldsymbol{A}}} \,\,\,\mathcal{J}(\mathbb{M},{\boldsymbol{\Psi}},{\boldsymbol{A}}) \,.\end{aligned}$$ This problem is non-convex and hard to solve directly due to the interdependence between $\mathbb{M}$, ${\boldsymbol{\Psi}}$ and ${\boldsymbol{A}}$. Nevertheless, a local stationary point can be found using an Alternating Least Squares (ALS) strategy, which minimizes  successively with respect to one variable at a time [@xu2013blockCoordinateDescent]. The ALS approach allows us to break  into three simpler problems which are solved sequentially, consisting of: $$\begin{aligned} \label{eq:als_sketch} \hspace{-0.1cm} \begin{split} & \!a) \, \text{minimize } \mathcal{J}(\mathbb{M}|{\boldsymbol{A}},{\boldsymbol{\Psi}}) \text{ w.r.t. } \mathbb{M} \text{ with } {\boldsymbol{A}}\text{ and } {\boldsymbol{\Psi}}\text{ fixed} \\ & \!b) \, \text{minimize } \mathcal{J}({\boldsymbol{\Psi}}|{\boldsymbol{A}},\mathbb{M}) \text{ w.r.t. } {\boldsymbol{\Psi}}\text{ with } {\boldsymbol{A}}\text{ and } \mathbb{M} \text{ fixed} \\ & \!c) \, \text{minimize } \mathcal{J}(\!{\boldsymbol{A}}|\,\mathbb{M},{\boldsymbol{\Psi}}) \text{ w.r.t. } {\boldsymbol{A}}\text{ with } \mathbb{M} \text{ and } {\boldsymbol{\Psi}}\text{ fixed} \end{split}\end{aligned}$$ where $\mathcal{J}({\boldsymbol{B}}_1|{\boldsymbol{B}}_2,{\boldsymbol{B}}_3)$ denotes a cost function $\mathcal{J}$ in which ${\boldsymbol{B}}_1$ is considered a variable and ${\boldsymbol{B}}_2$, ${\boldsymbol{B}}_3$ are fixed and Although this strategy yields a local minimum of the non-convex problem  by solving a sequence of convex optimization problems, it is still computationally intensive, specially due to the abundance estimation problem. This is because the spatial regularization term ${\mathcal{R}}({\boldsymbol{A}})$ in  imposes interdependency among the different pixels of ${\boldsymbol{A}}$, what also happens when the TV regularization is employed [@drumetz2016blindUnmixingELMMvariability; @shi2014surveySpatialRegUnmixing]. Each of the optimization subproblems of the ALS strategy in  will be treated in detail in the next section. Furthermore, in Section \[sec:probA\_reformulation\] we will present a multiscale formulation that eliminates the interdependency of the abundance estimation problem between the different image pixels, allowing the solution to be computed faster and in parallel. Formulation and solutions to the optimization problems in  {#sec:algorithm} ========================================================== We now detail the solution to each of the optimization problems in the ALS strategy outlined in . Although the solutions to the minimization problems w.r.t. $\mathbb{M}$ and ${\boldsymbol{\Psi}}$ are relatively straightforward and directly amenable to paralel or efficient implementations, optimizing  w.r.t. ${\boldsymbol{A}}$ proves to be significantly more challenging due to the multiscale spatial regularization term $\mathcal{R}({\boldsymbol{A}})$. Nevertheless, by making some approximations in Section \[sec:probA\_reformulation\], we will reformulate this optimization problem as a function of the multiscale representations ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ of the abundances. This will allow the extension of the ALS strategy to consider separate minimization steps w.r.t. ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$, leading to a simple and parallelizable solution. The complete algorithm including all optimization steps will be detailed in Section \[sec:mua\_sv\_alg\]. Optimizing with respect to $\mathbb{M}$ at the $i$-th iteration --------------------------------------------------------------- The cost function in this case is $\mathcal{J}(\mathbb{M}\,|{\boldsymbol{A}},{\boldsymbol{\Psi}})$, where $\mathbb{M}$ is a variable and ${\boldsymbol{A}}$ and ${\boldsymbol{\Psi}}$ are fixed at the solutions obtained in the previous iteration. Then, $$\begin{aligned} \label{eq:opt_subprob_M_i} & \mathcal{J}(\mathbb{M}\,|{\boldsymbol{A}},{\boldsymbol{\Psi}}) \nonumber \\ & {}={} \frac{1}{2} \! \sum_{n=1}^N \Big(\|{\boldsymbol{y}}_n - {\boldsymbol{M}}_n {\boldsymbol{a}}_n\|^2_2 + \lambda_M \|{\boldsymbol{M}}_n-{\boldsymbol{M}}_0 \,{\mathrm{diag}}({\boldsymbol{\psi}}_n)\|_F^2 \Big) \nonumber \\ & \hspace{5ex}\text{subject to } {\boldsymbol{M}}_n\geq{\boldsymbol{0}}, \, n=1,\ldots, N\end{aligned}$$ Similarly to [@drumetz2016blindUnmixingELMMvariability], we compute an approximate solution to minimize  for each image pixel as $$\begin{aligned} \label{eq:opt_subprob_M_ii} & \widehat{\!{\boldsymbol{M}}}_n \\ & \hspace{1ex} \nonumber = \mathcal{P}_{\!+}\big(\big({\boldsymbol{y}}_n{\boldsymbol{a}}_n^\top + \lambda_M{\boldsymbol{M}}_0\,{\mathrm{diag}}({\boldsymbol{\psi}}_n)\big) \big({\boldsymbol{a}}_n{\boldsymbol{a}}_n^\top + \lambda_M {\boldsymbol{I}}\big)^{-1}\big)\end{aligned}$$ where $\mathcal{P}_{\!+}(\cdot)$ is an operator that projects each element of a matrix onto the nonnegative orthant by thresholding any negative element to zero. Optimizing with respect to ${\boldsymbol{\Psi}}$ at the $i$-th iteration ------------------------------------------------------------------------ The cost function in this case is $\mathcal{J}({\boldsymbol{\Psi}}\,|\,\mathbb{M},{\boldsymbol{A}})$, where ${\boldsymbol{\Psi}}$ is a variable and ${\boldsymbol{A}}$ and $\mathbb{M}$ are fixed at the solutions obtained in the previous iteration. Then, $$\begin{aligned} \label{eq:opt_subprob_psi_i} \mathcal{J}({\boldsymbol{\Psi}}| \mathbb{M},\!{\boldsymbol{A}}) \!=\! \frac{\lambda_M}{2} \! \sum_{n=1}^N \|{\boldsymbol{M}}_n-{\boldsymbol{M}}_0{\boldsymbol{\psi}}_n\|_F^2 + \lambda_{{\boldsymbol{\Theta}}} \mathcal{R}({\boldsymbol{\Psi}}).\end{aligned}$$ We follow the approach detailed in [@drumetz2016blindUnmixingELMMvariability Eqs. (20)-(23)] to minimize . Optimizing with respect to ${\boldsymbol{A}}$ at the $i$-th iteration {#sec:A_opt} --------------------------------------------------------------------- The cost function in this case is $\mathcal{J}({\boldsymbol{A}}\,|\,\mathbb{M},{\boldsymbol{\Psi}})$, where ${\boldsymbol{A}}$ is a variable and ${\boldsymbol{\Psi}}$ and $\mathbb{M}$ are fixed at the solutions obtained in the previous iteration. Then, $$\begin{aligned} \label{eq:opt_prob_a_minusOne} \mathcal{J}({\boldsymbol{A}}\,|\,\mathbb{M},{\boldsymbol{\Psi}}) & {}={} \frac{1}{2} \big\| {\boldsymbol{Y}}- \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ] \big\|_F^2 \nonumber\\& + \frac{\rho\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{C}}} \|_F^2 + \frac{\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{D}}} \|_F^2 \\ \nonumber & \text{subject to } {\boldsymbol{A}}\geq{\boldsymbol{0}}, \, {\boldsymbol{1}}^\top {\boldsymbol{A}}= {\boldsymbol{1}}^\top \\ \nonumber & \hspace{9.5ex} {\boldsymbol{A}}_{\!{\mathcal{C}}}={\boldsymbol{A}}{\boldsymbol{W}}, \, {\boldsymbol{A}}_{\!{\mathcal{D}}}={\boldsymbol{A}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^*)\end{aligned}$$ Using the multiscale transformation ${\mathcal{W}}$ to write  as a function of the observed hyperspectral images ${\boldsymbol{Y}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{Y}}_{\!{\mathcal{D}}}$ represented at the coarse and detail scales yields $$\begin{aligned} \label{eq:opt_prob_a_zero} \mathcal{J}({\boldsymbol{A}}&\,|\,\mathbb{M},{\boldsymbol{\Psi}}) {}={} \frac{1}{2} \big\|{\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ] {\boldsymbol{W}}{\boldsymbol{W}}^\ast \big\|_F^2 \nonumber \\ & + \frac{1}{2} \big\|{\boldsymbol{Y}}_{\!{\mathcal{D}}} - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ]({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast)\big\|_F^2 \nonumber\\ & + {\mathrm{tr}}\Big\{ \big({\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ] {\boldsymbol{W}}{\boldsymbol{W}}^\ast \big)^\top \nonumber \\ & \hspace{0.75cm} \cdot \big( {\boldsymbol{Y}}_{\!{\mathcal{D}}} - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N ]({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast)\big)\Big\} \nonumber \\ & + \frac{\rho\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{C}}} \|_F^2 + \frac{\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{D}}} \|_F^2 \\ \nonumber & \text{subject to } {\boldsymbol{A}}\geq {\boldsymbol{0}}, \,\, {\boldsymbol{1}}^\top {\boldsymbol{A}}= {\boldsymbol{1}}^\top \\ \nonumber & \hspace{9.5ex} {\boldsymbol{A}}_{\!{\mathcal{C}}}={\boldsymbol{A}}{\boldsymbol{W}}, \, {\boldsymbol{A}}_{\!{\mathcal{D}}}={\boldsymbol{A}}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^*) \nonumber\end{aligned}$$ where ${\mathrm{tr}}(\cdot)$ is the matrix trace operator. Cost function  is neither separable with respect to the abundance matrices ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ in the coarse and detail scales, nor with respect to the image pixels. This can severely impact the required computational load and the convergence time to a meaningful result. To mitigate this issue, in the following section we propose to use few reasonable approximations to turn the minimization of  into an optimization problem separable in ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$. This will remove the interdependency between the different image pixels, and allow the extension of the ALS strategy to consider the optimization w.r.t. ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ successively, instead of w.r.t. ${\boldsymbol{A}}$. Modification and solution to the optimization problem w.r.t. ${\boldsymbol{A}}$ {#sec:probA_reformulation} =============================================================================== Initially, we note that the cost function  does not depends on the endmember variability model $f({\boldsymbol{M}}_0,{\boldsymbol{\theta}}_n)$. Hence, the derivations presented in this section are not limited to the ELMM, and can be equally applied to other models without loss of generality. Residuals inner product {#sec:residuals} ----------------------- To proceed, we first denote by $RE_{{\mathcal{C}}}$ and $RE_{{\mathcal{D}}}$ the residuals/reconstruction errors in each image scale ${\mathcal{C}}$ and ${\mathcal{D}}$, where $RE_{{\mathcal{C}}}$ and $RE_{{\mathcal{D}}}$ are given by $$\begin{aligned} \begin{split} RE_{{\mathcal{C}}} & {}={} {\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N \big] {\boldsymbol{W}}{\boldsymbol{W}}^\ast \\ RE_{{\mathcal{D}}} & {}={} {\boldsymbol{Y}}_{\!{\mathcal{D}}} - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_1 \ldots {\boldsymbol{M}}_N{\boldsymbol{a}}_N \big]({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast) \,. \end{split}\end{aligned}$$ It follows from the above definition that the third term in the cost function  consists of the inner product $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle$ between the residuals/reconstruction errors at the coarse and detail scales. This inner product, however, usually contributes a small value to the cost function, and can be neglected under the following assumption: - **Zero-mean, uncorrelated residuals:** We assume that for ${\boldsymbol{A}}$ a critical point of , $RE_{{\mathcal{C}}}$ and $RE_{{\mathcal{D}}}$ are spatially zero-mean and uncorrelated across scales. This is reasonable if the observation/mixing model given by the ELMM in  represents the data with reasonable accuracy, in which case the main contribution towards the residual error comes from the observation noise ${\boldsymbol{e}}_n$, which is white and spatially uncorrelated. If is satisfied, then the term $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle$ can be neglected when compared to the first two terms without significantly altering the critical point. Although neglecting the third term of  simplifies the optimization problem, the first two terms still encompass intricate relationships between the abundances at different pixels due to the action of the multiscale transformation ${\boldsymbol{W}}$. Furthermore, the optimization problem still involves terms depending on both ${\boldsymbol{A}}$ and the pair $({\boldsymbol{A}}_{\!{\mathcal{C}}}$, ${\boldsymbol{A}}_{\!{\mathcal{D}}})$, which are related through ${\mathcal{W}}$, and thus cannot be easily solved in this form. In order to proceed, we make the following assumption: - **Spatially smooth endmember signatures:** We assume that the pixel-by-pixel endmember signatures ${\boldsymbol{M}}_n$ are similar in small, compact spatial neighborhoods. More precisely, if $\mathcal{N}$ is a set of pixels comprising a compact spatial neighborhood, we assume that the endmember signature of any pixel in $\mathcal{N}$ does not deviate significantly from the average signature, so that the quantity $$\bigg\| {\boldsymbol{M}}_j - \frac{1}{|\mathcal{N}|} \sum_{n\in\mathcal{N}} {\boldsymbol{M}}_n \bigg\|_F$$ is small for all $j\in\mathcal{N}$, where $|\mathcal{N}|$ is the cardinality of $\mathcal{N}$. We show in the following that this assumption leads to the separation of the optimization w.r.t. ${\boldsymbol{A}}$ in  into two optimization steps, one w.r.t. ${\boldsymbol{A}}_{\!{\mathcal{C}}}$, and the other w.r.t. ${\boldsymbol{A}}_{\!{\mathcal{D}}}$. For numerical verification of the reasonability of and , see the supplemental material, also available in [@Borsoi_multiscaleVar_2018]. Approximate Mixture Model ------------------------- Consider  after neglecting its third term. Both ${\boldsymbol{W}}$ (in the first term) and ${\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast$ (in the second term) act upon all the products ${\boldsymbol{M}}_n{\boldsymbol{a}}_n$, instead of just upon ${\boldsymbol{a}}_n$, for $n = 1, \dots, N$. This precludes the separation of  in a sum of non-negative functions exclusively dependent on ${\boldsymbol{A}}_{\mathcal{C}}$ or ${\boldsymbol{A}}_{\mathcal{D}}$, which could be independently minimized. However, combining and the fact that the transformation ${\boldsymbol{W}}$ groups pixels that are in spatially adjacent regions, we now propose an approximate separable mixing model. We initially express each pixel ${\boldsymbol{y}}_{{\mathcal{C}}_i}$ and ${\boldsymbol{y}}_{{\mathcal{D}}_i}$ of  as $$\begin{aligned} \label{eq:model_decomposed_ii_c} {\boldsymbol{y}}_{{\mathcal{C}}_i} & = \sum_{j=1}^N W_{j,i} \, {\boldsymbol{M}}_j {\boldsymbol{a}}_j + {\boldsymbol{e}}_{{\mathcal{C}}_i}\end{aligned}$$ and $$\begin{aligned} \label{eq:model_decomposed_ii_d} {\boldsymbol{y}}_{{\mathcal{D}}_i} & = {\boldsymbol{M}}_i{\boldsymbol{a}}_i - \sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{M}}_{\ell} {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i}\end{aligned}$$ where $W_{j,i}$ and $W^\ast_{j,i}$ are the $(j,i)$-th elements of ${\boldsymbol{W}}$ and ${\boldsymbol{W}}^\ast$, respectively, and ${\boldsymbol{e}}_{{\mathcal{C}}_i}$ and ${\boldsymbol{e}}_{{\mathcal{D}}_i}$ denote the $i$-th columns of ${\boldsymbol{E}}_{{\mathcal{C}}}$ and ${\boldsymbol{E}}_{{\mathcal{D}}}$. Then, using and the fact that ${\boldsymbol{W}}$ is a localized decomposition, we approximate every endmember matrix ${\boldsymbol{M}}_j$ in  by $${\boldsymbol{M}}_j \approx {\boldsymbol{M}}_{\!{\mathcal{C}}_i} = \sum_{\ell=1}^N \frac{\mathbbm{1}_{W_{\ell,i}}}{|\emph{supp}_{\ell}(W_{\ell,i})|} {\boldsymbol{M}}_{\ell}$$ where $\mathbbm{1}_{W_{j,i}}$ is the indicator function of $W_{j,i}$ (i.e. $\mathbbm{1}_{W_{j,i}}=1$ if $W_{j,i}\neq0$ and $\mathbbm{1}_{W_{j,i}}=0$ otherwise), and $|\emph{supp}_{\ell}(f)|$ denotes the cardinality of the support of $f$ as a function of $\ell$. Equivalently, we approximate every matrix ${\boldsymbol{M}}_{\ell}$ in  by $$\label{eq:mci_star} {\boldsymbol{M}}_{\ell} \approx {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast} = \sum_{n=1}^S\sum_{m=1}^N \frac{\mathbbm{1}_{W_{n,i}^\ast} \mathbbm{1}_{W_{m,n}}}{|\emph{supp}_{n,m}(W_{n,i}^\ast W_{m,n})|} {\boldsymbol{M}}_m$$ where $|\emph{supp}_{m,n}(f)|$ denotes the cardinality of the support of $f$ as a function of both $m$ and $n$. Thus,  and  can be approximated as (details in Appendix \[app:model\]) $$\begin{aligned} \label{eq:model_decomposed_iii_c} {\boldsymbol{y}}_{{\mathcal{C}}_i} & \approx {\boldsymbol{M}}_{\!{\mathcal{C}}_i} {\boldsymbol{a}}_{\!{\mathcal{C}}_i} + {\boldsymbol{e}}_{{\mathcal{C}}_i} $$ and $$\begin{aligned} \label{eq:model_decomposed_iii_d} {\boldsymbol{y}}_{{\mathcal{D}}_i} & \approx {\boldsymbol{M}}_i{\boldsymbol{a}}_{\!{\mathcal{D}}_i} + {\boldsymbol{M}}_{\!{\mathcal{D}}_i} \big[{\boldsymbol{A}}_{{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_i + {\boldsymbol{e}}_{{\mathcal{D}}_i} $$ where $[\cdot]_i$ denotes the $i-$th column of a matrix, and ${\boldsymbol{M}}_{\!{\mathcal{D}}_i} = {\boldsymbol{M}}_i - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}$ reflects the variability of ${\boldsymbol{M}}_i$ with respect to ${\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}$, the average endmember matrix of its neighborhood. According to , ${\boldsymbol{M}}_{\!{\mathcal{D}}_i}\approx {\boldsymbol{0}}$. Note that, since the transformation ${\boldsymbol{W}}$ only groups together pixels that lie inside a single superpixel, we average ${\boldsymbol{a}}_{n}$ and ${\boldsymbol{M}}_n$ only in small spatial neighborhoods where their variability is small. Selecting ${\boldsymbol{W}}$ and ${\boldsymbol{W}}^*$ according to the superpixels decomposition, we have that: - ${\boldsymbol{M}}_{\!{\mathcal{C}}_i}$ is the average of all ${\boldsymbol{M}}_j$ inside the $i$-th superpixel. - ${\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}$ is the average of all ${\boldsymbol{M}}_j$ inside the superpixel that contains the $i$-th pixel. Thus, if pixel $i$ belongs to the $k$-th superpixel, ${\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}$ is the average of all ${\boldsymbol{M}}_j$ inside the $k$-th superpixel. Note that ${\boldsymbol{W}}^\ast$ is also a localized transform, as it attributes the superpixel value to all pixels in the original domain that lie inside that superpixel, which encompasses a compact spatial neighborhood. Writting  and  for all pixels, we write and  in the matrix form as: $$\begin{aligned} \label{eq:model_decomposed_iv} \begin{split} {\boldsymbol{Y}}_{\!{\mathcal{C}}} {}={} & \big[{\boldsymbol{M}}_{{\mathcal{C}}_1}{\boldsymbol{a}}_{{\mathcal{C}}_1},\ldots,{\boldsymbol{M}}_{{\mathcal{C}}_S}{\boldsymbol{a}}_{{\mathcal{C}}_S}\big] + \widetilde{\!{\boldsymbol{E}}}_{{\mathcal{C}}} \\ {\boldsymbol{Y}}_{\!{\mathcal{D}}} {}={} & \big[{\boldsymbol{M}}_{\!{\mathcal{D}}_1} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_1,\ldots,{\boldsymbol{M}}_{\!{\mathcal{D}}_N} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_N\big] \\ & + \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_{{\mathcal{D}}_1},\ldots,{\boldsymbol{M}}_N{\boldsymbol{a}}_{{\mathcal{D}}_N}\big] + \widetilde{\!{\boldsymbol{E}}}_{{\mathcal{D}}} \end{split}\end{aligned}$$ where $\widetilde{\!{\boldsymbol{E}}}_{{\mathcal{C}}}$ and $\widetilde{\!{\boldsymbol{E}}}_{{\mathcal{D}}}$ include additive noise and modeling errors. ### Abundance constraints {#sec:constraints} The two constraints in  are functions of ${\boldsymbol{A}}$, and thus must be considered in the optimization with respect to ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$. Assuming ${\mathcal{W}}$ in to be of full row rank, the sum-to-one constraint can be expressed as $$\begin{aligned} \label{eq:sum_to_one_transf} &\mathbf{1}^\top{\boldsymbol{A}}{\mathcal{W}}= \mathbf{1}^\top{\mathcal{W}}\nonumber \\ & \Longleftrightarrow \mathbf{1}^\top{\boldsymbol{A}}_{\!{\mathcal{C}}} = \mathbf{1}^\top{\boldsymbol{W}}\,,\quad \mathbf{1}^\top{\boldsymbol{A}}_{\!{\mathcal{D}}} = \mathbf{1}^\top ({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast).\end{aligned}$$ Considering the positivity constraint we have $$\begin{aligned} \label{eq:abundances_pos_constr} & {\boldsymbol{A}}\geq{\boldsymbol{0}} \Rightarrow {\boldsymbol{A}}{\mathcal{W}}{\mathcal{W}}^\dagger \geq{\boldsymbol{0}} \nonumber \\ & \Longleftrightarrow [{\boldsymbol{A}}_{\!{\mathcal{C}}} \,\,\, {\boldsymbol{A}}_{\!{\mathcal{D}}}]{\mathcal{W}}^\dagger \geq {\boldsymbol{0}} \Longleftrightarrow {\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^\ast + {\boldsymbol{A}}_{\!{\mathcal{D}}} \geq {\boldsymbol{0}} \,.\end{aligned}$$ If ${\boldsymbol{W}}^\ast \ge {\boldsymbol{0}}$, which is true if ${\boldsymbol{W}}$ is selected as the superpixel decomposition, we can further state that $$\begin{aligned} \label{eq:SimplifiedConstraint} {\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^\ast \geq{\boldsymbol{0}} \Longleftrightarrow {\boldsymbol{A}}_{\!{\mathcal{C}}}\geq{\boldsymbol{0}}\end{aligned}$$ what simplifies the constraint by removing possible interdependencies between different pixels, and makes the problem separable for all pixels in the coarse scale ${\mathcal{C}}$. ### The updated optimization problem Using the results obtained in Sections \[sec:residuals\] to \[sec:constraints\], minimizing  with respect to ${\boldsymbol{A}}$ can be restated as determining ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ that minimize $$\begin{aligned} \label{eq:sec_opt_A_approx_gl} \widetilde{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{C}}}&,{\boldsymbol{A}}_{\!{\mathcal{D}}}|\,\mathbb{M},{\boldsymbol{\Psi}}) \nonumber \\ & {}={} \frac{1}{2} \Big\|{\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast - \big[{\boldsymbol{M}}_{{\mathcal{C}}_1}{\boldsymbol{a}}_{{\mathcal{C}}_1},\ldots,{\boldsymbol{M}}_{{\mathcal{C}}_S}{\boldsymbol{a}}_{{\mathcal{C}}_S}\big] {\boldsymbol{W}}^\ast \Big\|_F^2 \nonumber \\ & + \frac{1}{2} \Big\|{\boldsymbol{Y}}_{\!{\mathcal{D}}} - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_{{\mathcal{D}}_1},\ldots,{\boldsymbol{M}}_N{\boldsymbol{a}}_{{\mathcal{D}}_N}\big] \nonumber \\ & \qquad - \big[{\boldsymbol{M}}_{\!{\mathcal{D}}_1} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_1,\ldots,{\boldsymbol{M}}_{\!{\mathcal{D}}_N} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_N\big] \Big\|_F^2 \nonumber \\ & + \frac{\rho\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{C}}} \|_F^2 + \frac{\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{D}}} \|_F^2 \nonumber \\ & \text{subject to } {\boldsymbol{A}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast+{\boldsymbol{A}}_{\!{\mathcal{D}}}\geq {\boldsymbol{0}}, \,\, {\boldsymbol{1}}^\top {\boldsymbol{A}}_{\!{\mathcal{C}}} = {\boldsymbol{1}}^\top{\boldsymbol{W}},\,\, \nonumber \\ & \hspace{10ex} {\boldsymbol{1}}^\top{\boldsymbol{A}}_{\!{\mathcal{D}}} = {\boldsymbol{1}}^\top({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast).\end{aligned}$$ Optimization problem  is amenable to an efficient solution, as detailed in the following section. Solution to the optimization problem  ------------------------------------- This section details the proposed solution of the optimization problem  w.r.t. ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$. ### Optimizing with respect to ${\boldsymbol{A}}_{\mathcal{C}}$ at the $i$-th iteration {#sec:als_opt_ac} The cost function in this case is $\widetilde{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{C}}}|{\boldsymbol{A}}_{\!{\mathcal{D}}},\mathbb{M},{\boldsymbol{\Psi}})$, where ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ is a variable and ${\boldsymbol{A}}_{\!{\mathcal{D}}}$, $\mathbb{M}$ and ${\boldsymbol{\Psi}}$ are fixed at the solutions obtained in the previous iteration. Note that this problem is still not separable with respect to each pixel in ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ since the second term of  includes products between ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ and ${\boldsymbol{W}}^*$. However, this cost function can be simplified to yield a separable problem by making the following considerations using assumption : 1. implies that the entries of ${\boldsymbol{M}}_{\!{\mathcal{D}}_i}$ are small when compared to those of ${\boldsymbol{M}}_n$; 2. also implies that the entries of ${\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast$ are usually much larger than the entries of ${\boldsymbol{Y}}_{\!{\mathcal{D}}}$. These considerations imply that the contribution of the terms ${\boldsymbol{M}}_{\!{\mathcal{D}}_i} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]$ in the second term of  can be neglected when compared to ${\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast$. Using this approximation and , the optimization with respect to ${\boldsymbol{A}}_{\!{\mathcal{C}}}$ can be stated as the minimization of $$\begin{aligned} \label{eq:opt_AC_1} \overline{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{C}}}&|{\boldsymbol{A}}_{\!{\mathcal{D}}},\mathbb{M},{\boldsymbol{\Psi}}) \nonumber \\ & {}={} \frac{1}{2} \big\|{\boldsymbol{Y}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast - \big[{\boldsymbol{M}}_{{\mathcal{C}}_1}{\boldsymbol{a}}_{{\mathcal{C}}_1},\ldots,{\boldsymbol{M}}_{{\mathcal{C}}_S}{\boldsymbol{a}}_{{\mathcal{C}}_S}\big] {\boldsymbol{W}}^\ast \big\|_F^2 \nonumber \\ & + \frac{\rho\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{C}}} \|_F^2 \nonumber \\ & \text{subject to } {\boldsymbol{A}}_{\!{\mathcal{C}}}\geq {\boldsymbol{0}}, \,\, {\boldsymbol{1}}^\top {\boldsymbol{A}}_{\!{\mathcal{C}}} = {\boldsymbol{1}}^\top{\boldsymbol{W}}\end{aligned}$$ For ${\boldsymbol{W}}$ based on the superpixel decomposition, ${\boldsymbol{W}}^\ast$ assigns to each pixel in the original image domain the value of the superpixel to which it belongs. Using this property, the cost function  simplifies to $$\begin{aligned} \label{eq:SimplifiedACOptimization} \overline{\mathcal{J}} ({\boldsymbol{A}}_{\!{\mathcal{C}}}&|{\boldsymbol{A}}_{\!{\mathcal{D}}},\mathbb{M},{\boldsymbol{\Psi}}) \\ & \!\!\! {}={} \frac{1}{2} \sum_{n=1}^S \Omega_s^2(n) \Big( \|{\boldsymbol{y}}_{{\mathcal{C}}_n} - {\boldsymbol{M}}_{\!{\mathcal{C}}_n}{\boldsymbol{a}}_{{\mathcal{C}}_n}\|_2^2 + \frac{\widetilde{\rho}(n)\lambda_A}{2} \|{\boldsymbol{a}}_{{\mathcal{C}}_n}\|_2^2 \Big) \nonumber \\ & \text{subject to} \,\,\, {\boldsymbol{a}}_{{\mathcal{C}}_n}\geq{\boldsymbol{0}}, \, {\boldsymbol{1}}^\top{\boldsymbol{a}}_{{\mathcal{C}}_n} = {\boldsymbol{1}}^\top [{{\boldsymbol{W}}}]_n ,\,\, n=1,\ldots,S \nonumber $$ where $[{{\boldsymbol{W}}}]_n$ is the $n$-th column of ${\boldsymbol{W}}$, $\Omega_s(n)$ is the number of pixels contained in the $n$-th superpixel and $\widetilde{\rho}(n)=\rho\Omega_s^{-2}(n)$, $n=1,\ldots,S$ is a superpixel-dependent regularization parameter that controls the balance between both terms in the cost function for each superpixel. For simplicity, in the following we replace $\widetilde{\rho}(n)$ by a weighting term $\widetilde{\rho}_0=\rho S^2/N^2$ that is constant for all superpixels. This further simplifies the optimization problem since $S$ is specified a priori by the user. Furthermore, since the optimization is independent for each pixel, we can also move the $\Omega_s^2(n)$ factor outside the summation in  without changing the critical point of the cost function. Doing this results in the following cost function that can be minimized individually for each pixel: $$\begin{aligned} \label{eq:simplified_cost_on_Ac} \widehat{\mathcal{J}} ({\boldsymbol{A}}_{\!{\mathcal{C}}}&|{\boldsymbol{A}}_{\!{\mathcal{D}}},\mathbb{M},{\boldsymbol{\Psi}}) \\ & {}={} \frac{N^2}{2S^2} \sum_{n=1}^S \Big( \|{\boldsymbol{y}}_{{\mathcal{C}}_n} - {\boldsymbol{M}}_{\!{\mathcal{C}}_n}{\boldsymbol{a}}_{{\mathcal{C}}_n}\|_2^2 + \frac{\widetilde{\rho}_0\lambda_A}{2} \|{\boldsymbol{a}}_{{\mathcal{C}}_n}\|_2^2 \Big) \nonumber \\ & \text{subject to} \,\,\, {\boldsymbol{a}}_{{\mathcal{C}}_n}\geq{\boldsymbol{0}}, \, {\boldsymbol{1}}^\top{\boldsymbol{a}}_{{\mathcal{C}}_n} = {\boldsymbol{1}}^\top [{{\boldsymbol{W}}}]_n ,\,\,n=1,\ldots,S. \nonumber $$ Note that  is equivalent to a a standard FCLS problem, which can be solved efficiently. ### Optimizing with respect to ${\boldsymbol{A}}_{\mathcal{D}}$ at the $i$-th iteration {#sec:als_opt_ad} The cost function in this case is $\widetilde{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{D}}}|{\boldsymbol{A}}_{\!{\mathcal{C}}},\mathbb{M},{\boldsymbol{\Psi}})$, where ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ is a variable and ${\boldsymbol{A}}_{\!{\mathcal{C}}}$, $\mathbb{M}$ and ${\boldsymbol{\Psi}}$ are fixed at the solutions obtained in the previous iteration. Then, considering only the terms and constraints in  that depend on ${\boldsymbol{A}}_{\!{\mathcal{D}}}$ yields $$\begin{aligned} \label{eq:opt_cf_cald_iii_prev} \widetilde{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{D}}}&|{\boldsymbol{A}}_{\!{\mathcal{C}}},\mathbb{M},{\boldsymbol{\Psi}}) \nonumber \\ & {}={} \frac{1}{2} \Big\|{\boldsymbol{Y}}_{\!{\mathcal{D}}} - \big[{\boldsymbol{M}}_1{\boldsymbol{a}}_{{\mathcal{D}}_1},\ldots,{\boldsymbol{M}}_N{\boldsymbol{a}}_{{\mathcal{D}}_N}\big] \nonumber \\ & \qquad - \big[{\boldsymbol{M}}_{\!{\mathcal{D}}_1} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_1,\ldots,{\boldsymbol{M}}_{\!{\mathcal{D}}_N} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_N\big] \Big\|_F^2 \nonumber \\ & + \frac{\lambda_A}{2} \| {\boldsymbol{A}}_{\!{\mathcal{D}}} \|_F^2 \\ & \text{subject to } {\boldsymbol{A}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast+{\boldsymbol{A}}_{\!{\mathcal{D}}}\geq 0, \nonumber \\ & \hspace{10ex} {\boldsymbol{1}}^\top{\boldsymbol{A}}_{\!{\mathcal{D}}} = {\boldsymbol{1}}^\top({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast). \nonumber\end{aligned}$$ Since matrix ${\boldsymbol{A}}_{{\mathcal{C}}}$ is fixed, this problem can be decomposed for each pixel. This results in the minimization of the following cost function: $$\begin{aligned} \label{eq:opt_cf_cald_iii} \widetilde{\mathcal{J}}({\boldsymbol{A}}_{\!{\mathcal{D}}}&|{\boldsymbol{A}}_{\!{\mathcal{C}}},\mathbb{M},{\boldsymbol{\Psi}}) \nonumber \\ & {}={} \frac{1}{2} \sum_{n=1}^N \Big( \big\|{\boldsymbol{y}}_{{\mathcal{D}}_n} - {\boldsymbol{M}}_n{\boldsymbol{a}}_{{\mathcal{D}}_n} - {\boldsymbol{M}}_{\!{\mathcal{D}}_n} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^\ast\big]_n \big\|_2^2 \nonumber \\ & + \lambda_A \|{\boldsymbol{a}}_{{\mathcal{D}}_n}\|_2^2 \Big) \\ & \text{subject to} \,\,\, \big[{\boldsymbol{A}}_{\!{\mathcal{C}}}{\boldsymbol{W}}^{\ast}\big]_n + {\boldsymbol{a}}_{{\mathcal{D}}_n}\geq0 \nonumber \\ & \hspace{10ex} \mathbf{1}^\top{\boldsymbol{a}}_{{\mathcal{D}}_n} = \mathbf{1}^\top \big[{\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^\ast\big]_n \nonumber \\ & \hspace{10ex} n=1,\ldots,N \nonumber \end{aligned}$$ where matrices ${\boldsymbol{M}}_{\!{\mathcal{D}}_n}$ are given in  and . Note that this cost function is again equivalent to a standard FCLS problem, which can be solved efficiently. The MUA-SV unmixing algorithm {#sec:mua_sv_alg} ============================= Considering the solutions to the optimization subproblems derived in the previous sections, the global unmixing procedure can be directly derived by setting the fixed variables of each subproblem with the estimates obtained from the previous iteration. The MUA-SV algorithm is presented in Algorithm \[alg:global\_opt\]. \[bth\] Compute the superpixel decomposition of the hyperspectral image ${\boldsymbol{Y}}$ and the corresponding transformation matrices ${\boldsymbol{W}}$, ${\boldsymbol{W}}^\ast$, ${\mathcal{W}}$ and ${\mathcal{W}}^\dagger$ using the SLIC algorithm [@achanta2012slicPAMI] Compute the decomposition of ${\boldsymbol{Y}}$ into approximation and detail domains ${\boldsymbol{Y}}_{\!\mathcal{C}}$, and ${\boldsymbol{Y}}_{\!\mathcal{D}}$ using  and  Set ${\boldsymbol{A}}_{\!{\mathcal{D}}}^{(0)}={\boldsymbol{A}}^{(0)}({\boldsymbol{I}}-{\boldsymbol{W}}{\boldsymbol{W}}^*)$ Set $i=1$ $\widehat{\!{\boldsymbol{A}}}={\boldsymbol{A}}^{(i-1)}$,  $\widehat{\mathbb{M}}=\mathbb{M}^{(i-1)}$,  $\widehat{{\boldsymbol{\Psi}}}={\boldsymbol{\Psi}}^{(i-1)}$ Results {#sec:results} ======= In this section, we compare the unmixing performances achieved using the proposed MUA-SV algorithm, the Fully Constrained Least Squares (FCLS), the Scaled Constrained Least Squares (SCLS), the PLMM-based solution [@thouvenin2016hyperspectralPLMM] and the ELMM-based solution [@drumetz2016blindUnmixingELMMvariability], the latter two designed to tackle spectral variability. The SCLS algorithm is a particular case of the ELMM model that employs the same scaling factors ${\boldsymbol{\psi}}_n$ for all endmembers in each pixel (i.e. ${\boldsymbol{M}}_n=\psi_n{\boldsymbol{M}}_0$, where $\psi_n\in\mathbb{R}_+$) [@Nascimento2005doesICAplaysRole]. It is a low complexity algorithm that can be used as a baseline method to account for spectral variability. For all simulations, the reference endmember signatures ${\boldsymbol{M}}_0$ were extracted from the observed image using the Vertex Component Analysis (VCA) algorithm [@nascimento2005vca]. The abundance maps were initialized with the SCLS result for all algorithms. The scaling factors ${\boldsymbol{\Psi}}$ for ELMM and MUA-SV were initialized with ones. The matrix ${\boldsymbol{M}}$ for the PLMM was initialized with the results from the VCA. The alternating least squares loop in Algorithm \[alg:global\_opt\] is terminated when the norm of the relative variation of the three variables between two successive iterations is smaller than $\epsilon_A=\epsilon_{\Psi}=\epsilon_M=2\times10^{-3}$. Experiments were performed for three synthetic and two real data sets. For the synthetic data, the regularization parameters were selected for each algorithm to provide the best abundance estimation performance. The complete set of parameters, comprising the SLIC ($S$ and $\gamma$) and the regularization parameters ($\rho$, $\lambda_M$, $\lambda_A$, and $\lambda_\psi$), were searched in appropriate intervals. For instance, $\gamma\in\{0.001,\, 0.0025,\, 0.005,\, 0.01,\, 0.025,\, 0.05\}$, $S$ assumed an integer value in the interval $[2,9]$, $\rho$ was selected so that $\rho S^2/N^2 \in\{0.001,\, 0.01,\, 0.025,\, 0.05,\, 0.1,\, 0.15,\, \allowbreak 0.2,\, 0.25,\, 0.35,\, 0.5\}$, while $\lambda_M$, $\lambda_A$, and $\lambda_\psi$ were searched in the range $[5\times10^{-4},100]$, with 12 points sampled uniformly. The algorithms were implemented on a desktop computer equipped with an Intel I7 4.2 Ghz processor with 4 cores and 16 Gb RAM. ELMM, PLMM and SLIC were implemented using the codes made available by the respective authors. We did not employ parallelism when implementing the MUA-SV algorithm, so as to reduce the influence of the hardware platform when evaluating the performance gains achieved through the proposed simplifications. If parallelism is employed, the execution times can be even smaller. [0.3]{} {width="5.5cm"}    [0.3]{} {width="5.5cm"}    [0.3]{} {width="5.5cm"} Synthetic data sets ------------------- Three synthetic data sets were built. The first data cube (DC1) was built from the ELMM model to verify how MUA-SV performs when the actual endmembers closely follows the adopted model. The second data cube (DC2) was built using the more challenging additive perturbation model of [@thouvenin2016hyperspectralPLMM]. The third data cube (DC3) was based on a realistic simulation of endmember variability caused by illumination conditions following the Hapke’s model [@Hapke1981]. The data cube DC1 contains $50\times50$ pixels and three materials selected randomly from the USGS library and used as the reference endmember matrix ${\boldsymbol{M}}_0$, with 224 spectral bands. The abundance maps are piecewise smooth images generated by sampling from a Gaussian Random field [^7] [@kozintsev1999computationsGaussianFields], and are depicted in Fig. \[fig:abundances\_DC1\]. Spectral variability was added to the reference endmembers using the same model as in [@drumetz2016blindUnmixingELMMvariability], where the endmember instances for each pixel were generated by applying a constant scaling factor to the reference endmembers with amplitude limited to the interval $[0.75,1.25]$. Finally, a white Gaussian noise with a 25 dB SNR was added to the already scaled endmembers. The true scaling factors applied to each endmember were generated using a Gaussian Random field, and thus exhibit spatial correlation. The data cube DC2 contains $70\times70$ pixels and three materials, also randomly selected from the USGS spectral library to compose matrix ${\boldsymbol{M}}_0$ with 224 spectral bands. The abundance maps (shown in Fig. \[fig:abundances\_DC2\]) are composed by square regions distributed uniformly over a background, containing pure pixels (first row) and mixtures of two and three endmembers (second and third rows). The background pixels are mixtures of the same three endmembers, with abundances $0.2744$, $0.1055$ and $0.62$. Spectral variability was added following the model proposed in [@thouvenin2016hyperspectralPLMM], which considered a per-pixel variability given by random piecewise linear functions to scale individually the spectrum of each endmember by a factor in the interval $[0.8,1.2]$. Such a variability model does not match the ELMM, as it yields different variabilities across the spectral bands, and is not designed to produce spatial correlation. Nevertheless, it provides a good ground for comparison with more flexible models such as the PLMM. ![Discrete terrain model used with the Hapke model in the data cube DC3, provided by [@drumetz2016blindUnmixingELMMvariability].[]{data-label="fig:terrain_hapke_ex"}](figures/terrain-crop){width="5cm"} The data cube DC3 contains $50\times50$ pixels and three materials, and is based on a simulation originally presented in [@drumetz2016blindUnmixingELMMvariability][^8]. This data cube is devised to realistically represent the spectral variability introduced due to changes in the illumination conditions caused by the topography of the scene, and is generated according to a physical model proposed by Hapke [@Hapke1981]. Hapke’s model is able to represent the reflectance of a material as a function of its single scattering albedo, photometric parameters and geometric characteristics of the scene, namely, the incidence, emergence and azimuth angles during acquisition [@drumetz2016blindUnmixingELMMvariability; @Hapke1981]. Thus, pixel dependent reflectance signatures for each endmember can be obtained given its single scattering albedo and the scene topography. In this example, the scene was composed of three materials, namely, basalt, palagonite and tephra, which are frequently present on small bodies of the Solar System, and contained 16 spectral bands. Afterwards, a digital terrain model simulating a hilly region was generated, which is shown in Fig. \[fig:terrain\_hapke\_ex\], and from this model the acquisition angles associated with each pixel were derived (as a function of the scene topography) by considering the angle between the sun and the horizontal plane as $18^\circ$, and the sensor to be placed vertically downward. Finally, the pixel dependent endmember signatures for the scene were generated from the single scattering albedo of the materials, and from the geometric characteristics of the scene using Hapke’s model. The abundance maps used for DC2 were the same used for DC1, as shown in Fig. \[fig:abundances\_DC3\]. The resulting hyperspectral images for all data cubes were generated from the pixel-dependent endmember signatures and abundance maps following the LMM, and were later contaminated by white Gaussian noise, with signal-to-noise ratios (SNR) of 20, 30, and 40 dB. The regularization parameters for all algorithms and all examples were selected using a grid search procedure in order to provide best abundance estimation performance, and are presented in the supplemental material and in [@Borsoi_multiscaleVar_2018]. The unmixing accuracy metrics used are the abundances mean squared error (MSE) $$\text{MSE}_{{\boldsymbol{A}}} = {\frac{1}{NP}\|{\boldsymbol{A}}- \widehat{\!{\boldsymbol{A}}}\|_F^2} \,,$$ the mean squared error of the estimated spectra $$\text{MSE}_{{\mathbb{M}}} = {\frac{1}{NLP} \sum_{n=1}^N \|{\boldsymbol{M}}_n - \widehat{\!{\boldsymbol{M}}}_n\|_F^2} \,,$$ and the mean squared reconstruction error $$\text{MSE}_{{\boldsymbol{Y}}} = \frac{1}{NL} \sum_{n=1}^N \|{\boldsymbol{y}}_n - \widehat{\!{\boldsymbol{M}}}_n\widehat{{\boldsymbol{a}}}_n\|^2 \,.$$ We also evaluate the estimates of the endmember signatures using the average Spectral Angle Mapper (SAM), defined by $$\text{SAM}_{{\mathbb{M}}} = \frac{1}{N}\sum_{n=1}^{N}\sum_{k=1}^{P}\arccos\left(\frac{{\boldsymbol{m}}_{k,n}^\top\widehat{{\boldsymbol{m}}}_{k,n}}{\|{\boldsymbol{m}}_{k,n}\|\|\widehat{{\boldsymbol{m}}}_{k,n}\|}\right).$$ where ${\boldsymbol{m}}_{k,n}$ and $\widehat{{\boldsymbol{m}}}_{k,n}$ are the $k$-th columns of ${\boldsymbol{M}}_n$ and $\widehat{\!{\boldsymbol{M}}}_n$, respectively. The quantitative results achieved by all algorithms are displayed in Table \[tab:quantitative\_results\] for all tested SNR values. The reconstructed abundance maps for the three data cubes and an SNR of 30 dB are shown in Figs. \[fig:abundances\_DC1\], \[fig:abundances\_DC2\] and \[fig:abundances\_DC3\] for a qualitative comparison. The computational complexity of the algorithms was evaluated through their execution times, which are shown in Table \[tab:alg\_exec\_time\]. \[!ht\] ----- -------- ----------------------------------- --------------------------- ----------------------------- --------------------------------- ----------------------------------- ----------------------------- ----------------------------- --------------------------------- ----------------------------------- ----------------------------- ----------------------------- --------------------------------- -- SNR Method $\text{MSE}_{\!{\boldsymbol{A}}}$ $\text{MSE}_{\mathbb{M}}$ $\text{SAM}_{\!\mathbb{M}}$ $\text{MSE}_{{\boldsymbol{Y}}}$ $\text{MSE}_{\!{\boldsymbol{A}}}$ $\text{MSE}_{\!\mathbb{M}}$ $\text{SAM}_{\!\mathbb{M}}$ $\text{MSE}_{{\boldsymbol{Y}}}$ $\text{MSE}_{\!{\boldsymbol{A}}}$ $\text{MSE}_{\!\mathbb{M}}$ $\text{SAM}_{\!\mathbb{M}}$ $\text{MSE}_{{\boldsymbol{Y}}}$ FCLS 21.97 $\times$ $\times$ 6.91 66.47 $\times$ $\times$ 6.45 74.14 $\times$ $\times$ 2.63 SCLS 28.79 6.87 190.50 6.86 73.35 4.07 171.01 6.20 73.18 3.02 214.56 0.50 PLMM 24.64 5.42 188.80 3.50 85.65 3.19 174.35 3.33 39.07 **1.44** **122.66** 0.39 ELMM 17.81 5.34 **186.70** 5.59 65.11 **3.09** **170.85** 6.69 59.54 2.80 317.44 **0.0001** MUA-SV **12.90** **5.24** 212.20 **1.56** **29.80** 3.36 185.67 **3.28** **28.11** 1.84 308.57 0.0002 FCLS 28.10 $\times$ $\times$ 1.76 60.28 $\times$ $\times$ 0.93 172.31 $\times$ $\times$ 1.41 SCLS 12.37 4.53 187.60 1.63 62.23 3.84 **161.20** 0.71 21.41 2.42 68.73 0.05 PLMM 19.61 4.88 173.00 0.86 49.38 3.95 162.54 0.41 38.00 **1.53** **68.53** 0.10 ELMM 10.71 3.70 170.20 0.59 40.16 3.05 177.91 **0.001** 18.47 1.73 101.51 **0.00002** MUA-SV **7.07** **3.46** **166.90** **0.35** **24.30** **2.83** 161.52 0.33 **14.70** 1.75 68.62 0.07 FCLS 20.04 $\times$ $\times$ 1.23 71.37 $\times$ $\times$ 0.44 256.20 $\times$ $\times$ 1.39 SCLS 7.38 3.88 186.30 1.10 69.48 3.52 160.10 0.17 8.98 2.40 30.90 **0.01** PLMM 13.44 3.64 170.30 0.56 44.73 3.02 **140.74** 0.11 34.38 1.47 74.15 0.08 ELMM 5.36 **2.51** **149.70** **0.02** 46.83 **2.63** 159.21 **0.0002** 8.12 **1.28** 43.14 **0.01** MUA-SV **3.98** 2.52 149.90 **0.02** **26.01** 2.97 155.96 0.31 **7.94** 1.81 **30.66** 0.02 ----- -------- ----------------------------------- --------------------------- ----------------------------- --------------------------------- ----------------------------------- ----------------------------- ----------------------------- --------------------------------- ----------------------------------- ----------------------------- ----------------------------- --------------------------------- -- \[tab:quantitative\_results\] FCLS SCLS ELMM PLMM MUA-SV --------- ------- -------- --------- ---------- -------- DC1 0.14s 0.42s 14.76s 16.17s 2.57s DC2 0.27s 0.83s 37.52s 149.91s 18.29s DC3 0.17s 0.35s 15.82s 63.07s 9.59s Houston 0.82s 2.31s 174.53s 484.02s 36.29s Cuprite 6.63s 15.61s 527.89s 7998.02s 95.54s : Execution time (in seconds) of the unmixing algorithms, averaged for all SNR values considered \[tab:alg\_exec\_time\] FCLS SCLS ELMM PLMM MUA-SV --------- ------- ------- ------- ------- -------- Houston 2.283 0.037 0.010 0.190 0.014 Cuprite 0.050 0.044 0.040 0.079 0.050 : Reconstruction errors ($\text{MSE}_{{\boldsymbol{Y}}}$) for the Houston and Cuprite data sets (all values are multiplied by $10^3$). \[tab:reconstr\_err\_real\_img\] ### Discussion Table \[tab:quantitative\_results\] shows a significantly better $\text{MSE}_{{\boldsymbol{A}}}$ performance of MUA-SV for all three data cubes and SNR values when compared with the other algorithms. This indicates that MUA-SV effectively exploits the spatial properties of the abundance maps, even when the actual spectral variability does not follow exactly the model in . Figs. \[fig:abundances\_DC1\], \[fig:abundances\_DC2\] and \[fig:abundances\_DC3\] show the true and reconstructed abundance maps for all algorithms and 30 dB SNR. As expected, models accounting for spectral variability tend to yield better reconstruction quality than FCLS, with EELMM yielding piecewise smooth solutions. In general, the solution provided by MUA-SV approaches better the ground-truth, in that it estimates the intensity of the abundance maps with better accuracy than the other algorithms. This can be most clearly seen for the results for DC2 (Fig. \[fig:abundances\_DC2\]), where the regions with pure pixels are better represented by MUA-SV. Regarding the spectral performances, as measured by the $\text{MSE}_{\mathbb{M}}$ and $\text{SAM}_{\mathbb{M}}$, the results varied among the algorithms, with no method performing uniformly better than the others. There is also a significant discrepancy between the Euclidean metric and the spectral angle in many examples, highlighting the different characteristics of the two metrics. The ELMM model yielded the smallest reconstruction error $\text{MSE}_{{\boldsymbol{Y}}}$ in most cases (6), followed by MUA-SV (4 cases). However, the connection between the reconstruction error $\text{MSE}_{{\boldsymbol{Y}}}$ and the abundance estimation performance $\text{MSE}_{{\boldsymbol{A}}}$ of the unmixing methods that address spectral variability is not clear, as can be attested from Table \[tab:quantitative\_results\]. The execution times shown in Table \[tab:alg\_exec\_time\] indicate that MUA-SV is 2.2 times faster than ELMM and 7.5 times faster than PLMM, a significant gain in computational efficiency. This difference is more accentuated when processing larger datasets, as will be verified in the following. Sensitivity analysis {#sec:sensitivity} -------------------- To evaluate the sensitivity of MUA-SV $\text{MSE}_{\!{\boldsymbol{A}}}$ to variations in the algorithm parameters[^9], we initially set all regularization parameters ($\lambda_M$, $\lambda_A$, $\lambda_\Psi$ and $\rho$) equal to their optimal values[^10]. Then, we varied one parameter at a time within a range from $-95\%$ to $+95\%$ of its optimal value. Fig. \[fig:sensitivity\_i\] presents the $\text{MSE}_{\!{\boldsymbol{A}}}$ values obtained by varying each parameter. It can be seen that small variations about the optimal values do not affect the $\text{MSE}_{\!{\boldsymbol{A}}}$ significantly, and that the maximum values obtained for the whole parameter ranges tested are still lower than those achieved by the other algorithms. To evaluate the sensitivity of the MUA-SV results to variations in the SLIC parameters, we plotted the resulting $\text{MSE}_{{\boldsymbol{A}}}$ as a function of $\sqrt{N/S}$ and $\gamma$, with the algorithm parameters $\lambda_M$, $\lambda_A$, $\lambda_\Psi$ and $\rho$ fixed at their optimal values. The results are shown in Fig. \[fig:sensitivity\_i\]. It is seen that the $\text{MSE}_{{\boldsymbol{A}}}$ performance does not deviate significantly from its optimal value unless the superpixel size $\sqrt{N/S}$ becomes too large. This is expected since very large superpixels may contain semantically different pixels, hindering the capability of the transform ${\boldsymbol{W}}$ to adequately capture coarse scale information. Furthermore, large values of $\sqrt{N/S}$ may violate hypothesis , which has been used thoroughly in the derivation of the MUA-SV algorithm, and thus represent a bad design choice. \[!htbp\] ![$\text{MSE}_{{\boldsymbol{A}}}$ variation due to relative changes in each parameter value about its optimal value (left) and $\text{MSE}_{{\boldsymbol{A}}}$ as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right).[]{data-label="fig:sensitivity_i"}](figures/finer_sensitivity_DC1_SNR30c){width="1\linewidth"} ![Reconstructed fractional abundance maps for the Houston data set.[]{data-label="fig:abundances_houston"}](figures/results/abundances_houston-crop.pdf){width="7.5cm"} ![Reconstructed fractional abundance maps for the Cuprite data set.[]{data-label="fig:abundances_cuprite"}](figures/results/abundances_cuprite4.pdf){width="7.5cm"} Simulations with real images ---------------------------- In this experiment, we consider two data sets obtained from real hyperspectral images. The first data set is comprised of a 152$\times$108 pixels subset of the Houston hyperspectral image, with 144 spectral bands. The second data set is a 250$\times$191 pixels subregion of the Cuprite image, with 188 spectral bands. Spectral bands presenting water absorption and low SNR were removed from both images. The parameters of the algorithms are shown in the supplemental material and in [@Borsoi_multiscaleVar_2018]. They were selected empirically for the proposed method, and set identically to those reported in [@drumetz2016blindUnmixingELMMvariability] for the ELMM and PLMM. The number of endmemebrs was selected as $P=4$ for the Houston data set, and as $P=14$ for the Cuprite data set, following the observations in [@drumetz2016blindUnmixingELMMvariability]. The endmembers were extracted using the VCA algorithm [@nascimento2005vca]. Since the true abundance maps are unavailable for those hyperspectral images, we make a qualitative assessment of the recovered abundance maps based on knowledge of materials present in prominent fashion in those scenes. The reconstructed abundance maps for the Houston data set are depicted in Fig. \[fig:abundances\_houston\]. The four materials which are prominently present in this dataset are vegetation, red metallic roofs, concrete stands, and asphalt. It can be seen that ELMM and MUA-SV yield the best results for the overall abundances of all materials, with smaller proportion indeterminacy in regions known to have mostly pure materials such as the football field, the square metallic roofs and the concrete stands in the stadium. However, MUA-SV provides better results, more clearly observed in the purer areas such as the concrete stands of the stadium, which appear to be more mixed with the asphalt abundances in the ELMM results. This evidences the better performance of the MUA-SV algorithm. The reconstructed abundance maps for the Alunite, Sphene, Buddingtonite and Muscovite materials of the Cuprite data set are depicted in Fig. \[fig:abundances\_cuprite\]. Although all methods provide abundance maps which generally agree with previous knowledge about their distribution in this image [@nascimento2005vca], the MUA-SV results show abundances for all endmembers in Fig. \[fig:abundances\_cuprite\] that are more homogeneous and clearly delineated in the regions where the materials are present. Moreover, these results show significantly smaller contributions due to outliers in the background regions of the abundance maps. The reconstruction errors for all algorithms and both data sets are shown in Table \[tab:reconstr\_err\_real\_img\]. For the Houston data, the ELMM and MUA-SV results are very close and significantly smaller than those of the other methods, what agrees with their better representation of the abundance maps. For the Cuprite data, the errors are small and comparable for all algorithms, except for a slightly larger PLMM error. This goes in line with the fact that the abundance maps generally agree with the known distribution of these materials in the scene. However, reconstruction error results should be taken with proper care, as observed in the examples using synthetic data. Their correlation with the quality of abundance estimation is far from straightforward. The execution times for all methods, shown in Table \[tab:alg\_exec\_time\], illustrate again the significantly smaller computational load of MUA-SV when compared to other methods addressing spectral variability, as it performed, on average, 5.3 times faster than ELMM and 64.3 times faster than PLMM. Conclusions {#sec:conclusions} =========== In this paper we proposed a new data-dependent multiscale model for spectral unmixing accounting for spectral variability of the endmembers. Using a multiscale transformation based on the superpixel decomposition, spatial contextual information was incorporated into the unmixing problem through the decomposition of the observation model into two models in different domains, one capturing coarse image structures and another representing fine scale details. This facilitated the characterization of spatial regularity. Under reasonable assumptions, the proposed method yields a fast iterative algorithm, in which the abundance estimation problem is solved only once in each scale. Simulation results with both synthetic and real data show that the proposed MUA-SV algorithm outperforms other methods addressing spectral variability, both in accuracy of the reconstructed abundance maps and in computational complexity. Derivation of the approximated mixing model {#app:model} =========================================== Given the coarse pixel model in  can be approximated using hypothesis as $$\begin{aligned} \label{app_eq:model_decomposed_iii_c} {\boldsymbol{y}}_{{\mathcal{C}}_i} & \approx \sum_{\ell=1}^N \frac{\mathbbm{1}_{W_{\ell,i}}}{|\emph{supp}_{\ell}(W_{\ell,i})|} {\boldsymbol{M}}_{\ell} \sum_{j=1}^N W_{j,i} \,{\boldsymbol{a}}_i + {\boldsymbol{e}}_{{\mathcal{C}}_i} \nonumber \\ & = \sum_{\ell=1}^N \frac{\mathbbm{1}_{W_{\ell,i}}}{|\emph{supp}_{\ell}(W_{\ell,i})|} {\boldsymbol{M}}_{\ell} \,\, {\boldsymbol{a}}_{\!{\mathcal{C}}_i} + {\boldsymbol{e}}_{{\mathcal{C}}_i} \nonumber \\ & = {\boldsymbol{M}}_{\!{\mathcal{C}}_i} {\boldsymbol{a}}_{\!{\mathcal{C}}_i} + {\boldsymbol{e}}_{{\mathcal{C}}_i} $$ where ${\boldsymbol{a}}_{\!{\mathcal{C}}_i} = \sum_{j=1}^N W_{j,i}{\boldsymbol{a}}_i$. The detail model in  can be approximated as $$\begin{aligned} \label{app_eq:model_decomposed_iii_d} {\boldsymbol{y}}_{{\mathcal{D}}_i} & = {\boldsymbol{M}}_i{\boldsymbol{a}}_i - \sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{M}}_{\ell}\, {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & \approx {\boldsymbol{M}}_i{\boldsymbol{a}}_i - \bigg(\sum_{n=1}^S\sum_{m=1}^N \frac{\mathbbm{1}_{W_{n,i}^\ast} \mathbbm{1}_{W_{m,n}}}{|\emph{supp}_{n,m}(W_{n,i}^\ast W_{m,n})|} {\boldsymbol{M}}_m\bigg) \nonumber \\ & \qquad \cdot \sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i{\boldsymbol{a}}_i - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast} \,\sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i} $$ and straightforward computations leads to $$\begin{aligned} {\boldsymbol{y}}_{{\mathcal{D}}_i} & \approx {\boldsymbol{M}}_i{\boldsymbol{a}}_i - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast} \,\sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i({\boldsymbol{a}}_{\!{\mathcal{D}}_i}+\sum_{j=1}^S W^\ast_{j,i} \, {\boldsymbol{a}}_{\!{\mathcal{C}}_j}) \nonumber \\ & \qquad - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast} \,\sum_{j=1}^S \sum_{\ell=1}^N W^\ast_{j,i} \, W_{\ell,j} \, {\boldsymbol{a}}_{\ell} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i\bigg({\boldsymbol{a}}_{\!{\mathcal{D}}_i}+\sum_{j=1}^S W^\ast_{j,i} \, {\boldsymbol{a}}_{\!{\mathcal{C}}_j}\bigg) - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast} \,\sum_{j=1}^S W^\ast_{j,i} \, {\boldsymbol{a}}_{\!{\mathcal{C}}_j} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i{\boldsymbol{a}}_{\!{\mathcal{D}}_i} + \bigg({\boldsymbol{M}}_i - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}\bigg) \sum_{j=1}^S W^\ast_{j,i} \, {\boldsymbol{a}}_{\!{\mathcal{C}}_j} + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i{\boldsymbol{a}}_{\!{\mathcal{D}}_i} + \bigg({\boldsymbol{M}}_i - {\boldsymbol{M}}_{\!{\mathcal{C}}_i^\ast}\bigg) \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_i + {\boldsymbol{e}}_{{\mathcal{D}}_i} \nonumber \\ & = {\boldsymbol{M}}_i{\boldsymbol{a}}_{\!{\mathcal{D}}_i} + {\boldsymbol{M}}_{\!{\mathcal{D}}_i} \big[{\boldsymbol{A}}_{\!{\mathcal{C}}} {\boldsymbol{W}}^{\ast}\big]_i + {\boldsymbol{e}}_{{\mathcal{D}}_i}. $$ where ${\boldsymbol{a}}_{\!{\mathcal{D}}_i} = {\boldsymbol{a}}_i-\sum_{j=1}^S W_{j,i}^*{\boldsymbol{a}}_{\!{\mathcal{C}}_j}$. [Ricardo Augusto Borsoi (S’18)]{} received the MSc degree in electrical engineering from Federal University of Santa Catarina (UFSC), Florianópolis, Brazil, in 2016. He is currently working towards his doctoral degree at Université Côte d’Azur (OCA) and at UFSC. His research interests include image processing, tensor decomposition, and hyperspectral image analysis. [Tales Imbiriba (S’14, M’17)]{} received his Doctorate degree from the Department of Electrical Engineering (DEE) of the Federal University of Santa Catarina (UFSC), Florianópolis, Brazil, in 2016. He served as a Postdoctoral Researcher at the DEE–UFSC and is currently a Postdoctoral Researcher at the ECE dept. of the Northeastern University, Boston, MA, USA. His research interests include audio and image processing, pattern recognition, kernel methods, adaptive filtering, and Bayesian Inference. [José Carlos M. Bermudez (S’78,M’85,SM’02)]{} received the B.E.E. degree from the Federal University of Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil, the M.Sc. degree in electrical engineering from COPPE/UFRJ, and the Ph.D. degree in electrical engineering from Concordia University, Montreal, Canada, in 1978, 1981, and 1985, respectively. He joined the Department of Electrical Engineering, Federal University of Santa Catarina (UFSC), Florianopolis, Brazil, in 1985. He is currently a Professor of Electrical Engineering at UFSC and a Professor at Catholic University of Pelotas (UCPel), Pelotas, Brazil. He has held the position of Visiting Researcher several times for periods of one month at the Institut National Polytechnique de Toulouse, France, and at Université Nice Sophia-Antipolis, France. He spent sabbatical years at the Department of Electrical Engineering and Computer Science, University of California, Irvine (UCI), USA, in 1994, and at the Institut National Polytechnique de Toulouse, France, in 2012. His recent research interests are in statistical signal processing, including linear and nonlinear adaptive filtering, image processing, hyperspectral image processing and machine learning. Prof. Bermudez served as an Associate Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING in the area of adaptive filtering from 1994 to 1996 and from 1999 to 2001. He also served as an Associate Editor of the EURASIP Journal of Advances on Signal Processing from 2006 to 2010, and as a Senior Area Editor of the IEEE TRANSACTIONS ON SIGNAL PROCESSING from 2015 to 2019. He is the Chair of the Signal Processing Theory and Methods Technical Committee of the IEEE Signal Processing Society. Prof. Bermudez is a Senior Member of the IEEE. [Supplemental Material: A Data Dependent Multiscale Model]{} [for Hyperspectral Unmixing With Spectral Variability]{} SLIC Superpixels for HIs ======================== The SLIC superpixel decomposition consists of an extension of the k-means algorithm, with properly initialized cluster centers and a suitable distance function, defined as \[S1\] $$D_{SLIC} = \sqrt{d_{spectral}^2 + \gamma^2 d_{spatial}^2 S/N}$$ where $d_{spatial}$ and $d_{spectral}$ are the spatial and spectral distances, respectively. Although the SLIC algorithm was initially designed to work with color (3 bands) images, it can be extended to HIs straightforwardly by considering $d_{spectral}$ to be the Euclidean distance between reflectance vectors (HI pixels) and adjusting the normalization factor $\gamma$ accordingly. The superpixel transform requires the number of clusters $S$ and their regularity $\gamma$ as parameters to compute the transformation ${\boldsymbol{Y}}{\boldsymbol{W}}$. Nevertheless, we found that it is often easier to design the transform using the parameter $\sqrt{N/S}$ instead of $S$ since it is invariant to the image size and corresponds to the average sampling interval in the irregular domain. This quantity changes only slightly on a relatively short interval between the different simulations. Numerical verification of the simplifying hypothesis ==================================================== Although hypothesis and impose some limitation to the MUA-SV algorithm, they are reasonable and are satisfied in many practical circumstances. Below, we present a more thorough analysis of each of these hypotheses. Hypothesis consists of assuming that the inner product $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle$ between the residuals/reconstruction errors $RE_{{\mathcal{C}}}$ and $RE_{{\mathcal{D}}}$ in the coarse and detail image scales is comparatively small, when compared to the first two terms of the cost function (22). To illustrate the validity of this claim, we compare here the values of $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle$ with those of the first two terms of the cost function, given by $\|RE_{{\mathcal{C}}}\|_F^2$ and $\|RE_{{\mathcal{D}}}\|_F^2$, for some practical examples. We considered the result of unmixing DC1, DC2 and DC3 with an SNR of 30 dB presented in Section VIII using the ELMM model. The results are presented below in Table \[tab:hypothesis\_A1\_ver\]. $\|RE_{{\mathcal{C}}}\|_F^2+\|RE_{{\mathcal{D}}}\|_F^2$ $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle$ ----- --------------------------------------------------------- -------------------------------------------------------- DC1 328.35 -1.316$\times10^{-15}$ DC2 0.5605 2.845$\times10^{-16}$ DC3 7.105$\times10^{-4}$ 1.948$\times10^{-19}$ : Comparison between the residuals inner product and the first two terms of the cost function \[tab:hypothesis\_A1\_ver\] It can be seen that the quadratic norms exceed this inner product in value by several orders of magnitude. Thus, the latter can be reasonably neglected, i.e. $\langle RE_{{\mathcal{C}}},RE_{{\mathcal{D}}}\rangle\approx0$. Hypothesis basically states that the endmember signatures for each pixel ${\boldsymbol{M}}_n$ do not deviate much from the average endmember signature in its neighborhood, i.e. ${\boldsymbol{M}}_n$ is similar to $\frac{1}{|\mathcal{N}_n|}\sum_{j\in\mathcal{N}_n}{\boldsymbol{M}}_j$ where $\mathcal{N}_n$ contains indexes of pixels that are spatially close to pixel $n$. This is an assumption about the underlying physical model that is reasonable in practical scenarios. To illustrate this, we consider two experiments, one based on the Hapke model and another based on real data. For instance, using synthetic data generated using the Hapke model \[S2\] we can represent spectral variability due to topographic variations of the scene. Consider the discrete terrain model and reference endmember signatures presented in Figure \[fig:refEms\_terrain\_hapke\_A2\] below, extracted from \[S3\]. From this data and using the Hapke model, one can generate a set of pixel dependent endmember signatures which can be used to evaluate the spatial characteristics of spectral variability. For simplicity, we measure the similarity between the reference and the pixel dependent endmember signatures using both the Euclidean distance and the spectral angle, for all materials. The results are shown in Figure \[fig:hypothesis\_A2\_ver\_ii\] below, where it can be seen that there these deviations show significant spatial correlation. ![Reference endmember signatures (left) and discrete terrain model (right) used with the Hapke model in the data cube DC3 to generate the pixel-dependent endmember signatures (data provided by [@drumetz2016blindUnmixingELMMvariability]).[]{data-label="fig:refEms_terrain_hapke_A2"}](figs_review/endmembers-crop "fig:"){width="6cm"} ![Reference endmember signatures (left) and discrete terrain model (right) used with the Hapke model in the data cube DC3 to generate the pixel-dependent endmember signatures (data provided by [@drumetz2016blindUnmixingELMMvariability]).[]{data-label="fig:refEms_terrain_hapke_A2"}](figures/terrain-crop "fig:"){width="7.5cm" height="5cm"} ![Measures of endmember spatial variability in the Hapke model. Top row: Euclidean distance between the soil spectral signature of each pixel and the reference signature. Bottom row: Spectral angle between the soil spectral signature of each pixel and the reference signature.[]{data-label="fig:hypothesis_A2_ver_ii"}](figs_review/euc_em_refs-crop.pdf "fig:"){width="10cm"}\ ![Measures of endmember spatial variability in the Hapke model. Top row: Euclidean distance between the soil spectral signature of each pixel and the reference signature. Bottom row: Spectral angle between the soil spectral signature of each pixel and the reference signature.[]{data-label="fig:hypothesis_A2_ver_ii"}](figs_review/sam_em_refs-crop.pdf "fig:"){width="10cm"} Spectral variability occurring due to intrinsic variations of the material spectra (e.g. soil or vegetation) can also show significant spatial correlation (see \[S4\]), since endmember spectra usually depends on physical quantities that are correlated in space. Many experimental studies support this claim, including geostatistical works evaluating the spatial distribution and variability of soil’s physico-chemical properties (e.g. for grass crop terrain (see \[S5\]), calcareous soils (see \[S6\]), rice fields (see \[S7\]) and tobacco plantations (see \[S8\]), and also measurements of mineral spectra due to the presence of spatially correlated grain sizes and impurity concentrations (see \[S9\],\[S10\]). To illustrate this effect, we performed an experiment considering real data using the Samsom image. We considered a subregion containing pure pixels of the soil material, shown in Figure \[fig:hypothesis\_A2\_ver\_iii\]-(a) below. We considered these pixels as pixel dependent endmember signatures and evaluated the similarity between them and the average endmember spectra for all these pixels. The results, shown in Figures \[fig:hypothesis\_A2\_ver\_iii\]-(b) and \[fig:hypothesis\_A2\_ver\_iii\]-(c), are similar to the Hapke data, and illustrate that the variability shows considerable spatial correlation. ![(a) Samson hyperspectral image with a subimage containing soil highlighted. (b) Euclidean distance between the soil spectral signature of each pixel and their average value. (c) Spectral angle between the soil spectral signature of each pixel and their average value.[]{data-label="fig:hypothesis_A2_ver_iii"}](figs_review/samson_highlight-crop.pdf "fig:"){height="5cm"}\ (a) \   ![(a) Samson hyperspectral image with a subimage containing soil highlighted. (b) Euclidean distance between the soil spectral signature of each pixel and their average value. (c) Spectral angle between the soil spectral signature of each pixel and their average value.[]{data-label="fig:hypothesis_A2_ver_iii"}](figs_review/euc_distance_mean_soil-crop.pdf "fig:"){height="4.75cm"}\ (b)   ![(a) Samson hyperspectral image with a subimage containing soil highlighted. (b) Euclidean distance between the soil spectral signature of each pixel and their average value. (c) Spectral angle between the soil spectral signature of each pixel and their average value.[]{data-label="fig:hypothesis_A2_ver_iii"}](figs_review/sam_distance_mean_soil-crop.pdf "fig:"){height="4.75cm"}\ (c) Estimated scaling factors for the ELMM and MUA-SV algorithms ============================================================ We have plotted the scaling factors $\psi_n$, $n=1,\ldots,N$ for ELMM and for the MUA-SV algorithms, for data cubes DC1, DC2 and DC3 and an SNR of 30 dB. They are shown in Figure \[fig:scaling\_facts\_example\]. It can be seen that the overall spatial variations of the scaling factors generally occur in the same regions for both algorithms, except for some endmembers such as EMs 1 and 2 for DC2 and EM2 in DC3. This difference is most easily relatable to the abundance estimation in the case of EM 1 in DC2, where the abundances estimated by the ELMM (shown in Figure \[fig:scaling\_facts\_example\_abundances\]) deviate significantly from the ground truth in a pattern that is very similar to the estimated scaling factors, with a different overall scaling and a significantly smaller amplitude in the upper-left square. ![Comparison between the scaling factors of the ELMM and MUA-SV algorithms for (a) DC1, (b) DC2 and (c) DC3, for an SNR of 30dB.[]{data-label="fig:scaling_facts_example"}](figs_review/estim_variability2_DC1_SNR30_tght.pdf "fig:"){height="5cm" width="9cm"}\ (a) \ ![Comparison between the scaling factors of the ELMM and MUA-SV algorithms for (a) DC1, (b) DC2 and (c) DC3, for an SNR of 30dB.[]{data-label="fig:scaling_facts_example"}](figs_review/estim_variability2_DC2_SNR30_tght.pdf "fig:"){height="5cm" width="9cm"}\ (b) \ ![Comparison between the scaling factors of the ELMM and MUA-SV algorithms for (a) DC1, (b) DC2 and (c) DC3, for an SNR of 30dB.[]{data-label="fig:scaling_facts_example"}](figs_review/estim_variability2_DC3_SNR30_tght.pdf "fig:"){height="5cm" width="9cm"}\ (c) ![Abundance maps estimated by the ELMM and MUA-SV algorithms for DC2 with an SNR of 30dB.[]{data-label="fig:scaling_facts_example_abundances"}](figs_review/estim_abund_ELMMvsSPPX_DC2_tght.png){height="7cm" width="9cm"} Parameter Selection =================== For the synthetic data, the parameters were selected by exhaustive search within the range of values used by the respective authors in the original papers, aiming at achieving the minimum MSE for the reconstructed abundances. They are depicted in Table \[tab:alg\_param\_dc1\_dc2\_optA\] for all data cubes and all SNRs. For the real data, the parameters for the MUA-SV were selected in order to produce coherent abundance maps. For the other methods the parameters were extracted from \[S3\]. All parameters used with real data simulations are displayed in Table \[tab:alg\_param\_realData\]. \[!ht\] [c||c|c]{}\ SNR & Method & Parameters\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=1000$, $\gamma=1.5$\ & ELMM & $\lambda_M=5$, $\lambda_{{\boldsymbol{\Psi}}}=0.0005$, $\lambda_A=0.5$\ & MUA-SV & $\lambda_M=0.5$, $\lambda_{{\boldsymbol{\Psi}}}=10$, $\lambda_A=1$, $\rho\,S^2/N^2=0.5$, $\sqrt{N/S}=5$, $\gamma=0.005$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=10^3$, $\gamma=1.5$\ & ELMM & $\lambda_M=1$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.01$\ & MUA-SV & $\lambda_M=0.5$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=1$, $\rho\,S^2/N^2=0.1$, $\sqrt{N/S}=3$, $\gamma=0.001$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=1000$, $\gamma=1.5$\ & ELMM & $\lambda_M=0.1$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.005$\ & MUA-SV & $\lambda_M=0.1$, $\lambda_{{\boldsymbol{\Psi}}}=1$, $\lambda_A=0.5$, $\rho\,S^2/N^2=0.1$, $\sqrt{N/S}=3$, $\gamma=0.01$\ \ SNR & Method & Parameters\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.000005$, $\beta=1000$, $\gamma=1.5$\ & ELMM & $\lambda_M=50$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.5$\ & MUA-SV & $\lambda_M=1$, $\lambda_{{\boldsymbol{\Psi}}}=50$, $\lambda_A=100$, $\rho\,S^2/N^2=0.005$, $\sqrt{N/S}=9$, $\gamma=0.01$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=0.05$, $\gamma=1.5$\ & ELMM & $\lambda_M=0.005$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.05$\ & MUA-SV & $\lambda_M=0.5$, $\lambda_{{\boldsymbol{\Psi}}}=50$, $\lambda_A=50$, $\rho\,S^2/N^2=0.005$, $\sqrt{N/S}=9$, $\gamma=0.01$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.00005$, $\beta=10$, $\gamma=1.5$\ & ELMM & $\lambda_M=0.005$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.05$\ & MUA-SV & $\lambda_M=1$, $\lambda_{{\boldsymbol{\Psi}}}=50$, $\lambda_A=100$, $\rho\,S^2/N^2=0.005$, $\sqrt{N/S}=9$, $\gamma=0.01$\ \ SNR & Method & Parameters\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=50$, $\gamma=1.5$\ & ELMM & $\lambda_M=0.005$, $\lambda_{{\boldsymbol{\Psi}}}=0.0005$, $\lambda_A=0.01$\ & MUA-SV & $\lambda_M=0.01$, $\lambda_{{\boldsymbol{\Psi}}}=0.05$, $\lambda_A=0.005$, $\rho\,S^2/N^2=0.001$, $\sqrt{N/S}=4$, $\gamma=0.05$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=50$, $\gamma=1$\ & ELMM & $\lambda_M=0.005$, $\lambda_{{\boldsymbol{\Psi}}}=0.01$, $\lambda_A=0.01$\ & MUA-SV & $\lambda_M=5$, $\lambda_{{\boldsymbol{\Psi}}}=0.05$, $\lambda_A=0.01$, $\rho\,S^2/N^2=0.001$, $\sqrt{N/S}=4$, $\gamma=0.01$\ & FCLS & $\times$\ & SCLS & $\times$\ & PLMM & $\alpha=0.01$, $\beta=50$, $\gamma=1$\ & ELMM & $\lambda_M=0.1$, $\lambda_{{\boldsymbol{\Psi}}}=0.5$, $\lambda_A=0.0005$\ & MUA-SV & $\lambda_M=5$, $\lambda_{{\boldsymbol{\Psi}}}=0.01$, $\lambda_A=0.01$, $\rho/\sqrt{N/S}^4=0.001$, $\sqrt{N/S}=2$, $\gamma=0.0005$\ \[tab:alg\_param\_dc1\_dc2\_optA\] \[!ht\] Dataset Method Parameters --------- -------- --------------------------------------------------------------------------------------------------------------------------------- FCLS $\times$ SCLS $\times$ PLMM $\alpha=0.0014$, $\beta=500$, $\gamma=1$ ELMM $\lambda_M=0.4$, $\lambda_{{\boldsymbol{\Psi}}}=0.001$, $\lambda_A=0.005$ MUA-SV $\lambda_M=0.5$, $\lambda_{{\boldsymbol{\Psi}}}=0.001$, $\lambda_A=0.001$, $\rho\,S^2/N^2=0.35$, $\sqrt{N/S}=5$, $\gamma=0.001$ FCLS $\times$ SCLS $\times$ PLMM $\alpha=0.00031$, $\beta=500$, $\gamma=1$ ELMM $\lambda_M=0.4$, $\lambda_{{\boldsymbol{\Psi}}}=0.005$, $\lambda_A=0.005$ MUA-SV $\lambda_M=5$, $\lambda_{{\boldsymbol{\Psi}}}=0.01$, $\lambda_A=0.05$, $\rho\,S^2/N^2=0.01$, $\sqrt{N/S}=6$, $\gamma=0.001$ \[tab:alg\_param\_realData\] Sensitivity Analysis {#sensitivity-analysis} ==================== The simulations discussed in Section V.B in the manuscript are replicated here for all datasets DC1, DC2 and DC3, and all SNR values 20, 30 and 40dB. Figures \[fig:supp\_sensitivity\_1\], \[fig:supp\_sensitivity\_2\] and \[fig:supp\_sensitivity\_3\] present the sensitivity for the DC1 data cube, Figures \[fig:supp\_sensitivity\_4\], \[fig:supp\_sensitivity\_5\] and \[fig:supp\_sensitivity\_6\] for the DC2 data cube, and Figures \[fig:supp\_sensitivity\_7\], \[fig:supp\_sensitivity\_8\] and \[fig:supp\_sensitivity\_9\] for the DC3 data cube, for SNRs of 20, 30 and 40dB respectively. The results corroborate the discussion presented in Section V.B of the manuscript. \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC1 with an SNR of 20dB.[]{data-label="fig:supp_sensitivity_1"}](figures/sensitivity/DC1_SNR20){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC1 with an SNR of 30dB.[]{data-label="fig:supp_sensitivity_2"}](figures/sensitivity/DC1_SNR30){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC1 with an SNR of 40dB.[]{data-label="fig:supp_sensitivity_3"}](figures/sensitivity/DC1_SNR40){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC2 with an SNR of 20dB.[]{data-label="fig:supp_sensitivity_4"}](figures/sensitivity/DC2_SNR20){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC2 with an SNR of 30dB.[]{data-label="fig:supp_sensitivity_5"}](figures/sensitivity/DC2_SNR30){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC2 with an SNR of 40dB.[]{data-label="fig:supp_sensitivity_6"}](figures/sensitivity/DC2_SNR40){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC3 with an SNR of 20dB.[]{data-label="fig:supp_sensitivity_7"}](figures/sensitivity/DC3_SNR20){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC3 with an SNR of 30dB.[]{data-label="fig:supp_sensitivity_8"}](figures/sensitivity/DC3_SNR30){width="1\linewidth"} \[!htbp\] ![MSE variation due to relative changes in each parameter value about its optimal value (left) and MSE as a function of SLIC parameters $\sqrt{N/S}$ and $\gamma$ (right) for data cube DC3 with an SNR of 40dB.[]{data-label="fig:supp_sensitivity_9"}](figures/sensitivity/DC3_SNR40){width="1\linewidth"} REFERENCES - R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua, and S. Süsstrunk, “SLIC superpixels compared to state-of-the-art superpixel methods,” IEEE transactions on pattern analysis and machine intelligence, vol. 34, no. 11, pp. 2274–2282, 2012. - B. Hapke, Theory of Reflectance and Emittance Spectroscopy. Cambridge University Press, 1993. - L. Drumetz, M.-A. Veganzones, S. Henrot, R. Phlypo, J. Chanussot, and C. Jutten, “Blind hyperspectral unmixing using an extended linear mixing model to address spectral variability,” IEEE Transactions on Image Processing, vol. 25, no. 8, pp. 3890–3905, 2016. - R. Webster, P. Curran, and J. Munden, “Spatial correlation in reflected radiation from the ground and its implications for sampling and mapping by ground-based radiometry,” Remote sensing of environment, vol. 29, no. 1, pp. 67–78, 1989. - E. Tola, K. Al-Gaadi, R. Madugundu, A. Zeyada, A. Kayad, and C. Biradar, “Characterization of spatial variability of soil physicochemical properties and its impact on rhodes grass productivity,” Saudi journal of biological sciences, vol. 24, no. 2, pp. 421–429, 2017. - A. Najafian, M. Dayani, H. R. Motaghian, and H. Nadian, “Geostatistical assessment of the spatial distribution of some chemical properties in calcareous soils,” Journal of Integrative Agriculture, vol. 11, no. 10, pp. 1729–1737, 2012. - Y.-C. Wei, Y.-L. Bai, J.-Y. Jin, F. Zhang, L.-P. Zhang, and X.-Q. Liu, “Spatial variability of soil chemical properties in the reclaiming marine foreland to yellow sea of china,” Agricultural Sciences in China, vol. 8, no. 9, pp. 1103–1111, 2009. - J. Hou-Long, L. Guo-Shun, W. Xin-Zhong, S. Wen-Feng, Z. Rui-Na, Z. Chun-Hua, H. Hong-Chao, and L. Yan-Tao, “Spatial variability of soil properties in a long-term tobacco plantation in central china,” Soil Science, vol. 175, no. 3, pp. 137–144, 2010. - J. K. Crowley, “Visible and near-infrared spectra of carbonate rocks: Reflectance variations related to petrographic texture and impurities,” Journal of Geophysical Research: Solid Earth, vol. 91, no. B5, pp. 5001–5012, 1986. - R. N. Clark, “Spectroscopy of rocks and minerals, and principles of spectroscopy,” in Remote Sensing for the Earth Sciences: Manual of Remote Sensing, A. N. Rencz, Ed. New York, NY, USA: Wiley, 1999, vol. 3, pp. 3–58. [^1]: This work has been supported by the National Council for Scientific and Technological Development (CNPq) under grants 304250/2017-1, 409044/2018-0, 141271/2017-5 and 204991/2018-8, and by the Brazilian Education Ministry (CAPES) under grant PNPD/1811213. [^2]: The authors would like to thank Lucas Drumetz and his collaborators for providing part of the data used in the experimental section of the manuscript. [^3]: R.A. Borsoi is with the Department of Electrical Engineering, Federal University of Santa Catarina (DEE–UFSC), Florianópolis, SC, Brazil, and with the Lagrange Laboratory, Université Côte d’Azur, Nice, France. e-mail: . T. Imbiriba was with DEE–UFSC, Florianópolis, SC, Brazil, and is with the ECE department of the Northeastern University, Boston, MA, USA. e-mail: . J.C.M. Bermudez is with the DEE–UFSC, Florianópolis, SC, Brazil, and with the Graduate Program on Electronic Engineering and Computing, Catholic University of Pelotas (UCPel) Pelotas, Brazil. e-mail: . [^4]: This paper has supplementary downloadable material available at http://ieeexplore.ieee.org., provided by the authors. The material includes more detailed experimental validations. Contact for further questions about this work. [^5]: Manuscript received Month day, year; revised Month day, year. [^6]: Note that the definition of a pure material depends on convention and may change depending on the problem. [^7]: Generated using the code in http://www.ehu.es/ccwintco/index.php/ Hyperspectral\_Imagery\_Synthesis\_tools\_for\_MATLAB [^8]: Most of the data for this simulation was generously provided by Lucas Drumetz and his collaborators. [^9]: For conciseness, we present only the results for the DC1 data cube with a 30 dB SNR. The results for the other data cubes and SNRs are described in the supplemental material, also available in [@Borsoi_multiscaleVar_2018], and corroborate the conclusions presented in this section. [^10]: Note that the operating point of MUA-SV is not optimal for this case due to the relatively coarse grid employed in the parameter search procedure.

ArXiv

--- abstract: 'Recently, many convolutional neural network (CNN) methods have been designed for hyperspectral image (HSI) classification since CNNs are able to produce good representations of data, which greatly benefits from a huge number of parameters. However, solving such a high-dimensional optimization problem often requires a large amount of training samples in order to avoid overfitting. Additionally, it is a typical non-convex problem affected by many local minima and flat regions. To address these problems, in this paper, we introduce naive Gabor Networks or Gabor-Nets which, for the first time in the literature, design and learn CNN kernels strictly in the form of Gabor filters, aiming to reduce the number of involved parameters and constrain the solution space, and hence improve the performances of CNNs. Specifically, we develop an innovative phase-induced Gabor kernel, which is trickily designed to perform the Gabor feature learning via a linear combination of local low-frequency and high-frequency components of data controlled by the kernel phase. With the phase-induced Gabor kernel, the proposed Gabor-Nets gains the ability to automatically adapt to the local harmonic characteristics of the HSI data and thus yields more representative harmonic features. Also, this kernel can fulfill the traditional complex-valued Gabor filtering in a real-valued manner, hence making Gabor-Nets easily perform in a usual CNN thread. We evaluated our newly developed Gabor-Nets on three well-known HSIs, suggesting that our proposed Gabor-Nets can significantly improve the performance of CNNs, particularly with a small training set.' author: - 'Chenying Liu, Jun Li, Lin He, Antonio Plaza, Shutao Li, and Bo Li [^1]' bibliography: - 'References.bib' title: Naive Gabor Networks for Hyperspectral Image Classification --- Hyperspectral images (HSIs), convolutional neural networks (CNNs), naive Gabor networks (Gabor-Nets). Introduction {#sec:intro} ============ Over the past two decades, hyperspectral imaging has witnessed a surge of interest for Earth Observations due to its capability to detect subtle spectral information using hundreds of continuous and narrow bands, thus making it promising for some applications such as classification [@Li201DL-HSI-ClassificationReview; @He2018Review]. In the early stages of HSI classification, most techniques were devoted to analyzing the data exclusively in the spectral domain, disregarding the rich spatial-contextual information contained in the scene [@Fauvel2013Review]. Then, many approaches were developed to extract spectral-spatial features prior to classification to overcome this limitation, such as morphological profiles (MPs) [@Benediktsson2005EMP], spatial-based filtering techniques [@He2017DLRGF; @Jia2018GaborHSI], etc., which generally adopts hand-crafted features following by a classifier with predefined hyperparameters. Recently, inspired by the great success of deep learning methods [@Liu2019CNNSAR; @Kim2019CNNBlind], CNNs have emerged as a powerful tool for spectral and spatial HSI classification [@Paoletti2019Pyramidal; @Hamouda2019AdaptiveSize]. Different from traditional methods, CNNs jointly learn the information for feature extraction and classification with a hierarchy of convolutions in a data-driven context, which is capable to capture features in different levels and generate more robust and expressive feature representations than the hand-crafted ones. Furthermore, the parameters can be optimized in accordance with data characteristics, leading to more effective models. However, CNN methods often require a large number of training samples in order to avoid overfitting, due to the huge number of free parameters involved. This is particularly challenging in the context of HSI classification, where manual annotation of samples is difficult, expensive and time-consuming. Moreover, solving the kernels of a CNN is a typical non-convex problem, affected by many local minima and flat regions [@Blum1993NPComplete], which is usually addressed with a local search algorithm, such as the gradient descend algorithm under a random initialization scheme, making the kernels very likely converge to a bad/spurious local minimum [@Sinom2018SupriousLocalMinima]. To tackle these issues, a recent trend is to embed *a priori* knowledge into deep methods to refine model architectures. For example, Shamir [@Shamir2016GaussianInput] and Tian [@Tian2017GaussianInput] showed that the adoption of Gaussian assumptions on the input distribution can assist the successful training of neural networks. Chen *et al.* [@Chen2018MMDP] overcame the contradiction between a small training size and a large parameter space through the integration of Bayesian modeling into neural networks. These previous works reveal that *a priori* knowledge exhibits good potential in improving the reliability and generalization of deep models. More specifically, for CNNs some attempts have been made to reinforce model robustness via redesigning convolutional kernels using certain *a priori* knowledge. For instance, circular harmonics are employed to equip CNNs with both translation and rotation equivariant kernels [@Worrall2017HNet]. However, the construction of such rotation-equivariant kernels is somewhat complicated, where each filter is a combination of a set of filter bases. Besides, the complex-valued convolution operations require a new CNN thread and increase the calculation burden in both the forward and backward propagations when using the same number of kernels. Apart from circular harmonics, Gabor filters offer another type of *a priori* knowledge that can be used to reinforce convolutional kernels of CNNs. Gabor filters are able to achieve optimal joint time-frequency resolution from a signal processing perspective [@Gabor1946], thus being appropriate for low-level and middle-level feature extractions (which are exactly the functions of the bottom layers of CNNs). Furthermore, researches have revealed that the shape of Gabor filters is similar to that of receptive fields of simple cells in the primary visual cortex [@Hubel1965CatCortex; @Jones1987EvaluationCat; @Pollen1983Bio; @Alex2012AlexNet], which means that using Gabor filters to extract low-level and middle-level features can be associated with a biological interpretation. In fact, as illustrated by Fig. \[fig:1stlayerKer\], many filters in CNNs (especially those in the first several layers), look very similar to Gabor filters. Inspired by these aspects, some attempts have been made to utilize the Gabor *a priori* knowledge to reinforce CNN kernels, i.e. by replacing some regular kernels with fixed Gabor filters to reduce the number of parameters [@Jiang2018GCNNBinary; @Calderon2003GCNNHandwritten; @Sarwar2017GCNNEnergy], initializing regular kernels in the form of Gabor filters [@Jiang2018GCNNBinary; @Chang2014GCNNSpeech], and modulating regular kernels with predefined Gabor filters [@Luan2018GCN]. Their good performance indicates a promising potential of Gabor filters in promoting the capacity of CNN models. However, traditional Gabor filtering is complex-valued, while CNNs are usually fulfilled with real-valued convolutions. Therefore, most Gabor-related CNNs only utilize the real parts of Gabor filters to form CNNs, which means that they only collect local low-frequency information in the data while disregarding (possibly useful) high-frequency information. To mitigate these issues, in analogy with some shallow-learning based Gabor approaches [@He2017DLRGF], Jiang *et al.* [@Jiang2018GCNNBinary] used the direct concatenation of the real and imaginary parts in CNNs. However, this approach is unable to tune the relationship between these two parts when extracting Gabor features. Most importantly, all these Gabor-related methods still manipulate hand-crafted Gabor filters, whose parameters are empirically set and remain unchanged during the CNN learning process. That is, the Gabor computation (although involved in these existing Gabor-related CNN models) does not play a significant role and, hence, is independent of the CNN learning. The remaining question is how to conveniently and jointly utilize the Gabor representation and the learning ability of CNNs so as to generate more effective features in a data-driven fashion. ![Filters extracted from the first convolutional layer of a well-trained CNN using 100 training samples per class for a hyperspectral image collected over Pavia University, Italy.[]{data-label="fig:1stlayerKer"}](CNN_FirstLayerV2.eps "fig:"){width="3.5in"}\ In this work, we introduce a new concept of *naive Gabor Networks* (or Gabor-Nets) for HSI classification where *naive* refers to the fact that we straightforwardly replace regular convolutional kernels of CNNs with Gabor kernels, which is based on the following intuitions: - First, Gabor filtering can be fulfilled with a linear convolution [@Arya2018ReviewGaborFuzzySVM], which implies that Gabor filtering can be naturally extended to implement the basic convolution operations in CNNs. - Second, transforming the problem of solving CNN kernels to that of finding the optimal parameters of Gabor kernels tends to reduce the number of free parameters. If Gabor kernels, instead of regular CNN kernels, are used in a CNN, then the parameters to solve in each kernel will be transformed from the CNN kernel elements to Gabor parameters, such as the frequency magnitude, frequency direction and scale. - Although usually CNNs require real-valued computations (while Gabor filtering involves complex-valued computations related to the real and imaginary parts), there is a possibility to design flexible Gabor representations computed in a real-valued fashion without missing any information on the real and imaginary parts. This is because the local cosine harmonic and the local sinusoidal harmonic in a Gabor filter can be connected with each other by a phase offset term. It is noteworthy that remotely sensed images are mainly composed of a series of geometrical and morphological features, i.e., low-level and middle-level features. Therefore, our networks (with a few Gabor convolutional layers) are expected to be able to extract representative features for HSI processing. To the best of our knowledge, this is the first attempt in the literature to both design and learn CNN convolutions strictly in the form of Gabor filters. More specifically, the innovative contributions of our newly developed Gabor-Nets can be summarized as follows: - Gabor-Nets operate in a twofold fashion. On the one hand, using Gabor filtering to perform convolutions in CNNs tends to reduce the number of the parameters to learn, thus requiring a smaller training set and achieving faster convergence during the optimization. On the other hand, the free parameters of Gabor filters can be automatically determined by the forward- and backward-propagations of CNNs. - Gabor-Nets are built on novel phase-induced Gabor kernels. The Gabor kernels, induced with a kernel phase term, exhibit two important properties. First, they have potential to adaptively collect both the local cosine and the local sinusoidal harmonic characteristics of the data. Second, the kernels can be used for real-valued convolutions. Thus, Gabor-Nets implemented with the new kernels are able to perform similarly to CNNs while generating more representative features. - Gabor-Nets adopt a well-designed initialization scheme. Specifically, the parameters used to construct usually-used hand-crafted Gabor filter banks are initialized based on the Gabor *a priori* knowledge, while the kernel phases utilize a random initialization in order to increase the diversity of kernels. Such an initialization scheme can not only make Gabor-Nets inherit the advantages of traditional Gabor filters, but also equip Gabor-Nets with some superior properties such as the ability against the gradient vanishing problem often arising in CNNs. The remaining of the paper is organized as follows. Section \[sec:relatedWorks\] gives the general formulation of Gabor harmonics and simultaneously reviews some related works on Gabor filtering. Section \[sec:meth\] introduces the proposed Gabor-Nets constructed with an innovative phase-induced Gabor kernel in detail. Section \[sec:exp\] describes the experimental validation using three real hyperspectral datasets. Finally, section \[sec:con\] concludes the paper with some remarks and hints at plausible future research lines. Related work {#sec:relatedWorks} ============ Let $(x, y)$ denote the space domain of an image. A [general]{} 2-D Gabor filter can be mathematically formulated by a Gaussian envelope modulated sinusoid harmonic, as follows: $$\label{eq:2Dgabor:general2D} \begin{aligned} \mathbf{G}(x,y) = & \frac{1}{2\pi\sigma_x\sigma_y}\exp{\bigg\{\!\!-\frac{1}{2}\Big(\frac{x_r^2}{\sigma_x^2}+\frac{y_r^2}{\sigma_y^2}\Big)\bigg\}} \\ & \times\exp{\{j(x\omega_x+y\omega_y)\}}, \end{aligned}$$ where $\sigma_x$ and $\sigma_y$ are the scales along the two spatial axes of the Gaussian envelope, $x_r=x\cos\phi+y\sin\phi$ and $y_r=-x\sin\phi+y\cos\phi$ are the rotated coordinates of $x$ and $y$ with a given angle $\phi$, $\omega_x=|\bm{\omega}|\cos\theta$ and $\omega_y=|\bm{\omega}|\sin\theta$ are the projections of a given angular frequency $\bm{\omega}$ onto $x$ and $y$-directions, respectively, $\theta$ is the angle between $\bm{\omega}$ and the $x$-direction, $|\bm{\omega}|=(\omega_x^2+\omega_y^2)^\frac{1}{2}$ is the magnitude of $\bm{\omega}$ (hereinafter replaced by $\omega$) and $j$ is the imaginary unit. To simplify the gradient calculation of $\theta$, we utilize the rotation-invariant Gaussian envelope under unrotated coordinates with $\phi=0$ and $\sigma_x\!=\!\sigma_y\!=\!\sigma$. With Euler’s relation, let $M=x\omega_x+y\omega_y$, we can rewrite the 2-D Gabor filter in the following complex form: $$\label{eq:2Dgabor:euler} \begin{aligned} \mathbf{G}(x,y) & = K \times\exp{\{jM\}}\\ &= K\cos{M} +jK\sin{M}\\ &= \Re\{\mathbf{G}(x,y)\} +j\Im\{\mathbf{G}(x,y)\}, \end{aligned}$$ where $K =\frac{1}{2\pi\sigma^2}\exp{\{-\frac{x^2+y^2}{2\sigma^2}\}}$ is the rotation-invariant Gaussian envelope. Specifically, the local cosine harmonic $\Re\{\mathbf{G}(x,y)\}$ is associated with the local low-frequency component in the image, and the local sinusoidal harmonic $\Im\{\mathbf{G}(x,y)\}$ is connected to the local high-frequency component [@He2017DLRGF], thus enabling Gabor filtering to access the local harmonic characteristics of the data. In the following, we review some existing works relevant to the Gabor filtering. Traditional hand-Crafted Gabor filters {#sec:relatedWorks:GaborFilter} -------------------------------------- From a signal processing perspective, Gabor harmonics maximize joint time/frequency or space/frequency resolutions [@Gabor1946], making them ideal for computer vision tasks. Hand-crafted Gabor filters have achieved a great success in many applications, such as texture classification [@Idrissa2002GaborTexture], face and facial expression recognition [@See2017GaborFace], palmprint recognition [@Younesi2017GaborPalm], edge detection [@Namuduri1992GaborEdge], and several others [@Sun2005GaborVehicle]. Regarding HSI data interpretation, Bau *et al.* [@Bau2010RealGabor] used the real part of 3D Gabor filters to extract the energy features of regions for HSI classification, suggesting the effectiveness of Gabor filtering in feature extraction. He *et al.* [@He2017DLRGF] proposed a novel discriminative low-rank Gabor filtering (DLRGF) method able to generate highly discriminative spectral-spatial features with high computational efficiency, thus greatly improving the performance of Gabor filtering on HSIs. Jia *et al.* [@Jia2018GaborHSI] also achieved good classification results using the phase of complex-valued Gabor features. These hand-crafted Gabor features can be regarded as single-layer features extracted by Gabor filter banks. The involved parameters are empirically set following a “search strategy” where the orientations and spatial frequencies obey certain uniform distributions, aimed to cover as much optimal parameters as possible. Gabor-Related CNNs {#sec:relatedWorks:GaborCNN} ------------------ [m[1.1cm]{}&lt;|m[2.5cm]{}&lt;||m[5.0cm]{}&lt;||m[2.5cm]{}&lt;||m[3.5cm]{}&lt;]{} & Functions of Gabor filters in CNNs & &\ & Using Gabor features & Feature extraction for inputs[@Hosseini2018InputGaborAge; @Yao2016InputGaborOR; @Lu2017InputGaborFace; @Rizvi2016GCNNOR; @Chen2017InputGaborHSI; @Shi2018InputGaborShip] & &\ & & Fixed filters in shallow layers [@Jiang2018GCNNBinary; @Calderon2003GCNNHandwritten; @Sarwar2017GCNNEnergy]; & &\ & Using Gabor filters & Initialization of kernels [@Chang2014GCNNSpeech]; & &\ & & Modulation to kernels [@Luan2018GCN] & &\ & Convolutional kernels & Learnable & Tunable\ Recently, some attempts have been made to incorporate Gabor harmonics into CNNs, in order to reduce the number of parameters and equip CNNs with orientation and frequency selectivity. The existing Gabor-related CNNs can be roughly categorized into two groups, i.e., those using Gabor features and those using Gabor filters. The former category uses Gabor features only as the inputs of networks; while in the latter category, predefined Gabor filters with fixed parameters are used in some convolutional layers. Researches show that using hand-crafted Gabor features could help mitigate the negative effects introduced by a lack of training samples in CNNs. For example, Hosseini *et al.* [@Hosseini2018InputGaborAge] utilized additional Gabor features as inputs for CNN-based age and gender classification, and obtained better results. Yao *et al.* [@Yao2016InputGaborOR] achieved a higher recognition rate by using Gabor features to pre-train CNNs before fine-tuning. Similar works can be found in [@Lu2017InputGaborFace; @Rizvi2016GCNNOR]. In the field of remote sensing, Chen *et al.* [@Chen2017InputGaborHSI] fed the Gabor features extracted on the first several principal components into CNNs for HSI classification. Shi *et al.* [@Shi2018InputGaborShip] complemented CNN features with Gabor features in ship classification. Their experimental results indicate that Gabor features are able to improve the performance of CNNs. Another trend is to manipulate certain layers or kernels of CNNs with Gabor filters. For example, Jiang *et al.* [@Jiang2018GCNNBinary] replaced the kernels in the first layer of a CNN with a bank of Gabor filters under predefined orientations and spatial frequencies. These first-layer Gabor filters can be fixed, as explained in [@Calderon2003GCNNHandwritten], or be tuned at each kernel element, like [@Chang2014GCNNSpeech] where, in fact, Gabor filters were used for initialization purposes. Moreover, to reduce the training complexity, Sarwar *et al.*, [@Sarwar2017GCNNEnergy] replaced some kernels in the intermediate layers with fixed Gabor filters and yielded better results. More recently, Luan *et al.* [@Luan2018GCN] utilized Gabor filters to modulate regular convolutional kernels, thus making the network capable to capture more robust features with regards to orientation and scale changes. However, Gabor convolutional networks (GCNs) still learned the regular convolutional kernels, i.e., GCNs in fact utilized Gabor-like kernels. These Gabor-related works, either using Gabor features or using Gabor filters, manipulate the hand-crafted Gabor filters without Gabor feature learning, which means that their parameters are empirically set (and remain unchanged) during the learning process. That is, in these existing Gabor-related CNNs, the Gabor computation does not play any relevant role in the CNN learning. In contrast, as illustrated in Table \[table:diff\], our proposed Gabor-Nets directly use Gabor kernels with free parameters as CNN kernels, and automatically determine the Gabor parameters with forward- and backward-propagations of CNNs in a data-driven fashion, thus being able to not only use the Gabor *a priori* knowledge, but also to fulfill Gabor feature learning, therefore being adaptive to specific datasets and able to reduce the human supervision. Proposed Method {#sec:meth} =============== In this section, we first introduce an innovative phase-induced Gabor kernel, followed by a discussion on its superior frequency properties. Then, we describe the proposed Gabor-Nets in detail. Phase-induced Gabor {#sec:meth:gkernel} ------------------- The real and imaginary parts of commonly-adopted Gabor filters are associated with the local low-frequency and high-frequency information in the data, respectively [@He2017DLRGF]. Some Gabor methods use only the real part to extract features, which obviously discards the possibility of exploiting local high-frequency information. In order to integrate the two components, other methods utilize the amplitude feature [@He2017DLRGF] $$\label{eq:2Dgabor:amplitude} \begin{aligned} \|\mathbf{G}(x,y)\| = \sqrt{\Re^2\{\mathbf{G}(x,y)\} +\Im^2\{\mathbf{G}(x,y)\}}, \end{aligned}$$ the phase feature [@Jia2018GaborHSI] $$\label{eq:2Dgabor:phase} \begin{aligned} \sphericalangle\mathbf{G}(x,y) = \frac{\Im\{\mathbf{G}(x,y)\}}{\Re\{\mathbf{G}(x,y)\}}, \end{aligned}$$ or the direct concatenation of the real and the imaginary parts [@Jiang2018GCNNBinary]. The latter case, though considering the real and the imaginary parts simultaneously (and hence synthesizing both the low-frequency and high-frequency information), is under a formulation where there is no parameter able to tune their relationship. Additionally, as aforementioned, traditional Gabor filtering is connected to a complex-valued computation, whereas the standard CNNs are based on real-valued computations. Therefore, when applying Gabor kernels to CNNs, this difference has to be handled carefully. In the following, we design an innovative phase-induced Gabor filter to deal with these problems. Let $P$ denote the phase offset of the sinusoidal harmonic. With $P$ added, the usually-used 2-D Gabor filtering formulated in (\[eq:2Dgabor:euler\]) becomes $$\label{eq:2Dgabor:general2DP} \begin{aligned} \mathbf{G}_{P}(x,y) & = K \times\exp{\{j(M+P)\}}\\ & = K\cos{(M+P)}+jK\sin{(M+P)}\\ & = \Re\{\mathbf{G}_{P}(x,y)\} +j\Im\{\mathbf{G}_{P}(x,y)\},\\ \end{aligned}$$ where we find, $$\label{eq:equi:im} \begin{aligned} K\sin(M+P) = K\cos\Big(M+(P-\frac{\pi}{2})\Big), \end{aligned}$$ that is, $$\label{eq:equi:relation} \Im\{\mathbf{G}_P(x,y)\} = \Re\{\mathbf{G}_{(P-\frac{\pi}{2})}(x,y)\}.$$ As illustrated in (\[eq:equi:relation\]) and Fig. \[fig:gkernel:ReIm\], there is a one-to-one correspondence in terms of $P$ between the real and the imaginary parts of $\mathbf{G}_{P}(x,y)$, i.e., the imaginary part with $P$ is the same as its real counterpart with a phase offset of $(P-\pi/2)$. Based on this observation, we develop a new Gabor filtering to serve as Gabor kernels in CNNs as follows, $$\label{eq:gkernel:2D} \begin{aligned} \mathbf{G}(x,y) = K\cos{(M+P)}. \end{aligned}$$ It can be observed that the Gabor filters with $P$=$0$ and $P$=$-\pi/2$ above are exactly the real and the imaginary parts of the traditional Gabor filters in (\[eq:2Dgabor:euler\]), i.e., the low-frequency and the high-frequency components, respectively. This indicates that, with $P$ added, the Gabor filtering in (\[eq:gkernel:2D\]), though only formulated with the cosine harmonic, can be equipped with different frequency properties as $P$ varies. Thus, we name this newly developed Gabor filtering as the phase-induced Gabor filtering. Obviously, this new Gabor filtering is fulfilled with a real-valued convolution, which means that, if we utilize such Gabor filters as Gabor kernels of a CNN, the traditional complex-valued Gabor computation can be avoided, hence allowing us to directly use Gabor kernels in a usual CNN thread. In this work, we refer to $P$ as the kernel phase of Gabor kernels. ![[Real and imaginary parts of complex-valued Gabor filters using $\theta=\pi/4$, $\omega=\pi/100$, $\sigma=30$, where the imaginary part with a certain phase offset $P$ is just the corresponding real one with $(P-\pi/2)$.]{}[]{data-label="fig:gkernel:ReIm"}](Gabor_Re_and_Im.eps "fig:"){height="1.3in"}\ For clarity, hereinafter we utilize $\mathbf{G}'(x,y)$ to represent the Gabor filters without a phase offset term in (\[eq:2Dgabor:general2D\])-(\[eq:2Dgabor:phase\]), with their real and the imaginary parts denoted as $\Re\{\mathbf{G}'(x,y)\}$ and $\Im\{\mathbf{G}'(x,y)\}$, respectively. Adaptive Frequency Response Property {#sec:meth:properties} ------------------------------------ To obtain a deeper inspection of our innovative phase-induced Gabor kernel, we decompose (\[eq:gkernel:2D\]) using the trigonometric formula as follows: $$\label{eq:gkernel:sepP} \begin{aligned} \mathbf{G}(x,y) & = \cos{P}\cdot K\cos{M}-\sin{P}\cdot K\sin{M}\\ & = \cos{P}\cdot \Re\{\mathbf{G}'(x,y)\}-\sin{P}\cdot \Im\{\mathbf{G}'(x,y)\}. \end{aligned}$$ As can be observed, $\mathbf{G}(x,y)$ is actually a linear combination of $\Re\{\mathbf{G}'(x,y)\}$ and $\Im\{\mathbf{G}'(x,y)\}$ in (\[eq:2Dgabor:euler\]), which involves the weights $\cos P$ and $\sin P$ controlled by the kernel phase $P$. Utilizing the forward- and backward- propagations of CNNs, the free parameter $P$ can be tuned following the data. Recall that $\Re\{\mathbf{G}'(x,y)\}$ and $\Im\{\mathbf{G}'(x,y)\}$ are associated with the local low-frequency and high-frequency components, respectively [@He2017DLRGF]. Thus, by involving the kernel phase $P$, we can develop the CNN constructed by Gabor kernels, which is able to adaptively process both the low-frequency and high-frequency characteristics of the data. Reconsidering the decomposition of (\[eq:gkernel:2D\]), it can be found that the cosine harmonic is formed by the coupling of $x$ and $y$. If we decouple $x$ and $y$ via the trigonometric formula and separate the Gaussian envelope $K$ along the $x$ and $y$-directions, respectively, (\[eq:gkernel:2D\]) turns to [^2]: $$\label{eq:gkernel:decomp2} \begin{aligned} \mathbf{G}=g_{c,p}^{(x)} \cdot g_{c}^{(y)}-g_{s,p}^{(x)} \cdot g_{s}^{(y)}, \end{aligned}$$ where $$\label{eq:gkernel:xpcos} \begin{aligned} g_{c,p}^{(x)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{x^2}{2\sigma^2}\Big)}\cos{(x\omega_x+P)}, \end{aligned}$$ $$\label{eq:gkernel:ycos} \begin{aligned} \hspace{-0.7cm}g_{c}^{(y)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{y^2}{2\sigma^2}\Big)}\cos{(y\omega_y)}, \end{aligned}$$ $$\label{eq:gkernel:xpsin} \begin{aligned} g_{s,p}^{(x)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{x^2}{2\sigma^2}\Big)}\sin{(x\omega_x+P)}, \end{aligned}$$ and $$\label{eq:gkernel:ysin} \begin{aligned} \hspace{-0.7cm}g_{s}^{(y)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{y^2}{2\sigma^2}\Big)}\sin{(y\omega_y)}. \end{aligned}$$ As proven in [@He2017DLRGF], $g_{c}^{(y)}$ and $g_{s}^{(y)}$, without the kernel phase $P$, are low-frequency pass and low-frequency resistant filters, respectively. Regarding the other two components, i.e., $g_{c,p}^{(x)}$ and $g_{s,p}^{(x)}$, given $\omega_0$ of $\omega_x$, their mathematical representations in the frequency domain are calculated as follows, $$\label{eq:FT:cos} \begin{aligned} \widehat{g}_{c,p}(\omega) = \frac{1}{2}(A+B)\cos{P} + \frac{1}{2}j(A-B)\sin{P}, \end{aligned}$$ and $$\label{eq:FT:sin} \begin{aligned} \widehat{g}_{s,p}(\omega) = \frac{1}{2}(A+B)\sin{P} - \frac{1}{2}j(A-B)\cos{P}, \end{aligned}$$ where $A\!=\!\exp\!{\big(\!-\!\frac{\sigma^2(\omega-\omega_0)^2}{2}\!\big)}$, and $B\!=\!\exp\!{\big(\!-\!\frac{\sigma^2(\omega+\omega_0)^2}{2}\!\big)}$. Then their corresponding squared magnitudes of frequency can be obtained as follows, $$\label{eq:FT:cos:mag} \begin{aligned} |\widehat{g}_{c,p}(\omega)|^2 & = \frac{1}{4}\!\exp{\big(\!-\!{\sigma^2(\omega\!-\!\omega_0)^2}\big)} \!+\! \frac{1}{4}\!\exp{\big(\!-\!{\sigma^2(\omega\!+\!\omega_0)^2}\big)}\\ &\;\;\;\;+ \frac{1}{2}\cos{(2P)}\exp{\big(-\sigma^2(\omega^2+\omega_0^2)\big)}, \end{aligned}$$ $$\label{eq:FT:sin:mag} \begin{aligned} |\widehat{g}_{s,p}(\omega)|^2 & = \frac{1}{4}\!\exp{\big(\!-\!{\sigma^2(\omega\!-\!\omega_0)^2}\big)}\!+\!\frac{1}{4}\!\exp{\big(\!-\!{\sigma^2(\omega\!+\!\omega_0)^2}\big)}\\ &\;\;\;\;- \frac{1}{2}\cos{(2P)}\exp{\big(-\sigma^2(\omega^2+\omega_0^2)\big)}. \end{aligned}$$ If we set $\omega$ in (\[eq:FT:cos:mag\]) and (\[eq:FT:sin:mag\]) to be zero, we have $$\label{eq:FT:cos:mag0} \begin{aligned} |\widehat{g}_{c,p}(0)|^2 & = \frac{1}{2}[1+\cos(2P)]\exp{\big(-\sigma^2\omega_0^2\big)}, \end{aligned}$$ $$\label{eq:FT:sin:mag0} \begin{aligned} |\widehat{g}_{s,p}(0)|^2 & = \frac{1}{2}[1-\cos(2P)]\exp{\big(-\sigma^2\omega_0^2\big)}. \end{aligned}$$ As shown in (\[eq:FT:cos:mag0\]) and (\[eq:FT:sin:mag0\]), when $\cos(2P)$ is approaching $-1$, i.e., $P$ is approaching $\pi/2$, $|\widehat{g}_{c,p}(0)|^2$ and $|\widehat{g}_{s,p}(0)|^2$ are decreasing to and increasing away from $0$, respectively (which implies that the low-frequency resistance of $g_{c,p}^{(x)}$ is being enforced, while $g_{s,p}^{(x)}$ behaves more like a low-pass filter). The situation is opposite when $\cos(2P)$ is approaching $1$, i.e., $P$ is approaching 0. Fig. \[fig:phase:mag\] shows the appearance of the squared magnitudes in the frequency domain with varying values of $P$. Clearly, [the frequency response characteristics]{} of $g_{c,p}^{(x)}$ and $g_{s,p}^{(x)}$ significantly change as $P$ varies. [p[3.7cm]{}&lt;p[3.7cm]{}&lt;]{} ![Squared frequency magnitudes of Gaussian enveloped (a) cosine and (b) sinusoidal harmonics with varying values of $P$.[]{data-label="fig:phase:mag"}](FreqMag_cos.eps "fig:"){height="1.2in"} & ![Squared frequency magnitudes of Gaussian enveloped (a) cosine and (b) sinusoidal harmonics with varying values of $P$.[]{data-label="fig:phase:mag"}](FreqMag_sin.eps "fig:"){height="1.2in"}\ (a) & (b) For comparison, we employ the same strategy to decompose (\[eq:2Dgabor:euler\]) as follows, $$\label{eq:2Dgabor:noPdecompRe} \begin{aligned} \Re\{\mathbf{G}^{'}(x,y)\} = g_{c}^{(x)}\!\cdot\!g_{c}^{(y)}-g_{s}^{(x)}\!\cdot\!g_{s}^{(y)}, \end{aligned}$$ $$\label{eq:2Dgabor:noPdecompIm} \begin{aligned} \Im\{\mathbf{G}^{'}(x,y)\} = g_{s}^{(x)}\!\cdot\!g_{c}^{(y)}+g_{c}^{(x)}\!\cdot\!g_{s}^{(y)}, \end{aligned}$$ where $$\label{eq:gkernel:xcos} \begin{aligned} g_{c}^{(x)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{x^2}{2\sigma^2}\Big)}\cos{(x\omega_x)}, \end{aligned}$$ $$\label{eq:gkernel:xsin} \begin{aligned} g_{s}^{(x)}=\frac{1}{\sqrt{2\pi}\sigma}\exp{\Big(-\frac{x^2}{2\sigma^2}\Big)}\sin{(x\omega_x)}. \end{aligned}$$ It can be observed that, (\[eq:2Dgabor:euler\]) is composed of a series of components which all have an explicit frequency nature. Accordingly, if we utilize the Gabor filtering without $P$ in (\[eq:2Dgabor:euler\]) as Gabor kernels, the fundamental properties of kernels, i.e., the frequency response characteristics, can hardly be changed though other parameters are adaptively tuned in the learning process. {width="0.90\linewidth"} To conclude the above two parts, the roles of the kernel phase $P$, which is crucial in our phase-induced Gabor kernel and therefore the newly developed Gabor-Nets, can be summarized as follows: - The kernel phase $P$ endows Gabor kernels with the ability to adaptively collect both the local cosine and the local sinusoidal harmonic characteristics of the data, via adjusting the their linear combination. - With the kernel phase $P$, the traditional complex-valued Gabor kernel can be fulfilled in a real-valued manner, therefore making it possible to directly (and conveniently) utilize our phase-induced Gabor kernel to construct a real-valued CNN. Gabor-Nets {#sec:meth:gabornet} ---------- The proposed Gabor-Nets directly use phase-induced Gabor kernels to fulfill CNN convolutions. Then, the parameters to solve in each convolutional kernel are transformed from the kernel elements *per se* to the Gabor parameters of a phase-induced Gabor kernel: $\{\theta, \omega, \sigma, P\}$, i.e., the angle between the angular frequency and the $x$-direction $\theta$, the magnitude of the angular frequency $\omega$, the scale $\sigma$, and the kernel phase $P$. Let $k$ denote the kernel size. A phase-induced Gabor kernel has only four parameters to learn no matter how $k$ varies, whereas a regular kernel has $k^2$ elements to solve. In the situation with a smallest kernel size (the $1\times1$ kernels are not considered in this work), i.e., when $k=3$, the numbers of free parameters of a Gabor kernel and a regular kernel are 4 and 9, respectively. With the increase of $k$, the difference between those parameter numbers increases. For simplicity, we utilize $\mathbf{G}$ in place of $\mathbf{G}(x,y)$, and $\{\theta_0, \omega_0, \sigma_0, P_0\}$ to represent the initializations of corresponding Gabor parameters hereinafter. As illustrated in Fig. \[fig:GaborNet\], the number of output features in the $l$th convolutional layer of Gabor-Nets is determined as $N_o=N_t \times N_m$, where $N_t$ and $N_m$ are the predefined numbers of $\theta_0$s and $\omega_0$s, respectively. Then, with $N_i$ input features, the kernels in the $l$th layer are defined as $$\label{eq:GaborNet:kernels} \begin{aligned} \mathbf{G}^{(l)}=\{ \mathbf{G}_{1}^{(l)},\mathbf{G}_{2}^{(l)},\cdots,\mathbf{G}_{N_o}^{(l)}\}, \end{aligned}$$ where $\mathbf{G}_{o}^{(l)}=\{\mathbf{G}_{o,1}^{(l)},\cdots,\mathbf{G}_{o,N_i}^{(l)} \}, o = 1,2,\cdots,N_o$ is the $o$th kernel, i.e., a set of $N_i$ Gabor filters corresponding to $N_i$ input features used to generate the $o$th output feature. Within a kernel, the $N_i$ filters are initialized with the same $\theta_0$ and $\omega_0$, and then are fine-tuned in a data-driven context during the training process. As a result, we can obtain the output features as follows, $$\label{eq:GaborNet:output} \begin{aligned} \mathbf{O}^{(l)}=\{\mathbf{O}_{1}^{(l)},\mathbf{O}_{2}^{(l)},\cdots,\mathbf{O}_{N_o}^{(l)} \}, \end{aligned}$$ where $\mathbf{O}_{o}^{(l)}=\sum_i^{N_i} \mathbf{I}_i^{(l)} \ast \mathbf{G}_{o,i}^{(l)}$ for $o=1,2,\cdots,N_o$, and $\mathbf{I}^{(l)}=\{\mathbf{I}_1^{(l)},\cdots,\mathbf{I}_{N_i}^{(l)}\}$ are the input features of the $l$th layer. For the first layer, $\mathbf{I}^{(l)}$ are the initial input features of the network, otherwise $\mathbf{I}^{(l)}=\mathbf{O}^{(l-1)}$. Notice that the key difference between the proposed Gabor-Nets and regular CNNs is the designed form of convolutional kernels. Therefore, it is very easy to incorporate other CNN elements or tricks into Gabor-Nets, such as pooling, batch normalization, activation functions, etc. ### Initialization of Gabor kernels In order to guarantee the effectiveness of Gabor kernels, we provide a generally reliable initialization scheme for Gabor-Nets. First, following the “search strategy” used for the settings of hand-crafted Gabor filter banks, the $\theta_0$s are predefined as an evenly spaced sequence of $[0,\pi)$ based on $N_t$, and the $\omega_0$s are set as a geometric sequence with an initial value of $(\pi/2)$ and a geometric progression of $(1/2)$. For example, as shown in Fig. \[fig:GaborNet\], to construct a Gabor convolutional layer using $N_t=4$ and $N_m=2$, we set $\theta_0$s to be 0, $(\pi/4)$, $(\pi/2)$, $(3\pi/4)$, and $\omega_0$s to be $(\pi/2)$, $(\pi/4)$, respectively. On the one hand, the “search strategy” has been proven effective in traditional hand-crafted Gabor feature extraction by covering as many orientations and frequencies as possible. Then in Gabor-Nets, although each kernel is initially specific to one orientation and one frequency, different orientations and frequencies couple with each other as the layer goes deeper. On the other hand, such initializations are in accordance with the common observation that an HSI contains the information in many directions, while the discriminative information tends to appear on low frequencies [@He2017DLRGF]. The initialization of $\sigma$s is relatively empirical among the four parameters. As stated above, $\sigma$ controls the localization scale of the filter. In hand-crafted Gabor filter design, $\sigma$ is always set to be one quarter of the kernel size. Taken into consideration the fact that CNNs generate features via multi-layer convolutions, we initialize $\sigma$s to be one eight of the kernel size in our work. Regarding the kernel phase $P$, we adopt a random initialization of $P$ in order to increase the diversity of Gabor kernels, aimed at promoting the robustness of Gabor-Nets. As indicated in (\[eq:gkernel:sepP\]), $P$ dominates the harmonic characteristics of Gabor kernels via $\cos{P}$ and $\sin{P}$. Therefore, we randomly initialize $P_0$s within $[0,2\pi)$, i.e., both $\sin{P_0}$ and $\cos{P_0}$ within $[-1,1]$ in each layer. ### Updating of Gabor kernels In the back-propagation stage of Gabor-Nets, we update the convolutional kernels as a whole by solving the aforementioned Gabor parameters, the gradients of which are aggregated from all the elements of the kernel as follows: $$\label{eq:GaborNet:error} \begin{aligned} &\delta_\tau = \frac{\partial L}{\partial \tau}=\sum_{x,y}\mathbf{\delta}_{\mathbf{G}}\circ \frac{\partial \mathbf{G}}{\partial \tau},\\ &\tau\leftarrow\tau-\delta_\tau, \;\;\;\;\text{for}\; \tau = \{\theta, \omega\,\sigma, P\}, \end{aligned}$$ where $\mathbf{\delta}_{\mathbf{G}}$ is [the gradient of the training loss $L$ w.r.t. $\mathbf{G}$]{}, $\circ$ is the Hadamard product, and $$\label{eq:GaborNet:PGrad} \begin{aligned} \hspace{-0.3cm}\frac{\partial \mathbf{G}}{\partial P}&= -\frac{1}{2\pi\sigma^2}\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\sin{(x\omega_x+y\omega_y+P)} \\ &=-K\sin{(x\omega_x+y\omega_y+P)}, \end{aligned}$$ $$\label{eq:GaborNet:thGrad} \begin{aligned} \hspace{-0.3cm}\frac{\partial \mathbf{G}}{\partial \theta} &= -\frac{1}{2\pi\sigma^2}\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\sin{(x\omega_x+y\omega_y+P)}\\ &\hspace{0.5cm} \cdot (-x\omega\sin\theta+y\omega\cos\theta)\\ &= \frac{\partial \mathbf{G}}{\partial P}\circ(-x\omega_y+y\omega_x), \end{aligned}$$ $$\label{eq:GaborNet:omgGrad} \begin{aligned} \hspace{-0.3cm}\frac{\partial \mathbf{G}}{\partial \omega} &= -\frac{1}{2\pi\sigma^2}\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\sin{(x\omega_x+y\omega_y+P)}\\ &\hspace{0.5cm}\cdot(x\cos\theta+y\sin\theta)\\ &= \frac{\partial \mathbf{G}}{\partial P}\circ(x\cos\theta+y\sin\theta), \end{aligned}$$ $$\label{eq:GaborNet:sigGrad} \begin{aligned} \hspace{-0.3cm}\frac{\partial \mathbf{G}}{\partial \sigma}&= -\frac{1}{\pi\sigma^3}\cdot\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\cos{(x\omega_x+y\omega_y+P)} \\ &\hspace{0.8cm}+\frac{1}{2\pi\sigma^2}\cdot\frac{x^2+y^2}{\sigma^3}\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\\ &\hspace{1.5cm}\cdot\cos{(x\omega_x+y\omega_y+P)}\\ &=(\frac{x^2+y^2}{\sigma^3}-\frac{2}{\sigma})\cdot \frac{1}{2\pi\sigma^2}\exp{\big(-\frac{x^2+y^2}{2\sigma^2}\big)}\\ &\hspace{0.5cm}\cdot\cos{(x\omega_x+y\omega_y+P)}\\ &=\mathbf{G}\circ(\frac{x^2+y^2}{\sigma^3}-\frac{2}{\sigma}). \end{aligned}$$ Experiments {#sec:exp} =========== In this section, we first present the experimental settings for the sake of reproduction, following which the proposed Gabor-Nets are evaluated using three real hyperspectral datasets, i.e. the Pavia University scene, the Indian Pines scene, and the Houston scene[^3]. After that, we investigate some properties of Gabor-Nets via relevant experiments. Experimental settings {#sec:exp:settings} --------------------- To conduct the pixel-wise HSI classification with CNNs, patch generation is a widely used strategy in the preprocessing stage to prepare the inputs of networks [@Plaza20183DCNN; @Lee2017DCCNN; @Liu2018SCNN]. Let $S_p$ denote the patch size. The input patch is defined as $S_p\times S_p$ neighboring pixels centered on the given pixel. As shown in Fig. \[fig:patchGene\], taking $S_p$ of 5 for example, the patch of the given pixel A is the surrounding square area, each side of which is 5 pixels long (the red box in Fig. \[fig:patchGene\]). Accordingly, the label/output of this patch is that of A. ![Graphical illustration of the patch generation. Take the patch size of 5 for example.[]{data-label="fig:patchGene"}](Illustration_For_Patch_GenerationV2.eps "fig:"){width="1.8in"}\ ![(a) Unit convolutional block (CV Block) and (b) fully connected block (FC Block) used for HSI classification network construction, where $S_p$ is the patch size, $c$ is the index of CV Blocks, $N_i$ and is the number of inputs, $N_t$ and $N_m$ are the numbers of $\theta_0$s and $\omega_0$s of Gabor convolutional layers, $N_o$ is the number of outputs for the layers/blocks.[]{data-label="fig:para:struct"}](Structure_Block.eps "fig:"){height="1.85in"}\ $\sharp$para Regular CNNs Gabor-Nets -------------- ------------------------------------------------ ---------------------------------------------- Conv1 $\hspace{0.2cm}k^2\times N_i \times N_o + N_o$ $\hspace{0.2cm}4\times N_i \times N_o + N_o$ Conv2 $\hspace{0.2cm}k^2\times N_o \times N_o$ $\hspace{0.2cm}4\times N_o \times N_o$ BN $2\times N_o$ $2\times N_o$ Total $k^2(N_i+N_o)N_o+3N_o$ $4(N_i+N_o)N_o+3N_o$ : Numbers of parameters used in a CV Block of the regular CNNs and Gabor-Nets, respectively, where $k$ is the kernel size.[]{data-label="table:cvbpara"} Regarding the architecture of networks, we utilized the unit convolutional block (CV Block) and the fully connected block (FC Block) illustrated in Fig. \[fig:para:struct\] to construct the basic network architectures for regular CNNs and Gabor-Nets used in our experiments. Notice that regular CNNs and Gabor-Nets shared the same architectures, yet utilized different types of convolutional kernels. The former used regular kernels, while the latter used the proposed Gabor kernels. As shown in Fig. \[fig:para:struct\], each CV Block contains two convolutional (Conv) layers, one rectified linear unit (ReLU) nonlinearity layer, and one Batch Normalization (BN) layer. The CV Block is designed in accordance to [@Worrall2017HNet]. We utilized 16 kernels in each convolutional layers of the first CV Block. Each additional CV Blocks doubled the kernel number. For the initialization of Gabor-Nets, the number of $\theta_0$s of the first CV Block was set to be 4, and then doubled as more CV Blocks were added. The number of $\omega_0$s remained to be 4 in all the CV Blocks. The input number was the number of bands for the first CV block, otherwise equalled the output number of the last CV Block. No pooling layers were utilized in CV Blocks, since the patch size was relatively small in our experiments. As reported in Table \[table:cvbpara\], for each CV Block, regular CNNs contain $(k^2-4)(N_i+N_o)N_o$ more parameters than Gabor-Nets. The difference becomes larger as $k$, $N_i$ and $N_o$ increase. On top of CV Blocks is one FC Block with two fully connected layers, one global average pooling layer, and one ReLU layer. In the FC Block, the global average pooling layer is first utilized to rearrange $N_i$ input feature maps into a vector of $N_i$ elements, in order to reduce the number of parameters of fully connected layers. The output number of the first fully connected layer is twice its input number, while that of the second fully connected layer equals the number of predefined classes for classification purposes. The FC Block is completely the same for Gabor-Nets and regular CNNs since it contains no Conv layers. The number of parameters in an FC Block is $(N_i\cdot 2N_i+2N_i)+(2N_i\cdot N_c+N_c)=2{N_i}^2+2N_i+2{N_i}{N_c}+N_c$. In our experiments, We used the cross-entropy loss and Adam optimizer. The learning rate was initially set to be 0.0076, decaying automatically at a rate of 0.995 at each epoch. The total number of epochs is 300. Except for the regular CNNs, we also considered some other state-of-the-art deep learning based HSI classification methods for comparison. The first one used the hand-crafted Gabor features as the inputs of the CNNs (Gabor as inputs) [@Chen2017InputGaborHSI], where the Gabor features were generated from the first several principal components of HSI data. The hand-crafted parameters were set in accordance to the initializations of the first convolutional layer of Gabor-Nets. The second one is the deep contextual CNN (DC-CNN) [@Lee2017DCCNN], leveraging the residual learning to build a deeper and wider network, and simultaneously using a multi-scale filter bank to jointly exploit spectral and spatial information of HSIs. The next is the CNN with pixel-pair features (CNN-PPF) [@Li2017CNNPPF], which is a spectral-feature based method, using a series of pixel pairs as inputs. Similar to CNN-PPF, the siamese convolutional neural network (S-CNN) [@Liu2018SCNN] also took pairs of samples as inputs, yet used the pairs of patches to extract deep features, on which the support vector machine (SVM) was utilized for classification purposes. The last one is a 3-D CNN proposed in [@Plaza20183DCNN], which actually extracted the spectral-spatial information with 2-D kernels. As indicated in [@Plaza20183DCNN], keeping all the bands of an HSI as inputs could provide CNNs with full potential in spectral information mining, following which we also fed all the bands into the networks in our experiments. Furthermore, two traditional supervised classification algorithms were implemented on hand-crafted Gabor features. The first one is the multinomial logistic regression (MLR) via variable splitting and augmented Lagrangian algorithm [@Bioucas2009LORSAL], and the other one is the probabilistic SVM, which estimates the probabilities via combining pairwise comparisons [@Lin2007ProbSVM]. Both methods have been proven successful when dealing with high-dimensional data. Classification Results {#sec:exp:classif} ---------------------- In the following, we describe the obtained experimental results in detail. ### Experiments with the Pavia University Scene {height="1.55in"}\ This scene is a benchmark hyperspectral dataset for classification, which was collected by the Reflective Optics Imaging Spectrometer System (ROSIS) sensor over the urban area of the University of Pavia, Italy, in 2003 (see Fig. \[fig:pu:maps\] (a)). The image contains $610\times340$ samples with a spatial resolution of 1.3m, and 103 spectral bands ranging from 0.43 to 0.86 $\mu$m. The ground truth contains 42776 labeled samples within 9 classes of interest, where the numbers of samples corresponding to C1-C9 are 6631, 18649, 2099, 3064, 1345, 5029, 1330, 3682 and 947, respectively. The training samples were randomly selected from each class, and the rest were taken for test. To test the performance of Gabor-Nets with a small number of training samples, we evaluated their performance using 50, 100, and 200 training samples per class, respectively. We argue that 50 training samples per class is not a small training set for usual methods, but for deep learning based methods this number of training samples is actually limited with respect to the large number of model parameters. The patch size and and the kernel size were empirically set to 15 and 5, respectively. The regular CNNs and Gabor-Nets were constructed using two CV Blocks and one FC Block. In order to guarantee statistical significance, the results are reported by averaging five Monte Carlo runs corresponding to independent training sets. First of all, we report the test accuracies obtained with different numbers of training samples for the Pavia University scene in Table \[table:pu:acc\]. As can be observed, the proposed Gabor-Nets obtained very competitive results when compared to the other tested methods. The improvements were quite significant, especially in the case of 50 training samples per class. The 3-D CNN, Gabor as inputs and regular CNNs could obtain very close results to Gabor-Nets under the circumstance of 200 training samples per class. Nevertheless, Gabor-Nets outperformed 3-D CNN, Gabor as inputs and regular CNNs with accuracy gains of 5%, 2% and 10%, respectively, when using 50 training samples per class. Furthermore, we implemented a data augmentation strategy for regular CNNs and Gabor-Nets, by mirroring each of them across the horizontal, vertical, and diagonal axes, respectively [@Lee2017DCCNN]. As shown in Table \[table:pu:acc\], the data augmentation strategy benefitted regular CNNs much more than Gabor-Nets, indicating that Gabor-Nets were less negatively affected by a lack of training samples than regular CNNs. This is expected since Gabor-Nets involve much less free parameters than regular CNNs. Besides, Gabor filters are able to achieve optimal resolution in both space and frequency domains, thus suitable for feature extraction purposes. Therefore, Gabor-Nets based on Gabor kernels can still yield representative features to some extent with limited training samples. [l|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;]{} $\sharp$ Train & 50/class & 100/class & 200/class\ Gabor-MLR & 92.41$\pm$0.66 & 94.96$\pm$0.30 & 96.73$\pm$0.25\ Gabor-SVM & 90.99$\pm$0.72 & 92.93$\pm$0.91 & 95.79$\pm$0.38\ CDCNN [@Lee2017DCCNN]$^\ast$ & - & - & 95.97\ CNN-PPF[@Li2017CNNPPF]$^\ast$ & - & - & 96.48\ S-CNN[@Liu2018SCNN]$^\ast$ & - & - & 99.08\ 3-D CNN [@Plaza20183DCNN]$^\ast$ & 90.22$\pm$1.78 & 94.37$\pm$1.10 & 98.06$\pm$0.13\ Gabor as inputs [@Chen2017InputGaborHSI] & 93.20$\pm$1.55 & 96.14$\pm$0.75 & 98.79$\pm$0.29\ Regular CNNs & 85.52$\pm$1.51 & 95.43$\pm$0.60 & 98.12$\pm$0.18\ **Gabor-Nets** & **95.91$\pm$1.53** & **98.40$\pm$0.31** & **99.22$\pm$0.19**\ CNNs+Aug. & 94.67$\pm$1.07 & 97.53$\pm$0.20 & 99.10$\pm$0.18\ **Gabor-Nets+Aug.** & **97.28$\pm$1.09** & **98.65$\pm$0.38** & **99.48$\pm$0.06**\ Fig. \[fig:acc:process\] plots the training accuracies and losses obtained by the Regular CNNs and Gabor-Nets in the first 150 epochs. It can be seen that Gabor-Nets initially yielded a higher training accuracy and a smaller loss, and then converged faster, which indicates that Gabor-Nets constructed by Gabor kernels are able to constrain the solution space of CNNs, thus playing a positive role in the learning of CNNs. [p[3.8cm]{}&lt;p[3.8cm]{}&lt;]{} ![(a) Training accuracies and (b) losses as functions of the number of epochs obtained using 100 training samples per class by Gabor-Nets and the regular CNNs, respectively. The initial values are marked with circles.[]{data-label="fig:acc:process"}](PU_Train_AccV2.eps "fig:"){height="1.2in"}& ![(a) Training accuracies and (b) losses as functions of the number of epochs obtained using 100 training samples per class by Gabor-Nets and the regular CNNs, respectively. The initial values are marked with circles.[]{data-label="fig:acc:process"}](PU_Train_LossV2.eps "fig:"){height="1.2in"}\ (a) Training accuracy & (b) Loss\ Some of the classification maps obtained using 50 training samples per class are shown in Fig. \[fig:pu:maps\]. It can be seen that the classification map obtained by Gabor-Nets is smoother than those obtained by other methods. In contrast, the maps obtained by traditional methods are negatively affected by the appearance of noises. It is known that CNNs extract features via multi-layer convolutions, while traditional shallow filtering methods convolve the image using a single-layer strategy, which makes CNNs have a better ability to remove noises. However, this also tends to make CNNs over-smooth HSIs sometimes, which leads to the information loss especially of small ground objects. Then, we investigate the relationship between the patch size and the classification performance of Gabor-Nets. We set the patch sizes varying from 7 to 23 with an increasing interval of 2 pixels, and illustrate the obtained test accuracies along with standard deviations in Fig. \[fig:pu:patch\]. It can be observed that very small patches had a negative effect on the classification accuracies and the robustness of Gabor-Nets. As the patch size increased, Gabor-Nets performed better. However, when the patch size became very large, the performance decreased again. We argue that the patches can be regarded as a series of local spatial dependency systems [@He2018Review], using the patch size to define the neighborhood coverage. According to *Tobler’s first law of geography*, the similarity between two objects on the same geographical surface has an inverse relationship with their distance. Therefore, those samples located at a distance away from the central one are not helpful (and even will confuse the classifier), whereas small patches are unable to provide relevant information, thus limiting the potential of networks. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Test accuracies (along with standard deviations) as a function of patch sizes for the Pavia University scene using 100 training samples per class.[]{data-label="fig:pu:patch"}](PU_TestAcc_Patch_SizeV2.eps "fig:"){width="1.8in"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [p[2.0cm]{}&lt;|p[2.4cm]{}&lt;|p[2.4cm]{}&lt;|p[2.4cm]{}&lt;|p[2.6cm]{}&lt;|p[3.2cm]{}&lt;]{} & 1 & 2 & 3 & 4\ & 16-16 & 16-16-32-32 & 16-16-32-32-64-64 & 16-16-32-32-64-64-128-128\ Regular CNNs & Test Accuracies ($\sharp$para) & 92.47$\pm$1.68 (48K) & 95.43$\pm$0.60 (89K) & 92.96$\pm$1.44 (249K) & 86.93$\pm$3.28 (890K)\ Gabor-Nets & Test Accuracies ($\sharp$para) & 96.37$\pm$1.10 (8K) & 98.40$\pm$0.31 (17K) & 98.46$\pm$0.22 (48K) & 98.19$\pm$0.74 (172K)\ Finally, we test the performance of Gabor-Nets with different numbers of CV Blocks. From Table \[table:pu:moreBlocks\] we can observe that Gabor-Nets exhibit better robustness when using different numbers of CV Blocks. In this case, Gabor-Nets were able to achieve more reliable results with more CV Blocks added, whereas the performance of regular CNNs degraded severely, as a result of overfitting caused by a sharp rise in parameter numbers. Additionally, both Gabor-Nets and regular CNNs performed worse when the number of CV Blocks decreased to 1, partly due to the decline in the representation ability of networks. Yet the drop in test accuracies of Gabor-Nets is around 1% less than that of regular CNNs, which suggests the superiority of Gabor-Nets employing the Gabor *a priori*. ![The false color map along with the ground truth (GT) of 8, 9 and 16 classes, respectively, for the Indian Pines scene.[]{data-label="fig:indian:data"}](Indian_gts.eps "fig:"){height="1.85in"}\ ### Experiments with the Indian Pines Scene The second dataset used in our experiments is the well-known Indian Pines scene, collected over a mixed agricultural/forest area in North-western Indiana, USA, by the Airborne Visible Infrared Imaging Spectrometer (AVIRIS) sensor in 1992. This scene is composed of 220 spectral bands with wavelength varying from 0.4 to 2.5 $\mu$m, and 145$\times$145 pixels with a spatial coverage of 20m$\times$20m. In our experiments, we removed 20 bands due to noises and water absorption, resulting in 200 bands. This scene is challenging for traditional HSI classification methods due to the fact that most of the samples are highly mixed. As shown in Fig. \[fig:indian:data\] (d) and Table \[table:indian:16tnum\], the available ground truth (GT) contains 10249 labeled samples belonging to 16 unbalanced classes. To tackle this problem, Liu *et al.* [@Liu2018SCNN] and Li *et al.* [@Li2017CNNPPF] removed C1, C4, C7, C9, C13, C15, and C16 from the original GT, leaving 9 classes. Lee *et al.* [@Lee2017DCCNN] removed C6 besides the above seven classes, leaving 8 classes. Furthermore, Paoletti *et al.* [@Plaza20183DCNN] balanced the number of training samples of each class in accordance with their sample sizes via a stratified sampling strategy. For comparison purposes, we considered all the three circumstances in our experiments, and randomly selected 50, 100, and 200 samples per class for training, leaving the remains for test. Specifically, Table \[table:indian:16tnum\] presents the numbers of training and test samples using the 16-class GT [@Plaza20183DCNN]. We utilized a similar network architecture to the previous one, i.e., two CV Blocks and one FC Block, with the patch size of 15 and the filter size of 5. We conducted five Monte Carlo runs and reported the average results in the following. [c|c|c|c|c|c|c|c]{} & & & &\ & & $\sharp$Train & $\sharp$Test & $\sharp$Train & $\sharp$Test & $\sharp$Train & $\sharp$Test\ C1 & 46 & 33 & 13 & 33 & 13 & 33 & 13\ C2 & 1428 & 50 & 1378 & 100 & 1328 & 200 & 1228\ C3 & 830 & 50 & 780 & 100 & 730 & 200 & 630\ C4 & 237 & 50 & 187 & 100 & 137 & 181 & 56\ C5 & 483 & 50 & 433 & 100 & 383 & 200 & 283\ C6 & 730 & 50 & 680 & 100 & 630 & 200 & 530\ C7 & 28 & 20 & 8 & 20 & 8 & 20 & 8\ C8 & 478 & 50 & 428 & 100 & 378 & 200 & 278\ C9 & 20 & 14 & 6 & 14 & 6 & 14 & 6\ C10 & 972 & 50 & 922 & 100 & 872 & 200 & 772\ C11 & 2455 & 50 & 2405 & 100 & 2355 & 200 & 2255\ C12 & 593 & 50 & 543 & 100 & 493 & 200 & 393\ C13 & 205 & 50 & 155 & 100 & 105 & 143 & 62\ C14 & 1265 & 50 & 1215 & 100 & 1165 & 200 & 1065\ C15 & 386 & 50 & 336 & 100 & 286 & 200 & 186\ C16 & 93 & 50 & 43 & 75 & 18 & 75 & 18\ Total & 10249 &717 & 9532 &1342 & 8907 &2466 & 7783\ Tables \[table:ip8c:acc\]-\[table:ip16c:acc\] list the test accuracies obtained using the ground truth of 8, 9 and 16 classes, respectively. Clearly, Gabor-Nets outperformed the other methods in all the considered cases, especially when only 50 training samples per class were utilized, indicating that Gabor-Nets have a capability to deal with limited training samples. Additionally, Gabor-Nets yielded better classification results than Gabor as inputs, from which we can infer that Gabor-Nets, via adjusting the Gabor parameters in a data-driven manner, were able to generate more effective features than hand-crafted Gabor filters. Furthermore, most deep learning based methods outperformed the traditional ones, showing the potential of CNNs in HSI classification tasks. {height="1.00in"}\ [l|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;]{} $\sharp$Train & 50/class & 100/class & 200/class\ Gabor-MLR & 86.70$\pm$0.84 & 93.63$\pm$0.57 & 97.04$\pm$0.39\ Gabor-SVM & 83.79$\pm$0.98 & 91.08$\pm$0.54 & 96.22$\pm$0.68\ CDCNN [@Lee2017DCCNN]$^\ast$ & - & - & 93.61$\pm$0.56\ Gabor as inputs[@Chen2017InputGaborHSI] & 93.45$\pm$1.11 & 97.34$\pm$1.04 & 98.86$\pm$0.31\ Regular CNNs & 91.83$\pm$3.31 & 96.50$\pm$0.78 & 99.12$\pm$0.31\ **Gabor-Nets** & **94.33$\pm$0.42** & **97.58$\pm$0.43** & **99.24$\pm$0.33**\ [l|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;]{} $\sharp$Train & 50/class & 100/class & 200/class\ Gabor-MLR & 88.34$\pm$1.08 & 93.78$\pm$0.61 & 96.99$\pm$0.41\ Gabor-SVM & 84.39$\pm$0.51 & 91.44$\pm$0.58 & 96.11$\pm$0.51\ CNN-PPF[@Li2017CNNPPF]$^\ast$ & - & - & 94.34\ S-CNN[@Liu2018SCNN]$^\ast$ & - & - & 99.04\ Gabor as inputs[@Chen2017InputGaborHSI] & 93.65$\pm$1.07 & 97.55$\pm$0.31 & 98.84$\pm$0.31\ Regular CNNs & 92.41$\pm$2.45 & 96.42$\pm$0.83 & 98.73$\pm$0.44\ **Gabor-Nets** & **94.76$\pm$0.46** & **97.54$\pm$0.16** & **99.05$\pm$0.19**\ [l|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;|p[1.4cm]{}&lt;]{} $\sharp$Train & 50/class & 100/class & 200/class\ Gabor-MLR & 87.63$\pm$0.79 & 94.05$\pm$0.71 & 97.18$\pm$0.32\ Gabor-SVM & 85.15$\pm$0.43 & 92.23$\pm$0.36 & 96.18$\pm$0.37\ 3-D CNN[@Plaza20183DCNN]$^\ast$ & 88.78$\pm$0.78 & 95.05$\pm$0.28 & 98.37$\pm$0.17\ Gabor as inputs[@Chen2017InputGaborHSI] & 93.29$\pm$1.09 & 96.91$\pm$0.63 & 98.67$\pm$0.39\ Regular CNNs & 92.74$\pm$0.67 & 96.42$\pm$0.47 & 98.28$\pm$0.58\ **Gabor-Nets** & **94.05$\pm$0.79** & **97.01$\pm$0.52** & **98.75$\pm$0.38**\ Fig. \[fig:indian:maps\] shows some of the classification maps obtained with 50 training samples per class using the 16-class GT. As illustrated, the assignments by Gabor-Nets are more accurate, and the corresponding classification map looks smoother. However, the maps by CNN methods are somewhat over-smoothed on this scene, partly due to their multi-layer feature extraction strategy, which makes CNNs prone to over-smooth hyperspectral images, especially when the interclass spectral variability is low, as in the case of the Indian Pines scene. ### Experiments with the Houston Scene The Houston scene was acquired by the Compact Airborne Spectrographic Imagery from the ITRES company (ITRES-CASI 1500) over the area of University of Houston campus and the neighboring urban area in 2012. It was first known and distributed as the hyperspectral image provided for the 2013 IEEE Geoscience and Remote Sensing Society (GRSS) data fusion contest. This scene is composed of 349$\times$1905 pixels at a spatial resolution of 2.5m and 144 spectral bands ranging from 380nm to 1050nm. The public ground truth contains 15029 labeled samples of 15 classes, including 2832 training samples (198, 190, 192, 188, 186, 182, 196, 191, 193, 191, 181, 192, 184, 181 and 187 corresponding to C1-C15, respectively) and 12197 test samples (1053, 1064, 505, 1056, 1056, 143, 1072, 1053, 1059, 1036, 1054, 1041, 285, 247 and 473 corresponding to C1-C15, respectively) as shown in Fig. \[fig:houston:maps\]. This dataset is a typical urban scene with a complex spatial appearance containing many natural or artificial ground fabrics, based on which, we utilized three CV Blocks and one FC Block to mine deeper feature representations, and empirically set the patch size and filter size as 13 and 3, respectively. We utilized the publicly available training set and repeated the tested CNN-based methods five times w.r.t different random initializations. ![The false color map, training set, test set, and classification maps for the Houston scene.[]{data-label="fig:houston:maps"}](HST_CMaps.eps "fig:"){width="7.8cm"}\ First and foremost, we quantitatively evaluate the classification performance in Table \[table:houston:acc\], where the proposed Gabor-Nets obtained the highest test accuracies. However, other deep learning based methods could not outperform the traditional ones as much as they did on the previous scenes. These observations indicate that Gabor-Nets also exhibit potential when dealing with complex urban scenarios. [p[3.5cm]{}|p[1.8cm]{}&lt;]{} Methods & Test accuracies\ Gabor-MLR & 79.86\ Gabor-SVM & 79.38\ CNN-PPF[@Li2017CNNPPF][@Li201DL-HSI-ClassificationReview]$^\ast$ & 81.38\ S-CNN[@Liu2018SCNN][@Li201DL-HSI-ClassificationReview]$^\ast$ & 82.34\ Gabor as inputs [@Chen2017InputGaborHSI] & 79.43$\pm$0.91\ Regular CNNs & 78.55$\pm$0.99\ **Gabor-Nets** & **85.57$\pm$1.18**\ A visual comparison can be found in Fig. \[fig:houston:maps\], where the classification map generated by Gabor-Nets looks smoother than the others, also with clear roads. Furthermore, more details below the cloud are revealed on the map by Gabor-Nets than those by Gabor as inputs and regular CNNs, which suggests the effectiveness of the proposed Gabor kernels. Besides, the maps by traditional methods are severely affected by the appearance of noises, though they yielded very close statistic results to CNNs and Gabor as inputs. Next, we evaluated the performance of Gabor-Nets with varying patch sizes in Fig. \[fig:houston:patch\]. Similar to the experiments on the Pavia University scene, too small and too big patches both had a negative effect on the performance of Gabor-Nets. This scene is more sensitive to big patches. As mentioned before, this scene is quite complex spatially. As a result, big patches tend to damage its underlying spatial structure. Finally, we investigate the relationship between the number of CV Blocks and the classification performance of the proposed Gabor-Nets. As illustrated in Table \[table:houston:moreBlocks\], for all the considered architectures, the number of required parameters of Gabor-Nets was only around half of that of regular CNNs. Also, Gabor-Nets performed better than regular CNNs, no matter what architecture was utilized. The Gabor-Nets were able to maintain its superiority as more CV Blocks were involved into the architectures, whereas an obvious degradation can be observed in the obtained test accuracies of regular CNNs when the number of CV Blocks increased from 3 to 4. --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Test accuracies (along with standard deviations) as a function of patch sizes for the Houston scene using the public training samples.[]{data-label="fig:houston:patch"}](HST_TestAcc_Patch_SizeV2.eps "fig:"){width="1.8in"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [p[2.0cm]{}&lt;|p[2.4cm]{}&lt;|p[2.4cm]{}&lt;|p[2.4cm]{}&lt;|p[2.6cm]{}&lt;|p[3.2cm]{}&lt;]{} & 1 & 2 & 3 & 4\ & 16-16 & 16-16-32-32 & 16-16-32-32-64-64 & 16-16-32-32-64-64-128-128\ Regular CNNs & Test Accuracies ($\sharp$para) & 71.90$\pm$2.84 (24K) & 80.59$\pm$3.67 (40K) & 78.55$\pm$0.99 (103K) & 72.38$\pm$2.67 (351K)\ Gabor-Nets & Test Accuracies ($\sharp$para) & 77.37$\pm$1.65 (11K) & 83.01$\pm$0.88 (20K) & 85.57$\pm$1.18 (51K) & 85.43$\pm$1.34 (177K)\ {height="1.75in"}\ Random initializations – a b c abc ------------------------ ---------------- ---------------- ---------------- ---------------- ---------------- Test accuracies (%) 96.37$\pm$1.10 96.07$\pm$1.37 95.27$\pm$1.05 88.07$\pm$3.63 90.17$\pm$0.84 Model Insight ------------- To further analyze the mechanism behind Gabor-Nets, we investigate some of the properties of Gabor-Nets. For illustrative purposes, we focus on the experiments with the Pavia University scene (using 100 randomly selected training samples per class without the augmentation strategy). ### Visualizations of first-layer features To help readers understand more about what Gabor-Nets learn at the bottom layers, we visualize their first-layer features along with those of regular CNNs in Fig. \[fig:pavia:FeatVisual\], which are extracted from two patches (15$\times$15) of the Pavia University scene. As illustrated, both regular CNNs and Gabor-Nets can extract features at certain orientations, which indicates that their low-layer features share some similar characteristics. However, the features extracted by regular CNNs are somewhat blurred and in various shapes. In contrast, the boundaries are depicted clearly on the feature maps by Gabor kernels, and each feature map reflects the information specific to an orientation and a frequency. This confirms that, although the variety of the features obtained by Gabor kernels is not as high as that obtained by regular kernels, Gabor-Nets can extract more compact and representative features, since the underlying features of HSIs are relatively simple, mainly composed of a series of geometrical and morphological features, in which case Gabor filters have been proven effective. ### Initialization scheme In this work, we design an initialization scheme in accordance to Gabor *a priori* knowledge for Gabor-Nets to guarantee their performances. To verify the reliability of our initialization scheme, we conducted some experiments using the network architecture of one CV Block and one FC Block with random initializations of $\theta$s, $\omega$s and $\sigma$s for the Pavia University scene. Let $\tilde{\mu}$ and $\tilde{\sigma}$ denote the mean and the standard deviation of the normal distribution. The random initializations of $\theta$s, $\omega$s and $\sigma$s adopted in our experiments are as follows: a\. $\theta_0$s obeying a uniform distribution within $[0,2\pi)$; b\. $\omega_0$s obeying a normal distribution with $\tilde{\mu}$=$0$, $\tilde{\sigma}$=$\pi/4$; c\. $\sigma_0$s obeying a normal distribution with $\tilde{\mu}$=$0$, $\tilde{\sigma}$=$5/8$. We utilize the above random initializations to replace the corresponding original initializations in the proposed initialization scheme, and report the obtained results in Table \[table:OA:randInit\], from which we can observe that the initialization scheme can guarantee Gabor-Nets to yield reliable results. [p[3.8cm]{}&lt;p[3.8cm]{}&lt;]{} ![Training accuracies obtained with an initial learning rate of (a) 0.0076 and (b) 0.02, respectively, as functions of the number of epochs, where three types of Gabor-Nets are considered: the one that we proposed (red), the one with $P$s initialized to 0, i.e., $P_0=0$ (blue), and the one without $P$, i.e., $P=0$ (black). The dashed lines indicate their test accuracies.[]{data-label="fig:phase:process"}](Phase_Training_Acc_New_lr_0076.eps "fig:"){height="1.2in"}& ![Training accuracies obtained with an initial learning rate of (a) 0.0076 and (b) 0.02, respectively, as functions of the number of epochs, where three types of Gabor-Nets are considered: the one that we proposed (red), the one with $P$s initialized to 0, i.e., $P_0=0$ (blue), and the one without $P$, i.e., $P=0$ (black). The dashed lines indicate their test accuracies.[]{data-label="fig:phase:process"}](Phase_Training_Acc_New_lr_02.eps "fig:"){height="1.2in"}\ (a) & (b) ### Phase Offsets As stated above, the kernel phase $P$ is crucial in Gabor-Nets, which controls the frequency characteristics of Gabor kernels. To test the role of $P$, we implemented two variants of Gabor-Nets, i.e., the one with all $P$s initialized to 0 ($P_0=0$); and the one without $P$ ($P=0$). Fig. \[fig:phase:process\] shows the training accuracies obtained with initial learning rates of 0.0076 and 0.02, respectively, as functions of the number of epochs. Remarkably, randomly initializing $P$ in $[0,2\pi)$ can make Gabor-Nets achieve better performance and higher robustness with different learning rates in comparison to the two variants, where Gabor-Nets yielded results of around 98% using both the considered initial learning rates. Quite opposite, the two variants performed much worse when using the initial learning rate of 0.0076. Recall that the gradient descend back propagation is a local search algorithm, in which reducing the learning rate will bring to smaller adjustments to the parameters, thus easily leading to the gradient vanishing phenomenon. Therefore, it can be inferred that Gabor-Nets with randomly initialized $P$s can resist against this phenomenon to some extent. The differences of test accuracies between the Gabor-Nets and two variants still exist when the learning rate increases, although the two variants performed better. Furthermore, the variant without $P$ yielded the worst results among the three models, which indicates that without the kernel phase term $P$, the ability of Gabor-Nets will be restricted since the frequency properties of Gabor kernels cannot adaptively follow the data. Regarding $P_0=0$, the fixed initialization will also harm the potential of Gabor-Nets due to a lack of diversity. In another experiment, we investigate the learned angular frequencies of Gabor-Nets and the two variants. Fig. \[fig:phase:freq\] shows the finally learned frequencies of Gabor kernels in the first layer (in terms of their angles and magnitudes), with an initial learning rate of 0.0076. Noticeably, the learned frequencies of Gabor-Nets tend to cover the whole semicircle region, while those of the two variants are only distributed at some local narrow regions, i.e., their $\theta$s and $\omega$s changed rarely in the learning process, which suggests that the two variants suffered from the gradient vanishing problem. Thus, from these results, we can infer that Gabor-Nets with randomly initialized kernel phases could resist against the gradient vanishing problem to some extent and positively affect the learning process of other parameters. {height="1.05in"}\ {height="2in"}\ ### Parameters in Traditional Gabor Filters Here we analyze other parameters used in often-used hand-crafted Gabor filter construction, i.e, the frequency angle $\theta$, the frequency magnitude $\omega$, and the scale $\sigma$ in Gabor-Nets. Fig. \[fig:para:other\] shows the learned angular frequencies determined by $\theta$s and $\omega$s, and the histograms of the learned $\sigma$s of the Gabor kernels in the first layer, where each color in each column corresponds to a kernel bank used to generate an output feature, i.e., $\mathbf{G}_{o}^{(1)}$. As shown in the first row of Fig. \[fig:para:other\], almost all the points gather in a $\theta_0$-centric sector region with an angle range of $(\pi/4)$, where those marked with different colors are well distributed around the arc corresponding to their $\omega_0$s. Namely, although the points in Fig. \[fig:phase:freq\] (b) tend to cover the whole semicircle region, the points representing different kernel banks barely overlap with each other. This means that, around $\theta_0$ and $\omega_0$, the Gabor kernel banks can extract the features with varying $\theta$s and $\omega$s rather than those intended for a single predetermined frequency (as the hand-crafted Gabor filters do), thus making Gabor filters in Gabor-Nets more powerful. Furthermore, as shown in the second row of Fig. \[fig:para:other\], the $\sigma$s are also automatically adjusted following the data characteristics during the learning process. Conclusions and Future Lines {#sec:con} ============================ We have introduced the naive Gabor Networks (or Gabor-Nets) for HSI classification which, for the first time in the literature, design and learn convolutional kernels strictly in the form of Gabor filters – with much less parameters in comparison with regular CNNs, thus requiring a smaller training set and achieving faster convergence. By exploiting the kernel phase term, we develop an innovative phase-induced Gabor kernel, with which Gabor-Nets are capable to tune the convolutional kernels for data-driven frequency responses. Additionally, the newly developed phase-induced Gabor kernel is able to fulfill the traditional Gabor filtering in a real-valued manner, thus making it possible to directly and conveniently use Gabor kernels in a usual CNN thread. Another important aspect is that, since we only manipulate the way of kernel generation, Gabor-Nets can be easily implemented with other CNN tricks or structures. Our experiments on three real HSI datasets show that Gabor kernels can significantly improve the convergence speed and the performance of CNNs, particularly in scenarios with relatively limited training samples. However, the classification maps generated by Gabor-Nets tend to be over-smoothed sometimes, especially if ground objects are small and the interclass spectral variability is low. In the future, we will develop some edge-preservation strategies for Gabor-Nets to alleviate this negative effects. Furthermore, we will explore new kinds of filters that can be used as kernels in networks, which provides a plausible future research line for CNN-based HSI classification. [^1]: This paper has been accepted by IEEE Transactions on Neural Networks and Learning Systems (IEEE TNNLS). This work was supported in part by National Science Foundation of China under Grants 61771496, 61571195, and 61901208, in part by National Key Research and Development Program of China under Grant 2017YFB0502900, in part by Guangdong Provincial Natural Science Foundation under Grants 2016A030313254, 2016A030313516, and 2017A030313382, in part by Natural Science Foundation of Jiangxi China under Grant 20192BAB217003. (*Corresponding authors: Jun Li; Lin He*). Chenying Liu and Jun Li are with the Guangdong Provincial Key Laboratory of Urbanization and Geo-simulation, School of Geography and Planning, Sun Yat-sen University, Guangzhou, 510275, China (*e-mails: [email protected]; [email protected]*). Lin He is with the School of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, China (*e-mail: [email protected]*). Shutao Li is with the College of Electrical and Information Engineering, Hunan University, Changsha 410082, China (*e-mail: shutao\[email protected]*). Antonio Plaza is with the Hyperspectral Computing Laboratory, Department of Technology of Computers and Communications, Escuela Politécnica, University of Extremadura, Cáceres, E-10071, Spain (*e-mail: [email protected]*). Bo Li is with the Beijing Key Laboratory of Digital Media, School of Computer Science and Engineering, and the State Key Laboratory of Virtual Reality Technology and Systems, Beihang University, Beijing 100191, China (*e-mail: [email protected]*). [^2]: It is identical to allocate the phase offset $P$ with $x$ or $y$. [^3]: The Pavia University scene and the Indian Pines scene can be downloaded from <https://www.sipeo.bgu.tum.de/downloads>. The Houston scene can be downloaded from <https://www.grss-ieee.org/community/technical-committees/data-fusion/2013-ieee-grss-data-fusion-contest/>

ArXiv

--- abstract: 'The formation of compact stellar-mass binaries is a difficult, but interesting problem in astrophysics. There are two main formation channels: In the field via binary star evolution, or in dense stellar systems via dynamical interactions. The Laser Interferometer Gravitational-Wave Observatory (LIGO) has detected black hole binaries (BHBs) via their gravitational radiation. These detections provide us with information about the physical parameters of the system. It has been claimed that when the Laser Interferometer Space Antenna (LISA) is operating, the joint observation of these binaries with LIGO will allow us to derive the channels that lead to their formation. However, we show that for BHBs in dense stellar systems dynamical interactions could lead to high eccentricities such that a fraction of the relativistic mergers are not audible to LISA. A non-detection by LISA puts a lower limit of about $0.005$ on the eccentricity of a BHB entering the LIGO band. On the other hand, a deci-Hertz observatory, like DECIGO or Tian Qin, would significantly enhance the chances of a joint detection, and shed light on the formation channels of these binaries.' author: - Xian Chen - 'Pau Amaro-Seoane' title: | Revealing the formation of stellar-mass black hole binaries:\ The need for deci-Hertz gravitational wave observatories --- [*Introduction.*]{}–The first LIGO events, GW150914 and GW151226 [@ligo16a; @ligo16b], are consistent with mergers of General-Relativity black holes (BHs). Data analysis reveal that the orbits started at a semi-major axis of $a\sim10$ Schwarzschild radii ($R_S$) with an eccentricity of $e<0.1$. The BH masses are about $M_1\simeq36$ and $M_2\simeq29~M_\odot$ for GW150914 and $M_1\simeq14$ and $M_2\simeq7.5~M_\odot$ for GW151226. The detections can be used to infer new, more realistic event rates, of about $9-240~{\rm Gpc^{-3}~yr^{-1}}$ [@ligo16rate]. This rate agrees with two formation channels: (i) evolution of a binary of two stars in the field of the host galaxy, where stellar densities are very low (e.g [@belczynski16]) or (ii) via exchange of energy and angular momentum in dense stellar systems, where the densities are high enough for stellar close encounters to be common (e.g. [@rodriguez16]). LIGO and other ground-based gravitational wave (GW) observatories, such as Virgo, are, however, blind with regarding the formation channels of BH binaries (BHBs). Both channels predict populations in the $10-10^3~{\rm Hz}$ detector band with similar features, i.e. masses larger than the nominal $10\,M_{\odot}$, a mass ratio ($q\equiv M_2/M_1$) of about $1$, low spin, and nearly circular orbits [@ligo16astro; @ama16]. It has been suggested that a joint detection with a space-borne observatory such as LISA [@Amaro-SeoaneEtAl2012; @Amaro-SeoaneEtAl2013; @Amaro-SeoaneEtAl2017] could allow us to study different moments in the evolution of BHBs on their way to coalescence: LISA can detect BHBs when the BHs are still $10^2-10^3~R_S$ apart, years to weeks before they enter the LIGO/Virgo band [@miller02; @ama10; @kocsis12levin; @kocsis13; @sesana16; @seto16; @vitale16]. At such a separation, the orbital eccentricity bears the imprint of the formation channel because (i) BHBs in dense stellar systems form on systematically more eccentric orbits and (ii) the GW radiation at this stage is too weak to circularize the orbits [@miller02; @wen03; @gultekin04; @gultekin06; @oleary06]. Therefore, circular binaries typically form in the field, while eccentric ones through the dynamical channel. Recent studies further predict that those BHBs with an eccentricity of $e>0.01$ in the LISA band preferentially originate from the dynamical channel [@kyutoku16; @nishizawa16a; @nishizawa16b; @breivik16; @seto16]. In this letter we prove that eccentric BHBs originating in dense stellar environments have a large chance to elude the LISA band. *Inaudible black hole binaries*–Non-circular BHBs have two distinct properties. (i) Eccentricity damps the characteristic amplitude ($h_c$) of each GW harmonic, as compared to a circular BHB. In Figure \[fig.harmonics\] we depict two sources similar to GW150914 but originating from two distinct channels, i.e. with two different initial eccentricities. In the low-eccentricity case, the $n=2$ harmonic predominates and it is strong enough to be jointly detected by LISA and LIGO/Virgo. In the (very) eccentric case, however, the amplitudes of the harmonics are orders of magnitude below the noise level of LISA, so that a joint detection is ruled out. When the eccentricity has been significantly damped, about one hour before the merger, the dominant harmonic starts to converge to the $n=2$ one, and later, upon entering the LIGO band, becomes indistinguishable from that in the circular case. Therefore, the imprint about the formation channel is lost. ![ Characteristic amplitude $h_c$ of the first four harmonics (indicated with numbers) emitted by a BHB with masses $M_1=M_2=30\,M_{\odot}$ and at a luminosity distance of $D=500~{\rm Mpc}$. The amplitude is calculated as described in [@barack04] and the orbital evolution as in [@peters64]. We display a BHB starting at a semi-major axis of $a_0=0.1$ AU and with initially two very different eccentricities, so as to illustrate the main idea of this article: (i) $e_0=0.05$ (thin colored lines), and (ii) an extreme case, $e_0=0.999$ (thick colored lines). Along the harmonics we mark several particular moments with dots, where the labels show the time before the coalescence of the binary and the corresponding orbital eccentricities. The two black solid curves depict the noise curves ($\sqrt{f\,S_h(f)}$) for LISA and LIGO in its advanced configuration. Although we have chosen a very high eccentricity for the second case in this example, we note that lower eccentricities can also be inaudible to LISA (see discussion). \[fig.harmonics\]](harmonics.eps){width="1\columnwidth"} \(ii) Increasing the eccentricity shifts the peak of the relative power of the GW harmonics towards higher frequencies (see Fig. 3 of [@Peters63]). Hence, more eccentric orbits emit their maximum power at frequencies farther away from LISA. More precisely, when $e=0$, all the GW power is radiated through the $n=2$ harmonic, so that the GWs have a single frequency of $2/P$, where $P=2\pi(GM_{12}/a^3)^{-1/2}$ is the orbital period and $M_{12}=M_1+M_2$. On the other hand, when $e\simeq1$, the $n=2.16(1-e)^{-3/2}$ harmonic becomes predominant [@farmer03], so most GW power is radiated at a frequency of $f_{\rm peak}=2.16(1-e)^{-3/2}P^{-1}$. ![ Different detectors’ bands for a binary of $M_1=M_2=30~M_\odot$. We have considered four types of detectors: (i) a ground-based interferometer like LIGO and Virgo (pink stripe), with the minimum and maximum observable frequencies $(f_1,\,f_2)\sim(10,10^3)~{\rm Hz}$ [@abbott09; @accadia10], (ii) a space-borne solar-orbit interferometer such as the DECi-hertz Interferometer Gravitational Wave Observatory (DECIGO, blue) with $(f_1,f_2)\sim(0.1,10)~{\rm Hz}$ [@kawamura11], (iii) a geocentric space observatory like the Tian Qin project (TQ hereafter, orange) with T$(f_1,f_2)\sim(10^{-2},0.3)~{\rm Hz}$ [@luo16], and (iv) another solar-orbit interferometer but with million-kilometer baseline, like LISA or Tai Ji (TJ hereafter, shown as cyan), which operates at milli-Hz, $(f_1,f_2)\sim(10^{-3},0.1)~{\rm Hz}$ [@Amaro-SeoaneEtAl2013; @gong15] The upper, horizontal limit in the color stripes corresponds to an orbital period of one week for LIGO/Virgo/DECIGO, one month for TQ, and one year for LISA/TJ, as imposed by the restrictions in the search of the different data streams. The green solid lines show the evolutionary tracks of a binary evolving only due to GW emission, in the approximation of Keplerian ellipses [@peters64]. The dashed, black lines are isochrones displaying the time to relativistic merger in the same approximation ($t_{\rm gw}$, see text), provided that the evolution is driven only by GWs. The thick gray stripe displays the last stable orbit, below which the two BHs will merge within one orbital period. We also display with red stars the positions of the eccentric BHB in Figure \[fig.harmonics\] at different stages, to illustrate the process. \[fig:detectors\]](detectors.eps){width="1\columnwidth"} In Figure \[fig:detectors\] we display the $a-(1-e)$ plane for a BHB. The boundaries of the stripes have been estimated by looking at the minimum and maximum frequencies audible by the detectors, $f_1$ and $f_2$, and letting $f_1<\,f_{\rm peak}<\,f_2$, with $f_{\rm peak}$ defined before. If a BHB is evolving only due to GW emission, it will evolve parallel to the green lines. These track are parallel to the stripes because as long as $e\simeq1$, the pericenter distance, $r_p=a\,(1-e)$, is almost constant during the evolution [@peters64], and a constant $r_p$ corresponds to a constant $f_{\rm peak}$. Because of this parallelism, a BHB cannot evolve into the band of a GW detector if it initially lies below the detector stripe. Hence, we can see that some binaries will fully miss the LISA/TJ range. A good example is the eccentric BHB we chose for Figure \[fig.harmonics\]. A detector operating at higher frequencies, such as TQ or DECIGO, can however cover the relevant part of the phase-space, so that a joint search is possible. These detectors could alert LIGO/Virgo decades to hours before an event is triggered, as one can read from the isochrones of Figure \[fig:detectors\]. [*Dense stellar environments.–*]{}BHBs such as the one we have used for our last example completely miss the LISA/TJ band. Eccentric binaries typically originate from dense stellar systems such as globular clusters (GCs) and nuclear star clusters (NSC), as shown by a number of authors in a number of publications [@miller02; @wen03; @gultekin04; @gultekin06; @oleary06; @nishizawa16a; @nishizawa16b; @breivik16]. In these systems, BHs diffuse towards the center via a process called mass segregation [see e.g. @Peebles72; @BW76; @ASEtAl04; @FAK06a; @AlexanderHopman09; @PretoAmaroSeoane10]. To model it, we adopt a Plummer model [@Plummer11], and we assume that the mean stellar density is $\rho_*=5\times10^{5}~M_\odot~{\rm pc^{-3}}$ and the one-dimensional velocity dispersion is $\sigma_*=15~{\rm km~s^{-1}}$. These values correspond to a typical GC with a final mass of $M_{\rm GC}\approx10^5~M_\odot$ and a half-mass radius of $R_h\approx0.5$ pc. We note, however, that the main conclusions derived in this work do not significantly change for a NSC. The two driving and competing mechanisms in the evolution of any BHB in the center of the cluster are (i) interaction with other stars, “interlopers”, which come in at a rate of $\Gamma\sim2\pi G\rho_*a(M_{12}/M_*)/\sigma_*$, with $M_*=10~M_\odot$ the mean mass of the interlopers because the cluster has gone through mass segregation, and (ii) gravitational radiation, which shrinks the orbital semi-major axis at a rate of $$\dot{a}_{\rm gw}=-\frac{8\,c\,R_S^3q\,(1+q)}{5a^3(1-e^2)^{7/2}} \left(1+\frac{73}{24}e^2+\frac{37}{96}e^4\right),$$ [@peters64]. We can readily separate the phase-space in two distinct regimes according to these two competing processes by equating their associated timescales: $t_{\rm int}:=1/\Gamma$ and $t_{\rm gw}:=(1/4)\,\left|a/\dot{a}_{\rm gw}\right|$, which defines the threshold shown as the thick, black line in Figure \[fig:GC\]. The reason for the $1/4$ factor is given in [@peters64]. Below the curve, BHBs will evolve due to GW emission. Above it, close encounters with interlopers are the main driving mechanism, so that BHBs can be scattered in both directions in angular momentum in a random-walk fashion. The scattering in energy is less significant but also present (see [@alexander17] and discussion in [@Amaro-SeoaneLRR2012]). ![ Phase space structure of a BHB with $M_1=M_2=30~M_\odot$. The top-right box fences in the birthplace of 95% of a thermal distribution of primordial binaries, i.e. those binaries formed not dynamically but via binary stellar evolution. In this box, but limited within the radii $a_h$ and $a_{\rm ej}$, the hard and ejection radius, which end at the boundary of the dynamical region because of the absence of interlopers, we also find the vast majority of binaries formed dynamically, i.e. the 95% of their thermal distribution. The colored, dashed lines depict the birthplaces of BHBs formed via three different processes which we explain in the main text. The green lines display the evolutionary tracks of a BHB entering the LIGO/Virgo band at two different eccentricities, $e=0.1$ (lower) and $e=5\times10^{-3}$ (upper). The first LIGO detections have an eccentricity $e\lesssim0.1$, meaning that they have formed between the lower green line and the upper thick, black line. \[fig:GC\]](GC.eps){width="1\columnwidth"} [*Possible ways of forming relativistic BHBs.–*]{} Different mechanisms have been proposed in the literature to form a BHB which eventually might end up emitting detectable GWs. \(1) Primordial binaries: In stellar dynamics this term refers to binaries already present in the cluster which form via stellar evolution. Population synthesis models predict that these binaries populate the area of phase-space displayed as the grey thick-dashed box of Figure \[fig:GC\] (see e.g. [@belczynski04]). We note that only a small fraction of them are in the LISA/TJ band. \(2) Dynamics: (2.1) Close encounters of multiple single, i.e. initially not bound, objects also form BHBs (see e.g. [@kulkarni93; @sigurdsson93; @miller02hamilton; @ivanova05]). Their formation follows a thermal distribution in $e$ (e.g. [@antognini16]), like primordial binaries, but the distribution of $a$ is better constrained: When the binding energy of the binary, $E_b=GM_1M_2/(2a)$ becomes smaller than the mean kinetic energy of the interlopers, $E_*=3M_*\sigma_*^2/2$, the binary ionizes [@BT08]. The threshold condition $E_b=E_*$ can be expressed in terms of a “hard radius”, $a_h=GM_1M_2/(3M_*\sigma_*^2)$. These “hard” binaries heat up the system, meaning that they deliver energy to the rest of the stars interacting with them: Binaries with $a<a_h$ impart on average an energy of $\Delta E\simeq kG\mu M_*/a$ to each interloper, where $\mu$ is the reduced mass of the binary and $k$ is about $0.4$ when $M_1\simeq M_2\simeq M_*$ [@heggie75]. The interloper hence is re-ejected into the stellar system with a higher velocity because of the extra energy, $v\sim\left(3\sigma_*^2+2kG\mu/a\right)^{1/2}$, and the center-of-mass of the BHB recoils at a velocity of $v_b\sim M_*v/(M_1+M_2)$. Occasionally, the BHB will leave the system if this velocity exceeds the escape velocity of the GC, $v_{\rm esc}=\sqrt{2.6GM_{\rm GC}/R_h}$ [@rodriguez16]. The threshold for this to happen is defined by the condition $v_b=v_{\rm esc}$, i.e. the binary must have a semi-major axis smaller than the “ejection radius”, $a_{\rm ej}$. Therefore, all of these BHBs are confined in $a_h<a<a_{\rm ej}$ of Figure \[fig:GC\]. Because of their thermal distribution, we have that $95\%$ of them have $e<0.975$. Therefore, they populate an even smaller area than those primordial binaries. (2.2) Binary-single interactions: Initially we have a hard BHB which interacts with a single object in a chaotic way. During the interaction the interloper might excite the eccentricity of the inner binary to such high values that the binary is on an almost head-on-collision orbit, to soon merge and emit a detectable burst of GWs [@gultekin06; @samsing14; @ama16]. This happens only if $t_{\rm gw}$ is shorter than the period of the captured interloper $P_{\rm int}$. The event rate for BHBs has not been calculated for this scenario but earlier calculations for neutron-star binaries find it to be $1~{\rm Gpc^{-3}~yr^{-1}}$ [@samsing14]. We derive now the eccentricities of these BHBs: Suppose the semi-major axis of a BHB changes from $a$ (with, of course, $a_{\rm ej}<\,a<\,a_h$) to $a'$, and $e$ to $e'$ during the three-body interaction, and the final orbit of the interloper around the center-of-mass of the BHB has a semi-major axis of $a_{\rm int}$. Energy conservation results in the following relations, $a'>a$ and $a_{\rm int}\simeq 2a/(1-a/a')$ (see [@samsing14]), where we neglect the initial energy of the interloper because the BHB is assumed to be hard. Then using a conservative criterion for a successful inspiral, $t_{\rm gw}(a',\,e')=P_{\rm int}(a_{\rm int})$, we derive $e'$ for the BHB, which allows us to confine the range of eccentricities as the dashed, blue curve of Figure \[fig:GC\]. (2.3) Hierarchical triple: This is similar to the previous configuration, but now we only consider $1<a'/a<1.5$, because this requires that $a_{\rm int} > 6\,a$, in which case the configuration is stable [@mardling01]. This leads to a secular evolution of the orbital eccentricity of the inner BHB which is known as the Lidov-Kozai resonance (see [@lidov62; @kozai62] and also [@miller02; @wen03; @oleary06; @naoz13; @antognini14; @liu15]). The inner BHB will decouple via GW emission and merge at a critical eccentricity, and the merger rate has been estimated to be $0.3-6~{\rm Gpc^{-3}~yr^{-1}}$ [@antonini14; @antonini16BS; @kimpson16; @silsbee16]. We follow the scheme of [@antonini14] of isolated hierarchical triples but impose four additional requirements which are fundamental for a realistic estimation of the threshold eccentricity in our work: (a) The BHB has $a_{\rm ej}<\,a<\,a_h$. (b) The third body orbiting the BHB has a mass of $M_{\rm int}=10~M_\odot$ because of mass segregation, and an eccentricity of $e_{\rm int}=2/3$, which corresponds to the mean of a thermal distribution [@antognini16]. (c) The outer binary, i.e. the third object and the inner BHB, is also hard, so that $a_{\rm int}<GM_{12}/(3\sigma_*^2)$. (d) The pericenter distance of the outer binary, $a_{\rm int}(1-\,e_{\rm int})$ should meet the criterion for a stable triple (Eq. 90 in Ref. [@mardling01]). These conditions delimit the range of eccentricities as shown by the dashed, orange lines in Figure \[fig:GC\]. \(3) Gravitational braking: There is a small probability that two single BHs come to such a close distance that GW radiation dissipates a significant amount of the orbital energy, leaving the two BHs gravitationally bound [@turner77; @quinlan87; @kocsis06; @oleary09; @lee10; @hong15]. For GCs, and using optimistic assumptions, these binaries contribute an event rate of $0.06-20~{\rm Gpc^{-3}~yr^{-1}}$ in the LIGO band [@lee10; @antonini16BS], while in NSCs it has been estimated to range between $0.005-0.02~{\rm Gpc^{-3}~yr^{-1}}$ [@tsang13]. The boundaries in Figure \[fig:GC\] for BHBs formed via this mechanism can be calculated using the formulae of [@oleary09]. For that, we choose an initial relative velocity $v$ in the range $\sigma_*<v<3\sigma_*$ and an initial impact parameter $b$ in the range $0.3b_{\rm max}<b<0.99b_{\rm max}$ to account for the majority of the encounters, because the encounter probability is proportional to $b^2$, and $b_{\rm max}$ is the maximum impact parameter that leads to a bound binary. The first LIGO detections, had they been originated via this mechanism, should originate from the red area above the green line. [*Discussions and conclusions.–*]{}A joint detection of BHBs with LIGO/Virgo and LISA/TJ would be desirable because of the science payback. In this paper we show that the actual number of BHBs to be coincidentally detected is very uncertain. As Figure \[fig:detectors\] shows, LISA/TJ is already deaf to mildly eccentric BHBs: For example, a BHB at milli-Hertz orbital frequencies starting at $a \sim 10^{-3}$ AU and $0.7\lesssim e\lesssim0.9$ will also be missed by LISA/TJ, but later be detectable by LIGO/Virgo. BHBs can form via the five mechanisms which we discussed in the list of possible formations. This allows us to pinpoint the regions in phase-space which produce BHBs that eventually will merge via gravitational radiation. The total area of these five regions is a small subset of phase-space. It is an error to assume that all binaries born in this subset are jointly detectable by LIGO/Virgo and LISA/TJ. Only a subset of that subset of phase-space will lead to successful joint detections. This sub-subset depends on the masses of the BHBs. We can see this in Figures \[fig:GC\] and \[fig:GC2\]. While in the first figure the hierarchical triple gets into the LISA/TJ band, it does not in the second one. On the other hand, up to 95% of primordial and dynamical binaries (1 and 2.1 in the list of possible formations) are produced in the box delimited by grey dashed lines. In that box, and in principle, the BHBs can lead to sources jointly detectable by LIGO/Virgo and LISA/TJ. However, exceptions might occur if a scatter results in a BHB jumping towards high eccentricities. This probability has not been fully addressed. It requires dedicated numerical scattering experiments with relativistic corrections (e.g. [@ama10]), as well as a proper star-cluster model to screen out BHBs that can decouple from the stellar dynamics (e.g. our model as presented in Figures \[fig:GC\] and \[fig:GC2\]). We have shown that mergers in GCs produced by the mechanisms (2.2), (2.3), and (3) are inaudible to LISA. The event rates corresponding to these mergers have been largely discussed in the literature, but are uncertain, due to questionable parameters, such as the cosmic density of GCs and the number of BHs in them. Nevertheless, it has been estimated that the rate could be as large as $20~{\rm Gpc^{-3}~yr^{-1}}$ [@lee10], while the current LIGO detections infer a total event rate of $9-240~{\rm Gpc^{-3}~yr^{-1}}$. Moreover, these mergers could also originate in NSCs, [@kocsis06; @MillerLauburg09; @oleary09; @tsang13; @hong15; @antonini12; @addison15; @antonini16], and the event rates there are higher, up to $10^2~{\rm Gpc^{-3}~yr^{-1}}$ [@VL16]. Therefore, future multi-band GW astronomy should prepare for LIGO/Virgo BHBs that do not have LISA/TJ counterparts. A non-detection by LISA/TJ is also useful in constraining astrophysics: It puts a lower limit on the eccentricities of the LIGO/Virgo sources, which according to Figures \[fig:GC\] and \[fig:GC2\] is about $0.005$. A deci-Hz detector, by covering the gap in frequencies between LISA/TJ and LIGO/Virgo, would drastically enhance the number of jointly detectable binaries. [*Acknowledgement.*]{}–This work is supported partly by the Strategic Priority Research Program “Multi-wavelength gravitational wave universe” of the Chinese Academy of Sciences (No. XDB23040100) and by the CAS President’s International Fellowship Initiative. PAS acknowledges support from the Ram[ó]{}n y Cajal Programme of the Ministry of Economy, Industry and Competitiveness of Spain. We thank Bence Kocsis and Fukun Liu for many fruitful discussions, and Eric Peng for a thorough reading of our manuscript. 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ArXiv

--- abstract: 'We define two algorithms for propagating information in classification problems with pairwise relationships. The algorithms are based on contraction maps and are related to non-linear diffusion and random walks on graphs. The approach is also related to message passing algorithms, including belief propagation and mean field methods. The algorithms we describe are guaranteed to converge on graphs with arbitrary topology. Moreover they always converge to a unique fixed point, independent of initialization. We prove that the fixed points of the algorithms under consideration define lower-bounds on the energy function and the max-marginals of a Markov random field. The theoretical results also illustrate a relationship between message passing algorithms and value iteration for an infinite horizon Markov decision process. We illustrate the practical application of the algorithms under study with numerical experiments in image restoration, stereo depth estimation and binary classification on a grid.' author: - | Pedro F. Felzenszwalb[^1]\ Brown University\ Providence, RI, USA\ [[email protected]]{} - | Benar F. Svaiter[^2]\ IMPA\ Rio de Janeiro, RJ, Brazil\ [[email protected]]{} bibliography: - 'prop.bib' title: Diffusion Methods for Classification with Pairwise Relationships --- Introduction ============ In many classification problems there are relationships among a set of items to be classified. For example, in image reconstruction problems adjacent pixels are likely to belong to the same object or image segment. This leads to relationships between the labels of different pixels in an image. Energy minimization methods based on Markov random fields (MRF) address these problems in a common framework [@Besag74; @WJ08; @KF09]. Within this framework we introduce two new algorithms for classification with pairwise information. These algorithms are based on contraction maps and are related to non-linear diffusion and random walks on graphs. The setting under consideration is as follows. Let $G=(V,E)$ be an undirected simple graph and $L$ be a set of labels. A labeling of $V$ is a function $x : V \to L$ assigning a label from $L$ to each vertex in $V$. Local information is modeled by a cost $g_i(a)$ for assigning label $a$ to vertex $i$. Information on label compatibility for neighboring vertices is modeled by a cost $h_{ij}(a,b)$ for assigning label $a$ to vertex $i$ and label $b$ to vertex $j$. The cost for a labeling $x$ is defined by an energy function, $$F(x) = \sum_{i \in V} g_i(x_i) + \sum_{\{i,j\} \in E} h_{ij}(x_i,x_j).$$ In the context of MRFs the energy function defines a Gibbs distribution on random variables $X$ associated with the vertices $V$, $$\begin{aligned} p(X=x) & = \frac{1}{Z} \exp(-F(x)).\end{aligned}$$ Minimizing the energy $F(x)$ corresponds to maximizing $p(X=x)$. This approach has been applied to a variety of problems in image processing and computer vision [@FZ11]. A classical example involves restoring corrupted images [@GG84; @Besag86]. In image restoration there is a grid of pixels and the problem is to estimate an intensity value for each pixel. To restore an image $I$ one looks for an image $J$ that is similar to $I$ and is smooth almost everywhere. Similarity between $I$ and $J$ is defined by local costs at each pixel. The smoothness constraint is defined by pairwise costs between neighboring pixels in $J$. Basic Definitions and Overview of Results ----------------------------------------- Let $G=(V,E)$ be an undirected, simple, connected graph, with more than one vertex. For simplicity let $V=\{1,\dots,n\}$. Let $N(i)$ and ${\mathrm{d}}(i)$ denote respectively the set of neighbors and the degree of vertex $i$, $$\begin{aligned} N(i) = \{j \in V \;|\; \{i,j\} \in E\}, \quad {\mathrm{d}}(i)=|N(i)|.\end{aligned}$$ Let $L$ be a set of labels. For each vertex $i \in V$ we have a non-negative cost for assigning label $a$ to vertex $i$, denoted by $g_i(a)$. These costs capture local information about the label of each vertex. For each edge $\{i,j\} \in E$ we have a non-negative cost for assigning label $a$ to vertex $i$ and label $b$ to vertex $j$, denoted equally by $h_{ij}(a,b)$ or $h_{ji}(b,a)$. These costs capture relationships between labels of neighboring vertices. - $g_i:L\to [0,\infty)$ for $i \in V$; - $h_{ij},h_{ji}:L^2 \to [0,\infty)$ for $\{i,j\} \in E$ with $h_{ij}(a,b) = h_{ji}(b,a)$ Let $x \in L^V$ denote a labeling of $V$ with labels from $L$. A cost for $x$ that takes into account both local information at each vertex and the pairwise relationships can be defined by an energy function $F:L^V\to{\mathbb{R}}$, $$\begin{aligned} \label{eq:F} F(x)=\sum_{i\in V} g_i(x_i) +\sum_{\{i,j\}\in E} h_{ij}(x_i,x_j).\end{aligned}$$ This leads to a natural optimization problem where we look for a labeling $x$ with minimum energy. Throughout the paper we assume $L$ is finite. The optimization problem defined by $F$ is NP-hard even when $|L|=2$ as it can be used to solve the independent set problem on $G$. It can also be used to solve coloring with $k$ colors when $|L|=k$. The optimization problem can be solved in polynomial time using dynamic programming when $G$ is a tree [@BB72]. More generally dynamic programming leads to polynomial optimization algorithms when the graph $G$ is chordal (triangulated) and has bounded tree-width. Min-sum (max-product) belief propagation [@WJ08; @KF09] is a local message passing algorithm that is equivalent to dynamic programming when $G$ is a tree. Both dynamic programming and belief propagation aggregate local costs by sequential propagation of information along the edges in $E$. For $i \in V$ we define the value function $f_i:L\to{\mathbb{R}}$, $$\begin{aligned} \label{eq:f} f_i(\tau) = \min_{\substack{x \in L^V\\ x_i=\tau}} F(x).\end{aligned}$$ In the context of MRFs the value functions are also known as *max-marginals*. The value functions are also what is computed by the dynamic programming and belief propagation algorithms for minimizing $F$ when $G$ is a tree. Each value function defines a cost for assigning a label to a vertex that takes into account the whole graph. If $x^*$ minimizes $F(x)$ then $x_i^*$ minimizes $f_i(\tau)$, and when $f_i(\tau)$ has a unique minimum we can minimize $F(x)$ by selecting $$x^*_i = \operatorname*{argmin}_{\tau} f_i(\tau).$$ A local belief is a function $\gamma:L\to{\mathbb{R}}$. A field of beliefs specifies a local belief for each vertex in $V$, and is an element of $$\begin{aligned} {({\mathbb{R}}^L)^V}= \{\varphi=(\varphi_1,\ldots,\varphi_N)\;|\; \varphi_i:L\to{\mathbb{R}}\}.\end{aligned}$$ We define two algorithms in terms of maps, $$\begin{aligned} T:{({\mathbb{R}}^L)^V}\to{({\mathbb{R}}^L)^V},\\ S:{({\mathbb{R}}^L)^V}\to{({\mathbb{R}}^L)^V}.\end{aligned}$$ The maps $T$ and $S$ are closely related. Both maps are contractions, but each of them has its own unique fixed point. Each of these maps can be used to define an algorithm to optimize $F(x)$ based on fixed point iterations and local decisions. For $z\in\{T,S\}$ we start from an initial field of beliefs $\varphi^0$ and sequentially compute $$\varphi^{k+1}=z(\varphi^k).$$ Both $S^k(\varphi^0)$ and $T^k(\varphi^0)$ converge to the unique fixed points of $S$ and $T$ respectively. After convergence to a fixed point $\varphi$ (or a bounded number of iterations in practice) we select a labeling $x$ by selecting the label minimizing the belief at each vertex (breaking ties arbitrarily), $$\begin{aligned} x_i = \operatorname*{argmin}_\tau \varphi_i(\tau).\end{aligned}$$ The algorithms we consider depend on parameters $p \in (0,1)$, $q=1-p$ and weights $w_{ij} \in [0,1]$ for each $i \in V$ and $j \in N(i)$. The weights from each vertex are constrained to sum to one, $$\label{eq:wsum} \sum_{j \in N(i)} w_{ij} = 1, \qquad \forall i\in V.$$ These weights can be interpreted in terms of transition probabilities for a random walk on $G$. In a uniform random walk we have $w_{ij} = 1/{\mathrm{d}}(i)$. Non-uniform weights can be used to capture additional information about an underlying application. For example, in the case of stereo depth estimation (Section \[sec:stereo\]) we have used non-uniform weights that reflect color similarity between neighboring pixels. We note, however, that while we may interpret the results of the fixed point algorithms in terms of transition probabilities in a random walk, the algorithms we study are deterministic. The maps $S$ and $T$ we consider are defined as follows, \[df:maps\] $$\begin{aligned} \label{eq:T} &(T \varphi)_i(\tau) = p g_i(\tau) + \sum_{j\in N(i)} \min_{u_j\in L}\; \dfrac{p}{2}h_{ij}(\tau,u_j) +q w_{ji} \varphi_j(u_j) \\ \label{eq:S} &(S \varphi)_i(\tau) = p g_i(\tau) + \sum_{j\in N(i)} w_{ij} \min_{u_j\in L}\; p h_{ij}(\tau,u_j) +q\varphi_j(u_j) \end{aligned}$$ The map defined by $T$ corresponds to a form of non-linear diffusion of beliefs along the edges of $G$. The map defined by $S$ corresponds to value iteration for a Markov decision process (MDP) [@Bertsekas05] defined by random walks on $G$. We show that both $S$ and $T$ are contractions. Let $\bar{\varphi}$ be the fixed point of $T$ and $\hat{\varphi}$ be the fixed point of $S$. We show $\bar{\varphi}$ defines a lower bound on the energy function $F$, and that $\hat{\varphi}$ defines lower bounds on the value functions $f_i$, $$\begin{aligned} \sum_{i \in V} \bar{\varphi}_i(x_i) & \leq F(x),\qquad \forall x\in L^V, \\ \hat{\varphi}_i(\tau) & \le f_i(\tau),\qquad \forall i\in V,\; \tau\in L.\end{aligned}$$ In Section \[sec:T\] we study the fixed point iteration algorithm defined by $T$ and the relationship between $\bar{\varphi}$ and $F$. To the extent that $\sum_{i \in V} \bar{\varphi}_i(x_i)$ approximates $F(x)$ this justifies selecting a labeling $x$ by minimizing $\bar{\varphi}_i$ at each vertex. This approach is related to mean field methods and variational inference with the Gibbs distribution $p(X=x)$ [@WJ08; @KF09]. In Section \[sec:S\] we study the algorithm defined by $S$ and the relationship between $\hat{\varphi}_i$ and $f_i$. To the extent that $\hat{\varphi}_i(\tau)$ approximates $f_i(\tau)$ this justifies selecting a labeling $x$ by minimizing $\hat{\varphi}_i$ at each vertex. We also show a connection between the fixed point $\hat{\varphi}$ and optimal policies of a Markov decision process. The process is defined in terms of random walks on $G$, with transition probabilities given by the weights $w_{ij}$. Examples -------- ![The fixed points of $T$ on two problems defined on the graph above. In this case $L=\{1,2\}$. In both cases the local costs $g_i$ are all zero except for vertex 1 who has a preference towards label 2. In [**(a)**]{} the pairwise costs encourage neighboring vertices to take the same label. In [**(b)**]{} the pairwise costs encourage neighboring vertices to take different labels.[]{data-label="fig:example"}](example/graph.pdf){height="1in"} --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![The fixed points of $T$ on two problems defined on the graph above. In this case $L=\{1,2\}$. In both cases the local costs $g_i$ are all zero except for vertex 1 who has a preference towards label 2. In [**(a)**]{} the pairwise costs encourage neighboring vertices to take the same label. In [**(b)**]{} the pairwise costs encourage neighboring vertices to take different labels.[]{data-label="fig:example"}](example/propT1.pdf "fig:"){width="3in"} ![The fixed points of $T$ on two problems defined on the graph above. In this case $L=\{1,2\}$. In both cases the local costs $g_i$ are all zero except for vertex 1 who has a preference towards label 2. In [**(a)**]{} the pairwise costs encourage neighboring vertices to take the same label. In [**(b)**]{} the pairwise costs encourage neighboring vertices to take different labels.[]{data-label="fig:example"}](example/propT2.pdf "fig:"){width="3in"} [**(a)**]{} Attractive relationships [**(b)**]{} Repulsive relationships --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Figure \[fig:example\] shows two examples of fixed points of $T$ when the graph $G=(V,E)$ is a cycle with 5 vertices. In this case we have a binary labeling problem $L=\{1,2\}$. The local costs are all zero except that vertex 1 has a preference for label 2. This is encoded by a cost for label 1, $$\begin{aligned} g_1(1) & = 1, \\ g_1(2) & = 0, \\ g_i(a) & = 0, \qquad \forall i \neq 1, \; a\in L.\end{aligned}$$ In example (a) we have pairwise costs that encourage equal labels for neighboring vertices, $$\begin{aligned} h_{ij}(a,b) = \begin{cases} 0 \qquad a = b \\ 1 \qquad a \neq b, \end{cases}\end{aligned}$$ In example (b) we have pairwise costs that encourage different labels for neighboring vertices, $$\begin{aligned} h_{ij}(a,b) = \begin{cases} 1 \qquad a = b \\ 0 \qquad a \neq b, \end{cases}\end{aligned}$$ Figure \[fig:example\] shows a graphical representation of the local costs for each vertex, and the value of $\bar{\varphi}$, the fixed point of $T$, on each example. In (a) local selection of $x_i$ minimizing $\bar{\varphi}_i$ leads to $x=(2,2,2,2,2)$. In (b) local selection of $x_i$ minimizing $\bar{\varphi}_i$ leads to $x=(2,1,2,2,1)$. In both examples the resulting labeling $x$ is the global minimum of $F(x)$. For these examples we used $p=0.1$ and $w_{ij} = 1/{\mathrm{d}}(i)$. Of course in general local minimization of $\bar{\varphi}$ does not lead to a labeling minimizing $F(x)$ and it would be interesting to characterize when this happens. Related Work ------------ For general graphs $G$, when the pairwise costs $h_{ij}(a,b)$ define a metric over $L$ there are polynomial time approximation algorithms for the optimization problem defined by $F$ [@KT02]. In some important cases the optimization problem can be solved using graph cuts and maximum flow algorithms [@GPS89; @BVZ01; @Boros02; @KZ04]. This includes in particular the case of MAP estimation for an Ising model with an external field [@GPS89]. The algorithms we study are closely related to message passing methods, in particular to min-sum (or equivalently max-product) belief propagation (BP) [@WJ08; @KF09]. When the graph $G$ is a tree, BP converges and solves the optimization problem defined by $F$. Unfortunately BP is not guaranteed to converge and it can have multiple fixed points for general graphs. Some form of dampening can help BP converge in practice. The algorithms we study provide a simple alternative to min-sum belief propagation that is guaranteed to converge to a unique fixed point, regardless of initialization. The algorithms are also guaranteed to converge “quickly”. One approach for solving the optimization problem defined by $F$ involves using a linear program (LP) relaxation. The optimization problem can be posed using a LP with a large number of constraints and relaxed to obtain a tractable LP over the *local polytope* [@WJW05]. Several message passing methods have been motivated in terms of this LP [@MGW09]. There are also recent methods which use message passing in the inner loop of an algorithm that converges to the optimal solution of the local polytope LP relaxation [@RAW10; @SSKS12]. In Section \[sec:lp\] we characterize the fixed point of $S$ using a different LP. The mean-field algorithm [@WJ08; @KF09] is an iterative method for approximating the Gibbs distribution $p(x)$ by a factored distribution $q(x)$, $$q(x) = \prod_{i \in V} q_i(x_i).$$ The mean-field approach involves minimization of the KL divergence between $p$ and $q$ using fixed point iterations that repeatedly update the factors $q_i$ defining $q$. A drawback of the approach is that the fixed point is not unique and the method is sensitive to initialization. The algorithm defined by $T$ is related to the mean-field method in the sense that the fixed points of $T$ appear to approximate $F(x)$ by a function $H(x)$ that is a sum of local terms, $$H(x) = \sum_{i \in V} \bar{\varphi}_i(x_i).$$ We do not know, however, if there is a measure under which the resulting $H(x)$ is an optimal approximation to $F(x)$ within the class of functions defined by a sum of local terms. Preliminaries ============= The algorithms we study are efficient in the following sense. Let $m=|E|$ and $k=|L|$. Each iteration in the fixed point algorithm involves evaluating $T$ or $S$. This can be done in $O(mk^2)$ by “brute-force” evaluation of the expressions in Definition \[df:maps\]. In many applications, including in image restoration and stereo matching, the pairwise cost $h_{ij}$ has special structure that allows for faster computation using the techniques described in [@FH12]. This leads to an $O(mk)$ algorithm for each iteration of the fixed point methods. Additionally, the algorithms are easily parallelizable. The fixed point algorithms defined by $T$ and $S$ converge quickly because the maps are contractions in ${({\mathbb{R}}^L)^V}$. Let $z:{\mathbb{R}}^K \to {\mathbb{R}}^K$ and ${\|x\|}$ be a norm in ${\mathbb{R}}^K$. For $\gamma \in (0,1)$, $z$ is a $\gamma$-contraction if $${\|z(x)-z(y)\|} \le \gamma{\|x-y\|}.$$ When $z$ is a contraction it has a unique fixed point $\bar{x}$. It also follows directly from the contraction property that fixed point iteration $x_k = z(x_{k-1})$ converges to $\bar{x}$ quickly, $${\|x_k-\bar{x}\|} \le \gamma^k {\|x_0-\bar{x}\|}.$$ The weights $w_{ij}$ in the definition of $T$ and $S$ define a random process that generates random walks on $G$. We have a Markov chain with state space $V$. Starting from a vertex $Q_0$ we generate an infinite sequence of random vertices $(Q_0,Q_1,\ldots)$ with transition probabilities $$p(Q_{t+1}=j|Q_t=i) = w_{ij}.$$ A natural choice for the weights is $w_{ij} = 1/{\mathrm{d}}(i)$, corresponding to moving from $i$ to $j$ with uniform probability over $N(i)$. This choice leads to uniform random walks on $G$ [@Lovasz93]. We consider in ${({\mathbb{R}}^L)^V}$ the partial order $$\begin{aligned} \varphi \leq \psi \iff \varphi_i(\tau) \leq \psi_i(\tau) \;\; \forall i\in V,\; \forall \tau\in L.\end{aligned}$$ It follows trivially from the definitions of $T$ and $S$ that both maps preserve order in ${({\mathbb{R}}^L)^V}$, $$\varphi \leq \psi \Rightarrow T\varphi \leq T\psi,\; S\varphi \leq S\psi. \label{eq:order}$$ We claim that for any $\alpha\in{\mathbb{R}}^V$, $$\begin{aligned} \label{eq:fsum} \sum_{i\in V}\sum_{j\in N(i)} w_{ji} \alpha_j = \sum_{j\in V} \alpha_j.\end{aligned}$$ This follows from re-ordering the double summation and the constraints that the weights out of each vertex sum to one, $$\begin{aligned} \sum_{i\in V}\sum_{j\in N(i)} w_{ji} \alpha_j = \sum_{j\in V} \sum_{i \in N(j)} w_{ji} \alpha_j = \sum_{j \in V} \alpha_j\end{aligned}$$ We note that the algorithms defined by $T$ and $S$ are related in the following sense. For a regular graph with degree $d$, if we let $w_{ij} = 1/d$ the maps $T$ and $S$ are equivalent up to rescaling if the costs in $T$ and $S$ are rescaled appropriately. Algorithm defined by $T$ (Diffusion) {#sec:T} ==================================== In this section we study the fixed point algorithm defined by $T$. We show that $T$ is a contraction in ${({\mathbb{R}}^L)^V}$ and that the fixed point of $T$ defines a “factored” lower bound on $F$. We start by showing that $T$ is a contraction with respect to the norm on ${({\mathbb{R}}^L)^V}$ defined by $$\begin{aligned} {\|\varphi\|}_{\infty,1}=\sum_{i\in V}{\|\varphi_i\|}_\infty.\end{aligned}$$ \[lm:ctT\] (Contraction) For any $\varphi,\psi\in {({\mathbb{R}}^L)^V}$ $$\begin{aligned} \label{eq:ct1} {\|(T\varphi)_i - (T\psi)_i\|}_\infty & \le q \sum_{j \in N(i)} w_{ji} {\|\varphi_j - \psi_j\|}_\infty \qquad \forall i\in V,\\ \label{eq:ct2} {\|(T\varphi)-(T\psi)\|}_{\infty,1} & \leq q {\|\varphi-\psi\|}_{\infty,1}. \end{aligned}$$ Take $i\in V$ and $\tau \in L$. For any $x\in L^V$ $$\begin{aligned} (T\varphi)_i(\tau) & = p g_i(\tau)+ \sum_{j\in N(i)} \min_{u_j \in L} \dfrac{p}{2}h_{ij}(\tau,u_j) + q w_{ji} \varphi_j(u_j) \\ & \leq p g_i(\tau)+ \sum_{j\in N(i)}\dfrac{p}{2}h_{ij}(\tau,x_j) + q w_{ji} \varphi_j(x_j) \\ & \leq p g_i(\tau) + \sum_{j\in N(i)}\dfrac{p}{2}h_{ij}(\tau,x_j) + q w_{ji} (\psi_j(x_j) + {|\varphi_j(x_j)-\psi_j(x_j)|}) \\ & \leq p g_i(\tau) + \sum_{j\in N(i)}\dfrac{p}{2}h_{ij}(\tau,x_j) + q w_{ji} (\psi_j(x_j) + {\|\varphi_j-\psi_j\|}_\infty) \end{aligned}$$ Since the inequality defined by the first and last terms above holds for any $x$, it holds when $x$ minimizes the last term. Therefore $$(T\varphi)_i(\tau)\leq (T\psi)_i(\tau) + q \sum_{j\in N(i)} w_{ji} {\|\varphi_j-\psi_j\|}_\infty.$$ Since this inequality holds interchanging $\varphi$ with $\psi$ we have $${|(T\varphi)_i(\tau)-(T\psi)_i(\tau)|} \leq q \sum_{j\in N(i)} w_{ji} {\|\varphi_j-\psi_j\|}_\infty.$$ Taking the $\tau$ maximizing the left hand side proves (\[eq:ct1\]). To prove (\[eq:ct2\]), we sum the inequalities (\[eq:ct1\]) for all $i \in V$ and use (\[eq:fsum\]). The contraction property above implies the fixed point algorithm defined by $T$ converges to a unique fixed point independent on initialization. It also implies the distance to the fixed point decreases quickly, and we can bound the distance to the fixed point using either the initial distance to the fixed point or the distance between consecutive iterates (a readily available measure). \[th:Tconv\] The map $T$ has a unique fixed point $\bar{\varphi}$ and for any $\varphi \in {({\mathbb{R}}^L)^V}$ and integer $k \ge 0$, $$\begin{aligned} {\|\bar{\varphi} - T^k \varphi\|}_{\infty,1} & \le q^k {\|\bar{\varphi}-\varphi\|}_{\infty,1}, \\ {\|\bar{\varphi}-\varphi\|}_{\infty,1} & \le \dfrac{1}{p} {\|T\varphi - \varphi\|}_{\infty,1}. \end{aligned}$$ Existence and uniqueness of the fixed point, as well as the first inequality follows trivially from Lemma \[lm:ctT\]. To prove the second inequality observe that since $T^k\varphi$ converges to $\bar{\varphi}$, $${\|\bar{\varphi}-\varphi\|}_{\infty,1} \le \sum_{k=0}^\infty {\|T^{k+1}\varphi - T^k\varphi\|}_{\infty,1} \le \sum_{k=0}^\infty q^k {\|T\varphi - \varphi\|}_{\infty,1}.$$ Now note that since $p \in (0,1)$ and $p+q=1$, $$\sum_{k=0}^\infty q^k p = 1 \implies \sum_{k=0}^\infty q^k = \frac{1}{p}.$$ The map $T$ and the energy function $F$ are related as follows. \[pr:TF\] For any $\varphi\in{({\mathbb{R}}^L)^V}$ and $x\in L^V$ $$\begin{aligned} \sum_{i\in V}(T\varphi)_i(x_i)\leq p F(x) + q \sum_{i \in V} \varphi_i(x_i). \end{aligned}$$ Direct use of the definition of $T$ yields $$\begin{aligned} \sum_{i\in V}(T\varphi)_i(x_i) & = \sum_{i\in V} pg_i(x_i)+\sum_{j\in N(i)} \min_{u_j\in L} \frac{p}{2} h_{ij}(x_i,u_j) + q w_{ji} \varphi_j(u_j) \\ & \le \sum_{i\in V} pg_i(x_i)+\sum_{j\in N(i)} \frac{p}{2} h_{ij}(x_i,x_j) + q w_{ji} \varphi_j(x_j) \\ & = p \left( \sum_{i\in V}g_i(x_i) + \sum_{i\in V}\sum_{j\in N(i)} \frac{1}{2} h_{ij}(x_i,x_j) \right) + q \sum_{i\in V} \sum_{j \in N(i)} w_{ji} \varphi_j(x_j) \\ & = p F(x) + q \sum_{j\in V}\varphi_j(x_j), \end{aligned}$$ where the last equality follows from the fact that $h_{ij}(x_i,x_j)=h_{ji}(x_j,x_i)$ and Equation (\[eq:fsum\]). Now we show the fixed point of $T$ defines a lower bound on $F$ in terms of a sum of local terms. \[th:b1\] Let $\bar{\varphi}$ be the fixed point of $T$ and $$H(x) = \sum_{i \in V} \bar{\varphi}_i(x_i).$$ Then $0 \leq \bar{\varphi}$ and $H(x) \le F(x)$. The fact that $H(x) \le F(x)$ follows directly from Proposition \[pr:TF\]. To prove $0 \le \bar{\varphi}$ consider the sequence $(0,T0,T^20,\ldots)$. The non-negativity of $g_i$ and $h_{ij}$ implies $0 \le T0$. Since $T$ is order preserving (\[eq:order\]) it follows by induction that $T^k0 \le T^{k+1}0$ for all $k\ge0$. Since the sequence is pointwise non-decreasing and converges to $\bar{\varphi}$ we have $0 \le \bar{\varphi}$. Theorem \[th:b1\] allows us to compute both a lower and an upper bound on the optimal value of $F$, together with a solution where $F$ attains the upper bound. \[cr:bracket\] Let $\bar\varphi$ be the fixed point of $T$ and $$\begin{aligned} \bar x_i = \operatorname*{argmin}_{\tau}\bar\varphi_i(\tau) \;\; \forall i\in V, \end{aligned}$$ then for any $x^*$ minimizing $F$, $$\begin{aligned} \sum_{i\in V}\bar\varphi_i(\bar x_i)\leq F(x^*)\leq F(\bar x). \end{aligned}$$ If $x^*$ is a minimizer of $F$, then the inequality $F(x^*)\leq F(\bar x)$ holds trivially. We can use the definition of $\bar x$ to conclude that $$\begin{aligned} \sum_{i\in V}\bar\varphi_i(\bar x_i) \leq \sum_{i\in V}\bar\varphi_i(x^*_i) \leq F(x^*), \end{aligned}$$ where the second inequality follows from Theorem \[th:b1\] Linear Programming Formulation {#sec:lp} ------------------------------ Here we provide a LP characterizing for the fixed point of $T$. We note that the LP formulation described here is different from the standard LP relaxation for minimizing $F(x)$ which involves the local polytope described in [@WJW05]. Consider the following LP which depends on a vector of coefficients $a$ in ${({\mathbb{R}}^L)^V}$, $$\begin{aligned} & \max_\varphi a^T \varphi \\ & \varphi_i(u_i) \le pg_i(u_i) + \sum_{j \in N(i)} \frac{p}{2} h_{ij}(u_i,u_j) + q w_{ji} \varphi_j(u_j) & \qquad \forall i\in V, \forall u\in L^V. \end{aligned}$$ Note that the constraints in the LP are equivalent to $\varphi \le T\varphi$. Next we show that this LP has a unique solution which equals the fixed point of $T$ whenever every coefficient is positive, independent of their specific values. For example, $\bar \varphi$ is the optimal solution when $a$ is the vector of ones. \[tr:lp\] If $a$ is a non-negative vector the fixed point of $T$ is an optimal solution for the LP. If $a$ is a positive vector the fixed point of $T$ is the unique optimal solution for the LP. Let $\bar{\varphi}$ be the fixed point of $T$. First note that $\bar{\varphi}$ is a feasible solution since $\bar{\varphi} \le T\bar{\varphi}$. Let $\varphi \in {({\mathbb{R}}^L)^V}$ be any feasible solution for the LP. The linear constraints are equivalent to $\varphi \le T\varphi$. Since $T$ preserves order it follows by induction that $T^k \varphi \le T^{k+1} \varphi$ for all $k \ge 0$. Since the sequence $(\varphi,T\varphi,T^2\varphi,\ldots)$ converges to $\bar{\varphi}$ and it is pointwise non-decreasing we conclude $\varphi \le \bar{\varphi}$. If $a$ is non-negative we have $a^T\varphi \le a^T \bar{\varphi}$ and therefore $\bar{\varphi}$ must be an optimal solution for the LP. If $a$ is positive and $\varphi \neq \bar{\varphi}$ we have $a^T\varphi < a^T\bar{\varphi}$. This proves the fixed point is the unique optimal solution for the LP. Algorithm defined by $S$ (Optimal Control) {#sec:S} ========================================== In this section we study the algorithm defined by $S$. We start by showing that $S$ corresponds to value iteration for an infinite horizon discounted Markov decision process (MDP) [@Bertsekas05]. An infinite horizon discounted MDP is defined by a tuple $(Q,A,c,t,\gamma)$ where $Q$ is a set of states, $A$ is a set of actions and $\gamma$ is a discount factor in ${\mathbb{R}}$. The cost function $c:Q \times A \to {\mathbb{R}}$ specifies a cost $c(s,a)$ for taking action $a$ on state $s$. The transition probabilities $t:Q \times A \times Q \to {\mathbb{R}}$ specify the probability $t(s,a,s')$ of moving to state $s'$ if we take action $a$ in state $s$. Let $o$ be an infinite sequence of state and action pairs, $o=((s_1,a_1),(s_2,a_2),\ldots) \in (Q \times A)^\infty$. The (discounted) cost of $o$ is $$c(o) = \sum_{k=0}^\infty \gamma^k c(s_k,a_k).$$ A policy for the MDP is defined by a map $\pi:Q \rightarrow A$, specifying an action to be taken at each state. The value of a state $s$ under the policy $\pi$ is the expected cost of an infinite sequence of state and action pairs generated using $\pi$ starting at $s$, $$v_\pi(s) = E[c(o) | \pi, s_1=s].$$ An optimal policy $\pi^*$ minimizes $v_\pi(s)$ for every starting state. Value iteration computes $v_{\pi^*}$ as the fixed point of ${\cal L}:\mathbb{R}^Q \to \mathbb{R}^Q$, $$({\cal L} v)(s) = \min_{a \in A} c(s,a) + \gamma \sum_{s' \in Q} t(s,a,s') v(s').$$ The map ${\cal L}$ is known to be a $\gamma$-contraction [@Bertsekas05] with respect to the ${\|\cdot\|}_\infty$ norm. Now we show that $S$ is equivalent to value iteration for an MDP defined by random walks on $G$. Intuitively we have states defined by a vertex $i \in V$ and a label $a \in L$. An action involves selecting a different label for each possible next vertex, and the next vertex is selected according to a random walk defined by the weights $w_{ij}$. \[lm:MDP\] Define an MDP $(Q,A,c,t,\gamma)$ as follows. The states are pairs of vertices and labels $Q = V \times L$. The actions specify a label for every possible next vertex $A = L^V$. The discount factor is $\gamma = q$. The transition probabilities and cost function are defined by $$\begin{aligned} t((i,\tau),u,(j,\tau')) & = \begin{cases} w_{ij} & j \in N(i),\;\tau' = u_j\\ 0 & \text{otherwise} \end{cases} \\ c((i,\tau),u) & = p g_i(\tau) + \sum_{j \in N(i)} p w_{ij}h_{ij}(\tau,u_j) \end{aligned}$$ The map $S$ is equivalent to value iteration for this MDP. That is, if $\varphi_i(\tau) = v((i,\tau))$ then $$(S\varphi)_i(\tau) = ({\cal L}v)((i,\tau)).$$ The result follows directly from the definition of the MDP, ${\cal L}$ and $S$. $$\begin{aligned} ({\cal L} v)((i,\tau)) & = \min_{u \in L^V} c((i,\tau),u) + \gamma \sum_{(j,\tau') \in Q} t((i,\tau),u,(j,\tau'))v(j,\tau') \\ & = \min_{u \in L^V} pg_i(\tau) + \sum_{j \in N(i)} pw_{ij}h_{ij}(\tau,u_j) + q \sum_{j \in N(i)} w_{ij} v(j,u_j) \\ & = pg_i(\tau) + \sum_{j \in N(i)} w_{ij} \min_{u_j \in L} ph_{ij}(\tau,u_j) + q v(j,u_j) \\ & = (S\varphi)_i(\tau)\end{aligned}$$ The relationship to value iteration shows $S$ is a contraction and we have the following results regarding fixed point iterations with $S$. The map $S$ has a unique fixed point $\hat \varphi$ and for any $\varphi \in {({\mathbb{R}}^L)^V}$ and integer $k \ge 0$, $$\begin{aligned} {\|\hat{\varphi}-S^k\varphi\|}_\infty & \leq q^k {\|\hat{\varphi}-\varphi\|}_\infty, \\ {\|\hat{\varphi}-\varphi\|}_\infty & \leq \frac{1}{p} {\|S\varphi-\varphi\|}_\infty. \end{aligned}$$ The first inequality follows directly from Lemma \[lm:MDP\] and the fact that ${\cal L}$ is a $\gamma$-contraction with $\gamma=q$. The proof of the second inequality is similar to the proof of the analogous result for the map $T$ in Theorem \[th:Tconv\]. Random Walks ------------ The formalism of MDPs is quite general, and encompasses the fixed point algorithm defined by $S$. In this section we further analyze this fixed point algorithm and provide an interpretation using one-dimensional problems defined by random walks on $G$. The weights $w_{ij}$ define a random process that generates infinite walks on $G$. Starting from some vertex in $V$ we repeatedly move to a neighboring vertex, and the probability of moving from $i \in V$ to $j \in N(i)$ in one step is given by $w_{ij}$. An infinite walk $\omega=(\omega_1,\omega_2,\ldots) \in V^\infty$ can be used to define an energy on an infinite sequence of labels $z=(z_1,z_2,\ldots) \in L^\infty$, $$F_\omega(z) = \sum_{t = 0}^\infty pq^t g_{\omega_t}(z_t) + pq^t h_{\omega_t \omega_{t+1}}(z_t,z_{t+1}).$$ The energy $F_\omega(z)$ can be seen as the energy of a pairwise classification problem on a graph $G'=(V',E')$ that is an infinite path, $$\begin{aligned} V'&=\{1,2,\ldots\}, \\ E'&=\{\{1,2\},\{2,3\},\ldots\}.\end{aligned}$$ The graph $G'$ can be interpreted as a one-dimensional “unwrapping” of $G$ along the walk $\omega$. This unwrapping defines a map from vertices in the path $G'$ to vertices in $G$. Consider a policy $\pi : V \times L \times V \to L$ that specifies $z_{k+1}$ in terms of $\omega_k$, $z_k$ and $\omega_{k+1}$, $$z_{k+1} = \pi(\omega_k,z_k,\omega_{k+1}).$$ Now consider the expected value of $F_\omega(z)$ when $\omega$ is a random walk starting at $i \in V$ and $z$ is a sequence of labels defined by the policy $\pi$ starting with $z_1=\tau$, $$v_\pi(i,\tau) = E[F_\omega(z)|\omega_1=i,z_1=\tau,z_{k+1} = \pi(\omega_k,z_k,\omega_{k+1})].$$ There is an optimal policy $\pi^*$ that minimizes $v_\pi(i,\tau)$ for every $i \in V$ and $\tau \in L$. Let $\hat{\varphi}$ be the fixed point of $S$. Then $\hat{\varphi}_i(\tau) = v_{\pi^*}(i,\tau)$. This follows directly from the connection between $S$ and the MDP described in the last section. Bounding the Value Functions of $F$ ----------------------------------- Now we show that $\hat \varphi$ defines lower bounds on the value functions (max-marginals) $f_i$ defined in (\[eq:f\]). We start by showing that $f_i$ can be lower bounded by $f_j$ for $j \in N(i)$. \[pr:flowerbound\] Let $i \in V$ and $j \in N(i)$. $$\begin{aligned} f_i(u_i) & \ge pg_i(u_i) + \min_{u_j} ph_{ij}(u_i,u_j) + qf_j(u_j), \\ f_i(u_i) & \ge pg_i(u_i) + \sum_{j \in N(i)} w_{ij} \min_{u_j} ph_{ij}(u_i,u_j) + qf_j(u_j). \end{aligned}$$ The second inequality follows from the first one by taking a convex combination over $j \in N(i)$. To prove the first inequality note that, $$\begin{aligned} f_i(u_i) & = \min_{\substack{x \in L^V\\ x_i=u_i}} F(x) \\ & = \min_{u_j \in L} \min_{\substack{x \in L^V\\ x_i=u_i, x_j=u_j}} F(x) \\ & = \min_{u_j \in L} \min_{\substack{x \in L^V\\ x_i=u_i, x_j=u_j}} pF(x)+qF(x) \\ & \ge pg_i(u_i) + \min_{u_j \in L} ph_{ij}(u_i,u_j) + \min_{\substack{x \in L^V\\ x_i=u_i, x_j=u_j}} qF(x) \\ & \ge pg_i(x_i) + \min_{u_j \in L} ph_{ij}(u_i,u_j) + \min_{\substack{x \in L^V\\ x_j=u_j}} qF(x) \\ & = pg_i(x_i) + \min_{u_j \in L} ph_{ij}(u_i,u_j) + qf_j(u_j). \end{aligned}$$ The first inequality above follows from $F(x) \ge g_i(x_i) + h_{ij}(x_i,x_j)$ since all the terms in $F(x)$ are non-negative. The second inequality follows from the fact that we are minimizing $F(x)$ over $x$ with fewer restrictions. The map $S$ and the value functions are related as follows. \[pr:Sf\] Let $f = (f_1,\ldots,f_N) \in {({\mathbb{R}}^L)^V}$ be a field of beliefs defined by the value functions. $$\begin{aligned} Sf \le f. \end{aligned}$$ The result follows directly from Proposition \[pr:flowerbound\]. Now we show that the fixed point of $S$ defines lower bounds on the value functions. Let $\hat{\varphi}$ be the fixed point of $S$. Then $$0 \le \hat{\varphi}_i(\tau) \le f_i(\tau).$$ Since the costs $g_i$ and $h_{ij}$ are non-negative we have $0 \le S0$. Using the fact that $S$ preserves order we can conclude $0 \le \hat{\varphi}$. Since $Sf\leq f$ and $S$ preserves order, $S^kf\leq f$ for all $k$. To end the proof, take the limit $k\to \infty$ at the left hand-side of this inequality. Numerical Experiments ===================== In this section we illustrate the practical feasibility of the proposed algorithms with preliminary experiments in computer vision problems. We also evaluate the fixed point algorithms defined by $S$ and $T$ and other methods on random binary classification problems on a grid. Image Restoration ----------------- The goal of image restoration is to estimate a clean image $z$ from a noisy, or corrupted, version $y$. A classical approach to solve this problem involves looking for a piecewise smooth image $x$ that is similar to $y$ [@GG84; @BZ87]. In the weak membrane model [@BZ87] the local costs $g_i(a)$ penalize differences between $x$ and $y$ while the pairwise costs $h_{ij}(a,b)$ penalize differences between neighboring pixels in $x$. In this setting, the graph $G=(V,E)$ is a grid in which the vertices $V$ correspond to pixels and the edges $E$ connect neighboring pixels. The labels $L$ are possible pixel values and a labeling $x$ defines an image. For our experiments we use $L=\{0,\ldots,255\}$ corresponding to the possible values in an 8-bit image. To restore $y$ we define the energy $F(x)$ using $$\begin{aligned} g_i(x_i) &= (y_i-x_i)^2; \\ h_{ij}(x_i,x_j) &= \lambda \min((x_i-x_j)^2,\tau).\end{aligned}$$ The local cost $g_i(x_i)$ encourages $x_i$ to be similar to $y_i$. The pairwise costs depend on two parameters $\lambda,\tau \in {\mathbb{R}}$. The cost $h_{ij}(x_i,x_j)$ encourages $x_i$ to be similar to $x_j$ but also allows for large differences since the cost is bounded by $\tau$. The value of $\lambda$ controls the relative weight of the local and pairwise costs. Small values of $\lambda$ lead to images $x$ that are very similar to the noisy image $y$, while large values of $\lambda$ lead to images $x$ that are smoother. Figure \[fig:restore\] shows an example result of image restoration using the algorithm defined by $T$. The example illustrates the algorithm is able to recover a clean image that is smooth almost everywhere while at the same time preserving sharp discontinuities at the boundaries of objects. For comparison we also show the results of belief propagation. In this example the noisy image $y$ was obtained from a clean image $z$ by adding independent noise to each pixel using a Gaussian distribution with standard deviation $\sigma=20$. The input image has 122 by 179 pixels. We used $\lambda = 0.05$ and $\tau = 100$ to define the pairwise costs. For the algorithm defined by $T$ we used uniform weights, $w_{ij} = 1/{\mathrm{d}}(i)$ and $p = 0.001$. Both the algorithms defined by $T$ and belief propagation were run for 100 iterations. We based our implementations on the belief propagation code from [@FH06], which provides efficient methods for handling truncated quadratic discontinuity costs. The algorithm defined by $T$ took 16 seconds on a 1.6Ghz Intel Core i5 laptop computer while belief propagation took 18 seconds. ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Image restoration using the fixed point algorithm defined by $T$ and BP. The algorithms were run for 100 iterations.[]{data-label="fig:restore"}](results-revision/penguin-truth "fig:"){width="1.5in"} ![Image restoration using the fixed point algorithm defined by $T$ and BP. The algorithms were run for 100 iterations.[]{data-label="fig:restore"}](results-revision/penguin-noisy "fig:"){width="1.5in"} ![Image restoration using the fixed point algorithm defined by $T$ and BP. The algorithms were run for 100 iterations.[]{data-label="fig:restore"}](results-revision/penguin-outT "fig:"){width="1.5in"} ![Image restoration using the fixed point algorithm defined by $T$ and BP. The algorithms were run for 100 iterations.[]{data-label="fig:restore"}](results-revision/penguin-outbp "fig:"){width="1.5in"} Original Image Noisy Image Output of $T$ Output of BP RMS error = 8.9 RMS error = 10.7 Energy = 1519837 Energy = 650296 ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The goal of restoration is to recover a clean image $z$. We evaluate the restored image $x$ by computing the root mean squared error (RMSE) between $x$ and $z$. We see in Figure \[fig:restore\] that when $\lambda=0.05$ and $\tau = 100$ the result of $T$ has lower RMSE value compared to the result of BP, even though the result of $T$ has significantly higher energy. We also evaluate the results of $T$, $S$ and BP using different values of $\lambda$ in Table \[tb:restoration\]. For all of these experiments we used $\tau = 100$ and ran each algorithm for 100 iterations. The minimum RMSE obtained by $T$ and $S$ is lower than the minimum RMSE obtained by BP considering different values for $\lambda$, even though $T$ and $S$ alwayd find solutions that have higher energy compared to BP. This suggests the algorithms we propose do a good job aggregating local information using pairwise constraints, but the energy minimization problem defined by $F(x)$ may not be the ideal formulation of the restoration problem. ----------- --------- ------ --------- ------ --------- ------ $\lambda$ Energy RMSE Energy RMSE Energy RMSE 0.01 659210 10.1 842459 9.1 211646 15.8 0.02 943508 9.0 1220572 8.7 337785 13.4 0.05 1519837 8.9 1873560 10.9 650296 10.7 0.10 2089415 11.0 2506230 14.8 1080976 10.1 0.20 2700193 14.8 2942392 17.7 1730132 12.9 ----------- --------- ------ --------- ------ --------- ------ : Results of restoration using $T$, $S$ and belief propagation (BP). The goal of restoration is to recover the original image $z$. We show the energy of the restored image $x$ and the root mean squared error (RMSE) between $x$ and $z$. We show the results of the different algorithms for different values of the parameter $\lambda$. Both $T$ and $S$ obtain lower RMSE compared to BP even though BP generally obtains results with significantly lower energy.[]{data-label="tb:restoration"} Stereo Depth Estimation {#sec:stereo} ----------------------- In stereo matching we have two images $I_l$ and $I_r$ taken at the same time from different viewpoints. Most pixels in one image have a corresponding pixel in the other image, being the projection of the same three-dimensional point. The difference in the coordinates of corresponding pixels is called the disparity. We assume the images are rectified such that a pixel $(x,y)$ in $I_l$ matches a pixel $(x-d,y)$ in $I_r$ with $d \ge 0$. For rectified images the distance of a three-dimensional point to the image plane is inversely proportional to the disparity. In practice we consider the problem of labeling every pixel in $I_l$ with an integer disparity in $L=\{0,\ldots,D\}$. In this case a labeling $x$ is a disparity map for $I_l$. The local costs $g_i(a)$ encourage pixels in $I_l$ to be matched to pixels of similar color in $I_r$. The pairwise costs $h_{ij}(a,b)$ encourage piecewise smooth disparity maps. The model we used in our stereo experiment is defined by, $$\begin{aligned} g_{i}(a) &= \min(\gamma, ||I_l(i)-I_r(i-(a,0))||_1); \\ h_{ij}(a,b) &= \begin{cases} 0 \qquad a = b, \\ \alpha \qquad |a-b| = 1, \\ \beta \qquad |a-b| > 1. \end{cases}\end{aligned}$$ Here $I_l(i)$ is the value of pixel $i$ in $I_l$ while $I_r(i-(a,0))$ is the value of the corresponding pixel in $I_r$ assuming a disparity $a$ for $i$. The $\ell_1$ norm $||I_l(i)-I_r(i-(a,0))||_1$ defines a distance between RGB values (matching pixels should have similar color). The color distance is truncated by $\gamma$ to allow for some large color differences which occur due to specular reflections and occlusions. The pairwise costs depend on two parameter $\alpha, \beta \in \mathbb{R}$ with $\alpha < \beta$. The pairwise costs encourage the disparity neighboring pixels to be similar or differ by 1 (to allow for slanted surfaces), but also allows for large discontinuities which occur at object boundaries. Figure \[fig:stereo\] shows an example result of disparity estimation using the fixed point algorithm defined by $S$. In this example we used non-uniform weights $w_{ij}$ to emphasize the relationships between neighboring pixels of similar color, since those pixels are most likely to belong to the same object/surface. The parameters we used for the results in Figure \[fig:stereo\] were defined by, $$\begin{aligned} w_{ij} \propto 0.01 + e^{-0.2 ||I_l(i)-I_l(j)||_1},\end{aligned}$$ $p = 0.0001$, $\alpha = 500$, $\beta = 1000$ and $\gamma = 20$. The input image has 384 by 288 pixels and the maximum disparity is $D=15$. The fixed point algorithm was run for 1,000 iterations which took 13 seconds on a laptop computer. [cc]{} ![Stereo disparity estimation using the fixed point algorithm defined by $S$ on the Tsukuba image pair. The algorithm was run for 1,000 iterations.[]{data-label="fig:stereo"}](results/tsukuba1.png "fig:"){width="2.5in"} & ![Stereo disparity estimation using the fixed point algorithm defined by $S$ on the Tsukuba image pair. The algorithm was run for 1,000 iterations.[]{data-label="fig:stereo"}](results/tsukuba2.png "fig:"){width="2.5in"}\ $I_l$ & $I_r$\ \ ![Stereo disparity estimation using the fixed point algorithm defined by $S$ on the Tsukuba image pair. The algorithm was run for 1,000 iterations.[]{data-label="fig:stereo"}](results/tsukuba-truth.png "fig:"){width="2.5in"} & ![Stereo disparity estimation using the fixed point algorithm defined by $S$ on the Tsukuba image pair. The algorithm was run for 1,000 iterations.[]{data-label="fig:stereo"}](results/tsukuba-out.png "fig:"){width="2.5in"}\ Ground truth & Result of $S$\ We note that the results in Figure \[fig:stereo\] are similar to results obtained min-sum belief propagation shown in [@FH06]. Binary Classification on a Grid ------------------------------- Let $G=(V,E)$ be a $K$ by $K$ grid and $L = \{-1,+1\}$. We can define a classification problem on $G$ using energy functions of the form, $$F(x) = \sum_{i \in V} \alpha_i x_i + \sum_{\{i,j\} \in E} \beta_{ij} x_i x_j.$$ To evaluate different algorithms we generated random classification problems by independently selecting each $\alpha_i$ from a uniform distribution over $[-1,+1]$ and each $\beta_{ij}$ from a uniform distribution over $[-\lambda,+\lambda]$. The parameter $\lambda$ controls the relative strength of the pairwise relations and the local information associated with each vertex. Let $x^*$ be the global minimum of $F(x)$. When $K=10$ we can compute $x^*$ using dynamic programming over the columns of $G$. In our experiments we quantify the quality of a potential solution $x$ using two different measures: (1) the value of $F(x)$, and (2) the Hamming distance between $x$ and $x^*$. Table \[tb:glass10\] shows the results of various algorithms we evaluated on 20 random problem instances. We compare the results of the following algorithms: - Dynamic programming over the columns of $G$. Dynamic programming finds the global minimum of $F(x)$ but has a runtime that is exponential in $K$. - Iterative conditional modes [@Besag86]. This is a simple local search technique that considers changing the label of a single vertex at a time. - The fixed point algorithm defined by $S$ with $p=0.01$ and $w_{ij} = 1/{\mathrm{d}}(i)$. - The fixed point algorithm defined by $T$ with $p=0.01$ and $w_{ij} = 1/{\mathrm{d}}(i)$. - Loopy belief propagation with a damping factor of $0.5$. - Sequential tree-reweighted message passing [@K06]. - Adaptive Diminishing Smoothing [@SSKS12]. This algorithm is based on the local polytope LP relaxation of optimization problem defined by $F$. All of the algorithms were implemented in C++. For BP, TRWS and ADSAL we used the implementation available in OpenGM2 [@opengm2]. We also consider the result of applying local-search (ICM) to the output of each algorithm. All iterative algorithms were run for 1000 iterations. We found that post-processing solutions of different algorithms with local-search (ICM) often gives a substantial improvement. In particular the solutions found by the algorithms defined by $S$ and $T$ are not very good in terms of their energy value when compared to the alternatives. However, the solutions found by $S$ and $T$ improve in energy substantially after post-processing with ICM. We also found that lower energy solutions are sometimes further from the global minimum, in terms of Hamming distance, when compared to higher energy solutions. For example, on average, the energy of the solutions found by BP+ICM is lower the energy of the solutions found by T, but the solutions found by T are closer to the global minimum. After post-processing with ICM the solutions found by $S$ and $T$ and are often closer to the global optimum when compared to the alternatives. [|l|rr|rr|]{}\ Algorithm & &\ & Mean & $\sigma$ & Mean & $\sigma$\ DP & -727.133222 & 34.035102 & 0.000000 & 0.000000\ ICM & -621.519900 & 43.942530 & 47.750000 & 15.112495\ S & -389.791776 & 81.379435 & 37.750000 & 6.744442\ S + ICM & -650.013870 & 36.927557 & 35.600000 & 11.275637\ T & -401.615127 & 68.749218 & 37.950000 & 6.924413\ T + ICM & -650.204712 & 40.796138 & 36.050000 & 10.883359\ BP & -595.403707 & 79.575241 & 42.350000 & 18.224366\ BP + ICM & -672.816406 & 46.895083 & 42.150000 & 18.850133\ TRWS & -660.259499 & 37.781695 & 42.650000 & 20.857313\ TRWS + ICM & -689.407548 & 39.445723 & 41.100000 & 22.666936\ ADSAL & -667.080187 & 36.400576 & 35.700000 & 19.657314\ ADSAL + ICM & -693.317195 & 36.631941 & 34.650000 & 21.036338\ In terms of running time the algorithms defined by $S$ and $T$ are more efficient than the alternatives. Each iteration using $S$ or $T$ requires updating a belief *for each vertex* of $G$. In contrast, each BP iteration requires updating *two* messages *for each edge* of $G$. Other message passing methods like TRWS are similar to BP. Both $S$ and $T$ require less memory than these other algorithms because they work with beliefs associated with vertices instead of messages associated with edges. Moreover, we find that the fixed point algorithms defined by $S$ and $T$ converge faster than the alternatives. Table \[tb:convergence\] compares the results of different algorithms when using a limited number of iterations. The ADSAL algorithm solves a sequence of problems using TRWS as a subroutine. In this case the number of iterations refers to the iterations of the TRWS subroutine. Algorithm 100 Iterations 1000 Iterations ------------- ---------------- ----------------- S + ICM 35.6 35.6 T + ICM 36.0 36.0 BP + ICM 45.9 42.1 TRWS + ICM 41.6 41.1 ADSAL + ICM 37.3 34.6 : Comparison of different algorithms when using a limited number of iterations, in terms of the mean Hamming distance between the solution found and the global optimum $x^*$.[]{data-label="tb:convergence"} Conclusion and Future Work ========================== The experimental results in the last section illustrate the practical feasibility of the algorithms under study. Our theoretical results prove these algorithms are guaranteed to converge to unique fixed points on graphs with arbitrary topology and with arbitrary pairwise relationships. This includes the case of repulsive interactions which often leads to convergence problems for message passing methods. Our results can be extended to other contraction maps similar to $T$ and $S$ and alternative methods for computing the fixed points of these maps. Some specific directions for future work are as follows. 1. *Asynchronous updates*. It is possible to define algorithms that update the beliefs of a single vertex at a time in any order. As long as all vertices are updated infinitely many times, the resulting algorithms converge to the same fixed point as the parallel update methods examined in this work. We conjecture that in a *sequential* computation, the sequential update of vertices in a “sweep” would converge faster than a “parallel” update. Moreover, after a sequential update of all vertices, the neighbors of those vertices with greater change should be the first ones to be updated in the next “sweep”. 2. *Non-backtracking random walks*. The algorithms defined by $S$ and $T$ can be understood in terms of random walks on $G$. It is possible to define alternative algorithms based on non-backtracking random walks. In particular, starting with the MDP in Section \[sec:S\] we can increase the state-space $Q$ to keep track of the last vertex visited in the walk and define transition probabilities that avoid the previous vertex when selecting the next one. The resulting value iteration algorithm becomes very similar to belief propagation and other message passing methods that involve messages defined on the edges of $G$. Acknoledgements {#acknoledgements .unnumbered} =============== We thank the anonymous reviewers for many helpful comments and suggestions. [^1]: Partially supported by NSF under grant 1161282 and the Brown-IMPA collaboration program. [^2]: Partially supported by CNPq grants 474996/2013-1, 302962/2011-5 and FAPERJ grant E-26/201.584/2014.

ArXiv

--- abstract: | The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. Especially, for the weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in the presence of a sharp barrier. In contrast to the non-adiabatic case, the ET kinetics can be, however considerably influenced by the medium dynamics.\ In this paper, the problem of the escape time over a dichotomously fluctuating cusp barrier is discussed with its relevance to the high temperature ET reactions in condensed media. author: - 'Bart[ł]{}omiej' - Ewa - 'Pawe[ł]{} F.' title: Implication of Barrier Fluctuations on the Rate of Weakly Adiabatic Electron Transfer --- Introduction ============ Mechanism of the electron transfer (ET) in condensed and biological media goes beyond universal nonadiabatic approach of the Marcus theory.[@Marcus1; @Marcus2; @Ulstrup; @Kuznetsov; @Chandler; @Makarov] In particular, relaxation properties of medium may slow down the overall ET kinetics and lead to an adiabatic dynamics.[@Hynes] An excess electron appearing in the medium introduces local fluctuations of polarization, that in turn contribute to the change of Gibbs energy. Equilibration of those fluctuations leads to a new state with a localized position of a charge. In chemical reactions, the electron may change its location passing from a donoring to an accepting molecule, giving rise to the same scenario of Gibbs energy changes that allows to discriminate between the (equilibrium) states “before” and “after” the transfer (see Fig. \[et\_co\]). The free energy surfaces for “reactants” and “products” are usually multidimensional functions which intersect at the transition point. The deviation from it, or the Gibbs energy change, can be calculated from the reversible work done along the path that forms that state, so that by use of a simple thermodynamic argument, one is able to associate a change in the Gibbs energy with the change of multicomponent “reaction coordinate” that describes a response of the system to the instantaneous transfer of a charge from one site to another. ![\[et\_co\] Schematic energy profiles of reactant ([**R**]{}) and product ([**P**]{}) states of the electron transfer reaction coordinate. The reorganization energy $E_r$ is the sum of the reaction energy $\Delta E$ and the optical excitation energy $E^*$. $\Delta$ stands for energy separation between the energy surfaces due to electronic coupling of [**R**]{} and [**P**]{} states. Thermal ET occurs at nuclear configurations characteristic to the intersection of the parabolas.](rys_et.eps){width="8.5cm" height="8.5cm"} ET reactions involve both classical and quantum degrees of freedom. Quantum effects are mostly related to electronic degrees of freedom. Because of the mass difference between electrons and nuclei, it is frequently assumed that the electrons follow the nuclear motion adiabatically (Born-Oppenheimer approximation). The interaction between two different electronic states results in a splitting energy $\Delta$. The reaction from reactants to products is then mediated by an interplay of two parameters: time of charge fluctuations between the two neighbouring electronic states and a typical time within which the nuclear reaction coordinate crosses the barrier region. When the electronic “uncertainty” time of charge fluctuations is shorter than the nuclear dynamics, the transition is adiabatic with the overall dynamics evolving on an adiabatic ground-state surface. For small splitting between electronic states $\Delta\approx 0.1-1$, this ground-state adiabatic surface is often characterized by a cusp-shaped potential.[@Marcus1; @Ulstrup; @Kuznetsov; @Hynes] In fact, it is often argued[@Hynes] that majority of natural ET reactions are in what we term weakly adiabatic regime, when the reaction is still adiabatic but the barrier is quite sharp and characterized by the barrier frequency $\omega_a$ roughly by an order of magnitude higher than the medium relaxation frequency $\omega_0$. The dynamics of the reaction coordinate in either of the potential wells can be estimated by use of a generalized Langevin equation with a friction term mimicking dielectric response of the medium. We are thus left with a standard model of a Brownian particle in a (generally time-dependent) medium.[@Hynes; @Hynes1] In the case of a cusp-shaped potential, a particle approaching the top of a barrier with positive velocity will almost surely be pulled towards the other minimum. The kinetic rate is then determined[@Hynes; @Hanggi] by the reciprocal mean first passage time (MFPT) to cross the barrier wall with positive velocity $\dot{x}>0$ and in a leading order in the barrier height $\Delta E/k_BT$ yields the standard transition state theory (TST) result $$k^{cusp}=\frac{\omega_0}{2\pi}\exp(-\Delta E/k_BT). \label{cusp}$$ As discussed by Talkner and Braun,[@Talk] the result holds also for non-Markovian processes with memory friction satisfying the fluctuation-dissipation theorem. The TST formula follows from the Kramers rate for the spatial-diffusion-controlled rate of escape at moderate and strong frictions $\eta$ $$k_{R \rightarrow P}=\frac{(\eta^2/4+\omega_a^2)^{1/2}-\eta/2}{\omega_a}\frac{\omega_0}{2\pi} \exp(-\Delta E/k_BT).$$ Here $\omega_a$ stays for the positive-valued angular frequency of the unstable state at the barrier, and $\omega_0$ is an angular frequency of the metastable state at $x=R$. For strong friction, $\eta>>\omega_a$ the above formula leads to a common Kramers result $$k_{R \rightarrow P}=\frac{\omega_0 \omega_a}{2\pi\eta}\exp(-\Delta E/k_BT),$$ and reproduces the TST result eq. (\[cusp\]) after letting the barrier frequency tend to infinity $\omega_a\rightarrow\infty$ with $\eta$ held fixed. Moreover, as pointed out by Northrup and Hynes,[@Hynes1] in a weakly adiabatic case, the barrier “point” is a negligible fraction of the high energy barrier region, so that the rate can be influenced by medium relaxation in the wells. The full rate constant for a symmetric reaction is then postulated in the form $$k_{WA}=\left(1+2k^a_{WA}/k_D\right)^{-1}k_{WA}^a$$ where $k_{WA}^a\approx k^{cusp}$ and $k_D$ is the well (solvent or medium) relaxation rate constant which for a harmonic potential and a high barrier ($\Delta E\ge 5k_BT$) within 10% of accuracy simplifies to $$k_D=\frac{m_0 \omega_0^2 D}{k_BT}\left (\frac{\Delta E}{\pi k_BT}\right)^{1/2} e^{-\Delta E/k_BT}.$$ In the above equation, the diffusion constant $D$ is related[@Ulstrup] [*via*]{} linear response theory to the longitudinal dielectric relaxation time $\tau_L$ and for symmetric ET reaction ($m_0^2\omega_0^2=2E_r$) reads $$D=\frac{k_BT}{m_0\omega_0^2\tau_L}=\frac{k_BT}{2E_r \tau_L}.$$ Existence of a well defined rate constant for a chemical reaction requires that the relaxation time of all the degrees of freedom involved in the transformation, other than the reaction coordinate, is fast relative to motion along the reaction coordinate. If the separation of time scales were not present, the rate coefficient would have a significant frequency dependence reflecting various modes of relaxation. Such a situation can be expected in complex media, like non-Debeye solvents or proteins, where there are many different types of degrees of freedom with different scales of relaxation. Although in these cases the rate “constant” can no longer be defined, the overall electron transfer can be described in terms of the mean escape time that takes into account noisy character of a potential surface along with thermal fluctuations. The time effect of the surroundings (“environmental noises”), expressed by different time constants for polarization and the dielectric reorganization has not been so far explored in detail, except in photochemical reaction centers[@Chandler] where molecular dynamics studies have shown that the slow components of the energy gap fluctuations are, most likely, responsible for the observed nonexponential kinetics of the primary ET process. The latter will be assumed here to influence the activation free energy of the reaction $\Delta E=E_r/4$ and will be envisioned as a barrier alternating processes. In consequence, even small variations $\delta E$ in $\Delta E$ can greatly modulate the escape kinetics (passage from reactants’ to products’ state) in the system. If the barrier fluctuates extremely slowly, the mean first passage time to the top of the barrier is dominated by those realizations for which the barrier starts in a higher position and thus becomes very long. The barrier is then essentially quasistatic throughout the process. At the other extreme, in the case of rapidly fluctuating barrier, the mean first passage time is determined by the “average barrier”. For some particular correlation time of the barrier fluctuations, it can happen however, that the mean kinetic rate of the process exhibits an extremum[@doe; @iwa; @bork] that is a signature of resonant tuning of the system in response to the external noise. In this contribution we present analytical and numerical results for the escape kinetics over a fluctuating cusp for the high temperature model ET system with a harmonic potential subject to dichotomous fluctuations. We perform our investigations of the average escape time as a function of the correlation rate of the dichotomous noise in the barrier height at fixed temperatures. In particular, we examine variability of the mean first passage time for different types of barrier switching[@doe; @zurcher; @boguna] when the barrier changes either between the “on-off” position, flips between a barrier or a well, or it varies between two different heights. By use of a Monte Carlo procedure we determine probability density function (pdf) of escape time in the system and investigate the degree of nonexponential behavior in the decay of a primary state located in the reactants’ well. Generic Model System ==================== At high temperatures it is permissible to treat the low frequency medium modes classically. The medium coordinates are continuous and it is useful to draw a one-dimensional schematic representation of the system (Fig. \[et\_co\]) with the reaction proceeding almost exclusively at the intersection energy. As a model of the reaction coordinate kinetics, we have considered an overdamped Brownian particle moving in a potential field between absorbing and reflecting boundaries in the presence of noise that modulates the barrier height. The evolution of the reaction coordinate $x(t)$ is described in terms of the Langevin equation $$\frac{dx}{dt}=-V'(x)+\sqrt{2T}\xi(t)+g(x)\eta(t)= -V'_{\pm}(x)+\sqrt{2T}\xi(t). \label{lang}$$ Here $\xi(t)$ is a Gaussian process with zero mean and correlation $<\xi(t)\xi(s)>=\delta(t-s)$ ([*i.e.*]{} the Gaussian white noise arising from the heat bath of temperature $T$), $\eta(t)$ stands for a dichotomous (not necessarily symmetric) noise taking on one of two possible values $a_\pm$ and prime means differentiation over $x$. Correlation time of the dichotomous process has been set up to ${1\over 2\gamma}$ with $\gamma$ expressing the flipping frequency of the barrier fluctuations. Both noises are assumed to be statistically independent, [*i.e.*]{} $<\xi(t)\eta(s)>=0$. Equivalent to eq. (\[lang\]) is a set of the Fokker-Planck equations describing evolution of the probability density of finding the particle at time $t$ at the position $x$ subject to the force $-V'_{\pm}(x)=-V'(x)+a_{\pm}g(x)$ $$\begin{aligned} \partial_t {P}(x,a_\pm,t)& =& \partial_x \left[V'_{\pm}(x)+T\partial_x \right]P(x,a_\pm,t) \nonumber \\ & -& \gamma P(x,a_\pm,t)+\gamma P(x,a_\mp,t). \label{schmidr}\end{aligned}$$ In the above equations time has dimension of $[\mathrm{length}]^2/\mathrm{energy}$ due to a friction constant that has been “absorbed” in the time variable. We are assuming absorbing boundary condition at $x=L$ and a reflecting boundary at $x=0$ $$P(L,a_\pm,t)=0, \label{bon0}$$ $$\left[V'_{\pm}(x) +T\partial_x\right]P(x,a_\pm,t)|_{x=0}=0. \label{bon}$$ The initial condition $$P(x,a_+,0)=P(x,a_-,0)=\frac{1}{2}\delta(x)$$ expresses equal choice to start with any of the two configurations of the barrier. The quantity of interest is the mean first passage time $$\begin{aligned} \mathrm{MFPT} & =& \int\limits_0^\infty dt\int\limits_0^L\left[P(x,a_+,t)+P(x,a_-,t)\right]dx \nonumber \\ & = & \tau_+(0)+\tau_-(0)\end{aligned}$$ with $\tau_+$ and $\tau_-$ being MFPT for $(+)$ and $(-)$ configurations, respectively. MFPTs $\tau_+$ and $\tau_-$ fulfill the set of backward Kolmogorov equations[@bork] $$-{1\over 2}=-\gamma\tau_\pm (x)+\gamma\tau_\mp(x)-{dV_\pm (x)\over dx}{d\tau_\pm (x)\over dx}+T{d^2\tau_\pm (x)\over dx^2} \label{mr_uklad}$$ with the boundary conditions ([*cf.*]{} eq. (\[bon0\]) and (\[bon\])) $$\tau'_{\pm}(x)|_{x=0}=0, \qquad \tau_{\pm}(x)|_{x=L}=0.$$ Although the solution of (\[mr\_uklad\]) is usually unique,[@molenaar] a closed, “ready to use” analytical formula for the MFPT can be obtained only for the simplest cases of the potentials (piecewise linear). More complex cases, like even piecewise parabolic potential $V_\pm$ result in an intricate form of the solution to eq. (\[mr\_uklad\]). Other situations require either use of approximation schemes,[@rei] perturbative approach[@iwa] or direct numerical evaluation methods.[@rec; @gam] In order to examine MFPT for various potentials a modified program[@musn] applying general shooting methods has been used. Part of the mathematical software has been obtained from the [*Netlib*]{} library. Solution and Results ==================== Equivalent to equation (\[mr\_uklad\]) is a set of equations $$\left[\begin{array}{c} {du(x)\over dx}\\ {dv(x)\over dx}\\ {dp(x)\over dx}\\ {dq(x)\over dx}\\ \end{array}\right]= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & {{d\over dx}\left[V_+(x)+V_-(x)\right]\over 2T} & {{d\over dx}\left[V_+(x)-V_-(x)\right]\over 2T} \\ 0 & {2\gamma\over T} & {{d\over dx}\left[V_+(x)-V_-(x)\right]\over 2T} & {{d\over dx}\left[V_+(x)+V_-(x)\right]\over 2T}\\ \end{array}\right] \left[\begin{array}{c} u(x)\\ v(x)\\ p(x)\\ q(x)\\ \end{array}\right]+ \left[\begin{array}{c} 0\\ 0\\ -{1\over T}\\ 0\\ \end{array}\right], \label{muklad}$$ where new variables have been introduced $$\left\{\begin{array}{c} u(x)=\tau_+(x)+\tau_-(x) \\ v(x)=\tau_+(x)-\tau_-(x) \end{array}\right., \qquad \left\{\begin{array}{l} {du(x)\over dx}=p(x) \\ {dv(x)\over dx}=q(x) \end{array}\right..$$ Since $u$ does not enter the right-hand side of any of the above equations, the system can be further converted to $$\left[\begin{array}{c} {dv(x)\over dx}\\ {dp(x)\over dx}\\ {dq(x)\over dx}\\ \end{array}\right]= \left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & {{d\over dx}\left[V_+(x)+V_-(x)\right]\over 2T} & {{d\over dx}\left[V_+(x)-V_-(x)\right]\over 2T} \\ {2\gamma\over T} & {{d\over dx}\left[V_+(x)-V_-(x)\right]\over 2T} & {{d\over dx}\left[V_+(x)+V_-(x)\right]\over 2T}\\ \end{array}\right] \left[\begin{array}{c} v(x)\\ p(x)\\ q(x)\\ \end{array}\right]+ \left[\begin{array}{r} 0\\ -1/T\\ 0\\ \end{array}\right] \label{mr_6}$$ [*i.e.*]{} it has a form of $${d\vec{f}(x)\over dx}=\hat{A}(x)\vec{f}(x)+\vec{\beta}(x). \label{vecf}$$ A unique solution to (\[vecf\]) exists[@molenaar] and reads $$\begin{aligned} \vec{f}(x)&=&\exp\left\{\int\limits_0^x\hat{A}(x')dx'\right\}\vec{f}(0) \nonumber \\ &+& \exp\left\{\int\limits_0^x\hat{A}(x')dx'\right\}\int\limits_0^x \exp\left\{-\int\limits_0^{x'}\hat{A}(x'')dx''\right\} \vec{\beta}(x')dx' \nonumber \\ &=& {\bf{A}}(x)\vec{f}(0)+{\bf{A}}(x)\vec{B}(x) = {\bf{A}}(x)\vec{f}(0)+\vec{C}(x) \label{rozw}\end{aligned}$$ with boundary conditions leading to $$\vec{f}(0)= \left[ \begin{array}{c} \tau_+-\tau_- \\ 0 \\ 0 \\ \end{array} \right], \qquad \vec{f}(L)= \left[ \begin{array}{c} 0 \\ p(L) \\ q(L) \\ \end{array} \right]. \label{abs}$$ MFPT is the quantity of interest $$\tau=\tau(0)=u(0),$$ which can be obtained from $$p(x)={du(x)\over dx},$$ and $$\int\limits_0^Lp(x)dx=u(L)-u(0)=0-u(0)=-u(0),$$ with $$u(0)=\tau=-\int\limits_0^Lp(x)dx. \label{mr_7}$$ For the parabolic potential $V_+(x)=-V_-(x)={Hx^2\over L^2}\equiv 2E_r x^2$ the above procedure leads to $$\hat{A}(x)=\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & {2Hx\over L^2T}\\ {2\gamma\over T} & {2Hx\over L^2T}& 0\\\end{array}\right], \qquad \int\limits_0^x\hat{A}(x)dx=x\left[\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & {Hx\over L^2T}\\ {2\gamma\over T} & {Hx\over L^2T}& 0\\\end{array}\right],$$ and $$\tau=\int\limits_0^L\int\limits_0^L\Phi(x,y)dxdy,$$ where $$\begin{aligned} \Phi(x,y)& =& \left[-{L^2H\over 2}{x[\varphi(x)-2]\over\rho(x)} -{LH[\varphi(L)-2]\over4[H^2+\gamma TL^2\varphi(L)]}{4\gamma L^4T+H^2x^2\varphi(x)\over \rho(x)} \right. \nonumber \\ & + &\left. {\sqrt{\rho(L)}\xi(L)H \over 4[H^2+\gamma TL^2\varphi(L)]}{x\xi(x)\over\sqrt{\rho(x)}}\right] \times H\gamma L^2{y[\varphi(y)-2]\over\rho(y)} \nonumber \nonumber \\ % & + & \left[{H^2L^4\gamma y[\varphi(y)-2]\over 2\rho(y)}{x[\varphi(x)-2]\over \rho(x)} - {H^2y\xi(y)\over 4T\sqrt{\rho(y)}}{x\xi(x)\over\sqrt{\rho(x)}} \right. \nonumber \\ & + & \left.{4\gamma L^4T+H^2y^2\varphi(y)\over 4T\rho(y)} {4\gamma L^4T+H^2x^2\varphi(x)\over \rho(x)} \right] \times\theta(y-x),\end{aligned}$$ $$\rho(x)=H^2x^2+2\gamma L^4T,$$ $$\varphi(x)=\exp\left[{{\sqrt{\rho(x)}x\over L^2T}}\right]+\exp\left[{-{\sqrt{\rho(x)}x\over L^2T}}\right]=2\cosh\left[{{\sqrt{\rho(x)}x\over L^2T}}\right],$$ $$\xi(x)=\exp\left[{{\sqrt{\rho(x)}x\over L^2T}}\right]-\exp\left[{-{\sqrt{\rho(x)}x\over L^2T}}\right]=2\sinh\left[{{\sqrt{\rho(x)}x\over L^2T}}\right].$$ ![\[flat\] MFPT as a function of the correlation rate of the dichotomous noise for parabolic potential barriers switching between different heights $H_{\pm}$. Full lines: analytical results; symbols stay for the results from MC simulations with $\Delta t=10^{-5}$ and ensemble of $N=10^4$ trajectories. For simplicity, the parametrization $L=T=1$ has been used.](flat.eps){width="8.5cm" height="8.5cm"} Fig. \[flat\] displays calculated MFPT as a function of switching frequency $\gamma$. Analytical solutions are presented along with results from Monte Carlo simulations of eq. (\[lang\]). As a choice for various configurations of the potential, we have probed $H_{\pm}=\pm 8T; H_+=8T, H_-=0$ and $H_+=8T, H_-=4T$ that set up reorganization energies ($2E_r=H$) and heights of barrier in the problem of interest. The distinctive characteristics of resonant activation is observed with the average escape time initially decreasing, reaching a minimum value, and then increasing as a function of switching frequency $\gamma$. At slow dynamics of the barrier height, [*i.e.*]{} for values of the rate $\gamma$ less than $\tau_+^{-1}$, the average escape time approaches the value $\tau=(\tau_--\tau_+)/2$ predicted by theory[@doe; @boguna; @rei; @bier] and observed in experimental investigations of resonant activation.[@Mantegna] For fast dynamics of the barrier height the average escape time reaches the value associated with an effective potential characterized by an average barrier. In comparison to the “on-off” switching of the barrier, the region of resonant activation flattens for dichotomic flipping between the barrier and a well, and in the case of the Bier-Astumian model ($H_+=8T,H_-=4T$) when the barrier changes its height between two different values. The resonant frequency shifts from the lowest value for the Bier-Astumian model to higher values for the “on-off” and the “barrier-well” scenarios, respectively. This observation is in agreement with former studies[@boguna] aimed to discriminate between characteristic features of resonant activation for models with “up-down” configurations of the barrier and models with the “up” configuration but fluctuating between different heights. The “up-down” switching of the barrier heights produces shorter MFPT and in consequence, higher value of crossing rates for resonant frequencies than two other models of barrier switching. For each of the above situations we have evaluated probability density function for first escape times. Pdfs have been obtained as a result of MC simulations on $N=10^4$ trajectories by use of histograms and kernel density estimation methods. In the resonant activation regime, pdf of escape times has an exponential slope, that suggests that the reactants’ population follows preferentially the kinetics through the state with the lowest barrier. Similarly, the exponential decay times of reactants’ are observed in the high frequency limit ($\gamma\approx 10^9$), when the system experiences an effective potential with an average barrier.[@doe; @zurcher; @boguna; @rei; @gam; @Mantegna; @reihan; @pechukas] Apparent nonexponential decay of the initial population is observed at low frequencies ($\gamma\approx 10^{-6}$), when the flipping rate becomes less than $\tau_-^{-1}$. ![\[scal\]Semilog plot of the pdf for first escape times in the system. The differences between lines fitted to the slopes reflect various $\tau_-$ values for a given type of barrier switching: ($+$) for $H_+=8T,\;H_-=-8T$; ($\times$) for $H_+=8T,\;H_-=0$ and ($\ast$) for $H_+=8T,\;H_-=4T$ ([*cf.*]{} Table 1).](low.eps){width="8.5cm" height="8.5cm"} As it is clearly demonstrated in Fig. \[scal\] and summarized in Table 1, passage times at low frequencies are roughly characterized by two distinct time scales that correspond to $\tau_-$ (different for various switching barrier scenarios - [*cf.*]{} first three rows of Table 1) and $\tau_+$ (bottom three rows of Table 1), respectively. The inset of Fig \[scal\] shows pdf zoomed for part of the main plot, where the differences between various, model dependent $\tau_-$ values can be well distinguished. $H_+$ $H_-$ fitted value static barrier value ----------- ------- ------- ---------------- ---------------------- 8T -8T $0.09\pm0.01$ 0.12 $\tau_- $ 8T 0 $0.46\pm0.01$ 0.50 8T 4T $4.51\pm0.05$ 3.43 8T -8T $68.73\pm0.89$ 63.01 $\tau_+$ 8T 0 $71.33\pm3.04$ 63.01 8T 4T $79.11\pm1.98$ 63.01 : Relaxation time for parabolic potential barriers switching between different heights $H_\pm$ at low frequencies. \[tablel\] Summary ======= In the foregoing sections we have considered the thermally activated process that can describe classical ET kinetics. The regime of dynamical disorder where fluctuations of the environment can interplay with the time scale of the reaction itself is not so well understood. The examples where such a physical situation can happen are common to nonequilibrium chemistry and, in particular to ET reactions[@Chandler] in photosynthesis. As a toy model for describing the nonexponential ET kinetics[@Chandler] we have chosen a generic system displaying the resonant activation phenomenon. The reaction coordinate has been coupled to an external noise source that can describe polarization and depolarization processes responsible for the height of barrier between the reactants and products wells. We have assumed that the driving forces for the ET process interconverse at a rate $\gamma$ reflecting dynamic changes in the transition state. The best tuning of the system and its highest ET rate can be achieved within the resonant frequency region. On the other hand, nonexponential ET kinetics can be attributed to long, time-persisting correlations in barrier configuration that effectively change a Poissonian character of escape events demonstrating, in general, multiscale time-decay of initial population. This project has been partially supported by the Marian Smoluchowski Institute of Physics, Jagellonian University research grant (E.G-N). The contribution is Authors’ dedication to 50th birthday anniversary of Prof. Jerzy Łuczka. [10]{} , Biochim. Biophys. Acta **811**, 265 (1985). , J. Phys. Chem. **98**, 7170 (1994). , [*Charge Transfer in Condensed Media*]{} (Springer Verlag, Berlin, 1979). , Eds. [*Advances in Chemical Physics: Electron Transfer – From Isolated Molecules to Biomolecules, Vol. 106*]{} (Wiley, New York, 1999). , [Science]{} **263**, 499 (1994). , [Proc. Natl. Acad. Sci. USA]{} **93**, 3926 (1996). , J. Phys. Chem. **90**, 3701 (1986). , J. Chem. Phys. **73**, 2700 (1980). , [Rev. Mod. Phys.]{} **62**, 251 (1990). , J. Chem. Phys. **88**, 7357 (1988). , Phys. Rev. Lett. **69**, 2318 (1992). J. Iwaniszewski, Phys. Rev. E **54**, 3173 (1996). , Phys. Rev. Lett. **73**, 2772 (1994). , Phys. Rev. E **47**, 3862 (1993). , Phys. Rev. E **57**, 3990 (1998). , [*Ordinary Differential Equations in Theory and Practice*]{} (John Wiley and Sons, Chichester, 1996). , Chem. Phys. **235**, 11 (1998). , [*Numerical Recipes. The art of scientific computing*]{} (Cambridge University Press, Cambridge, 1992). , Rev. Mod. Phys. **70**, 223 (1998). , [*MUS.F program for solving general two point boundary problems*]{} (http://www.netlib.org). , Phys. Rev. Lett. **72**, 1766 (1994). , Phys. Rev. Lett. **84**, 3025 (2000). , in [*Stochastic Dynamics, Lecture Notes in Physics, Vol. 484,*]{} ed. L. Schimansky Geier and T. Pöschel (Springer Verlag, Berlin, 1997), p. 127-139. , Phys. Rev. E **66**, 026123 (2002).

ArXiv

--- abstract: 'A critique of the singularity theorems of Penrose, Hawking, and Geroch is given. It is pointed out that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator that can accelerate particles to energies inconceivable in any terrestrial particle accelerator, and that when the energy $E$ of the particles comprising matter in a black hole is $\sim 10^{2} GeV$ or more, or equivalently, the temperature $T$ is $\sim 10^{15} K$ or more, the entire matter in the black hole is converted into quark-gluon plasma permeated by leptons. As quarks and leptons are fermions, it is emphasized that the collapse of a black-hole to a space-time singularity is inhibited by Pauli’s exclusion principle. It is also suggested that ultimately a black hole may end up either as a stable quark star, or as a pulsating quark star which may be a source of gravitational radiation, or it may simply explode with a mini bang of a sort.' author: - 'R. K. Thakur' date: 'Received: date / Accepted: date' title: 'Can a Black Hole Collapse to a Space-time Singularity? ' --- [example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore Introduction ============ When all the thermonuclear sources of energy of a star are exhausted, the core of the star begins to contract gravitationally because, practically, there is no radiation pressure to arrest the contraction, the pressure of matter being inadequate for this purpose. If the mass of the core is less than the Chandrasekhar limit ($\sim 1.44 \msol$), the contraction stops when the density of matter in the core, $\rho > 2 \times 10^{6} \gcmcui$; at this stage the pressure of the relativistically degenerate electron gas in the core is enough to withstand the force of gravitation. When this happens, the core becomes a stable white dwarf. However, when the mass of the core is greater than the Chandrasekhar limit, the pressure of the relativistically degenerate electron gas is no longer sufficient to arrest the gravitational contraction, the core continues to contract and becomes denser and denser; and when the density reaches the value $\rho \sim 10^{7}\gcmcui$, the process of neutronization sets in; electrons and protons in the core begin to combine into neutrons through the reaction $p + e^{-} \rightarrow n + \nu_{e}$ The electron neutrinos $\nu_{e}$ so produced escape from the core of the star. The gravitational contraction continues and eventually, when the density of the core reaches the value $\rho \sim 10^{14} \gcmcui$, the core consists almost entirely of neutrons. If the mass of the core is less than the Oppenheimer-Volkoff limit ($\sim 3\msol$), then at this stage the contraction stops; the pressure of the degenerate neutron gas is enough to withstand the gravitational force. When this happens, the core becomes a stable neutron star. Of course, enough electrons and protons must remain in the neutron star so that Pauli’s exclusion principle prevents neutron beta decay $n \rightarrow p + e^{-} + \overline \nu_{e}$ Where $\overline \nu_{e}$ is the electron antineutrino (Weinberg 1972a). This requirement sets a lower limit $\sim 0.2 \msol$ on the mass of a stable neutron star.\ If, however, after the end of the thermonuclear evolution, the mass of the core of a star is greater than the Chandrasekhar and Oppenheimer-Volkoff limit, the star may eject enough matter so that the mass of the core drops below the Chandrasekhar and Oppenheimer-Volkoff limit as a result of which it may settle as a stable white dwarf or a stable neutron star. If not, the core will gravitationally collapse and end up as a black hole. As is well known, the event horizon of a black hole of mass $M$ is a spherical surface located at a distance $r = r_{g} = 2GM/c^{2}$ from the centre, where $G$ is Newton’s gravitational constant and $c$ the speed of light in vacuum; $r_{g}$ is called gravitational radius or Schwarzschild radius. An external observer cannot observe anything that is happening inside the event horizon, nothing, not even light or any other electromagnetic signal can escape outside the event horizon from inside. However, anything that enters the event horizon from outside is swallowed by the black hole; it can never escape outside the event horizon again. Attempts have been made, using the general theory of relativity (GTR), to understand what happens inside a black hole. In so doing, various simplifying assumptions have been made. In the simplest treatment (Oppenheimer and Snyder 1939; Weinberg 1972b) a black hole is considered to be a ball of dust with negligible pressure, uniform density $\rho = \rho(t)$, and at rest at $t=0$. These assumptions lead to the unique solution of the Einstein field equations, and in the comoving co-ordinates the metric inside the black hole is given by $$\begin{aligned} ds^2 = dt^2 -R^2(t)\bbs \frac{dr^2}{1-k\,r^2} + r^2 d\theta^2 + r^2\sin^2\theta\,d\phi^2\ebs\end{aligned}$$ in units in which speed of light in vacuum, c=1, and where $k$ is a constant. The requirement of energy conservation implies that $\rho(t)R^3(t)$ remains constant. On normalizing the radial co-ordinate $r$ so that $$\begin{aligned} R(0) = 1\end{aligned}$$ one gets $$\begin{aligned} \rho(t) = \rho(0)R^{-3}(t)\end{aligned}$$ The fluid is assumed to be at rest at $t=0$, so $$\begin{aligned} \dot{R}(0) = 0\end{aligned}$$ Consequently, the field equations give $$\begin{aligned} k = \frac{8\pi\,G}{3} \rho(0)\end{aligned}$$ Finally, the solution of the field equations is given by the parametric equations of a cycloid : $$\begin{aligned} \nonumber t = \bb \frac{\psi + \sin\,\psi}{2\sqrt{k}} \eb \\ R = \frac{1}{2} \bb 1 + \cos\,\psi \eb\end{aligned}$$ From equation $(6)$ it is obvious that when $\psi = \pi $. i.e., when $$\begin{aligned} t = t_{s} = \frac{\pi}{2\sqrt{k}} = \frac{\pi}{2} \bb \frac{3}{8\pi\,G \rho(0)} \eb^{1/2}\end{aligned}$$ a space-time singularity occurs; the scale factor $R(t)$ vanishes. In other words, a black hole of uniform density having the initial values $\rho(0)$, and zero pressure collapses from rest to a point in $3$ - subspace, i.e., to a $3$ - subspace of infinite curvature and zero proper volume, in a finite time $t_{s}$; the collapsed state being a state of infinite proper energy density. The same result is obtained in the Newtonian collapse of a ball of dust under the same set of assumptions (Narlikar 1978). Although the black hole collapses completely to a point at a finite co-ordinate time $t=t_{s}$, any electromagnetic signal coming to an observer on the earth from the surface of the collapsing star before it crosses its event horizon will be delayed by its gravitational field, so an observer on the earth will not see the star suddenly vanish. Actually, the collapse to the Schwarzschild radius $r_{g}$ appears to an outside observer to take an infinite time, and the collapse to $R=0$ is not at all observable from outside the event horizon. The internal dynamics of a non-idealized, real black hole is very complex. Even in the case of a spherically symmetric collapsing black hole with non-zero pressure the details of the interior dynamics are not well understood, though major advances in the understanding of the interior dynamics are now being made by means of numerical computations and analytic analyses. But in these computations and analyses no new features have emerged beyond those that occur in the simple uniform-density, free-fall collapse considered above (Misner,Thorne, and Wheeler 1973). However, using topological methods, Penrose (1965,1969), Hawking (1996a, 1966b, 1967a, 1967b), Hawking and Penrose (1970), and Geroch (1966, 1967, 1968) have proved a number of singularity theorems purporting that if an object contracts to dimensions smaller than $r_{g}$, and if other reasonable conditions - namely, validity of the GTR, positivity of energy, ubiquity of matter and causality - are satisfied, its collapse to a singularity is inevitable. A critique of the singularity theorems ====================================== As mentioned above, the singularity theorems are based, inter alia, on the assumption that the GTR is universally valid. But the question is : Has the validity of the GTR been established experimentally in the case of strong fields ? Actually, the GTR has been experimentally verified only in the limiting case of week fields, it has not been experimentally validated in the case of strong fields. Moreover, it has been demonstrated that when curvatures exceed the critical value $C_{g} = 1/L_{g}^4$, where $L_{g} = \bb \hbar\,G/c^{3} \eb^{1/2} = 1.6 \times 10^{-33} \cm $ corresponding to the critical density $\rho_{g} = 5 \times 10^{93} \gcmcui$, the GTR is no longer valid; quantum effects must enter the picture (Zeldovich and Novikov 1971). Therefore, it is clear that the GTR breaks down before a gravitationally collapsing object collapses to a singularity. Consequently, the conclusion based on the GTR that in comoving co-ordinates any gravitationally collapsing object in general, and a black hole in particular, collapses to a point in 3-space need not be held sacrosanct, as a matter of fact it may not be correct at all. Furthermore, while arriving at the singularity theorems attention has mostly been focused on the space-time geometry and geometrodynamics; matter has been tacitly treated as a classical entity. However, as will be shown later, this is not justified; quantum mechanical behavior of matter at high energies and high densities must be taken into account. Even if we regard matter as a classical entity of a sort, it can be easily seen that the collapse of a black hole to a space-time singularity is inhibited by Pauli’s exclusion principle. As mentioned earlier, a collapsing black hole consists, almost entirely, of neutrons apart from traces of protons and electrons; and neutrons as well as protons and electrons are fermions; they obey Pauli’s exclusion principle. If a black hole collapses to a point in 3-space, all the neutrons in the black hole would be squeezed into just two quantum states available at that point, one for spin up and the other for spin down neutron. This would violate Pauli’s exclusion principle, according to which not more than one fermion of a given species can occupy any quantum state. So would be the case with the protons and the electrons in the black hole. Consequently, a black hole cannot collapse to a space-time singularity in contravention to Pauli’s exclusion principle. Besides, another valid question is : What happens to a black hole after $t > t_{s}$, i.e., after it has collapsed to a point in 3-space to a state of infinite proper energy density, if at all such a collapse occurs? Will it remain frozen forever at that point? If yes, then uncertainties in the position co-ordinates of each of the particles - namely, neutrons, protons, and electrons - comprising the black hole would be zero. Consequently, according to Heisenberg’s uncertainty principle, uncertainties in the momentum co-ordinates of each of the particles would be infinite. However, it is physically inconceivable how particles of infinite momentum and energy would remain frozen forever at a point. From this consideration also collapse of a black hole to a singularity appears to be quite unlikely. Earlier, it was suggested by the author that the very strong ’hard-core’ repulsive interaction between nucleons, which has a range $l_{c} \sim 0.4 \times 10^{-13} \cm $, might set a limit on the gravitational collapse of a black hole and avert its collapse to a singularity (Thakur 1983). The existence of this hard-core interaction was pointed out by Jastro (1951) after the analysis of the data from high energy nucleon-nucleon scattering experiments. It has been shown that this very strong short range repulsive interaction arises due to the exchange of isoscalar vector mesons $\omega$ and $\phi$ between two nucleons ( Scotti and Wong 1965). Phenomenologically, that part of the nucleon-nucleon potential which corresponds to the repulsive hard core interaction may be taken as $$\begin{aligned} V_{c}(r) = \infty~~~~~~~~~for~~~ r < l_{c}\end{aligned}$$ where $r$ is the distance between the two interacting nucleons. Taking this into account, the author concluded that no spherical object of mass M could collapse to a sphere of radius smaller than $R_{min} = 1.68 \times 10^{-6} M^{1/3} \cm$, or of the density greater than $\rho_{max} = 5.0 \times 10^{16} \gcmcui$. It was also pointed out that an object of mass smaller than $M_{c} \sim 1.21 \times 10^{33} \gm$ could not cross the event horizon and become a black hole; the only course left to an object of mass smaller than $M_{c}$ was to reach equilibrium as either a white dwarf or a neutron star. However, one may not regard these conclusions as reliable because they are based on the hard core repulsive interaction (8) between nucleons which has been arrived at phenomenologically by high energy nuclear physicists while accounting for the high energy nucleon-nucleon scattering data; but it must be noted that, as mentioned above, the existence of the hard core interaction has been demonstrated theoretically also by Scotti and Wong in 1965. Moreover, it is interesting to note that the upper limit $M_{c} \sim 1.21 \times 10^{33} \g = 0.69 \msol$ on the masses of objects that cannot gravitationally collapse to form black holes is of the same order of magnitude as the Chandrasekhar and the Oppenheimer- Volkoff limits. Even if we disregard the role of the hard core, short range repulsive interaction in arresting the collapse of a black hole to a space-time singularity in comoving co-ordinates, it must be noted that unlike leptons which appear to be point-like particles - the experimental upper bound on their radii being $10^{-16} \cm$ (Barber 1979) -nucleons have finite dimensions. It has been experimentally demonstrated that the radius $r_0$ of the proton is about $10^{-13} \cm$(Hofstadter & McAllister 1955). Therefore, it is natural to assume that the radius $r_0$ of the neutron is also about $10^{-13} \cm$. This means the minimum volume $v_{min}$ occupied by a neutron is $\frac{4\pi}{3}{r_0}^3$. Ignoring the “mass defect” arising from the release of energy during the gravitational contraction (before crossing the event horizon), the number of neutrons $N$ in a collapsing black hole of mass $M$ is, obviously, $\frac{M}{m_{n}}$ where $m_{n}$ is the mass of the neutron. Assuming that neutrons are impregnable particles, the minimum volume that the black hole can occupy is $V_{min} = Nv_{min} = v_{min} \frac{M}{m_{n}} $, for neutrons cannot be more closely packed than this in a black hole. However, $V_{min} = \frac{4\pi R_{min}^3}{3}$ where $R_{min}$ is the radius of the minimum volume to which the black hole can collapse. Consequently, $R_{min} = r_{0} {\bb\frac{M}{m_{n}}\eb}^{1/3}$. On substituting $10^{-13} \cm$ for $r_{0}$ and $1.67 \times 10^{-24} \g$ for $m_n$ one finds that $R_{min} = 8.40 \times 10^{-6} M^{1/3}$. This means a collapsing black hole cannot collapse to a density greater than $\rho_{max} = \frac{M}{V_{min}} = \frac{Nm_{n}}{4/3 \pi r_{0}^{3} N} = 3.99 \times 10^{14} \gcmcui$. The critical mass $M_{c}$ of the object for which the gravitational radius $R_{g} = R_{min}$ is obtained from the equation $$\begin{aligned} \frac{2GM_{c}}{c^{2}} = r_{0} \bb \frac{M_{c}}{m_{n}} \eb^{1/3}\end{aligned}$$ This gives $$\begin{aligned} M_{c} = 1.35 \times 10^{34} \g = 8.68 \msol\end{aligned}$$ Obviously, for $M > M_{c}, R_{g} > R_{min} $, and for $M < M_{c}, R_{g} < R_{min} $. Consequently, objects of mass $M < M_{c}$ cannot cross the event horizon and become a black hole whereas those of mass $M > M_{c}$ can. Objects of mass $M < M_{c}$ will, depending on their mass, reach equilibrium as either white dwarfs or neutron stars. Of course, these conclusions are based on the assumption that neutrons are impregnable particles and have radius $r_{0} = 10^{-13} cm $ each. Also implicit is the assumption that neutrons are [ *fundamental* ]{} particles; they are not composite particles made up of other smaller constituents. But this assumption is not correct; neutrons as well as protons and other hadrons are [*not fundamental*]{} particles; they are made up of smaller constituents called [*quarks* ]{} as will be explained in section 4. In section 5 it will be shown how, at ultrahigh energy and ultrahigh density, the entire matter in a collapsing black hole is eventually converted into quark-gluon plasma permeated by leptons. Gravitationally collapsing black hole as a particle accelerator =============================================================== We consider a gravitationally collapsing black hole. On neglecting mutual interactions the energy E of any one of the particles comprising the black hole is given by $E^2 = p^2 + m^2 > p^2$, in units in which the speed of light in vacuum $c= 1$, where $p$ is the magnitude of the 3-momentum of the particle and $m$ its rest mass. But $p = \frac{h}{\lambda}$, where $\lambda$ is the de Broglie wavelength of the particle and $h$ Planck’s constant of action. Since all lengths in the collapsing black hole scale down in proportion to the scale factor $R(t)$ in equation $(1)$, it is obvious that $\lambda \propto R(t) $. Therefore it follows that $p \propto R^{-1}(t)$, and hence $p = a\,R^{-1}(t)$, where [*a*]{} is the constant of proportionality. From this it follows that $E > a/R$. Consequently, $E$ as well as $p$ increases continually as R decreases. It is also obvious that $E$ and $p$, the magnitude of the 3-momentum, $\rightarrow \infty$ as $R \rightarrow 0$. Thus, in effect, we have an ultra-high energy particle accelerator, [*so far inconceivable in any terrestrial laboratory*]{}, in the form of a collapsing black hole, which can, in the absence of any physical process inhibiting the collapse, accelerate particles to an arbitrarily high energy and momentum without any limit. What has been concluded above can also be demonstrated alternatively, without resorting to GTR, as follows. As an object collapses under its selfgravitation, the interparticle distance $s$ between any pair of particles in the object decreases. Obviously, the de Broglie’s wavelength $\lambda$ of any particle in the object is less than or equal to $s$, a simple consequence of Heisenberg’s uncertainty principle. Therefore, $s \geq h/p $, where $h$ is Planck’s constant and $p$ the magnitude of 3-momentum of the particle. Consequently, $p \geq h/s $ and hence $E \geq h/s $. Since during the collapse of the object $s$ decreases, the energy $E$ as well as the momentum $p$ of each of the particles in the object increases. Moreover, from $E \geq h/s $ and $p \geq h/s $ it follows that $E$ and $p \rightarrow \infty$ as $ s \rightarrow 0$. Thus, any gravitationally collapsing object in general, and a black hole in particular, acts as an ultrahigh energy particle accelerator. It is also obvious that $\rho$, the density of matter in the black hole, increases as it collapses. In fact, $\rho \propto R^{-3} $, and hence $\rho \rightarrow \infty$ as $R \rightarrow 0$. Quarks: The building blocks of matter ===================================== In order to understand eventually what happens to matter in a collapsing black hole one has to take into account the microscopic behavior of matter at high energies and high densities; one has to consider the role played by the electromagnetic, weak, and strong interactions - apart from the gravitational interaction - between the particles comprising the matter. For a brief account of this the reader is referred to Thakur(1995), for greater detail to Huang(1992), or at a more elementary level to Hughes(1991). As has been mentioned in Section 2, unlike leptons, hadrons are not point-like particles, but are of finite size; they have structures which have been revealed in experiments that probe hadronic structures by means of electromagnetic and weak interactions. The discovery of a very large number of [*apparently elementary (fundamental)*]{} hadrons led to the search for a pattern amongst them with a view to understanding their nature. This resulted in attempts to group together hadrons having the same baryon number, spin, and parity but different strangeness $S$ ( or equivalently hypercharge $Y = B + S$, where $B$ is the baryon number) into I-spin (isospin) multiplets. In a plot of $Y$ against $I_{3}$ (z- component of isospin I), members of I-spin multiplets are represented by points. The existence of several such hadron (baryon and meson) multiplets is a manifestation of underlying internal symmetries. In 1961 Gell-Mann, and independently Neémann, pointed out that each of these multiplets can be looked upon as the realization of an irreducible representation of an internal symmetry group $SU(3)$ ( Gell-Mann and Neémann 1964). This fact together with the fact that hadrons have finite size and inner structure led Gell-Mann, and independently Zweig, in 1964 to hypothesize that hadrons [*are not elementary particles*]{}, rather they are composed of more elementary constituents called [*quarks ($q$)*]{} by Gell-Mann (Zweig called them [*aces*]{}). Baryons are composed of three quarks ($q\,q\,q$) and antibaryons of three antiquarks ($\overline q\,\overline q\, \overline q$) while mesons are composed of a quark and an antiquark each. In the beginning, to account for the multiplets of baryons and mesons, quarks of only three flavours, namely, u(up), d (down), and s(strange) were postulated, and they together formed the basic triplet $\left( \begin{array}{c}u\\d\\ s \end{array} \right)$ of the internal symmetry group $SU(3)$. All these three quarks u, d, and s have spin 1/2 and baryon number 1/3. The u quark has charge $2/3\,e$ whereas the d and s quarks have charge $-1/3\,e$ where $e$ is the charge of the proton. The strangeness quantum number of the u and d quarks is zero whereas that of the s quark is -1. The antiquarks ($\overline u\,,\overline d\,,\overline s$) have charges $-2/3\,e, 1/3\,e,1/3\,e$ and strangeness quantum numbers 0, 0, 1 respectively. They all have spin 1/2 and baryon number -1/3. Both u and d quarks have the same mass, namely, one third that of the nucleon, i.e., $\simeq 310 MeV/c^2 $ whereas the mass of the $s$ quark is $\simeq 500 MeV/c^2$. The proton is composed of two up and one down quarks (p: uud) and the neutron of one up and two down quarks (n: udd). Motivated by certain theoretical considerations Glashow, Iliopoulos and Maiani (1970) proposed that, in addition to $u$, $d$, $s$ quarks, there should be another quark flavour which they named [*charm*]{} $(c)$. Gaillard and Lee (1974) estimated its mass to be $\simeq 1.5 GeV/c^{2}$. In 1974 two teams, one led by S.C.C. Ting at SLAC (Aubert 1974) and another led by B. Richter at Brookhaven (Augustin 1974) independently discovered the $J/\Psi$, a particle remarkable in that its mass ($3.1 GeV/c^{2}$) is more than three times that of the proton. Since then, four more particles of the same family, namely, $\psi (3684)$, $\psi (3950)$, $\psi (4150)$, $\psi (4400)$ have been found. It is now established that these particles are bound states of [*charmonium*]{} ($\overline c c$), $J/\psi$ being the ground state. On adopting non-relativistic independent quark model with a linear potential between $c$ and $\overline c$, and taking the mass of $c$ to be approximately half the mass of $J/\psi$, $1.5 GeV/c^{2}$, one can account for the $J/\psi$ family of particles. The $c$ has spin $1/2$, charge $2/3$ $e$, baryon number $1/3$, strangeness $-1$, and a new quantum number charm $(c)$ equal to 1. The $u$, $d$, $s$ quarks have $c=0$. It may be pointed out here that charmed mesons and baryons, the bound states like ($c\overline d$), and ($cdu$) have also been found. Thus the existence of the $c$ quark has been established experimentally beyond any shade of doubt. The discovery of the $c$ quark stimulated the search for more new quarks. An additional motivation for such a search was provided by the fact that there are three generations of lepton [*weak*]{} [*doublets*]{}: ${\nu_{e}\choose e}$, ${\nu_{\mu}\choose \mu},$ and ${\nu_{\tau}\choose \tau}$ where $\nu_{e}$, $\nu_{\mu},$ and $\nu_{\tau}$ are electron ($e$), muon ($\mu$), and tau lepton ($\tau$) neutrinos respectively. Hence, by analogy, one expects that there should be three generations of quark [*weak*]{} [*doublets*]{} also: ${u \choose d}$, ${c \choose s}$, and ${? \choose ?}$. It may be mentioned here that weak interaction does not distinguish between the upper and the lower members of each of these doublets. In analogy with the isopin $1/2$ of the [*strong doublet*]{} ${p \choose n}$, the [*weak doublets*]{} are regarded as possessing [*weak isopin*]{} $I_{W} = 1/2$, the third component $(I_{W})_{3}$ of this [*weak isopin*]{} being + 1/2 for the upper components of these doublets and - 1/2 for the lower components. These statements apply to the left-handed quarks and leptons, those with negative helicity (with the spin antiparallel to the momentum) only. The right-handed leptons and quarks, those with positive helicity (with the spin parallel to the momentum), are [*weak singlets*]{} having [*weak isopin*]{} zero. The discovery, at Fermi Laboratory, of a new family of vector mesons, the upsilon family, starting at a mass of $9.4 GeV/c^{2}$ gave an evidence for a new quark flavour called [*bottom*]{} or [*beauty*]{} $(b)$ (Herb 1997; Innes 1977). These vector mesons are in fact, bound states of bottomonium $(\overline b b)$. These states have since been studied in detail at the Cornell electron accelerator in an electron-positron storage ring of energy ideally matched to this mass range. Four such states with masses $9.46, 10.02, 10.35,$ and $10.58$ $GeV/c^{2}$ have been found, the state with mass $9.46 GeV/c^{2}$ being the ground state (Andrews 1980). This implies that the mass of the $b$ quark is $\simeq 4.73 GeV/c^{2}$. The $b$ quark has spin $1/2$ and charge $-1/3\ e$. Furthermore, the $b$ flavoured mesons have been found with exactly the expected properties (Beherend 1983). After the discovery of the $b$ quark, the confidence in the existence of the sixth quark flavour called [*top*]{} or [*truth*]{} $(t)$ increased and it became almost certain that, like leptons, the quarks also occur in three generations of weak isopin doublets, namely, ${u \choose d}$, ${c \choose s}$, and ${t\choose b}$. In view of this, intensive search was made for the $t$ quark. But the discovery of the $t$ quark eluded for eighteen years. However, eventually in 1995, two groups, the CDF (Collider Detector at Fermi lab) Collaboration (Abe 1995) and the $D\phi$ Collaboration (Abachi 1995) succeeded in detecting [*toponium*]{} $\overline t t$ in very high energy $\overline p p$ collisions at Fermi Laboratory’s $1.8 TeV$Tevetron collider. The [*toponium*]{} $\overline t t$ is the bound state of $t$ and $\overline t$. The mass of $t$ has been estimated to be $176.0\pm2.0 GeV/c^{2}$, and thus it is the most massive elementary particle known so far. The $t$ quark has spin $1/2$ and charge $2/3\ e$. Moreover, in order to account for the apparent breaking of the spin-statistics theorem in certain members of the $J^{p}=\frac{3^{+}}{2}$ decuplet (spin 3/2,parity even), , $\bigtriangleup^{++}$ $(uuu)$, and $\Omega^{-}$ $(sss)$, Greenberg $(1964)$ postulated that quark of each flavour comes in three [*colours*]{}, namely, [*red*]{}, [*green*]{}, and [*blue*]{}, and that real particles are always [*colour singlets*]{}. This implies that real particles must contain quarks of all the three colours or colour-anticolour combinations such that they are overall [*white*]{} or [*colourless*]{}. [*White*]{} or [*colourless*]{} means all the three primary colours are equally mixed or there should be a combination of a quark of a given colour and an antiquark of the corresponding anticolour. This means each baryon contains quarks of all the three colours(but not necessarily of the same flavour) whereas a meson contains a quark of a given colour and an antiquark having the corresponding anticolour so that each combination is overall white. Leptons have no colour. Of course, in this context the word ‘colour’ has nothing to do with the actual visual colour, it is just a quantum number specifying a new internal degree of freedom of a quark. The concept of colour plays a fundamental role in accounting for the interaction between quarks. The remarkable success of quantum electrodynamics (QED) in explaining the interaction between electric charges to an extremely high degree of precision motivated physicists to explore a similar theory for strong interaction. The result is quantum chromodynamics (QCD), a non-Abelian gauge theory (Yang-Mills theory), which closely parallels QED. Drawing analogy from electrodynamics, Nambu (1966) postulated that the three quark colours are the charges (the Yang-Mills charges) responsible for the force between quarks just as electric charges are responsible for the electromagnetic force between charged particles. The analogue of the rule that like charges repel and unlike charges attract each other is the rule that like colours repel, and colour and anticolour attract each other. Apart from this, there is another rule in QCD which states that different colours attract if the quantum state is antisymmetric, and repel if it is symmetric under exchange of quarks. An important consequence of this is that if we take three possible pairs, red-green. green-blue, and blue-red, then a third quark is attracted only if its colour is different and if the quantum state of the resulting combination is antisymmetric under the exchange of a pair of quarks thus resulting in red-green-blue baryons. Another consequence of this rule is that a fourth quark is repelled by one quark of the same colour and attracted by two of different colours in a baryon but only in antisymmetric combinations. This introduces a factor of 1/2 in the attractive component and as such the overall force is zero, i.e., the fourth quark is neither attracted nor repelled by a combination of red-green-blue quarks. In spite of the fact that hadrons are overall colourless, they feel a residual strong force due to their coloured constituents. It was soon realized that if the three colours are to serve as the Yang-Mills charges, each quark flavour must transform as a triplet of $SU_{c}(3)$ that causes transitions between quarks of the same flavour but of different colours ( the SU(3) mentioned earlier causes transitions between quarks of different flavours and hence may more appropriately be denoted by $SU_{f}(3)$). However, the $SU_{c}(3)$ Yang-Mills theory requires the introduction of eight new spin 1 gauge bosons called [*gluons*]{}. Moreover, it is reasonable to stipulate that the gluons couple to [*left-handed*]{} and [*right-handed*]{} quarks in the same manner since the strong interactions do not violate the law of conservation of parity. Just as the force between electric charges arise due to the exchange of a photon, a massless vector (spin 1) boson, the force between coloured quarks arises due to the exchange of a gluon. Gluons are also massless vector (spin 1) bosons. A quark may change its colour by emitting a gluon. For example, a [*red*]{} quark $q_{R}$ may change to a blue quark $q_{B}$ by emitting a gluon which may be thought to have taken away the [ *red (R) colour* ]{} from the quark and given it the [*blue (B)*]{} colour, or, equivalently, the gluon may be thought to have taken away the [*red (R)*]{} and the [*antiblue ($\overline B$)*]{} colours from the quark. Consequently, the [*gluon $G_{RB}$*]{} emitted in the process $q_{R} \rightarrow q_{B}$ may be regarded as the composite having the [*colour R $\overline B$* ]{} so that the emitted gluon $G_{RB} = q_{R}\overline q_{B}$. In general, when a quark $q_{i}$ of [*colour i*]{} changes to a quark $q_{j}$ of [*colour j*]{} by emitting a gluon $G_{ij}$, then $G_{ij}$ is the composite state of $q_{i}$ and $\overline q_{j}$, i.e., $G_{ij} = q_{i} \overline q_{j}$. Since there are three [*colours*]{} and three[*anticolours*]{}, there are $3 \times 3 = 9$ possible combinations ([*gluons*]{})of the form $G_{ij} = q_{i} \overline q_{j}$. However, one of the nine combinations is a special combination corresponding to the [*white colour*]{}, namely, $G_{W} = q_{R} \overline q_{R} = q_{G} \overline q_{G} = q_{B} \overline q_{B}$. But there is no interaction between a [*coloured*]{} object and a [*white (colourless)*]{} object. Consequently, gluon $G_{W}$ may be thought not to exist. This leads to the conclusion that only $9 - 1 = 8$ kinds of gluons exist. This is a heuristic explanation of the fact that $SU_{c}(3)$ Yang-Mills gauge theory requires the existence of eight gauge bosons, i.e., the gluons. Moreover, as the gluons themselves carry colour, gluons may also emit gluons. Another important consequence of gluons possessing colour is that several gluons may come together and form [*gluonium*]{} or [*glue balls*]{}. Glueballs have integral spin and no colour and as such they belong to the meson family. Though the actual existence of quarks has been indirectly confirmed by experiments that probe hardronic structure by means of electromagnetic and weak interactions, and by the production of various quarkonia ($\overline q q $) in high energy collisions made possible by various particle accelerators, no [*free*]{} quark has been detected in experiments at these accelerators so far. This fact has been attributed to the [*infrared slavery*]{} of quarks, i.e., to the nature of the interaction between quarks responsible for their [*confinement*]{} inside hadrons. Perhaps enormous amount of energy , much more than what is available in the existing terrestrial accelerators, is required to liberate the quarks from confinement. This means the force of attraction between quarks increases with increase in their separation. This is reminiscent of the force between two bodies connected by an elastic string. On the contrary, the results of deep inelastic scattering experiments reveal an altogether different feature of the interaction between quarks. If one examines quarks at very short distances ($ < 10^{-13}$ cm ) by observing the scattering of a nonhadronic probe, e.g., an electron or a neutrino, one finds that quarks move almost freely inside baryons and mesons as though they are not bound at all. This phenomenon is called the [ *asymptotic freedom*]{} of quarks. In fact Gross and Wilczek (1973 a,b) and Politzer (1973) have shown that the running coupling constant of interaction between two quarks vanishes in the limit of infinite momentum (or equivalently in the limit of zero separation). Eventually what happens to matter in a collapsing black hole? ============================================================= As mentioned in Section 3 the energy $E$ of the particles comprising the matter in a collapsing black hole continually increases and so does the density $\rho$ of the matter whereas the separation $s$ between any pair of particles decreases. During the continual collapse of the black hole a stage will be reached when $E$ and $\rho$ will be so large and $s$ so small that the quarks confined in the hadrons will be liberated from the [*infrared slavery*]{} and will enjoy [*asymptotic freedom*]{}, i.e., the quark [*deconfinement*]{} will occur. In fact, it has been shown that when the energy $E$ of the particle $\sim 10^{2}$ GeV ($s \sim 10^{-16}$ cm) corresponding to a temperature $T \sim 10^{15} K$ all interactions are of the Yang-Mills type with $SU_{c}(3) \times SU_{I_W}(2) \times U_{Y_W}(1)$ gauge symmetry, where c stands for colour, $I_{W}$ for weak isospin, and $Y_{W}$ for weak hypercharge, and at this stage quark deconfinement occurs as a result of which matter now consists of its fundamental constituents : spin 1/2 leptons, namely, the electrons, the muons, the tau leptons, and their neutrinos, which interact only through the electroweak interaction(i.e., the unified electromagnetic and weak interactions); and the spin 1/2 quarks, u, d, s, c, b, t, which interact eletroweakly as well as through the colour force generated by gluons(Ramond, 1983). In other words, when $E \geq 10^{2}$ GeV ($s \leq 10^{-16}$ cm) corresponding to $T \geq 10^{15} K$, the entire matter in the collapsing black hole will be in the form of qurak-gluon plasma permeated by leptons as suggested by the author earlier (Thakur 1993). Incidentally, it may be mentioned that efforts are being made to create quark-gluon plasma in terrestrial laboratories. A report released by CERN, the European Organization for Nuclear Research, at Geneva, on February 10, 2000, said that by smashing together lead ions at CERN’s accelerator at temperatures 100,000 times as hot as the Sun’s centre, i.e., at $T \sim 1.5 \times 10^{12} K$, and energy densities never before reached in laboratory experiments, a team of 350 scientists from institutes in 20 countries succeeded in isolating tiny components called quarks from more complex particles such as protons and neutrons. “A series of experiments using CERN’s lead beam have presented compelling evidence for the existence of a new state of matter 20 times denser than nuclear matter, in which quarks instead of being bound up into more complex particles such as protons and neutrons, are liberated to roam freely ” the report said. However, the evidence of the creation of quark gluon plasma at CERN is indirect, involving detection of particles produced when the quark-gluon plasma changes back to hadrons. The production of these particles can be explained alternatively without having to have quark-gluon plasma. Therefore, Ulrich Heinz at CERN is of the opinion that the evidence of the creation of quark-gluon plasma at CERN is not enough and conclusive. In view of this, CERN will start a new experiment, ALICE, soon (around 2007-2008) at its Large Hadron Collider (LHC) in order to definitively and conclusively creat QGP. In the meantime the focus of research on quark-gluon plasma has shifted to the Relativistic Heavy Ion Collider (RHIC), the worlds newest and largest particle accelerator for nuclear research, at Brookhaven National Laboratory in Upton, New York. RHIC’s goal is to create and study quark-gluon plasma. RHIC’s aim is to create quark-gluon plasma by head-on collisions of two beams of gold ions at energies 10 times those of CERN’s programme, which ought to produce a quark-gluon plasma with higher temperature and longer lifetime thereby allowing much clearer and direct observation. RHIC’s quark-gluon plasma is expected to be well above the transition temperature for transition between the ordinary hadronic matter phase and the quark-gluon plasma phase. This will enable scientists to perform numerous advanced experiments in order to study the properties of the plasma. The programme at RHIC began in the summer of 2000 and after two years Thomas Kirk, Brookhaven’s Associate Laboratory Director for High Energy Nuclear Physics, remarked, “It is too early to say that we have discovered the quark-gulon plasma, but not too early to mark the tantalizing hints of its existence.” Other definitive evidence of quark-gluon plasma will come from experimental comparisons of the behavior in hot, dense nuclear matter with that in cold nuclear matter. In order to accomplish this, the next round of experimental measurements at RHIC will involve collisions between heavy ions and light ions, namely, between gold nuclei and deuterons. Later, on June 18, 2003 a special scientific colloquium was held at Brcokhaven Natioal Laboratory (BNL) to discuss the latest findings at RHIC. At the colloquium, it was announced that in the detector system known as STAR ( Solenoidal Tracker AT RHIC ) head-on collision between two beams of gold nuclei of energies of 130 GeV per nuclei resulted in the phenomenon called “jet quenching“. STAR as well as three other experiments at RHIC viz., PHENIX, BRAHMS, and PHOBOS, detected suppression of “leading particles“, highly energetic individual particles that emerge from nuclear fireballs, in gold-gold collisions. Jet quenching and leading particle suppression are signs of QGP formation. The findings of the STAR experiment were presented at the BNL colloquium by Berkeley Laboratory’s NSD ( Nuclear Science Division ) physicist Peter Jacobs. Collapse of a black hole to a space-time singularity is inhibited by Pauli’s exclusion principle ================================================================================================ As quarks and leptons in the quark-gluon plasma permeated by leptons into which the entire matter in a collapsing black hole is eventually converted are fermions, the collapse of a black hole to a space-time singularity in a finite time in a comoving co-ordinate system, as stipulated by the singularity theorems of Penrose, Hawking and Geroch, is inhibited by Pauli’s exclusion principle. For, if a black hole collapses to a point in 3-space, all the quarks of a given flavour and colour would be squeezed into just two quantum states available at that point, one for spin up and the other for spin down quark of that flavour and colour. This would violate Pauli’s exclusion principle according to which not more than one fermion of a given species can occupy any quantum state. So would be the case with quarks of each distinct combination of colour and flavour as well as with leptons of each species, namely, $e, \mu, \tau, \nu_{e}, \nu_{\mu}$ and $\nu_{\tau}$. Consequently, a black hole cannot collapse to a space-time singularity in contravention to Pauli’s exclusion principle. Then the question arises : If a black hole does not collapse to a space-time singularity, what is its ultimate fate? In section 7 three possibilities have been suggested. Ultimately how does a black hole end up? ======================================== The pressure $P$ inside a black hole is given by $$\begin{aligned} P = P_{r} + \sum_{i,j}P_{ij} + \sum_{k}P_{k} + \sum_{i,j} \overline P_{ij} + \sum_{k} \overline P_{k} \end{aligned}$$ where $P_{r}$ is the radiation pressure, $P_{ij}$ the pressure of the relativistically degenerate quarks of the $i^{th}$ flavour and $j^{th}$ colour, $P_{k}$ the pressure of the relativistically degenerate leptons of the $k^{th}$ species, $\overline P_{ij}$ the pressure of relativistically degenerate antiquarks of the $i^{th}$ flavour and $j^{th}$ colour, $\overline P_{k}$ that of the relativistically degenerate antileptons of the $k^{th}$ species. In equation (11) the summations over $i$ and $j$ extend over all the six flavours and the three colours of quarks, and that over $k$ extend over all the six species of leptons. However, calculation of these pressures are prohibitively difficult for several reasons. For example, the standard methods of statistical mechanics for calculation of pressure and equation of state are applicable when the system is in thermodynamics equilibrium and when its volume is very large, so large that for practical purpose we may treat it as infinite. Obviously, in a gravitationally collapsing black hole, the photon, quark and lepton gases cannot be in thermodynamic equilibrium nor can their volume be treated as infinite. Moreover, at ultrahigh energies and densities, because of the $SU_{I_W}$(2) gauge symmetry, transitions between the upper and lower components of quark and lepton doublets occur very frequently. In addition to this, because of the $SU_{f}$(3) and $SU_{c}$(3) gauge symmetries transitions between quarks of different flavours and colours also occur. Furthermore, pair production and pair annihilation of quarks and leptons create additional complications. Apart from these, various other nuclear reactions may as well occur. Consequently, it is practically impossible to determine the number density and hence the contribution to the overall pressure $P$ inside the black hole by any species of elementary particle in a collapsing black hole when $E \geq 10^2$ Gev ($s \leq 10^{-16}$ cm), or equivalently, $T \geq 10^{15} K$. However, it may not be unreasonable to assume that, during the gravitational collapse, the pressure $P$ inside a black hole increases monotonically with the increase in the density of matter $\rho$. Actually, it might be given by the polytrope, $P = k\rho^{\frac{(n+1)}{n}}$, where $K$ is a constant and $n$ is polytropic index. Consequently, $P \rightarrow \infty$ as $ \rho \rightarrow \infty$, i.e., $P \rightarrow \infty$ as the scale factor $R(t) \rightarrow 0$ (or equivalently $s \rightarrow 0$). In view of this, there are three possible ways in which a black hole may end up. 1\. During the gravitational collapse of a black hole, at a certain stage, the pressure $P$ may be enough to withstand the gravitational force and the object may become gravitationally stable. Since at this stage the object consists entirely of quark-gluon plasma permeated by leptons, it means it would end up as a stable quark star. Indeed, such a possibility seems to exist. Recently, two teams - one led by David Helfand of Columbia University, NewYork (Slane, Helfand, and Murray 2002) and another led by Jeremy Drake of Harvard-Smithsonian Centre for Astrophysics, Cambridge, Mass. USA (Drake 2002) studied independently two objects, 3C58 in Cassiopeia, and RXJ1856.5-3754 in Corona Australis respectively by combining data from the NASA’s Chandra X-ray Observatory and the Hubble Space Telescope, that seemed, at first, to be neutron stars, but, on closer look, each of these objects showed evidence of being an even smaller and denser object, possibly a quark star. 2\. Since the collapse of a black hole is inhibited by Pauli’s exclusion principle, it can collapse only upto a certain minimum radius, say, $r_{min}$. After this, because of the tremendous amount of kinetic energy, it would bounce back and expand, but only upto the event horizon, i.e., upto the gravitational (Schwarzschild ) radius $r_g$ since, according to the GTR, it cannot cross the event horizon. Thereafter it would collapse again upto the radius $r_{min}$ and then bounce back upto the radius $r_g$. This process of collapse upto the radius $r_{min}$ and bounce upto the radius $r_g$ would occur repeatedly. In other words, the black hole would continually pulsate radially between the radii $r_{min}$ and $r_g$ and thus become a pulsating quark star. However, this pulsation would cause periodic variations in the gravitational field outside the event horizon and thus produce gravitational waves which would propagate radially outwards in all directions from just outside the event horizon. In this way the pulsating quark star would act as a source of gravitational waves. The pulsation may take a very long time to damp out since the energy of the quark star (black hole) cannot escape outside the event horizon except via the gravitational radiation produced outside the event horizon. However, gluons in the quark-gluon plasma may also act as a damping agent. In the absence of damping, which is quite unlikely, the black hole would end up as a perpetually pulsating quark star. 3\. The third possibility is that eventually a black hole may explode; a [*mini bang*]{} of a sort may occur, and it may, after the explosion, expand beyond the event horizon though it has been emphasized by Zeldovich and Novikov (1971) that after a collapsing sphere’s radius decreases to $r < r_g$ in a finite proper time, its expansion into the external space from which the contraction originated is impossible, even if the passage of matter through infinite density is assumed. Notwithstanding Zeldovich and Novikov’s contention based on the very concept of event horizon, a gravitationally collapsing black hole may also explode by the very same mechanism by which the big bang occurred, if indeed it did occur. This can be seen as follows. At the present epoch the volume of the universe is $\sim 1.5 \times 10^{85} \cm^3$ and the density of the galactic material throughout the universe is $\sim 2 \times 10^{-31} \gcmcui$ (Allen 1973). Hence, a conservative estimate of the mass of the universe is $\sim 1.5 \times 10^{85} \times 2 \times 10^{-31} \g = 3 \times 10^{54} \g $. However, according to the big bang model, before the big bang, the entire matter in the universe was contained in an [*ylem*]{} which occupied very very small volume. The gravitational radius of the ylem of mass $3 \times 10^{54} g $ was $ 4.45 \times 10^{21} \km$ (it must have been larger if the actual mass of the universe were taken into account which is greater than $3 \times 10^{54} \g $). Obviously, the radius of the ylem was many orders of magnitude smaller than its gravitational radius, and yet the ylem exploded with a big bang, and in due course of time crossed the event horizon and expanded beyond it upto the present Hubble distance $c/H_0 \sim 1.5 \times 10^{23} \km$ where $c$ is the speed of light in vacuum and $H_0$ the Hubble constant at the present epoch. Consequently, if the ylem could explode in spite of Zeldovich and Novikov’s contention, a gravitationally collapsing black hole can also explode, and in due course of time expand beyond the event horizon. The origin of the big bang, i.e., the mechanism by which the ylem exploded, is not definitively known. However, the author has, earlier proposed a viable mechanism (Thakur 1992) based on supersymmetry/supergravity. But supersymmetry/supergravity have not yet been validated experimentally. Conclusion ========== From the foregoing three inferences may be drawn. One, eventually the entire matter in a collapsing black hole is converted into quark-gluon plasma permeated by leptons. Two, the collapse of a black hole to a space - time singularity is inhibited by Pauli’s exclusion principle. Three, ultimately a black hole may end up in one of the three possible ways suggested in section 7. The author thanks Professor S. K. Pandey, Co-ordinator, IUCAA Reference Centre, School of Studies in Physics, Pt. Ravishankar Shukla University, Raipur, for making available the facilities of the Centre. He also thanks Sudhanshu Barway, Mousumi Das for typing the manuscript. Abachi S., , 1995, PRL, 74, 2632 Abe F., , 1995, PRL, 74, 2626 Allen C. W., 1993, [**]{}, The Athlone Press, University of London, 293 Andrew D., , 1980, PRL, 44, 1108 Aubert J. J., , 1974, PRL, 33, 1404 Augustin J. E., , 1974, PRL, 33, 1406 Barber D.P.,,1979, PRL, 43, 1915 Beherend S., , 1983, PRL, 50, 881 Drake J. , 2002, ApJ, 572, 996 Gaillard M. K., Lee B. 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ArXiv

--- abstract: 'The ground state properties including radii, density distribution and one neutron separation energy for C, N, O and F isotopes up to the neutron drip line are systematically studied by the fully self-consistent microscopic Relativistic Continuum Hartree-Bogoliubov (RCHB) theory. With the proton density distribution thus obtained, the charge-changing cross sections for C, N, O and F isotopes are calculated using the Glauber model. Good agreement with the data has been achieved. The charge changing cross sections change only slightly with the neutron number except for proton-rich nuclei. Similar trends of variations of proton radii and of charge changing cross sections for each isotope chain is observed which implies that the proton density plays important role in determining the charge-changing cross sections.' address: - '${}^{1}$Department of Technical Physics, Peking University, Beijing 100871, China' - '${}^{2}$Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China' - '${}^{3}$Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou 730000, China' - '${}^{4}$The Institute of Physical and Chemical Research (RIKEN), Hirosawa 2-1, Wako-shi, Saitama 351-0198, Japan' author: - 'J. Meng$^{1-3}$[^1], S.-G. Zhou$^{1-3}$ and I. Tanihata$^{4}$' --- =10000 Recent progresses in the accelerator and detection techniques all around the world have made it possible to produce and study the nuclei far away from the stability line — so called “EXOTIC NUCLEI”. Based on the measurement of interaction cross section with radioactive beams at relativistic energy, novel and entirely unexpected features has appeared: e.g., the neutron halo and skin as the rapid increase in the measured interaction cross-sections in the neutron-rich light nuclei [@THH.85b; @HJJ.95]. Systematic investigation of interaction cross sections for an isotope chain or an isotone chain can provide a good opportunity to study the density distributions over a wide range of isospin [@Suz.95; @MTY.97]. However the contribution from proton and neutron are coupled in the measurement of interaction cross section. To draw conclusion on the differences in proton and neutron density distributions definitely, a combined analysis of the interaction cross section and other experiment on either proton or neutron alone are necessary. The charge-changing cross section which is the cross section for all processes which result in a change of the atomic number for the projectile can provide good opportunity for this purpose. In Ref.[@Chu.00], the total charge-changing cross section $\sigma_{\rm cc}$ for the light stable and neutron-rich nuclei at relativistic energy on a carbon target were measured. We will study $\sigma_{\rm cc}$ theoretically by using the fully self-consistent and microscopic relativistic continuum Hartree-Bogoliubov (RCHB) theory and the Glauber Model in the present letter. The RCHB theory [@ME.98; @MR.96; @MR.98], which is an extension of the relativistic mean field (RMF) [@SW86; @RE89; @RI96] and the Bogoliubov transformation in the coordinate representation, can describe satisfactorily the ground state properties for nuclei both near and far from the $\beta$-stability line and from light to heavy or super heavy elements, as well as for the understanding of pseudo-spin symmetry in finite nuclei [@MSY.98; @Meng993; @Meng992; @Meng00]. A remarkable success of the RCHB theory is the self-consistent reproduction of the halo in $^{11}$Li [@MR.96] and the prediction of giant halo [@MR.98]. In combination with the Glauber model, the RCHB theory successfully reproduces the interaction cross section in Na isotopes [@MTY.97]. These successes encourage us to apply the RCHB theory to calculate the charge changing cross section of the C, N, O, F isotopes (ranging from the $\beta$-stability line to the neutron drip line) on the target of $^{12}$C reported in Ref.[@Chu.00]. With the density distribution provided by RCHB theory, the total charge-changing cross section can be calculated based on the Glauber model and compared with the data directly [@Chu.00], as was done in Ref.[@MTY.97] for the interaction cross section. Since the theory used here is fully microscopic and basically parameter free, we hope it provide us more reliable information on both the proton and neutron distribution. The ground state properties of C, N, O and F isotopes up to neutron drip line are studied first, including single neutron separation energies, density distributions and radii. Then the total charge-changing cross sections will be calculated from the Glauber model with the density obtained from RCHB calculations. The basic ansatz of the RMF theory is a Lagrangian density whereby nucleons are described as Dirac particles which interact via the exchange of various mesons (the scalar sigma ($\sigma$), vector omega ($\omega$) and iso-vector vector rho ($\rho$)) and the photon. The $\sigma$ and $\omega$ meson provide the attractive and repulsive part of the nucleon-nucleon force, respectively. The necessary isospin asymmetry is provided by the $\rho$ meson. The scalar sigma meson moves in a self-interacting field having cubic and quadratic terms with strengths $g_2$ and $g_3$ respectively. The Lagrangian then consists of the free baryon and meson parts and the interaction part with minimal coupling, together with the nucleon mass $M$ and $m_\sigma$ ($g_\sigma$), $m_\omega$ ($g_\omega$), and $m_\rho$ ($g_\rho$) the masses (coupling constants) of the respective mesons: $$\begin{array}{rl} {\cal L} &= \bar \psi (i\rlap{/}\partial -M) \psi + \,{1\over2}\partial_\mu\sigma\partial^\mu\sigma-U(\sigma) -{1\over4}\Omega_{\mu\nu}\Omega^{\mu\nu}\\ \ &+ {1\over2}m_\omega^2\omega_\mu\omega^\mu -{1\over4}{\vec R}_{\mu\nu}{\vec R}^{\mu\nu} + {1\over2}m_{\rho}^{2} \vec\rho_\mu\vec\rho^\mu -{1\over4}F_{\mu\nu}F^{\mu\nu} \\ &- g_{\sigma}\bar\psi \sigma \psi~ -~g_{\omega}\bar\psi \rlap{/}\omega \psi~ -~g_{\rho} \bar\psi \rlap{/}\vec\rho \vec\tau \psi -~e \bar\psi \rlap{/}A \psi \end{array}$$. For the proper treatment of the pairing correlations and for correct description of the scattering of Cooper pairs into the continuum in a self-consistent way, one needs to extend the present relativistic mean-field theory to the RCHB[@ME.98; @MR.96; @MR.98]: $$\left(\begin{array}{cc} h-\lambda & \Delta \\ -\Delta^* & -h^*+\lambda \end{array}\right) \left(\begin{array}{r} U \\ V\end{array}\right)_k~=~ E_k\,\left(\begin{array}{r} U \\ V\end{array}\right)_k, \label{RHB}$$ $E_k$ is the quasi-particle energy, the coefficients $U_k(r)$ and $V_k(r)$ are four-dimensional Dirac spinors, and $h$ is the usual Dirac Hamiltonian $$h = \left[ {\mbox{\boldmath$\alpha$}} \cdot {\bf p} + V( {\bf r} ) + \beta ( M + S ( {\bf r} ) ) \right], \label{h-field}$$ with the vector and scalar potentials calculated from: $$\begin{aligned} \left\{ \begin{array}{lll} V( {\bf r} ) &=& g_\omega\rlap{/}\omega({\bf r}) + g_\rho\rlap{/}\mbox{\boldmath$\rho$}({\bf r})\mbox{\boldmath$\tau$} + \displaystyle\frac{1}{2}e(1-\tau_3)\rlap{\,/}{\bf A}\mbox{\boldmath$\tau$}({\bf r}) , \\ S( {\bf r} ) &=& g_\sigma \sigma({\bf r}). \\ \end{array} \right. \label{vaspot}\end{aligned}$$ The chemical potential $\lambda$ is adjusted to the proper particle number. The meson fields are determined as usual in a self-consistent way from the Klein Gordon equations in [*no-sea*]{}-approximation. The pairing potential $\Delta$ in Eq. (\[RHB\]) is given by $$\Delta_{ab}~=~\frac{1}{2}\sum_{cd} V^{pp}_{abcd} \kappa_{cd} \label{gap}$$ It is obtained from the pairing tensor $\kappa=U^*V^T$ and the one-meson exchange interaction $V^{pp}_{abcd}$ in the $pp$-channel. As in Ref. [@ME.98; @MR.96; @MR.98] $V^{pp}_{abcd}$ in Eq. (\[gap\]) is the density dependent two-body force of zero range: $$V(\mbox{\boldmath $r$}_1,\mbox{\boldmath $r$}_2) ~=~\frac{V_0 }{2}(1+P^\sigma) \delta(\mbox{\boldmath $r$}_1-\mbox{\boldmath$r$}_2) \left(1 - \rho(r)/\rho_0\right). \label{vpp}$$ The ground state $|\Psi\rangle$ of the even particle system is defined as the vacuum with respect to the quasi-particle: $\beta_{\nu} |\Psi\rangle=0$, $|\Psi\rangle = \prod_\nu \beta_{\nu} |-\rangle$, where $|-\rangle$ is the bare vacuum. For odd system, the ground state can be correspondingly written as: $|\Psi \rangle_{\mu} = \beta_{\mu}^\dagger\prod_{\nu \ne \mu} \beta_{\nu} | - \rangle$, where $\mu$ is the level which is blocked. The exchange of the quasiparticle creation operator $\beta_{\mu}^\dagger$ with the corresponding annihilation operator $\beta_{\mu}$ means the replacement of the column $\mu$ in the $U$ and $V$ matrices by the corresponding column in the matrices $V^*$, $U^*$ [@RS.80]. The RCHB equations (\[RHB\]) for zero range pairing forces are a set of four coupled differential equations for the quasi-particle Dirac spinors $U(r)$ and $V(r)$. They are solved by the shooting method in a self-consistent way as [@ME.98]. The detailed formalism and numerical techniques of the RCHB theory can be found in Ref.[@ME.98] and the references therein. In the present calculations, we follow the procedures in Ref.[@ME.98; @MR.98; @MTY.97] and solve the RCHB equations in a box with the size $R=20$ fm and a step size of 0.1 fm. The parameter set NL-SH [@SNR.93] is used, which aims at describing both the stable and exotic nuclei. The density dependent $\delta$-force in the pairing channel with $\rho_0=0.152$ fm$^{-3}$ is used and its strength $V_0$ is fixed by the Gogny force as in Ref.[@ME.98]. The contribution from continua is restricted within a cut-off energy $E_{cut}\sim 120$MeV. Systematic calculations with RCHB theory has been carried out for the C, N, O and F isotopes. The one neutron separation energies $S_{\rm n}$ predicted by RCHB and their experimental counterparts [@AUD.93] for the nuclei $^{11-22}$C, $^{13-24}$N, $^{15-26}$O and $^{17-25}$F are given in Fig.\[fig.s\] as open and solid circles respectively. For carbon isotopes, the theoretical one neutron separation energies for $^{11-18,20,22}$C are in agreement with the data. The calculated $S_{\rm n}$ is less than 0 (-0.003 MeV) for the odd-$A$ nucleus $^{19}$C which is bound from experiment. While for the experimentally unbound nucleus $^{21}$C, the predicted value of $S_{\rm n}$ is positive. From the neutron-deficient side to the neutron drip line, excellent agreement has been achieved for the nitrogen isotopes. Just as the other relativistic mean field approaches, the RCHB calculations overestimate binding for $^{25,26}$O which are unstable in experiment. For fluorine isotopes, the $S_{\rm n}$ in $^{17,26-29}$F are overestimated in contrast with the underestimated one in $^{18}$F. The neutron drip line nucleus is predicted as $^{30}$F. In general, the RCHB theory reproduces the $S_{\rm n}$ data well considering that this is a microscopic and almost parameter free model. There are some discrepancies between the calculations and the empirical values for some of the studied isotopes. This may be due to deformation effects, which has been neglected here. The proton density distributions predicted by RCHB for the nuclei $^{10-22}$C,$^{12-24}$N,$^{14-26}$O and $^{16-25}$F are given in Fig.\[fig.d\] in logarithm scale. The change in the density distributions for each isotopes chain in Fig.\[fig.d\] occurs only at the tail or in the center part as the proton number is constant. Because the density must be multiplied by a factor $4\pi r^2$ before the integration in order to give proton number or radii, the large change in the center does not matter very much. What important is the density distribution in the tail part. Compared with the neutron-rich isotopes, the proton distribution with less $N$ has higher density at the center, lower density in the middle part ( $2.5 < r < 4.5 $fm ), a larger tail in the outer part ( $r > 4.5 $ fm ) which gives rise to the increase of $r_{\rm p}$ and $\sigma_{\rm cc}$ for the proton rich nuclei as will be seen in the following. The neutron and proton rms radii predicted by RCHB for the nuclei $^{10-22}$C, $^{12-24}$N,$^{14-26}$O and $^{16-25}$F are given in Fig.\[fig.r\]. The neutron radii for nuclei in each isotope chain increase steadily. While the corresponding proton radii remains almost constant with neutron number for nuclei in each isotope chain except for the proton rich ones. To compare the charge-changing cross sections $\sigma_{\rm cc}$ directly with experimental measured values, the densities $\rho_{\rm p}(r)$ of the target $^{12}$C and the C, N, O and F isotopes obtained from RCHB ( see Fig.\[fig.d\] ) were used. The cross sections were calculated in Glauber model by using the free nucleon-nucleon cross section [@Ray.79] for the proton and neutron respectively. The total charge-changing cross sections $\sigma_{\rm cc}$ of the nuclei $^{10-22}$C,$^{12-24}$N,$^{14-26}$O and $^{16-25}$F on a carbon target at relativistic energy are given in Fig. \[fig.c\]. The open circles are the result of RCHB combined with the Glauber Model and the available experimental data [@Chu.00] are given by solid circles with their error-bars. The agreement between the calculated results and measured ones are fine. The charge changing cross sections change only slightly with the neutron number except for proton-rich nuclei, which means the proton density plays important role in determining the charge-changing cross sections $\sigma_{\rm cc}$. A gradual increase of the cross section has been observed towards the neutron drip line. However, the big error bars of the data can not help to conclude anything here yet. It is shown clearly that the RCHB theory, when combined with the Glauber model, can provide reliable description for not only interaction cross section but also charge changing cross section. From comparison of this figure and Fig.\[fig.r\], we can find similar trends of variations of proton radii and of charge changing cross sections for each isotope chain which implies again that the proton density plays important role in determining the charge-changing cross sections. Summarizing our investigations, the ground state properties for C, N, O and F isotopes have been systematically studied with a microscopic model — the RCHB theory, where the pairing and blocking effect have been treated self-consistently. The calculated one neutron separation energies $S_{\rm n}$ are in good agreement with the experimental values available with some exceptions due to deformation effects which is not included in the present study. A Glauber model calculation for the total charge-changing cross section has been carried out with the density obtained from the RCHB theory. A good agreement was obtained with the measured cross sections for $^{12}$C as a target. Another important conclusion here is that, contrary to the usual impression, the proton density distribution is less sensitive to the proton and neutron ratio. Instead it is almost unchanged from stability to the neutron drip-line. The influence of the deformation, which is neglected in the present investigation, is also interesting to us, more extensive study by extending the present study to deformed cases are in progress. This work was partly supported by the Major State Basic Research Development Program Under Contract Number G2000077407 and the National Natural Science Foundation of China under Grant No. 10025522, 19847002 and 19935030. I. Tanihata, Prog. Part. Nucl. Phys.,[**35**]{}(1995) 505. P.G. Hansen, A.S. Jensen, and B. Jonson, Ann. Rev. Nucl. Part. Sci. [**45**]{} (1995) 591 T. Suzuki, et al., Phys. Rev. Lett. [**75**]{} (1995) 3241 J. Meng, I. Tanihata and S. Yamaji, Phys. Lett. [**B419**]{} (1998) 1. L.V. Chulkov, et al., Nucl. Phys. [**A674**]{} (2000) 330 J. Meng, Nucl. Phys. [**A635**]{} (1998) 3. J. Meng, and P. Ring, Phys. Rev. Lett. [**77**]{} (1996) 3963. J. Meng, and P. Ring, Phys. Rev. Lett. [**80**]{} (1998) 460. B.D. Serot and J.D. Walecka, Adv. Nucl. Phys. [**16**]{}, 1 (1986). P-G. Reinhard, Rep. Prog. Phys. [**52**]{}, 439 (1989). P. Ring, Prog. Part. Nucl. Phys. [**37**]{}, 193 (1996). J. Meng, K. Sugawara-Tanabe, S. Yamaji, P. Ring and A. Arima, Phys. Rev. C58 (1998) R628. J. Meng, K. Sugawara-Tanabe, S. Yamaji and A. Arima, Phys. Rev. C 59 (1999) 154-163. J. Meng, and I. Tanihata, Nucl. Phys. A 650 (1999) 176-196. J. Meng, and N. Takigawa, Phys. Rev. C61 (2000) 064319. P. Ring and P. Schuck, [*The Nuclear Many-body Problem*]{}, Springer Verlag, Heidelberg (1980) M. Sharma, M. Nagarajan and P. Ring, Phys. Lett. [**B312**]{} (1993) 377 G. Audi, and A.H. Wapstra, Nucl. Phys. [**A565**]{} (1993) 1 L. Ray, Phys. Rev. [**C**]{} 20 ( 1979 ) 1857 [^1]: e-mail: [email protected]

ArXiv

--- abstract: 'We show that charge noise $S_Q$ in Josephson qubits can be produced by fluctuating two level systems (TLS) with electric dipole moments in the substrate using a flat density of states. At high frequencies the frequency and temperature dependence of the charge noise depends on the ratio $J/J_c$ of the electromagnetic flux $J$ to the critical flux $J_c$. It is not widely appreciated that TLS in small qubits can easily be strongly saturated with $J/J_c\gg 1$. Our results are consistent with experimental conclusions that $S_Q\sim 1/f$ at low frequencies and $S_Q\sim f$ at high frequencies.' author: - 'Clare C. Yu$^1$, Magdalena Constantin$^1$, and John M. Martinis$^2$' title: Effect of Two Level System Saturation on Charge Noise in Josephson Junction Qubits --- Noise and decoherence are a major obstacle to using superconducting Josephson junction qubits to construct quantum computers. Recent experiments [@Simmonds2004; @Martinis2005] indicate that a dominant source of decoherence is two level systems (TLS) in the insulating barrier of the tunnel junction as well as in the dielectric material used to fabricate the circuit. It is believed that these TLS fluctuators lead to low frequency $1/f$ charge noise $S_Q(f)$ [@Martinis1992; @Mooij1995; @Zorin1996; @Kenyon2000; @Astafiev2006]. However, at high frequencies, one experiment finds that the charge noise increases linearly with frequency [@Astafiev2004]. This has prompted some theorists to use a TLS density of states linear in energy [@Shnirman2005] which is contrary to the constant density of states that has been so successful in explaining the low temperature properties of glasses such as the specific heat that is linear in temperature [@Phillips]. A linear distribution has been proposed in conjunction with a Cooper pair tunneling into a pair of electron traps [@Faoro2005], and with electron hopping between Kondo–like traps to account for the charge noise [@Faoro2006]. However, these previous theoretical efforts have neglected the important issue of the saturation of the two level systems. Dielectric (ultrasonic) experiments on insulating glasses at low temperatures have found that when the electromagnetic (acoustic) energy flux $J$ used to make the measurements exceeds the critical flux $J_c$, the dielectric (ultrasonic) power absorption by the TLS is saturated, and the attenuation decreases [@Golding1976; @Arnold1976; @Schickfus1977; @Graebner1983; @Martinis2005]. The previous theoretical efforts to explain the linear increase in the charge noise in Josephson junctions assumed that the TLS were not saturated, i.e., that $J\ll J_c$. This seems sensible since the charge noise experiments were done in the limit where the qubit absorbed only one photon. However, stray electric fields could saturate TLS in the dielectric substrate as the following simple estimate shows. We can estimate the voltage $V$ across the capacitor associated with the substrate and ground plane beneath the Cooper pair box by setting $CV^2/2=\hbar\omega$ where $\hbar\omega$ is the energy of the microwave photon. We estimate the capacitance $C=\varepsilon_o\varepsilon A/L \sim 7$ aF using the area $A=40\times 800$ nm$^2$ [@Astafiev2004] of the Cooper pair box, the thickness $L=400$ nm of the substrate [@Astafiev2004], and the dielectric constant $\varepsilon=10$. Using $\omega/2\pi=10$ GHz, we obtain a voltage of $V\sim 1.4\;$mV. The substrate thickness $L$ of 400 nm yields an electric field of $E\sim 3.4\times 10^{3}$ V/m. For amorphous SiO$_2$ at $f=7.2$ GHz and SiN$_{x}$ at $f=4.7$ GHz, the critical rms voltage $V_{c} \sim 0.2\;\mu$V [@Martinis2005], and with a capacitor thickness of 300 nm, the critical field is $E_c\sim 0.7$ V/m at $T=25$ mK. So $E/E_{c}\sim 5\times 10^{3}$, and $J/J_{c}=\left(E/E_{c}\right)^{2}\sim 2\times 10^{7}\gg 1$. A similar estimate shows that a single photon would even more strongly saturate resonant TLS in the insulating barrier of the tunnel junction, again resulting in $J/J_{c}\gg 1$. However, there are only a few TLS in the oxide barrier of a small tunnel junction. (For a parallel plate capacitor with a specific capacitance of 60 fF/$\mu$m$^2$, $L=1.5$ nm, $A=1\;\mu$m$^2$ and a dielectric constant of 10, the volume is $\Omega=1.5\times 10^{-21}$ m$^3$. With a density of states $P_0\simeq 10^{45}/\left(Jm^{3}\right)\simeq663/h$GHz$\mu$m$^3$, there are only 2 TLS with an energy splitting less than 10 GHz.) A single fluctuator would have a Lorentian noise spectrum. The presence of $1/f$ noise implies many more than 2 fluctuators. It is likely that these additional fluctuators are in the substrate. Our main point is that TLS in small devices are easily saturated. In this letter we explore the consequences of this saturation. We find that at high frequencies ($\hbar\omega\gg k_B T$), the frequency and temperature dependence of the charge noise depends on the ratio $J/J_c(\omega,T)$ of the electromagnetic flux $J$ to the critical value $J_c(\omega,T)$ which is a function of frequency $\omega$ and temperature $T$. Starting from the fluctuation-dissipation theorem, we show that the charge noise is proportional to the dielectric loss tangent $\tan\delta$. We then calculate the dielectric loss tangent due to fluctuating TLS with electric dipole moments [@Phillips; @Classen1994; @Arnold1976]. At low frequencies we recover $1/f$ noise. At high frequencies $\tan\delta$ is proportional to $1/\sqrt{1+\left(J/J_c(\omega,T)\right)}$. In the saturation regime ($J\gg J_c(\omega,T)$), $\tan\delta$ and hence, the charge noise, are proportional to $\sqrt{J_c(\omega,T)/J}$. Some TLS experiments [@Bachellerie1977; @Bernard1979; @Graebner1979] indicate that $J_c(\omega,T)\sim\omega^{2}T^{2}$ which implies that at high frequencies the charge noise and the dielectric loss tangent would increase linearly in frequency if $J\gg J_c(\omega,T)$. Unlike previous theoretical efforts, we use the standard TLS density of states that is independent of energy, and can still obtain charge noise that increases linearly with frequency in agreement with the conclusions of Astafiev [*et al.*]{} [@Astafiev2004]. In applying the standard model of two level systems to Josephson junction devices, we consider a TLS that sits in the insulating substrate or in the tunnel barrier, and has an electric dipole moment ${\bf p}$ consisting of a pair of opposite charges separated by a distance $d$. The electrodes are located at $z=0$ and $z=L$ and kept at the same potential. The angle between ${\bf p}$ and $z$–axis which lies perpendicular to the plane of the electrodes is $\theta_0$. The dipole flips and induces charge fluctuations on the electrodes. These induced charges are proportional to the $z$-component of the dipole moment, i.e., $Q=|p\cos \theta_0/L|$. The TLS is in a double–well potential with a tunneling matrix element $\Delta_0$ and an asymmetry energy $\Delta$ [@Phillips]. The Hamiltonian of a TLS in an external ac field can be written as $H=H_0+H_1$, where $H_0=\frac{1}{2}(\Delta \sigma_z+\Delta_0 \sigma_x)$, and $H_1=-\sigma_z {\bf p}\cdot\bm{\xi}_{ac}(t)$. Here $\sigma_{x,z}$ are the Pauli spin matrices and $\bm {\xi}_{ac}(t)=\bm {\xi}_{ac}\mbox{cos}\omega t$ is a small perturbing ac field of frequency $\omega$ that couples to the TLS electric dipole moment. After diagonalization, the TLS Hamiltonian becomes $H_0=\frac{1}{2}E\sigma_z$, where $E$ is the TLS energy splitting (i.e., $E=\sqrt{\Delta^2 + \Delta_{o}^2}$) and, in the new basis, the perturbing Hamiltonian is $H_1=-(\sigma_z\Delta/E+\sigma_x\Delta_0/E){\bf p}\cdot\bm{\xi}_{ac}(t)$. The complete TLS Hamiltonian is similar to the Hamiltonian $H_S = -\gamma{\bf S}\cdot{\bf B}$ for a spin $1/2$ particle in a magnetic field given by ${\bf B}(t)={\bf B}_0+{\bf B}_1(t)$, where the static field is $-\hbar \gamma {\bf B}_0=(0,0,E)$ and the rotating field is $\hbar \gamma {\bf B}_1(t)=(2\Delta_0/E,0,2\Delta/E) {\bf p}\cdot \bm {\xi}_{ac}\mbox{cos}\omega t$ [@herve]. $\gamma$ is the gyromagnetic ratio and ${\bf S}=\hbar\bm {\sigma}/2$. Therefore, the equations of motion of the expectation values of the spin components are given by the Bloch equations [@Slichter] with the longitudinal and the transverse relaxation times, $T_1$ and $T_2$. In the standard model, an excited two-level system decays to the ground state by emitting a phonon. The longitudinal relaxation rate, $T_1^{-1}$, is given by [@Phillips] $$T_1^{-1}=a E \Delta_o^{2}\coth\left(\frac{E}{2k_BT}\right), \label{eq:T1inv}$$ where $a=\left[\gamma_d^2/\left(2\pi\rho\hbar^4\right)\right] \left[\left(1/c_{\ell}^{5}\right)+\left(2/c_{t}^{5}\right)\right]$ where $\rho$ is the mass density, $c_{\ell}$ is the longitudinal speed of sound, $c_{t}$ is the transverse speed of sound, and $\gamma_d$ is the deformation potential. The distribution of TLS parameters can be expressed in terms of $E$ and $T_1$: $P(E,T_1)=P_0/(2T_1\sqrt{1-\tau_{min}(E)/T_1})$ [@Phillips1987; @Phillips] where $P_0$ is a constant. The minimum relaxation time $\tau_{min}(E)$ corresponds to a symmetric double–well potential (i.e., $E=\Delta_0$). The transverse relaxation time $T_2$ represents the broadening of levels due to their mutual interaction [@Black1977]. [*General Expression for Charge Noise:*]{} We derive a general expression valid at all frequencies for the charge noise in terms of the dielectric loss tangent $\tan\delta(\omega)=\varepsilon^{\prime\prime}(\omega)/ \varepsilon^{\prime}(\omega)$, where $\varepsilon^{\prime}(\omega)$ and $\varepsilon^{\prime\prime}(\omega)$ are the real and imaginary parts of the permittivity. According to the Wiener-Khintchine theorem, the charge spectral density $S_Q(\omega)$ is twice the Fourier transform $\Psi_Q(\omega)$ of the autocorrelation function of the fluctuations in the charge. From the fluctuation-dissipation theorem, the charge noise is given by: $$S_{Q}(\bm{k},\omega)=\frac{4\hbar}{1-\mbox{e}^{-\hbar \omega/k_BT}} \chi_{Q}^{\prime\prime}(\bm{k},\omega), \label{eq:S_Q}$$ where $Q$ is the induced (bound) charge and $\chi_{Q}^{\prime\prime}(\bm{k},\omega)$ is the Fourier transform of $\chi_{Q}^{\prime\prime}(\bm{r},t;\bm{r}^{\prime},t^{\prime})= \langle\left[Q(\bm{r},t),Q(\bm{r}^{\prime},t^{\prime})\right]\rangle/2\hbar$. We use $Q=\int \bm{P}\cdot d\bm{A}$, where $\bm{P}$ is the electric polarization density, and choose $P_z$ and $d\bm{A} \| \hat{z}$ since $Q\sim |p_z|$ to find $\chi_{Q}^{\prime\prime}(\bm{k},\omega)=\varepsilon_{o}A^2 \chi_{P_z}^{\prime\prime}(\bm{k},\omega)$, where $\varepsilon_0$ is the vacuum permittivity, $A$ is the area of a plate of the parallel plate capacitor with capacitance $C$, and $\chi_{P_z}^{\prime\prime}(\bm{k},\omega)$ is the imaginary part of the electric susceptibility. Setting $\bm{k}=0$, and using $\varepsilon_o\chi_{P_z}^{\prime\prime}(\omega)= \varepsilon^{\prime}(\omega)\tan\delta(\omega)$, and $C=\varepsilon^{\prime}A/L$, we find $$S_Q(\omega)=\frac{4\hbar C}{1-\mbox{e}^{-\hbar \omega/k_BT}} \tan\delta(\omega), \label{eq:S_Q_Tan_delta}$$ where $S_Q(\omega)\equiv S_Q(\bm{k}=0,\omega)/\Omega$, the volume of the capacitor is $\Omega=AL$, and $\varepsilon^{\prime}(\omega)= \varepsilon^{\prime}+\varepsilon_{TLS}(\omega)\simeq\varepsilon^{\prime}= \varepsilon_0 \varepsilon_r$ where $\varepsilon_r$ is the relative permittivity. The frequency dependent $\varepsilon_{TLS}(\omega)$ produced by TLS is negligible compared to the constant permittivity $\varepsilon^{\prime}$ [@Phillips]. The dynamic electric susceptibilities ($\chi^{\prime}(\omega)$, $\chi^{\prime\prime}(\omega)$), and hence the dielectric loss tangent, can be obtained by solving the Bloch equations [@Jackle1975; @Arnold1976; @Graebner1983]. (Shnirman [*et al.*]{} [@Shnirman2005] gave a reformulation of the Bloch equations for TLS in terms of density matrices and the Bloch-Redfield theory.) The electromagnetic dispersion and attenuation due to TLS has two contributions. First there is the relaxation process due to the modulation of the TLS energy splitting by the incident wave resulting in readjustment of the equilibrium level population. This is described by $\chi_{z}(\omega)$ that comes from solving the Bloch equations for $S_z$ which is associated with the population difference of the two levels. The second process is the resonant absorption by TLS of phonons with $\hbar \omega=E$. Resonance is associated with $\chi_{\pm}(\omega)=\chi_{x}(\omega) \pm i\chi_{y}(\omega)$ since $\sigma_x$ and $\sigma_y$ are associated with transitions between the two levels. The total dielectric loss tangent is the sum of these two contributions: $\tan \delta=\tan \delta_{REL} +\tan \delta_{RES}$. The steady-state solution of the Bloch equations and the resulting dielectric loss tangent for TLS are well known [@Jackle1975; @Arnold1976; @Graebner1983; @herve]. In the steady-state regime the experimental values for the relaxation times $T_1$ and $T_2$ are considered small compared to the electromagnetic pulse duration, $t_p$. One might ask if one should use the transient solution [@herve] of the Bloch equations since the pulse applied to superconducting qubits is often extremely short, $t_p\sim10^{-10}$ sec [@Astafiev2004]. We find that the transient $z$–component of the magnetization [@herve], $S_z^{0}(t)$, decays exponentially to the equilibrium value denoted by $S_{z,eq}^0$, i.e.: $S_z^{0}(t)=S_{z,eq}^0+\mbox{exp}(-t/T^{\star})[S_z^{0}(0)-S_{z,eq}^0]$, where $S_z^{0}(0)$ is the initial value of $S_z^{0}$ and $S_{z,eq}^0=-\hbar\mbox{tanh}(E/2k_BT)/2$. The transient relaxation time $T^{\star}$ is given by $$\begin{aligned} &T^{\star}=\frac{T_1}{1+\left(J/J_c(\omega,T)\right) \times g(\omega,\omega_0,T_2)},~~~ \mbox{where} \nonumber\\ &g(\omega,\omega_0,T_2)=\frac{1}{1+T_2^2(\omega-\omega_0)^2}+ \frac{1}{1+T_2^2(\omega+\omega_0)^2}.\end{aligned}$$ Here $J/J_c(\omega,T)=({\bf p^{\prime}}\cdot \bm {\xi}_{ac} /\hbar)^2 T_1 T_2$, ${\bf p^{\prime}}=(\Delta_0/\epsilon){\bf p}$ represents the induced TLS dipole moment, and $\omega_0=\gamma B_0=-E/\hbar$. In the saturated regime, for $J\gg J_c(\omega,T)$, we find that $T^{\star}\approx [T_2({\bf p^{\prime}}\cdot \bm {\xi}_{ac}/\hbar)^2]^{-1}$. Using $p^{\prime}=3.7$ D, $\xi_{ac}=3.4\times 10^3$ V/m, and $T_2=8$ $\mu$s at $T=0.1$ K [@Bernard1979; @herve] yields $T^{\star}\simeq 8\times10^{-13}$ sec at resonance when $\omega/2\pi\simeq \omega_0/2\pi=10$ GHz. This value is much shorter than the typical pulse length used in the Josephson junction qubits experiments and therefore the results from steady-state saturation theory can be used. However, in the unsaturated regime, $T^{\star}\approx T_1\approx 8 \times 10^{-8}$ sec at $T=0.1$ K, so transient effects can be important. [*High Frequency Charge Noise:*]{} At high frequencies (HF) ($\hbar\omega\gg k_BT$) the dielectric loss tangent is dominated by resonant (RES) absorption processes (i.e., $\tan\delta_{HF}\simeq\tan\delta_{RES}$) [@Arnold1976; @Phillips; @comment:REL_abs]: $$\tan \delta_{HF}(\omega,T) = \frac{\pi p^{2} P_0}{3 \varepsilon'} \tanh(\hbar\omega/2k_BT)\frac{1}{\sqrt{1+J/J_c(\omega,T)}}, \label{HFloss}$$ where $J_c(\omega,T)=3\hbar^2\varepsilon^{\prime}v/(2p^2T_1 T_2)$ and $v$ is the speed of light in the solid. Eq. (\[HFloss\]) comes from integrating over the TLS distribution [@Arnold1976]. However, if no integration is done due to the small number $N_0$ of TLS, e.g., in the tunnel junction barrier, $\tan \delta_{HF}(\omega,T)=N_0 p^{2}T_2/(3\varepsilon^{\prime}\hbar \Omega) \times \mbox{tanh}(\hbar \omega/2k_BT)[1+J/J_c(\omega,T)]^{-1}$. So for high intensities the $(J/J_c)^{-1/2}$ dependence of $\tan \delta$ becomes a $(J/J_c)^{-1}$ dependence. The frequency and temperature dependence of $\tan\delta$, and hence of $S_{Q}(\omega)$, depends on $J/J_c(\omega,T)$. At low intensities ($J\ll J_c(\omega,T)$) in the unsaturated steady-state resonant absorption regime, the dielectric loss tangent is constant: $$\tan \delta_{HF} \simeq \pi p^{2} P_0/\left(3\varepsilon^{\prime}\right) \label{IntrinsicLoss}$$ For $\varepsilon_r=10$, $P_0 \approx 10^{45}$(Jm$^3$)$^{-1}$ [@ccyjjf; @comment:P0], and $p=3.7$ D (which corresponds to the dipole moment of an OH$^-$ impurity [@Golding1979]), we estimate $\tan\delta_{HF}\approx 1.8\times 10^{-3}$. This result agrees well with the value of $\delta\simeq 1.6\times 10^{-3}$ reported in Ref. [@Martinis2005]. In this regime the charge noise is constant: $S_Q/e^2\simeq 4\pi\hbar C p^{2} P_0/(3e^2 \varepsilon^{\prime})$. For $C=$ 7 aF, $S_Q/e^2\simeq 2\times10^{-16}$ Hz$^{-1}$. For high field intensities $J\gg J_c(\omega,T)$, we obtain $\tan \delta_{HF}=\pi p^{2} P_0/(3 \varepsilon^{\prime}) \times\sqrt{J_c(\omega,T)/J}$. The $J^{-1/2}$ dependence of $\tan \delta$ has been found for materials such as amorphous SiO$_2$ and amorphous SiN$_x$ [@Martinis2005]. In the saturated resonant absorption regime the charge noise is given by $$\frac{S_Q(\omega,T)}{e^2}\simeq \frac{4\hbar C}{e^2}\frac{\pi p^{2} P_0} {3\varepsilon^{\prime}}\sqrt{\frac{J_c(\omega,T)}{J} }. \label{eq:S_Q_HF_HI}$$ So the frequency and temperature dependence of the noise is determined by $T_1$ and $T_2$: $S_Q(\omega,T)\sim \sqrt{J_c(\omega,T)}\sim \left(T_1 T_2\right)^{-1/2}$. Experiments and theory find that $T_2^{-1}\sim T^m$ where $m$ ranges from 1 to 2.2 [@Black1977; @Graebner1979; @Bernard1979; @Golding1982; @Hegarty1983; @Schickfus1988; @herve; @Enss1996short]. $T_2$ decreases with increasing frequency [@Graebner1979] but the exact frequency dependence is not known, so we will ignore it in what follows. $T_1$ in the symmetric case with $\hbar\omega=E=\Delta_o$ is given by Eq. (\[eq:T1inv\]): $T_1^{-1}\sim \omega^3$, implying that at high frequencies and intensities $S_Q(\omega,T)\sim \omega^{3/2}T^{m/2}$, where $m/2$ varies between 0.5 and 1.1. If there are only a very few TLS, then $S_Q(\omega,T)\sim J_c(\omega,T)\sim \left(T_1 T_2\right)\sim \omega^3 T^{m}$. Although the experimental frequency dependence was reported to be linear [@Astafiev2004], the scatter in the data is large enough to allow for a steeper frequency dependence. In fact, the dependence is much steeper for data at the degeneracy point. Experimental measurements [@Schickfus1977] on SiO$_2$ at $f=10$ GHz and 0.4 $< T < $1 K find $J_c(\omega,T)=25$ mW/cm$^2\times (T/0.4 $K$)^4$, and that $J/J_c(\omega,T)$ varies between $10^{-2}$ and $10^{4}$. This implies that the charge noise $S_Q/e^2$ should vary between $2\times 10^{-16}$ Hz$^{-1}$ and $2\times 10^{-18}$ Hz$^{-1}$ at $T=0.1$ K. However, other measurements have found $J_c(\omega,T)\sim \omega^nT^2$ [@Arnold1976], where $n$ is equal to either 0 [@Arnold1974; @Bernard1979] or 2 [@Graebner1979; @Bachellerie1977]. If $J_c(\omega,T)\sim\omega^{2}$, then $\tan\delta(\omega)\sim\omega$, and $S_{Q}(\omega)\sim\omega$ at high frequencies and high intensities, which agrees with the recent experiments by Astafiev [*et al.*]{} [@Astafiev2004]. It would be interesting measure the temperature dependence of $S_{Q}$ experimentally. While currently there are no direct experimental measurements of the HF charge noise, in Ref. [@Astafiev2004] $S_Q(\omega)$ has been deduced by measuring the qubit relaxation rate $\Gamma_1$ versus the gate induced charge $q$ for a Cooper pair box with a capacitance $C_b$, Josephson energy $E_J$ (in the GHz range), and electrostatic energy $U=2eq/C_b$. From $\Gamma_1=\pi S_U(\omega) \sin^2(\theta)/2\hbar^2$, where $\sin\theta=E_J^2/(E_J^2+U^2)$ and $\hbar\omega$ equals the qubit energy splitting ($\hbar\omega=\left(U^2+E^2_J\right)^{1/2}$), and $S_U(\omega)=(2e/C_b)^2 S_q(\omega)$, they obtained the charge noise $S_q(\omega)$ at high frequency [@Astafiev2004]. We can compare our results with these experiments by reversing this procedure to find $\Gamma_1(q)$ from $S_q(\omega)$. We find for saturated TLS that $\Gamma_1$ at the maximum ($q=0$) is of order $10^{8}$ s$^{-1}$ [@comment:Gamma1] and increases as $E_J^2$ in good quantitative agreement with the experimental results from Fig. 2 and 3 of Astafiev [*et al.*]{} [@Astafiev2004]. We find that $\Gamma_{1,max}$, the maximum value of $\Gamma_1$ (at $q=0$), increases with the frequency $f=E_J/h$ in the saturated regime but is independent of frequency at low intensities $J\ll J_c(\omega,T)$. [*Low Frequency Charge Noise:*]{} We now show that we can recover the low frequency $1/f$ charge noise using Eq. (\[eq:S\_Q\_Tan\_delta\]). At low frequencies (LF) where $\hbar\omega\ll k_B T$, only the relaxation absorption process contributes: $\tan\delta_{LF}\simeq\tan\delta_{REL} = \pi p^2 P_0/(6\varepsilon')$ [@Classen1994]. Eq. (\[eq:S\_Q\_Tan\_delta\]) gives [@Faoro2006; @Kogan96] $$\frac{S_Q(f)}{e^2}=\frac{2k_BT}{e^2/2C}\tan \delta_{LF}\frac{1}{2\pi f} =\frac{1}{3}\Omega P_0 k_BT \Bigr(\frac{p}{eL}\Bigr)^2\frac{1}{f}. \label{eq:SQ_LF}$$ To estimate the value of $S_Q$, we use $p=3.7$ D, $P_0 \approx 10^{45}$ (Jm$^3$)$^{-1}$, $L=400$ nm, and $A=40\times 800$ nm$^2$. At $T=100$ mK and $f=1$ Hz, we obtain $S_Q/e^2=2\times10^{-7}$ Hz$^{-1}$, which is comparable to the experimental value of $4\times 10^{-6}$ Hz$^{-1}$ deduced from current noise [@Astafiev2006]. As Eq. (\[eq:SQ\_LF\]) shows, the standard TLS distribution gives low frequency $1/f$ charge noise that is linear in temperature, while experiments find a quadratic temperature dependence [@Kenyon2000; @Astafiev2006]. This implies that at low frequencies and temperatures contributions from other mechanisms may dominate the charge noise [@Faoro2005]. To conclude, we have shown that the frequency and temperature dependence of high frequency charge noise in Josephson junction devices depends on the ratio $J/J_c(\omega,T)$ of the electromagnetic flux to the critical flux. Using the standard theory of two level systems with a flat density of states, we find that the charge noise at high frequencies can increase linearly with frequency and temperature if $J/J_c(\omega,T)\gg 1$. This agrees with the conclusions of recent experiments on the high frequency charge noise in Josephson junction qubits [@Astafiev2004] which our estimates show are in the strongly saturated limit. This work was supported by ARDA through ARO Grant W911NF-04-1-0204, and by DOE grant DE-FG02-04ER46107. M. C. wishes to thank Fred Wellstood for useful discussions. 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ArXiv

--- abstract: | We consider the problem of designing a packet-level congestion control and scheduling policy for datacenter networks. Current datacenter networks primarily inherit the principles that went into the design of Internet, where congestion control and scheduling are distributed. While distributed architecture provides robustness, it suffers in terms of performance. Unlike Internet, data center is fundamentally a “controlled” environment. This raises the possibility of designing a centralized architecture to achieve better performance. Recent solutions such as Fastpass [@perry2014fastpass] and Flowtune [@perry17flowtune] have provided the proof of this concept. This raises the question: what is theoretically optimal performance achievable in a data center? We propose a centralized policy that guarantees a per-flow end-to-end flow delay bound of $O$(\#hops $\times$ flow-size $/$ gap-to-capacity). Effectively such an end-to-end delay will be experienced by flows even if we removed congestion control and scheduling constraints as the resulting queueing networks can be viewed as the classical [*reversible*]{} multi-class queuing network, which has a product-form stationary distribution. In the language of [@harrison2014bandwidth], we establish that [*baseline*]{} performance for this model class is achievable. Indeed, as the key contribution of this work, we propose a method to [*emulate*]{} such a reversible queuing network while satisfying congestion control and scheduling constraints. Precisely, our policy is an emulation of Store-and-Forward (SFA) congestion control in conjunction with Last-Come-First-Serve Preemptive-Resume (LCFS-PR) scheduling policy. address: - - author: - - bibliography: - 'datacenter.bib' title: Centralized Congestion Control and Scheduling in a Datacenter --- Introduction ============ With an increasing variety of applications and workloads being hosted in datacenters, it is highly desirable to design datacenters that provide high throughput and low latency. Current datacenter networks primarily employ the design principle of Internet, where congestion control and packet scheduling decisions are distributed among endpoints and routers. While distributed architecture provides scalability and fault-tolerance, it is known to suffer from throughput loss and high latency, as each node lacks complete knowledge of entire network conditions and thus fails to take a globally optimal decision. The datacenter network is fundamentally different from the wide-area Internet in that it is under a single administrative control. Such a single-operator environment makes a centralized architecture a feasible option. Indeed, there have been recent proposals for centralized control design for data center networks [@perry17flowtune; @perry2014fastpass]. In particular, Fastpass [@perry2014fastpass] uses a centralized arbiter to determine the path as well as the time slot of transmission for each packet, so as to achieve zero queueing at switches. Flowtune [@perry17flowtune] also uses a centralized controller, but congestion control decisions are made at the granularity of a flowlet, with the goal of achieving rapid convergence to a desired rate allocation. Preliminary evaluation of these approaches demonstrates promising empirical performance, suggesting the feasibility of a centralized design for practical datacenter networks. Motivated by the empirical success of the above work, we are interested in investigating the theoretically optimal performance achievable by a centralized scheme for datacenters. Precisely, we consider a centralized architecture, where the congestion control and packet transmission are delegated to a centralized controller. The controller collects all dynamic endpoints and switches state information, and redistributes the congestion control and scheduling decisions to all switches/endpoints. We propose a packet-level policy that guarantees a per-flow end-to-end flow delay bound of $\ensuremath{O}(\#\text{hops}\ensuremath{\times}\text{flow-size}\ensuremath{/}\text{gap-to-capacity})$. To the best of our knowledge, our result is the first one to show that it is possible to achieve such a delay bound in a network with congestion control and scheduling constraints. Before describing the details of our approach, we first discuss related work addressing various aspects of the network resource allocation problem. Related Work ------------ There is a very rich literature on congestion control and scheduling. The literature on congestion control has been primarily driven by bandwidth allocation in the context of Internet. The literature on packet scheduling has been historically driven by managing supply-chain (multi-class queueing networks), telephone networks (loss-networks), switch / wireless networks (packet switched networks) and now data center networks. In what follows, we provide brief overview of representative results from theoretical and systems literature. **Job or Packet Scheduling:** A scheduling policy in the context of classical multi-class queueing networks essentially specifies the service discipline at each queue, i.e., the order in which waiting jobs are served. Certain service disciplines, including the last-come-first-serve preemptive-resume (LCFS-PR) policy and the processor sharing discipline, are known to result in quasi-reversible multi-class queueing networks, which have a product form equilibrium distribution [@kelly1979reversibility]. The crisp description of the equilibrium distribution makes these disciplines remarkably tractable analytically. More recently, the scheduling problems for switched networks, which are special cases of stochastic processing networks as introduced by Harrison [@harrison2000brownian], have attracted a lot of attention starting [@tassiulas1992maxweight] including some recent examples [@walton2014concave; @shah2014SFA; @maguluri2015heavy]. Switched networks are queueing networks where there are constraints on which queues can be served simultaneously. They effectively model a variety of interesting applications, exemplified by wireless communication networks, and input-queued switches for Internet routers. The MaxWeight/BackPressure policy, introduced by Tassiulas and Ephremides for wireless communication [@tassiulas1992maxweight; @mckeown1996achieving], have been shown to achieve a maximum throughput stability for switched networks. However, the provable delay bounds of this scheme scale with the number of queues in the network. As the scheme requires maintaining one queue per route-destination at each one, the scaling can be potentially very bad. For instance, recently Gupta and Javidi [@gupta2007routing] showed that such an algorithm can result in very poor delay performance via a specific example. Walton [@walton2014concave] proposed a proportional scheduler which achieves throughput optimality as the BackPressure policy, while using a much simpler queueing structure with one queue per link. However, the delay performance of this approach is unknown. Recently Shah, Walton and Zhong [@shah2014SFA] proposed a policy where the scheduling decisions are made to approximate a queueing network with a product-form steady state distribution. The policy achieves optimal queue-size scaling for a class of switched networks.In a recent work, Theja and Srikant [@maguluri2015heavy] established heavy-traffic optimality of MaxWeight policy for input-queued switches. **Congestion control:** A long line of literature on congestion control began with the work of Kelly, Maulloo and Tan [@kelly1998rate], where they introduced an optimization framework for flow-level resource allocation in the Internet. In particular, the rate control algorithms are developed as decentralized solutions to the utility maximization problems. The utility function can be chosen by the network operator to achieve different bandwidth and fairness objectives. Subsequently, this optimization framework has been applied to analyze existing congestion control protocols (such as TCP) [@low2002vegas; @low2002internet; @mo2000fair]; a comprehensive overview can be found in [@srikant_book]. Roberts and Massoulié [@massoulie2000bandwidth] applied this paradigm to settings where flows stochastically depart and arrive, known as bandwidth sharing networks. The resulting proportional fairness policies have been shown to be maximum stable [@bonald2001fairness; @massoulie2007fairness]. The heavy traffic behavior of proportional fairness has been subsequently studied [@shah2014qualitative; @kang2009diffusion]. Another bandwidth allocation of interest is the store-and-forward allocation (SFA) policy, which was first introduced by Massoulié (see Section 3.4.1 in [@proutiere_thesis]) and later analyzed in the thesis of Proutière [@proutiere_thesis]. The SFA policy has the remarkable property of insensitivity with respect to service distributions, as shown by Bonald and Proutière [@bonald2003insensitive], and Zachary[@zachary2007insensitivity]. Additionally, this policy induces a product-form stationary distribution [@bonald2003insensitive]. The relationship between SFA and proportional fairness has been explored [@massoulie2007fairness], where SFA was shown to converge to proportional fairness with respect to a specific asymptote. **Joint congestion control and scheduling:** More recently, the problem of designing joint congestion-control and scheduling mechanisms has been investigated [@eryilmaz2006joint; @lin2004joint; @stolyar2005maximizing]. The main idea of these approaches is to combine a queue-length-based scheduler and a distributed congestion controller developed for wireline networks, to achieve stability and fair rate allocation. For instance, the joint scheme proposed by Eryilmaz and Srikant combines the BackPressure scheduler and a primal-dual congestion controller for wireless networks. This line of work focuses on addressing the question of the stability. The Lyapunov function based delay (or queue-size) bound for such algorithm are relatively very poor. It is highly desirable to design a joint mechanism that is provably throughput optimal and has low delay bound. Indeed, the work of Moallemi and Shah [@moallemi2010flow] was an attempt in this direction, where they developed a stochastic model that jointly captures the packet- and flow-level dynamics of a network, and proposed a joint policy based on $\alpha$-weighted policies. They argued that in a certain asymptotic regime (critically loaded fluid model) the resulting algorithm induces queue-sizes that are within constant factor of optimal quantities. However, this work stops short of providing non-asymptotic delay guarantees. **Emulation:** In our approach we utilize the concept of emulation, which was introduced by Prabhakar and Mckeown [@prabhakar1999speedup] and used in the context of bipartite matching. Informally, a network is said to emulate another network, if the departure processes from the two networks are identical under identical arrival processes. This powerful technique has been subsequently used in a variety of applications [@jagabathula2008delay_scheduling; @shah2014SFA; @gamal2006throughput_delay; @chuang1999matching]. For instance, Jagabathula and Shah designed a delay optimal scheduling policy for a discrete-time network with arbitrary constraints, by emulating a quasi-reversible continuous time network [@jagabathula2008delay_scheduling]; The scheduling algorithm proposed by Shah, Walton and Zhong [@shah2014SFA] for a single-hop switched network, effectively emulates the bandwidth sharing network operating under the SFA policy. However, it is unknown how to apply the emulation approach to design a joint congestion control and scheduling scheme. **Datacenter Transport:** Here we restrict to system literature in the context of datacenters. Since traditional TCP developed for wide-area Internet does not meet the strict low latency and high throughput requirements in datacenters, new resource allocation schemes have been proposed and deployed [@alizadeh2010DCTCP; @alizadeh2013pfabric; @nagaraj2016numfabric; @hong2012pdq; @perry17flowtune; @perry2014fastpass]. Most of these systems adopt distributed congestion control schemes, with the exception of Fasspass [@perry2014fastpass] and Flowtune [@perry17flowtune]. DCTCP [@alizadeh2010DCTCP] is a delay-based (queueing) congestion control algorithm with a similar control protocol as that in TCP. It aims to keep the switch queues small, by leveraging Explicit Congestion Notification (ECN) to provide multi-bit feedback to the end points. Both pFabric [@alizadeh2013pfabric] and PDQ [@hong2012pdq] aim to reduce flow completion time, by utilizing a distributed approximation of the shortest remaining flow first policy. In particular, pFabric uses in-network packet scheduling to decouple the network’s scheduling policy from rate control. NUMFabric [@nagaraj2016numfabric] is also based on the insight that utilization control and network scheduling should be decoupled. In particular, it combines a packet scheduling mechanism based on weighted fair queueing (WFQ) at the switches, and a rate control scheme at the hosts that is based on the network utility maximization framework. Our work is motivated by the recent successful stories that demonstrate the viability of centralized control in the context of datacenter networks [@perry2014fastpass; @perry17flowtune]. Fastpass [@perry2014fastpass] uses a centralized arbiter to determine the path as well as the time slot of transmission for each packet. To determine the set of sender-receiver endpoints that can communicate in a timeslot, the arbiter views the entire network as a single input-queued switch, and uses a heuristic to find a matching of endpoints in each timeslot. The arbiter then chooses a path through the network for each packet that has been allocated timeslots. To achieve zero-queue at switches, the arbiter assigns packets to paths such that no link is assigned multiple packets in a single timeslot. That is, each packet is arranged to arrive at a switch on the path just as the next link to the destination becomes available. Flowtune [@perry17flowtune] also uses a centralized controller, but congestion control decisions are made at the granularity of a flowlet, which refers to a batch of packets backlogged at a sender. It aims to achieve fast convergence to optimal rates by avoiding packet-level rate fluctuations. To be precise, a centralized allocator computes the optimal rates for a set of active flowlets, and those rates are updated dynamically when flowlets enter or leave the network. In particular, the allocated rates maximize the specified network utility, such as proportional fairness. **Baseline performance:** In a recent work Harrison et al. [@harrison2014bandwidth] studied the *baseline performance* for congestion control, that is, an achievable benchmark for the delay performance in flow-level models. Such a benchmark provides an upper bound on the optimal achievable performance. In particular, baseline performance in flow-level models is exactly achievable by the store-and-forward allocation (SFA) mentioned earlier. On the other hand, the work by Shah et al. [@shah2014SFA] established baseline performance for scheduling in packet-level networks. They proposed a scheduling policy that effectively emulates the bandwidth sharing network under the SFA policy. The results for both flow- and packet-level models boil down to a product-form stationary distribution, where each component of the product-form behaves like an $M/M/1$ queue. However, no baseline performance has been established for a hybrid model with flow-level congestion control and packet scheduling. This is precisely the problem we seek to address in this paper. The goal of this paper is to understand what is the best performance achievable by centralized designs in datacenter networks. In particular, we aim to establish baseline performance for datacenter networks with congestion control and scheduling constraints. To investigate this problem, we consider a datacenter network with a tree topology, and focus on a hybrid model with simultaneous dynamics of flows and packets. Flows arrive at each endpoint according to an exogenous process and wish to transmit some amount of data through the network. As in standard congestion control algorithms, the flows generate packets at their ingress to the network. The packets travel to their respective destinations along links in the network. We defer the model details to Section \[sec:Model-and-Notation\]. Our approach ------------ The control of a data network comprises of two sub-problems: congestion control and scheduling. On the one hand, congestion control aims to ensure fair sharing of network resources among endpoints and to minimize congestion inside the network. The congestion control policy determines the rates at which each endpoint injects data into the internal network for transmission. On the other hand, the internal network maintains buffers for packets that are in transit across the network, where the queues at each buffer are managed according to some packet scheduling policy. Our approach addresses these two sub-problems simultaneously, with the overall architecture shown in Figure \[fig:Overview\]. The system decouples congestion control and in-network packet scheduling by maintaining two types of buffers: *external* buffers which store arriving flows of different types, and *internal* buffers for packets in transit across the network. In particular, there is a separate external buffer for each type of arriving flows. Internally, at each directed link $l$ between two nodes $(u,v)$ of the network, there is an internal buffer for storing packets waiting at node $u$ to pass through link $l$. Conceptually, the internal queueing structure corresponds to the output-queued switch fabric of ToR and core switches in a datacenter, so each directed link is abstracted as a queueing server for packet transmission. Our approach employs independent mechanisms for the two sub-problems. The congestion control policy uses only the state of external buffers for rate allocation, and is hence decoupled from packet scheduling. The rates allocated for a set of flows that share the network will change only when new flows arrive or when flows are admitted into the internal network for transmission. For the internal network, we adopt a packet scheduling mechanism based on the dynamics of internal buffers. ![Overview of the congestion control and scheduling scheme.\[fig:Overview\]](overview){width="0.99\columnwidth"} Figure \[fig:congestion\_control\] illustrates our congestion control policy. The key idea is to view the system as a bandwidth sharing network with flow-level capacity constraints. The rate allocated to each flow buffered at source nodes is determined by an online algorithm that only uses the queue lengths of the external buffers, and satisfies the capacity constraints. Another key ingredient of our algorithm is a mechanism that translates the allocated rates to congestion control decisions. In particular, we implement the congestion control algorithm at the granularity of flows, as opposed to adjusting the rates on a packet-by-packet basis as in classical TCP. We consider a specific bandwidth allocation scheme called the store-and-forward algorithm (SFA), which was first considered by Massoulié and later discussed in [@bonald2003insensitive; @kelly2009resource; @proutiere_thesis; @walton2009fairness]. The SFA policy has been shown to be insensitive with respect to general service time distributions [@zachary2007insensitivity], and result in a reversible network with Poisson arrival processes [@walton2009fairness]. The bandwidth sharing network under SFA has a product-form queue size distribution in equilibrium. Given this precise description of the stationary distribution, we can obtain an explicit bound on the number of flows waiting at the source nodes, which has the desirable form of $\ensuremath{O}(\#\text{hops}\ensuremath{\times}\text{flow-size}\ensuremath{/}\text{gap-to-capacity})$. Details of the congestion control policy description and analysis are given in Section \[sec:congestion\_control\] to follow. ![A congestion control policy based on the SFA algorithm for a bandwidth sharing network.\[fig:congestion\_control\]](congestion){width="0.9\columnwidth"} We also make use of the concept of emulation to design a packet scheduling algorithm in the internal network, which is operated in discrete time. In particular, we propose and analyze a scheduling mechanism that is able to emulate a continuous-time *quasi-reversible* network, which has a highly desirable queue-size scaling. Our design consists of three elements. First, we specify the granularity of timeslot in a way that maintains the same throughput as in a network without the discretization constraint. By appropriately choosing the granularity, we are able to address a general setting where flows arriving on each route can have arbitrary sizes, as opposed to prior work that assumed unit-size flows. Second, we consider a continuous-time network operated under the Last-Come-First-Serve Preemptive-Resume (LCFS-PR) policy. If flows on each route are assumed to arrive according a Poisson process, the resulting queueing network is quasi-reversible with a product-form stationary distribution. In this continuous-time setting, we will show that the network achieves a flow delay bound of $\ensuremath{O}(\#\text{hops }\ensuremath{\times}\text{ flow-size }\ensuremath{/}\text{ gap-to-capacity})$. Finally, we design a feasible scheduling policy for the discrete-time network, which achieves the same throughput and delay bounds as the continuous-time network. The resulting scheduling scheme is illustrated in Figure \[fig:scheduling\]. ![An adapted LCFS-PR scheduling algorithm \[fig:scheduling\]](scheduling){width="0.7\columnwidth"} Our Contributions ----------------- The main contribution of the paper is a centralized policy for both congestion control and scheduling that achieves a per-flow end-to-end delay bound $\ensuremath{O}(\#\text{hops }\ensuremath{\times}$ $\text{flow-size }\ensuremath{/}\text{ gap-to-capacity})$. Some salient aspects of our result are: 1. The policy addresses both the congestion control and scheduling problems, in contrast to other previous work that focused on either congestion control or scheduling. 2. We consider flows with variable sizes. 3. We provide per-flow delay bound rather than an aggregate bound. 4. Our results are non-asymptotic, in the sense that they hold for any admissible load. 5. A central component of our design is the emulation of continuous-time quasi-reversible networks with a product-form stationary distribution. By emulating these queueing networks, we are able to translate the results therein to the network with congestion and scheduling constraints. This emulation result can be of interest in its own right. Organization ------------ The remaining sections of the paper are organized as follows. In Section \[sec:Model-and-Notation\] we describe the network model. The main results of the paper are presented in Section \[sec:results\]. The congestion control algorithm is described and analyzed in Section \[sec:congestion\_control\]. Section \[sec:scheduling\] details the scheduling algorithm and its performance properties. We discuss implementation issues and conclude the paper in Section \[sec:discussion\].

ArXiv

--- author: - | Simon Donig [](https://orcid.org/0000-0002-1741-466X)\ Chair for Digital Humanities\ University Passau, Germany\ [[email protected]]([email protected])\ Maria Christoforaki\ Chair for Data Science\ Institute for Computer Science\ University of St.Gallen, Switzerland\ [[email protected]]([email protected])\ Bernhard Bermeitinger [](https://orcid.org/0000-0002-2524-1850)\ Chair for Data Science\ Institute for Computer Science\ University of St.Gallen, Switzerland\ [[email protected]]([email protected])\ Siegfried Handschuh\ Chair for Data Science\ Institute for Computer Science\ University of St.Gallen, Switzerland\ [[email protected]]([email protected])\ bibliography: - 'references.bib' date: December 2019 title: | Multimodal Semantic Transfer\ from Text to Image.\ Fine-Grained Image Classification\ by Distributional Semantics. --- Introduction ============ In the last years, image classification processes like neural networks in the area of art-history and *Heritage Informatics* have experienced a broad distribution [@lang_AttestingSimilaritySupportingOrganizationStudy_2018]. These methods face several challenges, including the handling of comparatively small amounts of data as well as high-dimensional data in the Digital Humanities. In most cases, these methods map the classification task to flat target space. This “flat” surface loses several relevant dimensions in the search for ontological uniqueness, including taxonomical, mereological, and associative relationships between classes, or the non-formal context, respectively. The proposed solution by @donig_VomBildTextUndWieder_2019a [@donig_VomBildTextUndWieder_2019a] to expand the capabilities of visual classifiers is to take advantage of the greater expressiveness of text-based models. Here, a *Convolutional Neural Network* (CNN) is used that output is not as usual a series of flat text labels but a series of semantically loaded vectors. These vectors result from a *Distributional Semantic Model* (DSM) ([@lenci_DistributionalModelsWordMeaning_2018a]) which is generated from an in-domain text corpus. Here, we propose an early implementation of this method and analyze the results. The conducted experiment is based on the collation of two corpora: one text-based and a visual. From the text, a DSM is created and then queried for a list of target words that are functionally the labels that are manually given to the images. The result is a list of vectors that correspond to the target words leading to images that are annotated not only with a label but also with a unique vector. The images and vectors are used as training data for a CNN that, afterward, should be able to predict a vector for an unseen image. This prediction vector can be converted back to a word by the DSM using a nearest-neighbor algorithm. We are looking for richer representation in this process, so we choose the five nearest neighbors. The similarity measure is the cosine similarity for high-dimensional vector spaces between the given target vector and the prediction vector. We derive a positive classification result if the target label is within the list of five nearest neighbors of the prediction vector. Moreover, we compare the results between the proposed classification method and a conventional classification method using the same CNN as for the vector-based experiment but a list of flat labels. Finally, we can show that the vector-based approach (judging from classification metrics) is equally performant or even better than the label-based version. Experiment Structure ==================== The experiment is based on one text and one visual corpus from the area of material culture research with a focus on neo-classical artifacts. Text Corpus ----------- The text corpus is compiled out of 44 sources that are available under a free and permissive license. It contains English specialist publications on furniture and spatial art, published from the end of the 19^th^ century to the middle of the 20^th^ century. In multiple steps, the corpus is cleaned and preprocessed. First, a series of standard NLP methods are applied like tokenization, sentence- and word splitting, normalization of numbers, and named entity recognition (NER). Since we used retro-digitized material from a different source, we implemented manual corrections for the most common errors (such as ligatures like `TT` that were misinterpreted as `U`). Another level of preprocessing consists of content-related augmentations. In particular, we normalized compound words and synonyms according to a specified list, which is based on an ontology, the *Neoclassica-Ontology* [@donig_NeoclassicaMultilingualDomainOntology_2016]. This resulted in the final text corpus of total words comprised of basic word forms. The DSM is created using the *Indra Frameworks* [@sales_IndraWordEmbeddingSemanticRelatedness_2018a] with a vector size of , a word window size of , and minimal word count of . We used *Skipgram* as the *Word2Vec* model [@mikolov_EfficientEstimationWordRepresentationsVector_2013] with negative sampling. Image Corpus ------------ The image corpus consists of images of neoclassical furniture in their entirety, which are permissive licensed[^1]. The images are either historical pictorial material or photographs from the modern as-built documentation. They are split into different classes. Combined Corpus --------------- The nature of the experiment is *proof-of-concept*, so we used a VGG-like architecture of a “simple” convolutional neural network[^2]. The independence and robustness of the train/test split are guaranteed with 5-fold cross-validation on on the full corpus from which are used for training and for testing. The remaining are treated as an evaluation set. ![Class distribution of the image corpus[]{data-label="fig:class_distribution"}](class_distribution){height=".45\textheight"} The dataset is unbalanced, as can be seen in Fig. \[fig:class\_distribution\]. During training, more prominent classes are weighted down and underrepresented classes are given a higher weight [@johnson_SurveyDeepLearningClassImbalance_2019 p. 27]. Apart from dropout during training for regularization, *Early Stopping* was used to prevent overfitting. Results ======= Results for CNN trained on Word Vectors --------------------------------------- The Top-5 true-positive rate is , meaning that the gold label from the annotations is in of the cases within the list of the five nearest neighbors. However, the mathematical quality metric in itself represents only part of the overall picture. For this reason, a qualitative analysis of the results was performed. A few true-positives show that, for example, the classification is by no means random but that the top 5 terms originate from the same semantic neighborhoods. They express several relationships of a taxonomical and associative nature. As an example, the *Roentgen* desk from the *Victoria & Albert* inventory in Fig. \[fig:classification\_same\_object\_1\] is associated with the labels `dressing_table`, `writing_table` and `work_table`. This triad is meaningful because many of these artifacts were multifunctional and fulfilled several of these functions. Besides, those artifacts that decidedly served only one purpose are constructively similar to the other types of furniture. The similarity of the three concepts thus emerges both on a semantic level (the distance of the words in the DSM, which in turn is the product of real-world distance) and on a visual level in the CNN (visual similarity of the form). Another image of the same object (see Fig. \[fig:classification\_same\_object\_2\]) shows, on the other hand, that the method is consisted in itself—the top 4 nearest neighbors of the predicted vector are identical, although the photograph is taken from a different perspective—and, on the other hand, that the visual features within the CNN also affect the classification process. Since writing cabinets (*secrétaires à abbatants*) are often displayed frontally, upright and with an open flap or drawer, their appearance in the image seems to have triggered a classification as a secretary. In the first image, however, the presence of drawers could have led to a classification as a chest-of-drawers, which is associated with drawers on a semantic level. ![Differences in the classification of the same object (1)[]{data-label="fig:classification_same_object_1"}](classification_same_object_1){height=".4\textheight"} ![Differences in the classification of the same object (2)[]{data-label="fig:classification_same_object_2"}](classification_same_object_2){height=".4\textheight"} While the labels considered so far reflect the taxonomic relationships and all originate from the target words derived from the ontology, Fig. \[fig:medici\_vase\] shows that the procedure can also generate labels for itself, purely data-driven. The crater vase shown was originally classified as an urn. The Top-2 words therefore also reflect taxonomic relationships (“urn”, “vase”). The other concepts reflect associative relationships. The label `bell` is a leftover from the data cleaning process of the text corpus to describe this kind of artifact. “Ovoid” refers to the egg stick decoration of the upper bead, which is often described with this adjective. This ornamentation seems at the same time to have an association with the rosette (patera). In this way, the target word `patera_element` appears among the top-5 nearest neighbors, although only whole artifacts were annotated in the image corpus but not their decoration. ![A sèvres copy of a Medici vase results in a classification of associative labels.[]{data-label="fig:medici_vase"}](medici_vase){height=".6\textheight"} An effect of the visual classifier cannot be excluded, as shown in Fig. \[fig:misclassification\]. The misclassification of the object, a small sewing table, led to consistent attributions in the area of seating and reclining furniture. Looking at the outer form of the artifact on a more abstract level, visual proximity to e.g. a (double)-camel-back sofa can be easily derived. ![Misclassification of a sewing table in the proximity of seating furniture hierarchy.[]{data-label="fig:misclassification"}](misclassification){height=".6\textheight"} Comparison to a CNN with flat labels ------------------------------------ To better estimate the differences between the two approaches (conventional classification by flat labels vs. vector-based approach), we present a comparison between different metrics for both approaches. Metric Vector-based Label-based --------------------------- -------------- ------------- Top-1 Accuracy Top-1 Precision Top-1 Recall Top-1 F1-Score Top-5 True-Positive-Rate Top-5 False-Positive-Rate : Results of different Metrics.[]{data-label="tab:results"} As shown in Table \[tab:results\] the metrics are comparable. Thus, the proposed approach not only improves the accuracy but also provides a richer description of the image. Conclusion and Outlook ====================== In this paper, we have presented a new multimodal approach for the classification of images based on the combination of NLP methods with image classification techniques. The goal was to classify objects not only according to a scheme of flat labels but in a more context-appropriate way, whereby the context informed by relevant domain-historical publications. This classification method offers access to the multidimensional embedding of artifacts in the real-world and their linguistic reflection. This circumstance is particularly useful for classifying multifunctional objects without having to resort to several classifiers and a complex annotation process with several labels. The results are encouraging. Even with a very simple CNN, we achieve an accuracy of . As a next step, we want to train with a deeper CNN and an extended image corpus—to reduce known problems like overfitting. The comparative experiment with a conventional flat-label approach has shown that from an efficiency point-of-view, i.e., indirect metric comparison, our method not only provides comparable results but also provides a richer description of the image. We will continue to work to better understand how a particular body of text is reflected int he labels that the DSM automatically assigns and that are not part of the list of target words. A better understanding of these processes seems particularly relevant given the relatively manageable text corpora that can be collated into specific topic complexes in the humanities. Last but not least, for this reason, we will consider the use of thesauri and dictionaries to create synonym lists for target words. Similarly, we are considering combining named entities into URIs. This would allow us to associate specific entities (e.g., workshops, ebenists, owners) with specific objects. We think that multimodal access provides particularly efficient access to humanities and cultural studies corpora that are small and domain-restricted compared to corpora of other disciplines in the natural and social sciences. [^1]: The corpus was compiled from collections from the Metropolitan Museum, New York, the Victoria & Albert Museum, London, the Wallace Collection, London, and several contemporary pattern books. [^2]: The CNN is built from three convolutional blocks with two consecutive Convolutional Layers each with 32/64/64 filters of size $ 3 \times 3 $. Each block is followed by a Maximum Pooling Layer with a size of $ 2 \times 2 $ and a Dropout Layer for regularization with a dropout rate of . After the Convolutional Block, there are two Fully Connected Layers with nodes and a Dropout Layer with a dropout rate of . The weights and biases for the Convolutional and the Fully Connected Layers are initialized randomly. Their activation function is *ReLU*. The last layer, the classification layer, is a Fully Connected Layer with output nodes and a linear activation function. The optimization function of the chosen *mean absolute error* function is *RMSprop*. The implementation is done with Keras (<https://keras.io>) and TensorFlow (<https://tensorflow.org>)

ArXiv

--- abstract: | We present an overview of scalable load balancing algorithms which provide favorable delay performance in large-scale systems, and yet only require minimal implementation overhead. Aimed at a broad audience, the paper starts with an introduction to the basic load balancing scenario – referred to as the *supermarket model* – consisting of a single dispatcher where tasks arrive that must immediately be forwarded to one of $N$ single-server queues. The supermarket model is a dynamic counterpart of the classical balls-and-bins setup where balls must be sequentially distributed across bins. A popular class of load balancing algorithms are so-called power-of-$d$ or JSQ($d$) policies, where an incoming task is assigned to a server with the shortest queue among $d$ servers selected uniformly at random. As the name reflects, this class includes the celebrated Join-the-Shortest-Queue (JSQ) policy as a special case ($d = N$), which has strong stochastic optimality properties and yields a mean waiting time that *vanishes* as $N$ grows large for any fixed subcritical load. However, a nominal implementation of the JSQ policy involves a prohibitive communication burden in large-scale deployments. In contrast, a simple random assignment policy ($d = 1$) does not entail any communication overhead, but the mean waiting time remains constant as $N$ grows large for any fixed positive load. In order to examine the fundamental trade-off between delay performance and implementation overhead, we consider an asymptotic regime where the diversity parameter $d(N)$ depends on $N$. We investigate what growth rate of $d(N)$ is required to match the optimal performance of the JSQ policy on fluid and diffusion scale, and achieve a vanishing waiting time in the limit. The results demonstrate that the asymptotics for the JSQ($d(N)$) policy are insensitive to the exact growth rate of $d(N)$, as long as the latter is sufficiently fast, implying that the optimality of the JSQ policy can asymptotically be preserved while dramatically reducing the communication overhead. Stochastic coupling techniques play an instrumental role in establishing the asymptotic optimality and universality properties, and augmentations of the coupling constructions allow these properties to be extended to infinite-server settings and network scenarios. We additionally show how the communication overhead can be reduced yet further by the so-called Join-the-Idle-Queue (JIQ) scheme, leveraging memory at the dispatcher to keep track of idle servers. author: - Mark van der Boor - 'Sem C. Borst' - 'Johan S.H. van Leeuwaarden' - Debankur Mukherjee title: | Scalable Load Balancing in Networked Systems:\ Universality Properties and Stochastic Coupling Methods --- Introduction ============ In the present paper we review scalable load balancing algorithms (LBAs) which achieve excellent delay performance in large-scale systems and yet only involve low implementation overhead. LBAs play a critical role in distributing service requests or tasks (e.g. compute jobs, data base look-ups, file transfers) among servers or distributed resources in parallel-processing systems. The analysis and design of LBAs has attracted strong attention in recent years, mainly spurred by crucial scalability challenges arising in cloud networks and data centers with massive numbers of servers. LBAs can be broadly categorized as static, dynamic, or some intermediate blend, depending on the amount of feedback or state information (e.g. congestion levels) that is used in allocating tasks. The use of state information naturally allows dynamic policies to achieve better delay performance, but also involves higher implementation complexity and a substantial communication burden. The latter issue is particularly pertinent in cloud networks and data centers with immense numbers of servers handling a huge influx of service requests. In order to capture the large-scale context, we examine scalability properties through the prism of asymptotic scalings where the system size grows large, and identify LBAs which strike an optimal balance between delay performance and implementation overhead in that regime. The most basic load balancing scenario consists of $N$ identical parallel servers and a dispatcher where tasks arrive that must immediately be forwarded to one of the servers. Tasks are assumed to have unit-mean exponentially distributed service requirements, and the service discipline at each server is supposed to be oblivious to the actual service requirements. In this canonical setup, the celebrated Join-the-Shortest-Queue (JSQ) policy has several strong stochastic optimality properties. In particular, the JSQ policy achieves the minimum mean overall delay among all non-anticipating policies that do not have any advance knowledge of the service requirements [@EVW80; @Winston77]. In order to implement the JSQ policy however, a dispatcher requires instantaneous knowledge of all the queue lengths, which may involve a prohibitive communication burden with a large number of servers $N$. This poor scalability has motivated consideration of JSQ($d$) policies, where an incoming task is assigned to a server with the shortest queue among $d \geq 2$ servers selected uniformly at random. Note that this involves exchange of $2 d$ messages per task, irrespective of the number of servers $N$. Results in Mitzenmacher [@Mitzenmacher01] and Vvedenskaya [*et al.*]{} [@VDK96] indicate that even sampling as few as $d = 2$ servers yields significant performance enhancements over purely random assignment ($d = 1$) as $N$ grows large, which is commonly referred to as the “power-of-two” or “power-of-choice” effect. Specifically, when tasks arrive at rate $\lambda N$, the queue length distribution at each individual server exhibits super-exponential decay for any fixed $\lambda < 1$ as $N$ grows large, compared to exponential decay for purely random assignment. As illustrated by the above, the diversity parameter $d$ induces a fundamental trade-off between the amount of communication overhead and the delay performance. Specifically, a random assignment policy does not entail any communication burden, but the mean waiting time remains *constant* as $N$ grows large for any fixed $\lambda > 0$. In contrast, a nominal implementation of the JSQ policy (without maintaining state information at the dispatcher) involves $2 N$ messages per task, but the mean waiting time *vanishes* as $N$ grows large for any fixed $\lambda < 1$. Although JSQ($d$) policies with $d \geq 2$ yield major performance improvements over purely random assignment while reducing the communication burden by a factor O($N$) compared to the JSQ policy, the mean waiting time *does not vanish* in the limit. Thus, no fixed value of $d$ will provide asymptotically optimal delay performance. This is evidenced by results of Gamarnik [*et al.*]{} [@GTZ16] indicating that in the absence of any memory at the dispatcher the communication overhead per task *must increase* with $N$ in order for any scheme to achieve a zero mean waiting time in the limit. We will explore the intrinsic trade-off between delay performance and communication overhead as governed by the diversity parameter $d$, in conjunction with the relative load $\lambda$. The latter trade-off is examined in an asymptotic regime where not only the overall task arrival rate is assumed to grow with $N$, but also the diversity parameter is allowed to depend on $N$. We write $\lambda(N)$ and $d(N)$, respectively, to explicitly reflect that, and investigate what growth rate of $d(N)$ is required, depending on the scaling behavior of $\lambda(N)$, in order to achieve a zero mean waiting time in the limit. We establish that the fluid-scale and diffusion-scale limiting processes are insensitive to the exact growth rate of $d(N)$, as long as the latter is sufficiently fast, and in particular coincide with the limiting processes for the JSQ policy. This reflects a remarkable universality property and demonstrates that the optimality of the JSQ policy can asymptotically be preserved while dramatically lowering the communication overhead. We will extend the above-mentioned universality properties to network scenarios where the $N$ servers are assumed to be inter-connected by some underlying graph topology $G_N$. Tasks arrive at the various servers as independent Poisson processes of rate $\lambda$, and each incoming task is assigned to whichever server has the shortest queue among the one where it appears and its neighbors in $G_N$. In case $G_N$ is a clique, each incoming task is assigned to the server with the shortest queue across the entire system, and the behavior is equivalent to that under the JSQ policy. The above-mentioned stochastic optimality properties of the JSQ policy thus imply that the queue length process in a clique will be ‘better’ than in an arbitrary graph $G_N$. We will establish sufficient conditions for the fluid-scaled and diffucion-scaled versions of the queue length process in an arbitrary graph to be equivalent to the limiting processes in a clique as $N \to \infty$. The conditions reflect similar universality properties as described above, and in particular demonstrate that the optimality of a clique can asymptotically be preserved while markedly reducing the number of connections, provided the graph $G_N$ is suitably random. While a zero waiting time can be achieved in the limit by sampling only $d(N) = o(N)$ servers, the amount of communication overhead in terms of $d(N)$ must still grow with $N$. This may be explained from the fact that a large number of servers need to be sampled for each incoming task to ensure that at least one of them is found idle with high probability. As alluded to above, this can be avoided by introducing memory at the dispatcher, in particular maintaining a record of vacant servers, and assigning tasks to idle servers, if there are any. This so-called Join-the-Idle-Queue (JIQ) scheme [@BB08; @LXKGLG11] has gained huge popularity recently, and can be implemented through a simple token-based mechanism generating at most one message per task. As established by Stolyar [@Stolyar15], the fluid-scaled queue length process under the JIQ scheme is equivalent to that under the JSQ policy as $N \to \infty$, and this result can be shown to extend the diffusion-scaled queue length process. Thus, the use of memory allows the JIQ scheme to achieve asymptotically optimal delay performance with minimal communication overhead. In particular, ensuring that tasks are assigned to idle servers whenever available is sufficient to achieve asymptotic optimality, and using any additional queue length information yields no meaningful performance benefits on the fluid or diffusion levels. Stochastic coupling techniques play an instrumental role in the proofs of the above-described universality and asymptotic optimality properties. A direct analysis of the queue length processes under a JSQ($d(N)$) policy, in a load balancing graph $G_N$, or under the JIQ scheme is confronted with unsurmountable obstacles. As an alternative route, we leverage novel stochastic coupling constructions to relate the relevant queue length processes to the corresponding processes under a JSQ policy, and show that the deviation between these two is asymptotically negligible under mild assumptions on $d(N)$ or $G_N$. While the stochastic coupling schemes provide a remarkably effective and overarching approach, they defy a systematic recipe and involve some degree of ingenuity and customization. Indeed, the specific coupling arguments that we develop are not only different from those that were originally used in establishing the stochastic optimality properties of the JSQ policy, but also differ in critical ways between a JSQ($d(N)$) policy, a load balancing graph $G_N$, and the JIQ scheme. Yet different coupling constructions are devised for model variants with infinite-server dynamics that we will discuss in Section \[bloc\]. The remainder of the paper is organized as follows. In Section \[spec\] we discuss a wide spectrum of LBAs and evaluate their scalability properties. In Section \[jsqd\] we introduce some useful preliminaries, review fluid and diffusion limits for the JSQ policy as well as JSQ($d$) policies with a fixed value of $d$, and explore the trade-off between delay performance and communication overhead as function of the diversity parameter $d$. In particular, we establish asymptotic universality properties for JSQ($d$) policies, which are extended to systems with server pools and network scenarios in Sections \[bloc\] and \[networks\], respectively. In Section \[token\] we establish asymptotic optimality properties for the JIQ scheme. We discuss somewhat related redundancy policies and alternative scaling regimes and performance metrics in Section \[miscellaneous\]. Scalability spectrum {#spec} ==================== In this section we review a wide spectrum of LBAs and examine their scalability properties in terms of the delay performance vis-a-vis the associated implementation overhead in large-scale systems. Basic model ----------- Throughout this section and most of the paper, we focus on a basic scenario with $N$ parallel single-server infinite-buffer queues and a single dispatcher where tasks arrive as a Poisson process of rate $\lambda(N)$, as depicted in Figure \[figJSQ\]. Arriving tasks cannot be queued at the dispatcher, and must immediately be forwarded to one of the servers. This canonical setup is commonly dubbed the *supermarket model*. Tasks are assumed to have unit-mean exponentially distributed service requirements, and the service discipline at each server is supposed to be oblivious to the actual service requirements. In Section \[bloc\] we consider some model variants with $N$ server pools and possibly finite buffers and in Section \[networks\] we will treat network generalizations of the above model. Asymptotic scaling regimes {#asym} -------------------------- An exact analysis of the delay performance is quite involved, if not intractable, for all but the simplest LBAs. Numerical evaluation or simulation are not straightforward either, especially for high load levels and large system sizes. A common approach is therefore to consider various limit regimes, which not only provide mathematical tractability and illuminate the fundamental behavior, but are also natural in view of the typical conditions in which cloud networks and data centers operate. One can distinguish several asymptotic scalings that have been used for these purposes: (i) In the classical heavy-traffic regime, $\lambda(N) = \lambda N$ with a fixed number of servers $N$ and a relative load $\lambda$ that tends to one in the limit. (ii) In the conventional large-capacity or many-server regime, the relative load $\lambda(N) / N$ approaches a constant $\lambda < 1$ as the number of servers $N$ grows large. (iii) The popular Halfin-Whitt regime [@HW81] combines heavy traffic with a large capacity, with $$\label{eq:HW} \frac{N - \lambda(N)}{\sqrt{N}} \to \beta > 0 \mbox{ as } N \to \infty,$$ so the relative capacity slack behaves as $\beta / \sqrt{N}$ as the number of servers $N$ grows large. (iv) The so-called non-degenerate slow-down regime [@Atar12] involves $N - \lambda(N) \to \gamma > 0$, so the relative capacity slack shrinks as $\gamma / N$ as the number of servers $N$ grows large. The term non-degenerate slow-down refers to the fact that in the context of a centralized multi-server queue, the mean waiting time in regime (iv) tends to a strictly positive constant as $N \to \infty$, and is thus of similar magnitude as the mean service requirement. In contrast, in regimes (ii) and (iii), the mean waiting time decays exponentially fast in $N$ or is of the order $1 / \sqrt{N}$, respectively, as $N \to \infty$, while in regime (i) the mean waiting time grows arbitrarily large relative to the mean service requirement. In the present paper we will focus on scalings (ii) and (iii), and occasionally also refer to these as fluid and diffusion scalings, since it is natural to analyze the relevant queue length process on fluid scale ($1 / N$) and diffusion scale ($1 / \sqrt{N}$) in these regimes, respectively. We will not provide a detailed account of scalings (i) and (iv), which do not capture the large-scale perspective and do not allow for low delays, respectively, but we will briefly revisit these regimes in Section \[miscellaneous\]. Random assignment: N independent M/M/1 queues {#random} --------------------------------------------- One of the most basic LBAs is to assign each arriving task to a server selected uniformly at random. In that case, the various queues collectively behave as $N$ independent M/M/1 queues, each with arrival rate $\lambda(N) / N$ and unit service rate. In particular, at each of the queues, the total number of tasks in stationarity has a geometric distribution with parameter $\lambda(N) / N$. By virtue of the PASTA property, the probability that an arriving task incurs a non-zero waiting time is $\lambda(N) / N$. The mean number of waiting tasks (excluding the possible task in service) at each of the queues is $\frac{\lambda(N)^2}{N (N - \lambda(N))}$, so the total mean number of waiting tasks is $\frac{\lambda(N)^2}{N - \lambda(N)}$, which by Little’s law implies that the mean waiting time of a task is $\frac{\lambda(N)}{N - \lambda(N)}$. In particular, when $\lambda(N) = N \lambda$, the probability that a task incurs a non-zero waiting time is $\lambda$, and the mean waiting time of a task is $\frac{\lambda}{1 - \lambda}$, independent of $N$, reflecting the independence of the various queues. A slightly better LBA is to assign tasks to the servers in a Round-Robin manner, dispatching every $N$-th task to the same server. In the large-capacity regime where $\lambda(N) = N \lambda$, the inter-arrival time of tasks at each given queue will then converge to a constant $1 / \lambda$ as $N \to \infty$. Thus each of the queues will behave as an D/M/1 queue in the limit, and the probability of a non-zero waiting time and the mean waiting time will be somewhat lower than under purely random assignment. However, both the probability of a non-zero waiting time and the mean waiting time will still tend to strictly positive values and not vanish as $N \to \infty$. Join-the-Shortest Queue (JSQ) {#ssec:jsq} ----------------------------- Under the Join-the-Shortest-Queue (JSQ) policy, each arriving task is assigned to the server with the currently shortest queue (ties are broken arbitrarily). In the basic model described above, the JSQ policy has several strong stochastic optimality properties, and yields the ‘most balanced and smallest’ queue process among all non-anticipating policies that do not have any advance knowledge of the service requirements [@EVW80; @Winston77]. Specifically, the JSQ policy minimizes the joint queue length vector in a stochastic majorization sense, and in particular stochastically minimizes the total number of tasks in the system, and hence the mean overall delay. In order to implement the JSQ policy however, a dispatcher requires instantaneous knowledge of the queue lengths at all the servers. A nominal implementation would involve exchange of $2 N$ messages per task, and thus yield a prohibitive communication burden in large-scale systems. Join-the-Smallest-Workload (JSW): centralized M/M/N queue {#ssec:jsw} --------------------------------------------------------- Under the Join-the-Smallest-Workload (JSW) policy, each arriving task is assigned to the server with the currently smallest workload. Note that this is an anticipating policy, since it requires advance knowledge of the service requirements of all the tasks in the system. Further observe that this policy (myopically) minimizes the waiting time for each incoming task, and mimics the operation of a centralized $N$-server queue with a FCFS discipline. The equivalence with a centralized $N$-server queue yields a strong optimality property of the JSW policy: The vector of joint workloads at the various servers observed by each incoming task is smaller in the Schur convex sense than under any alternative admissible policy [@FC01]. The equivalence with a centralized FCFS queue means that there cannot be any idle servers while tasks are waiting. In our setting with Poisson arrivals and exponential service requirements, it can therefore be shown that the total number of tasks under the JSW policy is stochastically smaller than under the JSQ policy. At the same time, it means that the total number of tasks under the JSW policy behaves as a birth-death process, which renders it far more tractable than the JSQ policy. Specifically, given that all the servers are busy, the total number of waiting tasks is geometrically distributed with parameter $\lambda(N) / N$. Thus the total mean number of waiting tasks is $\Pi_W(N, \lambda(N)) \frac{\lambda(N)}{N - \lambda(N)}$, and the mean waiting time is $\Pi_W(N, \lambda(N)) \frac{1}{N - \lambda(N)}$, with $\Pi_W(N, \lambda(N)$ denoting the probability of all servers being occupied and a task incurring a non-zero waiting time. This immediately shows that the mean waiting time is smaller by at least a factor $\lambda(N)$ than for the random assignment policy considered in Subsection \[random\]. In the large-capacity regime $\lambda(N) = N \lambda$, it can be shown that the probability $\Pi_W(N, \lambda(N))$ of a non-zero waiting time decays exponentially fast in $N$, and hence so does the mean waiting time. In the Halfin-Whitt heavy-traffic regime , the probability $\Pi_W(N, \lambda(N))$ of a non-zero waiting time converges to a finite constant $\Pi_W^\star(\beta)$, implying that the mean waiting time of a task is of the order $1 / \sqrt{N}$, and thus vanishes as $N \to \infty$. Power-of-d load balancing (JSQ(d)) {#ssec:powerd} ---------------------------------- As mentioned above, the achilles heel of the JSQ policy is its excessive communication overhead in large-scale systems. This poor scalability has motivated consideration of so-called JSQ($d$) policies, where an incoming task is assigned to a server with the shortest queue among $d$ servers selected uniformly at random. Results in Mitzenmacher [@Mitzenmacher01] and Vvedenskaya [*et al.*]{} [@VDK96] indicate that even sampling as few as $d = 2$ servers yields significant performance enhancements over purely random assignment ($d = 1$) as $N \to \infty$. Specifically, in the fluid regime where $\lambda(N) = \lambda N$, the probability that there are $i$ or more tasks at a given queue is proportional to $\lambda^{\frac{d^i - 1}{d - 1}}$ as $N \to \infty$, and thus exhibits super-exponential decay as opposed to exponential decay for the random assignment policy considered in Subsection \[random\]. As illustrated by the above, the diversity parameter $d$ induces a fundamental trade-off between the amount of communication overhead and the performance in terms of queue lengths and delays. A rudimentary implementation of the JSQ policy ($d = N$, without replacement) involves $O(N)$ communication overhead per task, but it can be shown that the probability of a non-zero waiting time and the mean waiting *vanish* as $N \to \infty$, just like in a centralized queue. Although JSQ($d$) policies with a fixed parameter $d \geq 2$ yield major performance improvements over purely random assignment while reducing the communication burden by a factor O($N$) compared to the JSQ policy, the probability of a non-zero waiting time and the mean waiting time *do not vanish* as $N \to \infty$. In Subsection \[univ\] we will explore the intrinsic trade-off between delay performance and communication overhead as function of the diversity parameter $d$, in conjunction with the relative load. We will examine an asymptotic regime where not only the total task arrival rate $\lambda(N)$ is assumed to grow with $N$, but also the diversity parameter is allowed to depend on $N$. As will be demonstrated, the optimality of the JSQ policy ($d(N) = N$) can be preserved, and in particular a vanishing waiting time can be achieved in the limit as $N \to \infty$, even when $d(N) = o(N)$, thus dramatically lowering the communication overhead. Token-based strategies: Join-the-Idle-Queue (JIQ) {#ssec:jiq} ------------------------------------------------- While a zero waiting time can be achieved in the limit by sampling only $d(N) = o(N)$ servers, the amount of communication overhead in terms of $d(N)$ must still grow with $N$. This can be countered by introducing memory at the dispatcher, in particular maintaining a record of vacant servers, and assigning tasks to idle servers as long as there are any, or to a uniformly at random selected server otherwise. This so-called Join-the-Idle-Queue (JIQ) scheme [@BB08; @LXKGLG11] has received keen interest recently, and can be implemented through a simple token-based mechanism. Specifically, idle servers send tokens to the dispatcher to advertise their availability, and when a task arrives and the dispatcher has tokens available, it assigns the task to one of the corresponding servers (and disposes of the token). Note that a server only issues a token when a task completion leaves its queue empty, thus generating at most one message per task. Surprisingly, the mean waiting time and the probability of a non-zero waiting time vanish under the JIQ scheme in both the fluid and diffusion regimes, as we will further discuss in Section \[token\]. Thus, the use of memory allows the JIQ scheme to achieve asymptotically optimal delay performance with minimal communication overhead. Performance comparison {#ssec:perfcomp} ---------------------- We now present some simulation experiments that we have conducted to compare the above-described LBAs in terms of delay performance. Specifically, we evaluate the mean waiting time and the probability of a non-zero waiting time in both a fluid regime ($\lambda(N) = 0.9 N$) and a diffusion regime ($\lambda(N) = N - \sqrt{N}$). The results are shown in Figure \[differentschemes\]. We are especially interested in distinguishing two classes of LBAs – ones delivering a mean waiting time and probability of a non-zero waiting time that vanish asymptotically, and ones that fail to do so – and relating that dichotomy to the associated overhead. ![Simulation results for mean waiting time $\mathbb{E}[W^N]$ and probability of a non-zero waiting time $p_{\textup{wait}}^N$, for both a fluid regime and a diffusion regime.[]{data-label="differentschemes"}](DS.pdf){width="\linewidth"} #### JSQ, JIQ, and JSW. JSQ, JIQ and JSW evidently have a vanishing waiting time in both the fluid and the diffusion regime as discussed in Subsections \[ssec:jsq\], \[ssec:jsw\] and \[ssec:jiq\]. The optimality of JSW as mentioned in Subsection \[ssec:jsw\] can also be clearly observed. However, there is a significant difference between JSW and JSQ/JIQ in the diffusion regime. We observe that the probability of a non-zero waiting time *approaches a positive constant* for JSW, while it *vanishes* for JSQ/JIQ. In other words, the mean of all positive waiting times is of a larger order of magnitude in JSQ/JIQ compared to JSW. Intuitively, this is clear since in JSQ/JIQ, when a task is placed in a queue, it waits for at least a residual service time. In JSW, which is equivalent to the M/M/$N$ queue, a task that cannot start service immediately, joins a queue that is collectively drained by all the $N$ servers #### Random and Round-Robin. The mean waiting time does not vanish for Random and Round-Robin in the fluid regime, as already mentioned in Subsection \[random\]. Moreover, the mean waiting time grows without bound in the diffusion regime for these two schemes. This is because the system can still be decomposed, and the loads of the individual M/M/1 and D/M/1 queues tend to 1. #### JSQ(${\bf d}$) policies. Three versions of JSQ($d$) are included in the figures; $d(N)=2\not\to \infty$, $d(N)=\lfloor\log(N)\rfloor\to \infty$ and $d(N)=N^{2/3}$ for which $\frac{d(N)}{\sqrt{N}\log(N)}\to \infty$. Note that the graph for $d(N)=\lfloor \log(N) \rfloor$ shows sudden jumps when $d(N)$ increases by $1$. The variants for which $d(N)\to\infty$ have a vanishing waiting time in the fluid regime, while $d = 2$ does not. The latter observation is a manifestation of the results of Gamarnik [*et al.*]{} [@GTZ16] mentioned in the introduction, since JSQ($d$) uses no memory and the overhead per task does not increase with $N$. Furthermore, it follows that JSQ($d$) policies outperform Random and Round-Robin, while JSQ/JIQ/JSW are better in terms of mean waiting time.\ In order to succinctly capture the results and observed dichotomy in Figure \[differentschemes\], we provide an overview of the delay performance of the various LBAs and the associated overhead in Table \[table\], where $q_i^\star$ denotes the stationary fraction of servers with $i$ or more tasks. [|C[3cm]{}|C[3cm]{}|C[2.5cm]{}|C[3cm]{}|C[1.5cm]{}|]{} Scheme & Queue length & Waiting time (fixed $\lambda < 1$) & Waiting time ($1 - \lambda \sim 1 / \sqrt{N}$) & Overhead per task\ Random & $q_i^\star = \lambda^i$ & $\frac{\lambda}{1 - \lambda}$ & $\Theta(\sqrt{N})$ & 0\ JSQ($d$) & $q_i^\star = \lambda^{\frac{d^i - 1}{d - 1}}$ & $\Theta$(1) & $\Omega(\log{N})$ & $2 d$\ & same as JSQ & same as JSQ & ?? & $2d(N)$\ $\frac{d(N)}{\sqrt{N} \log(N)}\to \infty$ & same as JSQ & same as JSQ & same as JSQ & $2d(N)$\ JSQ & $q_1^\star = \lambda$, $q_2^\star =$ o(1) & o(1) & $\Theta(1 / \sqrt{N})$ & $2 N$\ JIQ & same as JSQ & same as JSQ & same as JSQ & $\leq 1$\ JSQ(d) policies and universality properties {#jsqd} =========================================== In this section we first introduce some useful preliminary concepts, then review fluid and diffusion limits for the JSQ policy as well as JSQ($d$) policies with a fixed value of $d$, and finally discuss universality properties when the diversity parameter $d(N)$ is being scaled with $N$. As described in the previous section, we focus on a basic scenario where all the servers are homogeneous, the service requirements are exponentially distributed, and the service discipline at each server is oblivious of the actual service requirements. In order to obtain a Markovian state description, it therefore suffices to only track the number of tasks, and in fact we do not need to keep record of the number of tasks at each individual server, but only count the number of servers with a given number of tasks. Specifically, we represent the state of the system by a vector ${\mathbf{Q}}(t) := \left(Q_1(t), Q_2(t), \dots\right)$, with $Q_i(t)$ denoting the number of servers with $i$ or more tasks at time $t$, including the possible task in service, $i = 1, 2 \dots$. Note that if we represent the queues at the various servers as (vertical) stacks, and arrange these from left to right in non-descending order, then the value of $Q_i$ corresponds to the width of the $i$-th (horizontal) row, as depicted in the schematic diagram in Figure \[figB\]. \ In order to examine the asymptotic behavior when the number of servers $N$ grows large, we consider a sequence of systems indexed by $N$, and attach a superscript $N$ to the associated state variables. The fluid-scaled occupancy state is denoted by ${\mathbf{q}}^N(t) := (q_1^N(t), q_2^N(t), \dots)$, with $q_i^N(t) = Q_i^N(t) / N$ representing the fraction of servers in the $N$-th system with $i$ or more tasks as time $t$, $i = 1, 2, \dots$. Let ${\mathcal{S}}= \{{\mathbf{q}}\in [0, 1]^\infty: q_i \leq q_{i-1} \forall i = 2, 3,\dots\}$ be the set of all possible fluid-scaled states. Whenever we consider fluid limits, we assume the sequence of initial states is such that ${\mathbf{q}}^N(0) \to {\mathbf{q}}^\infty \in {\mathcal{S}}$ as $N \to \infty$. The diffusion-scaled occupancy state is defined as $\bar{{\mathbf{Q}}}^N(t) = (\bar{Q}_1^N(t), \bar{Q}_2^N(t), \dots)$, with $$\label{eq:diffscale} \bar{Q}_1^N(t) = - \frac{N - Q_1^N(t)}{\sqrt{{N}}}, \qquad \bar{Q}_i^N(t) = \frac{Q_i^N(t)}{\sqrt{{N}}}, \quad i = 2,3, \dots.$$ Note that $-\bar{Q}_1^N(t)$ corresponds to the number of vacant servers, normalized by $\sqrt{N}$. The reason why $Q_1^N(t)$ is centered around $N$ while $Q_i^N(t)$, $i = 2,3, \dots$, are not, is because for the scalable LBAs that we pursue, the fraction of servers with exactly one task tends to one, whereas the fraction of servers with two or more tasks tends to zero as $N \to \infty$. Fluid limit for JSQ(d) policies ------------------------------- We first consider the fluid limit for JSQ($d$) policies with an arbitrary but fixed value of $d$ as characterized by Mitzenmacher [@Mitzenmacher01] and Vvedenskaya [*et al.*]{} [@VDK96]. [*The sequence of processes $\{{\mathbf{q}}^N(t)\}_{t \geq 0}$ has a weak limit $\{{\mathbf{q}}(t)\}_{t \geq 0}$ that satisfies the system of differential equations*]{} $$\label{fluid:standard} \frac{{\ensuremath{\mbox{d}}}q_i(t)}{{\ensuremath{\mbox{d}}}t} = \lambda [(q_{i-1}(t))^d - (q_i(t))^d] - [q_i(t) - q_{i+1}(t)], \quad i = 1, 2, \dots.$$ The fluid-limit equations may be interpreted as follows. The first term represents the rate of increase in the fraction of servers with $i$ or more tasks due to arriving tasks that are assigned to a server with exactly $i - 1$ tasks. Note that the latter occurs in fluid state ${\mathbf{q}}\in {\mathcal{S}}$ with probability $q_{i-1}^d - q_i^d$, i.e., the probability that all $d$ sampled servers have $i - 1$ or more tasks, but not all of them have $i$ or more tasks. The second term corresponds to the rate of decrease in the fraction of servers with $i$ or more tasks due to service completions from servers with exactly $i$ tasks, and the latter rate is given by $q_i - q_{i+1}$. The unique fixed point of  for any $d \geq 2$ is obtained as $$\label{eq:fixedpoint1} q_i^\star = \lambda^{\frac{d^i-1}{d-1}}, \quad i = 1, 2, \dots.$$ It can be shown that the fixed point is asymptotically stable in the sense that ${\mathbf{q}}(t) \to {\mathbf{q}}^\star$ as $t \to \infty$ for any initial fluid state ${\mathbf{q}}^\infty$ with $\sum_{i = 1}^{\infty} q_i^\infty < \infty$. The fixed point reveals that the stationary queue length distribution at each individual server exhibits super-exponential decay as $N \to \infty$, as opposed to exponential decay for a random assignment policy. It is worth observing that this involves an interchange of the many-server ($N \to \infty$) and stationary ($t \to \infty$) limits. The justification is provided by the asymptotic stability of the fixed point along with a few further technical conditions. Fluid limit for JSQ policy {#ssec:jsqfluid} -------------------------- We now turn to the fluid limit for the ordinary JSQ policy, which rather surprisingly was not rigorously established until fairly recently in [@MBLW16-3], leveraging martingale functional limit theorems and time-scale separation arguments [@HK94]. In order to state the fluid limit starting from an arbitrary fluid-scaled occupancy state, we first introduce some additional notation. For any fluid state ${\mathbf{q}}\in {\mathcal{S}}$, denote by $m({\mathbf{q}}) = \min\{i: q_{i + 1} < 1\}$ the minimum queue length among all servers. Now if $m({\mathbf{q}})=0$, then define $p_0(m({\mathbf{q}}))=1$ and $p_i(m({\mathbf{q}}))=0$ for all $i=1,2,\ldots$. Otherwise, in case $m({\mathbf{q}})>0$, define $$\label{eq:fluid-gen} p_i({\mathbf{q}}) = \begin{cases} \min\big\{(1 - q_{m({\mathbf{q}}) + 1})/\lambda,1\big\} & \quad\mbox{ for }\quad i=m({\mathbf{q}})-1, \\ 1 - p_{m({\mathbf{q}}) - 1}({\mathbf{q}}) & \quad\mbox{ for }\quad i=m({\mathbf{q}}), \end{cases}$$ and $p_i({\mathbf{q}})=0$ otherwise. The coefficient $p_i({\mathbf{q}})$ represents the instantaneous fraction of incoming tasks assigned to servers with a queue length of exactly $i$ in the fluid state ${\mathbf{q}}\in \mathcal{S}$. [*Any weak limit of the sequence of processes $\{{\mathbf{q}}^N(t)\}_{t \geq 0}$ is given by the deterministic system $\{{\mathbf{q}}(t)\}_{t \geq 0}$ satisfying the following system of differential equations*]{} $$\label{eq:fluid} \frac{{\ensuremath{\mbox{d}}}^+ q_i(t)}{{\ensuremath{\mbox{d}}}t} = \lambda p_{i-1}({\mathbf{q}}(t)) - (q_i(t) - q_{i+1}(t)), \quad i = 1, 2, \dots,$$ [*where ${\ensuremath{\mbox{d}}}^+/{\ensuremath{\mbox{d}}}t$ denotes the right-derivative.*]{} The unique fixed point ${\mathbf{q}}^\star = (q_1^\star,q_2^\star,\ldots)$ of the dynamical system in  is given by $$\label{eq:fpjsq} q_i^\star = \left\{\begin{array}{ll} \lambda, & i = 1, \\ 0, & i = 2, 3,\dots. \end{array} \right.$$ The fixed point in , in conjunction with an interchange of limits argument, indicates that in stationarity the fraction of servers with a queue length of two or larger under the JSQ policy is negligible as $N \to \infty$. Diffusion limit for JSQ policy {#ssec:diffjsq} ------------------------------ We next describe the diffusion limit for the JSQ policy in the Halfin-Whitt heavy-traffic regime , as recently derived by Eschenfeldt & Gamarnik [@EG15]. [*For suitable initial conditions, the sequence of processes $\big\{\bar{{\mathbf{Q}}}^N(t)\big\}_{t \geq 0}$ as in  converges weakly to the limit $\big\{\bar{{\mathbf{Q}}}(t)\big\}_{t \geq 0}$, where $(\bar{Q}_1(t), \bar{Q}_2(t),\ldots)$ is the unique solution to the following system of SDEs*]{} $$\label{eq:diffusionjsq} \begin{split} {\ensuremath{\mbox{d}}}\bar{Q}_1(t) &= \sqrt{2}{\ensuremath{\mbox{d}}}W(t) - \beta{\ensuremath{\mbox{d}}}t - \bar{Q}_1(t){\ensuremath{\mbox{d}}}t + \bar{Q}_2(t){\ensuremath{\mbox{d}}}t-{\ensuremath{\mbox{d}}}U_1(t), \\ {\ensuremath{\mbox{d}}}\bar{Q}_2(t) &= {\ensuremath{\mbox{d}}}U_1(t) - (\bar{Q}_2(t)-\bar{Q}_3(t)){\ensuremath{\mbox{d}}}t, \\ {\ensuremath{\mbox{d}}}{\bar{Q}}_i(t) &= - ({\bar{Q}}_i(t) - {\bar{Q}}_{i+1}(t)){\ensuremath{\mbox{d}}}t, \quad i \geq 3, \end{split}$$ [*for $t \geq 0$, where $W(\cdot)$ is the standard Brownian motion and $U_1(\cdot)$ is the unique nondecreasing nonnegative process satisfying*]{} $\int_0^\infty \mathbbm{1}_{[\bar{Q}_1(t) < 0]} {\ensuremath{\mbox{d}}}U_1(t) = 0$. The above diffusion limit implies that the mean waiting time under the JSQ policy is of a similar order $O(1 / \sqrt{N})$ as in the corresponding centralized M/M/$N$ queue. Hence, we conclude that despite the distributed queueing operation a suitable load balancing policy can deliver a similar combination of excellent service quality and high resource utilization in the Halfin-Whitt regime  as in a centralized queueing arrangement. It it important though to observe a subtle but fundamental difference in the distributional properties due to the distributed versus centralized queueing operation. In the ordinary M/M/$N$ queue a fraction $\Pi_W^\star(\beta)$ of the customers incur a non-zero waiting time as $N \to \infty$, but a non-zero waiting time is only of length $1 / (\beta \sqrt{N})$ in expectation. In contrast, under the JSQ policy, the fraction of tasks that experience a non-zero waiting time is only of the order $O(1 / \sqrt{N})$. However, such tasks will have to wait for the duration of a residual service time, yielding a waiting time of the order $O(1)$. Heavy-traffic limits for JSQ(d) policies ---------------------------------------- Finally, we briefly discuss the behavior of JSQ($d$) policies for fixed $d$ in a heavy-traffic regime where $(N - \lambda(N)) / \eta(N) \to \beta > 0$ as $N \to \infty$ with $\eta(N)$ a positive function diverging to infinity. Note that the case $\eta(N)= \sqrt{N}$ corresponds to the Halfin-Whitt heavy-traffic regime . While a complete characterization of the occupancy process for fixed $d$ has remained elusive so far, significant partial results were recently obtained by Eschenfeldt & Gamarnik [@EG16]. In order to describe the transient asymptotics, we introduce the following rescaled processes ${\bar{Q}}_i^N(t) := (N-Q_i^N(t)) / \eta(N)$, $i = 1, 2, \ldots$. Then, [*for suitable initial states, on any finite time interval, $\{\bar{{\mathbf{Q}}}^N(t)\}_{t \geq 0}$ converges weakly to a deterministic system $\{\bar{{\mathbf{Q}}}(t)\}_{t \geq 0}$ that satisfies the following system of ODEs*]{} $$\frac{{\ensuremath{\mbox{d}}}{\bar{Q}}_i(t)}{{\ensuremath{\mbox{d}}}t} = - d [{\bar{Q}}_i(t) - {\bar{Q}}_{i-1}(t)] - [{\bar{Q}}_i(t) - {\bar{Q}}_{i+1}(t)], \quad i = 1, 2, \ldots,$$ [*with the convention that ${\bar{Q}}_0(t) \equiv 0$.*]{} It is noteworthy that the scaled occupancy process loses its diffusive behavior for fixed $d$. It is further shown in [@EG16] that with high probability the steady-state fraction of queues with length at least $\log_d(N/\eta(N)) - \omega(1)$ tasks approaches unity, which in turn implies that with high probability the steady-state delay is [*at least*]{} $\log_d(N/\eta(N)) - O(1)$ as $N \to \infty$. The diffusion approximation of the JSQ($d$) policy in the Halfin-Whitt regime , starting from a different initial scaling, has been studied by Budhiraja & Friedlander [@BF17]. Recently, Ying [@Ying17] introduced a broad framework involving Stein’s method to analyze the rate of convergence of the scaled steady-state occupancy process of the JSQ($2$) policy when $\eta(N) = N^\alpha$ with $\alpha>0.8$. The results in [@Ying17] establish that in steady state, most of the queues are of size $\log_2(N/\eta(N))+O(1),$ and thus the steady-state delay is of order $\log_2(N/\eta(N))$. Universality properties {#univ} ----------------------- We now further explore the trade-off between delay performance and communication overhead as a function of the diversity parameter $d$, in conjunction with the relative load. The latter trade-off will be examined in an asymptotic regime where not only the total task arrival rate $\lambda(N)$ grows with $N$, but also the diversity parameter depends on $N$, and we write $d(N)$, to explicitly reflect that. We will specifically investigate what growth rate of $d(N)$ is required, depending on the scaling behavior of $\lambda(N)$, in order to asymptotically match the optimal performance of the JSQ policy and achieve a zero mean waiting time in the limit. The results presented in this subsection are based on [@MBLW16-3], unless specified otherwise. [(Universality fluid limit for JSQ($d(N)$))]{} \[fluidjsqd\] If $d(N)\to\infty$ as $N\to\infty$, then the fluid limit of the JSQ$(d(N))$ scheme coincides with that of the ordinary JSQ policy given by the dynamical system in . Consequently, the stationary occupancy states converge to the unique fixed point in . [(Universality diffusion limit for JSQ($d(N)$))]{} \[diffusionjsqd\] If $d(N) /( \sqrt{N} \log N)\to\infty$, then for suitable initial conditions the weak limit of the sequence of processes $\big\{\bar{{\mathbf{Q}}}^{ d(N)}(t)\big\}_{t \geq 0}$ coincides with that of the ordinary JSQ policy, and in particular, is given by the system of SDEs in . The above universality properties indicate that the JSQ overhead can be lowered by almost a factor O($N$) and O($\sqrt{N} / \log N$) while retaining fluid- and diffusion-level optimality, respectively. In other words, Theorems \[fluidjsqd\] and \[diffusionjsqd\] thus reveal that it is sufficient for $d(N)$ to grow at any rate and faster than $\sqrt{N} \log N$ in order to observe similar scaling benefits as in a corresponding centralized M/M/$N$ queue on fluid scale and diffusion scale, respectively. The stated conditions are in fact close to necessary, in the sense that if $d(N)$ is uniformly bounded and $d(N) /( \sqrt{N} \log N) \to 0$ as $N \to \infty$, then the fluid-limit and diffusion-limit paths of the system occupancy process under the JSQ($d(N)$) scheme differ from those under the ordinary JSQ policy, respectively. In particular, if $d(N)$ is uniformly bounded, the mean steady-state delay does not vanish asymptotically as $N \to \infty$. #### High-level proof idea. The proofs of both Theorems \[fluidjsqd\] and \[diffusionjsqd\] rely on a stochastic coupling construction to bound the difference in the queue length processes between the JSQ policy and a scheme with an arbitrary value of $d(N)$. This S-coupling (‘S’ stands for server-based) is then exploited to obtain the fluid and diffusion limits of the JSQ($d(N)$) policy under the conditions stated in Theorems \[fluidjsqd\] and \[diffusionjsqd\]. A direct comparison between the JSQ$(d(N))$ scheme and the ordinary JSQ policy is not straightforward, which is why the ${\mbox{CJSQ}}(n(N))$ class of schemes is introduced as an intermediate scenario to establish the universality result. Just like the JSQ$(d(N))$ scheme, the schemes in the class ${\mbox{CJSQ}}(n(N))$ may be thought of as “sloppy” versions of the JSQ policy, in the sense that tasks are not necessarily assigned to a server with the shortest queue length but to one of the $n(N)+1$ lowest ordered servers, as graphically illustrated in Figure \[fig:sfigCJSQ\]. In particular, for $n(N)=0$, the class only includes the ordinary JSQ policy. Note that the JSQ$(d(N))$ scheme is guaranteed to identify the lowest ordered server, but only among a randomly sampled subset of $d(N)$ servers. In contrast, a scheme in the ${\mbox{CJSQ}}(n(N))$ class only guarantees that one of the $n(N)+1$ lowest ordered servers is selected, but across the entire pool of $N$ servers. It may be shown that for sufficiently small $n(N)$, any scheme from the class ${\mbox{CJSQ}}(n(N))$ is still ‘close’ to the ordinary JSQ policy. It can further be proved that for sufficiently large $d(N)$ relative to $n(N)$ we can construct a scheme called JSQ$(n(N),d(N))$, belonging to the ${\mbox{CJSQ}}(n(N))$ class, which differs ‘negligibly’ from the JSQ$(d(N))$ scheme. Therefore, for a ‘suitable’ choice of $d(N)$ the idea is to produce a ‘suitable’ $n(N)$. This proof strategy is schematically represented in Figure \[fig:sfigRelation\]. [.5]{} ![(a) High-level view of the ${\mbox{CJSQ}}(n(N))$ class of schemes, where as in Figure \[figB\], the servers are arranged in nondecreasing order of their queue lengths, and the arrival must be assigned through the left tunnel. (b) The equivalence structure is depicted for various intermediate load balancing schemes to facilitate the comparison between the JSQ$(d(N))$ scheme and the ordinary JSQ policy. []{data-label="fig:strategy"}](sfigCJSQ.pdf "fig:") [.5]{} ![(a) High-level view of the ${\mbox{CJSQ}}(n(N))$ class of schemes, where as in Figure \[figB\], the servers are arranged in nondecreasing order of their queue lengths, and the arrival must be assigned through the left tunnel. (b) The equivalence structure is depicted for various intermediate load balancing schemes to facilitate the comparison between the JSQ$(d(N))$ scheme and the ordinary JSQ policy. []{data-label="fig:strategy"}](sfigRelation.pdf "fig:") In order to prove the stochastic comparisons among the various schemes, the many-server system is described as an ensemble of stacks, in a way that two different ensembles can be ordered. This stack formulation has also been considered in the literature for establishing the stochastic optimality properties of the JSQ policy [@towsley; @Towsley95; @Towsley1992]. However, it is only through the stack arguments developed in [@MBLW16-3] that the comparison results can be extended to any scheme from the class CJSQ. Blocking and infinite-server dynamics {#bloc} ===================================== The basic scenario that we have focused on so far involved single-server queues. In this section we turn attention to a system with parallel server pools, each with $B$ servers, where $B$ can possibly be infinite. As before, tasks must immediately be forwarded to one of the server pools, but also directly start execution or be discarded otherwise. The execution times are assumed to be exponentially distributed, and do not depend on the number of other tasks receiving service simultaneously. The current scenario will be referred to as ‘infinite-server dynamics’, in contrast to the earlier single-server queueing dynamics. As it turns out, the JSQ policy has similar stochastic optimality properties as in the case of single-server queues, and in particular stochastically minimizes the cumulative number of discarded tasks [@STC93; @J89; @M87; @MS91]. However, the JSQ policy also suffers from a similar scalability issue due to the excessive communication overhead in large-scale systems, which can be mitigated through JSQ($d$) policies. Results of Turner [@T98] and recent papers by Karthik [*et al.*]{} [@KMM17], Mukhopadhyay [*et al.*]{} [@MKMG15; @MMG15], and Xie [*et al.*]{} [@XDLS15] indicate that JSQ($d$) policies provide similar “power-of-choice” gains for loss probabilities. It may be shown though that the optimal performance of the JSQ policy cannot be matched for any fixed value of $d$. Motivated by these observations, we explore the trade-off between performance and communication overhead for infinite-server dynamics. We will demonstrate that the optimal performance of the JSQ policy can be asymptotically retained while drastically reducing the communication burden, mirroring the universality properties described in Section \[univ\] for single-server queues. The results presented in the remainder of the section are extracted from [@MBLW16-4], unless indicated otherwise. Fluid limit for JSQ policy {#ssec:jsqfluid-infinite} -------------------------- As in Subsection \[ssec:jsqfluid\], for any fluid state ${\mathbf{q}}\in {\mathcal{S}}$, denote by $m({\mathbf{q}}) = \min\{i: q_{i + 1} < 1\}$ the minimum queue length among all servers. Now if $m({\mathbf{q}})=0$, then define $p_0(m({\mathbf{q}}))=1$ and $p_i(m({\mathbf{q}}))=0$ for all $i=1,2,\ldots$. Otherwise, in case $m({\mathbf{q}})>0$, define $$\label{eq:fluid-prob-infinite} p_{i}({\mathbf{q}}) = \begin{cases} \min\big\{m({\mathbf{q}})(1 - q_{m({\mathbf{q}}) + 1})/\lambda,1\big\} & \quad \mbox{ for } \quad i=m({\mathbf{q}})-1, \\ 1 - p_{ m({\mathbf{q}}) - 1}({\mathbf{q}}) & \quad \mbox{ for } \quad i=m({\mathbf{q}}), \end{cases}$$ and $p_i({\mathbf{q}})=0$ otherwise. As before, the coefficient $p_i({\mathbf{q}})$ represents the instantaneous fraction of incoming tasks assigned to servers with a queue length of exactly $i$ in the fluid state ${\mathbf{q}}\in \mathcal{S}$. [*Any weak limit of the sequence of processes $\{{\mathbf{q}}^N(t)\}_{t \geq 0}$ is given by the deterministic system $\{{\mathbf{q}}(t)\}_{t \geq 0}$ satisfying the following of differential equations*]{} $$\label{eq:fluid-infinite} \frac{{\ensuremath{\mbox{d}}}^+ q_i(t)}{{\ensuremath{\mbox{d}}}t} = \lambda p_{i-1}({\mathbf{q}}(t)) - i (q_i(t) - q_{i+1}(t)), \quad i = 1, 2, \dots,$$ [*where ${\ensuremath{\mbox{d}}}^+/{\ensuremath{\mbox{d}}}t$ denotes the right-derivative.*]{} Equations  and are to be contrasted with Equations  and . While the form of  and the evolution equations  of the limiting dynamical system remains similar to that of  and , respectively, an additional factor $m({\mathbf{q}})$ appears in  and the rate of decrease in  now becomes $i (q_i - q_{i+1})$, reflecting the infinite-server dynamics. Let $K := \lfloor \lambda \rfloor$ and $f := \lambda - K$ denote the integral and fractional parts of $\lambda$, respectively. It is easily verified that, assuming $\lambda<B$, the unique fixed point of the dynamical system in  is given by $$\label{eq:fixed-point-infinite} q_i^\star = \left\{\begin{array}{ll} 1 & i = 1, \dots, K \\ f & i = K + 1 \\ 0 & i = K + 2, \dots, B, \end{array} \right.$$ and thus $\sum_{i=1}^{B} q_i^\star = \lambda$. This is consistent with the results in Mukhopadhyay [*et al.*]{} [@MKMG15; @MMG15] and Xie [*et al.*]{} [@XDLS15] for fixed $d$, where taking $d \to \infty$ yields the same fixed point. The fixed point in , in conjunction with an interchange of limits argument, indicates that in stationarity the fraction of server pools with at least $K+2$ and at most $K-1$ active tasks is negligible as $N \to \infty$. Diffusion limit for JSQ policy {#ssec:jsq-diffusion-infinite} ------------------------------ As it turns out, the diffusion-limit results may be qualitatively different, depending on whether $f = 0$ or $f > 0$, and we will distinguish between these two cases accordingly. Observe that for any assignment scheme, in the absence of overflow events, the total number of active tasks evolves as the number of jobs in an M/M/$\infty$ system, for which the diffusion limit is well-known. For the JSQ policy, it can be established that the total number of server pools with $K - 2$ or less and $K + 2$ or more tasks is negligible on the diffusion scale. If $f > 0$, the number of server pools with $K - 1$ tasks is negligible as well, and the dynamics of the number of server pools with $K$ or $K + 1$ tasks can then be derived from the known diffusion limit of the total number of tasks mentioned above. In contrast, if $f = 0$, the number of server pools with $K - 1$ tasks is not negligible on the diffusion scale, and the limiting behavior is qualitatively different, but can still be characterized. We refer to [@MBLW16-4] for further details. Universality of JSQ(d) policies in infinite-server scenario {#ssec:univ-infinite} ----------------------------------------------------------- As in Subsection \[univ\], we now further explore the trade-off between performance and communication overhead as a function of the diversity parameter $d(N)$, in conjunction with the relative load. We will specifically investigate what growth rate of $d(N)$ is required, depending on the scaling behavior of $\lambda(N)$, in order to asymptotically match the optimal performance of the JSQ policy. [(Universality fluid limit for JSQ($d(N)$))]{} \[fluidjsqd-infinite\] If $d(N)\to\infty$ as $N\to\infty$, then the fluid limit of the JSQ$(d(N))$ scheme coincides with that of the ordinary JSQ policy given by the dynamical system in . Consequently, the stationary occupancy states converge to the unique fixed point in . In order to state the universality result on diffusion scale, define in case $f > 0$, $f(N):=\lambda(N)-K(N)$, $$\bar{Q}_i^{d(N)}(t) := \dfrac{N - Q_i^{d(N)}(t)}{\sqrt{N}} \: (i \leq K), \ \bar{Q}_{K+1}^{d(N)}(t) := \dfrac{Q_{K+1}^{d(N)}(t) - f(N)}{\sqrt{N}}, \ \bar{Q}_i^{d(N)}(t) := \frac{Q_i^{d(N)}(t)}{\sqrt{N}}\geq 0 \: (i \geq K + 2),$$and otherwise, if $f = 0$, assume $(KN-\lambda(N))/\sqrt{N}\to\beta\in \R$ as $N\to\infty$, and define $$\footnotesize {\hat{Q}}_{K-1}^{d(N)}(t) := \sum_{i=1}^{K-1} \dfrac{N - Q_i^{d(N)}(t)}{\sqrt{N}}, \ {\hat{Q}}_K^{d(N)}(t) := \dfrac{N - Q_K^{d(N)}(t)}{\sqrt{N}}, \ {\hat{Q}}_i^{d(N)}(t) := \dfrac{Q_i^{d(N)}(t)}{\sqrt{N}} \geq 0 \: (i \geq K + 1).$$ \[diffusionjsqd-infinite\] Assume $d(N) / (\sqrt{N} \log N) \to \infty$. Under suitable initial conditions\ [(i)]{} If $f>0$, then ${\bar{Q}}_i^{d(N)}(\cdot)$ converges to the zero process for $i\neq K+1$, and ${\bar{Q}}^{d(N)}_{K+1}(\cdot)$ converges weakly to the Ornstein-Uhlenbeck process satisfying the SDE $d\bar{Q}_{K+1}(t)=-\bar{Q}_{K+1}(t)dt+\sqrt{2\lambda}dW(t)$, where $W(\cdot)$ is the standard Brownian motion.\ [(ii)]{} If $f=0$, then ${\hat{Q}}_{K-1}^{d(N)}(\cdot)$ converges weakly to the zero process, and $({\hat{Q}}_{K}^{d(N)}(\cdot), {\hat{Q}}_{K+1}^{d(N)}(\cdot))$ converges weakly to $({\hat{Q}}_{K}(\cdot), {\hat{Q}}_{K+1}(\cdot))$, described by the unique solution of the following system of SDEs: $$\begin{aligned} {\ensuremath{\mbox{d}}}{\hat{Q}}_{K}(t) &= \sqrt{2 K} {\ensuremath{\mbox{d}}}W(t) - ({\hat{Q}}_K(t) + K {\hat{Q}}_{K+1}(t)){\ensuremath{\mbox{d}}}t + \beta {\ensuremath{\mbox{d}}}t + {\ensuremath{\mbox{d}}}V_1(t) \\ {\ensuremath{\mbox{d}}}{\hat{Q}}_{K+1}(t) &= {\ensuremath{\mbox{d}}}V_1(t) - (K + 1) {\hat{Q}}_{K+1}(t){\ensuremath{\mbox{d}}}t,\end{aligned}$$ where $W(\cdot)$ is the standard Brownian motion, and $V_1(\cdot)$ is the unique nondecreasing process satisfying $\int_0^t {\ensuremath{\mathbbm{1}_{\left[{\hat{Q}}_K(s)\geq 0\right]}}} {\ensuremath{\mbox{d}}}V_1(s) = 0$. Given the asymptotic results for the JSQ policy in Subsections \[ssec:jsqfluid-infinite\] and \[ssec:jsq-diffusion-infinite\], the proofs of the asymptotic results for the JSQ$(d(N))$ scheme in Theorems \[fluidjsqd-infinite\] and \[diffusionjsqd-infinite\] involve establishing a universality result which shows that the limiting processes for the JSQ$(d(N))$ scheme are ‘$g(N)$-alike’ to those for the ordinary JSQ policy for suitably large $d(N)$. Loosely speaking, if two schemes are $g(N)$-alike, then in some sense, the associated system occupancy states are indistinguishable on $g(N)$-scale. The next theorem states a sufficient criterion for the JSQ$(d(N))$ scheme and the ordinary JSQ policy to be $g(N)$-alike, and thus, provides the key vehicle in establishing the universality result. \[th:pwr of d\] Let $g: {\mathbbm{N}}\to \R_+$ be a function diverging to infinity. Then the JSQ policy and the JSQ$(d(N))$ scheme are $g(N)$-alike, with $g(N) \leq N$, if\ [(i)]{} $d(N) \to \infty$ for $g(N) = O(N)$, [(ii)]{} $d(N) \left(\frac{N}{g(N)}\log\left(\frac{N}{g(N)}\right)\right)^{-1} \to \infty$ for $g(N)=o(N)$. The proof of Theorem \[th:pwr of d\] relies on a novel coupling construction, called T-coupling (‘T’ stands for task-based), which will be used to (lower and upper) bound the difference of occupancy states of two arbitrary schemes. This T-coupling [@MBLW16-4] is distinct from and inherently stronger than the S-coupling used in Subsection \[univ\] in the single-server queueing scenario. Note that in the current infinite-server scenario, the departures of the ordered server pools cannot be coupled, mainly since the departure rate at the $m^{\rm th}$ ordered server pool, for some $m = 1, 2, \ldots, N$, depends on its number of active tasks. The T-coupling is also fundamentally different from the coupling constructions used in establishing the weak majorization results in [@Winston77; @towsley; @Towsley95; @Towsley1992; @W78] in the context of the ordinary JSQ policy in the single-server queueing scenario, and in [@STC93; @J89; @M87; @MS91] in the scenario of state-dependent service rates. Universality of load balancing in networks {#networks} ========================================== In this section we return to the single-server queueing dynamics, and extend the universality properties to network scenarios, where the $N$ servers are assumed to be inter-connected by some underlying graph topology $G_N$. Tasks arrive at the various servers as independent Poisson processes of rate $\lambda$, and each incoming task is assigned to whichever server has the smallest number of tasks among the one where it arrives and its neighbors in $G_N$. Thus, in case $G_N$ is a clique, each incoming task is assigned to the server with the shortest queue across the entire system, and the behavior is equivalent to that under the JSQ policy. The stochastic optimality properties of the JSQ policy thus imply that the queue length process in a clique will be better balanced and smaller (in a majorization sense) than in an arbitrary graph $G_N$. Besides the prohibitive communication overhead discussed earlier, a further scalability issue of the JSQ policy arises when executing a task involves the use of some data. Storing such data for all possible tasks on all servers will typically require an excessive amount of storage capacity. These two burdens can be effectively mitigated in sparser graph topologies where tasks that arrive at a specific server $i$ are only allowed to be forwarded to a subset of the servers ${\mathcal N}_i$. For the tasks that arrive at server $i$, queue length information then only needs to be obtained from servers in ${\mathcal N}_i$, and it suffices to store replicas of the required data on the servers in ${\mathcal N}_i$. The subset ${\mathcal N}_i$ containing the peers of server $i$ can be naturally viewed as its neighbors in some graph topology $G_N$. In this section we focus on the results in [@MBL17] for the case of undirected graphs, but most of the analysis can be extended to directed graphs. The above model has been studied in [@G15; @T98], focusing on certain fixed-degree graphs and in particular ring topologies. The results demonstrate that the flexibility to forward tasks to a few neighbors, or even just one, with possibly shorter queues significantly improves the performance in terms of the waiting time and tail distribution of the queue length. This resembles the “power-of-choice” gains observed for JSQ($d$) policies in complete graphs. However, the results in [@G15; @T98] also establish that the performance sensitively depends on the underlying graph topology, and that selecting from a fixed set of $d - 1$ neighbors typically does not match the performance of re-sampling $d - 1$ alternate servers for each incoming task from the entire population, as in the power-of-$d$ scheme in a complete graph. If tasks do not get served and never depart but simply accumulate, then the scenario described above amounts to a so-called balls-and-bins problem on a graph. Viewed from that angle, a close counterpart of our setup is studied in Kenthapadi & Panigrahy [@KP06], where in our terminology each arriving task is routed to the shortest of $d \geq 2$ randomly selected neighboring queues. The key challenge in the analysis of load balancing on arbitrary graph topologies is that one needs to keep track of the evolution of number of tasks at each vertex along with their corresponding neighborhood relationship. This creates a major problem in constructing a tractable Markovian state descriptor, and renders a direct analysis of such processes highly intractable. Consequently, even asymptotic results for load balancing processes on an arbitrary graph have remained scarce so far. The approach in [@MBL17] is radically different, and aims at comparing the load balancing process on an arbitrary graph with that on a clique. Specifically, rather than analyzing the behavior for a given class of graphs or degree value, the analysis explores for what types of topologies and degree properties the performance is asymptotically similar to that in a clique. The proof arguments in [@MBL17] build on the stochastic coupling constructions developed in Subsection \[univ\] for JSQ($d$) policies. Specifically, the load balancing process on an arbitrary graph is viewed as a ‘sloppy’ version of that on a clique, and several other intermediate sloppy versions are constructed. Let $Q_i(G_N, t)$ denote the number of servers with queue length at least $i$ at time $t$, $i = 1, 2, \ldots$, and let the fluid-scaled variables $q_i(G_N, t) := Q_i(G_N, t) / N$ be the corresponding fractions. Also, in the Halfin-Whitt heavy-traffic regime , define the centered and diffusion-scaled variables $\bar{Q}_1(G_N,t) := - (N - Q_1(G_N,t)) / \sqrt{N}$ and $\bar{Q}_i(G_N,t) := Q_i(G_N,t) / \sqrt{N}$ for $i = 2, 3, \ldots$, analogous to .\ The next definition introduces two notions of *asymptotic optimality*. \[def:opt\] A graph sequence ${\mathbf{G}}= \{G_N\}_{N \geq 1}$ is called ‘asymptotically optimal on $N$-scale’ or ‘$N$-optimal’, if for any $\lambda < 1$, the scaled occupancy process $(q_1(G_N, \cdot), q_2(G_N, \cdot), \ldots)$ converges weakly, on any finite time interval, to the process $(q_1(\cdot), q_2(\cdot),\ldots)$ given by . Moreover, a graph sequence ${\mathbf{G}}= \{G_N\}_{N \geq 1}$ is called ‘asymptotically optimal on $\sqrt{N}$-scale’ or ‘$\sqrt{N}$-optimal’, if in the Halfin-Whitt heavy-traffic regime , on any finite time interval, the process $({\bar{Q}}_1(G_N, \cdot), {\bar{Q}}_2(G_N, \cdot), \ldots)$ converges weakly to the process $({\bar{Q}}_1(\cdot), {\bar{Q}}_2(\cdot), \ldots)$ given by . Intuitively speaking, if a graph sequence is $N$-optimal or $\sqrt{N}$-optimal, then in some sense, the associated occupancy processes are indistinguishable from those of the sequence of cliques on $N$-scale or $\sqrt{N}$-scale. In other words, on any finite time interval their occupancy processes can differ from those in cliques by at most $o(N)$ or $o(\sqrt{N})$, respectively. Asymptotic optimality criteria for deterministic graph sequences ---------------------------------------------------------------- We now develop a criterion for asymptotic optimality of an arbitrary deterministic graph sequence on different scales. We first introduce some useful notation, and two measures of *well-connectedness*. Let $G = (V, E)$ be any graph. For a subset $U \subseteq V$, define ${\text{\textsc{com}}}(U) := |V\setminus N[U]|$ to be the set of all vertices that are disjoint from $U$, where $N[U] := U\cup \{v \in V:\ \exists\ u \in U \mbox{ with } (u, v) \in E\}$. For any fixed $\varepsilon > 0$ define $$\label{def:dis} {\text{\textsc{dis}}}_1(G,\varepsilon) := \sup_{U\subseteq V, |U|\geq \varepsilon |V|}{\text{\textsc{com}}}(U), \qquad {\text{\textsc{dis}}}_2(G,\varepsilon) := \sup_{U\subseteq V, |U|\geq \varepsilon \sqrt{|V|}}{\text{\textsc{com}}}(U).$$ The next theorem provides sufficient conditions for asymptotic optimality on $N$-scale and $\sqrt{N}$-scale in terms of the above two well-connectedness measures. \[th:det-seq\] For any graph sequence ${\mathbf{G}}= \{G_N\}_{N \geq 1}$, [(i)]{} ${\mathbf{G}}$ is $N$-optimal if for any $\varepsilon > 0$, ${\text{\textsc{dis}}}_1(G_N, \varepsilon) / N \to 0$ as $N \to \infty$. [(ii)]{} ${\mathbf{G}}$ is $\sqrt{N}$-optimal if for any $\varepsilon > 0$, ${\text{\textsc{dis}}}_2(G_N, \varepsilon) / \sqrt{N} \to 0$ as $N \to \infty$. The next corollary is an immediate consequence of Theorem \[th:det-seq\]. Let ${\mathbf{G}}= \{G_N\}_{N \geq 1}$ be any graph sequence. Then [(i)]{} If $d_{\min}(G_N) = N - o(N)$, then ${\mathbf{G}}$ is $N$-optimal, and [(ii)]{} If $d_{\min}(G_N) = N - o(\sqrt{N})$, then ${\mathbf{G}}$ is $\sqrt{N}$-optimal. We now provide a sketch of the main proof arguments for Theorem \[th:det-seq\] as used in [@MBL17], focusing on the proof of $N$-optimality. The proof of $\sqrt{N}$-optimality follows along similar lines. First of all, it can be established that if a system is able to assign each task to a server in the set ${\mathcal{S}}^N(n(N))$ of the $n(N)$ nodes with shortest queues, where $n(N)$ is $o(N)$, then it is $N$-optimal. Since the underlying graph is not a clique however (otherwise there is nothing to prove), for any $n(N)$ not every arriving task can be assigned to a server in ${\mathcal{S}}^N(n(N))$. Hence, a further stochastic comparison property is proved in [@MBL17] implying that if on any finite time interval of length $t$, the number of tasks $\Delta^N(t)$ that are not assigned to a server in ${\mathcal{S}}^N(n(N))$ is $o_P(N)$, then the system is $N$-optimal as well. The $N$-optimality can then be concluded when $\Delta^N(t)$ is $o_P(N)$, which is demonstrated in [@MBL17] under the condition that ${\text{\textsc{dis}}}_1(G_N, \varepsilon) / N \to 0$ as $N \to \infty$ as stated in Theorem \[th:det-seq\]. Asymptotic optimality of random graph sequences ----------------------------------------------- Next we investigate how the load balancing process behaves on random graph topologies. Specifically, we aim to understand what types of graphs are asymptotically optimal in the presence of randomness (i.e., in an average-case sense). Theorem \[th:inhom\] below establishes sufficient conditions for asymptotic optimality of a sequence of inhomogeneous random graphs. Recall that a graph $G' = (V', E')$ is called a supergraph of $G = (V, E)$ if $V = V'$ and $E \subseteq E'$. \[th:inhom\] Let ${\mathbf{G}}= \{G_N\}_{N \geq 1}$ be a graph sequence such that for each $N$, $G_N = (V_N, E_N)$ is a super-graph of the inhomogeneous random graph $G_N'$ where any two vertices $u, v \in V_N$ share an edge with probability $p_{uv}^N$. 1. If for each $\varepsilon>0$, there exists subsets of vertices $V_N^\varepsilon\subseteq V_N$ with $|V_N^\varepsilon|<\varepsilon N$, such that $\inf\ \{p^N_{uv}: u, v\in V_N^\varepsilon\}$ is $\omega(1/N)$, then ${\mathbf{G}}$ is $N$-optimal. 2. If for each $\varepsilon>0$, there exists subsets of vertices $V_N^\varepsilon\subseteq V_N$ with $|V_N^\varepsilon|<\varepsilon \sqrt{N}$, such that $\inf\ \{p^N_{uv}: u, v\in V_N^\varepsilon\}$ is $\omega(\log(N)/\sqrt{N})$, then ${\mathbf{G}}$ is $\sqrt{N}$-optimal. The proof of Theorem \[th:inhom\] relies on Theorem \[th:det-seq\]. Specifically, if $G_N$ satisfies conditions (i) and (ii) in Theorem \[th:inhom\], then the corresponding conditions (i) and (ii) in Theorem \[th:det-seq\] hold. As an immediate corollary to Theorem \[th:inhom\] we obtain an optimality result for the sequence of Erdős-Rényi random graphs. \[cor:errg\] Let ${\mathbf{G}}= \{G_N\}_{N \geq 1}$ be a graph sequence such that for each $N$, $G_N$ is a super-graph of ${\mathrm{ER}}_N(p(N))$, and $d(N) = (N-1) p(N)$. Then [(i)]{} If $d(N) \to \infty$ as $N \to \infty$, then ${\mathbf{G}}$ is $N$-optimal. [(ii)]{} If $d(N) / (\sqrt{N} \log N) \to \infty$ as $N\to\infty$, then ${\mathbf{G}}$ is $\sqrt{N}$-optimal. The growth rate condition for $N$-optimality in Corollary \[cor:errg\] (i) is not only sufficient, but necessary as well. Thus informally speaking, $N$-optimality is achieved under the minimum condition required as long as the underlying topology is suitably random. Token-based load balancing {#token} ========================== While a zero waiting time can be achieved in the limit by sampling only $d(N) =o(N)$ servers as Sections \[univ\], \[bloc\] and \[networks\] showed, even in network scenarios, the amount of communication overhead in terms of $d(N)$ must still grow with $N$. As mentioned earlier, this can be avoided by introducing memory at the dispatcher, in particular maintaining a record of only vacant servers, and assigning tasks to idle servers, if there are any, or to a uniformly at random selected server otherwise. This so-called Join-the-Idle-Queue (JIQ) scheme [@BB08; @LXKGLG11] can be implemented through a simple token-based mechanism generating at most one message per task. Remarkably enough, even with such low communication overhead, the mean waiting time and the probability of a non-zero waiting time vanish under the JIQ scheme in both the fluid and diffusion regimes, as we will discuss in the next two subsections. Asymptotic optimality of JIQ scheme ----------------------------------- We first consider the fluid limit of the JIQ scheme. Let $q_i^N(\infty)$ be a random variable denoting the process $q_i^N(\cdot)$ in steady state. It was proved in [@Stolyar15] for the JIQ scheme (under very broad conditions), $$\label{eq:fpjiq} q_1^N(\infty) \to \lambda, \qquad q_i^N(\infty) \to 0 \quad \mbox{ for all } i \geq 2, \qquad \mbox{ as } \quad N \to \infty.$$ The above equation in conjunction with the PASTA property yields that the steady-state probability of a non-zero wait vanishes as $N \to \infty$, thus exhibiting asymptotic optimality of the JIQ scheme on fluid scale.\ We now turn to the diffusion limit of the JIQ scheme. [(Diffusion limit for JIQ)]{} \[diffusionjiq\] In the Halfin-Whitt heavy-traffic regime , under suitable initial conditions, the weak limit of the sequence of centered and diffusion-scaled occupancy process in  coincides with that of the ordinary JSQ policy given by the system of SDEs in . The above theorem implies that for suitable initial states, on any finite time interval, the occupancy process under the JIQ scheme is indistinguishable from that under the JSQ policy. The proof of Theorem \[diffusionjiq\] relies on a coupling construction as described in greater detail in [@MBLW16-1]. The idea is to compare the occupancy processes of two systems following JIQ and JSQ policies, respectively. Comparing the JIQ and JSQ policies is facilitated when viewed as follows: (i) If there is an idle server in the system, both JIQ and JSQ perform similarly, (ii) Also, when there is no idle server and only $O(\sqrt{N})$ servers with queue length two, JSQ assigns the arriving task to a server with queue length one. In that case, since JIQ assigns at random, the probability that the task will land on a server with queue length two and thus JIQ acts differently than JSQ is $O(1/\sqrt{N})$. Since on any finite time interval the number of times an arrival finds all servers busy is at most $O(\sqrt{N})$, all the arrivals except an $O(1)$ of them are assigned in exactly the same manner in both JIQ and JSQ, which then leads to the same scaling limit for both policies. Multiple dispatchers {#multiple} -------------------- So far we have focused on a basic scenario with a single dispatcher. Since it is not uncommon for LBAs to operate across multiple dispatchers though, we consider in this subsection a scenario with $N$ parallel identical servers as before and $R \geq 1$ dispatchers. (We will assume the number of dispatchers to remain fixed as the number of servers grows large, but a further natural scenario would be for the number of dispatchers $R(N)$ to scale with the number of servers as considered by Mitzenmacher [@Mitzenmacher16], who analyzes the case $R(N) = r N$ for some constant $r$, so that the relative load of each dispatcher is $\lambda r$.) Tasks arrive at dispatcher $r$ as a Poisson process of rate $\alpha_r \lambda N$, with $\alpha_r > 0$, $r = 1, \dots, R$, $\sum_{r = 1}^{R} \alpha_r = 1$, and $\lambda$ denoting the task arrival rate per server. For conciseness, we denote $\alpha = (\alpha_1, \dots, \alpha_R)$, and without loss of generality we assume that the dispatchers are indexed such that $\alpha_1 \geq \alpha_2 \geq \dots \geq \alpha_R$. When a server becomes idle, it sends a token to one of the dispatchers selected uniformly at random, advertising its availability. When a task arrives at a dispatcher which has tokens available, one of the tokens is selected, and the task is immediately forwarded to the corresponding server. We distinguish two scenarios when a task arrives at a dispatcher which has no tokens available, referred to as the [*blocking*]{} and [*queueing*]{} scenario respectively. In the blocking scenario, the incoming task is blocked and instantly discarded. In the queueing scenario, the arriving task is forwarded to one of the servers selected uniformly at random. If the selected server happens to be idle, then the outstanding token at one of the other dispatchers is revoked. In the queueing scenario we assume $\lambda < 1$, which is not only necessary but also sufficient for stability. Denote by $B(R, N, \lambda, \alpha)$ the steady-state blocking probability of an arbitrary task in the blocking scenario. Also, denote by $W(R, N, \lambda, \alpha)$ a random variable with the steady-state waiting-time distribution of an arbitrary task in the queueing scenario. Scenarios with multiple dispatchers have received limited attention in the literature, and the scant papers that exist [@LXKGLG11; @Mitzenmacher16; @Stolyar17] almost exclusively assume that the loads at the various dispatchers are strictly equal. In these cases the fluid limit, for suitable initial states, is the same as that for a single dispatcher, and in particular the fixed point is the same, hence, the JIQ scheme continues to achieve asymptotically optimal delay performance with minimal communication overhead. As one of the few exceptions, [@BBL17a] allows the loads at the various dispatchers to be different. #### Results for blocking scenario. For the blocking scenario, it is established in [@BBL17a] that, $$B(R,N,\lambda,\alpha) \to \max\{1-R\alpha_R,1-1/\lambda\} \quad \mbox{ as } N \to \infty.$$ This result shows that in the many-server limit the system performance in terms of blocking is either determined by the relative load of the least-loaded dispatcher, or by the aggregate load. This indirectly reveals that, somewhat counter-intuitively, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck. #### Results for queueing scenario. For the queueing scenario, it is shown in [@BBL17a] that, for fixed $\lambda < 1$ $$\mathbb{E}[W(R, N, \lambda, \alpha)] \to \frac{\lambda_2(R, \lambda, \alpha)}{1 - \lambda_2(R, \lambda, \alpha)} \quad \mbox{ as } N \to \infty,$$ where $\lambda_2(R,\lambda,\alpha) = 1 - \frac{1 - \lambda \sum_{i=1}^{r^\star} \alpha_i}{1 - \lambda r^\star / R}$, with $r^\star = \sup\big\{r \big| \alpha_r > \frac{1}{R} \frac{1 - \lambda \sum_{i=1}^{r}\alpha_i}{1 - \lambda r/R}\big\}$, may be interpreted as the rate at which tasks are forwarded to randomly selected servers.\ When the arrival rates at all dispatchers are strictly equal, i.e., $\alpha_1 = \dots = \alpha_R = 1 / R$, the above results indicate that the stationary blocking probability and the mean waiting time asymptotically vanish as $N \to \infty$, which is in agreement with the observations in [@Stolyar17] mentioned above. However, when the arrival rates at the various dispatchers are not perfectly equal, so that $\alpha_R < 1 / R$, the blocking probability and mean waiting time are strictly positive in the limit, even for arbitrarily low overall load and an arbitrarily small degree of skewness in the arrival rates. Thus, the ordinary JIQ scheme fails to achieve asymptotically optimal performance for heterogeneous dispatcher loads. In order to counter the above-described performance degradation for asymmetric dispatcher loads, [@BBL17a] proposes two enhancements. Enhancement A uses a non-uniform token allotment: When a server becomes idle, it sends a token to dispatcher $r$ with probability $\beta_r$. Enhancement B involves a token exchange mechanism: Any token is transferred to a uniformly randomly selected dispatcher at rate $\nu$. Note that the token exchange mechanism only creates a constant communication overhead per task as long as the rate $\nu$ does not depend on the number of servers $N$, and thus preserves the scalability of the basic JIQ scheme. The above enhancements can achieve asymptotically optimal performance for suitable values of the $\beta_r$ parameters and the exchange rate $\nu$. Specifically, the stationary blocking probability in the blocking scenario and the mean waiting time in the queueing scenario asymptotically vanish as $N \to \infty$, upon using Enhancement A with $\beta_r = \alpha_r$ or Enhancement B with $\nu \geq \frac{\lambda}{1 - \lambda}(\alpha_1 R - 1)$. Redundancy policies and alternative scaling regimes {#miscellaneous} =================================================== In this section we discuss somewhat related redundancy policies and alternative scaling regimes and performance metrics. #### Redundancy-d policies. So-called redundancy-$d$ policies involve a somewhat similar operation as JSQ($d$) policies, and also share the primary objective of ensuring low delays [@AGSS13; @VGMSRS13]. In a redundancy-$d$ policy, $d \geq 2$ candidate servers are selected uniformly at random (with or without replacement) for each arriving task, just like in a JSQ($d$) policy. Rather than forwarding the task to the server with the shortest queue however, replicas are dispatched to all sampled servers. Two common options can be distinguished for abortion of redundant clones. In the first variant, as soon as the first replica starts service, the other clones are abandoned. In this case, a task gets executed by the server which had the smallest workload at the time of arrival (and which may or may not have had the shortest queue length) among the sampled servers. This may be interpreted as a power-of-$d$ version of the Join-the-Smallest Workload (JSW) policy discussed in Subsection \[ssec:jsw\]. In the second option the other clones of the task are not aborted until the first replica has completed service (which may or may not have been the first replica to start service). While a task is only handled by one of the servers in the former case, it may be processed by several servers in the latter case. #### Conventional heavy traffic. It is also worth mentioning some asymptotic results for the classical heavy-traffic regime as described in Subsection \[asym\] where the number of servers $N$ is fixed and the relative load tends to one in the limit. The papers [@FS78; @Reiman84; @ZHW95] establish diffusion limits for the JSQ policy in a sequence of systems with Markovian characteristics as in our basic model set-up, but where in the $K$-th system the arrival rate is $K \lambda + \hat\lambda \sqrt{K}$, while the service rate of the $i$-th server is $K \mu_i + \hat\mu_i \sqrt{K}$, $i = 1, \dots, N$, with $\lambda = \sum_{i = 1}^{N} \mu_i$, inducing critical load as $K \to \infty$. It is proved that for suitable initial conditions the queue lengths are of the order O($\sqrt{K}$) over any finite time interval and exhibit a state-space collapse property. Atar [*et al.*]{} [@AKM17] investigate a similar scenario, and establish diffusion limits for three policies: the JSQ($d$) policy, the redundancy-$d$ policy (where the redundant clones are abandoned as soon as the first replica starts service), and a combined policy called Replicate-to-Shortest-Queues (RSQ) where $d$ replicas are dispatched to the $d$-shortest queues. #### Non-degenerate slowdown. Asymptotic results for the so-called non-degenerate slow-down regime described in Subsection \[asym\] where $N - \lambda(N) \to \gamma > 0$ as the number of servers $N$ grows large, are scarce. Gupta & Walton [@GW17] characterize the diffusion-scaled queue length process under the JSQ policy in this asymptotic regime. They further compare the diffusion limit for the JSQ policy with that for a centralized queue as described above as well as several LBAs such as the JIQ scheme and a refined version called Idle-One-First (I1F), where a task is assigned to a server with exactly one task if no idle server is available and to a randomly selected server otherwise. It is proved that the diffusion limit for the JIQ scheme is no longer asymptotically equivalent to that for the JSQ policy in this asymptotic regime, and the JIQ scheme fails to achieve asymptotic optimality in that respect, as opposed to the behavior in the large-capacity and Halfin-Whitt regimes discussed in Subsection \[ssec:jiq\]. In contrast, the I1F scheme does preserve the asymptotic equivalence with the JSQ policy in terms of the diffusion-scaled queue length process, and thus retains asymptotic optimality in that sense. #### Sparse-feedback regime. As described in Section \[ssec:jiq\], the JIQ scheme involves a communication overhead of at most one message per task, and yet achieves optimal delay performance in the fluid and diffusion regimes. However, even just one message per task may still be prohibitive, especially when tasks do not involve big computational tasks, but small data packets which require little processing. Motivated by the above issues, [@BBL17b] proposes a novel class of LBAs which also leverage memory at the dispatcher, but allow the communication overhead to be seamlessly adapted and reduced below that of the JIQ scheme. Specifically, in the proposed schemes, the various servers provide occasional queue status notifications to the dispatcher, either in a synchronous or asynchronous fashion. The dispatcher uses these reports to maintain queue estimates, and forwards incoming tasks to the server with the lowest queue estimate. The results in [@BBL17b] demonstrate that the proposed schemes markedly outperform JSQ($d$) policies with the same number of $d \geq 1$ messages per task and they can achieve a vanishing waiting time in the limit when the update frequency exceeds $\lambda / (1 - \lambda)$. In case servers only report zero queue lengths and suppress updates for non-zero queues, the update frequency required for a vanishing waiting time can in fact be lowered to just $\lambda$, matching the one message per task involved in the JIQ scheme. #### Scaling of maximum queue length. So far we have focused on the asymptotic behavior of LBAs in terms of the number of servers with a certain queue length, either on fluid scale or diffusion scale, in various regimes as $N \to \infty$. A related but different performance metric is the maximum queue length $M(N)$ among all servers as $N \to \infty$. Luczak & McDiarmid [@LM06] showed that for fixed $d \geq 2$ the steady-state maximum queue length $M(N)$ under the JSQ($d$) policy is given by $\log(\log(N)) / \log(d) + O(1)$ and is concentrated on at most two adjacent values, whereas for purely random assignment ($d=1$), it scales as $\log(N) / \log(1/\lambda)$ and does not concentrate on a bounded range of values. This is yet a further manifestation of the “power-of-choice” effect. The maximum queue length $M(N)$ is the central performance metric in balls-and-bins models where arriving items (balls) do not get served and never depart but simply accumulate in bins, and (stationary) queue lengths are not meaningful. In fact, the very notion of randomized load balancing and power-of-$d$ strategies was introduced in a balls-and-bins setting in the seminal paper by Azar [*et al.*]{} [@ABKU94]. [10]{} G. Ananthanarayanan, A. Ghodsi, S. Shenker, and I. Stoica. . In [*NSDI ’13*]{}, pages 185–198, 2013. R. Atar. . , 60(2):490–500, 2012. 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ArXiv

--- author: - 'Robert J. Perry' --- 0[[H]{}\^\_0]{} [*v*]{}\^ [H]{}\^ LIGHT-FRONT QCD: A CONSTITUENT PICTURE OF HADRONS ================================================= MOTIVATION AND STRATEGY ----------------------- We seek to derive the structure of hadrons from the fundamental theory of the strong interaction, QCD. Our work is founded on the hypothesis that a constituent approximation can be [*derived*]{} from QCD, so that a relatively small number of quark [*and gluon*]{} degrees of freedom need be explicitly included in the state vectors for low-lying hadrons. To obtain a constituent picture, we use a Hamiltonian approach in light-front coordinates. I do not believe that light-front Hamiltonian field theory is extremely useful for the study of low energy QCD unless a constituent approximation can be made, and I do not believe such an approximation is possible unless cutoffs that [it violate]{} manifest gauge invariance and covariance are employed. Such cutoffs [*inevitably*]{} lead to relevant and marginal effective interactions ([*i.e.*]{}, counterterms) that contain functions of longitudinal momenta. It is [it not]{} possible to renormalize light-front Hamiltonians in any useful manner without developing a renormalization procedure that can produce these non-canonical counterterms. The line of investigation I discuss has been developed by a small group of theorists who are working or have worked at Ohio State University and Warsaw University. Ken Wilson provided the initial impetus for this work, and at a very early stage outlined much of the basic strategy we employ. I make no attempt to provide enough details to allow the reader to start doing light-front calculations. The introductory article by Harindranath is helpful in this regard. An earlier version of these lectures also provides many more details. ### A Constituent Approximation Depends on Tailored Renormalization If it is possible to derive a constituent approximation from QCD, we can formulate the hadronic bound state problem as a set of coupled few-body problems. We obtain the states and eigenvalues by solving $$H_\Lambda \mid \Psi_\Lambda\rangle = E \mid \Psi_\Lambda \rangle,$$ where, $$\mid \Psi_\Lambda \rangle = \phi^\Lambda_{q\bar{q}} \mid q\bar{q} \rangle + \phi^\Lambda_{q\bar{q}g} \mid q\bar{q}g \rangle + \cdot\cdot\cdot.$$ where I use shorthand notation for the Fock space components of the state. The full state vector includes an infinite number of components, and in a constituent approximation we truncate this series. We derive the Hamiltonian from QCD, so we must allow for the possibility of constituent gluons. I have indicated that the Hamiltonian and the state both depend on a cutoff, $\Lambda$, which is critical for the approximation. This approach has no chance of working without a renormalization scheme [*tailored to it.*]{} Much of our work has focused on the development of such a renormalization scheme. In order to understand the constraints that have driven this development, seriously consider under what conditions it might be possible to truncate the above series without making an arbitrarily large error in the eigenvalue. I focus on the eigenvalue, because it will certainly not be possible to approximate all observable properties of hadrons (, wee parton structure functions) this way. For this approximation to be valid, [*all*]{} many-body states must approximately decouple from the dominant few-body components. We know that even in perturbation theory, high energy many-body states do not simply decouple from few-body states. In fact, the errors from simply discarding high energy states are infinite. In second-order perturbation theory, for example, high energy photons contribute an arbitrarily large shift to the mass of an electron. This second-order effect is illustrated in Figure 1, and the precise interpretation for this light-front time-ordered diagram will be given below. The solution to this problem is well-known, renormalization. We [*must*]{} use renormalization to move the effects of high energy components in the state to effective interactions[^1] in the Hamiltonian. It is difficult to see how a constituent approximation can emerge using any regularization scheme that does not employ a cutoff that either removes degrees of freedom or removes direct couplings between degrees of freedom. A Pauli-Villars “cutoff," for example, drastically increases the size of Fock space and destroys the hermiticity of the Hamiltonian. In the best case scenario we expect the cutoff to act like a resolution. If the cutoff is increased to an arbitrarily large value, the resolution increases and instead of seeing a few constituents we resolve the substructure of the constituents and the few-body approximation breaks down. As the cutoff is lowered, this substructure is removed from the state vectors, and the renormalization procedure replaces it with effective interactions in the Hamiltonian. Any “cutoff" that does not remove this substructure from the states is of no use to us. This point is well-illustrated by the QED calculations discussed below$\!.$ There is a window into which the cutoff must be lowered for the constituent approximation to work. If the cutoff is too large, atomic states must explicitly include photons. After the cutoff is lowered to a value that can be self-consistently determined [*a-posteriori*]{}, photons are removed from the states and replaced by the Coulomb interaction and relativistic corrections. The cutoff cannot be lowered too far using a perturbative renormalization group, hence the window. Thus, if we remove high energy degrees of freedom, or coupling to high energy degrees of freedom, we should encounter self-energy shifts leading to effective one-body operators, vertex corrections leading to effective vertices, and exchange effects leading to explicit many-body interactions not found in the canonical Hamiltonian. We naively expect these operators to be local when acting on low energy states, because simple uncertainty principle arguments indicate that high energy virtual particles cannot propagate very far. Unfortunately this expectation is indeed naive, and at best we can hope to maintain transverse locality. I will elaborate on this point below. The study of perturbation theory with the cutoffs we must employ makes it clear that it is [*not enough*]{} to adjust the canonical couplings and masses to renormalize the theory. It is not possible to make significant progress towards solving light-front QCD without fully appreciating this point. Low energy many-body states do not typically decouple from low energy few-body states. The worst of these low energy many-body states is the vacuum. This is what drives us to use light-front coordinates. Figure 2 shows a pair of particles being produced out of the vacuum in equal-time coordinates $t$ and $z$. The transverse components $x$ and $y$ are not shown, because they are the same in equal-time and light-front coordinates. The figure also shows light-front time, $$x^+=t+z\;,$$ and the light-front longitudinal spatial coordinate, $$x^-=t-z\;.$$ In equal-time coordinates it is kinematically possible for virtual pairs to be produced from the vacuum (although relevant interactions actually produce three or more particles from the vacuum), as long as their momenta sum to zero so that three-momentum is conserved. Because of this, the state vector for a proton includes an arbitrarily large number of particles that are disconnected from the proton. The only constraint imposed by relativity is that particle velocities be less than or equal to that of light. In light-front coordinates, however, we see that all allowed trajectories lie in the first quadrant. In other words, light-front longitudinal momentum, $p^+$ (conjugate to $x^-$ since $a\cdot b={{\scriptstyle{{1\over 2}}}}(a^+ b^- + a^- b^+) - {\bf a}_\perp \cdot {\bf b}_\perp$), is always positive, $$p^+ \ge 0 \;.$$ We exclude $p^+=0$, forcing the vacuum to be trivial because it is the only state with $p^+=0$. Moreover, the light-front energy of a free particle of mass $m$ is $$p^-={{\bf p}_\perp^2+m^2 \over p^+} \;.$$ This implies that all free particles with zero longitudinal momentum have infinite energy, unless their mass and transverse momentum are identically zero. Replacing such particles with effective interactions should be reasonable. - [Is the vacuum really trivial?]{} - [What about confinement?]{} - [What about chiral symmetry breaking?]{} - [What about instantons?]{} - [What about the job security of theorists who study the vacuum?]{} The question of how one should treat “zero modes," degrees of freedom (which may be constrained) with identically zero longitudinal momentum, divides the light-front community. Our attitude is that explicitly including zero modes defeats the purpose of using light-front coordinates, and we do not believe that significant progress will be made in this direction, at least not in $3+1$ dimensions. The vacuum in our formalism is trivial. We are forced to work in the “hidden symmetry phase" of the theory, and to introduce effective interactions that reproduce all effects associated with the vacuum in other formalisms. The simplest example of this approach is provided by a scalar field theory with spontaneous symmetry breaking. It is possible to shift the scalar field and deal explicitly with a theory containing symmetry breaking interactions. In the simplest case $\phi^3$ is the only relevant or marginal symmetry breaking interaction, and one can simply tune this coupling to the value corresponding to spontaneous rather than explicit symmetry breaking. Ken Wilson and I have also shown that in such simple cases one can use coupling coherence to fix the strength of this interaction so that tuning is not required. I will make an additional drastic assumption in these lectures, an assumption that Ken Wilson does not believe will hold true. I will assume that all effective interactions we require are local in the transverse direction. If this is true, there are a finite number of relevant and marginal operators, although each contains a function of longitudinal momenta that must be determined by the renormalization procedure.[^2] There are [*many more*]{} relevant and marginal operators in the renormalized light-front Hamiltonian than in ${\cal L}_{\rm QCD}$. If transverse locality is violated, the situation is much worse than this. The presence of extra relevant and marginal operators that contain functions tremendously complicates the renormalization problem, and a common reaction to this problem is denial, which may persist for years. However, this situation may make possible tremendous simplifications in the final nonperturbative problem. For example, few-body operators must produce confinement manifestly! Confinement cannot require particle creation and annihilation, flux tubes, etc. This is easily seen using a variational argument. Consider a color neutral quark-antiquark pair that are separated by a distance $R$, which is slowly increased to infinity. Moreover, to see the simplest form of confinement assume that there are no light quarks, so that the energy should increase indefinitely as they are separated if the theory possesses confinement. At each separation the gluon components of the state adjust themselves to minimize the energy. But this means that the expectation value of the Hamiltonian for a state with no gluons must exceed the energy of the state with gluons, and therefore must diverge even more rapidly than the energy of the true ground state. This means that there must be a two-body confining interaction in the Hamiltonian. If the renormalization procedure is unable to produce such confining two-body interactions, the constituent picture will not arise. Manifest gauge invariance and manifest rotational invariance require all physical states to contain an arbitrarily large number of constituents. Gauge invariance is not manifest since we work in light-cone gauge with the zero modes removed; and it is easy to see that manifest rotational invariance requires an infinite number of constituents. Rotations about transverse axes are generated by dynamic operators in interacting light-front field theories, operators that create and annihilate particles. No state with a finite number of constituents rotates into itself or transforms as a simple tensor under the action of such generators. These symmetries seem to imply that we can not obtain a constituent approximation. To cut this Gordian knot we employ cutoffs that violate gauge invariance and covariance, symmetries which then must be restored by effective interactions, and which need not be restored exactly. A familiar example of this approach is supplied by lattice gauge theory, where rotational invariance is violated by the lattice. ### Simple Strategy We have recently employed a conceptually simple strategy to complete bound state calculations. The first step is to use a perturbative similarity renormalization group and coupling coherence to find the renormalized Hamiltonian as an expansion in powers of the canonical coupling: $$H^\Lambda = h_0 + g_\Lambda h_1^\Lambda + g_\Lambda^2 h_2^\Lambda + \cdot \cdot \cdot$$ We compute this series to a finite order, and to date have not required any [*ad hoc*]{} assumptions to uniquely fix the Hamiltonian. No operators are added to the hamiltonian, so the hamiltonian is completely determined by the underlying theory to this order. The second step is to employ bound state perturbation theory to solve the eigenvalue problem. The complete Hamiltonian contains every interaction (although each is cut off) contained in the canonical Hamiltonian, and many more. We separate the Hamiltonian, $$H^\Lambda=H_0^\Lambda+V^\Lambda \;,$$ treating $H_0^\Lambda$ nonperturbatively and computing the effects of $V^\Lambda$ in bound state perturbation theory. We must choose $H_0^\Lambda$ and $\Lambda$ so that $H_0^\Lambda$ is [*manageable*]{} and to minimize corrections from higher orders of $V^\Lambda$ within a constituent approximation. If a constituent approximation is valid [*after*]{} $\Lambda$ is lowered to a critical value which must be determined, we may be able to move all creation and annihilation operators to $V^\Lambda$. $H_0^\Lambda$ will include many-body interactions that do not change particle number, and these interactions should be primarily responsible for the constituent bound state structure. There are several obvious flaws in this strategy. Chiral symmetry-breaking operators, which must be included in the Hamiltonian since we work entirely in the hidden symmetry phase of the theory, do not appear at any finite order in the coupling. These operators must simply be added and tuned to fit spectra or fixed by a non-perturbative renormalization procedure. In addition, there are perturbative errors in the strengths of all operators that do appear. We know from simple scaling arguments that when $\Lambda$ is in the scaling regime: - [small errors in relevant operators exponentiate in the output,]{} - [small errors in marginal operators produce comparable errors in output,]{} - [small errors in irrelevant operators tend to decrease exponentially in the output.]{} This means that even if a relevant operator appears (, a constituent quark or gluon mass operator), we may need to tune its strength to obtain reasonable results. We have not had to do this, but we have recently studied some of the effects of tuning a gluon mass operator. To date this strategy has produced well-known results in QED through the Lamb shift, and reasonable results for heavy quark bound states in QCD. The primary objective of the remainder of these lectures is to review these results. I first use the Schwinger model as an illustration of a light-front bound state calculation. This model does not require renormalization, so before turning to QED and QCD I discuss the renormalization procedure that we have developed. LIGHT-FRONT SCHWINGER MODEL --------------------------- In this section I use the Schwinger model, massless QED in $1+1$ dimensions, to illustrate the basic strategy we employ [*after*]{} we have computed the renormalized Hamiltonian. No model in $1+1$ dimensions illustrates the renormalization problems we must solve before we can start to study QCD$_{3+1}$. The Schwinger model can be solved analytically. Charged particles are confined because the Coulomb interaction is linear and there is only one physical particle, a massive neutral scalar particle with no self-interactions. The Fock space content of the physical states depends crucially on the coordinate system and gauge, and it is only in light-front coordinates that a simple constituent picture emerges. The Schwinger model was first studied in Hamiltonian light-front field theory by Bergknoff. My description of the model follows his closely, and I recommend his paper to the reader. Bergknoff showed that the physical boson in the light-front massless Schwinger model in light-cone gauge is a pure electron-positron state. This is an amazing result in a strong-coupling theory of massless bare particles, and it illustrates how a constituent picture may arise in QCD. The electron-positron pair is confined by the linear Coulomb potential. The light-front kinetic energy vanishes in the massless limit, and the potential energy is minimized by a wave function that is flat in momentum space, as one might expect since a linear potential produces a state that is as localized as possible (given kinematic constraints due to the finite velocity of light) in position space. In order to solve this theory I must first set up a large number of details. I recommend that for a first reading these details be skimmed, because the general idea is more important than the detailed manipulations. The Lagrangian for the theory is $${\cal L} = \overline{\psi} \bigl( i \not{{\hskip-.08cm}\partial} -m \bigr) \psi - e \overline{\psi} \gamma_\mu \psi A^\mu - {1 \over 4} F_{\mu \nu} F^{\mu \nu} \;,$$ where $F_{\mu \nu}$ is the electromagnetic field strength tensor. I have included an electron mass, $m$, which is taken to zero later. I choose light-cone gauge, $$A^+=0 \;.$$ In this gauge we avoid ghosts, so that the Fock space has a positive norm. This is absolutely essential if we want to apply intuitive techniques from many-body quantum mechanics. Many calculations are simplified by the use of a chiral representation of the Dirac gamma matrices, so in this section I will use: $$\gamma^0=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right)~~,~~~ \gamma^1=\left(\begin{array}{cc} 0 & i \\ i & 0 \end{array}\right)\;,$$ which leads to the light-front coordinate gamma matrices, $$\gamma^+=\left(\begin{array}{cc} 0 & 0 \\ 2 i & 0 \end{array}\right)~~,~~~ \gamma^-=\left(\begin{array}{cc} 0 & -2 i \\ 0 & 0 \end{array}\right)\;.$$ In light-front coordinates the fermion field $\psi$ contains only one dynamical degree of freedom, rather than two. To see this, first define the projection operators, $$\Lambda_+={1 \over 2} \gamma^0 \gamma^+=\left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)~~~~~ \Lambda_-={1 \over 2} \gamma^0 \gamma^-=\left(\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}\right)\;.$$ Using these operators split the fermion field into two components, $$\psi=\psi_+ + \psi_-=\Lambda_+ \psi + \Lambda_- \psi \;.$$ The two-component Dirac equation in this gauge is $$\biggl( {i \over 2} \gamma^+ \partial^- + {i \over 2} \gamma^- \partial^+ -m - {e \over 2} \gamma^+ A^- \biggr) \psi = 0 \;;$$ which can be split into two one-component equations, $$i \partial^- \psit_+ = -i m \psit_- + e A^- \psit_+ \;,$$ $$i \partial^+ \psit_- = i m \psit_+ \;.$$ Here $\psit_\pm$ refers to the non-zero component of $\psi_\pm$. The equation for $\psi_+$ involves the light-front time derivative, $\partial^-$; so $\psi_+$ is a dynamical degree of freedom that must be quantized. On the other hand, the equation for $\psi_-$ involves only spatial derivatives, so $\psi_-$ is a constrained degree of freedom that should be eliminated in favor of $\psi_+$. Formally, $$\psit_-={m \over \partial^+} \psit_+ \;.$$ This equation is not well-defined until boundary conditions are specified so that $\partial^+$ can be inverted. I will eventually define this operator in momentum space using a cutoff, but I want to delay the introduction of a cutoff until a calculation requires it. I have chosen the gauge so that $A^+=0$, and the equation for $A^-$ is $$-{1 \over 4} \bigl(\partial^+\bigr)^2 A^- = e \psi_+^\dagger \psi_+ \;.$$ $A^-$ is also a constrained degree of freedom, and we can formally eliminate it, $$A^-=-{4 e \over \bigl(\partial^+\bigr)^2} \psi_+^\dagger \psi_+ \;.$$ We are now left with a single dynamical degree of freedom, $\psi_+$, which we can quantize at $x^+=0$, $$\bigl\{\psi_+(x^-),\psi_+^\dagger(y^-)\bigr\} = \Lambda_+ \delta(x^--y^-) \;.$$ We can introduce free particle creation and annihilation operators and expand the field operator at $x^+=0$, $$\psit_+(x^-) = \int_{k^+ > 0} {dk^+ \over 4\pi} \Biggl[ b_k e^{-i k \cdot x} + d_k^\dagger e^{i k \cdot x} \Biggr] \;,$$ with, $$\bigl\{b_k,b_p^\dagger\} = 4 \pi \delta(k^+-p^+) \;.$$ In order to simplify notation, I will often write $k$ to mean $k^+$. If I need $k^-=m^2/k^+$, I will provide the superscript. The next step is to formally specify the Hamiltonian. I start with the canonical Hamiltonian, $$H = \int dx^- \Bigl( H_0 + V \Bigr) \;,$$ $$H_0 = \psi_+^\dagger \Biggl({m^2 \over i\partial^+}\Biggr) \psi_+ \;,$$ $$V= -2 e^2 \psi_+^\dagger \psi_+ \Biggl({1 \over \partial^+}\Biggr)^2 \psi_+^\dagger \psi_+ \;.$$ To actually calculate: - [replace $\psi_+$ with its expansion in terms of $b_k$ and $d_k$]{}, - [normal-order]{}, - [throw away constants]{}, - [drop all operators that require $b_0$ and $d_0$]{}. The free part of the Hamiltonian becomes $$H_0=\int_{k>0} {dk \over 4\pi} \Biggl({m^2 \over k}\Biggr) \bigl(b_k^\dagger b_k+d_k^\dagger d_k\bigr) \;.$$ When $V$ is normal-ordered, we encounter new one-body operators, $$H'_0={e^2 \over 2\pi} \int_{k>0} {dk \over 4\pi} \Biggl[\int_{p>0} dp \biggl( {1 \over (k-p)^2} - {1 \over (k+p)^2} \biggr)\Biggr] \bigl(b_k^\dagger b_k+d_k^\dagger d_k\bigr) \;.$$ This operator contains a divergent momentum integral. From a mathematical point of view we have been sloppy and need to carefully add boundary conditions and define how $\partial^+$ is inverted. However, I want to apply physical intuition and even though no physical photon has been exchanged to produce the initial interaction, I will act as if a photon has been exchanged and everywhere an ‘instantaneous photon exchange’ occurs I will cut off the momentum. In the above integral I insist, $$|k-p|>\epsilon \;.$$ Using this cutoff we find that $$H'_0={e^2 \over \pi} \int{dk \over 4\pi} \biggl({1 \over \epsilon} - {1 \over k} + {\cal O}(\epsilon) \biggr) \bigl(b_k^\dagger b_k + d_k^\dagger d_k \bigr) \;.$$ Comparing this result with the original free Hamiltonian, we see that a divergent mass-like term appears; but it does not have the same dispersion relation as the bare mass. Instead of depending on the inverse momentum of the fermion, it depends on the inverse momentum cutoff, which cannot appear in any physical result. There is also a finite shift in the bare mass, with the standard dispersion relation. The normal-ordered interactions are $$\begin{aligned} V'= 2 e^2 \int {dk_1 \over 4\pi}\cdot\cdot\cdot{dk_4 \over 4\pi} 4\pi\delta(k_1+k_2-k_3-k_4) \nonumber \\ ~~~~~~~~~~~ \Biggl\{ -{2 \over (k_1-k_3)^2} b_1^\dagger d_2^\dagger d_4 b_3 + {2 \over (k_1+k_2)^2} b_1^\dagger d_2^\dagger d_3 b_4 \nonumber \\ ~~~~~~~~~~~~~~-{1 \over (k_1-k_3)^2} \bigl(b_1^\dagger b_2^\dagger b_3 b_4 + d_1^\dagger d_2^\dagger d_3 d_4\bigr) +\cdot\cdot\cdot \Biggr\} \;.\end{aligned}$$ I do not display the interactions that involve the creation or annihilation of electron/positron pairs, which are important for the study of multiple boson eigenstates. The first term in $V'$ is the electron-positron interaction. The longitudinal momentum cutoff I introduced above requires $|k_1-k_3| > \epsilon$, so in position space we encounter a potential which I will naively define with a Fourier transform that ignores the fact that the momentum transfer cannot exceed the momentum of the state, $$\begin{aligned} v(x^-) &=& 4 q_1 q_2 \int_{-\infty}^\infty {dk \over 4\pi}\; {1 \over k^2}\; \theta(|k|-\epsilon) \; {\rm exp}\bigl(-{i \over 2} k x^-\bigr) \nonumber \\ &=& {q_1 q_2 \over \pi} \; \Biggl[ {2 \over \epsilon} - {\pi \over 2} |x^-| + {\cal O}(\epsilon) \Biggr] \;.\end{aligned}$$ This potential contains a linear Coulomb potential that we expect in two dimensions, but it also contains a divergent constant that is negative for unlike charges and positive for like charges. In charge neutral states the infinite constant in $V'$ is [*exactly*]{} canceled by the divergent ‘mass’ term in $H'_0$. This Hamiltonian assigns an infinite energy to states with net charge, and a finite energy as $\epsilon \rightarrow 0$ to charge zero states. This does not imply that charged particles are confined, but the linear potential prevents charged particles from moving to arbitrarily large separation except as charge neutral states. The confinement mechanism I propose for QCD in 3+1 dimensions shares many features with this interaction. I would also like to mention that even though the interaction between charges is long-ranged, there are no van der Waals forces in 1+1 dimensions. It is a simple geometrical calculation to show that all long range forces between two neutral states cancel exactly. This does not happen in higher dimensions, and if we use long-range two-body operators to implement confinement we must also find many-body operators that cancel the strong long-range van der Waals interactions. Given the complete Hamiltonian in normal-ordered form we can study bound states. A powerful tool for the initial study of bound states is the variational wave function. In this case, we can begin with a state that contains a single electron-positron pair, $$|\Psi(P)\rangle = \int_0^P {dp \over 4\pi} \phi(p) b_p^\dagger d_{P-p}^\dagger |0\rangle \;.$$ The norm of this state is $$\langle \Psi(P')|\Psi(P)\rangle = 4\pi P \delta (P'-P) \Biggl\{{1 \over P} \int_0^P {dp \over 4\pi} |\phi(p)|^2 \Biggr\}\;,$$ where the factors outside the brackets provide a covariant plane wave normalization for the center-of-mass motion of the bound state, and the bracketed term should be set to one. The expectation value of the one-body operators in the Hamiltonian is $$\langle\Psi|H_0+H'_0|\Psi\rangle = {1 \over P} \int {dk \over 4\pi} \Biggl[{m^2-e^2/\pi \over k}+ {m^2-e^2/\pi \over P-k}+ {2 e^2 \over \pi \epsilon} \Biggr] |\phi(k)|^2 \;,$$ and the expectation value of the normal-ordered interactions is $$\langle\Psi|V'|\Psi\rangle = -{4 e^2 \over P} \int' {dk_1 \over 4\pi} {dk_2 \over 4\pi} \Bigl[{1 \over (k_1-k_2)^2}-{1 \over P^2}\Biggr] \phi^*(k_1) \phi(k_2) \;,$$ where I have dropped the overall plane wave norm. The prime on the last integral indicates that the range of integration in which $|k_1-k_2|<\epsilon$ must be removed. By expanding the integrand about $k_1=k_2$, one can easily confirm that the $1/\epsilon$ divergences cancel. With $m=0$ the energy is minimized when $$\phi(k)=\sqrt{4\pi} \;,$$ and the invariant-mass is $$M^2={e^2 \over \pi} \;.$$ This type of simple analysis can be used to show that this electron-positron state is actually the [*exact ground*]{} state of the theory with momentum $P$, and that bound states do not interact with one another. The primary purpose of introducing the Schwinger model is to illustrate that bound state center-of-mass motion is easily separated from relative motion in light-front coordinates, and that standard quantum mechanical techniques can be used to analyze the relative motion of charged particles once the Hamiltonian is found. It is intriguing that even when the fermion is massless, the states are constituent states in light-cone gauge and in light-front coordinates. This is not true in other gauges and coordinate systems. The success of light-front field theory in 1+1 dimensions can certainly be downplayed, but it should be emphasized that no other method on the market is as powerful for bound state problems in 1+1 dimensions. The most significant barriers to using light-front field theory to solve low energy QCD are not encountered in 1+1 dimensions. The Schwinger model is super-renormalizable, so we completely avoid serious ultraviolet divergences. There are no transverse directions, and we are not forced to introduce a cutoff that violates rotational invariance, because there are no rotations. Confinement results from the Coulomb interaction, and chiral symmetry is not spontaneously broken. This simplicity disappears in realistic $3+1$-dimensional calculations, which is one reason there are so few $3+1$-dimensional light-front field theory calculations. LIGHT-FRONT RENORMALIZATION GROUP: SIMILARITY TRANSFORMATION AND COUPLING COHERENCE ----------------------------------------------------------------------------------- As argued above, in $3+1$ dimensions we must introduce a cutoff on energies, $\Lambda$, and we never perform explicit bound state calculations with $\Lambda$ anywhere near its continuum limit. In fact, we want to let $\Lambda$ become as small as possible. In my opinion, any strategy for solving light-front QCD that requires the cutoff to explicitly approach infinity in the nonperturbative part of the calculation is useless. Therefore, we must set up and solve $$P^-_\Lambda \mid \Psi_\Lambda(P) \rangle = {{\bf P}_\perp^2 + M^2 \over P^+} \mid \Psi_\Lambda(P) \rangle \;.$$ Physical results, such as the mass, $M$, can not depend on the arbitrary cutoff, $\Lambda$, [*even*]{} as $\Lambda$ approaches the scale of interest. This means that $P^-_\Lambda$ and $\mid \Psi_\Lambda\rangle$ must depend on the cutoff in such a way that $\langle \Psi_\Lambda \mid P^-_\Lambda \mid \Psi_\Lambda \rangle$ does not. Wilson based the derivation of his renormalization group on this observation, and we use Wilson’s renormalization group to compute $P^-_\Lambda$. It is difficult to even talk about how the Hamiltonian depends on the cutoff without having a means of changing the cutoff. If we can change the cutoff, we can explicitly watch the Hamiltonian’s cutoff dependence change and fix its cutoff dependence by insisting that this change satisfy certain requirements (, that the limit in which the cutoff is taken to infinity exists). We introduce an operator that changes the cutoff, $$H(\Lambda_1) = T[H(\Lambda_0)] \;,$$ where I assume that $\Lambda_1 < \Lambda_0$. To simplify the notation, I will let $H(\Lambda_l)=H_l$. To renormalize the hamiltonian we study the properties of the transformation. Figure 3 displays two generic cutoffs that might be used. Traditionally theorists have used cutoffs that remove high energy states, as shown in Figure 3a. This is the type of cutoff Wilson employed in his initial work and I have studied its use in light-front field theory. When a cutoff on energies is reduced, all effects of couplings eliminated must be moved to effective operators. As we will see explicitly below, when these effective operators are computed perturbatively they involve products of matrix elements divided by energy denominators. Expressions closely resemble those encountered in standard perturbation theory, with the second-order operator involving terms of the form $$\delta V_{ij} \sim {\langle \phi_i \mid V \mid \phi_k \rangle \langle \phi_k \mid V \mid \phi_j \rangle \over \epsilon_i-\epsilon_k}\;.$$ This new effective interaction replaces missing couplings, so the states $\phi_i$ and $\phi_j$ are retained and the state $\phi_k$ is one of the states removed. The problem comes from the shaded, lower right-hand corner of the matrix, where the energy denominator vanishes for states at the corner of the remaining matrix. In this corner we should use nearly degenerate perturbation theory rather than perturbation theory, but to do this requires solving high energy many-body problems nonperturbatively before solving the low energy few-body problems. An alternative cutoff, which does not actually remove any states and which can be run by a similarity transformation[^3] is shown in Figure 3b. This cutoff removes couplings between states whose free energy differs by more than the cutoff. The advantage of this cutoff is that the effective operators resulting from it contain energy denominators which are never smaller than the cutoff, so that a perturbative approximation for the effective Hamiltonian may work well. I discuss a conceptually simple similarity transformation that runs this cutoff below. Given a cutoff and a transformation that runs the cutoff, we can discuss how the Hamiltonian depends on the cutoff by studying how it changes with the cutoff. Our objective is to find a “renormalized" Hamiltonian, which should give the same results as an idealized Hamiltonian in which the cutoff is infinite and which displays all of the symmetries of the theory. To state this in simple terms, consider a sequence of Hamiltonians generated by repeated application of the transformation, $$H_0 \rightarrow H_1 \rightarrow H_2 \rightarrow \cdot \cdot \cdot \;.$$ What we really want to do is fix the final value of $\Lambda$ at a reasonable hadronic scale, and let $\Lambda_0$ approach infinity. In other words, we seek a Hamiltonian that survives an infinite number of transformations. In order to do this we need to understand what happens when the transformation is applied to a broad class of Hamiltonians. Perturbative renormalization group analyses typically begin with the identification of at least one fixed point, $H^*$. A [*fixed point*]{} is defined to be any Hamiltonian that satisfies the condition $$H^*=T[H^*] \;.$$ For perturbative renormalization groups the search for such fixed points is relatively easy. If $H^*$ contains no interactions (, no terms with a product of more than two field operators), it is called [*Gaussian*]{}. If $H^*$ has a massless eigenstate, it is called [*critical*]{}. If a Gaussian fixed point has no mass term, it is a [*critical Gaussian*]{} fixed point. If it has a mass term, this mass must typically be infinite, in which case it is a [*trivial Gaussian*]{} fixed point. In lattice QCD the trajectory of renormalized Hamiltonians stays ‘near’ a critical Gaussian fixed point until the lattice spacing becomes sufficiently large that a transition to strong-coupling behavior occurs. If $H^*$ contains only weak interactions, it is called [*near-Gaussian*]{}, and we may be able to use perturbation theory both to identify $H^*$ and to accurately approximate ‘trajectories’ of Hamiltonians near $H^*$. Of course, once the trajectory leaves the region of $H^*$, it is generally necessary to switch to a non-perturbative calculation of subsequent evolution. Consider the immediate ‘neighborhood’ of the fixed point, and assume that the trajectory remains in this neighborhood. This assumption must be justified [*a posteriori*]{}, but if it is true we should write $$H_l=H^*+\delta H_l \;,$$ and consider the trajectory of small deviations $\delta H_l$. As long as $\delta H_l$ is ‘sufficiently small,’ we can use a perturbative expansion in powers of $\delta H_l$, which leads us to consider $$\delta H_{l+1}= L \cdot \delta H_l + N[\delta H_l] \;.$$ Here $L$ is the linear approximation of the full transformation in the neighborhood of the fixed point, and $N[\delta H_l]$ contains all contributions to $\delta H_{l+1}$ of $\order(\delta H_l^2)$ and higher. The object of the renormalization group calculation is to compute trajectories and this requires a representation for $\delta H_l$. The problem of computing trajectories is one of the most common in physics, and a convenient basis for the representation of $\delta H_l$ is provided by the eigenoperators of $L$, since $L$ dominates the transformation near the fixed point. These eigenoperators and their eigenvalues are found by solving $$L \cdot O_m=\lambda_m O_m \;.$$ If $H^*$ is Gaussian or near-Gaussian it is usually straightforward to find $L$, and its eigenoperators and eigenvalues. This is not typically true if $H^*$ contains strong interactions, but in QCD we hope to use a perturbative renormalization group in the regime of asymptotic freedom, and the QCD ultraviolet fixed point is apparently a critical Gaussian fixed point. For light-front field theory this linear transformation is a scaling of the transverse coordinate, the eigenoperators are products of field operators and transverse derivatives, and the eigenvalues are determined by the transverse dimension of the operator. All operators can include both powers and inverse powers of longitudinal derivatives because there is no longitudinal locality. Using the eigenoperators of $L$ as a basis we can represent $\delta H_l$, $$\delta H_l = \sum_{m\in R} \mu_{m_l}O_m +\sum_{m\in M} g_{m_l}O_m+ \sum_{m\in I} w_{m_l}O_m \;.$$ Here the operators $O_m$ with $m\in R$ are [*relevant*]{} (, $\lambda_m>1$), the operators $O_m$ with $m\in M$ are [*marginal*]{} (, $\lambda_m=1$), and the operators with $m\in I$ are either [*irrelevant*]{} (, $\lambda_m<1$) or become irrelevant after many applications of the transformation. The motivation behind this nomenclature is made clear by considering repeated application of $L$, which causes the relevant operators to grow exponentially, the marginal operators to remain unchanged in strength, and the irrelevant operators to decrease in magnitude exponentially. There are technical difficulties associated with the symmetry of $L$ and the completeness of the eigenoperators that I ignore. For the purpose of illustration, let me assume that $\lambda_m=4$ for all relevant operators, and $\lambda_m=1/4$ for all irrelevant operators. The transformation can be represented by an infinite number of coupled, nonlinear difference equations: $$\mu_{m_{l+1}}=4 \mu_{m_l} + N_{\mu_m}[\mu_{m_l}, g_{m_l}, w_{m_l}] \;,$$ $$g_{m_{l+1}}=g_{m_l} + N_{g_m}[\mu_{m_l}, g_{m_l}, w_{m_l}] \;,$$ $$w_{m_{l+1}}={1 \over 4} w_{m_l} + N_{w_m}[\mu_{m_l}, g_{m_l}, w_{m_l}] \;.$$ Sufficiently near a critical Gaussian fixed point, the functions $N_{\mu_m}$, $N_{g_m}$, and $N_{w_m}$ should be adequately approximated by an expansion in powers of $\mu_{m_l}$, $g_{m_l}$, and $w_{m_l}$. The assumption that the Hamiltonian remains in the neighborhood of the fixed point, so that all $\mu_{m_l}$, $g_{m_l}$, and $w_{m_{l}}$ remain small, must be justified [*a posteriori*]{}. Any precise definition of the neighborhood of the fixed point within which all approximations are valid must also be provided [*a posteriori*]{}. Wilson has given a general discussion of how these equations can be solved, but I will use coupling coherence to fix the Hamiltonian. This is detailed below, so at this point I will merely state that coupling coherence allows us to fix all couplings as functions of the canonical couplings and masses in a theory. The renormalization group equations specify how all of the couplings run, and coupling coherence uses this behavior to fix the strength of all of the couplings. But the first step is to develop a transformation. ### Similarity Transformation Stan G[ł]{}azek and Ken Wilson studied the problem of small energy denominators which are apparent in Wilson’s first complete non-perturbative renormalization group calculations, and realized that a similarity transformation which runs a different form of cutoff (as discussed above) avoids this problem. Independently, Wegner developed a similarity transformation which is easier to use than that of G[ł]{}azek and Wilson. In this section I want to give a simplified discussion of the similarity transformation, using sharp cutoffs that must eventually be replaced with smooth cutoffs which require a more complicated formalism. Suppose we have a Hamiltonian, $$H^\lzero=H_0^\lzero+V^\lzero \;,$$ where $H^\lzero_0$ is diagonal. The cutoff $\lzero$ indicates that $\langle \phi_i|V^\lzero|\phi_j\rangle=0$ if $|E_{0 i}-E_{0 j}|>\lzero$. I should note that $\lzero$ is defined differently in this section from later sections. We want to use a similarity transformation, which automatically leaves all eigenvalues and other physical matrix elements invariant, that lowers this cutoff to $\lone$. This similarity transformation will constitute the first step in a renormalization group transformation, with the second step being a rescaling of energies that returns the cutoff to its original numerical value.[^4] The transformed Hamiltonian is $$H^\lone=e^{i R}\bigl(H_0^\lzero+V^\lzero\bigr)e^{-i R} \;,$$ where $R$ is a hermitian operator. If $H^\lzero$ is already diagonal, $R=0$. Thus, if $R$ has an expansion in powers of $V$, it starts at first order and we can expand the exponents in powers of $R$ to find the perturbative approximation of the transformation. We must adjust $R$ so that the matrix elements of $H^\lone$ vanish for all states that satisfy $\lone<|E_{0 i}-E_{0 j}|<\lzero$. We insist that this happens to each order in perturbation theory. Consider such a matrix element, $$\begin{aligned} \langle \phi_i|H^\lone|\phi_j\rangle &=& \langle \phi_i| e^{i R}\bigl(H_0^\lzero+V^\lzero\bigr)e^{-i R} |\phi_j\rangle \nonumber \\ &=& \langle \phi_i| (1+i R+\cdot\cdot\cdot) (H_0^\lzero+V^\lzero\bigr) (1 -i R-\cdot\cdot\cdot) |\phi_j\rangle \nonumber \\ &=& \langle \phi_i|H_0^\lzero |\phi_j\rangle + \langle \phi_i|V^\lzero+i\Bigl[R,H_0^\lzero\Bigr] |\phi_j\rangle + \cdot\cdot\cdot \;.\end{aligned}$$ The last line contains all terms that appear in first-order perturbation theory. Since $\langle \phi_i|H_0^\lzero |\phi_j\rangle =0$ for these off-diagonal matrix elements, we can satisfy our new constraint using $$\langle \phi_i|R |\phi_j\rangle = {i \langle \phi_i|V^\lzero |\phi_j\rangle \over E_{0 j}-E_{0 i}} \;+\;\order(V^2) \;.$$ This fixes the matrix elements of $R$ when $\lone<|E_{0 i}-E_{0 j}|<\lzero$ to first order in $V^\lzero$. I will assume that the matrix elements of $R$ for $|E_{0i}-E_{0j}|<\lone$ are zero to first order in $V$, and fix these matrix elements to second order below. Given $R$ we can compute the nonzero matrix elements of $H^\lone$. To second order in $V^\lzero$ these are $$\begin{aligned} H^\lone_{ab} &=& \langle \phi_a| H_0+V + i\Bigl[R,H_0\Bigr] +i\Bigl[R,V \Bigr] -{1 \over 2} \Bigl\{R^2,H_0\Bigr\} + R H_0 R + \order(V^3) |\phi_b\rangle \nonumber \\ &=& \langle \phi_a|H_0+V + i\Bigl[R_2,H_0\Bigr]|\phi_b\rangle \nonumber \\ && -{1 \over 2} \sum_k \Theta_{a k} \Theta_{k b} V_{ak} V_{kb} \Biggl[ {1 \over E_{0k}-E_{0a}}+{1 \over E_{0k}-E_{0b}} \Biggr] \nonumber \\ &&-\sum_k\Biggl[\Theta_{ak}\bigl(1-\Theta_{kb}\bigr) {V_{ak} V_{kb} \over E_{0k}-E_{0a}} + \Theta_{kb}\bigl(1-\Theta_{ak}\bigr) {V_{ak} V_{kb} \over E_{0k}-E_{0b}} \Biggr]+\order(V^3) \nonumber \\ &=& \langle \phi_a|H_0+V+ i\Bigl[R_2,H_0\Bigr] |\phi_b\rangle \nonumber \\&&+{1 \over 2} \sum_k \Theta_{a k} \Theta_{k b} V_{ak} V_{kb} \Biggl[ {1 \over E_{0k}-E_{0a}}+{1 \over E_{0k}-E_{0b}} \Biggr] \times \nonumber \\ &&~~~~~\Biggl[\theta(|E_{0a}-E_{0k}|-|E_{0b}-E_{0k}|) - \theta(|E_{0b}-E_{0k}|- |E_{0a}-E_{0k}|)\Biggr] \nonumber \\ &&-\sum_k V_{ak} V_{kb} \biggl[ \Theta_{ak} { \theta(|E_{0a}-E_{0k}|-|E_{0b}-E_{0k}|) \over E_{0k} - E_{0a} } + \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~ \Theta_{kb} { \theta(|E_{0b}-E_{0k}|- |E_{0a}-E_{0k}|) \over E_{0k} - E_{0b} } \Biggr] \;.\end{aligned}$$ I have dropped the $\lzero$ superscript on the right-hand side of this equation and used subscripts to indicate matrix elements. The operator $\Theta_{ij}$ is one if $\lone<|E_{0i}-E_{0j}|<\lzero$ and zero otherwise. It should also be noted that $V_{ij}$ is zero if $|E_{0i}-E_{0j}|>\lzero$. All energy denominators involve energy differences that are at least as large as $\lone$, and this feature persists to higher orders in perturbation theory; which is the main motivation for choosing this transformation. $R_2$ is second-order in $V$, and we are still free to choose its matrix elements; however, we must be careful not to introduce small energy denominators when choosing $R_2$. The matrix element $\langle \phi_a| i\Bigl[R_2, H_0 \Bigr] |\phi_b \rangle = i (E_{0b}-E_{0a}) \langle \phi_a| R_2 | \phi_b \rangle $ must be specified. I will choose this matrix element to cancel the first sum in the final right-hand side of Eq. (55). This choice leads to the same result one obtains by integrating a differential transformation that runs a step function cutoff. To cancel the first sum in the final right-hand side of Eq. (55) requires $$\begin{aligned} \langle \phi_a| R_2 | \phi_b \rangle &&= {i \over 2} \sum_k \Theta_{ak} \Theta_{kb} V_{ak} V_{kb} \Biggl[ {1 \over E_{0k}-E_{0a}}+{1 \over E_{0k}-E_{0b}} \Biggr] \times \nonumber \\ &&\Biggl[\theta(|E_{0a}-E_{0k}|-|E_{0b}-E_{0k}|) - \theta(|E_{0b}-E_{0k}|- |E_{0a}-E_{0k}|)\Biggr] \;.\end{aligned}$$ No small energy denominator appears in $R_2$ because it is being used to cancel a term that involves a large energy difference. If we tried to use $R_2$ to cancel the remaining sum also, we would find that it includes matrix elements that diverge as $E_{0a}-E_{0b}$ goes to zero, and this is not allowed. The non-vanishing matrix elements of $H^{\Lambda_1}$ are now completely determined to $\order(V^2)$, $$\begin{aligned} H^\lone_{ab} &=& \langle \phi_a|H_0+V^\lzero|\phi_b\rangle \nonumber \\&& - \sum_k V_{ak} V_{kb} \biggl[ \Theta_{ak} { \theta(|E_{0a}-E_{0k}|-|E_{0b}-E_{0k}|) \over E_{0k} - E_{0a} } + \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~ \Theta_{kb} { \theta(|E_{0b}-E_{0k}|- |E_{0a}-E_{0k}|) \over E_{0k} - E_{0b} } \Biggr] \;.\end{aligned}$$ Let me again mention that $V^\lzero_{ij}$ is zero if $|E_{0i}- E_{0j}|>\lzero$, so there are implicit cutoffs that result from previous transformations. As a final word of caution, I should mention that the use of step functions produces long-range pathologies in the interactions that lead to infrared divergences in gauge theories. We must replace the step functions with smooth functions to avoid this problem. This problem will not show up in any calculations detailed in these lectures, but it does affect higher order calculations in QED and QCD. ### Light-Front Renormalization Group In this section I use the similarity transformation to form a perturbative light-front renormalization group for scalar field theory. If we want to stay as close as possible to the canonical construction of field theories, we can: - Write a set of ‘allowed’ operators using powers of derivatives and field operators. - Introduce ‘free’ particle creation and annihilation operators, and expand all field operators in this basis. - Introduce cutoffs on the Fock space transition operators. Instead of following this program I will skip to the final step, and simply write a Hamiltonian to initiate the analysis. $$\begin{aligned} H & = &\qquad \int \dqt_1 \; \dqt_2 \; (16 \pi^3) \delta^3(q_1-q_2) \; u_2(-q_1,q_2) \;a^\dagger(q_1) a(q_2) \nonumber \\ &&+{1 \over 6} \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1+q_2+q_3-q_4) \nonumber \Theta(q_4^--q_3^--q_2^--q_1^-) \\ &&\qquad\qquad\qquad\qquad\qquad u_4(-q_1,-q_2,-q_3,q_4)\; a^\dagger(q_1) a^\dagger(q_2) a^\dagger(q_3) a(q_4) \nonumber \\ &&+{1 \over 4} \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1+q_2-q_3-q_4) \Theta(q_4^-+q_3^--q_2^--q_1^-) \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad u_4(-q_1,-q_2,q_3,q_4)\; a^\dagger(q_1) a^\dagger(q_2) a(q_3) a(q_4) \nonumber \\ &&+{1 \over 6} \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1-q_2-q_3-q_4) \Theta(q_4^-+q_3^-+q_2^--q_1^-) \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad u_4(-q_1,q_2,q_3,q_4)\; a^\dagger(q_1) a(q_2) a(q_3) a(q_4) \nonumber \\ &&+ \qquad {\cal O}(\phi^6) \;,\end{aligned}$$ where, $$\dqt={dq^+ d^2q_\perp \over 16\pi^3 q^+} \;,$$ $$q_i^-={q_{i\perp}^2 \over q_i^+} \;,$$ and, $$\Theta(Q^-)=\theta\Biggl({\Lambda^2 \over P^+}-|Q^-| \Biggr) \;.$$ I assume that no operators break the discrete $\phi \rightarrow -\phi$ symmetry. The functions $u_2$ and $u_4$ are not yet determined. If we assume locality in the transverse direction, these functions can be expanded in powers of their transverse momentum arguments. Note that to specify the cutoff both transverse and longitudinal momentum scales are required, and in this case the longitudinal momentum scale is independent of the particular state being studied. Note also that $P^+$ breaks longitudinal boost invariance and that a change in $P^+$ can be compensated by a change in $\Lambda^2$. This may have important consequences, because the Hamiltonian should be a fixed point with respect to changes in the cutoff’s longitudinal momentum scale, since this scale invariance is protected by Lorentz covariance. I have specified the similarity transformation in terms of matrix elements, and will work directly with matrix elements, which are easily computed in the free particle Fock space basis. In order to study the renormalization group transformation I will assume that the Hamiltonian includes only the interactions shown above. A single transformation will produce a Hamiltonian containing products of arbitrarily many creation and annihilation operators, but it is not necessary to understand the transformation in full detail. I will [*define*]{} the full renormalization group transformation as: (i) a similarity transformation that lowers the cutoff in $\Theta$, (ii) a rescaling of all transverse momenta that returns the cutoff to its original numerical value, (iii) a rescaling of the creation and annihilation operators by a constant factor $\zeta$, and (iv) an overall constant rescaling of the Hamiltonian to absorb a multiplicative factor that results from the fact that it has the dimension of transverse momentum squared. These rescaling operations are introduced so that it may be possible to find a fixed point Hamiltonian that contains interactions. To find the critical Gaussian fixed point we need to study the linearized approximation of the full transformation, as discussed above. In general the linearized approximation can be extremely complicated, but near a critical Gaussian fixed point it is particularly simple in light-front field theory with zero modes removed, because tadpoles are excluded. We have already seen that the similarity transformation does not produce any first order change in the Hamiltonian (see Eq. (57)), so the first order change is determined entirely by the rescaling operation. If we let $$\Lambda_1 = \eta \Lambda_0 \;,$$ $${\bf p}_{i\perp} = \eta {\bf p}_{i\perp}' \;,$$ and, $$a_p = \zeta a_{p'} \;,\;\;\;a_p^\dagger = \zeta a_{p'}^\dagger \;,$$ to first order the transformed Hamiltonian is $$\begin{aligned} H & = &\qquad \zeta^2 \int \dqt_1 \; \dqt_2 \; (16 \pi^3) \delta^3(q_1-q_2) \; u_2(q_i^+, \eta q_i^\perp) \;a^\dagger(q_1) a(q_2) \nonumber \\ &&+{1 \over 6} \eta^4 \zeta^4 \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1+q_2+q_3-q_4) \nonumber \Theta(q_4^--q_3^--q_2^--q_1^-) \\ &&\qquad\qquad\qquad\qquad\qquad u_4(q_i^+, \eta q_i^\perp)\; a^\dagger(q_1) a^\dagger(q_2) a^\dagger(q_3) a(q_4) \nonumber \\ &&+{1 \over 4} \eta^4 \zeta^4 \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1+q_2-q_3-q_4) \Theta(q_4^-+q_3^--q_2^--q_1^-) \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad u_4(q_i^+, \eta q_i^\perp)\; a^\dagger(q_1) a^\dagger(q_2) a(q_3) a(q_4) \nonumber \\ &&+{1 \over 6} \eta^4 \zeta^4 \int \dqt_1\; \dqt_2\; \dqt_3\; \dqt_4 \; (16 \pi^3) \delta^3(q_1-q_2-q_3-q_4) \Theta(q_4^-+q_3^-+q_2^--q_1^-) \nonumber \\ &&\qquad\qquad\qquad\qquad\qquad u_4(q_i^+, \eta q_i^\perp)\; a^\dagger(q_1) a(q_2) a(q_3) a(q_4) \nonumber \\ &&+ \qquad \order(\phi^6) \;,\end{aligned}$$ where I have simplified my notation for the arguments appearing in the functions $u_i$. An overall factor of $\eta^2$ that results from the engineering dimension of the Hamiltonian has been removed. The Gaussian fixed point is found by insisting that the first term remains constant, which requires $$\zeta^2 u_2^*(q^+, \eta {\bf q}_\perp)=u_2^*(q^+,{\bf q}_\perp) \;.$$ I have used the fact that $u_2$ actually depends on only one momentum other than the cutoff. The solution to this equation is a monomial in ${\bf q}_\perp$ which depends on $\zeta$, $$u_2^*(q^+,{\bf q}_\perp)=f(q^+)\bigl({\bf q}_\perp \bigr)^n \;,\;\;\; \zeta=\Biggl({1 \over \eta}\Biggr)^{(n/2)} \;.$$ The solution depends on our choice of $n$ and to obtain the appropriate free particle dispersion relation we need to choose $n=2$, so that $$u_2^*(q^+,{\bf q}_\perp)=f(q^+) {\bf q}_\perp^2 \;\;,\;\;\; \zeta={1 \over \eta} \;.$$ $f(q^+)$ is allowed because the cutoff scale $P^+$ allows us to form the dimensionless variable $q^+/P^+$, which can enter the one-body operator. We will see that this happens in QED and QCD. Note that the constant in front of each four-point interaction becomes one, so that their scaling behavior is determined entirely by $u_4$. If we insist on transverse locality (which may be violated because we remove zero modes), we can expand $u_4$ in powers of its transverse momentum arguments, and discover powers of $\eta q_i^\perp$ in the transformed Hamiltonian. Since we are lowering the cutoff, $\eta<1$, and each power of transverse momentum will be suppressed by this factor. This means increasing powers of transverse momentum are increasingly [*irrelevant*]{}. I will not go through a complete derivation of the eigenoperators of the linearized approximation to the renormalization group transformation about the critical Gaussian fixed point, but the derivation is simple. Increasing powers of transverse derivatives and increasing powers of creation and annihilation operators lead to increasingly irrelevant operators. The irrelevant operators are called ‘non-renormalizable’ in old-fashioned Feynman perturbation theory. Their magnitude decreases at an exponential rate as the cutoff is lowered, which means that they [*increase*]{} at an exponential rate as the cutoff is raised and produce increasingly large divergences if we try to follow their evolution perturbatively in this exponentially unstable direction. The only relevant operator is the mass operator, $$\mu^2 \int \dqt_1 \; \dqt_2 \; (16 \pi^3) \delta^3(q_1-q_2) \; \;a^\dagger(q_1) a(q_2) \;,$$ while the fixed point Hamiltonian is marginal (of course), and the operator in which $u_4=\lambda$ (a constant) is marginal. A $\phi^3$ operator would also be relevant. The next logical step in a renormalization group analysis is to study the transformation to second order in the interaction, keeping the second-order corrections from the similarity transformation. I will compute the correction to $u_2$ to this order and refer the interested reader to Ref. [@perryrg] for more complicated examples. The matrix element of the one-body operator between single particle states is $$\langle p'|h|p\rangle = \langle 0|\;a(p')\; h\; a^\dagger(p)\;|0\rangle = (16 \pi^3) \delta^3(p'-p)\;u_2(-p,p) \;.$$ Thus, we easily determine $u_2$ from the matrix element. It is easy to compute matrix elements between other states. We computed the matrix elements of the effective Hamiltonian generated by the similarity transformation when the cutoff is lowered in Eq. (57), and now we want to compute the second-order term generated by the four-point interactions above. There are additional corrections to $u_2$ at second order in the interaction if $u_6$, [*etc.*]{} are nonzero. Before rescaling we find that the transformed Hamiltonian contains $$\begin{aligned} &&(16 \pi^3)\delta^3(p'-p)\;\delta v_2(-p,p)= \nonumber \\ &&~~~~~ {1 \over 3!} \int \dkt_1 \dkt_2 \dkt_3 \; \theta\bigl (\Lambda_0 -| p^- - k_1^- -k_2^- -k_3^-|\bigr) \nonumber \\ &&~~~~~ \theta\bigl( | p^- - k_1^- -k_2^- -k_3^-| - \eta \Lambda_0\bigr) ~ {\langle p'| V|k_1,k_2,k_3\rangle \langle k_1,k_2,k_3|V|p\rangle \over p^--k_1^--k_2^--k_3^-} \;,\end{aligned}$$ where $p^-={\bf p}_\perp^2/p^+$, [*etc.*]{} One can readily verify that $$\langle p'|V|k_1,k_2,k_3\rangle = (16 \pi^3) \delta^3(p'-k_1-k_2-k_3) \; \Theta\bigl(p^--k_1^--k_2^--k_3^-\bigr)\; u_4(-p',k_1,k_2,k_3) \;,$$ $$\langle k_1,k_2,k_3|V|p\rangle= (16 \pi^3) \delta^3(p-k_1-k_2-k_3) \; \Theta\bigl(p^--k_1^--k_2^--k_3^-\bigr)\; u_4(-k_1,-k_2,-k_3,p) \;.$$ This leads to the result, $$\begin{aligned} \delta v_2(-p,p)&=&{1 \over 3!} \int \dkt_1 \dkt_2 \dkt_3 \; (16 \pi^3)\delta^3(p-k_1-k_2-k_3) \nonumber \\ &&\theta\bigl(\Lambda_0 -| p^- - k_1^- -k_2^- -k_3^-|\bigr)~ \theta\bigl( | p^- - k_1^- -k_2^- -k_3^-| - \eta \Lambda_0\bigr) \nonumber \\ &&~~~~~~~ {u_4(-p,k_1,k_2,k_3)\; u_4(-k_1,-k_2,-k_3,p) \over p^--k_1^--k_2^--k_3^-} \;.\end{aligned}$$ To obtain $\delta u_2$ from $\delta v_2$ we must rescale the momenta, the fields, and the Hamiltonian. The final result is $$\begin{aligned} \delta u_2(-p,p)&=&{1 \over 3!} \int \dkt_1 \dkt_2 \dkt_3 \; (16 \pi^3)\delta^3(p-k_1-k_2-k_3) \nonumber \\ &&\theta\Biggl({\Lambda_0 \over \eta} -| p^- - k_1^- -k_2^- -k_3^-|\Biggr)~ \theta\bigl( | p^- - k_1^- -k_2^- -k_3^-| - \Lambda_0\bigr) \nonumber \\ &&~~~~~~~ {u_4(p',k_1',k_2',k_3')\; u_4(-k_1',-k_2',-k_3',p') \over p^--k_1^--k_2^--k_3^-} \;,\end{aligned}$$ where $p^{+'}=p^+$, $p^{\perp'}=\eta p^\perp$, $k_i^{+'}=k_i^+$, and $k_i^{\perp'}=\eta k_i^\perp$. ### Coupling Coherence The basic mathematical idea behind coupling coherence was first formulated by Oehme, Sibold, and Zimmerman. They were interested in field theories where many couplings appear, such as the standard model, and wanted to find some means of reducing the number of couplings. Wilson and I developed the ideas independently in an attempt to deal with the functions that appear in marginal and relevant light-front operators. The puzzle is how to reconcile our knowledge from covariant formulations of QCD that only one running coupling constant characterizes the renormalized theory with the appearance of new counterterms and functions required by the light-front formulation. What happens in perturbation theory when there are effectively an infinite number of relevant and marginal operators? In particular, does the solution of the perturbative renormalization group equations require an infinite number of independent counterterms (, independent functions of the cutoff)? Coupling coherence provides the conditions under which a finite number of running variables determines the renormalization group trajectory of the renormalized Hamiltonian. To leading nontrivial orders these conditions are satisfied by the counterterms introduced to restore Lorentz covariance in scalar field theory and gauge invariance in light-front gauge theories. In fact, the conditions can be used to determine all counterterms in the Hamiltonian, including relevant and marginal operators that contain functions of longitudinal momentum fractions; and with no direct reference to Lorentz covariance, this symmetry is restored to observables by the resultant counterterms in scalar field theory. A coupling-coherent Hamiltonian is analogous to a fixed point Hamiltonian, but instead of reproducing itself exactly it reproduces itself in form with a limited number of independent running couplings. If $g_\Lambda$ is the only independent coupling in a theory, in a coupling-coherent Hamiltonian [*all other couplings are invariant functions of $g_\Lambda$, $f_i(g_\Lambda)$*]{}. The couplings $f_i(g_\Lambda)$ depend on the cutoff only through their dependence on the running coupling $g_\Lambda$, and in general we demand $f_i(0)=0$. This boundary condition on the dependent couplings is motivated in our calculations by the fact that it is the combination of the cutoff and the interactions that force us to add the counterterms we seek, so the counterterms should vanish when the interactions are turned off. Let me start with a simple example in which there is a finite number of relevant and marginal operators , and use coupling coherence to discover when only one or two of these may independently run with the cutoff. In general such conditions are met only when an underlying symmetry exists. Consider a theory in which two scalar fields interact, $$V(\phi)={\lambda_1 \over 4!}\phi_1^4+{\lambda_2 \over 4!}\phi_2^4+ {\lambda_3 \over 4!}\phi_1^2 \phi_2^2 \;.$$ Under what conditions will there be fewer than three independent running coupling constants? We can use a simple cutoff on Euclidean momenta, $q^2<\Lambda^2$. Letting $t=\ln(\Lambda/\Lambda_0)$, the Gell-Mann–Low equations are $${\partial \lambda_1 \over \partial t} = 3 \zeta \lambda_1^2 + {1 \over 12} \zeta \lambda_3^2 + \order(2\;{\rm loop}) \;,$$ $${\partial \lambda_2 \over \partial t} = 3 \zeta \lambda_2^2 + {1 \over 12} \zeta \lambda_3^2 + \order(2\;{\rm loop}) \;,$$ $${\partial \lambda_3 \over \partial t} = {2 \over 3} \zeta \lambda_3^2 + \zeta \lambda_1 \lambda_3+\zeta \lambda_2 \lambda_3 + \order(2\;{\rm loop}) \;;$$ where $\zeta=\hbar/(16\pi^2)$. It is not important at this point to understand how these equations are derived. First suppose that $\lambda_1$ and $\lambda_2$ run separately, and ask whether it is possible to find $\lambda_3(\lambda_1,\lambda_2)$ that solves Eq. (79). To one-loop order this leads to $$\Bigl(3 \lambda_1^2 + {1 \over 12} \lambda_3^2 \Bigr) \; {\partial \lambda_3 \over \partial \lambda_1} \;+\; \Bigl(3 \lambda_2^2 + {1 \over 12} \lambda_3^2 \Bigr) \; {\partial \lambda_3 \over \partial \lambda_2} = {2 \over 3} \lambda_3^2 + \lambda_1 \lambda_3+ \lambda_2 \lambda_3 \;.$$ If $\lambda_1$ and $\lambda_2$ are independent, we can equate powers of these variables on each side of Eq. (80). If we allow the expansion of $\lambda_3$ to begin with a constant, we find a solution to Eq. (80) in which all powers of $\lambda_1$ and $\lambda_2$ appear. In this case a constant appears on the right-hand-sides of Eqs. (77) and (78), and there will be no Gaussian fixed points for $\lambda_1$ and $\lambda_2$. We are generally not interested in the possibility that a counterterm does not vanish when the canonical coupling vanishes, so we will simply discard this solution both here and below. We are interested in the conditions under which one variable ceases to be independent, and the appearance of such an arbitrary constant indicates that the variable remains independent even though its dependence on the cutoff is being reparameterized in terms of other variables. If we do not allow a constant in the solution, we find that $\lambda_3=\alpha \lambda_1+\beta \lambda_2+\order(\lambda^2)$. When we insert this in Eq. (80) and equate powers on each side, we obtain three coupled equations for $\alpha$ and $\beta$. These equations have no solution other than $\alpha=0$ and $\beta=0$, so we conclude that if $\lambda_1$ and $\lambda_2$ are independent functions of $t$, $\lambda_3$ will also be an independent function of $t$ unless the two fields decouple. Assume that there is only one independent variable, $\lt=\lambda_1$, so that $\lambda_2$ and $\lambda_3$ are functions of $\lt$. In this case we obtain two coupled equations, $$\Bigl(3 \lt^2+{1 \over 12} \lambda_3^2 \Bigr) \; {\partial \lambda_2 \over \partial \lt} = 3 \lambda_2^2+{1 \over 12} \lambda_3^2 \;,$$ $$\Bigl(3 \lt^2+{1 \over 12} \lambda_3^2 \Bigr) \; {\partial \lambda_3 \over \partial \lt} = {2 \over 3} \lambda_3^2 + \lt \lambda_3+ \lambda_2 \lambda_3 \;.$$ If we again exclude a constant term in the expansions of $\lambda_2$ and $\lambda_3$ we find that the only non-trivial solutions to leading order are $\lambda_2=\lt$, and either $\lambda_3 = 2 \lt$ or $\lambda_3=6 \lt$. If $\lambda_3=2\lt$, $$V(\phi)={\lt \over 4!}\;\bigl(\phi_1^2+\phi_2^2 \bigr)^2 \;,$$ and we find the $O(2)$ symmetric theory. If $\lambda_3=6\lt$, $$V(\phi)={\lt \over 2 \cdot 4!}\;\Bigl[\bigl(\phi_1+\phi_2\bigr)^4+ \bigl(\phi_1-\phi_2\bigr)^4\Bigr] \;,$$ and we find two decoupled scalar fields. Therefore, $\lambda_2$ and $\lambda_3$ do not run independently with the cutoff if there is a symmetry that relates their strength to $\lambda_1$. The condition that a limited number of variables run with the cutoff does not only reveal symmetries broken by the regulator, it may also be used to uncover symmetries that are broken by the vacuum. I will not go through the details, but it is straightforward to show that in a scalar theory with a $\phi^3$ coupling, this coupling can be fixed as a function of the $\phi^2$ and $\phi^4$ couplings only if the symmetry is spontaneously broken rather than explicitly broken. This example is of some interest in light-front field theory, because it is difficult to reconcile vacuum symmetry breaking with the requirements that we work with a trivial vacuum and drop zero-modes in practical non-perturbative Hamiltonian calculations. Of course, the only way that we can build vacuum symmetry breaking into the theory without including a nontrivial vacuum as part of the state vectors is to include symmetry breaking interactions in the Hamiltonian and work in the hidden symmetry phase. The problem then becomes one of finding all necessary operators without sacrificing predictive power. The renormalization group specifies what operators are relevant, marginal, and irrelevant; and coupling coherence provides one way to fix the strength of the symmetry-breaking interactions in terms of the symmetry-preserving interactions. This does not solve the problem of how to treat the vacuum in light-front QCD by any means, because we have only studied perturbation theory; but this result is encouraging and should motivate further investigation. For the QED and QCD calculations discussed below, I need to compute the hamiltonian to second order, while the canonical coupling runs at third order. To determine the generic solution to this problem, I present an oversimplified analysis in which there are three coupled renormalization group equations for the independent marginal coupling ($g$), in addition to dependent relevant ($\mu$) and irrelevant ($w$) couplings. $$\mu_{m_{l+1}}=4 \mu_{m_l} + c_\mu g_l^2 + \order(g_l^3) \;,$$ $$g_{m_{l+1}}=g_{m_l} + c_g g_l^3 + \order(g_l^4) \;,$$ $$w_{m_{l+1}}={1 \over 4} w_{m_l} + c_w g_l^2 + \order(g_l^3) \;.$$ I assume that $\mu_l = \zeta g_l^2 + \order(g_l^3)$ and $w_l=\eta g_l^2 + \order(g_l^3)$, which satisfy the conditions of coupling coherence. Substituting into the renormalization group equations yields (dropping all terms of $\order(g_l^3)$), $$\zeta g_{l+1}^2 = \zeta g_l^2 = 4 \zeta g_l^2 + c_\mu g_l^2 \;,$$ $$\eta g_{l+1}^2 = \eta g_l^2 = {1 \over 4} \eta g_l^2 + c_w g_l^2 \;.$$ The solutions are $\zeta=-{1 \over 3} c_\mu$ and $\eta = {4 \over 3} c_w$. These are [*exactly*]{} the coefficients in a Taylor series expansion for $\mu(g)$ and $w(g)$ that reproduce themselves. This observation suggests an alternative way to find the coupling coherent Hamiltonian without explicitly setting up the renormalization group equations. Although this method is less general, we only need to find what operators must be added to the Hamiltonian so that at $\order(g^2)$ it reproduces itself, with the only change being the change in the specific value of the cutoff. Coupling coherence allows us to substitute the running coupling in this solution, but it is not until third order that we would explicitly see the coupling run. This is how I will fix the QED and QCD hamiltonians to second order. QED and QCD Hamiltonians ------------------------ In order to derive the renormalized QED and QCD Hamiltonians I must first list the canonical Hamiltonians. I follow the conventions of Brodsky and Lepage. There is no need to be overly rigorous, because coupling coherence will fix any perturbative errors. I recommend the papers of Zhang and Harindranath for a more detailed discussion from a different point of view. The reader who is not yet concerned with details can skip this section. I will use $A^+=0$ gauge, and I drop zero modes. I use the Bjorken and Drell conventions for gamma matrices. The gamma matrices are $$\gamma^0=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)~,~~~~ \gamma^k=\left(\begin{array}{cc} 0 & \sigma_k \\ -\sigma_k & 0 \end{array}\right)\;,$$ where $\sigma_k$ are the Pauli matrices. This leads to $$\gamma^+=\left(\begin{array}{cc} 1 & \sigma_3 \\ -\sigma_3 & -1 \end{array}\right)~,~~~~ \gamma^-=\left(\begin{array}{cc} 1 & -\sigma_3 \\ \sigma_3 & -1 \end{array}\right)\;.$$ Useful identities for many calculations are $\gamma^+ \gamma^- \gamma^+=4 \gamma^+$, and $\gamma^- \gamma^+ \gamma^-=4 \gamma^-$. The operator that projects onto the dynamical fermion degree of freedom is $$\Lambda_+= {1 \over 2} \gamma^0 \gamma^+={1 \over 4} \gamma^- \gamma^+= {1 \over 2} \left(\begin{array}{cc} 1 & \sigma_3 \\ \sigma_3 & 1 \end{array}\right) \;,$$ and the complement projection operator is $$\Lambda_-= {1 \over 2} \gamma^0 \gamma^-={1 \over 4} \gamma^+ \gamma^-= {1 \over 2} \left(\begin{array}{cc} 1 & -\sigma_3 \\ -\sigma_3 & 1 \end{array}\right) \;.$$ The Dirac spinors $u(p,\sigma)$ and $v(p,\sigma)$ satisfy $$(\pslash-m) u(p,\sigma)=0 \;,\;\;\; (\pslash+m) v(p,\sigma)=0 \;,$$ and, $$\overline{u}(p,\sigma)u(p,\sigma')=-\overline{v}(p,\sigma) v(p,\sigma')=2 m \delta_{\sigma \sigma'} \;,$$ $$\overline{u}(p,\sigma) \gamma^\mu u(p,\sigma')= \overline{v}(p,\sigma) \gamma^\mu v(p,\sigma')= 2 p^\mu \delta_{\sigma \sigma'} \;,$$ $$\sum_{\sigma=\pm {1 \over 2}} u(p,\sigma) \overline{u}(p,\sigma) = \pslash+m \;,\;\;\; \sum_{\sigma=\pm {1 \over 2}} v(p,\sigma) \overline{v}(p,\sigma)= \pslash-m \;.$$ There are only two physical gluon (photon) polarization vectors, $\epsilon_{1\perp}$ and $\epsilon_{2\perp}$; but it is sometimes convenient (and dangerous once covariance and gauge invariance are violated) to use $\epsilon^\mu$, where $$\epsilon^+=0 \;,\;\;\;\epsilon^-={2 {\bf q}_\perp \cdot \epsilon_\perp \over q^+} \;.$$ It is often possible to avoid using an explicit representation for $\epsilon_\perp$, but completeness relations are required, $$\sum_{\lambda} \epsilon^\mu_\perp(\lambda) \epsilon^{*\nu}_\perp(\lambda) = -g_\perp^{\mu \nu} \;,$$ so that, $$\sum_{\lambda} \epsilon^\mu(\lambda) \epsilon^{*\nu}(\lambda) = -g_\perp^{\mu \nu} + {1 \over q^+} \bigl(\eta^\mu q_\perp^\nu + \eta^\nu q_\perp^\mu \bigr) + { {\bf q}_\perp^2 \over (q^+)^2} \eta^\mu \eta^\nu \;,$$ where $\eta^+=\eta^1=\eta^2=0$ and $\eta^-=2$. One often encounters diagrammatic rules in which the gauge propagator is written so that it looks covariant; but this is dangerous in loop calculations because such expressions require one to add and subtract terms that contain severe infrared divergences. The QCD Lagrangian density is $${\cal L}=-{1 \over 2} Tr F^{\mu \nu} F_{\mu \nu} + \overline{\psi} \Bigl(i \Dslash -m\Bigr)\psi \;,$$ where $F^{\mu \nu}=\partial^\mu A^\nu-\partial^\nu A^\mu+i g \bigl[A^\mu,A^\nu\bigr]$ and $i D^\mu=i \partial^\mu-g A^\mu$. The SU(3) gauge fields are $A^\mu=\sum_a A^\mu_a T^a$, where $T^a$ are one-half the Gell-Mann matrices, $\lambda^a$, and satisfy $Tr~ T^a T^b= 1/2~ \delta^{ab}$ and $\bigl[T^a,T^b\bigr]=i f^{abc} T^c$. The dynamical fermion degree of freedom is $\psi_+=\Lambda_+ \psi$, and this can be expanded in terms of plane wave creation and annihilation operators at $x^+=0$, $$\psi_+^r(x)=\sum_{\sigma=\pm 1/2} \int_{k^+>0} {dk^+ d^2k_\perp \over 16\pi^3 k^+} \Bigl[b^r(k,\sigma) u_+(k,\sigma) e^{-ik\cdot x} + d^{r\dagger}(k,\sigma) v_+(k,\sigma) e^{ik\cdot x}\Bigr] \;,$$ where these field operators satisfy $$\Bigl\{\psi_+^r(x),\psi_+^{s\dagger}(y)\Bigr\}_{x^+=y^+=0}=\Lambda_+ \delta_{rs} \delta^3(x-y) \;,$$ and the creation and annihilation operators satisfy $$\Bigl\{b^r(k,\sigma),b^{s\dagger}(p,\sigma')\Bigr\}= \Bigl\{d^r(k,\sigma),d^{s\dagger}(p,\sigma')\Bigr\} = 16\pi^3 k^+ \delta_{rs} \delta_{\sigma \sigma'} \delta^3(k-p) \;.$$ The indices $r$ and $s$ refer to SU(3) color. In general, when momenta are listed without specification of components, as in $\delta^3(p)$, I am referring to $p^+$ and ${\bf p}_\perp$. The transverse dynamical gluon field components can also be expanded in terms of plane wave creation and annihilation operators, $$A_\perp^{ic}(x)=\sum_\lambda\int_{k^+>0} {dk^+ d^2k_\perp \over 16\pi^3 k^+} \Bigl[a^c(k,\lambda) \epsilon_\perp^i(\lambda) e^{-ik\cdot x} +a^{c\dagger}(k,\lambda) \epsilon_\perp^{i*}(\lambda) e^{ik\cdot x} \Bigr] \;.$$ The superscript $i$ refers to the transverse dimensions $x$ and $y$, and the superscript $c$ is for SU(3) color. If required the physical polarization vector can be represented $${\bf \epsilon}_\perp(\uparrow)=-{1 \over \sqrt{2}} (1,i) \;,\;\;\; {\bf \epsilon}_\perp(\downarrow)={1 \over \sqrt{2}} (1,-i) \;.$$ The quantization conditions are $$\Bigl[A^{ic}_{\perp}(x),\partial^+ A^{jd}_{\perp}(y) \Bigr]_{x^+=y^+=0} =i \delta^{ij} \delta_{cd} \delta^3(x-y) \;,$$ $$\Bigl[a^c(k,\lambda),a^{d\dagger}(p,\lambda')\Bigr] = 16\pi^3 k^+ \delta_{\lambda \lambda'} \delta_{cd} \delta^3(k-p) \;.$$ The classical equations for $\psi_-=\Lambda_- \psi$ and $A^-$ do not involve time-derivatives, so these variables can be eliminated in favor of dynamical degrees of freedom. This formally yields $$\begin{aligned} \psi_-&=&{1 \over i\partial^+} \Bigl[i {\bf \alpha}_\perp \cdot {\bf D}_\perp+\beta m\Bigr] \psi_+ \nonumber \\ &=&\psit_- - {g \over i\partial^+} {\bf \alpha}_\perp \cdot {\bf A}_\perp \psi_+ \;,\end{aligned}$$ where the variable $\psit_-$ is defined on the second line to separate the interaction-dependent part of $\psi_-$; and $$\begin{aligned} A^{a-}&=&{2 \over i\partial^+} i {\bf \partial}_\perp \cdot {\bf A}^a_\perp + {2 i g f^{abc} \over (i\partial^+)^2}\Biggl\{\bigl(i\partial^+ A^{bi}_\perp\bigr)A^{ci}_\perp +2 \psi_+^\dagger T^a \psi_+\Biggr\} \nonumber \\ &=&\At^{a-}+{2 i g f^{abc} \over (i\partial^+)^2} \Biggl\{\bigl(i\partial^+ A^{bi}_\perp\bigr) A^{ci}_\perp +2 \psi_+^\dagger T^a \psi_+ \Biggr\} \;,\end{aligned}$$ where the variable $\At^-$ is defined on the second line to separate the interaction-dependent part of $A^-$. Given these replacements, we can follow a canonical procedure to determine the Hamiltonian. This path is full of difficulties that I ignore, because ultimately I will use coupling coherence to refine the definition of the Hamiltonian and determine the non-canonical interactions that are inevitably produced by the violation of explicit covariance and gauge invariance. For my purposes it is sufficient to write down a Hamiltonian that can serve as a starting point: $$H=H_0+V \;,$$ $$\begin{aligned} H_0&=&\int d^3x \Bigl\{ Tr\bigl(\partial^i_\perp A^j_\perp \partial^i_\perp A^j_\perp\bigr)+\psi_+^\dagger \bigl(i \alpha^i_\perp \partial^i_\perp+\beta m\bigr) \psi_+\Bigr\} \nonumber \\ &=&\sum_{colors} \int {dk^+ d^2k_\perp \over 16\pi^3 k^+} \Biggl\{ \sum_\lambda {{\bf k}_\perp^2 \over k^+} a^\dagger(k,\lambda) a(k,\lambda) \nonumber \\ &&~~~~~~~~~~~+ \sum_\sigma {{\bf k}_\perp^2+m^2 \over k^+} \Bigl(b^\dagger(k,\sigma) b(k,\sigma)+d^\dagger(k,\sigma) d(k,\sigma) \Bigr) \Biggr\} \;.\end{aligned}$$ In the last line the ‘self-induced inertias’ ([*i.e.*]{}, one-body operators produced by normal-ordering $V$) are not included. It is difficult to regulate the field contraction encountered when normal-ordering in a manner exactly consistent with the cutoff regulation of contractions encountered later. Coupling coherence avoids this issue and produces the correct one-body counterterms with no discussion of normal-ordering required. The interactions are complicated and are most easily written using the variables, $\psit=\psit_- + \psi_+$, and $\At$, where $\At^+=0$, $\At^-$ is defined above, and $\At^i_\perp=A^i_\perp$. Using these variables we have $$\begin{aligned} V&=\int d^3x\Biggl\{& g \overline{\psit} \gamma_\mu \At^\mu \psit + 2 g~ Tr\Bigl(i\partial^\mu \At^\nu \Bigl[ \At_\mu, \At_\nu \Bigr] \Bigr) \nonumber \\ &&-{g^2 \over 2} Tr\Bigl( \Bigl[\At^\mu,\At^\nu\Bigr] \Bigl[ \At_\mu, \At_\nu \Bigr] \Bigr) + g^2 \overline{\psit} \gamma_\mu \At^\mu {\gamma^+ \over 2i\partial^+} \gamma_\nu \At^\nu \psit \nonumber \\ &&+{g^2 \over 2} \overline{\psi} \gamma^+ T^a \psi {1 \over (i\partial^+)^2} \overline{\psi} \gamma^+ T^a \psi \nonumber \\ &&-g^2 \overline{\psi} \gamma^+ \Biggl({1 \over (i\partial^+)^2} \Bigl[i\partial^+ \At^\mu, \At_\mu \Bigr] \Biggr) \psi \nonumber \\ &&+g^2 ~Tr \Biggl(\Bigl[i\partial^+ \At^\mu, \At_\mu \Bigr] {1 \over (i\partial^+)^2} \Bigl[i\partial^+ \At^\nu, \At_\nu \Bigr] \Biggr) ~~~\Biggr\}\;.\end{aligned}$$ The commutators in this expression are SU(3) commutators only. The potential algebraic complexity of calculations becomes apparent when one systematically expands every term in $V$ and replaces: $$\psit^- \rightarrow {1 \over i\partial^+}\Bigl(i {\bf \alpha}_\perp \cdot {\bf \partial}_\perp+\beta m\Bigr) \psi_+ \;,$$ $$\At^- \rightarrow {2 \over i\partial^+} i{\bf \partial}_\perp \cdot {\bf A}_\perp \;;$$ and then expands $\psi_+$ and ${\bf A}_\perp$ in terms of creation and annihilation operators. It rapidly becomes evident that one should avoid such explicit expansions if possible. LIGHT-FRONT QED --------------- In this section I will follow the strategy outlined in the first section to compute the positronium spectrum. I will detail the calculation through the leading order Bohr results and indicate how higher order calculations proceed. The first step is to compute a renormalized cutoff Hamiltonian as a power series in the coupling $e$. Starting with the canonical Hamiltonian as a ‘seed,’ this is done with the similarity renormalization group and coupling coherence. The result is an apparently unique perturbative series, $$H^\Lambda_N=h_0 + e_\Lambda h_1 +e_\Lambda^2 h_2 + \cdot\cdot\cdot + e_\Lambda^N h_N \;.$$ Here $e_\Lambda$ is the running coupling constant, and all remaining dependence on $\Lambda$ in the operators $h_i$ must be explicit. In principle I must also treat $m_\Lambda$, the electron running mass, as an independent function of $\Lambda$; but this will not affect the results to the order I compute here. We must calculate the Hamiltonian to a fixed order, and systematically improve the calculation later by including higher order terms. Having obtained the Hamiltonian to some order in $e$, the next step is to split it into two parts, $$H^\Lambda=\H_0+\V \;.$$ As discussed before, $\h0$ must be accurately solved non-perturbatively, producing a zeroth order approximation for the eigenvalues and eigenstates. The greatest ambiguities in the calculation appear in the choice of $\h0$, which requires one of science’s most powerful computational tools, trial and error. In QED and QCD I [*assume*]{} that all interactions in $\H_0$ preserve particle number, with all interactions that involve particle creation and annihilation in $\V$. This assumption is consistent with the original hypothesis that a constituent picture will emerge, but it should emerge as a valid approximation. The final step before the loop is repeated, starting with a more accurate approximation for $H^\Lambda$, is to compute corrections from $\V$ in bound state perturbation theory. There is no reason to compute these corrections to arbitrarily high order, because the initial Hamiltonian contains errors that limit the accuracy we can obtain in bound state perturbation theory. In this section I: (i) compute $H^\Lambda$ to ${\cal O}(e^2)$, (ii) assume the cutoff is in the range $\alpha m^2 < \Lambda^2 < \alpha^2 m^2$ for non-perturbative analyses, (iii) include the most infrared singular two-body interactions in $\H_0$, and (iv) estimate the binding energy for positronium to ${\cal O}(\alpha^2 m)$. Since $\H_0$ is assumed to include interactions that preserve particle number, the zeroth order positronium ground state will be a pure electron-positron state. We only need one- and two-body interactions; [*i.e.*]{}, the electron self-energy and the electron-positron interaction. The canonical interactions can be found in Eq. (113), and the second-order change in the Hamiltonian is given in Eq. (57). The shift due to the bare electron mixing with electron-photon states to lowest order (see Figure 1) is $$\delta \Sigma_p=e^2 \int {dk_1^+ d^2k_{1\perp} \over 16 \pi^3} {\theta(k_1^+) \theta(k_2^+) \over k_1^+ k_2^+} {\overline{u} (p) D_{\mu\nu}(k_1) \gamma^\mu \bigl(\kslash_2+m\bigr) \gamma^\nu u(p) \over p^--k_1^--k_2^-} \;,$$ where, $$k_2^+=p^+-k_1^+ \;,\;\;\;{\bf k}_{2\perp}={\bf p}_\perp-{\bf k}_{1\perp} \;,$$ $$k_i^-={k_{i\perp}^2 + m_i^2 \over k_i^+} \;,$$ $$D_{\mu\nu}(k)=-g_{\perp \mu \nu} + {k_\perp^2 \over (k^+)^2} \eta_\mu \eta_\nu + {1 \over k^+} \Bigl(k_{\perp \mu} \eta_\nu + k_{\perp \nu} \eta_\mu\Bigr) \;,$$ $$\eta_\mu a^\mu = a^+ \;.$$ I have not yet displayed the cutoffs. To evaluate the integrals it is easiest to use Jacobi variables $x$ and ${\bf s}$ for the relative electron-photon motion, $$k_1^+=x p^+\;,\;\;\;{\bf k}_{1\perp}=x {\bf p}_\perp+{\bf s} \;,$$ which implies $$k_2^+=(1-x) p^+\;,\;\;\;{\bf k}_{2\perp}=(1-x) {\bf p}_\perp-{\bf s} \;.$$ The second-order change in the electron self-energy becomes $$\begin{aligned} \delta \Sigma_p &=& -{e_\Lambda^2 \over p^+} \int {dx d^2s \over 16\pi^3} \theta\Biggl(y \Lambda_0^2-{s^2 +x^2 m^2 \over x (1-x)}\Biggr) \theta\Biggl({s^2 +x^2 m^2 \over x (1-x)}-y \Lambda_1^2\Biggr) \nonumber \\ &&~~~\Biggl({1 \over s^2+x^2 m^2}\Biggr)~ \overline{u}(p,\sigma) \Biggl\{ 2 (1-x) \pslash -2 m + \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~{\gamma^+ \over p^+} \Biggl[ {2 s^2 \over x^2}+{2s^2 \over 1-x}+{x (2-x) m^2 \over 1-x} \Biggr] \Biggr\} u(p,\sigma) \;,\end{aligned}$$ where $y=p^+/P^+$. It is straightforward in this case to determine the self-energy required by coupling coherence. Since the electron-photon coupling does not run until third order, to second order the self-energy must exactly reproduce itself with $\Lambda_0 \rightarrow \Lambda_1$. For the self-energy to be finite we must assume that $\delta \Sigma$ reduces a positive self-energy, so that $$\begin{aligned} \Sigma^{\Lambda}_{coh}(p)&=& {e_\Lambda^2 \over p^+} \int {dx d^2s \over 16\pi^3} \theta\bigl(yx-\epsilon\bigr) \theta\Biggl(y \Lambda^2-{s^2 +x^2 m^2 \over x (1-x)}\Biggr) \Biggl({1 \over s^2+x^2 m^2}\Biggr) \nonumber \\&& \overline{u}(p,\sigma) \Biggl\{ 2 (1-x) \pslash -2 m +{\gamma^+ \over p^+} \Biggl[ {2 s^2 \over x^2}+{2s^2 \over 1-x}+ {x (2-x) m^2 \over 1-x} \Biggr] \Biggr\} u(p,\sigma) \nonumber \\ &=&{e_\Lambda^2 \over 8\pi^2 p^+} \Biggl\{2 y \Lambda^2 \ln\Biggl( { y^2 \Lambda^2 \over (y\Lambda^2 + m^2)\epsilon}\Biggr) -{3 \over 2} y \Lambda^2+{1 \over 2} {ym^2 \Lambda^2 \over y \Lambda^2+m^2} \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~+ 3 m^2 \ln\Biggl( {m^2 \over y \Lambda^2 + m^2} \Biggr) \Biggl\} + {\cal O}(\epsilon/y) \;.\end{aligned}$$ I have been forced to introduce [*a second cutoff*]{}, $$x p^+ > \epsilon P^+ \;,$$ because after the ${\bf s}$ integration is completed we are left with a logarithmically divergent $x$ integration. Other choices for this second infrared cutoff are possible and lead to similar results. This second cutoff must be taken to zero and no new counterterms can be added to the Hamiltonian, so all divergences must cancel before it is taken to zero. The electron and photon (quark and gluon) ‘mass’ operators, are a function of a longitudinal momentum scale introduced by the cutoff, and there is an exact scale invariance required by longitudinal boost invariance. Here I mean by ‘mass operator’ the one-body operator when the transverse momentum is zero, even though this does not agree with the free mass operator because it includes longitudinal momentum dependence. The cutoff violates boost invariance and the mass operator is required to restore this symmetry. We must interpret this new infrared divergence, because we have no choice about whether it is in the Hamiltonian if we use coupling coherence. We can only choose between putting the divergent operator in $\H_0$ or in $\V$. I make different choices in QED and QCD, and the arguments are based on physics. The [*divergent*]{} electron ‘mass’ is a complete lie. We encounter a term proportional to $e_\Lambda^2 \Lambda^2 \ln(1/\epsilon)/P^+$ when the scale is $\Lambda$; however, we can reduce this scale as far as we please in perturbation theory. Photons are massless, so the electron will continue to dress itself with small-x photons to arbitrarily small $\Lambda$. Since I believe that this divergent self-energy is exactly canceled by mixing with small-x photons, and that this mixing can be treated perturbatively in QED, I simply put the divergent electron self-energy in $\V$, which is treated perturbatively. There are two time-ordered diagrams involving photon exchange between an electron with initial momentum $p_1$ and final momentum $p_2$, and a positron with initial momentum $k_1$ and final momentum $k_2$. These are shown in Figure 4, along with the instantaneous exchange diagram. Using Eq. (57), we find the required matrix element of $\delta H$, $$\begin{aligned} \delta H &=& -{e^2 \over q^+} D_{\mu \nu}(q) \overline{u}(p_2,\sigma_2) \gamma^\mu u(p_1,\sigma_1) \overline{v}(k_1,\lambda_1) \gamma^\nu v(k_2,\lambda_2) \nonumber \\ &&~~\theta\bigl(|q^+|-\epsilon P^+\bigr) \theta\Biggl({\Lambda_0^2 \over P^+} - \mid p_1^- -p_2^- -q^- \mid\Biggr) \theta\Biggl({\Lambda_0^2 \over P^+} - \mid k_2^- -k_1^- -q^- \mid\Biggr) \nonumber \\ &&~~\Biggl[ {\theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda_1^2 / {\cal P}^+ \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \over p_1^- -p_2^- -q^-} \nonumber \\ &&~~~~ +{\theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda_1^2 / {\cal P}^+ \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \over k_2^- -k_1^- -q^-} \Biggr] \nonumber \\ &&~~~~~~~~~~~\theta\Biggl(\Lone-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \;, \end{aligned}$$ where $q^+=p_1^+-p_2^+\;,\;\;{\bf q}_\perp={\bf p}_{1\perp} - {\bf p}_{2\perp}$, and $q^-=q_\perp^2/q^+$. I have used the second cutoff on longitudinal momentum that I was forced to introduce when computing the change in the self-energy. We will see in the section on confinement that it is essential to include this cutoff everywhere consistently. In QED this point is not immediately important, because all infrared singular interactions, including the infrared divergent self-energy, are put in $\V$ and treated perturbatively. Divergences from higher orders in $\V$ cancel. -3mm To determine the interaction that must be added to the Hamiltonian to maintain coupling coherence, we must again find an interaction that when added to $\delta V$ reproduces itself with $\Lambda_0 \rightarrow \Lambda_1$ everywhere. The coupling coherent interaction generated by the first terms in $\delta H$ are not uniquely determined at this order. There is some ambiguity because we can obtain coupling coherence either by having $\delta H$ increase the strength of an operator by adding additional phase space strength, or we can have $\delta H$ reduce the strength of an operator by subtracting phase space strength. The ambiguity is resolved in higher orders, so I will simply state the result. If an instantaneous photon-exchange interaction is present in $H$, $\delta H$ cancels part of this marginal operator and increases the strength of a new photon-exchange interaction. This new interaction reproduces the effects of high energy photon exchange removed by the cutoff. The result is $$\begin{aligned} V_{coh}^{\Lambda} &=& - {e_{\Lambda}^2 \over q^+} D_{\mu \nu}(q) \overline{u}(p_2,\sigma_2) \gamma^\mu u(p_1,\sigma_1) \overline{v}(k_1,\lambda_1) \gamma^\nu v(k_2,\lambda_2) \nonumber \\ &&~~~~~~~\theta\bigl(|q^+|-\epsilon P^+\bigr)~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \nonumber \\ &&~~\Biggl[ {\theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \over p_1^- -p_2^- -q^-} \nonumber \\ &&~~~~ +{\theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \over k_2^- -k_1^- -q^-} \Biggr] \;. \nonumber \\\end{aligned}$$ This matrix element exactly reproduces photon exchange above the cutoff. The cutoff removes the direct coupling of electron-positron states to electron-positron-photon states whose energy differs by more than the cutoff, and coupling coherence dictates that the result of this mixing should be replaced by a direct interaction between the electron and positron. We could obtain this result by much simpler means at this order by simply demanding that the Hamiltonian produce the ‘correct’ scattering amplitude at $\order(e^2)$ with the cutoffs in place. Of course, this procedure requires us to provide the ‘correct’ amplitude, but this is easily done in perturbation theory. $V^\Lambda_{coh}$ is non-canonical, and we will see that it is responsible for producing the Coulomb interaction. We need some guidance to decide which irrelevant operators are most important. We find [*a posteriori*]{} that differences of external transverse momenta, and differences of external longitudinal momenta are both proportional to $\alpha$. This allows us to identify the dominant operators by expanding in powers of these [*implicit*]{} powers of $\alpha$. This indicates that it is the [*most infrared singular*]{} part of $V_{coh}$ that is important. As explained above, this operator receives substantial strength only from the exchange of photons with small longitudinal momentum; so we expect inverse $q^+$ dependence to indicate ‘strong’ interactions between low energy pairs. So the part of $V_{coh}$ that is included in $\H_0$ is $$\begin{aligned} \tilde{V}_{coh}^{\Lambda} &=& - e_{\Lambda}^2 {q_\perp^2 \over (q^+)^3}~ \overline{u}(p_2,\sigma_2) \gamma^+ u(p_1,\sigma_1) \overline{v}(k_1,\lambda_1) \gamma^+ v(k_2,\lambda_2) \nonumber \\ &&~~~~~~~\theta\bigl(|q^+|-\epsilon P^+\bigr)~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \nonumber \\ &&~~\Biggl[ {\theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \over p_1^- -p_2^- -q^-} \nonumber \\ &&~~~~ +{\theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \over k_2^- -k_1^- -q^-} \Biggr] \nonumber \\ &=& - 4 e_{\Lambda}^2 \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} {q_\perp^2 \over (q^+)^3} \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&~~~~~~~\theta\bigl(|q^+|-\epsilon P^+\bigr)~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \nonumber \\ &&~~\Biggl[ {\theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \over p_1^- -p_2^- -q^-} \nonumber \\ &&~~~~ +{\theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \over k_2^- -k_1^- -q^-} \Biggr] \;. \nonumber \\\end{aligned}$$ The Hamiltonian is almost complete to second order in the electron-positron sector, and only the instantaneous photon exchange interaction must be added. The matrix element of this interaction is $$\begin{aligned} V_{instant} &=& - e_{\Lambda}^2 \Biggl({1 \over q^+}\Biggr)^2 \overline{u}(p_2,\sigma_2) \gamma^+ u(p_1,\sigma_1) \overline{v}(k_1,\lambda_1) \gamma^+ v(k_2,\lambda_2) \nonumber \\ &&~~~~~~~~~\times \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \nonumber \\ &=& - 4 e_{\Lambda}^2 \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} \Biggl({1 \over q^+}\Biggr)^2 \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&~~~~~~~~~~~~\times \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \;.\end{aligned}$$ The only cutoff that appears is the cutoff directly run by the similarity transformation that prevents the initial and final states from differing in energy by more than $\Lambda^2/P^+$. This brings us to a final subtle point. Since there are no cutoffs in $V_{instant}$ that directly limit the momentum exchange, the matrix element diverges as $q^+ \rightarrow 0$. Consider $\tilde{V}_{coh}$ in this same limit, $$\begin{aligned} \tilde{V}_{coh}^{\Lambda} &\rightarrow& 4 e_{\Lambda}^2 \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} \Biggl({1 \over q^+}\Biggr)^2 \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&~~~~ \times \theta\Biggl(\mid q^- \mid - {\Lambda^2 \over P^+} \Biggr) ~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \;.\end{aligned}$$ This means that as $q^+ \rightarrow 0$, $V_{coh}$ partially screens $V_{instant}$, leaving the original operator multiplied by $\theta(\Lambda^2/P^+-|q^-|)$. However, even after this partial screening, the matrix elements of the remaining part of $V_{instant}$ between bound states diverge and we must introduce the same infrared cutoff used for the self-energy to regulate these divergences. This is explicitly shown in the section on confinement. However, all divergences from $V_{instant}$ are exactly canceled by the exchange of massless photons, which persists to arbitrarily small cutoff. This cancellation is exactly analogous to the cancellation of the infrared divergence of the self-energy, and will be treated in the same way. The portion of $V_{instant}$ that is not canceled by $\tilde{V}_{coh}$ will be included in $\V$, the perturbative part of the Hamiltonian. We will not encounter this interaction until we also include photon exchange below the cutoff perturbatively, so all infrared divergences should cancel in this bound state perturbation theory. I repeat that this is not guaranteed for arbitrary choices of $\H_0$, and we are not free to simply cancel these divergent interactions with counterterms because coupling coherence completely determines the Hamiltonian. We now have the complete interaction that I include in $\H_0$. Letting $\H_0=h_0+\v_0$, where $h_0$ is the free hamiltonian, I add parts of $V_{instant}$ and $V_{coh}$ to obtain $$\begin{aligned} \v_0 &=& - 4 e_{\Lambda}^2 \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \times \nonumber \\ && \Biggl\{ \Biggl[ {q_\perp^2 \over (q^+)^3} { 1 \over p_1^- -p_2^- -q^-} + {1 \over (q^+)^2} \Biggr] \theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda^2 / {\cal P}^+ \bigr) \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \nonumber \\ &&+ \Biggl[ {q_\perp^2 \over (q^+)^3} {1 \over k_2^- -k_1^- -q^-} + {1 \over (q^+)^2} \Biggr] \theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda^2 / {\cal P}^+ \bigr) \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \times \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \Biggr\} \;.\end{aligned}$$ In order to present an analytic analysis I will make assumptions that can be justified $a~ posteriori$. First I will assume that the electron and positron momenta can be arbitrarily large, but that in low-lying states their relative momenta satisfy $$|{\bf p}_\perp-{\bf k}_\perp| \ll m \;,$$ $$|p^+-k^+| \ll p^++k^+ \;.$$ It is essential that the condition for longitudinal momenta not involve the electron mass, because masses have the scaling dimensions of transverse momenta and not longitudinal momenta. As above, I use $p$ for the electron momenta and $k$ for the positron momenta. To be even more specific, I will assume that $$|{\bf p}_\perp-{\bf k}_\perp| \sim \alpha m \;,$$ $$|p^+-k^+| \sim \alpha (p^++k^+) \;.$$ This allows us to use power counting to evaluate the perturbative strength of operators for small coupling, which may prove essential in the analysis of QCD. Note that these conditions allow us to infer $$|{\bf p}_{1\perp}-{\bf p}_{2\perp}| \sim \alpha m \;,$$ $$|p_1^+-p_2^+| \sim \alpha (p^++k^+) \;.$$ Given these order of magnitude estimates for momenta, we can drastically simplify the free energies in the kinetic energy operator and the energy denominators in $\v_0$. We can use transverse boost invariance to choose a frame in which $$p_i^+=y_i P^+ \;,\;\;p_{i\perp}=\kappa_i \;,\;\;\;\;k_i^+=(1-y_i)P^+ \;,\;\;k_{i\perp}=-\kappa_i \;,$$ so that $$\begin{aligned} P^+(p_1^--p_2^--q^-) &=& {\kappa_1^2+m^2 \over y_1} - {\kappa_2^2+m^2 \over y_2} - {(\kappa_1-\kappa_2)^2 \over y_1-y_2} \nonumber \\&=&-4 m^2 (y_1-y_2) - {(\kappa_1-\kappa_2)^2 \over y_1-y_2} + \order(\alpha^2 m^2) \;.\end{aligned}$$ To leading order all energy denominators are the same. Each energy denominator is $\order(\alpha m^2)$, which is large in comparison to the binding energy we will find. This is important, because the bulk of the photon exchange that is important for the low energy bound state involves intermediate states that have larger energy than the differences in constituent energies in the region of phase space where the wave function receives most of its strength. This allows us to use a perturbative renormalization group to compute the dominant effective interactions. There are similar simplifications for all energy denominators. After making these approximations we find that the matrix element of $\v_0$ is $$\begin{aligned} \v_0&=&4 e_\Lambda^2 \sqrt{y_1 y_2 (1-y_1) (1-y_2)} ~\theta\Bigl( 4 m^2 |y_1-y_2| + {(\kappa_1-\kappa_2)^2 \over |y_1-y_2|} -\Lambda^2\Bigr) \nonumber \\ &&~~~~~\theta\Bigl(\Lambda^2-4\mid \kappa_1^2+4 m^2(1-2y_1)^2- \kappa_2^2-4m^2(1-2y_2)^2 \mid\Bigr) \nonumber \\ &&~~~~~~~~~~~~~~~~~~~ {(\kappa_1-\kappa_2)^2 \over (y_1-y_2)^2} \Biggl[{1 \over 4 m^2 (y_1-y_2)^2+(\kappa_1 - \kappa_2)^2}-{1 \over (\kappa_1-\kappa_2)^2}\Biggr] \nonumber \\ &=&-16 e_\Lambda^2 m^2 \sqrt{y_1 y_2 (1-y_1) (1-y_2)}~ \theta\Bigl( 4 m^2 |y_1-y_2| + {(\kappa_1-\kappa_2)^2 \over |y_1-y_2|}-\Lambda^2\Bigr) \nonumber \\ &&~~~~\theta\Bigl(\Lambda^2-4\mid \kappa_1^2+4 m^2(1-2y_1)^2- \kappa_2^2-4m^2(1-2y_2)^2 \mid\Bigr) \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~\Biggl[ {1 \over 4 m^2 (y_1-y_2)^2+(\kappa_1-\kappa_2)^2} \Biggr] \;.\end{aligned}$$ In principle the electron-positron annihilation graphs should also be included at this order, but the resultant effective interactions do not diverge as $q^+ \rightarrow 0$, so I include such effects perturbatively in $\V$. At this point we can complete the zeroth order analysis of positronium using the state, $$\begin{aligned} |\Psi(P)\rangle &=& \sum_{\sigma \lambda} \int {dp^+ d^2p_\perp \over 16\pi^3 p^+} {dk^+ d^2k_\perp \over 16\pi^3 k^+} \sqrt{p^+ k^+} 16\pi^3 \delta^3(P-p-k) \nonumber \\ &&~~~~~~~~~~~~~~\phi(p,\sigma;k,\lambda) b^\dagger(p,\sigma) d^\dagger(k,\lambda) |0\rangle \;,\end{aligned}$$ where $\phi(p,\sigma;k,\lambda)$ is the wave function for the relative motion of the electron and positron, with the center-of-mass momentum being $P$. We need to choose the longitudinal momentum appearing in the cutoff, and I will use the natural scale $P^+$. The matrix element of $\H_0$ is $$\begin{aligned} &&\langle \Psi(P)|\H_0|\Psi(P')\rangle = 16\pi^3 \delta^3(P-P') \times \nonumber \\ &&~~~\Biggl\{ \int {dy d^2\kappa \over 16\pi^3} \Bigl[4m^2+4\kappa^2+4m^2(1-2y)^2\Bigr] |\phi(\kappa,y)|^2 \nonumber \\ &&~~~~~-16 e^2 m^2 \int {dy_1 d^2\kappa_1 \over 16\pi^3} {dy_2 d^2\kappa_2 \over 16\pi^3} \theta\Bigl(4 m^2|y_1-y_2| + {(\kappa_1 - \kappa_2)^2 \over |y_1-y_2|}-\Lambda^2\Bigr) \nonumber \\ &&~~~~~~~~~~~\theta\Bigl(\Lambda^2-4\mid \kappa_1^2+4 m^2(1-2y_1)^2- \kappa_2^2-4m^2(1-2y_2)^2 \mid\Bigr) \nonumber \\ &&~~~~~~~~~~~~~~~\Biggl[{1 \over 4m^2|y_1-y_2|^2+(\kappa_1-\kappa_2)^2} \Biggr] \phi^*(\kappa_2,y_2) \phi(\kappa_1,y_1) \Biggr\} \;.\end{aligned}$$ I have chosen a frame in which $P_\perp=0$ and used the Jacobi coordinates defined above, and indicated only the electron momentum in the wave function since momentum conservation fixes the positron momentum. I have also dropped the spin indices because the interaction in $\H_0$ is independent of spin. If we vary this expectation value subject to the constraint that the wave function is normalized we obtain the equation of motion, $$\begin{aligned} M^2 \phi(\kappa_1,y_1) &=& (4 m^2 - 4 m E + E^2) \phi(\kappa_1,y_1) \nonumber \\&=& \Bigl[4m^2+4\kappa_1^2+4m^2(1-2y_1)^2\Bigr] \phi(\kappa_1,y_1) \nonumber \\ &&-16 e^2 m^2 \int {dy_2 d^2\kappa_2 \over 16\pi^3} \theta\Bigl(4 m^2|y_1-y_2| + {(\kappa_1 - \kappa_2)^2 \over |y_1-y_2|} -\Lambda^2\Bigr) \nonumber \\ &&~~~~~~\theta\Bigl(\Lambda^2-4\mid \kappa_1^2+4 m^2(1-2y_1)^2- \kappa_2^2-4m^2(1-2y_2)^2 \mid\Bigr) \nonumber \\ &&~~~~~~~~~~\Biggl[{1 \over 4m^2|y_1-y_2|^2+(\kappa_1-\kappa_2)^2} \Biggr] \phi(\kappa_2,y_2) \;.\end{aligned}$$ $E$ is the binding energy, and we can drop the $E^2$ term since it will be $\order(\alpha^4)$. I do not think that it is possible to solve this equation analytically with the cutoffs in place, and with the light-front kinematic constraints $-1 \le 1-2y_i \le 1$. In order to determine the binding energy to leading order, we need to evaluate the regions of phase space removed by the cutoffs. If we want to find a cutoff for which the ground state is dominated by the electron-positron component of the wave function, we need the first cutoff to remove the important part of the electron-positron-photon phase space. Using the ‘guess’ that $|\kappa|=\order(\alpha m)$ and $1-2y=\order(\alpha)$, this requires $$\Lambda^2<\alpha m^2 \;.$$ On the other hand, we cannot allow the cutoff to remove the region of the electron-positron phase space from which the wave function receives most of its strength. This requires $$\Lambda^2>\alpha^2 m^2 \;.$$ While it is not necessary, the most elegant way to proceed is to introduce ‘new’ variables, $$\kappa_i = k_{\perp i} \;,$$ $$y_i={1 \over 2}+{k_z \over 2 \sqrt{{\bf k}_\perp^2 +k_z^2+m^2}} \;.$$ This change of variables can be ‘discovered’ in a number of ways, but they basically take us back to equal time coordinates, in which both boost and rotational symmetries are kinematic after a nonrelativistic reduction. For cutoffs that satisfy $\alpha m^2 > \Lambda^2 > \alpha^2 m^2$, Eq. (145) simplifies tremendously when all terms of higher order than $\alpha^2$ are dropped. Using the scaling behavior of the momenta, and the fact that we will find $E^2$ is $\order(\alpha^4)$, Eq. (145) reduces to: $$-E \phi({\bf k}_1) = { {\bf k}_1^2 \over m} \phi({\bf k}_1) -\alpha \int {d^3 k_2 \over (2 \pi)^3} {1 \over \bigl({\bf k}_1^2 - {\bf k}_2^2\bigr)} \phi({\bf k}_2) \;.$$ The step function cutoffs drop out to leading order, leaving us with the familiar nonrelativistic Schr[ö]{}dinger equation for positronium in momentum space. The solution is $$\phi({\bf k}) = {{\cal N} \over \bigl({\bf k}^2+m E\bigr)^2} \;,$$ $$E = {1 \over 4} \alpha^2 m \;.$$ ${\cal N}$ is a normalization constant. This is the Bohr energy for the ground state of positronium, and it is obvious that the entire nonrelativistic spectrum is reproduced to leading order. Beyond this leading order result the calculations become much more interesting, and in any Hamiltonian formulation they rapidly become complicated. There is no analytic expansion of the binding energy in powers of $\alpha$, since negative mass-squared states appear to signal vacuum instability when $\alpha$ is negative ; but our simple strategy for performing field theory calculations can be improved by taking advantage of the weak coupling expansion. We can expand the binding energy in powers of $\alpha$, and as is well known we find that powers of ${\rm log}(\alpha)$ appear in the expansion at $\order(\alpha^5)$. We have taken the first step to generate this expansion by expanding the effective Hamiltonian in the explicit powers of $\alpha$ which appear in the renormalization group analysis. The next step is to take advantage of the fact that all bound state momenta are proportional to $\alpha$, which allows us to expand each of the operators in the Hamiltonian in powers of momenta. The renormalization group analysis justifies an expansion in powers of transverse momenta, and the nonrelativistic reduction leads to an expansion in powers of longitudinal momentum differences. The final step is to regroup terms appearing in bound state perturbation theory. For example, when we compute the first order correction in bound state perturbation theory, we find all powers of $\alpha$, and these must be grouped order-by-order with terms that appear at higher orders of bound state perturbation theory. The leading correction to the binding energy is $\order(\alpha^4)$, and producing these corrections is a much more serious test of the renormalization procedure than the calculation shown above. To what order in the coupling must the Hamiltonian be computed to correctly reproduce all masses to $\order(\alpha^4)$? The leading error can be found in the electron mass itself. With the Hamiltonian given above, two-loop effects would show errors in the electron mass that are $\order(\alpha^2 \Lambda^2)$. This would appear to present a problem for the calculation of the binding energy to $\order(\alpha^2)$, but remembering that the cutoff must be lowered so that $\alpha m^2 > \Lambda^2 > \alpha^2 m^2$, we see that the error in the electron mass is actually of $\order(\alpha^{3+\delta} m^2)$. This means that to compute masses correctly to $\order(\alpha^4)$ we would have to compute the Hamiltonian to $\order(\alpha^4)$, which requires a fourth-order similarity calculation for QED. Such a calculation has not yet been completed. However, if we compute the splitting between bound state levels instead, errors in the electron mass cancel and we find that the Hamiltonian computed to $\order(\alpha^2)$ is sufficient. In Ref. [@BJ97a] we have shown that the fine structure of positronium is correctly reproduced when the first- and second-order corrections from bound state perturbation theory are added. This is a formidable calculation, because the exact Coulomb bound and scattering states appear in second-order bound state perturbation theory[^5] A complete calculation of the Lamb shift in hydrogen would also require a fourth-order similarity calculation of the Hamiltonian; however, the dominant contribution to the Lamb shift that was first computed by Bethe can be computed using a Hamiltonian determined to $\order({\alpha})$. In this calculation a Bloch transformation was used rather than a similarity transformation because the Bloch transformation is simpler and small energy denominator problems can be avoided in analytic QED calculations. The primary obstacle to using our light-front strategy for precision QED calculations is algebraic complexity. We have successfully used QED as a testing ground for this strategy, but these calculations can be done much more conveniently using other methods. The theory for which we believe our methods are best suited is QCD. LIGHT-FRONT QCD --------------- This section relies heavily on the discussion of positronium, because we only require the QCD Hamiltonian determined to $\order(\alpha)$ to discuss a simple confinement mechanism which appears naturally in light-front QCD and to complete reasonable zeroth order calculations for heavy quark bound states. To this order the QCD Hamiltonian in the quark-antiquark sector is almost identical to the QED Hamiltonian in the electron-positron sector. Of course the QCD Hamiltonian differs significantly from the QED Hamiltonian in other sectors, and this is essential for justifying my choice of $\H_0$ for non-perturbative calculations. The basic strategy for doing a sequence of (hopefully) increasingly accurate QCD bound state calculations is almost identical to the strategy for doing QED calculations. I use coupling coherence to find an expansion for $H^\Lambda$ in powers of the QCD coupling constant to a finite order. I then divide the Hamiltonian into a non-perturbative part, $\h0$, and a perturbative part, $\V$. The division is based on the physical argument that adding a parton in an intermediate state should require more energy than indicated by the free Hamiltonian, and that as a result these states will ‘freeze out’ as the cutoff approaches $\Lambda_{QCD}$. When this happens the evolution of the Hamiltonian as the cutoff is lowered further changes qualitatively, and operators that were consistently canceled over an infinite number of scales also freeze, so that their effects in the few parton sectors can be studied directly. A one-body operator and a two-body operator arise in this fashion, and serve to confine both quarks and gluons. The simple confinement mechanism I outline is certainly not the final story, but it may be the seed for the full confinement mechanism. One of the most serious problems we face when looking for non-perturbative effects such as confinement is that the search itself depends on the effect. A candidate mechanism must be found and then shown to self-consistently produce itself as the cutoff is lowered towards $\Lambda_{QCD}$. Once we find a candidate confinement mechanism, it is possible to study heavy quark bound states with little modification of the QED strategy. Of course the results in QCD will differ from those in QED because of the new choice of $\H_0$, and in higher orders because of the gluon interactions. When we move on to light quark bound states, it becomes essential to introduce a mechanism for chiral symmetry breaking. I will discuss this briefly at the end of this section. When we compute the QCD Hamiltonian to $\order(\alpha)$, several significant new features appear. First are the familiar gluon interactions. In addition to the many gluon interactions found in the canonical Hamiltonian, there are modifications to the instantaneous gluon exchange interactions, just as there were modifications to the electron-positron interaction. For example, a Coulomb interaction will automatically arise at short distances. In addition the gluon self-energy differs drastically from the photon self-energy. The photon develops a self-energy because it mixes with electron-positron pairs, and this self energy is $\order(\alpha \Lambda^2/P^+)$. When the cutoff is lowered below $4 m^2$, this mass term vanishes because it is no longer possible to produce electron-positron pairs. For all cutoffs the small bare photon self-energy is exactly canceled by mixing with pairs below the cutoff. I will not go through the calculation, but because the gluon also mixes with gluon pairs in QCD, the gluon self-energy acquires an infrared divergence, just as the electron did in QED. In QCD both the quark and gluon self-energies are proportional to $\alpha \Lambda^2 \ln(1/\epsilon)/P^+$, where $\epsilon$ is the secondary cutoff on parton longitudinal momenta introduced in the last section. This means that even when the primary cutoff $\Lambda^2$ is finite, the energy of a single quark or a single gluon is infinite, because we are supposed to let $\epsilon \rightarrow 0$. One can easily argue that this result is meaningless, because the relevant matrix elements of the Hamiltonian are not even gauge invariant; however, since we must live with a variational principle when doing Hamiltonian calculations, this result may be useful. In QED I argued that the bare electron self-energy was a complete lie, because the bare electron mixes with photons carrying arbitrarily small longitudinal momenta to cancel this bare self-energy and produce a finite mass physical electron. However, in QCD there is no reason to believe that this perturbative mixing continues to arbitrarily small cutoffs. There are [*no*]{} massless gluons in the world. In this case, the free QCD Hamiltonian is a complete lie and [*cannot be trusted*]{} at low energies. On the other hand, coupling coherence gives us no choice about the quark and gluon self-energies as computed in perturbation theory. These self-energies appear because of the behavior of the theory at extremely high energies. The question is not whether large self-energies appear in the Hamiltonian. The question is whether these self-energies are canceled by mixing with low energy multi-gluon states. I argue that this cancellation does not occur, and that the infrared divergent quark and gluon self-energies [*should be included*]{} in $\H_0$. The transverse scale for these energies is the running scale $\Lambda$, and over many orders of magnitude we should see the self-energies canceled by mixing. However, as the cutoff approaches $\Lambda_{QCD}$, I speculate that these cancellations cease to occur because perturbation theory breaks down and a mass gap between states with and without extra gluons appears. But if the quark and gluon self-energies diverge, and the divergences cannot be canceled by mixing between sectors with an increasingly large number of partons, how is it possible to obtain finite mass hadrons? The parton-parton interaction also diverges, and the infrared divergence in the two-body interaction [*exactly cancels*]{} the infrared divergence in the one-body operator for color singlet states. Of course, the cancellation of infrared divergences is not enough to obtain confinement. The cancellation is exact regardless of the relative motion of the partons in a color singlet state, and confinement requires a residual interaction. I will show that the $\order(\alpha)$ QCD Hamiltonian produces a logarithmic potential in both longitudinal and transverse directions. I will not discuss whether a logarithmic confining potential is ‘correct,’ but to the best of my knowledge there is no rigorous demonstration that the confining interaction is linear, and a logarithmic potential may certainly be of interest phenomenologically for heavy quark bound states. I would certainly be delighted if a better light-front calculation produces a linear potential, but this may not be necessary even for light hadron calculations. The calculation of how the quark self-energy changes when a similarity transformation lowers the cutoff on energy transfer is almost identical to the electron self-energy calculation. Following the steps in the section on positronium, we find the one-body operator required by coupling coherence, $$\begin{aligned} \Sigma^{\Lambda}_{coh}(p)&=& {g^2 C_F \over 8\pi^2 p^+} \Biggl\{2 y \Lambda^2 \ln\Biggl({ y^2 \Lambda^2 \over (y \Lambda^2+m^2) \epsilon } \Biggr) -{3 \over 2} y \Lambda^2+{1 \over 2} {y m^2 \Lambda^2 \over y \Lambda^2+m^2} \nonumber \\ &&~~~~~~~~~~~~~~~+ 3 m^2 \ln\Biggl( {m^2 \over y \Lambda^2 + m^2} \Biggr) \Biggl\} + {\cal O}(\epsilon/y) \;,\end{aligned}$$ where $C_F=(N^2-1)/(2N)$ for a SU(N) gauge theory. The calculation of the quark-antiquark interaction required by coupling coherence is also nearly identical to the QED calculation. Keeping only the infrared singular parts of the interaction, as was done in QED, $$\begin{aligned} \tilde{V}_{coh}^{\Lambda} &=& - 4 g_{\Lambda}^2 C_F \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} {q_\perp^2 \over (q^+)^3} \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&~~~~~~~\theta\bigl(|q^+|-\epsilon P^+\bigr)~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \nonumber \\ &&~~\Biggl[ {\theta\bigl(|p_1^- -p_2^- -q^-| -\Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) \over p_1^- -p_2^- -q^-} \nonumber \\ &&~~~~ +{\theta\bigl(|k_2^- -k_1^- -q^-| - \Lambda^2 / {\cal P}^+ \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \over k_2^- -k_1^- -q^-} \Biggr] \;. \nonumber\\\end{aligned}$$ The instantaneous gluon exchange interaction is $$\begin{aligned} V^\Lambda_{instant} &=& - 4 g_{\Lambda}^2 C_F \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} \Biggl({1 \over q^+}\Biggr)^2 \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&~~\times \theta\bigl(|q^+|-\epsilon P^+\bigr)~ \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \;.\end{aligned}$$ Just as in QED the coupling coherent interaction induced by gluon exchange above the cutoff partially cancels instantaneous gluon exchange. For the discussion of confinement the part of $V_{coh}$ that remains is not important, because it produces the short range part of the Coulomb interaction. However, the part of the instantaneous interaction that is not canceled is $$\begin{aligned} \tilde{V}^\Lambda_{instant} &=& - 4 g_{\Lambda}^2 C_F \sqrt{p_1^+ p_2^+ k_1^+ k_2^+} \Biggl({1 \over q^+}\Biggr)^2 \delta_{\sigma_1 \sigma_2} \delta_{\lambda_1 \lambda_2} \nonumber \\ &&\times \theta\Biggl(\Lam-\mid p_1^-+k_1^--p_2^--k_2^- \mid \Biggr) \theta\bigl(|p_1^+-p_2^+|-\epsilon P^+\bigr) \nonumber \\ &&\times \Biggl[ \theta\bigl(\Lam -|p_1^- -p_2^- -q^-| \bigr) \;\; \theta\bigl(|p_1^- -p_2^- -q^-|- |k_2^- -k_1^- -q^-| \bigr) + \nonumber \\ &&\theta\bigl(\Lam-|k_2^- -k_1^- -q^-| \bigr) \;\; \theta\bigl( |k_2^- -k_1^- -q^-| - |p_1^- -p_2^- -q^-| \bigr) \Biggr] \;.\end{aligned}$$ Note that this interaction contains a cutoff that projects onto exchange energies below the cutoff, because the interaction has been screened by gluon exchange above the cutoffs. This interaction can become important at long distances, if parton exchange below the cutoff is dynamically suppressed. In QED I argued that this singular long range interaction is exactly canceled by photon exchange below the cutoff, because such exchange is not suppressed no matter how low the cutoff becomes. Photons are massless and experience no significant interactions, so they are exchanged to arbitrarily low energies as effectively free photons. This cannot be the case for gluons. For the discussion of confinement, I will place only the most singular parts of the quark self-energy and the quark-antiquark interaction in $\h0$. To see that all infrared divergences cancel and that the residual long range interaction is logarithmic, we can study the matrix element of these operators for a quark-antiquark state, $$\begin{aligned} |\Psi(P)\rangle &=& \sum_{\sigma \lambda} \sum_{rs} \int {dp^+ d^2p_\perp \over 16\pi^3 p^+} {dk^+ d^2k_\perp \over 16\pi^3 k^+} \sqrt{p^+ k^+} 16\pi^3 \delta^3(P-p-k) \nonumber \\ &&~~~~~~~~~~~~~~\phi(p,\sigma,r;k,\lambda,s) b^{r\dagger}(p,\sigma) d^{s\dagger}(k,\lambda) |0\rangle \;,\end{aligned}$$ where $r$ and $s$ are color indices and I will choose $\phi$ to be a color singlet and drop color indices. The cancellations we find do not occur for the color octet configuration. The matrix element is, $$\begin{aligned} \langle \Psi(P)|\H_0|\Psi(P')\rangle &=& 16\pi^3 P^+ \delta^3(P-P') \times \nonumber \\ &\Biggl\{& \int {dy d^2\kappa \over 16\pi^3} \Biggl[{g_\Lambda^2 C_F \Lambda^2 \over 2\pi^2 P^+} \; \ln\bigl({1 \over \epsilon}\bigr) \Biggr] |\phi(\kappa,y)|^2 \nonumber \\ &-&{4 g_\Lambda^2 C_F \over P^+} \int {dy_1 d^2\kappa_1 \over 16\pi^3} {dy_2 d^2\kappa_2 \over 16\pi^3} \theta\Biggl(\Lambda^2-\mid {\kappa_1^2+m^2 \over y_1(1-y_1)}-{\kappa_2^2+m^2 \over y_2(1-y_2)} \mid\Biggr) \nonumber \\ &&~~\times \Biggl[ \theta\Biggl(\Lambda^2- \mid {\kappa_1^2+m^2 \over y_1} - {\kappa_2^2 +m^2 \over y_2}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|}\mid \Biggr) \nonumber \\ &&~~~~~~ \times \theta\Biggl(\mid {\kappa_1^2+m^2 \over y_1} - {\kappa_2^2 +m^2 \over y_2}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|}\mid - \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~ \mid {\kappa_2^2+m^2 \over 1-y_2} - {\kappa_1^2 +m^2 \over 1-y_1}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|} \mid\Biggr) \nonumber \\ && ~~~~~~ + \theta\Biggl(\Lambda^2-\mid {\kappa_2^2+m^2 \over 1-y_2} - {\kappa_1^2 +m^2 \over 1-y_1}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|} \mid\Biggr) \nonumber \\ && ~~~~~~ \times \theta\Biggl(\mid {\kappa_2^2+m^2 \over 1-y_2} - {\kappa_1^2 +m^2 \over 1-y_1}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|}\mid - \nonumber \\ &&~~~~~~~~~~~~~~~~~~~~~~~~~~ \mid {\kappa_1^2+m^2 \over y_1} - {\kappa_2^2 +m^2 \over y_2}-{(\kappa_1-\kappa_2)^2 \over |y_1-y_2|}\mid \Biggr) \Biggr] \nonumber \\ &&~~ \times \theta\Bigl(\mid y_1-y_2 \mid -\epsilon\Bigr) \Biggl({1 \over y_1-y_2 }\Biggr)^2 \phi^*(\kappa_2,y_2) \phi(\kappa_1,y_1) \Biggr\}.\end{aligned}$$ Here I have chosen a frame in which the center-of-mass transverse momentum is zero, assumed that the longitudinal momentum scale introduced by the cutoffs is that of the bound state, and used Jacobi coordinates, $$p_i^+=y_i P^+\;,\;\;p_{i\perp}=\kappa_i\;\;;\;\;\; k_i^+=(1-y_i)P^+\;, \;\; k_{i\perp}=-\kappa_i \;.$$ The first thing I want to do is show that the last term is divergent and the divergence exactly cancels the first term. My demonstration is not elegant, but it is straightforward. The divergence results from the region $y_1 \sim y_2$. In this region the second and third cutoffs restrict $(\kappa_1-\kappa_2)^2$ to be small compared to $\Lambda^2$, so we should change variables, $$Q={\kappa_1+\kappa_2 \over 2} \;,\;\;Y={y_1+y_2 \over 2} \;\;;\;\;\; q=\kappa_1-\kappa_2 \;,\;\;y=y_1-y_2 \;.$$ Using these variables we can approximate the above interaction near $q=0$ and $y=0$. The double integral becomes $$\begin{aligned} {-4 g_\Lambda^2 C_F \over P^+} \int {dY d^2Q \over 16\pi^3} {dy d^2q \over 16\pi^3} \theta(1-Y) \theta(Y) |\phi(Q,Y)|^2 \nonumber \\ ~~~~~\theta\Biggl(\Lambda^2-{q^2 \over |y|}\Biggr) \theta\bigl( |y|-\epsilon\bigr) \theta\bigl(\eta-|y|\bigr) ~ \Biggl({1 \over y}\Biggr)^2 \;,\end{aligned}$$ where $\eta$ is an arbitrary constant that restricts $|y|$ from becoming large. Completing the $q$ and $y$ integration we get $$-{g_\Lambda^2 C_F \Lambda^2 \over 2\pi^2 P^+} \ln\Biggl({1 \over \epsilon}\Biggr) \int {dY d^2Q \over 16\pi^3} \theta(1-Y) \theta(Y) |\phi(Q,Y)|^2 \;.$$ The divergent part of this exactly cancels the first term on the right-hand side of Eq. (122). This cancellation occurs for any state, and this cancellation is unusual because it is between the expectation value of a one-body operator and the expectation value of a two-body operator. The cancellation resembles what happens in the Schwinger model and is easily understood. It results from the fact that a color singlet has no color monopole moment. If the state is a color octet the divergences are both positive and cannot cancel. Since the cancellation occurs in the matrix element, [we can let $\epsilon \rightarrow 0$ before diagonalizing]{} $\H_0$. The fact that the divergences cancel exactly does not indicate that confinement occurs. This requires the residual interactions to diverge at large distances, which means small momentum transfer. Equivalently, we need the color dipole self-energy to diverge if the color dipole moment diverges because the partons separate to large distance. My analysis of the residual interaction is neither elegant nor complete. For a more complete analysis see Ref.[@Br96a]. I show that the interaction is logarithmic in the longitudinal direction at zero transverse separation and logarithmic in the transverse direction at zero longitudinal separation, and I present the full angle-averaged interaction without derivation. In order to avoid the infrared divergence, which is canceled, I compute spatial derivatives of the potential. First consider the potential in the longitudinal direction. Given a momentum space expression, set $x_\perp=0$ and the Fourier transform of the longitudinal interaction requires the transverse momentum integral of the potential, $${\partial \over \partial z} V(z) = \int {d^3 q \over (2\pi)^3} i q_z V(q_\perp,q_z) e^{i q_z z} \;.$$ We are interested only in the long range potential, so we can assume that $q_z$ is arbitrarily small during the analysis and approximate the step functions accordingly. For our interaction this leads to $$\begin{aligned} {\partial \over \partial x^-} V(x^-) &=& -4 g_\Lambda^2 C_F P^+ \int {dq^+ d^2q_\perp \over 16\pi^3} \theta\bigl(P^+-|q^+|\bigr) \nonumber \\ &&~~~~~\theta\bigl(\Lam-{q_\perp^2 \over |q^+|}\bigr) \Bigl( {1 \over q^+} \Bigr)^2 (i q^+) e^{i q^+ x^-} \;.\end{aligned}$$ Completing the $q_\perp$ integration we have $$\begin{aligned} {\partial \over \partial x^-} V(x^-) &=& - {i g_\Lambda^2 C_F \Lambda^2 \over 4\pi^2} \int dq^+ \theta\bigl(P^+-|q^+|\bigr) {q^+ \over |q^+|} e^{i q^+ x^-} \nonumber \\ &=& {g_\Lambda^2 C_F \Lambda^2 \over 2\pi^2} \int_0^{P^+} dq^+ \sin\bigl({q^+ x^-}\bigr) \nonumber \\ &=& {g_\Lambda^2 C_F \Lambda^2 \over 2\pi^2} \Biggl( {1 \over x^-} - {\cos\bigl( P^+ x^- \bigr) \over x^-}\Biggr) \nonumber \\ &=& {g_\Lambda^2 C_F \Lambda^2 \over 2\pi^2} {\partial \over \partial x^-} \; \ln\bigl(|x^-|) + {\rm short~range}\;.\end{aligned}$$ To see that the term involving a cosine in the next-to-last line produces a short range potential, simply integrate it. At large $|x^-|$, which is the only place we can trust our approximations for the original integrand, this yields a logarithmic potential, as promised. Next consider the potential in the transverse direction. Here we can set $x^-=0$ and get $$\begin{aligned} {\partial \over \partial x_\perp^i} V(x_\perp) &=& -4 g_\Lambda^2 C_F P^+ \int {dq^+ d^2 q_\perp \over 16\pi^3} \theta\bigl(P^+-|q^+|\bigr) \nonumber \\ &&~~~~~\theta\Biggl(\Lam-{q_\perp^2 \over |q^+|}\Biggr) \Biggl( {1 \over q^+}\Biggr)^2 (i q_\perp^i) e^{i {\bf q}_\perp \cdot {\bf x}_\perp} \;.\end{aligned}$$ Here I have used the fact that the integration is dominated by small $q^+$ to simplify the integrand again. Completing the $q^+$ integration this becomes $$\begin{aligned} {\partial \over \partial x_\perp^i} V(x_\perp) &=& -{i g_\Lambda^2 C_F \Lambda^2 \over 2\pi^3} \int d^2q_\perp {q_\perp^i \over q_\perp^2} e^{ i {\bf q}_\perp \cdot {\bf x}_\perp}~+~{\rm short~range}\nonumber \\ &=& {g_\Lambda^2 C_F \Lambda^2 \over \pi^2} {x_\perp^i \over x_\perp^2} \nonumber \\ &=& {g_\Lambda^2 C_F \Lambda^2 \over \pi^2} {\partial \over \partial x_\perp^i} \;\ln(|{\bf x_\perp}|) \;.\end{aligned}$$ Once again, this is the derivative of a logarithmic potential, as promised. The strength of the long-range logarithmic potential is not spherically symmetrical in these coordinates, with the potential being larger in the transverse than in the longitudinal direction. Of course, there is no reason to demand that the potential is rotationally symmetric in these coordinates. The angle-averaged potential for two heavy quarks with mass $M$ is $${g_\Lambda^2 C_F \Lambda^2 \over \pi^2} \Bigl(\ln({\cal R}) - {\rm Ci}({\cal R}) + 2 {{\rm Si}({\cal R}) \over {\cal R}} - {(1- {\rm cos}({\cal R})) \over {\cal R}^2} + {{\rm sin}({\cal R}) \over {\cal R}} - {5 \over 2}+\gamma_E\Bigr) \;,$$ where, $${\cal R} = {\Lambda^2 \over 2M} r \;.$$ I have assumed the longitudinal momentum scale in the cutoff equals the longitudinal momentum of the state. The full potential is not naively rotationally invariant. Of course, the two body interaction in QED is also not rotationally invariant. The leading term in an expansion in powers of momenta yields the rotationally invariant Coulomb interaction, but higher order terms are not rotationally invariant. These higher order terms do not spoil rotational invariance of physical results because they must be combined with higher order interactions in the effective Hamiltonian, and with corrections involving photons at higher orders in bound state perturbation theory. There is no reason to expect, and no real need for a potential that is rotationally invariant; however, to proceed we need to decide which part of the potential must be treated non-perturbatively. The simplest reasonable choice is the angle-averaged potential, which is what we have used in heavy quark calculations. Had we computed the quark-gluon or gluon-gluon interaction, we would find essentially the same residual long range two-body interaction in every Fock space sector, although the strengths would differ because different color operators appear. In QCD gluons have a divergent self-energy and experience divergent long range interactions with other partons if we use coupling coherence. In this sense, the assumption that gluon exchange below some cutoff is suppressed is consistent with the Hamiltonian that results from this assumption. To show that gluon exchange is suppressed when $\Lambda \rightarrow \Lambda_{QCD}$, rather than some other scale ([*i.e.*]{}, zero as in QED), a non-perturbative calculation of gluon exchange is required. This is exactly the calculation bound state perturbation theory produces, and bound state perturbation theory suggests how the perturbative renormalization group calculation might be modified to generate these confining interactions self-consistently. If perturbation theory, which produced this potential, continues to be reasonable, the long range potential will be exactly canceled in QCD just as it is in QED. We need this exact cancellation of new forces to occur at short distances and turn off at long distances, if we want asymptotic freedom to give way to a simple constituent confinement mechanism. At short distances the divergent self-energies and two-body interactions cannot be ignored, but they should exactly cancel pairwise if these divergences appear only when gluons are emitted and absorbed in immediately successive vertices, as I have speculated. The residual interaction must be analyzed more carefully at short distances, but in any case a logarithmic potential is less singular than a Coulomb interaction, which does not disturb asymptotic freedom. This is the easy part of the problem. The hard part is discovering how the potential can survive at any scale. A perturbative renormalization group will not solve this problem. The key I have suggested is that interactions between quarks and gluons in the intermediate states required for cancellation of the potential will eventually produce a non-negligible energy gap. I am unable to detail this mechanism without an explicit calculation, but let me sketch a naive picture of how this might happen. Focus on the quark-antiquark-gluon intermediate state, which mixes with the quark-antiquark state to screen the long range potential. The free energy of this intermediate state is always higher than that of the quark-antiquark free energy, as is shown using a simple kinematic argument. However, if the gluon is massless, the energy gap between such states can be made arbitrarily small. As we use a similarity transformation to run a vertex cutoff on energy transfer, mixing persists to arbitrarily small cutoffs since the gap can be made arbitrarily small. Wilson has suggested using a gluon mass to produce a gap that will prevent this mixing from persisting as the cutoff approaches $\Lambda_{QCD}$, and I am suggesting a slightly different mechanism. If we allow two-body interactions to act in both sectors to all orders, even the Coulomb interaction can produce quark-antiquark and quark-antiquark-gluon bound states. In this respect QCD again differs qualitatively from QED because the photon cannot be bound and the energy gap is always arbitrarily small even when the electron-positron interaction produces bound states. If we assume that a fixed energy gap arises between the quark-antiquark bound states and the quark-antiquark-gluon bound states, and that this gap establishes the important scale for non-perturbative QCD, these states must cease to mix as the cutoff goes below the gap. An important ingredient for any calculation that tries to establish confinement self-consistently is a [*seed mechanism*]{}, because it is possible that it is the confining interaction itself which alters the evolution of the Hamiltonian so that confinement can arise. I have proposed a simple seed mechanism whose perturbative origin is appealing because this allows the non-perturbative evolution induced by confinement to be matched on to the perturbative evolution required by asymptotic freedom. I provide only a brief summary of our heavy quark bound state calculations, and refer the reader to the original articles for details. We follow the strategy that has been successfully applied to QED, with modifications suggested by the fact that gluons experience a confining interaction. We keep only the angle-averaged two-body interaction in $\H_0$, so the zeroth order calculation only requires the Hamiltonian matrix elements in the quark-antiquark sector to $\order(\alpha)$. All of the matrix elements are given above. For heavy quark bound states we can simplify the Hamiltonian by making a nonrelativistic reduction and solving a Schr[ö]{}dinger equation, using the potential in Eq. (168). We must then choose values for $\Lambda$, $\alpha$, and $M$. These should be chosen differently for bottomonium and charmonium. The cutoff for which the constituent approximation works well depends on the constituent mass, as in QED where it is obviously different for positronium and muonium. In order to fit the ground state and first two excited states of charmonium, we use $\Lambda=2.5 GeV$, $\alpha=0.53$, $M_c=1.6 GeV$. In order to fit these states in bottomonium we use $\Lambda=4.9 GeV$, $\alpha=0.4$, and $M_b=4.8 GeV$.[^6] Violations of rotational invariance from the remaining parts of the potential are only about 10%, and we expect corrections from higher Fock state components to be at least of this magnitude for the couplings we use. These calculations show that the approach is reasonable, but they are not yet very convincing. There are a host of additional calculations that must be done before the success of this approach can be judged, and most of them await the solution of several technical problems. In order to test the constituent approximation and see that full rotational invariance is emerging as higher Fock states enter the calculation, we must be able to include gluons. If gluons are massless (which need not be true since the gluon mass is a relevant operator that must in principle be tuned), we cannot continue to employ a nonrelativistic reduction. In any case, we are primarily interested in light hadrons and must learn how to perform relativistic calculations to study them. The primary difficulty is that evaluation of matrix elements of the interactions given above involves high dimensional integrals which display little symmetry in the presence of cutoffs. Worse than this is the fact that the confinement mechanism requires cancellation of infrared divergences which have prevented us from using Monte Carlo methods to date. These difficulties are avoided when a nonrelativistic reduction is made, but there is little more that we can do for which such a reduction is valid. I conclude this section with a few comments on chiral symmetry breaking. While quark-quark, quark-gluon, and gluon-gluon confining interactions appear in the hamiltonian at [O]{}($\alpha$), chiral symmetry is not broken at any finite order in the perturbative expansion; so symmetry-breaking operators must be allowed to appear [*ab initio*]{}. Light-front chiral symmetry differs from equal-time chiral symmetry in several interesting and important aspects. For example, quark masses in the kinetic energy operator do not violate light-front chiral symmetry; and the only operator in the canonical hamiltonian that violates this symmetry is a quark-gluon coupling linear in the quark mass. Again, the primary technical difficulty is the need for relativistic bound state calculations, and real progress on chiral symmetry cannot be made before we are able to perform relativistic calculations with constituent gluons. If transverse locality is maintained, simple renormalization group arguments indicate that chiral symmetry should be broken by a relevant operator in order to preserve well-established perturbative results at high energy. The only relevant operator that breaks chiral symmetry involves gluon emission and absorption, leading to the conclusion that the pion contains significant constituent glue. This situation is much simpler than what we originally envisioned, because it does not require the addition of any operators that cannot be found in the canonical QCD hamiltonian; and we have long known that relevant operators must be tuned to restore symmetries. ### Acknowledgments I would like to thank the staff of the Newton Institute for their wonderful hospitality, and the Institute itself for the support that made this exceptional school possible. 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Supp. [**86**]{}, 215 (1986). S. J. Brodsky and G. P. Lepage, in [*Perturbative Quantum Chromodynamics*]{}, A. H. Mueller, Ed. (World Scientific, Singapore, 1989). M. Brisudov[á]{} and R. J. Perry, Phys. Rev. [**D54**]{}, 6453 (1996). H. A. Bethe, Phys. Rev. [**72**]{}, 339 (1947). C. Quigg and J.L. Rosner, Phys. Lett. [**71B**]{}, 153 (1977). Chris Quigg, [*Realizing the Potential of Quarkonium.*]{} Preprint hep-ph/9707493. [^1]: These include one-body operators that modify the free dispersion relations. [^2]: These functions imply that there are effectively an infinite number of relevant and marginal operators; however, their dependence on fields and transverse momenta is extremely limited. [^3]: In deference to the original work I will call this a similarity transformation even though in all cases of interest to us it is a unitary transformation. [^4]: The rescaling step is not essential, but it avoids exponentials of the cutoff in the renormalization group equations. [^5]: There is a trick which allows this calculation to be performed using only first-order bound state perturbation theory. The trick basically involves using a Melosh rotation. [^6]: There are several minor errors in Ref. [@Br97a], which are discussed in Ref.[@Sz97a]. I have also chosen ${\cal P}^+=P^+$, and in those papers ${\cal P}^+=P^+/2$; and I have absorbed this change in a redefinition of $\Lambda$.

ArXiv

Astro2020 APC White Paper The Dark Energy Spectroscopic Instrument (DESI) **Thematic Areas:** $\square$ Planetary Systems $\square$ Star and Planet Formation $\square$ Formation and Evolution of Compact Objects Cosmology and Fundamental Physics Stars and Stellar Evolution Resolved Stellar Populations and their Environments Galaxy Evolution $\square$ Multi-Messenger Astronomy and Astrophysics **Principal Authors:** Michael E. Levi (Lawrence Berkeley National Laboratory)\ & Lori E. Allen (National Optical Astronomy Observatory) **Email:** [email protected], [email protected] **Co-authors:** Anand Raichoor (EPFL, Switzerland), Charles Baltay (Yale University), Segev BenZvi (University of Rochester), Florian Beutler (University of Portsmouth, UK), Adam Bolton (NOAO), Francisco J. Castander (IEEC, Spain), Chia-Hsun Chuang (KIPAC), Andrew Cooper (National Tsing Hua University, Taiwan), Jean-Gabriel Cuby (Aix-Marseille University, France), Arjun Dey (NOAO), Daniel Eisenstein (Harvard University), Xiaohui Fan (University of Arizona), Brenna Flaugher (FNAL), Carlos Frenk (Durham University, UK), Alma X. González-Morales (Universidad de Guanajuato, México), Or Graur (CfA), Julien Guy (LBNL), Salman Habib (ANL), Klaus Honscheid (Ohio State University), Stephanie Juneau (NOAO), Jean-Paul Kneib (EPFL, Switzerland), Ofer Lahav (UCL, UK), Dustin Lang (Perimter Institute, Canada), Alexie Leauthaud (UC Santa Cruz), Betta Lusso (Durham University, UK), Axel de la Macorra (UNAM, Mexico), Marc Manera (IFAE, Spain), Paul Martini (Ohio State University), Shude Mao (Tsinghua University, China), Jeffrey A. Newman (University of Pittsburgh), Nathalie Palanque-Delabrouille (CEA, France), Will J. Percival (University of Waterloo, Canada), Carlos Allende Prieto (IAC, Spain), Constance M. Rockosi (UC Santa Cruz), Vanina Ruhlmann-Kleider (CEA, France), David Schlegel (LBNL), Hee-Jong Seo (Ohio University), Yong-Seon Song (KASI, South Korea), Greg Tarlé (University of Michigan), Risa Wechsler (Stanford University), David Weinberg (Ohio State University), (Christophe Yèche (CEA, France), Ying Zu (Shanghai Jiao Tong University, China)\ **Abstract:** We present the status of the Dark Energy Spectroscopic Instrument (DESI) and its plans and opportunities for the coming decade. DESI construction and its initial five years of operations are an approved experiment of the U.S. Department of Energy and is summarized here as context for the Astro2020 panel. Beyond 2025, DESI will require new funding to continue operations. We expect that DESI will remain one of the world’s best facilities for wide-field spectroscopy throughout the decade. More about the DESI instrument and survey can be found at https://www.desi.lbl.gov. An Overview of DESI: 2020-2025 ============================== DESI is an ambitious multi-fiber optical spectrograph sited on the Kitt Peak National Observatory Mayall 4m telescope, funded to conduct a Stage IV spectroscopic dark energy experiment. DESI featuers 5000 robotically positioned fibers in an 8 deg$^2$ focal plane, feeding a bank of 10 triple-arm spectrographs that measure the full bandpass from 360 nm to 980 nm at spectral resolution of 2000 in the UV and over 4000 in the red and IR (Martini et al. 2018). DESI is designed for efficient operations and exceptionally high throughput, anticipated to peak at over 50% from the top of the atmosphere to detected photons, not counting obscuration of the telescope or aperture loss from the $1.5''$ diameter fibers. More information is in Table \[tab:desi\]. As of this writing in July 2019, DESI construction is nearly complete and the instrument is being installed at the Mayall telescope. The new prime-focus corrector was operated on sky in April/May 2019 and confirmed to produce sharp images. All ten petals of the robotic positioners and all fibers have been constructed; these are being installed on the telescope in July (Figure \[fig:petal\]). Six of the ten spectrographs are installed and entering off-sky commissioning (Figure \[fig:spect\] and \[fig:res\]); the other four should arrive in fall 2019. We anticipate spectroscopic first-light in October 2019, with commissioning running through January 2020. The collaboration will then operate a 4-month Survey Validation program in spring 2020 and begin the 5-year survey in summer 2020. {width="95.00000%"} [[**Key Science Goals:** ]{}]{} The DESI Collaboration will use this facility to conduct a 5-year survey of galaxies and quasars, covering 14,000 deg$^2$ and yielding 34 million redshifts. The mission-need science of this survey is the study of dark energy through the measurement of the cosmic distance scale with the baryon acoustic peak method as a standard ruler and through the study of the growth of structure with redshift-space distortions. The survey will further allow measurement of other cosmological quantities, such as neutrino mass and primordial non-Gaussianity, as well as studies of galaxies, quasars, and stars. The DESI survey uses a sequence of target classes to map the large-scale structure of the Universe from redshift 0 to 3.5 (Aghamousa et al. 2016). In dark and grey time, DESI will utilize quasars, emission-line galaxies, and luminous red galaxies. Over 4M luminous red galaxy sample will cover $0.3<z<1$, including coverage to $z\sim0.8$ at a density twice that of SDSS-III BOSS. The emission-line galaxy sample is the largest set, 18M, covering $0.6<z<1.6$ and providing the majority of the distance scale precision. 2.4M quasars selected from their WISE excess will extend the map. Importantly, these will yield Lyman $\alpha$ forest measurements along 600K lines-of-sight, from which we will measure the acoustic oscillations at $z>2$. In bright time, DESI will conduct a flux-limited survey of 10M galaxies to $r\approx19.5$, with a median redshift around 0.2. This will allow dense sampling of a volume over 10 times that of the SDSS MAIN and 2dF GRS surveys, which we expect will spur development of cosmological probes of the non-linear regime of structure formation. In addition to extragalactic targets, DESI will observe many millions of stars. About 10M stars at $16<G<19$ will fill unused fibers in the bright time program, and we will conduct a backup program of brighter stars when observing conditions (clouds, moon, and/or seeing) prevent useful data from being collected on extragalactic targets. Because DESI is a bench-mounted spectrograph with sub-degree temperature stability, we anticipate velocity precision to $\sim1$ km/s. The DESI Collaboration plans for release of annual data sets, including survey selection functions and mock catalogs suitable for clustering analyses, following completion of its cosmology key projects. [[**Target Selection:** ]{}]{} In preparation of target selection for DESI, the Collaboration has played a leading role in the execution of the Legacy Survey imaging program, using nearly 1000 nights on the Blanco, Mayall, and Bok telescopes to image 15,000 deg$^2$ to $g=24$, $r=23.4$, and $z=22.5$ depth, co-reduced with 5 years of WISE satellite imaging (Dey et al. 2019). This is the deepest coverage of the full high-latitude sky in the Northern hemisphere (Figure \[fig:foot\]). The imaging data and catalogs have had 8 data releases, available at [http://legacysurvey.org]{}, the last of which reaches over 19,000 deg$^2$ by inclusion of the 5-year Dark Energy Survey. Hence, DESI has already provided an extensive data product for the general astronomical community. ![\[fig:petal\]The first of 10 focal plate petals installed at the Mayall prime focus (June 26, 2019).](petal_install-6.jpg){width="95.00000%"} As regards this first five-year survey with DESI, we stress the opportunity of this U.S.-led project to conduct cutting-edge dark energy research, both in its own right and in coordination with optical, millimeter, and X-ray imaging data sets. As well illustrated by the SDSS, the combination of spectroscopy and imaging unlocks a wide range of applications. ![\[fig:spect\]Six of the ten 3-armed DESI spectrographs, installed in their thermal enclosure.](specs6.jpg){width="95.00000%"} [[**Organization:** ]{}]{} DESI is being built by the DESI Collaboration with primary funding from the U.S. Department of Energy, and additional funding from the National Science Foundation, the Science and Technologies Facilities Council of the United Kingdom, the Gordon and Betty Moore Foundation, the Heising-Simons Foundation, the National Council of Science and Technology of Mexico, the French Alternative Energies and Atomic Energy Commission (CEA), and by the DESI Member Institutions. The DESI Collaboration currently has more than 600 total members from 79 institutions from 13 countries around the world. Of those, $\sim$200 are senior members, and $\sim$400 are early career scientists. With DESI, the Mayall telescope is dedicated to this single instrument configuration (unlike DECam on the Blanco, which could also mount a secondary mirror). The DOE Office of Science High-Energy Physics division will be the primary funder of the DESI survey, including the operation of the Mayall telescope through the 5-year survey. The survey will utilize at least the darkest 21 nights per lunation, plus engineering time, and potentially may use all of the telescope time. ![\[fig:res\]Measured resolution of the six installed spectrographs. Dotted lines are the system requirements. The as-built results match exquisitely to the modeled performance.](resolution.png){width="75.00000%"} ![\[fig:foot\]The planned DESI survey footprint is shown by the shaded area; it is built on several existing imaging surveys and extends as far south as $\delta=-20^\circ$ in the SGC and $-10^\circ$ in the NGC.](footprint-desi.png){width="95.00000%"} [[**Cost:** ]{}]{} The DESI construction project has cost \$75M, with \$56M from the DOE and the balance by other partners and institutional buy-ins. Survey operations are budgeted at $\sim\!\$12$M/year, split approximately one-third site operations, and the rest supporting the instrument, the survey planning, and data processing and catalog creation. Beyond 2025 with DESI ===================== Beyond the end of the first 5-year survey, DESI will remain a state-of-the-art facility for wide-field surveys. New commitments for funding will be required. But given the time scale to construct a facility more powerful than DESI and with no such project yet approved[^1] we expect that the ground-based facility landscape in the second half of the 2020’s will look much like the first. See Table \[tab:facilities\] for a summary. We note that not only is DESI at the forefront of this generation, without it, the U.S. community will not have a facility to compete with ESO and Subaru. Further, we note that while Euclid and WFIRST will offer space-based platforms for slitless IR spectroscopy, optical spectroscopy remains a highly efficient way to get redshifts both at $z<1.5$ and $z>2$. Name Telescope \# Fibers FOV (deg$^2$) Bandpass (nm) Resolution -------- ---------------------- ----------- --------------- --------------- ----------------------- DESI Mayall 4-m 5000 8 360–980 mid PFS Subaru 8-m 2400 1.5 380–1260 mid 4MOST VISTA 4-m 2436 5 370–950 mid & high WEAVE WHT 4-m 960 3 370–960 mid & high SDSS-V Sloan & DuPont 2.5-m 1000 7 360–1700 mid (opt) & high (IR) : A brief comparison of multi-fiber facilities under construction. Mid-resolution is typically a few thousand; high-resolution is typically around 20K, but for a more limited bandpass. DESI will offer the highest multiplex and largest field of view of these next-generation facilities; only PFS has more instantaneous light-gathering power, but it is not a dedicated platform. Of the current generation of facilities, LAMOST is operating 4000 fibers in a 20 deg$^2$ field of view, but with performance limited to bright galaxies and stars.[]{data-label="tab:facilities"} [[**Key Science Goals:** ]{}]{} We anticipate that a second phase of DESI will continue to offer exciting survey opportunities. Certainly we will not have exhausted the supply of plausible targets on the sky. Imaging surveys from HSC, LSST, Euclid, SphereX, WFIRST, eROSITA, and others will yield improved isolation of valuable targets over areas of thousands to tens of thousands of deg$^2$. Spectroscopy can provide the key leverage to realize the science potential of these candidates, whether for redshifts or for more detailed characterization. DESI’s combination of field of view, multiplex, throughput, and resolution makes it a great complement to the coming generation of imaging surveys. There are at least 5 fertile areas of potential targets for such a survey: 1\) High-redshift emission-line targets: improved selection of $1<z<1.6$ emission-line galaxies from deeper imaging; selection of Ly$\alpha$ emission candidates from deep imaging in the blue; or follow-up of low-quality emission-line candidates from Euclid and SPHEREx. Such a survey would increase the sampling of the large volume available at higher redshift. 2\) Increased depth and sampling in the Lyman-$\alpha$ forest, reobserving known targets and adding fainter candidates from deeper imaging. 3\) A high-density galaxy survey at $z<1$, building on the DESI bright galaxy survey. These candidates are readily identified, but a high-density sample with precise spectroscopic redshifts would allow identification of groups and redshift-space distortions in the non-linear regime within the cosmic acceleration epoch. 4\) A high-multiplex survey of the Milky Way, with $O(100)$ million stars, to yield radial velocities and stellar abundances to pair with the exquisite Gaia astrometry. The rapid reconfiguration time of DESI ($<2$ minutes) makes short exposures an effective strategy. 5\) Time-domain spectroscopy and transient host spectroscopy, building on SDSS-V and time-domain imaging surveys such as ZTF, LSST, and TESS. [[**Technical Drivers:** ]{}]{} While the DESI instrument could continue to be usefully operated in the same configuration as the pre-2025 phase, there may be opportunities for augmentations. Notably, the spectrographs are modular and could be altered or replaced, subject to cost and space constraints, if the adopted science goals called for it. [[**Organization, Status, and Schedule:** ]{}]{} A science collaboration for post-2025 operations has not yet been formed, but we expect that many of the current participants would be interested in continuing. The Mayall telescope remains property of the National Science Foundation, while the DESI equipment is DOE property. We expect that planning for such surveys will pick up speed in 2021 with the arrival of early DESI data, which will solidify the on-sky performance and give a tactile sense of the target selections. We note that there has been mention of the idea of moving DESI south to the Blanco in 2025. On the plus side, such a move would increase sky overlap with LSST. On the down side, it is an expensive proposition: it has taken over 1.5 years to install DESI at the Mayall, and much of that work would need to re-occur. We therefore expect that a move would result in substantial downtime for both telescopes, along with financial cost. Such a decision will require a detailed cost-benefit analysis. Given the large amount of near-equatorial sky visible jointly from Arizona and Chile, we suspect that many post-2025 survey options could be well performed without a move, potentially in collaboration with an instrument of similar or even lesser capability in the south. [[**Cost Estimate:** ]{}]{} The budget for a second phase of DESI operations would depend on the survey choices made as well as on the assessment of costs of ongoing instrument support, presumably informed by experience in the coming years of operations. However, the cost is likely $O(\$10-15)$M/year (inclusive), comparable to those of other mid-scale facilities that deliver highly processed data products. Hence, we expect ongoing operations to fall in the Medium class of ground-based activities. In conclusion, we expect that the Mayall telescope with DESI will remain a world-class facility for high-multiplex optical mid-resolution spectroscopy in the latter half of this decade, offering the U.S. the opportunity to continue its leadership in spectroscopic wide-field surveys. **References**\ Dey, A., et al., ”Overview of the DESI Legacy Imaging Surveys,” ApJ, 157, 168 (2019) Martini, P., et al., “Overview of the Dark Energy Spectroscopic Instrument,” SPIE, 107021F, (2018). DESI Collaboration, Aghamousa, A., et al., “The DESI Experiment Part I: Science,Targeting, and Survey Design,” arXiv:1611.00036 (2016) DESI Collaboration, Aghamousa, A., et al., “The DESI Experiment Part II: Instrument Design,” arXiv:1611.00037 (2016) [^1]: We note that DESI (under the previous name BigBOSS) was identified in New Worlds New Horizons as an exemplar of the MidScale Innovation Program. Despite timely agency support (CD-0 approved in 2012 and CD-2 in 2015), non-federal funding to conduct long-lead procurements, and no major programmatic or technical interruptions, we will be in operations in 2020. We believe this is indicative of what projects of this complexity require, even in good outcomes!

ArXiv

--- abstract: 'In recent years, the increasing interest in Stochastic model predictive control (SMPC) schemes has highlighted the limitation arising from their inherent computational demand, which has restricted their applicability to slow-dynamics and high-performing systems. To reduce the computational burden, in this paper we extend the probabilistic scaling approach to obtain low-complexity inner approximation of chance-constrained sets. This approach provides probabilistic guarantees at a lower computational cost than other schemes for which the sample complexity depends on the design space dimension. To design candidate simple approximating sets, which approximate the shape of the probabilistic set, we introduce two possibilities: i) fixed-complexity polytopes, and ii) $\ell_p$-norm based sets. Once the candidate approximating set is obtained, it is scaled around its center so to enforce the expected probabilistic guarantees. The resulting scaled set is then exploited to enforce constraints in the classical SMPC framework. The computational gain obtained with the proposed approach with respect to the scenario one is demonstrated via simulations, where the objective is the control of a fixed-wing UAV performing a monitoring mission over a sloped vineyard.' author: - 'Martina Mammarella$^1$, Teodoro Alamo$^2$, Fabrizio Dabbene$^{1}$, and Matthias Lorenzen$^{3}$ [^1] [^2] [^3] [^4]' bibliography: - 'main.bib' title: ' **Computationally efficient stochastic MPC: a probabilistic scaling approach**' --- Introduction {#sec:intro} ============ In recent years, the performance degradation of model predictive control (MPC) schemes in the presence of uncertainty has driven the interest towards stochastic MPC, to overcome the inherent conservativeness of robust approaches. A probabilistic description of the disturbance or uncertainty allows to optimize the average performance or appropriate risk measures. Furthermore, allowing a (small) probability of constraint violation, by introducing so-called chance constraints, seems more appropriate in some applications. As highlighted in [@farina2016stochastic], current SMPC methods can be divided in two main groups, depending on the approach followed to solve the chance-constrained optimization problem: (i) analytic approximation methods; and (ii) randomized [@tempo2012randomized] and scenario-based methods. For the analytic approximation methods, the probabilistic properties of the uncertainty are exploited to reformulate the chance constraints in a deterministic form. For the second class of methods, the craved control performance and constraint satisfaction are guaranteed properly generating a sufficient number of uncertainty realizations and on the solution of a suitable constrained optimization problem, as proposed in [@calafiore2006scenario], [@schildbach2014scenario]. The main advantage of this class of stochastic MPC algorithms is given by the inherent flexibility to be applied to (almost) every class of systems, including any type of uncertainty and both state and input constraints, as long as the optimization problem is convex. On the other hand, they share two main drawback: i) slowness, which has limited their application to problems involving slow dynamics and where the sample time is measured in tens of seconds or minutes; and ii) a significant computational burden required for real-time implementation, narrowing the application domains to those involving low-computation assets. Some examples are [@grosso2017stochastic] for water networks, [@nasir2015randomised] for river flood control, [@van2006stochastic] for chemical processes, and [@vignali2017energy] for energy plants. An efficient solution to the aforementioned disadvantages was proposed in [@matthias1] where the SMPC controllers design is based on an *offline* sampling approach and only a predefined number of necessary samples are kept for online implementation. In this approach, the sample complexity is linearly dependent to the design space dimension and the sampling procedure allows to obtain offline an inner approximation of the chance-constrained set. This approach has been extended to a more generic setup in [@Mammarella:18:Control:Systems:Technology] and experimentally validated for the control of a spacecraft during rendezvous maneuvers. Beside the efficacy of the approach, the results highlighted the need to further reduce the computational load and the slowness of the proposed approach to comply with faster dynamics and low-cost, low-performance hardware. Among challenging applications, the control of unmanned aerial vehicles (UAVs) during assorted scenarios, have been triggering the attention of MPC community. These platforms are typically characterized by fast dynamics and equipped with computationally-limited autopilots. In the last decade, different receding horizon techniques have been proposed, see e.g. [@kamel; @Alexis2016; @stastny; @michel], including a stochastic approach by [@mammarella2018sample]. In this case, preliminary analysis have confirmed the effectiveness of the proposed offline sampling-based SMPC (OS-SMPC) strategy but the results highlighted also the need to further reduce the dimension of the optimization problem to comply with hardware requirements. The main contribution of this paper is to propose a new methodology that combines the probabilistic-scaling approach proposed in [@alamo2019safe], which allows to obtain a low-complexity inner approximation of the chance constrained set, with the SMPC approach of [@matthias1; @Mammarella:18:Control:Systems:Technology]. In [@alamo2019safe], authors show how to scale a given set of manageable complexity around its center to obtain, with a user-defined probability, a region that is included in the chance constrained set. In this paper, we extend the aforementioned approach showing how it is possible to reduce the sample complexity via probabilistic scaling exploiting so-called simple approximating sets (SAS). The starting point consists in obtaining a first simple approximation of the “shape” of the probabilistic set. To design a candidate SAS, we propose two possibilities. The first one is based on the definition of an approximating set by drawing a fixed number of samples. On the other hand, the second case envisions the use of $\ell_p$-norm based sets, first proposed in [@dabbene2010complexity]. In particular, we consider as SAS a $\ell_1$-norm *cross-polytope* and a $\ell_\infty$-norm *hyper-cube*. Solving a standard optimization problem, it is possible to obtain the center and the shape of the SAS, which will be later scaled to obtain the expected probabilistic guarantees following the approach described in [@alamo2019safe]. Then, the scaled SAS is used in the classical SMPC algorithm to enforce constraints. To validate the proposed approach, an agriculture scenario has been selected, because of the increasing interest of using drones in the agriculture 4.0 framework, as explained in [@sylvester2018agriculture], due to their great potential to support and address some of the most pressing dares in farming. And real-time quality data and crop monitoring are two of those challenges. In particular, UAVs could represent a favorable alternative to conventional farming machines, whenever clear advantages with respect to traditional methods, in terms of higher efficiency in operations, reduced environmental impact or enhanced human health and safety are sought. For this paper, the control objective envisions the proposed approach applied to a fixed-wing UAV performing a monitoring mission over a sloped vineyard, following a pre-defined snake path. The performance of the proposed approach in terms of tracking capabilities and computational load has been compared with those obtained exploiting the “classical” OS-SMPC scheme proposed in [@mammarella2018sample]. [*Notation*: The set $\mathbb{N}_{>0}$ denotes the positive integers, the set $\mathbb{N}_{\geq 0} = \left\{0\right\} \cup\mathbb{N}_{>0}$ the non-negative integers, and $\mathbb{N}_a^b$ the integers interval $[a,b]$. Positive (semi)definite matrices $A$ are denoted $A\succ 0$ $(A\succeq 0)$ and $\|x\|_A^2\doteq x^TAx$. For vectors, $x\succeq 0$ ($x\preceq 0$) is intended component-wise. $\mathsf{Pr}_a$ denotes the probabilistic distribution of a random variable $a$. Sequence of scalars/vectors are denoted with bold lower-case letters, i.e. $\textbf{v}$. ]{} Offline Sampling-based Stochastic MPC ===================================== In this section, we first recall the Stochastic MPC Framework proposed in [@matthias1; @Mammarella:18:Control:Systems:Technology]. Problem setup {#sec:setup} ------------- We consider the case of a discrete-time system subject to generic uncertainty $w_{k} \in \mathbb{R}^{n_{w}}$ $$x_{k+1} = A(w_{k})x_{k}+B(w_{k})u_{k}+a_{w}(w_{k}), \label{eq:sys}$$ with state $x_{k} \in \mathbb{R}^{n}$, control input $u_{k} \in \mathbb{R}^{m}$, and the vector valued function $a_w(w_{k})$ represent the additive disturbance affecting the systems states. The system matrices $A(w_{k})$ and $B(w_{k})$, of appropriate dimensions, are (possibly nonlinear) functions of the uncertainty $w_{k}$ at step $k$. The disturbances $(w_{k})_{k\in \mathbb{N}_{\geq 0}}$ are modeled as realizations of the stochastic process $(W_{k})_{k\in \mathbb{N}_{\geq 0}}$, on which take the following assumptions. \[bound\_rand\_dist\] The disturbances $W_{k}$, for $k\in \mathbb{N}_{\geq 0}$, are independent and identically distributed (i.i.d.), zero-mean random variables with support $\mathbb{W}\subseteq \mathbb{R}^{n_{w}}$. Moreover, let $\mathbb{G}=\left\{(A(w_{k}),B(w_{k}),a_{w}(w_{k})\right\}_{w_{k}\in \mathbb{W}}$, a polytopic outer approximation with $N_c$ vertexes $\bar{\mathbb{G}}\doteq co\left\{A^{j},B^{j},a_{w}^{j}\right\}_{j\in \mathbb{N}_{1}^{N_{c}}}\supseteq \mathbb{G}$ exists and is known. We can notice that the system can be augmented by a filter to model a specific stochastic processes of interest. The assumption of independent random variables in necessary to perform the offline computations discussed next while the need of a known outer bound is required to establish a safe operating region (see [@matthias1] for details). We remark that the system’s representation in is very general, and encompasses e.g. those in [@matthias1; @Mammarella:18:Control:Systems:Technology; @matthias2]. Given the model  and a realization of the state $x_k$ at time $k$, state predictions $l$ steps ahead are random variables, as well and are denoted $x_{l|k}$, to differentiate it from the realization $x_{l+k}$. Similarly $u_{l|k}$ denotes predicted inputs that are computed based on the realization of the state $x_k$. The system is subject to $p$ state and input chance constraints of the form[^5] $$\begin{aligned} &&\mathsf{Pr}_{\mathbf{w}}\left\{ [H_x]_j^T x_{l|k} + [H_u]_j^T u_{l|k} \le 1 |x_k \right\}\ge 1-\varepsilon_j,\nonumber\\ && \qquad l \in \mathbb{N}_{>0}, \, j \in \mathbb{N}_1^p, \label{eq:origConstr} \end{aligned}$$ with $\varepsilon_j \in (0,1)$, and $H_x\in {\mathbb{R}}^{p \times n}$, $H_u \in {\mathbb{R}}^{p \times m}$, where $[H]_j^T$ denotes the $j$-th row of matrix $H$. The probability $\mathsf{Pr}_{\mathbf{w}}$ is measured with respect to the sequence ${\mathbf{w}}=\{w_i\}_{i>k}$. Hence, equation  states that the probability of violating the linear constraint $[H_x]_j^T x + [H_u]_j^T u \le 1$ for any future realization of the disturbance should not be larger than $\varepsilon_j$. The objective is to derive an asymptotically stabilizing control law for the system  such that, in closed loop, the constraints  are satisfied. Stochastic Model Predictive Control ----------------------------------- To solve the constrained control problem, a stochastic MPC algorithm is considered. The approach is based on repeatedly solving a stochastic optimal control problem over a finite, moving horizon, but implementing only the first control action. Defined the control sequence as $\mathbf{u}_{k} = (u_{0|k}, u_{1|k}, ..., u_{T-1|k})$, the prototype optimal control problem that is to be solved at each sampling time is given minimizing the cost function $$\begin{gathered} J_T(x_k,\mathbf{u}_{k}) =\\ \mathbb{E}\left\{ \sum_{l=0}^{T-1} \left( x_{l|k}^\top Q x_{l|k} + u_{l|k}^\top Ru_{l|k} \right) + x_{T|k}^\top P x_{T|k} ~|~ x_k\right\} \label{eq:origCostFnc}\end{gathered}$$ with $Q\succ 0$, $R \succ 0$, and appropriately chosen $P \succ 0$, subject to the system dynamics  and constraints . The online solution of the stochastic MPC problem remains a challenging task but several special cases, which can be evaluated exactly, as well as methods to approximate the general solution have been proposed in the literature. The approach followed in this work was first proposed in [@matthias1; @Mammarella:18:Control:Systems:Technology], where an offline sampling scheme was introduced. Therein, with a prestabilizing input parametrization $$u_{l|k}=Kx_{l|k}+v_{l|k}, \label{eq:prestabilizingInput}$$ with suitably chosen control gain $K\in\mathbb{R}^{n\times m}$ and free optimization variables $v_{l|k} \in \mathbb{R}^m$, equation  is solved explicitly for the predicted states $x_{1|k},\ldots,x_{T|k}$ and predicted inputs $u_{0|k},\ldots,u_{T-1|k}$. In this case, the expected value of the finite-horizon cost  can be evaluated *offline*, leading to a quadratic cost function of the form $$J_{T}(x_{k},\mathbf{v}_{k})=[x_{k}^{T}\,\, \mathbf{v}_{k}^{T} \,\,\textbf{1}_{n}^{T}]\tilde{S}\begin{bmatrix} x_{k} \\ \textbf{v}_{k} \\ \textbf{1}_{n}\\ \end{bmatrix} \label{eq:cost_new_1}$$ in the deterministic variables $\mathbf{v}_{k} = (v_{0|k}, v_{1|k}, ..., v_{T-1|k})$ and $x_{k}$. The reader can refer to [@Mammarella:18:Control:Systems:Technology Appendix A] for a detailed derivation of the cost matrix $\tilde{S}$. Focusing now on the constraint definition, we can notice that by introducing the uncertainty sequence ${\mathbf{w}}_k=\{w_l\}_{l=k,...,k+T-1}$, we can rewrite the $j$-th chance constraint defined by equation  as $$\begin{gathered} {\mathbb{X}_\varepsilon}^j= \left\{\begin{bmatrix} x_k\\ \mathbf{v}_k \end{bmatrix}\in\mathbb{R}^{n+mT} ~|~ \right. \\ \left. \mathsf{Pr}_{{\mathbf{w}}_k}\left\{f_j^T({\mathbf{w}}_k) \begin{bmatrix} x_k\\ \mathbf{v}_k \end{bmatrix} \leq 1 \right\} \geq 1-\varepsilon \right\}, \label{eq:Xej}\end{gathered}$$ with $f_j$ being a function of the sequence of random variables ${\mathbf{w}}_k$. Again, the reader is referred to [@Mammarella:18:Control:Systems:Technology] for details on the derivation of $f_j$. The results in [@matthias1] show that, by exploiting results from statistical learning theory (cf. [@vidyasagar; @alamo2009randomized]), we can construct an inner approximation $\underline{{\mathbb{X}}}^{j}$ of the constraint set ${\mathbb{X}_\varepsilon}^j$ by extracting $N_{LT}$ i.i.d. samples ${\mathbf{w}}_k^{(i)}$ of ${\mathbf{w}}_k$ and taking the intersection of the sampled constraints, i.e. $$\begin{gathered} {\underline{{\mathbb{X}}}^{j}_{LT}}= \left\{\begin{bmatrix} x_k\\ \mathbf{v}_k \end{bmatrix}\in\mathbb{R}^{n+mT} ~|~ \right. \\ \left. f_j^T({\mathbf{w}}_k^{(i)})\begin{bmatrix} x_k\\ \mathbf{v}_k \end{bmatrix}\leq 1,\;i=1,\ldots,N_{LT}\right\},\end{gathered}$$ In particular, it has been shown in [@matthias1], that for given probabilistic levels $\delta\in(0,1)$ and $\varepsilon_j\in(0,0.14)$, choosing the sample complexity $$\begin{gathered} \label{eq:Ntilde} N_{LT}^j \ge \tilde{N}(n+mT,\varepsilon_j,\delta) \\ \doteq \frac{4.1}{\varepsilon}\Big(\ln \frac{21.64}{\delta}+4.39(n+mT)\,\log_{2}\Big(\frac{8e}{\varepsilon_j}\Big)\Big),\end{gathered}$$ guarantees that with probability at least $\delta$ the sample approximation ${\underline{{\mathbb{X}}}^{j}_{LT}}$ is included in the original chance constraint ${\mathbb{X}_\varepsilon}^j$, i.e. $$\mathsf{Pr}\left\{{\underline{{\mathbb{X}}}^{j}_{LT}}\subseteq {\mathbb{X}_\varepsilon}^j\right\}\geq 1-\delta, \quad j=1,.., p.$$ Hence, exploiting these results, we obtain that the stochastic MPC problem can be well approximated by the following linearly constrained quadratic program $$\begin{aligned} \min_{\mathbf{v}_k} ~&J_T(x_k \mathbf{v}_k) \\ \text{s.t. } & (x_k, \mathbf{v}_k) \in {\underline{{\mathbb{X}}}^{j}_{LT}}, \quad j=1,.., p\end{aligned}$$ While the result reduces the original stochastic optimization program to an efficiently solvable quadratic program, the ensuing number of constraints, equal to $$N_{LT}=\sum_{i=1}^T N_{LT}^j,$$ may still be too large. For instance, even for a moderately sized MPC problem with $n=5$ states, $m=2$ inputs and horizon of $T=10$, and for a reasonable choice of probabilistic $\varepsilon^j=0.05$, $\delta=10^{-6}$, we get $N_{LT}^j=20,604$. For this reason, in [@matthias1] a post-processing analysis of the constraint set was proposed for removing redundant constraints. While it is indeed true that all the cumbersome computations may be performed offline, it is still the case that in applications with stringent requirements on the solution time the final number of inequalities may easily become unbearable. This observation motivates the approach presented in the next section, which builds upon the results presented in [@alamo2019safe], showing how the probabilistic scaling approach leads to approximations of “controllable size," that can be directly used in applications. Complexity Reduction via Probabilistic Scaling {#sec:scaling} ============================================== In this section, we consider the very general problem of finding a decision variable vector $\xi$, restricted to a set $\Xi\subseteq{\mathbb{R}}^{n_\xi}$, subject $p$ uncertain linear inequalities. Formally, we consider uncertain inequalities of the form $$\label{ineq:F:g} F(q)\xi \le g({q})$$ where $F({q})\in\mathbb{R}^{p\times{n_\xi}}$ and $g(q)\in\mathbb{R}^{n_p}$ are continuous function of the uncertainty vector $q\in\mathbb{R}^{n_q}$. The uncertainty vector $q$ is assumed to be of random nature, with given probability distribution ${\mathsf{Pr}_{q}}$ and (possibly unbounded) support ${\mathbb{Q}}$. Hence, to each sample of $q$ corresponds a different set of linear inequalities. We aim at finding an approximation of the $\varepsilon$-chance-constraint set, defined as $$\label{XE} {\mathbb{X}_\varepsilon}\doteq \Bigl\{\xi\in\Xi\;|\; \mathsf{Pr}_q\left\{ F(q)\xi \le g(q) \right\}\ge 1-\varepsilon \Bigr\}$$ that represents the region of the design space $\Xi$ for which this probabilistic constraint is satisfied. Note that this captures exactly the SMPC setup discussed in the previous section. Indeed, the chance-constrained set in is a special instance of , with $\xi=[x_k^T\quad\mathbf{v}_k^T]^T$ and $q={\mathbf{w}}_k$. \[f:Xe2D1\] The characterization of the chance constrained set has several application in robust and stochastic control. A classical approach is to find inner convex approximation of the probabilistic set ${\mathbb{X}_\varepsilon}$, obtained for instance by means of applications of Chebyshev-like inequalities, see e.g. [@yan2018stochastic] and [@hewing2018stochastic]. A recent approach, which is the one applied in the previous section to the SMPC problem, is instead based on the derivation of probabilistic approximations of the chance constraints set ${\mathbb{X}_\varepsilon}$ through sampling of the uncertainty. That is, we aim at constructing a set $\underline{{\mathbb{X}}}$ which is contained in ${\mathbb{X}_\varepsilon}$ *with high probability*. Denote $F_j(q)$ and $g_j$ the $j$-th row of $F(q)$ and $j$-th component of $q$ respectively. Consider the binary functions $$h_j(\xi,q) \doteq \left\{ \begin{array}{rl} 0 & \mbox{ if } F_j(q)\xi \le g_j(q) \\ 1 & \mbox{ otherwise} \end{array}\right.,\; j=1,\ldots, p.$$ Now, if we define $$h(\xi,q) \doteq \Prod{j=1}{p} h_j(\xi,q).$$ we have that $h$ is an $(1,p)$-boolean function since it can be expressed as a function of $p$ boolean functions, each of them involving a polynomial of degree 1. See e.g. [@alamo2009randomized Definition 7] for a precise definition of this sort of boolean functions. Suppose that we draw $N$ i.i.d. samples $q^{(i)}$, $i=1,\ldots,N$. Then, we can consider the (empirical) region ${\mathbb{X}}_N$ defined as $${\mathbb{X}}_N\doteq \set{\xi \in {\mathbb{R}}^{n_\xi}}{h(\xi,q^{(i)})=0, \, i=1,\ldots,N}.$$ It has been proved in [@alamo2009randomized Theorem 8], that if $\epsilon\in(0,0.14)$ and $N$ is chosen such that[^6] $$N \geq \frac{4.1}{\epsilon} \left( \ln\frac{21.64}{\delta} + 4.39 n_\xi \log_2 \left( \frac{8ep}{\epsilon}\right) \right)$$ then ${\mathbb{X}}_N \subseteq {\mathbb{X}_\varepsilon}$ with a probability no smaller than $1-\delta$. We notice that ${\mathbb{X}}_N$ is a convex set, which is a desirable property in an optimization framework. However, the number of required samples $N$ might be prohibitive for a real-time application. To tackle this issue, in this paper we exploit an appealing alternative approach proposed in [@alamo2019safe], and we specialize it to the problem at hand. This work proposes a probabilistic scaling approach to obtain, with given confidence, an inner approximation of the chance constrained set ${\mathbb{X}_\varepsilon}$ avoiding the computational burden due to the sample complexity raising in other strategies. The main idea behind this approach consist in first obtaining a simple initial approximation of the “shape" of the probabilistic set ${\mathbb{X}_\varepsilon}$ by exploiting simple approximating sets of the form $$x_c\oplus{\underline{\mathbb{S}}}.$$ This set is not required to have *any* guarantees of probabilistic nature. Instead, to derive such probabilistic guaranteed set, a scaling procedure is devised. In particular, an optimal scaling factor $\gamma$ is derived so that the set scaled around its center $x_c$ $$\label{SASgamma} {\underline{\mathbb{S}}}(\gamma)\doteq x_c\oplus\gamma {\underline{\mathbb{S}}}.$$ is guaranteed to be an inner approximation of ${\mathbb{X}_\varepsilon}$ with the desired confidence level $\delta$. Simple Approximating Sets ------------------------- The idea at the basis of the proposed approach is to define Simple Approximating Sets (SAS), which represent specifically defined sets with a low – and pre-defined – number of constraints. First, we note that the most straightforward way to design a candidate SAS is to draw a fixed number $N_S$ of uncertainty samples, and to construct a sampled approximation as follows:\ **1. Sampled-poly** $${\underline{\mathbb{S}}}_S=\bigcap_{i=1}^{N_S}{\mathbb{X}}_i$$ where $$\label{Xi} {\mathbb{X}}_i\doteq\Bigl\{\xi\in\Xi\;|\; F(q^{(i)})\le g(q^{(i)}), \quad i=1,\ldots,N_{S} \Bigr\}$$ Clearly, if $N_S<<N_{LT}$, the probabilistic properties of ${\underline{\mathbb{S}}}_S$ before scaling will be very bad. However, at this point we do not care, since the probabilistic scaling proposed in Section \[sec:sas-scale\] will take care of this. A second way to construct a SAS considered in this paper exploits a class of $\ell_p$-norm based sets introduced in [@dabbene2010complexity] as follows $$\mathcal{A}(x_c,P) \doteq \left\{\xi \in \mathbb{R}^{n_{\xi}}\;|\;\xi=x_c+Pz,z\in\mathcal{B}_p\right\}, \label{eq:gener_xc}$$ where $\mathcal{B}_p \subset \mathbb{R}^{n_{\xi}}$ is the unit ball in the *p* norm, $x_c$ is the center and $P=P^T\succeq 0$ is the so-called *shape* matrix. In particular, we note that for $p=1,\infty$ these sets take the form of polytopes with fixed number of facets/vertices. Hence, we introduce the following two SAS:\ **2. $\ell_1$-poly** $${\underline{\mathbb{S}}}_1=\left\{\xi\in\mathbb{R}^{n_{\xi}}\;|\;\xi=x_c+Pz,\;\|z\|_1\leq 1 \right\},$$ defined starting from a *cross-polytope*, also known as *diamond*, of order $n_{\xi}$ with $2n_{\xi}$ vertices and $2^{n_{\xi}}$ facets.\ **3. $\ell_{\infty}$-poly** $${\underline{\mathbb{S}}}_{\infty}=\left\{\xi\in\mathbb{R}^{n_{\xi}}\;|\;\xi=x_c+Pz,\;\|z\|_{\infty}\leq 1 \right\},$$ defined starting from a *hyper-cube* of dimension $n_{\xi}$ with $2^{n_{\xi}}$ vertices and $2n_{\xi}$ facets.\ Hence, the problem becomes designing the center and shape parameters $(x_c,P)$ of the set ${\underline{\mathbb{S}}}_1$ (resp. ${\underline{\mathbb{S}}}_\infty$) so that they represent in the best possible way the set ${\mathbb{X}_\varepsilon}$. To this end, we start from a *sampled design polytope* $${\mathbb{D}}= \bigcap_{i=1}^{N_D}{\mathbb{X}}_i,$$ with a fixed number of samples $N_D$, and construct the largest set ${\underline{\mathbb{S}}}_1$ (resp. ${\underline{\mathbb{S}}}_\infty$) contained in ${\mathbb{D}}$. It is easily observed that to obtain the largest $\ell_1$-poly inscribed in ${\mathbb{D}}$, we need to solve the following convex optimization problem $$\begin{aligned} \label{eq:test} \max\limits_{x_c,C} \,\,\,\,& \text{tr}(P)\\ \text{s.t.}&\quad P\succeq 0,\nonumber\\ &\quad f_i^TPz^{[j]}\leq g_i-f_i^Tx_c\nonumber\\ &\quad \quad i=1,\ldots,N_D, \quad z^{[j]}\in\mathcal{V}_1, \end{aligned}$$ where $\mathcal{V}_1 =\{z^{[1]},\ldots,z^{[2n_\xi]}\}$ are the vertices of the unit cross-polytope while the vertices of the optimal $\ell_1$-poly can then be obtained as $$\xi^{[j]}=x_c+Pz^{[j]}, \quad j=1,\ldots,2n_\xi.$$ It should be remarked that, from these vertices, one could then recover the corresponding $2^{n_\xi}$ linear inequalities, each one defining a facet of the rotated diamond. However, this procedure, besides being computationally extremely demanding (going from a vertex-description to a linear inequality description of a polytope is known to be NP hard, [@kaibel2003some]), would lead to an exponential number of linear inequalities, thus rendering the whole approach not viable. Instead, we exploit the following equivalent formulation of , see e.g. [@dabbene2010complexity] for details $${\underline{\mathbb{S}}}_1=\left\{\xi\in\mathbb{R}^{n_\xi}\;|\;\|Mx-c\|_1\leq 1\right\}$$ where $M\doteq P^{-1}$ and $c\doteq P^{-1}x_c$. From a computational viewpoint, this second approach results to be more appealing. Indeed, using a slack variable $\zeta$, it is possible to obtain the following system of $3n_\xi+1$ linear inequalities $$\left\{ \begin{array}{ll} m_i^T\xi-c_i \leq \zeta_i,& i=1,\ldots,n_\xi\\ -m_i^T\xi+c_i \leq \zeta_i,& i=1,\ldots,n_\xi\\ \zeta_i\geq 0,& i=1,\ldots,n_\xi\\ \sum_i^{n_\xi}\zeta_i\leq 1, \end{array} \right.$$ The same convex optimization problem of could be solved to define the center and the shape of the *largest* $\ell_{\infty}$-poly inscribed in ${\mathbb{D}}$. However, this would involve an exponential number of vertices $2^{n_\xi}$. To avoid this, an approach based on Farkas lemma can be adopted, exploiting again a formulation in terms of linear inequalities. The details are not reported here due to space limitations. In this second case, obtained the center $x_c$ and the rotation matrix $P$, the corresponding $\mathcal{H}$-poly has only $2n_\xi$ hyper-planes, each one representing a different linear inequality. Once the initial SAS, ${\underline{\mathbb{S}}}_S$ and the $\ell_1$- and $\ell_{\infty}$-polys, i.e. ${\underline{\mathbb{S}}}_1$ and ${\underline{\mathbb{S}}}_{\infty}$ respectively, has been evaluated in terms of linear inequalities, the probabilistic scaling approach can be applied to determine the corresponding scaling factor $\gamma$. The scaling procedure is described in details in the paper [@alamo2019safe]. For the sake of completeness, in the next subsection we recall its basic ideas and illustrate its application to the SAS case. SAS probabilistic scaling {#sec:sas-scale} ------------------------- Given a candidate SAS set, the following simple algorithm can be used to guarantee with prescribed probability $1-\delta$ that the scaled set ${\underline{\mathbb{S}}}(\gamma)$ is a good inner approximation of ${\mathbb{X}_\varepsilon}$. Given probability levels $\varepsilon$ and $\delta$, let $$N_\gamma \ge \frac{7.67}{\varepsilon} \ln\frac{1}{\delta}\text{ and } r=\left\lceil \frac{\varepsilon N_\gamma}{2}\right\rceil.$$ Draw $N_\gamma$ samples of the uncertainty $q^{(1)},\ldots,q^{(N_\gamma)}$ Solve the optimization problem $$\begin{aligned} \gamma_i \doteq &\arg\max \gamma \\ &\text{s.t.}\quad {\underline{\mathbb{S}}}(\gamma) \subseteq {\mathbb{X}}_i \nonumber\end{aligned}$$ Return the $r$-th smallest value of $\gamma_i$. A few comments are at hand regarding the algorithm above. In step 4, for each uncertainty sample $q^{(i)}$ one has to solve a convex optimization problem, which amounts at finding the largest value of $\gamma$ such that ${\underline{\mathbb{S}}}(\gamma)$ is contained in the set ${\mathbb{X}}_i$ defined in . Then, in step 6, one has to reorder the set $\{ \gamma_1, \gamma_2, \ldots, \gamma_{N_\gamma}\}$ so that the first element is the smallest one, the second element is the second smallest one, and so on and so fort, and then return the $r$-th element of the reordered sequence. The following Lemma applies to Algorithm 1. 0.3cm Given a candidate SAS set in the form ${\underline{\mathbb{S}}}(\gamma)= x_c\oplus\gamma {\underline{\mathbb{S}}}$, assume that $x_c\in{\mathbb{X}_\varepsilon}$. Then, Algorithm 1 guarantees that $${\underline{\mathbb{S}}}(\gamma)\subseteq {\mathbb{X}_\varepsilon}$$ with probability at least $1-\delta$. 0.3cm Proof to Lemma 1 is reported in Appendix. \[f:Xe2D\] Illustrating Example -------------------- To better illustrate the proposed approach, and to highlight its main features, we first consider a simple three-dimensional examples ($n_\xi=3$), with scalar uncertain linear inequalities of the form $$f(q)^T \xi \le 1$$ with $f(q)=q_1 q_2$, with $q_1\in{\mathbb{R}}$ uniformly distributed in the interval $[0.5,1.5]$ and $q_2\in{\mathbb{R}}^3$ zero-mean Gaussian distribution. Note that, for $n_\xi=3$, the $\ell_1$ and $\ell_{\infty}$-polys have $10$ and $6$ facets, respectively, irrespective to the number of design samples $N_D$ used to preliminary obtain the generic polyhedron ${\mathbb{D}}$. However, as we will see, the number of constraints $N_D$ employed to design the initial SAS plays a significant role in the final outcome of the procedure. To show this, we performed two different tests, where the number of design samples was set to $N_D=100$ and $N_D=1,000$. The results are shown in Figures \[f:diam\_hyper\_100\] and \[f:diam\_hyper\_1000\] respectively, for both the $\ell_1$ (left) and $\ell_\infty$ (right) cases. Algorithm 1 was applied in all cases with $\varepsilon=0.05$ and $\delta=10^{-6}$, leading to $N_\gamma=2,063$ and $r=103$. For allowing a better comparison, the same set of samples where considered for the evaluation of the scaling factor in all examples. These samples lead to $N_\gamma$ random hyper-planes which define a polyhedron represented (in black) in the figures. It can be observed that when $N_D$ is small, the ensuing initial $\ell_1$- (resp. $\ell_\infty$-) poly is large, and Algorithm 1 returns a scaling factor $\gamma$ which is less than one (Fig. \[f:diam\_hyper\_100\]). Hence, the probabilistic scaling produces a “deflation” of the original set so to guarantee the probabilistic constraints. Vice-versa, for large $N_D$ (Fig. \[f:diam\_hyper\_1000\]), the scaling produces an inflation, returning a value of $\gamma$ larger than one. Finally, we compared the $\ell_1$- / $\ell_\infty$- polys with the naive approach based on sampled polytope ${\underline{\mathbb{S}}}_S$. Notice that, to allow a fair comparison, we should select a number of hyper-planes comparable with the number of linear inequalities defining ${\underline{\mathbb{S}}}_1$ and ${\underline{\mathbb{S}}}_\infty$. In Fig. \[f:poly\_N10\_16JAN2020\], we represent the initial and final polytopes. Then, we also generated two additional sampled-polys with $N_S=100$ and $N_S=1,000$, i.e. equal to the number of hyper-planes used to generate the design polyhedrons ${\mathbb{D}}$ for the previous case. \ These are depicted in Figs. \[f:poly\_N100\_16JAN2020\]-\[f:poly\_N1000\_16JAN2020\] while the volumes of the different SASs are reported in Table \[tab:volume\]. ${\underline{\mathbb{S}}}$ $N_S$ $N_D$ $V$ ------------------------------------- -------- -------- ---------- -- ${\underline{\mathbb{S}}}_{10}$ $10$ $-$ $0.0091$ ${\underline{\mathbb{S}}}_{100}$ $100$ $-$ $0.0403$ ${\underline{\mathbb{S}}}_{1000}$ $1000$ $-$ $0.0526$ ${\underline{\mathbb{S}}}_1$ $-$ $100$ $0.0176$ ${\underline{\mathbb{S}}}_1$ $-$ $1000$ $0.0258$ ${\underline{\mathbb{S}}}_{\infty}$ $-$ $100$ $0.0131$ ${\underline{\mathbb{S}}}_{\infty}$ $-$ $1000$ $0.0175$ : Volume of the different SASs considered in Example 1. \[tab:volume\] UAV control over a sloped vineyard {#sec:results} ================================== The selected application involves a fixed-wing UAV performing a monitoring mission over a Dolcetto vineyard at Carpeneto, Alessandria, Italy ($44^{\circ}40'55.6''\text{N}, 8^{\circ}37'28.1''\text{E}$). The Mission Planner of ArduPilot open source autopilot has been used to identify a grid pattern with a peculiar path orientation with respect to the grapevine rows, as shown in Fig. \[f:vineyard\_WPs1\]. ![Carpeneto vineyard, Piedmont, Italy (credit: Google).[]{data-label="f:vineyard_WPs1"}](figures/vineyard_WP1){width="1\columnwidth"} The main objective is to provide proper control capabilities to a fixed-wing UAV to guarantee a fixed relative altitude with respect to the terrain of $150$ m while following the desired optimal path defined by the guidance algorithm (described in detail in [@mammarella2019waypoint]), maintaining a constant airspeed, i.e. $V_{ref}=12$ m/s. The controllability of the aircraft shall be guaranteed despite the presence of external disturbance due to a fixed-direction wind turbulence, which intensity can randomly vary among $\pm1$ m/s. For validation purpose, the longitudinal control of the UAV has been provided exploiting both OS-SMPC and the new PS-SMPC approach. In this case study, we have that the state variable are the longitudinal component of the total airspeed in body axes $u$, the angle of attack $\alpha$, the pitch angle $\theta$, the pitch rate $q$, and the altitude $h$. On the other hand, the control variables are represented by the throttle command $\Delta T$ and the elevator deflection $\delta_e$. Hence, we have $n=5$ and $m=2$ while the prediction horizon $T$ has been set equal to $15$. Consequently, setting $\varepsilon=0.05$, $\delta=10^{-6}$, we get $N_{LT}=20,604$ and $N_\gamma=2,063$. On the other hand, the sample complexity selected for generating the $\ell_1$-poly has been set equal to $N_D=100$ obtaining $(n+mT)\cdot N_{D}=3,500$ hyper-planes but only $3(n+mT)+1=107$ linear constraints implemented online. ![Zoom-in on the behavior of controlled state variables, i.e. airspeed $u$, altitude $h$ and roll angle $\phi$, obtained exploiting OS-SMPC (blue lines) and PS-SMPC (red lines) with respect to corresponding reference signals (black lines), i.e. $u_{ref}$, $h_{ref}$ and $\phi_{ref}$.[]{data-label="f:u_h_phi_CCTA"}](figures/u_h_phi_CCTA.jpg){width="1\columnwidth"} The preliminary results are represented in Fig. \[f:3D\_traj\] as 3D trajectories and in Fig. \[f:u\_h\_phi\_CCTA\] as controlled states with respect to reference signals. We can notice that both MPC schemes provide acceptable tracking capabilities, despite larger (but still acceptable) oscillations can be observed in \[f:3D\_traj\_CCTA\_PSSMPC\] when the scaled set is exploited. More interesting results are reported in Tab. \[tab:t\_comp\] in terms of maximum and average values of the computational time required to solve *online* the finite-horizon optimal control problem, evaluated for $5$ different run each. The results show a significant reduction (about 100 times lower) of the computational load when a lower complexity constraint set is employed. This makes the stochastic MPC approach not only effective from a performance viewpoint but also presumably compliant with the computational constraint coming from autopilot hardware. [ccccc]{} n. & $t_{c_{MAX_{OS}}}$ & $t_{c_{AVG_{OS}}}$ & $t_{c_{MAX_{PS}}}$ & $t_{c_{AVG_{PS}}}$\ \ 1 & $2.0959$ & $0.4178$ & $0.0966$ & $0.0087$\ 2 & $0.5394$ & $0.3291$ & $0.0215$ & $0.0088$\ 3 & $5.1215$ & $0.4546$ & $0.1065$ & $0.0045$\ 4 & $2.1497$ & $0.5434$ & $0.2628$ & $0.0086$\ 5 & $2.9411$ & $0.5626$ & $0.7221$ & $0.0190$\ \[tab:t\_comp\] Conclusions {#sec:concl} =========== In this paper, we proposed a novel approach which exploits a probabilistic scaling technique recently proposed by some of the authors to derive a novel Stochastic MPC scheme. The introduced framework exhibits a lower computational complexity, while sharing the appealing probabilistic guarantees of off-line sampling. The proof of to Lemma 1 follows from Proposition 1 in [@alamo2019safe], which guarantees that, for given $r\ge 0$, $\mathsf{Pr}\{{\underline{\mathbb{S}}}(\gamma) \subseteq {\mathbb{X}_\varepsilon}\}$ is guaranteed if the scaling is performed on a number of samples such that $$\label{ineq:N} N \geq \frac{1}{\varepsilon} \left( r-1+\ln\frac{1}{\delta}+\sqrt{2(r-1)\ln\frac{1}{\delta}}\right).$$ Since $r=\lceil \frac{\varepsilon N}{2} \rceil$, we have that $r-1 \leq \frac{\varepsilon N}{2}$. Thus, inequality is satisfied if $$\begin{aligned} N &\geq& \frac{1}{\varepsilon} \left( \frac{\varepsilon N}{2}+\ln\frac{1}{\delta}+\sqrt{\varepsilon N\ln\frac{1}{\delta}}\right)\\ &=& \frac{ N}{2}+\frac{1}{\varepsilon} \ln\frac{1}{\delta}+\sqrt{N\frac{1}{\varepsilon} \ln\frac{1}{\delta}}.\end{aligned}$$ Letting[^7] $\nabla\doteq \sqrt{N}$ and $\alpha\doteq \sqrt{\frac{1}{\varepsilon} \ln\frac{1}{\delta}}$, the above inequality rewrites $ \nabla^2-2\alpha\nabla -2\alpha^2 \ge 0$, which has unique positive solution $\nabla\ge (1+\sqrt{3})\alpha$, which rewrites as $ N \ge \frac{(1+\sqrt{3})^2}{\varepsilon} \ln\frac{1}{\delta}$. The formula in Algorithm 1 follows by observing that $(1+\sqrt{3})^2<7.67$. [^1]: $^*$This work was funded by the Italian Institute of Technology (IIT) and the Italian Ministry of Education, University and Research (MIUR) within the 2017 Projects of National Interest (PRIN 2017 N. 2017S559BB). *Corresponding author*: [email protected] (Dabbene F.) [^2]: $^1$ Institute of Electronics, Computer and Telecommunication Engineering, National Research Council of Italy, Turin, Italy, [[email protected], [email protected]]{} [^3]: $^2$ Departamento de Ingeniería de Sistemas y Automática, Universidad de Sevilla, Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain, [[email protected]]{} [^4]: [[email protected]]{} [^5]: The case where one wants to impose *hard* input constraints can be also be formulated in a similar framework, see e.g. [@matthias1]. [^6]: Note the difference under the $\log_2$ with respect to . [^7]: Note that both quantities under square root are positive.

ArXiv

--- abstract: 'In this review we present an overview of observing facilities for solar research, which are planned or will come to operation in near future. We concentrate on facilities, which harbor specific potential for solar magnetometry. We describe the challenges and science goals of future magnetic measurements, the status of magnetic field measurements at different major solar observatories, and provide an outlook on possible upgrades of future instrumentation.' author: - Lucia Kleint - Achim Gandorfer bibliography: - 'journals.bib' - 'papers.bib' date: 'Received: date / Accepted: date' title: 'Prospects of solar magnetometry - from ground and in space' --- Introduction: Complementary worlds - the advantages and drawbacks of ground-based and space-borne instruments {#se:intro} ============================================================================================================= To nighttime astronomers it usually sounds as a paradox that solar magnetic measurements are photon-starved. Detecting four polarization states ($I$, $Q$, $U$, $V$) with high enough spatial resolution (sub-arcsecond) in a relatively large field-of-view (several dozen arcsec), in a short time (less than a second), plus in sufficient wavelengths in and around a spectral line poses strict limitations on the instrumentation. Night-time telescopes simply increase their aperture in order to collect more photons, with the currently largest aperture of 10.4 m at the Gran Telescopio Canarias (Grantecan) on La Palma, Spain. This is also desirable for solar telescopes, but the technical requirements are more complicated owing to the heat management and the required corrections for seeing variations by adaptive optics (AO). The construction of the world’s largest solar telescope, the 4 m DKIST, led by the National Solar Observatory, is currently underway with planned first-light in 2019. Compared to the current largest solar telescope, the 1.6 m New Solar Telescope (NST) at the Big Bear Solar Observatory, the photon collecting area will increase by a factor of more than six. A selection of effective aperture size of solar telescopes is shown in Fig. \[telsize\]. Only recently, with the exception of the McMath-Pierce telescope, the aperture sizes surpassed 1 m. {width=".95\textwidth"} The Need for Large Apertures {#lgap} ---------------------------- A large telescope serves two main purposes: to increase the number of collected photons and to increase the spatial resolution. Both obviously can be traded for another, depending on the science question that is being investigated. There are several science questions that currently cannot be answered because of the limited available resolutions and sensitivities - temporal, spatial, and polarimetric. For example, the need for a large number of photons is evident when searching for horizontal magnetic fields, which are observed in linear polarization. These small-scale fields are important for a better understanding of the surface magnetism, e.g. whether they are created by local dynamo processes. The answer to the long-standing question of whether small-scale magnetic fields are more horizontal or vertical seems to currently depend on the method of analysis, and especially how the noise in $Q$ and $U$ is dealt with [e.g., @litesetal2008; @andres2009; @stenflo2010; @borrerokobel2011; @orozcobellot2012; @steinerrezaei2012; @valentin2013]. Because the signals of linear polarization ($Q$, $U$) are much smaller than the circular polarization ($V$) in the photospheric quiet Sun, there is a bias to detect line-of-sight fields more easily. The solution would require a higher polarimetric sensitivity, which can be reached by a larger number of collected photons. In the chromosphere, the problem becomes even more complicated because the magnetic fields are generally weaker leading to even lower polarization signals, and because the selection of spectral lines is more limited, also in terms of Landé factors, again leading to lower polarization signals. Simulations of the chromospheric 8542 Å line have shown that a noise level below $\sim 10^{-3.5}$ I$_c$, I$_c$ being the continuum intensity, is required to detect quiet Sun magnetic fields [@jaimeetal2012]. Currently, this level is rarely reached, except for observations with the Zurich Imaging Polarimeter [ZIMPOL, @Povel1995; @Gandorferpovel1997; @Gandorferetal2004; @Kleintetal2011]. ZIMPOL modulates the polarization with frequencies of several kHz, which eliminates seeing-induced crosstalk and allows to reach polarimetric sensitivities of up to 10$^{-5}$ I$_c$. However, this is done at the expense of spatial resolution, which generally is $>$1pixel$^{-1}$, and requires averaging over a large part of the spectrograph slit. Another argument to aim for large numbers of photons are coronal measurements. The free energy stored in coronal magnetic fields is believed to drive solar flares and coronal mass ejections. But so far, it cannot be measured with the desired temporal and spatial resolution. The coronal intensity $I_{\rm cor}$ is about $10^{-5} - 10^{-6} $ I$_c$ and its polarimetric signal is even smaller: $10^{-3} - 10^{-4} $ I$_{\rm cor}$. Currently, it takes 70 minutes of exposure time at the 0.4 m Solar-C Haleakala telescope with a huge pixel size of 20$\times$ 20 to obtain the desired sensitivity of 1 G for coronal measurements [@linetal2004]. Scattering polarization is another prime candidate for photon-starved observations. Because of its low amplitudes, usually only small fractions of a percent of the continuum intensity, it is notoriously difficult to observe [@stenflokeller1997]. With integration times of several minutes and averaging over most of the length of the spectrograph slit (few dozen of arcsec), one obtains a puzzling result: it seems as if the field strength of the turbulent magnetic field depends on the spectral line that was observed. For example, measurements in the atomic Sr I line consistently give higher magnetic field values on the order of 100 G [@trujillobuenoetal2006], than molecular lines, such as CN ($\sim 80$ G) or C$_2$ ($\sim 10$ G) [@shapiroetal2011; @kleintetal2011b]. An explanation based on modeling was proposed by @trujillobueno2003, suggesting that Sr I may be formed in intergranular lanes, where the magnetic field is stronger and C$_2$ in granules, where the field is weaker. But for a conclusive explanation, one would need Hanle imaging without having to average over large spatial scales. Current solar telescopes are unable to collect a sufficient number of photons for the required polarimetric sensitivity and spatial resolution on time scales before the granulation changes. Flare observations would mostly benefit from an increased temporal resolution, which again is only feasible with more photons and faster instrumentation. A scan of a spectral line with full polarimetry takes about 15 seconds at the Dunn Solar Telescope with the IBIS instrument. The evolution and motion of flaring plasma is clearly noticeable during this time. The Stokes vector and possible changes of the magnetic fields are then hard to interpret. On the other hand, increased spatial resolution will also prove highly interesting to for example compare the Sun with state-of-the-art 3D radiative MHD simulations whose resolution reaches few km [for details see @rempelschlichenmaier2011], a factor of $\sim$5 better than current observations. Open questions about the origin of the Evershed flow, the overturning convection and small-scale features, such as umbral-dots and penumbral filament structure may then be resolved. In summary, one can identify many science topics, which will greatly benefit from a 4 m telescope, e.g.: - Magnetic field measurements throughout the solar atmosphere (including the corona) - Comparisons with high-resolution simulations, including fast polarimetry at maximum resolution - Turbulent magnetic fields and studies of the solar dynamo, which require Hanle imaging - Flares, small-scale dynamics and trigger mechanisms. Space weather. Advantages and Disadvantages of Ground- and Space-based Telescopes ------------------------------------------------------------------ There are many advantages of ground-based observations, one being the possibility to upgrade and to repair instruments. The flexibility of the instrumentation and the wavelength selection, e.g. by simply replacing a prefilter, are also important. The “unlimited” telemetry, which possibly is only limited by data transfer rates and the size of data storage, is another factor when comparing e.g. the SST (several TB during a good day) with SDO, the solar space mission with the currently highest data rates, which requires a dedicated ground-station to download its $\sim$1.5 TB/day [@sdo2012]. Another advantage of ground-based observations is the possibility of real-time target changes, especially important when targeting flares or rapidly changing features, for example. Additionally, the technical possibilities to launch a 4 m telescope currently do not exist and the costs are generally much lower for ground-based observations. There are more or less obvious reasons for launching telescopes into space. The most obvious one is the absorptance of wide portions of the solar spectrum by the Earth´s atmosphere; especially in the UV, observations of the Sun (e.g. the transition region) are restricted to space-borne instruments. Less obvious, but of increasing interest is the capability of space telescopes to observe the Sun from a different vantage point, offering complementary views on our star. The Stereo mission has impressively demonstrated the advantages of stereoscopic viewing [@harrisonetal2009; @aschwanden2011]. Of particular interest are observations of the solar polar regions, which can be obtained from the ecliptic with only marginal quality (up to 7degrees, thanks to the inclination of the solar rotation axis). Views from inclined orbits my greatly improve on this shortcoming. Solar Orbiter is the first mission carrying telescopes to an orbit, which is inclined with respect to the ecliptic by up to 34 degrees. But even “normal” observations, which could [*per se*]{} be done from ground, are worth being done from space: The fact that most solar observatories are located in low to mid geographic latitudes excludes a 24 hour view on our target, a severe drawback considering the dynamic timescales of solar activity. For a limited time of up to several days, stratospheric ballooning in high geographic latitudes essentially offers uninterrupted observations, as demonstrated by the Flare Genesis Experiment [@bernasconietal2000] and by Sunrise [@bartholetal2011; @solankietal2010]. But only space guarantees uninterrupted viewing for weeks, months, or even years, with constantly high data quality and without the effects of seeing. This comes, however, to a high prize, not only literally. Space instruments generally lack the flexibility to adjust for new observing strategies (especially the choice of observed spectral bands), and are notoriously short of telemetry, not accounting for potential loss of data due to radiation induced upsets. The need to build an instrument under severe restrictions in terms of mass, volume, power allocation, and often in combination with harsh radiation and varying thermal environment makes it necessary to agree on technical compromises, which often limit the performance of the instrument. Only thanks to the unrivaled stability of the observing conditions, these instruments can - for specific observations - outperform their ground-based counterparts. A drawback for space-based missions is their long lead time and often their electronics, such as cameras, are no longer state-of-the-art by the time they launch. Because space instruments generally have a fixed configuration and often are simpler in design than their ground-based counterparts in order to be fail-safe, data pipelines for space-based instruments are better developed and more stable. The importance of this issue has been recognized and there currently is an effort in the DKIST project to develop such pipelines for ground-based projects as well. In general, ground- and space-based observations ideally complement each other, and this will always be the case. The question is not, which route - ground or space - to take. The key point is how to make optimum use of the complementary aspects of both worlds. Current and future projects in ground-based solar physics ========================================================= In this section, we review a selection of current and future optical telescopes from all over the world and their measurements of magnetic fields. Access to observing time of most European telescopes (and to the DST) is possible through the SOLARNET consortium, a European FP7 Capacities Programme. Access to US-based publicly funded telescopes (DST and later DKIST) is possible through an open proposal process. {width=".8\textwidth"} GREGOR ------ [**Design and Instrumentation.** ]{} The 1.5 m GREGOR telescope on Tenerife, Spain, is the currently largest European solar telescope [e.g., @gregor2012an and further papers in the special issue of AN Vol. 333, Issue 9]. Its design started in 1998/1999 and the commissioning in 2012. The early science phase is ongoing from 2014-2015 with access mostly limited to the consortium, which is led by the German Kiepenheuer Institute. The first open call for proposals from SOLARNET was released in March 2015, but is restricted to EU and associated countries. The design consists of a Gregory telescope with three on-axis mirrors and an image derotator further down in the image path, which is planned to be installed in November 2015. The large primary mirror suffered from several fabrication problems with the originally planned silicon carbide (Cesic) material and is now made from light-weighted Zerodur with active cooling. High-order adaptive optics, which allow diffraction-limited observations at 0.08 at 500 nm for good seeing ($r_0 \ge 10$) are integrated into the path. Three (later four) post focus instruments allow to observe a very large spectral range from 350 nm to the near Infrared (several micron) and have a field-of-view of up to 150. The instruments include a broadband imager (no polarimetry), the GREGOR Fabry Perot Interferometer [GFPI, @gregorgfpi2013], a dual collimated Fabry Perot system, and the GREGOR Infrared Spectrograph [GRIS, @gris2012AN], which can be combined with the ZIMPOL system [@gregorzimpol2014]. A stellar spectrograph is planned to be installed in the future. [**Science.** ]{}The main science goals include high-resolution photospheric observations to investigate the structure and dynamics of sunspots and of granulation, small-scale magnetic field studies to investigate the presence of a local dynamo, and the largely unexplored chromospheric magnetic field. Fig \[gregor\] shows some first results of the fine structure of magnetic features obtained from an inversion of GRIS data of a sunspot with lightbridges (AR 12049). For another example, see Lagg et al. (this issue). GRIS has already shown a good performance in the infrared. [**Future.**]{} Proposals for the first open observing season in 2016 will be solicited at the end of 2015. It is also planned to replace the M2 mirror, currently made of Cesic, by a version made of Zerodur in the next couple of years, which is expected to improve the contrast and thus the use of the AO. Polarimetric capabilities are a possible upgrade for the stellar spectrograph. SST --- [**Design and Instrumentation.** ]{} The Swedish 1-m Solar Telescope [SST, @scharmeretal2003] on La Palma, Spain, has produced some of the highest-resolution magnetograms to date, both in the chromosphere and in the photosphere. Its design consists of a fused silica lens with a clear aperture of 97 cm, a turret with two 1.4 m flat mirrors, and an evacuated tube to minimize air turbulence caused by heating. Chromatic aberration induced by the front lens is corrected by the Schupmann system, which consists of a negative lens (being passed twice by the light) and a mirror, and creates an achromatic image at the secondary focus. With only 6 mirrors in the beam before the optical table, the SST’s throughput is very high. An Echelle-Littrow spectrograph (called TRIPPEL) is available for spectroscopy, simultaneously for three spectral windows with a resolution of R$\sim$200000 [@kiselmanetal2011]. For polarimetry, SST’s main instrument is CRISP, a telecentric dual Fabry Perot system with a spectral resolution of $\sim$60 mÅ at 6302 Å, and an image scale of 0.059 pixel$^{-1}$ [@scharmeretal2008; @jaime2015]. The incoming beam is split and the short wavelengths are sent to separate fast cameras, while CRISP records the red part of the light. The polarization modulation was recently upgraded to ferroelectric liquid crystals, which change modulation state in less than 1 ms. The modulation is thus limited by the camera exposure time and readout, giving a speed of $\sim$28 ms per state, i.e. 112 ms for a full polarization cycle. After the prefilter for the Fabry Perot, a part of the light is split off to a separate camera, serving as a broad-band reference for image reconstruction. The remainder of the light passes through the two Fabry Perot interferometers and the two orthogonal beams after the polarizing beamsplitter are recorded with two separate cameras. The SST group has pioneered the use of the multi-object multi-frame blind deconvolution (MOMFBD) technique [@vannoortetal2005], which is applied on the frames already before the demodulation. Coupled with exceptionally good seeing, it produces high spatial resolution, as illustrated in Fig. \[figsst\]. {width=".7\textwidth"} [**Science.** ]{} Recent scientific results with polarimetry include the discovery of opposite polarities in the penumbra [@scharmeretal2013] and a study of the effects of umbral flashes and running penumbral waves on the chromospheric magnetic field through inversions of the chromospheric Ca II 8542 Å line [@jaimewaves2013]. Chromospheric inversions were also used to demonstrate that fibrils often, but not always trace the magnetic field lines [@jaimehector2011]. [**Future.** ]{}The next planned upgrade at the SST is the CHROMIS instrument, a Fabry Perot Interferometer to observe the Ca II H (3934 Å) and K (3968 Å) lines. While its first version is planned without polarimetry, an upgrade would be possible later and it would be interesting to investigate the rather strong scattering polarization of these lines. It would be possible to upgrade TRIPPEL for polarimetry, enabling for example rasters of He 10830 to derive chromospheric magnetic fields, but there are no immediate plans. THEMIS ------ [**Design and Instrumentation.**]{} The THEMIS telescope on Tenerife, Spain, is a 90 cm Ritchey-Chrétien reflector with an alt-az mount. Belonging to the French Centre National de la Recherche Scientifique (CNRS), THEMIS was operated by France & Italy until 2009, and by France since then, but with a lower level of funding. Its specialty is being virtually polarization-free due to its symmetric design and the polarimeter being placed next to the primary focus, enabling high-accuracy spectropolarimetry. Seeing in and around the telescope is minimized by a helium-filled tube and a specially constructed dome whose small opening coincides and co-rotates with the telescope aperture. A highly complex system of transfer optics and vertically oriented spectrographs feeds light to the instruments. Its design with many optical elements however leads to light loss, especially towards the blue wavelengths. Long-slit multi-line observations are possible with the MTR (MulTi-Raies) instrument [@arturo2000mtr], which can observe up to 6 different wavelength regions, recorded by 12 cameras in a dual-beam setup. THEMIS pioneered several new instrument concepts: The MSDP [Multi Channel Subtractive Double Pass, @mein2002], allowed to observe two spectral lines simultaneously, with several windows per spectral line and two orthogonal polarization states recorded simultaneously through an elaborate double-pass through a grating. It is no longer available at THEMIS, but a new version is operating at the Meudon Observatory. TUNIS (Tunable Universal Narrowband Imaging Spectrograph) [@arturo2010tunis; @arturo2011tunis] was developed after MSDP’s idea and expanded through a so-called Hadamard mask, encoding full spectral information in the images. After 63 measurements shifting the Hadamard mask to predefined positions, the cube of $x$, $y$, $\lambda$ can be reconstructed, making it a fast instrument compared to regular scanning spectrographs, for example. [**Science.**]{} Due to its low instrumental polarization, THEMIS is well-suited for high-precision polarimetric studies. Studies have focused on magnetic fields in prominences [e.g., @arturoetal2006; @schmiederetal2014] and the second solar spectrum [@faurobertarnaud2003; @faurobertetal2009]. @lopezariste2012mercury investigated the scattering polarization of Na in the exosphere of Mercury, which could be used in the future to study its magnetic field. [**Future.** ]{}There are no observing campaigns in 2015 and 2016 to allow refurbishing THEMIS’ full optical path to enhance the “polarization-free” feature of the telescope. It is also planned to install an AO system to improve its spatial resolution. THEMIS plans to resume observations in 2017 and stay operational until the EST comes online. {width=".8\textwidth"} BBSO/NST -------- [**Design and Instrumentation.** ]{} The Big Bear Solar Observatory (BBSO) is located in Big Bear, California on a pier inside a lake and is operated by the New Jersey Institute of Technology. Its location results in less turbulent air motions, which is very beneficial for the seeing. The 1.6 m New Solar Telescope (NST) was inaugurated in 2009 [@goodeetal2010; @caoetal2010; @goodecao2012], with the high order adaptive system following in 2013 [@varsikbbso2014]. It is currently the world’s largest solar telescope and has provided some of the highest resolution images to date. Currently, there are no calls for proposals, but some data and overviews are available on their website http://www.bbso.njit.edu/ and they plan to accept observing proposals in the near future. NST’s design is an off-axis Gregorian system with a wavelength range of 390 nm - 5 $\mu$m. Its instrumentation consists of fast imaging (e.g. Broadband Filter Imagers - BFI - for G-band and TiO and the Visible Imaging Spectrometer - VIS - for H$\alpha$ and Na D2), spectrographs (e.g. Fast Imaging Solar Spectrograph - FISS - used for He 10830 Å and Ca II 8542 Å and the Cryogenic Infrared Spectrograph - Cyra - for observations in the infrared 1 - 5 $\mu$m), and polarimetry. Polarimetric observations are carried out with the Near InfraRed Imaging Spectropolarimeter [NIRIS, @bbsoniris2012]. NIRIS is a dual Fabry Perot system in a telecentric configuration with an 85 FOV with a wavelength coverage from 1.0 to 1.7 microns. As of 2015, the instrument is in routine operations and acquires full-Stokes images in the near infrared line pair at Fe I 1564.85 nm and 1565.29 nm [**Science.** ]{} During the past few years, NST studies have focused on fast imaging (no polarimetry yet in the following science highlights). For example, they studied rather rare three-ribbon flares [@wangetal2014], sunspot oscillations [@yurchyshynetal2015], flux emergence coupled with a jet [@zengetal2013; @vargasetal2014], the eruption of a flux rope [@wangetal2015], and the connection between small-scale events in the photosphere and subsequent coronal emission [@jietal2012]. From observations of a C-flare, @zengetal2014 concluded that the He 1083 nm triplet is formed primarily by photoionization of chromospheric plasma followed by radiative recombination. A first result from NIRIS is shown in Fig. \[figniris\]. [**Future.** ]{} Spectroscopy and spectropolarimetry in the He I 1083.0 nm are the next goal of NIRIS to investigate chromospheric magnetic fields in high resolution. Benefiting from an existing Lyot filter, the NST explored the scientific potential of such observations. First-light for this NIRIS upgrade is expected in 2016. VIS is also being upgraded to dual-FPIs with a diameter of 100 mm each to provide spectroscopic/ polarimetric measurements of the lines in a wavelength range of 550-860 nm. NSO/DST ------- The National Solar Observatory (NSO) operates several telescopes, one of them being the 76 cm Dunn Solar Telescope (DST) at Sacramento Peak in New Mexico [e.g. @zirker1998]. Inaugurated in 1969, it was the world’s leading telescope for many years. [**Design and Instrumentation.** ]{} The design of the DST is very peculiar, with a turret on top of a 41.5 m tower and a 250-ton optical system suspended on a liquid mercury bearing to counteract the solar image rotation caused by the alt-az mount. With its long history, instruments at the DST were constantly upgraded and are now some of the most versatile in the world. For polarimetric measurements, the Diffraction Limited Spectropolarimeter [DLSP, @sankarasubramanian2004], the Interferometric Bidimensional Spectropolarimeter [IBIS, @cavallini2006], the Facility Infrared Spectropolarimeter [FIRS, @jaegglietal2010], or the Spectro-Polarimeter for Infrared and Optical Regions [SPINOR, @socasnavarroetal2006] can be used, covering the range from visible to infrared and from spectroscopy to imaging. IBIS is currently one of the few instruments capable of imaging polarimetry in the chromospheric Ca II 8542 Å line. The ASP and DLSP helped in the development of the highly successful SP instrument onboard Hinode. The DST led the development of solar adaptive optics, and was the first telescope to be equipped with high-order adaptive optics [@rimmeleetal2004], enabling diffraction limited studies of small-scale features. [**Science.** ]{} The DST enabled many discoveries, whose list would exceed the scope of this paper. Some examples are sunspot and penumbral oscillations [@beckersschultz1972], the first study of the subsurface structure of sunspots [@thomasetal1982], the confirmation that the penumbral magnetic field is composed of two components using the ASP [@litesetal1993], the discovery of small-scale short-lived ($\sim$minutes) horizontal magnetic fields in the internetwork [@litesetal1996] and the first magnetic map of a prominence [@casinietal2003]. Using the multi-line capabilities of SPINOR, the magnetic field of the quiet Sun [@socasnavarroetal2008], of spicules [@socasnavarroelmore2005] and its 3D structure in sunspots [@socasnavarro2005] could be studied. FIRS allowed to confirm through spectropolarimetric observations in the He 10830 Å line that the superpenumbral fibrils trace the magnetic field [@schadetal2013]. More recently, polarimetric observations by IBIS allowed to study the inclination change of the magnetic field during the formation of a penumbra [@romanoetal2014], and a coordinated observing campaign with Hinode and IRIS was led from the DST, resulting in the “best-observed X-flare” [@kleintetal2015]. In recent years, the DST carried out 3 cycles of service mode observations. While common in night-time astronomy and for satellite missions, they were a first in solar ground-based observations and a preparation for significantly more efficient DKIST observing modes. [**Future.** ]{}NSO plans to cease operations of the DST by the end of 2017 while preparing for first light with their new 4-m DKIST on Maui, but negotiations with a consortium of universities and institutes that could operate the facility after 2017 are ongoing. NLST ---- The 2 m National Large Solar Telescope (NLST) is a proposed project in India [@hasan2012]. The design studies and site selection are complete, but the construction is not funded (yet). The proposal is awaiting formal clearance from the government of India. The fabrication will take about 3.5 years and it is expected to take place from late 2016 - 2020. The project is led by the Indian Institute of Astrophysics. [**Design and Instrumentation.** ]{} The NLST’s Gregorian design is similar to the GREGOR telescope, but with fewer mirrors, leading to a higher throughput. It is designed to observe from 380-2500 nm. The planned first-light instruments include a narrow band imager and a spectropolarimeter, whose design has not been finalized yet. After four years of site surveys, Hanle near the border to Tibet was selected as primary site. At an altitude of 4500 m, it has very low water vapor and favorable weather. Currently, India is commissioning the 50 cm off-axis MAST telescope. The integration of its 19 actuator AO and of two LiNbO$_3$ Fabry Perot interferometers to observe the photospheric 6173 Å line and the chromospheric 8542 Å line is ongoing [@bayannaetal2014]. The post-focus instruments include broad band and tunable Fabry-Perot narrow band imaging instruments; a high resolution spectropolarimeter and an echelle spectrograph for night time astronomy. NVST ---- The 98.5 cm New Vacuum Solar Telescope (NVST) at Fuxian Lake is a Chinese telescope, which has recently been upgraded for high-resolution observations [@yangetal2014; @liuetal2014]. [**Design and Instrumentation.** ]{} As a vacuum telescope, it contains an entrance window, which is followed by a modified Gregorian design. It is capable of observing from 300 to 2500 nm. A new high-order adaptive optics system with 151 actuators was developed in 2015 and great care is taken to reduce the local seeing, with the location of the telescope (lakeshore), by cooling the building’s roof with a shallow pool of water, with a wind screen, and with the vacuum inside the telescope tube. There are two optical paths after the AO, one leading to three broadband channels (H-$\alpha$, TiO and G-band), which are imaged by three separate cameras, and the other to a spectrograph system, which is being upgraded to polarimetric capabilities. A polarimeter for observations of the Fe I 5324 Å  line and the line pair at 5247–5250 Å has been developed and calibrations are ongoing and a near-IR magnetograph for the 1.56 $\mu$m line is planned. [**Future.** ]{}An ambitious Chinese project is the Chinese Giant Solar Telescope, currently in the planning phase with no final design yet [@liuetal2012; @liuetal2014cgst]. Current ideas include a ring of mirrors on a 8 m diameter aperture, which would have the light-gathering power equivalent to a 5-m telescope. The telescope is planned to work mostly in the infrared (0.3 – 15 $\mu$m), because the seeing variations are lower. One goal is to measure the magnetic field in lines near 10 $\mu$m, e.g. 10.4 $\mu$m. After 5 years of site surveys, two candidate sites have been selected (one lake and one mountain site), with no final decision expected in the near future. EST --- The European Solar Telescope (EST) is a joint project of several European institutions with the goal of building a 4 m telescope on La Palma in the Canary Islands [@est2013]. Its conceptual design study was finished successfully in 2011 and it is currently in the detailed design stage (2013-2017). The construction phase might take place earliest from 2018-2023, but its funding, estimated at 150 M€, is not approved yet. [**Design and Instrumentation.** ]{} EST’s design consists of an on-axis Gregorian on an alt-az mount, similar to GREGOR. The instruments are not defined yet, but will include a broad band imager, a narrow-band tunable imager and a grating spectrograph. The advantage of the EST compared to the DKIST is that the design is favorable for polarimetric observations (though at the expense of coronal observations) and one of EST’s requirements is high-resolution spectropolarimetry simultaneously in the photosphere and in the chromosphere. The telescope Mueller matrix is unity for all wavelengths and independent of elevation and azimuth and the calibration optics are located on axis, which is the ideal configuration for polarimetry. Another feature of EST is to include multi-conjugate adaptive optics (MCAO) in the beam, which so far has not been done before, even though first MCAO tests are ongoing. Distortions of the incoming wavefront through atmospheric turbulence occur at multiple heights. Current AO systems include one deformable mirror, which allows for a good correction within the so-called isoplanatic patch. In other words, at its edges, away from the AO lockpoint, the FOV is more variable and possibly blurry. MCAO consists of multiple deformable mirrors and would allow to correct for turbulence occurring at multiple heights, giving a more uniform FOV. Because the EST is on nearly the opposite side of the world than the DKIST, a near-continuous coverage of a target would be possible. DKIST ----- {width=".4\textwidth"} {width=".55\textwidth"} The 4 m Daniel K. Inouye Solar Telescope (DKIST, formerly ATST) led by NSO will be the world’s largest solar telescope when it is commissioned in 2019. The four meter aperture is a factor of more than 6 in light gathering power compared to the currently largest telescope, the NST. With a wavelength range of 380 - 5000 nm, and possibly up to 28000 nm (28 $\mu$m) for second generation instruments, the DKIST will provide unprecedented magnetic field measurements from the photosphere to the corona up to 1.5 solar radii. [**Design and Instrumentation.** ]{} The DKIST is currently under construction on Haleakala on Maui HI. The enclosure, which was shipped from Spain to Hawaii, will be installed in 2015. Its design allows for thermal control and dust mitigation, especially important for coronal observations where any scattering through dust on the mirrors needs to be avoided. The Telescope Mount Assembly will be installed in 2016-2017 and the optics system installation and integration is planned for 2017-2018. After the instrument commissioning, first light is planned for the middle of 2019. The off-axis Gregory setup was chosen in view of the low straylight for coronal observations. However, the off-axis design results in significant polarization, variable over the day, and will need to be calibrated carefully. There will be five first-generation instruments, four of which are capable of polarimetric measurements. Contrary to many other telescopes, most of them can be operated simultaneously (with the exception of the coronal instrument Cryo-NIRSP) by splitting spectral ranges off the incoming beam. The Visible Spectropolarimeter (VISP), led by the High Altitude Observatory, consists of an Echelle spectrograph from 380 - 900 nm, and will record three spectral ranges on different cameras simultaneously. The Visible Tunable Filter (VTF), led by the Kiepenheuer Institute, is a dual Fabry Perot with coatings that permit observations from 550 - 860 nm. The Diffraction Limited Near Infrared Spectro-Polarimeter (DL-NIRSP), led by the University of Hawaii, consists of a fiber-fed multi-slit spectrograph and a selection of cameras that cover the range of 900 - 2300 nm. Additionally to these polarimetric instruments, the Visible Broadband Imager (VBI), led by NSO, will record the intensity in selected filters from 390-860 nm and the images will be speckle-reconstructed directly after their acquisition to improve the spatial resolution. The fourth polarimetric instrument is the Cryogenic Near Infrared Spectro-Polarimeter (Cryo-NIRSP), led by the University of Hawaii, which is the only instrument that does not allow beam-sharing and which will not utilize the adaptive optics. Its wavelength range 1000-5000 nm is optimized for diffraction-limited coronal observations. Polarimetry is possible up to 4000 nm. For a more complete overview of the instruments, see @Elmoreetal2014. [**Science.** ]{}The DKIST will be the largest advance in ground-based solar observations in several decades. It is expected that its high spatial and temporal resolution and its polarimetric capabilities will lead to ground-breaking discoveries, especially for chromospheric magnetic fields, turbulent magnetic fields and local dynamo mechanisms, coronal magnetism, and the photospheric fine-structure, including that of sunspots. Next Generation Instrumentation =============================== {width=".95\textwidth"} Even with the significant advances of DKIST, some desired observations will still not be possible. For example, Hanle imaging to investigate turbulent magnetic fields, or extremely fast flare polarimetry are outside the realm of any currently planned DKIST instrument. While a 4 m telescope will provide the necessary light gathering power and the spatial resolution, possibly with MCAO, it would be desirable to record the cube $[x,y,\lambda,Stokes]$ quasi-simultaneously, or at least faster than any seeing-variation ($\sim$hundred Hz). This requires a sophisticated instrument design, plus very fast detectors that can modulate and read-out faster than $\sim$100 Hz. Spatial and spectral data simultaneously ---------------------------------------- For the instrument design, there are several possibilities to obtain $[x,y,\lambda]$ simultaneously, with two examples depicted in Fig. \[xylambda\]. One option is a fiber array (integral field unit) that subdivides the focal plane into parts that are then fed into a spectrograph and re-arranged onto the detector. An example for this type of instrument is the DL-NIRSP of DKIST, which is planned to have a mode with 19200 fibers that are fed into 5 slits. A slight disadvantage is the rather limited FOV (for good spatial resolution) and thus a requirement to scan to obtain a picture of e.g. a full sunspot. Another option is to re-image the focal plane with a microlens array, in principle shrinking the pixels. They are then dispersed by a spectrograph at a small rotation angle, so that they can fit on a detector. A prefilter needs to be used to avoid an overlap of spectra of different image elements. FOV and spectral range can be traded for another. A prototype of such an instrument is currently being tested by M. van Noort at the Max Planck Institute for Solar System Research. A third option would be to employ an image slicer that separates the FOV into equal slices that are re-imaged by mirrors onto a spectrograph. Within one of these slices, the spatial information is retained, which is not the case for principles a) and b). Such an option is currently being explored for the EST [@calcinesetal2013; @calcinesetal2014]. Fast Solar Polarimeter ---------------------- A very promising project for extremely fast modulation and readout is the *Fast Solar Polarimeter (FSP)*, led by A. Feller at the Max Planck Institute for Solar System Research [@felleretal2014]. The FSP overcomes several drawbacks of ZIMPOL, while reaching similar polarimetric sensitivity. ZIMPOL’s pixels are not square (due to the masked rows used for charge shifting), and rather large, which is not ideal for high-resolution images. The overall ZIMPOL throughput is also relatively low, even with the microlenses that focus light from the masked rows to the unmasked rows. Also, ZIMPOL’s overhead is rather large (above 100%) for short exposures, which is not ideal for fast changing solar features, such as flares. The FSP tackles these problems by a different design, and still reaches frame rates high enough to sufficiently suppress seeing-induced crosstalk. It has a specially designed pnCCD sensor, which can read out at speeds up to 400 fps. This is achieved by shifting the charges of each half of the CCD to one side (split frame transfer), and by parallel readout of all sensor columns. The polarization modulation is achieved with two ferro-electric liquid crystals, which operate at the same frame rates, reaching a duty cycle of $>95\%$. A polarimetric accuracy of $10^{-4} I{_c}$ at sub-arcsec resolution can be achieved by summing images below solar evolution time scales ($\sim$ minutes). Alternatively, about 1 restored set of Stokes images can be obtained per second with a typical S/N of order 300-500 by combining some 400 single frames. This allows to apply image restoration with a cadence down to about 1 s, which to our knowledge is not possible so far with any other ground-based solar polarimeter. The first prototype of the FSP with a 256x256 pixel pnCCD was tested at the VTT (see Fig. \[fsp\]). Currently, a 1024x1024 pnCCD is in development, which will be used in a dual beam setup. First observations are planned for 2016. The FSP may be the instrument, which will allow to clarify the question of the inconsistent field strengths derived from scattering polarization observations (cf Section \[lgap\]), or to enable fast polarimetry during flares. Future space initiatives and projects with relevance to solar magnetometry ========================================================================== In this section we present a brief overview of space missions with relevance to solar magnetometry. We restrict the list to space missions, which have not yet been launched, but rather are in the preparation or even planning phase. Details of some of these missions are also mentioned in A. Lagg et al. (this issue). CLASP - Chromosheric Lyman-alpha SpectroPolarimeter --------------------------------------------------- The observation and interpretation of linear polarization caused by scattering in the solar atmosphere, and its modification by magnetic fields via the Hanle effect, has been identified as a diagnostic tool for solar magnetic fields already early [see @stenflostenholm1976]. From a technological as well as from interpretational aspects, the field of scattering polarimetry has been boosted by the advent of highly sensitive polarimeters, mainly ZIMPOL in its various evolutionary generations. Ground based observations of the second solar spectrum, both with ZIMPOL and at the THEMIS telescope, stimulated the development of new theoretical concepts and numerical tools, which brought us into a situation, where we can finally use scattering polarization as a diagnostic tool for solar magnetometry. The first attempts to measure the scattering polarization in the Lyman alpha line of hydrogen and in this way retrieve information on the physical conditions in the upper chromosphere and transition region was undertaken by Stenflo and collaborators in the late 1970s. A small polarimeter for measuring linear polarization around 121 nm [@stenfloetal1976] was installed on the russian Intercosmos 16 satellite. Although the experiment failed due to contamination issues, it was able to set an upper limit to the polarization degree, however with a large uncertainty [@stenfloetal1980]. @stenfloetal1980 claim that “The average polarization of the Lyman alpha solar limb was found to be less than 1%. It is indicated that future improved VUV polarization measurements may be a diagnostic tool for chromospheric and coronal magnetic fields and for the three-dimensional geometry of the emitting structures.” Although the essential usefulness of UV polarimetry is widely accepted, and the development of UV polarization technology is explicitly recognized in the ASTRONET Infrastructure Roadmap as a “perceived gap” (see p. 73 of the Astronet Infrastructure Roadmap 2008), until today there has been no further instrument operating in this wavelength regime. Lyman alpha polarimeters have been proposed twice as strawman instruments for ESA missions, first for the COMPASS mission proposed in response to the call for the M2 slot [@fineschietal2007], later as part of the SOLMEX mission proposed in 2008 in response to ESAs M3 call [@peteretal2012]. It is obvious that - given the explorative character of such an instrument - a demonstration of the principle including the successful interpretation of the harvested data is mandatory. To this end, by initiative of the Japanese Space Agency JAXA, together with NASA and with European contributions, the CLASP mission was established. CLASP stands for Chromospheric Lyman-alpha SpectroPolarimeter and is an explorative sounding rocket experiment with the aim of demonstrating the capabilities of UV spectropolarimetry from space [c.f. @kanoetal2012; @kuboetal2014 and references therein]. CLASP consists of a 279 mm diameter Cassegrain telescope feeding a dual-beam spectrograph equipped with a rotating half-wave plate modulator for the measurement of Stokes I, Q, and U in and close to the Lyman alpha line in a wavelength range from 121.1nm - 122.1nm. The target is the chromosphere and transition region as seen on disk radially from the solar limb. The spatial resolution is 1.5 (set by the slit width) times 2.9 (along the slit) in a field of view of 1.5 times 400. With a spectral resolution of 0.01nm, in 5 min observing time, a noise level of 0.1% can be achieved when further binning the data by averaging along the slit. A context imager with an angular resolution of 2.2 images a field of view of 550$\times$ 550 in order to identify the structures, which are observed by the spectropolarimeter. The requirements on polarimetric sensitivity and accuracy are derived from models of the expected polarization signals [c.f. @ishikawaetal2014a and references therein], based on reported estimates of the field strength in chromospheric structures [@trujillobuenoetal2011]. For 5-50 G, the calculated polarization profiles indicate that a sensitivity of 0.1% - 0.2% is necessary. While in the line core a sensitivity of 0.1% in the linear polarization is aimed for, the sensitivity in the wings of the line can be decreased to 0.5% due to the larger polarization signals there. More recent 3-D simulations of scattering polarization and the Hanle effect in MHD chromospheric models have been undertaken by @stepan2015 and @stepanetal2015. The spectropolarimeter employs a spherical constant-line-spaced grating with 3000 grooves per mm and uses the plus and minus first diffraction orders for dual beam polarimetry. Two camera mirrors feed two cameras. Two reflective polarization analyzers are employed in front of the cameras. This setup maximizes the photon efficiency and allows for high sensitivity polarimetry. To this end, all polarimetric error sources must be controlled by a thorough calibration concept and a rigorous polarimetric error budget tracking [@ishikawaetal2014]. The linear polarization is modulated by a rotating wave plate made from MgF$_2$ with a retardance of half a wave, rotating at a period of 4.8 seconds. With sixteen exposures per round, the optimum exposure time of 0.3 seconds per frame is achieved. In order to cope with the demanding false-light suppression requirements and for thermal reasons, the primary mirror acts as a cold mirror. The coating of this mirror has high reflectivity only around 121 nm (54 % as measured on test samples), but is transparent in the visible, thus transmitting most of the incoming solar energy to a thermally insulated light trap. The international effort is led by Japan, which provides the experiment structure, telescope optics, polarimeter optics, slitjaw optics, and the waveplate rotation mechanism. The sounding rocket, flight operations, CCD cameras, and avionics are under responsibility of NASA. Spain contributes by modeling of the Hanle effect, while the modeling tools for the chromosphere are developed under Norwegian responsibility. France contributes to the mission by providing the diffraction grating. At the time of writing, CLASP was scheduled for launch in September 2015. During the manuscript review process, CLASP has been launched, but no results have been announced yet. The total mission observing time was 5 min. If the mission is successful, CLASP will pave the way for the application of UV polarimetry from space. Sunrise ------- Although not being real space missions according to classical definition, stratospheric balloons offer a valid alternative for a low cost access to a near space like environment [for a recent review see @gaskinetal2014 and references therein]. Since the technological aspects of balloon borne observatories are much closer to those of space instruments than to those of ground-based facilities, we list Sunrise here. Sunrise is the largest and most complex UV/Vis solar observatory up to date, which could escape the disturbing influences of the terrestrial atmosphere. Based on a 1-m aperture optical telescope and two scientific post focus instruments in the near UV and the visible range of the solar spectrum, it was designed for highest resolution observations of solar surface magnetic fields. Sunrise is an international mission led by the Max-Planck-Institute of Solar System Research (MPS) in Göttingen, Germany, together with the German Kiepenheuer Institute of solar physics in Freiburg, a Spanish consortium under leadership of the IAC (until 2012) and the IAA (since 2013), and the High Altitude Observatory in Boulder, USA. Also involved is the Lockheed Martin Solar and Astrophysics laboratory in Palo Alto. The ballooning aspects are under responsibility of NASA’s Columbia Ballooning Facility. In two 6-day flights from ESRANGE space center near Kiruna (Sweden) to Northern Canada in Summer 2009 and in summer 2013, respectively, Sunrise could demonstrate its unique potential. Despite of some technical issues in both flights, the magnetograms recorded of the extremely quiet Sun in 2009 and of active regions in 2013 represent some of the highest resolution seeing free magnetic field and Doppler maps of the photosphere. The co-spatial brightness maps of the photosphere and low chromosphere in the near UV (down to 220nm, which is not accessible from the ground) are the highest contrast images of the solar surface ever recorded [@hirzbergeretal2010], thanks to the absence of seeing and by exploiting the contrast transfer capabilities of a 1-m aperture telescope in the near UV. In this wavelength range, the sensitivity of the brightness to temperature fluctuations is very pronounced because of the steepness of the Planck curve; this helped in mapping the temperature structure and thus identifying magnetic bright points with high sensitivity [@riethmuelleretal2010], while the magnetic field and the flows were directly measured in the visible. A number of scientific insights into the physics of small-scale solar surface magnetism resulted from these observations [see e.g. @solankietal2011 and references therein]. Sunrise contains an instrument suite, which consists of a near UV filtergraph [SuFI, @gandorferetal2011] and an imaging magnetograph [IMaX, @valentinetal2011], which are both fed with light from a 1 m aperture diameter Gregory telescope by a light distribution unit [ISLiD, @gandorferetal2011]. This unit also provides the high resolution images for a correlation tracker and wave front sensor [CWS, @berkefeldetal2011], which reduces residual pointing errors of the telescope by a fast tip tilt mirror inside the light distribution unit, and keeps control of the telescope alignment. The IMaX magnetograph is based on two key technologies: wavelength selection is done by a tunable solid state etalon, and the polarization analysis makes use of two nematic liquid crystals. Both technologies have proven to be applicable in near-space conditions and represent a big step forward in our efforts to build compact and lightweight polarimeters for space applications. The Sunrise experience thus directly influences the development of the [*Polarimetric and Helioseismic Imager*]{} onboard [*Solar Orbiter*]{} (see below) and is considered a necessary and successful predecessor. Ballooning offers unique opportunities for testing new instrumental concepts for their usage in space. Most solar instruments have reached a technological complexity, which cannot be transferred to space in one step. This applies in particular to new detector architectures and the complex electronics systems associated with them. Special newly developed sensors like the one of the Fast Solar Polarimeter are today in a prototype status for ground-based demonstration of the concept. It is only natural to expand these efforts to use such sensors in Sunrise, and maybe - after successful qualification- some day in a new space observatory. Even without being affected by seeing, very high resolution polarimeters in space suffer from the internal vibrations of the spacecraft, which cannot be completely avoided. This becomes more and more the dominant cost driver in the development of satellite platforms for solar observations. Fast detectors greatly relax the requirements on the residual pointing errors and thus will some day enable polarimetry at extremely high resolution from space. The jitter spectrum of a stratospheric balloon is typically much more demanding and thus represents a worst case validation. Therefore, if Sunrise will fly again, it would be desirable to have a prototype of a near-space FSP-type sensor on board. Solar Orbiter and its polarimetric and helioseismic imager PHI -------------------------------------------------------------- More than twenty years after the launch of SoHO, a new European solar mission will be launched to space in 2018, Solar Orbiter, a collaborative mission of ESA and NASA. Solar Orbiter is conceived and designed to clarify the magnetic coupling from the photosphere throughout the solar atmosphere into the inner heliosphere, and aims at providing answers to questions like the following: How and where do the solar wind plasma and magnetic field originate in the corona? How do solar transients drive heliospheric variability? How do solar eruptions produce energetic particle radiation that fills the heliosphere? How does the solar dynamo work and drive the connections between the Sun and the heliosphere? Carrying a suite of remote sensing instruments and an in-situ analysis package, Solar Orbiter is an integrated and unique approach to heliophysics, since it combines aspects of a space solar observatory with the characteristics of an encounter mission. Solar Orbiter is not particularly focussed on magnetometry, but the mission will offer unique opportunities to study surface fields and to probe the solar dynamo [@loeptienetal2014]. Solar Orbiter’s science can be addressed thanks to a unique orbit design: During a 3.5 year transfer orbit the spacecraft undergoes several gravity assist maneuvers (GAM) at Venus and Earth, which help the spacecraft to lose orbital energy, and thus allow [*Solar Orbiter*]{} to come close to the Sun. After the second GAM at Venus, [*Solar Orbiter*]{} begins its operational phase. From then on its orbit is in a three-to-two resonance with Venus, such that after each third orbit the inclination of the orbital plane with respect to the ecliptical plane can be increased by a further gravity assist. This particular feature gives [*Solar Orbiter*]{} access to the high latitude regions of the Sun for the first time. While the in-situ instrument suite will be operational over the full orbit, the remote sensing instruments will be used in three distinct science phases per orbit, the perihelion passage, and the phases of maximum and minimum solar latitude. During the perihelion passages [*Solar Orbiter*]{} can follow the evolution of surface structures and solar features not only from close-by, but in addition under practically unchanged geometrical viewing conditions for several days, thanks to a corotating vantage point. The [*Solar Orbiter*]{} Instrumentation can be grouped in three major packages, each consisting of several instruments: - Field Package: Radio and Plasma Wave Analyzer and Magnetometer. - Particle Package: Energetic Particle Detector and Solar Wind Plasma Analyser - Solar remote sensing instrumentation: Visible-light Imager and Magnetograph, Extreme Ultraviolet Spectrometer, EUV Imager, Coronagraph, and Spectrometer/Telescope for Imaging X-rays, Heliospheric Imager. From all instruments onboard, the visible light imager and magnetograph, called “Polarimetric and Helioseismic Imager PHI” is of highest relevance for solar surface magnetometry. Together with the other high resolution instruments it observes the same target region on the solar surface with an identical angular sampling of 0.5 arcsec per pixel. By combining its observations with the other instruments on Solar Orbiter, PHI will address the magnetic coupling between the different atmospheric layers. Extrapolations of the magnetic field observed by PHI into the Sun’s upper atmosphere and heliosphere will provide the information needed for other optical and in-situ instruments to analyze and understand the data recorded by them in a proper physical context. The instruments on Solar Orbiter are very challenging to build. No space magnetograph has ever flown in such a difficult environment, with the spacecraft following a strongly elliptical trajectory, leading to significant thermal changes in the course of an orbit. The technical description below follows closely the more complete instrument descriptions by @gandorferetal2011 and @solankietal2015a. For the sake of completeness and readability, we present a shortened version here. PHI makes use of the Doppler- and Zeeman-effects in a single spectral line of neutral iron at 617.3 nm. The physical information is decoded from two-dimensional intensity maps at six wavelength points within this line, while four polarization states at each wavelength point are measured. In order to obtain the abovementioned observables, PHI is a diffraction limited, wavelength tunable, quasi-monochromatic, polarization sensitive imager with two telescopes, which (alternatively) feed a common filtergraph and focal plane array (for illustration see Fig. \[block\]): The High Resolution Telescope (HRT) provides a restricted FOV of 16.8 arcmin squared and achieves a spatial resolution that, near the closest perihelion pass, will correspond to about 200 km on the Sun. It is designed as a decentered Ritchey-Chrétien telescope with a pupil of 140 mm diameter. The all-refractive Full Disk Telescope (FDT), with a FOV of 2.1$^\circ$ in diameter and a pixel size corresponding to 730 km (at 0.28 AU), provides a complete view of the full solar disk during all orbital phases. These two telescopes are used alternatively and their selection is made by a feed selection mechanism. Both telescopes are protected from the intense solar flux by special heat-rejecting entrance windows, which are part of the heat-shield assembly of the spacecraft. These multilayer filters have more than 80%  transmittance in a narrow notch around the science wavelength, while effectively blocking the remaining parts of the spectrum from 200 nm to the far infrared by reflection. Only a fraction of the total energy is absorbed in the window, which acts a a passive thermal element by emitting part of the thermal radiation to cold space; emission of infrared radiation into the instrument cavity is minimized by a low emissivity coating on the backside of the window (acting at the same time as an anti-reflection coating for the science wavelength). Thus the heat load into the instruments can be substantially reduced, while preserving the high photometric and polarimetric accuracy of PHI. The filtergraph unit FG uses two key technologies with heritage from IMaX onboard Sunrise: A LiNbO$_3$ solid state etalon in a telecentric configuration selects a passband of 100  mÅ width. Applying a high voltage across the crystal allows changing the refractive index of the material and its thickness, and thus tuning the passband in wavelength across the spectral line. @gensemerfarrant2014 report on the fabrication technology for these etalons, whose absolute thickness, approximately 250 $\mu$m, is controlled to $<$10 nm, maintaining a thickness uniformity of $<$1 nm over the 60 mm aperture. A 3 Å wide prefilter acts as an order sorter for the Fabry-Perot channel spectrum. The polarimetric analysis is performed by two Polarization Modulation Packages (PMP) in each of the telescopes. Each PMP consists of two nematic liquid crystal retarders, followed by a linear polarizer as an analyzer. The liquid crystal variable retarders have been successfully qualified for the use in Solar Orbiter [@alvarezherreroetal2011]. Both, the FG and the PMPs, are thermally insulated from the Optics Unit and actively temperature stabilized. The opto-mechanical arrangement is designed to operate in a wide temperature range. To this end the optics unit structure consists of a combination of AlBeMet (an aluminum-beryllium alloy) and low expansion carbon-fibre reinforced plastic. An internal image stabilization system in the HRT channel system acts on the active secondary mirror, which greatly reduces residual pointing error by the spacecraft to levels compatible with high resolution polarimetry. The error signal for the piezo-driven mirror support is derived from a correlation tracker camera inside the HRT. For details on the Image stabilization system we refer to @carmonaetal2014. The focal plane assembly is built around a 2048 by 2048 pixel Active Pixel Sensor (APS), which is specially designed and manufactured for this instrument. It delivers 10 frames per second, which are read out in synchronism with the switching of the polarization modulators. Besides all the technological complexity of the cutting-edge hardware, the most critical aspect of the instrument lies, however, in the data reduction strategy: The extremely limited telemetry rate and the large amount of scientific information retrieved from the PHI instrument demand a sophisticated on-board data reduction and necessitate to employ a non-linear, least-square, inversion technique, which numerically solves the radiative transfer equation on board. This inversion is based on the Milne-Eddington approximation and is provided by the highly adaptable onboard s/w, two powerful reprogrammable FPGAs [@fietheetal2012], and a large non-volatile data storage unit. Solar-C ------- Solar-C [@watanabe2014] is an international vision for the ultimate future solar space observatory, initiated by Japan in the wake of the highly successful Hinode mission. Solar-C is a JAXA led solar space observatory with the overarching science goal of "understanding the Sun s magnetized atmosphere from bottom to top by - understanding the dynamic structuring and mass and energy loading of the solar atmosphere - understanding the basic plasma processes at work throughout the solar atmosphere - understand the causes of solar activity, which affects our natural and technical environment" [@solankietal2015prop]. To this end, Solar-C represents an extremely powerful space observatory in a geosynchronous orbit, with a payload that significantly improves our capabilities in imaging and spectropolarimetry in the UV, visible, and near infrared with respect to what is available today or foreseen in the near future. The mission concept and strawman instrumentation has been conceived and designed by the efforts of the ISAS/JAXA Solar-C working group, and a summary of the mission has been published by @watanabe2014. A strong European Contribution to Solar-C is envisaged. A first proposal to ESA in response to the call for the M4 mission [@solankietal2015prop] was very highly ranked but was not selected in this round. Solar-C aims at probing the different temperature regimes in the solar atmosphere with unprecedented angular resolution and at simultaneously covering the entire solar atmosphere from the solar surface to the outer corona. Three telescopes, working in distinct spectral regimes, are required for this task: -SUVIT, a 1.4 m aperture UV/VIS/NIR telescope for imaging and spectropolarimetry with an order of magnitude increase in photon collecting area over the largest solar telescope in space. With its instrumentation suite, which will be described below, this telescopes covers seamlessly the solar photosphere and chromosphere. - EUVST, a 30 cm aperture VUV telescope feeding a spectrometer for imaging spectroscopy covering line formation temperatures ranging from the chromosphere to the hottest parts of the corona. Over this extended temperature range, EUVST will have a resolution and effective collecting area an order of magnitude better than available today. EUVST is based on the LEMUR instrument described by @teriacaetal2012. - HCI, a next generation high resolution extreme-ultraviolet imager with 32 cm aperture operating at high cadence in multiple spectral lines sampling the chromosphere and corona. The telescope, which is of most relevance to magnetometry, is the Solar Ultraviolet and Visible Telescope SUVIT [@suematsuetal2014solarc]. SUVIT will for the first time measure chromospheric magnetic fields from space and is expected to obtain photospheric vector fields at very high resolution with the highest sensitivity ever for subarcsec resolution. SUVIT is a telescope with an aperture of 1.4 m, providing an angular resolution of 0.1 in combination with spectropolarimetry in wavelengths ranging from 280 nm to 1080 nm. With its aperture, SUVIT would have at least 10 times higher sensitivity than any other current or planned space mission. The increase in spatial resolving power and light gathering power directly results in an order of magnitude improvement in sensitivity of magnetic flux, electric current and magnetic and kinetic energy. Another major advantage of SUVIT is that it samples a large range of line formation heights almost seamlessly. In order to achieve this, the telescope feeds several highly dedicated post-focus instruments via a common optical interface unit. This interface unit provides each instrument with a collimated beam with the required spectral bands; it also contains the common tip-tilt correction mirror for stabilizing image motion caused by the residual pointing error of the S/C. A similar strategy - although for a much narrower spectral range - has been successfully implemented in the SOT of Hinode [@suematsuetal2008]. The post-focus instrumentation consists of a tunable filtergraph (FG) with polarimetric sensitivity for the spectral range from 525 nm to 1083 nm. The baseline design employs a Lyot filter equipped with liquid crystals as variable retardation plates, which allows tuning in wavelength without any moving or rotating parts. The classical echelle-type grating spectropolarimeter (SP) covers the same wavelength range and also possesses vector magnetometric capabilities in a variety of spectral lines. In order to achieve the key science goals of retrieving the magnetic field information over the full atmospheric height, nearly simultaneous observations in several lines are mandatory. This can be achieved in two different ways: Either the grating is moved and the spectral bands are recorded in series with a single camera, or - in order to obtain strictly simultaneous recordings - three cameras are employed; in both options interference filters act as order sorters for the echelle grating. While the three camera solution - to first sight - seems more complex, it should not be forgotten, that it spares two mechanisms, which is always beneficial in terms of reliability and microvibration control. The final decision on which option to follow must be done by a careful assessment during the study phase of the mission. Photospheric and chromospheric magnetograms with simultaneous two-dimensional coverage can be achieved only by an integral field unit (IFU). Such a unit is foreseen for the SP as well, it maps a 9“ squared patch of the field of view onto three entrance slits of the SP, located parallel to the ”normal" entrance slit. It can be based on an array of optical fibers (Lin 2012) or on micro image slicers [@suematsuetal2014slicer]. The capabilities of high-resolution imaging in the near UV for providing new insights into the photosphere and chromosphere have been established by the SUFI imager onboard the Sunrise balloon-borne stratospheric observatory. IRIS has unveiled the upper chromosphere in the light of the Mg II h& k lines, uncovering the dynamic and energetic phenomena by high resolution UV spectroscopy. In the wake of these achievements, the Ultraviolet Imager and Spectropolarimeter (UBIS) will further advance those observations and provide UV filtergrams at much higher spatial resolution. In addition, expanding the capabilities of IRIS, UBIS aims at spectropolarimetry in the Mg II h & k lines, allowing for magnetometry in the upper chromosphere. To this end, UBIS consists of two sub-units, a re-imager with filter wheels, and an optional spectropolarimeter behind it. The spectropolarimeter should be considered in this stage as an optional instrument, since the potential of spectropolarimetry in the Mg II lines still needs to be thoroughly demonstrated. Recent significant progress in forward modelling suggests that the chromospheric magnetic field can be reliably diagnosed with these lines [@belluzzijavier2012]. Outlook ======= We are at an exciting stage in solar magnetic measurements and the next major missions and telescopes are within reach. By 2020, we expect our newest ground- and space-based telescopes, DKIST and Solar Orbiter, to provide unprecedented data. The 4 m ground-based DKIST, currently in construction and scheduled for first light in 2019, will improve our current spatial resolution (diffraction limit) by a factor of 2.5. This will for example allow to compare features of sizes of fractions of arcseconds in sunspots, such as umbral dots, or penumbral grains to the most advanced simulations to investigate details of sunspot dynamics, formation and decay. The polarimetric sensitivity of better than 10$^{-4} I_c$, coupled with the large aperture will hopefully allow us to solve the mystery of the orientation of small-scale magnetic fields in the quiet Sun and thus to investigate the local dynamo processes. Employing fast polarimetry will enable us to study the magnetic structure of rapidly changing features, such as flares, filament eruptions, or dynamic flows. After Solar Orbiter’s launch, currently scheduled for 2018, and its cruise phase, we will for the first time have a polarimeter on a significantly inclined orbit around the Sun allowing us to study its polar regions. We will not only be able to study the mechanism of flux transport towards the poles in more detail, but also combine the observations with a whole suite of 9 additional instruments on Solar Orbiter, giving us a more complete picture of the different solar layers. Plans for the far future include the EST, a 4 m class ground-based telescope in Europe, with a design specifically optimized for polarimetric measurements and Solar-C, the successor of the Japanese Hinode satellite, that will hopefully enable us to study chromospheric polarimetry systematically and from space for the first time, shedding light on a highly dynamic and yet not well explored solar layer. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the experts of the telescopes and instruments for their advice and information, in particular, Wenda Cao, Gianna Cauzzi, Manolo Collados, Jaime de la Cruz Rodriguez, Alex Feller, Bernard Gelly, S S Hasan, Bruce Lites, Chang Liu, Arturo Lopez Ariste, Zhong Liu, Valentin Martinez Pillet, Rolf Schlichenmaier, Alexandra Tritschler, Michiel van Noort, and Haimin Wang.

ArXiv

--- author: - 'Jin-Beom Bae,' - 'Dongmin Gang,' - and Kimyeong Lee bibliography: - 'ref-AdS5.bib' title: 'Magnetically charged $AdS_5$ black holes from class $\CS$ theories on hyperbolic 3-manifolds' --- [KIAS-P19038]{}\ Introduction and Conclusion =========================== Microscopic understanding of black hole entropy is one of the prominent success of string theory. Indeed, it is well known that string theory can provide microscopic interpretation to the Bekenstein-Hawking entropy of asymptotically flat black hole [@Strominger:1996sh]. A lot of work has been done [@Dijkgraaf:1996it; @Shih:2005uc; @David:2006yn] in order to analyze black hole entropy involving quantum corrections, based on 2d field theory approach. Meanwhile, the entropy of black hole in AdS$_3$ has been analyzed via AdS/CFT correspondence [@Maldacena:1997re], by counting microscopic states of 2d conformal field theory(CFT) [@Kraus:2006nb]. It has been believed that entropy of higher-dimensional supersymmetric black hole in AdS$_d$($d>3$) can be understood from boundary superconformal field theory(SCFT) using AdS/CFT. Recently, there has been remarkable progresses in this direction. In [@Benini:2015noa; @Benini:2015eyy], the entropy of static dyonic BPS black hole in AdS$_4 \times S^7$ is shown to agree with the topologically twisted index of 3d ABJM model [@Aharony:2008ug] with $k=1$. More recently, many works have been done regarding entropy of black holes in AdS$_5$ [@Cabo-Bizet:2018ehj; @Choi:2018hmj; @Choi:2018vbz; @Benini:2018ywd; @Honda:2019cio; @ArabiArdehali:2019tdm; @Kim:2019yrz; @Cabo-Bizet:2019osg] using refined 4d superconformal index. In this paper, our goal is to understand microscopic origin of a magnetically charged black hole [@Nieder:2000kc] in AdS$_5$ using twisted index of 4d $\mathcal{N}=2$ SCFT on $S^1 \times M_3$, where $M_3$ is a closed hyperbolic 3-manifold. The entropy of magnetically charged black holes of our interest is not easy to analyze quantitatively via localization technique, as we consider the 4d SCFT on closed hyperbolic 3-manifolds. To circumvent this technical difficulty, we suggest alternative way of computing twisted index of a certain class of 4d $\mathcal{N}=2$ SCFTs on $S^1 \times M_3$. We start at 6d $(2,0)$ theory on $S^1 \times M_3 \times \Sigma_g$, where $\Sigma_g$ denote Riemann surface with genus $g$. As we shrinking the size of $\Sigma_g$, the 6d mother theory is reduced to 4d class $\CS$ SCFT [@Gaiotto:2008cd; @Gaiotto:2009we] on $S^1 \times M_3$. On the other hand, one can arrive to a 3d $\mathcal{N}=2$ class $\CR$ SCFT [@Terashima:2011qi; @Dimofte:2011ju] on $S^1 \times \Sigma_g$ by reducing the size of $M_3$ in 6d theory. Since the 6d twisted index is invariant under continous supersymmetry preserving deformations, we expect the equality between the 4d twisted index for class $\CS$ theory and 3d twisted index for class $\CR$ theory. Fortunately, the 3d twisted index was already studied in [@Gang:2018hjd; @Gang:2019uay] using the 3d-3d relation. Thus our main claim in this paper is that one can utilize the twisted index computation to analyze the entropy of magnetically charged black hole in AdS$_5$. At large-$N$ limit, we checked our speculation using supergravity solution in [@Nieder:2000kc; @Maldacena:2000mw]. One interesting future problem is to widen our understanding of the higher-dimensional black hole entropy up to subleading order in the $1/N$ expansion. We believe that the magnetically charged black hole in AdS$_5$ is an excellent testground for it, because subleading correction of twisted index already computed in [@Gang:2018hjd; @Gang:2019uay]. It would be interesting to understand the meaning of those subleading correction at the M-theory side. Class $\CS$ theories on hyperbolic 3-manifold ============================================= Twisted index of class $\CS$ theories on 3-manifold $M_3$ --------------------------------------------------------- A 4d class $\mathcal{S}$ theory $\CT_N[\Sigma_g]$ associated to a compact Riemann surface $\Sigma_g$ of genus $g$ is defined as [@Gaiotto:2008cd; @Gaiotto:2009we] $$\begin{aligned} \begin{split} \CT_N [\Sigma_g] &:= (\textrm{4d $\mathcal{N}=2$ SCFT at the infra-red (IR) fixed point of } \\ &\qquad \textrm{a twisted compactification of 6d $A_{N-1}$ (2,0) theory along $\Sigma_g$})\;. \end{split}\end{aligned}$$ Using $SO(2)$ subgroup of $SO(5)$ R-symmetry of the 6d theory, we perform a partial twisting along $\Sigma_g$. It means that the following background gauge fields $A^{\rm background}_{SO(2)}$ coupled to the $SO(2)$ R-symmetry are turned on $$\begin{aligned} \textrm{topological twisting : }A^{\rm background}_{SO(2)} = \omega (\Sigma_g),\end{aligned}$$ where $\omega (\Sigma_g)$ is the spin connection on the Riemann surface. The topological twisting preserves 8 supercharges (16 supercharges) out of original 16 supercharges for $g > 1$ ($g=1$). For $g\geq 1$, the system is believed to flow to a non-trivial SCFT under the renormalization group. Especially when $g>1$, the $SO(5)$ R-symmetry of 6d theory is broken to $SO(2)\times SO(3)$ due to the topological twisting. The remaining symmetry can be identified with $u(1)_R \times su(2)_R$ R-symmetry of the 4d $\CT_N[\Sigma_g]$ theory. $$\begin{aligned} \begin{split} &SO(5) \textrm{ R-symmetry of 6d $A_{N-1}$ (2,0) theory } \\ & \xrightarrow{\quad \textrm{twisted compactification} \quad } SO(2)\times SO(3) \textrm{ R-symmetry of 4d $\CT_N[\Sigma_g]$ theory } \end{split} \end{aligned}$$ For sufficiently large $N$, the system does not have any emergent IR symmetry and the 4d $\CT_N[\Sigma_{g> 1}]$ is a $\CN=2$ SCFT without any (non-R) flavor symmetry. Then, the superconformal $u(1)_R$ symmetry can be identified with the compact $SO(2) \subset SO(2)\times SO(3)$.[^1] We consider following twisted index of the $\CT_N[\Sigma_g]$ theory that is defined on closed hyperbolic 3-manifold $M_3 = \mathbb{H}^3/\Gamma$. $$\begin{aligned} \begin{split} \mathcal{I}_{M_3} (\CT_{N}[\Sigma_g]) :=& \mathcal{Z}_{BPS}(\textrm{4d $\CT_{N}[\Sigma_g]$ on $S^1\times M_3$}) \\ =& \textrm{Tr} (-1)^R\; \end{split}\end{aligned}$$ Here, the trace is taken over the Hilbert-space of $\CT_N[\Sigma_g]$ on $M_3$. $\mathcal{Z}_{BPS}(\textrm{$\CT$ on $\mathbb{B}$})$ denotes the partition function of a theory $\CT$ on a supersymmetric background $\mathbb{B}$ while $R$ denote the charge of IR superconformal $u(1)_R$ R-symmetry, which is normalized as $$\begin{aligned} R (Q) = \pm 1\;, \quad \textrm{for supercharge $Q$}\;.\end{aligned}$$ Note that the topological twisting is performed along the $M_3$ using the $su(2)_R$ symmetry of the 4d SCFT, to preserve some supercharges. The twisted index can be defined for arbitrary 4d $\CN=2$ SCFTs using $su(2)_R \times u(1)_R$ R-symmetry. For general 4d $\CN=2$ SCFTs, the charge $R$ is not integer valued and thus the index is complex valued. For $\CT_N[\Sigma_g]$ theory with sufficiently large $N$, on the other hand, the twisted index is an integer because the $u(1)_R$ symmetry comes from compact $SO(5)$ R-symmetry of 6d theory. The index is invariant under the continuous supersymmetric deformations of the 4d theory, thus it only depends on the topology (not on the metric choice) of $M_3$. $$\begin{aligned} \mathcal{I}_{M_3} (\mathcal{T}_N[\Sigma_g]) \;:\; \textrm{a topological invariant of $M_3$}\;\end{aligned}$$ In the next section, we express the twisted index $\CI_{M_3}(\mathcal{T}_N[\Sigma_g])$ in terms of previously known topological invariants on 3-manifold, [*analytic torsion*]{} twisted by irreducible flat connections. Twisted Index computation using 6d picture ------------------------------------------- For a generic 4d $\CN=2$ SCFT, which we will denote as $\CT_{4d \;\CN=2}$, the computation of the twisted index $\CI_{M_3}(\CT_{4d\;\CN=2})$ on a hyperbolic manifold $M_3$ is quite challenging. Unlike 3d cases [@Nekrasov:2014xaa; @Benini:2015noa; @Benini:2016hjo; @Closset:2016arn; @Closset:2017zgf], there is no developed localization formula for the 4d twisted index on hyperbolic 3-manifolds. Since the landscape of hyperbolic 3-manifolds is much wilder than hyperbolic Riemann surface, obtaining localization formula for general $M_3$ might be very challenging task. For class $\CS$ theories $\CT_N [\Sigma_g]$, we can bypass these difficulties using 6d picture. #### A 4d-3d relation The twisted index for 4d class $\mathcal{S}$ theory $\CT_N[\Sigma_g]$ on $M_3$ can be obtained from the supersymmetric partition function of 6d $A_{N-1}$ (2,0) theory on $S^1\times M_3 \times \Sigma_g$ by shrinking the size the $\Sigma_g$: $$\begin{aligned} \begin{split} &\mathcal{Z}_{BPS}(\textrm{6d $A_{N-1}$ (2,0) theory on $S^1\times M_3\times \Sigma_g$}) \\ & \xrightarrow{\quad \Sigma_g \rightarrow 0 \quad } \mathcal{Z}_{BPS}(\textrm{4d $\CT_N[\Sigma_g]$ on $S^1\times M_3$}) \;. \end{split}\end{aligned}$$ As usual Witten index, the 6d partition function is invariant under the continuous supersymmetric deformations. One possible supersymmetric deformation is the overall size change of the compact Riemman surface $\Sigma_g$. Thus, we expect $$\begin{aligned} \begin{split} &\mathcal{Z}_{BPS}(\textrm{6d $A_{N-1}$ (2,0) theory on $S^1\times M_3\times \Sigma_g$}) \\ & = \mathcal{Z}_{BPS}(\textrm{4d $\CT_N[\Sigma_g]$ on $S^1\times M_3$}) \;. \label{6d-4d relation} \end{split}\end{aligned}$$ On the other hand, one may consider a limit where the size of $M_3$ shrinks. In the case, from the same argument above we expect $$\begin{aligned} \begin{split} &\mathcal{Z}_{BPS}(\textrm{6d $A_{N-1}$ (2,0) theory on $S^1\times M_3\times \Sigma_g$}) \\ & = \mathcal{Z}_{BPS}(\textrm{3d $\CT_N[M_3]$ on $S^1\times \Sigma_g$}) \;. \label{6d-3d relation} \end{split}\end{aligned}$$ Here $\CT_N[M_3]$ is a 3d class $\mathcal{R}$ theory associated to a hyperbolic 3-manifold $M_3$: $$\begin{aligned} \begin{split} \CT_N [M_3] &:= (\textrm{3d $\mathcal{N}=2$ SCFT at the IR fixed point of a twisted compactification } \\ &\qquad \textrm{of 6d $A_{N-1}$ (2,0) theory along $M_3$})\;. \end{split} \end{aligned}$$ In the partial topological twisting along $M_3$, we use a $SO(3)$ subgroup of $SO(5)$ R-symmetry of the 6d theory and the twisting preserves 1/4 supercharges. Therefore, the resulting low-energy theory is described by a 3d $\mathcal{N}=2$ SCFT denoted as $\CT_N[M_3]$. Combining and , we have a following equality between the 4d/3d twisted indices $$\begin{aligned} \textrm{`4d-3d relation' : }\mathcal{I}_{M_3} ( \CT_N[\Sigma_g]) = \mathcal{I}_{\Sigma_g} (\CT_{N}[M_3])\;. \label{4d-3d relation}\end{aligned}$$ Here, the 3d twisted index $\CI_{\Sigma_g} (\CT_N[M_3])$ is defined as $$\begin{aligned} \begin{split} \mathcal{I}_{\Sigma_g} (\CT_{N}[M_3]) :=& \mathcal{Z}_{BPS}(\textrm{3d $\CT_{N}[M]$ on $S^1\times \Sigma_g$}) \\ =&\textrm{Tr} (-1)^R\;, \end{split}\end{aligned}$$ where the trace is taken over the Hilbert-space of $\CT_N[M_3]$ on $\Sigma_g$. In summary, the 4d-3d relation is depicted in the diagram below : $$\boxed{ \xymatrix{ \mathcal{Z}_{BPS}(\textrm{6d $A_{N-1}$ (2,0) theory on $S^1\times M_3\times \Sigma_g$}) \ar[d]^{M_3 \rightarrow 0} \ar[dr]^{\Sigma_g \rightarrow 0} & \\ \mathcal{Z}_{BPS}(\textrm{3d $\CT_N[M_3]$ on $S^1\times \Sigma_g$}) & \mathcal{Z}_{BPS}(\textrm{4d $\CT_N[\Sigma_g]$ on $S^1\times M_3$}) } }$$ See [@Gukov:2016gkn] for previous study on the relations among supersymmetric partition functions in various dimensions originated from the same 6d picture. #### $\mathcal{I}_{\Sigma_g} ( \CT_N[M_3])$ from twisted analytic torsions on $M_3$ The quantity on the RHS of is much easier to handle than the quantity on the LHS. First, we have an explicit field theoretic description for 3d theory $\CT_N[M_3]$ for general $N \geq 2$ and closed hyperbolic 3-manifold $M_3$ [@Dimofte:2011ju; @Dimofte:2013iv; @Gang:2018wek]. Second, we have general localization formula for the 3d twisted index. Combining these developments, recently it was found that the twisted index $\mathcal{I}_{\Sigma_g}(T_N[M_3])$ is simply given as [@Gang:2018hjd; @Gang:2019uay]: $$\begin{aligned} \mathcal{I}_{\Sigma_g} (\CT_{N}[M_3]) = \sum_{\CA^\alpha \in \chi^{\rm irred}(M_3;N)} (N \times {\bf Tor}^{\alpha}_{M_3})^{g-1}\;. \label{index from torsion}\end{aligned}$$ For a technical reason, the above relation holds only for closed hyperbolic 3-manifold with vanishing $H_1 (M_3, \mathbb{Z}_N)$ [@Gang:2019uay].[^2] The summation is over irreducible $PGL(N,\mathbb{C})$ flat connections on $M_3$: $$\begin{aligned} \chi^{\rm irred}(M_3;N) := \frac{\{ \textrm{irreducible $PGL(N,\mathbb{C})$ flat-connections on $M_3$}\}}{(\textrm{gauge quotient})}\;.\end{aligned}$$ The set is finite for generic choice of $M_3$. ${\bf Tor}^{\alpha}_{M_3}$ is a topological invariant called analytic (or Ray-Singer) torsion twisted by a flat connection $\mathcal{A}^\alpha$. The topological quantity is defined as follows [@ray1971r; @gukov2008sl] $$\begin{aligned} {\bf Tor}^\alpha_{M_3} := \frac{[\det ' \Delta_1 (\mathcal{A}^\alpha)]^{1/4}}{[\det ' \Delta_0 (\mathcal{A}^\alpha)]^{3/4}}\;.\end{aligned}$$ where $\Delta_n (\mathcal{A}^\alpha)$ is a Laplacian acting on $pgl(N,\mathbb{C})$-valued $n$-form twisted by a flat connection $\mathcal{A}^\alpha$: $$\begin{aligned} \Delta_n (\mathcal{A}) = d_A * d_A * +*d_A *d_A\;, \quad d_A = d+ \mathcal{A}\wedge\;.\end{aligned}$$ One non-trivial prediction of the above relation is that the topological quantity on RHS is an integer. The integrality has been checked for various examples in [@Gang:2019uay], which gives a strong consistency check for the 3d-3d relation . From and , we finally have following simple expression for the twisted index of 4d class S theories $\CT_N[\Sigma_g]$ on closed hyperbolic 3-manifold $M_3$ with $H_1(M_3, \mathbb{Z}_N)=0$: $$\begin{aligned} \mathcal{I}_{M_3} ( \CT_N[\Sigma_g]) = \sum_{\CA^\alpha \in \chi^{\rm irred}(M_3;N)} (N \times {\bf Tor}^{\alpha}_{M_3})^{g-1}\;. \label{4d twisted index from torsions} \end{aligned}$$ Large $N$ twisted index and a magnetically charged $AdS_5$ BH ============================================================= Via AdS$_5$/CFT$_4$, the twisted index is expected to count the microstates of a magnetically charged black hole in the holographic dual AdS$_5$ gravity with [*signs*]{}: $$\begin{aligned} \begin{split} \CI_{M_3}(\CT_N[\Sigma_g]) &=d^{+}_{\rm micro} - d^{-}_{\rm micro}, \end{split}\end{aligned}$$ here $d^{+}_{\rm micro}$ and $d^{-}_{\rm micro}$ stand for the number of microstates with even $R$-charge and odd $R$-charge, respectively. Unless there is highly fine-tuned cancellation between $d^{+}_{\rm micro}$ and $d^{-}_{\rm micro}$, the twisted index can see the Bekenstein-Hawking entropy $S_{BH}$ of the black hole at large $N$: $$\begin{aligned} \begin{split} &\log \CI_{M_3}(\CT_N[\Sigma_g]) = \log (d^{+}_{\rm micro}-d^{-}_{\rm micro}) \simeq \log (d^{+}_{\rm micro}+d^{-}_{\rm micro}) \simeq S_{BH}, \;\; \\ &\textrm{unless}\;\; \bigg{|} \frac{d^{+}_{\rm micro}-d^{-}_{\rm micro}}{d^{+}_{\rm micro}+d^{-}_{\rm micro}} \bigg{|} < e^{ -\kappa N^3 }\; \textrm{at sufficiently large $N$ for some positive, finite $\kappa$}. \end{split}\end{aligned}$$ Here the equivalence relation $\simeq$ is defined as $$\begin{aligned} f(N)\simeq g(N) \;\; \textrm{if } \lim_{N\rightarrow \infty }\frac{f- g}{N^3} =0\;. \end{aligned}$$ Since the black hole is made of $N$ M5-branes, $S_{BH}$ is expected to scale as $o(N^3)$. In this section, we consider the large $N$ limit of the 4d twisted index $\CI_{M_3}(\CT_N[\Sigma_g])$ using the relation in . For $g > 1$, indeed the twisted index behaves as $N^3$ at large $N$ limit. Furthermore, we will show that the leading term of $\log \mathcal{I}_{M_3}(\CT_{N}[\Sigma_g])$ perfectly agree with the Bekenstein-Hawking entropy of the magnetically charged black hole in $AdS_5$. Full perturbative $1/N$ expansion of the index ---------------------------------------------- The $1/N$ pertubative expansion of the 3d twisted index $ \CI_{\Sigma_g}(\CT_N[M_3])$ is studied in [@Gang:2018hjd; @Gang:2019uay] using the 3d-3d relation. From the 4d-3d relation , the 4d twisted index $\CI_{\Sigma_g}(\CT_N[M_3])$ is expected to share the same $1/N$ expansion. Let us summarize the process of taking the large $N$ limit to , in order to figure out leading large $N$ terms of the $\CI_{\Sigma_g}(\CT_N[M_3])$. In the large $N$ limit, only two irreducible flat connections, $\mathcal{A}^{\rm geom}$ and $\mathcal{A}^{\rm \overline{geom}}$ are expected to give exponentially dominant contribution to the summation in . $$\begin{aligned} \begin{split} \mathcal{I}_{M_3} (T_{N}[\Sigma_g]) \xrightarrow{\qquad N\rightarrow \infty \quad } &\; (N \times {\bf Tor}^{ \textrm{geom}}_{M_3} )^{g-1} + (N \times {\bf Tor}^{ \overline{\rm geom}}_{M_3} )^{g-1} \\ &+ \textrm{(exponentially smaller terms at large $N$)}\;. \end{split}\end{aligned}$$ The two dominant flat-connections can be constructed from the unique unit hyperbolic structure on $M_3$: $$\begin{aligned} \mathcal{A}^{\rm geom}_{N} := \rho_N (\omega +i e)\;, \quad \mathcal{A}^{\overline{\rm geom} } := \rho_N (\omega- i e)\;.\end{aligned}$$ Here $\omega$ and $e$ are spin-connection and dreibein of the hyperbolic structure on $M_3$. Both of them can be thought as $so(3)$-valued 1-form on $M_3$ and they form two $PGL(2,\mathbb{C})= SO(3)_{\mathbb{C}}$ flat connections $\mathcal{A}^{\textrm{geom},\overline{\rm geom}}_{N=2} = \omega \pm i e$ related to each other by complex conjugation. $\rho_N$ is the principal embedding from $PGL(2,\mathbb{C})$ to $PGL(N, \mathbb{C})$. Using mathematical results [@muller2012asymptotics; @park2019reidemeister], we obtain following asymptotic expansion of the twisted index at large $N$ [@Gang:2018hjd; @Gang:2019uay] $$\begin{aligned} \begin{split} \mathcal{I}_{M_3} (T_{N}[\Sigma_g]) & = \left(e^{i \theta(N,M_3)} +e^{-i \theta (N,M_3)}\right) \; \\ & \ \times \exp \bigg{[}(g-1) \left( \frac{\textrm{vol}(M_3)}{6\pi } (2N^3-N-1) + \log N\right) \bigg{]} \; \\ & \ \times \exp \bigg{[}(g-1) \left( \sum_{\gamma}\sum_{m=1}^{N-1} \sum_{k=m+1}^{\infty} \log |1-e^{-k \ell_{\mathbb C}(\gamma)} | \right) \bigg{]} \; \\ &+ (\textrm{exponentially smaller terms at large $N$}) \;. \label{large N expansion} \end{split}\end{aligned}$$ The exponentially smaller terms come from contributions of irreducible flat connections other than $\mathcal{A}^{\rm geom}_{N}$ and $\mathcal{A}^{\overline{\rm geom}}_{N}$. Again, the above expansion only holds for closed hyperbolic 3-manifold with vanishing $H_1(M_3, \mathbb{Z}_N)$. $\theta (N,M_3)$ is an undetermined angle due to the relative phase differences of the contributions from two dominant flat-connections, $\mathcal{A}^{\rm geom}$ and $\mathcal{A}^{\rm \overline{geom}}$. $\sum_{\gamma}$ is summation over the nontrivial primitive conjugacy classes of the fundamental group $\pi_1(M_3)$. $\ell_{\mathbb{C}}(\gamma)$ is the complexified geodesic length of the $\gamma$, which is defined by following relation: $$\begin{aligned} \textrm{Tr} P\exp \left(-\oint_{\gamma} \CA_{N=2}^{\rm geom} \right)= e^{\frac{1}2 \ell_{\mathbb{C}}(\gamma)}+ e^{-\frac{1}2 \ell_{\mathbb{C}}(\gamma)}\;, \quad \mathfrak{Re} [\ell_{\mathbb{C}}]>0\;.\end{aligned}$$ The term $\Sigma_\gamma (\ldots)$ in the above can be decomposed into two parts $$\begin{aligned} \begin{split} &\sum_{\gamma}\sum_{m=1}^{N-1} \sum_{k=m+1}^{\infty} \log |1-e^{-k \ell_{\mathbb C}(\gamma)} | \\ &= -\mathfrak{Re} \sum_\gamma \sum_{s=1}^{\infty}\frac{1}s \left(\frac{e^{-s \ell_{\mathbb{C}}}}{1-e^{-s \ell_{\mathbb{C}}}}\right)^2 + \mathfrak{Re} \sum_\gamma \sum_{s=1}^{\infty}\frac{1}s \left(\frac{e^{-\frac{s(N+1)}{2} \ell_{\mathbb{C}}}}{1-e^{-s \ell_{\mathbb{C}}}}\right)^2 \end{split}\end{aligned}$$ The first term is $N$-independent while the second term is exponentially suppressed at large $N$. Note that the leading order behavior of the twisted index only depend on vol$(M_3)$ while subleadings depend on both vol$(M_3)$ and length spectrum $\{\ell_{\mathbb{C}}(\gamma) \}$. A magnetically charged $AdS_5$ black hole ------------------------------------------ In [@Nieder:2000kc], a magnetically charged black hole solution in 5d $\mathcal{N}=4$ gauged supergravity is numerically constructed. The black hole interpolates $$\begin{aligned} \begin{split} &\textrm{UV : $AdS_5$ with $\mathbb{R}\times M_3$ conformal boundary} \\ &\textrm{IR : $AdS_2 \times M_3$ near-horizon} \end{split}\end{aligned}$$ The 11d uplift of the solution is studied in [@Gauntlett:2007sm]. Including the internal direction, the near horizon geometry is a warped product of $AdS_2 \times M_3 \times \Sigma_g \times S^4$ [@Gauntlett:2001jj]. The Bekenstein-Hawking entropy of the $AdS_5$ black hole is universally given as [@Bobev:2017uzs] $$\begin{aligned} S_{BH} = a_{4d}\frac{\textrm{vol}(M_3)}{\pi}\;.\end{aligned}$$ Here $a_{4d}$ is the a-anomaly coefficient of the dual 4d $\mathcal{N}=2$ SCFT. For class $\CS$ theory $\CT_{N}[\Sigma_g]$, the anomaly coefficient is given by [@Gaiotto:2009gz] $$\begin{aligned} a_{4d}(\CT_{N}[\Sigma_g]) = \frac{1}3 (g-1) N^3 + \textrm{subleading},\end{aligned}$$ thus the Bekenstein-Hawking entropy for the magnetically charged black hole inside $AdS_5$ dual [@Maldacena:2000mw; @Gaiotto:2009gz] of 4d $\CT_{N}[\Sigma_g]$ read $$\begin{aligned} S_{BH} = \frac{(g-1) N^3 \textrm{vol}(M_3)}{3\pi}\;. \label{S-BH}\end{aligned}$$ Here $\textrm{vol}(M_3)$ is the volume of the 3-manifold measured using unit hyperbolic metric normalized as $$\begin{aligned} R_{\mu\nu} = -2 g_{\mu\nu}\;.\end{aligned}$$ Locally, the unit hyperbolic metric is given as $$\begin{aligned} ds^2 (\mathbb{H}^3) = d\phi^2+\sinh^2 \phi (d\theta^2+ \sin^2 \theta d\nu^2)\;.\end{aligned}$$ According to Mostow’s rigidity theorem [@mostow1968quasi], there is an unique unit hyperbolic metric on $M_3$ and the volume is actually a topological invariant. It turns out that the large $N$ leading part of the $\log \mathcal{I}_{M_3}(\CT_{N}[\Sigma_g])$ nicely matches with the Bekenstein-Hawking entropy : $$\begin{aligned} \log \mathcal{I}_{M_3}(\CT_{N}[\Sigma_g]) = \frac{(g-1) N^3 \textrm{vol}(M_3)}{3\pi} + (\textrm{subleadings})\end{aligned}$$ From , the full perturbative $1/N$ corrections can be summarized as follows $$\begin{aligned} \begin{split} &\textrm{a term with $\textrm{vol}(M_3)$ \; :\; }(g-1) \frac{\textrm{vol}(M_3)}{6\pi } (2N^3-N-1)\;, \\ &\textrm{a term involving length spectrum : } (1-g)\times \mathfrak{Re} \sum_\gamma \sum_{s=1}^{\infty}\frac{1}s \left(\frac{e^{-s \ell_{\mathbb{C}}}}{1-e^{-s \ell_{\mathbb{C}}}}\right)^2 \;, \\ &\textrm{logarithmic correction : } (g-1)\log N\;. \label{perturbative corrections} \end{split}\end{aligned}$$ Since we do not have understanding of contributions from other irreducible flat-connections, it is very difficult to determine the full non-perturbative corrections. From the above analysis, we could identify following non-perturbative correction: $$\begin{aligned} \textrm{a non-perturbative correction : }(g-1)\mathfrak{Re} \sum_\gamma \sum_{s=1}^{\infty}\frac{1}s \left(\frac{e^{-\frac{s(N+1)}{2} \ell_{\mathbb{C}}}}{1-e^{-s \ell_{\mathbb{C}}}}\right)^2\;. \label{non-perturbative correction}\end{aligned}$$ It would interesting future work to identify these subleading corrections in and from the quantum computation in the holographic dual M-theory. Especially, the logarithmic subleading correction should be reproduced from zero mode analysis on the 11d uplift of the $AdS_5$ black hole as done in [@Bhattacharyya:2012ye; @Liu:2017vbl; @Gang:2019uay] for AdS$_4$/CFT$_3$ examples. Acknowledgments {#acknowledgments .unnumbered} =============== This work was partially done while one of authors (DG) was visiting APCTP, Pohang for a workshop “Frontiers of Physics Symposium”, 13-14 May 2019. We thank APCTP for hospitality. The researches of DG and KL were supported in part by the National Research Foundation of Korea Grant NRF grant 2019R1A2C2004880 and NRF-2017R1D1A1B06034369, respectively. This work was benefited from the 2019 Pollica summer workshop, which was supported in part by the Simons Foundation (Simons Collaboration on the Non-perturbative Bootstrap) and in part by the INFN. [^1]: For small $N$, there could be accidental IR symmetries in $\CT_{N}[\Sigma_g]$ theory. [^2]: Generalization to general $M_3$ is proposed in [@Benini:2019dyp].

ArXiv

--- address: 'Lawrence Livermore National Laboratory, 7000 East Ave., Livermore, California, 94551, USA; $^\dagger$ Perceptive Software, Shawnee, KS 66226 ' author: - 'Michael P. Surh, Jess B. Sturgeon$^\dagger$, and Wilhelm G. Wolfer' bibliography: - 'CoalMethods4.bib' title: 'Void Nucleation, Growth, and Coalescence in Irradiated Metals' --- Abstract ======== A novel computational treatment of dense, stiff, coupled reaction rate equations is introduced to study the nucleation, growth, and possible coalescence of cavities during neutron irradiation of metals. Radiation damage is modeled by the creation of Frenkel pair defects and helium impurity atoms. A multi-dimensional cluster size distribution function allows independent evolution of the vacancy and helium content of cavities, distinguishing voids and bubbles. A model with sessile cavities and no cluster-cluster coalescence can result in a bimodal final cavity size distribution with coexistence of small, high-pressure bubbles and large, low-pressure voids. A model that includes unhindered cavity diffusion and coalescence ultimately removes the small helium bubbles from the system, leaving only large voids. The terminal void density is also reduced and the incubation period and terminal swelling rate can be greatly altered by cavity coalescence. Temperature-dependent trapping of voids/bubbles by precipitates and alterations in void surface diffusion from adsorbed impurities and internal gas pressure may give rise to intermediate swelling behavior through their effects on cavity mobility and coalescence. Introduction ============ Irradiation of metals has long been known to culminate in macroscopic property changes including void swelling [@CAWTHORNE:1967]. Characteristic stable voids and steady volumetric swelling develop for a range of temperatures and fluxes, independent of whether radiation bombardment damage occurs as disseminated Frenkel pairs or as small defect clusters. This can occur whether or not helium is generated along with atomic displacements. In either case, small, unstable voids, loops, and other defect clusters will develop almost immediately within the irradiated material. Their subsequent evolution determines the fluence required to create stable voids and achieve steady swelling; this so-called incubation dose includes most of the dependence on radiation environment [@GarnerWolfer:1984; @Okita:2000; @Okita:2002]. The processes that govern microstructure evolution include thermally-activated motion of small defect clusters, mutual impingement, and annihilation or coalescence reactions along with micro-chemical changes from nuclear transmutation and displacements or diffusion of pre-existing impurities. Radiation simulations should ideally encompass all of these processes. Typically, existing models have included only particular types of defects and reactions or have made other numerical approximations in order to obtain a solution. At the least, simulations of early irradiation must account for void nucleation and growth processes, since annihilation, aggregation, and cluster ripening take place concurrently. Transient and steady-state swelling behavior due to these processes have been studied recently [@Surh:2004; @Surh:2004b; @Surh:2005; @Surh:ERR]. However, only void reactions with vacancy or interstitial monomers are included in these studies. This minimal model of void nucleation gives reasonable swelling behavior as a function of temperature and flux [@Surh:2005; @Surh:ERR], viz. an observed steady swelling rate around 1$\%$/dpa in austenitic stainless steels and an important flux-effect on the measured incubation times [@Garner:1998; @Okita:2001b]. While the results are encouraging, these calculations neglect many of the processes believed to shape the microstructure. For example, the generation and aggregation of helium impurities is not explicitly modeled. Size-dependent void diffusion [@GREENWOOD:1963; @GRUBER:1967] is neglected, and thus direct void-void coalescence is not included. Dislocation loop formation, migration, and coalescence is not explicitly simulated, either. The model can be considered to combine the production and biased diffusion of small vacancy and interstitial clusters into [*effective*]{} generation and reaction rates for monomer species alone, but it is unclear a priori how a coarse-grained treatment of these processes affects microstructure evolution. It is now clear that the model must presuppose a ready supply of gas impurity atoms (e.g., oxygen and helium [@COGHLAN:1983]) to promote the formation of small voids from the radiation-induced, supersaturated vacancy population. In practice, reasonable corrections to void energies may reproduce the approximate void number density observed in irradiated steel [@Surh:ERR]. Ultimately, however, crude models for the apportionment of impurities among the defect clusters should be supplanted by a detailed accounting of multicomponent aggregation and coalescence reactions and their influence on the non-equilibrium cluster size distribution. Such problems are widely addressed in the literature, including gelation, polymerization, and formation of aerosols and precipitates in solid or fluid media [@MARCUS:1968; @GILLESPIE:1972; @LUSHNIKOV:1978; @VODE; @SMITH:1998; @BABOVSKY:1999; @GILLESPIE:2000; @GILLESPIE:2001; @EIBECK:2001; @HASELTINE:2002; @FRIESEN:2003; @LAURENZI:2003; @MUKHERJEE:2003; @ALEXOPOULOS:2004; @FILBET:2004; @SALIS:2005; @KRAFT:2005]. The numerical methods developed for such problems may also be fruitfully applied to radiation swelling. Here, a hybrid numerical approach that can encompass the full range of possible cluster compositions and cluster reactions in mean field is introduced, a Livermore Microstructure Evolution program, [*LiME*]{}. As a first application, the method is applied to the nucleation and growth of voids with a two-component distribution of cluster compositions, examining the evolution of helium-vacancy clusters [@COGHLAN:1983], while continuing to treat oxygen adsorption by simply reducing the cavity surface energy by a constant (temperature-independent) factor. The method predicts realistic swelling behavior for ferritic steel in reactor environments. As before, the void distribution function is partitioned into overlapping regions [@Surh:2004], treating small clusters with the Master Equation (ME domain) and large ones with Monte Carlo methods (MC domain). This allows self-consistent evolution of the full void population with no truncation of the size domain, no assumptions as to the critical void size or the nature of the nucleation process, and no approximations for the overall nucleation rate or duration of the nucleation stage. Monomer concentrations are included in the ME region, where they may either be treated separately by a quasi-stationary approximation or evolved along with the small clusters through coupled nonlinear reaction rate equations. The formation and evolution of dislocation loops is not explicitly modeled; network dislocations and loops are already described by a single, time-dependent density parameter rather than a detailed size distribution function [@WOLFER:1985]. However, the methods used for void evolution would be easily generalizable to other defect species and reactions, provided that suitable mean field rate coefficients are specified for their reaction rate equations. In particular, future calculations will consider the formation, unfaulting, and migration of dislocation loops; loop coalescence and annihilation; and incorporation of loops in the dislocation network. The remainder of this paper first describes the coupled, stiff, non-linear evolution equations for void nucleation, growth, and coalescence. It presents the microscopic rate theory model, gives an overview of the computational scheme, details the various numerical methods employed in the calculations, and makes a preliminary application to void nucleation in irradiated stainless steel. The simulations include vacancy, interstitial, and helium generation, aggregation and, annihilation, with or without cluster coalescence. The results are sensitive to the effects of absorbed impurity atoms on cavity surface energy. They also expose a substantial influence of small, unstable, [*mobile*]{} clusters on the formation of critical-sized voids via direct cluster-cluster coalescence. Realistic incubation and swelling behavior cannot be obtained over wide ranges of temperature and flux without including cluster mobility and coalescence. Rate Theory Model {#RTM} ================= Allowable microstructure reactions (either aggregation or annihilation) are assumed to occur whenever two defects, ${\mathbf m}$ and ${\mathbf n}$, come into contact. Within the mean field continuum approximation, the collision rate is proportional to their relative diffusivity, $D_{{\mathbf m},{\mathbf n}}$, and effective collision cross-section, $A_{{\mathbf m},{\mathbf n}}$. As before [@Surh:2004; @Surh:ERR], a bias factor $Z_{{\mathbf m},{\mathbf n}}$ includes the effect of long-range interactions [@SNIEGOWSKI:1983; @SurhWolfer:TBP] on the binary reaction rates, $K({{\mathbf m},{\mathbf n}}) \rho_{\mathbf {\mathbf m}} \rho_{\mathbf n}$, where $\rho$ are densities of reactant species ${\mathbf m}\ne {\mathbf n}$ and the rate coefficients are: $$K({{\mathbf m},{\mathbf n}}) = Z_{{\mathbf m},{\mathbf n}} A_{{\mathbf m},{\mathbf n}} D_{{\mathbf m},{\mathbf n}} \label{KEQN}$$ Note that an additional factor of 1/2 may be required when ${\mathbf m}={\mathbf n}$, to prevent double-counting of unique pairs of identical reactant particles. This factor is not explicitly shown in the definition of $K$. Microstructure defect species are limited here to self-interstitials and -vacancies, substitutional and interstitial helium, voids/bubbles, and network dislocations. Vacancy and helium monomers as well as clusters are characterized by their composition, ${\mathbf n}=(n_{vac},n_{hel})$, in a two dimensional space. Self-vacancies and interstitials are also specially identified by $(1,0)=v$ and $(-1,0)=i$, respectively; substitutional and interstitial helium by $(1,1)=vh$ and $(0,1)=h$; and network dislocations by $d$. Monomer densities evolve according to: $$\begin{aligned} {{d\rho_i}\over{dt}} = g_i & -\sum_{\mathbf m \not\in\{h,i\} } K({\mathbf m},i) \rho_{\mathbf m} \rho_i - K(d,i) \rho_{d}\rho_i \cr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {{d\rho_h}\over{dt}} = g_h & -\sum_{{\mathbf m \not\in\{ h,i\} }} K({\mathbf m},h) \rho_{{\mathbf m}}\rho_h + K(vh,i) \rho_{vh} \rho_i\cr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {{d\rho_v}\over{dt}} = g _v & - \sum_{{\mathbf m }} K({{\mathbf m},v}) \rho_{\mathbf m} \rho_v + \sum_{\mathbf m} \bigl(K({{\mathbf m}-v,v}) c_v^{[{\mathbf m}]}\bigr) \rho_{\mathbf m} \cr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% & +K({v_2,i}) \rho_{v_2} \rho_i - K({d,v}) \rho_d\rho_v + (K({d,v}) c_v^{[eq]}) \rho_d \cr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {{d\rho_{vh}}\over{dt}} = g_{vh} & -\sum_{\mathbf m} K({{\mathbf m},vh}) \rho_{\mathbf m} \rho_{vh} +K({v,h})\rho_{v} \rho_{h} \cr &+ (K({vh,v}) c_v^{[v_2h]})\rho_{v_2h} + K({v_2h,i}) \rho_{v_2h} \rho_i \label{MEQN}\end{aligned}$$ The vacancy-vacancy aggregation term, $(Z_{v,v} A_{v,v} D_{v,v}) \rho_v \rho_v$, within the first summation for ${d\rho_v}/{dt}$ in Eq. \[MEQN\] includes that two vacancies are consumed by the reaction, that there is a factor of 1/2 to prevent double-counting of unique pairs of vacancies from the population $\rho_v$, and that the relative diffusivity is twice $D_v$. The net rate is identical to that used in a previous study [@Surh:2004]. Similar considerations also apply to pairs of substitutional helium and to thermal dissociation of vacancy dimers. Cluster (${\mathbf n}\not\in \{ v,vh,h,i\}$) densities evolve as: $$\begin{aligned} {{d\rho_{\mathbf n}}\over{dt}} = g_{\mathbf n} & + \biggl \{ \sum_{{\mathbf m}\in(v,vh,h)} K({{\mathbf m},{\mathbf n}-{\mathbf m}}) \rho_{\mathbf m} \rho_{{\mathbf n}-{\mathbf m}} \enskip \bigl (1-{\delta_{\mathbf m,\mathbf n-\mathbf m}\over2}\bigr) \bigl (1-U({\mathbf m}-{\mathbf n}) \bigr) \cr & -\sum_{{\mathbf m}\in(v,vh,h)} K({{\mathbf m},{\mathbf n}}) \rho_{\mathbf m} \rho_{{\mathbf n}} \cr & +K({{\mathbf i},{\mathbf n}+v}) \rho_{\mathbf i} \rho_{{\mathbf n}+v} -K({{\mathbf i},{\mathbf n}}) \rho_{\mathbf i} \rho_{{\mathbf n}} \enskip U(\mathbf n-v) \cr & +(K({{\mathbf n},v}) c_v^{[{{\mathbf n}+v}]})\rho_{{\mathbf n}+v} -(K({{\mathbf n}-v,v}) c_v^{[{\mathbf n}]})\rho_{{\mathbf n}} \enskip U(\mathbf n-v) \biggr\} \cr &+\biggl \{ - \sum_{{\mathbf m}\not\in(i,v,h,vh)} K({{\mathbf m},{\mathbf n}}) \rho_{\mathbf m}\rho_{\mathbf n} \cr & + {1\over2}{\sum}^\prime K({{\mathbf m},{\mathbf n-\mathbf m}}) \rho_{\mathbf m} \rho_{{\mathbf n}-{\mathbf m}} \bigl (1-\enskip U({\mathbf m}-{\mathbf n})\bigr ) \biggr\} \label{RATEEQN}\end{aligned}$$ in terms of any direct generation of clusters in the radiation damage cascade, $g_{\mathbf n}$; reactions of existing clusters with monomers (in brackets) that consume or create ${\mathbf n}$-mers including thermal emission of vacancies, and cluster-cluster reactions (in the second set of brackets) that consume or create ${\mathbf n}$-mers. Factors of 1/2 in the first and last summations prevent double counting of indistinguishable reactant pairs, and $\delta_{\mathbf m,\mathbf n} = \delta_{m_{v},n_{v}}\delta_{m_{h},n_{h}}$ where the right hand side consists of the usual Kronecker deltas, $\delta_{i,j}=\begin{cases}1& i= j\\0&i\ne j \end{cases}$. The primed summation is restricted to all pairs of reactants with ${\mathbf m}, {\mathbf {n-m}} \not\in\{v,vh,h,i\}$. Defects $\mathbf n - \mathbf m$ ($\mathbf n - v$, etc.) are restricted to the domain of allowed compositions by a step function: $U({\mathbf n}) = U(n_{v})U(n_{h})$, where $U(n)=\begin{cases}1& n\geq0\\0&n<0\end{cases}$. Finally, clusters never undergo fission in this model, only the thermal emission of single vacancies. Radiation damage deposition is approximated by the creation of disseminated monomers, so $g_{{\mathbf n}}\equiv0$ for ${{\mathbf n}}\not\in\{v,vh,h,i\}$. In this case, $g_i = \phi \xi$, in terms of the atomic displacement rate, $\phi$, and the damage production efficiency, $\xi$. The total helium production is $g_h+g_{vh}$, with the ratio of interstitial to substitutional depending on the model. (Here, it is assumed that the helium is all deposited as substitutional defects.) Conservation of host atoms (including transmutation products) requires $g_v + g_{vh} \equiv g_i$. Helium impurities are added with a temperature-independent, gradual activation of $\alpha$-emitters. This is modeled for a Fe-Ni-Cr steel undergoing neutron bombardment according a two-step activation process, in analogy to the $^{58}$Ni(n,$\gamma$)$^{59}$Ni(n,$\alpha$) reaction. Model transmutation rates are treated as free parameters and are fit to the experimental helium content in HFIR-irradiated nickel [@GARNER:2003; @SCHALDACH:2003]. The parameters are $\gamma$, $\alpha$, and $\delta$ for the rates of (respectively) $^{58}$Ni(n,$\gamma$), $^{59}$Ni(n,$\alpha$), and the sum of all transmutations that consume $^{59}$Ni. In terms of the cumulative radiation dose in dpa, $x=\int \phi(t) dt$ (for radiation flux, $\phi$): $$% I copied this expression from Mathematica notebook "Time-Dependent Helium Generation", by using TeXForm {d\over{dx}}\left( \begin{array}{c} \rho _{58} \\ \rho _{59} \end{array}\right) \text{=} \left(\begin{array}{cc} -\gamma & 0 \\ +\gamma & -\delta \end{array} \right)\left( \begin{array}{c} \rho _{58} \\ \rho _{59} \end{array} \right) \label{EQ3}$$ The $^{59}$Ni content, $\rho_{59}$, is obtained from Eq. \[EQ3\] by transforming to the eigenbasis, where $\rho_A(x) = \rho_{58}(x)$ and $\rho_B(x) = \rho_{59}(x) + {\gamma\over{\gamma-\delta}} \rho_{58}(x)$ are solved, and then transforming back. The helium generation rate is given by: $${{d \rho_{He}}\over{d x}} = \alpha \rho_{59}(x) = \alpha {\gamma\over{\gamma-\delta}} \rho_{58}(0) \bigl [ e^{-\delta x} - e^{-\gamma x} \bigr ]$$ assuming that only $^{59}$Ni(n,$\alpha$) produces $\alpha$-particles. The fit parameters are $\gamma=0.0255$, $\alpha=0.0711$, and $\delta=0.297$ dpa$^{-1}$. Pristine type 316 stainless steel is approximately 14$\%$ nickel, with 68.08% of that $^{58}$Ni and with no naturally-occurring $^{59}$Ni. Other relevant materials parameters for type-316 stainless steel are listed in Table \[TableOne\]. Non-interacting diffusion (independent random walks) implies $D_{\mathbf m,\mathbf n} = D_{\mathbf m}+D_{\mathbf n}$. Defect collision cross-sections are simply given by $$\begin{aligned} A_{{\mathbf m},{\mathbf n}} &= 4\pi(r_{\mathbf m} + r_{\mathbf n}) &{\rm for }\enskip {\mathbf m}\not\in\{v,vh,h,i\} \enskip {\rm and } \enskip {\mathbf n}\not\in\{v,vh,h,i\}\cr A_{{\mathbf m},{\mathbf n}} &= r_{\mathbf m} + b &{\rm for} \enskip {\mathbf n}\in\{v,vh,h,i\} \label{XSection}\end{aligned}$$ in terms of radii for (spherical) defects, $r_{\mathbf n}=\sqrt{ {(3n_{v}\Omega)^2}\over{4\pi} }$ (except for interstitial monomers, where $r_i=r_h=r_v$). For consistency with earlier work, cross-sections involving monomers are defined using the Burgers vector magnitude in place of a monomer radius. Bias factors between voids and the four defect monomers are calculated from a mean field solution of the diffusion including stress-mediated interactions [@SurhWolfer:TBP]. The infinite series describing the image interaction [@MOONPAO:1967] is fit by a simple analytic form, while the modulus interaction [@WOLFERASHKIN:1973] is treated analytically. The numerical results are tabulated for small voids and computed as needed for larger ones. Long range void-void interactions are presently neglected, so $Z_{{\mathbf m},{\mathbf n}}=1$ for ${\mathbf m},{\mathbf n}\not\in\{v,vh,h,i\}$. In principle, the effect of any long-range interactions or net drift velocities (e.g., from external stress or temperature gradients [@COTTRELL:2002]) can be incorporated in the void-void reaction rates, so the mean field reaction kernel, $K$, has general applicability. Thermal emission from vacancy clusters is evaluated by a detailed balance argument. Equating vacancy emission and absorption for the ${\mathbf n}$-mer identifies the chemical potential, $\mu_v^{[{\mathbf n}]} = F^{[{\mathbf n}]} - F^{[{\mathbf n-v}]}$, in terms of the $\mathbf n$-mer and $({\mathbf n}-v)$-mer (i.e., void minus one vacancy) free energies. Rewriting in terms of void internal energies, $E$, and the inert gas pressure, $P$: $$c_v^{[\mathbf n]}= c_v^{[eq]} e^{(E^{[{\mathbf n}]}-E^{[{\mathbf n}-v]}-P\Omega)/kT} \label{CVEQN}$$ Gas pressure is described with a non-ideal equation of state for helium versus density and temperature [@Wolfer:1988]. No volume relaxation is included (i.e., the void volume is $n_{v}\Omega$). In the absence of surface-adsorbed impurity atoms, $E^{[{\mathbf n}]}$ is parametrized in terms of an effective surface energy, $\gamma^{[{\mathbf n}]} $, and the surface area of a spherical cavity of volume $n_v \Omega$ $$E^{[{\mathbf n}]} = \gamma^{[{\mathbf n}]} 4\pi r_{\mathbf n}^2 = \Lambda\gamma^0(T) \biggl( 1 - {0.8\over{n_v+2}}\biggr)4\pi r_{\mathbf n}^2 \label{SIZDE}$$ In the continuum limit, $\gamma^{[{\mathbf n}]} $ approaches that of a flat, clean surface, $\gamma_0(T)$, while it approaches the results of atomic calculations in the limit of small voids [@ADAMS:1989]. This surface energy is then further reduced by an constant scale factor, $\Lambda$, to reflect the presence of adsorbed oxygen impurities [@English:1987] (see Table \[TableOne\]). Finally, the emission rate is obtained from $c_v^{[{\mathbf n}]}$ and the vacancy-cluster reaction parameters for the $({\mathbf n}-v)$-mer. For straight, jogged dislocation segments, $c_v^{[d]} =c_v^{[eq]}$, the thermal equilibrium concentration. Emission rate coefficients in Eq. \[MEQN\] are represented as unary reactions, by including the defect-dependent $c_v^{[{\mathbf n}]}$ within the rate coefficient. At some maximum density, an over-pressurized bubble would begin to emit self-interstitials via loop punching [@Wolfer:1988]. Such a possibility is not considered here; instead, an artificial constraint is imposed on the helium density in a reactant cluster, $n_{h}\leq 2 n_{v}$. Any reactions that would yield a higher density are disallowed. Thermal dissociation of substitutional helium into a self-vacancy plus interstitial helium is similarly assumed to be energetically impossible at temperatures of interest. Note that self-interstitial and interstitial helium aggregation is excluded since interstitial loops are effectively part of the dislocation density model. Mixed interstitial clusters can develop in principle [@WILSON:1983]. Void diffusivity $D_{\mathbf n}=D_v/{n_{v}}^{4/3}$ for ${\mathbf n}=(n_{v},n_{h})$. This gives both the correct monomer value and size-dependence for large cluster diffusion [@GREENWOOD:1963; @GRUBER:1967], although the activation energy for void migration should more properly be that for surface diffusion. This diffusivity takes no account of the effect of reversible pinning [@NELSON:1966], or internal gas pressure on the migration [@MIKHLIN:1979], or radiation-enhanced diffusion [@ALEXANDER:1992], or, e.g., that vacancy dimer diffusion may be $D_{v_2}\simeq D_v$. Trapping at dislocations and grain boundaries are not considered. Such features would be straightforward to incorporate in the future. The dislocation model reproduces measured dislocation densities versus dose and temperature [@WOLFER:1985]. It includes separate source and annealing terms in terms of the biased flow of radiation-induced vacancies and interstitials. There is one adjustable parameter, $l$, representing a characteristic dislocation pinning length [@WOLFER:1985]. This is taken to be independent of the density of voids/bubbles in the matrix, because the pinning length in stainless steels is more determined by carbide nano-precipitates than by the density of voids/bubbles. Numerical Method ================ Overview -------- Once the temperature- and radiation-environment are specified and initial conditions for the microstructure are fixed, the Master Equations \[MEQN\] and \[RATEEQN\] completely determine the void/bubble size distribution function . Such stiff, non-linear coupled rate equations can be integrated numerically [@VODE], although this becomes intractable for a large domain of cluster sizes. The number of distinct species may be reduced by grouping similar clusters together [@GOLUBOV:2001], but the direct approach still becomes intractable for multi-dimensional distributions. Monte Carlo schemes for discrete coalescence events [@GILLESPIE:1972; @GILLESPIE:2000] can naturally encompass large voids of arbitrary composition; however, they are inefficient for simulating nucleation from sub-critical clusters. Here, the advantages of both methods are combined by partitioning the cluster composition domain into two overlapping regions. Separate sub-distributions are defined for each, labeled ME and MC for treatment by Master Equation and Monte Carlo, with $P = P^{ME} + P^{MC}$. Each sub-distribution is composed of discrete ensembles of identical clusters, represented by $(\mathbf n,\rho)$ for the paired multi-dimensional cluster composition, $\mathbf n$, and the ensemble density, $\rho$. The distribution $P^{ME}=\{ ({\mathbf n},\rho) \}^{ME}$ includes interstitials, $i$, and vacancy-helium clusters, $\mathbf n$, with $0 \leq n_{v} \leq N_{v}^{ME}$ and $0 \leq n_{h} \leq N_{h}^{ME}$. There is exactly one element for each ME species, for a total of $N^{ME}$. Only the densities of the ME elements evolve over time. A sparse, random set, $\{( {\mathbf n},\rho)\}^{MC}$, approximates $P^{MC}$ for all $0 \leq n_{v}, n_{h} <\infty$. The total number of elements, $N^{MC}$, is variable, and there may be none, one, or many MC elements for a given ${\mathbf n}$ (each with potentially different values of $\rho$). Both the densities and the compositions of the MC elements evolve with time. Such split distribution functions have been used before in a Fokker-Planck treatment of void growth [@Surh:2004], and in non-equilibrium chemistry [@HASELTINE:2002; @SALIS:2005], and plasma physics applications [@SOLOVYEV:1999]. In essence, the elements of $P^{MC}$ also constitute so-called “macroparticles”, already in wide use for non-equilibrium plasma physics problems [@MACRO]. ME-ME reactions (those processes with reactants and product among the elements of $P^{ME}$) are evaluated in a continuum approximation, using the Master Equation [@VODE]. Discrete MC-MC reactions are performed stochastically using a Markov Monte Carlo procedure [@GILLESPIE:1972]. ME-MC cross-reactions are included using either the Markov Monte Carlo method or Poisson-distributed random walks [@GILLESPIE:2000; @GILLESPIE:2001] for $P^{MC}$, and using average sink or source terms in the rate equations for $P^{ME}$. There are also procedures to transfer clusters between the two sub-distributions and to regulate the number of elements and their ensemble densities in $P^{MC}$, in order to control statistical errors and computational cost. This mixed algorithm is elaborate, so the different approaches for each of the various components are described in detail in the following sections. The material microstructure is evolved over time-step, $\tau$, by operator splitting into five stages. First, ME-MC reactions for rapidly evolving MC clusters (i.e., those with small $n_{v}$) are included (Sec. \[MEMCtext\]) with a Markov chain method. Second, the ME-MC reactions for the large, slowly-evolving clusters are evaluated by Poisson-distributed random walks in composition space for each possible reaction with ME species (Sec. \[MEMCtext\]). Third, all MC-MC reactions are evaluated with the Markov Monte Carlo method. (Sec. \[MCMCtext\]) This completes the evolution of $P^{MC}$ over $\tau$. The fourth stage integrates the ME including the [*average*]{} source and sink terms from MC defects and dislocations (Sec \[MEMEtext\]). This completes the evolution for the void/bubble $P$. At this point, clusters may be exchanged between $P^{ME}$ and $P^{MC}$, without affecting the instantaneous total $P$ in any way (Sec. \[ME2MCSec\]). This procedure may create new MC elements or eliminate existing ones, in order to control the growth of $N^{MC}$ versus time. Fifth and finally, dislocation evolution is performed using a previously-described model [@WOLFER:1985]. Overall numerical accuracy is monitored through the conservation of host and helium atoms. Operator splitting of the evolution equations causes first-order time-step errors. However, conservation errors are dominated by differences between the ME and MC treatment of reactions in Sec. \[MEMCtext\] (i.e., continuous reactions versus discrete, stochastic events). These artifact statistical fluctuations are most important at low temperatures and especially during incubation, when $N^{MC}$ is smaller, defect annihilation dominates, and little net swelling occurs. They must be carefully controlled, since the transient period represents a sort of barrier-crossing problem, with nucleation of stable voids and concomitant, self-consistent changes in the vacancy/interstitial populations as the barrier. Any artificial Monte Carlo noise must not spuriously affect the crossing into the steady-state. In other words, $N^{MC}$-dependent fluctuations in the net vacancy content must not significantly promote or inhibit void nucleation. In practice, stable cavities form naturally under the vacancy supersaturation and essentially irreversible aggregation of helium, and the volumetric swelling behavior is not unduly sensitive to $N^{MC}$ for the situations considered here. ME-ME reactions {#MEMEtext} --------------- Small defect clusters develop at high concentrations under irradiation, and so they dominate the system of reactions. However, they quickly reach a quasi-stationary distribution wherein further reactions cause little change in their densities; i.e., the majority of their reactions subsequently cancel one another. It is much more efficient to treat the net reaction rates in a continuum approximation rather than to explicitly account for individual reactions. The ordinary differential equation solver, VODE, provides an optimized treatment of stiff, nonlinear reaction equations [@VODE], given $f_n={{d\rho_n}\over{dt}}$ (Eqs. \[MEQN\] and \[RATEEQN\]) and the Jacobian, $J_{nm} = {{\partial f_n}\over{\partial \rho_m}}$ for all species. The computational cost increases rapidly with the number of coupled equations, hence the cluster domain is limited to $0\le n_{vac}\le N_{vac}^{ME}$ and $0\le n_{hel} \le N_{hel}^{ME}$. Typically, $N_{vac}^{ME}$ = 10-100 and $N_{hel}^{ME}$ = 2-10. Some terms are excluded from the Master Equation so that all reaction products remain within this finite domain. Clusters with $ 0\le n_{vac}\le N_{vac}^{ME}/2$ and $ 0\le n_{hel} \le N_{hel}^{ME}/2$ may undergo any mutual reactions, but no other ME clusters may undergo any reactions. These latter clusters are frozen in size, so their density only increases as reaction products accumulate. Frozen clusters eventually transfer to the MC distribution as described in Section \[ME2MCSec\], after which they will undergo reactions normally. With reaction constraints and separate ME and MC distributions, the vacancy Eq. \[MEQN\] becomes: $$\begin{aligned} {{d\rho_v}\over{dt}} =& \enskip g_v(t) +K({ v_2,i})\rho_{v_2}(t)\rho_i(t) \cr %- \beta_v\rho_v % remember X2 X1/2 X 2 (and last X2 is hidden in D+D) %+\theta_2 \rho_2 % remember X2 X12 X2 (and last X2 is hidden in D+D) &+ \sum_{\mathbf n\in ME} \biggl[- K({{\mathbf n},v})\rho_{\mathbf n}(t) \rho_v(t) + K({{\mathbf n},0})\rho_{\mathbf n}(t)\biggr] U\bigl({1\over2}{\mathbf N}^{ME}-{\mathbf n}\bigr) \cr %-\beta_d(t)\thinspace \rho_{d}(t_i) +\theta_d(t) \thinspace \rho_{d}(t_i) \cr & -\biggl( \overline{S^{fast}_{v}}+S^{slow}_{v}(t_0)\biggr)\thinspace\rho_v(t) + \biggl(\overline{S^{fast}_{0}} +S^{slow}_{0}(t_0)\biggr) \label{VTRUNC}\end{aligned}$$ restricting the sums over $\mathbf n\in{ME}$ to reactive defects. Eq. \[KEQN\] also parametrizes unary vacancy emission reactions as the ${\mathbf n}$-null reaction, $K({{\mathbf n},0})=Z_{\mathbf n-v,v} A_{\mathbf n-v,v}D_{\mathbf n-v,v} c_v^{[\mathbf n-v]}$. $S$ includes the external source and sink terms for reactive elements of $P^{ME}$; it accounts for ME reactions with defects in $P^{MC}$ and with dislocations. Vacancy absorption at MC defects and dislocations is parametrized by $S_v$, and vacancy emission by $S_0$. The vacancy sinks and sources include separate terms that either evolve slowly or rapidly with time. The coefficients are obtained in Sec. \[MEMCtext\]. The rest of Eq. \[MEQN\] takes similar form, with sinks $S_i$, $S_{vh}$, or $S_{h}$. (Only vacancies can be thermally emitted from defect clusters, so $S_0$ is the only source term.) Operator splitting over the time-step, $\tau$, is such that external source and sink terms $S$ are held constant as $P^{ME}$ evolves. $S$ is divided into terms that evolve slowly or rapidly with time. The bar indicates an average of the sink strength over the time-step, from $t_0$ to $t_0+\tau$, useful for rapidly evolving MC clusters, while slowly-evolving dislocations and large MC voids are simply evaluated at the beginning $t_0$ (see also Sec. \[MEMCtext\] for further details). The constrained coalescence Eq. \[RATEEQN\] becomes: $$\begin{aligned} {{d\rho_{\mathbf n}}\over{dt}} = g_{\mathbf n}(t) & +\biggl \{ \sum_{\mathbf m \in\{v,vh,h\}} K(\mathbf m,{\mathbf n}-{\mathbf m}) \rho_\mathbf m(t) \rho_{{\mathbf n}-\mathbf m}(t) \bigl (1-{\delta_{\mathbf m,\mathbf n-\mathbf m}\over2}\bigr) \biggl( 1- U\bigl({\mathbf m}-\mathbf n\bigr ) \biggr) \cr & \hskip 3truecm \times U\biggl({1\over2}{\mathbf N}^{ME}-({\mathbf n}-\mathbf m)\biggr) U\biggl({1\over2}{\mathbf N}^{ME}-\mathbf m\biggr) \cr & -\sum_{{\mathbf m}\in\{v,vh,h,i\}} K({{\mathbf m},{\mathbf n}})\rho_{{\mathbf m}}(t) \rho_{\mathbf n}(t) U\biggl({1\over2}{\mathbf N}^{ME}-{\mathbf n}\biggr) \cr & +\biggl[ K(i,{\mathbf n}+v) \rho_i(t) \rho_{{\mathbf n}+v}(t) +K({{\mathbf n}+v,0}) \rho_{{\mathbf n}+v}(t) \biggr] U\biggl({1\over2}{\mathbf N}^{ME}-({\mathbf n}+v)\biggr) \cr & -\biggl[ K(i,{\mathbf n}) \rho_i(t) \rho_{{\mathbf n}}(t) + K({{\mathbf n},0}) \rho_{{\mathbf n}}(t) \biggr] \enskip U\bigl({\mathbf n}-v\bigr) U\biggl({1\over2}{\mathbf N}^{ME}-{\mathbf n}\biggr) \cr &+ \biggl\{ - \sum_{\mathbf m\not\in\{v,vh,h,i\}} K({\mathbf m},{\mathbf n}) \rho_{\mathbf m}(t) \rho_{\mathbf n}(t) \enskip U\biggl( {{ {\mathbf N}^{ME} }\over2}-{\mathbf n}\biggr) U\biggl( {{ {\mathbf N}^{ME} }\over2}-{\mathbf m}\biggr) \cr & + {1\over2} {\sum_{\mathbf m\not\in \{v,vh,h,i\}}}^{\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace\negthinspace \prime} K({\mathbf n}-{\mathbf m},{\mathbf m}) \rho_{{\mathbf n}-{\mathbf m}}(t) \rho_{\mathbf m}(t) \enskip \biggl (1-U\bigl({\mathbf m}-\mathbf n\bigl) \biggr) \cr & \hskip 3truecm \times U\biggl({{{\mathbf N}^{ME}}\over2}-({\mathbf n}-{\mathbf {\mathbf m}})\biggr) U\biggl({{{\mathbf N}^{ME}}\over2}-{\mathbf m}\biggr) \biggr\} \cr & - \biggl(\overline{S^{fast}_{\mathbf n}}\thinspace +S^{slow}_{\mathbf n}(t_0) \biggr) \thinspace \rho_{\mathbf n}(t) \label{TRUNC}\end{aligned}$$ for clusters $\mathbf m, \mathbf n\in {\rm ME}$, and $\mathbf n \not\in\{v,vh,h,i\}$. The primed summation excludes $\mathbf n - \mathbf m \in \{v,vh,h,i\}$, since the monomer reactions are evaluated separately. $S$ includes any reactions of the $\mathbf n$-mer with the MC clusters and with dislocations. There are no reactions that consume frozen clusters, so their concentration increases with time. A subset of the disallowed reactions would produce clusters that still lie within the ME domain. These have been excluded, for simplicity and to better resemble an earlier scheme for monomer aggregation [@Surh:2004]. Specifically, a homogeneous boundary condition is imposed on the Fokker-Planck equation in Ref. [@Surh:2004], at $n=N_{vac}^{ME}$. Clusters that grow to the boundary are removed from the Master Equation treatment and accumulated separately, during which time they are prevented from changing size. This is equivalent to keeping those $N_{vac}^{ME}$-sized clusters within $P^{ME}$ but disabling all of their reactions. Frozen clusters are then intermittently transferred to $P^{MC}$, where they are no longer constrained [@Surh:2004]. Ideally, the ME domain will encompass all non-zero generation terms, $g_{\mathbf n}$, and include as many sub-critical or transient defect cluster species as possible. A relatively small domain of $N_{v}^{ME} \simeq 60$, $N_{h}^{ME} \simeq 4$ is chosen here, reflecting the computational demands that coalescence imposes. Similarly to [@Surh:2004], the solution is recorded at exponentially-increasing intervals. This time-step is irrelevant to the ME evolution itself, which advances by adaptive sub-steps. However, $\tau$ controls errors from operator splitting of the evolution equations, and it governs the creation of MC elements, as described below. Because the sink/source terms, $S$, are evaluated by a discrete MC method, they introduce a fictitious noise to the continuum rate equations. This partly manifests as step-function discontinuities in the sink strength over successive time-steps, which in turn causes transient relaxation in the concentrations of the ME species. The numerical solution tries to accurately follow the transients, potentially making the fully coupled, non-linear evolution inefficient, when large time-steps are otherwise possible. Rather than faithfully simulating these spurious transients at late times, it may be preferable to solve the monomer concentrations (Eqs. \[MEQN\],  \[VTRUNC\], etc.) in the quasi-stationary approximation after any real transient behavior (due to the abrupt onset of irradiation or other changes in environmental parameters) has concluded. Eq. \[TRUNC\] for dimers and larger clusters may then be solved while holding the monomer concentrations fixed over the time-step. In practice, after a brief transient, the results are comparable to those obtained from the full, coupled, non-linear ME solution. Transfer between ME and MC domains {#ME2MCSec} ---------------------------------- A majority of the ME elements in a small multi-dimensional domain will lie near its boundary, and so the majority of the ME cluster species will be artificially frozen. The constraints on the defect clusters are only lifted after they are transferred to $P^{MC}$. There are three desiderata to this transfer process. Foremost, it must minimize any systematic, constraint-induced errors, therefore the density of frozen clusters must be small compared to the rest of $P$. Secondly, the MC computational cost must be controlled, therefore $N^{MC}$ must be kept small. Rather than increasing $N^{MC}$ at every opportunity, frozen clusters at ${\mathbf n}\in{\rm ME}$ are allowed to accumulate until exceeding a spawning threshold density, $\rho_{\mathbf n}^{ME} > \rho_{sp}$, as in [@Surh:2004]. At the end of that time-step, a portion of the accumulated density is removed from $P^{ME}$ and transferred to a new element of $P^{MC}$, incrementing $N^{MC}$. $$({\mathbf n}, \rho_{{\mathbf n}} )^{ME} \rightarrow ({\mathbf n}, \rho_{{\mathbf n}} -\delta\rho)^{ME} + ({\mathbf n}; {{\delta\rho}}) ^{MC}$$ with the ME and MC compositions coinciding. If the accumulated $\rho_{\mathbf n}^{ME} > \rho_{sp}$ after each time-step, then the accumulating clusters are effectively never constrained. Finally, it is imperative to minimize any $N^{MC}$-dependent Monte Carlo statistical error. Individual MC elements with the largest $\rho$ will contribute the most to this error. Therefore, if $\rho_{{\mathbf n}}^{ME} \gg \rho_{sp}$ at the end of a time-step, then $\Delta N > 1$ new MC elements are created, as: $$({\mathbf n}, \rho_{{\mathbf n}} ) ^{ME} \rightarrow ({\mathbf n}, \rho_{{\mathbf n}} -\delta\rho)^{ME} + \Delta N \times({\mathbf n}; {{\delta\rho}\over{\Delta N}}) ^{MC}$$ Equivalently, MC elements with large $\rho$ may be split into identical macroparticles with smaller densities. The chosen values for $\tau$, $\rho_{sp}$, and the functional dependence of $\Delta N$ on $\delta \rho$ control the $N^{MC}$-related statistical error and computational cost for a simulation. Typically, ${\rm log}_2(\Delta N)= {\rm Int}({\rm log_{30}}(\delta\rho/\rho_{sp}))$. For example, the distribution in Fig. \[Fig2\] shows the production of many MC macroparticles containing 2-4 helium atoms; these react and form a plume that extends to $n_{v}\simeq100$. The ME domain used in this example also includes frozen cluster species with 5-9 helium; these species have not yet reached the threshold density. They eventually spawn MC elements, but at a much slower rate than for the near-critical sizes of 2-4 helium. Even at this early time, the total density of constrained ME clusters is small compared to the MC population so constraint errors are minimized. Since $\rho_{sp}$ cannot be made arbitrarily small in practice, it is useful to add a second transfer mechanism. When a pre-existing MC element at $\mathbf n$ falls inside the frozen ME domain, the change: $$({\mathbf n},\rho_{{\mathbf n}} )^{ME}+({\mathbf n},\rho_{{\mathbf n}} ) ^{MC} \rightarrow ({\mathbf n},\rho_{{\mathbf n}} -\delta\rho)^{ME}+({\mathbf n},\rho_{{\mathbf n}} +\delta\rho)^{MC}$$ leaves the total distribution unchanged. $N^{MC}$ remains constant, so the calculation remains tractable. In practice, the maximum amount $\delta\rho\leq \rho_{\mathbf n}$ is transferred until the receiving MC element reaches a cutoff density, $\rho_{\mathbf n}^{MC}+\delta \rho \leq \rho_{max}$ (where typically, $\rho_{max}\simeq 2\rho_{sp}$ to $10\rho_{sp}$.) The cutoff prevents over-weighting of individual Monte Carlo elements so as to control the statistical error. At low temperatures, a very high density of small bubbles can coexist with a moderate density of large, low-pressure voids. Such distributions are most efficiently treated by making $\rho_{max}$ size-dependent, so that the maximum macroparticle densities are high in the region of bubbles, but low in the region of voids. Macroparticles can freely wander between the two regions. Accordingly, if macroparticle $A$ moves to a region where $\rho_A>\rho_{max}$, it may be split into two identical parts; or if two MC elements at the same coordinate have $\rho_A+\rho_B < \rho_{max}$, they may be united into one. In problems of reversible nucleation and growth, small MC clusters may shrink and disappear. It is computationally inefficient to follow unstable clusters by Monte Carlo methods. Accordingly, macroparticles of the smallest vacancy clusters (with both $n_{vac}<N_{vac}^{min}<N_{vac}^{ME}$ and $n_{hel}=0$) are deleted at the end of each time-step and their density returned to the corresponding element of $P_{ME}$. (The numerical solution of the ME automatically accommodates any subsequent transients by adjusting its internal time-steps.) The minimum MC size should be large enough that macroparticles at the threshold only rarely shrink to monomer sizes during the interval $\tau$. It should also be far enough from $N_{vac}^{ME}/2$ that the cycle ME$\rightarrow$MC$\rightarrow$ME (involving creation of a new macroparticle, shrinkage of the constituent clusters, and transfer of that element back to $P^{ME}$) is infrequent. In practice, ${\mathbf N}^{min}={\mathbf N}^{ME}/4$ is used, and these two criteria are accomodated by taking the largest possible ${\mathbf N}^{ME}$. Helium clusters are never returned from MC to ME distribution; helium emission is not permitted, so the clusters will only grow along the helium axis. In the examples considered here, all ME clusters are sub-critical for $N_{vac}^{ME}\simeq60$, so that newly-created MC particles frequently shrink and are annihilated. This is especially true at low temperatures, when the proliferation of small voids favors vacancy/interstitial recombination. Here, this “rare event problem” for nucleation of stable voids from small vacancy clusters is at least improved from conventional kinetic Monte Carlo methods, where even the monomers would be treated stochastically. Ultimately, direct application shows this mixed scheme is suitable for radiation damage calculations to high doses. MC-MC reactions {#MCMCtext} --------------- Coalescence problems are frequently treated by a Markov Monte Carlo method [@GILLESPIE:1972]. A straightforward approach defines a finite volume, $V$, containing $N$ (i.e., $N^{MC}$) discrete clusters of sizes $\{\mathbf n\}$ that stochastically evolve to a new $N - 1$ population $\{{\mathbf n}^\prime\}$ through the binary coalescence of any pair of particles. The average rate of reaction between the $i$th and $j$th particles is simply $K({\mathbf n_i,\mathbf n_j})/V^2$ per unit volume. The total rate of reaction of all $N$ clusters is $R_{N}$, where $$R_i = \sum_{k=1}^{i} R_{k,N} \label{RVECTOR}$$ and $$R_{i,j} = \sum_{k=1}^{j} {1\over2} K({\mathbf n_i,\mathbf n_k})/V \label{RROWS}$$ in terms of the sum over reactions in the entire volume, $V$, assuming they are uncorrelated and occur in parallel. $R_{i,N}$ is proportional to the rate at which cluster $i$ reacts with all other clusters. A stochastic sequence of discrete reactions may be constructed from these parameters. The random interval, $\xi$, to the next reaction is obtained from a uniform variate, $x\in(0,1]$, as [@MACKEOWN]: $$\xi = -{\rm ln}(x)/R_{N}$$ The first cluster of the random reaction pair, $i$, is selected with a probability proportional to $R_{i,N}$, from $y\in(0,1]$ and $${R_{i-1}\over{R_{N}}} < y \leq {{R_i}\over{R_{N}}} \label{ISELECT}$$ where $R_{0} \equiv 0$. Finally, the reaction counterpart, $j$, is identified from $z\in(0,1]$ and $${{R_{i,j-1}}\over{R_{i,N}}} < z \leq {{R_{i,j}}\over{R_{i,N}}} \label{JSELECT}$$ with $R_{i,0}\equiv 0$. This selects $j$ with a probability proportional to ${1\over2}K({\mathbf n_i,\mathbf n_j})/V$. The procedure repeatedly selects new $x$, $y$, and $z$ for the next event, increments the system time by $\xi$, performs the reaction $i+j$, and recalculates $R$ for the next iteration. This repeats until the elapsed time exceeds $\tau$. Since the last reaction falls outside the desired interval, it is discarded without being performed. The procedure may then be started anew for the next time-step. The choice of two random numbers to select the pair, $i, j$, differs from the usual scheme, where the pair is selected from a single value. In either case, the search for $i$ and $j$ takes $o(log_2(N))$ operations using the method of bisection [@NUMREC3]. However, separate selection of $i$ and $j$ makes it possible to record all $R_{m}$ with $o(N)$ storage space and a one-time computational effort of $o({N}^2 )$. Once $i$ is determined, $R_{i,m}$ may be tabulated with $o(N)$ effort for all $m$, so the full matrix need not be stored. Finally, after $i$ and $j$ react, the $R_m$ may be updated with $o(N)$ effort by replacing only those terms involving the old clusters $i$ and $j$ with the results for a single new, coalesced cluster, and re-indexing to account for the lost cluster. Since $R_{N}$ is an extensive quantity for a given total density, evolution of $N$ particles over a finite interval requires $o({N}^2 )$ effort and $o(N)$ storage. Specifying the binary reaction rate coefficients, $K$, as a half-triangular matrix increases the efficiency marginally. This MC scheme has difficulty modeling widely varying concentrations of reactants (e.g., the monomer density is typically orders of magnitude higher than the large clusters for radiation damage). Also, $N$ decreases after every coalescence, which increases the statistical noise. There are methods that preserve $N$ [@SMITH:1998; @MUKHERJEE:2003], but it is possible to encompass a wider range of densities at the same time. In the approach taken here, the discrete MC elements are macroparticles, widely used in, e.g., non-equilibrium simulations of plasma physics [@MACRO]. (This is distinct from related, “weighted particle” schemes for coagulation [@BABOVSKY:1999; @EIBECK:2001; @SABLEFELD:1996].) Here, the $j$-th macroparticle in the system consists of an ensemble of clusters all of the same composition $(\mathbf n_j,\rho_j)$. Consistent with the Gillespie procedure, macroparticle reactions are evaluated discretely, so clusters in an ensemble react simultaneously but otherwise stochastically. However, here reactants will generally have different ensemble densities, $\rho_L < \rho_H$, which are independent of their sizes/compositions, $\mathbf n_L,\mathbf n_H$. The lower-density macroparticle, $L$, reacts completely, leaving behind an unchanged portion $\rho_H-\rho_L$ of clusters from the higher-density ensemble, $H$. The total cluster density declines, but $N$ stays constant, and $N$-dependent errors remain steady over time. Macroparticle reaction rates (analogous to Eq. \[RROWS\]) are defined so as to reproduce the continuum limit as $N\rightarrow\infty$. Pairs $i$ and $j$, with $\rho_i<\rho_j$, react according to: $$\begin{alignedat}{5} &(\mathbf n_i, \rho_i) + (\mathbf n_j, \rho_j) \rightarrow (\mathbf n_i + \mathbf n_j, \rho_i) + (\mathbf n_j, \rho_j-\rho_i) \label{MCMC1} \end{alignedat}$$ at an average rate of $K({\mathbf n_i,\mathbf n_j}) \rho_j $. Two macroparticles of the same density ($ \rho_i =\rho_j=\rho;\enskip i \neq j$) react as: $$\begin{alignedat}{5} &(\mathbf n_i,\rho) + (\mathbf n_j,\rho) \rightarrow (\mathbf n_i+\mathbf n_j,\rho/2) + (\mathbf n_i+\mathbf n_j,\rho/2) \end{alignedat}$$ at an average rate of $K ({\mathbf n_i,\mathbf n_j}) \rho$. The product is simply split into two equal pieces so that $N$ remains constant. Finally, the individual clusters within a single macroparticle ensemble may coalesce with one another, so there is also a unary reaction process: $$\begin{alignedat}{5} &(\mathbf n_i,\rho_i) \rightarrow (2 \mathbf n_i,\rho_i/2) \quad \end{alignedat}$$ which also proceeds at an average rate of $K({\mathbf n_i,\mathbf n_i}) \rho_i$. This possibility modifies Eq. \[RROWS\] to include a non-zero reaction rate for $i=j$. Macroparticle dynamics never corresponds to an atomistic simulation for finite $N$. Instead, this is a distinct, approximative discretization of the continuum equations themselves, in the same spirit as earlier approaches [@Surh:2004]. Again, $P(t)$ is approximated here by a sparse set of elements without arbitrarily imposing some coarse-graining of finite difference equations for the distribution. Since the computational cost scales as $o(N^2)$ for a dense reaction matrix, the method is also efficient. This is especially advantageous in higher dimensions, e.g., in describing helium-vacancy-impurity clusters. ME-MC reactions {#MEMCtext} --------------- Additional schemes are required for treating reactions between ME and MC elements. In the continuum approximation, reaction with external entities, $\mathbf n \not\in $ ME, introduces unary sink terms to the rate equation for $\mathbf m \in{\rm ME}$, cf. Eqs. \[VTRUNC\], \[TRUNC\]: $$S_{\mathbf m}(t) \rho_{\mathbf m}(t)= \biggl[\sum_{\mathbf n\in MC}K\bigl(\mathbf m,\mathbf n(t)\bigr) \rho_{\mathbf n}(t) +K(\mathbf m,d)\rho_d(t) \biggr]\rho_{\mathbf m}(t) U(\mathbf N^{ME}/2 - \mathbf m) \label{MEMC1}$$ where the summation includes all elements $\{(\mathbf n(t),\rho_{\mathbf n}(t)\}^{MC}$ at time $t$ and where $K(\mathbf m,d)$ includes reactions with network dislocations. The sink term, $S$, is identically zero for constrained ME defects. At present, $K(\mathbf m,d)$ is only nonzero for $\mathbf m=(1,0), (-1,0)$ and for vacancy emission $K(\mathbf 0,d)$. The counterpart to Eq. \[MEMC1\] is expressed for $\mathbf n\in{\rm MC}$ in the macroparticle scheme by: $$\begin{alignedat}{5} &({\mathbf n},\rho_{\mathbf n})^{MC}\rightarrow (\mathbf m+\mathbf n,\rho_{\mathbf n})^{MC} \label{MEMC2} \end{alignedat}$$ as a discrete reaction with an average rate of $K(\mathbf m,\mathbf n) \rho_{\mathbf m}$. A stochastic sequence of reactions at these average rates will approach the continuum behavior of Eq. \[MEMC1\] in the limit $N^{MC}\rightarrow\infty$. A single reaction can change a macroparticle size, cross-section, and reaction rate substantially, if $\mathbf m$ is comparable in size to $\mathbf n$. Accordingly, ME-MC reactions for such “small” MC clusters are included by the Markov Monte Carlo scheme described above, and the reaction parameters are immediately updated to reflect the change, before evaluating the next reaction. Reaction events are randomly performed from the $N^{MC}\times N^{ME}$ matrix of reaction rates, at overall rate $Q$. If the next event occurs within the desired interval, the $i$th MC element is selected as a reactant with probability $Q_i/Q$, where: $$Q_{i} = \sum_j K(\mathbf n_i,\mathbf m_j)\rho_{\mathbf m_j} \label{QVECTOR}$$ for reactive elements $\mathbf m_j\in ME$ and: $$Q = \sum_{i=1}^{N^{MC}} Q_i$$ The $j$th ME element is selected as a reactant with probability $K(\mathbf n_i,\mathbf m_j)\rho_{\mathbf m_j}/Q_i$. Finally, the time index is updated, the reaction is performed, and $Q$ is revised. This is analogous to the Markov procedure for MCMC reactions, except that the reaction matrix is full-rectangular rather than half-triangular and that the rates are always proportional to the density of the ME reactant. As for the corresponding evolution of $P^{ME}$, the instantaneous source/sink terms, Eq. \[MEMC1\], change after each discrete reaction event in $P^{MC}$, possibly multiple times during the interval $\tau$. It is not computationally practical to evolve $P_{ME}$ over each individual Markov sub-step, $\xi$, to account for this. Instead, $P_{ME}$ is evolved over the full time-step $\tau$ by operator splitting, after all ME-MC and MC-MC reactions in $P_{MC}$ are performed. To minimize any convergence error, the instantaneous sink strength can be replaced with a weighted time average over the interval: $$\begin{aligned} \overline{S^{fast}_{\mathbf m}} &=& {1\over\tau} \int_{t_0}^{t_0+\tau} dt \bigg\lbrack \sum_{\mathbf n\in MC}K(\mathbf m,\mathbf n(t)) \rho_{\mathbf m}(t)\bigg\rbrack \\ &=& {1\over\tau} \sum_j \xi_j \bigg\lbrack \sum_{\mathbf n}K(\mathbf m,\mathbf n(t_{j-1})) \rho_{\mathbf m}(t_{j-1})\bigg\rbrack \label{SPRIME} \end{aligned}$$ finally expressed as a sum over sub-intervals, $\xi_j$, between discrete reaction times, $t_j$. Such attention to detail is unnecessary for large MC clusters (and for network dislocations), where rapid reactions with highly mobile defects (i.e., small $\mathbf m $) do not substantially change the sink strength over short intervals. Thus, it is sufficient to update parameters for the large $\mathbf n$ clusters at the end of each time-step. In this case, MC clusters are evolved using a Poisson-distributed random variate, $P(x)$, [@GILLESPIE:2000; @NUMREC5] for the number of reactions that occurs during $\tau$. These MC elements are only updated at $t_0+\tau$, with all reactions accumulated in each of the $N_{ME}$ channels: $$(\mathbf n,\rho_{\mathbf n}) \rightarrow \bigg(\mathbf n + \sum_{\mathbf m\in ME} \mathbf m P\big\lbrack \tau K({\mathbf m,\mathbf n}) \rho_{\mathbf m}\big\rbrack, \enskip \rho_{\mathbf n}\bigg) \label{MCPOISSON}$$ Equation \[MCPOISSON\] is the discrete analogue of the Gaussian-distributed random walk used previously [@Surh:2004]. The corresponding ME sink term is: $$S^{slow}_{\mathbf m}(t_0) = \sum_{\mathbf n \in MC}K\big(\mathbf m,\mathbf n(t_0)\big) \rho_{\mathbf n}(t_0) + K(\mathbf m,d)\rho_d(t_0) \label{SPOISSON}$$ including the dislocation contribution, assuming that $\rho_d(t)$ is slowly changing. Finally, discrete reactions could also be evaluated by a rejection method, given a Majorant kernel $M(\mathbf m,\mathbf n)\ge K(\mathbf m,\mathbf n)$ [@SABLEFELD:1996]. For example, the reaction rates, $M$, can be evaluated on a coarse grid of ${\mathbf n}_i$ and all reactants ${\mathbf n}_i\le{\mathbf n}\leq{\mathbf n}_i+1$ be treated alike. In another approach, $M$ may be chosen to be a sum of products [@EIBECK:2001], $$\begin{aligned} M(\mathbf m,\mathbf n) & = \enskip \vec {\cal M}(\mathbf m) \cdot \vec {\cal M}(\mathbf n) \label{P1}\end{aligned}$$ It is then only required to evaluate $N^{MC}$ vectors, $\cal M$, (of one or more dimensions) and to take dot products. Either approach is easier than directly computing $N^{MC}(N^{MC}+1)/2$ binary rate coefficients for Eqs. \[RVECTOR\], \[RROWS\]. The Majorant kernel is selected to be easy to evaluate and to predict a faster (or equal) event rate than the true system. To correct for any overestimate, the time index is updated according to the usual Markov Monte Carlo procedure, but the reaction is only performed if a uniform variate, $w\in(0,1]$ also satisfies $w\le K(\mathbf m,\mathbf n)/M(\mathbf m,\mathbf n)$. Thus, excess events predicted by $M$ are rejected (with the required probability $1-K/M$). At present, the full reaction rate coefficients from Eq. \[KEQN\] can be evaluated very efficiently, so this method is not employed here. However, it is expected to be advantageous when biased cavity-cavity, cavity-loop, and loop-loop reactions are included in the future. Results ======= Monomer aggregation model ------------------------- A high density of trapping/recombination centers is believed to delay the onset of void swelling [@NELSON:1966; @MANSUR:1986; @MANSUR:1990]. Traps hinder void diffusion and coalescence and prolong the incubation stage. The simplest trapping model assumes that all dimers and larger clusters are immobile: $D_{\mathbf n}\equiv0$ for all ${\mathbf n}\not\in\{v,vh,h,i\}$, so that the last two summations in Eq. \[RATEEQN\] are zero. If Eq. \[MEQN\] is solved separately from the remainder of the Master Equation (Eq. \[RATEEQN\]) in a quasi-stationary approximation, then that problem may be solved by existing methods [@Surh:2004; @WEHNER:1985]. However, here the problem is simply treated as a limit case of Smoluchowski’s coagulation equation. Initial cluster populations are shown in Figs. \[Fig2\]-\[Fig4\] for type-316 stainless steel irradiated to low doses at $10^{-6}$ dpa/s and 300, 500, and 700 C. It is well-known that helium-vacancy clusters may be separated into distinct species (of equilibrium bubbles and stable or unstable voids), according to their size-dependent free energies. Accordingly, the figures are marked with black lines where the net [*average*]{} vacancy addition rate for the defect clusters approaches zero. The leftmost black line in Figs. \[Fig2\]-\[Fig4\] represents a hard wall for over-pressurized bubbles: by fiat, bubbles cannot reach densities greater than 2 helium per vacant site. Here, this is imposed by disallowing further reactions with helium- and self-interstitials. Other lines separate clusters that add or lose vacancies on average. Small, over-pressurized bubbles tend to add vacancies until reaching the next line in the Figures, where stable helium bubbles are in dynamic equilibrium with the vacancy and interstitial population. (This approximates the line of bubbles with $P\simeq \gamma/2r$, which would be in equilibrium in the absence of a vacancy and interstitial supersaturation.) The stability of these bubbles is reflected by their elevated concentration in that region, especially visible in Fig. \[Fig4\]. Stable bubbles tend to grow along the equilibrium line as they accumulate helium, while adjusting their vacancy content [*on average*]{} to remain in equilibrium. Finally, bubbles cannot exceed some critical helium content - larger clusters are stable voids that tend to add vacancies and grow to arbitrary size. This is seen in Fig. \[Fig3\]; there the clusters grow along the line of stable bubbles until reaching a critical helium content (11 heliums), at which point they grow by adding vacancies in excess of helium, forming a plume of rapidly-growing voids in the size distribution. Voids are here simply taken to be cavities with higher vacancy/helium ratio than any bubble species of the same helium content. An approximately parabolic region under the black curves bounds a set of small, unstable voids that tend to lose vacancies and shrink back towards the equilibrium bubble configuration. For example, this ranges from the origin to vacancy/helium compositions of (19,11) and (94,0) in Fig. \[Fig3\]. The rightmost solid line identifies the critical or unstable equilibrium void compositions; larger voids tend to add net vacancies with time. Note that a percentage of equilibrium bubbles in Fig. \[Fig3\] are able to fluctuate in vacancy content across the barrier of unstable voids. That is, they become stable voids without having first reached the critical helium content. Similarly, helium dimers are readily able to cross the barrier of unstable voids in Fig. \[Fig2\]. Very large voids ultimately become neutral (unbiased) sinks, adding helium/vacancies at an average rate of 1:200 (based on anticipated asymptotic swelling of 1$\%$/dpa and model helium generation around 50 appm/dpa). Thus, voids approach a line of constant composition. Except for a brief transient at the onset of irradiation, the vacancy monomer concentration decreases monotonically with time as the total sink strength of the microstructure rises with dose. After a few dpa, production of $\alpha$-particles also peaks, and the helium monomer concentration also declines. During this extended period, equilibrium bubbles continue to grow by adding helium, they continue to reach the critical size, and they continue to become voids. However, the critical size for equilibrium bubbles increases with time (as a function of declining $\rho_v$), and the rate of formation of new helium dimer nuclei and bubble growth rates decrease (as a function of declining $\rho_h+\rho_{hv}$). This causes the rate of void formation to decrease gradually with time, giving a broad void size distribution. Eventually, the larger stable bubbles become TEM-visible, and the overall size distribution becomes bimodal. The time-dependent volumetric swelling for this model is shown at a series of temperatures in Fig. \[NoCoalSwell\]. The low temperature system is initially dominated by large numbers of transient, unstable vacancy clusters (Fig. \[Fig2\]) that serve as recombination centers and suppress swelling. So many defect centers form that helium/vacancy ratios are kept low, and helium plays a reduced role in the initial evolution. As a result, the visible cavity density ($r>0.5$ nm) is sensitive to the surface energy parameter, $\gamma$: $\rho_{vis} = 5\times10^{23}$ m$^{-3}$ for $\gamma(T)=0.8\gamma_0(T)$ and $1\times10^{24}$ m$^{-3}$ for $\gamma(T)=0.5\gamma_0(T)$. Eventually, some vacancy clusters acquire significant amounts of helium, and the system is filled with a high concentration of small equilibrium bubbles. These function as recombination centers; they keep the vacancy supersaturation low so that few, if any, bubbles grow into stable voids. They also keep the asymptotic swelling rate small. At and above 500 C, swelling is more a matter of helium bubble formation and growth towards critical sizes (Figs. \[Fig3\] and \[Fig4\]). The cavity density and swelling rates are therefore insensitive to $\gamma$. The steady swelling rate of 0.8 %/dpa at 500 C is consistent with void swelling in austenitic stainless steel [@Surh:2005; @Surh:ERR]. At higher temperatures, the increased helium mobility results in fewer cavities (7-8$\times10^{20}$ m$^{-3}$ at 700 C), and a smaller density of bubbles escape to become stable voids and contribute to steady swelling. Cluster coalescence model ------------------------- The other simplification of defect trapping is to neglect it entirely and assume that clusters diffuse freely according to their size. The predicted void size distribution changes significantly when coalescence is included. This is seen in Figs. \[Fig7\] and \[Fig8\], for the same temperatures as in Figs. \[Fig2\] and \[Fig3\]. Coalescence reactions continually, preferentially consume the smaller, more mobile clusters. The largest voids grow an order of magnitude larger through coalescence, making the distribution of stable void sizes substantially broader than before. Very large voids achieve such low diffusivities as to be effectively immobile; this results again in a terminal void population. At low temperatures, the removal of small unstable or equilibrium defect clusters reduces the number of recombination centers, suppresses damage annihilation, and speeds the formation of large, stable voids. This enhances low temperature swelling. At high temperatures, this same coalescence of small clusters greatly reduces the total number of helium bubbles and voids, so that the total void sink strength is kept small and the asymptotic swelling rate is diminished compared to the monomer aggregation model (Fig. \[SwellCoal\]). Small clusters are absorbed as rapidly as new ones form, which prevents the formation of a bimodal distribution of small equilibrium bubbles and large voids. These differences suggest that competition between trapping and coalescence of very small (mostly TEM-invisible) clusters significantly shapes the microstructure in real irradiated materials. When coalescence is included, the terminal void density and swelling rate remain sensitive to $\gamma$ up to 500 C. The predicted void density at this temperature increases from $7\times10^{19}$ m$^{-3}$ for $\gamma = 0.75\gamma_0$ to $7\times10^{20}$ m$^{-3}$ for $\gamma = 0.5\gamma_0$. The swelling rate for the former case is only 0.3%/dpa at 50 dpa but reaches 0.8%/dpa for the latter. This suggests that either the cavity surface energy is substantially smaller than the value for the clean metal or that the vacancy clusters have much smaller mobilities than are modeled here. The swelling behavior finally becomes insensitive to the surface energy by 700C. In this limit, coalescence reduces the terminal void density to 4-5$\times10^{18}$ m$^{-3}$. Conclusions =========== This paper introduces a mixed Master Equation/Monte Carlo treatment of rate theory calculations in a mean field continuum approximation. This enables flexible treatment of the defect density variables, using different algorithms to treat the various reactions as efficiently as possible. The approximately quasi-stationary distribution of small, unstable or transient clusters is treated (as much as possible) by solving continuum rate equations. This eliminates the need to evaluate rapid individual reactions that mostly cancel one another. Larger clusters are treated by Monte Carlo methods, which treats clusters of arbitrary size and composition without the need for a fixed grid or artificial discretization of the defect distribution. A Markov method for smaller clusters accurately simulates rapid fluctuations in size and in the reaction parameters, and a Poisson-distributed random walk efficiently treats the more gradual evolution of the largest clusters. Finally, a macroparticle approach is introduced to encompass large differences in species densities in the Monte Carlo distribution. This hybrid scheme readily treats void/bubble evolution to high cumulative fluxes for temperatures and dose rates that are characteristic of real reactor systems. Calculations demonstrate that void coalescence provides an important channel for consolidating vacancy defects into large, stable voids, controlling the duration of incubation and the terminal void density. It is expected that thermal and radiation-induced micro-chemical evolution of solute and precipitate distributions will influence the cluster mobility and thereby the macroscopic incubation and steady-swelling behavior. Some degree of void/bubble trapping seems to be required in order to obtain a bimodal bubble/void size distribution, while some coalescence may be needed to give a realistically low terminal void density at higher temperature. The cavity surface energy determines the barrier for nucleation of stable voids and hence also affects the incubation behavior; this contribution becomes temperature- and time-dependent if oxygen is explicitly modeled. All of these effects can be addressed, in principle, by extensions of the method described here. These calculations also suggest the importance of additional, competing processes that are not evaluated at present, such as interstitial-interstitial aggregation or cluster annihilation from void-dislocation reaction. The methods described here can be extended to treat coalescence of loops as easily as voids, given a suitable binary reaction kernel. Such reactions should be included for reasons of consistency, besides their likely contribution to transient and steady swelling behavior. They would be especially important if radiation damage were introduced as a variety of pre-formed defect clusters. Based on the preliminary findings for cavity coalescence, more general defect cluster reactions are expected to have a significant influence on radiation swelling behavior. Acknowledgements ================ This work performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. MPS acknowledges V.V. Bulatov for an early introduction to Markov chain Monte Carlo methods, e.g., Ref. [@CAI:1999]. ![A portion of the void/bubble distribution for a model with mobile monomer defects and sessile clusters, $\gamma=0.8 \gamma_0(T)$, at T=300 C, $10^{-6}$ dpa/s, and $32\times10^{-3}$ dpa. The largest void in the distribution contains 110 vacancies. The solid lines display the loci of stable and unstable equilibrium cluster compositions, based on average vacancy accumulation rates. This distribution has not been smoothed - the pixellated appearance reflects discrete cluster compositions. (Fig2)[]{data-label="Fig2"}](T300_t32021_dpa6.jpg){width="12cm"} ![A portion of the void/bubble distribution as in Fig. \[Fig2\], but at T=500 C, $10^{-6}$ dpa/s, and $16\times10^{-3}$ dpa. The distribution has been smoothed for the large clusters, where Monte Carlo data is increasingly sparse. The solid lines display the stable and unstable equilibrium cluster compositions. []{data-label="Fig3"}](T500_t16009_dpa6.jpg){width="12cm"} ![The full void/bubble distribution as in Fig. \[Fig3\], but at T=700 C. The curved solid line locates the stable equilibrium bubbles; the critical void size is not visible on this scale.[]{data-label="Fig4"}](T700_t16009_dpa6.jpg){width="12cm"} ![Volumetric swelling curves versus dose in the model that excludes void-void coalescence. The cavity surface energy is fixed at $\gamma(T)=0.4\gamma_0(T)$[]{data-label="NoCoalSwell"}](coal_no.jpg){width="12cm"} ![A portion of the void/bubble distribution as in Fig. \[Fig2\] (300 C), but including void coalescence and with $\gamma(T) = 0.5\gamma_0(T)$. The distribution has been smoothed for the large clusters, where Monte Carlo data is sparse. []{data-label="Fig7"}](CoalT300_t32012_dpa6.jpg){width="12cm"} ![The full void/bubble distribution as in Fig. \[Fig4\] (500 C), but including void-void coalescence and with $\gamma(T) = 0.5\gamma_0(T)$.[]{data-label="Fig8"}](CoalT500_t16009_dpa6.jpg){width="12cm"} ![Volumetric swelling curves versus dose in the model that includes void-void coalescence. The cavity surface energy is set to $\gamma(T)=0.4\gamma_0(T)$.[]{data-label="SwellCoal"}](coal_yes.jpg){width="12cm"} -- --------------------------------------- -------------------------------------------- Lattice constant $a_0$ $3.639\times10^{-10}$ m Burgers vector $b$ $a_0/\sqrt{2}$ Atomic volume $\Omega$ ${a_0}^3/4$ Shear modulus $\mu$ $8.295\times10^{10}$ Pa Poisson’s ratio $\nu$ 0.264 Cascade efficiency $\xi_{Frenkel}$ 0.1 Relaxation volume -0.2 $\Omega$ Migration energy $E_{m}$ $2.08\times10^{-19}$ J Formation energy $E_f$ $2.88\times10^{-19}$ J Formation entropy $S_f$ 1.5 $k_B$ Pre-exponential factor $1.29\times10^{-6}$ m$^2$/s Shear polarizability $ -2.4\times10^{-18}$ Relaxation volume 1.50 $\Omega$ Migration energy $E_m$ $0.320\times10^{-19}$ J Pre-exponential factor $1.29\times10^{-6}$ m$^2$/s Shear polarizability $-2.535\times10^{-17} $ Relaxation volume 0.60 $\Omega$ Migration energy $E_m$ $0.320\times10^{-19}$ J Pre-exponential factor $1.29\times10^{-6}$ m$^2$/s Shear polarizability $-2.535\times10^{-17} $ Relaxation volume -0.2 $\Omega$ Migration energy $E_{m}$ $2.08\times10^{-19}$ J Pre-exponential factor $1.29\times10^{-6}$ m$^2$/s Shear polarizability $ -2.4\times10^{-18}$ Relaxation volume 0 Surface energy $\gamma_0(T=0)$ 2.408 J/m$^2$ Temperature derivative $d\gamma_0/dT$ $0.440\times10^{-3}$ J/m$^2$/K Adsorption factor $\Lambda$ 0.45-0.80 Migration energy $E_{m}$ $2.08\times10^{-19}$ J Pre-exponential factor $1.29\times10^{-6}$ m$^2$/s /$n_{v}^{4/3}$ Temperature $T$ 300-700 C Flux $\phi$ $10^{-6}$ dpa/s Damage efficiency $\xi$ 0.1 -- --------------------------------------- -------------------------------------------- : Model material parameters for type-316 stainless steel. \[TableOne\]

ArXiv

--- abstract: 'We point out a surprising consequence of the usually assumed initial conditions for cosmological perturbations. Namely, a spectrum of Gaussian, linear, adiabatic, scalar, growing mode perturbations not only creates acoustic oscillations of the kind observed on very large scales today, it also leads to the production of shocks in the radiation fluid of the very early universe. Shocks cause departures from local thermal equilibrium as well as creating vorticity and gravitational waves. For a scale-invariant spectrum and standard model physics, shocks form for temperatures $1$ GeV$<T<10^{7}$ GeV. For more general power spectra, such as have been invoked to form primordial black holes, shock formation and the consequent gravitational wave emission provides a signal detectable by current and planned gravitational wave experiments, allowing them to strongly constrain conditions present in the primordial universe as early as $10^{-30}$ seconds after the big bang.' author: - 'Ue-Li Pen' - Neil Turok title: Shocks in the Early Universe --- Over the past two decades, observations have lent powerful support to a very simple model of the early universe: a flat, radiation-dominated Friedmann-Lemaître-Robinson-Walker (FLRW) background cosmology, with a spectrum of small-amplitude, growing perturbations. In this Letter we study the evolution of these perturbations on very small scales and at very early times. The simplest and most natural possibility is that their spectrum was almost scale-invariant, with the rms fractional density perturbation $\epsilon \sim10^{-4}$ on all scales. However, more complicated spectra are also interesting to consider. For example, LIGO’s recent detection of $\sim 30 M_{\odot}$ black holes [@LIGO] motivated some to propose a bump in the primordial spectrum with $\epsilon \sim 10^{-1}$ on the relevant comoving scale. High peaks on this scale would have collapsed shortly after crossing the Hubble horizon, at $t\sim 10^{-4}$ seconds, to form $30 M_{\odot}$ black holes in sufficient abundance to constitute the cosmological dark matter today [@Bird]. Here we focus on the evolution of acoustic waves inside the Hubble horizon. In linear theory, they merely redshift away as the universe expands. However, higher order calculations [@Gielen] revealed that perturbation theory fails due to secularly growing terms. We explain this here by showing, both analytically and numerically, in one, two and three dimensions, that small-amplitude waves steepen and form shocks, after $\sim \epsilon^{-1}$ oscillation periods [@earlierrefs]: for movies and supplementary materials see [@weblink]. Furthermore, shock collisions would generate gravitational waves. As we shall later explain, the scenario of Ref. [@Bird], for example, would produce a stochastic gravitational wave background large enough to be detected by existing pulsar timing array measurements. More generally, planned and future gravitational wave detectors will be sensitive to gravitational waves generated by shocks as early as $10^{-30}$ seconds after the big bang [@penturoklong]. -.6in ![ Simulation showing cosmological initial conditions (left) evolving into shocks (right). The magnitude of the gradient of the energy density is shown in greyscale. The initial spectrum is scale-invariant and cut off at ${1\over 128}$ the box size, with rms amplitude $\epsilon=.02$. Movie available at [@weblink].[]{data-label="fig:movie"}](shockspanel1.pdf "fig:") -.5in -.1in Shock formation also has important thermodynamic consequences. In a perfect fluid, entropy is conserved and the dynamics is reversible. The presence of a spectrum of acoustic modes means that the entropy is lower than that of the homogeneous state but, within the perfect fluid description, the entropy cannot increase. Shock formation leads to the breakdown of the fluid equations, although the evolution is still determined by local conservation laws. Within this description, shocks generate entropy, allowing the maximum entropy, thermal equilibrium state to be achieved. Shock collisions also generate vorticity, a process likewise forbidden by the fluid equations. Both effects involve strong departures from local equilibrium and are of potential relevance to early-universe puzzles including the generation of primordial magnetic fields and baryogenesis [@penturoklong]. Of course, the perfect fluid description is not exact and dissipative processes operate on small scales. In fact, the shock width $L_s$ is set by the shear viscosity $\eta$, and the density jump $\epsilon\, \rho$ across the shock. For a relativistic equation of state, [*i.e.*]{}, $P=c_s^2 \rho$, with $c_s=1/\sqrt{3}$, we find $L_s=9\sqrt{2}\,\eta/(\epsilon \,\rho)$  [@LL; @penturoklong]. For shocks to form, $L_s$ must be smaller than the scale undergoing non-linear steepening, of order $\epsilon$ times the Hubble radius $H^{-1}$. In the standard model, at temperatures above $\sim 100 $ GeV, the right handed leptons, coupling mainly through weak hypercharge, dominate the viscosity, yielding $\eta\sim 16 /g'^4\ln(1/g') \sim 400 \,T^3$  [@moore]. Using $\rho=(\pi^2/30){\cal N} T^4$, with ${\cal N}=106.75$, we find $L_s$ falls below $\epsilon H^{-1}$ and hence shocks form when $T$ falls below $\sim \epsilon^2 10^{15}$ GeV, [*i.e.*]{}, $10^7$ GeV for $\epsilon=10^{-4}$. At the electroweak temperature viscous effects are negligible both in shock formation and, as we discuss later, shock decay. However, once $T$ falls below the electroweak scale, the Higgs field gains a vev $v$ and the neutrino mean free path grows as $\sim v^4/T^5$, exceeding $10^{-4} H^{-1}$ when $T$ falls below $\sim 1$ GeV for $\epsilon=10^{-4}$ or $\sim 100$ MeV for $\epsilon=10^{-1}$ . At lower temperatures, acoustic waves are damped away by neutrino scattering before they steepen into shocks. This Letter is devoted to the early, radiation-dominated epoch in which shocks form. We assume standard, adiabatic, growing mode perturbations. Their evolution is shown in Fig. \[modesfig\]: as a mode crosses the Hubble radius, the fluid starts to oscillate as a standing wave, and the associated metric perturbations decay. Thereafter, the fluid evolves as if it is in an unperturbed FLRW background. The tracelessness of the stress-energy tensor means that the evolution of the fluid is identical, up to a Weyl rescaling, to that in Minkowski spacetime, where the conformal time and comoving cosmological coordinates are mapped to the usual Minkowski coordinates. In flat spacetime, the fluid equations read $\partial_\mu T^{\mu \nu}=0$. For a constant equation of state, $P=c_s^2 \rho$, we have $T^{\mu \nu}=(1+c_s^2) \rho \, u^\mu u^\nu +c_s^2 \rho\, \eta^{\mu \nu}$, with $u^\mu=\gamma_{v}(1,\vec{v})$ the fluid 4-velocity. In linear theory, the fractional density perturbation $\delta$ and velocity potential $\phi$ (with $\vec{v}= \vec{\nabla} \phi$) obey the continuity equation $\dot{\delta}=-(1+c_s^2) \vec{\nabla}^2 \phi$ and the acceleration equation $\dot{\phi}=-c_s^2/(1+c_s^2)\, \delta$. Setting $\delta(t,\vec{x})=\sum_{\vec{k}} \delta^{(1)}_{\vec{k}} (t) e^{i\vec{k}.\vec{x}}$, for scale-invariant, Gaussian cosmological perturbations on sub-Hubble scales the statistical ensemble is completely characterized by \^[(1)]{}\_(t)\^[(1)]{}\_[’]{}(t’) =\_[+’,]{}[2 \^2 [A]{}k\^3 V]{} (k c\_s t) (k c\_s t’) \[eq1\] where ${\cal A} \equiv \epsilon^2$ is the variance per log interval in $k$ and $V$ is a large comoving box. From Planck measurements, we determine $\epsilon\approx 6\times 10^{-5}$ [@foottilt]. [*Wave steepening:*]{} The fluid energy-momentum tensor $T^{\mu \nu}$ depends on four independent variables, $\rho$ and $\vec{v}$. So the spatial stresses $T^{ij}$ may be expressed in terms of the four $T^{\mu0}$ and the four equations, $\partial_\mu T^{\mu \nu}=0$ used to determine the evolution of the fluid. For small-amplitude perturbations, we expand in $T^{0i}/\overline{T^{00}}$, where bar denotes spatial average, obtaining $T^{ij}\approx c_s^2 T^{00}\delta^{ij} +(T^{i 0}T^{j 0}-c_s^2\delta^{ij} T^{0k} T^{0k})/((1+c_s^2)\overline{T^{00}})$ at second order. ![The growing mode perturbation, in a radiation-dominated universe, in conformal Newtonian gauge. The density perturbation $\delta_k(t)$ (black), the Newtonian potential $\Phi$ (red) and the flat spacetime approximation to $\delta_k(t)$ (blue) are plotted against $kc_s t $.[]{data-label="modesfig"}](modesfig.pdf) At the linearized level, a standing wave is the sum of a left-moving and a right-moving wave. Assuming planar symmetry, we define $\Pi\equiv T^{01}/\overline{T^{00}}$. Consider a right-moving linearized wave $\Pi^{(1)}(u)$, where $u\equiv x-c_s t$, $v\equiv x+c_s t$. To second order, the fluid equations read $2\kappa \partial_u \partial_v \Pi +(\partial_u^2-\partial_v^2)\Pi^2=0$, with $\kappa=2 c_s(1+c_s^2)/(1-c_s^2)$. For the given initial condition, $v$ derivatives are suppressed relative to $u$ derivatives by one power of $\Pi$. Hence we can drop the $\partial_v^2\Pi^2$ term and integrate once in $u$ to obtain Burger’s equation, \_v + \_u =0, \[eq2\] for which, as is well-known, generic smooth initial data $\Pi(0,u)$ develop discontinuities in finite time $v$. Equation (\[eq2\]) may be solved exactly by the method of characteristics: the solution propagates along straight lines, so that $\Pi(v,u+\Pi(0,u) v/\kappa)=\Pi(0,u)$, where $\Pi(0,u)$ is initial data at $v=0$. Consider a standing wave $\delta=-\sqrt{2} \epsilon \sin k x \cos k c_s t$, with initial variance $\epsilon^2$. Decomposing it into left- and right-moving waves, the latter is $\delta^{(1)}=- \epsilon \sin(k u)/\sqrt{2} $ and, correspondingly, $\Pi^{(1)}=- c_s \epsilon \sin (k u)/\sqrt{2}.$ The characteristic lines first intersect at $u=0$ and $v=\sqrt{2}\kappa/(k c_s\epsilon)$, [*i.e.*]{}, when $t=\kappa /(k c^2_s \epsilon)$. Setting $c_s=1/\sqrt{3}$, we conclude that shocks form when $k c_s t \epsilon \sim \sqrt{8}$ or after $\sqrt{2}/ (\pi \epsilon)$ oscillation periods. The wave steepening effect is also seen in perturbation theory. From (\[eq2\]) one finds $\Pi^{(2)}=-c_s^2 \epsilon^2 (k v/4\kappa) \sin (2 k u)$, steepening $\Pi$ around its zero at $u=0$, with the second order contribution to the gradient equalling the first order contribution precisely when $k c_s t \epsilon \sim \sqrt{8}$. [*Characteristic rays:*]{} In higher dimensions, we can likewise gain insight into shock formation by following characteristic rays. These are the trajectories followed by small amplitude, short-wavelength disturbances [@LL], moving in the background provided by the perturbed fluid. For a perfect fluid with $c_s=1/\sqrt{3}$, if the 3-vorticity $\vec{\nabla}\wedge (\rho^{1\over 4} \vec{u})$ is initially zero, it remains zero for all time. We can then write $\rho^{1\over 4} \vec{u} = \overline{\rho}^{1\over 4} \vec{\nabla} \phi$, with $\phi$ a potential, at least until shocks form. We write the perturbed density and potential as: $\rho=\overline{\rho}(1+\delta_b+d \delta)$ and $\phi=\delta \phi_{b}+ d\phi$, where $\delta_{b}$ and $\delta \phi_{b}$ represent a background of linearized waves and $d \delta$ and $d \phi$ represent short-wavelength disturbances. The evolution of $d \delta $ and $d \phi$ is governed by the second order perturbation equations, $\partial_t d \delta+{4\over 3} \vec{\nabla}^2 d \phi +{1\over 3} \vec{\nabla}\cdot (\delta_{b}\vec{\nabla}d\phi+d \delta \,\vec{\nabla}\phi_{b} )=0$ and $\partial_t d \phi +{1\over 4} d \delta - {1\over 16} \delta _{b} d\delta+\vec{\nabla}\phi_{b}\cdot\vec{\nabla} d\phi=0$. These may be solved in the stationary phase approximation: we set $d\phi=A_\phi e^{i {S}}$ and $d\delta=A_\delta e^{i {S}}$ and assume that $A_\phi$ and $A_\delta$ vary slowly so that the variation of the phase $S$ controls the wave fronts. The leading (imaginary) part of the equations of motion yields a linear eigenvalue problem for $A_\phi$ and $A_\delta$, with $i\partial_t S$ the eigenvalue. We obtain \_t [ S]{}=--[23]{} (S \_[b]{}), \[eq4\] the Hamilton-Jacobi equation for a dynamical system with Hamiltonian ${\cal H}(\vec{p},\vec{x},t) =-\partial_t {S}(t,\vec{x})$, where ${ S}(t,\vec{x}(t))$ is the action calculated on a natural path, [*i.e*]{}, a solution of the equations of motion. The Hamiltonian (,,t) =[||]{}+[23]{} \_[b]{}(t,) \[eq5\] and the ray trajectories $\vec{x}(t)$ obey Hamilton’s equations: = +[23]{} \_[b]{}, =-[23]{} (\_i-n\_i(n\_j\_j))(\_[b]{}), \[eq6\] where $\vec{n}\equiv \vec{p}/|\vec{p}|$. Note that (due to scale invariance) ${\cal H}$ is homogeneous of degree unity in $\vec{p}$. It follows that (i) the ray velocities $\dot{\vec{x}}$ depend only on the direction of $\vec{p}$, not its magnitude, and (ii) the phase of the wave on the stationary-phase wavefront, ${\cal S}=\int dt( \vec{p}\cdot\dot{\vec{x}} -{\cal H}),$ is a constant as a consequence of Hamilton’s equations. Hence, when characteristic rays cross there are no diffractive or interference phenomena. [*Caustics and Shocks:*]{} The set of all characteristic rays is obtained by solving the equations of motion (\[eq6\]), for all possible initial positions and directions, $\vec{q}\equiv \vec{x}(0)$ and $\vec{m}\equiv \vec{n}(0)$. The solutions provide a mapping from $(\vec{q},\vec{m})$ to $(\vec{x},\vec{n})$, at each time $t$, which can become many to one through the formation of caustics [@Arnold]. If so, the solution to the fluid equations can be expected to acquire discontinuities such as shocks. The signature of the mapping becoming many to one is the vanishing of the Jacobian determinant $J\equiv|\partial (\vec{x}, \vec{n})/\partial (\vec{q} ,\vec{m})|$ at some $(\vec{q},\vec{m})$. We compute the change in this determinant in linear theory, and extrapolate to determine when it might vanish. The dominant effect comes from the deviation in the ray position which grows linearly in time (whereas the deviation in the ray direction does not). Thus we may approximate $\delta J \approx \delta |\partial \vec{x}/\partial \vec{q} |$. Setting $\vec{x}(t,\vec{q})=\vec{x}_0(t)+\vec{\psi}(t, \vec{q})$, where $\vec{x}_0(t)\equiv \vec{q}+\vec{m}\,t/\sqrt{3}$ is the unperturbed trajectory and $\vec{\psi}(t,\vec{q})$ is the displacement, we integrate (\[eq6\]) in the approximation that $\psi$ is small, so that the spatial argument of $ \phi_b$ may be taken as $\vec{x}_0(t)$. To first order in $\psi$, $\delta J \approx \vec{\nabla}_{\vec{q}} \cdot \vec{\psi}(t,\vec{q})$. A rough criterion for $J$ to develop zeros and thus for shocks to form, in abundance, is that the variance $\langle(\delta J)^2\rangle$, computed in the Gaussian ensemble of linearized perturbations for $\phi_b$, attains unity. In these approximations, from (\[eq6\]) we obtain J \_0\^t dt’ (\_\^2 -[t-t’ ]{} ) \_[b]{}(t’,\_0(t’)), \[eq7\] where $\hat{ O}\equiv \left(\vec{\nabla}_{\vec{q}}^2-(\vec{m}\cdot\vec{\nabla}_{\vec{q}})^2\right)(\vec{m}\cdot \vec{\nabla}_{\vec{q}})$. The term involving $\hat{O}$ (which only exists for $d>1$) dominates at large $t$. It describes how gradients in the background fluid velocity deflect the ray direction $\vec{n}$, with each “impulse” on $\vec{n}$ contributing a linearly growing displacement to $\vec{\psi}$. We compute the variance $\langle (\delta J)^2 \rangle$ from (\[eq7\]) by taking the ensemble average using the $\phi_b$ correlator implied by (\[eq1\]). The contribution of modes with $k<k_c$ is given by (J)\^2(k\_c c\_s t )\^2 [332]{} & d=1 & d=2 & d=3, \[eq8\] so that, for example, in the 3d ensemble, at any time $t$ shocks form on a length scale $\lambda_s\approx (\pi/\sqrt{6}) \epsilon \, c_s t$. [*Simulations*]{} We have implemented a fully relativistic TVD hydro code to solve the non-linear conservation equations in 1, 2 and 3 dimensions (always using $c_s=1/\sqrt{3}$). The code is a slight modification of [@trac] to relativistic fluids, and parallelizes on a single node under OpenMP. For the initial conditions, $T_{00}$ was taken to be perturbed with a scale-invariant Gaussian random field, and $T^{0i}$ was set zero, consistent with cosmological initial conditions. The initial power was truncated at $N$ times the fundamental mode where, for example, $N=128$ for $1024^3$ simulations in 3d and $N=256$ for $4096^2$ simulations in 2d. Various initial perturbation amplitudes were simulated in order to check consistency with the analytical discussion provided above and below. Further details and movies of the simulations are provided at [@weblink]. [*Thermalization:*]{} Consider the effect of an initially static density perturbation, $\rho(\vec{x})\rightarrow \overline{\rho}\left(1+\delta_i(\vec{x})\right)$, where $\overline{\rho}$ is the mean energy density. The fluid energy density is $T^{00}= {4\over 3} \rho \gamma_{v}^2-{1\over 3} \rho$, where $\vec{v}$ is the fluid velocity. Expanding to quadratic order in the perturbations, we find $T^{00}(\vec{x})=\overline{\rho}(1+\delta+{4\over 3} \vec{v}^2)$. At the initial moment, $\vec{v}(x)$ is zero everywhere and the spatial average $\overline{\delta_i}$ is zero by definition, hence $\overline{T^{00}} =\overline{\rho}$. However, once $\delta$ starts oscillating, a virial theorem holds, connecting the average variances: $\langle \vec{v}^2\rangle ={3\over 16}\langle \delta^2\rangle$. Thus, energy conservation implies that $\overline{\delta}$ falls by ${1\over 4} \langle \delta^2 \rangle$, to compensate for the kinetic energy in the oscillating modes. The system is not, however, in local thermal equilibrium. The entropy density is given, up to a constant, by $\rho^{3\over 4} \gamma_{v}\approx \overline{\rho}^{3\over 4} (1+{3\over 4} \delta -{3\over 32} \delta^2 +{1\over 2} \vec{v}^2),$ to second order in the perturbations. Using energy conservation and the virial theorem, the fractional deficit in the mean entropy density is thus $-{3\over 16} \langle \delta^2 \rangle=-{3\over 32} \langle \delta_i^2 \rangle$, where $\delta_i$ is the initial density perturbation. For a scale-invariant spectrum of initial perturbations, the fractional entropy deficit contributed by waves of wavelengths $\lambda_{1}<\lambda<\lambda_{2}$ is $-{3\over 32} \epsilon^2 \int_{\lambda_{1}}^{\lambda_{2}}(d\lambda/\lambda)=-{3\over 32} \epsilon^2\ln(\lambda_{2}/\lambda_{1})$. -.5in -.42in ![Entropy, in units of its equilibrium value, versus the time $t$, in units of the sound-crossing time, for $512^3$ simulations of a perfect radiation fluid with cosmological initial conditions as in (\[eq1\]). The red dashed curve is a fit to the $\epsilon=.05$ curve using (\[eq9\]) with $C={1\over 4}$. For $\epsilon=0.1$, $t$ has been doubled and the entropy deficit rescaled by a quarter to verify good agreement with (\[eq9\]). []{data-label="entropyfig"}](entropymodelnewf11.pdf "fig:") -3in Once shocks form, they generate entropy at a rate which may be computed as follows [@LL2]. Local energy-momentum conservation requires that the incoming and outgoing energy and momentum flux balance in the shock’s its rest frame. This determines the incoming fluid velocity $v_0$ and the outgoing velocity $v_1$ in terms of the fractional increase $\Delta$ in the density across the shock. One finds $v_0=\sqrt{(4+3 \Delta)/(4+\Delta)}/\sqrt{3}$ and $v_1=\sqrt{(4+\Delta)/( 4+3 \Delta)}/\sqrt{3}$. Next, the rest-frame entropy density is directly related to the rest-frame energy density and is therefore enhanced behind the shock front by a factor of $(1+\Delta)^{3\over 4}$. Therefore, the outgoing entropy flux is enhanced relative to the incoming flux by $(1+\Delta)^{3\over 4} (\gamma_1 v_1)/(\gamma_0 v_0) = (1+\Delta)^{1\over 4} \sqrt{(4+\Delta)/(4+3\Delta)}\approx 1 +{1\over 64} \Delta^3$, for small $\Delta$. The entropy density behind the shock is larger than that in front by the same factor. Entropy production results in the dissipation of shocks. Consider a sinusoidal density perturbation of initial amplitude $\epsilon$ which forms left- and right-moving shocks of strength $\Delta= \epsilon$. Averaging over space, the entropy deficit per unit volume is $-{3\over 64}\Delta^2 s_0$, where $s_0$ is the equilibrium entropy density. The rate of change of this deficit equals the rate at which the shocks generate entropy, which is ${1\over 64}c_s \Delta^3 s_0/\lambda_s$, where $\lambda_s$ is the mean shock separation. Hence, we obtain $\dot{\Delta}=-{1\over 6} (c_s/\lambda_s) \Delta^2$ so that shocks of amplitude $\epsilon$ decay in a time $t_d\sim 6 \lambda_s/(c_s \epsilon)$, larger than the shock formation time by a numerical factor (which, in our simplified model, is $\sqrt{3} \pi\approx 5$ in $d=3$). The shock amplitude decay introduces a short wavelength cutoff in the entropy deficit: ss\_0(1 -[332]{} \^2 (\_[2]{}/(C c\_s t)), \[eq9\] with $C$ a constant (equal to ${1\over 6}$ in our simplified model). We have checked this picture in detailed numerical simulations in one, two and three dimensions. Fig. \[entropyfig\] shows a full 3d numerical simulation compared with the prediction of Eq. (\[eq9\]), with excellent agreement. Not only do shocks generate entropy, shock-shock interactions generate vorticity, in a precisely calculable amount. For example, one can find a stationary solution representing two shocks intersecting on a line, leaving behind a “slip sheet” across which the tangential component of the velocity is discontinuous. In such steady-state flows the strength of the tangential discontinuity (and hence the vorticity) is proportional to $\Delta^3$ with $\Delta$ the shock amplitude. More generally, non-stationary configurations can generate parametrically larger vorticity and indeed, it is conceivable that in rare localized regions fully developed turbulence may occur. Finally, let us return to the production of gravitational waves from larger-amplitude perturbations such as have been invoked to explain the formation of black holes in the early universe. In second order perturbation theory, adiabatic perturbations with amplitude $\epsilon$ lead to a stochastic background of gravitational waves, produced at the Hubble radius, with spectral density $\Omega_g(f) h^2 \sim \epsilon^4 \Omega_\gamma h^2$ where $\Omega_\gamma h^2 \sim 4.2\times 10^{-5}$ is the fractional contribution of radiation to the critical density today [@baumann]. As we shall show elsewhere [@penturoklong], shock collisions generate a parametrically similar contribution to the stochastic gravitational wave background, also on Hubble horizon scales. But because shocks form later, they emit gravitational waves at longer wavelengths, with frequencies which are lower by a factor of $\epsilon$. In the scenario of Ref. [@Bird], $30 M_{\odot}$ primordial black holes would form at a time $t\sim 10^{-4}$ seconds from high peaks in perturbations with rms amplitude $\epsilon\sim 10^{-1}$. At second order in perturbation theory these contribute a stochastic gravitational wave background with $\Omega_g(f) h^2 \sim 4\times 10^{-9}$, at frequencies of $\sim 30$ nHz today. This is outside the exclusion window of the European Pulsar Timing Array, $\Omega_g(f) h^2 < 1.1 \times 10^{-9}$ at frequencies $f\sim 2.8$ nHz [@EPTA]. However, for $\epsilon\sim 0.1$, the gravity wave background due to shocks peaks at $\sim 3$ nHz, inside the exclusion window, potentially ruling out the scenario of Ref. [@Bird]. Gravitational wave detectors are now operating or planned over frequencies from nHz to tens of MHz (see, [*e.g.*]{}, Ref. [@holom]), corresponding to gravitational waves emitted on the Hubble horizon at times from $10^{-4}$ to $10^{-30}$ seconds. In combination with detailed simulations of the nonlinear evolution of the cosmic fluid and consequent gravitational wave emission, these experiments will revolutionize our ability to constrain the physical conditions present in the primordial universe, an exciting prospect indeed. [*Acknowledgments.*]{} — We thank John Barrow, Dick Bond, Job Feldbrugge, Steffen Gielen, Luis Lehner, Jim Peebles, Dam Son and Ellen Zweibel for valuable discussions and correspondence. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. [99]{} B. P. Abbott [*et al.*]{} (LIGO Scientific Collaboration, Virgo Collaboration), Phys. Rev. Lett. 116, 061102 (2016), arXiv:1602.03837; see also arXiv:1606.04856 (2016). S. Bird [*et al.*]{}, Phys. Rev. Lett. [**116**]{}, 201301 (2016). S. Gielen and N. Turok, Phys. Rev. Lett.  [**117**]{}, 021301 (2016). E.P.T. Liang, Ap.J., [**211**]{}, 361 (1977); [**216**]{}, 206 (1977), studied exact, planar “simple wave" solutions discovered by A.H. Taub, Phys. Rev. [**74**]{}, 328 (1948), focusing on large-amplitude shocks which would have formed much later, near matter-radiation equality. P.J.E. Peebles, in [*The Large Scale Structure of the Universe*]{}, Princeton, 1980, p. 332-341, extended this to generic distributions of linearized waves, but missed the dominant term (the second term in (\[eq7\])) and incorrectly concluded that the shock formation time scales as $\epsilon^{-2}$ instead of $\epsilon^{-1}$. https://www.github.com/PerimeterInstitute/shocks-in-the-early-universe/wiki J. Feldbrugge, U-L. Pen and N. Turok, in preparation (2016). P. B. Arnold, G. D. Moore and L. G. Yaffe, JHEP [**0011**]{}, 001 (2000) \[hep-ph/0010177\]. We ignore the modest effect of scalar tilt on the power spectrum. L. D. Landau and E. M. Lifshitz, [*Fluid Mechanics*]{}, Pergamon 1987, Sec. 103. , Sec. 135. 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ArXiv

--- abstract: 'Representing domain knowledge is crucial for any task. There has been a wide range of techniques developed to represent this knowledge, from older logic based approaches to the more recent deep learning based techniques (i.e. embeddings). In this paper, we discuss some of these methods, focusing on the representational expressiveness tradeoffs that are often made. In particular, we focus on the the ability of various techniques to encode ‘partial knowledge’ - a key component of successful knowledge systems. We introduce and describe the concepts of *ensembles of embeddings* and *aggregate embeddings* and demonstrate how they allow for partial knowledge.' author: - 'R.V.Guha' bibliography: - 'emt.bib' title: Partial Knowledge in Embeddings --- Motivation {#motivation .unnumbered} ========== Knowledge about the domain is essential to performing any task. Representations of this knowledge have ranged over a broad spectrum in terms of the features and tradeoffs. Recently, with the increased interest in deep neural networks, work has focussed on developing knowledge representations based on the kind of structures used in such networks. In this paper, we discuss some of the representational expressiveness tradeoffs that are made, often implicitly. In particular we focus on the loss of the ability to encode partial knowledge and explore two different paths to regain this ability. Logic Based Representations {#logic-based-representations .unnumbered} --------------------------- Beginning with McCarthy’s Advice Taker [@advicetaker], logic has been the formal foundation for a wide range of knowledge representations. These have ranged across the spectrum of formal models to the more experimental languages created as part of working systems. The more formal work started with McCarthy and Hayes’s Situation Calculus [@sitcalc], the various flavors of non-monotonic logics [@reiter], [@circumscription] and included proposed axiomatizations such as those suggested in the ‘Naive physics manifesto’ [@naivephysics] and Allen’s temporal representation [@allen]. There were a number of less formal approaches, that also had their roots in logic. Notable amongst these include Minsky’s frames [@frames], EMycin [@emycin], KRL [@krl] and others. The representation language used by Cyc, CycL [@cyc] is a hybrid of these approaches. Most of these representation systems can be formalized as variants of first order logic. Inference is done via some form of theorem proving. One of the main design issues in these systems is the tradeoff between expressiveness and inferential complexity. More recently, systems such as Google’s ‘Knowledge Graph’ have started finding use, though their extremely limited expressiveness makes them much more like simple databases than knowledge bases. Though a survey of the wide range of KR systems that have been built is beyond the scope of this paper, we note that there are some very basic abilities all of these systems have. In particular, - They are all relational, i.e., are built around the concept of entities that have relations between them - They can be updated incrementally - The language for representing ’ground facts’ is the same as the language for representing generalizations. There is no clear line demarcating the two. - They can represent partial knowledge. It is this feature we focus on in this paper. Feature Vectors {#feature-vectors .unnumbered} --------------- One of the shortcomings of the logic based representations was that they largely assumed that everything the system knew was manually given to the system, i.e., learning from ground data was not a central part of the design. The complexity of the representational formalisms and wide range of possible functions or expressions have made machine learning rather difficult in traditional knowledge representation systems. The rise of machine learning as the primary mechanism for creating models of the domain have lead to the use of much simpler representations. In particular, we notice, - The language for representing ground facts is distinct from the language for representing generalizations or models. The ground facts about the domain, i.e., the training data, is usually represented as a set of feature vectors. - The language for representing generalizations or models is also usually highly restricted. Each family of models (e.g., linear regression, logistic regression, support vector machines, has a function template with a number of parameters that the learning algorithm computes. Recent work on neural networks attempts to capture the generality of Turing machines with deep networks, but the structure of the function learnt by these systems is still uniform. - The language for representing ground facts is propositional, i.e., doesn’t have the concept of entities or relations. This constraint makes it very difficult to use these systems for modeling situations that are relational. Many problems, especially those that involve reasoning about people, places, events, etc. need the ability to represent these entities and the relations between them. - Most of these systems allow for partial knowledge in their representation of ground facts. i.e., some of the features in the training data may be missing for some of the instances of the training data. The inability of feature vectors to represent entities and relations between them has lead to work in embeddings, which try to represent entities and relations in a language that is more friendly to learning systems. However, as we note below, these embedding based representations leave out an important feature of classical logic based representations — a feature we argue is very important. We first review embedding based representations, show how they are incapable of capturing partial information Embeddings {#embeddings .unnumbered} ---------- Recent work on distributed representations \[[@socher], [@manning], [@bordes], [@bordes2014semantic], [@quoc]\] has explored the use of embeddings as a representation tool. These approaches typically ’learn an embedding’, which maps terms and statements in a knowledge base (such as Freebase [@freebase]) to points in an N-dimensional vector space. Vectors between points can then be interpreted as relations between the terms. A very attractive property of these distributed representations is the fact that they are learnt from a set of examples. Weston, Bordes, et. al. [@bordes] proposed a simple model (TransE) wherein each entity is mapped to a point in the N-dimensional space and each relation is mapped to a vector. So, given the triple $r(a, b)$, we have the algebraic constraint $\overrightarrow{\boldsymbol{a}} - \overrightarrow{\boldsymbol{b}} = \overrightarrow{\boldsymbol{r}} + \epsilon$, where $\epsilon$ is an error term. Given a set of ground facts, TransE picks coordinates for each entity and vectors for each relation so as to minimize the cumulative error. This simple formulation has some problems (e.g., it cannot represent many to many relations), which has been fixed by subsequent work ([@wang2014knowledge]). However, the core representational deficiency of TransE has been retained by these subsequent systems. The goal of systems such as TransE is to learn an embedding that can predict new ground facts from old ones. They do this by dimensionality reduction, i.e., by using a low number of dimensions into which the the ground facts are embedded. Each triple is mapped into an algebraic constraint of the form $\overrightarrow{\boldsymbol{a}} -\overrightarrow{\boldsymbol{b}} = \overrightarrow{\boldsymbol{r}} + \epsilon$ and an optimization algorithm is used determine the vectors for the objects and relations such that the er- ror is minimized. If the number of dimensions is sufficiently large, the $\epsilon$s are all zero and no generalizations are made. As the number of dimensions is reduced, the objects and relation vectors get values that minimize the εs, in effect learning generalizations. Some of the generalizations learnt may be wrong, which contribute to the non-zero $\epsilon$s. As the number of dimensions is reduced further, the number of wrong generalizations increases. This is often referred to as ‘KB completion’. We believe that the value of such embeddings goes beyond learning simple generalizations. These embeddings are a representation of the domain and should be usable by an agent to encode its knowledge of the domain. Further, any learning task that takes descriptions of situations that are best represented using a graph now has a uniform representation in terms of this embedding. When it is used for this purpose, it is very important that the embedding accurately capture what is in the training data (i.e., the input graph). In such a case, we are willing to forgo learning in order to minimize the overall error and can pick the smallest number of dimensions that accomplishes this. In the rest of this paper, we will focus on this case. Ignorance or Partial Knowledge {#ignorance-or-partial-knowledge .unnumbered} ------------------------------ In a logic based system that is capable of representing some proposition $P$ (relational or propositional), it is trivial for the system to not know whether $P$ is true or not. I.e., its knowledge about the world is partial with respect to $P$. However, when a set of triples is converted to an embedding, this ability is lost. Consider a KB with the entities $Joe$, $Bob$, $Alice$, $John$, $Mary$. It has a single relation $friend$. The KB specifies that $Joe$ and $Bob$ are friends and that $Alice$ and $John$ are friends and that $Mary$ and $John$ are $not$ friends. It does not say anything about whether $Mary$ and $Alice$ are friends or not friends. This KB can be said to have partial knowledge about the relation between $Mary$ and $Alice$. When this KB is converted into an embedding, to represent the agent’s knowledge about the domain, it is important that this aspect of the KB be preserved. Unfortunately, in an embedding, $Mary$ and $Jane$ are assigned particular coordinates. Either $\overrightarrow{\boldsymbol{Mary}} - \overrightarrow{\boldsymbol{Jane}}$ is equal to $\overrightarrow{\boldsymbol{friend}}$ or it is not. If it is, then, according to the embedding, they are friends and if it is not, then, according to the embedding, they are not friends. The embedding is a complete world, i.e., every proposition is either true or false. There is no way of encoding ’unknown’ in an embedding. If the task is knowledge base completion, especially of an almost complete knowledge base, then it may be argued that this deficiency is excusable. However, if the KB is very incomplete (as most real world KBs are) and if such as KB is being used as input training data, or as the basis for an agents engagement with the world, this forced completion could be problematic. We now explore two alternatives for encoding partial knowledge in embeddings. Encoding Partial Knowledge {#encoding-partial-knowledge .unnumbered} -------------------------- Logic based formalisms distinguish between a knowledge base (a set of statements) and what it might denote. The object of the denotation is some kind of structure (set of entities and ntuples in the case of first order logic or truth assignments in the case of propositional logic). The KB corresponds not to a single denotation, but set of $possible$ denotations. Something is true in the KB if it holds in every possible denotation and false if it does not hold in any of the possible denotations. If it holds in some denotations and does not hold in some, then it is neither true nor false in the KB. ### Ensemble of Embeddings {#ensemble-of-embeddings .unnumbered} In other words, the key in logic based systems to partial knowledge is the distinction between the KB and its denotation and the use of a set of possible denotations of a KB. Note that in logic based KR systems, these possible denotations (or possible worlds) are almost always in the meta-theory of the system. They are rarely actually instantiated. We could also imagine a KR system which does instantiate an ensemble (presumbaly representative) of possible denotations and determine if a proposition is true, false or unknown based on whether it holds in all, none of some of these models. We follow this approach with embeddings to encode partial knowledge. Instead of a single embedding, we can use an ensemble of embeddings to capture a KB. If we use a sufficient number of dimensions, we should be able to create embeddings that have zero cumulative error. Further, different initial conditions for the network will give us different embeddings. These different embeddings correspond to the different possible denotations. A ensemble of such embeddings can be used to capture partial knowledge, to the extent desired. While this approach is technically correct in some sense, it also defeats the purpose. The reason for creating the embeddings was in part to create an encoding that could be used as input to a learning system. When we go from a single embedding to an ensemble of embeddings, the learning algorithm gets substantially complicated. One approach to solving this problem is to develop a more compact encoding for an ensemble of embeddings. Remember that we don’t need to capture every single possible embedding corresponding to the given KB. All we need is sample that is enough to capture the essential aspects of the partiality of the KB. ### Aggregate Models {#aggregate-models .unnumbered} Consider the set of points across different models corresponding to a particular term. Consider a cluster of these points (from an ensemble of embeddings) which are sufficiently close to each other. This cluster or cloud of points (each of which is in a different embeddingl), corresponds to $an$ aggregate of possible interpretations of the term. We can extend this approach for all the terms in the language. We pick a subset of models where every term forms such a cluster. The set of clusters and the vectors between them gives us the aggregate model. Note that in vectors corresponding to relations will also allow amount of variation. If a model satisfies the KB, any linear transform of the model will also satisfy the KB. In order to keep these transforms from taking over, no two models that form an aggregate should be linear transforms of each other. In both aggregate models, each object corresponds to a cloud in the N-dimensional space and the relation between objects is captured by their approximate relative positions. The size of the cloud corresponds to the vagueness/approximateness (i.e., range of possible meanings) of the concept. Partial knowledge is captured by the fact that while some of the points in the clouds corresponding to a pair of terms may have coordinates that maps to a given relation, other points in the clouds might not. In the earlier example, we now have a set of points corresponding to $Mary$ and $Jane$. Some of these are such that $\overrightarrow{\boldsymbol{Mary}} - \overrightarrow{\boldsymbol{Jane}} = \overrightarrow{\boldsymbol{friend}}$, while others are such that $\overrightarrow{\boldsymbol{Mary}} - \overrightarrow{\boldsymbol{Jane}} \ne \overrightarrow{\boldsymbol{friend}}$. Thus, these aggregates can encode partial knowledge. Conclusions {#conclusions .unnumbered} ----------- The ability to encode partial knowledge is a very important aspect of knowledge representation systems. While recent advances to KR using embeddings offer many attractions, the current approaches are lacking in this important aspect. We argue that an agent should be aware of what it doesn’t know and should use representations that are capable of capturing this. We described one possible approach to extending embeddings to capture partial knowledge. While there is much work to be done before embeddings can be used as practical knowledge representation systems, we believe that with additions like the one described here, embeddings may turn out to be another useful addition to the knowledge representation tool chest.

ArXiv

--- abstract: 'The notion of Haar null set was introduced by J. P. R. Christensen in 1973 and reintroduced in 1992 in the context of dynamical systems by Hunt, Sauer and Yorke. During the last twenty years this notion has been useful in studying exceptional sets in diverse areas. These include analysis, dynamical systems, group theory, and descriptive set theory. Inspired by these various results, we introduce the topological analogue of the notion of Haar null set. We call it Haar meager set. We prove some basic properties of this notion, state some open problems and suggest a possible line of investigation which may lead to the unification of these two notions in certain context.' address: 'Department of Mathematics University of Louisville Louisville, KY 40292,USA' author: - 'Udayan B. Darji' title: On Haar Meager Sets --- \[section\] \[defn\][Theorem]{} \[defn\][Example]{} \[defn\][Lemma]{} \[defn\][Remark]{} \[defn\][Proposition]{} \[defn\][Corollary]{} \[defn\][Conjecture]{} \[defn\][Exercise]{} [H]{} ¶[[P]{}]{} ß introduction ============ Often in various branches of mathematics one would like to show that a certain phenomenon occurs very rarely or occurs rather frequently. This is particularly the case in analysis and probability theory. Hence we have the notion of measure zero set and the notion of almost everywhere or almost sure. Each locally compact group admits a translation invariant regular Borel measure which is finite on compact sets and positive on open sets. Such a measure is unique, up to multiplication by a constant, and is called Haar measure. More precisely, if $\mu$ and $\nu$ are two such measure and $\nu$ is not identically zero, then $\mu = c \nu$ for some $c \ge 0$. Of course, in $\R ^n$ this is the Lebesgue measure. If one has no prior bias towards any particular set of points one uses this measure. However, Haar measures only exist on locally compact groups. Therefore, in many important spaces such as $C([0,1])$, the space of continuos real-valued functions on $[0,1]$ and $S_{\infty}$, the group of permutations on $\N$, one does not have a suitable notions of smallness. As these objects are complete topological groups with natural topologies, one can use the notion of meagerness as the notion of smallness. One of the earliest instance of such a result is a theorem of Banach which says that the set of all functions in $C([0,1])$ which are differentiable at some point is meager in $C([0,1])$. The notion of meagerness is a topological one. Often it fails to capture the essence of certain properties studied in analysis, dynamical systems, etc. To remedy this, J. P. R. Christensen in 1973 [@christensen] introduced what he called “Haar null" sets. The beauty of this concept is that in locally compact group it is equivalent to the notion of measure zero set under the Haar measure and at the same time it is meaningful in Polish groups in general. Unaware of the result of Christensen, Hunt, Sauer and Yorke [@hsy] found this notion essential in the context of dynamical systems, reintroduced it and gave some applications of it in dynamics. Since then this notion has found applications in many diverse places. We only list a handful here. For applications in dynamics see [@hsy], [@oy]. Its relations to descriptive set theory and group theory can be found in [@dm], [@solecki], [@do]. Some applications of this notion in real analysis can be found in [@ds],[@hunt], [@kolar], [@z]. Throughout $X$ is a fixed abelian Polish group. Following Christensen [@christensen] we say that $A \subseteq X$ is [**Haar null**]{} if there is a Borel probability measure $\mu$ on $X$ and a Borel set $B \subseteq X$ such that $A \subseteq B$ and $\mu(x +B )=0$ for all $x \in X$. By using standard tools of Borel measures on Polish groups, it can be shown that $\mu$ can be chosen so that its support is a Cantor set, i.e., a $0$-dimensional, compact metric space with no isolated points. In the original paper, Christensen showed that the set of Haar null sets forms a $\sigma$-ideal and in locally compact groups the notion of Haar null is equivalent to having measure zero under the Haar measure. Often in the literature Haar null sets are called [**shy sets**]{} and the complement of Haar null sets are called [**prevalent sets**]{}. There are explicit examples of Haar null sets which are also meager sets. For example, Hunt [@hunt] showed that the set of functions in $C([0,1])$ which are differentiable at some point is Haar null, complementing the classical result of Banach that this set is meager. Kolar [@kolar] introduced a notion of small sets which he called [**HP-small**]{}. This notion captures the concept of Haar null as well as $\sigma$-porosity. (All $\sigma$-porous sets are meager. See [@z2] for more on $\sigma$-porosity.) Recently, Elekes and Steprāns investigated some deep foundational properties of Haar null sets and analogous problems concerning meager sets. We refer the interested reader to their paper [@es]. In this note we introduce what we call Haar meager sets. The purpose of this definition is to have a topological notion of smallness which is analogous to Christensen’s measure theoretic notion of smallness. We show that Haar meager sets form a $\sigma$-ideal. We show every Haar meager set is meager. Example \[ex1\] shows that some type of definability is necessary in the definition of Haar null to obtain this result. We next show that in locally compact groups the notions Haar meagerness and meagerness coincide. Using a result of Solecki, we observe that every non locally compact Polish group admits a closed nowhere dense set which is not Haar meager. We also formulate a criteria equivalent to Haar meagerness which may be useful in applications. In the next section we prove the main results. We end this note in the third section by stating some problems and suggesting some line of investigation. main results ============ Throughout, $X$ is an abelian Polish group. A set $A \subseteq X$ is [**Haar meager**]{} means that there is a Borel set $B$ with $A \subseteq B$, a compact metric space $K$ and a continuous function $f: K \rightarrow X$ such that $f^{-1}(B +x)$ is meager in $K$ for all $x \in X$. We call $f$ [*a witness function*]{} for $A$. The collection of all Haar meager sets is denoted by $\hm$. \[haarimpmeager\] Let $A \ss X$ be Haar meager. Then, $A$ is meager. Since subsets of meager sets are meager, it suffices to prove the theorem for sets which are Borel and Haar meager. Let $A$ be a Borel Haar meager set. Let $K$ be a compact set and $f: K \rightarrow X$ be a continuous function which witnesses that $A$ is Haar meager. Let $\Sigma _A =\{(x,y) \in X \times K: f(y) \in (-x+ A) \}$. As $A$ is Borel, $\Sigma _A$ is Borel and hence has the Baire property. Also, for each $x \in X$, $\{y \in K: (x,y) \in \Sigma_ A \}$ is meager in $K$ since $A$ is Haar meager. By Kuratowski-Ulam Theorem $\Sigma_A$ is meager in $X \times K$. Again by Kuratowski-Ulam theorem we may choose $y \in K$ such that $B=\{x: (x,y) \in \Sigma_A\} $ is meager in $X$. Note that $B=\{x: f(y) \in (-x+A)\} =\{x: x \in (-f(y)+A)\} = -f(y)+A$. Hence $-f(y)+A$ is meager in $X$. Since meager sets are translation invariant in a Polish group, we have that $A$ is meager in $X$. The following example shows that some sort of definability condition is necessary in the definition of Haar meager sets in order for Haar meager sets to meager. \[ex1\] (Assume the Continuum Hypothesis) There is a subset $A \ss \mathbb R$ and a compact set $K \ss \mathbb R$ such that $(A +x) \cap K $ is countable for all $x \in \mathbb R$, yet $A$ is not meager. Let $K$ be the standard middle third Cantor set. Let $\{K_{\alpha}\}_{\alpha < \omega _1}$ be an enumeration of all translates of $K$ and let $\{C_{\alpha}\}_{\alpha < \omega _1}$ be an enumeration of closed nowhere dense subsets of $\mathbb R$. For each $\alpha < \omega _1$, let $p_{\alpha} \in {\mathbb R} \setminus \bigcup_{\beta \le \alpha} \left ( K_{\beta} \cup C_{\beta} \right ) $. Let $A=\{p_{\alpha}: \alpha < \omega _1\}$. Then, $A$ is the desired set. If $X$ is locally compact and $A \subset X$ is meager, then $A$ is Haar meager. Let $U \ss X$ be an open set such that $\overline {U}$ is compact. That $A$ is Haar meager is witnessed by $\overline{U}$ and the identity function on $\overline{U}$. \[meagereqhaar\] In a locally compact group $X$ a set is Haar meager iff it is meager. Every nonlocally compact Polish group $X$ has a closed meager set which is not Haar meager. This simply follows from the following result of Solecki. (Solecki [@solecki]) Let $X$ be a nonlocally compact Polish group. Then, there is a closed set $F \ss X$ and a continuous function $f:F \rightarrow \{0,1\}^{\omega}$ such that each $f^{-1}(x)$, $x\in \{0,1\}^{\omega}$, contains a translate of every compact subset of $X$. Since $X$ is separable, there is $x \in \{0,1\} ^{\omega}$ such that $f^{-1}(x)$ is nowhere dense. This set is clearly not Haar meager. Our next goal is to prove that collection of Haar meager sets forms a $\sigma$-ideal. We prove an intermediate lemma first. \[smallwitness\] Let $A$ be Haar meager and $\e>0$. Then, there is a witness function whose range is contained in an $\e$ ball around the origin. Let $K$ be a compact set and $f: K \rightarrow X$ be a witness function for $A$. Let $U$ be an open subset of $X$ with diameter less than $\frac{\e}{2}$ such that $U \cap f(K) \neq \emptyset$. Let $y \in U \cap f(K)$ and $L =\overline {f^{-1}(U)}$. Define $g:L \rightarrow X$ by letting $g(x)=f(x)-y$ for all $x \in L$. This $g$ satisfies the required property. \[sigmaideal\] $\hm$ is a $\sigma$-ideal. That $\hm$ is hereditary follows from the definition of Haar meager. We will show that $\hm$ is closed under countable union. To this end, let $A_1, A_2, \dots$ be Borel Haar meager sets. For each $i$, let $K_i$ be a compact metric space and $g_i:K_i \rightarrow X$ be witness function for $A_i$. By Lemma \[smallwitness\], we may assume that $g_i(K_i)$ is contained in the ball centered at the origin with radius $2^{-i}$. Simply let $K =\prod_{i=1}^{\infty}K_i$. Now we define $g:K \rightarrow X$ as follows. $g(x)=\sum_{i=1}^{\infty} g_i(x_i)$ where $x=(x_1,x_2,\ldots)$. Clearly, $g$ is well-defined and continuous. To complete the proof of the theorem, it will suffice to show that $g$ is a witness for each $A_i$. To this end, fix an $i \in \N$ and $t \in X$. Note that $g^{-1}(A_i+t) = \{(x_1, x_2, x_3,\ldots)\in \Pi_{i=1}^{\infty} K_i: \sum_{i=1}^{\infty} g_i(x_i) \in (A_i +t)\}$. Fix $x_1,x_2,\ldots,x_{i-1},x_{i+1}, \ldots$. Since $g_i$ is a witness for $A_i$, we have that $$\{a \in K_i: g_i(a) \in A_i +t - \sum_{k \neq i} g_k(x_k) \}$$ is meager in $K_i$. As $A_i$ is Borel and $g$ is continuous, we have that $g^{-1}(A_i+t)$ has the Baire property. Applying the Kuratowski-Ulam Theorem, we obtain that\ $g^{-1}(A_i+t)$ is meager in $\Pi_{i=1}^{\infty} K_i$, completing the proof. We next show that in the definition of Haar meager set, one can choose $K \subseteq X$. However, we prove a lemma first. \[cantoroncompact\] Let $K$ be a Cantor set and $M$ be any compact metric space. Then, there is a continuous function $f$ from $K$ onto $M$ such that if $A$ is meager in $M$ then $f^{-1}(A)$ is meager in $K$. Using a standard construction from general topology, one can obtain a continuous function $f$ from $K$ onto $M$ such that if $U$ is nonempty and open in $K$ then $f(U)$ contains a nonempty open subset of $M$. We claim this $f$ has the desired property. To obtain a contradiction, assume that there is a meager set $A \ss M$ such that $f^{-1}(A)$ is not meager. Let $F_1, F_2,\ldots$ be a sequence of nowhere dense closed sets such that $A \ss \cup_{i=1}^{\infty} F_i$. Since $f^{-1}(A)$ is not meager, $f^{-1}(F_i)$ contains a nonempty open set for some $i$. This contradicts that the image of an open set under $f$ must contain a nonempty open set. \[equivalentdef\] Let $A \subseteq X$. Then the following are equivalent. - Set $A$ is Haar meager. - There is a compact set $K \subseteq X$ and continuous function $g:K \rightarrow X$ which is a witness function for $A$. Only ($\Rightarrow$) needs a proof as ($\Leftarrow$) is obvious. Let $A \subseteq X$ be a Borel Haar meager set. If $X$ is countable, then the only Haar meager subset of $X$ is the empty set and any set $K \subseteq X$ and $g:K \rightarrow X$ will do. If $X$ is uncountable, then it contains a copy of the Cantor space. Let $K$ be one such copy. Let $M$ be a compact metric space and $g:M \rightarrow X$ witness the fact that $A$ is Haar meager. Let $f:K \rightarrow M$ be a function of the type in Lemma \[cantoroncompact\]. Then $g\circ f$ is the desired function. Remarks and Problems ==================== In this section we make some remarks, state some open problems concerning Haar meager sets and suggest possible lines of investigation which may be fruitful. Is the Continuum Hypothesis necessary in Example \[ex1\]? Can such an example be constructed in ZFC? Next problems concern the witness function in the definition of Haar meager set. \[one\] Suppose $A \subseteq X$ is Haar meager. Is there a compact set $K \subseteq X$ such that $(A+t) \cap K $ is meager in $K$ for all $t \in X$? A negative answer to Problem \[one\], would beg the following question. \[two\] Does the collection of all sets from Problem \[one\] form a $\sigma$-ideal? The next set of problems concerns concrete examples of Haar meager sets. As noted earlier, Kolar [@kolar] defined the notion of HP-small set. This notion is finer than the notion of Haar null set and the notion of Haar meager set. As a corollary to his main result we obtain that the set of functions in $C([0,1])$ which have finite derivative at some point is HP-small. What we would like to do is distinguish the notion of Haar null set from the notion of Haar meager set. Are there concrete examples in analysis, topology, etc of sets which are Haar null but not Haar meager? Or vice versa? As was observed earlier, every Polish group which is not locally compact admits meager sets which are not Haar meager. Concrete examples of such sets in analysis are given by Zajíček [@z] and Darji and White [@ds]. Zajíček [@z] showed that the set of functions in $C([0,1])$ for which $f'(0) = \infty$ has the property that it contains a translate of every compact subset of $C([0,1])$. Inspired by the result of Zajíček, Darji and White [@ds] gave an example of an uncountable, pairwise disjoint collection of subsets of $C([0,1])$ with the property that each element is meager yet contains a translate of every compact subset of $C([0,1])$. What are some natural examples in topology, dynamical systems, analysis of sets of natural interest which are meager but not Haar meager? We note that the notions of Haar meager is the topological mirror of the notion of Haar null. In Christensen’s definition of Haar null, one can choose the test measure $\mu$ as a push forward of some measure defined on some Cantor set in $X$. However, there are many instances in which these two notions coincide. Are there some general conditions that one can put on a set so that Haar null $+$ the conditions implies Haar meager? The same question with the role of Haar meager and Haar null is reversed. Of course, Kolar’s notion of HP-small implies both Haar null and Haar meager. What we are looking for some external condition of sets which allows transference principle. acknowledgement =============== The author would like to thank the anonymous referee for valuable suggestions that improved the exposition of the paper. [9]{} To appear in the Canadian Journal of Mathematics.

ArXiv

--- abstract: 'Incorporating hierarchical structures like constituency trees has been shown to be effective for various natural language processing (NLP) tasks. However, it is evident that state-of-the-art (SOTA) sequence-based models like the Transformer struggle to encode such structures inherently. On the other hand, dedicated models like the Tree-LSTM, while explicitly modeling hierarchical structures, do not perform as efficiently as the Transformer. In this paper, we attempt to bridge this gap with “Hierarchical Accumulation” to encode parse tree structures into self-attention at constant time complexity. Our approach outperforms SOTA methods in four IWSLT translation tasks and the WMT’14 English-German translation task. It also yields improvements over Transformer and Tree-LSTM on three text classification tasks. We further demonstrate that using hierarchical priors can compensate for data shortage, and that our model prefers phrase-level attentions over token-level attentions.' author: - | Xuan-Phi Nguyen$^\ddagger$[^1] , Shafiq Joty$^{\dagger \ddagger}$, Steven C.H. Hoi$^{\dagger}$, Richard Socher$^{\dagger}$\ $^\dagger$Salesforce Research\ $^\ddagger$Nanyang Technological University\ `[email protected],{sjoty,shoi,rsocher}@salesforce.com` bibliography: - 'refs.bib' title: 'Tree-Structured Attention with Hierarchical Accumulation' --- Introduction {#sec:intro} ============ Tree-based Attention {#sec:method} ==================== Experiments {#sec:experiments} =========== Conclusion ========== We presented a novel approach to incorporate constituency parse trees as an architectural bias to the attention mechanism of the Transformer network. Our method encodes trees in a bottom-up manner with constant parallel time complexity. We have shown the effectiveness of our approach on various NLP tasks involving machine translation and text classification. On machine translation, our model yields significant improvements on IWSLT and WMT translation tasks. On text classification, it also shows improvements on Stanford and IMDB sentiment analysis and subject-verb agreement tasks. Appendix ======== [^1]: Work done during an internship at Salesforce Research Asia, Singapore.

ArXiv

--- abstract: 'We study the dynamical stability of planetary systems consisting of one hypothetical terrestrial mass planet ($1~ $ or $10 \mearth$) and one massive planet ($10 \mearth - 10 \mjup$). We consider masses and orbits that cover the range of observed planetary system architectures (including non-zero initial eccentricities), determine the stability limit through N-body simulations, and compare it to the analytic Hill stability boundary. We show that for given masses and orbits of a two planet system, a single parameter, which can be calculated analytically, describes the Lagrange stability boundary (no ejections or exchanges) but which diverges significantly from the Hill stability boundary. However, we do find that the actual boundary is fractal, and therefore we also identify a second parameter which demarcates the transition from stable to unstable evolution. We show the portions of the habitable zones of $\rho$ CrB, HD 164922, GJ 674, and HD 7924 which can support a terrestrial planet. These analyses clarify the stability boundaries in exoplanetary systems and demonstrate that, for most exoplanetary systems, numerical simulations of the stability of potentially habitable planets are only necessary over a narrow region of parameter space. Finally we also identify and provide a catalog of known systems which can host terrestrial planets in their habitable zones.' author: - 'Ravi kumar Kopparapu, Rory Barnes' title: Stability analysis of single planet systems and their habitable zones --- Introduction {#sec1} ============ The dynamical stability of extra-solar planetary systems can constrain planet formation models, reveal commonalities among planetary systems and may even be used to infer the existence of unseen companions. Many authors have studied the dynamical stability of our solar system and extra-solar planetary systems [see @Wisdom1982; @Laskar1989; @RasioFord1996; @Chambers1996; @LaughlinChambers2001; @Gozdziewski2001; @Ji2002; @BQ04; @Ford2005; @Jones2006; @raymond09 for example]. These investigations have revealed that planetary systems are close to dynamical instability, illuminated the boundaries between stable and unstable configurations, and identified the parameter space that can support additional planets. From an astrobiological point of view, dynamically stable habitable zones (HZs) for terrestrial mass planets ($0.3 \mearth < M_{p} < 10 \mearth$) are the most interesting. Classically, the HZ is defined as the circumstellar region in which a terrestrial mass planet with favorable atmospheric conditions can sustain liquid water on its surface [@Huang1959; @Hart1978; @Kasting1993; @Selsis2007 but see also Barnes et al.(2009)]. Previous work [@Jones2001; @MT2003; @Jones2006; @Sandor2007] investigated the orbital stability of Earth-mass planets in the HZ of systems with a Jupiter-mass companion. In their pioneering work, [@Jones2001] estimated the stability of four known planetary systems in the HZ of their host stars. [@MT2003] considered the dynamical stability of 100 terrestrial mass-planets (modelled as test particles) in the HZs of the then-known 85 extra-solar planetary systems. From their simulations, they generated a tabular list of stable HZs for all observed systems. However, that study did not systematically consider eccentricity, is not generalizable to arbitrary planet masses, and relies on numerical experiments to determine stability. A similar study by [@Jones2006] also examined the stability of Earth-mass planets in the HZ. Their results indicated that $41 \%$ of the systems in their sample had “sustained habitability”. Their simulations were also not generalizable and based on a large set of numerical experiments which assumed the potentially habitable planet was on a circular orbit. Most recently, [@Sandor2007] considered systems consisting of a giant planet with a maximum eccentricity of $0.5$ and a terrestrial planet (modelled as a test particle initially in circular orbit) They used relative Lyapunov indicators and fast Lyapunov indicators to identify stable zones and generated a stability catalog, which can be applied to systems with mass-ratios in the range $10^{-4} - 10^{-2}$ between the giant planet and the star. Although this catalog is generalizable to massive planets between a Saturn-mass and $10 \mjup$, it still assumes the terrestrial planet is on a circular orbit. These studies made great strides toward a universal definition of HZ stability. However, several aspects of each study could be improved, such as a systematic assessment of the stability of terrestrial planets on eccentric orbits, a method that eliminates the need for computationally expensive numerical experiments, and wide coverage of planetary masses. In this investigation we address each of these points and develop a simple analytic approach that applies to arbitrary configurations of a giant-plus-terrestrial planetary system. As of March 2010, 376 extra-solar planetary systems have been detected, and the majority (331, $\approx 88 \%$) are single planet systems. This opens up the possibility that there may be additional planets not yet detected, in the stable regions of these systems. According to [@wright2007] more than $30 \%$ of known single planet systems show evidence for additional companions. Furthermore, [@marcy2005] showed that the distribution of observed planets rises steeply towards smaller masses. The analyses of [@wright2007] & [@marcy2005] suggests that many systems may have low mass planets[^1]. Therefore, maps of stable regions in known planetary systems can aid observers in their quest to discover more planets in known systems. We consider two definitions of dynamical stability: (1) [*Hill stability*]{}: A system is Hill-stable if the ordering of planets is conserved, even if the outer-most planet escapes to infinity. (2) [*Lagrange stability*]{}: In this kind of stability, every planet’s motion is bounded, i.e, no planet escapes from the system and exchanges are forbidden. Hill stability for a two-planet, non-resonant system can be described by an analytical expression [@MB1982; @Gladman1993], whereas no analytical criteria are available for Lagrange stability so we investigate it through numerical simulations. Previous studies by [@BG06; @BG07] showed that Hill stability is a reasonable approximation to Lagrange stability in the case of two approximately Jupiter-mass planets. Part of the goal of our present work is to broaden the parameter space considered by [@BG06; @BG07]. In this investigation, we explore the stability of hypothetical $1 \mearth$ and $10 \mearth$ planets in the HZ and in the presence of giant and super-Earth planets. We consider nonzero initial eccentricities of terrestrial planets and find that a modified version of the Hill stability criterion adequately describes the Lagrange stability boundary. Furthermore, we provide an analytical expression that identifies the Lagrange stability boundary of two-planet, non-resonant systems. Utilizing these boundaries, we provide a catalog of fractions of HZs that are Lagrange stable for terrestrial mass planets in all the currently known single planet systems. This catalog can help guide observers toward systems that can host terrestrial-size planets in their HZ. The plan of our paper is as follows: In Section \[sec2\], we discuss the Hill and Lagrange stability criteria, describe our numerical methods, and present our model of the HZ. In Section \[sec3\], we present our results and explain how to identify the Lagrange stability boundary for any system with one $\ge 10 \mearth$ planet and one $\le 10 \mearth$ planet. In Section \[sec4\], we apply our results to some of the known single planet systems. Finally, in Section \[sec5\], we summarize the investigation, discuss its importance for observational programs, and suggest directions for future research. Methodology {#sec2} =========== According to [@MB1982], a system is Hill stable if the following inequality is satisfied: $$\begin{aligned} - \frac{2 M}{G^{2} M^{3}_{\star}} ~ c^{2} h > 1 + 3^{4/3} \frac{m_{1} m_{2}}{m^{2/3}_{3} (m_{1} + m_{2})^{4/3}} + . . . \label{bbc}\end{aligned}$$ where $M$ is the total mass of the system, $G$ is the gravitational constant, $M_{\star} = m_{1} m_{2} + m_{2} m_{3} + m_{3} m_{1}$, $c$ is the total angular momentum of the system, $h$ is the total energy, $m_{1}$, $m_{2}$ and $m_{3}$ are the masses of the planets and the star, respectively. We call the left hand side of Eq.(\[bbc\]) $\beta$ and the right hand-side $\beta_{crit}$. If $\bbc > 1$, then a system is definitely Hill stable, if not the Hill stability is unknown. Studies by [@BG06; @BG07] found that for two Jupiter mass planets, if $\bbc \gsim 1$ (and no resonances are present), then the system is Lagrange stable. Moreover, Barnes et al. (2008a) found that systems tend to be packed if $\bbc \lesssim 1.5$ and not packed when $\bbc \gtrsim 2$. [@BG07] pointed out that the vast majority of two-planet systems are observed with $\bbc < 1.5$ and hence are packed. Recently, [@KRB09] proposed that the HD 47186 planetary system, with $\bbc = 6.13$ the largest value among known, non-controversial systems that have not been affected by tides[^2], may have at least one additional (terrestrial mass) companion in the HZ between the two known planets. To determine the dynamically stable regions around single planet systems, we numerically explore the mass, semi-major axis and eccentricity space of model systems, which cover the range of observed of extra-solar planets. In all the models (listed in Table \[table1\]), we assume that the hypothetical additional planet is either $1 \mearth$ or $10 \mearth$ and consider the following massive companions, (which we presume are already known to exist): (1) $10 \mjup$, (2) $5.6 \mjup$ (3) $3 \mjup$, (4) $1.77 \mjup$ (5) $1 \mjup$, (6) $1.86 \msat$ (7) $1 \msat$, (8) $56 \mearth$ (9) $30 \mearth$ (10) $17.7 \mearth$ and (11) $10 \mearth$. Most simulations assume that the host star has the same mass, effective temperature ($T_\mathrm{eff}$) and luminosity as the Sun. Orbital elements such as the longitude of periastron $\varpi$ are chosen randomly before the beginning of the simulation (Eq. (1) only depends weakly on them). For “known” Saturns and super-Earths, we fix semi-major axis $a$ at $0.5$ AU (and the HZ is exterior) or at $2$ AU (the HZ interior). For super-Jupiter and Jupiter mass, $a$ is fixed either at $0.25$ AU or at $4$ AU. These choices allow at least part of the HZ to be Lagrange stable. Although we choose configurations that focus on the HZ, the results should apply to all regions in the system. We explore dynamical stability by performing a large number of N-body simulations, each with a different initial condition. For the known planet, we keep $a$ constant and vary its initial eccentricity, $e$, from $0 - 0.6$ in steps of $0.05$. We calculate $\bbc$ from Eq.(\[bbc\]), by varying the hypothetical planet’s semi-major axis and initial eccentricity. In order to find the Lagrange stability boundary, we perform numerical simulations along a particular $\bbc$ curve, with [Mercury]{} [@Chambers1999], using the hybrid integrator. We integrate each configuration for $10^{7}$ years, long enough to identify unstable regions [@BQ04]. The time step was small enough that energy is conserved to better than 1 part in $10^{6}$. A system is considered Lagrange unstable if the semi-major axis of the terrestrial mass planet changes by 15% of the initial value or if the two planets come within $3.5$ Hill radii of each other[^3]. In total we ran $\sim 70,000$ simulations which required $\sim 35,000$ hours of CPU time. We use the definition of the “eccentric habitable zone” (EHZ; [@Barnes2008]), which is the HZ from Selsis et al (2007), with $50 \%$ cloud cover, but assumes the orbit-averaged flux determines surface temperature (Williams & Pollard 2002). In other words, the EHZ is the range of orbits for which a planet receives as much flux over an orbit as a planet on a circular orbit in the HZ of [@Selsis2007]. Results: Dynamical Stability in and around Habitable Zones {#sec3} ========================================================== Jupiter mass planet with hypothetical Earth mass planet {#sec3.1} -------------------------------------------------------- In Figs. 1 & 2, we show representative results of our numerical simulations from the Jupiter mass planet with hypothetical Earth mass planet case discussed in Section \[sec2\]. In all panels of Figs. 1 & 2, the blue squares and red triangles represent Lagrange stable and unstable simulations respectively, the black circle represents the “known” planet and the shaded green region represents the EHZ. For each case, we also plot $\bbc$ contours calculated from Eq. (\[bbc\]). In any given panel, as $a$ increases, the curves change from all unstable (all red triangles) to all stable (all blue squares), with a transition region in between. We designate a particular $\bbc$ contour as $\tau_\mathrm{s}$, beyond which (larger values) a hypothetical terrestrial mass planet is stable for all values of $a$ and $e$, for at least $10^{7}$ years. We tested $\tau_\mathrm{s}$ is the first $\bbc$ (close to the known massive planet) that is completely stable (only blue squares). For $\bbc$ curves below $\tau_\mathrm{s}$, all or some locations along those curves may be unstable; hence, $\tau_\mathrm{s}$ is a conservative representation of the Lagrange stability boundary. Similarly, we designate $\tau_\mathrm{u}$ as the largest value of $\bbc$ for which all configurations are unstable. Therefore, the range $\tau_{u} < \bbc < \tau_{s}$ is a transition region, where the hypothetical planet’s orbit changes from unstable ($\tau_\mathrm{u}$) to stable ($\tau_\mathrm{s}$). Typically this transition occurs over $10^{-3} \bbc$. Although Figs. 1 & 2 only show curves in this transition region, we performed many more integrations at larger and smaller values of $\bbc$, but exclude them from the plot to improve the readability. For all cases, [*all our simulations with $\bbc > \tau_\mathrm{s}$ are stable, and all with $\bbc < \tau_\mathrm{u}$ are unstable.*]{} These figures show that the Lagrange stability boundary significantly diverges from Hill stability boundary, as the eccentricity of the known Jupiter-mass planet increases. Moreover, $\tau_\mathrm{s}$ is more or less independent (within $0.1 \%$) of whether the Jupiter mass planet lies at $0.25$ AU or at $4$ AU. If an extra-solar planetary system is known to have a Jupiter mass planet, then one can calculate $\bbc$ over a range of $a$ & $e$, and those regions with $\bbc > \tau_\mathrm{s}$ are stable. We show explicit examples of this methodology in Section 4. We also consider host star masses of $0.3 M_\odot$ and performed additional simulations. We do not show our results here, but they indicate the mass of the star does not effect stability boundaries. Lagrange stability boundary as a function of planetary mass & eccentricity. {#analtau} ---------------------------------------------------------------------------- In this section we consider the broader range of “known” planetary masses discussed in Section 2 and listed in Table \[table1\]. Figures \[JEfig\_in\] & \[JEfig\_out\] show that as the eccentricity of the “known” planet increases, $\tau_\mathrm{s}$ and $\tau_\mathrm{u}$ appear to change monotonically. This trend is apparent on all our simulations, and suggests $\tau_\mathrm{s}$ and $\tau_\mathrm{u}$ may be described by an analytic function of the mass and eccentricity of the known planet. Therefore, instead of plotting the results from these models in $a-e$ space, as shown in Figs. \[JEfig\_in\] & \[JEfig\_out\], we identified these analytical expressions that relate $\tau_\mathrm{s}$ and $\tau_\mathrm{u}$ to mass $m_{1}$ and eccentricity $e_{1}$ of the known massive planet. Although these fits were made for planets near the host star, these fits should apply in all cases, irrespective of it’s distance from the star. In the following equations, the parameter $x = \log [m_{1}]$, where $m_{1}$ is expressed in Earth masses and $y = e_{1}$. The stability boundaries for systems with hypothetical $1 \mearth$ and $10 \mearth$ mass companion are: $$\begin{aligned} \tau_\mathrm{j} &=& c_{1}+ \frac{c_{2}}{x} + c_{3}~y + \frac{c_{4}}{x^{2}} + c_{5}~y^{2} + c_{6}~\frac{y}{x} + \frac{c_{7}}{x^{3}} + c_{8}~y^{3} + c_{9}~\frac{y^{2}}{x} + c_{10}~\frac{y}{x^{2}} \label{tauearth}\end{aligned}$$ where $j=s,u$ indicate stable or unstable and the coefficients for each case are given Table 2. The coefficients in the above expression were obtained by finding a best fit curve to our model data that maximizes the $R^{2}$ statistic, $$\begin{aligned} %R^{2} &=& 1 - \frac{\displaystyle\sum_{i}^n}{2} R^{2} &=& 1 - \frac{\displaystyle\sum_{i}^n (\tau^{model}_{i} - \tau^{fit}_{i})^2}{\displaystyle\sum_{i}^n (\tau^{model}_{i} - \overline{\tau^{model}})^2}\end{aligned}$$ where $\tau^{model}_{i}$ is the $i^{th}$model value of $\tau$ from numerical simulations, $\tau^{fit}_{i}$ is the corresponding model value from the curve fit, $\overline{\tau^{model}}$ is the average of all the $\tau_{model}$ values and $n=572$ is the number of models (including mass, eccentricity and locations of the massive planet). Values close to $1$ indicate a better quality of the fit. In Fig. \[SHB\_jup\], the top panels (a) $\&$ (b) show contour maps of $\tau_\mathrm{s}$ as a function of $\log [m_{1}]$ and $e_{1}$ between model data (solid line) and best fit (dashed line). The $R^{2}$ values for $1 \mearth$ companion (Fig. \[SHB\_jup\](a)) and $10 \mearth$ companion (Fig. \[SHB\_jup\](b)) are $0.99$ and $0.93$, respectively, for $\tau_\mathrm{s}$. In both the cases, the model and the fit deviates when the masses of both the planets are near terrestrial mass. Therefore our analysis is most robust for more unequal mass planets. The residuals between the model and the predicted $\tau_{s}$ values are also shown in Fig. \[SHB\_jup\](c) ($1 \mearth$ companion) $\&$ Fig. \[SHB\_jup\](d) ($10 \mearth$ companion). The standard deviation of these residuals is $0.0065$ and $0.0257$ for $1 \mearth$ and $10 \mearth$, respectively, though the $1 \mearth$ case has an outlier which does not significantly effect the fit. The maximum deviation is $0.08$ for $1 \mearth$ and $0.15$ for $10 \mearth$ cases. The expression given in Eq. (\[tauearth\]) can be used to identify Lagrange stable regions ($\bbc > \tau_\mathrm{s}$) for terrestrial mass planets around stars with one known planet with $e \le 0.6$ and may provide an important tool for the observers to locate these planets[^4]. Once a Lagrange stability boundary is identified, it is straightforward to calculate the range of $a$ and $e$ that is stable for a hypothetical terrestrial mass planet, using Eq.(\[bbc\]). In the next section, we illustrate the applicability of our method for selected observed systems. Application to observed systems {#sec4} =============================== The expressions for $\tau_{s}$ given in §\[analtau\] can be very useful in calculating parts of HZs that are stable for all currently known single planet systems. In order to calculate this fraction, we used orbital parameters from the Exoplanet Data Explorer maintained by California Planet Survey consortium[^5], and selected all $236$ single planet systems in this database with masses in the range $10 \mjup$ to $10 \mearth$ and $e \leq 0.6$. Table \[table2\] lists the properties of the example systems that we consider in §\[rhocrb\]-§\[hd7924\], along with the orbital parameters of the known companions and stellar mass. The procedure to determine the extent of the stable region for a hypothetical $1 \mearth$ and $10 \mearth$ is as follows: (1) Identify the mass ($m_{1}$) and eccentricity ($e$) of the known planet. (2) Determine $\tau_\mathrm{s}$ from Eq. \[tauearth\] with coefficients from Table 2. (3) Calculate $\bbc$ over the range of orbits ($a$ and $e$) around the known planet using Eq. \[bbc\]. (4) The Lagrange stability boundary is the $\bbc = \tau_{s}$ curve. Rho CrB {#rhocrb} ------- As an illustration of the internal Jupiter + Earth case, we consider the Rho CrB system. Rho CrB is a G0V star with a mass similar to the Sun, but with greater luminosity. [@Noyes1997] discovered a Jupiter-mass planet orbiting at a distance of $0.23$ AU with low eccentricity ($e=0.04$). Since the current inner edge of the circular HZ of this star lies at $0.90$ AU, there is a good possibility for terrestrial planets to remain stable within the HZ. Indeed, [@Jones2001] found that stable orbits may be prevalent in the present day [*circular*]{} HZ of Rho CrB for Earth mass planets. Fig. \[observed\]a shows the EHZ (green shaded) assuming a 50% cloud cover in the $a-e$ space of Rho CrB. The Jupiter mass planet is the blue filled circle. Corresponding $\tau_\mathrm{s}$ values calculated from Eq.(\[tauearth\]) for $1 \mearth$ companion ($0.998$, dashed magenta line) and $10 \mearth$ companion ($1.009$, black solid line) are also shown. These two contours represent the stable boundary beyond which an Earth-mass or super-Earth will remain stable for all values of $a$ and $e$ (cf. Figs. \[JEfig\_in\]a and \[JEfig\_in\]b). The fraction of the HZ that is stable for $1 \mearth$ is $72.2 \%$ and for $10 \mearth$ is $77.0 \%$. Therefore the Lagrange stable stable region is larger for a larger terrestrial planet. We conclude that the HZ of rho CrB can support terrestrial-mass planets, except for very high eccentricity ($e > 0.6$). These results are in agreement with the conclusion of [@Jones2001] and [@MT2003], who found that a planet with a mass equivalent to the Earth-moon system, when launched with $e=0$ within the HZ of Rho CrB, can remain stable for $\sim 10^{8}$ years. They also varied the mass of Rho CrB b up to $8.8 \mjup$ and still found that the HZ is stable. Our models also considered systems with $3 \mjup$, $5 \mjup$ and $10 \mjup$ and our results show that even for these high masses, if the initial eccentricity of the Earth-mass planet is less than $0.3$, then it is stable. To show the detectability of a $10 \mearth$ planet, we have also drawn a radial velocity (RV) contour of 1 ms$^{-1}$ (red curve), which indicates that a $10 \mearth$ planet in the HZ is detectable. A similar contour for an Earth mass planet is not shown because the precision required is extremely high. HD 164922 {#hd164922} ---------- [@Butler2006] discovered a Saturn-mass planet ($0.36 \mjup$) orbiting HD 164922 with a period of 1150 days ($a = 2.11$ AU) and an eccentricity of $0.1$. Although it has a low eccentricity, the uncertainty (0.14) is larger than the value itself. Therefore, the appropriate Saturn mass cases could legitimately use any $e$ in the range $0.0 < e < 0.25$ cases, but we use $e=0.1$. Figure \[observed\]b, shows the stable regions in the EHZ (green shaded) of HD 164922, for hypothetical Earth (magenta) and super-Earth (black) planets. The Saturn-mass planet (blue filled circle) is also shown at 2.11 AU. About $28 \%$ of the HZ in HD 164922 is stable for a $10 \mearth$ planet (for eccentricities $ \lsim 0.6$), whereas for Earth mass planets only $10 \%$ of the HZ is stable. We again show the detection limit for a $10 \mearth$ case. GJ 674 {#gj674} ------- GJ 674 is an M-dwarf star with a mass of $0.35 M_\odot$ and an effective temperature of $3600 $ K. [@bonfils2007] found a $12 \mearth$ with an orbital period and eccentricity of $4.69$ days ($a = 0.039$ AU) and $0.20$, respectively. Fig. \[observed\]c shows the EHZ of GJ 674 in $a - e$ space. Also shown are the known planet GJ 674 b (filled blue circle), EHZ (green shaded), and detection limit for an Earth-mass planet (red curve). The values of $\tau_\mathrm{s}$ for $1 \mearth$ and $10 \mearth$ planets, from Eq.\[tauearth\], are $0.973$ (magenta) and $1.0$ (black), respectively. Notice that the fraction of the HZ that is stable for $1 \mearth$ is slightly greater ($79.1 \%$) than $10 \mearth$ planet ($78.8 \%$), which differs from previous systems we considered here. A similar behavior can be seen in another system (HD 7924) that is discussed in the next section. It seems that when the planet mass ratio is approaching $1$, the HZ of a $10 \mearth$ mass planet offers less stability at high eccentricities ($> 0.6$) than a $1 \mearth$ planet. But as noted in §\[analtau\], this analysis should be weighted with the fact that our fitting procedure is not as accurate for a $10 \mearth$ planet than a $1 \mearth$ planet. HD 7924 {#hd7924} -------- Orbiting a K0 dwarf star at $0.057$ AU, the super-Earth HD 7924 b was discovered by NASA-UC Eta-Earth survey by the California Planet Search (CPS) group [@Howard2009], in an effort to find planets in the mass range of $3-30 M_\oplus$. It is estimated to have an $M \sin i = 9.26 M_\oplus$ with an eccentricity of $0.17$. Fig. \[observed\]d shows $\tau_\mathrm{s}$ values for hypothetical $10 M_\oplus$ (magenta) and $1 M_\oplus$ (black line) planets are $1.00$ and $0.98$, respectively. Unlike GJ 674, where only part of the HZ is stable, around $94 \%$ of HD 7924’s HZ is stable for these potential planets. Furthermore, we have also plotted an RV contour of $1$ ms$^{-1}$ arising from the $10 M_\oplus$ planet (red curve). This indicates that this planet may lie above the current detection threshold, and may even be in the HZ. [@Howard2009] do find some additional best-fit period solutions with very high eccentricities ($ e > 0.45$), but combined with a false alarm probability (FAP) of $> 20\%$, they conclude that these additional signals are probably not viable planet candidates. Further monitoring may confirm or forbid the existence of additional planets in this system. Fraction of stable HZ {#subfraction} --------------------- For astrobiological purposes, the utility of $\tau_\mathrm{s}$ is multi-fold. Not only is it useful in identifying stable regions within the HZ of a given system, but it can also provide (based on the range of $a$ & $e$) what fraction of the HZ is stable. We have calculated this fraction for all single planet systems in the Exoplanet Data Explorer, as of March 25 2010. The distribution of fractions of currently known single planet systems is shown in Fig. \[fraction\] and tabulated in Table \[fractable\]. A bimodal distribution can clearly be seen. Nearly $40 \%$ of the systems have more than $90 \%$ of their HZ stable and $38 \%$ of the systems have less than $10 \%$ of their HZ stable. Note that if we include systems with masses $> 10 \mjup$ and also $e > 0.6$ (which tend to have $a \sim 1$ AU [@wright2009]), the distribution will change and there will be relatively fewer HZs that are fully stable. Summary {#sec5} ======= We have empirically determined the Lagrange stability boundary for a planetary system consisting of one terrestrial mass planet and one massive planet, with initial eccentricities less than 0.6. Our analysis shows that for two-planet systems with one terrestrial like planet and one more massive planet, Eq.(\[tauearth\]) defines Lagrange stable configurations and can be used to identify systems with HZs stable for terrestrial mass planets. Furthermore, in Table \[fractable\] we provide a catalog of exoplanets, identifying the fraction of HZ that is Lagrange stable for terrestrial mass planets. A full version of the table is available in the electronic edition of the journal[^6]. In order to identify stable configurations for a terrestrial planet, one can calculate a stability boundary (denoted as $\tau_\mathrm{s}$ from Eq.(\[tauearth\])) for a given system (depending on the eccentricity and mass of the known planet), and calculate the range of $a$ & $e$ that can support a terrestrial planet, as shown in Section 4. For the transitional region between unstable to stable ($\tau_\mathrm{u} < \bbc < \tau_\mathrm{s}$), a numerical integration should be made. Our results are in general agreement with previous studies [@MT2003], [@Jones2006] & [@Sandor2007], but crucially our approach does not (usually) require a large suite of N-body integrations to determine stability. We have only considered two-planet systems, but the possibility that the star hosts more, currently undetected planets is real and may change the stability boundaries outlined here. However, the presence of additional companions will likely decrease the size of the stable regions shown in this study. Therefore, those systems that have fully unstable HZs from our analysis will likely continue to have unstable HZs as more companions are detected (assuming the mass and orbital parameters of the known planet do not change with these additional discoveries). The discovery of an additional planet outside the HZ that destabilizes the HZ is also an important information. As more extra-solar planets are discovered, the resources required to follow-up grows. Furthermore, as surveys push to lower planet masses, time on large telescopes is required, which is in limited supply. The study of exoplanets seems poised to transition to an era in which systems with the potential to host terrestrial mass planets in HZs will be the focus of surveys. With limited resources, it will be important to identify systems that can actually support a planet in the HZ. The parameter $\tau_{s}$ can therefore guide observers as they hunt for the grand prize in exoplanet research, an inhabited planet. Although the current work focuses on terrestrial mass planets, the same analysis can be applied to arbitrary configurations that cover all possible orbital parameters. Such a study could represent a significant improvement over the work of [@BG07]. The results presented here show that $\bbc=1$ is not always the Lagrange stability boundary, as they suggested. An expansion of this research to a wider range of planetary and stellar masses and larger eccentricities could provide an important tool for determining the stability and packing of exoplanetary systems. Moreover, it could reveal an empirical relationship that describes the Lagrange stability boundary for two planet systems. As new planets are discovered in the future, the stability maps presented here will guide future research on the stability of extra-solar planetary systems. R. K gratefully acknowledges the support of National Science Foundation Grants No. PHY 06-53462 and No. PHY 05-55615, and NASA Grant No. NNG05GF71G, awarded to The Pennsylvania State University. R.B. acknowledges funding from NASA Astrobiology Institutes’s Virtual Planetary Laboratory lead team, supported by NASA under Cooperative Agreement No. NNH05ZDA001C. This research has made use of the Exoplanet Orbit Database and the Exoplanet Data Explorer at exoplanets.org. The authors acknowledge the Research Computing and Cyberinfrastructure unit of Information Technology Services at The Pennsylvania State University for providing HPC resources and services that contributed to the research results reported in this paper. URL: http://rcc.its.psu.edu. 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A. 2009, arXiv0903.0652 Wright, J, T., et al. 2007, ApJ, 657, 533 Wright, J, T., et al. 2010, ApJ, 693, 1084 [cc]{} 10 M$_\mathrm{jup}$ & (0.25, 4)\ \ 5.6 M$_\mathrm{jup}$ & (0.25, 4)\ \ 3 M$_\mathrm{jup}$ & (0.25, 4)\ \ 1.77 M$_\mathrm{jup}$ & (0.25, 4)\ \ 1 M$_\mathrm{jup}$ & (0.25, 4)\ \ 1.86 M$_\mathrm{Sat}$ & (0.5, 2)\ \ 1 M$_\mathrm{Sat}$ & (0.5, 2)\ \ 56 $\mearth$ & (0.5, 2)\ \ 30 $\mearth$ & (0.5, 2)\ \ 17.7 $\mearth$ & (0.5, 2)\ \ 10 $\mearth$ & (0.5, 2)\ \ \[table1\] [|l|l|l||l|l|]{}\ Coefficients & $\tau_\mathrm{s}$ & $\tau_\mathrm{u}$ & $\tau_\mathrm{s}$ & $\tau_\mathrm{u}$\ $c_{1}$ & 1.0018 & 1.0098 &0.9868 & 1.0609\ &&&&\ $c_{2}$ & -0.0375 & -0.0589 &0.0024 & -0.3547\ &&&&\ $c_{3}$ & 0.0633 & 0.04196 &0.1438 & 0.0105\ &&&&\ $c_{4}$ & 0.1283 & 0.1078 &0.2155 & 0.6483\ &&&&\ $c_{5}$ & -1.0492 & -1.0139 &-1.7093 & -1.2313\ &&&&\ $c_{6}$ & -0.2539 & -0.1913 &-0.2485 & -0.0827\ &&&&\ $c_{7}$ & -0.0899 & -0.0690 &-0.1827 & -0.4456\ &&&&\ $c_{8}$ & -0.0316 & -0.0558 &0.1196 & -0.0279\ &&&&\ $c_{9}$ & 0.2349 & 0.1932 &1.8752 & 0.9615\ &&&&\ $c_{10}$ & 0.2067 & 0.1577 &-0.0289 & 0.1042\ &&&&\ $R^{2}$ & 0.996 & 0.997 &0.931 & 0.977\ &&&&\ $\sigma$ & 0.0065 & 0.0061 &0.0257 & 0.0141\ &&&&\ Max. dev. & 0.08 & 0.08 &0.15 & 0.05\ \[table0\] [ccccc]{} Rho CrB & 1.06 $M_\mathrm{jup}$ & 0.23 & 0.06 ($\pm$ 0.028) & 0.97\ \ HD 164922 & 0.36 $M_\mathrm{jup}$ & 2.11 & 0.05 ($\pm$ 0.14) & 0.94\ \ GJ 674 & 12 $\mearth$ & 0.039 & 0.20 ($\pm$ 0.02) & 0.35\ \ HD 7924 & 9.26 $\mearth$ & 0.057 & 0.17 ($\pm$ 0.16) & 0.832\ \ \[table2\] [cccccccccc]{} HD 142b & 1.3057 & 1.04292 & 0.26 & 0.9347 & 0.9323 & 0.000 & 0.9552 & 0.9320 & 0.000\ \ HD 1237 & 3.3748 & 0.49467 & 0.51 & 0.7407 & 0.7401 & 0.213 & 0.7549 & 0.7450 & 0.000\ \ HD 1461 & 0.0240 & 0.06352 & 0.14 & 0.9920 & 0.9780 & 0.976 & 1.0200 & 0.9200 & 0.959\ \ WASP-1 & 0.9101 & 0.03957 & 0.00 & 1.0022 & 0.9990 & 0.990 & 1.0200 & 0.9980 & 0.991\ \ HIP 2247 & 5.1232 & 1.33884 & 0.54 & 0.7138 & 0.7111 & 0.000 & 0.7490 & 0.7387 & 0.000\ \ \[fractable\] [^1]: [@Wittenmyer2009] did a comprehensive study of 22 planetary systems using the Hobby-Eberly Telescope (HET; [@Ramsey98]) and found no additional planets, but their study had a radial velocity (RV) precision of just $10 \sim 20$ ms$^{-1}$, which can only detect low-mass planets in tight orbits. [^2]: See <http://xsp.astro.washington.edu> for an up to date list of $\bbc$ values for the known extra-solar multiple planet systems. [^3]: A recent study by [@CY2009] notes that the Hill-radius criterion for ejection of a Earth mass planet around a giant planet may not be valid. Our stability maps shown here are, therefore, accurate to within the constraint highlighted by that study. [^4]: Note that a more thorough exploration of mass and eccentricity parameter space may indicate regions of resonances on both sides of the stability. Hence, we advice caution in applying our expression in those regions. [^5]: <http://exoplanets.org/> [^6]: Updates to this catalog is available at <http://gravity.psu.edu/~ravi/planets/>.

ArXiv

--- author: - Remya Nair - and Takahiro Tanaka title: 'Synergy between ground and space based gravitational wave detectors II: Localisation' --- Introduction ============ We are in an exciting era of gravitational wave (GW) astronomy. After the multiple GW detections by the LIGO-VIRGO network [@gw_det], and the successful pathfinder mission of Laser Interferometer Space Antenna (LISA) [@lisa_pf], we can now look forward to a future where multiple GW detections by both the space and ground based interferometers will be a norm. Astronomers rely on various cosmological observations to probe our Universe, some of which include: the type Ia supernovae, the baryon acoustic oscillations, the cosmic microwave background, gravitational lensing, etc. Requiring consistency between these measurements and combining them helped us converge on what we now know as the standard model of cosmology. Consistency checks between different observations of the same physical phenomenon help us identify the systematic effects. On the other hand, combining measurements aids parameter estimations by removing degeneracies in the parameter space and reducing the errors on the parameter estimates. The work we present here is an extension of our earlier work, where we demonstrated the advantage of combining measurements of ground and space based GW interferometers in estimating parameters of a compact binary coalescence [@synergy1]. Coalescing compact binaries which are composed of neutron stars (NS) - NS, NS- black hole (BH), or BH - BH produce GW signals during their inspiral, merger and ringdown phases. The merger and ringdown phases of at least five such events have been recorded by the LIGO-VIRGO GW detector network so far [@gw_det]. These ground based detectors are sensitive in the frequency range from a few tens of Hz to a few 1000 Hz. Till now we have seen BH-BH binary mergers with total mass ranging from $\sim 20~M_{\odot}$ to $\sim 70~ M_{\odot}$. There are already plans for third generation detectors like the Einstein Telescope (ET) of the European mission and the Cosmic explorer (CE) [@next_genGW]. ET and CE will detect binary coalescence with a higher signal to noise ratio (SNR) and possibly at lower frequencies than the second generation detectors like advanced LIGO , advanced VIRGO and KAGRA [@refET]. LISA on the other hand, would aim to observe supermassive BH binaries in the frequency range 0.1 mHz - 1Hz. Many studies have been performed to estimate the binary parameters (and additionally testing gravity theories) with LISA like detectors [@berti; @yagi_tanaka]. Recently there has been a lot of interest in the possibility of doing cosmography with LISA [@kyutoku_seto; @pozzo]. There is also a proposal for a Japanese space mission, Deci-Hertz Interferometer Gravitational Wave Observatory (DECIGO) for observing GW around $f \sim 0.1 - 10$ Hz. In such a scenario, one can ask how these detectors can complement each other. GW signals from the coalescing binaries that have passed beyond the LISA band will sweep through the DECIGO band before entering the frequency range of the ground based interferometers. Hence DECIGO can act as a follow up for the LISA mission and as a precursor for ground based detectors. DECIGO alone can determine the location of NS-NS sources to about an arcminute [@nakamura_decigo], which can aid the electro-magnetic follow up of these events by ground based detectors. Low frequency space detectors may also help in confirming those GW signals for which only the ringdown signals are detected by the ground-based detectors. In this spirit we studied non-spinning NS-BH compact binaries in [@synergy1] and for simplicity we ignored the information of the location of the source and its orientation with respect to the detector. Instead we used pattern averaged waveforms to show that low frequency space based interferometers like DECIGO can complement the observations of ground based measurements from third generation detectors like ET. In the present paper we extend our earlier work by incorporating information about the location and orientation of the source and the effect of spin to the GW phase. We further analyse two configurations for the detector orbit of DECIGO, one which is helio-centric and one which is geo-centric. Here we consider BH-BH binaries and focus on the improvement in the localisation accuracy obtained by combining space and ground measurements. In the absence of an electro-magnetic counterpart, such improvements will help us in identifying the host galaxies of these GW sources. In cases where we can identify host galaxies, these GW measurements will be further useful for cosmological studies. The paper is organized as follows. In §\[sec:non-spin\] and §\[sec:spin\] we briefly outline the expressions used for the GW phase in case of non-spinning binaries and spinning (non-precessing) binaries respectively. In §\[sec:error\] we discuss the basics of error estimation and the detector noise curves used in this study, and briefly outline the detector orbits in §\[det\_orbit\] (more details are available in the appendix). We report the results of our analysis in §\[result\], and discuss implications of our results and some future directions in §\[sec:impli\]. Non-spinning compact binaries {#sec:non-spin} ============================= Within general relativity, the post-Newtonian (PN) formalism is used to model the inspiral part of the binary evolution and the gravitational waveform. Physical quantities like the conserved energy, flux etc. are written as expansions in a small parameter $(v/c)$, where $v$ is the characteristic speed of the binary system and $c$ is the speed of light [@luc]. Corrections of [*O*]{}$((v/c)^n)$ (counting from the leading order) are referred to as a $(n/2)$PN order terms in the standard convention. For non-spinning systems, the GW amplitude is known up to 3PN order, whereas the phase and binary dynamics are known up to 3.5PN and 4PN order respectively (please refer to [@luc; @luc02; @luc04; @luc08; @dam14] and references therein). Spin corrections have been calculated to 2.5 PN order in phase and 2 PN order in amplitude in [@arun09] and Marsat et al., calculated 3 PN and 3.5 PN order spin-orbit phase corrections [@marsat13]. In an extension to our previous work [@synergy1], we now consider error estimations without averaging over the relative orientation of the binaries with respect to the detectors, to focus on the sky localisation. First we introduce two Cartesian reference frames following [@cutler_98]. One is a barred barycentric frame $(\bar{x},\bar{y},\bar{z})$ tied to the ecliptic and centered in the solar system barycentre. In this frame $\bar{\bm{e}}_z$ is normal to the ecliptic and $\bar{x}\bar{y}$-plane is aligned with the ecliptic plane. The second frame (unbarred) is the detector frame $(x,y,z)$ and is centered in the barycentre of the detector. In this frame the direction of the $z$ axis, $\bm{e}_z$, is normal to the detector plane (see Fig. 1 in [@YT]). We assume that the detector output of DECIGO consists of two independent interferometer outputs, much like LISA, and we introduce the standard mass variables to write the waveform. ${\cal M} = M\nu^{3/5}$ is the chirp mass, written in terms of the total mass $M=m_1+m_2$ and the symmetric mass ratio $\nu = (m_1 m_2)/M^2$ , where $m_1$ and $m_2$ are the component masses of the two compact objects in the binary. We begin by writing the frequency domain GW signal $h(f)$ under the stationary phase approximation, for non-spinning compact binaries which are in quasi-circular orbits. Here we use the *restricted PN waveforms*, which keeps the higher order terms in phase but only takes the leading order terms for the amplitude [@cutler]. This simplification is valid in the non-spinning/aligned-spin case, but for binaries with misaligned spins, the amplitude modulation on precession time scale may also be important to determine the spin parameters [@veccio]. Under the stationary phase approximation, the Fourier component of the waveform $h_{\alpha}(f)$ ($\alpha=1,2$ is the detector index) can be written as [@non_spin_PN] $$h_{\alpha}(f)= C {\cal A} f^{-7/6} e^{i\Psi(f)} \left\lbrace \frac{5}{4} A_{\alpha}(t(f))\right\rbrace e^{-i\left(\varphi_{p,\alpha}(t(f))+\varphi_D(t(f)) \right)}, \label{waveform}$$ where $C=\sqrt{3}/2$ and 1 for the case of DECIGO and ET, respectively. The amplitude ${\cal A}$ and the phase $\Psi(f)$ are given by $${\cal A} = \frac{1}{\sqrt{30} \pi^{2/3}} \frac{{\cal M}^{5/6}}{D_L},$$ $$\begin{aligned} \Psi(f) &= 2 \pi f t_c - \phi_c + \frac{3}{128} (\pi {\cal M}f)^{-5/3} \left\lbrace 1+ \left(\frac{3715}{756}+\frac{55}{9} \nu\right) \nu^{-2/5} (\pi {\cal M} f)^{2/3} - 16\pi \nu^{-3/5}(\pi {\cal M} f) \right. \nonumber \\ &+\left. \left(\frac{15293365}{508032}+ \frac{27145}{504} \nu+\frac{3085}{72}\nu^2\right)\nu^{-4/5}(\pi {\cal M} f)^{4/3}+ \pi\left(\frac{38645}{756} -\frac{65\nu}{9}\right) \right. \nonumber \\ & \times \left. \left(1+\log(6^{3/2}\nu^{-3/5}(\pi {\cal M} f))\right) \nu^{-1}(\pi {\cal M} f)^{5/3} + \left(\frac{11583231236531}{4694215680} -\frac{640}{3} \pi^2 - \frac{6848}{21}\gamma_E \right. \right. \nonumber\\ &+ \left. \left. \left[-\frac{15737765635}{3048192} + \frac{2255}{12} \pi^2\right]\nu+ \frac{76055}{1728}\nu^2 - \frac{127825}{1296} \nu^3 - \frac{6848}{63} \log(64 \nu^{-3/5}(\pi {\cal M} f)) \right) \right. \nonumber \\ & \times \left. \nu^{-6/5}(\pi {\cal M} f)^2 + \pi \left(\frac{77096675}{254016} + \frac{378515}{1512} \nu - \frac{74045}{756}\nu^2\right)\nu^{-7/5}(\pi {\cal M} f)^{7/3} \right\rbrace ,\end{aligned}$$ and the time evolution of the GW is given by $$\begin{aligned} t(f) &= t_c - \frac{5}{256} {\cal M} (\pi {\cal M}f)^{-8/3} \left\lbrace1+\frac{4}{3} \left(\frac{743}{336}+\frac{11}{4} \nu\right)\nu^{-2/5} (\pi {\cal M} f) ^{2/3} - \frac{32\pi}{5}\nu^{-1} (\pi {\cal M} f) \right. \nonumber \\ &+\left. 2\left(\frac{3058673}{1016064} + \frac{5429}{1008} \nu+\frac{617}{144}\nu^2\right) \nu^{-4/5}(\pi {\cal M} f)^{4/3} + \left(\frac{13\pi}{3} \nu - \frac{7729\pi}{252}\right)\nu^{-1}(\pi {\cal M} f)^{5/3} \right. \nonumber \\ &+ \left. 15\left(-\frac{10817850546611}{93884313600} + \left[\frac{15335597827}{60963840}+ \frac{1223992}{27720} - \pi^2 \frac{451}{48} -\frac{1041128}{27720}\right]\nu - \frac{15211}{6912}\nu^2 \right.\right. \nonumber \\ &+ \left. \left. \frac{25565}{5184}\nu^3 + \frac{1712}{105} \gamma_E + \frac{32}{3}\pi^2 + \frac{3424}{1575}\log\left(32768 \nu^{-2/5}(\pi {\cal M} f) ^{2/3}\right) \right) \nu^{-6/5} (\pi {\cal M} f)^2 \right. \nonumber \\ &+ \left. \left(\frac{14809\pi}{378} \nu^2 -\frac{75703\pi}{756} \nu -\frac{15419335\pi}{127008} \right) \nu^{-7/5} (\pi {\cal M} f)^{7/3} \right\rbrace.\end{aligned}$$ $D_L$ is the luminosity distance to the binary, $\gamma_E=0.577216 \cdots$ is the Euler’s constant, $t_c$ and $\phi_c$ are the time and phase at coalescence, respectively. The waveform polarization phase $\varphi_{p,\alpha}(t)$ and the polarisation amplitude $A_{\alpha}(t)$ are defined as: $$\begin{aligned} A_{\alpha}(t)&=&\sqrt{(1+(\hat{\bm{L}}\cdot\hat{\bm{N}})^2)^2F_{\alpha}^{+}(t)^2+4(\hat{\bm{L}}\cdot\hat{\bm{N}})^2F_{\alpha}^{\times}(t)^2}, \label{Apol} \\ \cos(\varphi_{\mathrm{p},\alpha}(t))&=&\frac{(1+(\hat{\bm{L}}\cdot\hat{\bm{N}})^2)F^{+}_{\alpha}(t)}{A_{\alpha}(t)}, \label{phipol1} \\ \sin(\varphi_{\mathrm{p},\alpha}(t))&=&\frac{2(\hat{\bm{L}}\cdot\hat{\bm{N}}) F^{\times}_{\alpha}(t)}{A_{\alpha}(t)}, \label{phipol2}\end{aligned}$$ where $\hat{\bm{L}}$ is the unit vector parallel to the orbital angular momentum and $\hat{\bm{N}}$ is the unit vector pointing toward the centre of mass of the binary system. $F_{\alpha}^+$ and $F_{\alpha}^{\times}$ are the beam pattern functions for the plus and cross polarisation modes for the detectors (see Appendix \[app\_detResponse\]). $(\bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}})$ represents the direction of the source in the barred-barycentric frame. We discuss the frequency cut-offs and the detector orbits in sections §\[sec:error\] and §\[det\_orbit\] respectively. Spinning compact binaries {#sec:spin} ========================= The efficiency of detection of GW signals from inspiraling binaries and the accuracy of the parameter estimation depend crucially on the accuracy of the templates used for matched filtering. Hence, while constructing these templates it is important to consider all the physical parameters which may effect the GW signal. The non-spinning or small spin approximation may work in some systems but it is important to consider the effect of spin on the waveform. In this work we restrict ourselves to spin-aligned (or antialigned), nonprecessing BH-binary systems. There are studies that show that including precession breaks the degeneracies in the parameter space and improves parameter estimation [@Lang06; @Vecc04; @Lang11]. We will consider this effect in a future publication. We now give the expression for the gravitational waveform used for the spin-aligned (anti-aligned) systems. Again we only take the post-Newtonian corrections in phase. Here we include spin corrections to the phase which include spin-orbit corrections at 1.5PN, 2.5PN, 3PN and 3.5PN order, and spin-spin corrections at 2PN order [@wade; @arun09; @marsat13]. $$\begin{aligned} \Psi^{\rm spin}(f) &= 2 \pi f t_c - \phi_c + \frac{3}{256} (\pi {\cal M}f)^{-5/3} \left\{1+\left(\frac{3715}{756} + \frac{55}{9}\nu\right) \nu^{-2/5} (\pi {\cal M}f)^{2/3} + \left(4 \beta - 16 \pi\right) \nu^{-3/5} \right. \nonumber \\ \nonumber &\times \left. (\pi {\cal M}f)+ \left(\frac{15293365}{508032} + \frac{27145}{504}\nu + \frac{3085}{72}\nu^2 - 10 \sigma\right) \nu^{-4/5} (\pi {\cal M}f)^{4/3} + \left(\frac{38645\pi}{756} - \frac{65\pi}{9}\nu - \gamma\right) \right. \\ \nonumber &\times \left. \left(1+3 \ln\left( \nu^{-1/5} (\pi {\cal M}f) \right)\right) \nu^{-1} (\pi {\cal M}f)^{5/3} + \left[\frac{11583231236531}{4694215680} - \frac{6848}{21} \gamma_{\rm E} - \frac{640 \pi^2}{3} \right. \right. \\ \nonumber &+ \left. \left. \left(\frac{2255 \pi^2}{12} - \frac{15737765635}{3048192}\right) \nu \right . + \frac{76055}{1728} \nu^2 - \frac{127825}{1296} \nu^3\left . - \frac{6848}{21} \ln\left(4 v_k\right) + \left(160 \pi \beta - 20 \xi\right) \right] \right. \\ \nonumber &\times \left. \nu^{-6/5} (\pi {\cal M}f)^{2} \left[ \frac{77096675\pi}{254016} + \frac{378515\pi\nu }{1512} - \frac{74045\pi\nu^2}{756} + \alpha \left(-20 \zeta + \gamma \left(-\frac{2229}{112} - \frac{99\nu}{4}\right) \right. \right . \right .\\ \nonumber &+\left. \left . \left . \beta \left(\frac{43939885}{254016}+\frac{259205\nu}{504} + \frac{10165 \nu^2}{36} \right)\right) \right] \nu^{-7/5} (\pi {\cal M}f)^{7/3} \right \} \ , \label{spinphase}\end{aligned}$$ with $$\begin{aligned} \beta &=& \sum_{i=1}^2\left(\frac{113}{12} \left(\frac{m_i}{M}\right)^2 + \frac{25}{4}\nu\right) \vec \chi_i \cdot \hat{\bm{L}}\ , \nonumber \\ \sigma &=& \nu \left[\frac{721}{48} \left(\vec \chi_1 \cdot \hat{\bm{L}}\right) \left(\vec \chi_2 \cdot \hat{\bm{L}} \right) - \frac{247}{48}\left( \vec \chi_1 \cdot \vec \chi_2\right) \right ] \sum_{i=1}^2 \left \{\frac{5}{2} \left(\frac{m_i}{M}\right)^2 \left[3 \left(\vec \chi_i \cdot \hat{\bm{L}} \right)^2 - \chi_i^2\right] \right . \nonumber \\ &&\left .+ \frac{1}{96} \left(\frac{m_i}{M}\right)^2\left[7 \chi_i^2 - \left( \vec \chi_i \cdot \hat{\bm{L}} \right)^2\right]\right \} \ , \nonumber \\ \gamma &=& \sum_{i=1}^2 \left [ \left ( \frac{732985}{2268} + \frac{140}{9} \nu\right)\left(\frac{m_i}{M}\right)^2 \right .\left. + \nu \left(\frac{13915}{84} - \frac{10}{3} \nu\right)\right]\vec \chi_i \cdot \hat{\bm{L}} \ , \nonumber \\ \xi &=& \sum_{\i=1}^2 \left[ \frac{75 \pi}{2} \left(\frac{m_i}{M}\right)^2 + \frac{151 \pi}{6} \nu\right] \vec \chi_i \cdot \hat{\bm{L}} \ , \nonumber \\ \zeta &=& \sum_{i=1}^2 \left[ \left(\frac{m_i}{M}\right)^2 \left (\frac{130325}{756} - \frac{796069}{2016} \nu + \frac{100019}{864} \nu^2 \right) + \nu \left(\frac{1195759}{18144} - \frac{257023}{1008} \nu \right. \right. \nonumber \\ && \left. \left. + \frac{2903}{32} \nu^2 \right) \right ] \vec \chi_i \cdot \hat{\bm{L}},\end{aligned}$$ where $\sigma$ is a spin-spin correction and $\beta$, $\gamma$, $\xi$ and $\zeta$ are spin-orbit corrections. $\alpha$ is either 1 or 0 to turn on or off the 3PN and 3.5PN order spin corrections to the phase. Here, $\vec \chi_i = \vec S_i / m_i^2$ are the dimensionless spins of the $i$th compact object of the binary. For BHs, the dimensionless spin parameters $\vec \chi_i$ is smaller than unity, while for NS, they can be larger in principle but are thought to be typically much smaller than unity. One can decompose the component spins $\chi_i$ into a symmetric and an antisymmetric combination, $$\begin{aligned} \vec \chi_s &=& \frac{1}{2} \left (\vec \chi_1 +\vec \chi_2\right), \\ \label{chi} \vec \chi_a &=& \frac{1}{2} \left( \vec \chi_1 - \vec \chi_2 \right) \ .\end{aligned}$$ Here, $\vec \chi_a \cdot \hat{\bm{L}} = \pm |\vec \chi_a|$ and $\vec \chi_s \cdot \hat{\bm{L}} = \pm |\vec \chi_s|$. Hence in addition to ${\cal M},\nu,t_c,\phi_c$, and the four angles specifying the location and orientation of the binary with respect to the detector, we now also have spin correction parameters. In the following section we introduce the Fisher matrix approach we use for error estimation, and provide the noise curves used in our analysis. Error estimation {#sec:error} ================ GW signals coming from the inspiral of compact binaries are very weak. To look for these signals in the noisy output of the GW interferometers, the technique of matched filtering is used [@bAllen]. Waveforms in a template bank are fitted to the detector output to extract signals that may match them. As can be expected, the effectiveness of such a method relies on accurate modeling of the GW signals as incorrect modeling can lead to systematic errors in the parameter estimations or missing the signal altogether. Next, to estimate statistical errors in the parameter estimates, the standard Fisher Matrix method can be used. We briefly give an overview of this method in this section but please refer to [@cutler; @finn; @valli08] for excellent reviews and details ([*and*]{} shortcomings) of the method. We start by assuming that the GW signal depends on the parameter vector $\boldsymbol{\theta}$. So in the non-spinning case $\boldsymbol{\theta}=\lbrace \log{\cal M}, \nu, t_c, \phi_c, \bar{\theta}_{\mathrm{L}}, \bar{\theta}_{\mathrm{S}}, \bar{\phi}_{\rm{L}}, \bar{\phi}_{\mathrm{S}} \rbrace$. For the spinning case we find that only the coefficient at the leading order spin corrections at 1.5PN and 2PN are enough to specify the higher order terms, assuming aligned spins. Hence we focus only on the leading order corrections here and we have $\boldsymbol{\theta}=\lbrace \log{\cal M}, \nu, t_c, \phi_c, \bar{\theta}_{\mathrm{L}}, \bar{\theta}_{\mathrm{S}}, \bar{\phi}_{\rm{L}}, \bar{\phi}_{\mathrm{S}}, \beta,\sigma \rbrace$. We study BH-BH binaries with component masses 30 $M_{\odot} +$ 40 $M_{\odot}$ located at a distance of 3 Gpc. The fiducial values of other parameters are $t_c=\phi_c=\beta=\sigma=0$. Choices for angles $\bar{\theta}_{\mathrm{L}}, \bar{\theta}_{\mathrm{S}}, \bar{\phi}_{\rm{L}}, {\rm and} ~\bar{\phi}_{\mathrm{S}}$ are explained in §\[nonS\]. Now we write the standard expressions used for obtaining the Fisher matrix. The noise weighted inner product of two waveforms $h_1(t)$ and $h_2(t)$ is defined as $$(h_1,h_2)=2 \int _0 ^{\infty} \frac{\tilde{h}_1^*(f) \tilde{h}_2(f)+ \tilde{h}_2^*(f) \tilde{h}_1(f)}{S_n(f)}df. \label{in_pro}$$ $\tilde{h}_1(f)$ and $\tilde{h}_2(f)$ are the Fourier transforms of $h_1(t)$ and $h_2(t)$, respectively, and “$*$” represents the complex conjugation. To account for the frequency dependent sensitivity of the GW interferometers, the outputs are weighted by the power spectral density of detector noise $S_n(f)$. The Fisher matrix is defined as [@cutler] $$\Gamma_{ij}\equiv \left(\frac{\partial { h}}{\partial \theta_i}, \frac{\partial { h}}{\partial \theta_j}\right). \label{fishm}$$ In the limit of large SNR, the probability that the signal is characterized by the chosen parameters $\boldsymbol{\theta}$ is given by $$P(\Delta\theta^i)\propto e^{-\Gamma_{ij}\Delta\theta^i\Delta\theta^j /2}.$$ In the limit of large SNR, and stationary Gaussian noise, the inverse of the Fisher matrix gives the error covariance matrix $\Sigma$ of the parameters. The diagonal elements of this covariance matrix give the root mean square error in the estimate of the parameters: $$\sqrt{\left\langle(\Delta\theta^i)^2\right\rangle}=\sqrt{\Sigma^{ii}}.$$ The angular resolution $\Delta \Omega$ is defined as $$\Delta \Omega \equiv 2 \pi |\sin \bar{\theta}_{\mathrm{S}}| \sqrt{\Sigma_{\bar{\theta}_{\mathrm{S}},\bar{\theta}_{\mathrm{S}}} \Sigma_{ \bar{\phi}_{\mathrm{S}}, \bar{\phi}_{\mathrm{S}}} - \Sigma^2_{ \bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}}}}.$$ Note that the Fisher matrix formalism is limited to high SNR cases (see [@cornish06; @rodri13; @valli11; @cho13; @cho14] for case studies where the Fisher matrix formalism fails). In spite of its limitations, the Fisher matrix method is the simplest and one of the most inexpensive ways to infer parameter uncertainties from future surveys. In this work our main aim is to study the synergy (or lack thereof) between ground-space detectors in a qualitative way and Fisher matrix method is accurate enough for this purpose. Moreover, we only study high SNR cases, SNR$>650$ for DECIGO and SNR$>20$ for B-DECIGO and the use of Fisher matrix for error estimation is justified. Hence we will adopt the Fisher matrix method for our error estimations. It is fairly straightforward to extend the Fisher matrix method to joint measurements. One merely needs to add the Fisher matrices of the individual measurements, $\Gamma_{\rm Combined} = \Gamma_{1}+\Gamma_{2}$, and then invert the summed matrix. This is how we obtain combined estimates while analyzing the synergy effect between DECIGO and ET. The covariance matrix for the combined measurement and the corresponding error estimate is given as $$\begin{aligned} \Sigma_{\rm Combined} &=& \Gamma_{\rm Combined}^{-1} ~,\\ \Delta \theta_{\rm Combined}^{i} &=& \sqrt{\Sigma_{\rm Combined}^{ii}} ~. \label{Ccomb}\end{aligned}$$ Combining measurements may help in resolving the degeneracy between parameters and hence improve the parameter estimates. Noise curves {#secnoise} ------------ The output of a GW interferometer $s(t)$, is composed of two components: the signal $h(t)$ and the detector noise $n(t)$, $s(t)=h(t)+n(t)$. We will assume that the detector noise is stationary and Gaussian, with zero mean $\left\langle \tilde{n} \right\rangle =0$ (note that this is not the case in actual observations). Here angular brackets denote average over different noise realizations. The assumption of stationarity ensures that the different Fourier components of the noise are uncorrelated. The (one-sided) noise power spectral density $S_n(f)$ is then given by $$\left\langle \tilde{n}(f)\tilde{n}(f') \right\rangle = \frac{1}{2}\delta(f-f')S_n(f).$$ The square root of the power spectral density is commonly used to describe the sensitivity of a GW interferometer. When $S_n(f)$ is integrated over positive frequencies, it gives mean square amplitude of the noise in the detector [@moore]. Below we give the expressions of the noise spectral densities of DECIGO and ET used in this work. ### DECIGO {#decigo .unnumbered} The Decihertz Interferometer Gravitational Wave Observatory (DECIGO) is a future plan of a space mission initially proposed by Seto et al. [@Seto:2001qf], with an aim of detecting GWs in the frequency range $f \sim 0.1-10$ Hz. Owing to its sensitivity range, DECIGO would be able to observe inspiral sources that have advanced beyond the frequency band of space based detector like LISA, but which have not yet entered the ground detector band. The following form for the DECIGO noise curve is adopted from Yagi and Seto [@seto]: $$\begin{split} S_n(f)=7.05 \times 10^{-48} \left[ 1+ \left(\frac{f}{f_p}\right)^2 \right] + 4.8 \times 10^{-51} \left(\frac{f}{1 \mbox{Hz}}\right)^{-4} \frac{1} {1+\left(\frac{f}{f_p}\right)^2}\\+5.33 \times 10^{-52}\left(\frac{f}{1 \mbox{Hz}}\right)^{-4} \mbox{Hz}^{-1}, \label{decigo_noise} \end{split}$$ where $f_p = 7.36$ Hz. ### ET {#et .unnumbered} The Einstein Telescope (ET) is a European commission project. The aim here is to develop a third generation GW observatory and achieve high SNR GW events at distances that are comparable with the sight distance of electromagnetic telescopes. We adopt the noise curve given by Keppel and Ajith [@ajith] which was obtained by assuming ET to be a single L-shaped interferometer with a 90$\degree$ opening angle and arm length of 10 km (ET-B): $$\begin{split} S_n(f)=10^{-50} \left[2.39 \times 10^{-27}\left(\frac{f}{f_0}\right)^{-15.64} +0.349 \left(\frac{f}{f_0}\right)^{-2.145} + 1.76 \left(\frac{f}{f_0}\right)^{-0.12} \right. \\ \left. +0.409 \left(\frac{f}{f_0}\right)^{1.1} \right]^2 \mbox{Hz}^{-1}, \end{split}$$ where $f_0=100$ Hz. The noise curves for the GW detectors are plotted in Fig. \[nc\] for easy reference. Varying design sensitivity for B-DECIGO --------------------------------------- There is a proposal for a precursor mission for DECIGO called B-DECIGO [@bdecigo]. The design sensitivity of B-DECIGO is still to be decided conditional on the scientific gain. Thus, in addition to assuming the noise sensitivity as given in Eq., we also study the synergy effect between the space detector and ET by varying the sensitivity of DECIGO. For brevity, we simply call it B-DECIGO. The detector sensitivity is changed by scaling the design sensitivity curve uniformly over all frequencies: $$S_n(f)^{\rm B-DECIGO}={\cal K} S_n(f)^{\rm DECIGO}, \label{sdecigo}$$ with a constant ${\cal K}$. Frequency cutoffs for the integral in Fisher Matrix {#sec:freq_cut} --------------------------------------------------- Now we discuss our choice of the frequency cutoffs, ($f_{{\rm in}},f_{{\rm fin}}$), for the integral in Eq. . First we introduce cutoff frequencies ($f_{{\rm low}},f_{{\rm high}}$) for the two GW detectors considered in this work. For DECIGO we choose $f_{{\rm low}} =0.01$ Hz, $f_{{\rm high }}=20$ Hz and for ET we choose $f_{{\rm low }} =10$ Hz, $f_{{\rm high }}=100$ Hz. We assume one year of observation in the space based band and the lower cutoff frequency for the integral is chosen as $f_{{\rm in }}={\rm max}(f_{{\rm year }},f_{{\rm low }})$, where $f_{{\rm year }}$ is the GW frequency 1 year before merger. We choose the upper cutoff frequency of the integral $f_{{\rm fin}}$ as $f_{{\rm fin}}={\rm min}(f_{{\rm high }},f_{{\rm LSO }})$, where $f_{{\rm LSO }}=1/(6^{3/2}\pi M)$ is the approximate frequency corresponding to the last stationary orbit. For the binary system we consider here, $f_{{\rm LSO }}\approx 62.9$ Hz. There is another proposed space based mission called LISA or eLISA ([@lisa_noise]). eLISA design is most sensitive at milli-Hertz frequencies, and hence more suitable for observing inspirals with much larger total mass. The binary systems that can be observed by the ground detectors do not have an extremely large total mass. We do not expect to observe high SNR events from the inspiral phase of such binaries using eLISA. If one considers $\sim$1 year of observation time, the SNR wouldn’t be enough to observe these low mass binaries using eLISA (multiband astronomy with eLISA may be possible with 5-10 years of observation, see [@elisa_multi]). In the following section we give a brief overview of the detector orbits for ET and DECIGO. Specific expressions for detector orbits are provided in appendix \[app\_detOrbit\]. Detector orbits {#det_orbit} =============== There are three detector orbits we consider here. The terrestrial detector, ET, would rotate, according the the Earth’s rotation, once every day and would orbit around the Sun every year. Similarly, if DECIGO is in a geo-centric orbit, it will reorient by orbiting around the Earth every $T_E$ (few hours) and orbit the Sun once a year. On the other hand if DECIGO is in a helio-centric orbit, it will orbit the Sun once per year. Both these orbital motions (helio-centric and geo-centric) contain information about the location of the source. For ET, the reorientation of the detector (with Earth’s rotation) has more contribution than its motion around the Sun (which is negligible). We briefly highlight the detector orbits in this section and for the expressions corresponding to the various detector orbits and terms to be used in -, we refer the readers to appendix \[app\_detOrbit\]. Orbit for ET {#orbit-for-et .unnumbered} ------------ The detector orbit of ET is obtained by assuming that it will be situated near Sardinia, Italy (lattitude 39$\degree$ N). As mentioned earlier, for such an orbit we account for the re-orientation of the detector with the Earth’s rotation and the Doppler effect due to the Earth’s motion around the Sun. Helio-centric orbit for DECIGO {#helio-centric-orbit-for-decigo .unnumbered} ------------------------------ We refer the readers to [@Seto:2001qf] for the original plan for DECIGO. This orbit is helio-centric much like LISA, but the interferometer arm-lengths are much shorter. Geo-centric orbit for B-DECIGO {#geo-centric-orbit-for-b-decigo .unnumbered} ------------------------------ The orbit of B-DECIGO is not fully determined yet. A possible alternative to the helio-centric orbit is a geo-centric orbit where the detector orbits the Earth in the same way (record plate orbit) as in the case of helio-centric orbit proposed in [@Seto:2001qf]. We further assume a Sun-synchronous orbit (which allows the detector to receive sunlight constantly). The orbital plane will precess because of the spin-orbit coupling and there will be added perturbations to the orbit because of the Moon, but the effect would be negligible. For this test case, we fix the distance of the detector from the Earth’s surface at $\sim$2600 km ($T_e \sim2.3$ hours) with the detector plane having an inclination of $\epsilon \sim 85.6 \degree$ with the ecliptic plane. The precession of the orbital plane of the record plate orbit is neglected. We account for the re-orientation of the detector with the rotation around the Earth in addition to the Doppler effect due to the Earth motion around the Sun. In the next section we discuss our findings. Results obtained when assuming a geo-centric orbit for DECIGO are labeled with ‘G’, while those from the helio-centric orbit configuration are labeled with ‘H’. Results {#result} ======= In this section we present the results of our analysis for the non-spinning and spin-aligned BH-BH binaries. We would like to emphasize that our results should be understood in a qualitative way. Since we only use the information coming from the inspiral phase of the binary coalescence, our results will contradict with studies that use the information from the inspiral-merger-ringdown evolution. Also, with ET-only measurements, we do not expect any localisation information and it is not possible to put constraints on the parameters. One way to get around this issue is to use a multi-detector terrestrial network composed of ET, LIGO-VIRGO, LIGO-INDIA and KAGRA. We leave this discussion for a future publication. Nonetheless, ET measurements help to remove degeneracy between parameters when combined with DECIGO measurements, thereby improving the error estimates. The error estimates obtained from DECIGO in helio-centric and geo-centric orbit configurations are similar in magnitude but we note that the synergy between the ground and space based measurements is larger for the case of helio-centric orbit, especially for the localisation errors. This is because the distance between the ground and space detector is larger in the case of the helio-centric orbit which aids in the parameter estimation, especially for sky localisation. In the following sections we report DECIGO-only and joint DECIGO-ET estimates for the no-spin and aligned-spin cases. Non-spinning BH-BH binary {#nonS} ------------------------- [Detector]{} [$\Delta t_c$]{} [$\Delta \phi_c$ ]{} [$\Delta {\cal M}/{\cal M} (\%) $ ]{} [$\Delta \nu/\nu (\%) $]{} [$\Delta \Omega~(\rm arcmin^2)$]{} [SNR]{} -------------- ---------------------- ---------------------- --------------------------------------- ---------------------------- ------------------------------------ ------------ DECIGO (H) $1.4 \times 10^{-1}$ $1.7 \times 10^{-2}$ $2.8 \times 10^{-6}$ $4.4 \times 10^{-3}$ $1.8 \times 10^{-1}$ $\sim 650$ Joint (H) $2.4 \times 10^{-3}$ $1.4 \times 10^{-2}$ $8.1 \times 10^{-7}$ $3.3 \times 10^{-3}$ $1.8 \times 10^{-3}$ DECIGO (G) $1.0 \times 10^{-1}$ $1.5 \times 10^{-2}$ $2.1 \times 10^{-6}$ $4.3 \times 10^{-3}$ $1.6\times 10^{-1}$ $\sim 677$ Joint (G) $1.0 \times 10^{-1}$ $1.1 \times 10^{-2}$ $2.0 \times 10^{-6}$ $3.2 \times 10^{-3}$ $1.4 \times 10^{-1}$ : Errors estimates for the binary coalescence parameters from DECIGO-only and joint DECIGO-ET measurements are reported for the non-spinning case. These are calculated for BH-BH binaries with masses $30~M_{\odot}$+$40~M_{\odot}$ with the distance fixed to 3 Gpc. The fiducial values of the parameters are chosen as: $t_c=\phi_c=0$ and, choices for the angles $\bar{\theta}_{\mathrm{L}}, \bar{\theta}_{\mathrm{S}}, \bar{\phi}_{\rm{L}}, {\rm and} ~\bar{\phi}_{\mathrm{S}}$ are explained in §\[nonS\]. The frequency cut-offs used for calculating the Fisher matrices are mentioned in §\[sec:freq\_cut\]. Corresponding to these cut-offs, the signal duration in the space and ground detector is $\sim$1 year and $\sim$4 seconds respectively. The first two rows of the table correspond to the helio-centric orbit (H) while the last two rows correspond to a geo-centric orbit (G) for DECIGO. \[tab1\] [Detector]{} [$\Delta t_c$]{} [$\Delta \phi_c$ ]{} [$\Delta {\cal M} /{\cal M} (\%) $ ]{} [$\Delta \nu /\nu (\%)$]{} [$\Delta \Omega~(\rm arcmin^2)$]{} [SNR]{} -------------- ---------------------- ----------------------- ---------------------------------------- ---------------------------- ------------------------------------ ----------- B-DECIGO (H) $4.4$ $5.5 \times 10^{-1} $ $8.8 \times 10^{-5}$ $1.4 \times 10^{-1}$ $1.8 \times 10^{2}$ $\sim 20$ Joint (H) $6.3 \times 10^{-2}$ $1.2 \times 10^{-1}$ $2.5 \times 10^{-5}$ $4.1 \times 10^{-2}$ $1.3 $ B-DECIGO (G) $3.3$ $4.9 \times 10^{-1} $ $6.7 \times 10^{-5}$ $1.3 \times 10^{-1}$ $1.6 \times 10^{2}$ $\sim 22$ Joint (G) $2.7$ $1.0 \times 10^{-1}$ $5.5 \times 10^{-5}$ $4.1 \times 10^{-2}$ $1.1 \times 10^{2}$ : Similar to Table \[tab1\] but with scaled-DECIGO (B-DECIGO) errors shown along with joint error estimates ($S_n(f)^{\rm B-DECIGO}=10^3 S_n(f)^{\rm DECIGO}$). \[tab2\] Now we tabulate and visualize the expected errors in the parameter estimation when we combine measurements of ground and space based detectors, ET and DECIGO. As mentioned earlier, we have considered a BH-BH binary systems with component masses $30~M_{\odot}+40~M_{\odot}$ and with the distance fixed at 3 Gigaparsecs (Gpc). To obtain the error estimates we uniformly distribute $10^4$ BH-BH sources over the sky. $\bar{\phi}_{\mathrm{S}}$ and $\bar{\phi}_{\mathrm{L}}$ are randomly generated in the range $\left[0, 2\pi \right]$ and $\cos \bar{\theta}_{\mathrm{S}}$ and $\cos \bar{\theta}_{\mathrm{L}}$ are randomly generated in the range $\left[-1, 1 \right]$. After computing the parameter errors for each such system, we group them into bins in a logarithmic distribution following Berti et.al., [@berti]. A source is assumed to belong to the $j$th bin if the error on some parameter $\theta$ satisfies $$\left[ \ln(\theta_{\min})+\frac{(j-1)[\ln(\theta_{\max})-\ln(\theta_{\min})]}{N_{\mathrm{bins}}} \right] < \ln(\theta) <\left[ \ln(\theta_{\min})+\frac{j[\ln(\theta_{\max})-\ln(\theta_{\min})]}{N_{\mathrm{bins}}} \right],$$ where $j=1,2,3...N$. Here $N$ is the total number of bins which we fix to 50. Once the errors are binned using the above relation, binaries in each bin are normalized (by dividing sources in each bin with the total number of binaries) to get a probability distribution of the error. Plots obtained from these histograms for each parameter error are shown in Fig. \[figD\_mc\_nu\] and \[figD\_omega\]. We report the results of combining the ET-DECIGO measurements in Table \[tab1\] and we see that although the improvement for most parameters is not significant (DECIGO dominates the error budget) the localisation is improved from $\sim$ 0.18 arcmin$^2$ to $\sim$ 1.8 $\times 10^{-3}$ arcmin$^2$ (first two rows). Similar results are obtained for the case of geo-centric DECIGO orbit (bottom two rows), but unlike the helio-centric case, here we see almost no improvement in localisation. Results obtained for B-DECIGO are shown in Figs. \[fig\_var\_mc\_nu\] and \[fig\_var\_omega\] where the variation in the error estimates are plotted with varying B-DECIGO sensitivity. We cut off this curve where the SNR for B-DECIGO falls below $\sim$10 (this happens when $S_n(f)^{\rm B-DECIGO}$ / $S_n(f)^{\rm DECIGO} \sim 10^{3}$). In Table \[tab2\] we report the error estimates for the case where we have maximum synergy between ET and B-DECIGO (with SNR threshold at $\sim$10). We see that one can obtain better estimates for all the parameters considered, and the maximum improvement is seen for the time of coalescence and localisation in the helio-centric DECIGO case. The localisation from B-DECIGO of $\sim$ $1.8 \times 10^2$ arcmin$^2$ is reduced to $\sim 1.3$ arcmin$^2$ when the ET measurements are included (top two rows in Table. \[tab2\]). Spin-(anti)aligned BH-BH binary ------------------------------- In this section we comment on the results for the spinning (non-precessing) case. As mentioned earlier, we have a 10 dimensional parameter space in this case. We forecast the parameter errors in a similar way as in the non-spinning case by constructing the Fisher matrix and subsequently inverting it to obtain the parameter errors. To include all the spin corrections (instead of just the leading order $\beta$ and $\sigma$) one can write the correction terms using the variables $s_1 = \vec \chi_s \cdot \hat{{\rm L}}$ and $s_2 = \vec \chi_a \cdot \hat{{\rm L}}$. The spin corrections are then given as: $$\begin{aligned} \beta &=& \left(\frac{113}{12} -\frac{19}{3} \nu \right) s_ 1+ \frac{113}{12} s_2 \delta \ , \nonumber \\ \sigma &=& \frac{81}{16}\left(s_1^2 + s_2^2 \right) - \frac{1}{4} \nu s_1^2 -20 \nu s_2^2 +\frac{81}{8} s_1 s_2 \delta \ , \nonumber \\ \gamma &=& \left ( \frac{732985}{2268} - \frac{24260}{81} \nu - \frac{340}{9} \nu^2 \right) s_1 + \left(\frac{732985}{2268} + \frac{140}{9} \nu \right) s_2 \delta \ , \nonumber \\ \xi &=& \left( \frac{75}{2} -\frac{74}{3}\nu \right) \pi s_1 + \frac{75}{2} \pi s_2 \delta \ , \nonumber \\ \zeta &=& \left (\frac{130325}{756} - \frac{1575529}{2592} \nu + \frac{341753}{864} \nu^2 -\frac{10819}{216} \nu^3 \right) s_1 + \left(\frac{130325}{756} - \frac{796069}{2016} \nu + \frac{100019}{864} \nu^2 \right) s_2 \delta , \nonumber \\ \hfill\end{aligned}$$ where $\delta = (m_1-m_2)/(m_1+m_2) = \sqrt{1-4 \nu}$. In this work we only report the results for the leading order spin-spin ($\sigma$) and spin-orbit ($\beta$) corrections. We find that the estimates for the chirp mass, symmetric mass ratio and the sky localisation are worsened, compared to the non-spinning case, if we account for $\sigma$ and $\beta$ (dot-dashed and dotted curves in Figs. \[figD\_mc\_nu\] and \[figD\_omega\_s\_b\]). We further note that although the cutoff frequency depends also on spins, here it is determined by the total mass only and our results may change if the effect of spins on $f_{\rm LSO}$ is taken into account. [Detector]{} [$\Delta t_c$]{} [$\Delta \phi_c$ ]{} [$\Delta {\cal M}/{\cal M}(\%) $ ]{} [$\Delta \nu/\nu (\%) $]{} [$\Delta \Omega~(\rm arcmin^2)$]{} [$\Delta \sigma$]{} [$\Delta \beta$]{} [SNR]{} -------------- ---------------------- ---------------------- -------------------------------------- ---------------------------- ------------------------------------ ---------------------- ---------------------- ------------ DECIGO (H) $4.2 \times 10^{-1}$ $1.6 \times 10^{-1}$ $6.2 \times 10^{-5}$ $2.4 \times 10^{-1}$ $4.8 \times 10^{-1}$ $9.6 \times 10^{-2}$ $2.2 \times 10^{-3}$ $\sim 650$ Joint (H) $2.6 \times 10^{-3}$ $5.2 \times 10^{-2}$ $2.1 \times 10^{-5}$ $8.6\times 10^{-2}$ $2 \times 10^{-3}$ $3.4 \times 10^{-2}$ $9.1 \times 10^{-4}$ DECIGO (G) $2.2 \times 10^{-1}$ $1.4 \times 10^{-1}$ $4.4 \times 10^{-5}$ $1.8 \times 10^{-1}$ $3.1 \times 10^{-1}$ $7.8 \times 10^{-2}$ $2.2 \times 10^{-3}$ $\sim 678$ Joint (G) $2.1 \times 10^{-1}$ $4.9 \times 10^{-2}$ $3.2 \times 10^{-5}$ $1.1 \times 10^{-1}$ $2.8 \times 10^{-1}$ $4.2 \times 10^{-2}$ $8.7 \times 10^{-4}$ : Errors estimates for the binary coalescence parameters from DECIGO-only and joint DECIGO-ET measurements are reported for the aligned-spin case. These are calculated for BH-BH binaries with masses $30~M_{\odot}$+$40~M_{\odot}$ with the distance fixed to 3 Gpc. The fiducial values of the parameters are chosen as: $t_c=\phi_c=\sigma=\beta=0$ and, choices for the angles $\bar{\theta}_{\mathrm{L}}, \bar{\theta}_{\mathrm{S}}, \bar{\phi}_{\rm{L}}, {\rm and} ~\bar{\phi}_{\mathrm{S}}$ are explained in §\[nonS\]. The frequency cut-offs used for calculating the Fisher matrices are mentioned in §\[sec:freq\_cut\]. Corresponding to these cut-offs, the signal duration in the space and ground detector is $\sim$1 year and $\sim$4 seconds respectively. The first two rows correspond to the helio-centric orbit (H) while the last two rows correspond to a geo-centric orbit (G) for DECIGO. \[tab3\] [Detector]{} [$\Delta t_c$]{} [$\Delta \phi_c$ ]{} [$\Delta {\cal M}/{\cal M}(\%)$ ]{} [$\Delta \nu/\nu(\%)$]{} [$\Delta \Omega~(\rm arcmin^2)$]{} [$\Delta \sigma$]{} [$\Delta \beta$]{} [SNR]{} -------------- ---------------------- ---------------------- ------------------------------------- -------------------------- ------------------------------------ ---------------------- ---------------------- ----------- B-DECIGO (H) $1.3 \times 10^{1}$ $4.9$ $1.9 \times 10^{-3}$ $7.5 $ $4.8 \times 10^{2}$ $3.0$ $7.0 \times 10^{-2}$ $\sim 20$ Joint (H) $6.5 \times 10^{-2}$ $4.3 \times 10^{-1}$ $5.5 \times 10^{-4}$ $2.0$ $1.4 $ $7.0 \times 10^{-1}$ $1.0 \times 10^{-2}$ B-DECIGO (G) $7.1 $ $4.5$ $1.4 \times 10^{-3}$ $5.7$ $3.3 \times 10^{2}$ $2.4$ $6.9 \times 10^{-2}$ $\sim 22$ Joint (G) $4.6$ $4.5 \times 10^{-1}$ $7.3 \times 10^{-4}$ $2.5$ $1.9 \times 10^{2}$ $8.5 \times 10^{-1}$ $1.1 \times 10^{-2}$ : Similar to Table \[tab3\] but with scaled-DECIGO errors shown along with joint error estimates ($S_n(f)^{\rm B-DECIGO}=10^3 S_n(f)^{\rm DECIGO}$). \[tab4\] The histograms for all the parameter errors are obtained in a similar manner as the non-spinning case and are shown in Figs. \[figD\_mc\_nu\] and \[figD\_omega\_s\_b\] (dot-dashed and dotted curves). Here also, we vary the B-DECIGO sensitivity to study the synergy between ground and space detector, and we find that similar to the non-spinning case, maximum synergy is obtained when $S_n(f)^{{\rm B-DECIGO}} = 10^3 S_n(f)^{{\rm DECIGO}}$ (with SNR threshold $>$10). For the helio-centric DECIGO case, the error estimates improve for all parameters when the measurements are combined, and again, as in the case of non-spinning binaries, we see that the maximum improvement is in time of coalescence and localisation. For ET-DECIGO pair we see that the localisation improves from $\sim 4.8 \times 10^{-1}$ arcmin$^2$ to $\sim 2 \times 10^{-3}$ arcmin$^2$ and in ET- (B-DECIGO) case it is improved from $\sim 4.8 \times 10^2$ arcmin$^2$ to $\sim$ 1.4 arcmin$^2$. For the geo-centric case we see that when the ET measurements are combined, better constraints are obtained for all the parameters except for localisation. As also mentioned earlier, compared to the geo-centric DECIGO orbit, the synergy between the space and ground detector is mildly larger in the case of helio-centric DECIGO orbit. For the symmetric mass ratio and chirp mass the difference is not very remarkable (Fig. \[fig\_var\_mc\_nu\]). But, for the case of localisation error, this difference is quite significant as seen in Fig. \[fig\_var\_omega\_s\_b\](a), where $\Delta \Omega^{\rm B-DECIGO}/\Delta \Omega^{\rm Joint}\sim1$ from geo-centric DECIGO. Again, this is because compared to the geo-centric DECIGO, the distance between the ground and space detector is larger in the case of the helio-centric DECIGO and this helps to improve localisation. Implications {#sec:impli} ============ In this paper we have assessed the expected synergy effects between ground and space based detectors, in the determination of binary coalescence parameters. For this, we study the estimated errors on parameters of these systems by considering $30$ $M_{\odot}$ + $40$ $M_{\odot}$ BH-BH binaries. This mass range corresponds to the first GW detection (GW150914), and is compatible with the mass ranges of subsequent BH-BH detections. We studied two cases: non spinning BH-BH binaries and spin-aligned (non-precessing) BH-BH binaries, with two different detector configuration for DECIGO (helio-centric and geo-centric). For the helio-centric DECIGO orbit we found that combining measurements with ET gave us better error estimates for all parameters and the gain was most significant for the time of coalescence and the localisation of the source. This improvement in localisation is very crucial for the future of GW astronomy, as it gives us a chance to identify the host galaxy. We did not find large synergy between the space and ground based detectors for the geo-centric DECIGO orbit case. Binary BH mergers are not expected to be accompanied by an electro-magnetic event but if the localisation of the source is good enough, one can expect to identify the host galaxies through galaxy catalogs or through dedicated survey of the localisation area. The large synoptic survey telescope which will begin science operation around 2022, will survey nearly 18,000 square degrees of the sky [@lsst] and the proposed BigBOSS survey is an all sky galaxy redshift survey, spanning redshift from $0.2<z<3.5$ [@bigb]. Hence, there is hope that by the time we achieve $\sim$sub-arcmin$^2$ accuracy on localisation, as is seen for the ET+(B-)DECIGO joint measurements, such surveys would have determined redshifts for a large fraction of the possible host galaxies (in a good fraction of the sky). Note that there are selection effects in all electro-magnetic surveys as there are mass (luminosity) cuts, and we are making an optimistic assumption that host galaxies would be seen by these surveys. In case the merger happens in galaxies that are not observed by these surveys, we would not be able to identify the host galaxy even if the sky localisation is small. Identifying the host galaxies of GW events would further give us a chance to do cosmological studies with GW observations. Here we highlight how this may be achieved by doing joint measurements. If we make some simple estimates for milky-way type galaxies, similar to those quoted in [@nissanke], we find that combining measurements from DECIGO and ET for the aligned-spin case reduces the number of possible host galaxies in the localisation region from $\sim$9 to $ \sim 1$, and for B-DECIGO+ET this number reduces from $\sim 8.6 \times 10^3$ galaxies to $\sim$24 galaxies (both numbers for helio-centric case with $S_n(f)^{\rm B-DECIGO}=10^3 S_n(f)^{\rm DECIGO}$). To get these numbers, we have assumed the Schechter luminosity function [@schechter] which provides a description of the density of galaxies as a function of their luminosity: $\rho_{gal}(x)dx = \phi^{*}x^a\exp^{-x}dx$, where $x=L/L^*$ and $L^*$ is some characteristic luminosity where the power law character of the function truncates. After making the same assumptions for $\phi^*$, $a$ and $L^*$ (from B-band measurements) as in [@nissanke], we get the galaxy density above $x_{1/2}$ (for $a = -1.07$, half of the luminosity density is contributed by galaxies with $x_{1/2}> 0.626$) as $2.35 \times 10^{-3}$ Mpc$^{-3}$. To get the numbers quoted above we multiply this density by a volume element $\Delta V = (4/3) ~\pi ~D_L^3 (\Delta \Omega/4 \pi)$. These numbers may further decrease if we take into account the distance estimate and the corresponding error since the volume element would significantly decrease ($\Delta V = \Delta \Omega~ D_L^3 ~(2 \times \Delta D_L/D_L)$) when we have good distance estimates. As an example we find that for the spin-aligned case, $\Delta D_L/D_L \sim 6.4 \%$ for (helio-centric) B-DECIGO measurements of binaries located at 3 Gpc. Incorporating this distance error estimate we find that the number of galaxies in the volume element reduces from $\sim 3000$ for B-DECIGO measurements to $\sim 9$ galaxies for B-DECIGO+ET measurements. For BH-BH binaries at a distance $\sim$400 Mpc (GW150914) these numbers would further reduce by a factor of $\sim$1000. Namely, at lower distances we can expect the number of potential host galaxies to reduce to just a few, even in the case of B-DECIGO+ET measurements. In cases where there are many galaxies within the sky localisation region, one can use the distance information obtained from GW measurements with a well motivated distance-redshift relation to rule out those galaxies which are at the right position on the sky but have significantly different redshifts [@cutler_holz]. We stress again that though these numbers may not be very robust, they are indicative of what can be observed in the future and this is very encouraging for the future of GW astronomy. In this work we only accounted for the leading order spin-spin and spin-orbit corrections to the phase of the GW waveform. Also, for simplicity we considered spin-aligned (non-precessing) waveforms. Including precession is very important for unequal-mass systems (NS-BH binaries) as unequal-mass systems precess more than equal-mass systems. There are many studies in the literature that explore the effect of including precession on the parameter estimation (for space-based detectors) [@Lang06; @Vecc04; @Lang11], and they have found that including precession can improves parameter estimation by breaking parameter degeneracies. Including eccentricity may also effect the parameter estimation ([@yagi_tanaka]), but since we focus on very late phase of the inspiral in the space based band, we have neglected the effect of eccentricity in our study, and used waveforms for quasi-circular orbits. In a recent paper, authors studied multi-band measurements of non-spinning binary NS and aligned-spin BH-BH systems with B-DECIGO/LISA and advanced LIGO/ET detectors. Neglecting the sky localisation information, they focused on the parameter estimation accuracy of mass, NS Love numbers, and the BH spins [@nakano]. One can also study the improvement in localisation of GW sources when a ground based detector network is considered instead of a single detector. It will be interesting to study the modifications in our results due to these effects (higher spin corrections, precession, eccentricity, ground based detector network) and we will consider them in future publications. Acknowledgment ============== R. N is an international research fellow of the Japan Society for the Promotion of Science (JSPS) and acknowledges support from JSPS grant No. 16F16025. T. T acknowledges support in part by MEXT Grant-in-Aid for Scientific Research on Innovative Areas, Nos. 17H06357 and 17H06358, and by Grant-in-Aid for Scientific Research Nos. 26287044 and 15H02087. Authors thank Chandra Kant Mishra and Nathan Johnson-McDaniel for discussions. Detector response to inspiraling binary signals {#app_detResponse} =============================================== In this section we briefly layout the expressions used to obtain the GW waveform. The beam pattern function in Eq. are given by: $$\begin{aligned} F_{\mathrm{I}}^{+}(\theta_{\mathrm{S}},\phi_{\mathrm{S}},\psi_{\mathrm{S}}) &=&\frac{1}{2}(1+\cos^2 \theta_{\mathrm{S}}) \cos(2\phi_{\mathrm{S}}) \cos (2\psi_{\mathrm{S}}) -\cos(\theta_{\mathrm{S}}) \sin(2\phi_{\mathrm{S}}) \sin(2\psi_{\mathrm{S}}), \\ F_{\mathrm{I}}^{\times}(\theta_{\mathrm{S}},\phi_{\mathrm{S}},\psi_{\mathrm{S}}) &=&\frac{1}{2}(1+\cos^2 \theta_{\mathrm{S}}) \cos(2\phi_{\mathrm{S}}) \sin (2\psi_{\mathrm{S}}) +\cos(\theta_{\mathrm{S}}) \sin(2\phi_{\mathrm{S}}) \cos(2\psi_{\mathrm{S}}). \label{beam-pattern}\end{aligned}$$ Here $(\theta_{\mathrm{S}},\phi_{\mathrm{S}})$ represents the direction of the source in the detector frame and $\psi_{\mathrm{S}}$ is the polarisation angle defined as $$\tan\psi_{\mathrm{S}} =\frac{\hat{\bm{L}}\cdot\hat{\bm{z}}-(\hat{\bm{L}}\cdot\hat{\bm{N}})(\hat{\bm{z}}\cdot\hat{\bm{N}})}{\hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})}. $$ DECIGO, since it has three arms, corresponds to having two individual detectors. Therefore, it is possible to measure both polarisations with one detector. One can reduce DECIGO to two independent interferometers with an equilateral triangle shape. If such an equilateral triangle is placed symmetrically inside the 90$\degree$ interferometer then the beam-pattern functions for the two detectors are the same as for a single detector, (except for the factor $\sqrt{3}/2$). The beam pattern function for the second detector output is given by $$\begin{aligned} F_{\mathrm{II}}^{+}(\theta_{\mathrm{S}},\phi_{\mathrm{S}},\psi_{\mathrm{S}})&=&F_{\mathrm{I}}^{+}(\theta_{\mathrm{S}},\phi_{\mathrm{S}}-\pi/4,\psi_{\mathrm{S}}), \\ F_{\mathrm{II}}^{\times}(\theta_{\mathrm{S}},\phi_{\mathrm{S}},\psi_{\mathrm{S}})&=&F_{\mathrm{I}}^{\times}(\theta_{\mathrm{S}},\phi_{\mathrm{S}}-\pi/4,\psi_{\mathrm{S}}).\end{aligned}$$ Detector orbit {#app_detOrbit} ============== While performing parameter estimation we take the direction of the source $(\bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}})$ and the direction of the orbital angular momentum $(\bar{\theta}_{\mathrm{L}},\bar{\phi}_{\mathrm{L}})$, both in the solar barycentric frame. Therefore we need to express the waveforms (especially $\hat{\bm{L}}\cdot\hat{\bm{N}}$ and the beam-pattern functions $F_{\alpha}^{+}$ and $F_{\alpha}^{\times}$ which appear in Eqs. (\[Apol\])-(\[phipol2\])) in terms of the barred coordinates $\bar{\theta}_{\mathrm{S}},\bar{\phi}_{\mathrm{S}},\bar{\theta}_{\mathrm{L}}$ and $\bar{\phi}_{\mathrm{L}}$. We express $\theta_{\mathrm{S}}(t)$, $\phi_{\mathrm{S}}(t)$ and other required quantities in terms of the barred coordinates, for the different detectors in the following section. ### ET {#et-1 .unnumbered} In the expressions below $\delta=39\degree$ specifies the location of the detector on the Earth (latitude), $\epsilon = 23.4 \degree$ is the inclination of Earth’s equator with respect to the ecliptic plane, $R_E$ is the radius of the Earth, $R_{AU}$ is the astronomical unit, $\phi_E = 2 \pi t [ (1/T_E)-(1/T) ]$ and $\bar{\phi}(t) = 2 \pi t/T$, where $T$ and $T_E$ correspond to 1 year and 1 day respectively. $$\begin{aligned} \cos \theta_{\mathrm{S}}(t)&=&\cos \bar{\theta}_{\mathrm{S}} \left(\cos \delta \cos \epsilon - \sin \delta \sin \epsilon \cos \phi_E \right) \nonumber \\ &+& \sin \bar{\theta}_{\mathrm{S}} \left[\cos \bar{\phi}_{\mathrm{S}} \left(\cos \delta \cos \epsilon + \sin \delta \cos \epsilon \cos \phi_E \right) -\sin \bar{\phi}_{\mathrm{S}} \cos \delta \sin \phi_E \right] , \\ \phi_{\mathrm{S}}(t)&=&\tan^{-1} \left( \frac{y_s}{x_s} \right),\end{aligned}$$ where $$\begin{aligned} x_s&=& \cos \bar{\theta}_{\mathrm{S}} \left(\sin \delta \cos \epsilon - \cos \delta \sin \epsilon \cos \phi_E \right) \nonumber \\ &&+ \sin \bar{\theta}_{\mathrm{S}} \left[ \cos \bar{\phi}_{\mathrm{S}} \left(\sin \delta \sin \epsilon + \cos \delta \cos \epsilon \cos \phi_E \right) + \sin \bar{\phi}_{\mathrm{S}} \cos \delta \sin \phi_E \right], \nonumber \\ y_s&=& \cos \bar{\theta}_{\mathrm{S}} \sin \epsilon \sin \phi_E + \sin \bar{\theta}_{\mathrm{S}} \left[ - \cos \bar{\phi}_{\mathrm{S}} \cos \epsilon \sin \phi_E +\sin \bar{\phi}_{\mathrm{S}} \cos \phi_E \right].\end{aligned}$$ The polarisation angle $\psi_{\mathrm{S}}$ is given as: $$\tan\psi_{\mathrm{S}}=\frac{\hat{\bm{L}}\cdot\hat{\bm{z}}-(\hat{\bm{L}}\cdot\hat{\bm{N}})(\hat{\bm{z}}\cdot\hat{\bm{N}})} {\hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})}.$$ where $\hat{\bm{z}}\cdot\hat{\bm{N}}=\cos\theta_{\mathrm{S}}$ and since in this work we neglect the spin precessional effects, $\hat{\bm{L}}$ is a constant. $\hat{\bm{L}}\cdot\hat{\bm{z}}$, $\hat{\bm{L}}\cdot\hat{\bm{N}}$, and $\hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})$ are given in terms of the barred coordinates by: $$\begin{aligned} \hat{\bm{L}}\cdot\hat{\bm{z}}&=&\cos \bar{\theta}_{\mathrm{L}} \left(\cos \delta \cos \epsilon - \sin \delta \sin \epsilon \cos \phi_E \right) \nonumber \\ &&+ \sin \bar{\theta}_{\mathrm{L}} \left[\cos \bar{\phi}_{\mathrm{L}} \left(\cos \delta \sin \epsilon + \sin \delta \cos \epsilon \cos \phi_E \right) +\sin \bar{\phi}_{\mathrm{L}} \sin \delta \sin \phi_E \right] \nonumber \label{lz} \\ \hat{\bm{L}}\cdot\hat{\bm{N}}&=&\cos\bar{\theta}_{\mathrm{L}}\cos\bar{\theta}_{\mathrm{S}} +\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}}\cos(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}), \nonumber \label{ln} \\ \hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})&=&\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}} \sin(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}) \left(\cos \delta \cos \epsilon + \sin \delta \sin \epsilon \cos \phi_E \right) \\ &&+\sin \delta \sin \phi_E (\cos\bar{\theta}_{\mathrm{S}}\cos\bar{\phi}_{\mathrm{L}}\sin \bar{\theta}_{\mathrm{L}} -\cos\bar{\theta}_{\mathrm{L}}\cos\bar{\phi}_{\mathrm{S}}\sin \bar{\theta}_{\mathrm{S}}) \nonumber \\ && + \left(\cos \delta \sin \epsilon + \sin \delta \cos \epsilon \cos \phi_E \right) (\cos\bar{\theta}_{\mathrm{L}}\sin \bar{\phi}_{\mathrm{S}} \sin \bar{\theta}_{\mathrm{S}} -\sin \bar{\theta}_{\mathrm{L}} \sin \bar{\phi}_{\mathrm{L}}\cos \bar{\theta}_{\mathrm{S}}) \nonumber \label{nlz}\end{aligned}$$ The Doppler phase which contains angular information is given by: $$\begin{aligned} \phi_D &=& 2 \pi f \left\lbrace R_{\rm AU} \sin \bar{\theta}_{\mathrm{S}} \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] + R_E \cos \theta_{\mathrm{S}}(t) \right \rbrace\end{aligned}$$ ### Helio-centric DECIGO {#helio-centric-decigo .unnumbered} For details on the the detector configuration we refer the readers to [@Seto:2001qf]. In the following expressions, $R_{AU}$ is the astronomical unit and $\bar{\phi}(t) = 2 \pi t/T$ where $T$ is equal to 1 year. We assume that the detector follows the same helio-centric orbit as the Earth, but keeping its position $\pi/9$ radians behind it. The location of the binary source is written in terms of the barred coordinates as: $$\begin{aligned} \cos \theta_{\mathrm{S}}(t)&=&\frac{1}{2}\cos \bar{\theta_{\mathrm{S}}} -\frac{\sqrt{3}}{2}\sin \bar{\theta_{\mathrm{S}}} \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}], \\ \phi_{\mathrm{S}}(t)&=&\tan^{-1} \left( \frac{\sqrt{3}\cos{\bar{\theta_{\mathrm{S}}}} +\sin \bar{\theta_{\mathrm{S}}}\cos [\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}]} {2\sin \bar{\theta_{\mathrm{S}}}\sin [\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}]} \right).\end{aligned}$$ The Doppler phase $\phi_D =2\pi f R_{AU}\sin \bar{\theta}_{\mathrm{S}} \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] $. Terms required to define the polarisation angle $\psi_{\mathrm{S}}$ are given as: $$\begin{aligned} \hat{\bm{L}}\cdot\hat{\bm{z}}&=&\frac{1}{2}\cos\bar{\theta}_{\mathrm{L}} -\frac{\sqrt{3}}{2}\sin\bar{\theta}_{\mathrm{L}} \cos[\bar{\phi}(t)-\bar{\phi}_{\mathrm{L}}], \label{lz} \\ \hat{\bm{L}}\cdot\hat{\bm{N}}&=&\cos\bar{\theta}_{\mathrm{L}}\cos\bar{\theta}_{\mathrm{S}} +\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}}\cos(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}), \label{ln} \\ \hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})&=&\frac{1}{2}\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}} \sin(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}) \notag \\ &&+\frac{\sqrt{3}}{2} \left \lbrace \cos\bar{\theta}_{\mathrm{L}} \sin\bar{\theta}_{\mathrm{S}} \sin [\bar{\phi}(t)- \bar{\phi}_{\mathrm{S}}] -\cos\bar{\theta}_{\mathrm{S}}\sin\bar{\theta}_{\mathrm{L}} \sin [ \bar{\phi}(t)- \bar{\phi}_{\mathrm{L}}] \right \rbrace\end{aligned}$$ ### Geo-centric DECIGO {#geo-centric-decigo .unnumbered} In the expressions below $\epsilon \sim 85.6 \degree$ is the angle between the detector plane and ecliptic plane, $R_E \sim 9000$km is the distance of the detector from the Earth, $R_{\rm AU}$ is the astronomical unit, $\bar{\phi}(t) = 2 \pi t/T$ and $\phi_E = 2 \pi t/T_e$, where $T$ and $T_e$ correspond to 1 year and $\sim$2.36 hours respectively. $$\begin{aligned} \cos \theta_{\mathrm{S}}(t)&=&\cos \bar{\theta}_{\mathrm{S}} \left(\frac{1}{2} \cos \epsilon + \frac{\sqrt{3}}{2} \sin \epsilon \cos \phi_E \right) \nonumber \\ &&+ \sin \bar{\theta}_{\mathrm{S}} \left[\left(-\frac{1}{2} \sin \epsilon + \frac{\sqrt{3}}{2} \cos \epsilon \cos \phi_E \right) \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] \right . \nonumber \\ &&\left .-\frac{\sqrt{3}}{2} \sin \phi_E \sin[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}]\right] , \\ \phi_{\mathrm{S}}(t)&=&\tan^{-1} \left( \frac{y_s}{x_s} \right),\end{aligned}$$ where $$\begin{aligned} x_s&=&\cos \bar{\theta}_{\mathrm{S}} \left(-\frac{\sqrt{3}}{2} \cos \epsilon + \frac{1}{2} \sin \epsilon \cos \phi_E \right) \nonumber \\ &&+ \sin \bar{\theta}_{\mathrm{S}} \left[\left(\frac{\sqrt{3}}{2} \sin \epsilon + \frac{1}{2} \cos \epsilon \cos \phi_E \right) \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] \right . \nonumber \\ &&\left .-\frac{1}{2} \sin \phi_E \sin[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}]\right], \nonumber \\ y_s&=&-\cos \bar{\theta}_{\mathrm{S}} \sin \epsilon \sin \phi_E - \sin \bar{\theta}_{\mathrm{S}} \left[ \cos \epsilon \sin \phi_E \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] \right . \nonumber \\ &&\left .+ \cos \phi_E \sin[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}]\right].\end{aligned}$$ Terms required to define the polarisation angle $\psi_{\mathrm{S}}$ are given as: $$\begin{aligned} \hat{\bm{L}}\cdot\hat{\bm{z}}&=&\cos \bar{\theta}_{\mathrm{L}} \left(\frac{1}{2} \cos \epsilon + \frac{\sqrt{3}}{2} \sin \epsilon \cos \phi_E \right) \nonumber \\ &&+ \sin \bar{\theta}_{\mathrm{L}} \left[\left(-\frac{1}{2} \sin \epsilon + \frac{\sqrt{3}}{2} \cos \epsilon \cos \phi_E \right) \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{L}}}] \right . \nonumber \\ &&\left .-\frac{\sqrt{3}}{2} \sin \phi_E \sin[\bar{\phi}(t)-\bar{\phi_{\mathrm{L}}}]\right] \nonumber \label{lz} \\ \hat{\bm{L}}\cdot\hat{\bm{N}}&=&\cos\bar{\theta}_{\mathrm{L}}\cos\bar{\theta}_{\mathrm{S}} +\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}}\cos(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}), \nonumber \label{ln} \\ \hat{\bm{N}}\cdot(\hat{\bm{L}}\times\hat{\bm{z}})&=&\sin\bar{\theta}_{\mathrm{L}}\sin\bar{\theta}_{\mathrm{S}} \sin(\bar{\phi}_{\mathrm{L}}-\bar{\phi}_{\mathrm{S}}) \left(\frac{1}{2} \cos \epsilon + \frac{\sqrt{3}}{2} \sin \epsilon \cos \phi_E \right) \\ &&+\cos\bar{\phi}(t)\left[\frac{\sqrt{3}}{2} \sin \phi_E (\cos\bar{\theta}_{\mathrm{S}}\cos\bar{\phi}_{\mathrm{L}}\sin \bar{\theta}_{\mathrm{L}} -\cos\bar{\theta}_{\mathrm{L}}\cos\bar{\phi}_{\mathrm{S}}\sin \bar{\theta}_{\mathrm{S}}) \right . \nonumber \\ &&\left . + \left(-\frac{1}{2} \sin \epsilon + \frac{\sqrt{3}}{2} \cos \epsilon \cos \phi_E \right) (\cos\bar{\theta}_{\mathrm{L}}\sin \bar{\phi}_{\mathrm{S}}\sin \bar{\theta}_{\mathrm{S}} -\sin \bar{\theta}_{\mathrm{L}} \sin \bar{\phi}_{\mathrm{L}}\cos \bar{\theta}_{\mathrm{S}}) \right] \nonumber \\ &&+\sin\bar{\phi}(t)\left[\frac{\sqrt{3}}{2} \sin \phi_E (\cos\bar{\theta}_{\mathrm{S}}\sin\bar{\phi}_{\mathrm{L}}\sin \bar{\theta}_{\mathrm{L}} -\cos\bar{\theta}_{\mathrm{L}}\sin\bar{\phi}_{\mathrm{S}}\sin \bar{\theta}_{\mathrm{S}}) \right . \nonumber \\ &&\left . + \left(-\frac{1}{2} \sin \epsilon + \frac{\sqrt{3}}{2} \cos \epsilon \cos \phi_E \right) (\sin\bar{\theta}_{\mathrm{L}}\cos \bar{\phi}_{\mathrm{L}}\cos \bar{\theta}_{\mathrm{S}} -\cos \bar{\theta}_{\mathrm{L}} \cos \bar{\phi}_{\mathrm{S}}\sin \bar{\theta}_{\mathrm{S}}) \right] \nonumber \label{nlz},\end{aligned}$$ and the Doppler phase is given by: $$\begin{aligned} \phi_D &=& 2 \pi f \left\lbrace R_E\cos \bar{\theta}_{\mathrm{S}} \sin \epsilon \cos \phi_E \right. \nonumber \\ && \left.+ \sin \bar{\theta}_{\mathrm{S}} \left[ \left( R_{AU}+ R_E \cos \epsilon \cos \phi_E \right) \cos[\bar{\phi}(t)-\bar{\phi_{\mathrm{S}}}] \right. \right. \nonumber \\ &&\left .\left. - 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ArXiv

--- abstract: 'In this paper, we discuss the distribution of the t-statistic under the assumption of normal autoregressive distribution for the underlying discrete time process. This result generalizes the classical result of the traditional t-distribution where the underlying discrete time process follows an uncorrelated normal distribution. However, for AR(1), the underlying process is correlated. All traditional results break down and the resulting t-statistic is a new distribution that converges asymptotically to a normal. We give an explicit formula for this new distribution obtained as the ratio of two dependent distribution (a normal and the distribution of the norm of another independent normal distribution). We also provide a modified statistic that is follows a non central t-distribution. Its derivation comes from finding an orthogonal basis for the the initial circulant Toeplitz covariance matrix. Our findings are consistent with the asymptotic distribution for the t-statistic derived for the asympotic case of large number of observations or zero correlation. This exact finding of this distribution has applications in multiple fields and in particular provides a way to derive the exact distribution of the Sharpe ratio under normal AR(1) assumptions.' author: - 'Eric Benhamou [^1] ^,^ [^2] ^,^ [^3]' bibliography: - 'mybib.bib' title: 'T-statistic for Autoregressive process' --- *AMS 1991 subject classification:* 62E10, 62E15 *Keywords*: t-Student, Auto regressive process, Toeplitz matrix, circulant matrix, non centered Student distribution Introduction ============ Let $X_1, \ldots, X_n$ be a random sample from a cumulative distribution function (cdf) $F(·)$ with a constant mean $\mu$ and let define the following statistics referred to as the t-statistic $$\label{tstatistic} T_n = T(X_n) = \frac{\sqrt{n} ( \bar X_n - \mu ) }{s_n}$$ where $\bar X_n $ is the empirical mean, $s_n^2$ the empirical Bessel corrected empirical variance, and $X_n$ the regular full history of the random sample defined by: $$\bar{X}_n =\frac{1}{n}\sum_{i=1}^{n}X_i, \quad s_n^2 = \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X}_n)^2, \quad X_n = (X_1, \ldots, X_n)^T$$ It is well known that if the sample comes from a normal distribution, $N(0, \sigma)$, $T_n$ has the Student t-distribution with $n - 1$ degrees of freedom. The proof is quite simple (we provide a few in the appendix section in \[t\_student\]). If the variables $(X_i)_{i=1..n}$ have a mean $\mu$ non equal to zero, the distribution is referred to as a non-central t-distribution with non centrality parameter given by $$\eta = \sqrt n \quad \frac{\mu}{\sigma}$$ Extension to weaker condition for the t-statistics has been widely studied. Mauldon [@Mauldon_1956] raised the question for which pdfs the t-statistic as defined by \[tstatistic\] is t-distributed with $d - 1$ degrees of freedom. Indeed, this characterization problem can be generalized to the one of finding all the pdfs for which a certain statistic possesses the property which is a characteristic for these pdfs. [@Kagan_1973], [@Bondesson_1974] and [@Bondesson_1983] to cite a few tackled Mauldon’s problem. [@Bondesson_1983] proved the necessary and sufficient condition for a t-statistic to have Student’s t-distribution with $d - 1$ degrees of freedom for all sample sizes is the normality of the underlying distribution. It is not necessary that $X_1,...,X_n$ is an independent sample. Indeed consider $X_1,...,X_n$ as a random vector $X_n = (X_1,...,X_n)^T$ each component of which having the same marginal distribution function, $F(·)$. [@Efron_1969] has pointed out that the weaker condition of symmetry can replace the normality assumption. Later, [@Fang_2001] showed that if the vector $X_n$ has a spherical distribution, then the t-statistic has a t-distribution. A natural question that gives birth to this paper was to check if the Student resulting distribution is conserved in the case of an underlying process $(X_i)_{i=1,\ldots }$ following an AR(1) process. This question and its answer has more implication than a simple theoretical problem. Indeed, if by any chance, one may want to test the statistical significance of a coefficient in a regression, one may do a t-test and rely upon the fact that the resulting distribution is a Student one. If by any chance, the observations are not independent but suffer from auto-correlation, the building blocks supporting the test break down. Surprisingly, as this problem is not easy, there has been few research on this problem. Even if this is related to the Dickey Fuller statistic (whose distribution is not closed form and needs to be computed by Monte Carlo simulation), this is not the same statistics. [@Mikusheva_2015] applied an Edgeworth expansion precisely to the Dickey Fuller statistics but not to the original t-statistic. The neighboring Dickey Fuller statistic has the great advantage to be equal to the ratio of two known continuous time stochastic process making the problem easier. In the sequel, we will first review the problem, comment on the particular case of zero correlation and the resulting consequence of the t-statistic. We will emphasize the difference and challenge when suddenly, the underlying observations are not any more independent. We will study the numerator and denominator of the t-statistic and derive their underlying distribution. We will in particular prove that it is only in the case of normal noise in the underlying AR(1) process, that the numerator and denominator are independent. We will then provide a few approximation for this statistic and conclude. AR(1) process ============= The assumptions that the underlying process (or observations) $(X_i)_{i=1,\ldots; n }$ follows an AR(1) writes : $$\label{AR_assumptions} \left\{ { \begin{array}{l l l l } X_t & = & \mu + \epsilon_t & \quad t \geq 1 ; \\ \epsilon_t & = & \rho \epsilon_{t-1} + \sigma v_t & \quad t \geq 2 ; \end{array} } \right.$$ where $v_t$ is an independent white noise processes (i.i.d. variables with zero mean and unit constant variance). To assume a stationary process, we impose $$\lvert {\rho} \rvert \leq 1$$ It is easy to check that equation \[AR\_assumptions\] is equivalent to $$\begin{array}{l l l l } X_t & = & \mu + \rho ( X_{t-1} - \mu )+ \sigma v_t & \quad t \geq 2 ; \end{array}$$ We can also easily check that the variance and covariance of the returns are given by $$\label{moment2} \begin{array}{l l l l } V(X_t) & = & \frac {\sigma^2} {1-\rho^2} \; \;\;\;\; \text{for} \; t \geq 1 \\ Cov(X_t, X_u ) & = & \frac {\sigma^2 \rho^{ \lvert {t -u} \rvert }} {1-\rho^2} \; \;\;\;\; \text{for} \; t,u \geq 1 \end{array}$$ Both expressions in \[moment2\] are independent of time $t$ and the covariance only depends on $\lvert {t -u} \rvert$ implying that $X_t$ is a stationary process. Case of Normal errors --------------------- If in addition, we assume that $v_t$ are distributed according to a normal distribution, we can fully characterize the distribution of $X$ and rewrite our model in reduced matrix formulations as follows: $$\label{reduced_matrix} X = \left( \begin{array}{c} X_1 \\ \vdots \\ X_n \end{array} \right) = \mu \cdot \mathbbm{1}_n + \sigma \cdot \epsilon = \mu \left( \begin{array}{c} 1 \\ \vdots \\ 1 \end{array} \right) + \sigma \left( \begin{array}{c} \epsilon_1 \\ \vdots \\ \epsilon_n \end{array} \right)$$ where $\epsilon \sim N \left( 0, \Omega = \left( \frac{ \rho ^{ | i - 1 | }}{1-\rho^2}\right)_{ij} \right)$, hence, $X \sim N \left( \mu \cdot \mathbbm{1}_n, \sigma^2 \Omega \right)$. The $\Omega$ matrix is a Toeplitz circulant matrix defined as $$\label{Omega} \Omega = \frac{1}{1 - \rho^2} \left( \begin{array}{l l l l l} {1} & {\rho} & \ldots & {\rho^{n-2}} & {\rho^{n-1}} \\ {\rho} & 1 & \ldots & {\rho^{n-3}} & {\rho^{n-2}} \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\rho^{n-2}} & {\rho^{n-3}} & \ldots & 1 & {\rho} \\ {\rho^{n-1}} & {\rho^{n-2}} & \ldots & {\rho} &{1} \end{array} \right) = M^T M$$ Its Chlolesky decomposition is given by $$\begin{aligned} \label{OmegaSqrt} M &= &\frac{1}{\sqrt{ 1- \rho^2} } \left( \begin{array}{l l l l l} {1} & 0 & \ldots & 0 & 0 \\ {\rho} & \sqrt{1 -\rho^2} & \ldots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\rho^{n-2}} & \rho^{n-3} \sqrt{1 -\rho^2} & \ldots & \sqrt{1 -\rho^2} & 0 \\ {\rho^{n-1}} & \rho^{n-2} \sqrt{1 -\rho^2} & \ldots & \rho \sqrt{1 -\rho^2} & \sqrt{1 -\rho^2} \end{array} \right) \end{aligned}$$ It is worth splitting $M$ into $I_n$ and another matrix as follows: $$\begin{aligned} M &= &\left( \begin{array}{l l l l l} 1+\frac{1-\sqrt{1-\rho^2}}{\sqrt{1-\rho^2}} & 0 & \ldots & 0 & 0 \\ \frac{\rho}{\sqrt{1-\rho^2}} & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \frac{\rho^{n-2}}{\sqrt{1-\rho^2}} & \rho^{n-3} & \ldots & 1 & 0 \\ \frac{\rho^{n-1}}{\sqrt{1-\rho^2}} & \rho^{n-2} & \ldots & \rho & 1 \end{array} \right) = \underbrace{I_n + \left( \begin{array}{l l l l l} \frac{1-\sqrt{1-\rho^2}}{\sqrt{1-\rho^2}} & 0 & \ldots & 0 & 0 \\ \frac{\rho}{\sqrt{1-\rho^2}} & 0 & \ldots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ \frac{\rho^{n-2}}{\sqrt{1-\rho^2}} & \rho^{n-3} & \ldots & 0 & 0 \\ \frac{\rho^{n-1}}{\sqrt{1-\rho^2}} & \rho^{n-2} & \ldots & \rho & 0 \end{array} \right) }_{I_n \; + \; \hspace{2.3cm} N \hspace{3cm}}\end{aligned}$$ The inverse of $\Omega$ is given by $$\label{OmegaInv} A = \Omega^{-1} = \left( \begin{array}{l l l l l} {1} & {-\rho} & \ldots & 0 & 0 \\ {-\rho} & {1+ \rho ^2 } & \ldots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & {1+ \rho ^2 } & {-\rho} \\ 0 & 0 & \ldots & {-\rho} & 1 \end{array} \right) = L^{T} L$$ Its Cholesky decomposition $L$ is given by $$\label{OmegaInvSqrt} L = \left( \begin{array}{l l l l l} \sqrt{1 -\rho^2} & 0 & \ldots & 0 & 0 \\ {-\rho} & 1 & \ldots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & 1 & 0 \\ 0 & 0 & \ldots & {-\rho} & 1 \end{array} \right)$$ Notice in the various matrix the dissymmetry between the first term and the rest. This shows up for instance in the first diagonal term of $L$ which is $\sqrt{1 -\rho^2}$, while all other diagonal terms are equal to 1. Similarly, in the matrix $N$, we can notice that the first column is quite different from the other ones as it is a fraction over $\sqrt{1 -\rho^2}$. T-statistics issue ------------------ The T-statistic given by equation \[tstatistic\] is not easy to compute. For the numerator, we have that $ \bar X_n - \mu$ follows a normal distribution. The proof is immediate as $\bar X_n $ is a linear combination of the Gaussian vector generated by the AR(1) process. We have $\bar X_n = \frac{1}{n} \mathbbm{1}_n \cdot X$. It follows that $\bar X_n \sim N( \mu, \frac{ \sigma^2 }{n^2} \mathbbm{1}_n^T \cdot \Omega \cdot \mathbbm{1}_n )$ (for a quick proof of the fact that any linear combination of a Gaussian vector is normal, see \[corr\_normal\]). In section \[Computations\], we will come back on the exact computation of the characteristics of the distribution of the numerator and denominator as this will be useful in the rest of the paper. As for the denominator, for a non null correlation $\rho$, the distribution of $s_n^2$ is not a known distribution.\ \ The distributions of the variables $\left(Y_i=X_i - \bar{X}_n \right)_{i=2, \ldots, n}$ are normal given by $ Y_ i \sim N( 0, \sigma_{Y_i} )$ with $\sigma_{Y_i} = (\delta_i - \frac 1 n \mathbbm{1}_n)^T \cdot \Omega \cdot (\delta_i - \frac 1 n \mathbbm{1}_n)$. where $ \delta_i = \underbrace{\left( 0, 0, \ldots, 1, \ldots, 0, 0 \right)^T }_{\text{1 at ith position}}$ Hence the square of these normal variables $Z_i=Y_i ^2$ is the sum of Gamma distributions. However, we cannot obtain a closed form for the distribution $s_n^2$ as the variance of the different terms are different and the terms are neither independent. If the correlation is null, and only in this specific case, we can apply the Cochran’s Theorem to prove that $s_n^2$ follows a Chi square distribution with $n-1$ degree of freedom. However, in the general case, we need to rely on approximation that will be presented in the rest of the paper. Another interesting result is to use the Cholesky decomposition of the inverse of the covariance matrix of our process to infer a modified t-statistic that has now independent terms and is defined as follows Let us take the modified process defined by $$U = L X$$ The variables $U$ is distributed according to a normal $U \sim N( \mu L \mathbbm{1}, Id_n )$. We can compute the modified T-statistic $\tilde{T}_n$ on $U$ as follows: $$\label{tstatistic2} \tilde{T}_n = \frac{\sqrt{n} ( \bar U_n - \mu ) }{\tilde{s}_n}$$ where $$\bar{U}_n =\frac{1}{n}\sum_{i=1}^{n}U_i, \quad \tilde{s}_n^2 = \frac{1}{n-1}\sum_{i=1}^{n}(U_i - \bar{U}_n)^2$$ In this specific case, the distribution of $\tilde{T}_n$ is a Student distribution of degree $n-1$. We will now work on the numerator and denominator of the T-statistic in the specific case of AR(1) with a non null correlation $\rho$. Expectation and variance of numerator and denominator {#Computations} ===================================================== The numerator of the T-statistic writes $$\begin{aligned} \sqrt{n} (\bar{X}_n - \mu) =\frac{1}{\sqrt{n}}\sum_{i=1}^{n}( X_i - \mu),\end{aligned}$$ Its expectation is null as each term is of zero expectation. Its variance is given by \[lemma\_var\_num\] $$\begin{aligned} \text{Var}(\sqrt n (\bar{X}_n-\mu)) &=& \frac{ \sigma^2}{1 - \rho^2} \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} \right] \label{lemma_var_num_eq1} \\ & =& \frac{ \sigma^2}{ (1 - \rho)^2} \left[ 1 - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)(1+\rho)} \right] \label{lemma_var_num_eq2}\end{aligned}$$ : See \[proof\_var\_num\] The proposition \[lemma\_var\_num\] is interesting as it states that the sample mean variance converges to $\cfrac{ \sigma^2}{(1 - \rho)^2}$ for large $n$. It is useful to keep the two forms of the variance. The first one (equation (\[lemma\_var\_num\_eq1\])) is useful in following computation as it shares the denominator term $1 - \rho^2$. The second form (equation \[lemma\_var\_num\_eq2\]) gives the asymptotic form. The denominator writes: $$\begin{aligned} s_n = \sqrt{ \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X}_n)^2},\end{aligned}$$ In the following, we denote by $Y_i = X_i - \mu$ the zero mean variable and work with these variables to make computation easier. We also write $Y_i^{\perp}$ the variable orthogonal to $Y_i$ whose variance (we sometimes refer to it as its squared norm to make notation easier) is equal to the one of $Y_i$ : ${\left\lVertY_i\right\rVert}^2 = {\left\lVertY_i^{\perp}\right\rVert}^2$. To see the impact of correlation, we can write for any $j>i$, $Y_j=\rho^{j-i} Y_i + \sqrt{ 1 - \rho^{2(j-i)} } Y_i^{\perp}$. As studying this denominator is not easy because of the presence of the square root, it is easier to investigate the properties of its squared given by $$\begin{aligned} s_n^2 = \frac{\sum_{i=1}^{n}(Y_i - \bar{Y}_n)^2}{n-1}= \frac{\sum_{i=1}^{n}Y_i^2 - n\bar{Y}_n^2}{n-1}\end{aligned}$$ We have that the mean of $\bar{Y}_n$ is zero while proposition \[lemma\_var\_num\] gives its variance : $$\begin{aligned} \text{Var}(\bar{Y}_n) & =& \frac{ \sigma^2}{n (1 - \rho^2)} \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} \right] = \frac{ \sigma^2}{ n (1 - \rho)^2} \left[ 1 - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)(1+\rho)} \right] \end{aligned}$$ \[lemma\_covar\] The covariance between $\bar{Y}_n$ and each stochastic variable $Y_j$ is useful and given by $$\begin{aligned} \label{lemma_covar_eq1} \text{Cov}(\bar{Y}_n, Y_j) = \frac{ \sigma^2}{ n (1 - \rho^2)} \left[ \frac{ 1 + \rho -\rho^{n+1-j}-\rho^{j}} {1-\rho} \right] \end{aligned}$$ In addition, we have a few remarkable identities $$\begin{aligned} \label{lemma_covar_eq2} \sum_{j=1}^n \text{Cov}(\bar{Y}_n, Y_j) = \frac{ \sigma^2}{ (1 - \rho^2)} \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} \right]\end{aligned}$$ $$\begin{aligned} \label{lemma_covar_eq3} \sum_{j=1}^n \left( \text{Cov}(\bar{Y}_n, Y_j) \right)^2= \frac{ \sigma^4}{ (1 - \rho^2)^2} \left[ \frac{ (1+\rho)^2 + 2 \rho^{n+1} } { (1-\rho)^2 } \frac{1}{n} - \frac{4 (1+\rho)^2 \rho (1-\rho^n) - 2 \rho^2 (1-\rho^{2n})} {(1-\rho)^2 (1-\rho^2)} \frac{1}{n^2} \right]\end{aligned}$$ : See \[proof\_covar\] We can now compute easily the expectation and variance of the denominators as follows \[prop\_denom\_expectation\] The expectation of $s_n^2$ is given by: $$\begin{aligned} \mathbb{E} {s_n^2} = \frac{ \sigma^2}{1 - \rho^2} \left( 1 - \frac{ 2 \rho }{(1-\rho) (n-1)} + \frac{ 2 \rho( 1-\rho^{n}) }{n (n-1) (1-\rho)^2} \right) \end{aligned}$$ : See \[proof\_denom\_expectation\] \[second\_moment\_denom\] The second moment of $s_n^2$ is given by: $$\begin{aligned} \mathbb{E}[s_n^4] & = \frac{\sigma^4}{(1-\rho^2)^2} \frac{1}{(n-1)^2} \left[ n^2-1 + \rho \left(n A_1 + A_2 + \frac{1}{n} A_3 + \frac{1}{n^2} A_4 \right) \right]\end{aligned}$$ with $$\begin{aligned} A_1 & = \frac{-4}{1-\rho^2} \\ A_2 & = \frac{- 2 \left(3 + 9 \rho + 11 \rho^2 + 3 \rho^3 + 6 \rho^n + 12 \rho^{n+1} + 6 \rho^{n+2}-2 \rho^{2n+2}\right)} {(1-\rho^2)^2} \\ A_3 & = \frac{ 4 (1 - \rho^n) (1 - 3 \rho + 4 \rho^2 - 8 \rho^{n+1})}{(1 -r)^3 (1 + r)} \\ A_4 & = \frac{12 \rho (1-\rho^{n})^2}{(1-\rho)^4} \end{aligned}$$ : See \[proof\_second\_moment\] Combining the two results leads to \[second\_moment\_denom\] The variance of $s_n^2$ is given by: $$\begin{aligned} \text{Var}[s_n^4] & = \frac{\sigma^4}{(1-\rho^2)^2} \frac{1}{(n-1)} \left[ 2 + \frac{ \rho}{n-1} \left( n B_1 + B_2 + \frac{1}{n} B_3 + \frac{1}{n^2} B_4 \right) \right]\end{aligned}$$ with $$\begin{aligned} B_1 & = \frac{-2}{1+ \rho} \\ B_2 & =-\frac{2}{1-\rho } -\frac{4 \rho^2}{(1-\rho )^2} -\frac{2 \left(1-\rho ^n\right)}{(1-\rho )^2} \\ & -\frac{2 \left(12 \rho^{n+1}+6 \rho ^{n+2}-2 \rho ^{2 n+2}+6 \rho ^n+3 \rho ^3+11 \rho^2+9 \rho +3\right)}{\left(1-\rho ^2\right)^2} \\ B_3 & = \frac{ (1-\rho^n)(13- 4 \rho+15 \rho^2-\rho^n - 32 \rho^{n+1} + \rho^{n+2}) }{(1 -r)^3 (1 + r)} \\ B_4 & = \frac{- 4 (1-3\rho) (1-\rho^{n})^2}{(1-\rho)^4} \end{aligned}$$ : See \[proof\_var\_moment\] It is worth noting that a direct approach as explained in [@Benhamou_2018_SampleVariance] could also give the results for the first, second moments and variance for the numerator and denominator. Resulting distribution ====================== The previous section shows that under the AR(1) assumptions, the t-statistic is no longer a Student distribution but the ratio of a normal whose first and second moments have been given above and the norm of a Gaussian whose moments have also been provided. To go further, one need to rely on numerical integration. This is the subject of further research. Conclusion ========== In this paper, we have given the explicit first, second moment and variance of the numerator of the t statistic under the assumption of AR(1) underlying process. We have seen that these moments are very sensitive to the correlation $\rho$ assumptions and that the distribution is far from a Student distribution. Various Proofs for the Student density ====================================== Deriving the t-student density {#t_student} ------------------------------ Let us first remark that in the T-statistic, the $\sqrt n$ factor cancels out to show the degree of freedom $\sqrt{n-1}$ as follows: $$T_n = \frac{\bar{X}\,-\,\mu}{s_n/\sqrt{n}} = \frac{\bar{X}\,-\,\mu}{\frac{\sigma}{\sqrt{n}}} \frac{1}{\frac{s_n}{\sigma}} = U \,\frac{1}{\frac{s_n}{\sigma}} = \sqrt{n-1} \frac{U}{\sqrt{\frac{\sum(X_i-\bar X)^2}{\sigma^2}}} = \sqrt{n-1} \frac{U}{V}$$ In the above expression, it is well know that if $X\sim \,\,\small N(\mu, \sigma)$, then the renormalized variable $U=\frac{(\bar{X}-\mu)}{\sigma/\sqrt{n}}\,\,\sim \,\,\small N(0,1)$ and $V = \sqrt{\frac{\sum(X_i-\bar X)^2}{\sigma^2}} \sim \,\,\small \chi_{(n-1)}^2$ as well as $U$ and $V$ are independent. Hence, we need to prove that the distribution of $T= U/{\sqrt{V/k}}$ is a Student distribution with $U\sim N(0,1),$ and $V\sim\chi^2_k$ mutually independent, and $k$ is the degree of freedom of the chi squared distribution.\ The core of the proof relies on two steps that can be proved by various means.\ **Step 1** is to prove that the distribution of $T$ is given by $$\label{t_student:step1_eq1} f_T(t) = \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \int_0^\infty e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} w^{\frac{ k-1}{2} } dw$$ **Step 2** is to compute explicitly the integral in equation \[t\_student:step1\_eq1\]\ Step 1 can be done by transformation theory using the Jacobian of the inverse transformation or the property of the ratio distribution. Step 2 can be done by Gamma function, Gamma distribution properties, Mellin transform or Laplace transform. Proving step 1 -------------- ### Using transformation theory The joint density of $U$ and $V$ is: $$f_{U,V}(u,v) = \underbrace{\frac{1}{(2\pi)^{1/2}} e^{-u^2/2}}_{\text{pdf } N(0,1)}\quad \underbrace{\frac{1}{\Gamma(\frac{k}{2})\,2^{k/2}}\,v^{(k/2)-1}\, e^{-v/2}}_{\text{pdf }\chi^2_k}$$ with the distribution support given by $-\infty < u < \infty$ and $0 < v < \infty$. Making the transformation $t=\frac{u}{\sqrt{v/ k}}$ and $w=v$, we can compute the inverse: $u=t\,\left(\frac{w}{k}\right)^{1/2}$ and $v=w$. The Jacobian [^4] is given by $$J(t,w)= \begin{vmatrix} \left(\frac{w}{k}\right)^{1/2} & \frac{t}{ 2 \left( k w \right)^{1/2} }\\ 0 & 1 \end{vmatrix}$$ whose value is $(w/ k)^{1/2}$. The marginal pdf is therefore given by: $$\begin{aligned} f_T(t) & = & \displaystyle\int_0^\infty \,f_{U,V}\bigg(t\,(\frac{w}{k})^{1/2},w\bigg) J(t,w) \,\mathrm{d} w \\ & = & \displaystyle\int_0^\infty \frac{1}{(2\pi)^{1/2}} e^{-(t^2 \, \frac{w}{k}) /2} \frac{1}{\Gamma(\frac{k}{2})\,2^{k/2}} w^{(k/2)-1}\, e^{-w/2} (w/ k )^{1/2} \,\mathrm{d} w \\ & = & \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \int_0^\infty e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} w^{\frac{ k-1}{2} } dw \end{aligned}$$ which proves the result\ \ ### Using ratio distribution The square-root of $V$, $\sqrt V \equiv \hat V$ is distributed as a chi-distribution with $k$ degrees of freedom, which has density $$f_{\hat V}(\hat v) = \frac {2^{1-\frac k 2}}{\Gamma\left(\frac {k}{2}\right)} \hat v^{k-1} \exp\Big \{{-\frac {\hat v^2}{2}} \Big\} \label{t_student:step1_eq21}$$ Define $X \equiv \frac {\hat V}{\sqrt k}$. Then by change-of-variable, we can compute the density of $X$: $$\begin{aligned} f_{X}(x) &= & f_{\hat V}(\sqrt k x)\Big |\frac {\partial \hat V}{\partial X} \Big| \\ &= & \frac {2^{1-\frac k 2}}{\Gamma\left(\frac {k}{2}\right)} k^{\frac k 2}x^{k-1} \exp\Big \{{-\frac {k \, x^2}{2}} \Big\} \label{t_student:step1_eq22}\end{aligned}$$ The student’s t random variable defined as $T = \frac {Z} {X}$ has a distribution given by the ratio distribution: $$f_T(t) = \int_{-\infty}^{\infty} |x|f_U(xt)f_X(x)dx$$ We can notice that $f_X(x) = 0$ over the interval $[-\infty, 0]$ since $X$ is a non-negative random variable. We are therefore entitled to eliminate the absolute value. This means that the integral reduces to $$\begin{aligned} f_T(t) & =& \int_{0}^{\infty} xf_U(xt)f_X(x)dx \\ & = & \int_{0}^{\infty} x \frac{1}{\sqrt{2\pi}}\exp \Big \{{-\frac{(xt)^2}{2}}\Big\}\frac {2^{1-\frac k 2}}{\Gamma\left(\frac {k}{2}\right)} k^{\frac k 2}x^{k-1} \exp\Big \{{-\frac {k}{2}x^2} \Big\}dx \\ & = & \frac{1}{\sqrt{2\pi}}\frac {2^{1-\frac k 2}}{\Gamma\left(\frac {k}{2}\right)} k^{\frac k 2}\int_{0}^{\infty} x^k \exp \Big \{-\frac 12 (k +t^2) x^2\Big\} dx \label{proof1:student3}\end{aligned}$$ To conclude, we make the following change of variable $x=\sqrt{\frac w k}$ that leads to $$f_T(t) = \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \int_0^\infty e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} w^{\frac{ k-1}{2} } dw$$ Proving step 2 -------------- The first step is quite relevant as it proves that the integral to compute takes various form depending on the change of variable done. ### Using Gamma function Using the change of variable $w = \frac{ 2 k u}{t^2 + k }$ and knowing that $\Gamma(n) =\int_0^\infty e^{-u}u^{n-1}\ du$, we can easily conclude as follows: $$\begin{aligned} f_T(t) &=& \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \int_0^\infty e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} w^{\frac{ k-1}{2} } dw \\ &= &\frac{1}{\Gamma(\frac{k}{2})2^{\frac{k+1}{2}}\sqrt{\pi k}} \bigg(\frac{2 k}{t^2+k} \bigg)^{\frac{k+1}{2}}\int_0^\infty e^{-u}u^{\frac{k+1}{2}-1}\ du \\ &= & \frac{1}{\Gamma(\frac{k}{2})2^{\frac{k+1}{2}}\sqrt{\pi k}} \bigg(\frac{2 k}{t^2+k} \bigg)^{\frac{k+1}{2}}\Gamma\Big(\frac{k+1}{2}\Big) \\ & =& \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})}\frac{1}{\sqrt{\pi k}}\bigg( \frac{k }{t^2+ k}\bigg)^{\frac{k+1}{2}}\end{aligned}$$ ### Using Gamma distribution properties Another way to conclude is to notice the kernel of a gamma distribution pdf given by $x^{\alpha-1}\,e^{x\,\lambda}$ in the integral of \[t\_student:step1\_eq1\] with parameters $\alpha=(k+1)/2,\,\lambda=(1/2)(1+t^2/k)$. The generic pdf for the gamma distribution is $\large \frac{\lambda^\alpha}{\Gamma(\alpha)}\,x^{\alpha-1}\,e^{x\,\lambda}$ and it sums to one over $\left[ 0, \infty \right]$, hence $$\begin{aligned} f_T(t) &=& \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \int_0^\infty e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} w^{\frac{ k-1}{2} } dw \\ & = & \frac{1}{\Gamma(\frac{k}{2}) 2^{\frac{k+1}{2}}\sqrt{\pi k}} \frac{ \Gamma(\frac{k+1}{2}) }{ (\frac{t^2+ k}{2 k})^{\frac{k+1}{2} } } \\ & =& \frac{\Gamma(\frac{k+1}{2})}{\Gamma(\frac{k}{2})}\frac{1}{\sqrt{\pi k}}\bigg( \frac{k }{t^2+ k}\bigg)^{\frac{k+1}{2}}\end{aligned}$$ ### Using Mellin transform The integral of equation \[t\_student:step1\_eq1\] can be seen as a Mellin transform for the function $g(x) = e^{-w (\frac{t^2}{2 k}+\frac{1}{2})} $, whose solution is well known and given by $$\begin{aligned} \mathcal{M}_g(\frac{ k+1}{2} ) \equiv \int_0^{\infty} x^{\frac{ k+1}{2} -1 } g(x) dx = \frac{ \Gamma(\frac{k+1}{2}) }{ (\frac{t^2+ k}{2 k})^{\frac{k+1}{2} } } \label{proof1:student4} \end{aligned}$$ Like previously, this concludes the proof. ### Using Laplace transform We can use a result of Laplace transform for the function $ f(u) = u^{\alpha}$ as folllows: $$\mathcal{L}_{f}(s) = \int_0^\infty e^{-u s }u^{\alpha} du = \frac{ \Gamma(\alpha + 1 ) }{ s ^{ \alpha + 1 }}$$ Hence the integral $\int_0^\infty e^{-u}u^{\frac{k+1}{2}-1}\ du$ is simply the the value of the Laplace transform of the polynomial function taken for $s=1$, whose value is $\Gamma\Big(\frac{k+1}{2}\Big)$. Making the change of variable $w = \frac{ 2 k u}{t^2 + k }$ in equation \[t\_student:step1\_eq1\] enables to conclude similarly to the proof for the Gamma function ### Using other transforms Indeed, as the Laplace transform is related to other transform, we could also prove the result with Laplace–Stieltjes, Fourier, Z or Borel transform. Sum of independent normals {#sum_indep_normal} -------------------------- We want to prove that if $X_i \sim N(0,1)$ then $\frac 1 {n-1} \sum_{i=1}^n (X_i-\bar X_n)^2 \sim \chi^2_{n-1}$. There are multiple proofs for this results: - Recursive derivation - Cochran’s theorem ### Recursive derivation Let us remind a simple lemma: - If $Z$ is a $N(0, 1)$ random variable, then $Z^2 \sim \chi^2_1$; which states that the square of a standard normal random variable is a chi-squared random variable. - If $X_1, \ldots, X_n$ are independent and $X_i \sim \chi^2_{p_i}$ then $X_1 + \ldots + X_n \sim \chi^2_{p_1+ \ldots + p_n}$, which states that independent chi-squared variables add to a chi-squared variable with its degree of freedom equal to the sum of individual degree of freedom. The proof of this simple lemma can be established with variable transformations for the fist part and by moment generating function for the second part. We can now prove the following proposition If $X_1, \ldots, X_n$ is a random sample from a $N( \mu, \sigma^2)$ distribution, then - $\bar{X}_n$ and $s_n^2$ are independent random variables. - $\bar{X}_n$ has a $N( \mu, \sigma^2 / n )$ distribution where $N$ denotes the normal distribution. - $(n-1) s_n^2 / \sigma^2 $ has a chi-squared distribution with $n - 1$ degrees of freedom. Without loss of generality, we assume that $\mu = 0$ and $\sigma = 1$. We first show that $s_n$ can be written only in terms of $\left( X_i - \bar{X}_n \right)_{i=2, \ldots, n}$. This comes from: $$\begin{aligned} s_n^2 & = & \frac{1}{n-1}\sum_{i=1}^{n}(X_i - \bar{X}_n)^2 = \frac{1}{n-1} \left[ (X_1 - \bar{X}_n)^2 + \sum_{i=2}^{n}(X_i - \bar{X}_n)^2 \right] \\ & = & \frac{1}{n-1} \left[ (\sum_{i=2}^{n}(X_i - \bar{X}_n))^2 + \sum_{i=2}^{n}(X_i - \bar{X}_n)^2 \right] \end{aligned}$$ where we have use the fact that $\sum_{i=1}^{n}(X_i - \bar{X}_n)= 0$, hence $X_1 - \bar{X}_n = - \sum_{i=2}^{n}(X_i - \bar{X}_n)$. We now show that $s_n^2$ and $\bar{X}_n$ are independent as follows: The joint pdf of the sample $X_1, \ldots, X_n$ is given by $$f(x_1, \ldots, x_n) = \frac 1 {(2 \pi )^{n/2}} e^{- \frac 1 2 \sum_{i=1}^{n} x_i^2}, \quad -\infty < x_i < \infty.$$ We make the $$\begin{aligned} y_1 &= & \bar x \\ y_2 & = & x_2 - \bar x \\ \vdots \\ y_n & = & x_n - \bar x \end{aligned}$$ The Jacobian of the transformation is equal to $1/n$. Hence $$\begin{aligned} f(y_1, , \ldots, y_n) & =& \frac n {(2 \pi )^{n/2}} e^{- \frac 1 2 (y_1 - \sum_{i=2}^{n} y_i)^2 } e^{- \frac 1 2 \sum_{i=2}^{n} (y_i + y_1)^2}, \quad -\infty < x_i < \infty \\ & = & [ (\frac n {2 \pi} ) ^{1/2} e^{- \frac n 2 y_1^2 } ] [ \frac {n^{1/2}}{ (2 \pi) ^{(n-1)/2} } e ^{ - \frac 1 2 [ \sum_{i=2}^{n} y_i^ 2 + (\sum_{i=2}^{n} y_i)^2 ] } ] \end{aligned}$$ which proves that $Y_1 = \bar X_n$ is independent of $Y_2, \ldots, Y_n$, or equivalently, $\bar{X}_n$ is independent of $s_n^2$. To finalize the proof, we need to derive a recursive equation for $s_n^2$ as follows: We first notice that there is a relationship between $\bar x_n$ and $\bar x_{n+1}$ as follows: $$\begin{aligned} \bar x_{n+1} = \frac{\sum_{i=1}^{n+1} x_i }{n + 1} = \frac{x_{n+1} + n \bar{x}_n}{n + 1} = \bar x_n + \frac{1}{n + 1} (x_{n+1}- \bar x_n),\end{aligned}$$ We have therefore: $$\begin{aligned} n s^2_{n+1} &= & \sum_{i=1}^{n+1} (x_i - \bar x_{n+1})^2 = \sum_{i=1}^{n+1} [ (x_i - \bar x_{n}) - \frac{1}{n+1}( x_{n+1} - \bar x_n) ] ^2 \\ &=& \sum_{i=1}^{n+1} [ (x_i - \bar x_{n})^2 - 2 (x_i - \bar x_n) ( \frac{ x_{n+1}-\bar x_n }{n+1} ) + \frac{1}{(n+1)^2} (x_{n+1} - \bar x_n)^2 ] \\ &=& \sum_{i=1}^{n+1} (x_i - \bar x_{n})^2 + (x_{n+1} - \bar x_{n})^2 - 2 \frac{(x_{n+1} - \bar x_n)^2}{n+1} + \frac{(n + 1)}{(n + 1)^2} (x_{n+1} - \bar x_n)^2 \\ &=& (n - 1) s_{n}^2 + \frac{n}{n + 1} (x_{n+1} - \bar x_n)^2\end{aligned}$$ We can now get the result by induction. The result is true for $n=2$ since $s_2^2 = \frac{ (x_2 -x_1)^2}{2}$ with $ \frac{ x_2 -x_1}{\sqrt 2} \sim N(0,1)$, hence $s_2^2 \sim \chi^2_1$. Suppose it is true for $n$, that is $(n - 1) s_{n}^2 \sim \chi^2_{n-1}$, then since $n s^2_{n+1} = (n - 1) s_{n}^2 + \frac{n}{n + 1} (x_{n+1} - \bar x_n)^2$, $s^2_{n+1}$ is the sum of a $\chi^2_{n-1}$ and $ \frac{n}{n + 1} (x_{n+1} - \bar x_n)^2$ which is independent of $s_{n}$ and distributed as $\chi^2_1$ since $x_{n+1} - \bar x_n\sim N(0, \frac{n+1}{n}$. Using our lemma, this means that $n s^2_{n+1} \sim \chi_{n}^2$. This concludes the proof. ### Cochran’s theorem We define the sub vectorial space $F$ spanned by the vector $\mathbbm{1}_n = (1, \ldots, 1)^2$ which is one for each coordinate. Its projection matrix is given by $P_F = \mathbbm{1}_n (\mathbbm{1}_n^T \mathbbm{1}_n)^{-1} \mathbbm{1}_n^T = \frac 1 n \mathbbm{1}_n \mathbbm{1}_n^T$. The orthogonal sub vectorial space of $\mathbb{R}_n$, denoted by $F^{\bot}$, has its projection matrix given by $P_{F^{\bot}} = Id_n - P_F$. The projection of the $(x_i)_{i=1, \dots, n}$ over $F$ (respectively $F^{\bot}$) is given by $(\hat x_n, \ldots, \hat x_n)^T$ (respectively) $(x_1 - \hat x_n, \ldots, x_n - \hat x_n)^T$. The Cochran’s theorem states that these two vectors are independent and that $|| P_{F^{\bot}} X ||^2 = (n-1) s_n ^2 \sim \chi^2(n-1)$ Various proof around Normal =========================== Linear combination of Correlated Normal {#corr_normal} --------------------------------------- For any $d$-dimensional multivariate normal distribution $X\sim N_{d}(\mu,\Sigma)$ where $N_d$ stands for the multi dimensional normal distribution, $\mu=(\mu_1,\dots,\mu_d)^T$ and $\Sigma_{jk}=cov(X_j,X_k)\;\;j,k=1,\dots,d$, the characteristic function is given by: $$\begin{aligned} \varphi_{X}({\bf{t}}) & = & E\left[\exp(i{\bf{t}}^TX)\right]=\exp\left(i{\bf{t}}^T\mu-\frac{1}{2}{\bf{t}}^T\Sigma{\bf{t}}\right) \\ & =& \exp\left(i\sum_{j=1}^{d}t_j\mu_j-\frac{1}{2}\sum_{j=1}^{d}\sum_{k=1}^{d}t_jt_k\Sigma_{jk}\right)\end{aligned}$$ For a new random variable $Z={\bf{a}}^TX=\sum_{j=1}^{d}a_jX_j$, the characteristic function for $Z$ writes: $$\begin{aligned} \varphi_{Z}(t)& = & E\left[\exp(itZ)\right]=E\left[\exp(it{\bf{a}}^TX)\right]=\varphi_{X}(t{\bf{a}}) \\ &= & \exp\left(it\sum_{j=1}^{d}a_j\mu_j-\frac{1}{2}t^2\sum_{j=1}^{d}\sum_{k=1}^{d}a_ja_k\Sigma_{jk}\right)\end{aligned}$$ This proves that $Z$ is normally distributed with mean given by $\mu_Z=\sum_{j=1}^{d}a_j\mu_j$ and variance given by $\sigma^2_Z=\sum_{j=1}^{d}\sum_{k=1}^{d}a_ja_k\Sigma_{jk}$. We can simplify the expression for the variance since $\Sigma_{jk}=\Sigma_{kj}$ get: $$\sigma^2_Z=\sum_{j=1}^{d}a_j^2\Sigma_{jj}+2\sum_{j=2}^{d}\sum_{k=1}^{j-1}a_ja_k\Sigma_{jk}$$ Variance of the sample mean in AR(1) process {#proof_var_num} -------------------------------------------- The computation is given as follows $$\begin{aligned} \text{Var}(\sqrt n (\bar{X}_n-\mu)) &= & \frac{1}{n}\text{Var}( \sum_{i=1}^{n} (X_i-\mu) ) = \frac{1}{n} \mathbb{E} \left[ ( \sum_{i=1}^{n} (X_i - \mu)) ^2 \right] \\ &=& \frac{1}{n} \mathbb{E} \left[ \sum_{i=1}^{n} (X_i- \mu ) ^2 + 2 \sum_{i=1..n, j=1...i-1} (X_i - \mu) (X_j - \mu) \right] \end{aligned}$$ We have $$\begin{aligned} \mathbb{E} \left[ \sum_{i=1}^{n} (X_i- \mu )^2 \right] & = & \frac{ n \sigma^2}{(1 - \rho^2)} \\ \nonumber \\ \text{and} \quad \quad \mathbb{E} \left[ \sum_{i=1..n, j=1...i-1} (X_i - \mu) (X_j - \mu)\right] & = & \frac{ \sigma^2}{(1 - \rho^2)} \sum_{\substack{i=1..n \\ j=1...i-1}} \rho^{i-j} \\ & = & \frac{ \sigma^2}{(1 - \rho^2)} \sum_{i=1..n} (n-i) \rho^{i} \end{aligned}$$ At this stage, we can use rules about geometric series. We have $$\begin{aligned} \sum_{i=1}^{n} \rho^{i} & =\rho \sum_{i=0}^{n-1} \rho^{i} & \sum_{i=1}^{n} i \rho^{i} & = \rho \frac{\partial}{\partial \rho} \sum_{i=0}^{n} \rho^{i} \\ & =\rho \frac{1-\rho^n}{1-\rho} & & = \rho \frac{ 1 - (n+1) \rho^{n} + n \rho^{n+1}}{(1-\rho)^2} \end{aligned}$$ This leads in particular to $$\begin{aligned} \sum_{i=1}^{n} (n-i ) \rho^{i} = \rho \frac{ n (1-\rho) - (1-\rho^{n}) }{(1-\rho)^2}\end{aligned}$$ Hence, $$\begin{aligned} \text{Var}(\sqrt n (\bar{X}_n-\mu)) &= &\frac{ \sigma^2}{(1 - \rho^2)n} \left[ n + 2 \left( n \rho \frac{1-\rho^{n}}{1-\rho} - \rho \frac{ 1- (n+1) \rho^{n} + n \rho^{n+1}}{(1-\rho)^2} \right) \right] \\ &=& \frac{ \sigma^2}{(1 - \rho^2)n} \left[ n + 2 \left( \frac{ n \rho (1 - \rho) - \rho (1-\rho^{n})}{(1-\rho)^2} \right) \right] \\ &=& \frac{ \sigma^2}{1 - \rho^2} \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} \right] \\ &=& \frac{ \sigma^2}{(1 - \rho)^2} \left[1 - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)(1+\rho)} \right] \end{aligned}$$ Variance of the sample mean in AR(1) process {#proof_covar} -------------------------------------------- The computation of \[lemma\_covar\_eq1\] is easy and given by $$\begin{aligned} \text{Cov}(\bar{Y}_n, Y_j) & =& \frac{1}{n} \mathbb{E}\left[ \sum_{i=1}^{n}Y_i Y_j \right] \\ & = & \frac{\sigma^2}{ n (1 - \rho^2)} \left[ \sum_{i=1}^{n} \rho^{ \vert i - j \vert } \right] \\ & =& \frac{ \sigma^2}{ n (1 - \rho^2)} \left[ \sum_{i=0}^{n-j} \rho^{i} + \sum_{i=0}^{j-1} \rho^{i} -1 \right] \\ &= & \frac{ \sigma^2}{ n (1 - \rho^2)} \left[ \frac{ 1-\rho^{n+1-j}} {1-\rho} + \frac{ 1-\rho^{j}} {1-\rho} -1 \right] \\ &= & \frac{ \sigma^2}{ n (1 - \rho^2)} \left[ \frac{ 1 + \rho -\rho^{n+1-j}-\rho^{j}} {1-\rho} \right] \end{aligned}$$ The second equation \[lemma\_covar\_eq2\] is trivial as $$\bar{Y}_n = \frac{\sum_{i=1}^n Y_i}{n}.$$Hence $$\begin{aligned} \sum_{j=1}^n \text{Cov}(\bar{Y}_n, Y_j) = n \text{Var}(\bar{Y}_n) = \frac{ \sigma^2}{ (1 - \rho)^2} \left[ 1 - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)(1+\rho)} \right] \end{aligned}$$ For the last equation, we can compute and get the result as follows $$\begin{aligned} \sum_{j=1}^n \left( \text{Cov}(\bar{Y}_n, Y_j) \right)^2 &=& \sum_{j=1}^n \frac{ \sigma^4}{n^2 (1 - \rho^2)^2} \left[ \frac{ 1 + \rho -\rho^{n+1-j}-\rho^{j}} {1-\rho} \right] ^2 \\ &=& \frac{ \sigma^4}{n^2 (1 - \rho^2)^2 (1-\rho)^2} \sum_{j=1}^n \left[ 1 + \rho -\rho^{n+1-j}-\rho^{j} \right]^2\end{aligned}$$ Expanding the square leads to $$\sum_{j=1}^n \left[ 1 + \rho -\rho^{n+1-j}-\rho^{j} \right]^2 = \sum_{j=1}^n ( 1 + \rho )^2 + (\rho^2)^{n+1-j} + (\rho^{2})^{j}- 2 (1+\rho) \rho^{n+1-j} - 2 (1+\rho) \rho^j + 2 \rho^{n+1} \\$$ Denoting by $S$ the summation, computing the different terms and summing them up leads to $$S = n \left( (1+\rho)^2 + 2 \rho^{n+1} \right) + 2 \rho^2 \frac{1-\rho^{2n}}{1-\rho^2} - 4 (1+\rho) \rho \frac{1-\rho^{n}}{1-\rho}$$ since $$\sum_{j=1}^n \rho^{j}+\rho^{n+1-j} = 2 \rho \frac{1 - \rho^2}{1-\rho}$$ Regrouping all the terms leads to $$\begin{aligned} \sum_{j=1}^n \left( \text{Cov}(\bar{Y}_n, Y_j) \right)^2 & =& \frac{ \sigma^4}{ (1 - \rho^2)^2} \left[ \frac{ (1+\rho)^2 + 2 \rho^{n+1} } { (1-\rho)^2 } \frac{1}{n} - \frac{4 (1+\rho)^2 \rho (1-\rho^n) - 2 \rho^2 (1-\rho^{2n})} {(1-\rho)^2 (1-\rho^2)} \frac{1}{n^2} \right]\end{aligned}$$ Expectation of denominator {#proof_denom_expectation} -------------------------- Lemma \[lemma\_var\_num\]) states that $$\begin{aligned} \mathbb{E} \left[ n \bar{Y}_n^2 \right] = \frac{ \sigma^2}{1 - \rho^2} \left( \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho ( 1- \rho^{n}) }{n(1-\rho)^2} \right)\end{aligned}$$ We can compute as follows: $$\begin{aligned} \mathbb{E} {s_n^2} &= & \frac{1}{n-1} \mathbb{E} \left[ \sum_{i=1}^{n} Y_i^2 - n \bar{Y}_n^2 \right] \\ &=& \frac{ \sigma^2}{(n-1)(1 - \rho^2)} \left[ n - \left( \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho ( 1- \rho^{n}) }{n(1-\rho)^2} \right) \right ] \\ &=& \frac{ \sigma^2}{(n-1)(1 - \rho^2)} \left[ (n - 1) - \left( \frac{ 2 \rho }{1-\rho} - \frac{2 \rho (1- \rho^{n}) }{n(1-\rho)^2} \right) \right ] \\ &=& \frac{ \sigma^2}{1 - \rho^2} \left( 1 - \frac{ 2 \rho }{(1-\rho) (n-1)} ( 1 -\frac{ 1-\rho^{n} }{n (1-\rho)} \right) \\ &=& \frac{ \sigma^2}{1 - \rho^2} \left( 1 - \frac{ 2 \rho }{(1-\rho) (n-1)} + \frac{ 2 \rho( 1-\rho^{n}) }{n (n-1) (1-\rho)^2} \right) \end{aligned}$$ Second moment of denominator {#proof_second_moment} ---------------------------- We have $$\begin{aligned} \mathbb{E} {s_n^4}& = & \frac{1}{(n-1)^2} \mathbb{E} \left[ (\sum_{i=1}^{n} Y_i^2 - n \bar{Y}_n^2 )^2\right] \\ &= & \frac{1}{(n-1)^2} \mathbb{E} \left[ (\sum_{i=1}^{n} Y_i^2)^2 + n^2 \bar{Y}_n^4 -2 n \bar{Y}_n^2 (\sum_{i=1}^{n} Y_i^2 ) \right] \\ &= & \frac{1}{(n-1)^2} \mathbb{E} \left[ \sum_{i=1}^{n} Y_i^4 + 2 \sum_{i=1,k=i+1}^{n} Y_i ^2 Y_k^2 + n^2 \bar{Y}_n^4 -2 n \bar{Y}_n^2 (\sum_{i=1}^{n} Y_i^2 ) \right] \end{aligned}$$ We can compute the fourth moment successively. Since both $Y_i$ and $\bar{Y}_n$ are two normal, their fourth moment is three times the squared variance. This gives: $$\begin{aligned} \mathbb{E} \left[ \sum_{i=1}^{n} Y_i^4 \right] &= & 3 n \times \frac{ \sigma^4 }{(1 - \rho^2)^2} \\ \mathbb{E} \left[ n^2 \bar{Y}_n^4 \right] &= & 3 \times \frac{ \sigma^4}{(1 - \rho^2)^2} \left[ \frac{1+\rho}{1-\rho}-\frac{2 \rho (1-\rho^{n})}{n(1-\rho)^2} \right]^2 \end{aligned}$$ The cross terms between $Y_i$ $Y_k$ is more involved and is computed as follows: $$\begin{aligned} \mathbb{E} \bigg[ \sum_{\substack{ i=1 \\ k = i+1}}^{n} Y_i ^2 Y_k^2 \bigg] &= & \mathbb{E} \bigg[ \sum_{\substack{ i=1 \\ k = i+1}}^{n} \rho^{2(k-i)} Y_i ^4 + (1- \rho^{2(k-i)}) Y_i^2 (Y_i^{\perp})^2 \bigg] \\ & =& \frac{ \sigma^4 }{(1 - \rho^2)^2} \bigg[ \sum_{\substack{ i=1 \\ k = i+1}}^{n} 2 \rho^{2(k-i)} + 1 \bigg] \\ & =& \frac{ \sigma^4 }{(1 - \rho^2)^2} \bigg[ 2 \sum_{i=1}^{n} (n-i) \rho^{2i} + \frac{ n (n-1)}{2} \bigg] \\ & =& \frac{ \sigma^4 }{(1 - \rho^2)^2} \bigg[ 2 \rho^2 \frac{ n (1-\rho^2) - (1-\rho^{2n}) }{(1-\rho^2)^2} + \frac{ n (n-1)}{2} \bigg] \end{aligned}$$ For the cross term between $\bar{Y}_n^2$ and $\sum_{i=1}^{n} Y_i^2 $, we wan use the fact that $\bar{Y}_n$ and $Y_i$ are two correlated Gaussians. Remember that for two Gaussians, $\mathbb{E}[U^2 V^2] = \mathbb{E}[U^2] \mathbb{E}[V^2] + 2 (\text{Cov}(U,V))^2$. We apply this trick to get: $$\begin{aligned} \mathbb{E} \left[ \bar{Y}_n^2 (\sum_{i=1}^{n} Y_i^2 ) \right] &= & \sum_{i=1}^{n} \mathbb{E} \left[ \bar{Y}_n^2 Y_i^2 \right] \\ &= & \sum_{i=1}^{n} \mathbb{E} \left[ \bar{Y}_n^2 \right] \mathbb{E} \left[ Y_i^2 \right] + 2 (\text{Cov}( \bar{Y}_n, Y_i ))^2 \\ &= & \mathbb{E} \left[ \bar{Y}_n^2 \right] \sum_{i=1}^{n} \mathbb{E} \left[ Y_i^2 \right] + 2 \sum_{i=1}^{n} (\text{Cov}( \bar{Y}_n, Y_i ))^2 \end{aligned}$$ The first term is given by $$\begin{aligned} \mathbb{E} \left[ \bar{Y}_n^2 \right] \sum_{i=1}^{n} \mathbb{E} \left[ Y_i^2 \right] & = & \frac{ \sigma^4}{ (1 - \rho^2)^2 } \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} \right] \end{aligned}$$ The second term is given by $$\begin{aligned} 2 \sum_{i=1}^{n} (\text{Cov}( \bar{Y}_n, Y_i ))^2 & = & \frac{ 2 \sigma^4}{ (1 - \rho^2)^2} \left[ \frac{ (1+\rho)^2 + 2 \rho^{n+1} } { (1-\rho)^2 } \frac{1}{n} - \frac{4 (1+\rho)^2 \rho (1-\rho^n) - 2 \rho^2 (1-\rho^{2n})} {(1-\rho)^2 (1-\rho^2)} \frac{1}{n^2} \right]\end{aligned}$$ Summing up all quantities leads to $$\begin{aligned} (n-1)^2 \mathbb{E} {s_n^4} & = \frac{ \sigma^4}{ (1 - \rho^2)^2} \bigg[ 3n + n (n-1) + 4 \rho^2 \frac{ n (1-\rho^2) - (1-\rho^{2n}) }{(1-\rho^2)^2} + 3 \left[ \frac{1+\rho}{1-\rho}-\frac{2 \rho (1-\rho^{n})}{n(1-\rho)^2} \right]^2 \\ & -2 n \left[ \frac{ 1 + \rho}{1-\rho} - \frac{2 \rho (1 - \rho^{n}) }{n(1-\rho)^2} + \frac{ (1+\rho)^2 + 2 \rho^{n+1} } { (1-\rho)^2 } \frac{2}{n} - \frac{ (1+\rho)^2 \rho (1-\rho^n) - 2 \rho^2 (1-\rho^{2n})} {(1-\rho)^2 (1-\rho^2)} \frac{8}{n^2} \right] \bigg] \end{aligned}$$ Regrouping all the terms leads to $$\begin{aligned} \mathbb{E}[s_n^4] & = \frac{\sigma^4}{(1-\rho^2)^2} \frac{1}{(n-1)^2} \left[ n^2-1 + \rho \left(n A_1 + A_2 + \frac{1}{n} A_3 + \frac{1}{n^2} A_4 \right) \right]\end{aligned}$$ with $$\begin{aligned} A_1 & = \frac{-4}{1-\rho^2} \\ A_2 & = \frac{- 2 \left(3 + 9 \rho + 11 \rho^2 + 3 \rho^3 + 6 \rho^n + 12 \rho^{n+1} + 6 \rho^{n+2}-2 \rho^{2n+2}\right)} {(1-\rho^2)^2} \\ A_3 & = \frac{ 4 (1 - \rho^n) (1 - 3 \rho + 4 \rho^2 - 8 \rho^{n+1})}{(1 -r)^3 (1 + r)} \\ A_4 & = \frac{12 \rho (1-\rho^{n})^2}{(1-\rho)^4} \end{aligned}$$ Variance of denominator {#proof_var_moment} ----------------------- The result is obtained from meticulously computing the variance knowing that $$\begin{aligned} \text{Var}[s_n^4] & = \mathbb{E}[s_n^4] - (\mathbb{E}[s_n^2] )^2\end{aligned}$$ The terms for the part $ \mathbb{E}[s_n^4]$ have already been computed in proposition \[second\_moment\_denom\]. As for the term coming from the square of the expectation, they write as: $$\begin{aligned} (\mathbb{E}[s_n^2] )^2 & = \frac{ \sigma^4}{(1 - \rho^2)^2} \left( 1 - \frac{ 2 \rho }{(1-\rho) (n-1)} + \frac{ 2 \rho( 1-\rho^{n}) }{n (n-1) (1-\rho)^2} \right)^2 \\ & = \frac{ \sigma^4}{(n-1)^2 (1 - \rho^2)^2} \left( (n-1) - \frac{ 2 \rho }{1-\rho} + \frac{ 2 \rho( 1-\rho^{n}) }{n (1-\rho)^2} \right)^2\end{aligned}$$ Let us write the square as $E1 = \left( (n-1) - \frac{ 2 \rho }{1-\rho} + \frac{ 2 \rho( 1-\rho^{n}) }{n (1-\rho)^2} \right)^2 $. We can expand the square as follows $$\begin{aligned} E1 & = (n-1)^2 + \frac{ 4 (n-1) \rho( 1-\rho^{n}) }{n (1-\rho) } - \frac{ 4 \rho (n-1) }{1-\rho }+ \frac{ 2 \rho}{(1-\rho)^2} \left(\frac{ 1-\rho^{n}}{n(1-\rho)} -1 \right)^2 \end{aligned}$$ Rearranging the terms leads then to the final result. [^1]: A.I. SQUARE CONNECT, 35 Boulevard d’Inkermann 92200 Neuilly sur Seine, France [^2]: LAMSADE, Université Paris Dauphine, Place du Maréchal de Lattre de Tassigny,75016 Paris, France [^3]: E-mail: [email protected], [email protected] [^4]: determinant of the Jacobian matrix of the transformation

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--- --- =-0.75cm -.25cm 16.5cm 23.0cm CERN-TH/98-360 $ $\ **Duality symmetry of Reggeon interactions in multicolour QCD** L.N. Lipatov [$^{*}$\ Petersburg Nuclear Physics Institute,\ Gatchina, 188350, St.Petersburg, Russia]{} 15.0pt **Abstract** The duality symmetry of the Hamiltonian and integrals of motion for Reggeon interactions in multicolour QCD is formulated as an integral equation for the wave function of compound states of $n$ reggeized gluons. In particular the Odderon problem in QCD is reduced to the solution of the one-dimensional Schrödinger equation. The Odderon Hamiltonian is written in a normal form, which gives a possibility to express it as a function of its integrals of motion. ------------------------------------------------------------------------ $^{*}$ [*Supported by the CRDF, INTAS and INTAS-RFBR grants: RP1-253, 1867-93, 95-0311*]{} Introduction ============ The hadron scattering amplitude at high energies $\sqrt{s}$ in the leading logarithmic approximation (LLA) of the perturbation theory is obtained by calculating and summing all contributions $\left( g^{2}\ln (s)\right) ^{n}$, where $g$ is the coupling constant. In this approximation the gluon is reggeized and the BFKL Pomeron is a compound state of two reggeized gluons \[1\]. Next-to-leading corrections to the BFKL equation were also calculated \[2\], which gives a possibility to find its region of applicability. In particular the Möbius invariance of the equation valid in LLA \[1\] turns out to be violated after taking into account next-to-leading terms. The asymptotic behaviour $\propto s^{j_{0}}$ of scattering amplitudes is governed by the $j$-plane singularities of the $t$-channel partial waves $% f_{j}(t)$. The position of these singularities $\omega _{0}=j_{0}-1$ for the Feynman diagrams with $n$ reggeized gluons in the $t$-channel is proportional to eigenvalues of a Schrödinger-like equation \[3\]. For the multicolour QCD $N_{c}\rightarrow \infty $ the colour structure and the coordinate dependence of the eigenfunctions are factorized \[4\]. The wave function $f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},% \overrightarrow{\rho _{2}},...,\overrightarrow{\rho _{n}};\overrightarrow{% \rho _{0}})$ of the colourless compound state $O_{m\widetilde{,m}}(% \overrightarrow{\rho _{0}})$ depends on the two-dimensional impact parameters $\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},...,% \overrightarrow{\rho _{n}}$ of the reggeized gluons. It belongs to the basic series of unitary representations of the Möbius group transformations $$\rho _{k}\rightarrow \frac{a\,\rho _{k}+b}{c\,\rho _{k}+d}\,,$$ where $\rho _{k}=x_{k}+iy_{k},\,\rho _{k}^{*}=x_{k}-iy_{k}$ and $a,b,c,d$ are arbitrary complex parameters \[1\]. For this series the conformal weights $$m=1/2+i\nu +n/2,\,\,\widetilde{m}=1/2+i\nu -n/2$$ are expressed in terms of the anomalous dimension $\gamma =1+2i\nu $ and the integer conformal spin $n$ of the composite operators $O_{m\widetilde{,m}}(% \overrightarrow{\rho _{0}})$. They are related to the eigenvalues $$M^{2}f_{m,\widetilde{m}}=m(m-1)f_{m,\widetilde{m}}\,,\,\,\,M^{*2}f_{m,% \widetilde{m}}=\widetilde{m}(\widetilde{m}-1)f_{m,\widetilde{m}}\,$$ of the Casimir operators $M^{2}$ and $M^{*2}$: $$M^{2}=\left( \sum_{k=1}^{n}M_{k}^{a}\right) ^{2}=\sum_{r<s}2\,M_{r}^{a}\,M_{s}^{a}=-\sum_{r<s}\rho _{rs}^{2}\partial _{r}\partial _{s}\,,\,\, M^{*2}=(M^{2})^{*}.$$ Here $M_{k}^{a}$ are the Möbius group generators $$M_{k}^{3}=\rho _{k}\partial _{k}\,,\,\,\,M_{k}^{-}=\partial _{k}\,,\,\,\,M_{k}^{+}=-\rho _{k}^{2}\partial _{k}$$ and $\partial _{k}=\partial /(\partial \rho _{k})$. The wave function $f_{m,\widetilde{m}}$ satisfies the Schrödinger equation \[4\]: $$E_{m,\widetilde{m}}\,f_{m,\widetilde{m}}=H\,f_{m,\widetilde{m}}.$$ Its eigenvalue $E_{m,\widetilde{m}}$ is proportional to the position $\omega _{m,\widetilde{m}}=j-1$ of a $j$-plane singularity of the $t$-channel partial wave: $$\omega _{m,\widetilde{m}}\,=-\frac{g^{2}N_{c}}{8\,\pi ^{2}}\,E_{m,\widetilde{% m}}$$ governing the $n$-Reggeon asymptotic contribution to the total cross-section $\sigma _{tot}\sim s^{\omega _{m,\widetilde{m}}}$. In the particular case of the Odderon, being a compound state of three reggeized gluons with the charge parity $C=-1$ and the signature $P_{j}=-1$, the eigenvalue $\omega _{m,\widetilde{m}}^{(3)}$ is related to the high-energy behaviour of the difference of the total cross-sections $\sigma _{pp}$ and $\sigma _{p\overline{p}}$ for interactions of particles $p$ and antiparticles $\overline{p}$ with a target: $$\sigma _{pp}-\sigma _{p\overline{p}}\sim s^{\omega _{m,\widetilde{m}}^{(3)}}.$$ The Hamiltonian $H$ in the multicolour QCD has the property of the holomorphic separability \[4\]: $$H=\frac{1}{2}(h+h^{*}),\,\,\,\left[ h,h^{*}\right] =0\,,$$ where the holomorphic and anti-holomorphic Hamiltonians $$h=\sum_{k=1}^{n}h_{k,k+1\,},\,h^{*}=\sum_{k=1}^{n}h_{k,k+1\,}^{*}$$ are expressed in terms of the BFKL operator \[4\] : $$h_{k,k+1}=\log (p_{k})+\log (p_{k+1})+\frac{1}{p_{k}}\log (\rho _{k,k+1})p_{k}+\frac{1}{p_{k+1}}\log (\rho _{k,k+1})p_{k+1}+2\,\gamma \,.$$ Here $\rho _{k,k+1}=\rho _{k}-\rho _{k+1}\,,\,p_{k}=i\,\partial /(\partial \rho _{k}),\,p_{k}^{*}=i\,\partial /(\partial \rho _{k}^{*})\,$, and $\gamma =-\psi (1)$ is the Euler constant. Owing to the holomorphic separability of $h$, the wave function $f_{m,% \widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},...,% \overrightarrow{\rho _{n}};\overrightarrow{\rho _{0}})$ has the property of the holomorphic factorization \[4\]: $$f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}}% ,...,\overrightarrow{\rho _{n}};\overrightarrow{\rho _{0}}% )=\sum_{r,l}c_{r,l}\,f_{m}^{r}(\rho _{1},\rho _{2},...,\rho _{n};\rho _{0})\,f_{\widetilde{m}}^{l}(\rho _{1}^{*},\rho _{2}^{*},...,\rho _{n}^{*};\rho _{0}^{*})\,,$$ where $r$ and $l$ enumerate degenerate solutions of the Schrödinger equations in the holomorphic and anti-holomorphic sub-spaces: $$\epsilon _{m}\,f_{m}=h\,f_{m}\,,\,\,\,\epsilon _{\widetilde{m}}\,f_{% \widetilde{m}}=h^{*}\,f_{\widetilde{m}}\,,\,\,\,E_{m,\widetilde{m}}=\epsilon _{m}+\epsilon _{\widetilde{m}}\,.$$ Similarly to the case of two-dimensional conformal field theories, the coefficients $c_{r,l}$ are fixed by the single-valuedness condition for the function $f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{% \rho _{2}},...,\overrightarrow{\rho _{n}};\overrightarrow{\rho _{0}})$ in the two-dimensional $\overrightarrow{\rho }$-space. There are two different normalization conditions for the wave function \[4\],\[5\]: $$\left\| f\right\| _{1}^{2}=\int \prod_{r=1}^{n}d^{2}\rho _{r}\,\left| \prod_{r=1}^{n}\rho _{r,r+1}^{-1}\,\,f\right| ^{2}\,,\,\,\,\left\| f\right\| _{2}^{2}=\int \prod_{r=1}^{n}d^{2}\rho _{r}\left| \prod_{r=1}^{n}p_{r}\,\,f\right| ^{2}$$ compatible with the hermicity properties of $H$. Indeed, the transposed Hamiltonian $h^{t}$ is related with $h$ by two different similarity transformations \[5\]: $$h^{t}=\prod_{r=1}^{n}p_{r}\,h\,\prod_{r=1}^{n}p_{r}^{-1}=\prod_{r=1}^{n}\rho _{r,r+1}^{-1}\,h\,\prod_{r=1}^{n}\rho _{r,r+1.}$$ Therefore $h$ commutes $$\left[ h,A\right] =0$$ with the differential operator \[4\] $$A=\rho _{12}\rho _{23}...\rho _{n1}\,p_1p_2...p_n\,.$$ Furthermore \[5\], there is a family $\{q_{r}\}$of mutually commuting integrals of motion: $$\left[ q_{r},q_{s}\right] =0\,,\,\,\,\left[ q_{r},h\right] =0.$$ They are given as $$q_{r}=\sum_{i_{1}<i_{2}<...<i_{r}}\rho _{i_{1}i_{2}\,}\rho _{i_{2}i_{3}}...\rho _{i_{r}i_{1}}\,p_{i_{1}}\,p_{i_{2}}...p_{i_{r}}.$$ In particular $q_{n}$ is equal to $A$ and $q_{2}$ is proportional to $M^{2}$. The generating function for these integrals of motion coincides with the transfer matrix $T$ for the $XXX$ model \[5\]: $$T(u)=tr\,(L_{1}(u)L_{2}(u)...L_{n}(u))=\sum_{r=0}^{n}u^{n-r}\,q_{r},$$ where the $L$-operators are $$L_{k}(u)=\left( \begin{array}{cc} u+\rho _{k}\,p_{k} & p_{k} \\ -\rho _{k}^{2}\,p_{k} & u-\rho _{k}\,p_{k} \end{array} \right) =u\,\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) +\left( \begin{array}{c} 1 \\ -\rho _{k} \end{array} \right) \,\left( \begin{array}{cc} \rho _{k} & 1 \end{array} \right) \,p_{k}\,.$$ The transfer matrix is the trace of the monodromy matrix $t(u)$ \[6\]: $$T(u)=tr\,(t(u)),\,\,t(u)=L_{1}(u)L_{2}(u)...L_{n}(u)\,.$$ It can be verified that $t(u)$ satisfies the Yang-Baxter equation \[5\],\[6\]: $$t_{r_{1}^{\prime }}^{s_{1}}(u)\,t_{r_{2}^{\prime }}^{s_{2}}(v)\,l_{r_{1}r_{2}}^{r_{1}^{\prime }r_{2}^{\prime }}(v-u)=l_{s_{1}^{\prime }s_{2}^{\prime }}^{s_{1}s_{2}}(v-u)\,t_{r_{2}}^{s_{2}^{\prime }}(v)\,t_{r_{1}}^{s_{1}^{\prime }}(u)\,,$$ where $l(w)$ is the $L$-operator for the well-known Heisenberg spin model: $$l_{s_{1}^{\prime }s_{2}^{\prime }}^{s_{1}s_{2}}(w)=w\,\delta _{s_{1}^{\prime }}^{s_{1}}\,\delta _{s_{2}^{\prime }}^{s_{2}}+i\,\delta _{s_{2}^{\prime }}^{s_{1}}\,\delta _{s_{1}^{\prime }}^{s_{2}}\,.$$ The commutativity of $T(u)$ and $T(v)$ $$\lbrack T(u),T(v)]=0$$ is a consequence of the Yang-Baxter equation. If one will parametrize $t(u)$ in the form $$t(u)=\left( \begin{array}{cc} j_{0}(u)+j_{3}(u) & j_{-}(u) \\ j_{+}(u) & j_{0}(u)-j_{3}(u) \end{array} \right) ,$$ this equation is reduced to the following Lorentz-covariant relations for the currents $j_{\mu }(u)$: $$\left[ j_{\mu }(u),j_{\nu }(v)\right] =\left[ j_{\mu }(v),j_{\nu }(u)\right] =\frac{i\,\epsilon _{\mu \nu \rho \sigma }}{2(u-v)}\left( j^{\rho }(u)j^{\sigma }(v)-j^{\rho }(v)j^{\sigma }(u)\right) .$$ Here $\epsilon _{\mu \nu \rho \sigma }$ is the antisymmetric tensor ($% \epsilon _{1230}=1$) in the four-dimensional Minkovski space and the metric tensor $g^{\mu \nu }$ has the signature ($1,-1,-1,-1$). This form follows from the invariance of the Yang-Baxter equations under the Lorentz transformations. The generators for the spacial rotations coincide with that of the Möbius transformations $$\overrightarrow{M}=\sum_{k=1}^n\overrightarrow{M_k}\,,$$ $$M_{k}^{3}=\rho _{k}\,\partial _{k}\,,\,\,M_{k}^{1}=\frac{1}{2}\,(1-\rho _{k}^{2})\,\partial _{k}\,,\,\,M_{k}^{2}=\frac{i}{2}\,(1+\rho _{k}^{2})\,\partial _{k}\,\,.$$ The commutation relations for the Lorentz algebra are given below: $$\left[ M^{s},M^{t}\right] =i\epsilon _{stu}\,M^{u},\,\,\,\left[ M^{s},N^{t}\right] =i\epsilon _{stu}\,N^{u},\,\,\,\left[ N^{s},N^{t}\right] =i\epsilon _{stu}\,M^{u}\,,$$ where $\overrightarrow{N}$ are the Lorentz boost generators. The commutativity of the transfer matrix $T(u)$ with the local hamiltonian $% h $ \[5\],\[7\] $$\left[ T(u),h\right] =0$$ is a consequence of the relation: $$\left[ L_{k}(u)\,L_{k+1}(u),h_{k,k+1}\right] =-i\,(L_{k}(u)-L_{k+1}(u))$$ for the pair Hamiltonian $h_{k,k+1}$. In turn, this relation follows from the Möbius invariance of $h_{k,k+1}$ and the identity: $$\left[ h_{k,k+1},\left[ \left( \overrightarrow{M_{k,k+1}}\right) ^{2},% \overrightarrow{N_{k,k+1}}\right] \right] =4\,\overrightarrow{N_{k,k+1}}\,,$$ where $$\overrightarrow{M_{k,k+1}}=\overrightarrow{M_{k}}+\,\overrightarrow{M_{k+1}}% \,,\,\,\,\,\,\overrightarrow{N_{k,k+1}}=\overrightarrow{M_{k}}-\,% \overrightarrow{M_{k+1}}$$ are the Lorentz group generators for the two gluon state. Because the pair hamiltonian $h_{k,k+1}$ depends only on the Casimir operator $\left( \overrightarrow{M_{k,k+1}}\right) ^{2}$, it is diagonal $$h_{k,k+1}\left| m_{k,k+1}\right\rangle =\left( \psi (m_{k,k+1})+\psi (1-m_{k,k+1})-2\,\psi (1)\right) \left| m_{k,k+1}\right\rangle$$ in the conformal weight representation: $$\left( \overrightarrow{M_{k,k+1}}\right) ^2\left| m_{k,k+1}\right\rangle =m_{k,k+1}(m_{k,k+1}-1)\left| m_{k,k+1}\right\rangle \,.$$ Using the commutation relations between $\overrightarrow{M_{k,k+1}}$ and $% \overrightarrow{N_{k,k+1}}$ and taking into account that $\left( \overrightarrow{M_{k}}\right) ^{2}=0$, one can verify that the operator $% \overrightarrow{N_{k,k+1}}$ has non-vanishing matrix elements only between the states $\left| m_{k,k+1}\right\rangle $ and $\left| m_{k,k+1}\pm 1\right\rangle $. It means that the above identity for $h_{k,k+1}$ is a consequence of the known recurrence relations for the $\psi $-functions: $$\psi (m)=\psi (m-1)+1/(m-1)\,,\,\,\,\,\psi (1-m)=\psi (2-m)+1/(m-1).$$ The pair Hamiltonian $h_{k,k+1}$ can be expressed in terms of the small-$u$ asymptotics $$\widehat{L}_{k,k+1}(u)=P_{k,k+1}(1+i\,u\,h_{k,k+1}+...)$$ of the fundamental $L$-operator $\widehat{L}_{k,k+1}(u)$ acting on functions $f(\rho _{k},\rho _{k+1})$ \[6\]. Here $P_{k,k+1}$ is defined by the relation: $$P_{k,k+1}\,\,f(\rho _{k},\rho _{k+1})=f(\rho _{k+1},\rho _{k}).$$ The operator $\widehat{L}_{k,k+1}$ satisfies the linear equation \[6\] $$L_{k}(u)\,L_{k+1}(v)\,\,\widehat{L}_{k,k+1}(u-v)=\widehat{L}% _{k,k+1}(u-v)\,\,L_{k+1}(v)\,L_{k}(u).$$ This equation can be solved in a way similar to that for $h_{k,k+1}$$$\widehat{L}_{k,k+1}(u)\sim \,P_{k,k+1}\sqrt{\frac{\Gamma (\widehat{m}% _{k,k+1}+iu)\Gamma (1-\widehat{m}_{k,k+1}+iu)}{\Gamma (\widehat{m}% _{k,k+1}-iu)\Gamma (1-\widehat{m}_{k,k+1}-iu)}}\frac{\Gamma (1-iu)}{\Gamma (1+iu)}\,,$$ where the integral operator $\widehat{m}_{k,k+1}$ is defined by the relation $$\widehat{m}_{k,k+1}\,(\widehat{m}_{k,k+1}-1)=\left( \overrightarrow{M_{k}}+% \overrightarrow{M_{k+1}}\right) ^{2}$$ and the proportionality constant, being a periodic function of $\widehat{m}% _{k,k+1}$ with an unit period, is fixed from the triangle relation $$\widehat{L}_{13}(u)\,\widehat{L}_{23}(v)\,\,\widehat{L}_{12}(u-v)=\widehat{L}% _{12}(u-v)\,\,\widehat{L}_{23}(v)\,\widehat{L}_{13}(u)\,.$$ To find a representation of the Yang-Baxter commutation relations, the algebraic Bethe anzatz is used \[6\]. To begin with, in the above parametrization of the monodromy matrix $t(u)$ in terms of the currents $% j_{\mu }(u)$, one should construct the pseudovacuum state $|0\rangle $ satisfying the equations $$j_{+}(u)\,|0\rangle =0.$$ However, these equations have a non-trivial solution only if the above $L$-operators are regularized as $$L_{k}^{\delta }(u)=\left( \begin{array}{cc} u+\rho _{k}\,p_{k}-i\,\delta & p_{k} \\ -\rho _{k}^{2}\,p_{k}+2\,i\,\rho _{k}\,\delta & u-\rho _{k}\,p_{k}+i\delta \end{array} \right)$$ by introducing an infinitesimally small conformal weight $\delta \,\,$$% \rightarrow 0$ for reggeized gluons (another possibility is to use the dual space corresponding to $\delta =-1$ \[7\]). For this regularization the pseudovacuum state is $$|\delta \rangle =\prod_{k=1}^{n}\rho _{k}^{2\delta }$$ It is also an eigenstate of the transfer matrix: $$T(u)\,|\delta \rangle =2\,j_{0}(u)\,|\delta \rangle =\left( (u-i\,\delta )^{n}+(u+i\,\delta )^{n}\right) \,|\delta \rangle .$$ Furthermore, all excited states are obtained by applying the product of the operators $j_{-}(v)$ to the pseudovacuum state: $$|v_{1}v_{2}...v_{k}\rangle =j_{-}(v_{1})\,j_{-}(v_{2})...j_{-}(v_{k})\,\,|\delta \rangle .$$ They are eigenfunctions of the transfer matrix $T(u)$ with the eigenvalues: $$\widetilde{T}(u)=(u+i\delta )^{n}\prod_{r=1}^{k}\frac{u-v_{r}-i}{u-v_{r}}% +(u-i\delta )^{n}\prod_{r=1}^{k}\frac{u-v_{r}+i}{u-v_{r}}\,,$$ providing that the spectral parameters $v_{1},v_{2},...,v_{k}$ are solutions of the set of Bethe equations: $$\left( \frac{v_{s}-i\delta }{v_{s}+i\delta }\right) ^{n}=\prod_{r\neq s}% \frac{v_{s}-v_{r}-i}{v_{s}-v_{r}+i}$$ for $s=1,2...k$. Due to above relations the function $$Q(u)=\prod_{r=1}^{k}(u-v_{r})\,$$ satisfies the Baxter equation \[6,7\]: $$\widetilde{T}(u)\,Q(u)=(u-i\delta )^{n}\,Q(u+i)+(u+i\delta )^{n}\,Q(u-i)\,,$$ where $\widetilde{T}(u)$ is an eigenvalue of the transfer matrix. Its corresponding eigenfunctions can be expressed in terms of the solution $% Q^{(k)}(u)$ of this equation as follows \[7\] $$|v_{1}v_{2}...v_{k}\rangle =Q^{(k)}(\widehat{u}_{1})\,\,Q^{(k)}(\widehat{u}% _{2})\,...\,Q^{(k)}(\widehat{u}_{n-1})\,|\delta \rangle \,,$$ where the integral operators $\widehat{u}_{r}$ are zeros of the current $% j_{-}(u)$: $$j_{-}(u)=c\prod_{r=1}^{n-1}(u-\widehat{u}_{r})\,.$$ Eigenvalues $\epsilon $ of the holomorphic Hamiltonian $h$ also can be expressed in terms of $Q(u)$ \[7\]. Up to now the Baxter equation was solved only for the case of the BFKL Pomeron ($n=2$). This is the reason why we use below another approach, based on the diagonalization of the transfer matrix. Duality of Reggeon interactions at large N$_{c}$ ================================================ The differential operators $q_{r}$ and the Hamiltonian $h$ are invariant under the cyclic permutation of gluon indices $i\rightarrow i+1$ ($i=1,2...n$), corresponding to the Bose symmetry of the Reggeon wave function at $% N_{c}\rightarrow \infty $. It is remarkable that above operators are invariant also under the more general canonical transformation: $$\rho _{i-1,i}\rightarrow p_{i}\rightarrow \rho _{i,i+1}\,,\,$$ combined with reversing the order of the operator multiplication. This invariance is obvious for the Hamiltonian $h$ if we write it in the form: $$h=h_{p}+h_{\rho }\,,$$ where $$h_{p}=\sum_{k=1}^{n}\left( \ln (p_{k})+\frac{1}{2}\sum_{\lambda =\pm 1} \rho _{k,k+\lambda }\,\ln (p_{k})\,\rho _{k,k+\lambda }^{-1}+\gamma \right) \,,$$ $$h_{\rho }=\sum_{k=1}^{n}\left( \ln (\rho _{k,k+1})+\frac{1}{2}\sum_{\lambda =\pm 1}p_{k+(1+\lambda )/2}^{-1}\,\ln (\rho _{k,k+1})\,p_{k+(1+\lambda )/2}+\gamma \right) .$$ Here $\gamma =-\psi (1)$ is the Euler constant. The invariance of the transfer matrix can be verified using two equivalent representations for $q_{r}$: $$q_{r}=\sum_{i_{1}<i_{2}<...i_{r}}\prod_{l=1}^{r}\left( \sum_{k=i_{l}+1}^{i_{l+1}}\rho _{k-1,k}\,\,\,\,p_{i_{l}}\right) =\sum_{i_{1}<i_{2}<...i_{r}}\prod_{l=1}^{r}\left( \rho _{i_{l},i_{l}+1}\,\sum_{k=i_{l}+1}^{i_{l+1}}p_{k}\right) \,.$$ Note, that the supersymmetry corresponds to an analogous generalization of translations to super-translations. Furthermore, the Kramers-Wannier duality in the Ising model and the popular electromagnetic duality $\overrightarrow{E% }\leftrightarrow \overrightarrow{H}$ can be considered as similar canonical transformations \[8\]. The above duality symmetry is realized as an unitary transformation only for the vanishing total momentum: $$\overrightarrow{p}=\sum_{r=1}^{n}\overrightarrow{p_{r}}=0.$$ In this case one can parametrize gluon momenta in terms of momentum transfers $k_{r}$ as follows $$p_{r}=k_{r}-k_{r+1}\,,$$ which gives a possibility to present the symmetry transformation in a simpler form: $$k_{r}\rightarrow \rho _{r}\rightarrow k_{r+1}\,,\,r=1,2...n\,.$$ Because the operators $q_{r}$ compose a complete set of invariants of the transformation, the Hamiltonian $h$ should be their function $$h=h(q_{2},q_{3},...,q_{n})\,,$$ fixed by the property of its locality. Furthermore, a common eigenfunction of $q_{r}\,\,(r=2,...,n)$ is simultaneously a solution of the Schrödinger equation, which means, that the duality symmetry gives an explanation of the integrability of the Reggeon model at $N_c \rightarrow \infty$. To formulate the duality as an integral equation we work in the two-dimensional impact parameter space $\overrightarrow{\rho }$, initially without taking into account the property of the holomorphic factorization of the Green functions. The wave function $\psi _{m,\widetilde{m}}$ of the composite state with $\overrightarrow{p}=0$ can be written in terms of the eigenfunction $f_{m\widetilde{m}}$ of a commuting set of the operators $q_{k} $ and $q_{k}^{*}$ for $k=1,2...n$ as follows $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})=\int \frac{d^{2}\rho _{0}}{2\,\pi }% \,f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}}% ,...,\overrightarrow{\rho _{n}};\overrightarrow{\rho _{0}})\,.$$ It is a highest-weight component of the Möbius group representation with the quantum numbers $m=1/2+i\nu +n/2$ and $\,\widetilde{m}=1/2+i\nu -n/2$ related with eigenvalues of the Casimir operators $$\left( \sum_{k=1}^{n}\overrightarrow{M_{k}}\right) ^{2}\psi _{m,\widetilde{m}% }=m(m-1)\psi _{m,\widetilde{m}}\,\,\,,\,\,\,\,\,\,\left( \sum_{k=1}^{n}% \overrightarrow{M_{k}^{*}}\right) ^{2}\psi _{m,\widetilde{m}}=\widetilde{m}(% \widetilde{m}-1)\psi _{m,\widetilde{m}}\,.$$ The other components of this highest-weight representation can be obtained by applying to $\psi _{m,\widetilde{m}}$ the Möbius group generators: $$\psi _{m,\widetilde{m}}^{r_1r_2}=\left( \sum_{k=1}^n\rho _k^2\,\partial _k\right) ^{r_1}\left( \sum_{k=1}^n\rho _k^{*2}\,\partial _k^{*}\right) ^{r_2}\psi _{m,\widetilde{m}}\,.$$ Note that, in accordance with the relations $$\widetilde{m}^{*}=1-m\,\,,\,\,\,m^{*}=1-\widetilde{m}$$ the conjugate function $f_{m,\widetilde{m}}^{*}$ is transformed as $f_{1-m,1-% \widetilde{m}}$: $$\left( f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},...,\overrightarrow{% \rho _{n}};\overrightarrow{\rho _{0}})\right) ^{*}\sim f_{1-m,1-\widetilde{m}% }(\overrightarrow{\rho _{1}},...,\overrightarrow{\rho _{n}};\overrightarrow{% \rho _{0}}).$$ Moreover, because of the reality of the Möbius group, the complex-conjugated representations are linearly dependent: $$\left( f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},...,\overrightarrow{% \rho _{n}};\overrightarrow{\rho _{0}})\right) ^{*}\sim \int d^{2}\rho _{0^{\prime }}\,(\rho _{00^{\prime }})^{2m-2}(\rho _{00^{\prime }}^{*})^{2% \widetilde{m}-2}\,f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},...,% \overrightarrow{\rho _{n}};\overrightarrow{\rho _{0^{\prime }}}).$$ By considering the limit $\rho _{0}\rightarrow \infty $ of this equation, we obtain for $\psi _{m,\widetilde{m}}$ the new representation $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})\sim f_{1-m,1-\widetilde{m}}(% \overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},...,\overrightarrow{% \rho _{n}};\infty )\,.$$ Because of the relations $$\left( \psi _{\widetilde{m},m}(\overrightarrow{\rho _{12}},\overrightarrow{% \rho _{23}},...,\overrightarrow{\rho _{n1}})\right) ^{*}\sim \psi _{1-% \widetilde{m},1-m}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}}% ,...,\overrightarrow{\rho _{n1}})\,,\,$$ the functions $\psi _{\widetilde{m},m}^{*}$ and $\psi _{m,\widetilde{m}}$ have the same conformal spin $n=m-\widetilde{m}$. Taking into account the hermicity properties of the total Hamiltonian: $$H^{+}=\prod_{k=1}^{n}\left| \rho _{k,k+1}\right| ^{-2}H\prod_{k=1}^{n}\left| \rho _{k,k+1}\right| ^{2}=\prod_{k=1}^{n}\left| p_{k}\right| ^{2}H\prod_{k=1}^{n}\left| p_{k}\right| ^{-2},$$ the solution $\psi _{\widetilde{m},m}^{+}$ of the complex-conjugated Schrödinger equation for $\overrightarrow{p}=0$ can be expressed in terms of $\psi _{\widetilde{m},m}$ as follows : $$\psi _{\widetilde{m},m}^{+}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})=\prod_{k=1}^{n}\left| \rho _{k,k+1}\right| ^{-2}\left( \psi _{\widetilde{m},m}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})\right) ^{*}\,.$$ If one performs the Fourier transformation of $\psi _{\widetilde{m},m}^{+}$ to the momentum space $$\Psi _{m,\widetilde{m}}(\overrightarrow{p_{1}},\overrightarrow{p_{2}},...,% \overrightarrow{p_{n}})=\int \prod_{k=1}^{n-1}\frac{d^{2}\rho _{k-1,k}^{^{\prime }}}{2\pi }\prod_{k=1}^{n}e^{i\overrightarrow{p_{k}}\,% \overrightarrow{\rho _{k}^{\prime }}}\psi _{\widetilde{m},m}^{+}(% \overrightarrow{\rho _{12}^{\prime }},\overrightarrow{\rho _{23}^{\prime }}% ,...,\overrightarrow{\rho _{n1}^{\prime }})$$ with substituting the arguments $$\overrightarrow{p_{k}}\rightarrow \overrightarrow{\rho _{k,k+1}}\,,$$ the new expression $\Psi _{m,\widetilde{m}}(\overrightarrow{\rho _{12}},% \overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})$ will have the same properties as the initial function $\psi _{m,\widetilde{m}}(% \overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...\overrightarrow{% \rho _{n1}})$ under rotations and dilatations. Moreover, in accordance with the above duality symmetry it satisfies the same set of equations as $\psi _{m,\widetilde{m}}$ and therefore these two functions can be chosen to be proportional: $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,\overrightarrow{\rho _{n1}})=c_{m,\widetilde{m}}\,\Psi _{m,% \widetilde{m}}(\overrightarrow{\rho _{12}},\overrightarrow{\rho _{23}},...,% \overrightarrow{\rho _{n1}})\,.$$ The proportionality constant $c_{m,\widetilde{m}}$ is determined from the condition that the norm of the function $\psi _{m,\widetilde{m}}$: $$\left\| \psi _{m,\widetilde{m}}\right\| ^{2}=\int \prod_{k=1}^{n-1}\frac{% d^{2}\rho _{k,k+1}}{2\,\pi }\,\,\psi _{m,\widetilde{m}}^{+}\psi _{m,% \widetilde{m}}$$ is conserved after this transformation. Because $\psi _{m,\widetilde{m}}$ is also an eigenfunction of the integrals of motion $A$ and $A^{*}$ with their eigenvalues $\lambda _{m}$ and $\lambda _{m}^{*}=\lambda _{\widetilde{m}}$: $$A\,\psi _{m,\widetilde{m}}=\lambda _{m}\,\psi _{m,\widetilde{m}% }\,,\,\,\,\,A^{*}\,\psi _{m,\widetilde{m}}=\lambda _{\widetilde{m}}\,\psi _{m,\widetilde{m}}\,,\,\,\,\,A=\rho _{12}...\rho _{n1}p_{1}...p_{n}\,,\,\,$$ one can verify that, for the unitarity of the duality transformation, the constant $c_{m,\widetilde{m}}$ should be chosen as $$c_{m,\widetilde{m}}=\left| \lambda _{m}\right| \,\,2^{n}\,,$$ for an appropriate phase of $\psi _{m,\widetilde{m}}$. Here the factor $2^{n} $ appears due to the relation $\partial _{\mu }^{2}=4\partial \partial ^{*}$. This value of $c_{m,\widetilde{m}}$ is compatible also with the requirement that two subsequent duality transformations are equivalent to the cyclic permutation $i\rightarrow i+1$ of gluon indices. Thus, the duality symmetry takes the form of the following integral equation for $\psi _{m,\widetilde{m}}$: $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{12}},...,\overrightarrow{\rho _{n1}})=\left| \lambda _{m}\right| \,2^{n}\,\int \,\prod_{k=1}^{n-1}\frac{% d^{2}\rho _{k-1,k}^{\prime }}{2\pi }\,\prod_{k=1}^{n}\frac{e^{i% \overrightarrow{\rho _{k,k+1}}\,\overrightarrow{\rho _{k}^{\prime }}}}{% \left| \rho _{k,k+1}^{\prime }\right| ^{2}}\,\psi _{\widetilde{m},m}^{*}(% \overrightarrow{\rho _{12}^{\prime }},...,\overrightarrow{\rho _{n1}^{\prime }})\,.$$ Note that the validity of this equation in the Pomeron case $n=2$ can be verified from the relations $$f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}};% \overrightarrow{\rho _{0}})\sim \left( \frac{\rho _{12}}{\rho _{10}\rho _{20}% }\right) ^{m}\left( \frac{\rho _{12}^{*}}{\rho _{10}^{*}\rho _{20}^{*}}% \right) ^{\widetilde{m}}\,\,,\,\,\,\,\,\,\psi _{m,\widetilde{m}}(% \overrightarrow{\rho _{12}})\sim (\rho _{12})^{1-m}(\rho _{12}^{*})^{1-% \widetilde{m}}\,\,,$$ $$\left| \lambda _{m}\right| \,2^{2}\,\int \,\frac{d^{2}\rho _{12}^{\prime }}{% 2\pi }\,\frac{e^{i\overrightarrow{\rho _{12}}\,\overrightarrow{\rho _{12}^{\prime }}}}{\left| \rho _{12}^{\prime }\right| ^{4}}\,(\rho _{12}^{\prime })^{\widetilde{m}}(\rho _{12}^{\prime *})^{m}=e^{i\delta (m,% \widetilde{m})}{\rho _{12}}^{1-m}\,{\rho _{12}^{*}}^{1-\widetilde{m}% },\,\,\,\left| \lambda _{m}\right| =\left| m(1-m)\right| ,$$ $$e^{i\delta (m,\widetilde{m})}=2^{m+\widetilde{m}-1}(-i)^{\widetilde{m}-m}% \frac{\Gamma (1+m)}{\Gamma (2-\widetilde{m})}\,\left( \frac{m^{*}(m^{*}-1)}{% m(m-1)}\right) ^{1/2}\,.$$ Let us use for $f_{m,\widetilde{m}}$ the conformally covariant anzatz $$f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},...,\overrightarrow{\rho _{n}}% ;\overrightarrow{\rho _{0}})=\left( \prod_{i=1}^{n}\frac{\rho _{i,i+1}}{\rho _{i0}^{2}}\right) ^{m/n}\left( \prod_{i=1}^{n}\frac{\rho _{i,i+1}^{*}}{\rho _{i0}^{*2}}\right) ^{\widetilde{m}/n}f_{m,\widetilde{m}}(\overrightarrow{% x_{1}},...,\overrightarrow{x_{n}})\,,$$ where the anharmonic ratios $x_{r}$ ($r=1,2...n$) of the gluon coordinates are chosen as follows $$x_{r}=\frac{\rho _{r-1,r}\,\rho _{r+1,0}}{\rho _{r-1,0}\,\rho _{r+1,r}}% \,;\,\,\,\,\,\,\,\prod_{r=1}^{n}x_{r}=(-1)^{n}\,;\,\,\,\,\,% \sum_{r=1}^{n}(-1)^{r}\prod_{k=r+1}^{n}x_{k}=0\,.$$ The function $f_{m,\widetilde{m}}(\overrightarrow{x_{1}},...,\overrightarrow{% x_{n}})$ is invariant under certain modular transformations as a consequence of the Bose symmetry. For the physically interesting case $m,\widetilde{m}\rightarrow 1/2$ we can calculate $\psi _{m,\widetilde{m}}$, taking into account the logarithmic divergence of the integral at $\overrightarrow{\rho _{0}}\rightarrow \infty $: $$\lim_{m,\widetilde{m}\rightarrow 1/2}\psi _{m,\widetilde{m}}(\overrightarrow{% \rho _{12}},\overrightarrow{\rho _{23}},...\overrightarrow{\rho _{n1}})=% \frac{f(\overrightarrow{z_1},...\overrightarrow{z_n})}{1-m-\widehat{m}}% \,\prod_{k=1}^n\left| \rho _{k,k+1}\right|^{1/n} \,,\,\,z_r= \frac{\rho _{r-1,r}\,}{\,\rho _{r+1,r}}\,.$$ Let us change the variables $z_r$ to new ones $y_{k}$ as follows: $$y_{k}=\rho _{k+1,k}/\rho _{n,n-1}=(-1)^{n-k-1}\prod_{r=k+1}^{n-1}z_{r}\,\,,\,\,\,\,\,y_{n-1}=1\,,\,\,% \,\sum_{k=1}^{n}y_{k}=0\,.$$ The duality equation for the wave function $f(\overrightarrow{y_{1}},...,% \overrightarrow{y_{n-2}})$ can be written in the form: $$f(\overrightarrow{y_{1}},...,\overrightarrow{y_{n-2}})=\left| \lambda \right| \,\,\int \prod_{k=1}^{n-2}\frac{d^{2}y_{k}^{\prime }}{2\pi }\,K(% \overrightarrow{y};\overrightarrow{y^{\prime }})\prod_{k=1}^{n}\frac{\left| y_{k}\right| ^{1/n}}{\left| y_{k}^{\prime }\right| ^{2-1/n}}\,\,f(% \overrightarrow{y_{1}^{\prime }},...,\overrightarrow{y_{n-2}^{\prime }})\,.$$ The integral kernel $K(\overrightarrow{y};\overrightarrow{y^{\prime }})$ is given below: $$K(\overrightarrow{y};\overrightarrow{y^{\prime }})=2^{n}\int \frac{d^{2}\rho _{n,n-1}^{\prime }}{2\pi \left| \rho _{n,n-1}^{\prime }\right| ^{3}\left| \rho _{n,n-1}\right| }\,\exp \left( -i\,\sum_{k=1}^{n}\left( \overrightarrow{% \rho _{k}}\,,\,\overrightarrow{y_{k-1}^{\prime }\rho _{n,n-1}^{^{\prime }}}% \right) \right) \,$$ and is calculated analytically: $$K(\overrightarrow{y};\overrightarrow{y^{\prime }})=-2^{n}\left| \sum_{k=1}^{n}y_{k-1}^{\prime }\rho _{k}^{*}\right| \left| \rho _{n,n-1}\right| ^{-1}=-2^{n}\left| \sum_{k=1}^{n-1}y_{k-1}^{\prime }\sum_{r=n}^{k-1}y_{r}^{*}\right| .$$ To simplify the duality equation, we consider below the compound state of three reggeized gluons. Duality equation for the Odderon wave function ============================================== In the case of the Odderon the conformal invariance fixes the solution of the Schrödinger equation \[5\] $$f_{m,\widetilde{m}}(\overrightarrow{\rho _1},\overrightarrow{\rho _2},% \overrightarrow{\rho _3};\overrightarrow{\rho _0})=\left( \frac{\rho _{12}\,\rho _{23}\,\rho _{31}}{\rho _{10}^2\,\rho _{20}^2\,\rho _{30}^2}% \right) ^{m/3}\left( \frac{\rho _{12}^{*}\,\rho _{23}^{*}\,\rho _{31}^{*}}{% \rho _{10}^{*2}\,\rho _{20}^{*2}\,\rho _{30}^{*2}}\right) ^{\widetilde{m}% /3}f_{m,\widetilde{m}}(\overrightarrow{x})\,$$ up to an arbitrary function $f_{m,\widetilde{m}}(\overrightarrow{x})$ of one complex variable $x$ being the anharmonic ratio of four coordinates $$x=\frac{\rho _{12}\,\rho _{30}}{\rho _{10}\,\rho _{32}}\,.$$ Note that, owing to the Bose symmetry of the Odderon wave function, $f_{m,% \widetilde{m}}(\overrightarrow{x})$ has the following modular properties: $$f_{m,\widetilde{m}}(\overrightarrow{x})=(-1)^{(\widetilde{m}-m)/3}\,f_{m,% \widetilde{m}}(\overrightarrow{x}/\left| x\right| ^{2})\,=(-1)^{(\widetilde{m% }-m)/3}\,f_{m,\widetilde{m}}(\overrightarrow{1}-\overrightarrow{x})\,$$ and satisfies the normalization condition $$\left\| f_{m,\widetilde{m}}\right\| ^{2}= \int \frac{d^{2}x}{\left| x(1-x)\right| ^{4/3}}\left| f_{m,\widetilde{m}}(\overrightarrow{x})\right| ^{2}\,,$$ compatible with the modular symmetry. After changing the integration variable from $\rho _{0}$ to $x$ in accordance with the relations $$\rho _{0}=\frac{\rho _{31}\rho _{12}}{\rho _{12}-x\,\rho _{32}}+\rho _{1}\,,\,\,\,\,\,\,\,d\,\rho _{0}=\frac{\rho _{31}\rho _{12}\rho _{32}}{% (\rho _{12}-x\,\rho _{32})^{2}}\,\,dx$$ the Odderon wave function $\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{ij}% })$ at $\overrightarrow{q}=0$ can be written as $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{ij}})=\left( \frac{\rho _{23}% }{\rho _{12}\rho _{31}}\right) ^{m-1}\left( \frac{\rho _{23}^{*}}{\rho _{12}^{*}\rho _{31}^{*}}\right) ^{\widetilde{m}-1} \chi _{m,\widetilde{m}}(% \overrightarrow{z})\,\,,\,\,\,z=\frac{\rho _{12}}{\rho _{32}}\,,$$ where $$\chi _{m,\widetilde{m}}(\overrightarrow{z})=\int \frac{d^{2}x\,\,f_{m,% \widetilde{m}}(\overrightarrow{x})}{2\,\pi \left| x-z\right| ^{4}}\,\left( \frac{(x-z)^{3}}{x(1-x)}\right) ^{2m/3}\left( \frac{(x^{*}-z^{*})^{3}}{% x^{*}(1-x^{*})}\right) ^{2\widetilde{m}/3}.$$ In fact this function is proportional to $f_{1-m,1-\widetilde{m}}(% \overrightarrow{z})$: $$\chi _{m,\widetilde{m}}(\overrightarrow{z})\sim \left( x(1-x)\right) ^{2(m-1)/3}\left( x^{*}(1-x^{*})\right) ^{2(\widetilde{m}-1)/3}f_{1-m,1-% \widetilde{m}}(\overrightarrow{z})\,\,,$$ which is a realization of the discussed linear dependence between two representations $(m,\widetilde{m})$ and $(1-m,1-\widetilde{m})$. The corresponding reality property for the Möbius group representations can be presented in the form of the integral relation $$\chi _{m,\widetilde{m}}(\overrightarrow{z})=\int \frac{d^{2}x}{2\,\pi }% \,(x-z)^{2m-2}\,(x^{*}-z^{*})^{2\widetilde{m}-2}\,\chi _{1-m,1-\widetilde{m}% }(\overrightarrow{x})\,$$ for an appropriate choice of phases of the functions $\chi _{m,\widetilde{m}% } $ and $\chi _{1-m,1-\widetilde{m}}$. The function $\chi _{m,\widetilde{m}}(% \overrightarrow{z})$ satisfies the modular relations $$\chi _{m,\widetilde{m}}(\overrightarrow{z})=(-1)^{m-\widetilde{m}% }z^{2m-2}z^{*2\widetilde{m}-2}\chi _{m,\widetilde{m}}(\overrightarrow{z}% /\left| z\right| ^2)=(-1)^{m-\widetilde{m}}\chi _{m,\widetilde{m}}(% \overrightarrow{1}-\overrightarrow{z})\,.$$ The duality property of the wave function $\psi _{m,\widetilde{m}}(% \overrightarrow{\rho _{ij}})$ can be written in a form of the integral equation: $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{ij}})=\left| \lambda _{m}\right| \,2^{3}\,\int \frac{d^{2}\rho _{12}^{\prime }}{2\,\pi }\frac{% d^{2}\rho _{23}^{\prime }}{2\,\pi }\frac{\exp (i(\overrightarrow{\rho _{21}^{\prime }}\,\overrightarrow{\rho _{23}}+\overrightarrow{\rho _{31}^{\prime }}\,\overrightarrow{\rho _{31}}))}{\left| \rho _{12}^{\prime }\rho _{23}^{\prime }\rho _{31}^{\prime }\right| ^{2}}\,\psi _{\widetilde{m}% ,m}^{*}(\overrightarrow{\rho _{ij}^{\prime }})\,,$$ where $\left| \lambda _{m}\right| ^{2}$ is the corresponding eigenvalue of the differential operator $$\left| A\right| ^2=\left| \rho _{12}\rho _{23}\rho _{31}\,p_1p_2p_3\right| ^2\,.$$ The function $\psi _{\widetilde{m},m}^{*}(\overrightarrow{\rho _{ij}^{\prime }})$ is transformed as $\psi _{1-\widetilde{m},1-m}(\overrightarrow{\rho _{ij}^{\prime }})$: $$\psi _{\widetilde{m},m}^{*}(\overrightarrow{\rho _{ij}^{\prime }})\sim \psi _{1-\widetilde{m},1-m}(\overrightarrow{\rho _{ij}^{\prime }})\,.$$ In terms of $\chi _{m,\widetilde{m}}(\overrightarrow{z})$ the above duality equation looks as follows $$\frac{\chi _{m,\widetilde{m}}(\overrightarrow{z})}{\left| \lambda _{m}\right| }=\int \frac{d^{2}z^{\prime }\left( z^{\prime }(1-z^{\prime })\right) ^{\widetilde{m}-1}\left( z^{\prime *}(1-z^{\prime *})\right) ^{m-1}% }{2\pi \left( z(1-z)\right) ^{1-m}\left( z^{*}(1-z^{*})\right) ^{1-% \widetilde{m}}}\,R(\overrightarrow{z},\overrightarrow{z^{\prime }})\chi _{% \widetilde{m},m}^{*}(\overrightarrow{z^{\prime }})\,,$$ where $$R(\overrightarrow{z},\overrightarrow{z^{\prime }})=\int \frac{8\,d^{2}\rho _{32}^{\prime }/(2\pi )}{\left| \rho _{32}^{\prime }\right| ^{4}\left| \rho _{32}\right| ^{2}}\left( \rho _{32}\rho _{32}^{\prime *}\right) ^{m}\left( \rho _{32}^{*}\rho _{32}^{\prime }\right) ^{\widetilde{m}}\exp (i(% \overrightarrow{\rho _{32}}\overrightarrow{\rho _{12}^{\prime }}+% \overrightarrow{\rho _{13}}\overrightarrow{\rho _{13}^{\prime }}))\,.$$ This integral is calculated to be $$R(\overrightarrow{z},\overrightarrow{z^{\prime }})=C_{m,\widetilde{m}% }\,Z^{1-m}\,(Z^{*})^{1-\widetilde{m}}$$ with $$C_{m,\widetilde{m}}=\frac{2\,e^{i\delta (m,\widetilde{m})}}{\left| m(1-m)\right| }=-2^{\widetilde{m}+m}\frac{(-i)^{\widetilde{m}-m}}{\sin (\pi m)}\frac{\Gamma (1/2)}{\Gamma (2-m)}\frac{\Gamma (1/2)}{\Gamma (2-\widetilde{% m})}\,,$$ where the phase $\delta (m,\widetilde{m})$ was defined above and $$Z=zz^{\prime *}-z+1=z(z^{\prime *}-1+\frac{1}{z})\,.$$ Further, we introduce the new integration variable $z^{\prime }\rightarrow z^{\prime }{}^{*}$, which effectively leads to the substitution $\chi _{% \widetilde{m}.m}(\overrightarrow{z^{\prime }})\rightarrow \chi _{m,% \widetilde{m}}(\overrightarrow{z^{\prime }})$. By performing the modular transformation $1-\frac{1}{z}\rightarrow z$ and taking into account the modular relation $$\chi _{m,\widetilde{m}}(\overrightarrow{z})=(1-z)^{2m-2}(1-z^{*})^{2% \widetilde{m}-2}\chi _{m,\widetilde{m}}((\overrightarrow{1}-\overrightarrow{z% })/\left| 1-z\right| ^{2})\,,$$ one can rewrite the above duality equation for $\chi _{m,\widetilde{m}}(% \overrightarrow{z})$ as $$\frac{\chi _{m,\widetilde{m}}(\overrightarrow{z})}{\left| \lambda _{m}\right| \,C_{m,\widetilde{m}}}=\int \frac{d^{2}z^{\prime }}{2\pi }\left( \frac{z^{\prime }(1-z^{\prime })z(1-z)}{z^{\prime }-z}\right) ^{m-1}\left( \frac{z^{\prime }(1-z^{\prime })z(1-z)}{z^{\prime }-z}\right) ^{*\,% \widetilde{m}-1}\chi _{m,\widetilde{m}}^{*}(\overrightarrow{z^{\prime }}).$$ This equation for $\chi _{m,\widetilde{m}}(\overrightarrow{z})$ corresponds to the symmetry of the Odderon wave function under the involution $% p_{k}\leftrightarrow \frac{1}{2}\varepsilon _{klm}\rho _{lm}$. It can be written in the pseudo-differential form: $$z(1-z)\,(i\partial )^{2-m}\,z^{*}(1-z^{*})\,(i\partial ^{*})^{2-\widetilde{m}% }\,\varphi _{1-m,1-\widetilde{m}}(\overrightarrow{z})=\left| \lambda _{m,% \widetilde{m}}\right| \,\left( \varphi _{1-m,1-\widetilde{m}}(% \overrightarrow{z})\right) ^{*}\,,$$ where $$\varphi _{1-m,1-\widetilde{m}}(\overrightarrow{z})=2^{(1-m-\widetilde{m}% )/2}\,\left( z(1-z)\right) ^{1-m}\left( z^{*}(1-z^{*})\right) ^{1-\widetilde{% m}}\,\chi _{m,\widetilde{m}}(\overrightarrow{z})\,.$$ Note, that for a self-consistency of the above equation its right-hand side should be ortogonal to the zero modes of the operator $(i\partial )^{2-m}(i\partial ^{*})^{2-\widetilde{m}}$. Due to the Bose symmetry of the wave function it is enough to impose on it only one integral constraint: $$\int \frac{d^{2}z}{|z(1-z)|^{2}}\,\left( \varphi _{1-m,1-\widetilde{m}}(% \overrightarrow{z})\right) ^{*}=0\,.$$ The normalization condition for the wave function $$\left\| \varphi _{m,\widetilde{m}}\right\| ^{2}=\int \frac{d^{2}x}{\left| x(1-x)\right| ^{2}}\,\left| \varphi _{m,\widetilde{m}}(\overrightarrow{x}% )\right| ^{2}$$ is compatible with the duality symmetry. The holomorphic and anti-holomorphic factors of $f_{m,\widetilde{m}}(% \overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},\overrightarrow{\rho _{3}};\overrightarrow{\rho _{0}})$ are eigenfunctions of the integrals of motion $A$ and $A^{*}$: $$A\,f_{m}=\lambda _{m}\,f_{m}\,,\,\,\,\,A^{*}\,f_{\widetilde{m}}=\lambda _{% \widetilde{m}}\,f_{\widetilde{m}}\,\,,\,\,\,\,\lambda _{\widetilde{m}% }=(\lambda _{m})^{*}\,.$$ In the limit $m\rightarrow 1/2,\,\widetilde{m}\rightarrow 1/2$, corresponding to the ground state, the integral over $x$ for the wave function $\psi _{m,\widetilde{m}}$ at $q=0$ is calculated, since the main contribution appears from the singularity at $x=z$: $$\psi _{m,\widetilde{m}}(\overrightarrow{\rho _{ij}})\simeq \frac{1}{(m+% \widetilde{m}-1)}\,\left| \rho _{12}\rho _{23}\rho _{31}\right| ^{1/3}f(z,z^{*})\,\,,\,\,\,\,z=\rho _{12}/\rho _{32}\,,$$ where $f(x,x^{*})=f_{1/2,1/2}(x,x^{*})$. This means that one can obtain for $% f(x,x^{*})$ the following equation in the $x$-representation: $$\left| x(1-x)\right| ^{1/3}f(x,x^{*})=-\frac{4\left| \lambda \right| }{\pi }% \int \frac{d^{2}y\,\,\left| y-x\right| }{\left| y(1-y)\right| ^{5/3}}% \,\,f(y,y^{*})\,$$ for a definite choice of the phase of the ground-state function $f$. The common factor in front of the integral is in agreement with the condition of conservation of the norm of the wave function after this canonical transformation (only its sign can be changed). The above integral relation is reduced to the following pseudo-differential equation $$\left| x(1-x)\,p^{3/2}\right| ^{2}\varphi (x,x^{*})=\left| \lambda \right| \,\,\,\varphi (x,x^{*})\,,$$ where $p=i\,\partial /\partial (x)\,,\,p^{*}=i\,\partial /\partial (x^{*})$, if we introduce the new function $$\varphi (x,x^{*})=\left| x(1-x)\right| ^{1/3}f(x,x^{*})$$ and use the relation $$\left| p\right| ^{3}\,\left| x\right| =-\left| p\right| ^{3}\int_{-\pi /2}^{\pi /2}\frac{d\phi }{2\pi }\,\,\int_{-\infty }^{\infty }\frac{d\left| \overrightarrow{p}\right| }{\left| \overrightarrow{p}\right| ^{2}}\,\,\exp (-i\overrightarrow{p}\overrightarrow{x})=-\frac{\pi }{4}\,\,\delta ^{2}(% \overrightarrow{x})\,.$$ For self-consistency of the pseudo-differential equation one should impose on the wave function the following constraint: $$\int \,\frac{d^{2}x}{|x(1-x)|^{2}}\,\varphi (x,x^{*})=0\,.$$ Let us derive the duality equation for $\varphi _{m,\widetilde{m}}(% \overrightarrow{x})$ for general values of $m$ and $\widetilde{m}$ by using different arguments. We start with the conformally covariant anzatz for the holomorphic factor $\,f_{m}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})$: $$f_{m}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})=\left( \frac{\rho _{12}\,\rho _{23}\,\rho _{31}}{\rho _{10}^{2}\,\,\rho _{20}^{2}\,\,\rho _{30}^{2}}% \right) ^{m/3}f_{m}(x)\,\,,\,\,\,\,x=\frac{\rho _{12}\,\rho _{30}}{\rho _{10}\,\,\rho _{32}}\,.$$ In the $x$ representation the integral of motion $A=\rho _{12}\,\rho _{23}\,\rho _{31}\,p_{1}p_{2}p_{3}$ is an ordinary differential operator. When acting on $f_m (x)$, it can be presented in the following form $$\frac{i^{3}}{x}\left( \frac{m}{3}(x-2)+x(1-x)\partial \right) \frac{x}{1-x}% \left( \frac{m}{3}(1+x)+x(1-x)\partial \right) \frac{1}{x}\left( \frac{m}{3}% (1-2x)+x(1-x)\partial \right)$$ $$=X^{-m/3}\,A_{m}\,\,X^{m/3}\,\,,\,\,\, X=x\,(1-x)\,\,,\,\,\,\,A_{m}=a_{1-m}\,a_{m}\,\,,$$ where $$a_{m}=\,x\,\,(1-x)\,p^{m+1},\,\,\,\,\,\,a_{1-m}=\,x\,\,(1-x)\,\,p^{2-m}\,,\,% \,\,\,\,\,p=i\,\partial \,.$$ Therefore the differential equation $A\,\,f_{m}=\lambda _{m}\,\,f_{m}$ for eigenfunctions $f_{m}$ and eigenvalues $\lambda _{m}$ is equivalent to the system of two pseudo-differential equations $$a_{m}\,\varphi _{m}=l_{m}\,\,\varphi _{1-m}\,\,,\,\,\,\,\,\,\,a_{1-m}\,\varphi _{1-m}=l_{1-m}\,\varphi _{m}\,,$$ where we introduced the functions $\varphi _{m}$ and $\varphi _{1-m}$ in accordance with the definitions $$\varphi _{m}\equiv X^{m/3}\,f_{m}\,,\,\,\,\,\,\,\varphi _{1-m}\equiv X^{(1-m)/3}\,f_{1-m}\,$$ and the eigenvalues $l_{m}$ and $l_{1-m}$ are related with the eigenvalue $% \lambda _m$ of $A_m$ as follows $$\,\,l_{1-m}\,l_{m}=\lambda _{m}.$$ The function $\varphi _{1-m}(x)$ has the conformal weight equal to $1-m$. It is important that there is another relation between $\varphi _{m}$ and $% \varphi _{1-m}$$$\varphi _{1-m}=R_{m}(x,p)\,\varphi _{m}\,\,,$$ where $$R_{m}(x,p)=X^{-m+1}\,p^{1-2m}\,X^{-m}\,,\,\,\,\,\,\,R_{1-m}(x,p)=\left( R_{m}(x,p)\right) ^{-1}\,.$$ This follows from the fact that, for the Möbius group, the complex-conjugated representations $O_{m,\widetilde{m}}(\overrightarrow{\rho _{0}})$ and $O_{1-m,1-\widetilde{m}}(\overrightarrow{\rho _{0}})$ are linearly dependent. To obtain the above expression for $R_m(x,p)$ one should use for the Odderon wave function $f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},\overrightarrow{\rho _{3}};\overrightarrow{% \rho _{0}})$ the coformal anzatz and the property of holomorphic factorization. The transformation $$\varphi ^{\prime }(\overrightarrow{x})=U\,\varphi (\overrightarrow{x}% )\,,\,\,\,\,\,\,U=R_{m}(x,p)\,R_{\widetilde{m}}(x^{*},p^{*})$$ is unitary for our choice of the norm: $$\left\| \varphi ^{\prime }\right\| ^{2}=\left\| \varphi \right\| ^{2}\,.$$ Because $$\int \frac{d^{2}x}{\left| x(1-x)\right| ^{2}}\,(\varphi (\overrightarrow{x}% ))^{*}\,\,A_{m}\,(x)\,A_{\widetilde{m}}(x^{*})\,\varphi (\overrightarrow{x}% )=\left\| A_{m}\,\varphi \right\| ^{2}\,,$$ the eigenvalue of the operator $A_{\widetilde{m}}(x^{*})$ $\,$is complex conjugated to $\lambda _{m}$: $$A_{m}(x)\,\,\varphi (\overrightarrow{x})=\lambda _{m}\,\,\varphi (% \overrightarrow{x})\,\,,\,\,\,\,\,\,A_{\widetilde{m}}(x^{*})\,\varphi (% \overrightarrow{x})=\lambda _{m}^{*}\,\varphi (\overrightarrow{x})\,.$$ Note that due to its Möbius covariance $\varphi (\overrightarrow{x})$ is the sum of products of the eigenfunctions having opposite signs of their eigenvalues $\lambda $: $$\varphi (\overrightarrow{x})=\sum C_{ik}\left( \varphi _{m,\lambda }^{i}(x)\,\varphi _{\widetilde{m},\lambda ^{*}}^{k}(x^{*})+\varphi _{m,-\lambda }^{i}(x)\,\varphi _{\widetilde{m},-\lambda ^{*}}^{k}(x^{*})\,\right) \,\,.$$ Since under the simultaneous transformations $x\leftrightarrow x^{*}$ and$% \,\,m\leftrightarrow \widetilde{m}$ this function should be symmetric for fixed $\lambda $, we conclude, that eigenvalues satisfy one of two relations $$\lambda _{m}=\pm (\lambda _{m})^{*}$$ and therefore $\lambda _{m}$ can be purely real or imaginary. It turns out, that $\lambda _{m}$ is imaginary as a consequence of the modular invariance \[9\]. One can veryfy the validity of the following relation $$R_{m}(x,p)\,a_{1-m}\,a_{m}=a_{m}\,a_{1-m}\,R_{m}(x,p)\,,$$ if the following identity is used: $$X^{-m}a_{1-m}a_{m}X^{m}=x^{2}(1-x)^{2}p^{3}+2(1+m)\,x(1-x)\left( i\,p\,(1-2x)\,p-p\right) +$$ $$m(1+m)\,\left( x(1-x)\,p-(1-2x)^{2}\,p+i\,m\,(1-2x)\right) .$$ In particular this relation means that two eigenvalues of $A_{m}$ coincide $$\lambda _{m}=\lambda _{1-m}\,\,$$ for the eigenfunctions $\varphi _{m}$ and $\varphi _{1-m}$ linearly related by $R_{m}(x,p)$. Let us introduce the operators $$S_{m}(x,p)=R_{1-m}(x,p)\,a_{m}\,,\,\,\,\,\,\,S_{1-m}^{t}(x,p)=a_{1-m}% \,R_{m}(x,p)\,,\,$$ leaving the function $\varphi _{m}$ in the same space. Due to the above formulas these operators commute with one another: $$A_{m}=S_{1-m}^{t}(x,p)\,S_{m}(x,p)=S_{m}(x,p)\,S_{1-m}^{t}(x,p)\,.$$ Therefore we obtain two duality equations for the wave function whose conformal weight is equal to $m$: $$S_{m}(x,p)\,\varphi _{m}=\,L_{1}^{(m)}\,\varphi _{m}\,,\,\,\,\,\,\,S_{1-m}^{t}(x,p)\,\varphi _{m}=L_{2}^{(m)}\,\varphi _{m}\,.$$ As a consequence of the relation $$L_{1}^{(m)}L_{2}^{(m)}=\lambda \,$$ between the eigenvalues of $S_m$ and $S_{1-m}^t$, the second equation follows from the first one. In the particular case $m=1/2$, the equation for the holomorphic factors $% \varphi (x)$ looks especially simple: $$x(1-x)p^{3/2}\varphi _{\pm \sqrt{\lambda }}(x)=\pm \sqrt{\lambda }\varphi _{\pm \sqrt{\lambda }}(x)$$ and can be reduced in the $p$-representation to the Schrödinger equation with the potential $V(p)=p^{-3/2}$. For $m=\widetilde{m}=1/2$ the total odderon wave function $\varphi (x,x^{*})$, symmetric under the above canonical transformation, is a solution of the equation: $$\left| x(1-x)\right|^{2}\,|p|^3 \,\varphi (x,x^{*})=\left| \lambda \right| \,\varphi (x,x^{*}),$$ where in accordance with its hermicity properties the eigenvalue of the operator $\sqrt{A^{*}}$ for the anti-holomorphic factor $\varphi (x^{*})$ is taken to be equal to $\lambda ^{*}$. Single-valuedness condition =========================== There are three independent solutions $\varphi _{i}^{(m)}(x,\lambda )$ of the third-order ordinary differential equation $$a_{1-m}\,a_{m}\,\varphi =-ix(1-x)\left( x(1-x)\partial ^{2}+(2-m)\left( (1-2x)\partial -1+m\right) \right) \partial \,\varphi =\lambda \,\,\varphi$$ for each eigenvalue $\lambda $. In the region $x\rightarrow 0$ they can be chosen as follows $$\varphi _{r}^{(m)}(x,\lambda )=\sum_{k=1}^{\infty }d_{k}^{(m)}(\lambda )\,\,x^{k}\,\,,\,\,\,\,\,d_{1}^{(m)}(\lambda )=1\,.$$ $$\varphi _{s}^{(m)}(x,\lambda )=\sum_{k=0}^{\infty }a_{k}^{(m)}(\lambda )\,\,x^{k}+\,\varphi _{r}^{(m)}(x,\lambda )\,\ln x\,,\,\,\,\,\,a_{1}^{(m)}=0\,,$$ $$\varphi _{f}^{(m)}(x,\lambda )=\sum_{k=0}^{\infty }c_{k+m}^{(m)}(\lambda )\,\,x^{k+m}\,,\,\,\,\,\,c_{m}^{(m)}(\lambda )=1\,.$$ The appearance of $\ln x$ in $\varphi _{s}^{(m)}(x,\lambda )$ is related with the degeneracy of the differential equation in the small-$x$ region. There is an ambiguity in the definition of $\varphi _{s}^{(m)}(x,\lambda )$ because one can add to it the function $\varphi _{r}^{(m)}(x,\lambda )$ with an arbitrary coefficient. We have chosen $a_{1}^{m}=0$ to remove this uncertainty. Taking into account that for $\rho _{1}\rightarrow \rho _{2}$ the operator product expansion is applicable, the functions $\varphi _{i}^{(m)}(x,\lambda )$ can be considered as contributions of the holomorphic composite operators $O_{i}^{(M)}(\rho _{1})$ with the conformal weights $M=0,m$ and $1$ for $% i=s,\,f$ and $r$ correspondingly. In this interpretation the above degeneracy is related with the existence of the conserved vector current (for $m=1/2$ there is also a conserved fermion current). Due to the above differential equation the coefficients $a,c$ and $d$ satisfy the following recurrence relations $$\begin{aligned} i\lambda a_{k}^{(m)} &=&\left( a_{k+1}^{(m)}+d_{k+1}^{(m)}\frac{d}{d\,k}% \right) k(k+1)(k+1-m)-\left( a_{k}^{(m)}+d_{k}^{(m)}\frac{d}{d\,k}\right) k(k-m)(2k-m) \\ &&+\left( a_{k-1}^{(m)}+d_{k-1}^{(m)}\frac{d}{d\,k}\right) (k-1)(k-m)(k-1-m)\,\,,\end{aligned}$$ $$\begin{aligned} i\lambda c_{k+m}^{(m)} &=&(k+m)(k+m+1)(k+1)c_{k+m+1}^{(m)}-(k+m)k(2k+m)c_{k+m}^{(m)} \\ &&+(k+m-1)k(k-1)c_{k+m-1}^{(m)}\,,\end{aligned}$$ $$\begin{aligned} i\lambda d_{k}^{(m)} &=&k(k+1)(k+1-m)d_{k+1}^{(m)}-k(k-m)(2k-m)d_{k}^{(m)} \\ &&+(k-1)(k-m)(k-1-m)\,d_{k-1}^{(m)}.\end{aligned}$$ In particular, from the equation for $a_{k}^{(m)}$ at $k=0$, since $% d_{1}^{(m)}=1$, we obtain $$a_{0}^{(m)}=\frac{i}{\lambda }\,(m-1)\,.$$ The introduced functions have simple analytic properties in the vicinity of the point $x=0$. In particular, $\varphi _{r}^{(m)}(x,\lambda )$ is regular here and is transformed under the modular transformation $$x\rightarrow x^{\prime }=-x/(1-x)$$ as $$\varphi _{r}^{(m)}(x^{\prime },\lambda )=-(1-x)^{m}\,\varphi _{r}^{(m)}(x,-\lambda )\,.$$ The functions $\varphi _{s}^{(m)}(x,\lambda )$ and $\varphi _{f}^{(m)}(x,\lambda )$ have singularities at $x=0$, which leads to different results for their analytic continuations to negative values of $x$: $$\varphi _{s}^{(m)}(x^{\prime },\lambda )=-(1-x)^{m}\,\left( \varphi _{s}^{(m)}(x,-\lambda )\pm i\pi \,\varphi _{r}^{(m)}(x,-\lambda )\right) ,$$ $$\varphi _{f}^{(m)}(x^{\prime },\lambda )=\exp (\pm i\pi m)\,(1-x)^{m}\,\varphi _{f}^{(m)}(x,-\lambda ).$$ Therefore, from the Bose symmetry of the Odderon wave function $$f_{m,\widetilde{m}}(\overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},% \overrightarrow{\rho _{3}};\overrightarrow{\rho _{0}})=\left( \frac{\rho _{23}\,}{\,\rho _{20}\,\rho _{30}}\right) ^{m}\left( \frac{\,\rho _{23}^{*}\,% }{\,\rho _{20}^{*}\,\rho _{30}^{*}}\right) ^{\widetilde{m}}\varphi _{m,% \widetilde{m}}(x,x^{*})\,,\,$$ combined with the single-valuedness condition near $\overrightarrow{x}=0$, we obtain for the total wave function the following representation: $$\varphi _{m,\widetilde{m}}(x,x^{*})=\varphi _{f}^{(m)}(x,\lambda )\,\varphi _{f}^{(\widetilde{m})}(x^{*},\lambda ^{*}) +c_{1}\left( \varphi _{s}^{(m)}(x,\lambda )\,\varphi _{r}^{(\widetilde{m})}(x^{*},\lambda ^{*})+\varphi _{r}^{(m)}(x,\lambda )\,\varphi _{s}^{(\widetilde{m}% )}(x^{*},\lambda ^{*})\right)$$ $$+c_{2} \,\varphi _{r}^{(m)}(x,\lambda )\,\varphi _{r}^{(\widetilde{m}% )}(x^{*},\lambda ^{*})+\left( \lambda \rightarrow -\lambda \right) \, .$$ The complex coefficients $c_{1},c_{2}$ and the eigenvalues $\lambda $ are fixed from the conditions of the single-valuedness of $f_{m,\widetilde{m}}(% \overrightarrow{\rho _{1}},\overrightarrow{\rho _{2}},\overrightarrow{\rho _{3}};\overrightarrow{\rho _{0}})$ at $\overrightarrow{\rho _{3}}=% \overrightarrow{\rho _{i}}$ ($i=1,2$) and the Bose symmetry \[9\]. It is sufficient to require its invariance under the transformation $% \overrightarrow{\rho _{2}}\leftrightarrow \overrightarrow{\rho _{3}}$ corresponding to the symmetry of $\varphi _{m,\widetilde{m}}(x,x^{*})$ $$\varphi _{m,\widetilde{m}}(x,x^{*})=\varphi _{m,\widetilde{m}% }(1-x,1-x^{*})\,.$$ For this purpose, one should analytically continue the functions $\varphi _{i}^{(m)}(x)$ in the region near the point $x=1$ and calculate from the differential equation the monodromy matrix $C_{rk}^{(m)}\,\,$ defined by the relations $$\begin{aligned} \varphi _{r}^{(m)}(x,\lambda ) &=&\sum_{k}C_{rk}^{(m)}\,\varphi _{k}^{(m)}(1-x,-\lambda )\,,\, \\ \,\varphi _{r}^{(\widetilde{m})}(x^{*},\lambda ^{*}) &=&\sum_{k}C_{rk}^{(% \widetilde{m})}\,\varphi _{k}^{(\widetilde{m})}(1-x^{*},-\lambda ^{*})\,.\end{aligned}$$ Owing to the single-valuedness condition and the Bose symmetry of $f_{m,% \widetilde{m}}$ we obtain a set of linear equations for parameters $c_{1}$ and $c_{2}$ with coefficients expressed in terms of $C_{rk}^{(m)}$ and $% C_{rk}^{(\widetilde{m})}$. The spectrum of $\lambda $ is fixed by the condition of selfconsistency of these equations \[9\]. To derive the relations among parameters $c_1, c_2$ and $\lambda$ following from the duality symmetry, it is convenient to introduce the operators $$S_{m,\widetilde{m}}=S_m (x,p) \, S_{\widetilde{m}}(x^*,p^*)\,,\,\, S^{+}_{m,% \widetilde{m}}=S^{t}_{1-m} (x,p) \, S^{t}_{1-\widetilde{m}}(x^*,p^*)\, ,$$ where $S_m (x,p)$ and $S^{t}_{1-m}(x,p)$ were defined in the previous section. These operators are hermitially conjugated each to another and have the property: $$|A_m (x)|^2 = S^{+}_{m,\widetilde{m}}\,S_{m,\widetilde{m}}=S_{m,\widetilde{m}% } \, S^{+}_{m,\widetilde{m}} \,.$$ In particular it means, that they have common eigenfunctions $$S_{m,\widetilde{m}} \,\varphi _{m,\widetilde{m}}(x,x^{*})= \frac{|\lambda |^2% }{c_1}\,e^{i\theta}\,\varphi _{m,\widetilde{m}}(x,x^{*})\,\,, \,\,\,S^{+}_{m,% \widetilde{m}} \,\varphi _{m,\widetilde{m}}(x,x^{*})= c_1\,e^{-i\theta}\,\varphi _{m,\widetilde{m}}(x,x^{*})\,,$$ where $$e^{i\theta}=(-i)^{m-\widetilde{m}} \,\frac{\Gamma (m)}{\Gamma (1-\widetilde{m% })} \frac{\Gamma (1+m) \Gamma (1+\widetilde{m})}{\Gamma (2-m) \Gamma (2-% \widetilde{m})}\,.$$ The eigenvalues are obtained from the small-$x$ asymptotics of these equations. Because the operators $S_{m,\widetilde{m}}$ and $S_{m,\widetilde{m}}^{+}$ are hermitially conjugated, we have $$|c_{1}|=|\lambda |.$$ In the particular case $m=\widetilde{m}=1/2$, where $% S_{1/2,1/2}=|S_{1/2}|^{2}$, the coefficient $c_{1}$ is positive: $$c_{1}=|\lambda |\,.$$ Another relation can be derived if we shall take into account, that the complex conjugated representations $\varphi _{m,\widetilde{m}}$ and $\varphi _{1-m,1-\widetilde{m}}$ of the Möbius group are related by the unitary operator $U=R_m (x,p)R_{\widetilde{m}} (x^{*},p^{*})$, defined in the previous section: $$e^{i\gamma}(\varphi _{m,\widetilde{m}})^{*}= U \,\varphi _{m,\widetilde{m}% }\,,$$ where $e^{i\gamma}$ is an eigenvalue of this operator. By calculating the right hand side of this equation at $x \rightarrow 0$, we obtain $$e^{i\gamma}=(-1)^{m-\widetilde{m}}\, \frac{\Gamma (2-\widetilde{m}) \,\Gamma (2-m)}{\Gamma (1+\widetilde{m})\, \Gamma (1+m)} \,\frac{c_1}{c_1^{*}} \,\,,\,\,\, Im\,\frac{c_{2}}{c_{1}}= Im \,(m^{-1}+\widetilde{m}^{-1})\,.$$ One can verify from the numerical results of ref. \[9\] that both relations for $c_1$ and $c_2$ are fulfilled. For example, we have for the ground-state eigenfunction with $m=\widetilde{m}=1/2$: $$i \,\lambda =0.205257506 \,\,,\,\,\,c_1=0.205257506$$ and for one of the excited states with $m=\widetilde{m}=1/2+i \, 3/10$: $$i \, \lambda = 0.247227544 \,,\,\,c_1=0.247186043-i \,0.004529717 \,,\,\, c_2=-1.156524786-i \,0.415163678 ,$$ $$|c_1|=|\lambda| \,\,,\,\,\, Im \,\frac{c_2}{c_1}= 2 \,Im \,(1/2+i \, 3/10)^{-1}\,.$$ After the Fourier transformation of $\varphi _{m,\widetilde{m}}(% \overrightarrow{x})$ to the momentum space $\overrightarrow{p}$ the regular terms near the points $\overrightarrow{x}=0$ and $\overrightarrow{x}=1$ do not give any contribution to its asymptotic behaviour at $\overrightarrow{p}% \rightarrow \infty $. The requirement of the holomorphic factorization and single-valuedness of the wave function in the momentum space leads to the quantization of $\lambda $. We can obtain from the duality equation and reality condition also the representations for the coefficients $c_{1,2}$ in terms of integrals from $% \varphi _{m,\widetilde{m}}(x,x^{*})$ over the fundamental region of the modular group, where the expansion in $x$ is convergent. These relations allow one to calculate the coefficients $c_{1,2}$ without using the single-valuedness condition. Hamiltonian and integrals of motion =================================== The holomorphic Hamiltonian $h$ for the compound state of $n$ Reggeons for $% N_{c}\rightarrow \infty $ commutes with the transfer matrix $T(u)$ owing to the following relation for $h_{k,k+1}$: $$\left[ h_{k,k+1},T(u)\right] =-i\,tr\,\left( L_{1}(u)...L_{k-1}(u)\left( L_{k}(u)-L_{k+1}(u)\right) L_{k+2}(u)...L_{n}(u)\right) .$$ It can be considered as a linear equation for $h_{k,k+1}$. The formal solution of this equation can be written as $$h_{k,k+1}=\lim_{t\rightarrow \infty }\left( i\int_{0}^{t}d\,t\,^{^{\prime }}\,\exp (i\,T(u)\,t^{\prime })\left[ h_{k,k+1},T(u)\right] \exp (-i\,T(u)\,t^{\prime })+h_{k,k+1}(t)\right) \,,$$ where $h_{k,k+1}(t)$ is the time-dependent operator $$h_{k,k+1}(t)=\exp (i\,T(u)\,t)\,\,h_{k,k+1}\,\exp (-i\,T(u)\,t)\,\,.$$ Since the integral term is cancelled in the sum of $h_{k,k+1}$, we can substitute $$h_{k,k+1}\rightarrow h_{k,k+1}(t).$$ At $t\rightarrow \infty $ as a result of rapid oscillations of off-diagonal matrix elements, each pair Hamiltonian is diagonalized in the representation, where the transfer matrix is diagonal, and therefore it is a function of the integrals of motion $\widehat{q_{k}}$: $$h_{k,k+1}(\infty )=f_{k,k+1}(\widehat{q_{2}},\widehat{q_{3}},...\widehat{% q_{n}})\,.$$ Its dependence from the spectral parameter $u$ disappears in this limit and the total Hamiltonian is $$h=h(\widehat{q_2},\widehat{q_3},...\widehat{q_n})=\sum_{k=1}^nf_{k,k+1}(% \widehat{q_2},\widehat{q_3},...\widehat{q_n}).$$ All operators $O(t)$ satisfy the Heisenberg equations $$-i\,\frac{d}{d\,t}O(t)=\left[ T(u)\,,\,O(t)\right] \,$$ with certain initial conditions. In the case of the pair Hamiltonian the initial conditions are $$h_{k,k+1}(0)=\psi (\widehat{m}_{k,k+1})+\psi (1-\widehat{m}_{k,k+1})-2\psi (1)\,,$$ where the quantities $\widehat{m}_{k,k+1}$ are related to the pair Casimir operators as $$\widehat{m}_{k,k+1}(\widehat{m}_{k,k+1}-1)=M_{k,k+1}^{2}\,=-\rho _{k,k+1}^{2}\,\partial _{k\,}\,\partial _{k+1}.$$ In the case of the Odderon, $h$ does not depend on time if $h_{k,k+1}(t)$ is determined as $$h_{k,k+1}(t)=e^{itA}\,h_{k,k+1}\,e^{-itA}\,\,,$$ and $h_{k,k+1}(\infty )$ is a function of the total conformal momentum $% \overrightarrow{M}^{2}=\widehat{m}(\widehat{m}-1)$ and of the integral of motion $q_{3}=A$, which can be written as follows: $$A=\frac{i^3}2\left[ M_{12}^2\,,\,M_{13}^2\right] =\frac{i^3}2\left[ M_{23}^2\,,\,M_{12}^2\right] =\frac{i^3}2\left[ M_{13}^2\,,\,M_{23}^2\right] \,.$$ Using these formulas and the following relations among the Möbius group generators $\overrightarrow{M}_r$ $$M_{ir}^{2}-M_{kr}^{2}=2\,\left( \overrightarrow{M}_{i}-\overrightarrow{M}% _{k}\,,\,\overrightarrow{M}_{r}\right) \,,$$ $$\left[ h_{ik},\left[ M_{ik}^{2}\,,\,\overrightarrow{M}_{i}-\overrightarrow{M}% _{k}\right] \right] =4\left( \,\overrightarrow{M}_{i}-\overrightarrow{M}% _{k}\right) \,,$$ we can verify the commutation relations $$i\left[ h_{12}\,,\,A\right] =M_{13}^{2}-M_{23}^{2}\,,\,\,i\left[ h_{13}\,,\,A\right] =M_{23}^{2}-M_{12}^{2},\,\,i\left[ h_{23}\,,\,A\right] =M_{12}^{2}-M_{23}^{2}\,,$$ from which it is obvious that $A$ commutes with $h$. In a general case of $n$ reggeized gluons, one can use the Clebsch-Gordan approach, based on the construction of common eigenfunctions of the total momentum $\widehat{M}$ and a set $\left\{ \widehat{M}_{k}\right\} $ of the commuting sub-momenta, to find all operators $M_{k,k+1}^{2}$ in the corresponding representation. However to calculate $h$ we should perform an unitary transformation to the representation, where $T(u)$ is diagonal, because in this case for $t\rightarrow \infty $ the off-diagonal matrix elements of $M_{k,k+1}^{2}$ disappear and their diagonal elements depend only on $q_{r}$: $$\begin{aligned} f_{k,k+1}(q_2,q_3,...q_n)=\langle q_2,...q_n\mid h_{k,k+1}\mid q_2,...q_n\rangle \,,\end{aligned}$$ $$\widehat{q_k}\mid q_2,...q_n\rangle =q_k\mid q_2,...q_n\rangle \,.$$ Let us consider, for example, the interaction between particles $1$ and $2$. The transfer matrix, which should be diagonalized, can be written as follows $$T(u)=\left( u^{2}-\frac{1}{2}\overrightarrow{L}^{2}\right) d_{3...n}(u)+\left( i\,u\,\overrightarrow{L}-\frac{1}{4}\left[ \overrightarrow{L}^{2},\overrightarrow{N}\right] \right) \overrightarrow{d}% _{3...n}(u)\,\,,$$ where the differential operators $d_{3...n}(u)$ and $\overrightarrow{d}% _{3...n}(u)$ are independent of $\overrightarrow{\rho _{1}}$ and $% \overrightarrow{\rho _{2}}$. They are related to the monodromy matrix $% t_{3...n}(u)\,$ for particles $3,4,...,n$ as follows $$d_{3...n}(u)=tr\,t_{3...n}(u)\,,\,\overrightarrow{d}_{3...n}(u)=tr(% \overrightarrow{\sigma }t_{3...n}(u)),\,\,t_{3...n}(u)=L_{3}(u)...L_{n}(u)\,$$ and the matrix $t_{3...n}(u)$ satisfies the Yang-Baxter equations with a hidden Lorentz symmetry. The operators $\overrightarrow{L}$ and $\overrightarrow{N}$ are constructed in terms of the Möbius group generators of particles $1$ and $2$: $$\overrightarrow{L}=\overrightarrow{M}_1+\overrightarrow{M}_2\,,\,\,% \overrightarrow{N}=\overrightarrow{M}_1-\overrightarrow{M}% _2\,,\,\,M_k^z=\rho _k\partial _k\,,\,M_k^{+}=-\rho _k^2\partial _k\,,\,M_k^{-}=\partial _k\,.$$ They have the commutation relations, corresponding to the Lorentz algebra: $$\left[ L^{z},L^{\pm }\right] =\pm L^{\pm },\,\left[ L^{+},L^{-}\right] =2L^{\pm },\,\left[ L^{z},N^{\pm }\right] =\pm N^{\pm },\,$$ $$\left[ L^{+},N^{-}\right] =2N^{z}\,,\,\left[ N^{z},N^{\pm }\right] =\pm L^{\pm },\,\left[ N^{+},N^{-}\right] =2L^{z}\,.$$ Let us introduce the Polyakov basis for the wave function of the composite state of two gluons with the conformal weight $M$: $$\mid \rho _{0^{\prime }},M\rangle =\left( \frac{\rho _{12}}{\rho _{10^{\prime }}\rho _{20^{\prime }}}\right) ^{M}.$$ Here $\rho _{0^{\prime }}$ enumerates the components of the infinite-dimensional irreducible representation of the conformal group. One can verify that the representation of the generators $\overrightarrow{L}$ and $\overrightarrow{N}$ in this basis is given by $$\begin{aligned} M^z &\mid &\rho _{0^{\prime }},M\rangle =(\rho _1\partial _1+\rho _2\partial _2)\mid \rho _{0^{\prime }},M\rangle =-(\rho _{0^{\prime }}\partial _{0^{\prime }}+M)\mid \rho _{0^{\prime }},M\rangle , \\ M^{+} &\mid &\rho _{0^{\prime }},M\rangle =-(\rho _1^2\partial _1+\rho _2^2\partial _2)\mid \rho _{0^{\prime }},M\rangle =(\rho _{0^{\prime }}^2\partial _{0^{\prime }}+2M\rho _{0^{\prime }})\mid \rho _{0^{\prime }},M\rangle ,\end{aligned}$$ $$M^{-} \mid \rho _{0^{\prime }},M\rangle =(\partial _1+\partial _2)\mid \rho _{0^{\prime }},M\rangle =-\partial _{0^{\prime }}\mid \rho _{0^{\prime }},M\rangle$$ and $$\begin{aligned} \left( N^{z}-\rho _{0^{\prime }}\,N^{-}\right) &\mid &\rho _{0^{\prime }},M\rangle =\frac{M}{M-1}\,\,\partial _{0^{\prime }}\mid \rho _{0^{\prime }},M-1\rangle , \\ \left( N^{+}+2\rho _{0^{\prime }}\,N^{z}-\rho _{0^{\prime }}^{2}N^{-}\right) &\mid &\rho _{0^{\prime }},M\rangle =-2M\mid \rho _{0^{\prime }},M-1\rangle ,\end{aligned}$$ $$\frac{2M-1}{M(M-1)}N^{-}\mid \rho _{0^{\prime }},M\rangle =\,\mid \rho _{0^{\prime }},M+1\rangle +\frac{1}{(M-1)^{2}}\,\,\partial _{0^{\prime }}^{2}\mid \rho _{0^{\prime }},M-1\rangle .$$ Note that there is a simple relation among the generators, provided that they act on the state $\mid \rho _{0^{\prime }},M\rangle $: $$\left[ N^{z}-\rho _{0^{\prime }}\,N^{-}\,,\,N^{+}+2\rho _{0^{\prime }}\,N^{z}-\rho _{0^{\prime }}^{2}N^{-}\right] =M^{+}+2\rho _{0^{\prime }}\,M^{z}-\rho _{0^{\prime }}^{2}M^{-}=0\,.$$ The eigenfunction of the transfer matrix $T(u)$ can be written as a superposition of the states $\mid \rho _{0^{\prime }},M\rangle $ with various values of $\rho _{0^{\prime }}$ and $M$: $$f_{m}(\rho _{1},\rho _{2},...,\rho _{n};\rho _{0})=\sum_{M}\int d\rho _{0^{\prime }}\mid \rho _{0^{\prime }},M\rangle \,\,f_{m,M}(\rho _{0^{\prime }},\rho _{3},...,\rho _{n};\rho _{0})\,,$$ where $m$ is the conformal weight of the composite state. The function $% f_{m,M}(\rho _{0^{\prime }}...\rho _{n};\rho _{0})$ in accordance with the Möbius symmetry, has the form $$f_{m,M}(\rho _{0^{\prime }},\rho _{3},...,\rho _{n};\rho _{0})=\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M-1}\prod_{r=3}^{n}\left( \frac{\rho _{r0}}{% \rho _{r0^{^{\prime }}}}\right) ^{-\frac{m+M}{n-2}}\psi (x_{1},x_{2},...,x_{n-3}),$$ where $x_{r}$ are independent anharmonic ratios constructed from the coordinates $\rho _{0^{\prime }},\rho _{3},...,\rho _{n}$. Because of its Möbius invariance, the transfer matrix $T(u)$ after acting on $f_{m}(\rho _{1},...,\rho _{n};\rho _{0})$ gives again a superposition of the states $\mid \rho _{0^{\prime }},M\rangle $, but with the coefficients which are linear combinations of $f_{m,M}$ and $f_{m,M\pm 1} $. Therefore for its eigen function the coefficients satisfy some recurrence relations, and the problem of the diagonalization of the transfer matrix $% T(u)$ is reduced to the solution of these recurrence relations. For $n\geq 3$ in the sub-channel $\rho _{1,2}$ the recurrence relations depend on matrix elements of the operators $\overrightarrow{d}_{3...n}(u)$ and $d_{3...n}(u)$ between the wave functions $f_{m,M}(\rho _{0^{\prime }},\rho _{3},...,\rho _{n};\rho _{0})$ which should be chosen in such a way, to provide the property of $f_{m}(\rho _{1},\rho _{2},...,\rho _{n};\rho _{0})$ to be a representation of the cyclic group of transformations $i\rightarrow i+1$. In the appendix we consider these relations in the first non-trivial case $% n=3$. In the next section the relation between the Odderon Hamiltonian and its integral of motion $A$ is discussed from another point of view. Odderon Hamiltonian in the normal order ======================================= Let us write down the pair Hamiltonian as follows \[4\] $$h_{12}=\log (\rho _{12}^{2}\,\partial _{1})+\log (\rho _{12}^{2}\,\partial _{2})-2\,\log (\rho _{12}\,)-2\,\psi (1)\,.$$ This representation allows us to present the total Hamiltonian for $n$ reggeized gluons in the form invariant under the Möbius transformations $$h=\sum_{k=1}^{n}\left( \log \left( \frac{\rho _{k+2,0}\,\,\rho _{k,k+1}^{2}}{% \rho _{k+1,0}\,\,\rho _{k+1,k+2}}\,\partial _{k}\right) +\log \left( \frac{% \rho _{k-2,0}\,\,\rho _{k,k-1}^{2}}{\rho _{k-1,0}\,\,\rho _{k-1,k-2}}% \,\partial _{k}\right) -2\,\psi (1)\right) \,,$$ where $\rho _{0}$ is the coordinate of the composite state. We consider below in more detail the Odderon. Using for its wave function the conformal anzatz $$f_{m}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})=\left( \frac{\rho _{23}}{\rho _{20}\rho _{30}}\right) ^{m}\varphi _{m}(x)\,,\,\,\,\,x=\frac{\rho _{12}\rho _{30}}{\rho _{10}\rho _{32}}\,,\,\,$$ one can obtain the following Hamiltonian for the function $\varphi _{m}(x)$ in the space of the anharmonic ratio $x$ \[4\] $$h=6\gamma +\log \left( x^{2}\partial \right) +\log \left( (1-x)^{2}\partial \right) +\log \left( \frac{x^{2}}{1-x}((1-x)\partial +m)\right) +$$ $$\log \left( \frac{1}{1-x}((1-x)\partial +m))\right) +\,\log \left( \frac{% (1-x)^{2}}{x}(x\partial -m)\right) +\log \left( \frac{1}{x}(x\partial -m)\right) .$$ It is convenient to introduce the logarithmic derivative $P\equiv x\partial $ as a new momentum. Using the relations \[4\]: $$\begin{aligned} \log (x^{2}\partial ) &=&\log (x)+\psi (1-P)\,,\,\,\,\,\log (\partial )=-\log (x)+\psi (-P)\,\,,\,\, \\ \log (x^{2}\partial ) &=&\log (\partial )+2\log (x)-\frac{1}{P}\,,\end{aligned}$$ $$((1-x)\partial +m)=(1-x)^{1+m}\,\partial \,(1-x)^{-m}\,,\,\,\,\,\,x\partial -m=x^{1+m}\partial \,x^{-m}\,,$$ one can transform this Hamiltonian to the normal form: $$\frac{h}{2}=\,\,-\log (x)+\psi (1-P)+\psi (-P)+\psi (m-P)-3\psi (1)+\sum_{k=1}^{\infty }x^{k}\,f_{k}(P)\,,$$ where $$f_{k}(P)=-\frac{2}{k}+\frac{1}{2}\left( \frac{1}{P+k-m}+\frac{1}{P+k}\right) +\sum_{t=0}^{k}\frac{c_{t}(k)}{P+t}\,.$$ Here $$c_{t}(k)=\frac{(-1)^{k-t}\,\,\Gamma (m+t)\,\left( (t-k)\,(m+t)+m\,k/2\right) }{k\,\Gamma (m-k+t+1)\,\Gamma (t+1)\,\Gamma (k-t+1)}\,.$$ On the other hand the differential operators $a_{m}$ and $a_{1-m}$ can be written in terms of the quantities $P$ and $x$ as follows $$\begin{aligned} a_{m} &=&i^{-1-m}x^{-m}(1-x)\frac{\Gamma (m-P+1)}{\Gamma (-P)}\,,\,\, \\ a_{1-m} &=&i^{-2+m}x^{-1+m}(1-x)\frac{\Gamma (-P-m+2)}{\Gamma (-P)}\,.\end{aligned}$$ Using the above representation for $h$ and the following expression for the integral of motion: $$B=i\,a_{1-m}\,a_{m}=\frac{(1-x)}{x}\left( (1-x)P-1-x+xm\right) P\,(P-m) \,,$$ one can verify their commutativity $$\left[ h,B\right] =0\,.$$ Therefore $h$ is a function of $B$. In particular for large $B$ this function should have the form: $$\frac{h}{2}=\,\,\log (B)+3\gamma +\sum_{r=1}^{\infty }\frac{c_{r}}{B^{2r}}\,.$$ The first two terms of this asymptotic expansion were calculated in ref. \[4\]. The series is constructed in inverse powers of $B^{2}$, because $h$ should be invariant under all modular transformations, including the inversion $x\rightarrow 1/x$ under which $B$ changes its sign. The same functional relation should be valid for the eigenvalues $\varepsilon /2$ and $\mu =i\,\lambda $ of these operators: $$\frac{\varepsilon }{2}=\,\,\log (\mu )+3\gamma +\sum_{r=1}^{\infty }\frac{% c_{r}}{\mu ^{2r}}\,.$$ For large $\mu $ it is convenient to consider the corresponding eigenvalue equations in the $P$ representation, where $x$ is the shift operator $$x=\exp (-\frac{d}{dP})\,,$$ after extracting from eigenfunctions of $B$ and $h$ the common factor $$\varphi _{m}(P)=\Gamma (-P)\,\Gamma (1-P)\,\Gamma (m-P)\,\exp (i\pi P)\,\,\Phi _{m}(P).$$ The function $\Phi _{m}(P)$ can be expanded in series over $1/\mu $ $$\Phi _{m}(P)=\sum_{n=0}^{\infty }\mu ^{-n}\Phi _{m}^{n}(P)\,,\,\Phi _{m}^{0}(P)=1\,,$$ where the coefficients $\Phi _{m}^{n}(P)$ turn out to be the polynomials of order $4n$ satisfying the recurrence relation: $$\Phi _{m}^{n}(P)=\sum_{k=1}^{P}(k-1)\,(k-1-m)\,\left( (k-m)\,\Phi _{m}^{n-1}(k-1)+(k-2)\,\Phi _{1-m}^{n-1}(k-1-m)\right)$$ $$-\frac{1}{2}\sum_{k=1}^{m}(k-1)\,(k-1-m)\,\left( (k-m)\,\Phi _{m}^{n-1}(k-1)+(k-2)\,\Phi _{1-m}^{n-1}(k-1-m)\right) ,$$ valid due to the duality equation written below for a definite choice of the phase of $\Phi _{m}(P)$ $$x^{-m}\left( 1-x\,P\,(P-m)\,(P-m+1)\right) \,\Phi _{m}(P)=\mu ^{m}\,\Phi _{1-m}(P)\,$$ with the use of the substitution $x\,\mu \rightarrow x$. Note that the summation constants $\Phi _{m}^{n}(0)$ in the above recurrence relation have the anti-symmetry property $$\Phi _{m}^{n}(0)=-\Phi _{1-m}^{n}(0)\,,$$ which guarantees the fulfilment of the relation $$\Phi _{m}^{n}(m)=\Phi _{1-m}^{n}(0\,)\,$$ being a consequence of the duality relation $$\Phi _{m}^{n}(P)=\Phi _{1-m}^{n}(P-m)+(P-1)(P-m)(P-m-1)\Phi _{m}^{n-1}(P-1)\,.$$ The symmetric part of $\Phi _{m}^{n}(0)$ under the substitution $% m\leftrightarrow 1-m$ would simply modify the normalization constant for $% \Phi _{m}(P)$. The energy can be expressed in terms of $\Phi _{m}(P)$ as follows: $$\frac{\varepsilon }{2}=\,\,\log (\mu )+3\gamma +\frac{\partial}{\partial P}% \log \Phi _{m}(P)$$ $$+\left( \Phi _{m}(P)\right)^{-1} \, \sum_{k=1}^{\infty }\mu ^{-k}\,f_{k}(P-k)\,\Phi _{m}(P-k)\prod_{r=1}^{k}(P-r)(P-r+1)(P-r-m+1)\,$$ and it should not depend on $P$ due to the commutativity of $h$ and $B$. By solving the recurrence relations for $\Phi _{m}^{n}(P)$ and putting the result in the above expression, we obtain the following asymptotic expansion for $\varepsilon /2$: $$\frac{\varepsilon }{2}=\log (\mu )+3\gamma +\left( \frac{3}{448}+\frac{13}{% 120}(m-1/2)^{2}-\frac{1}{12}(m-1/2)^{4}\right) \frac{1}{\mu ^{2}}+$$ $$\left( -\frac{4185}{2050048}-\frac{2151}{49280}(m-1/2)^{2}+...\right) \frac{1% }{\mu ^{4}}+\left( \frac{965925}{37044224}+...\right) \frac{1}{\mu ^{6}}% +...\,\,\,.$$ This expansion can be used with a certain accuracy even for the smallest eigenvalue $\mu =0.20526$, corresponding to the ground-state energy $% \varepsilon =0.49434$ \[9\]. For the first excited state with the same conformal weight $m=1/2$, where $\varepsilon = 5.16930$ and $\mu = 2.34392$ \[9\], the energy can be calculated from the above asymptotic series with a good precision. The analytic approach, developed in this section, should be compared with the method based on the Baxter equation \[10\]. In the conclusion, we note that the remarkable properties of the Reggeon dynamics are presumably related with supersymmetry. In the continuum limit $% n\rightarrow \infty $ the above duality transformation coincides with the supersymmetric translation, which is presumably connected with the observation \[11\], that in this limit the underlying model is a twisted $N=2$ supersymmetric topological field theory. Additional arguments supporting the supersymmetric nature of the integrability of the reggeon dynamics were given in ref. \[12\]. Namely, the eigenvalues of the integral kernels in the evolution equations for quasi-partonic operators in the $N=4$ supersymmetric Yang-Mills theory are proportional to $\psi (j-1)$, which means that these evolution equations in the multicolour limit are equivalent to the Schrödinger equation for the integrable Heisenberg spin model similar to the one found in the Regge limit \[7\]. Note that at large $N_{c}$ the $N=4$ Yang-Mills theory is guessed to be related with the low-energy asymptotics of a superstring model \[13\]. $$$$ I want to thank the Universität Hamburg for its hospitality during my stay in Germany, where the basic part of this work was done. I thank G. Altarelli, J. Ellis and other participants of the CERN theory seminar for their interest in my talk. Fruitful discussions with L. Faddeev, A. Neveu, V. Fateev, A. Zamolodchikov, J. Bartels, A. Martin, B. Nicolescu, P. Gauron, E. Antonov, M. Braun, A. Bukhvostov, S. Derkachev, A. Manashov, G. Volkov, R. Kirschner, L. Szymanowski and J. Wosiek were helpful. $$$$ Appendix {#appendix .unnumbered} ======== Here we consider consequences of the conformal weight representation for the Odderon. In this case the total Hamiltonian is $$h(m,\lambda )\,=\,h_{12}+h_{23}+h_{31}\,.$$ The eigenvalue of $h$ is expressed in terms of its matrix elements: $$h(m,\lambda )=\sum_{k=1}^{3}\langle m,\lambda \mid h_{k,k+1}\mid m,\lambda \rangle \,,$$ where $\mid m,\lambda \rangle $ is a normalized eigenfunction of two commuting operators $$\left( \sum_{k=1}^{3}\overrightarrow{M_{k}}\right) ^{2}\mid m,\lambda \rangle =m(m-1)\mid m,\lambda \rangle \,,\,\,\,\,A\mid m,\lambda \rangle =\lambda \mid m,\lambda \rangle \,.$$ Let us consider for definiteness the interaction in the channel $12$, where $% M$ is the pair conformal weight. If one will construct the matrix $% V_{M}^{\lambda }(m)$ performing the unitary transformation between the $M$- and $\lambda $-representations $$\mid m,\lambda \rangle =\sum_{M}V_{M}^{\lambda }(m)\mid m,M\rangle \,,\,\,\,M_{12}^{2}\mid m,M\rangle =M(M-1)\mid m,M\rangle \,,\,\,\,\sum_{M}V_{M}^{\lambda }\,V_{\lambda ^{\prime }}^{M}=\delta _{\lambda ^{\prime }}^{\lambda },$$ then the diagonal matrix elements of the pair Hamiltonian $h_{12}$ can be calculated as $$\langle m,\lambda \mid h_{12}\mid m,\lambda \rangle =\sum_{M}h(M)\,V_{M}^{\lambda }\,(m)V_{\lambda }^{M}(m)\,,\,\,h(M)=\psi (M)+\psi (1-M)+2\gamma \,.$$ We shall derive below the recurrence relations for $V_{M}^{\lambda }$. To begin with, we note that, according to the commutation relations for the Lorentz algebra generators, there are non-vanishing matrix elements of the boost operator $\overrightarrow{N}$ only between the states $\mid M\rangle $ and $\mid M\pm 1\rangle $. This is valid also for the matrix elements of the operators $A$ and $M_{13}^{2}-M_{23}^{2}$ according to the relations $$\langle M^{\prime }\mid A\mid M\rangle =\frac{i^{3}}{4}(M^{\prime }-M)\,(M^{\prime }+M-1)\,\,\langle M^{\prime }\mid M_{13}^{2}-M_{23}^{2}\mid M\rangle \,,$$ $$\langle M\pm 1\mid M_{13}^{2}-M_{23}^{2}\mid M\rangle =2\overrightarrow{M_{3}% }\,\langle M\pm 1\mid \overrightarrow{N}\mid M\rangle .$$ Thus, for the common eigenfunctions $\mid m,M\rangle $ of two commuting Casimir operators $$\left( \sum_{k=1}^{3}\overrightarrow{M_{k}}\right) ^{2}\mid m,M\rangle =m(m-1)\mid m,M\rangle \,$$ and $$M_{12}^{2}\mid m,M\rangle =M(M-1)\mid m,M\rangle \,$$ we have $$A\mid m,M\rangle =\frac{i^3}2\left( MC_{m,M}^{+}\mid m,M+1\rangle -(M-1)C_{m,M}^{-}\mid m,M-1\rangle \right) \,,$$ where the coefficients $C_{m,M}^{\pm }$ are defined by the relations $$\left( M_{13}^2-M_{23}^2\right) \mid m,M\rangle =C_{m,M}^{+}\mid m,M+1\rangle +C_{m,M}^{-}\mid m,M-1\rangle \,.$$ One can obtain from the above equations the following recurrence relation for the unitary matrix $V_{M}^{\lambda }(m)$: $$\lambda \,V_M^\lambda (m)=\frac{i^3}2\left( (M-1)C_{m,M-1}^{+}\,V_{M-1}^\lambda (m)-MC_{m,M+1}^{-}V_{M+1}^\lambda (m)\right) .$$ To calculate the matrix elements $C_{m,M}^{\pm }$ of the operator $% M_{13}^{2}-M_{23}^{2}$, we use the above representation of the generators $% \overrightarrow{L}$ and $\overrightarrow{N}$ in the Polyakov basis and obtain $$\left( M_{13}^{2}-M_{23}^{2}\right) \mid \rho _{0^{\prime }},M\rangle =\left( 2N^{z}M_{3}^{z}+N^{+}M_{3}^{-}+N^{-}M_{3}^{+}\right) \,\mid \rho _{0^{\prime }},M\rangle =$$ $$2M\left( \frac{1}{M-1}\,\rho _{30^{\prime }}\partial _{0^{\prime }}-1\right) \mid \rho _{0^{\prime }},M-1\rangle \,\partial _{3}-\rho _{30^{\prime }}^{2}\,N^{-}\mid \rho _{0^{\prime }},M\rangle \,\partial _{3}\,.$$ Owing to the Möbius invariance, the three-gluon state $\mid m,M\rangle $, with the conformal weights $m$ and $M$, can be written as a superposition of the Polyakov functions $$f_{m,M}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})=\int_{L}d\rho _{0^{\prime }}\,\mid \rho _{0^{\prime }},M\rangle \,\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M-1}\left( \frac{\rho _{30}}{\rho _{30^{^{\prime }}}}% \right) ^{-M-m+1}$$ with various integration contours $L$. By integrating the terms in $M_{3}^{i} $ with derivatives of $\mid \rho _{0^{\prime }},M\rangle $ by parts and using the relations $$-\rho _{30^{\prime }}^{2}\partial _{3}\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M-1}\left( \frac{\rho _{30}}{\rho _{30^{^{\prime }}}}\right) ^{-M-m+1}=-(M+m-1)\,\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M}\,\left( \frac{\rho _{30}}{\rho _{30^{^{\prime }}}}\right) ^{-M-m},$$ $$\left( \frac{1}{1-2M}\,\partial _{0^{\prime }}^{2}\,\rho _{30^{\prime }}^{2}\,-2\,\partial _{0^{\prime }}\,\rho _{30^{\prime }}-2(M-1)\right) \,\partial _{3}\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M-1}\left( \frac{% \rho _{30}}{\rho _{30^{^{\prime }}}}\right) ^{-M-m+1}=$$ $$-(M+m-1)\,\frac{(m-M)(m-M+1)}{2M-1}\,\left( \rho _{0^{^{\prime }}0}\right) ^{-m+M-2}\left( \frac{\rho _{30}}{\rho _{30^{^{\prime }}}}\right) ^{-M-m+2},$$ one can obtain the recurrence relation for the function $f_{m,M}=\,f_{m,M}(% \rho _{1},\rho _{2},\rho _{3};\rho _{0})$: $$\left( M_{13}^{2}-M_{23}^{2}\right) \,f_{m,M}=\frac{M(M+m-1)}{1-2M}\left( (M-1)\,f_{m,M+1}+\frac{(m-M)(m-M+1)}{M-1}f_{m,M-1}\right) .$$ Due to its Möbius covariance $f_{m,M}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})$ can be presented in the form $$f_{m,M}(\rho _{1},\rho _{2},\rho _{3};\rho _{0})=\left( \frac{\rho _{12}\rho _{23}\rho _{31}}{\rho _{10}^{2}\rho _{20}^{2}\rho _{30}^{2}}\right) ^{m/3}f_{m,M}(x)\,,$$ where $x$ is the anharmonic ratio $$x=\frac{\rho _{12}\,\rho _{30}}{\rho _{10}\,\rho _{32}}\,.\,$$ By introducing the new integration variable $$x^{^{\prime }}=\frac{\rho _{12}\rho _{30^{^{\prime }}}}{\rho _{10^{^{\prime }}}\rho _{32}}\,$$ we obtain the following expression for $f_{m,M}(x)\,$: $$f_{m,M}(x)=\left( x(1-x)\right) ^{2m/3}\int dx^{^{\prime }}(1-x^{^{\prime }})^{-M}(x^{^{\prime }}-x)^{-m+M-1}\left( \frac{x}{x^{^{\prime }}}\right) ^{-M-m+1}.$$ This function satisfies the following differential equation $$M_{12}^{2}(x)\,f_{m,M}(x)=M(M-1)\,f_{m,M}(x)\,$$ and the recurrence relation $$\left( M_{13}^{2}(x)-M_{23}^{2}(x)\right) \,f_{m,M}(x)=$$ $$\frac{M(M+m-1)}{1-2M}\left( (M-1)\,f_{m,M+1}(x)+\frac{(m-M)(m-M+1)}{M-1}% f_{m,M-1}(x)\right) \,.$$ The pair Casimir operators in the $x$-representation are given below $$M_{12}^{2}(x)=\frac{x}{1-x}\left( \frac{m}{3}(1-2x)+x(1-x)\partial \right) \frac{1}{x}\left( \frac{m}{3}(1+x)+x(1-x)\partial \right) ,$$ $$M_{13}^{2}(x)=\frac{1-x}{x}\left( \frac{m}{3}(1-2x)+x(1-x)\partial \right) \frac{1}{1-x}\left( \frac{m}{3}(x-2)+x(1-x)\partial \right) ,$$ $$M_{23}^{2}(x)=-\frac{1}{x(1-x)}\left( \frac{m}{3}(x-2)+x(1-x)\partial \right) \left( \frac{m}{3}(1+x)+x(1-x)\partial \right)$$ and satisfy the relation: $$M_{12}^{2}(x)+M_{13}^{2}(x)+M_{23}^{2}(x)=m(m-1)\,.$$ The function $f_{m,M}(x)$ can be expressed for two different choices of the integration contour $L$ through the hypergeometric functions: $$f_{m,M}^1(x)=\frac{\Gamma (1-M)}{\Gamma (M)\Gamma (2-2M)}\,\left( x(1-x)\right) ^{2m/3}x^{1-M-m}F(m+1-M,1-M;2-2M;x)\, ,$$ $$f_{m,M}^2(x)=\frac{\Gamma (m+M)}{\Gamma (2M)\Gamma (1+m-M)}\left( x(1-x)\right) ^{2m/3}x^{M-m}F(m+M,M;2M;x)\,,$$ where $$F(a,b;c;x)=1+\frac{ab}{c}\frac{x}{1!}+\frac{a(a+1)b(b+1)}{c(c+1)}\frac{x^{2}% }{2!}+...\,\,.$$ Moreover, the above eigenvalue equation for $f_{m,M}(x)$ is equivalent to the hypergeometric equation for $F(a,b;c;x)$: $$x(1-x)\frac{d^2}{dx^2}F+\left( c-(a+b+1)x\right) \frac d{dx}F-abF=0\,,$$ because $$\left( x(1-x)\right) ^{-2m/3}x^{-M+m}M_{12}^2(x)\left( x(1-x)\right) ^{2m/3}x^{M-m}= M(M-1)+$$ $$x\,\left( x(1-x)\frac{d^2}{dx^2}+\left( 2M-(m+2M+1)x\right) \frac d{dx}-M(M+m)\right) \,.$$ In an analogous way, using the relation $$\left( x(1-x)\right) ^{-2m/3}x^{-M+m}\left( M_{13}^{2}(x)-M_{23}^{2}(x)\right) \left( x(1-x)\right) ^{2m/3}x^{M-m}=$$ $$\frac{1}{x}(M(2-x)+(1-m)x+(2-x)x\partial )\left( M(1-x)-m+x(1-x)\partial \right)$$ and extracting from it the hypergeometric differential operator with the coefficient chosen to cancel the term with the second derivative, we obtain that the above recurrence relation for $f_{m,M}(x)$ is equivalent to the following relation for the hypergeometric function $F_{M}(x)=F(M,M+m;2M;x)$: $$\left( m(m-1)+M(2m-1-M)+2(1-m)(1-x)\partial +\frac{2M(M-m)}{x}\right) F_{M}(x)$$ $$=(M-m)\left( \frac{(M-1)(M+m)(M+m-1)}{2(2M-1)(2M+1)}\,x\,F_{M+1}(x)+\frac{2M% }{x}F_{M-1}(x)\right) .$$ One can verify its validity in the $k$-th order of the expansion in $x$ by taking into account the algebraic identity: $$(m+2k)(m-1)+M(2m-1-M)+2\left( 1-m+\frac{M(M-m)}{k+1}\right) \frac{% (M+k)(M+m+k)}{2M+k}$$ $$=\frac{(M-m)(M+m-1)}{2M-1}\left( \frac{(M-1)k}{2M+k}+\frac{M(2M+k-1)}{k+1}% \right) \,.$$ Let us now return to the problem of finding the recurrence relations for the matrix $V^{\lambda }_M (m)$ performing the unitary transformation between the $M$- and $\lambda $-representations. As earlier, it is convenient to work with the function $\varphi _m(x)$ defined by the relation $$f_m (x) =(x(1-x))^{-m/3} \,\varphi _m (x) \,.$$ It satisfies the equation $$a_{1-m}\,a_m \varphi _m (x)=\lambda \varphi _m (x)\,,$$ where $$a_{m}=x(1-x)(i\partial )^{m+1}.$$ One can search the solution of the above eigenvalue equation as the linear combination $$\varphi _m(x)=\sum_{M}C_{M}(m)\,F_{M}^{m}(x)\,$$ of the eigenfunctions of the operator $M_{12}^{2}$. The coefficients $C_M (m) $ would coincide with the matrix $V^{\lambda }_M (m)$, if the functions $% F_{M}^{m} (x)$ would be normalized. However, in accordance with above formulas we define $F_{M}^{m}(x)$ by the following expression: $$F_{M}^{m}(x)=\frac{\Gamma (2M)\Gamma (1+m-M)}{\Gamma (m+M)}\left( x(1-x)\right) ^{m/3}f_{m,M}^{2}(x)\,=x^{M}(1-x)^{m}\,F(m+M,M;2M;x)\,.$$ The transition to the renormalized functions can be easily done. According to the above presentation the quantities $F_M^m(x)$ satisfy the following equality: $$a_{1-m}a_{m}\,F_{M}^{m}\,(x)=\frac{i^{3}}{2}M(M-1)(M-m)\left( \frac{% (M+m)(M+m-1)}{2(2M-1)(2M+1)}\,F_{M+1}^{m}(x)-2F_{M-1}^{m}(x)\right) .$$ Because the right-hand side of this equality is zero for $M=0,1$ and $m$, one can restrict the summation over $M$ in the eigenfunction $\varphi _{m}(x) $ to two series: $M=r$ and $M=m+r$ with $r=0,1,2,...$. However, in the first case the coefficient in front of $F_{1}^{m}(x)$ can not be calculated through the coefficient in front of $F_{0}^{m}(x)$. Due to this degeneracy of the equation one should introduce for $r\geq 1$ the more complicated function $\Phi _{r}^{m}(x)$: $$\Phi _{r}^{m}(x)=\lim_{M\rightarrow r}\frac{d}{dM}F_{M}^{m}(x)\,.$$ The recurrence relation for these functions can be obtained by differentiating the relation for $F_{M}^{m}(x)$ . In particular, for $r=1$ we obtain $$a_{1-m}a_{m}\,\Phi _{1}^{m}\,(x)=\frac{i^{3}}{2}(1-m)\left( \frac{(1+m)\,m}{6% }\,F_{2}^{m}(x)-2F_{0}^{m}(x)\right) \,.$$ Thus, in accordance with the small-$x$ behaviour of the eigenfunctions $% \varphi (x)$, discussed above, we write the linearly independent solutions in the form: $$\varphi _{r}^{(m)}(x,\lambda )=\sum_{k=1}^{\infty }\Delta _{k}^{m}(\lambda )\,F_{k}^{m}(x)\,.$$ $$\varphi _{s}^{(m)}(x,\lambda )=\sum_{k=0}^{\infty }\left( \alpha _{k}^{m}(\lambda )\,F_{k}^{m}(x)+\Delta _{k}^{m}(\lambda )\Phi _{k}^{m}\,(x)\right) \,,$$ $$\varphi _{f}^{(m)}(x,\lambda )=\sum_{k=0}^{\infty }\gamma _{k}^{m}(\lambda )\,F_{k+m}^{m}(x)\,,$$ The coefficients $\alpha _{k}^{m},\gamma _{k}^{m}$ and $\Delta _{k}^{m}$ satisfy the recurrence relations: $$i\lambda \,\alpha _{k}^{m}(\lambda )=\left( \alpha _{k+1}^{m}(\lambda )+\beta _{k+1}^{m}(\lambda )\,\frac{d}{dk}\right) \,k(k+1)(k-m+1)\,$$ $$-\frac{1}{4}\left( \alpha _{k-1}^{m}(\lambda )+\beta _{k-1}^{m}(\lambda )\,% \frac{d}{dk}\right) (k-1)(k-2)(k-m-1)\frac{(k+m-1)(k+m-2)}{(2k-3)(2k-1)}\,,$$ $$i\lambda \,\Delta _{k}^{m}(\lambda )=-\frac{1}{4}(k-1)(k-2)(k-m-1)\frac{% (k+m-1)(k+m-2)}{(2k-3)(2k-1)}\Delta _{k-1}^{m}(\lambda )$$ $$+k(k+1)(k-m+1)\,\Delta _{k+1}^{m}(\lambda )\,,\,\,\,\Delta _{1}^{m}(\lambda )=1\,;$$ $$i\lambda \,\gamma _{k}^{m}(\lambda )=-\frac{1}{4}(k+m-1)(k+m-2)(k-1)\frac{% (k+2m-1)(k+2m-2)}{(2k+2m-3)(2k+2m-1)}\gamma _{k-1}^{m}(\lambda )$$ $$+(k+m)(k+m+1)(k+1)\,\gamma _{k+1}^{m}(\lambda )\,,\,\,\,\gamma _{0}^{m}(\lambda )=1\,.$$ If we compare these relations with the derived above analogous recurrence relations for the coefficients of the expansion of $\varphi _{r}^{(m)}(x)$ in the series over $x$, it is obvious that the factors in front of the corresponding quantities with the index $k+1$ coincide. Furthermore, the quantities $\alpha _{k},\,\gamma _{k}$ and $\Delta _{k}$ are absent in the right-hand side of these relations contrary to the previous case, where the similar factors in front of $a_{k}\,,\,c_{k}$ and $d_{k}$ are non-zero. $$$$ References {#references .unnumbered} ========== $$$$ 1\. L.N. Lipatov, Sov. J. Nucl. Phys. ${\bf 23}$ (1976) 642; V.S. Fadin, E.A. Kuraev and L.N. Lipatov, Phys. Lett. ${\bf B60}$ (1975) 50; Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. ${\bf 28}$ (1978) 822; L.N. Lipatov, Sov. Phys. JETP ${\bf 63}$ (1986) 904; 2\. V.S. Fadin, L.N. Lipatov, Phys. Lett. ${\bf {B429}}$ (1998) 127; 3\. J.Bartels, Nucl. Phys. ${\bf {B175}}$ (1980) 365 ; J. Kwiecinski and M. Prascalowicz, Phys. Lett. ${\bf {B94}}$ (1980) 413; 4\. L.N. Lipatov, Phys. Lett. ${\bf {B251}}$ (1990) 284; ${\bf {B309}}$ (1993) 394 5\. L.N. Lipatov, hep-th/9311037, Padua preprint DFPD/93/TH/70, unpublished; 6\. R.J. Baxter, Exactly Solved Models in Statistical Mechanics, (Academic Press, New York, 1982); V.O. Tarasov, L.A. Takhtajan and L.D. Faddeev, Theor. Math. Phys. ${\bf {57}} $ (1983) 163; 7\. L.N. Lipatov, Sov. Phys. JETP Lett. ${\bf {59}}$ (1994) 571; L.D. Faddeev and G.P. Korchemsky, Phys. Lett. ${\bf {B342}}$ (1995) 311; 8\. C. Montonen and D. Olive, Phys. Lett. ${\bf 72}$ (1977) 117; N. Seiberg and E. Witten, Nucl. Phys. ${\bf 426}$ (1994) 19; 9\. L.N. Lipatov, Recent Advances in Hadronic Physics, Proceedings of the Blois conference (World Scientific, Singapore, 1997); R.Janik and J. Wosiek, hep-ph/9802100, Crakow preprint TPJU-2/98; M.A. Braun, hep-ph/9801352, St.Petersburg University preprints; M.A. Braun, P. Gauron and B. Nicolescu, preprint LPTPE/UP6/10/July 98; M. Praszalowicz and A. Rostworowski, hep-ph/9805245, Crakow preprint TPJU-8/98; 10\. R. Janik and J. Wosiek, Phys. Rev. Lett. ${\bf 79}$ (1997) 2935; 11\. J. Ellis and N.E. Mavromatos, preprint OUTP-98-51P, hep-ph/9807451; 12\. L.N. Lipatov, Perspectives in Hadronics Physics, Proceedings of the ICTP conference (World Scientific, Singapore, 1997). 13\. J. Maldacena, Adv. Theor. Math. Phys. ${\bf 2:231}$ (1998), hep-th/9711200.

ArXiv

--- abstract: 'The spectrum of energy levels is computed for all available angular momentum and parity quantum numbers in the SU(2)-Higgs model, with parameters chosen to match experimental data from the Higgs-$W$ boson sector of the standard model. Several multiboson states are observed, with and without linear momentum, and all are consistent with weakly interacting Higgs and $W$ bosons. The creation operators used in this study are gauge-invariant so, for example, the Higgs operator is quadratic rather than linear in the Lagrangian’s scalar field.' author: - Mark Wurtz and Randy Lewis title: Higgs and $W$ boson spectrum from lattice simulations --- Introduction ============ The complex scalar doublet of the standard model accommodates all of the necessary masses for elementary particles. A testable prediction of this theory is the presence of a fundamental scalar particle: the Higgs boson. Recently, ATLAS and CMS have discovered a Higgs-like boson with a mass near 125 GeV [@Aad:2012tfa; @Chatrchyan:2012ufa]. Lattice simulations of the scalar doublet with the SU(2) gauge part of the electroweak theory give a nonperturbative description of the Higgs mechanism. Early studies [@Fradkin:1978dv; @Osterwalder:1977pc; @Lang:1981qg; @Seiler:1982pw; @Kuhnelt:1983mw; @Montvay:1984wy; @Jersak:1985nf; @Evertz:1985fc; @Gerdt:1984ft; @Langguth:1985dr; @Montvay:1985nk; @Langguth:1987vf; @Evertz:1989hb; @Hasenfratz:1987uc] revealed two regions in the phase diagram: the Higgs region with three massive vector bosons and a single Higgs particle, and the confinement region with QCD-like bound states of the fundamental fields. These two regions are partially separated by a first-order phase transition, but are analytically connected beyond the phase transition’s end point. Subsequent lattice studies of the SU(2)-Higgs model have explored the electroweak finite-temperature phase transition [@Jansen:1995yg; @Rummukainen:1996sx; @Laine:1998jb; @Fodor:1999at] and recent work has incorporated additional scalar doublets [@Wurtz:2009gf; @Lewis:2010ps]. In the present work, we calculate the spectrum of the standard SU(2)-Higgs model at zero temperature in the Higgs region of the phase diagram. As already mentioned, there will be a Higgs boson ($H$) and three massive vector bosons ($W^1$, $W^2$ and $W^3$), but the spectrum contains much more than this. For comparison recall the well-known case of QCD, which has a small number of fields in the Lagrangian (gluons and quarks) and a huge number of particles in the spectrum (glueballs and hadrons). The glueballs and hadrons are created by gauge-invariant operators but the gluons and quarks correspond to gauge-dependent fields in the Lagrangian. The spectrum of the SU(2)-Higgs model is similar, at least in the confinement region: the Lagrangian contains gauge fields and a doublet of scalar fields, but lattice simulations suggest a dense spectrum of “W-balls” and “hadrons.” (For lattice studies of the spectrum in 2+1 dimensions, see Refs. [@Philipsen:1996af; @Philipsen:1997rq].) It is interesting to consider the spectrum in the Higgs region of the phase diagram. At weak coupling (which is directly relevant to the actual experimental situation), one might anticipate one Higgs boson, three vector bosons, and nothing else. On the other hand, since the Higgs region and the confinement region are truly a single phase, one might wonder whether the rich spectrum of confinement-region states will persist into the Higgs region, though smoothly rearranged in some way. An appealing view can be found in Refs. [@Philipsen:1996af; @Philipsen:1997rq] where the smooth transition from confinement region to Higgs region was observed for an SU(2)-Higgs model in 2+1 dimensions. Reference [@Philipsen:1997rq] describes the results in terms of a flux loop that is completely stable in the pure gauge theory but can decay in the confinement region of the SU(2)-Higgs phase diagram. When approaching the analytic pathway into the Higgs region, such decays become so rapid that the particle description loses its relevance, leaving the Higgs region with the simple spectrum of Higgs and $W$ bosons. Reference [@Philipsen:1997rq] concludes by emphasizing the usefulness of a future study of multiparticle states in the Higgs region. In practice, even a simple spectrum of four bosons ($W^1$, $W^2$, $W^3$, $H$) will be accompanied by a tower of multiparticle states ($WW$, $WH$, $HH$, $WWW$, …) that is consistent with conservation of weak isospin, angular momentum and parity. Therefore a thorough lattice study of the spectrum will always involve many states appearing with many different quantum numbers. In general, these could be bound states and/or scattering states, and there is a history of nonlattice attempts to determine whether a pair of Higgs bosons might form a bound state [@Cahn:1983vi; @Contogouris:1988rd; @Grifols:1991gw; @Rupp:1991bb; @DiLeo:1994xc; @Clua:1995ni; @DiLeo:1996aw; @Siringo:2000kh]. The existence of nonperturbative states for $\phi^4$ theory in 2+1 dimensions has support from lattice simulations [@Caselle:2000yx; @Caselle:1999tm]. Attempts for the 3+1 dimensional SU(2)-Higgs model [@Maas:2012tj; @Maas:2012zf] (see for example Fig. 3 of Ref. [@Maas:2012zf]) indicate that the task of computing the Higgs-region spectrum with sufficient precision to observe and identify more than the most basic states is a significant challenge. We have had success in this endeavor, which is the theme of the present work. Section \[sec:simulation\] describes the method used to create the lattice ensembles. Section \[sec:operators\] develops a set of creation operators that is able to probe all quantum numbers $I(\Lambda^P)$, where $I$ denotes weak isospin, $P$ is parity, and $\Lambda$ is a lattice representation corresponding to angular momentum. Section \[sec:analysis\] explains how the variational method was used for analysis of the lattice data. Section \[sec:spectrum\] presents the energy spectrum that was obtained from our lattice simulations. Section \[sec:biggerlattice\] examines the effects on the spectrum of increasing the lattice volume. Section \[sec:infiniteHiggs\] reports on a simulation with a much larger Higgs mass, so that changes in the energy spectrum can be observed and understood. Section \[Two-Particle Operators\] describes the construction of two-particle operators and uses them to extend the observed energy spectrum. Concluding remarks are contained in Sec. \[sec:conclusions\]. Simulation Details {#sec:simulation} ================== The discretized SU(2)-Higgs action used for lattice simulations is given by $$\begin{aligned} S[U,\phi] &=& \sum_x \left\{ \beta \sum_{\mu<\nu} \left[ 1 - \tfrac{1}{2} \operatorname{Tr} \left( U_\mu(x) U_\nu(x+\hat\mu)U_\mu^\dag(x+\hat\nu)U_\nu^\dag(x) \right) \right] \right. \nonumber \\ && + \tfrac{1}{2} \operatorname{Tr} \left(\phi^\dag(x)\phi(x)\right) + \lambda \left( \tfrac{1}{2} \operatorname{Tr} \left(\phi^\dag(x)\phi(x)\right) - 1 \right)^2 \nonumber \\ && \left. - \kappa \sum_{\mu=1}^4 \operatorname{Tr} \left( \phi^\dag(x)U_\mu(x) \phi(x+\hat\mu) \right) \right\} \,\, ,\end{aligned}$$ where $U_\mu(x) = e^{iag_0A_\mu(x)}$ is the gauge field, $\phi(x)$ is the scalar field in $2\times2$ matrix form, $\beta = \frac{4}{g_0^2}$ is the gauge coupling, $\kappa = \frac{1-2\lambda}{8+a^2\mu_0^2}$ is the hopping parameter (related to the inverse bare mass squared), and $\lambda = \kappa^2\lambda_0$ is the scalar self-coupling. The $2\times2$ complex scalar field contains only four degrees of freedom because of a relation involving a Pauli matrix, $$\begin{aligned} \sigma_2\phi(x)\sigma_2 = \phi^*(x) \,,\end{aligned}$$ and is written as $\phi(x) = \rho(x)\alpha(x)$, where $\rho(x)>0$ is called the scalar length and $\alpha(x)\in SU(2)$ is the scalar’s angular component. We refer to $\phi(x)$ as the scalar field rather than the Higgs field, reserving the “Higgs” label for the physical particle which, as discussed in Sec. \[sec:operators\], is quadratic in the scalar field. Our simulations are performed in the Higgs region of the phase diagram, with a gauge coupling near the physical value $g_0^2\approx\frac{4\pi\alpha}{\sin^2\theta_W}\approx\frac{4\pi\alpha}{1-m_W^2/m_Z^2}\approx 0.5$, corresponding to $\beta=8$, which is in the weak coupling region. The remaining parameters are tuned to $\kappa=0.131$ and $\lambda=0.0033$ to give a Higgs mass near the physical value of $\sim$125 GeV and a reasonable lattice spacing. The number of lattice sites is $20^3\times40$ (where the longer direction is Euclidean time) and $24^3\times48$, and the scale is set with the $W$ mass defined to be 80.4 GeV. For comparison, separate simulations are carried out with $\kappa=0.4$ and $\lambda=\infty$. Although $\phi^4$ theories are trivial, the standard model can be viewed as an effective field theory up to some finite cutoff. The calculations presented in this paper are at a cutoff of approximately $1/a=400$ GeV. Even though the continuum limit is problematic in a trivial theory, simulations at an appropriately-large cutoff are sufficient to produce phenomenological results. Standard heatbath and over-relaxation algorithms [@Creutz:1980zw; @Creutz:1984mg; @Kennedy:1985nu; @Creutz:1987xi; @Bunk:1994xs; @Fodor:1994sj] were used for the Monte Carlo update of the gauge and scalar fields. Define one sweep to mean an update at all sites across the lattice. Then our basic update step is one gauge heatbath sweep followed by two scalar heatbath sweeps followed by one gauge over-relaxation sweep followed by four scalar over-relaxation sweeps. Ten of these basic update steps are performed between the calculation of lattice observables. Any remaining autocorrelation is handled by binning the observable. Stout link smearing [@Morningstar:2003gk] and scalar smearing [@Bulava:2009jb; @Peardon:2009gh] are used to improve the lattice operators, reduce statistical fluctuations, and construct a large basis of operators. For the gauge links, one stout-link iteration is given by $$\begin{aligned} U^{(n+1)}_\mu(x) &= \exp\left\{-r_{\rm stout} \, Q^{(n)}_\mu(x)\right\} U^{(n)}_\mu(x) \quad , \quad \mu \neq 4 \\ Q^{(n)}_\mu(x) &= \frac{1}{2} \sum_{\nu \neq \mu, \nu \neq 4} \left\{ U^{(n)}_\mu(x) U^{(n)}_\nu(x+\hat{\mu}) U^{(n)\dag}_\mu(x+\nu) U^{(n)\dag}_\nu(x) \right. \nonumber \\ & \left. \qquad\qquad + U^{(n)}_\mu(x) U^{(n)\dag}_\nu(x+\hat{\mu}-\hat{\nu}) U^{(n)\dag}_\mu(x-\hat{\nu}) U^{(n)}_\nu(x-\hat{\nu}) \right\} - \text{h.c.}\end{aligned}$$ where $r_{\rm stout}$ is the stout link smearing parameter. Only the spatial links are smeared, and only in the spatial direction. The final stout links $\tilde{U}$ are given after a number of successive smearing iterations $$\begin{aligned} U = U^{(0)} \rightarrow U^{(1)} \rightarrow U^{(2)} \rightarrow \cdots \rightarrow U^{(n_{\rm stout})} = \tilde{U} \,\, .\end{aligned}$$ The smearing for the scalar field uses the lattice Laplacian $\Delta$, $$\begin{aligned} \phi^{(n+1)}(x) &= \left(1 + r_{\rm smear}\Delta\right)\phi^{(n)}(x) \\ &= \phi^{(n)}(x) + r_{\rm smear}\sum_{\mu=1}^3\left\{ \tilde{U}_\mu(x)\phi^{(n)}(x+\hat{\mu}) - 2\phi^{(n)}(x) + \tilde{U}^\dag_\mu(x-\hat{\mu})\phi^{(n)}(x-\hat{\mu}) \right\} \,\, ,\end{aligned}$$ where $r_{\rm smear}$ is the scalar smearing parameter. Note that the stout links $\tilde{U}$ are used for scalar smearing, and only in spatial directions. The final smeared scalar fields $\tilde{\phi}$ are given by $$\begin{aligned} \phi = \phi^{(0)} \rightarrow \phi^{(1)} \rightarrow \phi^{(2)} \rightarrow \cdots \rightarrow \phi^{(n_{\rm smear})} = \tilde{\phi} \,\, .\end{aligned}$$ Primary operators {#sec:operators} ================= This study begins with two basic options for gauge-invariant operators, the first being two scalar fields connected by a string of gauge links, and the second being a closed loop of gauge links. Use of stout links and smeared scalar fields within those operators enables many different possible gauge link paths and scalar field separations to be included. To obtain information about continuum angular momentum from a lattice simulation, there is a well-known correspondence with irreducible representations (irreps) of the octahedral group of rotations [@Johnson:1982yq; @Berg:1982kp], as shown in Table \[irrep\_spin\_table\]. ----------- --- --- --- --- --- --- --- --------- $\Lambda$ 0 1 2 3 4 5 6 $\dots$ $A_1$ 1 0 0 0 1 0 1 $\dots$ $A_2$ 0 0 0 1 0 0 1 $\dots$ $E$ 0 0 1 0 1 1 1 $\dots$ $T_1$ 0 1 0 1 1 2 1 $\dots$ $T_2$ 0 0 1 1 1 1 2 $\dots$ ----------- --- --- --- --- --- --- --- --------- : The number of copies of each irreducible representation $\Lambda$ for each continuum spin $J$.[]{data-label="irrep_spin_table"} The simplest gauge-invariant operator that can be constructed from scalar fields is the Higgs length operator $$\begin{aligned} H(t) = \frac{1}{2} \operatorname{Tr}\sum_{\vec{x}}\phi^\dag(x)\phi(x) = \sum_{\vec{x}} \rho^2(x) \,\, ,\end{aligned}$$ where the sum includes all spatial sites at a single Euclidean time. The $H(t)$ operator transforms according to the $\Lambda^P=A_1^+$ irrep and thus couples to the spin-0 Higgs state. Notice that the Higgs operator is quadratic in the scalar field $\phi(x)$, as is familiar from the earliest SU(2)-Higgs model lattice simulations [@Fradkin:1978dv; @Osterwalder:1977pc; @Lang:1981qg; @Seiler:1982pw; @Kuhnelt:1983mw; @Montvay:1984wy; @Jersak:1985nf; @Evertz:1985fc; @Gerdt:1984ft; @Langguth:1985dr; @Montvay:1985nk; @Langguth:1987vf; @Evertz:1989hb; @Hasenfratz:1987uc]. The simplest operator that couples to the $W$ particle is the isovector gauge-invariant link $$\begin{aligned} W^a_\mu(t) = \frac{1}{2} \operatorname{Tr} \sum_{\vec{x}} -i\sigma^a \phi^\dag(x) U_\mu(x) \phi(x+\hat{\mu}) \,\, ,\end{aligned}$$ which belongs to the $\Lambda^P=T_1^-$ irrep. Notice that, in general, an isovector operator does not have definite charge conjugation. The operator $W^a_\mu(t)$, for example, transforms under charge conjugation as $(W^1_\mu,W^2_\mu,W^3_\mu) \rightarrow (-W^1_\mu,+W^2_\mu,-W^3_\mu)$. Clearly, if the operator $W_\mu^a$ is given an arbitrary isospin rotation it will not be an eigenfunction of charge conjugation. Therefore charge conjugation is not helpful for the present work. Other irreps can be obtained by considering more complicated operators. The gauge-invariant link operator $$\begin{aligned} L^\phi_{\mu\nu\rho}(t) = \sum_{\vec{x}} \phi^\dag(x) U_\mu(x) U_\mu(x+\hat{\mu}) U_\nu(x+2\hat{\mu}) U_\rho(x+2\hat{\mu}+\hat{\nu}) \phi(x+2\hat{\mu}+\hat{\nu}+\hat{\rho}) \,, \label{asymtwistedlinkphi}\end{aligned}$$ shown in Fig. \[figure\_link\], ![Sketch of the two-scalar-field operator $L_{\mu\nu\rho}$. The two dots at the ends of $L_{\mu\nu\rho}$ represent the scalar fields.[]{data-label="figure_link"}](figure_link.eps) has 48 possible orientations and is one of the simplest two-scalar-field operators that couples to all of the $I(\Lambda^P)$ channels. Also considered is the gauge-invariant link constructed using SU(2)-“angular” components of the scalar field $$\begin{aligned} L^\alpha_{\mu\nu\rho}(t) = \sum_{\vec{x}} \alpha^\dag(x) U_\mu(x) U_\mu(x+\hat{\mu}) U_\nu(x+2\hat{\mu}) U_\rho(x+2\hat{\mu}+\hat{\nu}) \alpha(x+2\hat{\mu}+\hat{\nu}+\hat{\rho}) \,\, , \label{asymtwistedlinkalpha}\end{aligned}$$ which has exactly the same rotational symmetries as $L^\phi_{\mu\nu\rho}(t)$. Useful linear combinations of $L_{\mu\nu\rho}(t)$ (dropping the $\phi$, $\alpha$ and $t$ symbols for brevity) are given by $$\begin{aligned} A^+_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} + L_{+\mu+\nu-\rho} + L_{+\mu-\nu+\rho} + L_{+\mu-\nu-\rho} \nonumber \\ &+ L_{-\mu+\nu+\rho} + L_{-\mu+\nu-\rho} + L_{-\mu-\nu+\rho} + L_{-\mu-\nu-\rho} \label{operatorA+} \\ A^-_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} - L_{+\mu+\nu-\rho} - L_{+\mu-\nu+\rho} + L_{+\mu-\nu-\rho} \nonumber \\ &- L_{-\mu+\nu+\rho} + L_{-\mu+\nu-\rho} + L_{-\mu-\nu+\rho} - L_{-\mu-\nu-\rho} \label{operatorA-} \\ B^+_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} - L_{+\mu+\nu-\rho} - L_{+\mu-\nu+\rho} + L_{+\mu-\nu-\rho} \nonumber \\ &+ L_{-\mu+\nu+\rho} - L_{-\mu+\nu-\rho} - L_{-\mu-\nu+\rho} + L_{-\mu-\nu-\rho} \label{operatorB+} \\ B^-_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} + L_{+\mu+\nu-\rho} + L_{+\mu-\nu+\rho} + L_{+\mu-\nu-\rho} \nonumber \\ &- L_{-\mu+\nu+\rho} - L_{-\mu+\nu-\rho} - L_{-\mu-\nu+\rho} - L_{-\mu-\nu-\rho} \label{operatorB-} \\ C^+_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} + L_{+\mu+\nu-\rho} - L_{+\mu-\nu+\rho} - L_{+\mu-\nu-\rho} \nonumber \\ &- L_{-\mu+\nu+\rho} - L_{-\mu+\nu-\rho} + L_{-\mu-\nu+\rho} + L_{-\mu-\nu-\rho} \label{operatorC+} \\ C^-_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} + L_{+\mu+\nu-\rho} - L_{+\mu-\nu+\rho} - L_{+\mu-\nu-\rho} \nonumber \\ &+ L_{-\mu+\nu+\rho} + L_{-\mu+\nu-\rho} - L_{-\mu-\nu+\rho} - L_{-\mu-\nu-\rho} \label{operatorC-} \\ D^+_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} - L_{+\mu+\nu-\rho} + L_{+\mu-\nu+\rho} - L_{+\mu-\nu-\rho} \nonumber \\ &- L_{-\mu+\nu+\rho} + L_{-\mu+\nu-\rho} - L_{-\mu-\nu+\rho} + L_{-\mu-\nu-\rho} \label{operatorD+} \\ D^-_{\mu\nu\rho} &= L_{+\mu+\nu+\rho} - L_{+\mu+\nu-\rho} + L_{+\mu-\nu+\rho} - L_{+\mu-\nu-\rho} \nonumber \\ &+ L_{-\mu+\nu+\rho} - L_{-\mu+\nu-\rho} + L_{-\mu-\nu+\rho} - L_{-\mu-\nu-\rho} \label{operatorD-}\end{aligned}$$ and Table \[operator\_table\] shows how to construct operators of any irrep and parity. Note that operators $A^+_{\mu\nu\rho}$, $B^+_{\mu\nu\rho}$, $C^+_{\mu\nu\rho}$ and $D^+_{\mu\nu\rho}$ are even under parity, whereas $A^-_{\mu\nu\rho}$, $B^-_{\mu\nu\rho}$, $C^-_{\mu\nu\rho}$ and $D^-_{\mu\nu\rho}$ are odd. The operators $A^\pm_{\mu\nu\rho}$ belong to the $A_1$, $A_2$ and $E$ irreps, whereas $B^\pm_{\mu\nu\rho}$, $C^\pm_{\mu\nu\rho}$ and $D^\pm_{\mu\nu\rho}$ belong to the $T_1$ and $T_2$ irreps. $\Lambda^P$ mult$(\Lambda^P)$ ------------- ------------------- ------------------------------------------------------------------------------------------------------ $A_1^{+}$ 1 $A^+_{123} + A^+_{231} + A^+_{312} + A^+_{132} + A^+_{213} + A^+_{321}$ $A_1^{-}$ 1 $A^-_{123} + A^-_{231} + A^-_{312} - A^-_{132} - A^-_{213} - A^-_{321}$ $A_2^{+}$ 1 $A^+_{123} + A^+_{231} + A^+_{312} - A^+_{132} - A^+_{213} - A^+_{321}$ $A_2^{-}$ 1 $A^-_{123} + A^-_{231} + A^-_{312} + A^-_{132} + A^-_{213} + A^-_{321}$ $E^{+}$ 2 $\left\{(A^+_{123} - A^+_{231} + A^+_{132} - A^+_{213}) / \sqrt{2}, \right.$ $\left. (A^+_{123} + A^+_{231} -2A^+_{312} + A^+_{132} + A^+_{213} -2A^+_{321}) / \sqrt{6} \right\}$ $\left\{(A^+_{123} - A^+_{231} - A^+_{132} + A^+_{213}) / \sqrt{2}, \right.$ $\left. (A^+_{123} + A^+_{231} -2A^+_{312} - A^+_{132} - A^+_{213} +2A^+_{321}) / \sqrt{6} \right\}$ $E^{-}$ 2 $\left\{(A^-_{123} - A^-_{231} + A^-_{132} - A^-_{213}) / \sqrt{2}, \right.$ $\left. (A^-_{123} + A^-_{231} -2A^-_{312} + A^-_{132} + A^-_{213} -2A^-_{321}) / \sqrt{6} \right\}$ $\left\{(A^-_{123} - A^-_{231} - A^-_{132} + A^-_{213}) / \sqrt{2}, \right.$ $\left. (A^-_{123} + A^-_{231} -2A^-_{312} - A^-_{132} - A^-_{213} +2A^-_{321}) / \sqrt{6} \right\}$ $T_1^{+}$ 3 $\left\{ B^+_{123} - B^+_{132} \,,\, B^+_{231} - B^+_{213} \,,\, B^+_{312} - B^+_{321} \right\}$ $\left\{ C^+_{123} - C^+_{213} \,,\, C^+_{231} - C^+_{321} \,,\, C^+_{312} - C^+_{132} \right\}$ $\left\{ D^+_{123} - D^+_{321} \,,\, D^+_{231} - D^+_{132} \,,\, D^+_{312} - D^+_{213} \right\}$ $T_1^{-}$ 3 $\left\{ B^-_{123} + B^-_{132} \,,\, B^-_{231} + B^-_{213} \,,\, B^-_{312} + B^-_{321} \right\}$ $\left\{ C^-_{123} + C^-_{321} \,,\, C^-_{231} + C^-_{132} \,,\, C^-_{312} + C^-_{213} \right\}$ $\left\{ D^-_{123} + D^-_{213} \,,\, D^-_{231} + D^-_{321} \,,\, D^-_{312} + D^-_{132} \right\}$ $T_2^{+}$ 3 $\left\{ B^+_{123} + B^+_{132} \,,\, B^+_{231} + B^+_{213} \,,\, B^+_{312} + B^+_{321} \right\}$ $\left\{ C^+_{123} + C^+_{213} \,,\, C^+_{231} + C^+_{321} \,,\, C^+_{312} + C^+_{132} \right\}$ $\left\{ D^+_{123} + D^+_{321} \,,\, D^+_{231} + D^+_{132} \,,\, D^+_{312} + D^+_{213} \right\}$ $T_2^{-}$ 3 $\left\{ B^-_{123} - B^-_{132} \,,\, B^-_{231} - B^-_{213} \,,\, B^-_{312} - B^-_{321} \right\}$ $\left\{ C^-_{123} - C^-_{321} \,,\, C^-_{231} - C^-_{132} \,,\, C^-_{312} - C^-_{213} \right\}$ $\left\{ D^-_{123} - D^-_{213} \,,\, D^-_{231} - D^-_{321} \,,\, D^-_{312} - D^-_{132} \right\}$ : Linear combinations of operators that give any irrep and parity. The multiplicity, mult$(\Lambda^P)$, is shown for each case.[]{data-label="operator_table"} The operator $L_{\mu\nu\rho}$ consists of four gauge-invariant real components: one is an isoscalar, $$\frac{1}{2}{\rm Tr}(L_{\mu\nu\rho}) \,,$$ and the other three form an isovector, $$\frac{1}{2}{\rm Tr}(-i\sigma^aL_{\mu\nu\rho}) \,.$$ In addition to the gauge-invariant link, which contains two scalar fields, there are operators that contain only gauge fields. A Wilson loop is a gauge-invariant operator in which the path of gauge links returns to itself to form a closed loop. A particular Wilson loop that couples to all available irreps is shown in Fig. \[figure\_wilson\]. ![The Wilson loop operator $W_{\mu\nu\rho}$ of Eq. (\[wilsonloop\]).[]{data-label="figure_wilson"}](figure_wilson.eps) Mathematically, it is $$\begin{aligned} W_{\mu\nu\rho}(t) = \frac{1}{2} \operatorname{Tr} & \sum_{\vec{x}} U_\mu(x) U_\mu(x+\hat{\mu}) U_\nu(x+2\hat{\mu}) U^\dag_\mu(x+\hat{\mu}+\hat{\nu}) \nonumber \\ & \quad \times U_\rho(x+\hat{\mu}+\hat{\nu}) U^\dag_\mu(x+\hat{\nu}+\hat{\rho}) U^\dag_\rho(x+\hat{\nu}) U^\dag_\nu(x) \label{wilsonloop}\end{aligned}$$ which is operator \#4 in Table 3.2 of Ref. [@Berg:1982kp] and has 48 different orientations. A Polyakov loop is also a gauge-invariant closed loop, but it wraps around a boundary of the periodic lattice. All irreps can be obtained from a Polyakov loop that contains a “kink,” denoted by $K_{\mu\nu\rho}$, such as the one shown in Fig. \[figure\_polyakov\] which is ![The “kinked” Polyakov loop operator $P_{\mu\nu\rho}$ of Eq. (\[polyakovloop\]).[]{data-label="figure_polyakov"}](figure_polyakov.eps) $$\begin{aligned} P_{\mu\nu\rho}(t) &= \frac{1}{2} \operatorname{Tr} \sum_{\vec{x}} \left\{ \prod_{y_\mu < x_\mu} U_{\mu}(x+(y_\mu-x_\mu)\hat{\mu}) \right\}K_{\mu\nu\rho}(x) \left\{ \prod_{y_\mu > x_\mu} U_{\mu}(x+(y_\mu-x_\mu)\hat{\mu}) \right\} \,, \label{polyakovloop} \\ K_{\mu\nu\rho}(x) &= U_\nu(x) U^\dag_\mu(x+\hat{\nu}-\hat{\mu}) U_\rho(x+\hat{\nu}-\hat{\mu}) U_\mu(x+\hat{\nu}-\hat{\mu}+\hat{\rho}) \nonumber \\ &\times U_\mu(x+\hat{\nu}+\hat{\rho}) U^\dag_\rho(x+\hat{\nu}+\hat{\mu}) U^\dag_\nu(x+\hat{\mu}) \,,\end{aligned}$$ and has 48 different orientations. The kink $K_{\mu\nu\rho}$ is inserted to fill the gap between points $x$ and $x+\hat{\mu}$ of an otherwise normal Polyakov loop. All possible irreps and parities for $W_{\mu\nu\rho}$ and $P_{\mu\nu\rho}$ can be obtained from Table \[operator\_table\] simply by replacing $L_{\mu\nu\rho}$ with $W_{\mu\nu\rho}$ or $P_{\mu\nu\rho}$ in Eqs.  to . Since a Pauli matrix cannot be inserted into the trace of a closed loop operator made entirely of gauge links without destroying gauge invariance, there are no isovector Wilson or Polyakov loop operators. ![Effective masses of the $I(\Lambda^P)=0(A_1^+)$ gauge-invariant link operators $L^\phi_{\mu\nu\rho}$ and $L^\alpha_{\mu\nu\rho}$, Wilson loop $W_{\mu\nu\rho}$ and Polyakov loop $P_{\mu\nu\rho}$ operators on a $20^3\times 40$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$.[]{data-label="graph_effmass_0A1+"}](graph_effmass_0A1+.eps) ![Effective masses of the $I(\Lambda^P)=0(A_1^-)$ gauge-invariant link operators $L^\phi_{\mu\nu\rho}$ and $L^\alpha_{\mu\nu\rho}$, Wilson loop $W_{\mu\nu\rho}$ and Polyakov loop $P_{\mu\nu\rho}$ operators on a $20^3\times 40$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$.[]{data-label="graph_effmass_0A1-"}](graph_effmass_0A1-.eps) To illustrate the efficacy of the operators, consider effective masses[^1] $$\begin{aligned} m_\text{eff}(t) = - \log\left(\frac{\langle{\cal O}(t+1){\cal O}(0)\rangle}{\langle{\cal O}(t){\cal O}(0)\rangle}\right)\end{aligned}$$ where ${\cal O}(t)$ is a gauge-invariant operator with its vacuum expectation value subtracted, $$\begin{aligned} {\cal O}(t) = O(t) - \left<O(t)\right> \,\, . \label{subtractedO}\end{aligned}$$ Figures \[graph\_effmass\_0A1+\] and \[graph\_effmass\_0A1-\] show effective mass plots for the $I(\Lambda^P)=0(A_1^+)$ and $0(A_1^-)$ channels of four operators: two gauge-invariant links, a Wilson loop, and a Polyakov loop. The stout link and smearing parameters are $n_{\rm stout}=n_{\rm smear}=200$ and $r_{\rm stout}=r_{\rm smear}=0.1$. For $0(A_1^+)$, the $L^\alpha_{\mu\nu\rho}$ and $P_{\mu\nu\rho}$ operators have nearly identical effective mass plots despite being conceptually very different operators. The mass is near 0.4 in lattice units. The $L^\phi_{\mu\nu\rho}$ operator with identical quantum numbers produces a different effective mass (near 0.3), and the $W_{\mu\nu\rho}$ operator gives another (noisier) result. This is an indication that the $0(A_1^+)$ spectrum (corresponding to $J=0$ in the continuum) contains more than a lone Higgs boson. A more sophisticated analysis method is presented in Sec. \[sec:analysis\] and applied in subsequent sections. For $0(A_1^-)$, Fig. \[graph\_effmass\_0A1-\] provides four effective mass plots that collectively indicate a mass near 0.6 in lattice units. Again this is $J=0$ in the continuum, and of course neither a single Higgs nor a single $W$ has $J^P=0^-$. Our complete analysis of this and all other channels is discussed below. Correlation Matrix and Variational Method {#sec:analysis} ========================================= Particle energies, $E_n$, are extracted from lattice simulations by observing the exponential decay of correlation functions, $$\begin{aligned} C_{ij}(t) = \left< {\cal O}_i(t) {\cal O}_j(0) \right> &= \sum_n \left<0\right|{\cal O}_i\left|n\right> \left<n\right|{\cal O}_j\left|0\right> \exp\left(-E_n t \right) \\ &= \sum_n a_i^n a_j^n \exp\left(-E_n t \right) \,\, ,\end{aligned}$$ where ${\cal O}_i(t)$ is a Hermitian gauge-invariant operator with its vacuum expectation value subtracted as in Eq. (\[subtractedO\]). The choice of operator determines the quantum numbers $I(\Lambda^P)$ of the states $\left|n\right>$ that are present in the correlation function and also determines the coupling strength, $a_i^n$, to each. The operators are calculated for eight different levels of smearing, $n_{\rm stout}=n_{\rm smear}=0$, 5, 10, 25, 50, 100, 150, and 200. The smearing parameters are held fixed at $r_{\rm stout}=r_{\rm smear}=0.1$. Each of these different smearing levels produces a unique operator ${\cal O}_i$ in the correlation matrix $C_{ij}(t)$. The energy spectrum is extracted using the variational method [@Kronfeld:1989tb; @Luscher:1990ck]. To begin, the eigenvectors $\vec{v}_n$ and eigenvalues $\lambda_n$ ($n=1,...,M$) of the correlation matrix are found at a single time step $C_{ij}(t_0)$ ($i,j=1,...,N$), where $N$ is the number of operators, $M$ is the number of statistically nonzero eigenvalues, which corresponds to the number of states that can be resolved, and $M\leq N$. The value of $t_0$ is typically chosen to be small, e.g. $t_0=1$, where the signal-to-noise ratio is large. The correlation matrix is changed from the operator basis to the eigenvector basis by $$\begin{aligned} \widetilde{C}_{nm}(t) = \frac{\vec{v}_n^T C(t) \vec{v}_m}{\sqrt{\lambda_n\lambda_m}} \,\, . \label{eigenstate_correlation_matrix}\end{aligned}$$ The correlation function for the $k\text{th}$ ($k=1,...,M$) state is then given by $$\begin{aligned} C_{k}(t) = \vec{R}_{k}^T \widetilde{C}(t) \vec{R}_{k} \,\, ,\end{aligned}$$ where $\vec{R}_k$ is a set of orthonormal vectors chosen such that the energies from $C_k(t)$ are ordered from smallest to largest for increasing $k$. $\vec{R}_{k}$ is determined recursively by a variational method as follows: $\vec{R}_1$ maximizes $C_1(t_1)$, the correlation function of the smallest energy at a time step $t_1>t_0$. The normalization of Eq.  ensures that $C_{k}(t_0)=1$, thus maximizing $C_1(t_1)$ ensures that $\vec{R}_1$ projects out the state with smallest energy while minimizing contamination from higher-energy states. In practice, the time step $t_1$ is taken to be $t_0+1$. The optimization of $C_1(t_1)$ reduces to solving the eigenproblem $$\begin{aligned} \widetilde{C}(t_1) \vec{x}_1 = \mu_1 \vec{x}_1 \,\, , \label{lagrange_multiplier}\end{aligned}$$ where the eigenvalue $\mu_1$ is the Lagrange multiplier for the constraint $\vec{R}_1^T\vec{R}_1=1$, and the solution for $\vec{R}_1$ is given by the eigenvector $\vec{x}_1$ that maximizes $C_1(t_1)$. The correlation function $C_2(t)$ of the next-smallest-energy state can be found by calculating $\vec{R}_2$ in the same way as above, given that $\vec{R}_2$ must be orthonormal to $\vec{R}_1$. This is accomplished by defining $\vec{R}_2$ as the vector $$\begin{aligned} \vec{x}_2=\sum_{n=1}^{M-1} a_{n}\vec{x}_{1,n} \label{vec_x2}\end{aligned}$$ that maximizes $C_2(t_1)$, where $\vec{R}_1=\vec{x}_{1,M}$ and $\vec{x}_{1,n}$ ($n=1,...,M-1$) are the remaining eigenvectors from Eq. . The eigenproblem resulting from the maximization of $C_2(t_1)$ is $$\begin{aligned} X_1^T\widetilde{C}(t_1)X_1 \vec{a} = \mu_2 \vec{a} \,\, ,\end{aligned}$$ where the matrix $X_1=(\vec{x}_{1,1},...,\vec{x}_{1,M-1})$, the vector $\vec{a}^T=(a_{1},...,a_{M-1})$ contains the coefficients from Eq.  and the vector $\vec{R}_2~=~ X_1\vec{a}$ is calculated from the eigenvector $\vec{a}$ that maximizes $C_2(t_1)$. The calculation can continue recursively up to the $M\text{th}$ case, where the eigenproblem becomes trivial. The energy can then be extracted by a $\chi^2$-minimizing fit to a single exponential using $$\begin{aligned} C_{k}(t) = A_k \exp\left(-E_k t \right) \,\, .\end{aligned}$$ Spectrum at the physical point {#sec:spectrum} ============================== Using the methods described above, an ensemble of 20,000 configurations was created on a $20^3\times40$ lattice with $\beta=8$, $\lambda=0.0033$, and $\kappa=0.131$. Figure \[graph\_mass\_link\] shows the energy levels for isospins 0 and 1 as obtained from the gauge-invariant link operators $L^\phi_{\mu\nu\rho}$ and $L^\alpha_{\mu\nu\rho}$, and Fig. \[graph\_mass\_wilson\_polyakov\] shows the energy levels for isospin 0 as obtained from the Wilson loop and Polyakov loop operators. (Wilson/Polyakov loops cannot produce isospin 1, and lattice results for isospins higher than 1 are not considered in this work.) As expected, the lightest state in the spectrum has $I(\Lambda^P)=1(T_1^-)$ corresponding to a single $W$ boson. The mass is near 0.2 in lattice units (with a tiny statistical error) and identification with the experimentally known $W$ mass allows us to infer the lattice spacing in physical units. ![Energy spectrum extracted from correlation functions of the gauge-invariant link operators $L^\phi_{\mu\nu\rho}$ and $L^\alpha_{\mu\nu\rho}$ for all isoscalar and isovector channels on a $20^3\times 40$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. These parameters put the theory very close to the experimental Higgs and $W$ boson masses. Data points are lattice results with statistical errors; horizontal lines are the expectations from Eq. .[]{data-label="graph_mass_link"}](graph_mass_link.eps) ![Energy spectrum extracted from correlation functions of the Wilson loop and Polyakov loop operators $W_{\mu\nu\rho}$ and $P_{\mu\nu\rho}$ for all isoscalar channels on a $20^3\times 40$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. These parameters put the theory very close to the experimental Higgs and $W$ boson masses. Data points are lattice results with statistical errors; horizontal lines are the expectations from Eq. .[]{data-label="graph_mass_wilson_polyakov"}](graph_mass_wilson_polyakov.eps) The next energy level above the single $W$ has an energy near 0.3 and is observed in the $0(A_1^+)$ channel, exactly as expected for the Higgs boson. Our lattice parameters were tuned to put this mass near its experimental value; the result from our simulation is $122\pm1$ GeV. Notice that neither the single $W$ nor the single Higgs is observed from the Wilson loop or Polyakov loop, but both are seen from the gauge-invariant link operators. Moreover, notice that the Higgs boson $H$ has not been created by just a single $\phi(x)$ but rather by gauge-invariant operators that can never contain any odd power of $\phi(x)$. Much like QCD, physical particles in the observed spectrum do not present any obvious linear one-to-one correspondence with fields in the Lagrangian. For a recent discussion in the context of a gauge-fixed lattice study, see Refs. [@Maas:2012tj; @Maas:2012zf]. Continuing upward in energy within Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\], we see a signal with energy at $2m_W$ in four specific channels. These are exactly the four channels that correspond to the allowed quantum numbers of a pair of [*stationary*]{} $W$ bosons. In the continuum, the wave function for such a pair of spin-1 $W$ bosons would be the product of a spin part and an isospin part. The total wave function must be symmetric under particle interchange. This permits just two continuum states with isospin 0 \[$0(0^+)$ and $0(2^+)$\], and a single continuum state with isospin 1 \[$1(1^+)$\]. Note that the parity of a $W$ pair is always positive in the absence of orbital angular momentum. A glance at Table \[irrep\_spin\_table\] reveals that these continuum states match the lattice observations at energy $2m_W$ perfectly. An energy shift away from $2m_W$ would represent binding energy or a scattering state, but no shift is visible in our lattice simulation at this weak coupling value. The next state in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] has an energy of $m_H+m_W$ and is another pair of stationary bosons. Because the Higgs boson is $0(0^+)$, the Higgs-$W$ pair should have the quantum numbers of the $W$. The lattice data show that the Higgs-$W$ pair does indeed appear in exactly the same $I(\Lambda^P)$ channels as does the single $W$. Two states are expected to appear with an energy near 0.6 because this corresponds to $2m_H\approx3m_W$. A pair of stationary Higgs bosons should have the same quantum numbers as a single Higgs, i.e. $I(J^P)=0(0^+)$, but no such signal appears in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\]. To see this two-Higgs state we will need a different creation operator; Sec. \[Two-Particle Operators\] introduces this operator and uses it to observe the two-Higgs state within our lattice simulations. A collection of three stationary $W$ bosons must have a wave function that is symmetric under interchange of any pair, and must be built from a spin part and an isospin part. The $I=0$ case has an antisymmetric isospin part and the only available antisymmetric spin part is $J=0$. The $I=1$ case is of mixed symmetry and can combine with $J=1$, 2, or 3 (but not $J=0$) to form a symmetric wave function. These continuum options, i.e. $0(0^-)$, $1(1^-)$, $1(2^-)$ and $1(3^-)$, can be converted into lattice channels easily by using Table \[irrep\_spin\_table\] and the result is precisely the list of channels observed in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\], i.e.$0(A_1^-)$, $1(T_1^-)$, $1(E^-)$, $1(T_2^-)$, and $1(A_2^-)$. The next energy level is $m_H+2m_W$ which should have identical $I(\Lambda^P)$ options to the pair of stationary $W$ bosons discussed above. Figure \[graph\_mass\_link\] verifies this expectation, having signals for $0(A_1^+)$, $0(E^+)$, $0(T_2^+)$, and $1(T_1^+)$, although errors bars are somewhat larger for this high energy state. The next energy level in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] is a pair of [*moving*]{} $W$ bosons with vanishing [*total*]{} momentum. Recall that our operators were defined to have zero total momentum, but this still permits a two-particle state where the particles have equal and opposite momenta. Momentum components along the $x$, $y$ or $z$ axes of the lattice can have integer multiple values of $2\pi/L$, where $L$ is the spatial length of the lattice. The lattice dispersion relation for a boson with mass $m$ and momentum $\vec{p}$ is $$\begin{aligned} \sinh^2\left(\frac{aE(\vec{p})}{2}\right) = \sinh^2\left(\frac{am}{2}\right) + \sum_{i=1}^3 \sin^2\left(\frac{ap_i}{2}\right) \label{lattice_dispersion}\end{aligned}$$ which reduces to the continuum relation, $E(\vec{p}) = \sqrt{m^2 + \vec{p}^2}$, as the lattice spacing $a$ goes to zero. Given the lattice spacing and statistical precision used in this paper, the difference between Eq.  and the continuum relation is noticeable. The energy of a state of two noninteracting bosons is simply $E_1(\vec{p}_1)+E_2(\vec{p}_2)$, with energies from Eq. . Two particles with relative motion can also have orbital angular momentum $L$; the allowed $I(J^P)$ for Higgs-Higgs, Higgs-$W$ and $W$-$W$ states are listed in Table \[orbital\_angular\_momentum\]. Higgs-Higgs Higgs-$W$ ------- ------------- ------------------------- ----------------------------------- ---------------------------  $L$  $I=0$ $I=1$ $I=0$ $I=1$ 0 $0^+$ $1^-$ $0^+$, $2^+$ $1^{+}$ 1 — $0^{+}$, $1^+$, $2^{+}$ $1^-$, $2^-$, $3^-$ $0^{-}$, $1^{-}$, $2^{-}$ 2 $2^+$ $1^{-}$, $2^-$, $3^{-}$ $0^+$, $1^+$, $2^+$, $3^+$, $4^+$ $1^{+}$, $2^{+}$, $3^{+}$ 3 — $2^{+}$, $3^+$, $4^{+}$ $1^-$, $2^-$, $3^-$, $4^-$, $5^-$ $2^{-}$, $3^{-}$, $4^{-}$ : $I(J^P)$ quantum numbers for Higgs-Higgs, Higgs-$W$ and $W$-$W$ states with orbital angular momentum $L$. Higgs-Higgs states must have positive parity due to Bose statistics.[]{data-label="orbital_angular_momentum"} There is no way to specify $L$ with lattice operators because it is not a conserved quantum number; only the total momentum $J$ can be specified, which corresponds to $\Lambda$ in a lattice simulation. For two moving $W$ particles, all quantum numbers with $I=0$ or 1 are possible except $0(0^-)$ and $1(0^+)$. Therefore a signal could appear in all $I(\Lambda^P)$ channels, even $0(A_1^-)$ and $1(A_1^+)$ because of $J=4$ states. As evident from Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\], our lattice simulation produced signals in many channels, but not in all. Section \[Two-Particle Operators\] provides the explanation for why this particular subset of channels did not show a signal. Beyond this large energy, we are approaching the limit of the reach of this set of operators. A few data points are shown at even higher energies (in the neighborhood of $4m_W$) in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\], but a confident interpretation of those will require further computational effort that is presented in Secs. \[sec:biggerlattice\] and \[sec:infiniteHiggs\]. To conclude this section, it is interesting to notice a clear qualitative distinction between the Wilson/Polyakov loop operators and the gauge-invariant link operators: the former (Fig. \[graph\_mass\_wilson\_polyakov\]) found only pure $W$ boson states whereas the latter (Fig. \[graph\_mass\_link\]) found additional states containing one Higgs boson. States containing two Higgs bosons must wait until Sec. \[Two-Particle Operators\]. Spectrum on a larger lattice {#sec:biggerlattice} ============================ To confirm that several of the states in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] are truly multiparticle states with linear momentum, the simulations of the previous section are repeated using a larger lattice volume. Since momentum on a lattice is given by integer multiples of $2\pi/L$, where $L$ is the spatial length of the lattice, increasing the lattice volume should cause the energies of states with linear momentum to decrease by a predictable amount. Here the lattice parameters are set to $\beta = 8$, $\lambda = 0.0033$, $\kappa = 0.131$, which is the same as the previous section, but now the lattice volume is $24^3\times 48$. An ensemble of 20,000 configurations is used. ![The same as Fig. \[graph\_mass\_link\] but using a $24^3\times 48$ lattice.[]{data-label="graph_mass_link_24x24x24x48"}](graph_mass_link_24x24x24x48.eps) ![The same as Fig. \[graph\_mass\_wilson\_polyakov\] but using a $24^3\times 48$ lattice.[]{data-label="graph_mass_wilson_polyakov_24x24x24x48"}](graph_mass_wilson_polyakov_24x24x24x48.eps) The energy spectrum, extracted by a variational analysis, is shown in Figs. \[graph\_mass\_link\_24x24x24x48\] and \[graph\_mass\_wilson\_polyakov\_24x24x24x48\]. The Higgs and $W$ masses remain virtually unchanged, with a Higgs mass of $123\pm 1$ GeV. This stability indicates that finite volume artifacts are negligible. The data points that lie at 0.65 in lattice units correspond perfectly to two $W$ particles with the minimal nonzero linear momentum. This physics appears in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] at a larger energy, and the energy shift is in numerical agreement with the change in energy due to changing the lattice volume. Also, the four data points at 0.8 in Fig. \[graph\_mass\_link\] were numerically compatible with (a) a Higgs-$W$ pair moving back-to-back with the minimal momentum or (b) a collection of four $W$ bosons all at rest. This physics has energy 0.73 in Fig. \[graph\_mass\_link\_24x24x24x48\] which cannot be a four-$W$ state but is in good agreement with a back-to-back Higgs-$W$ pair. From Table \[orbital\_angular\_momentum\] all $J^P$ quantum numbers except $0^-$ are allowed for a moving Higgs-$W$ pair, but these lattice operators have found a signal in only a few channels. Section \[Two-Particle Operators\] addresses the issue of missing irreducible representations for multiparticle states with momentum. It is noteworthy that some states consisting of three stationary $W$ particles, $1(T_1^-)$ in Fig. \[graph\_mass\_link\_24x24x24x48\] and $0(A_1^-)$ in Fig. \[graph\_mass\_wilson\_polyakov\_24x24x24x48\], as well as the $0(A_1^+)$ Higgs-$W$-$W$ state in Fig. \[graph\_mass\_link\_24x24x24x48\], were not detected in the larger lattice volume. This is because the variational analysis cannot resolve these states from the current basis of operators. When the lattice volume was increased, the spectral density increased as more multiparticle states became detectable in the correlation functions. As a result, states with a small overlap with the basis of operators could not be successfully extracted, even though they had been observed for the smaller lattice volume. Of course, these states could be seen again if the basis of operators was improved, for example, by increasing the number of operators. Spectrum with a heavy Higgs {#sec:infiniteHiggs} =========================== A simple method to confirm which of the multiparticle states in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] contain a Higgs boson is to change the Higgs mass and leave everything else unchanged. Here we choose the extreme case of an infinite quartic coupling, corresponding to the maximal Higgs mass [@Hasenfratz:1987uc; @Langguth:1987vf; @Hasenfratz:1987eh]. The lattice parameters are set to $\beta = 8$, $\lambda = \infty$, $\kappa=0.40$, and the geometry is $20^3\times40$. An ensemble of 20,000 configurations is used. With these parameters, the $W$ mass in lattice units is nearly identical to the value in Fig. \[graph\_mass\_link\]. The energy spectrum, extracted by a variational analysis as usual, is shown in Figs. \[graph\_mass\_link\_lambda\_inf\] and \[graph\_mass\_wilson\_polyakov\_lambda\_inf\]. ![The same as Fig. \[graph\_mass\_link\] but using $\kappa=0.40$ and $\lambda=\infty$. The Higgs mass is off the graph because of its large value.[]{data-label="graph_mass_link_lambda_inf"}](graph_mass_link_lambda_inf.eps) ![The same as Fig. \[graph\_mass\_wilson\_polyakov\] but using $\kappa=0.40$ and $\lambda=\infty$.[]{data-label="graph_mass_wilson_polyakov_lambda_inf"}](graph_mass_wilson_polyakov_lambda_inf.eps) The spectrum of states containing $W$ particles remains essentially the same as in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\] but all states with Higgs content are no longer visible. This is consistent with the notion that the Higgs mass is now so large that all states with Higgs content have been pushed up to a higher energy scale. To test this expectation of a large Higgs mass, a simultaneous fit of the entire $0(A_1^+)$ gauge-invariant-link correlation matrix was performed. (For a comparison of this method to the variational analysis in a different lattice context, see Ref. [@Lewis:2011ti].) A three-state fit to time steps $t\ge2$ provided a good description of the lattice data, with a $\chi^2/\text{d.o.f.}=0.84$. The smallest energy corresponds to a pair of stationary $W$ bosons, the next energy is a pair of $W$ bosons moving back-to-back with vanishing total momentum, and the third energy is $1.8\pm0.2$ in lattice units which is $720\pm70$ GeV. This third energy is consistent with the maximal Higgs energy found in early lattice studies [@Hasenfratz:1987uc; @Langguth:1987vf; @Hasenfratz:1987eh]. Lattice artifacts will be significant for this Higgs mass, since it is larger than unity in lattice units. For our purposes it is sufficient to conclude that the Higgs mass is much larger than the low-lying spectrum of multiparticle $W$-boson states. This study of the spectrum in a heavy-Higgs world reinforces our understanding of which states in the spectrum contain a Higgs boson. Two-Particle Operators {#Two-Particle Operators} ====================== The operators used in previous sections of this work were, at most, quadratic in the field $\phi(x)$. They led to excellent results for several states in the SU(2)-Higgs spectrum, including multiboson states, but additional operators can accomplish even more. In particular, recall that the two-Higgs state was not observed in previous sections, the two-$W$ state with internal linear momentum was missing from a few $I(\Lambda^P)$ channels, and the Higgs-$W$ state with internal linear momentum was similarly missing from some $I(\Lambda^P)$ channels. Presently, multiparticle operators will be constructed and the allowed irreducible representations will be compared to the results in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\]. A two-particle operator ${\cal O}^{AB}(t)$ can be obtained by multiplying two operators with the following vacuum subtractions: $$\begin{aligned} {\cal O}^{AB}(t) &= {\cal O}^{A}(t) {\cal O}^{B}(t) - \left<{\cal O}^{A}(t) {\cal O}^{B}(t)\right> \,\, , \\ {\cal O}^{A}(t) &= {O}^{A}(t) - \left<{O}^{A}(t)\right> \,\, , \\ {\cal O}^{B}(t) &= {O}^{B}(t) - \left<{O}^{B}(t)\right> \,\, ,\end{aligned}$$ where ${\cal O}^{A}(t)$ and ${\cal O}^{B}(t)$ each couple predominantly to a single-particle state. The two-particle correlation function is then simply $$\begin{aligned} C^{AB}(t) = \left<{\cal O}^{AB}(t) {\cal O}^{AB\dag}(0)\right> \,\, .\end{aligned}$$ Note that ${\cal O}^{AB}(t)$ is not strictly a two-particle operator because all states with the same quantum numbers as ${\cal O}^{AB}(t)$ can be created by it, including single-particle states. However, this construction will result in a much stronger overlap with the two-particle states, such as Higgs-Higgs which was not found using the operators in Sec. \[sec:operators\]. A three-particle operator is defined similarly: $$\begin{aligned} {\cal O}^{ABC}(t) &= {\cal O}^{A}(t) {\cal O}^{B}(t) {\cal O}^{C}(t) - \left<{\cal O}^{A}(t) {\cal O}^{B}(t) {\cal O}^{C}(t)\right> \,\, .\end{aligned}$$ In this section we have written the correlation function using the Hermitian conjugate because we intend to use operators with nonzero momentum, whereas in the previous sections all operators were strictly Hermitian. This does not affect our variational method because all of our correlation functions are real; to be precise, the imaginary component of each correlation function is equal to zero within statistical fluctuations. The single-particle operators for the Higgs and $W$ are given by $$\begin{aligned} H(\vec{p}) &= \sum_{\vec{x}} \frac{1}{2} \operatorname{Tr}\left\{\phi^\dag(x)\phi(x)\right\} \, \exp\left\{i\vec{p}\cdot\vec{x}\right\} \,\, , \label{Hp} \\ W^a_{\mu}(\vec{p}) &= \sum_{\vec{x}} \frac{1}{2} \operatorname{Tr}\left\{-i\sigma^a \phi^\dag(x) U_\mu(x) \phi(x+\hat{\mu})\right\} \, \exp\left\{i\vec{p}\cdot\left(\vec{x}+\tfrac{1}{2}\hat{\mu}\right)\right\} \,\, , \label{Wp}\end{aligned}$$ where $\vec{p}$ is the momentum and has components given by integer multiples of $2\pi/L$ in the $x$, $y$ or $z$ directions, with $L$ being the spatial length of the lattice. Combining the $W$ operators requires some additional care due to the isospin indices. $W$-$W$ eigenstates of $I$ are obtained using the scalar and vector products $$\begin{aligned} &I=0: \quad \vec{W}_\mu\cdot\vec{W}_\nu = W_\mu^aW_\nu^a \,\, , \\ &I=1: \quad \vec{W}_\mu\times\vec{W}_\nu = \epsilon^{abc}W_\mu^bW_\nu^c \,\, ,\end{aligned}$$ where the repeated $a$, $b$, $c$ indices are summed. Combinations of $W$ operators with $I>1$ are not considered in this paper. The irreducible representations of the $W$-$W$ operators with $\vec{p}=\vec0$ are given by $$\begin{aligned} 0(A_1^{+}) :& \quad W_1^aW_1^a+W_2^aW_2^a+W_3^aW_3^a \\ 0(E^{+}) :&\quad \frac{W_1^aW_1^a-W_2^aW_2^a}{\sqrt{2}}, \frac{W_1^aW_1^a+W_2^aW_2^a-2W_3^aW_3^a}{\sqrt{6}}\\ 0(T_2^{+}) :&\quad W_1^aW_2^a,W_2^aW_3^a,W_3^aW_1^a \\ 1(T_1^{+}) :&\quad \epsilon^{abc}W_1^bW_2^c,\epsilon^{abc}W_2^bW_3^c,\epsilon^{abc}W_3^bW_1^c\end{aligned}$$ which correspond to the allowed continuum spins. The isospin combinations for three $W$’s with $I=0$ or $1$ are $$\begin{aligned} &I=0; \quad \vec{W}_\mu\cdot\left(\vec{W}_\nu\times\vec{W}_\rho\right) = \epsilon^{abc}W_\mu^aW_\nu^bW_\rho^c \,\, , \\ &I=1: \quad \vec{W}_\mu\left(\vec{W}_\nu\cdot\vec{W}_\rho\right) = W_\mu^aW_\nu^bW_\rho^b \,\, , \\ &I=1: \quad \vec{W}_\mu\times\left(\vec{W}_\nu\times\vec{W}_\rho\right) = \epsilon^{abc}\epsilon^{cde}W_\mu^bW_\nu^dW_\rho^e \,\, .\end{aligned}$$ (Unnecessary for our purposes is another $I=1$ triple-$W$ operator, formed by combining an $I=2$ pair with the third $W$.) Operator $~I~$ $A_1^+$ $A_2^+$ $E^+$ $T_1^+$ $T_2^+$ $A_1^-$ $A_2^-$ $E^-$ $T_1^-$ $T_2^-$ ------------------------------------------------------ ------- --------- --------- ------- --------- --------- --------- --------- ------- --------- --------- -- $HH$ 0 1 0 0 0 0 0 0 0 0 0 $HW_\mu^a$ 1 0 0 0 0 0 0 0 0 1 0 $W_\mu^aW_\mu^a$ 0 1 0 1 0 0 0 0 0 0 0 $W_\mu^aW_\nu^a$ 0 0 0 0 0 1 0 0 0 0 0 $\epsilon^{abc}W_\mu^bW_\nu^c$ 1 0 0 0 1 0 0 0 0 0 0 $\epsilon^{abc}W_\mu^aW_\nu^bW_\rho^c$ 0 0 0 0 0 0 1 0 0 0 0 $W_\mu^aW_\mu^bW_\mu^b$ 1 0 0 0 0 0 0 0 0 1 0 $W_\mu^aW_\mu^bW_\nu^b$ 1 0 0 0 0 0 0 0 0 1 1 $W_\mu^aW_\mu^bW_\mu^b$ 1 0 0 0 0 0 0 0 0 1 1 $W_\mu^aW_\nu^bW_\rho^b$ 1 0 0 0 0 0 0 1 1 0 0 $\epsilon^{abc}\epsilon^{cde}W_\mu^bW_\mu^dW_\nu^e$ 1 0 0 0 0 0 0 0 0 1 1 $\epsilon^{abc}\epsilon^{cde}W_\mu^bW_\nu^dW_\rho^e$ 1 0 0 0 0 0 0 0 1 0 0 : Octahedral group multiplicities of Higgs-Higgs, Higgs-$W$, $W$-$W$ and $W$-$W$-$W$ operators built of the operators in Eqs. (\[Hp\]) and (\[Wp\]) with $\vec p=\vec 0$. Repeated SU(2) indices $a$, $b$, $c$ are summed, but Lorentz indices $\mu$, $\nu$, $\rho$ are not. The indices $\mu$, $\nu$, $\rho$ are not equal to one another.[]{data-label="zero_momentum_multiplicities"} ![Energy spectrum extracted from correlation functions of Higgs-Higgs, Higgs-$W$ and $W$-$W$ operators built from Eqs. (\[Hp\]) and (\[Wp\]) with $\vec p=\vec 0$ on a $20^3\times 40$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. Data points are lattice results with statistical errors; horizontal lines are the expectations from Eq. .[]{data-label="graph_mass_hh_hw_ww"}](graph_mass_hh_hw_ww.eps) Table \[zero\_momentum\_multiplicities\] shows the multiplicities for Higgs-Higgs, Higgs-$W$, $W$-$W$ and $W$-$W$-$W$ operators built entirely of $\vec p=\vec 0$ operators. The energy spectrum obtained from the two-boson operators by variational analysis is displayed in Fig. \[graph\_mass\_hh\_hw\_ww\]. The two-Higgs state, absent until now, is seen quite precisely. The $W$-$W$ and Higgs-$W$ signals are also excellent. Even three-boson and four-boson states are observed. (Readers of Sec. \[sec:biggerlattice\] might wonder whether the four-$W$ states in Fig. \[graph\_mass\_hh\_hw\_ww\] could instead be a Higgs-$W$ state with momentum. Recall, though, that a Higgs-$W$ state cannot have isospin 0.) Another success worth noticing is that the single Higgs does not appear at all and the single $W$ couples only weakly; that is a success because the operators were intended to be multiparticle operators. The operators $H(\vec{p})$ and $W^a_\mu(\vec{p})$ from Eqs.  and were calculated for momenta given by $\left|\vec{p}\right|=2\pi/L$, $\left|\vec{p}\right|=\sqrt{2}(2\pi/L)$ and $\left|\vec{p}\right|=\sqrt{3}(2\pi/L)$. Figure \[graph\_mass\_h\_w\_momentum\] shows the spectrum obtained from a variational analysis of the single Higgs and $W$ operators versus momentum. Both Higgs and $W$ operators contain an excited state which is a two-$W$ state, where one $W$ is stationary and the other has momentum. Notice that the two-$W$ energy does not form a straight line since its continuum relation is $E = m + \sqrt{m^2+\vec{p}^2}$. ![Energy spectrum extracted from correlation functions of $H(\vec{p})$ and $W^a(\vec{p})$ operators from Eqs. (\[Hp\]) and (\[Wp\]) as a function of momentum $\vec{p}$ on a $24^3\times 48$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. Data points are lattice results with statistical errors; solid curves are based on the continuum dispersion relation $E^2=m^2+\vec{p}^2$; empty boxes are the expectations from the lattice dispersion relation Eq. .[]{data-label="graph_mass_h_w_momentum"}](graph_mass_h_w_momentum.eps) Operator $~I~$ $A_1^+$ $A_2^+$ $E^+$ $T_1^+$ $T_2^+$ $A_1^-$ $A_2^-$ $E^-$ $T_1^-$ $T_2^-$ ------------------------------------------------------------ ------- --------- --------- ------- --------- --------- --------- --------- ------- --------- --------- -- $H(\vec{p}_\mu)H(-\vec{p}_\mu)$ 0 1 0 1 0 0 0 0 0 0 0 $H(\vec{p}_\mu)W_\mu^a(-\vec{p}_\mu)$ 1 1 0 1 0 0 0 0 0 1 0 $H(\vec{p}_\mu)W_\nu^a(-\vec{p}_\mu)$ 1 0 0 0 1 1 0 0 0 1 1 $W_\mu^a(\vec{p}_\mu)W_\mu^a(-\vec{p}_\mu)$ 0 1 0 1 0 0 0 0 0 0 0 $W_\nu^a(\vec{p}_\mu)W_\nu^a(-\vec{p}_\mu)$ 0 1 1 2 0 0 0 0 0 0 0 $W_\mu^a(\vec{p}_\mu)W_\nu^a(-\vec{p}_\mu)$ 0 0 0 0 1 1 0 0 0 1 1 $W_\nu^a(\vec{p}_\mu)W_\rho^a(-\vec{p}_\mu)$ 0 0 0 0 0 1 1 0 1 0 0 $\epsilon^{abc}W_\mu^b(\vec{p}_\mu)W_\mu^c(-\vec{p}_\mu)$ 1 0 0 0 0 0 0 0 0 1 0 $\epsilon^{abc}W_\nu^b(\vec{p}_\mu)W_\nu^c(-\vec{p}_\mu)$ 1 0 0 0 0 0 0 0 0 1 1 $\epsilon^{abc}W_\mu^b(\vec{p}_\mu)W_\nu^c(-\vec{p}_\mu)$ 1 0 0 0 1 1 0 0 0 1 1 $\epsilon^{abc}W_\nu^b(\vec{p}_\mu)W_\rho^c(-\vec{p}_\mu)$ 1 0 0 0 1 0 0 1 1 0 0 : Octahedral group multiplicities of Higgs-Higgs, Higgs-$W$ and $W$-$W$ operators built of the operators in Eqs. (\[Hp\]) and (\[Wp\]) with $\vec p\neq\vec 0$, where $\vec{p}_1=\tfrac{2\pi}{L}(1,0,0)$, $\vec{p}_2=\tfrac{2\pi}{L}(0,1,0)$ and $\vec{p}_3=\tfrac{2\pi}{L}(0,0,1)$. Repeated SU(2) indices $a$, $b$, $c$ are summed, but Lorentz indices $\mu$, $\nu$, $\rho$ are not. The indices $\mu$, $\nu$, $\rho$ are not equal to one another.[]{data-label="momentum1_multiplicities"} Operator $~I~$ $A_1^+$ $A_2^+$ $E^+$ $T_1^+$ $T_2^+$ $A_1^-$ $A_2^-$ $E^-$ $T_1^-$ $T_2^-$ ----------------------------------------------------------------------- ------- --------- --------- ------- --------- --------- --------- --------- ------- --------- --------- -- $H(\vec{p}_{\mu\nu})H(-\vec{p}_{\mu\nu})$ 0 1 0 1 0 1 0 0 0 0 0 $H(\vec{p}_{\mu\nu})W_\mu^a(-\vec{p}_{\mu\nu})$ 1 1 1 2 1 1 0 0 0 2 2 $H(\vec{p}_{\mu\nu})W_\rho^a(-\vec{p}_{\mu\nu})$ 1 0 0 0 1 1 0 1 1 1 0 $W_\mu^a(\vec{p}_{\mu\nu})W_\mu^a(-\vec{p}_{\mu\nu})$ 0 1 1 2 1 1 0 0 0 0 0 $W_\rho^a(\vec{p}_{\mu\nu})W_\rho^a(-\vec{p}_{\mu\nu})$ 0 1 0 1 0 1 0 0 0 0 0 $W_\mu^a(\vec{p}_{\mu\nu})W_\nu^a(-\vec{p}_{\mu\nu})$ 0 1 0 1 0 1 0 0 0 1 1 $W_\mu^a(\vec{p}_{\mu\nu})W_\rho^a(-\vec{p}_{\mu\nu})$ 0 0 0 0 2 2 1 1 2 1 1 $\epsilon^{abc}W_\mu^b(\vec{p}_{\mu\nu})W_\mu^c(-\vec{p}_{\mu\nu})$ 1 0 0 0 0 0 0 0 0 1 1 $\epsilon^{abc}W_\rho^b(\vec{p}_{\mu\nu})W_\rho^c(-\vec{p}_{\mu\nu})$ 1 0 0 0 0 0 0 0 0 1 1 $\epsilon^{abc}W_\mu^b(\vec{p}_{\mu\nu})W_\nu^c(-\vec{p}_{\mu\nu})$ 1 0 1 1 1 0 0 0 0 1 1 $\epsilon^{abc}W_\mu^b(\vec{p}_{\mu\nu})W_\rho^c(-\vec{p}_{\mu\nu})$ 1 0 0 0 2 2 1 1 2 1 1 : Octahedral group multiplicities of Higgs-Higgs, Higgs-$W$ and $W$-$W$ operators built of the operators in Eqs. (\[Hp\]) and (\[Wp\]) with $\vec p\neq\vec 0$, where $\vec{p}_{12}=\tfrac{2\pi}{L}(1,1,0)$, $\vec{p}_{23}=\tfrac{2\pi}{L}(0,1,1)$, $\vec{p}_{31}=\tfrac{2\pi}{L}(1,0,1)$, $\vec{p}_{1-2}=\tfrac{2\pi}{L}(1,-1,0)$, $\vec{p}_{2-3}=\tfrac{2\pi}{L}(0,1,-1)$ and $\vec{p}_{3-1}=\tfrac{2\pi}{L}(-1,0,1)$. Repeated SU(2) indices $a$, $b$, $c$ are summed, but Lorentz indices $\mu$, $\nu$, $\rho$ are not. The indices $\mu$, $\nu$, $\rho$ are not equal to one another.[]{data-label="momentum2_multiplicities"} Operator $~I~$ $A_1^+$ $A_2^+$ $E^+$ $T_1^+$ $T_2^+$ $A_1^-$ $A_2^-$ $E^-$ $T_1^-$ $T_2^-$ ----------------------------------------------------------------------------- ------- --------- --------- ------- --------- --------- --------- --------- ------- --------- --------- -- $H(\vec{p}_{\mu\nu\rho})H(-\vec{p}_{\mu\nu\rho})$ 0 1 0 0 0 1 0 0 0 0 0 $H(\vec{p}_{\mu\nu\rho})W_\mu^a(-\vec{p}_{\mu\nu\rho})$ 1 1 0 1 1 2 0 1 1 2 1 $W_\mu^a(\vec{p}_{\mu\nu\rho})W_\mu^a(-\vec{p}_{\mu\nu\rho})$ 0 1 0 1 1 2 0 0 0 0 0 $W_\mu^a(\vec{p}_{\mu\nu\rho})W_\nu^a(-\vec{p}_{\mu\nu\rho})$ 0 1 0 1 1 2 1 0 1 1 2 $\epsilon^{abc}W_\mu^b(\vec{p}_{\mu\nu\rho})W_\mu^c(-\vec{p}_{\mu\nu\rho})$ 1 0 0 0 0 0 0 1 1 2 1 $\epsilon^{abc}W_\mu^b(\vec{p}_{\mu\nu\rho})W_\nu^c(-\vec{p}_{\mu\nu\rho})$ 1 0 1 1 2 1 0 1 1 2 1 : Octahedral group multiplicities of Higgs-Higgs, Higgs-$W$ and $W$-$W$ operators built of the operators in Eqs. (\[Hp\]) and (\[Wp\]) with $\vec p\neq\vec 0$, where $\vec{p}_{123}=\tfrac{2\pi}{L}(1,1,1)$, $\vec{p}_{-123}=\tfrac{2\pi}{L}(-1,1,1)$, $\vec{p}_{1-23}=\tfrac{2\pi}{L}(1,-1,1)$ and $\vec{p}_{12-3}=\tfrac{2\pi}{L}(1,1,-1)$. Repeated SU(2) indices $a$, $b$, $c$ are summed, but Lorentz indices $\mu$, $\nu$, $\rho$ are not. The indices $\mu$, $\nu$, $\rho$ are not equal to one another.[]{data-label="momentum3_multiplicities"} Tables \[momentum1\_multiplicities\], \[momentum2\_multiplicities\] and \[momentum3\_multiplicities\] show the multiplicities for Higgs-Higgs, Higgs-$W$ and $W$-$W$ operators with the nonzero internal momentum, $\left|\vec{p}\right|=2\pi/L$, $\left|\vec{p}\right|=\sqrt{2}(2\pi/L)$ and $\left|\vec{p}\right|=\sqrt{3}(2\pi/L)$, respectively. The list of allowed $W$-$W$ representations for $\left|\vec{p}\right|=2\pi/L$ agrees completely with the states that were found in Figs. \[graph\_mass\_link\] and \[graph\_mass\_wilson\_polyakov\]. This shows why the $W$-$W$ signal was absent from other channels in those graphs. In general, the direction of the internal momentum on the lattice will affect the allowed irreducible representations of multiparticle states [@Moore:2006ng; @Moore:2005dw]. Application of the variational analysis to the two-Higgs, Higgs-$W$ and two-$W$ operators with back-to-back momenta $\left|\vec{p}\right|=2\pi/L$, $\left|\vec{p}\right|=\sqrt{2}(2\pi/L)$ and $\left|\vec{p}\right|=\sqrt{3}(2\pi/L)$ produced Figs. \[graph\_mass\_hh\_hw\_momentum\] and \[graph\_mass\_ww\_momentum\]. ![Energy spectrum extracted from correlation functions of Higgs-Higgs and Higgs-$W$ operators built from Eqs. (\[Hp\]) and (\[Wp\]) with $\left|\vec{p}\right|=2\pi/L$, $\left|\vec{p}\right|=\sqrt{2}(2\pi/L)$ and $\left|\vec{p}\right|=\sqrt{3}(2\pi/L)$ on a $24^3\times 48$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. Data points are lattice results with statistical errors; horizontal lines are the expectations from Eq. .[]{data-label="graph_mass_hh_hw_momentum"}](graph_mass_hh_hw_momentum.eps) The single-$W$ states (near energy 0.2) and two-stationary-$W$ states (near 0.4) were detected in a few channels but, as intended, these operators couple strongly to a pair with internal momentum. Comparison of Tables \[momentum1\_multiplicities\], \[momentum2\_multiplicities\] and \[momentum3\_multiplicities\] with Figs. \[graph\_mass\_hh\_hw\_momentum\] and \[graph\_mass\_ww\_momentum\] shows that signals are observed in precisely the expected subset of $I(\Lambda^P)$ channels in each case. ![Energy spectrum extracted from correlation functions of $W$-$W$ operators built from Eq. (\[Wp\]) with $\left|\vec{p}\right|=2\pi/L$, $\left|\vec{p}\right|=\sqrt{2}(2\pi/L)$ and $\left|\vec{p}\right|=\sqrt{3}(2\pi/L)$ on a $24^3\times 48$ lattice with $\beta=8$, $\kappa=0.131$ and $\lambda=0.0033$. Data points are lattice results with statistical errors; horizontal lines are the expectations from Eq. .[]{data-label="graph_mass_ww_momentum"}](graph_mass_ww_momentum.eps) Conclusions {#sec:conclusions} =========== The particle spectrum of the SU(2)-Higgs model has been computed thoroughly, using lattice simulations with all parameters tuned to experimental values. Three conceptually different classes of operators were used to extract the energy spectrum: gauge-invariant links, Wilson loops and Polyakov loops. Particular spatial shapes were chosen for these operators to provide access to all irreducible representations of angular momentum and parity, for both isospin 0 and 1. Varying levels of stout-link and scalar smearing were applied to improve the operators and to generate a basis for a variational analysis of the correlation matrices. The energies computed from the variational analysis comprise a vast multi-particle spectrum that is completely consistent with collections of almost-noninteracting Higgs and $W$ bosons. No states were found beyond this simple picture. Of course the interactions between bosons are not expected to be strictly zero, but such tiny deviations from zero are not attainable using the lattice studies presented here. Simulations with a stronger gauge coupling – but still in the Higgs region of the phase diagram – might provide information about interactions, and the fact that the SU(2)-Higgs model is a single phase implies an analytic connection from strong coupling to the physical point. It also implies an analytic connection to the confinement region of the phase diagram with its seemingly very different spectrum. Therefore future lattice studies, similar to what we have done but at stronger gauge coupling, could be of significant value. 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[^1]: In general one would use ${\cal O}^\dagger(0)$ rather than ${\cal O}(0)$, but in our SU(2) study the $\vec p=\vec 0$ operators are Hermitian and (as discussed in Sec. \[Two-Particle Operators\]) even the $\vec p\neq\vec 0$ correlation functions are statistically real.

ArXiv

--- abstract: 'Let $C/K$ be a curve over a local field. We study the natural semilinear action of Galois on the minimal regular model of $C$ over a field $F$ where it becomes semistable. This allows us to describe the Galois action on the $l$-adic Tate module of the Jacobian of $C/K$ in terms of the special fibre of this model over $F$.' address: - 'Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK' - 'King’s College London, Strand, London WC2R 2LS, UK' - 'King’s College London, Strand, London WC2R 2LS, UK' author: - 'Tim and Vladimir Dokchitser, Adam Morgan' title: Tate module and bad reduction --- -1cm Introduction ============ Let $C/K$ be a curve[^1] of positive genus over a non-Archimedean local field, with Jacobian $A/K$. Our goal is to describe the action of the absolute Galois group $G_K$ on the $l$-adic Tate module $T_l A$ in terms of the reduction of $C$ over a field where $C$ becomes semistable, for $l$ different from the residue characteristic. Fix a finite Galois extension $F/K$ over which $C$ is semistable [@DM]. Write $\mathcal{O}_F$ for the ring of integers of $F$, $k_F$ for the residue field of $F$, $I_F$ for the inertia group, $\cC/\mathcal{O}_F$ for the minimal regular model of $C/F$, and $\cC_{k_F}/k_F$ for its special fibre. Grothendieck defined a canonical filtration by $G_F$-stable $\Z_l$-lattices [@SGA7I §12], $$\label{eq1} 0\subset T_l(A)^t \subset T_l(A)^{I_F} \subset T_l(A);$$ $T_l(A)^t$ is sometimes referred to as the “toric part”. He showed that its graded pieces are unramified $G_F$-modules and are, canonically, $$\label{eq2} H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1), \qquad T_l \Pic^0 \tilde\cC_{k_F}, \qquad H_1(\Upsilon,\Z) \tensor_\Z\Z_l,$$where $\tilde\cC_{k_F}$ is the normalisation of $\cC_{k_F}$, $\Upsilon$ is the dual graph of $\cC_{{\bar k}_F}$ (a vertex for each irreducible component and an edge for every ordinary double point) and $H^1, H_1$ are singular (co)homology groups. Here the middle piece may be further decomposed as[^2] $$\label{eqind} T_l \Pic^0 (\tilde \cC_{k_F}) \iso \bigoplus_{\Gamma\in \JJ/G_F} \Ind_{\Stab(\Gamma)}^{G_F} T_l\Pic^0(\Gamma),$$ where $\JJ$ is the set of geometric connected components of $\tilde\cC_{k_F}$. In particular, the above discussion determines the first $l$-adic étale cohomology group of $C$ as a $G_F$-module: $$\label{eq3} \H(C_{\Kbar},\Q_l) \>\>\iso\>\> H^1(\Upsilon,\Z)\!\tensor\!\Sp_2 \>\oplus\> \H(\tilde\cC_{k_F},\Q_l),$$ where $\Sp_2$ is the 2-dimensional ‘special’ representation (see [@Ta 4.1.4]). In this paper we describe the full $G_K$-action on $T_l(A)$ in terms of this filtration, even though $C$ may not be semistable over $K$. \[main\] The filtration of $T_l(A)$ is independent of the choice of $F/K$ and is $G_K$-stable. Moreover, $G_K$ acts semilinearly[^3] on $\mathcal{C}/\mathcal{O}_F$, inducing actions on $\cC_{k_F}$, $\Upsilon$, $\Pic^0 \cC_{k_F}$ and $\Pic^0 \tilde\cC_{k_F}$, with respect to which identifies the graded pieces as $G_K$-modules and extends to a $G_K$-isomorphism $$T_l \Pic^0 (\tilde \cC_{k_F}) \iso \bigoplus_{\Gamma\in \JJ/G_K} \Ind_{\Stab(\Gamma)}^{G_K} T_l\Pic^0(\Gamma).$$ The action of $\sigma \in G_K$ on $\cC_{k_F}$ is uniquely determined by its action on non-singular points, where it is given by $$\qquad \cC_{k_F}({\bar k}_F)_{\ns} \!\overarrow{\text{lift}}\! \cC(\mathcal{O}_{\bar F})=C(\bar K) \!\overarrow{\sigma}\! C(\bar K) =\cC(\mathcal{O}_{\bar F}) \!\overarrow{\text{reduce}}\! \cC_{k_F}({\bar k}_F)_{\ns}.$$ There is an isomorphism of $G_K$-modules $$\begin{array}{llllll} \H(C_{\Kbar},\Q_l) &\iso& H^1(\Upsilon,\Z)\!\tensor\!\Sp_2 \>\oplus\> \H(\tilde\cC_{{\bar k}_F},\Q_l) \\[5pt] &\iso& \displaystyle H^1(\Upsilon,\Z)\!\tensor\!\Sp_2 \>\oplus\> \bigoplus_{\Gamma\in \JJ/G_K} \Ind_{\Stab(\Gamma)}^{G_K} \H(\Gamma_{{\bar k}_F},\Q_l). \end{array}$$ Suppose $\sigma\in \Stab_{G_K}(\Gamma)$ acts on $\bar{k}_F$ as a non-negative integer power of Frobenius $x\mapsto x^{|k_K|}$. Its (semilinear) action on the points of $\Gamma(\bar{k}_F)$ coincides with the action of a $k_F$-linear morphism (see Remark \[rmk-frob\]). In particular, one can determine trace of $\sigma$ on $\H(\Gamma_{{\bar k}_F},\Q_l)=\Hom(T_l\Pic^0(\Gamma),\Q_l)$ using the Lefschetz trace formula and counting fixed points of this morphism on $\Gamma(\bar{k}_F)$. See [@hq §6] for an explicit example. For the background in the semistable case see [@SGA7I §12.1-§12.3, §12.8] when $k=\bar k$ and [@BLR §9.2] or [@Pap] in general. In the non-semistable case, the fact that the inertia group of $F/K$ acts on $A$ by geometric automorphisms goes back to Serre–Tate [@ST Proof of Thm. 2], and [@CFKS pp. 12–13] explains how to extend this to a semilinear action of the whole of $G_K$. We also note that in [@BW Thm. 2.1] the $I_K$-invariants of $T_l A$ ($A$ a Jacobian) are described in terms of the quotient curve by the Serre–Tate action. We now illustrate how one might use Theorem \[main\] in two simple examples. For applications to the arithmetic of curves, in particular hyperelliptic curves, we refer the reader to [@hq §6] and [@M2D2 §10]. Let $p>3$ be a prime. Fix a primitive 3rd root of unity $\zeta\in \bar \Q_p$ and $\pi=\sqrt[3]p$, and let $F=\Q_p(\zeta,\pi)$. Consider the elliptic curve $$%\phantom{hahahahaha} E/\Q_p \colon y^2=x^3+p^2 %\qquad \qquad (p>3).$$ It has additive reduction over $\Q_p$, and acquires good reduction over $F$. The substitution $x'= \frac{x}{\pi^2}, y'= \frac{y}{p}$ shows that the special fibre of its minimal model is the curve $$\bar{E}/\F_p: y^2=x^3+1.$$ The Galois group $G_{\Q_p}$ acts on $\bar{E}$ by semilinear morphisms, which by Theorem \[main\] are given on $\bar{E}(\bar\F_p)$ by the “lift-act-reduce” procedure. Explicitly, we compute the action of $\sigma \in G_{\Q_p}$ on a point $(x,y)\in\bar{E}(\bar\F_p)$, with lift $(\tilde x,\tilde y)$ to the model of $E$ with good reduction, as $$(x,y) \!\to\! (\tilde x, \tilde y) \!\to\! (\pi^2 \tilde x, p \tilde y) \!\to\! (\sigma (\pi^2\tilde x), p \sigma \tilde y) \!\to\! (\tfrac{\sigma\pi^2}{\pi^2} \sigma \tilde x, \sigma\tilde y) \!\to\! (\zeta^{2\chi(\sigma)}\bar\sigma x,\bar\sigma y),$$ where $\bar\sigma$ is the induced action of $\sigma$ on the residue field and $\frac{\sigma(\pi)}{\pi}\equiv \zeta^{\chi(\sigma)}\mod \pi$. In particular, $\sigma$ in the inertia group of $\mathbb{Q}_p$ acts as the geometric automorphism $(x,y)\mapsto (\zeta^{2\chi(\sigma)}x,y)$ of $\bar{E}$. By Theorem \[main\], $T_l(E)$ with the usual Galois action is isomorphic to $T_l(\bar{E})$ with the action induced by the semilinear automorphisms. In particular, we see that the action factors through $\Gal(F^{nr}/\Q_p)$, the Galois group of the maximal unramified extension of $F$. Moreover the inertia subgroup acts by elements of order 3 (as expected from the Néron–Ogg–Shafarevich criterion), and the usual actions of $G_{\Q_p(\pi)}$ on $T_l(E)$ and $T_l(\bar{E})$ agree under the reduction map. Let $p$ be an odd prime and $E/\Q_p$ the elliptic curve $$E/\Q_p \colon py^2=x^3+x^2+p.$$ It has additive reduction over $\Q_p$ that becomes multiplicative over $\Q_p(\sqrt{p})$, where the special fibre of its minimal regular model is the nodal curve $$\bar{E}/\F_p \colon y^2= x^3 + x^2.$$ The dual graph $\Upsilon$ of the special fibre is one vertex with a loop: (\#1)[ (\#1) edge\[out=155,in=90\] ($(#1)-(1.3,0)$); (\#1) edge\[out=210,in=270\] ($(#1)-(1.3,0)$); ]{} ; (1) We compute analagously to the previous example that $\sigma\in G_{\Q_p}$ acts on the non-singular points of $\bar{E}(\bar\F_p)$ by $$\phantom{hahahahahahaha} (x,y)\mapsto (\bar \sigma x, \epsilon_\sigma\bar \sigma y), \qquad \qquad \epsilon_\sigma=\frac{\sqrt{p}}{\sigma\sqrt{p}}\in\{\pm 1\}.$$ This formula describes a semilinear morphism of $\bar{E}$ which induces the map $(x,y)\mapsto (\bar \sigma x, \epsilon_\sigma \bar \sigma y)$ on the normalisation $\tilde{E}:y^2=x+1$ which permutes the two points above the singular point (and hence the orientation of the edge in $\Upsilon$) if and only if $\epsilon_\sigma=-1$. Thus $H_1(\Upsilon,\Z)$ is isomorphic to $\Z$ with $G_{\Q_p}$ acting by the order 2 character $\sigma\mapsto \epsilon_\sigma$. By Theorem \[main\], $G_{\Q_p}$ acts on $T_l(E)$ as $ \sigma \mapsto \smallmatrix {\epsilon_\sigma\chi_{cyc}(\sigma)}*0{\epsilon_\sigma}, $ where $\chi_{cyc}$ is the cyclotomic character. Layout {#layout .unnumbered} ------ To prove Theorem \[main\], we review semilinear actions in §\[semilinear actions section\], and prove a general theorem (\[geommain\]) for models of schemes that are sufficiently ‘canonical’ to admit a unique extension of automorphisms of the generic fibre; in particular, this applies to minimal regular models and stable models of curves, and Néron models of abelian varieties (this again goes back to [@ST Proof of Thm. 2]). We then apply this result in §\[curves and jacobians sect\] to obtain Theorem \[main\]. In fact, all our results are slightly more general, and apply to $K$ the fraction field of an arbitrary Henselian DVR, and not just for the Galois action but also the action of other (e.g. geometric) automorphisms. Acknowledgements {#acknowledgements .unnumbered} ---------------- We would like to thank the University of Warwick and Baskerville Hall where parts of this research were carried out. This research is supported by grants EP/M016838/1 and EP/M016846/1 ‘Arithmetic of hyperelliptic curves’. The second author is supported by a Royal Society University Research Fellowship. Semilinear actions {#semilinear actions section} ================== For schemes $X/S$ and $S'/S$ we denote by $X_{S'}/S'$ the base change $X\times_S S'$. For a scheme $T/S$ we write $X(T)=\textup{Hom}_S(T,X)$ for the $T$-points of $X$. For a ring $R$, by an abuse of notation we write $X(R)=X(\textup{Spec }R)$. Semilinear morphisms -------------------- If $S$ is a scheme, $\alpha\in\Aut S$, and $X$ and $Y$ are $S$-schemes, a morphism $f: X\to Y$ is *$\alpha$-linear* (or simply *semilinear*) if the following diagram commutes: $$\begin{CD} X @>f>> Y \\ @VVV @VVV \\ S @>\alpha>> S \\ \end{CD}$$ \[twisted scheme\] For a scheme $X/S$ and an automorphism $\alpha\in\Aut(S)$, write $X_\alpha$ for $X$ viewed as an $S$-scheme via $X\to S\overarrow{\alpha} S$. \[remtwist\] An $\alpha$-linear morphism $X\to X$ is the same as an $S$-morphism $X_\alpha\to X$. Note further that - $X_{\alpha \beta}=(X_\alpha)_\beta$ canonically; - an $S$-morphism $f: X\to X$ induces an $S$-morphism $\alpha(f): X_\alpha\to X_\alpha$, which is the same map as $f$ on the underlying schemes. \[rembasechange\] Equivalently, $X_\alpha=X\times_{S,\alpha^{-1}}S$ viewed as an $S$-scheme via the second projection, where the notation indicates that we are using the morphism $\alpha^{-1}:S\rightarrow S$ to form the fibre product. More precisely, the first projection gives an isomorphism of $S$-schemes $X\times_{S,\alpha^{-1}}S\rightarrow X_\alpha$. \[lemproj\] Let $X$, $Y$, $S'$ be $S$-schemes, $\alpha\in\Aut S$ and suppose we are given an $\alpha$-linear morphism $f: X\to Y$ and an $\alpha$-linear automorphism $\alpha': S'\to S'$. 1. There is a unique $\alpha'$-linear morphism $f\times_\alpha \alpha': X_{S'}\to Y_{S'}$ such that $\pi_Y\ring (f\times_\alpha \alpha') = f\ring \pi_X$, where $\pi_X$ and $\pi_Y$ are the projections $X_{S'}\rightarrow X$ and $Y_{S'}\rightarrow Y$ respectively. 2. Given another $S$-scheme $Z$, $\beta\in \Aut S$, $g: Y\to Z$ a $\beta$-linear morphism and $\beta': S'\to S'$ a $\beta$-linear automorphism, we have $$(f\times_{\alpha} {\alpha'})\ring (g\times_{\beta} {\beta'}) = (f\ring g)\times_{\alpha\ring \beta} ({\alpha'}\ring {\beta'}).$$ \(1) By the universal property of the fibre product $Y_{S'}$ applied to the morphisms $f\circ\pi_X: X_{S'}\to Y$ and $\alpha'\circ\pi_{S'}: X_{S'}\to S'$ there is a unique morphism $X_{S'}\to Y_{S'}$ with the required properties. \(2) Follows from the uniqueness of the morphisms afforded by (1). Semilinear actions {#semilinear-actions} ------------------ For a group $G$ acting on a scheme $X$, for each $\sigma\in G$ we write $\sigma_X$ (or just $\sigma$) for the associated automorphism of $X$. All actions considered are left actions. \[def:semilinearaction\] Let $G$ be a group and $S$ a scheme on which $G$ acts. We call an action of $G$ on a scheme $X/S$ *semilinear* if for each $\sigma \in G$, the automorphism $\sigma_X$ is $\sigma_S$-linear. \[cocycle\] Specifying a semilinear action of $G$ on $X/S$ is equivalent to giving $S$-isomorphisms $c_\sigma: X_{\sigma}\to X$ for each $\sigma\in G$, satisfying the cocycle condition $ c_{\sigma \tau}=c_\sigma \ring \sigma(c_\tau) $ (cf. Remark \[remtwist\]). \[defpointact\] Given a semilinear action of $G$ on $X/S$ and $T/S$, $G$ acts on $X(T)$ via $$P \quad\longmapsto\quad \sigma_X\ring P\ring \sigma_T^{-1}.$$ Suppose $G$ acts semilinearly on $X/S$. Then given $S'/S$ and a semilinear action of $G$ on $S'$, we get a semilinear *base change action* of $G$ on $X_{S'}$ by setting, for $\sigma \in G$, $$\sigma_{X_{S'}}=\sigma_X\times_{\sigma_S}\sigma_{S'}.$$ \[lemactpoints\] Suppose $G$ acts semilinearly on $X/S$ and $T/S$. 1. If $G$ acts semilinearly on $Y/S$ and $f:X\rightarrow Y$ is $G$-equivariant, then so is the natural map $X(T)\rightarrow Y(T)$ given by $P\mapsto f\circ P$. 2. If $G$ acts semilinearly on $T'/S$ and $f:T'\rightarrow T$ is $G$-equivariant, then so is the natural map $X(T)\to X(T')$ given by $P\mapsto P\circ f$. 3. If $G$ acts semilinearly on $S'/S$ then the natural map $X(T)\to X_{S'}(T_{S'})$ given by $P\mapsto P\times_{\id}\id$ is equivariant for the action of $G$, where $G$ acts on $X_{S'}(T_{S'})$ via base change. \(1) Clear. (2). Clear. (3) Denoting by $\phi$ the map $X(T)\to X_{S'}(T_{S'})$ in the statement, for each $\sigma\in G$ we have by Lemma \[lemproj\] (2) that $$\sigma\cdot \phi(P) = (\sigma_X\times_{\sigma_S}\sigma_{S'}) \ring (P\times_{\id}\id) \ring (\sigma_{T}\times_{\sigma_S}\sigma_{S'})^{-1} = (\sigma_X\ring P\ring \sigma_T^{-1}) \times_{\id} \id = \phi(\sigma\cdot P)$$ as desired. Let $X$ be an $S$-scheme and $G=\Aut_S X$. Then the natural action of $G$ on $X$ is semilinear for the trivial action on $S$. Given $T/S$ with trivial $G$-action, the induced action of $\sigma \in G$ on $X(T)$ recovers the usual action $ P \mapsto \sigma\ring P$. \[galois example1\] Let $K$ be a field, $G=G_K$ and $S=\Spec K$ with trivial $G$ action. Let $T=\Spec\Kbar$ with $\sigma\in G$ acting via $$(\sigma^{-1})^*: \Spec \Kbar\to \Spec \Kbar.$$ Then for any scheme $X/K$, letting $G$ act trivially on $X$, the action on $X(\bar{K})$ is $ P \mapsto P\ring \sigma^* $, which is just the usual Galois action on points. Now let $F/K$ be Galois, so that the $G$-action on $\textup{Spec }\bar{K}$ restricts to an action on $\textup{Spec }F$. We obtain an example of a genuinely semilinear action by considering the base change action of $G$ on $X_F$, so that here the action on the base $\textup{Spec }F$ is through $(\sigma^{-1})^*$. The natural map $X(\bar K)\to X_F(\bar K)$ is an equality, and identifies the $G$-actions by Lemma \[lemactpoints\]. Geometric action over local fields {#geom act section} ================================== Let $\mathcal{O}$ be a Henselian DVR, $K$ its field of fractions, $F/K$ a finite Galois extension, $\mathcal{O}_F$ the integral closure of $\mathcal{O}$ in $F$, and $k_F$ the residue field of $\mathcal{O}_F$. Let $G$ be a group equipped with a homomorphism $\theta:G\rightarrow G_K$ (in our applications we will either take $G=G_K$, or $\theta$ the zero-map). This induces an action of $G$ on $\textup{Spec }\bar{K}$ via $\sigma \mapsto (\theta(\sigma)^{-1})^*$, which restricts to actions on $\textup{Spec }F$, $\textup{Spec }\mathcal{O}_F$, etc. Now let $X/K$ be a scheme and suppose that $G$ acts semilinearly on the base-change $X_F/\textup{Spec }F$ with respect to the above action on $\Spec F$. \[geommain\] Suppose $\cX/\mathcal{O}_F$ is a model[^4] of $X_F$ such that for each $\sigma\in G$ the semilinear morphism $\sigma_{X_F}$ extends uniquely to a semilinear morphism $\sigma_{\cX}: \cX\to \cX$. Then 1. The map $\sigma \mapsto \sigma_{\cX}$ defines a semilinear action of $G$ on $\cX/\mathcal{O}_F$. In particular, it induces by base-change a semilinear action of $G$ on the special fibre $\cX_{k_F}$, and also induces actions on $\cX(\OFB)$ and $\cX_{k_F}(\bar k_F)$. 2. The natural maps on points $\cX(\OFB)\to X(\FB)$ and $\cX(\OFB)\to \cX_{k_F}(\bar k_F)$ are $G$-equivariant. 3. Suppose $\cX(\OFB)\to X(\FB)$ is bijective, and let $I$ be the image of $\cX(\OFB)\to \cX_{k_F}(\bar k_F)$. Then the action of $\sigma\in G$ on $I$ is given by $$I \overarrow{\text{lift}} \cX(\OFB) \overarrow{=} X(\FB) \overarrow{\sigma} X(\FB) \overarrow{=} \cX(\OFB) \overarrow{\text{reduce}} I.$$ \(1) Follows from uniqueness of the extension of the $\sigma_{X_F}$ to $\cX$. \(2) Follows from Lemma \[lemactpoints\] applied to the maps $\OFB\tensor_{\mathcal{O}_F} F=\FB$ and $\OFB\tensor_{\mathcal{O}_F} k_F\to k_{\FB}$. (3) Follows from (2). The assumption on the uniqueness of the extensions of the $\sigma_{X_F}$ is automatic if $\cX/\mathcal{O}_F$ is separated. The assumption that $\cX(\mathcal{O}_{\bar F})\to X(\bar F)$ is bijective in (3) is automatic if $\cX/\mathcal{O}_F$ is proper. \[rmk-frob\] Suppose $\vchar k_F=p>0$ and $\sigma\in G$ acts on $\bar k_F$ as $x\mapsto x^{p^n}$ for some $n\geq 0$. Let $\Fr$ denote the $p^n$-power absolute Frobenius. Note that $\Fr:\cX_{k_F}\to \cX_{k_F}$ is $\Fr=\sigma_{\textup{Spec }k_F}^{-1}$-linear whilst $\sigma_{\cX_{k_F}}$ is $\sigma_{\textup{Spec }k_F}$-linear, so that $\psi_\sigma=\sigma_{\cX_{k_F}}\circ \Fr$ is a $k_F$-morphism. Moreover, since absolute Frobenius commutes with all scheme morphisms, for any $P\in \cX_{k_F}(\bar k_F)$ we have $$\psi_\sigma(P) = \sigma_{\cX_{k_F}}\circ \Fr \circ P = \sigma_{\cX_{k_F}} \circ P \circ \Fr = \sigma_{\cX_{k_F}} \circ P \circ \sigma_{\textup{Spec }k_F}^{-1} = \sigma \cdot P.$$ In particular, the action of $\sigma$ on the $\bar k_F$-points of $\cX_{k_F}$ agrees with that of a $k_F$-morphism, even though the action of $\sigma$ on $k_F$ may be non-trivial. \[assumptionshold\] The assumptions of Theorem \[geommain\], including (3), hold in the following situations: - $X/K$ a curve and $\cX/\mathcal{O}_F$ the minimal regular model of $X/F$. - $X/K$ a curve which becomes semistable over $F$, and $\cX/\mathcal{O}_F$ the stable model of $X/F$. - $X/K$ an abelian variety, and $\cX/\mathcal{O}_F$ the Néron model of $X/F$. To see that the assumption of the theorem is satisfied, use Remark \[cocycle\]: in all three cases, for any $\sigma \in G$, $\cX_\sigma$ is again a model of $X_\sigma$ of the same type as $\cX$, and the universal properties that these models satisfy guarantee the existence and uniqueness of the extensions. Regarding (3), $\cX(\mathcal{O}_{\bar F})=\cX_F(\bar F)$ by properness in (i),(ii) and the Néron mapping property in (iii). Moreover, the image $I$ of the reduction map contains all non-singular points since $\mathcal{O}_F$ is Henselian. Curves and Jacobians {#curves and jacobians sect} ==================== As in §\[geom act section\], let $\mathcal{O}$ be a Henselian DVR, $K$ its field of fractions, $F/K$ a finite Galois extension, $\mathcal{O}_F$ the integral closure of $\mathcal{O}$ in $F$ and $k_F$ the residue field of $\mathcal{O}_F$. Fix a curve $C/K$ that becomes semistable over $F$, and let $A/K$ be the Jacobian of $C$. Let $\cC/\mathcal{O}_F$ be the minimal regular model of $C/F$ (which is semistable since $C/F$ is). Let $\A/\mathcal{O}_F$ be the Néron model of $A/F$ with special fibre $\Abar/k_F$, and let $\A^o/\mathcal{O}_F$ be its identity component with special fibre $\Abar^o/k_F$. Now let $G$ be a group equipped with a homomorphism $\theta:G\rightarrow G_K$, acting on $\textup{Spec }\bar{K}$ via $\sigma \mapsto (\theta(\sigma)^{-1})^*$, and hence also on $\textup{Spec }F$, $\textup{Spec }\mathcal{O}_F$, etc. Suppose that our curve $C/K$ is equipped with a semilinear action of $G$ on $C_F/F$, whence $A_F/F$ inherits a semilinear action of $G$ via $\sigma \mapsto (\sigma_{C_F}^{-1})^*$. If we take $G=G_K$ then this is the usual Galois action on the Jacobian on $C$. Theorem \[geommain\] and Remark \[assumptionshold\] provides semilinear actions of $G$ on $\cC/\mathcal{O}_F$ and $\A/\mathcal{O}_F$, and these induce semilinear actions on $\cC_{k_F}$, $\A^o/\mathcal{O}_F$, $\Abar/k_F$, and $\Abar^o/k_F$ also. Finally, let $ \Pic^0_{\cC/\mathcal{O}_F} $ denote the relative Picard functor. This inherits a semilinear action $\sigma \mapsto (\sigma_{\cC}^{-1})^*$ of $G$ induced from that on $\mathcal{C}/\mathcal{O}_F$ as we now explain. By Remark \[rembasechange\] and the fact that the relative Picard functor commutes with base change, it also commutes with twisting in the sense of Definition \[twisted scheme\]: for all $\sigma \in G$ we have $ \Pic^0_{\cC_\sigma/\mathcal{O}_F} = (\Pic^0_{\cC/\mathcal{O}_F})_\sigma. $ Functoriality of $\Pic^0_{\cC/\mathcal{O}_F}$ combined with Remark \[remtwist\] gives the sought automorphism $(\sigma_{\cC}^{-1})^*$. Note that the action on $\Pic^0_{\cC/\mathcal{O}_F}$ induces by base-change a semilinear action of $G$ on the special fibre $\Pic^0_{\cC_{k_F}/k_F}$, and that this is compatible with the action of $G$ on $\mathcal{C}_{k_F}$ in the sense that $\sigma\in G$ acts as $(\sigma_{\cC_{k_F}}^{-1})^*$. \[compatibility of geom actions\] For any $\sigma \in G$, the following diagram commutes $$\begin{CD} \A^o @>\sigma_{\A}>> \A^o \\ @VV\iso V @VV\iso V \\ \Pic^0_{\cC/\mathcal{O}_F} @>(\sigma_{\cC}^{-1})^*>> \Pic^0_{\cC/\mathcal{O}_F}, \\ \end{CD}$$ with the vertical isomorphisms provided by [@BLR Thm. 9.5.4]. Since $\A^o$ and $\Pic^0_{\cC/\mathcal{O}_F}$ are separated over $\mathcal{O}_F$ it suffices to check that the diagram commutes on the generic fibre, where it does by the definition of the action of $G$ on $A$. We now turn to the $G$-action on the Tate module of $A/K$. \[limit of ST cor\] For every $m\ge 1$ coprime to $\vchar k_F$, $$A[m]^{I_F}\iso \Abar[m]$$ as $G$-modules. For every $l\ne\vchar k_F$, $$T_l A^{I_F} \iso T_l \Abar = T_l \Abar^o \iso T_l \Pic^0_{\cC_{k_F}/k_F}$$ as $G$-modules. \(1) Note that $A[m]^{I_F}=A(F^{\textup{nr}})[m]$ is a $G$-submodule of $A[m]$ since $G$ acts on $F^{\textup{nr}}$. By [@ST Lemma 2], under the reduction map $A[m]^{I_F}$ is isomorphic to $\Abar[m]$ as abelian groups, and this map is $G$-equivariant for the given actions by Theorem \[geommain\] (2). (2) Pass to the limit in (1) and apply Lemma \[compatibility of geom actions\] for the final isomorphism. The following theorem describes the $G$-module $T_l \Pic^0_{\cC_{k_F}/k_F}$. We begin by explaining how $G$ acts on certain objects associated to $\cC_{k_F}$. \[rem:graphact\] Let $Y=\cC_{\bar k_F}$. Combining the action of $G$ on $\mathcal{C}_{k_F}$ with the action on $\bar{k}_F$ coming from the homomorphism $\theta:G\rightarrow G_K$ we obtain by base-change a semilinear action of $G$ on $Y$. This moreover induces a semilinear action on the normalisation $\tilde Y$ of $Y$ (any automorphism of $Y$, semilinear or otherwise, lifts uniquely to $\tilde{Y}$ and the lifts of the $\sigma_Y$ are easily checked to define a semilinear action of $G$). Write [llllll]{} $\nmap$ &=& normalisation map $\Cbarn\to\Cbar$,$\II$ &=& set of singular (ordinary double) points of $\Cbar$,$\JJ$ &=& set of connected components of $\Cbarn$,$\KK$ &=& $\nmap^{-1}(\II)$; this comes with two canonical maps&& $\phi: \KK\to \II$, $P\mapsto\nmap(P),$&& $\psi: \KK\to \JJ$, $P\mapsto$ component of $\Cbarn$ on which $P$ lies. The dual graph $\Upsilon$ of $Y$ has vertex set $\JJ$ and edge set $\II$. $\KK$ is the set of edge endpoints, and the maps $\phi$ and $\psi$ specify adjacency (note that loops and multiple edges are allowed). A graph automorphism of $\Upsilon$ (which we allow to permute multiple edges and swap edge endpoints) is precisely the data of bijections $\KK\to\KK$, $\II\to\II$ and $\JJ\to\JJ$ that commute with $\phi$ and $\psi$. In this way, the action of $G$ on $\tilde{Y}$ induces an action of $G$ on $\Upsilon$, and hence also on $H_1(\Upsilon,\Z)$ and $H^1(\Upsilon,\Z).$ \[dual graph theorem\] We have an exact sequence of $G$-modules $$0 \lar H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1) \lar T_l \Pic^0_{\cC_{k_F}/k_F} \lar T_l \Pic^0_{\tilde \cC_{k_F}/k_F} \lar 0$$ where $\Upsilon$ is the dual graph of $\cC_{\bar k_F}$ and $\tilde \cC_{k_F}$ the normalisation of $\cC_{k_F}$. Moreover, $$T_l \Pic^0_{\tilde \cC_{k_F}/k_F}\iso \bigoplus_{\Gamma\in \JJ/G} \Ind_{\Stab(\Gamma)}^{G} T_l\Pic^0(\Gamma)$$ where $\JJ$ is the set of geometric connected components of $\tilde\cC_{\bar k_F}$. (The action of $G$ on $\Z_l(1)$ is via the map $\theta:G\rightarrow G_K$.) We follow [@SGA7I pp. 469–474] closely, except our sequences (\[PicSeq\]) and (\[mayer\]) are slightly tweaked from the ones appearing there, and we must check $G$-equivariance of all maps appearing. Write $k=k_F$, $Y=\cC_{\bar k_F}$ and let $\tilde{Y}$, $n$, $\mathcal{I}$, $\mathcal{J}$, $\mathcal{K}$, $\phi$, $\psi$ be as in Remark \[rem:graphact\]. The normalisation map $\nmap$ is an isomorphism outside $\II$, and yields an exact sequence of sheaves on $\Cbar$ $$1 \lar \mathcal{O}_{\Cbar}^\times \lar \nmap_* \mathcal{O}_{\Cbarn}^\times \lar \Ish \lar 0,$$ with $\Ish$ concentrated in $\II$. Consider the long exact sequence on cohomology $$0 \to H^0(\Cbar,\mathcal{O}_{\Cbar}^\times) \to H^0(\Cbarn,\mathcal{O}_{\Cbarn}^\times) \to H^0(\Cbar,\Ish) \to H^1(\Cbar,\mathcal{O}_{\Cbar}^\times) \to H^1(\Cbarn,\mathcal{O}_{\Cbarn}^\times) \to 0$$ which is surjective on the right since $\Ish$ is flasque. Writing $(\kbar^\times)^\II$ for the set of functions $\II\to\kbar^\times$, and similarly for $\JJ$ and $\KK$, we have $$H^0(\Cbar,\Ish) = \coker((\kbar^\times)^\II\overarrow{\phi^*}(\kbar^\times)^\KK),$$ where $\phi^*$ takes a function $\II\to \kbar^\times$ to $\KK\to \kbar^\times$ by composing with $\phi$. With $\psi^*$ defined in the same way, the exact sequence above becomes $$\label{PicSeq} 0 \lar \kbar^\times \lar (\kbar^\times)^\JJ \overarrow{\psi^*} \frac{(\kbar^\times)^\KK}{\phi^*((\kbar^\times)^\II)} \lar \Pic \Cbar(\kbar) \lar \Pic \Cbarn (\kbar) \lar 0.$$ Write the dual graph $\Upsilon$ as the union $\Upsilon=U\cup V$, where $U$ is the union of open edges, and $V$ is the union of small open neighbourhoods of the vertices. Then the Mayer-Vietoris sequence reads $$\label{mayer} 0 \lar H_1(\Upsilon,\Z) \lar \Z^\KK \overarrow{(\phi,\psi)} \Z^\II\times\Z^\JJ \lar \Z \lar 0,$$ since $H_0(U)=\Z^\II$, $H_0(V)=\Z^\JJ$, $H_0(U\cap V)=\Z^\KK$ and the higher homology groups of $U$, $V$ and $U\cap V$ all vanish. Now take $\sigma\in G$. Since the semilinear action of $G$ on $\tilde{Y}$ lifts that on $Y$, the natural maps $\mathcal{O}_\Cbar\to (\sigma_Y)_* \mathcal{O}_\Cbar$ and $\mathcal{O}_\Cbarn\to (\sigma_{\tilde{Y}})_* \mathcal{O}_\Cbarn$ give the left two vertical maps in the commutative diagram $$\begin{CD} 0 @>>> \mathcal{O}_\Cbar^\times @>>> n_* \mathcal{O}_\Cbarn^\times @>>> \Ish @>>> 0 \\ @. @VVV @VVV @VVV \\ 0 @>>> (\sigma_Y)_* \mathcal{O}_\Cbar^\times @>>> (\sigma_Y)_* n_* \mathcal{O}_\Cbarn^\times @>>> (\sigma_Y)_*\Ish @>>> 0 \\ \end{CD}$$ with the rightmost vertical map coming for free. Taking the long exact sequences for cohomology associated to this diagram we find that is an exact sequence of $G$-modules (note that as $\sigma_Y$ is an isomorphism, for any sheaf $\cF$ on $Y$ the natural pullback map on cohomology identifies $H^i(Y,(\sigma_Y)_*\cF)$ with $H^i(Y,\cF)$ for all $i$). On the level of Tate modules $T_l$ ($l\ne\vchar k$), then yields an exact sequence of $G$-modules $$0 \!\lar\! \Z_l(1) \!\lar\! \Z_l[\II](1)\oplus \Z_l[\JJ](1) \!\lar\! \Z_l[\KK](1) \!\lar\! T_l \Pic \Cbar \!\lar\! T_l \Pic \Cbarn \!\lar\! 0$$ with $G$ acting on $\Z_l(1)$ via the map $\theta:G\rightarrow G_K$ and on $\mathcal{I}, \mathcal{J}$ and $\mathcal{K}$ by permutation. On the other hand, applying $\Hom(-,\Z_l(1))$ to yields an exact sequence of $G$-modules $$0 \lar \Z_l(1) \lar \Z_l[\II](1)\oplus \Z_l[\JJ](1) \lar \Z_l[\KK](1) \lar H^1(\Upsilon,\Z)\tensor_\Z\Z_l(1) \lar 0.$$ The first claim follows. For the second claim, note that $T_l\Pic^0\Cbarn=\bigoplus_{\Gamma\in\JJ}T_l\Pic^0\Gamma$ abstractly, and that once the $G$-action is accounted for the right hand side becomes the asserted direct sum of induced modules. \[rem:toric part\] Under the Serre–Tate isomorphism $T_l \Pic^0_{\cC_{k_F}/k_F} \iso T_l(A)^{I_F}$, the subspace $H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1)$ maps onto $T_l(A)^t$. To see this, let $\cF$ be the image of $H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1)$ in $T_l(A)$. Both $\cF$ and $T_l(A)^t$ are saturated since quotients by them are free, by and Theorem \[dual graph theorem\]. They have the same $\Z_l$-rank by and Theorem \[dual graph theorem\], so it is enough to check $\cF\subseteq T_l(A)^t$. When $K$ is a local field, by Theorem \[dual graph theorem\] the eigenvalues of Frobenius on $\cF$ have absolute value $|k_F|$ and hence $\cF$ is contained in $T_l(A)^t$ by , . For general $K$ one can use Deligne’s Frobenius weights argument in [@SGA7I I,§6] to reduce to this case. \[corGsplit\] The canonical filtration $0\subset T_l(A)^t \subset T_l(A)^{I_F} \subset T_l(A)$ in is $G$-stable and its graded pieces are, as $G$-modules, $$H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1), \qquad T_l \Pic^0 (\tilde \cC_{k_F}), \qquad H_1(\Upsilon,\Z) \tensor_\Z\Z_l.$$ It follows from Lemma \[limit of ST cor\], Theorem \[dual graph theorem\] and Remark \[rem:toric part\] that the filtration is $G$-stable and that the first two graded pieces are as claimed. Now by Grothendieck’s orthogonality theorem [@SGA7I Theorem 2.4], $T_l(A)^{I_F}$ is the orthogonal complement of $T_l(A)^t$ under the Weil pairing $$\label{weil pairing} T_l(A)\times T_l(A)\rightarrow \Z_l(1)$$ (here we use the canonical principal polarisation to identify $A$ with its dual). Since the Weil pairing is $G$-equivariant this identifies the quotient $T_l(A)/T_l(A)^{I_F}$ with $$\textup{Hom}_{\Z_l}(H^1(\Upsilon,\Z) \tensor_\Z\Z_l(1),\Z_l(1))=H_1(\Upsilon,\Z) \tensor_\Z\Z_l$$ which completes the proof. That the filtration is independent of $F$ follows from its characterisation in terms of the identity component of the Néron model in [@SGA7I §12], combined with the fact that the identity component of the Néron model of a semistable abelian variety commutes with base change. (Alternatively, this can also be seen by considering Frobenius eigenvalues on the graded pieces.) To deduce our main theorem, we take $G=G_K$ acting as in Example \[galois example1\] throughout this section: this gives the claimed description of the graded pieces and the Tate module decomposition. The explicit formula for the action on non-singular points of $\cC_{k_F}(\bar{k}_F)$ follows from Theorem \[geommain\](3). [CFKS]{} S.Bosch, W.Lütkebohmert, M.Raynaud, Néron Models, Erg. 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Tate, Good reduction of abelian varieties, Annals of Math. 68 (1968), 492–517. J. Tate, Number theoretic background, in: Automorphic forms, representations and L-functions, Part 2 (ed. A. Borel and W. Casselman), Proc. Symp. in Pure Math. 33 (AMS, Providence, RI, 1979) 3-26. [^1]: smooth, proper, geometrically connected [^2]: here $\Ind_H^G(\cdot)$ stands for ${\Z_l[G]}\tensor_{\Z_l[H]}(\cdot)$ [^3]: see Definition \[def:semilinearaction\] [^4]: For our purposes, a *model* $\cX/\mathcal{O}_F$ of $X_F$ is simply a scheme over $\mathcal{O}_F$ with a specified isomorphism $i: \cX\times_{\mathcal{O}_F}F\overarrow{\iso} X_F$.

ArXiv

--- abstract: 'We report results of lattice Boltzmann simulations of a high-speed drainage of liquid films squeezed between a smooth sphere and a randomly rough plane. A significant decrease in the hydrodynamic resistance force as compared with that predicted for two smooth surfaces is observed. However, this force reduction does not represent slippage. The computed force is exactly the same as that between equivalent smooth surfaces obeying no-slip boundary conditions, but located at an intermediate position between peaks and valleys of asperities. The shift in hydrodynamic thickness is shown to depend on the height and density of roughness elements. Our results do not support some previous experimental conclusions on very large and shear-dependent boundary slip for similar systems.' author: - Christian Kunert - Jens Harting - 'Olga I. Vinogradova' title: 'Random-roughness hydrodynamic boundary conditions' --- [**Introduction.–**]{} It has been recently well recognized that the famous no-slip boundary condition, for more than a hundred years applied to model experiments in fluid mechanics, reflected mostly a macroscopic character and insensitivity of old style experiments. Modern experiments concluded that although the no-slip postulate is valid for molecularly smooth hydrophilic surfaces down to contact [@vinogradova:03; @charlaix.e:2005; @vinogradova.oi:2009], for many other systems it does not apply when the size of a system is reduced to micro- and nano scales. The changes in hydrodynamic behavior are caused by an impact of interfacial phenomena, first of all hydrophobicity and roughness, on the flow. The effect of hydrophobicity on the flow past smooth surfaces is reasonably clear and suggests an amount of slippage described by the condition $v_s = b \partial v / \partial z$ where $v_s$ is the slip velocity at the wall, $b$ the slip length, and the axis $z$ is normal to the surface. The assumption is justified theoretically [@vinogradova:99; @barrat:99; @andrienko.d:2003; @bib:jens-kunert-herrmann:2005] and was confirmed by surface force apparatus (SFA) [@charlaix.e:2005], atomic force microscope (AFM) [@vinogradova:03], and fluorescence cross-correlation (FCS) [@vinogradova.oi:2009] experiments. Despite some remaining controversies in the data and amount of slip (cf. [@lauga2007]), a concept of hydrophobic slippage is now widely accepted. If a liquid flows past a rough hydrophobic (i.e. superhydrophobic) surface, roughness may favor the formation of trapped gas bubbles, resulting in a large slip length [@bib:joseph_superhydrophobic:2006; @ou2005; @feuillebois.f:2009; @sbragaglia-etal-06; @kusumaatmaja-etal-08b; @jari-jens-08]. For rough hydrophilic surfaces the situation is much less clear, and opposite experimental conclusions have been made: one is that roughness generates extremely large slip [@bonaccurso.e:2003], and one is that it decreases the degree of slippage [@granick.s:2003; @granick:02]. More recent experimental data suggests that the description of flow near rough surfaces has to be corrected, but for a separation, not slip [@vinogradova.oi:2006]. The theoretical description of such a flow represents a difficult, nearly insurmountable, problem. It has been solved only approximately, and only for a case of the periodic roughness and far-field flow with a conclusion that it may be possible to approximate the actual surface by a smooth one with the apparent slip boundary condition [@sarkar.k:1996; @lecoq.n:2004; @kunert-harting-07]. In this letter we address the fundamental, but still open questions (i) whether the effect of random roughness on the flow may be represented by replacing the no-slip condition on the exact boundary by an effective condition on the equivalent smooth surface, (ii) where this smooth surface is located, depending on geometric parameters of roughness, and (iii) does this effective condition represent that of slip or no-slip? We will quite generally assume that the flow near and far from the interface is a stable, laminar flow field. [**General idea and models.–**]{} To address these issues we analyze the hydrodynamic interaction between a smooth sphere of radius $R$ and a rough plane (see Fig. \[fig:picture\]). Beside its significance as a geometry of SFA/AFM dynamic force experiments, this allows us to explore both far and near-field flows in a single “experiment”. As an initial application we study roughness elements of a fixed height $r$ that are distributed at random uncorrelated positions with a given probability $\phi$. Such a surface mimics a situation explored in recent experiments [@bonaccurso.e:2003; @granick.s:2003; @granick:02]. In Cartesian coordinates ${\bf x}=(x_1,x_2,x_3)$, a separation $h$ is defined on top of the roughness, $x_2=r$, which finds its definition in the AFM experiment [@bonaccurso.e:2003; @vinogradova.oi:2006]. The exact solution, valid for an arbitrary separation, for a sphere approaching a smooth plane is given by theoretical solutions of Brenner and Maude [@Brenner:1961; @Maude:1961], $$\begin{aligned} \label{eq:maudea} &\!\!\!\!\!\!\!\!\!\!\!\frac{F_1}{F_{St}}=-\frac{1}{3} \sinh \xi \\ &\!\!\!\!\!\!\!\!\!\!\!\times\left(\sum^{\infty}_{n=1} \frac{n(n+1)\left [8e^{(2n+1)\xi}+2(2n+3)(2n-1)\right ]}{(2n-1)(2n+3)[4\sinh^2(n+\frac{1}{2})\xi-(2n+1)^2\sinh^2\xi]}\right.\nonumber\\ &\!\!\!\!\!\!\!\left.-\sum^{\infty}_{n=1} \frac{n(n+1)\left [ (2n+1)(2n-1)e^{2\xi}-(2n+1)(2n+3)e^{-2\xi} \right ]}{(2n-1)(2n+3)[4 \sinh^2(n+\frac{1}{2})\xi-(2n+1)^2\sinh^2\xi]}\right),\nonumber\end{aligned}$$ with $F_{St}=6 \pi \mu R v$, where $\mu$ is the dynamic viscosity, $v$ is the velocity, and $\cosh \xi=h/R$, $\xi<0$. The leading term of this expression can be evaluated as $$\label{firstorder} \frac{F_2}{F_{St}}\sim 1+\frac{9}{8}\frac{R}{h}.$$ At large separations, $h \gg R$, the hydrodynamic force on a sphere turns to the Stokes formula, but at small distances, $h \ll R$, the drag force is inversely proportional to the gap, $F_2/F_{St} \to 9 R/(8 h)$. A consequence of this lubrication effect is that the sphere would never touch the wall in a finite time. The flow in the vicinity of a rough surface should deviate from these predictions. A possible assumption is that the boundary condition at the plane $x_2=r$ should be written as a slip condition [@bonaccurso.e:2003]. To investigate this scenario we suggest to present a force as a product of Eq. \[firstorder\] and a correction for slip $$\label{firstorder_slip} \frac{F_3}{F_{St}}\sim \left(1+\frac{9}{8}\frac{R}{h}\right) f^{\ast},$$ where this correction, $f^{\ast}$, is taken to be equal as predicted for a lubrication force between a no-slip surface and a surface with partial slip [@vinogradova:95]. $$\label{model2} f^{\ast} = \frac{1}{4} \left( 1 + \frac{3 h}{2 b}\left[ \left( 1 + \frac{h}{4 b} \right) \ln \left( 1 + \frac{4 b}{h} \right) - 1 \right]\right).\nonumber$$ Another assumption would be that the rough surface is hydrodynamically equivalent to a smooth one located somewhere between the top and bottom of rugosities (at $x_2=r_{\rm eff}=r-s$). As found in [@vinogradova.oi:2006]), the force can be represented as $$\label{approximation} \frac{F_4}{F_{St}}\sim 1+\frac{9}{8}\frac{R}{h+s}.$$ At small $h$ expressions \[firstorder\_slip\] and \[approximation\] give different asymptotic behavior of a drag force, $F_3/F_{St} \to 9R/(32 h)$, and $F_4/F_{St} \to 9 R/(8 s)$. While the second scenario allows a sphere to touch a plane, in the first model this is impossible since the drag force diverges (but differs from the standard lubrication asymptotics by a factor of 4). Thus, a drainage study allows to distinguish between these two models of hydrodynamic flow past rough surfaces. [**Simulation method.–**]{} We apply the lattice Boltzmann (LB) method to simulate the flow field between a smooth sphere approaching a rough plane [@bib:higuera-succi-benzi; @2002RSPTA.360..437D; @ladd01]. The method allows precise measurements of the force acting on the sphere and to explore the very large range of parameters. Besides that, in our simulations we can consider a “clean” situation of a hydrodynamic force and avoid effects of surface forces which significantly complicate the analysis of SFA/AFM data. Since the method is well established, we only shortly describe it here. By using a discretized and linearized version of Boltzmann’s equation $$n_i({\bf x}+{\bf c}_i,t+1) - n_i({\bf x},t) = \sum_{j}\Lambda_{ij}(n^{\rm eq}_j-n_j({\bf x},t)), \label{eq:LBE}$$ the LB approach allows to fully resolve the hydrodynamics [@bib:succi-01]. Positions $\bf{x}$ are discretized on a 3D lattice with $19$ discrete velocities ${\bf c}_i$ pointing to neighboring sites. Each ${\bf c}_i$ relates to a single particle distribution function $n_i({\bf x},t)$ which is advected to neighboring sites at every time step. Then, $n_i({\bf x},t)$ is relaxed towards a local equilibrium $n_i^{\rm eq}(\rho,{\bf j})$ with a rate given by the matrix elements $\Lambda_{ij}$. Mass $\rho$ and momentum ${\bf j}$ as given by moments of $n_i({\bf x},t)$ are conserved. We use the natural units of the system, i.e. the lattice constant $\delta{\rm x}$ for the length and the time step $\delta{\rm t}$ for time. Massive particles are described by a continuously moving boundary which is discretized on the lattice. Momentum from the particle to the fluid is transfered such that the fluid velocity at the boundary equals the particle’s surface velocity. Since the momentum transferred from the fluid to the particle is known, the hydrodynamic force can be recorded. If not stated otherwise, the $256^3\delta{\rm x^3}$ system contains a sphere with radius $R=16\delta{\rm x}$ which is moved in $y$ direction at constant velocity $v=10^{-3}\delta{\rm x}/\delta{\rm t}$. The fluid density is kept constant and the kinematic viscosity is $\mu/\rho=0.1$ resulting in a Reynolds number $Re=0.16$. No-slip surfaces are described by mid-grid bounce back boundaries and a slip boundary is implemented by a repulsive mean-field force acting between fluid and surface [@bib:jens-kunert-herrmann:2005; @jari-jens-08]. We carefully checked the influence of system size, radius, and separation to insure that finite size and resolution effects are negligible [@kunert-harting-09]. Also, by testing different resolutions we assured that a lateral width of roughness elements of $1\delta{\rm x}$ is sufficient. [**Results and discussion.–**]{} We test our method by measuring the hydrodynamic interaction between smooth surfaces. Fig. \[fig:forceflat\] shows the normalized hydrodynamic force for two simulation sets. In the first one, a sphere of $R=8\delta{\rm x}$ is driven with $v=10^{-4}\delta{\rm x}/\delta{\rm t}$. In the second run the sphere is twice as large, $R=16\delta{\rm x}$, and the driving velocity is an order of magnitude larger, $v=10^{-3}\delta {\rm x}/\delta{\rm t}$. Fig. \[fig:forceflat\] includes the exact theoretical curve, Eq. \[eq:maudea\]. The fit is excellent for all separations, indicating that large shear rates do not induce any slip, a conclusion which does not support recent experimental data [@lauga2007]. Note that the first-order approximation, Eq. \[firstorder\], practically coincides with the exact solution. These simulations demonstrate that finite size effects and resolution effects can be well controlled: for $h<2R$ a $256^3\delta{\rm x^3}$ system is found to be sufficient to avoid artefacts at large separations $h$ [@kunert-harting-09]. Separations $<1\delta{\rm x}$ are excluded from the analysis since the finite resolution leads to larger deviations. Also included in Fig. \[fig:forceflat\] is a normalized force measured near a rough wall ($\phi=4\%$, $r=10\delta{\rm x}$), which at small distances is much smaller than predicted by Eq. \[eq:maudea\]. This is qualitatively consistent with the AFM observations [@bonaccurso.e:2003; @vinogradova:03], but in contrast to the SFA data [@granick:02], which likely reflects a different way of a definition of zero separation in the SFA (at the bottom of asperities). To examine the significance of roughness more closely, the force curves from Fig. \[fig:forceflat\] are reproduced in Fig. \[fig:analysis1\] in different coordinates. Figs. \[fig:analysis1\]a and b are intended to indicate that both near field and far field theoretical asymptotics for smooth surfaces are well reproduced in simulations. Figs. \[fig:analysis1\]c and d show that simulation data for a rough surface ($\phi=4\%$, $r=10\delta{\rm x}$) show deviations from the behavior predicted by Eq. \[eq:maudea\]. A possible explanation for this discrepancy is that we invoke slippage at the wall, as modeled by Eq. \[firstorder\_slip\]. This is illustrated in Figs. \[fig:analysis1\]c and d, where the simulation data are compared with another theoretical calculation in which a constant slip length of $b=2.55\delta{\rm x}$, obtained from the best possible fit of the force curve, is incorporated in the model. This has the effect of decreasing the force, and it provides a reasonable fit to the data down to $h/R \sim 3$, but at smaller gap the simple model of slip fails to describe simulation data, by predicting a larger force and its different asymptotic behavior. This suggests, that it can only be considered as a first approximation, valid at large distances from the wall. This conclusion is consistent with early results obtained for a far field situation [@sarkar.k:1996; @lecoq.n:2004; @kunert-harting-07], but does not support recent AFM data [@bonaccurso.e:2003]. However, as shown by the simulation data, Eq. \[firstorder\_slip\] is well applicable in the case of a slippery wall. An alternative explanation for the smaller force compared to the theory for smooth surfaces, Eq. \[eq:maudea\], can be obtained if we assume that the location of an equivalent effective wall, where no-slip boundary conditions are applied, should be shifted, as modeled by Eq. \[approximation\]. A corresponding theoretical calculation of the drag force is shown in Figs. \[fig:analysis1\]c and d. This estimate requires knowledge of the effective wall position $r_{\rm eff}$. The value $r_{\rm eff}=7.86\delta{\rm x}$ was obtained from the fit of the measured force curve and is enough to give a good match to the data at very small distances, which confirms the conclusions of a recent experiment [@vinogradova.oi:2006]. ![\[fig:percent\] Effective height $r_{\rm eff}$ normalized by the maximum height $r$ as a function of a density of roughness elements, $\phi$, for $r=10\delta{\rm x}$ and $r=20\delta{\rm x}$ plotted in different scales. ](fig4.eps "fig:"){width="17.00000%"} ![\[fig:percent\] Effective height $r_{\rm eff}$ normalized by the maximum height $r$ as a function of a density of roughness elements, $\phi$, for $r=10\delta{\rm x}$ and $r=20\delta{\rm x}$ plotted in different scales. ](fig5.eps "fig:"){width="17.00000%"} By performing similar fits for a variety of drainage runs with different $\phi$ and for surfaces with different height of roughness elements ($r=10\delta{\rm x}$ and $r=20\delta{\rm x}$) as well as its different lateral width ($\delta{\rm x}$ and $2\delta{\rm x}$) we find that the same conclusion is valid for all situations, but $r_{\rm eff}/r$ is itself a function of $\phi$ (being surprisingly insensitive to the value of $r$). In Fig. \[fig:percent\] we examine this in more detail. The simulation data show that $r_{\rm eff}$ required to fit each run increases from 0 to $r$ very rapidly, so that at $\phi=20\%$ it is already above $0.9 r$, and at $\phi =50\%$ it is almost equal to $r$. This is illustrated by including the data obtained for a larger density of roughness elements ($\phi=50\%$, $r=10\delta{\rm x}$) in Fig. \[fig:analysis1\]c, that do not show a discernible deviation from the theoretical predictions for smooth surfaces. Thus, a small number of roughness elements has enormous influence on film drainage, confirming earlier theoretical ideas [@fan.th:2005]. [**Conclusion.–**]{} We have presented lattice Boltzmann simulations describing the drainage of a liquid confined between a smooth sphere and a randomly rough plate. The measured force is smaller than predicted for two smooth surfaces if the standard no-slip boundary conditions are used in the calculation. What our results show, however, is that at small separations the force is even weaker and shows different asymptotics than expected if we invoke slippage at the smooth fluid-solid interfaces. To explain this we use the model of a no-slip wall, located at an intermediate position (controlled by the density of roughness elements) between top and bottom of asperities. Calculations based on this model provide an excellent description of the simulation data. Besides this, by proving a correctness of this simple model to describe flow past a randomly rough surface, we have suggested a validity of a number of simple formulas for a hydrodynamic drag force. Although formally they can only be considered as first-order approximations, their accuracy is confirmed by simulation. Our results open the possibility of solving quantitatively many fundamental hydrodynamic problems involving randomly-rough interfaces, including contact angle dynamics, coagulation and more. We acknowledge A.J.C. Ladd for his hospitality (C. Kunert) and access to his simulation code, the DFG for financial support (grants Vi 243/1-3 and Ha 4382/2-1), and SSC Karlsruhe for computing time. [10]{} O. I. Vinogradova and G. E. Yakubov. , 19:1227, 2003. C. Cottin-Bizonne, B. Cross, A. Steinberger, and E. Charlaix. , 94:056102, 2005. O. I. Vinogradova, K. Koynov, A. Best, and F. Feuillebois. , 102:118302, 2009. O. I. Vinogradova. , 56:31 – 60, 1999. J. L. Barrat and L. Bocquet. , 82:4671 – 4674, 1999. D. Andrienko, B. Dünweg, and O. I. Vinogradova. , 119:13106, 2003. J. Harting, C. Kunert, and H. Herrmann. , 75:328, 2006. E. Lauga, M. P. Brenner, and H. A. Stone. In C. Tropea, A. Yarin, and J. F. Foss, editors, [*Handbook of Experimental Fluid Dynamics*]{}, chapter 19, pp 1219–1240. Springer, NY, 2007. P. Joseph, C. Cottin-Bizonne, J. M. Benoi, C. Ybert, C. Journet, P. Tabeling, and L. Bocquet. , 97:156104, 2006. J. [Ou]{} and J. P. [Rothstein]{}. , 17:103606, October 2005. F. Feuillebois, M. Z. Bazant, and O. I. Vinogradova. , 102:026001, 2009. J. Hyväluoma and J. Harting. , 100:246001, 2008. M. Sbragaglia, R. Benzi, L. Biferale, S. Succi, and F. Toschi. , 97:204503, 2006. H. Kusumaatmaja, M. L. Blow, A. Dupuis, and J. M.Yeomans. , 91:36003, 2008. E. Bonaccurso, H.-J. Butt, and V. S. J. Craig. , 90:144501, 2003. S. Granick, Y. Zhu, and H. Lee. , 2:221 – 227, 2003. Y. X. Zhu and S. Granick. , 88:106102, 2002. O. I. Vinogradova and G. E. Yakubov. , 73:045302(R), 2006. K. Sarkar and A. Prosperetti. , 316:223, 1996. N. Lecoq, R. Anthore, B. Cichocki, P. Szymczak, and F. Feuillebois. , 513:247, 2004. C. Kunert and J. Harting. , 99:176001, 2007. H. Brenner. , 16:242–251, 1961. A. D. Maude. , 12:293–295, 1961. O. I. Vinogradova. , 11:2213 – 2220, 1995. A. J. C. Ladd and R. Verberg. , 104:1191, 2001. D. d’Humi[è]{}res, I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo. , 360:437, 2002. F. J. Higuera, S. Succi, and R. Benzi. , 9:345, 1989. S. Succi. . Oxford University Press, 2001. C. Kunert and J. Harting. , 2009. T. H. Fan and O. I. Vinogradova. , 72:066306, 2005.

ArXiv

--- author: - 'Dietmar Klemm$^{a,b}$' - and Andrea Maiorana$^a$ title: Fluid dynamics on ultrastatic spacetimes and dual black holes --- Introduction ============ The AdS/CFT correspondence has provided us with a powerful tool to get insight into the dynamics of certain field theories at strong coupling by studying classical gravity solutions. In the long wavelength limit, where the mean free path is much smaller then any other scale, one expects that these interacting field theories admit an effective hydrodynamical description. In fact, it was shown in [@Bhattacharyya:2008jc][^1] that the five-dimensional Einstein equations with negative cosmological constant reduce to the Navier-Stokes equations on the conformal boundary of AdS$_5$. The analysis of [@Bhattacharyya:2008jc] is perturbative in a boundary derivative expansion, in which the zeroth order terms describe a conformal perfect fluid. The coefficient of the first subleading term yields the shear viscosity $\eta$ and confirms the famous result $\eta/s=1/(4\pi)$ by Policastro, Son and Starinets [@Policastro:2001yc], which was obtained by different methods. Subsequently, the correspondence between AdS gravity and fluid dynamics (cf. [@Rangamani:2009xk] for a review) was extended in various directions, for instance to include forcing terms coming from a dilaton [@Bhattacharyya:2008ji] or from electromagnetic fields (magnetohydrodynamics) [@Hansen:2008tq; @Caldarelli:2008ze]. The gravitational dual of non-relativistic incompressible fluid flows was obtained in [@Bhattacharyya:2008kq]. In addition to providing new insights into the dynamics of gravity, the map between hydrodynamics and AdS gravity has contributed to a better understanding of various issues in fluid dynamics. One such example is the role of quantum anomalies in hydrodynamical transport [@Son:2009tf]. Moreover, it has revealed beautiful and unexpected relationships between apparently very different areas of physics, for instance it was argued in [@Caldarelli:2008mv] that the Rayleigh-Plateau instability in a fluid tube is the holographic dual of the Gregory-Laflamme instability of a black string[^2]. The hope is that eventually the fluid/gravity correspondence may shed light on fundamental problems in hydrodynamics like turbulence. Another possible application is the quark-gluon plasma created in heavy ion collisions, where perturbative QCD does not work, and lattice QCD struggles with dynamic situations, cf. e.g. [@Myers:2008fv]. We will come back to this point in section \[fin-rem\]. Here we will use fluid dynamics to make predictions on which types of black holes can exist in four-dimensional Einstein-Maxwell-AdS gravity. In particular, we shall classify all possible stationary equilibrium flows on ultrastatic manifolds with constant curvature spatial sections, and then use these results to predict (and explicitely construct) new black hole solutions. The remainder of this paper is organized as follows: In the next section, we briefly review the basics of conformal hydrodynamics. In section \[stat flow ultrastatic st\] we consider shearless and incompressible stationary fluids on ultrastatic manifolds, and show that the classification of such flows is equivalent to classifying the isometries of the spatial sections $(\Sigma,\bar g)$[^3]. This is then applied to the three-dimensional case with constant curvature spatial sections, i.e., to fluid dynamics on $\mathbb{R}\times\text{S}^2$, $\mathbb{R}\times\text{H}^2$ and Minkowski space $\mathbb{R}\times\text{E}^2$. It is shown that, up to isometries, the flow on the 2-sphere is unique, while there are three non-conjugate Killing fields on the hyperbolic plane and two on the Euclidean plane. In almost all cases, it turns out that the fluid can cover only a part of the manifold, since there exist regions where the fluid velocity exceeds the speed of light. This property is quite obvious for rigid rotations on $\text{H}^2$ or $\text{E}^2$: Here there is a certain radius where the velocity reaches the speed of light, and thus the fluid can cover only the region within this radius. Due to the diverging gamma factor at the boundary of the fluid, the global thermodynamic variables like energy, angular momentum, entropy and electric charge are infinite in these cases. Nevertheless, we show that a local form of the first law of thermodynamics still holds. At the end of section \[stat flow ultrastatic st\], we transform the rigidly rotating conformal fluid on the open Einstein static universe to $\mathbb{R}\times\text{S}^2$ and to Minkowski space (this is possible since both are conformal to $\mathbb{R}\times\text{H}^2$), and shew that this yields contracting or expanding vortex configurations. In section \[dual-AdS-BH\], the gravity duals of the hydrodynamic flows considered in \[stat flow ultrastatic st\] are identified. Although they all lie within the Carter-Plebański class [@Carter:1968ks; @Plebanski:1975], many of them have never been studied in the literature before, and represent thus in principle new black hole solutions in AdS$_4$. Quite remarkably, it turns out that the boundary of these black holes are conformal to exactly that part of $\mathbb{R}\times\text{S}^2$, $\mathbb{R}\times\text{H}^2$ or Minkowski space in which the fluid velocity does not exceed the speed of light. Thus, the correspondence between AdS gravity and hydrodynamics automatically eliminates the unphysical region. We conclude in section \[fin-rem\] with some final remarks. In appendix \[app-CP\], our results are extended to establish a precise mapping between possible flows on ultrastatic spacetimes (with constant curvature spatial sections) and the parameter space of the Carter-Plebański solution to Einstein-Maxwell-AdS gravity. The proofs of some propositions are relegated to appendix \[app-proof\]. Note that a related, but slightly different approach was adopted in [@Mukhopadhyay:2013gja], where uncharged fluids in Papapetrou-Randers geometries were considered. In these flows, the fluid velocity coincides with the timelike Killing vector of the spacetime (hence the fluid is at rest in this frame), and the Cotton-York tensor has the form of a perfect fluid (so-called ‘perfect Cotton geometries’). We will see below that there is some overlap between the bulk geometries dual to such flows, constructed explicitely in [@Mukhopadhyay:2013gja], and the solutions obtained here. Throughout this paper we use calligraphic letters ${\cal T},{\cal V},{\cal S},\ldots$ to indicate local thermodynamic quantities, whereas $T,V,S,\ldots$ refer to the whole fluid configuration. $\mu$ and $\phi_{\text e}$ are local and global electric potentials respectively. Conformal hydrodynamics {#conf-hydro} ======================= Consider a charged fluid on a $d$-dimensional spacetime. The equations of hydrodynamics are simply the conservations laws for the stress tensor $T^{\mu\nu}$ and the charge current $J^{\mu}$, $$\nabla_\mu T^{\mu\nu}=0\,, \qquad \nabla_\mu J^{\mu} = 0\,.$$ Since fluid mechanics is an effective description at long distances, valid when the fluid variables vary on scales much larger than the mean free path, it is natural to expand the energy-momentum tensor, charge current and entropy current $J^{\mu}_S$ in powers of derivatives. At zeroth order in this expansion, one has the perfect fluid form [@Andersson:2006nr] $$T^{\mu\nu}_{\text{perf}} = (\rho+{\cal P})u^\mu u^\nu+{\cal P} g^{\mu\nu}\,, \qquad J^\mu_{\text{perf}} = \rho_{\text e}u^\mu\,, \qquad {J^\mu_S}_{\text{perf}} = s u^\mu\,,$$ where $u$ denotes the velocity profile, and $\rho$, ${\cal P}$, $\rho_{\text e}$ and $s$ are the energy density, pressure, charge density and entropy density respectively, measured in the local rest frame of the fluid. At first subleading order, one obtains the dissipative contributions [@Andersson:2006nr] $$\label{T-diss} T^{\mu\nu}_{\text{diss}}=-\zeta\vartheta P^{\mu\nu}-2\eta\sigma^{\mu\nu}+(q^\mu u^\nu+ q^\nu u^\mu)\,, \qquad J^\mu_{\text{diss}} = q^\mu_{\text e}\,, \qquad {J^\mu_S}_{\text{diss}} = \frac{q^\mu - \mu q^\mu_{\text e}}{\cal T}\,,$$ where $$P^{\mu\nu}=g^{\mu\nu}+u^\mu u^\nu\,,$$ and $$\label{a-theta-sigma} a^\mu=u^\nu\nabla_\nu u^\mu\,, \qquad \vartheta=\nabla_\mu u^\mu\,, \qquad \sigma^{\mu\nu}=\frac{1}{2}(P^{\mu\rho}\nabla_\rho u^\nu+P^{\nu\rho}\nabla_\rho u^\mu)-\frac{1}{d-1}\vartheta P^{\mu\nu}\,,$$ $$q^\mu=-\kappa P^{\mu\nu} (\partial_\nu + a_\nu){\cal T}\,, \qquad q_{\text e}^\mu = -D P^{\mu\nu}\partial_\nu\frac{\mu}{\cal T} \label{subl-currents}$$ denote the acceleration, expansion, shear tensor, heat flux and diffusion current respectively. Moreover, ${\cal T}$ and $\mu$ are the local temperature and electric potential, $\zeta$ is the bulk viscosity, $\eta$ the shear viscosity, $\kappa$ the thermal conductivity and $D$ the diffusion coefficient. Note that the equations are the relativistic generalizations of Fourier’s law of heat condution and Fick’s first law. At first order in the derivative expansion, the entropy current is no longer conserved, but obeys [@Andersson:2006nr] $$\label{div-JS} {\cal T}\nabla_\mu J_S^\mu=\frac{q_\mu q^\mu}{\kappa {\cal T}} + \frac{\cal T}{D} q_{\text e}^\mu q_{\text{e}\mu}+\zeta\vartheta^2+2\eta\sigma_{\mu\nu}\sigma^{\mu\nu}\,.$$ If the coefficients satisfy certain non-negativity conditions, this implies $\nabla_\mu J_S^\mu\geq 0$, and thus entropy is always non-decreasing. In equilibrium, $ J^\mu_S$ must be conserved, which is the case if and only if $q^\mu$, $q^\mu_{\text e}$, $\vartheta$ and $\sigma^{\mu\nu}$ all vanish[^4]. Since we consider fluids on curved manifolds, we could add to $T^{\mu\nu}$ also terms constructed from the curvature tensors. In fact, at second order in a derivative expansion, there is a term proportional to the Weyl tensor of the boundary, cf. equation (2.10) of [@Bhattacharyya:2008mz]. However, in all explicit examples considered here, the boundary is three-dimensional, and thus its Weyl tensor vanishes. Note that in three dimensions there is a possible third order contribution from the Cotton tensor [@Mukhopadhyay:2013gja], but since our boundary geometries are conformally flat for vanishing NUT-parameter, this contribution vanishes as well. In what follows, we specialize to conformal fluids[^5]. Upon a Weyl rescaling $\tilde{g}_{\mu\nu}=\Omega^2 g_{\mu\nu}$, the energy-momentum tensor must transform as $\tilde T^{\mu\nu}=\Omega^w T^{\mu\nu}$ for some weight $w$, and hence $$\tilde{\nabla}_\mu\tilde{T}^{\mu\nu}=\Omega^w\nabla_\mu T^{\mu\nu} + \Omega^{w-1}\left((w+d+2) T^{\mu\nu}\partial_\mu\Omega - T^\mu{}_\mu\,\partial^\nu\Omega\right)\,,$$ from which we learn that $w=-(d+2)$ and $T^\mu{}_\mu=0$ in order for $\tilde{T}$ to be conserved. The tracelessness of $T$ implies the equation of state $\rho=(d-1){\cal P}$ and requires the bulk viscosity $\zeta$ to be zero. The transformation laws for the fluid variables are $$\tilde{u}=\Omega^{-1}u\,, \quad \tilde{\rho}=\Omega^{-d}\rho\,, \quad \tilde{\cal P}= \Omega^{-d}{\cal P}\,, \quad \tilde\rho_{\text e} = \Omega^{-(d-1)}\rho_{\text e}\,, \quad \tilde s = \Omega^{-(d-1)}s\,, \quad \tilde{\cal T} = \Omega^{-1}{\cal T}\,.$$ Furthermore, the charge- and entropy current transform as $$\tilde{J}^\mu=\Omega^{-d}J^\mu\,, \qquad \tilde{J}_S^\mu=\Omega^{-d}J_S^\mu\,.$$ Note that, if the charged fluid moves in an external electromagnetic field $F_{\mu\nu}$, its stress tensor is no more conserved, and the equations of motion become $$\nabla_\mu T^{\mu\nu}=F^\nu{}_\mu J^\mu\,, \label{eq:MHD}$$ where the rhs represents the Lorentz force density. This scenario was studied in full generality in [@Caldarelli:2008ze]. According to the AdS/CFT dictionary, such an external field is related to the magnetic charge of the dual black hole. Quite surprisingly, it turns out that for all the magnetically charged black holes considered here, there is no net Lorentz force acting on the dual fluid, since the electric and magnetic forces exactly cancel. One has thus $F^\nu{}_\mu J^\mu=0$, hence $T^{\mu\nu}$ is conserved. At the end of this section, we briefly review the constraints imposed on the thermodynamics by conformal invariance. First of all, define the grand-canonical potential $$\Phi = {\cal E} - {\cal T}{\cal S} - \mu{\cal Q}_{\text{e}}\,,$$ which satisfies the first law $$\mathrm{d}\Phi = -{\cal S}\mathrm{d}{\cal T} - {\cal P}\mathrm{d}{\cal V} - {\cal Q}_{\text e}\mathrm{d}\mu\,. \label{first-law}$$ Conformal invariance and extensivity imply that $\Phi$ must have the form [@Bhattacharyya:2007vs] $$\Phi = -{\cal V}{\cal T}^d h(\psi)\,,$$ for some function $h(\psi)$, where $\psi:=\mu/{\cal T}$. The remaining thermodynamic quantities are then easily obtained using , $$\label{remaining-h} {\cal P} = \frac{\rho}{d-1} = {\cal T}^d h(\psi)\,, \quad \rho_{\text e} = \frac{{\cal Q}_{\text e}}{\cal V} = {\cal T}^{d-1} h'(\psi)\,, \quad s = \frac{\cal S}{\cal V} = {\cal T}^{d-1}(dh(\psi) - \psi h'(\psi))\,.$$ Equilibrium flows in ultrastatic spacetimes {#stat flow ultrastatic st} =========================================== We will now focus on conformal fluids in ultrastatic spacetimes, and explain how the equilibrium flows can be classified using the isometries of the spatial sections. A $d$-dimensional spacetime $(M,g)$ is said to be *ultrastatic* if there are a timelike Killing field $\xi$ such that $\xi_\mu\xi^\mu=-1$ and a hypersurface $\Sigma$ orthogonal to $\xi$. In such a spacetime one can always choose a coordinate system such that $$g=-\mathrm{d}t^2+\bar{g}_{ij}\mathrm{d}x^i \mathrm{d}x^j\,,$$ where $\bar{g}$ is the induced metric on $\Sigma$. The velocity field $u$ for a flow on $M$ can be written as $$\label{u=gamma(1,v)} u^\mu=\gamma(1,v^i)\,,$$ and the constraint $u_\mu u^\mu=-1$ implies that $\gamma^2=1/(1-v^2)$, where $v^2:=\bar{g}_{ij}v^i v^j$. We assume that the fluid is stationary in the frame $(t,\vec{x})$, that is $\partial_t u^\mu=0$. Equ.  defines then a vector field $v$ on $\Sigma$. Note that the property of ultrastaticity is not conserved under general Weyl rescalings. Thus when we say that a conformal spacetime $(M,[g])$ is ultrastatic, we mean that it has *some* metric representative $g$ which is ultrastatic. As was explained in section \[conf-hydro\], a fluid is in equilibrium when the entropy current is conserved, which implies that the flow must be shearless and incompressible. The classification of such flows becomes quite easy if we use the following proposition, proven in appendix \[app-proof\]. \[prop-killing\] $\sigma^{\mu\nu}=0$ and $\vartheta=0$ $\Leftrightarrow$ $v$ is a Killing field for $(\Sigma,\bar{g})$. The classification of shearless and incompressible flows on ultrastatic manifolds is thus equivalent to classifying the isometries of the spatial sections $(\Sigma,\bar g)$. In equilibrium, the dissipative contribution to $T_{\mu\nu}$ in vanishes, and the stress tensor is just $$\label{d-dim conf fl stress tens} T^{\mu\nu} = {\cal P}(d\, u^\mu u^\nu+g^{\mu\nu})\,,$$ where we used the equation of state $\rho=(d-1){\cal P}$ of conformal fluids. The solution of the Navier-Stokes equations becomes then particularly simple: \[prop-NS\] When $\sigma^{\mu\nu}=\vartheta=0$ and ${\cal P},u^\mu$ are independent of $t$, the stress tensor satisfies $\nabla_\mu T^{\mu\nu}=0$ if and only if $$\label{sol eq NS} {\cal P}={\cal P}_0\gamma^d$$ for some constant ${\cal P}_0$. Consider now the heat flux $q^\mu$ and the diffusion current $q_{\text e}^\mu$, given in . \[prop-heat-flux\] If the flow is stationary, incompressible and shearless, then $q^\mu=0$ implies ${\cal T} = \tau\gamma$ for some constant $\tau$. \[prop-diff-curr\] For a stationary flow, $q_{\text e}^\mu=0$ implies $\mu=\psi{\cal T}$, where $\psi$ is a constant. The proofs of propositions \[prop-NS\]-\[prop-diff-curr\] are again given in appendix \[app-proof\]. With \[prop-NS\], \[prop-heat-flux\] and \[prop-diff-curr\], becomes $${\cal P}_0 = \tau^d h(\psi)\,, \qquad \rho_{\text e} = \tau^{d-1}\gamma^{d-1} h'(\psi)\,, \qquad s = \tau^{d-1}\gamma^{d-1}(dh(\psi) - \psi h'(\psi))\,. \label{P_0-rho_e-s}$$ Finally, the second of these equations implies that the charge current $J^\mu=\rho_{\text e}u^\mu$ is conserved. At this point, some comments are in order: We have shown that we can construct all stationary shearless and incompressible fluid configurations on the spacetime $M$, if we know the Killing fields on the spatial sections $(\Sigma,\bar g)$. A Killing field $v$ is defined on the whole manifold $\Sigma$, but it gives a physically meaningful flow only on the subset $U\subset\Sigma$ in which $v^2<1$. Notice that $v^2$ is constant along the integral curves of $v$, and therefore the flow does not cross the boundary of $U$, where the fluid moves at the speed of light. Moreover, we do not need to consider the flow arising from each Killing field $v$, since different flows can be isometric. Suppose in fact that we have two Killing fields $v,\tilde{v}$ which are related by an isometry $\Psi$, i.e., $\tilde{v}\circ\Psi=\mathrm{d}\Psi\circ v$. In terms of the 1-parameter groups of isometries $\Phi^{(v)},\Phi^{(\tilde{v})}$ that these fields generate, this condition reads $$\label{Psi-related 1-par groups} \Phi^{(\tilde{v})}_{\lambda}\circ\Psi=\Psi\circ\Phi^{(v)}_{\lambda}\,.$$ When this holds, the flows arising from $v$ and $\tilde{v}$ are physically equivalent. Now, to the Killing fields $v,\tilde{v}$ correspond two elements $A,\tilde{A}$ in the Lie algebra $i(\Sigma)$ of the isometry group $I(\Sigma)$, namely the generators of the 1-parameter subgroups $\lambda\mapsto\Phi^{(v)}_{\lambda}$ and $\lambda\mapsto\Phi^{(\tilde{v})}_{\lambda}$, for which equ.  becomes $$\label{Psi-related Lie algebra elements} \tilde{A}=\textup{Ad}_\Psi(A)\,,$$ where $\textup{Ad}$ is the adjoint representation of $I(\Sigma)$ on $\mathfrak{i}(\Sigma)$. This reduces the problem of finding inequivalent flows to the study of the properties of the Lie algebra $\mathfrak{i}(\Sigma)$ under the adjoint representation. Stationary conformal fluid on the 2-sphere {#stat conf fluid sph} ------------------------------------------ Let us first study the case (partially considered in [@Bhattacharyya:2007vs; @Caldarelli:2008ze]) in which the conformal fluid lives on the ultrastatic spacetime $\mathbb{R}\times\text{S}^2$, with metric given by $$\label{metric sph spacetime} g=-\mathrm{d}t^2+\ell^2(\mathrm{d}\theta^2+\sin^2\theta\mathrm{d}\varphi^2)\,.$$ As was explained above, the 3-velocity of the fluid is $u^\mu=\gamma(1,v^i)$, where $v$ is a Killing field of $\text{S}^2$. By a rotation, $v$ can be brought to a multiple of any other Killing field, say $\partial_\varphi$. Thus we can take $v=\omega\partial_\varphi$, with $\omega\in\mathbb{R}$, without loss of generality. Hence $$u=\gamma(\partial_t+\omega\partial_\varphi)\,,$$ where $\gamma=(1-\omega^2\ell^2\sin^2\theta)^{-1/2}$. This means that the motion of the fluid in equilibrium is just a rigid rotation on $\text{S}^2$. The physical constraint $v^2<1$ limits the fluid to polar caps at $|\omega|\ell\sin\theta<1$. Thus, if we restrict $|\omega|\ell<1$, the physical region $U$ is the whole sphere. The stress tensor of the fluid is $T^{\mu\nu}=(\rho+{\cal P})u^\mu u^\nu+{\cal P} g^{\mu\nu}$ (dissipative terms vanish because of equilibrium), where $\rho=2{\cal P}$. This gives $$\label{fluid on sphere stress tensor} T^{\mu\nu}={\cal P}\begin{pmatrix} 3\gamma^2-1 & 0 & 3\gamma^2\omega\\ 0 & \frac{1}{\ell^2} & 0\\ 3\gamma^2\omega & 0 & \frac{3\gamma^2-2}{\ell^2\sin^2\theta} \end{pmatrix}\,,$$ which is conserved if $${\cal P} = {\cal P}_0\gamma^3\,, \label{P_0gamma^3}$$ where we used . The heat flux $q^\mu$ and diffusion current $q_{\text e}^\mu$ vanish by virtue of propositions \[prop-heat-flux\] and \[prop-diff-curr\]. We now want to compute the conserved charges associated to the stress tensor $T^{\mu\nu}$ and the currents $J^\mu$, $J^\mu_S$. These are well-defined only for $|\omega|\ell<1$, since otherwise the physical region $U$ has a boundary where the Lorentz factor $\gamma$ diverges. We consider the foliation of spatial surfaces $\Sigma_t\simeq\text{S}^2$ of constant $t$, with induced metric $\bar g$. In the case $|\omega|\ell<1$ the electric charge and entropy are given respectively by $$Q_{\text e} = \int_{\Sigma_t} d^2x\sqrt{\bar g}J^t = \frac{4\pi\ell^2\tau^2 h'(\psi)}{1 - \omega^2\ell^2}\,, \quad S = \int_{\Sigma_t} d^2x\sqrt{\bar g}J^t_S = \frac{4\pi\ell^2\tau^2(3h(\psi) - \psi h'(\psi))} {1 - \omega^2\ell^2}\,, \label{Q-S}$$ while the total energy $E$ and angular momentum $L$ read $$E = -\int_{\Sigma_t} d^2x\sqrt{\bar g}{T^t}_\mu\xi^\mu = \frac{8\pi\ell^2\tau^3 h(\psi)}{(1-\omega^2 \ell^2)^2}\,, \label{E}$$ $$L = -\int_{\Sigma_t} d^2x\sqrt{\bar g}{T^t}_\mu\chi^\mu = \frac{8\pi\ell^4\tau^3\omega h(\psi)} {(1-\omega^2\ell^2)^2}\,, \label{L}$$ where we used the Killing vectors $\xi=\partial_t$ and $\chi=-\partial_{\varphi}$. The charges - were obtained for the first time in [@Bhattacharyya:2007vs]. The volume $V=4\pi\ell^2$ is fixed and not considered as a thermodynamical variable. It is straightforward to verify that $E$, $L$, $S$, $Q_{\text e}$, which are functions of the parameters $\omega,\tau,\psi$, satisfy the first law $$\mathrm{d}E=\tau\mathrm{d}S+\omega\mathrm{d}L+\tau\psi\mathrm{d}Q_{\text e}\,.$$ As a consequence, the intensive variables conjugate to $S,L,Q_{\text e}$ are respectively $$T = \left(\frac{\partial E}{\partial S}\right)_{L,Q_{\text e}} = \tau\,, \qquad \Omega = \left(\frac{\partial E}{\partial L}\right)_{S,Q_{\text e}} = \omega\,, \qquad \phi_{\text e} = \left(\frac{\partial E}{\partial Q_{\text e}}\right)_{S,L} = \tau\psi\,. \label{T-Omega-phi}$$ Finally, the grandcanonical potential $G=E-TS-\Omega L-\phi_{\text e}Q_{\text e}$ reads $$G = -\frac{4\pi\ell^2\tau^3 h(\psi)}{1-\omega^2\ell^2}\,,$$ where $\psi=\phi_{\text e}/T$. Stationary conformal fluid on a plane {#sec:rot-plane} ------------------------------------- We now consider a conformal fluid on three-dimensional Minkowski space $\mathbb{R}\times\text{E}^2$, with metric $$g=-\mathrm{d}t^2+\mathrm{d}x^2+\mathrm{d}y^2\,.$$ The Killing fields on the plane $\text{E}^2$ are linear combinations of $$\xi^{(R)} = -y\partial_x+x\partial_y\,, \qquad \xi^{(T_1)} = \partial_x\,, \qquad \xi^{(T_2)} = \partial_y\,.$$ By using the commutation relations $$[R,T_1] = T_2\,, \qquad [R,T_2] = -T_1\,, \qquad [T_1,T_2] = 0$$ of the Euclidean group $\text{ISO}(2)$, it is easy to shew that $$\label{Ad_T(S)} e^{a\hat{\bf m}\cdot{\bf T}}Re^{-a\hat{\bf m}\cdot{\bf T}} = R + a(m^2T_1 - m^1T_2)\,,$$ where $a$ is a constant, $\hat{\bf m}=(m^1,m^2)$ denotes a unit vector, and ${\bf T}=(T_1,T_2)$. If we choose $$a = \frac{\beta}{\omega}\,, \qquad m^1 = -\frac{\beta^2}{\beta}\,, \qquad m^2 = \frac{\beta^1}{\beta}\,, \qquad \beta := \sqrt{(\beta^1)^2 + (\beta^2)^2}\,,$$ implies that $\omega R + \beta^1T_1 + \beta^2T_2$ is in the same orbit as $\omega R$ under $\text{ISO}(2)$, as long as $\omega\neq 0$. For $\omega=0$ the spatial fluid velocity is $v=\beta^1\partial_x + \beta^2\partial_y$, i.e., one has a purely translating fluid on $\mathbb{R}\times\text{E}^2$, which is dual to a boosted Schwarzschild-AdS black hole with flat horizon. We shall thus assume $\omega\neq 0$ in what follows. In this case, as just explained, it is (up to isometries) sufficient to consider a fluid that rotates around the origin. If we introduce polar coordinates $r,\varphi$, the 3-velocity becomes $$u=\gamma(\partial_t+\omega\partial_\varphi)\,,$$ where $\gamma=(1-\omega^2 r^2)^{-1/2}$. Note that the flow is well-defined only for $r<1/\omega$. At $\omega r=1$ the fluid rotates at the speed of light. The stress tensor of this configuration is given by $$T^{\mu\nu} = {\cal P}\begin{pmatrix} 3\gamma^2-1 & 0 & 3\gamma^2\omega\\ 0 & 1 & 0\\ 3\gamma^2\omega & 0 & \frac{3\gamma^2-2}{r^2} \end{pmatrix}\,,$$ which is again conserved if holds. Stationary conformal fluid on hyperbolic space {#sec:fluid-H2} ---------------------------------------------- The last example that we consider is a conformal fluid in equilibrium on $\mathbb{R}\times\text{H}^2$, with metric given by $$\label{metric hyperb spacetime} g=-\mathrm{d}t^2+\ell^2(\mathrm{d}\theta^2+\sinh^2\theta\mathrm{d}\varphi^2)\,.$$ To begin with a simple scenario, one can just follow what we did for the 2-sphere in subsection \[stat conf fluid sph\], taking the fluid in rigid rotation on the hyperboloid. Most of the results reflect what we found for the spherical flow. However, there are also some differences: As we shall see, no matter how small the angular velocity is, there always exists a certain critical distance from the center of rotation where the fluid moves at the speed of light, and hence the physical region $U$ is always smaller than the whole hyperboloid $\text{H}^2$. As a consequence, one cannot analyze the global thermodynamic properties of the system, since the extensive variables diverge. Anyway, we will show that for this fluid configuration one can make a local thermodynamical analysis to find some results comparable with those of subsection \[stat conf fluid sph\]. While in the spherical case the rigidly rotating flux is the only solution in equilibrium, for the hyperbolic plane there are different, inequivalent solutions, since this space admits non-conjugate Killing fields. (The isometry group $\text{SL}(2,\bR)$ of $\text{H}^2$ has parabolic, hyperbolic and elliptic elements). We denote the generators of $\text{SL}(2,\bR)$ by $R,B_1,B_2$. These obey $$[R,B_1] = B_2\,, \qquad [R,B_2] = -B_1\,, \qquad [B_1,B_2] = -R\,,$$ and are represented on the Poincaré disk by $$\xi^{(R)} = i(z\partial_z - \bar z\partial_{\bar z})\,, \quad \xi^{(B_1)} = \frac12(1 - z^2)\partial_z + \frac12(1 - \bar z^2)\partial_{\bar z}\,, \quad \xi^{(B_2)} = \frac i2(1 + z^2)\partial_z - \frac i2(1 + \bar z^2)\partial_{\bar z}\,.$$ The complex coordinate $z$ is related to $\theta,\varphi$ by $z=e^{i\varphi}\tanh\frac{\theta}2$. One easily shows that $$e^{\alpha R}(\omega R + \beta B_1)e^{-\alpha R} = \omega R + \beta(B_1\cos\alpha + B_2\sin\alpha)\,,$$ and thus a general linear combination $\omega R + \beta^1 B_1 + \beta^2 B_2$ is conjugate to $\omega R + \beta B_1$, so we can drop $B_2$ without loss of generality. Moreover, one has $$e^{\chi B_2} R e^{-\chi B_2} = R\cosh\chi + B_1\sinh\chi\,. \label{Ad_B2(R)}$$ If $\omega^2>\beta^2$, we can put $\tanh\chi=\beta/\omega$, and implies that $\omega R+\beta B_1$ is conjugate to $\tilde\omega R$, where $\tilde\omega:=\omega\sqrt{1-\beta^2/\omega^2}$. This case corresponds to an elliptic element of $\text{SL}(2,\bR)$, and describes a fluid in rigid rotation on $\text{H}^2$. For $\omega^2<\beta^2$ (hyperbolic element), use $$e^{\chi B_2} B_1 e^{-\chi B_2} = R\sinh\chi + B_1\cosh\chi\,, \qquad \tanh\chi=\omega/\beta\,,$$ to show that $\omega R+\beta B_1$ is in the same orbit as $\tilde\beta B_1$, with $\tilde\beta:=\beta\sqrt{1-\omega^2/\beta^2}$. Finally, for $\omega^2=\beta^2$ (parabolic element), one can set $\omega=\beta$ without loss of generality, since the case $\omega=-\beta$ is related to this by the discrete isometry $J$ obeying $$JRJ^{-1} = -R\,, \qquad JB_1J^{-1} = B_1\,, \qquad JB_2J^{-1} = -B_2\,.$$ In the complex coordinates $z,\bar z$, the transformation $J$ acts as $z\to\bar z$. As representative in this last case we can thus take the Killing vector $\omega(\xi^{(R)} + \xi^{(B_1)})$. Notice that due to $$e^{\chi B_2}(R+B_1)e^{-\chi B_2} = e^{\chi}(R+B_1)\,,$$ the absolute value of $\omega$ can be set equal to $1/\ell$ without loss of generality[^6]. The integral curves of the fluid two-velocity $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ are visualized in figure \[integral-curves\]. For $\omega^2>\beta^2$ the stream lines are closed and the flow has one fixed point. For $\omega^2<\beta^2$ there are two fixed points lying on the boundary of the Poincaré disk (which does not belong to the manifold itself). If $\omega^2=\beta^2$, these fixed points coincide. Of course, the cases $(\omega,\beta)=(1,0.5)$ and $(0.2,0.4)$ are isometric to $(1,0)$ and $(0,0.5)$ respectively. ![Integral curves (stream lines) of the vector field $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ on the Poincaré disk, for different values of $\omega$ and $\beta$. The white area denotes the physical region, where the fluid velocity does not exceed the speed of light.[]{data-label="integral-curves"}](PDa.pdf "fig:") ![Integral curves (stream lines) of the vector field $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ on the Poincaré disk, for different values of $\omega$ and $\beta$. The white area denotes the physical region, where the fluid velocity does not exceed the speed of light.[]{data-label="integral-curves"}](PDb.pdf "fig:") ![Integral curves (stream lines) of the vector field $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ on the Poincaré disk, for different values of $\omega$ and $\beta$. The white area denotes the physical region, where the fluid velocity does not exceed the speed of light.[]{data-label="integral-curves"}](PDc.pdf "fig:") ![Integral curves (stream lines) of the vector field $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ on the Poincaré disk, for different values of $\omega$ and $\beta$. The white area denotes the physical region, where the fluid velocity does not exceed the speed of light.[]{data-label="integral-curves"}](PDd.pdf "fig:") ![Integral curves (stream lines) of the vector field $v=\omega\xi^{(R)} + \beta\xi^{(B_1)}$ on the Poincaré disk, for different values of $\omega$ and $\beta$. The white area denotes the physical region, where the fluid velocity does not exceed the speed of light.[]{data-label="integral-curves"}](PDe.pdf "fig:") In what follows, we shall analyze each of the three distinct cases separately. ### Rigid rotation {#fluid rig rot hyp} As was explained above, for $\omega^2>\beta^2$ one can take $\beta=0$ without loss of generality. The 3-velocity of the fluid is then given by $$\label{rotating flux on the hyp} u=\gamma(\partial_t+\omega\partial_\varphi)\,,$$ where $\omega\in\mathbb{R}$ and $\gamma=(1-\omega^2\ell^2\sinh^2\theta)^{-1/2}$. Note that the flow is well-defined only in the region $U=\{(\theta,\varphi)\,|\,|\omega|\ell\sinh\theta<1\}$. At the boundary of $U$, the fluid rotates at the speed of light. Since $v=\omega\partial_\varphi$ is a Killing field of $\text{H}^2$, this configuration is shearless and incompressible. The stress tensor is given by $$\label{fluid on hyperboloid stress tensor} T^{\mu\nu} = {\cal P}\begin{pmatrix} 3\gamma^2-1 & 0 & 3\gamma^2\omega\\ 0 & \frac{1}{\ell^2} & 0\\ 3\gamma^2\omega & 0 & \frac{3\gamma^2-2}{\ell^2\sinh^2\theta} \end{pmatrix}\,,$$ which is conserved once is satisfied. Moreover, the heat flux $q^\mu$ and diffusion current $q^\mu_{\text e}$ vanish by virtue of propositions \[prop-heat-flux\] and \[prop-diff-curr\]. Since the fluid velocity tends to the speed of light at the boundary of $U$, $\gamma$ diverges there and the total energy and angular momentum are infinite. Thus, unlike in the spherical case, we cannot define global thermodynamical variables here, and have to consider instead only their densities. These are 1. the energy density $\varepsilon=T_{tt}={\cal P}_0\gamma^3(3\gamma^2-1)$, 2. the angular momentum density $l=-T_{t\varphi}=3{\cal P}_0\ell^2\omega\gamma^5\sinh^2\theta$, 3. the entropy density $\sigma=J_S^t=\tau^2\gamma^3(3h(\psi)-\psi h'(\psi))=\gamma s$, 4. the charge density $\varrho_{\text e}=J^t=\tau^2\gamma^3h'(\psi)=\gamma\rho_{\text e}$. We remark that these densities are evaluated in the frame $(t,\theta,\varphi)$, in which the fluid is moving, while the densities $\rho,s,\rho_{\text e}$ are measured in the local rest frame of the fluid. Pointwise, $\varepsilon,l,\sigma$ and $\varrho_{\text e}$ are functions of the free parameters $\omega,\tau,\psi$. Calculating their differentials one finds a local form of the first law, $$\mathrm{d}\varepsilon = \tau\mathrm{d}\sigma + \omega\mathrm{d}l + \tau\psi\mathrm{d}\varrho_{\text e}\,, \label{1st-law-local}$$ which implies that the intensive variables conjugate to $\sigma,l$ and $\varrho_{\text e}$ are respectively given by $$\frac{\partial \varepsilon(\sigma,l,\varrho_{\text e})}{\partial \sigma} = \tau\,, \qquad \frac{\partial \varepsilon(\sigma,l,\varrho_{\text e})}{\partial l} = \omega\,, \qquad \frac{\partial \varepsilon(\sigma,l,\varrho_{\text e})}{\partial \varrho_{\text e}} = \tau\psi\,.$$ The local grandcanonical potential $\mathpzc{g}=\varepsilon-\tau\sigma-\omega l-\tau\psi\varrho_{\text e}$ reads $$\mathpzc{g} = -\tau^3\gamma^3 h(\psi) = -\frac{\tau^3 h(\psi)}{(1-\omega^2\ell^2\sinh^2 \theta)^{3/2}}\,.$$ is of course a consequence of local thermodynamical equilibrium. ### Purely translational flow {#sec:transl-flow} Now we consider the case $\omega^2<\beta^2$, in which one can take $\omega=0$ without loss of generality. This flow is visualized in the last figure of \[integral-curves\]. In this case it is convenient to use the coordinates $$X = \sinh\theta\cos\varphi\,, \qquad Y = \sinh\theta\sin\varphi\,,$$ in which the metric of the spacetime is given by $$\label{RxH2:XY} g=-\mathrm{d}t^2+\frac{\ell^2}{1+X^2+Y^2}\left((1+Y^2)\mathrm{d}X^2+(1+X^2)\mathrm{d}Y^2-2XY\mathrm{d}X\mathrm{d}Y\right)\,,$$ and the the fluid moves along the $X$ direction, $$v=\beta\sqrt{1+X^2+Y^2}\partial_X\,.$$ Since $v^2=\beta^2\ell^2(1+Y^2)$, the physical region $U$ is vertically narrowed by the condition $$Y^2<\frac{1}{\beta^2\ell^2}-1\,,$$ which also shows that the flow exists only for $\beta^2<\ell^{-2}$. The 3-velocity reads $$u=\gamma(\partial_t+\beta\sqrt{1+X^2+Y^2}\partial_X)\,,$$ where $\gamma=(1-\beta^2\ell^2(1+Y^2))^{-1/2}$. Notice that the lower two figures of \[integral-curves\] look very reminiscent of the black funnels constructed in [@Fischetti:2012vt] to study heat transport in holographic CFT’s. This raises the question whether the bulk duals of the fluid flows in hyperbolic space considered here could be used as toy models for the gravity side of the construction in [@Fischetti:2012vt]. In this context, one should note however that the black funnels of [@Fischetti:2012vt] contain a single connected bulk horizon that extends to meet the conformal boundary. Thus the induced boundary metric has smooth horizons as well. In our case instead, it turns out that the bulk horizon does not extend to meet the boundary, although the boundary metric itself may be considered to contain a horizon, since $\mathbb{R}\times\text{H}^2$ is conformal to the static patch of three-dimensional de Sitter space [@Fischetti:2012vt], which has a cosmological horizon. ### Mixed flow: $\omega^2=\beta^2$ {#sec:mixed-flow} Finally, in the parabolic case $\omega^2=\beta^2$ one can choose $\omega=\beta$, as was explained above. The Killing vector $v$ becomes then $$v = \beta\left(iz + \frac12(1 - z^2)\right)\partial_z + \text{c.c.}$$ It proves useful to introduce new coordinates $A,B$ defined by $$A = \ln\frac{1 - z\bar z}{z\bar z + i(z - \bar z) + 1}\,, \qquad B = \frac{z + \bar z}{z\bar z + i(z - \bar z) + 1}\,,$$ such that $v=\beta\partial_B$ and $$g=-\mathrm{d}t^2+\ell^2(\mathrm{d}A^2 + e^{-2A}\mathrm{d}B^2)\,.$$ The 3-velocity becomes $$u = \gamma(\partial_t + \beta\partial_B)\,,$$ with the Lorentz factor $\gamma=(1-\beta^2\ell^2e^{-2A})^{-1/2}$. The physical region $U$ is thus given by $1-\beta^2\ell^2e^{-2A}>0$. Fluid in rigid rotation on $\text{H}^2$ seen on the sphere or plane {#H2-to-plane} ------------------------------------------------------------------- The manifolds $\mathbb{R}\times\text{S}^2$ and $\mathbb{R}\times\text{H}^2$, with metrics and , are conformally flat. This means that each of them can be brought by a combined diffeomorphism plus Weyl rescaling into a part of the other or into a part of three-dimensional Minkowski space $\mathbb{M}^3$. One might thus ask how a fluid in one of these spaces appears when seen in the others after a conformal transformation. Since one may be interested in the description of hyperbolic AdS black holes in terms of hydrodynamics on Minkowski space or on $\mathbb{R}\times\text{S}^2$, we study as an example the rigidly rotating fluid on $\mathbb{R}\times\text{H}^2$ analyzed in subsection \[fluid rig rot hyp\] to see how it looks like on $\mathbb{M}^3$ or on the closed Einstein static universe. We will see that this leads to interesting dynamical fluid configurations. The coordinate transformation $$\label{H^2-to-Mink} T = \ell e^{\frac t{\ell}}\cosh\theta\,, \qquad X = \ell e^{\frac t{\ell}}\sinh\theta\cos\varphi\,, \qquad Y = \ell e^{\frac t{\ell}}\sinh\theta\sin\varphi\,,$$ combined with a conformal rescaling $\tilde{g}=\Omega^2 g$, where $$\Omega=e^\frac{t}{\ell}=\frac{\sqrt{T^2-X^2-Y^2}}{\ell}\,,$$ brings to the flat metric $$\tilde{g}=-\mathrm{d}T^2+\mathrm{d}X^2+\mathrm{d}Y^2\,.$$ Now consider the rigidly rotating fluid in \[fluid rig rot hyp\], which has 3-velocity $$u=\gamma(\partial_t+\omega\partial_\varphi)= \frac{\gamma T}{\ell}\left(\partial_T+\frac{X-\omega\ell Y}{T}\partial_X+\frac{Y+\omega\ell X}{T}\partial_Y\right)\,,$$ where $$\gamma=(1-\omega^2\ell^2\sinh^2\theta)^{-\frac{1}{2}}=\sqrt{\frac{T^2-X^2-Y^2}{T^2-(1+\omega^2\ell^2)(X^2+Y^2)}}\,.$$ Recall that the flow is defined only for $|\omega|\ell\sinh\theta<1$. In the coordinates $(T,X,Y)$, this condition becomes $(1+\omega^2\ell^2)(X^2+Y^2)<T^2$. Notice also that maps $\mathbb{R}\times\text{H}^2$ to the inside of the future light cone $X^2+Y^2<T^2$, $T>0$. The conformal rescaling transforms $u$ into $$\tilde{u} = \Omega^{-1}u = \frac{T}{\sqrt{T^2-(1+\omega^2\ell^2)(X^2+Y^2)}} \left(\partial_T+\frac{X-\omega\ell Y}{T}\partial_X+\frac{Y+\omega\ell X}{T}\partial_Y\right)\,.$$ This flow is plotted in coordinates $(T,X,Y)$ in figure \[grafici R x H2 visto in R x E2\]. ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the plane in coordinates $X,Y$, at times $T=1$, $T=2$ and $T=3$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x E2"}](FrE2a.pdf "fig:") ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the plane in coordinates $X,Y$, at times $T=1$, $T=2$ and $T=3$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x E2"}](FrE2b.pdf "fig:") ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the plane in coordinates $X,Y$, at times $T=1$, $T=2$ and $T=3$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x E2"}](FrE2c.pdf "fig:") We see that the rigidly rotating fluid in $\mathbb{R}\times\text{H}^2$ appears in Minkowski space as an expanding vortex. Let us now transform the same fluid configuration to the closed Einstein static universe $\mathbb{R}\times\text{S}^2$. To this aim, introduce new coordinates $$\label{H^2-to-S^2} \tau = -\ell \arctan\frac{\cosh\theta}{\sinh\frac{t}{\ell}}\,, \qquad \Theta = \arctan\frac{\sinh\theta}{\cosh\frac{t}{\ell}}\,, \qquad \Phi=\varphi\,,$$ where $\tau\in(-\ell\frac{\pi}{2},0)$, $\Theta\in(0,\frac{\pi}{2})$ and $\Phi\in(0,2\pi)$. The inverse of is $$t = \ell{\mathrm{arsinh}}\frac{\cos\frac{\tau}{\ell}}{\sqrt{\cos^2\Theta - \cos^2\frac{\tau}{\ell}}}\,, \qquad \theta = {\mathrm{arsinh}}\frac{\sin\frac{\tau}{\ell}}{\sqrt{\cos^2\Theta - \cos^2\frac{\tau}{\ell}}}\,,$$ hence one has the additional restriction $\Theta<-\frac{\tau}{\ell}$. Subsequently, rescale as $\tilde{g}=\Omega^2 g$, where $$\Omega = \sqrt{\cos^2\Theta-\cos^2\frac{\tau}{\ell}}\,,$$ to get $$\tilde{g}=-\mathrm{d}\tau^2+\ell^2(\mathrm{d}\Theta^2+\sin^2\Theta\mathrm{d}\Phi^2)\,.$$ Now the 3-velocity of the rigidly rotating fluid on $\text{H}^2$ is mapped into $$\tilde{u} = \Omega^{-1}u = \frac{-\sin\frac{\tau}{\ell}\cos\Theta}{\sqrt{\sin^2\frac{\tau}{\ell}-(1+\omega^2\ell^2)\sin^2\Theta}} \left(\partial_\tau+\frac{\tan\Theta}{\ell\tan\frac{\tau}{\ell}}\partial_\Theta+\frac{\omega}{-\sin\frac{\tau}{\ell}\cos\Theta}\partial_\Phi\right)\,.$$ In the coordinates $(\tau,\Theta,\Phi)$, the constraint $|\omega|\ell\sinh\theta<1$, limiting the region where the fluid is located, becomes $$\sin\Theta<\frac{-\sin\frac{\tau}{\ell}}{\sqrt{1+\omega^2\ell^2}}\,.$$ This flow is plotted in figure \[grafici R x H2 visto in R x S2\] at different times $\tau$ projected on the equatorial plane of $\text{S}^2$ and viewed from the north pole. ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the 2-sphere (from the north pole and projected on the equatorial plane), at times $\tau\simeq -\ell\frac{\pi}{2}$, $\tau=-\ell$ and $\tau=-\frac{\ell}{2}$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x S2"}](FrS2a.pdf "fig:") ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the 2-sphere (from the north pole and projected on the equatorial plane), at times $\tau\simeq -\ell\frac{\pi}{2}$, $\tau=-\ell$ and $\tau=-\frac{\ell}{2}$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x S2"}](FrS2b.pdf "fig:") ![Fluid in rigid rotation on $\text{H}^2$ with $\omega=\ell=1$, seen on the 2-sphere (from the north pole and projected on the equatorial plane), at times $\tau\simeq -\ell\frac{\pi}{2}$, $\tau=-\ell$ and $\tau=-\frac{\ell}{2}$. The grey area is the region of spacetime where the flow is not defined.[]{data-label="grafici R x H2 visto in R x S2"}](FrS2c.pdf "fig:") Again, we encounter a dynamical fluid configuration that is a sort of contracting vortex on $\text{S}^2$. Note that a similar technique was applied in 3+1 dimensions in [@Gubser:2010ui]. There, it was shown (using the Weyl covariance of the stress tensor) that the dynamical solution of [@Gubser:2010ze] (which represents a generalization of Bjorken flow [@Bjorken:1982qr]) can be recast as a static flow in three-dimensional de Sitter space times a line. The simplicity of the de Sitter form enabled the authors of [@Gubser:2010ui] to obtain several generalizations of it, such as flows in other spacetime dimensions, second order viscous corrections, and linearized perturbations. Dual AdS black holes {#dual-AdS-BH} ==================== Now we want to identify the AdS black holes dual to the fluid configurations classified in section \[stat flow ultrastatic st\]. It turns out that these bulk spacetimes are all contained in the Carter-Plebański family [@Carter:1968ks; @Plebanski:1975], whose metric is given by $$\label{Carter-Plebanski} \mathrm{d}s^2 = \frac{p^2+q^2}{\mathsf{P}(p)}\mathrm{d}p^2+ \frac{\mathsf{P}(p)}{p^2+q^2}(\mathrm{d}\tau+q^2\mathrm{d}\sigma)^2+ \frac{p^2+q^2}{\mathsf{Q}(q)}\mathrm{d}q^2- \frac{\mathsf{Q}(q)}{p^2+q^2}(\mathrm{d}\tau-p^2\mathrm{d}\sigma)^2\,,$$ where $$\mathsf{P}(p)=\alpha-\mathrm{g}^2+2lp-\epsilon p^2+\frac{p^4}{\ell^2}\,, \qquad \mathsf{Q}(q)=\alpha+\mathrm{e}^2-2mq+\epsilon q^2+\frac{q^4}{\ell^2}\,. \label{PQ}$$ This solves the Einstein-Maxwell equations with cosmological constant $\Lambda=-3\ell^{-2}$ and electromagnetic field $$A=-\frac{\mathrm{e}\,q}{p^2+q^2}(\mathrm{d}\tau-p^2\mathrm{d}\sigma) -\frac{\mathrm{g}\,p}{p^2+q^2}(\mathrm{d}\tau+q^2\mathrm{d}\sigma)\,, \label{A-CP}$$ whose field strength is $$F=\frac{\mathrm{e}(p^2-q^2)+2\mathrm{g}\,pq}{(p^2+q^2)^2}\mathrm{d}q\wedge(\mathrm{d}\tau-p^2\mathrm{d}\sigma) -\frac{\mathrm{g}(p^2-q^2)-2\mathrm{e}\,pq}{(p^2+q^2)^2}\mathrm{d}p\wedge(\mathrm{d}\tau+q^2\mathrm{d}\sigma)\,. \label{F-CP}$$ can be obtained by a scaling limit from the Plebański-Demiański spacetime [@Plebanski:1976gy][^7], which is the most general known Petrov-type D solution to the Einstein-Maxwell equations with cosmological constant. Other references studying algebraically special spacetimes and their fluid duals include [@deFreitas:2014lia], where the AdS/CFT interpretation of the Robinson-Trautman (RT) solution to vacuum AdS gravity was investigated. This is slightly different from our case, since the boundary metric of the RT geometry is in general time-dependent [@deFreitas:2014lia]. The metric $\hat g$ on the conformal boundary of can be obtained by setting $q=\text{const.}\to\infty$ and rescaling with $\ell^2/q^2$. This leads to $$\label{conf-bdry-CP} {\hat g} = -\mathrm{d}\tau^2+ \frac{\ell^2}{\mathsf{P}(p)}\mathrm{d}p^2+ (\ell^2\mathsf{P}(p)-p^4)\mathrm{d}\sigma^2+2p^2\mathrm{d}\tau\mathrm{d}\sigma\,.$$ Notice that for vanishing NUT-parameter $l$ this metric is conformally flat[^8]. In what follows, we shall consider the case $l=0$ only[^9]. Using standard holographic renormalization techniques [@Balasubramanian:1999re], one can compute the holographic stress tensor associated to , with the result $$\hat{T}_{\mu\nu}=\frac{m}{8\pi\ell^2}(\gamma_{\mu\nu}+3u_\mu u_\nu)\,,$$ where $u=\partial_\tau$. $\hat{T}$ describes thus a conformal fluid in equilibrium, at rest in the frame $(\tau,p,\sigma)$. The external electromagnetic field $\hat F$ and the $\text{U}(1)$ current $\hat J$ dual to on the conformal boundary of are found to be respectively $$\hat{F} = \mathrm{g}\,\mathrm{d}p\wedge\mathrm{d}\sigma\,, \qquad \hat{J} = \frac{\mathrm{e}}{4\pi\ell^2}\partial_\tau = \frac{\mathrm{e}}{4\pi\ell^2}u\,.$$ The last equation shows that the fluid has a constant charge density $\mathrm{e}/(4\pi\ell^2)$. Note also that the current $\hat{J}$ is conserved, ${\hat\nabla}_\mu\hat{J}^\mu=0$, where $\hat\nabla$ denotes the Levi-Civita connection of $\hat g$. Moreover, since $\hat{F}_{\mu\nu}\hat{J}^\nu=0$, the Lorentz force exerted by the field $\hat{F}$ on the charged fluid vanishes, and thus $\hat T$ is conserved as well, ${\hat\nabla}_\mu\hat{T}^{\mu\nu}=0$. Notice that the solution , enjoys the scaling symmetry $$\begin{aligned} &&p\to \lambda p\,, \qquad q \to \lambda q\,, \qquad \tau \to \tau/\lambda\,, \qquad \sigma \to \sigma/\lambda^3\,, \qquad \alpha\to \lambda^4\alpha\,, \nonumber \\ &&\mathrm{g}\to \lambda^2\mathrm{g}\,, \qquad \mathrm{e} \to \lambda^2\mathrm{e}\,, \qquad m\to \lambda^3 m\,, \qquad l \to \lambda^3 l\,, \qquad \epsilon \to \lambda^2\epsilon\,, \label{scaling-symm}\end{aligned}$$ that can be used to eliminate one unphysical parameter. The line element describes a black hole whose event horizon $\cal H$ is located at the largest root of the polynomial $\mathsf{Q}(q)$. As we shall see below, the horizon geometry depends crucially on the choice of parameters contained in the function $\mathsf{P}(p)$, which determine the number of real roots of $\mathsf{P}$. In what follows, we will discuss more in detail some subcases of the Carter-Plebański family, which are dual to the fluid configurations classified in section \[stat flow ultrastatic st\]. Spherical and hyperbolic Kerr-Newman-AdS$_4$ black holes {#Kerr-Newman-AdS4 BH} -------------------------------------------------------- If we set $$\alpha = ka^2 + \mathrm{g}^2\,, \quad \epsilon = k + \frac{a^2}{\ell^2}\,, \quad \tau = \frac{t - a\varphi}{\Xi}\,, \quad q = r\,, \quad p = a c_k(\theta)\,, \quad \sigma = -\frac{\varphi}{a\Xi}\,,$$ where $$k = \pm 1\,, \qquad \Xi = 1 - \frac{ka^2}{\ell^2}\,, \qquad c_k(\theta) = \frac{d s_k(\theta)}{d\theta}\,, \qquad s_k(\theta)=\left\{\begin{matrix} \sin\theta\,, \quad k=1\,, \\ \sinh\theta\,, \quad k=-1\,, \end{matrix}\right.$$ the metric becomes $$\mathrm{d}s^2 = -\frac{\Delta_r}{\Xi^2\rho^2}\left(\mathrm{d}t-ka\,s_k^2(\theta)\mathrm{d} \varphi\right )^2 + \rho^2\left(\frac{\mathrm{d}r^2}{\Delta_r}+ \frac{\mathrm{d}\theta^2}{\Delta_\theta}\right )+ \frac{\Delta_\theta}{\Xi^2\rho^2}\left(a\mathrm{d}t-(r^2+a^2)\mathrm{d}\varphi\right )^2 s_k^2(\theta)\,, \label{sph-hyp KNAdS4}$$ with $$\rho^2 = r^2+a^2 c_k^2(\theta)\,, \qquad \Delta_r = (r^2+a^2)\left(k+\frac{r^2}{\ell^2}\right) - 2mr + \mathrm{e}^2 + \mathrm{g}^2\,, \qquad \Delta_\theta = 1 - \frac{ka^2}{\ell^2}c_k^2(\theta)\,.$$ For $k=1$ this is the Kerr-Newman-AdS$_4$ black hole, while for $k=-1$ one has the rotating hyperbolic solution constructed in [@Klemm:1997ea]. Note also that in the spherical case $(k=1)$ the rotation parameter $a$ is bounded by $a^2<\ell^2$ in order for $\Delta_{\theta}$ to be positive, while it can take any value if $k=-1$. The metric on the conformal boundary of reads $$\label{metric conf bound s-h rotating bh} {\hat g} = -\frac{\mathrm{d}t^2}{\Xi^2}+\frac{\ell^2\mathrm{d}\theta^2}{\Delta_{\theta}}+ \frac{\ell^2}{\Xi}s_k^2(\theta)\mathrm{d}\varphi^2+2\frac{ak}{\Xi^2}s_k^2(\theta)\mathrm{d}t\mathrm{d}\varphi\,.$$ Since this is conformally flat there exist coordinates in which, after a conformal rescaling, it takes the ultrastatic spherical or hyperbolic form (like in eqns.  and ). These are given by $$\label{coord change conf bound} \tau = \frac{t}{\Xi}\,, \qquad c_k(\Theta) = c_k(\theta)\sqrt{\frac{\Xi}{\Delta_\theta}}\,, \qquad \Phi = \varphi+\frac{kat}{\ell^2\Xi}\,.$$ Notice that $\Theta$ ranges in $(0,\pi)$ when $k=1$ and in $(0,\text{arsinh}(\ell/|a|))$ when $k=-1$, cf. fig. \[grafici Theta(theta)\], where $\Theta(\theta)$ is plotted for different values of $a/\ell$. ![Graphs of $\Theta(\theta)$ for $k=1$ (left) and $k=-1$ (right), for different values of $a/\ell$.[]{data-label="grafici Theta(theta)"}](Theta1.pdf "fig:") ![Graphs of $\Theta(\theta)$ for $k=1$ (left) and $k=-1$ (right), for different values of $a/\ell$.[]{data-label="grafici Theta(theta)"}](Theta-1.pdf "fig:") In the new coordinates, the boundary metric takes the form $${\hat g} = \frac{\Delta_\theta}{\Xi}\left(-\mathrm{d}\tau^2+\ell^2(\mathrm{d}\Theta^2+s_k^2(\Theta)\mathrm{d}\Phi^2)\right)\,,$$ such that, after a Weyl rescaling $${\hat g}\rightarrow\tilde{g}=\Omega^2{\hat g}\,, \qquad \Omega^2=\Xi/\Delta_{\theta}\,, \label{Weyl-KNAdS}$$ one obtains the desired metric $$\label{rescaled metr on bound} \tilde{g} = -\mathrm{d}\tau^2+\ell^2(\mathrm{d}\Theta^2+s_k^2(\Theta)\mathrm{d}\Phi^2)\,.$$ Thus the boundary of the Kerr-AdS$_4$ black hole is conformal to $\mathbb{R}\times\text{S}^2$ for $k=1$, and to the part of $\mathbb{R}\times\text{H}^2$ with $\sinh\Theta<\ell/|a|$ for $k=-1$. If we identify $|\omega|=|a|/\ell^2$, this is exactly the part of $\mathbb{R}\times\text{H}^2$ on which a fluid in rigid rotation with angular velocity $\omega$ does not exceed the speed of light. We will have to say more on this below. The holographic stress tensor associated to can be written in the form $$\label{black hole stress tensor} \hat{T}_{\mu\nu}=\frac{m}{8\pi\ell^2}\left({\hat g}_{\mu\nu}+3u_\mu u_\nu \right )\,,$$ where $u=\Xi\partial_t$. This is the stress tensor of a conformal fluid at rest in the coordinate frame $(t,\theta,\varphi)$, with pressure ${\cal P}=m/(8\pi\ell^2)$. After the diffeomorphism and the subsequent Weyl rescaling (recall that $\hat T$ transforms as $\tilde T^{\mu\nu}=\Omega^{-d-2}\hat T^{\mu\nu}$) one obtains $$\tilde{T}^{\mu\nu}=\frac{m\gamma^3}{8\pi\ell^2} \begin{pmatrix} 3\gamma^2-1 & 0 & \frac{3ka \gamma^2}{\ell^2}\\ 0 & \frac{1}{\ell^2} & 0\\ \frac{3ka \gamma^2}{\ell^2} & 0 & \frac{3\gamma^2-2}{\ell^2 s_k^2(\Theta)} \end{pmatrix}\,, \label{tilde-T-KNAdS}$$ with $\gamma:=(1-a^2s_k^2(\Theta)/\ell^2)^{-1/2}$. This can also be rewritten as $\tilde{T}^{\mu\nu}=\tilde{\cal P}\left(\tilde{g}^{\mu\nu}+3\tilde{u}^\mu \tilde{u}^\nu\right )$, where $$\tilde{\cal P}=\Omega^{-3}{\cal P} = \frac{m\gamma^3}{8\pi\ell^2}\,, \qquad \tilde{u} =\Omega^{-1}u = \gamma(\partial_\tau+\frac{ka}{\ell^2}\partial_\Phi)\,.$$ For $k=1$, $\tilde{T}$ is exactly the stress tensor of the stationary conformal fluid on the 2-sphere, if we identify $$\label{spher corr rules} {\cal P}_0 = \frac{m}{8\pi\ell^2}\,, \qquad \omega = \frac{a}{\ell^2}\,.$$ On the other hand, for $k=-1$, coincides with the stress tensor of the rigidly rotating conformal fluid on the hyperbolic plane, after the identifications $$\label{hyp corr rules} {\cal P}_0= \frac{m}{8\pi\ell^2}\,, \qquad \omega = -\frac{a}{\ell^2}\,.$$ The KNAdS black hole is thus dual to a fluid in rigid rotation on $\text{S}^2$ for $k=1$ and on $\text{H}^2$ for $k=-1$. In the spherical case, this is of course well-known [@Bhattacharyya:2007vs; @Caldarelli:2008ze]. The result for hyperbolic black holes is new, and it is remarkable how the conformal transformation , maps the boundary geometry of the rotating hyperbolic black hole precisely to the region of $\mathbb{R}\times\text{H}^2$ on which a fluid in rigid rotation does not exceed the speed of light. The electromagnetic field and electric current on the boundary are given respectivey by $$\hat{F} = \frac{k\mathrm{g}s_k(\theta)}{\Xi}\mathrm{d}\theta\wedge\mathrm{d}\varphi\,, \qquad \hat{J} = \frac{\mathrm{e}\Xi}{4\pi\ell^2}\partial_t = \frac{\mathrm{e}}{4\pi\ell^2}u\,.$$ After the coordinate change and Weyl rescaling , they become[^10] $$\tilde{F} = k\mathrm{g}\gamma^3 s_k(\Theta)\,\mathrm{d}\Theta\wedge(\mathrm{d}\Phi- \frac{ka}{\ell^2}\mathrm{d}\tau)\,, \qquad \tilde{J} = \frac{\mathrm{e}\gamma^3}{4\pi\ell^2}(\partial_\tau+\frac{ka}{\ell^2}\partial_\Phi) = \frac{\mathrm{e}\gamma^2}{4\pi\ell^2}\tilde{u}\,,$$ and thus $\tilde J$ coincides with the hydrodynamical expression if the charge density of the fluid is $$\rho_{\text e} = \frac{\mathrm{e}\gamma^2}{4\pi\ell^2}\,. \label{rho_e}$$ Note that in the coordinate system $(\tau,\Theta,\Phi)$ there is also an electric field. Moreover, one has $\tilde{F}^\nu{}_\mu\tilde{J}^\mu=0$, so there is no net Lorentz force acting on the charged fluid due to an exact cancellation of electric and magnetic forces[^11]. In the orthonormal frame $$e^0 = \mathrm{d}\tau\,, \qquad e^1 = \ell\mathrm{d}\Theta\,, \qquad e^2 = \ell s_k(\Theta) \mathrm{d}\Phi\,,$$ the electric field in $1$-direction and the spatial current in $2$-direction are $$E^1 = \frac{\mathrm{g}\gamma^3 s_k(\Theta)a}{\ell^3}\,, \qquad \tilde{J}^{\,2} = \frac{\mathrm{e}\gamma^3 s_k(\Theta)ka}{4\pi\ell^3} = \sigma^{21}E^1\,,$$ with the Hall conductivity $$\sigma^{21} = \frac{\mathrm{e}k}{4\pi\mathrm{g}}\,.$$ In the spherical case $k=1$, it was furthermore shown in [@Bhattacharyya:2007vs] that, in the large black hole limit where fluid dynamics provides an accurate description of the dual conformal field theory, the black hole electric charge, entropy, mass and angular momentum coincide precisely with the conserved charges , and computed in fluid mechanics, if we identify the Hawking temperature $T$ with the global fluid temperature in [^12]. On the other hand, for $k=-1$, we already saw in the previous section that the conserved charges are ill-defined in fluid mechanics. The same problem is encountered on the gravity side: If one tries to compute for instance the entropy of the solution with $k=-1$, one has to integrate over the noncompact variable $\theta$, which makes the result divergent. A possible way out could be to consider only excitations above some ‘ground state’, which may have finite energy, but we shall not attempt to do this here. In spite of these difficulties, we saw in section \[fluid rig rot hyp\] that a local form of the first law of black hole mechanics holds. Boosting AdS$_4$ black holes {#boosting charged bh} ---------------------------- In the previous subsection it was shown that the spherical and hyperbolic KNAdS$_4$ black holes are holographically dual to conformal fluids in rigid rotation on $\mathbb{R}\times\text{S}^2$ and $\mathbb{R}\times\text{H}^2$ respectively. While a rigid rotation is (up to isometries) the only possible equilibrium configuration for a stationary conformal fluid on a sphere, the same is not true for hyperbolic space: We saw in \[sec:fluid-H2\] that on the hyperbolic plane one can also have purely translational (‘boosting’) or mixed flows, which are not isometric to rotations. In this section we describe a family of black holes, obtained by analytically continuing the hyperbolic KNAdS$_4$ metric, whose dual fluid is translating on the hyperbolic plane. We will call these solutions ‘boosting black holes’. Consider the KNAdS$_4$ metric with $k=-1$, and analytically continue $$\label{anal cont KNAdS metric} a\rightarrow ib\,, \qquad \theta\rightarrow\theta-\frac{i\pi}{2}\,, \qquad \varphi\rightarrow i\varphi\,.$$ This leads to $$\mathrm{d}s^2 = -\frac{\Delta_r}{\Xi^2\rho^2}\left(\mathrm{d}t+b\cosh^2\theta\mathrm{d}\varphi \right )^2 + \rho^2\left(\frac{\mathrm{d}r^2}{\Delta_r} + \frac{\mathrm{d}\theta^2}{\Delta_\theta} \right ) + \frac{\Delta_\theta\cosh^2\theta}{\Xi^2\rho^2}\left(b\mathrm{d}t-(r^2-b^2) \mathrm{d}\varphi\right )^2\,,\label{boostingAdS4}$$ where now $$\rho^2 = r^2+b^2\sinh^2\theta\,, \quad \Delta_r = (r^2-b^2)\left(-1+\frac{r^2}{\ell^2}\right) - 2m r + \mathrm{e}^2 + \mathrm{g}^2\,, \quad \Delta_{\theta} = 1+\frac{b^2}{\ell^2}\sinh^2\theta\,,$$ and $\Xi=1-b^2/\ell^2$. Alternatively, can be obtained directly from the Carter-Plebański solution by setting $$\alpha = b^2 + \mathrm{g}^2\,, \quad l = 0\,, \quad \epsilon = -1-\frac{b^2}{\ell^2}\,, \quad \tau = \frac{t+b\varphi}{\Xi}\,, \quad q = r\,, \quad p = b\sinh\theta\,, \quad \sigma = -\frac{\varphi}{b\Xi}\,.$$ The electromagnetic 1-form potential becomes then $$A = -\frac{\mathrm{e}r}{\Xi\rho^2}\left(\mathrm{d}t+b\cosh^2\theta\mathrm{d}\varphi\right) -\frac{\mathrm{g}\sinh\theta}{\Xi\rho^2}\left(b\mathrm{d}t-(r^2-b^2)\mathrm{d}\varphi\right)\,. \label{A-boostingAdS4}$$ Notice that now $\theta,\varphi$ are not polar coordinates on $\text{H}^2$ (in that case it would not be possible to extend the 1-form $\cosh\theta\,\mathrm{d}\varphi$ to $\theta=0$), but they are rather Cartesian-type coordinates on a plane, possibly compactified to a cylinder by periodic identifications of $\varphi$[^13]. The metric on the conformal boundary of is given by $${\hat g} = -\frac{\mathrm{d}t^2}{\Xi^2}+\frac{\ell^2}{\Delta_\theta}\mathrm{d}\theta^2+\frac{\ell^2} {\Xi}\cosh^2\theta\mathrm{d}\varphi^2-2\frac{b}{\Xi^2}\cosh^2\theta\mathrm{d}t\mathrm{d}\varphi\,,$$ from which we see that $\partial_{\varphi}$ is spacelike only for $b^2<\ell^2$. Now introduce the ultrastatic coordinates $$\label{coord change conf bound mod} T = \frac{t}{\Xi}\,, \qquad X = \frac{\cosh\theta}{\sqrt{\Delta_\theta}}\sinh\left(\varphi-\frac{bt} {\ell^2\Xi}\right)\,, \qquad Y = \sqrt{\frac{\Xi}{\Delta_\theta}}\sinh\theta\,,$$ where $T,X\in\mathbb{R}$ and $Y$ is bounded by $Y^2<\ell^2/b^2-1$, and perform a Weyl rescaling $\tilde{g}=\Omega^2{\hat g}$ with $$\Omega = \sqrt{\frac{\Xi}{\Delta_\theta}} = \sqrt{1-\frac{b^2}{\ell^2}(1+Y^2)}\,. \label{Weyl-boost}$$ This yields $$\label{Gamma mod} \tilde{g} = -\mathrm{d}T^2+\frac{\ell^2}{1+X^2+Y^2}\left((1+Y^2)\mathrm{d}X^2+(1+X^2) \mathrm{d}Y^2-2XY\mathrm{d}X\mathrm{d}Y\right)\,,$$ which is the slice $Y^2<\ell^2/b^2-1$ of the spacetime $\mathbb{R}\times\text{H}^2$, cf. . In the frame $(t,\theta,\varphi)$ the holographic stress tensor on the conformal boundary is found to be $$\hat{T}_{\mu\nu}=\frac{m}{8\pi\ell^2}\left(\gamma_{\mu\nu}+3u_\mu u_\nu \right )\,,$$ with $u=\Xi\partial_t$. After the diffeomorphism and the Weyl rescaling , the stress tensor becomes $$\tilde{T}_{\mu\nu} = \tilde{\cal P}\left(\tilde{g}_{\mu\nu}+3\tilde{u}_\mu\tilde{u}_\nu\right )\,,$$ where $$\tilde{\cal P} = \frac{m\gamma^3}{8\pi\ell^2}\,, \qquad \tilde{u} = \gamma\left(\partial_T-\frac{b}{\ell^2}\sqrt{1+X^2+Y^2}\partial_X\right)\,, \qquad \gamma = \Omega^{-1} = \left(1-\frac{b^2}{\ell^2}(1+Y^2)\right)^{-1/2}\,.$$ This is exactly the energy-momentum tensor and 3-velocity of a conformal fluid translating on the hyperbolic plane studied in section \[sec:transl-flow\], after the identifications $${\cal P}_0 = \frac{m}{8\pi\ell^2}\,, \qquad \beta = -\frac{b}{\ell^2}\,.$$ The gravity dual of the ‘boosting’ fluid on $\text{H}^2$ is thus given by the black hole solution , [^14]. Although the latter is contained in the general Carter-Plebańksi solution, it is in principle new, since its physical properties have not been discussed in the literature so far. Note again the remarkable fact that the conformal transformation , maps the boundary geometry of precisely to the region of $\mathbb{R}\times\text{H}^2$ in which the fluid velocity does not exceed the speed of light. Black holes dual to mixed (parabolic) flow on the hyperbolic plane ------------------------------------------------------------------ Consider now the following choice for the parameters and coordinates of the Carter-Plebański solution : $$\alpha = \mathrm{g}^2\,, \qquad l = 0\,, \qquad \epsilon = -1\,, \qquad q = r\,, \qquad p = aP\,, \qquad \sigma = -\frac{\varphi}{a}\,,$$ which leads to $$\label{gravdual-mixed} \mathrm{d}s^2 = -\frac{\Delta_r}{\rho^2}\left(\mathrm{d}\tau + aP^2\mathrm{d}\varphi\right)^2 + \rho^2\left(\frac{\mathrm{d}r^2}{\Delta_r} + \frac{\mathrm{d}P^2}{\Delta_P}\right) + \frac{\Delta_P}{\rho^2}\left(a\mathrm{d}\tau - r^2\mathrm{d}\varphi\right)^2\,,$$ $$A = -\frac{\mathrm{e}r}{\rho^2}\left(\mathrm{d}\tau+a P^2\mathrm{d}\varphi\right) -\frac{\mathrm{g}P}{\rho^2}\left(a\mathrm{d}\tau-r^2\mathrm{d}\varphi\right)\,,$$ where $$\rho^2 = r^2+a^2P^2\,, \qquad \Delta_r = r^2\left(\frac{r^2}{\ell^2}-1\right) - 2mr + \mathrm{e}^2 + \mathrm{g}^2\,, \qquad \Delta_P = P^2\left(1+\frac{a^2}{\ell^2}P^2\right)\,.$$ The metric on the conformal boundary of reads $${\hat g} = -\mathrm{d}\tau^2+\frac{\ell^2}{\Delta_P}\mathrm{d}P^2+\ell^2P^2\mathrm{d}\varphi^2 -2aP^2\mathrm{d}\tau\mathrm{d}\varphi\,,$$ and the holographic stress tensor takes the usual form $$\label{T-usual} \hat{T}_{\mu\nu} = \frac{m}{8\pi\ell^2}\left({\hat g}_{\mu\nu}+3u_\mu u_\nu\right)\,,$$ with 3-velocity $u=\partial_{\tau}$. Like in the previous cases, one can introduce ultrastatic coordinates on the conformal boundary, defined by $$\label{ultrastat-coord-mixed} T = \tau\,, \qquad A = \frac12\ln\frac{\Delta_P}{P^4}\,, \qquad B = \varphi-\frac{a\tau}{\ell^2}\,,$$ where $A>\frac{1}{2}\ln\frac{a^2}{\ell^2}$ and $T,B\in\mathbb{R}$. After a subsequent Weyl rescaling $\tilde{g}=\Omega^2{\hat g}$ with $$\Omega = \frac{P}{\sqrt{\Delta_P}} = \sqrt{1-\frac{a^2}{\ell^2}e^{-2A}}\,, \label{Weyl-mixed}$$ one gets the metric $$\tilde{g} = -\mathrm{d}T^2+\ell^2(\mathrm{d}A^2+e^{-2A}\mathrm{d}B^2)\,.$$ Thus we have shown that the boundary geometry of is conformal to the subset $A>\frac{1}{2}\ln\frac{a^2}{\ell^2}$ of the spacetime $\mathbb{R}\times\text{H}^2$. After the coordinate transformation and the Weyl rescaling , the 3-velocity $u$ becomes $$\tilde{u} = \gamma(\partial_T-\frac{a}{\ell^2}\partial_B)\,,$$ where $$\gamma = \Omega^{-1} = \left(1-\frac{a^2}{\ell^2}e^{-2A}\right)^{-1/2}\,.$$ The transformed energy-momentum tensor $\tilde T$ and the 3-velocity $\tilde u$ coincide with the ones considered in section \[sec:mixed-flow\], if we identify $${\cal P}_0 = \frac{m}{8\pi\ell^2}\,, \qquad \beta = -\frac{a}{\ell^2}\,.$$ The gravity dual of the mixed (parabolic) flow on $\text{H}^2$ is thus given by the black hole solution . Again, the conformal transformation , maps the boundary of exactly to the region $1-\beta^2\ell^2e^{-2A}>0$ where the fluid velocity is smaller than the speed of light. If $\varphi$ is compactified, $B$ becomes also a compact coordinate. The flow in this case is visualized in fig. \[Grafico PD ident\]. ![Poincaré disk compactified by identifications of the coordinate $B$; the two thick black lines have to be identified. The grey area is the unphysical region where the fluid velocity exceeds the speed of light. The fluid is located in the red region.[]{data-label="Grafico PD ident"}](RegPD.pdf) Black holes dual to rotating fluid on the Euclidean plane --------------------------------------------------------- The family of black holes dual to a rotating fluid in Minkowski space $\mathbb{R}\times\text{E}^2$ is obtained by making the following choice for the parameters and coordinates of the Carter-Plebański solution : $$\alpha = \mathrm{g}^2\,, \qquad l = 0\,, \qquad \epsilon = \frac{a^2}{\ell^2}\,, \qquad q = r\,, \qquad p = aP\,, \qquad \sigma = -\frac{\varphi}{a}\,,$$ which leads to $$\label{gravdual-rotE2} \mathrm{d}s^2 =-\frac{\Delta_r}{\rho^2}\left(\mathrm{d}\tau+a P^2\mathrm{d}\varphi\right)^2 + \rho^2\left(\frac{\mathrm{d}r^2}{\Delta_r} + \frac{\mathrm{d}P^2}{\Delta_P}\right) + \frac{\Delta_P}{\rho^2}\left(a\mathrm{d}\tau - r^2\mathrm{d}\varphi\right)^2\,,$$ $$A = -\frac{\mathrm{e}r}{\rho^2}\left(\mathrm{d}\tau + a P^2\mathrm{d}\varphi\right) -\frac{\mathrm{g}P}{\rho^2}\left(a\mathrm{d}\tau - r^2\mathrm{d}\varphi\right)\,,$$ where $$\rho^2 = r^2+a^2P^2\,, \qquad \Delta_r = \frac{r^2}{\ell^2}(r^2+a^2)-2mr + \mathrm{e}^2 + \mathrm{g}^2\,, \qquad \Delta_P = \frac{a^2}{\ell^2}P^2(P^2-1)\,,$$ and $P>1$. In the uncharged case ($\mathrm{e}=\mathrm{g}=0$), the solution appeared in (C.10) of [@Mukhopadhyay:2013gja]. Notice that, unlike in the previous cases, the Killing vector $\partial_\varphi$ becomes timelike for large $r$. Hence, to avoid closed timelike curves, we shall not compactify $\varphi$. Instead we replace the coordinates $\tau,\varphi$ with $T,\Phi$ defined by $$T = \tau+a\varphi\,, \qquad \Phi=\frac{a\tau}{\ell^2}\,. \label{T-Phi-rotE2}$$ Since $\partial_\Phi$ is spacelike everywhere outside the horizon (located at the largest root of $\Delta_r$), we compactify $\Phi\sim\Phi+2\pi$. This choice is done in order that the conformal boundary has the topology of $\mathbb{R}$ times a disk, as we will see shortly. The boundary geometry of is given by $${\hat g} = -\mathrm{d}\tau^2 + \frac{\ell^2}{\Delta_P}\mathrm{d}P^2 -a^2P^2\mathrm{d}\varphi^2 - 2aP^2\mathrm{d}\tau\mathrm{d}\varphi\,,$$ and the holographic stress tensor has the usual form , where $u=\partial_\tau$. Now consider the coordinate transformation , supplemented by $$R=\frac{\ell^2}{a}\sqrt{1-P^{-2}}\,, \qquad 0 < R < \frac{\ell^2}{a}\,,$$ and perform a Weyl rescaling $\tilde{g}=\Omega^2{\hat g}$ with $$\Omega = P^{-1} = \sqrt{1-\frac{a^2 R^2}{\ell^4}}\,. \label{Weyl-rotE2}$$ This yields the metric $$\tilde{g} = -\mathrm{d}T^2+\mathrm{d}R^2+R^2\mathrm{d}\Phi^2\,.$$ Thus we have shown that the conformal boundary is the subset $R<\ell^2/a$ of the spacetime $\mathbb{R}\times\text{E}^2$, i.e., the real line times a disk. After the coordinate change to $(T,R,\Phi)$ and the conformal rescaling , the energy momentum tensor becomes $$\label{stress-rotE2} \tilde{T}^{\mu\nu} = \tilde{\cal P}\left(\tilde{g}^{\mu\nu}+3\tilde{u}^\mu\tilde{u}^\nu\right)\,,$$ where $$\tilde{\cal P} = \frac{m\gamma^3}{8\pi\ell^2}\,, \qquad \tilde{u} = \gamma\left(\partial_T + \frac{a}{\ell^2}\partial_\Phi\right)\,, \qquad \gamma = \Omega^{-1} = \left(1-\frac{a^2 R^2}{\ell^4} \right)^{-1/2}\,.$$ The stress tensor and the 3-velocity $\tilde u$ coincide with the ones considered in section \[sec:rot-plane\], if we identify $${\cal P}_0 = \frac{m}{8\pi\ell^2}\,, \qquad \omega = \frac{a}{\ell^2}\,.$$ The gravity dual of the rotating fluid on $\text{E}^2$ is thus given by the black hole solution , and the conformal transformation that we used here maps the boundary geometry of to a line times the disk $R<\ell^2/a$, where the fluid flow is well-defined. Super-rotating hyperbolic black holes ------------------------------------- We saw in section \[Kerr-Newman-AdS4 BH\] that the spherical KNAdS black hole is dual to a rotating fluid on $\text{S}^2$ if the angular velocity of the latter is limited by $|\omega|\ell<1$, which translates on the gravity side into $a^2<\ell^2$. For $|\omega|\ell>1$ the constraint $v^2<1$ restricts the rotating fluid to the polar caps $|\omega|\ell\sin\theta<1$. It turns out that in this case the dual black hole can be obtained from the KNAdS$_4$ metric with $k=1$ by the analytical continuation $\theta\rightarrow i\theta$, which leads to $$\mathrm{d}s^2 = -\frac{\Delta_r}{\Xi^2\rho^2}\left(\mathrm{d}t + a\sinh^2\theta\mathrm{d}\varphi \right )^2 + \rho^2\left(\frac{\mathrm{d}r^2}{\Delta_r} + \frac{\mathrm{d}\theta^2}{\Delta_\theta} \right ) + \frac{\Delta_\theta\sinh^2\theta}{\Xi^2\rho^2}\left(a\mathrm{d}t - (r^2+a^2)\mathrm{d}\varphi\right)^2\,, \label{hyp super KNAdS4}$$ $$A = -\frac{\mathrm{e}r}{\Xi\rho^2}\left(\mathrm{d}t + a\sinh^2\theta\mathrm{d}\varphi\right) - \frac{\mathrm{g}\cosh\theta}{\Xi\rho^2}\left(a\mathrm{d}t - (r^2+a^2)\mathrm{d}\varphi\right)\,,$$ where $$\rho^2 = r^2+a^2\cosh^2\theta\,, \quad \Delta_r = (r^2+a^2)\left(1+\frac{r^2}{\ell^2}\right) - 2mr + \mathrm{e}^2 + \mathrm{g^2}\,, \quad \Delta_\theta = \frac{a^2}{\ell^2}\cosh^2\theta - 1\,,$$ and $\Xi=a^2/\ell^2-1$. Note that there is a lower bound on the rotation parameter $a$: Positivity of $\Delta_\theta$ requires $a^2>\ell^2$, so that these black holes exist only above some minimum amount of rotation and thus have no static limit. The metric is again a special case of the Carter-Plebański family, obtained by setting $$\gamma = a^2+\mathrm{g}^2\,, \quad l = 0\,, \quad \epsilon = 1+\frac{a^2}{\ell^2}\,, \quad \tau = \frac{t-a\varphi}{\Xi}\,, \quad q = r\,, \quad p = a\cosh\theta\,, \quad \sigma = -\frac{\varphi}{a\Xi}\,.$$ The metric on the conformal boundary of is given by $$\label{metric conf bound sup rotating bh} {\hat g} = -\frac{\mathrm{d}t^2}{\Xi^2} + \frac{\ell^2}{\Delta_\theta}\mathrm{d}\theta^2 + \frac{\ell^2}{\Xi}\sinh^2\theta\mathrm{d}\varphi^2 - \frac{2a}{\Xi^2}\sinh^2\theta\mathrm{d}t \mathrm{d}\varphi\,.$$ Now introduce new coordinates $\tau,\Theta,\Phi$ defined by $$\label{coord change conf bound suprot} \tau = \frac{t}{\Xi}\,, \qquad \sin\Theta = \frac{\sinh\theta}{\sqrt{\Delta_\theta}}\,, \qquad \Phi = \varphi-\frac{at}{\ell^2\Xi}\,,$$ where $0<\Theta<\arcsin(\ell/|a|)$, and perform a Weyl rescaling $\tilde{g}=\Omega^2{\hat g}$ with $$\Omega = \sqrt{\frac{\Xi}{\Delta_\theta}}\,. \label{Weyl-suprot}$$ This gives $${\tilde g} = -\mathrm{d}\tau^2 + \ell^2(\mathrm{d}\Theta^2 + \sin^2\Theta\mathrm{d}\Phi^2)\,,$$ and thus the conformal boundary of is, up to conformal transformations, the polar cap $\Theta<\arcsin(\ell/|a|)$ of $\mathbb{R}\times \text{S}^2$. After the conformal transformation , , the holographic stress tensor associated to the spacetime becomes $$\tilde{T}_{\mu\nu} = \tilde{\cal P}\left(\tilde{g}_{\mu\nu} + 3\tilde{u}_\mu \tilde{u}_\nu\right)\,,$$ where $$\tilde{\cal P} = \frac{m\gamma^3}{8\pi\ell^2}\,, \qquad \tilde{u} = \gamma\left(\partial_\tau - \frac{a}{\ell^2}\partial_\Phi\right)\,, \qquad \gamma = \Omega^{-1} = \left(1 - \frac{a^2}{\ell^2} \sin^2\Theta\right)^{-1/2}\,.$$ $\tilde{T}$ is exactly the stress tensor of the stationary conformal fluid on the 2-sphere with $|\omega|\ell>1$, if we identify $${\cal P}_0 = \frac{m}{8\pi\ell^2}\,, \qquad \omega = -\frac{a}{\ell^2}\,.$$ Final remarks {#fin-rem} ============= In this paper, we used hydrodynamics in order to make predictions on the possible types of black holes in Einstein-Maxwell-AdS gravity. In particular, we classified the stationary equilibrium flows on ultrastatic manifolds with spatial sections of constant curvature, and then used these results to identify the dual black hole solutions. Although these are all contained in the Carter-Plebański family, only a few of them have been studied in the literature before, so that the major part is in principle new. The following table summarizes the results, relating to each spacetime the corresponding dual fluid configuration. It would be interesting to study more in detail the physics of these new black holes. Another possible direction for future work is to repeat our analysis for hydrodynamics in four dimensions (cf. e.g. [@Gubser:2010ui] for work in this direction) and to see if the dual metrics still enjoy any sort of algebraic speciality. Some remaining open questions concern for instance the boundary geometries of the Carter-Plebański metric that have either no irrotational Killing field $\xi$ ($\Delta<0$ in app. \[app-CP\]), or where $\xi$ is lightlike ($\Delta=0$). These cases include the black holes with noncompact horizons but finite entropy constructed recently in [@Gnecchi:2013mja; @Klemm:2014rda], as well as the cylindrical (or planar) solutions of [@Klemm:1997ea]. Although the boundary metric is still conformally flat for $\Delta\le 0$ (if the NUT charge vanishes), we were not able to find the coordinate transformation that makes this manifest. However, the explicit form of this diffeomorphism would be needed in order to quantitatively determine the hydrodynamic flow that is dual to these black holes. Another intriguing point is the absence of a net Lorentz force acting on the charged fluid on the boundary, as we saw in section \[dual-AdS-BH\]. It would be very interesting to see if this can be relaxed and, if so, what the holographic duals of such fluid configurations are. For instance, one might ask which gravity dual corresponds to a charged fluid rotating in a plane, with only a magnetic field orthogonal to that plane. In section \[H2-to-plane\] we saw that a conformal fluid in rigid rotation on hyperbolic space looks completely different when transformed to the 2-sphere or the plane: There it becomes highly dynamical, and takes the form of an expanding or contracting vortex. There is thus no need to have dynamical spacetimes (which are notoriously difficult to construct) in order to build holographic models of nonstationary (conformal) fluids. This raises the question if bulk geometries of the type considered here can have applications in a holographic description of the (dynamical) quark-gluon plasma produced in heavy-ion collisions, cf. [@McInnes:2013wba] for first attempts in this direction. We hope to come back to some of these points in the future. This work was partially supported by INFN. The authors would like to thank M. M. Caldarelli for useful comments on the manuscript. Notes on the Carter-Plebański metric {#app-CP} ==================================== In this section, we present a systematic classification of the possible types of black holes contained in the Carter-Plebański family , with a particular emphasis on the geometries that can arise on the conformal boundary. The various cases are distinguished by the number of real roots of the function $\mathsf{P}(p)$. This function must be positive in order for the induced metric on the horizon to have the right signature. We consider the case of vanishing NUT charge only, $l=0$, and define $\Gamma=\alpha-\mathrm{g}^2$, such that $\mathsf{P}(p)$ in boils down to $$\mathsf{P}(p) = \frac{p^4}{\ell^2}-\epsilon p^2+\Gamma\,.$$ Consider the discriminant $\Delta=\epsilon^2-4\Gamma/\ell^2$. For $\Delta\geq0$ one has $$\mathsf{P}(p) = \frac{1}{\ell^2}(p^2-\alpha_+)(p^2-\alpha_-)\,, \label{P-alpha-pm}$$ where $\alpha_{\pm}=\ell^2(\epsilon\pm\sqrt\Delta)/2$. We have then the following subcases: 1. If $\Gamma>0,\ \epsilon>2\sqrt\Gamma/\ell$, then $\Delta>0$, $\sqrt\Delta<\epsilon$, so $\alpha_{\pm}>0$, and $\mathsf{P}$ has 4 real roots, $$\mathsf{P}(p)=\frac{1}{\ell^2}(p-\sqrt\alpha_+)(p+\sqrt\alpha_+)(p-\sqrt\alpha_-)(p+\sqrt\alpha_-)\,.$$ $\mathsf{P}$ is positive for $|p|>\sqrt\alpha_+$ or $|p|<\sqrt\alpha_-$. In the latter region, use the scaling symmetry to set $\alpha_+=\ell^2$ without loss of generality, and define $a^2:=\alpha_-$. This gives the spherical KNAdS solution ( with $k=1$). In the range $|p|>\sqrt\alpha_+$, use to set $\alpha_-=\ell^2$, and define $a^2:=\alpha_+$, which leads to the super-rotating black hole . 2. If $\Gamma>0,\ \epsilon=2\sqrt\Gamma/\ell$, then $\Delta=0$, so $\alpha_{\pm}=\ell\sqrt\Gamma$, and $$\mathsf{P}(p) = \frac{1}{\ell^2}\left(p-\sqrt{\ell\sqrt\Gamma}\right)^2\left(p+\sqrt{\ell\sqrt\Gamma} \right)^2\,.$$ $\mathsf{P}$ is positive for $p\neq\pm\sqrt{\ell\sqrt\Gamma}$. By virtue of one can always take $\epsilon=2$, i.e., $\sqrt{\ell\sqrt{\Gamma}}=\ell$. Then, for $|p|<\ell$, we get the black holes that have a noncompact horizon with finite area, constructed recently in [@Gnecchi:2013mja; @Klemm:2014rda]. For $|p|>\ell$ one obtains new solutions that have not been discussed in the literature so far. 3. If $\Gamma>0,\ -2\sqrt\Gamma/\ell<\epsilon<2\sqrt\Gamma/\ell$, then $\Delta<0$, so $\mathsf{P}$ has no real roots and is always positive. These solutions are new, except the case $\epsilon=0$, which corresponds (with the definition $a^2:=\Gamma$) to the cylindrical black holes found in [@Klemm:1997ea]. 4. If $\Gamma>0,\ \epsilon=-2\sqrt\Gamma/\ell$, then $\Delta=0$, so $\alpha_{\pm}=-\ell\sqrt\Gamma$. $\mathsf{P}$ has no real roots and is always positive, $$\mathsf{P}(p)=\frac{1}{\ell^2}(p^2+\ell\sqrt\Gamma)^2\,.$$ Also this case has not been considered in the literature yet. 5. If $\Gamma>0,\ \epsilon<-2\sqrt\Gamma/\ell$, then $\Delta>0$, $\epsilon<-\sqrt\Delta$, so $\alpha_{\pm}<0$. $\mathsf{P}$ has no real roots and is given by . It is easy to see that one can always use to set $\epsilon=-1-\Gamma/\ell^2$. If we define $b^2:=\Gamma$, we obtain the boosting AdS$_4$ back holes . 6. If $\Gamma=0,\ \epsilon>0$, then $\Delta>0$, $\alpha_+=\epsilon\ell^2$, $\alpha_-=0$, and $$\mathsf{P}(p)=\frac{p^2}{\ell^2}(p-\ell\sqrt\epsilon)(p+\ell\sqrt\epsilon)\,,$$ which is positive for $|p|>\ell\sqrt\epsilon$. This case yields the solution , dual to a rotating fluid on $\mathbb{R}\times\text{E}^2$, with rotation parameter $a$ given by $\epsilon=a^2/\ell^2$. 7. If $\Gamma=0,\ \epsilon=0$, then $\Delta=0$, $\alpha_+=\alpha_-=0$, and $\mathsf{P}(p)=p^4/\ell^2$. This is again a hitherto undiscussed geometry. 8. If $\Gamma=0,\ \epsilon<0$, then $\Delta>0$, $\alpha_+=0$, $\alpha_-=\epsilon\ell^2$, and $$\mathsf{P}(p)=\frac{p^2}{\ell^2}(p^2-\epsilon\ell^2)\,,$$ which is positive for $p\neq 0$. By means of one can scale $\epsilon=-1$, and gets the solution , dual to a mixed flow on $\mathbb{R}\times\text{H}^2$. 9. If $\Gamma<0$, then $\Delta>0$, $\sqrt\Delta>|\epsilon|$, $\alpha_+>0$, $\alpha_-<0$, and $$\mathsf{P}(p)=\frac{1}{\ell^2}(p-\sqrt{\alpha_+})(p+\sqrt{\alpha_+})(p^2-\alpha_-)\,.$$ $\mathsf{P}$ is positive for $|p|>\sqrt{\alpha_+}$. Use to set $\alpha_-=-\ell^2$ and define $a$ by $a^2=\alpha_+$. This leads to the hyperbolic KNAdS black hole, i.e., with $k=-1$. The static Killing fields of the conformal boundary {#stat-kill-bdry} --------------------------------------------------- The metric $\hat g$ on the conformal boundary of the Carter-Plebański family is given by . The only Killing fields $\xi$ of $\hat g$ are linear combinations of $\partial_\tau$ and $\partial_\sigma$, i.e., $\xi=A\partial_\tau+B\partial_\sigma$, and the orthogonal distribution of $\xi$ is generated by the fields $f\partial_\tau+\partial_\sigma$ and $\partial_p$, where $$f=\frac{A p^2+B(\ell^2\mathsf{P}(p)-p^4)}{A-B p^2}\,.$$ $\xi$ is irrotational if and only if $\Delta\geq0$ and $A=\alpha_{\pm}B$. With this choice, the function $f$ reduces to $f_\pm=\alpha_\mp$. To see this, consider the orthogonal distribution of $\xi$, which is involutive if and only if $[f\partial_\tau+\partial_\sigma,\partial_p]=-\partial_p f\partial_\tau$ belongs to it, which happens when $\partial_pf$ vanishes, i.e. when $A^2-\epsilon\ell^2 AB+B^2\ell^2\Gamma=0$. This equation has solutions $A$ for $\Delta\ge 0$; these are $A_\pm=\ell^2B(\epsilon\pm\sqrt{\Delta})/2=\alpha_\pm B$. Plugging $A_\pm$ into $f$ yields $f_\pm=\alpha_\mp$. The only irrotational Killing fields are thus multiples of $$\xi_\pm=\alpha_\pm\partial_\tau+\partial_\sigma\,,$$ and the orthogonal distribution of $\xi_\pm$ is generated by $\xi_\mp$ and $\partial_p$. Now introduce, for $\Delta\geq 0$, the functions $$\Psi_\pm(p) = {\hat g}(\xi_\pm,\xi_\pm)=\pm\ell^2\sqrt\Delta(p^2-\alpha_\pm )\,.$$ We have then: - For $\Delta<0$ (case 3) there are no irrotational Killing fields. - For $\Delta=0$ (cases 2,4,7), $\xi_+=\xi_-$ is lightlike, so there is no static Killing field. - For $\Gamma>0$ and $\epsilon>2\sqrt\Gamma/\ell$ (case 1), one has $\alpha_\pm>0$, so $\xi_+$ is timelike for $|p|<\sqrt{\alpha_+}$ and spacelike for $|p|>\sqrt\alpha_+$, while $\xi_-$ is timelike for $|p|>\sqrt{\alpha_-}$ and spacelike for $|p|<\sqrt{\alpha_-}$. - If $\Gamma>0$ and $\epsilon<-2\sqrt\Gamma/\ell$ (case 5), then $\alpha_\pm<0$, and thus $\xi_+$ is always spacelike, whereas $\xi_-$ is always timelike. - If $\Gamma=0$ and $\epsilon>0$ (case 6), then $\alpha_+>0$, $\alpha_-=0$, hence $\xi_+$ is timelike for $|p|<\sqrt{\alpha_+}$ and spacelike for $|p|>\sqrt{\alpha_+}$, while $\xi_-$ is timelike for $p\neq 0$ and never spacelike. - When $\Gamma=0$ and $\epsilon<0$ (case 8), then $\alpha_+=0$, $\alpha_-<0$, so $\xi_+$ is spacelike for $p\neq 0$ and never timelike, whereas $\xi_-$ is always timelike. - For $\Gamma<0$ (case 9), we have $\alpha_+>0$, $\alpha_-<0$, and thus $\xi_+$ is timelike for $|p|<\sqrt{\alpha_+}$ and spacelike for $|p|>\sqrt{\alpha_+}$, while $\xi_-$ is always timelike. In each case with $\Delta>0$, in the regions where $\mathsf{P}(p)>0$ either $\xi_+$ is spacelike and $\xi_-$ is timelike or vice versa. $\xi_\pm$ do not change their causal character inside these regions, and therefore the spacetime is static. Moreover, $\xi_+,\xi_-,\partial_p$ form an orthogonal frame. Now introduce coordinates $\tau_\pm$ such that $\partial_{\tau_\pm}=\xi_\pm$, given by $$\label{static coordinates on CP boundary} \tau_\pm = \pm\frac{1}{\ell^2\sqrt\Delta}(\tau-\alpha_\mp\sigma)\,.$$ In these coordinates the boundary metric reads $${\hat g} = \Psi_+(p)\mathrm{d}\tau_+^2+\Psi_-(p)\mathrm{d}\tau_-^2+\frac{\ell^2}{\mathsf{P}(p)}\mathrm{d}p^2\,.$$ If $\xi_+$ is timelike ($\Psi_+<0$), then we rescale $\hat g$ with $\Omega^2=-\kappa/\Psi_+(p)$ (where $\kappa\in\mathbb{R}_+$ has been introduced for later convenience) to get the ultrastatic metric $$\tilde g = -\kappa\mathrm{d}\tau_+^2-\frac{\kappa}{\Psi_+(p)}\left(\frac{\ell^2}{\mathsf{P}(p)} \mathrm{d}p^2+\Psi_-(p)\mathrm{d}\tau_-^2\right)\,. \label{metr:ultra:+}$$ Note that the sections of constant $\tau_+$ have constant scalar curvature $R=2\alpha_+\Delta/\kappa$. If $\xi_-$ is timelike ($\Psi_-<0$), then we rescale $\hat g$ with $\Omega^2=-\kappa/\Psi_-(p)$ to get the ultrastatic metric $$\tilde g = -\kappa\mathrm{d}\tau_-^2-\frac{\kappa}{\Psi_-(p)}\left(\frac{\ell^2}{\mathsf{P}(p)} \mathrm{d}p^2+\Psi_+(p)\mathrm{d}\tau_+^2\right)\,, \label{metr:ultra:-}$$ whose $\tau_-=\text{constant}$ sections have scalar curvature $R=2\alpha_-\Delta/\kappa$. We have thus shown that in the cases 1,5,6,8,9 the spacetime has one static Killing field, and is conformal to an ultrastatic manifold with spatial sections of constant curvature. In what follows, we shall consider each of these cases separately, and show that they correspond precisely to the equilibrium flows considered in section \[stat flow ultrastatic st\]. - Case 1, region $p\in(-\sqrt{\alpha_-},\sqrt{\alpha_-})$ Consider case 1 ($\Gamma>0$, $\epsilon>2\sqrt\Gamma/\ell$). Take the region $p\in(-\sqrt{\alpha_-},\sqrt{\alpha_-})$, where $\mathsf{P}(p)>0$, $\Psi_+(p)<0$, $\Psi_-(p)>0$, and thus the boundary metric is conformal to . If we choose $\kappa=\ell^2\alpha_+\Delta$ (which is positive), the sections of constant $\tau_+$ have scalar curvature $R=2/\ell^2$. Now introduce new coordinates $(T,\Theta,\Phi)$ defined by $$\label{spherical coordinates in case 1.1} T = \ell\sqrt{\alpha_+\Delta}\tau_+\,, \qquad \cos\Theta = \sqrt{\frac{\alpha_+ - \alpha_-}{\alpha_-(\alpha_+-p^2)}}p\,, \qquad \Phi=-\sqrt{\alpha_-\Delta}\tau_-\,,$$ where $\Theta$ ranges in $(0,\pi)$. Then simplifies to $$\label{g-tilde:case1.1} \tilde g = -\mathrm{d}T^2+\ell^2\left(\mathrm{d}\Theta^2+\sin^2\Theta\mathrm{d}\Phi^2\right)\,.$$ In section \[dual-AdS-BH\] it was found that the 3-velocity of the fluid dual to the Carter-Plebanski geometry is given by $u=\partial_\tau$. In the coordinates and after the conformal rescaling with $\Omega^2=-\kappa/\Psi_+(p)$, this becomes $$\label{u-tilde:case1.1} \tilde{u}=\frac{1}{\sqrt{1-\omega^2\ell^2\sin^2\Theta}}\left(\partial_T+\omega\partial_\Phi\right)\,,$$ with $\omega=\sqrt{\frac{\alpha_-}{\alpha_+}}\frac{1}{\ell}$. Notice that $\omega\in(0,\frac{1}{\ell})$. This is precisely the flow on $\mathbb{R}\times\text{S}^2$ considered in section \[stat conf fluid sph\]. - Case 1, region $p\in(-\infty,-\sqrt{\alpha_+})\cup(\sqrt{\alpha_+},+\infty)$ Consider still case 1, but this time take the region $|p|>\sqrt{\alpha_+}$, where $\mathsf{P}(p)>0$, $\Psi_+(p)>0$, $\Psi_-(p)<0$, and thus the boundary metric is conformal to . If we choose $\kappa=\ell^2\alpha_-\Delta$ (which is positive), the scalar curvature of the constant $\tau_-$ sections becomes $R=2/\ell^2$. Now introduce new coordinates $(T,\Theta,\Phi)$ according to $$\label{spherical coordinates in case 1.2} T = -\ell\sqrt{\alpha_-\Delta}\tau_-\,, \qquad \sin\Theta = \sqrt{\frac{\alpha_-(p^2-\alpha_+)}{\alpha_+(p^2-\alpha_-)}}\,, \qquad \Phi=\sqrt{\alpha_+\Delta}\tau_+\,,$$ where now $\Theta$ ranges in $(0,\arcsin\sqrt{\frac{\alpha_-}{\alpha_+}})$ when $p\in(\sqrt{\alpha_+},+\infty)$ and $(\pi-\arcsin\sqrt{\frac{\alpha_-}{\alpha_+}},\pi)$ when $p\in(-\infty,-\sqrt{\alpha_+})$. Then the metric becomes again , and the 3-velocity $u=\partial_\tau$ of the fluid is still transformed into , but this time with $\omega=\sqrt{\frac{\alpha_+}{\alpha_-}}\frac{1}{\ell}$, which satisfies $\omega>1/\ell$. Moreover, $\Theta$ is now restricted to the polar caps $\omega\ell\sin\Theta<1$. This is again the flow on $\mathbb{R}\times\text{S}^2$ considered in section \[stat conf fluid sph\], but with $\omega>1/\ell$. - Case 5 In this case ($\Gamma>0$, $\epsilon<-2\sqrt\Gamma/\ell$) we have, for each $p\in\mathbb{R}$, $\mathsf{P}(p)>0$, $\Psi_+(p)>0$, $\Psi_-(p)<0$, hence the boundary metric is conformal to . If we choose $\kappa=-\ell^2\alpha_-\Delta$ (which is positive), the sections of constant $\tau_-$ have scalar curvature $R=-2/\ell^2$. After the coordinate change $$\label{hyperbolic coordinates in case 5} T = -\ell\sqrt{-\alpha_-\Delta}\tau_-\,, \qquad \sinh\Theta = \sqrt{\frac{\alpha_+ - \alpha_-}{-\alpha_+(p^2-\alpha_-)}}p\,, \qquad \Phi = \sqrt{-\alpha_+\Delta}\tau_+\,,$$ where $|\Theta|<\text{arcosh}\sqrt{\frac{\alpha_-}{\alpha_+}}$, the metric boils down to $$\tilde g = -\mathrm{d}T^2+\ell^2\left(\mathrm{d}\Theta^2+\cosh^2\Theta\mathrm{d}\Phi^2\right)\,,$$ while the fluid velocity becomes $$\tilde{u} = \frac{1}{\sqrt{1-\beta^2\ell^2\cosh^2\Theta}}\left(\partial_T+\beta\partial_\Phi\right)\,,$$ with $\beta=\sqrt{\frac{\alpha_+}{\alpha_-}}\frac{1}{\ell}$. Notice that $\beta\in(0,1/\ell)$ and $\beta\ell\cosh\Theta<1$. This is the purely translational flow on $\mathbb{R}\times\text{H}^2$ of section \[sec:transl-flow\][^15]. - Case 6 In this case ($\Gamma=0$, $\epsilon>0$), $\mathsf{P}(p)$ is positive for $|p|>\sqrt{\alpha_+}$, where $\Psi_+(p)>0$, $\Psi_-(p)<0$. The boundary metric is thus conformal to , and (since $\alpha_-=0$) the scalar curvature of the constant $\tau_-$ sections vanishes. Now put $\kappa=\ell^4\Delta$ and introduce new coordinates $(T,R,\Phi)$ defined by $$\label{polar coordinates in case 6} T = -\ell^2\sqrt{\Delta}\tau_-\,, \qquad R = \frac{\ell^2}{\sqrt{\alpha_+}}\sqrt{1-\frac{\alpha_+}{p^2}}\,, \qquad \Phi=\frac{\sqrt{\alpha_+^3}}{\ell^2}\tau_+\,,$$ where $0<R<\ell/\sqrt{\epsilon}$. Then turns into $$\tilde g = -\mathrm{d}T^2+\mathrm{d}R^2+R^2\mathrm{d}\Phi^2\,,$$ and the 3-velocity of the fluid becomes $$\tilde{u} = \frac{1}{\sqrt{1-\omega^2 R^2}}\left(\partial_T+\omega\partial_\Phi\right)\,,$$ with $\omega=\sqrt{\alpha_+}/\ell^2$. Notice that $R<1/\omega$. This is the rigidly rotating fluid on Minkowski space considered in \[sec:rot-plane\]. - Case 8 Here we have $\Gamma=0$, $\epsilon<0$, and $\mathsf{P}(p)>0$ for $p\neq 0$. Moreover, $\Psi_+(p)>0$ and $\Psi_-(p)<0$, and thus the boundary metric is conformal to . If we choose $\kappa=-\ell^2\alpha_-\Delta$ (which is positive), the scalar curvature of the sections $\tau_-=\text{constant}$ becomes $R=-2/\ell^2$. Now introduce new coordinates $(T,A,B)$ according to $$\label{special coordinates in case 8.1} T = -\ell\sqrt{-\alpha_-\Delta}\tau_-\,, \qquad A = \frac{1}{2}\log\left(1-\frac{\alpha_-}{p^2}\right)+\ln\ell\beta\,, \qquad B = \ell\beta\sqrt{-\alpha_-\Delta}\tau_+\,,$$ where we have introduced an arbitrary parameter $\beta>0$, which can be chosen as $\beta=1/\ell$ without loss of generality. Note that $\ln\ell\beta<A<\infty$. This casts into the form $$\tilde g = -\mathrm{d}T^2+\ell^2(\mathrm{d}A^2+e^{-2A}\mathrm{d}B^2)\,,$$ while the 3-velocity $u=\partial_\tau$, after the conformal rescaling $\tilde g=\Omega^2\hat g$, becomes $$\tilde{u} = \frac{1}{\sqrt{1-\beta^2\ell^2 e^{-2A}}}\left(\partial_T+\beta\partial_B\right)\,.$$ This corresponds to the mixed (parabolic) flow on $\mathbb{R}\times\text{H}^2$ of section \[sec:mixed-flow\]. - Case 9 The last case is $\Gamma<0$. The polynomial $\mathsf{P}(p)$ is positive for $p>\sqrt{\alpha_+}$, where $\Psi_+(p)>0$, $\Psi_-(p)<0$. Therefore the boundary metric is conformal to . If we choose $\kappa=-\ell^2\alpha_-\Delta$ (which is positive), the constant $\tau_-$ sections have scalar curvature $R=-2/\ell^2$. After the coordinate change $$\label{hyperbolic coordinates in case 9.1} T = -\ell\sqrt{-\alpha_-\Delta}\tau_-\,, \qquad \sinh\Theta = \sqrt{\frac{-\alpha_-(p^2-\alpha_+)}{\alpha_+(p^2-\alpha_-)}}\,, \qquad \Phi=\sqrt{\alpha_+\Delta}\tau_+\,,$$ where $\Theta$ ranges in $(0,\text{arcsinh}\sqrt{-\alpha_-/\alpha_+})$, the metric boils down to $$\tilde g = -\mathrm{d}T^2+\ell^2\left(\mathrm{d}\Theta^2+\sinh^2\Theta\mathrm{d}\Phi^2\right)\,,$$ and the 3-velocity of the fluid is $$\tilde{u} = \frac{1}{\sqrt{1-\omega^2\ell^2\sinh^2\Theta}}\left(\partial_T+\omega\partial_\Phi\right)\,,$$ with $\omega=\sqrt{-\alpha_+/\alpha_-}/\ell$. Notice that $\omega\ell\sinh\Theta<1$. This corresponds to the rigidly rotating fluid on $\mathbb{R}\times\text{H}^2$, considered in \[fluid rig rot hyp\]. Note that in all cases where the positivity region of $\mathsf{P}(p)$ consists of two disconnected parts (1b,6,8,9), the corresponding coordinate transformations map both the branch where $p$ is positive and the one where $p$ is negative to the same spacetime. (In case 1b up to isometries, since the region $p<-\sqrt{\alpha_+}$ maps to the lower polar cap, while $p>\sqrt{\alpha_+}$ maps to the upper polar cap). Proof of propositions {#app-proof} ===================== [**Prop. \[prop-killing\]:**]{} Equ.  implies that $$\label{theta} \nabla_t u^\mu=0\,,\ \ \ \ \ \nabla_\mu u^t=\partial_\mu u^t\,,\ \ \ \ \ \nabla_i u^j=v^j\partial_i\gamma+\gamma\bar{\nabla}_i v^j\,, \ \ \ \ \ \vartheta = v^i\partial_i\gamma+\gamma\bar{\nabla}_i v^i\,,$$ where $\bar{\nabla}$ denotes the Levi-Civita connection of $(\Sigma,\bar{g})$. Moreover $$\label{Dgamma} \partial_i\gamma=\gamma^3 v_j\bar{\nabla}_i v^j\,.$$ These expressions can be used in to compute $\sigma^{\mu\nu}$, with the result &\^[tt]{} = (d-1-v\^2)v\^i\_i - |\_iv\^i, \[sigma\^tt\]\ &\^[ti]{} = \^2 v\^iv\^j\_j+|[g]{}\^[ij]{}\_j+\^3(v\^j|\_j v\^i-v\^i|\_j v\^j )\[sigma\^ti\],\ &\^[ij]{} = (|[g]{}\^[ik]{}v\^j+|[g]{}\^[jk]{}v\^i-|[g]{}\^[ij]{}v\^k )\_k+ (|\^iv\^j+|\^jv\^i-|[g]{}\^[ij]{}|\_k v\^k )\ &      +\^2 v\^iv\^jv\^k\_k+ \^3(v\^iv\^k|\_k v\^j+v\^jv\^k|\_k v\^i-v\^iv\^j|\_k v\^k ).\[sigma\^ij\] Putting together eqns.  and we find that &\^[ij]{}= v\^i\^[tj]{}+v\^j\^[ti]{}-\^2 v\^iv\^jv\^k\_k+ \^3 v\^iv\^j|\_k v\^k&\ &     -|[g]{}\^[ij]{}(v\^k\_k+|\_k v\^k)+(|\^iv\^j+|\^jv\^i).\[sigma\^ij (2)\] Now let $\vartheta=0$ and $\sigma^{\mu\nu}=0$. From one gets $$\label{vDgamma} v^i\partial_i\gamma=\frac{\gamma v^2}{d-1-v^2}\bar{\nabla}_iv^i\,.$$ Using the last equ. of , we obtain then $$0 = \vartheta = v^i\partial_i\gamma +\gamma\bar{\nabla}_iv^i = \gamma\frac{d-1}{d-1-v^2}\bar{\nabla}_iv^i\,,$$ and thus $$\label{Div v=0} \bar{\nabla}_iv^i=0\,.$$ Plugging into yields $$0=\sigma^{ij}= \frac{\gamma\bar{\nabla}_k v^k}{(d-1)(d-1-v^2)}\left((d-1)v^iv^j-\gamma^2 v^2 v^iv^j-(d-1)\bar{g}^{ij} \right )+\frac{\gamma}{2}(\bar{\nabla}^iv^j+\bar{\nabla}^jv^i)\,,$$ and hence, by , $$\label{v Killing} \bar{\nabla}^iv^j+\bar{\nabla}^jv^i=0\,,$$ i.e., $v$ is Killing. Viceversa, suppose that $v$ is a Killing field: Taking the trace of gives ; moreover, eqns.  and give $$\label{vDgamma=0} v^i\partial_i\gamma=\gamma^3v^iv^j\bar{\nabla}_iv_j=0\,,$$ so that $\vartheta=0$ by equ. . Now leads to $\sigma^{tt}=0$, while equ.  becomes, using , and , $$\sigma^{ti}=\frac{1}{2}\bar{g}^{ij}\partial_j\gamma+\frac{\gamma^3}{2}v^j\bar{\nabla}_j v^i=\frac{\gamma^3}{2}\bar{g}^{ij}v^k(\bar{\nabla}_j v_k+\bar{\nabla}_k v_j)=0\,.$$ Finally, using these results in we find $\sigma^{ij}=0$, which completes the proof.\ [**Prop. \[prop-NS\]:**]{} Since $\vartheta=0$ and $\partial_t {\cal P}=0$, we have $$\nabla_\mu T^{\mu\nu}=\partial_i{\cal P}(d\, u^i u^\nu+g^{i\nu})+{\cal P}d\,u^\mu\nabla_\mu u^\nu\,.$$ Using eqns. , , one gets $$u^\mu\nabla_\mu u^t=u^i\partial_i u^t=\gamma v^i\partial_i\gamma=0\,, \qquad u^\mu\nabla_\mu u^j=u^i\nabla_i u^j=\gamma v^i(v^j\partial_i\gamma+\gamma\bar{\nabla}_iv^j) = \gamma^2v^i\bar{\nabla}_iv^j\,,$$ and thus $$\nabla_\mu T^{\mu t}=d\,\gamma^2 v^i\partial_i {\cal P}\,, \qquad \nabla_\mu T^{\mu j}=d\,\gamma^2 v^i v^j \partial_i {\cal P} + \partial^j {\cal P}+ {\cal P}d\gamma^2 v^i\bar{\nabla}_i v^j\,.$$ The vanishing of these two expressions is equivalent to[^16] $$\label{equiv-equ} \partial_j {\cal P} + {\cal P}d\gamma^2 v^i\bar{\nabla}_i v_j=0\,,$$ which can be rewritten as[^17] $\partial_i\ln{\cal P}=d\partial_i\ln\gamma$.\ [**Prop. \[prop-heat-flux\]:**]{} Using we get $$a_t = -a^t = -\gamma v^i\partial_i\gamma\,, \qquad a_i = \bar{g}_{ij}a^j = \gamma v^k (v_i\partial_k\gamma + \gamma\bar{\nabla}_kv_i)\,.$$ Owing to $\partial_i\gamma=\gamma^3v_j\bar{\nabla}_iv^j$ one has moreover $$P^{t\nu}a_\nu = \gamma^4 v^i v^j\bar{\nabla}_i v_j\,, \qquad P^{i\nu}a_\nu = \gamma^2 v^k \bar{\nabla}_k v^i + \gamma v^iv^k\partial_k\gamma\,.$$ The components of the heat flux in become thus &q\^t = -((\^2-1)\_t[T]{}+\^2 v\^i\_i[T]{}+[T]{}\^4 v\^i v\^j|\_iv\_j),\ &q\^i = -(\^2v\^i\_t[T]{}+(|[g]{}\^[ij]{}+\^2 v\^iv\^j) \_j[T]{}+[T]{}(\^2v\^k|\_kv\^i+v\^iv\^k\_k) ). \[comp-heat-flux\] Since our assumptions imply $\partial_t{\cal T}=0$, $\bar{\nabla}_iv_j+\bar{\nabla}_jv_i=0$, $v^i\partial_i\gamma=0$, $\partial_i\gamma=-\gamma^3v^k\bar{\nabla}_kv_i$, boils down to $$q^t = -\kappa\gamma^2 v^i\partial_i{\cal T}\,, \qquad q^i = -\kappa{\cal T}\partial^i\ln\frac{\cal T}{\gamma}+v^iq^t\,.$$ The vanishing of $q^\mu$ gives thus $\partial^i\ln({\cal T}/\gamma)=0$, i.e., ${\cal T}/\gamma=\tau$, where $\tau$ is a constant.\ [**Prop. \[prop-diff-curr\]:**]{} The components of the diffusion current in read $$q_{\text e}^t = -D\left((\gamma^2-1)\partial_t\frac{\mu}{\cal T}+\gamma^2 v^i\partial_i \frac{\mu}{\cal T}\right)\,, \quad q_{\text e}^i = -D\left(\gamma^2 v^i\partial_t\frac{\mu}{\cal T}+(\bar{g}^{ij}+\gamma^2 v^iv^j)\partial_j\frac{\mu}{\cal T}\right)\,. \label{comp-diff-curr}$$ Stationarity implies $\partial_t(\mu/{\cal T})=0$, hence reduces to $$q_{\text e}^t = -D\gamma^2 v^i\partial_i\frac{\mu}{\cal T}\,, \qquad q_{\text e}^i = -D\partial^i\frac{\mu}{\cal T}+v^i q_{\text e}^t\,.$$ $q_{\text e}^\mu=0$ leads thus to $\partial^i(\mu/{\cal T})=0$, i.e., $\mu/{\cal T}=\psi$, with $\psi$ constant. 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[^2]: Note in this context that the instability of the effective fluid that describes higher-dimensional asymptotically flat black branes, analyzed in [@Camps:2010br], is not of the Rayleigh-Plateau type, but rather one in the sound modes. [^3]: Since every static spacetime is conformally ultrastatic, these results extend of course to arbitrary static spacetimes in the case of conformal hydrodynamics. [^4]: In the case of zero viscosities, $\zeta=\eta=0$, one can in principle allow for nonvanishing expansion and shear tensor. In particular, for conformal fluids (cf. below), the bulk viscosity vanishes, and therefore the third term on the rhs of is zero without imposing $\vartheta=0$. [^5]: For a Weyl-covariant formalism that simplifies the study of conformal hydrodynamics cf. [@Loganayagam:2008is]. [^6]: This corresponds to the choice made in case 8 of appendix \[stat-kill-bdry\]. [^7]: This scaling limit eliminates the acceleration parameter. [^8]: The nonvanishing components of the Cotton tensor $C_{\mu\nu\rho}$ of $\hat g$ are given by $$C_{\tau p \sigma}=-C_{p\tau \sigma}=C_{\sigma \tau p}=-C_{\tau \sigma p}=\frac{2l}{\ell^2}\,, \qquad C_{\sigma p \tau}=-C_{p \sigma \tau}=\frac{4l}{\ell^2}\,, \qquad C_{p \sigma \sigma}=-C_{\sigma p \sigma}=\frac{6l p^2}{\ell^2}\,.$$ [^9]: Holographic fluids that are dual to geometries with NUT charge were considered in [@Leigh:2011au; @Caldarelli:2012cm; @Mukhopadhyay:2013gja]. [^10]: One has $\tilde F_{\mu\nu}=\hat F_{\mu\nu}$ and $\tilde J^\mu=\Omega^{-d}\hat J^\mu$. In this way, the MHD equations are conformally invariant. [^11]: In the case of the spherical KNAdS black hole this fact was first noticed in [@Caldarelli:2008ze]. [^12]: The remaining fluid parameters $\omega$ and $\psi$ are fixed by and (combined with $\rho_{\text e}=\tau^2\gamma^2h'(\psi)$, cf. ) respectively. The function $h(\psi)$ determining the hydrodynamic grandcanonical potential is that of the unrotating black hole, given by eqns. (3.19) and (3.20) of [@Caldarelli:2008ze]. [^13]: In the latter case the dual fluid lives on a quotient space of $\text{H}^2$. [^14]: Strictly speaking, in order to prove this rigorously, one would have to apply the map (4.1) of [@Bhattacharyya:2008mz], and show that this yields (up to second order in the boundary derivative expansion) the metric . We leave this for future work. In this context, note also that [@Bhattacharyya:2008mz] deals only with the uncharged case. We are not aware of a magnetohydrodynamical generalization of the results of [@Bhattacharyya:2008mz]. [^15]: Set $X=\cosh\Theta\sinh\Phi$, $Y=\sinh\Theta$ to compare with section \[sec:transl-flow\]. [^16]: Contracting with $v^j$ yields $v^j\partial_j {\cal P}=0$ by . [^17]: Use $\partial_j\gamma=\gamma^3 v^i\bar{\nabla}_j v_i= -\gamma^3 v^i\bar{\nabla}_i v_j$, which follows from and .

ArXiv

--- abstract: 'Dark energy can modify the dynamics of dark matter if there exists a direct interaction between them. Thus a measurement of the structure growth, e.g., redshift-space distortions (RSD), can provide a powerful tool to constrain the interacting dark energy (IDE) models. For the widely studied $Q=3\beta H\rho_{de}$ model, previous works showed that only a very small coupling ($\beta\sim\mathcal{O}(10^{-3})$) can survive in current RSD data. However, all of these analyses had to assume $w>-1$ and $\beta>0$ due to the existence of the large-scale instability in the IDE scenario. In our recent work \[Phys. Rev. D [**90**]{}, 063005 (2014)\], we successfully solved this large-scale instability problem by establishing a parametrized post-Friedmann (PPF) framework for the IDE scenario. So we, for the first time, have the ability to explore the full parameter space of the IDE models. In this work, we reexamine the observational constraints on the $Q=3\beta H\rho_{de}$ model within the PPF framework. By using the Planck data, the baryon acoustic oscillation data, the JLA sample of supernovae, and the Hubble constant measurement, we get $\beta=-0.010^{+0.037}_{-0.033}$ ($1\sigma$). The fit result becomes $\beta=-0.0148^{+0.0100}_{-0.0089}$ ($1\sigma$) once we further incorporate the RSD data in the analysis. The error of $\beta$ is substantially reduced with the help of the RSD data. Compared with the previous results, our results show that a negative $\beta$ is favored by current observations, and a relatively larger interaction rate is permitted by current RSD data.' author: - 'Yun-He Li' - 'Jing-Fei Zhang' - 'Xin Zhang[^1]' title: 'Exploring the full parameter space for an interacting dark energy model with recent observations including redshift-space distortions: Application of the parametrized post-Friedmann approach' --- Introduction {#sec:intro} ============ Dark energy and dark matter are the dominant sources for the evolution of the current Universe [@Ade:2013zuv]. Both are currently only indirectly detected via their gravitational effects. There might, however, exist a direct non-gravitational interaction between them that does not violate current observational constraints. Furthermore, such a dark sector interaction can provide an intriguing mechanism to solve the “coincidence problem” [@coincidence; @Comelli:2003cv; @Zhang:2005rg; @Cai:2004dk] and also induce new features to structure formation by exerting a nongravitational influence on dark matter [@Amendola:2001rc; @Bertolami:2007zm; @Koyama:2009gd]. In an interacting dark energy (IDE) scenario, the energy conservation equations of dark energy and cold dark matter satisfy $$\begin{aligned} \label{rhodedot} \rho'_{de} &=& -3\mathcal{H}(1+w)\rho_{de}+ aQ_{de}, \\ \label{rhocdot} \rho'_{c} &=& -3\mathcal{H}\rho_{c}+ aQ_{c},~~~~~~Q_{de}=-Q_{c}=Q,\end{aligned}$$ where $Q$ denotes the energy transfer rate, $\rho_{de}$ and $\rho_{c}$ are the energy densities of dark energy and cold dark matter, respectively, $\mathcal{H}=a'/a$ is the conformal Hubble expansion rate, a prime denotes the derivative with respect to the conformal time $\tau$, $a$ is the scale factor of the Universe, and $w$ is the equation of state parameter of dark energy. Several forms for $Q$ have been constructed and constrained by observational data [@He:2008tn; @He:2009mz; @He:2009pd; @He:2010im; @Boehmer:2008av; @Guo:2007zk; @Xia:2009zzb; @Wei:2010cs; @Li:2011ga; @Li:2013bya]. The common data sets used in these works are the cosmic microwave background (CMB), the baryon acoustic oscillation (BAO), the type Ia supernovae (SNIa), as well as the Hubble constant measurement. These observations constrain the IDE models mainly by the geometric measurement information, leading to a significant degeneracy between the constraint results of interaction and background parameters. This degeneracy results from the fact that the IDE model cannot be distinguished from the uncoupled dark energy model in the background evolution since the expansion history of the Universe given by an IDE model and an uncoupled dark energy model can mimic each other by adjusting the values of their free parameters. Fortunately, the dynamics of dark matter can be modified by dark energy in an IDE model, so any observation containing the structure formation information might be a powerful tool to break this degeneracy. Redshift-space distortions (RSD) arising from peculiar velocities of galaxies on an observed galaxy map provide a direct measurement of the linear growth rate $f(a)$ of the large-scale structure formation [@Peacock:2001gs; @Guzzo:2008ac]. Currently, a number of RSD data are available from a variety of galaxy surveys, such as 6dFGS, [@RSD6dF], 2dFGRS [@RSD2dF], WiggleZ [@RSDwigglez], SDSS LRG DR7 [@RSDsdss7], BOSS CMASS DR11 [@Beutler:2013yhm], and VIPERS [@RSDvipers]. These RSD measurements have been used to constrain the IDE models [@Honorez:2010rr; @Yang:2014gza; @yang:2014vza; @Wang:2014xca; @Yang:2014hea]. For the widely studied $Q=3\beta H\rho_{de}$ model, recent CMB+BAO+SNIa data give $\beta=0.209^{+0.0711}_{-0.0403}$ ($1\sigma$), while the fit result becomes $\beta=0.00372^{+0.00077}_{-0.00372}$ ($1\sigma$) once the RSD data are added to the analysis [@Yang:2014gza]. This result shows that a large interaction rate for the $Q=3\beta H\rho_{de}$ model is ruled out by the RSD data. However, the above results may not reflect the actual preference of the data sets, because the full parameter space cannot be explored in these works, due to the well-known large-scale instability existing in the IDE scenario. The cosmological perturbations will blow up on the large scales for the $Q\propto \rho_{de}$ model with the early-time $w<-1$ or $\beta<0$ [@Clemson:2011an; @He:2008si] and for the $Q\propto \rho_{c}$ model with the early-time $w>-1$ [@Valiviita:2008iv]. So to avoid this instability, one has to assume $w>-1$ and $\beta>0$ for the $Q\propto \rho_{de}$ model and $w<-1$ for the $Q\propto \rho_{c}$ model in the observational constraint analyses. In practice, the $Q\propto \rho_{c}$ model with $w<-1$ is not favored by the researchers, since $w<-1$ will lead to another instability of our Universe in a finite future. Thus, the $Q\propto \rho_{de}$ case with $w>-1$ and $\beta>0$ becomes the widely studied IDE model in the literature. The large-scale instability arises from the way of calculating the dark energy pressure perturbation $\delta p_{de}$. In the standard linear perturbation theory, dark energy is considered as a nonadiabatic fluid. Thus, $\delta p_{de}$ contains two parts, the adiabatic pressure perturbation in terms of the adiabatic sound speed and the intrinsic nonadiabatic pressure perturbation in terms of the rest frame sound speed. If dark energy interacts with dark matter, then the interaction term $Q$ will enter the expression of the nonadiabatic pressure perturbation of dark energy. For some specific values of $w$ and $\beta$, as mentioned above, the nonadiabatic mode grows fast at the early times and soon leads to rapid growth of the curvature perturbation on the large scales [@Valiviita:2008iv]. However, current calculation of $\delta p_{de}$ may not reflect the real nature of dark energy, since it can also bring instability when $w$ crosses the phantom divide $w=-1$ even for the uncoupled dark energy [@Vikman:2004dc; @Hu:2004kh; @Caldwell:2005ai; @Zhao:2005vj]. As it is, finding an effective theoretical framework to handle the cosmological perturbations of dark energy may be a good choice before we exactly know how to correctly calculate $\delta p_{de}$. The simplified version of the parametrized post-Friedmann (PPF) approach [@Hu:2008zd; @Fang:2008sn] is just an effective framework but is constructed for the uncoupled dark energy models. In our recent work [@Li:2014eha], we established a PPF framework for the IDE scenario. The large-scale instability problem in all the IDE models can be successfully solved within such a generalized PPF framework. As an example, we used the observational data to constrain the $Q=3\beta H\rho_{c}$ model without assuming any specific priors on $w$ and $\beta$. The fit result showed that the full parameter space of this model can be explored within the PPF framework (also see Ref. [@Richarte:2014yva] for a similar follow-up analysis). In this work, we focus on the widely studied $Q=3\beta H\rho_{de}$ model with a constant $w$. We use the PPF approach to handle its cosmological perturbations. As mentioned above, previous observational constraints on this model have to assume $w>-1$ and $\beta>0$ to avoid the large-scale instability. Within the PPF framework established in Ref. [@Li:2014eha], we, for the first time, have the ability to explore the full parameter space of this model. So it is of great interest to see how the constraint results change when we let the parameter space of this model fully free. We perform a full analysis on the $Q=3\beta H\rho_{de}$ model by using current observations including the Planck data, the seven data points of BAO, the recent released JLA sample of SNIa, the Hubble constant measurement, and the ten data points of RSD as well. We show that current observations actually favor a negative $\beta$ when $w<-1$ and $\beta<0$ are also allowed. Moreover, with the help of the RSD data, $\beta$ can be tightly constrained, but unlike the previously obtained results $\beta\sim\mathcal{O}(10^{-3})$ in Refs. [@Yang:2014gza; @yang:2014vza], a relatively larger absolute value of $\beta$ (about $\mathcal{O}(10^{-2})$) is favored by the RSD data. Our paper is organized as follows. In Sec. \[sec:pertur\], we give the general perturbation equations in the IDE scenario. The perturbations of dark matter are given by the standard linear perturbation theory, while those of dark energy are calculated by using the PPF approach established in Ref. [@Li:2014eha]. Some details of the PPF approach as a supplement of Ref. [@Li:2014eha] are also presented in this section. In Sec. \[sec:constraint\], we show how we use the observations to constrain the $Q=3\beta H\rho_{de}$ model, and give a detailed discussion on the fit results. Our conclusions are given in Sec. \[sec:conclusions\]. In Appendix \[app:fzeta\], we introduce how to calibrate the function $f_\zeta(a)$ of the PPF approach in a specific IDE model. Perturbation equations in the IDE scenario {#sec:pertur} ========================================== General equations ----------------- A dark sector interaction in a perturbed Universe will influence the scalar perturbation evolutions. So let us start with the scalar perturbation theory in an FRW universe. The scalar metric perturbations can be expressed in general in terms of four functions, $A$, $B$, $H_L$, and $H_T$ [@Kodama:1985bj; @Bardeen], $$\begin{aligned} &\delta {g_{00}} = -a^{2} (2 {{{A}}}Y),\qquad\delta {g_{0i}} = -a^{2} {{{B}}} Y_i, \nonumber\\ & \qquad\delta {g_{ij}} = a^{2} (2 {{{H}_L}} Y \gamma_{ij} + 2 {{{H}_T}Y_{ij}}), \label{eqn:metric}\end{aligned}$$ where $\gamma_{ij}$ denotes the spatial metric and $Y$, $Y_i$, and $Y_{ij}$ are the eigenfunctions of the Laplace operator, $\nabla^2Y=-k^2Y$, and its covariant derivatives, $Y_i=(-k)\nabla_iY$ and $Y_{ij}=(k^{-2}\nabla_i\nabla_j+\gamma_{ij}/3)Y$, with $k$ the wave number. Similarly, the perturbed energy-momentum tensor can also be expressed in terms of another four functions—energy density perturbation $\delta\rho$, velocity perturbation $v$, isotropic pressure perturbation $\delta p$, and anisotropic stress perturbation $\Pi$, $$\begin{aligned} & \delta{T^0_{\hphantom{0}0}} = - { \delta\rho}Y,\qquad\delta{T_0^{\hphantom{i}i}} = -(\rho + p){v}Y^i, \nonumber\\ & \qquad \delta {T^i_{\hphantom{i}j}} = {\delta p}Y \delta^i_{\hphantom{i}j} + p{\Pi Y^i_{\hphantom{i}j}}. \label{eqn:dstressenergy}\end{aligned}$$ With the existence of the dark sector interaction, the conservation laws become $$\label{eqn:energyexchange} \nabla_\nu T^{\mu\nu}_I = Q^\mu_I, \quad\quad \sum_I Q^\mu_I = 0,$$ where $Q^\mu_I$ is the energy-momentum transfer vector of $I$ fluid, which can be split in general as $$Q_{\mu}^I = a\big( -Q_I(1+AY) - \delta Q_IY,\,[ f_I+ Q_I (v-B)]Y_i\big),\label{eq:Qenergy}$$ where $\delta Q_I$ and $f_I$ denote the energy transfer perturbation and momentum transfer potential of $I$ fluid, respectively. In a perturbed FRW Universe, Eqs. (\[eqn:energyexchange\]) and (\[eq:Qenergy\]) lead to the following two conservation equations for the $I$ fluid [@Kodama:1985bj], $$\begin{aligned} &{\delta\rho_I'} + 3\mathcal{H}({\delta \rho_I}+ {\delta p_I})+(\rho_I+p_I)(k{v}_I + 3 H_L')=a(\delta Q_I-AQ_I),\label{eqn:conservation1}\\ & [(\rho_I + p_I)({{v_I}-{B}})]'+4\mathcal{H}(\rho_I + p_I)({{v_I}-{B}}) -k{ \delta p_I }+ {2 \over 3}k{c_K}p_I {\Pi_I} - k(\rho_I+ p_I) {A}=a[Q_I(v-B)+f_I],\label{eqn:conservation2}\end{aligned}$$ where $c_K = 1-3K/k^2$ with $K$ the spatial curvature. The PPF framework for the IDE scenario -------------------------------------- Now we discuss the perturbation evolutions for cold dark matter and dark energy, in the comoving gauge, $B=v_T$ and $H_T=0$, where $v_T$ denotes the velocity perturbation of total matters except dark energy. To avoid confusion, we use the new symbols, $\zeta\equiv H_L$, $\xi\equiv A$, $\rho\Delta\equiv\delta\rho$, $\Delta p\equiv\delta p$, $V\equiv v$, and $\Delta Q_I\equiv\delta Q_I$, to denote the corresponding quantities of the comoving gauge except for the two gauge independent quantities $\Pi$ and $f_I$. For cold dark matter, $\Delta p_c=\Pi_c=0$, thus the evolutions of the remaining two quantities $\rho_c\Delta_c$ and $V_c$ are totally determined by Eqs. (\[eqn:conservation1\]) and (\[eqn:conservation2\]). Note that $\Delta Q_{I}$ and $f_{I}$ can be got in a specific IDE model. For dark energy, we need an extra condition on $\Delta p_{de}$ besides $\Pi_{de}=0$ and Eqs. (\[eqn:conservation1\]) and (\[eqn:conservation2\]) to complete the dark energy perturbation system. A common practice is to treat dark energy as a nonadiabatic fluid and to calculate $\Delta p_{de}$ in terms of the adiabatic sound speed and the rest frame sound speed (see, e.g., Ref. [@Valiviita:2008iv]). However, this will induce the large-scale instability in the IDE scenario, as mentioned above. So we handle the perturbations of dark energy by using the generalized PPF framework established in Ref. [@Li:2014eha]. As shown in Ref. [@Li:2014eha], the key point to avoid the large-scale instability is establishing a direct relationship between $V_{de} - V_T$ and $V_T$ on the large scales instead of directly defining a rest-frame sound speed for dark energy and calculating $\Delta p_{de}$ in terms of it. This relationship can be parametrized by a function $f_\zeta(a)$ as [@Hu:2008zd; @Fang:2008sn] $$\lim_{k_H \ll 1} {4\pi G a^2\over \mathcal{H}^2} (\rho_{de} + p_{de}) {V_{de} - V_T \over k_H} = - {1 \over 3} {c_K}f_\zeta(a) k_H V_T,\label{eq:DEcondition}$$ where $k_H=k/\mathcal{H}$. This condition in combination with the Einstein equations gives the equation of motion for the curvature perturbation $\zeta$ on the large scales, $$\begin{aligned} \lim_{k_H \ll 1} \zeta' = \mathcal{H}\xi - {K \over k} V_T +{1 \over 3} {c_K}f_\zeta(a) k V_T. \label{eqn:zetaprimesh}\end{aligned}$$ On the small scales, the evolution of the curvature perturbation is described by the Poisson equation, $\Phi=4\pi G a^2\Delta_T \rho_T/( k^2{c_K})$, with $\Phi=\zeta+V_T/k_H$. The evolutions of the curvature perturbation at $k_H\gg1$ and $k_H\ll1$ can be related by introducing a dynamical function $\Gamma$ to the Poisson equation, such that $$\Phi+\Gamma = {4\pi Ga^2 \over k^2{c_K}} \Delta_T \rho_T \label{eqn:modpoiss}$$ on all scales. Then compared with the small-scale Poisson equation, Eq. (\[eqn:modpoiss\]) gives $\Gamma\rightarrow0$ at $k_H\gg1$. On the other hand, with the help of the Einstein equations and the conservation equations as well as the derivative of Eq. (\[eqn:modpoiss\]), Eq. (\[eqn:zetaprimesh\]) gives the equation of motion for $\Gamma$ on the large scales, $$\label{eq:gammadot} \lim_{k_H \ll 1} \Gamma' = S -\mathcal{H}\Gamma,$$ with $$\begin{aligned} S&={4\pi Ga^2 \over k^2 } \Big\{[(\rho_{de}+p_{de})-f_{\zeta}(\rho_T+p_T)]kV_T \nonumber\\ &\quad+{3a\over k_Hc_K}[Q_c(V-V_T)+f_c]+\frac{a}{c_K}(\Delta Q_c-\xi Q_c)\Big\},\nonumber\end{aligned}$$ where $\xi$ can be obtained from Eq. (\[eqn:conservation2\]), $$\xi = -{\Delta p_T - {2\over 3}{{c_K}p_T \Pi_T}+{a\over k}[Q_c(V-V_T)+f_c] \over \rho_T + p_T}. \label{eqn:xieom}$$ With a transition scale parameter $c_\Gamma$, we can take the equation of motion for $\Gamma$ on all scales to be [@Hu:2008zd; @Fang:2008sn] $$(1 + c_\Gamma^2 k_H^2) [\Gamma' +\mathcal{H} \Gamma + c_\Gamma^2 k_H^2 \mathcal{H}\Gamma] = S. \label{eqn:gammaeom}$$ Here we note that the prime in this paper is used to denote the derivative with respect to the conformal time $\tau$ (i.e., $'\equiv d/d\tau$), but in Ref. [@Li:2014eha], it is defined to be the derivative with respect to $\ln a$ (i.e., $'\equiv d/d\ln a$). This explains why $\mathcal{H}$ appears in Eqs. (\[eq:gammadot\]) and (\[eqn:gammaeom\]) (compared to the corresponding equations in Ref. [@Li:2014eha]). From the above equations, we can find that all of the perturbation quantities relevant to the equation of motion for $\Gamma$ are those of matters except dark energy. So we can solve the differential equation (\[eqn:gammaeom\]) without any knowledge of the dark energy perturbations. Once the evolution of $\Gamma$ is obtained, we can immediately get the energy density and velocity perturbations, $$\begin{aligned} &\rho_{de}\Delta_{de} =- 3(\rho_{de}+p_{de}) {V_{de}-V_{T}\over k_{H} }-{k^{2}{c_K}\over 4\pi G a^{2}} \Gamma,\\ \label{eqn:ppffluid} & V_{de}-V_{T} ={-k \over 4\pi Ga^2 (\rho_{de} + p_{de}) F} \nonumber \\ &\quad\quad\quad\times\left[ S - \Gamma' - \mathcal{H}\Gamma + f_{\zeta}{4\pi Ga^2 (\rho_{T}+p_{T}) \over k}V_{T} \right],\end{aligned}$$ with $F = 1 + 12 \pi G a^2 (\rho_T + p_T)/( k^2 {c_K})$. The IDE model {#subsec:modelequations} ------------- In the following, we get the evolution equations for the specific IDE model under study in this work. To achieve this, we need to construct a covariant interaction form whose energy transfer can reduce to $Q=3\beta H\rho_{de}$ in the background evolution. A simple physical choice is assuming that the energy-momentum transfer is parallel to the four-velocity of dark matter, so that the momentum transfer vanishes in the dark matter rest frame. Then, we have $$\label{eq:covQ} Q^{\mu}_c = -Q^{\mu}_{de}=-3\beta H\rho_{de} u^{\mu}_c,$$ with the dark matter four-velocity, $$u^\mu_c = a^{-1}\big(1-AY,\,v_cY^i \big),\quad u_\mu^c = a\big(-1-AY,\,(v_c -B)Y_i\big).$$ Comparing Eq. (\[eq:covQ\]) with Eq. (\[eq:Qenergy\]), we get $$\begin{aligned} &\delta Q_{de}= -\delta Q_c=3\beta H\rho_{de} \delta_{de},\nonumber\\ &f_{de}=-f_c=3\beta H\rho_{de} (v_c-v),\nonumber \\ & Q_{de}=-Q_c=3\beta H\rho_{de},\label{eqn:emtransfer} \end{aligned}$$ where we define the dimensionless density perturbation $\delta_I=\delta\rho_I/\rho_I$ for the $I$ fluid. Substituting Eq. (\[eqn:emtransfer\]) into Eqs. (\[rhodedot\]) and (\[rhocdot\]), we can obtain the background evolutions of dark energy and dark matter, $$\begin{aligned} &\rho_{de}=\rho_{de0}a^{-3(1+w-\beta)},\\ &\rho_{c}=\rho_{c0}a^{-3}\left[1+{\beta\over\beta-w}{\rho_{de0}\over\rho_{c0}}\left(1-a^{3\beta-3w}\right)\right],\label{rhocdot}\end{aligned}$$ where the subscript “0” denotes the value of the corresponding quantity at $a=1$ or $z=0$. For the dark sector perturbation evolutions, we obtain them in the synchronous gauge since most public numerical codes are written in this gauge. The synchronous gauge is defined by $A=B=0$, $\eta=-H_T/3-H_L$, and $h=6H_L$. Then Eqs. (\[eqn:conservation1\]) and (\[eqn:conservation2\]) reduce to $$\begin{aligned} &\delta_c'+kv_c +{h'\over2} ={3\beta{\cal H}\rho_{de}\over\rho_c}(\delta_c-\delta_{de}), \label{eq:dmdensity}\\ &v_c'+{\cal H}v_c=0,\label{eq:dmvelocity} \end{aligned}$$ for cold dark matter in the synchronous gauge. From Eq. (\[eq:dmvelocity\]), we can see that the momentum transfer vanishes for the IDE model. So there is no violation of the weak equivalence in this model. To get the dark energy perturbations in the synchronous gauge, we need to make a gauge transformation, since the PPF approach is written in the comoving gauge. The gauge transformation from the synchronous gauge to the comoving gauge is given by [@Hu:2008zd] $$\begin{aligned} &\rho_I\Delta_I = \delta_I \rho_I - \rho_I' v_T/k, \label{eq:transdelta}\\ &\Delta p_I = \delta p_I - p_I' v_T/k, \label{eq:transdp}\\ &V_I-V_T=v_I-v_T, \label{eq:transv}\\ &\zeta = -\eta - {v_{T}/k_{H}}. \label{eq:transzeta}\end{aligned}$$ By using Eq. (\[eq:transdelta\]), we can obtain $\Phi$ of Eq. (\[eqn:modpoiss\]) in terms of $\delta_T$ and $\Gamma$. Then combining Eq. (\[eq:transzeta\]) and the gauge relation $V_{T} = k_{H}(\Phi-\zeta)$ [@Hu:2008zd], we can get another useful transformation relation, $$V_{T} =v_{T}+{4 \pi G a^2\over \mathcal{H}kc_K}\left(\delta_T \rho_T - \rho_T' {v_T\over k}\right) +k_H\eta-k_H\Gamma.\label{eq:transvt}$$ With the help of Eqs. (\[eq:transdelta\])–(\[eq:transvt\]), we can rewrite all the equations of PPF approach in terms of the corresponding quantities in the synchronous gauge. For the IDE model under study, we have $$\begin{aligned} &\delta_{de} =- 3(1+w){v_{de}\over k_{H}}+3\beta{v_T\over k_{H}}-{k^{2}{c_K}\over 4\pi G a^{2}\rho_{de}} \Gamma,\label{eq:deltadesync}\\ & v_{de}-v_{T} ={-k \over 4\pi Ga^2 \rho_{de}(1+w) F} \nonumber \\ &\quad\quad\quad\times\left[ S - \Gamma' - \mathcal{H}\Gamma + f_{\zeta}{4\pi Ga^2 (\rho_{T}+p_{T}) \over k}(v_{T}+\sigma) \right], \label{eq:vdesync}\end{aligned}$$ in the synchronous gauge, where $$\sigma={4 \pi G a^2\over \mathcal{H}kc_K}\left[\delta\rho_T +3(\rho_T+p_T+\beta\rho_{de}) {v_T\over k_H}\right] +k_H\eta-k_H\Gamma.$$ The source term $S$ of Eq. (\[eqn:gammaeom\]) can be rewritten in the synchronous gauge as $$\begin{aligned} S&={4\pi Ga^2 \over k} \Big\{[\rho_{de}(1+w)-f_{\zeta}(\rho_T+p_T)](v_T+\sigma) \nonumber\\ &\quad+\frac{3\beta\rho_{de}}{k_Hc_K}\Big[\xi-\delta_{de}-3(w-\beta) {v_T\over k_H}\Big]\Big\},\nonumber\end{aligned}$$ where $$\xi = -{k\delta p_T -p_T'v_T- {2\over 3}k{{c_K}p_T \Pi_T}+3\beta\mathcal{H}\rho_{de}v_T \over k(\rho_T + p_T)}.\label{eq:xi}$$ Here note that due to our studied interaction model with $Q$ proportional to $\rho_{de}$, the dark energy density perturbation $\delta_{de}$ occurs in the expression of the source term $S$. Under such circumstance, how can we solve the equation of motion (\[eqn:gammaeom\]) for $\Gamma$ before $\delta_{de}$ is got from Eq. (\[eq:deltadesync\])? For this issue, we can utilize an iteration approach. For example, we can set an initial value for $v_{de}$ and get the value of $\delta_{de}$ from Eq. (\[eq:deltadesync\]). Then we can obtain $S$ and solve the differential equation (\[eqn:gammaeom\]). Finally, we can update the value of $v_{de}$ from Eq. (\[eq:vdesync\]) and start another iteration. The convergence speed of this iteration method is proven to be very quick from our tests. We also need to determine the parameter $c_\Gamma$ and the function $f_\zeta(a)$. For the value of $c_\Gamma$, we find that the perturbation evolutions of dark energy are insensitive to its value, so we follow Ref. [@Fang:2008sn] and choose it to be $0.4$. The function $f_\zeta(a)$ can be calibrated in a specific IDE model, but no one gave a concrete way to do this in the previous works. For simplicity, one often takes $f_\zeta(a)=0$ in the literature, because its effect may only be detected gravitationally in the future [@Fang:2008sn]. However, we still need, though not urgently, to give a general approach to calibrate $f_\zeta$ for the future high-precision observational data. Besides, we also want to know whether $f_\zeta(a)$ can affect current observations, such as the CMB temperature power spectrum. So we give a detailed process of calibrating $f_\zeta(a)$ in Appendix \[app:fzeta\]. Using the calibrated $f_\zeta(a)$ given by Eq. (\[eq:fzetafinal\]), we plot the CMB temperature power spectrum for the studied IDE model in Fig. \[fig:power\]. As a comparison, we also plot the case with $f_\zeta(a)=0$. In this figure, we fix $w=-1.05$, $\beta=-0.01$, and other parameters at the best-fit values from Planck. From Fig. \[fig:power\], we can see that the CMB temperature power spectrum does not have the ability to distinguish these two cases from each other. So it is appropriate to simply set $f_\zeta(a)=0$, currently. Nevertheless, we still utilize the calibrated $f_\zeta(a)$ in our calculations, because we find that taking $f_\zeta(a)$ to be the calibrated function can help to improve the convergence speed of the aforementioned iteration. ![The CMB temperature power spectrum for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. The red curve denotes the case with $f_\zeta(a)$ taken to be the calibrated function in Eq. (\[eq:fzetafinal\]), while the black curve is the case with $f_\zeta(a)=0$. We fix $w=-1.05$, $\beta=-0.01$, and other parameters at the best-fit values from Planck. The overlap of these two curves indicates that the CMB temperature power spectrum does not have the ability to detect the effect of the calibrated $f_\zeta(a)$.[]{data-label="fig:power"}](power.pdf){width="8cm"} Now, all the perturbation equations can be numerically solved. In Fig. \[perturbationevolve\], we show the matter and metric perturbation evolutions for the IDE model under study at $k=0.01\,\rm{Mpc^{-1}}$, $k=0.1\,\rm{Mpc^{-1}}$ and $k=1.0\,\rm{Mpc^{-1}}$. Here, we also fix $w=-1.05$, $\beta=-0.01$, and other parameters at the best-fit values from Planck. As mentioned above, the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model with $w=-1.05$ and $\beta=-0.01$ would be an unstable case if the dark energy perturbations are given by the standard linear perturbation theory. Now we can clearly see from Fig. \[perturbationevolve\] that all the perturbation evolutions are stable and normal within the PPF framework. {width="8cm"} {width="8cm"} {width="8cm"} {width="8cm"} Data and constraints {#sec:constraint} ==================== In this section, we use the latest observational data to constrain the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. We modify the [camb]{} code [@Lewis:1999bs] to include the background and perturbation equations given in Sec. \[subsec:modelequations\]. To explore the parameter space, we use the public [CosmoMC]{} package [@Lewis:2002ah]. The free parameter vector is: $\left\{\omega_b,\,\omega_c,\,H_0,\, \tau,\, w,\, \beta, n_{\rm{s}},\, {\rm{ln}}(10^{10}A_{\rm{s}})\right\}$, where $\omega_b$, $\omega_c$ and $H_0$ are the baryon density, dark matter density and Hubble constant of present day, respectively, $\tau$ denotes the optical depth to reionization, and ${\rm{ln}}(10^{10}A_{\rm{s}})$ and $n_{\rm{s}}$ are the amplitude and the spectral index of the primordial scalar perturbation power spectrum for the pivot scale $k_0=0.05\,\rm{Mpc}^{-1}$. Here note that we take $H_0$ as a free parameter instead of the commonly used $\theta_{\rm{MC}}$, because $\theta_{\rm{MC}}$ is dependent on a standard noninteracting background evolution. We set a prior \[$-0.15$, 0.15\] for the coupling constant $\beta$, and keep the priors of other free parameters the same as those used by Planck Collaboration [@Ade:2013zuv]. In our calculations, we fix $N_{\rm{eff}}=3.046$ and $\sum m_{\nu}=0$ eV for the three standard neutrino species. We use the following data sets in our analysis: Planck+WP: the CMB temperature power spectrum data from Planck [@Ade:2013zuv] combined with the polarization measurements from 9-year WMAP [@wmap9]; BAO: the latest BAO measurements from 6dFGS ($z=0.1$) [@6df], SDSS DR7 ($z=0.35$) [@sdss7], WiggleZ ($z=0.44$, 0.60, and 0.73) [@wigglez], and BOSS DR11 ($z=0.32$ and 0.57) [@boss]; JLA: the latest released 740 data points of SNIa from JLA sample [@Betoule:2014frx]; $H_0$: the Hubble constant measurement from HST [@Riess:2011yx]; RSD: the RSD measurements from 6dFGS ($z=0.067$) [@RSD6dF], 2dFGRS ($z=0.17$) [@RSD2dF], WiggleZ ($z=0.22$, 0.41, 0.60, and 0.78) [@RSDwigglez], SDSS LRG DR7 ($z=0.25$ and 0.37) [@RSDsdss7], BOSS CMASS DR11 ($z=0.57$) [@Beutler:2013yhm], and VIPERS ($z=0.80$) [@RSDvipers]. As mentioned in Sec. \[sec:intro\], RSD actually reflect the coherent motions of galaxies and hence provide information about the formation of large-scale structure. Due to the existence of the peculiar velocities of galaxies the observed overdensity field $\delta_g$ of galaxies in redshift space is enlarged by a factor of $1+f\mu^2/b$ [@Kaiser:1987], where $\mu$ is the cosine of the angle to the line of sight, $b\equiv\delta_g/\delta$ denotes the large-scale bias, and $f(a)\equiv d\ln D(a)/d\ln a$ is the linear growth rate, with the growth factor $D(a)=\delta(a)/\delta(a_{\rm{ini}})$. Thus, through precisely measuring the RSD effect from galaxy redshift surveys, one can obtain information of $f(a)$. However, this measurement of $f(a)$ is bias-dependent. To avoid this issue, Song and Percival [@Song:2008qt] suggested using a bias-independent combination, $f(z)\sigma_8(z)$, to extract information from the RSD data, where $\sigma_8(z)$ is the root-mean-square mass fluctuation in spheres with radius $8h^{-1}$ Mpc at redshift $z$. To use the RSD data, we need to do some modifications to the [CosmoMC]{} package. First, we add an extra subroutine to the [CAMB]{} code to output the theoretical values of $f(z)\sigma_8(z)$. Here the calculation of $\sigma_8(z)$ inherits from the existing subroutine of the [CAMB]{} code and $f(a)$ is calculated by $f(a)=d\ln \delta/d\ln a$ with $\delta=(\rho_c\delta_c+\rho_b\delta_b)/(\rho_c+\rho_b)$. Then, we transfer the obtained theoretical values of $f(z)\sigma_8(z)$ to the source files of the [CosmoMC]{} package and calculate the $\chi^2$ value of the RSD data. First, we constrain the IDE model under study by using the Planck+WP+BAO+JLA+$H_0$ data combination. This data combination can be safely used since the BAO and JLA data are well consistent with the Planck+WP data [@Ade:2013zuv; @Betoule:2014frx], and the tension between Planck data and $H_0$ measurement can be greatly relieved in a dynamical dark energy model [@Li:2013dha]. The fit results are shown in Table \[table1\] and Fig. \[contours\]. Obviously, the whole parameter space can be explored. By using this data combination, we get $w=-1.061\pm0.056$ and $\beta=-0.010^{+0.037}_{-0.033}$ at the $1\sigma$ level for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. The parameter space for $w$ and $\beta$ shifts left dramatically compared with the previous results, e.g., $w=-0.940^{+0.0158}_{-0.0599}$ and $\beta=0.209^{+0.0711}_{-0.0403}$ obtained by using a similar data combination but assuming $w>-1$ and $\beta>0$ [@Yang:2014gza]. This qualitative change indicates that compulsively assuming $w>-1$ and $\beta>0$ can induce substantial errors on the observational constraint results. So it is of great importance to use the PPF approach to get the correct, whole parameter space for the IDE models. Although the Planck+WP+BAO+JLA+$H_0$ data combination can give a good constraint result, there still exists a significant degeneracy between the coupling parameter $\beta$ and the background parameter $\Omega_m$ (see the green contours in Fig. \[contours\]). This degeneracy, as mentioned in Sec. \[sec:intro\], is hard to break by only using the geometric measurements. So next, we add the extra structure formation information from the RSD data into our analysis. The fit results can also be found in Table \[table1\] and Fig. \[contours\]. By using the Planck+WP+BAO+JLA+$H_0$+RSD data, we get $\beta=-0.0148^{+0.0100}_{-0.0089}$ and $\Omega_m=0.309^{+0.010}_{-0.011}$ at the $1\sigma$ level. Clearly, the errors of $\beta$ and $\Omega_m$ and the degeneracy between them are substantially reduced. Besides, we can also see that current RSD data favor a relatively larger interaction rate for the studied model. This result remarkably differs from that in Ref. [@Yang:2014gza] where a very small positive coupling constant, $\beta=0.00372^{+0.00077}_{-0.00372}$ (1$\sigma$), is obtained by the CMB+BAO+SNIa+RSD data combination. Since $w>-1$ and $\beta>0$ are assumed in Ref. [@Yang:2014gza], it can be deduced that such a small positive coupling constant just arises from the cut-off effect of the parameter space rather than reflecting the actual preference of the RSD data. So we conclude that our work gives the correct and tightest fit results for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. Parameters Planck+WP+BAO+JLA+$H_0$ +RSD ------------------------- --------------------------------- --------------------------------- $\omega_b$ $0.02209^{+0.00025}_{-0.00026}$ $0.02220^{+0.00025}_{-0.00024}$ $\omega_c$ $0.123\pm0.011$ $0.1226^{+0.0039}_{-0.0038}$ $H_0$ $69.4\pm1.0$ $68.5^{+1.0}_{-0.9}$ $\tau$ $0.089^{+0.012}_{-0.014}$ $0.087^{+0.012}_{-0.013}$ $w$ $-1.061\pm0.056$ $-1.009\pm0.045$ $\beta$ $-0.010^{+0.037}_{-0.033}$ $-0.0148^{+0.0100}_{-0.0089}$ ${\rm{ln}}(10^{10}A_s)$ $3.088^{+0.024}_{-0.026}$ $3.079^{+0.023}_{-0.026}$ $n_s$ $0.9601\pm0.0057$ $0.9638\pm0.0057$ $\Omega_\Lambda$ $0.700\pm0.024$ $0.691^{+0.011}_{-0.010}$ $\Omega_m$ $0.300\pm0.024$ $0.309^{+0.010}_{-0.011}$ $\sigma_8$ $0.846^{+0.051}_{-0.065}$ $0.808\pm0.016$ $\chi^2_{\rm{min}}$ 10508.090 10519.498 : \[table1\] The mean values and $1\sigma$ errors of all the free parameters and some derived parameters for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. It can be found that current observations actually favor a negative value of $\beta$. The errors of $\beta$ and $\Omega_m$ are substantially reduced once the RSD data are used. {width="17cm"} Finally, we make a comparison for the $Q=3\beta H\rho_{de}$ model and the $Q=3\beta H\rho_{c}$ model according to their cosmological constraint results. For the $Q=3\beta H\rho_{c}$ model, we got $\beta=-0.0013\pm0.0008$ ($1\sigma$) by using the CMB+BAO+SNIa+$H_0$ data in Ref. [@Li:2014eha]. Then, Ref. [@Richarte:2014yva] further studied this model by adding the RSD data into the analysis. However, the fit result $\xi_c=0.0014\pm0.0008$ ($\xi_c=-\beta$) obtained in Ref. [@Richarte:2014yva] shows that the extra information from the RSD data nearly has no contribution to the fit result of the coupling constant. This phenomenon is somewhat counter-intuitive but still can be understood. Since the coupling form is proportional to $\rho_{c}$ whose value is far greater than that of $\rho_{de}$ at the early times, the amplitude of dark energy perturbation is greatly enlarged by the coupling at the early times (see Fig. 1 of Ref. [@Li:2014eha]), inducing significant effect on the large-scale CMB power spectrum even for a small coupling constant. Thus CMB itself can provide tight constraint on the coupling constant for the $Q=3\beta H\rho_{c}$ model. This feature makes it easy to rule out the $Q=3\beta H\rho_{c}$ model in the future. However, there is no such issue for the $Q=3\beta H\rho_{de}$ model. From this point, we believe that the $Q=3\beta H\rho_{de}$ model should deserve more attention in the future works. Conclusion {#sec:conclusions} ========== Current astronomical observations provide us with substantial room to study the possible interaction between dark energy and dark matter. However, such an IDE scenario occasionally suffers from the well-known large-scale instability. In our previous work [@Li:2014eha], we successfully solved this instability problem by establishing a PPF framework for the IDE scenario for the first time. However, there are also some issues needing our further discussions. For example, how do we apply the PPF framework to the widely studied $Q\propto\rho_{de}$ model? How do we calibrate $f_\zeta$ in a specific IDE model? More importantly, how will the cosmological constraint results, especially by using the structure formation measurement from RSD, be changed in the widely studied $Q\propto\rho_{de}$ model once $w$ and $\beta$ can be fully let free within the PPF framework? To answer all of these questions, in this work we focus on the $Q=3\beta H\rho_{de}$ model with the momentum transfer vanishing in the dark matter rest frame. So the covariant interaction form in a perturbed Universe is $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$. We handle its cosmological perturbations by using the PPF framework established in Ref. [@Li:2014eha]. For the problem of how we can solve the equation of motion for $\Gamma$ before the exact value of $\delta_{de}$ is known, we introduce an iteration method. We give a concrete way to calibrate $f_\zeta$. We find that the effect of taking $f_\zeta$ to be the calibrated function cannot be detected by the CMB temperature power spectrum. However, the general calibration approach we provide in this paper may play a crucial role for the future high-precision observations. Finally, we perform a full analysis on this model with current observations. By using the Planck+WP+BAO+JLA+$H_0$ data combination, we get $w=-1.061\pm0.056$, $\beta=-0.010^{+0.037}_{-0.033}$, and $\Omega_m=0.300\pm0.024$ at the $1\sigma$ level. The fit results become $w=-1.009\pm0.045$, $\beta=-0.0148^{+0.0100}_{-0.0089}$, and $\Omega_m=0.309^{+0.010}_{-0.011}$ once we further incorporate the RSD data into the analysis. From the above results, we have the following conclusions: (1) Within the PPF framework, the full parameter space of $w$ and $\beta$ can be explored for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. (2) The fit results show that current observations actually favor a negative coupling constant once $w<-1$ and $\beta<0$ are also allowed by the PPF framework. (3) With the help of the RSD data, the errors of $\beta$ and $\Omega_m$ and the degeneracy between them are substantially reduced. (4) Compared with the previous works, our results show that a relatively larger absolute value of $\beta$ can survive in current RSD data. We believe that our work gives the correct and tightest fit results for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model. The shortage of our work is that we have not considered the $Q^{\mu}\propto u^{\mu}_{de}$ case. In this case, the Euler equation for dark matter is modified, and hence, the weak equivalence principle is broken. However, this breakdown of the weak equivalence principle might be detected by the future weak lensing measurement [@Koyama:2009gd]. For current observations, it has been found that the observational constraint results in these two cases are similar [@Clemson:2011an]. So we leave this analysis for future work. We acknowledge the use of [CosmoMC]{}. This work was supported by the National Natural Science Foundation of China (Grant No. 11175042) and the Fundamental Research Funds for the Central Universities (Grant No. N120505003). Calibration of the function $f_\zeta(a)$ {#app:fzeta} ======================================== The first step to construct the PPF approach is using a function $f_\zeta(a)$ to parametrize the large-scale velocity of dark energy in terms of the total velocity of other matters, as shown in Eq. (\[eq:DEcondition\]). This parametrization is based on $V_{de}-V_T={\cal O}(k_H^3 \zeta)$ and $V_T={\cal O}(k_H \zeta)$ at $k_H\ll1$ in the comoving gauge [@Hu:2004xd]. Thus, from Eq. (\[eq:DEcondition\]), $f_\zeta(a)$ can be calibrated by finding out the exact form of $C(a)\equiv(V_{de}-V_T)/(k_H^2V_T)$ at $k_H\ll1$ in the comoving gauge. Then Eq. (\[eq:DEcondition\]) gives $$f_\zeta(a)=-{12\pi Ga^2 \over c_K\mathcal{H}^2} (\rho_{de} + p_{de})C(a).\label{eq:fzeta}$$ In what follows, we show how to get $C(a)$ for the $Q^{\mu}=3\beta H\rho_{de}u^{\mu}_c$ model in detail. The function $C(a)$ can only be obtained by solving all the standard linear perturbation equations in the comoving gauge where dark energy is treated as a nonadiabatic fluid with its pressure perturbation, $$\Delta p_{de} = c_{s}^2\rho_{de}\Delta_{de} +\rho_{de}'(c_{s}^2-c_{a}^2){V_{de}-V_T \over k},\label{eq:deltap}$$ where $c_a$ and $c_s$ are the adiabatic sound speed and rest-frame sound speed of dark energy, respectively. In the following calculations, we take $c_a^2=p_{de}'/\rho_{de}'=w$ and $c_s^2=1$. Substituting Eq. (\[eq:deltap\]) and $\Pi_{de}=0$ into Eqs. (\[eqn:conservation1\]) and (\[eqn:conservation2\]), we get the following two conservation equations for dark energy in the comoving gauge, $$\begin{aligned} &\Delta_{de}'+3{\cal H}(1-w)\Delta_{de}+(1+w)kV_{de}+9{\cal H}^2(1-w^2){V_{de}-V_T \over k} +3(1+w)\zeta'=3\beta{\cal H}\left[3{\cal H}(1-w){V_{de}-V_T \over k}-\xi\right],\label{eq:Deltade}\\ &(V_{de} -V_T)'-2{\cal H}(V_{de} -V_T)-{k\over 1+w}\Delta_{de}-k\xi= {3\beta{\cal H}\over1+w}(V_c+V_T-2V_{de}).\end{aligned}$$ Similarly, substituting $p_c=\Delta p_c=\Pi_c=0$ into Eqs. (\[eqn:conservation1\]) and (\[eqn:conservation2\]), we obtain $$\begin{aligned} &\Delta_c'+kV_c +3\zeta' ={3\beta{\cal H}\rho_{de}\over\rho_c}(\Delta_c-\Delta_{de}+\xi), \\ &(V_c -V_T)'+{\cal H}(V_c -V_T)-k\xi= 0.\label{eq:VcvT}\end{aligned}$$ The linear perturbation equations of all the components, in principle, are hard to solve analytically. However, since we only focus on the perturbation evolution on the large scales where the period we care about is radiation-dominated one, the perturbation equations can be further simplified and solved analytically. In the early radiation dominated epoch, $\mathcal{H}=\tau^{-1}$, $k_H=k\tau$, $V_b=V_\gamma$ (tight coupling), and $\Pi_I=0$ for $I\neq\nu$, the solutions to the perturbation equations can be found by solving the following first-order differential matrix equation [@Doran:2003xq], $$\label{eq.dUdlnx_gen} \frac{d \bm{U}}{d \ln x} = \mathbf{A}(x) \bm{U}(x),$$ where $x=k\tau$, $\mathbf{A}(x)$ is the coefficient matrix and $\bm{U}(x)$ is the matter perturbation vector containing $\Pi_\nu$, $\Delta_I$ for $I=de$, $c$, $\gamma$, $b$, and $\nu$, and $V_I$ for $I=de$, $c$, $\gamma$, and $\nu$. Here the subscripts $\gamma$, $b$, and $\nu$ represent photons, baryons and neutrinos, respectively. As a matter of convenience, we use the following rescaled variables: $\tilde{\Pi}_\nu\equiv\Pi_\nu/x^2$, ${\tilde V}_T\equiv V_T/x$, ${\tilde\Delta}_{I}\equiv\Delta_{I}/x^2$, and ${\tilde V}_{I}\equiv(V_I-V_T)/x^3$ for $I=de$, $c$, $\gamma$, $b$, and $\nu$. Thus, our final matter perturbation vector is $$\bm{U}^T = \left\{ {\tilde\Delta}_c, \, \tilde{V}_c, \, {\tilde\Delta}_\gamma, \, \tilde{V}_\gamma, \, {\tilde\Delta}_b, \, {\tilde\Delta}_\nu,\, \tilde{\Pi}_\nu , \, {\tilde\Delta}_{de}, \, \tilde{V}_{de}, \,\tilde{V}_T \right\},$$ where we solve the differential equation of ${\tilde V}_T$ instead of that of ${\tilde V}_\nu$ so that we can directly get $C(x)={\tilde V}_{de}/{\tilde V}_T$ from the solutions. Note that $(\rho_T+p_T)V_T=\sum_{I=c,\,b,\,\gamma,\,\nu}(\rho_I+p_I)V_I$ and the differential equation for $V_T$ can be found from the second Einstein equation [@Hu:2008zd], $$V_T'+2\mathcal{H}V_T+k\xi + k\zeta= -{8\pi Ga^2 \over k} {p\Pi}.\label{eq:VTcom}$$ Using Eqs. (\[eq:Deltade\])–(\[eq:VcvT\]) and (\[eq:VTcom\]) and the perturbation equations of photons, baryons and neutrinos given by Ref. [@Doran:2003xq], we can easily obtain the following evolution equations in terms of the rescaled variables: $$\begin{aligned} &\frac{d {\tilde\Delta}_{c}}{d\ln x} =-2{\tilde\Delta}_{c} -x^2{\tilde V}_c-{\tilde V}_T +{3\beta\Omega_{de}\over\Omega_c}(x^{-2}\xi+{\tilde\Delta}_{c}-{\tilde\Delta}_{de})-3x^{-2}\frac{d \zeta}{d\ln x},\label{eq.Deltac_newnew}\\ &\frac{d {\tilde V}_c}{d\ln x} = -4{\tilde V}_c+{\xi\over x^2}, \label{eq.Vc_newnew}\\ &\frac{d {\tilde\Delta}_{\gamma}}{d\ln x} =-2{\tilde\Delta}_{\gamma} -\frac{4}{3} x^2{\tilde V}_{\gamma}-\frac{4}{3} {\tilde V}_T-4x^{-2}\frac{d \zeta}{d\ln x}, \label{eq.Dg_newnew}\\ &\frac{d{\tilde V}_{\gamma}}{d\ln x} = \frac{1}{4} {\tilde\Delta}_{\gamma} - 3{\tilde V}_\gamma, \label{eq.Vg_newnew}\\ &\frac{d {\tilde\Delta}_b}{d\ln x} = -2{\tilde\Delta}_b- x^2{\tilde V}_\gamma-{\tilde V}_T-3x^{-2}\frac{d \zeta}{d\ln x}, \label{eq.Db_newnew}\\ &\frac{d {\tilde\Delta}_{\nu} }{d\ln x}= -2{\tilde\Delta}_{\nu}-\frac{4}{3} x^2{\tilde V}_\nu-\frac{4}{3}{\tilde V}_T-4x^{-2}\frac{d \zeta}{d\ln x},\label{eq.Dn_newnew}\\ &\frac{d {\tilde\Pi}_{\nu}}{d\ln x} = \frac{8}{5} x^2{\tilde V}_\nu+\frac{8}{5}{\tilde V}_T-2{\tilde\Pi}_\nu, \label{eq.Pnu_newnew} \\ & \frac{d {\tilde\Delta}_{de}}{d\ln x} =-3({5\over3}-w){\tilde\Delta}_{de}-(1+w)x^2{\tilde V}_{de}-(1+w){\tilde V}_T -3(1+w)x^{-2}\frac{d \zeta}{d\ln x} \nonumber \\ &\quad\quad\quad~~-9(1-w^2){\tilde V}_{de} +3\beta[-x^{-2}\xi+3(1-w){\tilde V}_{de}], \label{eq.Dx_newnew} \\ &\frac{d {\tilde V}_{de}}{d\ln x}=-{\tilde V}_{de} +{{\tilde\Delta}_{de}\over 1+w}+x^{-2}\xi+{3\beta\over(1+w)}({\tilde V}_c-2{\tilde V}_{de}), \label{eq.Vx_newnew} \\ &\frac{d {\tilde V}_T}{d\ln x}=-3{\tilde V}_T-\Omega_\nu{\tilde \Pi}_\nu-\zeta-\xi,\label{eq:VTEOM} \end{aligned}$$ where $\Omega_I=8\pi Ga^2\rho_I/(3\mathcal{H}^2)$ denotes the dimensionless energy density of $I$ fluid, $$\begin{aligned} x^{-2}\xi = -{\sum_{I=c,\,b,\,\gamma,\,\nu}w_I\Omega_I{\tilde\Delta}_I - {2\over 9}c_K\Omega_\nu{\tilde\Pi}_\nu-3\beta\Omega_{de}{\tilde V}_c \over \sum_{I=c,\,b,\,\gamma,\,\nu}(1+w_I)\Omega_I}, \label{eq:xixeom}\end{aligned}$$ is derived from Eq. (\[eq:xi\]), and $$\begin{aligned} &\zeta=-{\tilde V}_T+ {3\over 2c_K}\left[\sum\Omega_I{\tilde\Delta}_I+3(1+w)\Omega_{de}{\tilde V}_{de}\right],\\ &x^{-2}\frac{d\zeta}{d\ln x} = x^{-2}\xi - {K\over k^2} {\tilde V}_T - {3\over 2} (1+w)\Omega_{de}{\tilde V}_{de}, \label{eqn:xieom}\end{aligned}$$ can be found from the first and third Einstein equations given in Ref. [@Hu:2008zd]. In the early radiation dominated epoch, $a=\mathcal{H}_0\sqrt{\Omega_{r0}}\tau$ and $\Omega_I\simeq\rho_I/\rho_r$ with $\rho_r=\rho_\gamma+\rho_\nu$, we have $$\begin{aligned} & \Omega_{de} =\frac{\Omega_{de0}}{\Omega_{r0}}\left(\frac{\sqrt{\Omega_{r0}}\mathcal{H}_0}{k}\right)^{1-3(w-\beta)} \, x^{1-3(w-\beta)}, \quad \quad \quad \Omega_{c} = \left(1+{\beta\over\beta-w}{\Omega_{de0}\over\Omega_{c0}}\right)\frac{\Omega_{c0}}{\sqrt{\Omega_{r0}}}\frac{\mathcal{H}_0}{k} \, x \,, \nonumber\\ &\quad \Omega_{b}=\frac{\Omega_{b0}}{\Omega_{r0}} \, a =\frac{\Omega_{b0}}{\sqrt{\Omega_{r0}}}\frac{\mathcal{H}_0}{k} \, x,\quad \quad \Omega_\nu = {\rho_\nu \over\rho_{r}} = R_\nu, \quad \quad \Omega_\gamma = 1- \Omega_{b} - \Omega_{c} - \Omega_{de} -\Omega_\nu \,. \label{eq:Omegas_expl}\end{aligned}$$ Now using Eqs. (\[eq:xixeom\])–(\[eq:Omegas\_expl\]), we can obtain the coefficient matrix $\mathbf{A}(x)$ from Eqs. (\[eq.Deltac\_newnew\])–(\[eq:VTEOM\]). Furthermore, since $x\ll1$, we can approximate $\mathbf{A}(x)$ by a constant matrix $\mathbf{A}_0$, as long as no divergence occurs when $x\rightarrow0$. For the $w<-1/3$ and small coupling $|\beta|<|w|$ case, there is no divergence term in the matrix $\mathbf{A}_0$ at $x=0$. Thus, we have $$\mathbf{A}_0= \left( \begin{array}{cccccccccc} -2 & 0 & \frac{-3}{4} \mathcal{N} & 0 & 0 & \frac{3 R_{\nu }}{4} & -\frac{R_{\nu }}{2} & 0 & 0 & -1 \\ 0 & -4 & \frac{1}{4} \mathcal{N} & 0 & 0 & -\frac{R_{\nu }}{4} & \frac{R_{\nu }}{6} & 0 & 0 & 0 \\ 0 & 0 & -R_{\nu }-1 & 0 & 0 & R_{\nu } & -\frac{2}{3} R_{\nu } & 0 & 0 & -\frac{4}{3} \\ 0 & 0 & \frac{1}{4} & -3 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{-3}{4} \mathcal{N} & 0 & -2 & \frac{3 R_{\nu }}{4} & -\frac{R_{\nu }}{2} & 0 & 0 & -1 \\ 0 & 0 & -\mathcal{N} & 0 & 0 & \mathcal{N}-1 & -\frac{2}{3} R_{\nu } & 0 & 0 & -\frac{4}{3} \\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & \frac{8}{5} \\ 0 & 0 & \frac{-3}{4} \mathcal{N}\mathcal{B} & 0 & 0 & \frac{3}{4} \mathcal{B} R_{\nu } & -\frac{1}{2} \mathcal{B} R_{\nu } & 3 w-5 & 9 (w-1) \mathcal{M} & -w-1 \\ 0 & \frac{3 \beta }{w+1} & \frac{1}{4} \mathcal{N} & 0 & 0 & -\frac{R_{\nu }}{4} & \frac{R_{\nu }}{6} & \frac{1}{w+1} & -\frac{w+6 \beta +1}{w+1} & 0 \\ 0 & 0 & \frac{3}{2} \mathcal{N} & 0 & 0 & \frac{-3}{2} R_{\nu } & -R_{\nu } & 0 & 0 & -2 \\ \end{array} \right),$$ where $\mathcal{N}=R_{\nu }-1$, $\mathcal{B}=w+\beta +1$ and $\mathcal{M}=w-\beta +1$. The eigenvalues of $\mathbf{A}_0$ can be obtained immediately, \_i = {0,-4,-3,-2,-2,-2,- - ,- + , \_[d]{}\^[-]{}, \_[d]{}\^[+]{} }, \[eq.eigenvals\] where \_[d]{}\^ = - . \[eq:lambdas\] The approximate solutions to the matrix equation (\[eq.dUdlnx\_gen\]) can be written as $\sum c_ix^{\lambda_i}\bm{U}_0^{(i)}$ with $c_i$ the dimensionless constant and $\bm{U}_0^{(i)}$ the eigenvector corresponding to eigenvalue $\lambda_i$. Obviously, the mode with negative ${\rm Re}(\lambda_i)$ will soon decay or oscillate. For our studied IDE model, if $w<-1$ or $\beta<0$, ${\rm Re}(\lambda_{d}^{\pm})\gg1$, the curvature perturbation will grow rapidly at the early times, leading to the large-scale instability. So we actually calibrate $f_\zeta(a)$ for the stable $w>-1$ and $\beta>0$ case. When $w>-1$ and $\beta>0$, the only largest eigenvalue in Eq. (\[eq.eigenvals\]) is zero, and the corresponding eigenvector is $$\bm{U}_0^T = \left\{ -\mathcal{P},\,\frac{1}{6}\mathcal{P}-\frac{1}{12},\,-\frac{4}{3}\mathcal{P},\,-\frac{1}{9}\mathcal{P},\,-\mathcal{P},-\frac{4}{3}\mathcal{P},\,\frac{4}{5},{\tilde\Delta}_{de}^{(0)},\, \tilde{V}_{de}^{(0)},\,1 \right\},$$ where $\mathcal{P}=(2 R_{\nu }+5)/5$ and $$\tilde{V}_{de}^{(0)}=\frac{4 R_{\nu } \left(-3 \beta +12 w^2+9 \beta w+4 w-8\right)+5 \left(-3 \beta +12 w^2+9 \beta w+16 w+4\right)}{60 \left(-21 \beta +12 w^2+9 \beta w-2 w-14\right)}.$$ This mode dominates the cosmological perturbation evolutions on the large scales, so from Eq. (\[eq:fzeta\]) we have $$f_\zeta(a)=-{12\pi Ga^2 \over c_K\mathcal{H}^2} (\rho_{de} + p_{de})\tilde{V}_{de}^{(0)}.\label{eq:fzetafinal}$$ Here note that $\tilde{V}_T^{(0)}=1$. [99]{} P. A. R. Ade [*et al.*]{} \[Planck Collaboration\], arXiv:1303.5076 \[astro-ph.CO\]. 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ArXiv

--- abstract: 'The Configuration Interaction (CI) method is applied to the calculation of the structures of a number of positron binding systems, including $e^+$Be, $e^+$Mg, $e^+$Ca and $e^+$Sr. These calculations were carried out in orbital spaces containing about 200 electron and 200 positron orbitals up to $\ell = 12$. Despite the very large dimensions, the binding energy and annihilation rate converge slowly with $\ell$, and the final values do contain an appreciable correction obtained by extrapolating the calculation to the $\ell \to \infty$ limit. The binding energies were 0.00317 hartree for $e^+$Be, 0.0170 hartree for $e^+$Mg, 0.0189 hartree for $e^+$Ca, and 0.0131 hartree for $e^+$Sr.' author: - 'M.W.J.Bromley' - 'J.Mitroy' title: Large dimension Configuration Interaction calculations of positron binding to the group II atoms --- Introduction ============ The ability of positrons to bind to a number of atoms is now well established [@mitroy02b; @schrader01a; @strasburger03a], and all of the group II elements of the periodic table are expected to bind a positron [@mitroy02b; @mitroy02f]. There have been two sets of calculations that are consistent, in that they tend to predict the same binding energy and annihilation rate. The first set of calculations were those undertaken on $e^+$Be and $e^+$Mg [@ryzhikh98c; @ryzhikh98e; @mitroy01c] with the fixed core stochastic variational method (FCSVM) [@ryzhikh98b; @ryzhikh98e; @mitroy02b]. Some time later, configuration interaction (CI) calculations were undertaken on $e^+$Be, $e^+$Mg, $e^+$Ca and $e^+$Sr [@bromley02a; @bromley02b]. The calculations for $e^+$Be and $e^+$Mg agreed to within the respective computational uncertainties, which were roughly about 5-10$\%$ for the binding energy. One feature common to all the CI calculations is the slow convergence of the binding energy and the annihilation rate. The attractive electron-positron interaction leads to the formation of a Ps cluster (i.e. something akin to a positronium atom) in the outer valence region of the atom [@ryzhikh98e; @dzuba99; @mitroy02b; @saito03a]. The accurate representation of a Ps cluster using only single particle orbitals centered on the nucleus requires the inclusion of orbitals with much higher angular momenta than a roughly equivalent electron-only calculation [@strasburger95; @schrader98; @mitroy99c; @dzuba99]. For example, the largest CI calculations on the group II positronic atoms and PsH have typically have involved single particles bases with 8 radial function per angular momenta, $\ell$, and inclusion of angular momenta up to $L_{\rm max} = 10$ [@bromley02a; @bromley02b; @saito03a]. Even with such large orbital basis sets, between 5-60$\%$ of the binding energy and some 30-80$\%$ of the annihilation rate were obtained by extrapolating from $L_{\rm max} = 10$ to the $L_{\rm max} = \infty$ limit. Since our initial CI calculations [@bromley00a; @bromley02a; @bromley02b], advances in computer hardware mean larger dimension CI calculations are possible. In addition, program improvements have removed the chief memory bottleneck that previously constrained the size of the calculation. As a result, it is now appropriate to revisit the group II atoms to obtain improved estimates of their positron binding energies and other expectation values. The new calculations that we have performed have orbital spaces more than twice as large as those reported previously. The estimated CI binding energies for all systems have increased, and furthermore the uncertainties resulting from the partial wave extrapolation have decreased. Calculation Method ================== The CI method as applied to atomic systems with two valence electrons and a positron has been discussed previously [@bromley02a; @bromley02b], and only a brief description is given here. All calculations were done in the fixed core approximation. The effective Hamiltonian for the system with $N_e = 2$ valence electrons and a positron was $$\begin{aligned} H &=& - \frac{1}{2}\nabla_{0}^2 - \sum_{i=1}^{N_e} \frac {1}{2} \nabla_{i}^2 - V_{\rm dir}({\bf r}_0) + V_{p1}({\bf r}_0) \nonumber \\ &+& \sum_{i=1}^{N_e} (V_{\rm dir}({\bf r}_i) + V_{\rm exc}({\bf r}_i) + V_{p1}({\bf r}_i)) - \sum_{i=1}^{N_e} \frac{1}{r_{i0}} \nonumber \\ &+& \sum_{i<j}^{N_e} \frac{1}{r_{ij}} - \sum_{i<j}^{N_e} V_{p2}({\bf r}_i,{\bf r}_j) + \sum_{i=1}^{N_e} V_{p2}({\bf r}_i,{\bf r}_0) \ .\end{aligned}$$ The index $0$ denotes the positron, while $i$ and $j$ denote the electrons. The direct potential ($V_{\rm dir}$) represents the interaction with the electronic core, which was derived from a Hartree-Fock (HF) wave function of the neutral atom ground state. The exchange potential ($V_{\rm exc}$) between the valence electrons and the HF core was computed without approximation. The one-body and two-body polarization potentials ($V_{p1}$ and $V_{p2}$) are semi-empirical with the short-range cut-off parameters derived by fitting to the spectra of their singly ionized ions. All details of the core-polarization potentials including the polarizabilities, $\alpha_d$, are given in [@bromley02a; @bromley02b]. Note that the functional form of the polarization potential,$V_{p1}$, was set to be the same for the electrons and the positron. The positronic atom wave function is a linear combination of states created by multiplying atomic states to single particle positron states with the usual Clebsch-Gordan coupling coefficients ; $$\begin{aligned} |\Psi;LS \rangle &=& \sum_{i,j} c_{i,j} \ \langle L_i M_i \ell_j m_j|L M_L \rangle \langle S_i M_{S_i} {\scriptstyle \frac{1}{2}} \mu_j|S M_S \rangle \nonumber \\ &\times& \Phi_i(Atom;L_iS_i) \phi_j({\bf r}_0) \ . \end{aligned}$$ In this expression $\Phi_i(Atom;L_i S_i)$ is an antisymmetric atomic wave function with good $L$ and $S$ quantum numbers. The function $\phi_j({\bf r}_0)$ is a single positron orbital. The single particle orbitals are written as a product of a radial function and a spherical harmonic: $$\phi({\bf r}) = P(r) Y_{lm}({\hat {\bf r}}) \ .$$ As the calculations were conducted in a fixed core model we used HF calculations of the neutral atom ground states to construct the core orbitals. These HF orbitals were computed with a program that can represent the radial wave functions as a linear combination of Slater Type Orbitals (STO) [@mitroy99f]. A linear combination of STOs and Laguerre Types Orbitals (LTOs) was used to describe the radial dependence of electrons occupying orbitals with the same angular momentum as those in the ground state. Orbitals that did not have any core orbitals with the same angular momentum were represented by a LTO set with a common exponential parameter. The STOs give a good representation of the wave function in the interior region while the LTOs largely span the valence region. The LTO basis [@bromley02a; @bromley02b] has the property that the basis can be expanded toward completeness without introducing any linear independence problems. The CI basis included all the possible $L = 0$ configurations that could be formed by letting the two electrons and positron populate the single particle orbitals subject to two selection rules, $$\begin{aligned} \max(\ell_0,\ell_1,\ell_2) & \le & L_{\rm max} \ , \\ \min(\ell_1,\ell_2) & \le & L_{\rm int} \ . \end{aligned}$$ In these rules $\ell_0$ is the positron orbital angular momentum, while $\ell_1$ and $\ell_2$ are the angular momenta of the electrons. A large value of $L_{\rm max}$ is necessary as the attractive electron-positron interaction causes a pileup of electron density in the vicinity of the positron. The $L_{\rm int}$ parameter was used to eliminate configurations involving the simultaneous excitation of both electrons into high $\ell$ states. Calculations on PsH and $e^+$Be had shown that the choice of $L_{\rm int} = 3$ could reduce the dimension of the CI basis by a factor of 2 while having an effect of about 1$\%$ upon the binding energy and annihilation rate [@bromley02a]. The present set of calculations were all performed with $L_{\rm int} = 4$. Various expectation values were computed to provide information about the structure of these systems. The mean distance of the electron and positron from the nucleus are denoted by $\langle r_e \rangle$ and $\langle r_p \rangle$. The $2\gamma$ annihilation rate for annihilation with the core and valence electrons was computed with the usual expressions [@neamtan62; @drachman95b; @ryzhikh99a]. The $2\gamma$ rate for the core ($\Gamma_c$) and valence ($\Gamma_v$) electrons are tabulated separately. Extrapolation issues -------------------- The feature that differentiates mixed electron-positron CI calculations from purely electron CI calculations is the slow convergence of the calculation with respect to $L_{\rm max}$, the maximum $\ell$ of any electron or positron orbital included in the CI basis. Typically, a calculation is made to $L_{\rm max} \approx 10$ (or greater), with various extrapolation techniques used to estimate the $L_{\rm max} \to \infty$ correction. For any expectation value one can write formally $$\langle X \rangle^{L_{\rm max}} = \sum_{L=0}^{L_{\rm max}} \Delta X^{L} \ , \label{XJ1}$$ where $\Delta X^{L}$ is the increment to the observable that occurs when the maximum orbital angular momentum is increased from $L\!- \! 1$ to $L$, e.g. $$\Delta X^{L} = \langle X \rangle^{L} \ - \ \langle X \rangle^{L-1} \ . \label{XJ3}$$ Hence, one can write formally $$\langle X \rangle = \langle X \rangle^{L_{\rm max}} \ + \sum_{L=L_{\rm max}+1}^{\infty} \Delta X^{L} \ . \label{XJ2}$$ However, it is quite easy to make substantial errors in estimating the $L_{\rm max} \to \infty$ correction [@mitroy05i; @mitroy06a; @mitroy06b]. There have been a number of investigations of the convergence of CI expansions for electronic and mixed electron-positron systems [@schwartz62a; @carroll79a; @hill85a; @kutzelnigg92a; @schmidt93a; @ottschofski97a; @gribakin02a; @mitroy02b; @bromley06a; @mitroy06a]. The reliability of the different methods to estimate the $L_{\rm max} \to \infty$ correction for the energy and annihilation rate has been assessed in detail elsewhere [@mitroy06a]. In this work, only the briefest description of the recommended methods are described. The recent computational investigations of helium [@bromley06a] and some positron-atom systems [@mitroy06a] suggest that usage of an inverse power series of the generic type $$\begin{aligned} \Delta X^{L_{\rm max}} &=& \frac{B_X}{(L_{\rm max}+{\scriptstyle \frac{1}{2}})^n} + \frac{C_X}{(L_{\rm max}+{\scriptstyle \frac{1}{2}})^{n+1}} \nonumber \\ & + & \frac{D_X}{(L_{\rm max}+{\scriptstyle \frac{1}{2}})^{n+2}} + \ldots \ , \label{Aseries} \end{aligned}$$ is the best way to determine the $L_{\rm max} \to \infty$ correction for the energy, $E$ and the 2$\gamma$ annihilation rate. A three term series with $n = 4$ is used for the energy. One needs four successive values of $E^{L}$ to determine the coefficients $B_E$, $C_E$ and $D_E$. Once the coefficients have been fixed, the inverse power series is summed to $J_{\rm max} = 100$,after which the approximate result $$\sum_{L=J_{\rm max}+1}^{\infty} \frac{1}{(L+{\scriptstyle \frac{1}{2}})^p } \approx \frac{1}{(p-1)(J_{\rm max}+1)^{p-1} } \ , \label{bettertail}$$ is used [@mitroy06b]. The correction to $\Gamma$ follows the same general procedure as the energy, but with two differences. The power in eq. (\[Aseries\]) is set to $n = 2$ and only 2-terms are retained in the series (requiring three successive values of $\Gamma^{L}$). The usage of the inverse power series is the preferred approach when the asymptotic form for $\Delta X^L$ has been established by perturbation theory. For other operators it is best to to use a single-term inverse power series with an indeterminate power, e.g $$\Delta X^L = \frac{A}{(L+ {\scriptstyle \frac{1}{2}})^p} .$$ The factors $A$ and $p$ can be determined from the three largest calculations using $$p = \ln \left( \frac {\Delta X^{L_{\rm max}-1}}{\Delta X^{L_{\rm max}}} \right) \biggl/ \ln \left( \frac{L_{\rm max}+{\scriptstyle \frac{1}{2}}}{L_{\rm max}-{\scriptstyle \frac{1}{2}}} \right) \ , \label{pdef}$$ and $$A = \Delta X^{L_{\rm max}} (L_{\rm max} +{\scriptstyle \frac{1}{2}})^{p} \ . \label{Avalue}$$ Once $p$ and $A$ are determined, the $L_{\rm max} \to \infty$ correction can be included using the same procedure as adopted for the multi-term fits to the energy and annihilation. This method is used in determination of the $L_{\rm max} \to \infty$ estimates of $\langle r_e \rangle$, $\langle r_p \rangle$ and $\Gamma_c$. However, the value of $p$ is computed for all operators since it is useful to know whether $p_E$ and $p_{\Gamma_v}$ are close to the expected values of 4 and 2 respectively. While the subdivision of the annihilation rate into core and valence components is convenient for physical interpretation, it was also done on mathematical grounds. The calculation of $\Gamma_c$ does not explicitly include correlations between the core electrons and the positron, and so the $\Delta \Gamma_c^L$ increments converge faster than the $\Delta \Gamma_v^L$ increments (i.e. $p_{\Gamma_c} > p_{\Gamma_v}$). Calculation Results =================== Improved FCSVM data for $e^+$Be and $e^+$Mg ------------------------------------------- The FCSVM [@ryzhikh98e; @mitroy02b] has also been applied to determine the structures of $e^+$Be and $e^+$Mg [@ryzhikh98e; @mitroy01c]. The FCSVM expands the wave function as a linear combination of explicitly correlated gaussians (ECGs), with the core orbitals taken from a HF calculation. One- and two-body polarization potentials are included while orthogonality of the active electrons with the core is enforced by the use of an orthogonalizing pseudo-potential [@ryzhikh98e; @mitroy99h; @mitroy02b]. The FCSVM model hamiltonians are very similar to those used in the CI calculations. But there are some small differences in detail that lead to the FCSVM hamiltonian giving slightly different energies. The best previous FCSVM wave function for $e^+$Be [@mitroy01c] gave a binding energy, 0.03147 hartree, and annihilation rate $0.420 \times 10^9$ sec$^{-1}$,that were close to convergence. Some extensive re-optimizations seeking to improve the quality of the wave function in the asymptotic region yielded only minor changes (of the order of 1$\%$) in the ground state properties [@mitroy05d]. Nevertheless, the latest energies and expectation values for the $e^+$Be ground state are tabulated in Tables \[Beptab\] and \[summary\]. These values should be converged to better than 1$\%$ with respect to further enlargement and optimization of the ECG basis. The more complex core for Mg does slow the convergence of the energy and other properties of $e^+$Mg considerably [@mitroy02b]. The best energy previously reported for this system was 0.016096 hartree [@mitroy05d]. The current best wave function, which is constructed from a linear combination of 1200 ECGs gives a binding energy of 0.016930 hartree and a valence annihilation rate of $1.0137 \times 10^9$ sec$^{-1}$. Other expectation values are listed in Table \[Beptab\]. Examination of the convergence pattern during the series of basis set enlargements and optimizations suggests that the binding energy and annihilation rate are converged to between 2$\%$ and 5$\%$. The FCSVM binding energies do have a weak dependence on one parameter in the calculation since the orthogonalizing pseudo-potential is actually a penalty function, viz $$\lambda {\hat P} = \sum_{i \ \in \ \text{core}} \lambda |\phi_i \rangle \langle \phi_i | \ , \label{OPP}$$ that was added to the hamiltonian. Choosing $\lambda$ to be large and positive means the energy minimization automatically acts to construct a wave function which has very small overlap with the core [@krasnopolsky74; @ryzhikh98e; @mitroy99h]. The FCSVM properties reported in Tables \[Beptab\] and \[summary\] were computed with $\lambda = 10^5$ hartree. The core overlap (i.e. the expectation value of ${\hat P}$) was $1.86 \times 10^{-11}$ for $e^+$Be and $1.61 \times 10^{-10}$ for $e^+$Mg. CI results for group II atoms ----------------------------- Table \[Beptab\] contains the results of the current series of calculations on the four positronic atoms. The size of the calculations for the four atoms were almost the same. The electron-electron angular momentum selector was set to $L_{\rm int} = 4$. For $\ell > 3$ at least 15 LTOs were included in the radial basis sets for the electron and positron orbitals. For $\ell \le 2$ the dimension of the orbital basis sets were slightly larger than 15 and the basis sets for electrons occupying orbitals with the same angular momentum as those in the core were typically a mix of STOs (to describe the electron close to nucleus) and LTOs. The calculations used basis sets with $L_{\rm max} = 9, 10, 11$ and 12. The calculations with $L_{\rm max} < 12$ had configuration spaces which were subsets of the $L_{\rm max} = 12$ and this expedited the computations since one list of radial matrix elements was initially generated for the $L_{\rm max} = 12$ basis and then reused for the smaller basis sets. The secular equations that arose typically had dimensions of about 500,000 and the diagonalizations were performed with the Davidson algorithm using a modified version of the program of Stathopolous and Froese-Fischer [@stathopolous94a]. Convergence was not very quick and about 16000 iterations were needed to achieve convergence in some cases. It was possible to speed up the diagonalization for $L_{\rm max} < 12$. An edited eigenvector from the $L_{\rm max} = 12$ calculation was used as the initial eigenvector estimate, and this often reduced the number of iterations required by 50$\%$. ### Results for $e^+$Be The lowest energy dissociation channel is the $e^+ + \text{Be}$ channel, which has an energy of $-1.01181167$ hartree with respect to the doubly ionized Be$^{2+}$ core. The agreement of the extrapolated CI binding energy of $\varepsilon = 0.003169$ hartree with the FCSVM binding energy of $\varepsilon =0.003180$ is better than 1$\%$. A similar level of agreement exists for the $\langle r_e \rangle$ and $\langle r_p \rangle$ expectation values. The only expectation value for which 1$\%$ level of agreement does not occur is the annihilation rate and here the extrapolated CI value of $0.4110 \times 10^9$ sec$^{-1}$ is only about $3.5\%$ smaller than the FCSVM of $0.4267 \times 10^9$ sec$^{-1}$. However, it is known that the convergence of the annihilation rate with respect to an increasing number of radial basis functions is slower than the convergence of the energy [@mitroy06a; @bromley06a]. This means that a CI type calculation has an inherent tendency to underestimate the annihilation rate. For example, a CI calculation on PsH of similar size to the present $e^+$Be calculation underestimated the annihilation rate by 6$\%$ [@mitroy06a]. That the exponent of the polar law decay, $p_{\Gamma_v} = 2.10$, is larger than the expected asymptotic value of $p = 2.0$ is consistent with this idea. A better estimate of the annihilation rate can be obtained by simply forcing $C_{\Gamma}$ to be zero in eq. (\[Aseries\]) and thus using $\Delta \Gamma^{12}$ to fit $B_{\Gamma}$. When this is done done the annihilation rate increases to $0.4178 \times 10^9$ sec$^{-1}$. [lcccccc]{} $L_{\rm max}$& $E$ & $\varepsilon$ & $\langle r_e \rangle$ & $\langle r_p \rangle$ & $\Gamma_c$ & $\Gamma_v$\ \ 10$^*$ & (-1.0143769)& (0.002533) & (2.639) & (10.746) & (0.001962) & (0.2411)\ 9 & -1.01435756 & 0.00254589 & 2.6388477 & 10.874256 & 0.00193993 & 0.24026720\ 10 & -1.01448318 & 0.00267151 & 2.6418168 & 10.699433 & 0.00198405 & 0.25651443\ 11 & -1.01457837 & 0.00276670 & 2.6441227 & 10.574208 & 0.00201619 & 0.27004634\ 12 & -1.01465138 & 0.00283971 & 2.6459282 & 10.482126 & 0.00204005 & 0.28140404\ $p$ & 3.1806 & 3.1806 & 2.9339 & 3.6871 & 3.5764 & 2.1006\ $\infty$ & -1.0149809 & 0.0031692 & 2.65673 & 10.09755 & 0.002144 & 0.410976\ FCSVM & -1.0151335 & 0.003180 & 2.654 & 10.048 & 0.00221 & 0.4267\ \ 10$^*$ & (-0.8473592)& (0.0145092)& (3.382) & (7.101) & (0.010845) & (0.5429)\ 9 & -0.84741494 & 0.01450067 & 3.3831320 & 7.116532 & 0.01079647 & 0.54089010\ 10 & -0.84790548 & 0.01499121 & 3.3936654 & 7.071950 & 0.01084944 & 0.57692369\ 11 & -0.84828090 & 0.01536663 & 3.4022694 & 7.040929 & 0.01087921 & 0.60775407\ 12 & -0.84857204 & 0.01565777 & 3.4093312 & 7.018703 & 0.01089568 & 0.63435278\ $p$ & 3.0496 & 3.0496 & 2.3690 & 3.9985 & 7.0953 & 1.7706\ $\infty$ & -0.8499543 & 0.0170400 & 3.47039 & 6.93657 & 0.010922 & 0.990069\ FCSVM & -0.849002 & 0.016930 & 3.447 & 6.923 & 0.0112 & 1.0137\ \ 10$^*$ & (-0.6986443)& (0.0123578)& (4.456) & (6.848) & (0.01355) & (0.7335)\ 9 & -0.69855551 & 0.01226895 & 4.4602428 & 6.863740 & 0.01343426 & 0.72709017\ 10 & -0.69975764 & 0.01347109 & 4.4873848 & 6.872414 & 0.01323075 & 0.78001274\ 11 & -0.70069553 & 0.01440898 & 4.5110869 & 6.885039 & 0.01304512 & 0.82640757\ 12 & -0.70143637 & 0.01514981 & 4.5315631 & 6.898804 & 0.01288316 & 0.86733542\ $p$ & 2.8286 & 2.8286 & 1.7546 & -1.0371 & 1.6361 & 1.5037\ $\infty$ & -0.7052160 & 0.0189295 & 4.86076 & — & 0.009780 & 1.478148\ \ 10$^*$ & (-0.6602186)& (0.0048689)& (4.850) & (7.056) & (0.01487) & (0.7488)\ 9 & -0.65997599 & 0.00462598 & 4.8638673 & 7.100141 & 0.01464684 & 0.73239378\ 10 & -0.66146709 & 0.00611708 & 4.8979559 & 7.123685 & 0.01432317 & 0.78845209\ 11 & -0.66263875 & 0.00728874 & 4.9283728 & 7.150071 & 0.01403253 & 0.83790890\ 12 & -0.66357065 & 0.00822064 & 4.9552753 & 7.176708 & 0.01377785 & 0.88177286\ $p$ & 2.7459 & 2.7459 & 1.4725 & -0.1134 & 1.5844 & 1.4393\ $\infty$ & -0.6684520 & 0.0131020 & 5.65380 & — & 0.008456 & 1.552589\ ### Results for $e^+$Mg The results of the calculations with $e^+$Mg are listed in Table \[Beptab\]. The lowest energy dissociation channel is to $e^+ + \text{Mg}$, which has an energy of $-0.83291427$ hartree with respect to the doubly ionized Mg$^{2+}$ core. The CI calculations, reported in Table \[Beptab\] for $L_{\rm max} =$ 9, 10, 11 and 12 are largely consistent with the FCSVM calculations. The largest explicit CI calculation gives a binding energy of 0.015658 hartree. Extrapolation to the $L_{\rm max} \rightarrow \infty$ limit adds about $10\%$ to the binding energy, and the final estimate was 0.017040 hartree. Despite the better than 1$\%$ agreement between the CI and FCSVM calculations, a further binding energy increase of about 1-2$\%$ would be conceivable if both calculations were taken to the variational limit. The slow convergence of $\Gamma_v$ with $L_{\rm max}$ is evident from Table \[Beptab\] and the extrapolation correction contributes about 36$\%$ to the overall annihilation rate. The present $L_{\rm max} \to \infty$ estimate can be expected to be too small by 5-10$\%$. All the other expectation values listed in Table \[Beptab\] lie with 1-2$\%$ of those of the FCSVM expectation values. As a general rule, inclusion of the $L_{\max} \to \infty$ corrections generally improves the agreement between the CI and FCSVM calculations. ### Results for $e^+$Ca The results of the calculations with $e^+$Ca are listed in Table \[Beptab\]. Since neutral calcium has an ionization potential smaller than the energy of Ps ground state (the present model potential and electron orbital basis gives -0.43628656 hartree for the Ca$^+$ energy and -0.65966723 hartree for the neutral Ca energy), its lowest energy dissociation channel is the $\text{Ps} + \text{Ca}^+$ channel. The present model potential gives this channel an energy of $-0.68628656$ hartree. The energies listed in Table \[Beptab\] indicate that $e^+$Ca is the positronic atom with the largest known binding energy, namely $\varepsilon = 0.018929$ hartree. The $L_{\rm max} \to \infty$ correction contributes 20 $\%$ of the binding energy. The partial wave series is more slowly convergent for $e^+$Ca than for $e^+$Mg (i.e. $p_E$ is smaller, and the coefficients $C_E$ and $D_E$ in eq. (\[bettertail\]) are larger). This is expected since calcium has a smaller ionization potential, and so the electrons are located a greater distance away from the nucleus. This makes it easier for the positron to attract the electrons, and the stronger pileup of electron density around the positron further from the nucleus requires a longer partial wave expansion to represent correctly. The slower convergence of the wave function with $L_{\rm max}$ makes an even larger impact on the annihilation rate. Some 41$\%$ of the annihilation rate of $\Gamma_v = 1.478 \times 10^9$ sec$^{-1}$ comes from the $L_{\rm \max} \to \infty$ correction. As mentioned earlier for $e^+$Mg, it is likely that this value is slightly smaller than the true annihilation rate. The extrapolation corrections for $\langle r_p \rangle$ and $\Gamma_c$ listed in Table \[Beptab\] are unreliable. The $e^+$Ca system, at large distances consists of Ca$^+$ + Ps. In other calculations of positron binding systems it has been noticed that systems that decay asymptotically into $\text{Ps} + \text{X}$ do not have an $\langle r_p \rangle$ that changes monotonically with $L_{\rm max}$ [@bromley00a; @bromley02a]. Initially, the positron becomes more tightly bound to the system as $L_{\rm max}$ increases, resulting in a decrease in $\langle r_p \rangle$. However, $\langle r_p \rangle$ tends to increase at the largest values of $L_{\rm max}$. The net result of all this is that $\Delta \langle r_p \rangle^{L}$ (and by implication $\Delta \Gamma_c^L$) approach their asymptotic forms very slowly. The best policy is to simply not to give any credence to the extrapolation corrections for either of these operators for $e^+$Ca (and $e^+$Sr). The small value of $p$ for $\Delta \langle r_e \rangle^{L}$ suggests that the reliability of the $L_{\rm max} \to \infty$ correction may be degraded for this expectation value as well. ### Results for $e^+$Sr The results of the calculations for $e^+$Sr are listed in Table \[Beptab\]. Since neutral strontium has an ionization potential smaller than the energy of Ps ground state (the present model potential and electron orbital basis gives -0.40535001 hartree for the Sr$^+$ energy and -0.61299101 hartree for the neutral Sr energy), its lowest energy dissociation channel is the $\text{Ps} + \text{Sr}^+$ channel, which has an energy of -0.65535001 hartree. The small ionization potential of 0.20764100 hartree means that the structure of the $e^+$Sr ground state will be dominated by a $\Psi(\text{Sr}^+$)$\Psi(\text{Ps})$ type configuration [@mitroy02b]. This leads to slower convergence of the ground state with $L_{\rm max}$ which is evident from Table \[Beptab\]. As expected, the binding energy of $e^+$Sr is smaller than that of $e^+$Ca. Previous investigations have indicated that positron binding energies should be largest for atoms with ionization potentials closest to 0.250 hartree (the Ps binding energy) [@mitroy99b; @mitroy02f]. There is obviously some uncertainty in the precise determination of the binding energy due to fact that $L_{\rm max} \to \infty$ correction constitutes some $37 \%$ of the binding energy of 0.013102 hartree. The net effect of errors due to the extrapolation correction are not expected to be excessive. Applying eq. (\[Aseries\]) with only the first two-terms retained (i.e. $D_E = 0$) results in a final energy 0.012764 hartree, which is 3$\%$ smaller than the value of 0.013102 Hartree. The present $e^+$Sr binding energy is some 30$\%$ larger than the energy of the previous CI calculation listed in Table \[summary\] [@bromley02b]. The final estimate of the valence annihilation rate was $ 1.553 \times 10^9$ sec$^{-1}$ and some 43$\%$ of the annihilation rate comes from the $L_{\rm \max} \to \infty$ correction. This value of $\Gamma_v$ could easily be 10$\%$ smaller than the true annihilation rate. The explicitly calculated expectation values for $\langle r_e \rangle$, $\langle r_p \rangle$ and $\Gamma_c$ at $L_{\rm max} =12$ should be preferred since the $L_{\rm \max} \to \infty$ corrections in these cases are likely to be unreliable. 3-body clustering ----------------- While the truncation of the basis to $L_{\rm int} = 4$ has little effect on the $e^+$Be system, its effect is larger for the $e^+$Sr system. The more loosely bound alkaline-earth atoms have their electrons localized further away from the nucleus, and this makes it easier for the positron to form something like a Ps$^{-}$ cluster [@mitroy02f; @mitroy04a]. When this occurs, correlations of the positron with [*both*]{} electrons increase in strength, and the inclusion of configurations with $L_{\rm int} > 4$ becomes more important. The relative size of of these neglected $L_{\rm int} > 4$ configurations can be estimated using techniques similar to those adopted for the $L_{\rm max} \to \infty$ corrections. Calculations for a succession of $L_{\rm int}$ values were performed in earlier works [@bromley02a; @bromley02b]. The assumption is made that the binding energy and annihilation rate increments scale as $A/(L_{\rm int}+{\scriptstyle \frac{1}{2}})^4$ (note, the power of 4 for the annihilation is used since $L_{\rm int}$ only has a directly effect on electron-electron correlations). The difference between an $L_{\rm int} = 2$ and $L_{\rm int} = 3$ is used to estimate $A$ and then eq.( \[bettertail\]) determines the $L_{\rm int} \to \infty$ correction (in the case of $e^+$Be calculations up to $L_{\rm int} = 10$ exist [@bromley02a]). Table \[summary\] contains a summary of the final binding energies obtained from the present CI calculations, and earlier binding energies obtained alternate methods. As part of this table, energies with an additional $L_{\rm int} \to \infty$ correction are also given. The size of the correction ranges from $1.8 \times 10^{-5}$ hartree for $e^+$Be to $21.9 \times 10^{-5}$ hartree for $e^+$Sr. Even though these estimations of the correction are not rigorous, they indicate that the underestimation in the binding energy resulting from a truncation of the configuration space to $L_{\rm int} < 4$ is most likely to be 2$\%$ or smaller. A similar analysis could be done for the annihilation rate but previous results indicate that $\Gamma_v$ is less sensitive than $\varepsilon$ to an increase in $L_{\rm int}$ [@bromley02a; @bromley02b]. The net increases in $\Gamma_v$ for $e^+$Be, $e^+$Mg, $e^+$Ca and $e^+$Sr were $0.0011 \times 10^9$ sec$^{-1}$, $0.0030 \times 10^9$ sec$^{-1}$, $0.0039 \times 10^9$ sec$^{-1}$ and $0.0039 \times 10^9$ sec$^{-1}$ respectively. All of these extra contributions to $\Gamma_v$ correspond to changes of less than 0.5$\%$. Calculation $e^+$Be $^+e$Mg $e^+$Ca $e^+$Sr ------------------------------- ----------- ------------ ---------- ---------- CI ($L_{\rm max}=12$) 0.002840 0.015658 0.015150 0.008221 CI ($L_{\rm max} \to \infty$) 0.003169 0.017040 0.018929 0.013102 CI ($L_{\rm int} \to \infty$) 0.003187 0.017099 0.019122 0.013321 Previous-CI 0.003083 0.01615 0.01650 0.01005 FCSVM 0.003161 0.016930 DMC 0.0012(4) 0.0168(14) SVM 0.001687 PO 0.00055 PO 0.00459 MBPT 0.0362 : \[summary\] Binding energies (in hartree) of positronic beryllium, magnesium, calcium and strontium. Only the latest calculations of a given type by a particular group are listed in this table. Summary and conclusions ======================= The summary of binding energies, produced by the current methods and other completely different approaches presented in Table \[summary\] shows that the only methods that consistently agree with each other are the CI and FCSVM calculations. Both these methods are variational in nature, both use realistic model potentials designed on very similar lines, and both have shown a tendency for the binding energies to slowly creep upwards as the calculation size is increased (refer to refs [@ryzhikh98e; @mitroy02b; @bromley02b] for examples of earlier and slightly smaller binding energies). The PO and MBPT approaches do not give reliable binding energies. The diffusion Monte Carlo method [@mella02a] gives an $e^+$Mg binding energy of 0.0168$\pm0.0014$ hartree which is very close to the present energy. This calculation was fully *ab-initio* and did not use the fixed core approximation. However, application of the same diffusion Monte Carlo method to $e^+$Be gave a binding energy which is only half the size of the present value. The present binding energies are all larger than those given previously [@bromley02a; @bromley02b] due to the usage of a radial basis which was almost twice the size of earlier calculations. In two cases, $^+$Ca and $e^+$Sr the increase in binding energy exceeds 10$\%$. The binding energies for $e^+$Be and $e^+$Mg are in agreement with those of FCSVM calculations to within their mutual uncertainties. Further enlargement of the basis could lead to the positron binding energies for Mg, Ca and Sr increasing by a few percent. Estimates of the annihilation rate have also been extracted from the CI wave functions. The present annihilation rates are certainly underestimates of the true annihilation rate. The annihilation rate converges very slowly with respect to the radial basis and similar sized calculations on PsH suggest that the present annihilation rates could easily be too small by at least 5$\%$ [@bromley06a; @mitroy06a; @mitroy06c]. The speed at which the partial wave expansion converges with respect to $L_{\rm max}$ is seen to decrease as the ionization energy of the parent atom decreases [@bromley02b; @mitroy02b]. In addition, the importance of 3-body clustering (i.e. convergence with respect to $L_{\rm int}$) was seen to increase as the ionization energy of the parent atom decreased [@mitroy02f]. The main factor limiting the size of the calculations now is the time taken to perform the diagonalizations. Although, the calculations were performed on a Linux/Myrinet-based cluster, the sheer number of iterations, (16000 in the worst case), used by the Davidson method, meant that it could take 30 days to perform a diagonalization using 24 CPUs. However, the main reason for adopting the Davidson method was the availability of a program that was easy to modify [@stathopolous94a]. Usage of the more general Lanczos method [@whitehead77a] might lead to a quicker diagonalization and thus permit even larger calculations. This work was supported by a research grant from the Australian Research Council. The calculations were performed on a Linux cluster hosted at the South Australian Partnership for Advanced Computing (SAPAC) with thanks to Grant Ward, Patrick Fitzhenry and John Hedditch for their assistance. The authors would like to thank Shane Caple for providing workstation maintenance and arranging access to additional computing resources. [43]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , , , ****, (). , in **, edited by (, , ), p. . , in **, edited by (, , ), p. . , ****, (). , ****, (). , , , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , , , ****, (). , in **, edited by , , , (, , ), vol. , p. . , ****, (). , ****, (). , p. (). , p. (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , p. 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ArXiv

--- abstract: 'The Hamiltonian actions for extreme and non-extreme black holes are compared and contrasted and a simple derivation of the lack of entropy of extreme black holes is given. In the non-extreme case the wave function of the black hole depends on horizon degrees of freedom which give rise to the entropy. Those additional degrees of freedom are absent in the extreme case.' address: | Centro de Estudios Científicos de Santiago, Casilla 16443, Santiago 9, Chile\ and\ Institute for Advanced Study, Olden Lane, Princeton, New Jersey 0854, USA. author: - 'Claudio Teitelboim [^1]' date: September 1994 title: 'Action and Entropy of Extreme and Non-Extreme Black Holes' --- It has been recently proposed [@hawking], [@horowitz] that extreme black holes have zero entropy [@wilczek]. The purpose of this note is to adhere to this claim by providing an economical derivation of it. The derivation also helps to set the result in perspective and to relate it to key issues in the quantum theory of gravitation, such as the Wheeler-De Witt equation. The argument is the application to the case of an extreme black hole of an approach to black hole entropy based on the dimensional continuation of the Gauss-Bonnet theorem developed in [@btz]. The approach in question had been previously applied to non extreme black holes only [@ct]. To put into evidence as clearly as possible the distinction between extreme and non extreme holes, we first perform the analysis for the non-extreme case and then see how it is modified in the extreme case. We will deal with gravitation theory in a spacetime of dimension $D$ with positive definite signature (Euclidean formulation). To present the argument in what we believe is its most transparent form for the purpose at hand, we will start with the Hamiltonian action and will only at the end discuss the connection with the Hilbert action. For non-extreme black holes the Euclidean spacetimes admitted in the action principle have the topology I$\!$R$^2 \times S^{D-2}$. It is useful to introduce a polar system of coordinates in the I$\!$R$^2$ factor of I$\!$R$^2 \times S^{D-2}$. The reason is that the black hole will have a Killing vector field –the Killing time– whose orbits are circles centered at the horizon. We will take the polar angle in I$\!$R$^2$ as the time variable in a Hamiltonian analysis. An initial surface of time $t_1$ and a final surface of time $t_2$ will meet at the origin. There is nothing wrong with the two surfaces intersecting. The Hamiltonian can handle that. The canonical action $$I_{can} = \int(\pi^{ij}\dot{g}_{ij} - N{\cal H} - N^i {\cal H}_i), \label{1}$$ [*without any surface terms added*]{} can be taken as the action for the wedge between $t_1$ and $t_2$ provided the following quantities are held fixed:\ (i) the intrinsic geometries $^{(D-1)} {\cal G} _1$, $^{(D-1)} {\cal G} _2$ of the slices $t=t_1$ , and $t=t_2$,\ (ii) the intrinsic geometry $^{(D-2)} {\cal G}$ of the $S^{D-2}$ at the origin\ (iii) the mass at infinity, with an appropiate asymptotic fall-off for the field. The term “mass” here refers to the conserved quantity associated with the time Killing vector at infinity. It is thus more general than the $P^0$ of the Poincaré group, which only exists when the spacetime is a symptotically flat. For example when there is a negative cosmological constant this mass is the value of a generator of the anti-de Sitter group. Note that we have listed the intrinsic geometry of the $S^{D-2}$ as a variable independent from the three-geometries of the Slices $t=t_1$ and $t=t_2$. This is because in the variation of the action (\[1\]) there is a separate term in the form of an integral over $S^{D-2}$, which contains the variation of $^{(D-2)} {\cal G}$. It should be observed that there will be no solution of the equations of motion satisfying the given boundary conditions if, for example, one fixes the mass at $t_2$ to be different from the mass at $t_1$. However in the quantum theory one [*can*]{} take $M_1 \neq M_2$, the path integral will then yield a factor $\delta(M_2 -M_1)$ in the amplitude. Similarly there will be no solution of the equations of motion unless the geometry of the $S^{D-2}$ at the origin as approached from the slice $t =t_1$, coincides with the one corresponding to $t=t_2$, and unless that common value also coincides with the one taken for the geometry of the $S^{D-2}$ at the origin. However these precautions need not be taken in the path integral, which will automatically enforce them by yielding appropiate $\delta$-functionals. This situation is the same as that arising with the action of a free particle in the momentum representation, where there is no clasical solution unless the initial and the final momenta are equal, but yet, one can (and must) compute the amplitude to go from any initial momentum to any final momentum. To the action (\[1\]) one may add any functional of the quantities held fixed and obtain another action appropiate for the same boundary conditions. In particular one may replace (\[1\]) by $$I= I_{can} + B[^{(D-2)} {\cal G}], \label{2}$$ where $B[^{(D-2)} {\cal G}]$ is any functional of the $(D-2)$-geometry at the origin. If we only look at the wedge $t_1 \leq t \leq t_2$ there is no privileged choice for $B$. However if we demand that the action we adopt should also be appropiate for the complete spacetime, then B is uniquely fixed. This is because when one deals with the complete spacetime the slices $t=t_1$ and $t=t_2$ are identified and neither $^{(D-1)} {\cal G}_1$ nor $^{(D-1)} {\cal G}_2$ nor $^{(D-2)} {\cal G}$ are held fixed. Now, unlike its Minkowskian signature continuation, the Euclidean black hole obeys Einstein’s equations [*everywhere*]{}. Thus it should be an extremum of the action with only the asymptotic data (mass) held fixed. The demand that the action should be such as to have the black hole as an extremum with respect to variations of $^{(D-2)} {\cal G}$ fixes $$B = 2\pi A(r_+)\;\;\;\; \mbox{(non-extreme case)}. \label{3}$$ where $A(r_+)$ is the area of the $S^{D-2}$ at the origin. Note that if one includes $B$ for the full spacetime one must include it for the wedge as well. This is because (i) the full spacetime is a particular case of the wedge, and (ii) the boundary term (\[3\]) depends only on the $(D-2)$ geometry at the origin and not on $t_1$ or $t_2$, The way in which (\[3\]) arises is the following. First one writes the metric near the origin in “Schwarzchild coordinates” as $$ds^2 = N^2(r, x^p) dt^2 +N^{-2}(r, x^p) dr^2 +\gamma_{m n}(r, x^p) dx^m dx^n, \label{4}$$ with $$(t_2 -t_1)N^2 = 2\Theta(x^p)(r-r_+) + O(r-r_+)^2\;\;\; \mbox{(non-extreme case)}. \label{5}$$ Here $r$ and $t$ are coordinates in I$\!$R$^2$ and $x^p$ are coordinates in $S^{D-2}$. The parameter $\Theta$ is the total proper angle (proper length divided by proper radius) of an arc of very small radius and coordinate angular opening $t_2 -t_1$ in the I$\!$R$^2$ at $x^p$. For this reason it is called the opening angle. When the sides of the wedge are identified $2\pi -\Theta$ becomes the deficit angle of a conical singularity in I$\!$R$^2$. Next, one evaluates the variation of the canonical action (\[1\]) to obtain $$\begin{aligned} \delta I_{can} & = & -\int_{S^{(D-2)}(r_+)} \Theta (x^p) \delta \gamma^{1/2}(x^p) d^{D-2}x + \beta \delta M + \nonumber \\ & & \int \pi^{ij} \delta g_{ij} \left|^2 _1 \right. + \mbox{(terms vanishing on shell)}. \label{6}\end{aligned}$$ Here $\beta$ is the Killing time separation at infinity. Last, one observes that when the slices $t=t_1$ and $t=t_2$ are identified, the term $\int \pi^{ij} \delta g_{ij} \left|^2 _1 \right.$ cancels out. Thus if $M$ and $J$ are kept fixed but $\gamma^{1/2}(x^p)$ is allowed to vary one must add (\[3\]) to (\[1\]) in order to obtain from the action principle that at the extremum. $$\Theta (x^p) = 2\pi\;\;\;\;\;\; \mbox{(complete spacetime, non-extreme case)}. \label{7}$$ Equation (\[7\]) must hold because otherwise there would be a conical singularity at $r_+$ and Einstein’s equations would be violated in the form of a $\delta$ -function source at the origin. Let us now turn to the extreme case. By definition of an extreme black hole the square lapse $N^2$ has a double root at the origin. Thus one must replace (\[5\]) by $$(t_2 -t_1)N^2 = O(r-r_+)^2\;\;\; \mbox{(extreme case)}. \label{8}$$ This means that one must have $$\Theta (x^p) =0 \;\;\;\; \mbox{(extreme case)}, \label{9}$$ instead of (\[7\]). It then follows that $$B =0 \;\;\;\; \mbox{(extreme case)}, \label{10}$$ so that the canonical action (\[1\]) is appropiate for extreme black holes. Note that equation (\[8\]) holds not only for the complete spacetime but also for a wedge of the extreme black hole geometry. This implies that (\[9\]) must hold also off-shell (for all configurations allowed in the action principle). This is so because for the wedge there is no way to obtain $\Theta =0$ by extremizing the action since $^{(D-2)} {\cal G}$ is held fixed. The difference between non-extreme and extreme cases has a topological origin. For all $\Theta$’s in the interval $$0< \Theta \leq 2\pi, \label{11}$$ the topology of the $t, r$ piece of the complete spacetime is that of a disk with the boundary at infinity. When $\Theta < 2\pi$ the disk has a conical singularity in the curvature at the origin with deficit angle $2\pi - \Theta$. When $\Theta = 2\pi$ the singularity is absent. However when $\Theta =0$ the topology is different. Indeed, what would appear naively to be a source at the origin in the form of a “fully closed cone” –as was misunderstood in [@btz]– is really the signal of a spacetime with different topology. As the cone closes, its apex recedes to give rise to the infinite throat of an extreme black hole. Thus the origin is effectively removed from the manifold whose $t, r$ piece is no longer a disk, but rather, an annulus whose inner boundary is at infinite distance. Now, one wants to include in the action principle fields of a given topology so that one can continuously vary from one to another. Therefore for the complete spacetime of the non-extreme case all fields obeying (\[11\]) are allowed so that (\[7\]) only holds on-shell. On the other hand, for the extreme case we must have (\[9\]) to also hold off-shell. This is so since if the origin is removed, and there is no place to put a conical singularity. We reach therefore an important conclusion: if we demand that the action should have an extremum on the black hole solution, then we must use a different action for extreme and non-extreme black holes. This means that these two kinds of black holes are to be regarded as drastically different physical objects, much in the same way as particle of however small but finite mass is drastically different from one of zero mass [@gibbons]. The discontinuous jump in the action is just the way that the geometrical theory at hand has to remind us that extreme and non-extreme black holes fall into different topological classes. The action may be rewritten as $$I = 2\pi \chi A(r_+) + I_{can}, \label{12}$$ and equations (\[5\]) and (\[8\]) may be summarized as $$(t_2 -t_1)N^2 = 2\chi \Theta(x^p)(r-r_+) + O(r-r_+)^2, \label{12.1}$$ where $\chi$ is the Euler characteristic of the $t, r$ factor of the complete black hole spacetime. For the non-extreme case one has $\chi =1$ (disk), and for the extreme case $\chi =0$ (annulus). Expression (\[12\]) had been anticipated in [@btz], where it emerged naturally from a study of the dimensional continuation of the Gauss-Bonnet theorem, but it was missed there that $\chi =0$ corresponds to extreme black holes. If one evaluates the action on the black hole solution one finds $$I_{can}(Black Hole) =0, \label{13}$$ because the black hole is stationary ($\dot{g}_{ij} =0$) and because the constraint equations ${\cal H} = {\cal H}_i =0$ hold. Thus one has $$I(Black Hole) =2\pi \chi A(r_+). \label{14}$$ Now, the action (\[12\]) is appropiate for keeping $M$ fixed. In statistical thermodynamics this corresponds to the microcanonical ensemble. Thus, for the entropy $S$ in the classical approximation one finds $$S= (8\pi G \hbar)^{-1}2\pi \chi A(r_+), \label{15}$$ where we have restored the universal constants. Thus one sees that extreme black holes ($\chi =0$) have zero entropy. A word is now in place about the relation of (\[12\]) with the Hilbert action $$I_H = \frac{1}{2} \int_M \sqrt{g} R d^Dx - \int_{\partial M} \sqrt{g} K d^{D-1}x, \label{16}$$ As was shown in [@btz], for the complete spacetime (\[12\]) and (\[16\]) just differ by a boundary term at infinity, which automatically regulates the divergent functional (\[16\]). This assertion is not valid for the wedge. In that case, as was also noted in [@btz], (\[12\]) and (\[16\]) differ not only by a boundary term at infinity but also by a boundary term at the origin. For the complete spacetime one has $$I = I_H -B_{\infty} , \label{16.1}$$ whereas for the wedge $$I = I_H + \pi(2\chi -1) A(r_+) -B_{\infty} - \pi A_{\infty}. \label{16.2}$$ For the reasons given above we adopt (\[12\]) and not (\[16\]) as the action for the wedge. The discontinuous change in the action between extreme and non-extreme black holes has dramatic consequences for the wave functional of the gravitational field in the presence of a black hole –which one may call for short the wave function of the black hole. Indeed, in the extreme case, the wave function has the usual arguments, namely, it may be taken to depend on the geometry of the spatial section and on the asymptotic time separation $\beta$, $$\Psi = \Psi[^{(D-1)} {\cal G}, \beta]. \label{17}$$ The dependence of $\Psi$ on the three geometry is governed by the Wheeler – De Witt equation $${\cal H} \Psi = 0, \label{18}$$ whereas the dependence on the asymptotic time $\beta$ is governed by the Schrödinger equation $$\frac{\partial \Psi}{\partial \beta} + M\Psi= 0, \label{19}$$ where $M$ is the mass as defined by Arnowitt, Deser and Misner (see for example [@ct2]). On the other hand, for the non-extreme case the wave function has an extra argument which may be taken to be the opening angle $\Theta$, $$\Psi = \Psi[^{(D-1)} {\cal G}, \beta, \Theta]. \label{21}$$ Since according to (\[6\]) $\Theta$ is canonically conjugate to $\gamma^{1/2}$, one has in addition to equations (\[18\]) and (\[19\]) the extra Schrödinger equation at the horizon [@carlip] $$\frac{\delta \Psi}{\delta \Theta(x)} - \gamma^{1/2}(x) \Psi =0. \label{22}$$ The additional horizon degree of freedom canonical pair $(\gamma^{1/2}, \Theta)$ may be regarded as responsible for the black-hole entropy in the non-extreme case. Indeed there is no entropy in the extreme case precisely because then the origin is absent and there is no place for $(\gamma^{1/2}, \Theta)$ to sit at. This agrees with a point of view previously expressed [@carlip], namely that, –in a way yet to be spelled– the black-hole entropy could be conjectured as arising from “counting conformal factors on the $S^{D-2}$ at $r_+$” or, in terms of the canonically conjugate statement “from counting two-dimensional geometries within a small disk at the horizon”. That disk is removed in the extreme case –and with it the entropy. ACKNOWLEDGMENTS The author is very grateful to J. Zanelli for many inlightening discussions and for much help in preparing this manuscript. Thanks are also expressed to Dr. A. Flisfisch for his kind interest in the author’s work. This work was supported in part by Grant 194.0203/94 from FONDECYT (Chile), by a European Communities contract, and by institutional support to the Centro de Estudios Científicos de Santiago provided by SAREC (Sweden), and a group of chilean private companies (COPEC, CMPC, ENERSIS, CGEI). This research was also sponsored by IBM and XEROX-Chile. Talk at the meeting on “Quantum Concepts in Space and Time”, Durham, July 1994. S. W. Hawking, G. T. Horowitz and S. F. Ross, “Entropy, Area, and Black Hole Pairs” (to be published). For previous evidence on the lack of thermodynamic behavior of extreme black holes, see for example, J. Preskill, P. Schwarz, A. Shapere, S. Trivedi and F. Wilczek, [*Mod. Phys. Lett. A*]{} [**6**]{} (1991), 2353; F. Wilczek “Lectures on Black Hole Quantum Mechanics”, in [Quantum Mechanics of Fundamental Systems IV ]{}, (Proceedings of a Meeting held in Santiago, Chile, December 1991), C. Teitelboim and J. Zanelli, eds. (Plenum, N.Y., in press). G. W. Gibbons and R. E. Kallosh, “Topology, the Gauss Bonnet Theorem and the Entropy of Extreme Black Holes” (to be published). M. Ba!ados, C. Teitelboim and J. Zanelli, [*Phys. Rev. Lett.*]{} [**72**]{} 957, (1994). See also C. Teitelboim, “Topological Roots of Black Hole Entropy”, in Proceedings of the Lanczos Centenary Conference, J. D. Brown, M. T. Chu, D. C. Ellison and R. J. Plemmons, eds. (SIAM, Philadelphia, in press). This paper was presented at the meeting on “Quantum Concepts in Space and Time”, Durham, July 1994. It was motivated by Hawking’s talk [@hawking] where the zero entropy of extreme black holes was derived in a different manner. G. Gibbons, private communication, Durham, 1994. This analogy is very much to the point. Indeed the number of states of a given spin –and with it the entropy– is also discontinuous ($2s+1$ states for $m\neq 0$ however small, 2 states for $m=0$). However this jump does not make the observable properties to be discontinuous as was discussed by L. Bass and E. Schrödinger, [*Proc. Roy. Soc. (London)*]{} [**232**]{} (1955), 1. C. Teitelboim, [*Phys. Rev.*]{} [**D28**]{} (1983), 310. S. Carlip and C. Teitelboim, “The Off-Shell Black Hole” (to be published). [^1]: Electronic address: [email protected]

ArXiv

--- abstract: 'We first obtained the spectrum of the diffuse Galactic light (DGL) at general interstellar space in 1.8-5.3 $\mu$m wavelength region with the low-resolution prism spectroscopy mode of the AKARI Infra-Red Camera (IRC) NIR channel. The 3.3 $\mu$m PAH band is detected in the DGL spectrum at Galactic latitude $\mid b \mid < 15^{\circ }$, and its correlations with the Galactic dust and gas are confirmed. The correlation between the 3.3 $\mu$m PAH band and the thermal emission from the Galactic dust is expressed not by a simple linear correlation but by a relation with extinction. Using this correlation, the spectral shape of DGL at optically thin region ($5^{\circ } < \mid b \mid < 15^{\circ }$) was derived as a template spectrum. Assuming that the spectral shape of this template spectrum is uniform at any position, DGL spectrum can be estimated by scaling this template spectrum using the correlation between the 3.3 $\mu$m PAH band and the thermal emission from the Galactic dust.' author: - 'Kohji <span style="font-variant:small-caps;">Tsumura</span>, Toshio <span style="font-variant:small-caps;">Matsumoto</span>, Shuji <span style="font-variant:small-caps;">Matsuura</span>, Itsuki <span style="font-variant:small-caps;">Sakon</span>, Masahiro <span style="font-variant:small-caps;">Tanaka</span>, and Takehiko <span style="font-variant:small-caps;">Wada</span>' title: 'Low-Resolution Spectrum of the Diffuse Galactic Light and 3.3 $\mu$m PAH emission with AKARI InfraRed Camera' --- Introduction ============ The Diffuse Galactic Light (DGL) comprises scattered starlight by dust particles in the interstellar space at $<$3 $\mu$m, and emissions from the dust particles with some band features at longer wavelengths[^1]. Thus observational studies of DGL is important to investigate the dust property in our Galaxy, and it is also important for deriving the extragalactic background light (EBL) since DGL is one of foregrounds for the EBL measurement. However, isolation of DGL from other diffuse emissions, especially the strongest zodiacal light (ZL) foreground, is very difficult due to its diffuse, extended nature. A commonly-used method to estimate DGL is the correlation with the dust column density estimated by the thermal emission of the dust from far-infrared (100 $\mu$m) observations, or the column density of HI and/or CO from radio observations. In the optical wavelengths, DGL brightness [@Witt2008; @Matsuoka2011; @Ienaka2013] and spectrum [@Brandt2012] are obtained by the correlation with the 100 $\mu$m dust thermal emission. However, observations of DGL at near-infrared (NIR) are limited and controversial. The presence of the infrared band features in DGL has been first confirmed for the 3.3 $\mu$m band by the AROME balloon experiment [@Giard1988]. Such ubiquitous Unidentified Infrared (UIR) bands are a series of distinct emission bands seen at 3.3, 5.3, 6.2, 7.7, 8.6, 11.2, and 12.7 $\mu$m, and they are supposed to be carried by the polycyclic aromatic hydrocarbons (PAH) [@Leger1984; @Allamandola1985]. They are excited by absorbing a single ultraviolet (UV) photon and release the energy with a number of infrared photons in cascade via several lattice vibration modes of aromatic C-H and C-C bonds [@Allamandola1989]. The 3.3 $\mu$m PAH band emission has been assigned to the stretching mode transition ($v=1-0$) of the C-H bond on aromatic rings. There is a quantitative model for DGL from interstellar dust including PAH [@Li2001]. The correlation between the 3.3 $\mu$m PAH band detected by the Near-Infrared Spectrometer (NIRS) on the Infrared Telescope in Space (IRTS) and the 100 $\mu$m thermal emission of the large dust grains by the Infrared Astronomical Satellite (IRAS) was confirmed at the Galactic plane region ($42^{\circ } < l < 55^{\circ }$, $\mid b \mid < 5^{\circ }$), implying that the PAH molecules are well mixed with the large dust grains at the Galactic plane [@Tanaka1996]. (160mm,100mm)[fig1.eps]{} In this paper, we describe DGL spectrum obtained with the low-resolution prism spectroscopy mode on the AKARI Infra-Red Camera (IRC) NIR channel in 1.8-5.3 $\mu$m wavelength region. Our idea to derive DGL spectrum at NIR in this paper is to use the 3.3 $\mu$m PAH band feature as a tracer of DGL combined with the correlation with the 100 $\mu$m dust thermal emission. The 3.3 $\mu$m PAH band is detected in this wavelength region at $\mid b \mid < 15^{\circ }$, and the correlation with the thermal emission of the large dust grains is also confirmed. Using this correlation, we developed a method to estimate the DGL spectrum at NIR at any location. This paper is organized as follows. In Section \[sec\_reduction\], we describe the data reduction. In Section \[sec\_PAH\], we describe the correlation of the 3.3 $\mu$m PAH band feature in DGL with Galactic latitude and the distribution of Galactic dust and gas. The method to estimate DGL spectrum using this correlation is shown in Section \[sec\_DGL\], and the summary of this paper is given in Section \[sec\_summary\]. There are two companion papers describing the spectrum of the infrared diffuse sky; ZL is described in @Tsumura2013a (hereafter Paper I) and EBL is described in @Tsumura2013c (hereafter Paper III) in which the foregrounds described in Paper I and this paper (Paper II) are subtracted. Data Selection and Reduction {#sec_reduction} ============================ AKARI is the first Japanese infrared astronomical satellite launched at February 2006, equipped with a cryogenically cooled telescope of 68.5 cm aperture diameter [@Murakami07]. IRC is one of two astronomical instruments of AKARI, and it covers 1.8-5.3 $\mu$m wavelength region with a 512$\times $412 InSb detector array in the NIR channel[^2] [@Onaka07]. It provides low-resolution ($\lambda /\Delta \lambda \sim 20$) slit spectroscopy for the diffuse radiation by a prism[^3] [@Ohyama2007]. One biggest advantage to the previous IRTS measurement [@Tanaka1996] is the higher angler resolution (1.46 arcseconds) of AKARI IRC which allows us to detect and remove fainter point sources, while the IRTS measurement was highly contaminated by bright stars at the Galactic plane because of its resolution (8 arcminutes). See Paper I for the details of the data selection and reduction. Here we simply note that 278 pointed data of diffuse spectrum were selected in this study distributing wide range of ecliptic and Galactic coordinates. Dark current was subtracted by the method specialized for the diffuse sky analysis described in @TsumuraWada2011. Stars brighter than $m_K(\textrm{Vega}) = 19$ were detected on the slit and masked for deriving the diffuse spectrum. It was confirmed that the brightness due to unresolved Galactic stars under this detection limit is negligible ($<$0.5 % of the sky brightness at 2.2 $\mu$m) by a Milky Way star counts model, TRILEGAL [@Girardi2005]. Cumulative brightness contributed by unresolved galaxies can be estimated by the deep galaxy counts, being $<$4 [nWm$^{-2}$sr$^{-1}$]{} at K band in the case of limiting magnitude of $m_K = 19$ [@Keenan10]. 3.3 $\mu$m PAH band {#sec_PAH} =================== Association to our Galaxy ------------------------- Figure \[spectrum\] shows examples of the spectra of the infrared diffuse sky used in this study. Although the obtained spectra are dominated by ZL except for the Galactic plane, the 3.3 $\mu$m PAH band is detectable at $\mid b \mid < 15^{\circ }$ in our dataset. The 3.3 $\mu$m PAH band is easy to be found at the bottom of the ZL spectrum because ZL spectrum has a local minimum at around 3.5 $\mu$m as shown in Figure \[spectrum\] (a). At the Galactic plane, DGL dominates the sky spectrum as shown in Figure \[spectrum\] (b). The spectral shape of the 3.3 $\mu$m band is asymmetry because other 3.4 and 3.5 $\mu$m PAH sub-band features are combined and detected together, which were separately detected by the high-resolution spectroscopy ($\lambda /\Delta \lambda \sim 120$) with the IRC grism mode at the Galactic plane [@Onaka2011]. The 3.3 $\mu$m PAH band was extracted from the sky spectrum by almost the same method used in @Tanaka1996. First, the continuum intensity at 3.3 $\mu$m ($\lambda I_{3.3\mu m}^{cont}$) was interpolated between 3.2 and 3.8 $\mu$m. Although the continuum was interpolated between 3.2 and 3.6 $\mu$m in @Tanaka1996, we used the intensity at 3.8 $\mu$m for the interpolation to avoid the contamination from the PAH sub-feature at 3.5 $\mu$m. Then, the total energy of the 3.3 $\mu$m PAH band feature ($E_{3.3}$) was calculated as the excess from the continuum; $$E_{3.3} = \frac{\Delta \lambda }{\lambda } [\lambda I_{3.3\mu m} - \lambda I_{3.3\mu m}^{cont})]+0.58$$ where $\Delta \lambda$ = 0.13 $\mu$m was employed for direct comparison to @Tanaka1996, and two data points around 3.3 $\mu$m were summed to compute $\lambda I_{3.3\mu m}$ for matching the wavelength resolution to IRTS and reducing the error. A small offset of 0.58 [nWm$^{-2}$sr$^{-1}$]{} is applied to correct the difference between the ZL continuum and the linear interpolation between 3.2 and 3.8 $\mu$m. PAH band at $\mid b \mid < 15^{\circ }$ was detected by this method. First, we show a general correlation of this PAH band with our Galaxy in Figure \[cosec\]. It means that the PAH dust is associated with DGL from our Galaxy, and this correlation can be expressed by $$E_{3.3} = (0.17^{+0.2}_{-0.1}) \cdot (\textrm{cosec} \mid b \mid ) ^{(1.03\mp 0.04)}$$ Since such correlation between the 3.3 $\mu$m PAH band and ecliptic latitude was not detected, we can conclude that the observed PAH features is not associated with ZL from the Solar system. (80mm,50mm)[fig2.eps]{} Association to the distribution of dust and gas ----------------------------------------------- Next, we tested the correlations of the PAH band with the distribution of Galactic dust and gas, which are not correlated simply with Galactic latitude. The 100 $\mu$m intensity map ($\lambda I_{100 \mu m}$) that is a reprocessed composite of the COBE/DIRBE and IRAS/ISSA maps (SFD map, see @Schlegel1998) is used as the dust distribution, and the column density map of HI obtained from the Leiden/Argentine/Bonn (LAB) Galactic HI survey [@Kalberla2005] is used as the gas distribution map. The good correlation between the dust and gas was reported [@Stark92; @Arendt98], and HI column density can be converted to the 100 $\mu$m intensity by $0.018 \pm 0.003$ ([$\mu $Wm$^{-2}$sr$^{-1}$]{})/($10^{20}$ atoms/cm$^2$) at HI $< 10^{22}$ atoms/cm$^2$ as adopted in @Matsuura2011. In previous works, the correlation between the 3.3 $\mu$m PAH band and the 100 $\mu$m thermal intensity was reported by @Giard1989 and @Tanaka1996 but they were limited at the Galactic plane ($\mid b \mid < 6^{\circ }$). AKARI data in this study provides the correlations up to $\mid b \mid = 15^{\circ }$ with higher angler resolution and higher sensitivity of point sources to remove foreground stars than the previous works. Figure \[E33\] shows the correlations of the 3.3 $\mu$m PAH band of our data set with the 100 $\mu$m thermal intensity from the SFD map [@Schlegel1998] and the column density of HI from the LAB survey [@Kalberla2005]. These correlations can be expressed by $$\frac{E_{3.3}}{\textrm{{nWm$^{-2}$sr$^{-1}$}}} = (2.9 \pm 1.0) \cdot \left( \frac{\lambda I_{100 \mu m}}{\textrm{{$\mu $Wm$^{-2}$sr$^{-1}$}}}\right)^{(0.91\mp 0.04)} \label{eq_correlation}$$ $$\frac{E_{3.3}}{\textrm{{nWm$^{-2}$sr$^{-1}$}}} = (0.7 \pm 0.3) \cdot \left( \frac{\textrm{HI}}{10^{21} \ \textrm{atoms/cm}^2}\right)^{(1.08\mp 0.05)} \label{eq_nH}$$ These correlations are better than the correlation with the Galactic latitude shown in Figure \[cosec\], and it means that PAH molecules, interstellar dust, and interstellar gas are well mixed in the interstellar space. Deviation of the correlation with HI in LAB survey data shown in Figure \[E33\] (b) is larger than the correlation with the 100 $\mu$m intensity in SFD map shown in Figure \[E33\] (a), because angular resolution of SFD map is better than LAB map allowing us better point-to-point correlation analysis. Thus advanced analysis described in the next sub-section is investigated only for the correlation with the 100 $\mu$m intensity. (160mm,100mm)[fig3.eps]{} The effect of extinction on the PAH band ---------------------------------------- @Tanaka1996 assumed a simple linear relation between $E_{3.3}$ and $\lambda I_{100 \mu m}$ obtaining a relation of $E_{3.3}/\lambda I_{100 \mu m}= (2.9\pm 0.9)\times 10^{-3}$, and systematic difference from the linear relation at high $\lambda I_{100 \mu m}$ region was concluded to be due to extinction. @Giard1989 also investigated the correlation between $E_{3.3}$ and $\lambda I_{100 \mu m}$ and found the extinction at the Galactic plane. The power of $0.91 \mp 0.04$ (smaller than unity) in our fitting in equation (\[eq\_correlation\]) is consistent with the extinction as mentioned in @Giard1989 and @Tanaka1996. Therefore, we conducted a fitting including the effect of extinction. When the source’s term in the transfer equation is proportional to extinction term all along the line of sight, extinction can be written as $(1-e^{-\tau_{\lambda}})/\tau_{\lambda}$, where $\tau_{\lambda}$ is the optical depth at a wavelength ${\lambda}$ [@Giard1989]. Then the correlation with extinction between the PAH emission ($E_{3.3}$) and the the 100 $\mu$m intensity ($\lambda I_{100 \mu m}$) can be written as $$\frac{E_{3.3}}{\textrm{{nWm$^{-2}$sr$^{-1}$}}} = \alpha \frac{1-e^{-\tau_{3.3} }}{\tau_{3.3} } \left( \frac{\lambda I_{100 \mu m}}{\textrm{{$\mu $Wm$^{-2}$sr$^{-1}$}}} \right ) \label{eq_SFD}$$ where $\alpha$ is a fitting parameter. Assuming that the 100 $\mu$m intensity is proportional to the number of PAH molecules associating with dust particles, and the number of dust particles is proportional to $\tau + \tau^2$ [@Rybicki1986], thus the optical depth $\tau_{3.3}$ can be obtained as a solution of this equation. $$\frac{\lambda I_{100 \mu m}}{\textrm{{$\mu $Wm$^{-2}$sr$^{-1}$}}} = \frac{1}{\beta} (\tau_{3.3} +\tau_{3.3} ^2) \label{eq_tau}$$ where $\beta$ is another fitting parameter. By fitting to our data, we obtained $\alpha =3.5 \pm 1.0$ and $\beta =0.10 \pm 0.04$. The curve of equation (\[eq\_SFD\]) with best fit parameters is shown as a solid curve in Figure \[E33\] (a). In the optically thin case ($\tau_{3.3} \ll 1$), the extinction term $(1-e^{-\tau_{3.3}})/\tau_{3.3}$ becomes unity. Thus we obtain the linear correlation. $$\frac{E_{3.3}}{\textrm{{nWm$^{-2}$sr$^{-1}$}}} = \alpha \left( \frac{\lambda I_{100 \mu m}}{\textrm{{$\mu $Wm$^{-2}$sr$^{-1}$}}} \right) \; \; \; (\tau_{3.3} \ll 1)$$ In the optically thick case ($\tau_{3.3} \gg 1$), the extinction term can be written as $1/\tau_{3.3}$ because the term $e^{-\tau_{3.3}}$ becomes zero. In addition, the optical depth $\tau_{3.3}$ can be written $\tau_{3.3} = \sqrt{\beta \cdot \lambda I_{100 \mu m}}$ from equation (\[eq\_tau\]) owing to $\tau_{3.3} \gg 1$. Combining these equations, we obtain $$\frac{E_{3.3}}{\textrm{{nWm$^{-2}$sr$^{-1}$}}} = \frac{\alpha}{\sqrt{\beta}} \sqrt{ \frac{\lambda I_{100 \mu m}}{\textrm{{$\mu $Wm$^{-2}$sr$^{-1}$}}} } \; \; \; (\tau_{3.3} \gg 1)$$ These two extreme cases are also shown as broken lines in Figure \[E33\] (a). The gradient of $\alpha =3.5 \pm 1.0$ is higher than the value from the IRAS result of $\alpha = 2.5 \pm 0.4$ [@Giard1994], but this result was determined based on the data averaged in the Galactic latitude range of $\mid b \mid < 1^{\circ }$ where the bright discrete sources are included. IRTS, with higher sensitivity than IRAS but still limited at $\mid b \mid < 5^{\circ }$, obtained the value of $\alpha = 2.9 \pm 0.9$ in @Tanaka1996, which is closer to our result. The 3.3 $\mu$m PAH band intensity deviates from the linearity at $\lambda I_{100 \mu m} > 10$ [$\mu $Wm$^{-2}$sr$^{-1}$]{} or $\mid b \mid < 1^{\circ }$. This is equivalent to $\tau _{3.3}=0.6$ at $\lambda I_{100 \mu m} > 10$ [$\mu $Wm$^{-2}$sr$^{-1}$]{} or $\mid b \mid < 1^{\circ }$ in our fitting, which is higher than the estimated value of $\tau _{3.3}=0.18$ at $\mid b \mid < 0.75^{\circ }$ based on the extinction low summarized in @Mathis1990 and optical depth at 240 $\mu$m from @Sodroski1994. (160mm,100mm)[fig4.eps]{} DGL spectrum {#sec_DGL} ============ Correlation method ------------------ In this section, we develop a method to derive the DGL spectrum at 1.8-5.3 $\mu$m by the correlation with the 100 $\mu$m intensity. The diffuse sky spectrum includes ZL, DGL, and EBL, i.e. $$SKY (\lambda ) = ZL (\lambda ) + DGL (\lambda ) +EBL(\lambda )$$ ZL is modeled in Paper I which can be subtracted based on the DIRBE ZL model [@Kelsall98], and EBL is the isotropic component. Therefore, only DGL has the correlation with the 100 $\mu$m intensity, so DGL can be derived by the correlation by assuming a linear correlation with the 100 $\mu$m intensity. $$SKY(\lambda ) - ZL(\lambda ) = a(\lambda ) \cdot \lambda I_{100 \mu m} + b(\lambda )$$ where $ a(\lambda ) \cdot \lambda I_{100 \mu m} $ is equivalent to $DGL(\lambda )$ and $b(\lambda )$ is equivalent to $EBL(\lambda )$. Figure \[correlation\] shows the correlation between $SKY(\lambda ) - ZL(\lambda )$ in our dataset and the 100 $\mu$m intensity from the SFD map [@Schlegel1998]. The data for this correlation analysis were selected by a criteria of $ \lambda I_{100 \mu m} < 3$ [$\mu $Wm$^{-2}$sr$^{-1}$]{}, equivalent to the Galactic latitude $\mid b \mid >5^{\circ }$, to trace low dust density regions owing to the assumption of the linear correlation. The gradients as a function of wavelength in Figure \[correlation\], $a(\lambda )$, correspond to the spectral shape of DGL. Normalized spectrum of the obtained DGL spectrum is shown in Figure \[DGL\] (a), and the 3.3 $\mu$m PAH band feature in DGL was detected. The error of the obtained DGL spectrum by this correlation method is 5 % at $<$3.8 $\mu$m, 15 % between 3.8 $\mu$m and 4.2 $\mu$m, and 20 % at $>$4.2 $\mu$m. Since the spectral shape of DGL may vary depending on environments, it is a representative spectrum of DGL at low dust density regions, typically $5^{\circ }< \mid b \mid <15^{\circ }$, and the variance of environments is included in these errors. (160mm,100mm)[fig5.eps]{} In this correlation method, we assumed the linear correlation between DGL and the 100 $\mu$m intensity, but we have already showed the non-linear correlation between the 3.3 $\mu$m PAH band and the 100 $\mu$m intensity owing to extinction as shown in Figure \[E33\] (a). Thus we modify this DGL estimation method by combining it with the 3.3 $\mu$m PAH band as a tracer for scaling the DGL spectrum at any general sky. First, the template DGL spectrum $DGL_{\textrm{temp}}(\lambda )$ is defined as the derived DGL spectrum by this correlation method normalized to be $E_{3.3}=1$, $$DGL_{\textrm{temp}}(\lambda) = \frac{a(\lambda)}{E_{3.3}(a(\lambda))}$$ This template DGL spectrum is shown in Figure \[DGL\] (a). Assuming that the spectral shape of this template DGL spectrum does not change at any location, we can estimate the DGL spectrum at any place by scaling this template DGL spectrum by $E_{3.3}$ which can be obtained as a function of the 100 $\mu$m thermal intensity using the correlation shown in Figure \[E33\] (a), i.e., $$DGL(\lambda ) = E_{3.3}(I_{100 \mu m}) \cdot DGL_{\textrm{temp}}(\lambda )$$ Figure \[DGL\] (b) shows the the resultant DGL spectrum with other DGL estimations [@Mattila2006; @Flagey2006] scaling to $\lambda I_{100 \mu m}= 0.1$ [$\mu $Wm$^{-2}$sr$^{-1}$]{} or HI = $5 \times 10^{20}$ atoms/cm$^2$. Pointing ID Galactic longitude $l$ Galactic latitude $b$ $\lambda I_{100 \mu m}$ [$\mu $Wm$^{-2}$sr$^{-1}$]{} HI atoms/cm$^2$ ------------- ------------------------ ----------------------- ------------------------------------------------------ ----------------------- 410017.1 30.76 0.36 59.00 $1.89 \times 10^{22}$ 410018.1 31.01 0.12 111.09 $1.95 \times 10^{22}$ 410021.1 30.99 -0.04 115.25 $1.95 \times 10^{22}$ 410022.1 31.24 -0.27 47.65 $1.66 \times 10^{22}$ Our DGL spectrum is lower than the other previous estimations, but our method gives a better DGL estimation at the general interstellar space with low dust density. In the previous works, DGL was estimated by scaling using HI based on only a limited number of specific dense regions with HI $\sim 2 \times 10^{22}$ atoms/cm$^2$. However, the ratio of DGL/HI at dense regions ($>10^{22}$ atoms/cm$^2$) is higher than that in the general interstellar fields as shown in Figure \[E33\] (b), which leads high DGL estimations in the previous works. On the other hand, our estimation is based on a number of the wide-spread data points at the general interstellar fields with low dust density with higher spacial resolution to remove stars, and scaling is based on the 100 $\mu$m intensity which has tighter correlation to DGL as shown in Figure \[E33\] (a). Therefore our estimate gives more reliable estimation of the DGL spectrum especially at low dust density regions. Although our DGL estimation is lower than previous estimations, it may still overestimates the 3.3 $\mu$m intensity at high Galactic latitude regions. The 3.3 $\mu$m PAH band was detected only at the region of $\mid b \mid <15^{\circ }$ in our dataset, and we assumed that the obtained spectral shape of DGL does not change at any location in this method, but there is no guarantee that this assumption is valid at high Galactic latitude regions. For example, UV radiation field at high Galactic latitude regions is weaker than that at Galactic plane [@Seon2011], therefore the PAH molecules are less excited at high Galactic latitude than Galactic plane. In such a case, our method overestimates the 3.3 $\mu$m intensity in DGL at high Galactic latitude as implied in Paper III. DGL spectrum in the Galactic plane ---------------------------------- We compared the obtained DGL spectrum by the correlation method ($5^{\circ } < \mid b \mid < 15^{\circ }$) with the spectrum at the Galactic plane, because the diffuse sky spectrum at the Galactic plane is dominated by DGL. For example, DGL brightness at Galactic plane in this wavelength region is several thousands [nWm$^{-2}$sr$^{-1}$]{}, while ZL brightness is a few hundreds [nWm$^{-2}$sr$^{-1}$]{}, which is less than 5 % as shown in Figure \[spectrum\]. Therefore, the spectral shape of DGL can be evaluated by the diffuse sky spectra at the Galactic plane. We selected four spectra with the strongest 100 $\mu$m dust thermal emission and HI column density in our data set, located at the Galactic plane ($l$, $b$)$\sim$(31$^{\circ }$, 0$^{\circ }$) summarized in Table \[DGL\_table\], where the brightness of ZL contribution is $<$5 %. These spectra at the Galactic plane are similar with each other with $\sim$15 % dispersion, and the broken curve in Figure \[DGL\] (a) shows the average spectrum of these selected spectra normalized to be $E_{3.3}=1$. The 3.3 $\mu$m PAH band is the most distinctive among others, and the second outstanding feature at 5.25 $\mu$m is also the PAH feature [@Allamandola1989b; @Cohen1989; @Boersma2009]. Br-$\alpha $ at 4.05 $\mu$m and Pf-$\beta $ at 4.65 $\mu$m are also detected, which can be useful information to estimate ionizing temperature and extinction. One example of such study in M17 case with AKARI IRC high-resolution spectroscopy mode ($\lambda /\Delta \lambda \sim 120$) can be found in @Onaka2011. An excess continuum emission at the Galactic plane was confirmed at $>$3.5 $\mu$m as shown in Figure \[DGL\] (a). This excess continuum emission was first reported in visual reflection nebulae [@Sellgren1983], and then found in galaxies [@Lu2003; @Onaka2010] and DGL [@Flagey2006]. The emission process of this excess continuum is still unknown, but @Flagey2006 suggests the PAH fluorescence excited by UV photons. This excess continuum emission may be a reason why the previous works overestimated DGL from high dust density regions as shown in Figure \[DGL\] (b). Summary {#sec_summary} ======= The 3.3 $\mu$m PAH band is detected in the diffuse sky spectrum of the interstellar space at $\mid b \mid < 15^{\circ }$, and this band intensity is correlated with the 100 $\mu$m thermal intensity of interstellar dust and HI column density. We modeled the correlation between the 3.3 $\mu$m PAH band and the 100 $\mu$m thermal intensity with extinction. We also introduce a method to estimate the DGL spectrum at 1.8-5.3 $\mu$m. This is the first estimation of DGL spectrum in the general sky at NIR based on observation. In this method, the spectral shape of DGL is derived by the correlation with the 100 $\mu$m thermal intensity, and it is scaled by the correlation of the 3.3 $\mu$m PAH band brightness as a tracer. DGL spectrum estimated by our method is lower than the previous estimations, but our result is more reliable for the regions with low dust density regions because our result is based on a wide range of general interstellar field although previous result is based on some specific region with high dust density. In addition, we found the excess continuum emission at Galactic plane at 3-5 $\mu$m as reported by previous works. This research is based on observations with AKARI, a JAXA project with the participation of ESA. This research is also based on significant contributions of the IRC team. We thank Dr. Mori-Ito Tamami (The university of Tokyo) and Mr. Arimatsu Ko (ISAS/JAXA) for discussion about PAH. The authors acknowledge support from Japan Society for the Promotion of Science, KAKENHI (grant number 21111004 and 24111717). Allamandola, L. J., Tielens, A. G. G. M., & Barker J. R. 1985, , 290, L25 Allamandola, L. J., Tielens, A. G. G. M., & Barker J. R. 1989a, , 71, 733 Allamandola, L. J., Bregman, J. D., Sandford, S. A., Tielens, A. G. G. M., Witteborn, F. C., Wooden, D. H., & Rank, D. 1989b, , 345, L59 Arendt, R. G., et al. 1998, , 508, 74 Brandt, T. D. & Draine, B. T. 2012, , 744, 129 Boersma, C., Mattioda, A. L., Bauschlicher, C. W., Peeters, E., Tielens, A. G. G. M., & Allamandola, L. J. 2009, , 690, 1208 Cohen, M., Tielens, A. G. G. M., Bregman, J. D., Witteborn, F. C., Rank, D. M., Allamandola, L. J., Wooden, D., & de Muizon, M. 1989, , 341, 246 Flagey, N., Boulanger, F., Verstraete, L., Miville Desch$\hat{\textrm{e}}$nes, M. A., Noriega Crespo, A., & Reach, W. T. 2006, , 453, 969 Giard, M., Pajot, F., Lamarre, J. M., Serra, G., Caux, E., Gispert, R., L$\acute{\textrm{e}}$ger, A. & Rouan, D. 1988, , 201, L1 Giard, M., Pajot, F., Lamarre, J. M., Serra, G., & Caux, E. 1989, , 215, 92 Giard, M., Lamarre, J. M., Pajot, & F., Serra, 1994, , 286, 203 Girardi, L., Groenewegen, M. A. T., Hatziminaoglou, E., & da Costa, L. 2005, , 436, 895 Ienaka, N., Kawara, K., Matsuoka, Y., Sameshima, H., Oyabu, S., Tsujimoto, T., & Peterson, B. A. 2013, , 767, 80 Kalberla, P. M. W., Burton, W. B., Hartmann, Dap, Arnal, E. M., Bajaja, E., Morras, R., & Pöppel, W. G. L., 2005, , 440, 775 Keenen, R. C., Barger, A. J., Cowie, L. L., & Wang, W. -H. 2010, , 723, 40 Kelsall, J. L, et al. 1998, , 508, 44 L$\acute{\textrm{e}}$ger, A. & Puget, J. L. 1984, , 137, L5 Li, A, & Draine, B., T. 2001, , 554, 778 Lu, N., et al. 2003, , 588, 199 Mathis, J. K. 1990, , 28, 37 Matsuoka, Y., Ienaka, N., Kawara, K., & Oyabu, S. 2011, , 736, 119 Matsuura, S., et al. 2011, , 737, 2 Mattila, K. 2006, , 372, 1253 Murakami, M., et al. 2007, , 59, S369 Ohyama, Y., et al. 2007, , 59, S411 Onaka, T., et al. 2007, , 59, S401 Onaka, T., Matsumoto, H., Sakon, I., & Kaneda, H. 2010, , 514, 15 Onaka, T., Sakon, I., Ohsawa, R., Shimonishi, T., Okada, Y., Tanaka, M., & Kaneda, H. 2011, EAS publication Series, 46, 55 Rybicki, G. B., & Lightman, A. P. 1986, Radiative Processes in Astrophysics (New York: Wiley) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, , 500, 525 Sellgren, K., Wernre, M., & Dinerstein, H. L. 1983, , 271, L13 Seon, K. -I., et al. 2011, , 196, 15 Sodroski, T. J. et al. 1994, , 428, 638 Stark, A. A. et al. 1992, , 79, 77 Tanaka, M., Matsumoto, T., Murakami, H., Kawada, M., Noda, M., & Matsuura, S. 1996, , 48, L53 Tsumura, K., & Wada, T. 2011, , 63, 755 Tsumura, K., Matsumoto, T., Matsuura, J., Pyo, S., Sakon, I., & Wada, T. 2013a, , in press, arXiv:1306.6191 (Paper I) Tsumura, K., Matsumoto, T., Matsuura, S., Sakon, I., & Wada, T. 2013c, , in press (Paper III) Witt, A. N., Mandel, S., Sell, P. H., Dixon, T., & Vijh, U. P. 2008, , 679, 497 [^1]: Although the term DGL sometimes indicates only the scattered starlight component, the term DGL indicates both scattered starlight and emission components in this paper. [^2]: IRC has two other channels covering 5.8-14.1 $\mu$m in the MIR-S channel and 12.4-26.5 $\mu$m in the MIR-L channel. [^3]: High-resolution spectroscopy ($\lambda /\Delta \lambda \sim 120$) with a grism is also available.

ArXiv

--- author: - Christian Feuersänger title: 'Test with mode list and make, automatic Basefilename' --- tikzexternaltest.sharedpreamble.tex

ArXiv

--- abstract: 'Random transformations are commonly used for augmentation of the training data with the goal of reducing the uniformity of the training samples. These transformations normally aim at variations that can be expected in images from the same modality. Here, we propose a simple method for transforming the gray values of an image with the goal of reducing cross modality differences. This approach enables segmentation of the lumbar vertebral bodies in CT images using a network trained exclusively with MR images. The source code is made available at <https://github.com/nlessmann/rsgt>' title: Random smooth gray value transformations for cross modality learning with gray value invariant networks --- Introduction ============ Detection and segmentation networks are typically trained for a specific type of images, for instance MR images. Networks that reliably recognize an anatomical structure in those images most often completely fail to recognize the same structure in images from another imaging modality. However, a lot of structures arguably *look* similar across modalities. We therefore hypothesize that a network could recognize those structures if the network was not specialized to the gray value distribution of the training images but was invariant to absolute and relative gray value variations. Invariance to certain aspects of the data is commonly enforced by applying random transformations to the training data, such as random rotations to enforce rotational invariance, or similar transformations [@Roth16; @Khal19]. In this paper, we present a simple method for randomly transforming the gray values of an image while retaining most of the information in the image. We demonstrate that this technique enables cross modality learning by training a previously published method for segmentation of the vertebral bodies [@Less19a] with a set of MR images and evaluating its segmentation performance on a set of CT images. Method ====== We define a transformation function $y(x)$ that maps a gray value $x$ to a new value. This function is a sum of $N$ sines with random frequencies, amplitudes and offsets. A sum of sines is a straightforward way of creating a smooth and continuous but non-linear transformation function. A continuous function ensures that gray values that are similar in the original image are also similar in the transformed image so that homogeneous structures remain homogeneous despite the presence of noise. The transformation function is therefore defined as: $$y(x) = \sum_{i=1}^{N} A_i \cdot \sin(f_i \cdot (2\pi \cdot x + \varphi_i)). \label{eq:y}$$ The frequencies $f_i$ are uniformly sampled from $[f_{\min}, f_{\max}]$. This range of permitted frequencies and the number of sines $N$ determine the aggressiveness of the transformation, i.e., how much it deviates from a simple linear mapping. The amplitudes $A_i$ are uniformly sampled from $[\nicefrac{-1}{f}, \nicefrac{1}{f}]$, which ensures that low frequency sines dominate so that the transformation function is overall relatively calm. The offsets $\varphi_i$ are uniformly sampled from $[0, 2\pi]$. A few examples of randomly transformed MR images are shown in Figure \[fig:examples\]. For simplicity, we fix the input range of the neural network to values in the range $[0,1]$. All input images regardless of modality need to be normalized to this range during training and inference. CT images were normalized by mapping the fixed range $[-1000, 3000]$ to $[0,1]$, for MR images this range was based on the and percentiles of the image values. Values below 0 or above 1 were clipped. During training, the images were additionally randomly transformed according to Equation \[eq:y\] and again scaled to $[0,1]$. Results ======= To evaluate the cross modality learning claim, we trained a method for segmentation of the vertebral bodies [@Less19a] with a set of T2-weighted lumbar spine MRI scans. These scans and corresponding reference segmentations of the vertebral bodies were collected from two publicly available datasets [@Chu15; @Zuki14]. We used in total 30 scans for training, 3 for validation and kept 7 scans for testing. Additionally, we collected a second test set consisting of 3 contrast-enhanced CT scans from another publicly available dataset [@Yao16]. This dataset contains reference segmentations of the entire vertebrae, and the images show all thoracic and lumbar vertebrae. We therefore first cropped the images to the lumbar region and then manually removed the posterior parts of the vertebrae from the masks to create reference segmentations of the vertebral bodies. Three experiments were performed: (1) The segmentation network was trained without any gray value modifications other than normalization to the $[0,1]$ input range. (2) The segmentation network was trained with randomly inverted gray values so that dark structures like bone appear bright in every other training sample and vice versa. (3) The segmentation network was trained with the proposed random smooth gray value transformations applied to the training samples. We used frequencies from $f_{\min} = 0.2$ to $f_{\max} = 1.6$, and the transformation functions were sums of $N = 4$ sine functions. For the MR test set, the Dice scores were virtually identical in all three experiments, the addition of the data augmentation step did neither negatively nor positively influence the segmentation performance (1: , 2: , 3: ). For the CT test set, the segmentation was only successful when random smooth gray value transformations were applied during training (1: , 2: , 3: ). Examples are shown in Figure \[fig:results\]. These results demonstrate that the proposed gray value transformations can enable training of gray value invariant networks.

ArXiv

--- author: - 'Xiao-Wei Duan,' - 'Min Zhou,' - 'Tong-Jie Zhang' title: Testing consistency of general relativity with kinematic and dynamical probes --- Introduction {#sec:intro} ============ Einstein’s theory of general relativity (GR) has, unchangeably, remained evergreen at the heart of astrophysics over almost one century after its formulation. The testing of the general relativity theory takes the central position all the time in the modern physics[@Berti2015]. The theory has already passed the precisive experimental tests to data on the scale of the Solar System and smaller ones with flying colors. Naturally, testing general relativity on cosmological scales[@Peebles2004] will be the current and future target for gravitational physics. The cosmological observations, testing the general relativity, include two traditional classes of probes[@Perivolaropoulos2010]. One class is the so-called “geometric probes”, which includes Type Ia supernovae (as standard candles), Baryon Acoustic Oscillations (BAO) and geometric properties of weak lensing. These probes can determinate the Hubble parameter $H(z)$ as a function of the redshift $z$ through angular or luminosity distances. As the ramification of geometric probes, we call $H(z)$ a “kinematic probe” because of its kinematic origin. The Hubble parameter $H(z)$ is defined as $H = \dot{a}/a$, where $a$ denotes the cosmic scale factor and $\dot{a}$ is its rate of change with respect to the cosmic time (the age of the universe when the observed photon is emitted). Moreover, the cosmic scale factor $a$ is related to the redshift $z$ by the formula $a(t )/a(t_0) = 1/(1+z)$, where $t_0$ denotes the current time which is taken to be a constant. The observational $H(z)$ data (OHD) are directly related to the expansion history of the universe. The other class is the so-called “dynamical probes”, including weak lensing, galaxy clustering and redshift-space distortions. The dynamical probes can measure the gravitational law on cosmological scales, main of which is the evolution of linear density perturbations $\delta(z)$, where $\delta(z) = (\rho - \bar{\rho})/\rho$ is the overdensity. In order to obtain observable data, the growth rate can be defined as a combination of the structure growth, $f(z) = dln\delta/dlna$, and the redshift-dependent rms fluctuations of the linear density field, $\sigma_8(z)$, $R(z)(f\sigma_8(z))\equiv f(z)\sigma_8(z)$. For the cosmic large scale structure, both of them can be measured from cosmological data. Therefore, the kinematic probes and dynamical probes provide complementary measurements of the nature of the observed cosmic acceleration[@Lue2004; @Lue2006; @Heavens2007; @Zhang2007]. In this paper, we will investigate consistency relations between the kinematic and dynamical probes in the framework of general relativity. Our work is partly motivated by the consistency tests in cosmology proposed by Knox et al.[@Knox2006], Ishak et al.[@Ishak2006], T. Chiba and R. Takahashi[@Chiba2007] and Yun Wang[@YW2008]. The consistency relation from T. Chiba and R. Takahashi, which is constructed theoretically between the luminosity distance and the density perturbation, relies on the matter density parameter today, and it is hard to achieve using available observed data. Linder et al. [@Linder2005] introduced the gravitational growth index $\gamma$ which is defined by $d\rm{ln}(\delta/a)/d\rm{ln}a=\Omega_m(a)^\gamma-1$ to finish the model-testing[@Huterer2007; @Linder2007; @Pouri2013]. It was initially considered not very insensitive to the equation of state of dark energy[@Chiba2007], but the measurement of $\gamma$ shows us that it can be used to distinguish between different models and it is clear that the growth of matter perturbations can be an efficient dynamical probe to show us the nature of dark energy[@Polarski2008; @Gannouji2008]. Inspired by these works, we wonder whether we can step a little further. Is there a more direct way to set the test? We understand that the most accurate information about the cosmic expansion rate, $H(z)$, can provide the unique prediction for the growth rate data. The obtained $data_{cal}(z)$ must be consistent with the observed $data_{obs}(z)$ if general relativity is the correct theory of gravity in the universe. Given that the assumptions used in the methods to obtain observed data may make specific cosmological models involved in, we choose $f\sigma_8$ for test implementing, which are produced using almost model-independent way. Using the kinematic probe $H(z)$, our testing methods are independent of any cosmological constant from theoretical calculation according to general relativity. The presence of significant deviations from the consistency can be used as the signature of a breakdown of general relativity on cosmological scales. It is hoped to gain new knowledge about testing the general relativity. This paper is organized as follows. In Section II we briefly describe the observable data and the relation between Hubble parameter and growth rate, particularly how to establish the consistency relation equations. The analysis methods, calculated results and discussion are presented in Section III. Furthermore, in Section IV, we try to check the validity of this test. And finally, the summary is given in Section V. 0.5cm Observable data and relation between Hubble parameter and growth rate ===================================================================== The kinematic probe $H(z)$ directly measure the cosmic metric while dynamical probes not only measure the cosmic metric but also the gravitational law concurrently on cosmological scales. Converging these two kinds of probes indicates that our universe is accelerating in its expansion. In our work, we choose $H(z)$, the kinematic probe and the growth rate deduced from density fluctuation as the dynamical one to establish the consistency equation. Kinematic Probes: Observational $H(z)$ Data ------------------------------------------- The charming potential of OHD to constrain cosmological parameters and distinguish different models has been surfacing over in recent years. $H(z)$ can be produced by model-independent direct observations, and three methods have been developed to measure them up to now [@Zhang2010]: galaxy differential age, radial BAO size and gravitational-wave standard sirens methods. In practice, $H(z)$ is usually utilized as a function of the redshift $z$, getting its expression as: $$\begin{aligned} H(z)=-\frac{1}{1+z}\frac{dz}{dt}.\end{aligned}$$ According this expression, $H(z)$ got its initial measuring method: differential age method. It was firstly put forward by Jimenez & Loeb [@Jimenez2002] in 2002 using relative galaxy ages, demonstrated by Jimenez et al.[@Jimenez2003] with the first $H(z)$ measurement at $z\approx1$ reported. Soon afterwards, Simon et al.[@Simon2005] provided eight additional $H(z)$ data points in the redshift value range from 0.17 to 1.75 from the relative ages of passively evolving galaxies (henceforward ¡°SVJ05¡±) and use them to constrain the redshift dependency of DE potential. In 2010, Stern et al. [@Stern2010] expanded the dataset (henceforward ¡°SJVKS10¡±) with two new determinations and constrain DE parameters and the spatial curvature via combining the dataset with CMB data. Moreover, Moresco et al. [@Moresco2012] utilized the differential spectroscopic evolution of early-type, massive, red elliptical galaxies (can be used as standard cosmic chronometers) and got eight new data points. As time goes by, bigger and better instruments are used to make cosmological observations. Chuang & Wang [@ChuangWang2012] measured $H(z)$ at $z=0.35$ with luminous red galaxies from Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7). Also with applying the galaxy differential age method to SDSS DR7, Zhang et al. [@Zhang2014] obtained four new measurements. Moresco [@Moresco2015] reported two latest data points with near-infrared spectroscopy of high redshift galaxies. And, recently, they has enriched the dataset again with 5 points through the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 9, which yield a 6% accuracy constraint of $H(z=0.4293)=91.8\pm5.3kms^{-1}Mpc^{-1}$[@Moresco2016]. Apart from the differential method, there is another method to obtain $H(z)$ : detection of radial BAO features. It was first utilized by Gazta$\tilde{\rm{n}}$aga et al. [@Gaztanaga2009] to get two new $H(z)$ data points in 2009. They used the BAO peak position as a standard ruler in the radial direction. From then on, the dataset on $H(z)$ is expanded: 3 data points from Blake et al. [@Blake2012] by combining the measurements of BAO peaks and the Alcock-Paczynski distortion; one data point from Samushia et al. [@Samushia2013] using the BOSS DR9 CMASS sample; one data point from Xu et al. [@Xu2013] by means of BAO signals from the SDSS DR7 luminous red galaxy sample. Busca et al. [@Busca2013], Font-Ribera et al. [@Font-Ribera2014] and Delubac et al. [@Delubac2015] provided respectively one data point on $H(z)$ by BAO features in the Lyman-$\alpha$forest of SDSS-III quasars. Further more, the detection of gravitational wave also provides a way to get $H(z)$: gravitational-wave standard sirens. With future gravitational wave detectors, it will be able to measure source luminosity distances, as integrated quantities of $H^{-1}(z)$, previously out to $z\sim5$ [@Bonvin2006; @Nishizawa2010]. The quantity of the available data will promisingly be improved dramatically in the near future. As for this work, we put all the available $H(z)$ data up to now into use. After getting rid of overlapped data points with same sources, table \[tab:Hubble\] lists the precise numerical values of $H(z)$ data points with the corresponding errors, which is collected by Meng et al. [@Meng2015] and added with [@Moresco2016]. The correspondence of the columns is as follows: redshift, observed Hubble parameter ($km s^{-1} Mpc^{-1}$), used method (I: differential age method, II: BAO method) and references. [$z$]{} $H(z)$ Method Ref. ---------- ---------------- -------- --------------------------------------------------------- $0.0708$ $69.0\pm19.68$ I Zhang et al. (2014)-[@Zhang2014] $0.09$ $69.0\pm12.0$ I Jimenez et al. (2003)-[@Jimenez2003] $0.12$ $68.6\pm26.2$ I Zhang et al. (2014)-[@Zhang2014] $0.17$ $83.0\pm8.0$ I Simon et al. (2005)-[@Simon2005] $0.179$ $75.0\pm4.0$ I Moresco et al. (2012)-[@Moresco2012] $0.199$ $75.0\pm5.0$ I Moresco et al. (2012)-[@Moresco2012] $0.20$ $72.9\pm29.6$ I Zhang et al. (2014)-[@Zhang2014] $0.240$ $79.69\pm2.65$ II Gazta$\tilde{\rm{n}}$aga et al. (2009)-[@Gaztanaga2009] $0.27$ $77.0\pm14.0$ I Simon et al. (2005)-[@Simon2005] $0.28$ $88.8\pm36.6$ I Zhang et al. (2014)-[@Zhang2014] $0.35$ $84.4\pm7.0$ II Xu et al. (2013)-[@Xu2013] $0.352$ $83.0\pm14.0$ I Moresco et al. (2012)-[@Moresco2012] $0.3802$ $83.0\pm13.5$ I Moresco et al. (2016)-[@Moresco2016] $0.4$ $95\pm17.0$ I Simon et al. (2005)-[@Simon2005] $0.4004$ $77.0\pm10.2$ I Moresco et al. (2016)-[@Moresco2016] $0.4247$ $87.1\pm11.2$ I Moresco et al. (2016)-[@Moresco2016] $0.43$ $86.45\pm3.68$ II Gaztanaga et al. (2009)-[@Gaztanaga2009] $0.44$ $82.6\pm7.8$ II Blake et al. (2012)-[@Blake2012] $0.4497$ $92.8\pm12.9$ I Moresco et al. (2016)-[@Moresco2016] $0.4783$ $80.9\pm9.0$ I Moresco et al. (2016)-[@Moresco2016] $0.48$ $97.0\pm62.0$ I Stern et al. (2010)-[@Stern2010] $0.57$ $92.4\pm4.5$ II Samushia et al. (2013)-[@Samushia2013] $0.593$ $104.0\pm13.0$ I Moresco et al. (2012)-[@Moresco2012] $0.6$ $87.9\pm6.1$ II Blake et al. (2012)-[@Blake2012] $0.68$ $92.0\pm8.0$ I Moresco et al. (2012)-[@Moresco2012] $0.73$ $97.3\pm7.0$ II Blake et al. (2012)-[@Blake2012] $0.781$ $105.0\pm12.0$ I Moresco et al. (2012)-[@Moresco2012] $0.875$ $125.0\pm17.0$ I Moresco et al. (2012)-[@Moresco2012] $0.88$ $90.0\pm40.0$ I Stern et al. (2010)-[@Stern2010] $0.9$ $117.0\pm23.0$ I Simon et al. (2005)-[@Simon2005] $1.037$ $154.0\pm20.0$ I Moresco et al. (2012)-[@Moresco2012] $1.3$ $168.0\pm17.0$ I Simon et al. (2005)-[@Simon2005] $1.363$ $160.0\pm33.6$ I Moresco (2015)-[@Moresco2015] $1.43$ $177.0\pm18.0$ I Simon et al. (2005)-[@Simon2005] $1.53$ $140.0\pm14.0$ I Simon et al. (2005)-[@Simon2005] $1.75$ $202.0\pm40.0$ I Simon et al. (2005)-[@Simon2005] $1.965$ $186.5\pm50.4$ I Moresco (2015)-[@Moresco2015] $2.34$ $222.0\pm7.0$ II Delubac et al. (2015)-[@Delubac2015] : \[tab:Hubble\] The currently available OHD dataset. In order to set up a model-independent consistency relation to test general relativity, firstly we need to obtain a good expression on $H(z)$. From the Friedmann equation, $H(z)$ can be written as $$\begin{aligned} \label{eq:modeleq} \frac{H^2}{H^2_0} & = & \Omega_{m_0}(1+z)^3+\Omega_{r_0}(1+z)^4+\Omega_{k_0}(1+z)^2 \nonumber \\ & & +\Omega_{x}{\rm{exp}}[3\int^z_0(1+\omega_x(z))d{\rm{ln}}(1+z)].\end{aligned}$$ Moreover, when $\omega=-1$ (vacuum energy), it reduces to $$\begin{aligned} \label{eq:modeleq1} \frac{H^2}{H^2_0} & = & \Omega_{m_0}(1+z)^3+\Omega_{r_0}(1+z)^4 \nonumber \\ & & +\Omega_{k_0}(1+z)^2+\Omega_{\Lambda_0}.\end{aligned}$$ For lower redshift and flat model, it further reduces to $$\begin{aligned} \label{eq:modeleq2} \frac{H^2}{H^2_0} & = & \Omega_{m_0}(1+z)^3+\Omega_{\Lambda_0}.\end{aligned}$$ It occurred to us that $H^2$ can be parameterized. If we use 4th degree polynomial to fit $H^2$, we can see that this expression can directly degenerate into Eq. \[eq:modeleq1\] and this method is a very general way of data modeling. But based on the current observed data, the fitting result of it has a negative coefficient for fourth power term. It will lead fitting curve of $H^2$ to be negative when the redshift getting bigger, which is unreasonable. Polynomial fitting suffers from the imprecision of current observed data. So we adopted a general choice: $$\begin{aligned} \label{eq:Hz} H^2(z)=a(1+z)^b+c,\end{aligned}$$ which set exponent into free. It may give more favor to $\Lambda$CDM model but it keeps a more convincing trace when fitting current data. Therefore, we set it as the parametric method (Method A) in our test. Visualization of our fitting curve is shown in Figure 1 (the blue solid line) and the error range is derived from simultaneous predicting method [@Matlabd1]. ![\[fig:H1\] Visualization of $\chi^2$ approach to the observational $H(z)$ dataset (Method A) and model-independent reconstruction of it using Gaussian processes (Method B). The solid curve represents the best fitting result in method A and the dashed line refers to the reconstruction with GaPP in method B. The points with error bars denote OHD.](Figure1.eps){width="90.3mm" height="7cm"} ![\[fig:H2\] Visualization of model-independent reconstruction of the first derivative of $H(z)$ using Method A and B. The solid curve represents result in method A and the dashed line refers to the reconstruction with GaPP in method B. Here is also $H(z)$ reconstruction in $\Lambda$CDM plotted with green line for reference.](Figure2.eps){width="90.3mm" height="7cm"} As for our second method (Method B), we use Gaussian Processes (GP), which is a non-parametric method for smoothing the observational data. The advantage of this method is that it is also model-independent as Method A and it can perform a reconstruction of a function from data without assuming a parametrization of the function. A Gaussian process is the generalization of a Gaussian distribution on a random variable. It describes a distribution over functions. The freedom in Gaussian process comes in the choice of covariance function $k(z,\tilde{z})$, which determines how smooth the process is and how nearby points such as $z$ and $\tilde{z}$ are correlated. The covariance function $k(z,\tilde{z})$ depends on a set of hyperparameters $\ell$ and $\sigma_f$, which can be determined by Gaussian process with the observed data. The detailed analysis and descriptions of Gaussian process method can be found in [@Seikel2012], [@Seikel20121] and [@Seikel2013]. Here we use the Gaussian processes in Python (GaPP) [@Seikel2012] to reconstruct the Hubble parameter as a function of the redshift from the observational Hubble data. We use 38 $H(z)$ measurements to obtain the model-independent reconstruction of $H(z)$ and the first derivative of it using Gaussian processes. The redshift range of observational $H(z)$ data is \[0.0708,2.36\] so that we set $z$ variable as \[0,2.5\]. Our results for the non-parametric approach are shown in Figure \[fig:H1\] and \[fig:H2\]. Figure 1 also gives the observational $H(z)$ data with their respective errorbars and makes the simulated $H(z)$ data obtained from the two methods we use compared with each other. The blue solid line refers to the fitting result in Method A, with the dash line denoting production from Method B. It is obvious in Figure 1 that method A and B capture data correctly though there is a slight difference between them. The reconstructions of first derivative of $H(z)$ are displayed in Figure 2. There is also $H(z)$ reconstruction in $\Lambda$CDM plotted with green line for reference. It is obvious that confined by the quality of current data, the trend of $dH(z)/dz$ is not very reasonable. Nevertheless, Gaussian Processes have the limitation that they can’t keep reconstructing data exactly all the time as the argument getting bigger. Dynamical Probes: the Growth Rate deduced from Density Fluctuation ------------------------------------------------------------------ Through the observations of distant supernovae, we know that the Universe is in a phase of accelerated expansion now. Different models, though producing similar expansion rates, are corresponding to different growth rate of large-scale structure with cosmic time [@Peebles1980]. The growth of cosmic structure is driven by the motion of matter, while galaxies act as “tracer particles”. Coherent galaxy motions, reconstructed by galaxy redshift surveys, can introduce a radial anisotropy in the clustering pattern [@LMBS2008]. Moreover, in linear perturbation theory it is possible to establish an expression to describe the growth of a generic small-amplitude density fluctuation $\delta(z)$, which is defined as $\delta_m\equiv\delta\rho_m/\rho_m$, by general relativity. It is expressed as [@LGuzzo2008]]: $$\begin{aligned} \label{eq:defidensitydelta} \ddot{ \delta}+2H\dot{\delta}-4{\pi}G\rho_M{\delta} &=& 0,\end{aligned}$$ where a dot represents the derivative with respect to the cosmic time $t$ and $\rho_M$ denotes the matter density. The linear growth rate $f(z)$ can be defined as $$\begin{aligned} \label{eq:fdefinition} f(z)\equiv\frac{dln\delta}{dlna},\end{aligned}$$ to measure how rapidly the structure is being assembled according to different redshifts. However, most of the observed data on $f(z)$ we obtained up to now let specific cosmological models (mostly $\Lambda$CDM) play quite big role in the process to produce them. So we turned our eyes toward another growth rate dataset: $R(z)$, which is almost a model-independent way to express the observed growth history of the universe [@Song2009]. It is combined by the growth rate of structure $f(z)$ and the redshift-dependent rms fluctuations of the linear density field $\sigma_8$ as $$\begin{aligned} \label{eq:fsigma8definition} R(z)(f\sigma_8(z))\equiv f(z)\sigma_8(z),\end{aligned}$$ where $\sigma_8(z)=\sigma_8(0)\frac{\delta(z)}{\delta(0)}$. The used dataset is shown in Table \[tab:R\], which summarizes the used numerical values of the observational data on $R(z)$ with the corresponding errors. The correspondence of the columns in it is as follows: redshift, observed growth rate (henceforward $R_{obs}(z)$), references. [$z$]{} $R_{obs}(z)$ Ref. --------- ------------------- ---------------------------- $0.02$ $0.360\pm0.040$ [@Hudson2012] $0.067$ $0.423\pm0.055$ [@Beutler2012] $0.17$ $0.510\pm0.060$ [@Song2009] $0.25$ $0.3512\pm0.0583$ [@Samushia2012] $0.30$ $0.407\pm0.055$ [@Tojeiro2012] $0.32$ $0.384\pm0.095$ [@Chuang201312] $0.35$ $0.440\pm0.050$ [@Song2009] $0.35$ $0.445\pm0.097$ [@Chuang2013] $0.37$ $0.4602\pm0.0378$ [@Samushia2012] $0.40$ $0.419\pm0.041$ [@Tojeiro2012] $0.44$ $0.413\pm0.080$ [@Blake2012] $0.50$ $0.427\pm0.043$ [@Tojeiro2012] $0.57$ $0.423\pm0.052$ [@Samushia2014] $0.60$ $0.433\pm0.067$ [@Tojeiro2012] $0.60$ $0.390\pm0.063$ [@Blake2012] $0.73$ $0.437\pm0.072$ [@Blake2012] $0.77$ $0.490\pm0.018$ [@Song2009],[@LGuzzo2008]] $0.80$ $0.470\pm0.080$ [@Torre2013] : \[tab:R\] The growth data $R_{obs}(z)$. Establishing the Equation of Consistency Relation ------------------------------------------------- In order to test general relativity on cosmological scales, T. Chiba and R. Takahashi[@Chiba2007] provided theoretically a consistency relation in general relativity which relates the luminosity distance and the density perturbation and relies on the matter density parameter today. In this paper, we would like to provide a new way to test the consistency between the kinematic and dynamical probes in the framework of general relativity. Solving Eq. \[eq:defidensitydelta\], we can obtain an analytical solution on $\delta(z)$ given by $$\begin{aligned} \label{eq:deltasolution} \delta(z)=\frac{5\Omega_{M0}E(z)}{2}\int^\infty_z\frac{1+z}{E^3(z)}dz,\end{aligned}$$ where $E(z)=H(z)/H_0$. According to the definition of growth rate $f(z)$ and $f\sigma_8$ (Eq. \[eq:fdefinition\] and Eq. \[eq:fsigma8definition\]), we can derive the more specific forms: $$\begin{aligned} \label{eq:Handfsigma8} f\sigma_8(z)&=&\frac{5\Omega_{M0}H^2_0\sigma_8(0)}{2\delta(0)}(1+z)[\frac{1+z}{H^2(z)}%\nonumber\\ -\frac{dH(z)}{dz}\int^\infty_z\frac{1+z}{H^3(z)}dz],\end{aligned}$$ $$\begin{aligned} \label{eq:Handf} f(z)&=&-(1+z)[\frac{\frac{dH(z)}{dz}}{H(z)}+\frac{\frac{-(1+z)}{H^3(z)}}{\int^\infty_z\frac{1+z}{H^3(z)}dz}],\end{aligned}$$ where we could see that there is no annoying parameter to be preset at the beginning for the expression of $f(z)$ and it can be used for predicting. As for our consistency testing, we changed the appearance of Eq. \[eq:Handfsigma8\] to make the terms whose value can be confirmed by simulation moved to one side of the equation: $$\begin{aligned} \label{eq:Handfsigma8co1} C(z)&=&\frac{f\sigma_8(z)}{H^2_0(1+z)[\frac{1+z}{H^2(z)}-\frac{dH(z)}{dz}\int^\infty_z\frac{1+z}{H^3(z)}dz]}%\nonumber\\ =\frac{5\Omega_{M0}\sigma_8(0)}{2\delta(0)}=\lambda,\end{aligned}$$ We refer to it as the first consistency relation we get, using the kinematic and dynamical probes in the framework of general relativity. It is easy for comprehension that in this equation $R(z) (f\sigma_8(z))$ act as the $data_{obs}$ and $H^2_0(1+z)[\frac{1+z}{H^2(z)}-\frac{dH(z)}{dz}\int^\infty_z\frac{1+z}{H^3(z)}dz]$ is treated as $data_{cal}$. The ratio of them is the constant $\lambda$ but there is no need to preset it. The detailed processes will be illuminated in the next section. $C(z)$ have to be constant for all redshifts, as a null-test. $H^2_0$ has also been moved to left because it was a part of the simulated $H(z)$ data. We also provided another type of consistency relation. The integral term can be set alone as: $$\begin{aligned} \label{eq:Handfsigma8co2pro} I(z)&=&\int^\infty_z\frac{1+z}{H^3(z)}dz%\nonumber\\ =\frac{1+z}{H'(z)H^2(z)}-\frac{f\sigma_8(z)}{\lambda H^2_0(1+z)H'(z)},\end{aligned}$$ $$\begin{aligned} \label{eq:Handfsigma8co2pro2} I(z_2)-I(z_1)&=&\int^{z_1}_{z_2}\frac{1+z}{H^3(z)}dz=[\frac{1+z}{H'(z)H^2(z)}-\frac{f\sigma_8(z)}{\lambda H^2_0(1+z)H'(z)}]\mid^{z_2}_{z_1},\end{aligned}$$ $$\begin{aligned} \label{eq:Handfsigma8co2} C(z_1;z_2)&=&\frac{\frac{f\sigma_8(z)}{H^2_0(1+z)H'(z)}\mid^{z_2}_{z_1}}{[\frac{1+z}{H'(z)H^2(z)}+\int\frac{1+z}{H^3(z)}dz]\mid^{z_2}_{z_1}}%\nonumber\\ =\frac{5\Omega_{M0}\sigma_8(0)}{2\delta(0)}=\lambda,\end{aligned}$$ It is obvious that this equation enjoys a pleasing advantage that the calculated result can only be dominated by data in observed range, which means that the errors introduced when tracing $H(z)$ of $z\rightarrow\infty$ can be avoided. However, it would suffer from the newborn errors coming from the increased calculation steps. The similar problem occurs when we change Eq. \[eq:Handfsigma8co2pro2\] to $$\begin{aligned} \label{eq:Handfsigma8co3pro} \lim\limits_{\Delta z\rightarrow0}{\frac{\int^{z_1}_{z_2}\frac{1+z}{H^3(z)}dz}{z_2-z_1}}&=&\lim\limits_{ \Delta z\rightarrow0}{\frac{[\frac{1+z}{H'(z)H^2(z)}-\frac{f\sigma_8(z)}{\lambda H^2_0(1+z)H'(z)}]\mid^{z_2}_{z_1}}{z_2-z_1}},\end{aligned}$$ in which $\Delta z=z_2-z_1$ and $\lambda=\frac{5\Omega_{M0}\sigma_8(0)}{2\delta(0)}$, $$\begin{aligned} \label{eq:Handfsigma8co3pro2} -\frac{1+z}{H^3(z)}&=&\frac{d[\frac{1+z}{H'(z)H^2(z)}-\frac{f\sigma_8(z)}{\lambda H^2_0(1+z)H'(z)}]}{dz},\end{aligned}$$ $$\begin{aligned} \label{eq:Handfsigma8co3} C(z)&=&\frac{d[\frac{f\sigma_8(z)}{H^2_0(1+z)H'(z)}]/dz}{d[\frac{1+z}{H'(z)H^2(z)}]/dz+\frac{1+z}{H^3(z)}}=\lambda,\end{aligned}$$ where the first derivative of $f\sigma_8(z)$ would be involved in to finish the calculation. It appears to be a noticeable problem when the quality of observed data still needs improvement. To be clear: errors produced by tracing $H(z)$ of $z\to\infty$ would not make test based on Eq. \[eq:Handfsigma8co1\] lose credibility, because $\frac{1+z}{H^3(z)}$ drops quickly as $z$ getting larger and the high-$z$ integral value is a negligible portion of the total result. In this work, we put Eq. \[eq:Handfsigma8co1\] and Eq. \[eq:Handfsigma8co2\] into practice. We give them code names “CRO (Consistency Relation using Original formula, for Eq. \[eq:Handfsigma8co1\])” and “CRI (Consistency Relation using Interval integral method, for Eq. \[eq:Handfsigma8co2\])” for future reference. Furthermore, It is worthy discussing that how should we realize the CRI testing. Binning all data would smooth the fluctuation and reduce the sensitivity. So we choose to bin the last two points (because they are quite close to each other and represent the value of higher redshift) and set the result as the base point $(z_1,f\sigma_8(z_1))$ used for subtracting process. We calculate $C(z)$(henceforward $C_{obs}(z))$ with its theoretical uncertainty $\sigma_{C_{obs}}$ by means of $f\sigma_8$ and Hubble parameter $H(z)$ obtained from above mentioned two methods. The uncertainty $\sigma_{C_{obs}}$ is given as $$\begin{aligned} \label{eq:deltaCtheo} \sigma^2_{C_{obs}}&=&(\frac{\partial C}{\partial H})^2\sigma_H^2+(\frac{\partial C}{\partial H'})^2\sigma_{H'}^2+(\frac{\partial C}{\partial {f\sigma_8}})^2\sigma_{f\sigma_8}^2+(\frac{\partial C}{\partial {H_0}})^2\sigma_{H_0}^2\nonumber\\ &&+2(\frac{\partial C}{\partial H})(\frac{\partial C}{\partial {H'}})Cov(H,H')\nonumber\\ &&+2(\frac{\partial C}{\partial H})(\frac{\partial C}{\partial {f\sigma_8}})Cov(H,{f\sigma_8})\nonumber\\ &&+2(\frac{\partial C}{\partial H'})(\frac{\partial C}{\partial {f\sigma_8}})Cov(H',{f\sigma_8}).\end{aligned}$$ Since we have already get the expressions of the consistency relation with the kinematic and dynamical probes, in the next section, we will discuss quantitatively how to use them to test general relativity on cosmological scales. 0.5cm Analysis methods, results and discussion {#section:analysis methods} ========================================== To quantify the relation between the kinematic and dynamical probes, we perform the method of parameterizations which has witnessed boom in recent years. Originally it is in the works about the state parameter of dark energy models[@Huterer2001; @Efstathiou1999; @Chevallier2001; @Linder2003]. Inspired by their work, Holanda et al.[@Holanda2011] assumed $D_L(1+z)^{-2}/{D_A}=\eta(z)$, where $\eta(z)$ quantifies a possible epoch-dependent departure from the standard photon conserving scenario ($\eta=1$), and set two parametric presentations of $\eta(z)$ as $1+\eta_0z$ and $1+\eta_0z/(1+z)$ [@Chevallier2001] to test the DD relation. The former is a continuous and smooth linear expansion and the latter includes a possible epoch-dependent correction, making the value of $\eta$ become bounded when redshift goes higher. The parametric expressions enjoy many advantages. They have good sensitivity to observational data and avoid the divergence at extremely high $z$, which make them more useful when data in higher redshift become available. Since then, the method has been developed with more expressions and into two-dimension [@Zhengxiang; @2011; @Remya; @2011; @Meng; @2012]. They helped to deal with a lot of significant work. We have already seen that if general relativity is the correct theory of gravity in the universe, the equation $$\begin{aligned} C_{cal}(z)=\frac{5\Omega_{M0}\sigma_8(0)}{2\delta(0)}=constant,\end{aligned}$$ should hold. It is also clear that the value of this constant can’t be obtained based on the existing information in our work. With a view to the limitation from one-dimensional parameterization, two-dimensional parameterizations of $C(z)$, which is more general are enabled. For the possible redshift dependence of the consistency relation, we parameterize $C(z)$ as following: \[eta12\] $$\begin{aligned} \label{eq:eta12 1} C(z)&=C_1+C_2z ,\\%\label{eq:eta12 1}\\ \label{eq:eta12 2} C(z)&=C_1+C_2z/(1+z),\\ %C(z)&=&C_1+C_2z/(1+z)^2 ,\\ \label{eq:eta12 4} C(z)&=C_1-C_2{\rm{ln}}(1+z),\\ \label{eq:eta12 5} C(z)&=C_1-C_2{\rm{sin}}(z).\end{aligned}$$ where the expression Eq. \[eq:eta12 5\] using trigonometric function is firstly proposed in the testing by us. As for all the expressions, we have $C=C_1$ when $z\ll1$. Moreover, the charming waving character of trigonometric function make it deserved to be tried in consistency parameterizations. To determine the most probable values of the parameters in $C(z)$, maximum likelihood estimation is employed, via $L{\propto}e^{\chi^2}$ and $$\begin{aligned} \label{eq:kai} \chi^2=\sum_{z}\frac{[C(z)-C_{obs}(z)]^2}{\sigma^2_{C_{obs}}},\end{aligned}$$ and the uncertainty $\sigma_{C_{obs}}$ is given by Eq. \[eq:deltaCtheo\]. If the consistency relation holds using the kinematic and dynamical probes in the framework of general relativity, the likelihood $e^{-\chi^2/2}$ would peak at $(C_1,C_2)=(\lambda,0)$, where $\lambda=\frac{5\Omega_{m0}\sigma_8(0)}{\delta(0)}$ for two-dimensional parameterizations. To draw the likelihood contours at 1, 2, 3$\sigma$ confidence level (CL), there is $\Delta\chi^2=$2.3, 6.17 and 11.8 for 1, 2 and 3$\sigma$ CL respectively. In order to obtain the value of $\Omega_{m0}\frac{\sigma_8(0)}{\delta(0)}$ simultaneously, we take the average of the $\lambda_{obs}$s (the value of $C_1$ where $e^{-\chi^2/2}$ peak at) as: $$\begin{aligned} \label{eq:binlambda} \bar{\lambda}_{obs} &=&\frac{\sum_{i}(\lambda_{obsi}/\sigma^2_{\lambda_{obsi}})}{\sum_{i}1/\sigma^2_{\lambda_{obsi}}},\sigma^2_{\bar{\lambda_{obs}}}=\frac{1}{\sum_{i}1/\sigma^2_{\lambda_{obsi}}},\end{aligned}$$ where $\lambda_{obsi}$ represents the $i\rm{th}$ result of the $\lambda_{obs}$, and $\sigma_{\lambda_{obsi}}$ denotes its observational uncertainty. [|lcc|]{} & $\chi^2/d.o.f.$ & $C_1 \& C_2$\ $$& Method A &$$\ $C=C_1+C_2z$ & $0.1818$ & $0.6226\pm 0.0830$\ $ $ & $ $ & $0.0232\pm 0.1655$\ $C=C_1+C_2z/(1+z)$ & $0.1830$ & $0.6320\pm 0.1046$\ $ $ & $ $ & $0.0044\pm 0.3285$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.1826$ & $0.6267\pm 0.0933$\ $ $ & $ $ & $-0.0181\pm 0.2362$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.1824$ & $0.6256\pm 0.0879$\ $ $ & $ $ & $-0.0177\pm 0.1870$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.2533\pm 0.0540$ & $$&$$ $$\\ \hline$$ & Method B& $$\ $C=C_1+C_2z$ & $0.1505$ & $0.5054\pm 0.1000$\ $ $ & $ $ & $0.2930\pm 0.2071$\ $C=C_1+C_2z/(1+z)$ & $0.1786$ & $0.4839\pm 0.1281$\ $ $ & $ $ & $0.5123\pm 0.4112$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.1641$ & $0.4937\pm 0.1133$\ $ $ & $ $ & $-0.3946\pm 0.2954$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.1597$ & $0.5012\pm 0.1062$\ $ $ & $ $ & $-0.3179\pm 0.2335$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.2546\pm 0.0625$ & $$&$$ $$\ [|lcc|]{} & $\chi^2/d.o.f.$ & $C_1 \& C_2$\ $$& Method A &$$\ $C=C_1+C_2z$ & $0.1226$ & $0.7098\pm 0.2048$\ $ $ & $ $ & $-0.6219\pm 0.6169$\ $C=C_1+C_2z/(1+z)$ & $0.1303$ & $0.7233\pm 0.2287$\ $ $ & $ $ & $-0.9098\pm 0.9631$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.1263$ & $0.7174\pm 0.2168$\ $ $ & $ $ & $0.7602\pm 0.7774$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.1242$ & $0.7117\pm 0.2086$\ $ $ & $ $ & $0.6448\pm 0.6479$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.2105\pm 0.1391$ & $$&$$ $$\\ \hline$$ & Method B & $$\ $C=C_1+C_2z$ & $0.0204$ & $0.5808\pm 0.3419$\ $ $ & $ $ & $-0.4406\pm 1.0140$\ $C=C_1+C_2z/(1+z)$ & $0.0217$ & $0.5907\pm 0.3801$\ $ $ & $ $ & $-0.6496\pm 1.5967$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.0211$ & $0.5864\pm 0.3612$\ $ $ & $ $ & $0.5407\pm 1.2794$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.0207$ & $0.5823\pm 0.3481$\ $ $ & $ $ & $0.4578\pm 1.0659$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.1799\pm 0.2417$ & $$&$$ $$\ ----------------------------------------------------- ----------------------------------------------------- {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} ----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------------- ----------------------------------------------------- {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} ----------------------------------------------------- ----------------------------------------------------- Using the maximum likelihood described above, we have constrained the parameter values in parameterizations of $C(z)$. As for CRO and CRI, using the parametric method (Method A) and Gaussian Process (Method B) respectively, the $\chi^2$ per degrees of freedom ($\chi^2 /d.o.f.$) [@Press; @1994] and the best-fit value of $(C_1,C_2)$ with their errors are summarized in Table \[table:parameterizationCRO\],\[table:parameterizationCRI\]. The normalized likelihood distribution functions are plotted in Figure 3,4, where the different color curves correspond to the results from four two-dimensional parameterizations respectively, with the principle of correspondence as $C(z)=C_1+C_2z\rightarrow$ red, $C(z)=C_1+C_2z/(1+z)\rightarrow$ orange, $C(z)=C_1-C_2\rm{ln}(1+z)\rightarrow$ green, $C(z)=C_1-C_2\rm{sin}(z)\rightarrow$ blue. According to the results shown in Figure 3,4 and Table \[table:parameterizationCRO\],\[table:parameterizationCRI\], the consistency is clear between the kinematic and dynamical probes for all parameterizations within $1\sigma$ CL. CRI shows better result than CRO does, but it can’t be ignored that its error gets larger due to the extra calculating steps. It means that in terms of the present condition of data, CRO implements the stricter constraint, and the calculation using our mock data for high-$z$ Hubble data doesn’t deviate much from the real one. In conclusion, all two-dimensional parameterizations give a substantial support to the consistency relation between the kinematic and dynamical probes within $1\sigma$ CL. \[fig:cocoplot\] ----------------------------------------------------- ----------------------------------------------------- {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} ----------------------------------------------------- ----------------------------------------------------- In order to understand behaviour of the used parameterizations in a quantitative manner, we plot the result of CRO, which does stricter constraint, using its best fit values along with $C_{obs}(z)$ and $\sigma_{C_{obs}}$ in Figure 5 from the two methods: Method A and B. For the parametric method (Method A), fitting curves are pleasant thanks to good parametrization of $H(z)$, so there is not obvious difference. By means of the non-parametric method (Method B) from Gaussian Processes (GP), it is clear from the Figures 5 that parametrization forms $C(z)\sim z/(1+z)$ (Eq. \[eq:eta12 2\]) and $C(z)\sim {\rm{sin}}(z)$ (Eq. \[eq:eta12 5\]) are good choices for modeling the consistency relation because other two parameterizations gradually deviate from $\bar{\lambda}_{obs}$ (Eq. \[eq:binlambda\]) as the increase of redshift. As for $C(z)=C_1+C_2z$ (Eq. \[eq:eta12 1\]), there is a large slope, and no chance for it to be convergent. $C(z)=C_1-C_2{\rm{ln}}(1+z)$ (Eq. \[eq:eta12 4\]) inclines too much as well. The four of them give approached departures at $z=0$ (which for Eq. \[eq:eta12 1\] and Eq. \[eq:eta12 4\] are slightly smaller). In sum, as seen in Figure 5, $C(z)\sim z/(1+z)$ and $C(z)\sim {\rm{sin}}(z)$ perform better than the other two. What is noteworthy is that the parametrization model using trigonometric function is considered aesthetically very beautiful. It provides an error model which is always dipping and heaving around the expected value. We can see that it performs well in this work, providing a tight constraint to the expected value. Therefore, in consideration of the pivotal role of growth data in high redshift region, we suggest to employ the parametrization forms $C(z)\sim z/(1+z)$ and $C(z)\sim {\rm{sin}}z$ in future testing of consistency relations between the kinematic and dynamical probes in the framework of general relativity. 0.5cm Validity check for the testing method with mock data ==================================================== It is worth stressing that our method testing general relativity on cosmological scale is independent of any cosmological model. But $f\sigma_8$, the dynamical probe data we used, which can be considered as “almost model-independent”, are also not evaluated in completely model-independent processes. That is to say, the testing may prefer the models which participated in the method of providing data (for $f\sigma_8$ it is $\Lambda$CDM model)[@Nesseris; @2015]. It occurred to us that, in order to confirm that the validity of this test wasn’t whelmed by model-dependency, we should have it checked. The main weak point of cosmological models based on the validity of general relativity lies in involving fined tuned parameters, such as standard cold dark matter (SCDM)[@Nesseris; @2004], cold dark matter with cosmological constant $\Lambda$ ($\Lambda$CDM, also called LCDM model)[@Peebles; @2003], quiessence model (QUIES)[@Alam; @2003], dark energy with equation of state $w(z) = w_0 + w_1z$ (Linear)[@Huterer2001], $w(z) = w_0 + w_1z/(1 + z)$ (CPL)[@Chevallier2001]. Since all of them are based on general relativity, they should enjoy significant advantage compared to other models which disfavor general relativity, like DGP (Dvali-Gabadadze-Poratti) model. Table \[table:modelcomparing\] presents the best-fit parameters of above-mentioned six cosmological models using currently available OHD dataset. The comparison of the observed and theoretical evolution of the Hubble parameter $H(z)$ is given in the top-left panel in Fig. 7. Because of the significantly larger $\chi^2_{min}/d.o.f$, we conclude that SCDM does not provide a good fit to $H(z)$ data. By means of the best-fit parameters of cosmological models, Eq. \[eq:Handf\] and Eq. \[eq:Handfsigma8co1\], the theoretical results of the growth rate $f(z)$ and $f\sigma_8/\lambda$ are visualized in Fig. 6. It is indicated that four cosmological models except of SCDM are visually consistent with the reasonable trend. It can be deduced that, if the validity was whelmed by interference from the model ($\Lambda$CDM, which $f\sigma_8$ depends on), the success of the test would depend on the similarity between the $H(z)$ data we used and those predicted by $\Lambda$CDM model. Potential inconsistencies between the predicted growth data and the observed ones are introduced, which are not due to failure of what we test but the inconsistent use of $H(z)$. In order to check the validity, we make DGP model involved in. Moreover, in Fig. 7 we compared its fitting result of $H(z)$ with the parametrization of $H(z)$ of method A (Eq. \[eq:Hz\]), and four models which we found more reasonable in the earlier paragraph. We can see that fitting result of $H(z)$ in DGP model is quite approached to the $\Lambda$CDM case. However, since DGP is base on a different principle from general relativity, the consistency relations they use wouldn’t be the same. Furthermore, according to predictions of future observation[@Weinberg2013], the relative uncertainty of these data can be reduced up to 1% to set stricter constraints. Taking these into consideration, we test $H(z)$ fitting result in DGP model with principles of general relativity and DGP model respectively, to see how much difference there would be. DGP (Dvali-Gabadadze-Poratti) model is a model of brane world where three-dimensional brane is embedded in an infinite five-dimensional spacetime (bulk). The action for this five-dimensional theory is: $$\begin{aligned} \label{eq:DGPS} S&=&\frac{1}{2}M^3_{(5)}\int d^4xdy\sqrt{-g_{(5)}}R_{(5)}\nonumber\\ &&\frac{1}{2}M^2_{(4)}\int d^4x\sqrt{-g_{(4)}}R_{(4)}+S_m,\end{aligned}$$ in which subscripts 4 represents the quantities on the brane and subscripts 5 denotes those in the bulk. $M_{(4)}$ and $M_{(5)}$ denote four and five-dimensional reduced Planck mass respectively. $S_m$ represents the action of the matter on the brane. The consistency relations based on general relativity hold on cosmological scales when dark energy has anisotropic pressure or interaction with dark matter[@Kunz; @2007]. The modification of gravity theories affects the gravitational instability. As for Eq. \[eq:defidensitydelta\], it gets the third term, the self-gravity of density perturbations modified. In scalar-tensor theories of gravity, Eq. \[eq:defidensitydelta\] is modified as: $$\begin{aligned} \label{eq:defidensitydeltaDGP} \ddot{ \delta}+2H\dot{\delta}-4{\pi}G_{eff}\rho_M{\delta} &=& 0,\end{aligned}$$ in which $G_{eff}$ denotes the effective local gravitational “constant”, which is time dependent, measured by Cavendish-type experiment. In general it may be written as: $$\begin{aligned} \label{eq:defidensitydeltaDGP2} \ddot{ \delta}+2H\dot{\delta}-4{\pi}G\rho_M(1+\frac{1}{3\beta}){\delta} &=& 0,\end{aligned}$$ in which $\beta$ in general depends on time. Once we specify the modified gravity theory, it is determined. The evolution of density perturbations during time is also modified. Furthermore, the Friedmann equation is changed to: $$\begin{aligned} \label{eq:FreDGP1} H^2+\frac{K}{a^2}=\frac{8\pi G}{3}(\sqrt{\rho_m+\rho_{r_c}}+\sqrt{\rho_{r_c}})^2,\end{aligned}$$ in which $\rho_{r_c}=3/(32\pi Gr^2_c)$ and $r_c$ can be given as $r_c=M^2_{(4)}/2M^3_{(5)}$. $\beta$ of Eq. \[eq:defidensitydeltaDGP2\] can be obtained by: $$\begin{aligned} \label{eq:DGPbeta} \beta=1-2r_cH(1+\frac{\dot{H}}{3H^2}),\end{aligned}$$ If $\Omega_{r_c}=\frac{1}{4r^2_cH^2_0}$ involved in, Eq. \[eq:FreDGP1\] changes to[@Wan; @2007]: $$\begin{aligned} \label{eq:DGPHz} H^2(z)/H^2_0&=&\Omega_{k0}(1+z)^2+[\sqrt{\Omega_{r_c}}\nonumber\\ &&+\sqrt{\Omega_{r_c}+\Omega_{m0}(1+z)^3}]^2,\end{aligned}$$ and $$\begin{aligned} \label{eq:DGPHz1} \Omega_{k0}+[\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}}]^2=1,\end{aligned}$$ which can be obtained when $z=0$. Eq. \[eq:DGPHz\] and Eq. \[eq:DGPHz1\] can be used to get fitting result with observational Hubble data. We choose CRO (Eq. \[eq:Handfsigma8co1\]) to be “modified” with the “correction factor”: $(1+\frac{1}{3\beta})$, since it directly set comparison between $f\sigma_{8obs}$ and the calculated value of $f\sigma_{8}/\lambda$. As for solving Eq. \[eq:defidensitydeltaDGP2\], we use iterative computing method by the fourth-order Runge-Kutta scheme. Since we are interested in the influence from the change of testing principle, the computing process shares the same initial conditions of matter density perturbations with $\lambda$CDM model[@Hirano; @2015]. The comparison between the results of $\Lambda$CDM, “DGP in GR”(which uses $H(z)$ fitting result in DGP and testing principle of general relativity) and DGP model are illustrated with Fig. 7 and detailed analysis information is listed in Table \[table:parameterizationABC\]. What we can see is that $\Lambda$CDM and “DGP in GR” pass the test with obvious advantage while DGP model gets apparent deterioration in the result compared with “DGP in GR”. In other words, based on the same expression of $H(z)$ which provides quite similar data with $\Lambda$CDM model, the change of testing principle leads to significantly different results. Furthermore, because of the consistency between the $H(z)$ fitting result of $\Lambda$CDM and DGP model, the failure of DGP model to pass the test can’t be imputed to the inconsistent use of $H(z)$. In conclusion, our testing principle, the consistency relation is the one which dominates the result. [$Model$]{} $H(z)$ $\chi^2_{min}$ $\chi^2_{min}/d.o.f.$ Best fit parameters --------------- ------------------------------------------------------------------------------- ---------------- ----------------------- -------------------------------- $ $ $ $ $ $ $ $ $ $ $Linear$ $H^2(z)=H^2_0[\Omega_{0m}(1+z)^3+(1-\Omega_{0m})$ $18.3781 $ $0.5405 $ $H_0=71.032\pm4.4611,$ $ $ $ \times(1+z)^{3(1+w_0-w_1)}e^{3w_1z}]$ $ $ $ $ $\Omega_{0m}=0.2223\pm0.0210,$ $ $ $ $ $ $ $ $ $w_0=-1.0081\pm0.2688,$ $ $ $ $ $ $ $ $ $w_1=0.2454\pm0.1196$ $ $ $ $ $ $ $ $ $ $ $CPL$ $H^2(z)=H^2_0[\Omega_{0m}(1+z)^3+(1-\Omega_{0m})$ $18.0807 $ $0.5318 $ $H_0=72.714\pm5.6131,$ $ $ $ \times(1+z)^{3(1+w_0+w_1)}e^{3w_1(1/(z+1)-1)}]$ $ $ $ $ $\Omega_{0m}=0.1870\pm0.1390,$ $ $ $ $ $ $ $ $ $w_0=-1.1378\pm0.3693,$ $ $ $ $ $ $ $ $ $w_1=1.0845\pm0.9859$ $ $ $ $ $ $ $ $ $ $ $QUIES$ $H^2(z)=H^2_0[\Omega_{0m}(1+z)^3+(1-\Omega_{0m})$ $18.4592$ $0.5274$ $H_0=70.540\pm4.5097,$ $ $ $\times(1+z)^{3(1+w)}]$ $ $ $ $ $\Omega_{0m}=0.2529\pm0.0261,$ $ $ $ $ $ $ $ $ $w=-1.0013\pm0.2388$ $ $ $ $ $ $ $ $ $ $ $\Lambda CDM$ $H^2(z)=H^2_0[\Omega_{0m}(1+z)^3+(1-\Omega_{0m})]$ $18.4592$ $0.5128$ $H_0=70.519\pm1.8259,$ $ $ $ $ $ $ $ $ $\Omega_{0m}=0.2529\pm0.0251$ $ $ $ $ $ $ $ $ $ $ $SCDM$ $H^2(z)=H^2_0(1+z)^3$ $175.0028$ $4.7298$ $H_0=45.701\pm0.6292$ $ $ $ $ $ $ $ $ $ $ $DGP$ $H^2(z)=H^2_0[\Omega_{k0}(1+z)^2+(\sqrt{\Omega_{r_c}}$ $18.4486 $ $0.5271 $ $H_0=70.053\pm1.6963,$ $ $ $ +\sqrt{\Omega_{r_c}+\Omega_{m0}(1+z)^3}]$ $ $ $ $ $\Omega_{m0}=0.2688\pm0.0208,$ $ $ $ ps: \Omega_{k0}+(\sqrt{\Omega_{r_c}}+\sqrt{\Omega_{r_c}+\Omega_{m0}})^2=1$ $ $ $ $ $\Omega_{r_c}=0.1970\pm0.0151$ : \[table:modelcomparing\] Marginal mean and standard deviation of model parameters in $H(z)$ expressions for various models as inferred from $\chi^2/d.o.f.$. ----------------------------------------------------- ----------------------------------------------------- {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} ----------------------------------------------------- ----------------------------------------------------- [|lcc|]{} & $\chi^2/d.o.f.$ & $C_1 \& C_2$\ $$ & $\Lambda$CDM & $$\ $C=C_1+C_2z$ & $0.3671$ & $0.6320\pm 0.0564$\ $ $ & $ $ & $0.0818\pm 0.1037$\ $C=C_1+C_2z/(1+z)$ & $0.3852$ & $0.6360\pm 0.0674$\ $ $ & $ $ & $0.1164\pm 0.2020$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.3768$ & $0.6334\pm 0.0618$\ $ $ & $ $ & $-0.1003\pm 0.1468$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.3741$ & $0.6337\pm 0.0591$\ $ $ & $ $ & $-0.0836\pm 0.1171$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.2689\pm 0.0389$ & $$&$$ $$\ $\frac{\sigma_8(0)}{\delta(0)}=1.0632\pm 0.1742 $ & $$&$$ $$\\ \hline$$ & DGP in GR & $$\ $C=C_1+C_2z$ & $0.3417$ & $0.6587\pm 0.0580$\ $ $ & $ $ & $0.0905\pm 0.1064$\ $C=C_1+C_2z/(1+z)$ & $0.3587$ & $0.6603\pm 0.0686$\ $ $ & $ $ & $0.1384\pm 0.2058$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.3506$ & $0.6591\pm 0.0632$\ $ $ & $ $ & $-0.1144\pm 0.1501$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.3483$ & $0.6599\pm 0.0606$\ $ $ & $ $ & $-0.0943\pm 0.1200$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.2813\pm 0.0407$ & $$&$$ $$\ $\frac{\sigma_8(0)}{\delta(0)}=1.0464\pm 0.1717$ & $$&$$ $$\\ \hline$$ & DGP & $$\ $C=C_1+C_2z$ & $0.3417$ & $0.6588\pm 0.5799$\ $ $ & $ $ & $0.0905\pm 0.1064$\ $C=C_1+C_2z/(1+z)$ & $0.3587$ & $0.6603\pm 0.0686$\ $ $ & $ $ & $0.1384\pm 0.2058$\ $C=C_1-C_2{\rm{ln}}(1+z)$ & $0.3506$ & $0.6591\pm 0.0632$\ $ $ & $ $ & $-0.1144\pm 0.1501$\ $C=C_1-C_2{\rm{sin}}(z)$ & $0.3483$ & $0.6599\pm 0.0606$\ $ $ & $ $ & $-0.0943\pm 0.1200$\ $\Omega_{M0}\frac{\sigma_8(0)}{\delta(0)}=0.3369\pm 0.0496$ & $$&$$ $$\ $\frac{\sigma_8(0)}{\delta(0)}=1.2532\pm 0.2085$ & $$&$$ $$\ \[fig:mocking\] ----------------------------------------------------- ----------------------------------------------------- {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} {width="75.3mm" height="6cm"} ----------------------------------------------------- ----------------------------------------------------- Summary ======== In this paper, we construct consistency relations between a kinematic probe, Hubble parameter $H(z)$, and a dynamical probe, the growth rate $R(z) (f\sigma_8(z))$ deduced from density fluctuation $\delta(z)$, and test their confidence level. The consistency relation should hold if general relativity is the correct theory of gravity in the universe, and the presence of significant deviations from the consistency can be used as the signature of a breakdown of general relativity at cosmological scales. To quantify the relations between the kinematic and dynamical probes, we set up three consistency relations and test two of them in data experiment using parametric and non-parametric method. To model any departure from consistency relations, we employ two-dimensional parameterizations for a possible redshift dependence of the consistency relation to avoid making uncertain parameter $\Omega_{m0}\frac{\sigma_8(0)}{\delta(0)}$ involved in. Moreover, we propose trigonometric functions as efficient tools for parameterizations. As for both parametric method (Method A) and non-parametric method (Method B) for Hubble parameter, the theoretical results show us that in all two-dimensional parameterizations in the testing of CRO and CRI, there is no departure from the consistency relation in the $1\sigma$ region. In sum, the present observational Hubble parameter data and growth data favor reasonably that the general relativity is the correct theory of gravity on cosmological scales. Furthermore, in order to confirm the validity of our test, we introduced a model of modified gravity, DGP model in it. It favors different gravitational theories from general relativity and the consistency relations based on the function of density perturbations would be changed. The fitting result of theoretical expression of $H(z)$ in DGP model with observational Hubble data has quite similar trace with that of $\Lambda$CDM model. Moreover, we compared the testing results of $\Lambda$CDM, “DGP in GR” and DGP model with CRO, which has shown tighter constraint in former test, and its modified version with fourth order Runge-Kutta method for DGP model. Finally we got that, $\Lambda$CDM and “DGP in GR” model passes the test with high consistency, while the result of DGP model obviously degrades. That is to say, based on the same expression of $H(z)$ which provides quite similar data with $\Lambda$CDM model, the change of testing principle leads to apparently disparate results. Considering that the fitting trace of $H(z)$ data in DGP model is quite similar to $\Lambda$CDM, the failure of its testing can’t be blamed upon the inconsistency use of $H(z)$. It’s due to the change of principle. It can be seen that it is the establishing of consistency relations which dominates the results of the testing. Overall, according to the present observational Hubble data and growth rate data $f\sigma_8$, the general relativity is the correct theory of gravity on cosmological scales. It is desirable that the precise growth rate data in the high redshift region will refine the consistency relation between the kinematic and dynamical probes and we hope that our results would provide better directional information on testing general relativity. It’s indubitable that with the quality and quantity of cosmological observations improving and unremitting efforts, the ultimate truth will be more and more clear. Thanks to all predecessors’ marvelous work, of which the reverberations echo still. Thanks for all helpful comments and discussions. We appreciate the help so much with Dr. Xiao-Lei Meng, Dr. Shuo Cao and our research group for improvement of the paper. 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ArXiv

--- abstract: 'To study subregions of a turbulence velocity field, a long record of velocity data of grid turbulence is divided into smaller segments. For each segment, we calculate statistics such as the mean rate of energy dissipation and the mean energy at each scale. Their values significantly fluctuate, in lognormal distributions at least as a good approximation. Each segment is not under equilibrium between the mean rate of energy dissipation and the mean rate of energy transfer that determines the mean energy. These two rates still correlate among segments when their length exceeds the correlation length. Also between the mean rate of energy dissipation and the mean total energy, there is a correlation characterized by the Reynolds number for the whole record, implying that the large-scale flow affects each of the segments.' author: - Hideaki Mouri - Akihiro Hori - Masanori Takaoka title: Fluctuations of statistics among subregions of a turbulence velocity field --- Introduction {#s1} ============ For locally isotropic turbulence, Kolmogorov [@k41] considered that small-scale statistics are uniquely determined by the kinematic viscosity $\nu$ and the mean rate of energy dissipation $\langle \varepsilon \rangle$. The Kolmogorov velocity $u_{\rm K} = (\nu \langle \varepsilon \rangle)^{1/4}$ and the Kolmogorov length $\eta = (\nu ^3 / \langle \varepsilon \rangle)^{1/4}$ determine the statistics of velocity increment $\delta u_r = u(x+r)-u(x)$ at scale $r$ as $$\frac{\langle \delta u_r^n \rangle}{u_{\rm K}^n} = F_n \left( \frac{r}{\eta} \right) \quad \mbox{for} \ \ n=2,3,4,....$$ Here $\langle \cdot \rangle$ denotes an average over position $x$, and $F_n$ is a universal function. The universality is known to hold well. While $\langle \delta u_r^n \rangle$ at each $r$ is different in different velocity fields, $\langle \varepsilon \rangle$ and hence $u_{\rm K}^n$ and $\eta$ are accordingly different. That is, $\langle \varepsilon \rangle$ is in equilibrium with the mean rate of energy transfer that determines $\langle \delta u_r^n \rangle$. However, the universality of small-scale statistics might not be exact. To argue against the exact universality, Landau[@ll59] pointed out that the local rate of energy dissipation $\varepsilon$ fluctuates over large scales. This fluctuation is not universal and is always significant.[@po97; @c03; @mouri06] In fact, the large-scale flow or the configuration for turbulence production appears to affect some small-scale statistics.[@mouri06; @k92; @pgkz93; @ss96; @sd98; @mininni06] Obukhov[@o62] discussed that Kolmogorov’s theory[@k41] still holds in an ensemble of “pure” subregions where $\varepsilon$ is constant at a certain value. Then, the $\varepsilon$ value represents the rate of energy transfer averaged over those subregions. For the whole region, small-scale statistics reflect the large-scale flow through the large-scale fluctuation of the $\varepsilon$ value. The idea that turbulence consists of some elementary subregions is of interest even now.[@ss96] We study statistics among subregions in terms of the effect of large scales on small scales, by using a long record of velocity data obtained in grid turbulence. Experiment {#s2} ========== The experiment was done in a wind tunnel of the Meteorological Research Institute. Its test section had the size of 18, 3, and 2m in the streamwise, spanwise, and floor-normal directions. We placed a grid across the entrance to the test section. The grid consisted of two layers of uniformly spaced rods, with axes in the two layers at right angles. The cross section of the rods was $0.04 \times 0.04$m$^2$. The separation of the axes of adjacent rods was 0.20m. On the tunnel axis at 4m downstream of the grid, we simultaneously measured the streamwise ($U+u$) and spanwise ($v$) velocities. Here $U$ is the average while $u(t)$ and $v(t)$ are fluctuations as a function of time $t$. We used a hot-wire anemometer with a crossed-wire probe. The wires were made of platinum-plated tungsten, 5$\mu$m in diameter, 1.25mm in sensing length, 1mm in separation, oriented at $\pm 45^{\circ}$ to the streamwise direction, and 280$^{\circ}$C in temperature. The signal was linearized, low-pass filtered at 35kHz, and then digitally sampled at $f_s = 70$kHz. We obtained as long as $4 \times 10^8$ data. The calibration coefficient, with which the flow velocity is proportional to the anemometer signal, depends on the condition of the hot wires and thereby varied slowly in time. We determine the coefficient so as to have $U = 21.16$ms$^{-1}$ for each segment with $4 \times 10^6$ data. Within each segment, the coefficient varied by $\pm 0.4$% at most. Also varied slowly in time the flow temperature and hence the kinematic viscosity $\nu$. We adopt $\nu = 1.42 \times 10^{-5}$m$^2$s$^{-1}$ based on the mean flow temperature, 11.8$^{\circ}$C. The temperature variation, $\pm1.2$$^{\circ}$C, corresponds to the $\nu$ variation of $\pm 0.7$%. These variations are small and ignored here. Taylor’s frozen-eddy hypothesis, i.e., $x = -Ut$, is used to obtain $u(x)$ and $v(x)$ from $u(t)$ and $v(t)$. This hypothesis requires a small value of $\langle u^2 \rangle^{1/2}/U$. The value in our experiment, 0.05, is small enough. Since $u(t)$ and $v(t)$ are stationary, $u(x)$ and $v(x)$ are homogeneous, although grid turbulence decays along the streamwise direction in the wind tunnel. We are mostly interested in scales up to about the typical scale for energy-containing eddies, which is much less than the tunnel size. Over such scales, fluctuations of $u(x)$ and $v(x)$ correspond to spatial fluctuations that were actually present in the wind tunnel.[@note0] Those over the larger scales do not. They have to be interpreted as fluctuations over long timescales described in terms of large length scales.[@note1] Quantity Value ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------- $\langle \varepsilon \rangle = 15 \nu \langle (\partial _x v)^2 \rangle /2$ 7.98m$^2$s$^{-3}$ $\langle u^2 \rangle ^{1/2}$ 1.10ms$^{-1}$ $\langle v^2 \rangle ^{1/2}$ 1.06ms$^{-1}$ $u_{\rm K} = (\nu \langle \varepsilon \rangle)^{1/4} $ 0.103ms$^{-1}$ $\langle u^4 \rangle / \langle u^2 \rangle ^2$ 3.02 $\langle v^4 \rangle / \langle v^2 \rangle ^2$ 3.00 $L_u = \int^{\infty}_{0} \langle u(x+r) u(x) \rangle dr / \langle u^2 \rangle$ 17.9cm $L_v = \int^{\infty}_{0} \langle v(x+r) v(x) \rangle dr / \langle v^2 \rangle$ 4.69cm $L_{\varepsilon} = \int^{\infty}_{0} \langle \varepsilon (x+r) \varepsilon(x) - \langle \varepsilon \rangle ^2 \rangle dr / \langle \varepsilon ^2 - \langle \varepsilon \rangle ^2 \rangle$ 0.469cm $\lambda = [ 2 \langle v^2 \rangle / \langle (\partial _x v )^2 \rangle ]^{1/2}$ 0.548cm $\eta = (\nu ^3 / \langle \varepsilon \rangle )^{1/4}$ 0.0138cm Re$_{\lambda} = \langle v^2 \rangle ^{1/2} \lambda / \nu$ 409 : \[t1\] Turbulence parameters: mean energy dissipation rate $\langle \varepsilon \rangle$, rms velocity fluctuations $\langle u^2 \rangle ^{1/2}$ and $\langle v^2 \rangle ^{1/2}$, Kolmogorov velocity $u_{\rm K}$, flatness factors $\langle u^4 \rangle / \langle u^2 \rangle ^2$ and $\langle v^4 \rangle / \langle v^2 \rangle ^2$, correlation lengths $L_u$, $L_v$, and $L_{\varepsilon}$, Taylor microscale $\lambda$, Kolmogorov length $\eta$, and microscale Reynolds number Re$_{\lambda}$. Turbulence parameters are listed in Table \[t1\]. Here and hereafter, $\langle \cdot \rangle$ is used to denote an average over the whole record. The derivative was obtained as $\partial _x v = [ 8 v(x+ \delta x)- 8 v(x- \delta x)-v(x+ 2 \delta x)+v(x- 2 \delta x)]/ 12 \delta x$ with $\delta x = U/f_s$. The local rate of energy dissipation was obtained as $\varepsilon = 15 \nu (\partial _x v)^2 /2$ instead of usual $15 \nu (\partial _x u)^2$, in order to avoid possible spurious correlations with $\delta u_r$ over small $r$ for analyses in the next section. Figure \[f1\] shows $\langle \delta u_r^2 \rangle / u_{\rm K}^2$, the $u$, $v$, and $\varepsilon$ correlations, and also the correlation lengths $L_u$, $L_v$, and $L_{\varepsilon}$. We see the inertial range, albeit narrow, where $\langle \delta u_r^2 \rangle$ roughly scales with $r^{2/3}$. The $u$ and $v$ correlations are significant up to $r \simeq 10^4 \eta$, which corresponds to the scale of largest eddies. The correlation length $L_u$ corresponds to the typical scale for energy-containing eddies. Since $\varepsilon$ belongs to small scales, its correlation decays quickly. Results and discussion {#s3} ====================== The data record is now divided into segments with length $R$. They correspond to subregions considered by Obukhov.[@o62] For each segment, we have statistics such as $$\begin{aligned} &&(\partial _x v)^2_R(x) = \frac{1}{R} \int ^{x+R/2}_{x-R/2} \left[ \frac{\partial v(x')}{\partial x'} \right]^2 dx', \nonumber \\ &&\delta u_{r,R}^n(x) = \frac{1}{R-r} \int ^{x+R/2-r}_{x-R/2} \delta u_r^n(x') dx', \\ &&v(x+r)v(x)_R = \frac{1}{R-r} \int ^{x+R/2-r}_{x-R/2} v(x'+r)v(x') dx'. \nonumber\end{aligned}$$ Here $x$ is the center of the segment, and $r < R$. The mean rate of energy dissipation is $\varepsilon_R = 15 \nu (\partial _x v)^2_R /2$, which yields the Kolmogorov velocity $u_{{\rm K},R} = (\nu \varepsilon _R)^{1/4}$ and the Kolmogorov length $\eta _R =(\nu ^3 / \varepsilon _R)^{1/4}$. We also have the mean total energy, $v^2_R = v(x+r)v(x)_R$ for $r=0$, and the microscale Reynolds number, Re$_{\lambda,R} = 2^{1/2} v^2_R / \nu [(\partial _x v)^2_R]^{1/2}$. The mean rate of energy transfer, however, is not available from our experimental data. Fig. \[f1\](a) shows an example of $\delta u_{r,R}^2/u_{{\rm K},R}^2$, which differs from $\langle \delta u_r^2 \rangle / u_{\rm K}^2$. Distribution of fluctuation {#s31} --------------------------- Over a range of $R$, we study statistics of $\delta u_{r,R}^2/u_{{\rm K},R}^2$ among segments. The scale $r$ is fixed at $10\eta _R$ in the dissipation range and $100\eta _R$ in the inertial range. Since $\delta u_r$ is available only at discrete scales $r$ that are multiples of the sampling interval $U/f_s$, $\delta u_{10\eta _R,R}^2$ and $\delta u_{100\eta _R,R}^2$ are obtained through interpolation by incorporating the fluctuation of $\eta_R$ among segments. Figure \[f2\](a) shows the standard deviation. The fluctuation of $\delta u_{r,R}^2/u_{{\rm K},R}^2$ at a fixed $r/\eta_R$ is significant even when $R$ is large. In individual segments, Kolmogorov’s theory[@k41] does not hold. The mean rate of energy transfer that determines $\delta u_{r,R}^2$ is not in equilibrium with the mean rate of energy dissipation $\varepsilon_R$ that determines $u_{{\rm K},R}^2$ and $\eta_R$. The degree of this nonequilibrium fluctuates among segments and thereby induces the observed fluctuation of $\delta u_{r,R}^2/u_{{\rm K},R}^2$. The mean rate of energy transfer also fluctuates among scales in each segment, which does not necessarily have the inertial-range scaling $\delta u_{r,R}^2 \propto r^{2/3}$ \[Fig. \[f1\](a)\].[@note2] If each segment had this scaling, the fluctuation of $\delta u_{r,R}^2/u_{{\rm K},R}^2$ among segments at a fixed $r/\eta_R$ in the inertial range would correspond to the fluctuation of the Kolmogorov constant $\delta u_{r,R}^2/(r \varepsilon_R)^{2/3}$. Figures \[f2\](b) and \[f2\](c) show the skewness and flatness factors of $\ln (\delta u _{r,R}^2 /u_{{\rm K},R}^2)$ (open triangles). They are close to the Gaussian values of 0 and 3. Thus, at least as a good approximation, the distribution of $\delta u _{r,R}^2 /u_{{\rm K},R}^2$ is lognormal. This is also the case in $\varepsilon _R$ (filled circles) and at $R \gtrsim 10^3 \eta \simeq L_u$ in $v^2_R$ and Re$_{\lambda,R}$ (filled squares and diamonds), while the mean rate of energy transfer should not have a lognormal distribution because it changes its sign. Examples of the probability density functions are shown in Fig. \[f3\]. The lognormal distribution of $\varepsilon_R$ was discussed as a tentative model by Obukhov.[@o62] A lognormal distribution stems from some multiplicative stochastic process, e.g., a product of many independent stochastic variables with similar variances. To its logarithm, if not too far from the average, the central limit theorem applies. For the lognormal distributions observed here, the process is related with the energy transfer. While the mean energy transfer is to a smaller scale and is significant between scales in the inertial range alone, the local energy transfer is either to a smaller or larger scale and is significant between all scales.[@mouri06; @mininni06; @ok92] Any scale is thereby affected by itself and by many other scales. They involve large scales because the lognormal distributions are observed up to large $R$. There is no dominant effect from a few specific scales, in order for the central limit theorem to be applicable. At $R \gtrsim 10^5 \eta \simeq 10^2 L_u$, there are alternative features.[@mouri06; @kg02] The standard deviations scale with $R^{-1/2}$ \[Fig. \[f2\](a)\]. The skewness and flatness factors of $\delta u _{r,R}^2 /u_{{\rm K},R}^2$, $\varepsilon _R$, $v^2_R$, and Re$_{\lambda,R}$ are close to the Gaussian values \[Figs. \[f2\](b) and \[f2\](c): filled triangles, open circles, open squares, and open diamonds; see also Fig. \[f3\]\]. Their distributions are regarded as Gaussian rather than lognormal, although this has to be confirmed in future using high-order moments or probability densities at the tails for the larger number of segments. The $R^{-1/2}$ scaling and Gaussian distribution are typical of fluctuations in thermodynamics and statistical mechanics,[@ll79] for which no correlation is significant at scales of interest as in our case at $r \gtrsim 10^5 \eta$ \[Fig. \[f1\](b)\]. Correlation between fluctuations {#s32} -------------------------------- The fluctuations among segments at $R \gtrsim L_u$ have interesting correlations. Since these correlations are weak, they are extracted by following Obukhov,[@o62] i.e., by averaging over segments with similar $\varepsilon _R$. Specifically, we use conditional averages, denoted by $\langle \cdot \rangle _{\varepsilon}$, for ranges of $\varepsilon _R$ separated at $\langle \varepsilon \rangle /4$, $\langle \varepsilon \rangle /2$, $\langle \varepsilon \rangle$, $2\langle \varepsilon \rangle$, and $4 \langle \varepsilon \rangle$. Figure \[f4\] shows $\langle \delta u_{r,R}^2/u_{{\rm K},R}^2 \rangle _{\varepsilon}$ as a function of $r/\eta_R$. When $R = 10^3 \eta \simeq L_u$ \[Fig. \[f4\](b)\], $\langle \delta u_{r,R}^2/u_{{\rm K},R}^2 \rangle _{\varepsilon}$ is independent of $\langle \varepsilon _R \rangle _{\varepsilon}$. The former varies only by a factor of 1.5 while the latter varies by a factor of 20. When $R = 10^2 \eta \simeq 10^{-1} L_u$ \[Fig. \[f4\](a)\], $\langle \delta u_{r,R}^2/u_{{\rm K},R}^2 \rangle _{\varepsilon}$ is not independent of $\langle \varepsilon _R \rangle _{\varepsilon}$. The implication of the above result is that the mean rate of energy transfer that determines $\delta u_{r,R}^2$ correlates with the mean rate of energy dissipation $\varepsilon_R$ that determines $u_{{\rm K},R}^2$ and $\eta_R$ among segments with $R \gtrsim L_u$. Here $L_u$ is the typical scale for energy-containing eddies. Most of the energy of such an eddy is transferred through scales and dissipated within its own volume. Thus, each energy-containing eddy tends toward equilibrium between the mean rates of energy transfer and dissipation. This tendency does not exist at $R \lesssim L_u$.[@note3] Within an energy-containing eddy, the spatial distribution of $\varepsilon$ is not homogeneous. In fact, the $\varepsilon$ correlation is significant at $r \lesssim L_u$ \[Fig. \[f1\](b)\]. Therefore, in order for statistics such as $\delta u_{r,R}^2/u_{{\rm K},R}^2$ and $\langle \delta u_{r,R}^2/u_{{\rm K},R}^2 \rangle _{\varepsilon}$ to have physical meanings, the minimum segment length is about $L_u$. We are to discuss that such segments individually reflect the large-scale flow. Figure \[f5\](a) shows $\langle v^2_R \rangle _{\varepsilon} / \langle v^2 \rangle$ as a function of $\langle \varepsilon _R \rangle _{\varepsilon} / \langle \varepsilon \rangle$. The former varies with the latter. When $R \ge 10^3 \eta \simeq L_u$, the microscale Reynolds number $\propto \langle v^2_R \rangle _{\varepsilon} / \langle \varepsilon _R \rangle ^{1/2}_{\varepsilon}$ is almost constant at the value for the whole record, Re$_{\lambda} = 409$. Thus, segments obey a correlation between $v^2_R$ and $\varepsilon_R$ characterized by the Re$_{\lambda}$ value. Since Re$_{\lambda}$ is determined by the large-scale flow, it follows that the large-scale flow affects each of the segments. A consistent result is obtained for $\langle \mbox{Re}_{\lambda,R} \rangle _{\varepsilon}$ in Fig. \[f5\](b). The above tendency toward a constant microscale Reynolds number originates in energy-containing scales. In general, through an empirical relation $\langle \varepsilon \rangle \propto \langle v^2 \rangle ^{3/2} /L_v$, Re$_{\lambda}$ is related to the Reynolds number for energy-containing scales as ${\rm Re}_{\lambda} \propto (\langle v^2 \rangle ^{1/2} L_v / \nu)^{1/2}$. Then, Re$_{\lambda}$ is constant if $L_v \propto \langle v^2 \rangle^{-1/2}$. Fig. \[f6\] shows $\langle v(x+r)v(x)_R \rangle _{\varepsilon} / \langle v^2_R \rangle _{\varepsilon}$, which extends to larger scales for smaller $\langle v_R^2 \rangle _{\varepsilon}$. The correlation length $\int^{\infty}_{0} \langle v(x+r) v(x)_R \rangle _{\varepsilon} dr / \langle v^2_R \rangle _{\varepsilon}$ should be accordingly larger. The process for each segment to reflect the large-scale flow could be related to the energy transfer. As noted before, the energy transfer couples all scales. The energy transfer itself is affected by the large-scale flow. This is because energy is transferred between two scales via an interaction with some other scale. When the interaction occurs with a large scale, the energy transfer is strong.[@mininni06; @ok92] Unimportance of $\varepsilon$ fluctuation {#s33} ----------------------------------------- Obukhov[@o62] discussed that, through the fluctuation of the $\varepsilon_R$ value, the large-scale flow affects small-scale statistics for the whole region such as $\langle \delta u_r^n \rangle / u_{\rm K}^n$. The reason is that $\langle \delta u_r^n \rangle$ is obtained at a fixed $r$, regardless of the fluctuations of $u_{{\rm K},R}^n$ and $\eta_R$ induced by the fluctuation of $\varepsilon_R$. We are to discuss that the fluctuation of the $\varepsilon_R$ value is not important so far as $R \gtrsim L_u$. This condition on $R$ is required for the $\varepsilon_R$ value to correlate with and thus statistically represent the mean rate of energy transfer that determines $\delta u_{r,R}^n$. Figure \[f7\] compares $\langle \delta u_r^n \rangle/u_{\rm K}^n$ to $\langle \delta u_{r,R}^n/u_{{\rm K},R}^n \rangle$, i.e., average of $\delta u_{r,R}^n/u_{{\rm K},R}^n$ at each $r/\eta_R$ over all segments, with $R = 10^3 \eta \simeq L_u$. Even for $n = 6$ \[Fig. \[f7\](b)\], they are not distinguishable \[see also Fig. \[f4\](b)\]. Although $\varepsilon _R$ and hence $u_{K,R}$ and $\eta_R$ fluctuate among segments, these fluctuations are not large enough for $\langle \delta u_r^n \rangle/u_{\rm K}^n$ to differ from $\langle \delta u_{r,R}^n/u_{{\rm K},R}^n \rangle$. This conclusion is general. In various flows,[@c03; @po97; @mouri06] including an atmospheric boundary layer at Re$_{\lambda} \simeq 9000$, the standard deviation of $\varepsilon _R/\langle \varepsilon_R \rangle$ at $R \simeq L_u$ is close to the value obtained here \[Fig. \[f2\](a)\]. Hence, through the fluctuation of the $\varepsilon_R$ value, the large-scale flow does not affect small-scale statistics for the whole record. It was suggested that the large-scale flow does affect $\langle \delta u_r^n \rangle/u_{\rm K}^n$, even in the scaling exponents.[@k92; @sd98; @mininni06] If this is the case, the effect is already inherent in the individual segments. They are unlikely to be “pure[@o62]” or elementary. Concluding remarks {#s4} ================== Using segments of a long record of velocity data obtained in grid turbulence, we have studied fluctuations of statistics such as $\delta u_{r,R}^2/u_{{\rm K},R}^2$, $\varepsilon_R$, and $v_R^2$. The fluctuations are significant and have lognormal distributions at least as a good approximation (Figs. \[f2\] and \[f3\]). In each segment, the mean rate of energy transfer that determines $\delta u_{r,R}^2$ is not in equilibrium with the mean rate of energy dissipation $\varepsilon_R$ that determines $u_{{\rm K},R}^2$ and $\eta_R$. These two rates still correlate among segments with $R \gtrsim L_u$ (Fig. \[f4\]), which tend toward equilibrium between the two rates. Also between $\varepsilon_R$ and $v_R^2$, there is a correlation characterized by Re$_{\lambda}$ for the whole record (Fig. \[f5\]). Thus, the large-scale flow affects each of the segments. The observed fluctuations depend on $L_u$ and Re$_{\lambda}$, which in turn depend on the configuration for turbulence production, e.g., boundaries such as the grid used in our experiment. Nevertheless, the significance of those fluctuations implies that they have been developed in turbulence itself. Their lognormal distributions are explained by a multiplicative stochastic process in turbulence, which is related with the energy transfer among scales. The correlations among the fluctuations are also explained in terms of the energy transfer. Previous studies[@mouri06; @k92; @pgkz93; @ss96; @sd98; @mininni06] suggested that the large-scale flow affects some small-scale statistics, although this has to be confirmed at higher Re$_{\lambda}$ where large and small scales are more distinct.[@ab06] If the effect really exists, it is inherent individually in the segments. Our study was motivated by Obukhov’s discussion.[@o62] It implies the presence of equilibrium between the mean rates of energy transfer and dissipation in an ensemble of segments with similar values of $\varepsilon_R$. This is the case at $R \gtrsim L_u$ (Fig. \[f4\]).[@note4] Also as discussed by Obukhov, the distribution of $\varepsilon_R$ is lognormal at least as a good approximation (Figs. \[f2\] and \[f3\]). However, although Obukhov discussed that the large-scale flow affects small-scale statistics through the fluctuation of the $\varepsilon_R$ value, this is not the case (Fig. \[f7\]). The lognormal distributions observed here have to be distinguished from those proposed by Kolmogorov.[@k62] While he was interested in small-scale intermittency and studied $\varepsilon_r$ and $\delta u_r^n$ at small $r$ to obtain their scaling laws, we are interested in large-scale fluctuations and have studied $\varepsilon_R$ and $\delta u_{r,R}^n$ at small $r$ but at large $R$. The scaling laws of $\varepsilon_R$ and $\delta u_{r,R}^n$ are not necessary. In addition, the lognormality has been attributed to a different process. Hence, our study is not necessarily concerned with the well-known problems of Kolmogorov’s lognormal model, e.g., violation of Novikov’s inequality[@n71] for scaling exponents. Still exists a possibility that small-scale intermittency is affected by large-scale fluctuations. The study of this possibility is desirable. There were no studies of statistics among segments with large $R$. Hence, we have focused on grid turbulence, which is simple and thus serves as a standard. For flows other than grid turbulence, the fluctuations of statistics among segments are expected to be significant as well. In fact, regardless of the flow configuration and the Reynolds number, the large-scale fluctuation of $\varepsilon_R$ is significant.[@po97; @c03; @mouri06] Those fluctuations are also expected to have lognormal distributions and mutual correlations as observed here because they are due to the energy transfer in turbulence itself. However, grid turbulence is free from shear. It was previously found that $\delta u_r$ correlates with $u$ in shear flows such as a boundary layer but not in shear-free flows.[@pgkz93; @ss96; @sd98] The fluctuations of statistics among segments might be somewhat different in a shear flow. To this and other flow configurations, it is desirable to apply our approach. We are grateful to K. R. Sreenivasan for inspiring this study and for helpful comments and also to M. Tanahashi for helpful comments. [999]{} A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” Dokl. Akad. Nauk SSSR [**30,**]{} 301 (1941). L. D. Landau and E. M. Lifshitz, [*Fluid Mechanics*]{} (Pergamon, London, 1959), Chap. 3. A. Praskovsky and S. Oncley, “Comprehensive measurements of the intermittency exponent in high Reynolds number turbulent flows,” Fluid Dyn. Res. [**21,**]{} 331 (1997). J. Cleve, M. Greiner, and K. R. Sreenivasan, “On the effects of surrogacy of energy dissipation in determining the intermittency exponent in fully developed turbulence,” Europhys. Lett. [**61,**]{} 756 (2003). H. Mouri, M. Takaoka, A. Hori, and Y. Kawashima, “On Landau’s prediction for large-scale fluctuation of turbulence energy dissipation,” Phys. Fluids [**18,**]{} 015103 (2006). V. R. Kuznetsov, A. A. Praskovsky, and V. A. Sabelnikov, “Fine-scale turbulence structure of intermittent shear flows,” J. Fluid Mech. [**243,**]{} 595 (1992). A. A. Praskovsky, E. B. Gledzer, M. Y. Karyakin, and Y. Zhou, “The sweeping decorrelation hypothesis and energy-inertial scale interaction in high Reynolds number flows,” J. Fluid Mech. [**248,**]{} 493 (1993). K. R. Sreenivasan and G. Stolovitzky, “Statistical dependence of inertial range properties on large scales in a high-Reynolds-number shear flow,” Phys. Rev. Lett. [**77,**]{} 2218 (1996). K. R. Sreenivasan and B. Dhruva, “Is there scaling in high-Reynolds-number turbulence?” Prog. Theor. Phys. Suppl. [**130,**]{} 103 (1998). P. D. Mininni, A. Alexakis, and A. Pouquet, “Large-scale flow effects, energy transfer, and self-similarity on turbulence,” Phys. Rev. E [**74,**]{} 016303 (2006). A. M. Obukhov, “Some specific features of atmospheric turbulence,” J. Fluid Mech. [**13,**]{} 77 (1962). If we had obtained snapshots of the velocity field in the wind tunnel, we should have observed various eddies. Their strengths should have been random but on average decayed along the streamwise direction. Over scales along the streamwise direction, statistics of velocity fluctuations should have suffered from the decay, which is not relevant to our study. Over scales along the spanwise direction, so far as they were not too large, statistics of velocity fluctuations should have been identical to those for $u(x)$ and $v(x)$. The alternative interpretation is that such fluctuations correspond to spatial fluctuations of some virtual turbulence. Local values of $u(x)$ and $v(x)$ represent local regions of actual turbulence, and they are continuously connected up to largest scales. Throughout the scales, $u(x)$ and $v(x)$ obtained here are consistent with those measured at a certain time in spatially homogeneous but temporally decaying turbulence (see Appendix of Ref. ). The reader might consider that $\delta u_{r,R}^2 \propto r^{2/3}$ is absent in some segments because their Re$_{\lambda,R}$ values are not high enough. This is not the case. For example, the segment shown in Fig. \[f1\](a) has Re$_{\lambda,R} \simeq 1000$. K. Ohkitani and S. Kida, “Triad interactions in a forced turbulence,” Phys. Fluids A [**4,**]{} 794 (1992). K. Kajita and T. Gotoh, ”Statistics of the energy dissipation rate in turbulence,” in [*Statistical Theories and Computational Approaches to Turbulence*]{}, edited by Y. Kaneda and T. Gotoh (Springer, Tokyo, 2003), p. 260. L. D. Landau and E. M. Lifshitz, [*Statistical Physics,*]{} 3rd ed. (Pergamon, Oxford, 1979), Part 1, Chap. 12. For such a segment, statistics do not strictly have local isotropy, and hence $\varepsilon_R = 15 \nu (\partial _x v)^2_R /2$ is not strictly exact. The associated uncertainty is nevertheless small and not serious to our study. In fact, among segments with $R = 10^2 \eta$ and $10^3 \eta$, respectively, the correlation coefficients are as high as 0.89 and 0.95 between $15 \nu (\partial _x v)^2_R /2$ and $15 \nu [(\partial _x u)^2_R+(\partial _x v)^2_R/2]/2$. The value of the latter tends to be more close to the true $\varepsilon_R$ value (see Ref. ), although it is not used in our study because of possible spurious correlations with the values of $\delta u_r$ over small $r$. R. A. Antonia and P. Burattini, “Approach to the 4/5 law in homogeneous isotropic turbulence,” J. Fluid Mech. [**550,**]{} 175 (2006). To examine Obukhov’s discussion in a strict manner, we have to use $\langle \delta u_{r,R}^2 \rangle _{\varepsilon}/(\nu \langle \varepsilon_R \rangle_{\varepsilon})^{1/2}$ instead of $\langle \delta u_{r,R}^2/u_{{\rm K},R}^2 \rangle _{\varepsilon}$. Even in this case, the result is almost the same as in Fig. \[f4\]. An analogous situation is seen in Fig. \[f5\], where the Reynolds number obtained from $\langle v_R^2 \rangle _{\varepsilon}$ and $\langle \varepsilon_R \rangle_{\varepsilon}$ is almost the same as $\langle \mbox{Re}_{\lambda,R} \rangle _{\varepsilon}$. A. N. Kolmogorov, “A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number,” J. Fluid Mech. [**13,**]{} 82 (1962). E. A. Novikov, “Intermittency and scale similarity in the structure of a turbulent flow,” J. Appl. Math. Mech. [**35,**]{} 231 (1971).

ArXiv

--- abstract: 'We construct a new class of positive indecomposable maps in the algebra of $d \times d$ complex matrices. These maps are characterized by the ‘weakest’ positivity property and for this reason they are called atomic. This class provides a new reach family of atomic entanglement witnesses which define important tool for investigating quantum entanglement. It turns out that they are able to detect states with the ‘weakest’ quantum entanglement.' author: - | Dariusz Chruściński and Andrzej Kossakowski\ Institute of Physics, Nicolaus Copernicus University,\ Grudziadzka 5/7, 87–100 Toruń, Poland title: '**A class of positive atomic maps**' --- Introduction ============ One of the most important problems of quantum information theory [@QIT] is the characterization of mixed states of composed quantum systems. In particular it is of primary importance to test whether a given quantum state exhibits quantum correlation, i.e. whether it is separable or entangled. For low dimensional systems there exists simple necessary and sufficient condition for separability. The celebrated Peres-Horodecki criterium [@Peres; @PPT] states that a state of a bipartite system living in $\mathbb{C}^2 {{\,\otimes\,}}\mathbb{C}^2$ or $\mathbb{C}^2 {{\,\otimes\,}}\mathbb{C}^3$ is separable iff its partial transpose is positive. Unfortunately, for higher-dimensional systems there is no single universal separability condition. The most general approach to separability problem is based on the following observation [@Horodeccy-PM]: a state $\rho$ of a bipartite system living in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is separable iff $\mbox{Tr}(W\rho) \geq 0$ for any Hermitian operator $W$ satisfying $\mbox{Tr}(W P_A {{\,\otimes\,}}P_B)\geq 0$, where $P_A$ and $P_B$ are projectors acting on $\mathcal{H}_A$ and $\mathcal{H}_B$, respectively. Recall, that a Hermitian operator $W \in \mathcal{B}(\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B)$ is an entanglement witness [@Horodeccy-PM; @Terhal1] iff: i) it is not positively defined, i.e. $W \ngeq 0$, and ii) $\mbox{Tr}(W\sigma) \geq 0$ for all separable states $\sigma$. A bipartite state $\rho$ living in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$ is entangled iff there exists an entanglement witness $W$ detecting $\rho$, i.e. such that $\mbox{Tr}(W\rho)<0$. Clearly, the construction of entanglement witnesses is a hard task. It is easy to construct $W$ which is not positive, i.e. has at leat one negative eigenvalue, but it is very difficult to check that $\mbox{Tr}(W\sigma) \geq 0$ for all separable states $\sigma$. The separability problem may be equivalently formulated in terms positive maps [@Horodeccy-PM]: a state $\rho$ is separable iff $(\oper {{\,\otimes\,}}\Lambda)\rho$ is positive for any positive map $\Lambda$ which sends positive operators on $\mathcal{H}_B$ into positive operators on $\mathcal{H}_A$. Due to the celebrated Choi-Jamio[ł]{}kowski [@Jam; @Choi1] isomorphism there is a one to one correspondence between entanglement witnesses and positive maps which are not completely positive: if $\Lambda$ is such a map, then $W_\Lambda:=(\oper {{\,\otimes\,}}\Lambda)P^+$ is the corresponding entanglement witness ($P^+$ stands for the projector onto the maximally entangled state in $\mathcal{H}_A {{\,\otimes\,}}\mathcal{H}_B$). Unfortunately, in spite of the considerable effort, the structure of positive maps (and hence also the set of entanglement witnesses) is rather poorly understood \[7–44\]. Now, among positive linear maps the crucial role is played by indecomposable maps. These are maps which may detect entangled PPT states. Among indecomposable maps there is a set of maps which are characterized by the ‘weakest positivity’ property: they are called [*atomic maps*]{} and they may be used to detect states with the ‘weakest’ entanglement. The corresponding entanglement witnesses we call indecomposable and atomic, respectively. There are only few examples of indecomposable maps in the literature (for the list see e.g. the recent paper [@OSID-W]). The set of atomic ones is considerably smaller. Interestingly, Choi first example [@Choi1] of indecomposable positive map turned out to be an atomic one. Recently, Hall [@Hall] and Breuer [@Breuer] considered a new family of indecomposable maps (they were applied by Breuer [@Breuer-bis] in the study of rotationally invariant bipartite states, see also [@Remigiusz]). In this paper we show that these maps are not only indecomposable but also atomic. Moreover, we show how to generalize this family to obtain a large family of new positive maps. We study which maps within this family are indecomposable and which are atomic. The paper is organized as follows: in the next Section we introduce a natural hierarchy of positive convex cones in the space of (unnormalized) states of bipartite $d {{\,\otimes\,}}d$ quantum systems and recall basis notions from the theory of entanglement witnesses and positive maps. Section \[SEC-BH\] discusses properties of the recently introduced indecomposable maps [@Hall; @Breuer] and provides the proof that these maps are atomic. Finally, Section \[NEW\] introduces a new class of indecomposable maps and studies which maps within this class are atomic. A brief discussion is included in the last section. Quantum entanglement vs. positive maps ====================================== Let $M_d$ denote a set of $d \times d$ complex matrices and let $M_d^+$ be a convex set of semi-positive elements in $M_d$, that is, $M_d^+$ defines a space of (unnormalized) states of $d$-level quantum system. For any $\rho \in (M_d {{\,\otimes\,}}M_d)^+$ denote by $\mathrm{SN}(\rho)$ a Schmidt number of $\rho$ [@SN]. This notion enables one to introduce the following family of positive cones: $$\label{} \mathrm{V}_r = \{\, \rho \in (M_d {{\,\otimes\,}}M_d)^+\ |\ \mathrm{SN}(\rho) \leq r\, \} \ .$$ One has the following chain of inclusions $$\label{V-k} \mathrm{V}_1 \subset \ldots \subset \mathrm{V}_d \equiv (M_d {{\,\otimes\,}}M_d)^+\ .$$ Clearly, $\mathrm{V}_1$ is a cone of separable (unnormalized) states and $V_d \smallsetminus V_1$ stands for a set of entangled states. Note, that a partial transposition $(\oper_d {{\,\otimes\,}}\tau)$ gives rise to another family of cones: $$\label{} \mathrm{V}^l = (\oper_d {{\,\otimes\,}}\tau)\mathrm{V}_l \ ,$$ such that $ \mathrm{V}^1 \subset \ldots \subset \mathrm{V}^d$. One has $\mathrm{V}_1 = \mathrm{V}^1$, together with the following hierarchy of inclusions: $$\label{} \mathrm{V}_1 = \mathrm{V}_1 \cap \mathrm{V}^1 \subset \mathrm{V}_2 \cap \mathrm{V}^2 \subset \ldots \subset \mathrm{V}_d \cap \mathrm{V}^d \ .$$ Note, that $\mathrm{V}_d \cap \mathrm{V}^d$ is a convex set of PPT (unnormalized) states. Finally, $\mathrm{V}_r \cap \mathrm{V}^s$ is a convex subset of PPT states $\rho$ such that $\mathrm{SN}(\rho) \leq r$ and $\mathrm{SN}[(\oper_d {{\,\otimes\,}}\tau)\rho] \leq s$. Consider now a set of positive maps $\varphi : M_d \longrightarrow M_d$, i.e. maps such that $\varphi(M_d^+) \subseteq M_d^+$. Following St[ø]{}rmer definition [@Stormer1], a positive map $\varphi$ is $k$-positive iff $$\label{} (\oper {{\,\otimes\,}}\varphi)(\mathrm{V}_k) \subset (M_d {{\,\otimes\,}}M_d)^+\ ,$$ and it is $k$-copositive iff $$\label{} (\oper {{\,\otimes\,}}\varphi)(\mathrm{V}^k) \subset (M_d {{\,\otimes\,}}M_d)^+\ .$$ Denoting by $\mathrm{P}_k$ ($\mathrm{P}^k$) a convex cone of $k$-positive ($k$-copositive) maps one has the following chains of inclusions $$\label{P-k} \mathrm{P}_d \subset \mathrm{P}_{d-1} \subset \ldots \subset \mathrm{P}_2 \subset \mathrm{P}_1 \ ,$$ and $$\label{P=k} \mathrm{P}^d \subset \mathrm{P}^{d-1} \subset \ldots \subset \mathrm{P}^2 \subset \mathrm{P}^1 \ ,$$ where $\mathrm{P}_d$ ($\mathrm{P}^d$) stands for a set of completely positive (copositive) maps. A positive map $\varphi : M_d \longrightarrow M_d$ is [*decomposable*]{} iff $\varphi \in P_d + P^d$, that is, $\varphi$ can be written as $\varphi = \varphi_1 + \varphi_2$, with $\varphi_1 \in P_d$ and $\varphi_2 \in P^d$. Otherwise $\varphi$ is [*indecomposable*]{}. Indecomposable maps can detect entangled state from $V_d \cap V^d \equiv $ PPT, that is, bound entangled states. Finally, a positive map is [*atomic*]{} iff $\varphi \notin \mathrm{P}_2 + \mathrm{P}^2$. The importance of atomic maps follows from the fact that they may be used to detect the ‘weakest’ bound entanglement, that is, atomic maps can detect states from $V_2 \cap V^2$. Actually, St[ø]{}rmer definition [@Stormer1] is rather difficult to apply in practice. Using the Choi-Jamio[ł]{}kowski isomorphism [@Jam; @Choi1] we may assign to any linear map $\varphi : M_d \rightarrow M_d$ the following operator $\widehat{\varphi} \in M_d {{\,\otimes\,}}M_d$: $$\label{} \widehat{\varphi} = (\oper_d {{\,\otimes\,}}\varphi) P^+ \in M_d {{\,\otimes\,}}M_d\ ,$$ where $P^+$ stands for (unnormalized) maximally entangled state in $\Cd {{\,\otimes\,}}\Cd$. If $e_i$ $(i=1,\ldots,d)$ is an orthonormal basis in $\Cd$, then $$\label{J} \widehat{\varphi} = \sum_{i,j=1}^d e_{ij} {{\,\otimes\,}}\varphi(e_{ij})\ ,$$ where $e_{ij} = |i\>\<j|$ defines a basis in $M_d$. It is clear that if $\varphi$ is a positive but not completely positive map then the corresponding operator $\widehat{\varphi}$ is an entanglement witness. Now, the space of linear maps $\mathcal{L}(M_d,M_d)$ is endowed with a natural inner product: $$\label{} (\varphi,\psi) = \mathrm{Tr} \Big( \sum_{\alpha=1}^{d^2}\, \varphi(f_\alpha)^* \psi(f_\alpha) \Big)\ ,$$ where $f_\alpha$ is an arbitrary orthonormal basis in $M_d$. Taking $f_\alpha = e_{ij}$, one finds $$\begin{aligned} \label{} (\varphi,\psi) &=& \mathrm{Tr} \Big( \sum_{i,j=1}^{d}\, \varphi(e_{ij})^* \psi(e_{ij}) \Big) \ = \ \mathrm{Tr} \Big( \sum_{i,j=1}^{d}\, \varphi(e_{ij})\psi(e_{ji}) \Big)\ .\end{aligned}$$ The above defined inner product is compatible with the standard Hilbert-Schmidt product in $M_d {{\,\otimes\,}}M_d$. Indeed, taking $\widehat{\varphi}$ and $\widehat{\psi}$ corresponding to $\varphi$ and $\psi$, one has $$\label{} (\widehat{\varphi},\widehat{\psi})_{\rm HS} = \mathrm{Tr} ({\widehat{\varphi}}^{\,*}\widehat{\psi})$$ and using (\[J\]) one easily finds $$\label{} (\varphi,\psi) = (\widehat{\varphi},\widehat{\psi})_{\rm HS}\ ,$$ that is, formula (\[J\]) defines an inner product isomorphism. This way one establishes the duality between maps from $\mathcal{L}(M_d,M_d)$ and operators from $M_d {{\,\otimes\,}}M_d$ [@Eom]: for any $\rho \in M_d {{\,\otimes\,}}M_d$ and $\varphi \in \mathcal{L}(M_d,M_d)$ one defines $$\label{DUAL} \< \rho, \varphi\> := (\rho,\widehat{\varphi})_{\rm HS} \ .$$ In the space of entanglement witnesses $\mathbf{W}$ one may introduce the following family of subsets $\mathbf{W}_r \subset M_d {{\,\otimes\,}}M_d$: $$\label{} \mathbf{W}_r = \{\, W\in M_d {{\,\otimes\,}}M_d\ |\ \mathrm{Tr}(W\rho) \geq 0\ , \ \rho \in \mathrm{V}_r\, \}\ .$$ One has $$\label{} (M_d {{\,\otimes\,}}M_d)^+ \equiv \mathbf{W}_d \subset \ldots \subset \mathbf{W}_1 \ .$$ Clearly, $\mathbf{W} = \mathbf{W}_1 \smallsetminus \mathbf{W}_d$. Moreover, for any $k>l$, entanglement witnesses from $\mathbf{W}_l \smallsetminus \mathbf{W}_k$ can detect entangled states from $\mathrm{V}_k \smallsetminus V_l$, i.e. states $\rho$ with Schmidt number $l < \mathrm{SN}(\rho) \leq k$. In particular $W \in \mathbf{W}_k \smallsetminus \mathbf{W}_{k+1}$ can detect state $\rho$ with $\mathrm{SN}(\rho)=k$. Consider now the following class $$\label{} \mathbf{W}_r^s = \mathbf{W}_r + (\oper {{\,\otimes\,}}\tau)\mathbf{W}_s\ ,$$ that is, $W \in \mathbf{W}_r^s$ iff $$\label{} W = P + (\oper {{\,\otimes\,}}\tau)Q\ ,$$ with $P \in \mathbf{W}_r$ and $Q \in \mathbf{W}_s$. Note, that $\mathrm{Tr}(W\rho) \geq 0$ for all $\rho \in \mathrm{V}_r \cap \mathrm{V}^s$. Hence such $W$ can detect PPT states $\rho$ such that $\mathrm{SN}(\rho) \geq r$ or $\mathrm{SN}[(\oper_d {{\,\otimes\,}}\tau)\rho] \geq s$. Entanglement witnesses from $\mathbf{W}_d^d$ are called decomposable [@optimal]. They cannot detect PPT states. One has the following chain of inclusions: $$\label{} \mathbf{W}_d^d\, \subset\, \ldots\, \subset\, \mathbf{W}^2_2\, \subset\, \mathbf{W}^1_1\, \equiv\, \mathbf{W}\ .$$ The ‘weakest’ entanglement can be detected by elements from $\mathbf{W}_1^1 \smallsetminus \mathbf{W}_2^2$. We shall call them [*atomic entanglement witnesses*]{}. It is clear that $W$ is an atomic entanglement witness if there is an entangled state $\rho \in \mathrm{V}_2 \cap \mathrm{V}^2$ such that $\mathrm{Tr}(W \rho) <0$. The knowledge of atomic witnesses, or equivalently atomic maps, is crucial: knowing this set we would be able to distinguish all entangled states from separable ones. A class of atomic maps of Breuer and Hall {#SEC-BH} ========================================= Recently Breuer and Hall [@Breuer; @Hall] analyzed the following class of positive maps $\varphi : M_d \longrightarrow M_d$ $$\label{B-H} \varphi^d_U(X) = \mbox{Tr}(X)\, \mathbb{I}_d - X - UX^TU^* \ ,$$ where $U$ is an antisymmetric unitary matrix in $\mathbb{C}^d$ which implies that $d$ is necessarily even and $d\geq 4$ (for $d=2$ the above map is trivial $\varphi^d_U(X)=0$). One may easily add a normalization factor such that $$\label{BH-norm} \widetilde{\varphi}^d_U = \frac{1}{d-2}\ \varphi^d_U\ ,$$ is unital, that is, $\widetilde{\varphi}^d_U(\mathbb{I}_d)=\mathbb{I}_d$. The characteristic feature of these maps is that for any rank one projector $P$ its image under $\varphi^d_U$ reads as follows $$\label{} \varphi^d_U(P) = \mathbb{I}_d - P - Q \ ,$$ where $Q$ is again rank one projector satisfying $PQ=0$. Hence $\varphi^d_U(P)\geq 0$ which proves positivity of $\varphi^d_U$. It was shown [@Breuer; @Hall] that these maps are not only positive but also indecomposable. Interestingly, maps considered by Breuer and Hall are closely related to a positive map introduced long ago by Robertson [@Robertson1]–[@Robertson4]. The Robertson map $\varphi_R : M_4 \longrightarrow M_4$ is defined as follows $$\label{} \varphi_R\left( \begin{array}{c|c} X_{11} & X_{12} \\ \hline X_{21} & X_{22} \end{array} \right) = \frac 12 \left( \begin{array}{c|c} \mathbb{I}_2\, \mbox{Tr} X_{22} & X_{12} + R(X_{21}) \\ \hline X_{21} + R(X_{12}) & \mathbb{I}_2\, \mbox{Tr} X_{11} \end{array} \right) \ ,$$ where $X_{kl} \in M_2$ and $R : M_2 \longrightarrow M_2$ is defined by $$\label{} R(a) = \mathbb{I}_2\, \mbox{Tr}a - a \ ,$$ that is, $R$ is nothing but the reduction map. Introducing an orthonormal basis $(e_1,\ldots,e_4)$ in $\mathbb{C}^4$ and defining $e_{ij} = |e_i\>\<e_j|$, one easily finds the following formulae: $$\begin{aligned} \label{} \varphi_R(e_{11}) &=& \varphi_4(e_{22}) = \frac 12 ( e_{33} + e_{44}) \ , \nonumber \\ \varphi_R(e_{33}) &=& \varphi_4(e_{44}) = \frac 12 ( e_{11} + e_{22}) \ , \nonumber \\ \varphi_R(e_{13}) &=& \frac 12 (e_{13} + e_{42})\ , \nonumber \\ \varphi_R(e_{14}) &=& \frac 12 (e_{14} - e_{32})\ , \\ \varphi_R(e_{23}) &=& \frac 12 (e_{23} - e_{41})\ , \nonumber \\ \varphi_R(e_{24}) &=& \frac 12 (e_{24} + e_{31})\ , \nonumber \\ \varphi_R(e_{12}) &=& \varphi_R(e_{34})\ =\ 0 \ . \nonumber\end{aligned}$$ Note, that the Robertson map is unital, i.e. $\varphi_R(\mathbb{I}_4)=\mathbb{I}_4$. \[BH-R\] The normalized Breuer-Hall map $\widetilde{\varphi}^4_U$ in $d=4$ is unitary equivalent to the Robertson map $\varphi_R$, that is $$\label{} \widetilde{\varphi}^4_U(X) = U_1 \varphi_R(U_2^*XU_2)U_1^*\ ,$$ for some unitaries $U_1$ and $U_2$. [**Proof**]{}: Let us observe that $$\label{} \Gamma \varphi_R(X) \Gamma^* = \widetilde{\varphi}^4_0(X) \ ,$$ where $\Gamma$ is the following $4\times 4$ unitary matrix $$\label{} \Gamma = \left( \begin{array}{c|c} \mathbb{I}_2 & 0 \\ \hline 0 & - \mathbb{I}_2 \end{array} \right) \ ,$$ and $\widetilde{\varphi}^4_0$ is a normalized Breuer-Hall map (\[B-H\]) corresponding to $4 \times 4$ antisymmetric unitary diagonal matrix[^1] $$\label{} U_0 = i\, \mathbb{I}_2 {{\,\otimes\,}}\sigma_2 \ .$$ Now, any antisymmetric unitary matrix $U$ may be represented as $$\label{UV} U = VU_0V^T\ ,$$ for some orthogonal matrix $V$. It shows that a general Breuer-Hall map $\varphi^4_U$ is unitary equivalent to $\varphi^4_0$ $$\label{} \varphi^4_U(X) = V\varphi^4_0(V^TXV)V^T\ ,$$ and hence (after normalization) to the Robertson map $$\label{} \widetilde{\varphi}^4_U(X) = (V\Gamma)\varphi_R(V^TXV)(V\Gamma)^T\ ,$$ with $U_1 = V\Gamma$ and $U_2 = V$. $\Box$ Note, that for $V = \mathbb{I}_4$, one obtains $$\begin{aligned} \label{} \widetilde{\varphi}^4_{\mathbb{I}}(e_{ii}) &=& \varphi_R(e_{ii}) \ ,\\ \widetilde{\varphi}^4_{\mathbb{I}}(e_{ij}) &=& - \varphi_R(e_{ij}) \ , \ \ \ i \neq j \ ,\end{aligned}$$ It was already shown by Robertson [@Robertson3] that $\varphi_R$ is indecomposable. However, it turns out that one may prove the following much stronger property: Robertson map $\varphi_R$ is atomic. [**Proof**]{}: to prove atomicity of $\varphi_R$ one has to construct a PPT state $\rho \in (M_4 {{\,\otimes\,}}M_4)^+$ such that: 1) both $\rho$ and its partial transpose $\rho^\tau$ are of Schmidt rank two, and 2) entanglement of $\rho$ is detected by the corresponding entanglement witness $$W_R = (\oper {{\,\otimes\,}}\varphi_R)P^+_4 = \sum_{i,j=1}^4 e_{ij} {{\,\otimes\,}}\varphi_R(e_{ij})\ .$$ One easily finds $$\label{} W_R = \frac 12 \left( \begin{array}{cccc|cccc|cccc|cccc} \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& 1\\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot\\ \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& 1 \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ 1& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\ \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ 1& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \end{array} \right)\ ,$$ where to maintain more transparent form we replace all zeros by dots. Note, that $W_R$ has single negative eigenvalue ‘$-1$’, ‘$0$’ (with multiplicity 10) and ‘$+1$’ (with multiplicity 5). Consider now the following state constructed by Ha [@Ha1]: $$\label{Ha} \rho_{\rm Ha}\ =\ \frac 17\ \left( \begin{array}{cccc|cccc|cccc|cccc} 1 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& \cdot& \cdot & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& 1 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \end{array} \right)\ .$$ It turns out [@Ha1] that $\rho_{\rm Ha}$ is PPT, and both $\rho_{\rm Ha}$ and $(\oper {{\,\otimes\,}}\tau)\rho_{\rm Ha}$ have Schmidt rank 2. One easily finds $$\label{Ha-R} \mbox{Tr}(W_R \rho_{\rm Ha}) = -1/14<0\ ,$$ which proves atomicity of $\varphi_R$.[^2] $\Box$ The Breuer-Hall map $\varphi^4_U$ is atomic. [**Proof**]{}: using the relation between $\varphi^4_U$ and the Roberston map $\varphi_R$ $$\label{} \varphi^4_U(X) = U_1 \varphi_R(U_2^* XU_2)U_1^* \ ,$$ let us compute $\mbox{Tr}(\rho W_U)$, where $$\label{} W^4_U = (\oper {{\,\otimes\,}}\varphi^4_U)P^+_4\ ,$$ and $\rho$ is an arbitrary state in $4 {{\,\otimes\,}}4$. One obtains $$\begin{aligned} \label{} \mbox{Tr}(\rho W^4_U) = \frac 14\, \mbox{Tr}\left(\rho\cdot \sum_{i,j=1}^4 e_{ij} {{\,\otimes\,}}\varphi^4_U(e_{ij}) \right) = \frac 14\, \mbox{Tr}\left(\rho \cdot\sum_{i,j=1}^4 e_{ij} {{\,\otimes\,}}U_1\varphi_R(U_2^* e_{ij} U_2 )U_1^* \right)\ .\end{aligned}$$ Now, introducing $\widetilde{e}_i = U_2^* e_i$, one has $$\begin{aligned} \label{} \mbox{Tr}(\rho W_U) &=& \frac 14\, \mbox{Tr}\left(\rho \cdot\sum_{i,j=1}^4 U_2 \widetilde{e}_{ij} U_2^* {{\,\otimes\,}}U_1\varphi_R(\widetilde{e}_{ij})U_1^* \right) \nonumber\\ & =& \mbox{Tr} \Big(\rho \cdot (U_2 {{\,\otimes\,}}U_1)(\oper {{\,\otimes\,}}\varphi_R)P^+_4 (U_2 {{\,\otimes\,}}U_1)^* \Big) \nonumber\\ &=& \mbox{Tr}\Big( (U_2 {{\,\otimes\,}}U_1)^* \rho (U_2 {{\,\otimes\,}}U_1) \cdot W_R\Big) \ .\end{aligned}$$ Hence, if $\rho_{\rm Ha}$ witnesses atomiticity of $\varphi_R$, then $(U_2 {{\,\otimes\,}}U_1) \rho_{\rm Ha} (U_2 {{\,\otimes\,}}U_1)^*$ witnesses atomiticity of $\varphi^4_U$. $\Box$ The above result may be immediately generalized as follows If a positive map $\varphi : \mathcal{B}(\mathcal{H}_1) \longrightarrow \mathcal{B}(\mathcal{H}_2)$ is atomic, then $\widetilde{\varphi} : \mathcal{B}(\mathcal{H}_1) \longrightarrow \mathcal{B}(\mathcal{H}_2)$ defined by $$\label{} \widetilde{\varphi}(X) := U_1 \varphi(U_2^* XU_2)U_1^* \ ,$$ is atomic for arbitrary unitary operators $U_1$ and $U_2$ ($U_k : \mathcal{H}_k \longrightarrow \mathcal{H}_k$; $k=1,2$). \[TH-BHA\] The Breuer-Hall map $\varphi^d_U : M_d \longrightarrow M_d$ with even $d$ is atomic. [**Proof**]{}: let $\Sigma$ be a 4-dimensional subspace in $\mathbb{C}^d$. It is clear that $U_\Sigma := U|_\Sigma$ gives rise to the Breuer-Hall map in 4 dimensions $$\varphi_{U_\Sigma}^4 : \mathcal{B}(\Sigma) \longrightarrow \mathcal{B}(U(\Sigma))\ .$$ This map is atomic and hence it is witnessing by a $4 \times 4$ density matrix supported on $\Sigma$, such that $\rho$ is PPT, Schmidt rank of $\rho$ and its partial transposition equals 2, and such that $\mbox{Tr}(\rho W_{U_\Sigma}^4) < 0$. Let us extend the $4 \times 4$ state $\rho$ into the following $d {{\,\otimes\,}}d$ state: $$\label{} \widehat{\rho}_{ij,kl} = \left\{ \begin{array}{cc} \rho_{ij,kl} \ , & \ \ \ i,j,k,l \leq 4 \\ 0 & \ \ \ {\rm otherwise} \end{array} \ , \right.$$ where we take a basis $(e_1,\ldots,e_d)$ such that $e_1,\ldots,e_4 \in \Sigma$. It is clear that extended $\widehat{\rho}$ is PPT in $d {{\,\otimes\,}}d$ and Schmidt rank of $\widehat{\rho}$ and $(\oper {{\,\otimes\,}}\tau)\widehat{\rho}$ equals again 2. Moreover $$\label{} \mbox{Tr}( \widehat{\rho} W^d_U) = \mbox{Tr}( \rho W_{U_\Sigma}^4) < 0 \ ,$$ which proves atomicity of $\varphi^d_U$. $\Box$ Let us observe that $d$ needs not be even. Indeed, let $d \geq 4$ and let $U$ be antisymmetric unitary operator $U : \Sigma \longrightarrow \Sigma$, where $\Sigma$ denotes an arbitrary even-dimensional subspace of $\mathbb{C}^d$. One extends $U$ to an operator $\widehat{U}$ in $\mathbb{C}^d$ by $$\label{} \widehat{U}(x,y) = (U x,0) \ ,$$ where $x\in \Sigma$ and $y\in \Sigma^\perp$, and hence, $\widehat{U}$ is still antisymmetric but no longer unitary in $\mathbb{C}^d$. Finally, let us define $$\label{BH-gen} \varphi_{\widehat{U}}^d(X) = \mbox{Tr}(X)\, \mathbb{I}_d - X - \widehat{U}X^T\widehat{U}^* \ ,$$ that is, it acts as the standard Breuer-Hall map on $\mathcal{B}(\Sigma)$ only. Note, that $$\label{} \varphi_{\widehat{U}}^d(\mathbb{I}_d) = (d-2)\mathbb{I}_d + P^\perp\ ,$$ where $P^\perp$ denotes a projector onto $\Sigma^\perp$. Therefore, the normalized map reads as follows $$\label{} \widetilde{\varphi}_{\widehat{U}}^d(X) = [(d-2)\mathbb{I}_d + P^\perp]^{-1/2} \cdot \varphi_{\widehat{U}}^d(X) \cdot [(d-2)\mathbb{I}_d + P^\perp]^{-1/2}\ ,$$ and has much more complicated form than (\[BH-norm\]). \[BH-arbitrary\] The formula (\[BH-gen\]) with arbitrary $d\geq 4$ and even dimensional subspace $\Sigma$ $($with ${\rm dim}\,\Sigma \geq 4)$ defines a positive atomic map. [**Proof**]{}: let $d > {\rm dim}\,\Sigma =2k \geq 4 $. It is clear that $$\label{} \varphi^{2k}_U := \varphi_{\widehat{U}}^d\Big|_{\mathcal{B}(\Sigma)}\ ,$$ defines the standard Breuer-Hall map in $\mathcal{B}(\Sigma)$. Now, due to Theorem \[TH-BHA\] the map $\varphi^{2k}_U$ is atomic. If $\rho$ is a $2k {{\,\otimes\,}}2k$ state living in $\Sigma {{\,\otimes\,}}\Sigma$ witnessing atomicity of $\varphi^{2k}_U$, then trivially extended $\widehat{\rho}\,$ in $\,\mathbb{C}^d {{\,\otimes\,}}\mathbb{C}^d$ witnesses atomicity of $\varphi_{\widehat{U}}^d$. $\Box$ New classes of atomic maps {#NEW} ========================== Now we are ready to propose the generalization of the class of positive maps considered by Hall [@Hall]: $$\label{H1} \varphi(X) = \sum_{k<l} \sum_{m<n} \, c_{kl,mn}\, A_{kl} \, X^T\, A_{mn}^*\ ,$$ where $$\label{} A_{kl} = e_{kl} - e_{lk} \ ,$$ with $c_{kl,mn}$ being a $d \times d$ Hermitian matrix. One example of such a map is a Breuer-Hall one $$\label{} \varphi^d_U (X) = \mathbb{I}_d\, \mbox{Tr}\, X - X - UX^TU^*\ ,$$ which is shown to be atomic. Moreover, the well know reduction map $$\label{} R(X) = \mathbb{I}_d\, \mbox{Tr}\, X - X\ ,$$ belongs to (\[H1\]). This map is completely co-positive and hence decomposable. Finally, denote by $\varepsilon$ the following map $$\label{} \varepsilon(X) = \mathbb{I}_d\, \mbox{Tr}\, X \ ,$$ which is completely positive and does not belong to (\[H1\]). Now, let us introduce the new class which is defined by the following convex combination: $$\label{} \phi^U_x(X) = x\varphi^d_U(X) + (1-x)R(X) = \mathbb{I}_d\, \mbox{Tr}\, X - X - x UX^TU^*\ .$$ It is clear that for $x \in [0,1]$ the above formula defines a positive map from the class (\[H1\]). Note, that if $\mbox{rank}\, U = 2k<d$, then the matrix $[c_{kl,mn}]$ possesses a negative eigenvalue ‘$1-xk$’ for $x$ satisfying $$\label{} \frac 1k < x \leq 1\ ,$$ and hence $\phi^U_x(X)$ is indecomposable. Similarly, a family $$\label{} \psi_y(X) = y\varepsilon(X) + (1-y)R(X) = \mathbb{I}_d\, \mbox{Tr}\, X - yX\ ,$$ define for $y\in [0,1]$ decomposable maps from (\[H1\]). Finally, consider $$\label{chi} \chi^U_{x,y}(X) = y\phi^U_x(X) + (1-y)\psi_y(X) = \mathbb{I}_d\, \mbox{Tr}\, X - yX - x UX^TU^*\ .$$ Now, we are going to establish the range of $(x,y) \in [0,1] \times [0,1]$ for which $\chi^U_{x,y}$ is atomic. \[7/4\] A positive map $\chi^U_{x,y}$ is atomic if $ x+y > 7/4$. [**Proof:**]{} let us start with $d=4$ and $\Sigma = \mathbb{C}^4$ and consider $\chi^U_{x,y}$ with $U=\mathbb{I}$: $$\label{chi} \chi^\mathbb{I}_{x,y}(X) = \mathbb{I}_4\, \mbox{Tr}\, X - yX - x X^T \ .$$ Let $W^\mathbb{I}_{x,y}$ be the corresponding entanglement witness: $$\begin{aligned} \label{} \lefteqn{ W^\mathbb{I}_{x,y} \ = \ (\oper {{\,\otimes\,}}\chi^U_{x,y})P^+_4 \ = \ } && \\ && \frac 12 \left( \begin{array}{cccc|cccc|cccc|cccc} 1-x& \cdot& \cdot& \cdot& \cdot& y-x& \cdot& \cdot& \cdot& \cdot& -x& \cdot& \cdot& \cdot& \cdot& -x\\ \cdot& 1-y& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& y& \cdot& \cdot\\ \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& -y& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& 1-y& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ y-x& \cdot& \cdot& \cdot& \cdot& 1-x& \cdot& \cdot& \cdot& \cdot& -x& \cdot& \cdot& \cdot& \cdot& -x \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& -y& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& y& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& y& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& -y& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ -x& \cdot& \cdot& \cdot& \cdot& -x& \cdot& \cdot& \cdot& \cdot& 1-x& \cdot& \cdot& \cdot& \cdot& y-x \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1-y& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -y& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\ \cdot& \cdot& y& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1-y& \cdot \\ -x& \cdot& \cdot& \cdot& \cdot& -x& \cdot& \cdot& \cdot& \cdot& y-x& \cdot& \cdot& \cdot& \cdot& 1-x \end{array} \right)\ , \nonumber\end{aligned}$$ It is easy to show that $$\label{} \mbox{Tr}((\Gamma\rho_{\rm Ha}\Gamma^*) W^\mathbb{I}_{x,y}) = \frac{1}{7}\, (7 - 4x-4y) \ ,$$ where $\rho_{\rm Ha}$ is defined in (\[Ha\]). Hence, if $7-4(x+y) < 0$, then $\chi^\mathbb{I}_{x,y}$ is atomic. Now, it is clear from the proofs of Theorems \[BH-R\] and \[BH-arbitrary\] that the same result applies for arbitrary $d$ and arbitrary $U$. $\Box$ Similarly, we may find a region in $(x,y)$ square where $\chi^U_{x,y}$ is indecomposable. One has A positive map $\chi^U_{x,y}$ is indecomposable if $ x+y > 3/2$. [**Proof:**]{} similarly, as in the proof of the previous theorem, one computes $$\label{3/2} \mbox{Tr}((\Gamma\rho_{\rm new}\Gamma^*) W^\mathbb{I}_{x,y}) = \frac{1}{24}\, (24 - 16x-16y) \ ,$$ where $\rho_{\rm new}$ is defined by $$\label{} \rho_{\rm new} = \frac{1}{24} \left( \begin{array}{cccc|cccc|cccc|cccc} 2& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& -1\\ \cdot& 2& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot\\ \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& -1 \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ -1& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2 & \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot& \cdot \\ \cdot& \cdot& 1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2& \cdot \\ -1& \cdot& \cdot& \cdot& \cdot& -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 2 \end{array} \right)\ ,$$ and turns out to be PPT.[^3] It is therefore clear that for $x + y > 3/2$, that map $\chi^\mathbb{I}_{x,y}$ is indecomposable. Using the same techniques as in the proof of Theorem \[7/4\] we prove that $x + y > 3/2$ guaranties indecomposability for arbitrary $d$ and $U$. $\Box$ The regions of indecomposability $(x + y > 3/2)$ and of atomicity $(x+y> 7/4$) are displayed on Figure 1. We stress that these regions are derived by using specific states: $\rho_{\rm new}$ and $\rho_{\rm Ha}$, respectively. It is interesting to look for other states which are ‘more optimal’ and enable us to enlarge these regions. Note, that the same analysis applies for the maps defined by $$\label{H0} \varphi(X) = \sum_{k<l} \sum_{m<n} \, c_{kl,mn}\, A_{kl} \, X\, A_{mn}^*\ .$$ Conclusions =========== We provided a new large class of positive atomic maps in the matrix algebra $M_d$. These maps generalize a class of maps discussed recently by Breuer [@Breuer] and Hall [@Hall]. The importance of atomic maps follows from the fact that they may be used to detect the ‘weakest’ bound entanglement, that is, atomic maps can detect entangled states from $V_2 \cap V^2$. By duality, these maps provide new class of atomic entangled witnesses. Note, that if $\varphi$ is atomic and $(\oper {{\,\otimes\,}}\varphi)\rho \ngeq 0$, then $\rho \in V_2 \cap V^2$ and hence $\rho$ may be used as a test for atomicity of positive indecomposable maps. Since we know only few examples of quantum states belonging to $V_2 \cap V^2$ any new example of this kind is welcome. It is hoped that new maps provided in this paper find applications in the study of ‘weakly’ entangled PPT states. For example in recent papers [@PPT-nasza] and [@CIRCULANT] we constructed very general classes of PPT states in $d {{\,\otimes\,}}d$. It would be interesting to search for entangled states within these classes by applying our new family of indecomposable and atomic maps. 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[^1]: Actually, $U_0$ may be multiplied by a unitary block-diagonal matrix $$U_0 \longrightarrow U_\Lambda = \left( \begin{array}{c|c} e^{i\lambda_1}\mathbb{I}_2 & 0 \\ \hline 0 & e^{i\lambda_2}\mathbb{I}_2 \end{array} \right) \cdot U_0 \ ,$$ but the arbitrary phases $\lambda_1$ and $\lambda_2$ do not enter the game. [^2]: Note, that $\rho_{\rm Ha}$ is trivially extended from the following state in $3 {{\,\otimes\,}}3$: $$\label{} \frac 17\ \left( \begin{array}{ccc|ccc|ccc} 1 & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& -1& \cdot\\ \cdot& 1 &\cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot\\ \cdot& \cdot& \cdot & \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& 1 & \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1 & \cdot& \cdot \\ \hline \cdot& \cdot& \cdot& \cdot& \cdot& 1& 1& \cdot& \cdot \\ -1& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1& \cdot \\ \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& \cdot& 1 \end{array} \right)\ ,$$ which, therefore, provides an example of a bound entangled state. [^3]: Actually, we originally constructed $\rho_{\rm new}$ to ‘beat’ (\[Ha-R\]). One finds $$\label{} \mbox{Tr}(W_R \rho_{\rm new}) = -1/6\ ,$$ which is ‘much better’ that $-1/14$. We conjecture, that $\rho_{\rm new}$ is ‘optimal’ in the following sense: $$\label{} \min_{\rho\in {\rm PPT}} \mbox{Tr}(W_R \rho) = -1/6\ ,$$ that is, $\rho_{\rm new}$ minimizes $\mbox{Tr}(W_R \rho)$ among all PPT states.

ArXiv

--- abstract: '[We show that an analogue of the Ball-Box Theorem holds true for a class of corank 1, non-differentiable tangent subbundles that satisfy a geometric condition. In the final section of the paper we give examples of such bundles and an application to dynamical systems.]{}' author: - Sina Türeli title: 'The Ball-Box Theorem for a Class of Non-differentiable Tangent Subbundles ' --- Introduction ============ Sub-Riemannian geometry is a generalization of Riemannian geometry which is motivated by very physical and concrete problems. It is the language for formalizing questions like: “Can we connect two thermodynamical states by adiabatic paths?"[@Car09], “Can a robot with a certain set of movement rules reach everywhere in a factory?"[@Agr04], “Can a business man evade tax by following the rules that were set to avoid tax evasion?", “By adjusting the current we give to a neural system, can we change the initial phase of the system to any other phase we want?"[@Li10]. However one drawback of current sub-Riemannian geometry literature is that it almost exclusively focuses on the study of “smooth systems, which is sometimes too much to ask for a mathematical subject that has close connections with physical sciences. For instance, one place where non-differentiable objects appear in a physically motivated mathematical branch (and which is the main motivation of the authors) is the area of dynamical systems. More specifically in (partially and uniformly) hyperbolic dynamics, bundles that are only Hölder continuous are quite abundant and their sub-Riemannian properties (i.e. accessibility and integrability) play an important role in the description and classification of the dynamics. The aim of this paper is to give a little nudge to sub-Riemannian geometry in the direction of non-differentiable objects. To get into more technical details we need some definitions. Let $\Delta$ be a $C^r$ tangent subbundle defined on a smooth manifold $M$ and $g$ a metric on $\Delta$ (the triple $(M,\Delta,g)$ is called a $C^r$ sub-Riemannian manifold). We will always assume that $\Delta$ is corank $1$ and $dim(M)=n+1$ with $n \geq 2$. A piecewise $C^1$ path $\gamma$ is said to be ***admissible*** if it is a.e everywhere tangent to $\Delta$ (i.e. $\dot{\gamma}(t) \in \Delta_{\gamma(t)}$ for $t$ a.e). We let $\mathcal{C}_{pq}$ denote the set of length parameterized (i.e. $g(\dot{\gamma}(t),\dot{\gamma}(t))=1$ for $t$ a.e) admissible paths between $p$ and $q$. If $\mathcal{C}_{pq} \neq \emptyset$ for all $p,q \in U\subset M$ then $\Delta$ is said to be ***controllable*** or ***accessible*** on $U$. For smooth bundles, the ***Chow-Rashevskii Theorem*** says that if $\Delta$ is everywhere completely non-integrable (i.e. if the smallest Lie algebra $Lie(\Gamma(\Delta))$ generated by smooth sections of $\Delta$ is the whole tangent space at every point), then it is accessible [@Agr04]. In particular if $\Gamma(\Delta)(p) + [\Gamma(\Delta),\Gamma(\Delta)](p) =T_pM$ then such a bundle is called called step 2, completely non-integrable at $p$ [^1]. These are the $C^1$ analogues of the non-differentiable bundles that we will be interested in this paper. We denote $$d_{\Delta}(p,q) = \inf_{\gamma \in\mathcal{C}_{pq}}\ell(\gamma),$$ where $\ell(\cdot)$ denotes length with respect to the given metric $g$ on $\Delta$. $d_{\Delta}$ is called the ***sub-Riemannian metric***. We let $B_{\Delta}(p,\epsilon)$ denote the ball of radius $\epsilon$ around $p$ with respect to the metric $d_{\Delta}$. Given $p \in M$ and a coordinate neighbourhood $U$, coordinates $z=(x^1,\dots,x^n,y)$ are called ***adapted*** for $\Delta$ if, $p=0$, $\Delta_0 = \text{span}<\frac{\partial}{\partial x^i}|_0>_{i=1}^n$ and $\Delta_q$ is transverse to $\mathcal{Y}_q = \text{span}<\frac{\partial}{\partial y}|_q>$ for every $q \in U$. Given such coordinates and $\alpha>0$, we define the coordinate weighted box as: $$B_{\alpha}(0,\epsilon)=\{z=(x,y) \in U \quad | \quad |x^i| \leq \epsilon, \quad |y| \leq \epsilon^{\alpha}\},$$ where $|\cdot|$ denotes the Euclidean norm with respect to the given adapted coordinates. A specialization of the fundamental the ***Ball-Box Theorem*** [@Bel96; @Mon02], says that if $\Delta$ is a smooth, step 2, completely non-integrable bundle at a point $p$, then, given any smooth adapted coordinate system defined on some small enough neighborhood of $p$, there exists constants $K_1, K_2, \epsilon_0 >0$ such that for all $\epsilon < \epsilon_0$, $$B_2(0,K_1\epsilon) \subset B_{\Delta}(0,\epsilon) \subset B_2(0, K_2\epsilon).$$ Note that here the lower inclusion $B_2(0,K_1\epsilon) \subset B_{\Delta}(0,\epsilon)$ implies accessibility around $p$. This specialized case is also known to hold true for $C^1$ bundles, see [@Gro96]. There are several works ( [@Kar11], [@MonMor12], [@Sim10]) that try to generalize the Ball-Box Theorem to the setting of less regular bundles. Each has their own set of assumptions about the regularity or geometric properties of the bundle. In [@Sim10], they generalize the Ball-Box Theorem to Hölder continuous bundles following a proof for $C^1$ bundles given in [@Gro96]. The extra assumption is that the bundle $\Delta$ is accessible to start with. Under this assumption, they prove that for the case of accessible, $\theta-$Hölder, codimension 1 bundle, the inclusion “$B_{\Delta}(0,\epsilon) \subset B_{\alpha}(0, K_2\epsilon)$” holds true with $\alpha=1+\theta$ (this inclusion translates as a certain “lower bound" in the way they choose to express his results in [@Sim10]). Although this result does not assume any regularity beyond being Hölder (which is the weakest regularity assumption in the works we compare), what is lacking is that there is no criterion for accessibility and without the lower inclusion one has no qualitative information about the shape or the volume of the sub-Riemannian ball. In [@MonMor12] they prove the full Ball-Box Theorem under certain Lipschitz continuity assumptions for commutators of the vector-fields involved. In particular working with a collection of vector-fields $\{X_i\}_{i=1}^n$, they say that these vector-fields are completely non-integrable of step $s$ if their Lie derivatives $X_I=[X_{i_1},[\dots,[X_{i_{s-1}},X_{s}]]\cdots]$ up to $s$ iterations are defined and Lipschitz continuous and these $\{X_I\}_{I \in \mathcal{I}}$ span the whole tangent space at a point. And then under these assumptions (and some more less significant technical assumptions) they obtain the usual Ball-Box Theorem for step $s$, completely non-integrable collection of vector-fields. In [@Kar11], they consider $C^{1+\alpha}$ bundles with $\alpha>0$. In this case the bundle itself is differentiable and therefore the Lie derivatives $[X_i,X_j]$ are defined although only Hölder continuous. Therefore many of the tools of classical theory such as Baker-Hausdorff-Campbell formula are not applicable and extension of the theory already becomes non-trivial. Although the case of step 2, completely non-integrable $C^1$ bundles is already dealt with in [@Gro96], the cited paper considers the general case and does not only study the Ball-Box theorem but several other important theorems from sub-Riemannian geometry. This paper makes progress toward extending the Ball-Box Theorem to continuous bundles. We establish an analogue of the Ball-Box Theorem (of the step 2 case) and therefore Chow-Rashevskii Theorem for a certain class of non-integrable, continuous, corank 1 bundles that satisfy a geometric condition (explained in the next section). In particular studying continuous bundles allows us to analyze what are the important features for giving volume and shape to a sub-Riemannian ball. In the end we can conclude that certain geometric features are sufficient for obtaining lower bounds on the volume while regularity also plays an important role for the shape. The authors believe that the methods presented in this paper can become useful for answering these questions in more generality and this is discussed in section \[sec-generalizations\]. After the proof of the main theorems, we give examples of bundles that satisfy these geometric properties and yet are not differentiable (nor Hölder), we present an application to dynamical systems and we also study some interesting properties of this class of bundles and pose some questions related to possible generalizations including measurable bundles (measurable in terms of space variables, not just the time variables which is already completely covered by classical control theory). After the examples we also compare our results with the other results discussed. But we can before hand say that all the results are somehow related to each other but are not completely covered by neither (see in particular the discussion following Proposition \[prop-example\]). *Acknowledgments:* The author is greatly thankful to the anonymous referee for a lot of improvements, in particular several corrections and useful remarks regarding the distinction between step 2, completely non-integrable and other cases in corank 1 case. The author was supported by ERC AdG grant no: 339523 RGDD. All the figures were created using *Apache OpenOffice Draw*. Statement of the Theorems ------------------------- We assume that $\Delta$ is a corank 1, continuous tangent subbundle defined on a $n+1$ dimensional smooth manifold $M$ with a given metric $g$ on $\Delta$. Except for some certain general definitions, we will carry out the proof in coordinate neighborhoods. The domain of the coordinate will be possibly a smaller Euclidean box where we work and all the supremums and infumums of the functions defined on this coordinate neighbourhood will be over this domain. Henceforth we denote the domain of any chosen coordinate system by $U$. By $|\cdot|$, we denote the Euclidean norm given by the coordinates and we identify the tangent spaces with $\mathbb{R}^{n+1}$. We denote by $\mathcal{A}^n_0(\Delta)(U)$ the space of continuous differential $n$-forms over $U$ that annihilate $\Delta$, which is seen as a module over $C(U)$. We let $\Omega^n_{r}(U)$ denote the space of $C^r$ differential n-forms over $U$, again as a module over $C^r(U)$. With this notation $\mathcal{A}^n_0(\Delta)(U)$ is a submodule of $\Omega^n_{0}(U)$. Given a submodule $\mathcal{E}\subset \Omega^n_{r}(U)$, a local basis for this submodule on $U$ is a finite collection of elements from $\mathcal{E}$, which are linearly independent over $C^r(U)$ and which span $\mathcal{E}$ over $C^r(U)$. We use the induced norm on these spaces coming from the Euclidean norm to endow them with a Banach space structure. We use $|\cdot|_{\infty}$ and $|\cdot|_{\inf}$ denote the supremum and infimum of the norms of an object over the domain $U$ we are working with. More precisely if $\alpha$ is some $k-$differential form, $|\alpha|_{\infty}=\sup_{q \in U}|\alpha_q|$ and $|\alpha|_{\inf}=\inf_{q \in U}|\alpha_q|$ where generally a subscripted point denotes evaluation at that point. If $Y_1,\dots,Y_k$ are vector-fields then $\alpha(Y_1,\dots,Y_k)$ denotes the function obtained by contracting $\alpha$ with the given vector-fields. Therefore $|\alpha(Y_1,\dots,Y_k)|_{\infty}$ and $|\alpha(Y_1,\dots,Y_k)|_{\inf}$ denotes the supremum and infimum over $U$ of the absolute value of this function. Occasionally when the need arises, we might make the distinction of evaluation points by the notation $\alpha_q(Y_1(q),\dots,Y_k(q))$ or even by $\alpha_q(Y_1(p),\dots,Y_k(p))$ when we work in coordinates and identify tangent spaces with $\mathbb{R}^{n+1}$. Finally given a sub-bundle $\Delta$ of $TU$, we denote $$|\alpha|_{\Delta}|_{\infty} = \sup_{q \in U, \ v_i \in \Delta_q, \ |v_1 \wedge \dots \wedge v_k|=1} |\alpha_q(v_1 \wedge \dots \wedge v_k)|,$$ $$m(\alpha|_{\Delta})_{\inf} = \inf_{q \in U, \ v_i \in \Delta_q, \ |v_1 \wedge \dots \wedge v_k|=1} |\alpha_q(v_1 \wedge \dots \wedge v_k)|.$$ The first expression is the supremum over $U$ of the norms of $\alpha_q$ seen as linear maps acting on $\bigwedge^k (\Delta_q)$ while the second is the infimum over $U$ of the conorms of $\alpha_q$. On the passing we note that given any $v_1,\dots,v_k \in \Delta_q$, $$m(\alpha|_{\Delta})_{\inf} \leq \frac{|\alpha_q(v_1,\dots,v_k)|}{|v_1 \wedge \dots \wedge v_k|} \leq |\alpha|_{\Delta}|_{\infty}.$$ The next definition is the fundamental geometric regularity property that we require of our non-differentiable bundles in order to endow them with other nice geometric and analytic properties: \[defn-continuousexteriorderivative\] A continuous differential $k$-form $\eta$ is said to have a continuous exterior differential if there exists a continuous differential $k$+1-form $\beta$ such that for every $k$-cycle $Y$ and $k$+1 chain $H$ bounded by it, one has that $$\int_Y \eta = \int_H \beta.$$ If such a $\beta$ exists, we suggestively denote it as $d\eta$. A rank $k$ subbundle $\mathcal{E} \subset \Omega^n_{0}(M)$ is said to be equipped with continuous exterior differential at $p \in M$, if there exists a neighbourhood $V$ of $p$ on which $\{\eta_i\}_{i=1}^k$ is a local basis of sections of $\mathcal{E}$ on $V$ and $\{d\eta_i\}_{i=1}^k$ are their continuous exterior differentials. We will occasionally refer to above property as Stokes property and also denote this triple by $\{V,\eta_i,d\eta_i\}_{i=1}^k$. We denote by $\Omega^k_{d}(M)$ the space of all differential $k$-forms that have continuous exterior differentials. Obviously $\Omega^k_{0}(M) \supset \Omega^k_{d}(M) \supset \Omega^k_{1}(M)\supset \Omega^k_{2}(M)\supset \dots \ $. It will be the purpose of section \[sec-continuousexteriordifferential\] to give non-trivial examples (i.e. non-differentiable and non-Hölder) of such differential forms for $k=1$ and discuss their properties to illustrate their geometric significance. However, our main theorems only deal with sub-Riemannian properties of tangent subbundles defined by such differential 1-forms. \[defn-integrable\]Let $\Delta$ be a corank $k$, continuous, tangent subbundle. Assume $\mathcal{A}^1_0(\Delta)(M)$ is equipped with continuous exterior differential $\{V,\eta_i,d\eta_i\}_{i=1}^k$ at $p_0$. We say that $\Delta$ is non-integrable at $p_0$ if $$(\eta_1 \wedge \eta_2 \dots \wedge \eta_k \wedge d\eta_{\ell})_{p_0} \neq 0,$$ for some $\ell \in \{1, \dots ,k\}$. Note that if the bundle $\Delta$ was $C^1$ and corank $1$, then this condition would imply that $\Delta_{p_0} + [\Delta_{p_0},\Delta_{p_0}] =T_{p_0}M$, and therefore would be a step 2, completely non-integrable subbundle at $p_0$. Therefore the corank $1$, non-integrable, continuous bundles defined above can be thought as of continuous analogues of these step $2$, completely non-integrable subbundles. First of our theorems is the analogue of the Chow-Rashevskii Theorem for continuous bundles whose annihilators are equipped with continuous exterior differentials: \[thm-chow\] Let $\Delta$ be a corank 1, continuous tangent subbundle. Let $p_0 \in M$ and assume $\mathcal{A}^1_0(\Delta)(M)$ is equipped with a continuous exterior differential $\{V,\eta,d\eta\}$ at $p_0$. If $\Delta$ is non-integrable at $p_0$ then it is accessible in some neighborhood of $p_0$. A direct corollary is Let $\Delta$ be a corank 1, continuous tangent subbundle defined on a connected manifold $M$. If $\mathcal{A}^1_0(\Delta)(M)$ is equipped with a continuous exterior differential and non-integrable at every $p \in M$, then $\Delta$ is accessible on $M$. Our next theorem will be about metric properties of such a bundle, namely we will give an analogue of the Ball-Box Theorem (specialized to case of differentiable step 2, completely non-integrable tangent subbundles). Theorem \[thm-chow\] will then be a consequence of this theorem. We say that a bundle $\Delta$ has modulus of continuity $\omega: s \rightarrow \omega_s$ if in every coordinate neighborhood there exists a constant $C>0$ such that, it has a basis of sections $\{Z_i\}_{i=1}^n$ whose elements have modulus of continuity ${C}\omega_s$ with respect to the Euclidean norm of the coordinates. More explicitly, these basis of sections satisfy, in coordinates, $$|Z_i(p)-Z_i(q)| \leq C\omega_{|p-q|},$$ for all $i=1,\dots,n$. We assume that moduli of continuity are increasing and therefore $\omega_t = \sup_{s\leq t}\omega_s$. Now we remind the notion of adapted coordinates that was introduced informally in the beginning of this section: \[def-adaptedcoordinates\]Given a bundle $\Delta$, a coordinate system $(x^1,\dots ,x^n,y^1,\dots,$ $y^m)$ with a domain $U$ around $p\in M$ is said to be adapted to $\Delta$ if $p=0$, $\Delta_0= \text{span}<\frac{\partial}{\partial x^i}|_0>_{i=1}^n$ and $\Delta_q$ is transverse to $\text{span}<\frac{\partial}{\partial y^i}|_q>_{i=1}^m$ for all $q \in U$. Then, given some adapted coordinates $(x^1,\dots,x^n,y)$ and the adapted basis for the corank $1$ bundle $\Delta$, we define: $$D^{\omega}_{2}(0,K_1,\epsilon)=\{z=(x,y) \in U \quad | \quad |x| + \sqrt{K_1(|x|\tilde{C}\omega_{2|x|} +|y|)} \leq \epsilon,$$ $$H^{\omega}_{2}(0,K_2,\epsilon)=\{z=(x,y) \in U \quad | \quad |x| \leq \epsilon, \quad |y| \leq K_2\epsilon^{2}+ |x|\tilde{C}\omega_{2|x|}\},$$ $$\mathcal{B}_{2}(0,K_3,\epsilon)=\{z=(x,y) \in U \quad | \quad |x| \leq \epsilon, \quad |y| \leq K_3\epsilon^{2}\},$$ where $|\cdot|$ is the Euclidean norm given by the coordinates and $|x| = \sum_{i=1}^n |x_i|$. Here $H$ stands for hourglass and $D$ for diamond. Note that in the case $\omega_t = t^{\theta}$ (i.e. Hölder continuous) the shape of $H_2^{\omega}$ indeed looks like an hourglass and $D^{\omega}_{2}$ looks like a diamond with sides that are bent inwards and becomes more linear as $x$ increases (since $K_1^{\frac{1}{2}}|x|^{\frac{1+{\theta}}{2}}$ dominates $|x|$ near $0$, see figure \[fig-hourglass\]).The ball $\mathcal{B}_{2}(0,K_3,\epsilon)$ is an analogue of the usual box in smooth sub-Riemannian geometry with the exception $y$ direction is allowed to have its own scaling factors $K_3$. We belive that with a careful geometric analysis, these constants turn out to have geometric significance and that is why we decided to define a more generalized ball like this. In an adapted coordinate system with a domain $U$, we can also define a basis of sections $\Delta$ of the form $$X_i = \frac{\partial}{\partial x^i} + a_i(x,y)\frac{\partial}{\partial y},$$ where $a_i(x,y)$ are continuous functions. If $\Delta$ has modulus of continuity $\omega$, then it is possible to show that, the functions $a_i$ also have modulus of continuity $\tilde{C}\omega$ on $U$ with respect to $|\cdot|$, with some multiplicative constant $\tilde{C}>1$ possibly depending on $U$ and on the chosen coordinates. The assumption of non-integrability at $p_0$ would then mean that there exists $i,j \in \{1,\dots,n\}$, $i \neq j$ and a domain $U$ such that $|d\eta(X_i,X_j)|_{\inf}>0$. Therefore we also have $ m(d\eta|_{\Delta})_{\inf}>0. $ We will also need to define a constant $c$. This constant is later explained in Lemma \[lem-Gromov\] (and the remark \[rem-Gromov\] that follows), which is an independent Lemma from [@Gro83] (sublemma 3.4B, see also Corollary $2.3$ in [@Sim10]). Finally for a fixed adapted coordinate system with its Euclidean norm, we let $d_g\geq 1$ denote a constant such that, for all $v \in \Delta_p$ and for all $p \in U$: $$\frac{1}{d_g}\sqrt{g(v,v)} \leq |v| \leq d_g \sqrt {g(v,v)}.$$ Then, we can state the next main theorem: \[thm-ballbox\] Let $\Delta$ be a corank 1, continuous bundle with modulus of continuity $\omega$. Let $p_0 \in M$ and assume $\mathcal{A}^1_0(\Delta)(M)$ is equipped with a continuous exterior differential $\{V,\eta,d\eta\}$ at $p_0$ and that $\Delta$ is non-integrable at $p_0$. Then, for any adapted coordinate system, there exists a domain $U$ and constants $\epsilon_0,K_1,K_2>0$ such that, for all $\epsilon < \frac{\epsilon_0}{2nd_g}$: $$\label{eq-inclusions1} D^{\omega}_2(0, \frac{1}{K_1},\frac{1}{4d_g}\epsilon) \subset B_{\Delta}(0,\epsilon) \subset H^{\omega}_2(0,K_2, 2nd_g\epsilon),$$ where $K_1,K_2>0$ are constants given by $$\frac{1}{K_1} = 42\frac{|\eta(\partial_y)|_{\infty}}{m(d\eta|_{\Delta})_{\inf}},$$ $$K_2 = 42(1+2n)^2c \frac{ |d\eta|_{\Delta}|_{\infty}}{|\eta(\partial_y)|_{\inf}}.$$ Moreover for each such smooth adapted coordinate system, there exists a $C^1$ adapted coordinate system such that, $$\mathcal{B}_2(0,K_1, \frac{\epsilon}{4d_g}) \subset B_{\Delta}(0,\epsilon) \subset \mathcal{B}_2(0,K_2,2nd_g\epsilon).$$ This theorem is a generalization of the smooth Ball-Box Theorem on codimension 1, step 2, completely non-integrable bundle case. Indeed if $\Delta$ is smooth then $\mathcal{A}^1_0(\Delta)(M)$ is equipped with continuous exterior differential and the non-integrability definition given in \[defn-integrable\] coincides with the step 2, completely non-integrable case. Also since $\omega_{2|x|} = 2|x|$, one can check the following: $$\mathcal{B}_2(0,K_1,\epsilon) \subset D_2^{\omega}(0, \frac{1}{K_1}, (1+\sqrt{\frac{2\tilde{C}}{K_1}+1}) \epsilon),$$ $$H_2(0,K_2,\epsilon) \subset \mathcal{B}_2(0,K_2+2\tilde{C},\epsilon).$$ So one gets in the case of $C^1$ bundles: $$\mathcal{B}_2(0,\frac{1}{K_1}, (1+\sqrt{\frac{2\tilde{C}}{K_1}+1})^{-1}\frac{\epsilon}{4d_g}) \subset B_{\Delta}(0,\epsilon) \subset \mathcal{B}_2(0,K_2+2\tilde{C},2nd_g\epsilon),$$ which is the usual Ball-Box relations in the smooth sub-Riemannian geometry (apart from the fact that we use a generalized version of the usual boxes which contain and are contained in usual boxes with different constants). Note also that if $\Delta$ is a Lipschitz continuous bundle then again we have that $\omega_t =t$. Therefore this theorem also says that if a Lipschitz continuous bundle has an annihilator equipped with a continuous exterior differential, then the usual Ball-Box relations also hold true for this bundle. We also would like to note that the statement about the existence of $C^1$ adapted coordinates has a much more geometric interpretation. However we can only explain it after certain objects are constructed. This is explored in subsubsection \[ssection-spread\] (see figure \[fig-ballbox\]). The smooth sub-Riemannian geometry is usually shy of explicit constants and the explicit description of the neighbourhood $U$ for the Ball-Box Theorem. This makes the results particularly hard to apply on a sequence of $C^1$ bundles, which might be used to approximate a continuous bundle. Therefore we believe that this theorem can also be seen as a version of the Ball-Box Theorem with explicit constants. The explicit constants by themselves are not enough however, but it is also essential to understand how the size of $U$ depends on regularity properties of $\Delta$. The explicit relations are listed in subsection \[sssection-fixU\]. As far as we are aware this is one of the few proofs that pays particular attention to these details. ![Pictorial representation of Theorem \[thm-ballbox\] with $\omega(t)=t^{\theta}$.[]{data-label="fig-hourglass"}](hourglass){width="300px"} Organization of the Paper ------------------------- In this subsection we describe the layout of the paper and the main ideas of the proofs. First note that Theorem \[thm-ballbox\] implies Theorem \[thm-chow\] therefore it is sufficient to prove the former. Section \[sec-proof\] contains the proof of Theorem \[thm-ballbox\]. The proof of this theorem has two main ingredients. These are the fundamental tools that we use repeatedly and are therefore are proven in a seperate subsections \[ssec-uniqueness\] and \[ssec-geometric\]. First ingredient is Proposition \[prop-uniquebasis\], where we prove that the adapted basis is uniquely integrable. This is only due to existence of a continuous exterior differential. The second ingredient is Proposition \[prop-stokessubriemann\] which quantifies the amount admissible curves travel in the $\partial_y$ direction by a certain surface integral of $d\eta$. This again only assumes existence of the continuous exterior differential and Proposition \[prop-uniquebasis\]. This proposition can be seen as a generalization of certain beautiful ideas from [@Arn89] (see Section 36 of Chapter 7 and Appendix 4). Then, in subsection \[subsec-ballboxproof\] we prove Theorem \[thm-ballbox\] using Propositions \[prop-uniquebasis\] and \[prop-stokessubriemann\]. The main idea is to first construct certain, accessible $n$ dimensional manifolds $\mathcal{W}_{\epsilon}$ (see Lemma \[lem-propertiesofW\]) and study how the sub-Riemannian balls are spread around these manifolds (see Lemma \[lem-distancefromW\]). To summarize the main ideas of the proof of Theorem \[thm-ballbox\], we can say - The existence of $d\eta$ and the condition that $\eta \wedge d\eta \neq 0$ gives volume to the sub-Riemannian ball. - The loss of regularity in the bundle may cause the sub-Riemannian ball to bend which results in the outer sub-Riemannian ball being distorted and the inner one getting smaller. The two main tools that we repeatedly use (Proposition \[ssec-uniqueness\] and \[ssec-geometric\]) are obtained via application of Stokes property, thus establishing it as the germ of many geometric and analytic properties of vector fields and bundles. In section \[sec-continuousexteriordifferential\], we give some examples of bundles whose annihilators are equipped with continuous exterior differentials and which are non-integrable on neighbourhoods where they are non-differentiable. We also study some of the properties of such bundles to emphasize that having a continuous exterior differential is a geometrically very relevant property, yet not as strong as being $C^1$ in terms of regularity. We compare the results of this paper to the several other results we have explained in the introduction. Finally in section \[sec-generalizations\] we sketch some thoughts on some possible generalizations that relax the conditions required for the theorems and some comments on how to possibly proceed with the proof in special cases of higher corank bundles. The Proof {#sec-proof} ========= In the next two subsections we prove the two technical propositions: Proposition \[prop-uniquebasis\] and Proposition \[prop-stokessubriemann\]. Proposition \[prop-uniquebasis\] {#ssec-uniqueness} --------------------------------- Lets remind the definition of $\{X_i\}_{i=1}^n$. Given $p_0 \in M$, assume we are given any adapted coordinates $\{x^1,\dots,x^n,y\}$ with some domain $U$ such that $\frac{\partial}{\partial y}$ is everywhere transverse to $\Delta$ on $U$. Occasionally we will denote $\partial_i = \frac{\partial}{\partial x_i}$, $\partial_y = \frac{\partial}{\partial y}$, $\mathcal{X}^k_p= \text{span}\frac{\partial}{\partial x^k}|_p$ and $\mathcal{Y}_p= \text{span}\frac{\partial}{\partial y}|_p$ . Then, it is easy to show that in this domain, sections of $\Delta$ admit a basis of the form $$\label{eq-adapted} X_i = \frac{\partial}{\partial x^i}+a_i(x,y)\frac{\partial}{\partial y},$$ where $a_i$ have the same modulus of continuity as $\Delta$ up to multiplication by some constant $\tilde{C}>0$. Note that the adapted coordinate assumption also means $X_i(0)=\partial_i$. We call such a basis an adapted basis. It is also easy to show that such an adapted basis also exists in higher coranks but of the form $$X_i = \frac{\partial}{\partial x^i}+\sum_{j=1}^m a_{ij}(x,y)\frac{\partial}{\partial y^j}.$$ If $X$ is any vector-field defined on some neighborhood $U_1$, we call it integrable if for all $p \in U_1$ there exists $\epsilon_p$ and a $C^1$ curve $\gamma: [-\epsilon_p, \epsilon_p] \rightarrow U_1$ such that $\gamma(0)=p$ and $\dot{\gamma}(t) = X(\gamma(t))$ for all $t \in [-\epsilon_p, \epsilon_p]$ (these curves are called integral curves passing through $p$). By Peano’s Theorem, continuous vector-fields are always integrable. We call it uniquely integrable if it is integrable and if $\gamma_1$ and $\gamma_2$ are two integral curves which intersect, then each intersection point is contained in a relatively open (in both integral curves) set. In this case, there exists a unique maximal integral curve of $X$ (not to be confused with maximal and minimal solutions of non-uniquely integrable vector-fields), starting at $q$ and defined on the interval $[-\epsilon_q,\epsilon_q]$. We denote this integral curve by $t \rightarrow e^{tX}(q)$. Unique integrability is more stringent and commonly known sufficient conditions are $X$ being Lipschitz or $X$ satisfying the Osgood criterion. We say that an integrable vector-field $X$ defined on $U_1$ has $C^r$ family of solutions if there exists some $U_2 \subset U_1$, an $\epsilon_0$ such that, for all $p \in U_2$, there exists an integral curve passing through $p$ with $\epsilon_p \geq \epsilon_0$ and such that this choice of solutions seen as maps from $[-\epsilon_0,\epsilon_0] \times U_2 \rightarrow U_1$ are $C^r$. \[prop-uniquebasis\] Assume $\Delta$ is a co-rank $1$, continuous tangent subbundle such that, at $p_0$, $\mathcal{A}^1_0(\Delta)(M)$ is equipped with a continuous exterior differential $\{V,\eta, d\eta\}$. Then, for any adapted coordinate $\{x^1,\dots$ $,x^n,y\}$ around $p_0$ with an adapted basis $\{X_i\}_{i=1}^{n}$, there exists some domain $U$ on which $X_i$ are uniquely integrable and have $C^1$ family of solutions. Pick any adapted coordinate system and adapted basis with some domain $U \subset V$. Assume by contradiction that there exists an $X_k$ which is not uniquely integrable. The there exists two integral curves $\gamma_i:(0,\epsilon_i) \rightarrow U$ which intersect at some point $p=\gamma_i(\tau_i)$ which is not contained in a relatively open set in one of the curves. This means there exists an interval $[\tau_i,\kappa_i]$ (WLOG assume $\kappa_i > \tau_i$) on which $\gamma_i$ do not coincide but are defined and such that $\gamma_1(\tau_1)=\gamma_2(\tau_2)$. By shifting and restricting to a smaller interval we can then assume we have $\gamma_i: [0, \epsilon_0] \rightarrow U$ such that $\gamma_1(0) = \gamma_2(0)$ but that they are not everywhere equal. Now take any $q = \gamma_1(\epsilon_1)$ that is not in $\gamma_2$. Therefore they also do not coincide on some interval around $\epsilon_1$. Denote $\epsilon_2= \sup_{0 \leq t \leq \epsilon_1}\{t \quad \text{such that} \quad \gamma_1(t)=\gamma_2(t)\}$. $\epsilon_2$ exists since we know at least that $\gamma_1(0)=\gamma_2(0)$. We have that clearly $\epsilon_2 < \epsilon_1$ and between $\epsilon_2$ and $\epsilon_1$ $\gamma_1,\gamma_2$ can not intersect. So by restricting everything to $[\epsilon_2,\epsilon_1]$ and reparametrizing we obtain two integral curves of $X_k$ defined on some $[0,\epsilon]$ such that $\gamma_1(0)=\gamma_2(0)$ and that $\gamma_1(t) \neq \gamma_2(t)$ for all $0<t\leq \epsilon$ for some $\epsilon$. Due to the specific form of $X_k$ both curves lie in the $\mathcal{X}^k_p-\mathcal{Y}_p$ plane (whose coordinate we will denote as $(x,y)$) and have the form: $$\gamma_j(t) = (t, d_j(t)).$$ Without loss of generality we can assume $d_1(t)> d_2(t)$ for all $0<t\leq \epsilon$. We are going to show that existence of exterior differential forces $d_1(t)=d_2(t)$ for all $t \leq \epsilon$ leading to a contradiction. To this end let $h(t) = d_1(t)-d_2(t)$. Before continuing with the proof we make one elementary remark. By our choice of coordinates $\eta$ will have the form $$\eta = a_0(x,y)dy + \sum_{i=1}^n a_i(x,y)dx^i,$$ with $\inf_{q \in U}|a_0(q)|= |\eta(\partial_y)|_{\inf}>0$ (since $\Delta$ is always transverse $y-$direction, it can not contain $\partial_y$ and therefore $b$ can not be $0$) and so in particular $a_0(q)$ has constant sign. So if $\alpha(t)$ is any (non-singularly parametrized) curve whose tangent vectors lie in $\mathcal{Y}_{\gamma(t)}$ axis, one has that $\eta(\dot{\alpha})$ always has constant sign and therefore $$\bigg|\int_{\gamma}\alpha \bigg| = \int_{\gamma}\bigg|\alpha\bigg| \geq |\eta(\partial_y)|_{\inf}| \ |\alpha|,$$ where $|\alpha|$ is the Euclidean length of the curve $\alpha$. Let $v_t$ be the straight line segment that lies in the $\mathcal{Y}_{d_2(t)}$ axis and which starts at $d_2(t)$ and ends at $d_1(t)$. We let $\gamma_t$ be the loop that is formed by composing $\gamma_2, v_t$ and then $\gamma_1$ backwards. We also let $\Gamma_t$ be the surface in $\mathcal{X}^k_p-\mathcal{Y}_p$ plane that is bounded by this curve. Note that since $\eta(\dot{\gamma}_i)=0$, $\int_{\gamma_t}\eta = \int_{v_t}\eta$. But $\dot{v}_t=\partial_y$ and $\partial_y$ is always transverse to $\Delta$ on $U$. Therefore $\eta(\dot{v}_t)$ is never $0$ and so it never changes sign. So we have that $$\label{eq-segmentlength} \bigg|\int_{\gamma_t}\eta\bigg| = \bigg|\int_{v_t}\eta\bigg| > |\eta(\partial_y)|_{\inf} h(t).$$ Since $\eta$ has the continuous exterior differential $d\eta$, we have using Stokes property and equation $$\bigg|\int_{\Gamma_t}d\eta\bigg|=\bigg|\int_{\gamma_t}\eta\bigg| \geq |\eta(\partial_y)|_{\inf} h(t).$$ But $|\int_{\Gamma_t}d\eta| \leq |\Gamma_t||d\eta|_{\infty}$ (where $|\Gamma_t|$ denote the Euclidean area). Therefore we have $$\label{eq-inequni} h(t) \leq \frac{|d\eta|_{\infty}}{|\eta(\partial_y)|}|\Gamma_t|.$$ We are going to show that this leads to a contradiction for $t$ small enough. Let $t_n$ be a sequence of times such that $t_n \leq \epsilon$, $t_n \rightarrow 0$ as $n\rightarrow \infty$ and $$\label{eq-assumption} h(t_n) \geq \sup_{t<t_n}h(t),$$ which is possible since $h(t)$ is continuous and $0$ at $0$. Let $S_n$ be the strip obtained by parallel sliding the segment $v_{t_n}$ along the curve $\gamma_1$. By our assumption in equation , $S_n$ contains the surface $\Gamma_{t_n}$ (see figure \[fig-strip\]). ![Strip $S_n$.[]{data-label="fig-strip"}](strip){width="150px"} \[lem-striparea\]$|S_n|= t_n h(t_n)$. Consider the transformation $(x,y)\rightarrow (x, y-\gamma_1(x))$ on its maximally defined domain (which includes $S_n$ and which is differentiable since $\gamma_1(t)$ is differentiable in the $t$ variable). It takes the strip $S_n$ to a rectangle with two sides of length $t_n$ and $h(t_n)$ so in particular it has area $t_n h(t_n)$. The Jacobian of this transformation is $1$ and therefore it preserves area so the strip itself has area $t_n h(t_n)$. Since this strip contains $\Gamma_n$ we see that $|\Gamma_n| \leq t_n h(t_n)$. Then, using equation we obtain for all $t_n$ $$h(t_n) \leq \frac{|d\eta|_{\infty}}{|\eta(\partial_y)|_{\inf}} t_n h(t_n) \Rightarrow \frac{|\eta(\partial_y)|_{\inf}}{|d\eta|_{\infty}} \leq t_n,$$ which leads to a contradiction since $t_n$ tends to $0$. This concludes the proof of uniqueness. For being $C^1$, note that we can always find a neighborhood $\tilde{U} \varsubsetneq U$ and some $\epsilon_0$ such that for all $q \in \tilde{U}$ and for all $|t| \leq \epsilon_0$, $e^{tX_k}(q) \in U$ and is well defined (the size of $\tilde{U}$ and $\epsilon_0$ depend on each other and on $|X_k|_{\infty}$). Therefore we obtain the map $[-\epsilon_0, \epsilon_0] \times \tilde{U} \rightarrow U$. Then, for being $C^1$ we use a result from [@Har64] (the theorem stated there is more general so we state the specialized version)[^2]: Let $f(t,y): U\subset \mathbb{R}^{n+1} \rightarrow \mathbb{R}^n$ (with $y \in \mathbb{R}^n$) be continuous. Then, the ODE $\dot{y}=f(t,y)$ has unique and $C^1$ solutions $y= \eta(t,t_0,y_0)$ for all $(t_0,y_0) \in U$ if for any $p \in U$, there exists a neighborhood $U_p$ and a non-singular matrix $A(t,y)$ such that the $1-$forms $\eta^i = \sum_{i=1}^n A^{ij} (dy^i - f^idt)$ have continuous exterior differentials. Note first that the unique integrability of the non-autonomous ODE with $C^1$ solutions above is equivalent to unique integrability of the vector-field $X=\frac{\partial}{\partial t} + \sum_{i=1}^n f^i(t,y,z)\frac{\partial}{\partial y^i}$ with $C^1$ family of integral curves which would be given by $t \rightarrow (t, \eta(t,t_0,y_0)) \subset U$ for $t$ small enough. Second let $\mathbb{X}$ be the bundle spanned by this vector-field in the $(t,y)$ space. Then, this bundle is the intersection of the kernel of the 1-forms $\eta^i=dy^i - f^i dt$. Therefore $\eta^i$ is a basis of sections for $\mathcal{A}_0^1(\mathbb{X})$. In particular then the condition of this theorem about the existence of such a non-singular $A(t,y)$ simply means that there must exist a basis of sections for $\mathcal{A}_0^1(\mathbb{X})$ which has continuous exterior differentials. In our case for each $X_k$ we have explicitly built that basis of sections which is given by $\eta, dx^1,\dots,dx^{k-1},dx^{k+1},\dots,dx^n$. It is possible to prove stronger versions of this theorem but they use approximations to $\Delta$ rather than $\Delta$ itself and $C^1$ regularity is interchanged with Lipschitzness. We refer the interested readers to [@Har64; @LuzTurWar16] for the generalizations. Proposition \[prop-stokessubriemann\] {#ssec-geometric} ------------------------------------- For the following, given some $p \in U$, let $\bar{X}_p$ be the space spanned by $X_i(p)$ at the point $p$ , $|X|_{\infty}=\max_{i=1,\dots, n}|X_i|_{\infty}$ , $|\wedge X|_{\inf} = |X_1 \wedge \dots \wedge X_n \wedge \partial_y|_{\inf}$ and $\Pi_x$ be the projection to the $x$ coordinates. \[prop-stokessubriemann\] Let $\gamma_i: [0,\varepsilon_i] \rightarrow U$ for $i=1,2$ be two $\Delta-$admissible curves with lengths $\ell_{i}$ such that $\gamma_1(0)=\gamma_2(0)=q$, $\Pi_x\gamma_1(\varepsilon_1) = \Pi_x\gamma_2(\varepsilon_2)$ (that is they start on the same point and end at the same $\partial_y$ axis). Let $\ell = \max\{\ell_1,\ell_2\}$, $\varepsilon=\max\{\varepsilon_1,\varepsilon_2\}$, $\xi = \max_{k=1,2} n \frac{(n|X|_{\infty})^{n}}{|\wedge X|_{\inf}}\sup_{t \leq \varepsilon_k} |\dot{\gamma}_k(t)|\tilde{C}\omega_{\ell} $ and $\beta$ be the segment in the $\partial_y$ direction that connects $\gamma_1(\varepsilon_1)$ to $\gamma_2(\varepsilon_2)$. Assume moreover that $B(q, 2\ell) \subset U$. Then, for any 2-chain $P \subset \bar{X}_{q} \cap U$ whose boundary is the projection of $\gamma^{-1}_1 \circ \gamma_2$ along $\partial_y$ to $\bar{X}_{q}$ we have that $$\label{eq-mainprop} \frac{1}{|\eta(\partial_y)|_{\infty}}\bigg(\bigg|\int_P d\eta \bigg| - |c|\bigg) \leq |\gamma_1(\varepsilon_1) - \gamma_2(\varepsilon_2)| \leq \frac{1}{|\eta(\partial_y)|_{\inf}}\bigg(\bigg|\int_P d\eta\bigg| + |c|\bigg),$$ and $$\label{eq-mainprop2} sign\bigg(\int_{\beta}\eta\bigg) = sign\bigg(\int_P d\eta + c\bigg),$$ where $$|c| \leq 4 \ell \varepsilon \xi |d\eta|_{\infty}.$$ Denote $\gamma_1(\varepsilon_1)=q_1$, $\gamma_2(\varepsilon_2)=q_2$. Assume wlog that $q_1 \geq q_2$ with respect to the order given by the positive orientation of $\partial_y$ direction. We first define the projections if $\gamma_i$ to $\bar{X}_q$. Since $\gamma_i$ are admissible curves we have that $$\dot{\gamma}_i(t)= \sum_{k=1}^n u_i^k(t)X_k(\gamma_i(t)) = \sum_{k=1}^n u_i^k(t)(\partial_k + a_k(\gamma_i(t))\partial_y) ,$$ $t$ a.e for some piecewise $C^1$ functions $u_i^k(t)$. Define the following non-autonomous vector-fields $$Z_i(t,p) = \sum_{k=1}^n u_i^k(t)X_k(q)= \sum_{k=1}^n u_i^k(t)(\partial_k + a_k(q)\partial_y),$$ which are constant in the $p$ variable. Therefore they admit unique solutions starting at $t=0$ and $q$ which we denote as $\alpha_i(t,q,0) = e^{tZ_i}(q)$ that are inside $\bar{X}(q)$. We will denote the images of these curves as $\alpha_{i}$. Its clear that $\Pi_x(\gamma_i(t))=\Pi_x(\alpha_i(t))$. Since $\bar{X}_q$ is transversal to $\partial_y$ direction, these are the unique projections of $\gamma_i$ to $\bar{X}_q$ alogn $\partial_y$ direction. The condition that $B(q,2\ell) \subset U$ also implies that they are inside $U$. We can build some 2 chains inside $U$ bounded by these 1 chains as follows [^3] $$v_{\ell}(t,s) = \alpha_{\ell}(s) + t(\gamma_{\ell}(s)-\alpha_{\ell}(s)),$$ from $[0,1]\times [0,\epsilon]$ to $U$. Since $\alpha_{\ell}(s)$ and $\gamma_{\ell}(s)$ are piecewise $C^1$ in the $s$ variable, the domain of this map can be partitioned into smaller rectangles on which $v_{\ell}(t,s)$ are differentiable and therefore whose images are $2$ cells. Then, the images of $v_{\ell}(t,s)$ become 2 chains which we denote as $C_{\ell}$. Note that $v_{1}(t,0) = v_{2}(t,0)=q$ for all $t$. And also let the image of $v_{2}(t,\varepsilon_2)$ be a curve $\tau$. Since $q_1 \geq q_2$ then image of $v_{1}(t,\varepsilon_2)$ is $\beta \circ \tau$. It is also clear that $v_{\ell}(0, s) = \alpha_{\ell}(s)$ and $v_{\ell}(1,s)=\gamma_{\ell}(s)$. Then, we orient these curves and $C_i$ such that: $$\partial C_1 = \gamma_1 -\beta - \tau -\alpha_1,$$ $$\partial C_2 = \tau -\gamma_2 + \alpha_2.$$ Let $\Gamma$ be any 2 chain in $U$ bounded by concatenating $\gamma_1, \gamma_2$ and $\beta$ (whose composition is a 1 cycle in a contractible space so it always bounds a chain) in the right orientation so that $$\partial \Gamma = \beta -\gamma_1 + \gamma_2.$$ Finally also orient $P$ so that $$\partial P = \alpha_1 - \alpha_2.$$ Then, $\Gamma, C_1,C_2$ and $P$ form a closed 2 chain $CC$ (see figure \[fig-cycle\]). Using Stokes property and the fact that $\partial CC = \emptyset$ we get ![The Closed 2-Chain C.[]{data-label="fig-cycle"}](cycles){width="150px"} $$\int_{\Gamma}d\eta = -\bigg(\int_{P}d\eta + \int_{C_1 \cup C_2}d\eta \bigg).$$ Moreover since $\Gamma$ is bounded by $\gamma_1,\gamma_2$ and $\beta$ and $\eta_{\gamma_i(t)}(\dot{\gamma}_i(t))=0$ we get again by Stokes property $$\label{eq-finaleq} \int_{\beta}\eta=\int_{\Gamma}d\eta = -\bigg(\int_{P}d\eta + \int_{C_1 \cup C_2}d\eta \bigg).$$ Defining $c= \int_{C_1 \cup C_2}d\eta$, we require one more final lemma to finish the proof, For $\beta,c$ as defined above $$\label{eq-Cintegral} |c| \leq 4\xi\ell \varepsilon|d\eta|_{\infty},$$ $$\label{eq-Bintegral} |q_2-q_1||\eta(\partial_y)|_{\inf} \leq \bigg|\int_{\beta} \eta \bigg| \leq |\eta(\partial_y)|_{\infty}|q_2-q_1|.$$ For the first inequality with $\dot{\alpha}(s)$ and $\dot{\gamma}(s)$ a.e defined we can write, $$\begin{aligned} \bigg|\int_{C_i}d\eta \bigg| &= \bigg|\int_0^1 dt \int_0^{\epsilon}ds \ d\eta_{v_i(t,s)} (\gamma_i(s)-\alpha_i(s), (1-t)\dot{\alpha}_i(s) + t\dot{\gamma}_{i}(s)) \bigg|, \\ & \leq \int_0^1 dt \int_0^{\epsilon}ds \ |d\eta|_{\infty} \sup_{s \leq \epsilon}|\alpha_i(s)-\gamma_i(s)|((1-t)|\dot{\alpha}_i(s) | + t|\dot{\gamma}_i(s)|), \\ & \leq 2\ell|d\eta|_{\infty} \sup_{s \leq \epsilon}|\alpha_i(s)-\gamma_i(s)|.\\ \end{aligned}$$ Therefore we need to obtain an estimate on the maximum distance between $\gamma_i$ and $\alpha_i$. For this, we have that $$|\alpha_i(t) - \gamma_i(t)| \leq n \varepsilon_i \sup_{t \leq \varepsilon_i, \ell=1,\dots,n}|u_i^{\ell}(t)||X_{\ell}(\gamma_1(t))-X_{\ell}(q)|.$$ Since $|X_{\ell}(\gamma_i(t))-X_{\ell}(q)| \leq \tilde{C}\omega_{\ell}$ we have that $$|\alpha_i(t) - \gamma_i(t)| \leq n \varepsilon_i \sup_{t \leq \varepsilon_i, \ell=1,\dots,n}|u_i^{\ell}(t)|\tilde{C}\omega_{\ell}.$$ Now we need to estimate $\max_{\ell=1,\dots,n}|u_i^{\ell}(t)|$. Let $L_p$ be the linear transformation that takes $X_i(p)$ to $\partial_i$ and fixes $\partial_y$ (which is a matrix whose columns in the Euclidean basis are $X_i(p)$ and $\partial_y$). Then, $det(L^i_p) \geq |X_1 \wedge \dots \wedge X_n \wedge \partial_y|_{\inf}$ for all $p\in U$ which is non-zero by the form of $X_i$. Let $s_1(p)=m(L_p),s_2(p),\dots,s_{n+1}(p)=|L_p|$ be the singular values of $L_p$. Then, $det(L_p) = s_1(p)\dots s_n(p) \leq m(L_p)|L_p|^n$. Since $L_p$ is a matrix whose each column has norm less than $|X|_{\infty}>1=|\partial_y|$ one has that $|L_p| \leq n|X|_{\infty}$ for all $p\in U$ so, $m(L_p) \geq \frac{|\wedge X|_{\inf}}{(n|X|_{\infty})^{n}}$. Then, $$|\dot{\gamma}_i(t)| = |\sum_{k=1}^n u^k_i(t)L_{\gamma_k(t)}\partial_k| \geq m(L_{\gamma_i(t)})\max_{k=1,\dots,n}|u^k_i(t)|.$$ So $$\sup_{t < \varepsilon_i, k=1,\dots,n }|u^k_i(t)| \leq\frac{(n|X|_{\infty})^{n}}{|\wedge X|_{\inf}}\sup_{t < \varepsilon_i}|\dot{\gamma}_i(t)|.$$ This gives $$|\alpha_i(t) - \gamma_i(t)| \leq \varepsilon n \frac{(n|X|_{\infty})^{n}}{|\wedge X|_{\inf}}\sup_{t < \varepsilon_i}|\dot{\gamma}_i(t)|\tilde{C}\omega_{\ell} \leq \varepsilon \xi.$$ With this estimate in hand, we have $|\int_{C_i}d\eta| \leq 2\ell \varepsilon \xi$, and so $|c| =|\int_{C_1 \cup C_2}d\eta| \leq 4\ell \varepsilon \xi$. For the second note that $\beta$ is a curve whose tangent vector is always parallel to $\partial_y$. But since $\eta$ annihiliates $\Delta$ which is transverse to $\partial_y$, we have that $\eta_p(\partial_y)$ is never $0$ in $U$ and never changes sign. Then, similarly for any non-singular parametrization $\eta_{\beta(t)}(\dot{\beta}(t))$ also never changes sign. So assuming unit length parametrization $$\bigg|\int_{\eta}\beta\bigg| = \int_{0}^{|q_2-q_1|}|\eta_{\beta(t)}(\dot{\beta}(t))|dt,$$ which gives $$|\eta(\partial_y)|_{\inf}|q_2-q_1| \leq \bigg|\int_{\eta}\beta\bigg| \leq |\eta(\partial_y)|_{\infty}|q_2-q_1|.$$ Now using equations , and we get that $$\frac{1}{|\eta(\partial_y)|_{\infty}}\bigg(\int_P d\eta - c\bigg) \leq |q_2 - q_1| \leq \frac{1}{|\eta(\partial_y)|_{\inf}}\bigg(\int_P d\eta + c\bigg),$$ where $$|c|=\bigg|\int_{C_1 \cup C_2}d\eta\bigg|\leq 4\xi\ell\varepsilon|d\eta|_{\infty}.$$ The claim about the sign is also an immediate consequence of equations and . Fixing $U$ and $\epsilon_0$ {#sssection-fixU} --------------------------- Now, before carrying out the rest of the proof of Theorem \[thm-ballbox\], we will fix $\epsilon$ and $U$ once and for all. Assume we are given any adapted coordinate system with $p_0 \rightarrow 0$ and an adapted basis. Choose the domain $U$ containing $0$ so that - There exist some $i,j \in \{1,\dots,n\}$ such that for all $q \in U$ $$\label{eq-noninvolutive} d\eta_q(X_i,X_j) \neq 0.$$ This is possible due to assumption of non-integrability of at $0$ and of the continuity of $\eta$ and $d\eta$. This also implies $m(d\eta|_{\Delta})_{\inf}>0$. - For all $\ell,k=1,\dots,n$, $$\label{eq-normX} \frac{1}{2} \leq |X_{\ell}|_{\inf} \leq |X_{\ell}|_{\infty} \leq 2,$$ $$\label{eq-normX2} \frac{1}{2} \leq |X_{\ell} \wedge X_k|_{\inf} \leq |X_{\ell} \wedge X_k|_{\infty} \leq 2,$$ and $$\label{eq-normX3} \frac{1}{1.75} \leq |X_1\wedge \dots X_n\wedge \partial_y|_{\inf} \leq|X_1\wedge \dots X_n\wedge \partial_y|_{\infty} \leq 2.$$ These are possible since at the origin $X_{\ell}(0) = \partial_{\ell}$ and so the norms above are equal to $1$ at the point $0$. - The vector-fields $\{X_{\ell}\}_{\ell=1}^{n}$ are uniquely integrable on $U$ and have $C^1$ family of solutions $e^{tX_{\ell}}(q)$ defined on $[-\epsilon_0, \epsilon_0] \times \tilde{U}$ for some $\epsilon_0$ and $\tilde{U} \varsubsetneq U$ containing $0$. Finally fix $\epsilon_0$ so that, - For all $q \in \tilde{U}$ $$\label{eq-remaininside1} B((13 + (c+1)(2+6n)d_g)\epsilon_0,q) \subset U,$$ where $B(r,q)$ is defined with respect to the Euclidean norm and $c>0$ is the constant appearing in Lemma \[lem-Gromov\] and the remark \[rem-Gromov\] that follows. Although this inequality will be used in various places, one immediate consequence that we state now is that for all $q \in \tilde{U}$, $|t_{i_k}| \leq \epsilon_0$, where $i_k \in \{1, \dots ,n\}$ and $k=1,\dots ,n+4$, one has that $$\label{eq-remaininside2} e^{t_{i_k}X_{i_k}}\circ \dots \circ e^{t_{i_1}X_{i_1}}(q) \in U.$$ That is starting at $q$, even we apply the flows consecutively $n+4$ times up to time $\epsilon_0$ we still stay in the domain $U$ (for this $B(q, 2(n+4))$ is sufficient). This also guarantees us that on $[-\epsilon_0,\epsilon_0] \times \tilde{U}$ this composition of flows is a $C^1$ map with respect to $q$ and $t_{i_k}$. - Also the following are satisfied: $$\label{eq-estimate1} |d\eta|_{\infty}\tilde{C}\omega_{13(c+1)\epsilon_0}(4 + \tilde{C}\omega_{13(c+1)\epsilon_0})(2n)^{n+7}d^2_g < \frac{1}{4}|d\eta(X_i,X_j)|_{\inf},$$ $$\label{eq-estimate3} 3\epsilon_0 \leq \delta.$$ where $\delta>0$, $c>0$ are the constants appearing in Lemma \[lem-Gromov\] (and the remark \[rem-Gromov\] that follows). These are possible since $\omega_0=0$ and $\omega_t$ is continuous, and for all $q\in U$ we have that $d\eta_q(X_i,X_j) > 0$. \[rem-start\]Some of these conditions were chosen for convenience and if one digs carefully into the proof, it can be seen that the numerical constants appearing in the expressions are not sharp. However sharp constants are not relevant for our purposes so in order not to introduce additional complexity to the exposition, we will not attempt to sharpen them. Also we note that several similar looking conditions were combined into a single condition in equation using the ”worst” one, in order to decrease the number of conditions. Proof of Theorem \[thm-ballbox\] {#subsec-ballboxproof} -------------------------------- The part of the Theorem \[thm-ballbox\] about smooth adapted coordinates is divided into two seperate parts. First given $\epsilon$, we will construct a certain $n-$dimensional manifold $\mathcal{W}_{\epsilon}$ using the adapted basis $X_i$, which is admissible and transverse to $\partial_y$ direction. This is carried out in subsubsection \[ssec-partI\]. Then, next we will describe how the sub-Riemannian ball spreads around these manifolds in subsubsection \[ssection-spread\]. The results we obtain in these sections will then quickly lead us to the proof of the theorem. The construction of $\mathcal{W}_{\epsilon}$ will be based on the Proposition \[prop-uniquebasis\] while the description of the spread of the sub-Riemannian ball will use Proposition \[prop-stokessubriemann\]. The proofs will make it clear that the regularity of the bundle plays an important role in the shape of the manifolds $\mathcal{W}_{\epsilon}$ but the spread of the sub-Riemannian ball depends mainly only on geometric properties of $\Delta$, in particular the non-involutivity amount. By assumptions of the theorem, we have a $p_0 \in M$ where the condition of non-integrability is satisfied and that $\mathcal{A}_0^1(\Delta)(M)$ is equipped with a continuous exterior differential $\{V,\eta,d\eta\}$. We assume we are given an adapted coordinate system with a domain $U$ and an adapted basis which satisfies the properties listed in section \[sssection-fixU\]. We let $\Pi_x$ denote the projection to the $x$ coordinates and $\Pi_y$ to $y$ coordinates. ### Part I: Construction of $\mathcal{W}_{\epsilon}$ and Its Properties {#ssec-partI} Given any $|\epsilon| \leq \epsilon_0$ we can define the function $T_{\epsilon}: (-\epsilon,\epsilon)^n \rightarrow V$ by $$T_{\epsilon}: (t_1,\dots,t_n) \rightarrow e^{t_nX_n}\circ \dots \circ e^{t_1X_1}(0).$$ This function is $C^1$ by condition . Notice also that it is 1-1 since $X_i$ are uniquely integrable and since due to their form, an integral curve of $X_i$ can intersect an integral curve of $X_j$ for $i\neq j$ only once. Therefore the image of $T$ which we denote as $\mathcal{W}_{\epsilon}$ is a $C^1$ surface. Moreover, every point on it is obviously accessible. Finally it can be given as a graph over $(x_1,\dots,x_n)$, in fact due to the form of the vector-fields $T_{\epsilon}(x_1,\dots,x_n) = (x_1,\dots,x_n, a(x_1,\dots,x_n))$ for some $C^1$ function $a$. In particular note that if $(x,y)= T_{\epsilon}(t_1,\dots,t_n)$ then $|x| = |t|$. By the condition , $\mathcal{W}_{\epsilon} \subset U$. Two important properties that we will use often are given in the next lemma: \[lem-propertiesofW\] Let $\mathcal{W}_{\epsilon}$ be as defined above with $\epsilon \leq \epsilon_0$. Then, - For any $p=(x,y) \in U$ with $|x^i| \leq \epsilon$, there exists a unique $q \in \mathcal{W}_{\epsilon}$ with the same $x$ coordinates as $p$. - For any $q=(x,y) \in \mathcal{W}_{\epsilon}$ one has that $|y| \leq |x|\tilde{C}\omega_{2|x|}$. For the first item simply note that $T_{\epsilon}(t_1,\dots,t_n) = (t_1,\dots,t_n, a(t_1,$ $\dots,t_n))$ which is a map defined for $|t_i| \leq \epsilon$. So if $|x^i| \leq \epsilon$ then we have $T_{\epsilon}(x^1,\dots,x^n)=(x,a(x))$. Uniqueness then follows from the observation that it is a graph over the $x$ coordinates. For the second let $q=T(t_1,\dots,t_n)$ and $\tilde{X}_i = sign(t_i)X_i$. Then, $q$ lies at the end of a piecewise $C^1$ curve $\tau :[0,|t|] \rightarrow U$ which is a concatenation of the curves $s\rightarrow e^{s \tilde{X}_i}(e^{t_{i-1} X_{i-1}}(\dots e^{t_1 X_1}(0)))$ for $0 \leq s \leq |t_i|$. So for a.e $s \leq |t|$, $\dot{\tau}(s) = \tilde{X}_{\ell(s)}(\tau(s))$ for some piecewise constant function $\ell(t)$. Consider the vector-field $Z(s,p) = sign(t_{\ell(s)})\partial_{\ell(s)}$ with an integral curve $v(s)$ starting at $0$ which is a curve in the $x$ plane. We have $\Pi_x(\tau(s)) = \Pi_x(v(s))$ and $\Pi_y(v(s)) = 0$. So $$|\tau(s)-v(s)| = |\Pi_y(\tau(s)-v(s))|=|\Pi_y(\tau(s))| .$$ But since $|t| =|x|$, $$|\tau(s)-v(s)| \leq \int_0^{s}| X_{\ell(w)}(\tau(w))- \partial_{\ell(w)}(v(w))|dw \leq |x|\sup_{w \leq t}|X_{\ell(w)}(\tau(w))- \partial_{\ell(w)}(v(w))|.$$ Then, using $\partial_{\ell(w)}(v(w)) = X_{\ell(w)}(0)$ and $|\tau(w)| \leq |\tau| \leq 2|x|$ (by condition ), one has that $$|\tau(s)-v(s)| \leq |x| \tilde{C}\omega_{2|x|},$$ which implies $|y| \leq |x| \tilde{C}\omega_{2|x|}$. ### PartII: Description of the Spread of $B_{\Delta}(0,\epsilon)$ around $\mathcal{W}_{\epsilon_0}$ {#ssection-spread} Define for $p=(x,y)$ with $|x| \leq \epsilon_0$, $d_y(p,\mathcal{W}_{\epsilon_0})$ to be the distance of $p$ to the point $q$ on $\mathcal{W}_{\epsilon_0}$ with the same $x$ coordinate as $p$ (by Lemma \[lem-propertiesofW\], there is a unique such point). For $\epsilon \leq \epsilon_0$ we can define the box around $\mathcal{W}_{\epsilon_0}$ (using the given smooth adapted coordinates) as follows: $$\mathcal{BW}_{\epsilon_0}(K_1, \epsilon) = \{ p=(x,y) \in U, \ |x| \leq \epsilon \ \ s.t \ \ d_y(p,\mathcal{W}_{\epsilon_0}) \leq K_1\epsilon^2\}.$$ Then, the lemma we will prove in this section using Proposition \[prop-stokessubriemann\] is the following (see figure \[fig-ballbox\]): \[lem-distancefromW\]For $\mathcal{BW}_{\epsilon_0}(K, \epsilon)$ as defined above and $K_1,K_2$ constants given in Theorem \[thm-ballbox\], we have $$\mathcal{BW}_{\epsilon_0}(K_1, \frac{\epsilon}{4d_g}) \subset B_{\Delta}(0,\epsilon) \subset \mathcal{BW}_{\epsilon_0}(K_2, 2nd_g\epsilon).$$ ![Pictorial representation of emma \[lem-distancefromW\].[]{data-label="fig-ballbox"}](ballbox){width="200px"} \[Proof of $\mathcal{BW}_{\epsilon_0}(K_1, \frac{1}{4d_g}\epsilon) \subset B_{\Delta}(0,\epsilon)$\] Given $\epsilon < \frac{\epsilon_0}{2nd_g}$, to prove the inclusion we need to show that if $p=(x,y)$ is such that $|x| \leq \frac{1}{4d_g}\epsilon$ and $d(p,\mathcal{W}_{\epsilon_0}) \leq K_1\frac{1}{16d^2_g}\epsilon^2$, then $d_{\Delta}(x,0) \leq \epsilon$. For this it is enough to find a path $\psi$ between $p$ and $0$ which is admissible and $\ell(\psi) \leq \epsilon$, since $d_{\Delta}(x,0) \leq \ell(\psi)$ where here $\ell(\cdot)$ is the length defined with respect to the metric $g$ defined on $\Delta$. This path $\psi$ will be written as the composition of two paths $\psi = \gamma \circ \tau$ where $\tau$ is a path that allows to travel along $x$ direction and $\gamma$ is a path which will allow us to travel in the $y$ direction and for which we will apply the Proposition \[prop-stokessubriemann\]. First we construct $\tau$. Let $q_1$ be the point in $\mathcal{W}_{\epsilon_0}$ that has the same $x$ coordinates as $p$ (since $|x| \leq \frac{\epsilon}{4d_g} < \epsilon_0$ such a point exists). Let $\tau$ be the curve in $\mathcal{W}_{\epsilon_0}$ described in Lemma \[lem-propertiesofW\] that connects $0$ to $q_1$. It is curve defined from $[0,|x|]$ to $U$. We have that by condition $$\label{eq-taulength} |\tau| \leq 2|x| \leq \frac{\epsilon}{2d_g}$$ Using equation , we have that $\ell(\tau) \leq d_g |\tau| \leq \frac{\epsilon}{2}$. Now we need to build the other admissible curve $\gamma$ that starts at $q_1$ and ends at $p$ and such that $\ell(\gamma) \leq \frac{\epsilon}{2}$. If we build this curve then we are done since both $\tau$ and $\gamma$ are admissible and $\ell(\gamma \circ \tau) \leq \epsilon$. The amount that we need to travel in the $\partial_y$ direction is given by the condition $d(p,\mathcal{W}_{\epsilon_0}) \leq K_1\frac{1}{16d^2_g}\epsilon^2$ which tells us that $$\label{eq-neededyrise} |q_1 - p| \leq K_1\frac{1}{16d^2_g}\epsilon^2.$$ Now we start building $\gamma$. The idea is similar to the one employed in [@Arn89] (see Section 36 of Chapter 7 and Appendix 4) which defines exterior differential of 1-forms in terms of loops tangent to the bundles that they annihilate. For any $\tilde{\epsilon} \leq \epsilon_0$, consider the curves defined on $[0,\tilde{\epsilon}]$ (the indexing of points below are chosen so that they are in direct alignment with Proposition \[prop-stokessubriemann\]): $$\kappa_1(s)=e^{sX_i}(q_1) \quad \quad \kappa_2(s)= e^{sX_j}( \kappa_1(\tilde{\epsilon})) \quad \quad \kappa_2(\tilde{\epsilon})= q,$$ $$\kappa_3(s)=e^{-sX_i}(q) \quad \quad \kappa_4(s)= e^{-sX_j}(\kappa_3(\tilde{\epsilon})) \quad \quad \kappa_4(\tilde{\epsilon})= q_2.$$ These $X_i$ and $X_j$ are the vector-fields that satisfy $|d\eta(X_i,X_j)|_{\inf}>0$. We let $\gamma(s;\tilde{\epsilon},q_1)$ be the parametrization for the curve obtained as the concatenation $\kappa_4\circ \kappa_3 \circ \kappa_2 \circ \kappa_1$ so that $\dot{\gamma}(s)=X_{\ell(s)}(\gamma(s;\tilde{\epsilon},q_1))$ for a.e $s$ with $\ell(s)=i,j$. Also $q_2=\gamma(4\tilde{\epsilon}; \tilde{\epsilon}, q_0)$ becomes the end point of this curve (see figure \[fig-part1\]). This curve is defined on $[0,4\tilde{\epsilon}]$ to $U$. Moreover by the condition , we have that $\gamma(s;\tilde{\epsilon},q_0)$ is continuous with respect to $\tilde{\epsilon}$ since it is equal to $\gamma(s;\tilde{\epsilon},\tau(\tilde{\epsilon}))$ which is just a composition of $n+4$ integral curves of some $\pm X_{\ell}$ with integration times less than $\tilde{\epsilon}_0 \leq \epsilon_0$. Denoting the image of this curve as $\gamma_{\tilde{\epsilon}}$ we have $\ell(\gamma_{\tilde{\epsilon}}) \leq d_g|\gamma_{\tilde{\epsilon}}| \leq 8d_g\tilde{\epsilon}$. Therefore the following lemma is sufficient to get $\gamma$. ![Trying to get $p=q_2$.[]{data-label="fig-part1"}](part1){width="200px"} \[lem-final\]There exists some $\tilde{\epsilon}$ such that $\gamma(4\tilde{\epsilon};\tilde{\epsilon},q_1) = p$ with $\tilde{\epsilon} \leq \frac{\epsilon}{16d_g}$ If we can show this we are done since then setting $\gamma=\gamma_{\tilde{\epsilon}}$, $\ell(\gamma) \leq d_g|\gamma| \leq 8d_g\tilde{\epsilon} \leq \frac{1}{2}\epsilon$. To prove the lemma it is sufficient to show the following: - There exists some $\bar{\epsilon}$ with $\bar{\epsilon} \leq \frac{\epsilon}{16d_g}$ such that $|\gamma(4\bar{\epsilon};\bar{\epsilon},q_1)-q_1| \geq K_1\frac{1}{16d^2_g}\epsilon^2.$ - By reverting either $X_i$ or $X_{j}$ we can go in the opposite direction along the $\partial_y$. The last item allows us to travel in both directions (with respect to $\partial_y$) while the second item guarantees that we can travel more than $|p-q_1|$. Then, since $\gamma(4{\epsilon};{\epsilon},q_1)$ continuous with respect to ${\epsilon}$ and it always lies on the one dimensional $\partial_y$ axis passing through $p$ and $q_1$ and satisfies $\gamma(0;0,q_1)=q_1$, there exists some $\tilde{\epsilon} \leq \bar{\epsilon}$ (so that automatically $\tilde{\epsilon} \leq \frac{\epsilon}{16d_g}$ ) for which $q_2=\gamma(4\tilde{\epsilon};\tilde{\epsilon},q_1)=p$ and we are done with the lemma. So we will now prove these items. To apply Proposition \[prop-stokessubriemann\], set $$\gamma_1 = \kappa_1^{-1} \circ \kappa_2^{-1}, \quad \quad \gamma_2 = \kappa_3 \circ \kappa_4,$$ so that $\gamma_1(0)=\gamma_2(0) = q$ and $\gamma_1(\varepsilon_1)=q_1$, $\gamma_2(\varepsilon_2) = q_2$ which have the same $x$ coordinates due to the form of the vector-fields $X_i,X_j$. We also have (in the terminology of Proposition \[prop-stokessubriemann\]) $\varepsilon \leq 2\epsilon$, $\ell \leq 4\epsilon$. By conditions , , we have $$\xi \leq (2n)^{n+2}\tilde{C}\omega_{8\epsilon}.$$ Then, by condition we have that $B(2\ell ,q_0) \subset U$. Now it remains to build a surface $P$ at the point $q$ so that $P \subset \bar{X}_{q} \cap U$. Note that projection of $\gamma_i$ along $\partial_y$ to $\bar{X}_{q}$ is simply the boundary of a parallelogram $P$ whose sides are given by the vectors $X_i(q)$,$X_j(q)$ and have length less than $\ell \leq 4\epsilon$. Therefore given any point $z \in P$, $|z| \leq |z-q| + |q_1 - q| + |q_1|$. But $|q_1| \leq 2|x| \leq \frac{\epsilon}{2}$, $|q_1 - q| \leq \ell \leq 4\epsilon$, and $|z-q| \leq 2\ell \leq 8\epsilon$. So $|p| \leq 13\epsilon$ which by condition gives us that $P \subset U$. Therefore all the conditions required for application of \[prop-stokessubriemann\] is satisfied. So $$\frac{1}{|\eta(\partial_y)|_{\infty}}\bigg(\bigg|\int_P d\eta \bigg| - c\bigg) \leq |\gamma_1(\epsilon_1) - \gamma_2(\epsilon_2)|=|q_2-q_1|,$$ and $$sign\bigg(\int_{\beta}\eta \bigg) = sign\bigg(\int_P d\eta + c\bigg),$$ where also by condition \[eq-estimate1\], $$\label{eq-cnorm} |c| \leq 4 \ell \varepsilon \xi |d\eta|_{\infty} \leq (2n)^{n+7} \epsilon^2\tilde{C}\omega_{8\epsilon}|d\eta|_{\infty} \leq \frac{1}{4}|d\eta(X_i,X_j)|_{\inf}\epsilon^2.$$ To apply these result we need to calculate $\int_P d\eta$ which is the content of the next lemma: For $P$ as defined above, $$\label{eq-parallelogramintegral} \bigg|\int_{P}d\eta \bigg| \geq \frac{3}{4}\epsilon^2 |d\eta(X_i,X_j)|_{\inf}.$$ $$\label{eq-parallelogramintegral2} sign\bigg(\int_{P}d\eta \bigg) =sign(d\eta_{q}(X_i(q),X_j(q))$$ Note that $P$ is a parallelogram based at $q$ and always tangent to $X_i(q)$ and $X_j(q)$. We will denote by $P(s_1,s_2)$ a parametrization for $P$ such that $\frac{\partial P}{\partial s_1}(s_1,s_2)=X_i(q)$ and $\frac{\partial P}{\partial s_2}(s_1,s_2)=X_j(q)$ for all $0 \leq s_1,s_2 \leq \epsilon$. Then, we have in this parametrization, $$\int_{P}d\eta = \int_{0}^{\epsilon}ds_1 \int_{0}^{\epsilon}ds_2 \ \ d\eta_{P(s_1,s_2)}(X_i(q),X_j(q)).$$ But $$\begin{aligned} d\eta_p(X_i(q),X_j(q)) &= d\eta_p(X_i(p),X_j(p)) \\ &+ d\eta_p(X_i(q)-X_i(p),X_j(q)) + d\eta_p(X_i(q),X_j(q)-X_j(p))\\ &+d\eta_p(X_i(q)-X_i(p),X_j(q)-X_j(p)).\\ \end{aligned}$$ Note that $|X_{\ell}(q)-X_{\ell}(p)| \leq \tilde{C}\omega_{|q-p|}$ and for $p \in P$, $|q-p| \leq 13\epsilon$. So $$\begin{aligned} |d\eta_p(X_i(q)&-X_i(p),X_j(q)) + d\eta_p(X_i(q),X_j(q)-X_j(p))\\ &+d\eta_p(X_i(q)-X_i(p),X_j(q)-X_j(p))| \leq |d\eta|_{\infty}\tilde{C}\omega_{13\epsilon}(4 + \tilde{C}\omega_{13\epsilon}),\\ \end{aligned}$$ and therefore $$\label{eq-parallelogramestimate} d\eta_p(X_i(q),X_j(q)) \geq d\eta_p(X_i(p),X_j(p)) - |d\eta|_{\infty}\tilde{C}\omega_{13\epsilon}(4 + \tilde{C}\omega_{13\epsilon}),$$ and $$\label{eq-parallelogramestimate2} d\eta_p(X_i(q),X_j(q)) \leq d\eta_p(X_i(p),X_j(p)) + |d\eta|_{\infty}\tilde{C}\omega_{13\epsilon}(4 + \tilde{C}\omega_{13\epsilon}).$$ Then, using the condition we have that $$\label{eq-bounds} \frac{3}{4}d\eta_p(X_i(p),X_j(p)) \leq d\eta_p(X_i(q),X_j(q)) \leq \frac{5}{4}d\eta_p(X_i(p),X_j(p)).$$ So for all $p \in P$, $sign(d\eta_p(X_i(q),X_j(q)) )=sign(d\eta_p(X_i(p),X_j(p)) )$. But in $U$, $d\eta_p(X_i(p),X_j(p))$ is never $0$ and therefore never changes sign so this proves equation . This also means that $d\eta_p(X_i(q),X_j(q))$ never changes sign and using equation we get, $$\bigg|\int_{P}d\eta \bigg| = \int_{0}^{\epsilon}ds_1 \int_{0}^{\epsilon}ds_\frac{3}{4} |d\eta_{P(s_1,s_2)}(X_i(q),X_j(q))| \geq \epsilon^2 |d\eta(X_i,X_j)|_{\inf}.$$ Now we have that $sign(\int_{\beta}\eta) = sign(\int_P d\eta + c)$ where by the equation in the previous lemma and equation $$|c| \leq \frac{1}{4}|d\eta(X_i,X_j)|_{\inf}\epsilon^2\leq \frac{1}{3}\bigg |\int_{P}d\eta \bigg |.$$ So $sign(\int_P d\eta + c)=sign(\int_Pd\eta) = sign(d\eta_{q}(X_i,X_j))$ (which can be reverted by changing the direction of, say $X_i$). Moreover $$|\alpha(4\epsilon;\epsilon,q_1)-q_1| \geq \frac{1}{|\eta(\partial_y)|_{\infty}}\bigg(\bigg|\int_{P}d\eta \bigg| - |c|\bigg) \geq \frac{8}{12}\epsilon^2 \frac{|d\eta(X_i,X_j)|_{\inf}}{|\eta(\partial_y)|_{\infty}}.$$ Since by condition \[eq-normX2\] we have $|d\eta(X_i,X_j)|_{\inf} \geq \frac{1}{1.75}m(d\eta|_{\Delta})_{\inf}$, replacing now $\epsilon$ with $\bar{\epsilon}$ $$\label{eq-endpoint} |\alpha(4\bar{\epsilon};\bar{\epsilon},q_1)-q_1| \geq \frac{8}{21}\bar{\epsilon}^2 \frac{m(d\eta|_{\Delta})_{\inf}}{|\eta(\partial_y)|_{\infty}}.$$ We require this quantity to be bigger than $ K_1\frac{1}{16d^2_g}\epsilon^2$ $\geq K_1\frac{1}{16d^2_g}256 d_g^2\bar{\epsilon}^2$=$ 16K_1\bar{\epsilon}^2$ So we need to satisfy $$\frac{1}{42}\bar{\epsilon}^2 \frac{m(d\eta|_{\Delta})_{\inf}}{|\eta(\partial_y)|_{\infty}} \geq K_1\bar{\epsilon}^2.$$ This is satisfied since $$K_1 = \frac{1}{42} \frac{m(d\eta|_{\Delta})_{\inf}}{|\eta(\partial_y)|_{\infty}}$$ This finishes the proof of the inclusion $\mathcal{BW}_{\epsilon_0}(K_1,\frac{\epsilon}{4d_g}) \subset B_{\Delta}(0,\epsilon) $. \[Proof of $B_{\Delta}(0,\epsilon) \subset \mathcal{BW}_{\epsilon_0}(K_2, 2nd_g\epsilon)$\] To prove this inclusion we need to prove that if $p=(x,y) \in U$ such that $d_{\Delta}(p,0) \leq \epsilon$, then $|x| \leq 2nd_g\epsilon$ and $d(p,\mathcal{W}_{\epsilon_0}) \leq K_24n^2d_g^2\epsilon^2$ $d_{\Delta}(p,0) \leq \epsilon$ means that there exists an admissible curve $\gamma_1$ that connects $0$ to $p$ such that $\ell(\gamma_1) \leq 2\epsilon$. Then $|\gamma_1| \leq d_g\ell(\gamma_1) \leq 2d_g\epsilon$. Since for all $i$ $|x^i| \leq |\gamma_1|$ we get $|x| \leq 2nd_g\epsilon $ trivially. So it remains to estimate $d(p,\mathcal{W}_{\epsilon_0})$. The rest of the proof follows the method given in [@Gro96] by Gromov. The only essential difference is that the term $|d\eta|_{\infty}$ appearing there is replaced by $|d\eta|_{\Delta}|_{\infty}$ which will be thanks to Proposition \[prop-stokessubriemann\]. We first state a lemma due to Gromov, ([@Gro83], Sublemma $3.4$.B, see also Corollary $2.3$ in [@Sim10]) specialized to lower dimension: \[lem-Gromov\]For every compact Riemannian manifold $S$, there exists constants $\delta_S,c_S>0$ such that for any 1-cycle $\gamma$ in $S$ with length less than $\delta_S$, there exists a 2-chain $\Gamma$ in $S$, bounded by $\gamma$ such that $$|\Gamma| \leq c_S |\gamma|^2,$$ and $\Gamma$ is contained in the $\varrho$ neighbourhood of $\gamma$, where $\varrho=c_S|\gamma|$. \[rem-Gromov\] Here $|\cdot|$ denotes the norm of the chosen metric on $S$ and the constants $\delta_S,c_S$ depend on this metric. We will always apply this result to closures of precompact open submanifolds of $n$ dimensional Euclidean space which are given by $S = U\cap \text{span}\{\frac{\partial}{\partial x^i}\}|_{q}$ for some $q \in U$. They will have the usual Euclidean inner product. These spaces are affine translates of each other (since $U$ is a box) all equipped with Euclidean metrics for which these affine translations are isometries. Therefore it is clear that whatever constant $c_S$ and $\delta_S$ works for one, also works for the other. We denote these constants simply by $\delta$ and $c$. Actually note that the proof of this lemma ([@Gro83], Sublemma $3.4$.B), starts by first finding an embedding $\phi$ of $S$ into some $\mathbb{R}^N$. Then, a 2-chain with the required properties is constructed inside $\mathbb{R}^N$ and normally projected back to $S$. So the constant $c_S$ and $\delta_S$ depend only on the embedding and $N$. In our case since we are working with affine translates of open subsets of $\mathbb{R}^n$, the constants $c$ and $\delta$ depend only on the dimension $n$ since the embedding becomes an isometry and normal projection is not required. Now since $2nd_g\epsilon \leq \epsilon_0$, there exists a point $q$ on $\mathcal{W}_{\epsilon_0}$ which has the same $x$ coordinates as $p$. This means $d(p,\mathcal{W}_{\epsilon_0})=|q-p|$ and so we need to show that $|q-p| \leq K_24n^2d_g^2\epsilon^2$. We can connect $p$ to $0$ by first going from $0$ along $\mathcal{W}_{\epsilon_0}$ with an admissible path $\gamma_2$ to $q$ and then going in the $\partial_y$ direction with a length parametrized segment $\beta$ (see figure \[fig-part2\]). Then, $\gamma_1, \gamma_2, \beta$ forms a 1-cycle. ![The Curve $\gamma_2\circ \beta \circ \gamma^{-1}_1$.[]{data-label="fig-part2"}](part2){width="200px"} Now to apply Proposition \[prop-stokessubriemann\] we have $\gamma_1(0)=\gamma_2(0)=0$ and $\gamma_1(\epsilon_1)=p$, $\gamma_2(\epsilon_2)=q$ which have the same $x$ coordinates. Then $$\ell_1 \leq 2d_g\epsilon \leq \epsilon_0, \quad \quad \ell_2 \leq 2|x| \leq 4nd_g\epsilon \leq 2\epsilon_0, \quad \quad \ell \leq 4nd_g\epsilon \leq 2\epsilon_0,$$ $$\varepsilon_1 \leq 2d_g\epsilon \leq \epsilon_0, \quad \quad \varepsilon_2 \leq |x| \leq 2nd_g\epsilon \leq \epsilon_0, \quad \quad \varepsilon \leq \epsilon_0.$$ Also by conditions and we have that $$\xi \leq (2n)^{n+2}\tilde{C}\omega_{4\epsilon_0}.$$ Therefore by condition , $B(0, 2\ell) \subset U$. The projection of $\gamma_1$ and $\gamma_2$ along $\partial_y$ to $\bar{X}_{0}$ formes a 1-cycle $\alpha=\alpha^{-1}_1 \circ \alpha_2$ which has length less than $2d_g\epsilon +4nd_g\epsilon \leq 3\epsilon_0$. By the condition this is less than $\delta$ so by Lemma \[lem-Gromov\], $\alpha$ bounds a 2-cycle $P \subset \bar{X}_{0}$ such that $|P| \leq c(2+4n)^2d^2_g\epsilon^2$ and which is in the $c(2+4n)d_g\epsilon$ neighbourhood of $\alpha$. We have that for any $z \in P$ $$\begin{aligned} d(z,0) &\leq |\alpha| + c(2+4n)d_g \\ & \leq (c+1)(2+6n)d_g\epsilon,\\ \end{aligned}$$ so by condition we have that $P \subset U \cap \bar{X}_{0}$. Then, the requirements of the Proposition \[prop-stokessubriemann\] are satisfied. Since $4\ell\varepsilon\xi \leq (2n)^{n+7}d_g^2\tilde{C}\omega_{8\epsilon_0}\epsilon^2$, we have $$\label{eq-lastestimate} |p-q| \leq \frac{1}{|\eta(\partial_y)|_{\inf}}\bigg(\bigg|\int_P d\eta \bigg| + 4\ell\varepsilon\xi |d\eta|_{\infty}\bigg) \leq \frac{1}{|\eta(\partial_y)|_{\inf}}\bigg( \bigg |\int_P d\eta \bigg| + (2n)^{n+7}d_g^2\tilde{C}\omega_{8\epsilon_0}\epsilon^2|d\eta|_{\infty}\bigg).$$ We need to estimate $|\int_P d\eta|$. For $P$ as defined above, $|\int_P d\eta| \leq 4n^2c(2+4n)^2d^2_g\epsilon^2 |d\eta|_{\Delta}|_{\infty}.$ Since $P$ is inside $\bar{X}_{0}$, it is everywhere tangent to $X_i(0)=\partial_i$ so $|\int_P d\eta| \leq |P| |d\eta|_{\Delta_0}|_{\infty}$. But for all $p\in U$, $$|d\eta_p|_{\Delta_0}|_{\infty} \leq n^2 \sup_{\ell,k=1,\dots,n}|d\eta_p(\partial_{\ell},\partial_k)|.$$ Since $X_{\ell}(0) = \partial_{\ell}$, then for any $z \in P$ we have as in equation , $$|d\eta_z(\partial_{\ell},\partial_k)| \leq |d\eta_z(X_{k}(z),X_l(z))| + |d\eta|_{\infty}\tilde{C}\omega_{|z|}(4 + \tilde{C}\omega_{|z|}).$$ Then, since $|z| \leq (c+1)(2+6n)d_g\epsilon \leq 4(c+1)\epsilon_0$ by we get by condition $$|d\eta_p(\partial_{\ell},\partial_k)| \leq 2|d\eta_p(X_{\ell}(p),X_l(p))| \leq 4 |d\eta|_{\Delta}|_{\infty},$$ which implies $$|d\eta_p|_{\Delta_0}|_{\infty} \leq 8n^2 |d\eta|_{\Delta}|_{\infty}.$$ So since $|P| \leq c(2+4n)^2d^2_g\epsilon^2$, $$\bigg|\int_P d\eta \bigg| \leq 4n^2c(2+4n)^2d^2_g\epsilon^2|d\eta|_{\Delta}|_{\infty}.$$ Then this lemma and equation gives $$|p-q| \leq \frac{1}{|\eta(\partial_y)|_{\inf}}(4n^2c(2+4n)^2d^2_g\epsilon^2 |d\eta|_{\Delta}|_{\infty} + (2n)^{n+7}d_g^2\tilde{C}\omega_{8\epsilon_0}\epsilon^2|d\eta|_{\infty}).$$ Condition implies then $$|p-q| \leq 5n^2c(2+4n)^2d^2_g\epsilon^2 \frac{ |d\eta|_{\Delta}|_{\infty}}{|\eta(\partial_y)|_{\inf}}.$$ So to be able to satisfy $|p-q| \leq 4K_2n^2d_g^2\epsilon^2$, it is sufficient to satisfy $$9n^2c(2+4n)^2d^2_g\epsilon^2 \frac{ |d\eta|_{\Delta}|_{\infty}}{|\eta(\partial_y)|_{\inf}} \leq 4K_2d_g^2\epsilon^2,$$ which is satisfied with $$K_2 = 42(1+2n)^2c \frac{ |d\eta|_{\Delta}|_{\infty}}{|\eta(\partial_y)|_{\inf}}.$$ ### Rest of the Proof of Theorem \[thm-ballbox\] Now it is easy to prove the rest using Lemmas \[lem-propertiesofW\] and \[lem-distancefromW\]. The latter one says that $$\mathcal{BW}_{\epsilon_0}(K_1, \frac{\epsilon}{4d_g}) \subset B_{\Delta}(0,\epsilon) \subset \mathcal{BW}_{\epsilon_0}( K_2, 2nd_g\epsilon),$$ while the former one says that for $q=(x,y) \in \mathcal{W}_{\epsilon_0}$ we have that $$|y| \leq |x|\tilde{C}\omega_{2|x|}.$$ First we prove that $\mathcal{BW}_{\epsilon_0}( K_2, 2d_g\epsilon) \subset H_2^{\omega}(0,K_2,2d_g\epsilon)$. If $(x,y) \in \mathcal{BW}_{\epsilon_0}($ $K_2, 2nd_g\epsilon)$, then $|x| \leq 2nd_g\epsilon$ and $d(p,\mathcal{W}_{\epsilon_0}) \leq 4K_2n^2d_g^2\epsilon^2$. This means there exists $q=(x,z) \in \mathcal{W}_{\epsilon_0}$ such that $|z-y| \leq 4K_2n^2d_g^2\epsilon^2$. But we know that $|z| \leq |x|\tilde{C}\omega_{2|x|}$. So $|y| \leq |z| + |z-y| \leq 4K_2n^2d_g^2\epsilon^2 + |x|\tilde{C}\omega_{2|x|}$ which means that $(x,y) \in H_2^{\omega}(0,K_2,2nd_g\epsilon)$. Now we prove $D_2^{\omega}(0,\frac{1}{K_1}, \frac{\epsilon}{4d_g}) \subset \mathcal{BW}_{\epsilon_0}(K_1, \frac{\epsilon}{4d_g})$. If $p=(x,y) \in D_2^{\omega}(0,\frac{1}{K_1},$ $\frac{\epsilon}{4d_g})$, then $|x| + \sqrt{\frac{1}{K_1}(|x|\omega_{2|x|} + |y|)} \leq \frac{\epsilon}{4d_g}$. So we have $|x| \leq \frac{\epsilon}{4d_g}$ and $(|x|\omega_{2|x|} + |y|) \leq K_1\frac{\epsilon^2}{16d_g^2}$. Let $q=(x,z) \in \mathcal{W}_{\epsilon_0}$ with $|z| \leq |x|\tilde{C}\omega_{2|x|}$. Therefore $|y| + |z| \leq K_1\frac{\epsilon^2}{16d_g^2}$. So $d(p,\mathcal{W}_{\epsilon_0})= |p-q|=|y-z| \leq |y| + |z| \leq K_1\frac{\epsilon^2}{16d_g^2}$. This implies that $(x,y) \in \mathcal{BW}_{\epsilon_0}(K_1, \frac{\epsilon}{4d_g})$. Now finally we prove the statement about the existence of $C^1$ adapted coordinates. We will build the $C^1$ coordinate system using the manifold $\mathcal{W}_{\epsilon_0}$. Note that for each $\epsilon$, $\mathcal{W}_{\epsilon}$ are $C^1$ surfaces given as the image of the map $T_{\epsilon}(t_1,\dots,t_n)$ with $|t_i| \leq \epsilon$ and for $\epsilon_1 < \epsilon_2$, $\mathcal{W}_{\epsilon_1} \subset \mathcal{W}_{\epsilon_2}$. Define the transformation $\phi: V \rightarrow U$ on some appropriately sized domain $V\subset U$ such that $$\label{eq-c1adapted} \phi(x,y) = (x,y-T_{\epsilon_0}(x)).$$ Then, it is clear that this map is a $C^1$ diffeomorphism onto its image and takes each $\mathcal{W}_{\epsilon} \cap V$ (for $\epsilon \leq \epsilon_0$) to the $x$ plane. It also maps $X_{\ell}$ restricted to $\mathcal{W}_{\epsilon}$ to $\partial_{\ell}$ on the $x$ plane. This is again an adapted coordinate system. In particular in this adapted coordinate system $(x,y) \in \mathcal{BW}_{\epsilon_0}( K, \epsilon)$ simply implies $|x| \leq \epsilon$ and $|y| \leq K\epsilon^2$. So we get $\mathcal{BW}_{\epsilon_0}(K_1, \frac{\epsilon}{4d_g}) = \mathcal{B}(0,K_1, \frac{\epsilon}{4d_g})$ and $\mathcal{BW}_{\epsilon_0}( K_2, 2nd_g\epsilon)=\mathcal{B}_2(0,K_2,2nd_g\epsilon)$, which finishes the proof. Applications {#sec-continuousexteriordifferential} ============ In this section we first present interesting properties of the “continuous exterior differential" object and give some examples of bundles that satisfy the requirements of our main theorems. We also discuss the relations between our theorems and the works in [@MonMor12] and [@Kar11]. In the second part we present a tentative application to dynamical systems. Continuous Exterior Differential -------------------------------- In this subsection we study some important properties of $\Omega^k_d(M)$, including of course showing that there are some non-integrable examples inside $\Omega^1_d(M) \setminus \Omega^1_1(M)$, so that we cover some examples that were not covered by $C^1$ sub-Riemannian geometry. We start with an alternative characterization that already exists in [@Har64]: \[prop-char\] A differential n-form $\eta$ has a continuous exterior differential if and only if there exists a sequence of differential n-forms $\eta^k$ such that $\eta^k$ converges in $C^0$ topology to $\eta$ and $d\eta^k$ converges to some differential $n+1$ form which becomes the exterior differential of $\eta$. The sufficiency part of this proposition is quite direct since uniform convergence of the differential forms involved also means uniform convergence of the Stokes relation. The necessity can be obtained by locally mollifying the differential forms to $\eta^k=\phi_k\circ \eta$ (where $\phi_k$ are mollifiers). Then, under the integral $d\eta^k = d(\phi_k\circ \eta)= \phi^k d\eta$ (thanks to the fact that $\phi_k$ are compactly supported) so the Stokes relations convergence. Since Stokes relation holds for every surface and its boundary, it is enough to obtain that $d\eta^k$ themselves converge to $d\eta$. We start with two examples defined on $U \subset \mathbb{R}^n$ and then we show how to paste local differential forms with continuous exterior differential to obtain global ones on manifolds. This is an example from [@Har64]. Let $f$ be any $C^1$ function so that $df$ is $C^0$. Then setting $\eta = df$, we have that $\eta$ has continuous exterior differential $0$. Therefore $\eta \wedge d\eta=0$ and this gives us integrable examples inside $\Omega^1_d(M) \setminus \Omega^1_1(M)$. The following proposition allows us to construct examples that both have continuous exterior differentials and are non-integrable: \[prop-example\] Let $\eta =a(x,y,z)dy - b(x,y,z)dx - c(x,y,z)dz$ where $b$ and $c$ are continuous functions that are $C^1$ in $y$ variable, $b$ is $C^1$ in $z$ variable while $c$ is $C^1$ in $x$ variable and $a$ is a continuous function which is $C^1$ in the $x,z$ variables and $a>0$ everywhere. Assume moreover that for some $p \in \mathbb{R}^3$, $$(-(a_x + b_y)c + (a_z + c_y)b - (b_z-c_x)a)(p)>0.$$ Then, $\ker(\eta)=\Delta$ satisfies the conditions of Theorem \[thm-ballbox\] at $p$. By assumptions on regularity of the functions, we can find $C^1$ functions (by mollification) $a^k$, $b^k$ and $c^k$ that converge in $C^0$ topology to $a$, $b$ and $c$ such that $b$ and $c$’s partial derivatives with respect to $y$, partial derivative of $b$ with respect to $z$ and partial derivative of $c$ with respect to $x$ also converges (to the respective derivatives of $b$ and $c$) and the partial derivative of $a^k$ with respect to $x,z$ converge to the respective partial derivatives of $a$. Then we define $$\eta^k = a^kdy + b^kdx + c^kdz,$$ so that $$d\eta^k = a^k_xdx \wedge dy + a^k_z dz \wedge dy +b^k_y dy \wedge dx + c^k_y dy \wedge dz +b^k_z dz \wedge dx +c^k_x dx \wedge dz.$$ By assumption $\eta^k$ converges in $C^0$ topology to $\eta$ and $d\eta^k$ converges to $$d\eta = a_xdx \wedge dy + a_z dz \wedge dy +b_y dy \wedge dx + c_y dy \wedge dz +b_z dz \wedge dx +c_x dx \wedge dz.$$ Then we have that $$\eta \wedge d\eta(p) =(-(a_x + b_y)c + (a_z + c_y)b - (b_z-c_x)a)(p) dx^1 \wedge dx^2 \wedge dy.$$ The condition given in the theorem is then the non-integrability assumption required. Now we give a simple example: To create a non-integrable and non-differentiable example we have to satisfy $ (c_x- b_z+ b_yc - c_yb)(p)>0$ at some point where $\eta$ is non-differentiable. Consider $$b(x,y,z) = \text{sin}(y)e^{x^{\frac{1}{2}}}z \quad \quad c(x,y,z) = \text{cos}(y)e^{(z+2)^{\frac{2}{3}}}x,$$ which gives $$\begin{aligned} (c_x- b_z+ b_yc - c_yb)&= \\ \text{cos}(y)e^{(z+2)^{\frac{2}{3}}}- \text{sin}(y)e^{x^{\frac{1}{2}}} &+ e^{(z+2)^{\frac{2}{3}}} e^{x^{\frac{1}{2}}}zx.\\ \end{aligned}$$ Then for instance, $\eta$ (or any of its product with a differentiable function) is non-differentiable at $x=0, z=0, y=0$ but $\eta \wedge d\eta(0) = e^{2^{\frac{2}{3}}} dx \wedge dy \wedge dy$. Therefore there exists neighbourhoods on which $\eta$ is non-differentiable and yet satisfies the sub-Riemannian properties mentioned in this paper. To create a non-Hölder example out of this, we can replace for instance $e^{x^{\frac{1}{2}}} $ with the function $f(x) = \frac{1}{\text{log}(x)}$ (setting it to be $0$ at $0$) and we have an example that is non-Hölder at $x=0, z=0, y=0$ and also non-integrable. To compare our results to the results given in [@MonMor12] and [@Kar11] we need to work with a certain basis of $\Delta$. The most canonical one is the ones that we have been working with, the adapted basis. In the case when $a=1$, the bundle is spanned by two vector-fields of the form $X_1 = \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}, X_2 = \frac{\partial}{\partial z} + c \frac{\partial}{\partial y}$ and the differentiability assumptions above mean that $[X_1,X_2]$ exists and is continuous but not Lipschitz continuous. If $a$ is non-constant then we have $X_1 = \frac{\partial}{\partial x} + \frac{b}{a}\frac{\partial}{\partial y}$, $X_2 = \frac{\partial}{\partial z} + \frac{c}{a}\frac{\partial}{\partial y}$ and it is not a priori clear that whether if the Lie derivatives can be defined since $a$ is not differentiable in $y$. However we see that we can define it via: $$[X_1,X_2]= d\eta(X_2,X_1)\frac{1}{a}\partial_y = -(c_x + b_z - \frac{ca_x + bc_y - ba_z +cb_y}{a}) \frac{\partial}{\partial_y}.$$ One can check that this is the same expression as one would obtain in the $C^1$ case. The derivatives of $a$ with respect to $y$ disappear due to the symmetry in the vector-fields. And so if one mollifies $X_i$ to $X^{\epsilon}_i$, $[X^{\epsilon}_1,X^{\epsilon}_2]$ converges to $[X_1,X_2|$ above since also $\eta^{\epsilon}\rightarrow \eta$ and $d\eta^{\epsilon}\rightarrow d\eta$. Therefore we obtain continuous vector-fields which have continuous exterior differentialss. This example is perhaps related to the question ”Does there exists Carnot manifolds such that $\Delta$ is $C^1$, and commutators of its vector-fields are lienar combinations of $C^1$-smooth basis vectors with continuous coefficients?” posed in [@Kar11]. In this example the smooth basis vector is ${\partial_y}$ with the continuous coefficients as described above. The way we choose to define the Lie brackets above is due to the fact that because of the form of $X_i$ any ”loop” that we construct using their integral curve ends up in the same $y$ axis as the starting point and therefore the Lie derivatives end up being tangent to $\partial_y$ direction. It is also not clear how one would go about defining the Lie derivatives of arbitrary basis vectors-fields. It is also interesting to address the converse, that when can we construct a continuous exterior differential out of continuous Lie brackets. For this we need to give a definition for two continuous vector-fields to have a continuous Lie derivative, without using continuous exterior differentials. There may be several definitions the more geometric one being via loops. We will use another condition though. Let $\eta = a_0(x,y)dy + \dots$ and $\Delta = ker(\eta)$ as before with $a_0>0$. Assume we have a sequence of approximations $ker(\eta^k)=\Delta^k$ with any sequence of basis of sections $\{Z_i\}_{i=1}^n$ that converges to a basis of sections $\{Z_i\}_{i=1}^n$ of $\Delta$ and $\eta^k = a^k_0(x,y)dy + \dots$ such that $a^k_0>0$. Then, the regularity conditions we impose are the following: - Locally the functions $\eta^k([Z^k_{\ell},Z^k_{j}])$ converge uniformly to some functions (which can be seen as a weaker form of requiring existence of continuous Lie derivatives), - Locally the functions $Z^k_i(a^k_0)$ converge uniformly to some functions (which can be seen as a certain regularity assumption for the bundle $\Delta$ along its basis vector-fields), - Locally the functions $\eta([Z^k_i,\partial_y])$ converges uniformly to some functions (this will imply that $\Delta$ is differentiable along the $\partial_y$ direction). Then, the functions $$d\eta^k(Z^k_i,Z^k_j) = \eta^k([Z^k_j,Z^k_i]),$$ $$d\eta^k(\partial_y,Z^k_i) = \eta^k([Z^k_i,\partial_y]) - Z^k_i(a^k_0),$$ all converge and using bilinearity over smooth functions, we can show that for any 2 vector-fields $Z,Y$, $d\eta^k(Z,Y)$ converges which implies that $d\eta^k$ converges to a differential 2-form which is the requirement for the existence of continuous exterior differential. The conditions above can be stated in a more cordinate free way as follows: There exists a smooth vector-field $Y$, transverse to $\Delta$ such that all the derivatives $[Z_i,Z_j]$, $[Z_i,Y]$ and $Z_i(\eta(Y))$ exist (using the definition above via some approximations). Then, the 2-form $d\eta$ is defined in a similar way. So this demonstrates that with some reasonable definition of a continuous Lie derivative for continuous vector-fields and some regularity assumptions one can get existence of continuous exterior differential more geometrically. However the existence of continuous Lie derivatives alone might not be enough. Note that the last two items are required in order to define $d\eta^k(\partial_y,Z^k_i)$. But actually there is a more geometric way to obtain control over the transversal behavior of $d\eta^k$ via contact structures. Let us elaborate. Assume we have a sequence of contact structures $\Delta^k$ which approximate $\Delta$. Then, the Reeb vector-fields of the approximations can help us control the transversal behavior of $d\eta^k$. The main idea is that if $R^k$ are the Reeb vector-fields of $\Delta^k = ker(\eta^k)$, then $d\eta^k(R^k,\cdot)=0$ to start with and all the regularity assumptions above about $Z^k_i(a^k_0)$ and $\eta([Z^k_i,\partial_y])$ are not required. For certain other reasons, one requires $R^k$ not to converge to $\Delta$ in the limit however, which can be seen as a sort of non-involutivity condition that fits very naturally to our setting. Therefore we then only require convergence of $\eta^k([Z^k_{\ell},Z^k_{j}])$. Hence in this setting existence of continuous Lie brackets defined as above seems to be equivalent to existence of continuous exterior differential. At this moment it is useful to note that if we have a continuous exterior differential and non-integrability, then we automatically obtain a local sequence of contact structures that approximate our bundle. Therefore it becomes meaningful to ask whether if the existence of continuous exterior differential can be completely replaced by the existence of a sequence of approximating contact structures. Given the natural relation between contact structures and step 2 completely non-integrable bundles, it seems like a worth while direction to explore. So coming back to comparisons, for the particular examples that one can construct with Proposition \[prop-example\], the results of [@MonMor12] can not be applied since they require $X_1,X_2$ and $[X_1,X_2]$ to be Lipschitz continuous. Lipschitzness for instance, grants the critical property of existence of unique solutions for these vector-fields which is an essential tool in Lipschitz continuous analysis and which we have obtained through more geometric means thanks to Stokes property (for $X_i$). However we can not at the moment also say that our results cover all the step $2$ systems covered by their conditions. The conditions they have stated in their paper probably does not necessitate the existence of an exterior differential as described above. We would only have that the sequences of functions $\eta^k([X^k_{\ell},X^k_{j}])$, $X^k_i(a^k_0)$, $\eta^k([X^k_i,\partial_y])$ obtained via mollifications have uniformly bounded norms with respect to $k$ and converge a.e. We do believe that there is hope to extend the theory present in this paper to that direction though, and this is discussed in the final section. Considering the paper [@Kar11], the step 2 case of the Ball-Box result given there, although can be covered by our theorem, is already covered by results given in [@Gro96]. Our main motivation here is to in fact work with non-differentiable bundles (due to our interest in non-differentiable bundles that arise in dynamical systems) so the results of [@Kar11] do not directly apply to the class of examples we are interested in. However the fact that the authors there can simply work with continuous Lie derivatives and still get the results for all steps is already quite remarkable and can be seen as an extension of results of [@Gro96] to arbitrary step cases. Also the existence of continuous Lie derivatives is definitely related to existence of continuous exterior differentials as discussed above. Therefore, in some sense, our result can be seen as a step 2 version of some of the results obtained in [@Kar11] in which the differentiability assumption for the bundle is removed and the continuous Lie derivatives assumption is replaced with continuous exterior differential assumption. Now we are back to studying further properties of continuous exterior differentiability. In particular we will build examples of such bundles on manifolds and not just local neighborhoods. We now give a lemma from [@Har64] that is helpful in generating new examples of elements of $\Omega^1_d(M) \setminus \Omega^1_1(M)$ from given ones. Let $\eta$ be an element of $\Omega^1_d(M)$. Then, given any $C^1$ function $\phi$ one has that $\phi\eta$ is an element of $\Omega^1_d(M)$ with continuous exterior differential $d\phi \wedge \eta + \phi d\eta$. Of course if $\phi$ is nowhere $0$ from the point of view of integrability this construction does not change anything since $\eta \wedge d\eta >0$ implies $\phi \eta \wedge d(\phi \eta) >0$ and similarly for being equal to $0$. The importance of this lemma however lies in the fact that it allows us to paste together local elements of $\Omega^1_d(M)$ (which were shown to exist above). We now explain this. Let $\{U_i\}_{i=1}^{\infty}$ be a collection of local coordinate neighbourhoods that cover $M$. Assume they are equipped with local differential forms $\alpha_i$ defined on $U_i$ which are elements of $\Omega^1_d(V_i) \setminus \Omega^1_1(V_i)$ for some $V_i \subset U_i$ and a partition of unity $\{\psi_i\}_{i=1}^{\infty}$ such that $\psi_i|_{V_i}=1$. As a direct corollary of previous lemma (and the finiteness of overlapping partition of unity cover elements) we obtain The 1-form defined by $\eta = \sum_{i=1}^{\infty} \psi_i \eta_i$ is an element of $\Omega^1_d(M)$ $\setminus \Omega^1_1(M)$ with continuous exterior differential $d\eta = \sum_{i=1}^{\infty} d\psi_i \wedge \eta_i + \psi_i d\eta_i$. Of course even if every $\alpha_i$ is everywhere non-integrable on each $U_i$ we can only guarantee that $\eta$ would be non-integrable at certain points and not everywhere on $M$. This is similar to not being able to paste together local contact structures to form a global one (in general). It would be indeed very interesting to have an example of an element of $\Omega^1_d(M) \setminus \Omega^1_1(M)$, for some $M$, which is everywhere non-integrable. It would then also make more sense to generalize fundamental theorems of contact geometry to this setting. A good place to start would be Anosov flows as it is known that the continuous center-stable and center-unstable bundles of Anosov flows can be approximated by smooth contact structures [@Mit95]. We now prove an analytic property. For this we define the function $|\cdot|_d: \Omega^k_d(M) \rightarrow \mathbb{R}$, $|\beta|_d = \max\{|\beta|_{\infty}, |d\beta|_{\infty}\}$ where we assume that $M$ is compact (or if not that we work with only compactly supported differential forms). The space $\Omega^n_d(M)$ equipped with $|\cdot|_d$ is a Banach space over $\mathbb{R}$. It is easy to establish that $|\cdot|_d$ is a norm. Now assume $\beta^k \in \Omega^n_d(M)$ is a Cauchy sequence with respect to the given norm. This means that $\beta^k$ and $d\beta^k$ are Cauchy sequences with respect to supnorm over $M$. This means that $\beta^k$ converges uniformly to some $n-$form $\beta$ and $d\beta^k$ converges uniformly to some $n+1-$form $\alpha$. Then for any $n+1$ chain $S$ bounded by some $n$ chain $c$, we have by uniform convergence $$\int_c \beta = \lim_{k \rightarrow \infty} \int_c \beta^k = \lim_{k \rightarrow \infty} \int_S d\beta^k = \int_S \alpha.$$ This means that $\alpha$ is the continuous exterior differential of $\beta$ so $\beta \in (\Omega^n_d(M), |\cdot|_{d})$. Finally we prove an algebraic property. \[lem-dd=0\] $d$ maps $\Omega^n_d(M)$ to $\Omega^{n+1}_d(M)$ such that $d^2=0$. In particular the sequence $0 \rightarrow \dots \Omega^n_d(M) \rightarrow \Omega^{dim(M)}_d(M) \rightarrow 0$ is exact. Let $\eta$ be in $\Omega^n_d(M)$ and $d\eta$ be its continuous exterior differential. Then there exists a sequence of $C^1$ $k+1$ forms $d\eta^k$ that converges to $d\eta$. Moreover $dd\eta^k=0$. Therefore by Proposition \[prop-char\] $d\eta$ has continuous exterior differential $dd\eta=0$. So $d\eta \in \Omega^{n+1}_d(M)$ and $dd\eta=0$. Integrability of Bunched Partially Hyperbolic Systems ----------------------------------------------------- In this subsection we give a tentative application to dynamical systems. It is tentative because although we state a novel integrability theorem for a class of bundles that arise in dynamical systems, we can not yet construct any examples that satisfy the properties. However we decided to include it in this paper first of all because it conveys the potential of continuous sub-Riemannian geometry for applications and secondly if the generalizations stated in Section \[sec-generalizations\] can be carried out then it will most likely be possible to improve this theorem and construct examples of dynamical systems that satisfy it. Let $M$ be a compact Riemannian manifold of dimension $n+1$ and $f: M \rightarrow M$ is a diffeomorphism. Assume moreover that there exists a continuous splitting $T_xM = E^s_x \oplus E^c_x \oplus E^u_x$, each of which is invariant under $Df_x$. This splitting is called partially hyperbolic if there exists constants $K, \lambda_{\sigma}, \mu_{\sigma}>0$ for $\sigma = s,c,u$ such that $\mu_{s} < \lambda_{c}$, $\mu_{s} \leq 1$, $\mu_c < \lambda_u$, $\lambda_u>1$ and for all $\sigma = s,c,u$, $\ i>0 \ (i \in \mathbb{Z}^+),$ $x\in M$ and $v_{\sigma} \in T_xM$ such that $|v_{\sigma}|=1$ $$\frac{1}{K}\mu^i_{s} \leq |Df^i v_{\sigma}| \leq K\lambda^i_{\sigma}.$$ This basically means that under iteration of $f$, the $Df$ expands $E^u$ exponentially and contracts $E^s$ exponentially while the behavior of $E^c$ is in between the two. Although these bundles might be just Hölder continuous (see [@HasWil99]), a well known property of such a system is that $E^u$ and $E^s$ are uniquely integrable into what are called as unstable and stable manifolds. Given $p \in U$, we denote the connected component of the stable and unstable manifold in $U$ that contains $p$ as $\mathcal{W}^u_p$ and $\mathcal{W}^s_p$ and call them local stable and unstable manifolds. In general $E^{c}$, $E^{cs}=E^{c} \oplus E^s$ or $E^{cu}=E^{c} \oplus E^u$ maybe be non-integrable both in the case the bundles are continuous (see [@HerHerUre15]) and or differentiable (see [@BurWil07]). The integrability of these bundles play an important role in classification of the dynamics, see for instance [@HamPot13]. It is our aim to apply now Theorem \[thm-ballbox\] to get a novel criterion for integrability of these bundles under additional assumptions on geometry and dynamics. A dynamical assumption that we will make is center bunching. Conditions like center bunching appears quite commonly in studies of dynamical systems. See for instance [@BurWil10] where it plays an important role for ergodicity. A system is called center-bunched if $\frac{\lambda^2_c}{\mu_u}<1$. It means that the expansion in the unstable direction strongly dominates the expansion in center (as opposed to the definition of partially hyperbolic system where one only has $\frac{\lambda_c}{\mu_u}<1$). In [@BurWil07] Theorem 4.1, the authors prove that if $E^c$ and $E^s$ are $C^1$ and center bunched, partially hyperbolic then $E^{cs}$ is uniquely integrable. s far as we are aware there is no general result on integrability of such continuous bundles that do not make any assumptions on the differentiable and topological properties of the manifold $M$ (see for instance [@BriBurIva09] where they assume $M$ is a torus or [@HamPot13] where they have assumptions on the fundamental group of the manifold). An integrability theorem for continuous bundles that only make assumptions on the constants above would indeed be quite useful. Since being a partially hyperbolic system and center bunching are preserved under $C^1$ perturbations of $f$ they constitute an open set of examples (in $C^1$ topology) inside partially hyperbolic systems. Our aim is to make one small step towards an integrability condition for continuous bundles that relies only on the constants. The only place where differentiability is required in the proof of the theorem in [@BurWil07] is where certain sub-Riemannian properties (more specifically the smaller box inclusion in the Ball-Box Theorem) are required. Thus once these properties are guaranteed then the proof easily carries through. Assume $f: M \rightarrow M$ is a diffeomorphism of a compact manifold which admits a center bunched partially hyperbolic splitting $T_xM = E^s_x \oplus E^c_x \oplus E^u_x$ where $dim(E^u_x)=1$. Assume moreover that $\mathcal{A}^1_0(E^{cs})$ admits a continuous exterior differential. Then $E^{cs}$ is uniquely integrable with a $C^1$ foliation. As in [@BurWil07], one starts by assuming that there exists a point $p$ where $\eta \wedge d\eta(p)>0$ where $E^{cs}= ker(\eta)$ and $d\eta$ is the continuous exterior differential. Then by Theorem \[thm-ballbox\], there exists a $C^1$ adapted coordinate system and a neighbourhood $U$ of $p$ on which every point $q$ on the local unstable manifold $\mathcal{W}^u_p$ of $p$ can be connected to $p$ by a length parametrized admissible path $\kappa(t)$ such that $\kappa(0)=p$, $\kappa(\ell(\kappa))=q$ and $d(p,q) \geq c|\kappa|^2$. To show that this can be done, we choose first a smooth adapted coordinate system at $p$ for $E^{sc}$ so that the unstable bundle is very close to the $\partial_y$ direction (since both are transverse to $E^{sc}$ this is possible). By choosing it close enough, we can make sure that when we pass to the $C^1$ adapted coordinates using the transformation given in equation , the unstable direction and $\partial_y$ direction are still close enough so that in a small enough neighborhood $U$ and for any $q=(x,y) \in \mathcal{W}^u_p$, $|x^i| \leq \delta|y|$ where $\delta<\frac{1}{2n}$. Then to apply the Ball-Box Theorem in this $C^1$ adapted coordinate system, for $\epsilon$ small enough, we pick $q=(x,y) \in \mathcal{B}_2(0,K_1,\epsilon)$ so that $|y| = K_1\epsilon^2$. But the Ball-Box Theorem tells us that there exists a length parametrized admissible curve $\kappa$ such that $\ell(\kappa) \leq \epsilon$, $\kappa(0)=p$ and $\kappa(\ell(\kappa))=q$. Since $|x^i| \leq \frac{K_1}{2n}|y|$, we have $d(q,p) \geq \frac{1}{2}|y| = \frac{K_1}{2}{\epsilon^2}$ (where in this coordinate system we remind that $p=0$). Therefore for some constant $c$, $d(q,p) \geq c|\kappa|^2$. Thus the conditions 1 to 4 appearing in the proof of Theorem 4.1 of [@BurWil07] are fully satisfied and the rest of the analysis only depends on the dynamics of $f$. So by the same contradiction obtained there we get that $\eta \wedge d\eta=0$ everywhere. Then by the integrability theorem of Hartman in [@Har64], this means that $E^{cs}$ integrates to a unique $C^1$ foliation. Some Perspectives Regarding Generalizations {#sec-generalizations} =========================================== In this section we ask some questions that are related to generalizations of the theorems stated in this paper. Relaxing Existence of Continuous Exterior Differential: Exterior Regularity --------------------------------------------------------------------------- One meaningful way to relax the condition on existence of a continuous exterior differential of $\eta$ is to require the following - There exists a sequence of $C^1$ differential forms $\eta^k$ which converge uniformly to $\eta$ such that $|d\eta^k|_{\infty} \leq C$ for all $k$. Then the non-integrability at $p$ condition could be stated as - There exists a constant $c>0$ and a neighbourhood $U$ of $p$ such that for all $k$ $(\eta^k \wedge d\eta^k)_q>c$ for all $q \in U$. In this case we will be working with a sequence of differential forms $\eta^k$ and $d\eta^k$ acting on objects from $\Gamma(\Delta)$ all of which is defined on an adapted coordinate system with respect to $\Delta$ on some neighbourhood $U$. Once $U$ is fixed, one requires that its size does not change with respect to $\eta^k$. That is we should be able to satisfy the conditions given in subsection \[sssection-fixU\] on a fixed $U$ for all $\eta^k$. This is the first reason why we require non-integrality on a fixed neighbourhood $U$ of $p$ since otherwise $\eta^k$ could be non-integrable on smaller and smaller domains whose size shrink to $0$ forcing us to decrease the size of $U$ as well. We also see that the condition $|d\eta^k|_{\infty} \leq C$ is important in being able to satisfy the other requirements given in subsection \[sssection-fixU\] with respect to all $\eta^k$ on a fixed domain $U$. It has already been shown in previous work [@LuzTurWar16] that under this condition one of the crucial lemmas which is \[prop-uniquebasis\] holds true. Moreover one has that for every k-cycle $Y$ and k+1 chain $H$ bounded by it $\int_Y \eta = \lim_{k\rightarrow \infty}\int_H d\eta^k$ and also since $\eta^k$ converges to $\eta$ and $|d\eta^k|_{\infty}$ is uniformly bounded then $(\eta \wedge d\eta^k)_q$ can be made arbitrarily close to $(\eta^k \wedge d\eta^k)_q$ by taking $k$ large enough and hence non-zero. So one simply replaces $d\eta(X_i,X_j)$ with $d\eta^k(X_i,X_j)$. Although $\eta^k$ does not annihilate curves tangent to $X_i$, since it converges to $\eta$, this difference can be made arbitrarily small by taking $k$ large enough and the analysis will carry through. This generalization, if done, may allow one to replace the example \[prop-example\] which was $\eta = a(x,y,z)dy + b(x,y,z)dx + c(x,y,z)dz$, by a more general one where $b$ is only Lipschitz in $y$ and $z$, $c$ is only Lipschitz in $y$ and $x$ and $a$ is only Lipschitz in $x$ and $z$ (instead of the $C^1$ assumption). But note that the non-integrability condition will be more tricky to check. Higher Coranks -------------- Assume now that $\Delta$ is a corank $m$ tangent subbundle in a $m+n$ dimensional manifold. As pointed out after equation , we can still find an adapted basis $X_i$ for such a bundle. We also assume that on some $U$, $\mathcal{A}^1_0(\Delta)$ is spanned by $\{\eta_i\}_{i=1}^m$ with exterior differential $\{d\eta^i\}_{i=1}^m$. The case where $$\label{eq-nonint} (\eta^1 \wedge \dots \eta^m \wedge d\eta^{\ell})(p)>0,$$ for all $\ell = 1,\dots ,m$ represents the case of higher co-rank but still step 2 completely non-integrable case. In this case the transversal direction will not be one dimensional so the proof may become conceptually harder to carry out. But it seems to the authors that the main change appearing will only be the replacement of terms $|\eta(\partial_y)|_{\infty}$ and $|\eta(\partial_y)|_{\inf}$ with $|\eta_1\wedge \dots \wedge \eta_n|_{\infty}$ and $m(\eta_1\wedge \dots \wedge \eta_n)_{\inf}$. Higher step cases might be impossible to carry out in full generality however we believe that there might be a subclass on which this approach may be generalized. Assume $\Delta$ is a corank 1 bundle so that $\mathcal{A}_1^0(\Delta)$ is equipped with a continuous exterior differential $\{V,\eta_i, d\eta_i\}$ for $i=1,..,m$. Let $\{Y_i\}_{i=1}^m$ be a set of vector-fields that are inside $\text{span}\{\frac{\partial}{\partial y^i}\}_{i=1}^m$ and such that $\eta_j(Y_i)=\delta_{ij}$. Let also $\{X_i\}_{i=1}^n$ be the usual adapted basis. Then we can define the Lie bracket of $X_i$ via $$\label{eq-continousliebracket} [X_i,X_j] = -\sum_{i=1}^m d\eta_i(X_j,Z_i)Y_i.$$ Note that again by the form the adapted basis, any loop constructed from any pair of such vector-fields always stays in the $y$ plane. Now define $\Delta_0=\Delta$, $\Delta_1 = \Delta_{0} + [\Delta_0,\Delta_0]$. Assume $\mathcal{A}_1^0(\Delta_1)$ is equipped with an exterior differential $\{V,\eta_i, d\eta_i\}$ for $i=1,\dots,m-k_1$ for $1 \leq k_1 \leq m$. Then one can also define $[\Delta_1,\Delta_1]$ as above and then define $\Delta_2 = \Delta_{1} + [\Delta_1,\Delta_1]$. Proceeding inductively with always the assumption of existence of continuous exterior differentials and the strict inclusion $\Delta_{i+1} \varsupsetneq \Delta_i$ (since $k_i \neq 0$) we obtain a chain $\Delta_0 \varsubsetneq \Delta_1 \varsubsetneq \dots \varsubsetneq \Delta_{\ell}$ which terminates for some $\ell$ such that $\Delta_{\ell} = TM$. The meaningful question to ask then is whether if analogues of The Ball-Box and Chow-Rashevskii Theorem hold true in this case. This is very much akin to the requirements in [@MonMor12], where for higher step cases, one requires higher order Lie brackets to be Lipschitz continuous. Of course finding an example of a bundle that satisfies the properties above will be substantially harder, so one might first try to find such an example based on the examples given in this paper before embarking on trying to prove the generalization. More Generally, Hölder Continuous Bundles {#subsec-holder} ----------------------------------------- Now we explain a fundamentally more difficult generalization which the authors think is true but are not able to prove yet. We want to remove both the existence of $d\eta$ and boundedness of $|d\eta^k|$ explained in the previous section all together so that the applicability range of this theorem increases greatly. Namely, assuming corank 1, we only want to impose the following: There exists a sequence of differential 1-forms $\eta^k$ that converge to $\eta$ uniformly and for some neighbourhood $U$ of $p$, $(\eta^k \wedge d\eta^k)_q > 0$ for all $q \in U$. Note that we still have one fundamental equality satisfied: For every k-cycle $Y$ and k+1 chain $H$ bounded by it $\int_Y \eta = \lim_{k\rightarrow \infty}\int_H d\eta^k$. This is of course just one important step of the analysis. We lose one crucial property, we lose the fact that the adapted basis $\{X_i\}$ are uniquely integrable. This brings about the problem of choosing certain integral curves to build something similar to the surface $\mathcal{W}_{\epsilon}$ that was used in the construction of the accessible neighbourhood. At this part, in corank $1$ the notion of maximal and minimal solutions can be of help to determine in a well defined way some objects similar to $\mathcal{W}_{\epsilon}$. The problems don’t end here however. Note that in application of Stokes property with $d\eta^k$ we will need an estimate on objects like $d\eta^k(X_i,X_j)$. The fact that $|d\eta^k|$ might not be bounded may cause problems in conditions required in subsection \[sssection-fixU\]. However it seems likely that with some restrictive relations between how fast $\eta^k$ converges and how fast $|d\eta^k|$ may blow up these conditions can still be satisfied in certain cases. At this stage using for instance mollifications as the approximation could be useful as one can write down the relation between such terms more precisely, as was done in [@LuzTurWar16]. Although the authors are hopeful about this generalization, they are not completely sure whether if the analysis carries through or not. It will be subject of future works. Sina Türeli\ <span style="font-variant:small-caps;">Imperial College, South Kensington, London</span>\ *Email address:* `[email protected]` [1]{} *A.Agrachev, D. Barilari, U. Boscain* *A.Agrachev, Yu.L. Sachkov* *V.I Arnol’d* *A. Bellaïche* . *M. Brin, D. Burago, and S. Ivanov.* *K. Burns, A. Wilkinson* *K. Burns, A. Wilkinson* *C. Carathéodory* *M. Gromov* *M. Gromov* *R. Potrie, A. Hammerlindl* *P. Hartman*, *P. Hartman*, *B. Hasselblatt, A. Wilkinson* *F. R. Hertz, J.R. Herz, R. Ures* **A non-dynamically coherent example on T3** *Annales de l’Institut Henri Poincare (C) Non Linear Analysis.* *M. Karmanova* *S. Luzzatto, S. Türeli, K.M.War* *Y. Mitsumatsu* *R. Montgomery* *A. Montanari and D. Morbidelli* *L. Shin* *S. N. Simić* Sina Türeli\ <span style="font-variant:small-caps;">Imperial College London, South Kensington Campus, London</span>\ *Email address:* `[email protected]` [^1]: The accessibility theorem for corank $1$, step 2, completely non-integrable differentiable bundles was actually already formulated in 1909 by Carath$\acute{\text{e}}$odory with the aim of studying adiabatic paths in thermodynamical systems [@Car09]). [^2]: One can prove that the solutions are $C^1$ using Stokes Theorem on a sequence of approximations $\eta^k$ built in a certain way but it gets very lengthy and technical. [^3]: We remind that a $n$ cell in $U$ is a differentiable mapping from a convex n-polyhedron in $\mathbb{R}^n$ (with an orientation) to $U$ and a $n$ chain is a formal sum of $n$ cells over integers.

ArXiv

--- abstract: | [We first analyse the effect of a square root transformation to the time variable on the convergence of the Crank-Nicolson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio, $\lambda$, of the time step to space step held constant and the value of $\lambda$ controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed partial differential equation now always converges and that $\lambda$ controls the order of convergence, quadratic convergence being achieved for $\lambda$ below a critical value. Numerical results indicate that the time change used with an appropriate value of $\lambda$ also results in quadratic convergence for the calculation of the price, delta and gamma for standard European and American options without the need for Rannacher start-up steps.]{}\ *Keywords.* [Heat equation; Crank-Nicolson scheme; convergence; Black-Scholes; European option; American option; asymptotics; time change.]{} address: 'Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, U.K.' author: - 'C. Reisinger' - 'A. Whitley' title: 'The impact of a natural time change on the convergence of the Crank-Nicolson scheme' --- Introduction ============ The Crank-Nicolson scheme is a popular time stepping scheme for the numerical approximation of diffusion equations and is particularly heavily used for applications in computational finance. This is due to a combination of favourable properties: its simplicity and ease of implementation (especially in one dimension); second order accuracy in the timestep for sufficiently regular data; unconditional stability in an $L_2$ sense. However, there are well-documented problems which arise from the fact that the scheme is at the cusp of stability in the sense that it is *A-stable* [see @S] but does not dampen so-called stiff components effectively (i.e., is not *strongly A-stable*). Specifically, a straightforward Fourier analysis of the scheme applied to a standard finite difference discretisation of the heat equation shows that high wave number components are reflected in every time step but asymptotically do not diminish over time. This gives rise to problems for applications with non-smooth data, where the behaviour of these components over time plays a crucial role in the smoothing of solutions. Examples where this is relevant include Dirac initial data, as they appear naturally for adjoint equations [see @GS], and piecewise linear and step function payoff data in the computation of option prices [see @WHD]. There, the situation is exacerbated if sensitivities of the solution to its input data (so-called ‘Greeks’) are needed; see @SHAW for an early discussion of issues with time stepping schemes in this context. There is a sizeable body of literature concerned with schemes with better stability properties, in particular *L-stable* schemes [see @S] which share with the underlying continuous equation the property that high wave number components decay increasingly fast in time. Examples for schemes with this property include sub-diagonal Padé schemes based on rational approximations of the exponential function. In @RANNACHER, Rannacher proposes to replace the first $2 \mu$ steps of a higher-order Padé scheme of order $(\mu,\mu)$ (e.g., the Crank-Nicolson scheme for $\mu=1$) by a low-order sub-diagonal Padé scheme with order $(\mu-1,\mu)$ (e.g., backward Euler for $\mu=1$), and shows that this restores optimal order for diffusion equations; the case $\mu=1$ is already analysed in @LR. It is demonstrated in @PVF how this procedure can be used to obtain stable and accurate approximations to option values and their sensitivities for various non-smooth payoff functions. This has been widely adopted in financial engineering practice. @KVY, Khaliq extend this to more general Padé schemes and demonstrate numerically the stability and accuracy in practice for options with exotic payoffs. Further adaptations are possible, for instance, where @WKYVD give an application to discretely sampled barrier options, where discontinuities are introduced at certain points in time and a new ‘start-up’ is needed (see already @RANNACHER for the provision of such restarts in an abstract context). @GC analyse the behaviour of the Crank-Nicolson numerical scheme when used to solve a convection-diffusion equation with constant coefficients and Dirac delta function initial conditions. The numerical solution obtained by this method diverges as the time step goes to zero with the ratio $\lambda=k/h$ of the time step, $k$, to the space step, $h$, held constant. Their analysis in the frequency domain shows that there are errors associated with high wave numbers which behave as $O(h^{-1})$ and which will eventually overwhelm the $O(h^2)$ errors associated with low wave numbers as the time step goes to zero, thus explaining why the errors in this scheme will eventually increase as the time step is reduced. Keeping the ratio $\lambda$ constant is desirable because the Crank-Nicolson central difference scheme is second order consistent in both space and time and the numerical scheme is more efficient if this property is exploited. The authors extended their analysis to show why the incorporation of a small number of initial fully implicit time steps (i.e., the Rannacher scheme, @LR [@RANNACHER]) eliminates this divergence. In this paper, an alternative method of avoiding the divergence is proposed and analysed. The idea is to introduce a time change into the partial differential equation (PDE) by transforming the original time variable, $t$, to $$\tilde t =\sqrt{t}$$ and solving the equation numerically using the Crank-Nicolson scheme in the new variables. The time change will be applied to the heat equation (\[HE\]) on $\mathbb{R}\:\times\:[0,T]$ and can be considered natural as the heat equation with suitable initial data admits similarity solutions which depend on $x/\sqrt{t}$. From a probabilistic angle, the heat equation is the Kolmogorov forward equation for the transition density of Brownian motion whose standard deviation at time $t$ is $\sqrt{t}$.\ The main result of the paper is given by the following Theorem. \[T:MainTheorem\] The Crank-Nicolson central difference scheme with uniform time steps exhibits second order convergence (in the maximum norm) for the time-changed heat equation, with Dirac initial data, as the time step $k$ tends to zero with $\lambda=k/h$ held fixed, provided that $\lambda \leq 1/\sqrt{2}$. For $\lambda \geq 1/\sqrt{2}$ the scheme is still convergent with order $1/\lambda^2$. The peculiar dependence of the convergence rate on the mesh ratio is in fact seen to be sharp up to a logarithmic factor. It also follows from the analysis of the heat equation that the computational complexity for an optimal choice of the mesh ratio is lower than for the Rannacher scheme in its optimal refinement regime. As an added benefit, we obtain experimental second order convergence for the value, delta and gamma of European and American options. @FV observe that at-the-money prices of European and American options computed by a Crank-Nicolson finite difference scheme exhibit a reduced convergence order of 1 in the time step $k$, which improves to 2 for Rannacher start-up in the case of European options, but only to 3/2 for American options. This last observation is rationalised by the square-root behaviour of the value and exercise boundary for short time-to-expiry. A heuristic adaptive time-stepping scheme based on this observation is shown there to restore second order convergence. We show here, by means of numerical tests, that the square root time change introduced above provides a similar remedy even without Rannacher start-up. This is not surprising bearing in mind the relation between the time change and a non-uniform time mesh in the original time variable. Precisely, the time-changed scheme with constant steps of size $\sqrt{T}/N$, where $N$ is the total number of time steps for time up to $T$, is equivalent to a non-uniform scheme in the original time variable with time points $t_m=(km)^2$ for $m=0,...,\sqrt{T}/k$. The step size hence increases in proportion to $2 \sqrt{t}$ and the smallest (the first) step is of size $T/N^2$. The remainder of this article is organised as follows. In Section \[S:CNScheme\], we apply the time change transformation to the heat equation, describe the transformed numerical scheme, and calculate the discrete Fourier transform of the numerical solution. Then, in Section \[S:AsympFT\], we perform asymptotic analysis of the error between the transform and the transform of the true solution, identifying four wave number regimes which will then be used in the later analysis. In Section \[S:AsympErrors\], we use the results of the asymptotic analysis to determine the asymptotic behaviour of the errors between the numerical solution and the true solution, from which we deduce the convergence behaviour of the transformed scheme and hence Theorem \[T:MainTheorem\]. Section \[sec:numerical\] contains numerical results illustrating these findings and compares the computational complexity to the Rannacher scheme. In Section \[S:BSApp\], the time change transformation is applied to the Black-Scholes equation and the solution method is used to calculate the gamma of a European call option. The behaviour of the resulting errors is described and explained in terms of the values of the strike and volatility. Finally, in Section \[S:AmericPut\], the time change transformation is applied to a penalty method used to calculate the price of an American put, as described in @FV. Section \[sec:concl\] concludes. The Crank-Nicolson scheme applied to the transformed heat equation and its behaviour in the Fourier domain {#S:CNScheme} ========================================================================================================== To simplify the analysis, we restrict attention to the heat equation (i.e., we do not consider any convection terms) $$u_t= \frac{1}{2} \, u_{xx}, \label{HE}$$ for $t \in [0,T]$, $x \in \mathbb{R}$, where the solution, $u$, satisfies $u(x,0)=\delta(x)$, where $\delta$ is the Dirac delta function and which, after the change of time variable, becomes $$u_{\tilde t}= \tilde t\: u_{xx},$$ where $\tilde t = \sqrt{t}$. The transformed equation is then discretised using the Crank-Nicolson method, leading to the following numerical scheme $$%\begin{multiline} \frac{U^{n+1}_j-U^{n}_j}{k}=\frac{1}{2}\: \tilde t^{n}\:\frac{U^{n}_{j+1}-2U^{n}_j+U^{n}_{j-1}}{h^2}+\frac{1}{2}\: \tilde t^{n+1}\:\frac{U^{n+1}_{j+1}-2U^{n+1}_j+U^{n+1}_{j-1}}{h^2}, \label{EQ1} %\end{multiline}$$ where $U^{n}_j$ is the numerical solution at the spatial grid node $j$ at the ‘time’ step $n$, where the space grid, in principle, extends from $-\infty$ to $+\infty$, with $x_j=jk$ for $j \in \mathbb{Z}$. In the transformed ‘time’ direction, we have $\tilde t^i=ik$ for $1 \leq i \leq N$ where $N$ is the number of timesteps. We can simplify this to $$(1+(n+1)\lambda^2)U^{n+1}_j-\frac{1}{2}(n+1)\lambda^2U^{n+1}_{j+1}-\frac{1}{2}(n+1)\lambda^2U^{n+1}_{j-1}=(1-n\lambda^2)U^{n}_j+\frac{1}{2}n\lambda^2U^{n}_{j+1}+\frac{1}{2}n\lambda^2U^{n}_{j-1},$$ where $\lambda=k/h.$ To study the behaviour of this scheme in the Fourier domain we apply the discrete Fourier transform to this equation (e.g., see @GC and @STRANG). Multiplying equation  by $e^{i s x_j}$, summing over $j$ and simplifying, we obtain the following recurrence for the Fourier transform $$\widehat{U}^n(s)=h \sum ^{j=+\infty}_{j=-\infty} U_j^n e^{i s x_j}$$ at successive time steps, $$\left(1+2(n+1)\lambda^2 \sin^2\left(\frac{sh}{2}\right)\right)\widehat{U}^{n+1}(s)=\left(1-2n\lambda^2 \sin^2\left(\frac{sh}{2}\right)\right)\widehat{U}^{n}(s).$$ We note that $\widehat{U}^0(s)\equiv1$, as the initial condition is a delta function at $(0,0)$, so we have $$\label{EQ2} \widehat{U}^{N}(s)=\frac{1(1-\xi)(1-2\xi) \ldots (1-(N-1)\xi)}{(1+\xi)(1+2\xi) \ldots (1+N\xi)},$$ where $\xi = 2 \lambda^2 \sin^2(sh/2)$ and we want to study the behaviour of $\widehat{U}^{N}(s)$ as $N \rightarrow \infty$ with $kN$ and $k/h$ held fixed. In the subsequent analysis, it is often simpler to describe a condition on $s$, by stating the related condition on $\xi$, but it should be noted that the relationship between $s$ and $\xi$ depends on $h$. In what follows, and to simplify the notation, we will fix $T=1$, so that $$N=\frac{1}{k}=\frac{1}{h \lambda}.$$ We note that the exact solution of the PDE is given by $$u(x,t)=\frac{1}{\sqrt{2 \pi t}}\exp\left(-\frac{x^2}{2t}\right)$$ and its Fourier transform is given by $\widehat{u}(s,t)=\exp\left(-s^2 t/2\right)$. So, for $t=\tilde t=1$, we have $\widehat{u}(s,1)=\exp\left(-s^2/2\right)$. As in @GC we can estimate the errors in the numerical solution (at time $t=\tilde t=1$), i.e. the differences between the values of $U^{N}_j$ and $u(x_j,1)$, by applying the inverse Fourier transform to $\widehat{U}^N(s)-\widehat{u}(s,1)$. Asymptotic analysis of the Fourier transform {#S:AsympFT} ============================================ In this section we identify four wave number regimes for the Fourier transform and obtain useful asymptotic estimates for $\widehat{U}^N(s)$, in each of these regimes. Initially, we start by identifying three regimes, the low, intermediate and high wave number regimes (as in @GC), but then find it useful to subdivide the high wave number regime into two further regimes. In essence, the wave number regimes are determined by the locations of the real zeros of the equation $$\widehat{U}^{N}(s)=0,$$ which are at $s_m$, for $m=m^*,\ldots,N-1$, for some integer $m^* \geq 1$, which depends only on $\lambda$, where $$2 \lambda^2 \sin^2\left(\frac{s_mh}{2}\right)=\frac{1}{m}.$$ and we note that $m^* \geq \frac{1}{2\lambda^2}$. Note that each $s_m$ is a function of $h$ and $s_m \rightarrow \infty$ as $h \rightarrow 0$, so these wave number boundaries increase in absolute terms while always being less than or equal to $\pi/h$. The low wave number regime (which we will refer to as regime I) is, in part, defined by the requirement that all the terms in the numerator of the expression for $\widehat{U}^{N}(s)$ in equation (\[EQ2\]) should be positive. This requires that $\xi < 1/(N-1)$. If we assume the stronger condition that $\xi < 1/N$, we can write a truncated Taylor expansion for $$\log(\widehat{U}^N(s))=\sum^{N-1}_{m=1} \log(1-m\xi) - \sum^{N}_{m=1} \log(1+m\xi)$$ from which we can later derive an asymptotic estimate for $\widehat{U}^N(s)$. As in @GC, the low wave number regime is further restricted by the condition that $s<h^{-r}$, for a value $r$ such that $r<\frac{1}{3}$, which ensures that the remainder terms in the Taylor expansion go to zero as $h \rightarrow 0$, so that we can derive a useful approximation for $\widehat{U}^N(s)-\widehat{u}(s,1)$. As this condition also ensures that $\xi < 1/N$ asymptotically, this condition alone is sufficient to define the low wave number regime. Next, there exists an intermediate wave number regime (regime II) defined by the range of wave numbers for which $\xi < 1/N$ but which are not in regime I (cf. @GC). The high wave number regime is defined by the requirement that $\xi \geq 1/N$. If this condition on $\xi$ is satisfied then some of the terms in the numerator of the expression for $\widehat{U}^{N}(s)$ in equation (\[EQ2\]) will be negative. There will be an integer $m > 0$ such that $\xi \in [1/(m+1),1/m]$ and the number of positive terms will remain fixed as $N$ increases while the number of negative terms will increase with $N$. A particular negative factor $(1-r\xi)$ can be rewritten as $-r\xi(1-1/r\xi)$, with $1/r\xi<1$, and we can then write a truncated Taylor expansion for $\widehat{U}^{N}(s)$ in terms of $1/\xi$ which will then lead to an asymptotic value for $\widehat{U}^{N}(s)$. At first sight, it might seem necessary to treat separately all the $\xi$-intervals of the form, $[1/(m+1),1/m]$, giving rise to an ever-growing set of wave number regimes but, in fact, it suffices for the results we wish to prove, to treat the last of these intervals, $[1/m^*,2\lambda^2]$, separately and combine all the other intervals into a single regime. So we define wave number regime III to correspond to the interval $[1/N,1/m^*]$ and wave number regime IV to correspond to the interval $[1/m^*,2\lambda^2]$. These ranges are equivalent to $s_N \leq s \leq s_{m^*}$ and $s_{m^*} < s \leq \pi/h$ respectively. We will obtain a uniform bound on the magnitude of $\widehat{U}^{N}(s)$ for $\xi \in [1/(m+1), 1/m]$ in regime III and an asymptotic value for the magnitude of $\widehat{U}^{N}(s)$ in the regime IV. Table  \[tab:TableReg\] summarises the wave number regimes giving their boundaries in terms of both $\xi$ and $s$. (Only the regimes for positive wave numbers are shown. By symmetry, the regimes for the negative wave numbers are just the negatives of these intervals). The results of carrying out asymptotic expansions in the four wave number regimes are summarised in the following Lemma which gives the behaviour of the error between the two transforms, $\widehat{E}(s)=\widehat{U}^N(s)-\widehat{u}(s,1)$. \[L:LemmaA\] The following formulae give the errors in the Fourier transform in each wave number regime (see Table \[tab:TableReg\]). 1. In wave number regime I, $$\widehat{E}(s)=\widehat{u}(s,1)\left(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2 s^4\right)h^2(1+o(1)).$$\ 2. In wave number regime II, $$\widehat{E}(s)=o(h^q)$$ for any $q > 0$. 3. In wave number regime III, $$|\widehat{E}(s)| \leq \frac{((m+1)!)^2(N-m-1)!}{(2m+2)(N+m)!}$$ whenever $\xi=2\lambda^2\sin^2(\frac{sh}{2}) \in [1/{(m+1)},1/{m}]$ and, everywhere in this regime, $$\widehat{E}(s) = O(h^{2m^* + 1})$$ for some $m^*\ge 1$. 4. Finally, in wave number regime IV, there exists a positive, continuous, and hence bounded, function $S(\cdot,m^*)$ on $[1/m^*,2\lambda^2]$, such that $$|\widehat{E}(s)| = S(\xi,m^*)h^{(1+\frac{2}{\xi})}(1 + O(h))).$$ The key features of the proof are given here while further details are included in the Appendix.\ We first consider wave number regime I.Inspection of equation shows that all the terms in the numerator are positive if $\xi \leq 1/N<1/(N-1)$. We can then write $\log(U^N(s))$ as $$\label{EQ3} \log(\widehat{U}^N(s))=\sum^{N-1}_{k=1} \left(\log(1-k\xi) - \log(1+k\xi)\right) - \log(1+N\xi)$$ and expand the component terms, $\log(1 \pm k\xi)$, by first expanding $\xi$ in powers of $h$. The calculations are very similar to those in @GC and the details of the expansions are left to the Appendix. We find that $$\label{expapp} \log(\widehat{U}^N(s))=-\frac{1}{2}s^2 +\left(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2 s^4\right)h^2+O(s^8h^3).$$ we find we require that the second term converges to zero as $h \rightarrow 0$ and the last term is asymptotically dominated by the second term. These conditions are satisfied by requiring that $s<h^{-q}$ with $q < \frac{1}{3}$ and this condition on $s$ defines the upper limit of the wave number regime I. We then have, with $\widehat{u}(s,1)=e^{-\frac{1}{2}s^2}$, $$\widehat{U}^N(s)-\widehat{u}(s,1)=\widehat{u}(s,1)\left(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2 s^4\right)h^2(1+o(1))$$ as we set out to prove.\ We now consider wave number regime II.\ The analysis is again similar to that in @GC. The lower limit of wave number regime II is $O(h^{-\frac{1}{3}})$ (which corresponds to the upper limit of wave number regime I) and the upper limit of the regime corresponds to the first zero of $\widehat{U}^N(s)$, i.e., $s=(2/h)\sin^{-1}\sqrt{(1/2 \lambda^2(N-1))}$, which is $O(h^{-\frac{1}{2}})$. In this regime each of the factors $(1-m\xi)/(1+(m+1)\xi)$ in $\widehat{U}^N(s)$ is positive and decreasing as $s$ increases.\ Consequently, the value of $\widehat{U}^N(s)$ will be less than or equal to its value at $s=h^{-r}$ for $0<r<1/3$. However, at $s=h^{-r}$ we have, for any $q > 0$, $$\frac{\widehat{U}^N(s)}{h^q}= \frac{ e^{ -\frac{1}{2} h^{-2r}} }{h^q} \exp{\left(h^{2(1-2r)}\left(\frac{1}{24}+\frac{\lambda^2}{8}\right)-h^{2(1-3r)}\frac{1}{48}\lambda^2+ o(1)\right)}\rightarrow 0$$ as $h \rightarrow 0$, so $\widehat{U}^N(s)=o(h^q)$ at $s=h^{-\frac{1}{3}}$ and so we have $\widehat{U}^N(s)=o(h^q)$ throughout this regime.\ Next we consider the high wave number regime which is split into two parts.\ \ It is in wave number regimes III and IV that our analysis differs significantly from that in @GC where there are repeated factors in the corresponding expression for $\widehat{U}^N(s)$, which results in a simpler expansion in terms of $1/\xi$. In our case, $\widehat{U}^N(s)$ is a product of many more factors, so the analysis and expansion is more complicated. However, we can obtain our key results by breaking the high wave number regime into two parts and we will find that we only need to perform an expansion in regime IV.\ For wave number regime III, which extends from $s_N=(2/h)\sin^{-1}\sqrt{(1/2\lambda^2 N)}$ to $s_{m^*}$, corresponding to $\xi$ running from $1/N$ to $1/m^*$, with $m^* \geq 1$, we investigate the behaviour of $\widehat{U}^N(s)$ for values of $s$ corresponding to values of $\xi$ in the intervals $[1/(m+1), 1/m]$, where $m^* \leq m \leq N-1$. We write $\widehat{U}^N(s)$ as $$\begin{aligned} \label{uhatn} \widehat{U}^N(s)= \left\{ \frac{ 1(1-\xi)(1-2\xi)\ldots(1-m\xi) }{ (1+\xi)(1+2\xi)\ldots(1+(m+1)\xi) } \right\} . \left\{\frac{ (1-(m+1)\xi)\ldots(1-(N-1)\xi) }{ (1+(m+2)\xi)\ldots(1+N\xi) }\right\}.\end{aligned}$$ For the range of values of $\xi \in [1/(m+1), 1/m]$ being considered, the left-hand expression is a product of non-negative factors, $(1-\nu\xi)/(1+(\nu+1)\xi)$, for $\nu = 0,...,m$, each of which is decreasing in $\xi$, so the left-hand expression is bounded by the value of the expression evaluated at $\xi=1/(m+1)$, i.e., $((m+1)!)^2 /(2m+2)!$\ Similarly, the right-hand expression is a product of factors $(1-\nu\xi)/(1+(\nu+1)\xi)$, for $\nu=m+1,...,N-1$, which are all negative and whose magnitude is increasing with $\xi$. So the magnitude of the right-hand expression is bounded by the magnitude of the expression evaluated at $\xi=1/m$, i.e. $(N-m-1)!(2m+1)!/(N+m)!$\ Therefore, the magnitude of $\widehat{U}^N(s)$ for $1/(m+1) \leq \xi \leq 1/m$ is bounded by $$W_{N,m}= \frac{((m+1)!)^2(N-m-1)!}{(2m+2)(N+m)!},$$ and we consider the behaviour of this expression as $m$ decreases from $N-1$ to $m^*$. We have $$\frac{ W_{N,m-1}}{ W_{N,m}}=\frac{N^2-m^2}{m(m+1)},$$ and this expression is less than one for $$N^2-m^2 < m(m+1),$$ which corresponds approximately to $m > N/\sqrt{2}$. This implies that, as $m$ decreases from $N-1$, $W_{N,m}$ first decreases, reaching a minimum in the vicinity of $N/\sqrt{2}$. So the maximum of $|\widehat{U}^N(s)|$ will occur either at $m=N-1$ or at $m=m^*$.\ We now compare the values of $W_{N,m}$ at these values of $m$, by calculating the ratio $$\frac{ W_{N,m^*}}{ W_{N,N-1}}=\frac{((m^*+1)!))^2}{(2m^*+2)}\frac{(N-m^*-1)!}{(N+m^*)!}\frac{(2N)!}{(N!)^2}.$$\ By Stirling’s formula, we have the approximation $$\frac{(2N)!}{(N!)^2}\sim \frac{(2N)^{2N+\frac{1}{2}}e^{-2N}}{((N)^{N+\frac{1}{2}}e^{-N})^2}\sim \frac{2^{2N+\frac{1}{2}}}{N^{\frac{1}{2}}}$$ and we see that the exponential term dominates the other powers of $N$ and $W_{N,m^*}/W_{N,N-1} \rightarrow \infty$ as $N \rightarrow \infty $. So, for large enough $N$, $W_{N,m}$ is maximised at $m=m*$ and it then follows that the expression $$\frac{((m^*+1)!)^2}{(2m^*+2)}\frac{(N-m^*-1)!}{(N+m^*)!} = O(N^{-2m^*-1}) = O(h^{2m^*+1})$$ gives a uniform bound on $|\widehat{U}^N(s)|$ in wave number regime III.\ \ Finally, we consider wave number regime IV.\ \ In this regime, we will find that we must perform an expansion of $\widehat{U}^N(s)$ from (\[uhatn\]) and we write $$\big|\widehat{U}^N(s)\big| = M(\xi,m^*)\left|\frac{ (1-(m^*+1)\xi)\ldots(1-(N-1)\xi) }{ (1+(m^*+1)\xi)\ldots(1+(N-1)\xi) }\right|(1+N\xi)^{-1},$$ where $$M(\xi,m^*) = \left| \frac{(1-\xi)(1-2\xi)\ldots(1-m^*\xi) }{ (1+\xi)(1+2\xi)\ldots(1+m^*)\xi) } \right|,$$ so $$\label{secexp} \log\big(\big|\widehat{U}^N(s)\big|\big)=\log(M(\xi,m^*))+\log\left\{\frac{(1-\frac{1}{\xi(m^*+1)})\ldots(1-\frac{1}{\xi(N-1)})} {(1+\frac{1}{\xi(m^*+1)})\ldots(1+\frac{1}{\xi (N-1)})}\right\}+\log((1+N\xi)^{-1})$$ and we seek an expansion of the left-hand side of this equation in terms of $1/\xi$ to find, as described in the Appendix, that $$\begin{aligned} \label{expandapp} \big|\widehat{U}^N(s)\big|=S(\xi,m^*)h^{({2}/{\xi}+1)}(1+O(h)),\end{aligned}$$ where $S$ is a function of $\xi$ and $m^*$, which is positive and continuous on $[1/m^*,2\lambda^2]$, as we wanted to show. Error contributions from the various wave number regimes ========================================================  \[S:AsympErrors\] The error in the numerical scheme at $x_j$ is found by using the inverse Fourier transform $$E_j=\frac{1}{2\pi}\int ^{s=\frac{\pi}{h}}_{s=-{\frac{\pi}{h}}} \widehat{E}(s) \exp(-is x_j) \; ds,$$ where $\widehat{E}(s)$ is the error in the Fourier transform. So decomposing the error, applying the inverse Fourier transform and using symmetry, we can write $$E_j=\sum_{i=1}^{4} E^i_j,$$ where $$\begin{aligned} \label{eij} E^i_j=\frac{1}{\pi}\int _{s \in I_i}\widehat{E}(s) \cos(s x_j) \; ds,\end{aligned}$$ where $I_i$ is wave number region for $i\in \{I,II,III,IV\}$. We then have the following Lemma which makes use of the results from Lemma  \[L:LemmaA\] in Section 3. \[L:LemmaB\] With $E^i = (E_j^i)_{j\in \mathbb{Z}}$ defined in (\[eij\]), the contribution to the error of the numerical scheme from the four wave number regimes is given by $$\begin{aligned} \begin{array}{llll} E^{I} &=& O(h^2), &\\ E^{II} &=& o(h^q) &\quad \mathit{ for}\: \mathit{any } \: q>0, \\ E^{III} &=& O(h^{2m^*}) &\quad \mathit{ for}\: \mathit{some} \: m^*\geq 1, \\ E^{IV} &=& O\left(\frac{h^\frac{1}{\lambda^2}}{\sqrt{\log{\frac{1}{h}}}}\right).& \end{array}\end{aligned}$$ Again, only key features of the proof are given here while further details are included in the Appendix.\ In wave number regime I, the analysis is similar to that in @GC. The key observation is that the term $\widehat{u}(s,1)(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2 s^4)$ has a readily identifiable inverse Fourier transform, and we deduce that the error contribution is $O(h^2)$.\ In wave number regime II, we again follow the analysis in [@GC] to conclude that the error contribution from this regime will be dominated by $h^q$ for any $q > 0 $.\ In wave number regime III, we can write $$\label{EQ4} \int _{s_{N-1}}^{s_{m^*}} \big|\widehat{U}^N(s) \cos(s x_j)\big| \; ds \leq \int _{0}^{\pi/h} \big|\widehat{U}^N(s)\big| \; ds,$$ which is $\pi/h \cdot O(h^{2 m^* + 1})$, i.e., $O(h^{2m^*})$.\ Finally, we consider the error contribution from wave number regime IV, i.e., where $s_{m^*} \leq s \leq \pi/h$.\ We need to determine the behaviour, as $h \rightarrow 0$, of $$\int^{\frac{\pi}{h}}_{s=s_{m^*}} S(\xi,m^*)h^{(1+\frac{2}{\xi})} \cos(s x_j)\; ds,$$ the magnitude of which is maximised at $j=0$ (this is because $S$ is positive in this wave number regime). So we only need to consider the value of $$F(h)=\int^{\frac{\pi}{h}}_{s=s_{m^*}} S(\xi,m^*) h^{(1+\frac{2}{\xi})} \; ds.$$\ From $\xi=2\lambda^2\sin^2(\frac{sh}{2})$, we find that $$F(h)=\int^{2\lambda^2}_{\xi=\xi_{m^*}} \frac{ S(\xi,m^*)h^{\frac{2}{\xi}} }{ \lambda\sqrt{2}\sqrt{\xi}\sqrt{1-\frac{\xi}{2\lambda^2}} } \; d\xi.$$ As the lowest (i.e., dominant) order in $h$ from $h^{\frac{2}{\xi}}$ occurs when $\xi=2\lambda^2$ we expect that this integral will be close to $O(h^{\frac{1}{\lambda^2}})$ and using careful, but elementary, analysis (see the Appendix) we find that $$\begin{aligned} \label{intapp} I=\frac{F(h)}{h^{\frac{1}{\lambda^2}}}=O\left(\frac{1}{ \sqrt{ \log{ 1/h } }}\right),\end{aligned}$$ and so the error contribution in regime IV is found to be $O\left({h^{\frac{1}{\lambda^2}}}/{ \sqrt{ \log{ \frac{1}{h} } }}\right)$. As a consequence of Lemma \[L:LemmaB\], we obtain Theorem \[T:MainTheorem\], as stated in the Introduction. This result follows from the observation that when $\lambda \leq 1/\sqrt{2}$, we have $m^* \geq 1$ and $1/\lambda^2 \geq 2$, so the convergence behaviour is eventually dominated by wave number regime I and the errors (in the maximum norm) will be $O(h^2)$. However, if $\lambda > 1/\sqrt{2}$, then $m^*=1$ and $1/\lambda^2< 2$, so the numerical error of the scheme is dominated by wave number regime IV. The scheme still converges but the errors will now be $O(h^{1/\lambda^2}/\sqrt{\log(1/h)})$. The logarithmic factor in this expression is slow growing and the error will be close to $O(h^{1/\lambda^2})$ which is worse than $O(h^2)$. So our result is that we always get convergence in the transformed numerical scheme whereas, without the time change, the errors eventually increase as $h \rightarrow 0$, with $\lambda$ controlling how small $h$ has to be for the divergence to manifest itself in practice. Numerical results and complexity considerations {#sec:numerical} =============================================== Numerical experiments have been performed to investigate the behaviour of the time-changed scheme used to solve the heat equation with delta-function initial conditions. The analysis used $T=1$ and the space domain was truncated at $x=\pm 10$. The numerical scheme was run with a largest value of $k = 0.01$, i.e., with 100 time steps in the transformed coordinates. The grid was repeatedly refined by dividing both $h$ and $k$ by 2 and the calculations were stopped when the number of time steps reached 3200.\ The error (in the maximum norm) of the scheme was analysed by comparing the numerical results with the exact solution and the slope of the log-log plot of the error against $h$ was calculated. The slope was then plotted against the value of $\lambda$ in Figure 1, along with the $h$ exponent for the error derived from the analysis above, where we ignore the effect of the $\log(\sqrt{1/h})$ term on the behaviour of the error in wave number regime IV and just use the expression $\min(2,1/\lambda^2)$. {width="5.5in" height="2.6in"} \[fig:ConvergenceOrder200912\] We see a good match between theory and experiment with the larger divergences being seen where $\lambda$ is closest to the critical value of $1/\sqrt{2}$. Figures 2 and 3 show the log-log plots of the errors versus $h$ for the cases $\lambda=0.5$ and $\lambda=1.0$, either side of $1/\sqrt{2}$. {width="5.5in" height="2.5in"} \[fig:LambdaHalf200912\] {width="5.5in" height="2.5in"} \[fig:LambdaOne200912\] It is not possible to determine experimentally the $h$ exponent with complete accuracy for two reasons. Firstly, as $h \rightarrow 0$, the time required for each solution increases rapidly so, for comparison purposes, all the calculations were terminated when the number of time steps reached 3200. As the overall error of the scheme is a mixture of errors with different $h$ exponents, the asymptotically dominating $h$ exponents will not be reached because the calculations are terminated early. In particular, for $\lambda$ slightly above $1/\sqrt{2}$, the contribution to the error from the highest wave number regime will be limited and lower values of $h$ need to be reached before this error clearly dominates the other errors. Note that our analysis does not identify the weighting factors for the errors of different order in $h$. Secondly, the additional term, $\sqrt{\log( 1/h )}$, in the error behaviour in wave number regime IV will distort the error plot and tend to increase the apparent $h$ exponent to some extent. We have also carried out a comparison of the error performance between the time change scheme and the Rannacher scheme. For $\lambda \leq 1/\sqrt{2}$, both schemes will show quadratic convergence and, for small values of $h$, we can write $$\widehat{U}^N_R(s)-\widehat{u}(s,1)=\widehat{u}(s,1)\left(\frac{1}{24}s^4+\frac{1}{8}\lambda^2 s^4-\frac{1}{96}\lambda^2 s^6\right)h^2(1+o(1))$$ and $$\widehat{U}^N_{TC}(s)-\widehat{u}(s,1)=\widehat{u}(s,1)\left(\frac{1}{24}s^4+\frac{1}{8}\lambda^2 s^4-\frac{1}{48}\lambda^2 s^6\right)h^2(1+o(1)),$$ where $\widehat{U}^N_R(s)$ and $\widehat{U}^N_{TC}(s)$ are the Fourier transforms for the Rannacher and time-changed scheme in wave number regime I. For $m \geq 1$, the $m$th derivative of the cumulative Normal distribution has the Fourier transform given by $$\widehat{N^{(m)}}(s)=(is)^{(m-1)}e^{-s^2/2}$$ so that the Fourier inverse of $(is)^{(m-1)} e^{-s^2/2}$ is $\frac{1}{\sqrt{2\pi}}N^{(m)}(x) $. We can then estimate the ratio of the errors of the two schemes (at $x=0$) as $$\frac{E_R}{E_{TC}}=\frac{3\left(\frac{1}{24}+\frac{1}{8}\lambda^2\right)-15\frac{1}{96}\lambda^2}{3\left(\frac{1}{24}+\frac{1}{8}\lambda^2 \right)-15\frac{1}{48}\lambda^2}$$ by evaluating $N^{(5)}(0)=3$ and $N^{(7)}(0)=-15$ from $$N^{(5)}(x)=\frac{1}{\sqrt{2\pi}}(x^4-6x^2+3)e^{-x^2/2}$$ and $$N^{(7)}(x)=\frac{1}{\sqrt{2\pi}}(-x^6-15x^4+45x^2-15)e^{-x^2/2}.$$ The ratio ${E_R}/{E_{TC}}$ has been plotted as a function of $\lambda$ in Figure \[fig:A\] and the results compared to the empirical observations. So for values of $\lambda$ close to the critical value, there is a modest reduction in the error of the numerical scheme due to the time change method. Note that for $\lambda \geq 1/\sqrt{2}$ the errors will no longer be comparable as the order of convergence of the time-changed scheme is no longer quadratic. {width="5.5in" height="2.6in"} \[fig:A\] We are also interested in the computational efficiency of the scheme, so we have investigated the relative performance of the two schemes when both are subject to a constraint on the computational cost, $C$. We can write $C\sim c(1/k)(1/h)$ for some constant $c$ and we then can approximate the errors as $$\begin{aligned} E_R &=& 3(1/24+\lambda^2/8)h^2-(15\lambda^2/96)h^2\\ %\[E_R=3(1/24)h^2+(3/8-15/96)\lambda^2h^2\] &=& (1/8)h^2+(21/96)k^2\end{aligned}$$ and similarly $$\begin{aligned} E_{TC}&=&(1/8)h^2+(3/96)k^2.\end{aligned}$$ Then $$\begin{aligned} E_R&=&hk((1/8)(h/k)+(21/96)(k/h))\\ &=&(c/C)((1/8)(1/\lambda)+(21/96) \lambda)\end{aligned}$$ and similarly $$\begin{aligned} E_{TC}&=&(c/C)((1/8)(1/\lambda)+(3/48) \lambda).\end{aligned}$$ The errors $E_R$ and $E_{TC}$ will be separately minimised as functions of $\lambda$ at\ $$\lambda^*_R=\sqrt{(1/8)(96/21)}\sim0.756$$ and $$\lambda^*_{TC}=\sqrt{(1/8)(48/3)}=\sqrt{2}.$$ {width="5.5in" height="2.6in"} \[fig:B\] Note that for the time change the analysis is only valid for $\lambda \leq 1/\sqrt{2}$, so the minimum occurs at $1/\sqrt{2}$.\ The plot (Figure \[fig:B\]) of the errors for $\lambda \leq 1/\sqrt{2}$, with the fixed computational cost, shows that the time change scheme has the lowest achievable error for the value of $\lambda = 1/\sqrt{2}$ and this error is, in turn, better than the error of Rannacher scheme at the value $\lambda^*_R$.\ The time change applied to the Black-Scholes equation =====================================================  \[S:BSApp\] In this section, we describe numerical results obtained by applying the time change to the Black-Scholes equation $$\begin{aligned} \label{bspde} \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0, \quad S>0, t \in (0,T].\end{aligned}$$ This is the equation that can be used to calculate, for example, the price, $V$, of a European option. Here, $S$ is the price of the underlying, $\sigma$ is the volatility, $r$ is the risk-free rate, and $t$ is time. For a call option, the terminal condition is given by the payoff, i.e., $\max(S-K,0)$, where $K$ is the strike of the option. Practically important sensitivities of the call price to the initial asset price, used in risk management, are given by the delta, $\Delta$, and gamma, $\Gamma$, defined as $$\Delta = \frac{\partial V}{\partial S}$$ and $$\Gamma = \frac{\partial^2 V}{\partial S^2}$$ respectively. Our investigation of convergence for the heat equation was based on Fourier analysis that relies on the PDE having constant coefficients whereas the Black-Scholes PDE used to price a European option does not have constant coefficients and so our analysis is not directly applicable in this case. Nevertheless we have implemented the natural time change method to study the convergence behaviour of the gamma calculated with the modified numerical scheme. In this context the natural analogue to the solution of the heat equation with Dirac delta initial conditions would be the PDE satisfied by the gamma, $$\frac{\partial \Gamma}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2\Gamma}{\partial S^2}+(rS+2\sigma^2)\frac{\partial \Gamma}{\partial S}+(r+\sigma^2)\Gamma =0,$$ where the terminal condition for this equation is given by the second derivative of the payoff with respect to $S$, i.e., $\Gamma(S,T) = \delta(S-K)$. Here we have also assumed that $\sigma$ and $r$ are constant. This PDE could be used to test the performance of the time change approach but a more useful approach (as it reflects industry practice) is to apply the time change directly to the Black-Scholes equation and then use finite differences to derive the values for the gamma from the calculated option values. (Note that the scheme was also tested with the gamma PDE and gave identical results). We first note that if we can write our PDE in the form $$\frac{\partial V}{\partial \tau}- \mathcal{L} V = 0,$$ where $\tau=T-t$ is the time to maturity and where the spatial differential operator $\mathcal{L}$ has time-independent coefficients, then the transformed equation for $\tilde{t} = \sqrt{\tau}$ is simply $$\frac{\partial V}{\partial \tilde t} - 2\:\tilde t \:\mathcal{L} V = 0.$$ If the coefficients in $\mathcal{L}$ are time-dependent, then these need to be rewritten in terms of $\tilde t$. This simple change to the equation results in very minimal changes to the solver code. The calculation done for a single time step in the original variables can be written as $$(I+L)V^{n+1}=(I-L)V^{n},$$ where $L$ is a matrix dependent on $k$, $h$, $S$, $\sigma$ and $r$ while $V^{n}$ is the solution vector at time step $n$. In the case of constant coefficients, and in the time-changed variables, this calculation simply becomes $$(I+ 2 \:\tilde t_{n+1}L)\widetilde{V}^{n+1}=(I - 2 \:\tilde t_{n}L) \widetilde{V}^{n},$$ where $\widetilde{V}^n$ serves as numerical approximation to $\widetilde{V}(\cdot,\tilde{t}_n) = V(\cdot,T-\tilde{t}_n\,\!^2)$. In particular, the transformed Black-Scholes equation is $$\label{bstrans} \frac{\partial \widetilde{V}}{\partial \tilde t} - 2\:\tilde t\:\left(\frac{1}{2}\sigma^2S^2\frac{\partial^2\widetilde{V}}{\partial S^2}+rS\frac{\partial \widetilde{V}}{\partial S}-r\widetilde{V}\right)=0$$ with the payoff function as the initial condition. The transformed equation for $V$ was solved using the Crank-Nicolson scheme and the gamma calculated by finite differences. The errors in gamma were computed from the analytic values and the convergence rate of the numerical error as a function of $\lambda$ was analysed. Although the scheme does not have the constant coefficients required for the application of Fourier analysis, the errors do behave in a very similar way to those for the heat equation (see Figure 4). {width="5.5in" height="2.5in"} \[fig:BSConvergenceOrder200912\] The volatility, $\sigma$, was set to 20$\%$ and the strike, $K$, was set to 100. The risk-free rate, $r$, was set to 5$\%$. The results of the analysis showed that there was a critical value of $\lambda$ below which the error in the maximum norm was $O(h^2)$ and above which the $h$ exponent declined in the same manner as for the heat equation. Also shown in Figure 4 is the plot of $\min(2,1/(\sigma^2 K^2 \lambda^2))$ and it is clear that this gives a reasonable description of the behaviour of the $h$ exponent although it is not quite as good as with the heat equation itself. We can justify the critical value of $\lambda$ by noting that, in the Black-Scholes equation, the factor $\frac{1}{2}\sigma^2S^2$ plays the same role as the factor $\frac{1}{2}$ in the heat equation and we also observe that if we had written the heat equation in the form $u_t= \frac{1}{2}D \, u_{xx}$, for some constant diffusivity $D$, we would have found the critical value of $\lambda$ to be $1/\sqrt{2D}$. Typically, the worst behaviour of a numerical scheme for the Black-Scholes equation with delta function initial conditions is seen at times close to maturity and in the vicinity of the strike and so the strike is an appropriate parameter to use to represent the coefficient of $\frac{\partial^2V}{\partial S^2}$ which then determines the critical value of $\lambda$. Even without the precise results available from Fourier analysis, we can still attempt a partial explanation of how the time change improves the convergence of the Crank-Nicolson numerical scheme. One consideration is that the truncation error of the numerical scheme has a leading order term proportional to the third derivative of the European option price and the behaviour of this derivative is much improved by the time change. The theta, $\Theta_C$, of the call option is given by (see @HAUG, p.64) $$\Theta_C=\frac{\partial C}{\partial t}=\frac{Sn(d_1)\sigma}{2\sqrt{T-t}}-rKe^{-r(T-t)}N(d_2),$$ where $C$ is the solution to (\[bspde\]) with $C(S,T) = \max(S-K,0)$, $N(x)$ is the cumulative Normal(0,1) distribution, $n(x)$ is its first derivative (the Gaussian function) and $$d_{1,2}=\frac{( \log(S/K) + (r\pm\frac{1}{2}\sigma^2)(T-t) ) }{ \sigma\sqrt{T-t} }.$$ Clearly $\Theta_C$ will become singular at the strike as $t \rightarrow T$ and any higher derivatives will also be singular there. However, if we change the time variable to $\tilde t = \sqrt{T-t}$, we find that $$\tilde \Theta_C=\frac{\partial C}{\partial \tilde t}=-2\tilde t\frac{\partial C}{\partial t}=-2\tilde t\left(\frac{Sn(d_1)\sigma}{2\tilde t}-rKe^{-r\tilde t^2}N(d_2)\right)$$ $$=-Sn(d_1)\sigma+2\tilde t rKe^{-r\tilde t^2}N(d_2)$$ and $\tilde \Theta_C$ is then not singular at $\tilde t=0$. Carrying on to the second ‘time’ derivative we see that $$\frac{\partial^2 C}{\partial \tilde t^2}= -Sn'(d_1)\sigma \frac{\partial d_1}{\partial \tilde t}+2rKe^{-r\tilde t^2}(1-2\:\tilde t\: r)N(d_2)-2\:\tilde t \:r Ke^{-r\tilde t^2}n(d_2)\frac{\partial d_2}{\partial \tilde t},$$ where we also have $$\frac{\partial d_{1,2}}{\partial \tilde t}=-\frac{\log{\frac{S}{K}}}{\sigma \:\tilde t^2}+\frac{r \pm \frac{\sigma^2}{2}}{\sigma}$$ so there are potentially singular terms to consider, i.e., those featuring inverse powers of $\tilde t$. However, if $S \neq K$ then both $d_1$ and $d_2 \rightarrow \pm\infty$ as $\tilde t \rightarrow 0$ so that the exponential factors in $n(d_i)=(1/\sqrt{2\pi})\exp{(-d_i^2/2)}$ will eventually overwhelm the inverse powers of $\tilde t$. Finally, if $S=K$ the term $\log(S/K)$ vanishes and the derivative is non-singular at the strike. It is clear that this analysis will extend to the higher derivatives of $C$ because the negative powers of $\tilde t$ will now only arise from differentiating $N$ and will then only appear as multipliers of the Gaussian function. These powers will only be present for $S \neq K$ where the Gaussian function will decay rapidly and overwhelm the inverse powers. So the higher derivatives will not become singular as $\tilde t \rightarrow 0$. It is also clear that the factor $\tilde{t}$ in (\[bstrans\]) causes the singular spatial truncation error, which contains the fourth derivative of $V$ with respect to $S$, to become more well-behaved for small $\tilde{t}$ in the vicinity of the strike. So the time change has, to some extent, regularised the behaviour of the truncation error. Although it is not entirely clear how to deduce convergence properties of the transformed numerical scheme directly from this, and particularly convergence of derivatives of the solution, it seems plausible that the improved regularity of the truncation error is advantageous when using initial conditions that are not smooth. American options ================  \[S:AmericPut\] @FV investigate the convergence properties of Crank-Nicolson time-stepping when used to calculate the price of an American put by a penalty method. The penalised Black-Scholes equation is given by $$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV + \rho \max( \max(K-S,0)-V,0)=0,$$ where $\rho>0$ is a penalty parameter. This non-linear equation is solved with the terminal condition set equal to the option payoff. The penalty term comes into play when the solution violates the requirement that $V$ should not fall below the payoff. As the penalty parameter $\rho$ increases, the solution of the equation converges to the value for the American option satisfying $$\max\left(\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV, \max(K-S,0)-V\right) = 0.$$ For the present purposes, we are interested solely in the convergence of the Crank-Nicolson scheme and choose $\rho$ large enough to make the penalty solution numerically indistinguishable from its limit. @FV implemented their solver using the Crank-Nicolson scheme with Rannacher start up and demonstrated that as the time step was reduced with the ratio of time step to space step held constant, the error in the calculated at the money (ATM) value of the American put tended to $O(h^{\frac{3}{2}})$ behaviour. Their analysis of the problem suggests that non-uniform time-stepping might restore second order convergence so they went on to describe and implement an adaptive time-stepping procedure that did indeed yield second order convergence for the option value. Their results suggest that the simple time change method described above might also be able to restore second order convergence. [|l|l|l|l|l|]{}\ \ $M$ & $N$ & Value & Delta & Gamma\  3200 & 640 & 3.98 & 3.92 & 3.91\  6400 & 1280 & 3.97 & 3.97 & 3.94\  12800 & 2560 & 3.97 & 3.97 & 3.96\  25600 & 5120 & 3.99 & 3.97 & 3.99\ \ [|l|l|l|l|l|]{}\ \ $M$ & $N$ & Value & Delta & Gamma\  3200 & 320 & 3.84 & 4.01 & 3.80\  6400 & 640 & 3.99 & 3.93 & 3.96\  12800 & 1280 & 3.94 & 3.93 & 3.89\  25600 & 2560 & 3.96 & 3.97 & 3.98\ \ [|l|l|l|l|l|]{}\ \ $M$ & $N$ & Value & Delta & Gamma\  3200 & 160 & 3.85 & 3.65 & 2.07\  6400 & 320 & 3.78 & 3.87 & 2.08\  12800 & 640 & 3.89 & 3.87 & 2.09\  25600 & 1280 & 3.89 & 3.90 & 2.09\ \ The penalty code was modified to incorporate the time change and numerical results were generated. Second order convergence was observed for the option value and delta for the ATM option and it was also observed that, for sufficiently small $\lambda$, the gamma showed second order convergence. Table  \[tab:Table1\] shows the ratios of successive differences for mesh refinements with $\lambda$ equal to 0.0125, 0.025 and 0.05. As with the European option discussed above, and with the strike, $K=100$, and the volatility, $\sigma=20\%$, the critical value of $\lambda$ appears to be in the vicinity of $1/(\sigma K \sqrt{2})=0.035$. It was also observed that the plots of the calculated gamma revealed instability around the exercise boundary, behaviour which was also seen in @FV, where the authors found that their adaptive time-stepping approach resulted in instability of the gamma at the exercise boundary unless fully-implicit time-stepping was used. This behaviour is a separate phenomenon independent of the short-time asymptotics addressed here. In the previous section we suggested that the improved convergence properties of the Crank-Nicolson scheme might be related to improved regularity of the third time derivative of the value function of the European put in the vicinity of the strike and close to maturity. We would like to be able to perform a similar analysis for the American put but, of course, no simple closed form expression for the value function is known. However, @MA do present an asymptotic expansion for the value function which can be decomposed into the value function of the European put plus the sum of an asymptotic series representing the early exercise premium. In @MA, some changes of variable are used to define the final form of the expansion. We write $S=Ke^x$ and $\xi=x/(2\sqrt{\tau_s})$, where $\tau_s$ is the rescaled time to maturity, equal to $2(T-t)/\sigma^2$. The transformed value function is then given by $$v(x,\tau_s)=\frac{P(S,t)+S-K}{K},$$ where $P(S,t)$ is the value of the American put as a function of the original variables. We can then write that $$\begin{aligned} \label{americanexp} v(x,\tau_s)=v^E(x,\tau_s)+\sum_{n=2}^{\infty} \sum_{m=1}^{\infty}\tau_s^{n/2}(-\log \tau_s)^{-m}F^{m}_{n}(\xi),\end{aligned}$$ where $F^{m}_{n}$ are functions of the similarity variable $\xi$ and $v^E(x,\tau_s)$ is the value of the European put expressed in the new variables. In @MA, only a few of the functions $F^{m}_{n}$ are fully identified and discussion of those is restricted to $m=1$. Those with $n \geq 0$ and $m=1$ are, in principle, tractable but only partial results are known for the cases where $m>1$. This limits our ability to draw conclusions from the behaviour of the derivatives of the early exercise premium. Nevertheless, we will illustrate some aspects of the behaviour of the derivatives by describing a single term of the early exercise premium and its form and behaviour after the time change. For example we find that the leading order coefficient of the early exercise premium is given by $$F^{1}_{2}(\xi) = \mu(\xi \exp{(-\xi^2/2) }/\sqrt{\pi} + (2\xi^2+1)\mathrm{erfc}(\xi))$$ for some constant $\mu$. After the time change this early exercise term in (\[americanexp\]) is $$G = \mu \tilde t^2(-2\log \tilde t)^{-1}F^{1}_{2}(\eta),$$ where $\eta=x/(2\tilde t)$. At the strike, we have $$\begin{aligned} \frac{\partial G}{\partial \tilde t} &=& \mu(2\tilde t(-2\log \tilde t)^{-1} -2\tilde t(-2\log \tilde t)^{-2}), \\ \frac{\partial^2 G}{\partial \tilde t^2} &=& \mu(2(-2\log \tilde t)^{-1} +4(-2\log \tilde t)^{-2})+8(-2\log \tilde t)^{-3})\end{aligned}$$ and $$\begin{aligned} \frac{\partial^3 G}{\partial \tilde t^3}=\frac{\mu}{\tilde t}(4(-2\log \tilde t)^{-2} +16(-2\log \tilde t)^{-3})+48(-2\log \tilde t)^{-4}).\end{aligned}$$ Acknowledging that this calculation only addresses a single term of the expansion of the early exercise premium we see that there is an improvement in the regularity of the derivatives of the put but unfortunately this does not obviously extend to the third derivative terms such as ${\partial^3 G}/{\partial \tilde t^3}$. So, although we are encouraged by the empirical performance of the time change method applied to the American put, further analysis will be required to explain fully the improvement in convergence. Conclusions {#sec:concl} =========== In this paper we have presented the analysis of a simple and natural change of time variable that improves the convergence behaviour of the Crank-Nicolson scheme. When applied to the solution of the heat equation with Dirac delta initial conditions, the numerical solution of the time changed PDE is always convergent with rate of convergence determined by the ratio, $\lambda$, of time step to space step (held constant during grid refinement). This behaviour contrasts strongly to that of the Crank-Nicolson scheme when applied to the original PDE, which is always divergent and where $\lambda$ controls how small the time step must be before the divergence appears. A proof of the behaviour of the time-changed scheme is given and numerical results presented to support the theoretical analysis. Although the Fourier analysis used to prove this result cannot be applied directly to the Black-Scholes equation as it does not have constant coefficients, numerical experiments for the price and Greeks of a European call indicate that the time change method also leads to a convergent Crank-Nicolson scheme for this problem, without the need to introduce Rannacher steps, and quadratic convergence can be obtained if $\lambda$ is chosen appropriately. The change of convergence order at the value of $\lambda$ which gives optimal complexity for given error tolerance is in line with the sensitivity of the error to $\lambda$ observed in @GC2. The present analysis does not support the conjecture made in @GC2, however, that the convergence of the Rannacher scheme with non-uniform (square-root) time-stepping is never of second order, the argument being that for small initial time-steps the smoothing effect diminishes. In fact, we see here that under square-root time smoothing is not necessary at all. The experimentally observed second order convergence for the time-changed scheme extends to the computation of American option values which satisfy a linear complementarity problem with singular behaviour of the free boundary. These results are in line with those of @FV for a time-adaptive Crank-Nicolson scheme with Rannacher start-up. It can be seen as an advantage of the time-changed scheme proposed here that it provides a remedy for both the reduced convergence order of the Crank-Nicolson scheme for European and American option values and for the divergence of their sensitivities by a single, simple modification. Future work in this area will investigate other applications of the time variable change to option pricing and will consider whether the method offers efficiency gains over the standard approaches. Appendix A. Proof of Lemma \[L:LemmaA\], (\[expapp\]) and (\[expandapp\]) {#app:lemma1 .unnumbered} ========================================================================= We analyse the behaviour of the Fourier transform $\widehat{U}^N(s)$ in each of the four wavenumber regimes. **Proof of (\[expapp\])**: {#app:exp .unnumbered} -------------------------- We start from (\[EQ3\]), $$\log(\widehat{U}^N(s))=\sum^{N-1}_{k=1} \left(\log(1-k\xi) - \log(1+k\xi)\right) - \log(1+N\xi).$$ We seek to expand the logarithmic terms. We begin by first expanding $\xi=2 \lambda^2 \sin^2(\frac{sh}{2})$ in powers of $h$ to obtain $$1-k\xi=1-\frac{k\lambda^2 s^2}{2!} h^2 +\frac{k\lambda^2 s^4}{4!} h^4 -\frac{k\lambda^2 s^6}{6!} h^6 + \frac{k\lambda^2 s^8}{8!}h^8 \theta_k \equiv \sum_{i=0}^3 A_i h^{2i} \, + \, A_4 h^8,$$\ where $|\theta_k| \leq 1$. Setting $\delta=h^2$, we define $$\begin{aligned} \notag g(\delta) \equiv 1-k\xi= \sum_{i=0}^3 A_i \delta^{i} \, + \, A_4 \delta^4\end{aligned}$$ and $$\begin{aligned} \notag f(\delta)=\log(1-k\xi)=\log{(g(\delta))}\end{aligned}$$ and we can then expand $f(\delta)$ in terms of $\delta$ to obtain $$\log(1-k\xi)=-\frac{k\lambda^2}{2!}s^2\delta+\frac{1}{2!}\left(\frac{2 k \lambda^2}{4!}-\frac{k^2 \lambda^4}{(2!)^2}\right)s^4\delta^2+\frac{1}{3!}\left(-\frac{6 k \lambda^2}{6!}-\frac{2k^3 \lambda^6}{(2!)^3}+\frac{6k^2 \lambda^4}{2!4!}\right)s^6\delta^3+Z_ks^8\delta^4$$ and we have to ensure that the remainder $Z_k$ is well-behaved. We do this by analysing the behaviour of $$f^{(4)}(\delta)=\frac{p(\delta)}{g(\delta)^4},$$\ where\ $$p(\delta)=g(\delta)^3 g^{(4)}(\delta)-4g(\delta)g'(\delta)g^{(3)}(\delta)+6(g'(\delta))^2g(\delta)g''(\delta)-3(g''(\delta))^2g(\delta)-6(g'(\delta))^4.$$\ Then $g(\delta) \rightarrow 1$ as $h \rightarrow 0$ if we ensure that $s<h^{-m}$, where $m < \frac{1}{2}$, because, for example, $$|A_1 \delta| = \frac{k \lambda^2 s^2 h^2}{2} < \frac{N\lambda^2 s^2 h^2}{2} = \frac{ \lambda s^2 h}{2} \rightarrow 0.$$ Then $h$ can be chosen small enough (or $N$ large enough) so that $$|1+A_1 \delta +A_2 \delta^2 +A_3 \delta^3 +A_4 \delta^4| > \frac{1}{2}$$ so the denominator of $f^{(4)}(\delta)$ is bounded away from zero. Next we can show that $$|g^{(i)}(\delta)|<\alpha_i|A_i|$$ for some constants, $\alpha_i$ for $i=0,1,2,3$ and $4$, because of the constraint on $s$. We can finally conclude that\ $$|f^{(4)}(\psi\delta)|\delta^4<(2\alpha_4|A_4|+32\alpha_1\alpha_3|A_1||A_3|+48\alpha_1^2 \alpha_2|A_1|^2 |A_2|+24\alpha_2^2|A_2|^2+96\alpha_1^4|A_1|^4)h^8.$$\ and it follows that we can estimate asymptotically the remainder by just retaining the term proportional to $|A_1|^4 h^8$ which is, in turn, proportional to $k^4 \lambda^8s^8h^8$.\ By changing the sign of $k$, we can obtain a similar expression for $\log(1+k\xi)$ hence find that $$\log(1-k\xi)-\log(1+k\xi)=-k \lambda^2 s^2 h^2 +\frac{1}{12}k \lambda^2 s^4 h^4-\frac{1}{360}k \lambda^2 s^6 h^6-\frac{1}{12}k^3 \lambda^6 s^6 h^6 +W_k k^4 s^8 h^8,$$ where $|W_k|<K$ for some constant, $K$, and so $$\begin{aligned} \notag \log(\widehat{U}^N(s))&=\left(-\lambda^2 s^2 h^2 +\frac{1}{12}\lambda^2 s^4 h^4-\frac{1}{360}\lambda^2 s^6 h^6\right)\left(\sum_{k=1}^{k=N-1}k\right)%\\ %\notag -\frac{1}{12}\lambda^6 s^6 h^6 \left(\sum_{k=1}^{k=N-1}k^3\right)\\ \notag &+\left(-\frac{\lambda^2 s^2 h^2}{2}+\frac{1}{24}\lambda^2 s^4 h^4-\frac{1}{720}\lambda^2 s^6 h^6\right)N%\\ %\notag +\frac{1}{8}\lambda^4 s^4 h^4 N^2- \frac{1}{48}\lambda^4 s^6 h^6 N^2 \\ \notag &- \frac{1}{24}\lambda^6 s^6 h^6 N^3 +O(s^8h^3),\end{aligned}$$ because $(\sum_{k=1}^{k=N}k^4)\lambda^8s^8h^8=O(N^5\lambda^8s^8h^8)=O(\lambda^3s^8h^3)$. So $$\begin{aligned} \notag \log(\widehat{U}^N(s))&=\left(-\lambda^2 s^2 h^2 +\frac{1}{12}\lambda^2 s^4 h^4-\frac{1}{360}\lambda^2 s^6 h^6\right)\left(\frac{1}{2}(N-1)N+\frac{N}{2}\right)\\ \notag &-\frac{1}{12}\lambda^6 s^6 h^6 \left(\left(\frac{1}{2}(N-1)N\right)^2+\frac{N^3}{2}\right)+O(s^8h^3) %\notag %&+\frac{1}{8}\lambda^4 s^4 h^4 N^2- \frac{1}{48}\lambda^4 s^6 h^6 N^2 +O(s^8h^3).\end{aligned}$$ as the remaining terms are dominated by $O(s^8h^3)$. Substituting $N=\frac{1}{h\lambda}$ we get $$\begin{aligned} \notag \log(\widehat{U}^N(s))&=-\frac{1}{2}s^2 +\left(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2s^4\right)h^2-\left(\frac{1}{720}s^6+\frac{1}{48}\lambda^2 s^6+\frac{1}{48}\lambda^4 s^6\right)h^4+O(s^8h^3)\end{aligned}$$ and hence $$\log(\widehat{U}^N(s))=-\frac{1}{2}s^2 +\left(\frac{1}{24}s^4-\frac{1}{48}\lambda^2 s^6+\frac{1}{8}\lambda^2 s^4\right)h^2+O(s^8h^3),$$ because the $s^6h^4$ term of the expansion is dominated by the term $O(s^8h^3)$.\ **Proof of (\[expandapp\])**: {#app:expand .unnumbered} ----------------------------- We start from (\[secexp\]), $$\log\big(\big|\widehat{U}^N(s)\big|\big)=\log(M(\xi,m^*))+\log\left\{\frac{(1-\frac{1}{\xi(m^*+1)})\ldots(1-\frac{1}{\xi(N-1)})} {(1+\frac{1}{\xi(m^*+1)})\ldots(1+\frac{1}{\xi (N-1)})}\right\}+\log((1+N\xi)^{-1}),$$ and write $$\log{\frac{(1-\frac{1}{\xi(m^*+1)})\ldots(1-\frac{1}{\xi(N-1)})} {(1+\frac{1}{\xi(m^*+1)})\ldots(1+\frac{1}{\xi (N-1)})}}=\sum_{k=m^*+1}^{k=N-1} \left(\log\left(1-\frac{1}{k\xi}\right) -\log\left(1+\frac{1}{k\xi}\right)\right).$$ We can expand the logarithmic terms for $\xi\in[1/m^*,2\lambda^2]$ to get $$\begin{aligned} \notag \sum_{k=m^*+1}^{k=N-1} \left(\log(1-\frac{1}{k\xi}) -\log(1+\frac{1}{k\xi})\right)&=\sum_{k=m^*+1}^{k=N-1}\left(\sum_{j=1}^{\infty}\frac{(-1)}{j(k\xi)^j}-\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j(k\xi)^j}\right)\\ \notag &=-\sum_{j=0}^{\infty}\frac{2}{(2j+1)\xi^{2j+1}}\left(\sum_{k=m^*+1}^{k=N-1}\frac{1}{k^{2j+1}}\right)\end{aligned}$$ and, for any $N$, this series converges absolutely in $1/\xi$ as we have just rearranged a finite sum of absolutely convergent series. We can write this as $$\begin{aligned} \notag \sum_{k=m^*+1}^{k=N-1} \left(\log(1-\frac{1}{k\xi}) -\log(1+\frac{1}{k\xi})\right)&=-\frac{2}{\xi}\left(\sum_{k=m^*+1}^{k=N-1}\frac{1}{k}\right) -\sum_{j=1}^{\infty}\frac{2}{(2j+1)\xi^{2j+1}}\left(\sum_{k=m^*+1}^{k=N-1}\frac{1}{k^{2j+1}}\right)\\ \notag &=-\frac{2}{\xi}\left(\sum_{k=m^*+1}^{k=N-1}\frac{1}{k}\right)+A(\xi,m^*)+B(\xi,N),\end{aligned}$$ where\ $$A(\xi,m^*)=-\sum_{j=1}^{\infty}\frac{2}{(2j+1)\xi^{2j+1}}\left(\sum_{k=m^*+1}^{\infty}\frac{1}{k^{2j+1}}\right)$$\ and $$B(\xi,N)=\sum_{j=1}^{\infty}\frac{2}{(2j+1)\xi^{2j+1}}\left(\sum_{k=N}^{\infty}\frac{1}{k^{2j+1}}\right),$$\ and we must establish the convergence and size of the functions $A(\xi,m^*)$ and $B(\xi,N)$ (see below).\ We also have\ $$\begin{aligned} \notag -\log(1+N\xi)=-\log{N\xi}-\log{(1+{1}/{N\xi})}&=-\log{N\xi}-\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j(N\xi)^j}\\ \notag &=-\log{N\xi}-\frac{1}{N\xi}+C(\xi,N),\end{aligned}$$ where $$C(\xi,N)=-\sum_{j=2}^{\infty}\frac{(-1)^{j+1}}{j(N\xi)^j}$$ so that $$\begin{aligned} \notag\ \log\big(\big|\widehat{U}^N(s)\big|\big)&=\log(M(\xi,m^*))-\frac{2}{\xi}\left(\sum_{k=m^*+1}^{k=N-1}\frac{1}{k}\right)+A(\xi,m^*)+B(\xi,N)-\log{N\xi}-\frac{1}{N\xi}+C(\xi,N)\\ \notag\ &=\log(M(\xi,m^*))-\frac{2}{\xi}\left(\sum_{k=m^*+1}^{k=N}\frac{1}{k}\right)+A(\xi,m^*)+B(\xi,N)-\log{N\xi}+\frac{1}{N\xi}+C(\xi,N)\\ \notag\ &=P(\xi,m^*)-\frac{2}{\xi}\left(\sum_{k=1}^{k=N}\frac{1}{k}\right)+D(\xi,N)-\log{N}+O(N^{-1}),\end{aligned}$$ where $$P(\xi,m^*)=\log(M(\xi,m^*))-\frac{2}{\xi}\left(\sum_{k=1}^{k=m^*}\frac{1}{k}\right)+A(\xi,m^*)-\log{\xi}$$ and $$D(\xi,N)=B(\xi,N)+C(\xi,N).$$ Then we can write $$\begin{aligned} \notag\ \log\big(\big|\widehat{U}^N(s)\big|\big)&=P(\xi,m^*)-\frac{2}{\xi}\left(\log{N}+\gamma+O(N^{-1})\right)+D(\xi,N)-\log{N}+O(N^{-1})\\ \notag\ &=Q(\xi,m^*)-\left( \frac{2}{\xi}+1 \right) \log{N} + O(N^{-1}) +D(\xi,N),\end{aligned}$$ where $\gamma$ is the Euler-Macheroni constant [see @TT], and where $$Q(\xi,m^*)=P(\xi,m^*)-\frac{2}{\xi}\gamma.$$ Then using $N=1/(h\lambda)$ and assuming for now $D(\xi,N)=O(N^{-1})$, which we will show later, we get $$\begin{aligned} \notag\ \log\big(\big|\widehat{U}^N(s)\big|\big)&=R(\xi,m^*)+\left(\frac{2}{\xi}+1\right)\log{h}+O(h),\end{aligned}$$ where $$R(\xi,m^*)=Q(\xi,m^*)+\left(\frac{2}{\xi}+1\right)\log{ \lambda }.$$\ Then we finally have $$\begin{aligned} \notag\ \big|\widehat{U}^N(s))\big|&=S(\xi,m^*)h^{\left(\frac{2}{\xi}+1\right)}(1+O(h))\\\end{aligned}$$ as we were required to prove. We note that, written out in full, we have $$S(\xi,m^*)= M(\xi,m^*) \lambda^{ (\frac{2}{\xi}+1) } \exp{ \left( -\frac{2}{\xi} \left(\gamma+ \sum_{k=1}^{k=m^*}\frac{1}{k} \right) \right) } e^{A(\xi,m^*)}$$ and we note that $S(\xi,m^*)$ is continuous and hence bounded on $[1/m^*,2\lambda^2]$.\ We still need to prove that $A(\xi,m^*)+B(\xi,N)$ is a valid rearrangement and proceed by proving that $A(\xi,m^*)$ is convergent. It will then follow that $B(\xi,N)$ is also convergent. In addition we will show that $D(\xi,N)=B(\xi,N)+C(\xi,N)=O(N^{-2})$.\ By a result in @Timofte, we have, for $r \geq 1$, $$\sum_{k=m^*+1}^{k=\infty}\frac{1}{k^{2r+1}}=\frac{1}{2r(m^*+\theta_{m^*})^{2r}}$$ for some $\theta_{m^*} \geq \frac{1}{2}$. Therefore $$\begin{aligned} |A(\xi,m^*)| &<& \sum_{j=1}^{\infty}\frac{2}{(2j+1)}\frac{1}{\xi^{2j+1}}\frac{1}{2j(m^*+\frac{1}{2})^{2j}} \\ &=& \left(m^*+\frac{1}{2}\right)\sum_{j=1}^{\infty}\frac{1}{j(2j+1)}\frac{1}{(\xi(m^*+\frac{1}{2}))^{2j+1}} \\ &<& \frac{1}{3}\left(m^*+\frac{1}{2}\right)\eta^3\sum_{j=0}^{\infty}\eta^{2j},\end{aligned}$$ where $\eta=1/(\xi(m^*+\frac{1}{2}))<1$ and so $A(\xi,m^*)$ is convergent. It follows that $B(\xi,N)$ is convergent because $A(\xi,m^*)+B(\xi,N)$ is convergent.\ Next we find out how $B(\xi,N)$ depends on $N$. We have $$\begin{aligned} |B(\xi,N)| &<& 2\sum_{j=1}^{\infty}\frac{1}{(2j+1)}\frac{1}{\xi^{2j+1}}\frac{1}{2j(N-1+\frac{1}{2})^{2j}} \\ &=& 2N\sum_{j=1}^{\infty}\frac{1}{(2j+1)}\frac{1}{(N\xi)^{2j+1}}\frac{N^{2j}}{2j(N-\frac{1}{2})^{2j}},\end{aligned}$$ and as $N/(N-\frac{1}{2}) \leq 2$, we have $$|B(\xi,N)|<2N\sum_{j=1}^{\infty}\frac{1}{(2j+1)}\frac{1}{(N\xi)^{2j+1}}\frac{2^{2j+1}}{4j} < \frac{1}{12}N \sum_{j=1}^{\infty}\left(\frac{2}{N\xi}\right)^{2j+1}.$$ Then because we have $2/\xi \leq 2m^*$, we find that for sufficiently large $N$, we have $2/N\xi<c<1$ for a constant $c$ and the geometric series converges and we find that $$|B(\xi,N)| < \frac{1}{6}N\left(\frac{2}{N\xi}\right)^3\frac{1}{\left(1-\frac{2}{N\xi}\right)}=O(N^{-2})$$ and similarly for $C$. It then follows that $D(\xi,N)=O(N^{-2})$. Appendix B. Proof of Lemma \[L:LemmaB\], (\[intapp\]) {#app:int .unnumbered} ===================================================== We want to show that $$I %=\frac{F(h)}{h^{\frac{1}{\lambda^2}}} =\int^{2\lambda^2}_{\xi=\xi_{m^*}} \frac{ S(\xi,m^*)h^{(\frac{2}{\xi}-\frac{1}{\lambda^2})}}{ \lambda\sqrt{2}\sqrt{\xi}\sqrt{1-\frac{\xi}{2\lambda^2}} } \, d\xi =O\left(\frac{1}{ \sqrt{ \log{ \frac{1}{h} } }}\right),$$ where $S(\cdot,m^*)$ is continuous. We make the substitution $z^2=1-{\xi}/{2\lambda^2}$ to obtain $$I=4\lambda^2\int_{z=0}^{z=\sqrt{B}}S^*(z,m^*)h^{\left(\frac{1}{\lambda^2}\frac{z^2}{1-z^2}\right)} \, dz,$$ where $B=1-\frac{\xi_{m^*}}{2\lambda^2}$, and where $S^*(z,m^*)={S(\xi,m^*)}/\sqrt{1-z^2}$ is a continuous function on $[0,\sqrt{B}]$ as $B < 1$. We want to show that this integral is concentrated around $z=0$ for small $h$ and write $$I=I_1+I_2,$$ where $$I_1=4\lambda^2\int_{z=0}^{z=\sqrt{A}}S^*(z,m^*)h^{\left(\frac{1}{\lambda^2}\frac{z^2}{1-z^2}\right)} \,dz$$ and $$I_2=4\lambda^2\int_{z=\sqrt{A}}^{z=\sqrt{B}}S^*(z,m^*)h^{\left(\frac{1}{\lambda^2}\frac{z^2}{1-z^2}\right)} \, dz,$$ where $A$ is chosen so that, asymptotically, $I_1$ dominates $I_2$. Firstly, we consider $I_2$. For $h < 1$, the second factor of the integrand is decreasing in $z$, so $$|I_2|<4\lambda^2 S^*_{max} h^{\left(\frac{1}{\lambda^2}\frac{A}{1-A}\right)},$$ where $S^*_{max}$ is a bound for $S^*$ on $[\xi_{m^*},2\lambda^2]$. We now take $$\frac{A}{1-A}=\frac{1}{\sqrt{\log{{1}/{h}}}}$$ and note that $$\log{h^{\left(\frac{1}{\lambda^2}\frac{A}{1-A}\right)}}=-\frac{1}{\lambda^2}\frac{A}{1-A}\log{\frac{1}{h}}=-\frac{1}{\lambda^2}\sqrt{\log{\frac{1}{h}}}$$ so $$|I_2|<4\lambda^2 S^*_{max} e^{ -\frac{1}{\lambda^2} \sqrt{ \log{ \frac{1}{h} } }} \rightarrow 0 \quad \mathrm{ as } \;\;\; h \rightarrow 0.$$ Now we want to show that $$\lim_{h \rightarrow 0} I_1 \sqrt{ \log{ \frac{1}{h} } } = \lim_{h \rightarrow 0} \left(4\lambda^2\sqrt{\log{\frac{1}{h}}} \int_{z=0}^{z=\sqrt{A}} S^*(z,m^*)h^{\left(\frac{1}{\lambda^2}\frac{z^2}{1-z^2}\right)} \, dz\right)\ =O(1)$$\ and we note that $$A=\frac{1}{1+\sqrt{\log{{1}/{h}}}} \rightarrow 0 \quad \mathrm{as} \;\;\: h \rightarrow 0.$$ We make a final substitution $$\eta=\frac{z}{\lambda}\sqrt{\log{\frac{1}{h}}},$$ to get $$\lim_{h \rightarrow 0}I_1\sqrt{\log{\frac{1}{h}}}=4\lambda^2\lim_{h \rightarrow 0} \int_{\eta=0}^{\eta=\eta^*}S^*(\lambda \eta/\sqrt{\log{{1}/{h}}},m^*)\exp{\left(-\frac{\eta^2}{1-\lambda^2 \eta^2/\log{(1/h)}}\right)}\, d\eta,$$ where the upper limit $$\eta^*=\frac{\sqrt{A}}{\lambda}\sqrt{\log{\frac{1}{h}}}=\frac{ \sqrt{ \log{ {1}/{h} } } }{ \lambda \sqrt{ 1+\sqrt{ \log{{1}/{h}} } } }$$ tends to infinity as $h \rightarrow 0$. As $S^*(\cdot,m^*)$ is bounded and continuous at 0 we see that $$\lim_{h \rightarrow 0} \int_{\eta=0}^{\eta=\eta^*}S^*(\lambda \eta/\sqrt{\log{{1}/{h}}},m^*)\exp{ \left(-\frac{\eta^2}{1-\lambda^2 \eta^2/\log{(1/h)}}\right) }\, d\eta=S^*(0,m^*)\int_{\eta=0}^{\eta=\infty}\exp{\left(-\eta^2\right)}\, d\eta$$\ so we have $I_1\sqrt{\log{{1}/{h}}}=O(1)$ as $h \rightarrow 0$.\ From this we see that $I_1=O({1}/{ \sqrt{ \log{ \frac{1}{h} } }})$ and we also have $$\lim_{h \rightarrow 0}\left|\frac{I_2}{I_1}\right|=\lim_{h \rightarrow 0}\sqrt{ \log{ \frac{1}{h} } }e^{ -\frac{1}{\lambda^2} \sqrt{ \log{ \frac{1}{h} } }}= 0$$ so we have established that $I_1$ asymptotically dominates $I_2$ and so $I=O({1}/{ \sqrt{ \log{ {1}/{h} } }})$.

ArXiv

--- abstract: 'Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantities (such as limiting variances of different estimators) related to the first likelihood ratio can also be approximated by those related to the second one. Finally, we discuss the asymptotics of the large values of the parameter and illustrate the results by numerical simulations.' author: - | Sergueï <span style="font-variant:small-caps;">Dachian</span>\ Laboratoire de Mathématiques\ Université Blaise Pascal\ 63177 Aubière CEDEX, France\ [email protected] title: | On Limiting Likelihood Ratio Processes\ of some Change-Point Type Statistical Models --- **Keywords**: non-regularity, change-point, limiting likelihood ratio process, Bayesian estimators, maximum likelihood estimator, limiting distribution, limiting variance, asymptotic efficiency **Mathematics Subject Classification (2000)**: 62F99, 62M99 Introduction ============ Different change-point type models encountered in statistical inference for stochastic processes give rise to different limiting likelihood ratio processes. In this paper we consider two of these processes. The first one is the random process $Z_\rho$ on ${\mathbb{R}}$ defined by $$\label{proc1} \ln Z_\rho(x)=\begin{cases} \vphantom{\Big)}\rho\,\Pi_+(x)-x, &\text{if } x{\geqslant}0,\\ \vphantom{\Big)}-\rho\,\Pi_-(-x)-x, &\text{if } x{\leqslant}0,\\ \end{cases} $$ where $\rho>0$, and $\Pi_+$ and $\Pi_-$ are two independent Poisson processes on ${\mathbb{R}}_+$ with intensities $1/(e^\rho-1)$ and $1/(1-e^{-\rho})$ respectively. We also consider the random variables $$\label{vars1} \zeta_\rho=\frac{\int_{{\mathbb{R}}}x\,Z_\rho(x)\;dx}{\int_{{\mathbb{R}}}\,Z_\rho(x)\;dx} \quad\text{and}\quad\xi_\rho={\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}}Z_\rho(x) $$ related to this process, as well as to their second moments $B_\rho={\mathbf{E}}\zeta_\rho^2$ and $M_\rho={\mathbf{E}}\xi_\rho^2$. The process $Z_\rho$ (up to a linear time change) arises in some non-regular, namely change-point type, statistical models as the limiting likelihood ratio process, and the variables $\zeta_\rho$ and $\xi_\rho$ (up to a multiplicative constant) as the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator respectively. In particular, $B_\rho$ and $M_\rho$ (up to the square of the above multiplicative constant) are the limiting variances of these estimators, and the Bayesian estimators being asymptotically efficient, the ratio $E_\rho=B_\rho/M_\rho$ is the asymptotic efficiency of the maximum likelihood estimator in these models. The main such model is the below detailed model of i.i.d. observations in the situation when their density has a jump (is discontinuous). Probably the first general result about this model goes back to Chernoff and Rubin [@CR]. Later, it was exhaustively studied by Ibragimov and Khasminskii in [@IKh Chapter 5] $\bigl($see also their previous works [@IKh2] and [@IKh4]$\bigr)$. **Model 1.** Consider the problem of estimation of the location parameter $\theta$ based on the observation $X^n=(X_1,\ldots,X_n)$ of the i.i.d. sample from the density $f(x-\theta)$, where the known function $f$ is smooth enough everywhere except at $0$, and in $0$ we have $$0\ne\lim_{x\uparrow0}f(x)=a\ne b=\lim_{x\downarrow0}f(x)\ne0.$$ Denote ${\mathbf{P}}_{\theta}^n$ the distribution (corresponding to the parameter $\theta$) of the observation $X^n$. As $n\to\infty$, the normalized likelihood ratio process of this model defined by $$Z_n(u)=\frac{d{\mathbf{P}}_{\theta+\frac un}^n}{d{\mathbf{P}}_{\theta}^n}(X^n)= \prod_{i=1}^{n}\frac{f\left(X_i-\theta-\frac un\right)}{f(X_i-\theta)}$$ converges weakly in the space $\mathcal{D}_0(-\infty ,+\infty)$ (the Skorohod space of functions on ${\mathbb{R}}$ without discontinuities of the second kind and vanishing at infinity) to the process $Z_{a,b}$ on ${\mathbb{R}}$ defined by $$\ln Z_{a,b}(u)=\begin{cases} \vphantom{\bigg)}\ln(\frac ab)\,\Pi_b(u)-(a-b)\,u, &\text{if } u{\geqslant}0,\\ \vphantom{\bigg)}-\ln(\frac ab)\,\Pi_a(-u)-(a-b)\,u, &\text{if } u{\leqslant}0,\\ \end{cases}$$ where $\Pi_b$ and $\Pi_a$ are two independent Poisson processes on ${\mathbb{R}}_+$ with intensities $b$ and $a$ respectively. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are given by $$\zeta_{a,b}=\frac{\int_{{\mathbb{R}}}u\,Z_{a,b}(u)\;du}{\int_{{\mathbb{R}}}\,Z_{a,b}(u)\;du} \quad\text{and}\quad\xi_{a,b}={\mathop{\rm argsup}\limits}_{u\in{\mathbb{R}}}Z_{a,b}(u)$$ respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, ${\mathbf{E}}\zeta_{a,b}^2$ and ${\mathbf{E}}\xi_{a,b}^2$ are the limiting variances of these estimators, and ${\mathbf{E}}\zeta_{a,b}^2/{\mathbf{E}}\xi_{a,b}^2$ is the asymptotic efficiency of the maximum likelihood estimator. Now let us note, that up to a linear time change, the process $Z_{a,b}$ is nothing but the process $Z_\rho$ with $\rho={\left|\ln(\frac ab)\right|}$. Indeed, by putting $u=\frac {x}{a-b}$ we get $$\begin{aligned} \ln Z_{a,b}(u)&=\begin{cases} \vphantom{\bigg)}\ln(\frac ab)\,\Pi_b(\frac {x}{a-b})-x, &\text{if } \frac {x}{a-b}{\geqslant}0,\\ \vphantom{\bigg)}-\ln(\frac ab)\,\Pi_a(-\frac {x}{a-b})-x, &\text{if } \frac {x}{a-b}{\leqslant}0,\\ \end{cases}\\ &=\ln Z_\rho(x)=\ln Z_\rho\bigl((a-b)\,u\bigr). $$ So, we have $$\zeta_{a,b}=\frac{\zeta_\rho}{a-b}\quad\text{and}\quad \xi_{a,b}=\frac{\xi_\rho}{a-b}\,,$$ and hence $${\mathbf{E}}\zeta_{a,b}^2=\frac{B_\rho}{(a-b)^2}\;,\quad {\mathbf{E}}\xi_{a,b}^2=\frac{M_\rho}{(a-b)^2}\quad\text{and}\quad \frac{{\mathbf{E}}\zeta_{a,b}^2}{{\mathbf{E}}\xi_{a,b}^2}=E_\rho\,.$$ Some other models where the process $Z_\rho$ arises occur in the statistical inference for inhomogeneous Poisson processes, in the situation when their intensity function has a jump (is discontinuous). In Kutoyants [@Kut Chapter 5] $\bigl($see also his previous work [@Kut2]$\bigr)$ one can find several examples, one of which is detailed below. **Model 2.** Consider the problem of estimation of the location parameter $\theta\in\left]\alpha,\beta\right[$, $0<\alpha<\beta<\tau$, based on the observation $X^T$ on $[0,T]$ of the Poisson process with $\tau$-periodic strictly positive intensity function $S(t+\theta)$, where the known function $S$ is smooth enough everywhere except at points $t^*+\tau k$, $k\in{\mathbb{Z}}$, with some $t^*\in\left[0,\tau\right]$, in which we have $$0\ne\lim_{t\uparrow t^*}S(t)=S_-\ne S_+=\lim_{t\downarrow t^*}S(t)\ne0.$$ Denote ${\mathbf{P}}_{\theta}^T$ the distribution (corresponding to the parameter $\theta$) of the observation $X^T$. As $T\to\infty$, the normalized likelihood ratio process of this model defined by $$Z_T(u)=\frac{d{\mathbf{P}}_{\theta+\frac uT}^T}{d{\mathbf{P}}_{\theta}^T}(X^T)=\exp \biggl\{\int_{0}^{T}\!\!\ln\frac{S_{\theta+\frac uT}(t)}{S_{\theta}(t)}\,dX(t) -\int_{0}^{T}\!\!\left[S_{\theta+\frac uT}(t)-S_\theta(t)\right]\!dt\biggr\}$$ converges weakly in the space $\mathcal{D}_0(-\infty ,+\infty)$ to the process $Z_{\tau,S_-,S_+}$ on ${\mathbb{R}}$ defined by $$\ln Z_{\tau,S_-,S_+}=\begin{cases} \vphantom{\bigg)}\ln\bigl(\frac{S_+}{S_-}\bigr)\,\Pi_{S_-}\bigl(\frac u\tau\bigr)-(S_+-S_-)\,\frac u\tau\,, &\text{if } u{\geqslant}0,\\ \vphantom{\bigg)}-\ln\bigl(\frac{S_+}{S_-}\bigr)\,\Pi_{S_+}\bigl(-\frac u\tau\bigr)-(S_+-S_-)\,\frac u\tau\,, &\text{if } u{\leqslant}0,\\ \end{cases}$$ where $\Pi_{S_-}$ and $\Pi_{S_+}$ are two independent Poisson processes on ${\mathbb{R}}_+$ with intensities $S_-$ and $S_+$ respectively. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are given by $$\zeta_{\tau,S_-,S_+}=\frac{\int_{{\mathbb{R}}}u\,Z_{\tau,S_-,S_+}(u)\;du} {\int_{{\mathbb{R}}}\,Z_{\tau,S_-,S_+}(u)\;du}\quad\text{and}\quad \xi_{\tau,S_-,S_+}={\mathop{\rm argsup}\limits}_{u\in{\mathbb{R}}}Z_{\tau,S_-,S_+}(u)$$ respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, ${\mathbf{E}}\zeta_{\tau,S_-,S_+}^2$ and ${\mathbf{E}}\xi_{\tau,S_-,S_+}^2$ are the limiting variances of these estimators, and ${\mathbf{E}}\zeta_{\tau,S_-,S_+}^2/{\mathbf{E}}\xi_{\tau,S_-,S_+}^2$ is the asymptotic efficiency of the maximum likelihood estimator. Now let us note, that up to a linear time change, the process $Z_{\tau,S_-,S_+}$ is nothing but the process $Z_\rho$ with $\rho=\bigl|\ln\bigl(\frac{S_+}{S_-}\bigr)\bigr|$. Indeed, by putting $u=\frac{\tau x}{S_+-S_-}$ we get $$Z_{\tau,S_-,S_+}(u)=Z_\rho(x)= Z_\rho\left(\frac{S_+-S_-}{\tau}\,u\right).$$ So, we have $$\zeta_{\tau,S_-,S_+}=\frac{\tau\,\zeta_\rho}{S_+-S_-}\quad\text{and}\quad \zeta_{\tau,S_-,S_+}= \frac{\tau\,\xi_\rho}{S_+-S_-}\,,$$ and hence $${\mathbf{E}}\zeta_{\tau,S_-,S_+}^2=\frac{\tau^2\,B_\rho}{(S_+-S_-)^2}\;,\quad {\mathbf{E}}\xi_{\tau,S_-,S_+}^2=\frac{\tau^2\,M_\rho}{(S_+-S_-)^2}\quad\text{and}\quad \frac{{\mathbf{E}}\zeta_{\tau,S_-,S_+}^2}{{\mathbf{E}}\xi_{\tau,S_-,S_+}^2}=E_\rho\,.$$ The second limiting likelihood ratio process considered in this paper is the random process $$\label{proc2} Z_0(x)=\exp\left\{W(x)-\frac12{\left|x\right|}\right\},\quad x\in{\mathbb{R}}, $$ where $W$ is a standard two-sided Brownian motion. In this case, the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator (up to a multiplicative constant) are given by $$\label{vars2} \zeta_0=\frac{\int_{{\mathbb{R}}}x\,Z_0(x)\;dx}{\int_{{\mathbb{R}}}\,Z_0(x)\;dx} \quad\text{and}\quad\xi_0={\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}}Z_0(x) $$ respectively, and the limiting variances of these estimators (up to the square of the above multiplicative constant) are $B_0={\mathbf{E}}\zeta_0^2$ and $M_0={\mathbf{E}}\xi_0^2$. The models where the process $Z_0$ arises occur in various fields of statistical inference for stochastic processes. A well-known example is the below detailed model of a discontinuous signal in a white Gaussian noise exhaustively studied by Ibragimov and Khasminskii in [@IKh Chapter 7.2] $\bigl($see also their previous work [@IKh5]$\bigr)$, but one can also cite change-point type models of dynamical systems with small noise $\bigl($see Kutoyants [@Kut2] and [@Kut3 Chapter 5]$\bigr)$, those of ergodic diffusion processes $\bigl($see Kutoyants [@Kut4 Chapter 3]$\bigr)$, a change-point type model of delay equations $\bigl($see Küchler and Kutoyants [@KK]$\bigr)$, an i.i.d. change-point type model $\bigl($see Deshayes and Picard [@DP]$\bigr)$, a model of a discontinuous periodic signal in a time inhomogeneous diffusion $\bigl($see Höpfner and Kutoyants [@HK]$\bigr)$, and so on. **Model 3.** Consider the problem of estimation of the location parameter $\theta\in\left]\alpha,\beta\right[$, $0<\alpha<\beta<1$, based on the observation $X^\varepsilon$ on $\left[0,1\right]$ of the random process satisfying the stochastic differential equation $$dX^\varepsilon(t)=\frac1\varepsilon\,S(t-\theta)\,dt+dW(t),$$ where $W$ is a standard Brownian motion, and $S$ is a known function having a bounded derivative on $\left]-1,0\right[\cup\left]0,1\right[$ and satisfying $$\lim_{t\uparrow0}S(t)-\lim_{t\downarrow0}S(t)=r\ne0.$$ Denote ${\mathbf{P}}_{\theta}^\varepsilon$ the distribution (corresponding to the parameter $\theta$) of the observation $X^\varepsilon$. As $\varepsilon\to0$, the normalized likelihood ratio process of this model defined by $$\begin{aligned} Z_\varepsilon(u)&=\frac{d{\mathbf{P}}_{\theta+\varepsilon^2u}^\varepsilon} {d{\mathbf{P}}_{\theta}^\varepsilon}(X^\varepsilon)\\ &=\exp\biggl\{\frac1\varepsilon\int_{0}^{1} \bigl[S(t-\theta-\varepsilon^2u)-S(t-\theta)\bigr]\,dW(t)\\ &\phantom{=\exp\biggl\{}-\frac1{2\,\varepsilon^2}\int_{0}^{1} \bigl[S(t-\theta-\varepsilon^2u)-S(t-\theta)\bigr]^2\,dt \biggr\} $$ converges weakly in the space $\mathcal{C}_0(-\infty ,+\infty)$ (the space of continuous functions vanishing at infinity equipped with the supremum norm) to the process $Z_0(r^2u)$, $u\in{\mathbb{R}}$. The limiting distributions of the Bayesian estimators and of the maximum likelihood estimator are $r^{-2}\zeta_0$ and $r^{-2}\xi_0$ respectively. The convergence of moments also holds, and the Bayesian estimators are asymptotically efficient. So, $r^{-4}B_0$ and $r^{-4}M_0$ are the limiting variances of these estimators, and $E_0$ is the asymptotic efficiency of the maximum likelihood estimator. Let us also note that Terent’yev in [@Ter] determined explicitly the distribution of $\xi_0$ and calculated the constant $M_0=26$. These results were taken up by Ibragimov and Khasminskii in [@IKh Chapter 7.3], where by means of numerical simulation they equally showed that $B_0=19.5\pm0.5$, and so $E_0=0.73\pm0.03$. Later in [@Gol], Golubev expressed $B_0$ in terms of the second derivative (with respect to a parameter) of an improper integral of a composite function of modified Hankel and Bessel functions. Finally in [@RS], Rubin and Song obtained the exact values $B_0=16\,\zeta(3)$ and $E_0=8\,\zeta(3)/13$, where $\zeta$ is Riemann’s zeta function defined by $$\zeta(s)=\sum_{n=1}^{\infty}\frac1{n^s}\;.$$ The random variables $\zeta_\rho$ and $\xi_\rho$ and the quantities $B_\rho$, $M_\rho$ and $E_\rho$, $\rho>0$, are much less studied. One can cite Pflug [@Pfl] for some results about the distribution of the random variables $${\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}_+}Z_\rho(x)\quad\text{and}\quad{\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}_-}Z_\rho(x)$$ related to $\xi_\rho$. In this paper we establish that the limiting likelihood ratio processes $Z_\rho$ and $Z_0$ are related. More precisely, we show that as $\rho\to 0$, the process $Z_\rho(y/\rho)$, $y\in{\mathbb{R}}$, converges weakly in the space $\mathcal{D}_0(-\infty ,+\infty)$ to the process $Z_0$. So, the random variables $\rho\,\zeta_\rho$ and $\rho\,\xi_\rho$ converge weakly to the random variables $\zeta_0$ and $\xi_0$ respectively. We show equally that the convergence of moments of these random variables holds, that is, $\rho^2 B_\rho \to 16\,\zeta(3)$, $\rho^2 M_\rho \to 26$ and $E_\rho \to 8\,\zeta(3)/13$. These are the main results of the present paper, and they are presented in Section \[MR\], where we also briefly discuss the second possible asymptotics $\rho\to+\infty$. The necessary lemmas are proved in Section \[PL\]. Finally, some numerical simulations of the quantities $B_\rho$, $M_\rho$ and $E_\rho$ for $\rho\in\left]0,\infty\right[$ are presented in Section \[NS\]. Main results {#MR} ============ Consider the process $X_\rho(y)=Z_\rho(y/\rho)$, $y\in{\mathbb{R}}$, where $\rho>0$ and $Z_\rho$ is defined by . Note that $$\frac{\int_{{\mathbb{R}}}y\,X_\rho(y)\;dy}{\int_{{\mathbb{R}}}\,X_\rho(y)\;dy}=\rho \,\zeta_\rho \quad\text{and}\quad{\mathop{\rm argsup}\limits}_{y\in{\mathbb{R}}}X_\rho(y)=\rho\,\xi_\rho\,,$$ where the random variables $\zeta_\rho$ and $\xi_\rho$ are defined by . Remind also the process $Z_0$ on ${\mathbb{R}}$ defined by  and the random variables $\zeta_0$ and $\xi_0$ defined by . Recall finally the quantities $B_\rho={\mathbf{E}}\zeta_\rho^2$, $M_\rho={\mathbf{E}}\xi_\rho^2$, $E_\rho=B_\rho/M_\rho$, $B_0={\mathbf{E}}\zeta_0^2=16\,\zeta(3)$, $M_0={\mathbf{E}}\xi_0^2=26$ and $E_0=B_0/M_0=8\,\zeta(3)/13$. Now we can state the main result of the present paper. \[T1\] The process $X_\rho$ converges weakly in the space $\mathcal{D}_0(-\infty ,+\infty)$ to the process $Z_0$ as $\rho \to 0$. In particular, the random variables $\rho\,\zeta_\rho$ and $\rho\,\xi_\rho$ converge weakly to the random variables $\zeta_0$ and $\xi_0$ respectively. Moreover, for any $k>0$ we have $$\rho^k\,{\mathbf{E}}\zeta_\rho^k \to {\mathbf{E}}\zeta_0^k\quad\text{and}\quad \rho^k\,{\mathbf{E}}\xi_\rho^k \to {\mathbf{E}}\xi_0^k,$$ and in particular $\rho^2 B_\rho \to 16\,\zeta(3)$, $\rho^2 M_\rho \to 26$ and $E_\rho \to 8\,\zeta(3)/13$. The results concerning the random variable $\zeta_\rho$ are direct consequence of Ibragimov and Khasminskii [@IKh Theorem 1.10.2] and the following three lemmas. \[L1\] The finite-dimensional distributions of the process $X_\rho$ converge to those of $Z_0$ as $\rho \to 0$. \[L2\] For all $\rho>0$ and all $y_1,y_2\in{\mathbb{R}}$ we have $${\mathbf{E}}\left|X_\rho^{1/2}(y_1)-X_\rho^{1/2}(y_2)\right|^2{\leqslant}\frac14{\left|y_1-y_2\right|}.$$ \[L3\] For any $c\in\left]\,0\,{,}\;1/8\,\right[$ we have $${\mathbf{E}}X_\rho^{1/2}(y){\leqslant}\exp\bigl(-c{\left|y\right|}\bigr)$$ for all sufficiently small $\rho$ and all $y\in{\mathbb{R}}$. Note that these lemmas are not sufficient to establish the weak convergence of the process $X_\rho$ in the space $\mathcal{D}_0(-\infty ,+\infty)$ and the results concerning the random variable $\xi_\rho$. However, the increments of the process $\ln X_\rho$ being independent, the convergence of its restrictions (and hence of those of $X_\rho$) on finite intervals $[A,B]\subset{\mathbb{R}}$ $\bigl($that is, convergence in the Skorohod space $\mathcal{D}[A,B]$ of functions on $[A,B]$ without discontinuities of the second kind$\bigr)$ follows from Gihman and Skorohod [@GS Theorem 6.5.5], Lemma \[L1\] and the following lemma. \[L4\] For any $\varepsilon>0$ we have $$\lim_{h\to 0}\ \lim_{\rho\to 0}\ \sup_{{\left|y_1-y_2\right|} < h} {\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\}=0.$$ Now, Theorem \[T1\] follows from the following estimate on the tails of the process $X_\rho$ by standard argument. \[L5\] For any $b\in\left]\,0\,{,}\;3/40\,\right[$ we have $${\mathbf{P}}\biggl\{\sup_{{\left|y\right|}>A} X_\rho(y) > e^{-bA}\biggr\} {\leqslant}2\,e^{-bA}$$ for all sufficiently small $\rho$ and all $A>0$. All the above lemmas will be proved in the next section, but before let us discuss the second possible asymptotics $\rho\to+\infty$. One can show that in this case, the process $Z_\rho$ converges weakly in the space $\mathcal{D}_0(-\infty ,+\infty)$ to the process $Z_\infty(u)=e^{-u}\,{\mathbb{1}}_{\{u>\eta\}}$, $u\in{\mathbb{R}}$, where $\eta$ is a negative exponential random variable with ${\mathbf{P}}\{\eta<t\}=e^t$, $t{\leqslant}0$. So, the random variables $\zeta_\rho$ and $\xi_\rho$ converge weakly to the random variables $$\zeta_\infty= \frac{\int_{{\mathbb{R}}}u\,Z_\infty(u)\;du}{\int_{{\mathbb{R}}}\,Z_\infty(u)\;du}=\eta+1 \quad\text{and}\quad \xi_\infty={\mathop{\rm argsup}\limits}_{u\in{\mathbb{R}}}Z_\infty(u)=\eta$$ respectively. One can equally show that, moreover, for any $k>0$ we have $${\mathbf{E}}\zeta_\rho^k\to{\mathbf{E}}\zeta_\infty^k \quad\text{and}\quad {\mathbf{E}}\xi_\rho^k\to{\mathbf{E}}\xi_\infty^k,$$ and in particular, denoting $B_\infty\!={\mathbf{E}}\zeta_\infty^2$, $M_\infty\!={\mathbf{E}}\xi_\infty^2$ and $E_\infty\!=B_\infty/M_\infty$, we finally have $B_\rho \to B_\infty\!={\mathbf{E}}(\eta+1)^2=1$, $M_\rho \to M_\infty\!={\mathbf{E}}\eta^2=2$ and $E_\rho \to E_\infty\!=1/2$. Let us note that these convergences are natural, since the process $Z_\infty$ can be considered as a particular case of the process $Z_\rho$ with $\rho=+\infty$ if one admits the convention $+\infty\cdot0=0$. Note also that the process $Z_\infty$ (up to a linear time change) is the limiting likelihood ratio process of Model 1 (Model 2) in the situation when $a\cdot b=0$ ($S_-\cdot S_+$=0). In this case, the variables $\zeta_\infty=\eta+1$ and $\xi_\infty=\eta$ (up to a multiplicative constant) are the limiting distributions of the Bayesian estimators and of the maximum likelihood estimator respectively. In particular, $B_\infty=1$ and $M_\infty=2$ (up to the square of the above multiplicative constant) are the limiting variances of these estimators, and the Bayesian estimators being asymptotically efficient, $E_\infty=1/2$ is the asymptotic efficiency of the maximum likelihood estimator. Proofs of the lemmas {#PL} ==================== First we prove Lemma \[L1\]. Note that the restrictions of the process $\ln X_\rho$ (as well as those of the process $\ln Z_0$) on ${\mathbb{R}}_+$ and on ${\mathbb{R}}_-$ are mutually independent processes with stationary and independent increments. So, to obtain the convergence of all the finite-dimensional distributions, it is sufficient to show the convergence of one-dimensional distributions only, that is, $$\ln X_\rho(y)\Rightarrow\ln Z_0(y)=W(y)-\frac{{\left|y\right|}}{2}={\mathcal{N}}\Bigl(-\frac{{\left|y\right|}}{2}\,,{\left|y\right|}\Bigr)$$ for all $y\in{\mathbb{R}}$. Here and in the sequel “$\Rightarrow$” denotes the weak convergence of the random variables, and ${\mathcal{N}}(m,V)$ denotes a “generic” random variable distributed according to the normal law with mean $m$ and variance $V$. Let $y>0$. Then, noting that $\ds\Pi_+\Bigl(\frac{y}{\rho}\Bigr)$ is a Poisson random variable of parameter $\ds\lambda=\frac{y}{\rho\,(e^\rho-1)}\to\infty$, we have $$\begin{aligned} \ln X_\rho(y)&=\rho\,\Pi_+\Bigl(\frac{y}{\rho}\Bigr)-\frac{y}{\rho} =\rho\,\sqrt{\frac{y}{\rho\,(e^\rho-1)}}\ \frac {\Pi_+\bigl(\frac{y}{\rho}\bigr)-\lambda}{\sqrt{\lambda}} +\frac{y}{e^\rho-1}-\frac{y}{\rho}\\ &=\sqrt{y}\,\sqrt{\frac{\rho}{e^\rho-1}}\ \frac {\Pi_+\bigl(\frac{y}{\rho}\bigr)-\lambda}{\sqrt{\lambda}} -y\,\frac{e^\rho-1-\rho}{\rho\,(e^\rho-1)}\Rightarrow {\mathcal{N}}\left(-\frac{y}{2}\,,y\right), $$ since $$\frac{\rho}{e^\rho-1}=\frac{\rho}{\rho+o(\rho)}\to1,\qquad \frac{e^\rho-1-\rho}{\rho\,(e^\rho-1)}= \frac{\rho^2/2+o(\rho^2)}{\rho\,\bigr(\rho+o(\rho)\bigr)}\to\frac{1}{2}$$ and $$\frac{\Pi_+\bigl(\frac{y}{\rho}\bigr)-\lambda}{\sqrt{\lambda}}\Rightarrow {\mathcal{N}}(0,1).$$ Similarly, for $y<0$ we have $$\begin{aligned} \ln X_\rho(y)&=-\rho\,\Pi_-\Bigl(\frac{-y}{\rho}\Bigr)-\frac{y}{\rho} =\rho\,\sqrt{\frac{-y}{\rho\,(1-e^{-\rho})}}\ \frac {\lambda'-\Pi_-\bigl(\frac{-y}{\rho}\bigr)}{\sqrt{\lambda'}} -\frac{-y}{1-e^{-\rho}}-\frac{y}{\rho}\\ &=\sqrt{-y}\,\sqrt{\frac{\rho}{1-e^{-\rho}}}\ \frac {\lambda'-\Pi_-\bigl(\frac{-y}{\rho}\bigr)}{\sqrt{\lambda'}} +y\,\frac{e^{-\rho}-1+\rho}{\rho\,(1-e^{-\rho})}\Rightarrow {\mathcal{N}}\left(\frac{y}{2}\,,-y\right), $$ and so, Lemma \[L1\] is proved. Now we turn to the proof of Lemma \[L3\] (we will prove Lemma \[L2\] just after). For $y>0$ we can write $${\mathbf{E}}X_\rho^{1/2}(y)={\mathbf{E}}\exp\left(\frac\rho2\,\Pi_+\Bigl(\frac y\rho\Bigr)-\frac y{2\rho}\right)=\exp\left(-\frac y{2\rho}\right)\, {\mathbf{E}}\exp\left(\frac\rho2\,\Pi_+\Bigl(\frac y\rho\Bigr)\right).$$ Note that $\ds\Pi_+\Bigl(\frac{y}{\rho}\Bigr)$ is a Poisson random variable of parameter $\ds\lambda=\frac{y}{\rho\,(e^\rho-1)}$ with moment generating function $M(t)=\exp\bigl(\lambda\,(e^t-1)\bigr)$. So, we get $$\begin{aligned} {\mathbf{E}}X_\rho^{1/2}(y)&=\exp\left(-\frac y{2\rho}\right)\,\exp\left(\frac y{\rho\,(e^\rho-1)}\,(e^{\rho/2}-1)\right)\\ &=\exp\left(-\frac y{2\rho}+\frac y{\rho\,(e^{\rho/2}+1)}\right) =\exp\left(-y\,\frac{e^{\rho/2}-1}{2\rho\,(e^{\rho/2}+1)}\right)\\ &=\exp\left(-y\,\frac{e^{\rho/4}-e^{-\rho/4}} {2\rho\,(e^{\rho/4}+e^{-\rho/4})}\right)=\exp \left(-y\,\frac{\tanh(\rho/4)}{2\rho}\right). $$ For $y<0$ we obtain similarly $$\begin{aligned} {\mathbf{E}}X_\rho^{1/2}(y)&={\mathbf{E}}\exp\left(-\frac\rho2\,\Pi_-\Bigl(\frac {-y}\rho\Bigr)-\frac y{2\rho}\right)\\ &=\exp\left(-\frac y{2\rho}\right)\,\exp \left(\frac{-y}{\rho\,(1-e^{-\rho})}\,(e^{-\rho/2}-1)\right)\\ &=\exp\left(-\frac y{2\rho}+\frac y{\rho\,(1+e^{-\rho/2})}\right)=\exp \left(y\,\frac{1-e^{-\rho/2}}{2\rho\,(1+e^{-\rho/2})}\right)\\ &=\exp\left(y\,\frac{\tanh(\rho/4)}{2\rho}\right). $$ Thus, for all $y\in{\mathbb{R}}$ we have $$\label{Eroot} {\mathbf{E}}X_\rho^{1/2}(y)=\exp\left(-{\left|y\right|}\frac{\tanh(\rho/4)}{2\rho}\right), $$ and since $$\frac{\tanh(\rho/4)}{2\rho}=\frac{\rho/4+o(\rho)}{2\rho}\to\frac18$$ as $\rho\to0$, for any $c\in\left]\,0\,{,}\;1/8\,\right[$ we have ${\mathbf{E}}X_\rho^{1/2}(y){\leqslant}\exp\bigl(-c{\left|y\right|}\bigr)$ for all sufficiently small $\rho$ and all $y\in{\mathbb{R}}$. Lemma \[L3\] is proved. Further we verify Lemma \[L2\]. We first consider the case $y_1,y_2\in{\mathbb{R}}_+$ (say $y_1{\geqslant}y_2$). Using  and taking into account the stationarity and the independence of the increments of the process $\ln X_\rho$ on ${\mathbb{R}}_+$, we can write $$\begin{aligned} {\mathbf{E}}\left|X_\rho^{1/2}(y_1)-X_\rho^{1/2}(y_2)\right|^2&={\mathbf{E}}X_\rho(y_1)+{\mathbf{E}}X_\rho(y_2)-2\, {\mathbf{E}}X_\rho^{1/2}(y_1)X_\rho^{1/2}(y_2)\\ &=2-2\,{\mathbf{E}}X_\rho(y_2)\,{\mathbf{E}}\frac{X_\rho^{1/2}(y_1)}{X_\rho^{1/2}(y_2)}\\ &=2-2\,{\mathbf{E}}X_\rho^{1/2}(y_1-y_2)\\ &=2-2\exp\left(-{\left|y_1-y_2\right|}\frac{\tanh(\rho/4)}{2\rho}\right)\\ &{\leqslant}{\left|y_1-y_2\right|}\frac{\tanh(\rho/4)}{\rho}{\leqslant}\frac14{\left|y_1-y_2\right|}. $$ The case $y_1,y_2\in{\mathbb{R}}_-$ can be treated similarly. Finally, if $y_1y_2{\leqslant}0$ (say $y_2{\leqslant}0{\leqslant}y_1$), we have $$\begin{aligned} {\mathbf{E}}\left|X_\rho^{1/2}(y_1)-X_\rho^{1/2}(y_2)\right|^2&=2-2\,{\mathbf{E}}X_\rho^{1/2}(y_1)\, {\mathbf{E}}X_\rho^{1/2}(y_2)\\ &=2-2\exp\left(-{\left|y_1\right|}\frac{\tanh(\rho/4)}{2\rho}- {\left|y_2\right|}\frac{\tanh(\rho/4)}{2\rho}\right)\\ &=2-2\exp\left(-{\left|y_1-y_2\right|}\frac{\tanh(\rho/4)}{2\rho}\right)\\ &{\leqslant}\frac14{\left|y_1-y_2\right|}, $$ and so, Lemma \[L2\] is proved. Now let us check Lemma \[L4\]. First let $y_1,y_2\in{\mathbb{R}}_+$ (say $y_1{\geqslant}y_2$) such that $\Delta={\left|y_1-y_2\right|}<h$. Then $$\begin{aligned} {\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\} &{\leqslant}\frac1{\varepsilon^2}\,{\mathbf{E}}\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|^2\\ &=\frac1{\varepsilon^2}\,{\mathbf{E}}\bigl|\ln X_\rho(\Delta)\bigr|^2\\ &=\frac1{\varepsilon^2}\,{\mathbf{E}}\left|\rho\,\Pi_+ \Bigl(\frac{\Delta}{\rho}\Bigr)-\frac{\Delta}{\rho}\right|^2\\ &=\frac1{\varepsilon^2}\left(\rho^2(\lambda+\lambda^2) +\frac{\Delta^2}{\rho^2}-2\lambda\Delta\right)\\ &=\frac1{\varepsilon^2}\left(\beta(\rho)\,\Delta +\gamma(\rho)\,\Delta^2\right)\\ &<\frac1{\varepsilon^2}\left(\beta(\rho)\,h+\gamma(\rho)\,h^2\right), $$ where $\lambda=\frac{\Delta}{\rho\,(e^\rho-1)}$ is the parameter of the Poisson random variable $\Pi_+\bigl(\!\frac{\Delta}{\rho}\!\bigr)$, $$\begin{aligned} \beta(\rho)&=\frac{\rho}{(e^\rho-1)}=\frac{\rho}{\rho+o(\rho)}\to1\\ \intertext{and} \gamma(\rho)&=\frac{1}{(e^\rho-1)^2}+\frac{1}{\rho^2}-\frac{2}{\rho\,(e^\rho-1)} =\left(\frac{1}{\rho}-\frac{1}{e^\rho-1}\right)^2\\ &=\biggl(\frac{e^\rho-1-\rho}{\rho\,(e^\rho-1)}\biggr)^2=\biggl( \frac{\rho^2/2+o(\rho^2)}{\rho\,\bigl(\rho+o(\rho)\bigr)}\biggr)^2\to\frac14 $$ as $\rho\to 0$. So, we have $$\begin{aligned} \lim_{\rho\to 0}\ \sup_{{\left|y_1-y_2\right|} < h}{\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\} &{\leqslant}\lim_{\rho\to 0} \frac1{\varepsilon^2}\left(\beta(\rho)\,h+\gamma(\rho)\,h^2\right)\\ &=\frac1{\varepsilon^2}\left(h+\frac{h^2}4\right), $$ and hence $$\lim_{h\to 0}\ \lim_{\rho\to 0}\ \sup_{{\left|y_1-y_2\right|} < h}{\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\}=0,$$ where the supremum is taken only over $y_1,y_2\in{\mathbb{R}}_+$. For $y_1,y_2\in{\mathbb{R}}_-$ such that $\Delta={\left|y_1-y_2\right|}<h$ one can obtain similarly $$\begin{aligned} {\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\} &{\leqslant}\frac1{\varepsilon^2}\,{\mathbf{E}}\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|^2\\ &=\frac1{\varepsilon^2}\left(\beta'(\rho)\,\Delta +\gamma'(\rho)\,\Delta^2\right)\\ &<\frac1{\varepsilon^2}\left(\beta'(\rho)\,h+\gamma'(\rho)\,h^2\right), $$ where $$\begin{aligned} \beta'(\rho)&=\frac{\rho}{(1-e^{-\rho})}=\frac{\rho}{\rho+o(\rho)}\to1\\ \intertext{and} \gamma'(\rho)&=\biggl(\frac{e^{-\rho}-1+\rho}{\rho\,(1-e^{\rho})}\biggr)^2= \biggl(\frac{\rho^2/2+o(\rho^2)}{\rho\,\bigl(\rho+o(\rho)\bigr)}\biggr)^2\to \frac14 $$ as $\rho\to 0$, which will yield the same conclusion as above, but with the supremum taken over $y_1,y_2\in{\mathbb{R}}_-$. Finally, for $y_1y_2{\leqslant}0$ (say $y_2{\leqslant}0{\leqslant}y_1$) such that ${\left|y_1-y_2\right|}<h$, using the elementary inequality $(a-b)^2{\leqslant}2(a^2+b^2)$ we get $$\begin{aligned} {\mathbf{P}}\Bigl\{\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|>\varepsilon\Bigr\} &{\leqslant}\frac1{\varepsilon^2}\,{\mathbf{E}}\bigl|\ln X_\rho(y_1)-\ln X_\rho(y_2)\bigr|^2\\ &{\leqslant}\frac2{\varepsilon^2}\left({\mathbf{E}}\bigl|\ln X_\rho(y_1)\bigr|^2+ {\mathbf{E}}\bigl|\ln X_\rho(y_2)\bigr|^2\right)\\ &=\frac2{\varepsilon^2}\left(\beta(\rho)y_1\!+\!\gamma(\rho)y_1^2 \!+\!\beta'(\rho)\!{\left|y_2\right|}\!+\!\gamma'(\rho)\!{\left|y_2\right|}^2\right)\\ &{\leqslant}\frac2{\varepsilon^2}\Bigl(\bigl(\beta(\rho)+\beta'(\rho)\bigr)\,h +\bigl(\gamma(\rho)+\gamma'(\rho)\bigr)\,h^2\Bigr), $$ which again will yield the desired conclusion. Lemma \[L4\] is proved. It remains to verify Lemma \[L5\]. Clearly, $${\mathbf{P}}\biggl\{\sup_{{\left|y\right|}>A} X_\rho(y) > e^{-bA}\biggr\}{\leqslant}{\mathbf{P}}\biggl\{\sup_{y>A} X_\rho(y) > e^{-bA}\biggr\}+ {\mathbf{P}}\biggl\{\sup_{y<-A} X_\rho(y) > e^{-bA}\biggr\}.$$ In order to estimate the first term, we need two auxiliary results. \[L6\] For any $c\in\left]\,0\,{,}\;3/32\,\right[$ we have $${\mathbf{E}}X_\rho^{1/4}(y){\leqslant}\exp\bigl(-c{\left|y\right|}\bigr)$$ for all sufficiently small $\rho$ and all $y\in{\mathbb{R}}$. \[L7\] For all $\rho>0$ the random variable $$\eta_\rho=\sup_{t\in{\mathbb{R}}_+}\bigl(\Pi_\lambda(t)-t\bigr),$$ where $\Pi_\lambda$ is a Poisson process on ${\mathbb{R}}_+$ with intensity $\lambda=\rho/(e^\rho-1)\in\left]0,1\right[$, verifies $${\mathbf{E}}\exp\left(\frac\rho4\,\eta_\rho\right){\leqslant}2.$$ The first result can be easily obtained following the proof of Lemma \[L3\], so we prove the second one only. For this, let us remind that according to Shorack and Wellner [@ShW Proposition 1 on page 392] $\bigl($see also Pyke [@Pyke]$\bigr)$, the distribution function $F_\rho(x)={\mathbf{P}}\{\eta_\rho<x\}$ of $\eta_\rho$ is given by $$1-F_\rho(x)={\mathbf{P}}\{\eta_\rho{\geqslant}x\}=(1-\lambda)\,e^{\lambda x} \sum_{n>x}\frac{(n-x)^n}{n!}\,\bigl(\lambda\,e^{-\lambda}\bigr)^n$$ for $x>0$, and is zero for $x{\leqslant}0$. Hence, for $x>0$ we have $$\begin{aligned} 1-F_\rho(x)&{\leqslant}(1-\lambda)\,e^{\lambda x}\,\sum_{n>x} \frac{(n-x)^n}{\sqrt{2\pi n}\,n^n\,e^{-n}}\, \bigl(\lambda\,e^{-\lambda}\bigr)^n\\ &=\frac{1-\lambda}{\sqrt{2\pi}}\,e^{\lambda x}\,\sum_{n>x}\frac1{\sqrt{n}} \left(1-\frac{x}{n}\right)^n \bigl(\lambda\,e^{1-\lambda}\bigr)^n\\ &{\leqslant}\frac{1-\lambda}{\sqrt{2\pi}}\,e^{\lambda x}\,\sum_{n>x} e^{-x}\frac{\bigl(\lambda\,e^{1-\lambda}\bigr)^n}{\sqrt{n}}\\ &{\leqslant}\frac{1-\lambda}{\sqrt{2\pi}}\,e^{(\lambda-1)x}\, \bigl(\lambda\,e^{1-\lambda}\bigr)^x\sum_{n>x} \frac{\bigl(\lambda\,e^{1-\lambda}\bigr)^{n-x}}{\sqrt{n-x}}\\ &=\frac{1-\lambda}{\sqrt{2\pi}}\,\lambda^x \sum_{k>0}\frac{\bigl(\lambda\,e^{1-\lambda}\bigr)^k}{\sqrt{k}}{\leqslant}\frac{1-\lambda}{\sqrt{2\pi}}\,\lambda^x \int_{{\mathbb{R}}_+}\frac{\bigl(\lambda\,e^{1-\lambda}\bigr)^t}{\sqrt{t}}\;dt\\ &=\frac{1-\lambda}{\sqrt{2\pi}}\,\lambda^x \,\frac{\Gamma(1/2)}{\sqrt{-\ln\bigl(\lambda\,e^{1-\lambda}\bigr)}} =\frac{1-\lambda}{\sqrt{-2\ln\bigl(\lambda\,e^{1-\lambda}\bigr)}} \left(\frac{\rho}{e^\rho-1}\right)^x\\ &{\leqslant}\left(\frac{\rho\,e^{-\rho/2}}{e^{\rho/2}-e^{-\rho/2}}\right)^x =\left(\frac{\rho\,e^{-\rho/2}}{2\sinh(\rho/2)}\right)^x{\leqslant}e^{-\rho x/2}, $$ where we used Stirling inequality and the inequality $1-\lambda{\leqslant}\sqrt{-2\ln\bigl(\lambda\,e^{1-\lambda}\bigr)}$, which is easily reduced to the elementary inequality $\ln(1-\mu){\leqslant}-\mu-\mu^2/2$ by putting $\mu=1-\lambda$. So, we can finish the proof of Lemma \[L7\] by writing $$\begin{aligned} {\mathbf{E}}\exp\left(\frac\rho4\,\eta_\rho\right)&=\int_{{\mathbb{R}}}e^{\,\rho x/4}\; dF_\rho(x)\\ &=\Bigl[e^{\,\rho x/4}\bigl(F_\rho(x)-1\bigr)\Bigr]_{-\infty}^{+\infty}-\; \frac\rho4\int_{{\mathbb{R}}}e^{\,\rho x/4}\bigl(F_\rho(x)-1\bigr)\;dx\\ &=\frac\rho4\int_{{\mathbb{R}}_-}e^{\,\rho x/4}\;dx+ \frac\rho4\int_{{\mathbb{R}}_+}e^{\,\rho x/4}\bigl(1-F_\rho(x)\bigr)\;dx\\ &{\leqslant}1+\frac\rho4\int_{{\mathbb{R}}_+}e^{-\rho x/4}\;dx=2. $$ Now, let us get back to the proof of Lemma \[L5\]. Using Lemma \[L7\] and taking into account the stationarity and the independence of the increments of the process $\ln X_\rho$ on ${\mathbb{R}}_+$, we obtain $$\begin{aligned} {\mathbf{P}}\biggl\{\sup_{y>A} X_\rho(y) > e^{-bA}\biggr\}&{\leqslant}e^{\,bA/4}\; {\mathbf{E}}\sup_{y>A} X_\rho^{1/4}(y)\\ &=e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A)\; {\mathbf{E}}\sup_{y>A}\frac{X_\rho^{1/4}(y)}{X_\rho^{1/4}(A)}\\ &=e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A)\;{\mathbf{E}}\sup_{z>0} X_\rho^{1/4}(z)\\ &=e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A)\;{\mathbf{E}}\sup_{z>0} \left(\exp\Bigl(\frac\rho4\,\Pi_+(z/\rho)-\frac{z}{4\rho}\Bigr)\right)\\ &=e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A)\;{\mathbf{E}}\exp\left(\sup_{t>0}\Bigl( \frac\rho4\bigl(\Pi_{\textstyle\frac\rho{e^{\rho}-1}}(t)-t\bigr)\Bigr)\right)\\ &=e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A)\;{\mathbf{E}}\exp\left(\frac\rho4\,\eta_\rho\right) {\leqslant}2\,e^{\,bA/4}\;{\mathbf{E}}X_\rho^{1/4}(A). $$ Hence, taking $b\in\left]\,0\,{,}\;3/40\,\right[$, we have $5b/4\in\left]\,0\,{,}\;3/32\,\right[$ and, using Lemma \[L6\], we finally get $$\begin{aligned} {\mathbf{P}}\biggl\{\sup_{y>A} X_\rho(y) > e^{-bA}\biggr\}&{\leqslant}2\,e^{\,bA/4}\,\exp\Bigl(-\frac{5b}{4}A\Bigr)=2\,e^{-bA} $$ for all sufficiently small $\rho$ and all $A>0$, and so the first term is estimated. The second term can be estimated in the same way, if we show that for all $\rho>0$ the random variable $$\eta_\rho'=\sup_{t\in{\mathbb{R}}_+}\bigl(-\Pi_{\lambda'}(t)+t\bigr) =-\inf_{t\in{\mathbb{R}}_+}\bigl(\Pi_{\lambda'}(t)-t\bigr),$$ where $\Pi_{\lambda'}$ is a Poisson process on ${\mathbb{R}}_+$ with intensity $\lambda'=\rho/(1-e^{-\rho})\in\left]0,1\right[$, verifies $${\mathbf{E}}\exp\left(\frac\rho4\,\eta_\rho'\right){\leqslant}2.$$ For this, let us remind that according to Pyke [@Pyke] $\bigl($see also Cramér [@Cram]$\bigr)$, $\eta_\rho'$ is an exponential random variable with parameter $r$, where $r$ is the unique positive solution of the equation $$\lambda'(e^{-r}-1)+r=0.$$ In our case, this equation becomes $$\frac{\rho}{1-e^{-\rho}}\,(e^{-r}-1)+r=0,$$ and $r=\rho$ is clearly its solution. Hence $\eta_\rho'$ is an exponential random variable with parameter $\rho$, which yields $${\mathbf{E}}\exp\left(\frac\rho4\,\eta_\rho'\right)=\frac43<2,$$ and so, Lemma \[L5\] is proved. Numerical simulations {#NS} ===================== In this section we present some numerical simulations of the quantities $B_\rho$, $M_\rho$ and $E_\rho$ for $\rho\in\left]0,\infty\right[$. Besides giving approximate values of these quantities, the simulation results illustrate both the asymptotics $$B_\rho\sim\frac{B_0}{\rho^2}\,,\quad M_\rho\sim\frac{M_0}{\rho^2} \quad\text{and}\quad E_\rho\to E_0 \quad\text{as}\quad \rho\to 0,$$ with $B_0=16\,\zeta(3)\approx19.2329$, $M_0=26$ and $E_0=8\,\zeta(3)/13\approx 0.7397$, and $$B_\rho\to B_\infty,\quad M_\rho\to M_\infty \quad\text{and}\quad E_\rho\to E_\infty \quad\text{as}\quad \rho\to\infty,$$ with $B_\infty=1$, $M_\infty=2$ and $E_\infty=0.5$. First, we simulate the events $x_1,x_2,\ldots$ of the Poisson process $\Pi_+$ $\bigl($with the intensity $1/(e^\rho-1)\bigr)$, and the events $x'_1,x'_2,\ldots$ of the Poisson process $\Pi_-$ $\bigl($with the intensity $1/(1-e^{-\rho})\bigr)$. Then we calculate $$\begin{aligned} \zeta_\rho&=\frac{\ds\int_{{\mathbb{R}}}x\,Z_\rho(x)\;dx} {\ds\int_{{\mathbb{R}}}\,Z_\rho(x)\;dx}\\ &=\frac {\ds\sum_{i=1}^{\infty}x_i\,e^{\rho i-x_i} + \sum_{i=1}^{\infty} e^{\rho i-x_i} - \sum_{i=1}^{\infty}x'_i\,e^{\rho-\rho i+x'_i} + \sum_{i=1}^{\infty}e^{\rho-\rho i+x'_i}}{\ds\sum_{i=1}^{\infty}e^{\rho i-x_i} + \sum_{i=1}^{\infty}e^{\rho-\rho i+x'_i}}\\ \intertext{and} \xi_\rho&={\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}}Z_\rho(x)=\begin{cases} x_k, &\text{if } \rho k-x_k > \rho-\rho\ell+x'_\ell,\\ -x'_\ell, &\text{otherwise}, \end{cases} $$ where $$k={\mathop{\rm argmax}\limits}_{i{\geqslant}1}\,(\rho i-x_i)\quad\text{and}\quad \ell={\mathop{\rm argmax}\limits}_{i{\geqslant}1}\,(\rho-\rho i+x'_i),$$ so that $$x_k={\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}_+}Z_\rho(x)\quad\text{and}\quad -x'_\ell={\mathop{\rm argsup}\limits}_{x\in{\mathbb{R}}_-}Z_\rho(x).$$ Finally, repeating these simulations $10^7$ times (for each value of $\rho$), we approximate $B_\rho={\mathbf{E}}\zeta_\rho^2$ and $M_\rho={\mathbf{E}}\xi_\rho^2$ by the empirical second moments, and $E_\rho=B_\rho/M_\rho$ by their ratio. The results of the numerical simulations are presented in Figures \[fig1\] and \[fig2\]. The $\rho\to 0$ asymptotics of $B_\rho$ and $M_\rho$ can be observed in Figure \[fig1\], where besides these functions we also plotted the functions $\rho^2 B_\rho$ and $\rho^2 M_\rho$, making apparent the constants $B_0\approx19.2329$ and $M_0=26$. ![$B_\rho$ and $M_\rho$ ($\rho\to 0$ asymptotics)[]{data-label="fig1"}](fig1){width="60.00000%"} In Figure \[fig2\] we use a different scale on the vertical axis to better illustrate the $\rho\to\infty$ asymptotics of $B_\rho$ and $M_\rho$, as well as both the asymptotics of $E_\rho$. Note that the function $E_\rho$ appear to be decreasing, so we can conjecture that bigger is $\rho$, smaller is the efficiency of the maximum likelihood estimator, and so, this efficiency is always between $E_\infty=0.5$ and $E_0\approx 0.7397$. ![[]{data-label="fig2"}](fig2){width="60.00000%"} [99]{} <span style="font-variant:small-caps;">Chernoff, H</span> and <span style="font-variant:small-caps;">Rubin, H</span>, “The estimation of the location of a discontinuity in density”, *Proc. 3rd Berkeley Symp.* **1**, pp. 19–37, 1956. <span style="font-variant:small-caps;">Cramér, H.</span>, “On some questions connected with mathematical risk”, *Univ. California Publ. Statist.* **2**, pp. 99–123, 1954. <span style="font-variant:small-caps;">Deshayes, J.</span> and <span style="font-variant:small-caps;">Picard, D.</span>, “Lois asymptotiques des tests et estimateurs de rupture dans un modèle statistique classique”, *Ann. Inst. H. Poincaré Probab. Statist.* **20**, no. 4, pp. 309–327, 1984. <span style="font-variant:small-caps;">Gihman, I.I.</span> and <span style="font-variant:small-caps;">Skorohod, A.V.</span>, “*The theory of stochastic processes I.*”, Springer-Verlag, New York, 1974. <span style="font-variant:small-caps;">Golubev, G.K.</span>, “Computation of the efficiency of the maximum-likelihood estimator when observing a discontinuous signal in white noise”, *Problems Inform. Transmission* **15**, no. 3, pp. 61–69, 1979. <span style="font-variant:small-caps;">Höpfner, R.</span> and <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “Estimating discontinuous periodic signals in a time inhomogeneous diffusion”, preprint, 2009.\ `http://www.mathematik.uni-mainz.de/~hoepfner/ssp/zeit.html` <span style="font-variant:small-caps;">Ibragimov, I.A.</span> and <span style="font-variant:small-caps;">Khasminskii, R.Z.</span>, “On the asymptotic behavior of generalized Bayes’ estimator”, *Dokl.Akad. Nauk SSSR* **194**, pp. 257–260, 1970. <span style="font-variant:small-caps;">Ibragimov, I.A.</span> and <span style="font-variant:small-caps;">Khasminskii, R.Z.</span>, “The asymptotic behavior of statistical estimates for samples with a discontinuous density”, *Mat. Sb.* **87 (129)**, no. 4, pp. 554–558, 1972. <span style="font-variant:small-caps;">Ibragimov, I.A.</span> and <span style="font-variant:small-caps;">Khasminskii, R.Z.</span>, “Estimation of a parameter of a discontinuous signal in a white Gaussian noise”, *Problems Inform. Transmission* **11**, no. 3, pp. 31–43, 1975. <span style="font-variant:small-caps;">Ibragimov, I.A.</span> and <span style="font-variant:small-caps;">Khasminskii, R.Z.</span>, “*Statistical estimation. Asymptotic theory*”, Springer-Verlag, New York, 1981. <span style="font-variant:small-caps;">Küchler, U.</span> and <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “Delay estimation for some stationary diffusion-type processes”, *Scand. J. Statist.* **27**, no. 3, pp. 405–414, 2000. <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “*Parameter estimation for stochastic processes*”, Armenian Academy of Sciences, Yerevan, 1980 (in Russian), translation of revised version, Heldermann-Verlag, Berlin, 1984. <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “*Identification of dynamical systems with small noise*”, Mathematics and its Applications **300**, Kluwer Academic Publishers Group, Dordrecht, 1994. <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “*Statistical Inference for Spatial Poisson Processes*”, Lect. Notes Statist. **134**, Springer-Verlag, New York, 1998. <span style="font-variant:small-caps;">Kutoyants, Yu.A.</span>, “*Statistical inference for ergodic diffusion processes*”, Springer Series in Statistics, Springer-Verlag, London, 2004. <span style="font-variant:small-caps;">Pflug, G.Ch.</span>, “On an argmax-distribution connected to the Poisson process”, in *Proceedings of the Fifth Prague Conference on Asymptotic Statistics*, eds. P. Mandl and H. Hušková, pp. 123–130, 1993. <span style="font-variant:small-caps;">Pyke, R.</span>, “The supremum and infimum of the Poisson process”, *Ann. Math. Statist.* **30**, pp. 568–576, 1959. <span style="font-variant:small-caps;">Rubin, H.</span> and <span style="font-variant:small-caps;">Song, K.-S.</span>, “Exact computation of the asymptotic efficiency of maximum likelihood estimators of a discontinuous signal in a Gaussian white noise”, *Ann.Statist.* **23**, no. 3, pp. 732–739, 1995. <span style="font-variant:small-caps;">Shorack, G.R.</span> and <span style="font-variant:small-caps;">Wellner, J.A.</span>, “*Empirical processes with applications to statistics*”, John Wiley & Sons Inc., New York, 1986. <span style="font-variant:small-caps;">Terent’yev, A.S.</span>, “Probability distribution of a time location of an absolute maximum at the output of a synchronized filter”, *Radioengineering and Electronics* **13**, no. 4, pp. 652–657, 1968.

ArXiv

--- abstract: 'We consider the problem of estimating the location of a single change point in a network generated by a dynamic stochastic block model mechanism. This model produces community structure in the network that exhibits change at a single time epoch. We propose two methods of estimating the change point, together with the model parameters, before and after its occurrence. The first employs a least squares criterion function and takes into consideration the full structure of the stochastic block model and is evaluated at each point in time. Hence, as an intermediate step, it requires estimating the community structure based on a clustering algorithm at every time point. The second method comprises of the following two steps: in the first one, a least squares function is used and evaluated at each time point, but [*ignores the community structure*]{} and just considers a random graph generating mechanism exhibiting a change point. Once the change point is identified, in the second step, all network data before and after it are used together with a clustering algorithm to obtain the corresponding community structures and subsequently estimate the generating stochastic block model parameters. The first method, since it requires knowledge of the community structure and hence clustering at every point in time, is significantly more computationally expensive than the second one. On the other hand, it requires a significantly less stringent identifiability condition for consistent estimation of the change point and the model parameters than the second method; however, it also requires a condition on the misclassification rate of mis-allocating network nodes to their respective communities that may fail to hold in many realistic settings. Despite the apparent stringency of the identifiability condition for the second method, we show that networks generated by a stochastic block mechanism exhibiting a change in their structure can easily satisfy this condition under a multitude of scenarios, including merging/splitting communities, nodes joining another community, etc. Further, for both methods under their respective identifiability and certain additional regularity conditions, we establish rates of convergence and derive the asymptotic distributions of the change point estimators. The results are illustrated on synthetic data. In summary, this work provides an in depth investigation of the novel problem of change point analysis for networks generated by stochastic block models, identifies key conditions for the consistent estimation of the change point and proposes a computationally fast algorithm that solves the problem in many settings that occur in applications. Finally, it discusses challenges posed by employing clustering algorithms in this problem, that require additional investigation for their full resolution.' author: - Monika Bhattacharjee - Moulinath Banerjee - George Michailidis title: Change Point Estimation in a Dynamic Stochastic Block Model --- **Key words and phrases.** stochastic block model, Erdős-Rényi random graph, change point, edge probability matrix, community detection, estimation, clustering algorithm, convergence rate Introduction ============ The modeling and analysis of network data has attracted the attention of multiple scientific communities, due to their ubiquitous presence in many application domains; see [@newman2006structure], [@kolaczyk2014statistical], [@crane2018probabilistic] and references therein. A popular and widely used statistical model for network data is the Stochastic Block Model (SBM) introduced in [@HLL1983]. It is a special case of a random graph model, where the nodes are partitioned into $K$ disjoint groups (communities) and the edges between them are drawn independently with probabilities that only depend on the community membership of the nodes. This leads to a significant reduction in the dimension of the parameter space, from $\mathcal{O}(m^2)$ for the random graph model, with $m$ being the number of nodes in the network, to $\mathcal{O}(K^2)$ ($K<<m$). There has been a lot of technical work on the SBM, including (i) estimation of the underlying community structure and the corresponding community connection probabilities, e.g. [@C2014; @J2015; @J2016; @LR2015; @RCY2011; @SB2015; @ZLZ2012]; (ii) establishing the minimax rate for estimating the SBM parameters - e.g. [@G2015rate; @K2017]- and the community structure - e.g. [@ZZ2016; @GMZZ2015]- under the assumption that the assignment problem of nodes to communities can be solved [*exactly*]{}. However, the latter problem is computationally NP-hard and hence estimates of the community structure and connection probabilities based on easy to compute procedures compromise the minimax rate - see [@ZLZ2015]. There has also been some recent work on understanding the evolution of community structure over time, based on observing a sequence of network adjacency matrices - e.g. [@D2016a; @D2016b; @H2015; @K2010; @MM2017; @M2015b; @X2010; @Xu2015; @Y2011; @bao2018core]. Various modeling formalism are employed including Markov random field models, low rank plus sparse decompositions and dynamic versions of SBM (DSBM). These studies focus primarily on fast and scalable computational procedures for identifying the evolution of community structure over time. Some work that investigated theoretical properties of the DSBM and more generally graphon models assuming that the node assignment problem can be solved exactly includes [@P2016s], while the theoretical performance of spectral clustering for the DSBM was examined in [@B2017] and [@P2017]. The last two studies estimate the edge probability matrices by either directly averaging adjacency matrices observed at different time points, or by employing a kernel-type smoothing procedure and extract the group memberships of the nodes by using spectral clustering. The objective of this paper is to examine the [*offline estimation*]{} of a single change point under a DSBM network generating mechanism. Specifically, a sequence of networks is generated independently across time through the SBM mechanism, whose community connection probabilities exhibit a change at some time epoch. Then, the problem becomes one of identifying the change point epoch based on the observed sequence of network adjacency matrices, detecting the community structures before and after the change point and estimating the corresponding SBM model parameters. Existence of change points and their estimation has been well-studied for many univariate statistical models evolving independently over time and with shifts in mean and in variance structures. A broad overview of the corresponding literature can be found in [@BD2013noncp] and [@CH1997]. However, in many applications, multivariate (even high dimensional) signals are involved, while also exhibiting dependence across time - see review article by [@AH2013] and references therein. Further, the problem of change point detection in high dimensional stochastic network models has been recently considered in [@peel2015detecting], [@wang2017hierarchical]. The latter studies consider a generalized hierarchical random graph model. However, to the best of our knowledge, work on change point detection for the dynamic SBM is largely lacking. Therefore, the key contributions of this paper are threefold: first, the development of a computational strategy for solving the problem and establishing its theoretical properties under suitable regularity conditions, including (i) establishing the rate of convergence for the least squares estimate of the change point and (ii) the DSBM parameters, as well as (iii) deriving the asymptotic distribution of the change point. An important step in the strategy for obtaining an estimate of the change point involves clustering the nodes to communities, for which we employ a spectral clustering algorithm that exhibits cubic computational cost in the number of edges in the adjacency matrix. However, the theoretical analysis of the first method which involves clustering *at every time point* requires imposing a rather stringent assumption on the rate of misclassifying nodes to communities. For these reasons, the second key contribution of this work is the introduction of a two-step computational strategy, wherein the first step, the change point is estimated based on a procedure that [*ignores*]{} the community structure, while in the second step the pre- and post-change point model parameters are estimated using a spectral clustering algorithm, but at *a single time point*. It is established that this strategy yields consistent estimates for the change point and the community connection probabilities, at linear computational cost in the number of edges. However, the procedure requires a stronger identifiability condition compared to the first strategy. Naturally, no additional condition on controlling the rate of misclassifying nodes to communities during the first step is required. The third contribution of the paper is to show that the more stringent identifiability condition under the second strategy is easily satisfied in a number of scenarios by the DSBM, including splitting/merging communities and reallocating nodes to other communities before and after the change point. Overall, this work provides valuable insights into the technical challenges of change point analysis for DSBM and also an efficient computational strategy that delivers consistent estimates of all the model parameters. Nevertheless, the challenges identified require further investigation for their complete resolution, as discussed in Section \[sec:discuss\]. The remainder of the paper is organized as follows. In Section \[sec: DSBM\], we introduce the DSBM model together with necessary definitions and notation for technical development. Subsequently, we present the strategy to detect the change point that involves a [*community detection step at each time point*]{}, followed by estimation of the DSBM model parameters together with the asymptotic properties of the estimators. In Section \[sec: 2step\], we introduce a 2-step computational strategy for the DSBM change point detection problem, which is computationally significantly less expensive and discuss consistency of these estimators. Section \[sec: compare\] involves a comparative study between the two change point detection strategies previously presented and also provides many realistic settings where the computationally fast 2-step algorithm is provably consistent. The numerical performance of the two strategies based on synthetic data is illustrated in Section \[subsec: simulation\]. Note that in Sections \[sec: DSBM\]-\[sec: compare\], community detection is based on the clustering algorithm discussed in [@B2017]. We briefly discuss other community detection methods for DSBM involving a single change point in Section \[sec:discuss\]. Finally, the asymptotic distribution of the change point estimates along with a data driven procedure for identifying the correct limiting regime is presented in Section \[sec: ADAP\]. Some concluding remarks are drawn in Section \[sec:concluding-remarks\]. All proofs and additional technical material are presented in Section \[sec: proofs\]. The dynamic stochastic block model (DSBM) {#sec: DSBM} ========================================= The structure of the SBM is determined by the following parameters: (i) $m$ the number of nodes or vertices in the network, (ii) a symmetric $K \times K$ matrix $\Lambda = ((\lambda_{ij}))_{K \times K}$ containing the community connection probabilities and (iii) a partition of the node set $\{1,2,\ldots, m\}$ into $K$ communities, which is represented by a many-to-one onto map $z: \{1,2,\ldots, m\} \to \{1,2,\ldots, K\}$ for some $K \leq m$. Hence, for each $1 \leq i \leq m$, the community of node $i$ is determined by $z(i)$, or equivalently $$\begin{aligned} \text{$l$-th community} = \mathcal{C}_l = \{i \in \{1,2,\ldots, m\}: z(i)=l\}\ \forall 1 \leq l \leq K. \nonumber\end{aligned}$$ The map $z$ determines the community structure under the SBM. The observed edge set of the network is obtained as follows: any two nodes $i \in \mathcal{C}_l$ and $j \in \mathcal{C}_{l^\prime}$ are connected by an edge with probability $\lambda_{ll^\prime}I(i \neq j)$, independent of any other pair of nodes. Self-loops are not considered and hence the probability of having an edge between nodes $i$ and $j$ is $0$ whenever $i=j$. Henceforth, we use SBM$(z,\Lambda)$ for denoting an SBM with community structure $z$ and community connection probability matrix $\Lambda$. Next, let $$\begin{aligned} \text{Ed}_{z}(\Lambda) = ((\lambda_{z(i)z(j)}I(i \neq j)))_{m \times m}, \nonumber\end{aligned}$$ which is the edge probability matrix whose $(i,j)$-th entry represents the probability of having an edge between nodes $i$ and $j$. Note that we are dealing with undirected networks and thus $\Lambda$ and Ed$_{z}(\Lambda)$ are symmetric matrices. The data come in the form of an observed square symmetric matrix $A = ((A_{ij}))_{m \times m}$ with entries $$\begin{aligned} \nonumber A_{ij} = \begin{cases} 1,\ \ \text{if an edge between nodes $i$ and $j$ is observed} \\ 0,\ \ \text{otherwise}. \end{cases}\end{aligned}$$ An adjacency matrix $A$ is said to be generated according to SBM$(z,\Lambda)$, if $A_{ij} \sim$\ $\text{Bernoulli}(\Lambda_{z(i)z(j)}I(i \neq j))$ independently, and is denoted by $A \sim \text{SBM}(z,\Lambda)$. It is easy to see that all diagonal entries of $A$ are $0$. In a DSBM, we consider a sequence of stochastic block models evolving [*independently*]{} over time, with $A_{t,n} = ((A_{ij,(t,n)}))_{m \times m}$ denoting the adjacency matrix at time point $t$. Hence, $A_{t,n} \sim \text{SBM}(z_t,\Lambda_t)$ independently, with $n$ being the total number of time points available, and note $\{A_{t,n}: 1 \leq t \leq n, n \geq 1\}$ forms a triangular sequence of adjacency matrices. Further, in the technical analysis we assume that both the number of nodes and communities scale with time: i.e., $m = m_n \to \infty$ and $K= K_n \to \infty$ as $n \to \infty$. We are interested in a DBSM exhibiting a single change point and its estimation in an offline manner. For presenting the results, we embed all the time points in the $[0,1]$ interval, by dividing them by $n$. Suppose $\tau_n \in (c^{*},1-c^{*})$ corresponds to the change point epoch in the DSBM. Hence, $$\begin{aligned} A_{t,n} \sim \begin{cases} \text{SBM}(z,\Lambda),\ \ \text{if $1 \leq t \leq \lfloor n\tau_n \rfloor$} \\ \text{SBM}(w,\Delta),\ \ \text{if $\lfloor n\tau_n \rfloor < t <n$}, \label{eqn: dsbmmodel} \end{cases}\end{aligned}$$ and $z \neq w$ and/or $\Lambda \neq \Delta$. Note that $z$ and $w$ correspond to the pre- and post-change point community structures, respectively. Similarly $\Lambda$ and $\Delta$ are the pre- and post-change point community connection probability matrices, respectively. Further, note that $z, w, \Lambda$ and $\Delta$ may depend on $n$. Our objective is to estimate the model parameters $\tau_n, z,w,\Lambda$ and $\Delta$. Throughout this paper, we assume that $0 < c^{*} <\tau_n <1 - c^{*} <1$ with $c^{*}$ being known to avoid unecessary technical complications if the true change point is located arbitrarily close to boundary time points. We also assume that the total number of communities before and after the change point are equal. *Even if they are different, our results continue to hold after replacing $K_n$ by the maximum of the number of communities to the left and the right.* To identify the change point, we employ the following least squares criterion function $$\begin{aligned} \tilde{L} (b,z,w,\Lambda,\Delta) &=& \frac{1}{n}\sum_{i,j=1}^{m} \bigg[\sum_{t=1}^{nb} (A_{ij,(t,n)} - \lambda_{z(i)z(j)})^2 + \sum_{t=nb+1}^{n} (A_{ij,(t,n)} - \delta_{w(i)w(j)})^2 \bigg]. \hspace{1 cm}\label{eqn: estimateccc}\end{aligned}$$ \[use-likelihood-criterion\] Note that the Bernoulli likelihood function criterion could also be used that will yield similar results for the change point estimators, but it will need stronger assumptions and will involve more technicalities compared to the least squares criterion function adopted. More details on the likelihood criterion function for detecting a change point in a random graph model can be found in [@BBM2017]. Results on the maximum likelihood estimator of the change point in a random graph model are consequences of the results in [@BBM2017]. However, in DSBM one also needs to address the problem of assigning nodes to their respective communities, which makes the problem technically more involved as shown next. However, the main message of this paper will remain the same, irrespective of employing a likelihood or a least squares criterion. Let $$\begin{aligned} S_{u,z} &=& \{i \in \{1,2\ldots, m\}: z(i) = u\},\ \ \ s_{u,z} = |S_{u,z}|, \nonumber \\ S_{u,w} &=& \{i \in \{1,2\ldots, m\}: w(i) = u\},\ \ \ s_{u,w} = |S_{u,w}| \label{eqn: blocksize}\end{aligned}$$ denote the $u$-th block and block size under the community structures $z$ and $w$. Also define, $$\begin{aligned} \tilde{\Lambda}_{z,(b,n)} &=& ((\tilde{\lambda}_{uv,z,(b,n)}))_{m \times m},\ \ \tilde{\lambda}_{uv,z,(b,n)} = \frac{1}{nb} \frac{1}{s_{u,z} s_{v,z}} \sum_{t=1}^{nb} \sum_{\stackrel{i: z(i) =u}{j: z(j)=v}} A_{ij,(t,n)},\ \ \ \ \nonumber \\ \tilde{\Delta}_{w,(b,n)} &=& ((\tilde{\delta}_{uv,w,(b,n)}))_{m \times m},\ \ \tilde{\delta}_{uv,w,(b,n)} = \frac{1}{n(1-b)} \frac{1}{s_{u,w}s_{v,w}} \sum_{t=nb+1}^{n} \sum_{\stackrel{i: w(i)=u}{j: w(j)=v}} A_{ij,(t,n)}. \ \ \ \label{eqn: estimateb} \end{aligned}$$ We start our analysis by assuming that the [**community structures**]{} $z$ and $w$ are [**known**]{}. In that case, an estimate of the change point can be obtained by solving $$\begin{aligned} \tilde{\tau}_{n} &=& \arg \min_{b \in (c^{*},1-c^{*})} \tilde{L}(b,z,w,\tilde{\Lambda}_{z,(b,n)},\tilde{\Delta}_{w,(b,n)}). \label{eqn: estimateintro1}\end{aligned}$$ The following condition guarantees that the change-point is identifiable under a known community structure. **SNR-DSBM:** $\frac{n}{K^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 \to \infty$ Intuitively, it implies that the total signal per connection probability parameter needs to grow faster that $1/\sqrt{n}$, which is in accordance with identifiability conditions for other change point problems (e.g. see [@kosorok2008introduction] Section $14.5.1$). The following Theorem establishes asymptotic properties for $\tilde{\tau}_{n}$. Its proof is similar in nature (albeit much simpler in structure) to the proof of Theorem \[lem: b1\], where we deal with unknown community structures. Hence, it is omitted. \[thm: dsbmknown\] **(Convergence rate of ${\tilde{\tau}}_{n}$)** Suppose SNR-DSBM holds. Then, $$n ||\text{Ed}_z( \Lambda) - \text{Ed}_w(\Delta) ||_{F}^2 ({\tilde{\tau}}_{n} - \tau_n) = O_{\text{P}}(1).$$ In the ensuing Theorem \[lem: b1\], we will establish that the DSBM change point estimator $\tilde{\tau}_{n}$ with an [*unknown community structure*]{} (that needs to be estimated from the available data) has exactly the same convergence rate as the one posited in Theorem \[thm: dsbmknown\]. However, a much stronger identifiability condition to SNR-DSBM is needed, since more parameters are involved. Recall the estimates in (\[eqn: estimateb\]) given by $\tilde{\Lambda} = (( \tilde{\lambda}_{ab,z,(\tilde{\tau}_{n}, n)}))_{K \times K}$ and $\tilde{\Delta} = (( \tilde{\delta}_{ab,w,(\tilde{\tau}_{n}, n)}))_{K \times K}$. The edge probability matrices $\text{Ed}_{z}(\Lambda)$ and $\text{Ed}_{w}(\Delta)$ can also be estimated by $\text{Ed}_{z}(\tilde{\Lambda})$ and $\text{Ed}_{w}(\tilde{\Delta})$, respectively. The following Theorem provides the convergence rate of the corresponding estimators. Its proof is similar (and structurally simpler) to the proof of Theorem \[thm: c3\] where we deal with unknown community structures and hence omitted. \[thm: b2\] **(Convergence rate of edge probabilities when $z$ and $w$ are known)**\ Suppose SNR-DSBM holds. Let $\mathcal{S}_n = \min(\min_{u} s_{u,z}, \min_{u} s_{u,w})$. Then $$\begin{aligned} && \frac{1}{K^2}||\tilde{\Lambda} - \Lambda||_F^2,\ \ \frac{1}{K^2}||\tilde{\Delta} - \Delta||_F^2 = O_{\text{P}}\left( \frac{I(n>1)}{n^2 ||\text{Ed}_z(\Lambda)-\text{Ed}_w(\Delta) ||_F^4} + \frac{\log K}{n \mathcal{S}_n^2}\right),\ \ \nonumber \\ && \frac{1}{m^2} ||\text{Ed}_{z}(\tilde{\Lambda}) - \text{Ed}_{z}(\Lambda)||_F^2, \ \ \frac{1}{m^2}||\text{Ed}_{w}(\tilde{\Delta}) - \text{Ed}_{w}(\Delta)||_F^2 \nonumber \\ && \hspace{4cm}= O_{\text{P}}\left(\frac{I(n>1)}{n^2 ||\text{Ed}_z(\Lambda)-\text{Ed}_w(\Delta) ||_F^4}+\frac{\log m}{n \mathcal{S}_n^2}\right).\end{aligned}$$ Note that $\tilde{\Lambda} = (( \tilde{\lambda}_{ab,z,(\tilde{\tau}_{n}, n)}))_{K \times K}$. To compute the rate for $\frac{1}{K^2}||\tilde{\Lambda} - \Lambda||_F^2$, we have $(\tilde{\lambda}_{ab,z,(\tilde{\tau}_{n}, n)} - \lambda_{ab})^2 \leq 2(\tilde{\lambda}_{ab,z,(\tilde{\tau}_{n}, n)}-\tilde{\lambda}_{ab,z,({\tau}_{n}, n)})^2 + 2(\tilde{\lambda}_{ab,z,({\tau}_{n}, n)}-\lambda_{ab})^2 \equiv T_1 + T_2$. It is easy to see that the first term $T_1$ is dominated by $(\tilde{\tau}_n -\tau_n)^2$ and thus by Theorem \[thm: dsbmknown\], $(\tilde{\tau}_n -\tau_n)^2 = O_{\text{P}}\left( \frac{I(n>1)}{n^2 ||\text{Ed}_z(\Lambda)-\text{Ed}_w(\Delta) ||_F^4}\right)$. Moreover, the rate of $T_2$ is $\frac{\log K}{n \mathcal{S}_n^2}$. Details are given in Section \[subsec: c3\]. Similar arguments work for the other matrices present in Theorem \[thm: b2\]. \[rem: parameteroracle\] **(Rate for $n=1$)**. If $n=1$, then there is no change point and $T_1$ does not appear. In this case, we have only one community structure (say) $z$ and one community connection matrix (say) $\Lambda$. Moreover, the number of communities $K=K_m$ and the minimum block size $\mathcal{S}_m = \min_{u} s_{u,z}$ depend only on $m$. Estimation of $\Lambda$ for $n=1$ is studied in [@ZLZ2015] when community structure $z$ is unknown. In this remark, we assume that $z$ is known. We estimate $\Lambda$ by $\tilde{\Lambda} = (( \tilde{\lambda}_{ab,z,(1/n, n)}))_{K \times K}$. Then, $$\begin{aligned} && \frac{1}{K^2}||\tilde{\Lambda} - \Lambda||_F^2 = O_{\text{P}}\left( \frac{\log K}{\mathcal{S}_m^2}\right),\ \ \frac{1}{m^2} ||\text{Ed}_{z}(\tilde{\Lambda}) - \text{Ed}_{z}(\Lambda)||_F^2 = O_{\text{P}}\left(\frac{\log m}{ \mathcal{S}_m^2}\right). \nonumber\end{aligned}$$ Similar results for the case of unknown communities are discussed in Remark \[rem: parameter\] and Section \[sec:discuss\]. In Theorem \[thm: c3\], we establish the results for the same quantities in the case of unknown community structures. It will be seen that the convergence rate of $\tilde{\Lambda}$ and $\tilde{\Delta}$ given above, is much sharper compared to the case of unknown community structures, despite using repeated observations in the latter one; see also discussion in Remark \[rem: paraestimation\]. The real problem of interest is when the community structure is [**unkown**]{} and needs to be estimated from the observed sequence of adjacency matrices along with the change point. A standard strategy in change point analysis is to optimize the least squares criterion function $\tilde{L}(b,z,w,\Lambda,\Delta)$ posited above with respect to [*all*]{} the model parameters. This becomes challenging both computationally since one needs to find a good assignment of nodes to communities, and technically, since for any time point away from the true change point the node assignment problem needs to be solved under a misspecified model; namely, the available adjacency matrices are generated according to both the pre- and post-change point community connection probability matrices. A natural estimator of $\tau_n$ can be obtained by solving $$\begin{aligned} \tilde{\tilde{\tau}}_{n} &=& \arg \min_{b \in (c^{*},1-c^{*})} \tilde{L} (b,\tilde{z}_{b,n},\tilde{w}_{b,n},\tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)},\tilde{\Delta}_{\tilde{w}_{b,n},(b,n)}), \label{eqn: estimateintro}\end{aligned}$$ where $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$ are obtained using the clustering algorithm from [@B2017] (details below). While other clustering algorithms can also be employed, and are discussed in Section \[sec:discuss\], all clustering algorithms incur some degree of misclassification (while assigning nodes to communities) which must be suitably controlled by an appropriate assumption. The employed clustering algorithm requires a simpler and somewhat easier assumption on the missclassification rate, compared to other available clustering methods. **Clustering Algorithm I**: (proposed in [@B2017]) 1. Obtain sums of the adjacency matrices before and after $b$ as $B_1 = \sum_{t=1}^{nb} A^{(t)}$ and $B_2 = \sum_{t=nb +1}^{n} A^{(t)}$ respectively. 2. Obtain $\hat{U}_{m \times K}$ and $\hat{V}_{m \times K}$ consisting of the leading $K$ eigenvectors of $B_1$ and $B_2$, respectively, corresponding to its largest absolute eigenvalues. 3. Use an $(1+\epsilon)$ approximate $K$-means clustering algorithm on the row vectors of $\hat{U}$ and $\hat{V}$ to obtain $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$ respectively. Note that in Step $3$ above, an $(1+\epsilon)$ approximate $K$-means clustering procedure is employed, instead of the $K$-means. It is known that finding a global minimizer for the $K$-means clustering problem is NP-hard (see, e.g., [@aloise2009np]). However, efficient algorithms such as $(1+\epsilon)$ approximate $K$-means clustering provide an approximate solution, with the value of the objective function being minimized to within a constant fraction of the optimal value (for details, see [@kumar2004simple]). [*Computational complexity of the procedure:*]{} Note that for each $b \in (c^{*},1-c^{*})$, the complexity of computing $\tilde{z}_{b,n}$ (or $\tilde{w}_{b,n}$) and $\tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)}$ (or $\tilde{\Delta}_{\tilde{w}_{b,n},(b,n)}$) is $O(m^3)$ and $O(m^2n)$, respectively. Hence, $\tilde{L}(b,\tilde{z}_{b,n},\tilde{w}_{b,n},\tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)},\tilde{\Delta}_{\tilde{w}_{b,n},(b,n)})$ at $b$ has computational complexity $O(m^3+m^2n)$. Some calculations show that only finitely many binary operations are needed to update $\tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)}$ and $\tilde{\Delta}_{\tilde{w}_{b,n},(b,n)}$ for the next available time point. However, computing $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$ requires $O(m^3)$ operations for each time point. Therefore, the computational complexity for obtaining $\tilde{L}(b,\tilde{z}_{b,n},\tilde{w}_{b,n},\tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)},\tilde{\Delta}_{\tilde{w}_{b,n},(b,n)})$ for $n$-many time points is $O(m^3n + m^2 n)=O(m^3n)$. To establish consistency results for $\tilde{\tilde{\tau}}_{n}$, an additional assumption on the misclassification rate of $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$ is needed, given next. We start with the following definition. **(Misclassification rate)** A node is considered as misclassified, if it is not allocated to its true community. The misclassification rate corresponds to the fraction of misclassified nodes. Let $\mathcal{M}_{(z,\tilde{z}_{b,n})}$ and $\mathcal{M}_{(w,\tilde{w}_{b,n})}$ be the misclassification rates of estimating $z$ and $w$ by $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$, respectively. Then, $$\begin{aligned} \mathcal{M}_{(z,\tilde{z}_{b,n})} &=& \min_{\pi \in S_k} \sum_{i=1}^{m} \frac{I(\tilde{z}_{b,n}(i) \neq \pi(z(i)))}{s_{z,\pi(z(i))}},\nonumber \\ \mathcal{M}_{(w,\tilde{w}_{b,n})} &=& \min_{\pi \in S_k} \sum_{i=1}^{m} \frac{I(\tilde{w}_{b,n}(i) \neq \pi(w(i)))}{s_{w,\pi(w(i))}} \label{eqn: mdefine}\end{aligned}$$ where $S_K$ denotes the set of all permutations of $\{1,2,\ldots,K\}$. Define $$\mathcal{M}_{b,n} = \max(\mathcal{M}_{(z,\tilde{z}_{b,n})},\mathcal{M}_{(w,\tilde{w}_{b,n})}).$$ Consider the following assumption. **(NS)** $\Lambda$ and $\Delta$ are non-singular. (NS) implies that there are exactly $K$ non-empty communities in DSBM and hence we can use an $(1+\epsilon)$ approximate $K$-clustering algorithm. If (NS) does not hold, then we have $K^\prime$ $(<K)$ non-empty communities and an $(1+\epsilon)$ approximate $K^\prime$-clustering algorithm performs better. The following theorem provides the convergence rate of $\mathcal{M}_{b,n}$. Its proof is given in Section \[subsec: mismiscluster\]. Let $\nu_{m}$ denote the minimum between the smallest non-zero singular values of $\text{Ed}_{z}(\Lambda)$ and $\text{Ed}_{w}(\Delta)$. \[thm: mismiscluster\] Suppose (NS) holds. Then, for all $b \in (c^*,1-c^*)$, we have $$\begin{aligned} \mathcal{M}_{b,n} = O_{\text{P}}\left( \frac{K}{n\nu_m^2}(\tau m + |\tau-b|\ ||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2)\right). \nonumber\end{aligned}$$ To establish consistency of $\tilde{\tilde{\tau}}_{n}$, we require that the missclassification rate $\mathcal{M}_{b,n}$ decays faster than $n^{-1}||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F$; see the proof of Theorem \[lem: b1\] and Remark \[rem: proof\] for technical details. By the identifiability condition SNR-DSBM and Theorem \[thm: mismiscluster\], we have $$\begin{aligned} \mathcal{M}_{b,n}n||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^{-1} \leq C \frac{K}{\nu_m^2} (\frac{m\sqrt{n}}{K}o(1) + m) \leq C ( \frac{m\sqrt{n}}{\nu_m^2}o(1) + \frac{Km}{\nu_m^2})\end{aligned}$$ holds with probability tending to $1$. Consistency of $\tilde{\tilde{\tau}}_{n}$ can be achieved under the SNR-DSBM condition and the following assumption (A1). **(A1)** $\frac{Km}{ \nu_m^2} \to 0$ and $\frac{m\sqrt{n}}{\nu_m^2} = O(1)$ We note that (A1) is compatible with the clustering algorithm employed in our procedure. Other clustering algorithms may also be used which would lead to modifications of (A1), as discussed in Section \[sec:discuss\]. Theoretical properties of $\tilde{\tilde{\tau}}_{n}$ {#sec:theory-clustering} ---------------------------------------------------- Our first result establishes the convergence rate of the proposed estimate of the change point. \[lem: b1\] **(Convergence rate of $\tilde{\tilde{\tau}}_{n}$)**\ Suppose SNR-DSBM, (NS) and (A1) hold. Then, $$n ||\text{Ed}_z( \Lambda) - \text{Ed}_w(\Delta) ||_{F}^2 (\tilde{\tilde{\tau}}_{n} - \tau_n) = O_{\text{P}}(1).$$ The proof of the theorem is given in Section \[subsec: b1\]. The next result focuses on the misclassification rate for $\tilde{\tilde{z}} = \tilde{z}_{\tilde{\tilde{\tau}}_{n},n}$ and $\tilde{\tilde{w}} = \tilde{w}_{\tilde{\tilde{\tau}}_{n},n}$, respectively. \[thm: cluster\] **(Rate of misclassification)**\ Suppose SNR-DSBM, (NS) and (A1) hold. Then, $$\begin{aligned} \mathcal{M}_{(z,\tilde{\tilde{z}})},\mathcal{M}_{(w,\tilde{\tilde{w}})} = O_{\text{P}}\left( \frac{Km}{n\nu_m^2}\right). \nonumber\end{aligned}$$ The proof of the Theorem is immediate from Theorems \[thm: mismiscluster\] and \[lem: b1\]. Let $\tilde{\tilde{\Lambda}} = (( \tilde{\lambda}_{ab,\tilde{\tilde{z}},(\tilde{\tilde{\tau}}_n, n)}))_{K \times K}$ and $\tilde{\tilde{\Delta}} = (( \tilde{\delta}_{ab,\tilde{\tilde{w}},(\tilde{\tilde{\tau}}_n, n)}))_{K \times K}$. The final result obtained is on the convergence rate of the community connection probability matrices $\tilde{\tilde{\Lambda}}$ and $\tilde{\tilde{\Delta}}$, respectively. Let ${\mathcal{S}}_{n,\tilde{\tilde{z}}} = \min_{u} s_{u,\tilde{\tilde{z}}}$, $\mathcal{S}_{n,\tilde{\tilde{w}}} = \min_{u} s_{u,\tilde{\tilde{w}}}$ and $\tilde{\mathcal{S}}_n = \min(\mathcal{S}_{n,\tilde{\tilde{z}}},\mathcal{S}_{n,\tilde{\tilde{w}}})$. \[thm: c3\] **(Convergence rate of edge probabilities $\tilde{\tilde{\Lambda}}$ and $\tilde{\tilde{\Delta}}$)**\ Suppose SNR-DSBM, (NS) and (A1) hold. Further, for some positive sequence $\{\tilde{C}_n\}$, we have that $\tilde{\mathcal{S}}_n \geq \tilde{C}_n\ \forall n$ with probability $1$. Then, $$\begin{aligned} \frac{1}{m^2} ||\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^2 &=& O_{\text{P}}\left( \left(\frac{Km}{n\nu_m^2}\right)^2 + \frac{I(n>1)}{n^2 ||\text{Ed}_z(\Lambda)-\text{Ed}_w(\Delta) ||_F^4}+ \frac{\log m}{n \tilde{C}_n^2} \right),\nonumber \\ \frac{1}{m^2} ||\text{Ed}_{\tilde{\tilde{w}}}(\tilde{\tilde{\Delta}}) - \text{Ed}_{w}(\Delta)||_F^2 &=& O_{\text{P}}\left(\left(\frac{Km}{n\nu_m^2}\right)^2 + \frac{I(n>1)}{n^2 ||\text{Ed}_z(\Lambda)-\text{Ed}_w(\Delta) ||_F^4}+ \frac{\log m}{n \tilde{C}_n^2} \right). \nonumber \end{aligned}$$ The proof of the Theorem is given in Section \[subsec: c3\]. \[rem: paraestimation\] Note that the first term in the convergence rate of $\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}})$, which is the square of the misclassification rate obtained in Theorem \[thm: cluster\], measures the closeness of $\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}})$ to $\text{Ed}_{{z}}(\tilde{\tilde{\Lambda}})$. On the other hand, the second term is the convergence rate of $\text{Ed}_{{z}}(\tilde{\tilde{\Lambda}})$ for $\text{Ed}_{{z}}(\Lambda)$ and coincides with the convergence rate of the edge probability matrix estimator when the communities are known - see Theorem \[thm: b2\] for details. As expected, the convergence rate of $\tilde{\tilde{\Lambda}}$ and $\tilde{\tilde{\Delta}}$, given in Theorem \[thm: c3\], is slower than the rate of $\tilde{\Lambda}$ and $\tilde{\Delta}$ when the communities are known. The reason is that the former estimates involve the misclassification rate of estimating $z$ and $w$ by $\tilde{\tilde{z}}$ and $\tilde{\tilde{w}}$, respectively. \[rem: parameter\] **(Rate for $n=1$)**. For $n=1$, we go back to the setup of Remark \[rem: parameteroracle\]. Suppose $z$ is unknown. We estimate $z$ and $\Lambda$ respectively by $\tilde{\tilde{z}}$ and $\tilde{\tilde{\Lambda}} = (( \tilde{\lambda}_{ab,\tilde{\tilde{z}},(1/n, n)}))_{K \times K}$. Further, for some positive sequence $\{\tilde{C}_m\}$, suppose we have that $\tilde{\mathcal{S}}_m \geq \tilde{C}_m\ \forall m$ with probability $1$. Then $$\begin{aligned} \frac{1}{m^2} ||\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^2 &=& O_{\text{P}}\left( \left(\frac{Km}{\nu_m^2}\right)^2 + \frac{\log m}{\tilde{C}_m^2} \right). \nonumber $$ The above rate of convergence is slower than the rate obtained in Remark \[rem: parameteroracle\] where communities are known. This rate of convergence varies with different clustering methods employed for estimating $z$. [@ZLZ2015] used a clustering algorithm so that the square of misclassification rate is $\sqrt{\frac{\log m}{m}}$ and $\tilde{C}_m^2 = \sqrt{m\log m}$. A detailed discussion on the impact of the clustering algorithm is provided in Section \[sec:discuss\]. On the condition (A1) {#sec:A1} --------------------- As seen from the results in Section \[sec:theory-clustering\], condition (A1) plays a critical role. Next, we discuss examples where it holds - Examples \[example: misclassnew\] and \[example: misclass1new\] - and where it fails to do so - Example \[example: misclassnew2\]. \[example: misclassnew\] Suppose we have $K$ balanced communities of size $m/K$. Let $\Lambda = (p_1 - q_1)I_K + q_1J_K$ and $\Delta = (p_2 - q_2)I_K + q_2J_K$, where $I_K$ is the identity matrix of order $K$ and $J_K$ is the $K \times K$ matrix whose entries are all equal to $1$. Also assume $|p_1 - q_1|, |p_2-q_2| > \epsilon$ for some $\epsilon >0$. Then, the smallest non-zero singular value of $\text{Ed}_{z}(\Lambda)$ and $\text{Ed}_{w}(\Delta)$ are $\frac{m}{K}|p_1-q_1|$ and $\frac{m}{K}|p_2-q_2|$, respectively. Therefore, $\nu_m = O(\frac{m}{K})$ and (A1) reduces to $$\begin{aligned} \label{eqn: A1reduce1} \frac{K^3}{m} \to 0\ \ \text{and}\ \ \ \frac{K^2 \sqrt{n}}{m} = O(1).\end{aligned}$$ If $K$ is finite, then we need $n = O(m^2)$, which is a rather stringent requirement for most real applications. If $K = \sqrt{m}$, the condition does not hold as $n \to \infty$. If $K = Cm^{0.5 - \delta}$ for some $C, \delta>0$, then (\[eqn: A1reduce1\]) holds if $m^{0.5-3\delta} \to 0$ and $n = O(m^{4\delta})$. In summary, if $K = Cm^{0.5 - \delta}$, $n = O(m^{4\delta})$ for some $C>0$ and $\delta> 1/6$, then (A1) holds. Next, define $$\begin{aligned} m_{\max,z}, m_{\max,w} &=& \text{largest community size of $z$ and $w$ respectively} \nonumber \\ m_{\min,z}, m_{\min,w} &=& \text{smallest community size of $z$ and $w$ respectively}. \nonumber $$ The above conclusion also holds if we have $\lim \frac{m_{\max,z}}{m_{\min,z}} = \lim \frac{m_{\max,w}}{m_{\min,w}}= 1$ instead of having balanced communities. \[example: misclassnew2\] Consider the same model as in Example \[example: misclassnew\] with $|p_1-q_1| = |p_2-q_2| = n^{-\delta}$ for some $\delta>0$. Suppose $K$ is finite. Then, (A1) reduces to $$\begin{aligned} \label{eqn: A1reduce} \frac{n^{2\delta}}{m} \to 0\ \ \text{and}\ \ \ \frac{{n^{1/2+2\delta}}}{m} = O(1).\end{aligned}$$ In this case, (A1) does not hold if $m = C\sqrt{n}$ for some $C>0$. \[example: misclass1new\] Let $\Lambda = p_1 I_K$ and $\Delta = p_2 I_K$. Assume that there is $\epsilon>0$ such that $p_1, p_2 > \epsilon$. Then, the smallest non-zero singular values of $\text{Ed}_{z}(\Lambda)$ and $\text{Ed}_{w}(\Delta)$ are $m_{\min,z}p_1$ and $m_{\min,w}p_2$ respectively. Let $m_{\min} = \min \{m_{\min,z},m_{\min,w}\}$. Therefore, $\nu_m = O(m_{\min})$. Let $\rho_m = \frac{m_{\min}}{m}$. Thus, (A1) reduces to $$\begin{aligned} \frac{K}{m \rho_m^2} \to 0\ \ \text{and}\ \ \ \frac{\sqrt{n}}{m\rho_m^2} = O(1). \label{eqn: 2} \end{aligned}$$ Suppose $K =Cm^{\lambda}$ and $m_{\min} = Cm^{\delta}$ for some $\lambda, \delta \in [0,1]$. Then, $\rho _m = m^{\delta-1}$ and (\[eqn: 2\]) reduces to $$\begin{aligned} \frac{1}{m^{2\delta -\lambda-1}} \to 0 \ \ \text{and}\ \ \ \frac{\sqrt{n}}{m^{2\delta-1}} = O(1). \nonumber\end{aligned}$$ Thus, (A1) holds if $K =Cm^{\lambda}$, $m_{\min} = Cm^{\delta}$, $n = m^{4\delta-2}$ for some $\lambda, \delta \in [0,1]$ and $2\delta - \lambda-1 >0$. \[rem: nc\] Note that computation of $\tilde{\tilde{\tau}}_n$ involves estimation of communities at every time point. The necessity of clustering at every time-point leads us to consider condition (A1). One may note though that since for theoretical considerations the change point needs to be contained in the interval $(c^*,1-c^*)$, the following alternative approach can be employed: use Clustering Algorithm I for the first $[nc^{*}]$ and the last $[nc^{*}]$ time points for estimating $z$ and $w$, respectively. Denote the corresponding estimators by $z^{*}$ and $w^{*}$. Then, the corresponding change point estimator $\tau^*_n$ can be obtained by $$\begin{aligned} \tau_{n}^* &=& \arg \min_{b \in (c^{*},1-c^{*})} \tilde{L} (b,{z}^{*},{w}^{*},\tilde{\Lambda}_{{z}^{*},(b,n)},\tilde{\Delta}_{{w}^{*},(b,n)}). \nonumber $$ Since we are using order $n$-many time points in the clustering step and also for estimating the true change point $\tau_n \in (c^*,1-c^*)$, the misclassification rates for $z^*$ and $w^*$ are similar to those of $\tilde{\tilde{z}}$ and $\tilde{\tilde{w}}$ obtained in Theorem \[thm: cluster\]. As pointed out in the discussion preceding the statement of assumption (A1), clustering at every time point requires the misclassification rate $\mathcal{M}_{b,n}$ to decay faster than $n^{-1}||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F$. However, when computing $\tau^*_n$, we use the same estimates $z^*$ and $w^*$ for all time points. As will be seen later in Remark \[rem: proof\], if we use the same clustering solution (assignment of nodes to communities) through all the time points, we only require the misclassification rate to decay faster than $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F$ (instead of $n^{-1}||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F$) for consistency of the change point estimator. As a consequence, a weaker assumption (A1\*) $\frac{m}{\sqrt{n}\nu_m} = O(1)$ and the SNR-DSBM condition are needed to establish the consistency of $\tau^*_n$. The upshot is that if node assignments $z^*$ and $w^*$ are employed, assumption (A1) becomes weaker. To further illustrate the latter point, note that in Example \[example: misclassnew\], (A1\*) reduces to $K^2 = O(m\sqrt{n})$. Therefore, if $K = Cm^{\delta}$ and $n =Cm^{\lambda}$ for some $\delta \in (0,1)$ and $\lambda>0$, then (A1\*) reduces to $1-2\delta+\lambda/2 \geq 0$. For Example \[example: misclassnew2\], (A1\*) boils down to $n^{2\delta} = O(m)$. Finally, in Example \[example: misclass1new\], (A1\*) holds if $m_{\min} = Cm^{\delta}$, $n = m^{\lambda}$ for some $\lambda >0$, $\delta \in [0,1]$ and $2\delta + \lambda/2 -1 \geq 0$. Note that in practice this strategy requires that $c^*$ be known, which may not be the case in most applications. If $c^*$ is not known, a reasonable practical alternative is to use only the first and last time points to estimate $z$ and $w$, respectively. Further, $\frac{m\sqrt{n}}{\nu_m^2} = O(1)$ is required for consistency of the change point estimator. This is stronger than (A1\*), but weaker than (A1). One can argue that, in principle, the value of $c^*$ is needed to compute the change point, since for establishing the theoretical results the search to identify it is restricted in the interval $(c^*,1-c^*)$. However, in practice one always searches throughout the entire interval and hence the practical alternative of using the first and last time points to estimate $z$ and $w$ is compatible with it. Finally, note that this alternative, i.e. known stretches of points that belong to only a single regime, is viable for estimating a single change point, but no longer so when multiple change points are involved. In the latter case, one would still assume that the first and last change points are separated away from the boundary by some fixed amount, but no such restrictions on the locations of the intermediate change points can be imposed, and hence a full search strategy (see for example the algorithm proposed in [@auger1989algorithms]) combined with clustering is unavoidable. This is the reason that our analysis focuses on the “every time-point clustering algorithm," since it provides insights on where challenges will arise for the case of multiple change points, an appropriate treatment of which is however beyond the scope of this paper. A fast $2$-step procedure for change point estimation in the DSBM {#sec: 2step} ================================================================= The starting point of our exposition is the fact that the SBM is a special form of the Erdős-Rényi random graph model. The latter is characterized by the following edge generating mechanism. Let $p_{ij}$ be the probability of having an edge between nodes $i$ and $j$ and let $P$ be the $m\times m$ corresponding connectivity probability matrix. We denote this model by ER$(P)$. An adjacency matrix $A$ is said to be generated according to ER$(P)$, if $A_{ij} \sim \text{Bernoulli}(p_{ij})$ [*independently*]{} and we denote it by $A \sim \text{ER}(P)$. Clearly $A \sim \text{SBM}(z,\Lambda)$ implies $A \sim \text{ER}(\text{Ed}_{z}(\Lambda))$. The DSBM with single change point in (\[eqn: dsbmmodel\]) can be represented as a random graph model as follows: there is a sequence $\tau_n \in (0,1)$ such that for all $n \geq 1$, $$\begin{aligned} A_{t,n} \sim \begin{cases} \text{ER}(\text{Ed}_{z}(\Lambda)),\ \ \text{if $1 \leq t \leq \lfloor n\tau_n \rfloor$} \\ \text{ER}(\text{Ed}_{w}(\Delta)),\ \ \text{if $\lfloor n\tau_n \rfloor < t <n$} \label{eqn: dERmodel} \end{cases}\end{aligned}$$ and $\Lambda \neq \Delta$ and/or $z \neq w$. In general, without any structural assumptions, a dynamic Erdős-Rényi model with a single change point has $m(m+1)+1$ many unknown parameters, the $0.5 m(m+1)$ pre- and post- change point edge probabilities and $1$ change point. An estimate of $\tau_n$ can be obtained by optimizing the following least squares criterion function. $$\begin{aligned} \hat{\tau}_{n} &=& {\arg \min}_{b \in (c^{*},1-c^{*})} L(b)\ \ \ \text{where} \nonumber \\ L(b) &=& \frac{1}{n}\sum_{i,j=1}^{m} \bigg[\sum_{t=1}^{nb} (A_{ij,(t,n)} - \hat{p}_{ij,(b,n)})^2 + \sum_{t=nb+1}^{n} (A_{ij,(t,n)} - \hat{q}_{ij,(b,n)})^2 \bigg], \nonumber \\ \hat{p}_{ij,(b,n)} &=& \frac{1}{nb} \sum_{t=1}^{nb} A_{ij,(t,n)}\ \ \text{and}\ \ \hat{q}_{ij,(b,n)} = \frac{1}{n(1-b)} \sum_{t=nb+1}^{n} A_{ij,(t,n)}. \label{eqn: estimatea1}\end{aligned}$$ Next, we present our 2-step algorithm.\ **2-Step Algorithm**: **Step 1:** In this step, we ignore the community structures and assume $z(i) = w(i) = i$ for all $1 \leq i \leq m$. We compute the least squares criterion function $L(\cdot)$ given in (\[eqn: estimatea1\]) and obtain the estimate ${\hat{\tau}}_{n} = \arg \min_{b \in (c^{*},1-c^{*})} L(b)$. **Step 2:** This step involves estimation of other parameters in DSBM. We estimate $z$ and $w$ by $\hat{\hat{z}} = \tilde{z}_{\hat{\tau}_n,n}$ and $\hat{\hat{w}} = \tilde{w}_{\hat{\tau}_n,n}$, respectively, and subsequently $\Lambda$ and $\Delta$ by $\hat{\hat{\Lambda}} = \tilde{\Lambda}_{\hat{z},(\hat{\tau}_{n},n)}$ and $\hat{\hat{\Delta}} = \tilde{\Delta}_{\hat{w},(\hat{\tau}_{n},n)}$, respectively. [*Computational complexity of the 2-Step Algorithm*]{}\ It can easily be seen that Step $1$ requires $O(m^2n)$ operations, while Step 2 due to performing clustering requires $O(m^3)$ operations. Thus, the total computational complexity of the entire algorithm is $O(m^3 + m^2n) \sim O(m^2\max(m,n))$, which is significantly smaller than that of obtaining $\tilde{\tilde{\tau}}_{n}$ in (\[eqn: estimateintro\]). Theoretical Results for $\hat{\tau}_{n}$ {#sec:theory-ER} ---------------------------------------- The following identifiability condition is required. **SNR-ER:** $\frac{n}{m^2} ||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 \to \infty.$ It requires that the signal per edge parameter grows faster than $1/\sqrt{n}$. Clearly SNR-ER is stronger than SNR-DSBM, as expected since the ER model involves $m^2$ parameters, as opposed to $K^2$ parameters for the DSBM. The following Theorem provides asymptotic properties for the estimates of the DSBM parameters obtained from the 2-Step Algorithm. Its proof is given in Section \[subsec: 2step\]. \[thm: 2step\] Suppose that SNR-ER holds. Then, the conclusion of Theorem \[lem: b1\] holds for $\hat{\tau}_n$. Similarly, the conclusions of Theorems \[thm: cluster\] and \[thm: c3\] continue to hold for $\hat{z}$, $\hat{w}$, $\hat{\hat{\Lambda}}$ and $\hat{\hat{\Delta}}$. Comparison of the “Every time point clustering algorithm"’ vs the 2-step algorithm {#sec: compare} ================================================================================== Our analysis up to this point has highlighted the following key findings. If the total signal is strong enough (i.e. SNR-ER holds), then it is beneficial to use the $2$-step algorithm that provides consistent estimates of [*all*]{} DSBM parameters at [*reduced computational cost*]{}. On the other hand, if the signal is not adequately strong (i.e. SNR-ER fails to hold, but SNR-DSBM holds) then the only option available is to use the computationally expensive “every time point algorithm", provided that (A1) also holds. Our discussion in Section \[sec:A1\] indicates that (A1) is not an innocuous condition and may fail to hold in real application settings. For example, consider a DSBM with $m=60$ nodes, $K=2$ communities and $n=60$ time points. Suppose that there is a break at $n\tau_n = 30$, due to a change in community connection probabilities. Further, assume that the community connection probabilities before and after the change point are given by $ \Lambda = \left(\begin{array}{cc} 0.6 & 0.3 \\ 0.3 & 0.6 \end{array} \right)$ and $\Delta = \Lambda + \frac{1}{n^{1/4}}J_2$, respectively. Finally, suppose that there is no change in community structures and $z(i) = w(i) = I(1 \leq i \leq m/2) + 2I(m/2 +1 \leq i \leq m)$. In this case, one can check that $\frac{n}{m^2}||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_2^2 =7.75$, $\frac{Km}{\nu_m^2}= 1.48$, $\frac{m\sqrt{n}}{\nu_m^2} = 5.7$ and hence SNR-ER holds but (A1) fails. Figure \[fig: 2\] plots the least squares criterion function against time scaled by $1/n$, corresponding to the 2-step, known communities, and “every time point" algorithms, respectively. The plots show that the trajectory of the least squares criterion function is much smoother and the change point is easily detectable when known community structures are assumed. It is also the case for the 2-step algorithm, albeit with more variability. However, since (A1) fails to hold, the objective function depicted in Figure \[fig: 2\] (bottom middle panel) clearly illustrates that the change point is not detectable for the “every time point" algorithm. ![ \[fig: 2\] Plot of the least squares criterion function against time scaled by $1/n$. Top left and right panels correspond to the 2-step and known communities algorithm, while the bottom middle depicts the “every time point" algorithm, respectively.](1.pdf "fig:"){height="70mm" width="74mm"} ![ \[fig: 2\] Plot of the least squares criterion function against time scaled by $1/n$. Top left and right panels correspond to the 2-step and known communities algorithm, while the bottom middle depicts the “every time point" algorithm, respectively.](2.pdf "fig:"){height="70mm" width="74mm"} The next question to address is [*“How stringent is SNR-ER"*]{} under the DSBM model. As the following discussion shows, reallocation of nodes to new communities generates strong enough signal, and therefore SNR-ER may be easier to satisfy in practice than originally thought. **Sufficient conditions for SNR-ER under the DSBM model**. Next, we examine a number of settings where SNR-ER holds under the DSBM network generating mechanisms and hence the 2-step algorithm can be employed. Specifically, the following proposition provides sufficient conditions for SNR-ER to hold. Let $$A(\epsilon,\delta_1) = \{(i,j):\ |\lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| > \epsilon n^{-\delta_1 /2}\}.$$ Hence, $A(\epsilon,\delta_1)$ corresponds to the set of all edges for which the connection probability changes at least by an $\epsilon n^{-\delta_1 /2}$ amount. \[example1\] Suppose $|A(\epsilon,\delta_1)| \geq Cm^2 n^{-\delta_2}$ for some $C,\epsilon>0$ and $0 \leq \delta_1 + \delta_2 <1$. Then, SNR-ER holds. The above proposition follows since $\frac{n}{m^2}||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^2 \geq \frac{n}{m^2}C\epsilon^2 m^{2} n^{-\delta_1 -\delta_2} =$\ $C\epsilon^2 n^{1-\delta_1-\delta_2} \to \infty$. This implies that at least $Cm^2 n^{-\delta_2}$-many edges need to change their connection probability by at least $\epsilon n^{-\delta_1 /2}$ for SNR-ER to be satisfied. This leads us to the following scenarios that often arise in practice. \(A) **Reallocation of nodes**: Suppose that the pre- and post-community connection probabilities are the same; i.e. $\Lambda = \Delta$. This also implies that the total number of communities before and after the change point are equal. Suppose that some of the nodes are reallocated to new communities after the change point epoch.\ A motivating example for this scenario comes from voting patterns of legislative bodies as analyzed in [@bao2018core]. In this setting, one is interested in identifying when voting patterns of legislators change significantly. By considering the legislators as the nodes of the network, an edge between two of them indicates voting similarly on a legislative measure (e.g. bill, resolution), while the communities reflect their political affiliations, it can be seen that after an election the composition of the communities may be altered - reassignment of nodes. In this situation, SNR-ER holds if the entries of $\Lambda$ (or $\Delta$) are adequately separated and enough nodes are reallocated. Specifically, for some $\epsilon, C>0$ and $0 \leq \delta_1 + \delta_2 <1$, suppose we have $|\Lambda_{ij} - \Lambda_{i^\prime j^\prime}| > \epsilon n^{-\delta_1 /2}\ \forall (i,j) \neq (i^\prime,j^\prime)$ and $Cm^2 n^{-\delta_2}$-many nodes change their community after time $n\tau_n$. Then, by Proposition \[example1\], SNR-ER holds. \(B) **Change in connectivity**: Suppose that the community structures remain the same before and after the change point (i.e. $z = w$), but their community connection probabilities change (i.e. $\Lambda \neq \Delta$). This scenario is motivated by the following examples: in transportation networks, when service is reduced or even halted between two service locations, in social media platforms (e.g. Facebook) when a new online game launches, or in collaboration networks when a large scale project is completed. Then, SNR-ER holds if entries of $\Lambda$ are adequately separated from those of $\Delta$. Specifically, for some $\epsilon>0$ and $0 \leq \delta <1$, suppose we have $|\lambda_{ij} -\delta_{ij}| > \epsilon n^{-\delta/2}\ \forall i,j = 1,2,\ldots,K$. Then, by Proposition \[example1\], SNR-ER holds. \(C) **Merging Communities**: Sometimes, when two user communities cover the same subject matter and share similar contributors, they may wish to merge their communities to push their efforts forward in a desired direction. Suppose that the $1$st and the $K$th communities in $z$ merge into the $1$st community in $w$. In this situation, SNR-ER holds if the pre-connection probability between the $1$st and the $K$-th communities and the post-connection probability within the $1$st community are adequately separated and if the sizes of the $1$st and the $K$-th communities are large before the change. Precisely, suppose $|\lambda_{1k} - \delta_{11}| > Cn^{-\delta_1/2}$, $s_{1,z} s_{K,z} \geq Cm^{2}n^{-\delta_2}$ for some $C>0$ and $0 \leq \delta_1 + \delta_2 < 1$. Then, by Proposition \[example1\], SNR-ER holds. \(D) **Splitting communities**: One community often splits into two communities when conflicts and disagreements arise among its members. Suppose that the $1$st community in $z$ splits into the $1$st and $K$th communities in $w$. In this case, SNR-ER holds if the pre-connection probability within the $1$st community and the post-connection probability between the $1$st and the $K$th communities are adequately separated and the size of the $1$st and the $K$th communities are large after the change. Suppose $|\lambda_{11} - \delta_{1K}| > Cn^{-\delta_1/2}$, $s_{1,w} s_{K,w} \geq Cm^{2}n^{-\delta_2}$ for some $C>0$ and $0 \leq \delta_1 + \delta_2 < 1$. Then, by Proposition \[example1\], SNR-ER holds. \[rem: sparsecomp\] Examples (A)-(D) above and Proposition \[example1\] hold for both dense and sparse networks. However, for sparse networks, a large enough number of time points $n$ is required compared to the total number of nodes $m$. This is due to the fact that in a sparse network, there are relatively few edges ($|A(\epsilon,\delta_1)| = O(\log m)$) to contribute to the total signal in Proposition \[example1\]. Hence, $\frac{n}{m^2}||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 = \frac{n^{1-\delta_1}\log m}{m^2}$. Thus, SNR-ER holds if $\frac{m^2}{n^{1-\delta_1}\log m} \to 0$. Next, we discuss two examples where the SNR-ER fails to hold, but SNR-DSBM does. \(E) If most edges change their connection probabilities by an amount of $C_1/\sqrt{n}$ for some $C_1>0$, then SNR-ER does not hold, but SNR-DSBM does. Specifically, let $A(C_1) = \{(i,j):\ |\lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| = C_1/\sqrt{n}\}$. Suppose $|A(C_1)| = C_2m^2$ for some $C_1,C_2>0$, $|\lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| = 0\ \forall (i,j) \in A^c$ and $\displaystyle \min (\min_{u} s_{u,z},\min_{u} s_{u,w}) \to \infty$. Then $\frac{n}{m^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = C_1^2 C_2 \centernot\longrightarrow \infty$ but $\frac{n}{K^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = C_1^2C_2\frac{m^2}{K^2} \to \infty$. \(F) If the connection probabilities between the smallest community and the remaining ones change by $C/\sqrt{n}$ for some $C>0$, then for an appropriate choice of $K$ and smallest community size, SNR-ER does not hold, but SNR-DSBM does. Specifically, suppose $z=w$, $K = C_1 m^{\delta_1/2}$, $\displaystyle \min_{u} s_{u,z} = s_{1,z} = C_2m^{\delta_2 /2}$, $|\lambda_{1j} - \delta_{1j}| = C_3/\sqrt{n}\ \forall j$ and $|\lambda_{ij} - \delta_{ij}| = 0\ \forall i \neq 1, j \neq 1$ for some $C_1,C_2,C_3 >0$ and $0 < \delta_1 + \delta_2 \leq 2$, $\delta_1<\delta_2$. Then $\frac{n}{m^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = C_3^2C_2m^{-(2-\delta_2)} \to 0$ but $\frac{n}{K^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = C_3^2 C_2 C_1^{-2} m^{\delta_2 - \delta_1} \to \infty$. Note that examples (E)-(F) only deal with the SNR-DBSM condition and do not address the equally important (A1) condition for the “every time point clustering algorithm" to work. The next example, provides a setting where SNR-ER does not hold, but both SNR-DSBM and (A1) hold. \(G) Consider the model and assumptions in Example \[example: misclass1new\]. Suppose $p_2 = p_1 + \frac{1}{\sqrt{n}}$. Then, $mn^{-1} \leq ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 \leq m^2 n^{-1}$. Hence, SNR-ER does not hold. Further, if $K^2 = o(m)$, then SNR-DSBM holds. Thus, SNR-DSBM and (A1) hold if $K =Cm^{\lambda}$, $m_{\min} = Cm^{\delta}$ and $n = m^{4\delta-2}$ for some $\lambda \in [0,0.5), \delta \in [0,1]$ and $2\delta - \lambda-1 >0$. The upshot of the above examples is that due to the structure of the DSBM, there are many instances arising in real settings where SNR-ER holds. On the other hand, as example (G) illustrates, some rather special settings are required for SNR-ER to fail, while both SNR-DSBM and (A1) hold. Thus, it is relatively safe to assume that the 2-step algorithm is applicable across a wide range of network settings, making it a very attractive option to practitioners. Numerical Illustration {#subsec: simulation} ---------------------- Next, we discuss the performance of the two change point estimates $\hat{\tau}_n$ and $\tilde{\tilde{\tau}}_n$ based on synthetic data generated according to the following mechanism, focusing on the impact of the parameters $m$, $n$ and small community connection probabilities on their performance. **Effect of $m$ and $n$:** We simulate from the following DSBMs (I), (II), (III) for three choices of $(m,n, \tau_n) = (60,60,30), (500,20,10), (500,100,50)$. These results are presented in Tables $1-3$. Although the following DSBMs satisfy the assumptions in Proposition \[example1\], SNR-ER may be small for dealing with finite samples. \[Note that by Proposition \[example1\] and for finite number of communities (K is finite), $\frac{n}{m^2}||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = O(n^{1-\delta_1-\delta_2})$ and $\frac{n}{K^2}||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 = O(m^2 n^{1-\delta_1-\delta_2})$ and for balanced community and $\nu_m = O(\frac{mn^{-\delta}}{K})$ ($\delta \geq 0$), we have $\frac{Km}{\nu_m^2} = O(n^{2\delta}/m)$ and $\frac{m\sqrt{n}}{\nu_m^2} = O(\frac{n^{0.5+2\delta}}{m})$. Thus, for small $n$ and large $m$, SNR-ER becomes small, but SNR-DSBM and (A1) hold. Moreover, (A1) is not satisfied for large $\delta$ and small $m$.\] (I) **Reallocation of nodes**: $K=2$, $z(i) = I(1 \leq i \leq m/2) + 2I(m/2 +1 \leq i \leq m)$, $w(2i-1) = 1,\ w(2i)=2\ \forall 1 \leq i \leq m/2$. $ \Lambda = \Delta = \left(\begin{array}{cc} 0.6 & 0.6 - \frac{1}{n^\delta} \\ 0.6 - \frac{1}{n^\delta} & 0.6 \end{array} \right)$ for $\delta = 1/20,1/10,1/4$. (II) **Change in connectivity**: $K=2$, $z(i) = w(i) = I(1 \leq i \leq m/2) + 2I(m/2 +1 \leq i \leq m)$, $ \Lambda = \left(\begin{array}{cc} 0.6 & 0.3 \\ 0.3 & 0.6 \end{array} \right)$, $\Delta = \Lambda + \frac{1}{n^{1/4}}J_2$. (III) **Merging communities**: $K=3$, $z(i) = I(1 \leq i \leq 20) + 2I(21 \leq i \leq 40) + 3I(41 \leq i \leq 60)$, $w(i) = I(1 \leq i \leq 20,\ 41 \leq i \leq 60) + 2I(21 \leq i \leq 40)$, $\Lambda = \left(\begin{array}{ccc} 0.6 & 0.3 & 0.6 - \frac{1}{n^{1/20}} \\ 0.3 & 0.6 & 0.3 \\ 0.6 - \frac{1}{n^{1/20}} & 0.3 & 0.6 \end{array} \right)$, $\Delta = \left(\begin{array}{ccc} 0.6 & 0.3 & 0 \\ 0.3 & 0.6 & 0 \\ 0 & 0 & 0 \end{array} \right)$.\ Splitting communities and merging communities are similar once we interchange $z$, $w$ and $\Lambda$, $\Delta$. \[h!\] \[table: 1\] [| m[1.9cm]{} | m[1.9cm]{} | m[1.9cm]{} | m[2.4cm]{} | m[2.4cm]{}| m[2.4cm]{}| ]{} & & Change , in (2) & Merging , in (3)\ & $\delta = 1/20$ & $\delta = 1/10$ & $\delta = 1/4$ & &\ $F_n$ & $1195.246$ & $793.6742$ & $232.379$ & $464.758$ & $531.2205$\ $\frac{n}{m^2}F_n$ & $19.92077$ & $13.2279$ & $3.873$ & $7.745967$ & $8.8537$\ $\frac{n}{K^2}F_n$ & $17928.69$ & $11905.11$ & $3485.685$ & $6971.37$ & $3541.47$\ $\frac{Km}{\nu_m^2}$ & $0.198$ & $0.3$ & $1.2$ & $1.48$ & $3.11$\ $\frac{m\sqrt{n}}{\nu_m^2}$ & $0.7777$ & $1.1712$ & $4$ & $5.738$ & $8.03$\ $\hat{\tau}_{n}$ & $30$($88$), $28(5)$, $31(3)$, $34(4)$ & $30$($83$), $29(3)$, $28(7)$, $31(7)$ & $30(80)$, $29(9)$, $28(7)$, $31(4)$ & $30(83)$, $28(10)$, $31(7)$ & $30(88)$, $29(8)$, $28(4)$\ $\tilde{\tilde{\tau}}_{n}$ & $30(85)$, $28(5)$, $31(6)$, $32(4)$ & $30(80)$, $28(8)$, $31(6)$, $32(4)$, $33(2)$ & $30(34)$, $22(42)$, $25(10)$, $33(14)$ & $30(40)$, $21(30)$, $28(18)$, $26(12)$ & $30(21)$, $19(10)$, $23 (48)$, $26(14)$, $38(7)$\ \[h!\] \[table: 2\] [| m[1.9cm]{} | m[1.9cm]{} | m[1.9cm]{} | m[2.4cm]{} | m[2.4cm]{}| m[2.4cm]{}| ]{} & & Change , in (2) & Merging , in (3)\ & $\delta = 1/20$ & $\delta = 1/10$ & $\delta = 1/4$ & &\ $F_n$ & $92641.81$ & $68660.03$ & $27950.85$ & $53901.7$ & $41174.14$\ $\frac{n}{m^2}F_n$ & $7.411$ & $5.49$ & $2.24$ & $4.472$ & $3.294$\ $\frac{n}{K^2}F_n$ & $463209$ & $343300.2$ & $139754.2$ & $279508.5$ & $91498.08$\ $\frac{Km}{\nu_m^2}$ & $0.0216$ & $0.0291$ & $0.0716$ & $0.1778$ & $0.3732$\ $\frac{m\sqrt{n}}{\nu_m^2}$ & $0.0483$ & $0.0651$ & $0.16$ & $0.3975$ & $0.576$\ $\hat{\tau}_{n}$ & $10$($90$), $8(6)$, $11(4)$ & $10$($88$), $8(5)$, $11(7)$ & $10(39)$, $3(23)$, $7(30)$, $13(8)$ & $10(85)$, $9(7)$, $8(8)$ & $10(83)$, $9(7)$, $8(4)$ $11(4)$, $12(2)$\ $\tilde{\tilde{\tau}}_{n}$ & $10(85)$, $9(7)$, $11(5)$, $12(3)$ & $10(82)$, $8(6)$, $11(5)$, $12(7)$ & $10(77)$, $8(11)$, $9(4)$, $11(8)$ & $10(83)$, $9(9)$, $8(4)$, $11(4)$ & $10(80)$, $8(9)$, $9(7)$ $11(4)$\ \[h!\] \[table: 3\] [| m[1.9cm]{} | m[1.9cm]{} | m[1.9cm]{} | m[2.4cm]{} | m[2.4cm]{}| m[2.4cm]{}| ]{} & & Change , in (2) & Merging , in (3)\ & $\delta = 1/20$ & $\delta = 1/10$ & $\delta = 1/4$ & &\ $F_n$ & $78869.67$ & $49763.4$ & $12500$ & $25000$ & $35053.19$\ $\frac{n}{m^2}F_n$ & $31.548$ & $19.905$ & $5$ & $10$ & $14.0213$\ $\frac{n}{K^2}F_n$ & $1971742$ & $1244085$ & $312500$ & $625000$ & $1389479.8$\ $\frac{Km}{\nu_m^2}$ & $0.02536$ & $0.04019$ & $0.16$ & $0.1778$ & $0.3732$\ $\frac{m\sqrt{n}}{\nu_m^2}$ & $0.1268$ & $0.201$ & $0.8$ & $0.889$ & $1.244$\ $\hat{\tau}_{n}$ & $50$($92$), $48(3)$, $51(5)$ & $50$($88$), $49(7)$, $48(3)$, $47(1)$, $51(1)$ & $50(82)$, $49(5)$, $47(3)$, $52(7)$, $53(3)$ & $50(84)$, $49(4)$, $47(2)$, $51(6)$, $52(4)$ & $50(87)$, $49(6)$, $51(3)$, $52(4)$\ $\tilde{\tilde{\tau}}_{n}$ & $50(87)$, $49(7)$, $48(5)$, $91(1)$ & $50(88)$, $48(6)$, $47(2)$, $51(4)$ & $50(82)$, $49(7)$, $48(5)$, $51(4)$, $52(2)$ & $50(82)$, $49(7)$, $48(5)$, $52(6)$ & $50(87)$, $49(4)$, $47(3)$, $51(6)$\ The following conclusions are in accordance with the results presented in Tables 1 through 3. (a) SNR-ER holds for large $n$ and large signal $||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2$. We observe large SNR-ER and consequently good performance of $\hat{\tau}_n$, throughout Tables $1-3$ except Column $3$ in Table $2$, which involves a small $n$, leading to poor performance of $\hat{\tau}_n$. (b) SNR-ER implies SNR-DSBM and thus a large SNR-DSBM is observed throughout Tables \[table: 1\]-\[table: 3\]. Moreover, if $\nu_m = O(\frac{mn^{\delta}}{K})$ for some $\delta >0$, then (A1) holds for small $\delta$, $n$, small $K$ and large $m$. Thus, (A1) holds and $\tilde{\tilde{\tau}}_n$ exhibits good performance throughout Tables $1-3$ except Columns $3-5$ in Table $1$ where $\delta$ is large and and $m$ small. The above numerical results amply demonstrate the competitive nature of the computationally inexpensive 2-step algorithm under the settings posited. However, note that the connection probabilities assumed are in general strong that leads to a large $F_n$ signal. Next, we illustrate the performance for the case of excessively small connection probabilities. **Effect of excessively small connection probabilities:** In this paper, we assume that the entries of $\Lambda$ and $\Delta$ are bounded away from $0$ and $1$, in order to establish results on the asymptotic distribution of the change point estimators (see Section \[sec: ADAP\]). This assumption is not needed for establishing consistency of the estimators. Next, we consider DSBMs with small entries in $\Lambda$ and $\Delta$ and illustrate their effect on the performance of the change point estimators based on simulated results. For DSBMs (IV) and (V), we consider $(m,n,\tau_n) =(60,60,30)$. \(4) **Reallocation of nodes**: Let $K=2$, $z(i) = I(1 \leq i \leq m/2) + 2I(m/2 +1 \leq i \leq m)$, $w(2i-1) = 1,\ w(2i)=2\ \forall 1 \leq i \leq m/2$. Further, $ \Lambda = \Delta = \left(\begin{array}{cc} \frac{1}{n^\lambda} & \frac{1}{n^\lambda} - \frac{1}{n^\delta} \\ \frac{1}{n^\lambda} - \frac{1}{n^\delta} & \frac{1}{n^\lambda} \end{array} \right)$ for $(\delta,\lambda) = (3/4,1/2), (7/8,5/8)$. (5) **Change in connectivity**: Let $K=2$, $z(i) = w(i) = I(1 \leq i \leq m/2) + 2I(m/2 +1 \leq i \leq m)$, $ \Lambda = \left(\begin{array}{cc} \frac{2}{n^\lambda} & \frac{1}{n^\lambda} \\ \frac{1}{n^\lambda} & \frac{2}{n^\lambda} \end{array} \right)$, $\Delta = \Lambda + \frac{1}{n^{1/4}}J_2$, $\lambda = 1/2,5/8$. The results are presented in Table $4$. For models (IV) and (V), the SNR-ER is proportional to $n^{-2\delta}$. The choice of $\delta$ taken in (IV) is large enough (as connection probabilities in $\Lambda$ and $\Delta$ are small) for making SNR-ER small. Thus, $\hat{\tau}_n$ does not perform well in Columns $1$ and $2$ of Table \[table: 4\]. Moreover, $\delta = 1/4$ in (V) is adequate to induce a large SNR-ER. Hence, $\hat{\tau}_n$ estimates $\tau_n$ very well in Columns $3$ and $4$ of Table \[table: 4\]. On the other hand, $m/K$ is large enough to satisfy SNR-DSBM. However, $\nu_m$ is proportional to $n^{-\lambda}$ and the choice of $\lambda$ in models (IV) and (V) is large; as a consequence (A1) does not hold for the settings depicted in Table \[table: 4\]. Therefore, the performance of $\tilde{\tilde{\tau}}_n$ suffers. \[h!\] \[table: 4\] [| m[1cm]{} | m[3cm]{} | m[3cm]{} | m[3cm]{} | m[3cm]{} | ]{} & &\ & $(\delta,\lambda) = (3/4,1/2)$ & $(\delta,\lambda) = (7/8,5/8)$ & $\lambda = 1/2$ & $\lambda=5/8$\ $F_n$ & $3.873$ & $1.3915$ & $464.758$ & $464.758$\ $\frac{n}{m^2}F_n$ & $0.0645$ & $0.0232$ & $7.746$ & $7.746$\ $\frac{n}{K^2}F_n$ & $58.095$ & $20.874$ & $6971.37$ & $6971.37$\ $\frac{Km}{\nu_m^2}$ & $61.968$ & $172.466$ & $8$ & $22.265$\ $\frac{m\sqrt{n}}{\nu_m^2}$ & $240$ & $667.96$ & $30.984$ & $86.232$\ $\hat{\tau}_{n}$ & $30$($13$), $27(13)$, $25(49)$, $24(13)$, $33(12)$ & $30$($2$), $28(9)$, $24(22)$, $21(28)$, $35(21)$, $37(8)$, $39(10)$ & $30(85)$, $29(7)$, $28(3)$, $32(5)$ & $30(84)$, $28(7)$, $31(6)$, $32(3)$\ $\tilde{\tilde{\tau}}_{n}$ & $28(30)$, $27(43)$, $34(9)$,$35(6)$, $37(12)$ & $30(5)$, $26(15)$, $22(23)$, $32(17)$, $37(22)$, $40(18)$ & $30(14)$, $28(2)$, $23(25)$, $34(24)$, $38(30)$, $39(5)$ & $30(15)$, $26(13)$, $23(30)$, $21(20)$, $36(14)$, $37(8)$\ **Simulation on setting (G):** Consider the setup in setting (G) previously presented and let $n=20$, $\tau_n = 10$, $m = 20$, $K=2$, $z(i) = w(i) = I(1 \leq i \leq 9) + 2I(10 \leq i \leq 20)$, $p_1^2=0.8$, $p_2 = p_1 + 1/\sqrt{n}$, $\Lambda = p_1I_2$, $\Delta=p_2I_2$. Simulation results are given in Table \[table: 5\]. In this case, both SNR-DSBM and (A1) hold. Hence, $\tilde{\tilde{\tau}}_n$ performs well as expected. However, due to the failure of SNR-ER to hold, the performance of $\hat{\tau}_n$ suffers. \[h!\] \[table: 5\] [| m[1cm]{} | m[1cm]{} | m[1cm]{} | m[1cm]{} | m[1cm]{} | m[1cm]{} | m[3cm]{}| m[3cm]{} | ]{} & $F_n$ & $\frac{n}{m^2}F_n$ & $\frac{n}{K^2}F_n$ & $\frac{Km}{\nu_m^2}$ & $\frac{m\sqrt{n}}{\nu_m^2}$ & $\hat{\tau}_n$ & $\tilde{\tilde{\tau}}_n$\ & $10.1$ & $0.51$ & $50.5$ & $0.62$ & $1.38$ & &\ Discussion {#sec:discuss} ========== A key step in using the “all time point clustering" algorithm involves clustering. A specific clustering procedure proposed in [@B2017] was used to identify the communities and for locating the change point in Section \[sec: DSBM\]. Nevertheless, other clustering algorithms proposed in the literature \[[@P2017; @RCY2011]\] could be employed. For these alternative clustering algorithms the following statements hold. (a) The conclusions of Theorems \[lem: b1\], \[thm: cluster\] and \[thm: c3\] hold once we replace (NS) and (A1) by **(A9)** $n^2\mathcal{M}_{b,n}^2 ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-2} \to 0, \ \forall \ b\in (c^*,1-c^*)$, where $\mathcal{M}_{b,n}$ is the maximum misclassification error that $\tilde{z}_{b,n}$ and $\tilde{w}_{b,n}$ in estimating $z$ and $w$, respectively, given in (\[eqn: mdefine\]). (b) Suppose (A9) and SNR-DSBM hold and in addition $\mathcal{M}_{\tilde{\tilde{\tau}}_{n},n}^2 = O_{\text{P}}(E_n)$ for some sequence $E_n \to 0$. Moreover, assume that for some positive sequence $\{\tilde{C}_n\}$, $\hat{\mathcal{S}}_n \geq \tilde{C}_n\ \forall \ n$ with probability $1$ and $n\tilde{C}_n^{-2}\log m ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^4 \geq I(n > 1)$. Then $$\begin{aligned} && \frac{1}{m^2} ||\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^2,\ \ \frac{1}{m^2} ||\text{Ed}_{\tilde{\tilde{w}}}(\tilde{\tilde{\Delta}}) - \text{Ed}_{w}(\Delta)||_F^2 \nonumber \\ && \hspace{4.5 cm}= O_{\text{P}}\left( E_n + \frac{I(n>1)}{n^2|\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^4} + \frac{\log m}{n \tilde{C}_n^2} \right). \label{eqn: remremove}\end{aligned}$$ The proofs of statements (a) and (b) follow immediately from those of Theorems \[lem: b1\]$-$\[thm: c3\] and Remark \[rem: proof\]. [@ZLZ2015] considered a single SBM (i.e. $n=1$, $\tau_n =1/n$, $\Lambda = \Delta$ and $z=w$) and proposed a method of community estimation with $E_n = \sqrt{\frac{\log m}{m}}$ and $\tilde{C}_n^2 = \sqrt{m \log m}$. Therefore by (\[eqn: remremove\]), $m^{-2}||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^2 = O_{\text{P}}(\sqrt{\frac{\log m}{m}})$. This convergence rate is the same as that derived in [@ZLZ2015]. Moreover, when we observe an SBM with the same parameters independently over time, i.e. $n >1$, by (\[eqn: remremove\]) the convergence rate becomes sharper compared to the $n=1$ case. However, for these results to hold, the corresponding misclassification rate needs to satisfy $\mathcal{M}_{b,n} n ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-1} \to 0$ for consistency of the estimators. Next, we elaborate on these alternative clustering algorithms. **Clustering Algorithm II**. Instead of doing a spectral decomposition of the average adjacency matrices $B_1$ and $B_2$, the spectral decompositon is applied to their corresponding Laplcian matrices. An appropriate modification of the Proof of Theorem $2.1$ in [@RCY2011] implies $$\begin{aligned} \label{eqn: c2rem} \mathcal{M}_{b,n} = O_{\text{P}} \left(\frac{P_n}{\xi_{K_n}^4} \left(\frac{(\log m)^2}{nm} + m^2|\tau-b| ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{2} \right)\right)\,, \end{aligned}$$ where $P_n = \max \{s_{u,z}, s_{u,w}: u=1,2,\ldots, K\}$ is the maximum community size and $\xi_{K_n}$ is the minimum between the $K_n$-th smallest eigenvalue of the Laplacians of $B_1$ and $B_2$. A proof is given in Section \[subsec: remc2\]. Therefore, to satisfy $\mathcal{M}_{b,n} n ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-1} \to 0$ for the above spectral clustering, we need $\frac{n^{1/2} P_n (\log m)^2}{\xi_{K_n}^4 m K} = O(1)$ and $\frac{P_n m^3 n}{\xi_{K_n}^4} \to 0$. However, the latter condition seems excessively stringent in practical settings. For example, suppose $\Lambda = (p_1-q_1)I_K + q_1J_K$ and $\Delta = (p_2-q_2)I_K + q_2J_K$ where $I_K$ is the identity matrix of order $K$ and $J_K$ is the $K \times K$ matrix whose entries all equal $1$. Further, suppose $0 < C < p_1, q_1, p_2, q_2 < 1-C<1$ and that the communities are of equal size. Then, $P_n = O(m/K)$. Moreover, [@RCY2011] established that $\xi_{K} = O(K^{-1})$. Hence, $\frac{n^{1/2} P_n (\log m)^2}{\xi_{K_n}^4 m K} = O(\sqrt{n}K^2 (\log m)^2) \to \infty$ and $\frac{P_n m^3 n}{\xi_{K_n}^4} = O(m^4 n K^3) \to \infty$. On the other hand, as we have seen in Example \[example: misclassnew\], (A1) is satisfied for this example. **Clustering Algorithm III**. In this case, the following modification of [@RCY2011]’s algorithm for community detection is employed as follows. Define $$\begin{aligned} \label{eqn: deglapmatrix} D_{i,(t,n)} &=& \sum_{j=1}^{m} A_{ij,(t,n)}, \ \ \ D_{(t,n)} = \text{Diag}\{D_{i,(t,n)}: 1 \leq i \leq n\},\ \ \\ L_{(t,n)} &=& D_{(t,n)}^{-1/2} A_{t,n} D_{(t,n)}^{-1/2},\ \ \ L_{\Lambda, (b,n)} = \frac{1}{nb}\sum_{t=1}^{nb}L_{(t,n)},\ \ L_{\Delta, (b,n)} = \frac{1}{n(1-b)}\sum_{t=nb+1}^{n} L_{(t,n)}. \nonumber\end{aligned}$$ Note that $I - L_{(t,n)}$ is the Laplacian of $A_{t,n}$. Next, run the spectral clustering algorithm introduced in [@RCY2011] after replacing $L$ respectively by $L_{\Lambda, (b,n)}$ and $L_{\Delta, (b,n)}$ for estimating $z$, $w$. In this case, $$\begin{aligned} \label{eqn: remc3} \mathcal{M}_{b,n} = O_{\text{P}}\left(\frac{P_n}{\xi_{K_n}^4} \left(\frac{(\log m\sqrt{n})^2}{\sqrt{n}m} + m^2|\tau-b| ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{2} + |\tau-b| \frac{(\log m)^2}{m}\right)\right),\end{aligned}$$ where $P_n$ and $\xi_{K_n}$ are as described after (\[eqn: c2rem\]). A proof is given in Section \[subsec: remc3\]. Therefore, to satisfy $\mathcal{M}_{b,n} n ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-1} \to 0$ for this variant of the spectral clustering algorithm, we require $\frac{n^{3/2} P_n (\log m)^2}{\xi_{K_n}^4 m K} = O(1)$, $\frac{n P_n (\log m\sqrt{n})^2}{\xi_{K_n}^4 m K} = O(1)$ and $\frac{P_n m^3 n}{\xi_{K_n}^4} \to 0$. However, these are much stronger conditions that the one required for Clustering Algorithm II. The upshot of the previous discussion is that Clustering Algorithm I requires a milder assumption (A1) on the misclassification rate compared to Clustering Algorithms II and III. This is the reason that the results established in Sections \[sec: DSBM\] and \[sec: 2step\] leverage the former algorithm. Finally, one may wonder regarding settings where SNR-DSBM holds, but neither (A1) nor SNR-ER do. The following Examples \[example: v10new1\] and \[example: v10new2\] introduce such settings in the context of changes in the connection probabilities and in the community structures, respectively. \[example: v10new1\] **(Change in connection probabilities)** Consider a DSBM where $$\begin{aligned} z=w\hspace{1 cm}\text{and}\hspace{1 cm} \Lambda = \Delta - \frac{1}{\sqrt{n}}. \label{eqn: v10new1}\end{aligned}$$ In this case $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 = \frac{m^2}{n}$. Therefore, SNR-ER does not hold. However, SNR-DSBM holds if $K=o(m)$. Cases (a)-(c) presented below provide settings where (A1) does not hold, but SNR-DSBM does. (a) Consider the setting in Example \[example: misclassnew\] with $K = Cm^{0.5-\delta}$ (i.e. $K=o(m)$) and $n = Cm^{4\delta}$ for some $C>0$ and $\delta < 1/6$. It can easily be seen that (A1) does not hold, but SNR-DSBM does. (b) Suppose all assumptions in Example \[example: misclassnew2\] hold, $K$ is finite and $m = Cn^{2\delta}$. In this case, (A1) does not hold, but SNR-DSBM does. (c) Finally, consider the setup in Example \[example: misclass1new\] with $K = o(m)$, $m_{\min} = Cm^{\delta}$, $n = m^{\lambda}$ for some $\lambda >0$, $\delta \in [0,1]$ and $-\lambda/2 \leq 2\delta - 1 < \lambda/2$. The same conclusion on (A1) failing to hold, while SNR-DSBM holding is reached. Therefore, in each of the (a)-(c) cases, together with (\[eqn: v10new1\]) do not satisfy (A1) and SNR-ER, whereas SNR-DSBM holds. \[example: v10new2\] **(Change in communities)** Consider a DSBM where for $0<p<1$, $$\begin{aligned} K=2, && z(i) = \begin{cases} 1\ \ \text{if $i$ is odd} \\ 2\ \ \text{if $i$ is even}, \end{cases} \ \ w(i) = \begin{cases} 1\ \ \text{if $1 \leq i \leq [m/2]$} \\ 2\ \ \text{if $[m/2]<i\leq m$}, \end{cases} \\ &&\Lambda = \Delta= \left(\begin{array}{cc} p & p-\frac{1}{\sqrt{n}} \\ p-\frac{1}{\sqrt{n}} & p \end{array} \right). \nonumber \end{aligned}$$ This gives $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 = \frac{m^2}{n}$. Hence, SNR-ER does not hold, but SNR-DSBM does. Also suppose $m = Cn^{\delta}$ for some $C>0$ and $\delta \in [1,1.5)$. In this case (A1) is not satisfied. The methods discussed in Sections \[sec: DSBM\] and \[sec: 2step\] fail to detect the change point under the above presented settings. Therefore, alternative strategies not based on clustering and hence assumption (A1) need to be investigated. One possibility for the case of a single change point being present was discussed in Remark \[rem: nc\]. \[example: v10new3\] As the true change point $\tau_n \in (c^*,1-c^*)$, we can use $\tau^*_n$ to estimate $\tau_n$ and its consistency follows from SNR-DSBM and (A1\*) $\frac{m}{\sqrt{n}\nu_m^2} = O(1)$ which is much weaker than (A1). As we have seen before, (A1) and SNR-ER do not hold in Examples \[example: v10new2\] and \[example: v10new3\] whereas SNR-DSBM is satisfied. Based on the discussion in Remark \[rem: nc\], it is easy to see that (A1\*) holds for these examples. Therefore, for the settings posited in Examples \[example: v10new2\] and \[example: v10new3\], $\tau^*_n$ estimates $\tau_n$ consistently. Nevertheless, as mentioned in Remark \[rem: nc\], this strategy is not easy to extend to a setting involving multiple change points. Another setting that does not require clustering is presented next and builds on the model discussed in [@G2015rate]. \[example: v10new4\] Consider a DSBM with $K=2$ communities. Further, let $B_{1z}$ and $B_{1w}$ be the blocks where node $1$ belongs to under $z$ and $w$, respectively, and let $\Lambda = \left(\begin{array}{cc} a_1 & d_1 \\ d_1 & a_1 \end{array} \right)$, $\Delta = \left(\begin{array}{cc} a_2 & d_2 \\ d_2 & a_2 \end{array} \right)$ with $0<c<a_1,a_2,d_1,d_2<1-c<1$, $a_1>d_1, a_2>d_2$, $a_1-d_1 = a_2-d_2$ and the true change point $\tau \in (c^*,1-c^*)$. Recall $\hat{p}_{ij,(b,n)}$ and $\hat{q}_{ij,(b,n)}$ from (\[eqn: estimatea1\]). Let $\gamma_j = \hat{p}_{11,(b,n)}-\hat{p}_{1j,(b,n)}$ and $\delta_j = \hat{q}_{11,(b,n)}-\hat{q}_{1j,(b,n)}$. One can use the following algorithm to detect communities. Chose $B, B^*>0$ and $\delta \in (0,1)$ such that $\frac{B}{\sqrt{n^\delta}} \leq \frac{c^*}{1-c^*} (a_1-d_1)$. 1. If $\gamma_j \leq \frac{B}{\sqrt{n^\delta}}$ and $\delta_j \leq \frac{B}{\sqrt{n^\delta}}$, then put node $j$ in $B_{1z}\cap B_{1w}$. 2. If $\gamma_j \leq \frac{B}{\sqrt{n^\delta}}$ and $\delta_j > \frac{B}{\sqrt{n^\delta}}$, then put node $j$ in $B_{1z}\cap B_{1w}^{c}$. 3. If $\gamma_j > \frac{B}{\sqrt{n^\delta}}$ and $\delta_j \leq \frac{B}{\sqrt{n^\delta}}$, then put node $j$ in $B_{1z}^{c}\cap B_{1w}$. 4. If $\gamma_j > \frac{B}{\sqrt{n^\delta}}$ and $\delta_j > \frac{B}{\sqrt{n^\delta}}$, then we need further investigation. (4a) If $\frac{\gamma_j}{\delta_j} \leq 1-\frac{B^*}{\sqrt{n^\delta}}$, then put node $j$ in $B_{1z}\cap B_{1w}^{c}$. (4b) If $\frac{\gamma_j}{\delta_j} > 1+\frac{B^*}{\sqrt{n^\delta}}$, then put node $j$ in $B_{1z}^{c}\cap B_{1w}$. (4c) If $\frac{\gamma_j}{\delta_j} \in (1-\frac{B^*}{\sqrt{n^\delta}}, 1+ \frac{B^*}{\sqrt{n^\delta}})$, then put node $j$ in $B_{1z}^{c}\cap B_{1w}^{c}$. In this algorithm, it is easy to see that $\text{P}(\text{no node is misclassifed}) \to 1$. Therefore, an alternative condition (A9) is satisfied (see details about it in Section \[subsec: examplev10new4\]) and $\tilde{\tilde{\tau}}_n$ estimates $\tau_n$ consistently. However, the setting in Example \[example: v10new4\] is very specific involving two parameters only for each connection probability matrix), which in turn allows one to use statistics based on the degree connectivity of each node and thus avoid using a clustering algorithm. Nevertheless, a generally applicable strategy is currently lacking for the regime where SNR-DSBM holds, but neither SNR-ER or (A1) do. This constitutes an interesting direction of further research. Asymptotic distribution of change point estimators and adaptive inference {#sec: ADAP} ========================================================================= Up to this point, the analysis focused on establishing consistency results for the derived change point estimators and the corresponding convergence rates. Nevertheless, it is also of interest to provide confidence intervals, primarily for the change point estimates. This issue is addressed next for $\tilde{\tau}_n$, $\tilde{\tilde{\tau}}_n$ and $\hat{\tau}_n$, and as will be shortly seen the distributions are different depending on the behavior of the norm difference of the parameters before and after the change point. Since this norm difference is not usually known a priori, we solve this problem through a data based adaptive procedure to determine the quantiles of the asymptotic distribution, irrespective of the specific regime pertaining to the data at hand. Form of asymptotic distribution {#subsec: asympdist} ------------------------------- For ease of presentation, we focus on $\hat{\tau}_{n}$, but analogous results hold for $\tilde{\tau}_n$ and $\tilde{\tilde{\tau}}_n$. As previously mentioned, there are three different regimes for its asymptotic distribution depending on:– (I) $||\text{Ed}_z(\Lambda) - \text{Ed}_w(\Delta)||_F^2 \to \infty$, (II) $||\text{Ed}_z(\Lambda) - \text{Ed}_w(\Delta)||_F^2 \to 0$ and (III) $||\text{Ed}_z(\Lambda) - \text{Ed}_w(\Delta)||_F \to c>0$. Assuming SNR-ER holds, $\hat{\tau}_{n}$ degenerates in Regime I. We need additional regularity assumptions (A2)-(A7) for the other regimes. Assumption (A2) stated below ensures that the connection probabilities are bounded away from $0$ and $1$, which gives rise to a dense graph and ensures positive asymptotic variance of the change point estimators. **(A2)** For some $c>0$, $0< c < \inf_{u,v} \lambda_{uv}, \inf_{u,v} \delta_{uv} \leq \sup_{u,v} \lambda_{uv}, \sup_{u,v} \delta_{uv} < 1-c<1$. The precise statements of (A3)-(A7) are given in Section \[subsec: assumption\], but a brief discussion of their roles is presented below. Assumption (A3) is required in Regime II and guarantees the existence of the asymptotic variance of the change point estimator. In Theorem \[lem: b2lse\](b), this variance is denoted by $\gamma^2$. In Regime III, we consider the following set of edges $$\begin{aligned} \mathcal{K}_n = \{(i,j): 1 \leq i,j \leq m,\ \ |\lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| \to 0\}\end{aligned}$$ and treat edges in $\mathcal{K}_n$ and $\mathcal{K}_0 = \mathcal{K}_n^c$ separately. Note that in Regime II, $\mathcal{K}_n = \{(i,j):\ 1\leq i,j \leq m\}$ is the set of all edges. Hence, we can treat $\mathcal{K}_n$ in a similar way as in Regime II. The role of (A4) in Regime III is analogous to that of (A3) in Regime II. In the limit, $\mathcal{K}_n$ contributes a Gaussian process with a triangular drift term. (A4) ensures the existence of the asymptotic variance $\tilde{\gamma}^2$ of the limiting Gaussian process as well as the drift $c_1^2$. (A5) is a technical assumption and is required for establishing asymptotic normality on $\mathcal{K}_n$. Moreover, $\mathcal{K}_0$ is a finite set. (A6) guarantees that $\mathcal{K}_0$ does not vary with $n$. (A7) guarantees that $\tau_n \to \tau^{*}$ for some $\tau^{*} \in (c^{*},1-c^{*})$, $\lambda_{z(i)z(j)} \to a_{ij,1}^{*}$ and $\delta_{w(i)w(j)} \to a_{ij,2}^{*}$ for all $(i,j) \in \mathcal{K}_0$. Consider the collection of independent Bernoulli random variables $\{A_{ij,l}^{*}: (i,j) \in \mathcal{K}_0, l=1,2\}$ with $E(A_{ij,l}^{*}) = a_{ij,l}^{*}$. Then, (A7) implies $A_{ij,(\lfloor nf \rfloor, n)} \stackrel{\mathcal{D}}{\to} A_{ij,1}^{*}I(f<\tau^{*}) + A_{ij,2}^{*}I(f > \tau^{*})\ \forall (i,j) \in \mathcal{K}_0$. The following Theorem summarizes the asymptotic distribution results. \[lem: b2lse\] Suppose SNR-ER holds. Then, the following statements are true. [**(a)**]{} If $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 \to \infty$, then $\lim_{n \to \infty} P(\hat{\tau}_{n}=\tau_n) =1.$ [**(b)**]{} If (A2) and (A3) hold and $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 \to 0$, then $$\begin{aligned} n||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2 (\hat{\tau}_{n}-\tau_n) \stackrel{\mathcal{D}}{\to} \gamma^2 \arg \max_{h \in \mathbb{R}} (-0.5|h| + B_h),\ \ \end{aligned}$$ where $B_h$ denotes the standard Brownian motion. [**(c)**]{} Suppose (A2), (A4)-(A7) hold and $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to c >0$, then $$\begin{aligned} n (\hat{{\tau}}_{n} -\tau_n) &\stackrel{\mathcal{D}}{\to}& \arg \max_{h \in \mathbb{Z}} (D(h) + C(h) + A(h)) \nonumber $$ where for each ${h} \in \mathbb{Z}$, $$\begin{aligned} D (h+1)-D(h) &=& 0.5 {\rm{Sign}}(-h) c_1^2, \label{eqn: msethm2d}\\ C(h+1) - C(h) &=& \tilde{\gamma} W_{{h}},\ \ W_{{h}} \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1), \ \ \ \ \ \ \ \ \label{eqn: msethm2c}\\ A(h+1) - A(h) &=& \sum_{k \in \mathcal{K}_0} \bigg[(Z_{ij, {h}} -a_{ij,1}^{*})^2 -(Z_{ij,{h}} -a_{ij,2}^{*})^2 \bigg],\ \label{eqn: msethm2a}\end{aligned}$$ $\{Z_{ij, {h}}\}$ are independently distributed with $Z_{ij, {h}} \stackrel{d}{=} A_{ij,1}^{*}I({h} < 0) + A_{ij,2}^{*}I({h} \geq 0)$ for all $(i,j) \in \mathcal{K}_0$. The conclusions in (a)-(c) continue to hold for $\tilde{\tilde{\tau}}_n$ after replacing SNR-ER by SNR-DSBM, (NS) and (A1). The conclusions in (a)-(c) continue to hold for $\tilde{\tau}_n$ after replacing SNR-ER by SNR-DSBM. \[rem: sparsesmall\] As we have already noted, consistency of the change point estimators holds for both dense and sparse graphs. The same conclusion holds for the asymptotic distribution under Regime I. However, (A2) is a crucial assumption for establishing the asymptotic distribution of the change point estimator under Regimes II and III. (A2) implies that the random graph is dense. The different statistical and probabilistic aspects of sparse random graphs constitute a growing area in the recent literature. Most of the results in the sparse setting do not follow from the dense case and different tools and techniques are needed for their analysis; see Remark \[rem: sparse\] for examples. Though the convergence rate results established in Sections \[sec: DSBM\] and \[sec: 2step\] hold for the sparse setting, deriving the asymptotic distribution of the change point estimator under Regimes II and III in sparse random graphs will need separate attention and further investigation. Adaptive Inference {#subsec: ADAP} ------------------ Next, we present a data adaptive procedure that does [*not*]{} require a priori knowledge of the limiting regime. Recall the estimators ${\hat{\tau}}_{n}$, $\hat{\hat{\Lambda}}$, $\hat{\hat{\Delta}}$, $\hat{z}$ and $\hat{w}$ of the parameters in the DSBM model given in (\[eqn: dsbmmodel\]). We generate independent $m \times m$ adjacency matrices $A_{t,n,\text{DSBM}}$, $1 \leq t \leq n$, where $$\begin{aligned} A_{t,n,\text{DSBM}} = ((A_{ij,(t,n),\text{DSBM}} )) \sim \begin{cases}\text{SBM}(\hat{z},\hat{\hat{\Lambda}}),\ \ \text{if $1 \leq t \leq \lfloor n{\hat{\tau}}_{n} \rfloor$} \\ \text{SBM}(\hat{w},\hat{\hat{\Delta}}),\ \ \text{if $\lfloor n{\hat{\tau}}_{n} \rfloor < t <n$}. \label{eqn: dsbmmodeladap} \end{cases}\end{aligned}$$ Obtain $$\begin{aligned} \hat{h}_{\text{DSBM}} = \arg \max_{h \in (n(c^{*}- {\hat{\tau}}_{n}),n(1-c^{*}-{\hat{\tau}}_{n}))} \tilde{L}^{*} ({\hat{\tau}}_{n}+h/n,\hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}})\end{aligned}$$ where $$\begin{aligned} \tilde{L}^* ({\hat{\tau}}_{n}+h/n,\hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) &=& \frac{1}{n}\sum_{i,j=1}^{m} \bigg[\sum_{t=1}^{n{\hat{\tau}}_{n}+h} (A_{ij,(t,n),\text{DSBM}} - \hat{\hat{\lambda}}_{\hat{z}(i),\hat{z}(j)})^2 \nonumber \\ && \hspace{-0 cm}+ \sum_{t=n{\hat{\tau}}_{n}+h+1}^{n} (A_{ij,(t,n),\text{DSBM}} - \hat{\hat{\delta}}_{\hat{w}(i),\hat{w}(j)})^2 \bigg]. \hspace{1 cm}\label{eqn: estimatecccadap}\end{aligned}$$ Theorem \[thm: adapdsbm\] states the asymptotic distribution of $\hat{h}_{\text{DSBM}}$ under a stronger identifiability condition. Specifically, **SNR- ER-ADAP**: $\frac{\sqrt{n}}{m^2 \sqrt{\log m}} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 \to \infty$ It is easy to show that SNR-ER-ADAP holds if all assumptions in Proposition \[example1\] hold and **(AD)** $m = e^{n^{\delta_3}}$ for some $\delta_1,\delta_2,\delta_3>0$ and $0 <\delta_1 + \delta_2 + \delta_3 /2 < 1/2$ ($\delta_1, \delta_2$ are as in Proposition \[example1\]) is satisfied. Specifically, Examples (A)-(D) in Section \[sec: compare\] satisfy SNR-ER-ADAP in the presence of condition (AD). We also need the following condition to ensure that $\hat{z}$ and $\hat{w}$ are consistent estimates for $z$ and $w$, respectively. **(A1-ADAP)** $\frac{Km}{n\nu_m^2} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_2^{-1} \to 0$. Under SNR-ER-ADAP, one can reduce A1-ADAP to $\frac{K}{(n^3\log m)^{1/4}\nu_m^2 } \to O(1)$. This holds whenever within and between community connection probabilities are equal (i.e. $\lambda_{ij} = q_1, \delta_{ij}=q_2\ \forall\ i \neq j$ and $\lambda_{ii}=p_1, \delta_{ii}=p_2\ \forall i$), balanced communities of size $O(m/K)$ are present, and their number is $K = O(m^{2/3})$. This is because the first two conditions implies $\nu_m = O(m/k)$ (for example see Example \[example: misclassnew\]). We also require $\log m = o(\sqrt{n})$, so that the entries of $\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})$ and $\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})$ are bounded away from $0$ and $1$. Note that this assumption implies $0<\delta_3 < 1/2$ in (AD). \[thm: adapdsbm\] **(Asymptotic distribution of $\hat{h}_{\text{DSBM}}$)** Suppose (A2), SNR-ER-ADAP and A1-ADAP hold and $\log m = o(\sqrt{n})$. Then, the following results are true. $(a)$ If $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F \to \infty$, then $\lim_{n \to \infty} P(\hat{h}_{\text{DSBM}}=0) =1$ $(b)$ If (A3) holds and $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to 0$, then $$\begin{aligned} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 \hat{h}_{\text{DSBM}} \stackrel{\mathcal{D}}{\to} \gamma^{-2}\arg \max_{h \in \mathbb{R}} (-0.5|h| + B_h)\ \ \ \ \end{aligned}$$ where $B_h$ corresponds to a standard Brownian motion. $(c)$ If (A4)-(A7) hold and $||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F \to c >0$, then $$\begin{aligned} \hat{h}_{\text{DSBM}} &\stackrel{\mathcal{D}}{\to}& \arg \max_{h \in \mathbb{Z}} (D(h) + C(h) + A(h)), \nonumber \end{aligned}$$ where $D(\cdot)$, $C(\cdot)$ and $A(\cdot)$ are same as (\[eqn: msethm2d\])-(\[eqn: msethm2a\]). The proof of the Theorem is given in Section \[subsec: adapdsbm\]. Note that the asymptotic distribution of $\hat{h}_{n,\text{DSBM}}$ is identical to the asymptotic distribution of ${\hat{\tau}}_{n}$. Therefore, in practice we can simulate $\hat{h}_{n,\text{DSBM}}$ for a large number of replicates and use their empirical quantiles as estimates of the quantiles of the limiting distribution under the (unknown) true regime. Similar conclusions hold for $\tilde{\tilde{\tau}}_{n}$. Moreover, adaptive inference is a computationally expensive procedure and comes at a certain cost, namely the stronger assumption SNR-DSBM-ADAP. Concluding Remarks {#sec:concluding-remarks} ================== In this paper, we have addressed the change point problem in the context of DSBM. We establish consistency of the change point estimator under a suitable identifiability condition and a second condition that controls the misclassification rate arising from using clustering for assigning nodes to communities and discuss the stringency of the latter condition. Further, we propose a fast computational strategy that ignores the underlying community structure, but provides a consistent estimate of the change point. The latter is then used to split the time points into two regimes and solve the community assignment problem for each regime separately. The latter strategy requires a substantially more stringent identifiability condition compared to the first one that utilizes the full structure of the DSBM model. Nevertheless, we provide sufficient conditions for that condition to hold that are rather easy to satisfy when a sufficient number of nodes change community membership, or communities merge/split after the change point. In summary, the proposed strategy proves broadly applicable in numerous practical settings. In addition, this work identifies an interesting issue that requires further research; namely, a range of models where the SNR-DSBM identifiability condition holds, but the misclassification rate condition (A1) needed for the ‘every time point clustering algorithm’ and the identifiability condition (SNR-ER) of the alternative strategy fail to hold. In that range, no general strategy for solving the change point problem for DSBM seems to be currently available. ***Acknowledgment***. We are thankful to Dr. Daniel Sussman for valuable comments and suggestions. Proofs and Other Technical Material {#sec: proofs} =================================== Throughout this section, $C$ is a generic positive constant. Proof of Theorem \[thm: mismiscluster\] {#subsec: mismiscluster} --------------------------------------- Without loss of generality, assume $\tau<b$. By Lemmas $5.1$ and $5.3$ of [@LR2015], with probability tending to $1$, we have $$\begin{aligned} \mathcal{M}_{b,n} &\leq & C\frac{K}{n\nu_m^2} \bigg[ ||\frac{1}{n}\sum_{t=1}^{n\tau}(A_{t,n} - \text{Ed}_{z}(\Lambda)) ||_F^2 + ||\frac{1}{n}\sum_{t=n\tau +1}^{nb}(A_{t,n} -\text{Ed}_{w}(\Delta)) ||_F^2 \nonumber \\ && \hspace{2cm}+ ||\frac{1}{n}\sum_{t=n\tau +1}^{nb} (\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta))||_F^2 \bigg] \nonumber \\ &=& C\frac{K}{n\nu_m^2} (A_1+A_2 + |\tau-b|\ ||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2),\ \text{say}. \nonumber\end{aligned}$$ Now by Theorem $5.2$ of [@LR2015], $A_1, A_2 = O_{\text{P}}(m)$. Thus, $$\begin{aligned} \mathcal{M}_{b,n} = O_{\text{P}}\left(\frac{K}{n\nu_m^2} ( m + |\tau-b|||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2)\right). \nonumber\end{aligned}$$ This completes the proof of Theorem \[thm: mismiscluster\]. Selected useful lemmas ---------------------- The following two lemmas directly quoted from [@Wellner1996empirical] are needed to establish Theorems \[lem: b1\] and \[lem: b2lse\]. \[lem: wvan1\] For each $n$, let $\mathbb{M}_n$ and $\tilde{\mathbb{M}}_n$ be stochastic processes indexed by a set $\mathcal{T}$. Let $\tau_n\ \text{(possibly random)} \in \mathcal{T}_n \subset \mathcal{T}$ and $d_n(b,\tau_n)$ be a map (possibly random) from $\mathcal{T}$ to $[0,\infty)$. Suppose that for every large $n$ and $\delta \in (0,\infty)$ $$\begin{aligned} && \sup_{\delta/2 < d_n(b,\tau_n) < \delta,\ b \in \mathcal{T}} (\tilde{\mathbb{M}}_n(b) - \tilde{\mathbb{M}}_n(\tau_n)) \leq -C\delta^2, \label{eqn: lemcon1} \\ && E\sup_{\delta/2 < d_n(b,\tau_n) < \delta,\ b \in \mathcal{T}} \sqrt{n} |\mathbb{M}_n(b) - \mathbb{M}_n(\tau_n) - (\tilde{\mathbb{M}}_n(b) - \tilde{\mathbb{M}}_n(\tau_n))| \leq C\phi_{n}(\delta), \label{eqn: lemcon2}\end{aligned}$$ for some $C>0$ and for function $\phi_n$ such that $\delta^{-\alpha}\phi_n(\delta)$ is decreasing in $\delta$ on $(0,\infty)$ for some $\alpha <2$. Let $r_n$ satisfy $$\begin{aligned} \label{eqn: lemrn} r_n^2 \phi(r_n^{-1}) \leq \sqrt{n}\ \ \text{for every $n$}.\end{aligned}$$ Further, suppose that the sequence $\{\hat{\tau}_n\}$ takes its values in $\mathcal{T}_n$ and satisfies $\mathbb{M}_n(\hat{\tau}_n) \geq \mathbb{M}_n(\tau_n) - O_P (r_n^{-2})$ for large enough $n$. Then, $r_n d_{n}(\hat{\tau}_n,\tau_n) = O_P (1)$. \[lem: wvandis1\] Let $\mathbb{M}_n$ and $\mathbb{M}$ be two stochastic processes indexed by a metric space $\mathcal{T}$, such that $\mathbb{M}_n \Rightarrow \mathbb{M}$ in $l^{\infty}(\mathcal{C})$ for every compact set $\mathcal{C} \subset \mathcal{T}$ i.e., $$\begin{aligned} \sup_{h \in \mathcal{C}} |\mathbb{M}_n(h) - \mathbb{M}(h)| \stackrel{P}{\to} 0.\end{aligned}$$ Suppose that almost all sample paths $h \to \mathbb{M}(h)$ are upper semi-continuous and possess a unique maximum at a (random) point $\hat{h}$, which as a random map in $\mathcal{T}$ is tight. If the sequence $\hat{h}_n$ is uniformly tight and satisfies $\mathbb{M}_n(\hat{h}_n) \geq \sup_{n} \mathbb{M}_n(h) - o_{P}(1)$, then $\hat{h}_n \stackrel{\mathcal{D}}{\to} \hat{h}$ in $\mathcal{T}$. The following lemma is needed in the proof of Theorem \[thm: adapdsbm\]. \[lem: adaplem\] Suppose SNR-ER-ADAP, A1-ADAP holds and $\log m = o(\sqrt{n})$. Then, the following statements hold. (a) $\frac{||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}}) ||_F^2}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2} \stackrel{\text{P}}{\to} 1$. (b) If $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2 \to 0$, then $$\begin{aligned} \frac{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2 \lambda_{z(i)z(j)}(1-\lambda_{z(i)z(j)})}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2} & \stackrel{\text{P}}{\to} & \gamma^2, \nonumber \\ \frac{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2 \delta_{w(i)w(j)}(1-\delta_{w(i)w(j)})}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2} \stackrel{\text{P}}{\to} \gamma^2. \nonumber\end{aligned}$$ (c) If $||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2 \to c^2>0$, then $$\begin{aligned} \sum_{i,j\in \mathcal{K}_n}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2 \lambda_{z(i)z(j)}(1-\lambda_{z(i)z(j)}) \stackrel{\text{P}}{\to} \tilde{\gamma}^2, \nonumber \\ \sum_{i,j\in \mathcal{K}_n}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2 \delta_{w(i)w(j)}(1-\delta_{w(i)w(j)}) \stackrel{\text{P}}{\to} \tilde{\gamma}^2.\nonumber\end{aligned}$$ We only show the proof of part (a), since parts (b) and (c) follow employing similar arguments. $$\begin{aligned} \bigg|\frac{||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}}) ||_F^2}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2}-1 \bigg| &=& \frac{|||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}}) ||_F^2-||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2|}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2} \nonumber \\ & \leq & \frac{||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) -\text{Ed}_{z}(\Lambda) - \text{Ed}_{\hat{w}}(\hat{\hat{\Delta}}) + \text{Ed}_{w}(\Delta) ||_F^2}{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2} \nonumber \\ & \leq & \frac{||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) -\text{Ed}_{z}(\Lambda)||_F^2 +||\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}}) - \text{Ed}_{w}(\Delta) ||_F^2 }{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta) ||_F^2}\nonumber \end{aligned}$$ Therefore, part (a) follows from Theorem \[thm: c3\], SNR-ER-ADAP, A1-ADAP and $\log m = o(\sqrt{n})$. Proof of Theorem \[lem: b1\] {#subsec: b1} ---------------------------- Throughout this proof, we use the following simplified notation for ease of exposition: $A_{ijt} = A_{ij,(t,n)}, z_1 = \tilde{z}_{b,n}, z_2 = \tilde{z}_{\tau_n,n}, w_1 = \tilde{w}_{b,n}, w_2 = \tilde{w}_{\tau_n,n}$, $\Lambda_1 = \tilde{\Lambda}_{\tilde{z}_{b,n},(b,n)}$, $\Lambda_2 = \tilde{\Lambda}_{\tilde{z}_{\tau_n,n},(\tau,n)}$, $\Lambda_3 = \tilde{\Lambda}_{\tilde{z}_{\tau_n,n},(b,n)}$, $\Delta_1 = \tilde{\Delta}_{\tilde{w}_{b,n},(b,n)}$, $\Delta_2 = \tilde{\Delta}_{\tilde{w}_{\tau_n,n},(\tau_n,n)}$, $\Delta_w = \tilde{\Delta}_{w,(b,n)}$, $\lambda_{uv,1} = \tilde{\lambda}_{uv,\tilde{z}_{b,n},(b,n)}$, $\lambda_{uv,2} = \tilde{\lambda}_{uv,\tilde{z}_{\tau_n,n},(\tau_n,n)}$, $\lambda_{uv,3} = \tilde{\lambda}_{uv,\tilde{z}_{\tau_n,n},(b,n)}$, $\delta_{uv,1} = \tilde{\delta}_{uv,\tilde{w}_{b,n},(b,n)}$, $\delta_{uv,2} = \tilde{\delta}_{uv,\tilde{w}_{\tau_n,n},(\tau_n,n)}$, $\delta_{uv,w} = \tilde{\delta}_{uv,w,(b,n)}$. Suppose $b<\tau_n$. Similar arguments work when $b>\tau_n$. Note that $$\begin{aligned} \tilde{\tilde{\tau}}_n = \arg \min_{b \in (c^*,1-c^*)} \tilde{L}(b,z_1,w_1,\Lambda_1,\Delta_1) \nonumber \end{aligned}$$ where $$\begin{aligned} \tilde{L} (b,z_1,w_1,\Lambda_1,\Delta_1) = \frac{1}{n} \sum_{i,j=1}^{m} \bigg[ \sum_{t=1}^{nb}(A_{ijt}-{\lambda}_{z_1(i)z_1(j),1})^2 + \sum_{t=nb+1}^{n} (A_{ijt}-\hat{\delta}_{w_1(i) w_1(i),1})^2 \bigg].\end{aligned}$$ To prove Theorem \[lem: b1\], we need Lemma \[lem: wvan1\] quoted from [@Wellner1996empirical]. For our purpose, we make use of the above lemma with $\mathbb{M}_n (\cdot) = \tilde{L} (\cdot,\tilde{z}_{\cdot,n},\tilde{w}_{\cdot,n},\tilde{\Lambda}_{\tilde{z}_{\cdot,n},(\cdot,n)},\tilde{\Delta}_{\tilde{w}_{\cdot,n},(\cdot,n)})$, $\tilde{\mathbb{M}}_n(\cdot) = E\tilde{L} (\cdot,\tilde{z}_{\cdot,n},\tilde{w}_{\cdot,n},\tilde{\Lambda}_{\tilde{z}_{\cdot,n},(\cdot,n)},\tilde{\Delta}_{\tilde{w}_{\cdot,n},(\cdot,n)})$, $\mathcal{T} = [0,1]$, $\mathcal{T}_n = \{1/n,2/n,\ldots, (n-1)/n,1\} \cap [c^{*},1-c^{*}]$, $d_n(b,\tau_n) = ||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F \sqrt{|b - \tau_n|}$, $\phi_n(\delta) = \delta$, $\alpha = 1.5$, $r_n = \sqrt{n}$ and $\hat{\tau}_n = \tilde{\tilde{\tau}}_{n}$. Thus, to prove Theorem \[lem: b1\], it suffices to establish that for some $C>0$, $$\begin{aligned} && E(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n)) \leq -C||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 |b - \tau_n|\ \ \text{and} \label{eqn: lsecon1} \\ && E\sup_{\delta/2 < d_n(b,\tau_n) < \delta,\ b \in \mathcal{T}} |\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n) - E(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n))| \leq C\frac{\delta}{\sqrt{n}}. \label{eqn: lsecon2}\end{aligned}$$ As the right side of (\[eqn: lsecon1\]) and (\[eqn: lsecon2\]) are independent of $z_1,z_2,w_1,w_2$, it suffices to show $$\begin{aligned} && E^{*}(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n)) \leq -C||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 |b - \tau_n|\ \ \text{and} \label{eqn: lsecon1f} \\ && E^{*}\sup_{\delta/2 < d_n(b,\tau_n) < \delta,\ b \in \mathcal{T}} |\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n) - E(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n))| \leq C\frac{\delta}{\sqrt{n}}. \label{eqn: lsecon2f}\end{aligned}$$ where $E^{*}(\cdot) = E(\cdot| z_1,w_1,z_2,w_2)$. Similarly denote $V^* = V(\cdot|z_1,z_2,w_1,w_2)$ and $\text{Cov}^{*}(\cdot) = \text{Cov}(\cdot|z_1,z_2,w_1,w_2)$. Note that the left hand side of (\[eqn: lsecon2f\]) is dominated by $$\begin{aligned} \left(E^{*}\sup_{\delta/2 < d_n(b,\tau_n) < \delta,\ b \in \mathcal{T}} (\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n) - E(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n)))^2\right)^{1/2}. \label{eqn: doobprelse}\end{aligned}$$ By Doob’s martingale inequality, (\[eqn: doobprelse\]) is further dominated by $$\begin{aligned} (\text{V}^{*}(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n)))^{1/2}\ \ \text{where $d_n(b,\tau_n) = \delta$}.\end{aligned}$$ Thus, to prove Theorem \[lem: b1\], it suffices to show that for some $C>0$, $$\begin{aligned} \text{V}^{*}(\mathbb{M}_n (b) - \mathbb{M}_n (\tau_n)) \leq Cn^{-1} d_n^2(b,\tau_n). \label{eqn: lsecon2final}\end{aligned}$$ Hence, it sufficies to prove (\[eqn: lsecon1f\]) and (\[eqn: lsecon2final\]) to establish Theorem \[lem: b1\]. We shall prove these for $b<\tau_n$. Similar arguments work when $b \geq \tau_n$. Denote by $L_1 = \tilde{L} (b,z_1,w_1,\Lambda_1,\Delta_1)$ and $L_2 = \tilde{L} (\tau,z_2,w_2,\Lambda_2,\Delta_2)$. Hence, $$\begin{aligned} L_1 - L_2 = A(b) + B(b) + D(b), \label{eqn: abd1}\end{aligned}$$ where $$\begin{aligned} A(b) &=& \frac{1}{n} \sum_{i,j=1}^{m} \sum_{t=1}^{nb} \bigg[(A_{ijt}-\lambda_{z_1(i)z_1(j),1})^2-(A_{ijt} -\lambda_{z_2(i)z_2(j),2})^2 \bigg], \nonumber \\ B(b) &=& \frac{1}{n} \sum_{i,j=1}^{m} \sum_{t=nb+1}^{n\tau} \bigg[(A_{ijt}-\delta_{w_1(i)w_1(j),1})^2 - (A_{ijt}-\delta_{z_2(i)z_2(j),2})^2 \bigg],\nonumber \\ D(b) &=& \frac{1}{n} \sum_{i,j=1}^{m} \sum_{t=n\tau+1}^{n} \bigg[ (A_{ijt}-\delta_{w_1(i)w_1(j),1})^2-(A_{ijt}-\delta_{w_2(i)w_2(j),2})^2\bigg]. \nonumber \end{aligned}$$ Consider the first term of $A(b)$ as follows. $$\begin{aligned} && \frac{1}{nb}\sum_{t=1}^{nb}\sum_{i,j=1}^{m}(A_{ijt}-\lambda_{z_1(i)z_1(j),1})^2 \nonumber \\ &=& \frac{1}{nb}\sum_{t=1}^{nb}\sum_{i,j=1}^{m} A_{ijt}^2 + \sum_{u,v=1}^{K} s_{u,z_1}s_{v,z_1}(\lambda_{uv,1})^2 - \sum_{u,v=1}^{K}s_{u,z_1}s_{v,z_1} (\lambda_{z_1(i)z_1(j),1}) \frac{1}{nb}\sum_{\stackrel{i: z_1(i)=u}{j: z_1(j)=v}} \sum_{t=1}^{nb} A_{ijt} \nonumber \\ &=& \frac{1}{nb}\sum_{t=1}^{nb}\sum_{i,j=1}^{m} A_{ijt}^2 - \sum_{u,v=1}^{K} s_{u,z_1}s_{v,z_1}(\lambda_{uv,1})^2. \nonumber\end{aligned}$$ Similarly, the second term of $A(b)$ is $$\begin{aligned} \frac{1}{nb}\sum_{t=1}^{nb} \sum_{i,j=1}^{m}(A_{ijt} -\lambda_{z_2(i)z_2(j),2})^2 &=& \frac{1}{nb}\sum_{t=1}^{nb}\sum_{i,j=1}^{m} A_{ijt}^2 + \sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}(\lambda_{uv,2})^2 \nonumber \\ && \hspace{3 cm}- 2\sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}\lambda_{uv,2}\lambda_{uv,3}. \nonumber \end{aligned}$$ Therefore, $$\begin{aligned} bE^* (A(b)) &=& - \sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}E^* (\lambda_{uv,2})^2 - b \sum_{u,v=1}^{K} s_{u,z_1}s_{v,z_1}E^* (\lambda_{uv,1})^2 \nonumber \\ && \hspace{3 cm}+ 2\sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}E^{*}(\lambda_{uv,2}\lambda_{uv,3}). \nonumber \end{aligned}$$ Let $S((u,v,f),(a,b,g))$ be the total number of edges which connect communities $u$ and $v$ under community structure $f$, and also communities $a$ and $b$ under community structure $g$. Therefore, $$\begin{aligned} E^* (\lambda_{uv,1})^2 &=& V^{*} (\lambda_{uv,1}) + (E(\lambda_{uv,1}))^2 \nonumber \\ &=& \frac{1}{nb} \frac{1}{(s_{u,z_1}s_{v,z_1})^2} \sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ &&\hspace{1 cm} + \left(\frac{1}{s_{u,z_1}s_{v,z_1}} \sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab} \right)^2, \nonumber \\ E^* (\lambda_{uv,2})^2 &=& V^{*} (\lambda_{uv,2}) + (E(\lambda_{uv,2}))^2 \label{eqn: e2} \\ &=& \frac{1}{n\tau} \frac{1}{(s_{u,z_2}s_{v,z_2})^2} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ &&\hspace{1 cm} + \left(\frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab} \right)^2, \nonumber \\ E^* (\lambda_{uv,2}\lambda_{uv,3}) &=& \text{Cov}^* (\lambda_{uv,2},\lambda_{uv,3}) + (E(\lambda_{uv,2}))(E(\lambda_{uv,3})) \nonumber \\ &=& \frac{1}{n\tau} \frac{1}{(s_{u,z_2}s_{v,z_2})^2} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ &&\hspace{1 cm} + \left(\frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab} \right)^2. \nonumber \end{aligned}$$ Hence, $$\begin{aligned} bE^*(A(b)) &=& b(A_1(b) + A_2(b)) \nonumber \end{aligned}$$ where $$\begin{aligned} A_1(b) &=& -\sum_{u,v=1}^{K} \frac{1}{nb} \frac{1}{s_{u,z_1}s_{v,z_1}} \sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ && + \sum_{u,v=1}^{K} \frac{1}{n\tau} \frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}(1-\lambda_{ab}), \nonumber \\ A_2(b) &=& -\sum_{u,v=1}^{K} \frac{1}{s_{u,z_1}s_{v,z_1}} \left(\sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab}\right)^2 \nonumber \\ && + \sum_{u,v=1}^{K} \frac{1}{s_{u,z_2}s_{v,z_2}} \left(\sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}\right)^2. \label{eqn: v10a2}\end{aligned}$$ Note that $$\begin{aligned} A_1(b) &\geq & -\sum_{u,v=1}^{K} \left(\frac{1}{nb} - \frac{1}{n\tau}\right)\frac{1}{s_{u,z_1}s_{v,z_1}} \sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \label{eqn: com1} \\ && -\sum_{u,v=1}^{K} \frac{1}{n\tau}\frac{1}{s_{u,z_1}s_{v,z_1}} \sum_{a,b=1}^{K} S((u,v,z_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ && + \sum_{u,v=1}^{K} \frac{1}{n\tau} \frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ &\geq & -C \left(\frac{1}{nb} - \frac{1}{n\tau}\right)K^2 \nonumber \\ &&-C\sum_{u,v=1}^{K} \frac{1}{n} \frac{1}{s_{u,z_1}s_{v,z_1}} \sum_{(a,b)\neq (u,v)} S((u,v,z_1),(a,b,z)) \nonumber \\ && - C\sum_{u,v=1}^{K} \frac{1}{n} \frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{(a,b)\neq (u,v)} S((u,v,z_2),(a,b,z))\nonumber \\ && - C\frac{1}{n}\sum_{u,v=1}^{K} (S((u,v,z_1),(u,v,z))) |(s_{u,z_1}s_{v,z_1})^{-1} - (s_{u,z}s_{v,z})^{-1}| \nonumber \\ && - C\frac{1}{n}\sum_{u,v=1}^{K} (S((u,v,z_2),(u,v,z))) |(s_{u,z_2}s_{v,z_2})^{-1} - (s_{u,z}s_{v,z})^{-1}| \nonumber \\ && -C\frac{1}{n}\sum_{u,v=1}^{K} (s_{u,z}s_{v,z})^{-1} |(S((u,v,z_1),(u,v,z))) - (S((u,v,z_2),(u,v,z))| \nonumber \\ & \geq & -C(\tau-b)\frac{K^2}{n} - C(\tau-b)\mathcal{M}_{b,n}^2. \end{aligned}$$ Further, $$\begin{aligned} A_2(b) &\geq & -C\sum_{u,v=1}^{K} \frac{1}{s_{u,z_1}s_{v,z_1}} \left(\sum_{(a,b) \neq (u,v)} S((u,v,z_1),(a,b,z))\right)^2 \nonumber \\ && - C\sum_{u,v=1}^{K} \frac{1}{s_{u,z_1}s_{v,z_1}} \left( S((u,v,z_1),(u,v,z))\right)^2 \nonumber \\ && -C\sum_{u,v=1}^{K} \frac{1}{s_{u,z_2}s_{v,z_2}} \left(\sum_{(a,b) \neq (u,v)} S((u,v,z_2),(a,b,z))\right)^2 \nonumber \\ && - C\sum_{u,v=1}^{K} \frac{1}{s_{u,z_2}s_{v,z_2}} \left( S((u,v,z_2),(u,v,z))\right)^2 \nonumber \\ & \geq & -C(\tau-b)n^2 \mathcal{M}_{b,n}^2 - C\sum_{u,v=1}^{K} (S((u,v,z_1),(u,v,z)))^2 |(s_{u,z_1}s_{v,z_1})^{-1} - (s_{u,z}s_{v,z})^{-1}| \nonumber \\ && - C\sum_{u,v=1}^{K} (S((u,v,z_2),(u,v,z)))^2 |(s_{u,z_2}s_{v,z_2})^{-1} - (s_{u,z}s_{v,z})^{-1}| \nonumber \\ && -C\sum_{u,v=1}^{K} (s_{u,z}s_{v,z})^{-1} |(S((u,v,z_1),(u,v,z)))^2 - (S((u,v,z_2),(u,v,z))^2| \nonumber \\ & \geq & -C(\tau-b)n^2 \mathcal{M}_{b,n}^2. \label{eqn: v10a21}\end{aligned}$$ This proves $$\begin{aligned} E^*(A(b)) \geq -C(\tau-b) (\frac{K^2}{n}+n^2\mathcal{M}_{b,n}^2). \label{eqn: compa}\end{aligned}$$ Next, consider $B(b)$. Define $$\begin{aligned} \mu_{uv,1} &=& \frac{1}{n(\tau-b)}\sum_{t=nb+1}^{n\tau} \frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{\stackrel{i: z_2(i)=u}{j: z_2(j)=v}} A_{ijt}, \nonumber \\ \mu_{uv,2} &=& \frac{1}{n(\tau-b)}\sum_{t=nb+1}^{n\tau} \frac{1}{s_{u,w_1}s_{v,w_1}} \sum_{\stackrel{i: w_1(i)=u}{j: w_1(j)=v}} A_{ijt}. \nonumber \end{aligned}$$ Note that $$\begin{aligned} B(b) &=& \frac{1}{n} \sum_{t=nb+1}^{n\tau}\sum_{i,j=1}^{m}\bigg[(A_{ijt}-\delta_{w_1(i)w_1(j),1})^2 - (A_{ijt}-\lambda_{z_2(i)z_2(j),2})^2 \bigg] \nonumber \\ &=& \frac{1}{n} \sum_{t=nb+1}^{n\tau} \sum_{i,j=1}^{m} \bigg[ (\delta_{w_1(i)w_1(j),1})^2 - (\lambda_{z_2(i)z_2(j),2})^2 - 2 A_{ijt} (\delta_{w_1(i)w_1(j),1}) + 2A_{ijt}(\lambda_{z_2(i)z_2(j),2})) \bigg] \nonumber \\ &=& (\tau-b) \sum_{u,v=1}^{K} (s_{u,w_1}s_{v,w_1} (\delta_{uv,1})^2 - s_{u,z_2}s_{v,z_2}(\lambda_{uv,2})^2 -2\mu_{uv,2}\delta_{uv,1} + 2\mu_{uv,1}\lambda_{uv,2} ). \nonumber \end{aligned}$$ Therefore, $$\begin{aligned} E^{*}(B(b)) &=& B_1(b) + B_2(b) \label{eqn: compb4}\end{aligned}$$ where $$\begin{aligned} B_1(b) &=& (\tau-b) \sum_{u,v=1}^{K} \bigg[ s_{u,w_1}s_{v,w_1} V^{*}(\delta_{uv,1}) - s_{u,z_2}s_{v,z_2}V^{*}(\lambda_{uv,2}) - 2s_{u,w_1}s_{v,w_1} \text{Cov}^{*}(\mu_{uv,2},\delta_{uv,1}) \nonumber \\ && \hspace{9 cm}+ 2s_{u,z_2}s_{v,z_2}\text{Cov}^{*}(\mu_{uv,1},\lambda_{uv,2}) \bigg], \nonumber \\ B_2(b) &=& (\tau-b) \sum_{u,v=1}^{K} \bigg[s_{u,w_1}s_{v,w_1} (E^{*}(\delta_{uv,1}))^2 - s_{u,z_2}s_{v,z_2}(E^{*}(\lambda_{uv,2}))^2 - 2s_{u,w_1}s_{v,w_1} E^{*}(\mu_{uv,2})E^{*}(\delta_{uv,1}) \nonumber \\ && \hspace{9 cm}+ 2s_{u,z_2}s_{v,z_2}E^{*}(\mu_{uv,1})E^{*}(\lambda_{uv,2}) \bigg] \nonumber \\ &=& (\tau-b) (B_{21} + B_{22} + B_{23} + B_{24}). \label{eqn: compb3}\end{aligned}$$ Now, $V^{*}(\lambda_{uv,2})$ is given in (\[eqn: e2\]) and $$\begin{aligned} V^{*}(\delta_{uv,1}) &=& \frac{1}{(n(1-b))^2} \bigg[n(\tau-b) \frac{1}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab}) \nonumber \\ && \hspace{2 cm}+ n(1-\tau)\frac{1}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,w))\delta_{ab}(1-\delta_{ab}) \bigg], \nonumber \\ \text{Cov}^{*}(\mu_{uv,2},\delta_{uv,1}) &=& \frac{1}{n^2(\tau-b)(1-b)} \bigg[n(\tau-b) \frac{1}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,z))\lambda_{ab}(1-\lambda_{ab})\bigg], \nonumber \\ \text{Cov}^{*}(\mu_{uv,1},\lambda_{uv,2}) &=& \frac{1}{n^2(\tau-b)\tau} \bigg[n(\tau-b) \frac{1}{s_{u,z_2}s_{v,z_2}} \sum_{a,b=1}^{K} S((u,v,z_2),(a,b,z))\lambda_{ab}(1-\lambda_{ab})\bigg]. \nonumber \end{aligned}$$ Using similar calculations as in (\[eqn: com1\]), we obtain $$\begin{aligned} B_1(b) \geq -C(\tau-b)\left( \frac{K^2}{n} + \mathcal{M}_{b,n}^2\right). \label{eqn: compb5}\end{aligned}$$ Next, consider $B_2(b)$. $$\begin{aligned} B_{21} & = & \sum_{u,v=1}^{K} (s_{u,w_1}s_{v,w_1}-s_{u,w}s_{v,w}) (E^{*}(\delta_{uv,1}))^2 + \sum_{u,v=1}^{K} s_{u,w}s_{v,w} ((E^*(\delta_{uv,1}))^2 - (E^*(\delta_{uv,w}))^2) \nonumber \\ && \hspace{9 cm} + \sum_{u,v=1}^{K} s_{u,w}s_{v,w} (E^*(\delta_{uv,w}))^2 \nonumber \\ & \geq & -C\mathcal{M}_{b,n}^2 - C\sum_{u,v=1}^{K} s_{u,w}s_{v,w} |(E^*(\delta_{uv,1})) - (E^*(\delta_{uv,w}))| + \sum_{u,v=1}^{K}s_{u,w}s_{v,w} \delta_{uv}^2 \nonumber \\ & \geq & -C\mathcal{M}_{b,n}^2 - B_{211} + \sum_{u,v=1}^{K}s_{u,w}s_{v,w} \delta_{uv}^2. \label{eqn: compb1}\end{aligned}$$ We then get $$\begin{aligned} B_{211} = && \sum_{u,v=1}^{K} s_{u,w}s_{v,w} |(E^*(\delta_{uv,1})) - (E^*(\delta_{uv,w}))| \nonumber \\ & \leq & \sum_{u,v=1}^{K} s_{u,w}s_{v,w} \bigg | \frac{1}{n(1-b)} \bigg[ n(\tau-b) \frac{1}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,z))\lambda_{ab} \nonumber \\ && + n(1-\tau) \frac{1}{s_{u,w_1}s_{v,w_1}}\sum_{a,b=1}^{K} S((u,v,w_1),(a,b,w))\delta_{ab} \bigg] - \delta_{uv} \bigg | \nonumber \\ & \leq & C\frac{\tau-b}{1-b} \bigg| \sum_{u,v=1}^{K} \bigg[\frac{s_{u,w}s_{v,w}}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,z))\lambda_{ab} - s_{u,w}s_{v,w} \delta_{uv}\bigg] \bigg| \nonumber \\ && + C\frac{1-\tau}{1-b} \bigg| \sum_{u,v=1}^{K} \bigg[\frac{s_{u,w}s_{v,w}}{s_{u,w_1}s_{v,w_1}} \sum_{a,b=1}^{K} S((u,v,w_1),(a,b,w))\delta_{ab} - s_{u,w}s_{v,w} \delta_{uv}\bigg] \bigg| \nonumber \\ & = & B_{211a} + B_{211b},\ \ \text{say}. \label{eqn: compb2}\end{aligned}$$ It is easy to see that $$\begin{aligned} B_{211a} &\leq & C\sum_{u,v=1}^{K} \bigg|\frac{s_{u,w}s_{v,w}}{s_{u,w_1}s_{v,w_1}}-1\bigg| s_{u,w_1}s_{v,w_1} + C\sum_{u,v=1}^{K}\sum_{a,b=1}^{K} |S((u,v,w_1),(a,b,z))-S((u,v,w),(a,b,z))| \nonumber \\ &\leq & C\mathcal{M}_{b,n}^2. \nonumber \end{aligned}$$ Similarly, $B_{211b} \leq C\mathcal{M}_{b,n}^2$. Thus, by (\[eqn: compb1\]) and (\[eqn: compb2\]), we get $$\begin{aligned} B_{21} \geq -C\mathcal{M}_{b,n}^2 +\sum_{u,v=1}^{K}s_{u,w}s_{v,w}\delta_{uv}^2. \nonumber \end{aligned}$$ Using similar arguments as above, we also have $$\begin{aligned} B_{22} &\geq & -C\mathcal{M}_{b,n}^2 -\sum_{u,v=1}^{K}s_{u,z}s_{v,z}\lambda_{uv}^2, \nonumber \\ B_{23} &\geq & -C\mathcal{M}_{b,n}^2 - 2\sum_{u,v=1}^{K} s_{u,w}s_{v,w}\delta_{uv} \sum_{a,b=1}^{K}S((u,v,w), (a,b,z))\lambda_{ab} \nonumber \\ & \geq & -C\mathcal{M}_{b,n}^2 - 2 \sum_{i,j=1}^{m} \lambda_{z(i)z(j)}\delta_{w(i)w(j)}, \nonumber \\ B_{24} & \geq & -C\mathcal{M}_{b,n}^2 + 2\sum_{u,v=1}^{K}s_{u,z}s_{v,z}\lambda_{uv}^2. \nonumber\end{aligned}$$ Hence, by (\[eqn: compb3\]) $$\begin{aligned} B_2(b) \geq -C(\tau-b)\mathcal{M}_{b,n}^2 + C(\tau-b)||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^2. \nonumber \end{aligned}$$ Consequently, by (\[eqn: compb4\]) and (\[eqn: compb5\]), we have $$\begin{aligned} E^{*}(B(b)) \geq -C(\tau-b)\frac{K^2}{n} -C(\tau-b)\mathcal{M}_{b,n}^2 + C(\tau-b)||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^2. \label{eqn: compbb}\end{aligned}$$ Recall $D(b)$ in (\[eqn: abd1\]). Similar arguments as above also lead us to conclude $$\begin{aligned} E^*(D(b)) \geq -C(\tau-b)\frac{K^2}{n} -C(\tau-b)n^2\mathcal{M}_{b,n}^2. \label{eqn: compd}\end{aligned}$$ Hence by (\[eqn: abd1\]), (\[eqn: compa\]), (\[eqn: compbb\]) and (\[eqn: compd\]), we have $$\begin{aligned} E^* (L_1-L_2) &\geq & -C(\tau-b)\frac{K^2}{n} -C(\tau-b)n^2\mathcal{M}_{b,n}^2 + C(\tau-b)||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^2 \nonumber \\ &\geq & -C(\tau-b)||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^2, \ \ \text{by SNR-DSBM and (A1)}. \label{eqn: compe}\end{aligned}$$ This proves (\[eqn: lsecon1f\]). Next, we compute variances. By (\[eqn: abd1\]), $$\begin{aligned} V^{*}(L_1-L_2) = V^{*}(A(b)) + V^{*}(B(b)) + V^{*}(D(b)). \nonumber\end{aligned}$$ We only show the computation for $V^{*}(A(b))$. Other terms can be handled similarly. $$\begin{aligned} V^* (A(b)) & \leq & C \sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}V^* (\lambda_{uv,2})^2 + C \sum_{u,v=1}^{K} s_{u,z_1}s_{v,z_1}V^* (\lambda_{uv,1})^2 \nonumber \\ && \hspace{3 cm}+ 2C\sum_{u,v=1}^{K} s_{u,z_2}s_{v,z_2}V^{*}(\lambda_{uv,2}\lambda_{uv,3}). \label{eqn: vara1}\end{aligned}$$ Let $\text{Cum}_r(X)$ denote the $r$-th order cumulant of $X$. Then, $$\begin{aligned} V^* (\lambda_{uv,2})^2 & \leq & \frac{C}{n^4 (s_{u,z_2}s_{v,z_2})^4}E \bigg[\sum_{t=1}^{n\tau} \sum_{\stackrel{i: z_2(i)=u}{j: z_2(j)=v}} (A_{ijt} -EA_{ijt}) \bigg]^4 \nonumber \\ & \leq & \frac{C}{n^4 (s_{u,z_2}s_{v,z_2})^4} \bigg[ \sum_{t=1}^{n\tau} \sum_{\stackrel{i: z_2(i)=u}{j: z_2(j)=v}} \text{Cum}_4(A_{ijt} -EA_{ijt}) \nonumber \\ && + \bigg( \sum_{t=1}^{n\tau} \sum_{\stackrel{i: z_2(i)=u}{j: z_2(j)=v}} \text{Cum}_4(A_{ijt} -EA_{ijt})\bigg)\bigg( \sum_{t=1}^{n\tau} \sum_{\stackrel{i: z_2(i)=u}{j: z_2(j)=v}} \text{Cum}_4(A_{ijt} -EA_{ijt})\bigg)\bigg] \nonumber \\ & \leq & \frac{C}{n^2(s_{u,z_2}s_{v,z_2})^2}. \nonumber \end{aligned}$$ Similarly, $V^* (\lambda_{uv,1}) \leq \frac{C}{n^2(s_{u,z_1}s_{v,z_1})^2}$ and $V^{*}(\lambda_{uv,2}\lambda_{uv,3}) \leq \frac{C}{n^2(s_{u,z_2}s_{v,z_2})^2}$. Hence, $$\begin{aligned} V^{*}(A(b)) \leq C\frac{K^2}{n^2} \leq C(\tau-b)\frac{K^2}{n}.\nonumber \end{aligned}$$ Using similar arguments as above, we also have $$\begin{aligned} V^{*}(B(b)), V^{*}(D(b)) \leq C\frac{K^2}{n^2} \leq C(\tau-b)\frac{K^2}{n}.\nonumber \end{aligned}$$ Hence, $$\begin{aligned} V^{*}(L_1-L_2) \leq C\frac{K^2}{n^2} \leq C(\tau-b)\frac{||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^2}{n}.\label{eqn: Varcal1}\end{aligned}$$ This proves (\[eqn: lsecon2final\]). Therefore, by Lemma \[lem: wvan1\] the proof of Theorem \[lem: b1\] is complete. \[rem: proof\] Following the proof of Theorem \[lem: b1\] (see (\[eqn: compe\])), it is easy to see that Assumption (A9) $n^2\mathcal{M}_{b,n}^2 ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-2} \to 0\ \forall b\in (c^*,1-c^*)$ on the misclassification rate due to clustering is required for achieving consistency of $\tilde{\tilde{\tau}}_n$. The rate of $\mathcal{M}_{b,n}$ varies for different clustering procedures. For Clustering Algorithm I presented in Section \[sec: DSBM\], the rate of $\mathcal{M}_{b,n}^2$ is given in Theorem \[thm: mismiscluster\] and hence (A9) reduces to (A1). Details are given before stating (A1). Two variants together with Assumption (A9) and their corresponding misclassification error rates have also been presented and discussed in Section \[sec:discuss\]. Note that Assumption (A9) is needed when used in conjunction with the every time point algorithm. However, the assumption can be weakened if we only cluster the nodes once before and after the change pont. Note that we assume that the change point lies in the interval $ (c^*,1-c^*)$, which implies that we can cluster nodes using all time points before $c$ and obtain $z$ and similarly cluster nodes using all time points after ($1-c^*$ to obtain $w$. Then, $A_2(b) = 0$ and as a consequence $E^*(A(b)) \geq -C(\tau-b) (\frac{K^2}{n}+\mathcal{M}_{b,n}^2)$ holds which is a sharper lower bound for $E^*(A(b))$ than the one provided in (\[eqn: compa\]). Analogously, for $w$ we get $E^*(D(b)) \geq -C(\tau-b) (\frac{K^2}{n}+\mathcal{M}_{b,n}^2)$, This provides $E^* (L_1-L_2) \geq -C(\tau-b)\frac{K^2}{n} -C(\tau-b)\mathcal{M}_{b,n}^2 + C(\tau-b)||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^2$ and a weaker version (A9\*) is needed \[$\mathcal{M}_{b,n}^2 ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{-2} \to 0\ \forall \ b\in (c^*,1-c^*)$ (instead of (A9))\] along with SNR-DSBM to establish (\[eqn: lsecon1f\]). Proof of Theorem \[thm: c3\] {#subsec: c3} ---------------------------- Next, we focus on establishing the convergence rate for $\tilde{\tilde{\Lambda}}$, while analogous arguments are applicable for $\tilde{\tilde{\Delta}}$. Without loss of generality, assume $\tilde{\tilde{\tau}}_n > \tau_n$. For some clustering function $f$ and $b \in (c^{*},1-c^{*})$, recall that $\tilde{\lambda}_{uv,f,(b,n)} = \frac{1}{nb}\sum_{t=1}^{nb} \frac{1}{s_{u,f}s_{v,f}} \sum_{\stackrel{f(i)=u}{f(j)=v}} A_{ij,(t,n)}$. For some $C>0$, we have $$\begin{aligned} ||\text{Ed}_{\tilde{\tilde{z}}}(\tilde{\tilde{\Lambda}}) - \text{Ed}_{z}(\Lambda)||_F^2 &=& \sum_{i,j=1}^{m}(\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tilde{\tilde{\tau}}_n,n)} - \lambda_{z(i)z(j)})^2 \nonumber \\ & \leq & 2\sum_{i,j=1}^{m}(\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tilde{\tilde{\tau}}_n,n)} -\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)})^2 + 2\sum_{i,j=1}^{m} (\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)} - \lambda_{z(i)z(j)})^2 \nonumber \\ & \leq & Cm^2(\hat{\hat{\tau}}_n - \tau_n)^2 + C\sum_{i,j=1}^{m} (\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)} - E^{*}\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)})^2 \nonumber \\ && \hspace{3.5 cm} C\sum_{i,j=1}^{m} (E^{*}\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)}-\lambda_{z(i)z(j)})^2 \nonumber \\ &=& \mathbb{T}_1 + \mathbb{T}_2 + \mathbb{T}_3\ \ \text{(say)}. \nonumber \end{aligned}$$ Note that by Theorem \[lem: b1\] we have $m^{-2}\mathbb{T}_1 = O_{\text{P}}(I(n>1)n^{-2}||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta) ||_F^{-4})$. Let $P^{*}(\cdot) = P(\cdot|\tilde{\tilde{z}},\tilde{\tilde{w}})$. By the sub-Gaussian property of Bernoulli random variables and since for some positive sequence $\{\tilde{C}_n\}$, $\hat{\mathcal{S}}_n \geq \tilde{C}_n\ \forall \ n$ with probability $1$, we get $$\begin{aligned} P^{*}(m^{-2}\mathbb{T}_2 \geq t) &=& P^{*}\left(\frac{1}{m^2}\sum_{i,j=1}^{m} (\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)} - E^{*}\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)})^2 \geq t \right) \nonumber \\ & \leq & \sum_{i,j=1}^{m} P^{*} \left(|\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)} - E^{*}\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)}| \geq C\sqrt{t} \right) \nonumber \\ & \leq & m^2 C_1 e^{-C_2 n \hat{\mathcal{S}}_n^2 t} \leq m^2 C_1 e^{-C_2 n \tilde{C}_n^2 t}. \nonumber\end{aligned}$$ Therefore, $P\left(m^{-2}\mathbb{T}_2 \geq t \right) \leq m^2 C_1 e^{-C_2 n \tilde{C}_n t} \to 0$ for $t = \frac{\log m}{n \tilde{C}_n^2}$. Hence, $m^{-2} \mathbb{T}_2 = O_{\text{P}} \left( \frac{\log m}{n \tilde{C}_n^2} \right)$. Finally, $$\begin{aligned} m^{-2}\mathbb{T}_3 &=& \frac{1}{m^2}\sum_{i,j=1}^{m} (E^{*}\tilde{\lambda}_{\tilde{\tilde{z}}(i)\tilde{\tilde{z}}(j),\tilde{\tilde{z}},(\tau_n,n)}-\lambda_{z(i)z(j)})^2 \nonumber \\ &=& \frac{1}{m^2}\sum_{i,j=1}^{m} \bigg( \frac{1}{s_{\tilde{\tilde{z}}(i),\tilde{\tilde{z}}(j)}s_{\tilde{\tilde{z}}(j),\tilde{\tilde{z}}}} \sum_{a,b=1}^{K} S((\tilde{\tilde{z}}(i),\tilde{\tilde{z}}(j),\tilde{\tilde{z}}),(a,b,z)) (\lambda_{ab} -\lambda_{z(i)z(j)})\bigg)^2 \nonumber \\ & \leq & C\mathcal{M}_{\tilde{\tilde{\tau}}_n,n}^2 = O_{\text{P}} \left( \left(\frac{Km}{n\nu_m^2}\right)^2\right). \nonumber \end{aligned}$$ Thus, combining the convergence rate of $\mathbb{T}_1$, $\mathbb{T}_2$ and $\mathbb{T}_3$ derived above, establishes the convergence rate of $\text{Ed}_z(\tilde{\tilde{\Lambda}})$ when $\tilde{\tilde{\tau}}_n > \tau_n$. Similar arguments work for $\tilde{\tilde{\tau}}_n \leq \tau_n$. This completes the proof of Theorem \[thm: c3\]. Proof of Theorem \[thm: 2step\] {#subsec: 2step} ------------------------------- To prove Theorem \[thm: 2step\], note that the proof of Theorem \[lem: b1\] in Section \[subsec: b1\] goes through once we use $\mathcal{M}_{b,n} =0$, $z_1=z_2=z$, $w_1=w_2=w$ and $K=m$. In this case, (\[eqn: compe\]) and (\[eqn: Varcal1\]) implies $$\begin{aligned} E^{*}(L_1-L_2) & \geq & - C(\tau-b)\frac{m^2}{n} + C(\tau-b)||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2, \nonumber \\ V^{*}(L_1-L_2) &\leq & C(\tau-b)\frac{m^2}{n}. \nonumber\end{aligned}$$ Therefore, by SNR-ER and Lemma \[lem: wvan1\], Theorem \[thm: 2step\] follows. Justification of (\[eqn: c2rem\]) {#subsec: remc2} --------------------------------- Let $\mathcal{L}(A)$ denote the Laplacian of $A$. Also without loss of generality, assume $b >\tau$. Using similar arguments as in Appendix B, C and D of [@RCY2011], we can easily show that for some $C > 0$ and with probability tending $1$, $$\begin{aligned} \mathcal{M}_{b,n} &\leq & C\frac{P_n}{\xi_{K_n}^4} ||(\mathcal{L}(\frac{1}{n}\sum_{t=1}^{nb}A_{t,n}))^2 - (\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2||_F^2 \nonumber \\ & \leq & C\frac{P_n}{\xi_{K_n}^4}\bigg( ||(\mathcal{L}(\frac{1}{n}\sum_{t=1}^{n\tau}A_{t,n}))^2 - (\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2||_F^2 + ||(\mathcal{L}(\frac{1}{n}\sum_{t=n\tau+1}^{nb}A_{t,n}))^2 - (\mathcal{L}(\text{Ed}_{z}(\Delta)))^2||_F^2 \nonumber \\ && \hspace{2 cm}+ |\tau-b|\ ||(\mathcal{L}(\text{Ed}_{z}(\Delta)))^2 - (\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2 ||_F^2 \nonumber \\ &=& C\frac{P_n}{\xi_{K_n}^4} (A_1 +A_2+A_3),\ \text{say}. \end{aligned}$$ Then,using similar arguments as in Lemma $A.1$ of [@RCY2011], we obtain $$\begin{aligned} A_1, A_2 &=& O_{\text{P}}(\frac{(\log m)^2}{mn}). \nonumber\end{aligned}$$ Define, $$\begin{aligned} \mathcal{D}_{i,\Lambda} = \sum_{j=1}^{m}\lambda_{z(i)z(j)},\ \ \mathcal{D}_{\Lambda} = \text{Diag}\{\mathcal{D}_{i,\Lambda}:\ 1 \leq i \leq m\}, \nonumber \\ \mathcal{D}_{i,\Delta} = \sum_{j=1}^{m}\delta_{w(i)w(j)},\ \ \mathcal{D}_{\Delta} = \text{Diag}\{\mathcal{D}_{i,\Delta}:\ 1 \leq i \leq m\}. \nonumber\end{aligned}$$ Then, $$\begin{aligned} A_3 &\leq & Cm||\mathcal{L}(\text{Ed}_{z}(\Lambda)) - \mathcal{L}(\text{Ed}_{w}(\Delta))||_F^2 \nonumber \\ &\leq & Cm ||\mathcal{D}_{\Lambda}^{-1/2}\text{Ed}_{z}(\Lambda) \mathcal{D}_{\Lambda}^{-1/2} - \mathcal{D}_{\Delta}^{-1/2} \text{Ed}_{z}(\Delta)\mathcal{D}_{\Delta}^{1/2} ||_F^2\nonumber \\ & \leq & Cm \bigg[ ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2 ||\mathcal{D}_{\Lambda}^{1/2}||_F^4 + 2 ||\mathcal{D}_{\Lambda}^{-1/2} - \mathcal{D}_{\Delta}^{-1/2}||_F^2||\text{Ed}_{z}(\Delta)||_F^2||\mathcal{D}_{\Delta}^{-1/2}||_F^2 \bigg] \nonumber \\ & \leq & Cm (||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2 + \frac{C}{m} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2 m^2) \nonumber \\ & \leq & Cm^2||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2. \nonumber\end{aligned}$$ Hence, $$\begin{aligned} \mathcal{M}_{b,n} = O_{\text{P}}\left(\frac{P_n}{\xi_{K_n}^4} \left( \frac{(\log m)^2}{mn} +|\tau-b| m^2 ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2\right) \right). \nonumber\end{aligned}$$ This completes the justification of (\[eqn: c2rem\]). Justification of (\[eqn: remc3\]) {#subsec: remc3} --------------------------------- Using similar arguments to those presented in Section \[subsec: remc2\], with probability tending to $1$, we have $$\begin{aligned} \mathcal{M}_{b,n} &\leq & C \frac{P_n}{\xi_{K_n}^4} ||(L_{\Lambda,(b,n)})^2-(\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2 ||_F^2 \nonumber \\ & \leq & C \frac{P_n}{\xi_{K_n}^4} \bigg[||(L_{\Lambda,(\tau,n)})^2-(\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2 ||_F^2 + \frac{1}{n}\sum_{t=n\tau +1}^{nb} ||(\mathcal{L}(A_{t,n}))^2-(\mathcal{L}(\text{Ed}_{w}(\Delta)))^2 ||_F^2 \nonumber \\ & & \hspace{2 cm} +|\tau-b| ||(\mathcal{L}(\text{Ed}_{z}(\Delta)))^2 - (\mathcal{L}(\text{Ed}_{z}(\Lambda)))^2 ||_F^2\bigg] \nonumber \\ &\leq & C \frac{P_n}{\xi_{K_n}^4} (A_1+A_2 + |\tau - b| m^2 ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{z}(\Delta)||_F^2), \ \text{say}. \nonumber \end{aligned}$$ Then, by Theorem $2.1$ in [@RCY2011], we have $$\begin{aligned} A_1 = O_{\text{P}}(\frac{(\log m\sqrt{n})^2}{m\sqrt{n}})\ \ \text{and}\ \ A_2 = O_{\text{P}}(|\tau-b|\frac{(\log m)^2}{m}). \nonumber\end{aligned}$$ Hence, $$\begin{aligned} \nonumber \mathcal{M}_{b,n} = O_{\text{P}}\left(\frac{P_n}{\xi_{K_n}^4} \left(\frac{(\log m\sqrt{n})^2}{\sqrt{n}m} + m^2|\tau-b| ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^{2} + |\tau-b| \frac{(\log m)^2}{m}\right)\right). \end{aligned}$$ This completes the ustification of (\[eqn: remc3\]). justification of Example \[example: v10new4\] {#subsec: examplev10new4} --------------------------------------------- It is easy to see that the following results (a)-(d) hold under the assumptions in Example \[example: v10new4\]. (a) $\gamma_j, \delta_j = O_{\text{P}}(\frac{1}{\sqrt{n}})$ when $j \in B_{1z}\cap B_{1w}$. (b) $\gamma_j - \frac{b-\tau}{b} (a_2-d_2),\ \ \delta_j - (a_2-d_2),\ \ \frac{\gamma_j}{\delta_j} - \frac{b-\tau}{b} = O_{\text{P}}(\frac{1}{\sqrt{n}})$ when $j \in B_{1z} \cap B_{1w}^{c}$. (c) $\gamma_j - \frac{\tau}{b}(a_1-d_1), \ \ \delta_j = O_{\text{P}}(\frac{1}{\sqrt{n}})$ when $j \in B_{1z}^{c}\cap B_{1w}$. (d) $\gamma_j - (a_1-d_1),\ \ \delta_j - (a_1-d_1),\ \ \frac{\gamma_j}{\delta_j} -1 = O_{\text{P}}(\frac{1}{\sqrt{n}})$ when $j \in B_{1z}^c \cap B_{1w}^{c}$. Using the above results, we have (a) $P(j\ \text{is classified in}\ B_{1z} \cap B_{2w}\ |\ j \in B_{1z} \cap B_{2w})\ \leq \ P(\gamma_j < \frac{B}{\sqrt{n^{\delta}}}, \delta_j < \frac{B}{\sqrt{n^{\delta}}}\ |\ j \in B_{1z} \cap B_{2w}) \to 1$. (b) $P(j\ \text{is classified in}\ B_{1z}^{c} \cap B_{2w}\ |\ j \in B_{1z}^{c} \cap B_{2w}) \leq P(\gamma_j > \frac{B}{\sqrt{n^{\delta}}}, \delta_j < \frac{B}{\sqrt{n^{\delta}}}\ |\ j \in B_{1z}^{c} \cap B_{2w}) + P(\frac{\gamma_j}{\delta_j} > 1 + \frac{B^{*}}{\sqrt{n^\delta}}\ |\ j \in B_{1z}^{c} \cap B_{2w}) \to 1$. (c) $P(j\ \text{is classified in}\ B_{1z} \cap B_{2w}^{c}\ |\ j \in B_{1z} \cap B_{2w}^{c}) \leq P(\gamma_j < \frac{B}{\sqrt{n^{\delta}}}, \delta_j > \frac{B}{\sqrt{n^{\delta}}}\ |\ j \in B_{1z} \cap B_{2w}^{c}) + P(\frac{\gamma_j}{\delta_j} < 1 - \frac{B^{*}}{\sqrt{n^\delta}}\ |\ j \in B_{1z} \cap B_{2w}^{c}) \to 1$. (d) $P(j\ \text{is classified in}\ B_{1z}^{c} \cap B_{2w}^{c}\ |\ j \in B_{1z}^{c} \cap B_{2w}^{c}) \leq P(\frac{\gamma_j}{\delta_j} \in (1 - \frac{B^{*}}{\sqrt{n^\delta}}, 1 + \frac{B^{*}}{\sqrt{n^\delta}})\ |\ j \in B_{1z}^{c} \cap B_{2w}^{c}) \to 1$. These all together implies $P(\text{no node is missclassified}) \to 1$. Proof of Theorem \[lem: b2lse\] {#subsec: b2lse} ------------------------------- Next, we prove Theorem \[lem: b2lse\] for $\tilde{\tilde{\tau}}_n$.Note that the proof for $\hat{\tau}_n$ is much simpler, once we use $z_1=z_2=z$, $w_1=w_2=w$, $K=m$ and $\mathcal{M}_{b,n} =0$ in the following proof. Suppose $||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F \to \infty$. Then by Theorem \[lem: b1\], it is easy to see that $P(\tilde{\tilde{\tau}}_{n} = \tau_n) \to 1$. Lemma \[lem: wvandis1\] from [@Wellner1996empirical] proves useful for establishing the asymptotic distribution of the change point estimate, when $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to c \geq 0$. Next, suppose $||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F \to c \geq 0$. Take $h = n|\tau -b|||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F ^2$. Recall the definitions of $A(b)$ and $D(b)$ from (\[eqn: abd1\]). Using expectations in (\[eqn: compa\]) and (\[eqn: compd\]), it is easy to see that by SNR-DSBM, (A1) and as $||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F \to c \geq 0$, we have $$\begin{aligned} E\sup_{h \in \mathcal{C}}|nA(b)|, E\sup_{h \in \mathcal{C}}|nD(b)| \leq C\frac{K^2}{n||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F^2} + C \frac{n \mathbb{M}_{b,n}^2}{|\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F^2} \to 0 \nonumber \end{aligned}$$ for some compact set $\mathcal{C} \subset \mathbb{R}$. This establishes that if $||\text{Ed}_z(\lambda) -\text{Ed}_w(\Delta)||_F \to c \geq 0$ and SNR-DSBM and (A1) hold, then $$\begin{aligned} \label{eqn: distnt1t2} \sup_{h \in \mathcal{C}}|nA(b)|,\ \ \sup_{h \in \mathcal{C}}|nD(b)| \stackrel{\text{P}}{\to} 0. \end{aligned}$$ Next, recall the definition of $B(b)$ from (\[eqn: abd1\]). Using similar arguments in Section \[subsec: b1\], it is easy to show that $$\begin{aligned} B(b) &=& \sum_{t=nb+1}^{n\tau} \sum_{i,j=1}^{m} (2A_{ijt} -2\lambda_{z(i)z(j)})(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)}) \nonumber \\ && \hspace{2 cm} + n|\tau-b| ||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F^2 + R(b) \nonumber \end{aligned}$$ where $R(b) \leq C|\tau-b|n^2 \mathcal{M}_{b,n}^2$. Therefore, by (A1) $$\begin{aligned} \label{eqn: t32t33} \sup_{h \in \mathcal{C}}|R(b)| \stackrel{\text{P}}{\to} 0. \end{aligned}$$ Suppose $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to 0$. Applying the Central Limit Theorem, it is easy to see that $$\begin{aligned} \label{eqn: t31} \sup_{h \in \mathcal{C}}|B(b)-R(b) + |h| + 2\gamma B_h| \stackrel{\text{P}}{\to} 0. \end{aligned}$$ Thus, by (\[eqn: distnt1t2\])-(\[eqn: t31\]) and Lemma \[lem: wvandis1\], $$\begin{aligned} n ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 (\tilde{\tilde{\tau}}_n-\tau_n) \stackrel{\mathcal{D}}{\to} \arg \min_{h \in \mathbb{R}} (|h|+2\gamma B_h) & \stackrel{\mathcal{D} } {=} & \arg \max_{h \in \mathbb{R}} (-0.5|h|+\gamma B_h) \nonumber \\ & \stackrel{\mathcal{D} } {=} & \gamma^2 \arg \max_{h \in \mathbb{R}} (-0.5|h|+ B_h). \nonumber\end{aligned}$$ This proves Part(b) of Theorem \[lem: b2lse\]. Suppose $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to c>0$. Then, $$\begin{aligned} B(b) - R(b) &=& \sum_{i,j=1}^{m} \sum_{t=nb+1}^{n\tau} \bigg[- (A_{ij,(t,n)} - {\lambda}_{z(i)z(j)})^2 + (A_{ij,(t,n)} - \delta_{w(i)w(j)})^2\bigg] \nonumber \\ &=& \sum_{i,j \in \mathcal{K}_n} \sum_{t=nb+1}^{n\tau} \bigg[ - (A_{ij,(t,n)} - \lambda_{z(i)z(j)})^2 + (A_{ij,(t,n)} - \delta_{w(i)w(j)})^2\bigg] \nonumber \\ && + \sum_{i,j\in \mathcal{K}_0} \sum_{t=nb+1}^{n\tau} \bigg[ -(A_{ij,(t,n)} -\lambda_{z(i)z(j)})^2 + (A_{ij,(t,n)} - \delta_{w(i)w(j)})^2\bigg] \nonumber \\ &=& T_{a} + T_{b} \ \ (\text{say}). \label{eqn: t311312}\end{aligned}$$ By (A6) and (A7) and if $||\text{Ed}_{z}(\Lambda) -\text{Ed}_{w}(\Delta)||_F \to c>0$, we obtain $$\begin{aligned} \sup_{h \in \mathcal{C}} |T_{b} - A^{*}(h)| \stackrel{\text{P}}{\to} 0 \label{eqn: t312}\end{aligned}$$ where each $h \in \mathbb{Z}$, $A^{*}(c^2(h+1)) - A^{*}(c^2h) = \sum_{k \in \mathcal{K}_0} \bigg[(Z_{ij, {h}} -a_{ij,1}^{*})^2 -(Z_{ij,{h}} -a_{ij,2}^{*})^2 \bigg]$ and $\{Z_{ij, {h}}\}$ are independently distributed with $Z_{ij, {h}} \stackrel{d}{=} A_{ij,1}^{*}I({h} < 0) + A_{ij,2}^{*}I({h} \geq 0)$ for all $(i,j) \in \mathcal{K}_0$. Next, $T_{a} = 2\sum_{t=nb+1}^{n\tau}\sum_{i,j \in \mathcal{K}_n} (A_{ij,(t,n)}-\lambda_{z(i)z(j)})(\delta_{w(i)w(j)}-\lambda_{z(i)z(j)}) +|h|$. An application of the Central Limit Theorem together with (A4) and (A5) yields $$\begin{aligned} \label{eqn: t311} \sup_{h \in \mathcal{C}}|T_{a} - D^{*}(h)-C^{*}(h)| \stackrel{\text{P}}{\to} 0. \end{aligned}$$ where for each ${h} \in \mathbb{Z}$, $D^{*} (c^2 (h+1))-D^{*}(c^2 h) = 0.5 {\rm{Sign}}(-h) c_1^2$ and $C^{*}(c^2(h+1)) - C^{*}(c^2 h) = \tilde{\gamma}_{_{\text{LSE}}} W_{{h}},\ \ W_{{h}} \stackrel{\text{i.i.d.}}{\sim} \mathcal{N}(0,1)$. Therefore, by (\[eqn: distnt1t2\]), (\[eqn: t32t33\]), (\[eqn: t312\]), (\[eqn: t311\]) and Lemma \[lem: wvandis1\], Part (c) of Theorem \[lem: b2lse\] is established. This completes the proof of Theorem \[lem: b2lse\]. Proof of Theorem \[thm: adapdsbm\] {#subsec: adapdsbm} ---------------------------------- Suppose $h>0$. Then, $$\begin{aligned} \tilde{L}^{*}(\hat{\tau}_n + h/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) &=& \frac{1}{n} \sum_{i,j=1}^{m} \bigg[\sum_{t=1}^{n\hat{\tau}_n +h} (A_{ij,(t,n),\text{DSBM}} - \hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)})^2 \nonumber \\ && \hspace{1.5 cm}+ \sum_{t=n\hat{\tau}_n + h +1}^{n} (A_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})^2 \bigg]. \nonumber \end{aligned}$$ We then have $$\begin{aligned} && \tilde{L}^{*}(\hat{\tau}_n + h/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) - \tilde{L}^{*}(\hat{\tau}_n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) \nonumber \\ & = & \frac{1}{n} \sum_{i,j=1}^{m} \sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \bigg[(A_{ij,(t,n),\text{DSBM}} - \hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)})^2 - (A_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})^2\bigg] \nonumber \\ & = & \frac{1}{n} \sum_{i,j=1}^{m} \sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \bigg[(\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)} - \hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)})^2 + 2 (A_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})(\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)} - \hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)})\bigg]. \nonumber\end{aligned}$$ Let $E^{**}(\cdot) =E(\cdot| \hat{z},\hat{w})$, $V^{**} = V(\cdot|\hat{z},\hat{w})$ and $\text{Cov}^{**}(\cdot) = \text{Cov}(\cdot|\hat{z},\hat{w})$. Therefore, $$\begin{aligned} E^{**}(\tilde{L}^{*}(\hat{\tau}_n + h/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) - \tilde{L}^{*}(\hat{\tau}_n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}})) &=& \frac{h}{n} ||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) - \text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^2. \nonumber \end{aligned}$$ Note that all entries of $\hat{\hat{\Lambda}}$ and $\hat{\hat{\Delta}}$ are bounded away from $0$ and $1$, since $\log m = o(\sqrt{n})$. Therefore, $$\begin{aligned} V^{**}(\tilde{L}^{*}(\hat{\tau}_n + h/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) - \tilde{L}^{*}(\hat{\tau}_n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}})) &=& \frac{h}{n^2} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})^2 \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}(1-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) \nonumber \\ & \leq & \frac{h}{n^2} ||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) - \text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^2. \nonumber\end{aligned}$$ Hence, by Lemma \[lem: wvan1\] and similar arguments to those made at the beginning of Section \[subsec: b1\], we have $$\begin{aligned} ||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) - \text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^2 h = O_{\text{P}}(1).\nonumber \end{aligned}$$ Then, by Lemma \[lem: adaplem\](a), $$\begin{aligned} ||\text{Ed}_{z}(\Lambda) - \text{Ed}_{w}(\Delta)||_F^2 h = O_{\text{P}}(1).\nonumber \end{aligned}$$ This implies Theorem \[thm: adapdsbm\](a). Next, we establish Theorem \[thm: adapdsbm\](b). Note that $$\begin{aligned} && n(\tilde{L}^{*}(\hat{\tau}_n + h||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^{-2}/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) - \tilde{L}^{*}(\hat{\tau}_n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}})) \nonumber \\ &=& -|h| -2 \sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n + h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) + o_{\text{P}}(1), $$ Further, note that given $\{A_{t,n}\}$, $\{\sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\}$ is a collection of independent random variables. By Lemma \[lem: adaplem\](b), we have $$\begin{aligned} && E \sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\nonumber \\ & =& E\bigg[\sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) E^{**}({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\bigg] = 0, \end{aligned}$$ $$\begin{aligned} && \text{V}\bigg(\sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\bigg) \nonumber \\ &=& hE\bigg(||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}}) - \text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^{-2} \sum_{i,j=1}^{m} (\hat{\hat{\Lambda}}_{\hat{z}(i)\hat{z}(i)} -\hat{\hat{\Delta}}_{\hat{w}(i)\hat{w}(i)})^2 \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(i)}(1-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(i)}) \bigg) \to h \gamma^2, \end{aligned}$$ $$\begin{aligned} && \frac{E\bigg[\sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\bigg]^3}{\bigg[\text{V}\bigg(\sum_{t=n\hat{\tau}_n+1}^{n\hat{\tau}_n+h} \sum_{i,j=1}^{m} (\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}) ({A}_{ij,(t,n),\text{DSBM}}-\hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)})\bigg)\bigg]^{3/2}} \nonumber \\ & \leq & C E\left(\frac{\sum_{i,j=1}^{m}|\hat{\hat{\lambda}}_{\hat{z}(i)\hat{z}(j)} - \hat{\hat{\delta}}_{\hat{w}(i)\hat{w}(j)}|^3}{||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^{2}}\right) \nonumber \\ &\leq & C (E||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^{2})^{1/2} \nonumber \\ &\leq & C\left( E(||\text{Ed}_{z}(\Lambda) - \text{Ed}_{\hat{z}}{\hat{\hat{\Lambda}}}||_F^2) + E(||\text{Ed}_{w}(\Delta)- \text{Ed}_{\hat{w}}({\hat{\hat{\Delta}}})||_F^2)+ ||\text{Ed}_{z}(\Lambda)-\text{Ed}_{w}(\Delta)||_F^{2} \right) \to 0. \nonumber \end{aligned}$$ Hence, an application of Lyapunov’s Central Limit Theorem together with (A1)-(A4) and SNR\*\*-DSBM-ADAP yields $$\begin{aligned} n(\tilde{L}^{*}(\hat{\tau}_n + h||\text{Ed}_{\hat{z}}(\hat{\hat{\Lambda}})-\text{Ed}_{\hat{w}}(\hat{\hat{\Delta}})||_F^{-2}/n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}}) - \tilde{L}^{*}(\hat{\tau}_n, \hat{z},\hat{w},\hat{\hat{\Lambda}},\hat{\hat{\Delta}})) \Rightarrow -|h| + \gamma B_h \label{eqn: adaplseb1}\end{aligned}$$ Similar arguments are applicable for the case of $h<0$. Finally, (\[eqn: adaplseb1\]) in conjunction with Lemma \[lem: wvandis1\] establish Theorem \[thm: adapdsbm\](b). An analogous argument to that in the proof of Theorem \[lem: b2lse\](c) together with similar approximations as in the proof of Theorem \[thm: adapdsbm\](b) establish Theorem \[thm: adapdsbm\](c) and hence they are omitted. Hence, Theorem \[thm: adapdsbm\] is established. Assumptions for asymptotic distribution of change point estimators {#subsec: assumption} ------------------------------------------------------------------ Next, we provide precise statements of Assumptions (A3)-(A7) required for establishing the asymptotic distribution of the change point estimators in Theorem \[lem: b2lse\]. A brief comment on these assumptions is given after stating them. We refer to [@BBM2017] for more in depth explanation. For Regime II, we define $$\begin{aligned} \gamma^2 &=& \lim \frac{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2\lambda_{z(i)z(j)}(1-\lambda_{z(i)z(j)})}{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2} \nonumber \\ &=& \lim \frac{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2\delta_{w(i)w(j)}(1-\delta_{w(i)w(j)})}{\sum_{i,j=1}^{m}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2}, \nonumber \end{aligned}$$ and assume that **(A3)** $\gamma^2$ exists. In Regime II, the asymptotic variance of the change point estimator is proportional to $\gamma^2$. Hence, we require (A3) for its existence and (A2) for the non-degeneracy of the asymptotic distribution. In Regime III, we consider the following set of edges $$\begin{aligned} \mathcal{K}_n = \{(i,j): 1 \leq i,j \leq m,\ \ |\lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| \to 0\}.\end{aligned}$$ Define $$\begin{aligned} c_1^2 &=& \lim \sum_{i,j \in \mathcal{K}_n} (\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2 \ \ \text{and}\ \label{eqn: c1defn} \\ \tilde{\gamma}^2 &=& \lim \sum_{i,j \in \mathcal{K}_n}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2\lambda_{z(i)z(j)}(1-\lambda_{z(i)z(j)}) \nonumber \\ &=& \lim \sum_{i,j \in \mathcal{K}_n}(\lambda_{z(i)z(j)}-\delta_{w(i)w(j)})^2\delta_{w(i)w(j)}(1-\delta_{w(i)w(j)}). \nonumber \end{aligned}$$ Consider the following assumptions.\ **(A4)** $c_1$ and $\tilde{\gamma}$ exist.\ **(A5)** $\sup_{ij \in \mathcal{K}_n} | \lambda_{z(i)z(j)} - \delta_{w(i)w(j)}| \to 0$.\ **(A6)** $\mathcal{K}_0 = \mathcal{K}_n^c$ does not vary with $n$.\ **(A7)** For some $\tau^{*} \in (c^{*},1-c^{*})$, $\tau_n \to \tau^{*}$ as $n \to \infty$. Suppose $\lambda_{z(i)z(j)} \to a_{ij,1}^{*}$ and $\delta_{w(i)w(j)} \to a_{ij,2}^{*}$ for all $(i,j) \in \mathcal{K}_0$. In Regime III, we need to treat edges in $\mathcal{K}_n$ and $\mathcal{K}_0 = \mathcal{K}_n^c$ separately. Note that in Regime II, $\mathcal{K}_n = \{(i,j):\ 1\leq i,j \leq m\}$ is the set of all edges. Hence, we can treat $\mathcal{K}_n$ in a similar way as in Regime II and hence we need (A4) in Regime III analogous to (A3) in Regime II. (A5) is a technical assumption and is required for establishing asymptotic normality on $\mathcal{K}_n$. Moreover, $\mathcal{K}_0$ is a finite set. (A6) guarantees that $\mathcal{K}_0$ does not vary with $n$. Consider the collection of independent Bernoulli random variables $\{A_{ij,l}^{*}: (i,j) \in \mathcal{K}_0, l=1,2\}$ with $E(A_{ij,l}^{*}) = a_{ij,l}^{*}$. (A7) ensures that $A_{ij,(\lfloor nf \rfloor, n)} \stackrel{\mathcal{D}}{\to} A_{ij,1}^{*}I(f<\tau^{*}) + A_{ij,2}^{*}I(f > \tau^{*})\ \forall (i,j) \in \mathcal{K}_0$. \[rem: sparse\] Note that (A2) is a crucial assumption for establishing the asymptotic distribution of the change point estimator. It indicates that the resulting random graphi’s topology. However, another regime of interest is that where the expected degree of each node grows slower than the total number of nodes in the graph, which gives rise to a [*sparse regime*]{}. A number of technical results both from probabilistic and statistical viewpoints have been considered in the recent literature - see, e.g. [@SB2015] and [@LLV2017]. Note that results strongly diverge in their conclusions under these two regimes. For example, [@O2009] showed that the inhomogeneous Erdős-Rényi model satisfies $$\begin{aligned} ||L(A) - L(EA)|| = O\left(\sqrt{\frac{\log m}{d_0}} \right) \nonumber \end{aligned}$$ with high probability, where $m$ is the total number of nodes in the graph, $d_0 = \min_{i} \sum_{j=1}^{m} EA_{ij}$, $A$ is the observed adjacency matrix, $L(\cdot)$ is the Laplacian and $||\cdot||$ is the operator norm. Therefore, if the expected degrees are growing slower than $\log m$, $L(A)$ will not be concentrated around $L(EA)$. [@LLV2017] established a different concentration inequality for the case $d_0 = o(\log m)$ after appropriate regularization on the Laplacian and the edge probability matrix. [@SB2015] also established the convergence rate of the eigenvectors of the Laplacian for SBM with two communities, which deviates from existing results for dense random graphs. The upshot is that results for the sparse regime are markedly different than those for the dense one. 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ArXiv

--- abstract: 'Robotic and animal mapping systems share many challenges and characteristics: they must function in a wide variety of environmental conditions, enable the robot or animal to navigate effectively to find food or shelter, and be computationally tractable from both a speed and storage perspective. With regards to map storage, the mammalian brain appears to take a diametrically opposed approach to all current robotic mapping systems. Where robotic mapping systems attempt to solve the data association problem to minimise representational aliasing, neurons in the brain intentionally break data association by encoding large (potentially unlimited) numbers of places with a single neuron. In this paper, we propose a novel method based on supervised learning techniques that seeks out regularly repeating visual patterns in the environment with mutually complementary co-prime frequencies, and an encoding scheme that enables storage requirements to grow sub-linearly with the size of the environment being mapped. To improve robustness in challenging real-world environments while maintaining storage growth sub-linearity, we incorporate both multi-exemplar learning and data augmentation techniques. Using large benchmark robotic mapping datasets, we demonstrate the combined system achieving high-performance place recognition with sub-linear storage requirements, and characterize the performance-storage growth trade-off curve. The work serves as the first robotic mapping system with sub-linear storage scaling properties, as well as the first large-scale demonstration in real-world environments of one of the proposed memory benefits of these neurons.' author: - 'Litao Yu, Adam Jacobson and Michael Milford [^1]' bibliography: - 'reference.bib' title: '**Rhythmic Representations: Learning Periodic Patterns for Scalable Place Recognition at a Sub-Linear Storage Cost**' --- Introduction ============ Visual place recognition - recognising whether a current camera image matches to those stored in a map or database - is a fundamental component of most robotic mapping and navigation systems[@TR:SLAM_SURVEY]. These mapping systems are typically developed and evaluated based on the quality of the map they can produce, the robustness of representations and their associated computational requirements. Much emphasis has been placed on solving the “data association” problem - making sure that there are no incorrectly or aliased map-landmark associations. Navigation neurons found in the brain of many mammals such as rodents, known as “grid cells” [@SCIENCE:TF] (see Fig. \[fig:intuation\]), have highly aliased data associations with locations in the environment - each cell encodes an arbitrary number of physical locations laid out in a triangular tesselating grid [@NATURE:M; @ARN:PGB]. There has been much interest in the theoretical advantages of such a neural representation including implications for memory storage, error correction [@NATUREN:GC] and scalability that could revolutionize how artificial systems including robots are developed. ![Neurons in the mammalian brain known as grid cells intentionally alias their representation of the environment; each neuron represents an arbitrary number of places in a regularly repeating pattern. []{data-label="fig:intuation"}](fig/fig1.jpg){width="30.00000%"} In this paper, we propose a novel approach to discover regularly repeating visual patterns in an environment, and to encode these regularly repeated patterns in frame sequences (see Fig. \[fig:intuation\]). We adopt a supervised learning approach and take advantage of statistical properties to identify periodicity in the world being mapped. To perform place recognition, a global location estimate is reconstructed from identifying the phase of these learned patterns in a current camera image. In this way, the storage requirements scale up sub-linearly with the number of encoded places in the environment (or better). Extensive experiments on three real-world datasets demonstrate successful place recognition while retaining sub-linear storage growth. We present new research that significantly extends a pilot study [@IROS17:DEJAVU] by developing a number of new contributions that address scalability to large, challenging real world environments, including: - A method for actively finding and learning periodic visual patterns from frame sequences using supervised learning techniques and best practice for maximizing their utility in sub-linear mapping, - Techniques for optimizing for minimum storage requirements that balance the number of periodic patterns, their period lengths and their separability, - Development of a multi-exemplar training scheme that improves place recognition performance in perceptually challenging environments where multiple training examples are available, while maintaining sub-linear storage growth, - Visual data augmentation techniques for improving performance when multiple-examples are not available, and - Comprehensive performance evaluation on several large benchmark datasets including characterizations of the tradeoff between storage scalability and place recognition performance, and analysis of the benefits of multi-exemplar and augmentation-based training. Together these contributions represent a significant step towards enabling a sub-linear, highly scalable map encoding scheme for autonomous systems, and provide for the first time a real-world data-based test of one of the primary postulated memory benefits of this universal spatial encoding scheme found in the mammalian brain. The paper proceeds as follows. Section \[sec:background\] provides an overview of data compression in signal processing with relevance to the approach presented here. Section \[sec:approach\] describes the components of our proposed approach in detail. Experimental results and analysis are presented in section \[sec:experiment\], with Section \[sec:conclusions\] discussing the findings and future areas of research. Background {#sec:background} ========== Data compression has a broad range of applications in signal processing, which is to encode data into compact representations by taking advantage of perceptual and statistical properties of data to provide superior analytical results. In image processing, we can use cosine transform to compress a BitMap (BMP) image as a JPEG format with a tolerable information loss but a much smaller data size. In computer vision, images and videos are usually represented as high-dimensional visual feature vectors. The goal of encoding images into compact codes is to simultaneously reduce the storage cost and accelerate the computation. To achieve this, the most discriminant information contained in high-dimensional data is usually embedded into a lower-dimensional space for further analysis. Usually, the embeddings are in a discrete format such as hashing [@TPAMI:HASH]. However, the discrete data representations suffer from data collisions when data size is large, so it is not the best option for unique mapping in visual place recognition. To avoid data collision, visual information can also be embedded in continuous, rather than discrete lower-dimensional spaces [@RAS:KDE]. For multimedia data compression, there are two encoding families based on machine learning techniques: binary embedding and vector quantization, both of which are designed to compress continuous sensor data into discrete feature spaces. The idea of binary embedding is to represent feature vectors as compact binary codes, so the Euclidean distance between two vectors could be approximated by Hamming distance in the binary space [@TPAMI:HASH]. The advantage of binary embedding is due to the efficient Hamming distance computation, which can be implemented by the XOR and POPCOUNT operations. Different from binary embedding, vector quantization (VQ) adopts a codebook as a dictionary to quantize the feature vectors into a set of codewords, and the distances between any two codewords are pre-computed and stored in a lookup table [@BOOK:VQ]. When the original feature space is decomposed into the Cartesian product of several low-dimensional subspaces, vector quantization becomes product quantization (PQ) [@TPAMI:PQ; @TPAMI:OPQ; @TIP:BOPQ]. Compared with binary embedding, PQ based encoding methods have a lower information loss and thus can achieve a better accuracy, at the cost of a slightly lower computational speed. Both binary embedding and vector quantization are effective encoding techniques with regards to calculation and storage costs. Although binary embedding and vector quantization have different encoding strategies, the underlying mechanism is clustering, i.e., similar statistical patterns have the same codes. Both of the two techniques suffer from code collisions as the data size increases, and they have linear storage growth with the number of data instances. Computational and storage requirements are of particular importance for mobile robotic and autonomous systems. It is important to differentiate between at least three different goals - achieving highly compact but ultimately linear storage growth, achieving sub-linear computational requirements (but with linear or worse storage growth), and achieving sub-linear storage growth. For the first goal, various techniques have been applied in robotic applications. For example, Locality Sensitive Hashing (LSH) is used to deal with the problem of stereo correspondence estimation [@ICRA15:LSH], multi-model fusion techniques are adopted in humanoid robots to process large-scale language data [@ICRA12:BIGRAM], and Local Difference Binary (LDB) descriptor is applied to obtain a robust global image description for place recognition and loop closure detection [@IROS2014:BINARY]. In [@ICRA16:TACTILE] the authors proposed to compress sensory data from tactile skins. Similarly, the distributed sensor data with high-frequencies can be compressed as coresets for streaming motion [@IPSN12:CORESET]. To achieve the sub-linear computation, [@ICRA17:SQLSLAM] builds an index on the original database to reduce the computation cost. While significant effort has been devoted towards efficient computation with low absolute storage costs, there has been little work examining how sub-linear storage growth might be achieved for SLAM systems, or examining how natural systems achieve this sub-linear storage growth with a unique one-to-many neuronal mapping system. Approach {#sec:approach} ======== In this section, we describe our proposed encoding model for scalable place recognition, based on supervised learning techniques. The system comprises a periodic template learning phase, a database encoding phase, and a global place reconstruction phase. ![An illustrative scenario with only two visual patterns available: buildings and trees, and both of them appear periodically. Each column represents a frame, so the frame sequence simulates a virtual camera moving forward. The combination of the two landmarks can represent at most $3\times 4=12$ distinct locations.[]{data-label="fig:fake_road"}](fig/fake_road.png){width="45.00000%"} Learning Periodic Patterns from Frame Sequences ----------------------------------------------- For clarity, we start with a toy example, since in the actual system we are looking for the underlying visual patterns that are often not intuitive. In Fig. \[fig:fake\_road\], we show an illustrative scenario with only two visual patterns: different buildings and different trees. In this example, each column represents a frame in a sequence. By observing the frame sequence, we can see the style of building cycles in every 3 frames, and the type of the tree regularly changes in every 4 frames, respectively. Consequently, the combination of the two ideal periodic patterns can uniquely represent at most $3\times 4 = 12$ different locations. In place recognition systems, if there are more than two periodic patterns with different lengths, only storing the pattern information is sufficient for global place estimation. For example, we can use periodic landmarks and their positions to describe a frame, just like the visual templates used to detect interesting points as SIFT descriptors in image processing, and these descriptors can be then aggregated into visual feature vectors for further analysis. However, we will show that it is possible to learn latent periodic patterns from a wide variety of data. To enable the SLAM system to automatically analyse the feature vectors, in [@IROS17:DEJAVU] the authors proposed to apply spectrogram to find the regularly repeated patterns in frame sequences. The spectral methods assume the signal is composed of phase-shifted sine and cosine curves with scaled factors and offsets, but such a strong assumption is not valid in most real applications. Also, the thresholds need to be neatly set for signal discretization and template matching, because a high threshold may lead to the loss of matched templates, while a low threshold usually results in mismatches or redundancies. In our proposed data compression model, temporally periodic patterns are learned from the frame sequence in a database. Given an (integer) period $\tau$, our system will look for a linear separation for each possible (integer) phase of that period that separates frames in that phase from frames in all other phases. This allows us to train completely distinct classifiers for each phase of a period, which is much less restrictive than training a single template for all phases of that period, as is implicit in spectral methods. Let a location database be represented as a frame sequence as $\mathbf{X}=\{\mathbf{x}_1, \mathbf{x}_2, \ldots, \mathbf{x}_N\}$ where $N$ is the total number of data instances, and $\mathbf{x}_i$ is the $i$-th frame in the database ($1\le i \le N$). If each frame is represented as a $d$-dimensional visual vector, i.e., $\mathbf{x}_i\in\mathbb{R}^d$, the size of the database is $\mathbb{R}^{N\times d}$. When there is a cyclic visual pattern with the length $\tau$, $\tau$ templates are generated in each period. For the $j$-th template within a period ($1\le j \le \tau$), we assign a binary label $y_i^{(j)}\in\{1, -1\}$ for each frame $\mathbf{x}_i$ to indicate if it can match the $j$-th template: $$\label{eq:lbl}\small y_i^{(j)} = \begin{cases} 1 & \quad i\mod\tau = j - 1, \\ -1 & \quad \text{otherwise}. \\ \end{cases}$$ Consequently, the task of determining whether a frame $\mathbf{x}_i$ can match the $j$-th template in a period becomes a binary classification problem. The weight vector $\mathbf{w}_j$ and bias $b_j$ can be simply obtained by solving a linear SVM as follows: $$\begin{aligned} \small \min_{\mathbf{w}_j,b_j,\xi_{ij}} \quad& \frac{1}{2}\|\mathbf{w}_j\|^2+C\sum^N_{i=1} \xi_{ij}, \nonumber \\ \mathbf{s.t.} \quad& y_i^{(j)}(\mathbf{w}_j^{\top}\mathbf{x}_i + b_j) \ge 1 - \xi_{ij}, \nonumber \\ & \xi_{ij} \ge 0, 1\le i \le N, \end{aligned}$$ where $C$ is the penalty parameter to balance the hinge loss and functional margin. Here we mainly focus on the loss function to make the templates as linearly separable as possible, and we can simply set $C=\log{N}$. Simultaneously considering $\tau$ templates within a period, these binary classifiers can be integrated into a multi-class SVM model: $$\begin{aligned} \small \min_{\mathbf{w}_j,b_j,\xi_{ij}} \quad& \frac{1}{2}\sum_{j=1}^{\tau}\|\mathbf{w}_j\|^2+C\sum_{j=1}^{\tau}\sum^N_{i=1} \xi_{ij}, \nonumber \\ \mathbf{s.t.} \quad& y_i^{(j)}(\mathbf{w}_j^{\top}\mathbf{x}_i + b_j) \ge 1 - \xi_{ij}, \nonumber \\ & \xi_{ij} \ge 0, 1\le i \le N, 1\le j \le \tau.\end{aligned}$$ This multi-class SVM can be efficiently computed by some toolboxes such as scikit-learn[^2]. The statistical property of the weight vector $\mathbf{w}_j$ is straightforward: when all sequenced frames within the database are represented by $\lfloor N/\tau \rfloor$ periods and can be perfectly segmented by $\tau$ classifiers, each classifier has the minimum covariance with its positively classified data instances. Thus, these classifiers can be just considered as the templates within a period. The optimised weight vector $\mathbf{w}_j$ and bias $b_j$ ($1\le j \le \tau$) are able to determine if a frame is at the $j$-th position within a period with length $\tau$, i.e., the template that can be matched by a frame $\mathbf{x}$ is calculated by: $$\label{eq:index}\small f (\mathbf{x} | \tau)=\arg \max_j (\mathbf{w}^{\top}_j \mathbf{x} + b_j), \quad 1\le j \le \tau.$$ Note that in order to keep the sub-linearity for data compression, we do not use kernel SVMs. In the kernel case, the weight vector in the decision function $f(\cdot)$ is represented as a linear combination of support vectors. Although the kernel decision function is more discriminative than the linear one, it cannot achieve the sub-linear data compression, because when either the dimension or the size of database increases, the number of support vectors also increases accordingly. Database Encoding {#subsec:compression} ----------------- As we have seen, we can use linear SVMs to learn $\tau$ periodic templates given a period tau and the $N$ frames of a dataset in which we wish to localise. However, unless $\tau > N$, this is obviously not sufficient to uniquely identify a frame. The core idea of our method is to learn two or more such cyclic patterns $\{\tau_1, \tau_2, \ldots\}$, such that frames can be uniquely identified. In this subsection, we show how the periods $\tau$ can be chosen to allow for unique identification while minimising storage requirements. Assuming there are several candidates for $\tau$ available, we simply select $r$ cyclic patterns with periods $\tau_1,\tau_2,\ldots,\tau_r$ to estimate a frame position within a database when given an arbitrary frame $\mathbf{x}$ from it. The position of $\mathbf{x}$ could be determined by the [*phase matches*]{}, which are represented as a candidate set $\{j_1, j_2, \ldots, j_r\}$, $1\le j_k\le \tau_k$. The possible index $i$ is calculated by: $$\label{EQ:IDENT} i=a_k\cdot\tau_k+j_k,$$ where $k\in\{1,\ldots,r\}$, and $a_k$ is a natural number. To identify $\mathbf{x}_i$ with $\{j_1, j_2, \ldots, j_r\}$, its index $i$ needs to be the unique solution of Eq. (\[EQ:IDENT\]). Thus the selections of $\tau_1,\tau_2,\ldots,\tau_r$ should satisfy: $$\label{EQ:LCM} lcm(\tau_1,\tau_2,\ldots,\tau_r)\ge N,$$ where $lcm(\cdot)$ is the least common multiple operator. This condition guarantees the index mapping is unique, and there are sufficient “slots" to store all frames in the database. If we manage to make $\tau_1,\tau_2,\ldots,\tau_r$ co-prime, then Eq. (\[EQ:LCM\]) will be equivalent to $\prod\limits^{r}_{k=1}\tau_k\ge N$ . Given this constraint and if $r=2$, when given a $\tau_1$, $\tau_2$ needs to be at least $N/\tau_1$. In Fig. \[fig:taus\], we illustrate how the selection of the period lengths affects the total number of templates if $N=100$ and $r=2$. For each $\tau_1$ on the x-axis, we found the smallest $\tau_2$ that is both co-prime with $\tau_1$ and that satisfies $\tau_1\times\tau_2\ge N$. Then on the y-aixs we report $\tau_1+\tau_2$, which is proportional to the storage cost. We can see the minimum storage for 100 place estimations is 21 when $r=2$, so $\tau_1=10$ and $\tau_2=11$ would be the best period pair. ![The minimum number of templates required when $N=100$ and $r=2$.[]{data-label="fig:taus"}](fig/taus.png){width="25.00000%"} Assuming that our place recognition algorithm could correctly identify the phase matches $j_k$ (see Eq. (\[EQ:IDENT\])) corresponding to a query $\mathbf{x}$, the memory requirements for our system are to store the weight vectors $\mathbf{w}^{(k)}_1,\mathbf{w}^{(k)}_2,\ldots,\mathbf{w}^{(k)}_j$ and biases $b^{(k)}_1,b^{(k)}_2,\ldots,b^{(k)}_j$ for each possible $k$. In other words, we need to allocate memory for $\sum\limits_{k=1}^{r}\tau_k$ vectors of size $d+1$. Thus, the minimal storage requirements are achieved when $\prod\limits_{k=1}^{r} \tau_k \ge N$. In unconstrained cases, the solution is $\sqrt[r]{N}$, but $\tau_1,\tau_2,\ldots,\tau_r$ also need to satisfy they are coprime integers. Since we have not found a closed-form solution to this constrained problem, we instead propose to sample candidates from $\lceil\sqrt[r]{N}\rceil, \lceil\sqrt[r]{N}\rceil+1, \ldots,\lceil\sqrt[r]{N}\rceil+m$ with the least training errors. The training error $e$ is the fraction of misclassified training samples of $\tau$ linear models among the whole training set. Therefore, only storing $r$ groups of periodic templates can reduce the space complexity from $O(N)$ to $O(r\sqrt[r]{N})$. Reconstructing a Global Place Estimate {#subsec:search} -------------------------------------- The localisation of an arbitrary frame from the database $\mathbf{x}$ is implemented by the intersection operation. We first calculate the phase matches of $r$ periodic patterns: $j_k=f(\mathbf{x}|\tau_k)$ for $k\in\{1,2,\ldots, r\}$ by applying Eq. (\[eq:index\]) , and then generate $r$ candidate sets $P_1, P_2, \ldots, P_r$, where $P_k=\{f(\mathbf{x}|\tau_k), f(\mathbf{x}|\tau_k)+\tau_k, \ldots, f(\mathbf{x}|\tau_k)+\lfloor N/\tau_k \rfloor\}$. The index of $\mathbf{x}$ in the original frame database is calculated by $f(\mathbf{x}|\tau_1,\tau_2,\ldots,\tau_r)=\bigcap\limits_{l=1}^{r} P_k$. Before the online searching phase, given a query image, the system should first determine if the location that the image represents can be found in the database. Some appearance-based SLAM systems such as [@ICRA08:FABMAP], proposed to set a lower bound of likelihood, which is calculated in the training procedure and set by users. The low likelihood that falls below the lower bound means the query image cannot match any places in the database. Our proposal is a deterministic approach, and there would be at least one phase match when given a query even if it is actually an outlier. So a lower bound of decision value can also be set to determine if a query can match a template in a periodic pattern. Alternatively, an auxiliary classifier could be trained when there are negative exemplars available, which are not descriptive to any locations in the database. The retrieval of our method consists of $r-1$ 1d intersections of $r$ sets with size $N/\tau$, which is lower than $\sqrt[r]{N}$. Since the sets are already sorted, the 1d intersection can be achieved in $O(r\sqrt[r]{N})$: the intersection of a pair of sorted sets is linear in the sum of the sizes of both sets, so its time complexity is $O(\sqrt[r]{N})$, and we only need to perform $r-1$ such intersections on all sets. Improving Robustness Through Multi-Exemplar Training and Augmentation {#subsec:data_env} --------------------------------------------------------------------- A common method for improving the robustness of recognition techniques is to use, where available, multiple different examples of an object or place in training. For example, ImageNet has over ten million images with one thousand categories [@IJCV:IMGNET]. With large-scale and well-labelled training images, the classifiers trained on such datasets have near-human performance in very challenging recognition tasks. For the mobile place localisation systems, the robot should be able to memorise the scenes by revisiting the same places multiple times from different perspectives, or under distinct appearance conditions, to improve their discriminative power. In this case, the periodic patterns are essentially learned in a shared space rather than the original feature space. Since our proposed data compression model for scalable place recognition is also based on supervised learning techniques, using frames taken under different appearance conditions has the potential to improve recognition accuracy and robustness. However, there is not always multi-exemplar training data available. In this case, we can apply image augmentation methods such as Gaussian blur, flipping, random cropping and elastic transformation to simulate the multi-exemplar data environment. Experiment and analysis {#sec:experiment} ======================= In this section, we describe the datasets used, the image preprocessing methods, the evaluation metrics and the experimental results. We also provide analysis of our proposed model, breaking down the performance contributions of the core system, enhancements including multi-exemplar training and augmentation, and provide an analysis of the trade-off between performance and storage scaling. Datasets and Experiment Settings -------------------------------- To evaluate our proposed data compression model for scalable visual place recognition, we experimented with three different datasets: Nordland Train dataset, Aerial Brisbane dataset and Oxford RobotCar dataset. ### Nordland Train dataset The Nordland Line[^3] is a 729-kilometre railway between Trondheim and Bodø, Norway. This dataset contains four long videos captured by placing a camera at the front of a train facing forward along the railway track. The four videos describe the front views in four seasons, and each video is about ten hours long. This dataset was first used for visual navigation across seasons[@ECMR13:ACLCS], but we only use it to test the compressibility of our proposed model. In the first part of our evaluation, the queries are all from the reference data, and the model aims to find the exact positions of them in the database. To pre-process the video data, we first extracted the keyframes, then used the optical flow of the ground directly in front of the train to estimate the velocity then normalised it. Finally, the four subsets contain 10,713, 7,403, 9,267 and 7,276 frames, respectively. ### Aerial Brisbane dataset The Aerial Brisbane dataset is generated by taking a snapshot from NearMap[^4], which describes the Brisbane region in Queensland, Australia. The total size of the image is $7526\times6562$, and each pixel is an actual geographic area of $4.777\times 4.777$ square metres. The image was then segmented to $224\times 224$-pixel frames with 112-pixel strides, so in our setting the dataset can represent 3,705 different places. We use this dataset to test if our model can recognise locations in visual changing environment. To simulate the environment, the query images and reference data are from different sources. We collected several snapshots of the aerial map taken at different times ranging from 19/05/2013 to 24/06/2017. One image was selected as the query to search the absolute locations of the patches on the map, and the rest reference frames are used to train the model. ### Oxford RobotCar dataset The Oxford RobotCar dataset [@IJRR:RCD] contains over 100 repetitions of a consistent route through Oxford, UK. The dataset captures many different combinations of weather, traffic and pedestrians. For our testing, we used 5 subsets of the Oxford RobotCar dataset generated from a fixed route captured at different times of day using images captured by the Point Grey Bumblebee XB3. This dataset describes a small area with a limited number of locations. Since the geographic positions are described in consecutive northing and easting values as GPS data, we applied KMeans on the normalised coordinates to generate 100 clusters, so the GPS coordinates falling into the same cluster are considered as a unique location on the map. We ran our place recognition model to test if the periodic encoding can successfully capture the cyclic properties of the frame sequences to enable accurate reconstruct the location estimation. For all of the three datasets, we utilised the deep visual features extracted from popular ConvNet architecture to describe the frames. Specifically, we used the second output of the fully-connected layer of the VGG16 model [@ARXIV:VGG] and then applied L2 normalisation. Thus each frame is represented by a 4,096d visual feature vector. On the Oxford RobotCar dataset, each location is visually represented by multiple frames. To reduce the noise and fit class conditional densities to the data with multiple exemplars, we further applied the Linear Discriminant Analysis (LDA) to reduce the dimensionality to 64. In our experiment, we compared our model with KD-Tree[@COMM75:KD_TREE], Iterative Quantization (ITQ)[@TPAMI:ITQ] and Optimised Product Quantization (OPQ)[@TPAMI:OPQ] on Nordland Train and Aerial Brisbane datasets. KD-Tree is a well-known method for approximate nearest neighbour search, which builds a binary search tree as the index for a fixed-sized database. Although a KD-Tree can effectively accelerate the computation, this technique does not compress the data. ITQ and OPQ are discrete embedding approaches, which encode the high-dimensional data into compact codes for fast computing. However, they cannot achieve unique mapping for place recognition because code collision is inevitable. Furthermore, these techniques compress data in an absolute way, i.e., the compressed data size is proportional to the actual data size, which is not sub-linear. Let $T$ be the level of the period length, which is the minimum value for $\tau$ in data encoding. Assume we set $T=2\sqrt{N}$ and $r=2$ for our model, and use a 256-bit binary vector and a 256d integer vector to encode a 4,096d visual instance for ITQ and OPQ, respectively. When data size is comparably small, our proposed model has a higher memory cost, but it increases sub-linearly when the database becomes extremely large. We used the compression ratio to demonstrate how our model can achieve sub-linear storage and used the accuracy metric to evaluate place recognition performance. We also compared the computational speed of our model with the baseline models. Our experiment was conducted on a desktop with Intel(R) i7-7700K CPU 4.20GHz with 4 processors, 32GB RAM, and Windows 10 operating system with a Python 3.6 computational environment. Place Recognition Results When All Queries Are from Reference Data ------------------------------------------------------------------ We first investigated how the period length $\tau$ affects the training error. We tested different lengths of periods and trained the linear SVMs on the Aerial Brisbane dataset, with the training errors illustrated in Fig. \[fig:period\_err\]. We can see longer periods lead to a lower training error rate $e$ and a higher compression ratio when $r$ is fixed. In the extreme case, when a whole frame database has only one period, i.e., $r=1$ and $\tau=N$, no data compression is implemented, and the model reverts to brute-force search. ![The training errors on Aerial Brisbane dataset when setting different period lengths.[]{data-label="fig:period_err"}](fig/period_err.png){width="25.00000%"} We then investigate the data compression results for Nordland Train and Aerial Brisbane datasets when only two periodic patterns are available, i.e., $r=2$. We also tested the training errors at two periodic levels: $T=\sqrt{N}$ and $T=2\sqrt{N}$, respectively. For each subset of Nordland Train dataset, as well as the Aerial Brisbane dataset, we tried seven different values of $\tau$ in training the linear SVMs and recorded the error rates, and then the system automatically selected the best period pair. Based on the selection of periods, we analyse the storage cost when applying our proposed data compression approach and selecting the length of period $T$ at the level of $\sqrt{N}$ on the two datasets. In a 32-bit operating system, the memory cost for a float number is 4 bytes. The storage comparison of the datasets is summarised in Table \[tab:data\_compression\]. From the table, we can see that our proposed model is able to encode very large frame databases with high compression ratios. If the data size increases linearly, applying two-period values and several templates can make the storage increase in a sub-linear manner. When the number of frames is more than 10,000, our model only takes about 1/50 memory to store all data instances. [**Dataset**]{} [**Original size**]{} [**Original storage**]{} [**Compressed size**]{} [**Compressed storage**]{} [**Compression ratio**]{} ------------------- --------------------------------- -------------------------- ------------------------------- ---------------------------- --------------------------- Norland (spring) $\mathbb{R}^{10713\times 4096}$ 175,521,792 bytes $\mathbb{R}^{211\times 4097}$ 3,457,868 bytes 0.0207 Norland (summer) $\mathbb{R}^{7403\times 4096}$ 121,290,752 bytes $\mathbb{R}^{181\times 4097}$ 2,966,228 bytes 0.0246 Nordland (fall) $\mathbb{R}^{9267\times 4096}$ 151,830,528 bytes $\mathbb{R}^{201\times 4097}$ 3,293,988 bytes 0.0217 Nordland (winter) $\mathbb{R}^{7276\times 4096}$ 119,209,984 bytes $\mathbb{R}^{179\times 4097}$ 2,933,452 bytes 0.0246 Aerial Brisbane $\mathbb{R}^{3705\times 4096}$ 60,702,720 bytes $\mathbb{R}^{125\times 4097}$ 2,048,500 bytes 0.0337 We used the frames from the reference data as queries and applied our models for location estimation. We show the place recognition results of Nordland Train and Aerial Brisbane datasets in Table \[tab:acc\_nordland\_aerial\]. In all of the four subsets of the Nordland Train dataset, none of the accuracies falls below 98% even when we apply the extreme compression method. For example, in the spring subset, 10,713 frames record different views of places along the Norland railway. Applying our proposed data compression model can still achieve 99.46% accuracy. If we set longer periods, i.e., $T=2\sqrt{N}$, the compression ratio doubles, but the recognition accuracy is higher, which is very close to 100%. On the Aerial Brisbane dataset, our system achieved the very near-perfect accuracy of 99.92% (only 3 mismatches) when $T=\sqrt{N}$. When the length of period doubles, the recognition accuracy is 100%. ------------------ ---------- ------------------------ ---------- ------------------------ ---------- ---------- ------------------------ ---------- ------------------------ ---------- $\tau_1$ $e$ ($\times 10^{-3}$) $\tau_2$ $e$ ($\times 10^{-3}$) Accuracy $\tau_1$ $e$ ($\times 10^{-3}$) $\tau_2$ $e$ ($\times 10^{-3}$) Accuracy Norland (spring) 105 2.99 106 2.89 0.9946 211 0.84 212 0.65 0.9990 Norland (summer) 88 8.91 93 8.65 0.9846 179 2.70 180 2.30 0.9960 Norland (fall) 98 4.64 103 4.64 0.9912 197 0.54 200 0.22 0.9992 Norland (winter) 87 1.51 92 1.52 0.9971 174 0.27 175 0.00 0.9996 Aerial Brisbane 62 0.54 63 0.27 0.9992 122 0.00 123 0.00 1.0000 ------------------ ---------- ------------------------ ---------- ------------------------ ---------- ---------- ------------------------ ---------- ------------------------ ---------- We conducted an experiment using both Nordland Train and Aerial Brisbane datasets evaluating the performance of the KD-Tree, ITQ, OPQ, and the brute-force search techniques, recording the average search time. The comparison is displayed in Table \[tab:time\]. By using a few learned periodic templates and the matching approach introduced in section \[subsec:search\], the computational efficiency is significantly increased compared to the exhaustive search, although it is a lower than KD-Tree, ITQ and OPQ. However, KD-Tree only builds an index on the database but does not implement the data compression at all. Both ITQ and OPQ could compress the data in an absolute manner, but they cannot achieve the unique mapping required for place recognition, even when the distance between a query and its matched frame is zero. Considering the compression ratio, search accuracy and speed, it is worth applying our proposed model to large-scale place recognition systems. [**Method**]{} [**Compressed scale**]{} [**Search time**]{} [**Unique mapping**]{} ---------------- -------------------------- --------------------- ------------------------ KD-Tree No compression 0.000215 Yes ITQ Linear 0.000116 No OPQ Linear 0.000241 No Brute-force No compression 0.125948 Yes Ours Sub-linear 0.000503 Yes : The comparisons of different search methods (the seach time is recorded on Aerial Brisbane dataset). []{data-label="tab:time"} Then we used more than two periods by setting $r=3$ and $r=5$ respectively and re-ran our model on Nordland Train and Aerial Brisbane datasets. As is discussed in section \[subsec:compression\], we can choose different available periods for data compression. A larger value of $r$ means the system can achieve a lower compression ratio. Fig. \[fig:compressions\] summarises the compression ratios and the accuracy comparisons. From the two figures, it can be seen that although applying more periodic patterns can achieve an even higher compression ratio, the recognition accuracy falls significantly, even though the queries are all from reference data. The reason is that the value of $\tau$ is in direct proportion to the number of negative data instances in Eq. (\[eq:lbl\]), i.e. when the number of templates increases within a period, there are fewer positive-labelled instances in the learning process, which makes the periodic templates more linearly separable. By contrast, the training error rate is higher on a more “balanced” dataset. We tested our system using data from different times of day for the Aerial Brisbane and Oxford Robotcar datasets. For Aerial Brisbane dataset, we used one image from this dataset as queries, and used another image taken at a different time as the reference. For Oxford RobotCar dataset, we used the frames taken on a distinct date as queries to search their locations. When applying the brute-force search, the accuracy is 0.9582 on Aerial Brisbane dataset, and 0.236 on Oxford RobotCar dataset, respectively. Using our model to compress the reference data, the accuracy dropped to 0.6835 and 0.183 on the two datasets. The result signifies solely applying the data compression model cannot achieve a satisfactory recognition accuracy under different appearance conditions. As is introduced in Section \[subsec:data\_env\], we adopted different data augmentation approaches, including Gaussian blur, random cropping, flipping, elastic transformation, contrast normalisation, etc. We separately experimented with these augmented data sources, then merged them as a whole training set to train a unified place recognition model. Note that for Oxford RobotCar dataset, we did not use channel invert and grey-scale augmentations because the raw image data is grey-scale. The accuracies are displayed in Fig. \[fig:data\_aug\] for the two datasets. It can be seen that although each single augmented data source has limited power to help obtain a discriminative model, their combination can effectively boost the accuracy by 2% and 1% on the two datasets, respectively. Results in Visually Changing Environments ----------------------------------------- Next, we trained the models with the multi-reference set, where these frames are taken under different appearance conditions. We used different combinations of reference sources and evaluated their performance on Aerial Brisbane and Oxford RobotCar datasets, and the accuracy curves are plotted in Fig. \[fig:multi\_src\]. The accuracy improves steadily as the number of data sources increases and learning with the multi-source data and image augmentation enables the data compression model to better deal with the various appearance conditions while keeping the storage sub-linear. As is discussed in Section \[subsec:compression\], we set longer periods to further reduce the training error $e$, but with a lower data compression ratio. We tested different lengths of periods by setting $T=\sqrt{N}$, $T=2\sqrt{N}$, $T=3\sqrt{N}$, and $T=4\sqrt{N}$ respectively, and the accuracy curve is plotted in Fig. \[fig:multi\_period\_len\]. We can conclude that setting a longer period can improve the recognition accuracy, with the sacrifice of the compression rate. In this real application scenario, if we set $T=4\sqrt{N}$, the data compression ratio is 0.131 on Aerial Brisbane dataset and 0.81 on Oxford RobotCar dataset, respectively, but the recognition performance is boosted. Discussions and conclusions {#sec:conclusions} =========================== We have presented a novel image-based map encoding scheme that deliberately seeks out and learns mutually supportive visual pattern frequencies in the environment to enable place recognition with sub-linear storage growth as the environment size increases. The system is based on the nature of neural mapping systems in the mammalian brain that does not appear to approach the data association problem central to most robotic mapping systems the same way; instead each neural map “unit” is associated with an arbitrarily large number of places in the environment distributed at regular intervals. Results on large real-world datasets show that the fundamental premise is valid and that high-performance place recognition can be achieved with a mapping system whose map storage scales sub-linearly with environment size. The system is agnostic of any particular types of features or feature frequencies and its performance across a range of environments shows that, perhaps surprisingly, repetitive visual patterns can usually be found. We applied the data augmentation and multi-source training data, which are generic methods for visual recognition tasks, to make our model more applicable under different appearance conditions. In future work, we could also design a more sophisticated system by integrating some advanced machine learning techniques to better capture the spatial properties of the periodic patterns and improve the recognition performance. Alternatively, we could apply some existing matching schemes such as SeqSLAM [@ICRA12:SEQSLAM], to further improve the stability by taking consideration of multi-frame integration. Acknowledgements {#sec:acknowledgements .unnumbered} ================ This work was supported by an Asian Office of Aerospace Research and Development Grant FA2386-16-1-4027 and an ARC Future Fellowship FT140101229 to MM. [^1]: Litao Yu ([email protected]), Adam Jacobson ([email protected]) and Michael Milford ([email protected]) are with the School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, QLD, Australia. MM also with the Australian Centre for Robotic Vision. [^2]: http://scikit-learn.org/ [^3]: https://nrkbeta.no/2013/01/15/nordlandsbanen-minute-by-minute-season-by-season/ [^4]: http://maps.au.nearmap.com/

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--- author: - 'Federico R. Urban,' - 'Stefano Camera,' - and David Alonso bibliography: - 'references.bib' title: 'Detecting ultra-high energy cosmic ray anisotropies through cross-correlations' --- Introduction {#sec:intro} ============ Ultra-high energy cosmic rays (UHECRs), impacting the atmosphere of the Earth with energies in excess of $1\,\mathrm{EeV}$ ($10^{18}\,\mathrm{eV}$), have remained a mystery since their discovery 59 years ago [@Linsley:1961kt; @AlvesBatista:2019tlv]. We do not know what they are: observational data can not yet fully distinguish between several variants of pure and mixed primary compositions [@Castellina:2019huz; @Bergman:2019aaa]. We do not know where they come from: the astrophysical sources that generate and accelerate UHECRs have not been identified yet; the type of acceleration mechanism that is responsible for their formidable energies has not been discovered, either [@Kotera:2011cp]. What we do know is that the highest energy rays are most likely extra-Galactic. First, if UHECRs were produced within the Galaxy, their arrival directions in the sky would be very different from what we observe [@Tinyakov:2015qfz; @Abbasi:2016kgr; @Aab:2017tyv]. Second, banning a cosmic conspiracy that puts an end to the injection spectrum at that very energy, UHECR interactions with cosmological background photons produce a sharp cutoff (the Greisen-Zatsepin-Kuzmin limit) in the spectrum corresponding to $\sim60\,\mathrm{EeV}$ [@Greisen:1966jv; @Zatsepin:1966jv], and a cutoff is indeed observed in the data [@Abbasi:2007sv; @Abraham:2008ru]. If the sources of UHECRs are extra-Galactic, they most probably correlate with the large-scale distribution of matter (large-scale structure, or LSS). The interactions with the background cold photons limit UHECR propagation to circa $100\,\mathrm{Mpc}$ (for a review, see Ref. [@Kotera:2011cp]). Therefore, the UHECR flux distribution in the sky should be to some extent anisotropic, since $100\,\mathrm{Mpc}$ is roughly comparable with the scale of homogeneity expected in the standard cosmological model [@Pan:2000yg; @Scrimgeour:2012wt; @Alonso:2014xca]. How the anisotropy of UHECR sources manifests itself in the observed flux on Earth then depends on the original anisotropy of the sources, the UHECR chemical composition, and the properties of intervening magnetic fields – Galactic (GMF) and extra-Galactic (xGMF) – that deflect UHECRs and distort the original anisotropic patterns. Chemical composition and magnetic fields are degenerate when it comes to UHECRs deflections, since the latter depends on $ZB/E$, where $Z$ is the atomic number, $B$ the strength of the magnetic field, and $E$ is the UHECR energy: doubling the field strength is equivalent to doubling the charge (or halving the energy). Chemical composition instead is the only factor that determines the UHECR propagation length at a given energy: different nuclei come from a different portion of the Universe and carry a different anisotropic imprint, but the relationship between the two is non-monotonic and non-trivial (see, e.g., [@dOrfeuil:2014qgw; @diMatteo:2017dtg]). To a large extent, the statistics of the anisotropies in the distribution of UHECRs can be characterized by the UHECR angular auto-correlation (AC), which, in harmonic space, takes the form of angular the power spectrum coefficients $C_\ell$. Here, the $\ell$-th multipole quantifies the variance of the anisotropies on angular scales $\theta\sim\pi/\ell$ [@Sommers:2000us; @Tinyakov:2014fwa] (see Appendix \[app:pk\_cl\] for further details). To date, the number of UHECRs collected at the highest energies is low – of the order of a hundred above the cutoff [@AlvesBatista:2019tlv]. Because of this, the UHECR flux is dominated by Poisson statistics: the AC is mostly determined by shot noise, making the underlying correlation with the LSS very hard to detect. Indeed, the indications for anisotropy in the data are tenuous: save for a low-energy dipole [@Aab:2017tyv] and a high-energy hot-spot [@Abbasi:2014lda], the angular distribution of UHECR arrival directions appears to be nearly isotropic [@diMatteo:2020dlo]. Moreover, no anisotropies have been detected at small scales $\ell\gtrsim10$ [@Deligny:icrc2015; @diMatteo:2018vmr]. In this work, we quantify the possibility of detecting the anisotropy in the UHECR flux through the harmonic-space power spectrum of the cross-correlation (XC) between UHECR counts and the distribution of galaxies. Such XC technique was previously proposed to study the anisotropy of the $\gamma$-ray sky by Refs. [@Camera:2012cj; @Fornengo:2013rga; @Pinetti:2019ztr], and proved successful for several tracers of the LSS [@Fornengo:2014cya; @Cuoco:2015rfa; @Branchini:2016glc; @Ammazzalorso:2019wyr]. If UHECR sources statistically trace the LSS, then the positions of these sources, and the arrival directions of UHECRs (if not strongly affected by intervening magnetic fields) should have a non-zero correlation with a galaxy sample up to a given distance. Therefore, the detection or non-detection of the XC signal with galaxies at different redshifts would allow us to test whether UHECR sources are distributed according to the LSS, and to quantify to which extent the UHECR transfer function, determined by energy losses and intervening magnetic fields, does not depend on direction. There are at least three features that differentiate the XC from other methods (see for instance [@Koers:2008ba] and references therein). First, systematic uncertainties of different ‘messengers’, or observables, should not cross-correlate, and, under some conditions, statistical noise should also not strongly cross-correlate. This is because different experiments are different machines exploiting different physical effects. However, within a single dataset, for instance the set of arrival directions of UHECRs, the AC of the noise and systematic errors for that set are certainly non-zero, and contribute to hiding any underlying ‘true’ signal. Thus, in this sense the XC is an experimentally cleaner observable. Secondly, in the limit where the UHECR sources are numerous, but UHECR detections themselves are not, we can assume that we observe at most one UHECR per source (as seems to be the case given to the lack of obvious UHECR multiplets [@Abreu:2011md]). The much higher number of galaxies leads to a significant improvement in the signal-to-noise ratio of this cross-correlation (see the discussion in section \[sec:results\]). This effectively allows us to probe the anisotropies on smaller scales through the XC than the AC. There are several reasons why those smaller scales ($\ell>10$) are interesting. First of all, the experimental angular resolution of UHECR events is around $1^\circ$, which corresponds to $\ell\sim200$: from an experimental perspective we are not fully taking advantage of the data we already have. Furthermore, small-scale power in the LSS angular distribution is comparable to that at large scales: if UHECRs bear the imprint of the LSS this small-scale power is distorted but not strongly suppressed by the GMF [@Dundovic:2017vsz]; moreover, the sub-structures of the GMF themselves imprint small-scale signatures in UHECR anisotropies [@Mertsch:2013pua]. Lastly, small-scale anisotropies can be separately detected in different regions of the sky, allowing us to probe, for example, different GMF structures independently. Third, while most analyses have looked at the real-space correlation between UHECRs and the large-scale structure [@Kashti:2008bw; @Oikonomou:2012ef; @Abreu:2010ab; @PierreAuger:2014yba]), we will express our results here in terms of harmonic-space power spectra. These are common observables in cosmological studies, based on a natural decomposition of the celestial sphere. They also allow for a straightforward visualization of the main components of the astrophysical model (radial kernels, details of the galaxy-matter connection), which is one of the main novel aspects of this work. In this paper, we will introduce a formalism to model the AC and XC, and apply it to a vanilla proton-only model for UHECR injection in order to quantify the differences between the two observables and the detectability of the anisotropies on different scales with existing experimental facilities. We defer the more detailed discussion of the dependence of the XC on UHECR injection and source properties, a realistic treatment of the UHECR experimental setup, such as non-uniform sky coverage, as well as a full treatment of the effects of the GMF and xGMF on the signal, to upcoming work. This paper is organized as follows. In Section \[sec:model\] we introduce the formalism to describe the UHECR flux, the distribution of galaxies, and the AC and XC. We apply this formalism to a hypothetical UHECR dataset in Section \[sec:results\], where we obtain and compare the AC and the XC. We summarize our findings and conclude with an outlook for future work in Section \[sec:conclusions\]. Appendix \[app:pk\_cl\] collects useful standard formulæ pertaining to angular power spectra. Theoretical model {#sec:model} ================= UHECR flux {#ssec:model.flux} ---------- Let $\mathcal{E}({E_\text{inj}})$ be the (angle-integrated, isotropic) emissivity[^1] of cosmic rays (CRs) for a given galaxy (number of CRs of energy ${E_\text{inj}}$ emitted per unit energy, per unit time): $$\begin{aligned} \mathcal{E}[{E_\text{inj}}]{\coloneqq}\frac{{\mathrm{d}}{N_\text{inj}}}{{\mathrm{d}}{E_\text{inj}}\,{\mathrm{d}}{t_\text{inj}}} \,.\end{aligned}$$ The subscript ‘inj’ (injection) here indicates quantities evaluated in the rest frame of the emitting source. Due to the expansion of the Universe and to interactions between CRs and cosmic background light, the injected energy of a CR, whose energy at detection is $E$, is given by ${E_\text{inj}}(E,z)$ with $z$ the redshift of the source. In the absence of scattering processes the energy losses are adiabatic ${E_\text{inj}}= (1+z)E$. The differential emissivity (i.e., per unit solid angle) is $\epsilon{\coloneqq}\mathcal{E}/4\pi$ assuming isotropic emission. We will parameterize the emissivity as a power-law of energy: $$\begin{aligned} \label{eq:plaw} \mathcal{E}[{E_\text{inj}}]\propto{E_\text{inj}}^{-\gamma} \,.\end{aligned}$$ Energies will always be expressed in EeV for convenience. The quantity measured on Earth is the observed number of events per unit time, energy interval, detector area, solid angle on the sky and (assuming source redshifts can be measured), redshift interval. We can relate this number to the emissivity through $$\frac{{\mathrm{d}}N}{{\mathrm{d}}E\,{\mathrm{d}}t\,{\mathrm{d}}A\,{\mathrm{d}}\Omega\,{\mathrm{d}}z}=\frac{ n_{\rm s,c}\,\mathcal{E}({E_\text{inj}})}{4\pi\,(1+z)\,H(z)}\frac{{\mathrm{d}}{E_\text{inj}}}{{\mathrm{d}}E} \,,$$ where $H(z)$ is the Hubble parameter, $n_{\rm s,c}$ is the volumetric number density of CR sources, and we have ignored subdominant lightcone and relativistic effects [@Challinor:2011bk; @Bonvin:2011bg]. We will be interested in the number of UHECRs detected above a given energy threshold ${E_\text{cut}}$ (defined in the observer’s frame) and integrated over source redshifts, from the direction ${\hat{\bm n}}$: $$\begin{aligned} \nonumber \Phi({E_\text{cut}},{\hat{\bm n}})&{\coloneqq}\int_0^\infty {\mathrm{d}}z\,\int_{{E_\text{cut}}}^\infty {\mathrm{d}}E\,\frac{{\mathrm{d}}N}{{\mathrm{d}}E\,{\mathrm{d}}t\,{\mathrm{d}}A\,{\mathrm{d}}\Omega\,{\mathrm{d}}z}\\ &=\int \frac{{\mathrm{d}}z}{(1+z)H(z)}\frac{n_{\rm s,c}(z,\chi{\hat{\bm n}})}{4\pi}\,\int_{{E_\text{cut}}}^\infty {\mathrm{d}}E\,\frac{{\mathrm{d}}{E_\text{inj}}}{{\mathrm{d}}E}\,\mathcal{E}({E_\text{inj}}) \,,\end{aligned}$$ where $\chi(z)$ is the radial comoving distance. We can write the number density of sources as $n_{\rm s,c}(z,\chi{\hat{\bm n}})=\bar{n}_{\rm s,c}(z)\,[1+\delta_{\rm s}(z,\chi{\hat{\bm n}})]$, where $\delta_{\rm s}$ is the galaxy overdensity. Assuming a non-evolving galaxy population, and a power-law UHECR spectrum (as in Eq. \[eq:plaw\]) we obtain: $$\begin{aligned} \Phi({E_\text{cut}},{\hat{\bm n}})&\propto\frac{\bar{n}_{\rm s,c}}{4\pi}\int \frac{{\mathrm{d}}\chi}{(1+z)}\frac{{E_\text{inj}}^{1-\gamma}({E_\text{cut}},z)}{1-\gamma}\,\left[1+\delta_{\rm s}(z,\chi{\hat{\bm n}})\right] \,.\end{aligned}$$ ### Attenuation {#sssec:model.flux.attenuation} The *attenuation* factor $\alpha({E_\text{cut}},z;\gamma,Z)$ is defined as the number of events reaching the Earth with $E>{E_\text{cut}}$ divided by the number of events which would have reached the Earth if there were no energy losses at a given distance: $$\alpha(z,{E_\text{cut}};\gamma,Z){\coloneqq}\frac{{E_\text{inj}}^{1-\gamma}({E_\text{cut}},z)}{{E_\text{cut}}^{1-\gamma}}\,.$$ The attenuation $\alpha$ is a function of the energy cut and redshift, as well as the injection spectral slope and chemical composition. In terms of $\alpha$, the direction-dependent integral flux is $$\begin{aligned} \Phi({E_\text{cut}},{\hat{\bm n}})\propto\frac{\bar{n}_{\rm s,c}{E_\text{cut}}^{1-\gamma}}{4\pi\,(1-\gamma)}\int {\mathrm{d}}\chi\,\frac{\alpha(z,{E_\text{cut}})}{(1+z)}\,\left[1+\delta_{\rm s}(z,\chi{\hat{\bm n}})\right].\end{aligned}$$ In this paper, to introduce the formalism, we choose to work with a toy proton-only model with injection slope $\gamma=2.6$ as in model (4) of [@dOrfeuil:2014qgw], or model (i) of [@diMatteo:2017dtg]. In order to obtain the attenuation factor for our injection model we have followed $10^6$ events with *SimProp* v2r4 [@Aloisio:2017iyh] with energies above $E=10\,\mathrm{EeV}$ (with an upper cutoff of $E=10^{5}\,\mathrm{EeV}$), for redshifts up to $z=0.3$, and counted the number of events reaching the Earth with $E>{E_\text{cut}}$ for different values of ${E_\text{cut}}$. With *SimProp* we have accounted for all energy losses, adiabatic and interactions with cosmic microwave background (CMB) photons and extra-Galactic background photon according to the model [@Stecker:2005qs]. The UHECR radial kernels, defined in the next section, obtained from the attenuation factor $\alpha$ for different energies are shown in Fig. \[fig:kernels\]. ![Radial kernels for the two observables under consideration. The solid black line shows the approximate redshift distribution of galaxies in the 2MRS sample using the fit found by [@Ando:2017wff]. The red, yellow, and blue lines show the radial kernel for the UHECR flux (Eq. (\[eq:window\_cr\])) for the three energy thresholds studied here ($40\,\mathrm{EeV}$, $63\,\mathrm{EeV}$, and $100\,\mathrm{EeV}$ respectively). []{data-label="fig:kernels"}](Fig1){width="75.00000%"} Note that we assume that cosmic ray energy losses are to first order isotropic, that is, we ignore angular anisotropies in in the CMB and extra-Galactic background light, which are completely negligible for our analysis. Moreover, for simplicity here we work with full-sky uniform coverage, but the analysis can be readily generalized to non-uniform and partial sky coverage. ### Anisotropies {#sssec:model.flux.anisotropies} We will define the anisotropy in the UHECRs distribution as the over-density of rays detected as a function of sky position ${\hat{\bm n}}$ as $$\Delta_{\rm CR}({\hat{\bm n}},E_{\rm cut}) {\coloneqq}\frac{\Phi({\hat{\bm n}},E_{\rm cut})-\bar{\Phi}(E_{\rm cut})}{\bar{\Phi}(E_{\rm cut})} \,,$$ where $\bar{\Phi}({E_\text{cut}})$ is the sky-averaged UHECR flux. From the results in the previous section, this quantity is related to the three-dimensional overdensity of UHECR sources $\delta_{\rm s}(z,\chi{\hat{\bm n}})$ through $$\label{eq:delta_cr} \Delta_{\rm CR}({\hat{\bm n}},{E_\text{cut}})=\int {\mathrm{d}}\chi\,\phi_{\rm CR}(\chi)\,\delta_{\rm s}(z,\chi{\hat{\bm n}}) \,,$$ where the UHECR window function is $$\label{eq:window_cr} \phi_{\rm CR}(z){\coloneqq}\left[\int {\mathrm{d}}\tilde z \frac{\alpha(\tilde z)}{H(\tilde z)(1+\tilde z)}\right]^{-1}\frac{\alpha(z)}{(1+z)} \,.$$ Figure \[fig:kernels\] shows the radial kernels for UHECRs with energy thresholds ${E_\text{cut}}=40,\,63$, and $100\,\mathrm{EeV}$; as expected the lower the energy the farther UHECRs propagate. Galaxies {#ssec:model.gals} -------- We will consider the AC of the UHECR anisotropy, Eq. , and its XC with the galaxy number count fluctuations. In particular, we will work with the projected overdensity of sources for a given galaxy sample, $$\Delta_{\rm g}({\hat{\bm n}}){\coloneqq}\frac{N_{\rm g}({\hat{\bm n}})-\bar{N}_{\rm g}}{\bar{N}_{\rm g}},$$ where $N_{\rm g}({\hat{\bm n}})$ is the number of galaxies in a given direction ${\hat{\bm n}}$, and $\bar{N}_{\rm g}$ its average over the celestial sphere. This is related to the three-dimensional galaxy overdensity $\delta_{\rm g}(z,\chi\,{\hat{\bm n}})$ via $$\label{eq:delta_g} \Delta_{\rm g}({\hat{\bm n}})=\int {\mathrm{d}}\chi\,\phi_{\rm g}(\chi)\,\delta_{\rm g}(z,\chi\,{\hat{\bm n}}) \,,$$ where $\phi_{\rm g}(\chi)$ is the weighted distribution of galaxy distances. In general, we will assume that we have redshift information for all galaxies in the catalog, and that we can use that information to apply a distance-dependent weight $w(\chi)$. In that case, the galaxy overdensity kernel $\phi_{\rm g}(\chi)$ is given by $$\begin{aligned} \label{eq:window_g} \phi_{\rm g}(\chi){\coloneqq}\left[\int {\mathrm{d}}\tilde\chi \tilde\chi^2\,w(\tilde\chi)\,\bar{n}_{\rm g,c}(\tilde\chi)\right]^{-1}\,\chi^2\,w(\chi)\,\bar{n}_{\rm g,c}(\chi) \,,\end{aligned}$$ where $\bar{n}_{\rm g,c}$ is the comoving number density of galaxies in the sample. If no weights are applied – namely, $w(\chi)=1$ – then $$\label{eq:n3d_n2d} \int {\mathrm{d}}\chi\,\chi^2\,w(\chi)\,\bar{n}_{\rm g,c}(\chi) = \bar{N}_{\Omega,{\rm g}} \,,$$ where $\bar{N}_{\Omega,{\rm g}}$ is the angular number density of galaxies (i.e., number of galaxies per steradian). Figure \[fig:kernels\] shows the radial kernel for a low-redshift galaxy survey, modelled after the 2MASS Redshift Survey (2MRS) [@2012ApJS..199...26H]. This constitutes one of the most complete full-sky spectroscopic low-redshift surveys, and we will use it as our fiducial galaxy sample in this paper. In this work we consider full-sky datasets for simplicity, but the generalization of our results for an incomplete sky coverage is straightforward. In the case of a realistic setup based on 2MRS, a sky coverage around 70% will only degrade the signal by a factor of $\sqrt{0.7}\simeq0.86$. Power spectra {#ssec.model.cls} ------------- We are interested in detecting the intrinsic anisotropies in the distribution of UHECRs by considering the different two-point functions built from $\Delta_{\rm CR}$ and $\Delta_{\rm g}$. A given observation of any of these fields will consist of both signal ${\text{\textsc s}}$ and noise ${\text{\textsc n}}$: $\Delta_a={\text{\textsc s}}_a+{\text{\textsc n}}_a$ (where $a,\,b\,\in\{{\rm CR},g\}$). Assuming signal and noise to be uncorrelated, the corresponding power spectra can be split into both components, namely $$\begin{aligned} C_\ell{\coloneqq}{\mathcal{S}}_\ell+{\mathcal{N}}_\ell \,,\end{aligned}$$ where ${\mathcal{S}}_\ell$ and ${\mathcal{N}}_\ell$ are the power spectra of ${\text{\textsc s}}$ and ${\text{\textsc n}}$ respectively. In our case, the signal is the intrinsic clustering of both UHECRs and galaxies due to the underlying large-scale structure, while the noise is sourced by the discrete nature of both tracers as Poisson noise. A brief review of the mathematics behind angular power spectra is given in Appendix \[app:pk\_cl\]. ### Signal power spectra {#sssec:model.cls.sls} The angular power spectrum ${\mathcal{S}}_\ell^{ab}$ between two projected quantities $\Delta_a$ and $\Delta_b$ is related to their three-dimensional power spectrum $P_{ab}(z,k)$ by $$\label{eq:cl_limber} {\mathcal{S}}^{ab}_\ell=\int \frac{{\mathrm{d}}\chi}{\chi^2}\,\phi_a(\chi)\,\phi_b(\chi)\,P_{ab}\left(z,k=\frac{\ell+1/2}{\chi}\right) \,,$$ where $\phi_a$ and $\phi_b$ are the radial kernels of both quantities. The final piece of information needed in order to estimate the expected AC and XC signals is the power spectrum of the three-dimensional overdensities $\delta_{\rm s}$ and $\delta_{\rm g}$. In general, the clustering properties of galaxies and UHECR sources will depend on the specifics of the relation between galaxies and dark matter, and on the astrophysical properties of the UHECR sources. To simplify the discussion, here we will assume that all UHECR sources are also galaxies of the 2MASS sample (i.e. $\delta_{\rm s}=\delta_{\rm g}$). At this point, one might be tempted to use a linear bias prescription [@Mo:1995cs] to relate the galaxy and matter power spectra. However, as we show in Section \[sec:results\], since the UHECR radial kernel peaks at $z=0$ and covers only low redshifts, the cosmic ray flux auto-correlation probes mostly sub-halo scales ($r\lesssim1\,{\rm Mpc}$), for which a non-perturbative description of structure formation is necessary. To achieve this, we use here a halo model prescription [@Peacock:2000qk], based on the halo occupation distribution (HOD) model used by Ref. [@Ando:2017wff] to describe the 2MRS sample. In this model, the galaxy power spectrum is given by two contributions, $$P_{\rm g\,g}(z,k)=P_{\rm g\,g}^{1{\rm h}}(z,k)+P_{\rm g\,g}^{2{\rm h}}(z,k) \,,$$ being the so-called 1-halo and 2-halo terms. The former dominates on small scales and describes the distribution of galaxies within the halo, while the latter is governed by the clustering properties of dark matter halos. The HOD is then based on a prescription to assign central and satellite galaxies to halos of different masses. We refer the reader to [@Ando:2017wff] and references therein for further details about the specifics of the HOD model used. ### Shot noise {#sssec:model.cls.nls} Both projected overdensities, $\Delta_{\rm CR}$ and $\Delta_{\rm g}$, are associated to discrete point processes, represented by the angular positions of the UHECRs and the galaxies in each sample. In that case, even in the absence of intrinsic correlations between the different fields, their power spectra receive a non-zero white contribution, given by $${\mathcal{N}}^{ab}_\ell=\frac{\bar{N}_{\Omega,a\bigwedge b}}{\bar{N}_{\Omega,a}\,\bar{N}_{\Omega,b}} \,,$$ where $\bar{N}_{\Omega,a}$ ($\bar{N}_{\Omega,b}$) is the angular number density of points in sample $a$ or $b$, and $\bar{N}_{\Omega,a\bigwedge b}$ is the angular number density of points shared in common. In our case this would correspond to the number of UHECRs originating from galaxies in the galaxy sample. For simplicity we will assume that the galaxy survey under consideration is sufficiently complete, so that all UHECRs are associated to an observed galaxy. In this case, the shot-noise contributions to the power spectra are $$\begin{aligned} \label{eq:shot_noises} {\mathcal{N}}^{{\rm CR\,CR}}_\ell&=\left(\bar{N}_{\Omega,{\rm CR}}\right)^{-1},\\ {\mathcal{N}}^{\rm g\,g}_\ell={\mathcal{N}}^{{\rm g\,CR}}_\ell&=\left(\bar{N}_{\Omega,{\rm g}}\right)^{-1} \,.\end{aligned}$$ Since typically $\bar{N}_{\Omega,{\rm CR}}\ll \bar{N}_{\Omega,{\rm g}}$, then ${\mathcal{N}}^{{\rm g\,CR}}_\ell\ll {\mathcal{N}}^{{\rm CR\,CR}}_\ell$, and therefore we will neglect ${\mathcal{N}}^{{\rm g\,CR}}_\ell$ in what follows. We have explicitly checked that indeed the cross-noise can be safely neglected in all our estimates and numerical results. Note that, when non-flat weights are applied to the galaxy catalog, the resulting noise power spectrum reads $$\begin{aligned} {\mathcal{N}}^{\rm g\,g}_\ell = \frac{\int {\mathrm{d}}\chi\,\chi^2\,w^2(\chi)\,\bar{n}_{\rm g,c}(\chi)}{\left[\int {\mathrm{d}}\chi\, \chi^2\,w(\chi)\,\bar{n}_{\rm g,c}(\chi)\right]^2} \,.\label{eq:noise_gopt}\end{aligned}$$ For $w(\chi)=1$, Eq.  holds, and we recover the result in Eq. . ### Optimal weights {#sssec:model.cls.weight} We can use the results in this section to derive optimal weights $w(\chi)$ to maximize the signal-to-noise of the galaxy-UHECR cross-correlation. Let us pixelize the celestial sphere and consider the UHECRs in a given pixel $p$, $\Phi_p$, as well as the vector $N_{p,i}$ containing the number of galaxies along the same pixel in intervals of distance $\chi_i$. The optimal weights $w_i{\coloneqq}w(\chi_i)$ can be found by maximising the likelihood of $\Phi_p$ given $N_{p,i}$ [@Alonso:2020mva], and are given by the so-called *Wiener filter*, i.e.$$w_i = \sum_j {\sf Cov}^{-1}(N_{p,i},N_{p,j})\,{\sf Cov}(\Phi_p,N_{p,j}) \,.$$ Here ${\sf Cov}(x,y)$ is the covariance matrix of two vectors $x$ an $y$. Assuming Poisson statistics, we can use the results from the previous section to show that $$w(\chi) = \frac{\alpha[z(\chi),E_{\rm cut};\gamma,Z]}{[1+z(\chi)]\chi^2 \bar{n}_{\rm g,c}(\chi)} \,.$$ In hindsight, this result is obvious: by inspecting Eqs. (\[eq:window\_cr\]) and (\[eq:window\_g\]), we see that the optimal weights modify the radial galaxy kernel $\phi_{\rm g}$ to make it identical to the UHECR kernel $\phi_{\rm CR}$, thereby building the most likely estimate of the UHECR flux map from the galaxy positions. As we will see, this involves up-weighting galaxies at low redshifts, from where it is more likely that UHECRs that reach the Earth originate, but few galaxies can be found due to volume effects. We will show how the use of optimal weights can improve the signal-to-noise ratio for the XC in section \[ssec:results.cls\]. Intervening magnetic fields {#ssec:model.MFs} --------------------------- The Milky Way is host to a magnetic field of a few $\mu$G [@Boulanger:2018zrk], which is the screen that befogs UHECR sources. The variety of parametric models of the GMF, which disagree on the GMF functional forms and parameters, particularly on GMF substructures, reflects the complexity of the GMF, and, at the moment, cannot be taken at face value [@Boulanger:2018zrk; @Unger:2017kfh]. As a guideline, we can expect the GMF to deflect a UHECR with energy $E=100\,\mathrm{EeV}$ by less than a degree for the most part of the sky, and to up to a few degrees in certain directions close to the Galactic plane (see also Ref. [@Pshirkov:2013wka]). UHECRs will also be affected by any intervening xGMF, whose strength, shape, and filling factors varying by several orders of magnitude for different models and estimates [@Subramanian:2015lua]; the xGMF, however, it is believed to have a subdominant effect on large-scale UHECR propagation [@Pshirkov:2015tua]. In order to best introduce the method, in this first theoretical work we take a pragmatic approach and we neglect the effects of intervening magnetic fields. We will investigate their impact on the XC in future work. The GMF is generally split into a coherent, large-scale field and a stochastic, small-scale field; the latter sometimes split further into a turbulent, statistically isotropic field and a striated field, whose direction varies on small scales, but whose orientation does not. We can describe the effect of the small-scale GMF as randomly perturbing the arrival direction of each UHECR. To mimic this effect, we can multiply the signal part of the power spectrum by a Gaussian beam $B_\ell=\exp[-\ell(\ell+1)\sigma^2_d/2]$, where $\sigma_d$ is the typical deflection angle in radians. Specifically, using the notation in Section \[sssec:model.cls.sls\], we can write the power spectra as follows: $${\mathcal{S}}^{{\rm CR\,CR}}_\ell\,\,\,\longrightarrow\,\,\,{\mathcal{S}}^{{\rm CR\,CR}}_\ell B^2_{\ell} \,,\hspace{12pt} {\mathcal{S}}^{{\rm g\,CR}}_\ell\,\,\,\longrightarrow\,\,\,{\mathcal{S}}^{{\rm g\,CR}}_\ell B_{\ell} \,.$$ In practice this amounts to washing out the intrinsic UHECR anisotropy on angular scales smaller than the typical deflection angle, which is not well known (and direction-dependent, see [@Pshirkov:2013wka]). UHECRs will also be affected by any intervening xGMF, whose strength, shape, and filling factors varying by several orders of magnitude for different models and estimates [@Subramanian:2015lua]; the xGMF, however, it is believed to have a subdominant effect on large-scale UHECR propagation [@Pshirkov:2015tua] Results {#sec:results} ======= Signal-to-noise ratio {#ssec:results.fisher} --------------------- We estimate the signal-to-noise ratio (SNR) of the UHECR anisotropies as the square root of the Fisher matrix element corresponding to an effective amplitude parameter $A_{\rm CR}$ multiplying the signal component of $\Delta_{\rm CR}$ with a fiducial value $A_{\rm CR}=1$ [@2009arXiv0906.0664H], namely $$\begin{aligned} {\rm SNR}^2&{\coloneqq}\sum_{\ell=\ell_{\rm min}}^{\ell_{\rm max}}\left(\frac{\partial{\mathcal{S}}_\ell}{\partial A_{\rm CR}}\right)^{\sf T}{\sf Cov}^{-1}_{\ell\ell^\prime}\frac{\partial{\mathcal{S}}_\ell}{\partial A_{\rm CR}},\\ &=\sum_{\ell=\ell_{\rm min}}^{\ell_{\rm max}}\left({\rm SNR}_\ell\right)^2,\end{aligned}$$ where ${\bm S}_\ell$ is a vector containing the signal contribution to the power spectra under consideration, ${\sf Cov}$ is the covariance matrix of those power spectra, and ${\rm SNR}_\ell$ is the SNR of a single $\ell$ mode. If the fields being correlated are Gaussian ($\Delta_{\rm CR}$, $\Delta_{\rm g}$ in our case), the covariance matrix can be estimated using Wick’s theorem to be $${\sf Cov}\left(C^{ab}_\ell,C^{cd}_\ell\right)=\frac{C^{ac}_\ell C^{bd}_\ell+C^{ad}_\ell C^{bc}_\ell}{(2\ell+1)\Delta\ell}\delta_{\ell\ell^\prime} \,,\label{eq:cov}$$ with $\Delta\ell$ the size of the multipole bin. At this point we can consider three different cases: 1. [**AC only.**]{} In this case we only have a measurement of the UHECR AC, $C^{{\rm CR\,CR}}_\ell$. The SNR is given by $$\label{eq:sn_auto} {\rm SNR}^{\rm CR\,CR}=\sqrt{\sum_{\ell=\ell_{\rm min}}^{\ell_{\rm max}}2(2\ell+1)\left(\frac{{\mathcal{S}}^{{\rm CR\,CR}}_\ell}{{\mathcal{S}}^{{\rm CR\,CR}}_\ell+{\mathcal{N}}^{{\rm CR\,CR}}_\ell}\right)^2} \,.$$ 2. [**XC only.**]{} In this case we only use the XC, $C^{{\rm g\,CR}}_\ell$. The SNR is given by $$\label{eq:sn_cross} {\rm SNR}^{{\rm g\,CR}}=\sqrt{\sum_{\ell=\ell_{\rm min}}^{\ell_{\rm max}}(2\ell+1)\frac{({\mathcal{S}}^{{\rm g\,CR}}_\ell)^2}{C^{\rm g\,g}_\ell C^{{\rm CR\,CR}}_\ell+(C^{{\rm g\,CR}}_\ell)^2}} \,.$$ 3. [**All data.**]{} We use all available data, i.e., a data vector ${\bm S}_\ell=({\mathcal{S}}^{{\rm CR\,CR}}_\ell,\,{\mathcal{S}}^{{\rm g\,CR}}_\ell,\,{\mathcal{S}}^{\rm g\,g}_\ell)$. Although this is the manifestly optimal scenario, XCs are arguably safer than ACs in terms of systematic errors, and therefore it is interesting to quantify the loss of information if only XCs are used. Studying these three cases allows us to explore the benefits of using XCs vs ACs, as well as the relative amount of information in each of the different two-point function. Given the relatively small number of UHECRs currently measured, shot noise in the UHECR flux is the dominant contribution to the uncertainties. Comparing Eqs.  and , we can see that the SNR scales like $N_{\rm CR}^{-1}$ and $N_{\rm CR}^{-1/2}$ for cases 1 and 2 respectively, highlighting the potential of XCs to achieve a detection. Power spectra and signal-to-noise {#ssec:results.cls} --------------------------------- The energy ${E_\text{cut}}$ at which we choose to cut the UHECR integral spectrum determines the UHECR propagation horizon, which in turns determines the strength of the anisotropy. Moreover, the choice of ${E_\text{cut}}$, for a given UHECR spectrum, also determines the number of UHECR events we have to sample the anisotropic angular distribution. We expect a trade-off between the two. At low energies the UHECR sample contains many more events than at high energies because the UHECR spectrum is very steep (soft/red); however, for the range of energies we are interested in, the galaxy sample is much larger, so this does not have as strong an effect for the XC as it does for the traditional AC (whose noise is determined by the number of UHECT events). Moreover, at low energies UHECRs propagate further, and the larger line-of-sight averaging can dilute the expected anisotropy. Lastly, the effects of intervening magnetic fields are stronger. At high energies the UHECR horizon is smaller, UHECRs undergo smaller deflections, and the anisotropy should be more pronounced, but the number of events drops dramatically. In order to determine at which energy we have the best chances of detecting the XC we chose to work with three energy cuts at: ${E_\text{cut}}=10^{19.6}\,\mathrm{eV}\simeq40\,\mathrm{EeV}$, ${E_\text{cut}}=10^{19.8}\,\mathrm{eV}\simeq63\,\mathrm{EeV}$, and ${E_\text{cut}}=10^{20}\,\mathrm{eV}=100\,\mathrm{EeV}$. In a realistic scenario, based on data currently available [@AlvesBatista:2019tlv], we can expect to have about $N_{\rm CR}=1000$, $N_{\rm CR}=200$, and $N_{\rm CR}=30$ over the full sky, for the three energy cuts defined above, respectively. ![Angular AC and XC power spectra considered in this work. Dotted and dashed curves respectively refer to the 1- and 2-halo contribution to the total signal (solid curves). []{data-label="fig:cl_nl"}](Fig2_HOD){width="\textwidth"} In Fig. \[fig:cl\_nl\], we show the expected signal for the AC (left panel) and the XC (right panel). Colours refer to the three energy cuts discussed above, namely red for ${E_\text{cut}}\simeq40\,\mathrm{EeV}$, yellow for ${E_\text{cut}}\simeq63\,\mathrm{EeV}$, and blue for ${E_\text{cut}}=100\,\mathrm{EeV}$. The dashed and dotted curves show the 1-halo and 2-halo contributions to the total power spectrum, with the sum of both shown by the solid curves. For simplicity, we have not included any beam smoothing in the plot. We can see how the signal for the XC is lower than the AC, as is expected from the fact that the XC mixes two different radial kernels. If we employ optimal weights for the XC the signal would become identical to that of the AC. In our simplistic linear treatment of perturbations, this happens because the UHECR and galaxy kernels would be identical. The statistical uncertainties for both correlation functions, however, would be different, given their different shot-noise levels. ![Expected power spectra and $\ell$-binned 1$\sigma$ uncertainties (shaded boxes) including a $1^\circ$ Gaussian smoothing beam to account for the angular resolution of UHECR experiments (solid curves). For reference, horizontal lines in the leftmost plots denote shot noise levels and the dashed curves show the beam-free prediction.[]{data-label="fig:cl_el"}](Fig3_HOD){width="\textwidth"} To understand better the role of the different uncertainties on the theoretical signal, in Fig. \[fig:cl\_el\] we show again the expected signal as in Fig. \[fig:cl\_nl\] (solid curves, same colour code) and include a $1^\circ$ Gaussian smoothing beam to account for the angular resolution of UHECR experiments (for reference, we also show the beam-free prediction as dashed lines). On top of it, we present the corresponding $\ell$-binned 1$\sigma$ error bars as shaded boxes for 20 log-spaces multipole bins between $\ell_{\rm min}=2$ and $\ell_{\rm max}=1000$. If we compare the leftmost and central panels, namely AC vs XC, it is easy to see how the range of multipoles where error bars are small enough to allow a detection is larger for XC than for AC for the ${E_\text{cut}}\simeq40\,\mathrm{EeV}$ and ${E_\text{cut}}\simeq63\,\mathrm{EeV}$ cases. However, for the sparser UHECR sample with $E_{\rm cut}=100\,\mathrm{EeV}$ the opposite applies; more precisely, the detectable range of multipoles for the XC is smaller and pushed towards higher $\ell$ compared to the AC. This is due to a combination of two factors: for the higher end of UHECR energies the propagation horizon of UHECRs is small, and the UHECR sky looks more anisotropic, boosting the AC. At the same time, the mismatch in kernels is prominent, the more so the higher the energy, and it drives the XC signal down. Combined with the larger shot noise in the UHECR data, this can explain the performance of the $100\,\mathrm{EeV}$ case – indeed, the UHECR shot noise is the main factor that prevents a detection of the signal at mid-$\ell$ values (the per-$\ell$ signal is 1$\sigma$ compatible with zero, see below). In the rightmost panel of Fig. \[fig:cl\_el\] we show the XC signal when we apply theoretical optimal weights. In this case the highest energy set performs the best, and this is expected from the previous arguments: the signal is boosted back up to the same level of the AC because the kernels of galaxies and UHECRs now coincide. Additionally, while the uncertainty increases with energy as both samples become sparser, it is not large enough to hide the XC signal. It is worth noticing that the increase in galaxy power that we expect towards lower redshifts, is significantly less relevant than the matching of the radial kernels. In practice, using optimal weights may not be possible given the uncertainties in the radial kernel for UHECRs (we do not know yet the actual injection spectrum). The availability of redshift information in the galaxy catalog, however, would allow us to turn this into an advantage: the UHECR kernel could be reconstructed by modifying the galaxy weights to maximize the signal-to-noise, essentially following the ‘clustering redshifts’ method used to reconstruct unknown redshift distributions in weak lensing data [@Newman:2008mb]. ![SNR for UHECR flux anisotropies from different combinations of data, namely UHECR AC in the leftmost panel, XC in the central panel, and the combination of all data in the rightmost panel. In each panel, the left half shows the cumulative SNR as a function of the maximum multipole, $\ell_{\rm max}$, whereas the right half is for the cumulative SNR as a function of the minimum multipole, $\ell_{\rm min}$. The horizontal dashed line mark the $3\sigma$ threshold for detection.[]{data-label="fig:sn"}](Fig4_HOD){width="\textwidth"} To quantify the improvement in detectability brought by the XC, in Fig. \[fig:sn\] we present the cumulative SNR for all the data combinations discussed in Sect. \[ssec:results.fisher\], viz. AC alone (leftmost panel), XC alone (central panel), and all the data combined in a single data vector $\bm S_\ell$ (rightmost panel). In each panel, the left half shows the cumulative SNR as a function of the maximum multipole, $\ell_{\rm max}$, whilst the right half is for the cumulative SNR as a function of the minimum multipole, $\ell_{\rm min}$. In both cases, the case with all the data combined has unsurprisingly the largest SNR, but the contributions from AC and XC come from different angular scales, which in turn are sensitive to different redshift ranges, depending upon ${E_\text{cut}}$, which sets the propagation depth for UHECRs. ![SNR per multipole, ${\rm SNR}_\ell$, for the AC signal, the XC signal with both normal and optimal weights, and their combination AC+XC (leftmost, central, and rightmost panel, respectively). Different colours refer to different energy cuts, and the three horizontal, dashed lines show the thresholds for $1,\,2$, and $3\sigma$ detection. []{data-label="fig:sn_el"}](Fig5_HOD){width="\textwidth"} The aforementioned sensitivity to different angular scales can be captured better by looking at Fig. \[fig:sn\_el\], where we show the contribution to the total SNR from each integer multipole, ${\rm SNR}_\ell$. The colour code is the same as throughout the paper, and we mark with horizontal dashed lines the thresholds corresponding to $1,\,2$ and $3\sigma$ evidence for a one-parameter amplitude fit. These panels can be interpreted as the evidence for anisotropy on a given scale, for which it is clear that the XC with galaxies helps to push the detectability of the signal to smaller scales, i.e., larger $\ell$ values. This per-$\ell$ $\text{SNR}_\ell$ is a useful quantity to assess whether the AC or the XC is the best observable to detect the anisotropy in UHECRs, assuming that UHECRs trace the LSS. Before closing this section, let us remark that in a real experiment there will be modelled and unmodelled sytematic errors to take into account. Systematic errors are expected to contribute to the AC much more significantly than to the XC, particularly on large scales (low $\ell$). This means that the SNR for the AC, once systematic effects are taken into account, will likely decrease more than that of the XC. This is one further motivation to explore the possibilities and improvements from the use of cross-correlations in UHECR anisotropy studies. Conclusions and outlook {#sec:conclusions} ======================= In this work, we have introduced a new observable for UHECR physics, the harmonic-space cross-correlation between the arrival directions of UHECRs and the distribution of the cosmic LSS as mapped by galaxies, Eq. . The focus of this work has been the development of the main theoretical tools that are necessary to model the signal and its uncertainties. The take-away points of this study are: - The cross-correlation can be easier to detect than the UHECR auto-correlation for a range of energies and multipoles (see Figs. \[fig:cl\_el\] and \[fig:sn\_el\]). This performance is mostly driven by the sheer number of galaxies that can trace the underlying LSS distribution, which is assumed to be the baseline distribution for both the UHECR flux and the galaxy angular distribution. - The cross-correlation is more sensitive to small-scale angular anisotropies than the auto-correlation. It can, therefore, be instrumental in understanding properties of UHECR sources that would not be accessible otherwise. - It is in principle possible to optimize the cross-correlation signal by assigning optimal redshift-dependent weights to sources in the galaxy catalog, to match the UHECR radial kernel as determined by UHECR energy losses. Since matching the kernels has a strong impact on the cross-correlation, it could be possible to use this effect to reverse-engineer the injection model (which defines the radial kernel). - The great disruptor of UHECR anisotropies is the GMF. The cross-correlation, with its higher signal-to-noise ratio and sensitivity to small angular scales, could be very useful in understanding the properties of the GMF (although we have not explored this angle here). Moreover, it may be possible, in the near future, to exploit a tomographic approach to disentangle the effects of intervening magnetic fields from different injection spectra, and study different regions of the sky separately. In our treatment, we do not take any experimental uncertainties into account, besides the experimental UHECR angular resolution. Moreover, we limit ourselves to a proton-only injection model and do not include the effects of the intervening magnetic fields. This choice was made in order to underline the physics behind our proposal and method, and can be readily generalized and extended to include the (theoretical and experimental) properties of the different galaxy and UHECR catalogs, different injection models, and to separate the number of events and energy cut, in order to best forecast the possibilities of present and upcoming UHECR data sets. Moreover, in this first work we have made the case for the XC between UHECRs and galaxies, but the logic and methods we have developed can be applied to other XCs with different matter tracers and different messengers. The distribution of visible matter in the sky can be traced not only by galaxies, but also by the thermal Sunyaev-Zeldovich effect. The latter is produced by the inverse Compton scattering of CMB photons by hot electrons along the line-of-sight. Because a thermal Sunyaev-Zeldovich map is a map of CMB photons, it is very accurate down to angles much smaller than a degree, and its signal peaks at low redshifts [@Erler:2017dok]. This cross-correlation could therefore be useful in further disentangling the astrophysical properties of UHECR sources. Charged UHECRs are not the only high-energy messengers whose production mechanisms and sources are not known. Recently, the IceCube collaboration has detected a few high energy astrophysical neutrinos, with energies above a PeV [@Aartsen:2014gkd]. Such neutrinos are expected to be produced in the same extreme astrophysical sources as UHECRs and/or in their immediate surroundings. The cross-correlations between neutrinos and the LSS will then inform about the properties of the highest-energy astrophysical engines. Without the use of cross-correlations, because of the very small number of neutrino events in present data, and in the forseeable future [@Sapienza:2020rte; @Allison:2015eky; @Nelles_2019], the detection of the anisotropic pattern could be challenging. Since neutrinos interact extremely weakly, they can propagate unhampered for long distances: their horizon is almost the entire visible Universe. Therefore, in addition to galaxies, complementary information could be extracted from cross-correlating neutrinos with other tracers, including CMB lensing [@Aghanim:2018oex] and cosmic shear surveys [@Mandelbaum:2017jpr], both of which trace the overall matter distribution in the Universe, including both its dark and luminous components, out to higher redshifts with broader kernels (see Refs. [@Fornengo:2014cya; @Cuoco:2015rfa; @Branchini:2016glc; @Ammazzalorso:2019wyr] for the analogous analysis with $\gamma$ rays). Measuring these cross-correlations could reveal whether the most energetic particle accelerators in the Universe preferentially reside in high-density visible or dark environments. FU wishes to thank A. di Matteo for useful correspondence. FU is supported by the European Regional Development Fund (ESIF/ERDF) and the Czech Ministry of Education, Youth and Sports (MEYS) through Project CoGraDS - `CZ.02.1.01/0.0/0.0/15_003/0000437`. SC is supported by the Italian Ministry of Education, University and Research (<span style="font-variant:small-caps;">miur</span>) through Rita Levi Montalcini project ‘<span style="font-variant:small-caps;">prometheus</span> – Probing and Relating Observables with Multi-wavelength Experiments To Help Enlightening the Universe’s Structure’, and by the ‘Departments of Excellence 2018-2022’ Grant awarded by <span style="font-variant:small-caps;">miur</span> (L. 232/2016). DA acknowledges support from the Beecroft Trust, and from the Science and Technology Facilities Council through an Ernest Rutherford Fellowship, grant reference ST/P004474/1. We would like to acknowledge SARS-Cov-2 for the peace of spirit our quarantines in three different countries have given us to finish this work. Power spectra {#app:pk_cl} ============= Three dimensional fields $\delta_a({{\bm x}})$ can be decomposed into their Fourier coefficients $$\delta_a({{\bm k}}){\coloneqq}\int {\mathrm{d}}k^3\,\delta_a({{\bm x}})\,e^{-i{{\bm k}}\cdot{{\bm x}}} \,,$$ whose covariance is the power spectrum $P_{ab}(k)$, Assuming statistical homogeneity and isotropy, it is given by $$\left\langle \delta_a({{\bm k}})\delta_a^\ast({{\bm k}}')\right\rangle{\coloneqq}\delta({{\bm k}}-{{\bm k}}')\,P_{ab}(k) \,,$$ where the angle brackets denote averaging over ensemble realizations of the random fields inside them. Equivalently, two-dimensional fields $\Delta_a({\hat{\bm n}})$ can be decomposed into their harmonic coefficients $$a_{\ell m}^a{\coloneqq}\int {\mathrm{d}}\Omega\,Y^*_{\ell m}({\hat{\bm n}})\,\Delta_a({\hat{\bm n}}) \,,$$ where $Y_{\ell m}(\theta,\varphi)$ are the spherical harmonic functions, and $a=\left\{{\rm CR},g\right\}$. The covariance of the $a_{\ell m}$ is the angular power spectrum ${\mathcal{S}}^{ab}_\ell$, defined as $$\langle a^a_{\ell m}\,a^{b*}_{\ell'm'}\rangle{\coloneqq}\delta_{\ell\ell'}\delta_{mm'}{\mathcal{S}}^{ab}_\ell \,.$$ For two projected fields, $\Delta_a$ and $\Delta_b$, associated to three-dimensional fields $\delta_a$ and $\delta_b$ via radial kernels $\phi_a$ and $\phi_b$ (as in Eqs \[eq:delta\_cr\] and \[eq:delta\_g\]), their three-dimensional and angular power spectra are related through $${\mathcal{S}}^{ab}_\ell=\frac{2}{\pi}\int {\mathrm{d}}k\,k^2\int {\mathrm{d}}\chi_1\, \phi_a(\chi_1)j_\ell(k\chi_1)\int {\mathrm{d}}\chi_2\,\phi_b(\chi_2)j_\ell(k\chi_2)\,P_{ab}(k;z_1,z_2) \,,$$ where $j_\ell(x)$ are the spherical Bessel functions. For broad kernels, we can use the Limber approximation, $j_\ell(x)\sim\sqrt{\pi/(2\ell+1)}\delta(x-\ell-1/2)$, in which case the previous relation simplifies to Eq. . [^1]: Note that our definition of emissivity differs from the one used in, e.g., radio astronomy, which quantifies the *energy* (instead of *number*) emitted per unit time, volume and solid angle.

ArXiv

--- author: - | Juan Li\ [School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, P. R. China.]{}\ date: 'June 21, 2012' title: 'Note on stochastic control problems related with general fully coupled forward-backward stochastic differential equations' --- [**Abstract.**]{} In this paper we study stochastic optimal control problems of general fully coupled forward-backward stochastic differential equations (FBSDEs). In Li and Wei [@LW] the authors studied two cases of diffusion coefficients $\sigma$ of FSDEs, in one case when $\sigma$ depends on the control and does not depend on the second component of the solution $(Y, Z)$ of the BSDE, and in the other case $\sigma$ depends on $Z$ and doesn’t depend on the control. Here we study the general case when $\sigma$ depends on both $Z$ and the control at the same time. The recursive cost functionals are defined by controlled general fully coupled FBSDEs, then the value functions are given by taking the essential supremum of the cost functionals over all admissible controls. We give the formulation of related generalized Hamilton-Jacobi-Bellman (HJB) equations, and prove the value function is its viscosity solution. [[**Keywords.**]{}  Fully coupled FBSDEs; value functions; stochastic backward semigroup; dynamic programming principle; algebraic equation; viscosity solution.]{}\ Pardoux and Peng [@PaPe1] first introduced nonlinear backward stochastic differential equations (BSDEs) driven by a Brownian motion. Since then the theory of BSDEs develops very quickly, see El Karoui, Peng and Quenez [@ELPeQu], Peng [@Pe1], [@Pe2], [@Pe3], etc. Associated with the BSDE theory, the theory of fully coupled forward-backward stochastic differential equations (FBSDEs) develops also very quickly, refer to, Antonelli [@An], Cvitanic and Ma [@CM], Delarue [@D], Hu and Peng [@HP], Ma, Protter, and Yong [@MPY], Ma, Wu, Zhang, and Zhang [@MWZZ], Ma and Yong [@MY], Pardoux and Tang [@PaT], Peng and Wu [@PW], Wu [@W], Yong [@Y], [@Y2], and Zhang [@Z], etc. For more details on fully coupled FBSDEs, the reader is referred to the book of Ma and Yong [@MY]; also refer to Li and Wei [@LW] and the references therein. Pardoux and Tang [@PaT] studied fully coupled FBSDEs (but without controls), and gave an existence result for viscosity solutions of related quasi-linear parabolic PDEs, when the diffusion coefficient $\sigma$ of the forward equation does not depend on the second component of the solution $(Y, Z)$ of the BSDE. Wu and Yu [@WY], [@WY2] studied the case when the diffusion coefficient $\sigma$ of the forward equation depends on $Z$, but, the stochastic systems without controls. In Li and Wei [@LW] they studied the optimal control problems of fully coupled FBSDEs. They studied two cases of diffusion coefficients $\sigma$ of FSDEs, that is, in one case $\sigma$ depends on the control and does not depend on $Z$, and in the other case $\sigma$ depends on $Z$ and doesn’t depend on the control. They use a new method to prove that the value functions are deterministic, satisfied the dynamic programming principle (DPP), and were viscosity solutions to the related generalized Hamilton-Jacobi-Bellman (HJB) equations. The associated generalized HJB equations are related with algebraic equations when $\sigma$ depends on $Z$ and doesn’t depend on the control. They generalized Peng’s BSDE method, and in particular, the notion of stochastic backward semigroup in [@Pe4]. When $\sigma$ depends on $Z$ makes the stochastic control much more complicate and the related HJB equation is combined with an algebraic equation, which was inspired by Wu, Yu [@WY]. However, they use the continuation method combined with the fixed point theorem to prove very clearly that the algebraic equation has a unique solution, and, moreover, also give the representation for this solution. On the other hand, they also prove some new basic estimates for fully coupled FBSDEs under the monotonic assumptions. In particular, they prove under the Lipschitz and linear growth conditions that fully coupled FBSDEs have a unique solution on the small time interval, if the Lipschitz constant of $\sigma$ with respect to $z$ is sufficiently small. They also establish a generalized comparison theorem for such fully coupled FBSDEs. Here we want to study the general case, that is, when $\sigma$ depends on $Z$ and the control at the same time, for this general case what the associated HJB equations will be. Let us be more precise. We study a stochastic control problem related with fully coupled FBSDE. The cost functional is introduced by the following fully coupled FBSDE: \[ee1.1\] { [llll]{} dX\_s\^[t,x;u]{} & = & b(s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s)ds + (s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s) dB\_s,\ dY\_s\^[t,x;u]{} & = & -f(s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s)ds + Z\_s\^[t,x;u]{}dB\_s,      s,\ X\_t\^[t,x;u]{}& = & x,\ Y\_T\^[t,x;u]{} & = & (X\_T\^[t,x;u]{}), .\ where $T>0$ is an arbitrarily fixed finite time horizon, $B=(B_s)_{s\in[0,T]}$ is a d-dimensional standard Brownian motion, and $u=(u_s)_{s\in[t,T]}$ is an admissible control. Precise assumptions on the coefficients $b,\ \sigma, \ f, \ \Phi$ are given in the next section. Under our assumptions, (\[ee1.1\]) has a unique solution $(X_s^{t,x;u},Y_s^{t,x;u},Z_s^{t,x;u})_{s\in[t,T]}$ and the cost functional is defined by \[ee1.2\] J(t,x;u)=Y\_t\^[t,x;u]{}. We define the value function of our stochastic control problems as follows: \[ee1.3\] W(t,x):=\_[u\_[t,T]{}]{}J(t,x;u). The objective of our paper is to investigate this value function. The main results of the paper state that $W$ is deterministic (Proposition 2.1), continuous viscosity solution of the associated HJB equations (Theorem 3.1). The associated HJB equation is combined with an algebraic equation as follows: \[ee1.5\] { [ll]{} & W(t,x) + H\_V(t, x, W(t,x))=0,\ &V(t,x,u)=DW(t,x).(t,x,W(t,x),V(t,x,u),u),0.5cm (t,x)\[0,T)\^n , uU,\ & W(T,x) =(x),0.5cm x\^n. . In this case $$\begin{array}{lll} H_V(t, x, W(t,x))&=& \sup\limits_{u \in U}\{DW.b(t, x, W(t,x), V(t,x,u), u)+\frac{1}{2}tr(\sigma\sigma^{T}(t, x, W(t,x), V(t,x,u),u)D^2W(t,x))\\ && +f(t, x, W(t,x), V(t,x,u), u)\}, \end{array}$$ where $t\in [0, T], x\in{\mathbb{R}}^n.$ Our paper is organized as follows: Section 2 introduces the framework of the stochastic control problems. In Section 3, we prove that $W$ is a viscosity solution of the associated HJB equation described above. [Framework]{} ============== Let $(\Omega, {\cal{F}},$ P$)$ be the classical Wiener space, where $\Omega$ is the set of continuous functions from \[0, T\] to ${\mathbb{R}}^d$ starting from 0 ($\Omega= C_0([0, T];{\mathbb{R}}^d)$), $ {\cal{F}} $ is the completed Borel $\sigma$-algebra over $\Omega$, and P is the Wiener measure. Let B be the canonical process: $B_s(\omega)=\omega_s,\ s\in [0, T],\ \omega\in \Omega$. We denote by ${\mathbb{F}}=\{{\mathcal{F}}_s,\ 0\leq s \leq T\}$ the natural filtration generated by $\{B_t\}_{t\geq0}$ and augmented by all $P\mbox{-}$null sets, i.e., $${\cal{F}}_s=\sigma\{B_r,r\leq s\}\vee \mathcal {N}_P,\ \ \ \ s\in [0,T],$$ where $\mathcal {N}_P$ is the set of all $P\mbox{-}$null subsets and $T$ is a fixed real time horizon. For any $n\geq 1,$ $|z|$ denotes the Euclidean norm of $z\in {\mathbb{R}}^{n}$. We introduce the following two spaces of processes which will be used later: ${\cal{S}}^2(t_0, T; {\mathbb{R}^n})$ is the set of $\mathbb{R}^n\mbox{-valued}\ \mathbb{F}\mbox{-adapted continuous process}\ (\psi_t)_{t_0\leq t\leq T} $ which also satisfies $E[\sup\limits_{t_0\leq t\leq T}| \psi_{t} |^2]< +\infty; $   ${\cal{H}}^{2}(t_0,T;{\mathbb{R}}^{n})$ is the set of ${\mathbb{R}}^{n}\mbox{-valued}\ \mathbb{F} \mbox{-progressively measurable process}\ (\psi_t)_{t_0\leq t\leq T}$ which also satisfies $\parallel\psi\parallel^2=E[\int^T_{t_0}| \psi_t| ^2dt]<+\infty; $ where $t_0\in [0,T].$ First we introduce the setting for stochastic optimal control problems. We suppose that the control state space U is a compact metric space. ${\mathcal{U}}$ is the set of all U-valued $ {{\mathbb {F}} }$-progressively measurable processes. If $u\in \mathcal {U}$, we call $u$ an admissible control. For a given admissible control $u(\cdot)\in {\mathcal{U}}$, we regard $t$ as the initial time and $\zeta \in L^2 (\Omega ,{\mathcal{F}}_t, P;{\mathbb{R}}^n)$ as the initial state. We consider the following fully coupled forward-backward stochastic control system \[3.1\] { [llll]{} dX\_s\^[t,,u]{} & = & b(s,X\_s\^[t,;u]{},Y\_s\^[t,;u]{},Z\_s\^[t,;u]{},u\_s)ds + (s,X\_s\^[t,;u]{},Y\_s\^[t,;u]{},Z\_s\^[t,;u]{},u\_s) dB\_s,\ dY\_s\^[t,;u]{} & = & -f(s,X\_s\^[t,;u]{},Y\_s\^[t,;u]{},Z\_s\^[t,;u]{},u\_s)ds + Z\_s\^[t,;u]{}dB\_s,            s,\ X\_t\^[t,;u]{}& = & ,\ Y\_T\^[t,;u]{} & = & (X\_T\^[t,;u]{}), . where the deterministic mappings $b: [0,T] \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^d \times U \longrightarrow \mathbb{R}^n, \ \ \ \ \ \ \ \ \ \ \ \ \ \sigma: [0,T] \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^d \times U \longrightarrow \mathbb{R}^{n\times d}, $ $f: [0,T] \times \mathbb{R}^n \times \mathbb{R} \times \mathbb{R}^d \times U \longrightarrow \mathbb{R},\ \ \ \ \ \ \ \ \ \ \ \ \ \ \Phi: \mathbb{R}^n \longrightarrow \mathbb{R}$\ are continuous in $(t, u)\in [0, T]\times U$. In this paper we use the usual inner product and the Euclidean norm in $\mathbb{R}^n, \mathbb{R}^m$ and $\mathbb{R}^{m\times d},$ respectively. Given an $m \times n$ full-rank matrix G, we define: $$\lambda= \ \left(\begin{array}{c} x\\ y\\ z \end{array}\right) \ , \ \ \ \ \ \ \ \ \ \ A(t,\lambda)= \ \left(\begin{array}{c} -G^Tf\\ Gb\\ G\sigma \end{array}\right)(t,\lambda),$$where $G^T$ is the transposed matrix of $G$. We assume that (B1) (i) $A(t,\lambda)$ is uniformly Lipschitz with respect to $\lambda$, and for any $\lambda$, $A(\cdot,\lambda)\in $ 1.4cm${\cal{H}}^{2}(0,T;{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}\times{\mathbb{R}}^{m\times d});$         (ii) $\Phi(x)$ is uniformly Lipschitz with respect to $x\in \mathbb{R}^n$, and for any $x\in \mathbb{R}^n,\Phi(x)\in$ 1.4cm$ L^2(\Omega,\mathcal {F}_T,\mathbb{R}^m).$ The following monotonicity conditions are also necessary: (B2) (i) $\langle A(t,\lambda)-A(t,\overline{\lambda}),\lambda-\overline{\lambda} \rangle \leq -\beta_1|G\widehat{x}|^2-\beta_2(|G^T \widehat{y}|^2+|G^T \widehat{z}|^2),$ 0.8cm (ii) $ \langle \Phi(x)-\Phi(\overline{x}),G(x-\overline{x}) \rangle \geq \mu_1|G\widehat{x}|^2,\ \ \widehat{x}=x-\bar{x},\ \widehat{y}=y-\bar{y},\ \widehat{z}=z-\bar{z}$,\ where $\beta_1,\ \beta_2,\ \mu_1$ are nonnegative constants with $\beta_1 + \beta_2>0,\ \beta_2 + \mu_1>0$. Moreover, we have $\beta_1>0,\ \mu_1>0 \ (\mbox{resp., }\beta_2>0)$, when $m>n$ (resp., $m<n$). [(B3)-(ii)]{} $ \langle \Phi(x)-\Phi(\overline{x}),G(x-\overline{x}) \rangle \geq 0.$ The coefficients satisfy the assumptions (B1) and (B2), and also\ (B4) there exists a constant $K\geq 0$ such that, for all $t\in [0, T],\ u\in U,\ x_1,\ x_2\in\mathbb{R}^n, \ y_1,\ y_2\in\mathbb{R},\ z_1,\ z_2\in\mathbb{R}^d,$ $$|l(t,x_1,y_1,z_1,u)-l(t,x_2,y_2,z_2,u)|\leq K(|x_1-x_2|+|y_1-y_2|+|z_1-z_2|),$$       $l=b, \sigma, f$, respectively, and $|\Phi(x_1)-\Phi(x_2)|\leq K|x_1-x_2|$. Under our assumptions, it is obvious that there exists a constant $C\geq 0$ such that, $$|b(t,x,y,z,u)|+|\sigma(t,x,y,z,u)|+|f(t,x,y,z,u)|+|\Phi(x)|\leq C(1+|x|+|y|+|z|),$$ for all $(t,x,y,z,u)\in [0,T]\times \mathbb{R}^n\times\mathbb{R}\times\mathbb{R}^d\times U.$ Hence, for any $u(\cdot) \in \mathcal {U},$ from Lemma 2.4 in [@LW], FBSDE (\[3.1\]) has a unique solution. From Proposition 6.1 in [@LW], there exists $C \in \mathbb{R}^+$ such that, for any $t \in [0,T]$, $\zeta, \zeta' \in L^2(\Omega,\mathcal {F}_t,P;\mathbb{R}^n),$ $u(\cdot) \in \mathcal {U},$ we have, : \[3.2\] [llll]{} &E\[\_[tsT]{}|X\_s\^[t,;u]{}-X\_s\^[t,’;u]{}|\^2+\_[tsT]{}|Y\_s\^[t,;u]{}-Y\_s\^[t,’;u]{}|\^2+\_t\^T|Z\_s\^[t,;u]{}-Z\_s\^[t,’;u]{}|\^2ds\_t\] C|- ’|\^2,\ & E\[\_[tsT]{}|X\_s\^[t,;u]{}|\^2+\_[tsT]{}|Y\_s\^[t,;u]{}|\^2+ \_t\^T|Z\_s\^[t,;u]{}|\^2ds \_t\] C(1 +||\^2).\ Therefore, we get \[3.3\] [llll]{}&&[(i)]{}   |Y\_t\^[t,;u]{}| C(1+||), ;\ &&[(ii)]{}  |Y\_t\^[t,;u]{}-Y\_t\^[t,’;u]{}| C|- ’|, We now introduce the subspaces of admissible controls. An admissible control process $u=(u_r)_{r\in [t,s]}$ on $[t,s]$ is an $\mathbb{F}$-progressively measurable, $U$-valued process. The set of all admissible controls on $[t,s]$ is denote $\mathcal {U}_{t,s},\ t\leq s\leq T.$ For a given process $u(\cdot) \in \mathcal {U}_{t,T}$, we define the associated cost functional as follows: \[3.4\] J(t,x;u):=Y\_s\^[t,x;u]{}\_[s=t]{},   (t,x) \^n,where the process $Y^{t,x;u}$ is defined by FBSDE (\[3.1\]). From Theorem 6.1 in [@LW] we have, for any $t \in [0,T]$ and $\zeta \in L^2(\Omega,\mathcal {F}_t,P;\mathbb{R}^n),$ \[3.5\] J(t,;u)=Y\_t\^[t,;u]{}, For $ \zeta = x \in \mathbb{R}^n,$ we define the value function as \[3.6\] W(t,x) := \_[u\_[t,T]{}]{}J(t,x;u). From the assumptions (B1) and (B2), the value function $W(t,x)$ as the essential supremum over a family of ${\cal F}_t$-measurable random variables is well defined and it is a bounded ${\mathcal {F}}_t$-measurable random variable too. But it turns out to be even deterministic. In fact, inspired by the method in Buckdahn and Li [@BL], we can prove that $W$ is deterministic. Under the assumptions (B1) and (B2), for any $(t,x) \in [0,T] \times \mathbb{R}^n,$ $W(t,x)$ is a deterministic function in the sense that $W(t,x) = E[W(t,x)], \mbox{P-a.s.}$ The proof can be consulted in Li and Wei [@LW], Proposition 3.1.\ From (\[3.3\]) and (\[3.6\]) we get the following property of the value function $W(t,x)$: There exists a constant $C>0$ such that, for all $0 \leq t \leq T,\ x,x' \in \mathbb{R}^n,$ [llll]{}&& [(i)]{}  |W(t,x)-W(t,x’)| C|x-x’|;\ && [(ii)]{}  |W(t,x)| C(1+|x|). Under the assumptions (B1) and (B2), the cost functional $J(t,x;u),$ for any $u\in \mathcal {U}_{t,T}$, and the value function $W(t,x)$ are monotonic in the following sense: for each $x,\ \bar{x} \in \mathbb{R}^n,$ $t\in [0,T],$ $$\begin{array}{llll} &&{\rm(i)}\ \ \langle J(t,x;u)-J(t,\bar{x};u),\ G(x-\bar{x})\rangle\geq 0,\ \mbox{P-a.s.};\\ &&{\rm(ii)}\ \ \langle W(t,x)-W(t,\bar{x}),\ G(x-\bar{x})\rangle \geq 0.\end{array}$$ For the proof the reader is referred to Lemma 3.3 in Li and Wei [@LW]. (1) From (B2)-(i) we see that if $\sigma$ doesn’t depend on $z$, then $\beta_2=0$. Furthermore, we assume that:\ $\begin{array}{llll} &&{\rm{(B5)}}\ \mbox{the Lipschitz constant}\ L_\sigma\geq 0 \ \mbox{of}\ \sigma\ \mbox{with respect to}\ z\ \mbox{is sufficiently small, i.e., there exists some}\\ &&\ \ \ \ \ \ L_\sigma\geq 0\ \mbox{small enough such that}, \ \mbox{for all}\ t\in[0, T],\ u\in U,\ x_1,\ x_2\in\mathbb{R}^n,\ y_1,\ y_2\in\mathbb{R},\ z_1,\ z_2\in\mathbb{R}^d,\\ &&\ \ \ \ \ \ |\sigma(t,x_1,y_1,z_1,u)-\sigma(t,x_2,y_2,z_2,u)|\leq K(|x_1- x_2|+|y_1-y_2|)+L_\sigma|z_1-z_2|. \end{array}$\ (2) On the other hand, notice that when $\sigma$ doesn’t depend on $z$ it’s obvious that (B5) always holds true. The notation of stochastic backward semigroup was first introduced by Peng [@Pe4] and was applied to prove the DPP for stochastic control problems. Now we discuss a generalized DPP for our stochastic optimal control problem (\[3.1\]), (\[3.6\]). For this we have to adopt Peng’s notion of stochastic backward semigroup, and to define the family of (backward) semigroups associated with FBSDE (\[3.1\]). For given initial data $(t,x)$, a real $0<\delta \leq T-t,$ an admissible control process $u(\cdot)\ \in \ \mathcal {U}_{t,t+\delta}$ and a real-valued ${\cal F}_{t+\delta}\otimes {\cal B}(\mathbb{R}^n)$-measurable random function $\Psi: \Omega\times \mathbb{R}^n\rightarrow \mathbb{R}$, such that (B2)-(ii) holds, we put $$G_{s,t+\delta}^{t,x;u}[\Psi(t+\delta, \widetilde{X}_{t+\delta}^{t,x;u})]:=\widetilde{Y}_s^{t,x;u}, \ s \in [t,t+\delta],$$ where $(\widetilde{X}_s^{t,x;u},\widetilde{Y}_s^{t,x;u},\widetilde{Z}_s^{t,x;u})_{t \leq s \leq t+\delta}$ is the solution of the following FBSDE with time horizon $t+\delta$: \[3.8\] { [llll]{} d\_s\^[t,x;u]{} & = & b(s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)ds + (s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)dB\_s,\ d\_s\^[t,x;u]{} & = & -f(s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)ds + \_s\^[t,x;u]{}dB\_s,      s,\ \_t\^[t,x;u]{} & = & x,\ \_[t+]{}\^[t,x;u]{} & = & (t+, \_[t+]{}\^[t,x;u]{}). . From Lemmas 2.4 and 2.5 in [@LW] we know that, if $\Psi$ doesn’t depend on $x$, FBSDE (\[3.8\]) has a unique solution $(\widetilde{X}^{t,x;u},\widetilde{Y}^{t,x;u},\widetilde{Z}^{t,x;u}).$\ [(ii)]{} We also point out that if $\Psi$ is Lipschitz with respect to $x$, FBSDE (\[3.8\]) can be also solved under the assumptions (B4) and (B5) on the small interval $[t, t+\delta]$, for any $0\leq \delta\leq \delta_0,$ where the small parameter $\delta_0>0$ is independent of $(t, x)$ and the control $u$; see Proposition 6.4 in [@LW]. Since $\Phi$ satisfies (B2)-(ii), the solution $(X^{t,x;u},Y^{t,x;u},Z^{t,x;u})$ of FBSDE (\[3.1\]) exists and we get $$G_{t,T}^{t,x;u}[\Phi(X_T^{t,x;u})] = G_{t,t+\delta}^{t,x;u}[Y_{t+\delta}^{t,x;u}].$$ Moreover, we have \[3.9\] J(t,x;u)=Y\_t\^[t,x;u]{}=G\_[t,T]{}\^[t,x;u]{}\[(X\_T\^[t,x;u]{})\] =G\_[t,t+]{}\^[t,x;u]{}\[Y\_[t+]{}\^[t,x;u]{}\]=G\_[t,[t+]{}]{}\^[t,x;u]{}\[J(t+,X\_[t+]{}\^[t,x;u]{};u)\]. Under the assumptions (B1), (B2), (B4) and (B5), the value function $W(t,x)$ satisfies the following DPP: there exists a sufficiently small $\delta_0>0$, such that for any $0\leq \delta \leq \delta_0,\ t\in [0, T-\delta],\ x \in \mathbb{R}^n,$ $$W(t,x)=\esssup_{u\in \mathcal {U}_{t,t+\delta}}G_{t,{t+\delta}}^{t,x;u}[W(t+\delta,\widetilde{X}_{t+\delta}^{t,x;u})].$$ The proof refers to Theorem 3.1 in Li and Wei [@LW]. Notice that from the definition of our stochastic backward semigroup we know that $$G_{s,t+\delta}^{t,x;u}[W(t+\delta,\widetilde{X}_{t+\delta}^{t,x;u})]=\widetilde{Y}_s^{t,x;u}, \ s \in [t,t+\delta],\ u(\cdot)\ \in \ \mathcal {U}_{t,t+\delta},$$ where $(\widetilde{X}_s^{t,x;u},\widetilde{Y}_s^{t,x;u},\widetilde{Z}_s^{t,x;u})_{t \leq s \leq t+\delta}$ is the solution of the following FBSDE with time horizon $t+\delta$: \[3.10\] { [llll]{} d\_s\^[t,x;u]{} & = & b(s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)ds + (s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)dB\_s,\ d\_s\^[t,x;u]{} & = & -f(s,\_s\^[t,x;u]{},\_s\^[t,x;u]{},\_s\^[t,x;u]{},u\_s)ds + \_s\^[t,x;u]{}dB\_s,      s,\ \_t\^[t,x;u]{} & = & x,\ \_[t+]{}\^[t,x;u]{} & = & W(t+,\_[t+]{}\^[t,x;u]{}). . From Proposition 6.4 in [@LW] there exists a sufficiently small $\delta_0>0$, such that for any $0\leq \delta \leq \delta_0,$ the above equation (\[3.10\]) has a unique solution $(\widetilde{X}^{t,x;u},\widetilde{Y}^{t,x;u},\widetilde{Z}^{t,x;u})$ on the time interval $[t, t+\delta]$. From Lemma 2.2, we get the Lipschitz property of the value function $W(t,x)$ in $x$, uniformly in $t$. Now from Theorem 2.1 we can get the continuity property of $W(t,x)$ in $t$ and to conclude: Under the assumptions (B1), (B2), (B4) and (B5) $W(t,x)$ is continuous in $t$. Its proof can be found in Li and Wei [@LW], Theorem 3.2. Viscosity Solutions of HJB Equations ==================================== In this section we show that the value function $W(t, x)$ defined in (\[3.6\]) is a viscosity solution of the corresponding HJB equation. For this we use Peng’s BSDE approach [@Pe4] developed for stochastic control problems of decoupled FBSDEs. Let us consider equation (\[3.1\]): \[4.1\] { [llll]{} dX\_s\^[t,x;u]{} & = & b(s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s)ds + (s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s) dB\_s,\ dY\_s\^[t,x;u]{} & = & -f(s,X\_s\^[t,x;u]{},Y\_s\^[t,x;u]{},Z\_s\^[t,x;u]{},u\_s)ds + Z\_s\^[t,x;u]{}dB\_s,      s,\ X\_t\^[t,x;u]{}& = & x,\ Y\_T\^[t,x;u]{} & = & (X\_T\^[t,x;u]{}). . The related HJB equation is the following PDE combined with the algebraic equation: \[4.2\] { [ll]{} & W(t,x) + H\_V(t, x, W(t,x))=0,\ &V(t,x,u)=DW(t,x).(t,x,W(t,x),V(t,x,u),u),0.5cm (t,x)\[0,T)\^n , uU,\ & W(T,x) =(x),    x\^n, . where $$\begin{array}{lll} H_V(t, x, W(t,x))= \sup\limits_{u \in U}\{ DW(t,x).b(t, x, W(t,x), V(t,x,u), u)+f(t, x, W(t,x), V(t,x,u), u)\\ \hskip 3.5cm +\frac{1}{2}tr(\sigma\sigma^{T}(t, x, W(t,x), V(t,x,u),u)D^2W(t,x))\},\ t\in [0, T],\ x\in{\mathbb{R}}^n. \end{array}$$ We give the definition of viscosity solution for this kind of PDE. For more details on viscosity solution refer to Crandall, Ishii, Lions [@CIL]. A real-valued continuous function $W\in C([0,T]\times {\mathbb{R}}^n )$ is called\ [(i)]{} a viscosity subsolution of equation (\[4.2\]) if $W(T,x) \leq \Phi (x),\ \mbox{for all}\ x \in {\mathbb{R}}^n$, and if for all functions $\varphi \in C^3_{l, b}([0,T]\times {\mathbb{R}}^n)$ and for all $(t,x) \in [0,T) \times {\mathbb{R}}^n$ such that $W-\varphi $ attains a local maximum at $(t, x)$, $$\left \{\begin{array}{ll} &\!\!\!\!\! \frac{\partial \varphi}{\partial t} (t,x) + H_{\psi}(t,x,\varphi(t,x)) \geq 0,\\ &\!\!\!\!\!\mbox{where }\psi \mbox{ is the unique solution of the following algebraic equation:}\\ &\!\!\!\!\!\psi(t,x,u)=D\varphi(t,x).\sigma(t,x,\varphi(t,x),\psi(t,x,u),u),\ u\in U. \end{array}\right.$$ [(ii)]{} a viscosity supersolution of equation (\[4.2\]) if $W(T,x) \geq \Phi (x), \mbox{for all}\ x \in {\mathbb{R}}^n$, and if for all functions $\varphi \in C^3_{l, b}([0,T]\times {\mathbb{R}}^n)$ and for all $(t,x) \in [0,T) \times {\mathbb{R}}^n$ such that $W-\varphi $ attains a local minimum at $(t, x)$, $$\left \{\begin{array}{ll} &\!\!\!\!\! \frac{\partial \varphi}{\partial t} (t,x) + H_{\psi}(t,x,\varphi(t,x)) \leq 0,\\ &\!\!\!\!\!\mbox{where }\psi \mbox{ is the unique solution of the following algebraic equation:}\\ &\!\!\!\!\! \psi(t,x,u)=D\varphi(t,x).\sigma(t,x,\varphi(t,x,u),\psi(t,x),u),\ u\in U. \end{array}\right.$$ [(iii)]{} a viscosity solution of equation (\[4.2\]) if it is both a viscosity sub- and supersolution of equation (\[4.2\]). When $\sigma$ depends on $z$ we need the test function $\varphi$ in Definition 3.1 satisfies the monotonicity condition [(B3)-(ii)]{} and also the following technical assumptions:\ (B6) $\beta_2>0$;\ (B7) $G\sigma(s, x, y, z)$ is continuous in $(s, u)$, uniformly with respect to $(x, y, z)\in {\mathbb{R}}\times {\mathbb{R}}^n\times {\mathbb{R}}^d$. Under the assumptions (B1), (B2), (B4), (B5), (B6) and (B7), the value function $W$ is a viscosity solution of (\[4.2\]). We have the following important Representation Theorem for the solution of the algebraic equation. For any $s\in [0, T],\ \zeta\in\mathbb{R}^d,\ y\in \mathbb{R},\ \bar{x}\in \mathbb{R}^n,\ u\in U$, there exists a unique $z$ such that $z=\zeta+D\varphi(s,\bar{x})\sigma(s,\bar{x},y+\varphi(s,\bar{x}),z,u). $ That means, the solution $z$ can be written as $z=h(s, \bar{x},y, \zeta,u)$, where the function $h$ is Lipschitz with respect to $\ y, \ \zeta,$ and $|h(s, \bar{x},y, \zeta,u)|\leq C(1+|\bar{x}|+|y|+|\zeta|).$ The constant $C$ is independent of $s,\ \bar{x},\ y, \ \zeta,\ u$. And $z=h(s, \bar{x},y, \zeta,u)$ is continuous with respect to $(s,u)$. **Proof**. It’s obvious that we only need to consider the case when $\sigma$ depends on $z$. Then we have (B6) and (B7) hold. Furthermore, following the proof of Proposition 4.1 in [@LW] we can prove it. **Proof of Theorem 3.1**. Following the proof of Theorem 4.2 in [@LW] we can prove it. 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--- abstract: 'Preliminary results of the strong interaction shift and width in pionic hydrogen ($\pi H$) using an X-ray spectrometer with spherically bent crystals and CCDs as X-ray detector are presented. In the experiment at the Paul Scherrer Institute three different $(np\to1s)$ transitions in $\pi H$ were measured. Moreover the pion mass measurement using the $(5 \to 4)$ transitions in pionic nitrogen and muonic oxygen is presented' author: - | Martino Trassinelli [^1]\ *Laboratoire Kastler Brossel,Université P. et M. Curie, F-75252 Paris, France* title: ' PRECISION SPECTROSCOPY OF PIONIC ATOMS: FROM PION MASS EVALUATION TO TESTS OF CHIRAL PERTURBATION THEORY' --- =11.6pt Introduction ============ Pionic hydrogen atoms are unique systems to study the strong interaction at low energies[@gotta2004]. The influence of the strong interaction in pionic hydrogen can be extracted from the $(np\to1s)$ transitions. Compared to pure electromagnetic interaction, the 1s level is affected by an energy shift $\epsilon_{1s}$ and a line broadening $\Gamma_{1s}$. The shift and the broadening are related to the hadronic scattering lengths $a^h_{\pi^-\ p \to \pi^-\ p }$ and $a^h_{\pi^-\ p \to \pi^0\ n }$, by the Deser-type formulae [@deser]: $$\frac{\epsilon_{1s}}{E_{1s}}=-4\frac{1}{r_B}a^h_{(\pi^- \ p \to \pi^- \ p)}(1+\delta_\epsilon) \label{eq:deser1}$$ $$\frac{\Gamma_{1s}}{E_{1s}}=8 \frac{Q_0}{r_B}(1+\frac{1}{P})(a^h_{(\pi^- \ p \to \pi^0 \ n)}(1+\delta_\Gamma))^2 \label{eq:deser2}$$ where $\epsilon_{1s}$ is the strong interaction shift of the 1s level reflecting the $\pi\,p$ scattering process. $\Gamma_{1s}$ is the width of the ground state caused by the reactions $\pi^- + p \to \pi^0 + n$ and $\pi^- + p \to \pi^0 + \gamma $. $Q_0=0.1421~fm^{-1}$ is the kinetic center of mass momentum of the $\pi^0$ in $\pi^- + p \to \pi^0 + n$ reaction, and $P=1.546 \pm 0.009$[@spuller1977] is the branching ratio of the charge exchange and radiative capture (Panofsky ratio). $\delta_{\epsilon,\Gamma}$ are corrections that permit to connect the pure hadronic scattering lengths to the measurable shift and width[@gasser2002; @sigg1996th; @ericson2003]. The hadronic scattering lengths can be related to the isospin-even and isospin-odd scattering length, $a^+$ and $a^-$: $$a^h_{(\pi^- \ p \to \pi^- \ p)}= a^+ + a^-\qquad a^h_{(\pi^- \ p \to \pi^0 \ n)}=-\sqrt{2}\ a^-$$ The isospin scattering lengths can ben related to $\epsilon_{1s}$ and $\Gamma_{1s}$ in the framework of the Heavy Barion Chiral Perturbation Theory ($\chi$PT)[@lyubovitskij2000]. Scattering experiments are restricted to energies above 10 MeV and have to rely on an extrapolation to zero energy to extract the scattering lengths. Pionic hydrogen spectroscopy permits to measure this scattering length at almost zero energy (in the same order as the binding energies, i.e., some keV) and verify with high accuracy the $\chi$PT calculations. Moreover, the measurement of $\Gamma_{1s}$ allows an evaluation of the pion-nucleon coupling constant $f_{\pi N}$, which is related to $a^-$ by the Goldberger-Miyazawa-Oehme sum rule (GMO)[@GMO]. Pionic atom spectroscopy permits to measure another important quantity: the charged pion mass. Orbital energies of pionic atoms depend on the reduced mass of the system. These energies can be calculated with high accuracy using Quantum Electrodynamics. Measuring transition energies, not disturbed by strong interaction, allows to determine the reduced mass of the system and hence the mass of the pion. The accurate value of the pion mass is crucial to evaluate the upper bound of the mass of the muonic neutrino from a measurement of the pion decay[@nelms2002]. Description of the setup ======================== The pionic atoms are produced using the pion beam provided by the Paul Scherrer Institut[@proposals]. The beam momentum is 110 MeV/c with an intensity of $10^{8}~{\rm s}^{-1}$. The pions are captured and slowed down using a cyclotron trap[@simons1993]. The target is made of a cylindrical cell with Kapton walls, positioned in the center of the trap. In the target cell the decelerated pions are captured in bound atomic states. During the de-excitation X-rays are emitted. As the muons from the pion decay in the beam are present as well, it is possible to produce muonic atoms and pionic atoms at the same time. The X-ray transition energies are measured using a bent crystal spectrometer and a position sensitive detector. The reflection angle $\Theta_B$ between the crystal planes and the X-rays is related to the photon wavelength $\lambda = h c / E$ by the Bragg formula: $$n\ \lambda = 2\ d \sin \Theta_B$$ where [*n*]{} is the order of the reflection and $d$ is the spacing of the crystal planes.\ The detector is formed by an array of 6 CCDs composed each by $600 \times 600$ pixels[@nelms2002], the pixel size is $40~\mu m$. The 3-4 keV X-rays excite mostly one or two pixels. Larger clusters are due to charged particle or high-energy gamma radiation and can be eliminated by cluster analysis. Transitions of different energies result in different reflection lines on the detector. By measuring the distance between these lines it is possible to determine the energy difference. The resolution of the spectrometer is of the order of $0.4~eV$ at 3 keV. Extraction of the hadronic shift and width ========================================== The characteristics of the ground state of pionic hydrogen are evaluated measuring the X-ray transitions $np \to 1s$ (see fig.\[spectra\]). The line width is the result of the convolution of: the spectrometer resolution, the Doppler broadening effect from the non-zero atom velocity, the natural width of the ground state, and, of course, the hadronic broadening. A very accurate measurement of the response function of the crystal was performed using the $1s2s\,^{3}S_{1}\to1s^{2}\,^{1}S_{0}$ M1 transitions in He-like argon (with a natural line width less than 1 meV, Doppler broadening about 40 meV). For this measurement the cyclotron trap was converted into an Electron-Resonance Ion Trap (ECRIT)[@anagnostopoulos2003], with the crucial point that the geometry of the setup was preserved.\ The Doppler broadening effect in the pionic transitions can be studied by working at different pressures and with different transitions. With the help of a cascade model we can predict the kinetic energy distribution of the atoms and the corresponding Doppler broadening[@jensen2002a].\ A first series of measurements were completed in 2002. The hadronic broadening $\Gamma_{1s}$ extracted from the experimental line width is: $$\Gamma_{1s}= 0.80 \pm 0.03~eV$$ By varying the target density, we were able to prove that the formation of complex systems $\pi p + H_2 \to [(\pi p p ) p ] e e$[@jonsell1999], which can add an additional shift to the ground state, is negligible. Energy calibration for the $\pi H(3p \to 1s)$ transition is performed using the $6h \to 5g$ transition in pionic oxygen. Strong interaction and finite nucleus size effect are negligible for this transition. Orbital energies can be calculated with an accuracy of a few meV[@paul]. The result for the shift is: $$\epsilon_{1s}=7.120 \pm 0.017~eV$$ For the calculation of the shift a pure QED value of $E^{QED}_{3p-1s}=2878.809~eV$ was used. The above given errors include statistical accuracy and systematic effects [@maik2003]. The value of $\epsilon_{1s}$ is in agreement with the result of a previous experiment, where the energy calibration was performed with $K \alpha$ fluorescence X-rays[@schroder2001], but more precise by a factor of 3. ![*Left: $3p \to 1s$ transition measurement of pionic hydrogen. Right: $5g \to 4f$ transition in pionic oxygen and muonic hydrogen.*[]{data-label="spectra"}](piH_3p-1s.ps "fig:"){width="47.00000%"} ![*Left: $3p \to 1s$ transition measurement of pionic hydrogen. Right: $5g \to 4f$ transition in pionic oxygen and muonic hydrogen.*[]{data-label="spectra"}](fit-mix-txt.ps "fig:"){width="47.00000%"} Pion mass measurement ===================== The evaluation of the pion mass is obtained by the measurement of the transition energy of the $5g \to 4f$ transition in pionic nitrogen in 2000 (see fig.\[spectra\]). We used the analog transition in muonic oxygen as a reference line. The energy difference between the two lines depends on the ratio between the pion mass and the muon mass, which is known with 0.05 ppm accuracy. The expected accuracy for the pion mass is less than 2 ppm, to be compared with the actual value, which has an accuracy of 2.5 ppm. This value is the average of two measurements, obtained using two different techniques and which differ by 5.4 ppm[@PDG2004]. To reach this precision, we need a perfect understanding of the crystal spectrometer aberrations and the exact distance between pixels in the detector. For the second task an experiment was set up in September 2003 to measure the pixel distance at the working temperature of $-100^\circ$C. We used a nanometric grid composed by $21 \times 14$ lines, $20~\mu m$ thick, spaced by 2 mm with an accuracy of about $0.05~\mu m $. The mask, at room temperature, was illuminated by a point-like source at a distance of 6426.7 mm, and positioned at 37 mm from the CCD detector (see fig.\[mask\]). ![*Detail of the grid image on the CCD detector with the selected zones for the linear fit.* []{data-label="mask"}](WaferImage11_1_zones.ps){height="50.00000%"} Applying linear fits to the lines of the grid in the CCD image it was possible to provide an accurate measurement of the average pixel distance: $$pixel\ distance = 39.9943 \pm 0.0035~\mu m$$ Conclusions and outlook ======================= The strong interaction shift in pionic hydrogen has been determined with an accuracy of 0.2%. During spring-summer 2004 the crystals have been characterized with X-rays from the ECRIT. The measurement of the broadening in muonic hydrogen in November-December 2004, together with the cascade model, will allow us to reach an accuracy of 1% for $\Gamma_{1s}$. Acknowledgments =============== We thank the PSI staff, in particular the nanotechnology group, which provided the nanometric mask for the pixel measurement. [99]{} PSI experiment R-97.02 and R-98.01: http://pihydrogen.web.psi.ch D. Gotta, Prog. Part. Nucl. Phys.  [**52**]{}, 133 (2004) S. Deser [*et al.*]{}, Phys. Rev. [**96**]{}, 774 (1954),\ G. Rasche and W.S. Woolcock, Nucl. Phys. A [**381**]{}, 405 (1982) J. Spuller [*et al.*]{}, Phys. Lett. A [**67**]{}, 479 (1977) V. E. Lyubovitskij and A. Rusetsky, Phys. Lett. B [**494**]{}, 9 (2000) M.L. Goldberger, H. Miyazawa, R. Oehme, Phys. Rev. [**99**]{}, 986 (1955) J. Gasser [*et al.*]{}, Eur. Phys. J. C [**26** ]{}, 13 (2002) D. Sigg [*et al.*]{}, Nucl. Phys. A [**609**]{}, 310 (1996) T.E. Ericson [*et al.*]{}, arXiv:hep-ph/0310134v1 (2003) K. Assamagan [*et al.*]{}, Phys. Rev. D [**53**]{}, 6065 (1996) N. Nelms [*et al.*]{}, Nuc. Instr. Meth. A [**59**]{}, 419 (2002) L.M. Simons, Hyperfine Iteractions [**81**]{}, 253 (1993) D.F. Anagnostopoulos [*et al.*]{}, Nucl. Instrum. Meth. B [**205**]{}, 9 (2003) T.S. Jensen and V.E. Markushin, Eur. Phys. J. D [**19**]{}, 165 (2002) S. Jonsell [*et al.*]{}, Phys. Rev. A [**59**]{}, 3440 (1999) Paul Indelicato, private comunication. M. Hennebach, thesis Universität zu Köln, 2003 H.C. Schröder [*et al.*]{}, Eur. Phys. J. C [**21**]{}, 473 (2001) Particle Data Group, Phys. Lett. B [**592**]{}, 1 (2004) [^1]: On behalf of the PIONIC HYDROGEN and PION MASS collaboration[@proposals]

ArXiv

--- abstract: 'The growing number of extragalactic high-energy (HE, $E > 100$ MeV) and very-high-energy (VHE, $E > 100$ GeV) $\gamma$-ray sources that do not belong to the blazar class suggests that VHE $\gamma$-ray production may be a common property of most radio-loud Active Galactic Nuclei (AGN). In a previous paper, we have investigated the signatures of Compton-supported pair cascades initiated by VHE $\gamma$-ray absorption in monochromatic radiation fields, dominated by Ly$\alpha$ line emission from the Broad Line Region. In this paper, we investigate the interaction of nuclear VHE $\gamma$-rays with the thermal infrared radiation field from a circumnuclear dust torus. Our code follows the spatial development of the cascade in full 3-dimensional geometry. We provide a model fit to the broadband SED of the dust-rich, $\gamma$-ray loud radio galaxy Cen A and show that typical blazar-like jet parameters may be used to model the broadband SED, if one allows for an additional cascade contribution to the [*Fermi*]{} $\gamma$-ray emission.' author: - 'P. Roustazadeh and M. Böttcher' title: 'VHE Gamma-Ray Induced Pair Cascades in the Radiation Fields of Dust Tori of AGN: Application to Cen A' --- Introduction ============ Blazars are a class of radio-loud active galactic nuclei (AGNs) comprised of Flat-Spectrum Radio Quasars (FSRQs) and BL Lac objects. Their spectral energy distributions (SEDs) are characterized by non-thermal continuum spectra with a broad low-frequency component in the radio – UV or X-ray frequency range and a high-frequency component from X-rays to $\gamma$-rays, and they often exhibit substantial variability across the electromagnetic spectrum. In the VHE $\gamma$-ray regime, the time scale of this variability has been observed to be as short as just a few minutes [@albert07; @aharonian07]. While previous generations of ground-based Atmospheric Cherenkov Telescope (ACT) facilities detected almost exclusively high-frequency peaked BL Lac objects (HBLs) as extragalactic sources of VHE $\gamma$-rays (with the notable exception of the radio galaxy M87), in recent years, a number of non-HBL blazars and even non-blazar radio-loud AGN have been detected by the current generation of ACTs. This suggests that most blazars might be intrinsically emitters of VHE $\gamma$-rays. According to AGN unification schemes [@up95], radio galaxies are the mis-aligned parent population of blazars with the less powerful FR I radio galaxies corresponding to BL Lac objects and FR II radio galaxies being the parent population of radio-loud quasars. Blazars are those objects which are viewed at a small angle with respect to the jet axis. If this unification scheme holds, then, by inference, also radio galaxies may be expected to be intrinsically emitters of VHE $\gamma$-rays within a narrow cone around the jet axis. While there is little evidence for dense radiation environments in the nuclear regions of BL Lac objects — in particular, HBLs —, strong line emission in Flat Spectrum Radio Quasars (FSRQs) as well as the occasional detection of emission lines in the spectra of some BL Lac objects [e.g., @vermeulen95] indicates dense nuclear radiation fields in those objects. This is supported by spectral modeling of the SEDs of blazars using leptonic models which prefer scenarios based on external radiation fields as sources for Compton scattering to produce the high-energy radiation in FSRQs, LBLs and also some IBLs [e.g., @ghisellini98; @madejski99; @bb00; @acciari08]. If the VHE $\gamma$-ray emission is indeed produced in the high-radiation-density environment of the broad line region (BLR) and/or the dust torus of an AGN, it is expected to be strongly attenuated by $\gamma\gamma$ pair production [e.g. @pb97; @donea03; @reimer07; @liu08; @sb08]. [@akc08] have suggested that such intrinsic $\gamma\gamma$ absorption may be responsible for producing the unexpectedly hard intrinsic (i.e., after correction for $\gamma\gamma$ absorption by the extragalactic background light) VHE $\gamma$-ray spectra of some blazars at relatively high redshift. A similar effect has been invoked by [@ps10] to explain the spectral breaks in the [*Fermi*]{} spectra of $\gamma$-ray blazars. This absorption process will lead to the development of Compton-supported pair cascades in the circumnuclear environment [e.g., @bk95; @sb10; @rb10]. In [@rb10], we considered the full 3-dimensional development of a Compton-supported VHE $\gamma$-ray induced cascade in a monochromatic radiation field. This was considered as an approximation to BLR emission dominated by a single (e.g., Ly$\alpha$) emission line. In that work, we showed that for typical radio-loud AGN parameters rather small ($\sim \mu$G) magnetic fields in the central $\sim 1$ pc of the AGN may lead to efficient isotropization of the cascade emission in the [*Fermi*]{} energy range. We applied this idea to fit the [*Fermi*]{} $\gamma$-ray emission of the radio galaxy NGC 1275 [@abdo09a] under the plausible assumption that this radio galaxy would appear as a $\gamma$-ray bright blazar when viewed along the jet axis. In this paper, we present a generalization of the Monte-Carlo cascade code developed in [@rb10] to arbitrary radiation fields. In particular, we will focus on thermal blackbody radiation fields, representative of the emission from a circum-nuclear dust torus. In Section \[setup\] we will outline the general model setup and assumptions and describe the modified Monte-Carlo code that treats the full three-dimensional cascade development. Numerical results for generic parameters will be presented in Section \[parameterstudy\]. In Section \[CenA\], we will demonstrate that the broad-band SED of the radio galaxy Cen A, including the recent [*Fermi*]{} $\gamma$-ray data [@abdo09c], can be modeled with plausible parameters expected for a mis-aligned blazar, allowing for a contribution from VHE $\gamma$-ray induced cascades in the [*Fermi*]{} energy range. We summarize in Section \[summary\]. ![\[geometry\]Geometry of the model setup.](f1.eps){width="15cm"} \[setup\]Model Setup and Code Description ========================================= Figure \[geometry\] illustrates the geometrical setup of our model system. We represent the primary VHE $\gamma$-ray emission as a mono-directional beam of $\gamma$-rays propagating along the X axis, described by a power-law with photon spectrum index $\alpha$ and a high-energy cut-off at $E_{\gamma, max}$. For the following study, we assume that the primary $\gamma$-rays interact via $\gamma\gamma$ absorption and pair production with an isotropic thermal blackbody radiation field within a fixed boundary, given by a radius $R_{\rm ext}$, i.e., $$u_{\rm ext} (\nu, r, \Omega) = 2 h \nu^3 / c^3 \frac{A}{\exp(\frac{h\nu}{k T})-1} \, H(R_{\rm ext} - r) \label{uapprox}$$ where $A$ is a normalization factor chosen to obtain a total radiation energy density $u_{\rm ext}$ (see Eq. \[uext\] below), and $H$ is the Heaviside function, $H(x) = 1$ if $x > 0$ and $H(x) = 0$ otherwise. A magnetic field of order $\sim $mG is present. Without loss of generality, we choose the $y$ and $z$ axes of our coordinate system such that the magnetic field lies in the (x,y) plane. The input parameters to our model simulation describing the external radiation field are the integral of $u_{\rm ext} (\nu, r, \Omega)$ over all frequencies: $$u_{\rm ext}=4\pi\int_0 ^\infty u_{\rm ext} (\nu, r, \Omega) d\nu, \label{uext}$$ the blackbody temperature $T$, and the radial extent $R_{\rm ext}$. We have used the Monte-Carlo code developed by [@rb10]. This code evaluates $\gamma\gamma$ absorption and pair production using the full analytical solution to the pair production spectrum of [@bs97] under the assumption that the produced electron and positron travel along the direction of propagation of the incoming $\gamma$-ray. The trajectories of the particles are followed in full 3-D geometry. Compton scattering is evaluated using the head-on approximation, assuming that the scattered photon travels along the direction of motion of the electron/positron at the time of scattering. While the Compton energy loss to the elecron is properly accounted for, we neglect the recoil effect on the travel direction of the electron. In order to improve the statistics of the otherwise very few highest-energy photons, we introduce a statistical weight, $w$, inversely proportional to the square of the photon energy, $w = 1/\epsilon^2$. Where $\epsilon = \frac{E_{\gamma}}{m_e c^2}$. To save CPU time, we precalculate tables for the absorption opacity $\kappa_{\gamma\gamma}$, Compton scattering length $\lambda_{\rm IC}$, and Compton cross section for each photon energy, electron energy and interaction angle before the start of the actual simulation. The simulation produces a photon event file, logging the energy, statistical weight, and travel direction of every photon that escapes the region bounded by $R_{\rm ext}$. A separate post-processing routine is then used to extract angle-dependent photon spectra with arbitrary angular and energy binning from the log files. 1.5cm ![\[standardfig\] Cascade emission at different viewing angles ($\mu = \cos\theta_{\rm obs}$). Parameters: $B = 1 \,$ mG, $\theta_B = 5^o$; $u_{\rm ext} = 10^{-5}$ erg cm$^{-3}$, $R_{\rm ext} = 10^{18}$ cm, $T = 1000$ K, $\alpha = 2.5$, [$E_{\gamma, {\rm max}} = 5$ TeV]{}. ](f2.eps "fig:"){width="12cm"} \[parameterstudy\]Numerical Results =================================== For comparison with our previous study on mono-energetic radiation fields, we conduct a similar parameter study as presented in [@rb10], investigating the effects of parameter variations on the resulting angle-dependent photon spectra. Standard parameters for most simulations in our parameter study are: a magnetic field of $B = 1$ mG, oriented at an angle of $\theta_B = 5^o$ with respect to the X axis ($B_x = 1$ mG, $B_y = 0.1$ mG); an external radiation energy density of $u_{\rm ext} = 10^{-5}$ erg cm$^{-3}$, extended over a region of radius $R_{\rm ext} = 10^{18}$ cm; a blackbody temperature of $ T= 10^3$ K (corresponding to a peak of the blackbody spectrum at a photon energy of $E_s^{\rm pk}= 0.25$ eV). The incident $\gamma$-ray spectrum has a photon index of $\alpha = 2.5$ and extends out to $E_{\gamma, {\rm max}} = 5$ TeV. The emanating photon spectra for all directions have been normalized with the same normalization factor, corresponding to a characteristic flux level of a $\gamma$-ray bright ([*Fermi*]{}) blazar in the forward direction. Figure \[standardfig\] illustrates the viewing angle dependence of the cascade emission. The $\gamma\gamma$ absorption cut-off at an energy $E_c = (m_e c^2)^2 / E_s \sim 1$ TeV is very smooth in this simulation because of the broad thermal blackbody spectrum of the external radiation filed. In contrast, the $\delta$-function approximation for the external radiation field adopted in [@rb10] resulted in an artificially sharp cutoff in that work. At off-axis viewing angles, the observed spectrum is exclusively the result of the cascade. In the limit of low photon energies (far below the $\gamma\gamma$ absorption threshold) and neglecting particle escape from the cascade zone, one expects a low-frequency shape close to $\nu F_{\nu} \propto \nu^{1/2}$ due to efficient radiative cooling of secondary particles injected at high energies [@rb10]. However, with the typical parameters used for this parameter study, the assumption of negligible particle escape is not always justified. In order to estimate the possible suppression of the low-frequency cascade emission due to particle escape, we calculate the critical electron energy for which the Compton cooling time scale $\tau_{\rm IC} (\gamma)$ equals the escape time scale, $\tau_{\rm esc} = R_{\rm ext} / (c \, \cos\theta_B)$. Using a characteristic thermal photon energy of $\epsilon_{\rm Th} = 2.8 \, k T / (m_e c^2)$, the resulting Compton scattered photon energy, $\epsilon_{\rm esc}$ below which we expect to see the effects of particle escape and, hence, inefficient radiative cooling, is $$\epsilon_{\rm esc} = {9 \times 2.8 \, k T \, m_e c^2 \, \cos^2\theta_B \over 16 \, \sigma_T^2 \, u_{\rm ext}^2 \, R_{\rm ext}^2} \label{epsesc}$$ corresponding to an actual energy (in GeV) of $$E_{\rm esc} \approx 2 \; T_3 \, u_{-5}^{-2} \, R_{18}^{-2} \, \cos^2\theta_B \; {\rm GeV} \label{Eesc}$$ where $T = 10^3 \, T_3$ K, $u_{\rm ext} = 10^{-5} \, u_{-5}$ erg cm$^{-3}$, and $R = 10^{18} \, R_{18}$ cm. Therefore, for our standard parameters, we expect the low-frequency ($E \lesssim$ a few GeV) to be significantly affected by particle escape. This explains why the low-energy photon spectra shown in Figure \[standardfig\] are harder than $\nu^{1/2}$. The cascade emission is progressively suppressed at high energies with increasing viewing angle due to incomplete isotropization of the highest-energy secondary particles. This effect becomes important beyond the energy $E_{\rm IC, br}$ of Compton-scattered photons by secondary electrons/positrons for which the Compton-scattering length $\lambda_{IC}$ equals the distance travelled along the gyrational motion over an angle $\theta$, $\lambda_{IC}(\gamma) = \theta r_{\rm gyr}(\gamma)$, which is given by $$E_{\rm IC, br} = {3 \, e \, B \over 4 \, \sigma_T \, u_{\rm ext} \, \theta} \, E_s \; \sim \; 1.3 \; B_{-3} \, u_{-5}^{-1} \, T_3 \, \theta^{-1} \; {\rm GeV}. \label{ICbreak}$$ where $B = 10^{-3}$ mG [@rb10]. 1.5cm ![\[ufig\]The effect of a varying external radiation energy density. Parameters: $B_x = 10^{-3}$ G, $B_y = 1.3\times 10^{-4}$ G, $\theta_B = 7.4^0$; and other parameters are the same as for Figure \[standardfig\] in the angular bin $0.4\leq\mu\leq0.6$ ](f3.eps "fig:"){width="12cm"} Figure \[ufig\] shows the cascade spectra for different values of the external radiation field energy density $u_{\rm ext}$. For large energy densities $u_{\rm ext} \gtrsim 10^{-4}$ erg cm$^{-3}$, $\tau_{\gamma\gamma} \gg 1$ for photons above the pair production threshold $\gamma\gamma$ so that essentially all VHE photons will be absorbed and the photon flux from the cascade becomes independent of $u_{\rm ext}$. The figure confirms our discussion concerning escape and hence inefficient radiative cooling of low-energy particles above (see Eq. \[Eesc\]). For large values of $u_{\rm ext}$, the Compton loss time scale for all relativistic electrons producing $\gamma$-rays in the considered range, is much shorter than the escape time scale. Hence, the expected $\nu F_{\nu} \propto \nu^{1/2}$ shape results. In the low-$u_{\rm ext}$ case, escape affects even ultrarelativistic electrons, resulting in a substantial hardening of the low-energy photon spectrum. Figure \[Tfig\] illustrates the effect of a varying temperature of the external blackbody radiation field. As the temperature increases up to $1000$ K the cascade flux increases because the $\gamma\gamma$ absorption threshold energy decreases so that an increasing fraction of $\gamma$-rays can be absorbed. The isotropization turnover is almost independent of $T$. For temperatures $ T > 1000$ K the cascade flux decreases with increasing $T$ because $u_{\rm ext}$ remains fixed, leading to a decreasing photon number density and absorption opacity $\kappa_{\gamma\gamma}$ with increasing $T$ (and, hence, increasing $E_s$). The figure also confirms our expectation (Eq. \[Eesc\]) that an increasing blackbody temperature leads to an increasing suppression of the low-frequency portion of the cascade emission due to particle escape. 1.5cm ![\[Tfig\]The effect of a varying temperature of the external blackbody radiation field. $\theta_B = 15^o$ and other parameters are the same as for Figure \[standardfig\] in the angular bin $0.4\leq\mu\leq0.6$ ](f4.eps "fig:"){width="12cm"} 1.5cm ![\[Bfig\]The effect of a varying magnetic field strength, for a fixed angle of $\theta_B = 45^o$ between jet axis and magnetic field. All other parameters are the same as for Figure \[standardfig\] in the angular bin $0.4\leq\mu\leq0.6$](f5.eps "fig:"){width="12cm"} Figures \[Bfig\] and \[Bthetafig\] illustrate the effects of varying magnetic-field parameters (strength and orientation). As expected, the results are essentially the same as for cascades in monoenergetic radiation fields investigated in [@rb10]: The cascade development is extremely sensitive to the transverse magnetic field $B_y$. The limit in which even the highest-energy secondary particles are effectively isotropized before undergoing the first Compton scattering interaction, is easily reached for typical magnetic fields expected in the circum-nuclear environment of AGNs. 1.5cm ![\[Bthetafig\]The effect of a varying magnetic field orientation, for a fixed magnetic field strength of $B = 1$ mG. All other parameters are the same as for Figure \[standardfig\] in the angular bin $0.4\leq\mu\leq0.6$](f6.eps "fig:"){width="12cm"} \[CenA\]Application to Cen A ============================ The standard AGN unification scheme [@up95] proposes that blazars and radio galaxies are intrinsically identical objects viewed at different angles with respect to the jet axis. According to this scheme, FR I and FR II radio galaxies are believed to be the parent population of BL Lac objects and FSRQs, respectively. Hence, if most blazars, including LBLs and FSRQs, are intrinsically VHE $\gamma$-ray emitters potentially producing pair cascades in their immediate environments, the radiative signatures of these cascades might be observable in many radio galaxies. In fact, [*EGRET*]{} provided evidence for $> 100$ MeV $\gamma$-ray emission from three radio galaxies (Cen A: [@sreekumar99], 3C 111: [@hartman08], and NGC 6251: [@muk02]). These sources have been confirmed as high-energy $\gamma$-ray sources by [*Fermi*]{} [@abdo09c; @abdo10b], along with the detection of five more radio galaxies (NGC 1275: [@abdo09a], M 87: [@abdo09b], 3C 120, 3C 207, and 3C 380: [@abdo10b]). In this paper, we focus on the radio galaxy Cen A [@abdo09c]. The FR I Cen A is the nearest radio-loud active galaxy to Earth. It has a redshift of z=0.00183 at the distance of $D = 3.7 $ Mpc. Recently, the Auger collaboration reported that the arrival directions of the highest energy cosmic rays ($E \gtrsim 6 \times10^{19}$ eV) observed by the Auger observatory are correlated with nearby AGN, including Cen A [@abraham07; @abraham08]. This suggests that Cen A may be the dominant source of observed UHECR nuclei above the GZK cut off. ![\[fit\]Fit to the SED of Cen A. The green curve is a fit to the broad-band SED using the model of [@bc02], while the maroon curve is the cascade emission resulting from $\gamma\gamma$ absorption of the forward jet emission. The red curve is the sum of both contributions (viewed at an angle of $70^o$. ](f7.eps){width="10cm"} Cen A has an interesting radio structure on several size scales. The most prominent features are its giant radio lobes, which subtend $\sim 10^0$ on the sky, oriented primarily in the north-south direction. They have been imaged at $4.8$ GHz by the Parkes telescope [@junkes93] and studied at up to $\sim 60$ GHz by [@hardcastle09]. The radio lobes are the only extragalactic source structure that has so far been spatially resolved in GeV $\gamma$-rays by $\emph{Fermi}$ [@abdo10c]. The innermost region of Cen A has been resolved with VLBI and shown to have a size of $\sim 3 \times 10^{16}$ cm [@kellermann97; @horiuchi06]. Observations at shorter wavelengths also reveal a small core. VLT infrared interferometry resolves the core size to $\sim 6 \times 10^{17}$ cm [@meisen07]. The angle of the sub-parsec jet of Cen A to our line of sight is $\thicksim 50^0-80^0$ [@Tingay98] with a preferred value of $\sim 70^o$ [@steinle09]. The K-band nuclear flux with starlight subtracted is $F(K) \thicksim 38$ mJy, corresponding to $\nu L_{\nu} \thicksim 7 \times 10^{40}$ erg $s^{-1}$ for a distance of $3.5$ Mpc [@marconi00]. The mid-IR flux of $\thicksim 1.6$ Jy at $11.7 \, \mu$m corresponds to $\nu L_\nu \thicksim 6 \times 10^{41}$ erg $s^{-1}$. The broadband SED from radio to $\gamma$-rays has been fitted with a synchrotron self-Compton model by [@abdo10a]. In their model [see Table 2 of @abdo10a], a maximum electron Lorentz factor of $\gamma_{max}= 1\times10^8$ was required in order to produce the observed $\gamma$-ray emission. However, given the assumed magnetic field of $B=6.2$ G, this does not seem possible since for $\gamma = 10^8$ electrons, the synchrotron loss time scale is shorter than their gyro-timescale, which sets the shortest plausible acceleration time scale. Here, we therefore present an alternative interpretation of the SED, based on the plausible assumption that Cen A would appear as a VHE $\gamma$-ray emitting blazar in the jet direction, and cascading of VHE $\gamma$-rays on the nuclear infrared radiation field produces an observable off-axis $\gamma$-ray signature in the [*Fermi*]{} energy range. Figure \[fit\] illustrates a broadband fit to the SED of Cen A [data from @abdo10a], using the equilibrium version of the blazar radiation transfer code of [@bc02], as described in more detail in [@acciari09]. For this fit, standard blazar jet parameters were adopted, but the viewing angle was chosen in accord with the observationally inferred range. Specifically, we chose $\theta_{\rm obs} = 70^o$. Other model parameters include a bulk Lorentz factor of $\Gamma = 5$, a radius of the emission region of $R = 1 \times 10^{16}$ cm, a kinetic luminosity in relativistic electrons, $L_e = 9.4 \times 10^{43}$ erg s$^{-1}$, a co-moving magnetic field of $B = 11$ G, corresponding to a luminosity in the magnetic field (Poynting flux) of $L_B = 1.1 \times 10^{45}$ erg s$^{-1}$ and a magnetic-field equipartition fraction $\epsilon_B \equiv L_B / L_e = 12$, corresponding to a Poynting-flux dominated jet. Electrons are injected into the emission region at a steady rate, with a distribution characterized by low- and high-energy cutoffs at $\gamma_1 = 1.2 \times 10^3$ and $\gamma_2 = 1.0 \times 10^6$, respectively, and a spectral index of $q = 3.5$. The code finds a self-consistent equilibrium between particle injection, radiative cooling and escape, from which the final photon spectrum is calculated. The resulting broadband SED fit is illustrated by the solid green curve in Figure \[fit\]. The flux emanating in the forward direction ($\theta_{\rm obs} = 0^o$, i.e., the blazar direction) has been chosen as an input to our cascade simulation to evaluate the cascade emission in the nuclear infrared radiation field of Cen A observed at the given angle of $\theta_{\rm obs} = 70^o$. For the cascade simulation, we assumed a blackbody temperature of $2300$ K resulting in a peak frequency in the K-band. The external radiation field is parameterized through $u_{\rm ext} = 1.5 \times 10^{-3}$ erg cm$^{-3}$ and $R_{\rm ext} = 3 \times 10^{16}$ cm. These parameters combine to an IR luminosity of $L_{\rm BLR} = 4 \pi R_{\rm ext}^2 \, c \, u_{\rm ext} = 5 \times 10^{41}$ erg s$^{-1}$, in agreement with mid-IR flux observed for Cen A [@marconi00]. The magnetic field is $B = 1$ mG, oriented at an angle of $\theta_B = 4^o$. The cascade spectrum shown in Figure \[fit\] pertains to the angular bin $0.28 < \mu < 0.38$ (corresponding to $67^o \lesssim \theta \lesssim 73^o$), appropriate for the known orientation of Cen A and consistent with our broadband SED fit parameters. The cascade spectrum is shown by the maroon curve in Figure \[fit\], while the total observed spectrum is the solid red curve. The figure illustrates that the cascade contribution in the [*Fermi*]{} range substantially improves the fit, while still allowing physically reasonable parameters for the broadband SED fit. \[summary\]Summary ================== We investigated the signatures of Compton-supported pair cascades initiated by the interaction of nuclear VHE $\gamma$-rays with the thermal infrared radiation field of a circumnuclear dust torus in AGNs. 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ArXiv

--- abstract: 'We present the first evolved solutions to a computational task within the [*N*]{}euronal [*Org*]{}anism [*Ev*]{}olution model (***Norgev***) of artificial neural network development. These networks display a remarkable robustness to external noise sources, and can regrow to functionality when severely damaged. In this framework, we evolved a doubling of network functionality (double-NAND circuit). The network structure of these evolved solutions does not follow the logic of human coding, and instead more resembles the decentralized dendritic connection pattern of more biological networks such as the brain.' author: - 'Alan N. Hampton$^{1}$' - | Christoph Adami$^{1,2}$\ \ $^1$Digital Life Laboratory 136-93, California Institute of Technology, Pasadena, CA 91125\ $^2$Jet Propulsion Laboratory 126-347, California Institute of Technology, Pasadena, CA 91109\ [email protected] nocite: - '[@Dittrich01]' - '[@Astor00]' title: Evolution of Robust Developmental Neural Networks --- Introduction ============ The complexity of mammalian brains, and the animal behaviors they elicit, continue to amaze and baffle us. Through neurobiology, we have an almost complete understanding of how a single neuron works, to the point that simulations of a few connected neurons can be carried out with high precision. However, human designed neural networks have not fulfilled the promise of emulating these animal behaviors. The problem of designing the neural network [*structure*]{} can be generalized to the problem of designing complex computer programs because, in a sense, an artificial neural network is just a representation of an underlying computer program. Computer scientists have made substantial progress in this area, and routinely create increasingly complicated codes. However, it is a common experience that when these programs are confronted with unexpected situations or data, they stall and literally stop in their tracks. This is quite different from what happens in biological systems, where adequate reactions occur even in the rarest and most uncommon circumstances, as well as in noisy and incompletely known environments. It is for this property that some researchers have embraced evolution as a tool for arriving at robust computational systems. Darwinian evolution not only created systems that can withstand small changes in their external conditions and survive, but has also enforced [*functional modularity*]{} to enhance a species’ evolvability [@Gerhart98] and long-term survival. This modularity is one of the key features that is responsible for the evolved system’s robustness: one part may fail, but the rest will continue to work. Functional modularity is also associated with component re-use and developmental evolution [@Koza03]. The idea of evolving neural networks is not new [@Kitano90; @Koza91], but has often been limited to just adapting the network’s structure and weights with a bias to specific models (e.g., feed-forward) and using [*homogeneous*]{} neuron functions. Less constrained models have been proposed [@Belew93; @Eggenberger97; @Gruau95; @Nolfi95], most of which encompass some sort of implicit genomic encoding. In particular, developmental systems built on artificial chemistries (reviewed in Dittrich et al. 2001) represent the least constrained models for structural and functional growth, and thus offer the possibility of creating modular complex structures. Astor and Adami (2000) introduced the **Norgev** (***N***euronal ***Org***anism ***Ev***olution) model, which not only allows for the evolution of the developmental mechanism responsible for the [*growth*]{} of the neural tissue or artificial brain, but also has no *a priori* model for how the neuron computes or learns. This allows neural systems to be created that have the potential of evolving developmental robustness as found in nature. In this paper, we present evolved neural networks using the Norgev model, with inherent robustness and self-repair capabilities. Description of Norgev ===================== Norgev is, at heart, a simulation of an artificial wet chemistry capable of complex computation and gene regulation. The model defines the tissue substrate as a two-dimensional hexagonal grid on which proteins can diffuse through discrete stepped diffusion equations. On these hexagons, neural cells can exist, and carry out actions such as the production of proteins, the creation of new cells, the growth of axons, etc. Proteins produced by the cell can be external (diffusible), internal (confined within the cell and undiffusible) or neurotransmitters (which are injected through connected axons when the neuron is excited). Cells also produce a constant rate of cell-tag proteins, which identify them to other cells and diffuse across the substrate. Each neural cell carries a genome which encodes its behavior. Genomes consist of genes which can be viewed as a genetic program that can either be executed (expressed) or not, depending on a gene [*condition*]{} (see Fig. \[DNAexec\]). A gene condition is a combination of several condition [*atoms*]{}, whose values in turn depend on local concentrations of proteins. The gene condition can be viewed as the upstream regulatory region of the genetic program it is attached to, while the atoms can be seen as different binding modules within the regulatory region. Each gene is initially active (activation level $\theta=1$) and then each condition atom acts one after another on $\theta$, modifying it in the $[0,1]$ range, or totally suppressing it ($\theta=0$). Table \[cond1\] shows all the possible condition atoms and how they act on the gene expression level $\theta$ passed on to them. Once a gene activation value $\theta$ has been reached, each of the gene’s expression atoms are executed. Expression atoms can carry out simple actions such as producing a specific protein, or they can emulate complex actions such as cell division and axon growth. Table \[expr1\] contains a complete list of expression atoms used in Norgev. A more complete description of the Norgev model and its evolution operators (mutation and crossover) can be found in [@Astor00]. We know that in cellular biology, gene activation leads to the production of a specific protein that subsequently has a function of its own, ranging from enzymatic catalysis to the docking at other gene regulatory sites. In this model, the most basic expression element is the production of proteins (local or externally diffusible) through the PRD\[*PTx*\] atom. These can then interact and modulate the activation of other genes in the genome. In this sense, it can be argued that they are only regulatory proteins. However, at least abstractly, genes in this model need not only represent genes in biological cells but can also represent the logic behind enzyme interaction and their products. Thus, Norgev’s genome encodes a dynamical system that represents low level biological DNA processes, as well as higher level enzymatic processes including long-range interaction through diffusible substances like hormones. However, the objective is not to create a complete simulation of an artificial biochemistry, and thus other expression atoms are defined that represent more complex actions, actions that in real cells would need a whole battery of orchestrated protein interactions to be accomplished. Organism example ================ The best way to understand the model is probably to sit down and create by hand a functional organism. Here we will present a handwritten organism (Fig. \[sgenome\]) and explain how it develops into a fully connected neural network that computes a NAND logical function on its two inputs and sends the result to its output. The organism, which we named ***Stochastic***, relies on the random nature of the underlying chemical world to form its tissue structure. When an organism is first created, a tissue seed (type *CPT*) is placed in the center of the hexagonal grid, two sensor cells on the left of the grid and an actuator cell on the right. These then diffuse their marker proteins *CPT*, *SPTO*, *SPT1* and *APT0* respectively. In the first time step, only the first gene (Fig. \[sgenome\]) is active in the tissue seed and all the rest are suppressed. This gene will always be active and step after step will split off cells of type *ACPT0* until all the surrounding hexagons are occupied by these cells. After that, the seed does not execute any further function other than secrete its own cell type protein *CPT*. The new cells will, in turn, also split off more cells of type *ACPT0* (gene 5), and so make the tissue grow larger and larger (time=4 in Fig. \[cellSto\]). In a sense, these cells provide a cellular support for further development of the actual network, and could thus be called [*glial*]{}-type cells, in analogy to the supportive function glial cells have in real brains. These glial cells can split off three different types of neurons. If the signal from the actuator *APT0* is greater than the signal from the tissue seed *CPT*, then a neuron of type *ACPT3* will split off with probability $p>0$ (gene 2). On the other hand, if the external protein signal of sensor *SPT0* is strong compared to the external protein of sensor *SPT1*, then instead a neuron of type *ACPT1* will split off with $p>0$ (gene 3). Last of all, if the signal *SPT1* is greater than *SPT0*, then it is more likely that a neuron of type *ACPT2* will split off (gene 4). This is all that these glial cells of type *ACPT0* do: split off more glial cells, or any of three differentiated neuron types depending on how close they are to the sensors or the actuators. ----------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------- time = $4$ time = $24$ ![Successive stages in the developmental growth of the ***Stochastic*** neural tissue.[]{data-label="cellSto"}](LifeBubble_C_time1b.ps "fig:"){width="1.65in"} ![Successive stages in the developmental growth of the ***Stochastic*** neural tissue.[]{data-label="cellSto"}](LifeBubble_C_time6b.ps "fig:"){width="1.65in"} time = $40$ time = $120$ ![Successive stages in the developmental growth of the ***Stochastic*** neural tissue.[]{data-label="cellSto"}](LifeBubble_C_time10b.ps "fig:"){width="1.65in"} ![Successive stages in the developmental growth of the ***Stochastic*** neural tissue.[]{data-label="cellSto"}](LifeBubble_C_time30b.ps "fig:"){width="1.65in"} ----------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------- These three cell types (*ACPT1*, *ACPT2* and *ACPT3*), will then form the actual neural network that will do all the processing. Through gene 6, cells of type *ACPT1* will grow a dendrite towards sensor *SPT0* and define their default neurotransmitter as *NT1*. In the same way, cells of type *ACPT2* will have gene 7 active and will grow a dendrite towards sensor *SPT1* and define their neurotransmitter as *NT2*. Last of all, gene 8 is active in cells of type *ACTP3*, and will direct the growth of dendrites towards cells of type *ACPT1* and *ACPT2* and an axon towards the actuator *APT0*. In the end, each sensor *SPT0* and *SPT1* is connected to every neuronal cell *ACPT1* and *ACPT2*, and all the *ACPT3* neuronal cells are connected to the actuator *APT0* (time=120 in Fig. \[cellSto\]). However, which and how many *ACPT1* and *ACPT2* neurons connect to which and how many of the *ACPT3* neurons relies on stochastic axonal growth, preferably connecting neurons that are nearer on the hexagonal grid. Moreover, all neurons end up connected after the axonal growth process has finished, forming a fully functional NAND implementation. We still need to understand how the neurons actually process the signals passing through them. This is mediated through genes 9 and 10. Neurons *ACPT1* and *ACPT2* act as [*relays*]{} of the sensor signals through gene 10. That is, whenever they receive any neurotransmitter of type *eNT* (default sensor neurotransmitter) they will become excited and inject their gene-defined neurotransmitters through their axons. Neuronal cells of type *ACPT3* will then compute the NAND evaluative action on the amount of neurotransmitters *NT1* and *NT2* injected into their cell bodies and activate accordingly (gene 9). Their activity causes the default neurotransmitter to be injected into the actuator, thus finalizing the simulated input-output NAND computation. Robustness of *Stochastic* ========================== While *Stochastic*’s neural tissue will always look different every time it is grown because of the stochastic nature of neuronal splitting, it always forms a processing network that correctly computed the NAND function. This confers some robustness to the phenotype of the network in spite of the stochastic, but genetically directed, growth process. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![Robustness of ***Stochastic*** under cell death. Half the neural tissue from Fig. \[cellSto\] was removed (left). After $80$ time steps a different, but functional tissue arises (right).[]{data-label="RobustSto"}](LifeBubble_C_time31b.ps "fig:"){width="1.65in"} ![Robustness of ***Stochastic*** under cell death. Half the neural tissue from Fig. \[cellSto\] was removed (left). After $80$ time steps a different, but functional tissue arises (right).[]{data-label="RobustSto"}](LifeBubble_C_time50b.ps "fig:"){width="1.65in"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- However, the developmental process is far more robust than that. For example, we can manually kill (remove) neurons of a fully developed tissue and have a similar functional (but somewhat scarred) tissue grow back. Fig \[RobustSto\] shows an example where we even removed the tissue seed *CPT*, which has an important role in the organisms development (without its external signal, glial cells of type *ACPT0* do not proliferate). While the morphology of the self-repaired tissue has changed, it still computes the NAND function. More than anything, this observation helps illustrate the potential capabilities of developmental processes in artificial chemistries to create robust information processing neural tissues even under the breakdown of part of their structure. Note that the self-repair property of *Stochastic* was not evolved (or even hand-coded), but rather emerged as a property of the developmental process. Naturally, these robustness traits can be augmented and exploited under suitable evolutionary pressures. Evolution of organisms in Norgev ================================ Here, we present the evolutionary capabilities of Norgev, that is, how its genetic structure and chemistry model allow for the evolution of developmental neural networks that solve pre-specified tasks. In the previous section we presented the ***Stochastic*** organism, which grew into a neural tissue that computed a NAND function on its inputs. Our goal was to study how difficult it would be to *double* the tissue’s functionality and compute a double NAND on three inputs, and send the result to two outputs (Fig. \[doubleNAND\]). Because one of the mutational operators used in the Genetic Algorithm is [*gene doubling*]{} (see Astor and Adami, 2000), we surmised that there was an easy route through duplication and subsequent differentiation. Because of the universality of NAND, showing that more complex tissues can evolve from *Stochastic* suggests that arbitrary computational tissues can evolve in Norgev. The input signal was applied for four time steps (the time for the input to pass through the tissue and reach the output), and then the output was evaluated by a reward function $R=1-\sqrt{\sum_{i} (y_i(x) - t_i(x))^2}$ where $x$ is the input, $y$ the tissue’s output, and $t$ the expected output. Organisms were then selected according to a fitness function given by the average reward over 400 time steps, and a small pressure for small genome sizes and neuron numbers. Mutation rates were high and evolution was mainly asexual. Details of the experiments will appear elsewhere (Hampton and Adami, in preparation). ![Evolution objective: to *double* the functionality of the original organism.[]{data-label="doubleNAND"}](double_nand2.eps){width="2.4in"} We evolved organisms that obtained the double NAND functionality in two separate runs on massively parallel cluster computers, over several weeks. The two solutions were very different in both structure and algorithm. The simplest, ***Stochastic A***, evolved the fastest with the more straightforward morphology (Fig. \[cellStoA\]). Its genome is short (Fig. \[sgenomeA\]) when compared to evolved organisms in other runs, but is substantially more difficult to understand compared to its ancestor. ![***Stochastic A*** neural tissue expressing 6 different cell types. Most of the axonal connections that spread out from the central sensor are not utilized. Instead, the actual computation takes place in a compact area near the center.[]{data-label="cellStoA"}](12example3b.ps){width="3.25in"} After careful analysis of the genome, paired with an evaluation of the physical connections present in the neural tissue, we came to the conclusion that the organism had not reused [*any*]{} genomic material to double the NAND function, but had instead [*completely*]{} rewritten its code to implement a shorter and more efficient algorithm when compared to the ancestor we wrote. Let us embark once again in a quick step-by-step genome analysis. Gene 1 is active in the tissue seed, which then splits off a cell of type *ACPT0* and *APT2*. After this, the gene is forever shut off because of the repressive *NNY(apt2)* condition. Cell *ACPT0* then splits off cells of type *ACPT1*, *ACPT2* and *ACPT3* through gene 2. This gene is always active, and thus *ACPT0* cells are always in an inhibitive activation state (due to action atom *INH1*). Gene 6 makes *ACPT1* cells grow a dendrite to sensor *SPT0* and have same-type daughter cells. These are the cells that cover the whole substrate in Fig. \[cellStoA\]. Gene 7 causes *ACPT2* cells to grow a dendrite towards sensor *SPT1*, an axon towards actuator *APT1* and define its neurotransmitter as *NT2*. Through gene 8, *ACPT3* cells grow a dendrite to sensor *SPT2*, a dendrite to cells *ACPT2*, and an axon to actuator *APT1*. Gene 11 is the most cryptic. This gene is only active in the first $\sim$3 time steps of the organism’s life, and effectively makes cells of type *ACTP0*, *ACPT1* (only the ones in the center, not all the rest) and *ACPT2* grow an axon towards the actuator *APT0*. Once the tissue has developed, gene 10 is used by all cells for processing sensory information (neurotransmitter *eNT*), on which it performs a NOT function. ![Effective neural circuit grown by ***Stochastic A***. Dashed axonal connections grow due to gene 11, which is only active during the first moments of the organisms life. Axons and neurons that have no influence on the final computations are rendered in light gray.[]{data-label="wireStoA"}](stochasticA_net2.eps){width="2.7in"} The effective neural circuit is shown in Fig. \[wireStoA\]. The result is processed in three time steps instead of the incorrectly postulated minimum of four time steps. This is due to an implicit OR function computed by the actuator cells that we did not anticipate, but which was discovered and exploited by the organism. The neural tissue is applying a NOT function at a relay of its inputs, and then an OR on the actuators to arrive at the double NAND (Fig. \[wireStoA\_Logic\]). The resulting simplicity of the organism is apparent from the fact that only gene 10 is used for neural processing once the tissue has developed, and it thus has a structure more conducive to further function doubling. ![The computation carried out by ***Stochastic A**.* \[wireStoA\_Logic\]](stochasticA_func2.eps){width="2.4in"} Another organism that solved the problem was ***Stochastic B***, which took considerably longer to evolve, and that turned out to be highly complex and difficult to understand. In Fig. \[actvStoB\], cellular structures can clearly be seen in which stripe-like patterns of two different neural types succeed one another. These stripes were different for each organism, and reflect a stochastic development. The axonal connections linking all these neurons are so interwoven that it is difficult to believe that this organism is actually acting on its inputs instead of undergoing some recurrent neuronal oscillation. ------------------------------------------------- ------------------------------------------------ ------------------------------------------------- ------------------------------------------------- $000 \rightarrow 11$ $001 \rightarrow 11$ $010 \rightarrow 11$ $011 \rightarrow 10$ {width="1.5in"} {width="1.5in"} {width="1.5in"} {width="1.5in"} $100 \rightarrow 11$ $101 \rightarrow 11$ $110 \rightarrow 01$ $111 \rightarrow 00$ {width="1.5in"} {width="1.5in"} {width="1.5in"} {width="1.5in"} ------------------------------------------------- ------------------------------------------------ ------------------------------------------------- ------------------------------------------------- We were unable to describe the development and internal workings of this organism due to its complexity. However, a complete description is in principle always possible because of our access to all of the organism’s internal state variables, and more importantly, to its genetic code: the source of its dynamics. Taking the first steps in that direction, we studied the neuronal activation under each of the eight possible input configurations (Fig. \[actvStoB\]). We can clearly see neuronal activity that follows the striped pattern on the right-hand side of the tissue (for inputs of the form $x0x \rightarrow 11$). Remarkably, the left side of the tissue does not follow the same organization and thus we theorize that although they have the same cell type, they have differentiated internally even further depending on their position on the tissue. We came to the conclusion that this organism is not performing the same internal computation as ***Stochastic A***. We can see this by inspecting input $110 \rightarrow 01$, and noticing that no tissue neurons are activated, and thus there is no neuron performing the NOT function on the last input. Conclusions =========== Biology baffles us with the development of even seemingly simple organisms. We have yet to recreate insect neural brains that perform such feats as flight control. As an even simpler organism, the flatworm [*C. elegans*]{}, has a nervous system which consists of 302 neurons, highly interconnected in a specific (and mostly known) pattern, and 52 glial cells, but whose exact function we still do not understand. Within Norgev, we have shown that such structural biocomplexity can arise *in silico*, with dendritic connection patterns surprisingly similar to the seemingly random patterns seen in [*C. elegans*]{}. And we might have been baffled at the mechanism of development and function of our in silico neural tissue if it were not for our ability to probe every single neuron, study every neurotransmitter or developmental transcription factor, and isolate every part of the system to understand its behavior. Thus, we believe that evolving neural networks under a developmental paradigm is a promising avenue for the creation and understanding of robust and complex computational systems that, in the future, can serve as the nervous systems of autonomous robots and rovers. Acknowledgements ================ Part of this work was carried out at the Jet Propulsion Laboratory, California Institute of Technology, supported by the Physical Sciences Division of the National Aeronautics and Space Administration’s Office of Biological and Physical Research, and by the National Science Foundation under grant DEB-9981397. Evolution experiments were carried out on a OSX-based Apple computer cluster at JPL. Astor, J. and Adami, C. (2000). A developmental model for the evolution of artificial neural networks. , 6:189–218. Belew, R. R. (1993). Interposing an ontogenetic model between genetic algorithms and neural networks. In [*NIPS5 ed J Cowan (San Mateo), CA: Morgan Kaufmann*]{}. Dittrich, P., Ziegler, J., and Banzhaf, W. (2001). Artificial chemistries–[A]{} review. , 7:225–275. Eggenberger, P. (1997). Creation of neural networks based on developmental and evolutionary principles. In [*Proc. ICANN’97, Lausanne, Switzerland, October 8-10, 1997*]{}. Gruau, F. (1995). Automatic definition of modular neural networks. , 3:151–183. Kirschner, M. and Gerhart, J. (1998). Evolvability. , 95:8420–8427. Kitano, K. (1990). Designing neural network using genetic algorithm with graph generation system. , 4:461–476. Koza, J. R., Keane, M. A., and Streeter, M. J. (2003). The importance of reuse and development in evolvable hardware. In [*5th NASA/DoD Workshop on Evolvable Hardware, Chicago, IL, USA*]{}. IEEE Computer Society. Koza, J. R. and Rice, J. P. (1991). Genetic generation of both the weights and architecture for a neural network. , 2:397–404. Nolfi, S. and Parisi, D. (1995). Evolving artificial neural networks that develop in time. In [*Advances in Artificial Life, Proceedings of the Third European Conference on Artificial Life*]{}, pages 353–367. Springer.

ArXiv

--- abstract: 'Finding ways of creating, measuring and manipulating Majorana bound states (MBSs) in superconducting-semiconducting nanowires is a highly pursued goal in condensed matter physics. It was recently proposed that a periodic covering of the semiconducting nanowire with superconductor fingers would allow both gating and tuning the system into a topological phase while leaving room for a local detection of the MBS wavefunction. We perform a detailed, self-consistent numerical study of a three-dimensional (3D) model for a finite-length nanowire with a superconductor superlattice including the effect of the surrounding electrostatic environment, and taking into account the surface charge created at the semiconductor surface. We consider different experimental scenarios where the superlattice is on top or at the bottom of the nanowire with respect to a back gate. The analysis of the 3D electrostatic profile, the charge density, the low energy spectrum and the formation of MBSs reveals a rich phenomenology that depends on the nanowire parameters as well as on the superlattice dimensions and the external back gate potential. The 3D environment turns out to be essential to correctly capture and understand the phase diagram of the system and the parameter regions where topological superconductivity is established.' author: - 'Samuel D. Escribano' - Alfredo Levy Yeyati - Yuval Oreg - Elsa Prada bibliography: - 'superlattice.bib' title: Effects of the electrostatic environment on superlattice Majorana nanowires --- Introduction {#Introduction} ============ The appearance of Majorana bound states (MBSs) at the edges of topological superconductors in solid-state devices has attracted a great deal of attention both from theorists and experimentalists [@Hasan:RMP10; @Alicea:RPP12; @Beenakker:arxiv11; @Sato:JPSJ16; @Aguado:rnc17; @Lutchyn:NRM18]. These non-Abelian mid-gap zero energy modes are intriguing from a fundamental point of view and germane to topologically protected quantum computing applications [@Nayak:RMP08; @Aasen:PRX16; @Das:NPJ15]. Due to their relative simplicity, most of the scrutiny has fallen onto one-dimensional (1D) proposals such as hybrid superconducting-semiconducting nanowires with strong spin-orbit coupling [@Lutchyn:NRM18] and ferromagnetic atomic chains on a superconductor (SC) [@Nadj-Perge:Science14; @Ruby:PRL15; @Feldman:17; @Pawlak:17]. Tuning the system to appropriate conditions, experimentalists are able to find zero energy modes compatible with the existence of MBSs in the form of zero bias peaks in tunnelling spectroscopy experiments [@Mourik:Science12; @Deng:Science16; @Zhang:Nat17a; @Chen:Science17; @Deng:PRB18; @Vaitiekenas:PRL18; @Gul:NNano18; @Grivnin:arxiv18; @Vaitiekenas:arxiv18]. However, due to the possibility of alternative explanations for the observed zero bias peak, the actual nature of these low-energy states has been brought into question [@Setiawan:PRB17; @Liu:PRB17; @Reeg:PRB18b; @Moore:PRB18; @Avila:arxiv18; @Vuik:arxiv18]. A complementary measurement that could disperse the doubts would be to measure the actual zero mode probability density along the wire or chain, which should show for Majoranas an exponential decay from the edge towards its center with the Majorana localization length [@Klinovaja:PRB12]. Attempts in this direction, including simultaneous tunneling measurement at the the end and the bulk of the wire, were performed in Ref. . The zero mode probability profile could in principle be accessed with the help of a scanning tunneling microscope (STM) that explores the local density of states at a certain energy along the wire [@Ben-Sach:PRB15]. STM measurements of this type have been carried out in iron chains on lead [@Nadj-Perge:Science14; @Ruby:PRL15], but in this case it is difficult to control the parameters of the system as these are fixed by material properties. In contrast, the parameters and topological phase transition of semiconducting wires can be manipulated by external magnetic and electric fields [@Lutchyn:NRM18]. This is one of the reasons making the semiconducting wire platforms so popular in the attempts to engineer topological superconductivity and to pursue MBSs. In these wires the induced pairing is achieved by proximity to a SC that can be either deposited or grown epitaxially over the wire [@Chang:Nnano15]. In the last case, hard superconducting gaps have been reported in InAs [@Chang:Nnano15] and InSb [@Gul:NNano17] wires with epitaxial Al layers. These hybrid wires are subjected to an external in-plane magnetic field $B$ that generates a Zeeman energy for the electrons in the wire, $V_{\rm{Z}}=g\mu_{\rm{B}}B/2$, given in terms of the wire’s $g$-factor and the Bohr magneton $\mu_{\rm{B}}$. According to simple 1D effective models [@Lutchyn:PRL10; @Oreg:PRL10], these wires experience a phase transition to a topological state at Zeeman fields greater than $V_{\rm c}\equiv\sqrt{\Delta^2+\mu^2}$, where $\Delta$ is the induced gap and $\mu$ the wire’s chemical potential. The charge density inside the wire, and thus $\mu$, can in principle be controlled by the voltage applied to a back gate, $V_{\rm gate}$. Due to their tunability, it would be ideal to perform STM experiments on these wires, a task that can be carried out nowadays [@Beidenkopf:private]. ![(Colour online) Schematic 3D (top) and lateral view (bottom) representations of the two types of superlattice Majorana nanowires analysed in the text: the bottom-superlattice where the SC fingers are below (a) and the top-superlattice where they are on top of the nanowire (b). The nanowire is depicted in green, the SC superlattice in grey, the dielectric substrate in purple and the back gate in black. We choose the $x$-axis along the nanowire and the $z$-axis as the direction perpendicular to the back gate’s surface. Different materials have different dielectric constants and dimensions. $V_{\rm SC}$ is the wire’s conduction band offset to the metal Fermi level at the interface with the SC fingers, $\rho_{\rm surf}$ is the positive surface charge at the rest of the wire’s facets and $V_{\rm gate}$ is the back gate’s voltage. (c) and (d): examples of the self consistent solution of the Poisson-Schrödinger equations in the Thomas-Fermi approximation. The electrostatic potential energy profile (in red) and the charge density profile (in blue) are shown along the wire ($x$-direction at $z=30$nm) in (c) and across the wire ($z$-direction at $x=1\mu$m) in (d), for $V_{\rm gate}=-0.5$V, $V_{\rm SC}=0.2$V and $\rho_{\rm surf}=2\times 10^{17}e/cm^3$ in a surface layer of thickness $1nm$. Geometric parameters are: $L_{\rm wire}=2\mu$m, $L_{\rm cell}=500$nm, $L_{\rm SC}=250$nm, $W_{\rm Al}=10$nm, $W_{\rm SiO}=20$nm, $W_{\rm wire}=80$nm. Other parameters are given in Table \[Table\_parameters\].[]{data-label="Fig1"}](Fig1.pdf) Looking for an appropriate device to conduct such an experiment, Levine *et al.* [@Levine:PRB17] recently showed that it is possible to find topological superconductivity in these wires when the superconductor (SC) is deposited periodically, forming a superlattice structure instead of covering continuously the length of the wire. A configuration with a superlattice of SC fingers at the bottom enables the STM tip to approach the wire from above, where it is free of any metal, and to drive a current between the tip and each of the SC fingers. Due to the metal free regions between the fingers, the back gate is capable of changing the charge density inside the wire due to the reduced screening by the finite size fingers. In this case [@Levine:PRB17], the topological phase diagram becomes more complex than for the uniformly covered nanowire (due to the presence of longitudinal minibands created by the periodicity of the system), and extends over a wider region in parameter space (to lower Zeeman fields and higher values of the chemical potential). Levine et al. [@Levine:PRB17] considered a minimal 1D model for the nanowire superstructure, in a similar fashion to other previous studies [@Sau:PRL12; @Malard:PRB16; @Hoffman:PRB16; @Lu:PRB16] with related periodic structures. However, in the last couple of years it has been shown that the electrostatic environment and the three-dimensionality of these wires play an important role in all aspects concerning the trivial/topological phases and the appearance of MBSs. For instance, the electrostatic profile is not homogeneous along (and across) the wire, which creates a position-dependent chemical potential [@Prada:PRB12; @Kells:PRB12; @Liu:PRB17; @Setiawan:PRB17; @Moore:PRB18; @Liu:PRB18] that has consequences for the topological phase transition and the shape and the overlap of MBSs [@Dominguez:NPJ17; @Escribano:BJN18; @Penaranda:PRB18; @Fleckenstein:PRB18]. It also creates a position-dependent Rashba spin-orbit coupling [@Wojcik:PRB18; @Moor:NJP18; @Bommer:arxiv18]. Moreover, the charge density is not distributed uniformly across the wire and its location depends strongly on the external gate voltage [@Vuik:NJP16; @Moor:NJP18]. This has consequences for the induced proximity effect [@Antipov:PRX18; @Mikkelsen:PRX18] and the appearance of orbital magnetic effects [@Nijholt:PRB16; @Kiczek:JoP17]. All these aspects influence the topological phase diagram and the topological protection of the Majorana zero modes [@Winkler:arxiv18]. Motivated by the new possibilities afforded by the superlattice structures and the necessity to include electrostatic effects when analysing the performance of a particular device design, here we perform a detailed study of the systems shown in Fig. \[Fig1\]. We consider two types of generic superlattice Majorana nanowires, one with the superconducting fingers at the bottom, between the nanowire and the back gate used to control the wire’s charge density, see Fig. \[Fig1\](a), and the other with the fingers on top, further away from the back gate, see Fig. \[Fig1\](b). In this last case, the fingers themselves can play the role of local probes along the wire [@Grivnin:arxiv18; @Huang:PRB18]. In both scenarios we assume that the fingers are connected to a macroscopic SC or grounded, so that we can neglect charging effects. Note that there are other works[@Sau:NatCom12; @Fulga:NJP13; @Lu:PRB16; @Stenger:PRB18] with periodic structures in the form of coupled quantum dots where the charging effect could be essential. The physics of the setups analysed here is primarily affected by the periodic structure along the wire that creates, among other things, a periodic potential profile for the electrons, see Fig. \[Fig1\](c), a periodic spin-orbit coupling and a periodic induced pairing potential. These quantities are further dependent on the transverse coordinates, see Fig. \[Fig1\](d), which are in turn conditioned by the wire’s boundary conditions (discussed in the next section). All this gives rise to a rich phenomenology that has consequences for the topological phase diagram and the spectral properties of the wires. Fundamental parameters characterizing this phenomenology are the superlattice cell length $L_{\rm cell}$ and the SC coverage ratio $r_{\rm SC}=L_{\rm SC}/L_{\rm cell}$, where $L_{\rm SC}$ is the size along the wire of the SC fingers. Since the geometry and the resulting 3D electrostatic profile in each setup are different, we find notable differences between both of them with advantages and disadvantages. The bottom superlattice setup can be easily accessed from the top, for example with an STM tip as mentioned before, while its charge density is still controllable with the back gate thanks to the metal-free regions between the SC fingers. Nevertheless, the screening effect of the fingers is strong due to their vicinity to the gate, which produces sizeable potential oscillations for the electrons inside the wire. This in turn has negative consequences for the stability of the topological phase due to the appearance of localized states on top of the SC fingers that interact with the MBSs when they are present. Furthermore, the spin-orbit coupling changes sign along the wire with the periodicity of the superlattice, averaging to a small value. In contrast, in the top-superlattice device the charge density is more easily varied without the need of large back gate potentials and the topological phase is more readily accessible. The potential oscillations are thus softer and the spin-orbit coupling doesn’t change sign and averages to a larger value. In turn, there is less nanowire surface exposed to open air and it is in principle more difficult to access. In both setups the SC doesn’t cover continuously the wire and consequently there is less induced superconductivity than in a uniformly covered one. We find that this leads to a reduced topological protection, manifested in a smaller topological minigap (energy difference between the Majorana zero energy mode and the continuum of states for $V_{\rm Z}>V_c$) and in a larger overlap between Majoranas at opposite ends of the wire (as measured by the Majorana charge[@Ben-Sach:PRB15]). Interestingly, the Majorana localization length is not only dependent on the SC coherence length, Fermi wavelength and spin-orbit length, as in the uniform hybrid wire, but also on the superlattice length. To enhance the topological protection, at the end of the paper we propose an alternative configuration that combines a conventional hybrid Majorana nanowire (with one of its facets covered uniformly by a thin SC layer) and a superlattice of (normal or superconducting) fingers. This setup benefits from the advantages of the superlattice configuration while displaying a topological minigap and Majorana charge comparable to the uniform wire. The structure of the paper is the following. In Sec. \[Methods\] we describe the superlattice setups and the methodology employed to analyse them (further details on the numerical methods can be found in App. \[SI1\]). We use a numerical approach that combines the effect of the electrostatic environment through the Poisson’s equation and the wire’s charge density through the Schrödinger’s equation in a self-consistent manner. As in previous works where the electrostatic environment was considered [@Antipov:PRX18; @Mikkelsen:PRX18; @Vuik:NJP16; @Winkler:arxiv18; @Woods:PRB18], our calculations are very demanding computationally, more so here since we have a superlattice structure. For this reason, we perform a series of approximations. For instance, we treat the proximity effect by the SC superlattice as a rigid boundary condition on the nanowire, effectively integrating out other SC degrees of freedom. We also ignore the orbital effects of the magnetic field. As we argue later on, this approximation will be justified at low densities and when the electron’s wave function is pushed towards the SCs by the effect of the back gate. It is important to note that in these systems there are many parameters as well as many length scales playing a role. Thus, we analyse different aspects separately in the first sections. In Subsec. \[Potential\] we inspect the electrostatic potential profile along and across the wire for the two setups (further details in App. \[SI3\]). In Subsec. \[Rashba\] we analyse their inhomogeneous Rashba couplings. In Sec. \[Impact\] we examine the impact of the superlattice on the nanowire spectral properties. We consider separately the effect of the inhomogeneous electrochemical potential, Subsec. \[Impact-mu\], the role of the wire’s intrinsic doping, Subsec. \[Impact-Eint\], and the impact of the inhomogeneous induced pairing, Subsec. \[Impact-SC\]. In Subsec. \[Optimal\_parameters\] we present a diagram in superlattice parameter space where we summarize the different features having a role in the stability of the topological phase analysed in the previous sections. Finally, in Sec. \[3D\_results\] we consider all the previous ingredients together and analyse the behaviour of both setups for realistic superlattice nanowire parameters. In particular, we find the spectrum over an extended range of external gate’s voltages. We then focus on a particular longitudinal subband where the wire is topological and analyse the appearance of Majorana oscillations, the size of the topological minigap as well as the spatial profile of MBSs. An alternative configuration that enhances the topological protection is discussed in Subsec. \[Alternative\]. For these calculations we solve the Schrödinger-Poisson equation in the Thomas-Fermi approximation. To check its accuracy, we compare it with the full Schrödinger-Poisson problem for some specific values of back gate’s potential in App. \[SI2\]. Finally, we conclude in Sec. \[Conclusions\]. Setup and methodology {#Methods} ===================== Our aim is to study equilibrium properties of the superlattice Majorana nanowires of Figs. \[Fig1\](a,b) taking into account their electrostatic environment. To that end, we first compute the electrostatic potential by solving the Poisson’s equation along and across the wire, taking into account its 3D geometry and the electrostatic parameters of the different materials. Then, we introduce this potential into the system’s Bogoliubov-de Gennes Hamiltonian and diagonalize it to find its eigenvalues and eigenvectors (both for infinite and finite-length wires) as a function of external parameters such as the voltage applied to the back gate or the external magnetic field. Since the potential profile depends on the wire’s charge density according to the Poisson’s equation, and the charge density is calculated by diagonalizing the system’s Hamiltonian, to solve the full Poisson-Schrödinger problem one needs to iterate the two in a self-consistent manner until convergence. In order to simplify this procedure we will employ the Thomas-Fermi approximation to calculate the wire’s charge density, as explained below in this section. In doing so, and similarly to previous works [@Mikkelsen:PRX18; @Winkler:arxiv18], we assume that the potential is independent of the magnetic field (calculated at $B=0$). This is justified since the charge density only depends slightly on $B$ for the $B$ values considered in this work, as we prove in App. \[SI2\]. A fully realistic calculation of the three-dimensional (3D) device would require to include the SC superlattice in the Hamiltonian at the same level as the nanowire itself. This is an involved problem that has been tackled in Refs. . In general, it can be seen that the SC induces by proximity effect a renormalization of the wire’s parameters such as $\mu$, $\alpha$ or $g$. When this renormalization is strong, called a metallization of the wire [@Reeg:PRB18], it is detrimental for the appearance of a topological phase. Concerning the induced pairing, it is possible to find parameters (including the width of the SC layer [@Reeg:PRB17; @Mikkelsen:PRX18]) where it is good, but it is in general necessary to assume a certain degree of disorder [@Winkler:arxiv18] in the SC to obtain a hard induced gap in the nanowire that is close to the parent’s one. Here, and due to the complexity already introduced by the superlattice, we will treat the SC as a rigid boundary. Nonetheless, the SC superlattice width $W_{\rm Al}$ and its infinite dielectric constant will be taken into account when solving the electrostatic problem. We will assume good proximity effect described by a constant pairing amplitude $\Delta_0$, comparable to that of the SC bulk gap, at the sites in contact to the SC fingers (determined by the superlattice parameters $L_{\rm cell}$ and $L_{\rm SC}$). Good proximity in such superlattice devices could be achieved, for example, by using molecular beam epitaxy, either by shadowing techniques or by etching half-shell coated wires [@Nygard:private]. We model the superlattice Majorana nanowire generalizing the 1D Hamiltonian of Refs. to 3D space $$\begin{aligned} \label{Hamiltonian} H= \frac{1}{2}\int \psi^\dagger(\vec{r})\hat{H}(\vec{r})\psi(\vec{r})d\vec{r}, \nonumber \\ \hat{H}(\vec{r})= \left[\frac{\hbar^2 k^2}{2m^*}-e\phi(\vec{r})-E_{\rm F}\right]\hat{\sigma}_0\hat{\tau}_z - \nonumber \\ -\frac{i}{2}\hat{\vec{\sigma}}\cdot\left[\vec{k}\times\vec{\alpha}_{\rm R}(\vec{r})-\vec{\alpha}_{\rm R}(\vec{r})\times\vec{k}\right]\hat{\tau}_z+ \nonumber \\ +V_{\rm Z}\hat{\sigma}_x\hat{\tau}_z -i\Delta(\vec{r})\hat{\sigma}_y\hat{\tau}_y,\end{aligned}$$ where $\vec{r}=(x,y,z)$ and $\vec{k}=(k_x,k_y,k_z)$. Here $m^*$ is the effective mass of the conduction band of the InAs nanowire, $\phi(\vec{r})$ the electrostatic potential inside the wire, $E_{\rm F}$ the wire’s Fermi energy, $\vec{\alpha}_{\rm R}(\vec{r})$ the vector of Rashba couplings in the three spatial directions, $V_{\rm Z}$ the Zeeman energy produced by an external magnetic field in the $x$-direction, $\Delta(\vec{r})$ the induced superconducting pair potential, and $\sigma$ and $\tau$ the Pauli matrices in spin and electron-hole space, respectively. The specific wire, electrostatic and geometrical parameters used in our simulations are summarized in Table \[Table\_parameters\]. We note that there are three quantities entering the Hamiltonian as inhomogeneous functions: the potential profile $\phi$ (that controls the local wire’s band bottom), the spin-orbit coupling $\vec{\alpha}_{\rm R}$, and the induced pairing $\Delta$. On the other hand, we consider other quantities constant in space: the Zeeman splitting $V_{\rm Z}$, assuming that the applied magnetic field does not suffer from SC finger screening, and the effective mass $m^*$, which is taken as an effective renormalized parameter. In the remainder of this section we explain in detail how we model the spatial-dependent quantities. For a description of the precise numerical methods used to solve the Hamiltonian, see App. \[SI1\]. The electrostatic potential $\phi(\vec{r})$ is found by solving self-consistently the Poisson’s equation $$\begin{aligned} \vec{\nabla}(\epsilon(\vec{r})\cdot\vec{\nabla}\phi(\vec{r}))=\rho_{\rm tot}[\phi(\vec{r})], \label{Poisson}\end{aligned}$$ where $\epsilon(\vec{r})$ is the dielectric permittivity in the entire system and $\rho_{\rm tot}[\phi(\vec{r})]$ is the total charge density of the wire, which itself depends on $\phi(\vec{r})$. The two superlattice geometries considered in this work, Figs. \[Fig1\](a,b), are taken into account through piecewise functions of $\epsilon(\vec{r})$, where each material is characterized by a different constant permittivity, as shown in Fig. \[Fig1\](a), leading to abrupt changes at the interfaces. Following Ref. , we model the total charge density of the wire as $$\rho_{\rm tot}=\rho_{\rm surf}+\rho_{\rm mobile}. \label{charge_density}$$ Here $\rho_{\rm surf}$ represents the charge density of a thin layer of donor states that typically forms at the surface of the InAs wire exposed to air [@Olsson:PRL96]. It depends on the details of the surface chemistry and its precise value is difficult to know [@Thelander:Nano10]. We model it as a $1$nm layer of *positive* charge fixed at the wire’s surface that is independent of the applied gate voltage. We consider two possible values compatible with existent literature, one larger, $\rho_{\rm surf}/e=2\times 10^{18}$cm$^{-3}$, and the other smaller, $\rho_{\rm surf}/e=2\times 10^{17}$cm$^{-3}$. The main effect of this charge is to produce an accumulation of electrons in the wire close to the surface and thus an [*[intrinsic]{}*]{} average doping in the absence of applied gate voltage. Hence, it conditions the values of $V_{\rm gate}$ necessary to deplete or charge the wire. On the other hand, $\rho_{\rm mobile}$ represents the mobile charges inside the wire. For the range of $V_{\rm gate}$ values that we are going to explore in this work $\rho_{\rm mobile}=\rho_{\rm e}$, i.e., it is the charge density produced by the electrons in the InAs conduction band. Should we consider stronger (negative) gate voltages, we would need to also take into account mobile charges coming from the InAs heavy hole and light hole bands (separated from the conduction band by the semiconducting gap energy), but this is not the case here (see App. \[SI1\].1 for more details). The spatial distribution of $\rho_{\rm e}$ depends on $\phi(\vec{r})$, in contrast to the surface charge $\rho_{\rm surf}$ that is localized at the nanowire facets not covered by the Al, as explained before. In our calculations we use the Thomas-Fermi approximation for a 3D electron gas and take $$\rho_{\rm e}(\vec{r})=-\frac{e}{3\pi^2}\left(\frac{2m^*|e\phi(\vec{r})+E_{\rm F}|f(-(e\phi(\vec{r})+E_{\rm F}))}{\hbar^2}\right)^\frac{3}{2}, \label{Thomas-Fermi}$$ where $f$ is the Fermi-Dirac distribution (we assume $T=10$mK) and we set to zero the wire’s Fermi energy ($E_{\rm F}=0$). We use the Thomas-Fermi approximation instead of performing a full Schrödinger-Poisson calculation because it is less demanding computationally. It has nevertheless been shown recently [@Mikkelsen:PRX18] that this approximation gives results in good agreement with the full treatment in similar simulations of InAs/Al heterostructures. To check this, we perform Schrödinger-Poisson self-consistent calculations for some specific cases in App. \[SI2\] and quantify the deviations of the wire’s charge distribution between the two. We find that Thomas-Fermi approximation slightly overestimates the electron charge density close to the SC fingers and at the wire’s boundaries, but otherwise produce very similar results for the electrostatic potential. In the bottom-right panel of Fig. \[Fig1\](b) we show schematically the boundary conditions used in our simulations. A voltage $V_{\rm gate}$ is applied to the back gate that is at a distance from the SC fingers/nanowire structure given by the width of the substrate (which we take as SiO$_2$). This back gate is used to tune the average chemical potential inside the wire. We assume that $\rho_{\rm surf}$ covers all the wire’s facets except for those in direct contact to the SC fingers. The boundary condition between the nanowire and the SC superlattice depends on several microscopic details such as their material composition, their sizes, the type and quality of the interface, etc. Certainly, the proximity effect will also depend on these details. A detailed description of this problem is beyond the scope of this work. Concerning its electrostatic effect, we shall assume that there is a perfect Ohmic contact between the SC and the semiconductor that imposes a constant potential at the interface that we call $V_{\rm SC}$. It represents the band bending with respect to the Fermi level of the InAs conduction band in the vicinity of the SC-semiconductor interface due to the work function difference between both materials. For an extended epitaxial InAs-Al interface, this quantity has been recently analysed in Refs. . Following those studies, here we will take $V_{\rm SC}=0.2$eV. However, the precise number is not important for the qualitative analysis that we present here. It will create an accumulation of electrons close to the SCs very similar to the one created by $\rho_{\rm surf}$, contributing to the [*[intrinsic]{}*]{} doping of the wire in the absence of $V_{\rm gate}$. It will thus have an influence on the values of back gate’s voltages needed to deplete or charge the wire. To visualize the effect of SC superlattice and $\rho_{\rm surf}$, we show for the top-superlattice setup an example of the potential energy profile (in red) and the electron charge density profile (in blue) along the wire ($x$-direction) in Fig. \[Fig1\](c), and across the wire’s section ($z$-direction) in Fig. \[Fig1\](d). These curves are calculated with the self-consistent Thomas-Fermi approximation for some particular representative values of $V_{\rm gate}$, $V_{\rm SC}$ and $\rho_{\rm surf}$. As expected, the potential energy profile (that represents the local band bottom energy) oscillates along the wire with the periodicity set by the SC superlattice. It is minimum below the SC fingers and maximum between them. Conversely, the charge density profile is maximum below the fingers and minimum between them. In the transverse direction we can see that the charge density localizes close to the SC finger, right where the band-bottom energy is minimum, forming an electron accumulation layer. The second inhomogeneous quantity that enters the Hamiltonian of Eq. (\[Hamiltonian\]) is the spin-orbit coupling. We assume that it is locally proportional to the electric field $\vec{\alpha}(\vec{r})\propto\vec{E}(\vec{r})=-\vec{\nabla}\phi(\vec{r})$. According to Refs. and using an 8-band $k\cdot p$ theory [@Winkler:03], it can be modelled as $$\label{alpha} \vec{\alpha}(\vec{r})=\vec{\alpha}_{\rm int}+\frac{eP^2}{3}\left[\frac{1}{E_{\rm cv}^2}-\frac{1}{(E_{\rm cv}+E_{\rm vv})^2}\right] \vec{\nabla} \phi(\vec{r}),$$ where $P$ is the coupling between the lowest-energy conduction band and the highest-energy valence band, $E_{\mathrm{cv}}$ is the semiconductor gap (energy difference between the conduction and valence bands), and $E_{\mathrm{vv}}$ is the energy gap between the highest-energy and lowest-energy valence bands (split-off gap). For an InAs nanowire with wurzite crystal structure these values are [@Winkler:03] $P=919.7$meV$\cdot$nm, $E_{\mathrm{cv}}=418$meV and $E_{\mathrm{vv}}=380$meV. Additionally, since we are considering a wurtzite InAs nanowire, there is an intrinsic Rashba constant contribution in the x-direction [@Voon:PRB96; @Gmitra:PRB16] of the order of $\alpha_{\rm int}\simeq30$meV$\cdot$nm. Finally, the last inhomogeneous quantity is the induced superconducting pairing $\Delta(\vec{r})$, which we model as a telegraph function with a constant value $\Delta_0$ (of the order of the bulk gap in the parent superconductor) at the wire’s facets in contact to the SC fingers and zero otherwise. [0.47]{}[ |X|X|X| ]{}\ $m^*=0.023m_{\rm e}$ & $E_{\rm F}=0$ & $\alpha_{\rm int}=30$meV$\cdot$nm\ [0.48]{}[ |s|l|l| ]{}\ $\epsilon_{\mathrm{InAs}}=17.7\epsilon_0$ & $\epsilon_{\mathrm{SiO}}=5.5\epsilon_0$ & $\epsilon_{\mathrm{vacuum}}=\epsilon_0$\ $V_{\rm SC}=200$meV & $\rho_{\rm surf}^{(1)}=2\cdot10^{-3}\frac{e}{nm^3}$ & $\rho_{\rm surf}^{(2)}=2\cdot10^{-4}\frac{e}{nm^3}$\ [0.48]{}[ |X|X|X| ]{}\ $W_{\mathrm{InAs}}=80$nm & $W_{\mathrm{Al}}=10$nm & $W_{\mathrm{SiO}}=20$nm\ [0.32]{}[ |X|X| ]{}\ $\Delta_0=0.2$meV [@Chang:Nnano15] & $T=10$mK\ \[Table\_parameters\] ![(Colour online) Electrostatic potential profile created inside an InAs wire in contact to Al SC fingers due to the voltage applied to the back gate. Here $V_{\rm SC}=0$, $\rho_{\rm surf}=0$ and $\rho_{\rm e}$ is neglected. Two setups are considered, bottom-superlattice to the left and top-superlattice to the right, with $L_{\rm cell}=150$nm and $r_{\rm SC}=0.5$. (a,b) Sketches of both systems. (c,d) Electrostatic profile normalized to $V_{\rm gate}$ along the wire (top), and across the wire’s section (bottom), both for sections with SC finger (enclosed by a purple square) and between SC fingers (enclosed by a green square). A white dotted line is used in (c,d) to highlight the shape of the potential oscillations in each setup for one particular isopotential.[]{data-label="Fig2"}](Fig2.pdf) ![(Colour online) Electrostatic potential profile created inside an InAs wire in contact to Al SC fingers due to the wire’s band offset with respect to the Fermi level at the interface with the SC ($V_{\rm SC}=0.2$V) and the surface charge layer at the rest of the facets. Here $V_{\rm gate}=0$ and $\rho_{\rm e}$ is neglected. Two setups are considered, bottom-superlattice to the left and top-superlattice to the right, with $L_{\rm cell}=150$nm and $r_{\rm SC}=0.5$. (a,b) Sketches of both systems. (c,d) Electrostatic profile along the wire (top), and across the wire’s section (bottom) for a surface charge density of $\rho_{\rm surf}=2\cdot 10^{18}(e/cm^{3})$. (e,f) Same for $\rho_{\rm surf}=2\cdot 10^{17}(e/cm^{3})$.[]{data-label="Fig3"}](Fig3.pdf) Electrostatic effects {#Electrostatic} ===================== Electrostatic potential profile {#Potential} ------------------------------- We want to study the impact of a realistic electrostatic potential profile along and across the 3D wire on the topological phase diagram and the formation of MBSs. Since we are interested in understanding the effect of the superlattice structure, we consider throughout this work periodic boundary conditions in the $x$ direction (and thus ignore border effects in the electrostatic problem). Moreover, in this section we ignore the screening effect of the mobile charges inside the wire, $\rho_{\rm e}$, because we want to isolate the impact of the electrostatic environment on the wire’s potential profile (see App. \[SI1-1\] and Fig. \[FigSI1\]). Nevertheless, they are included self-consistently in Sec. \[3D\_results\]. In Fig. \[Fig2\] we plot the potential profile $\phi$ created by the bottom gate normalized to $V_{\rm gate}$, both for the bottom-superlattice device to the left and for the top-superlattice one to the right. In this case we ignore the presence of the Al-InAs band offset and the surface charge layer and take $V_{\rm SC}=0$ and $\rho_{\rm surf}=0$. The potential oscillates along the wire with the periodicity of the superlattice, but the oscillations are very different for each setup, see white dotted guidelines in Figs. \[Fig2\](c,d) that highlight some isopotentials. In the bottom-superlattice device the potential maximum oscillates between the top and the bottom of the wire depending on whether the wire’s section is between or on top of the SC fingers, while in the top-superlattice setup the maximum is always at the bottom of the wire, leading to smaller oscillations along the $x$ direction. This can be better appreciated in the bottom panels of Figs. \[Fig2\](c,d), where the potential profile across the wire’s section is depicted both for sections with a SC finger (purple squared) and between SC fingers (green squared). The oscillations thus produce stronger potential wells in the first setup and subsequent bound states localized over the SCs. When present, these states are detrimental for the stability of the topological phase as we will analyse in Sec. \[Impact\]. Another difference between the two setups is the ability of the gate to control the potential inside the wire (and, therefore, to produce a certain doping) in the presence of the electrostatic environment. Gating is more difficult in the bottom-superlattice device because the metallic fingers are closer to the gate and thus they screen its potential more efficiently. This is why $\phi/V_{\rm gate}$ is closer to zero (blue color) in Fig. \[Fig2\](c) whereas in Fig. \[Fig2\](d) the potential better approaches $V_{\rm gate}$ (red color) at the bottom of the wire, away from the SC fingers. Now we explore the electrostatic potential profile created by the surface charge density $\rho_{\rm surf}$ and the potential boundary condition at the interface with the SC fingers ($V_{\rm SC}=0.2$V). As illustrated in Fig. \[Fig3\], we perform this study setting the back gate potential to zero. As before, the potential oscillates along the wire with the periodicity of the superlattice and across the wire’s section it varies depending on whether that section is on or between the SC fingers. Since the potential profile times the electron charge $-e$ represents the wire’s conduction band bottom, the wire’s doping is proportional to the electrostatic potential. The main effect of the wire’s band-offset with respect to the Fermi level at the SC interface and the surface charge at the other interfaces is to increase the wire’s doping by a quantity that we call $\mu_{\rm int}$, which is the spatial average of the potential energy profile created by $V_{\rm SC}$ and $\rho_{\rm surf}$. This is more pronounced for the case with a larger $\rho_{\rm surf}$. We note that for realistic parameters $\mu_{\rm int}$ is always positive. On the other hand, the total doping of the wire $\mu$ coming both from the intrinsic charge and the gate voltage can be positive or negative depending on the sign and magnitude of $V_{\rm gate}$. ![(Colour online) Contribution of the back gate potential to the local longitudinal Rashba coupling inside the wire. $V_{\rm SC}$ and $\rho_{\rm surf}$ are fixed to zero, and $\rho_{\rm e}$ is neglected. Two setups are considered, bottom-superlattice to the left and top-superlattice to the right, with $L_{\rm cell}=150$nm and $r_{\rm SC}=0.5$. (a,b) Sketches of the two setups. (c,d) $\alpha_z$ along the wire (top) and across the wire’s section (bottom), both for sections on with SC finger (inside a purple square) and between SC fingers (inside a green square).[]{data-label="Fig4"}](Fig4.pdf) ![(Colour online) Contribution of the Al-InAs band offset ($V_{\rm SC}=0.2$V) and surface charge layer to the to the local longitudinal Rashba coupling inside the wire. Here $V_{\rm gate}=0$V and $\rho_{\rm e}$ is neglected. Two setups are considered, bottom-superlattice to the left and top-superlattice to the right, with $L_{\rm cell}=150$nm and $r_{\rm SC}=0.5$. (a,b) Sketches of the two setups. (c,d) $\alpha_z$ along the wire (top), and across the wire’s section (bottom) for a surface charge density of $\rho_{\rm surf}=2\cdot 10^{18}(e/cm^{3})$. (e,f) Same for $\rho_{\rm surf}=2\cdot 10^{17}(e/cm^{3})$.[]{data-label="Fig5"}](Fig5.pdf) Inhomogeneous Rashba coupling {#Rashba} ----------------------------- The inhomogeneous electrostatic potential calculated in the previous section creates an inhomogeneous electric field that, in turn, generates an inhomogeneous Rashba coupling along and across the wire. We assume that the Rashba coupling is locally proportional to the electric field, as explained in Sec. \[Methods\]. There are three Rashba couplings, $\alpha_{x,y,z}$, giving rise to six terms in the Hamiltonian of Eq. \[Hamiltonian\]. Considering that the magnetic field in our model points in the $x$ direction, only two of those terms contribute to the opening of a topological minigap. These are proportional to $\alpha_z \sigma_y k_x$ and $\alpha_y \sigma_z k_x$. The effect of the other four Rashba terms is basically to produce hybridization of the transverse subbands and the subsequent even-odd effect for the appearance of Majoranas [@Lim:PRB12]. It turns out that $\alpha_y$ is negligible in these wire setups (due to the back gate-superlattice parallel disposition). Thus, we focus here on analysing the spatial behaviour of the transverse $\alpha_z$ coupling, shown in Figs. \[Fig4\] and \[Fig5\]. Following the rationale of the previous section, in the first figure we explore the Rashba coupling behaviour against the back gate potential (normalized to $V_{\rm gate}$) setting $V_{\rm SC}=0$V and $\rho_{\rm surf}=0$. Conversely, in the second one we study the contribution of the Al-InAs band offset and surface charge density setting $V_{\rm gate}=0$V. For the top-superlattice setup we can see in Fig. \[Fig4\](d) that $\alpha_z$ exhibits some oscillations along the wire with the periodicity of the lattice, specially close to the SC fingers, but it is on average large and positive. This is beneficial for the formation of a robust topological minigap for $V_{\rm Z}>V_{\rm c}$. On the contrary, for the bottom-superlattice device $\alpha_z$ oscillates between positive and negative values along the $x$ direction, see Fig. \[Fig4\](d), averaging to a smaller number, which is detrimental for the protection of MBSs. The Rashba coupling produced by the back gate electric field has to be supplemented with the one created by the Al-InAs band offset and surface charge layer, shown in Fig. \[Fig5\]. On average this is proportional to the magnitude of $\rho_{\rm surf}$, see the different color bar ranges in (c,d) and (e,f). For the bottom-superlattice device shown in Figs. \[Fig5\](c,e), $\alpha_z$ oscillates along $x$ as before but with the same sign, giving a finite contribution to the topological gap (specially close to the SC fingers). This is also true for the top-superlattice device in the case of the smaller $\rho_{\rm surf}$, Fig. \[Fig5\](f), but it changes sign along and across the wire for the larger one, Fig. \[Fig5\](d). According to these results and unless there are other sources of electric fields, the Rashba spin-orbit coupling relevant for Majoranas in the bottom-superlattice setup is going to be dominated by boundary conditions rather than by the voltage applied to the back gate. On the contrary, for the top-superlattice device $\alpha_z$ is going to be dominated by the back gate except for small values of $V_{\rm gate}$, in which case its qualitative behaviour is strongly dependent on the magnitude of $\rho_{\rm surf}$. Impact of the superlattice on the nanowire spectral properties {#Impact} ============================================================== We focus now on the impact of the superlattice, in particular the inhomogeneous electrochemical potential and the inhomogeneous induced superconductivity, on the spectral properties of a finite-length nanowire. In the calculations of this section we consider that all the charge density is at the wire’s symmetry axis, so that we effectively solve a 1D problem. We do this for two reasons. One is that it is computationally less expensive and still useful to understand the impact of the superlattice on the formation of MBSs. It is also a way to isolate the effect of the longitudinal subbands created by the superlattice, which is what we seek here, from the transverse subbands, which introduce further phenomenology [@Stanescu:PRB11; @Lutchyn:PRL11; @Lim:PRB12] unrelated to the superlattice. Nevertheless, as explained in the Introduction, in the final section we will solve the complete 3D problem. Since we aim to understand qualitatively the effect of each [*[kind]{}*]{} of inhomogeneity, in the following subsections we study their contribution separately, fixing other parameters to constant homogeneous values. For example, to find the spectrum in Subsecs. \[Impact-mu\] and \[Impact-Eint\] we diagonalize the Hamiltonian of Eq. (\[Hamiltonian\]) for constant $\Delta$ and $\alpha_{\rm R}=\alpha_z$, but for the potential profile along $x$ calculated in Sec. \[Potential\], which is the result of a 3D Poisson calculation (but taken at $y=0$ and $z=0$). In Subsec. \[Impact-SC\] we consider an inhomogeneous induced pairing in $x$ and fix $\mu$ and again $\alpha_z$ to constant values. We have also analysed the case of an inhomogeneous superlattice Rashba coupling with other parameters constant (not shown), but the effect on the wire’s spectrum is small, although it does influence the Majorana wave function shape. ![(Colour online) (a) Dispersion relation for a 1D superlattice nanowire with superlattice parameters $L_{\rm cell}=400$nm and $r_{\rm SC}=0.5$. The electrostatic potential profile oscillates along the wire’s axis following a similar profile as the one shown in Fig. \[Fig2\] but evaluated at $(y,z)=(0,0)$. Here we take homogeneous in $x$ values for the induced SC pairing, Rashba coupling and intrinsic doping: $\Delta_0=0.2$meV, $\alpha_z=40$meV$\cdot$nm, and $\mu_{\rm int}=5$meV. (b) Corresponding topological phase diagram for the bulk system. (c) Lowest level energy for a finite-length nanowire of $L_{\rm wire}=1.2\mu$m. (d-f) The same but for $L_{\rm cell}=100$nm. The green dots mark the values of $V_{\rm Z}$ and $\mu=e\langle\phi(x)\rangle$ for which the top figures are plotted.[]{data-label="Fig6"}](Fig6.pdf) Impact of the inhomogeneous electrochemical potential {#Impact-mu} ----------------------------------------------------- We start by analysing the effect on the wire’s spectrum of the superlattice chemical potential. We take a similar potential profile as the ones of Figs. \[Fig2\](c,d) (but with different $L_{\rm cell}$ values), i.e. ignoring the inhomogeneous intrinsic doping of the wire, at (y,z)=(0,0). On the other hand, we take constant values for the induced pairing and Rashba coupling ($\Delta_0=0.2$meV and $\alpha_z=40$meV$\cdot$nm). Due to the superlattice structure, the real space unit cell is larger than for a homogeneous potential wire, leading to the formation of longitudinal subbands in the dispersion relation, see Figs. \[Fig6\](a,d) for two values of $L_{\rm cell}$. The number of these longitudinal subbands per unit energy increases with $L_{\rm cell}$. As stated in Ref. , only when the Fermi energy crosses an odd number of Fermi pair points, the system is topologically non-trivial (light blue regions). Otherwise it is trivial (light pink regions). The electrostatic potential can open a gap between longitudinal subbands, whose size depends on the strength of the potential oscillations, leading to energy ranges where the Fermi energy crosses no band [@Malard:PRB16] (see Fig. \[Fig6\](d)). This causes the wire to exit the topological phase. In Figs. \[Fig6\](b,e) we plot the wire’s phase diagram versus Zeeman field $V_{\rm Z}$ and chemical potential, given by the space average of the electrostatic potential times the electric charge $e\langle\phi(x)\rangle$. The green dots mark the values of these parameters for which the dispersion relations in (a,d) are plotted. This phase diagram is certainly more complex than the one of an homogeneous 1D Majorana nanowire, characterized by a single solid hyperbolic topological zone corresponding to one topological band (whose boundary is given by the condition $\mu=\pm\sqrt{V_{\rm Z}^2-\Delta^2}$). Here, since we have several longitudinal subbands, we have several more or less contiguous topological zones (with shapes that only slightly resemble the single-band hyperbolic one) separated by trivial regions whenever the Fermi energy crosses an even number of Fermi pair points, see Fig. \[Fig6\](b). Moreover, whenever the Fermi energy lies at the gaps between longitudinal subbands, the phase diagram develops trivial [*[holes]{}*]{} within the topological phase, see for instance the light pink region at the bottom-right corner in Fig. \[Fig6\](e). At the boundaries of this trivial holes we have the condition $\lambda_{\rm F}=L_{\rm cell}$, as pointed out in Refs. . Additionally, we note that a change in the back gate potential will not only move the subbands upwards or downwards in a rigid way, but it will also change the hybridization between the longitudinal subbands, leading to a change in the trivial hole sizes. It is known that, for a finite-length nanowire, Majorana zero modes appear in the wire’s spectrum in the topological phase. These states are localized at the edges of the wire and decay exponentially towards its center with the so called Majorana localization length, that is proportional to the SC coherence length [@Klinovaja:PRB12]. When the wire’s length is not much greater than the Majorana localization length, left and right MBSs overlap and their energy lifts from zero producing characteristic Majorana oscillations as a function of $V_{\rm Z}$ and $\mu$. The lowest level energy of a finite-length nanowire ($L_{\rm wire}=1.2\mu$m) is shown in Figs. \[Fig6\](c,f), where we see the impact of the electrostatic potential superlattice on the Majorana oscillations. As it can be observed, the regions where the lowest-energy modes approach zero energy in Figs. \[Fig6\](c,f), coincide (roughly) with the non-trivial regions in the phase diagrams of Figs. \[Fig6\] (b,e). ![(Colour online) Lowest level energy as a function of applied gate voltage and Zeeman field for a finite-length 1D bottom-superlattice nanowire in the presence of an inhomogeneous potential profile. This potential is taken from a 3D calculation with Al-InAs band offset $V_{\rm SC}=0.2$V and different surface charge density values, evaluated at $(y,z)=(0,0)$. Different superlattice cell sizes (with $r_{\rm SC}=0.5$) are considered. Topologically trivial regions are coloured in light pink, non-trivial regions are plotted in blue-red scale given by the color bar (where the MBS energy is normalized to $\Delta_0$), and the black dashed lines mark localized trivial zero energy modes. The Rashba coupling and induced pairing in the Hamiltonian are fixed to the homogeneous quantities $\alpha_{\rm R}=30$meV$\cdot$nm and $\Delta_0=0.2$meV. The length of the wire is $L_{\rm wire}=1.2\mu m$.[]{data-label="Fig7"}](Fig7.pdf) ![(Colour online) Same as Fig. \[Fig7\] but for a top-superlattice setup.[]{data-label="Fig8"}](Fig8.pdf) Role of the intrinsic doping {#Impact-Eint} ---------------------------- In this subsection we solve the same problem as in the previous one, but we now include the effect of the inhomogeneous doping $\mu_{\rm int}$ created by the SC-semiconductor band offset and the surface charge density. Fig. \[Fig7\] and Fig. \[Fig8\] show, for the bottom- and top-superlattice devices, the lowest level energy as a function of the Zeeman field and the back gate voltage for different superlattice cell sizes (with $r_{\rm SC}=0.5$) and for different surface charge densities. Note that trivial regions are coloured in light pink, as in the phase diagrams of Fig. \[Fig6\]. The different columns in Fig. \[Fig7\] and Fig. \[Fig8\] correspond to different sizes of $L_{\rm cell}$. Notice that the size of the topological regions increases as the superlattice cell decreases. Actually, for a large enough $L_{\rm cell}$ the topological phase is inexistent, see Figs. \[Fig7\], \[Fig8\] (d,g). For large superlattice cell sizes, topologically trivial localized states are present (black dashed lines), which may interfere with the MBSs. This effect is more pronounced in the bottom-superlattice setup because the back gate voltages needed to enter the topological phase are larger due to the screening of the SC fingers. This in turn produces stronger potential oscillations and subsequent localized states, as explained in Sec. \[Potential\]. At smaller $L_{\rm cell}$ sizes, the localized states disappear. For medium cell sizes $L_{\rm cell}$, which are probably more appropriate for experimental realization, we typically encounter the condition $\lambda_{\rm F}=L_{\rm cell}$ explained in the previous subsection and trivial holes appear in the topological phase, both in the bottom and top-superlattice setups. However, the top-superlattice setup develops larger topological regions and they are present for the two values of $\rho_{\rm surf}$ considered, see Figs. \[Fig8\](e,h). In the bottom-superlattice case no topological region is found for the larger $\rho_{\rm surf}$, see Fig. \[Fig7\](e). For small $L_{\rm cell}$ sizes the topological phase is more stable, meaning that there are no trivial holes. This is so because for small and short potential oscillations the electrons in the wire feel an effective homogeneous potential [@Levine:PRB17]. Moreover, the performance of both setups (top and bottom) is comparable, although the back gate voltages needed for the bottom one are much larger. Impact of inhomogeneous induced pairing {#Impact-SC} --------------------------------------- Finally, we consider the impact of the inhomogeneous superconductivity. For this purpose, we solve a 1D wire where we fix the chemical potential and Rashba coupling to constant values. The superconducting pairing amplitude is taken as a telegraph function that oscillates between $\Delta_0=0.2$meV and zero with a period given by $L_{\rm cell}$ and $r_{\rm SC}$. As done in the previous sections, this is a simplified model to understand qualitatively the effect of inhomogeneous superconductivity. When we consider the realistic 3D model later on, the induced pairing will be only present at the surface of the wire in the regions where it is close to the SC fingers. Fig. \[Fig9\] shows the energy gap (energy of the lowest-energy state at $k=0$) normalized to $\Delta_0$ for an infinite 1D wire against the superlattice parameters $L_{\rm cell}$ and $r_{\rm SC}=L_{\rm SC}/L_{\rm cell}$. For small coverage $r_{\rm SC}<0.5$ the induced superconductivity is poor and it improves as $r_{\rm SC}$ increases. For $r_{\rm SC}\rightarrow 1$ we recover a perfect induced gap $\Delta_0$ corresponding to a wire covered by an uniform SC at $V_{\rm Z}=0$. Interestingly, for strong spin-orbit coupling the gap energy basically does not depend on $L_{\rm cell}$, see Fig. \[Fig9\](b). However, for small $\alpha_{\rm R}$ the induced gap worsens considerably with $L_{\rm cell}$, as shown in Fig. \[Fig9\](a). ![(Colour online) Energy gap (at Zeeman energy $V_{\rm Z}=0$ and $k=0$) versus $L_{\rm cell}$ and $r_{\rm SC}=L_{\rm SC}/L_{\rm cell}$ for a 1D nanowire with a telegraph superconducting pairing that oscillates between $\Delta_0=0.2$meV and zero along $x$. The chemical potential and Rashba coupling are fixed to homogeneous values $\mu=0$ and (a) $\alpha_{\rm R}=10$meV$\cdot$nm, (b) $\alpha_{\rm R}=100$meV$\cdot$nm.[]{data-label="Fig9"}](Fig9.pdf) ![(Colour online) Approximate regions in superlattice parameter space $L_{\rm cell}$ and $r_{\rm SC}$ where different mechanisms that spoil the topological phase appear, such as the formation of longitudinal subband overlaps, longitudinal subband gaps and localized states; marked in brown, red and blue, respectively. We have taken $V_{\rm Z}=0.6$meV, $\langle\mu_{\rm int}\rangle=200$meV, $\langle\mu_{\rm V_{\rm gate}}\rangle\in[0,3]$meV, and $\langle\alpha_{z}\rangle\in[5,50]$meV$\cdot$nm.[]{data-label="Fig10"}](Fig10.pdf) Superlattice features in parameter space {#Optimal_parameters} ---------------------------------------- We can summarize our previous findings by plotting a diagram in parameter space that shows the different features caused by the superlattice and that interfere with the topological phase. This is done in Fig. \[Fig10\] versus $L_{\rm cell}$ and $r_{\rm SC}$ for $V_{\rm Z}=0.6$meV and $\Delta_0=0.2$meV, and taking the following (realistic) spatial average values for other parameters: $\langle\mu_{int}\rangle=200$meV, $\langle\mu_{\rm V_{\rm gate}}\rangle\in[0,3]$meV and $\langle\alpha_{z}\rangle\in[5,50]$meV$\cdot$nm. In the brown area we have values of $L_{\rm cell}$ and $r_{\rm SC}$ for which the Fermi energy crosses an even number of Fermi pair points in the nanowire dispersion relation. This happens when the level spacing between longitudinal subbands is smaller than the (energy) size of the topological phase ($\frac{\pi^2\hbar^2}{2mL_{\rm cell}^2}\le \sqrt{V_{\rm Z}^2-\Delta^2}$). In this case the topological regions of contiguous longitudinal subbands interfere and the system exits the topological phase (there is an annihilation of an even number of Majoranas at each wire’s edge). See, for instance, the upper subbands plotted in Figs. \[Fig6\](a,b). In the red area we have values of $L_{\rm cell}$ and $r_{\rm SC}$ for which there appear gaps between (the lowest) longitudinal subbands in the nanowire’s dispersion relation. As we explained in Sec. \[Impact-mu\], when the Fermi energy is within these gaps, trivial holes emerge in the topological regions of the phase diagram. See for example the bottom-right corner of Fig. \[Fig6\](e). This happens when there is a resonance between the Fermi wavelength $\lambda_{\rm F}$ and the superlattice length $L_{\rm cell}$. The red area is somewhat larger for the bottom-superlattice than for the top one. This is because the appearance and size of the longitudinal subbands gaps depends on the strength of the potential oscillations, which is larger for the bottom-superlattice due to the back gate’s screening by the metallic fingers. Finally, in the blue area localized states are formed. As we saw in Sec. \[Potential\], the superlattice of fingers creates potential oscillations along the wire. When the height of these oscillations is large enough ($\frac{\pi^2\hbar^2}{2mL_{\rm SC}^2}\le \frac{\sigma_{\rm \phi}}{<\phi>}\langle\mu_{\rm int}\rangle$), there appear potential wells for the electrons that create localized states (see App. \[SI3\] for the $L_{\rm cell}$-$r_{\rm SC}$ dependence of $\langle\phi\rangle$ and $\sigma_{\rm \phi}$). These states interfere with the MBSs detaching them from zero energy. Moreover, when the potential oscillations are very strong, they divide effectively the nanowire into regions of smaller length, destroying the Majoranas. Again, the blue area is slightly larger for the bottom-superlattice than for the top-one. This diagram gives us an idea of different detrimental mechanisms for a robust topological phase that may appear as a function of superlattice parameters. This does not mean that we cannot find topological regions for those $L_{\rm cell}$ and $r_{\rm SC}$ values, but that those regions will be interrupted at some points instead of extending more widely as a function of nanowire parameters. To complete this study we should also consider the size of the topological minigap. As we saw in Sec. \[Impact-SC\] (see Fig. \[Fig9\]), it decreases when the superconducting partial coverage $r_{\rm SC}$ does, which is additionally Rashba coupling dependent (see App. \[SI3\] for more details). Moreover, we have to bear in mind that the qualitative analysis of Fig. \[Fig10\] is performed for a 1D model of the nanowire. When a 3D wire is considered, several transverse modes can be occupied. In this case there will be an interplay between longitudinal and transverse subbands that will introduce further complexity to the determination of the optimal superlattice parameters. 3D results {#3D_results} ========== In this section we consider together all the different ingredients that have been analysed separately in the previous sections and for a realistic 3D nanowire. In particular, we take representative superlattice parameters $L_{\rm cell}=100$nm and $r_{\rm SC}=0.5$. To calculate the electrostatic potential profile we perform self-consistent Poisson simulations in the Thomas-Fermi approximation for the bottom and top-superlattice setups. We find the wire states by diagonalizing the Bogoliubov-de Gennes Hamiltonian for a $2\mu$m long wire using the previous potential. As mentioned in Sec. \[Impact-SC\], we model the induced pairing as a telegraph function with $\Delta_0=0.2$meV in the regions of the wire close to the SC fingers and zero otherwise. In particular, for these 3D calculations we consider $\Delta_0\neq 0$ for a certain depth ($\sim 25\%$ of the wire’s width) close to the SC fingers in the transverse direction.  In Fig. \[Fig11\] we show the low-energy spectrum as a function of back gate voltage for a particular value of Zeeman splitting, $V_{\rm Z}=0.6$meV, both for the bottom-superlattice in (a) and the top-superlattice setup in (b). We explore a wide range of $V_{\rm gate}$ values that corresponds to the first transverse occupied subband that develops topological states (seen as quasi-zero energy states whose energies split from zero in an oscillating manner). As explained before, this subband appears for larger negative values of $V_{\rm gate}$ in the bottom-superlattice case due to the screening effects of the SC fingers. We note that, strictly speaking, in these systems one cannot really label subbands as purely transverse or longitudinal because the spin-orbit term in the Hamiltonian mixes the two momenta. However, and due to the small cross-section of the wires, groups of subbands have still a dominant weight on a particular quasi-transverse subband. In these spectra we can observe all the phenomenology that we have been discussing in previous sections. For the most negative values of $V_{\rm gate}$, left part of Figs. \[Fig11\](a,b), the wire is almost empty except for very flat bands that appear at the quantum wells of the electrostatic potential oscillations. As a function of $V_{\rm gate}$ these create quick gap closings and reopenings and the topological phase cannot be developed. As $V_{\rm gate}$ is increased, middle part of Figs. \[Fig11\](a,b), different dispersing longitudinal subbands become populated. When the topological conditions are satisfied, we find extended $V_{\rm gate}$ regions with oscillating low energy modes separated by a minigap from the quasicontinuum of states (dark grey). These are the regions of interest because those oscillating states correspond to (more or less overlapping) MBSs localized at the left and right edges of the finite-length wire. The size of the oscillations and the minigap depends on the longitudinal subband. Sometimes, these topological regions are crossed by a localized state that closes the minigap at a certain $V_{\rm gate}$ point (see arrows in Figs. \[Fig11\](a,b)). The localized states disperse linearly with $V_{\rm gate}$ and cross zero energy displaying an $x$ shape. Other times we find trivial regions (without Majorana oscillations) between two topological ones due to topological subbands gaps or to topological subbands overlaps, as explained in Sec. \[Impact-mu\]. Finally, at the right-most values of $V_{\rm gate}$ an additional transverse subband crosses below the Fermi level and the spectrum becomes more intricate, with the even-odd effect playing a role (not shown). For comparison, we also show in Fig. \[Fig11\](c) the case of a nanowire homogeneously covered by a SC at the top of the wire. The range of $V_{\rm gate}$ values displayed in this case is chosen so that no hole states appear in the system. For more negative voltages the lower part of the nanowire becomes populated by hole quasiparticles from the valence band and a proper description of the system would require to consider an extended version of the model Hamiltonian of Eq. (\[Hamiltonian\]) where electrons and holes coexist. To avoid this complication, we analyse higher voltages for which several transverse subbands are already populated. Note that here there are no longitudinal subbands. At the left and right parts of panel (c) we observe the well known even-odd effect between overlapping topological regions of different subbands. In the middle part, however, and for a pretty wide range of gate voltages, we have a region with no subgap states that corresponds to the trivial phase developed between two well separated transverse subbands.  Now we focus more specifically on one of the topological regions and analyse the location and shape of its MBSs. In Fig. \[Fig12\] we show with more detail the low-energy spectrum as a function of back gate voltage for the regions marked by a red rectangle in Fig. \[Fig11\]. To understand their behaviour, in Fig. \[Fig13\] we plot the corresponding electrostatic potential, Rashba coupling $\alpha_z$ and charge density profiles for the $V_{\rm gate}$ voltage marked by a blue line in the corresponding spectrum. The topological minigap is somewhat larger for the top-superlattice setup than for the bottom one. In the top-superlattice device it reaches approximately $\Delta_0/2$, which corresponds to the maximum possible induced gap for a superlattice with $r_{\rm SC}=0.5$, see analysis of Fig. \[Fig9\]. This relatively large value can be understood by looking at the Rashba coupling profile in Fig. \[Fig13\](d). We see that $\alpha_z$ has a pretty homogeneous finite value all over the wire and it gets specially sizeable ($\sim -30$meV$\cdot$nm) below the SC fingers, precisely where most of the charge density is located according to Fig. \[Fig13\](f). However, the minigap in the bottom-superlattice is smaller than in the top’s one. In this case $\alpha_z$ strongly oscillates between positive and negative values along the wire’s axis, see Fig. \[Fig13\](c), resulting in a smaller average Rashba coupling. In the homogeneous case the minigap is the largest, close to $\Delta_0$ in the middle region of panel Fig. \[Fig12\](j). Here the induced effective gap has to be necessarily better since the SC covers the whole length of the wire. Moreover, there is a homogeneous and large Rashba coupling along the wire ($\sim -30$meV$\cdot$nm) close to the SC where the charge density concentrates (not shown here). Concerning the Majorana oscillations, they are pretty comparable for the two types of superlattices and definitively bigger than for the homogeneous case. In Figs. \[Fig12\](c,g,k) we plot the Majorana probability density of the different setups along and across the wire for the values of $V_{\rm gate}$ marked by the blue lines in (b,f,j), respectively. We find that in all cases the MBSs are localized at the edges of the wire, but with different longitudinal and transverse profiles. Across the wire’s section the wave function tends to be close to the SC fingers in the top-superlattice setup. This is consistent with the charge density profile of Fig. \[Fig13\](f). On the other hand, the probability density oscillates from top to bottom in the bottom-superlattice one, see lower panel of Fig. \[Fig12\](c). As we noticed in Sec. \[Potential\], this is related to the shape of the potential profile due to the strong gate voltages needed to deplete the wire in this setup. The probability density accommodates to the isopotential curves, which for the bottom-superlattice device oscillate from top to bottom in the $z$-direction as highlighted with a white guideline in Fig. \[Fig13\](a) for a particular $\phi$ value. Figures \[Fig12\](d,h,l) show longitudinal cuts of the probability density at the (y,z) cross-section values marked by arrows in Figs. \[Fig12\](c,g,k). As expected, in the homogeneous case the wave function decays exponentially towards the wire’s center with the Majorana localization length $\xi_{\rm M}$ [@Klinovaja:PRB12]. For the parameters of this case we obtain $\xi_{\rm M}=350$nm, which is consistent with panel (l). On the other hand, for the superlattice nanowires the decay length is characterized by the interplay between two scales, the Majorana length and the superlattice length $L_{\rm cell}$. The decay length in the homogeneous case is shorter and the probability density is pretty localized at the wire edges and almost zero at its center. This is not the case for the superlattices since their wave functions decay more slowly. To quantify this, we finally compute the absolute value of the Majorana charge $Q_{\rm M}$ that measures the wave function overlap between the right and the left Majoranas [@Ben-Sach:PRB15; @Escribano:BJN18; @Penaranda:PRB18] $$|Q_{\rm M}|=e\left|\int u_{\mathrm{L}}(\vec{r})u_{\mathrm{R}}(\vec{r}) dr^3\right|,$$ where $u_{\rm L,R}$ are the electron components of the left and right Majorana wavefunctions, given by $\gamma_{\rm L}=\psi_{+1}+\psi_{-1}$ and $\gamma_{\rm R}=-i(\psi_{+1}-\psi_{-1})$, being $\psi_{\pm1}$ the even/odd lowest-energy eigenstates. We get the values $\left|Q_{\rm M}^{\rm BS}\right|/e=0.93$, $\left|Q_{\rm M}^{\rm TS}\right|/e=0.88$ and $\left|Q_{\rm M}^{\rm h}\right|/e=0.63$ for the bottom-superlattice, top-superlattice and homogeneous cases, respectively. As expected, the Majorana charge is larger for both superlattice devices compared to the homogeneous case. ![(Colour online) Electrostatic potential (a,b), Rashba coupling $\alpha_z$ (c,d) and charge density profiles (e,f) for the same bottom and top-superlattice nanowires of Fig. \[Fig12\]. Here, $V_{\rm gate}=-2.142$V for the bottom-superlattice and $V_{\rm gate}=-0.376$V for the top one, marked by blue lines in Figs. \[Fig12\](b,f). The total wire’s charge is $Q_{\rm tot}/e=809$ for (e) and $Q_{\rm tot}/e=633$ for (f).[]{data-label="Fig13"}](Fig13.pdf) Alternative superlattice configuration {#Alternative} -------------------------------------- We have seen that the main inconvenience of the Majorana superlattice nanowires analysed in this work comes from the partial superconducting coverage produced by the SC superlattice (specially as $r_{\rm SC}$ diminishes). This leads to a reduced induced SC gap that, in turn, produces a smaller topological minigap and a larger Majorana charge. We could improve this scenario by covering one of the wire’s facets continuously with a thin SC layer, like in a conventional epitaxial Majorana nanowire, while still placing the hybrid structure on a superlattice. We analyse this alternative configuration in Fig. \[Fig14\] for the case of a bottom-superlattice setup. Now the superlattice can be either superconducting or normal (since the induced superconductivity is already provided by the SC layer). We choose here a set of normal metal fingers, such as gold, that could be used as tunneling local probes along the wire by driving a current between each finger and the SC homogeneous layer. The tunneling coupling in this case is advantageous because it leads to a smaller wire’s intrinsic doping and to a larger localization of the MBS wavefunctions close to the Al SC layer, where the electrostatic potential and induced pairing are larger. ![\[Fig14\] (Colour online) (a) Alternative superlattice nanowire configuration designed to increase the MBSs topological protection. It combines a semiconducting nanowire (green) with one facet covered uniformly by SC layer (grey) and a superlattice of (non-superconducting) fingers (brown). (b) Low-energy spectrum versus back gate voltage. (c) Probability density of the lowest-energy eigenstate at the voltage marked with a blue line in (b). (d) Longitudinal cut of the probability density of (c) at the (y,z) cross-section values marked by arrows. Parameters are the same as in Fig. \[Fig12\]: $L_{\rm wire}=2\mu$m, $L_{\rm cell}=100$nm, $W_{\rm Au}=W_{\rm Al}=10$nm, $r_{\rm SC}=0.5$, $V_{\rm Z}=0.6$meV, $\Delta_0=0.2$meV, $V_{\rm SC}=0.2$V and $\rho_{\rm surf}/e=2\cdot 10^{17}(cm)^{-3}$. We take $V_{\rm N}=0$V as the boundary condition for the fingers.](Fig14.pdf) In Fig. \[Fig14\](b) we show the low-energy spectrum of this setup for the same parameters of Fig. \[Fig12\] except for the boundary condition between the (normal) bottom superlattice and the wire, which we take as $V_{\rm N}\simeq0$V. The values of $V_{\rm gate}$ for which we find the first topological subbands are pretty negative since the continuous Al layer induces a large intrinsic doping in the wire. The structure of this spectrum is a combination of the homogeneous and superlattice ones. From $V_{\rm gate}\simeq-13.7$V to $\simeq-12.3$V one transverse topological subband is occupied. At that point a different transverse subband populates the wire and the even-odd effect destroys the topological phase (as it occurs in the homogeneous wire). However, at $V_{\rm gate}\simeq-11.3$V a zero energy mode appears again but without a (prominent) gap closing. This is the signature of a gap between different longitudinal subbands, which allows one of the last two transverse subbands to re-enter into the topological phase. The interplay between longitudinal and transverse subbands gives rise to a wider $V_{\rm gate}-V_{\rm Z}$ space where topological states emerge, in comparison to a homogeneous nanowire, as it was previously stated in Ref. . Now, as was our intention, in the topological regions we get a topological minigap that is comparable to the one of the homogeneous case, see Fig. \[Fig12\](j). The probability density of the lowest energy mode at the $V_{\rm gate}$ value marked with a blue line in (b) can be seen in (c). As expected, it is located close to the Al thin layer in the transverse direction. A longitudinal cut at the (z,y) values marked by arrows is shown in (d). The MBSs, that still display a doubling of the oscillating peaks characteristic of the superlattice, decay exponentially from the edges towards the wire’s center faster than for the top- and bottom-superlattices analysed before. The Majorana charge is now $\left|Q_{\rm M}\right|/e=0.71$, considerably smaller than for the bottom superlattice alone (0.93) and closer to that of the homogeneous case (0.61). The sizeable minigap in this case protects the system from quasiparticle excitations, separating the Majorana modes from the quasi-continuum of states and preventing transitions into it due to, e.g., temperature or out of equilibrium perturbations. To finish this section, we would like to mention that in this study and for simplicity we have ignored the orbital effects of the magnetic field. According to the literature (see for instance Ref. ), the orbital effects are important when the electron’s wave function is spread across the wire’s section, especially when it has a ring-like shape. In this case, the electrons circulate around the magnetic field that points along the wire and interference *orbital* effects appear. Furthermore, orbital effects are also enhanced for high electron densities, since most high transverse subbands have large angular momentum that couples strongly to the magnetic field. Conversely, the orbital effects diminish both for low transverse subbands and when the electron’s wave function is pushed towards the SCs (by the action of the back gate), since it then occupies only a small region of the wire’s section. We note that this is precisely the region in the spectrum that we focus on in Figs. \[Fig12\] and \[Fig14\]. We have explored the first occupied transverse subband (that displays MBSs) for the different superlattice structures, since it is the best behaved for Majorana purposes. For the back gate voltages involved, the wave function is indeed pushed towards the SC fingers (which is beneficial for the stability of the Majoranas since the induced pairing, and consequently the minigap, are larger there). Admittedly, this is not the case for the bottom superlattice setup, Fig. \[Fig12\](c), where the electron probability density oscillates from top to bottom in the transverse direction. Therefore, we expect that the orbital effects might be important in that case and beyond the current analysis performed in our work. Summary and Conclusions {#Conclusions} ======================= In this work we have analysed in detail the proposal of Levine *et al.*[@Levine:PRB17] to look for topological superconductivity in Majorana nanowires in which the induced superconductivity is achieved by proximity to a superlattice of SC fingers (instead of having the SC cover continuously the length of the semiconducting wire). This configuration can have practical benefits to manipulate the Majorana wave function and to measure it. For instance, one could use an STM tip to drive a current between the tip and each of the SC fingers to measure the local density of states along the wire. The fingers could also work as local probes themselves. Specifically, here we study the impact of the three-dimensionality of the system and the electrostatic environment on the spectral properties of the nanowire. To this end, we compute self-consistently the 3D Schrödinger-Poisson equations in the Thomas-Fermi approximation, where we include the Rashba coupling as a term locally proportional to the electric field. We consider two types of experimental setups, one in which the SC superlattice is on top of the nanowire and the other where it is below with respect to the back gate. We find that an accurate description of the nanowire boundary conditions and the surrounding media are crucial for a proper understanding of the system’s properties. In particular, the interface of the nanowire with the SC, vacuum or substrate, creates an accumulation of electrons around the wire’s cross-section. Its main effect is to contribute to the average intrinsic doping of the wire (that has to be compensated with an external gate when looking for the first populated subbands). On the other hand, the extrinsic doping produced by the applied gate voltage is dominated by the SC superlattice structure, giving rise to an inhomogeneous (oscillating) electrostatic potential. Depending on the location of the SC superlattice and the number and width of the SC fingers, we find a rich phenomenology that includes the emergence of trivial holes in the topological phase diagram and the formation of localized states near the SC fingers that may interfere with the topological states. Moreover, since the Rashba coupling is proportional to the electric field, the spin-orbit coupling also becomes an inhomogeneous quantity in this system. This results in a reduction of the topological minigap, specially in the bottom-superlattice device, owing to a lower spatial average Rashba value. In the same vein, the induced superconducting gap is smaller than in a conventional homogeneous Majorana nanowire due to the smaller superconducting coverage of the nanowire. In contrast, the system develops a wider topological phase as a function of magnetic field and average chemical potential as a consequence of the emergence of additional (longitudinal) subbands. In the topological regions, MBSs do appear at the edges of the superlattice nanowire. Their probability density across the wire’s section is concentrated close to the SC fingers in the top-superlattice setup. They extend further into the wire’s bulk in the bottom-superlattice one due to the stronger potential oscillations created in this case by the back gate. Along the wire, the MBSs decay exponentially towards its center with a decay length characterized by the interplay between the superconducting coherence length and the superlattice length. In general, we find that the performance of the two types of setups considered here is quite similar, although the bottom-superlattice nanowire is slightly worse because of the larger potential oscillations that appear in this case. In both cases, the main disadvantage is the poor topological protection of the MBSs (manifested in a small topological minigap and large left and right Majorana wave function overlap), arising essentially from the low superconducting coverage. This could be solved by covering one of the lateral wire’s facets with a continuous SC layer while still placing it on a superlattice of fingers (that could be superconducting or not). This kind of device benefits from the superlattice structure (with a wider topological phase in $V_{\rm gate}-V_{\rm Z}$ space and the possibility to use the fingers as probes), and furthermore displays a sizeable topological minigap and small Majorana charge comparable to those of a conventional homogeneous Majorana nanowire. We thus believe that the use of mixed setups of this type is probably the best route towards creating Majorana states in the presence of superlattices. The dataset and scripts required to plot the figures of this publication are available at the Zenodo repository [@Zenodo]. We thank Eduardo J. H. Lee, Haim Beidenkopf, Enrique G. Michel, Nurit Avraham, Hadas Shtrikman and Jesper Nygård for valuable discussions. Research supported by the Spanish MINECO through Grants FIS2016-80434-P, BES-2017-080374 and FIS2017-84860-R (AEI/FEDER, EU), the European Union’s Horizon 2020 research and innovation programme under the FETOPEN grant agreement No 828948 and grant agreement LEGOTOP No 788715, the Ramón y Cajal programme RYC-2011-09345, the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0377), the DFG (CRC/Transregio 183, EI 519/7- 1), the Israel Science Foundation (ISF) and the Binational Science Foundation (BSF). Numerical details {#SI1} ================= In this appendix we detail the numerical methods used to solve the Schrödinger-Poisson equation given by Eq. (\[Hamiltonian\]) and Eq. (\[Poisson\]) in the main text. As explained in Sec. \[Methods\], instead of solving the coupled equations, our general procedure consists of, first, computing self-consistently the electrostatic potential within the Thomas-Fermi approximation, and then, building and diagonalizing the Hamiltonian in order to obtain the eigenspectrum. The reliability of this procedure compared to a full Schrödinger-Poisson approach is discussed in App. \[SI2\]. Electrostatic potential {#SI1-1} ----------------------- To obtain the electrostatic potential, we solve the Poisson equation (given by Eq. (\[Poisson\]) in the main text) using a Partial Differential Equation solver for Python called *FEniCS* [@Logg:10; @Logg:12], which uses finite element techniques. We use a mesh with Lagrange elements with a discretization of 1nm. Regarding the boundary conditions of the semiconducting nanowire, we impose $V_{\rm gate}$ at the back gate, $V_{\rm SC}$ at the boundaries with the SC fingers, $V_{\rm N}$ at the normal metal boundaries (if applicable), and periodic boundary conditions at the nanowire ends. This last condition eliminates border effects, which are well-known [@Escribano:BJN18; @Penaranda:PRB18; @Fleckenstein:PRB18] and do not change the qualitative physics introduced by the superlattice structure. The different geometries studied in this work (i.e. the bottom- and top-superlattices, the continuously covered nanowire and their combinations) are taken into account through an inhomogeneous dielectric permittivity $\epsilon(\vec{r})$. We model it as a piecewise function: constant inside each material and with abrupt changes at the interfaces. The specific values used in our simulations for the dielectric permittivity can be found in Table \[Table\_parameters\] in the main text. The source term $\rho_{\rm tot}=\rho_{\rm surf}+\rho_{\rm mobile}$ of the Poisson equation (shown in Eq. (\[charge\_density\]) in the main text) has two independent terms. The first one is the surface charge layer, that we model as a fix superficial positive charge density $\rho_{\rm surf}$ placed in the points of the mesh localized at the InAs-vacuum and InAs-SiO interfaces. The second source term is the 3D mobile charge density inside the wire, $\rho_{\rm mobile}=\rho_{\rm e}+\rho_{\rm lh}+\rho_{\rm hh}$, which in principle includes the contributions of the conduction band $\rho_{\rm e}$, and the light hole $\rho_{\rm lh}$ and heavy hole $\rho_{\rm hh}$ bands. However, in this work we ignore the hole terms since they are not present for the gate potentials that we consider in our simulations. Specifically, they are relevant when $e\phi(x,y,z)\lesssim E_{\rm vv}$, that, for the specific geometries of this work, only occurs when $V_{\rm gate}<-3.5$V in the bottom-superlattice, $V_{\rm gate}<-0.8$V in the top one, $V_{\rm gate}<-1.8$V in the homogeneous nanowire, and $V_{\rm gate}<-15.7$V in the alternative configuration that combines a bottom (normal) superlattice and a continuous SC layer. Therefore, we only compute the electron charge density corresponding to the wire’s conduction band using to this end the Thomas-Fermi approximation for a 3D electron gas, as explained in Sec. \[Methods\] of the main text. As the charge density depends in turn on the potential, the Poisson equation must be solved self-consistently. For this purpose, we use an iterative method to obtain the charge density using the Anderson mixing $$\rho_{\rm mobile}^{(n)}=\beta\tilde{\rho}_{\rm mobile}^{(n)}+(1-\beta)\rho_{\rm mobile}^{(n-1)}, \label{Anderson_mixing}$$ where $n$ is the step of the procedure and $\beta$ is the Anderson coefficient. In the first step of the process (i.e. $n=0$) we take $\rho_{\rm mobile}^{(0)}=0$ and we compute the electrostatic potential of the system. At any other arbitrary step $n$, we compute the charge density $\tilde{\rho}_{\rm mobile}^{(n)}$ using the electrostatic potential found in the previous iteration $n-1$. Then, we compute the electrostatic potential at the $n$ step using $\rho_{\rm mobile}^{(n)}$, given by the Anderson mixing of Eq. \[Anderson\_mixing\]. This charge density mixing between the $n$ and $n-1$ steps ensures the convergence to the solution. We repeat the procedure until the cumulative error is below the 1%. Once the electrostatic potential is known, the Rashba coupling $\alpha_{\rm R}(\vec{r})$ is computed using Eq. (\[alpha\]) of the main text. To provide more insight into the electrostatic potential created by the gate, the superficial charge density and the mobile charge density, we show in Fig. \[FigSI1\](a) the potential profile produced by some particular values of $V_{\rm gate}$ and $V_{\rm SC}$ (in the absence of $\rho_{\rm surf}$) along the wire’s cross-section ($z$-direction) for a top-superlattice device. Separately, in Fig. \[FigSI1\](b) we show the effect of the surface charge layer for zero $V_{\rm gate}$ and $V_{\rm SC}$ for the same device. The solid curve corresponds to the self-consistent solution (in the Thomas-Fermi approximation), while for the dashed curve the presence of $\rho_{\rm mobile}$ in the Poisson equation has been ignored. The effect of $\rho_{\rm mobile}$ is small when the effective chemical potential is close to the bottom of the conduction band, as is the case of Fig. \[Fig1\](a). However, when this is not the case, the non self-consistent solution overestimates the band bottom displacement with respect to the Fermi level, see Fig. \[Fig1\](b). This happens because the screening effect of the mobile charges pushes the band bottom upwards reducing the wire’s average doping.  Representative examples of the electrostatic potential profile in a top-superlattice nanowire along the $z$-direction for $V_{\rm gate}=-0.5$V, $V_{\rm SC}=0.2$V and $\rho_{\rm acc}=0$ in (a); and for $\rho_{\rm acc}=2\times 10^{18}$e/cm$^3$ and $V_{\rm gate}=V_{\rm SC}=0$ in (b). The screening effect of the mobile charges inside the wire, $\rho_{\rm mobile}$, is ignored in the dashed line solution, whereas it is taken into account in the solid one. Geometric parameters are $W_{\rm Al}=10$nm, $W_{\rm SiO}=20$nm, $W_{\rm wire}=80$nm.](FigSI1.pdf) 3D Hamiltonian -------------- The 3D Hamiltonian of Eq. (\[Hamiltonian\]) in the main text is discretized using the finite difference method within the Bogoliubov-de Gennes formalism, using an inter-site distance (discretization) of $5$nm in the three directions. We find the eigenstates of the Hamiltonian using the ARPACK diagonalization tools implemented in the standard package *Scipy* of Python. In order to reduce the computational cost, we only compute the 10 lowest-energy eigenstates, which are the relevant ones for Majorana physics. 1D Hamiltonian and topological invariant ---------------------------------------- We build the finite 1D Hamiltonian following the same method as for the 3D case, but taking the electrostatic potential at the center of the wire (i.e. $\phi(x,y=0,z=0)$). We exploit the periodic nature of this Hamiltonian to build the infinite 1D Hamiltonian in k-space $H(k)$, as explained in Ref. . From there, we can compute the class D topological invariant [@Levine:PRB17] $\mathcal{Q}$ as $$\mathcal{Q}=\operatorname{sign}\left(\operatorname{Pf}\left\lbrace\Lambda H(k=0)\right\rbrace\right)\cdot \operatorname{sign}\left(\operatorname{Pf}\left\lbrace\Lambda H(k=\frac{\pi}{L})\right\rbrace\right),$$ where $\operatorname{Pf}\left\lbrace M\right\rbrace$ is the Pfaffian of a matrix $M$, which we compute numerically using the Python package *Pfaffian* provided by Ref. , and $\Lambda$ is the electron-hole symmetry matrix, that in our basis obeys $$\Lambda H^*(-k) \Lambda^{-1}=-H(k) \leftarrow \Lambda=\mathcal{I}_{\mathrm{site}}\otimes\sigma_0\otimes\tau_x.$$ Reliability of Thomas-Fermi approximation {#SI2} ========================================= The calculations shown in the main text have been performed using the Thomas-Fermi approximation for the charge density inside the wire. However, a more realistic and complete description consists of solving the coupled Schrödinger-Poisson equations, which requires to compute the charge density from the eigenspectrum of the Hamiltonian $$\rho_{\rm e}^{\rm (SP)}(\vec{r})=e\sum_{i>0} \left|u_i(\vec{r})\right|^2 f(E_i)+ \left|v_i(\vec{r})\right|^2 f(-E_i),$$ where $u_i(\vec{r})$ and $v_i(\vec{r})$ are the electron and hole components of the $i$-th eigenstate, $E_i$ its corresponding energy, $f(E)$ the Fermi-Dirac distribution, and the sum is done for every positive energy ($i>0$). Since the eigenspectrum is found by diagonalizing the Hamiltonian, which depends in turn on the charge density through the Poisson equation, the coupled Schrödinger-Poisson equations have to be solved self-consistently as well, following the same iterative procedure as described in App. \[SI1\]. Nevertheless, this process is computationally more expensive because the Hamiltonian is diagonalized in each self-consistent step. Hence, when both methods provide similar results, it is justified to use the Thomas-Fermi approximation to reduce the computational cost. It is a well-known fact that the Thomas-Fermi approximation ignores the kinetic terms, as well as the magnetic field dependence. Remarkably, some previous works [@Mikkelsen:PRX18] have shown that both approaches provide similar results, although for simplistic models of Majorana nanowires. However, this could not be true for superlattice ones since the superlattice leads to a stronger charge localization. In this appendix we compare the results obtained using both methods. First, we show that for $V_{\rm Z}=0$ both methods predict similar results for the lowest energy modes, in spite of ignoring the kinetic terms. And second, we show that the magnetic field dependence of the wire’s charge density can be neglected. Comparison between Thomas-Fermi approximation and full Schrödinger-Poisson calculation -------------------------------------------------------------------------------------- Fig. \[FigSI2-1\] shows a comparison between both methods for the bottom (a-b) and top (c-d) setups of Sec. \[3D\_results\], for the same parameters of Fig. \[Fig13\], except for $V_{\rm Z}=0$. The difference $\Delta\rho_{\rm e}$ between the charge densities computed using the Schrödinger-Poisson approach $\rho_{\rm e}^{\rm (SP)}$ and the Thomas-Fermi approximation $\rho_{\rm e}^{\rm (TF)}$ is shown in Figs. \[FigSI2-1\](a,c) for both devices. In both cases, the difference is a small positive quantity very close to the SC-InAs interface (more clearly seen in Fig. \[FigSI2-1\](a)), which means that the Thomas-Fermi approximation slightly overestimates the electron density close to this interface, where the majority of the charge is located. Conversely, it is slightly negative further away from the interface. Everywhere else $\Delta\rho_{\rm e}\approx 0$. The total charge obtained with both methods are $Q_{\rm tot}^{\rm (TF)}\simeq809e$ and $Q_{\rm tot}^{\rm (SP)}\simeq709e$ in the bottom-superlattice nanowire, and $Q_{\rm tot}^{\rm (TF)}\simeq633e$ and $Q_{\rm tot}^{\rm (SP)}\simeq624e$ in the top-superlattice one, which are pretty similar.  Difference between the electron charge densities inside the nanowire computed using the Schrödinger-Poisson approach and the Thomas-Fermi approximation, $\Delta\rho_{\rm e}$, for the bottom-superlattice setup (a) and the top-superlattice one (c). (b) and (d) show their corresponding electrostatic potential difference $\Delta\phi$. Parameters are the same as in Fig. \[Fig13\], except for $V_{\rm Z}=0$.](FigSI2-1.pdf) To obtain a quantitative estimation of the error made using the Thomas-Fermi approximation, we now analyse the electrostatic potential created by the charge density using both methods, which is the quantity that indeed enters into the Hamiltonian of Eq. (\[Hamiltonian\]). The electrostatic potential difference $\Delta\phi=\phi^{\rm (SP)}-\phi^{\rm (TF)}$ between both methods is plotted in Figs. \[FigSI2-1\](b,d) for each device. Since the bare electrostatic interaction given by the Poisson equation is long-ranged, $\Delta\phi$ is very small (or zero) close to the SCs, despite the finite charge density difference there. By contrast, the maximum $\Delta\phi$ in both cases is found far apart from the back gate. It is roughly $20$mV and homogeneous for the bottom-superlattice nanowire, and around $10$mV between SC fingers for the top one. Comparing with the electrostatic potential of Fig. \[Fig13\], which is computed using the Thomas-Fermi approximation for the same devices and for the same back gate voltages, we conclude that the average error is below 10%, justifying the use of the Thomas-Fermi approximation for the range of gate voltages used in our simulations. Accuracy of the zero magnetic field Thomas-Fermi approximation -------------------------------------------------------------- The previous analysis has been carried out for Zeeman splitting $V_{\rm Z}=0$, since the charge density computed using the Thomas-Fermi approximation (Eq. (\[Thomas-Fermi\])) in the main text ignores the magnetic field dependence. To obtain a quantitative estimation of the error made due to this approximation, we show in Figs. \[FigSI2-2\](a,c) the difference between the charge densities computed using the Schrödinger-Poisson approach with and without an applied magnetic field (for both geometries). In addition, Figs. \[FigSI2-2\](b,d) show their corresponding electrostatic potential difference. Comparing with Fig. \[Fig13\], one can see that the error is below 1%. This small difference is due to the fact that typical Zeeman splittings in these systems ($V_{\rm Z}\sim 1meV$) are much smaller than the electrochemical potentials ($e\phi-E_{\rm F}\sim 100meV$), so that last quantity dominates. Consequently, we conclude that neglecting the magnetic field dependency in the charge density is an adequate approximation for these calculations.  Difference between the charge densities computed using the Schrödinger-Poisson equation with magnetic field ($V_{\rm Z}=0.6meV$) and without it, for the bottom-superlattice nanowire (a) and for the top-superlattice one (c). (b) and (d) show their corresponding electrostatic potential difference. Parameters are the same as in Fig. \[Fig13\].](FigSI2-2.pdf) Electrostatic potential and Rashba coupling in superlattice parameter space {#SI3} =========================================================================== In this last appendix we show how the induced electrostatic potential and Rashba coupling behave versus the superlattice parameters $L_{\rm cell}-r_{\rm SC}$. This is relevant since, as we show below, we find that for some $L_{\rm cell}$ and $r_{\rm SC}$ values it is difficult to gate the wire due to screening effects, or the spin-orbit coupling induced by the back gate is negligible. We have (partially) used this information to plot Fig. \[Fig10\] in the main text. Figures \[FigSI3-1\](c,d) show the lever arm (in logarithmic scale) versus the superlattice parameters for the bottom- (c) and top-superlattice (d) devices. This quantity is defined as the back gate potential needed to change the spatially averaged electrostatic potential $\langle \phi\rangle$. Here, this variation is independent of $V_{\rm gate}$ because for simplicity we ignore the screening produced by $\rho_{\rm e}$. In both setups, when the partial coverage of the SC $r_{\rm SC}$ is small, the lever arm is a factor of the order of $10^0$-$10^1$. This means that, for example, to change $\langle \phi\rangle$ by 1meV we need to apply a voltage to the gate of 1-10meV. However, as $r_{\rm SC}$ increases, so does the lever arm and larger back gate potentials are needed to effectively deplete or fill the wire. This happens dramatically for the bottom-superlattice setup since the superlattice is placed between the back gate and the nanowire. Thus, for large $r_{\rm SC}$, the screening effect of the SC fingers is strong. By contrast, in the top-superlattice setup the lever arm converges to a finite small value corresponding to that of the continuously covered nanowire. ![\[FigSI3-1\] (Colour online) Variation of the spatially averaged electrostatic potential inside the wire due to the voltage applied to the back gate. Here $V_{\rm SC}=0$, $\rho_{\rm surf}=0$ and $\rho_{\rm e}$ is neglected. Two setups are considered, bottom-superlattice to the left and top-superlattice to the right. (a,b) Sketches of both systems. (c,d) Lever arm, defined as the back gate potential needed to change the spatially averaged electrostatic potential $\langle \phi\rangle$, versus superlattice parameters $L_{\rm cell}$ and $r_{\rm SC} = L_{\rm SC}/L_{\rm cell}$ in logarithmic scale. (e,f) Dispersion of the electrostatic potential variations along and across the wire, $\sigma_\phi=\sqrt{\left< \phi^2\right>-(\left< \phi\right>)^2}$.](FigSI3-1.pdf) Since the electrostatic potential close to the SC fingers is fixed by the boundary condition $V_{\rm SC}$, large lever arms lead to large electrostatic variations along the wire. This can be detrimental for the stability of Majorana states, since these large variations lead in turn to the formation of localized states (as explained in Sec. \[Impact\] of the main text). The standard deviation $\sigma_{\phi}$ of the electrostatic potential along and across the wire shown in Figs. \[FigSI3-1\](e,f) (for both setups) gives an idea of the size of these potential variations. As pointed out before, for small $r_{\rm SC}$, when the lever arm is small as well, the variations are negligible. However, for larger $r_{\rm SC}$, the variations are larger, specially for the bottom-superlattice setup. Since the Rashba coupling depends locally on the electric field, the spin-orbit strength also depends on the superlattice parameters. The average value of $\alpha_z$ along the wire is shown in Figs. \[FigSI3-2\](c,d) for both superlattice setups. We only consider the contribution of the back gate potential (fixing $V_{\rm SC}=0$ and $\rho_{\rm surf}=0$). For small $r_{\rm SC}$, the Rashba coupling is roughly 5meV$\cdot$nm when 1V is applied to the back gate (for both devices). As $r_{\rm SC}$ is increased, $\left<\alpha_z\right>$ decreases for the bottom-superlattice setup until it reaches zero, while it increases for the top-superlattice one until it reaches 15meV$\cdot$nm (when 1V is applied to the back gate), corresponding to the value of the mean Rashba coupling in the homogeneous nanowire. This qualitative difference is again due to the strong screening effect of the SC fingers in the bottom-superlattice setup. For completeness, we show the dispersion of the $\alpha_z$ spin-orbit coupling variation along the wire in Figs. \[FigSI3-2\](e,f). We find that the dispersion is constant in the top-superlattice setup (f) regardless of the superlattice parameters. However, the spin-orbit variations increase with $r_{\rm SC}$ in the bottom-superlattice one. ![\[FigSI3-2\] (Colour online) Similar analysis as in Fig. \[FigSI3-1\] but for the spin-orbit coupling. Two setups are considered, bottom-superlattice (a) to the left and top-superlattice (b) to the right. (a,b) Sketches of both systems. (c,d) Variation of the spatially averaged $\alpha_{\rm z}$ Rashba coupling inside the wire due to the voltage applied to the back gate. (e,f) Dispersion of $\alpha_{\rm z}$ variation defined as $\sigma_{\alpha_z}=\sqrt{\left< \alpha_z^2\right>-(\left< \alpha_z\right>)^2}$.](FigSI3-2.pdf)

ArXiv

--- abstract: 'Let $M$ be a compact Riemannian manifold with smooth boundary. We obtain the exact long time asymptotic behaviour of the heat kernel on abelian coverings of $M$ with mixed Dirichlet and Neumann boundary conditions. As an application, we study the long time behaviour of the abelianized winding of reflected Brownian motions in $M$. In particular, we prove a Gaussian type central limit theorem showing that when rescaled appropriately, the fluctuations of the abelianized winding are normally distributed with an explicit covariance matrix.' address: - ' ^1^ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.' - ' ^2^ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213.' author: - Xi Geng^1^ - Gautam Iyer^2^ bibliography: - 'refs.bib' title: Long Time Asymptotics of Heat Kernels and Brownian Winding Numbers on Manifolds with Boundary --- [^1] Introduction. ============= Consider a compact Riemannian manifold $M$ with boundary. We address the following questions in this paper: 1. What is the long time asymptotic behaviour of the heat kernel on abelian covering spaces of $M$, under mixed Dirichlet and Neumann boundary conditions? 2. What is the long time behaviour of the abelianized winding of trajectories of normally reflected Brownian motion $M$. Our main results are Theorem \[t:hker\] and Theorem \[t:winding\], stated in Sections \[s:mainthm\] and \[s:winding\] respectively. In this section we survey the literature and place this paper in the context of existing results. Long Time Behaviour of Heat Kernels on Abelian Covers. ------------------------------------------------------ The short time behaviour of heat kernels has been extensively studied and is relatively well understood (see for instance [@BerlineGetzlerEA92; @Grigoryan99] and the references therein). The exact long time behaviour, on the other hand, is subtly related to global properties of the manifold, and our understanding of it is far from being complete. There are several scenarios in which the long time asymptotics can be determined precisely. The simplest scenario is when the underlying manifold is compact, in which case the long time asymptotics is governed by the bottom spectrum of the Laplace-Beltrami operator. The problem becomes highly non-trivial for non-compact manifolds. Li [@Li86] determined the exact long time asymptotics on manifolds with nonnegative Ricci curvature, under a polynomial volume growth assumption. Lott [@Lott92] and Kotani-Sunada [@KotaniSunada00] determined the long time asymptotics on abelian covers of closed manifolds. In a very recent paper, Ledrappier-Lim [@LedrappierLim15] established the exact long time asymptotics of the heat kernel of the universal cover of a negatively curved closed manifold, generalizing the situation for hyperbolic space with constant curvature. We also mention that for non-compact Riemannian symmetric spaces, Anker-Ji [@AnkerJi01] established matching upper and lower bounds on the long time behaviour of the heat kernel. Since the work by Lott [@Lott92] and Kotani-Sunada [@KotaniSunada00] is closely related to ours, we describe it briefly here. Let $M$ be a closed Riemannian manifold, and $\hat M$ be an abelian cover (i.e. a covering space whose deck transformation group is abelian). The main idea in [@Lott92; @KotaniSunada00] is an exact representation of the heat kernel $\hat{H}(t,x,y)$, in terms of a compact family of heat kernels of sections of twisted line bundles over $M$. Precisely, the representation takes the form $$\hat{H}(t,x,y)=\int_{\mathcal{G}}H_{\chi}(t,x,y) \, d\chi,$$ where $\mathcal{G}$ is a compact Lie group, and $H_\chi(t,x,y)$ is the heat kernel on sections of a twisted line bundle $E_\chi$ over $M$. (This is described in detail in Section \[s:liftedRep\], below.) Since $M$ is compact, $H_\chi$ decays exponentially with rate $\lambda_{\chi, 0}$, the principal eigenvalue of the associated Laplacian $\lap_\chi$. Thus the long time behaviour of $\hat H$ can be determined from the behaviour of $\lambda_{\chi, 0}$ near its global minimum. For closed manifolds, it is easy to see that the global minimum of $\lambda_{\chi, 0}$ is $0$, and is attained at a non-degenerate critical point. In the present paper we study abelian covers of manifolds with boundary, and impose (mixed) Dirichlet and Neumann boundary conditions. Our main result determines the exact long time asymptotic behaviour of the heat kernel (Theorem \[t:hker\]) and is stated in Section \[s:mainthm\]. In this case, the main strategy in [@Lott92; @KotaniSunada00] can still be used, however, the minimum of $\lambda_{\chi, 0}$ need not be $0$. The main difficulty in the proof in our context is precisely understanding the behaviour of $\lambda_{\chi, 0}$ near the global minimum. Under a suitable transformation, the above eigenvalue minimization problem can be reformulated directly as follows. Let $\omega$ be a harmonic $1$-form on $M$ with Neumann boundary conditions, and consider eigenvalue problem $$-\lap\phi_{\omega}-4\pi i\omega\cdot\nabla\phi_{\omega}+4\pi^{2}|\omega|^{2}\phi_{\omega}=\mu_{\omega}\phi_{\omega}\,,$$ with mixed Dirichlet and Neumann boundary conditions. It turns out that in order to make the strategy of [@Lott92; @KotaniSunada00] work, one needs to show that the eigenvalue $\mu_\omega$ above attains the global minimum if and only if the integral of $\omega$ on closed loops is integer valued, and in this case the minimum is non-degenerate of second order. These are the two key ingredients of the proof, and they are formulated in Lemmas \[l:minLambdaChi\] and \[l:muBound\] below. Given these lemmas, the main result of this paper (Theorem \[t:hker\]) shows that $$\hat H(t, x, y) \approx \frac{C'_\mathcal I(x, y)}{t^{k/2}} \exp \paren[\Big]{ -\mu_0 t - \frac{d'_\mathcal I(x, y)^2 }{t} }\,, \quad\text{as } t \to \infty\,.$$ Here $k$ is the rank of the deck transformation group, and $C'_\mathcal I$, $d'_\mathcal I$ are explicitly defined functions. The Abelianized Winding of Brownian Motion on Manifolds. -------------------------------------------------------- We now turn our attention to studying the winding of Brownian trajectories on manifolds. The long time asymptotics of Brownian winding numbers is a classical topic which has been investigated in depth. The first result in this direction is due to Spitzer [@Spitzer58], who considered a Brownian motion in the punctured plane. If $\theta(t)$ denotes the total winding angle up to time $t$, then Spitzer showed $$\frac{2\theta(t)}{\log t} \xrightarrow[t\to \infty]{w} \xi \,,$$ where $\xi$ is a standard Cauchy distribution. The reason that the heavy tailed Cauchy distribution appears in the limit is because when the Brownian motion approaches the origin, it generates a large number of windings in a short period of time. If one looks at exterior disk instead of the punctured plane, then Rudnick and Hu [@RudnickHu87] (see also Rogers and Williams [@RogersWilliams00]) showed that the limiting distribution is now of hyperbolic type. In planar domains with multiple holes, understanding the winding of Brownian trajectories is complicated by the fact that it is inherently non-abelian if one wants to keep track of the order of winding around different holes. Abelianized versions of Brownian winding numbers have been studied in [@PitmanYor86; @PitmanYor89; @GeigerKersting94; @TobyWerner95], and various generalizations in the context of positive recurrent diffusions, Riemann surfaces, and in higher dimensional domains have been studied in [@GeigerKersting94; @LyonsMcKean84; @Watanabe00]. In this paper, we study the abelianized winding of trajectories of normally reflected Brownian motion on a compact Riemannian manifold with boundary. The techniques used by many of the references cited above are specific to two dimensions and relies on the conformal invariance of Brownian motion in a crucial way. Our approach studies Brownian winding on manifolds by lifting trajectories to a covering space, and then using the long time asymptotics of the heat kernel established in Theorem \[t:hker\]. Due to the limitations in Theorem \[t:hker\] we measure the winding of Brownian trajectories as a class in $\pi_1(M)_{\mathrm{ab}}$, the *abelianized* fundamental group of $M$. By choosing generators of $\pi_1(M)$, we measure the abelianized winding of Brownian trajectories as a $\Z^k$-valued process, denoted by $\rho$. We show (Theorem \[t:winding\], below) that $$\frac{\rho(t)}{t} \xrightarrow[t \to \infty]{p} 0 \qquad\text{and}\qquad \frac{\rho(t)}{\sqrt{t}} \xrightarrow[t \to \infty]{w} \mathcal N(0, \Sigma)\,,$$ for some explicitly computable matrix $\Sigma$. Here $\mathcal N(0, \Sigma)$ denotes a normally distributed random variable with mean $0$ and covariance matrix $\Sigma$. As a result, one can for instance, determine the long time asymptotics of the abelianized winding of Brownian trajectories around a knot in $\R^3$. We remark, however, that Theorem \[t:winding\] can also be proved directly by using a purely probabilistic argument. For completeness, we sketch this proof in Section \[s:windingDomain\]. The Non-abelian Case. --------------------- One limitation of our techniques is that they do not apply to the non-abelian situation. Studying the winding of Brownian trajectories without abelianization and the long time behaviour of the heat kernel on non-abelian covers (in particular, on non-amenable covers) are much harder questions. In the discrete context, Lalley [@Lalley93] (see also [@PittetSaloffCoste01]) showed that the $n$-step transition probability $\P(Z_n=g)$ of a finite range random walk $Z_n$ on the Cayley graph of a free group satisfies $$\P(Z_{n}=g)\approx C(g)n^{-{3 / 2}}R^{-n} \,, \quad \text{as } n\rightarrow\infty \,.$$ Here $R$ is the radius of convergence of the Greens function. In the continuous context, this suggests that the heat kernel on a non-amenable cover of $M$ decays exponentially faster than the heat kernel on $M$, which was shown by Chavel-Karp [@ChavelKarp91]. However, to the best of our knowledge, the exact long time asymptotics is only known in the case of the universal cover of a closed negatively curved manifold by the recent work of Ledrappier-Lim [@LedrappierLim15], and it remains open beyond the hyperbolic regime. #### Plan of this paper. In Section \[s:mainthm\] we state our main result concerning the long time asymptotics of the heat kernel on abelian covers of $M$ (Theorem \[t:hker\]). In Section \[s:winding\] we state our main result concerning the long time behaviour of winding of reflected Brownian motion on $M$ (Theorem \[t:winding\]). We prove these results in Sections \[s:mainproof\] and \[s:pfwinding\] respectively. Long Time Behaviour of the Heat Kernel on Abelian Covers. {#s:mainthm} ========================================================= Let $M$ be a compact Riemannian manifold with smooth boundary, and $\hat{M}$ be a Riemannian cover of $M$ with deck transformation group $G$ and covering map $\bpi$. We assume throughout this paper that $G$ is a finitely generated abelian group with rank $k \geq 1$, and $M \cong \hat M / G$. Let $G_T = \operatorname{tor}(G) \subseteq G$ denote the torsion subgroup of $G$, and let $G_F\defeq G/G_T$. Let $\lap$ and $\hat{\lap}$ denote the Laplace-Beltrami operator on $M$ and $\hat{M}$ respectively. Decompose $\partial M$, the boundary of $M$, into two pieces $\partial_N M$ and $\partial_D M$, and let $H(t,p,q)$ be the heat kernel of $\lap$ on $M$ with Dirichlet boundary conditions on $\partial_D M$ and Neumann boundary conditions on $\partial_N M$. Let $\partial_D \hat M = \bpi^{-1} (\partial_D M)$ and $\bpi^{-1} (\partial_N M)$, and let $\hat{H}(t,x,y)$ be heat kernel of $\hat{\lap}$ on $\hat{M}$ with Dirichlet boundary conditions on $\partial_D \hat M$, and Neumann boundary conditions on $\partial_N \hat M$. Let $\lambda_{0} \geq 0$ be the principal eigenvalue of $-\lap$ with the above boundary conditions. Since $M$ is compact, the long time asymptotic behaviour of $H$ can be obtained explicitly using standard spectral theory. The main result of this paper obtains the asymptotic long time behaviour of the heat kernel $\hat H$ on the non-compact covering space $\hat{M}$. \[t:hker\] There exist explicit functions $C_{\mathcal I}, d_{\mathcal I}\colon \hat M \times \hat M \to [0, \infty)$ (defined in  and , below), such that $$\label{e:hker} \lim_{t\to \infty} \paren[\Big]{ t^{k/2}e^{\lambda_{0}t}\hat{H}(t,x,y) -\frac{C_{\mathcal{I}}(x,y)}{|G_T|} \exp\paren[\Big]{-\frac{2\pi^{2}d_{\mathcal{I}}^{2}(x,y)}{t}}} =0 \,,$$ uniformly for $x, y \in \hat M$. In particular, for every $x,y\in\hat{M}$, we have $$\lim_{t\rightarrow\infty}t^{k/2}e^{\lambda_{0}t}\hat{H}(t,x,y)=\frac{C_{\mathcal{I}}(x,y)}{|G_T|} \,.$$ The definition of the functions $C_{\mathcal I}$ and $d_\mathcal I$ above requires the construction of an inner product on a certain space of harmonic $1$-forms over $M$. More precisely, let $\mathcal H^1$, defined by $$\mathcal H^1 \defeq \set{ \omega \in TM^* \st d \omega = 0,\ d^* \omega = 0, \text{ and } \omega \cdot \nu = 0 \text{ on } \partial M }\,,$$ be the space of harmonic $1$-forms on $M$ that are tangential on $\partial M$. Here $\nu$ denotes the outward pointing unit normal on $\partial M$, and depending on the context $x \cdot y$ denotes the dual pairing between co-tangent and tangent vectors, or the inner-product given by the metric. By the Hodge theorem we know that $\mathcal H^1$ is isomorphic to the first de Rham co-homology group on $M$. Now define $\mathcal H^1_G \subseteq \mathcal H^1$ by$$\label{e:H_G Def} \mathcal H^1_G = \set[\Big]{\omega \in \mathcal H^1 \st \oint_{\hat \gamma} \bpi^*(\omega) = 0 \text{ for all closed loops } \hat \gamma \subseteq \hat M}\,.$$ It is easy to see that $\mathcal H^1_G$ is naturally isomorphic[^2] to $\hom(G, \R)$, and hence $\dim(\mathcal H^1_G) = k$. Define an inner-product on $\mathcal H^1_G$ as follows. Let $\phi_{0}$ be the principal eigenfunction of $-\lap$ with boundary conditions $\phi_0 = 0$ on $\partial_D M$ and $\nu \cdot \grad \varphi_0 = 0$ on $\partial_N M$. Let $\lambda_0$ be the associated principal eigenvalue, and normalize $\phi_0$ so that $\phi_0 > 0$ in $M$ and $\norm{\phi_0}_{L^2} = 1$. Define the quadratic form $\mathcal{I} \colon \mathcal H^1_G \to \R$ by $$\label{e:Idef} \mathcal{I}(\omega) = 8\pi^{2}\int_{M} \abs{\omega}^{2}\phi_{0}^{2} + 8\pi\int_{M}\phi_{0} \omega\cdot\nabla g_\omega \,,$$ where $g_\omega$ is a[^3] solution to the equation $$\label{e:gomega} -\lap g_\omega - 4\pi \omega \cdot \grad \phi_{0} = \lambda_{0}g_\omega\,,$$ with boundary conditions $$\label{e:gomegaBC} g_\omega = 0 \quad\text{on } \partial_D M\,, \qquad\text{and}\qquad \nu \cdot \grad g_\omega = 0 \quad\text{on } \partial_N M\,.$$ In the course of the proof of Theorem \[t:hker\], we will see that $\mathcal I$ arises naturally as the quadratic form induced by the Hessian of the principal eigenvalue of a family of elliptic operators (see Lemma \[l:muBound\], below). Using $\mathcal I$, define an inner-product on $\mathcal H^1_G$ by $$\ip{ \omega,\tau}_{\mathcal{I}}\defeq\frac{1}{4}\paren[\big]{\mathcal{I}(\omega+\tau)-\mathcal{I}(\omega-\tau)},\ \ \ \omega,\tau\in\mathcal H^1_G\,.$$ We will show (Lemma \[l:muBound\], below) that the function $\mathcal{I}(\omega)$ is well-defined, and that $\ip{\cdot, \cdot}_{\mathcal I}$ is a positive definite inner-product on $\mathcal H^1_G$. We remark, however, that under Neumann boundary conditions (i.e. if $\partial_D M = \emptyset$), $\lambda_0 = 0$, $\phi_0$ is constant and $\lambda_0 = 0$. Hence, under Neumann boundary conditions $\ip{\cdot, \cdot}_\mathcal I$ is simply the (normalized) $L^2$ inner-product (see also Remark \[r:neumann\], below). Now, to define the distance function $d_{\mathcal{I}}:\hat{M}\times\hat{M}\rightarrow\R$ appearing in Theorem \[t:hker\], we take $x,y\in\hat{M}$ and define $\xi_{x,y}\in (\mathcal H^1_G)^* \defeq \hom (\mathcal H^1_G; \R)$ by $$\label{e:xiDef} \xi_{x,y}(\omega)\defeq\int_{x}^y \bpi^*(\omega) \,,$$ where the integral is taken over any any smooth path in $\hat M$ joining $x$ and $y$. By definition of $\mathcal H^1_G$, the above integral is independent of the choice of path joining $x$ and $y$. We will show that the function $d_{\mathcal{I}}:\hat{M}\times\hat{M}\rightarrow\R$ is given by $$\label{e:dIDef} d_{\mathcal{I}}(x,y) \defeq \norm{\xi_{x,y}}_{\mathcal{I}^*} = \sup_{\substack{\omega \in \mathcal H^1_G,\\ \norm{\omega}_\mathcal I = 1}} \xi_{x,y}(\omega)\,, \quad \text{for } x,y\in\hat{M} \,.$$ Here $\norm{\cdot}_{\mathcal I^*}$ denotes the norm on the dual space $(\mathcal H^1_G)^*$ obtained by dualising the inner product $\ip{\cdot, \cdot}_{\mathcal I}$. Finally, to define $C_\mathcal I$, we let $$\label{e:h1z} \mathcal H^1_\Z \defeq \set[\Big]{ \omega \in \mathcal H^1_G \st \oint_\gamma \omega \in\mathbb{Z},\ \text{for all closed loops } \gamma \subseteq M }\,.$$ Clearly $\mathcal H^1_\Z$ is isomorphic to $\Z^k$, and hence we can find $\omega_1, \dots, \omega_k\in\mathcal{H}^1_\mathbb{Z}$ which form a basis of $\mathcal H^1_\Z$. We will show that $C_\mathcal I$ is given by $$\label{e:CIdef} C_{\mathcal{I}}(x,y) = (2\pi)^{{k / 2}} \abs[\Big]{\det\paren[\Big]{\paren[\big]{\ip{\omega_{i},\omega_{j}}_{\mathcal{I}}}_{1\leq i,j\leq k}}}^{-1/2} \phi_{0}(\bpi(x))\phi_{0}(\bpi(y))\,.$$ Notice that the value of $C_\mathcal I(x, y)$ doe not depend on the choice of the basis $(\omega_1, \dots, \omega_k)$. Indeed, if $(\omega_1', \dots, \omega_k')$ is another such basis of the $\Z$-module $\mathcal H^1_\Z$, since the change-of-basis matrix belongs to $GL(k, \Z)$, it must have determinant $\pm 1$. We conclude this section by making a few remarks on simple and illustrative special cases. \[r:neumann\] If Neumann boundary conditions are imposed on all of $\partial M$ (i.e. $\partial_D M = \emptyset$), then the definitions of $C_\mathcal I$ and $d_\mathcal I$ simplify considerably. First, as mentioned earlier, under Neumann boundary conditions we have $$\lambda_0 = 0 \,, \qquad\text{and}\qquad \phi_{0} \equiv \operatorname{vol}(M)^{-1/2} \,,$$ and hence $$\label{e:INeumann} \ip{\omega, \tau}_{\mathcal I} = \frac{8\pi^{2}}{\operatorname{vol}(M)}\int_{M} \omega \cdot \tau \,, $$ is a multiple of the standard $L^2$ inner-product. Above $\omega \cdot \tau$ denotes the inner-product on $1$-forms inherited from the metric on $M$. In this case $$d_\mathcal I(x, y) = \paren[\Big]{ \frac{\operatorname{vol}(M)}{8 \pi^2} }^{1/2} \sup_{\substack{\omega \in \mathcal H^1_G\\ \norm{\omega}_{L^2(M)} = 1}} \int_x^y \bpi^*(\omega) \,,$$ and $$C_\mathcal I(x, y) = \frac{(2\pi)^{{k / 2}}}{\operatorname{vol}(M)} \abs[\Big]{\det\paren[\Big]{\paren[\big]{\langle\omega_{i},\omega_{j}\rangle_{\mathcal{I}}}_{1\leq i,j\leq k}}}^{-1/2} \,$$ is a constant independent of $x,y\in\hat{M}$. Note that under Neumann boundary conditions the heat kernel $\hat{H}(t,x,y)$ on the covering space $\hat M$ decays like $t^{-k/2}$ as $t \to \infty$. In contrast, if Dirichlet boundary conditions are imposed on part of the boundary (i.e. $\partial_D M \neq \emptyset$), then we know $\lambda_{0}>0$ and $\phi_0$ is not constant. In this case, $\ip{\cdot, \cdot}_{\mathcal I}$ is not a constant multiple of the standard $L^2$ inner product, and $\hat{H}(t,x,y)$ decays with rate $t^{-k/2}e^{-\lambda_{0}t}$. Let $H$ is the heat kernel of $\lap$ on $M$. Since $M$ is compact by assumption, the spectral decomposition of $-\lap$ shows that $$H(t,p,q)\approx e^{-\lambda_{0}t}\phi_{0}(p)\phi_{0}(q)\,, \quad\text{for } p,q\in M \,, \text{ as } t \to \infty\,.$$ Thus, using Theorem \[t:hker\] we see $$\lim_{t\rightarrow\infty} \frac{t^{{k / 2}}\hat{H}(t,x,y)}{H(t,\pi(x),\pi(y))} = \frac{(2\pi)^{{k / 2}}}{|G_T|} \; \abs[\Big]{\det\paren[\Big]{\paren[\big]{\langle\omega_{i},\omega_{j}\rangle_{\mathcal{I}}}_{1\leq i,j\leq k}}}^{-1/2} \,.$$ Namely, the heat kernel $\hat{H}(t,x,y)$ decays faster than $H(t,p,q)$ by exactly the polynomial factor $t^{-k/2}$. \[r:planar\] Suppose for now that $M$ is a bounded planar domain with $k$ holes excised, and $\operatorname{rank}(G_F) = k$. In this case, the basis $\set{\omega_{1},\cdots,\omega_{k}}$ can be constructed directly by solving some boundary value problems. Indeed, choose $(p_j, q_j)$ inside the $j^\text{th}$ excised hole and define the harmonic form $\tau_j$ by $$\label{e:taui} \tau_{j}\defeq \frac{1}{2\pi} \paren[\Big]{\frac{ (p - p_{j}) \, dq - (q - q_{j} ) \, dp }{ (p - p_j)^2 + (q - q_j)^2 }}\,.$$ Define $\phi_j \colon M \to \R$ to be the solution of the PDE $$\left\{ \begin{alignedat}{2} \span -\lap \phi_j = 0 &\qquad& \text{in } M\,, \\ \span \partial_\nu \phi_j = \tau_j \cdot \nu && \text{on } \partial M\,. \end{alignedat} \right.$$ Then $\omega_j$ is given by $$\omega_j = \tau_j + d\phi_j\,.$$ The Abelianized Winding of Brownian Motion on Manifolds. {#s:winding} ======================================================== We now study the asymptotic behaviour of the (abelianized) winding of trajectories of reflected Brownian motion on the manifold $M$. The winding of these trajectories can be naturally quantified by lifting them to the universal cover. More precisely, let $\bar M$ be the universal cover of $M$, and recall that the fundamental group $\pi_1(M)$ acts on $\bar M$ as deck transformations. Fix a fundamental domain $\bar U \subseteq \bar M$, and for each $g\in\pi_1(M)$ define $\bar U_g$ to be the image of $\bar U$ under the action of $g$. Also, define $\bar{\bm{g}} \colon \bar M \to \pi_1(M)$ by $$\bar{\bm{g}}(x) = g \quad \text{if } x \in U_g\,.$$ Now given a reflected Brownian motion $W$ in $M$ with normal reflection at the boundary, let $\bar W$ be the unique lift of $W$ to $\bar M$ starting in $\bar U$. Define $\bar \rho(t) = \bar{\bm{g}}( \bar W_t ) \in \pi_1(M)$. That is, $\bar \rho(t)$ is unique element of $\pi_1(M)$ such that $\bar W(t) \in \bar U_{\bar \rho(t)}$. Note that $\bar \rho(t)$ measures the winding of the trajectory of $W$ up to time $t$. Our main result of Theorem \[t:hker\] will enable us to study the asymptotic behaviour of the projection of $\bar \rho$ to the abelianized fundamental group $\pi_1(M)_{\mathrm{ab}}$. We know that $$G \defeq {}^{\textstyle \pi_1(M)_{\mathrm{ab}}} \Big/ \:_{\textstyle \operatorname{tor}(\pi_1(M)_{\mathrm{ab}})}$$ is a free abelian group of finite rank, and we let $k = \operatorname{rank}(G)$. Let $\pi_G\colon \pi_1(M) \to G$ be the projection of the fundamental group of $M$ onto $G$. Fix a choice of loops $\gamma_1$, …, $\gamma_k \in \pi_1(M)$ so that $\pi_G(\gamma_1)$, …, $\pi_G(\gamma_k)$ form a basis of $G$. \[d:winding\] The $\mathbb{Z}^k$-*valued winding number* of $W$ is defined to be the coordinate process of $\pi_G(\bar \rho(t))$ with respect to the basis $\pi_G(\gamma_1)$, …, $\pi_G(\gamma_k)$. Explicitly, we say $\rho(t) = (\rho_1(t), \dots, \rho_k(t)) \in \Z^k$ if $$\pi_G(\bar \rho(t)) = \sum_{i = 1}^k \rho_i(t) \pi_G(\gamma_i)\,.$$ Note that the $\Z^k$-valued winding number defined above depends on the choice of basis $\gamma_1$, …, $\gamma_k$. If $M$ is a planar domain with $k$ holes, we can choose $\gamma_i$ to be a loop that only winds around the $i^\text{th}$ hole once. In this case, $\rho_i(t)$ is the number of times the trajectory of $W$ winds around the $i^\text{th}$ hole in time $t$. Our main result concerning the asymptotic long time behaviour of $\rho$ can be stated as follows. \[t:winding\] Let $W$ be a normally reflected Brownian motion in $M$, and $\rho$ be its $\Z^k$ valued winding number (as in Definition \[d:winding\]). Then, there exists a positive definite, explicitly computable covariance matrix $\Sigma$ (defined in , below) such that $$\label{e:rhoLim} \frac{\rho(t)}{t} \xrightarrow{p} 0 \qquad\text{and}\qquad \frac{\rho(t)}{\sqrt{t}} \xrightarrow{w} \mathcal N(0, \Sigma)\,.$$ Here $\mathcal N(0, \Sigma)$ denotes a normally distributed random variable with mean $0$ and covariance matrix $\Sigma$. We now define the covariance matrix $\Sigma$ appearing in Theorem \[t:winding\]. Given $\omega, \in \mathcal H^1$ define the map $\varphi_\omega \in \hom(\pi_1(M), \R)$ by $$\varphi_\omega(\gamma) = \int_\gamma \omega \,.$$ It is well known that the map $\omega \mapsto \varphi_\omega$ provides an isomorphism between $\mathcal H^1$ and $\hom( \pi_1(M), \R )$. Hence there exists a unique dual basis $\omega_1$, …, $\omega_k \in \mathcal H^1$ such that $$\label{e:omegaiDef} \int_{\gamma_i} \omega_j = \delta_{i,j}\,.$$ Now, the covariance matrix $\Sigma$ appearing in Theorem \[t:winding\] is given $$\label{e:sigmadef} \Sigma_{i,j} \defeq \frac{1}{\operatorname{vol}M}\int_{M} \omega_{i} \cdot \omega_{j} \,.$$ The proof of Theorem \[t:winding\] follows quite easily from our heat kernel result Theorem \[t:hker\], which will be given in Section \[s:pfwinding\] below. We remark, however, that Theorem \[t:winding\] can also be proved directly by using a probabilistic argument. For completeness, we sketch this proof in Section \[s:windingDomain\]. A fundamental example of Theorem \[t:winding\] is the case when $M$ is a planar domain with multiple holes. In this case, in the limiting Gaussian distribution described in the proposition, the forms $\omega_i$ can be obtained quite explicitly following remark  \[r:planar\]. The winding of Brownian motion in planar domains is a classical topic which has been studied by many authors [@Spitzer58; @PitmanYor86; @PitmanYor89; @RudnickHu87; @RogersWilliams00; @LyonsMcKean84; @GeigerKersting94; @TobyWerner95; @Watanabe00]. The result by Toby and Werner [@TobyWerner95], in particular, obtains a law of large numbers type result for the time average of the winding number of an obliquely reflected Brownian motion in a bounded planar domain. Under normal reflection our result (Theorem \[t:winding\]) is a refinement of Toby and Werner’s result. Namely, we show that the long time average of the winding number is $0$, and we prove a Gaussian type central limit theorem for fluctuations around the mean. A more detailed comparison with the results of [@TobyWerner95] is in Section \[s:tobyWerner\], below. \[r:annulus\] When $M \subseteq \R^2$ is an annulus the covariance matrix $\Sigma$ can be computed explicitly. Explicitly, for $0 < r_1 < r_2$ and let $$A \defeq\set[\big]{ p\in\R^{2} \st r_{1}<|p|<r_{2}}$$ be the annulus with inner radius $r_{1}$ and outer radius $r_{2}$. In this case, $k=1$ and define $\rho(t)$ is simply the integer-valued winding number of the reflected Brownian motion in $A$ with respect to the inner hole. Now $k = 1$ and the one form $\omega_1$ can be obtained from Remark \[r:neumann\]. Explicitly, we choose $p_1 = q_1 = 0$, and define $\tau_1$ by . Now $\tau_1 \cdot \nu = 0$ on $\partial M$, forcing $\phi_1 = 0$ and hence $\omega_1 = \tau_1$. Thus Theorem \[t:winding\] shows that $\rho(t) / \sqrt{t} \to \mathcal N(0, \Sigma)$ weakly as $t \to \infty$. Moreover equation  and  show that $\Sigma$ is the $1 \times 1$ matrix $(\sigma^2)$ where $$\label{e:sigmaAnnulus} \sigma^{2} =\frac{1}{\operatorname{vol}A}\int_{A}\abs{\omega_1}^{2} =\frac{1}{2\pi^2(r_{2}^{2}-r_{1}^{2})}\log\paren[\Big]{\frac{r_{2}}{r_{1}}} \,.$$ We remark, however, that in this case a finer asymptotic result is available. Namely, Wen [@Wen17] shows that for large time $$\var( \rho(t) ) \approx \frac{1}{4\pi^2}\paren[\Big]{\ln^2 \paren[\big]{\frac{r_2}{r_1}} - \ln^2 \paren[\big]{\frac{r_1}{r_0}}} + \frac{\ln \paren{ {r_2 / r_1} }}{2\pi^2(r_2^2 - r_1^2)} \paren[\Big]{ t - \frac{r_2^2 - r_0^2}{2} + r_1^2 \ln \paren[\big]{ \frac{r_2}{r_0} }}$$ where $r_0 = \abs{W_0}$ is the radial coordinate of the starting point. Note Theorem \[t:hker\] only shows $\var \rho(t) / t \to \sigma^2$ as $t \to \infty$. Wen’s result above goes further by providing explicit limit for $\var\rho(t) - \sigma^2 t$ as $t \to \infty$. Another interesting example is the winding of 3D Brownian motion around knots. Recall that a knot $K$ is an embedding of $S^{1}$ into $\R^{3}$. A basic topological invariant of a knot $K$ is the fundamental group $\pi_{1}(\R^{3}{-}K)$ which is known as the *knot group* of $K$. The study of the fundamental group $\pi_{1}(\R^{3}{-}K)$ is important for the classification of knots and has significant applications in mathematical physics. It is well known that the abelianized fundamental group of $\R^{3}{-}K)$ is always $\Z$. Let $K$ be a knot in $\R^{3}$. Consider the domain $M=\Omega{-}N_{K}$, where $N$ is a small tubular neighborhood of $K$ and $\Omega$ is a large bounded domain (a ball for instance) containing $N_{K}$. Let $W(t)$ be a reflected Brownian motion in $M$, and define $\rho(t)$ to be the $\Z$-valued winding number of $W$ with respect to a fixed generator of $\pi_{1}(M)_{{\mathrm{ab}}}$. Now $\rho(t)$ contains information about the entanglement of $W(t)$ with the knot $K$. Theorem \[t:winding\] applies in this context, and shows that the long time behaviour of $\rho$ is Gaussian with mean $0$ and covariance given by . In some cases, the generator of $\pi_1(M)_{{\mathrm{ab}}}$ (which was used above in defining $\rho$) can be written down explicitly. For instance, consider the $(m,n)$-*torus knot*, $K=K_{m,n}$, defined by $S^{1}\ni z\mapsto(z^{m},z^{n})\in S^{1}\times S^{1}$ where $\gcd(m,n)=1$. Then $\pi_{1}(M)$ is isomorphic to the free group with two generators $a$ and $b$, modulo the relation $a^{m}=b^{n}$. Here $a$ represents a meridional circle inside the open solid torus and $b$ represents a longitudinal circle winding around the torus in the exterior. In this case, a generator of $\pi_1(M)_{{\mathrm{ab}}}$ is $a^{n'} b^{m'}$, where $m', n'$ are integers such that $m m'+n m'=1$. (The existence of such an $m'$ and $n'$ is guaranteed since $\gcd(m,n) = 1$ by assumption.) Now $a^{n'} b^{m'}$ represents a unit winding around the knot $K$, and $\rho(t)$ describes the total number of windings around $K$. Proof of Theorem \[t:hker\]. {#s:mainproof} ============================ The main tool used in the proof of Theorem \[t:hker\] is an integral representation due to Lott [@Lott92] and Kotani-Sunada [@KotaniSunada00]. Note that heat kernel $H$ on $M$ can be easily computed in terms of the heat kernel $\hat H$ on the cover $\hat M$ using the identity $$\label{e:HinTermsOfHatH} H(t,p,q)=\sum_{y\in\bpi^{-1}(q)}\hat{H}(t,x,y)\,,$$ for any $x\in\bpi^{-1}(p)$. Seminal work of Lott [@Lott92] and Kotani-Sunada [@KotaniSunada00] address an inverse representation where $\hat{H}(t,x,y)$ is expressed as the integral of a compact family of heat kernels on twisted bundles over $M$. Since $M$ is compact, the long time behaviour of the these twisted heat kernels is governed by the principal eigenvalue of the associated twisted Laplacian. Thus, using the integral representation in [@Lott92; @KotaniSunada00], the long time behaviour of $\hat H$ can be deduced by studying the behaviour of the above principal eigenvalues near the maximum. In the case where only Neumann boundary conditions are imposed on $\partial M$ (i.e. if $\partial_D M = \emptyset$), the proof in [@Lott92; @KotaniSunada00] can be adapted easily. If, however, there is a portion of the boundary where a Dirichlet boundary condition is imposed (i.e. if $\partial_D M \neq \emptyset$), then one requires finer spectral analysis than that is available in [@Lott92; @KotaniSunada00] The key new ingredient lies in understanding the behaviour of the principal eigenvalue of twisted Laplacians. #### Plan of this section. In Section \[s:liftedRep\] we describe the Lott / Kotani-Sunada representation of the lifted heat kernels. In Section \[ss:hkerProof\] we use this representation to prove Theorem \[t:hker\], modulo two key lemmas (Lemmas \[l:minLambdaChi\] and \[l:muBound\], below) concerning the principal eigenvalue of the twisted Laplacian. Finally in Sections \[s:lambdamin\] and \[s:mubound\] we prove Lemmas \[l:minLambdaChi\] and \[l:muBound\] respectively. A Representation of the Lifted Heat Kernel. {#s:liftedRep} ------------------------------------------- We begin by describing the Lott [@Lott92] / Kotani-Sunada [@KotaniSunada00] representation of the heat kernel $\hat H$. Let $S^1 = \set{ z \in \C \st \abs{z} = 1 }$ be the unit circle and let $$\mathcal{G}\defeq\hom (G; S^1)\,,$$ be the space of one dimensional unitary representations of $G$. We know that $\mathcal G$ is isomorphic to $(S^1)^k$, and hence is a compact Lie group with a unique normalized Haar measure. Given $\chi\in\mathcal{G}$, define an equivalence relation on $\hat M \times \C$ by $$(x, \zeta) \sim (g(x), \chi(g) \zeta) \quad\text{for all } g \in G\,,$$ and let $E_{\chi}$ be the quotient space $\hat{M}\times\C/ {\sim}$. Since the action of $G$ on fibers is transitive, it follows that $E_\chi$ is a complex line bundle on $M$. Let $C^{\infty}(E_{\chi})$ be the space of smooth sections of $E_{\chi}$. Note that elements of $C^{\infty}(E_{\chi})$ can be identified with smooth functions $s \colon \hat M \to \C$ which satisfy the twisting condition $$\label{e:twistingCondition} s(g(x))=\chi(g) s(x) \,, \quad \forall x\in\hat{M},\ g\in G \,.$$ Since $\bpi\colon \hat{M}\to M$ is a local isometry and $G$ acts on $\hat{M}$ by isometries, $E_{\chi}$ carries a natural connection induced by the Riemannian metric on $M$. Let $\lap_{\chi}$ be the associated Laplacian acting on sections of $E_{\chi}$. If we impose homogeneous Dirichlet boundary conditions on $\partial_D \hat M$ and homogeneous Neumann boundary conditions on $\partial_N \hat M)$, then the operator $-\lap_{\chi}$ is a self-adjoint positive-definite on $L^{2}(E_{\chi})$. To write this in terms of sections on $\hat M$, define the space $\mathcal D_\chi$ by $$\label{e:dchi} \begin{split} \mathcal D_\chi = \bigl\{ s \in C^\infty(\hat M, \C) \st[\big] & s \text{ satisfies~\eqref{e:twistingCondition}} \,, \ s = 0 \text{ on } \partial_D \hat M\,, \\ & \text{and } \nu \cdot \grad s = 0 \text{ on } \partial_N \hat M \bigr\}\,. \end{split}$$ Now $\lap_\chi$ is simply the restriction of the usual Laplacian $\hat \lap$ on $\hat M$, and the $L^2$ inner-product is given by $$\label{e:l2twisted} \ip{s_{1},s_{2}}_{L^{2}} \defeq \int_{M} s_{1}(x_p) \, \overline{s_{2}(x_p)} \, dp \,,$$ for $s_{1}, s_{2}\in\mathcal{D}_{\chi}$. Here for each $p\in M$, $x_{p}$ is a any point in the fiber $\bpi^{-1}(p)$ such that the function $p \mapsto x_p$ is measurable. The twisting condition ensures that is independent of the choice of $x_p$. When $\chi \equiv \one$ is the trivial representation, $E_{\chi}$ is the trivial line bundle $M\times\C$, and $\lap_{\chi}$ is the standard Laplacian $\lap$ on $M$. When $\chi \not\equiv \one$, $E_{\chi}$ is diffeomorphic to the trivial line bundle, as one can construct a non-vanishing section easily (c.f. below). However, $E_\chi$ is *not* isometric to the trivial line bundle, and the use of $E_\chi$ is in the structure of the twisted Laplacian $\lap_{\chi}$, which differs from the standard Laplacian on $M$. Let $H_{\chi}(t,x,y)$ be the heat kernel of $-\lap_{\chi}$ on $E_\chi$ (see [@BerlineGetzlerEA92] for the general construction of heat kernels on vector bundles). As before, we can view $H_\chi$ as a function on $(0,\infty)\times\hat{M}\times\hat{M}$ that satisfies the twisting conditions $$ H_{\chi}(t, g(x), y) = \chi(g) \, H_{\chi}(t,x,y)\,, \quad\text{and}\quad H_{\chi}(t, x, g(y) ) = \overline{\chi(g)} \, H_{\chi}(t,x,y)\,.$$ The Lott [@Lott92] and Kotani-Sunada [@KotaniSunada00] representation expresses $\hat H$ in terms of $H_\chi$, and allowing us to use properties of $H_\chi$ to deduce properties of $\hat H$. \[l:keyRepresentation\] The heat kernel $\hat H$ on $\hat M$ satisfies the identity $$\label{e:keyRepresentation} \hat{H}(t,x,y)=\int_{\mathcal{G}}H_{\chi}(t,x,y) \, d\chi \,,$$ where the integral is performed with respect to the normalized Haar measure $d\chi$ on $\mathcal G$. Since a full proof can be found in [@Lott92 Proposition 38], and [@KotaniSunada00 Lemma 3.1], we only provide a short formal derivation. Suppose $\hat H$ is defined by . Clearly $\hat H$ satisfies the heat equation with Dirichlet boundary conditions on $\partial_D \hat M$ and Neumann boundary conditions on $\partial_N \hat M)$. For initial data observe $$H_\chi(0, x, y) = \sum_{g \in G} \overline{\chi(g)} \, \delta_{g(x)} (y)\,,$$ where $\delta_{g(x)}$ denotes the Dirac delta function at $g(x)$. Integrating over $\mathcal G$ and using the orthogonality property $$\int_{\mathcal G} \chi(g) \, d\chi = \begin{cases} 1 & g = \mathrm{Id}\\ 0 & g \neq \mathrm{Id}\,, \end{cases}$$ we see that $\hat H(0, x, y) = \delta_x(y)$, and hence $\hat H$ must be the heat kernel on $\hat M$. The integral representation  is similar to Fourier transform and inversion. Indeed, for each $\chi \in \mathcal G$, it is easy to see that $$H_\chi(t, x, y) = \sum_{g \in G} \chi(g) \hat H(t, x, g(y))\,.$$ One can view $\mathcal{G}\ni\chi\mapsto H_\chi$ as a Fourier transform of $\hat{H}$, and equation  gives the Fourier inversion formula. Proof of the Heat Kernel Asymptotics (Theorem \[t:hker\]). {#ss:hkerProof} ---------------------------------------------------------- The representation  allows us to study the long time behaviour of $\hat H$ using the long time behaviour of $H_\chi$. Since $M$ is compact, the long time behaviour of the heat kernels $H_\chi$ can be studied by spectral theory. More precisely, the twisted Laplacian $\Delta_\chi$ admits a sequence of eigenvalues $$0\leqslant \lambda_{\chi,1}\leqslant\lambda_{\chi,2}\leqslant\cdots\leqslant\lambda_{\chi,j}\leqslant\cdots\uparrow\infty,$$and a corresponding sequence of eigenfunctions $\set{s_{\chi,j} \st j\geq0} \subseteq\mathcal{D}_{\chi}$ which forms an orthonormal basis of $L^{2}(E_{\chi})$. According to perturbation theory, $\lambda_{\chi,j}$ is smooth in $\chi$, and up to a normalization $s_{\chi,j}$ can be chosen to depend smoothly on $\chi$. The heat kernel $H_{\chi}(t,x,y)$ can now be written as $$\label{e:expansionHChi} H_{\chi}(t,x,y)=\sum_{j=0}^{\infty}e^{-\lambda_{\chi,j}t}s_{\chi,j}(x)\overline{s_{\chi,j}(y)} \,.$$Note that since $M$ is compact, the above heat kernel expansion is uniform in $x,y\in\hat{M}$ provided the boundary is smooth. This can be seen from the fact that the eigenfunction $s_{\chi,j}$ is uniformly bounded by a polynomial power of eigenvalue $\lambda_{\chi,j}$, together with Weyl’s law on the growth the eigenvalues. Combining  with Lemma \[l:keyRepresentation\], we obtain $$\label{e:hkerSpectral} \hat{H}(t,x,y)=\sum_{j=0}^{\infty}\int_{\mathcal{G}}e^{-\lambda_{\chi,j}t}s_{\chi,j}(x)\overline{s_{\chi,j}(y)}d\chi \,.$$ From , it is natural to expect that the long time behaviour of $\hat H$ is controlled by the initial term of the series expansion. In this respect, there are two key ingredients for proving Theorem \[t:hker\]. The first key point, which is the content of Lemma \[l:minLambdaChi\], will allow us to see that the integral $\int_{\mathcal{G}}e^{-\lambda_{\chi,0}t}s_{\chi,0}(x)\overline{s_{\chi,0}(y)}d\chi$ concentrates at the trivial representation $\chi=\bf{1}$ when $t$ is large. Having such concentration property, the second key point, which is the content of lemma \[l:muBound\], will then allow us to determine the long time asymptotics of $\hat{H}$ precisely from the rate at which $\lambda_{\chi,0}\rightarrow\lambda_0$ as $\chi\rightarrow\bf{1}\in\mathcal{G}$. Note that when $\chi = \one$ the corresponding eigenvalue $\lambda_{\one, 0}$ is exactly $\lambda_0$, the principal eigenvalue of $-\lap$ on $M$. \[l:minLambdaChi\] The function $\chi \mapsto \lambda_{\chi, 0}$ attains a unique global minimum on $\mathcal G$ at the trivial representation $\chi = \one$. We prove Lemma \[l:minLambdaChi\] in Section \[s:lambdamin\], below. Note that when $\chi=\one$, $\lap_{\chi}$ is simply the standard Laplacian $\lap$ acting on functions on $M$. If Neumann boundary conditions are imposed on all of $\partial M$ (i.e. when $\partial_D M = \emptyset$), $\lambda_{\one, 0} = 0$. In this case, the proof of Lemma \[l:minLambdaChi\] can be adapted from the arguments in [@Sunada89] (see also a direct proof in Section \[s:lambdamin\] in the Neumann boundary case). If, however, Dirichlet boundary conditions are imposed on a portion of $\partial M$ (i.e. $\partial_D M \neq \emptyset$), then $\lambda_{\one, 0} > 0$ and the proof of Lemma \[l:minLambdaChi\] requires some work. In view of   and Lemma \[l:minLambdaChi\], to determine the long time behaviour of $\hat H$ we also need to understand the rate at which $\lambda_{\chi, 0}$ approaches the global minimum as $\chi \to \one$. When $G$ is torsion free, we do this by transferring the problem to the linear space $\mathcal H^1_G$. Explicitly, given $\omega \in \mathcal H^1_G$, we define $\chi_\omega \in \mathcal G$ by $$\label{e:expmap} \chi_\omega(g) = \exp\paren[\Big]{ 2 \pi i \int_{x_0}^{g(x_0)} \bpi^*(\omega) }\,,$$ for some $x_0 \in \hat M$. The integral above is done over any smooth path in $\hat M$ joining $x_0$ and $g(x_0)$. Recall that (Section \[s:mainthm\]) for all $\omega \in \mathcal H^1_G$, this integrals is independent of both the path of integration and the choice of $x_0$. Note that when $G$ is torsion free, the map $\omega \mapsto \chi_\omega$ is a surjective homomorphism between $\mathcal H^1_G$ and $\mathcal G$ whose kernel is precisely $\mathcal H^1_\Z$ defined by . The space $\mathcal{H}^1_G$ can be identified with the Lie algebra of $\mathcal{G}$ and under this identification the map $\omega\mapsto\chi_\omega$ is exactly the exponential map. Now the rate at which $\lambda_{\chi, 0} \to \lambda_{0}$ as $\chi \to \one \in \mathcal G$ can be obtained from the rate at which $\lambda_{\chi_\omega, 0} \to \lambda_{0}$ as $\omega \to 0 \in \mathcal H^1_G$. In fact, we claim that the quadratic form induced by the Hessian of the map $\omega \mapsto \lambda_{\chi_\omega, 0}$ at $\omega = 0$ is precisely $\mathcal I(\omega)$ defined by , and this determines the rate at which $\lambda_{\chi_\omega, 0}$ approaches the global minimum $\lambda_0$. \[l:muBound\] For any $\epsilon > 0$, there exists $\delta > 0$ such that if $0 < \abs{\omega} < \delta$ we have $$\label{e:muBound} \abs[\Big]{\lambda_{\chi_\omega, 0} - \lambda_0-\frac{\mathcal{I}(\omega)}{2}} < \epsilon\norm{\omega}_{L^2(M)}^2\,,$$ where $\mathcal I(\omega)$ is defined in . Moreover, the map $\omega \mapsto \mathcal I(\omega)$ is a well defined quadratic form, and induces a positive definite inner product on $\mathcal H^1_G$. We point out that the positivity of the quadratic form $\mathcal{I}(\omega)$ is crucial. As mentioned earlier (Remark \[r:neumann\]), if only Neumann boundary condition is imposed on $\partial M$, $\mathcal I(\omega)$ is simply a multiple of the standard $L^2$ inner product on $1$-forms over $M$, whose positivity is straight forward. The positivity of $\mathcal{I}(\omega)$ in the case of Dirichlet boundary conditions requires some extra work. We prove Lemma \[l:muBound\] in Section \[s:mubound\]. Assuming Lemma \[l:minLambdaChi\] and Lemma  \[l:muBound\] for the moment, we can now prove Theorem \[t:hker\]. We first consider the case when $G$ is torsion free, and will later show how this implies the general case. Note first that Lemma \[l:minLambdaChi\] allows us to localize the integral in  to an arbitrarily small neighborhood of the trivial representation $\one$. More precisely, we claim that for any open neighborhood $R$ of $\one\in\mathcal{G}$, there exist constants $C_1>0$, such that $$\label{e:chiLocal} \sup_{x,y \in \hat M} \abs[\Big]{ e^{\lambda_0 t} \hat H(x, y, t) - \int_R \exp\paren[\Big]{-(\lambda_{\chi, 0} - \lambda_0) t } s_{\chi, 0}(x) \overline{s_{\chi, 0}(y)} \, d\chi } \leq e^{-C_1 t}.$$ This in particular implies that the long time behavior of $\hat{H}(t,x,y)$ is determined by the long time behavior of the integral representation around an arbitrarily small neighborhood of $\mathbf{1}\in\mathcal{G}$. To establish , recall that Rayleigh’s principle and the strong maximum principle guarantee that $\lambda_{\one,0}$ is simple. Standard perturbation theory (c.f. [@ReedSimon78], Theorem XII.13) guarantees that when $\chi$ is sufficiently close to $\one$, the eigenvalue $\lambda_{\chi,0}$ is also simple (i.e. $\lambda_{\chi,0}<\lambda_{\chi,1}$). Now, by Lemma \[l:minLambdaChi\], we observe $$\lambda'\defeq\min\set[\big]{ \inf\set{ \lambda_{\chi,1} \st \chi\in\mathcal{G}} \,, \ \inf\set{\lambda_{\chi,0} \st \chi\in\mathcal{G}{-}R} } > \lambda_{0}.$$ Hence by choosing $C_1 \in ( 0, \lambda'-\lambda_{0})$, we have $$\begin{gathered} \sup_{x,y\in\hat{M}}\Bigl( \abs[\Big]{\sum_{j=1}^{\infty}\int_{\mathcal{G}}e^{-(\lambda_{\chi,j}-\lambda_{0})t}s_{\chi,j}(x)\overline{s_{\chi,j}(y)} \, d\chi} \\ +\abs[\Big]{\int_{\mathcal{G}{-}R}e^{-(\lambda_{\chi,0}-\lambda_{0})t}s_{\chi,0}(x)\overline{s_{\chi,0}(y)}d\chi} \Bigr) \leq e^{-C_1 t} \end{gathered}$$ for all $t$ sufficiently large. This immediately implies . For any small neighborhood $R$ of $\one$ as before, our next task is to convert the integral over $R$ in  to an integral over a neighborhood of $0$ in $\mathcal H^1_G$ (the Lie algebra of $\mathcal{G}$) using the exponential map . To do this, recall $(\omega_1, \dots, \omega_k)$ was chosen to be a basis of $\mathcal H^1_\Z \subseteq \mathcal H^1_G$. Identifying $\mathcal H^1_G$ with $\R^k$ using this basis, we let $d\omega$ denote the pullback of the Lebesgue measure on $\R^k$ to $\mathcal H^1_G$. (Equivalently, $d\omega$ is the Haar measure on $\mathcal H^1_G$ normalized so that the parallelogram with sides $\omega_1$, …, $\omega_k$ has measure $1$.) Clearly $$\begin{gathered} \label{e:intR} \int_R \exp\paren[\Big]{-(\lambda_{\chi, 0} - \lambda_0) t } s_{\chi, 0}(x) \overline{s_{\chi, 0}(y)} \, d\chi \\ = \int_T \exp\paren[\Big]{-(\mu_\omega - \lambda_0) t } s_{\chi_\omega, 0}(x) \overline{s_{\chi_\omega, 0}(y)} \, d\omega\,. \end{gathered}$$ Here $\mu_\omega \defeq \lambda_{\chi_\omega, 0}$ and $T$ is the inverse image of $R$ under the map $\omega \mapsto \chi_\omega$. Recall the eigenfunctions $s_{\chi_\omega, 0}$ appearing above are sections of the twisted bundle $E_{\chi_\omega}$. They can be converted to functions on $M$ using some canonical section $\sigma_{\omega}$. Explicitly, let $x_{0}\in\hat{M}$ be a fixed point, and given $\omega\in \mathcal H^1_G$, define $\sigma_\omega\colon \hat M \to \C$ by $$\label{e:cannonicalSection} \sigma_{\omega}(x) \defeq \exp\paren[\Big]{2\pi i\int_{x_{0}}^{x} \bpi^*(\omega) } \,.$$ Here $\bpi^*(\omega)$ is the pullback of $\omega$ to $\hat{M}$ via the covering projection $\bpi$, and the integral above is performed along any smooth path in $\hat M$ joining $x_0$ and $x$. By definition of $\mathcal H^1_G$, this integral does not depend on the path of integration. Observe that for any $g\in G$ we have $$\label{e:sigmaOmegaDef} \sigma_\omega( g(x)) = \sigma_\omega(x) \exp\paren[\Big]{ 2 \pi i \int_x^{g(x)} \bpi^*(\omega) } = \chi_\omega(g) \sigma_\omega(x)\,,$$ where $\chi_\omega \in \mathcal G$ is defined in equation . Thus $\sigma_\omega$ satisfies the twisting condition  and hence can be viewed as a section of $E_{\chi_\omega}$. Now define $$\phi_\omega \defeq \overline{\sigma_\omega} \, s_{\chi_\omega, 0}$$ and notice that $\phi_\omega(g(x)) = \phi_\omega(x)$ for all $g \in \mathcal G$. This implies $\phi_\omega \circ \bpi = \phi_\omega$, and hence $\phi_\omega$ can be viewed as a (smooth) $\C$-valued function on $M$. Consequently, we can now rewrite  as $$\begin{gathered} \label{e:intR2} \int_R \exp\paren[\Big]{-(\lambda_{\chi, 0} - \lambda_0) t } s_{\chi, 0}(x) \overline{s_{\chi, 0}(y)} \, d\chi \\ = \int_T \exp\paren[\Big]{ -(\mu_\omega - \mu_0) t - 2\pi i \xi_{x,y}(\omega) } \phi_\omega(x) \overline{\phi_\omega(y)} \, d\omega\,. \end{gathered}$$ where $\xi_{x,y}(\omega)$ is defined in . (Of course, when $\omega = 0$, $\chi_\omega = \one$ and hence $\mu_0 = \lambda_0$.) Thus, using , we have $$\label{e:omegaLocal} \sup_{x,y\in\hat{M}}\abs[\Big]{e^{\lambda_{0}t}\hat{H}(x,y,t) - I_1} \leq e^{-C_{1}t}\,, \quad\text{for $t$ sufficiently large}\,.$$ Here $$I_1 \defeq \int_{T}\exp\paren[\big]{-\paren{\mu_{\omega}-\mu_{0}}t-2\pi i\xi_{x,y}(\omega)}\phi_{\omega}(x)\overline{\phi_{\omega}(y)} \, d\omega\,,$$ and $C_1$ is the constant appearing in , and depends on the neighborhood $R$. By making the neighborhood $R$ (and hence also $T$) small, we can ensure that $\phi_\omega$ close to $\phi_0$. Moreover, when $\omega$ is close to $0$, Lemma \[l:muBound\] implies $\mu_\omega - \mu_0 \approx \mathcal I(\omega)/2$. Thus we claim that for any $\eta > 0$, the neighborhood $R \ni \one$ can be chosen such that $$\label{e:triangle} \limsup_{t\to \infty} \sup_{x, y \in \hat M} t^{k/2} (I_1 - I_2) < \eta \,,$$ where $$I_2 \defeq \int_{\mathcal{H}_{G}^{1}}\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(\omega)t-2\pi i\xi_{x,y}(\omega)}\phi_{0}(x)\overline{\phi_{0}(y)} \, d\omega \,.$$ To avoid breaking continuity, we momentarily postpone the proof of . Now we see that  and  combined imply $$\label{e:HlimI2} \lim_{t\to \infty} \paren[\Big]{ t^{k/2} e^{\lambda_0 t} \hat H(t, x, y) - t^{k/2} I_2 } = 0$$ Thus to finish the proof we only need to evaluate $I_2$ and express it in the form in . To do this, write $\omega = \sum c_n \omega_n \in \mathcal H^1_G$ and observe $$\mathcal I(\omega) = \sum_{m, n \leq k} a_{m,n} c_m c_n\,, \qquad \text{where } a_{m,n} = \ip{\omega_m, \omega_n}_\mathcal I\,.$$ Let $A$ be the matrix $(a_{m,n})$, and $a_{m,n}^{-1}$ be the $(m,n)$ entry of the matrix $A^{-1}$. Consequently $$\begin{aligned} I_2 &= \phi_0(x) \, \overline{\phi_0(y)} \mathbin{\cdot} \\ &\qquad \int_{c \in \R^k} \exp\paren[\Big]{ - \sum_{m,n = 1}^k a_{m,n} c_m c_n t - 2\pi i \sum_{m=1}^k c_m \xi_{x,y}(\omega_m) } \, dc_1 \cdots dc_k \\ &= \phi_0(x) \, \overline{\phi_0(y)} \frac{(2\pi)^{k/2}}{t^{k/2}\det( a_{m,n} )^{1/2}} \exp\paren[\Big]{ - \frac{2\pi^2}{t} \sum_{m,n=1}^k a_{m,n}^{-1} \xi_{x,y}(\omega_m) \xi_{x,y}(\omega_n) } \\ &= \phi_0(x) \, \overline{\phi_0(y)} \frac{(2\pi)^{k/2}}{t^{k/2}\det( a_{m,n} )^{1/2}} \exp\paren[\Big]{ - \frac{2\pi^2}{t} \norm{\xi_{x,y}}^2_{\mathcal I^*} } \,, \end{aligned}$$ where the second equality followed from the formula for the Fourier transform of the Gaussian. Note that when $\omega = 0$, $\sigma_\omega \equiv \one$ and hence $\phi_0 = s_{\one, 0}$ is the principal eigenfunction of $-\lap$ on $M$, viewed as a function on $\hat M$. Hence $\phi_0$ is real, and so $\overline{\phi_0} = \phi_0$, and we have $$I_2 =t^{-{k / 2}}C_{\mathcal{I}}(x,y)\exp\paren[\Big]{-\frac{2\pi^{2}d_{\mathcal{I}}^{2}(x,y)}{t}} \,,$$ where $C_\mathcal{I}$ is defined by . Combined with  this finishes the proof of Theorem \[t:hker\] when $G$ is torsion free. It remains to prove . Since $\omega \mapsto \phi_\omega$ is continuous, there exists a neighborhood $T \ni 0$ such that $$\label{e:ctyPhi} \sup_{x\in\widehat{M}}\abs[\big]{\phi_{\omega}(x)-\phi_{0}(x)} < \eta \quad\text{for all } \omega \in T\,.$$ Now we know that holds with some constant $C_1 = C_1(\eta) >0$ when $t$ is large. Write $$t^{{k / 2}}(I_1-I_2)=J_{1}+J_{2}+J_{3} \,,$$ where $$\begin{gathered} J_{1}\defeq t^{{k / 2}}\int_{T}\paren[\Big]{e^{-\paren{\mu_{\omega}-\mu_{0}}t}-e^{-\mathcal{I}(\omega)t / 2}}\exp\paren[\big]{-2\pi i\xi_{x,y}(\omega)}\phi_{\omega}(x)\overline{\phi_{\omega}(y)} \, d\omega \,, \\ J_{2} \defeq t^{\frac{k}{2}}\int_{T}\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(\omega)t-2\pi i\xi_{x,y}(\omega)}\paren[\Big]{\phi_{\omega}(x)\overline{\phi_{\omega}(y)}-\phi_{0}(x)\overline{\phi_{0}(y)}} \, d\omega \,,\end{gathered}$$ and $$J_{3}\defeq t^{{k / 2}}\int_{\mathcal{H}_{G}^{1}{-}T}\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(\omega)t-2\pi i\xi_{x,y}(\omega)}\phi_{0}(x)\overline{\phi_{0}(y)} \, d\omega \,.$$ First, by Lemma \[l:muBound\], $\mathcal{I}(\omega)$ is a positive definite quadratic form, and hence the Gaussian tail estimate shows there exists $C_2 = C_2(\eta) >0$, such that $$|J_3|\leq e^{-C_{2}t}$$ uniformly in $x,y\in\hat{M}$, when $t$ is sufficiently large. Next, by and the positivity of the quadratic form $\mathcal{I}(\omega)$, we have $$\abs{J_{2}} \leq C_{3}\eta t^{{k / 2}}\int_{T}e^{-\mathcal{I}(\omega)t / 2} \, d\omega =C_{3}\eta\int_{\sqrt{t}\cdot T}e^{-\mathcal{I}(v) / 2} \, dv \leq C_{4}\eta \,,$$ uniformly in $x,y\in\hat{M}$. Finally, to estimate $J_1$, first choose $K\subseteq\mathcal{H}^1_G$ compact such that $$\int_{\mathcal{H}_{G}^{1}{-}K}\exp\paren[\Big]{-\frac{1}{4}\mathcal{I}(v)} \, dv<\eta \,.$$ By using the same change of variables $v=\sqrt{t}\omega$, we write $$J_{1}=J_{1}'+J_{1}'',$$ where $$\begin{aligned} J_{1}' &\defeq\int_{K}\paren[\Big]{\exp\paren[\Big]{-\paren[\Big]{\mu_{v/t^{1/2}}-\mu_{0}}t}-\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(v)}} \\ &\qquad\qquad\cdot\exp\paren[\Big]{-\frac{2\pi i}{\sqrt{t}}\xi_{x,y}(v)}\phi_{{v / t^{1/2}}}(x)\overline{\phi_{{v / \sqrt*{t}}}(y)} \, dv\end{aligned}$$ and $$\begin{gathered} J_{1}'' \defeq\int_{\sqrt{t}\cdot T{-}K}\paren[\Big]{\exp\paren[\Big]{-\paren[\Big]{\mu_{v/t^{1/2}}-\mu_{0}}t}-\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(v)}}\\ \cdot\exp\paren[\Big]{-\frac{2\pi i}{\sqrt{t}}\xi_{x,y}(v)}\phi_{{v / \sqrt*{t}}}(x)\overline{\phi_{{v / \sqrt*{t}}}(y)} \, dv\end{gathered}$$ respectively. By Lemma \[l:muBound\], we know that $$\lim_{t\rightarrow\infty}\paren[\big]{\mu_{v/t^{1/2}}-\mu_{0}}t=\frac{1}{2}\mathcal{I}(v) \,,$$ for every $v\in\mathcal{H}^1_G$. Therefore, by the dominated convergence theorem, we have $$\lim_{t\rightarrow\infty}\sup_{x,y\in\hat{M}}\abs{J_{1}'}=0 \,.$$ To estimate $J_1''$, choose $\epsilon>0$ such that $$\frac{1}{4}\mathcal{I}(\omega)\geq\epsilon\norm{\omega}_{L^{2}(M)}^{2} \,, \qquad\text{for all } \omega\in\mathcal{H}_{G}^{1} \,.$$ For this $\epsilon$, Lemma \[l:muBound\] allows us to further assume that $T$ is small enough so that $$\omega\in T\implies\mu_{\omega}-\mu_{0}\geq\frac{1}{2}\mathcal{I}(\omega)-\epsilon\norm{\omega}_{L^{2}(M)}^{2}\geq\frac{1}{4}\mathcal{I}(\omega).$$ In particular, we have $$v\in\sqrt{t}\cdot T\implies\paren[\big]{\mu_{{v / t^{1/2}}}-\mu_{0}}t\geq\frac{1}{4}\mathcal{I}(v).$$ It follows that $$\begin{aligned} J_{1}'' & \leq C_{5}\int_{\sqrt{t}\cdot T{-}K}\paren[\Big]{\exp\paren[\big]{-\paren[\big]{\mu_{v/t^{1/2}}-\mu_{0}}t}+\exp\paren[\Big]{-\frac{1}{2}\mathcal{I}(v)}} \, dv\\ & \leq2C_{5}\int_{\sqrt{t}\cdot T{-}K}\exp\paren[\Big]{-\frac{1}{4}\mathcal{I}(v)} \, dv\\ & \leq2C_{5}\int_{\mathcal{H}_{G}^{1}{-}K}\exp\paren[\Big]{-\frac{1}{4}\mathcal{I}(v)}dv\\ & \leq2C_{5}\eta \,,\end{aligned}$$ uniformly in $x,y\in\hat{M}$. Combining the previous estimates, we conclude $$\overline{\lim_{t\rightarrow\infty}}\sup_{x,y\in\hat{M}}\paren[\Big]{t^{k/2}(I_1-I_2)}\leq(C_{4}+2C_{5})\eta \,,$$ and $\eta$ with $\eta / (C_4 + 2 C_5)$ yields  as claimed. When $G$ is has a torsion subgroup, we prove Theorem \[t:hker\] factoring through an intermediate finite cover. Since $G$ can be (non-canonically) expressed as a direct sum $G_T\oplus G_F$, we define $M_1 = \hat M / G_F$. This leads to the covering factorization $$\label{e:torsionFactorization} \begin{tikzcd} \hat{M} \arrow[r,"\bpi_F"] \arrow[dr,"\bpi"'] & M_1 \defeq \hat{M}/G_F \arrow[d,"\bpi_T"] \\ & M\,, \end{tikzcd}$$ where $\bpi_{T}$ and $\bpi_{F}$ have deck transformation groups $G_T$ and $G_F$ respectively, and $M_{1}$ is compact. Recall that $\lambda_{0}$ is the principal eigenvalue of $-\lap$ on $M$, and $\phi_{0}$ is the corresponding $L^2$ normalized eigenfunction. Let $\Lambda_{0}$ be the principal eigenvalue of $-\lap_1$ on $M_1$, and $\Phi_{0})$ be the corresponding $L^2$ normalized eigenfunction. (Here $\lap_{1}$ is the Laplacian on $M_{1}$.) Notice that $\bpi_T^* \phi_0$, the pull back of $\phi_0$ to $M_1$, is an eigenfunction of $-\lap_1$ and $\norm{\bpi_T^* \phi_0}_{L^2(M)} = \abs{G_T}^{1/2}$. Thus $$\label{e:Lambda1EqLambda} \Lambda_0 = \lambda_0 \qquad\text{and}\qquad \Phi_{0}=\frac{\bpi_{T}^{*}\phi_{0}}{|G_T|^{1/2}}\,.$$ Let $\mathcal I_1(\omega_1)$ be the analogue of $\mathcal I$ (defined in equation ) for the manifold $M_1$. Explicitly, $$\mathcal{I}_1(\omega_1) =8\pi^{2}\int_{M_{1}}|\omega_1|^{2}\Phi_{0}^{2}+8\pi\int_{M_{1}}\Phi_{0} \, \omega_1\cdot\nabla g_{1}\,,$$ where $g_1$ is a solution of $$-\lap g_1 - 4 \pi \omega_1 \cdot \grad \Phi_0 = \Lambda_0 g_1\,,$$ with Dirichlet boundary conditions on $\bpi_T^{-1}(\partial_D M)$ and Neumann boundary conditions on $\bpi_T^{-1}(\partial_N M)$. Note that given $\omega_1 \in \mathcal H^1_G(M_1)$ we can find $\omega \in \mathcal H^1_G(M)$ such that $\bpi_T^*(\omega) = \omega_1$. Indeed, since $\dim(\mathcal H^1_G(M) ) = \dim( \mathcal H^1_G(M_1)) = k$ and $\bpi_T^* \colon \mathcal H^1_G(M) \to \mathcal H^1_G(M_1)$ is injective linear map, it must be an isomorphism. Now using  we observe that up to an addition of a scalar multiple of $\Phi_0$, we have $$g_{1} = \frac{\bpi_{T}^{*}g}{|G_T|^{1/2}} \,,$$ where $g = g_\omega$ is defined in . Thus, using  again we see $$\begin{aligned} \mathcal{I}_1(\omega_1) & =8\pi^{2}|G_T|\int_{M}|\omega|^{2}\frac{\phi_{0}^{2}}{|G_T|}+8\pi|G_T|\int_{M}\frac{\phi_{0}}{|G_T|^{1/2}}\omega\cdot\nabla\paren[\Big]{\frac{g}{|G_T|^{1/2}}}\nonumber \\ & =8\pi^{2}\int_{M}|\omega|^{2}\phi_{0}^{2}+8\pi\int_{M}\phi_{0} \, \omega\cdot\nabla g = \mathcal I(\omega)\,.\label{e:IonM} \end{aligned}$$ Since the deck transformation group of $\hat M$ as a cover of $M_1$ is torsion free, we may apply Theorem \[t:hker\] to $M_1$. Thus, we have $$\label{e:hkerTorsion} \lim_{t\to \infty} \paren[\Big]{ t^{k/2}e^{\Lambda_{0}t}\hat{H}(t,x,y) - C_{\mathcal{I}_1}(x,y) \exp\paren[\Big]{-\frac{2\pi^{2}d_{\mathcal{I}_1}^{2}(x,y)}{t}}}$$ uniformly on $\hat M$. Using  we see $d_{\mathcal I_1} = d_\mathcal I$. Using  and we see $$C_{\mathcal I_1}(x, y) = \frac{1}{\abs{G_T}} C_{\mathcal I}(x, y)\,,$$ and inserting this into  finishes the proof. The rest of this section is devoted to proving Lemma \[l:minLambdaChi\] and Lemma \[l:muBound\]. Minimizing the Principal Eigenvalue (Proof of Lemma \[l:minLambdaChi\]). {#s:lambdamin} ------------------------------------------------------------------------ Our aim in this subsection is to prove Lemma \[l:minLambdaChi\], which asserts that the function $\chi \mapsto \lambda_{\chi,0}$ attains a unique global minimum at $\chi = \one$. If only Neumann boundary condition is imposed on $\partial M$, Lemma \[l:minLambdaChi\] can be proved by adapting the argument in [@Sunada89]. This yields quantitative upper and lower bounds on the function $\chi\mapsto\lambda_{\chi,0}$ in addition to the global minimum.Since we only need the global minimum of $\lambda_{\chi, 0}$, there is a simple proof under Neumann boundary conditions. We present this first. We will subsequently provide an independent proof of Lemma \[l:minLambdaChi\] under mixed Dirichlet and Neumann boundary conditions. In this case we know that $\lambda_0 = \lambda_{\one,0}=0$, and the corresponding eigenfunction $s_{\one, 0}$ is constant. Thus to prove the lemma it suffices to show that $\lambda_{\chi,0}>0$ for all $\chi\neq\one$. To see this given $\chi\in\mathcal{G}$ let $s = s_{\chi, 0} \in\mathcal{D}_{\chi}$ be the principal eigenfunction of $-\lap_{\chi}$, and $\lambda = \lambda_{\chi, 0}$ be the principal eigenvalue. We claim that for any fundamental domain $U \subseteq \hat M$, the eigenvalue $\lambda$ satisfies $$\label{e:raleigh} \lambda \int_{U} \abs{s}^2 \, dx = \int_U \abs{\grad s}^2 \, dx\,.$$ Once  is established, one can quickly see that $\lambda > 0$ when $\chi \neq \one$. Indeed, if $\chi \neq \one$, $s(g(x)) = \chi(g) s(x)$ forces the function $s$ to be non-constant, and now equation  forces $\lambda > 0$. To prove  observe $$\label{e:stokesThmUg} \lambda \int_U \abs{s}^2 = -\int_{U} \bar{s} \lap_{\chi} s = \int_{U} \abs{\nabla s}^{2} -\int_{\partial U}\bar{s} \, \partial_\nu s \,.$$ Here, $\partial_\nu s = \nu \cdot \grad s$ is the outward pointing normal derivative on $\partial U$. We will show that the twisting condition  ensures that the boundary integral above vanishes. Decompose $\partial U$ as $$\partial U = \Gamma_{1}\cup\Gamma_{2} \,, \quad\text{where } \Gamma_{1} \defeq \partial U \cap \partial \hat M, \quad\text{and } \Gamma_{2} \defeq \partial U - \Gamma_1\,.$$ Note $\Gamma_1$ is the portion of $\partial U$ contained in $\partial\hat{M}$, and $\Gamma_2$ is the portion of $\partial U$ that is common to neighboring fundamental domains. Clearly, the Neumann boundary condition  implies $$\int_{\Gamma_{1}}\bar{s} \, \partial_\nu s =0 \,.$$ For the integral over $\Gamma_2$, let $(e_1, \dots, e_k)$ be a basis of $G$ and note that $\Gamma_2$ can be expressed as the disjoint union $$\Gamma_{2}=\bigcup_{j=1}^{k}\paren[\big]{\Gamma_{2,j}^{+}\cup\Gamma_{2,j}^{-}}\,,$$ where the $\Gamma_{2,j}^{\pm}$ are chosen so that $\Gamma_{2,j}^+ = e_{j}( \Gamma_{2,j}^{-})$. Using the twisting condition  and the fact that the action of $e_j$ reverses the direction of the unit normal on $\Gamma_{2,j}^-$, we see $$\begin{aligned} \int_{\Gamma_{2,j}^{+}}\overline{s(x)} \, \partial_\nu s (x) \,dx & =-\int_{\Gamma_{2,j}^{-}} \overline{s\paren[\big]{e_{j}(y)}} \, \partial_\nu s\paren[\big]{e_{j}(y)} \, dy \\ & =-\int_{\Gamma_{2,j}^{-}} \overline{\chi(e_{j})} \chi(e_{j}) \, \overline{s(y)} \, \paren[\big]{\partial_\nu s (y)} \, dy \\ & =-\int_{\Gamma_{2,j}^{-}}\overline{s(y)} \, \partial_\nu s(y) \, dy \,,\end{aligned}$$ Consequently, $$\int_{\Gamma_{2}}\overline{s} \, \partial_\nu s = \sum_{j=1}^{k}\paren[\Big]{ \int_{\Gamma_{2,j}^{+}}+\int_{\Gamma_{2,j}^{-}}}\overline{s} \, {\partial_\nu s} =0 \,.$$ and hence the boundary integral in  vanishes. Thus  holds, and the proof is complete. In the general case when $\partial_D M \neq \emptyset$, $\lambda_{\chi,0}>0$ for every $\chi\in\mathcal{G}$, and all eigenfunctions are non-constant. This causes the previous argument to break down and the proof involves a different idea. Before beginning the proof, we first make use of a canonical section to transfer the problem to the linear space $\mathcal{H}^1_G$. Let $\Omega$ be the space of $\C$-valued smooth functions $f\colon M \to \C$ such that $f = 0$ on $\partial_D M$ and $\ip{\grad f, \nu} = 0$ on $\partial_N M$. Let $\hat f = f \circ \bpi \colon \hat M \to \C$. Now given $\omega \in \mathcal H^1_G$, let $\sigma_\omega$ (defined in ) be the canonical section and $\chi_\omega \in \mathcal G$ be the exponential as defined in . Notice that the function $\sigma_\omega \hat f \in \mathcal D_{\chi_\omega}$ is a section on $E_{\chi_\omega}$. Clearly $\sigma_\omega \hat f = 0$ on $\partial_D \hat M$. Moreover, since $\omega \cdot \nu = 0$ on $\partial M$ we have $$\label{e:sOmegaNeumann} \nu \cdot \grad \sigma_\omega = 0 \qquad\text{on } \partial \hat M\,.$$ and hence $\nu \cdot \grad (\sigma_\omega \hat f) = 0$ on $\partial_N \hat M$. Thus $\sigma_\omega \hat f \in \mathcal D_{\chi_\omega}$, where $\mathcal D_{\chi_\omega}$ is defined in equation , and the map $f\mapsto\hat{f}\sigma_{\omega}$ defines a unitary isomorphism between $\Omega\subseteq L^{2}(M)$ and $\mathcal{D}_{\chi_{\omega}}\subseteq L^{2}(E_{\chi_{\omega}})$ respecting the imposed boundary conditions. Now, since $\omega$ and $\hat \omega \defeq \omega \circ \bpi$ are both harmonic, we compute $$\lap_{\chi_{\omega}}(\hat{f}\sigma_{\omega}) = \paren{ \paren{H_{\omega}f} \circ \bpi } \, \sigma_{\omega} \,,$$ where $H_{\omega}$ is the self-adjoint operator on $\Omega\subseteq L^{2}(M)$ defined by $$\label{e:hOmegaDef} H_{\omega}f\defeq\lap f+4\pi i \, \omega\cdot\nabla f-4\pi^{2}|\omega|^{2}f \,.$$ Here we used the Riemannian metric to identify the $1$-form $\omega$ with a vector field. The above shows that $\lap_{\chi_{\omega}}$ is unitarily equivalent to $H_{\omega}$. In particular, eigenvalues of $-H_{\omega}$, denoted by $\mu_{\omega, j}$ are exactly $\lambda_{\chi_\omega, j}$, the eigenvalues of $-\lap_{\chi_\omega}$. Moreover, the corresponding eigenfunctions, denoted by $\phi_{\omega, j}$, are given by $$\label{e:phiOmegaJ} \phi_{\omega,j}=\frac{s_{\chi_{\omega},j}}{\sigma_{\omega}} \,, \quad j\geq0 \,.$$ Note that $\phi_{\omega,j}$ is a well-defined function on $M$ that satisfies Dirichlet boundary conditions on $\partial_D M$ and Neumann boundary conditions on $\partial_N M$. We will now prove the general case of Lemma \[l:minLambdaChi\] by minimizing eigenvalues of the operator $-H_\omega$. Let $\omega\in \mathcal H^1_G$ and let $\chi_{\omega} = \exp(\omega)\in\mathcal{G}$ be the corresponding representation defined by . Let $\mu_{\omega} = \mu_{\omega, 0} = \lambda_{\chi_{\omega},0}$ and $\phi_{\omega} = \phi_{\omega, 0}$ where $\phi_{\omega, 0}$ is the principal eigenfunction of $-H_\omega$ as defined in  above. Using  we see $$\begin{gathered} \label{e:phiOmegaDef} -\lap\phi_{\omega}-4\pi i\omega\cdot\nabla\phi_{\omega}+4\pi^{2}|\omega|^{2}\phi_{\omega}=\mu_{\omega}\phi_{\omega}\,, \\ \label{e:phiOmega0Def} -\lap\phi_{0}=\mu_{0}\phi_{0}\,, \end{gathered}$$ with Dirichlet boundary conditions on $\partial_D \hat M$ and Neumann boundary conditions on $\partial_N \hat M$. Here $\mu_0$ and $\phi_0$ denote the principal eigenvalue and eigenfunction respectively when $\omega \equiv 0$. Note that when $\omega \in \mathcal H^1_\Z$, the corresponding representation $\chi_\omega$ is the trivial representation $\one$. We will show that $\mu_\omega$ above achieves a global minimum precisely when $\omega \in \mathcal H^1_\Z$ and $\chi_\omega = \one$. Now let $\epsilon > 0$ and write $$\overline{\phi_{\omega}} = (\phi_{0}+\epsilon) f \quad\text{where } f\defeq \frac{\overline{\phi_{\omega}}}{\phi_{0}+\epsilon}\,.$$ Multiplying both sides of  by $\overline{\phi_{\omega}} = (\phi_0 + \epsilon) f$ and integrating over $M$ gives $$\begin{aligned} -\int_{M}(\lap\phi_{\omega})(\phi_{0}+\epsilon)f & =\int_{M}\nabla\phi_{\omega}\cdot\paren[\big]{(\phi_{0}+\epsilon)\nabla f+f\nabla\phi_{0}} + \int_{\partial M} B_1\\ & =\int_{M}(\phi_{0}+\epsilon)\nabla\phi_{\omega}\cdot\nabla f\\ & \qquad-\int_{M}\phi_{\omega}\paren[\big]{\nabla f\cdot\nabla\phi_{0}+f\lap\phi_{0}} + \int_{\partial M} B_2\\ & =\int_{M}\paren[\big]{(\phi_{0}+\epsilon)\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}\cdot\nabla f\\ & \qquad \mathbin{+} \mu_{0}\int_{M}f\phi_{0}\phi_{\omega}+ \int_{\partial M} B_2\,, \end{aligned}$$ where $B_i \colon \partial M \to \C$ are boundary functions that will be combined and written explicitly below (equation ). (We clarify that even though the functions above are $\C$-valued, the notation $\grad \phi_\omega \cdot \grad f$ denotes $\sum_i \partial_i \phi_\omega \partial_i f$, and not the complex inner product.) Similarly, using the fact that $\omega$ is harmonic, we have $$\begin{aligned} \MoveEqLeft -4\pi i\int_{M}(\phi_{0}+\epsilon) f \omega\cdot\nabla\phi_{\omega} \\ & =-2\pi i\int_{M}(\phi_{0}+\epsilon)f\nabla\phi_{\omega}\cdot\omega \\ & \qquad + 2\pi i\int_{M}\phi_{\omega}\paren[\big]{(\phi_{0}+\epsilon)\nabla f+f\nabla\phi_{0}}\cdot\omega+ \int_{\partial M} B_3 \\ & =-2\pi i\int_{M}\paren[\big]{(\phi_{0}+\epsilon)\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}\cdot(f\omega) \\ & \qquad +2\pi i\int_{M}(\phi_{0}+\epsilon)\phi_{\omega}\nabla f\cdot\omega+\int_{\partial M} B_3 \,. \end{aligned}$$ Combining the above, we have $$\begin{gathered} \label{e:comparingEigenvalues} \mu_{\omega}-\mu_{0} \int_{M}f\phi_{0}\phi_{\omega} = \int_{M}\paren[\big]{(\phi_{0}+\epsilon)\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}\cdot\paren[\big]{\nabla f-2\pi if\omega} \\ +\int_{M}(\phi_{0}+\epsilon)\phi_{\omega}\paren[\big]{4\pi^{2}|\omega|^{2}f+2\pi i\nabla f\cdot\omega}+ \int_{\partial M} B_0 \,, \end{gathered}$$ where $$\label{e:boundaryTerms} B_0 =-\overline{\phi_{\omega}} \partial_\nu \phi_{\omega} + \phi_{\omega}f \partial_\nu \phi_{0} - 2\pi i (\phi_{0}+\epsilon)\phi_{\omega}f\omega\cdot\nu \,.$$ The boundary conditions imposed ensure that $B_0 = 0$ on both $\partial_D M$ and $\partial_N M$. Since $f=\overline{\phi_{\omega}}/(\phi_{0}+\epsilon)$, we have $$\nabla f=\frac{(\phi_{0}+\epsilon)\nabla\overline{\phi_{\omega}}-\overline{\phi_{\omega}}\nabla\phi_{0}}{(\phi_{0}+\epsilon)^{2}}.$$ Substituting this into the right hand side of , we obtain a perfect square: $$\label{e:comparingEvalsSquare} \mu_{\omega}-\mu_{0}\int_{M}f\phi_{0}\phi_{\omega}=\int_{M} \abs[\Big]{ 2\pi\phi_{\omega}\omega-\frac{i\paren{(\phi_{0}+\epsilon)\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}}{\phi_{0}+\epsilon}}^{2} \,.$$ In particular, $$\mu_{\omega}-\mu_{0}\int_{M}f\phi_{0}\phi_{\omega}=\mu_{\omega}-\mu_{0}\int_{M}\frac{\phi_{0}}{\phi_{0}+\epsilon}|\phi_{\omega}|^{2}\geq0.$$ Sending $\epsilon\to 0$, we obtain $\mu_{\omega}\geq\mu_{0}$, and so the function $\mathcal{G}\ni\chi\mapsto\lambda_{\chi,0}$ attains global minimum at $\chi=\one$. To see that $\chi=\one$ is the unique global minimum point, suppose that $\lambda_\chi = \lambda_0$ for some $\chi \in \mathcal G$. Writing $\chi = \chi_\omega$ for some $\omega \in \mathcal H^1_G$, this means $\mu_\omega = \mu_0$. Fatou’s lemma and  imply $$\begin{aligned} \MoveEqLeft \int_{M}\abs[\Big]{2\pi\phi_{\omega}\omega-\frac{i\paren[\big]{\phi_{0}\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}}{\phi_{0}}}^{2}\\ & \leq\liminf_{\epsilon\to 0}\int_{M}\abs[\Big]{2\pi\phi_{\omega}\omega-\frac{i\paren[\big]{(\phi_{0}+\epsilon)\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}}{\phi_{0}+\epsilon}}^{2}\\ & =\mu_{\omega}-\mu_{0} = 0\,, \end{aligned}$$ by assumption. Hence $$\label{e:integrandInPerfectSquare} 2\pi\phi_{\omega}\omega-\frac{i\paren{\phi_{0}\nabla\phi_{\omega}-\phi_{\omega}\nabla\phi_{0}}}{\phi_{0}}=0 \quad\text{in}\ M \,.$$ Since $\phi_{\omega}=s_{\chi ,0}/\sigma_{\omega}$, we compute $$\nabla\phi_{\omega}=\frac{\sigma_{\omega}\nabla s_{\chi,0}-2\pi i \sigma_{\omega}s_{\chi,0}\omega}{\sigma_{\omega}^{2}}.$$ Substituting this into , we see $$\phi_{0}\nabla s_{\chi,0}=s_{\chi,0}\nabla\phi_{0},$$ which implies that $$\nabla\paren[\Big]{\frac{s_{\chi,0}}{\phi_{0}}}=0.$$ Therefore, $s_{\chi,0}=c\phi_{0}$ for some non-zero constant $c$. However, the twisting conditions  for $\phi_0$ and $s_{\chi, 0}$ require $$\phi_0( g(x) ) = \phi_0(x) \qquad\text{and}\qquad s_{\chi, 0}( g(x)) = \chi(g) s_{\chi, 0} (x)\,,$$ for every $g \in \mathcal G$. This is only possible if $\chi(g)=1$ for all $g \in \mathcal G$, showing $\chi$ is the trivial representation $\one$. Positivity of the Hessian (Proof of Lemma \[l:muBound\]). {#s:mubound} --------------------------------------------------------- In this subsection we prove Lemma \[l:muBound\]. The main difficulty is proving positivity, which we postpone to the end. Given $\omega \in \mathcal H^1_G$, define $$\varphi_t = \phi_{t \omega} \qquad\text{and}\qquad h_t = \mu_{t\omega}\,,$$ where $\phi_{t \omega} = \phi_{t\omega, 0}$ is the principal eigenfunction of $-H_{t\omega}$ (equation ) and $\mu_{t\omega}$ is the corresponding principal eigenvalue. We claim that $$\label{e:hder} h'_0 = 0\,, \quad h''_0 = \mathcal I(\omega) \quad\text{and}\quad \operatorname{Re}\paren{\varphi'_0} = 0\,,$$ where $h'$, $\varphi'$ denote the derivatives of $h$ and $\varphi$ respectively with respect to $t$. This will immediately imply that at $\omega=0$ the quadratic form induced by the Hessian of the map $\omega\mapsto\mu_\omega$ is precisely $\mathcal{I}(\omega)$, hence proving  in the lemma. To establish , we first note that  implies $$\label{e:varphit} -\lap\varphi_{t}-4\pi it\omega\cdot\nabla\varphi_{t}+4\pi^{2}t^{2}|\omega|^{2}\varphi_{t}=h_t\varphi_{t} \,.$$ Conjugating both sides of  gives $$\label{e:varphiBar} -\lap\overline{\varphi_{t}}-4\pi i(-t)\omega\cdot\nabla\overline{\varphi_{t}}+4\pi^{2}(-t)^{2}|\omega|^{2}\overline{\varphi_{t}}=h_t\overline{\varphi_{t}} \,.$$ In other words, $\overline{\varphi_{t}}$ is an eigenfunction of $-H_{-t\omega}$ with eigenvalue $h_t$. Since $h_t = \mu_{t\omega}$ is the principal eigenvalue, this implies $h_{-t}\leq h_t$. By symmetry, we see that $h_{-t}=h_t$, and hence $h'_0=0$. To see that $\varphi_{0}'$ is purely imaginary, recall $h_t$ is a simple eigenvalue of $-H_{t\omega}$ when $t$ is small. Thus $$\label{e:phiPrime0imaginary} \overline{\varphi_{t}}=\zeta_{t}\varphi_{-t} \,,$$ for some $S^1$ valued function $\zeta_{t}$, defined for small $t$. Changing $t$ to $-t$, we get $$\overline{\varphi_{-t}}=\zeta_{-t}\varphi_{t}=\zeta_{-t}\overline{\zeta_{t}}\overline{\varphi_{-t}} \,.$$ Therefore, $\zeta_{-t}\overline{\zeta_{t}}=1$, which implies that $\zeta_{-t}=\zeta_{t}$. In particular, $\zeta'_{0}=0$. Differentiating and using the fact that $\zeta_{0}=1$, we get $$\overline{\varphi_{0}'}=-\varphi'_{0} \,,$$ showing that $\varphi_{0}'$ is purely imaginary as claimed. To compute $h''_0$, we differentiate  twice with respect to $t$. At $t=0$ this gives $$\label{e:phi0prime} -\lap\varphi_{0}'-4\pi i\omega\cdot\nabla\varphi_{0}=\lambda_{0}\varphi_{0}',$$ and $$\label{e:phi0doublePrime} -\lap\varphi_{0}''-8\pi i\omega\cdot\nabla\varphi_{0}'+8\pi^{2}|\omega|^{2}\phi_{0}=h''_{0}\phi_{0}+\lambda_{0}\varphi_{0}'' \,,$$ since $\varphi_0 = \phi_0$. Multiplying both sides of by $\phi_{0}$ and integrating over $M$ gives $$\label{e:hDouplePrime0} h_{0}''=\int_{M}\paren[\big]{8\pi^{2}|\omega|^{2}\phi_{0}^{2}-8\pi i\phi_{0}\omega\cdot\nabla\varphi_{0}'} \,.$$ Recalling that $\varphi_0'$ is purely imaginary, we let $g_\omega$ be the real valued function defined by $g_\omega = -i \varphi'_0$. Now equation  shows that $g_\omega$ satisfies . Moreover since $\varphi_0 = 0$ on $\partial_D M$ and $\nu \cdot \grad \varphi_0 = 0$ on $\partial_N M$, the function $g_\omega$ satisfies the boundary conditions . Therefore,  reduces to , showing that $h''_0 = \mathcal I(\omega)$ as claimed. Finally, we show that $\omega\mapsto\mathcal{I}(\omega)$ defined by  is a well defined positive definite quadratic form on $\mathcal{H}^1_G$. To see that $\mathcal I$ is well defined, we first note that in order for  to have a solution, we need to verify the solvability condition $$\int_M \phi_0 \paren[\big]{ 4 \pi \omega \cdot \grad \phi_0 } = 0\,.$$ This is easily verified as $$\label{e:intphi0DotStuff} \int_{M}\phi_{0}\omega\cdot\nabla\phi_{0}=\frac{1}{2}\int_{M}\omega\cdot\nabla\phi_{0}^{2}=0 \,.$$ Hence $g_\omega$ is uniquely defined up to the addition of a scalar multiple of $\phi_0$ (the kernel of $\lap + \lambda_0$). Now, using  again, we see that replacing $g_\omega$ with $g_\omega + \alpha \phi_0$ does not change the value of $\mathcal I(\omega)$. Thus, $\mathcal I(\omega)$ is a well defined function. The fact that $\mathcal I$ is a quadratic form  and the fact that $$g_{\tau+\omega} = g_{\tau}+g_{\omega}\quad \pmod{\phi_{0}} \,.$$ It remains to show that $\mathcal I$ is positive definite. Note that, in view of Lemma \[l:minLambdaChi\], we already know that $\mathcal I$ induces a positive *semi*-definite quadratic form on $\mathcal H^1_G$. For the convenience of notation, let $g = g_\omega = -i \varphi_0'$ as above. As before we write $$g = (\phi_0 + \epsilon) f_\epsilon\,, \quad\text{where } f_\epsilon \defeq \frac{g}{\phi_0 + \epsilon}\,,$$ and will multiplying both sides of  by $(\phi_0 + \epsilon) f_\epsilon$ and integrating. In preparation for this we compute $$\begin{gathered} -\int_{M}(\phi_{0}+\epsilon)f_{\epsilon}\lap g =\int_{M}\nabla g\cdot\paren[\Big]{f_{\epsilon}\nabla\phi_{0}+(\phi_{0}+\epsilon)\nabla f_{\epsilon}} \\ =\lambda_{0}\int_{M}\phi_{0}f_{\epsilon}g-\int_{M}g\nabla f_{\epsilon}\cdot\nabla\phi_{0}+\int_{M}(\phi_{0}+\epsilon)\nabla f_{\epsilon}\cdot\nabla g \,, \end{gathered}$$ and $$\begin{aligned} 4\pi\int_{M}(\phi_{0}+\epsilon)f_{\epsilon}\omega\cdot\nabla(\phi_{0}+\epsilon) & =2\pi\int_{M}f_{\epsilon}\omega\cdot\nabla(\phi_{0}+\epsilon)^{2}\\ & =-2\pi\int_{M}(\phi_{0}+\epsilon)^{2}\nabla f_{\epsilon}\cdot\omega \,. \end{aligned}$$ We remark that when integrating by parts above, the boundary terms that arise all vanish because of the boundary conditions imposed. Thus, multiplying  by $(\phi_0 + \epsilon) f_\epsilon$ and integrating gives $$\begin{aligned} \nonumber \lambda_{0}\int_{M}g^{2}\paren[\Big]{1-\frac{\phi_{0}}{\phi_{0}+\epsilon}}&= \int_{M}(\phi_{0}+\epsilon)\nabla f_{\epsilon}\cdot\nabla g-\int_{M}g\nabla f_{\epsilon}\cdot\nabla(\phi_{0}+\epsilon) \\ \label{eq: after multiplying g on both sides} & \qquad \mathbin{+} 2\pi\int_{M}(\phi_{0}+\epsilon)^{2}\nabla f_{\epsilon}\cdot\omega \,. \end{aligned}$$ Writing $\tau\defeq 2\pi\omega$ and adding the integral $$\begin{aligned} J_{\epsilon} & \defeq\int_{M}(\phi_{0}+\epsilon)\tau\cdot\nabla g-\int_{M}g\tau\cdot\nabla(\phi_{0}+\epsilon) +\int_{M}(\phi_{0}+\epsilon)^{2}|\tau|^{2} \end{aligned}$$ to both sides of , we obtain $$\begin{gathered} \label{eq: after adding J_epsilon on both sides} J_{\epsilon}+\lambda_{0}\int_{M}g^{2}\paren[\Big]{1-\frac{\phi_{0}}{\phi_{0}+\epsilon}} = \int_{M}(\phi_{0}+\epsilon)(\nabla f_{\epsilon}+\tau)\cdot\nabla g \\ -\int_{M}g(\nabla f_{\epsilon}+\tau)\cdot\nabla(\phi_{0}+\epsilon) +\int_{M}(\phi_{0}+\epsilon)^{2}(\nabla f_{\epsilon}+\tau)\cdot\tau \,. \end{gathered}$$ Now, since $g=(\phi_{0}+\epsilon)f_{\epsilon}$, we compute $$\nabla g=f_{\epsilon}\nabla(\phi_{0}+\epsilon)+(\phi_{0}+\epsilon)\nabla f_{\epsilon} \,.$$ Substituting this into gives $$\label{e:positivityOfI1} J_{\epsilon}+\lambda_{0}\int_{M}g^{2}\paren[\Big]{1-\frac{\phi_{0}}{\phi_{0}+\epsilon}}=\int_{M}(\phi_{0}+\epsilon)^{2}|\nabla f_{\epsilon}+\tau|^{2}\geq0 \,.$$ Using  we see $$\mathcal{I}(\omega) = 8\pi^{2}\int_{M}|\omega|^{2}\phi_{0}^{2}+4\pi\int_{M}\phi_{0}\omega\cdot\nabla g -4\pi\int_{M}g\omega\cdot\nabla\phi_{0} \,,$$ and hence it follows that $$\lim_{\epsilon\to 0}J_{\epsilon}=\frac{1}{2}\mathcal{I}(\omega) \,.$$ Also by the dominated convergence theorem, the second term on the left hand side of goes to zero as $\epsilon\to 0$. This shows $\mathcal{I}(\omega)\geq0$. It remains to show $\mathcal I(\omega) > 0$ if $\omega \neq 0$. Note that if $\mathcal{I}(\omega)=0$, then Fatou’s lemma and  imply $$\int_{M}\phi_{0}^{2}|\nabla f+\tau|^{2}\leq\liminf_{\epsilon{\to}0}\paren[\Big]{J_{\epsilon}+\lambda_{0}\int_{M}g^{2}\paren[\Big]{1-\frac{\phi_{0}}{\phi_{0}+\epsilon}}}=0 \,,$$ where $f\defeq g/\phi_{0}$. Therefore $\grad f + \tau = 0$ in $M$ and hence $\omega = - \grad f / (2\pi)$. Since $\omega \in \mathcal H^1_G \subseteq \mathcal H^1$, this forces $$\lap f = 0 \quad\text{in }M\,, \qquad\text{and}\qquad \nu \cdot \grad f = 0 \quad\text{on } \partial M\,.$$ Consequently $\grad f = 0$, which in turn implies $\omega = 0$. This completes the proof of the positivity of $\mathcal{I}$. Proof of the Winding Number Asymptotics (Theorem \[t:winding\]). {#s:pfwinding} ================================================================ In this section, we study the long time behaviour of the abelianized winding number of reflected Brownian motion on a manifold $M$. We begin by using Theorem \[t:hker\] to prove Theorem \[t:winding\] (Section \[s:windingProof\]). Next, in Section \[s:tobyWerner\] we discuss the connection of our results with those obtained by Toby and Werner [@TobyWerner95]. Finally, in Section \[s:windingDomain\], we outline a direct probabilistic proof of Theorem \[t:winding\]. Proof of Theorem \[t:winding\] {#s:windingProof} ------------------------------ We obtain the long time behaviour of the abelianized winding of reflected Brownian motion in $M$ by applying Theorem \[t:hker\] in this context. Let $\hat{M}$ be a covering space of $M$ with deck transformation group[^4] $\pi_1(M)_{{\mathrm{ab}}}$. In view of the covering factorization , we may, without loss of generality, assume that $\operatorname{tor}(\pi_1(M)_{{\mathrm{ab}}})=\{0\}$. Note that since the deck transformation group $G = \pi_1(M)_{\mathrm{ab}}$ by construction, we have $\mathcal H^1_G = \mathcal H^1$. Given $n \in \Z^k$ ($k=\rm{rank}(G)$), define $g_n \in G$ by $$g_n \defeq \sum_{i=1}^k n_i \pi_G(\gamma_i)\,, \quad\text{where } n = (n_1, \dots, n_k) \in \Z^k\,.$$ Here $(\pi_G(\gamma_1), \dots, \pi_G(\gamma_k))$ is the basis of $G$ chosen in Section \[s:winding\]. Clearly $n \mapsto g_n$ is an isomorphism between $G$ and $\Z^k$. \[l:dIgn\] For any $x, y \in \hat M$ and $n \in \Z^k$ we have $$d_\mathcal I(x, g_n(y) )^2 = (A^{-1}n) \cdot n + O(\abs{n})\,.$$ Here $A$ is the matrix $(a_{i,j})$ defined by $$\label{e:cov} a_{i,j} \defeq \ip{\omega_i, \omega_j}_\mathcal I = \frac{8 \pi^2}{\operatorname{vol}(M)} \int_M \omega_i \cdot \omega_j \,.$$ Given $\omega \in \mathcal H^1$ we compute $$\label{e:xix:gy} \xi_{x, g_n (y)}(\omega) = \int_{x}^y \bpi^*(\omega) + \int_y^{g_n(y)} \bpi^*(\omega) \,,$$ where the integrals are performed along any smooth path in $\hat M$ connecting the endpoints. By construction of $\hat M$, $\mathcal H^1_G = \mathcal H^1$, and hence both integrals above are independent of the path of integration. Moreover, the second integral is independent of $y$. Hence, if for any $g \in G$ we define $\psi_g\colon \mathcal H^1 \to \R$ by $$\psi_g(\omega) = \int_y^{g(y)} \bpi^*(\omega) \,,$$ then  becomes $$\xi_{x, g_n(y)} (\omega) = \xi_{x,y}(\omega) + \psi_{g_n}(\omega)\,.$$ From this we compute $$d_\mathcal I(x,g_n(y))^2 = d_\mathcal I(x,y)^2 + \sum_{i=1}^k n_i \ip{\psi_{\pi_G(\gamma_i)}, \xi_{x,y} }_{\mathcal I^*} + \sum_{i,j=1}^k n_i n_i \ip{\pi_G(\gamma_i), \pi_G(\gamma_j)}_{\mathcal I^*}\,.$$ Since $(\omega_1, \dots, \omega_k)$ is the dual basis to $(\pi_G(\gamma_1), \dots, \pi_G(\gamma_j))$, we have $$\ip{\pi_G(\gamma_i), \pi_G(\gamma_j)}_{\mathcal I^*} = (A^{-1})_{i,j}\,,$$ from which the first equality in  follows. The second equality follows from the fact that  holds under Neumann boundary conditions (Remark \[r:neumann\]). Now we prove Theorem \[t:winding\]. Recall in Section \[s:winding\] we decomposed the universal cover $\bar M$ as the disjoint union of fundamental domains $\bar U_g$ indexed by $g \in \pi_1(M)$. Projecting these domains to the cover $\hat M$ we write $\hat M$ as the disjoint union of fundamental domains $\bar U_g$ indexed by $g \in G$. Let $\hat W$ be the lift of the trajectory of $W$ to $\hat M$, and observe that if $\hat W(t) \in \hat U_{g_n}$, then $\rho(t) = n$. We use this to compute the characteristic function of $\rho(t) / \sqrt{t}$ as follows. Since the generator of $\hat W$ is $\frac{1}{2} \lap$, its transition density is given by $\hat H(t/2, \cdot, \cdot)$. Hence, for any $z \in \R^k$ we have $$\begin{gathered} \E^x \exp\paren[\Big]{ \frac{i z \cdot \rho(t)}{t^{1/2}} } = \sum_{n \in \Z^k} \exp\paren[\Big]{ \frac{i z \cdot n}{t^{1/2}} } \P^x( \hat W(t) \in \hat U_{g_n} ) \\ = \sum_{n \in \Z^k} \int_{\hat U_{g_n}} \hat H\paren[\Big]{ \frac{t}{2}, x, y } \exp\paren[\Big]{ \frac{i z \cdot n}{ t^{1/2} } } \, dy\,. \end{gathered}$$ By Theorem \[t:hker\] and Remark \[r:neumann\], this means that uniformly in $x \in \hat M$ we have $$\begin{aligned} \MoveEqLeft \lim_{t\to \infty} \E^x \exp\paren[\Big]{ \frac{i z \cdot \rho(t)}{t^{1/2}} } \\ &= C_\mathcal I \lim_{t\to \infty} \sum_{n \in \Z^k} \int_{\hat U_{g_n}} \frac{2^{k/2}}{t^{k/2}} \exp\paren[\Big]{ -\frac{4\pi^2 d_\mathcal I(x, g_n(y))^2 }{t} + \frac{i z \cdot n}{ t^{1/2} } } \, dy \\ &= C_\mathcal I \lim_{t\to \infty} \sum_{n \in \Z^k} \frac{2^{k/2}}{t^{k/2}} \exp\paren[\Big]{ -\frac{4\pi^2 (A^{-1} n) \cdot n}{t} + \frac{i z \cdot n}{ t^{1/2} } }\,. \end{aligned}$$ Here the last equality followed from Lemma \[l:dIgn\] above. Now the last term is the Riemann sum of a standard Gaussian integral, and hence $$\lim_{t\to \infty} \E^x \exp\paren[\Big]{ \frac{i z \cdot \rho(t)}{t^{1/2}} } = 2^{k/2} C_\mathcal I \int_{\zeta \in \R^k} \exp\paren[\Big]{ -4\pi^2 (A^{-1} \zeta) \cdot \zeta + i z \cdot \zeta } \, d\zeta\,.$$ This shows that as $t \to \infty$, $\rho(t) / \sqrt{t}$ converges to a normally distributed random variable with mean $0$ and covariance matrix $A / (8 \pi^2)$. By  and  we see that $\Sigma = A / (8 \pi^2 )$, which completes the proof of the second assertion in . The first assertion follows immediately from the second assertion and Chebychev’s inequality. This completes the proof of Theorem \[t:winding\]. Relation to the Work of Toby and Werner {#s:tobyWerner} --------------------------------------- Toby and Werner [@TobyWerner95] studied the long time behaviour of the winding of an obliquely reflected Brownian motion in bounded planar domains. In this case, we describe their result and relate it to Theorem \[t:winding\]. Let $\Omega \subseteq \R^2$ be a bounded domain with $k$ holes $V_1,\cdots,V_k$ of positive volume. Let $W_{t}$ be a reflected Brownian motion in $\Omega$ with a non-tangential reflecting vector field $u \in C^1(\partial \Omega)$. Let $p_{1},\cdots,p_{k}$ be $k$ distinct points in $\R^{2}$. For $1\leq j\leq k$, define $\rho(t,p_{j})$ to be the winding number of $W_{t}$ with respect to the point $p_{j}$. \[t:tobyWerner\] There exist constants $a_i$, $b_i$, depending on the domain $\Omega$, such that $$\label{e:TWrho} \frac{1}{t}\paren[\big]{\rho(t,p_{1}),\cdots,\rho(t,p_{k})} \xrightarrow[t \to \infty]{w} \paren[\big]{a_{1}C_{1}+b_{1},\cdots,a_{k}C_{k}+b_{k}}\,.$$ Here $C_{1}$, …, $C_{k}$ are standard Cauchy variables. Moreover, for any $j$ such that $p_{j}\notin \Omega$, we must have $a_{j}=0$. When $p_{j}\in \Omega$, the process $W$ can wind many times around $p_j$ when it gets close to $p_j$. This is why the heavy tailed Cauchy distribution arises in Theorem \[t:tobyWerner\], and the limiting process is non-degenerate precisely when each $p_j \in \Omega$. In the context of Theorem \[t:winding\] we require compactness of the domain. This will only be true when when $p_j \not \in \Omega$ for all $j$, in which case each $a_j = 0$. We now describe how the constants $b_{j}$ are computed in [@TobyWerner95]. Recall (see for instance Stroock-Varadhan [@StroockVaradhan71]) that reflected Brownian motion has the semi-martingale representation $$\label{e:smgRepBM} W_{t}=\beta_{t}+\int_{0}^{t}u(W_{s}) \, dL_{s}\,.$$ Here $\beta_{t}$ is a two dimensional Brownian motion, $u$ is the reflecting vector field on $\partial \Omega$, and $L_{t}$ is a continuous increasing process which increases only when $W_{t}\in\partial \Omega$. We also know that the process $W_{t}$ has a unique invariant measure, which we denote by $\mu$. Now, the constants $b_{j}$ are given by $$\label{e:TWb} b_{j} = \frac{1}{2\pi}\int_{p \in \Omega} \E^{p}\brak[\Big]{\int_{0}^{1}u_{j}(W_{s})dL_{s}} \, d\mu(p) \,,$$ where $u_{j} \colon \partial \Omega \to \R$ is defined by $$u_{j}(p)\defeq\frac{u(p) \cdot (p-p_{j})^\perp }{\abs{p-p_{j}}} \,.$$ Above the notation $\perp$ denotes the rotation of a point counter clockwise by an angle of $\pi/2$ about the origin. That is, if $q = (q_1, q_2) \in \R^2$, then $q^\perp = (-q_2, q_1)$. In the case that the reflection is normal, we claim that each $b_j = 0$. \[p:TWb\] Suppose $W_{t}$ is the normally reflected Brownian motion in $\Omega$, and $p_{j}\in V_{j}$ for each $j$. Then $b_{j}=0$ for all $j$, and consequently $$\lim_{t\rightarrow\infty}\frac{\rho(t, p_{j})}{t} \xrightarrow[t \to \infty]{p} 0\,.$$ Note that Proposition \[p:TWb\] is simply the first assertion in , and follows trivially from the second assertion (the central limit theorem). For completeness, we provide an independent proof of Proposition \[p:TWb\] directly using . Fix $1\leq j\leq k$. Let $w(t,p)$ be the solution to the following initial-boundary value problem: $$\label{e:w} \left\{ \begin{alignedat}{2} \span \partial_t w -\frac{1}{2}\lap w = 0 &\qquad& \text{in } (0,\infty)\times \Omega\,, \\ \span \nu \cdot \grad w =-u_{j} && \text{on } (0,\infty)\times\partial \Omega\,, \\ \span \lim_{t{\to}0}w(t,\cdot)=0 && \text{in}\ \Omega \,, \end{alignedat} \right.$$ where $\nu$ is the outward pointing unit normal on the boundary. By applying Itô’s formula to the process $[0,t-\epsilon]\ni s\mapsto w(t-s,W_{s})$ and using the semi-martingale representation of $W_{t}$, we get $$\begin{aligned} w(t,p)-\E^{p}\brak[\Big]{w(\epsilon,W_{t-\epsilon})} & =-\E^{p}\brak[\Big]{\int_{0}^{t-\epsilon} \nu \cdot \grad w (W_{s},t-s)dL_{s}}\\ & =\E^{p}\brak[\Big]{\int_{0}^{t-\epsilon}u_{j}(W_{s})dL_{s}},\end{aligned}$$ where in the last identity we have used the fact that $dL_{s}$ is carried by the set $\{s\geq0:W_{s}\in\partial \Omega\}$. Since $\P(B_{t}\in\partial U)=0$, sending $\epsilon{\to}0$ and using the dominated convergence theorem gives $$w(t,p)=\E^{p}\brak[\Big]{\int_{0}^{t}u_{j}(W_{s})dL_{s}} \,.$$ On the other hand, according to Harrison, Landau and Shepp [@HarrisonLandauEA85], Theorem 2.8, the invariant measure $\mu$ of $W_{t}$ is the unique probability measure on the closure $\bar{\Omega}$ of $\Omega$ that $\mu(\partial \Omega)=0$ and $$\int_{\Omega}\lap f(p) \, d\mu(p) \leq 0 \quad \text{for all $f\in C^{2}(\bar{\Omega})$ with $\nu \cdot \grad f \leq0$ on $\partial \Omega$.}$$ Stokes’ theorem now implies $\mu$ is the normalized Lebesgue measure on $\Omega$. Consequently, $$b_{j} =\frac{1}{2\pi\operatorname{vol}(\Omega)}\int_{\Omega}\E^{p}\brak[\Big]{\int_{0}^{1}u_{j}(W_{s}) \, dL_{s}} \, dp =\frac{1}{2\pi\operatorname{vol}(\Omega)}\int_{\Omega}w(1,p) \, dp \,.$$ Integrating  over $\Omega$ and using the boundary conditions yields $$\begin{aligned} 0 & = \partial_t \int_{\Omega}w \, dp-\int_{\Omega}\lap w \, dp\\ & =\partial_t \int_{\Omega}w \, dp+\int_{\partial \Omega}u_{j}(p) \, dp\\ & =\partial_t \int_{\Omega}w \, dp-\int_{\partial \Omega} \nu \cdot \frac{(p-p_{j})^\perp}{|p-p_{j}|} \, dp \,.\end{aligned}$$ Since when $p_{j}\in V_{j}$ the vector field $p\mapsto{(p-p_{j})^\perp / |p-p_{j}|}$ is a divergence free vector field on $\bar{\Omega}$, the last integral above above vanishes. Thus $$\partial_t \int_{\Omega}w \, dp=0\,,$$ and since $w=0$ when $t=0$, $w = 0$ for all $t \geq 0$, and hence $b_{j}=0$. Therefore, in the case with normal reflection and $p_j\in V_j$, the result of Toby and Werner becomes a law of large numbers and Theorem \[t:winding\] provides a central limit theorem. In this case, our result is a refinement of Theorem \[t:tobyWerner\]. The setting of Toby and Werner [@TobyWerner95] is more general. Namely they study obliquely reflected Brownian motion, and the case of punctured domains (i.e. when $z_{j}\in \Omega$) where the limiting behavior is the (heavy tailed) Cauchy distribution. A Direct Probabilistic Proof of Theorem \[t:winding\] {#s:windingDomain} ----------------------------------------------------- As mentioned earlier, Theorem \[t:winding\] can also be proved directly by using a probabilistic argument. The proof is particularly simple in the case of Euclidean domains with smooth boundary. On manifolds, however, there are a few details that need to be verified. While these are direct generalizations of their Euclidean counterparts, to our best knowledge, they are not readily available in the literature. First suppose $\gamma\colon [0, \infty) \to M$ is a smooth path. Let $\rho(t, \gamma)$ be the $\Z^k$-valued winding number of $\gamma$, as in Definition \[d:winding\]. Namely, let $\bar \gamma$ be the lift of $\gamma$ to the universal cover of $M$, and let $\rho(t, \gamma) = (n_1, \dots, n_k)$ if $$\pi_G\paren[\big]{ \bar{\bm{g}}(\bar \gamma(t))} = \sum_{i=1}^k n_i \pi_G(\gamma_i) \,.$$ By our choice of $(\omega_1, \dots, \omega_k)$ we see that $\rho_i(t, \gamma)$, the $i^\text{th}$ component of $\rho(t, \gamma)$, is precisely the integer part of $\theta_i(t, \gamma)$, where $$\label{e:gamma} \theta_i(t, \gamma) \defeq \int_{\gamma([0, t])} \omega_i = \int_0^t \omega_i(\gamma(s)) \, \gamma'(s) \, ds\,.$$ If $M$ is a planar domain with $k$ holes, and the forms $\omega_i$ are chosen as in Remark \[r:planar\], then $2 \pi \theta_i(t, \gamma)$ is the total angle $\gamma$ winds around the $k^\text{th}$ hole up to time $t$. In the case that $\gamma$ is not smooth, the theory of rough paths can be used to give meaning to the above path integrals. Moreover, when $\gamma$ is the trajectory of a reflected Brownian motion on $M$, we know that the integral obtained via the theory of rough paths agrees with the Stratonovich integral. To fix notation, let $W$ be a reflected Brownian motion in $M$, and $\rho(t)=(\rho_{1}(t),\cdots,\rho_{k}(t))$ to be the $\mathbb{Z}^{k}$-valued winding number of $W$ as in Definition \[d:winding\]. Then we must have $\rho_{i}(t)=\lfloor\theta_{i}(t)\rfloor$, where $\theta_{i}(t)$ is the Stratonovich integral $$\label{e:thetaiStrat} \theta_{i}(t)=\int_{0}^{t}\omega_{i}(W_{s})\circ dW_{s} \,.$$ In Euclidean domains, the long time behaviour of this integral can be obtained as follows. The key point to note is that the forms $\omega_i$ are chosen to be harmonic in $M$ and tangential on $\partial M$. Consequently, using the semi-martingale decomposition , we see that $\theta$ is a martingale with quadratic variation given by $$\label{e:qvTheta} \langle\theta_{i},\theta_{j}\rangle_{t}=\int_{0}^{t}\omega_{i}(W_{s})\cdot\omega_{j}(W_{s}) \, ds \, .$$ Moreover, by Harrison et. al. [@HarrisonLandauEA85], the unique invariant measure of $W_{t}$ is the normalized volume measure. Thus, by the ergodic theorem, $$\lim_{t\rightarrow\infty}\frac{1}{t}\langle\theta_{i},\theta_{j}\rangle_{t}=\frac{1}{{\rm vol}(M)}\int_{M}\omega_{i}\cdot\omega_{j}$$ for almost surely. Now, the martingale central limit theorem [@PavliotisStuart08 Theorem 3.33 and Corollary 3.34], implies conclude that $$\frac{\theta_t}{\sqrt{t}} \xrightarrow[w]{t \to \infty} \mathcal N(0, \Sigma)\,,$$ where the covariance matrix $\Sigma$ is given by . In order for the above argument to work on compact Riemannian manifolds, one needs to establish a few of the results used above in this setting. First, one needs to show the analogue of the semi-martingale decomposition  on manifolds with boundary. While this is a straightforward adaptation of [@StroockVaradhan71], there is (to the best of our knowledge) no easily available reference. Next, one needs to use the fact $\omega \in \mathcal H^1$ to show that $\theta_i$ is a martingale with quadratic variation . This can be done by breaking the Stratonovich integral defining $\theta_i$ (equation ) into pieces that are entirely contained in local coordinate charts, and using the analogue of  together with the fact that $\omega \in \mathcal H^1$. Now the rest of the proof is the same as in the case of Euclidean domains. Acknowledgements. {#acknowledgements. .unnumbered} ================= The authors wish to thank Jean-Luc Thiffeault for suggesting this problem to us and many helpful discussions. [^1]: This material is based upon work partially supported by the National Science Foundation under grant DMS-1252912 to GI, and the Center for Nonlinear Analysis. [^2]: The isomorphism between $\mathcal{H}^1_G$ and $\hom(G; \R)$, the dual of the deck transformation group $G$, can be described as follows. Given $g \in G$, pick a base point $p_0 \in M$, and a pre-image $x_0 \in \bpi^{-1}(p_0)$. Now define $$\varphi_\omega(g) = \int_{x_0}^{g(x_0)} \bpi^*(\omega)\,,$$ where the integral is done over any path connecting $x_0$ and $g(x_0)$. By definition of $\mathcal H^1_G$, the above integral is independent of the chosen path. Moreover, since $\bpi^*(\omega)$ is the pull-back of $\omega$ by the covering projection, it follows that $\varphi_\omega(g)$ is independent of the choice of $p_0$ or $x_0$. Thus $\omega \mapsto \varphi_\omega$ gives a canonical homomorphism between $\mathcal H^1_G$ and $\hom(G, \R)$. The fact that this is an isomorphism follows from the transitivity of the action of $G$ on fibers. [^3]: Note, since $\lambda_0$ manifestly belongs to the spectrum of $-\lap$, the function $g_\omega$ is not unique. Moreover, one has to verify a solvability condition to ensure that solutions to equation  exist. We do this in Lemma \[l:muBound\], which is proved in Section \[s:mubound\], below. [^4]: The existence of such a cover is easily established by taking the quotient of the universal cover $\bar{M}$ by the action of the commutator of $\pi_1(M)$.

ArXiv

--- author: - | Md Sarowar Morshed and Md Noor-E-Alam\ [Department of Mechanical & Industrial Engineering]{}\ [Northeastern University ]{}\ [360 Huntington Avenue, Boston, MA 02115, USA]{}\ [Email : [email protected]]{} bibliography: - 'aafs.bib' title: Generalized Affine Scaling Algorithms for Linear Programming Problems --- Abstract {#abstract .unnumbered} ======== Interior Point Methods are widely used to solve Linear Programming problems. In this work, we present two primal Affine Scaling algorithms to achieve faster convergence in solving Linear Programming problems. In the first algorithm, we integrate Nesterov’s restarting strategy in the primal Affine Scaling method with an extra parameter, which in turn generalizes the original primal Affine Scaling method. We provide the proof of convergence for the proposed generalized algorithm considering long step size. We also provide the proof of convergence for the primal and dual sequence without the degeneracy assumption. This convergence result generalizes the original convergence result for the Affine Scaling methods and it gives us hints about the existence of a new family of methods. Then, we introduce a second algorithm to accelerate the convergence rate of the generalized algorithm by integrating a non-linear series transformation technique. Our numerical results show that the proposed algorithms outperform the original primal Affine Scaling method. ***Key words:*** Linear Programming, Affine Scaling, Nesterov Acceleration, Dikin Process, Shanks Series Transformation Introduction {#sec:into} ============ The *Affine Scaling* (AFS) algorithm was introduced by Dikin [@dikin:1967], which remained unnoticed to the *Operations Research* (OR) community until the seminal work of Karmarkar [@karmarkar:1984]. Karmarkar’s work transformed the research in *Interior Point Methods* (IPMs) and induced a significant development in the theory of IPMs. As a result, several variants of AFS have been studied over the years by researchers (see [@jansen:1996], [@barnes:1986]). We refer to the books of Wright [@wright:1997], Ye [@ye:1997], Bertsimas [@bertsimas:1997] and Vanderbei [@vanderbei:1998] for more comprehensive discussion of these methods. Apart from the simplicity, convergence analysis of the AFS methods for generalized degenerate setup are considered difficult to analyze. Dikin first published a convergence proof with a non-degeneracy assumption in 1974 [@vanderbei:1990]. Both Vanderbei *et al.* [@vanderbei:1986] and Barnes [@barnes:1986] gave simpler proofs in their global convergence analysis but still assumed primal and dual non-degeneracy. First attempt to break out of the non-degeneracy assumption was made by Adler *et al.* [@adler:1991], who investigated the convergence of continuous trajectories of primal and dual AFS. Subsequently, assuming only dual non-degeneracy, Tsuchiya [@tsuchiya:19921] showed that under the condition of step size $\alpha < \frac{1}{8}$, the long-step version of AFS converges globally. In another work, Tsuchiya [@tsuchiya:1991] showed that the dual non-degeneracy condition is not a necessary condition for the convergence as assumed previously [@tsuchiya:19921]. Moreover, Tsuchiya [@tsuchiya:1991] introduced the idea of potential function, a slightly different function than the one provided by Karmarkar [@karmarkar:1984], for the analysis of the local behavior of the AFS near the boundary of the feasible region. Finally, using that potential function [@tsuchiya:1991], Dikin [@dikin:1991] and Tsuchiya *et al.* [@tsuchiya:1992] provided proofs for the global convergence of degenerate *Linear Programming* (LP) problems with $\alpha \footnote{$\alpha$ is the step size} < \frac{1}{2}$ and $\alpha \leqslant \frac{2}{3}$, respectively. Later, Hall *et al.* [@hall:1993] showed that the sequence of dual estimates will not always converge for $\alpha > \frac{2}{3}$. As a self-contained paper for a global convergence analysis for AFS, Monteiro *et al.* [@monteiro:1993] and Saigal [@saigal:1996] provided two simple proofs for the long-step AFS algorithms of degenerate LP’s. Subsequently, Saigal [@saigal:1997] introduced two step predictor corrector based methods to fasten the convergence of AFS. Besides, the chaotic analysis of AFS was first addressed by Castillo *et al.* [@barnes:2001]. Bruin *et al.* [@ross:2014] provided a proper chaotic explanation of the so called *Dikin Process* by showing the similarity of it with the logistic family in terms of chaotic behavior. In their work, they showed why the AFS algorithms behave differently when the step size $\alpha$ is close to $\frac{2}{3}$, which in general complies with the chaotic behavior of IPMs analyzed by several other researchers. There has been a significant development in applying the AFS techniques to solve various types of optimization problems: Semi Definite Programming [@vanderbei:1999], Nonlinear Smooth Programming [@wang:2009], Linear Convex Programming [@cunha:2011], Support Vector Machine [@maria:2011], Linear Box Constrained Optimization [@wang:2014], Nonlinear Box Constrained Optimization [@huang:2017]. Recently, Kannan *et al.* [@kannan:2012] applied the idea of AFS algorithm to generate a random walk for solving LP problems approximately. In a seminal work, Nesterov [@nesterov:1983] proposed an acceleration technique for the *Gradient Descent* that exhibits the worst-case convergence rate of $O(\frac{1}{k^2})$ for minimizing smooth convex functions compared to the original convergence rate of $O(\frac{1}{k})$. Since the inception of Nesterov’s work, there has been a body of work done on the theoretical development of first-order accelerated methods (for a detailed discussion see [@nesterov:2005], [@nesterov:2013] and [@nesterov:2014]). Furthermore, an unified summary of all of the methods of Nesterov can be found in [@tseng:2010]. Recently, Su *et al.* [@su:2016] carried out a theoretical analysis on the methods of Nesterov and showed that it can be interpreted as a finite difference approximation of a second-order *Ordinary Differential Equation* (ODE). **Motivation & Contribution** {#subsec:motiv} ----------------------------- We have seen from the literature that Nesterov’s restarting scheme is very successful in achieving faster convergence for *Gradient Descent* algorithms. However, to the best of our knowledge, the potential opportunity of Nesterov’s acceleration has not been yet explored to IPMs to solve LP problems. Motivating by the power of acceleration and to fill the research gap, as a first attempt, in this work we apply acceleration techniques in the Affine Scaling method. Affine Scaling method was chosen for the following three reasons: - *Historical Importance:* Affine Scaling method was the first discovered Interior Point algorithm discovered by Dikin in 1967 [@dikin:1967]. In 1984 Karmarkar introduced potential reduction method (Karmarkar method) based on Dikin’s original idea [@karmarkar:1984]. After that a significant amount of research was done in the field of IPM algorithms. Dikin’s original Affine Scaling method worked as a stepping stone for the development of further IPM algorithms for solving LP problems. - *Algorithmic Simplicity:* Affine Scaling method uses the simplest primal update whereas Path following/Barrier method and Karmarkar method use penalty function driven update formulas [@bertsimas:1997] (i.e., both Barrier method and Karmarkar method have different objectives in the EAP problem shown in equation \[eq:n\] of Section \[sec:conc\]). Note that in the EAP problem (equation and ) Affine Scaling method uses the objective function $c^Td$, whereas Karmarkar method uses the objective $ G(x+d,s)$ and Barrier method uses the objective $ B_\mu (x+d)$ ( where $ G(x,s)$ and $ B_\mu (x)$ are the potential function and barrier function respectively, see Section \[sec:conc\] for details). - *Generalized Dikin Process:* The chaotic behavior of Affine Scaling method can be explained by the so called *Dikin Process* [@ross:2014], which has a similarity to the logistic map in *Chaos Theory*. From a *Chaos Theory* perspective, our proposed algorithms may reveal a more generalized versions of *Dikin Process*, which can be represented as dynamic systems ([@adler:1991], [@barnes:2001], [@ross:2014]). Based on the above discussion in this work, we propose two algorithms: 1) Generalized AFS and 2) Accelerated AFS (see Section \[sec:afs\]). In the Generalized AFS algorithm, we propose to use the acceleration scheme discovered by Nesterov [@nesterov:1983; @nesterov:2013; @nesterov:2014; @nesterov:2005; @nesterov:2012] in the AFS framework to achieve super linear convergence rate as compared to the linear rate shown in [@Tsuchiya1996]. Note that, Nesterov’s accelerated scheme can be incorporated with Affine Scaling scheme by defining two new sequences $\{u_k\}$ and $\{v_k\} \ \footnote{Representation of this update formula is different than Algorithm 1 but they implies the same steps}$ as following: $$\begin{aligned} \label{eq:nes} & u_k = \alpha_k v_k + (1-\alpha_k) x_k \quad U_k = \textbf{diag} \left[(u_k)_1, (u_k)_2, ... (u_k)_n\right] \nonumber \\ & \bar{y_k} = \left(AU_k^2A^T\right)^{-1}AU_k^2c, \ \ \bar{s_k} = c- A^T\bar{y_k} \nonumber\\ & x_{k+1} = u_k - \theta_k \frac{U_k^2\bar{s_k}}{\|U_k\bar{s_k}\|}\\ & v_{k+1} = \beta_k v_k + (1- \beta_k) u_k - \gamma_k \frac{U_k^2\bar{s_k}}{\|U_k\bar{s_k}\|} \nonumber\end{aligned}$$ In equation (\[eq:nes\]), instead of using $\nabla f$ as in standard *Gradient Descent* we use $\frac{U_k^2\bar{s_k}}{\|U_k\bar{s_k}\|}$ and $\theta_k$ is the step-size. The main contribution for the above scheme is that it uses appropriate values for the parameters $\alpha_k, \beta_k$ and $\gamma_k$ [^1], which in turn yield better convergence in the context of standard Affine Scaling methods. Surprisingly, the generalized AFS algorithm also follows the same types of convergence scheme as the original AFS method (i.e., feasible step size for convergence is $\alpha \leq \frac{2}{3}$ for AFS, generalized AFS shows the same types of convergence $\alpha + \beta \leq \frac{2}{3}$), which leads to a more generalized method. We then provide a generalized proof of convergence of the proposed generalized AFS under sufficient conditions. To gain further acceleration, in the generalized AFS algorithm, we propose to exploit the entry-wise *Shanks Series Transformation* (SST) to the generalized update of the generalized AFS algorithm. We then carry out rigorous convergence analysis to provide guarantee for acceleration and show the effectiveness of the proposed algorithms through numerical experiments. This is the first time, Nesterov’s momentum method and SST are used to design better algorithms in the context of general IPMs. We believe our proposed acceleration scheme will facilitate the application of such acceleration techniques in Barrier method/Path following method and Karmarkar method. This scheme will also serve as a foundation for developing accelerated techniques for other more efficient but complex methods in the IPM family. In terms of theoretical contribution, our proposed algorithms reveal interesting properties about the convergence characteristics of Affine Scaling method. Based on our analysis, it is evident that the convergence criterion for AFS ’$\alpha \leq 2/3$’ is a universal bound as the proposed algorithms satisfy a more generalized bound ’$\alpha + \beta \leq 2/3$’. Finally, the proposed algorithms suggest availability of a more general family of numerically efficient and theoretically interesting AFS methods. The paper is organized as follows. In Section 2, we provide a preliminary idea of the original AFS algorithm, then we describe the proposed variant AFS algorithms. In Section 3, we show the convergence of primal sequence for the proposed algorithms. In section 4, we exploit the convergence rate of the accelerated AFS algorithm. In section 5, we present convergence of the dual sequence under sufficient conditions for both algorithms. In section 6, we present numerical results to demonstrate the effectiveness of the proposed algorithms. Finally, in Section 7, we conclude the paper with future research directions. Generalization of AFS {#sec:afs} ===================== Affine Scaling method uses the simple idea of reformulation, instead of minimizing over the whole interior, it generates a series of ellipsoids inside the interior of feasible region and moves according to *min.* cost direction. Consider the following standard LP and its dual, $$\begin{aligned} \label{eq:1} \textbf{(P):} \quad \min \ \ & c^Tx & \textbf{(D):} \quad \max \ \ & y^Tb \nonumber\\ & Ax= b & & A^Ty +s = c \\ & x \geqslant 0 & & s \geqslant 0 \nonumber \end{aligned}$$ Let $P = \left\{x \ | \ Ax = b, x \geqslant 0\right\}$ be the primal feasible set, then we call the set $\left\{x \in P \ | \ x > 0\right\}$ as the interior of $P$ and its elements as interior points. The basic idea of Affine Scaling method is that instead of minimizing over $P$, we solve a series of optimization problems over ellipsoids. Starting from an initial strictly feasible solution $x_0 >0$, we form an ellipsoid $S_0$ centered at $x_0$, which is contained in the interior of $P$. Then by minimizing $c^Tx$ over all $x \in S_0$, we find a new interior point $x_1$ and proceed in a similar way until stopping restrictions are satisfied. The AFS method can be easily formulated as the following problem: given a strictly feasible solution $x \in {\mathbb{R}}^n$, we need to find direction vector $d$ such that $\bar{x} = x+ \alpha d$ for some $\alpha \in (0,1)$ [^2] and it holds $\bar{x} \in P, \ c^T\bar{x} \leqslant c^Tx$. To integrate acceleration and generalization in the AFS, we have proposed the following two algorithms which are variant of original AFS algorithm: 1. **Generalized AFS algorithm (**GAFS**)** 2. **Accelerated AFS algorithm (**AAFS**)** In **GAFS**, we propose to use Nesterov’s restarting strategy with the original AFS to generalize AFS. To facilitate the convergence process of **GAFS**, we propose to use entry-wise *Shanks series transformation* (SST) introduced by Shanks [@shanks:1965] to **GAFS**. This integrated algorithm is referred as **AAFS** in this work. We explain details about these two algorithms below: **GAFS:** We followed the Nesterov’s restarting strategy and introduced a generalized version of AFS in a way that it will give us the original AFS in the absence of the other parameter. For doing so, we integrate an extra term from the original idea of Nesterov [@nesterov:1983]. Here this variant of AFS is refereed as GAFS. In the GAFS, we consider two strict feasible points $x,z$ with $ \ c^Tx < c^Tz $, instead of one point $x$ to find a direction vector $d \in {\mathbb{R}}^n$ such that $\bar{x} = x+ \alpha d+ \bar{\beta} (x-z)$ for some $\alpha, \beta \in (0,1), \ \bar{\beta} = \frac{\beta}{\|1-X^{-1}z\|_{\infty}}, \ X = \text{diag}(x) $, where $\beta$ is the generalization parameter. It allowed us to reformulate the problem as below: $$\begin{aligned} \label{eq:2} \textbf{min} \quad & w = c^Td \nonumber\\ \textbf{s.t} \ & Ad =0 \\ & \| X^{-1}d \| \leqslant \alpha \nonumber\end{aligned}$$ The above problem is known as *Ellipsoidal Approximating Problem* (EAP), see [@saigal:1996] and [@tsuchiya:1995] for more detailed information. Start with $A,\ b,\ c,\ \epsilon > 0, \ \alpha ,\beta \in (0,1), \ x_0 > 0, \ k =0$ $$\begin{aligned} & X_k = \textbf{diag} \left[(x_k)_1, (x_k)_2, ... (x_k)_n\right] \\ & y_k = \left(AX_k^2A^T\right)^{-1}AX_k^2c, \ \ s_k = c- A^Ty_k \\ & z_k = \begin{cases} \ x_k, & k =0\\ \ x_k + \beta \frac{\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}, & k > 0 \end{cases}\end{aligned}$$ **Stop**; the current $(x_k, y_k)$ are primal and dual $\epsilon$ optimal. **Stop**; the problem is unbounded $$\begin{aligned} x_{k+1} = z_k - \alpha \frac{X_k^2s_k}{\|X_ks_k\|}\end{aligned}$$ Start with $A,\ b,\ c,\ \epsilon > 0, \ \alpha \in (0,1), \ x_0 > 0, \ k =0$ $$\begin{aligned} & X_k = \textbf{diag} \left[(x_k)_1, (x_k)_2, ... (x_k)_n\right] \\ & y_k = \left(AX_k^2A^T\right)^{-1}AX_k^2c, \ \ s_k = c- A^Ty_k \\ & z_k = \begin{cases} \ x_k, & k =0\\ \ x_k + \beta \frac{\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}, & k > 0 \end{cases} \\ & (B(x_k))_j = \begin{cases} \ (x_k)_j, & k = 0, 1 \ j = 1,2,...n \\ \ (x_k)_j - \frac{((x_k)_j-(x_{k+1})_j)^2}{(x_k)_j-2(x_{k+1})_j+(x_{k+2})_j} , & k > 1, \ j = 1,2,...n \end{cases} \\ & B_k = \textbf{diag} \left[B(x_k)_1, B(x_k)_2, ... B(x_k)_n\right]\end{aligned}$$ **Stop**; the current $(B(x_k), y_k)$ are primal and dual $\epsilon$ optimal. **Stop**; the problem is unbounded $$\begin{aligned} x_{k+1} = z_k - \alpha \frac{X_k^2s_k}{\|X_ks_k\|}\end{aligned}$$ Since the generalization parameters do not affect the EAP problem, the main properties discussed in [@dikin:1991], [@saigal:1996] and [@tsuchiya:1995] are valid for (see \[appendix-sec2\] for more details). From now on, we denote $\delta(x_k) = x_k-x_{k-1}$. For all $ k \geqslant 0$, we constructed the sequence $z_k$ using the following update with $\alpha, \beta \in (0,1)$ and a strictly feasible point $x_0 > 0$ : $$\begin{aligned} \label{eq:3a} z_k = \begin{cases} \ x_k, & k =0 \\ \ x_k + \beta \frac{\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}, & k > 0 \end{cases}\end{aligned}$$ When the stopping criteria is not satisfied ($k \geqslant 0$), $x_{k+1}$ can be calculated using the following formula: $$\begin{aligned} \label{eq:3} x_{k+1} = z_k - \alpha \frac{X_k^2s_k}{\|X_ks_k\|}\end{aligned}$$ **AAFS:** In the AAFS, we integrated the SST with the GAFS to gain acceleration. Since the primal sequence generated by the GAFS converges as $ i $ goes to infinity (i.e., $\lim_{i \rightarrow \infty} x_i = x^*$, see Section \[sec:primal\]), it allowed us to write the following equation: $$\begin{aligned} x^*-x_0 = \sum\limits_{i = 0}^{\infty} (x_{i+1}-x_i)\end{aligned}$$ We denoted the entry-wise partial sum of the right hand side of above equation as $C_{k,j}$ as follows: $$\begin{aligned} C_{k,j} = \sum\limits_{i =0}^{k} (x_{i+1}-x_i)_j\end{aligned}$$ We see that $C_{k,j}+ (x_0)_j$ converges to $(x^*)_j$ as $k$ goes to infinity for all $j = 1,2,..,n$. This setup allowed us to introduce the entry-wise SST to the sequence $x_k$ generated by the GAFS. In the above algorithm, we define $(B(x_k))_j$ for all $k \geqslant 0$ and $j=1,2...,n$ as follows: $$\begin{aligned} \label{eq:4} (B(x_k))_j \overset{\underset{\mathrm{def}}{}}{=} \begin{cases} \ (x_k)_j, & k = 0, 1 \\ \ \Big | (x_k)_j - \frac{((x_k)_j-(x_{k+1})_j)^2}{(x_k)_j-2(x_{k+1})_j-(x_{k+2})_j} \Big |, & k > 1 \end{cases}\end{aligned}$$ As $(x_k)_j$ is approximated by $(B(x_k))_j$ for all $ j = 1,2,..,n$, we can modify the stopping criteria of AAFS algorithm with $e^T B_ks_k$. Convergence of the primal sequence {#sec:primal} ================================== In this section, we provided proof of convergence for the primal sequence $\{x_k\}$ generated by the GAFS and AAFS algorithms discussed in Section 2. We used some Lemmas related to the properties of the sequences $\{x_k\}, \{y_k\}, \{s_k\}, \{d_k \} =\{X_k^2s_k\}$ generated by the GAFS provided in the \[appendix-sec2\] for proving the next few Theorems of this section. We made the following assumptions before providing the proof of convergence of the primal sequence $\{x_k\}$ and the cost function sequence $\{c^Tx_k\}$: - The *Linear Program* has at least one interior point feasible solution. - The objective function $c^Tx$ is not constant over the feasible region of . - The matrix $A$ has rank $m$. - The *Linear Program* has an optimal solution. \[rem-1\] Note that, here we didn’t assume primal and dual non-degeneracy of the LP problem . For step size selection, we considered three well-known function defined for a vector $u$ as $$\gamma(u) = \max \left\{ u_i \ | \ u_i > 0\right\}, \quad \|u\|_{\infty} = \max_i |u_i|, \quad \|u\|_2 = \sqrt{\sum u_i^2}$$ Whereas the second and third terms are $l_{\infty}$ and $l_2$ norm, respectively. The 1st function is not a norm and not well defined as $\gamma(u)$, is undefined for a non-positive vector $u \leqslant 0$. The following relationship holds: $$\gamma(u) \leqslant \|u\|_{\infty} \leqslant \|u\|_2$$ For the generalization term, we considered only $l_{\infty}$ and $l_2$ norm as the first function is undefined for some cases, since there is no guarantee that $X_k^{-1}\delta(x_k) \geqslant 0$ will always hold for all $k \geqslant 1$. For our analysis, we select long-step size and long generalization parameter, i.e., we redefine the update formula for $k \geqslant 1$ as follows: $$\begin{aligned} \label{eq:5} x_{k+1} \overset{\underset{\mathrm{def}}{}}{=} x_k - \alpha \frac{X_k^2s_k}{\gamma(X_ks_k)} + \beta \frac{\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}; \ \ x_1 \overset{\underset{\mathrm{def}}{}}{=} x_0- \alpha \frac{X_0^2s_0}{\gamma(X_0s_0)}\end{aligned}$$ Now, by defining $\alpha_k \overset{\underset{\mathrm{def}}{}}{=} \frac{\alpha}{\gamma(X_ks_k)}, \ \beta_k \overset{\underset{\mathrm{def}}{}}{=} \frac{\beta}{\|X_k^{-1}\delta(x_k)\|_{\infty}}$, we get the modified update formula as follows: $$\begin{aligned} \label{eq:6} x_{k+1} = x_k - \alpha_k X_k^2s_k + \beta_k \delta(x_k); \ \ x_1 = x_0- \alpha_0 X_0^2s_0\end{aligned}$$ Let us assume $\lim_{k \to \infty} x_k = x^*$, then define sequences $\{u_k\}, \{\gamma_k\}, \{r_k\}$ and $\{p_k\}$ as follows: $$\begin{aligned} \label{eq:7} u_k \overset{\underset{\mathrm{def}}{}}{=} \frac{X_ks_k}{c^Tx_k-c^Tx^*} , \ \ \gamma_k \overset{\underset{\mathrm{def}}{}}{=} \prod_{j=1}^{k} \beta_k, \ \ r_k \overset{\underset{\mathrm{def}}{}}{=} \frac{\gamma_k}{c^Tx_k-c^Tx^*}, \ \ p_k \overset{\underset{\mathrm{def}}{}}{=} \gamma_k X_k^{-1}e\end{aligned}$$ \[theorem-1\] The sequences $\{x_{k+1}\}$ and $ \{x_k\}$ generated by the GAFS algorithm satisfy the following two identities for all $k \geqslant 0$: $$\begin{aligned} & \frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} = 1- \alpha \sum\limits_{j=0}^{k} \frac{\|u_j\|^2}{\gamma(u_j)}\frac{r_k}{r_j} \\ & \left(X_k^{-1}x_{k+1}\right)_j \ = \ \frac{(x_{k+1})_j}{(x_{k})_j} \ = \ 1- \alpha \sum\limits_{i=0}^{k} \frac{(u_i)_j}{\gamma(u_i)}\frac{(p_k)_j}{(p_i)_j} \end{aligned}$$ Taking inner product with $c$ in both sides of and using the definitions from , we can find the following relationship: $$\begin{aligned} c^Tx_{k}-c^Tx_{k-1} & = -\alpha_{k-1} \|X_{k-1}s_{k-1}\|^2 + \beta_{k-1} (c^Tx_{k-1}-c^Tx_{k-2}) \nonumber \\ & = -\alpha_{k-1} \|X_{k-1}s_{k-1}\|^2 - \beta_{k-1} \alpha_{k-2} \|X_{k-2}s_{k-2}\|^2 \nonumber \\ & \ \ \ + \beta_{k-1} \beta_{k-2} (c^Tx_{k-2}-c^Tx_{k-3}) \nonumber \\ & \vdots \nonumber \\ & = \gamma_{k-1} (c^Tx_1-c^Tx_0) - \sum\limits_{j=1}^{k-1} \frac{\gamma_{k-1}}{\gamma_j} \alpha_j \|X_js_j\|^2 \nonumber \\ & = -\alpha_0 \gamma_{k-1} \|X_0s_0\|^2 - \sum\limits_{j=1}^{k-1} \frac{\gamma_{k-1}}{\gamma_j} \alpha_j \|X_js_j\|^2 \nonumber \\ & = - \alpha \sum\limits_{j=0}^{k-1} \frac{\|u_j\|^2}{\gamma(u_j)}\frac{\gamma_{k-1}}{\gamma_j} (c^Tx_j-c^Tx^*) \label{eq:10}\end{aligned}$$ Now using the update formula , the definition and equation , we find the following equation, $$\begin{aligned} \frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} & = 1 - \alpha_k \ \frac{c^TX_k^2s_k}{c^Tx_k-c^Tx^*} - \beta_k \frac{c^Tx_{k-1}-c^Tx_{k}}{c^Tx_{k}- c^Tx^*} \nonumber \\ & = 1 -\alpha \frac{\|u_k\|^2}{\gamma(u_k)} - \alpha \sum\limits_{j=0}^{k-1} \frac{\|u_j\|^2}{\gamma(u_j)}\frac{\beta_k\gamma_{k-1}}{\gamma_j} \frac{c^Tx_j-c^Tx^*}{c^Tx_k-c^Tx^*} \\ & = 1- \alpha \sum\limits_{j=0}^{k} \frac{\|u_j\|^2}{\gamma(u_j)}\frac{r_k}{r_j} \label{eq:11}\end{aligned}$$ The above equation proves part (1) of Theorem \[theorem-1\]. Similarly, using equation and , we have, $$\begin{aligned} x_{k+1}-x_k & = -\alpha_k X_k^2s_k + \beta_k \delta(x_k) \nonumber\\ & = -\alpha_k X_k^2s_k - \beta_k \alpha_{k-1} X_{k-1}^2s_{k-1} + \beta_k \beta_{k-1} \delta(x_{k-1}) \nonumber \\ & \vdots \nonumber \\ & = \gamma_k \delta(x_1) - \sum\limits_{j=1}^{k} \frac{\gamma_k}{\gamma_j} \alpha_j X_j^2s_j \ = - \alpha \sum\limits_{j=0}^{k}\frac{\gamma_k}{\gamma_j} \frac{X_ju_j}{\gamma(u_j)} \label{eq:12}\end{aligned}$$ Then, multiplying both sides of by $X_k^{-1}$ and after simplification, we have for all $j =1,2,...,n.$, $$\begin{aligned} \left(X_k^{-1}x_{k+1}\right)_j \ = \ \frac{(x_{k+1})_j}{(x_{k})_j} \ &= \ 1 - \alpha \sum\limits_{i =0}^{k}\frac{(u_i)_j}{\gamma(u_i)}\frac{\gamma_k(x_i)_j}{\gamma_i (x_k)_j} \nonumber \\ & = \ 1- \alpha \sum\limits_{i=0}^{k} \frac{(u_i)_j}{\gamma(u_i)}\frac{(p_k)_j}{(p_i)_j} \label{eq:13}\end{aligned}$$ The above equations and prove part (1) and part (2) of Theorem \[theorem-1\], respectively. Now, for the rest of our analysis let us define the set $Q$ as below: $$Q \overset{\underset{\mathrm{def}}{}}{=} \{(\alpha, \beta) | \ 0 < \alpha < 1 ,\ 0 \leqslant \beta < \frac{1}{\phi}, \ \alpha +\beta \leqslant \frac{2}{3} \}$$ Where $\phi = 1.618...$ is the so called golden ratio. \[theorem-2\] For $\alpha, \beta \in Q$, starting from a strictly feasible point $x_0$, the sequence $x_{k+1}$ generated by the update formula has the following three properties for all $k \geqslant 0$: 1. $Ax_{k+1} =b$ 2. $x_{k+1} > 0$ 3. $c^Tx_{k+1} < c^Tx_k$ Since the sequence $\bar{v_j} = \frac{X_j^2s_j}{\gamma(X_js_j)}$ solves the EAP problem for all $j \geqslant 0$, we have $A\bar{v_j} = 0$ for all $j \geqslant 0$. As $Ax_0 = b$, using equation , we have, $$\begin{aligned} Ax_{k+1} & = Ax_0 + \sum\limits_{l=0}^{k} A \delta(x_{l+1}) \\ & = Ax_0 - \alpha \sum\limits_{l =0}^{k}\sum\limits_{j=0}^{l}\frac{\gamma_l}{\gamma_j} \frac{AX_ju_j}{\gamma(u_j)} \\ & = b - \alpha \sum\limits_{l =0}^{k}\sum\limits_{j=0}^{l}\frac{\gamma_l}{\gamma_j} \frac{AX_j^2s_j}{\gamma(X_js_j)} = b - \alpha *0 = b\end{aligned}$$ This proves part (1) of Theorem \[theorem-2\]. For the second part, let us evaluate the upper bound of $\|X_k^{-1}\delta(x_{k+1})\|_{\infty}$, $$\begin{aligned} \|X_k^{-1}\delta(x_{k+1})\|_{\infty} & = \Big\| - \alpha \frac{X_ks_k}{\gamma(X_ks_k)} +\beta \frac{X_k^{-1}\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \Big\|_{\infty} \\ & \leqslant \ \alpha \ \frac{\|X_ks_k\|_{\infty}}{\gamma(X_ks_k)} + \beta \ \frac{ \|X_k^{-1}\delta(x_k)\|_{\infty}}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \\ & \leqslant \ \alpha + \beta \ \leqslant \ \frac{2}{3} \ < \ 1\end{aligned}$$ In particular, for all $j = 1,2,...,n$, we have, $$\begin{aligned} \label{eq:14} \frac{|x_{k+1}^{j}-x_{k}^j|}{x_k^{j}} \ \leqslant \ \|X_k^{-1}\delta(x_{k+1})\|_{\infty} \ \leqslant \ \alpha + \beta \ \leqslant \ \frac{2}{3} \ < \ 1\end{aligned}$$ Which implies $x_{k+1}^j > 0$ for all $j$. Therefore, $x_{k+1} > 0$ for all $k \geqslant 0$. Now equation using we have, $$\begin{aligned} c^Tx_{k+1} & = c^Tx_k- \alpha \frac{c^TX_k^2s_k}{\gamma(X_ks_k)} + \beta \frac{c^T\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \nonumber \\ & \leqslant \ c^Tx_k- \alpha \|X_ks_k\|+ \beta \frac{c^T\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \nonumber \\ & < \ c^Tx_k +\beta_k \left(c^Tx_k-c^Tx_{k-1}\right) \nonumber \\ & < \ c^Tx_k + \gamma_k \left(c^Tx_1-c^Tx_0\right) \ = \ c^Tx_k - \alpha \gamma_k \frac{\|X_0s_0\|^2}{\gamma(X_0s_0)} \ < \ c^Tx_k \label{eq:15}\end{aligned}$$ Therefore, $c^Tx_{k+1} < c^Tx_k$ for all $k \geqslant 0$. \[theorem-3\] The following statements hold for the GAFS algorithm: 1. The sequence of objective function values $\{c^Tx_k\}$ generated by the GAFS strictly decreases and converges to a finite value. 2. $X_ks_k \rightarrow \textbf{0}$ as $k \rightarrow \infty$. As a consequence of Theorem \[theorem-2\], we know that the sequence $\{c^Tx_k\}$ is a decreasing sequence. For the part (1) of Theorem \[theorem-3\], we just need to show that the sequence $\{c^Tx_k\}$ is bounded. As per our assumption, $x^*$ is the optimal solution of the primal problem (P) in . However, it implies the following, $$c^Tx^* \leqslant \dots < c^Tx_{k+1} < c^Tx_k < ...< c^Tx_0$$ It means that the sequence $\{c^Tx_k\}$ is bounded. Therefore, using the *Monotone Convergence Theorem* we can conclude that the sequence $\{c^Tx_k\}$ is convergent and $\lim_{k \rightarrow \infty} c^Tx_k = c^Tx^*$. For the second part, we see that, $$\begin{aligned} 0 \ < \ \|X_ks_k\| & \ \leqslant \ \frac{1}{\alpha} \left[(c^Tx_{k}-c^Tx_{k+1})+ \beta_k (c^Tx_k-c^Tx_{k-1})\right] \nonumber \\ & < \ \frac{1}{\alpha} \left[(c^Tx_{k}-c^Tx_{k+1})+ \gamma_k (c^Tx_1-c^Tx_{0})\right] \label{eq:16}\end{aligned}$$ Now, by the properties of $\{c^Tx_k\}$, we have $ c^Tx_0 -c^Tx^* < \infty $ and $\bar{c} = c^Tx_0-c^Tx_1 < \infty $, also as a consequence of Lemma \[lemma-6\], we have $G = \sum\limits_{k =0}^{\infty} \gamma_k < \infty$. Combining these facts and equation , we can write the following equation, $$\begin{aligned} \sum\limits_{k =0}^{\infty}\|X_ks_k\| \ & < \ \frac{1}{\alpha} \left[\sum\limits_{k =0}^{\infty} (c^Tx_{k}-c^Tx_{k+1}) + \bar{c} \sum\limits_{k =0}^{\infty} \gamma_k \right] \nonumber \\ & = \ \frac{1}{\alpha} \left[c^Tx_0 -c^Tx^* + \bar{c} \ G\right] \ < \ \infty \label{eq:17}\end{aligned}$$ Now, equation allows us to write $ X_ks_k \rightarrow \textbf{0} \ \text{as} \ k \rightarrow \infty $. This proves second part of Theorem \[theorem-3\]. \[rem-2\] By using Theorem \[theorem-3\], we see that the complementary slackness condition holds in the limit since, $\lim_{k \rightarrow \infty} (x_k)_j(s_k)_j = 0$ for all $j = 1,2,3,...,n$. \[theorem-4\] The following statements hold for the GAFS algorithm: 1. The sequence $\{x_k\}$ converges to a point $x^*$, belongs to interior of the primal feasible region. 2. For all $k \geqslant 0$ there exists a $N = N(x,A) >0$ such that, $$\|x_k-x^*\| \ \leqslant \ M \left(c^Tx_k-c^Tx^*\right) +\frac{\|x_{1}-x_0\| }{\gamma_1} G(k) \ \leqslant \ (M+N) \left(c^Tx_k-c^Tx^*\right)$$ Denoting $t_k = (c^Tx_{k}-c^Tx_{k-1})$, as a direct consequence of equation and Lemma \[lemma-4\] we have, $$\begin{aligned} \|x_{k+1}-x_k\| \ & \leqslant \ \alpha M \frac{c^Td_k}{\|X_ks_k\|}+\frac{\beta}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \|x_{k}-x_{k+1}\| \nonumber \\ & = M \left(c^Tx_{k}-c^Tx_{k+1}\right) + \frac{M \beta \left(c^Tx_{k}-c^Tx_{k-1}\right)}{\|X_k^{-1}\delta(x_k)\|_{\infty}} + \frac{\beta \|x_{k}-x_{k+1}\|}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \nonumber \\ & = M \beta_k t_k - M t_{k+1}+ \beta_k \delta(x_{k+1}) \nonumber \\ & = M \sum \limits _{j= 1}^{k-1} \frac{\gamma_{k}}{\gamma_j} t_{j+1}- M \sum \limits _{j= 2}^{k} \frac{\gamma_{k}}{\gamma_j} t_{j+1} + \frac{\gamma_k}{\gamma_1} \delta(x_{1}) \nonumber \\ & = M \frac{\gamma_k}{\gamma_1} t_2 - M \frac{\gamma_k}{\gamma_k} t_{k+1} + \frac{\gamma_k}{\gamma_1} \delta(x_{1}) \nonumber \\ & = M \frac{\gamma_k}{\gamma_1} \left(c^Tx_{2}-c^Tx_{1}\right) + M \left(c^Tx_{k}-c^Tx_{k+1}\right) + \frac{\gamma_k}{\gamma_1} \|x_1-x_0\| \label{eq:19}\end{aligned}$$ Furthermore, from Lemma \[lemma-6\], we know that the sequence $\gamma_k$ converges to $0$ as $k \rightarrow \infty$, so we can assume that the sequence $\left\{\sum\limits_{k=1}^{m} \gamma_k \right\} $ converges to some finite value $G$ as $m$ goes to infinity, i.e., $$\sum\limits_{k=1}^{\infty} \gamma_k = G < \infty$$ Then from equation we have, $$\begin{aligned} \sum\limits_{k =0}^{\infty} & \|x_{k+1}-x_k\| = \|x_{1}-x_0\| + \sum\limits_{k =1}^{\infty} \|x_{k+1}-x_k\| \\ & \leqslant \|x_{1}-x_0\| + M \sum\limits_{k =1}^{\infty} \left(c^Tx_{k}-c^Tx_{k+1}\right) + \frac{M}{\gamma_1}\left(c^Tx_{2}-c^Tx_{1}\right) \\ & \ \ + \frac{\|x_{1}-x_0\|}{\gamma_1} \sum\limits_{k =1}^{\infty} \gamma_k \\ & = \|x_{1}-x_0\| + M \left(c^Tx_{1}-c^Tx^*\right) + \frac{G}{\gamma_1} \left[M \left(c^Tx_{2}-c^Tx_{1}\right) + \|x_{1}-x_0\| \right] \\ & < \infty\end{aligned}$$ The above identity shows that, $\{x_k\}$ is a Cauchy sequence, and therefore, it is a convergence sequence (i.e., every real Cauchy sequence is convergent). Now, for all $0 \leqslant k \leqslant l$, using equation we have, $$\begin{aligned} \|x_l-x_k\| & \ \leqslant \ \big \|\sum \limits_{j =k}^{l-1} \left(x_{j+1}-x_{j}\right)\big \| \ \leqslant \ \sum \limits_{j =k}^{l-1} \|x_{j+1}-x_{j}\| \nonumber \\ & \leqslant M \sum \limits _{j=k}^{l-1} \left(c^Tx_j-c^Tx_{k+1}\right)+ \frac{1}{\gamma_1} \left[M \left(c^Tx_{2}-c^Tx_{1}\right) + \|x_{1}-x_0\| \right]\sum\limits_{j=k}^{l-1} \gamma_j \nonumber \\ & \ \leqslant \ M \sum \limits _{j=k}^{l-1} \left(c^Tx_j-c^Tx_{k+1}\right)+ \frac{\|x_{1}-x_0\| }{\gamma_1}\sum\limits_{j=k}^{l-1} \gamma_j \label{eq:20}\end{aligned}$$ Now, letting $l \rightarrow \infty$ in and defining $G(k) \overset{\underset{\mathrm{def}}{}}{=} \sum\limits_{j = k}^{\infty} \gamma_j $, we have, $$\begin{aligned} \|x_k-x^*\| \ &\leqslant \ M \left(c^Tx_k-v^*\right) +\frac{\|x_{1}-x_0\| }{\gamma_1} G(k) \nonumber\\ & \leqslant M \left(c^Tx_k-c^Tx^*\right) + \frac{\|x_{1}-x_0\| }{\gamma_1} \bar{N} \left(c^Tx_k-c^Tx^*\right) \nonumber \\ & = (M+ N) \left(c^Tx_k-c^Tx^*\right) \label{eq:21}\end{aligned}$$ This is the required bound. In the last line, we used Lemma \[lemma-8\] with $N = \frac{\|x_{1}-x_0\| }{\gamma_1} \bar{N}$. \[theorem-x\] The sequence $\{B(x_k)\}$ generated by the AAFS algorithm converges to the same point $x^*$ and belongs to the interior of the primal feasible region. From Theorem \[theorem-4\], we know that the sequence $\{x_k\}$ generated by GAFS converges to $x^*$. Then using the definition and the basic idea of SST, we can immediately conclude that for all $j =1,2,...,n$, the following relation holds: $$\begin{aligned} \lim_{k \to \infty} (B(x_k))_j = (x^*)_j\end{aligned}$$ Since, this holds for all $j =1,2,...,n$, we can prove $\lim_{k \to \infty} B(x_k) = x^*$. The last part of Theorem \[theorem-x\] follows from the fact that $(B(x_k))_j > 0$, for all $k \geqslant 1$ and $j =1,2,...,n$. With $\lim_{k \rightarrow \infty}x_k = x^*$, let us define the sets $N$ and $B$ as follows, $$N \overset{\underset{\mathrm{def}}{}}{=} \{i \ | \ x_i^* = 0\}, \ \ B = \{i \ | \ x_i^* > 0\}, \ \ |N|= p$$ Now, we provide proof for an important property of the sequence $\{x_k\}$, which subsequently holds for original AFS algorithm. We showed that it holds for GAFS too with different constant. For the original AFS algorithm, the Theorem was proven by several authors in their work [@saigal:1996] and [@tsuchiya:1995]. \[theorem-5\] There exists a $\delta > 0$ and a $R > 0$ such that for each $k \geqslant 0$ $$\frac{c^Tx_k-c^Tx^*}{\|x_k-x^*\|} \ \geqslant \ \frac{1}{R}, \quad \frac{c^Tx_k-c^Tx^*}{\sum_{i \in N}(x_k)_i} \geqslant \delta, \quad \frac{c^Tx_k-c^Tx^*}{\sum_{i \in B}\big | (x_k)_i-(x^*)_i \big |} \geqslant \delta$$ Let, $R = M+N$, then from equation , we have for all $k \geqslant 0$, $$\begin{aligned} \frac{c^Tx_k-c^Tx^*}{\|x_k-x^*\|} \ \geqslant \ \frac{1}{M+N} \ = \ \frac{1}{R}\end{aligned}$$ It proves the first part of Theorem \[theorem-5\]. Similarly from equation , $$\begin{aligned} & c^Tx_k-c^Tx^* \ \geqslant \frac{\|x_k-x^*\|}{R} \geqslant \ \frac{\|x_{k,N}\|}{R} \geqslant \ \frac{\sum_{i \in N}(x_k)_i}{\sqrt{p} R} \\ & c^Tx_k-c^Tx^* \ \geqslant \frac{\|x_k-x^*\|}{R} \geqslant \ \frac{\|x_{k,B}-x_B^*\|}{R} \geqslant \ \frac{\sum_{i \in B}\big | (x_k)_i-(x^*)_i \big |}{\sqrt{n-p} R} \end{aligned}$$ By denoting $\delta = \min \{\frac{1}{\sqrt{p}R}, \frac{1}{\sqrt{n-p}R}\}$, we have the remaining results of Theorem \[theorem-5\]. If $\alpha, \ \beta \in Q$, then the following identities hold: \[theorem-6\] 1. For all $k \geqslant 1$, $$\|X_k^{-1}X_{k-1}\|_\infty \ < \ 5$$ 2. There exists a $L_2 \geqslant 1$ such that for all $k \geq L_2$, $$\begin{aligned} -\frac{\|X_ks_k\|}{c^Tx_k-c^Tx^*} \leqslant \frac{-1}{\|X_k^{-1}\left(x_k-x^*\right)\|} \leqslant \frac{-1}{\sqrt{n}} \end{aligned}$$ 3. For all $k \geqslant 1$, $$\begin{aligned} \frac{1}{\gamma_k} = \ \frac{1}{\beta_1 \beta_2 ... \beta_k} < \ \frac{\prod_{j=1}^{k} \|X_j^{-1}X_{j-1}\|}{\beta^k} (\alpha +\beta)^k \ < \ \left(\frac{5}{\beta}\right)^k\end{aligned}$$ From equation for all $j = 1,2,...,n$, we have, $$\begin{aligned} \label{eq:24} \frac{|(x_{k+1}-x_{k})_j|}{(x_k)_j} \ \leqslant \ \|X_k^{-1}\delta(x_{k+1})\|_{\infty} \ \leqslant \ \alpha + \beta \ \leqslant \ \frac{2}{3}\end{aligned}$$ Simplifying equation further for all $k \geqslant 1, \ j = 1,2,...,n$ we have, $$\begin{aligned} \label{eq:25} \frac{3}{5} \ \leqslant \ \frac{(x_{k-1})_j}{(x_k)_j} \ \leqslant \ 3\end{aligned}$$ Then using equation to the definition of maximum norm, we have, $$\begin{aligned} \|X_k^{-1}\delta(x_k)\|_{\infty} \ \leqslant \ \max_j \left\{1+\frac{(x_{k-1})_j}{(x_k)_j}\right\} \ \leqslant \ 4\end{aligned}$$ Therefore, we have, $$\begin{aligned} \label{eq:26} \|X_k^{-1}x_{k-1}\|_{\infty} \ \leqslant \ \|e - X_k^{-1}x_{k-1}\|_{\infty} + \|e\|_{\infty} \ \leqslant \ 4 + 1 = 5 \end{aligned}$$ Which proves part (1) of Theorem \[theorem-6\]. Part (2) of Theorem \[theorem-6\] is well studied in the literature ( see [@saigal:1996], [@tsuchiya:1995]). We can prove part (2) of this Theorem easily as the sequence $\frac{X_k^2s_k}{\|X_ks_k\|}$ generated by the GAFS algorithm solves the EAP problem provided in equation , i.e., there exists a $L_2$ such that for all $k > L_2$, $$\begin{aligned} -\frac{\|X_ks_k\|}{c^Tx_k-c^Tx^*} \ \leqslant \ \frac{-1}{\|X_k^{-1}\left(x_k-x^*\right)\|} \ \leqslant \ \frac{-1}{\sqrt{n}} \end{aligned}$$ For proving the last part, we first need an upper bound of $\frac{1}{\beta_k}$. For all $k \geqslant 1$, we have, $$\begin{aligned} \frac{1}{\beta_{k}} = \frac{\|X_{k}^{-1}\delta(x_k)\|_{\infty}}{\beta} &= \big \|\frac{X_{k}^{-1}\delta(x_{k-1})}{\|X_{k-1}^{-1}\delta(x_{k-1})\|_{\infty}} - \frac{\alpha}{\beta} \frac{X_{k}^{-1}X_{k-1}^2s_{k-1}}{\gamma(X_{k-1}s_{k-1})}\big \|_{\infty} \\ & \leqslant \|X_{k}^{-1}X_{k-1}\|_{\infty} \big \|\frac{X_{k-1}^{-1}\delta(x_{k-1})}{\|X_{k-1}^{-1}\delta(x_{k-1})\|_{\infty}} - \frac{\alpha}{\beta} \frac{X_{k-1}s_{k-1}}{\gamma(X_{k-1}s_{k-1})}\big \|_{\infty} \\ & \leqslant \|X_{k}^{-1}X_{k-1}\|_{\infty}\left(1+ \frac{\alpha}{\beta}\right) \ < \ \frac{5(\alpha + \beta)}{ \beta} \ < \ \frac{5 (\beta +2)}{3\beta} \ < \ \frac{5}{\beta} \end{aligned}$$ Here, we use the identity in equation . Then by the definition of $\gamma_k$ for all $k \geqslant 1$ we have, $$\begin{aligned} \frac{1}{\gamma_k} = \ \frac{1}{\beta_1 \beta_2 ... \beta_k} \ < \ \left(\frac{5}{\beta}\right)^k\end{aligned}$$ It proves the remaining parts of Theorem \[theorem-6\]. For the remaining sections, let us define sequences $\{u_k\}, \{v_k\}$ and $\{h_k\}$ as follows: $$\begin{aligned} \label{eq:30} u_k \overset{\underset{\mathrm{def}}{}}{=} \frac{X_ks_k}{c^Tx_k-c^Tx^*}, \ \ v_k \overset{\underset{\mathrm{def}}{}}{=} \frac{X_k^{-1}\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}(c^Tx_k-c^Tx^*) \ \ h_k \overset{\underset{\mathrm{def}}{}}{=} \frac{c^T\delta(x_k)}{\|X_k^{-1}\delta(x_k)\|_{\infty}}\end{aligned}$$ Convergence Rate {#sec:rate} ================ In this section, we measured the significance of AAFS over GAFS in terms of convergence rate given that GAFS converges linearly (see the following Theorem \[theorem-7\]). The following Theorem gives us the linear convergence rate of GAFS algorithm. \[theorem-7\] The following statements holds for the GAFS algorithm: 1. There exists a $L \geqslant 1$ such that for all $k \geqslant L$, $$\begin{aligned} \frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} \leqslant 1- \frac{\alpha}{\sqrt{n}}- \frac{\alpha}{\sqrt{n}} \left(\frac{\beta}{5}\right)^k\end{aligned}$$ 2. For $\alpha, \beta \in Q $ the following limit holds: $$\begin{aligned} \lim_{k \to \infty}\frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} = 1- \frac{\alpha}{\sqrt{n}} \ < \ 1\end{aligned}$$ We used Theorem \[theorem-1\] for proving part (1) of this Theorem. First, let us choose $L = L_2$ (part (2) of Theorem \[theorem-6\]), then using the update formula for all $k > L$, we have, $$\begin{aligned} \frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} & = 1- \frac{\alpha }{\gamma(X_ks_k)} \frac{c^TX_k^2s_k}{(c^Tx_{k}- c^Tx^*)} + \frac{\beta}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \frac{c^Tx_{k}- c^Tx_{k-1}}{(c^Tx_{k}- c^Tx^*)} \\ & \leqslant 1- \alpha \ \frac{\|X_ks_k\|}{c^Tx_{k}- c^Tx^*} + \beta_k \ \frac{c^Tx_{k}- c^Tx_{k-1}}{c^Tx_{k}- c^Tx^*} \\ & \leqslant 1- \frac{\alpha}{\|X_k^{-1}\left(x_k-x^*\right)\|} + \beta_k \gamma_{k-1} \ \frac{c^Tx_{1}- c^Tx_{0}}{c^Tx_{k}- c^Tx^*} \\ & \leqslant 1- \frac{\alpha}{\sqrt{n}} - \alpha \gamma_k \ \frac{\|X_0s_0\|}{c^Tx_{k}- c^Tx^*} \\ & \leqslant 1- \frac{\alpha}{\sqrt{n}} - \alpha \gamma_k \ \frac{\|X_ks_k\|}{c^Tx_{k}- c^Tx^*} \\ & \leqslant 1 - \frac{\alpha}{\sqrt{n}}- \frac{\alpha \gamma_k}{\sqrt{n}} \ \leqslant \ 1- \frac{\alpha}{\sqrt{n}}- \frac{\alpha}{\sqrt{n}} \left(\frac{\beta}{5}\right)^k \ < \ 1- \frac{\alpha}{\sqrt{n}}\end{aligned}$$ Here, we used the the fact that the sequence $\{\|X_ks_k\|\}$ is a decreasing sequence and converges to zero due to the property of complementary slackness, i.e., $\|X_0s_0\| \ \geqslant \ \|X_1s_1\| \ \geqslant ... \geqslant \ \|X_ks_k\|$ (see part (2) of Theorem \[theorem-3\]). Now, part (2) of Theorem \[theorem-7\] is a direct consequence of part (1) of Theorem \[theorem-7\] as the sequence $\{\left(\frac{\beta}{4 \sqrt{n}}\right)^k\}$ converges to zero as $k \rightarrow \infty$. Note that, Theorem \[theorem-7\] indicates that GAFS algorithm converges linearly. Next, we compared the convergence rates between the proposed algorithms. In other words, how good is the sequence $\{c^TB(x_k)\}$ compared to the sequence $\{c^Tx_k\}$ when the latter converges linearly. The next Theorem (Theorem \[theorem-ac1\]) shows that it is better, in the sense that it converges faster, meaning that the AAFS accelerates the convergence of GAFS. \[theorem-ac1\] Assume that the sequence $\{c^Tx_k\}$ converges to $c^Tx^*$ linearly. Let $B(x_k)$ be as in , then the sequence $\{c^TB(x_k)\}$, converges faster to $c^Tx^*$ than $\{c^Tx_k\}$ in the sense that, $$\begin{aligned} \lim_{k \rightarrow \infty} \frac{c^TB(x_k)-c^Tx^*}{c^Tx_k-c^Tx^*} = 0\end{aligned}$$ By virtue of Theorem \[theorem-7\], we can argue that there exists sequences $\{\sigma_j\}$ and $\{(\lambda_k)_j\}$ such that, $$\begin{aligned} \label{eq:31} \frac{c_j(x_{k+1})_j- c_j(x^*)_j}{c_j(x_{k})_j- c_j(x^*)_j} = \sigma_j + (\lambda_k)_j \ \ \forall \ j = 1,2,...,n, \ k \geq 1 \end{aligned}$$ And $\lim_{k \rightarrow \infty} (\lambda_k)_j = 0 \ \forall \ j$. Simplifying for $(k+1)$th and $(k+2)$th terms, we have, $$\begin{aligned} & c_j(x_{k+1})_j = c_j(x^*)_j + \left(\sigma_j + (\lambda_k)_j\right) \left(c_j(x_{k})_j- c_j(x^*)_j\right) \nonumber \\ & c_j(x_{k+2})_j = c_j(x^*)_j + \left(\sigma_j + (\lambda_{k+1})_j\right) \left(c_j(x_{k+1})_j- c_j(x^*)_j\right) \label{eq:32}\end{aligned}$$ for all $ j = 1,2,...,n, \ k \geq 1$. Now using and , we have, $$\begin{aligned} \frac{c^TB(x_k)-c^Tx^*}{c^Tx_k-c^Tx^*} & = \frac{c^Tx_k-c^Tx^*}{c^Tx_k-c^Tx^*} \\ & - \sum\limits_{j=1}^{n} \frac{[c_j(x_k)_j-c_j(x_{k+1})_j]^2}{[c^Tx_k-c^Tx^*][c_j(x_k)_j-2c_j(x_{k+1})_j+c_j(x_{k+2})_j]} \nonumber \\ = 1 & - \frac{1}{n} \sum\limits_{j=1}^{n} \frac{[c_j(x_k)_j-c_j(x_{k+1})_j]^2}{[c_j(x_k)_j-c_j(x^*)_j][c_j(x_k)_j-2c_j(x_{k+1})_j+c_j(x_{k+2})_j]} \nonumber \\ = 1 & - \frac{1}{n} \sum\limits_{j=1}^{n} \frac{\left[\frac{c_j(x_k)_j-c_j(x_{k+1})_j}{c_j(x_k)_j-c_j(x^*)_j}\right]^2}{\frac{c_j(x_k)_j-2c_j(x_{k+1})_j+c_j(x_{k+2})_j}{c_j(x_k)_j-c_j(x^*)_j}} \nonumber \\ = 1 & - \frac{1}{n} \sum\limits_{j=1}^{n} \frac{\left[\sigma_j + (\lambda_k)_j-1\right]^2}{\left[\sigma_j + (\lambda_k)_j\right]\left[\sigma_j + (\lambda_{k+1})_j\right]- 2 \left[\sigma_j + (\lambda_k)_j\right]+1} \label{eq:33}\end{aligned}$$ Taking the limit $k \rightarrow \infty$ in and using the property of sequence $\{(\lambda_k)_j\}$, we have, $$\begin{gathered} \lim_{k \rightarrow \infty} \frac{c^TB(x_k)-c^Tx^*}{c^Tx_k-c^Tx^*} \\ = 1- \lim_{k \rightarrow \infty} \frac{1}{n} \sum\limits_{j=1}^{n} \frac{\left[\sigma_j + (\lambda_k)_j-1\right]^2}{\left[\sigma_j + (\lambda_k)_j\right]\left[\sigma_j + (\lambda_{k+1})_j\right]- 2 \left[\sigma_j + (\lambda_k)_j\right]+1} \\ = 1- \frac{1}{n} \sum\limits_{j=1}^{n} \frac{(\sigma_j-1)^2}{\sigma_j^2-2 \sigma_j+1} \ = 1- 1 = 0\end{gathered}$$ This proves the required Theorem. Convergence of the Dual sequence {#sec:dual} ================================ In this section, we introduced a local version of potential function largely studied in the literature. For the convergence of the dual sequence, it is required to control both the step sizes $\alpha$ and $\beta$. For the original Affine Scaling method, it was first shown by Tsuchiya *et al.* [@tsuchiya:1995] that for the dual convergence we need to have $\alpha \leqslant \frac{2}{3}$. A simpler version of its proof is also available in Saigal [@saigal:1996]. Here, we proved that the dual sequence generated by GAFS method converges if we have $\alpha, \beta \in Q$. We see that the original result of AFS can be found with the choice of $ \beta = 0$ in our proof. At first, we introduced the local potential function (defined in [@tsuchiya:1991]). For any $x > 0 $ with $c^Tx- c^Tx^* > 0$ and $N = \{j \ | \ (x^*)_j = 0\}$ with $p = |N|$, let us define the following function: $$\begin{aligned} F_N(x) \overset{\underset{\mathrm{def}}{}}{=} p \log (c^Tx-c^Tx^*)- \sum\limits_{j \in N} \log (x)_j\end{aligned}$$ \[theorem-9\] For the sequence $\{x_k\}$ we can show that for all $k \geqslant 0$, $$\begin{gathered} F_N(x_{k+1})- F_N(x_k) = p \log (1- \theta \|w_{k,N}\|^2- \theta \sigma_k^2+ \theta w_{k,N}^T v_{k,N} \\ - \theta \left(\frac{2 \delta_k}{p} -\frac{2 \omega_k}{p}+ \epsilon_k + \phi h_k\right)) - \sum\limits_{j \in N} \log \left(1- \theta (w_k)_j - (\phi- \theta) (v_k)_j\right) \label{eq:35}\end{gathered}$$ ; where $w_{k,N} = u_{k,N}+ v_{k,N}-\frac{1}{p}e, \ \ \bar{\alpha} = \frac{\alpha}{\gamma(u_{k,N})}, \ \ \bar{\beta} = \frac{\beta}{\|v_{k,N}\|_{\infty}}, \ \ \theta = \frac{p\bar{\alpha}}{p- \bar{\alpha}}, \ \ \phi = \frac{p\bar{\beta}}{p- \bar{\alpha}}$,\ $\quad \quad \sum\limits_{k = L}^{\infty} |\epsilon_k| < \infty, \quad \sum\limits_{k = L}^{\infty} |\delta_k| < \infty, \quad \sum\limits_{k = L}^{\infty} |\omega_k| < \infty$ Using the update formula and definition , we have, $$\begin{aligned} \frac{c^Tx_{k+1}- c^Tx^*}{c^Tx_{k}- c^Tx^*} & = 1- \alpha \frac{c^TX_k^2s_k}{\gamma(X_ks_k) (c^Tx_k-c^Tx^*)} -\beta \frac{c^Tx_{k-1}-c^Tx_k}{\|X_k^{-1}\delta(x_k)\|_{\infty} (c^Tx_k-c^Tx^*)} \nonumber \\ & = 1- \alpha\frac{\|u_k\|^2}{\gamma(u_k)} - \beta \frac{h_k}{\|v_k\|_{\infty}} \ = \ 1 - \bar{\alpha} \|u_k\|^2 - \bar{\beta} h_k \label{eq:36}\end{aligned}$$ And also for all $j \in N$, we have, $$\begin{aligned} \frac{(x_{k+1})_j}{(x_{k})_j} \ & = \ 1- \alpha \frac{(X_ks_k)_j}{\gamma(X_ks_k)} - \beta \frac{\left[X_k^{-1}\delta(x_k)\right]_j}{\|X_k^{-1}\delta(x_k)\|_{\infty}} \nonumber \\ & = 1- \alpha\frac{(u_k)_j}{\gamma(u_k)} - \beta \frac{(v_k)_j}{\|v_k\|_{\infty}} \ = \ 1 - \bar{\alpha} (u_k)_j - \bar{\beta} (v_k)_j \label{eq:37}\end{aligned}$$ Now from part 2(a) of Lemma \[lemma-9\] , there exist a $L \geqslant 1$ such that for all $k \geqslant L$, we have, $$\begin{aligned} 1 - & \bar{\alpha} \|u_k\|^2 - \bar{\beta} h_k \ = \ 1 - \bar{\alpha} \ \| \ w_{k,N} - v_{k,N} + \frac{1}{p} e \ \|^2 -\bar{\alpha} \epsilon_k - \bar{\beta} h_k \nonumber \\ = \ & 1- \frac{\bar{\alpha}}{p} - \bar{\alpha} (\|w_{k,N}\|^2 +\|v_{k,N}\|^2) + \frac{2\bar{\alpha}}{p} e^T(v_{k,N}-w_{k,N}) \nonumber\\ & \qquad \qquad \qquad \qquad \qquad \qquad + 2 \bar{\alpha} v_{k,N}^T w_{k,N} - \bar{\alpha} \epsilon_k - \bar{\beta} h_k \nonumber \\ = \ & \frac{p-\bar{\alpha}}{p} \left[1- \theta \left(\|w_{k,N}\|^2 +\sigma_k^2 + \epsilon_k - 2 v_{k,N}^T w_{k,N} + \frac{2}{p} (\delta_k - \omega_k) \right) - \phi h_k\right] \label{eq:38}\end{aligned}$$ This is a simplification of equation . Also simplifying equation , we have, $$\begin{aligned} 1 - \bar{\alpha} (u_k)_j - \bar{\beta} (v_k)_j & = 1- \frac{\bar{\alpha}}{p} - \bar{\alpha} (w_k)_j + \bar{\alpha} (v_k)_j - \bar{\beta} (v_k)_j \nonumber \\ &= \frac{p-\bar{\alpha}}{p} \left[1- \theta (w_k)_j - (\phi- \theta) (v_k)_j \right] \label{eq:39}\end{aligned}$$ Using these two identities from and , we have the desired result of Theorem \[theorem-9\]. We know from the problem structure and assumptions that the sequence $\{c^Tx_k\}$ is bounded. In the next Theorem (Theorem \[theorem-10\]) we show that with $\alpha, \beta \in Q$, the dual sequence converges to the analytic center of the optimal face of dual polytope. As defined by Saigal [@saigal:1996], with $D \overset{\underset{\mathrm{def}}{}}{=} \{(y,s) : \ A_B^Ty = c_B, A_N^Ty+ s_N = c_N, s_B = 0\} $, we define the *Analytic Center Problem* (ACP) of the optimal dual face as the solution of $(y^*,s^*)$ to the following problem: $$\begin{aligned} \text{max} \ & \ \sum\limits_{j \in N} \log s_j \nonumber \\ & (y,s) \in D \label{eq:40} \\ & \ s_N > 0 \nonumber\end{aligned}$$ \[theorem-10\] If $\alpha, \beta \in Q$, then there exist vectors $x^*, y^*$ and $s^*$ such that the sequences $\{x_k\}$, $\{y_k\}$ and $\{s_k\}$ generated by the GAFS algorithm converges to $x^*, y^*$ and $s^*$, respectively, i.e., 1. $x_k \rightarrow x^*$, 2. $y_k \rightarrow y^*$, 3. $s_k \rightarrow s^*$ Where $x^*, y^*$ and $s^*$ are the optimal solutions of the respective primal and dual problems, and they also satisfy the strict complementary slackness property. Furthermore, the dual pair $(y^*,s^*)$ converges to the analytic center of the optimal dual face and the primal solution $x^*$ converges to the relative interior of the optimal primal face. Since, $ \log(1-a) < -a$, we can find a $L_1\geqslant 1$ such that for all $k \geqslant L_1$, $$\begin{gathered} F_N(x_{k+1})- F_N(x_k) \ \leqslant \ -p \theta \left[\|w_{k,N}\|^2 +\sigma_k^2 + \epsilon_k - 2 v_{k,N}^T w_{k,N} + \frac{2}{p} (\delta_k - \omega_k)\right] \\ - p \phi h_k - \sum\limits_{j \in N} \log \left(1- \theta (w_k)_j - (\phi- \theta) (v_k)_j\right) \label{eq:41}\end{gathered}$$ Now, we analyze equation for two cases based on the sign of $ \theta \gamma(w_{k,N}) + (\phi-\theta)\gamma(v_{k,N})$.\ **Case 1: $\ \ \theta \gamma(w_{k,N}) + (\phi-\theta)\gamma(v_{k,N}) \ \leqslant \ 0 $**\ Then we must have $\theta (w_k)_j + (\phi- \theta) (v_k)_j \leqslant 0$ for all $j \in N$, which implies, $$\begin{aligned} \log \left(1- \theta (w_k)_j - (\phi- \theta) (v_k)_j\right) \ \geqslant \ 0 \quad \text{for all} \quad j \in N \label{eq:42}\end{aligned}$$ Using part (d) of Theorem \[theorem-8\] and equation and , for all $k \geqslant L_1$ we have, $$\begin{aligned} F_N(x_{k+1}) & - F_N(x_k) \nonumber \\ & \leqslant -p \theta \|w_{k,N}\|^2 + 2p \theta v_{k,N}^T w_{k,N} - \theta (2 \delta_k + p \epsilon_k- 2\omega_k+ p \sigma_k^2) - p \phi h_k \nonumber \\ & \leqslant -p \theta \|w_{k,N}\|^2 + 2p \theta \epsilon \|w_{k,N}\|- \theta (2 \delta_k + p \epsilon_k- 2\omega_k+ p \sigma_k^2) - p \phi h_k \label{eq:43}\end{aligned}$$ **Case 2: $ \ \ \theta \gamma(w_{k,N}) + (\phi-\theta)\gamma(v_{k,N}) \ > \ 0 $**\ Let $\bar{\epsilon} > 0$, since $\gamma(v_{k,N}) = (c^Tx_k - c^Tx^*) \rightarrow 0$, there exist a $L_2 \geqslant 1$ such that for all $k \geqslant L_2$, $$\begin{aligned} \gamma(v_{k,N}) < \bar{\epsilon}\end{aligned}$$ Then, using the condition of Case 2, we have for all $k \geqslant L_2$, $$\begin{aligned} \gamma(w_{k,N}) - \gamma(v_{k,N}) & \ > \ (1-\frac{\phi}{\theta}) \gamma(v_{k,N}) - \gamma(v_{k,N}) \ = \ - \frac{\phi}{\theta} \gamma(v_{k,N}) \ > \ - \frac{\phi}{\theta} \bar{\epsilon} \label{eq:44}\end{aligned}$$ Since in equation , our choice of $\bar{\epsilon} > 0$ is arbitrary, this is true for any $\bar{\epsilon} > 0$, which implies for all $k \geqslant L_2$, we must have $\gamma(w_{k,N}) - \gamma(v_{k,N}) > 0$. Then, from the definition, we have, $$\begin{aligned} \label{eq:45} \gamma(u_{k,N}) = \gamma(w_{k,N}- v_{k,N}+ \frac{1}{p}e) \geqslant \frac{\gamma(w_{k,N}- v_{k,N})+1}{p} \geqslant \frac{\gamma(w_{k,N})- \gamma(v_{k,N})+1}{p} \end{aligned}$$ As a simple consequence of the definition and the condition $\alpha, \beta \in Q$, we have, $$\begin{aligned} \frac{\theta}{2(1-\theta \gamma(w_{k,N})-(\phi-\theta)\gamma(v_{k,n}))} & = \frac{\bar{\alpha}}{2(1-\alpha-\beta)} = \frac{\alpha}{2(1-\alpha-\beta)} \frac{1}{\gamma(u_{k,N})} \leqslant \frac{1}{\gamma(u_{k,N})} \nonumber \\ \frac{\phi}{2(1-\theta \gamma(w_{k,N})-(\phi-\theta)\gamma(v_{k,n}))} & = \frac{\bar{\beta}}{2(1-\alpha-\beta)} = \frac{\beta}{2(1-\alpha-\beta)} \frac{1}{\gamma(v_{k,N})} \leqslant \frac{1}{\gamma(v_{k,N})} \label{eq:46}\end{aligned}$$ Now, as $\theta (w_k)_j+ (\phi-\theta)(v_k)_j \leqslant \theta \gamma(w_{k,N}) + (\phi-\theta)\gamma(v_{k,N}) $ for all $j \in N$, using Lemma \[lemma-5\] and equation , we have, $$\begin{aligned} - \sum\limits_{j \in N} & \log \left(1- \theta (w_k)_j - (\phi- \theta) (v_k)_j\right) \nonumber \\ & \leqslant \ \theta \delta_k + (\phi-\theta) \omega_k + \frac{\|\theta w_{k,N}+ (\phi -\theta) v_{k,N}\|^2}{2(1-\theta \gamma(w_{k,N})-(\phi-\theta)\gamma(v_{k,n}))} \nonumber \\ & \leqslant \ \theta \delta_k + (\phi-\theta) \omega_k + \frac{\theta \|w_{k,N}\|^2 }{\gamma(u_{k,N})} + 2\frac{\epsilon(\phi-\theta) \|w_{k,N}\|}{\gamma(u_{k,N})} + \frac{(\phi-\theta)^2 \sigma_k^2 }{\theta \gamma(u_{k,N})} \label{eq:47}\end{aligned}$$ Now, by combining equation and , we have, $$\begin{gathered} F_N(x_{k+1})- F_N(x_k) \ \leqslant \ \theta \|w_{k,N}\|^2 [-p + \frac{1}{\gamma(u_{k,N})}] + 2 \epsilon \|w_{k,N}\| (p \theta + \frac{\phi -\theta}{\gamma(u_{k,N})}) \\ + \sigma_k^2 (-p \theta + \frac{(\phi -\theta)^2}{\theta \gamma(u_{k,N})})- p \theta \epsilon_k - p \theta \delta_k + (\phi + \theta) \omega_k - p \phi h_k \label{eq:48}\end{gathered}$$ Using the lower bound of equation , we can easily find, $$\begin{aligned} -p \theta + \frac{\theta}{\gamma(u_{k,N})} \ & \leqslant \ - p \theta \frac{\gamma(w_{k,N}) - \gamma(v_{k,N})}{1+\gamma(w_{k,N}) - \gamma(v_{k,N})} = - p \bar{a} \\ p \theta + \frac{\phi -\theta}{\gamma(u_{k,N})} \ & \leqslant \ p \frac{\phi + \theta(\gamma(w_{k,N}) - \gamma(v_{k,N}))}{1+\gamma(w_{k,N}) - \gamma(v_{k,N})} = p \bar{b}\end{aligned}$$ Where, by the definition of $\bar{a}$ and $\bar{b}$ given above, we can show that $\bar{a}, \bar{b} > 0$ are both finite constants. Now, from Theorem \[theorem-5\], we see that $\sum\limits_{k =L}^{\infty} (F_N(x_{k+1})- F_N(x_k)) > - \infty$. Also from Theorem \[theorem-9\], we get the following relation, $$\begin{aligned} \label{eq:49} \sum\limits_{k =L}^{\infty} \left(|\delta_k| + \epsilon_k + \omega_k + h_k + \sigma_k^2\right) < \infty\end{aligned}$$ Considering equations , and , for all $k \geqslant L$, we have, $$\begin{aligned} & \ \text{Case 1:} \quad \quad \sum\limits_{k =L}^{\infty} \|w_{k,N}\|^2 - 2 \epsilon \sum\limits_{k =L}^{\infty} \|w_{k,N}\| \ < \ \infty \label{eq:50} \\ & \ \text{Case 2:} \quad \quad \bar{a} \sum\limits_{k =L}^{\infty} \|w_{k,N}\|^2 - \bar{b} \sum\limits_{k =L}^{\infty} \|w_{k,N}\| \ < \ \infty \label{eq:51}\end{aligned}$$ Both of the above cases, equation and imply that either the sequence $\{\gamma(w_{k,N})\}$ has a strictly positive/negative cluster point or $\lim_{k \rightarrow \infty }\gamma(w_{k,N}) = 0 $. If $\{\gamma(w_{k,N})\}$ has a cluster point then we must have, $\|w_{k,N}\| \rightarrow 0$. Now, since $e^Tw_{k,N} = \delta_k$ and $\delta_k \rightarrow 0$, this implies whenever $\lim_{k \rightarrow \infty }\gamma(w_{k,N}) = 0 $, we must have, $w_{k,N} \rightarrow 0$. Either way, we have the following relationship, $$\begin{aligned} & \lim_{k \rightarrow \infty} w_{k,N} = 0 \ \ \Rightarrow \ \lim_{k \rightarrow \infty} u_{k,N} = \frac{1}{p} e \label{eq:52}\end{aligned}$$ Now, for each $j \in N$, consider the sequences $\{\frac{(x_k)_j}{c^Tx_k-c^Tx^*}\}, \{\frac{(x_k)_i-(x^*)_i}{c^Tx_k-c^Tx^*}\}, \{y_k\}$ and $\{s_k\}$ for each $i \in B$. Let $s_{p_k} \rightarrow s^*$ for some sub-sequence $\{p_k\}$ of $k$. Since all of them are bounded for all $1 \leqslant i,j \leqslant n$. Thus using equation , we have, $$\begin{aligned} y_{p_k} \rightarrow y^*, \quad s_{p_k} \rightarrow s^*, & \quad \frac{p(x_{p_k})_j}{c^Tx_{p_k}-c^Tx^*} \rightarrow a_j \quad \text{for each} \quad j \in N \\ & p\frac{(x_{p_k})_i-(x^*)_i}{c^Tx_{p_k}-c^Tx^*} \rightarrow b_j \quad \text{for each} \quad j \in B \end{aligned}$$ Considering equation , we know that $a_j > 0$ for all $j \in N$ and $(w_{p_k})_j \rightarrow 0$, for all $j \in N$ we have the following, $$\begin{aligned} (s_{p_k})_j = \frac{c^Tx_{p_k}-c^Tx^*}{(x_{p_k})_j} \left((w_{p_k})_j - \frac{1}{p}\right) \rightarrow \ \frac{1}{a_j}\end{aligned}$$ Since from the definition of $D$, $A_Nx_{k,N}+ A_Bx_{k,B} = A_Bx_B^* + A_N * 0 = A_Bx_B^*$ holds and taking the limits, we see that $A_Na+ A_B b = 0$. This implies that $s_j = \frac{1}{a_J}$, for each $j \in N$ and $x = [x_B, x_N] = [-a, -b], \ s =[0, s_N^*], \ y = y^*$ solve the corresponding *Karush Kahn Tucker* (KKT) conditions for the *Analytic Center Problem* . Thus, $s_{p_k,B}$ converges to the analytic center for each sub-sequence, which in turn proves part (2) and (3) of Theorem \[theorem-10\]. For proving the optimality, we notice that as $x^*, y^*$ and $s^*$ satisfies the primal and dual feasible criteria, respectively. They also satisfy the complementary slackness property. Thus, $x^*, y^*$ and $s^*$ are the optimal solutions for the respective primal and dual problems. Notice that, Theorem \[theorem-10\] is a generalization of the original AFS algorithm. In this Theorem, if we consider $\beta = 0$ (without acceleration term), then it gives us the condition $\alpha \leqslant \frac{2}{3}$, which is in fact the respective bound for the original AFS (see [@saigal:1996], [@tsuchiya:1995]). \[theorem-y\] If $\alpha, \beta \in Q$, then there exist vectors $x^*, y^*,s^*$ such that the sequences $\{B(x_k)\}$, $\{y_k\}$ and $\{s_k\}$ generated by the AAFS algorithm converges to $x^*, y^*$ and $s^*$ respectively, i.e., 1. $B(x_k) \rightarrow x^*$, 2. $y_k \rightarrow y^*$, 3. $s_k \rightarrow s^*$ Where $x^*, y^*$ and $s^*$ are the optimal solutions of the respective primal and dual problems, and they also satisfy the strict complementary slackness property. Furthermore, the dual pair $(y^*,s^*)$ converges to the analytic center of the optimal dual face and the primal solution $x^*$ converges to the relative interior of the optimal primal face. From Theorem \[theorem-10\], we know that the sequence $\{x_k\}, \{y_k\}$ and $\{s_k\}$ generated by the GAFS converges to $x^*, y^*$ and $s^*$, respectively. Then, using the definition and the basic idea of SST, we can conclude that for all $j =1,2,...,n$ and $\alpha, \ \beta \in Q$ the following relation holds, $$\begin{aligned} \lim_{k \to \infty} (B(x_k))_j = (x^*)_j, \quad \lim_{k \to \infty} y_k = y^*, \quad \lim_{k \to \infty} s_k = s^*\end{aligned}$$ Since, this holds for all $j =1,2,...,n$, we can prove $\lim_{k \to \infty} B(x_k) = x^*$. The last part is satisfied as we do not update the dual sequences at each iteration based on the sequence $\{B(x_k)\}$. Numerical Experiments {#sec:num} ===================== In this section, we verified the efficiency of the proposed variants of primal Affine Scaling algorithm presented in Section 2 through several numerical experiments. All of the experiments were carried out in a Intel Xeon Processor E5-2670, with double processors each with 20 MB cache, 2.60 GHz, 8.00 GT/s Intel QPI and 64 GB memory CPU. For simplicity of exposition, we considered three pairs of step sizes $(\alpha, \beta) = (0.4,0.2)$, $ (0.5,0.1)$ and $ (0.55,0.1)$, respectively for our experimental setup. We considered three types of LP problems: (1): *Randomized Gaussian LP*, (2): *Netlib Sparse LP* (real life instances [@netlib]) and (3): *Randomized LP* with increasing $n$, constant $m$. We evaluated the performance of GAFS and AAFS with a long-step version of classical AFS. We considered duality gap tolerance $\epsilon $ as $10^{-3}$, $10^{-4}$ and $10^{-7}$ respectively and compared the results of our algorithms with the commercial LP solver (CPLEX-dual simplex [@cplex]) and with the *MATLAB Optimization Toolbox* function *fmincon* [^3] [@fmincon]. The *fmincon* function allows us to select ‘Interior Point’ algorithm for a basic comparison as AFS is also an Interior Point Method. Comparison among AFS, GAFS and AAFS for dense data: {#subsec:1} --------------------------------------------------- The random dense data for these tests are generated as follows: All elements of the data matrix $A \in {\mathbb{R}}^{m \times n}$ and the cost vector $c \in {\mathbb{R}}^n$ are chosen to be *i.i.d.* Gaussian $ \sim \mathcal{N} (-9,9)$. The right hand side $b \in \mathbb{R}^m$ is generated at random from corresponding distribution, but we made sure that $b \in \mathcal{R}(\mathbf{A})$. For that, we generated two vectors $x_1, x_2 \in {\mathbb{R}}^n$ at random from the corresponding distributions, then multiplied them by $A$ and set $b$ as a convex combination of those two vectors. We also made sure that the generated problems have bounded feasible solutions. We ran all algorithms 15 times and reported the averaged performance. [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, dense data)[]{data-label="fig:1"}](040203.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, dense data)[]{data-label="fig:1"}](050103.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, dense data)[]{data-label="fig:1"}](550103.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, dense data)[]{data-label="fig:2"}](040204.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, dense data)[]{data-label="fig:2"}](050104.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, dense data)[]{data-label="fig:2"}](550104.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, dense data)[]{data-label="fig:3"}](040207.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, dense data)[]{data-label="fig:3"}](050107.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, dense data)[]{data-label="fig:3"}](550107.png "fig:"){width="\linewidth"} In the above figures (Figure \[fig:1\], \[fig:2\] and \[fig:3\]), we compared AFS, GAFS and AAFS for different sets of $\alpha, \beta$ with diferent duality gaps. Our results show that the proposed variant algorithms reduced the runtime significantly. Furthermore, the reduction of runtime increases as the size of the instance gets larger (see Figure \[fig:1\], \[fig:2\] and \[fig:3\]). From figure \[fig:1\], \[fig:2\] and \[fig:3\], we can conclude that the GAFS is faster than the original AFS irrespective of the size of the instances. Similarly, AAFS further accelerates the convergence of GAFS as the runtime decreases for all the instances. This is due to the integration of SST with the acceleration process and it converges much faster than the original AFS algorithm. Now, we compared the performance of our proposed algorithms and AFS with the standard LP solvers *fmincon* and ‘CPLEX-dual simplex’. For the comparison, we chose the best $\alpha, \beta$ pair from the above results ($\alpha = 0.55, \ \beta = 0.1$, validates our claim that ‘good result for larger $(\alpha+ \beta) \in Q$’) and compared them for the duality gaps. At first, we presented the comparison graph for all the algorithms (AFS, GAFS, AAFS, *fmincon* and CPLEX), where it is evident that performance of original AFS compare to *fmincon* and CPLEX solver is very poor. However, our proposed acceleration scheme has significantly improve this performance gap. For better understanding and fairness benchmark comparison, we compared our proposed AAFS with classical AFS and *fmincon* (this comparison is fair as *fmincon* uses the raw Barrier function method and CPLEX uses dual simplex [@cplex:h], [@cplex]). [0.32]{} ![Comparison with ‘*fmincon*’ and ‘CPLEX’ (dense data)[]{data-label="fig:4"}](randomall03.png "fig:"){width="\linewidth"} [0.32]{} ![Comparison with ‘*fmincon*’ and ‘CPLEX’ (dense data)[]{data-label="fig:4"}](randomall04.png "fig:"){width="\linewidth"} [0.32]{} ![Comparison with ‘*fmincon*’ and ‘CPLEX’ (dense data)[]{data-label="fig:4"}](randomall07.png "fig:"){width="\linewidth"} [0.32]{} ![Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (dense data)[]{data-label="fig:5"}](random03.png "fig:"){width="\linewidth"} [0.32]{} ![Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (dense data)[]{data-label="fig:5"}](random04.png "fig:"){width="\linewidth"} [0.32]{} ![Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (dense data)[]{data-label="fig:5"}](random07.png "fig:"){width="\linewidth"} Instances AFS GAFS AAFS *fmin* ----------- ------ ------ --------- --------- -------- -------- $m$ $n$ Time Time Time Time 1 400 650 9.02 7.96 7.11 5.07 2 700 1000 18.16 16.22 13.87 9.87 3 1000 2000 67.64 53.34 43.46 31.47 4 1500 2500 123.95 111.40 99.30 69.34 5 2000 3000 202.95 187.67 176.34 142.47 6 2500 3000 231.20 216.90 181.03 147.79 7 3000 3500 351.00 321.88 287.60 218.73 8 3500 4000 467.07 401.00 356.10 281.20 9 4000 4500 700.13 610.20 497.86 406.39 10 4500 5500 1281.00 1019.00 787.20 619.31 : Comparison among ‘GAFS’, AAFS’, ‘AFS and *fmin*’$^a$[]{data-label="table:1 duality = -03"} \ Based on the above figure (Figure \[fig:5\]) and Table \[table:1 duality = -03\], we concluded that AFS takes on average [^4] 75-80 $\%$ (*min.* 40 $\%$, *max.* 118 $\%$) more CPU time than *fmincon* (Barrier method). However, our proposed AAFS takes on average 25-30 $\%$ (*min.* 20 $\%$, *max.* 45 $\%$) more CPU time than *fmincon*. It is evident that the proposed acceleration reduced the CPU time consumption considerably (approximately on average 50 $\%$ reduction). The reason for this is that the classical AFS is an exponential time algorithm whereas the Barrier method is a polynomial time method. When we applied the proposed generalization and acceleration in AFS to generate GAFS and AAFS method respectively, they did well against Barrier function method as both methods uses the history information (i.e., AFS uses only $x_k$ to generate $x_{k+1}$, GAFS and AAFS uses $x_0,x_1,x_2,...,x_k$ to generate $x_{k+1}$). Comparison among AFS, GAFS and AAFS for sparse data {#subsec:2} --------------------------------------------------- In this subsection, we investigated the performance behaviour of classical AFS with the proposed GAFS and AAFS methods and also with MATLAB Optimization Toolbox function *fmincon* [@fmincon] for several *Netlib* LP instances (real life examples with sparse data [@netlib]). The experiment parameters remains the same as in the randomized instances. Figures \[fig:6\], \[fig:7\] and \[fig:8\] show the comparison graphs among AFS, GAFS and AAFS. In the following figures, we compared AFS, GAFS and AAFS with the same parameters for the *Netlib* LPs [^5]. Based on the figures below (Figure \[fig:6\], \[fig:7\] and \[fig:8\]), we can conclude that GAFS is faster than the original AFS and AAFS further accelerates the convergence of GAFS for all of the instances. Furthermore, one can notice that the runtime graphs do not follow the same trend as before (Figure \[fig:1\], \[fig:2\] and \[fig:3\]). The main reason for that is in these instances, we have another important parameter involved called sparsity (portion of nonzero entries in the matrix, $\delta = \frac{\text{nonzero entries of} \ A}{\text{total entries of} \ A}$) of the data matrix $A$. As shown in the following figures, sparsity affected the performance of the algorithms significantly. [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, sparse data)[]{data-label="fig:6"}](netlib040203.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, sparse data)[]{data-label="fig:6"}](netlib050103.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-3}$, sparse data)[]{data-label="fig:6"}](netlib550103.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, sparse data)[]{data-label="fig:7"}](netlib040204.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, sparse data)[]{data-label="fig:7"}](netlib050104.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-4}$, sparse data)[]{data-label="fig:7"}](netlib550104.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, sparse data)[]{data-label="fig:8"}](netlib040207.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, sparse data)[]{data-label="fig:8"}](netlib050107.png "fig:"){width="\linewidth"} [0.32]{} ![ Number of columns and rows ($m+n$) vs run time (duality gap $= 10^{-7}$, sparse data)[]{data-label="fig:8"}](netlib550107.png "fig:"){width="\linewidth"} Now, we evaluated the performance of our proposed algorithms and AFS with the standard solver *fmincon*. At first, we presented the comparison figure for all the algorithms (AFS, GAFS, AAFS and *fmincon*). Then finally, for better understanding, we compared our proposed AAFS with classical AFS and *fmincon* (we explain in subsection \[subsec:1\] that it is fair to compare with *fmincon* as AFS, GAFS, AAFS and *fmincon* uses the raw Interior Point Method whereas CPLEX uses dual simplex method [@cplex:h], [@cplex]). [0.32]{} ![ Comparison of ‘GAFS’ and ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:9"}](netlib03.png "fig:"){width="\linewidth"} [0.32]{} ![ Comparison of ‘GAFS’ and ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:9"}](netlib04.png "fig:"){width="\linewidth"} [0.32]{} ![ Comparison of ‘GAFS’ and ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:9"}](netlib07.png "fig:"){width="\linewidth"} [0.32]{} ![ Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:10"}](netliball03.png "fig:"){width="\linewidth"} [0.32]{} ![ Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:10"}](netliball04.png "fig:"){width="\linewidth"} [0.32]{} ![ Comparison of ‘AAFS’ with ‘AFS’ and ‘*fmincon*’ (sparse data)[]{data-label="fig:10"}](netliball07.png "fig:"){width="\linewidth"} ’ \[table:2 random duality -03\] [@ccccccccc@]{} Instances & & --------- Nonzero Entries --------- : Comparison of ‘GAFS’ and ‘AAFS’ with ‘AFS’ and ‘*fmin*$^a$ & Sparsity & AFS & GAFS & AAFS & *fmin*\ Title & $m$ & $n$ & $Z$ & $\delta$ & Time & Time & Time & Time\ lp\_blend & 377 & 114 & 1302 & 0.0303 & 6.83 & 3.98 & 2.57 & 1.54\ lp\_adlittle & 389 & 138 & 1206 & 0.0225 & 8.76 & 5.04 & 3.19 & 1.97\ lp\_stocfor1 & 565 & 165 & 1359 & 0.0146 & 9.70 & 5.94 & 3.72 & 2.12\ lp\_recipe & 591 & 204 & 1871 & 0.0155 & 10.75 & 7.13 & 5.47 & 3.19\ lp\_brandy & 1047 & 303 & 5012 & 0.0158 & 13.71 & 12.91 & 9.1 & 7.15\ lp\_bandm & 1555 & 472 & 6097 & 0.0083 & 29.30 & 25.74 & 20.95 & 19.79\ lp\_scorpion & 1709 & 466 & 4282 & 0.0054 & 18.01 & 14.64 & 11.16 & 9.61\ lp\_agg & 2207 & 615 & 7085 & 0.0052 & 25.37 & 20.48 & 16.81 & 14.47\ lp\_degen2 & 2403 & 757 & 10387 & 0.0057 & 14.41 & 12.70 & 9.71 & 7.67\ lp\_finnis & 3123 & 1064 & 8052 & 0.0024 & 31.59 & 25.73 & 22.59 & 19.34\ \ Based on the above figure (Figure \[fig:10\]) and Table \[table:2 random duality -03\], we conclude that AFS takes on average 160-170 $\%$ (*min.* 50 $\%$, *max.* 357 $\%$) more CPU time than *fmincon* (Barrier method). In comparison our proposed AAFS takes on average 30-35 $\%$ (*min.* 16 $\%$, *max.* 75 $\%$) more CPU time than *fmincon*. It is evident that the proposed acceleration reduced the CPU time consumption considerably (approximately on average 130-135 $\%$ reduction). The reason for this is that we use the history information (i.e., $x_0, x_1, ..., x_k$ to update $x_{k+1}$) to develop GAFS and we use acceleration to GAFS to develop AAFS. By the construction of AAFS and GAFS, they should accelerate the convergence of AFS based on our convergence analysis, presented in Section \[sec:primal\]. Another important consequence of acceleration is that the proposed AAFS is competitive with ‘*fmincon*’ for sparse instances (for some instances i.e., lp-bandm AAFS time consumption is much closer to ‘*fmincon*’ solver). Comparison between AFS, GAFS and AAFS for large $n$ and fixed $m$ {#subsec:3} ----------------------------------------------------------------- In this subsection, we considered the case where $m$ stays constant and $n$ grows exponentially. For better understanding, we consider $m = 50, 100$ and $n = 100,1000,10000,100000,1000000$ and run the experiments for duality gap $= 10^{-3}$ and $\alpha = 0.55, \beta = 0.1$. The comparison results of all algorithms (AFS, GAFS, AAFS and *fmincon*) for this special instance is shown in Figure \[fig:11\]. [0.45]{} ![ Comparison of GAFS, AAFS, AFS and *fmincon* while $m$ stays constant, $n$ increasing[]{data-label="fig:11"}](m03wplex50.png "fig:"){width="\linewidth"} [0.45]{} ![ Comparison of GAFS, AAFS, AFS and *fmincon* while $m$ stays constant, $n$ increasing[]{data-label="fig:11"}](m03wplex100.png "fig:"){width="\linewidth"} One interesting fact can be noted from Figure \[fig:11\] is that the runtime graph follows the logarithmic trend as opposed to the exponential trend obtained in randomized instances. The main reason for this phenomenon is that though $n$ is increasing and total number of entries in the data-set $A$ is also increasing, the computational cost for computing the term $AX(AX^2A^T)^{-1}XA^T$ is cheap ($A \in {\mathbb{R}}^{m \times n}, \ AA^T \in {\mathbb{R}}^{m \times m}$, for small $m$, $AA^T$ inversion is cheap compared to the other cases). Furthermore, AAFS and GAFS both algorithms outperformed the classical AFS which supports the claim of this work. For a better understanding of the comparison, we plotted AFS, AAFS and *fmincon* and the result is shown in Figure \[fig:12\]. Based on Figure \[fig:12\], we can conclude that AFS takes on average 130-140 $\%$ (*min.* 54 $\%$, *max.* 240 $\%$) more CPU time than *fmincon* (Barrier method). In comparison, our proposed AAFS takes on average 28-35 $\%$ (*min.* 12 $\%$, *max.* 52 $\%$) more CPU time than *fmincon*. It is evident that the proposed acceleration reduced the CPU time consumption considerably (approximately on average 100-105 $\%$ reduction). [0.45]{} ![ Comparison of GAFS, AFS and *fmincon* while $m$ stays constant, $n$ increasing[]{data-label="fig:12"}](m0350.png "fig:"){width="\linewidth"} [0.45]{} ![ Comparison of GAFS, AFS and *fmincon* while $m$ stays constant, $n$ increasing[]{data-label="fig:12"}](m03100.png "fig:"){width="\linewidth"} Our main goal of the numerical experiments is to show that the proposed GAFS and AAFS accelerate the classical AFS and support the claim proven in Section \[sec:primal\]. From the numerical results presented in Subsections \[subsec:1\], \[subsec:2\] and \[subsec:3\], it is evident that the GAFS works faster than the classical AFS and AAFS outperforms AFS and GAFS for all of the instances. Apart from that, we also compared the proposed algorithms with standard LP solvers like *fmincon* and CPLEX. Although the proposed GAFS and AAFS did not outperform the commercial LP solvers but in comparison with AFS, AAFS reduces the CPU time considerably (approximately on average 93-97 $\%$) for all of the instances. And almost for all of the instances, AAFS CPU time consumption is within (approximately 20-25 $\%$) of *fmincon* solver. Proposed accelerated work show evidence of potential opportunity for applying acceleration to other Interior Point methods (i.e., Barrier method). Furthermore, it is a natural question to ask whether GAFS and AAFS require more computational cost compared to the original AFS for the additional acceleration effort. Since, the extra term $\frac{\beta}{\|X_k^{-1}(x_k-x_{k-1})\|_{\infty}}(x_k-x_{k-1})$ in GAFS requires only $O(n^2)$ algebraic operations and the extra term $B(x_k)$ in AAFS requires only $O(n^3)$ algebraic operations, both GAFS and AAFS require at most $O(n^3)$ algebraic operations at each iterations which make them computationally cheap. Benefits gained from the proposed accelerated techniques offset this additional computational effort. While the original AFS algorithm uses the current update to find the next update, the proposed algorithms use all the previous updates to find the next update (see Theorem \[theorem-1\]) and thus the proposed generalized algorithm runs faster than the original algorithm. Conclusion {#sec:conc} ========== In this work, we proposed two Affine Scaling algorithms for solving LP problems. The first algorithm (GAFS) integrated Nesterov’s restarting strategy with the AFS method. Here, we introduced an additional residual term to the extrapolation step and determined the acceleration parameter $\beta$ adaptively. The proposed algorithm also generalizes the original AFS algorithm in the context of an extra parameter (i.e., the original AFS has $\beta =0$). The second algorithm (AAFS) integrated Shanks non-linear acceleration technique with the update of GAFS. Here, we introduced entry-wise SST to accelerate the process of GAFS. It is evident from our numerical experiments that the proposed AAFS and GAFS outperformed the classical AFS algorithm in comparison with standard LP solver *fmincon* (AFS takes approximately more than 121-130 $\%$ of CPU time compared to *fmincon* whereas in comparison AAFS takes approximately more than 20-25 $\%$). In terms of theoretical contribution, our proposed GAFS and AAFS revealed some interesting properties about the convergence characteristics of Affine Scaling method in general. Based on our analysis it is evident that the convergence criterion for AFS “$\alpha \leq \frac{2}{3}$” is a universal bound as GAFS and AAFS satisfy a more generalized bound “$\alpha + \beta \leq \frac{2}{3}$”. Finally, we believe that standard LP solvers can adapt acceleration in Barrier method (see below) for designing much more efficient solvers based on the theoretical and numerical results presented in this work.\ **Future research:** Based on our theoretical and numerical analysis of GAFS and AAFS, it is evident that in future it is possible to discover a more general family of numerically efficient and theoretically interesting Affine Scaling methods and our work can help researchers to look for these types of methods for other IPMs. Moreover, based on our theoretical analysis, we believe these types of efficient Affine Scaling variants can also be designed for the following class of problems: Semi Definite Programming [@vanderbei:1999], Nonlinear Smooth Programming [@wang:2009], Linear Convex Programming [@cunha:2011], Support Vector Machine [@maria:2011], Linear Box Constrained Optimization [@wang:2014], Nonlinear Box Constrained Optimization [@huang:2017]. This is due to the similarity of these methods in terms of algorithmic structure, i.e., the only difference between AFS and the above-mentioned AFS variants [@vanderbei:1999; @wang:2009; @cunha:2011; @maria:2011; @wang:2014; @huang:2017] is the defining formulas for the sequences $\{y_k\}$ and $\{s_k\}$ (see Algorithm \[alg:acc AFS\] in Section \[sec:afs\]). Furthermore, recent developments in optimization literature shows that Affine Scaling scheme is quite competitive with the state-of-the-art techniques for Linear Box Constrained Optimization, Nonlinear Box Constrained Optimization and Support Vector Machine problems [@wang:2014; @huang:2017; @maria:2011]. Our convergence analysis will enrich optimization literature and serve as a theoretical basis for applying the proposed acceleration schemes to the above-mentioned algorithms. Finally, the numerical results and the convergence analysis suggest that acceleration can also be applied to other efficient Interior Point methods (Barrier method/Path following method and Karmarkar method). Since AFS and Barrier method follow the same scheme (only difference is that the objective function defined in the EAP problem \[eq:2\] is different for Barrier method), the convergence analysis will also hold for Barrier method. Note that, for the Affine Scaling method, we defined the EAP problem (equation \[eq:2\] is Section \[sec:afs\]) as follows: $$\begin{aligned} \label{eq:n} \textbf{min} \quad & w = c^Td \nonumber\\ \textbf{s.t} \ & Ad =0 \\ & \| X^{-1}d \| \leqslant \alpha \nonumber\end{aligned}$$ The main difference among Affine Scaling, Karmarkar method and Path following/Barrier method is that the later two methods use the following objective functions $G(x+d,s)$ and $B_\mu (x+d)$ in the EAP problems respectively, $$\begin{aligned} w = G(x+d,s) & \overset{\underset{\mathrm{def}}{}}{=} q \log s^T(x+d) - \sum\limits_{j=1}^{n} \log (d_j+x_j)- \sum\limits_{j=1}^{n} \log s_j \\ w = B_{\mu}(x+d) & \overset{\underset{\mathrm{def}}{}}{=} c^T(x+d) - \mu \sum\limits_{j=1}^{n} \log (d_j+x_j)\end{aligned}$$ Here, $(x,s)$ is the primal dual pair and $q, \mu$ are parameters. Now, instead of using the original functions, we approximate the above penalty functions as follows: $$\begin{aligned} G(x+d,s) & \approx G(x,s) + \nabla^T G(x,s) d = G(x,s) + \frac{q}{s^Tx} s - X^{-1} =: w \ \\ B_{\mu}(x+d) & \approx B_{\mu}(x) + \left(c^T- \mu e^T X^{-1}\right) d + \frac{1}{2} \mu d X^{-2}d =: w\end{aligned}$$ We can easily find the respective optimal solution $d^*$ for the EAP problems for the Karmarkar method and the Barrier method as follows: $$\begin{aligned} \label{eq:d} d^*_{\text{Karmarkar}} & = - \alpha \frac{Xu}{\|u\|} ; \quad u = \left(I - XA^T \left(AX^2A^T\right)^{-1} AX\right) \left(\frac{q}{s^Tx} Xs - e\right) \nonumber\\ d^*_{Barrier}(\mu) & = \left(I - X^2A^T \left(AX^2A^T\right)^{-1}A\right) \left(Xe - \frac{1}{\mu}X^2 c\right) \end{aligned}$$ Finally, considering the optimal direction vectors in , we can write the update formula for the primal sequences as follows: $$\begin{aligned} & \text{Karmarkar:} \quad x_{k+1} = z_k + d^*_{\text{Karmarkar}} \\ & \text{Barrier:} \quad x_{k+1} = z_k + d^*_{\text{Barrier}}\end{aligned}$$ Here, the sequence $z_k$ is defined in . Using the similar setup of this work, we can construct the update formula for the respective dual sequences. We also believe that by designing special iterated sequences for $\alpha_k, \beta_k$ with appropriate choice of sequences $\{\pi_k\}$ and $\{\tau_k\}$, one can design efficient quadratically convergent algorithms, i.e., $$\begin{aligned} & \alpha_{k+1} =: \alpha_k + \pi_k \\ & \beta_{k+1} =: \beta_k + \tau_k\end{aligned}$$ Acknowledgements ================ We thank Mr. Tasnim Ibn Faiz, MIE, Northeastern University for his help and advice on programming in AMPL. We also thank Mr. Md Saiful Islam, MIE, Northeastern University for his help in the Numerical section. Finally, the authors are truly grateful to the anonymous referees and the editor for their valuable comments and suggestions in the revision process which have greatly improved the quality of this paper. Appendix A {#appendix-sec1 .unnumbered} ========== In this section, we presented some Lemmas and their proofs which are required for the convergence analysis (most of the Lemmas are well known in the literature but they need proof in our context). \[lemma-6\] For $\alpha, \beta \in Q$, the sequences $\beta_{k}$ and $\gamma_k$ defined in have the following properties: 1. There exists an $L \geqslant 1$ such that $\beta_k < 1$ for all $k \geqslant L$. 2. The sequence $\gamma_k \rightarrow 0$ as $k \rightarrow \infty$. Let, $\lim_{k \rightarrow \infty} x_k = x^*$ and $N = \{j \ | \ (x^*)_j = 0\}$. Then, we have $\lim_{k \rightarrow \infty} (x_k)_j = 0$, for all $j \in N$. As we know, $|\beta| < 1$, there exists a $M > 0$ and $L \geqslant 1$ such that for all $k \geqslant L$, we have, $$\begin{aligned} (x_{k-1})_j \ \geqslant \ M \beta^{k-1}, \ (x_{k})_j \ \leqslant \ M \beta^{k}\end{aligned}$$ Thus for all $j \in N$ and for all $k \geqslant L$, we have, $$\begin{aligned} \|X_k^{-1} \left(x_{k-1}-x_k\right)\|_{\infty} \ \geqslant \ |\frac{(x_{k-1})_j}{(x_k)_j} -1| \ \geqslant \ |\frac{ M \beta^{k-1}}{M \beta^{k}} -1| = \frac{1- \beta}{\beta}\end{aligned}$$ Since, $\beta \in Q$, thus $\beta < \frac{1}{\phi}$ gives us $\frac{\beta^2}{1-\beta} < 1$, therefore, we have, $$\begin{aligned} \beta_k = \frac{\beta}{\|X_k^{-1} \left(x_{k-1}-x_k\right)\|_{\infty}} \ < \ \frac{\beta^2}{1-\beta} \ < \ 1\end{aligned}$$ This proves part (1) of Lemma \[lemma-6\]. For the second part, let $\tilde{\beta} = \frac{\beta^2}{1-\beta} < 1$, then it is easy to see that for all $k \geqslant L, m > 0$, we have, $$\begin{aligned} \gamma_{k+m} \ = \ \prod_{i =1}^{k-1} \beta_{i}. \prod_{j =k}^{k+m} \beta_{j} \ < \ \prod_{i =1}^{k-1} \beta_{i}. \left(\tilde{\beta}\right)^{m} \rightarrow 0 \ \text{as} \ m \rightarrow \infty\end{aligned}$$ This proves $\gamma_k \rightarrow 0$ as $k \rightarrow \infty$. \[lemma-7\] There exists a $L_2 \geqslant 1$ such that for all $k \geqslant L_2$, $$\begin{aligned} \frac{\gamma_k}{c^Tx_k-c^Tx^*} \ < \ L_2\end{aligned}$$ From Lemma \[lemma-9\], we know that the sequence $\{u_k\}$ is bounded, it means there exists a $M_1 > 0$ such that for all $k$, we have $\|u_k\| \leqslant M_1$. Since, $\|X_ks_k\| > 0$, there exists a $\epsilon_2 > 0$ such that for all $k$, we have $ \|X_ks_k\| > \epsilon_2 $. Similarly, as $\gamma_k \rightarrow 0 $ as $k \rightarrow \infty$, there exists a $L_3\geqslant 1$ such that for all $k \geqslant L_3$, $$\begin{aligned} \gamma_k < \epsilon_2\end{aligned}$$ Combining these facts, for all $k \geqslant L_3$, we have, $$\begin{aligned} r_k \ = \ \frac{\gamma_k}{c^Tx_k-c^Tx^*} = \frac{\gamma_k \|u_k\|}{\|X_ks_k\|} \ \leqslant \ \frac{\epsilon_2 M_1}{\epsilon_2} = M_1 \end{aligned}$$ Let, $ L_2 = \max\{M_1, r_1, r_2,..., r_{L_3}\} $, then for all $k$, we have, $$\begin{aligned} r_k \ & \leqslant \ \max\{ r_1, r_2,..., r_{L_3}, r_{L_3+1}, ...\} \leqslant \ \max\{M_1, r_1, r_2,..., r_{L_3}\} = L_2\end{aligned}$$ Therefore, for all $k$ we have $r_k \ \leqslant \ L_2 $. \[lemma-8\] If we define $G(k) = \sum\limits_{j = k}^{\infty} \gamma_j$, then there exists a $\bar{N} > 0$ such that for all $k \geqslant 0$ the following relation holds: $$\begin{aligned} \label{eq:18} f_k \ = \ \frac{G(k)}{c^T x_k-c^Tx^*} \ \leqslant \ \bar{N}\end{aligned}$$ From Lemma \[lemma-6\], we know that there exists a $L \geqslant 1$ such that for all $k \geqslant L$, $\beta_k < 1$. Let $\tilde{\beta} = \max_{j \geqslant k}\{\beta_j\}$, then from the definition of $G(k)$ for all $k \geqslant L$, we have, $$\begin{aligned} f_k = \frac{G(k)}{c^T x_k-c^Tx^*} = & \frac{\gamma_k}{c^T x_k-c^Tx^*} (1+ \beta_{k+1}+ \beta_{k+1}\beta_{k+2}+ \beta_{k+1}\beta_{k+2}\beta_{k+3} + ...) \\ & \ < \ L_2 \sum\limits_{J=0}^{\infty} (\tilde{\beta})^j \ = \ \frac{L_2}{1-\tilde{\beta}}\end{aligned}$$ Let, $\bar{N} = \max \{f_1, f_2,..., f_{L}, \frac{L_2}{1-\tilde{\beta}}\}$, then for all $k \geqslant 0$, we have, $$\begin{aligned} f_k = \frac{G(k)}{c^T x_k-c^Tx^*} \leqslant \max \{f_1, f_2,..., f_{L}, \frac{L_2}{1-\tilde{\beta}}\} \ = \ \bar{N}\end{aligned}$$ This proves Lemma \[lemma-8\]. \[theorem-8\] The sequences $\{v_k\}$ and $\{h_k\}$ satisfy the following properties: Let $\epsilon > 0$, then there exists a $L \geqslant 1$ such that for all $k \geqslant L$, $$\begin{aligned} & (a) \quad \|v_{k,N}\|= \sigma_k, \ \ \sum\limits_{k = L}^{\infty} \sigma_k < \infty \quad \text{and} \quad \sum\limits_{k = L}^{\infty} \sigma_k^2 < \infty \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \\ & (b) \quad e^Tv_{k,N} = \omega_k, \ \ \sum\limits_{k = L}^{\infty} \omega_k < \infty \\ & (c) \quad \sum\limits_{k = L}^{\infty} h_k < \infty \\ & (d) \quad \text{There exists a} \ \ \epsilon_1 > 0 \ \text{such that} \ \epsilon_1 \ \leqslant \ \gamma(v_{k,N}) \ \leqslant \ \epsilon \\ & (e) \quad \|v_k\|_{\infty} = \|v_{k,N}\|_{\infty} = \gamma(v_{k}) = \gamma(v_{k,N})\end{aligned}$$ Part (a): We know from Theorem \[theorem-7\], there exist a $L_1 \geqslant 1$ such that for all $k \geqslant L_1$, $$\begin{aligned} c^Tx_{k}- c^Tx^* \leqslant \left(1- \frac{\alpha}{\sqrt{n}}\right) (c^Tx_{k-1}- c^Tx^*) \leqslant \left(1- \frac{\alpha}{\sqrt{n}}\right)^{k-L_1} (c^Tx_{L_1}- c^Tx^*) \label{eq:A1}\end{aligned}$$ As a consequence of , for all $k \geqslant L_1$, we have, $$\begin{aligned} \sum\limits_{k = L_1}^{\infty} \sigma_k = \sum\limits_{k = L_1}^{\infty} \|v_{k,N} \| & \ \leqslant \ \sum\limits_{k = L_1}^{\infty} \frac{\|X_k^{-1}(x_{k-1}-x_k)\|}{\|X_k^{-1}(x_k-x_{k-1})\|_{\infty}}(c^Tx_k-c^Tx^*) \\ & \ \leqslant \ \sqrt{n} \sum\limits_{k = L_1}^{\infty} (c^Tx_k-c^Tx^*)\\ & \ \leqslant \ \sqrt{n} (c^Tx_{L_1}- c^Tx^*) \sum\limits_{k = L_1}^{\infty} \left(1- \frac{\alpha}{\sqrt{n}}\right)^{k-L_1} \\ & \ = \ \frac{n(c^Tx_{L_1}- c^Tx^*) }{\alpha} \ < \ \infty\end{aligned}$$ Using the similar process, we can show that for all $k \geqslant L_1$, $$\begin{aligned} \sum\limits_{k = L_1}^{\infty} \sigma_k^2 \ = \ \sum\limits_{k = L_1}^{\infty} \|v_{k,N} \|^2 \ < \ \infty\end{aligned}$$ This proves part (a) of Theorem \[theorem-8\].\ Part (b): Using , for all $k \geqslant L_1$, we have, $$\begin{aligned} e^Tv_{k,N} = \frac{e^TX_{k,N}^{-1}(x_{k-1,N}-x_{k,N})}{\|X_k^{-1}(x_k-x_{k-1})\|_{\infty}}(c^Tx_k-c^Tx^*) \leqslant q \sqrt{n} ( c^Tx_{k}- c^Tx^* )\end{aligned}$$ Then we have, $$\begin{aligned} \sum\limits_{k = L_1}^{\infty} \omega_k \ = \ \sum\limits_{k = L_1}^{\infty} e^Tv_{k,N} \ \leqslant \ q \sqrt{n} \sum\limits_{k = L_1}^{\infty} ( c^Tx_{k}- c^Tx^* ) < \ \frac{nq(c^Tx_{L_1}- c^Tx^*) }{\alpha} \ < \ \infty\end{aligned}$$ This proves part (b) of Theorem \[theorem-8\].\ Part (c): From Lemma \[lemma-6\], we know there exist a $L_2 \geqslant 2$ such that for all $k \geqslant L_2$, $$\begin{aligned} \beta_k = \frac{\beta}{\|X_k^{-1} \left(x_{k-1}-x_k\right)\|_{\infty}} \ < \ \frac{\beta^2}{1-\beta} \ < \ 1\end{aligned}$$ As a consequence of the above relation, we have, $$\begin{aligned} \sum\limits_{k = L_2}^{\infty} h_k = \sum\limits_{k = L_2}^{\infty} \frac{c^T(x_{k-1}-x_k)}{\|X_k^{-1}(x_k-x_{k-1})\|_{\infty}} & = \sum\limits_{k = L_2}^{\infty} \frac{\beta_k}{\beta} (c^Tx_{k-1}-c^Tx_k) \\ & < \ \frac{\beta}{1-\beta} (c^Tx_{L_2-1}-c^Tx^*) \ < \ \infty\end{aligned}$$ This proves part (c) of Theorem \[theorem-8\].\ Part(d): Now, let $\epsilon > 0$. Since, $\lim_{k \rightarrow \infty} (c^Tx_k -c^Tx^*) = 0$ and $c^Tx_k -c^Tx^* > 0$, there exists an $\epsilon_1 > 0, L_3 \geqslant 1$ such that for all $k \geqslant L_3$, we have $\epsilon_1 < \gamma(v_{k,N}) < \epsilon$, which is exactly what we want for part (d) of Theorem \[theorem-8\].\ Part (e): Since, from the definition, we have $\lim_{k \rightarrow \infty} \frac{(x_{k-1})_j}{(x_{k})_j} = 1$, for all $j \in B$ and also $ \frac{(x_{k-1})_j}{(x_{k})_j} > 1$, for all $j \in N$, the required result follows as there exists a $L_4 \geqslant 1$ such that for all $k \geqslant L_4$, the following relation holds, $$\begin{aligned} \|v_k\|_{\infty} \ = \ \|v_{k,N}\|_{\infty} \ = \ \gamma(v_{k}) \ = \ \gamma(v_{k,N})\end{aligned}$$ Finally, if we choose $L = \max\{L_1, L_2,L_3,L_4\}$ and combine all of the identities, we get part (e) of Theorem \[theorem-8\]. Appendix B {#appendix-sec2 .unnumbered} ========== In this section, we provided some known results and their variants for the EAP problem given in . We discussed some properties of the sequences $\{x_k\}, \{y_k\}, \{s_k\}, \{d_k \} =\{X_k^2s_k\}$; generated by the GAFS discussed in Section \[sec:afs\]. We provided some required proofs here as well. These properties also hold for the original AFS. We also describe some of the properties of the solution of the EAP problem as per our formulation. The EAP problem is similar to the one well studied in the literature (see [@saigal:1996]). Since the generalization parameter doesn’t affect the EAP problem, we introduced some properties of EAP without providing proofs as most of the proofs are available in the literature (see [@dikin:1991], [@saigal:1996], [@tsuchiya:1995]). As shown in several works by Saigal [@saigal:1996], Vanderbei *et al.* [@vanderbei:1986] and Dikin [@dikin:1967], the solution $d^*$ of the EAP problem satisfies the following Lemma. \[lemma-1\] Assume that the rows of $A$ are linearly independent and $c$ is not a linear combination of the rows of $A$. Let $x$ and $z $ be some positive vectors with $Ax = Az =b$. Then the optimal solution $d^*$ of is given by $$d^*= - \alpha \frac{X^2(c-A^Ty)}{\|X(c-A^Ty)\|} = - \alpha\frac{X^2s}{\|Xs\|} \quad \text{with} \quad y = \left(AX^2A^T\right)^{-1}AX^2c$$ Furthermore, the vector $\bar{x} = x+ d^* + \bar{\beta} (x-z)$ satisfies $A \bar{x} =b$ and $c^T \bar{x} < c^T x$. We see that $d^*$ is the optimal solution of the EAP problem (2) (see [@vanderbei:1986]). Now, since $x$ and $z $ satisfy the condition, $Ax= Az = b$, then we have $A\bar{x} = Ax + Ad^* + \bar{\beta} (Ax-Az) = b$, which shows $A\bar{x} =b$. Now for the last part, we have, $$\begin{aligned} c^T \bar{x} = c^Tx - \alpha \frac{c^TX^2s}{\|Xs\|}+ \bar{\beta} (c^Tx-c^Tz) < c^Tx - \alpha \|Xs\| \ < \ c^Tx\end{aligned}$$ This proves the above Lemma. Here, we use the identity $c^TX^2s = \|Xs\|^2$ (see Lemma \[lemma-2\]). \[lemma-2\] \[Theorem 1 in [@saigal:1996]\] For all $k \geqslant 0$ the following identity holds: $$\begin{aligned} c^Td_k = c^TX_k^2s_k = \|X_ks_k\|^2 = \|X_k^{-1}d_k\|^2\end{aligned}$$ At first, let us denote $P_k = I - X_k A^T\left(AX_k^2A^T\right)^{-1}AX_k$, it can be easily shown that $P_k$ is a projection matrix, (i.e., $P_k = P_k^T = P_k^2$). Thus, we have, $$\begin{aligned} c^Td_k = c^TX_k^2s_k = & c^TX_k \left(I - X_k A^T\left(AX_k^2A^T\right)^{-1}AX_k\right)X_kc \\ & = \ c^TX_kP_kX_kc \ = \|P_kX_kc\|^2 \\ \text{And} \quad P_kX_kc \ = & \ X_k \left(c-A^T\left(AX_k^2A^T\right)^{-1}AX_k^2c\right) = X_ks_k\end{aligned}$$ The proof is complete. \[lemma-3\] \[Theorem 4 in [@saigal:1996]\] For all $x > 0$, there exists a $q(A) > 0$ such that, $$\begin{aligned} \|\left(AX^2A^T\right)^{-1}AX^2p\| \ \leqslant \ q(A) \|p\|\end{aligned}$$ \[lemma-4\] \[Corollary 6 in [@saigal:1996]\] For all $x > 0$, there exists a $p(A,c) > 0$ such that, if $\bar{d}$ solves EAP (2) then the following relationship holds: $$\begin{aligned} \|\bar{d}\| \ \leqslant \ p(A,c) \ c^T \bar{d} = M c^T \bar{d}\end{aligned}$$ \[lemma-5\] \[Lemma 8 in [@saigal:1996]\] Let $w \in {\mathbb{R}}^q$ and $0 <\lambda < 1$ be such that $w_j \leqslant \lambda$ then $$\begin{aligned} \sum\limits_{i =1}^{q} \log(1-w_i) \geqslant -e^Tw- \frac{\|w\|^2}{2(1-\lambda)}\end{aligned}$$ \[lemma-9\] It is a very well known Lemma in the literature for AFS methods (see Theorem 13 in [@saigal:1996], Lemma 3.8 and 3.11 in [@tsuchiya:1995]). The sequence $\{u_k\}$ has the following properties: 1. It is bounded 2. There exists a $L \geqslant 1$ such that for all $k > L$, $$\begin{aligned} & (a) \ \|u_k\|^2 = \|u_{k,N}\|^2+\epsilon_k, \ \ \sum\limits_{k = L}^{\infty} |\epsilon_k| < \infty \quad \quad \qquad \qquad \qquad \qquad \qquad \qquad \quad \quad \quad \quad \quad \quad \\ & (b) \ e^Tu_{k,N} = 1+ \delta_k, \ \ \sum\limits_{k = L}^{\infty} |\delta_k| < \infty \\ & (c) \ \frac{1}{\alpha} \ \geqslant \ \gamma(u_{k,N}) \ \geqslant \ \frac{1}{2p} \vspace{- 0.15 cm}\\ & (d) \ \gamma(u_{k}) \ = \ \gamma(u_{k,N}) \end{aligned}$$ The proof is similar to the one for AFS method given by Saigal [@saigal:1996], as the direction $\frac{X_k^2s_k}{\|X_ks_k\|}$ generated by GAFS algorithm also satisfies the EAP problem defined in equation (for a detailed proof see Tsuchiya *et al.* [@tsuchiya:1995]). [(Lemma 15 in [@saigal:1996])]{} \[lemma-11\] If the analytic center defined by the solution of problem exists, it is unique. [^1]: Used as a standard notation for Nesterov’s method, throughout the paper we used the same notations with new definition [^2]: $\alpha$ is the step size [^3]: For simplification, we used ‘fmin’ in all of the figures and tables [^4]: Note that, all comparison percentages reported in this work are calculated based on ‘*fmincon*’ CPU time [^5]: For some instances, we slightly modify parameters $b$ and $c$ to make the instances solvable by our setup, for instance in some cases $b$ vector had some values as infinity which our setup can’t handle, we replaced infinity with very large numbers

ArXiv

--- abstract: 'For a sequence $W$ we count the number $O_W(n)$ of minimal forbidden words no longer then $n$ and prove that $$\overline{\lim_{n \to \infty}} \frac{O_W(n)}{\log_3n} \geq 1.$$ [^1]' address: - 'Moscow Institute of Physics and Technology, Dolgoprudny, Russia' - 'C.N.R.S., École Normale Superieur, PSL Research University, France' author: - Igor Melnikov - Ivan Mitrofanov title: On cogrowth function of uniformly recurrent sequences --- Introduction ============ A language (or a subshift) can be defined by the list of [*forbidden subwords*]{}. The linearly equivalence class of the counting function for minimal forbidden words is an topological invariant of the corresponding symbolic dynamical system [@Beal]. G. Chelnokov, P. Lavrov and I. Bogdanov [@BogdCheln], [@Cheln1], [@Lavr1], [@Lavr2] estimated the minimum number of forbidden words that define a periodic sequence with a given length of period. We investigate a similar question for uniformly recurrent sequences and prove a logarithmic estimation for [*the cogrowth function*]{}. Preliminaries ============= An [*alphabet*]{} $A$ is a finite set of elements, [*letters*]{} are the elements of an alphabet. The finite sequence of letters of $A$ is called a [*finite word*]{} (or [*a word*]{}). An [*infinite word*]{}, or [*sequence*]{} is a map $\mathbb{N} \to A$. The [*length*]{} of a finite word $u$ is the number $|u|$ of letters in it. The [*concatenation*]{} of two words $u_1$ and $u_2$ is denoted by $u_1u_2$. A word $v$ is a [*subword*]{} of a word $u$ if $u = v_1vv_2 $ for some words $ v_1 $, $ v_2 $. If $ v_1 $ or $ v_2 $ is an empty word, then $ v $ is [*prefix*]{} or [*suffix*]{} of $u$ respectively. A sequence $W$ on a finite alphabet is called [*periodic*]{} if it has form $W=u^{\infty}$ for some finite word $u$. A sequence of letters $W$ on a finite alphabet is called [*uniformly recurrent*]{} if for any finite subword $u$ of $W$ there exists a number $C(u, W)$ such that any subword of $W$ with length $C(u, W)$ contains $u$. A finite word $u$ is called an [*obstruction*]{} for $W$ if it is not a subword of $W$ but any its proper subword is a subword of $W$. The [*cogrowth function*]{} $O_W(n)$ is the number of obstructions with length $\leqslant n$. Further we assume that the alphabet $A$ is binary, $A =\{\alpha, \beta\}$. The main result of this article is the following \[thm:ur\] Let $W$ be an uniformly recurrent non-periodic sequence on a binary alphabet. Then $$\overline{\lim_{n \to \infty}} \frac{O_W(n)}{\log_3n} \geq 1.$$ Note that if $F = \alpha \beta \alpha\alpha \beta\alpha \beta\alpha\alpha \beta \alpha \dots$ is the [*Fibonacci sequence*]{}, then $O_F(n) \sim \log_{\varphi}n$, where $\varphi = {\frac{\sqrt5 + 1}2}$ [@Beal]. Factor languages and Rauzy graphs ================================= A [*factor language*]{} $\mathcal{U} $ is a set of finite words such that for any $u\in \mathcal{U}$ all subwords of $u$ also belong to $\mathcal{U}$. A finite word $u$ is called an [*obstruction*]{} for $\mathcal{U}$ if $u\not \in \mathcal{U}$, but any its proper subword belongs to $\mathcal{U}$. For example, the set of all finite subwords of a given sequence $W$ forms a factor language denoted by $\mathcal{L}(W)$. Let $\mathcal{U}$ be a factor language and $k$ be an integer. The [*Rauzy graph*]{} $R_k(\mathcal{U})$ of order $k$ is the directed graph with the vertex set $\mathcal{U}_k$ and the edge set $\mathcal{U}_{k+1}$. Two vertices $u_1$ and $u_2$ of $R_k(\mathcal{U})$ are connected by an edge $u_3$ if and only if $u_1, u_2, u_3 \in \mathcal{U}$, $u_1$ is a prefix of $u_3$, and $u_2$ is a suffix of $u_3$. For a sequence $W$ we denote the language $R_k(\mathcal{L}(W))$ by $R_k(W)$. Further the word [*graph*]{} will always mean a directed graph, the word [*path*]{} will always mean a [*directed path*]{} in a directed graph. The [*length*]{} $|p|$ of a path $p$ is the number of its vertices, i.e. the number of edges plus one. If a path $p_2$ starts at the end of a path $p_1$, we denote their concatenation by $ p_1p_2 $. It is clear that $|p_1p_2| = |p_1| + |p_2| - 1$. Recall that a directed graph is [*strongly connected*]{} if it contains a directed path from $v_1$ to $v_2$ and a directed path from $v_2$ to $v_1$ for every pair of vertices $\{v_1,v_2\}$. Let $W$ be an uniformly recurrent non-periodic sequence. Then for any $k$ the graph $R_k(W)$ is strongly connected and is not a cycle. Let $u_1$, $u_2$ be two elements of $\mathcal{L}(W)_k$. Since $W$ is uniformly recurrent then ? $W$ contains a subword of form $u_1uu_2$. The subwords of $u_1uu_2$ of length $k+1$ form in $R_k(W)$ a path connecting $u_1$ and $u_2$. Assume that $R_k(W)$ is a cycle of length $n$. Then it is clear that $W$ is periodic and $n$ is the length of its period. If $H$ is a directed graph, its [*directed line graph*]{} $f(H)$ has one vertex for each edge of $H$. Two vertices of $f(H)$ representing directed edges $e_1$ from $v_1$ to $v_2$ and $e_2$ from $v_3$ to $v_4$ in $H$ are connected by an edge from $e_1$ to $e_2$ in $f(H)$ when $v_2 = v_3$. That is, each edge in the line digraph of $H$ represents a length-two directed path in $H$. Let $\mathcal{U}$ be a factor language. A path $p$ of length $m$ in $R_n(\mathcal{U})$ corresponds to a word of length $n + m - 1$. The graph $R_m (\mathcal{U}) $ can be considered as a subgraph of $f^{m-n}(R_n(\mathcal{U}))$. Moreover, the graph $R_{n+1}(\mathcal{U})$ is obtained from $f(R_{n}(\mathcal{U}))$ by deleting edges that correspond to obstructions of $\mathcal{U}$ of length $n+1$. We call a vertice $v$ of a directed graph $H$ [*a fork*]{} if $v$ has out-degree more than one. Further we assume that all forks have out-degrees exactly 2 (this is the case of a binary alphabet). For a directed graph $H$ we define its [*entropy regulator*]{}: $er(H)$ is the minimal integer such that any directed path of length $er(H)$ in $H$ contains at least one vertex that is a fork in $H$. Now we prove some facts about entropy regulators. Let $H$ be strongly connected digraph that is not a cycle, then ${\operatorname{er}}(H) < \infty$. Assume the contrary. Let $n$ be the total number of vertices in $H$. Consider a path of length $n + 1$ in $ H $ that does not contain forks. Note that this path visits some vertex $ v $ at least twice. This means that starting from $v$ it is possible to obtain only vertices of this cycle. Since the graph $H$ is strongly connected, $H$ coincides with this cycle. \[le:del\_edge\] Let $H$ be a strongly connected digraph, ${\operatorname{er}}(H) = L$, let $v$ be a fork in $H$, the edge $e$ starts at $v$. Let the digraph $H^*$ be obtained from $H$ by removing the edge $e$. Let $G$ be a subgraph of $H^*$ that consists of all vertices and edges reachable from $v$. Then $G$ is strongly connected digraph. Also $G$ is either a cycle of length at most $L$, or ${\operatorname{er}}(G) \leq 2L $. First we prove the digraph $G$ is strongly connected. Let $ v '$ be an arbitrary vertex of $G$, then there is a path in $G$ from $ v $ to $ v' $. Consider a path $p$ of minimum length from $ v '$ to $ v $ in $H$. Such path exists, otherwise $H$ is not strongly connected. The path $p$ does not contain the edge $ e $, otherwise it could be shortened. This means that $p$ connects $ v '$ with $ v $ in the digraph $ G $. From any vertex of $G$ we can reach the vertex $ v $, hence $ G $ is strongly connected. Consider an arbitrary path $ p $ of length $ 2L $ in the digraph $ G $, suppose that $p$ does not have forks. Since $ {\operatorname{er}}(H) = L $, then in $ p $ there are two vertices $ v_1 $ and $ v_2 $ such that they are forks in $H$ and there are no forks in $p$ between $v_1$ and $v_2$. The out-degrees of all vertices except $ v $ coincide in $ H $ and $ G $. If $v_1\neq v$ or $ v_2 \neq v $, then we find a vertex of $p$ that is a fork in $ G $. If $v_1 = v_2 = v$, then there is a cycle $C$ in $ G $ such that $|C| \leq L$ and $C$ does not contain forks of $G$. Since $ G $ is a strongly connected graph, it coincides with this cycle $C$. \[le:evol\] Let $H$ be a strongly connected digraph, ${\operatorname{er}}(H) = L$. Then ${\operatorname{er}}(f(H)) = L$. The forks of the digraph $ f (H) $ are edges in $H$ that end at forks. Consider $L$ vertices forming a path in $f(H)$. This path corresponds to a path of length $L + 1 $ in $ H $. Since ${\operatorname{er}}(H) \leq L$, there exists an edge of this path that ends at a fork. \[cor:er\] Let $ W $ be a binary uniformly recurrent non-periodic sequence; then for any $n$ $${\operatorname{er}}(R_{n-1}(W)) \leq 2^{O_W(n)}.$$ We prove this by induction on $n$. The base case $n=0$ is obvious. Let $ \ er (R_ {n-1} (W)) = L $ and suppose $ W $ has exactly $ a $ obstructions of length $ n + 1 $. These obstructions correspond to paths of length 2 in the graph $R_ {n-1} (W)$, i.e. edges of the graph $ H: = f (R_ {n-1} (W)) $. From Lemma \[le:evol\] we have that $ \ er (H) = L $. The graph $ R_ {n} (W) $ is obtained from the graph $ H $ by removing some edges $ e_1, e_2, \dots, e_a $. Since $ W $ is a uniformly recurrent sequence, the digraphs $ H $ and $ H - \{e_1, e_2, \dots, e_a \} $ are strongly connected. This means that the edges $ e_1, \dots, e_a $ start at different forks of $ H $. We also know that $ R_n (W) $ is not a cycle. The graph $ R_n (W) $ can be obtained by removing edges $e_i$ from $ H $ one by one. Applying Lemma \[le:del\_edge\] $a$ times, we show that $ {\operatorname{er}}(R_n (W)) \leq 2^aL $, which completes the proof. Proof of Theorem \[thm:ur\] =========================== \[pr:prol\] Let $H$ be a strongly connected digraph, let $ p $ be a path in $H$, let a fork $v$ be the starting point of the last edge of $p$. We call a path $t$ in $H$ *good* is $t$ does not contain $p$ as a sub-path. Then for any good path $ s $ there exists an edge $ e $ such that $ se $ is also a good path. Moreover, if the last vertex of $ s $ is a fork $ v '\neq v $, then there are two such edges. If the last vertex of $s$ is not $v$, then we can take any edge outgoing from it. If the last vertex of $s$ is $ v $, then 2 edges $ e_1 $ and $ e_2 $ go out of $v$. One of them is the last edge of the path $ p $, so we can take another edge. Let $H$ be a strongly connected digraph, ${\operatorname{er}}(H) = L$. Let $u$ be an arbitrary edge of the graph $f^{3L}(H)$, then the digraph $f^{3L}(H) - u$ contains a strongly connected subgraph $B$ such that ${\operatorname{er}}(B)\leq 3L$. The edge $u$ of the graph $f^{3L} (H) $ corresponds to the path $ p_u $ in the graph $H$, $|p_u| = 3L + 2 $. This path visits at least 3 forks (taking into account the number of visits). Next, we consider three cases. [**Case 1.**]{} Assume the path $ p_u $ visits at least two different forks of $ H $. Let $ v_1 $, $ v_2 $ be two different forks in $ H $, let $ p_ {1} e_2 $ be a sub-path of $ p_u $, where the path $ p_{1} $ starts at $ v_1 $ and ends at $ v_2 $ and does not contain forks other than $v_1$ and $v_2$, and let the edge $e_2$ go out of $ v_2 $. It is clear that the length of $ p_1 $ does not exceed $ L + 1 $. Lemma \[le:del\_edge\] implies that there is a strongly connected subgraph $G$ oh $H$ such that $G$ contains the vertex $ v_2 $ but does not contain the edge $e_2$. If $ G $ is not a cycle, then $ {\operatorname{er}}(G) \leq 2L $. Hence, the graph $ B: = f ^ {2L} (A) $ is a subgraph of $ f ^ {2L} (H) $, and from Lemma \[le:evol\] we have ${\operatorname{er}}(B) \leq 2L $. The edges of $ B $ are paths in $ G $ and do not contain $ e_2 $, which means that $ B $ does not contain the edge $ u $. If $ G $ is cycle, we denote it by $ p_{2} $ (we assume that $ v_2 $ is the first and last vertex of $ p_ {2} $). The length of $ p_2 $ does not exceed $ L $. Among the vertices $ p_2 $ there are no forks of $H$ besides $ v_2 $. Therefore, $ v_1 \not \in p_2 $. Call a path $t$ in $ H $ [*good*]{}, if $t$ does not contain the sub-path $ p_1e_2 $. Let us show that if $ s $ is a good path in $ H $, then there are two different paths $ s_1 $ and $ s_2 $ starting at the end of $ s $ such that $ | s_1 | = | s_2 | = 3L $ and the paths $ ss_1 $, $ ss_2 $ are also good. Proposition \[pr:prol\] says that for any good path we can add an edge an obtain a good path. There is a path $ t_1 $, $ | t_1 | <L $ such that $ st_1 $ is a good path and ends at some fork $v$. If $v\neq v_2 $, then two edges $ e_i $, $ e_j $ go out from $v$, the paths $ st_1e_i $ and $ st_2e_j $ are good, and each of them can be prolonged further to a good path of arbitrary length. If $v = v_2$, then the paths $st_1p_2p_2$ and $st_1p_2e_2$ are good. Consider in $f^{3L}(H)$ a subgraph that consists of all vertices and edges that are good paths in $ H $, let $B$ be a strongly connected component of this subgraph. We proved that $ {\operatorname{er}}(B) \leq 3L $. In addition, it is clear that $ B $ does not contain the edge $ u $. [**Case 2.**]{} Assume that the path $p_u$ visits exactly one fork $v_1$ (at least trice), but there are forks besides $v_1$ in $H$. There are two edges $ e_1 $ and $ e_2 $ that go out from $ v_1 $. Starting with these edges and and moving until forks, we obtain two paths $ p_1 $ and $ p_2 $. The edge $ e_1 $ is the first edge of $ p_1 $, the edge $ e_2 $ is the first of $ p_2 $, and $ | p_1 |, | p_2 | \leq L $. Since $ p_u $ goes through $ v_1 $ more than once and does not contain other forks, one of $p_1, p_2$ is a cycle. Without loss of generality, the path $p_1$ starts and ends at $ v_1 $. If $ p_2 $ also ends at $ v_1 $, then from $ v_1 $ it is impossible to reach any other fork. Therefore, $ p_2 $ ends at some fork $ v_2 \neq v_1 $. Since $ p_u $ visits $ v_1 $ at least three times and does not contain other forks, $p_u$ has sub-path $ p_1e_1 $. We call a path [*good*]{} if it does not contain $ p_1e_1 $. We show that if $ s $ is a good path in $ H $, then there are two different paths $ s_1 $ and $ s_2 $ starting at the end of $ s $ such that $ | s_1 | = | s_2 | = 3L $ and the paths $ ss_1 $, $ ss_2 $ are also good. There is a path $ t_1 $ such that $ | t_1 | <L $ and the path $ st_1 $ is a good path ending at some fork $v$. If $v = v_1 $, take $ t_2: = t_1p_2 $, otherwise we take $ t_2 = p_2 $. We see that $ | t_2 | \leq 2L $, the path $ st_2 $ is good and ends at some fork in $ v '\neq v_1 $. The proposition \[pr:prol\] shows that the path $st_2$ can be prolonged to the right at least in two ways. We complete the proof as the previous case. Consider in $f^{3L}$ a subgraph consisting of all vertices and edges corresponding to good paths in $H$ and take the strongly connected component $ B $ in this subgraph. [**Case 3.**]{} Assume that the path $p_u$ visits exactly one fork $v_1$ (at least trice), and there are no forks in $H$ besides $v_1$. The edges $ e_1 $ and $ e_2 $ go out from $ v_1 $, the cycles $ p_1 $ and $ p_2 $ start and end at $ v_1 $ and do not contain other forks, $ e_1 $ is the first edge of $ p_1 $, $ e_2 $ is the first edge of $ p_2 $, $ | p_1 |, | p_2 | \leq L $. The path $ p_u $ contains a sub-path of the form $p_ip_je_k $, where $ i, j, k \in \{1, 2 \} $. There are two cases. [**Case 3a**]{} Assume that $i = j$ or $j=k$. Without loss of generality we assume that $ p_u $ contains a sub-path $ p_1e_1 $. We call a path [*good*]{} if it does not contain a sub-path $ p_1e_1 $. We show that if $ s $ is a good path in $ H $, then there are two different paths $ s_1 $ and $ s_2 $ starting at the end of $ s $ such that $ | s_1 | = | s_2 | = 3L $ and the paths $ ss_1 $, $ ss_2 $ are also good. First we take a path $t_1$ such that $|t_1|< L$ and $st_1$ is a good path ending at $v_1$. The paths $st_1p_2e_2$ and $st_1p_2e_1$ are good. We complete the proof as in the previous cases. [**Case 3b**]{} Assume that $i \neq j$ and $j\neq k$. Without loss of generality we assume that $ p_u $ contains a sub-path $p_1p_2e_1$. We call a path in $H$ [*good*]{} if it does not contain a sub-path $p_1p_2e_1$. We show that if $ s $ is a good path in $ H $, then there are two different paths $ s_1 $ and $ s_2 $ starting at the end of $ s $ such that $ | s_1 | = | s_2 | = 3L $ and the paths $ ss_1 $, $ ss_2 $ are also good. First we take a path $t_1$ such that $|t_1|< L$ and $st_1$ is a good path ending at $v_1$. The paths $st_1p_2p_2e_1$ and $st_1p_2p_2e_2$ are good. Note that $|t_1p_2p_2e_1|= |t_1p_2p_2e_2| \leq 3L$. Again, we complete the proof as in the previous cases. \[cor:main\] Let $H$ be a strongly connected digraph, ${\operatorname{er}}(H) = L$, $k \geq 3L$. Let $u$ be an arbitrary edge of the graph $f^{k}(H)$, then the digraph $f^{k}(H) - u$ contains a strongly connected subgraph $B$ such that ${\operatorname{er}}(B)\leq 3L$. \[le:subseq\] Let $a_n$ be a sequence of positive numbers such that $$\underline{\lim}_{k\to \infty} \frac{\log_3 a_k}{k} > 1.$$ Then there exists $n_0$ such that for any $k>0$ $$a_{n_0 + k} - a_{n_0} > 4 \cdot 2^{n_0}\cdot 3^k$$ Let us denote $a_k/3^k$ by $b_k$. It is clear that $\lim_{k\to \infty}b_k = \infty$. Hence, there exists $n_0$ such that ??? $b_{n_0} > 10$ and $b_{n} \geq b_{n_0}$ for all $n > n_0$. Then for any $k > 0$ it holds $$a_{n_0 + k} - a_{n_0} = b_{n_0+k}3^{n_0 + k} - b_{n_0}3^{n_0} \geq b_{n_0}3^{n_0}(3^k - 1) > 4 \cdot 2^{n_0}\cdot 3^k.$$ Now we are ready to prove Theorem \[thm:ur\]. Arrange all the obstructions of the uniformly recurrent binary sequence $ W $ by their lengths: $$|u_1| \leq |u_2| \leq \dots$$ If $\underline{\lim}_{k\to \infty}\frac{\log_3 |u_k|}{k} \leq 1$, then the statement of the theorem holds. Assume the contrary. Lemma \[le:subseq\] says that there is $n_0$ such that for any positive integer $k$ it holds $$|u_{n_0 + k}| > |u_{n_0}| + 4 \cdot 2^{n_0} \cdot 3^k.$$ For $n > n_0$ denote the number $|u_{n_0}| + 4 \cdot 2^{n_0} \cdot 3^{n-n_0}$ by $b_n$. Let $b_n = |u_{n_0}|$. For all $n > n_0$ take a proper subword $v_n$ of $u_n$ such that $|v_n| = b_n$. Denote by $ \mathcal {U} $ the set of all finite binary words that do not contain as subwords the words $ u_i $ for $ 1 \leq i \leq n_0 $ and $ v_i $ for $ i> n_0 $ . We get a contradiction with the uniform recurrence of $ W $ if we show that the language $ \mathcal {U} $ is infinite. It is clear that the Rauzy graphs $R_{u_{n_0}-1}(\mathcal{U}) = R_{u_{n_0}-1}(W)$, and from Corollary \[cor:er\] we have $${\operatorname{er}}(R_{u_{n_0}-1}(\mathcal{L})) \leq 2^{n_0}.$$ By induction on $ k $, we show that the graph $ R_{b_{n_0 + k} -1} (\mathcal {U}) $ contains a strongly connected subgraph $ H_k $ such that $ {\operatorname{er}}(H_{k}) \leq 3^k \cdot 2^{n_0} $. We already have the base case $k=0$. The graph $R_{b_{n_0+k+1}-1}(\mathcal{U})$ contains a sub-graph $f^{b_{n_0+k+1}-b_{n_0+k}}(H_k)$ without at most one edge (that corresponds to the word $v_{n_0+k+1}$). Note that $$b_{n_0+k+1}-b_{n_0+k} > 3 \cdot {\operatorname{er}}(H_k).$$ hence we can apply Corollary \[cor:main\]. Then the digraph $R_{b_{n_0+k+1}-1}(\mathcal{U})$ has a strongly connected subgraph with entropy regulator at most $3^{k+1}\cdot 2^{n_0}$. We show that all the graphs $R_{b_n} (\mathcal {U}) $ are nonempty and, therefore, the language $ \mathcal {U} $ is infinite. On the other hand, all elements in $ \mathcal {U} $ are subwords of $ W $ and do not contain $v_{n_0 + 1}$. But the word $ v_ {n_0 + 1} $ is a proper subword of the obstruction $u_{n_0 + 1}$, and, therefore, $ v_ {n_0 + 1} $ is a subword of $ W $. This means that the infinity of the language $ \mathcal{U}$ contradicts the uniform recurrence of $ W $. [00]{} M.-P. B[é]{}al, *Forbidden Words in Symbolic Dynamics*. Advances in Applied Mathematics, 25, 163 – 193. Grigory R. Chelnokov, *On the number of restrictions defining a periodic sequence*. Model and analysis of inform. systems, 14:2 (2007), 12–16, in Russian P. A. Lavrov, *Number of restrictions required for periodic word in the finite alphabet*. Avaible online at arxiv.org/abs/1209.0220, 26p. P. A. Lavrov, *Minimal number of restrictions defining a periodic word*. Avaible online at arxiv.org/abs/1412.5201, 9p. Ilya I. Bogdanov, Grigory R. Chelnokov, *The maximal length of the period of a periodic word defined by restrictions*. Avaible online at arxiv.org/abs/1305.0460, 14p. [^1]: The paper was supported by Russian Science Foundation (grant no. 17-11-01377)

ArXiv

--- abstract: 'Worldwide, sewer networks are designed to transport wastewater to a centralized treatment plant to be treated and returned to the environment. This process is critical for the current society, preventing waterborne illnesses, providing safe drinking water and enhancing general sanitation. To keep a sewer network perfectly operational, sampling inspections are performed constantly to identify obstructions. Typically, a Closed-Circuit Television system is used to record the inside of pipes and report the obstruction level, which may trigger a cleaning operative. Currently, the obstruction level assessment is done manually, which is time-consuming and inconsistent. In this work, we design a methodology to train a *Convolutional Neural Network* for identifying the level of obstruction in pipes, thus reducing the human effort required on such a frequent and repetitive task. We gathered a database of videos that are explored and adapted to generate useful frames to fed into the model. Our resulting classifier obtains deployment ready performances. To validate the consistency of the approach and its industrial applicability, we integrate the Layer-wise Relevance Propagation explainability technique, which enables us to further understand the behavior of the neural network for this task. In the end, the proposed system can provide higher speed, accuracy, and consistency in the process of sewer examination. Our analysis also uncovers some guidelines on how to further improve the quality of the data gathering methodology.' author: - 'Mario A. Gutiérrez-Mondragón' - 'Dario Garcia-Gasulla\' - 'Sergio Alvarez-Napagao' - 'Jaume Brossa-Ordoñez' - 'Rafael Gimenez-Esteban' bibliography: - 'ecai.bib' title: Obstruction level detection of sewer videos using convolutional neural networks --- Introduction ============ In the US, there are roughly 1,200,000 kilometers of sewer lines [@sterling2010state]. That is more than three times the distance between the Earth and the Moon, considering only 4% of world population. The maintenance of such vasts networks of pipes is thus a real challenge world-wide. As of now, the most common approach is to have operators executing sampling inspections, trying to find obstructions before they can cause severe failures that would require urgent and expensive actions. The current approach is hardly scalable, as it is expensive and requires lots of human hours. Companies in charge of large wastewater networks face massive operational costs related to inspection and maintenance. The current environmental context brings added pressure to the topic, since episodes of heavy rainfall are becoming more common as a consequence of climate change [@donat2017addendum]. Within these episodes, obstructed wastewater networks may become the origin of sewer overflows and floods with an impact on urban environments and population. ![Sample frames from the videos database.[]{data-label="fig:sewer_samples"}](Images/sewers_grid_samples.png){width="49.00000%"} In an effort to increase the quality and efficiency of sewer maintenance, the industry is now looking into recent technological advancements in fields such as image recognition and unmanned aerial vehicles. In this paper, we tackle one of the challenges necessary for new methods to be functional: the automatic identification of obstructions in sewer pipes from image data. For this purpose we use real data from 6,590 inspection videos (samples shown in Figure \[fig:sewer\_samples\]), recorded and evaluated by operators. We post-process the videos to dissect and simplify the problem at hand. With this data we design, train and evaluate a convolutional neural network (CNN), for the task of predicting the level of obstruction of a sewer segment. The performance obtained in this work enables this technology to be directly applicable to the industrial challenge, increasing the efficiency of the procedure and enabling a more extensive maintenance. In this particular case, this is already in progress through CETaqua, industrial partner of this project, and part of the SUEZ group. Current Sewer Maintenance {#sec:maintenance} ========================= Regular sewer inspections are made for the operation and maintenance of the network. During inspections, videos of the inside of sewers are recorded by using a camera attached to a pole. Each of these videos will be carefully reviewed by an operator later, who must fill a report and deliver it to the inspection site or to the central offices. The report must include the level of obstruction of the sewer, categorized into five classes: clean; slightly dirty; dirty; very dirty; and obstructed. Cleaning operations prioritize their interventions based on these reports. Reviewing videos requires a significant amount of time from operators. This task is a major barrier for productivity because of its duration and repetitive nature; if the same operator dedicates too much time to this task, their performance will be affected. To avoid that, in practice, many different operators end up reviewing the same videos. While this is desirable for several reasons, it entails a significant variance in the evaluation criteria. Meanwhile, a reliable and consistent evaluation is critical for the efficient planning of maintenance. Our goal is to define and implement a system to automatically assess the obstruction on sewers from videos. This system has to provide a status on the volume of dirt or sedimentation in the pipes, to justify the cleaning needs. The deployment of this system in production will enable a more productive use of human resources, and will provide a unified model for guiding cleaning operations. State of the Art / Related Work =============================== The use of computer vision techniques in civil engineering applications has grown exponentially, as visual inspections are necessary to maintain the safety and functionality of basic infrastructures. To mitigate the costs derived from the manual interpretation of images or videos, diverse studies explore the use of computer vision techniques. Methods like feature extraction, edge detection, image segmentation, and object recognition have been considered to asses the condition of bridges, asphalt pavement, tunnels, underground concrete pipes, [*etc.*]{}[@abdel2003analysis; @zakeri2017image; @a_rose2014supervised]. Moreover, noise reduction [@yamaguchi2008image], and reconstruction and 3D visualization [@esquivel2009reconstruction; @lattanzi20143d; @huynh20163d; @belles2015kinect] have also been used to improve the precision and applicability of these techniques. In the most similar scenario to the one tackled in this paper, the automatic detection of cracks in sewers has been explored through image processing and segmentation methods [@halfawy2013efficient; @iyer2006segmentation]. Most of these related works are mainly focused on a single task, to detect cracks. Segmentation and classifications of pipe cracks, holes, laterals, joints and collapse surfaces are explored through mathematical morphology techniques in the work of S.K. Sinha and P. W. Fieguth [@sinha2006morphological]. A most recent study of L. M. Dang [*et al.*]{}[@dang2018utilizing] uses these morphological operations and other pre-processing techniques, like edge detection and binarisation, to identify the sewer defects by recognizing text displayed on the sewer video recording. Even though computer vision techniques have provided a significant improvement in the analysis of civil infrastructure, there are still several difficulties to overcome, such as the extensive pre-processing of the data that must be carried out, a high degree of expert knowledge in the design of complex features extractors, the treatment of noisy and low-quality data, among others. In this regard, CNNs require little image pre-processing, and more importantly, the feature extraction processes is learned automatically from the data through an optimization process. The performance of CNN models has been tested in several computer vision tasks, such as object detection or image classification. For instance, in the work of Y.-J. Cha [*et al.*]{}[@cha2017deep], an automated civil infrastructure damage system is presented, which is insensitive to the quality of the data and to camera specifications. Furthermore, CNNs use has demonstrated its efficiency in tunnel inspections [@montero2015past], revealing how the deep learning approach outperforms conventional methods. In the case of sewer inspections, the use of neural networks has been limited to defect detection. S.S. Kumar [*et al.*]{}proposes a convolutional neural network to identify root intrusions, deposits, and cracks in a set of sewer videos [@kumar2018automated]. This database is transformed into a sequence of RGB images and fed them to the model. The training methodology is very straightforward, all images comprising a particular defect are feed to the CNN so that discriminative features can be learned. To enhance the performance of its model they used data augmentation, simulating a variety of conditions and mitigating over-fitting. By doing so, the size of the dataset increases to millions of training samples for the model. However, and despite the good results, the model could not identify sub-classes, [*e.g.*, ]{}fine roots from medium roots. J. Cheng and M. Wang use a fast regional convolutional neural network (fast R-CNN) to detect different classes of sewer defects and also to identify the coarse category to which they belong [@cheng2018automated]. Their model is comprised of a set of images gathered from video sewer inspections which are fed to the model to generate both classification and bounding box regression of the defect. Despite is implemented data augmentation, the similarities in the geometry of the sewers and color gradients and intensity, penalised the model performance. So far, there are no studies that discuss the automated classification of sewer obstruction level using CNNs. Previous works focus on more general faulty elements in the sewer structure, [*e.g.*, ]{}roots, cracks or deposits. However, due to the nature of the sewer system we work with (Barcelona area), it is crucial to assess if the sewer is free from obstacles, so that wastewater can flow through it ordinarily [@chataigner2020arsi]. That being said, we can still use some of the insights found when tackling similar tasks. Sewer Data ========== CETaqua is a public-private research institution dedicated to the design of more sustainable water management services. Along the years, CETaqua has gathered a database of videos from 6,590 human-made inspections made in the sewers of the area of Barcelona. Each video has an associated label, obtained from the operator’s report. The original distribution of videos is shown in Table \[tab:video\_distribution\]. Label \# samples ----------------- ------------ clean 4720 slightly\_dirty 1146 dirty 235 very\_dirty 50 obstructed 98 Total: 6249 : Original video distribution.[]{data-label="tab:video_distribution"} Sewer videos were obtained by different operators following a shared set of guidelines: the operator brings the camera down into the sewer and starts recording the pipe from a static position. After a few seconds, the operator zooms in to look further into the end of the sewer, followed by a zoom out to the starting position. At that point, the video ends. Videos are mostly shot at 360x640 resolution, and 10Fps (frames per second). Prototypical sample frames for every obstruction level are shown in Figure \[fig:sewer\_samples\]. Since videos are recorded by different operators, there is a significant variance in their length. These range from \~[18]{} to \~[120]{} seconds. The distribution of video lengths is shown in Figure \[fig:frames per video\]. No video is excluded from this study because of its length. Notice the difference in number of samples between the most frequent and the least frequent classes is of two orders of magnitude. Such large class imbalance may handicap the learning process of many machine learning algorithms, including CNNs. To tackle this issue, and following the advise of use case industrial experts, we merge the two classes with less samples per class, “very dirty” and “obstructed”, which are very close in meaning. We can do that without affecting the performance of the system because both classes imply the same industrial response once they are identified ([*i.e.*, ]{}the prioritized cleaning of that particular sewer segment). After the merging, the number of elements in the minority class has increased (to 148), but the uneven data distribution remains relevant, which could lead to a severe bias in the model performance. To avoid that we balance the distribution of the data by randomly *under-sampling* them to the minority class. Before data is fed into the model, we will still need to perform some pre-processing, to enable the learning process. Dataset Engineering =================== The original task, as defined by the industrial requirements, is a video classification problem: Assign a given label to a given set of videos. However, we can reduce this to an aggregated image classification problem to simplify it, as the inherent temporal aspect of videos is mostly irrelevant for our case. Working with images also increases the number of training samples we can generate, as several frames from the same video become different (although not independent) training samples. With a larger training set we can improve the regularization and generalization of the CNN model. Before transforming videos into images, we need to specify our dataset splits. Its essential to do so at this point, to avoid having images from the same video on both the train and test partitions. This would introduce a significant bias into the model, and significantly affect the relevance of our evaluation. After the *under-sampling* process, the distribution of the videos is shown in table \[tab:video\_distribution\_2\]. We have split the videos in two subsets: 70% for the training process and the remaining 30% to validate the model. Label Train samples Validation samples ----------------- --------------- -------------------- clean 103 45 slightly\_dirty 103 45 dirty 104 44 very\_dirty 104 44 Total: 414 178 : Videos distribution per dataset split.[]{data-label="tab:video_distribution_2"} Frame Selection --------------- Of the full length of the video only a small portion of frames are usable for training. The zooming is digital on all cases, which means resolution is never increased, and some parts of the image are lost. For this reason, we gather the frames of the video where the camera is unzoomed. That is, from the beginning of the recording until the zooming in begins. To automatically locate this segment of interest, we used the VidStab video stabilization algorithm [^1] from the OpenCV library [@opencv_library]. This algorithm produces a smoothed trajectory of pixels through the use of key point detectors. Figure \[fig:smooth\_traj\] shows examples of these smoothed trajectories. ![Samples of pixels trajectories. Blue line shows change in pixel values. Red lines shows a smoothed version of the same function. Notice the significant scale variations in the vertical axis.[]{data-label="fig:smooth_traj"}](Images/grid_trajectories.png){width="49.00000%"} The top row examples of Figure \[fig:smooth\_traj\] are prototypical videos, where the zoom in and zoom out stages form a clear ’V’. Unfortunately, not all videos are like that. There is a significant variance and noise in the extracted trajectories, as shown in the bottom row of Figure \[fig:smooth\_traj\]. Our analysis of trajectories shows that most videos have at least 3 seconds of image stability. Thus, we capture 30 consecutive frames for all videos. We do not extract a variable number of frames per video to avoid biasing the dataset. Using several frames per video results in the dataset distribution of Table \[tab:image\_distribution\]. Even though the number of images per class seems remarkable, this is deceptive. All images come from a hundred videos per class, which constrains significantly the variance of our training set. This also makes unproductive the use generalization techniques like data augmentation, since there are already plenty of similar images with small variations in our dataset. Label Train samples Validation samples ----------------- --------------- -------------------- clean 3090 1350 slightly\_dirty 3090 1350 dirty 3120 1320 very\_dirty 3120 1320 Total: 12420 5340 : Images distribution per dataset split.[]{data-label="tab:image_distribution"} Input Pipeline -------------- Most frames have a resolution of 360x640. They also have a vertical border as seen in Figure \[fig:sewer\_samples\]. After removing it, images are at 360x480 resolution, as seen in Figure \[fig:bad\_classifications\]. For those few images that had a slightly higher resolution, we applied a central crop. During our experimentation we noticed that models had the same performance if the 360x480 resolution was scaled down to 150x150. This is coherent with the task: since no specific object has to be identified, fine-grained detail is unnecessary. For this reason, our final training dataset is composed by 150x150 images. Resizing the images also reduced the number of parameters needed and the training costs ([*i.e.*, ]{}time, power and money). Models ====== CNNs models are composed by a sequence of stacked layers which learn increasingly complex representations from the data. For image inputs, these representations are visual abstractions of shape, patterns, colors [*etc.*]{}which are used as building blocks for perception. In the context of our problem, where the goal is to identify the amount of obstruction, complex patterns are irrelevant for the CNN. In other words, we do not care if the obstruction is caused by a bicycle or by a pile of cement. What is important to learn for the CNN is what a clean pipe looks like, and how different alterations to that normality correspond to different levels of obstruction. Clearly, spatial information is essential for the task, as sediment may be distributed along the channel, or it may form an obstruction at the bottom of the sewer. A sense of depth is also desirable, to assess obstructions proportions (and thus size) correctly. While we will not enforce these priors into the CNN, we will take them into account in our architectural designs, and we will validate them in our later interpretability study. Transfer Learning ----------------- Fitting the many parameters found in deep CNNs to solve a task on high-dimensional inputs ([*i.e.*, ]{}images) requires many data samples. To mitigate this need, one can use transfer learning: Initializing the parameters from a state optimized for a different problem, instead of initializing from a random state. Transfer learning is based on the assumption that most image challenges share a given set of visual properties which can be reused, instead of re-learnt. This is particularly true for low-level descriptors ([*e.g.*, ]{}lines, angles [*etc.*]{}). Nevertheless, the transferability of features depends on the similarity between tasks, and in the variety and size of the data for which the pre-trained model was optimized [@yosinski2014transferable]. For this reason, the most popular source models for transfer learning are those containing a wide variety of patterns ([*e.g.*, ]{}VGG16 [@simonyan2014very]) for a wide variety of goals ([*e.g.*, ]{}ImageNet) [@azizpour2015factors]. Considering the limited number of samples available in our task (remember images come from a small set of videos), in this section we consider transfer learning as a potentially useful approach. We explore this hypothesis by using a VGG16 architecture trained on the ImageNet dataset. To adjust the VGG16 model to our needs, we start by removing the parameters of the original classifier ([*i.e.*, ]{}the two fully-connected layers), since these are too optimized for the original problem. We also addapt the output of the network (originally, a 1,000 classes problem) to fit our task. With this setting in place, we can now train the network through fine-tuning. When fine-tuning, one must decide which layers to freeze ([*i.e.*, ]{}fixing the weights), which to re-train ([*i.e.*, ]{}fine-tuning the weights) and which to replace ([*i.e.*, ]{}randomly initialized) or delete. The more layers we freeze, the more similar both tasks should be. Unfortunately, our industrial case is a rather unique one, even when compared with a broad classification task like ImageNet. In our experiments, we tried freezing a variable number of convolutional layers gradually bottom up (remember, the fully-connected layers are always randomly initialized and thus always replaced and optimized). Significantly, none of these experiments were successful. In all experiments, the model either overfitted to the data or failed to learn meaningful representations. We hypothesise that the particularity of our problem makes it hard to reusing patterns learned on general purpose datasets. Indeed, there is little in common between discriminating dog breeds and computing the level of obstruction of a sewer. On the other hand, the huge number of parameters in networks trained for large tasks like ImageNet is inadequate for a small problem like ours. Architecture proposed --------------------- Since transfer learning was unsuccessful, we decided to define an architecture design top to bottom for the sewer classification problem. This is motivated by the uniqueness of our problem. We started from a shallow architecture, and increased its size until underfitting was no longer an issue. At that point, we optimized the hyper-parameters to get the best working model. The Table \[tab:cnn\_small\] shows the final CNN design. Layer (type) Output Shape \# Param ------------------------------ ---------------- ---------- conv1 (Conv2D) (150, 150, 32) 896 pool1 (MaxPooling2D) (75, 75, 32) 0 conv2 (Conv2D) (75, 75, 32) 9248 pool2 (MaxPooling2D) (38, 38, 32) 0 conv3 (Conv2D) (38, 38, 64) 18496 pool3 (MaxPooling2D) (19, 19, 64) 0 flatten (Flatten) (23104) 0 fc1 (Dense) (1024) 23659520 dropout1 (Dropout) (1024) 0 logits (Dense) (4) 4100 Total params: 23,692,260 Trainable params: 23,692,260 : CNN architecture proposed.[]{data-label="tab:cnn_small"} Notice the relatively small size of the architecture. Increasing the number of filters and the kernel size provided no improvement, mostly because the variety of patterns to learn is small: the model does not have to recognize all possible objects and shapes that may obstruct the sewer. It must limit itself to learn what a clean sewer looks like, and what obstructions represent visually in that regard. Coherently, our experiments shows that a CNN with only three convolutional layers yields the best results. On top of that we add a fully-connected layer, accounting for 99.9% of the parameters of the CNN, to learn to discriminate between the different levels of obstruction. The size of this last layer was also optimized empirically. In our experiments we used the Cross-Entropy loss function, ADAM optimizer with a learning rate, and 0.5 dropout value. Evaluation and Results ====================== To evaluate the performance of the trained model, we start by the confusion matrix. This will allow us to understand the frequency and severity of the mistakes being made by the model. As shown in Figure \[fig:cm\], 53.7% of images are classified in the correct class. 34.7% of images are classified in a neighboring class ([*e.g.*, ]{}slightly dirty as dirty). The fact that mistakes are centered around the diagonal indicates that the model is properly learning the nature of the problem. It is also worth noticing how the most relevant classes for the industrial application (the dirty and very dirty ones) are the ones classified with the highest accuracy. ![Normalized image-wise confusion matrix for validation set.[]{data-label="fig:cm"}](Images/confusion_matrix.png){width="49.00000%"} The previous metric was computed image-wise, in the context of an image classification task. However, our final purpose is to provide a video classification tool. Based on the CNN image predictions, we generate a video classifier using a voting strategy, where each image from a video contributes with one vote towards the classification of the video itself. The confusion matrix of Figure \[fig:cm\_videos\] shows the video-wise classification results. In this case, 55.7% of images are classified in the correct class, 2% more than the image classifier. The images classified in a neighboring class decrease 0.7%, to 34.0%. The performance of this model fits the requirements of the industrial task, and is already profitable from a practical point of view. ![normalized video-wise confusion matrix on the validation set. From the voting of classified images.[]{data-label="fig:cm_videos"}](Images/confusion_matrix_videos.png){width="49.00000%"} Beyond the numeric analysis of the classifier outcome, we also explore the mistakes done by the model. Figure \[fig:bad\_classifications\] presents some representative examples of failed predictions. The first two rows contains examples of videos where the labeling seems to be erroneous, which we attribute to human error. These samples could be re-labeled to improve the training dataset quality and the model performance. The third row shows examples where the labeling criteria seems to be inconsistent, as a result of having multiple operators labeling videos. Although the model predictions in these cases count as miss-classifications, its criteria seems adequately coherent. Another complicating factor we found in our analysis of mistakes is rain. Examples of these are shown in the forth row of Figure \[fig:bad\_classifications\]. Rain introduces lots of noise in the images, which handicaps perception and model prediction. Finally, the last row shows cases where the perspective of the camera is not normative ([*i.e.*, ]{}centered in the pipe and looking towards the end of it). These variations confuse the model. To bypass this limitation more training data is needed. ![Samples of miss-predictions. *true* indicates ground truth. *pred* indicates model prediction.[]{data-label="fig:bad_classifications"}](Images/grid_bad_classifications_0.png){width="49.00000%"} Interpretability ---------------- So far we have gathered evidence that the model is learning properly. Nevertheless, trusting the predictions of a black-box is never ideal, no matter how confident we are on its performance. Explainability of the model is crucial for industrial risk assessment and regulation compliance. Thus, we take one more step into the validation of the model by looking at the visual patterns learned and used by the model to classify the data. This will provide interpretability to our system. The Layer-wise Relevance Propagation (LRP) is presented in Bach [*et al.*]{}[@bach2015pixel]. This algorithm works on a trained model, trying to identify which features of the input image have the highest relevance for the prediction of that image. Relevance is backpropagated from the output layer, assigning scores to the application of features, layer by layer until reaching the input. Each layer stores an equal amount of relevance, which is variably distributed among its features. The relevance of pixels in the input can be conveniently visualized through heatmaps. We integrate the LRP to our trained model, to explore its decision making process. A sample of the result can be seen in Figure \[fig:lrp\_bad\_classifications\]. [0.49]{} [0.49]{} [0.49]{} [0.49]{} [0.49]{} For visibility reasons, LRP values are not normalized among all plots ([*i.e.*, ]{}the same color on different LRP may indicate different relevance). The reference value for each LRP plot is shown above it (score relevance), and it depends on the confidence of that particular prediction. If all colors were normalized, colors from predictions with lower confidence would be barely visible. For this reason, plots of low probability predictions should not be over-interpreted. Let us first consider what evidence is used to predict obstructions. As shown in the second column of Figure \[fig:lrp\_bad\_classifications\], the main evidence used for justifying a high level of obstruction ([*i.e.*, ]{}*dirty* or *very dirty*) is located at the ground along the pipe. This seems adequate, since this center canal will be naturally occupied by most obstructions. The LRP plots also indicate that changes in illumination are taken as evidence of obstruction ([*e.g.*, ]{}third row, third column). Coherently, in a clean pipe light is smoothly distributed, while obstructed pipes contain segments of extreme illumination contrast. On the other hand, clean predictions seem to focus on circular shapes. These shapes are periodic within pipes, and their visibility is used by the CNN as evidence of cleanness. Among the circles the CNN is locating, we find an artificial guidance circle introduced to help the operator center the camera within the pipe. This visual aid is also being used by the CNN for prediction. With the current results we are unable to assert if the classifier would perform better without the visual aid or not. Regarding the use of circles for clean predictions, we find this a quite consistent policy, as any large obstruction would occlude these circles (in the cases of pipe circles), or would make them invisible due to changes in the illumination (in the case of the visual aid circle, as in the forth row). The use of both the ground path and illumination contrast as features for prediction explains the difficulties of the model for predicting images where there is either rain or changes in perspective. As shown in the bottom three rows of Figure \[fig:lrp\_bad\_classifications\], the model still focuses on these features, even though in these cases such features characterize noise instead of obstructions. Industrial Deployment {#Industrial_Deployment} ===================== Our purpose is to help improve the efficiency of cleaning operations. We propose to do so through an automated mechanism for the evaluation of sewer conditions, fueled by the CNN model previously described. In this section, we outline the rest of the necessary components for the implementation of the solution in the real environment. We design it so that the system keeps learning once deployed. The two main system components: a labeler API and a training pipeline. The labeler API is to be integrated into the IT systems of the maintenance department. It provides both automatic classification of videos, as well as a labeling interface for humans. Once a new video inspection is uploaded, the API is automatically requested for a classification. This will be done on a number of random frames from the static part of the video. The result, both classification and confidence, is processed by a rule-based system. This determines what to do with the video. Rules such as: If the classification is *dirty* or *very dirty* and the confidence is high, send it to the cleaning team with urgency. If the confidence is low, send it to the queue for human labeling. If classification is *slightly dirty* or *clean* with high confidence, send it to the queue with low priority. All video labeled by humans through the interface are automatically used in the training pipeline. The training pipeline is defined in a continuous integration server. When new videos arrive, these are fed into an object storage server. The storage server can triggers a series of jobs, after a minimum number of new samples are received. These jobs execute the following pipeline: 1. Dataset balancing and split 2. Video stabilization and frame selection 3. Frame resizing 4. Model training 5. Model evaluation The result of this pipeline is a model in *TensorFlow* along with a PDF document containing a sample of automatically labeled frames that are to be reviewed by an expert. If, according to this expert, the results are good –[*i.e.*, ]{}the classification of sewer images is adequate– the model is automatically deployed to the production API server, replacing the previous version. Every pipeline that generated every version of the model is stored along with the data used in it. This provides full reproducibility to the system. The system is designed for low-degree maintenance. Together with a heavy automatization we propose to scale resources to the cloud, accessing GPU resources only when training. That is periodically and for less than an hour. It is also designed for re-usability. The same pipeline could be potentially applied to any sewer system that shares strong similarities –both structural and sedimental– with the one we have worked with. If differences were significant the CNN model architecture should be reassessed. It is therefore our assumption that this solution could be deployed internationally to any sewer management that uses video sampling inspections. Beyond technical contributions, several improvements could be made to the process. For example, using higher resolution cameras, adding stabilization gear, giving more specific instructions to operators or applying heavier pre-processing techniques. However, relying on such improvements would reduce the generalization power of the model. Low-quality data, something frequent in sewer inspections, is something to be learned. Thus, we consider our current approach –dealing with our current datasets, as faulty as they might be– more beneficial for the industrial purpose in the long term. Conclusions =========== The proper operation of sewers is critical for current societies: It conveys domestic sewage, industrial wastewater, and storm-water runoff. The efficient and scalable identification of obstructions in sewer infrastructure is critical for their correct maintenance, given the sewer network length and the up-time requirements of the service. In this context, operators are put under severe pressure, forced to quickly record and evaluate inspections daily. In this work, we seek to alleviate the pressure on human performance through a CNN model trained to identify the level of obstruction of a sewer. We start by reducing the problem to an image classification one, as this is a more scalable and constrained approach. A pixel motion analysis allows us to measure the degree of noise in the dataset (which is high), and to define a unified frame extraction policy. Given a significant imbalance among target classes, we are forced to merge two similar classes and to down-sample the rest. In this setting we perform our experiments. In sight of the limited data availability, we decide to first use transfer learning, exploiting features learned for a different problem. This approach failed, most likely, due to the dissimilarities between tasks. Unfortunately this is a recurrent issue in industrial domains, where data follows a very particular distribution, with little in common with large, popular datasets. It remains to be seen if more flexible transfer learning mechanisms, like feature extraction where it is not needed to re-train the CNN [@garcia2018out], would be feasible in this setting. In our experiments the best results are obtained by a rather small and shallow architecture, consistently with the nature of the task: There is no need to learn any specific pattern, just an overall sense of space and obstruction. The evaluation indicates this model learns to solve the task satisfactorily, and illustrates the main reasons behind the failed predictions. Most frequently, inconsistent human labeling, variations in perspective and environmental noise like rain. We explore the behavior of the model by looking at the relevance of input pixels for output classification. This allows us to validate the visual features used by the model to make predictions. In particular, we notice how the center canal of the sewer is essential for the assessment of obstructions, how the visibility of circles around the pipe speaks for cleanness, and how changes in illumination and perspective can complicate the resolution of the problem. Based on these, we formulate two recommendations for current inspection protocols: to pay special attention to camera location before starting the recording and to avoid doing inspections under heavy rain. Two more complicating factors were identified in the data during the development work. First, human mistakes when labeling videos. These are unfortunately frequent and bias the performance measures obtained for the model. In other words, the model may be predicting better than what is measured. Second, the variability in labeling criteria. This is one of the motivating factors of this work, as an unstable policy reduces the quality and efficiency of maintenance interventions. All experimental outcomes suggest that the trained model has a more consistent behavior than human labeling. This already makes the solution appealing from an industrial perspective. Beyond the visual model, we propose an integral system design to deploy all desirable functionalities. This includes an API, through which videos can be automatically labeled by the model, while also providing a common interface for human labeling. It also includes a training pipeline, so that models can be periodically trained and deployed in production with minimal effort. To generate the deployment model we will retrain the system using all available data ([*i.e.*, ]{}including the validation set). Before that, we will try to reduce dataset noise by fixing some of the most obvious labeling mistakes, as well as removing videos with rain. For this last case, and while data availability remains limited, we consider best to train a model to discriminate between images with rain and images without rain, so that the system can automatically inhibit itself in favor of humans when asked to classify videos in rainy conditions. This work is partially supported by the Consejo Nacional de Ciencia y Tecnologia, No. CVU: 630716, by the RIS3CAT Utilities 4.0 SENIX project (COMRDI16-1-0055), cofounded by the European Regional Development Fund (FEDER) under the FEDER Catalonia Operative Programme 2014-2020. It is also partially supported by the Spanish Government through Programa Severo Ochoa (SEV-2015-0493), by the Spanish Ministry of Science and Technology through TIN2015-65316-P project, and by the Generalitat de Catalunya (contracts 2017-SGR-1414). [^1]: http://nghiaho.com/?p=2093

ArXiv

--- abstract: 'Using a $3.19~\mathrm{fb}^{-1}$ data sample collected at an $e^+e^-$ center-of-mass energy of $E_{\rm cm}=4.178$GeV with the BESIII detector, we measure the branching fraction of the leptonic decay $D_s^+\to\mu^+\nu_\mu$ to be $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}=(5.49\pm0.16_{\rm stat.}\pm0.15_{\rm syst.})\times10^{-3}$. Combining our branching fraction with the masses of the $D_s^+$ and $\mu^+$ and the lifetime of the $D_s^+$, we determine $f_{D_s^+}|V_{cs}|=246.2\pm3.6_{\rm stat.}\pm3.5_{\rm syst.}~\mathrm{MeV}$. Using the $c\to s$ quark mixing matrix element $|V_{cs}|$ determined from a global standard model fit, we evaluate the $D_s^+$ decay constant $f_{D_s^+}=252.9\pm3.7_{\rm stat.}\pm3.6_{\rm syst.}$MeV. Alternatively, using the value of $f_{D_s^+}$ calculated by lattice quantum chromodynamics, we find $|V_{cs}| = 0.985\pm0.014_{\rm stat.}\pm0.014_{\rm syst.}$. These values of $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}$, $f_{D_s^+}|V_{cs}|$, $f_{D_s^+}$ and $|V_{cs}|$ are each the most precise results to date.' title: '**Determination of the pseudoscalar decay constant $f_{D_s^+}$ via $D_s^+\to\mu^+\nu_\mu$** ' --- -0.2cm -0.2cm The leptonic decay $D^+_s\to \ell^+\nu_\ell$ ($\ell=e$, $\mu$ or $\tau$) offers a unique window into both strong and weak effects in the charm quark sector. In the standard model (SM), the partial width of the decay $D^+_s\to \ell^+\nu_\ell$ can be written as [@decayrate] $$\Gamma_{D^+_{s}\to\ell^+\nu_\ell}=\frac{G_F^2}{8\pi}|V_{cs}|^2 f^2_{D^+_{s}} m_\ell^2 m_{D^+_{s}} \left (1-\frac{m_\ell^2}{m_{D^+_{s}}^2} \right )^2,$$ where $f_{D^+_{s}}$ is the $D^+_{s}$ decay constant, $|V_{cs}|$ is the $c\to s$ Cabibbo-Kobayashi-Maskawa (CKM) matrix element, $G_F$ is the Fermi coupling constant, $m_\ell$ is the lepton mass, and $m_{D^+_{s}}$ is the $D^+_{s}$ mass. In recent years, much progress has been achieved in the measurements of $f_{D^+_{s}}$ and $|V_{cs}|$ with $D^+_s\to \ell^+\nu_\ell$ decays at the CLEO [@cleo2009; @cleo2009a; @cleo2009b], BaBar [@babar2010], Belle [@belle2013] and BESIII [@bes2016] experiments. However, compared to the precision of the most accurate lattice quantum chromodynamics (LQCD) calculation of $f_{D^+_s}$ [@FLab2018], the accuracy of the measurements is still limited. Improved measurements of $f_{D^+_{s}}$ and $|V_{cs}|$ are critical to calibrate various theoretical calculations of $f_{D^+_{s}}$ [@FLab2018; @LQCD; @etm2015; @ukqcd2017; @ukqcd2015; @milc2012; @hpqcd2010; @hpqcd2012; @milc2005; @hpqcd2008; @etm2012; @chen2014; @pacs2011; @qcdsf2007; @chiu2005; @ukqcd2001; @becirevic1999; @bordes2005; @narison2002; @badalian2007; @ebert2006; @cvetic2004; @choi2007; @salcedo2004; @wang2004; @amundson1993; @becirevic2013; @lucha2011; @hwang2010; @wang2015], such as those from quenched and unquenched LQCD, QCD sum rules, $etc.$, and to test the unitarity of the quark mixing matrix with better precision. In the SM, the ratio of the branching fraction (BF) of $D^+_s\to \tau^+\nu_\tau$ over that of $D^+_s\to \mu^+\nu_\mu$ is predicted to be 9.74 with negligible uncertainty and the BFs of $D_s^+\to\mu^+\nu_\mu$ and $D_s^-\to\mu^-\bar{\nu}_\mu$ decays are expected to be the same. However, hints of lepton flavor universality (LFU) violation in semileptonic $B$ decays were recently reported at BaBar, LHCb and Belle [@babar_1; @babar_2; @lhcb_1; @lhcb_kee_3; @belle_kee]. It has been argued that new physics mechanisms, such as a two-Higgs-doublet model with the mediation of charged Higgs bosons [@fajfer; @2hdm] or a Seesaw mechanism due to lepton mixing with Majorana neutrinos [@seesaw], may cause LFU or CP violation. Tests of LFU and searches for CP violation in $D^+_s\to\ell^+\nu_\ell$ decays are therefore important tests of the SM. In this Letter, we present an experimental study of the leptonic decay $D_s^+\to\mu^+\nu_\mu$ [@conjugate] by analyzing a 3.19fb$^{-1}$ data sample collected with the BESIII detector at an $e^+e^-$ center-of-mass energy of $E_{\rm cm}=4.178$GeV. At this energy, $D^+_s$ mesons are produced mainly through the process $e^+e^-\to D^+_sD_s^{*-}+c.c$. In an event where a $D_s^-$ meson (called a single-tag (ST) $D_s^-$ meson) is fully reconstructed, one can then search for a $\gamma$ or $\pi^0$ and a $D_s^+$ meson in the recoiling system (called a double-tag (DT) event). Details about the design and performance of the BESIII detector are given in Ref. [@BESCol]. The endcap time-of-flight (TOF) system was upgraded with multi-gap resistive plate chamber technology and now has a time resolution of 60 ps [@mrpc1; @mrpc2]. Monte Carlo (MC) events are generated with a [geant]{}4-based [@geant4] detector simulation software package [@boost], which includes both the geometrical description of the detector and the detector’s response. An inclusive MC sample is produced at $E_{\rm cm}=4.178$GeV, which includes all open charm processes, initial state radiation (ISR) production of the $\psi(3770)$, $\psi(3686)$ and $J/\psi$, and $q\bar{q}\,(q=u,d,s)$ continuum processes, along with Bhabha scattering, $\mu^+\mu^-$, $\tau^+\tau^-$ and $\gamma\gamma$ events. The open charm processes are generated using [ConExc]{} [@conexc]. The effects of ISR [@isr] and final state radiation (FSR) [@photons] are considered. The decay modes with known BF are generated using [EvtGen]{} [@evtgen] and the other modes are generated using [LundCharm]{} [@lundcharm]. The ST $D^-_s$ mesons are reconstructed from 14 hadronic decay modes, $D^-_s\to K^+K^-\pi^-$, $K^+K^-\pi^-\pi^0$, $K^0_SK^-$, $K^0_SK^-\pi^0$, $K^0_SK^0_S\pi^-$, $K^0_SK^+\pi^-\pi^-$, $K^0_SK^-\pi^+\pi^-$, $K^-\pi^+\pi^-$, $\pi^+\pi^-\pi^-$, $\eta_{\gamma\gamma}\pi^-$, $\eta_{\pi^0\pi^+\pi^-}\pi^-$, $\eta^\prime_{\eta_{\gamma\gamma}\pi^+\pi^-}\pi^-$, $\eta^\prime_{\gamma\rho^0}\pi^-$, and $\eta_{\gamma\gamma}\rho^-$, where the subscripts of $\eta^{(\prime)}$ represent the decay modes used to reconstruct $\eta^{(\prime)}$. All charged tracks except for those from $K_S^0$ decays must originate from the interaction point (IP) with a distance of closest approach less than 1 cm in the transverse plane and less than 10 cm along the $z$ axis. The polar angle $\theta$ of each track defined with respect to the positron beam must satisfy $|\cos\theta|<0.93$. Measurements of the specific ionization energy loss ($dE/dx$) in the main drift chamber and the TOF are combined and used for particle identification (PID) by forming confidence levels for pion and kaon hypotheses ($CL_\pi$, $CL_K$). Kaon (pion) candidates are required to satisfy $CL_{K(\pi)}>CL_{\pi(K)}$. To select $K_S^0$ candidates, pairs of oppositely charged tracks with distances of closest approach to the IP less than 20 cm along the $z$ axis are assigned as $\pi^+\pi^-$ without PID requirements. These $\pi^+\pi^-$ combinations are required to have an invariant mass within $\pm12$MeV of the nominal $K_S^0$ mass [@PDG2017] and have a decay length of the reconstructed $K_S^0$ larger than $2\sigma$ of the vertex resolution away from the IP. The $\pi^0$ and $\eta$ mesons are reconstructed via $\gamma\gamma$ decays. It is required that each electromagnetic shower starts within 700ns of the event start time and its energy is greater than 25(50)MeV in the barrel(endcap) region of the electromagnetic calorimeter (EMC) [@BESCol]. The opening angle between the shower and the nearest charged track has to be greater than $10^{\circ}$. The $\gamma\gamma$ combinations with an invariant mass $M_{\gamma\gamma}\in(0.115,\,0.150)$ and $(0.50,\,0.57)$GeV$/c^{2}$ are regarded as $\pi^0$ and $\eta$ mesons, respectively. A kinematic fit is performed to constrain $M_{\gamma\gamma}$ to the $\pi^{0}$ or $\eta$ nominal mass [@PDG2017]. The $\eta$ candidates for the $\eta\pi^-$ ST channel are also reconstructed via $\pi^0\pi^+\pi^-$ candidates with an invariant mass within $(0.53,\,0.57)~\mathrm{GeV}/c^2$. The $\eta^\prime$ mesons are reconstructed via two decay modes, $\eta\pi^+\pi^-$ and $\gamma\rho^0$, whose invariant masses are required to be within $(0.946,\,0.970)$ and $(0.940,\,0.976)~\mathrm{GeV}/c^2$, respectively. In addition, the minimum energy of the $\gamma$ from $\eta'\to\gamma\rho^0$ decays must be greater than 0.1GeV. The $\rho^0$ and $\rho^+$ mesons are reconstructed from $\pi^+\pi^-$ and $\pi^+\pi^0$ candidates, whose invariant masses are required to be larger than $0.5~\mathrm{GeV}/c^2$ and within $(0.67,\,0.87)~\mathrm{GeV}/c^2$, respectively. The momentum of any pion not originating from a $K_S^0$, $\eta$, or $\eta^\prime$ decay is required to be greater than 0.1GeV/$c$ to reject soft pions from $D^*$ decays. For $\pi^+\pi^-\pi^-$ and $K^-\pi^+\pi^-$ combinations, the dominant peaking backgrounds from $K^0_S\pi^-$ and $K_S^0K^-$ events are rejected by requiring the invariant mass of any $\pi^+\pi^-$ combination be more than $\pm 0.03$ GeV/$c^2$ away from the nominal $K^0_S$ mass [@PDG2017]. To suppress non-$D_s^+D^{*-}_s$ events, the beam-constrained mass of the ST $D_s^-$ candidate $$M_{\rm BC}\equiv\sqrt{(E_{\rm cm}/2)^2-|\vec{p}_{D_s^-}|^2}$$ is required to be within $(2.010,\,2.073)\,\mathrm{GeV}/c^2$, where $\vec{p}_{D_s^-}$ is the momentum of the ST $D_s^-$ candidate. This requirement retains $D_s^-$ mesons directly from $e^+e^-$ annihilation and indirectly from $D_s^{*-}$ decay (See Fig. 1 in Ref. [@supplemental]). In each event, we only keep the candidate with the $D_s^-$ recoil mass $$M_{\rm rec} \equiv \sqrt{ \left (E_{\rm cm} - \sqrt{|\vec p_{D^-_s}|^2+m^2_{D^-_s}} \right )^2 -|\vec p_{D^-_s}|^2}$$ closest to the nominal $D_s^{*+}$ mass [@PDG2017] per tag mode per charge. Figure \[fig:stfit\] shows the invariant mass ($M_{\rm tag}$) spectra of the accepted ST candidates. The ST yield for each tag mode is obtained by a fit to the corresponding $M_{\rm tag}$ spectrum. The signal is described by the MC-simulated shape convolved with a Gaussian function representing the resolution difference between data and MC simulation. For the tag mode $D^-_s\to K_S^0K^-$, the peaking background from $D^-\to K^0_S\pi^-$ is described by the MC-simulated shape and then smeared with the same Gaussian function used in the signal shape with its size as a free parameter. The non-peaking background is modeled by a second- or third-order Chebychev polynomial function. Studies of the inclusive MC sample validate this parametrisation of the background shape. The fit results on these invariant mass spectra are shown in Fig. \[fig:stfit\]. The events in the signal regions are kept for further analysis. The total ST yield in data is $N^{\rm tot}_{\rm ST}=388660\pm2592$ (see tag-dependent ST yields and background yields in the signal regions in Table I of Ref. [@supplemental]). ![Fits to the $M_{\rm tag}$ distributions of the accepted ST candidates. Dots with error bars are data. Blue solid curves are the fit results. Red dashed curves are the fitted backgrounds. The black dotted curve in the $K_S^0K^-$ mode is the $D^-\to K_S^0\pi^-$ component. The pairs of arrows denote the signal regions. []{data-label="fig:stfit"}](massDs.eps){width="48.00000%"} At the recoil sides of the ST $D_s^-$ mesons, the $D_s^+\to\mu^+\nu_\mu$ candidates are selected with the surviving neutral and charged tracks. To select the soft $\gamma(\pi^0)$ from $D_s^{*}$ and to separate signals from combinatorial backgrounds, we define two kinematic variables $$\Delta E \equiv E_{\rm cm}-E_{\rm tag}-E_{\rm miss}-E_{\gamma(\pi^0)}$$ and $$\begin{aligned} \mathrm{MM}^2&\equiv\left (E_{\rm cm}-E_{\rm tag}-E_{\gamma(\pi^0)}-E_{\mu}\right )^2\nonumber\\ &-|-\vec{p}_{\rm tag}-\vec{p}_{\gamma(\pi^0)}-\vec{p}_{\mu}|^2.\end{aligned}$$ Here $E_{\rm miss} \equiv \sqrt{|\vec{p}_{\rm miss}|^2+m_{D_s^+}^2}$ and $\vec{p}_{\rm miss} \equiv -\vec{p}_{\rm tag}-\vec{p}_{\gamma(\pi^0)}$ are the missing energy and momentum of the recoiling system of the soft $\gamma(\pi^0)$ and the ST $D_s^-$, where $E_i$ and $\vec p_i$ ($i=\mu,\gamma(\pi^0)$ or tag) denote the energy and momentum of the muon, $\gamma(\pi^0)$ or ST $D^-_s$, respectively. $\mathrm{MM}^2$ is the missing mass square of the undetectable neutrino. We loop over all remaining $\gamma$ or $\pi^0$ candidates and choose the one giving a minimum $|\Delta E|$. The events with $\Delta E\in(-0.05,\,0.10)$GeV are accepted. The muon candidate is required to have an opposite charge to the ST $D^-_s$ meson and a deposited energy in the EMC within $(0.0,\,0.3)$GeV. It must also satisfy a two dimensional (2D, e.g., $|\cos\theta_\mu|$ and momentum $p_{\mu}$) requirement on the hit depth ($d_\mu$) in the muon counter, as explained in Ref. [@muid]. To suppress the backgrounds with extra photon(s), the maximum energy of the unused showers in the DT selection ($E_{\mathrm{extra}~\gamma}^{\rm max}$) is required to be less than 0.4GeV and no additional charged track that satisfies the charged track selection criteria is allowed. To improve the $\mathrm{MM}^2$ resolution, the candidate tracks, plus the missing neutrino, are subjected to a 4-constraint kinematic fit requiring energy and momentum conservation. In addition, the invariant masses of the two $D_s$ mesons are constrained to the nominal $D_s$ mass, the invariant mass of the $D_s^-\gamma(\pi^0)$ or $D_s^+\gamma(\pi^0)$ combination is constrained to the nominal $D_s^*$ mass, and the combination with the smaller $\chi^2$ is kept. Figure \[fig:mm2fit\] shows the $\mathrm{MM}^2$ distribution for the accepted DT candidate events. ![Fit to the $\mathrm{MM}^2$ distribution of the $D^+_s\to \mu^+\nu_\mu$ candidates. Inset plot shows the same distribution in log scale. Dots with error bars are data. Blue solid curve is the fit result. Red dotted curve is the fitted background. Orange hatched and blue cross-hatched histograms are the BKGI component and the combined BKGII and BKGIII components, respectively (see text). []{data-label="fig:mm2fit"}](m2miss.eps){height="5cm"} To extract the DT yield, an unbinned constrained fit is performed to the $\mathrm{MM}^2$ distribution. In the fit, the background events are classified into three categories: events with correctly reconstructed ST $D_s^-$ and $\mu^+$ but an unmatched $\gamma(\pi^0)$ from the $D_s^{*-}$ (BKGI), events with a correctly reconstructed ST $D_s^-$ but misidentified $\mu^+$ (BKGII), and other events with a misreconstructed ST $D_s^-$ (BKGIII). The signal and BKGI shapes are modeled with MC simulation. The signal shape is convolved with a Gaussian function with its mean and width as free parameters. The ratio of the signal yield over the BKGI yield is constrained to the value determined with the signal MC events. The size and shape of the BKGII and BKGIII components are fixed by analyzing the inclusive MC sample. From the fit to the $\mathrm{MM}^2$ distribution, as shown in Fig. \[fig:mm2fit\], we determine the number of $D_s^+\to\mu^+\nu_\mu$ decays to be $N_{\rm DT}=1135.9\pm33.1$. The efficiencies for reconstructing the DT candidate events are determined with an exclusive MC sample of $e^+e^-\to D_s^+D^{*-}_s$, where the $D_s^-$ decays to each tag mode and the $D_s^+$ decays to $\mu^+\nu_\mu$. Dividing them by the ST efficiencies determined with the inclusive MC sample yields the corresponding efficiencies of the $\gamma(\pi^0)\mu^+\nu_\mu$ reconstruction. The averaged efficiency of finding $\gamma(\pi^0)\mu^+\nu_\mu$ is $(52.67\pm0.19)\%$ as determined from $$\varepsilon_{\gamma(\pi^0)\mu^+\nu_\mu}=f_{\mu\,\rm PID}^{\rm cor}\sum_{i}(N_{\rm ST}^i\varepsilon_{\rm DT}^i)/(N_{\rm ST}^{\rm tot}\varepsilon_{\rm ST}^i),$$ where $N_{\rm ST}^i$, $\varepsilon_{\rm ST}^i$, and $\varepsilon_{\rm DT}^i$ are the ST yield, ST efficiency and DT efficiency in the $i$-th ST mode, respectively. The factor $f_{\mu\,\rm PID}^{\rm cor}=0.897$ accounts for the difference between the $\mu^+$ PID efficiencies in data and MC simulation \[$ \varepsilon_{\mu\,\rm PID}^{\rm data\,(MC)}$\]. These efficiencies are estimated using $e^+e^-\to\gamma\mu^+\mu^-$ samples but reweighted by the $\mu^+$ 2D distribution of $D_s^+\to\mu^+\nu_\mu$. It is nonnegligible mainly due to the imperfect simulation of $d_\mu$ and its applicability in different topology environments is verified via three aspects: (1) Studies with signal MC events show that $\varepsilon_{\mu\,\rm PID}^{\rm MC}=(74.79\pm0.03)\%$ for $D_s^+\to\mu^+\nu_\mu$ signals can be well reproduced by the 2D reweighted efficiency $\varepsilon_{\mu\,\rm PID}^{\rm MC}=(74.91\pm0.10)\%$ with $e^+e^-\to\gamma\mu^+\mu^-$ samples. (2) Our nominal BF ($\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}$) obtained later can be well reproduced by removing the $d_\mu$ requirement, with negligible difference but obviously lower precision due to much higher background [@dstauv]. (3) The $\varepsilon_{\mu\,\rm PID}^{\rm data\,(MC)}$ for $e^+e^-\to\gamma_{\rm ISR}\psi(3686),\psi(3686)\to\pi^+\pi^-J/\psi,J/\psi\to\mu^+\mu^-$ events can be well reproduced by the corresponding 2D reweighted efficiencies with $e^+e^-\to\gamma\mu^+\mu^-$ samples (see Table II of Ref. [@supplemental]). The BF of $D_s^+\to\mu^+\nu_\mu$ is then determined to be $(5.49\pm0.16_{\rm stat.}\pm0.15_{\rm syst.})\times10^{-3}$ from $$\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}=f_{\rm cor}^{\rm rad}N_{\rm DT}/(N_{\rm ST}^{\rm tot}\varepsilon_{\gamma(\pi^0)\mu^+\nu_\mu}),$$ where the radiative correction factor $f_{\rm cor}^{\rm rad}=0.99$ is due to the contribution from $D^+_s\to \gamma {\mathcal D}^{*+}_s \to \gamma \mu^+\nu_\mu$ [@radiation], with ${\mathcal D}^{*+}_s$ as a virtual vector or axial-vector meson. This contribution is almost identical with our signal process for low energy radiated photons. We further examine the BFs measured with individual tags which have very different background levels, and a good consistence is found (see Table I of Ref. [@supplemental] for tag-dependent DT yields, $\varepsilon_{\gamma(\pi^0)\mu^+\nu_\mu}$ and $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}$). The systematic uncertainties in the BF measurement are estimated relative to the measured BF and are described below. For uncertainties in the event selection criteria, the $\mu^+$ tracking and PID efficiencies are studied with $e^+e^-\to\gamma\mu^+\mu^-$ events. After correcting the detection efficiency by $f^{\rm cor}_{\mu\,\rm PID}$, we assign 0.5% and 0.8% as the uncertainties in $\mu^+$ tracking and PID efficiencies, respectively. The photon reconstruction efficiency has been previously studied with $J/\psi\to\pi^+\pi^-\pi^0$ decays [@geff]. The uncertainty of finding $\gamma(\pi^0)$ is weighted according to the BFs of $D_s^{*+}\to\gamma D_s^+$ and $D_s^{*+}\to\pi^0D_s^+$ [@PDG2017] and assigned to be 1.0%. The efficiencies for the requirements of $E_{\mathrm{extra}~\gamma}^{\rm max}$ and no extra good charged track are studied with a DT hadronic sample. The systematic uncertainties are taken to be 0.3% and 0.9% considering the efficiency differences between data and MC simulation, respectively. The uncertainty of the $\Delta E$ requirement is estimated by varying the signal region by $\pm0.01$GeV, and the maximum change of the BF, 0.5%, is taken as the systematic uncertainty. To determine the uncertainty in the $\mathrm{MM}^2$ fit, we change the fit range by $\pm0.02~\mathrm{GeV}^2/c^4$, and the largest change of the BF is 0.6%. We change the signal shape by varying the $\gamma(\pi^0)$ match requirement and the maximum change is 0.2%. Two sources of uncertainty in the background estimation are considered. The effect of the background shape is obtained to be 0.2% by shifting the number of the main components of BKGII by $\pm 1\sigma$ of the uncertainties of the corresponding BFs [@PDG2017], and varying the relative fraction of the main components of BKGII by 50%. The effect of the fixed number of the BKGII and BKGIII is estimated to be 0.5% by varying the nominal numbers by $\pm 1\sigma$ of their uncertainties. To evaluate the uncertainty in the fixed ratio of signal and BKGI, we perform an alternative fit to the $\mathrm{MM}^2$ distribution of data without constraining the ratio of signal and BKGI. The change in the DT yield, 1.1%, is assigned as the relevant uncertainty. The uncertainty in the number of ST $D_s^-$ mesons is assigned to be 0.8% by examining the changes of the fit yields when varying the signal shape, background shape, bin size and fit range and considering the background fluctuation in the fit. The uncertainty due to the limited MC size is 0.4%. The uncertainty in the imperfect simulation of the FSR effect is estimated as 0.4% by varying the amount of FSR photons in signal MC events [@photons]. The uncertainty due to the quoted BFs of $D_s^{*-}$ subdecays from the particle data group (PDG) [@PDG2017] is examined by varying each subdecay BF by $\pm 1\sigma$. The efficiency change is found to be 0.4% and is taken as the associated uncertainty. The uncertainty in the radiative correction is assigned to be 1.0%, which is taken as 100% of its central value from theoretical calculation [@radiation]. The ST efficiencies in the inclusive and signal MC samples are slightly different with each other due to different track multiplicities in these two environments. This may cause incomplete cancellation of the uncertainties of the ST efficiencies. The associated uncertainty is assigned as 0.6%, by taking into account the differences of the efficiencies of tracking/PID of $K^\pm$ and $\pi^\pm$, as well as the selections of neutral particles between data and MC simulation in different environments. The total systematic uncertainty is determined to be 2.7% by adding all the uncertainties in quadrature. Combining our BF with the world average values of $G_F$, $m_\mu$, $m_{D^+_s}$ and the lifetime of $D_s^+$ [@PDG2017] in Eq. (1) yields $$f_{D_s^+}|V_{cs}|=246.2\pm3.6_{\rm stat.}\pm3.5_{\rm syst.}~\mathrm{MeV}.$$ Here the systematic uncertainties arise mainly from the uncertainties in the measured BF (1.5%) and the lifetime of the $D^+_s$ (0.4%). Taking the CKM matrix element $|V_{cs}|=0.97359_{-0.00011}^{+0.00010}$ from the global fit in the SM [@PDG2017] or the averaged decay constant $f_{D_s^+}=249.9\pm0.4~\mathrm{MeV}$ of recent LQCD calculations [@FLab2018; @etm2015] as input, we determine $$f_{D_s^+}=252.9\pm3.7_{\rm stat.}\pm3.6_{\rm syst.}~\mathrm{MeV}$$ and $$|V_{cs}|=0.985\pm0.014_{\rm stat.}\pm0.014_{\rm syst.}.$$ The additional systematic uncertainties according to the input parameters are negligible for $|V_{cs}|$ and 0.2% for $f_{D_s^+}$. The measured $|V_{cs}|$ is consistent with our measurements using $D\to\bar K\ell^+\nu_\ell$ [@bes3_kev; @bes3_ksev; @bes3_klev; @bes3_kmuv] and $D_s^+\to\eta^{(\prime)}e^+\nu_e$ [@bes3_etaev], but with much better precision. Combining the obtained $f_{D_s^+}|V_{cs}|$ and its counterpart $f_{D^+}|V_{cd}|$ measured in our previous work [@fdp], along with $|V_{cd}/V_{cs}|=0.23047\pm0.00045$ from the SM global fit [@PDG2017], yields $f_{D_s^+}/f_{D^+}=1.24\pm0.04_{\rm stat.}\pm0.02_{\rm syst.}$. It is consistent with the CLEO measurement [@cleo2009] within 1$\sigma$ and the LQCD calculation within 2$\sigma$ [@FLab2018]. Alternatively, with the input of $f_{D_s^+}/f_{D^+}=1.1749\pm0.0016$ calculated by LQCD [@FLab2018], we obtain $|V_{cd}/V_{cs}|^2=0.048\pm0.003_{\rm stat.}\pm0.001_{\rm syst.}$, which agrees with the one expected by $|V_{cs}|$ and $|V_{cd}|$ given by the CKMfitter within 2$\sigma$. Here, only the systematic uncertainty in the radiative correction is canceled since the two data samples were taken in different years. Based on our result for ${\mathcal{B}}_{D_s^+\to\mu^+\nu_\mu}$ and those measured at the CLEO [@cleo2009], BaBar [@babar2010] and Belle [@belle2013] experiments, along with a previous measurement at BESIII [@bes2016], the inverse-uncertainty weighted BF is determined to be $\bar {\mathcal{B}}_{D_s^+\to\mu^+\nu_\mu}=(5.49\pm0.17)\times10^{-3}$ [@bfweight]. The ratio of $\bar {\mathcal{B}}_{D_s^+\to\mu^+\nu_\mu}$ over the PDG value of $\mathcal{B}_{D_s^+\to\tau^+\nu_\tau}=(5.48\pm0.23)\%$ [@PDG2017] is determined to be $\frac{\mathcal{B}_{D_s^+\to\tau^+\nu_\tau}}{\bar {\mathcal{B}}_{D_s^+\to\mu^+\nu_\mu}}=9.98\pm0.52,$ which agrees with the SM predicted value of 9.74 within uncertainty. The BFs of $D_s^+\to\mu^+\nu_\mu$ and $D_s^-\to\mu^-\bar{\nu}_\mu$ decays are also measured separately. The results are ${\mathcal B}_{D_s^+\to\mu^+\nu_\mu}=(5.62\pm0.23_{\rm stat.})\times10^{-3}$ and ${\mathcal B}_{D_s^-\to\mu^-\bar \nu_\mu}=(5.40\pm0.23_{\rm stat.})\times10^{-3}$. The BF asymmetry is determined to be $A_{\rm CP}=\frac{{\mathcal B}_{D_s^+\to\mu^+\nu_\mu}-{\mathcal B}_{D_s^-\to\mu^-\bar{\nu}_\mu}}{{\mathcal B}_{D_s^+\to\mu^+\nu_\mu}+{\mathcal B}_{D_s^-\to\mu^-\bar{\nu}_\mu}}=(2.0\pm3.0_{\rm stat.}\pm1.2_{\rm syst.})\%,$ where the uncertainties in the tracking and PID efficiencies of the muon, the ST yields, the limited MC statistics, as well as the signal shape and fit range in $\mathrm{MM}^2$ fits for $D_s^+$ and $D_s^-$ have been studied separately and are not canceled. In summary, by analyzing 3.19 fb$^{-1}$ of $e^+e^-$ collision data collected at $E_{\rm cm}=4.178$GeV with the BESIII detector, we have measured $\mathcal{B}(D^+_s\to\mu^+\nu_\mu)$, the decay constant $f_{D_s^+}$, and the CKM matrix element $|V_{cs}|$. These are the most precise measurements to date, and are important to calibrate various theoretical calculations of $f_{D_s^+}$ and test the unitarity of the CKM matrix with better accuracy. We also search for LFU and CP violation in $D_s^+\to\ell^+\nu_\ell$ decays, and no evidence is found. The BESIII collaboration thanks the staff of BEPCII and the IHEP computing center for their strong support. This work is supported in part by National Key Basic Research Program of China under Contract No. 2015CB856700; National Natural Science Foundation of China (NSFC) under Contracts Nos. 11235011, 11335008, 11425524, 11505034, 11575077, 11625523, 11635010, 11675200, 11775230; the Chinese Academy of Sciences (CAS) Large-Scale Scientific Facility Program; the CAS Center for Excellence in Particle Physics (CCEPP); Joint Large-Scale Scientific Facility Funds of the NSFC and CAS under Contracts Nos. U1332201, U1532257, U1532258, U1632109; CAS under Contracts Nos. 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[**122**]{}, 011804 (2019). M. Ablikim [*et al.*]{} (BESIII Collaboration), arXiv:1901.02133. M. Ablikim [*et al.*]{} (BESIII Collaboration), Phys. Rev. D [**89**]{}, 051104 (2014). To be consistent with results at CLEO and BESIII, the BFs measured by Belle and BaBar have been further corrected by $f_{\rm cor}^{\rm rad}$. The uncertainty in $f_{\rm cor}^{\rm rad}$ is common for individual measurements, but will be taken into account in the averaged BF. For the previous measurement in BESIII, we quote the BF without the LFU constraint. Supplemental material {#supplemental-material .unnumbered} ===================== Figure \[fig:mBC\] shows the $M_{\rm BC}$ distributions of the ST $D_s^-$ candidates from $e^+e^-\to D_s^-D_s^{*+}$, $e^+e^-\to D_s^+D_s^{*-}$, and $e^+e^-\to D_s^+D_s^-$ processes based on MC simulation. Both $D_s^-$ mesons directly produced from $e^+e^-$ annihilation and indirectly produced from $D_s^{*-}$ decays are retained by our nominal $M_{\rm BC}$ requirement. Table \[tab:bf\] summarizes the ST yield $N_{\rm ST}$, the background yield $N_{\rm ST}^{\rm bkg}$ in the $M_{\rm tag}$ signal regions, the DT yield $N_{\rm DT}$, the signal efficiency $\varepsilon_{\gamma(\pi^0)\mu^+\nu_\mu}$ and the obtained $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}$ for each ST mode. Although the background levels for various ST modes are much different, the BFs measured with individual ST modes are consistent with each other. As an independent check, we further examine the $\mu^+$ PID efficiencies of data and MC simulation, $\varepsilon_{\mu~\mathrm{PID}}^{\rm data}$ and $\varepsilon_{\mu~\mathrm{PID}}^{\rm MC}$, by analyzing $e^+e^-\to\gamma_{\rm ISR}\psi(3686),\psi(3686)\to\pi^+\pi^-J/\psi,J/\psi\to\mu^+\mu^-$ events (sample I) and corresponding 2D reweighted efficiencies based on $e^+e^-\to\gamma\mu^+\mu^-$ samples (sample II). Two samples with much different topologies give consistent $\varepsilon_{\mu~\mathrm{PID}}^{\rm data}$, $\varepsilon_{\mu~\mathrm{PID}}^{\rm MC}$, and $f_{\mu~\mathrm{PID}}^{\rm cor}=\varepsilon_{\mu~\mathrm{PID}}^{\rm data}/\varepsilon_{\mu~\mathrm{PID}}^{\rm MC}$, as shown in Table \[tab:mupid\]. The obtained $f_{\mu~\mathrm{PID}}^{\rm cor}$ in these two samples are different with that in $D_s^+\to\mu^+\nu_\mu$ mainly due to much higher muon momentum. {height="7cm"} ST mode $N_{\rm ST}$ $N_{\rm ST}^{\rm bkg}$ $N_{\rm DT}$ $\varepsilon_{\gamma(\pi^0)\mu^+\nu_\mu}$ (%) $\mathcal{B}_{D_s^+\to\mu^+\nu_\mu}~(\times10^{-3})$ ---------------------------------------------------- ---------------- ------------------------ ---------------- ----------------------------------------------- ------------------------------------------------------ $K^+K^-\pi^+$ $133959\pm633$ 173160 $373.3\pm18.9$ $49.73\pm0.24$ $5.55\pm0.28$ $K^+K^-\pi^+\pi^0$ $41377\pm916$ 221099 $123.1\pm10.7$ $57.32\pm0.85$ $5.14\pm0.46$ $\pi^+\pi^+\pi^-$ $35966\pm913$ 300499 $90.0\pm9.9$ $51.21\pm0.53$ $4.84\pm0.55$ $K_S^0K^+$ $32039\pm291$ 18776 $79.7\pm9.0$ $49.77\pm0.36$ $4.95\pm0.56$ $K_S^0K^+\pi^0$ $11294\pm433$ 52788 $38.4\pm6.1$ $56.71\pm2.34$ $5.94\pm0.97$ $K^+\pi^+\pi^-$ $15877\pm872$ 246528 $45.6\pm7.2$ $51.21\pm1.30$ $5.55\pm0.93$ $K_S^0K_S^0\pi^+$ $4832\pm180$ 11274 $20.2\pm4.4$ $50.55\pm1.25$ $8.19\pm1.82$ $K_S^0K^-\pi^+\pi^+$ $14046\pm240$ 26873 $44.1\pm6.5$ $51.91\pm0.91$ $5.98\pm0.89$ $K_S^0K^+\pi^+\pi^-$ $7171\pm292$ 37456 $24.7\pm4.9$ $54.14\pm1.21$ $6.29\pm1.28$ $\eta_{\gamma\gamma}\pi^+$ $19323\pm725$ 53701 $63.5\pm8.1$ $52.72\pm0.62$ $6.17\pm0.82$ $\eta_{\pi^+\pi^-\pi^0}\pi^+$ $5508\pm202$ 11225 $20.2\pm4.5$ $54.00\pm1.13$ $6.73\pm1.51$ $\eta^\prime_{\pi^+\pi^-\eta_{\gamma\gamma}}\pi^+$ $9242\pm155$ 5002 $33.0\pm5.7$ $56.30\pm0.54$ $6.27\pm1.09$ $\eta^\prime_{\gamma\rho^0}\pi^+$ $25191\pm695$ 152363 $75.1\pm8.6$ $53.74\pm0.72$ $5.49\pm0.65$ $\eta_{\gamma\gamma}\rho^+$ $32835\pm1537$ 166324 $108.4\pm10.5$ $60.70\pm0.91$ $5.38\pm0.58$ Samples $\varepsilon_{\mu~\mathrm{PID}}^{\rm data}$ (%) $\varepsilon_{\mu~\mathrm{PID}}^{\rm MC}$ (%) $f_{\mu~\mathrm{PID}}^{\rm cor}$ --------- ------------------------------------------------- ----------------------------------------------- ---------------------------------- I $76.64\pm0.68$ $81.04\pm0.21$ $0.946\pm0.009$ II $76.85\pm0.30$ $81.66\pm0.11$ $0.941\pm0.004$

ArXiv

--- abstract: 'Complex Enriques surfaces with a finite group of automorphisms are classified into seven types. In this paper, we determine which types of such Enriques surfaces exist in characteristic 2. In particular we give a one dimensional family of classical and supersingular Enriques surfaces with the automorphism group $\Aut(X)$ isomorphic to the symmetric group $\mathfrak{S}_5$ of degree five.' address: - 'Faculty of Science and Engineering, Hosei University, Koganei-shi, Tokyo 184-8584, Japan' - 'Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan' author: - Toshiyuki Katsura - Shigeyuki Kondō title: On Enriques surfaces in characteristic 2 with a finite group of automorphisms --- [^1] Introduction {#sec1} ============ We work over an algebraically closed field $k$ of characteristic 2. Complex Enriques surfaces with a finite group of automorphisms are completely classified into seven types. The main purpose of this paper is to determine which types of such Enriques surfaces exist in characteristic 2. Recall that, over the complex numbers, a generic Enriques surface has an infinite group of automorphisms (Barth and Peters [@BP]). On the other hand, Fano [@F] gave an Enriques surface with a finite group of automorphisms. Later Dolgachev [@D1] gave another example of such Enriques surfaces. Then Nikulin [@N] proposed a classification of such Enriques surfaces in terms of the periods. Finally the second author [@Ko] classified all complex Enriques surfaces with a finite group of automorphisms, geometrically. There are seven types ${\I, \II,\ldots, \VII}$ of such Enriques surfaces. The Enriques surfaces of type ${\I}$ or ${\II}$ form an irreducible one dimensional family, and each of the remaining types consists of a unique Enriques surface. The first two types contain exactly twelve nonsingular rational curves, on the other hand, the remaining five types contain exactly twenty nonsingular rational curves. The Enriques surface of type ${\I}$ (resp. of type ${\VII}$) is the example given by Dolgachev (resp. by Fano). We call the dual graphs of all nonsingular rational curves on the Enriques surface of type $K$ the dual graph of type $K$ ($K = {\I, \II,..., \VII}$). In positive characteristics, the classification problem of Enriques surfaces with a finite group of automorphisms is still open. Especially the case of characteristic 2 is most interesting. In the paper [@BM2], Bombieri and Mumford classified Enriques surfaces in characteristic 2 into three classes, namely, singular, classical and supersingular Enriques surfaces. As in the case of characteristic $0$, an Enriques surface $X$ in characteristic 2 has a canonical double cover $\pi : Y \to X$, which is a separable ${\bf Z}/2{\bf Z}$-cover, a purely inseparable $\mu_2$- or $\alpha_2$-cover according to $X$ being singular, classical or supersingular. The surface $Y$ might have singularities, but it is $K3$-like in the sense that its dualizing sheaf is trivial. In this paper we consider the following problem: [*does there exist an Enriques surface in characteristic $2$ with a finite group of automorphisms whose dual graph of all nonsingular rational curves is of type ${\rm I, II,..., VI}$ or ${\rm VII}$ $?$*]{} Note that if Enriques surface $S$ in any characteristic has the dual graph of type $K$ ($K={\rm I, II,..., VII}$), then the automorphism group ${\rm Aut}(S)$ is finite by Vinberg’s criterion (see Proposition \[Vinberg\]). We will prove the following Table \[Table1\]: \[\] In Table \[Table1\], $\bigcirc$ means the existence and $\times$ means the non-existence of an Enriques surface with the dual graph of type $\I,..., \VII$. In case of type ${\I, \II, \VI}$, the construction of such Enriques surfaces over the complex numbers works well in characteristic 2 (Theorems \[Ithm\], \[IIthm\], \[VIthm\]). The most difficult and interesting case is of type ${\VII}$. We give a 1-dimensional family of classical and supersingular Enriques surfaces with a finite group of automorphisms whose dual graph is of type ${\VII}$ (Theorems \[main\], \[main2\]). We remark that this family is non-isotrivial (Theorem \[non-isotrivial\]). Recently the authors [@KK] gave a one dimensional family of classical and supersingular Enriques surfaces which contain a remarkable forty divisors, by using a theory of Rudakov and Shafarevich [@RS] on purely inseparable covers of surfaces. We employ here the same method to construct the above classical and supersingular Enriques surfaces with the dual graph of type ${\VII}$. It is known that there exist Enriques surfaces in characteristic 2 with a finite group of automorphisms whose dual graphs of all nonsingular rational curves do not appear in the case of complex surfaces (Ekedahl and Shepherd-Barron[@ES], Salomonsson[@Sa]). See Remark \[extra\]. The remaining problem of the classification of Enriques surfaces in characteristic 2 with a finite group of automorphisms is to determine such Enriques surfaces appeared only in characteristic 2. The plan of this paper is as follows. In section \[sec2\], we recall the known results on Rudakov-Shafarevich’s theory on derivations, lattices and Enriques surfaces. In section \[sec3\], we give a construction of a one dimensional family of classical and supersingular Enriques surfaces with the dual graph of type $\VII$. Moreover we show the non-existence of singular Enriques surfaces with the dual graph of type ${\rm VII}$ (Theorem \[non-existVII\]). In section \[sec4\], we discuss other cases, that is, the existence of singular Enriques surfaces of type $\I, \II, \VI$ and the non-existence of other cases (Theorems \[Ithm\], \[non-existI\], \[IIthm\], \[non-existII\], \[VIthm\], \[non-existVI\], \[non-existIII\]). In appendices A and B, we give two remarks. As appendix A, we show that the covering $K3$ surface of any singular Enriques surface has height $1$. As appendix B, we show that for each singular Enriques surface with the dual graph of type ${\rm I}$ its canonical cover is isomorphic to the Kummer surface of the product of two ordinary elliptic curves. [**Acknowledgement.**]{} The authors thank Igor Dolgachev for valuable conversations. In particular all results in Section \[sec4\] were obtained by discussion with him in Soeul and Kyoto, 2014. They thank him that he permits us to give these results in this paper. The authors also thank Matthias Schütt and Hiroyuki Ito for pointing out the non-existence of singular Enriques surfaces with the dual graph of nonsingular rational curves of type $\VII$. Preliminaries {#sec2} ============= Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $S$ be a nonsingular complete algebraic surface defined over $k$. We denote by $K_{S}$ a canonical divisor of $S$. A rational vector field $D$ on $S$ is said to be $p$-closed if there exists a rational function $f$ on $S$ such that $D^p = fD$. A vector field $D$ for which $D^p=0$ is called of additive type, while that for which $D^p=D$ is called of multiplicative type. Let $\{U_{i} = {\rm Spec} A_{i}\}$ be an affine open covering of $S$. We set $A_{i}^{D} = \{D(\alpha) = 0 \mid \alpha \in A_{i}\}$. Affine varieties $\{U_{i}^{D} = {\rm Spec} A_{i}^{D}\}$ glue together to define a normal quotient surface $S^{D}$. Now, we assume that $D$ is $p$-closed. Then, the natural morphism $\pi : S \longrightarrow S^D$ is a purely inseparable morphism of degree $p$. If the affine open covering $\{U_{i}\}$ of $S$ is fine enough, then taking local coordinates $x_{i}, y_{i}$ on $U_{i}$, we see that there exsit $g_{i}, h_{i}\in A_{i}$ and a rational function $f_{i}$ such that the divisors defined by $g_{i} = 0$ and by $h_{i} = 0$ have no common divisor, and such that $$D = f_{i}\left(g_{i}\frac{\partial}{\partial x_{i}} + h_{i}\frac{\partial}{\partial y_{i}}\right) \quad \mbox{on}~U_{i}.$$ By Rudakov and Shafarevich [@RS] (Section 1), divisors $(f_{i})$ on $U_{i}$ give a global divisor $(D)$ on $S$, and zero-cycles defined by the ideal $(g_{i}, h_{i})$ on $U_{i}$ give a global zero cycle $\langle D \rangle $ on $S$. A point contained in the support of $\langle D \rangle $ is called an isolated singular point of $D$. If $D$ has no isolated singular point, $D$ is said to be divisorial. Rudakov and Shafarevich ([@RS], Theorem 1, Corollary) showed that $S^D$ is nonsingular if $\langle D \rangle = 0$, i.e., $D$ is divisorial. When $S^D$ is nonsingular, they also showed a canonical divisor formula $$\label{canonical} K_{S} \sim \pi^{*}K_{S^D} + (p - 1)(D),$$ where $\sim$ means linear equivalence. As for the Euler number $c_{2}(S)$ of $S$, we have a formula $$\label{euler} c_{2}(S) = \deg \langle D \rangle - \langle K_{S}, (D)\rangle - (D)^2$$ (cf. Katsura and Takeda [@KT], Proposition 2.1). Now we consider an irreducible curve $C$ on $S$ and we set $C' = \pi (C)$. Take an affine open set $U_{i}$ above such that $C \cap U_{i}$ is non-empty. The curve $C$ is said to be integral with respect to the vector field $D$ if $g_{i}\frac{\partial}{\partial x_{i}} + h_{i}\frac{\partial}{\partial y_{i}}$ is tangent to $C$ at a general point of $C \cap U_{i}$. Then, Rudakov-Shafarevich [@RS] (Proposition 1) showed the following proposition: \[insep\] $({\rm i})$ If $C$ is integral, then $C = \pi^{-1}(C')$ and $C^2 = pC'^2$. $({\rm ii})$ If $C$ is not integral, then $pC = \pi^{-1}(C')$ and $pC^2 = C'^2$. A lattice is a free abelian group $L$ of finite rank equipped with a non-degenerate symmetric integral bilinear form $\langle . , . \rangle : L \times L \to {\bf Z}$. The signature of a lattice is the signature of the real vector space $L\otimes {\bf R}$ equipped with the symmetric bilinear form extended from the one on $L$ by linearity. A lattice is called even if $\langle x, x\rangle \in 2{\bf Z}$ for all $x\in L$. We denote by $U$ the even unimodular lattice of signature $(1,1)$, and by $A_m, \ D_n$ or $\ E_k$ the even [*negative*]{} definite lattice defined by the Cartan matrix of type $A_m, \ D_n$ or $\ E_k$ respectively. We denote by $L\oplus M$ the orthogonal direct sum of lattices $L$ and $M$. Let ${\rm O}(L)$ be the orthogonal group of $L$, that is, the group of isomorphisms of $L$ preserving the bilinear form. In characteristic 2, a minimal algebaic surface with numerically trivial canonical divisor is called an Enriques surface if the second Betti number is equal to 10. Such surfaces $S$ are divided into three classes (for details, see Bombieri and Mumford [@BM2], Section 3): - $K_{S}$ is not linearly equivalent to zero and $2K_{S}\sim 0$. Such an Enriques surface is called a classical Enriques surface. - $K_{S} \sim 0$, ${\rm H}^{1}(S, {{\mathcal{O}}}_{S}) \cong k$ and the Frobenius map acts on ${\rm H}^{1}(S, {{\mathcal{O}}}_S)$ bijectively. Such an Enriques surface is called a singular Enriques surface. - $K_{S} \sim 0$, ${\rm H}^{1}(S, {{\mathcal{O}}}_{S}) \cong k$ and the Frobenius map is the zero map on ${\rm H}^{1}(S, {{\mathcal{O}}}_S)$. Such an Enriques surface is called a supersingular Enriques surface. Let $S$ be an Enriques surface and let $\Num(S)$ be the quotient of the Néron-Severi group of $S$ by torsion. Then $\Num(S)$ together with the intersection product is an even unimodular lattice of signature $(1,9)$ (Cossec and Dolgachev [@CD], Chap. II, Theorem 2.5.1), and hence is isomorphic to $U\oplus E_8$. We denote by ${\rm O}(\Num(S))$ the orthogonal group of $\Num(S)$. The set $$\{ x \in \Num(S)\otimes {\bf R} \ : \ \langle x, x \rangle > 0\}$$ has two connected components. Denote by $P(S)$ the connected component containing an ample class of $S$. For $\delta \in \Num(S)$ with $\delta^2=-2$, we define an isometry $s_{\delta}$ of $\Num(S)$ by $$s_{\delta}(x) = x + \langle x, \delta\rangle \delta, \quad x \in \Num(S).$$ The isometry $s_{\delta}$ is called the reflection associated with $\delta$. Let $W(S)$ be the subgroup of ${\rm O}(\Num(S))$ generated by reflections associated with all nonsingular rational curves on $S$. Then $P(S)$ is divided into chambers each of which is a fundamental domain with respect to the action of $W(S)$ on $P(S)$. There exists a unique chamber containing an ample class which is nothing but the closure of the ample cone $D(S)$ of $S$. It is known that the natural map $$\label{coh-trivial} \rho : \Aut(S) \to {\rm O}(\Num(S))$$ has a finite kernel (Dolgachev [@D2], Theorems 4, 6). Since the image $\Im(\rho)$ preserves the ample cone, we see $\Im(\rho) \cap W(S) = \{1\}$. Therefore $\Aut(S)$ is finite if the index $[\O(\Num(S)) : W(S)]$ is finite. Thus we have the following Proposition (see Dolgachev [@D1], Proposition 3.2). \[finiteness\] If $W(S)$ is of finite index in ${\rm O}({\rm Num}(S))$, then ${\rm Aut}(S)$ is finite. Over the field of complex numbers, the converse of Proposition \[finiteness\] holds by using the Torelli type theorem for Enriques surfaces (Dolgachev [@D1], Theorem 3.3). Now, we recall Vinberg’s criterion which guarantees that a group generated by finite number of reflections is of finite index in ${\rm O}(\Num(S))$. Let $\Delta$ be a finite set of $(-2)$-vectors in $\Num(S)$. Let $\Gamma$ be the graph of $\Delta$, that is, $\Delta$ is the set of vertices of $\Gamma$ and two vertices $\delta$ and $\delta'$ are joined by $m$-tuple lines if $\langle \delta, \delta'\rangle=m$. We assume that the cone $$K(\Gamma) = \{ x \in \Num(S)\otimes {\bf R} \ : \ \langle x, \delta_i \rangle \geq 0, \ \delta_i \in \Delta\}$$ is a strictly convex cone. Such $\Gamma$ is called non-degenerate. A connected parabolic subdiagram $\Gamma'$ in $\Gamma$ is a Dynkin diagram of type $\tilde{A}_m$, $\tilde{D}_n$ or $\tilde{E}_k$ (see [@V], p. 345, Table 2). If the number of vertices of $\Gamma'$ is $r+1$, then $r$ is called the rank of $\Gamma'$. A disjoint union of connected parabolic subdiagrams is called a parabolic subdiagram of $\Gamma$. We denote by $\tilde{K_1}\oplus \tilde{K_2}$ a parabolic subdiagram which is a disjoint union of two connected parabolic subdiagrams of type $\tilde{K_1}$ and $\tilde{K_2}$, where $K_i$ is $A_m$, $D_n$ or $E_k$. The rank of a parabolic subdiagram is the sum of the rank of its connected components. Note that the dual graph of singular fibers of an elliptic fibration on $S$ gives a parabolic subdiagram. For example, a singular fiber of type ${\rm III}$, ${\rm IV}$ or ${\rm I}_{n+1}$ defines a parabolic subdiagram of type $\tilde{A}_1$, $\tilde{A}_2$ or $\tilde{A}_n$ respectively. We denote by $W(\Gamma)$ the subgroup of ${\rm O}(\Num(S))$ generated by reflections associated with $\delta \in \Gamma$. \[Vinberg\][(Vinberg [@V], Theorem 2.3)]{} Let $\Delta$ be a set of $(-2)$-vectors in $\Num(S)$ and let $\Gamma$ be the graph of $\Delta$. Assume that $\Delta$ is a finite set, $\Gamma$ is non-degenerate and $\Gamma$ contains no $m$-tuple lines with $m \geq 3$. Then $W(\Gamma)$ is of finite index in ${\rm O}(\Num(S))$ if and only if every connected parabolic subdiagram of $\Gamma$ is a connected component of some parabolic subdiagram in $\Gamma$ of rank $8$ [(]{}= the maximal one[)]{}. Finally we recall some facts on elliptic fibrations on Enriques surfaces. \[multi-fiber\][(Dolgachev and Liedtke [@DL], Theorem 4.8.3)]{} Let $f : S \to {\bf P}^1$ be an elliptic fibration on an Enriques surface $S$ in characteristic $2$. Then the following hold. $({\rm i})$ If $S$ is classical, then $f$ has two tame multiple fibers, each is either an ordinary elliptic curve or a singular fiber of additive type. $({\rm ii})$ If $S$ is singular, then $f$ has one wild multiple fiber which is a smooth ordinary elliptic curve or a singular fiber of multiplicative type. $({\rm iii})$ If $S$ is supersingular, then $f$ has one wild multiple fiber which is a supersingular elliptic curve or a singular fiber of additive type. As for the number of multiple fibers in each case, it is given in Bombieri and Mumford [@BM2], Proposition 11. Let $2G$ be a multiple fiber of $f : S \longrightarrow {\bf P}^1$. If $S$ is classical, the multiple fiber $2G$ is tame. Therefore, the normal bundle ${{\mathcal{O}}}_{G}(G)$ of $G$ is of order 2 (cf. Katsura and Ueno [@KU], p. 295, (1.7)). On the other hand, neither the Picard variety ${\rm Pic}^0({\bf G}_m)$ of the multiplicative group ${\bf G}_m$ nor ${\rm Pic}^0(E)$ of the supersingular elliptic curve $E$ has any 2-torsion point. Therefore, $G$ is either an ordinary elliptic curve or a singular fiber of additive type. Now, we consider an exact sequence: $$0 \longrightarrow {{\mathcal{O}}}_S(-G) \longrightarrow {{\mathcal{O}}}_S \longrightarrow {{\mathcal{O}}}_{G} \longrightarrow 0.$$ Then, we have the long exact sequence $$\rightarrow H^1(S, {{\mathcal{O}}}_S) \longrightarrow H^1(G, {{\mathcal{O}}}_G) \longrightarrow H^2(S, {{\mathcal{O}}}_S(-G)) \longrightarrow H^2(S, {{\mathcal{O}}}_S)\rightarrow 0.$$ If $S$ is either singular or supersingular, we have $H^1(S, {{\mathcal{O}}}_S)\cong H^2(S, {{\mathcal{O}}}_S)\cong k$. Note that in our case the canonical divisor $K_S$ is linearly equivalent to 0. Since $2G$ is a multiple fiber, by the Serre duality theorem, we have $$H^2(S, {{\mathcal{O}}}_S(-G)) \cong H^0(S, {{\mathcal{O}}}_S(K_S + G)) \cong H^0(S, {{\mathcal{O}}}_S(G))\cong k.$$ Therefore, we see that the natural homomorphism $$H^1(S, {{\mathcal{O}}}_S) \longrightarrow H^1(G, {{\mathcal{O}}}_G)$$ is an isomorphism. If $S$ is singular, then the Frobenius map $F$ acts bijectively on $H^1(S, {{\mathcal{O}}}_S)$. Hence, $F$ acts on $H^1(G, {{\mathcal{O}}}_G)$ bijectively. Therefore, $G$ is either an ordinary elliptic curve or a singular fiber of multiplicative type. If $S$ is supersingular, then the Frobenius map $F$ is the zero map on $H^1(S, {{\mathcal{O}}}_S)$. Hence, $F$ is also a zero map on $H^1(G, {{\mathcal{O}}}_G)$. Therefore, $G$ is either a supersingular elliptic curve or a singular fiber of additive type. Let $f : S \to {\bf P}^1$ be an elliptic fibration on an Enriques surface $S$. We use Kodaira’s notation for singular fibers of $f$: $${\rm I}_n,\ {\rm I}_n^*,\ {\rm II},\ {\rm II}^*,\ {\rm III},\ {\rm III}^*,\ {\rm IV},\ {\rm IV}^*.$$ \[singular-fiber\] Let $f : S \to {\bf P}^1$ be an elliptic fibration on an Enriques surface $S$ in characteristic $2$. Then the type of reducible singular fibers is one of the following: $$({\rm I}_3, {\rm I}_3, {\rm I}_3, {\rm I}_3), \ ({\rm I}_5, {\rm I}_5), \ ({\rm I}_9),\ ({\rm I}_4^*),\ ({\rm II}^*),\ ({\rm III}, {\rm I}_8),$$ $$({\rm I}_1^*, {\rm I}_4), \ ({\rm III}^*, {\rm I}_2),\ ({\rm IV}, {\rm IV}^*),\ ({\rm IV}, {\rm I}_2, {\rm I}_6),\ ({\rm IV}^*, {\rm I}_3).$$ Consider the Jacobian fibration $J(f) : R \to {\bf P}^1$ of $f$ which is a rational elliptic surface. It is known that the type of singular fibers of $f$ coincides with that of $J(f)$ (cf. Liu-Lorenzini-Raynaud [@LLR], Theorem 6.6). Now the assertion follows from the classification of singular fibers of rational elliptic surfaces in characteristic 2 due to Lang [@L1], [@L2] (also see Ito [@I]). Enriques surfaces with the dual graph of type VII {#sec3} ================================================= In this section, we construct Enriques surfaces in characteristic 2 whose dual graph of all nonsingular rational curves is of type VII. The method to construct them is similar to the one in Katsura and Kondo [@KK], §4. We consider the nonsingular complete model of the supersingular elliptic curve $E$ defined by $$y^2 + y = x^3 + x^2.$$ For $(x_1, y_1), (x_2, y_2) \in E$, the addition of this elliptic curve is given by, $$\begin{array}{l} x_{3} = x_1 + x_2 + \left(\frac{y_2 + y_1}{x_2 + x_1}\right)^2 + 1 \\ y_3 = y_1 + y_2 + \left(\frac{y_2 + y_1}{x_2 + x_1}\right)^3 + \left(\frac{y_2 + y_1}{x_2 + x_1}\right) + \frac{x_1y_2 +x_2y_1}{x_2 +x_1} + 1. \end{array}$$ The ${\bf F}_4$-rational points of $E$ are given by $$\begin{array}{l} P_{0} = \infty, P_{1} =(1, 0), P_{2} =(0, 0), P_{3} =(0, 1), P_{4} =(1, 1). \end{array}$$ The point $P_{0}$ is the zero point of $E$, and these points make the cyclic group of order five : $$P_{i} = iP_{1} \quad (i = 2, 3, 4),~P_{0} = 5P_{1}$$ Now we consider the relatively minimal nonsingular complete elliptic surface $\psi : R \longrightarrow {\bf P}^1$ defined by $$y^2 + sxy + y = x^3 + x^2 + s$$ with a parameter $s$. This surface is a rational elliptic surface with two singular fibers of type $\I_5$ over the points given by $s = 1, \infty$, and two singular fibers of type $\I_1$ over the points given by $t = \omega, \omega^2$. Here, $\omega$ is a primitive cube root of unity. We consider the base change of $\psi : R \longrightarrow {\bf P}^1$ by $s = t^2$. Then, we have the elliptic surface defined by $$(*)\quad \quad \quad y^2 + t^2xy + y = x^3 + x^2 + t^2.$$ We consider the relatively minimal nonsingular complete model of this elliptic surface : $$\label{pencil3} f : Y \longrightarrow {\bf P}^1.$$ The surface $Y$ is an elliptic $K3$ surface. From $Y$ to $R$, there exists a generically surjective purely inseparable rational map. We denote by $R^{(\frac{1}{2})}$ the algebraic surface whose coefficients of the defining equations are the square roots of those of $R$. Then, $R^{(\frac{1}{2})}$ is also a rational surface, and we have the Frobenius morphism $F : R^{(\frac{1}{2})}\longrightarrow R$. $F$ factors through a generically surjective purely inseparable rational map from $R^{(\frac{1}{2})}$ to $Y$. By the fact that $R^{(\frac{1}{2})}$ is rational we see that $Y$ is unirational. Hence, $Y$ is a supersingular $K3$ surface, i.e. the Picard number $\rho (Y)$ is equal to the second Betti number $b_{2}(Y)$ (cf. Shioda [@S], p.235, Corollary 1). The discriminant of the elliptic surface $f : Y \longrightarrow {\bf P}^1$ is given by $$\Delta = (t + 1)^{10}(t^2 + t + 1)^2$$ and the $j$-invariant is given by $$j = t^{24}/(t + 1)^{10}(t^2 + t + 1)^2.$$ Therefore, on the elliptic surface $f : Y \longrightarrow {\bf P}^1$, there exist two singular fibers of type $\I_{10}$ over the points given by $t = 1, \infty$, and two singular fibers of type $\I_2$ over the points given by $t = \omega, \omega^2$. The regular fiber over the point defined by $t = 0$ is the supersingular elliptic curve $E$. The elliptic $K3$ surface $f: Y \longrightarrow {\bf P}^1$ has ten sections $s_{i}, m_{i}$ $(i = 0, 1, 2, 3, 4)$ given as follows: $$\begin{array}{ll} s_0 : \mbox{the zero section} &\mbox{passing through}~P_{0}~\mbox{on}~E\\ s_1 : x = 1, y = t^2 &\mbox{passing through}~P_{1}~\mbox{on}~E\\ s_2 : x = t^2, y = t^2 &\mbox{passing through}~P_{2}~\mbox{on}~E\\ s_3 : x = t^2, y = t^4 + t^2 + 1&\mbox{passing through}~P_{3}~\mbox{on}~E\\ s_4 : x = 1, y = 1 &\mbox{passing through}~P_{4}~\mbox{on}~E\\ m_0 : x = \frac{1}{t^2}, y = \frac{1}{t^3} +\frac{1}{t^2} + t &\mbox{passing through}~P_{0}~\mbox{on}~E\\ m_1 : x = t^3 + t + 1,~ y = t^4 + t^3 + t &\mbox{passing through}~P_{1}~\mbox{on}~E\\ m_2 : x = t,~ y = t^3 &\mbox{passing through}~P_{2}~\mbox{on}~E\\ m_3 : x = t,~ y = 1&\mbox{passing through}~P_{3}~\mbox{on}~E\\ m_4 : x = t^3 + t + 1,~ y = t^5 + t^4 + t^2 + t + 1&\mbox{passing through}~P_{4}~\mbox{on}~E. \end{array}$$ These ten sections make the cyclic group of order 10, and the group structure is given by $$s_{i} = is_1,\ m_i =m_0 + s_i~(i = 0, 1, 2, 3, 4),\ 2m_0 = s_0$$ with $s_0$, the zero section. The images of $s_{i}$ (resp. $m_{i}$) ($i = 0, 1, 2, 3, 4$) on $R$ give sections (resp. multi-sections) of $\psi : R \longrightarrow {\bf P}^1$. The intersection numbers of the sections $s_i, m_i$ $(i = 0, 1, 2, 3, 4)$ are given by $$\label{int-sections} \langle s_i, s_j\rangle =-2\delta_{ij},~ \langle m_i, m_j\rangle~=-2\delta_{ij},~ \langle s_i, m_j\rangle = \delta_{ij},~$$ where $\delta_{ij}$ is Kronecker’s delta. On the singular elliptic surface $(*)$, we denote by $F_1$ the fiber over the point defined by $t = 1$. $F_1$ is an irreducible curve and on $F_1$ the surface $(*)$ has only one singular point $P$. The surface $Y$ is a surface obtained by the minimal resolution of singularities of $(*)$. We denote the proper transform of $F_1$ on Y again by $F_1$, if confusion doesn’t occur. We have nine exceptional curves $E_{1,i}$ $(i = 1,2, \ldots, 9)$ over the point $P$, and as a singular fiber of type $I_{10}$ of the elliptic surface $f : Y \longrightarrow {\bf P}^1$, $F_1$ and these nine exceptional curves make a decagon $F_1E_{1,1}E_{1,2}\ldots E_{1,9}$ clockwisely. The blowing-up at the singular point $P$ gives two exceptional curves $E_{1,1}$ and $E_{1,9}$, and they intersect each other at a singular point. The blowing-up at the singular point again gives two exceptional curves $E_{1,2}$ and $E_{1,8}$. The exceptional curve $E_{1,2}$ (resp. $E_{1,8}$) intersects $E_{1,1}$ (resp. $E_{1,9}$) transeversely. Exceptional curves $E_{1,2}$ and $E_{1,8}$ intersect each other at a singular point, and so on. By successive blowing-ups, the exceptional curve $E_{1,5}$ finally appears to complete the resolution of singularity at the point $P$, and it intersects $E_{1,4}$ and $E_{1,6}$ transeversely. Summerizing these results, we see that $F_1$ intersects $E_{1,1}$ and $E_{1,9}$ transversely, and that $E_{1,i}$ intersects $E_{1,i+ 1}$ $(i = 1, 2, \ldots, 8)$ transversely. We choose $E_{1,1}$ as the component which intersects the section $m_2$. Then, 10 sections above intersect these 10 curves transversely as follows: sections $s_0$ $s_1$ $s_2$ $s_3$ $s_4$ $m_0$ $m_1$ $m_2$ $m_3$ $m_4$ ----------- ------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- ----------- componets $F_1$ $E_{1,8}$ $E_{1,6}$ $E_{1,4}$ $E_{1,2}$ $E_{1,5}$ $E_{1,3}$ $E_{1,1}$ $E_{1,9}$ $E_{1,7}$ Here, the table means that the section $s_0$ intersects the singular fiber over the point defined by $t= 1$ with the component $F_1$, for example. The surface $Y$ has the automorphism $\sigma$ defined by $$(t, x, y) \mapsto (\frac{t}{t+1}, \frac{x + t^4 + t^2 + 1}{(t + 1)^4}, \frac{x + y +s^6 + s^2}{(s + 1)^6}).$$ The automorphism $\sigma$ is of order 4 and replaces the fiber over the point $t = 1$ with the one over the point $t = \infty$, and also replaces the fiber over the point $t =\omega$ with the one over the point $t = \omega^2$. The automorphism $\sigma$ acts on the ten sections above as follows: sections $s_0$ $s_1$ $s_2$ $s_3$ $s_4$ $m_0$ $m_1$ $m_2$ $m_3$ $m_4$ ------------------------ ------- ------- ------- ------- ------- ------- ------- ------- ------- ------- $\sigma^{*}$(sections) $s_0$ $s_2$ $s_4$ $s_1$ $s_3$ $m_0$ $m_2$ $m_4$ $m_1$ $m_3$ Using the automorphism $\sigma$, to construct the resolution of singularity on the fiber over the point $P_{\infty}$ defined by $t = \infty$, we use the resolution of singularity on the fiber over the point $P_{1}$ defined by $t = 1$. We attach names to the irreducible components of the fiber over $P_{\infty}$ in the same way as above. Namely, on the singular elliptic surface $(*)$, we denote by $F_{\infty}$ the fiber over the point defined by $t = \infty$. We also denote the proper transform of $F_{\infty}$ on $Y$ by $F_{\infty}$. We have 9 exceptinal curves $E_{\infty,i}$ $(i = 1,2, \ldots, 9)$ over the point $P_{\infty}$, and as a singular fiber of type $I_{10}$ of the elliptic surface $f : Y \longrightarrow {\bf P}^1$, $F_{\infty}$ and these 9 exceptional curves make a decagon $F_{\infty}E_{\infty, 1}E_{\infty, 2}\ldots E_{\infty, 9}$ clockwisely. $F_{\infty}$ intersects $E_{\infty, 1}$ and $E_{\infty, 9}$ transversely, and that $E_{\infty, i}$ intersects $E_{\infty, i+ 1}$ $(i = 1, 2, \ldots, 8)$ transversely. The singular fiber of $f : Y \longrightarrow {\bf P}^1$ over the point defined by $t= \omega$ (resp. $t = \omega^{2}$) consists of two irreducible components $F_{\omega}$ and $E_{\omega}$ (resp. $F_{\omega^{2}}$ and $E_{\omega^{2}}$), where $F_{\omega}$ (resp. $F_{\omega^{2}}$) is the proper transform of the fiber over the point $P_{\omega}$ (resp. $P_{\omega^{2}}$) in $(*)$. Then, the 10 sections above intersect singular fibers of elliptic surface $f : Y \longrightarrow {\bf P}^1$ as follows: \[\] sections $s_0$ $s_1$ $s_2$ $s_3$ $s_4$ $m_0$ $m_1$ $m_2$ $m_3$ $m_4$ ---------------- ---------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- ----------------- $t= 1$ $F_1$ $E_{1,8}$ $E_{1,6}$ $E_{1,4}$ $E_{1,2}$ $E_{1,5}$ $E_{1,3}$ $E_{1,1}$ $E_{1,9}$ $E_{1,7}$ $t = \infty$ $F_\infty$ $E_{\infty, 6}$ $E_{\infty, 2}$ $E_{\infty, 8}$ $E_{\infty, 4}$ $E_{\infty, 5}$ $E_{\infty, 1}$ $E_{\infty, 7}$ $E_{\infty, 3}$ $E_{\infty, 9}$ $t = \omega$ $F_\omega$ $F_\omega$ $F_\omega$ $F_\omega$ $F_\omega$ $E_{\omega}$ $E_{\omega}$ $E_{\omega}$ $E_{\omega}$ $E_{\omega}$ $t = \omega^2$ $F_{\omega^2}$ $F_{\omega^2}$ $F_{\omega^2}$ $F_{\omega^2}$ $F_{\omega^2}$ $E_{\omega^2}$ $E_{\omega^2}$ $E_{\omega^2}$ $E_{\omega^2}$ $E_{\omega^2}$ : []{data-label="Table2"} \[\] The surface $Y$ is a supersingular $K3$ surface with Artin invariant $1$. The elliptic fibration $(\ref{pencil3})$ has two singular fibers of type $\I_{10}$, two singular fibers of type $\I_2$ and ten sections. Hence the assertion follows from the Shioda-Tate formula (cf. Shioda [@Shio], Corollary 1.7). Incidentally, by the Shioda-Tate formula, we also see that the order of the group of the sections of $f : Y \longrightarrow {\bf P}^1$ is equal to 10 and so the group is isomorphic to ${\bf Z}/10{\bf Z}$. Now, we consider a rational vector field $$D' = (t - 1)(t - a)(t - b)\frac{\partial}{\partial t} + (1 + t^2x)\frac{\partial}{\partial x} $$ with $a, b \in k, \ a+b=ab, \ a^3\not=1$. Then, we have $D'^2 = t^2 D'$, that is, $D'$ is $2$-closed. On the surface $Y$, the divisorial part of $D'$ is given by $$\begin{array}{rl} (D') & = E_{1,1} + E_{1,3} + E_{1,5} + E_{1,7} + E_{1,9} + E_{\infty, 1} + E_{\infty, 3} + E_{\infty, 5} + E_{\infty, 7} \\ & + E_{\infty, 9} - E_{\omega} - E_{\omega^{2}} - 2(F_{\infty} + E_{\infty, 1} + E_{\infty, 2}+ E_{\infty, 3} + E_{\infty, 4} + E_{\infty, 5} \\ &+ E_{\infty, 6} + E_{\infty, 7} + E_{\infty, 8} + E_{\infty, 9}). \end{array}$$ We set $D = \frac{1}{t - 1}D'$. Then, $D^2=abD$, that is, $D$ is also 2-closed and $D$ is of additive type if $a=b=0$ and of multiplicative type otherwise. Moreover, we have $$\label{divisorial} \begin{array}{rl} (D) & = - (F_{1} + E_{1,2} + E_{1,4} + E_{1,6} + E_{1,8} + F_{\infty} + E_{\infty, 2} + E_{\infty, 4} \\ &+ E_{\infty, 6} + E_{\infty, 8} + E_{\omega} + E_{\omega^2}). \end{array}$$ From here until Theorem \[main\], the argument is parallel to the one in Katsura and Kondo [@KK], §4, and so we give just a brief sketch of the proofs for the readers’ convenience. The quotient surface $Y^{D}$ is nonsingular. Since $Y$ is a $K3$ surface, we have $c_{2}(Y) = 24$. Using $(D)^2 = -24$ and the equation (\[euler\]), we have $$24 = c_{2}(Y) = \deg \langle D\rangle - \langle K_{Y}, (D)\rangle - (D)^2 = \deg \langle D\rangle + 24.$$ Therefore, we have $\deg \langle D\rangle = 0$. This means that $D$ is divisorial, and that $Y^{D}$ is nonsingular. By the result on the canonical divisor formula of Rudakov and Shafarevich (see the equation (\[canonical\])), we have $$K_{Y} = \pi^{*} K_{Y^D} + (D).$$ \[exceptional\] Let $C$ be an irreducible curve contained in the support of the divisor $(D)$, and set $C' = \pi (C)$. Then, $C'$ is an exceptional curve of the first kind. By direct calculation, $C$ is integral with respect to $D$. Therefore, we have $C = \pi^{-1}(C')$ by Proposition \[insep\]. By the equation $2C'^2 = (\pi^{-1}(C'))^2 = C^2 = - 2$, we have $C'^2 = -1$. Since $Y$ is a $K3$ surface, $K_Y$ is linearly equivalent to zero. Therefore, we have $$2\langle K_{Y^D}, C'\rangle = \langle \pi^{*}K_{Y^D}, \pi^{*}(C')\rangle\\ = \langle K_Y - (D), C\rangle = C^2 = -2.$$ Therefore, we have $\langle K_{Y^D}, C'\rangle = -1$ and the arithmetic genus of $C'$ is equal to $0$. Hence, $C'$ is an exceptional curve of the first kind. We denote these 12 exceptional curves on $Y^{D}$ by $E'_{i}$ ($i = 1, 2, \ldots, 12$), which are the images of irreducible components of $-(D)$ by $\pi$. Let $$\varphi : Y^{D} \to X_{a,b}$$ be the blowing-downs of $E'_{i}$ ($i = 1, 2, \ldots, 12$). For simplicity, we denote $X_{a,b}$ by $X$. Now we have the following commutative diagram: $$\begin{array}{ccc}\label{maps} \quad Y^{D} & \stackrel{\pi}{\longleftarrow} & Y \\ \varphi \downarrow & & \downarrow f \\ \quad X=X_{a,b} & & {\bf P}^1 \\ g \downarrow & \quad \swarrow_{F}& \\ \quad {\bf P}^1 & & \end{array}$$ Here $F$ is the Frobenius base change. Then, we have $$K_{Y^D} = \varphi^{*}(K_{X}) + \sum_{i = 1}^{12}E'_{i}.$$ The canonical divisor $K_{X}$ of $X$ is numerically equivalent to $0$. As mentioned in the proof of Lemma \[exceptional\], all irreducible curves which appear in the divisor $(D)$ are integral with respect to the vector field $D$. For an irreducible component $C$ of $(D)$, we denote by $C'$ the image $\pi (C)$ of $C$. Then, we have $C = \pi^{-1}(C')$ by Proposition \[insep\]. Therefore, we have $$(D) = - \pi^{*}(\sum_{i = 1}^{12}E'_{i}).$$ Since $Y$ is a $K3$ surface, $$0 \sim K_{Y} = \pi^{*}K_{Y^D} + (D) = \pi^{*}( \varphi^{*}(K_{X}) + \sum_{i = 1}^{12}E'_{i}) + (D) = \pi^{*}(\varphi^{*}(K_{X}))$$ Therefore, $K_{X}$ is numerically equivalent to zero. The surface $X$ has $b_{2}(X) = 10$ and $c_{2}(X) = 12$. Since $\pi : {Y} \longrightarrow {Y}^{D}$ is finite and purely inseparable, the étale cohomology of $Y$ is isomorphic to the étale cohomology of $Y^{D}$. Therefore, we have $b_{1}(Y^{D}) = b_{1}(Y) = 0$, $b_{3}(Y^{D})= b_{3}(Y) = 0$ and $b_{2}(Y^{D}) = b_{2}(Y) = 22$. Since $\varphi$ is the blowing-downs of 12 exceptional curves of the first kind, we see $b_{0}(X) =b_{4}(X) = 1$, $b_{1}(X) =b_{3}(X) = 0$ and $b_{2}(X) = 10$. Therefore, we have $$c_{2}(X) = b_{0}(X) - b_{1}(X) + b_{2}(X) -b_{3}(X) + b_{4}(X) = 12.$$ \[main\] Under the notation above, the following statements hold. - The surface $X=X_{a,b}$ is a supersingular Enriques surface if $a = b = 0$. - The surface $X=X_{a,b}$ is a classical Enriques surface if $a + b = ab$ and $a \notin {\bf F}_{4}$. Since $K_X$ is numerically trivial, $X$ is minimal and the Kodaira dimension $\kappa (X) $ is equal to $0$. Since $b_2(X) = 10$, $X$ is an Enriques surface. Since $Y$ is a supersingular K3 surface, $X$ is either supersingular or classical. In case that $a= b = 0$, the integral fiber of the elliptic fibration $f : Y \longrightarrow {\bf P}^1$ with respect to $D$ exists only over the point $P_{0}$ defined by $t = 0$. Hence $g : X \longrightarrow {\bf P}^1$ has only one multiple fiber. Therefore, the multiple fiber is wild, and $X$ is a supersingular Enriques surface. In case that $a \not\in {\bf F}_4$, the integral fibers of the elliptic fibration $f : Y \longrightarrow {\bf P}^1$ with respect to $D$ exist over the points $P_{a}$ defined by $t = a$ and $P_b$ defined by $t = b$. Therefore, the multiple fibers are tame, and we conclude that $X$ is a classical Enriques surface. Recall that the elliptic fibration $f : Y \to {\bf P}^1$ given in (\[pencil3\]) has two singular fibers of type $\I_{10}$, two singular fibers of type $\I_2$ and ten sections. This fibration induces an elliptic fibration $$g : X\to {\bf P}^1$$ which has two singular fibers of type $\I_5$, two singular fibers of type $\I_1$, and ten 2-sections. Thus we have twenty nonsingular rational curves on $X$. Denote by ${\mathcal{E}}$ the set of curves contained in the support of the divisor $(D)$: $${\mathcal{E}}= \{F_{1}, E_{1,2}, E_{1,4}, E_{1,6}, E_{1,8}, F_{\infty}, E_{\infty, 2}, E_{\infty, 4}, E_{\infty, 6}, E_{\infty, 8}, E_{\omega}, E_{\omega^2}\}.$$ The singular points of four singular fibers of $g$ consist of twelve points denoted by $\{ p_1,..., p_{12}\}$ which are the images of the twelve curves in ${\mathcal{E}}$. We may assume that $p_{11}, p_{12}$ are the images of $E_{\omega}, E_{\omega^2}$ respectively. Then $p_{11}, p_{12}$ (resp. $p_1,..., p_{10}$) are the singular points of the singular fibers of $g$ of type $\I_1$ (resp. of type $\I_5$). Each of the twenty nonsingular rational curves passes through two points from $\{p_1,..., p_{12}\}$ because its preimage on $Y$ meets exactly two curves from twelve curves in ${\mathcal{E}}$ (see Table \[Table2\]). Let ${\mathcal{S}}_1$ be the set of fifteen nonsingular rational curves which are ten components of two singular fibers of $g$ of type ${\rm I}_5$ and five 2-sections which do not pass through $p_{11}$ and $p_{12}$, that is, the images of $s_0, s_1,..., s_4$. Then the dual graph of the curves in ${\mathcal{S}}_1$ is the line graph of the Petersen graph. For the Petersen graph, see Figure \[petersen\]. Here the line graph $L(G)$ of a graph $G$ is the graph whose vertices correspond to the edges in $G$ bijectively and two vertices in $L(G)$ are joined by an edge iff the corresponding edges meet at a vertex in $G$. In the following Figure \[enriques12\], we denote by ten dots the ten points $\{p_1,..., p_{10}\}$. The fifteen lines denote the fifteen nonsingular rational curves in ${\mathcal{S}}_1$. ![[]{data-label="enriques12"}](fano.eps){width="50mm"} On the other hand, let ${\mathcal{S}}_2$ be the set of curves which are the images of $m_0,..., m_4$. Then the dual graph of the curves in ${\mathcal{S}}_2$ is the complete graph with five vertices in which each pair of the vertices forms the extended Dynkin diagram of type $\tilde{A}_1$ because all of them pass through the two points $p_{11}$ and $p_{12}$. Each vertex in ${\mathcal{S}}_1$ meets exactly one vertex in ${\mathcal{S}}_2$ with multiplicity 2, because any component of the singular fibers of type $\I_{10}$ meets exactly one section from $m_0,..., m_4$ (see Table \[Table2\]) and $s_i$ meets only $m_i$ ($i=0,1,...,4)$ (see the equation (\[int-sections\])). On the other hand, the vertex in ${\mathcal{S}}_2$ meets three vertices in ${\mathcal{S}}_1$ with multiplicity 2, because $m_i$ meets one component of each singular fiber of type $\I_{10}$ and $s_i$. The dual graph $\Gamma$ of the twenty curves in ${\mathcal{S}}_1$ and ${\mathcal{S}}_2$ forms the same dual graph of nonsingular rational curves of the Enriques surfaces of type $\VII$ given in Figure \[Figure7-7\] (Fig. 7.7 in [@Ko]). ![[]{data-label="Figure7-7"}](Figure7-7.360x360pt.eps){width="60mm"} The 15 curves in ${\mathcal{S}}_1$ (resp. five curves in ${\mathcal{S}}_2$) correspond to $E_1, ..., E_{15}$ (resp. $K_1,..., K_5$) in Figure \[Figure7-7\]. It is easy to see that the maximal parabolic subdiagrams in $\Gamma$ are $$\tilde{A}_8,\ \tilde{A}_4\oplus \tilde{A}_4, \ \tilde{A}_5\oplus \tilde{A}_2\oplus \tilde{A}_1, \ \tilde{A}_7\oplus \tilde{A}_1$$ which are coresponding to elliptic fibrations of type $$({\rm I}_9),\ ({\rm I}_5, {\rm I}_5), \ ({\rm I}_6, {\rm IV}, {\rm I}_2), \ ({\rm I}_8, {\rm III}),$$ respectively. It follows from Vinberg’s criterion (Proposition \[Vinberg\]) that $W(X)$ is of finite index in $\O(\Num(X))$. The same argument in [@Ko], (3.7) implies that $X$ contains exactly twenty nonsingular rational curves in ${\mathcal{S}}_1, {\mathcal{S}}_2$. \[injectiv\] The map $\rho : {\rm Aut}(X) \to {\rm O}({\rm Num}(X))$ is injective. Let $\varphi \in \Ker(\rho)$. Then $\varphi$ preserves each nonsingular rational curve on $X$. Since each nonsingular rational curve meets other curves at least three points, $\varphi$ fixes all 20 nonsingular rational curves pointwisely. Now consider the elliptic fibration $g : X \to {\bf P}^1$. Since this fibration has has ten 2-sections, $\varphi$ fixes a general fiber of $g$ and hence $\varphi$ is identity. By Proposition \[finiteness\], we now have the following theorem. \[main2\] The automorphism group ${\rm Aut}(X)$ is isomorphic to the symmetric group $\mathfrak{S}_5$ of degree five and $X$ contains exactly twenty nonsingular rational curves whose dual graph is of type ${\rm VII}$. We have already showed that ${\rm Aut}(X)$ is finite and $X$ contains exactly twenty nonsigular rational curves whose dual graph $\Gamma$ is of type ${\rm VII}$. It follows Lemma \[injectiv\] that ${\rm Aut}(X)$ is a subgroup of $\Aut(\Gamma) \cong\mathfrak{S}_5$. Then by the same argument in [@Ko], (3.7), we see that ${\rm Aut}(\Gamma)$ is represented by automorpisms of $X$. \[non-isotrivial\] The one dimensional family $\{X_{a,b}\}$ is non-isotrivial. Denote by $\Gamma$ the dual graph of all nonsingular rational curves on $X$ which is given in Figure \[Figure7-7\]. $\Gamma$ contains only finitely many extended Dynkin diagrams (= the disjoint union of $\tilde{A}_m, \tilde{D}_n, \tilde{E}_k$), that is, $\tilde{A}_8, \tilde{A}_7\oplus \tilde{A}_1, \tilde{A}_4\oplus \tilde{A}_4, \tilde{A}_5\oplus \tilde{A}_2\oplus \tilde{A}_1$ (see also Kondo [@Ko], page 274, Table 2). Note that the elliptic fibrations on $X$ bijectively correspond to the extended Dynkin diagrams in $\Gamma$. This implies that $X$ has only finitely many elliptic fibrations. The $j$-invariant of the elliptic curve which appears as the fiber $E_{a}$ defined by $t = a$ of the elliptic fibration $f : Y \longrightarrow {\bf P}^{1}$ is equal to $a^{24}/(a + 1)^{10}(a^2 + a + 1)^2$ (cf. section 3). Consider the multiple fiber $2E'_{a}$ on the elliptic fibration on the Enriques surface $X$ which is the image of $E_a$. Since we have a purely inseparable morphism of degree 2 from $E_{a}$ to $E'_{a}$, we see that the $j$-invariant of $E'_{a}$ is equal to $a^{48}/(a + 1)^{20}(a^2 + a + 1)^4$. This implies the infiniteness of the number of elliptic curves which appear as the multiple fibers of the elliptic fibration on an Enriques surface in our family of Enriques surfaces with parameter $a$. Therefore, in our family of Enriques surfaces there are infinitely many non-isomorphic ones (see also Katsura-Kondō [@KK], Remark 4.9). The pullback of an elliptic fibration $\pi : X\to {\bf P}^1$ to the covering $K3$ surface $Y$ gives an elliptic fibration $\tilde{\pi} : Y\to {\bf P}^1$. The type of reducible singular fibers of $\tilde{\pi}$ is $({\rm I}_{10}, {\rm I}_{10}, {\rm I}_2, {\rm I_2})$ if $\pi$ is of type $\tilde{A}_4\oplus \tilde{A}_4$, $({\rm I}_{16}, {\rm I}_1^*)$ if $\pi$ is of type $\tilde{A}_7\oplus \tilde{A}_1$, $({\rm I}_{12}, {\rm III}^*, {\rm I}_4)$ if $\pi$ is of type $\tilde{A}_5\oplus \tilde{A}_2\oplus \tilde{A}_1$, and type $({\rm I}_{18}, {\rm I}_2, {\rm I}_{2}, {\rm I}_2)$ if $\pi$ is of type $\tilde{A}_8$, respectively. The following theorem is due to M. Schütt and H. Ito. \[non-existVII\] There are no singular Enriques surfaces with the dual graph of type ${\rm VII}$. Assume that there exists an Enriques surface $S$ with the dual graph of type ${\rm VII}$. In the dual graph of type [VII]{} there exists a parabolic subdiagram $\tilde{A}_5\oplus \tilde{A}_2 \oplus \tilde{A}_1$. By Proposition \[singular-fiber\], it corresponds to an elliptic fibration on $S$ with singular fibers of type $({\rm IV}, {\rm I}_2, {\rm I}_6)$. For example, the linear system $|2(E_1+E_2+E_{14})|$ defines a such fibration. Moreover the dual graph of type ${\rm VII}$ tells us that the singular fiber $E_1+E_2+E_{14}$ of type ${\rm IV}$ is a multiple fiber because $E_3$ is a 2-section of this fibration (see Figure \[Figure7-7\]). This contradicts to Proposition \[multi-fiber\], (ii). Examples of singular $K3$ surfaces with a finite automorphism group {#sec4} =================================================================== Type $\I$ {#type1} --------- Let $(x_0,x_1,x_2,x_3)$ be a homogeneous coodinate of ${\bf P}^3$. Consider the nonsingular quadric $Q$ in ${\bf P}^3$ defined by $$\label{type1quadric} x_0x_3 + x_1x_2=0$$ which is the image of the map $${\bf P}^1\times {\bf P}^1 \to {\bf P}^3,\quad ((u_0,u_1),(v_0,v_1)) \to (u_0v_0, u_0v_1, u_1v_0, u_1v_1).$$ The involution of ${\bf P}^1\times {\bf P}^1$ $$((u_0,u_1),(v_0,v_1)) \to ((u_1,u_0),(v_1,v_0))$$ induces an involution $$\label{typeIinv} \tau : (x_0,x_1,x_2,x_3) \to (x_3,x_2,x_1,x_0)$$ of $Q$ whose fixed point set on $Q$ is one point $(1,1,1,1)$. Consider four lines on $Q$ defined by $$L_{01}:x_0=x_1=0,\quad L_{02}: x_0=x_2=0,$$ $$L_{13}: x_1=x_3=0, \quad L_{23}: x_2=x_3=0,$$ and a $\tau$-invariant pencil of quadrics $$C_{\lambda,\mu} : \lambda (x_0+x_3)(x_1+x_2)+ \mu x_0x_3 =0$$ passing through the four vertices $$(1,0,0,0), \quad (0,1,0,0),\quad (0,0,1,0),\quad (0,0,0,1)$$ of the quadrangle $L_{01}, L_{02}, L_{13}, L_{23}$. Note that two conics $$Q_1: x_0+x_3=0, \quad Q_2: x_1+x_2=0$$ tangent to $C_{\lambda,\mu}$ at two vertices of the quadrangle. Obviously $$C_{1,0} = Q_1 +Q_2, \quad C_{0,1} = L_{01}+L_{02} + L_{13}+L_{23},$$ and $C_{\lambda,\mu}$ $(\lambda\cdot \mu\not=0)$ is a nonsingular elliptic curve. Thus we have the same configuration of curves given in [@Ko], Figure 1.1 except $Q_1$ and $Q_2$ tangent at $(1,1,1,1)$. Now we fix $(\lambda_0, \mu_0)\in {\bf P}^1$ $(\lambda_0\cdot \mu_0\not=0)$ and take Artin-Schreier covering $S \to Q$ defined by the triple $(L, a, b)$ where $L= {\mathcal{O}}_Q(2,2)$, $a \in H^0(Q,L)$ and $b\in H^0(Q,L^{\otimes 2})$ satisfying $Z(a) = C_{0,1}$ and $Z(b) = C_{0,1} + C_{\lambda_0,\mu_0}$. The surface $S$ has four singular points over the four vertices of quadrangle given locally by $z^2 +uvz + uv(u+v)=0$. In the notation in Artin’s list (see [@A], §3), it is of type $D^1_4$. Let $Y$ be the minimal nonsingular model of $S$. Then the exceptional divisor over a singular point has the dual graph of type $D_4$. The canonical bundle formula implies that $Y$ is a $K3$ surface. The pencil $\{C_{\lambda,\mu}\}_{(\lambda,\mu)\in {\bf P}^1}$ induces an elliptic fibration on $Y$. The preimage of $L_{01}+L_{02} + L_{13}+L_{23}$ is the singular fiber of type $\I_{16}$ and the preimage of $Q_1+Q_2$ is the union of two singular fibers of type $\III$. Note that the pencil has four sections. Thus we have 24 nodal curves on $Y$. Note that the dual graph of these 24 nodal curves coincide with the one given in [@Ko], Figure 1.3. The involution $\tau$ can be lifted to a fixed point free involution $\sigma$ of $Y$ because the branch divisor $C_{0,1}$ does not contain the point $(1,1,1,1)$. By taking the quotient of $Y$ by $\sigma$, we have a singular Enriques surface $X=Y/\langle \sigma \rangle$. The above elliptic fibration induces an elliptic pencil on $X$ with singular fibers of type $\I_8$ and of type $\III$. Since the ramification divisor of the covering $S\to Q$ is the preimage of $L_{01}+L_{02} + L_{13}+L_{23}$, the multiple fiber of this pencil is the singular fiber of type $\I_8$. By construction, $X$ contains twelve nonsingular rational curves whose dual graph coincides with the one given in [@Ko], Figure 1.4. It follows from Vinberg’s criterion (Proposition \[Vinberg\]) that $W(X)$ is of finite index in $\O(\Num(X))$, and hence the automorphism group $\Aut(X)$ is finite (Proposition \[finiteness\]). The same argument as in the proof of [@Ko], Theorem 3.1.1 shows that $\Aut(X)$ is isomorphic to the digedral group $D_4$ of order 8. Thus we have the following theorem. \[Ithm\] These $X$ form a one dimensional family of singular Enriques surfaces whose dual graph of nonsingular rational curves is of type ${\rm I}$. The automorphism group ${\rm Aut}(X)$ is isomorphic to the dihedral group $D_4$ of order $8$. \[non-existI\] There are no classical and supersingular Enriques surfaces with the dual graph of type ${\rm I}$. From the dual graph of type $\I$, we can see that such Enriques surface has an elliptic fibration with a multiple fiber of type $\I_8$. The assersion now follows from Proposition \[multi-fiber\]. \[typeInumtrivial\] In the above, we consider special quadrics $C_{\lambda, \mu}$ tangent to $Q_1, Q_2$. If we drop this condition and consider general $\tau$-invariant quadrics through the four vertices of the quadrangle $L_{01}, L_{02}, L_{13}, L_{23}$, we have a two dimensional family of singular Enriques surfaces $X$. The covering transformation of $Y \to S$ descends to a numerically trivial involution of $X$, that is, an involution of $X$ acting trivially on $\Num(X)$. In the appendix \[kummer\], we discuss Enriques surfaces with a numerically trivial involution. Type $\II$ {#type2} ---------- We use the same notation as in (\[type1\]). We consider a $\tau$-invariant pencil of quadrics defined by $$C_{\lambda,\mu} : \lambda (x_0+x_1+x_2+x_3)^2+ \mu x_0x_3 =0 $$ which tangents to the quadrangle $L_{01}, L_{02}, L_{13}, L_{23}$ at $(0,0,1,1)$, $(0,1,0,1)$, $(1,0,1,0)$, $(1,1,0,0)$ respectively. Let $$L_1: x_0+x_1=x_2+x_3=0,\quad L_2: x_0+x_2=x_1+x_3=0$$ be two lines on $Q$ which passes the tangent points of $C_{\lambda,\mu}$ and the quadrangle $L_{03}, L_{12}, L_{02}, L_{13}$. Note that $$C_{1,0}= 2L_1+ 2L_2, \quad C_{0,1}= L_{01}+L_{02} + L_{13}+L_{23},$$ and $C_{\lambda,\mu}$ $(\lambda\cdot \mu\not=0)$ is a nonsingular elliptic curve. Thus we have the same configuration of curves given in [@Ko], Figure 2.1. Now we fix $(\lambda_0, \mu_0)\in {\bf P}^1$ $(\lambda_0 \cdot \mu_0 \not=0)$ and take Artin-Schreier covering $S \to Q$ defined by the triple $(L, a, b)$ where $L= {\mathcal{O}}_Q(2,2)$, $a \in H^0(Q,L)$ and $b\in H^0(Q,L^{\otimes 2})$ satisfying $Z(a) = C_{0,1}$ and $Z(b) = C_{0,1} + C_{\lambda_0,\mu_0}$. The surface $S$ has four singular points over the four tangent points of $C_{\lambda_0,\mu_0}$ with the quadrangle and four singular points over the four vertices of the quadrangle. A local equation of each of the first four singular points is given by $z^2 +uz + u(u+v^2)=0$ and the second one is given by $z^2 + uvz + uv=0$. In the first case, by the change of coordinates $$t=z +\omega u + v^2,\quad s = z +\omega^2 u + v^2,\quad v = v$$ ($\omega^3=1$, $\omega\not= 1$), then we have $v^4 +ts=0$ which gives a rational double point of type $A_3$. In the second case, obviously, it is a rational double point of type $A_1$. Let $Y$ be the minimal nonsingular model of $S$. Then the exceptional divisor over a singular point in the first case has the dual graph of type $A_3$ and in the second case the dual graph of type $A_1$. The canonical bundle formula implies that $Y$ is a $K3$ surface. The pencil $\{C_{\lambda,\mu}\}_{(\lambda,\mu)\in {\bf P}^1}$ induces an elliptic fibration on $Y$. The preimage of $L_{01}+L_{02} + L_{13}+L_{23}$ is the singular fiber of type $\I_{8}$ and the preimage of $C_{1,0}$ is the union of two singular fibers of type $\I_1^*$. Note that the pencil has four sections. Thus we have 24 nodal curves on $Y$. Note that the dual graph of these 24 nodal curves coincide with the one given in [@Ko], Figure 2.3. The involution $\tau$ can be lifted to a fixed point free involution $\sigma$ of $Y$ because the branch divisor $C_{0,1}$ does not contain the point $(1,1,1,1)$. By taking the quotient of $Y$ by $\sigma$, we have a singular Enriques surface $X=Y/\langle \sigma \rangle$. The above elliptic fibration induces an elliptic pencil on $X$ with singular fibers of type $\I_4$ and of type $\I_1^*$. Since the ramification divisor of the covering $S\to Q$ is the preimage of $L_{01}+L_{02} + L_{13}+L_{23}$, the multiple fiber of this pencil is the singular fiber of type $\I_4$. By construction, $X$ contains twelve nonsingular rational curves whose dual graph $\Gamma$ coincides with the one given in [@Ko], Figure 2.4. The same argument as in the proof of [@Ko], Theorem 3.2.1 shows that $W(X)$ is of finite index in $\O(\Num(X))$ and $X$ contains only these twelve nonsingular rational curves. It now follows from Proposition \[finiteness\] that the automorphism group $\Aut(X)$ is finite. By the similar argument as in the proof of Lemma \[injectiv\], we see that the map $\rho : \Aut(X) \to \O(\Num(X))$ is injective. Moreover, by the same argument as in the proof of [@Ko], Theorem 3.2.1, $\Aut(X)$ is isomorphic to $\Aut(\Gamma) \cong \mathfrak{S}_4$. Thus we have the following theorem. \[IIthm\] These $X$ form a one dimensional family of singular Enriques surfaces whose dual graph of nonsingular rational curves is of type ${\rm II}$. The automorphism group ${\rm Aut}(X)$ is isomorphic to the symmetric group $\mathfrak{S}_4$ of degree four. \[non-existII\] There are no classical and supersingular Enriques surfaces with the dual graph of type ${\rm II}$. From the dual graph of type $\II$, we can see that such Enriques surface has an elliptic fibration with a multiple fiber of type $\I_4$. The assersion now follows from Proposition \[multi-fiber\]. Type $\VI$ {#type6} ---------- Over the field of complex numbers, the following example was studied by Dardanelli and van Geemen [@DvG], Remark 2.4. This surface $X$ is isomorphic to the Enriques surface of type $\VI$ given in [@Ko] (In [@DvG], Remark 2.4, they claimed that $X$ is of type $\IV$, but this is a misprint). Their construction works well in characteristic 2. Let $(x_1,\cdots , x_5)$ be a homogeneous coodinate of ${\bf P}^4$. Consider the surface $S$ in ${\bf P}^4$ defined by $$\sum_{i=1}^{5} x_i = \sum_{i=1}^5 {1/x_i} = 0.$$ Let $$\ell_{ij} : x_i=x_j=0 \ \ (1\leq i<j \leq 5),$$ $$p_{ijk} : x_i=x_j=x_k=0 \ \ (1\leq i<j<k\leq 5).$$ The ten lines $\ell_{ij}$ and ten points $p_{ijk}$ lie on $S$. By taking partial derivatives, we see that $S$ has ten nodes at $p_{ijk}$. Let $Y$ be the minimal nonsingular model of $S$. Then $Y$ is a $K3$ surface. Denote by $L_{ij}$ the proper transform of $\ell_{ij}$ and by $E_{ijk}$ the exceptional curve over $p_{ijk}$. The Cremonat transformation $$(x_i) \to \left({1/x_i}\right)$$ acts on $Y$ as an automorphism $\sigma$ of order 2. Note that the fixed point set of the Cremonat transformation is exactly one point $(1,1,1,1,1)$. Hence $\sigma$ is a fixed point free involution of $Y$. The quotient surface $X=Y/\langle \sigma \rangle$ is a singular Enriques surface. Obviously the permutation group $\mathfrak{S}_5$ acts on $S$ which commutes with $\sigma$. Therefore $\mathfrak{S}_5$ acts on $X$ as automorphisms. The involution $\sigma$ changes $L_{ij}$ and $E_{klm}$, where $\{i,j,k,l,m\} =\{1,2,3,4,5\}$. The images of twenty nonsingular rational curves $L_{ij}$, $E_{ijk}$ give ten nonsingular rational curves on $X$ whose dual graph is given by the following Figure \[petersen\]. Note that this graph is well known called the Petersen graph. ![[]{data-label="petersen"}](petersen.eps){width="60mm"} Here $\bar{L}_{ij}$ is the image of $L_{ij}$ (and $E_{klm}$). Note that $\mathfrak{S}_5$ is the automorphism group of the Petersen graph. The hyperplane section $x_i+x_j=0$ on $S$ is the union of the double line $2\ell_{ij}$ and two lines through $p_{klm}$ defined by $x_kx_l+x_kx_m+x_lx_m=0$. Thus we have additional twenty nodal curves on $Y$. Note that the Cremonat transformation changes two lines defined by $x_kx_l+x_kx_m+x_lx_m=0$. Thus $X$ contains twenty nonsingular rational curves whose dual graph $\Gamma$ coincides with the one of the Enriques surface of type [VI]{} (see Fig.6.4 in [@Ko]). It now follows from Proposition \[finiteness\] that the automorphism group $\Aut(X)$ is finite. The same argument as in the proof of [@Ko], Theorem 3.1.1 shows that $X$ contains only these 20 nonsingular rational curves. By a similar argument to the one in the proof of Lemma \[injectiv\], we see that the map $\rho : \Aut(X) \to \O(\Num(X))$ is injective. Since the classes of twenty nonsingular rational curves generate $\Num(X)\otimes {\bf Q}$, $\Aut(X)$ is isomorphic to $\Aut(\Gamma) \cong \mathfrak{S}_5$. Thus we have the following theorem. \[VIthm\] The surface $X$ is a singular Enriques surfaces whose dual graph of nonsingular rational curves is of type ${\rm VI}$. The automorphism group $\Aut(X)$ is isomorphic to the symmetric group $\mathfrak{S}_5$ of degree five. \[non-existVI\] There are no classical and supersingular Enriques surfaces with the dual graph of type ${\rm VI}$. A pentagon in the Figure \[petersen\], for example, $|\bar{L}_{12} + \bar{L}_{34}+ \bar{L}_{15}+ \bar{L}_{24}+\bar{L}_{35}|$, defines an elliptic fibration on $X$. The multiple fiber of this fibration is nothing but the pentagon, that is, of type $\I_5$. The assertion now follows from Proposition \[multi-fiber\]. \[type7\] Over the field of complex numbers, Ohashi found that the Enriques surface of type $\VII$ in [@Ko] is isomorphic to the following surface (see [@MO], §1.2). Let $(x_1,\cdots , x_5)$ be homogeneous coodinates of ${\bf P}^4$. Consider the surface in ${\bf P}^4$ defined by $$\sum_{i< j} x_i x_j = \sum_{i<j<k} x_ix_jx_k = 0$$ which has five nodes at coodinate points and whose minimal resolution is a $K3$ surface $Y$. The standard Cremonat transformation $$(x_i) \to \left({1/ x_i}\right)$$ acts on $Y$ as a fixed point free involution $\sigma$. Thus the quotient surface $X=Y/\langle \sigma \rangle$ is a complex Enriques surface. In characteristic 2, the involution $\sigma$ has a fixed point $(1,1,1,1,1)$ on $Y$, and hence the quotient is not an Enriques surface. Type $\III, \IV, \V$ {#type345} -------------------- In each case of type $\III$, $\IV$, $\V$, from the dual graph (cf. Kondo [@Ko], Figures 3.5, 4.4, 5.5) we can find an elliptic fibration which has two reducible multiples fibers. In fact, the parabolic subdiagram of type $\tilde{D}_6 \oplus \tilde{A}_1 \oplus \tilde{A}_1$ in case $\III$ (of type $\tilde{A}_3 \oplus \tilde{A}_3 \oplus \tilde{A}_1 \oplus \tilde{A}_1$ in case $\IV$, of type $\tilde{A}_5 \oplus \tilde{A}_2 \oplus \tilde{A}_1$ in case $\V$) defines such an elliptic fibration (see [@Ko], Table 2, page 274). Hence if an Enriques surface with the same dual graph of nodal curves exists in characteristic 2, then it should be classical (Proposition \[multi-fiber\]). On the other hand, in each case of type $\III$, $\IV$, $\V$, there exists an elliptic fibration which has a reducible multiple fiber of multiplicative type (see [@Ko], Table 2, page 274). However this is impossible because any multiple fiber of an elliptic fibration on a classical Enriques surface is nonsingular or singular of additive type (Proposition \[multi-fiber\]). Thus we have prove the following theorem. \[non-existIII\] There are no Enriques surfaces with the same dual graph as in case of type ${\rm III}$, ${\rm IV}$ or ${\rm V}$. Combining Theorems \[main2\], \[non-existVII\], \[Ithm\], \[non-existI\], \[IIthm\], \[non-existII\], \[VIthm\], \[non-existVI\], \[non-existIII\], we have the Table \[Table1\] in the introduction. \[extra\] In characteristic 2, there exist Enriques surfaces with a finite group of automorphisms whose dual graphs of all nonsingular rational curves do not appear in the case of complex surfaces. For example, it is known that there exists an Enriques surface $X$ which has a genus 1 fibration with a multiple singular fiber of type $\tilde{E}_8$ and with a 2-section (Ekedahl and Shepherd-Barron[@ES], Theorem A, Salomonsson[@Sa], Theorem 1). We have ten nonsingular rational curves on $X$, that is, nine components of the singular fiber and a 2-section, whose dual graph is given in Figure \[E10Dynkin\]. (120,30) (0, 20) (0, 30)[(0, 0)]{} (2, 20)[(1, 0)[15]{}]{} (20, 20) (20, 30)[(0, 0)]{} (22, 20)[(1, 0)[15]{}]{} (40, 20) (40, 30)[(0, 0)]{} (42, 20)[(1, 0)[15]{}]{} (60, 20) (60, 30)[(0, 0)]{} (62, 20)[(1, 0)[15]{}]{} (80, 20) (80, 30)[(0, 0)]{} (82, 20)[(1, 0)[15]{}]{} (100, 20) (100, 30)[(0, 0)]{} (102, 20)[(1, 0)[15]{}]{} (120, 20) (120, 30)[(0, 0)]{} (122, 20)[(1, 0)[15]{}]{} (140, 20) (140, 30)[(0, 0)]{} (142, 20)[(1, 0)[15]{}]{} (160, 20) (160, 30)[(0, 0)]{} (40, 17)[(0, -1)[15]{}]{} (40, 0) (50, 0)[(0, 0)]{} (120, 0)[(0, 0)]{} It is easy to see that they generate $\Num(X) \cong U\oplus E_8$. Moreover it is known that the reflection subgroup generated by reflections associated with these $(-2)$-vectors is of finite index in $\O(\Num(X))$ (Vinberg [@V], Table 4; also see Proposition \[Vinberg\]) and hence $\Aut(X)$ is finite (Proposition \[finiteness\]). The height of the covering $K3$ surfaces of singular Enriques surfaces {#height} ====================================================================== In this section we prove the following theorem. \[height\] In characteristic $2$, if $K3$ surface $Y$ has a fixed point free involution, then the height $h(Y)$ of the formal Brauer group of $Y$ is equal to $1$. \[height2\] Let $Y$ be the covering $K3$ surface of a singular Enriques surface. Then the height $h(Y) = 1$. Suppose $h = h(Y) \neq 1$. Since ${\rm H}^2(Y, {{\mathcal{O}}}_{Y})$ is the tangent space of the formal Brauer group of $Y$ (cf. Artin-Mazur [@AM], Corollary (2.4)), the Frobenius map $$F : {\rm H}^2(Y, {{\mathcal{O}}}_{Y}) \rightarrow {\rm H}^2(Y, {{\mathcal{O}}}_{Y})$$ is a zero map. Then, we have an isomorphism $${\rm id} - F : {\rm H}^2(Y, {{\mathcal{O}}}_{Y}) \rightarrow {\rm H}^2(Y, {{\mathcal{O}}}_{Y}).$$ Let $W_{i}({{\mathcal{O}}}_{Y})$ be the sheaf of ring of Witt vectors of length $i$ on $Y$. Assume ${\rm id} - F : {\rm H}^2(Y, W_{i-1}({{\mathcal{O}}}_{Y})) \longrightarrow {\rm H}^2(Y, W_{i-1}({{\mathcal{O}}}_{Y}))$ is an isomorphism. We have an exact sequence $$0 \rightarrow W_{i-1}({{\mathcal{O}}}_{Y}) \stackrel{V}{\longrightarrow} W_{i}({{\mathcal{O}}}_{Y}) \stackrel{R}{\longrightarrow} {{\mathcal{O}}}_{Y} \rightarrow 0,$$ where $V$ is the Verschiebung and $R$ is the restriction. Then, we have a diagram [$$\begin{array}{ccccccccc} 0 & \rightarrow & {\rm H}^2(Y, W_{i-1}({{\mathcal{O}}}_{Y})) & \stackrel{V}{\longrightarrow}& {\rm H}^2(Y, W_{i}({{\mathcal{O}}}_{Y})) &\stackrel{R}{\longrightarrow} & {\rm H}^2(Y, {{\mathcal{O}}}_{Y}) & \rightarrow & 0\\ & & {\rm id} - F \downarrow & & {\rm id} - F \downarrow & & {\rm id} - F \downarrow & & \\ 0 & \rightarrow & {\rm H}^2(Y, W_{i-1}({{\mathcal{O}}}_{Y})) & \stackrel{V}{\longrightarrow}& {\rm H}^2(Y, W_{i}({{\mathcal{O}}}_{Y})) &\stackrel{R}{\longrightarrow} & {\rm H}^2(Y, {{\mathcal{O}}}_{Y}) & \rightarrow & 0. \end{array}$$ ]{} By the assumption of induction, the first and the third downarrows are isomorphisms. Therefore, by the 5-lemma, we have an isomorphism $${\rm id} - F : {\rm H}^2(Y, W_{i}({{\mathcal{O}}}_{Y})) \cong {\rm H}^2(Y, W_i({{\mathcal{O}}}_{Y})).$$ Therefore, taking the projective limit, we have an isomorphism $${\rm id} - F : {\rm H}^2(Y, W({{\mathcal{O}}}_{Y})) \cong {\rm H}^2(Y, W({{\mathcal{O}}}_{Y}))$$ Therefore, denoting by $K$ the quotient field of the ring of Witt vectors $W(k)$ of infinite length, we have an isomorphism $${\rm id} - F : {\rm H}^2(Y, W({{\mathcal{O}}}_{Y}))\otimes K \cong {\rm H}^2(Y, W({{\mathcal{O}}}_{Y}))\otimes K.$$ Let ${\rm H}_{et}^2(Y, {\bf Q}_2)$ be the second 2-adic étale cohomology of $Y$. Then, we have an exact sequence $$0 \rightarrow {\rm H}_{et}^2(Y, {\bf Q}_2)\rightarrow {\rm H}^2(Y, W({{\mathcal{O}}}_{Y}))\otimes K \stackrel{{\rm id} - F}{\longrightarrow} {\rm H}^2(Y, W({{\mathcal{O}}}_{Y}))\otimes K \rightarrow 0$$ (cf. Crew [@Cr], (2.1.2) for instance). Therefore, we have ${\rm H}_{et}^2(Y, {\bf Q}_2)= 0$. On the other hand, we consider the quotient surface $X$ of $Y$ by the fixed point free involution. Then, $X$ is a singular Enriques surface and under the assumption Crew showed $\dim {\rm H}_{et}^2(Y, {\bf Q}_2)= 1$ for the $K3$ covering Y of $X$ (Crew [@Cr], p41), a contradiction. Enriques surfaces associated with Kummer surfaces {#kummer} ================================================= In this section we show that Enriques surfaces given in Remark \[typeInumtrivial\] are obtained from Kummer surfaces associated with the product of two ordinary elliptic curves. Let $E, F$ be two ordinary elliptic curves and let $\iota=\iota_E\times \iota_F$ be the inversion of the abelian surface $E\times F$. Let $\Km(E\times F)$ be the minimal resolution of the quotient surface $(E\times F)/\langle\iota\rangle$. It is known that $\Km(E\times F)$ is a $K3$ surface called Kummer surface associated with $E\times F$ (Shioda [@Shi], Proposition 1, see also Katsura [@Ka], Theorem B). The projection from $E\times F$ to $E$ gives an elliptic fibration which has two singular fibers of type $\I^*_4$ and two sections. Let $a \in E, \ b\in F$ be the unique non-zero 2-torsion points on $E$, $F$ respectively. Denote by $t$ the translation of $E\times F$ by the 2-torsion point $(a,b)$. The involution $(\iota_E\times 1_F)\circ t = t \circ (\iota_E\times 1_F)$ induces a fixed point free involution $\sigma$ of $\Km(E\times F)$. Thus we have an Enriques surface $S = \Km(E\times F)/\langle\sigma\rangle$. The involution $\iota_E\times 1_F$ (or $t$) induces a numerically trivial involution $\eta$ of $S$. \[nt\] The pair $(S, \eta)$ is isomorphic to an Enriques surface given in Remark [\[typeInumtrivial\]]{}. Let $$E: y^2+xy =x^3 +bx, \quad F: y'^2+x'y'=x'^3+b'x' \quad (b, b' \in k, bb'\not=0)$$ be two ordinary elliptic curves. The inversion $\iota_E$ is then expressed by $$(x,y) \to (x,y+x)$$ and the translation by the non-zero 2-torsion on $E$ is given by $$(x,y) \to (b/x, by/x^2 + b/x).$$ Then the function field of $(E\times F)/\langle \iota \rangle$ is given by $$k((E\times F)/\langle \iota\rangle) = k(x,x', z)$$ with the relation $$\label{shiodakummer} z^2 + xx'z= x^2(x'^3+b'x')+x'^2(x^3+bx)$$ where $z=xy'+x'y$ (see Shioda [@Shi], the equation (8)). The fixed point free involution $\sigma$ is expressed by $$\label{enriques-inv} \sigma (x,x',z) = (b/x, b'/x', bb'z/x^2x'^2 + bb'/xx'),$$ and the involution induced by $\iota_E\times 1_F$ on $\Km(E\times F)$ is given by $$\label{num-tri-inv} (x,x',z)\to (x,x', z+xx').$$ On the other hand, we consider the quadric $Q$ given in (\[type1quadric\]). Instead of $\tau$ in (\[typeIinv\]), we consider the involution given by $$\tau' : (x_0,x_1,x_2,x_3) \to (x_3,b'x_2,bx_1,bb'x_0)$$ whose fixed point is $(1,b',b,bb')$. The Artin-Schreier covering is defined by the equation $$z^2 + x_0x_3z = x_0x_3(x_1x_3 +b'x_0x_2+x_2x_3+bx_0x_1)$$ (in the example given in the subsection \[type1\], the term $\mu(x_0x_3)^2$ appears in the Artin-Schreier covering. If $\mu\not=0$, then changing $z$ by $z+ax_0x_3$ where $a^2+a+\mu=0$, we can delete this term). Now, by putting here $$x_0=u_0v_0, \ x_1=u_0v_1, \ x_2=u_1v_0, \ x_3=u_1v_1$$ and considering an affine locus $u_0\not=0, v_0\not=0$, we have $$z^2 + u_1v_1z=u_1v_1(u_1v_1^2+ u_1^2v_1+ bv_1+b'u_1)$$ which is the same as the equation given in (\[shiodakummer\]). Moreover the lifting of $\tau'$ and the covering transformation of the Artin-Schreier covering coincide with the ones given in (\[enriques-inv\]) and (\[num-tri-inv\]) respectively. Using appendix A, we see that the height of the formal Brauer group of Kummer surfaces associated with the product of two ordinary elliptic curves is equal to 1. All complex Enriques surfaces with cohomologically or numerically trivial automorphisms are classified by Mukai and Namikawa [@MN], Main theorem (0.1), and Mukai [@M], Theorem 3. There are three types: one of them is an Enriques surface associated with $\Km(E\times F)$ and the second one is mentioned in Remark \[typeInumtrivial\]. For the third one we refer the reader to Mukai [@M], Theorem 3. In positive characteristic, Dolgachev ([@D2], Theorems 4 and 6) determined the order of cohomologically or numerically trivial automorphisms. However, the explicit classification is not known. The above Theorem \[nt\] implies that two different type of complex Enriques surfaces with a numerically trivial involution coincide in characteristic 2. [99]{} M. Artin, , Iwanami Shoten, Publishers, Cambridge Univ. Press, 1977. 11–22. M. Artin and B. Mazur, (1977), 87–131. W. Barth and C. Peters, , E. Bombieri and D. Mumford, , F. Cossec and I. Dolgachev, R. M. Crew, E. Dardanelli and B. van Geemen, , I. Dolgachev, I. Dolgachev, I. Dolgachev and C. Liedtke, T. Ekedahl and N. I. Shepherd-Barron, G. Fano, , H. Ito, , T. Katsura, T. Katsura and S. Kondō, to appear in Pure Appl. Math. Q.; T. Katsura and Y. Takeda, T. Katsura and K. Ueno, S. Kondō, , W. Lang, , W. Lang, , Q. Liu, D. Lorenzini and M. Raynaud, , S. Mukai, , S. Mukai and Y. Namikawa, , S. Mukai and  H. Ohashi, , V. Nikulin, A. N. Rudakov and I. R. Shafarevich, P. Salomonsson, T. Shioda, T. Shioda, T. Shioda, E. B. Vinberg, , in [^1]: Research of the first author is partially supported by Grant-in-Aid for Scientific Research (B) No. 15H03614, and the second author by (S) No. 15H05738.

ArXiv

--- abstract: 'It is generally believed that the inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase appears in a color superconductor when the pairing between different quark flavors is under the circumstances of mismatched Fermi surfaces. However, the real crystal structure of the LOFF phase is still unclear because an exact treatment of 3D crystal structures is rather difficult. In this work we present a solid-state-like calculation of the ground-state energy of the body-centered cubic (BCC) structure for two-flavor pairing by diagonalizing the Hamiltonian matrix in the Bloch space without assuming a small amplitude of the order parameter. We develop a computational scheme to overcome the difficulties in diagonalizing huge matrices. Our results show that the BCC structure is energetically more favorable than the 1D modulation in a narrow window around the conventional LOFF-normal phase transition point, which indicates the significance of the higher-order terms in the Ginzburg-Landau approach.' author: - 'Gaoqing Cao,$^{1}$ Lianyi He,$^{2}$ and Pengfei Zhuang$^{1}$' title: 'Solid-state calculation of crystalline color superconductivity' --- Introduction ============ The ground state of exotic fermion Cooper pairing with mismatched Fermi surfaces is a longstanding problem in the theory of superconductivity  [@Casalbuoni2004]. In electronic superconductors, the mismatched Fermi surfaces are normally induced by the Zeeman energy splitting $2\delta\mu$ in a magnetic field. For $s$-wave pairing at weak coupling, it is known that, at a critical field $\delta\mu_1=0.707\Delta_0$ where $\Delta_0$ is the pairing gap at vanishing mismatch, a first-order phase transition from the gapped BCS state to the normal state occurs  [@CC1962]. Further theoretical studies showed that the inhomogeneous Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) state can survive in a narrow window $\delta\mu_1<\delta\mu<\delta\mu_2$, where the upper critical field $\delta\mu_2=0.754\Delta_0$ [@LO1964; @FF1964]. However, since the thermodynamic critical field is much lower than $\delta\mu_1$ due to strong orbit effect, it is rather hard to observe the LOFF state in ordinary superconductors [@CC1962]. In recent years, experimental evidences for the LOFF state in some superconducting materials have been reported [@Heavyfermion; @HighTc; @Organic; @FeSe]. On the other hand, exotic pairing phases have promoted new interest in the studies of dense quark matter under the circumstances of compact stars [@Alford2001; @Bowers2002; @Shovkovy2003; @Alford2004; @EFG2004; @Huang2004; @Casalbuoni2005; @Fukushima2005; @Ren2005; @Gorbar2006; @Anglani2014] and ultracold atomic Fermi gases with population imbalance [@Atomexp; @Sheehy2006]. Color superconductivity in dense quark matter appears due to the attractive interactions in certain diquark channels [@CSC01; @CSC02; @CSC03; @CSC04; @CSCreview]. Because of the constraints from Beta equilibrium and electric charge neutrality, different quark flavors ($u$, $d$, and $s$) acquire mismatched Fermi surfaces. Quark color superconductors under compact-star constraints as well as atomic Fermi gases with population imbalance therefore provide rather clean systems to realize the long-sought exotic LOFF phase. Around the tricritical point in the temperature-mismatch phase diagram, the LOFF phase can be studied rigorously by using the Ginzburg-Laudau (GL) analysis since both the gap parameter and the pair momentum are vanishingly small [@Casalbuoni2004]. It was found that the solution with two antipodal wave vectors is the preferred one [@Buzdin1997; @Combescot2002; @Ye2007]. However, the real ground state of the LOFF phase is still debated due to the limited theoretical approaches at zero temperature. So far rigorous studies of the LOFF phase at zero temperature are restricted to its 1D structures including the Fulde-Ferrell (FF) state with a plane-wave form $\Delta(z)=\Delta e^{2iqz}$ and the Larkin-Ovhinnikov (LO) state with an antipodal-wave form $\Delta(z)=2\Delta \cos(2qz)$. A recent self-consistent treatment of the 1D modulation [@Buballa2009] show that a solitonic lattice is formed near the lower critical field, and the phase transition to the BCS state is continuous. Near the upper critical field the gap function becomes sinusoidal, and the transition to the normal state is of first order. In addition to these 1D structures, there exist a large number of 3D crystal structures. The general form of a crystal structure of the order parameter can be expressed as $$\begin{aligned} \label{crystal} \Delta({\bf r})=\sum_{k=1}^P\Delta e^{2iq\hat{\bf n}_k\cdot{\bf r}}.\end{aligned}$$ A specific crystal structure corresponds to a multi-wave configuration determined by the $P$ unit vectors ${\bf n}_{k}$ ($k=1,2,...,P$). In general, we expect two competing mechanisms: Increasing the number of waves tends to lower the energy, but it may also causes higher repulsive interaction energy between different wave directions. In a pioneer work, Bowers and Rajagopal investigated 23 different crystal structures by using the GL approach [@Bowers2002], where the grand potential measured with respect to the normal state was expanded up to the order $O(\Delta^6)$, $$\begin{aligned} \label{GL} \frac{\delta\Omega(\Delta)}{{\cal N}_{\rm F}}= P\alpha\Delta^2+\frac{1}{2}\beta\Delta^4+\frac{1}{3}\gamma\Delta^6+O(\Delta^8)\end{aligned}$$ with ${\cal N}_{\rm F}$ being the density of state at the Fermi surface and the pair momentum fixed at the optimal value $q=1.1997\delta\mu$. Among the structures with $\gamma>0$, the favored one seems to be the body-centered cubic (BCC) with $P=6$ [@BCC]. Further, it was conjectured that the face-centered cubic (FCC) with $P=8$ [@FCC] is the preferred structure since its $\gamma$ is negative and the largest [@Bowers2002]. For BCC structure, the GL analysis up to the order $O(\Delta^6)$ predicts a strong first-order phase transition at $\delta\mu_*\simeq3.6\Delta_0$ with the gap parameter $\Delta\simeq0.8\Delta_0$ [@Bowers2002]. The prediction of a strong first-order phase transition may invalidate the GL approach itself. On the other hand, by using the quasiclassical equation approach with a Fourier expansion for the order parameter, Combescot and Mora [@Combescot2004; @Combescot2005] predicted that the BCC-normal transition is of rather weak first order: The upper critical field $\delta\mu_*$ is only about $4\%$ higher than $\delta\mu_2$ with $\Delta\simeq0.1\Delta_0$ at $\delta\mu=\delta\mu_*$. If this result is reliable, it indicates that the higher-order expansions in the GL analysis is important for quantitative predictions. To understand this intuitively, let us simply add the eighth-order term $\frac{\eta}{4}\Delta^8$ to the GL potential (\[GL\]). A detailed analysis of the influence of a positive $\eta$ on the phase transition is presented in Appendix A. We find that with increasing $\eta$, the first-order phase transition becomes weaker and the upper critical field $\delta\mu_*$ decreases. For $\eta\rightarrow+\infty$, the phase transition approaches second order and $\delta\mu_*\rightarrow\delta\mu_2$. Therefore, to give more precise predictions we need to study the higher-order expansions and the convergence property of the GL series, or use a different way to evaluate the grand potential without assuming a small value of $\Delta$. For a specific crystal structure given by (\[crystal\]), it is periodic in coordinate space. As a result, the eigenvalue equation for the fermionic excitation spectrum in this periodic pair potential, which is known as the Bogoliubov-de Gennes (BdG) equation, is in analogy to the Schrödinger equation of quantum particles in a periodic potential. This indicates that the fermionic excitation spectrum has a band structure, which can be solved from the BdG equation. The grand potential can be directly evaluated once the fermionic excitation spectrum is known [@Buballa2009]. In this work, we present a solid-state-like calculation of the grand potential of the BCC structure. Our numerical results show that the phase transition from the BCC state to the normal state is of rather weak first order, consistent with the work by Combescot and Mora [@Combescot2004; @Combescot2005]. This implies that it is quite necessary to evaluate the higher-order terms in the GL expansion to improve the quantitative predictions. Thermodynamic Potential ======================= To be specific, we consider a general effective Lagrangian for two-flavor quark pairing at high density and at weak coupling. The Lagrangian density is given by [@Casalbuoni2004] $${\cal L}=\psi^\dagger[i\partial_t-\varepsilon(\hat{\bf p})+\hat{\mu}]\psi+{\cal L}_{\rm int},$$ where $\psi=(\psi_{\rm u},\psi_{\rm d})^{\rm T}$ denotes the two-flavor quark field and $\varepsilon(\hat{\bf p})$ is the quark dispersion with the momentum operator $\hat{\bf p}=-i\mbox{\boldmath{$\nabla$}}$. In the momentum representation we have $\varepsilon({\bf p})=|{\bf p}|$. The quark chemical potentials are specified by the diagonal matrix $\hat{\mu}={\rm diag}(\mu_{\rm u},\mu_{\rm d})$ in the flavor space, where $$\begin{aligned} \mu_{\rm u}=\mu+\delta\mu,\ \ \ \ \ \mu_{\rm d}=\mu-\delta\mu.\end{aligned}$$ The interaction Lagrangian density which leads to Cooper pairing between different flavors can be expressed as [@Casalbuoni2004] $$\begin{aligned} {\cal L}_{\rm{int}}=g(\psi^\dagger\sigma_2\psi^*)(\psi^{\rm T}\sigma_2\psi),\end{aligned}$$ where $g$ is the coupling constant and $\sigma_2$ is the second Pauli matrix in the flavor space. Notice that we have neglected the antiquark degree of freedom because it plays no role at high density and at weak coupling. We have also neglected the color and spin degrees of freedom, which simply give rise to a degenerate factor. Color superconductivity is characterized by nonzero expectation value of the diquark field $\varphi(t,{\bf r})=-2ig\psi^{\rm T}\sigma_2\psi$. For the purpose of studying inhomogeneous phases, we set the expectation value of $\varphi(t,{\bf r})$ to be static but inhomogeneous, i.e., $\langle\varphi(t,{\bf r})\rangle=\Delta({\bf r})$. With the Nambu-Gor’kov spinor $\Psi=(\psi\ \ \psi^*)^{\rm T}$, the mean-field Lagrangian reads $$\begin{aligned} {\cal L}_{\rm{MF}}=\frac{1}{2}\Psi^\dagger\left(\begin{array}{cc} i\partial_t-\varepsilon(\hat{\bf p})+\hat{\mu} & -i\sigma_2\Delta({\bf r})\\ i\sigma_2\Delta^*({\bf r})& i\partial_t+\varepsilon(\hat{\bf p})-\hat{\mu} \end{array}\right)\Psi-\frac{|\Delta({\bf r})|^2}{4g}.\end{aligned}$$ The order parameters of the BCC and FCC structures can be expressed as $$\begin{aligned} \Delta({\bf r})=2\Delta\left[\cos\left(2qx\right) +\cos\left(2qy\right)+\cos\left(2qz\right)\right]\end{aligned}$$ and $$\begin{aligned} \Delta({\bf r})=8\Delta\cos\left(\frac{2qx}{\sqrt{3}}\right)\cos\left(\frac{2qy}{\sqrt{3}}\right)\cos\left(\frac{2qz}{\sqrt{3}}\right),\end{aligned}$$ respectively. Therefore, we consider a 3D periodic structure where the unit cell is spanned by three linearly independent vectors ${\bf a}_1=a{\bf e}_x$, ${\bf a}_2=a{\bf e}_y$, and ${\bf a}_3=a{\bf e}_z$ with $a=\pi/q$ for BCC and $a=\sqrt{3}\pi/q$ for FCC. The order parameter is periodic in space, i.e., $\Delta({\bf r})=\Delta({\bf r}+{\bf a}_i)$. It can be decomposed into a discrete set of Fourier components, $$\begin{aligned} \Delta({\bf r})=\sum_{{\bf G}}\Delta_{\bf G}e^{i{\bf G}\cdot {\bf r}}=\sum_{l,m,n=-\infty}^\infty \Delta_{[lmn]}e^{i{\bf G}_{[lmn]}\cdot {\bf r}},\end{aligned}$$ where the reciprocal lattice vector ${\bf G}$ reads $$\begin{aligned} {\bf G}={\bf G}_{[lmn]}=\frac{2\pi}{a}\left(l{\bf e}_x+m{\bf e}_y+n{\bf e}_z\right),\ \ \ l,m,n\in \mathbb{Z}.\end{aligned}$$ The Fourier component $\Delta_{\bf G}=\Delta_{[lmn]}$ can be evaluated as $$\begin{aligned} &&\Delta_{\bf G}=\Delta\ \Big[\left(\delta_{l,1}+\delta_{l,-1}\right)\delta_{m,0}\delta_{n,0} +\delta_{l,0}\left(\delta_{m,1}+\delta_{m,-1}\right)\delta_{n,0}\nonumber\\ &&\ \ \ \ \ \ \ \ \ \ +\ \delta_{l,0}\delta_{m,0}\left(\delta_{n,1}+\delta_{n,-1}\right)\Big]\end{aligned}$$ and $$\begin{aligned} \Delta_{\bf G}=\Delta \left(\delta_{l,1}+\delta_{l,-1}\right)\left(\delta_{m,1}+\delta_{m,-1}\right)\left(\delta_{n,1}+\delta_{n,-1}\right)\end{aligned}$$ for BCC and FCC structures, respectively. Then we consider a finite system in a cubic box defined as $x,y,z\in[-L/2,L/2]$ with the length $L=Na$. For convenience we impose the periodic boundary condition. The thermodynamic limit is reached by setting $N\rightarrow\infty$. Using the momentum representation, we have the Fourier transformation $$\begin{aligned} \Psi(\tau,{\bf r})=V^{-1/2}\sum_{\nu,{\bf p}}\Psi_{\nu{\bf p}}e^{-i(\omega_\nu\tau-{\bf p}\cdot{\bf r})}.\end{aligned}$$ Here $V$ is the volume of the system, $\omega_\nu=(2\nu+1)\pi T (\nu\in \mathbb{Z})$ is the fermion Matsubara frequency, and the quantized momentum ${\bf p}$ is given by $${\bf p}=\frac{2\pi}{L}(l{\bf e}_x+m{\bf e}_y+n{\bf e}_z)$$ with $l,m,n\in \mathbb{Z}$. The partition function of the system is given by $${\cal Z}=\int [d\Psi][d\Psi^\dagger]e^{-{\cal S}}$$ with the Euclidean action ${\cal S}=-\int_0^{1/T}d\tau\int_Vd^3{\bf r}{\cal L}$. The grand potential per volume reads $$\Omega=-\frac{T}{V}\ln{\cal Z}.$$ In the mean-field approximation, the action ${\cal S}$ is quadratic. Therefore, the partition function ${\cal Z}$ and grand potential $\Omega$ can be evaluated. Using the Fourier expansions for $\Psi$ and $\Delta$, we obtain the mean-field action $$\begin{aligned} {\cal S}_{\rm{MF}}=\frac{V}{T}\!\sum_{\bf G}\!\frac{|\Delta_{\bf G}|^2}{4g}-\frac{1}{2T}\!\sum_{\nu,{\bf p},{\bf p}^\prime}\! \Psi^\dagger_{\nu{\bf p}} \left(i\omega_\nu\delta_{{\bf p},{\bf p}^\prime} -{\cal H}_{{\bf p},{\bf p}^\prime}\right)\Psi_{\nu{\bf p}^\prime},\end{aligned}$$ where the effective Hamiltonian matrix ${\cal H}_{{\bf p},{\bf p}^\prime}$ reads $$\begin{aligned} {\cal H}_{{\bf p},{\bf p}^\prime}=\left(\begin{array}{cc} (|{\bf p}|-\hat{\mu})\delta_{{\bf p},{\bf p}^\prime} & i\sigma_2\sum_{\bf G}\Delta_{\bf G}\delta_{{\bf G},{\bf p}-{\bf p}^\prime} \\ -i\sigma_2\sum_{\bf G}\Delta_{\bf G}^*\delta_{{\bf G},{\bf p}^\prime-{\bf p}} & -(|{\bf p}|-\hat{\mu})\delta_{{\bf p},{\bf p}^\prime} \end{array}\right).\end{aligned}$$ The effective Hamiltonian ${\cal H}_{{\bf p},{\bf p}^\prime}$ is a huge matrix in Nambu-Gor’kov, flavor, and (discrete) momentum spaces. It is Hermitian and can in principle be diagonalized. Assuming that the eigenvalues of ${\cal H}_{{\bf p},{\bf p}^\prime}$ is denoted by $E_\lambda$, we can formally express the grand potential as $$\begin{aligned} \Omega=\frac{1}{4g}\sum_{\bf G}|\Delta_{\bf G}|^2-\frac{1}{2V}\sum_\lambda{\cal W}(E_\lambda),\end{aligned}$$ where the function ${\cal W}(E)=\frac{E}{2}+T\ln(1+e^{-E/T})$. The summation over ${\bf G}$ can be worked out as $\sum_{\bf G}|\Delta_{\bf G}|^2=P\Delta^2$. In practice, diagonalization of the matrix ${\cal H}_{{\bf p},{\bf p}^\prime}$ is infeasible. However, ${\cal H}$ can be brought into a block-diagonal form with $N^3$ independent blocks in the momentum space according to the famous Bloch theorem [@Buballa2009]. To understand this, we consider the eigenvalue equation for the fermionic excitation spectrum in the coordinate space, which is known as the BdG equation. For our system, the BdG equation reads $$\begin{aligned} \left(\begin{array}{cc} \varepsilon(-i\mbox{\boldmath{$\nabla$}})-\hat{\mu} & i\sigma_2\Delta({\bf r}) \\ -i\sigma_2\Delta^*({\bf r}) & -\varepsilon(-i\mbox{\boldmath{$\nabla$}})+\hat{\mu} \end{array}\right)\phi_\lambda({\bf r}) =E_\lambda\phi_\lambda({\bf r}).\end{aligned}$$ According to the Bloch theorem, the solution of the eigenfunction $\phi_\lambda({\bf r})$ takes the form of the so-called Bloch function. We have $$\phi_\lambda({\bf r})=e^{i{\bf k}\cdot{\bf r}}\phi_{\lambda{\bf k}}({\bf r}),$$ where ${\bf k}$ is the momentum in the Brillouin zone (BZ) and the function $\phi_{\lambda{\bf k}}({\bf r})$ has the same periodicity as the order parameter $\Delta({\bf r})$. We therefore have the similar Fourier expansion $$\phi_{\lambda{\bf k}}({\bf r})=\sum_{\bf G}\phi_{\bf G}({\bf k})e^{i{\bf G}\cdot {\bf r}}.$$ Substituting this expansion into the BdG equation, for a given ${\bf k}$ we obtain a matrix equation $$\begin{aligned} \sum_{{\bf G}^\prime}{\cal H}_{{\bf G},{\bf G}^\prime}({\bf k})\phi_{{\bf G}^\prime}({\bf k})=E_\lambda({\bf k}) \phi_{\bf G}({\bf k}),\end{aligned}$$ where the matrix ${\cal H}_{{\bf G},{\bf G}^\prime}({\bf k})$ is given by $$\begin{aligned} \left(\begin{array}{cc} (|{\bf k}+{\bf G}|-\hat{\mu})\delta_{{\bf G},{\bf G}^\prime} & i\sigma_2\Delta_{{\bf G}-{\bf G}^\prime} \\ -i\sigma_2\Delta^*_{{\bf G}^\prime-{\bf G}} & -(|{\bf k}+{\bf G}|-\hat{\mu})\delta_{{\bf G},{\bf G}^\prime} \end{array}\right).\end{aligned}$$ This shows that, for a given ${\bf k}$-point in the BZ, we can solve the eigenvalue spectrum $\{E_{\lambda}({\bf k})\}$ by diagonalizing the matrix ${\cal H}_{{\bf G},{\bf G}^\prime}({\bf k})$. Without loss of generality, the BZ can be chosen as $k_x,k_y,k_z\in[-\pi/a,\pi/a]$. For a quantized volume $V$ containing $N^3$ unit cells, we have $N^3$ allowed momenta ${\bf k}$ in the BZ. Accordingly, the grand potential is now given by $$\begin{aligned} \Omega=\frac{P\Delta^2}{4g}-\frac{1}{2V}\sum_{{\bf k}\in {\rm BZ}}\sum_\lambda{\cal W}[E_\lambda({\bf k})].\end{aligned}$$ In the thermodynamic limit $N\rightarrow\infty$, the summation $\frac{1}{V}\sum_{{\bf k}\in {\rm BZ}}$ is replaced by an integral over the BZ. The Hamiltonian matrix ${\cal H}_{{\bf G},{\bf G}^\prime}({\bf k})$ can be further simplified to lower the matrix size. After a proper rearrangement of the eigenvector $\phi_{\bf G}$, we find that ${\cal H}$ can be decomposed into two blocks. We have ${\cal H}={\cal H}_{\Delta,\delta\mu}\oplus{\cal H}_{-\Delta,-\delta\mu}$. The blocks can be expressed as ${\cal H}_{\Delta,\delta\mu}={\cal H}_\Delta-\delta\mu\ {\cal I}$ where ${\cal I}$ is the identity matrix and the matrix ${\cal H}_\Delta$ is given by $$\begin{aligned} \label{highdensity} ({\cal H}_\Delta)_{{\bf G},{\bf G}^\prime}=\left(\begin{array}{cc} (|{\bf k}+{\bf G}|-\mu)\delta_{{\bf G},{\bf G}^\prime} & \Delta_{{\bf G}-{\bf G}^\prime} \\ \Delta^*_{{\bf G}^\prime-{\bf G}} & -(|{\bf k}+{\bf G}|-\mu)\delta_{{\bf G},{\bf G}^\prime} \end{array}\right).\end{aligned}$$ The eigenvalues of ${\cal H}_{\Delta,\delta\mu}$ do not depend on the sign of $\Delta$. Moreover, replacing the $\delta\mu$ by $-\delta\mu$ amounts to a replacement of the eigenvalue spectrum $\{E_\lambda({\bf k})\}$ by $\{-E_\lambda({\bf k})\}$. Therefore, the two blocks contribute equally to the grand potential and we only need to determine the eigenvalues of ${\cal H}_{\Delta,\delta\mu}$. The Hamiltonian matrix (\[highdensity\]) represents the general problem of two-species pairing with mismatched Fermi surfaces. In the weak coupling limit, the pairing is dominated near the Fermi surfaces. Therefore, the physical result should be universal in terms of the pairing gap $\Delta_0$ at vanishing mismatch and the density of state ${\cal N}_{\rm F}$ at the Fermi surface. ![(Color online) The lower and upper critical fields (upper panel) and the size of the stability window $(\delta\mu_2-\delta\mu_1)/\Delta_0$ (lower panel) for the FF state as a function of $\Delta_0$ at $\mu=400$ MeV. The thin lines denote results in the weak-coupling limit. The blue solid and red dashed lines correspond to $\Lambda=400$ MeV and $\Lambda=800$ MeV, respectively. \[fig1\]](BCS-FF-N.eps "fig:"){width="8.6cm"} ![(Color online) The lower and upper critical fields (upper panel) and the size of the stability window $(\delta\mu_2-\delta\mu_1)/\Delta_0$ (lower panel) for the FF state as a function of $\Delta_0$ at $\mu=400$ MeV. The thin lines denote results in the weak-coupling limit. The blue solid and red dashed lines correspond to $\Lambda=400$ MeV and $\Lambda=800$ MeV, respectively. \[fig1\]](ddmu21.eps "fig:"){width="8.5cm"} In the following we shall focus on the zero-temperature case. The grand potential $\Omega$ is divergent and hence a proper regularization scheme is needed. Since we need to deal with the Bloch momentum ${\bf k}+{\bf G}$, the usual three-momentum cutoff scheme [@Alford2001; @Bowers2002] is not appropriate for numerical calculations. Moreover, we are interested in the grand potential $\delta\Omega$ measured with respect to the normal state. Therefore, we employ a Pauli-Villars-like regularization scheme, in which $\delta\Omega$ is well-defined [@Buballa2009]. The “renormalized" grand potential is given by [@note1] $$\begin{aligned} \delta\Omega(\Delta,q)=\Omega(\Delta,q)-\Omega(0,q),\end{aligned}$$ where $$\begin{aligned} \Omega(\Delta,q)=\frac{P\Delta^2}{4g}-\frac{1}{2}\int_{\rm BZ}\frac{d^3{\bf k}}{(2\pi)^3}\sum_\lambda \sum_{j=0}^2c_j\sqrt{E_\lambda^2({\bf k})+j\Lambda^2}\end{aligned}$$ with $c_0=c_2=1$ and $c_1=-2$. Here $\{E_\lambda(\bf k)\}$ denotes the eigenvalue spectrum of ${\cal H}_{\Delta,\delta\mu}$. The coupling constant $g$ can be fixed by the BCS gap $\Delta_0$ at $\delta\mu=0$. We expect that at weak coupling the physical results depend on the cutoff $\Lambda$ only through the BCS gap $\Delta_0$ [@Buballa2009]. In Fig. \[fig1\], we show the stability window for the FF state as a function of $\Delta_0$. In the weak coupling limit, the critical fields depend only on $\Delta_0$. For accuracy reason [@note2], we shall choose $\Delta_0\sim100$ MeV at $\mu=400$ MeV, which corresponds to the realistic value of $\Delta_0$ at moderate density [@CSC01]. Since the size of the FF window $(\delta\mu_2-\delta\mu_1)/\Delta_0$ depends very weakly on $\Delta_0$ and $\Lambda$, we can use the upper critical field $\delta\mu_2$ obtained in Fig. \[fig1\] to “calibrate" $\delta\mu$ and appropriately extrapolate the results to the weak coupling limit. Matrix Structure ================ For a given ${\bf k}$-point in the BZ, we can diagonalize the Hamiltonian matrix ${\cal H}_{\Delta,\delta\mu}$ to obtain its eigenvalue spectrum $\{E_\lambda({\bf k})\}$. The choice of the ${\bf k}$-points in the BZ should be dense enough to achieve the thermodynamic limit [@note3]. The eigenvalue equation can be rewritten as $$\begin{aligned} \sum_{{\bf G}^\prime}({\cal H}_{\Delta})_{{\bf G},{\bf G}^\prime}({\bf k})\phi_{{\bf G}^\prime}({\bf k}) =\left[E_\lambda({\bf k})+\delta\mu\right] \phi_{\bf G}({\bf k}),\end{aligned}$$ where the Hamiltonian matrix $({\cal H}_{\Delta})_{{\bf G},{\bf G}^\prime}({\bf k})$ reads $$\begin{aligned} ({\cal H}_{\Delta})_{{\bf G},{\bf G}^\prime}({\bf k})=\left(\begin{array}{cc} \xi_{{\bf k}+{\bf G}}\delta_{{\bf G},{\bf G}^\prime} & \Delta_{{\bf G}-{\bf G}^\prime} \\ \Delta_{{\bf G}-{\bf G}^\prime} & -\xi_{{\bf k}+{\bf G}}\delta_{{\bf G},{\bf G}^\prime} \end{array}\right).\end{aligned}$$ Here $\xi_{\bf p}=|{\bf p}|-\mu$ and we have used the fact $\Delta^*_{{\bf G}}=\Delta_{-{\bf G}}$. The eigenstate $\phi_{\bf G}$ includes two components $u_{\bf G}$ and ${\bf \upsilon}_{\bf G}$ as usual in the BCS theory. We have $$\begin{aligned} \phi_{\bf G}({\bf k})=\left(\begin{array}{cc} u_{\bf G}({\bf k}) \\ \upsilon_{\bf G}({\bf k}) \end{array}\right).\end{aligned}$$ We notice that $\delta\mu$ can be absorbed into the eigenvalues. It is easy to prove that the eigenvalues of ${\cal H}_{\Delta}$ do not depend on the sign of $\Delta$. Moreover, if $\varepsilon$ is an eigenvalue of ${\cal H}_{\Delta}$, $-\varepsilon$ must be another eigenvalue. Therefore, replacing the $\delta\mu$ by $-\delta\mu$ amounts to a replacement of the eigenvalue spectrum $\{E_\lambda({\bf k})\}$ by $\{-E_\lambda({\bf k})\}$. However, the matrix ${\cal H}_\Delta$ has infinite dimensions because the integers $l,m,n$ run from $-\infty$ to $+\infty$. Therefore, we have to make a truncation in order to perform a calculation. It is natural to make a symmetrical truncation, i.e., $$-D\leq l,m,n\leq D,\ \ \ (D\in \mathbb{Z}^+).$$ For sufficiently large $D$, the contribution from the high-energy bands becomes vanishingly small and the grand potential $\delta\Omega$ converges to its precise value. After making this truncation, the matrix equation can be expressed as $$\begin{aligned} {\bf H}\left(\begin{array}{cc} u \\ \upsilon \end{array}\right)=\left(\begin{array}{cc} {\bf H}_{11} & {\bf H}_{12} \\ {\bf H}_{21} & {\bf H}_{22} \end{array}\right)\left(\begin{array}{cc} u \\ \upsilon \end{array}\right)=(E+\delta\mu)\left(\begin{array}{cc} u \\ \upsilon \end{array}\right),\end{aligned}$$ where $u$ and $\upsilon$ are $(2D+1)^3$-dimensional vectors and ${\bf H}_{ij}$ are $(2D+1)^3\times(2D+1)^3$ matrices. The matrix elements of ${\bf H}_{ij}$ can be formally expressed as $$\begin{aligned} &&{\bf H}_{11}^{[l,m,n],[l^\prime,m^\prime,n^\prime]}=-{\bf H}_{22}^{[l,m,n],[l^\prime,m^\prime,n^\prime]} =\xi_{[l,m,n]}\delta_{l,l^\prime}\delta_{m,m^\prime}\delta_{n,n^\prime},\nonumber\\ &&{\bf H}_{12}^{[l,m,n],[l^\prime,m^\prime,n^\prime]}={\bf H}_{21}^{[l,m,n],[l^\prime,m^\prime,n^\prime]} =\Delta_{[l-l^\prime,m-m^\prime,n-n^\prime]}, \label{Elements}\end{aligned}$$ where $$\begin{aligned} \xi_{[l,m,n]}=\sqrt{\left(k_x+\frac{2\pi l}{a}\right)^2+\left(k_y+\frac{2\pi m}{a}\right)^2+\left(k_z+\frac{2\pi n}{a}\right)^2}-\mu.\end{aligned}$$ Here the matrix index $[l,m,n]$ corresponds to the reciprocal lattice vector ${\bf G}_{[lmn]}=(2\pi/a)(l{\bf e}_x+m{\bf e}_y+n{\bf e}_z)$. It shows that the blocks ${\bf H}_{11}$ and ${\bf H}_{22}$ are diagonal. The off-diagonal blocks ${\bf H}_{12}$ and ${\bf H}_{21}$ carry the information of the order parameter $\Delta$ and characterize the crystal structure. For a specific value of $D$, we can write down the explicit form of the vectors $u$ and $\upsilon$ and the matrices ${\bf H}_{ij}$. Here we use $D=1$ as an example. The vectors $u$ and $\upsilon$ are $27$-dimensional can be expressed as $$\begin{aligned} \label{Basis} u=\left(\begin{array}{cc} u_{[-1,-1]} \\ u_{[-1,0]} \\ u_{[-1,1]} \\ u_{[0,-1]} \\ u_{[0,0]} \\ u_{[0,1]} \\ u_{[1,-1]} \\ u_{[1,0]} \\ u_{[1,1]}\end{array}\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \upsilon=\left(\begin{array}{cc} \upsilon_{[-1,-1]} \\ \upsilon_{[-1,0]} \\ \upsilon_{[-1,1]} \\ \upsilon_{[0,-1]} \\ \upsilon_{[0,0]} \\ \upsilon_{[0,1]} \\ \upsilon_{[1,-1]} \\ \upsilon_{[1,0]} \\ \upsilon_{[1,1]} \end{array}\right),\end{aligned}$$ where $u_{[l,m]}$ and $\upsilon_{[l,m]}$ are defined as $$\begin{aligned} u_{[l,m]}=\left(\begin{array}{cc} u_{[l,m,-1]} \\ u_{[l,m,0]} \\ u_{[l,m,1]}\end{array}\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \upsilon_{[l,m]}=\left(\begin{array}{cc} \upsilon_{[l,m,-1]} \\ \upsilon_{[l,m,0]} \\ \upsilon_{[l,m,1]} \end{array}\right).\end{aligned}$$ In this representation, the off diagonal blocks ${\bf H}_{12}$ and ${\bf H}_{21}$ are given by $$\begin{aligned} \label{MBCC} {\bf H}_{12}=\left(\begin{array}{ccccccccc} \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 & 0 & 0 & 0 \\ \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 & 0 & 0\\ 0 & \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 & 0 \\ \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 \\ 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 \\ 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 \\ 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 & 0 \\ 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 & \mbox{\boldmath{$\Delta$}}_2 \\ 0 & 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_2 & 0 & \mbox{\boldmath{$\Delta$}}_2 & \mbox{\boldmath{$\Delta$}}_1 \end{array}\right)\end{aligned}$$ for BCC structure and $$\begin{aligned} \label{MFCC} {\bf H}_{12}=\left(\begin{array}{ccccccccc} 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 \\ 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 \\ \mbox{\boldmath{$\Delta$}}_1 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & \mbox{\boldmath{$\Delta$}}_1 \\ 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 \\ 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \mbox{\boldmath{$\Delta$}}_1 & 0 & 0 & 0 & 0 \end{array}\right)\end{aligned}$$ for FCC structure, respectively. Here the blocks $\mbox{\boldmath{$\Delta$}}_1$ and $\mbox{\boldmath{$\Delta$}}_2$ are defined as $$\begin{aligned} \mbox{\boldmath{$\Delta$}}_1=\left(\begin{array}{ccc} 0 & \Delta & 0 \\ \Delta & 0 & \Delta \\ 0 & \Delta & 0\end{array}\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{\boldmath{$\Delta$}}_2=\left(\begin{array}{ccc} \Delta & 0 & 0 \\ 0 & \Delta & 0 \\ 0 & 0 & \Delta\end{array}\right).\end{aligned}$$ In principle, the eigenvalue spectrum $\{E_\lambda({\bf k})\}$ can be obtained by diagonalizing the matrix ${\bf H}$ with a size $2(2D+1)^3$. We notice that the matrix size $2(2D+1)^3$ grows dramatically with increasing cutoff $D$. Therefore, for realistic diagonalization, it is better to reduce the size of the matrix. Here we find that, with a proper rearrangement of the basis $\phi$ or after a similarity transformation, the matrix ${\bf H}$ becomes block diagonal. We have $$\begin{aligned} {\bf H}\sim\left(\begin{array}{cc} {\bf H}_+ & 0 \\ 0 & {\bf H}_- \end{array}\right),\end{aligned}$$ where size of the blocks ${\bf H}_+$ and ${\bf H}_-$ are both $(2D+1)^3$. The eigenvector $\phi$ is now defined as $$\begin{aligned} \phi=\left(\begin{array}{cc} \phi_+ \\ \phi_- \end{array}\right).\end{aligned}$$ For $D=1$, the $27$-dimensional vectors $\phi_+$ and $\phi_-$ are given by $$\begin{aligned} \phi_+=\left(\begin{array}{cc} \upsilon_{[-1,-1,-1]} \\ u_{[-1,-1,0]} \\ \upsilon_{[-1,-1,1]} \\ u_{[-1,0,-1]} \\ \upsilon_{[-1,0,0]} \\ u_{[-1,0,1]} \\ \upsilon_{[-1,1,-1]} \\ u_{[-1,1,0]} \\ \upsilon_{[-1,1,1]} \\ u_{[0,-1,-1]} \\ \upsilon_{[0,-1,0]} \\ u_{[0,-1,1]} \\ \upsilon_{[0,0,-1]} \\ u_{[0,0,0]} \\ \upsilon_{[0,0,1]}\\ u_{[0,1,-1]} \\ \upsilon_{[0,1,0]} \\ u_{[0,1,1]} \\ \upsilon_{[1,-1,-1]} \\ u_{[1,-1,0]} \\ \upsilon_{[1,-1,1]} \\ u_{[1,0,-1]} \\ \upsilon_{[1,0,0]} \\ u_{[1,0,1]} \\ \upsilon_{[1,1,-1]} \\ u_{[1,1,0]} \\ \upsilon_{[1,1,1]}\end{array}\right), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phi_-=\left(\begin{array}{cc} u_{[-1,-1,-1]} \\ \upsilon_{[-1,-1,0]} \\ u_{[-1,-1,1]} \\ \upsilon_{[-1,0,-1]} \\u_{[-1,0,0]} \\ \upsilon_{[-1,0,1]} \\ u_{[-1,1,-1]} \\ \upsilon_{[-1,1,0]} \\ u_{[-1,1,1]} \\ \upsilon_{[0,-1,-1]} \\ u_{[0,-1,0]} \\ \upsilon_{[0,-1,1]} \\ u_{[0,0,-1]} \\ \upsilon_{[0,0,0]} \\ u_{[0,0,1]}\\ \upsilon_{[0,1,-1]} \\ u_{[0,1,0]} \\ \upsilon_{[0,1,1]} \\ u_{[1,-1,-1]} \\ \upsilon_{[1,-1,0]} \\ u_{[1,-1,1]} \\ \upsilon_{[1,0,-1]} \\ u_{[1,0,0]} \\ \upsilon_{[1,0,1]} \\ u_{[1,1,-1]} \\ \upsilon_{[1,1,0]} \\ u_{[1,1,1]}\end{array}\right),\end{aligned}$$ which is just a proper rearrangement of the original basis given by (\[Basis\]). The blocks ${\bf H}_+$ and ${\bf H}_-$ are given by $$\begin{aligned} {\bf H}_\pm=\pm{\bf H}_0+{\bf H}_{12},\end{aligned}$$ where ${\bf H}_{12}$ is given by (\[Elements\]) or (\[MBCC\]) and (\[MFCC\]) for $D=1$. ${\bf H}_0$ is a diagonal matrix containing the kinetic energies $\xi_{[l,m,n]}$. We have $$\begin{aligned} {\bf H}_0^{[l,m,n],[l^\prime,m^\prime,n^\prime]} =(-1)^{l+m+n}\xi_{[l,m,n]}\delta_{l,l^\prime}\delta_{m,m^\prime}\delta_{n,n^\prime}.\end{aligned}$$ For $D=1$, we obtain $$\begin{aligned} {\bf H}_0&=&{\rm diag}(-\xi_{[-1,-1,-1]},\xi_{[-1,-1,0]},-\xi_{[-1,-1,1]},\cdots,\nonumber\\ &&\ \ \ \ \ \ \ \ \xi_{[0,0,0]},\cdots,-\xi_{[1,1,-1]},\xi_{[1,1,0]},-\xi_{[1,1,1]}).\end{aligned}$$ {width="7.8cm"} {width="7.8cm"} {width="7.8cm"} {width="7.8cm"} It is easy to show that the eigenvalue spectra of ${\bf H}_+$ and ${\bf H}_-$ are dependent: If the eigenvalue spectrum of ${\bf H}_+$ is given by $\{\varepsilon_\lambda({\bf k})\}$, the eigenvalue spectrum of ${\bf H}_-$ reads $\{-\varepsilon_\lambda({\bf k})\}$. Therefore, we only need to diagonalize the matrix ${\bf H}_+$ or ${\bf H}_-$ which has a size $(2D+1)^3$. Once the eigenvalue spectrum of ${\bf H}_+$ is known, the eigenvalue spectrum of Hamiltonian matrix ${\cal H}_{\Delta,\delta\mu}$ is given by $$\begin{aligned} \{E_\lambda({\bf k})\}=\{\varepsilon_\lambda({\bf k})-\delta\mu\}\cup\{-\varepsilon_\lambda({\bf k})-\delta\mu\}.\end{aligned}$$ Therefore, we can in principle calculate the potential landscape $\delta\Omega(\Delta,q)$. The solution $(\Delta,q)$ of a specific crystal structure corresponds to the global minimum of the potential landscape. Computation and Results ======================= To achieve satisfying convergence we normally need a large cutoff $D$. However, the matrix size $(2D+1)^3$ and hence the computing time and cost grow dramatically with increasing $D$. The cutoff $D$ needed for convergence can be roughly estimated from the maximum momentum $k_{\rm max}$ in the matrix, $$k_{\rm max}=(2D+1)\frac{\pi}{a}$$ The value of $k_{\rm max}$ can be estimated from the LO state. The calculation of the LO state is much easier than 3D structures because the matrix size becomes $2D+1$. The details of the calculation of the LO state are presented in Appendix B. For $\Delta_0\sim100$ MeV we need $k_{\rm max}\simeq5$GeV [@note4]. Since we are interested in the region $\delta\mu/\Delta_0\in[0.7,0.8]$ and the optimal pair momentum is $q\sim\delta\mu$, we estimate $D\sim35$ for BCC and $D\sim60$ for FCC. These huge matrix sizes are beyond the capability of our current computing facilities. On the other hand, even though a supercomputer may be able to diagonalize these huge matrices, the computing time and cost are still enormous, which makes the calculation infeasible. ![(Color online) Comparison of the grand potentials of various phases: BCS (black solid), FF (blue dotted), LO (green dash-dotted), and BCC (red dashed). The horizontal axis has been “calibrated" by using the quantity $(\delta\mu-\delta\mu_2)/\Delta_0$. \[fig3\]](Omega.eps){width="8.7cm"} Since we are interested in the grand potential $\delta\Omega$ rather than the band structure (the eigenvalues), we can neglect a small amount of the off-diagonal couplings $\Delta$ in the matrix ${\bf H}_+$. By doing so, the huge matrix ${\bf H}_+$ can be decomposed into a number of blocks with size $(2d+1)^3$. For symmetry reason, we set the centers of these blocks at the reciprocal lattice vectors $${\bf G}_{[n_x,n_y,n_z]}=(2d+1)\frac{2\pi}{a}(n_x{\bf e}_x+n_y{\bf e}_y+n_z{\bf e}_z)$$ with $n_x,n_y,n_z\in\mathbb{Z}$. With increasing $d$, the grand potential converges to the result from exact diagonalization. Good convergence is normally reached at some value $d=d_0$. The details of our computational scheme are presented in Appendix C. If the block size $(2d_0+1)^3$ is within our computing capability, the calculation becomes feasible. Fortunately, we find that this computational scheme works for the BCC structure. At present, we are not able to perform a calculation for the FCC structure, since the value of $d_0$ needed for convergence is much larger. Note that the computing cost is still very large even though we have employed this effective computational scheme. We have performed calculations of the BCC structure for $\Delta_0=60,80,100$ MeV [@note2] at $\mu=400$ MeV [@note5]. For different values of $\Delta_0$, the results are almost the same in terms of the quantity $(\delta\mu-\delta\mu_2)/\Delta_0$. Therefore, we anticipate that our results can be appropriately extrapolated to the weak coupling limit $\Delta_0\rightarrow0$. In the following, we shall present the result for $\Delta_0=100$ MeV. For a given value of $\delta\mu/\Delta_0$, we calculate the potential curve $\delta\Omega(\Delta)$ at various values of $q$ and search for the optimal pair momentum and the minimum of the potential landscape. The potential curves at the optimal pair momenta for several values of $\delta\mu/\Delta_0$ are shown in Fig. \[fig2\]. With increasing value of $\delta\mu/\Delta_0$, the potential minimum gets shallower. At a critical value $\delta\mu_*-\delta\mu_2\simeq0.03\Delta_0$, the potential minimum approaches zero and a first-order phase transition to the normal state occurs. The comparison of the grand potentials of various phases are shown in Fig. \[fig3\]. For the LO state, its phase transition to the normal state occur almost at the same point as the FF state, $\delta\mu_2\simeq0.8\Delta_0$. At $\delta\mu=\delta\mu_2$, the grand potential of the BCC structure is negative, which indicates that the BCC structure is energetically favored around the FF-normal transition point. Well below the FF-normal transition point, the BCC state has higher grand potential than the LO state and hence is not favored. Near the BCS-LO transition, the solitionic state becomes favored [@Buballa2009]. However, this does not change our qualitative conclusion. Our result is qualitatively consistent with the GL analysis [@Bowers2002]. However, the quantitative difference is significant: The GL analysis predicts a strong first-order phase transition and a large upper critical field [@Bowers2002], while our result shows a weak first-order phase transition at which $\Delta\simeq0.1\Delta_0$. On the other hand, our result is quantitatively compatible with the quasiclassical equation approach [@Combescot2004; @Combescot2005], where it shows that the BCC structure is preferred in a narrow window around $\delta\mu=\delta\mu_2$ at zero temperature [@note6]. Therefore, the GL analysis up to the order $O(\Delta^6)$ may not be quantitatively sufficient. We notice that the LO state already shows the limitation of the GL analysis: While the GL analysis predicts a second-order phase transition, exact calculation shows a first-order phase transition [@Buballa2009] (see also Appendix B). In the future, it is necessary to study the higher-order expansions and the convergence property of the GL series, which would help to quantitatively improve the GL predictions. Summary ======= In summary, we have performed an solid-state-like calculation of the ground-state energy of a 3D structure in crystalline color superconductivity . We proposed a computational scheme to overcome the difficulties in diagonalizing matrices of huge sizes. Our numerical results show that the BCC structure is preferred in a small window around the conventional FF-normal phase transition point, which indicates that the higher-order terms in the GL approach are rather important. In the future it would be possible to perform a calculation for the FCC structure with stronger computing facilities and/or with better method of matrix diagonalization. This solid-state-like approach can also be applied to study the crystalline structures of the three-flavor color-superconducting quark matter [@Three-flavor] and the inhomogeneous chiral condensate [@Buballa2014]. *Acknowledgments* — We thank Profs. Mark Alford, Joseph Carlson, Roberto Casalbuoni, Stefano Gandolfi, Hui Hu, Xu-Guang Huang, Massimo Mannarelli, Sanjay Reddy, Armen Sedrakian, and Shiwei Zhang for useful discussions and comments. The work of G. C. and P. Z. was supported by the NSFC under Grant No. 11335005 and the MOST under Grant Nos. 2013CB922000 and 2014CB845400. The work of L. H. was supported by the US Department of Energy Topical Collaboration “Neutrinos and Nucleosynthesis in Hot and Dense Matter". L. H. also acknowledges the support from Frankfurt Institute for Advanced Studies in the early stage of this work. The numerical calculations were performed at Tsinghua National Laboratory for Information Science and Technology. Ginzburg-Landau Theory: Importance of Higher Order Expansions ============================================================= In the Ginzburg-Landau (GL) theory of crystalline color superconductors at zero temperature and at weak coupling, the grand potential measured with respect to the normal state, $\delta\Omega=\Omega-\Omega_{\rm N}$, is expanded as [@Bowers2002] $$\begin{aligned} \frac{\delta\Omega(\Delta)}{{\cal N}_{\rm F}}=P\alpha\Delta^2+\frac{1}{2}\beta\Delta^4 +\frac{1}{3}\gamma\Delta^6+\frac{1}{4}\eta\Delta^8+O(\Delta^{10}),\end{aligned}$$ where ${\cal N}_{\rm F}$ is the density of state at the Fermi surface. The coefficient $\alpha$ is universal for all crystal structures and is given by [@Bowers2002] $$\begin{aligned} \alpha=-1+\frac{\delta\mu}{2q}\ln\frac{q+\delta\mu}{q-\delta\mu}-\frac{1}{2}\ln\frac{\Delta_0^2}{4(q^2-\delta\mu^2)}.\end{aligned}$$ {width="7.8cm"} {width="7.8cm"} {width="7.8cm"} {width="7.8cm"} \ Let us consider the vicinity of the conventional LOFF-normal transition point $\delta\mu=\delta\mu_2$, where we have $$\begin{aligned} \frac{\delta\mu_2}{\Delta_0}=0.7544,\ \ \ \ \ \frac{q}{\delta\mu_2}=1.1997.\end{aligned}$$ At the pair momentum $q=1.1997\delta\mu$, we obtain $$\begin{aligned} \alpha=\ln\frac{\delta\mu}{\Delta_0}-\ln\frac{\delta\mu_2}{\Delta_0}=\ln\frac{\delta\mu}{\delta\mu_2}.\end{aligned}$$ For convenience, we make the GL potential dimensionless by using the variables $\delta\bar{\Omega}=\delta\Omega/(N_0\delta\mu^2)$, $\bar{\Delta}=\Delta/\delta\mu$, $\bar{\beta}=\beta\delta\mu^2$, $\bar{\gamma}=\gamma\delta\mu^4$, and $\bar{\eta}=\eta\delta\mu^6$. We have $$\begin{aligned} \delta\bar{\Omega}=P\alpha\bar{\Delta}^2+\frac{1}{2}\bar{\beta}\bar{\Delta}^4+\frac{1}{3}\bar{\gamma}\bar{\Delta}^6 +\frac{1}{4}\bar{\eta}\bar{\Delta}^8+O(\bar{\Delta}^{10}).\end{aligned}$$ The GL coefficients $\bar{\beta}$ and $\bar{\gamma}$ for a number of crystalline structures were first calculated by Bowers and Rajagopal [@Bowers2002]. The predictions for the nature of the phase transitions were normally based on the GL potential up to the sixth order ($\Delta^6$). To our knowledge, the higher order GL coefficients have never been calculated. Here we show that the higher-order GL expansions are important for the prediction of the phase transition. To be specific, let us consider the BCC structure. Its GL coefficients $\bar{\beta}$ and $\bar{\gamma}$ have been evaluated as [@Bowers2002] $$\begin{aligned} \bar{\beta}=-31.466,\ \ \ \ \ \bar{\gamma}=19.711.\end{aligned}$$ Since $\bar{\beta}<0$, the phase transition to the normal state should be of first order. If we employ the GL potential up to the sixth order, we predict a strong first-order phase transition at $\delta\mu=\delta\mu_*=3.625\Delta_0$. Let us turn on the eighth-order term and study how the size of the coefficient $\bar{\eta}$ influences the quantitative prediction of the phase transition. In Fig. \[GLBCC\], we show the GL potential curves for two different values of $\bar{\eta}$ at $\delta\mu=\delta\mu_2$. For vanishing $\bar{\eta}$, the potential curve develops a deep minimum $\delta\bar{\Omega}_{\rm min}\simeq-13.4$ at $\Delta\simeq0.95\Delta_0$, which indicates a strong first-order phase transition at $\delta\mu=\delta\mu_*\gg\delta\mu_2$. However, for a large value $\bar{\eta}=1000$, we find a shallow minimum $\delta\bar{\Omega}_{\rm min}\simeq-0.21$ located at $\Delta\simeq0.31\Delta_0$. In Fig. \[GLBCC2\], we show the GL potential curves at the first-order phase transition point $\delta\mu=\delta\mu_*$. For $\bar{\eta}=0$ we find a strong first-order phase transition at $\delta\mu=\delta\mu_*=3.625\Delta_0$, where the minima located at $\Delta=0$ and $\Delta=0.83\Delta_0$ become degenerate. For $\bar{\eta}=1000$, however, we observe a much weaker first-order phase transition at $\delta\mu=\delta\mu_*=0.951\Delta_0$, where the degenerate minima are located at $\Delta=0$ and $\Delta=0.28\Delta_0$. These results clearly show that, for larger $\bar{\eta}$, the first-order phase transition becomes weaker. For $\bar{\eta}\rightarrow+\infty$, we expect that $\delta\mu_*\rightarrow\delta\mu_2=0.754\Delta_0$. On the other hand, if $\bar{\eta}$ is small or even negative, then the next order $\Delta^{10}$ would become important. Calculation of the LO State =========================== The order parameter of the LO state is given by $$\begin{aligned} \Delta(z)=2\Delta\cos(2qz).\end{aligned}$$ It is periodic along the $z$ direction with the periodicity $a=\pi/q$. So it can be decomposed into a discrete set of Fourier components, $$\begin{aligned} \Delta(z)=\sum_{n=-\infty}^\infty\Delta_ne^{2niqz},\end{aligned}$$ The Fourier component $\Delta_n$ is given by $$\begin{aligned} \Delta_n=\frac{1}{a}\int_0^a dz\Delta(z)e^{-2niqz}=\Delta\left(\delta_{n,1}+\delta_{n,-1}\right).\end{aligned}$$ The matrix equation takes a similar form as the 3D structures. We have $$\begin{aligned} \sum_{n^\prime}({\cal H}_{\Delta})_{n,n^\prime}({\bf k})\phi_{n^\prime}({\bf k}) =\left[E_\lambda({\bf k})+\delta\mu\right] \phi_n({\bf k}),\end{aligned}$$ where the Hamiltonian matrix $({\cal H}_{\Delta})_{n,n^\prime}({\bf k})$ reads $$\begin{aligned} ({\cal H}_{\Delta})_{n,n^\prime}({\bf k})=\left(\begin{array}{cc} \xi_n\delta_{n,n^\prime} & \Delta_{n-n^\prime} \\ \Delta_{n-n^\prime} & -\xi_n\delta_{n,n^\prime} \end{array}\right)\end{aligned}$$ with $$\begin{aligned} \xi_n=\sqrt{{\bf k}_\perp^2+(k_z+2nq)^2}-\mu.\end{aligned}$$ We notice that only the motion in the $z$ direction becomes quantized. The BZ for $k_z$ can be defined as $-\pi/a<k_z<\pi/a$ or $-q<k_z<q$. The eigenstate $\phi_n$ includes two components $u_n$ and ${\bf \upsilon}_n$. We have $$\begin{aligned} \phi_n({\bf k})=\left(\begin{array}{cc} u_n({\bf k}) \\ \upsilon_n({\bf k}) \end{array}\right).\end{aligned}$$ If $\varepsilon$ is an eigenvalue of ${\cal H}_{\Delta}$, $-\varepsilon$ must be another eigenvalue. Therefore, replacing the $\delta\mu$ by $-\delta\mu$ amounts to a replacement of the eigenvalue spectrum $\{E_\lambda({\bf k})\}$ by $\{-E_\lambda({\bf k})\}$. After a truncation $-D<n<D$, we obtain a finite matrix equation $$\begin{aligned} {\bf H}\left(\begin{array}{cc} u \\ \upsilon \end{array}\right)=\left(\begin{array}{cc} {\bf H}_{11} & {\bf H}_{12} \\ {\bf H}_{21} & {\bf H}_{22} \end{array}\right)\left(\begin{array}{cc} u \\ \upsilon \end{array}\right) =(E+\delta\mu)\left(\begin{array}{cc} u \\ \upsilon \end{array}\right),\end{aligned}$$ where $u$ and $\upsilon$ are $(2D+1)$-dimensional vectors and ${\bf H}_{ij}$ are $(2D+1)\times(2D+1)$ matrices. For a specific value of $D$, we can write down the explicit form of the vectors $u$ and $\upsilon$ and the matrices ${\bf H}_{ij}$. Here we use $D=2$ as an example. The vectors $u$ and $\upsilon$ are $5$-dimensional can be expressed as $$\begin{aligned} u=\left(\begin{array}{cc} u_{-2} \\ u_{-1} \\ u_{0} \\ u_{1} \\ u_{2}\end{array}\right), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \upsilon=\left(\begin{array}{cc} \upsilon_{-2} \\ \upsilon_{-1} \\ \upsilon_{0} \\ \upsilon_{1} \\ \upsilon_2 \end{array}\right).\end{aligned}$$ The matrices ${\bf H}_{ij}$ can be explicitly written as $$\begin{aligned} &&{\bf H}_{11}=-{\bf H}_{22}=\left(\begin{array}{ccccc} \xi_{-2} & 0 & 0 & 0 & 0 \\ 0 & \xi_{-1} & 0 & 0 & 0 \\ 0 & 0 & \xi_0 & 0 & 0 \\ 0 & 0 & 0 & \xi_1 & 0 \\ 0 & 0 & 0 & 0 & \xi_2\end{array}\right), \nonumber\\ &&{\bf H}_{12}={\bf H}_{21}=\left(\begin{array}{ccccc} 0 & \Delta & 0 & 0 & 0 \\ \Delta & 0 & \Delta & 0 & 0 \\ 0 & \Delta & 0 & \Delta & 0 \\ 0 & 0 & \Delta & 0 & \Delta \\ 0 & 0 & 0 & \Delta & 0 \end{array}\right).\end{aligned}$$ The eigenvalue spectrum $\{E_\lambda({\bf k})\}$ can be obtained by diagonalizing the matrix ${\bf H}$ with a size $2(2D+1)$. With a proper rearrangement of the basis $\phi$ or a similarity transformation, we have $$\begin{aligned} {\bf H}\sim\left(\begin{array}{cc} {\bf H}_+ & 0 \\ 0 & {\bf H}_- \end{array}\right),\end{aligned}$$ where the sizes of ${\bf H}_+$ and ${\bf H}_-$ are both $2D+1$. The basis $\phi$ is now defined as $$\begin{aligned} \phi=\left(\begin{array}{cc} \phi_+ \\ \phi_- \end{array}\right).\end{aligned}$$ For $D=2$, the $5$-dimensional vectors $\phi_+$ and $\phi_-$ are given by $$\begin{aligned} \phi_+=\left(\begin{array}{cc} u_{-2} \\ \upsilon_{-1} \\ u_{0} \\ \upsilon_{1} \\ u_{2}\end{array}\right), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \phi_-=\left(\begin{array}{cc} \upsilon_{-2} \\ u_{-1} \\ \upsilon_{0} \\ u_{1} \\ \upsilon_2 \end{array}\right).\end{aligned}$$ The blocks ${\bf H}_+$ and ${\bf H}_-$ can be expressed as $$\begin{aligned} {\bf H}_\pm=\pm{\bf H}_0+{\bf H}_{12}.\end{aligned}$$ ${\bf H}_0$ is a diagonal matrix containing the kinetic energies. We have $$\begin{aligned} ({\bf H}_0)_{n,n^\prime}=(-1)^{n}\xi_{n}\delta_{n,n^\prime}.\end{aligned}$$ The eigenvalue spectra of ${\bf H}_+$ and ${\bf H}_-$ are dependent: If the eigenvalue spectrum of ${\bf H}_+$ is given by $\{\varepsilon_\lambda({\bf k})\}$, the eigenvalue spectrum of ${\bf H}_-$ reads $\{-\varepsilon_\lambda({\bf k})\}$. Therefore, we need only to diagonalize the matrix ${\bf H}_+$ or ${\bf H}_-$ which has a dimension $2D+1$. Once the eigenvalue spectrum of ${\bf H}_+$ is known, the eigenvalue spectrum of Hamiltonian matrix ${\cal H}_{\Delta,\delta\mu}$ is given by $\{E({\bf k})\}=\{\varepsilon_\lambda({\bf k})-\delta\mu\}\cup\{-\varepsilon_\lambda({\bf k})-\delta\mu\}$. {width="8cm"} {width="8cm"} The thermodynamic potential of the LO state at zero temperature can be expressed as $$\begin{aligned} \Omega_{\rm LO}=\frac{\Delta^2}{2H}-2\int\frac{d^2{\bf k}_\perp}{(2\pi)^2}\int_{\rm BZ}\frac{dk_z}{2\pi}\sum_\lambda|E_\lambda({\bf k}_\perp, k_z)|.\end{aligned}$$ Similar Pauli-Villas-like regularization scheme should be applied finally. In Fig. \[LO\] (a), we show the grand potential of the LO state for $\Delta_0=80$ MeV. The grand potential for the self-consistent 1D modulation for $\Delta_0=80$ MeV was also reported in [@Buballa2009]. We find that the results for the LO state and the self-consistent 1D modulation agrees with each other near the phase transition to the normal state. Near the BCS-LO transition point, the self-consistent 1D modulation has lower grand potential than the LO state. It was shown in [@Buballa2009] that the self-consistent 1D modulation forms a soliton lattice structure near the lower critical field, which lowers the grand potential of the system. Near the upper critical field the gap function becomes sinusoidal, and therefore the grand potentials of the LO state and the 1D modulation agree with each other. We notice that the phase transition from the LO state to the normal state is of first order, which is in contradiction to the prediction from the GL analysis. To understand the reason, we show in Fig. \[LO\] (b) the potential curve at $\delta\mu=0.775\Delta_0$ and at the optimal pair momentum $q=1.1613\delta\mu$. We find that the potential curve has two minima: a shallow minimum at $\Delta\simeq0.12\Delta_0$ and a deep minimum at $\Delta\simeq0.44\Delta_0$. Obviously, the shallow minimum is responsible for the GL theory which predicts a second-order phase transition. However, the deep global minimum, which cannot be captured by the GL theory up to the order $\Delta^6$, is responsible for the real first-order phase transition. Therefore, the LO state already shows the importance of the higher-order expansions in the GL theory. Calculation of the Grand Potential: Small Block Method ====================================================== The key problem in the numerical calculation is the diagonalization of the matrix ${\bf H}_+$ or ${\bf H}_-$ and obtaining all the eigenvalues. For BCC and FCC structures, we use a symmetrical truncation $-D<l,m,n<D$ with a large cutoff $D\in \mathbb{Z}^+$. However, the matrix size grows dramatically with increasing cutoff $D$, which makes the calculation infeasible because of not only the computing capability of current computing facilities but also the computing time and cost. Notice that we need to diagonalize the matrix ${\bf H}_+$ for various values of the momentum ${\bf k}$ in the BZ, the gap parameter $\Delta$, and the pair momentum $q$. We first estimate the size of $D$ needed for the convergence of the grand potential $\delta\Omega$. The matrix size $(2D+1)^3$ and hence the computing time and cost grow dramatically with increasing $D$. The cutoff $D$ is related to the maximum momentum $k_{\rm max}$ in each direction ($x$, $y$, and $z$). We have $$\begin{aligned} k_{\rm max}=(2D+1)\frac{\pi}{a}.\end{aligned}$$ This maximum momentum can be roughly estimated from the calculation of the LO state. For the LO state, the matrix size becomes $2D+1$ and exact diagonalization is possible. The regime of $\delta\mu$ we are interested in is $\delta\mu/\Delta_0\in[0.7-0.8]$ and the optimal pair momentum is located at $q\simeq\delta\mu$. From the calculation of the LO state at $\Delta_0\sim100$ MeV, we find that $k_{\rm max}$ must reach at least $5$GeV for convergence. Notice that we have $k_{\rm max}=(2D+1)q$ for BCC and $\sqrt{3}k_{\rm max}=(2D+1)q$ for FCC. Therefore, the cutoff $D$ for BCC can be estimated as $D\sim35$, which corresponds to a matrix size $\sim3\times10^5$. For FCC, the cutoff is even larger because of the factor $\sqrt{3}$. We have $D\sim60$ for FCC, which corresponds to a matrix size $\sim1.5\times10^6$. Notice that this is only a naive estimation. In practice, the cutoff needed for convergence may be smaller or larger. Exact diagonalization of such huge matrices to obtain all the eigenvalues are impossible with our current computing facility. We therefore need a feasible scheme to evaluate the grand potential $\delta\Omega$. Notice that decreasing the value of $\Delta_0$ does not reduce the size of the matrices. In this case, even though $k_{\rm max}$ becomes smaller, the pair momentum $q$ also gets smaller. Let us call an off-diagonal element $\Delta$ in ${\bf H}_+$ or ${\bf H}_-$ a “coupling". Because our goal is to evaluate the grand potential $\delta\Omega$ rather than to know exactly all the band dispersions (eigenvalues), we may neglect a small amount of couplings to lower the size of the matrices. By neglecting this small amount of couplings, the huge matrix ${\bf H}_+$ becomes block diagonal with each block having a much smaller size. In general, we expect that the omission of a small amount of couplings $\Delta$ induces only a perturbation to the grand potential $\delta\Omega$. We shall call this scheme small block method (SBM). {width="8.4cm"} {width="8.4cm"} To be specific, the size of the small blocks in our calculation is $(2d+1)^3$ with $d\in\mathbb{Z}^+$. In general, we have $d<D$. For symmetry reason, we require that the centers of these blocks are located at the reciprocal lattice vectors $$\begin{aligned} {\bf G}_{[n_x,n_y,n_z]}=(2d+1)\frac{2\pi}{a}(n_x{\bf e}_x+n_y{\bf e}_y+n_z{\bf e}_z)\end{aligned}$$ with $n_x,n_y,n_z\in\mathbb{Z}$. This scheme makes the SBM feasible even though $(2D+1)^3$ is not divisible by $(2d+1)^3$. In practice, we first choose a large cutoff $D$ which is sufficient for convergence. By increasing the value of $d$, we find that the grand potential $\delta\Omega$ finally converges. In practice, if the grand potentials evaluated at several values of $d$, i.e., $d_0-k$, $d_0-k+1$, ... , and $d_0$ ($k\in\mathbb{Z}^+$), are very close to each other, we identify that the grand potential converges to its precise value from exact diagonalization. At the converging value $d=d_0$, the block size $(2d_0+1)^3$ is normally much smaller than the total size $(2D+1)^3$. This scheme makes the calculation feasible and also saves a lot of computing time and cost. The matrices for the 3D structures are huge and cannot be written down here. For the sake of simplicity, let us use the LO state as a toy example for the SBM. In this case, the matrix size and the block size are $2D+1$ and $2d+1$, respectively. The centers of the blocks are located at the reciprocal lattice vectors $(2d+1)2qn_z{\bf e}_z$ with $n_z\in\mathbb{Z}$. For $D=10$, the matrix ${\bf H}_+$ reads $$\begin{aligned} \left(\begin{array}{ccccccccccccccccccccc} \varepsilon_{+10} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \Delta & \varepsilon_{+9} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \Delta & \varepsilon_{+8} & \color{red}{\bf\Delta} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \color{red}{\bf \Delta} & \varepsilon_{+7} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \Delta & \varepsilon_{+6} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+5} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+4} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+3} & \color{red}{\bf\Delta} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf\Delta} & \varepsilon_{+2} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+1} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_0 & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-1} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-2} & \color{red}{\bf\Delta} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf\Delta} & \varepsilon_{-3} & \Delta & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-4} & \Delta & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-5} & \Delta & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-6} & \Delta & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-7} & \color{red}{\bf\Delta} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf\Delta} & \varepsilon_{-8} & \Delta & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-9} & \Delta\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-10} \end{array} \right),\end{aligned}$$ where $\varepsilon_n=(-1)^n\left[\sqrt{{\bf k}_\perp^2 + (k_z +2nq)^2}- \mu\right]$. If we take $d=2$, we neglect the couplings $\Delta$ in red. In this case, the matrix ${\bf H}_+$ is approximated as $$\begin{aligned} \left( \begin{array}{ccccccccccccccccccccc} \varepsilon_{+10} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \Delta & \varepsilon_{+9} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & \Delta & \varepsilon_{+8} & \color{red}{\bf0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \color{red}{\bf0} & \varepsilon_{+7} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \Delta & \varepsilon_{+6} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+5} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+4} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+3} & \color{red}{\bf0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf0} & \varepsilon_{+2} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{+1} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_0 & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-1} & \Delta & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-2} & \color{red}{\bf0} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf0} & \varepsilon_{-3} & \Delta & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-4} & \Delta & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-5} & \Delta & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-6} & \Delta & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-7} & \color{red}{\bf0} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \color{red}{\bf0} & \varepsilon_{-8} & \Delta & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-9} & \Delta\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \Delta & \varepsilon_{-10} \end{array} \right).\end{aligned}$$ Therefore, by neglecting a small amount of couplings, we have made the large matrix ${\bf H}_+$ block-diagonal. Notice that this is only a toy example for the SBM. In practice, $D=10$ and $d=2$ is obviously not enough for convergence. For the LO state, exact diagonalization of the matrices at $q\simeq\delta\mu$ is quite easy because the size of the matrices is $2D+1$. We can therefore check the error induced by the SBM. The relative error induced by the SBM can be defined as $$\begin{aligned} R=\frac{|\delta\Omega_{\rm SBM}-\delta\Omega_{\rm EX}|}{\delta\Omega_{\rm EX}},\end{aligned}$$ where $\delta\Omega_{\rm SBM}$ and $\delta\Omega_{\rm EX}$ are the grand potentials obtained from the SBM and exact diagonalization, respectively. In Fig. \[ERROR\] (a), we show a numerical example of the relative error for the LO state at $\delta\mu/\Delta_0=0.77$ and $q/\delta\mu=1.16$. In the calculations, we use $D=50$ and $d=20$. We find that the relative error is very small, generally of order $O(10^{-3})$. The slightly larger error around $\Delta/\Delta_0=0.5$ is due to the fact that $\delta\Omega$ itself is very small there. For the BCC structure, we are not able to check the relative error at $q\simeq\delta\mu$ because it is impossible to exactly diagonalize the matrices with a huge size $(2D+1)^3$. However, we can check the $d$ dependence of the grand potential. For pair momentum around $q\simeq\delta\mu$, we choose a sufficiently large cutoff $D$ and increase the value of $d$. We evaluate the grand potentials for various values of $d$ (i.e., $d_0-k$, $d_0-k+1$, ... , and $d_0$). If they are very close to each other, we identify that the grand potential converges. Then the grand potential $\delta\Omega$ can be evaluated at $d=d_0$. In Fig. \[ERROR\], we show the $d$ dependence of the grand potential of the BCC structure at $\delta\mu/\Delta_0=0.75$, $q/\delta\mu=1.07$, and $\Delta/\Delta_0=0.167$. In the calculation we choose $D=50$ which is sufficiently large to guarantee the convergence at large ${\bf G}$. We find that for BCC structure, $d_0$ is normally within the range $10<d_0<15$, which is feasible for a calculation. For FCC structure, we do not find a satisfying convergence at these small values of $d$. [99]{}

ArXiv

--- abstract: 'In order to find counterparts of the detected objects in the [*AKARI*]{} Deep Field South (ADFS) in all available wavelengths, we searched public databases (NED, SIMBAD and others). Checking 500 sources brighter than 0.0482 Jy in the [*AKARI*]{} Wide-S band, we found 114 sources with possible counterparts, among which 78 were known galaxies. We present these sources as well as our first attempt to construct spectral energy distributions (SEDs) for the most secure and most interesting sources among them, taking into account all the known data together with the [*AKARI*]{} measurements in four bands.' author: - 'Katarzyna Ma[ł]{}ek$^1$, Agnieszka Pollo$^{2,3}$, Mai Shirahata$^4$, Shuji Matsuura$^{4}$, Mitsunobu Kawada$^5$, and Tsutomu T. Takeuchi$^5$' title: 'Identifications and SEDs of the detected sources from the [*AKARI*]{} Deep Field South' --- Introduction ============ The [*AKARI*]{} Deep Field South (ADFS) is one of the deep fields close to the Ecliptic Pole. The unique property of the ADFS is that the cirrus emission density is the lowest in the whole sky, i.e., the field is the most ideal sky area for far-infrared (FIR) extragalactic observations. Very deep imaging data were obtained down to $\sim 20$ mJy at $90\;\mu$m [for details, see @shirahata2009]. Catalog {#sec:catalog} ======= We cross-correlated the ADFS point source catalog (based on $90\;\mu$m) with other known and publicly available databases, mainly the SIMBAD and NED. For 500 sources brighter than 0.0482 Jy, we searched for their counterparts in other wavelengths within the radius of $40''$. In total, 110 counterparts for 114 ADFS sources were found. As shown in Figure \[fig:ddist\], the angular distance between the ADFS source and its counterpart is in most cases smaller than $20''$. It is plausible that the more distant identifications are caused by the contamination. In particular, all the three stars in the sample are most probably falsely identified because of the contamination (M. Fukagawa, private communication). Positional scatter map, shown in Figure \[fig:scatter\], displays a small but systematic bias of $\sim 4 ''$ in declination of the ADFS positions with respect to counterparts. We revealed that most of the detected bright sources are galaxies, and very few stars, quasars, or AGNs were found. As shown in Figure \[fig:zdist\], most of the identified objects are nearby galaxies at $z<0.1$, a large part of them belonging to a cluster DC 0428-53 at $z \sim 0.04$. The statistics of identified ADFS sources is presented in Table 1. [llc]{} Galaxies & & 78\ & Galaxy & 37\ & Galaxy in cluster of galaxies & 33\ & Pair or interacting galaxies & 4\ & Low surface brightness galaxy & 2\ & Seyfert 1 & 1\ & Starburst & 1\ Star & & 3\ Quasar & & 1\ X-ray source & & 3\ IR sources & & 24\ Spectral energy distributions ============================= Spectral energy distributions (SEDs) give first important clue to the physics of radiation of the sources. The deep image at the [*AKARI*]{} filter bands has significantly improved our understanding of the nature of the FIR emission of various sources and allows us to update the models of interstellar dust emission. We show six representative SEDs of galaxies with known redshifts in Figure \[fig:sed\]. We tried to fit four models of dust emission with the SEDs. First we tried a modified black body ($\nu^\gamma B_\nu(T)$ with $\gamma = 1.5$) to the dust emission part, and a black body to the stellar emission part in the galaxy SEDs ($\nu= 10^{13} \mbox{--} 10^{14}$ Hz). Since these galaxies are evolved, a single black body gives a poor fit to the observed SEDs for some galaxies. Using a more sophisticated stellar population synthesis model with realistic star formation history will be our next step. For dust emission, we then used models of @dale2002 and @li2001. These more refined models succeeded in reproducing the MIR part of the dust emission. By these fittings, we can calculate the mass and temperature of dust, as well as the PAH contribution to the total dust amount. NGC 1705 ======== One of the most interesting objects we found in the ADFS is a nearby dwarf starburst galaxy NGC 1705. This galaxy locates at a distance of $5.1\pm 0.6$ Mpc. It has a relatively low metallicity of $0.35\;Z_\odot$, and the star formation rate is estimated to be $0.3\;M_\odot \mbox{yr}^{-1}$. This galaxy is known to harbor the richest super star cluster (SSC) ever found [@cannon2006 and references therein]. The most striking feature of NGC 1705 is that it has completely hidden star formation only seen in the FIR, which does not correspond to the SSC observed at the optical. The [*AKARI*]{} data, as well as [*Spitzer*]{} and [*IRAS*]{} measurements enable more detailed studies on the dust emission, hidden star formation, ultraviolet radiation field strength and ISM physics of this galaxy. We thank Misato Fukagawa for sending the information about Vega-like star candidates. This work has been supported (in part) by the Polish Astroparticle Physics Network. AP was financed by the research grant of the Polish Ministry of Science PBZ/MNiSW/07/2006/34A. TTT has been supported by Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology, and the Grant-in-Aid for the Scientific Research Fund (20740105) commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. TTT has been partially supported from the Grand-in-Aid for the Global COE Program “Quest for Fundamental Principles in the Universe: from Particles to the Solar System and the Cosmos” from the MEXT. This research has made use of the NASA/IPAC Extragalactic Database (NED), operated by the Jet Propulsion Laboratory at Caltech, under contract with the NASA and the SIMBAD database, operated at CDS, Strasbourg, France Cannon, J. M., et al. 2006, , 647, 293 Dale, D. A., & Helou, G. 2002 , 576, 159 Li, A., & Draine, B. T. 2001, , 554, 778 Shirahata, M., Matsuura, S., Kawada, M., et al. 2009, this volume

ArXiv

--- abstract: 'In random networks decorated with Ising spins, an increase of the density of frustrations reduces the transition temperature of the spin-glass ordering. This result is in contradiction to the Bethe theory. Here we investigate if this effect depends on the small-world property of the network. The results on the specific heat and the spin susceptibility indicate that the effect appears also in spatial networks.' author: - 'Anna Mańka-Krasoń and Krzysztof Ku[ł]{}akowski' title: Frustration and collectivity in spatial networks --- [*PACS numbers:*]{} 75.30.Kz, 64.60.aq, 05.10.Ln\ [*Keywords:*]{} spatial networks; spin-glass; Introduction ============ A random network is an archetypal example of a complex system [@dgm]. If we decorate the network nodes with some additional variables, the problem can be mapped to several applications. In the simplest case, these variables are two-valued; these can be sex or opinion (yes or no) in social networks, states ON and OFF in genetic networks, ’sell’ and ’buy’ in trade networks and so on. Information on stationary states of one such system can be useful for the whole class of problems in various areas of science. The subject of this text is the antiferromagnetic network, where the variables are Ising spins $S_i=\pm 1/2$. As it was discussed in [@dgm], the ground state problem of this network can be mapped onto the MAX-CUT problem, which belongs to the class of NP-complete optimization problems. Also, the state of the antiferromagnetic network in a weak magnetic field gives an information on the minimal vertex cover of the network, which is another famous NP-complete problem [@dgm]. Further, in the ground state of the antiferromagnetic network all neighboring spins should be antiparallel, i.e. their product should be equal to -1. This can be seen as an equivalent to the problem of satisfiability of $K$ conditions imposed on $N$ variables, where $N$ is the number of nodes and $K$ is the number of links. The problem of satisfiability is known also to be NP-complete [@gar]. Here we are particularly interested in an ifluence of the network topology on the collective magnetic state of the Ising network. The topology is to be characterized by the clustering coefficient $C$, which is a measure of the density of triads of linked nodes in the network. In antiferromagnetic systems, these triads contribute to the ground state energy, because three neighboring spins of a triad cannot be antiparallel simultaneously to each other. This effect is known as spin frustration. When the frustration is combined with the topological disorder of a random network, the ground state of the system is expected to be the spin-glass state, a most mysterious magnetic phase which remains under dispute for more than thirty years [@by; @fh; @ns]. These facts suggest that a search on random network with Ising spins and antiferromagnetic interaction can be worthwhile.\ Here we are interested in an influence of the density of frustration on the phase transition from the disordered paramagnetic phase to the ordered spin-glass phase. In our Ising systems, the interaction is short-ranged and the dynamics is ruled by a simple Monte-Carlo heat-bath algorithm, with one parameter $J/k_BT$, i.e. the ratio of the exchange energy constant $J$ to the thermal energy $k_BT$ [@her]. Despite this simplicity, the low-temperature phase is very complex and multi-degenerate even in periodic systems, where the topological disorder is absent [@gos]. Current theory of Ising spin-glasses in random networks ignores the contribution of frustrations, reducing the network to a tree [@dgm]. In a ’tree-like structure’ closed loops as triads are absent. In the case of trees the Bethe theory is known to work well [@dgm; @bax]. In our considerations, the Bethe formula for the transition temperature $T_{SG}$ from the paramagnetic to the spin glass phase [@dgm] $$\frac{-2J}{T_{SG}}=\ln\frac{\sqrt{B}+1}{\sqrt{B}-1}$$ serves as a reference case without frustrations. Here $B=z_2/z_1$ is the ratio of the mean number of second neighbours to the mean number of the first neighbours. Then, the transition temperature $T_{SG}$ depends on the network topology. We note that in our network there is no bond disorder; all interactions are antiferromagnetic [@task].\ In our former texts, we calculated the transition temperature $T_{SG}$ of the Erdös-Rényi networks [@amk1] and of the regular network [@amk2]. The results indicated that on the contrary to the anticipations of the Bethe theory $T_{SG}$ decreases with the clustering coefficient $C$. However, in both cases we dealt with the networks endowed with the small-world property. It is not clear what dimension should be assigned to these networks, but it is well known that the dimensionality and in general the network topology influences the values of temperatures of phase transitions [@stan; @bax; @ho]. On the other hand, many real networks are embedded in the three-dimensional space - these are called spatial networks [@hbp]. In particular, magnetic systems belong obviously to this class. Therefore, the aim of this work is to calculate the phase transition temperature $T_{SG}$ again for the spatial networks. As in our previous texts [@amk1; @amk2] the clustering coefficient $C$ is varied as to investigate the influence of the density of frustrations on $T_{SG}$.\ In the next section we describe the calculation scheme, including the details on the control of the clustering coefficient. Third section is devoted to our numerical results. These are the thermal dependences of the magnetic susceptibility $\chi(T)$ and of the spacific heat $C_v(T)$. Final conclusions are given in the last section. Calculations ============ The spatial network is constructed as follows. Three coordinates of the positions of nodes are selected randomly from the homogeneous distribution between 0 and 1. Two nodes are linked if their mutual distance is not larger than some critical value $a$. In this way $a$ controls the mean number of neighbours, i.e. the mean degree $<k>$ of the network. In networks obtained in this way, the degree distribution $P(k)$ agrees with the Poisson curve. As a rule, the number of nodes $N=10^5$. Then, to obtain $<k>=4$ and $<k>=9$ we use $a=0.0212$ and $a=0.0278$. The mean degree $<k>$ is found to be proportional to $a^{2.91}$. This departure from the cubic function is due to the open boundary conditions. In two above cases, the values of the clustering coefficient $C$ are respectively 0.42 and 0.47.\ Now we intend to produce spatial networks with given mean degree $<k>$ and with enhanced clusterization coefficient $C$. This is done in two steps. First we adjust the radius $a$ to obtain a smaller $<k>$, than desired. Next we apply the procedure proposed by Holme and Kim [@hoki]: for each pair of neighbours of the same node a link between these neighbours is added with a given probability $p'$. This $p'$ is tuned as to obtain a desired value of the mean degree $<k>$. Simultaneously, the clustering coefficient $C$ is higher. In this way we obtain $C$ between 0.42 and 0.46 for $<k>=4$, and between 0.47 and 0.56 for $<k>=9$. The degree distribution $P(k)$ in the network with enhanced $C$ differs from the Poisson distribution, as shown in Fig. 1.\ The dynamics of the system is ruled by the standard Monte Carlo heat-bath algorithm [@her]. We checked that for temperature $T>0.5$, the system equilibrates after $10^3$ Monte Carlo steps; in one step each spin is checked. Sample runs ensured that after this time, the specific heat $C_v$ calculations from the thermal derivative of energy and from the energy fluctuations give - within the numerical accuracy - the same results. Results ======= In Fig. 2 we show the thermal dependence of the static magnetic susceptibility $\chi(T)$ for the network with mean degree $<k>=4$. Fig. 3 displays the magnetic specific heat $C_v(T)$ for the same network. The plots of the same quantities for $<k>=9$ are shown in Figs. 4 and 5. The positions of the maxima of $\chi(T)$ and $C_v(T)$ can be treated as approximate values of the transition temperature $T_{SG}$ [@mb; @by]. Most curves displayed show some maxima except two cases with highest $C$ for $<k>=4$, where the susceptibility for low temperatures does not decrease - this is shown in Fig. 2. Still it is clear that the observed maxima do not appear above $T=1.1$ for $<k=4>$ and above $T=1.7$ for $<k>=9$. Moreover, the data suggest that when the clustering coefficient $C$ increases, the positions of the maxima of $\chi(T)$ and $C_v(T)$ decrease. This is visible in particular for $<k>=9$, in Figs. 4 and 5.\ In Fig. 6 we show approximate values of the transition temperatures $T_{SG}$, as read from Figs. 2-5. These results are compared with the theoretical values of $T_{SG}$, obtained from Eq. 1. On the contrary to the numerical results, the Bethe theory indicates that $T_{SG}$ is almost constant or increases with $C$. This discrepancy is our main numerical result. It is also of interest that once the clustering coefficient $C$ increases, the susceptibility $\chi$ increases but the specific heat $C_v$ decreases. This can be due with the variation of the shape of the free energy, as dependent on temperature and magnetic field. Discussion ========== Our numerical results can be summarized as follows. The temperature $T_{SG}$ of the transition from the paramagnetic phase to the spin-glass phase decreases with the clustering coefficient $C$. We interpret this decrease as a consequence of the increase of the density of frustrations. More frustrated triads make the energy minima more shallow and then a smaller thermal noise is sufficient to throw the system from one to another minimum. This result is in contradiction to the Bethe theory. However, in this theory the frustration effect is neglected. Then the overall picture, obtained previously [@amk1; @amk2] for the small-world networks, is confirmed here also for the spatial networks.\ This interpretation can be confronted with recent numerical results of Herrero, where the transition temperature $T_{SG}$ increases with the clustering coefficient $C$ in the square lattice [@hero]. As it is shown in Fig. 7 of [@hero], $T_{SG}$ decreases from 2.3 to 1.7 when the rewiring probability $p$ increases from zero to 0.4. Above $p=0.4$, $T_{SG}$ remains constant or increases very slightly, from 1.7 to 1.72 when $p=1.0$. The overall dependence can be seen as a clear decrease of $T_{SG}$. On the other hand, the clustering coefficient $C$ does not increase remarkably with the rewiring probability $p$. The solution of this puzzle is that in the square lattice with rewiring the frustrations are not due to triads, but to two interpenetrating sublattices, which are antiferromagnetically ordered in the case when $p=0$. The conclusion is that it is the increase of the density of frustrations what always leads to a decrease of $T_{SG}$.\ A few words can be added on the significance of these results for the science of complexity, with a reference to the computational problem of satisfiability. In many complex systems we deal with a number of external conditions, when all of them cannot be fulfilled. Second premise is that in many complex systems a noise is ubiquitous. These are analogs of frustration and thermal noise. In the presence of noise and contradictive conditions, the system drives in its own way between temporally stable states, similarly to the way how the Ising spin glass wanders between local minima of energy. Once the number of contradictive tendencies or aspirations increases, the overall structure becomes less stable. [**Acknowledgements.**]{} We are grateful to Carlos P. Herrero for his comment. The calculations were performed in the ACK Cyfronet, Cracow, grants No. MNiSW /SGI3700 /AGH /030/ 2007 and MNiSW /SGI3700 /AGH /031/ 2007. [99]{} S. N. Dorogovtsev, A. V. Goltsev and J. F. F. Mendes, [*Critical phenomena in complex networks*]{}, Rev. Mod. Phys. [**80**]{} (2008) 1275-1335. M. R. Garey and D. S. Johnson, [*Computers and Intractability. A Guide to the Theory of NP-Completeness*]{}, W. H. Freeman and Comp., New York 1979. K. Binder and A. P. Young, [*Spin glasses: Experimental facts, theoretical concepts, and open questions*]{}, Rev. Mod. Phys. [**58**]{} (1986) 801-986. K. H. Fischer and J. A. Hertz, [*Spin Glasses*]{}, Cambridge UP, Cambridge 1991. C. M. Newman and D. L. Stein, [*Ordering and broken symmetry in short-ranged spin glasses*]{}, J. Phys.: Cond. Mat. [**15**]{} (2003) R1319-R1364. D. W. Heermann, [*Computer Simulation Methods in Theoretical Physics*]{}, Springer-Verlag, Berlin 1986. M. J. Krawczyk, K. Malarz, B. Kawecka-Magiera, A. Z. Maksymowicz and K. Ku[ł]{}akowski [*Spin-glass properties of an Ising antiferromagnet on the Archimedean $(3,12^2$) lattice*]{}, Phys. Rev. B [**72**]{} (2005) 24445. R. J. Baxter, [*Exactly Solved Models in Statistical Mechanics*]{}, Academic Press, London 1982. D. Stauffer and K. Ku[ł]{}akowski, [*Why everything gets slower?*]{}, TASK Quarterly [**7**]{} (2003) 257-262. A. Mańka, K. Malarz and K. Ku[ł]{}akowski, [*Clusterization, frustration and collectivity in random networks*]{}, Int. J. Mod. Phys. C [**18**]{} (2007) 1765-1773. A. Mańka-Krasoń and K. Ku[ł]{}akowski, [*Magnetism of frustrated regular networks*]{}, Acta Phys. Pol. B, in print (arXiv:0812.1128). H. E. Stanley, [*Introduction to Phase Transitions and Critical Phenomena*]{}, Clarendon Press, Oxford 1971. A. Aleksiejuk, J. A. Ho[ł]{}yst and D. Stauffer, [*Ferromagnetic phase transitions in Barabási-Albert networks*]{}, Physica A [**310**]{} (2002) 260-266. C. Herrmann, M. Barthélemy and M. Provero, [*Connectivity distribution of spatial networks*]{}, Phys. Rev. E [**68**]{} (2003) 026128. P. Holme and B. J. Kim, [*Growing scale-free networks with tunable clustering*]{}, Phys. Rev. E [**65**]{} (2002) 026107. J. Morgenstern and K. Binder, [*Magnetic correlations in three-dimensional Ising spin glasses*]{}, Z. Physik B [**39**]{} (1980) 227-232. C. P. Herrero, [*Antiferromagnetic Ising model in small-world networks*]{}, Phys. Rev. E [**77**]{} (2008) 041102.

ArXiv

--- abstract: 'If the large scale structure of the Universe was created, even partially, via Zeldovich pancakes, than the fluctuations of the CMB radiation should be formed due to bulk comptonization of black body spectrum on the contracting pancake. Approximate formulaes for the CMB energy spectrum after bulk comptonization are obtained. The difference between comptonized energy spectra of the CMB due to thermal and bulk comptonozation may be estimated by comparison of the plots for the spectra in these two cases.' author: - 'G.S. Bisnovatyi-Kogan [^1]' title: Spectral distortions in CMB by the bulk Comptonization due to Zeldovich pancakes --- Introduction ============ Fluctuations of CMB are created mainly by primordial quantum perturbations, growing due to inflation [@peeb93]. Also they may appear as a result of the interaction of CMB with a hot plasma appearing during the epoch of the formation of a large scale structure of the universe. Perturbations of CMB spectrum created by this interaction had been investigated in [@wey1], [@wey2], [@peeb70], [@sper17], and in the series of papers [@ZS1] - [@ZS4]. The review of theory of interaction of free electrons with a radiation is presented in [@rew]. A main process of interaction of hot plasma inside galaxy clusters with CMB is a Compton scattering, including a Doppler effect. Observations from the satellites WMAP and PLANCK had revealed an existence of CMB fluctuations in the directions of rich galactic clusters [@w1], [@w2], [@p8]-[@p], which had been interpreted in the frame of physical processes, investigated in [@wey1], [@wey2], [@ZS1], [@ZS2]. A study of interaction is based on the solution of the Kompaneets equation [@Komp], see also [@wey1], which is an approximate form of the radiative transfer equation with non-coherent scattering, when energy exchange $\Delta E_{e\gamma}$ between the electron and the photon in one scattering is much less than the photon energy $E_{\gamma}$, and kinetic energy (electron energy), $kT_e$ $$\frac{\Delta E_{e\gamma}}{E_{\gamma}}=\frac{4kT_e-E_{\gamma}}{m_ec^2}\ll 1. %\,E_{ek0},\, E_{\gamma 0}. \label{i1}$$ At such conditions, characteristic for the Thompson-Compton scattering in a non-relativistic plasma, the scattering term is reduced to the differential form [@Komp], [@wey1]. The relative value of spectral distortions induced by such scattering, is preserved during the universe expansion, because the perturbed and background photons have the same dependence on time and on the red shift, when moving without interactions. In addition to the scattering on hot electrons, another type of distortion of the photon spectrum takes place in the interaction of a photons with a bulk motion of a matter. This types of distortion had been investigated in application to the accretion of matter into neutron stars, stellar mass and AGN black holes, which are observed as X-ray, optical, and ultraviolet sources [@BP1], [@BP2], [@TZ], [@PL], [@PL2], [@T1], [@BW], [@CGF], [@BW2]. Among the objects, in which a bulk comptonizanion may be important, we should include Zeldovich pancakes which may be formed in the universe at first stages of a large-scale structure formation. [@ZPan], [@Z2], [@ZDSh]. The equation, including both thermal and bulk motion components, have been derived in [@BP1], [@PL] in different approximations. In this work we calculate distortions of the equilibrium Planck spectrum of CMB, induced by the bulk comptonization on the objects, collapsing to Zeldovich pancakes. We use an approximation of a uniform flat collapsing layer, which reproduces a motion of a non-spherical large-scale perturbation, with a gravity of a dark matter (DM) [@BK]. We suggest, that hydrogen baryonic matter, collapsing in the gravitational field of DM, is sufficiently ionised, to be able to produce a bulk comptonization. Observations of radio spectra in the region of the redshifted famous hydrogen radio line with a wavelength $\lambda\, =\,21$ cm, have lead to the detection of a flattened absorption profile in the sky-averaged radio spectrum, which is centred at a frequency of 78 megahertz, corresponding to the redshift $z\approx 17$ [@BNat]. The appearance of this profile was attributed to an action of the radiation of first stars [@BNat; @Funiv], which are responsible also for a secondary heating of the universe. Spectral observations of a distant quasar J1342 + 0928 with $z=7.54$ have shown strong evidence of absorption of the spectrum of the quasar redwards of the Lyman $\alpha$ emission line (the Gunn–Peterson damping wing), as would be expected if a significant amount (more than 10 per cent) of the hydrogen in the intergalactic medium surrounding this quasar is neutral [@BanNat]. The authors derive such a significant fraction of neutral hydrogen, although the exact fraction depends on the modelling. However, even in the most conservative analysis it was found a fraction the neutral hydrogen of more than 0.33 (0.11) at 68 per cent (95 per cent) probability, indicating that this redshift is well within the reionization epoch of the Universe. So we suggest that formation the main body of a large-scale structure in the model of Zeldovich pancakes happens after the period of the secondary ionization [@ts], [@tsb] in the universe. We consider a contractive flat layer of plasma surrounded by the radiation with equilibrium Planckian spectrum (CMB). When crossing the contractive layer the photons experience a Compton scattering on electrons, which velocities have a thermal (chaotic), and directed (bulk motion) components. For sufficiently low-temperature plasma the bulk motion comptonization becomes more important than the thermal one. In Section 2 we consider bulk motion comptonization by the “cold” contractive layer. We solve analytically the Kompaneets equation in a contractive layer, similar to the one for converging flow in [@BP1], [@BW], which is illuminated by an equilibrium radiation flux on its boundary. In the Section 3 we compare the spectra of a thermal, and bulk motion comptonization at parameters, when both distortions are quantitatively comparable, but have different spectral shapes. The bulk comptonization by the contractive self-gravitating layer ================================================================= Dynamics of the contractive self-gravitating layer -------------------------------------------------- There are several physical input parameters in the problem of calculation of resulting spectrum of radiation passing the flat contractive layer of plasma. We consider one-dimensional problem, when all function depend only on one space coordinate $x$. The temperature $T$ and density $\rho$ inside the layer are supposed to be uniform, depending only on time $t$. For adiabatic contraction the thermal state of the matter is characterized by a constant entropy $S$. For adiabatic contraction of the layer of non-relativistic ideal fully ionized plasma the equation of state, connecting pressure $P$ with density is written as $$\label{eos} P=\rho{\cal R}T=K(S)\rho^{5/3}.$$ The pressure $P_0$ and the density $\rho_0$ in the layer at the initial moment $t_0$ are given, determining the constant $K$, and the entropy $S(K)$. For any time dependent $\rho(t)$ we obtain $P(t,S)$ from (\[eos\]), and for a known gas constant ${\cal R}$ we obtain a time dependent temperature $$\label{T} T(t,S)=\frac{P}{\rho{\cal R}}.$$ We consider a self-gravitating layer with initial thickness $x_0$, initial density $\rho_0$, and initial velocity distribution over the layer as $$\begin{aligned} \upsilon_0(y) = - \upsilon_0\frac{y}{x_{0}}, \quad -\frac{x_0}{2}\le y \le \frac{x_0}{2},\nonumber \\ -\frac{\upsilon_0}{2}\le \upsilon_0(y)\le \frac{\upsilon_0}{2},\quad -\upsilon_0\left(\frac{x_0}{2}\right)=\upsilon_0\left(-\frac{x_0}{2}\right)=\frac{\upsilon_0}{2}. \label{v0}\end{aligned}$$ Here $\upsilon_0$ is the initial velocity of decreasing of the layer thickness $\frac{dx}{dt}(0)=-\upsilon_0$. The surface density of the layer $\sigma_0=\rho_0 x_0=\rho(t) x(t)=\sigma$ remains constant during contraction. The time dependent parameters of the cold $(S=K=0)$ uniform layer are obtained from solution of equations of motion and Poisson, for the gravitational potential $\varphi_g(y,t)$ and he gravitational force $F_{gy}$, and the equation for a time dependence of the thickness of the layer $x(t)$ $$\begin{aligned} \frac{\partial^2\varphi_g}{{\partial y}^2} = 4\pi G\rho, \,\, F_{gy}=-\frac{\partial\varphi_g}{\partial y}=-4\pi G\rho y,\,\, F_{gy}\left(\frac{x}{2}\right)=-2\pi G\rho x=-2\pi G\sigma_0. \label{pm} \\ \frac{\partial\upsilon_y}{\partial t}=F_{gy}=-4\pi G\rho y,\,\,\, \upsilon_y\left(\frac{x}{2}\right)=-2\pi G\rho xt-\frac{v_0}{2}=-2\pi G\sigma t-\frac{v_0}{2}. \label{pm1}\end{aligned}$$ Solving these equations with boundary and initial conditions, mentioned above, we obtain the solution, taking $ t_0=0$, as $$\begin{aligned} \upsilon(t,y) =( - \upsilon_0-4\pi G\sigma_0 t)\frac{y}{x}, \,\,\, \frac{dx}{dt}=-2\upsilon_y\left(\frac{x}{2}\right)= - \upsilon_0-4\pi G\sigma_0 t,\nonumber\\ \quad x=x_0- \upsilon_0 t -2\pi G\sigma_0 t^2 \quad {\rm at} \quad -\frac{x}{2}\le y \le \frac{x}{2}. \label{v}\end{aligned}$$ The thickness of the layer becomes $x=0$ at $t=t_1$, with $$t_1=-\frac{\upsilon_0}{4\pi G\sigma_0} + \sqrt{ \frac{\upsilon_0^2}{(4\pi G\sigma_0)^2}+\frac{x_{0}}{2\pi G\sigma_0}}. \label{th}$$ ![The layer collapsing to its midplane; $y$ is a variable across the layer, $x$ is a layer thickness.[]{data-label="fig:layer"}](Layer_collapse.eps) The bulk comptonization is more important at large contraction velocity. We consider only last stages, when on the surface $\upsilon$ remains close to $\upsilon_0$. It happens when $\upsilon_0^2\gg 8\pi G\sigma_0 x_0$. In this approximation we obtain the following formulae describing the dynamics of the cold layer $$t_1=\frac{x_0}{\upsilon_0},\quad \upsilon(y) = - \upsilon_0\frac{y}{x}, \quad x=x_0- \upsilon_0 t \quad {\rm at} \quad -\frac{x}{2}\le y \le \frac{x}{2}. \label{th1}$$ Equations determining the spectrum distortion due to bulk comptonization in the cold layer ------------------------------------------------------------------------------------------ We consider the problem of a bulk Comptonization of an incident radiation on a moving medium in diffusion approximation using Eq.(31) of [@BW] (see also [@BP1]). The scatterer is a flat contracting layer of cold electrons ($T_e \approx 0$), see Fig.\[fig:layer\]. The layer is consisting mainly of a non-collisional dark matter, where particles of two oppositely moving flows penetrate through each other, imitating an elastic bounce with respect to the behaviour of the gravitational potential. The flows with baryon and electrons collide, forming a shock wave with decreasing expanding velocities [@ZDSh]. Therefore, the bulk comptonization is effective only on the contracting stage of the pancake formation. Partial differential equation satisfied for the photon occupation number function $f(x,\varepsilon)$ is written as, (see [@BW; @TMK97]) $$\label{eq:sol1} \frac{\partial f}{\partial t} + \overrightarrow{\upsilon} \nabla f = \frac{\partial}{\partial y} \left(\frac{c}{3n_e\sigma_T} \frac{\partial f}{\partial y} \right) + \frac{1}{3} \left(\nabla \overrightarrow{\upsilon} \right)\nu\frac{\partial f}{\partial \nu}+j(y, \nu),$$ where $\nu$ is a photon frequency, $n_e$ is the electron density, $\sigma_T$ is the Thompson cross-section, $j(y, \nu)$ is the emisivity. Let us consider the respond of the system on the incoming stationary flux of monochromatic photons through both sides of the layer’s boundary, described by the Green function of the problem $f_G$. Introduce following non-dimensional variables $$\tilde{y} = n_e\sigma_Ty, \quad \tau = \int_{-x/2}^{x/2}n_e \sigma_Tdy = n_e\sigma_Tx,$$ $$\label{eq:dless} \tilde{\nu} = \frac{h\nu}{k_BT_r}, \quad \beta_0 = \frac{\upsilon_0}{c},$$ where $k_B$ is the Boltzmann constant, $T_r$ is the temperature of the incoming radiation, $c$ is velocity of light, $\tau$ is the constant optical depth of the contracting layer. Assuming the baryonic mass of the forming pancake at redshift $z\sim 10$ as $5\cdot 10^{14}\, M_\odot\,\approx\, 10^{48}\,g$, concentrated mainly inside the central radius $R_c\sim 0.1$ Mpc, we obtain its vertical optical depth as $$\label{eq:est2} \tau_v = \frac{M\sigma_T}{m_p\pi R_c^2}\approx 1.5$$ for fully ionized pancake plasma. This value of the optical depth is calculated for vertically falling background photons. For the photons falling with an inclination to pancake vertical the optical depth is larger. Therefore, qualitatively the solution for opaque pancake may be used for estimation of the bulk Comptonization effect at lower optical depths $\tau_v$. We consider the case, when the photon birth inside the layer is negligibly small ${j}({y},{\nu}) \approx 0$, and only scattering of the photons coming from the outer space is important. While the contracting velocity during the pancake formation is much smaller than the light velocity, and the vertical optical depth is of the order of unity, we may consider a time independent problem for a stationary contracting velocity distribution (\[v0\]), at fixed $\upsilon_0,\,\,\, x_0$. In the stationary one-dimensional case the equation (\[eq:sol1\]) is written in the form $$\label{eq:sol2} \frac{1}{3}\frac{\partial^2 f}{\partial\tilde{y}^2} + \frac{\beta_0\tilde{y}}{\tau}\frac{\partial f}{\partial \tilde{y}} - \frac{\beta_0}{3\tau} \tilde{\nu}\frac{\partial f}{\partial \tilde{\nu}} =0,$$ Approximate solution -------------------- To solve the partial differential equation (\[eq:sol2\]) approximately, we consider conditions, when perturbations of the CMB spectrum, produced by bulk comptonization are small, may represent the solution, similar to [@ZS1], in the form $$\label{bc1} f=C_f(f_0+f_1),\quad f_1 \ll f_0,\,\,\, C_f \,\,\,{\rm is\,\,\, constant},$$ where $f_0$ is a spatially uniform planck distribution $$\label{bc2} f_0(\tilde\nu) =\frac{1}{e^{\tilde\nu}-1}$$ Using these relations in (\[eq:sol2\]), we obtain $$\label{bc3} \frac{1}{3}\frac{\partial^2 f_1}{\partial\tilde{y}^2} + \frac{\beta_0\tilde{y}}{\tau}\frac{\partial f_1}{\partial \tilde{y}} - \frac{\beta_0}{3\tau} \tilde{\nu}\frac{\partial f_0}{\partial \tilde{\nu}} =0,$$ The reduce the partial differential equation (\[eq:sol2\]) to the ordinary equation, we approximate the space dependence of $f_1$ by profiling function $$\label{bc4} f_1(\tilde\nu,\tilde y)=\tilde{f_1}(\tilde\nu)\frac{4\tilde y^2-\tau^2}{\tau^2}$$ This function is symmetric relative to the plane $\tilde y$, and equal to zero on both boundaries of the layer, in accordance with boundary conditions. We have than $$\label{bc5} \frac{\partial f_1}{\partial \tilde{y}}=\tilde{f_1}\frac{8\tilde{y}}{\tau^2},\quad \frac{\partial^2 f_1}{\partial \tilde{y}^2}=\tilde{f_1}\frac{8}{\tau^2}, \qquad \frac{\partial f_0}{\partial\tilde{\nu}}=-\frac{e^{\tilde\nu}}{(e^{\tilde\nu}-1)^2}.$$ Substituting (\[bc5\]) into (\[bc3\]), we obtain $$\label{bc6} \frac{1}{3}\frac{8}{\tau^2}\tilde{f_1}+ \frac{\beta_0}{\tau}\frac{8\tilde{y}^2}{\tau^2}\tilde{f_1}+ \frac{\beta_0}{3\tau}\frac{\tilde{\nu}e^{\tilde\nu}}{(e^{\tilde\nu}-1)^2}=0.$$ Averaging (\[bc6\]) over the layer by integration $\int_{-\tau/2}^{\tau/2}d\tilde y$, we obtain $$\label{bc7} \frac{1}{3}\frac{8}{\tau}\tilde{f_1}+ \frac{2\beta_0}{3}\tilde{f_1}+ \frac{\beta_0}{3}\frac{\tilde{\nu}e^{\tilde\nu}}{(e^{\tilde\nu}-1)^2}=0.$$ We obtain from (\[bc7\]) $$\label{bc8} \tilde{f_1}=-\frac{1}{8} \frac{\beta_0\tau}{1+\frac{\beta_0\tau}{4}}\, \frac{\tilde{\nu}e^{\tilde\nu}}{(e^{\tilde\nu}-1)^2}.$$ Averaging (\[bc6\]) by the same integration we obtain finally $$\label{bc9} f_1(\tilde\nu)=-\frac{2}{3}\tilde{f_1}=\frac{1}{12}\frac{\beta_0\tau}{1+\frac{\beta_0\tau}{4}}\, \frac{\tilde{\nu}e^{\tilde\nu}}{(e^{\tilde\nu}-1)^2}.$$ To find the resulting spectrum after bulk comptonization process we should take into account, that in this process the number of photon is conserved, while for the resulting function $f_0+f_1$ the photon number density is larger than for the background function $f_0$. We should therefore consider the resulting function in the form $$\label{bc10} f(\tilde\nu)=C_f(f_0(\tilde\nu)+f_1(\tilde\nu)).$$ The number of photons does not change during the comptonization, what is represented by the relation $$\label{bc11} \int_0^\infty \tilde\nu^2 f(\tilde\nu)d\tilde\nu=\int_0^\infty \tilde\nu^2 f_0(\tilde\nu)d\tilde\nu.$$ From (\[bc10\])(\[bc11\]) we find the constant $C_f$, what uniquely define the photon distribution function after bulk comptonization $$\begin{aligned} \label{bc12} C_f=\frac{\int_0^\infty \tilde\nu^2 f_0(\tilde\nu)d\tilde\nu} {\int_0^\infty \tilde\nu^2 f_0(\tilde\nu)d\tilde\nu+\int_0^\infty \tilde\nu^2 f_1(\tilde\nu)d\tilde\nu},\\ \quad f(\tilde\nu)= \frac{C_f}{e^{\tilde\nu}-1}\left[1+\frac{1}{12}\frac{\beta_0\tau}{1+\frac{\beta_0\tau}{4}}\, \frac{\tilde{\nu}e^{\tilde\nu}}{e^{\tilde\nu}-1}\right]. \label{bc13}\end{aligned}$$ The influence of the bulk comptonization on the energy spectrum of CMB photons may be estimated by comparison of the distorted energy spectrum $f_{Eb}(\tilde\nu)$ with the planck spectrum $f_{E0}(\tilde\nu)$, defined as $$\begin{aligned} \label{bc14} f_{Eb}(\tilde\nu)= \frac{C_f \tilde\nu^3 }{e^{\tilde\nu}-1}\left[1+\frac{1}{12}\frac{\beta_0\tau}{1+\frac{\beta_0\tau}{4}}\, \frac{\tilde{\nu}e^{\tilde\nu}}{e^{\tilde\nu}-1}\right],\quad f_{E0}(\tilde\nu)= \frac{\tilde\nu^3 }{e^{\tilde\nu}-1}.\end{aligned}$$ Comparison with thermal comptonization effects ============================================== It was obtained in [@ZS1] that the relative distortion of the comptonized spectrum of CMB due to interaction with hot electrons is determined by the expression $$\label{bc15} \delta_{Th}= y\tilde{\nu}\frac{e^{\tilde\nu}} {e^{\tilde\nu}-1}\left[\frac{\tilde\nu}{\tanh(\tilde\nu/2)}-4\right].$$ Here the value $y$, for the flat universe with $\Omega=1$, is determined as [@ZS1] $$\label{bc16} y=n_{e0}\sigma_T c H_0^{-1}\int_0^{z_{max}}\frac{kT_e(z)}{m_e c^2}\sqrt{1+z}\,dz.$$ Equations (\[bc15\]),(\[bc16\]) are derived for the uniform expanding universe, with a present hot electron density $n_{e0}$, which had been heated at redshift $z_{max}$, and have a temperature dependence $T_e(z)$. When applying this formula for the CMB comptonization by hot gas in the galactic clusters, we may approximately take the value of $y$ in the form $$\label{bc17} y\approx n_e \sigma_T l \frac{kT_e}{m_e c^2} = \tau_{CL}\, \frac{kT_e}{m_e c^2},$$ where $n_e$, $T_e$, $l$, $\tau_{CL}$ are the number density, temperature, size, and the optical depth of hot electrons in the galactic cluster, respectively. As a result of thermal comptonization, the photon distribution function is written as $$\label{bc18} f_{Th}(\tilde\nu)=f_0(\tilde\nu)(1+\delta_{Th})=\frac{1}{e^{\tilde\nu}-1} \left\{1+ y \tilde\nu\frac{e^{\tilde\nu}}{e^{\tilde\nu}-1} \left[\frac{\tilde\nu}{\tanh(\tilde\nu/2)}-4\right] \right\}.$$ The distorted energy spectrum is written as $$\label{bc19} f_{ETh}(\tilde\nu)=\frac{\tilde\nu^3}{e^{\tilde\nu}-1} \left\{1+ y \tilde\nu\frac{e^{\tilde\nu}}{e^{\tilde\nu}-1} \left[\frac{\tilde\nu}{\tanh(\tilde\nu/2)}-4\right] \right\}.$$ Similar distortions due to bulk comptonization are obtained from (\[bc2\]), (\[bc10\])-(\[bc13\]), for small distortions with $C_f\approx 1$, $\beta_0 \tau\ll 1$, as $$\label{bc20} \delta_{Bulk}=\frac{f_1(\tilde\nu)}{f_0(\tilde\nu)}= \frac{1}{12}\frac{\beta_0\tau}{1+\frac{\beta_0\tau}{4}}\, \frac{\tilde{\nu}e^{\tilde\nu}}{e^{\tilde\nu}-1}.$$ For the Raleigh-Jeans low frequency side of the spectrum $h\nu < kT_r$ we have [@ZS1], using (\[bc15\]),(\[bc18\]), $$\label{bc21} \delta_{Th}\approx -2y=-\tau_{CL}\, \frac{kT_e}{m_e c^2}, \,\,\, \cite{ZS1} \qquad \delta_{Bulk} \approx \frac{\beta_0 \tau}{12},$$ A thermal comptonization leads to shifting the whole CMB spectrum to higher photon energy region, producing decrease of low-energy photons (see Fig.3). In the case of bulk comptonization the observed photons cross the whole collapsing layer. Half of the matter velocity is directed opposite to the photon trajectory, increasing its frequency , and another half is moving in the same direction with photons, which are loosing the energy. Actually, the resulting spectrum is moving to the higher frequency also (see Fig.4), but the deviations from the Planck are much smaller. The shift of the spectral maximum in both cases are comparable, but maximum itself is slightly increasing for the bulk comptonization, and is decreasing more for the thermal one. The distortion in the high energy Wien part of the spectrum due to comptonization is relatively large, and cannot be found correctly in the linear approximation, but its input in the energy is negligibly small. For thermal comptonization in the uniformly expanding universe it is found in [@ZS1]. The consideration of this case for a bulk comptonization in a contracting layer is more complicated, and will not be considered here. It seems, that for analysis of observations it is enough to know, that in both cases the high energy part of the spectrum is increasing. It is, probably, not possible to distinguish between two type of comptonization, by observations of high energy part because of substantial indefiniteness of our knowledge about the distribution of parameters of the hot gas in galactic clusters. The observations in the vicinity of maximum of the CMB spectrum could more distinctly indicate the mechanism of comptonization because of different sign of the spectrum distortion in this region. For qualitative comparison. energy spectrum distortions by comptonization are presented in Fig.3 for bulk comptonization at $\beta_0\tau=1.2$, $C_f=0.7744$, from (\[bc14\]), and for a thermal comptonization in Fig.4, for $y=1/60$. ![Spectral distortion due to comptonization on hot electrons at $y$=1/60, according to [@ZS1] and (\[bc19\]) Plotted are Planck curve (red line), and comptonized curve (green line)[]{data-label="therm"}](CThermal_fin.eps) ![Spectral distortion due to bulk comptonization on the contracting layer at $\beta_0\tau=1.2$, according to (\[bc14\]). Plotted are Planck curve (red line), and comptonized curve (green line)[]{data-label="bulk"}](CBulk_fin.eps) Conclusions =========== If the large scale structure of the Universe was created, even partially, via Zeldovich pancakes, than the fluctuations of the CMB radiation should be formed due to bulk comptonization of black body spectrum on the contracting pancake. Observations indicate [@BanNat], that the secondary ionization was started at early stages, at $z\sim 20$, so that contracting pancakes should have a high enough ionization level for the comptonization onset. It is not possible now to predict certainly the amplitude of these fluctuations, because of uncertainty in the epoch of formation, and masses of the pancakes. Nevertheless, measuring a spectrum of CMB fluctuations could permit to distinguish between thermal and bulk comptonization. In the first the deficit of low energy photons is accompanied by increase of high energy ones, because in the comptonization process the total number of photons is conserved, and decrease of the spectral maximum. In the bulk comptonization on the contracting pancake the excess is obtained also at higher energy side photons, with slight increasing of the maximum, while decrease in the lower part is less. May be in the observed spectrum where both types of comptonization are present: the bulk comptonization at the epoch of pancake formation, and the thermal comptonization at present time on the hot gas in the galaxy cluster. Acknowledgements {#acknowledgements .unnumbered} ================ The author is grateful to L.G. Titarchuk for useful discussions, and Ya.S. Lyakhova for cooperation. This work was partially supported by RFFI grants No.17-02-00760, 18-02-00619, and RAS Program of basic research 12 “Problems of Origin and Evolution of the Universe”. [99]{} Alonso D., Hill J.C., Hložek R., Spergel D.N. Physical Review D [**97**]{}, id.063514, 2018 Banados E., Venemans B.P., Mazzucchelli C., et al. Nature [**553**]{}, 473 (2018). Becker P.A. & Wolff M.T., ApJ, **630**, 465, 2005 Becker P.A. & Wolff M.T., ApJ, **654**, 435, 2007 Bielby, R. 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ArXiv

--- abstract: 'Learning with Fredholm kernel has attracted increasing attention recently since it can effectively utilize the data information to improve the prediction performance. Despite rapid progress on theoretical and experimental evaluations, its generalization analysis has not been explored in learning theory literature. In this paper, we establish the generalization bound of least square regularized regression with Fredholm kernel, which implies that the fast learning rate $O(l^{-1})$ can be reached under mild capacity conditions. Simulated examples show that this Fredholm regression algorithm can achieve the satisfactory prediction performance.' address: - 'Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China' - 'College of Engineering, Huazhong Agricultural University, Wuhan 430070, China' - 'College of Science, Huazhong Agricultural University, Wuhan 430070, China' author: - Yanfang Tao - Peipei Yuan - Biqin Song title: 'Error analysis of regularized least-square regression with Fredholm kernel' --- Fredholm learning, generalization bound, learning rate, data dependent hypothesis spaces Introduction {#section1} ============ Inspired from Fredholm integral equations, Fredholm learning algorithms are designed recently for density ratio estimation [@que1] and semi-supervised learning [@que2]. Fredholm learning can be considered as a kernel method with data-dependent kernel. This kernel usually is called as Fredholm kernel, and can naturally incorporate the data information. Although its empirical performance has been well demonstrated in the previous works, there is no learning theory analysis on generalization bound and learning rate. It is well known that generalization ability and learning rate are important measures to evaluate the learning algorithm [@cucker1; @zou1; @zou2]. In this paper, we focus on this theoretical theme for regularized least square regression with Fredholm kernel. In learning theory literature, extensive studies have been established for least square regression with regularized kernel methods, e.g., [@shi1; @sun1; @wu2]. Although the Fredholm learning in [@que2] also can be considered as a regularized kernel method, there are two key features: one is that Fredholm kernel is associated with the “inner" kernel and the “outer" kernel simultaneously, the other is that for the prediction function is double data-dependent. These characteristics induce the additional difficulty on learning theory analysis. To overcome the difficulty of generalization analysis, we introduce novel stepping-stone functions and establish the decomposition on excess generalization error. The generalization bound is estimated in terms of the capacity conditions on the hypothesis spaces associated with the “inner" kernel and the “outer" kernel, respectively. In particular, the derived result implies that fast learning rate with $\mathcal{O}(l^{-1})$ can be reached with proper parameter selection, where $l$ is the number of labeled data. To best of our knowledge, this is the first discussion on generalization error analysis for learning with Fredholm kernel. The rest of this paper is organized as follows. Regression algorithm with Fredholm kernel is introduced in Section \[section2\] and its generalization analysis is presented in Section \[section3\]. The proofs of main results are listed in Section \[section4\]. Simulated examples are provided in Section \[section5\] and a brief conclusion is summarized in Section \[section6\]. Regression with Fredholm kernel {#section2} =============================== Let $\mathcal{X}\subset\mathbb{R}^d$ be a compact input space and $\mathcal{Y}\subset[-M,M]$ for some constant $M>0$. The labeled data $\mathbf{z}=\{z_i\}_{i=1}^{l}=\{(x_i,y_i)\}_{i=1}^l$ are drawn independently from a distribution $\rho$ on $\mathcal{Z}:=\mathcal{X}\times\mathcal{Y}$ and the unlabeled data $\{x_{l+j}\}_{j=1}^u$ are derived random independently according to the marginal distribution $\rho_{\mathcal{X}}$ on $\mathcal{X}$. Given $\mathbf{z}, \mathbf{x}=\{x_i\}_{i=1}^{l+u}$, the main purpose of semi-supervised regression is to find a good approximation of the regression function $$\begin{aligned} f_{\rho}(x)=\int_{\mathcal{Y}}yd\rho(y|\mathcal{X})={\mathop{{\rm arg}\min}}_f\int_{\mathcal{Z}}(y-f(x))^2d\rho(x,y).\end{aligned}$$ In learning theory, $$\begin{aligned} \mathcal{E}(f):=\int_\mathcal{Z}(y-f(x))^2d\rho(x,y)\end{aligned}$$ and its discrete version $$\begin{aligned} \mathcal{E}_{\mathbf{z}}(f):=\frac{1}{l}\sum_{i=1}^l(y-f(x_i))^2\end{aligned}$$ are called as the expected risk and the empirical risk of function $f:\mathcal{X}\rightarrow\mathbb{R}$, respectively. Let $w(x,x')$ be a continuous bounded function on $\mathcal{X}^2$ with $\omega:=\sup\limits_{x,x'}w(x,x')<\infty$. Define the integral operator $L_w$ as $$\begin{aligned} L_wf(x)=\int_{\mathcal{X}} w(x,t)f(t)d\rho_{\mathcal{X}}(t), \forall f\in L_{\rho_{\mathcal{X}}}^2,\end{aligned}$$ where $L_{\rho_{\mathcal{X}}}^2$ is the space of square-integrable functions. Let $\mathcal{H}_K$ be a reproducing kernel Hilbert space (RKHS) associated with Mercer kernel $K:\mathcal{X}^2\rightarrow\mathbb{R}$. Denote $\|\cdot\|_K$ as the corresponding norm of $\mathcal{H}_K$ and assume the upper bound $\kappa:=\sup\limits_{x,x'\in\mathcal{X}}K(x,x')<\infty$. If choose $L_{w}\mathcal{H}=\{L_wf, f\in\mathcal{H}_K\}$ as the hypothesis space, the learning problem can be considered as to solve the Fredhom integral equation $L_wf(x)=y$. Sine the distribution $\rho$ is unknown, we consider the empirical version of $L_wf$ associated with $\mathbf{x}=\{x_i\}_{i=1}^{l+u}$, which is defined as $$\begin{aligned} L_{w,\mathbf{x}}f(x)=\frac{1}{l+u}\sum_{i=1}^{l+u}w(x,x_i)f(x_i).\end{aligned}$$ In the Fredholm learning framework, the prediction function is constructed from the data dependent hypothesis space $$\begin{aligned} L_{w,\mathbf{x}}\mathcal{H}=\{L_{w,\mathbf{x}}f, f\in\mathcal{H}_K\}.\end{aligned}$$ Given $\mathbf{z}, \mathbf{x}$, least-square regression with Fredholm kernel (LFK) can be formulated as the following optimization $$\begin{aligned} f_{\mathbf{z}}:=f_{\mathbf{z},\mathbf{x}}={\mathop{{\rm arg}\min}}\limits_{f\in\mathcal{H}_K}\{\mathcal{E}_{\mathbf{z}}(L_{w,\mathbf{x}}f)+\lambda\|f\|_{K}^2\}, \label{algorithm1}\end{aligned}$$ where $\lambda>0$ is a regularization parameter. Equation (\[algorithm1\]) can be considered as a discrete and regularized version of the Fredholm integral equation $L_wf=y$. When $w$ is the $\delta$-function, (\[algorithm1\]) becomes the regularized least square regression in RKHS $$\begin{aligned} \tilde{f}_{\mathbf{z}}={\mathop{{\rm arg}\min}}\limits_{f\in\mathcal{H}_K}\{\mathcal{E}_{\mathbf{z}}(f)+\lambda\|f\|_{K}^2\}. \label{algorithm2}\end{aligned}$$ When $\mathbf{x}=\{x_i\}_{i=1}^l$ and replacing $\|f\|_{K}^2$ with $\sum_{i=1}^l|f(x_i)|^q,q=1,2$, (\[algorithm1\]) is equivalent to the data-dependent coefficient regularization $$\begin{aligned} \tilde{f}_{\mathbf{z}}(x)=\sum_{i=1}^l\alpha_{\mathbf z,i}w(x,x_i),\end{aligned}$$ where $$\begin{aligned} \alpha_{\mathbf z} ={\mathop{{\rm arg}\min}}\limits_{\alpha\in\mathbb R^l}\Big\{\mathcal{E}_{\mathbf{z}}(\sum_{i=1}^l\alpha_iw(\cdot,x_i))+\lambda\sum_{i=1}^l| \alpha_i|^q \Big\}. \label{algorithm3}\end{aligned}$$ It is well known that (\[algorithm2\]) and (\[algorithm3\]) have been studied extensively in learning literatures, see, e.g. [@feng3; @shi1; @sun1]. These results relied on error analysis techniques for data independent hypothesis space [@cucker1; @cucker2; @zou2] and data dependent hypothesis space [@hong3; @sun1; @sun2; @feng3], respectively. Therefore, the Fredholm learning provides a novel framework for regression related with the data independent space $\mathcal{H}_K$ and the data dependent hypothesis space $L_{w,\mathbf{x}}\mathcal{H}$ simultaneously. Equation (\[algorithm1\]) involves the “inner" kernel $K$ and the “outer" kernel $w$. Denote $$\hat{K}(x,x')=\frac{1}{(l+u)^2}\sum_{i,j=1}^{l+u}w(x,x_i)K(x_i,x_j)w(x,x_j),$$ $\hat{\mathbf{K}}=(\hat{K}(x_i,x_j))_{i,j=1}^l$, and $\mathbf{Y}=(y_1,\cdots,y_l)^T$. It has been demonstrated in [@que2] that $$\begin{aligned} L_{w,\mathbf{x}}f_{\mathbf z}(x)=\frac{1}{l+u}\sum_{i=1}^{l+u}w(x,x_i)f_{\mathbf z}(x_i)=\sum_{s=1}^l\hat{K}(x,x_s)\alpha_s, \label{algorithm4}\end{aligned}$$ where $\alpha=(\alpha_1,\cdots,\alpha_l)^T=(\hat{\mathbf{K}}+\lambda I)^{-1}\mathbf{Y}$. Therefore, Fredholm regression in (\[algorithm1\]) can be implemented efficiently and the data-dependent kernel $\hat{K}(x,x')$ is called Fredholm kernel in [@que2]. Generalization bound {#section3} ==================== To provide the estimation on the excess risk, we introduce some conditions on the hypothesis space capacity and the approximation ability of Fredholm learning framework. For $R>0$, denote $$B_R=\{f\in\mathcal{H}_K:\|f\|_K\leq R\}$$ and $$\tilde{B}_R=\{f=\sum\limits_{i=1}^{l+u}\alpha_iw(\cdot,u_i): \sum\limits_{i=1}^{l+u}|\alpha_i|\leq R,u_i\in\mathcal X\}.$$ For any $\varepsilon>0$ and function space $\mathcal{F}$, denote $\mathcal{N}_{\infty}(\mathcal{F}, \varepsilon)$ as the covering number with $\ell_{\infty}$-metric. (Capacity condition) For the “inner" kernel $K$ and the “outer" kernel $w$, there exists positive constants $s$ and $p$ such that for any $\varepsilon>0$, $ \log\mathcal{N}_\infty( B_1,\varepsilon)\leq c_{s,K}\varepsilon^{-s} $ and $ \log\mathcal{N}_\infty( \tilde{B}_1,\varepsilon)\leq c_{p,w}\varepsilon^{-p}$, where $c_{s,K},c_{p,w}>0$ are constants independent of $\varepsilon$. \[condition1\] It is worthy notice that the capacity condition has been well studied in [@cucker1; @cucker2; @shi1]. In particular, this condition holds true when setting the “inner" and “outer" kernels as Gaussian kernel. For a function $f:\mathcal X\rightarrow\mathbb R$ and $q\in [1,+\infty)$, denote the $L^q$-norm on $\mathcal X$ as $$\begin{aligned} \|f\|_q:=\|f\|_{L_{\rho_{\mathcal X}}^q}=\Big(\int_{\mathcal X}|f(x)|^qd\rho_{\mathcal{X}}(x)\Big)^{\frac{1}{q}}.\end{aligned}$$ Define the data independent regularized function $$\begin{aligned} f_{\lambda}={\mathop{{\rm arg}\min}}\limits_{f\in{\mathcal{H}_K}}\{\|L_{w}f-f_{\rho}\|_2^2+\lambda\|f\|_K^2\}.\end{aligned}$$ The predictor associated with $f_\lambda$ is $$\begin{aligned} L_wf_\lambda=\int_{\mathcal{X}}w(x,t)f_{\lambda}(t)d\rho_{\mathcal X}(t)\end{aligned}$$ and the approximation ability of Fredholm scheme in $\mathcal{H}_K$ is characterized by $$\begin{aligned} D(\lambda)=\mathcal E(L_wf_{\lambda})-\mathcal E(f_{\rho})+\lambda\|f_{\lambda}\|_K^2.\end{aligned}$$ (Approximation condition) There exists a constant $\beta\in(0,1]$ such that $$\begin{aligned} D(\lambda)\leq c_\beta\lambda^{\beta},~~ \forall\lambda>0,\end{aligned}$$ where $c_\beta$ is a positive constant independent of $\lambda$.\[condition2\] This approximation condition relies on the regularity of $f_\rho$, and has been investigated extensively in [@cucker2; @sun1; @hong4]. To get tight estimation, we introduce the projection operator $$\begin{aligned} \pi(f)(x)= \left\{ \begin{array}{ll} M, & \hbox{if $f(x)>M$;} \\ f(x), & \hbox{if $|f(x)|\leq M$;} \\ -M, & \hbox{if $f(x)<-M$.} \end{array} \right.\end{aligned}$$ It is a position to present the generalization bound. Under Assumptions \[condition1\] and \[condition2\] , there exists $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho) \leq c\log^2(6/{\delta})(\lambda^{-\frac{s}{2+s}}l^{-\frac{s}{2+s}}+\lambda^{\beta} +\lambda^{\beta-1}l^{-\frac{s}{2+p}}),\end{aligned}$$ where $c$ is a positive constant independent of $l,\lambda,\delta$ \[theorem1\] The generalization bound in Theorem \[theorem1\] depends on the capacity condition , the approximation condition, the regularization parameter $\lambda$, and the number of labeled data. In particular, the labeled data is the key factor on the excess risk without the additional assumption on the marginal distribution. This observation is consistent with the previous analysis for semi-supervised learning [@belkin; @hong1]. To understand the learning rate of Fredholm regression, we present the following result where $\lambda$ is chosen properly. Under Assumptions \[condition1\] and \[condition2\], for any $0<\delta<1$, with confidence $1-\delta$, there exists some positive constant $\tilde c$ such that $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)\leq \tilde c\log^2(6/\delta)l^{-\theta},\end{aligned}$$ where $$\begin{aligned} \theta= \left\{ \begin{array}{ll} \min\{\frac{2\beta}{2+p},\frac{2}{2+s}-\frac{2s}{(2+s)(2+p)}\}, & \hbox{ $\lambda=l^{-\frac{2}{2+p}}$;} \\ \min\{\frac{2\beta}{2\beta+s\beta+s},\frac{(2\beta+s\beta+s)(\beta-1)}{2+s}-\frac{2}{2+p}\}, & \hbox{ $\lambda=l^{-\frac{2}{2\beta+s\beta+s}}$.} \\ \end{array} \right.\end{aligned}$$ \[theorem2\] Theorem \[theorem2\] tells us that Fredholm regression has the learning rate with polynomial decay. When $s=p$, there exists some constant $\bar{c}>0$ such that $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)\leq \bar c\log(6/\delta)l^{-\theta}\end{aligned}$$ with confidence $1-\delta$, where $$\begin{aligned} \theta= \left\{ \begin{array}{ll} \frac{2\beta}{2+s},& \beta\in(0,\frac{2}{2+s}]; \\ \frac{2\beta}{s+2\beta+s\beta},& \beta\in(\frac{2}{2+s},+\infty]. \\ \end{array} \right.\end{aligned}$$ and the rate is derived by setting $$\begin{aligned} \lambda= \left\{ \begin{array}{ll} l^{-\frac{2}{2+s}},& \beta\in(0,\frac{2}{2+s}]; \\ l^{-\frac{2}{s+2\beta+s\beta}},& \beta\in(\frac{2}{2+s},+\infty]. \\ \end{array} \right.\end{aligned}$$ This learning rate can be arbitrarily close to $\mathcal O(l^{-1})$ as $s$ tends to zero, which is regarded as the fastest learning rate for regularized regression in the learning theory literature. This result verifies the LFK in (\[algorithm1\]) inherits the theoretical characteristics of least square regularized regression in RKHS [@cucker2; @wu2] and in data dependent hypothesis spaces [@shi1; @sun2]. Error analysis {#section4} ============== We first present the decomposition on the excess risk $\mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)$, and then establish the upper bounds of different error terms. Error decomposition {#section4.1} ------------------- According to the definitions of $f_{\mathbf z}, f_\lambda$, we can get the following error decomposition. \[proposition1\] For $f_\mathbf z$ defined in (\[algorithm1\]), there holds $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)\leq E_1+E_2+E_3+D(\lambda),\end{aligned}$$ where $$\begin{aligned} E_1&=&\mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)-(\mathcal E_{\mathbf z}(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E_{\mathbf z}(f_\rho)),\\ E_2&=&\mathcal E_{\mathbf z}(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E_{\mathbf z}(f_\rho)-(\mathcal E(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(f_\rho)),\end{aligned}$$ and $$E_3=\mathcal E(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(L_{w}f_{\lambda}).$$ [**[Proof]{}**]{}: By introducing the middle function $L_{w,\mathbf{x}}f_{\lambda}$, we get $$\begin{aligned} &&\mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho)\\ &\leq& \mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E_{\mathbf z}(\pi(L_{w,\mathbf{x}}f_{\mathbf z})) +[\mathcal E_{\mathbf z}(L_{w,\mathbf{x}}f_{\mathbf z})+\lambda\|f_{\mathbf z}\|_K^2-(\mathcal E_{\mathbf z}(L_{w,\mathbf{x}}f_{\lambda})+\lambda\|f_\lambda\|_K^2)]\\ &&+\mathcal E_{\mathbf z}(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(L_{w,\mathbf{x}}f_{\lambda})+\mathcal E(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(L_{w}f_{\lambda}) +\mathcal E(L_{w}f_{\lambda})-\mathcal E(f_\rho)+\lambda\|f_{\lambda}\|_K^2 \\ &\leq& E_1+E_2+E_3+D(\lambda)\end{aligned}$$ where the last inequality follows from the definition $f_{\mathbf z}$. This completes the proof.$\blacksquare$ In learning theory, $E_1,E_2$ are called the sample error, which describe the difference between the empirical risk and the expected risk. $E_3$ is called the hypothesis error which reflects the divergence of expected risks between the data independent function $L_wf_\lambda$ and data dependent function $L_{w,\mathbf{x}}f_\lambda$. Estimates of sample error {#subsection4.2} ------------------------- We introduce the concentration inequality in [@wu1] to measure the divergence between the empirical risk and the expected risk. Let $\mathcal F$ be a measurable function set on $\mathcal Z$. Assume that, for any $f\in\mathcal F$, $\|f\|_{\infty}\leq B$ and $E(f^2)\leq cEf$ for some positive constants $B, c$. If for some $a>0$ and $s\in(0, 2)$, $\log \mathcal{N}_2(\mathcal F, \varepsilon)\leq a\varepsilon^{-s}$ for any $\varepsilon>0$, then there exists a constant $c_s$ such that for any $\delta\in(0, 1)$, $$\begin{aligned} \Big|Ef-\frac{1}{m}\sum_{i=1}^mf(z_i)\Big| \leq c_s\max\{c^{\frac{2-s}{2+s}}, B^{\frac{2-s}{2+s}}\} (\frac{a}{m})^{\frac{2}{2+s}} +\frac{1}{2}Ef+\frac{(2c+18B)\log(1/\delta)}{m}\end{aligned}$$ with confidence at least $1-2\delta$. \[lemma1\] To estimate $E_1$, we consider the function set containing $f_{\mathbf z}$ for any $\mathbf{z}\in \mathcal{Z}^l$, $\mathbf{u}\in \mathcal{X}^u$. The definition $f_{\mathbf z}$ in (\[algorithm1\]) tells us that $\|f_{\mathbf z}\|_K\leq\frac{M}{\sqrt\lambda}$. Hence, $\forall \mathbf{z}\in \mathcal{Z}^l,f_{\mathbf z}\in B_R$ with $R=\frac{M}{\sqrt\lambda}$ and $\|f_{\mathbf z}\|_{\infty}\leq\frac{\kappa M}{\sqrt\lambda}$. \[proposition2\] Under Assumption \[condition1\], for any $0<\delta<1$, $$\begin{aligned} E_1 \leq\frac{1}{2}(\mathcal E(\pi(L_{w,\mathbf{x}}f_{\mathbf z}))-\mathcal E(f_\rho))+c_1\lambda^{-\frac{s}{2+s}}m^{-\frac{s}{2+s}} +176M^2l^{-1}\log(1/\delta)\end{aligned}$$ with confidence $1-\delta$. [**[Proof]{}**]{}: For $f\in B_R,\mathbf{z}\in \mathcal{Z}^l, \mathbf{x}\in\mathcal \mathcal{X}^{l+u}$, denote $$\begin{aligned} G_R=\{g(z)=(y-\pi(L_{w,\mathbf{x}}f))^2-(y-f_\rho(x))^2\}.\end{aligned}$$ For any $z\in \mathcal{Z}$, $$\begin{aligned} |g(z)|\leq|2y-\pi(L_{w,\mathbf{x}}f)(x)-f_\rho(x)||\pi(L_{w,\mathbf{x}}f)-f_\rho(x)| \leq 8M^2.\end{aligned}$$ Moreover, $$\begin{aligned} Eg^2\leq 16M^2E(\pi(L_{w,\mathbf{x}}f)(x)-f_\rho(x))^2=16M^2Eg.\end{aligned}$$ For any $f_1,f_2\in B_R$, there exists $$\begin{aligned} |g_1(z)-g_2(z)| \leq \frac{4M}{l+u}\Big|\sum_{i=1}^{l+u}(f_1(x_i)-f_2(x_i))w(x,x_i)\Big| \leq 4M\omega\|f_1-f_2\|_{\infty}.\end{aligned}$$ This relation implies that $$\begin{aligned} \log\mathcal{N}_{\infty}( G_R,\varepsilon)\leq\log\mathcal{N}_{\infty}(B_1,\frac{\varepsilon}{4M\omega R})\leq c_{s,K}(4M\omega R)^s\varepsilon^{-s},\end{aligned}$$ where the last inequality from Assumption \[condition1\]. Applying the above estimates to Lemma \[lemma1\], we derive that $$\begin{aligned} && Eg-\frac{1}{l}\sum_{i=1}^{l}g(z_i)\\ &\leq& \frac{1}{2}Eg+\max\{16M^2\omega,8M^2\}^{\frac{2-s}{2+s}} c_{s,K}^{\frac{2}{2+s}}(4M\omega R)^{\frac{2s}{2+s}}l^{-\frac{2}{2+s}} + 176M^2l^{-1}\log(1/\delta)\end{aligned}$$ with confidence $1-\delta$. Considering $f_\mathbf{z}\in B_R$ with $R=\frac{M}{\sqrt\lambda}$, we obtain the desired result. $\blacksquare$ \[proposition3\] Under Assumption 1, with confidence $1-4\delta$, there holds $$\begin{aligned} E_2\leq\frac{1}{2}E_3+\frac{1}{2}D(\lambda)+c_2D(\lambda)\lambda^{-1}l^{-\frac{2}{2+p}} \log(1/\delta),\end{aligned}$$ where $c_2$ is a positive constant independent of $\lambda, m, \delta$. [**[Proof]{}**]{}: Denote $$\begin{aligned} \mathcal{G}=\{g_{\mathbf{v},\lambda}:g_{\mathbf{v},\lambda}(x)=L_{w,\mathbf{v}}f_{\lambda}(x), x,v_i\in\mathcal{X}\}.\end{aligned}$$ From the definition $f_{\lambda}$, we can deduce that $\forall g\in\mathcal{G}, g\in\tilde{B}_R$ with $R=\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}}$. For $z\in \mathcal{Z}, \mathbf{v}\in\mathcal{X}^{l+u}$, define $$\mathcal{H}=\{h(z)=(y-L_{w,\mathbf{v}}f_{\lambda}(x))^2-(y-f_\rho(x))^2\}.$$ It is easy to check that for any $z\in \mathcal{Z}$ $$\begin{aligned} |h(z)|&=&|2y-L_{w,\mathbf{v}}f_{\lambda}(x)-f_\rho(x)|\cdot|L_{w,\mathbf{v}}f_{\lambda}(x)-f_\rho(x)| \nonumber\\ &\leq& (3M+\omega\|f_\lambda\|_{\infty})^2\leq\Big(3M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}}\Big)^2. \label{p11}\end{aligned}$$ Then, $$\begin{aligned} Eh^2&=&E(2y-L_{w,\mathbf{v}}f_{\lambda}(x)-f_\rho(x))^2(L_{w,\mathbf{v}}f_{\lambda}(x)-f_\rho(x))^2 \nonumber\\ &\leq&\Big(3M+wk\sqrt{\frac{D(\lambda)}{\lambda}}\Big)^2Eh. \label{p22}\end{aligned}$$ For any $\mathbf{u},\mathbf{v}\in\mathcal{X}^{l+u}$, there exists $$\begin{aligned} \|h_1-h_2\|_{\infty} &=&\sup_{z}|(y-L_{w,\mathbf{u}}f_{\lambda}(x))^2-(y-L_{w,\mathbf{v}}f_{\lambda}(x))^2| \\ &\leq&2\Big(M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}}\Big)\|L_{w,\mathbf{u}}f_{\lambda}-L_{w,\mathbf{v}}f_{\lambda}\|_{\infty} \\ &=&2\Big(M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}}\Big)\|g_{\mathbf{u},\lambda}-g_{\mathbf{v},\lambda}\|_{\infty}.\end{aligned}$$ Then from Assumption \[condition1\], $$\begin{aligned} \log\mathcal{N}_{\infty}(\mathcal{H},\varepsilon) \leq\log\mathcal{N}_{\infty} \Big(\tilde{B}_R,\frac{\varepsilon}{2(M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}})}\Big) \leq4c_{p,w}\Big(M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}}\Big)^{2p} \varepsilon^{-p}. \label{p33}\end{aligned}$$ Combining (\[p11\])-(\[p33\]) with Lemma \[lemma1\], we get with confidence $1-\delta$ $$\begin{aligned} E_2\leq\frac{1}{2}(\mathcal E(L_{w,\mathbf{x}}f_{\lambda})-f_\rho)+ (M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}})^2l^{-\frac{2}{2+p}}\\ \cdot c_p(4c_{p,w})^{\frac{2}{2+p}}+\frac{20(3M+\omega\kappa\sqrt{\frac{D(\lambda)}{\lambda}})\log(1/\delta)}{l}.\end{aligned}$$ Considering $ \mathcal E(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(f_\rho) \leq E_3+D(\lambda)$, we get the desired result.$\blacksquare$ Estimate of hypothesis error {#section4.3} ---------------------------- The following concentration inequality with values in Hilbert space can be found in [@pinelis], which is used in our analysis. \[lemma2\] Let $\mathcal H$ be a Hilbert space and $\xi$ be independent random variable on $Z$ with values in $\mathcal H$. Assume that $\|\xi\|_{\mathcal H}\leq \tilde M<\infty$ almost surely. Let $\{z_i\}_{i=1}^m$ be independent random samples from $\rho$. Then, for any $\delta\in(0,1)$, $$\begin{aligned} \Big\|\frac{1}{m}\sum_{i=1}^m\xi(z_i)-E\xi\Big\|_{\mathcal H}\leq\frac{2\tilde M\log(\frac{1}{\delta})}{M} +\sqrt{\frac{2E\|\xi\|_{\mathcal H}^2\log(\frac{1}{\delta})}{m}}\end{aligned}$$ holds true with confidence $1-\delta$. Now we turn to estimate $E_3$, which reflects the affect of inputs $\mathbf{x}=\{x_i\}_{i=1}^{l+u}$ to the regularization function $f_{\lambda}$. \[proposition\] For any $0<\delta<1$, with confidence $1-\delta$, there holds $$\begin{aligned} E_3\leq 24\omega^2\kappa^2\log^2(\frac{1}{\delta})D(\lambda)\lambda^{-1}(l+u)^{-1}+D(\lambda).\end{aligned}$$ [**[Proof]{}**]{}: Note that $$\begin{aligned} &&\mathcal E(L_{w,\mathbf{x}}f_{\lambda})-\mathcal E(L_{w}f_{\lambda}) \nonumber \\ &\leq& \|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2\cdot(\|L_{w,\mathbf{x}}f_{\lambda}-f_{\rho}\|_2+ \|L_{w}f_{\lambda}-f_{\rho}\|_2)\nonumber\\ &\leq& \|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2(\|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2+ 2\|L_{w}f_{\lambda}-f_{\rho}\|_2)\nonumber\\ &\leq& 2\|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2^2+\|L_{w}f_{\lambda}-f_{\rho}\|_2^2\nonumber\\ &\leq& 2\|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2^2+D(\lambda). \label{p111}\end{aligned}$$ Denote $\xi(x_i)=f_{\lambda}(x_i)w(\cdot,x_i)$, which is continuous and bounded function on $\mathcal X$. Then $$\begin{aligned} L_{w,\mathbf{x}}f_{\lambda}=\frac{1}{l+u}\sum_{i=1}^{l+u}\xi(x_i)\end{aligned}$$ and $$\begin{aligned} L_{w}f_{\lambda}=\int w(\cdot,t)f_{\lambda}(t)d\rho_{\mathcal X}(t)=E\xi.\end{aligned}$$ We can deduce that $\|\xi\|_2\leq \omega\|f_{\lambda}\|_{\infty}\leq \omega\kappa\|f_{\lambda}\|_K$ and $E\|\xi\|_2^2\leq \omega^2\kappa^2\|f_{\lambda}\|_K^2$. From Lemma \[lemma2\], for any $0<\delta<1$, there holds with confidence $1-\delta$ $$\begin{aligned} \|L_{w,\mathbf{x}}f_{\lambda}-L_{w}f_{\lambda}\|_2 \leq\frac{2\omega\kappa\|f_{\lambda}\|_K\log(\frac{1}{\delta})}{l+u} +\sqrt{\frac{2\log(\frac{1}{\delta})}{l+u}}\omega\kappa\|f_{\lambda}\|_K. \label{p222}\end{aligned}$$ Combining (\[p111\]) and (\[p222\]), we get with confidence $1-\delta$, $$\begin{aligned} E_3&\leq& 2(\frac{2\omega\kappa\|f_\lambda\|_K\log(\frac{1}{\delta})}{l+u}+\omega\kappa\|f_\lambda\|_K\sqrt {\frac{2\log(\frac{1}{\delta})}{l+u}})^2+D(\lambda)\\ &\leq&\frac{16\omega^2\kappa^2\|f_\lambda\|_K^2\log^2(\frac{1}{\delta})}{(l+u)^2} +\frac{8\omega^2\kappa^2\|f_\lambda\|_K^2\log(\frac{1}{\delta})}{l+u}+D(\lambda).\end{aligned}$$ Then, the desired result follows from $\|f_\lambda\|_K^2\leq\frac{D(\lambda)}{\lambda}$. $\blacksquare$ Proofs of Theorem 1 and 2 {#section4.4} ------------------------- [**Proof of Theorem 1:**]{} Combining the estimations in Propositions 1-4, we get with confidence $1-6\delta$, $$\begin{aligned} &&\mathcal E(\pi(L_{w,\mathbf{x}}f_\mathbf z))-\mathcal E(f_\rho)\\ &\leq&\frac{1}{2}(\mathcal E(\pi(L_{w,\mathbf{x}}f_\mathbf z))-\mathcal E(f_\rho))+c_1\lambda^{-\frac{s}{2+s}}l^{-\frac{2}{2+s}} +176M^2l^{-1}\log(\frac{1}{\delta})\\ &&+ 3D(\lambda)+c_2D(\lambda)\lambda^{-1}l^{-\frac{2}{2+p}}\log(\frac{1}{\delta}) +\frac{36w^2k^2\log^2(\frac{1}{\delta})}{l+u}\frac{D(\lambda)}{\lambda}.\end{aligned}$$ Considering $u>0$, for $0<\delta<1$, we have with confidence $1-6\delta$ $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_\mathbf z))-\mathcal E(f_\rho) \leq c\log^2(\frac{1}{\delta}) [\lambda^{-\frac{s}{2+s}}l^{-\frac{s}{2+s}}+\lambda^{\beta}+\lambda^{\beta-1}l^{-\frac{2}{2+p}}],\end{aligned}$$ where $c$ is a constant independent of $l,\lambda,\delta$. [**Proof of Theorem 2:**]{} When setting $\lambda^{\beta}=\lambda^{\beta-1}l^{-\frac{2}{2+p}}$, we obtain $\lambda=l^{-\frac{2}{2+p}}$. Then, Theorem \[theorem1\] implies that $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_\mathbf z))-\mathcal E(f_\rho) \leq 3c\log^2(\frac{1}{\delta}) l^{-\min\{\frac{2\beta}{2+p},\frac{2}{2+s}-\frac{2s}{(2+s)(2+p)}\}}.\end{aligned}$$ When setting $\lambda^{\beta}=\lambda^{-\frac{s}{2+s}}l^{-\frac{2}{2+s}}$, we get $\lambda=l^{-\frac{2}{2\beta+s\beta+s}}$. Then, with confidence $1-6\delta$ $$\begin{aligned} \mathcal E(\pi(L_{w,\mathbf{x}}f_\mathbf z))-\mathcal E(f_\rho) \leq 3c\log^2(\frac{1}{\delta}) l^{-\min\{\frac{2\beta}{2\beta+s\beta+s},\frac{(2\beta+s\beta+s)(\beta-1)}{2+s} +\frac{2}{2+p}\}}.\end{aligned}$$ This complete the proof of Theorem 2. Empirical studies {#section5} ================== To verify the effectiveness of LFK in (\[algorithm1\]), we present some simulated examples for the regression problem. The competing method is support vector machine regression (SVM), which has been used extensively used in machine learning community (https://www.csie.ntu.edu.tw/ cjlin/libsvm/). The Gaussian kernel $K(x,t)=\exp\{-\frac{\|x-t\|_2^2}{2\sigma^2}\}$ is used for SVM. For LFK in (\[algorithm1\]), we consider the following “inner” and “outer” kernels: - LFK1: $w(x,z)=x^Tz$ and $K(x,z)=\exp\{-\frac{\|x-t\|_2^2}{\sigma^2}\}$. - LFK2: $w(x,z)=\exp\{-\frac{\|x-t\|_2^2}{\sigma^2}\}$ and $K(x,z)=x^Tz$. - LFK3: $w(x,z)=\exp\{-\frac{\|x-t\|_2^2}{\sigma^2}\}$ and $K(x,z)=\exp\{-\frac{\|x-t\|_2^2}{\sigma^2}\}$. Here the scale parameter $\sigma$ belongs to $[2^{-5}:2:2^5]$ and the regularization parameter belongs to $[10^{-5}:10:10^5]$ for LFK and SVM. These parameters are selected by 4-fold cross validation in this section. The following functions are used to generate the simulated data: $$\begin{aligned} f_1(x)&=&sin\Big(\frac{9\pi}{0.35x+1}\Big),~~x\in[0,10]\\ f_2(x)&=&xcos(x),~~x\in[0,10]\\ f_3(x)&=&\min(2|x|-1,1),~~x\in[-2,2]\\ f_4(x)&=&sign(x),~~x\in[-3,3].\\\end{aligned}$$ Note that $f_1$ is highly oscillatory, $f_2$ is smooth, $f_3$ is continuous not smooth, and $f_4$ is not even continuous. These functions have been used to evaluate regression algorithms in [@sun2]. [c|ccccc]{} Function &Number &SVM &LFK1 &LFK2 &LFK3\ $f_1$ &50 &$0.041\pm0.033$ &$0.434\pm0.032$ &$0.423\pm0.059$ &$\mathbf{0.036\pm0.053}$\ &300 &$0.044\pm0.006$ &$0.419\pm0.023$ &$0.404\pm0.021$ &$\mathbf{0.042\pm0.006}$\ $f_2$ &50 &$0.075\pm0.046$ &$18.52\pm1.30$ &$18.7\pm1.30$ &$\mathbf{0.060\pm0.028}$\ &300 &$\mathbf{0.011}\pm0.006$ &$17.10\pm0.941$ &$17.00\pm1.35$ &$0.012\pm\mathbf{0.004}$\ $f_3$ &50 &$0.013\pm 0.012$ &$0.670\pm0.034$ &$0.458\pm0.082$ &$\mathbf{0.010\pm0.005}$\ &300 &$0.004\pm0.001$ &$0.667\pm0.013$ &$0.427\pm0.020$ &$\mathbf{0.003\pm0.001}$\ $f_4$ &50 &$0.076\pm0.027$ &$0.260\pm\mathbf{0.012}$ &$0.194\pm0.040$ &$\mathbf{0.073}\pm0.021$\ &300 &$0.039\pm0.018$ &$0.251\pm0.017$ &$0.158\pm0.026$ &$\mathbf{0.032\pm0.009}$\ \[tab1\] In our experiment, Gaussian noise $N(0,0.01)$ is added to the data respectively. In each test, we first draw randomly 1000 samples according to the function and noise distribution, and then obtain a training set randomly with sizes $25, 50, 100, 200, 300$ respectively. Three hundred samples are selected randomly as the test set. The *Mean Squared Error* (MSE) is used to evaluate the regression results on synthetic data. To make the results more convincing, each test is repeated 10 times. Table \[tab1\] reports the average MSE and *Standard Deviation* (STD) with 50 training samples and 300 training samples respectively. Furthermore, we study the impact of the number of training samples on the final regression performance. Figure 1 shows the MSE for learning $f_1-f_4$ with numbers of training samples. These results illustrate that LFK has competitive performance compared with SVM. Conclusion {#section6} ========== This paper investigated the generalization performance of regularized least square regression with Fredholm kernel. Generalization bound is presented for the Fredholm learning model, which shows that the fast learning rate with $O(l^{-1})$ can be reached. In the future, it is interesting to investigate the leaning performance of ranking [@hong2] with Fredholm kernel. Acknowledgments {#acknowledgments .unnumbered} --------------- The authors would like to thank Prof.Dr.L.Q. Li for his valuable suggestions. This work was supported by the National Natural Science Foundation of China(Grant Nos. 11671161) and the Fundamental Research Funds for the Central Universities (Program Nos. 2662015PY046, 2014PY025). References {#references .unnumbered} ========== [99]{} M. Belkin, P. Niyogi, and V. Sindhwani, “Manifold regularization: A geometric framework for learning from labeled and unlabeled examples,” *J. Mach. Learn. Res.*, vol. 7, pp. 2399–2434, 2006. Q. Que and M. Belkin, “Inverse density as an inverse problem: the fredholm equation approach,” In *NIPS*, pp. 1484–1492, 2013. Q. Que, M. Belkin, and Y. 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Feng, S. Lv, H. Huang, and J. Suykens, “Kernelized elastic net reguularization: generalization bouunds and sparse recovery,” *Neural Comput.*, vol. 28, pp. 1–38, 2016. I. Pinelis, “Optimum bounds for the distribution of martingales in Banach spaces,” *Ann. Probab.*, vol. 22, pp. 1679–1706, 1994. L. Shi, Y. Feng, and D.X. Zhou, “Concentration estimates for learning with $\ell_1$-regularizer and data dependent hypothesis spaces,” *Appl. Comput. Harmon. Anal.*, vol. 31, no. 2, pp. 286–302, 2011. H. Sun and Q. Wu, “Least square regression with indefinite kernels and coefficient regularization,” *Appl. Comput. Harmon. Anal.*, vol. 30, no. 1, pp. 96–109, 2011. H. Sun and Q. Wu, “Sparse representation in kernel machines,” *IEEE Trans. Neural Netw. Learning Syst.*, vol. 26, no. 10, 2576–2582, 2015. Q. Wu, Y. Ying, and D.X. Zhou, “Multi-kernel regularized classfiers,” *J. Complexity*, vol. 23, pp. 108–134, 2007. Q. Wu, Y.M. Ying, and D.X. 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ArXiv

--- abstract: 'In this article we study the representations of general linear groups which arise from their action on flag spaces. These representations can be decomposed into irreducibles by proving that the associated Hecke algebra is cellular. We give a geometric interpretation of a cellular basis of such Hecke algebras which was introduced by Murphy in the case of finite fields. We apply these results to decompose representations which arise from the space of submodules of a free module over principal ideal local rings of length two with a finite residue field.' address: - 'Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105 Israel' - 'Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva 84105 Israel' author: - Uri Onn - 'Pooja Singla${}^\dag$' title: | Geometric Interpretation of Murphy\ Bases and an Application --- Introduction ============ Flags of vector spaces {#field} ---------------------- Let ${k}$ be a finite field and let $n$ be a fixed positive integer. Let $G={\text{GL}}_n({k})$ be the group of $n$-by-$n$ invertible matrices over ${k}$ and let $\Lambda_n$ stand for the set of partitions of $n$. For $\lambda=({{\lambda}}_i) \in \Lambda_n$, written in a non-increasing order, let $l(\lambda)$ denote its length, namely the number of non-zero parts. The set $\Lambda_n$ is a lattice under the opposite dominance partial order, defined by: ${{\lambda}}\le \mu$ if $\sum_{j=1}^i{{\lambda}}_j \ge\sum_{j=1}^i\mu_j$ for all $i \in \mathbb{N}$. Let $\vee$ and $\wedge$ denote the operations of join and meet, respectively, in the lattice $\Lambda_n$. We call a chain of ${k}$-vector spaces ${k}^n=x_{l(\lambda)} \supset x_{l(\lambda)-1} \supset \cdots \supset x_{0} = (0)$ a ${{\lambda}}$-flag if $\dim_{{k}}(x_{l({{\lambda}})-i+1}/x_{l({{\lambda}})-i}) = {{\lambda}}_i$ for all $1 \le i \le l({{\lambda}})$. Let $$X_\lambda=\{(x_{l(\lambda)-1},\cdots,x_{1}) \mid {k}^n = x_{l(\lambda)} \supset \cdots \supset x_{0} = (0)~\text{is a $\lambda$-flag} \},$$ be the set of all ${{\lambda}}$-flags in ${k}^n$. Let ${{\mathcal F}}_{{\lambda}}$ be the permutation representation of $G$ that arises from its action on $X_{{{\lambda}}}$ (${{\lambda}}\in \Lambda_n$). Specifically, ${{\mathcal F}}_{{{\lambda}}}={{\mathbb Q}}(X_\lambda)$ is the vector space of ${{\mathbb Q}}$-valued functions on $X_{{{\lambda}}}$ endowed with the natural $G$-action: $$\begin{split} \rho_{{{\lambda}}}: G& \rightarrow \text{Aut}_{{{\mathbb Q}}}({{\mathcal F}}_{{{\lambda}}}) \\ g& \mapsto [\rho_{{{\lambda}}}(g)f](x) = f(g^{-1}x). \end{split}$$ Let ${{\mathcal H}}_{{{\lambda}}} = {\text{End}}_G({{\mathcal F}}_{{{\lambda}}})$ be the Hecke algebra associated to ${{\mathcal F}}_{{\lambda}}$. The algebra ${{\mathcal H}}_\lambda$ captures the numbers and multiplicities of the irreducible constituents in ${{\mathcal F}}_\lambda$. The notion of Cellular Algebra, to be described in Section \[preliminaries\], was defined by Graham and Lehrer in [@MR1376244]. Proving that the algebra ${{\mathcal H}}_{{{\lambda}}}$ is cellular gives, in particular, a classification of the irreducible representations of ${{\mathcal H}}_{{\lambda}}$ and hence also gives the decomposition of ${{\mathcal F}}_{{{\lambda}}}$ into irreducible constituents. Murphy [@MR1194316; @MR1327362] gave a beautiful description of a cellular basis of the Hecke algebras of type $A_n$ denoted ${{\mathcal H}}_{R, q}(S_n)$; cf. [@MR1711316]. For $q=|k|$ one has ${{\mathcal H}}_{(1,...,1)} \simeq {{\mathcal H}}_{\mathbb Q, q}(S_n)$. Dipper and James [@MR812444] (see also [@MR1711316]) generalized this basis and constructed cellular bases for the Hecke algebras ${{\mathcal H}}_{{{\lambda}}}$. The first result in this paper is a new construction of this basis which is of geometric nature. More specifically, the characteristic functions of the orbits of the diagonal $G$-action on $X_{{\lambda}}\times X_\mu$ gives a basis of the Hecke modules ${{\mathcal N}}_{\mu{{\lambda}}}={\text{Hom}}_G({{\mathcal F}}_{{\lambda}},{{\mathcal F}}_\mu)$. For $\mu \leq {{\lambda}}$ we allocate a subset of these orbits denoted ${{\mathcal C}}_{\mu{{\lambda}}}$ such that going over all the compositions ${{\mathcal C}}^{\mathrm{op}}_{\mu{{\lambda}}} \circ {{\mathcal C}}_{\mu{{\lambda}}}$ and all $\mu \le {{\lambda}}$ gives the desired basis. The benefit of this description turns out to be an application in the following setting. Flags of $\mathfrak {o}$-modules {#ring} -------------------------------- Let $\mathfrak {o}$ be a complete discrete valuation ring. Let $\mathfrak{p}$ be the unique maximal ideal of $\mathfrak {o}$ and $\pi$ be a fixed uniformizer of $\mathfrak{p}$. Assume that the residue field $ {k}= \mathfrak {o}/\mathfrak{p}$ is finite. The typical examples of such rings are $\mathbf Z_{p}$ (the ring of $p$-adic integers) and $\mathbf F_q [[t]]$ (the ring of formal power series with coefficients over a finite field). We denote by $\mathfrak {o}_{\ell}$ the reduction of $\mathfrak {o}$ modulo $\mathfrak{p}^{\ell}$, i.e., $\mathfrak {o}_{\ell} = \mathfrak {o}/\mathfrak{p}^{\ell}$. Since $\mathfrak {o}$ is a principal ideal domain with a unique maximal ideal $\mathfrak{p}$, every finite $\mathfrak {o}$-module is of the form $\oplus_{i=1}^{j}\mathfrak {o}_{{{\lambda}}_{i}}$, where ${{\lambda}}_{i}$’s can be arranged so that $\lambda = ({{\lambda}}_1,\ldots,{{\lambda}}_j) \in \Lambda=\cup \Lambda_{n}$. Let ${\text{GL}_{n}({\mathfrak {o}}_{\ell})}$ denote the group of $n$-by-$n$ invertible matrices over $\mathfrak {o}_{\ell}$. We are interested in the following generalization of the discussion in §\[field\]. Let $${{\mathcal L}}^{(r)}={{\mathcal L}}^{(r)}({\ell^n})=\{(x_r,\cdots,x_{1}) \mid \mathfrak {o}_{\ell}^n \supset x_r \supset \cdots \supset x_{0} = (0),~\text{$x_i$ are $\mathfrak {o}$-modules}\}$$ be the space of flags of length $r$ of submodules in $\mathfrak {o}_{\ell}^n$. Let $\Xi \subset {{\mathcal L}}^{(r)}$ denote an orbit of the ${\text{GL}_{n}({\mathfrak {o}}_{\ell})}$-action on ${{\mathcal L}}^{(r)}$. Let ${{\mathcal F}}_\Xi=\mathbb{Q}(\Xi)$ be the corresponding permutation representation of ${\text{GL}_{n}({\mathfrak {o}}_{\ell})}$. One is naturally led to the following related problems: \[p1\] Decompose ${{\mathcal F}}_\Xi$ to irreducible representations. \[p2\] Find a cellular basis for the algebra ${{\mathcal H}}_\Xi={\text{End}}_{{\text{GL}_{n}({\mathfrak {o}}_{\ell})}}({{\mathcal F}}_\Xi)$. Few other cases, beside the field case ($\ell=1$) which is our motivating object, were treated in the literature. The Grassmannian of free $\mathfrak{o}_\ell$-modules, i.e., the case $r=1$ and $x_1 \simeq \mathfrak{o}_\ell^m$ is treated fully in [@BO2; @MR2283434]. The methods therein are foundational to the present paper. Another case which at present admits a very partial solution is the case of complete free flags in $\mathfrak{o}_\ell^3$; cf. [@MR2504482]. In this paper we treat the first case which is not free but we restrict ourselves to level $2$, that is, we look at the Grassmannian of arbitrary $\mathfrak{o}_2$-modules of type ($2^a1^b$) in $\mathfrak{o}_2^n$. To solve this problem we are naturally led to consider certain spaces of $2$-flags of $\mathfrak{o}_2$-modules as well. We give a complete solution to problems \[p1\] and \[p2\] in these cases. Preliminaries ============= Hecke algebras and Hecke modules -------------------------------- For ${{\lambda}},\mu \in \Lambda_n$, we let ${{\mathcal N}}_{{{\lambda}}\mu}={\text{Hom}}_G({{\mathcal F}}_\mu,{{\mathcal F}}_{{\lambda}})$ denote the ${{\mathcal H}}_{{\lambda}}$-${{\mathcal H}}_\mu$-bimodule of intertwining $G$-maps from ${{\mathcal F}}_\mu$ to ${{\mathcal F}}_{{\lambda}}$. The modules ${{\mathcal N}}_{{{\lambda}}\mu}$, and in particular the algebras ${{\mathcal H}}_\lambda$, have natural geometric basis  indexed by $X_\lambda \times_G X_\mu$, the space of $G$-orbits in $X_\lambda \times X_\mu$ with respect to the diagonal $G$-action. Specifically, for $\Omega \in X_{{{\lambda}}} \times_G X_{\mu}$, let $$\label{geometric.basis} \gb_\Omega f (x)= \sum_{y:(x,y) \in \Omega} f(y), \qquad f \in {{\mathcal F}}_\mu,\, x \in X_{{\lambda}}.$$ Then $\{\gb_\Omega \mid \Omega \in X_{{\lambda}}\times _G X_\mu \}$ is a basis of ${{\mathcal N}}_{{{\lambda}}\mu}$. Let ${{\mathcal M}}_{{{\lambda}}\mu}$ be the set of $l({{\lambda}})$-by-$l(\mu)$ matrices having non-negative integer entries with column sum equal to $\mu$ and row sum equal to ${{\lambda}}$, namely $$\label{intersection-matrix} {{\mathcal M}}_{{{\lambda}}\mu} = \{(a_{ij}) \mid a_{ij} \in {{\mathbb Z}}_{\geq 0},\,\sum_{i=1}^{l({{\lambda}})} a_{ij} = \mu_j, \sum_{j=1}^{l(\mu)} a_{ij} = {{\lambda}}_i \}.$$ Geometrically, the orbits in $X_{{{\lambda}}} \times_G X_{\mu}$ characterize the relative positions of ${{\lambda}}$-flags and $\mu$-flags in ${k}^n$ and hence are in bijective correspondence with the set ${{\mathcal M}}_{{{\lambda}}\mu}$. The bijection $$\label{orbits-matrices} X_{{{\lambda}}} \times_G X_{\mu} \longleftrightarrow {{\mathcal M}}_{{{\lambda}}\mu},$$ is obtained by mapping the pair $(x, y) \in X_{{{\lambda}}} \times X_{\mu}$ to its intersection matrix $(a_{ij}) \in {{\mathcal M}}_{{{\lambda}}\mu}$, defined by $$\label{intersection matrix} a_{ij} = \dim_{{k}}\left( \frac{x_i \cap y_j}{x_i \cap y_{j-1} + x_{i-1} \cap y_j}\right).$$ The RSK Correspondence {#subsec:Young} ---------------------- A Young diagram of a partition $\mu \in \Lambda_n$ is the set $[\mu] = \{(i,j) \mid 1 \leq j \leq \mu_i \, \mathrm{and} \, 1 \le i \le l(\mu) \} \subset \mathbb N \times \mathbb N$. One usually represent it by an array of boxes in the plane, e.g. if $\mu = (3,2)$ then $[\mu] = {\tiny \yng(3,2)}$. A $\mu$-tableau $\Theta$ is a labeling of the boxes of $[\mu]$ by natural numbers. The partition $\mu$ is called the shape of $\Theta$ and denoted $\mathrm{Shape}(\Theta)$. A Young tableau is called [*semistandard*]{} if its entries are increasing in rows from left to right and are strictly increasing in columns from top to bottom. A semistandard tableau of shape $\mu$ with $\sum \mu_i = n$ is called [*standard*]{} if its entries are integers from the set $\{1, 2, \ldots, n\}$, each appearing exactly once and strictly increasing from left to right as well. Given partitions $\nu$ and $\mu$, a tableau $\Theta$ is called of shape $\nu$ and type $\mu$  if it is of shape $\nu$ and each natural number $i$ occurs exactly $\mu_i$ times in its labeling. We denote by $\mathrm{std}(\nu)$ and ${\mathrm{sstd}}(\nu\mu)$ the set of all standard $\nu$-tableaux and set of semistandard $\nu$-tableaux of type $\mu$, respectively. We remark that the set ${\mathrm{sstd}}(\nu\mu)$ is nonempty if and only if $\nu \le \mu$. The RSK correspondence is an algorithm which explicitly defines a bijection $${{\mathcal M}}_{{{\lambda}}\mu} \longleftrightarrow \bigsqcup_{\nu \le {{\lambda}}\wedge \mu } {\mathrm{sstd}}(\nu {{\lambda}}) \times {\mathrm{sstd}}(\nu \mu),$$ where ${{\lambda}}\wedge \mu$ is the meet of ${{\lambda}}$ and $\mu$. For more details on this see [@MR0272654]. \[Embedding of flags\] We say that a $\nu$-flag $y$ is [*embedded*]{} in a $\mu$-flag $x$, denoted $y \hookrightarrow x$, if $l(\nu) \leq l(\mu)$ and $y_{l(\nu)-i} \subset x_{l(\mu)-i}$ for all $1\leq i \leq l(\nu)$. The intersection matrix of each embedding of $\nu$-flag into $\mu$-flag determines a $\nu$-tableau of type $\mu$ as follows: for any intersection matrix $E = (a_{ij}) \in {{\mathcal M}}_{\nu{{\lambda}}}$, construct the Young tableau with $a_{ij}$ many $i$’s in its $j^{\mathrm{th}}$ row. We call an embedding of $\nu$-flag into a $\mu$-flag [*permissible*]{} if the $\nu$-tableau obtained is semistandard. The set ${{\mathcal M}}_{\nu{{\lambda}}}^{\circ}$ denotes the subset of ${{\mathcal M}}_{\nu{{\lambda}}}$ consisting of intersection matrices that corresponds to permissible embeddings. The following gives a reformulation of the RSK correspondence purely in terms of intersection matrices: $$\label{the.RSK.in.terms.of.matrices} {{\mathcal M}}_{{{\lambda}}\mu} \longleftrightarrow \bigsqcup_{\nu \le {{\lambda}}\wedge \mu} {{\mathcal M}}_{\nu {{\lambda}}}^\circ \times {{\mathcal M}}_{\nu \mu}^\circ.$$ For partitions $\nu \le {{\lambda}}$ of $n$ we let $(X_\nu \times X_{{\lambda}})^\circ$ denote the subset of $X_\nu \times X_{{\lambda}}$ which consists of pairs $(z,x)$ such that $z$ is permissibly embedded in $x$. The orbits $(X_\nu \times_G X_{{\lambda}})^\circ$ are therefore parameterized by ${{\mathcal M}}_{\nu {{\lambda}}}^\circ$. This gives a purely geometric reformulation of the RSK correspondence: $$\label{geometric.RSK} X_{{{\lambda}}} \times_G X_{\mu} \tilde{\longrightarrow} \bigsqcup_{\nu \le {{\lambda}}\wedge \mu} \left(X_\nu \times_G X_{{\lambda}}\right)^\circ \times \left(X_\nu \times_G X_\mu\right)^\circ.$$ The gist of is that both sides have geometric interpretations. We remark that the above bijection is an important reason behind the cellularity of MDJ basis (see Section \[M-DJ bases\]). Cellular Algebras {#Cellularity} ----------------- Cellular algebras were defined by Graham and Lehrer in [@MR2283434]. We use the following equivalent formulation from Mathas [@MR1711316]. \[cellular algebra\] Let $K$ be a field and let $A$ be an associative unital $K$-algebra. Suppose that $(\zeta, \geq)$ is a finite poset and that for each $\tau \in \zeta$ there exists a finite set $\mathcal{T}(\tau)$ and elements $c_{st}^{\tau} \in A$ for all $s, t \in \mathcal{T}(\tau)$ such that $\mathcal{C} = \{ c_{st}^{\tau} \mid \tau \in \zeta \,\,\text{and}\,\,s, t \in \mathcal{T}(\tau) \}$ is a basis of $A$. For each $\tau \in \zeta$ let $\tilde{A}^{\tau}=\mathrm{Span}_K \{c_{uv}^{\omega} \mid \omega \in \zeta,\,\, \omega > \tau \,\,\text{and} \,\,u, v \in \mathcal{T}(\omega) \}$. The pair $(\mathcal{C}, \zeta)$ is a cellular basis of $A$ if 1. The $K$-linear map $\star : A \rightarrow A$ determined by $c_{st}^{\tau \star} = c_{ts}^{\tau}$ ($\tau \in \zeta, s,t \in \mathcal{T}(\tau)$) is an algebra anti-homomorphism of $A$; and, 2. for any $\tau \in \zeta$, $t \in \mathcal{T}(\tau)$ and $a \in A$ there exists $\{\alpha_{v} \in K \mid v \in {{\mathcal T}}(\tau)\}$ such that for all $s \in \mathcal{T}(\tau)$ $$\label{cellular condition} a \cdot c_{st}^{\tau} = \sum_{v \in {{\mathcal T}}(\tau)} \alpha_{v}c_{vt}^{\tau} \,\,\text{mod}\, \tilde{A}^{\tau}.$$ If $A$ has a cellular basis then $A$ is called a cellular algebra. The result about semisimple cellular algebras which we shall need is the following. Let $A$ be a semisimple cellular algebra with a fixed cellular basis $(\mathcal C = \{c_{st}^{\tau} \}, \zeta)$. For $\tau \in \zeta$ let $A^{\tau}$ be the $K$-vector space with basis $\{ c_{uv}^{\mu} \mid \mu \in \zeta, \mu \geq \tau ~\mathrm{and}~ u,v \in \mathcal T(\mu)\}$. Thus $\tilde A^{\tau} \subset A^{\tau}$ and $A^{\tau} / \tilde A^{\tau}$ has basis $c_{st}^{\tau} + \tilde A^{\tau}$ where $s, t \in \mathcal{T}(\tau)$. It is easy to prove that $A^{\tau}$ and $\tilde A^{\tau}$ are two sided ideals of $A$. Further if $s,t \in {{\mathcal T}}(\tau)$, then there exists an element $\alpha_{st} \in K$ such that for any $u, v \in {{\mathcal T}}(\tau)$ $$c_{us}^{\tau} c_{tv}^{\tau} = \alpha_{st} c_{uv}^{\tau} \,\, \mathrm{mod} \,\, \tilde A^{\tau}.$$ For each $\tau \in \zeta$ the cell modules $C^{\tau}$ is defined as the left $A$ module with $K$-basis $\{ b_{t}^{\tau} \mid t \in \mathcal {T}(\tau) \}$ and with the left $A$ action: $$a \cdot b_{t}^{\tau} = \sum_{v \in \mathcal{T}({{\lambda}}) } \alpha_{v} b_{v}^{\tau},$$ for all $a \in A$ and $\alpha_{v}$ are as given in the Definition \[cellular algebra\]. Furthermore, dual to $C^{\tau}$ there exists a right $A$-modules $C^{\tau *}$ which has the same dimension over $K$ as $C^{\tau}$, such that the $A$-modules $C^{\tau } \otimes_{K} C^{\tau *}$ and $A^{\tau}/\tilde{A}^{\tau}$ are canonically isomorphic. [@MR2283434 Lemma 2.2 and Theorem 3.8] Suppose $\zeta$ is finite. Then $\{ C^{\tau} \, | \, \tau \in \zeta \, \mathrm{and} \, C^{\tau} \neq 0 \}$ is a complete set of pairwise inequivalent irreducible $A$-modules. Let $\zeta^{+}$ be the set of elements $\tau \in \zeta$ such that $C^{\tau} \neq 0$. Then $A \cong \oplus_{\tau \in \zeta^{+}} C^{\tau } \otimes_{K} C^{\tau * }$. Another description of Murphy-Dipper-James bases ================================================ MDJ Bases {#M-DJ bases} --------- For a positive integer $n$, let $S_n$ be the symmetric group of $\{ 1,2, \ldots, n\}$. Let $S$ be the subset of $S_n$ consisting of the transpositions $(i, i+1)$. Let $R$ be a commutative integral domain and let $q$ be an arbitrary element of $R$. The Iwahori-Hecke algebra $\mathcal H_{R, q}(S_n)$ is the free $R$-module generated by $\{ T_{\omega} \mid \omega \in S_n \}$ with multiplication given by $$T_{w} T_s = \begin{cases} T_{ws} & \text{if} \;\; \ell(ws) > \ell(w), \\ qT_{ws} + (q-1)T_w & \text{otherwise}, \end{cases}$$ where $\ell(w)$ denotes the length of $w \in S_n$. Also $\star: \mathcal H_{R, q}(S_n) \rightarrow \mathcal H_{R, q}(S_n)$ denotes an algebra anti-involution defined by $T^{\star}_{\omega} = T_{\omega^{-1}}$. For a partition $\mu$, let $S_{\mu}$ be the subset of $S_n$ consisting of all permutations leaving the sets $\{\sum_{i=1}^{j-1} \mu_i +1,....,\sum_{i=1}^{j} \mu_i \}$ invariant for all $1 \leq j \leq l(\mu)$ and let $m_{\mu} = \sum_{\omega \in S_{\mu}} T_{\omega}$. Let ${{\mathcal N}}_{{{\lambda}}\mu}^q$ denote the free $R$-module $m_{{{\lambda}}} {{\mathcal H}}_{R, q}(S_n) m_{\mu}$.\ For each partition $\nu$ of $n$, let $\phi^{\nu}$ be the unique $\nu$-standard tableau in which the integers $\{1, 2, \ldots, n\}$ are entered in increasing order from left to right along the rows of $[\nu]$. For each $\nu$-standard tableau $\theta$ define the permutation matrix $d(\theta)$ by $\theta = \phi^{\nu} d(\theta)$. For any standard $\nu$-tableau $\theta$ and partition $\mu$ of $n$ such that $\nu \leq \mu$, we obtain a semistandard $\nu$-tableau of type $\mu$, denoted $\mu(\theta)$, by replacing each entry $i$ in $\theta$ by $r$ if $i$ appears in row $r$ of $\phi^{\mu}$. For given partitions $\mu$ and $\nu$, let $\Theta_1 \in \mathrm{sstd}(\nu, {{\lambda}})$ and $\Theta_2 \in \mathrm{sstd}(\nu, \mu)$, define $$m_{\Theta_1 \Theta_2} = \sum_{\theta_1, \theta_2} m_{\theta_1 \theta_2},$$ where $m_{\theta_1 \theta_2} = T^{\star}_{d(\theta_1)} m_{\nu} T_{d(\theta_2)}$ and the sum is over all pairs $(\theta_1,\theta_2)$ of standard $\nu$-tableau such that ${{\lambda}}(\theta_1) = \Theta_1$ and $\mu(\theta_2) = \Theta_2$. Let $$\mathbb M_{\mu{{\lambda}}} = \{m_{\Theta_1 \Theta_2} \mid \Theta_1 \in \mathrm{sstd}(\nu, {{\lambda}}), \Theta_2 \in \mathrm{sstd}(\nu, \mu), \nu \leq {{\lambda}}\wedge \mu \},$$ and for any partition $\mu \in \Lambda_n$, let $\Lambda_{\mu}=\{ \nu \in \Lambda_n \mid \nu \le \mu\}$. Then \[thm:M-DJ\] The set $(\mathbb M_{\mu{{\lambda}}}, \Lambda_{{{\lambda}}\wedge \mu})$ is an $R$-basis of the Hecke module ${{\mathcal N}}^q_{\mu{{\lambda}}}$. See Mathas [@MR1711316 Theorem 4.10, Corollary 4.12]. \[rk:M-DJ\] The following observation from the proof is important for us. For any semistandard $\nu$-tableau $\Theta$, let $\mathrm{first} (\Theta)$ be the unique row standard $\nu$-tableau such that ${{\lambda}}(\mathrm{first}(\Theta)) = \Theta$. For $\Theta \in \mathrm{sstd}(\nu, {{\lambda}})$, $$\label{remark from mathas} {\bold{G}}_{\Theta} := \sum_{\theta \in \mathrm{std}(\nu), \atop {{\lambda}}(\theta) = \Theta} m_{\nu}^{\star} T_{d(\theta)} = \sum_{\omega \in S_{\nu} \sigma S_{{{\lambda}}}} T_{\omega},$$ where $\sigma \in S_n$ is the unique permutation matrix satisfying $\sigma = d(\mathrm{first}(\Theta))$ (see also the Remark \[echelon form\]). Any partition $\delta = (\delta_i)$ associates $l(\delta)$ many $\delta$-row ($\delta$-column) submatrices with a given $n \times n$ matrix $A$ by taking its rows (columns) from $\sum_{i=0}^j {{\lambda}}_i +1$ to $\sum_{i=0}^j {{\lambda}}_{i+1}$ for all $0 \leq j \leq l(\delta)-1$. (${{\lambda}}\mu$-Echelon form) A matrix $A$ is called in ${{\lambda}}\mu $-Echelon form if its associated ${{\lambda}}$-row and $\mu$-column sub-matrices are in row reduced and column reduced Echelon form respectively. \[echelon form\] The matrices $\sigma_1$ and $\sigma_2$ appearing in the proof of Theorem \[thm:M-DJ\] and Remark \[rk:M-DJ\] are in ${{\lambda}}\nu$ and $\mu\nu$ Echelon form respectively. Geometric interpretation of the MDJ Bases ----------------------------------------- Recall RSK correspondence is an algorithm that explicitly defines the correspondence: $$\xymatrix{ {{\mathcal M}}_{{{\lambda}}\mu} \ar@{<->}[r] \ar@{<->}[d] & \bigsqcup_{\nu \in \Lambda_{{{\lambda}}\wedge \mu}} (X_{\nu} \times_G X_{{{\lambda}}})^{\circ} \times (X_{\nu} \times_G X_{\mu})^{\circ} \ar@{<->}[d] \\ \bigsqcup_{\nu \in \Lambda_{{{\lambda}}\wedge \mu}} {{\mathcal M}}_{\nu{{\lambda}}}^{\circ} \times {{\mathcal M}}_{\nu\mu}^{\circ} \ar@{<->}[r] & \bigsqcup_{\nu \in \Lambda_{{{\lambda}}\wedge \mu}} \mathrm{sstd}(\nu, {{\lambda}}) \times \mathrm{sstd}(\nu, \mu), }$$ where the upper left corner consists of intersection matrices, the upper right consists of orbits of permissible embeddings, the lower left corner consists of intersection matrices of permissible embedding and the lower right of semistandard tableaux. For $\nu \in \Lambda_{{{\lambda}}\wedge \mu}$ and orbits $ \Omega_1 \in \left(X_\nu \times_G X_{{\lambda}}\right)^\circ$, $\Omega_2 \in \left(X_{\nu} \times_G X_{\mu} \right)^{\circ}$ define $$\cb^\nu_{\Omega_1\Omega_2}:=\gb_{\Omega_1^{\text{op}}} \circ \gb_{\Omega_2},$$ where $[(x,y)]^{\text{op}}=[(y,x)]$. Clearly, $\cb^\nu_{\Omega_1\Omega_2} \in {{\mathcal N}}_{{{\lambda}}\mu}$. For any partition $\nu$, let $P_{\nu}$ be the stabilizer of standard $\nu$-flag in $G$ and $B$ be the Borel subgroup of $G$, that is the subgroup consisting of all invertible upper triangular matrices. Let ${{\mathcal C}}_{\mu {{\lambda}}} = \{\cb_{\Omega_1\Omega_2}^\nu \mid \nu \in \Lambda_{{{\lambda}}\wedge \mu},\,\,\Omega_1 \in \left(X_\nu \times_G X_{{\lambda}}\right)^\circ, \Omega_2 \in (X_{\nu} \times_G X_{\mu})^{\circ} \}$. \[thm:Murphy=our\] The set $({{\mathcal C}}_{\mu{{\lambda}}}, \Lambda_{{{\lambda}}\wedge \mu})$ is a $\mathbb Q$-basis of ${{\mathcal N}}_{{{\lambda}}\mu}$. We prove this by proving that if $q$ is the cardinality of field ${k}$ then the set $({{\mathcal C}}_{\mu{{\lambda}}}, \Lambda_{{{\lambda}}\wedge \mu})$ coincides with MDJ basis of ${{\mathcal N}}_{{{\lambda}}\mu}^q$ up to a scalar. That is, if the orbits $\Omega_1$ and $\Omega_2$ correspond to semistandard tableau $\Theta_1$ and $\Theta_2$ by RSK, then $$\cb_{\Omega_1 \Omega_2}^{\nu} = \frac{|P_{\nu}|}{|B|} m_{\Theta_1 \Theta_2}.$$ By definition of $m_{\Theta_1\Theta_2}$ and (\[remark from mathas\]), $$\begin{aligned} m_{\Theta_1 \Theta_2} = \sum_{\theta_1, \theta_2} m_{\theta_1 \theta_2} = \sum_{\theta_1, \theta_2} T^{\star}_{d(\theta_1)} m_{\nu} T_{d(\theta_2)}.\end{aligned}$$ By using the observations: 1. $m_{\nu}^{\star} = m_{\nu}$, $m_{\nu}^{2} = \sum_{w \in S_{\nu}} q^{l(w)} m_{\nu}$. 2. $\sum_{w \in S_{\nu}} q^{l(w)} = \frac{|P_{\nu}|}{|B|}$. We obtain $$m_{\Theta_1 \Theta_2} = \frac{|B|}{|P_{\nu}|} {\bold{G}}_{\Theta_1}^{\star} {\bold{G}}_{\Theta_2}.$$ We claim that if the semistandard tableau $\Theta_i$ corresponds to the orbit $\Omega_i$ by RSK then ${\bold{G}}_{\Theta_i} = \gb_{\Omega_i}$. We argue for $i=1$. We have an isomorphism $$\mathcal {H}_{\mathbb Q, q}(S_n) \cong \mathbb{Q} [B\backslash G / B],$$ such that a basis element $T_{\omega}$ in the Hecke algebra ${{\mathcal H}}_{{{\mathbb Q}}, q}(S_n)$ corresponds to the function $\bold{1}_{BwB} \in \mathbb Q[B\backslash G /B]$. The commutativity of the following diagram implies that the sum $\sum_{w \in S_\nu \sigma_1 S_{{{\lambda}}}} T_w$ corresponds to $\bold{1}_{P_{\nu} \sigma_1 P_{{{\lambda}}}}$. $$\begin{aligned} B\backslash G / B & \leftrightarrow & S_{n} \\ \downarrow & & \downarrow \\ P_{\nu} \backslash G / P_{{{\lambda}}} & \leftrightarrow & S_{\nu} \backslash S_n /S_{{{\lambda}}}\end{aligned}$$ Therefore ${\bold{G}}_{\Theta_1}$ belongs to $\mathrm{Hom}_G(\mathcal F_{{{\lambda}}}, \mathcal F_{\nu})$. Let orbit $\Omega_1$ corresponds to matrix $m = (m_{ij}) \in {{\mathcal M}}_{\nu{{\lambda}}}^{\circ}$ by RSK. Then by its definition, matrix $\sigma_1$ is the unique matrix in ${{\lambda}}\mu$-Echelon form such that when viewed as a block matrix having $(i,j)^{\mathrm{th}}$ block of size ${{\lambda}}_i \times \nu_j$ for $1 \leq i \leq l({{\lambda}})$ and $1 \leq j \leq l(\nu)$ then $m_{ij}$ = sum of entries of $(i,j)^{\mathrm{th}}$ block of $\sigma_1$. This implies that if $(y,x) \in \Omega_1$, then there exist full flags $\bar{y}$ and $\bar{x}$ that extend the flags $y$ and $x$ respectively such that the intersection matrix of $\bar{y}$ and $\bar{x}$ is $\sigma_1$. Therefore, $$\gb_{\Omega_1} = \bold{1}_{P_{\nu} \sigma_1 P_{{{\lambda}}}} = {\bold{G}}_{\Theta_1}.$$ Further, the anti-automorphism $\star$ on Hecke algebra ${{\mathcal H}}_{{{\mathbb Q}}, q}(S_n)$ coincides with ’op’ on $X_{{{\lambda}}} \times_G X_{{{\lambda}}}$, hence the result. For $\mu \in \Lambda$, let ${{\mathcal H}}_{{{\lambda}}}^{\mu} = \mathrm{Span}_{{{\mathbb Q}}} \{ \cb_{\Omega_1 \Omega_2}^{\nu} \mid \nu \leq \mu \} $ and ${{\mathcal H}}_{{{\lambda}}}^{\mu-} = \mathrm{Span}_{{{\mathbb Q}}} \{ \cb_{\Omega_1 \Omega_2}^{\nu} \mid \nu < \mu \} $. \[ideal structure\] 1. Let $f \in \mathrm{Hom}_G({{\mathcal F}}_{{{\lambda}}}, {{\mathcal F}}_{\mu})$ and $h \in \mathrm{Hom}_G({{\mathcal F}}_{\mu}, {{\mathcal F}}_{{{\lambda}}})$ then $h\circ f \in {{\mathcal H}}_{{{\lambda}}}^{\mu}$. 2. The spaces ${{\mathcal H}}_{{{\lambda}}}^{\mu}$ and ${{\mathcal H}}_{{{\lambda}}}^{\mu-}$ are two sided ideal of ${{\mathcal H}}_{{{\lambda}}}$. \(a) We prove this by induction on the partially ordered set $\Lambda_{{{\lambda}}}$. If $\delta = (n)$, the partition with only one part, then ${{\mathcal F}}_{\delta}$ is the trivial representation, and it is easily seen that ${{\mathcal N}}_{{{\lambda}}\delta}$ and ${{\mathcal N}}_{\delta{{\lambda}}}$ are one dimensional. It follows that ${{\mathcal H}}_{{{\lambda}}}^{\delta}={{\mathcal N}}_{{{\lambda}}\delta} \circ {{\mathcal N}}_{\delta {{\lambda}}}$ is one dimensional and spanned by $\cb_{[(x,0)],[(0,x)]}^\delta$. This established the basis for the induction. Now assume the assertion is true for any $\nu \in \lambda_{{{\lambda}}}$ such that $\nu < \mu$. We prove the result for $\mu$. Let $p_i$ for $1 \leq i \leq r$ be all the permissible embeddings of $\mu$-flags into ${{\lambda}}$-flags and let $\Omega_{p_i} \in (X_{\mu} \times_G X_{{{\lambda}}})^{\circ}$ be the orbits corresponding to these embeddings. The orbit corresponding to the identity mapping of $\mu$-flag into itself is denoted by $\Omega_{\mathrm{id}}$. Let ${{{\mathcal N}}}'_{{{\lambda}}\mu}$ be the subspace of ${{\mathcal N}}_{{{\lambda}}\mu}$ generated by the set $\{ \cb_{\Omega_{1} \Omega_2}^{\nu} \mid \nu < \mu, \Omega_1 \in (X_{\nu} \times_G X_{{{\lambda}}})^{\circ}, \Omega_2 \in (X_{\nu} \times_G X_{\mu})^{\circ}\}$. It follows that any $h \in \mathrm{Hom}_G({{\mathcal F}}_{\mu}, {{\mathcal F}}_{{{\lambda}}})$ can be written as linear combination $\sum_{i=1}^r \alpha_i \cb_{\Omega_{\mathrm{id}} \Omega_{p_i}}^{\mu} \;\; \mathrm{mod}\,\, {{{\mathcal N}}'_{{{\lambda}}\mu}}$. Therefore, it is enough to prove that $ \cb_{\Omega_{p_i} \Omega_{\mathrm{id}}}^{\mu} f \in {{\mathcal H}}_{{{\lambda}}}^{\mu}$ for all $1 \leq i \leq r$. Arguing similarly for $f$ it suffices to prove that $\cb_{\Omega_{p_{i}} \Omega_{id}}^{\mu} \cb_{\Omega_{\mathrm{id}} \Omega_{p_j}}^{\mu} \in {{\mathcal H}}_{{{\lambda}}}^{\mu}$ for all $1 \leq i,j \leq r$. But since $\cb_{\Omega_{p_{i}} \Omega_{\mathrm{id}}}^{\mu} \cb_{\Omega_{\mathrm{id}} \Omega_{p_j}}^{\mu} = \cb_{\Omega_{p_{i}} \Omega_{p_j}}^{\mu}$, the result follows. \(b) The fact that ${{\mathcal H}}_{{{\lambda}}}^{\mu}$ is a two-sided ideal follows immediately from (a) and the fact that $\{\cb_{\Omega_2 \Omega_1}^{\mu}\mid \mu \le {{\lambda}}, \Omega_1, \Omega_2\}$ is a basis by observing that compositions of basis elements of the form $\cb_{\Omega'_1\Omega'_2}^{\nu} \cb_{\Omega_1 \Omega_2}^{\mu}$ $$\xymatrix{ {{\mathcal F}}_{{{\lambda}}} \ar[dr]|{{\gb_{\Omega_2}}} & & {{\mathcal F}}_{{{\lambda}}} \ar[dr]|{{\gb_{\Omega_2'}}} & & {{\mathcal F}}_{{{\lambda}}} \\ & {{\mathcal F}}_{\mu} \ar[ur]|{{\gb_{\Omega_1^{\mathrm{op}}}}} & & {{\mathcal F}}_{\nu} \ar[ur]|{{\gb_{{\Omega_1'}^{\mathrm{op}}}}} & }$$ lies in ${{\mathcal H}}_{{\lambda}}^{\mu \wedge \nu}$. Finally, as ${{\mathcal H}}_{{{\lambda}}}^{\mu-}=\sum_{\nu < \mu}{{\mathcal H}}_{{{\lambda}}}^{\nu}$, the latter is an ideal as well. \[thm: cellular\] The Hecke algebra ${{\mathcal H}}_{{{\lambda}}}$ is cellular with respect to $({{\mathcal C}}_{{{\lambda}}{{\lambda}}}, \Lambda_{{{\lambda}}} )$. We have a natural anti-automorphism of the Hecke algebras ${{\mathcal H}}_{{{\lambda}}}$ defined as $$(\cb_{\Omega_1 \Omega_2}^{\mu})^{\star} = \cb_{\Omega_2 \Omega_1}^{\mu}.$$ Proposition \[ideal structure\] implies that the criterion \[cellular condition\] for cellularity is fulfilled as well. \[main\] There exists a collection $\{{{\mathcal U}}_{{{\lambda}}} \mid {{\lambda}}\in \Lambda_n\}$ of inequivalent irreducible representations of $\mathrm{GL}_n({k})$ such that 1. ${{\mathcal F}}_{{{\lambda}}} = \oplus_{\nu \leq {{\lambda}}} {{\mathcal U}}_{\nu}^{|{{\mathcal M}}_{\nu {{\lambda}}}^\circ|}$; 2. For every $\mu, \nu \leq {{\lambda}}$, one has $\dim_{{\mathbb Q}}\mathrm{Hom}_G \left( {{\mathcal U}}_{\nu}, {{\mathcal F}}_{\mu}\right) = |{{\mathcal M}}_{\nu\mu}^\circ|$. That is, the multiplicity of ${{\mathcal U}}_{\nu}$ in ${{\mathcal F}}_{\mu}$ is the number of non-equivalent permissible embeddings of a $\nu$-flag in a $\mu$-flag. In particular ${{\mathcal U}}_{\nu}$ appears in ${{\mathcal F}}_{\nu}$ with multiplicity one and does not appear in ${{\mathcal F}}_{\mu}$ unless $\nu \leq \mu$. We remark that part (2) of Theorem \[main\] gives a characterization of the irreducible representations ${{\mathcal U}}_\lambda$, that is, for each $\lambda \in \Lambda_n$, the representation ${{\mathcal U}}_\lambda$ is the unique irreducible representation which occurs in ${{\mathcal F}}_\lambda$ and do not occur in ${{\mathcal F}}_\mu$ for $\mu \leq \lambda$.\ General flags {#subsec: general flags} ------------- In this section we extend our results of previous section to the flags not necessarily associated with partitions. A tuple $c= (c_i)$ of positive integers such that $\sum c_{i} = n$ is called composition of $n$. The length of $c$, denoted $l(c)$ is the the number of its nonzero parts. By reordering parts of a composition in a decreasing order we obtain the unique partition associated with it. We shall use bar to denote the associated partition. For example if $c =(2,1,2)$, then $\bar{c} = (2,2,1)$. A chain of ${k}$-vector spaces $ x = ({k}^n = x_{l(c)} \supset x_{l(c)-1} \supset \cdots \supset x_1 \supset x_{0} = (0))$ is a $c$-flag if $\dim_{{k}}(x_{l(c)-i+1}/x_{l(c)-i}) = c_i$ for all $1 \le i \le l(c)$. Let $X_c$ be the space of all $c$-flags and ${{\mathcal F}}_{c} = \mathbb Q(X_c)$. By the theory of representation of symmetric groups and Bruhat decomposition, it follows that for any compositions $c_1$ and $c_2$, the Hecke algebras ${{\mathcal H}}_{c_1} = \mathrm{Hom}_G({{\mathcal F}}_{c_1}, {{\mathcal F}}_{c_2})$ and $ {{\mathcal H}}_{\bar{c}} = \mathrm{Hom}_G({{\mathcal F}}_{\bar{c_1}}, {{\mathcal F}}_{\bar{c_2}})$ are isomorphic as $G$-modules. By composing this isomorphism with the cellular basis of the Hecke algebra ${{\mathcal H}}_{\bar{c}}$, one obtains the cellular basis of the Hecke algebras ${{\mathcal H}}_{c}$. This implies that irreducible components of ${{\mathcal F}}_{c}$ are parameterized by the set of partitions ${{\lambda}}\in \Lambda$ such that ${{\lambda}}\leq \bar{c}$. In particular this gives the following bijection $$X_{c_1} \times_G X_{c_2} \longleftrightarrow \bigsqcup_{\nu \in \Lambda, \nu \leq \bar{c}} (X_{\nu} \times_G X_{c_1})^{\circ} \times (X_{\nu} \times_G X_{c_2})^{\circ}$$ for certain subsets $(X_{\nu} \times_G X_{c_1})^{\circ}$ and $(X_{\nu} \times_G X_{c_2})^{\circ}$ of $X_{\nu} \times_G X_{c_1}$ and $X_{\nu} \times_G X_{c_2}$ respectively. For any $(x,y) \in (X_{\nu} \times_G X_{c_1})^{\circ}$, we say $x$ has permissible embedding in $y$. Whenever we deal with compositions in later section, by cellular basis and permissible embedding we shall mean the general notions defined in this section. The Module Case =============== In this section $\mathfrak {o}$ denotes a complete discrete valuation ring with maximal ideal $\mathfrak{p}$ and fixed uniformizer $\pi$. Assume that the residue field $ {k}= \mathfrak {o}/\mathfrak{p}$ is finite. We denote by $\mathfrak {o}_{\ell}$ the reduction of $\mathfrak {o}$ modulo $\mathfrak{p}^{\ell}$, i.e., $\mathfrak {o}_{\ell} = \mathfrak {o}/\mathfrak{p}^{\ell}$. Since $\mathfrak {o}$ is a principal ideal domain with a unique maximal ideal $\mathfrak{p}$, every finite $\mathfrak {o}$-module is of the form $\oplus_{i=1}^{j}\mathfrak {o}_{{{\lambda}}_{i}}$, where the ${{\lambda}}_{i}$’s can be arranged so that $\lambda = ({{\lambda}}_i) \in \Lambda=\cup \Lambda_n$. The rank of an $\mathfrak {o}$-module is defined to be the length of the associated partition. Note that in this section we use arbitrary partitions rather than partitions of a fixed integer and parameterize different objects than in the previous sections: types of $\mathfrak{o}$-modules rather than types of flags of ${k}$-vector spaces. Let $\tau$ be the type map which maps each $\mathfrak {o}$-module to its associated partition. The group ${\text{GL}_{n}({\mathfrak {o}}_{\ell})}$ denotes the set of invertible matrices of order $n$ over the ring $\mathfrak {o}_{\ell}$. Let $${{\mathcal L}}^{(r)}={{\mathcal L}}^{(r)}({\ell^n})=\{(x_r,\cdots,x_{1}) \mid \mathfrak {o}_{\ell}^n \supset x_r \supset \cdots \supset x_{0} = (0),~\text{$x_i$ are $\mathfrak {o}$-modules}\}$$ be the space of flags of modules on length $r$ in $\mathfrak {o}_{\ell}^n$. There is a natural partial ordering on ${{\mathcal L}}^{(r)}$ defined by $\eta=(y_r,...,y_1) \le (x_r,...,x_1)=\xi$ if there exist embeddings $\phi_1,...,\phi_r$ such that the diagram $$\begin{matrix} x_r & \supset & x_{r-1} & \supset & \cdots & \supset & x_1 \\ \uparrow_{\phi_r} & & \uparrow_{\phi_{r-1}} & & \cdots & & \uparrow_{\phi_1} \\ y_r & \supset & y_{r-1} & \supset & \cdots & \supset & y_1 \end{matrix}$$ is commutative. Two flags $\xi$ and $\eta$ are called equivalent, denoted $\xi \sim \eta$, if the $\phi_i$’s in the diagram are isomorphisms. For any equivalence class $\Xi=[\xi]$ let ${{\mathcal F}}_\Xi=\mathbb{Q}(\Xi)$ denote the space of rational valued functions on $\Xi$ endowed with the natural ${\text{GL}_{n}({\mathfrak {o}}_{\ell})}$-action. We use the letter $\Xi$ to denote a set of flags as well as the [*type*]{} of the flags in this set. En route to developing the language and tools for decomposing the representations ${{\mathcal F}}_\Xi$ into irreducible representations we treat here the special case $\ell=2$ and give a complete spectral decomposition for the ${\text{GL}_{n}({\mathfrak {o}}_{2})}$-representations ${{\mathcal F}}_\Xi$ with $\Xi \subset {{\mathcal L}}^{(1)}({2^n})$. Recall $ \Xi \in {{\mathcal L}}^{(1)}(2^n)$ consists of all submodules $x \subset \mathfrak{o}_2^n$ with a fixed type ${{\lambda}}$. We shall also assume that $n \geq 2(\mathrm{Rank}(x))$ and $\mathrm{Rank}(x) \geq 2(\mathrm{Rank}(\pi x))$. We have a map $\iota: {{\mathcal L}}^{(1)} \to {{\mathcal L}}^{(2)}$ given by $y \mapsto (y,\pi y)$ which allows us to identify any module with a (canonically defined) pair of modules. We will see that to find and separate the irreducible constituents of ${{\mathcal F}}_{{{\lambda}}}$ with ${{\lambda}}\subset {{\mathcal L}}^{(1)}$ we need to use a specific set of representations ${{\mathcal F}}_\eta$ with $\eta \in {{\mathcal L}}^{(2)}$ such that $\eta \le \iota({{\lambda}})$. A similar phenomenon has been observed also in [@MR2283434]. We remark that for the groups ${\text{GL}_{n}({\mathfrak {o}}_{2})}$, it is known that the dimensions of complex irreducible representations and their numbers in each dimension depend only on the cardinality of residue field of $\mathfrak {o}$, see [@MR2684153]. For the current setting we shall prove that the numbers and multiplicities of the irreducible constituents of ${{\mathcal F}}_\lambda$ with ${{\lambda}}\subset {{\mathcal L}}^{(1)}$ are independent of the residue field as well, though this is not true in general, see [@MR2267575; @MR2504482]. In this section we shall use the notation $G$ to denote the group ${\text{GL}_{n}({\mathfrak {o}}_{2})}$, and the group of invertible matrices of order $n$ over the field ${k}$ is denoted by ${\text{GL}}_n({k})$. Parameterizing set ------------------ Let ${\mathcal{S}_{ \iota({{\lambda}})}}\subset {{\mathcal L}}^{(2)}$ be the set of tuples $(x_2, x_1)$ satisfying 1. The module $x_{1}$ has a unique embedding in $x_2$ (up to automorphism). 2. $(x_2,x_1) \le \iota(y)$, $\tau(y)={{\lambda}}$. 3. $\mathrm{Rank} (x_1) \leq \mathrm{Rank}(x_2/x_1)$. Let ${\mathcal{P}_{\iota({{\lambda}})}}= {\mathcal{S}_{ \iota({{\lambda}})}}/\!\sim$ be the set of equivalence classes in ${\mathcal{S}_{ \iota({{\lambda}})}}$. The uniqueness of embedding implies that $(x_2, x_1)$, $(y_2, y_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$ are equivalent if and only if $\tau(x_2) = \tau(y_2)$ and $\tau(x_1) = \tau(y_1)$. Therefore, $\xi = [(x_2, x_1)] \in {\mathcal{P}_{\iota({{\lambda}})}}$ may be identified with the pair $\mu^{(2)} \supset \mu^{(1)}$ where $\mu^{(2)} = \tau(x_2)$ and $\mu^{(1)} = \tau(x_1)$. Further, if $(x_2, x_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$ is such that $\tau(x_2) = {{\lambda}}$ and $x_1 = \pi x_2$, then the equivalence class of $(x_2, x_1)$ in ${\mathcal{P}_{\iota({{\lambda}})}}$ is also denoted by $\iota({{\lambda}})$. For $\xi \in {\mathcal{P}_{\iota({{\lambda}})}}$, let $$Y_{\xi} = \{ x \in {\mathcal{S}_{ \iota({{\lambda}})}}\mid [x] = \xi \}.$$ Then ${\mathcal{S}_{ \iota({{\lambda}})}}= \sqcup_{\xi \in {\mathcal{P}_{\iota({{\lambda}})}}} Y_{\xi}$. Let $F_{\xi} = \mathbb Q (Y_{\xi})$ be the space of rational valued functions on ${Y_{\xi}}$. As discussed earlier, the space ${{\mathcal F}}_{\iota({{\lambda}})}$ coincides with ${{\mathcal F}}_{{{\lambda}}}$. We shall prove that ${\mathcal{P}_{\iota({{\lambda}})}}$ parameterizes the irreducible representations of the space ${{\mathcal F}}_{{{\lambda}}}$ and in particular satisfies a relation similar to (\[geometric.RSK\]) (See Proposition \[RSK in modules\]). An analogue of the RSK correspondence ------------------------------------- For $a \in \mathfrak{o}$ and an $\mathfrak{o}$-module $x$, let $x[a]$ and $ax$ denote the kernel and the image, respectively, of the endomorphism of $x$ obtained by multiplication by $a$. For any $x = (x_2, x_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$, the flag of $\pi$-torsion points of $x$, denoted $x_{\pi}$, is the flag ${k}^n \supseteq x_2[\pi] \supset x_1 \supset \pi x_2$. In general this flag may not be associated with a partition but rather a composition. If $x, y \in {\mathcal{S}_{ \iota({{\lambda}})}}$ are such that $[x] = [y]$ then the compositions associated with the flags $x_{\pi}$ and $y_{\pi}$ are equal. Hence if $[x] = \xi$, then the composition associated with $x_{\pi}$ is denoted by $c(\xi)$. \[lem:reducing cosets to field\] There exists a canonical bijection between the sets $$\{[(x_2, x_1),(y_2, y_1)] \in Y_{\iota({{\lambda}})} \times_{{G}} Y_{\xi} \mid x_2 \cap y_2 \cong {k}^t, t \in \mathbb N \} \leftrightarrow X_{c(\iota({{\lambda}}) )} \times_{{\mathrm{GL}_n({k})}} X_{c(\xi)}$$ obtained by mapping $[(x_2, x_1),(y_2, y_1)] $ to $[(x_2, x_1)_{\pi}, (y_2, y_1)_{\pi}]$. Since all the pairwise intersections obtained from the modules $x_2$, $x_1$, $y_2$ and $y_1$ are ${k}$-vector spaces, by taking the flags of the $\pi$-torsion points we obtain a well-defined map from $Y_{\iota({{\lambda}})} \times_{{G}} Y_{\xi}$ to $X_{c(\iota({{\lambda}}))} \times_{{\mathrm{GL}_n({k})}} X_{c(\xi)}$. We first prove that this map is injective. Let $(x,y),(x',y')\in Y_{\iota({{\lambda}})} \times Y_{\xi}$ be such that $x_\pi=x'_\pi$ and $y_\pi=y'_\pi$. Assume that $[(x_\pi,y_\pi)]=[(x'_\pi,y'_\pi)]$. This means that there exists an isomorphism $h:\pi\mathfrak{o}_2^n \to \pi\mathfrak{o}_2^n$ such that $h(x_\pi) = x'_\pi$ and $h(y_\pi) = y'_\pi$. We need to extend $h$ to a map $\tilde{h}:\mathfrak{o}_2^n \to \mathfrak{o}_2^n$ such that $\tilde{h}(x)=x'$ and $\tilde{h}(y)=y'$. The elements $x$ and $y$ are tuples of the form $(x_2,x_1)$ and $(y_2,y_1)$, respectively. We choose maximal free $\mathfrak{o}_2$-submodules $x_3,x'_3,y_3,y'_3$ of $x_2,x'_2,y_2,y'_2$, respectively. Since $n \ge 2 (l({{\lambda}}))$, we can extend the map $h$ to maps $(x_3+\pi\mathfrak{o}_2^n) \to (x'_3+\pi\mathfrak{o}_2^n)$ and $(y_3+\pi\mathfrak{o}_2^n) \to (y'_3+\pi\mathfrak{o}_2^n)$ in a compatible manner such that these two extensions glue to a unique well-defined map $(x_3+y_3+\pi\mathfrak{o}_2^n) \to (x'_3+y'_3+\pi\mathfrak{o}_2^n)$. The latter can now be extended to an isomorphism $\tilde{h}:\mathfrak{o}_2^n \to \mathfrak{o}_2^n$ with the desired properties. To prove surjectivity we need to find a pair $(x,y) \in Y_{\iota({{\lambda}})} \times Y_{\xi}$ which maps to a given pair $(u,v) \in X_{c(\iota({{\lambda}}) )} \times X_{c(\xi)}$. This follows at once from the assumption $n \ge 2 (l({{\lambda}}))$. Let $x = ({k}^n = x_t \supset \cdots \supset x_1 \supset x_0 = (0))$ be a flag of ${k}$-vector spaces and $v$ be a $k$-vector space such that $x_i\supset v$ for all $i$, then $x$ modulo $v$, denoted as $x/v$ is the flag $x/v = ({k}^n /v = x_t/v \supset \cdots \supset x_1/v \supset (0))$. Let $x=(x_2, x_1), y=(y_2, y_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$ be such that $x \geq y$. Flags of our primal interest are $x_{\pi}/\pi y_2$ and $y_{\pi}/\pi y_2$. Observe that although the flag $y_{\pi}/\pi y_2$ is associated with a partition, the flag $x_{\pi}/\pi y_2$ may only be associated to a composition. We say that $y$ embeds into $x$ permissibly if $y \leq x$ and $y_{\pi}/\pi y_2$ embeds permissibly into $x_{\pi}/\pi y_2$ (see Section \[subsec: general flags\]). For $\eta \leq \xi$, let $ (Y_{\eta} \times_{{G}} Y_{\xi})^{\circ}$ denote the set of equivalence classes $[(y,x)] \in Y_{\eta} \times_{{G}} Y_{\xi}$ such that $y$ embeds permissibly in $x$. \[RSK in modules\]There exists a bijection between the following sets $$Y_{\iota({{\lambda}})} \times_{{G}} Y_{\xi} \longleftrightarrow \bigsqcup_{\eta \in {\mathcal{P}_{\iota({{\lambda}})}},\eta \leq \xi} (Y_{\eta} \times_{{G}} Y_{\iota({{\lambda}})})^{\circ} \times (Y_{\eta} \times_{{G}} Y_{\xi})^{\circ}.$$ Let $(x_2, x_1) \in Y_{\iota({{\lambda}})}$ and $(y_2, y_1) \in Y_{\xi}$ be elements such that $x_2 \cap y_2 = z_2 \oplus z_1$ such that $z_2 \cong {\mathfrak o}_2^s $ and $z_1 \cong {\mathfrak o}_1^t$, and denote $\Omega=[(x_2, x_1), (y_2, y_1)]$. Let $x'_2 = x_2/ z_2$, $y'_2 = y_2/ z_2$, $x'_1 = x_1/\pi(z_2)$ and $y'_1 = y_1/\pi(z_2)$ then $x'_2 \cap y'_2 \cong {\mathfrak o}_1^t$. By Lemma \[lem:reducing cosets to field\] and the RSK correspondence, we obtain that the double coset $[(x'_2, x'_1), (y'_2, y'_1)]$ corresponds to a $\delta$-flag $(z'_2, z'_1)$ for some partition $\delta$ of $n-s$ with its permissible embeddings $p_1$ and $p_2$ into the flags $(x'_2, x'_1)_{\pi}/\pi(z'_2)$ and $(y'_2, y'_1)_{\pi}/\pi(z'_2)$ respectively. By adjoining it with $({\mathfrak o}_2^s, \pi {\mathfrak o}_2^s)$, we obtain $(u_2, u_1) = ({\mathfrak o}_2^s \oplus z_2, \pi {\mathfrak o}_2^s \oplus z_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$ with permissible embeddings $p_1$ and $p_2$ in $(x_2, x_1)$ and $(y_2, y_1)$ respectively. The converse implication follows by combining the RSK correspondence with the definition of permissible embedding. \[omega notation\] Observe that if $\Omega = [(x_2, x_1), (y_2, y_1)] \in Y_{(\iota({{\lambda}}))} \times_G Y_{\xi}$ corresponds to permissible embeddings $p_1$, $p_2$ of $(z_2, z_1)$ in $(x_2, x_1)$ and $(y_2, y_1)$, respectively, then $\pi(z_2) \cong \pi(x_2 \cap y_2)$. If $[(z_2, z_1)] = (\nu^{(2)}, \nu^{(1)})$, we shall use the notation $\Omega_{\pi \nu}$ instead of $\Omega$ to specify this information. Geometric bases of modules -------------------------- The modules ${\text{Hom}}_{{G}}({{\mathcal F}}_{\xi}, {{\mathcal F}}_{\iota({{\lambda}})})$ for $\xi \in {\mathcal{P}_{\iota({{\lambda}})}}$, and in particular the Hecke algebras ${\mathcal{H}_{ \iota({{\lambda}}) }}= \mathrm{End}_G({\mathcal{F}_{ \iota({{\lambda}})}})$, have natural geometric bases indexed by ${Y_{\iota({{\lambda}})}}\times_{{G}} {Y_{\xi}}$, the space of ${G}$ orbits in ${Y_{\iota({{\lambda}})}}\times {Y_{\xi}}$ with respect to diagonal action of ${G}$. Specifically, let $$\label{geometric.basis.mod} \gb_\Omega f (x)= \sum_{y:(x,y) \in \Omega} f(y), \qquad f \in {\mathcal{F}_{\xi}},\, x \in {Y_{\iota({{\lambda}})}},$$ Then $\{ \gb_{\Omega} \mid \Omega \in {Y_{\iota({{\lambda}})}}\times_{{G}} {Y_{\xi}} \}$ is a basis of ${\text{Hom}}_{{G}}({\mathcal{F}_{ \iota({{\lambda}})}}, {\mathcal{F}_{\xi}})$. Cellular basis of the Hecke algebras ------------------------------------- In this section we determine the cellular basis of the Hecke algebras ${\mathcal{H}_{ \iota({{\lambda}}) }}$. Let $\mathcal {R}$ be a refinement of the partial order on ${\mathcal{S}_{ \iota({{\lambda}})}}$ given by: For any $(x_2, x_1)$, $(y_2, y_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$, $(x_2, x_1) \geq_{{{\mathcal R}}} (y_2, y_1)$ if either $(x_2, x_1) \geq (y_2, y_1)$ or $\pi x_2 > \pi y_2$. The set ${\mathcal{P}_{\iota({{\lambda}})}}$ inherits this partial order as well and is denoted by ${\mathcal{P}_{\iota({{\lambda}})}}^{{{\mathcal R}}}$ when considered as partially ordered set under ${{\mathcal R}}$. For $\eta \in {\mathcal{P}_{\iota({{\lambda}})}}$ and orbits $\Omega_1 \in ({Y_{\eta}} \times_{{G}} {Y_{\xi}})^{\circ}$, $\Omega_2 \in ({Y_{\eta}} \times_{{G}} {Y_{\iota({{\lambda}})}})^{\circ}$ define $$\cb_{\Omega_1 \Omega_2}^{\eta}:= \gb_{\Omega_1^{\mathrm{op}}} \gb_{\Omega_2}.$$ Then $\cb_{\Omega_1 \Omega_2}^{\eta} \in \mathrm{Hom}_{{G}}({\mathcal{F}_{ \iota({{\lambda}})}}, {\mathcal{F}_{\xi}})$. Let $$\mathcal {C}_{\iota({{\lambda}}) \xi} = \{ \cb_{\Omega_1 \Omega_2}^{\eta} \mid \eta \in {\mathcal{P}_{\iota({{\lambda}})}}, \eta \leq \xi, \Omega_1 \in ({Y_{\eta}} \times_{{G}} {Y_{\iota({{\lambda}})}})^{\circ} , \Omega_2 \in ({Y_{\eta}} \times_{{G}} {Y_{\xi}})^{\circ} \}.$$ \[module basis\] The set $\mathcal {C}_{\iota({{\lambda}}) \xi}$ is a $\mathbb Q$-basis of the Hecke module $\mathrm{Hom}_{{G}}({\mathcal{F}_{ \iota({{\lambda}})}}, {\mathcal{F}_{\xi}})$. We shall prove this proposition by proving that the transition matrix between the set ${{\mathcal C}}_{\iota({{\lambda}})\xi}$ and the geometric basis $\{\gb_{\Omega} \}$ is upper block diagonal matrix with invertible blocks on the diagonal. Wherever required we also use the notation $\Omega_{\pi \nu}$ in place of $\Omega$ (see Remark \[omega notation\]). We claim that $$\label{cell in module} \cb_{\Omega_1 \Omega_2}^{\eta} = \sum_{ \{\Delta_{\pi \chi} \in {Y_{\iota({{\lambda}})}}\times_{{G}} {Y_{\xi}} ~\mid~ \chi \geq_{\mathcal R} \eta \}} a_{\Delta_{\pi \chi}} \gb_{\Delta_{ \pi \chi}}.$$ Let $[(x,y)] = [(x_2, x_1), (y_2, y_1)] = \Delta_{\pi \chi}$. Indeed, from the definition of $\cb^{\eta}_{\Omega_1\Omega_2}$ and $\gb_{\Delta_{\pi \chi}}$, it is clear that the coefficient $a_{\Delta_{\pi \chi}}$ is given by $$a_{\Delta_{\pi \chi}} = |\{ z' \in {\mathcal{S}_{ \iota({{\lambda}})}}\mid [z'] = \eta, [(z',x)] = \Omega_1, [(z',y)] = \Omega_2 \}|.$$ Note that if $(z_2, z_1) \in {\mathcal{S}_{ \iota({{\lambda}})}}$ has permissible embedding in $(x_2, x_1)$ and $(y_2, y_1)$ then $\pi z_2$ embeds into $\pi (x_2 \cap y_2)$. For the case $\pi z_2 \cong \pi (x_2 \cap y_2)$, we claim that the coefficients $a_{\Delta_{\pi \chi}}$ are nonzero only if $\chi \geq \eta$. Let $z_\pi/\pi z_2$ be a $\delta$-flag and $\bar{\Omega}_1, \bar{\Omega}_2$ correspond to permissible embeddings of $z_\pi /\pi z_2$ in $x_\pi/\pi z_2$ and $y_\pi/ \pi z_2$, respectively. Assume that $x_\pi/\pi z_2$ is a $c_1$-flag and $y_\pi/\pi z_2$ is a $c_2$-flag for some compositions $c_1$ and $c_2$. If $\bar{\Delta}_{\chi} = [(x_\pi/ \pi z_2, y_\pi/ \pi z_2)]$, then the coefficient of $\gb_{\bar{\Delta}_{\pi \chi}}$ in the expression of $ c_{\bar{\Omega}_1 \bar{\Omega}_2}^{\gamma} \in \mathcal C_{c_2c_1}$ is given by $$\bar{a}_{\Delta_{\pi \chi}} = |\{ z' \in {{\mathcal F}}_{\gamma} \mid [(z' , x_\pi/ \pi z_2)] = \bar{\Omega}_1, [(z', y_\pi/ \pi z_2)] = \bar{\Omega}_2 \}|.$$ Since by definitions $a_{\Delta_{\pi \chi}}$ = $\bar{a}_{\Delta_{\pi \chi}}$, the coefficient $a_{\Delta_{\pi \chi}}$ is non-zero only if $\chi \geq \eta$. This implies $\chi \geq_{{{\mathcal R}}} \eta$ and completes the proof of (\[cell in module\]). By the discussion above we also obtain that by arranging the elements $\cb_{\Omega_1 \Omega_2}^{\eta}$ and $\gb_{\Delta_{\eta}}$ for $\eta \in {\mathcal{P}_{\iota({{\lambda}})}}$ in the relation ${{\mathcal R}}$, the obtained transition matrix between the set $$\{\cb_{\Omega_1 \Omega_2}^{\eta} \mid \eta \in {\mathcal{P}_{\iota({{\lambda}})}}, \Omega_1 \in ({Y_{\eta}} \times_G {Y_{\xi}})^{\circ}, \Omega_2 \in ({Y_{\eta}} \times_G {Y_{\iota({{\lambda}})}})^{\circ} \}$$ and $\{\gb_{\Omega} \mid \Omega \in {Y_{\iota({{\lambda}})}}\times_G {Y_{\xi}}\}$ is an upper block diagonal matrix with invertible diagonal blocks. Observe that the diagonal blocks are obtained as the transition matrix of certain cellular basis of Hecke algebras corresponding to the space of flags of ${k}$-vector spaces to the corresponding geometric basis. This implies that the set $\mathcal{C}_{\iota({{\lambda}}) \xi}$ is a $\mathbb Q$-basis of $\mathrm{Hom}_G({\mathcal{F}_{ \iota({{\lambda}})}}, {\mathcal{F}_{\xi}})$. The operation $(\cb_{\Omega_1 \Omega_2}^{\eta})^{\star} = \cb_{\Omega_2 \Omega_1}^{\eta}$ gives an anti-automorphism of ${\mathcal{H}_{ \iota({{\lambda}}) }}$. The ${{\mathbb Q}}$-basis of the modules $\mathrm{Hom}_{{G}}({\mathcal{F}_{\xi}}, {\mathcal{F}_{ \iota({{\lambda}})}})$ and $\mathrm{Hom}_{{G}}({\mathcal{F}_{ \iota({{\lambda}})}}, {\mathcal{F}_{\xi}})$ is given by Proposition \[module basis\]. This, combined with the arguments given in the proof of Theorem \[thm: cellular\] proves The set $(\mathcal {C}_{\iota({{\lambda}}) \iota({{\lambda}})}, {\mathcal{P}_{\iota({{\lambda}})}}^{{{\mathcal R}}})$ is a cellular basis of the Hecke algebra ${\mathcal{H}_{ \iota({{\lambda}}) }}$. \[main-modules\] There exists a collection $\{{{\mathcal V}}_{\eta} \mid \eta \in {\mathcal{P}_{\iota({{\lambda}})}}\}$ of inequivalent irreducible representations of ${\text{GL}_{n}({\mathfrak {o}}_{2})}$ such that $${\mathcal{F}_{ \iota({{\lambda}})}}= \oplus_{\eta \in {\mathcal{P}_{\iota({{\lambda}})}}} {{\mathcal V}}_{\eta}^{m_{\eta}},$$ where $m_{\eta} = |({Y_{\eta}} \times_G {Y_{\iota({{\lambda}})}})^{\circ}|$ is the multiplicity of ${{\mathcal V}}_{\eta}$. [10]{} U. [Bader]{} and U. [Onn]{}. . . U. Bader and U. Onn. Geometric representations of [${\rm GL}(n,R)$]{}, cellular [H]{}ecke algebras and the embedding problem. , 208(3):905–922, 2007. P. S. Campbell and M. Nevins. Branching rules for unramified principal series representations of [${\rm GL}(3)$]{} over a [$p$]{}-adic field. , 321(9):2422–2444, 2009. R. Dipper and G. James. Representations of [H]{}ecke algebras of general linear groups. , 52(1):20–52, 1986. J. J. Graham and G. I. Lehrer. Cellular algebras. , 123(1):1–34, 1996. D. E. Knuth. Permutations, matrices, and generalized [Y]{}oung tableaux. , 34:709–727, 1970. A. Mathas. , volume 15 of [*University Lecture Series*]{}. American Mathematical Society, Providence, RI, 1999. G. E. Murphy. On the representation theory of the symmetric groups and associated [H]{}ecke algebras. , 152(2):492–513, 1992. G. E. Murphy. The representations of [H]{}ecke algebras of type [$A_n$]{}. , 173(1):97–121, 1995. U. Onn, A. Prasad, and L. Vaserstein. A note on [B]{}ruhat decomposition of [${\rm GL}(n)$]{} over local principal ideal rings. , 34(11):4119–4130, 2006. P. Singla. On representations of general linear groups over principal ideal local rings of length two. , 324(9):2543–2563, 2010.

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--- author: - 'Tetsuo [Ohama]{}$^{1,}$[^1], Atsushi [Goto]{}$^2$, Tadashi [Shimizu]{}$^2$, Emi [Ninomiya]{}$^3$, Hiroshi [Sawa]{}$^3$, Masahiko [Isobe]{}$^4$ and Yutaka [Ueda]{}$^4$' title: 'Zigzag Charge Ordering in $\alpha''$-NaV$_2$O$_5$' --- Since the phase transition into a spin-gapped phase in  was reported, [@Isobe96] a lot of experimental efforts have been devoted to understand the nature of this transition. Although it was initially identified as a spin-Peierls transition, a recent room-temperature structural study [@Meetsma] questioned this interpretation: it concluded that all V ions are in a uniform oxidation state of V$^{4.5+}$ and form a quarter-filled trellis lattice composed of two-leg ladders. After that, $^{51}$V NMR measurements [@Ohama99] revealed charge ordering of V$^{4+}$ and V$^{5+}$ states below the transition temperature $T_{\rm C}\sim$ 34 K. Subsequent theoretical studies showed that long-range Coulomb interaction can induce charge ordering in a quarter-filled trellis lattice.[@Seo98; @Nishimoto; @Thalmeier; @Mostovoy] These studies suggest zigzag or linear chain ordering depending on the strength of the long-range Coulomb interactions. The proposed mechanism of charge ordering is similar to that for charge density wave in quarter-filled systems of low-dimensional organic compounds,[@SeoRev] suggesting some common physics to  and these systems. Soon after the finding of the transition, an x-ray diffraction measurement revealed superlattice formation of $2a\times 2b\times 4c$ in the charge-ordered phase,[@Fujii] but the detailed low-temperature structure has been unknown yet. Recently, two x-ray diffraction studies of the low-temperature structure were reported. These indicate almost the same structure of space group , but their assignments of V electronic states are different. One suggests a structure consists of half-filled (V$^{4+}$) and empty (V$^{5+}$) ladders. [@Luedecke] This charge distribution disagrees with a recent x-ray anomalous scattering measurement, which indicates charge modulation along $b$ axis.[@Nakao00] The other suggests a structure including three different electronic state of V$^{4+}$, V$^{5+}$ and V$^{4.5+}$. [@Boer] This structure is incompatible with the $^{51}$V NMR measurement, [@Ohama99] which clearly shows that all the V sites split into two groups of V$^{4+}$ and V$^{5+}$ states and that no V sites remains to be V$^{4.5+}$. Thus, the low-temperature structure and the charge ordering pattern are still under discussion. In this letter, we report $^{23}$Na NMR spectrum measurements with a single-crystalline sample. The obtained NMR spectra in the charge-ordered phase disagree with the space group , indicating lower symmetry. We will discuss possible low-temperature structures and charge ordering patterns, and will show that zigzag patterns are the most probable. The temperature variation of the NMR spectra near is incompatible with that of second-order transitions, suggesting that the charge ordering is first-order. The single-crystalline sample preparation was described in ref. . The NMR measurements were done using a high-resolution NMR spectrometer with a magnetic field $H\sim$ 63.382 kOe. The $^{23}$Na NMR spectra were obtained as power spectra by Fourier-transforming FID signals with the Gaussian multiplication for resolution enhancement and apodization. [@Sanders] Line width and intensity in the obtained spectra are thus inaccurate. NMR spectrum in a crystal is generally determined by the number of crystallographically inequivalent nuclear sites, and electrical field gradient and NMR shift tensors at each site. These are closely related to the crystal structure and the site symmetry. The room-temperature structure of  belongs to space group $Pmmn$,[@Meetsma; @Smolinski; @Schnering] in which structure Na ions occupy a unique atomic position of site symmetry . The principal axes of the electrical field gradient and NMR shift tensors are identical to the crystallographic axes accordingly. For the low-temperature structure of space group ,[@Luedecke] Na ions occupy six different atomic positions. Among 32 Na atoms in a unit cell, 16 atoms occupy four positions ($4a$) of site symmetry , and the other 16 do two positions ($8d$) of site symmetry . For the $4a$ sites, the principal axes are identical to the crystallographic axes, whereas for the $8d$ sites, one of the principal axes is identical to $b$ axis, but the other two are not determined only by the site symmetry. We present $^{23}$Na NMR spectra in the charge-ordered and uniform (above ) phases for $H\parallel b$, $c$, and $a$ in Figs. \[fig:specBC\](a), \[fig:specBC\](b), and \[fig:specA\], respectively. In the spectra in the uniform phase, three resonance lines (the central and two satellite lines) are observed corresponding to $^{23}$Na nuclear spins ($I=3/2$) at the unique Na position. Quadrupolar splitting, nuclear quadrupolar frequency $\nu_{\rm Q}$, and quadrupolar asymmetry parameter $\eta$ at 50 and 295 K are deduced (Table \[table:nuQ\_HT\]), in agreement with a previous measurement with a powder sample.[@Ohama97] ----- ----- ----- ----- --- ------ 50 682 99 581 2 0.85 295 641 120 520 1 0.62 ----- ----- ----- ----- --- ------ : Quadrupolar splitting and $\nu_{\rm Q}$ in kHz and $\eta$ in the uniform phase.[]{data-label="table:nuQ_HT"} Since electrical field gradient is a traceless tensor, and the principal axes at the Na site are identical with the crystallographic axes, $\nu_b - (\nu_a + \nu_c)$ should vanish. This is confirmed in Table \[table:nuQ\_HT\], indicating accurate alignments of the crystallographic axes along the magnetic field. We observed slight splitting of the resonance lines for $H\parallel b$ above . The splitting of the central line remains unchanged down to 10 K. We have no plausible explanation for its origin. For the Na site in the  structure, misalignment of the applied magnetic field from the crystallographic axes cannot cause any line splitting. Further, since the splitting of the central and satellite lines is comparable, it is not of quadrupolar origin and thus can be ascribed to neither inequivalent Na sites nor a twin sample. In the charge-ordered phase, each satellite line splits into eight lines for $H\parallel a$ and $b$, or four for $H\parallel c$. A similar result has been reported in ref. . As listed in Table \[table:nuQ\_LT\], these satellite lines can be assigned to eight Na sites with different quadrupolar splitting. site ------ ----- ----- ----- ---- -- A 752 193 563 -4 B 736 153 583 0 C 733 168 563 2 D 724 137 583 4 E 644 65 580 -1 F 640 36 608 -4 G 634 51 580 3 H 624 11 608 5 : Quadrupolar splitting in kHz at 10 K.[]{data-label="table:nuQ_LT"} Each of the four lines for $H\parallel c$ was considered as superposition of two lines, since these four lines have comparable intensities. This assignment in Table \[table:nuQ\_LT\] is not unique and other assignments of satellite lines for $H\parallel c$ are possible: two adjacent lines with $\nu_c$ of 580 and 583 kHz can be assigned to the sites B, D, E and G alternatively. The observation of the eight inequivalent Na sites disagrees with the low-temperature structures of , which has the six Na sites. Crystallographically equivalent sites, in general, can be observed as split resonance lines in NMR spectrum for some crystal structures. However, this is not the case for the Na sites in the structure with the magnetic field along the crystallographic axes. To explain this disagreement between the NMR and x-ray measurements, it is reasonable to suppose that the space group determined by the x-ray diffraction[@Luedecke] are correct as far as major atomic displacement is concerned, and further that the charge disproportionation of V ions is responsible for the disagreement, since usual x-ray diffraction measurements are sensitive to atomic displacement but not to charge disproportionation of V ions.[@Nakao00] This supposition is supported by the fact that contains all the subgroups of , the room-temperature structure, with the unit cell of $2a\times 2b\times 4c$. Since the observed atomic displacement through the phase transition is small, it is unlikely that a new symmetry element arises at the transition. Then the real low-temperature structure consistent with the present NMR result should belong to a proper subgroup of . order subgroups ------- ------------------------ 8 \(C) (C) (Z1) 4 (A, C, Z2) (C, Z2, Z3) (A, C, Z2) (C, Z2, Z3) : Possible subgroups of  (order 16) for low-temperature structure and charge ordering patterns. A: alternating chains along $a$ axis, C: four-V$^{4+}$ clusters, and Z1, Z2, Z3: zigzag chains.[]{data-label="table:SG"} In Table \[table:SG\], the subgroups compatible with the NMR spectra and possible charge ordering patterns in a plane of trellis lattice are listed. The following subgroups and patterns are here excluded: (1) the subgroups which are a proper subgroup of the subgroups of listed in Table \[table:SG\] and include only structures with more than eight Na sites, since there is no experimental evidence for such lower symmetry, (2) the patterns without charge modulation along $b$ axis, which disagree with the x-ray anomalous scattering measurements,[@Nakao00] (3) the patterns expected to contain V$^{4+}$ sites with very different magnetic properties, for example, a pattern containing both zigzag and linear chains, since all the V$^{4+}$ sites have the same nuclear relaxation rate well below .[@Ohama99] The remaining possible patterns are classified in two groups according to the arrangement of V$^{4+}$ and V$^{5+}$ sites in two-leg ladders: (1) the patterns in which a part of rungs of ladders have both the V sites occupied and consequently another part of rungs are empty. This group consists of several patterns of alternating chains along $a$ axis (A) and of clusters composed of four V$^{4+}$ sites (C). (2) the patterns which contain no doubly occupied rung. This group consists of three types of zigzag patterns (Z1, Z2, Z3). These zigzag patterns and examples of the patterns with doubly occupied rungs are shown in Figs. \[fig:CO\](a)–\[fig:CO\](e). The double occupancy of the rungs contained in the former group of charge ordering patterns is unfavorable because it has higher energy per rung by the order of the hopping parameter along the rung $t_\perp\sim$ 0.3 eV.[@Horsch; @Nishimoto] We therefore concluded that the zigzag patterns are the most probable. Although this conclusion is consistent with neutron scattering[@Yosi] and dielectric susceptibility[@Smirnov] measurements, it should be directly confirmed by an x-ray diffraction measurement. We next discuss complicated behavior experimentally observed near . Some experiments suggest the transition is second-order. X-ray diffraction measurements observed the superlattice reflection intensity continuously vanish at  as $(T_{\rm C}-T)^{\beta}$ ($\beta\sim$ 0.16).[@Ravy; @Nakao99] It was reported that $\nu_c$ for the $^{23}$Na NMR obeys the power law with $\beta\sim 0.19$.[@Fagot] On the contrary, Köppen [*et al*]{}. have found two adjacent phase transitions by a thermal-expansion measurement: a first-order transition at $T_h\sim 33.0$ K and the other at $T_l\sim 32.7$ K.[@Koeppen] In the $^{51}$V NMR spectrum, the resonance lines in the uniform and charge-ordered phases were observed to coexist in the temperature range between 33.4 and 33.8 K with a similar width to the separation between $T_h$ and $T_l$.[@Ohama99] Recent measurements of dielectric and magnetic properties under high pressure clearly show two separated transitions under pressure.[@Sekine] To investigate these characteristics of the transition, we measured the temperature variation of the NMR spectrum for $H\parallel a$ near  as shown in Fig. \[fig:specA\]. The experimental error of the absolute value of the temperature is less than $0.2$ K. We found that the lines in the uniform and charge-ordered phases coexist in a narrow temperature range between 33.4 and 33.8 K without any hysteresis. This temperature range agrees with the $^{51}$V NMR measurements with a different single-crystalline sample.[@Ohama99] In Fig. \[fig:split\], we show the temperature dependence of the satellite splitting for $H\parallel a$. We can fit this temperature dependence to the power law with $\beta\sim$ 0.29 and $T_{\rm C}\sim$ 34.04 K. However, the deduced  is too high, since the low-temperature signal disappears between 33.8 and 33.9 K. This indicates that the analysis for the ordinary second-order transition is inadequate. We therefore conclude that the transition at  is first-order. The present result does not rule out the other adjacent transition, but its existence is unclear. In summary, we have measured the $^{23}$Na NMR spectrum in  with a single-crystalline sample. We have observed eight Na sites in the charge-ordered phase in disagreement with the low-temperature structure of space group determined by the x-ray diffraction measurements. We discussed possible space groups and charge ordering patterns, and have concluded that zigzag patterns are the most probable. We also observed the resonance lines in the uniform and charge-ordered phase coexist near . Furthermore, the temperature dependence near  of the satellite line splitting is inconsistent with that of a second-order transition. We thus conclude that the transition is first-order. Acknowledgment {#acknowledgment .unnumbered} ============== We would like to thank H. Nakao, Y. Ohta, Y. Itoh and T. Yamauchi for helpful discussions. [99]{} M. Isobe and Y. Ueda: J. Phys. Soc. Jpn. [**65**]{} (1996) 1178. A. Meetsma, J.L. de Boer, A. Damascelli, J. Jegoudez, A. Revcolevschi and T.T.M. Palstra: Acta Cryst. C [**54**]{} (1998) 1558. The same structure was also reported in refs.  and . H.G. von Schnering, Y. Grin, M. Kaupp, M. Sommer, R.K. Kremer, O. Jepsen, T. Chatterji and M. Weiden: Z. Kristallogr. [**213**]{} (1998) 246. H. Smolinski, C. Gros, W. Weber, U. Peuchert, G. Roth, M. Weiden and C. Geibel: Phys. Rev. Lett. [**80**]{} (1998) 5164. T. Ohama, H. Yasuoka, M. Isobe, Y. Ueda: Phys. Rev. B [**59**]{} (1999) 3299. H. Seo and H. Fukuyama: J. Phys. Soc. Jpn. [**67**]{} (1998) 2602. S. Nishimoto and Y. Ohta: J. Phys. Soc. Jpn. [**67**]{} (1998) 2996. P. Thalmeier and P. Fulde: Europhys. Lett. [**44**]{} (1998) 242. M.V. Mostovoy and D.I. Khomskii: Solid State Commun. [**113**]{} (2000) 159. For a review, H. Seo and H. Fukuyama: Mater. Sci. Eng. B [**63**]{} (1999) 1. Y. Fujii, H. Nakao, T. Yosihama, M. Nishi, K. Nakajima, K. Kakurai, M. Isobe, Y. Ueda and H. Sawa: J. Phys. Soc. Jpn. [**66**]{} (1997) 326. J. Lüdecke, A. Jobst, S. van Smaalen, E. Moré, C. Geibel and H. Krane: Phys. Rev. Lett. [**82**]{} (1999) 3633. H. Nakao [*et al.*]{}: preprint. J.L. de Boer [*et al.*]{}: preprint. M. Isobe, C. Kagami, and Y. Ueda, J. Crystal Growth [**181**]{} 314 (1997). J.K.M. Sanders and B.K.Hunter: [*Modern NMR Spectroscopy: a Guide for Chemists 2nd ed.*]{} (Oxford University Press, New York, 1993). T. Ohama, M. Isobe, H. Yasuoka and Y. Ueda: J. Phys. Soc. Jpn. [**66**]{} (1997) 545. Y. Fagot-Revurat [*et al.*]{}: preprint (cond-mat/9907326). P. Horsch and F. Mack: Eur. Phys. J B [**5**]{} (1998) 367. T. Yosihama [*et al.*]{}: preprint. A.I. Smirnov, M.N. Popova, A.B. Sushkov, S.A. Golubchik, D.I. Khomskii, M.V. Mostovoy, A.N. Vasil’ev, M. Isobe and Y. Ueda: Phys. Rev. B [**59**]{} (1999) 14546. S. Ravy, J. Jegoudez and A. Revcolevschi: Phys. Rev. B [**59**]{} (1999) R681. H. Nakao, K. Ohwada, N. Takesue, Y. Fujii, M. Isobe and Y. Ueda: J. Phys. Chem. Solids [**60**]{} (1999) 1101. M. Köppen, D. Pankert, R. Hauptmann, M. Lang, M. Weiden, C. Geibel and F. Steglich: Phys. Rev. B [**57**]{} (1998) 8466. Y. Sekine, N. Takeshita, N. Môri, M. Isobe and Y. Ueda: unpublished. [^1]: E-mail: [email protected]

ArXiv

--- abstract: 'A concept of ultra-thin low frequency perfect sound absorber is proposed and demonstrated experimentally. To minimize non-linear effects, an high ratio of active area to total area is used to avoid large localized amplitudes. The absorber consists of three elements: a mass supported by a very flexible membrane, a cavity and a resistive layer. The resonance frequency of the sound absorber can be easily adjusted just by changing the mass or thickness of the cavity. A very large ratio between wavelength and material thickness is measured for a manufactured perfect absorber (ratio = 201) . It is shown that this high sub-wavelength ratio is associated with narrowband effects and that the increase in the sub-wavelength ratio is limited by the damping in the system.' author: - Yves Aurégan title: 'Ultra-thin low frequency perfect sound absorber with high ratio of active area' --- Sound absorption is of great interest to engineers who want to reduce sound reflection in room acoustics or reduce sound emissions. There is a need for innovative acoustic absorbent materials, effective in low frequencies while being able to deal with spatial constraints present in real applications. Innovative ultra-thin materials are also useful tools for the scientific community to manipulate sound waves and obtain negative refraction [@cummer2016controlling; @kaina2015negative], sub-wavelength imaging [@zhu2011holey; @qi2018ultrathin], cloaking [@faure2016experiments], etc. Traditional sound absorption structures use perforated and micro-perforated panels covering air or porous materials. [@allard2009; @maa1998]. These materials have a low reflection of normal incident waves at frequencies such that the wavelength ($\lambda = c_0/f$ where $c_0$ and $f$ are the sound velocity in air and the frequency) is about four times the thickness of the material $H$ leading to a sub-wavelength ratio $r_H = \lambda/H \simeq 4$. There has been a very significant reduction in the thickness of the absorbent materials [@yang2017optimal] by using space-coiling structures [@cai2014; @chen2017; @wang2018; @long2018multiband] of by using slow-sound inside the material [@groby2015; @yang2016; @jimenez2017]. Nevertheless, all these structures have a front plate with small holes leading to a very low open area ratio ($\sigma$ = area of the orifices on the total area). In this case, since the velocity in the orifices is equal to the acoustic velocity in the incident wave divided by $\sigma$, some non-linear effects can easily occur in the orifices when the amplitude of the incident wave becomes large enough [@ingard1967]. Moreover, in many engineering applications, a grazing flow is present and its effect on the impedance of the material is inversely proportional to $\sigma$ [@guess1975]. For instance, it was experimentally shown [@auregan2016low] that the efficiency of a thin slow-sound metamaterial with $\sigma=0.023$ was divided by 100 in the presence of a grazing flow with a Mach number of 0.2. Therefore, in the case of high sound levels or in the presence of flow, the additional constraint of having a high open area ratio must be added in the design of structures that are thin and absorbent at low frequency. From this perspective, the use of elastic membranes and decorated elastic membranes as sound absorbers is an interesting option. [@frommhold1994; @ma2014; @yang2017]. In most of the studied structures, the membrane represents a large part of the active surface and the added mass (platelets) is of smaller size. The maximum absorption appears for an hybrid resonance [@ma2014] due to the interaction between two modes of coupled system consisting of the membrane, the platelet and the air cavity. By a proper arrangement of the various parameters very impressive results can be obtained both in reflection [@ma2014] and in transmission [@wang2018acoustic]. At the resonance frequency, the three main parameters of membrane absorbers (equivalent mass, stiffness and damping) are very sensitive to the characteristics of the membrane. By changing one of the properties of the membrane (its tension, its mass density,...) or the size of the added mass, the three equivalent parameters are simultaneously changed. ![\[fig\_1\]Sketch of an ultra-thin low frequency (UTLF) resonator.](FIG1.eps){width="0.8\columnwidth"} This letter presents an ultra-thin low frequency (UTLF) resonator for which the three parameters (mass, rigidity and damping) can be modified independently of each other. A model has been measured and perfect absorption occurs for a sub-wavelength ratio $r_H=\lambda/H = 200$ which is, up to now, the highest sub-wavelength ratio experimentally demonstrated for a perfect absorber. This UTLF resonator is ideally composed of three elements displayed on Fig. \[fig\_1\]: A volume of air sealed in a cavity where the compression of the air acts as a spring, a mass that moves like a piston in the normal direction (without letting the air trapped in the cavity pass through with the help of a membrane) and a thin resistive layer that dissipates energy. The main difference compared to decorated elastic membranes absorbers [@ma2014] is that the membrane plays almost no role in the frequency of perfect absorption. The role of the very flexible membrane is just to guide the motion of the mass and seal the cavity. It will be demonstrated that the membrane also produces some damping. As the low frequency regime is considered (any axial dimension is much smaller than $\lambda=c_0/f$ ), the continuity of the acoustic flow rate is assumed across the resistance and the mass: $v_i = v_m=v_c $ where $v_i$, $v_m$ and $v_c $ are respectively the incident wave velocity, the velocity of the mass and the mean velocity entering in the cavity. For simplicity, $a$, the difference between the radius of the tube and the radius of the mass, is neglected compared to the tube radius $R$. The UTLF resonator can be described by its impedance $Z_{\mathrm{UTLF}}$ (normalized by the air characteristic impedance $Z_0=\rho_0 c_0$ where $\rho_0$ is the air density) which is the sum of three terms since the elements are mounted in series. The first term in the impedance is linked to the air compressibility in the cavity of thickness $B$ and can be written $Z_c =p_c/Z_0 v_c=c_0/{\mathrm{i}}\omega B= 1/{\mathrm{i}}\hat{\omega} C$ where $p_c$ is the uniform acoustic pressure in the cavity, $\omega=2 \pi f$, $\hat{\omega}=\omega/(2 \pi f_R)$ where $f_R$ is the resonance frequency of the UTLF resonator. The compressibility term $$C = \frac{2 \pi f_R B }{c_0} =\frac{2\pi }{r_B}$$ directly characterizes the sub-wavelength ratio $r_B = \lambda_R/B$ at resonance. This value of the cavity impedance is only valid at very low frequencies where $\omega B/c_0 \ll 1$. For higher frequencies or higher cavity thickness, a more exact value is $Z_c =1/{\mathrm{i}}\tan(\hat{\omega} C)$. The second term in the impedance is linked to the mass motion subjected to the pressure difference $\Delta p_m$ between its two faces. The associated impedance is $Z_m = \Delta p_m/Z_0 v_i ={\mathrm{i}}\omega m/\rho_0 c_0 S_m = {\mathrm{i}}\hat{\omega} L$ where $m=\rho_m S_m e$ is the mass of the moving part, $S_m=\pi (R-a)^2$ is the mass area, $\rho_m$ is the density of the mass material and $e$ is the mass thickness. It can be noted that the moving mass is connected to the fixed tube by a membrane that acts as a spring but this effect is neglected due to the low value of the membrane stiffness. The last part of the impedance is linked to the resistive layer and is $Z_R= \Delta p_R/Z_0 v_i = R_e$ where $\Delta p_R$ is the pressure drop through the resistive layer and $R_e$ is the normalized resistance of the resistive layer. As a result, the UTLF impedance can be written $Z_{\mathrm{UTLF}}=R_e+ {\mathrm{i}}\hat{\omega} L+1/{\mathrm{i}}\hat{\omega} C$. To be a perfect absorber when $ \hat{\omega}=1$, the UTLF impedance must perfectly match the characteristic impedance of the air. This is achieved when the two conditions $R=1$ and $LC=1$ are met. In this case, the normalized UTLF impedance can be written: $$Z_{\mathrm{UTLF}}=1+\frac{1}{C}\left({\mathrm{i}}\hat{\omega} +\frac{1}{{\mathrm{i}}\hat{\omega}} \right) \label{eq:Z_UTLF}$$ showing that the impedance depends only on the sub-wavelength ratio $r_B$. The absorption coefficient is defined by $\alpha=1-|(Z-1)/(Z+1)|^2$ and relation (\[eq:Z\_UTLF\]) shows that it is equal to 1 for $ \hat{\omega}=1$. The change rate of $\alpha$ around $ \hat{\omega}=1$ is linked to the slope of the imaginary part of the impedance which is given by $|\mathrm{d} Z_{\mathrm{UTLF}}/\mathrm{d} \hat{\omega}| = 2/C$ for $ \hat{\omega}=1$. Using a Taylor expansion near the resonance when $C$ is small, the frequency bandwidth $\Delta f$ for which $\alpha>0.75$ is given by $\Delta f /f_R = 2 C/\sqrt{3}$. Then, $\alpha$ has an increasingly sharp peak as the sub-wavelength ratio $r_B$ increases as it is shown in Fig. \[fig\_2\]. In principle, there is no theoretical upper limit for the sub-wavelength ratio but the price to pay to have highly sub-wavelength cavities is to have a very small bandwidth. From a practical point of view (as will be seen later), the increase in the sub-wavelength ratio $r_B$ is limited by the losses that always become large when $r_B$ increases. ![\[fig\_2\] *Absorption coefficient $\alpha$ of a perfect resonant absorber as a function of the dimensionless frequency $\hat{\omega}$ for two values of the sub-wavelength ratio: $r_B=10$ (dashed blue) and $r_B=200$ (continuous red).*](FIG2.eps){width="\columnwidth"} Figure \[fig\_3\](a) shows the different elements that composes a manufactured sample of UTLF. The first element is a resistive layer, $\underline{\mathbf{1}}$ on Fig. \[fig\_3\](a), consisting of a metal wire mesh glued to a short tube (inner radius $R $= 15 mm, outer radius 19 mm, 1 mm long). Many wire meshes are available with different resistances and the normalized resistance of the one used here was measured at $R_e$= 0.29. The second part is the mass element, $\underline{\mathbf{2}}$ on Fig. \[fig\_3\](a). The mass is a steel disk ($\rho_m = 7800$ kg.m$^{-3}$)with a radius of $R-a$ = 13 mm and a thickness of $e$ = 3 mm. A slightly tensioned latex membrane 20 $\mu$m thick is glued to a short tube (3 mm long) and then the mass is glued to this membrane. The cavity consists of tubes of various lengths $B$ and a plexiglas disk to close it, $\underline{\mathbf{3}}$ and $\underline{\mathbf{4}}$ on Fig. \[fig\_3\](a). ![\[fig\_3\] *Experimental results:(a) Picture of the elements of the UTLF: $\underline{\mathbf{1}}$ Resistive layer, $\underline{\mathbf{2}}$ Mass element, $\underline{\mathbf{3}}$ Spacer tube to make the cavity, $\underline{\mathbf{4}}$ plexiglass cover. (b) Measured absorption coefficient $\alpha$ of the UTLF as a function of frequency. (c) Zoom around the perfect absorption.* ](FIG3.eps){width="\columnwidth"} When a cavity is made with a tube of thickness $B$=12 mm, the theoretical value of the resonance frequency is given by $$f_R=\frac{c_0}{2\pi}\sqrt{\frac{\rho_0 }{\rho_m e B}} = 111 \: \mathrm{Hz}.$$ This value can be compared to the measured frequency given in \[fig\_3\](b). A nearly perfect absorption ($\alpha=$ 0.9991) is obtained for $f_R=107.25$ Hz. The 4% error made on the frequency value is small considering all the simplifications that have been made in the calculations. The value of the compressibility term is $C= 2 \pi/269$. The bandwidth for which $\alpha>0.75$ is $\Delta f$ = 2.5 Hz = $C f_R$. The total height of the material $H$ = 16 mm is the sum of the cavity height $B$, mass thickness $e$ and thickness of the resistive layer glued on its support (1 mm). Thus, the sub-wavelength ratio of this material is $r_H= \lambda/H$ = 201.6. Up to now, this ratio between wavelength and material thickness is the highest ever measured in an absorbent material. The resonance frequency is very easy to adjust since it depends only on the mass of the moving part and on the height of the cavity. The simplicity of the adjustment comes from the fact that membrane effects can be neglected to compute $f_R$ because of the low value of its equivalent stiffness. In the device under study, the resonance frequency of the mass element measured with the membrane alone (without the cavity) is about 35 Hz while it increases to 107.25 Hz in the presence of the cavity. This indicates that the equivalent stiffness of the membrane is 9 times lower than the one of the cavity with a thickness of $B$=12 mm. The membrane is involved in the dissipative part of the impedance which is shown in Fig. \[fig\_4\](a). The blue line in this figure gives the dissipative effect when the resistive layer is not present. In this case, the dissipative effects come solely from the membrane. First of all, it can be noted that this dissipation is relatively high (the minimum value of $\Re(Z)$ is 0.71 representing 71% of the target value). It has already been noted in [@ma2014] that even a small dissipation in the membrane material could result in significant absorption of the overall system. Second, it can be noted that the dissipation curve appears to result from two different effects. For $f>$140 Hz, the dissipation increases with frequency. This effect is the only one that is experimentally observed when there is no cavity. This dissipation can therefore be attributed to the mechanical damping resulting from the elongation of the membrane when the mass moves. For $f<$90 Hz a second phenomenon appears for which the resistance decreases with frequency. It has been experimentally observed that this second resistance increases as the cavity thickness decreases. One possible explanation is that, when the air trapped in the cavity becomes difficult to compress or expand (i.e. for small volume or low frequency), it inflates and aspirates the membrane as schematized in Fig. \[fig\_4\](b), which increases the dissipative phenomena. ![\[fig\_4\] *(a) Real part of the impedance (resistance) of the UTLF absorber without the resistive layer (blue) and with the resistive layer (black). (b) Schematic deformation of the membrane.* ](FIG4.eps){width="\columnwidth"} The combination of the two previous effects induces a minimum of resistance for a frequency close to resonance. When this minimum is smaller than 1, it is possible to add a resistive layer that shifts the resistance to higher values, see the red curve in Fig. \[fig\_4\](b) in order to adapt the resistance exactly to 1 at the resonant frequency. This is what has been done to obtain the perfect absorber whose absorption curve is presented on the Fig. \[fig\_3\](b). In the present case, the addition of a resistive layer only changes the maximum attenuation by a few percent. But for less extreme conditions ($\lambda/B \ll $200), the attenuation due to the membrane can be much lower and the addition of a resistive layer very useful to obtain a perfect absorption. To conclude, we have seen that it is possible to build a UTLF absorber with very simple elements: mass supported by a very flexible membrane, cavity closed by the membrane and resistive layer. This device can have a large open area to minimize non-linear effects or reduce dependence on grazing flows. The frequency of the UTLF absorber can be easily adjusted by simply changing the mass or thickness of the cavity. With this type of absorber, it is possible to obtain a very high ratio between the wavelength and the thickness of the material but, as a necessary counterpart, the bandwidth is very narrow. The energy dissipation comes partially from the membrane and partially from the resistive layer. When trying to further reduce the size of the cavity, the resistance of the absorber becomes greater than the characteristic impedance of the air due to the dissipation in the membrane. The dissipation is therefore the limiting factor in the decrease in the size of this type of absorbers. This remark is not specific to this absorber but can be generalized to many resonant sub-wavelength absorbers: It is the damping that limits the sub-wavelength ratio in resonant absorbing devices that are build with three in series elements (mass, spring and damping) like the classical Helmholtz resonator. One possibility to overcome this limit is to add some gain in the system by supplying external energy using, for example, electro-dynamical devices [@fleury2015invisible; @lissek2018toward], thermo-acoustic devices[@biwa2016experimental] or using the flow[@auregan2017p] which is present in many engineering applications. This work was supported by the “Agence Nationale de la Recherche” international project FlowMatAc No. ANR-15-CE22-0016-01. 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ArXiv

--- abstract: 'It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass $N>0$ satisfies $N<N^*$, where the finite constant $N^*$ is called the critical stellar mass. Lieb and Yau conjecture in \[Comm. Math. Phys., 1987\] that ground states of the pseudo-relativistic Boson stars are [*unique*]{} for each $N<N^*$. In this paper, we prove that the above uniqueness conjecture holds for the particular case where $N>0$ is small enough.' author: - 'Yujin Guo[^1] and Xiaoyu Zeng[^2]' title: 'The Lieb-Yau Conjecture for Ground States of Pseudo-Relativistic Boson Stars' --- 0.2truein [*Keywords:*]{} Uniqueness; Ground states; Boson stars; Pohozaev identity 0.2truein Introduction ============ Various models of pseudo-relativistic boson stars have attracted a lot of attention in theoretical and numerical astrophysics over the past few decades, see [@LY; @LY87] and references therein. In this paper, we are interested in ground states of pseudo-relativistic Boson stars in the mean field limit (cf. [@ES; @FJCMP; @LY]), which can be described by constraint minimizers of the following variational problem $$\label{def:eN} e(N):=\inf \Big\{ \mathcal{E}(u):\, u\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)\,\ \mbox{and}\ \int_{{{\mathbb{R}}}^3} |u(x)|^2dx=N\Big\},$$ where $N>0$ denotes the stellar mass of Boson stars, and the pseudo-relativistic Hartree energy functional $ \mathcal{E}(u)$ is of the form $$\label{f} \mathcal{E}(u):=\int_{{{\mathbb{R}}}^3} \bar u\big( \sqrt{-\Delta +m^2}-m\big)udx-\frac{1}{2}\int_{{{\mathbb{R}}}^3}\big(|x|^{-1}\ast |u|^2\big)|u|^2dx,\ \ m>0.$$ Here the operator $\sqrt{-\Delta +m^2}$ is defined via multiplication in the Fourier space with the symbol $\sqrt{|\xi|^2+m^2}$ for $\xi\in{{\mathbb{R}}}^3$, which describes the kinetic and rest energy of many self-gravitating and relativistic bosons with rest mass $m>0$, and the symbol $\ast$ stands for the convolution on ${{\mathbb{R}}}^3$. Because of the physical relevance, without special notations we always focus on the case $m>0$ throughout the whole paper. The main purpose of this paper is to prove the uniqueness of minimizers for $e(N)$, provided that $N>0$ is small enough. The variational problem $e(N)$ is essentially in the class of $L^2-$critical constraint minimization problems, which were studied recently in the nonrelativistic cases, e.g. [@Cao; @GLW; @GS; @GZZ] and references therein. Comparing with these mentioned works, it however deserves to remark that the analysis of $e(N)$ is more complicated in a substantial way, which is mainly due to the nonlocal nature of the pseudo-differential operator $\sqrt{-\Delta +m^2}$, and the convolution-type nonlinearity as well. Starting from the pioneering papers [@LY; @LY87], many works were devoted to the mathematical analysis of the variational problem $e(N)$ over the past few years, see [@ES; @FL; @FJ; @FJCMP; @FrL; @GZ; @L09; @N; @YY] and references therein. The existing results show that the analysis of $e(N)$ is connected well to the following Gagliardo-Nirenberg inequality of fractional type $$\label{GNineq} {\int_{\mathbb{R}^3}}\big(\frac{1}{|x|}\ast |u|^2\big)|u|^2 dx\le \frac 2 {\|w\|^2_2}\|(-\Delta )^{1/4}u\|^2_2\, \|u\|^2_2 ,\ \ u \in H^{\frac{1}{2}}({{\mathbb{R}}}^3),$$ where $w(x)=w(|x|)>0$ is a ground state, up to translations and suitable rescaling (cf. [@FJCMP; @LY]), of the fractional equation $$\sqrt{-\Delta}\,u+u-\Big(\frac{1}{|x|}\ast |u|^2\Big)u=0 \ \mbox{ in } \ {{\mathbb{R}}}^3,\ \mbox{ where }\ u\in H^{\frac{1}{2}}({{\mathbb{R}}}^3). \label{w:eqn}$$ By making full use of (\[GNineq\]), Lenzmann in [@L09] established the following interesting existence and analytical characters of minimizers for $e(N)$: 0.05truein [**Theorem A ([@L09 Theorem 1])**]{} *Under the assumption $m>0$, the following results hold for $e(N)$:* 1. $e(N)$ has minimizers if and only if $0<N<N^*$, where the finite constant $N^*$ is independent of $m$. 2. Any minimizer $u$ of $e(N)$ satisfies $u\in H^s({{\mathbb{R}}}^3)$ for all $s\ge \frac{1}{2}$. 3. Any nonnegative minimizer of $e(N)$ must be strictly positive and radially symmetric-decreasing, up to phase and translation. We remark that the existence of the critical constant $N^*>0$ stated in Theorem A, which is called the critical stellar mass of boson stars, was proved earlier in [@FJCMP; @LY]. Further, the dynamics and some other analytic properties of minimizers for $e(N)$ were also investigated by Lenzmann and his collaborators in [@FL; @FJ; @FJCMP; @FrL; @L; @L09] and references therein. Stimulated by [@GS; @GZZ], the related limit behavior of minimizers for $e(N)$ as $N\nearrow N^*$ were studied more recently in [@GZ; @N; @YY], where the Gagliardo-Nirenberg type inequality (\[GNineq\]) also played an important role. Note also from the variational theory that any minimizer $u$ of $e(N)$ satisfies the Euler-Lagrange equation $$\label{1:eq1.18} \big( \sqrt{-\Delta +m^2}-m\big)u- \Big(\frac{1}{|x|}\ast |u|^2\Big)u =-\mu u \ \ \text{in}\, \ \mathbb{R}^3,$$ where $\mu \in {{\mathbb{R}}}$ is the associated Lagrange multiplier. Following (\[1:eq1.18\]) and Theorem A, one can deduce that any minimizer of $e(N)$ must be either positive or negative, see [@GZ] for details. Therefore, it is enough to consider positive minimizers of $e(N)$, which are called ground states of $e(N)$ throughout the rest part of this paper. Whether a physics system admits a unique ground state or not is an interesting and fundamental problem. Lieb and Yau [@LY] conjectured in 1987 that for each $N<N^*$, there exists a unique ground state (minimizer) of $e(N)$. As expected by Lieb and Yau there, the analysis of this uniqueness conjecture is however challenging extremely. Actually, it is generally difficult to prove whether any two different ground states of $e(N)$ satisfy the equation (\[1:eq1.18\]) with [*the same*]{} Lagrange multiplier $\mu \in {{\mathbb{R}}}$. On the other hand, more difficultly, it seems very challenging to address the uniqueness of ground states for (\[1:eq1.18\]). Essentially, even though the uniqueness of ground states for the following fractional equation $$(-\Delta)^s\,u+u-u^{\alpha +1}=0 \ \mbox{ in } \ {{\mathbb{R}}}^N,\ u\in H^s({{\mathbb{R}}}^N) , \label{p:eqn}$$ where $0<s<1$, $\alpha >0$ and $N\ge 1$, was already proved in the celebrated works [@FL13; @FL], the uniqueness of ground states for (\[w:eqn\]) or (\[1:eq1.18\]) is still open, due to the nonlocal nonlinearity of Hartree type. Therefore, whether the above Lieb-Yau conjecture is true for all $0<N<N^*$ remains mainly open after three decades, except Lenzmann’s recent work [@L09] in 2009. As an important first step towards the Lieb-Yau conjecture, Lenzmann proved in [@L09] that for each $0<N\ll N^*$ and except for at most countably many $N$, the uniqueness of minimizers for $e(N)$ holds true. We emphasize that the additional assumption “except for at most countably many $N$" seems essential in Lenzmann’s proof, since the smoothness of the GP energy $e(N)$ with respect to $N$ was employed there. In this paper we intend to remove the above additional assumption and prove the Lieb-Yau conjecture in the particular case where $0<N<N^*$ is small enough. More precisely, our main result of this paper is the following uniqueness of minimizers for $e(N)$. \[th1\] If $N> 0$ is small enough, then the problem $e(N)$ admits a unique positive minimizer, up to phase and translation. The similar uniqueness results of Theorem \[th1\] were established recently in [@AF Theorem 2] and [@M Theorem 1.1] (see also [@GWZZ Corollary 1.1]) for the nonrelativistic Hartree minimization problems with trapping potentials, under the additional assumption that the associated nonrelativistic operator admits the first eigenvalue. We however emphasize that the arguments of [@AF; @M; @GWZZ] are not applicable for proving Theorem \[th1\], since the associated pseudo-relativistic operator $H:= \sqrt{-\Delta +m^2}-m$ does not admit any eigenvalue in our problem $e(N)$. Therefore, a different approach is needed for proving Theorem \[th1\]. Towards this purpose, since positive minimizers $u$ of $e(N)$ vanish uniformly as $N\searrow 0$, motivated by [@L09] we define $$\tilde u(x)=c^2u(cx),\ \ \mbox{where}\ \ c>0,$$ so that $$\label{G2:1} \mathcal{E}(u)=c^{-3}\mathcal{E}_c(\tilde u) \ \text{ and }{\int_{\mathbb{R}^3}}|\tilde u|^2dx=c{\int_{\mathbb{R}^3}}| u|^2dx,$$ where the energy functional $\mathcal{E}_c(\cdot)$ is given by $$\mathcal{E}_c(u):=\int_{{{\mathbb{R}}}^3} \bar{ u}\big( \sqrt{-c^2\Delta +c^4m^2}-c^2m\big) udx-\frac{1}{2}\int_{{{\mathbb{R}}}^3}\big(|x|^{-1}\ast |u|^2\big)|u|^2dx,\ \ m>0.$$ Consider the minimization problem $$\label{def:ec} \bar e(c):=\inf \Big\{ \mathcal{E}_c(u):\, u\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)\,\ \mbox{and}\ \int_{{{\mathbb{R}}}^3} |u(x)|^2dx=1\Big\}.$$ Note from Theorem A and (\[G2:1\]) that if $c>0$ is large enough, then $\bar e(c)$ in (\[def:ec\]) admits positive minimizers. More importantly, setting $c=N^{-1}>0$, studying positive minimizers of $e(N)$ as $N\searrow 0$ is then equivalent to investigating positive minimizers of the minimization problem (\[def:ec\]) as $c\nearrow \infty$. Therefore, to establish the uniqueness of Theorem \[th1\], it suffices to prove the following uniqueness theorem. \[th2\] If $c> 0$ is large enough, then $\bar e(c)$ in (\[def:ec\]) admits a unique positive minimizer, up to phase and translation. Suppose now that $Q_c>0$ is a minimizer of $\bar e(c)$ defined in (\[def:ec\]). Then there exists a Lagrange multiplier $\mu_c \in {{\mathbb{R}}}$ such that $Q_c>0$ solves $$\label{1D:eq1.18} \big( \sqrt{-c^2\Delta +c^4m^2}-c^2m\big)Q_c- \Big(\frac{1}{|x|}\ast |Q_c|^2\Big)Q_c =-\mu_c Q_c \ \ \text{in}\, \ \mathbb{R}^3.$$ Recall from [@L09 Proposition 1] that up to a subsequence if necessary, $\mu_c\in{{\mathbb{R}}}$ satisfies $$\label{1E:eq1.18} \mu_c\to -{\lambda}<0 \ \ \mbox{as}\ \ c\to \infty$$ for some constant ${\lambda}>0$. In order to prove Theorem \[th2\], associated to $Q_c$, we need to study the uniformly exponential decay of $\phi_c\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)$ as $c\to\infty$, where $\phi_c$ satisfies $$\label{1:e:1} \begin{split} &\big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)\phi_c -\Big(\frac{1}{|x|}\ast Q_c^2\Big)\phi_c -2k_1\Big\{\frac{1}{|x|}\ast \big(Q_c \phi_c \big)\Big\}Q_c\\ &-k_2(c)Q_c=-\mu_c \phi_c \ \text{ in}\ \ {{\mathbb{R}}}^3. \end{split}$$ Here $\mu_c\in{{\mathbb{R}}}$ satisfies (\[1E:eq1.18\]), $k_1\ge 0$ and $k_2(c)\in {{\mathbb{R}}}$ is bounded uniformly in $c>0$. As proved in Lemma \[lem:2.1\], we shall derive the following uniformly exponential decay of $\phi_c$ as $c\to\infty$: $$\label{D:eq1.09} |\phi_{c}(x)| \le Ce^{-\delta|x|} \ \ \mbox{in}\ \, {{\mathbb{R}}}^3$$ holds uniformly as $c\to\infty$, where the constants $\delta>0$ and $C=C(\delta)>0$ are independent of $c$. Since $\mu_c\in{{\mathbb{R}}}$ satisfies (\[1E:eq1.18\]), stimulated by [@AS; @FJCMP; @H; @SW], the proof of (\[D:eq1.09\]) is based on the uniformly exponential decay (\[E:e:3\]) of the Green’s function $G_c(\cdot)$ for $\big(\bar H_c+\mu _c\big)^{-1}$ as $c\to\infty$, where the operator $\bar H_c$ is defined by $$\bar H_c:=\sqrt{-c^2\Delta +m^2c^4}-mc^2.$$ As shown in Lemma \[lem:2.3\], it however deserves to remark that because $G_c(\cdot)$ depends on $c>0$, one needs to carry out more delicate analysis, together with some tricks, for addressing the uniformly exponential decay (\[E:e:3\]) of $G_c(\cdot)$ as $c\to\infty$. On the other hand, as a byproduct, the exponential decay (\[D:eq1.09\]) can be useful in analyzing the limiting procedure of solutions for Schrodinger equations involving the above fractional operator $\bar H_c$, which were investigated widely in [@CLM; @Choi; @CN] and references therein. Following (\[D:eq1.09\]) and the regularity of $Q_c$, in Section 2 we shall finally prove the following limit behavior $$\label{D:1.14} Q_c\to Q_\infty \ \mbox{ uniformly in }\ {{\mathbb{R}}}^3\ \text{ as }\ c \to \infty,$$ where $Q_\infty>0$ is the unique positive minimizer of (\[def:em\]) described below. Based on the refined estimates of Section 2, motivated by [@Deng; @Grossi; @GLW], we shall employ the nondegenerancy and uniqueness of $Q_\infty$ to complete the proof of Theorem \[th2\] by establishing Pohozaev identities. This paper is organized as follows. In Section 2 we shall address some refined estimates of $Q_c$ as $c\to\infty$, where Lemma \[lem:2.1\] is proved in Subsection 2.1. Following those estimates of Section 2, Section 3 is devoted to the proof of Theorems \[th2\] on the uniqueness of minimizers for $\bar e(c)$ as $c\to\infty$. Theorem \[th1\] then follows immediately from Theorem \[th2\] in view of the relation (\[G2:1\]). Analytical Properties of $Q_c$ as $c\to\infty$ ============================================== The main purpose of this section is to give some refined analytical estimates of $Q_c$ as $c\to\infty$, where $Q_c>0$ is a positive minimizer of $\bar e(c)$ defined in (\[def:ec\]). Note also from Theorem A and (\[G2:1\]) that $Q_c$ is radially symmetric in $|x|$. We first introduce the following limit problem associated to $\bar e(c)$: $$\label{def:em} { e}_m:=\inf \Big\{ {E}_m(u):\, u\in H^{1}({{\mathbb{R}}}^3)\,\ \mbox{and}\ \int_{{{\mathbb{R}}}^3} |u(x)|^2dx=1\Big\},$$ where the energy functional $E_m(\cdot)$ satisfies $${E}_m(u):=\frac{1}{2m}\int_{{{\mathbb{R}}}^3} |\nabla u|^2dx-\frac{1}{2}\int_{{{\mathbb{R}}}^3}\big(|x|^{-1}\ast |u|^2\big)|u|^2dx, \ \forall u\in H^1({{\mathbb{R}}}^3),\ \ m>0.$$ For any given $m>0$, it is well-known that, up to translations, problem (\[def:em\]) has a unique positive minimizer denoted by $Q_\infty(|x|)$, which must be radially symmetric, see [@L09; @L76] and references therein. Further, $Q_\infty>0$ solves the following equation $$\label{eq1.8} -\frac{1}{2m}\Delta Q_\infty+\lambda Q_\infty=(|x|^{-1}\ast |Q_\infty|^2)Q_\infty\ \ \mbox{in}\, \ {{\mathbb{R}}}^3,\ \ Q_\infty\in H^1({{\mathbb{R}}}^3),$$ where the Lagrange multiplier $\lambda >0$ depends only on $m$ and is determined uniquely by the constraint condition $\int_{{{\mathbb{R}}}^3} Q_\infty ^2dx=1$. Note from [@MS Theorem 3] that $Q_\infty>0$ is a unique positive solution of (\[eq1.8\]). Moreover, recall from [@L09 Theorem 4] that $Q_\infty$ is non-degenerate, in the sense that the linearized operator $L_+: \, H^2({{\mathbb{R}}}^3) \mapsto L^2({{\mathbb{R}}}^3)$, which is defined by $$\label{eq1.10} L_+\xi:=\Big(-\frac{1}{2m}\Delta+\lambda-|x|^{-1}\ast |Q_\infty|^2\Big)\xi-2\big(|x|^{-1}\ast(Q_\infty\xi)\big)Q_\infty,$$ satisfies $$\label{eq1.12} \ker L_+=\text{\rm span} \Big\{\frac{\partial Q_\infty}{\partial x_1}, \frac{\partial Q_\infty}{\partial x_2}, \frac{\partial Q_\infty}{\partial x_3}\Big\}.$$ As a positive minimizer of $\bar e(c)$, $Q_c>0$ satisfies the following equation $$\label{eq2.7} \big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)Q_c-(|x|^{-1}\ast Q_c^2)Q_c=\mu_c Q_c \ \text{ in}\,\ {{\mathbb{R}}}^3,$$ where $\mu_c\in {{\mathbb{R}}}$ is a suitable Lagrange multiplier. Recall from [@L09 Proposition 1] that up to a subsequence if necessary, the Lagrange multiplier $\mu_c$ of (\[eq2.7\]) satisfies $$\label{eq1.09C} \mu_c\to -{\lambda}<0 \ \, \text{ as }\ c\to \infty,$$ where ${\lambda}>0$ is the same as that of (\[eq1.8\]). Associated to the positive minimizer $Q_c>0$ of (\[eq2.7\]), we next define the linearized operator $$\label{e:1} \begin{split} \mathcal{L}_{k_1,k_2}\xi:=&\big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)\xi-\Big(\frac{1}{|x|}\ast Q_c^2\Big)\xi\\ &-2k_1\Big\{\frac{1}{|x|}\ast \big(Q_c \xi\big)\Big\}Q_c-k_2(c)Q_c \ \text{ in}\ \ {{\mathbb{R}}}^3, \end{split}$$ where the constants $m>0$ and $k_1\ge 0$, and $k_2(c)\in {{\mathbb{R}}}$ is bounded uniformly in $c>0$. \[lem:2.1\] Suppose $\varphi_c=\varphi_c(x)\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)$ is a solution of $$\label{e:2} \mathcal{L}_{k_1,k_2}\varphi_c=\mu_c \varphi_c \ \text{ in}\,\ {{\mathbb{R}}}^3,$$ where $\mu_c$ satisfies (\[eq1.09C\]), and the operator $\mathcal{L}_{k_1,k_2}$ is defined by (\[e:1\]) for some constants $m>0$ and $k_1\ge 0$, and $k_2(c)\in {{\mathbb{R}}}$ being bounded uniformly in $c>0$. Then there exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that $$\label{e:3} |\varphi_c(x)|\le Ce^{-\delta |x|}\ \text{ in}\,\ {{\mathbb{R}}}^3$$ uniformly for all sufficiently large $c>0$. Since the proof of Lemma \[lem:2.1\] is a little involved, we leave it to Subsection 2.1. Applying Lemma \[lem:2.1\], we next address the following estimates of $Q_c>0$ as $c\to \infty$, which are crucial for the proof of Theorem \[th2\]. Let $Q_{c}>0$ be a positive minimizer of $\bar e(c)$ defined in (\[def:ec\]) as $c\to\infty$. Then we have 1. There exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that $$\label{eq1.09} |Q_{c}(x)|,\, |\nabla Q_{c}(x)|\le Ce^{-\delta|x|} \ \ \mbox{in}\ \, {{\mathbb{R}}}^3$$ uniformly as $c\to\infty$. 2. $Q_{c}$ satisfies $$\label{eq1.09B} Q_{c}\to Q_\infty \ \mbox{ uniformly in }\ {{\mathbb{R}}}^3\ \text{ as }\ c\to \infty,$$ where $Q_\infty>0$ is the unique positive minimizer of (\[def:em\]). **Proof.** 1. Since $Q_{c}>0$ solves (\[eq2.7\]), the uniformly exponential decay (\[eq1.09\]) for $Q_{c}$ as $c\to\infty$ follows directly from (\[eF:3\]) below. Because $\frac{\partial Q_{c}}{\partial x_i}$ satisfies (\[e:2\]) for $k_1=1$ and $k_2(c)\equiv 0$, where $i=1, 2, 3$, the exponential decay (\[eq1.09\]) holds for $\nabla Q_{c}$ by applying Lemma \[lem:2.1\]. 2\. Following [@M Lemma 4.9] and references therein, we first recall from [@L09 Proposition 1] that $Q_{c}$ satisfies $$\label{eq1.9} Q_{c}\to Q_\infty \ \text{ in } H^1({{\mathbb{R}}}^3)\ \text{ as } c\to \infty,$$ where the convergence holds for the whole sequence of $\{c\}$, due to the uniqueness of $Q_\infty>0$. Rewrite (\[eq2.7\]) as $$\label{2A:2} \begin{split} \Big(-\frac{1}{2m}\Delta -\mu_{c}\Big)Q_{c}&=\big(|x|^{-1}\ast Q_{c}^2\big)Q_{c}- \Big(\sqrt{-c^2\Delta +m^2c^4}-mc^2+\frac{1}{2m}\Delta \Big)Q_{c}\\ &:= \big(|x|^{-1}\ast Q_{c}^2\big)Q_{c}- F_{c} (\nabla)Q_{c} \ \text{ in}\,\ {{\mathbb{R}}}^3, \end{split}$$ where we denote the pseudo-differential operator $$\label{2D:2} F_{c}(\nabla):=\sqrt{-c^2\Delta +m^2c^4}-mc^2-\frac{-\Delta}{2m}$$ with the symbol $$F_{c}(\xi):=\sqrt{c^2|\xi|^2 +m^2c^4}-mc^2-\frac{ |\xi|^2}{2m} ,\ \ \xi\in{{\mathbb{R}}}^3.$$ Recall from (\[eq1.09\]) that $Q_{c}$ decays exponentially as $|x|\to\infty$ for all sufficiently large $c>0$. Moreover, since the operator $\bar H_c:=\sqrt{-c^2\Delta +m^2c^4}-mc^2$ is uniformly bounded from below for all $c>0$, the similar argument of [@FJCMP Theorem 4.1(i)] or [@Choi Proposition 4.2] applied to (\[eq2.7\]) and (\[eq1.09C\]) yields that $$\label{B:M} Q_{c}\in H^s({{\mathbb{R}}}^3) \ \ \mbox{for all}\, \ s\ge \frac{1}{2},$$ which further implies the uniform smoothness of $Q_c$ in $c>0$, and $$\label{2A:5OK} Q_{c}\in L^\infty({{\mathbb{R}}}^3) \ \ \mbox{uniformly in}\, \ c>0.$$ Applying the Taylor expansion, we obtain from (\[2D:2\]) that for $|\xi|\ge \frac{mc}{2}$, $$\big|F_{c}(\xi)\big|=\Big|c|\xi|\sqrt{ 1+\big(\frac{mc}{|\xi|}\big)^2}-mc^2-\frac{ |\xi|^2}{2m} \Big|\le \sqrt 5\, c|\xi|+mc^2+\frac{ |\xi|^2}{2m} \le \frac{36 |\xi|^4}{m^3c^2},\ \ \xi\in{{\mathbb{R}}}^3,$$ and for $|\xi|\le \frac{mc}{2}$, $$\big|F_{c}(\xi)\big|=\Big|mc^2\sqrt{ 1+\big(\frac{|\xi|}{mc}\big)^2}-mc^2-\frac{ |\xi|^2}{2m} \Big|\le \frac{|\xi|^4}{8m^3c^2},\ \ \xi\in{{\mathbb{R}}}^3,$$ due to the fact that $|\sqrt{1+t}-1-\frac{1}{2}t|\le\frac{1}{8}t^2$ holds for all $0<t\le \frac{1}{2}$. Following above estimates, we then derive from (\[2D:2\]) and (\[B:M\]) that for sufficiently large $c>0$, $$\label{2D:2D} \|F_{c} (\nabla)Q_{c}\|_{H^s({{\mathbb{R}}}^3)}\le \frac{M_1}{m^3c^2}\|Q_c\|_{H^{s+4}({{\mathbb{R}}}^3)}<\frac{M_2}{m^3c^2} \ \ \mbox{for all}\ \ s\ge \frac{1}{2},$$ where $M_1>0$ and $M_2>0$ are independent of $c>0$. Also, since $$\label{2A:E1} \big\||x|^{-1}\ast Q_{c}^2\big\|_{L^\infty({{\mathbb{R}}}^3)}\le C\big(\big\|Q_{c}\big\|^2_{L^4({{\mathbb{R}}}^3)}+\big\|Q_{c}\big\|^2_{L^2({{\mathbb{R}}}^3)}\big),$$ we derive from (\[B:M\]) that for any $p\ge 2$, $$\label{2A:1M} \big\|\big(|x|^{-1}\ast Q_{c}^2\big)Q_{c}\big\|_{L^p({{\mathbb{R}}}^3)}\le C\big(\big\|Q_{c}\big\|^2_{L^4({{\mathbb{R}}}^3)}+\big\|Q_{c}\big\|^2_{L^2({{\mathbb{R}}}^3)}\big)\big\|Q_{c}\big\|_{L^p({{\mathbb{R}}}^3)}\le K_p,$$ where $K_p>0$ is independent of $c>0$. Employing (\[2D:2D\]) and (\[2A:1M\]), together with Sobolev imbedding theorem, the bootstrap argument applied to (\[2A:2\]) yields that $$\label{eq2.20} \text{$Q_{c}\to Q_\infty$ uniformly on any compact domain of ${{\mathbb{R}}}^3$ as $c\to\infty$.}$$ On the other hand, one can easily deduce from (\[eq1.8\]) that $Q_\infty$ decays exponentially as $|x|\to\infty$. Together with (\[eq1.09\]), this indicates that for any ${\varepsilon}>0$, there exists a constant $R_{\varepsilon}>0$, independent of $c>0$, such that $$|Q_c(x)|,\,|Q_\infty(x)|<\frac{{\varepsilon}}{4}\ \, \text{ for any }\ |x|>R_{\varepsilon},$$ and hence, $$\sup_{|x|>R_{\varepsilon}}|Q_c(x)-Q_\infty(x)|\leq\sup_{|x|>R_{\varepsilon}} (|Q_c(x)|+|Q_\infty(x)|)\leq \frac{{\varepsilon}}{2}.$$ Moreover, it follows from (\[eq2.20\]) that for sufficiently large $c>0$, $$\sup_{|x|\leq R_{\varepsilon}}|Q_c(x)-Q_\infty(x)|\leq\frac{{\varepsilon}}{2}.$$ The above two estimates thus yield that for sufficiently large $c>0$, $$\sup_{x\in{{\mathbb{R}}}^3}|Q_c(x)-Q_\infty(x)|\leq{\varepsilon},$$ which implies that (\[eq1.09B\]) holds true. The lemma is therefore proved. Uniformly exponential decay as $c\to\infty$ ------------------------------------------- In this Subsection, we address the proof of Lemma \[lem:2.1\] on the uniformly exponential decay as $c\to\infty$. We remark that even though the proof of Lemma \[lem:2.1\] is stimulated from [@AS; @FJCMP; @H; @SW], as shown in proving Lemma \[lem:2.3\] below, we need to carry out more delicate analysis together with some tricks. We first suppose that $\varphi_c\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)$ is a solution of $$\label{R:e:2} \big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)\varphi_c-\Big(\frac{1}{|x|}\ast \varphi_c^2\Big)\varphi_c=-{\lambda}_c \varphi_c \ \text{ in}\,\ {{\mathbb{R}}}^3,$$ where the constant $m>0$ and $$\label{R:e:2M} {\lambda}_c\to 2{\lambda}>0 \ \, \text{ as }\ c\to \infty$$ for some positive constant ${\lambda}$. Define $$\label{R:e:2D} \bar H_c:=\sqrt{-c^2\Delta +m^2c^4}-mc^2,\quad V(x):=-\frac{1}{|x|}\ast \varphi_c^2.$$ Therefore, $\varphi_c$ can be thought of as an eigenfunction of the Schrodinger operator $H:=\bar H_c+V(x)$. Moreover, the argument of [@FJCMP Theorem 4.1] or [@Choi Proposition 4.2] gives that $\varphi_c\in H^s({{\mathbb{R}}}^3)$ for all $s\ge \frac{1}{2}$, which implies the smoothness of $\varphi_c$. Further, the spectrum of $\bar H_c$ satisfies $$\sigma (\bar H_c)=\sigma _{ess}(\bar H_c)=[0,\infty)$$ for all $c>0$. Under the assumption (\[R:e:2M\]), then $\big(\bar H_c+{\lambda}_c \big)^{-1}$ exists for all sufficiently large $c>0$, and (\[R:e:2\]) can be rewritten as $$\varphi_c(x)=-\big(\bar H_c+ {\lambda}_c\big)^{-1}V(x)\varphi_c(x).$$ Note also from (\[R:e:2\]) and (\[R:e:2D\]) that $$\label{R:e:4} \varphi_c(x)=-{\int_{\mathbb{R}^3}}G_c(x-y)V(y)\varphi_c(y)dy,$$ where $G_c(x-y)$ is the Green’s function of $\big(\bar H_c+{\lambda}_c\big)^{-1}$ defined in (\[R:e:2D\]). The following lemma gives the uniformly exponential decay of $G_c(\cdot)$ as $c\to\infty$. \[lem:2.3\] Suppose ${\lambda}_c\in {{\mathbb{R}}}$ satisfies (\[R:e:2M\]) for some ${\lambda}>0$. Then for each $0<\delta < \min\{ \frac{m}{2},\sqrt{{\lambda}m}\}$, there exists a constant $M:=M(\delta)>0$, independent of $c>0$, such that the Green’s function $G_c(x-y)$ of $\big(\bar H_c+{\lambda}_c\big)^{-1}$ satisfies $$\label{E:e:3} |G_c(x-y)|\le M\frac{e^{-\delta |x-y|}}{|x-y|^2}\ \text{ in}\,\ {{\mathbb{R}}}^3$$ uniformly for all sufficiently large $c>0$. **Proof.** Under the assumption (\[R:e:2M\]), since $G_c(\cdot)$ is the Green’s function of $\big(\bar H_c+{\lambda}_c\big)^{-1}$ defined in (\[R:e:2D\]) for all sufficiently large $c>0$, we have $$\label{R:e:5} G_c(z)=f_c^{-1}(z), \ \, f_c(\mu )=\frac{1}{\sqrt{c^2|\mu|^2 +m^2c^4}-mc^2+{\lambda}_c}\quad \mbox{for} \ \ \mu\in {{\mathbb{R}}}^3,$$ and $f_c^{-1}: \mathcal{S'}\to \mathcal{S}'$ denotes the inverse Fourier transform of $f_c$. We obtain from (\[R:e:5\]) that for all sufficiently large $c>0$, $$\label{R:e:6} G_c(z)=f_c^{-1}(z)=(2\pi)^{-3/2}{\int_{\mathbb{R}^3}}\frac{e^{i\mu\cdot z}}{\sqrt{c^2|\mu|^2 +m^2c^4}-mc^2+{\lambda}_c}d\mu=\frac{1}{c}\,g_c^{-1}(z), \\ \quad$$ where $$g_c(\mu )=\frac{1}{\sqrt{ |\mu|^2 +m^2c^2}-mc+\frac{{\lambda}_c}{c}}, \ \ \mu\in {{\mathbb{R}}}^3.$$ In view of (\[R:e:6\]), we next define $$H_c+\frac{{\lambda}_c}{c}:=\sqrt{ -\Delta +m^2c^2}-mc+\frac{{\lambda}_c}{c},$$ so that $$\label{R:e:7} \Big( H_c+\frac{{\lambda}_c}{c}\Big)^{-1}=\int^\infty_0e^{-t\frac{{\lambda}_c}{c}}e^{-tH_c}dt=\int^\infty_0e^{-t(\frac{{\lambda}_c}{c}-mc) }e^{-t\sqrt{ p^2 +m^2c^2}}dt,\quad p=-i\nabla .$$ Note from pp. 183 of [@LL] that $$e^{-t\sqrt{ p^2 +m^2c^2}}(z)=\frac{m^2c^2}{2\pi ^2}\frac{t}{t^2+|z|^2}K_2\big(mc\sqrt{ t^2 +|z|^2}\big),\ \ z\in{{\mathbb{R}}}^3,$$ where $K_2(\cdot)>0$ denotes the modified Bessel function of the third kind. We then derive from above that $$\label{R:e:8} G_c(z)=\frac{1}{c}\,g_c^{-1}(z)=\frac{m^2c}{2\pi ^2}\int^\infty_0e^{-t(\frac{{\lambda}_c}{c}-mc) }\frac{t}{t^2+|z|^2}K_2\big(mc\sqrt{ t^2 +|z|^2}\big)dt.$$ Recall from [@H] that there exist positive constants $\bar M_1$ and $\bar M_2$, independent of $c>0$, such that $$\label{R:e:9} K_2(cmw)\le\arraycolsep=1.5pt\left\{\begin{array}{lll} \displaystyle\frac{\bar M_1}{c^2m^2w^2} \quad &\mbox{if} & \ \, \displaystyle w<\frac{2}{mc},\\[4mm] \displaystyle\frac{\bar M_2e^{-cmw}}{\sqrt{cmw}} \quad &\mbox{if}& \,\ \displaystyle w\ge\frac{1}{mc}, \end{array}\right.$$ where $w>0$ is a real number. We next follow (\[R:e:8\]) and (\[R:e:9\]) to complete the proof by discussing separately the following two cases, which involve very complicated estimates together with some tricks: 0.05truein (1). Case of $|z|\ge \frac{1}{mc}$. In this case, we have $\sqrt{t^2+|z|^2}\ge \frac{1}{mc}$ for all $t\ge 0$. We then obtain from (\[R:e:2M\]), (\[R:e:8\]) and (\[R:e:9\]) that for all sufficiently large $c>0$, $$\label{R:e:10}\arraycolsep=1.5pt\begin{array}{lll} \displaystyle\frac{2\pi ^2}{\bar M_2m^{\frac{3}{2}}}G_c(z)&\le & \displaystyle\sqrt c\int^\infty_0\frac{t}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-t(\frac{{\lambda}}{c}-mc) -mc\sqrt{ t^2 +|z|^2}} dt\\[3mm] &=&\displaystyle\int^\infty_{2\sqrt{\frac{m}{{\lambda}}}c^{\frac{3}{2}}|z|} A(t,z)dt+\displaystyle \int^{2\sqrt{\frac{m}{{\lambda}}}c^{\frac{3}{2}}|z|}_{c|z|} A(t,z)dt\\[4mm] &&+\displaystyle \int^{c|z|}_{\frac{1}{4}\sqrt{\frac{m}{{\lambda}}}c^{\frac{1}{2}}|z|} A(t,z)dt+\displaystyle \int^{\frac{1}{4}\sqrt{\frac{m}{{\lambda}}}c^{\frac{1}{2}}|z|}_0 A(t,z)dt\\[5mm] &:=&I_A(z)+I_B(z)+I_C(z)+I_D(z), \end{array}$$ where $A(t,z)$ satisfies $$\label{R:e:10A} A(t,z):=\frac{t\sqrt c}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-t(\frac{{\lambda}}{c}-mc) -mc\sqrt{ t^2 +|z|^2}}.$$ For $I_A(z)+I_D(z)$, we note that if $t\ge 0$ satisfies $$t\ge 2\sqrt{\frac{m}{{\lambda}}}\,c^{\frac{3}{2}}|z| \ \ \mbox{or}\ \ 0\le t\le \frac{1}{4}\sqrt{\frac{m}{{\lambda}}}\,c^{\frac{1}{2}}|z|,$$ then one can check that $$\label{R:e:11} \sqrt{ t^2 +|z|^2}\ge \sqrt{1-\frac{{\lambda}}{mc^2}}\ t+\sqrt{\frac{{\lambda}}{mc}}\,|z|\ge \Big(1-\frac{2{\lambda}}{3mc^2}\Big)t+\sqrt{\frac{{\lambda}}{mc}}\,|z|$$ for sufficiently large $c>0$. We thus obtain from (\[R:e:10A\]) and (\[R:e:11\]) that for sufficiently large $c>0$, $$\label{R:e:12}\arraycolsep=1.5pt\begin{array}{lll} I_A(z)+I_D(z)&\le &\displaystyle\int^\infty_0\frac{t\sqrt c}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-\frac{{\lambda}}{3c}t -\sqrt{ {\lambda}mc}|z|}dt\\[4mm] &\le &\displaystyle C_1\frac{e^{-\sigma |z|}}{\sqrt c \,|z|}\int^\infty_0\frac{t\sqrt c}{(t^2+|z|^2)^{\frac{5}{4}}}dt\le C_2\frac{e^{-\sigma |z|}}{|z|^\frac{3}{2}}, \end{array}$$ where $\sigma >0$ is arbitrary, and the constants $C_1>0$ and $C_2>0$ are independent of $c>0$. For $I_B(z)$, we observe that if $t> 0$ satisfies $$c|z|\le t\le 2\sqrt{\frac{m}{{\lambda}}}\,c^{\frac{3}{2}}|z|,$$ then we have $$\label{R:e:13} \sqrt{ t^2 +|z|^2}\ge \sqrt{1-\frac{{\lambda}}{mc^2}}\,t+\sqrt{\frac{{\lambda}}{mc^2}}\,|z|\ge \Big(1-\frac{2{\lambda}}{3mc^2}\Big)t+\sqrt{\frac{{\lambda}}{mc^2}}\,|z|$$ for sufficiently large $c>0$. We thus obtain from (\[R:e:10A\]) and (\[R:e:13\]) that for sufficiently large $c>0$, $$\label{R:e:14}\arraycolsep=1.5pt\begin{array}{lll} I_B(z)&\le &\displaystyle\int^{2\sqrt{\frac{m}{{\lambda}}}c^{\frac{3}{2}}|z|}_{c|z|}\frac{t\sqrt c}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-\frac{{\lambda}}{3c}t -\sqrt{ {\lambda}m}|z|}dt\\[4mm] &\le &\displaystyle C_3 e^{-\sqrt{ {\lambda}m}|z|} \int^\infty_{c|z|}\frac{t\sqrt c}{(t^2+|z|^2)^{\frac{5}{4}}}dt\le C_4\frac{e^{-\sqrt{ {\lambda}m}|z|}}{\sqrt{|z|}}, \end{array}$$ where the constants $C_3>0$ and $C_4>0$ are independent of $c>0$. As for $I_C(z)$, we get that if $t> 0$ satisfies $$C_0c^{\frac{1}{2}}|z|:=\frac{1}{4}\sqrt{\frac{m}{{\lambda}}}\,c^{\frac{1}{2}}|z|\le t\le c|z|,$$ then we have $$\label{R:e:15} \sqrt{ t^2 +|z|^2}=t\sqrt{ 1 +\Big(\frac{|z|}{t}\Big)^2}\ge t+\big(\frac{1}{2}-{\varepsilon}\big)\frac{|z|^2}{t}$$ for sufficiently large $c>0$, where $0<{\varepsilon}<\frac{1}{4}$ is arbitrary. We thus obtain from (\[R:e:10A\]) and (\[R:e:15\]) that for sufficiently large $c>0$, $$\label{R:e:16}\arraycolsep=1.5pt\begin{array}{lll} I_C(z)&\le &\sqrt c\displaystyle\int_{C_0c^{\frac{1}{2}}|z|}^{c|z|}\frac{t}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-\frac{{\lambda}}{c}t -(\frac{1}{2}-{\varepsilon})\frac{mc}{t}|z|^2}dt\\[4mm] &\le &-2\sqrt c\displaystyle \int_{C_0c^{\frac{1}{2}}|z|}^{c|z|}e^{-(\frac{1}{2}-{\varepsilon})\frac{mc}{t}|z|^2}dt^{-\frac{1}{2}}\\[4mm] &:=&2\displaystyle \sqrt{\frac{c}{|z|}}\displaystyle \int^{C_0^{-\frac{1}{2}}c^{-\frac{1}{4}}}_{c^{-\frac{1}{2}}}e^{-(\frac{1}{2}-{\varepsilon})mc|z| s^2}ds. \end{array}$$ Note that if $\frac{1}{\sqrt c}\le s$, then $2\sqrt cds\le cds^2$. We thus derive from (\[R:e:16\]) that for $\tau:=s^2>0$, $$\label{R:e:17} I_C (z)\le \displaystyle\frac{c}{\sqrt{|z|}} \int^{\frac{1}{c_0\sqrt c}}_{ \frac{1}{c}}e^{-(\frac{1}{2}-{\varepsilon})mc|z|\tau}d\tau\le C_5\frac{e^{-(\frac{1}{2}-{\varepsilon})m|z|}}{|z|^{\frac{3}{2}}},$$ where the constant $C_5>0$ is also independent of $c>0$, and $0<{\varepsilon}<\frac{1}{4}$ is arbitrary as before. Following (\[R:e:10\]), we now conclude from above that for $0<\delta_0:= \min\{ \big(\frac{1}{2}-{\varepsilon}\big)m,\sqrt{{\lambda}m}\}$, where $0<{\varepsilon}<\frac{1}{4}$ is arbitrary, there exists a constant $M_0:=M_0(\delta_0)>0$ such that for all sufficiently large $c>0$, $$\label{R:e:18} G_c(z) \le M_0\frac{e^{-\delta _0|z|}}{\min\{|z|^{\frac{1}{2}}, |z|^{\frac{3}{2}}\}},\ \ \mbox{if}\ \ |z|\ge \frac{1}{mc}.$$ This further implies that for each $0<\delta_1< \min\{ \frac{m}{2},\sqrt{{\lambda}m}\}$, there exists a constant $M_1:=M_1(\delta_1)>0$ such that for $|z|\ge \frac{1}{mc}$, $$\label{R:e:18A} G_c(z) \le M_1\frac{e^{-\delta _1|z|}}{ |z|^2}$$ uniformly for all sufficiently large $c>0$. 0.05truein (2). Case of $|z|\le \frac{1}{mc}$. In this case, we deduce from (\[R:e:2M\]), (\[R:e:8\]) and (\[R:e:9\]) that for all sufficiently large $c>0$, $$\label{R:e:19}\arraycolsep=1.5pt\begin{array}{lll} \displaystyle 2\pi ^2G_c(z)&\le & \displaystyle\bar M_2m^\frac{3}{2}\sqrt c\int^\infty_{\frac{1}{mc}}\frac{t}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-t(\frac{{\lambda}}{c}-mc) -mc\sqrt{ t^2 +|z|^2}} dt\\[4mm] &&+\displaystyle \frac{\bar M_1}{c} \int_0^{\frac{1}{mc}}\frac{t}{(t^2+|z|^2)^2}e^{-(\frac{{\lambda}}{c}-mc)t} dt\\[4mm] &:=&I_1(z)+I_2(z). \end{array}$$ Similar to (\[R:e:10\]) and (\[R:e:18\]), one can obtain that for all sufficiently large $c>0$, $$\label{R:e:20} I_1(z)\le C_6\sqrt c \int^\infty_0\frac{t}{(t^2+|z|^2)^{\frac{5}{4}}}e^{-t(\frac{{\lambda}}{c}-mc) -mc\sqrt{ t^2 +|z|^2}} dt \le \frac{M_2}{|z|^2},\ \ \mbox{if}\ \ |z|\le \frac{1}{mc},$$ where the constants $C_6>0$ and $M_2>0$ are independent of $c>0$. As for $I_2(z)$, we infer that $$\begin{split} I_2(z)&\le \frac{C_7}{c} \int_0^{\frac{1}{mc}}\frac{t}{(t^2+|z|^2)^2} dt=\frac{C_7}{2c|z|^2} \int_0^{\frac{1}{mc|z|}}\frac{1}{(s^2+1)^2} ds^2 \\ &= \frac{C_7}{2c|z|^2}\frac{(mc|z|)^2}{1+(mc|z|)^2}\le\frac{C_7}{2c|z|^2},\ \ \mbox{if}\ \ |z|\le \frac{1}{mc}, \end{split}$$ where the constant $C_7>0$ is independent of $c>0$. We therefore derive from (\[R:e:19\]) and above that for all sufficiently large $c>0$, $$\label{R:e:21} G_c(z) \le \frac{M_3}{|z|^2},\ \ \mbox{if}\ \ |z|\le \frac{1}{mc},$$ where the constant $M_3>0$ is also independent of $c>0$. We finally conclude from (\[R:e:18A\]) and (\[R:e:21\]) that (\[E:e:3\]) holds true, and we are done. 0.05truein **Proof of Lemma \[lem:2.1\].** We first prove that the positive solution $Q_c>0$ of (\[eq2.7\]), where the Lagrange multiplier $\mu_c$ is as in (\[eq1.09C\]), satisfies the following exponential decay $$\label{eF:3} |Q_c|\le Ce^{-\delta |x|}\ \text{ in}\,\ {{\mathbb{R}}}^3$$ uniformly for all sufficiently large $c>0$, where the constants $\delta>0$ and $C=C(\delta)>0$ are independent of $c>0$. Actually, recall from (\[eq2.7\]) that $Q_c$ can be rewritten as $$\label{R:e:22} Q_c(x)=-{\int_{\mathbb{R}^3}}G_c(x-y)V(y)Q_c(y)dy,$$ where $G_c(x-y)$ is the Green’s function of $\big(\bar H_c+(-\mu _c)\big)^{-1}$ defined by (\[R:e:2D\]), and the potential $V(x):=-(\frac{1}{|x|}\ast Q_c^2)(x)$ satisfies $V(x)\in C^0({{\mathbb{R}}}^3)$ and $\lim _{|x|\to\infty}V(x)=0$. Since $-\mu_c\to {\lambda}>0$ as $c\to\infty$ in view of (\[eq1.09C\]), $G_c(x-y)$ satisfies the exponential decay of Lemma \[lem:2.3\]. Following the above properties, the uniformly exponential decay (\[eF:3\]) as $c\to\infty$ can be proved in a similar way of [@FJCMP Appendix C] and [@H Theorem 2.1], where the Slaggie-Wichmann method (e.g. [@H]) is employed. To finish the proof of Lemma \[lem:2.1\], we next rewrite the solution $\varphi_c\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)$ of (\[e:2\]) as $$\label{R:e:23} \varphi_c(x)=\big(\bar H_c+(-\mu _c)\big)^{-1}\big(V_1\varphi_c+2k_1V_2(\varphi_c)+k_2(c)Q_c\big),$$ where the operator $\bar H_c$ satisfies (\[R:e:2D\]) as before and $$\label{R:e:24} V_1= \frac{1}{|x|}\ast Q_c^2\,,\quad V_2(\varphi_c)=\Big\{\frac{1}{|x|}\ast \big(Q_c \varphi_c\big)\Big\}Q_c.$$ Here $\mu_c\in{{\mathbb{R}}}$ satisfies (\[eq1.09C\]), $k_1\ge 0$, and $k_2(c)\in {{\mathbb{R}}}$ is bounded uniformly in $c>0$. Note from (\[R:e:23\]) that $\varphi_c\in H^{\frac{1}{2}}({{\mathbb{R}}}^3)$ solves $$\label{R:e:22} \varphi_c(x)={\int_{\mathbb{R}^3}}G_c(x-y)\big(V_1\varphi_c+k_1V_2(\varphi_c)+k_2(c)Q_c\big)(y)dy,$$ where the Green’s function $G_c(\cdot)$ of $\big(\bar H_c+(-\mu _c)\big)^{-1}$ satisfies as before the uniformly exponential decay of Lemma \[lem:2.3\] in view of (\[eq1.09C\]). Since $Q_c$ satisfies the uniformly exponential decay (\[eF:3\]) as $c\to\infty$, the uniformly exponential decay (\[e:3\]) of $\varphi_c$ as $c\to\infty$ can be further proved in a similar way of [@FJ Lemma 4.9]. We omit the detailed proof for simplicity. This completes the proof of Lemma \[lem:2.1\]. Proof of Theorem \[th2\] ======================== Following the refined estimates of previous section, in this section we shall complete the proof of Theorem \[th2\]. We begin with the following two lemmas. \[le2.1\] Suppose $Q_\infty$ is the unique radially symmetric positive solution of (\[eq1.8\]) and let the operator $L_+$ be defined by (\[eq1.10\]). Then we have $$L_+(x\cdot \nabla Q_\infty+2Q_\infty)=-2\lambda Q_\infty,$$ where ${\lambda}>0$ is as in (\[eq1.8\]). **Proof.** Direct calculations give that $$\begin{split} -\Delta(x\cdot \nabla Q_\infty)&=-\sum_{i=1}^3\partial_{i}\big(\partial_{i}(\sum_{j=1}^3x_j \partial_{j}Q_\infty)\big) =-\sum_{i,j=1}^3\partial_i\big(\delta_{ij}\partial_j Q_\infty+x_j\partial_{ij}Q_\infty\big)\\ &=-\sum_{i,j=1}^3\big(2\delta_{ij}\partial_{ij }Q_\infty+x_j\partial_{iij}Q_\infty\big)=-2\Delta Q_\infty-x\cdot\nabla(\Delta Q_\infty), \end{split}$$ and $$2Q_\infty\big[|x|^{-1}\ast (Q_\infty x\cdot\nabla Q_\infty)\big]=Q_\infty\big[|x|^{-1}\ast (x\cdot\nabla Q_\infty^2)\big].$$ We then have $$\label{eq2.2} \begin{split} L_+(x\cdot \nabla Q_\infty)&=-\frac{1}{m}\Delta Q_\infty-\frac{1}{2m}x\cdot\nabla\big(\Delta Q_\infty\big)+\lambda x\cdot \nabla Q_\infty\\ &-\big(|x|^{-1}\ast Q_\infty^2)(x\cdot \nabla Q_\infty)-Q_\infty\big[|x|^{-1}\ast (x\cdot\nabla Q_\infty^2)\big]. \end{split}$$ Taking the action $x\cdot \nabla $ on (\[eq1.8\]), we deduce that $$-\frac{1}{2m}x\cdot \nabla(\Delta Q_\infty)+\lambda x\cdot \nabla Q_\infty=\big(|x|^{-1}\ast Q_\infty^2\big)\big(x\cdot \nabla Q_\infty\big)+Q_\infty x\cdot\big(|x|^{-1}\ast \nabla Q_\infty^2\big).$$ Together with (\[eq2.2\]), this indicates that $$\label{eq2.3} \begin{split} L_+(x\cdot \nabla Q_\infty)&=-\frac{1}{m}\Delta Q_\infty+Q_\infty x\cdot\big(|x|^{-1}\ast \nabla Q_\infty^2\big)-Q_\infty\big[|x|^{-1}\ast (x\cdot\nabla Q_\infty^2)\big]. \end{split}$$ Since $$\label{eq2.33} \begin{split} & x\cdot\big(|x|^{-1}\ast \nabla Q_\infty^2\big)-|x|^{-1}\ast (x\cdot\nabla Q_\infty^2)\\ =&{\int_{\mathbb{R}^3}}\frac{x\cdot\nabla Q_\infty^2(y)}{|x-y|}dy-{\int_{\mathbb{R}^3}}\frac{y\cdot\nabla Q_\infty^2(y)}{|x-y|}dy \\ =&{\int_{\mathbb{R}^3}}\frac{(x-y)\cdot\nabla Q_\infty^2(y)}{|x-y|}dy=-\sum_{i=1}^3{\int_{\mathbb{R}^3}}\partial_{y_i}\Big(\frac{x-y}{|x-y|}\Big) Q_\infty^2(y)dy \\ =&2{\int_{\mathbb{R}^3}}\frac{Q_\infty^2(y)}{|x-y|}dy=2|x|^{-1}\ast Q_\infty^2, \end{split}$$ it follows from (\[eq2.3\]) that $$\label{eq2.4} \begin{split} L_+(x\cdot \nabla Q_\infty)&=-\frac{1}{m}\Delta Q_\infty+2\big(|x|^{-1}\ast Q_\infty^2\big)Q_\infty. \end{split}$$ Moreover, recall from (\[eq1.8\]) that $$\label{eq2.5} L_+ Q_\infty=\Big(-\frac{1}{2m}\Delta+\lambda-3|x|^{-1}\ast |Q_\infty|^2\Big)Q_\infty=-2(|x|^{-1}\ast Q_\infty^2)Q_\infty.$$ Combining (\[eq2.4\]) with (\[eq2.5\]) thus yields that $$L_+(x\cdot \nabla Q_\infty+2Q_\infty)=-\frac{1}{m}\Delta Q_\infty-2\big(|x|^{-1}\ast Q_\infty^2\big)Q_\infty=-2\lambda Q_\infty,$$ and the proof of this lemma is therefore complete. \[le2.2\] Let $Q_c$ be a radially symmetric positive minimizer of $\bar e(c)$ defined in (\[def:ec\]). Then we have the following Pohozaev identity $$\label{eq2.07} -m^2c^4 \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}Q_c,Q_c\big\rangle +mc^2{\int_{\mathbb{R}^3}}|Q_c|^2dx+\bar e(c)=0.$$ **Proof.** In the proof of this lemma, we denote $Q_c$ by $Q$ for convenience. We first note that $$x\cdot \nabla Q(x) =\frac{d}{d\lambda}Q_\lambda(x)\big|_{\lambda=1},\ \text{ where }Q_\lambda(x):=Q(\lambda x).$$ Multiplying $x\cdot \nabla Q(x)$ on both sides of (\[eq2.7\]) and integrating over ${{\mathbb{R}}}^3$, we have $$\label{eq2.8} \begin{split} & \big\langle\sqrt{-c^2\Delta+m^2c^4}Q,x\cdot \nabla Q\big\rangle \\ =&\frac{d}{d\lambda} \big\langle\sqrt{-c^2\Delta+m^2c^4}Q,Q_\lambda\big\rangle \Big|_{\lambda=1} \\ =&\frac{d}{d\lambda} \big\langle(-c^2\Delta+m^2c^4)^\frac{1}{4}Q,(-c^2\Delta+m^2c^4)^\frac{1}{4}Q_\lambda\big\rangle \Big|_{\lambda=1} \\ \quad\qquad \overset{\sqrt\lambda x=x'} = & \frac{d}{d\lambda} \lambda^{-1} \big\langle(-c^2\Delta+\frac{m^2c^4}{\lambda})^\frac{1}{4}Q\big(\frac{x}{\sqrt\lambda}\big), \big(-c^2\Delta+\frac{m^2c^4}{\lambda}\big)^\frac{1}{4}Q(\sqrt\lambda x)\big\rangle \Big|_{\lambda=1} \\ =&-{\int_{\mathbb{R}^3}}\big|(-c^2\Delta+m^2c^4)^\frac{1}{4}Q\big|^2dx\\ \quad &-\frac{m^2c^4}{2} \big\langle(-c^2\Delta+m^2c^4)^{-\frac{3}{4}}Q,(-c^2\Delta+m^2c^4)^\frac{1}{4}Q\big\rangle \\ =&- \big\langle\sqrt{-c^2\Delta+m^2c^4}Q,Q\big\rangle -\frac{m^2c^4}{2} \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}Q,Q\big\rangle . \end{split}$$ Moreover, we derive from the exponential decay (\[eq1.09\]) that $$\begin{aligned} &{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q(x\cdot \nabla Q)dx=\frac12{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)(x\cdot \nabla Q^2)dx\nonumber\\ &=-\frac{3}{2}{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx-\frac{1}{2}{\int_{\mathbb{R}^3}}\Big[(|x|^{-1}\ast \nabla Q^2)\cdot x\Big] Q^2dx\nonumber\\ &=-\frac{3}{2}{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx-\frac{1}{2}\Big[2{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2) Q^2dx\nonumber\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+\iint_{{{\mathbb{R}}}^3} \frac{y\cdot\nabla Q^2(y)}{|x-y|}Q^2(x)dydx\Big],\label{eq2.10}\end{aligned}$$ where the argument of deriving (\[eq2.33\]) is used in the last equality. Since $$\begin{split} \iint_{{{\mathbb{R}}}^3} \frac{y\cdot\nabla Q^2(y)}{|x-y|}Q^2(x)dydx ={\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)(x\cdot \nabla Q^2)dx, \end{split}$$ we obtain from (\[eq2.10\]) that $$\label{eq2.11} {\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q(x\cdot \nabla Q)dx=-\frac54{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx.$$ One can easily check that $${\int_{\mathbb{R}^3}}Q(x\cdot \nabla Q)dx=-\frac{3}{2}{\int_{\mathbb{R}^3}}Q^2dx.$$ Thus, it follows from (\[eq2.7\]), (\[eq2.8\]) and (\[eq2.11\]) that $$\begin{split} &- \big\langle\sqrt{-c^2\Delta+m^2c^4}Q,Q\big\rangle -\frac{m^2c^4}{2} \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}Q,Q\big\rangle \\ &=-\frac32(mc^2+\mu _c){\int_{\mathbb{R}^3}}Q^2dx-\frac54{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx. \end{split}\label{eq2.12}$$ By (\[eq2.12\]), multiplying $Q$ on both sides of (\[eq2.7\]) and integrating over ${{\mathbb{R}}}^3$ yield that $$\label{eq2.13} \begin{split} &-{m^2c^4} \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}Q,Q\big\rangle \\ &+(mc^2+\mu_c){\int_{\mathbb{R}^3}}|Q|^2dx+\frac12{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx=0. \end{split}$$ We also derive from (\[eq2.7\]) that $$\begin{split}\mu_c{\int_{\mathbb{R}^3}}|Q|^2dx&= \big\langle\sqrt{-c^2\Delta+m^2c^4}Q,Q\big\rangle -mc^2{\int_{\mathbb{R}^3}}|Q|^2dx-{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx\\ &=\bar e(c)-\frac{1}{2}{\int_{\mathbb{R}^3}}(|x|^{-1}\ast Q^2)Q^2dx, \end{split}$$ which therefore implies that (\[eq2.07\]) holds true by applying (\[eq2.13\]). .05truein Following previous estimates, we are now ready to finish the proof of Theorem \[th2\]. .05truein **Proof of Theorem \[th2\].** Up to the phase and translation, it suffices to prove that $\bar e(c)$ in (\[def:ec\]) admits a unique positive minimizer for sufficiently large $c> 0$. On the contrary, suppose that $Q_{1c}$ and $Q_{2c}$ are two different radially symmetric (about the origin) positive minimizers of problem (\[def:ec\]), where $m>0$ is fixed. Then $Q_{ic}\in H^s({{\mathbb{R}}}^3)$, where $s\ge \frac{1}{2}$, satisfies the following equation $$\label{eq2.14} \big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)Q_{ic}-(|x|^{-1}\ast Q_{ic}^2)Q_{ic}=\mu_{ic} Q_{ic} \ \text{ in }\, {{\mathbb{R}}}^3, \ i=1,2,$$ where $\mu_{ic} \in{{\mathbb{R}}}$ is the Lagrange multiplier associated to $Q_{ic}$ for $i=1,2$. Since $Q_{1c}\not\equiv Q_{2c}$ in ${{\mathbb{R}}}^3$, we define $$\label{eq2.15} w_c(x):=\frac{Q_{1c}(x)-Q_{2c}(x)}{\|Q_{1c}-Q_{2c}\|_{L^\infty({{\mathbb{R}}}^3)}} \ \text{ in }\, {{\mathbb{R}}}^3.$$ It then follows from (\[eq2.14\]) that $$\label{eq2.16} \begin{split} &\big(\sqrt{-c^2\Delta +m^2c^4}-mc^2\big)w_{c}-\big(|x|^{-1}\ast Q_{1c}^2\big)w_{c}-\Big\{|x|^{-1}\ast \big[(Q_{1c}+Q_{2c})w_c\big]\Big\}Q_{2c}\\ &=\mu_{2c} w_c+\frac{\mu_{1c}-\mu_{2c}}{\|Q_{1c}-Q_{2c}\|_{L^\infty({{\mathbb{R}}}^3)}}Q_{1c} \ \text{ in }\, {{\mathbb{R}}}^3. \end{split}$$ Recall from (\[eq1.09C\]) that $$\label{eq2.27} \lim_{c\to\infty}\mu_{ic}=-\lambda<0, \ i=1,2.$$ We also note from (\[eq2.14\]) that $$\label{eq2.17} \mu_{ic}=\bar e(c)-\frac{1}{2}{\int_{\mathbb{R}^3}}\big(|x|^{-1}\ast Q_{ic}^2\big)Q_{ic}^2dx, \ i=1,2,$$ which implies that $$\label{eq2.17:K} \begin{split} \frac{\mu_{1c}-\mu_{2c}}{\|Q_{1c}-Q_{2c}\|_{L^\infty({{\mathbb{R}}}^3)}} = -\frac{1}{2}{\int_{\mathbb{R}^3}}&\Big\{ (|x|^{-1}\ast Q_{1c}^2)(Q_{1c}+Q_{2c})w_{c}\\ &+\Big[|x|^{-1}\ast \big((Q_{1c}+Q_{2c})w_c\big)\Big]Q_{2c}^2\Big\}dx:=k_2(c)\in{{\mathbb{R}}}, \end{split}$$ where $k_2(c)$ is bounded uniformly in $c>0$ by (\[eq1.09\]) and (\[B:M\]). Applying Lemma \[lem:2.1\] to the equation (\[eq2.16\]), we then deduce from (\[eq2.27\]) and (\[eq2.17:K\]) that there exist $\delta>0$ and $C=C(\delta)>0$, which are independent of $c>0$, such that $$\label{e:3:3} |w_c|\le Ce^{-\delta |x|}\ \text{ in}\,\ {{\mathbb{R}}}^3$$ uniformly as $c\to\infty$. Similar to the proof of Lemma \[lem:2.1\], following (\[eq2.16\]) to consider the equation of $\frac{\partial w_c}{\partial x_i}$ ($i=1, 2, 3$), we further derive from (\[eq2.27\])–(\[e:3:3\]) that there exist $\delta _1>0$ and $C_1=C_1(\delta_1)>0$, which are independent of $c>0$, such that $$\label{e:3:3B} |\nabla w_c|\le C_1e^{-\delta _1|x|}\ \text{ in}\,\ {{\mathbb{R}}}^3$$ uniformly as $c\to\infty$. Rewrite the equation (\[eq2.16\]) as $$\label{B:eq2.16} \begin{split} \big(\bar{H}_c+\mu\big)w_{c}=&\Big\{\mu+\mu_{2c} +\big(|x|^{-1}\ast Q_{1c}^2\big)\Big\}w_c\\ &+k_2(c)Q_{1c} +\Big\{|x|^{-1}\ast \big[(Q_{1c}+Q_{2c})w_c\big]\Big\}Q_{2c}\ \text{ in }\, {{\mathbb{R}}}^3 \end{split}$$ for any $\mu\in {{\mathbb{R}}}$, where the uniformly bounded function $k_2(c)\in{{\mathbb{R}}}$ is as in (\[eq2.17:K\]), and the operator $\bar {H}_c$ satisfies $$\bar{H}_c:=\sqrt{-c^2\Delta +m^2c^4}-mc^2.$$ Since $\|w _c\|_{L^\infty({{\mathbb{R}}}^3)}\le 1$ for all $c>0,$ we obtain from (\[eq1.09\]), (\[B:M\]) and (\[2A:E1\]) that for sufficiently large $c>0$, $$\label{C:eq2.16} \big\| |x|^{-1}\ast Q_{1c}^2 \big\| _{L^\infty({{\mathbb{R}}}^3)}<M, \ \ \big\| |x|^{-1}\ast \big[(Q_{1c}+Q_{2c})w_c\big] \big\| _{L^\infty({{\mathbb{R}}}^3)}<M,$$ where $M>0$ is independent of $c>0$. In view of (\[B:M\]) and (\[C:eq2.16\]), the similar argument of [@FJCMP Theorem 4.1(i)] or [@Choi Proposition 4.2] applied to (\[B:eq2.16\]) and (\[eq2.27\]) yields that $$\label{F:M} w_{c}\in H^s({{\mathbb{R}}}^3) \ \ \mbox{for all}\, \ s\ge \frac{1}{2},$$ which further implies the uniform smoothness of $w_c$ in $c>0$. We next rewrite the equation (\[eq2.16\]) as $$\label{eq2.16K} \begin{split} & \Big(-\frac{1}{2m}\Delta -\mu_{2c}\Big)w_{c} -\big(|x|^{-1}\ast Q_{1c}^2\big)w_{c}-\Big\{|x|^{-1}\ast \big[(Q_{1c}+Q_{2c})w_c\big]\Big\}Q_{2c}\\ &=k_2(c)Q_{1c}- F_{c}(\nabla) w_c \ \text{ in }\, {{\mathbb{R}}}^3, \end{split}$$ where the uniformly bounded function $k_2(c)\in{{\mathbb{R}}}$ is again as in (\[eq2.17:K\]), and the pseudo-differential operator $F_{c}(\nabla)$ is the same as (\[2D:2\]) with the symbol $$F_{c}(\xi):=\sqrt{c^2|\xi|^2 +m^2c^4}-mc^2-\frac{ |\xi|^2}{2m}, \ \ \xi\in{{\mathbb{R}}}^3.$$ Note that $F_{c}(\nabla)$ satisfies the estimate (\[2D:2D\]). We then derive from (\[2D:2D\]) and (\[F:M\]) that $$\label{2K:2D} \|F_{c} (\nabla)w_{c}\|_{H^s({{\mathbb{R}}}^3)}\le \frac{M_1}{m^3c^2}\|w_c\|_{H^{s+4}({{\mathbb{R}}}^3)}<\frac{M_2}{m^3c^2} \ \ \mbox{for all}\ \ s\ge \frac{1}{2},$$ where $M_1>0$ and $M_2>0$ are independent of $c>0$. Using the uniformly exponential decays (\[eq1.09\]) and (\[e:3:3\]), by the standard elliptic regularity we derive from (\[B:M\]), (\[eq2.16K\]) and (\[2K:2D\]) that $\|w _c\|_{C^{\alpha }_{loc}({{\mathbb{R}}}^3)}\le M_3$ for some ${\alpha}\in (0,1)$, where the constant $M_3>0$ is independent of $c$. Therefore, there exists a function $w _0=w _0(x)$ such that up to a subsequence if necessary, we have $$w_c\to w_0\ \text{ in }\ C_{\rm loc}({{\mathbb{R}}}^3)\ \text{ as }\ c\to\infty.$$ Moreover, applying the estimates (\[eq2.27\]), (\[eq2.17:K\]) and (\[2K:2D\]), we deduce from (\[eq2.16K\]) that $w_0$ satisfies $$\label{eq2.18A} L_+ w_0 =-Q_\infty{\int_{\mathbb{R}^3}}\Big\{(|x|^{-1}\ast Q_{\infty}^2)Q_\infty w_{0}+\big[|x|^{-1}\ast (Q_{\infty}w_0)\big]Q_{\infty}^2\Big\}dx,$$ where the uniformly exponential decay (\[eq1.09\]) is also used. Applying (\[eq1.12\]) and Lemma 3.1, we now derive from (\[eq2.18A\]) that there exist constants $b_0$ and $c_i$ ($i=1, 2, 3$) such that $$w_0=b_0\big(x\cdot \nabla Q_\infty+2Q_\infty\big)+\sum ^3_{i=1}c_i\frac{\partial Q_\infty}{\partial x_i}.$$ Further, since $Q_{1c}$ and $Q_{2c}$ are both radially symmetric in $|x|$ for all $c>0$, the definition of $w_c$ implies that $w_0$ is also radially symmetric in $|x|$, i.e., $w_0\in L^2_{rad}({{\mathbb{R}}}^3)$. Applying [@L09 Proposition 2], it then follows from the above expression that $$\label{eq2.18} w_0=b_0\big(x\cdot \nabla Q_\infty+2Q_\infty\big)\ \text{ for some }\, b_0\in {{\mathbb{R}}}.$$ We next prove $b_0=0$ in (\[eq2.18\]) so that $w_0\equiv 0$ in ${{\mathbb{R}}}^3$. Indeed, multiplying (\[eq2.07\]) by $Q_{1c}$ and $Q_{2c}$ respectively, and integrating over ${{\mathbb{R}}}^3$, we obtain that $$\label{eq2.19} \begin{split} -mc^2 \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}w_c,Q_{1c}\big\rangle&-mc^2 \big\langle(-c^2\Delta+m^2c^4)^{-\frac{1}{2}}Q_{2c},w_c\big\rangle \\ &+{\int_{\mathbb{R}^3}}w_c(Q_{1c}+Q_{2c})dx=0. \end{split}$$ Moreover, since $\frac{1}{\sqrt{1+t}}=1-\frac{t}{2}+O(t^2)$ as $t\to0$ and $w_c$ is smooth for all $c>0$, we have $$\begin{split} mc^2\big(-c^2\Delta+m^2c^4\big)^{-\frac{1}{2}}= \Big(1-\frac{\Delta}{m^2c^2} \Big)^{-\frac{1}{2}}=\Big[1+\frac{\Delta}{2m^2c^2}+O(\frac{1}{m^4c^4})\Big]\ \ \mbox{as}\ \, c\to\infty. \end{split}$$ Putting it into (\[eq2.19\]) yields that $$-\frac{1}{2m}{\int_{\mathbb{R}^3}}\big(Q_{1c}\Delta w_c +w_c\Delta Q_{2c}\big)dx+O\Big(\frac{1}{m^3c^2}\Big)=0\ \ \mbox{as}\ \, c\to\infty,$$ where the exponential decays (\[eq1.09\]), (\[e:3:3\]) and (\[e:3:3B\]) are used again. Applying (\[eq1.09\]), (\[eq1.09B\]) and (\[F:M\]), it then follows from above that $$\label{eq2.31} {\int_{\mathbb{R}^3}}\nabla w_0\nabla Q_\infty dx =0.$$ We thus conclude from (\[eq2.18\]) and (\[eq2.31\]) that $$0=b_0{\int_{\mathbb{R}^3}}\nabla \big(x\cdot \nabla Q_\infty+2Q_\infty\big)\nabla Q_\infty dx=\frac{3}{2}b_0{\int_{\mathbb{R}^3}}|\nabla Q_\infty|^2 dx,$$ which therefore implies that $b_0=0$ in (\[eq2.18\]) and thus $w_0\equiv 0$ in ${{\mathbb{R}}}^3$. We are now ready to derive a contradiction. In fact, let $y_c\in {{\mathbb{R}}}^3$ be a point satisfying $|w_c(y_c)|=\|w_c \|_{L^\infty({{\mathbb{R}}}^3)}=1$. Since it follows from (\[e:3:3\]) that $w_c$ admits the exponential decay uniformly for all $c>0$, we have $|y_c|\le M$ uniformly in $c$ for some constant $M>0$. Therefore, we obtain that $w_c\to w_0\not\equiv 0$ uniformly on ${{\mathbb{R}}}^3$ as $c\to\infty$, which however contradicts to the fact that $w_0 \equiv 0$ on ${{\mathbb{R}}}^3$. 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--- abstract: 'In dialogues, an utterance is a chain of consecutive sentences produced by one speaker which ranges from a short sentence to a thousand-word post. When studying dialogues at the utterance level, it is not uncommon that an utterance would serve multiple functions. For instance, “Thank you. It works great.” expresses both gratitude and positive feedback in the same utterance. Multiple dialogue acts (DA) for one utterance breeds complex dependencies across dialogue turns. Therefore, DA recognition challenges a model’s predictive power over long utterances and complex DA context. We term this problem Concurrent Dialogue Acts (CDA) recognition. Previous work on DA recognition either assumes one DA per utterance or fails to realize the sequential nature of dialogues. In this paper, we present an adapted Convolutional Recurrent Neural Network (CRNN) which models the interactions between utterances of long-range context. Our model significantly outperforms existing work on CDA recognition on a tech forum dataset.' author: - Yue Yu - Siyao Peng - Grace Hui Yang bibliography: - 'citation.bib' title: | Modeling Long-Range Context for\ Concurrent Dialogue Acts Recognition --- Task Definition =============== The task is defined as a CDA recognition problem where for each utterance $u_t$ (the $t$-th utterance) in a dialogue, we predict a subset of DA labels $y_t$ that describes the functionality of the utterance from a candidate set of DA labels $\mathcal{L} = \{l_1, l_2,...,l_c\}$. For a dialog with $s$ utterances, the inputs to the algorithm is $\mathcal{U} = \{u_1, u_2,...,u_s\}$, and the output is $\mathcal{Y}=\{y_1, y_2,...,y_s\}$, where $y_t$ is the annotated DA label set for $u_t$, in which $y_t = \{y_t^{1}, y_t^{2},...,y_t^{c}\}$. Here, $y_t^{j} = \{1, 0\}$ denotes whether the $t$-th utterance of the dialog is labeled with DA label $l_j$ or not. When $\sum_{j=1}^c y_t^j > 1$, we say CDAs are recognized. Given a dialogue $\mathcal{U}$, the goal is to predict the DA sequence $\mathcal{Y}$ from the text. The Proposed Approach ===================== The challenge of this task lies in the complexity of dialogue structures in human conversations where an utterance can express multiple DAs. In this work, we improve CDA recognition with an adapted CRNN which models the interactions between long-range context. Convolutional Layer ------------------- The base of our architecture is a CNN module similar to @kim2014convolutional. The module works by ‘sliding’ through the embedding matrix of an utterance with various filter sizes to capture semantic features in differently ordered n-grams. A convolution operation is denoted as $$\begin{aligned} k_i &= \tanh(\mathbf{w} \cdot \mathbf{x}_{i:i+d-1}+b_k)\end{aligned}$$ where $k_i$ is the feature generated by the $i$-th filter with weights $\mathbf{w}$ and bias $b_k$. This filter of size $d$ is applied to an embedding matrix, which is the concatenation from the $i$-th to the $(i+d-1)$-th embedding vectors. This operation is applied to every possible window of words in an utterance of length $n$ and generates a feature map $\mathbf{k}$. $$\begin{aligned} \mathbf{k} &= [k_1, k_2,...,k_{n-d+1}]\end{aligned}$$ Dynamic $k$-Max Pooling ----------------------- A max-over-time pooling operation [@kim2014convolutional] is usually applied over the feature map and takes the maximum value as the feature corresponding to this particular filter. The idea is to capture the most important features of an utterance. However, this mechanism could be problematic when the utterance is long. We experimented with Dynamic $k$-Max Pooling [@liu2017extrememultilabel] to pool the most powerful features from $p$ sub-sequences of an utterance with $m$ words. This pooling scheme naturally deals with variable utterance length. $$\begin{aligned} p(\mathbf{k}) &= \left[\max \left\{\mathbf{k}_{1 : \lfloor\frac{m}{p}\rfloor}\right\}, \ldots, \max \left\{\mathbf{k}_{\lfloor m-\frac{m}{p}+1\rfloor : m}\right\}\right]\end{aligned}$$ Recurrent Layer --------------- Based on the local textual features extracted from each utterance, a bidirectional RNN is applied to gather features from a wider context for recognizing the DAs in the target utterance, $u_t$. We experimented with two variations of RNN: LSTM [@zaremba2014learning] and GRU [@cho2014learning], both of which utilize gating information to prevent the vanishing gradient problem. GRU is constructed similarly to LSTM but without using a memory cell. It exposes the full hidden state without any control, which may be computationally more efficient. We experimented with both since it is difficult to predict which one performs better on our task. Highway Connection ------------------ Although LSTM and GRU help capture wider context, the network training becomes more difficult with the additional recurrent layers. Inspired by the highway networks [@sanders2005highway], we propose to add a highway connection between the convolutional layer and the last fully connected layer. With this mechanism, the information about the target utterance, $u_t$, can flow across the recurrent layer without attenuation. The last fully connected layer learns from the outputs of both recurrent and convolutional layers. Dataset {#sec:dataset} ======= We use the MSDialog-Intent dataset [@qu2019user] to conduct experiments. In the dataset, each of the 10,020 utterances is annotated with a subset of 12 DAs. The abundance of information in a single utterance (avg. 72 tokens/utterance) breeds CDA (avg. 1.83 DAs/utterance). We observe a strong correlation between the number of DAs and utterance length, which necessitates a CDA model for forum conversations. The dataset includes plenty of metadata for each utterance, e.g., answer vote and user affiliation. For generalizability, our model only incorporates textual content of the dialogues. Besides, unlike @qu2019user, we keep all the DA annotations in the dataset to preserve the meaningful DA structures within and across utterances.[^1] Acknowledgements ================ This research is supported by U.S. National Science Foundation IIS-145374. Any opinions, findings, conclusions, or recommendations expressed in this paper are of the authors, and do not necessarily reflect those of the sponsor. [^1]: \[note1\]@qu2019user simplified the task by removing DA labels from certain utterances and reducing rare DA combinations to top $32$ frequent ones. We re-ran @qu2019user’s model on the dataset preserving all DA combinations.

ArXiv

--- abstract: | We present a detailed cross–correlation (CCF) and power spectrum re–analysis of the X–ray light curves of the bright Seyfert 1 galaxy NGC5548 obtained with [*EXOSAT*]{}. The 0.05–2 keV and 1–4 keV light curves are cross–correlated with the 1–9 keV and 4–9 keV light curves respectively. We discuss how spurious time lags can be introduced by systematic effects related to detector swapping as well as the switching on and off of the instruments. We also find strong evidence that one of the ME detectors was not working normally during the second part of the March 1986 observation. When these effects are taken into account, the CCF peaks are in all cases consistent with the absence of delays between X–ray variations at different energies. This is unlike the results found by several authors based on the same data. The power spectra of the 1–9 keV light curves are calculated and a detailed search for quasi periodic oscillations (QPOs) carried out on these spectra by using a new technique for the detection of periodic (or quasi–periodic) signals even in the presence of source noise variability. No significant peaks are found above the 95% confidence detection threshold, except during the second part of the March 1986 observation, most probably as a consequence of the ME detector malfunctioning. We discuss and compare our results with those of Papadakis & Lawrence (1993a). author: - '**G. Tagliaferri**' - '**G. Bao , G. L. Israel**' - '**L. Stella**' - '**A. Treves**' --- =6.0in =-0.5in =9.00in Introduction ============ NGC5548 is a bright, close–by (z=0.017) Seyfert 1 galaxy which was extensively studied in different bands of the electromagnetic spectrum. Large variability of both lines (optical–UV) and continuum have been reported making the source an important laboratory for exploring the mechanisms of spectral formation in AGNs. In particular the study of correlations and time lags between the various spectral components represent an important technique for constraining the geometry of the emitting regions (e.g. Mushotzky et al. 1993, and references therein). &gt;From a systematic study of IUE spectra (1200–3000 Å  Clavel et al. 1991) a strong correlation between emission lines and continuum variability was established, with lines responding to the continuum variations with delays of 10–70 days, depending on the degree of ionization of the species (a higher ionization corresponds to a smaller delay). Systematic UV–X–ray observations indicated a strong correlation in the continuum variability in the two bands with un upper limit of $\leq$ 6 days to any delay (Clavel et al. 1992). This, together with the simultaneous optical UV continuum variations, showed that at least a component of the optical–UV continuum should be generated by reprocessing of the X–rays, rather than by the intrinsic disk variability, which should be characterized by longer time–scales (Molendi, Maraschi & Stella 1992). ROSAT observations of a soft X–ray flare with correlated variability in the UV, but without a corresponding change in higher energy X–rays (Done et al. 1995, Walter et al. 1995) made apparent the complexity of the processes occurring in the object. New simultaneous observations at various wavelengths are currently being analysed (e.g. Korista et al. 1995 and references therein). The importance of reprocessing on both cold and warm material is confirmed by the observation of a Fe K fluorescent line at a centroid energy of $\sim$ $6.4$ keV , and of a Fe absorption edge at $\sim$ $8$ keV superimposed to a rather complex continuum (Nandra et al. 1991). This rich observational scenario may be accounted for by models where a hot corona above the accretion disk provides the hard X–ray photons. At the same time these photons are in part reprocessed by an accretion disk which in turn generates the photons for the Compton cooling of the electrons in the hot corona (e.g. Haardt & Maraschi 1993, $\dot{Z}$ycki et al. 1994). Because of its $\sim 4$ day orbital period, allowing long uninterrupted exposures and its wide spectral range (0.05–10 keV), the EXOSAT satellite was particular apt to study variability and reprocessing in the X–ray band on time scales of tens of hours or less. For a systematic study of the variability of Seyferts and QSO contained in the EXOSAT database we refer to Grandi et al. (1992) and Green, Mc Hardy & Lehto (1993). NGC 5548 was observed with EXOSAT 12 times in 1984–86 with a total exposure of $\sim$200 hrs. Flux variations up to a factor of 4 and 3 between the various observations and within each observation, respectively, were clearly seen. A detailed analysis of the EXOSAT light curves was performed by several authors. Variability on timescales of hours was studied by Kaastra & Barr (1989, hereafter KB), who searched for delays between variations in the soft (0.05–2 keV) and medium (2–6 keV) energy X–ray light curves, suggesting that soft X–rays variations lead by $\sim 4600\pm 1200$ [*s*]{}. This result, if confirmed, would have profound theoretical implications, favoring models, such as those mentioned above, where the medium energy X–rays are produced by Compton scattering off the UV/soft X–ray photons by electrons in a hot corona. Walter and Courvoisier (1990) re–examined the same data by using a different analysis technique substantially confirming the results of KB. Papadakis & Lawrence (1993a, hereafter PL) performed a power spectrum analysis the EXOSAT ME light curves and reported the likely detection of quasi–periodic oscillations (QPOs) in 5 out of 8 observations. They suggested that the frequency ($\sim$2 mHz) of the QPOs increases with source intensity, whereas the fractional root mean square amplitude of variability decreases as the source brightens. Since this behaviour is similar to that of e.g. compact galactic X–ray binaries, PL suggest that intensity–correlated QPOs in NGC5548 may also arise from instability or variability in an accretion disk around a massive black hole. Because of the quality of the EXOSAT light curves and the importance of the physical inferences derived from them, we feel justified in presenting a new and independent analysis of relatively old data. This is done in the light of some systematic uncertainties which may arise in the analysis of the EXOSAT light curves from relatively faint sources (such as most AGNs). These uncertainties are discussed in greater detail in a paper on the BL Lac object PKS 2155–304 (Tagliaferri et al. 1991, hereafter Paper I). In Section 2.1 we summarise the characteristics of the EXOSAT light curves of NGC5548. Our CCF analysis for different X–ray energy bands is described in Section 2.2. Details on our search for QPOs are given in section 2.3. The conclusions are in Section 3. Data Analysis ============== Light Curves ------------ The data considered here have been obtained through the EXOSAT database and refer to the low energy imaging telescope (LE) and the medium energy experiment (ME) (White & Peacock 1988). The Argon chambers of the ME experiment consisted of an array of 8 collimated proportional counters mainly sensitive to 1–20 keV X–rays. In order to monitor the background rates and spectrum the ME was generally operated with half of the detector array pointed at the target, and half at a nearby source–free region. The two halves were usually interchanged every 3–4 hours, a procedure indicated as [*array swap*]{} (hereafter [*AS*]{}). The LE telescope was used with a Channel Multiplier Array (CMA) in the focal plane. The CMA was sensitive to the 0.05–2.0 keV energy band and had no intrinsic energy resolution (De Korte et al. 1981); however, a set of filters with different spectral transmission could be interposed in front of the detector. The EXOSAT observations of NGC 5548 considered here are summarised in Table 1. Column 1 gives a letter identifying each observation, column 2 the observing date, column 3 the ME exposure time, column 4 the number of [*AS*]{}, [column 5 the r.m.s. dispersion calculated from the 1-9 keV light curves with a binning time of 1000 s, column 6 the expected r.m.s. dispersion from counting statistics]{} and column 7 the average 1–9 keV ME count rate (per ME array half). Columns 8 through 10 give the exposure times in the LE telescope used in conjunction with the Aluminium/Parylene (Al/P), Boron (Bor) and thin–Lexan (3Lex) filters, respectively. The ME data products (energy spectra and light curves) stored in the EXOSAT database have been given a quality flag ranging from 0 (unusable) to 5 (excellent). Data with quality flag between 3 and 5 are of sufficiently good quality for a detailed analysis (see [*The EXOSAT Database System: available databases*]{} (1991)). In our analysis we have considered all observations with the ME quality factor $\ge 3$; this excludes two observations carried out in March 1984 and January 1985 (not listed in Table 1) with exposure times of $\sim 36000$ and $\sim 22000$ s, respectively. An example of light curves is shown in Fig. 1. Cross Correlation Analysis -------------------------- In order to study the possible delays between the intensity variations in the various bands we calculated the CCF of the LE (0.05–2 keV) and ME light curves (1–9 keV); this is indicated as LE/ME. For the LE we considered only the light curves that were obtained with the 3Lex filter (which provided the highest photon throughput) and were longer than $\sim 15000$ s. This allows to search for delays longer than one hour. Observations I and J are the only suitable for the LE/ME analysis. We have also cross–correlated the 1–4 keV to the 4–9 keV ME light curves (ME/ME): This subdivision of the ME range for NGC 5548 provides comparable count rates in the two bands. The ME/ME CCF analysis was also performed for all the other observations given in Table 1. We used the standard CCF algorithm contained in the timing analysis package [*Xronos*]{} (Stella & Angelini 1992), which is well suited for equispaced and (nearly) continuous data, such as the EXOSAT light curves of NGC 5548. In the analysis of the EXOSAT light curves of PKS 2155–304 (Paper I) we identified a number of systematic effects that may alter the CCF. These are briefly summarized here. A possible problem is related to the [*AS*]{} procedure, which leaves an uncertainty of up to $\pm 0.5 \ cts \ s^{-1}$ in the level of the background subtraction. If the results of the CCF analysis change significantly by adding or subtracting one of the light curves across the [*AS*]{} with a constant value of up to $0.5 \ cts \ s^{-1}$, then these results should be considered with caution.[^1] Moreover, the presence of [*AS*]{} implies that the ME light curves are interrupted by gaps of typical duration of 15 minutes. Although these durations are short compared to the entire light curves, the discontinuity that they introduce in the ME light curves can have strong effects on the CCF. To reduce the effects of the gaps, we fill them with the running average of the light curve calculated over a duration of $\sim 1.5$ hour. With this choice the moving average follows the light curve behaviour on time scales of hours, while the statistical fluctuations are reduced due to the relatively high number of points used in the average. For a given observation, the start and end times of the LE and ME light curves usually differ from a few minutes to tens of minutes. This can alter the shape of the CCF, especially if standard algorithms are used as in our case (see Paper I). To avoid the problem altogether one should therefore make sure that the two light curves are strictly simultaneous, disregarding the data intervals in which only one light curve is available. In our analysis we rebinned the light curves in time bins of 1000 s. We excluded all bins with an exposure time of less than 50% . In some cases we used also intensity windows in order to exclude those bins in the original light curves (resolution of 4–10 s) which were clearly affected by an inadequate background subtraction. The CCF analysis of observation J, the longest and that for which KB report delays between soft and hard variations, was performed in two different ways both for the ME/LE and ME/ME cases. First we considered the entire LE and ME light curves (see Fig. 1), with the gaps bridged with the running mean [and did not exclude the non-simultaneous parts of the light curves.]{} A $\sim 6$ hr interruption is apparent close to the beginning of the observation, due to the switch off of the EXOSAT instruments at the perigee passage (see Fig. 1). If the $\sim 1.5$ hr long light curve interval that preceeds this long gap, is included in the analysis, than the LE/ME CCF shows a marked asymmetric peak centered around a delay of $\sim 7000-8000$ (Fig. 2). [If we exclude the gap or impose the simultaneity of the two light curves, then the delay is much shorter or not present at all. We note that the light curve shown in Fig. 2 of KB paper not only includes the gap, but in the LE it includes another 30 ks of data at the end of the NGC 5548 observation, when this source was seen serendipitously in the field of the BL Lac object 1E1415.6-2557 (Giommi et al. 1987). Of course there are no ME data for this additional interval. It appears that for their timing analysis KB used all the data shown in their Fig. 2, altough this is not explicitely stated. As we have shown the non simultaneity of the two data sets strongly affects the results of CCF analysis (see also paper I). Therefore, we did not consider the extra LE data on NGC 5548 during the EXOSAT observation of 1E1415.6-2557. Moreover, in the rest of all our analysis we considered only strictly simultaneous data.]{} We then [also]{} excluded the first $\sim7.5$ hr from our analysis of the light curves of observation J. The LE/ME and ME/ME CCFs calculated in this way are shown in Fig. 3. A peak is clearly present in the CCFs which is in both cases asymmetric and centered near zero time delay. To derive quantitative information on possible delays between the variations in the soft and hard X–ray light curves, we fitted the central peak of the CCF with a Gaussian function plus a constant. [In the case of the ME/ME CCF, a linear term was added to the fit, to account for the stronger asymmetry of the CCF (see Fig. 3).]{} The results for centroid of the peak are $+400$ s (90% confidence interval: $-800 \ +1500$ s) and $+1500$ s ($+500 \ +2700$ s) respectively. The second procedure consisted in dividing the light curve into three segments, the first two segments, about 6 and 7 hour long, containing only one [*AS*]{}, and the third segment, of about 11 hours, containing two [*AS*]{}. The CCF was calculated for each segment. This treatment reduced the possible effects of the [*AS*]{} discontinuity on the CCF; however it has the disadvantage of decreasing the longest detectable delay time (about 2–3 hours, a value still consistent with the delay time reported by KB). There is essentially no peak in the second intervals both for the LE/ME and ME/ME CCFs, [while in the first interval in both CCFs there is a weak peak centered on zero time delay]{} (Figs 4a–d). In the third segment a clear peak is present in the ME/ME CCF, while a feature with a negative and a positive component can be noted in the LE/ME CCF (Figs 4e,f). This feature is due to the presence of the two [*AS*]{}, indeed if we consider only one of the two [*AS*]{} each time, then the first [*AS*]{} gives rise only the negative peak, whereas the second [*AS*]{} causes only the positive peak. Moreover, by looking at the ME light curves it seems that the central parts (between the two [*AS*]{}, see fig 5) are not properly aligned with the other two. We tested how stable the CCF peaks are to the addition of a constant value to the central parts. For instance in the ME/ME CCF the peak disappears completely by adding 0.1 and 0.4 cts s$^{-1}$ to the central 1–4 and 4–9 keV light curves respectively (Figs 5a,b shows the two ME light curves before and after having added the above constant values to the central part). As a further test we performed an LE/ME and an ME/ME CCF analysis by considering only the first 13 hrs of this observations (i.e. 3 [*AS*]{}). Again no peak is present in either CCF. We can conclude that the segmented analysis does not provide evidence for delays between the LE/ME and ME/ME variations. Another problem emerged through a careful inspection of the ME light curves (Figs 1 and 5). One can see that the light curve intervals between the fourth and fifth [*AS*]{} and after the sixth [*AS*]{} are much noisier than the others (this is seen even more clearly in light curves with a somewhat shorter binning time). This is probably due to one of the three aligned detectors (first half of the ME in this case) not behaving normally. That this behaviour arises from one of the detectors (and not from the source) is confirmed [both by the lac of it in the LE light curve and by the fact that between the fifth and sixth array swap, when the relevant half of the ME array is offset, one of the detectors (detector B as reported in the “ME Obervation Log book" A. Parmar, private comunication) was switched off, due to malfunctioning. After the sixth array swap detector B was switched on again, but it was clearly not functioning properly, yet (see Figs 1 and 5).]{} In this case the malfunctioning detector should be excluded from the analysis, something that was not done in the automatic analysis that generated the ME database products for this observation. We conclude that detector B in the first ME half is most likely responsible for the extra variability in the ME light curves. We intended to repeat the analysis starting from the ME raw data. However we could not obtain the original data from ESA, since the relevant magnetic tape turned out to be unreadable (A. Parmar, private communication). For all other observations in Table 1, because of the shorter exposure times and therefore lower number of [*AS*]{} (see Tab. 1), we considered the cross correlation of the entire light curves with the data gaps bridged by the running mean. For observation I, the second longest, the LE/ME CCF is again flat, while the ME/ME CCF shows a strong peak centered around zero time delay (Fig 6). It can be seen from the figure, however, that the [*half width at half maximum*]{} of this peak is comparable to the duration of the light curve segments between [*AS*]{}; this suggests that the peak might be due to the systematic uncertainties in the ME background subtraction across the [*AS*]{}. To test the reliability of this CCF peak we subtracted 0.2 $cts~s^{-1}$ to the first part of both ME (1–4 and 4–9 keV) light curves (before the first [*AS*]{}) and added 0.2 $cts~s^{-1}$ to the second and third parts of the 4–9 keV light curve, trying to reduce the discontinuity due to [*AS*]{} visible in the light curves. Again this was sufficient to make the CCF peak disappear. Various other tests showed that by adding or subtracting 0.1–0.2 $cts~s^{-1}$ (which are well within the systematic uncertainties of the detector background subtraction) to selected segments of the ME light curves in between [*AS*]{}, the peak can become more pronounced or disappear altogether. [Also the r.m.s. dispersion is clearly affected by the [*AS*]{}; indeed if we add 0.3 and 0.5 $cts~s^{-1}$ to the 1-9 keV light curve before the first and after the last [*AS*]{}, the resulting r.m.s. dispersion is 0.28 to be compared with the value of 0.39 given in Table 1.]{} For all other observations, we calculated only the ME/ME CCF. No peak was detected in the CCFs of observations A, C, D, E, F and H, [consistent with the fact that no significant variability is present in the 1-9 keV light curves (see Table 1).]{} Instead a clear peak was detected in observations B and G (Figs 7,8). In both cases the peak is not centered around zero delay. This would indicate that the variations in the 1–4 keV light curves precede the variations in the 4–9 keV light curve by about 2000–3000 s. However, we believe that also these delays are spurious. In the case of observation G the peak is almost certainly due to background subtraction uncertainties across the [*AS*]{}. Fig. 9a gives the original ME light curves, while Fig. 9b shows the same light curves after the addition of a constant value of 0.2 cts s$^{-1}$ to the segments before the [*AS*]{}. The latter light curves show virtually no discontinuity across the [*AS*]{} and the resulting CCF is flat. [Again by adding 0.5 $cts~s^{-1}$ to the 1-9 keV light curve before the [*AS*]{}, the resulting r.m.s. dispersion is 0.18 to be compared with 0.34 of Table 1. We also used observation G to test whether the abrupt discontinuity introduced by the [*AS*]{} can cause the asymmetry seen in some of our CCFs. For instance, the CCF peak in Fig. 8 is steeper on the right hand side. If we subtract a constant from both ME light curves before the [*AS*]{} (increasing the [*AS*]{} discontinuity, see Fig. 9a) the peak asymmetry becames more prounanced. Instead by subtracting 0.5 $cts~s^{-1}$ from both ME light curves [*after*]{} the [*AS*]{} (changing the discontinuity from a step-up to a step-down), then an asymmetric peak, which is steeper on the left hand side, is obtained. This clearly shows that the discontinuities introduced by the [*AS*]{} procedure can also make the shape of the CCF peak asymmetric.]{} For observation B, that has no [*AS*]{}, the peak is probably due to instability in the background which are then reflected in the background–subtracted source light curves. Indeed if we cross–correlate either one of the source light curves in the two energy bands with the light curve of the background we find [a negative peak centered around zero time delay, which indicates an excess of background subtraction.]{} Search for QPOs --------------- We re–analysed the 1–9 keV ME light curves from the observations in Table 1, in order to carry out a detailed search for the QPOs with frequencies of $\sim 1-2.5 \times 10^{-3}$ Hz reported by PL. The 120 s binned light curve from each observation was divided in M consecutive intervals of $\sim 1-2$ hr duration and the average power spectrum calculated over the power spectra from individual intervals. This allowed to approximately reproduce the frequency range and resolution used by PL in their analysis. Values of M equal to 23, 17, 8, [10 and 17]{} were used for observations J, C, I, A–F–G and B–D, respectively. This method of analysis reduces by about one decade the low frequency end of the power spectra, such that only marginal evidence is found for the increase towards low frequencies, that reflects the [*red noise*]{} variability of the source. In any case, to search for QPOs we adopted a recently developed technique to detect significant power spectrum peaks even in the presence of “coloured" noise components arising from the source variability (Israel & Stella 1995; Stella et al. 1995). The technique relies upon a suitable smoothing algorithm in order to model the continuum power spectrum components underlying any possible peak. By dividing the power spectrum by the smoothed spectrum, a flat (white noise–like) spectrum is produced, the statistical properties of which are worked out as the ratio of two random variables of know distribution, namely the power spectrum and the smoothed spectrum. A search for oscillations is then carried out by looking for peaks in the divided power spectrum which exceed a given detection threshold. Selected average spectra and the corresponding $95\%$ confidence detection thresholds are shown in Fig 10. No significant peaks exceeding the threshold were found in the frequency range $\sim$4$\cdot$10$^{-4}$ –4$\cdot$ 10$^{-3}$ Hz for any of the power spectra from observations C, [B–D, A–F–G]{}, J and I. Observation J was also analysed in different time intervals, in consideration of the possible malfunctioning of one of the ME detectors during the second half of the observation (see Section 2.2). This was done by calculating a power spectrum for the source light curve and a power spectrum from the corresponding background light curve during each of the 4 array swap–free intervals in between the third array swap and the end of the observations. These power spectra and the corresponding $95 \%$ confidence detection thresholds are shown in Fig 11. Significant peaks are clearly detected in the second and fourth power spectra from the source at a frequency of about $1.8$ and $2.6 \times 10^{-3}$ Hz, respectively. It is very likely that these peaks were caused by some kind of quasi–periodic instability in the detector of the first half of the ME array that did not function properly during the second half of observation J. The LE light curves (0.05–2.0 keV), characterized by a poorer signal to noise ratio, were also searched for QPOs; only negative results were found. Our results argue against the detection of QPOs in the X–ray flux of NGC 5548 reported by PL. The power spectrum technique used by PL involves averaging the logarithm of the power spectra from different intervals therefore producing power estimates that approximately follow a Gaussian distribution (Papadakis & Lawrence 1993b). Model fitting can then be performed using standard least square techniques. The continuum power spectrum components are well fitted by a constant (representing the counting statistics noise) plus a power law (describing the source red noise). According to PL the grouped power spectra from the three longest observations (C, I and J) display a $95\%$ significant QPO peak (as estimated through an F–test after the addition of a Guassian to the model function). However, we have shown that the QPO during observation J very likely arise from a detector problem. PL devised also a test to evaluate the significance of power spectrum peaks from individual observations. The best fit model (a power law plus a constant) is used to estimate the continuum components. The power spectrum is then divided by the best fit model in order to produce a white noise power spectrum in which the presence of statistically significant peaks is tested. PL found, in 3 out of 5 cases, a peak in the $1.1-2.4 \times 10^{-3}$ Hz frequency range at a significance level of $>95\%$. However, PL did not take into account the statistical uncertainties introduced in the divided power spectrum by the uncertainties in the best fit model (as evidenced by the lack of any mention of them), therefore overestimating the significance of the peaks. To reassess this significance, we extracted the power spectra from Fig. 1 [of PL]{} fitted them with a constant, after excluding the power estimates corresponding to the peaks and the red noise. These constants together with their $1\sigma$ uncertainties on these averages, were then used to work out the distribution of the divided spectrum in a way that parallels the method of Israel & Stella (1995). Based on this distribution the significance of the peaks in Fig. 2 of PL was evaluated again. The divided power specta of observations A–F–G (G3 in Table 1 of PL) and observation I are characterised by a peak with a significance of $\sim 95 \%$ and $\sim 88\%$, respectively. These values are lower than those worked out by PL ($\sim 98 \%$ and $\sim 96\%$, respectively). The power spectrum of observation J, which formally contains the most significant peak, was disregarded in consideration of the detector problem discussed above. Conclusion ========== Our re–analysis of the CCFs of EXOSAT ME light curves of NGC5548 [does not confirm]{} the claim of KB of a $\sim 5000$ s delay between the medium and soft X–rays variations. This was considered as a strong argument in favour of models where medium energy X–rays are produced by scattering of softer photons. Our results do not exclude this possibility. We note however that the 1990 ROSAT observations (Nandra et al. 1993) detected a variability pattern hardly consistent with very soft X–ray variations (0.1–0.4 keV) preceding the variations of somewhat harder X–rays (1–2.5 keV). This indicates the complexity of physical processes occurring in the source. Our power spectrum analysis does not confirm the detection of QPOs in the mHz range reported by PL. In particular we have shown that the only power spectrum peak with a significance of $>95\%$ most probably results from the malfunctioning of one ME detector. The argument of PL according to which the black hole mass of NGC 5548 has an embarrassingly low value of a few hundred thousand solar masses, loses its validity. While the results of this paper are essentially “negative", we hope that our work contribute illustrating subtle effects which may yield spurious results in the analysis of X–ray light curves from AGNs. [lcccccllll]{}\ Obs & Date & Me exp. &[*rms*]{} disp. &expe. [*rms*]{} &No. of &Counts &Al/P&Bor&3Lex\ & &(s) &$cts~s^{-1}$ & $cts~s^{-1}$& AS & $cts~s^{-1}$&(s)&(s)&(s)\ A & 84/032 &17010 & 0.23 & 0.27 &0 &3.32$\pm$0.18 &1902 &7153 &3202\ B & 84/062 &32630 & 0.26 & 0.20 &0 &4.44$\pm$0.04 &3483 &3679 &2763\ C & 84/193 &59430 & 0.26 & 0.27 &3 &2.88$\pm$0.03 &11872&26488 &11053\ D & 85/062 &26800 & 0.22 & 0.20 &1 &3.65$\pm$0.04 &4434 &11413 &4605\ E & 85/159 &25750 & 0.30 & 0.26 &1 &1.43$\pm$0.05 &6675 &9915 &3824\ F & 85/173 &17020 & 0.30 & 0.20 &2 &3.03$\pm$0.05 &4252 &6672 &3311\ G & 85/186 &23370 & 0.34 & 0.20 &1 &1.82$\pm$0.04 &3613 &10963 &3095\ H & 85/195 &19020 & 0.22 & 0.19 &1 &1.35$\pm$0.05 &4040 & &2622\ I & 86/019 &59860 & 0.39 & 0.22 &3 &4.97$\pm$0.02 & &3469 &39037\ J & 86/062 &83830 & 0.45 & 0.29 &6 &3.82$\pm$0.02 &2392 &4017 &69809\ Clavel, J., et al. 1991, , 366, 64. Clavel, J., et al. 1992, , 393,113. De Korte, P. A. J., et al. 1981, , 30, 495. Done, C., Pounds, K. A., Nandra, K., & Fabian, A. 1995, , in press. Edelson, R. A., & Krolik. J. H. 1988, , 333, 646. Giommi, P., Barr, P., Garilli, B., Gioia, I.M., Maccacaro, T., Maccagni, D., Schild, R.E., 1987, , 322, 662 Grandi, P., Tagliaferri, G., Giommi, P., Barr, P., Palumbo, G.C. 1992, , 82, 93. Green, A. R., Mc Hardy, I. M., & Lehto, H. J. 1993, , 265, 664. Haardt, F., & Maraschi, L. 1993, , 413, 507. Kaastra, J. S., Barr, P. 1989, , 226, 59. Korista K. T., et al. 1995, , 97, 285. Israel, G. L. & Stella, L. 1995, submitted to . Molendi, S., Maraschi, L., Stella, L. 1992, , 255, 27 Mushotzky, R. F., Done, C., & Pounds, K. A. 1993, 31, 717. Nandra, K., Pounds, K. A., Stewart, G. C., George, I. M., Hayashida, K., Makino, F., & Ohashi, T. 1991, , 248, 760. Nandra, K., et al 1993, , 260, 504. Papadakis, I. E., & Lawrence, A. 1993a, Nature, 361, 233. Papadakis, I. E., & Lawrence, A. 1993b, , 261, 612. Parmar, A.N., & Izzo, C., 1986, The EXOSAT Express, no. 16, p. 21. Stella, L., & Angelini L. 1992, in “Data Analysis in Astronomy IV", Eds. V. Di Gesù, L. Scarsi, R. Buccheri, P. Crane, M.C. Maccarone & H.V. Zimmerman, (Plenum Press: New York), p. 59. Stella, L., Arlandi, E., Tagliaferri, G., & Israel, G. L. 1995 in “Time Series Analysis in Meterology and Astronomy", ed. S. Rao, in press. Tagliaferri, G., Stella, L., Maraschi, L, Treves, A., & Celotti, A. 1991, , 380, 78. Walter, R., & Courvoisier, T. 1990, , 233, 40. Walter, R., Courvoisier, T., Done, C., Maraschi, L., Pounds, K., & Urry, M. 1995, preprint. White, N. E., Peacock, A. 1988, in “X-ray Astronomy with EXOSAT", eds. R. Pallavicini & N. E. White, p. 7. $\dot{Z}$ycki, P. T., Krolik, J. H., Zdziarski, A. A., & Kallman, T. R. 1994, , 437, 597. Figure captions {#figure-captions .unnumbered} ================ [**Figure 1:**]{} NGC5548 LE (0.05–2 keV) and ME (1–4 and 4–9 keV) light curves during the longest EXOSAT observation (1986/062, observation J through the paper). The arrows show the array swaps of the ME detector halves, the corrspective data gaps of about 15 minutes are filled with the running mean (see text). Note the big gap at the beginning due to the switch off of the detectors at the satellite perigee passage. [[**Figure 2:**]{} LE/ME light curve cross correlations of observation J. All data shown in Fig. 1 has been used. Note that the peak centered around a delay of $\sim 7-8$ ks is also clearly asymmetric.]{} [**Figure 3:**]{} LE/ME (panel a) and ME/ME (panel b) light curve cross correlations of observation J. A clear peak around zero time lag is clearly present in both cases. [The Gaussian plus constant model fit to the central peak is also shown. The fit was carried out over a range of lags of $\pm 40$ ks and $\pm 20$ ks respectively. In the ME/ME case, a linear term was added to the fit, in order to account for the CCF asymmetry.]{} [**Figure 4:**]{} cross correlations of the light curves of observation J divided in three different segments (see text). A clear peak is present only in the third part (panels e–f). [**Figure 5:**]{} top panel: final part of the ME light curves of observation J, the arrows show the array swaps of the ME detector halves. Note the noisier light curves before and after the first and last array swap. Bottom panel the same light curves after having added 0.1 and 0.4 $cts~s^{-1}$ to the central 1–4 and 4–9 keV light curves respectively; the discontinuity due to the detector array swaps is clearly reduced. [**Figure 6:**]{} cross correlations of the ME light curves of observation I. Again a strong peak centered on zero delay is clearly present. [**Figure 7:**]{} cross correlations of the ME light curves of observation B. The peak is not consistent with zero delay time, and would imply that the variations in the 1–4 keV light curve precede the variations in the 4–9 keV light curve. However this result is probably spurious, see text. [**Figure 8:**]{} cross correlations of the ME light curves of observation G. The peak is not consistent with zero delay time, and would imply that the variations in the 1–4 keV light curve precede the variations in the 4–9 keV light curve. However this result is probably spurious, see text. [**Figure 9:**]{} top panel: ME light curves of observation G, the arrows show the array swap of the ME detector halves. Bottom panel: the same light curves after having added 0.2 $cts~s^{-1}$ to the two light curves before the array swap; the discontinuity due to the detector array swap is clearly reduced. [**Figure 10:**]{} Power spectra from the EXOSAT ME 1–9 keV light curves of observations [A–F–G (84/032, 85/173 and 85/186), B–D (84/062 and 85/062), C (84/193), and I (86/019)]{} (from top to bottom); the solid lines give the corresponding 95% confidence detection thresholds. [**Figure 11:**]{} EXOSAT ME 1–9 keV Power spectra of the ME light curves of NGC 5548 (left) and the background (right) during the second half of observation J. Each panel refers to an array swap free interval, starting from the third array swap (see text for details). The ME array half used in each panel is indicated. The solid lines give the 95% confidence detection thresholds. [^1]: Due to a misprint, paper I reports an uncertainty of up to $\pm 0.05 \ cts \ s^{-1}$ the correct value is the one reported here (Parmar & Izzo 1986; A.N. Parmar, private communication).

ArXiv

--- abstract: 'We show that the twisted traces of CM values of weak Maass forms of weight 0 are Fourier coefficients of vector valued weak Maass forms of weight 3/2. These results generalize work by Zagier on traces of singular moduli. We utilize a twisted version of the theta lift considered by Bruinier and Funke [@BrFu06].' address: 'Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstraße 7, D–64289 Darmstadt, Germany' author: - Claudia Alfes - Stephan Ehlen bibliography: - 'bib.bib' title: Twisted traces of CM values of weak Maass forms --- Introduction ============ The values of the modular invariant $j(z)$ at quadratic irrationalities, classically called “singular moduli”, are known to be algebraic integers. Their properties have been intensively studied since the 19th century. In an influential paper [@Zagier], Zagier showed that the (twisted) traces of singular moduli are Fourier coefficients of weakly holomorphic modular forms of weight $3/2$. Recall that these are meromorphic modular forms which are holomorphic on the complex upper half plane ${\mathbb{H}}= \{ z \in {\mathbb{C}};\ \Im(z) > 0\}$ with possible poles at the cusps. Throughout this paper, we let $N$ be a positive integer and we denote by ${\Gamma}$ the congruence subgroup ${\Gamma}=\Gamma_0(N)=\left\lbrace{\left(\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}\right)}\in {{\text {\rm SL}}}_2({\mathbb{Z}}); c \equiv 0 \bmod N \right\rbrace$. For a negative integer $D$ congruent to a square modulo $4N$, we consider the set $\mathcal{Q}_{D,N}$ of *positive definite* integral binary quadratic forms $\left[a,b,c\right]=ax^2+bxy+cy^2$ of discriminant $D=b^2-4ac$ such that $c$ is congruent to $0$ modulo $N$. If $N=1$, we simply write $\mathcal{Q}_{D}$. For each form $Q = \left[a,b,c\right] \in \mathcal{Q}_{D,N}$ there is an associated CM point $\alpha_Q=\frac{-b+i\sqrt{D}}{2a}$ in ${\mathbb{H}}$. The group ${\Gamma}$ acts on $\mathcal{Q}_{D,N}$ with finitely many orbits. Let $\Delta \in {\mathbb{Z}}$ be a fundamental discriminant (possibly 1) and $d$ a positive integer such that $-{\operatorname{sgn}}(\Delta)d$ and $\Delta$ are squares modulo $4N$. For a weakly holomorphic modular form $f$ of weight 0 for ${\Gamma}$, we consider the modular trace function $${\mathbf{t}}_\Delta(f;d)=\frac{1}{\sqrt{\Delta}}\sum\limits_{Q\in{\Gamma}\backslash\mathcal{Q}_{-d{\left\vert\Delta\right\vert},N}}\frac{\chi_{\Delta}(Q)}{{\left\vert\overline{\Gamma}_Q\right\vert}}f(\alpha_Q).$$ Here $\overline{{\Gamma}}_Q$ denotes the stabilizer of $Q$ in $\overline{{\Gamma}}$, the image of ${\Gamma}$ in ${{\text {\rm PSL}}}_2({\mathbb{Z}})$. The function $\chi_\Delta$ is a genus character, defined for $Q=[a,b,c] \in \mathcal{Q}_{-d{\left\vert\Delta\right\vert},N}$ by $$\label{intro:chi} \chi_\Delta(Q)= \begin{cases} {\left(\frac{\Delta}{n}\right)}, &\text{ if } (a,b,c,\Delta)=1 \text{ and } Q \text{ represents } n \text{ with } (n,\Delta)=1,\\ 0, &\text{otherwise}. \end{cases}$$ It is known that $\chi_\Delta(Q)$ is ${\Gamma}$-invariant [@GKZ]. Note that for $\Delta=1$ we have $\chi_\Delta(Q)=1$ for all $Q \in \mathcal{Q}_{-d,N}$. Let $J(z)=j(z)-744=q^{-1}+196884q+21493760q^2+\cdots$, $q:=e^{2\pi i z}$, be the normalized Hauptmodul for the group $\mathrm{PSL}_2({\mathbb{Z}})$. By the theory of complex multiplication it is known that ${\mathbf{t}}_\Delta(J;d)$ is a rational integer [@ShimAuto Section 5.4]. Zagier [@Zagier Theorem 6] proved that for $N=1$ and $\Delta > 0$ the “generating series” of these traces, $$\label{eq:intro1} g_\Delta(\tau) = q^{-\Delta} - \sum\limits_{d\geq 0} {\mathbf{t}}_\Delta(J;d) q^d,$$ is a weakly holomorphic modular form of weight $3/2$ for ${\Gamma}_0(4)$. Here, we set ${\mathbf{t}}_\Delta(J;0)=2$, if $\Delta=1$ and ${\mathbf{t}}_\Delta(J;0)=0$, otherwise. Using Hecke operators, it is also possible to obtain from this a formula for the traces of $J_m$, the unique weakly holomorphic modular function for ${{\text {\rm SL}}}_2({\mathbb{Z}})$ with principal part equal to $q^{-m}$, where $m$ is a positive integer. Namely, we have $$\label{eq:intro2} {\mathbf{t}}_\Delta(J_m;d) = \sum_{n \mid m} {\left(\frac{\Delta}{m/n}\right)} n\ {\mathbf{t}}_\Delta(J;d n^2).$$ Many authors [@BringmannOno; @DukeJenkins; @KimTwisted; @MillerPixton] worked on generalizations of these results, mostly also for modular curves of genus 0. Inspired by previous work [@Funke], Bruinier and Funke [@BrFu06] showed that the function $g_1(\tau)$ can be interpreted as a special case of a theta lift using a kernel function constructed by Kudla and Millson [@KM86]. They generalized Zagier’s result for $\Delta=1$ to traces of harmonic weak Maass forms of weight $0$ on modular curves of *arbitrary* genus [@BrFu06 Theorem 7.8]. The purpose of the present paper is to extend these results to the case $\Delta \neq 1$. We develop a systematic approach to twist vector valued modular forms transforming with a certain Weil representation of ${\text {\rm Mp}}_2({\mathbb{Z}})$. From a representation–theoretic point of view, this can be interpreted as an intertwining operator between two related representations. Then we define a twisted version of the theta lift studied by Bruinier and Funke and apply their results to obtain its Fourier expansion. To illustrate our main result, Theorem \[thm:main\], we consider the following special case. \[thm:intro\] Let $p$ be a prime or $p=1$ and let $f$ be a weakly holomorphic modular form of weight $0$ for ${\Gamma}_0(p)$. Assume that $f$ is invariant under the Fricke involution $z \mapsto -\frac{1}{pz}$ and write $f(z)=\sum_{n\gg-\infty}a(n)q^n$ for its Fourier expansion. Let $\Delta > 1$ and, if $p \neq 1$, assume that $(\Delta,2p)=1$. Then the function $$\sum\limits_{m>0}m\sum\limits_{n>0}\left(\frac{\Delta}{n}\right) a(-mn)q^{-{\left\vert\Delta\right\vert}m^2} - \sum\limits_{d>0}{\mathbf{t}}_\Delta^*(f;d)q^d$$ is a weakly holomorphic modular form of weight $3/2$ for ${\Gamma}_0(4p)$ contained in the Kohnen plus-space. Here, ${\mathbf{t}}_\Delta^*(f;d)={\mathbf{t}}_\Delta(f;d)/2$, for $p \neq 1$, and ${\mathbf{t}}_\Delta^*(f;d)={\mathbf{t}}_\Delta(f;d)$, for $p=1$. Note that in contrast to the case $\Delta=1$ [@BrFu06 Theorem 1.1] we do not get a constant term here and the lift is always weakly holomorphic. Setting $f=J_m$ and $p=1$, we also obtain the forms $g_\Delta(\tau)$ and the relation from Theorem \[thm:intro\]. Furthermore, note that our main theorem is also valid for $\Delta<0$. The paper is organized as follows. In Section 2 we review necessary background material on quadratic spaces and vector valued automorphic forms. In Section 3 we show that twisting can be regarded as an intertwining operation. Then we define a twisted Kudla-Millson theta kernel and in Section 5 we prove our main theorem. In Section 6 we compute the twisted theta lift for other types of automorphic forms following the examples studied by Bruinier and Funke. In particular, we show that a twisted intersection pairing à la Kudla, Rapoport, and Yang [@KRY] is given in terms of a weight $3/2$ Eisenstein series. We also explain how to deduce Theorem \[thm:intro\], and present a few computational examples. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Jan Bruinier for suggesting this project and for many valuable discussions. We thank Jens Funke for substantially improving the exposition of the proof of our main theorem. Moreover, we would like to thank Martin Hövel and Fredrik Strömberg for helpful comments on earlier versions of this paper. Preliminaries ============= For a positive integer $N$ we consider the rational quadratic space of signature $(1,2)$ given by $$V:=\left\{\lambda=\begin{pmatrix} \lambda_1 &\lambda_2\\\lambda_3& -\lambda_1\end{pmatrix}; \lambda_1,\lambda_2,\lambda_3 \in {\mathbb{Q}}\right\}$$ and the quadratic form $Q(\lambda):=N\text{det}(\lambda)$. The corresponding bilinear form is $(\lambda,\mu)=-N\text{tr}(\lambda \mu)$ for $\lambda, \mu \in V$. Let $G=\mathrm{Spin}(V) \simeq {{\text {\rm SL}}}_2$, viewed as an algebraic group over ${\mathbb{Q}}$ and write $\overline{\Gamma}$ for its image in $\mathrm{SO}(V)\simeq\mathrm{PSL}_2$. Let $D$ be the associated symmetric space realized as the Grassmannian of lines in $V({\mathbb{R}})$ on which the quadratic form $Q$ is positive definite, $$D \simeq \left\{z\subset V({\mathbb{R}});\ \text{dim}z=1 \text{ and } Q\vert_{z} >0 \right\}.$$ In this setting the group ${{\text {\rm SL}}}_2({\mathbb{Q}})$ acts on $V$ by conjugation $$g.\lambda :=g \lambda g^{-1},$$ for $\lambda \in V$ and $g\in{{\text {\rm SL}}}_2({\mathbb{Q}})$. In particular, $G({\mathbb{Q}})\simeq{{\text {\rm SL}}}_2({\mathbb{Q}})$. If we identify the symmetric space $D$ with the complex upper half plane ${\mathbb{H}}$ in the usual way, we obtain an isomorphism between ${\mathbb{H}}$ and $D$ by $$z \mapsto {\mathbb{R}}\lambda(z),$$ where, for $z=x+iy$, we pick as a generator for the associated positive line $$\lambda(z):=\frac{1}{\sqrt{N}y} \begin{pmatrix} -(z+\bar{z})/2 &z\bar{z} \\ -1 & (z+\bar{z})/2 \end{pmatrix}.$$ The group $G$ acts on ${\mathbb{H}}$ by linear fractional transformations and the isomorphism above is $G$-equivariant. In particular, $Q\left(\lambda(z)\right)=1$ and $g.\lambda(z)=\lambda(gz)$ for $g\in G({\mathbb{R}})$. Let $(\lambda,\lambda)_z=(\lambda,\lambda(z))^2-(\lambda,\lambda)$. This is the minimal majorant of $(\cdot,\cdot)$ associated with $z\in D$. We can view ${\Gamma}={\Gamma}_0(N)$ as a discrete subgroup of $\mathrm{Spin}(V)$. Write $M={\Gamma}\setminus D$ for the attached locally symmetric space. The set of isotropic lines $\mathrm{Iso}(V)$ in $V({\mathbb{Q}})$ can be identified with $P^1({\mathbb{Q}})={\mathbb{Q}}\cup \left\{ \infty\right\}$ via $$\psi: P^1({\mathbb{Q}}) \rightarrow \mathrm{Iso}(V), \quad \psi((\alpha:\beta)) = \mathrm{span}\left(\begin{pmatrix} \alpha\beta &\alpha^2 \\ -\beta^2 & -\alpha\beta \end{pmatrix}\right).$$ The map $\psi$ is a bijection and $\psi(g(\alpha:\beta))=g.\psi((\alpha:\beta))$. So the cusps of $M$ (i.e. the ${\Gamma}$-classes of $P^1({\mathbb{Q}})$) can be identified with the ${\Gamma}$-classes of $\mathrm{Iso}(V)$. If we set $\ell_\infty := \psi(\infty)$, then $\ell_\infty$ is spanned by $\lambda_\infty=\left(\begin{smallmatrix}0 & 1 \\ 0 & 0\end{smallmatrix}\right)$. For $\ell \in \mathrm{Iso}(V)$ we pick $\sigma_{\ell} \in{{\text {\rm SL}}}_2({\mathbb{Z}})$ such that $\sigma_{\ell}.\ell_\infty=\ell$. Furthermore, we orient all lines $\ell$ by requiring that $\lambda_\ell:=\sigma_{\ell}\lambda_\infty$ is a positively oriented basis vector of $\ell$. Let $\Gamma_{\ell}$ be the stabilizer of the line $\ell$. Then (if $-I\in \Gamma$) $$\sigma_{\ell} ^{-1}\Gamma_{\ell} \sigma_{\ell} = \left\{\pm \begin{pmatrix} 1 &k\alpha_{\ell} \\ & 1 \end{pmatrix}; k\in{\mathbb{Z}}\right\},$$ where $\alpha_{\ell} \in {\mathbb{Q}}_{>0}$ is the width of the cusp $\ell$ [@Funke]. In our case it does not depend on the choice of $\sigma_{\ell}$. For each $\ell$ there is a $\beta_{\ell} \in {\mathbb{Q}}_{>0}$ such that $\left(\begin{smallmatrix}0 & \beta_{\ell} \\ 0 & 0\end{smallmatrix}\right)$ is a primitive element of $\ell_\infty \cap\sigma_{\ell}L$. We write $\epsilon_{\ell} = \alpha_{\ell} /\beta_{\ell}$. Now Heegner points are given as follows. For $\lambda\in V({\mathbb{Q}})$ with $Q(\lambda)>0$ let $$D_{\lambda}= \mathrm{span}(\lambda) \in D.$$ For $Q(\lambda) \leq 0$ we set $D_{\lambda}=\emptyset$. We denote be the image of $D_{\lambda}$ in $M$ by $Z(\lambda)$. If $Q(\lambda)<0$, we obtain a geodesic $c_{\lambda}$ in $D$ via $$c_{\lambda}=\left\{z \in D; z \perp \lambda \right\}.$$ We denote $\Gamma_{\lambda} \backslash c_{\lambda}$ in $M$ by $c(\lambda)$. The stabilizer $\overline{\Gamma}_\lambda$ is either trivial or infinite cyclic. The geodesic $c(\lambda)$ is infinite if and only if the following equivalent conditions hold [@Funke]. 1. We have $Q(\lambda)\ \in\ -N({\mathbb{Q}}^{\times})^2$. 2. The stabilizer $\overline{\Gamma}_\lambda$ is trivial. 3. The orthogonal complement $\lambda^{\perp}$ is split over ${\mathbb{Q}}$. Thus if $c(\lambda)$ is an infinite geodesic, $\lambda$ is orthogonal to two isotropic lines $\ell_\lambda=\text{span}(\mu)$ and $\tilde{\ell}_\lambda=\text{span}(\tilde{\mu})$, with $\mu$ and $\tilde{\mu}$ positively oriented. We fix an orientation of $V$ and we say that $\ell_\lambda$ is the line associated with $\lambda$ if the triple $(\lambda,\mu,\tilde{\mu})$ is a positively oriented basis for $V$. In this case, we write $\lambda \sim \ell_\lambda$. A lattice related to ${\Gamma}_0(N)$ ------------------------------------ Following Bruinier and Ono [@BrOno], we consider the lattice $$L:=\left\{ \begin{pmatrix} b& -a/N \\ c&-b \end{pmatrix}; \quad a,b,c\in{\mathbb{Z}}\right\}.$$ The dual lattice corresponding to the bilinear form $(\cdot,\cdot)$ is given by $$L':=\left\{ \begin{pmatrix} b/2N& -a/N \\ c&-b/2N \end{pmatrix}; \quad a,b,c\in{\mathbb{Z}}\right\}.$$ We identify the discriminant group $L'/L=:{\mathcal{D}}$ with ${\mathbb{Z}}/2N{\mathbb{Z}}$, together with the ${\mathbb{Q}}/{\mathbb{Z}}$ valued quadratic form ${x \mapsto -x^2/4N}$. The level of $L$ is $4N$. We note that Bruinier and Ono [@BrOno] consider the same lattice together with the quadratic form $-Q$. For a fundamental discriminant $\Delta\in{\mathbb{Z}}$ we also consider the rescaled lattice $\Delta L$ together with the quadratic form $Q_\Delta(\lambda):=\frac{Q(\lambda)}{{\left\vert\Delta\right\vert}}$. The corresponding bilinear form is given by $(\cdot,\cdot)_\Delta = \frac{1}{{\left\vert\Delta\right\vert}} (\cdot,\cdot)$. The dual lattice of $\Delta L$ corresponding to $(\cdot,\cdot)_\Delta$ is equal to $L'$ as above, independent of $\Delta$. We denote the discriminant group $L'/\Delta L$ by ${{\mathcal{D}(\Delta)}}$. Note that ${\mathcal{D}}(1)={\mathcal{D}}$ and ${\left\vert{{\mathcal{D}(\Delta)}}\right\vert}={\left\vert\Delta\right\vert}^3 {\left\vert{\mathcal{D}}\right\vert} = 2N\, {\left\vert\Delta\right\vert}^3$. Note that ${\Gamma}_0(N) \subset \mathrm{Spin}(L)$ is a congruence subgroup of $\mathrm{Spin}(L)$ which takes $L$ to itself and acts trivially on the discriminant group ${\mathcal{D}}$. However, in general it does not act trivially on ${{\mathcal{D}(\Delta)}}$. For $m \in {\mathbb{Q}}$ and $h \in {\mathcal{D}}$, we will also consider the set $$L_{h,m} = \left\{ \lambda \in L+h; Q(\lambda)=m \right\}.$$ By reduction theory, if $m \neq 0$ the group $\Gamma$ acts on $ L_{h,m}$ with finitely many orbits. The Weil representation and vector valued automorphic forms ----------------------------------------------------------- We denote by ${\text {\rm Mp}}_2({\mathbb{Z}})$ the integral metaplectic group, which consists of pairs $(\gamma, \phi)$, where $\gamma = {{\left(\begin{smallmatrix}a & b \\ c & d\end{smallmatrix}\right)}\in {{\text {\rm SL}}}_2({\mathbb{Z}})}$ and $\phi:{\mathbb{H}}\rightarrow {\mathbb{C}}$ is a holomorphic function with $\phi^2(\tau)=c\tau+d$. The group ${\text {\rm Mp}}_2({\mathbb{Z}})$ is generated by $S=({\left(\begin{smallmatrix}0 & -1 \\ 1 & 0\end{smallmatrix}\right)},\sqrt{\tau})$ and $T=({\left(\begin{smallmatrix}1 & 1 \\ 0 & 1\end{smallmatrix}\right)}, 1)$. We consider the Weil representation $\rho_\Delta$ of ${\text {\rm Mp}}_2({\mathbb{Z}})$ corresponding to the discriminant group ${{\mathcal{D}(\Delta)}}$ on the group ring ${\mathbb{C}}[{{\mathcal{D}(\Delta)}}]$, equipped with the standard scalar product $\langle \cdot , \cdot \rangle$, conjugate-linear in the second variable. We simply write $\rho$ for $\rho_1$. For $\delta \in {{\mathcal{D}(\Delta)}}$, we write ${\mathfrak{e}}_\delta$ for the corresponding standard basis element of ${\mathbb{C}}[{{\mathcal{D}(\Delta)}}]$. The action of $\rho_\Delta$ on basis vectors of ${\mathbb{C}}[{{\mathcal{D}(\Delta)}}]$ can be described in terms of the following formulas [@BrHabil] for the generators $S$ and $T$ of ${\text {\rm Mp}}_2({\mathbb{Z}})$. In our special case we have $$\rho_\Delta(T) {\mathfrak{e}}_\delta = e(Q_\Delta(\delta)) {\mathfrak{e}}_\delta,$$ and $$\rho_\Delta(S) {\mathfrak{e}}_\delta = \frac{\sqrt{i}}{\sqrt{{\left\vert{{\mathcal{D}(\Delta)}}\right\vert}}} \sum_{\delta' \in {{\mathcal{D}(\Delta)}}} e(-(\delta',\delta)_\Delta) {\mathfrak{e}}_{\delta'}.$$ Let $k \in \frac{1}{2}{\mathbb{Z}}$, and let $A_{k,\rho_\Delta}$ be the vector space of functions $f: {\mathbb{H}}\rightarrow {\mathbb{C}}[{{\mathcal{D}(\Delta)}}]$, such that for $(\gamma,\phi) \in {\text {\rm Mp}}_2({\mathbb{Z}})$ we have $$f(\gamma \tau) = \phi(\tau)^{2k} \rho_\Delta(\gamma, \phi) f(\tau).$$ A twice continuously differentiable function $f\in A_{k,\rho_\Delta}$ is called a *(harmonic) weak Maass form of weight $k$ with respect to the representation $\rho_\Delta$* if it satisfies in addition: (i) $\Delta_k f=0$, (ii) there is a $C>0$ such that $f(\tau)=O(e^{Cv}) $ as $v \rightarrow \infty$. Here and throughout, we write $\tau=u+iv$ with $u,v \in {\mathbb{R}}$ and $\Delta_k=-v^2\left(\frac{\partial^2}{\partial u^2}+\frac{\partial^2}{\partial v^2}\right) +ikv\left(\frac{\partial}{\partial u}+i\frac{\partial}{\partial v}\right)$ is the weight $k$ Laplace operator. We denote the space of such functions by $H_{k,\rho_\Delta}$. Moreover, we let $H^{+}_{k,\rho_\Delta}$ be the subspace of functions in $H_{k,\rho_\Delta}$ whose singularity at $\infty$ is locally given by the pole of a meromorphic function. By $M^{\text{!}}_{k,\rho_\Delta} \subset H^{+}_{k,\rho_\Delta}$ we denote the subspace of weakly holomorphic modular forms. Similarly, we can define scalar valued analogues of these spaces of automorphic forms. In those cases, we require analogous conditions at all cusps of ${\Gamma}$ in $(ii)$. We denote the corresponding spaces by $H^{+}_{k}({\Gamma})$ and $M_k^{\text{!}}({\Gamma})$. Note that the Fourier expansion of any harmonic weak Maass form uniquely decomposes into a holomorphic and a non-holomorphic part [@BrFu04 Section 3]. For example, for $f\in H^{+}_0({\Gamma})$ we have $$\label{eq:fourierhmf0} f(\sigma_\ell \tau) = \sum_{\substack{n\in \frac{1}{\alpha_{\ell}}{\mathbb{Z}}\\n\gg -\infty}} a^{+}_{\ell}(n)e(n \tau) + \sum_{\substack{n\in \frac{1}{\alpha_{\ell}}{\mathbb{Z}}\\n<0}} a^{-}_{\ell}(n)e(n\bar{\tau}),$$ where $\alpha_{\ell}$ denotes the width of the cusp $\ell$ and the first summand is called the holomorphic part of $f$, the second one the non-holomorphic part. Twisting vector valued modular forms {#sec:twisting} ==================================== We now define a generalized genus character for $\delta = \left(\begin{smallmatrix} b/2N& -a/N \\ c&-b/2N \end{smallmatrix}\right) \in L'$. Let $\Delta\in{\mathbb{Z}}$ be a fundamental discriminant and $r\in{\mathbb{Z}}$ such that $\Delta \equiv r^2 \ (\text{mod } 4N)$. We let $$\label{def:chidelta} \chi_{\Delta}(\delta)=\chi_{\Delta}(\left[a,b,Nc\right]):= \begin{cases} {\left(\frac{\Delta}{n}\right)}, & \text{if } \Delta | b^2-4Nac \text{ and } (b^2-4Nac)/\Delta \text{ is a} \\ & \text{square modulo } 4N \text{ and } \gcd(a,b,c,\Delta)=1, \\ 0, &\text{otherwise}. \end{cases}$$ Here, $\left[a,b,Nc\right]$ is the integral binary quadratic form corresponding to $\delta$, and $n$ is any integer prime to $\Delta$ represented by $\left[a,b,Nc\right]$. The function $\chi_{\Delta}$ is invariant under the action of ${\Gamma}_0(N)$ and under the action of all Atkin-Lehner involutions. It can be computed by the following formula [@GKZ Section I.2, Proposition 1]: If $\Delta=\Delta_1\Delta_2$ is a factorization of $\Delta$ into discriminants and $N=N_1N_2$ is a factorization of $N$ into positive factors such that $(\Delta_1,N_1a)=(\Delta_2,N_2c)=1$, then $$\label{def:chi_Delta} \chi_{\Delta}(\left[a,b,Nc\right])=\left(\frac{\Delta_1}{N_1a}\right)\left(\frac{ \Delta_2}{N_2c}\right).$$ If no such factorizations of $\Delta$ and $N$ exist, we have $\chi_{\Delta}(\left[a,b,Nc\right])=0$. We note that $\chi_{\Delta}(\delta)$ depends only on $\delta \in L'$ modulo $\Delta L$. Therefore, we can view it as a function on the discriminant group ${{\mathcal{D}(\Delta)}}$. Using the function $\chi_\Delta$ we obtain an intertwiner of the Weil representations corresponding to ${\mathcal{D}}$ and ${{\mathcal{D}(\Delta)}}$ as follows. Denote by $\pi: {{\mathcal{D}(\Delta)}}\rightarrow {\mathcal{D}}$ the canonical projection. For $h \in {\mathcal{D}}$, we define $$\psi_{\Delta,r}({\mathfrak{e}}_h) := \sum_{\substack{\delta \in {{\mathcal{D}(\Delta)}}\\ \pi(\delta)=rh \\ Q_\Delta(\delta) \equiv {\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}} \chi_\Delta(\delta) {\mathfrak{e}}_\delta.$$ \[prop:intertwiner\] Let $\Delta \in {\mathbb{Z}}$ be a discriminant and $r \in {\mathbb{Z}}$ such that $\Delta \equiv r^2\ (4N)$. Then the map $\psi_{\Delta,r}: {\mathcal{D}}\rightarrow {{\mathcal{D}(\Delta)}}$ defines an intertwining linear map between the representations $\widetilde{\rho}$ and $\rho_\Delta$, where $\widetilde{\rho} = \rho$ if $\Delta>0$ and $\widetilde{\rho}=\bar\rho$ if $\Delta<0$. For $T \in {\text {\rm Mp}}_2({\mathbb{Z}})$ this is trivial. For the generator $S$, this follows from Proposition 4.2 in [@BrOno]; namely, we have the following identity of exponential sums $$\sum_{\substack{\delta\in {{\mathcal{D}(\Delta)}}\\ \pi(\delta) = rh \\Q_\Delta(\delta)\equiv{\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}} \hspace{-9mm}\chi_{\Delta}(\delta)\ e\left(\frac{(\delta,\delta')}{{\left\vert\Delta\right\vert}}\right) = \epsilon \left| \Delta \right|^{3/2} \chi_{\Delta} (\delta')\ \hspace{-9mm}\sum\limits_{\substack{h'\in {\mathcal{D}}\\ \pi(\delta')= rh' \\Q_\Delta(\delta')\equiv{\operatorname{sgn}}(\Delta) Q(h')\, ({\mathbb{Z}})}}\hspace{-9mm} e\left({\operatorname{sgn}}(\Delta)(h,h')\right),$$ where $\epsilon = 1$ if $\Delta>0$ and $\epsilon=i$ if $\Delta<0$. This, together with the unitarity of the Weil representation, directly implies the following \[cor:twisting\_mf\] Let $f \in A_{k,\rho_\Delta}$. Then the function $g: {\mathbb{H}}\rightarrow {\mathbb{C}}[{\mathcal{D}}]$, ${g=\sum_{h \in {\mathcal{D}}} g_h {\mathfrak{e}}_h}$ with $ {g_h := \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), f \right\rangle}$, is contained in $A_{k,\widetilde{\rho}}$. The twisted Kudla-Millson theta function {#theta} ======================================== We let $\delta \in {{\mathcal{D}(\Delta)}}$ and define the same theta function $\Theta_{\delta}(\tau,z)$ for $\tau, z \in {\mathbb{H}}$ as Bruinier and Funke [@BrFu06], here for the lattice $\Delta L$ with the quadratic form $Q_\Delta$. It is constructed using the Schwartz function $$\varphi^0_{\Delta}(\lambda,z) =\left(\frac{1}{{\left\vert\Delta\right\vert}}(\lambda,\lambda(z))^2-\frac{1}{2\pi} \right) e^{-2\pi R(\lambda,z)/|\Delta|}\omega,$$ where $R(\lambda,z):=\frac{1}{2}(\lambda,\lambda(z))^2-(\lambda,\lambda)$ and $\omega = \frac{i}{2}\frac{dz \wedge d\bar{z}}{y^2}$. Then let $\varphi(\lambda,\tau,z)=e^{2\pi i Q_\Delta(\lambda)\tau} \varphi^0_{\Delta}(\sqrt{v}\lambda,z)$ and define $$\label{ThetaDeltaOp} \Theta_{\delta}(\tau,z,\varphi)=\sum\limits_{\lambda\in \Delta L+\delta} \varphi (\lambda,\tau, z).$$ Since we are only considering the fixed Schwartz function $\varphi$ above, we will frequently drop the argument $\varphi$ and simply write $\Theta_{\delta}(\tau,z)$. The Schwartz function $\varphi$ has been constructed by Kudla and Millson [@KM86]. It has the crucial property that for $Q(\lambda)>0$ it is a Poincaré dual form for the Heegner point $D_\lambda$, while it is exact for $Q(\lambda)<0$. The vector valued theta series $$\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)=\sum\limits_{\delta\in {{\mathcal{D}(\Delta)}}} \Theta_{\delta}(\tau,z) \mathfrak{e}_{\delta}.$$ is a $C^{\infty}$-automorphic form of weight $3/2$ which transforms with respect to the representation $\rho_\Delta$ [@BrFu06]. Following Bruinier and Ono [@BrOno] we also define a twisted theta function. For $h \in {\mathcal{D}}$ the corresponding component is defined as $$\label{eq:twtheta} \Theta_{\Delta,r,h}(\tau,z) = \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), \overline{\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)} \right\rangle = \sum\limits_{\substack{\delta\in {{\mathcal{D}(\Delta)}}\\ \pi(\delta) = rh \\Q_\Delta(\delta)\equiv{\operatorname{sgn}}(\Delta)Q(h)\, ({\mathbb{Z}})}} \chi_{\Delta}(\delta)\Theta_{\delta}(\tau,z).$$ Using this, we obtain a ${\mathbb{C}}[{\mathcal{D}}]$-valued theta function by setting $$\Theta_{\Delta,r}(\tau,z) := \sum_{h \in {\mathcal{D}}} \Theta_{\Delta,r,h}(\tau,z) {\mathfrak{e}}_h.$$ Note that Bruinier and Ono actually introduce their twisted Siegel theta function in a more direct way. However, our interpretation makes it possible to apply this “method of twisting” directly to other modular forms and theta kernels. By Proposition \[cor:twisting\_mf\] we obtain the following transformation formula for $\Theta_{\Delta,r}(\tau,z)$. \[prop:twthetatrans\] The theta function $\Theta_{\Delta,r}(\tau,z)$ is a non-holomorphic ${\mathbb{C}}[{\mathcal{D}}]$-valued modular form of weight $3/2$ for the representation $\widetilde{\rho}$. Furthermore, it is a non-holomorphic automorphic form of weight 0 for ${\Gamma}_0(N)$ in the variable $z \in D$. In general $\Gamma_0(N)$ does not act trivially on ${{\mathcal{D}(\Delta)}}$. However, the ${\Gamma}_0(N)$ invariance of $\chi_\Delta$ implies $$\Theta_{\Delta,r}(\tau,\gamma z) = \sum_{h \in {\mathcal{D}}} \left\langle \sum_{\substack{\delta \in {{\mathcal{D}(\Delta)}}\\ \pi(\delta)=rh \\ Q_\Delta(\delta) \equiv {\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}}\!\!\!\!\!\!\! \chi_\Delta(\gamma^{-1}\delta) {\mathfrak{e}}_\delta,\ \overline{\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)} \right\rangle = \Theta_{\Delta,r}(\tau, z).$$ The twisted theta lift {#sec:mainresult} ====================== The twisted modular trace function ---------------------------------- Before we define the theta lift, we introduce a generalization of the twisted modular trace function given in the Introduction. The twisted Heegner divisor on $M$ is defined by $$Z_{\Delta,r}(h,m)= \sum\limits_{\lambda \in \Gamma \backslash L_{rh,m{\left\vert\Delta\right\vert}}}\frac{\chi_{\Delta}(\lambda)}{\left|\overline{\Gamma}_{\lambda}\right|} Z(\lambda)\in \mathrm{Div}(M)_{\mathbb{Q}}.$$ Note that for $\Delta=1$, we obtain the usual Heegner divisors [@BrFu06]. Let $f$ be a harmonic weak Maass form of weight $0$ in $H^{+}_0({\Gamma})$. If $m\in{\mathbb{Q}}_{>0}$ with $m \equiv {\operatorname{sgn}}(\Delta)Q(h)\ ({\mathbb{Z}})$ and $h\in {\mathcal{D}}$ we put $$\label{def:trace1} {\mathbf{t}}_{\Delta,r}(f;h,m) = \sum\limits_{z\in Z_{\Delta,r}(h,m)}f(z)=\sum\limits_{\lambda\in {\Gamma}\setminus L_{rh,{\left\vert\Delta\right\vert}m}} \frac{\chi_{\Delta}(\lambda)}{\left|\overline{\Gamma}_{\lambda}\right|} f(D_{\lambda}).$$ If $m=0$ or $m\in{\mathbb{Q}}_{<0}$ is not of the form $\frac{-Nk^2}{{\left\vert\Delta\right\vert}}$ with $k\in{\mathbb{Q}}_{>0}$ we let $${\mathbf{t}}_{\Delta,r}(f;h,m) = \begin{cases} -\frac{\delta_{h,0}}{2\pi} \int_{{\Gamma}\backslash{\mathbb{H}}}^{\text{reg}}f(z) \frac{dxdy}{y^2}, &\text{if } \Delta = 1 \\ 0, &\text{if } \Delta \neq 1. \end{cases}$$ Here the integral has to be regularized [@BrFu04 $(4.6)$]. Now let $m = -Nk^2/{\left\vert\Delta\right\vert}$ with $k\in{\mathbb{Q}}_{>0}$ and $\lambda\in L_{rh,m{\left\vert\Delta\right\vert}}$. We have $Q(\lambda) = -Nk^2$, which implies that $\lambda^{\perp}$ is split over ${\mathbb{Q}}$ and $c(\lambda)$ is an infinite geodesic. Choose an orientation of $V$ such that $$\sigma_{\ell_{\lambda}}^{-1}\lambda= \begin{pmatrix} m & s \\ 0 & -m \end{pmatrix}$$ for some $s\in{\mathbb{Q}}$. Then $c_{\lambda}$ is explicitly given by $$c_{\lambda}= \sigma_{\ell_{\lambda}} \left\{ z \in {\mathbb{H}}; \Re(z)=-s/2m\right\}.$$ Define the real part of $c(\lambda)$ by $\Re(c(\lambda))=-s/2m$. For a cusp $\ell_{\lambda}$ let $$\begin{aligned} \langle f, c(\lambda)\rangle &= -\sum\limits_{n\in{\mathbb{Q}}_{<0}}a^{+}_{\ell_{\lambda}}(n)e^{2\pi i\Re(c(\lambda))n} -\sum\limits_{n\in{\mathbb{Q}}_{<0}} a^{+}_{\ell_{-\lambda}}(n)e^{2\pi i\Re(c(-\lambda))n}.\end{aligned}$$ Then we define $$\label{def:trace4} {\mathbf{t}}_{\Delta,r}(f;h,m)= \sum\limits_{\lambda\in {\Gamma}\setminus L_{rh,{\left\vert\Delta\right\vert}m}} \chi_{\Delta}(\lambda) \langle f,c(\lambda)\rangle.$$ In order to describe the coefficients of the lift that are not given in terms of traces we introduce the following definitions. For $h \in {\mathcal{D}}$, we let $$\label{def:deltah} \delta_\ell(h)= \begin{cases} 1, & \text{if } \ell \cap (L+h) \neq \emptyset, \\ 0, & \text{otherwise.} \end{cases}$$ If $\delta_\ell(h)=1$, there is an $h_\ell$ such that $\ell \cap (L+h)={\mathbb{Z}}\lambda_\ell + h_\ell$. Now let $s \in {\mathbb{Q}}$ such that $h_\ell = s \lambda_\ell$. Write $s=\frac{p}{q}$ with $(p,q)=1$ and define $d(\ell,h):=q$, which depends only on $\ell$ and $h$. Moreover, we define $h'_\ell=\frac{1}{d(\ell,h)}\lambda_\ell$ which is well defined as an element of ${\mathcal{D}}$. \[def:dzero\] Let $h \in {\mathcal{D}}$ and $\ell \in \operatorname{Iso}(V)$. Then we let $$\operatorname{\mathfrak{d}}_{\Delta,r}(\ell,h) := \begin{cases} \delta_\ell(h), &\text{if } \Delta=1,\\ \chi_\Delta((rh)'_\ell), &\text{if } \Delta \neq 1,\ \delta_\ell(rh)=1 \text{ and } \Delta \mid d(\ell,rh),\\ 0, &\text{otherwise.} \end{cases}$$ If $\Delta \mid d(\ell,rh)$, then $\chi_\Delta((rh)'_\ell)$ is well defined because $(rh)'_\ell \in L'$. The fact that $d(\ell,0)=1$ implies that for $\Delta\neq 1$ and $rh=0$ we have $\operatorname{\mathfrak{d}}_{\Delta,r}(\ell,0)=0$. \[prop:dzero\] Assume that $\Delta \neq 1$. If $\operatorname{\mathfrak{d}}_{\Delta,r}(\ell,h) \neq 0$, then for every prime $p \mid \Delta$, we have that $p^2 \mid N$. In particular, for square-free $N$ and $\Delta \neq 1$, we always have $\operatorname{\mathfrak{d}}_{\Delta,r}(\ell,h) = 0$. We can assume that $rh \neq 0$ because otherwise $\operatorname{\mathfrak{d}}_{\Delta,r}(\ell,h) = 0$. Since $\lambda_\ell$ is primitive it is of the form $\lambda_\ell = \left(\begin{smallmatrix} b & -\frac{\Delta a}{N} \\ \Delta c & -b \end{smallmatrix}\right)$, with $a,b,c \in {\mathbb{Z}}$, $b\neq 0$, $(a,b,c)=1$ and $(\Delta,b)=1$. The facts that $\frac{1}{d(\ell,rh)}\lambda_\ell \in L'$ and $\Delta \mid d(\ell,rh)$ imply $\Delta \mid 2N$. Since $\lambda_\ell$ is isotropic, we have $\frac{\Delta^2 ac}{N} = b^2 \in {\mathbb{Z}}$. Now suppose that there exists a prime $p$ such that $p \mid \Delta$ but $p^2 \nmid N$. Then $\frac{\Delta^2 ac}{Np} \in {\mathbb{Z}}$ and thus $p \mid b$ which is a contradiction. The theta integral ------------------ Now we consider the integral $$I_{\Delta,r}(\tau,f)= \int_M f(z) \Theta_{\Delta,r}(\tau,z) = \sum_{h\in{\mathcal{D}}}\left(\int_M f(z) \Theta_{\Delta,r,h}(\tau,z)\right)\mathfrak{e}_h,$$ where $\Delta$ and $r$ are chosen as before. For the individual components, we write $$I_{\Delta,r,h}(\tau,f)=\int_M f(z) \Theta_{\Delta,r,h}(\tau,z).$$ This is a twisted version of the theta lift considered by Bruinier and Funke [@BrFu06], which we obtain as a special case when $\Delta=1$. Note that due to the rapid decay of the Kudla-Millson kernel the integral $I_{\Delta, r}(\tau,f)$ converges [@BrFu06 Proposition 4.1]. It defines a harmonic weak Maass form of weight $3/2$ transforming with the representation $\widetilde{\rho}$. \[thm:main\] Let $f\in H^{+}_{0}({\Gamma})$ with Fourier expansion as in . Assume that $f$ has vanishing constant term at every cusp of ${\Gamma}$. Then the Fourier expansion of $I_{\Delta,r, h}(\tau,f)$ is given by $$\label{thm:felift} I_{\Delta,r,h}(\tau,f) = \sum_{\substack{m\in{\mathbb{Q}}_{>0}\\ m\equiv {\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}}\!\!\!\!\!\! {\mathbf{t}}_{\Delta,r}(f;h,m)q^m \ + \!\!\!\!\!\!\!\! \sum\limits_{\substack{m\in{\mathbb{Q}}_{>0} \\ -N {\left\vert\Delta\right\vert} m^2 \equiv{\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}} \!\!\!\!\!\!\!\!\!\!\!\!\!\! {\mathbf{t}}_{\Delta,r}(f;h,-N{\left\vert\Delta\right\vert}m^2) q^{-N {\left\vert\Delta\right\vert} m^2}.$$ If the constant coefficients of $f$ at the cusps do not vanish, the following terms occur in addition: $$\begin{gathered} \frac{\sqrt{{\left\vert\Delta\right\vert}}}{2\pi\sqrt{Nv}} \sum_{\ell \in \Gamma\backslash \operatorname{Iso}(V) } \hspace{-2mm} \operatorname{\mathfrak{d}}_{\Delta,r}(\ell,h)\,\epsilon_\ell\, a^+_{\ell}(0)\\ + \sqrt{{\left\vert\Delta\right\vert}} \sum_{m>0}\sum_{\substack{\lambda \in \Gamma \backslash L_{rh,-Nm^2} \\ Q_\Delta(\lambda)\equiv{\operatorname{sgn}}(\Delta)Q(h)\, ({\mathbb{Z}})}} \chi_\Delta(\lambda) \frac{a^{+}_{\ell_{\lambda}}(0)+a^{+}_{\ell_{-\lambda}}(0)}{8\pi\sqrt{vN}m} \beta\left(\frac{4\pi vNm^2}{|\Delta|}\right)q^{-Nm^2/|\Delta|},\end{gathered}$$ where $\beta(s)=\int_1^{\infty}t^{-3/2}e^{-st}dt$. In particular, $I_{\Delta,r,h}(\tau,f)$ is contained in $H_{3/2,\widetilde{\rho}}$. If $N$ is square-free, Proposition \[prop:dzero\] implies that $I_{\Delta,r, h}(\tau,f)\in H^{+}_{3/2,\widetilde{\rho}}$. Since the traces of negative index essentially depend on the principal part of $f$, it is not hard to show the following Assume that all constant coefficients of $f\in H^{+}_0({\Gamma})$ vanish; then $$I_{\Delta, r}(\tau,f)\in M^{\text{!}}_{3/2,\widetilde{\rho}}.$$ The theorem can be proven in two different ways. It is possible to give a proof by explicitly calculating the contributions of the lattice elements of positive, negative, and vanishing norm, similarly to Bruinier and Funke [@BrFu06]. However, a substantially shorter proof is obtained by rewriting the twisted theta integral as a linear combination of untwisted ones. Throughout the proof, we may assume that $\Delta \neq 1$ because for $\Delta = 1$ the statement is covered by Bruinier and Funke [@BrFu06]. Replacing the theta function $\Theta_{\Delta,r}(\tau,z)$ by the expression in , we can write $$I_{\Delta,r}(\tau,f) = \int_{\mathcal{F}} f(z) \Theta_{\Delta,r}(\tau,z) = \sum_{h \in {\mathcal{D}}} \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), \int_{\mathcal{F}} f(z) \overline{\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)} \right\rangle {\mathfrak{e}}_h,$$ where $\mathcal{F}$ denotes a fundamental domain for the action of ${\Gamma}$ on ${\mathbb{H}}$. In general ${\Gamma}$ does not act trivially on ${{\mathcal{D}(\Delta)}}$. But $\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)$ is always invariant under the discriminant kernel ${\Gamma}_\Delta = \left\lbrace \gamma \in {\Gamma};\ \gamma \delta = \delta \text{ for all } \delta \in {{\mathcal{D}(\Delta)}}\right\rbrace \subset {\Gamma}$. Since $f(z)\Theta_{\Delta,r}(\tau,z)$ is ${\Gamma}$-invariant by Proposition \[prop:twthetatrans\], we obtain by a standard argument $$I_{\Delta,r}(\tau,f) = \frac{1}{[{\Gamma}:{\Gamma}_\Delta]}\ \sum_{h \in {\mathcal{D}}} \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), \int_{{\Gamma}_\Delta\backslash{\mathbb{H}}} f(z) \overline{\Theta_{{{\mathcal{D}(\Delta)}}}(\tau,z)} \right\rangle.$$ Now we are able to apply the result of Bruinier and Funke [@BrFu06 Theorem 4.5] to the integral above. For $m \in {\mathbb{Q}}$ we obtain that the $(h,m)$-th Fourier coefficient of the holomorphic part of $I_{\Delta,r, h}(\tau,f)$ is given by $1/[{\Gamma}:{\Gamma}_\Delta]$ times $$\label{eq:tracelincomb} \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), \sum_{\delta \in {{\mathcal{D}(\Delta)}}} \overline{{\mathbf{t}}(f;\delta,m)} {\mathfrak{e}}_\delta\right\rangle = \sum_{\substack{\delta \in {{\mathcal{D}(\Delta)}}\\ \pi(\delta)=rh \\ Q_\Delta(\delta) \equiv {\operatorname{sgn}}(\Delta) Q(h)\, ({\mathbb{Z}})}} \chi_\Delta(\delta)\ {\mathbf{t}}(f;\delta,m).$$ Here the traces are taken with respect to ${\Gamma}_\Delta$ and the discriminant group ${{\mathcal{D}(\Delta)}}$. Note that $${\mathbf{t}}(f;\delta,m) = \begin{cases} \sum\limits_{{\Gamma}_\Delta \backslash (\Delta L)_{\delta,m}} \frac{1}{{\left\vert\overline{{\Gamma}}_{\Delta,\lambda}\right\vert}} f(D_\lambda), & \text{if } m>0, \\ \sum\limits_{{\Gamma}_\Delta \backslash (\Delta L)_{\delta,m}} \langle f,c(\lambda)\rangle, & \text{if } m<0, \end{cases}$$ where $(\Delta L)_{\delta,m}=\left\lbrace \lambda\ \in\ \Delta L + \delta;\ Q_\Delta(\lambda) = m \right\rbrace$. For $m=0$, ${\mathbf{t}}(f;0,0)$ is defined as a regularized integral and we have ${\mathbf{t}}(f;\delta,m)=0$ for $\delta \neq 0$. Hence for $m = 0$ the left hand side in is equal to $\chi_\Delta(0)\ {\mathbf{t}}(f;0,0)$. Since $\chi_\Delta(0)=0$ for $\Delta \neq 1$, this quantity vanishes in our case. If $m \equiv {\operatorname{sgn}}(\Delta)Q(h) \pmod{{\mathbb{Z}}}$ the right hand side in is equal to ${\mathbf{t}}_{\Delta,r}(f;h,m)$ as in and . Otherwise it vanishes. Next, we consider the non-holomorphic part. It is again a straightforward calculation to obtain our result for the coefficients of negative index. It remains to evaluate $$\label{eq:proofmain} \frac{1}{2\pi \sqrt{N{\left\vert\Delta\right\vert} v}} \left\langle \psi_{\Delta,r}({\mathfrak{e}}_h), \sum_{\delta \in {{\mathcal{D}(\Delta)}}} \sum\limits_{\substack{\ell\in {\Gamma}\setminus\operatorname{Iso}(V)\\ \ell\cap \Delta L + \delta \neq \emptyset}} \chi_\Delta(\delta) a_\ell^+(0) \varepsilon_\ell \right\rangle.$$ Recall the definition of $\delta_{\ell}(rh)$ from . For $\delta_{\ell}(rh)=1$ there is an element $(rh)_\ell \in L + rh$ such that $\ell \cap (L + rh) = {\mathbb{Z}}\lambda_\ell + (rh)_\ell$, where $\lambda_\ell$ is a primitive element of $\ell \cap L$. Consequently, a system of representatives for all $\delta \in {{\mathcal{D}(\Delta)}}$ with $\pi(\delta)=rh$ such that $\ell \cap (\Delta L + \delta) \neq \emptyset$ is given by the set $\{ m\lambda_\ell + rh;\ m \mod \Delta \}$. So it boils down to computing $$\delta_{\ell}(rh)\sum\limits_{m \bmod \Delta} \chi_{\Delta}(m\lambda_\ell+(rh)_{\ell}).$$ To do this, we write $(rh)_\ell=\frac{n}{d(\ell,rh)}\lambda_\ell$ for some integer $n$ and $(rh)'_\ell=\frac{1}{d(\ell,rh)}\lambda_\ell$. So the inner product in equals $$\begin{aligned} \sum_{m \bmod \Delta} \chi_\Delta\left(\frac{d(\ell,rh)m+n}{d(\ell,rh)}\lambda_\ell\right) = \chi_\Delta((rh)'_\ell) \sum_{m \bmod \Delta} {\left(\frac{\Delta}{d(\ell,rh)m+n}\right)}.\end{aligned}$$ The latter sum vanishes unless $\Delta$ divides $d(\ell,rh)$, in which case it equals ${\left\vert\Delta\right\vert}$. Similarly to Bruinier and Funke [@BrFu06], we can give a more explicit description of the traces of negative index ${\mathbf{t}}_{\Delta,r}(f;h,-Nk^2/{\left\vert\Delta\right\vert})$. For this, sort the infinite geodesics according to the cusps from which they originate: For $k \in {\mathbb{Q}}_{>0}$, define $L_{h,-Nk^2,\ell}=\{ X \in L_{h,-Nk^2};\, X \sim \ell \}$ and let $$\nu_\ell(h,-Nk^2) := \#\Gamma_\ell\backslash L_{h,-Nk^2,\ell}.$$ We have that $\nu_\ell(h,-Nk^2) = 2k\epsilon_\ell$ if $L_{h,-Nk^2,\ell}$ is non-empty. Let $h \in {\mathcal{D}}$, $n \in {\mathbb{Z}}$ and $m=-Nk^2/{\left\vert\Delta\right\vert}$ with $k \in {\mathbb{Q}}_{>0}$, such that $\Delta \mid 2k\varepsilon_\ell$ and $\frac{2 k \varepsilon_\ell}{{\left\vert\Delta\right\vert}} \mid n$. We define $$\label{def:muell} \mu_\ell(h,m,n) = \frac{\nu_\ell(h,-Nk^2)}{\sqrt{{\left\vert\Delta\right\vert}}} \overline{\epsilon} \sum_{\substack{j \bmod \Delta \\ N\beta_\ell j - n'\, \equiv\, 0 \bmod \Delta}} {\left(\frac{\Delta}{j}\right)}\, \exp\left({\frac{4\pi i Nkr_\ell j}{{\left\vert\Delta\right\vert}}}\right).$$ Here, we let $r_\ell = \Re(c(\lambda))$ for any $\lambda \in L_{h,-Nk^2,\ell}$ and $n' = \frac{{\left\vert\Delta\right\vert}}{2k\varepsilon_\ell}n$. Moreover, $\epsilon=1$ for $\Delta > 0$ and $\epsilon=i$ for $\Delta < 0$. \[rem:explform\] The finite sum in can often be explicitly evaluated. For instance, if $N\beta_\ell$ is coprime to $\Delta$, it is equal to $${\left(\frac{\Delta}{N\beta_\ell n'}\right)}\, \exp\left({\frac{4 \pi i N k r_\ell n' s}{{\left\vert\Delta\right\vert}}}\right),$$ where $s$ denotes an integer such that $(N\beta_\ell) s \equiv 1 \bmod \Delta$. \[prop:tf-\] Let $f \in H^{+}_{0}({\Gamma})$ with Fourier expansion as in and let $m=\frac{-N {k}^2}{{\left\vert\Delta\right\vert}}$ for some $k \in {\mathbb{Q}}_{>0}$. Then ${\mathbf{t}}_{\Delta,r}(f;h,m)=0$ unless $k=\frac{{\left\vert\Delta\right\vert} k'}{2N}$ for some $k' \in {\mathbb{Z}}_{>0}$. In the latter case, we have $$\begin{aligned} {\mathbf{t}}_{\Delta,r}(f;h,m) &= - \sum_{\ell\in{\Gamma}\backslash\operatorname{Iso}(V)} \sum_{n \in \frac{2k}{{\left\vert\Delta\right\vert}\beta_\ell}{\mathbb{Z}}_{<0}} a^+_\ell(n)\, \mu_\ell(rh,m,n\, \alpha_\ell)\, e^{-2\pi i r_\ell n}\\ &\quad - {\operatorname{sgn}}(\Delta)\sum_{\ell\in{\Gamma}\backslash\operatorname{Iso}(V)} \sum_{n \in \frac{2k}{{\left\vert\Delta\right\vert}\beta_\ell}{\mathbb{Z}}_{<0}} a^+_\ell(n)\, \mu_\ell(-rh,m,n\, \alpha_\ell)\, e^{-2\pi i r'_\ell n}, \end{aligned}$$ with $r_\ell = \Re(c(\lambda))$ for any $\lambda \in L_{rh,{\left\vert\Delta\right\vert}m,\ell}$ and $r'_\ell = \Re(c(\lambda))$ for any $\lambda \in L_{-rh,{\left\vert\Delta\right\vert}m,\ell}$. Moreover, we have ${\mathbf{t}}_{\Delta,r}(f;h,-Nn^2/{\left\vert\Delta\right\vert}) = 0$ for $n \gg 0$. For the proof of the proposition we need the following \[lem:gmdelta\] For a fundamental discriminant $\Delta$, integers $a,b,n \in {\mathbb{Z}}$ and $M \in {\mathbb{Z}}_{>0}$, such that $\Delta \mid M$, we define the Gauss type sum $$g_M^\Delta(a,b;n) = \sum_{j \bmod M} {\left(\frac{\Delta}{aj+b}\right)}\, e^{\frac{2 \pi i j n}{M}}.$$ The sum $g_M^\Delta(a,b;n)$ vanishes unless $\frac{M}{\Delta} \mid n$. Then we obtain $$g_M^\Delta(a,b;n) = \epsilon^{-1} \frac{M}{\sqrt{{\left\vert\Delta\right\vert}}} \sum_{\substack{l \bmod \Delta \\ al+n' \equiv 0 \bmod \Delta}} {\left(\frac{\Delta}{l}\right)}\, e^{\frac{2\pi i b l}{{\left\vert\Delta\right\vert}}},$$ where $n':=\frac{{\left\vert\Delta\right\vert}}{M}n$. In many cases the sum above can also be evaluated more explicitly in a straightforward but tedious way. Replacing the Kronecker symbol by the finite exponential sum $${\left(\frac{\Delta}{aj+b}\right)} = \epsilon^{-1} \frac{1}{\sqrt{{\left\vert\Delta\right\vert}}} \sum_{l \bmod \Delta} {\left(\frac{\Delta}{l}\right)}\, e^{\frac{2\pi i (aj+b) l}{{\left\vert\Delta\right\vert}}}$$ yields $$\begin{aligned} g_M^\Delta(a,b;n) &= \epsilon^{-1} \frac{1}{\sqrt{{\left\vert\Delta\right\vert}}} \sum_{l \bmod \Delta} {\left(\frac{\Delta}{l}\right)}\, e^{\frac{2\pi i b l}{{\left\vert\Delta\right\vert}}} \sum_{j \bmod M} \exp\left({\frac{2 \pi i j \left(a\frac{M}{{\left\vert\Delta\right\vert}}l +n \right)}{M}}\right)\\ &= \epsilon^{-1} \frac{M}{\sqrt{{\left\vert\Delta\right\vert}}} \sum_{\substack{l \bmod \Delta \\ a\frac{M}{{\left\vert\Delta\right\vert}}l+n\, \equiv\, 0 \bmod M}} {\left(\frac{\Delta}{l}\right)}\, e^{\frac{2\pi i b l}{{\left\vert\Delta\right\vert}}}. \end{aligned}$$ The congruence condition above can only be satisfied if $\frac{M}{\Delta} \mid n$, which proves the statement of the lemma. Following Bruinier and Funke [@BrFu06 Proposition 4.7], we write $${\mathbf{t}}_{\Delta,r}(f;h,m) = \sum_{\ell\in{\Gamma}\backslash\operatorname{Iso}(V)} G_\Delta(rh,-Nk^2,\ell) + {\operatorname{sgn}}(\Delta)\sum_{\ell\in{\Gamma}\backslash\operatorname{Iso}(V)} G_\Delta(-rh,-Nk^2,\ell),$$ where $$G_\Delta(h,-Nk^2,\ell) = - \sum_{\lambda\in \Gamma_\ell \backslash L_{h,-Nk^2,\ell}} \chi_\Delta(\lambda) \sum_{n \in {\mathbb{Z}}_{<0}} a^+_\ell(n/\alpha_\ell)\, e^{\frac{2\pi i \Re(c(\lambda))n}{\alpha_\ell}}.$$ A set of representatives for $\Gamma_\ell \backslash L_{h,-Nk^2,\ell}$ is given by $$\left\{Y_j=\sigma_\ell\begin{pmatrix}k&-2kr_\ell-j\beta_\ell\\0&-k\end{pmatrix};j=0,\ldots,2k\epsilon_\ell-1 \right\}$$ for some $r\ell\in{\mathbb{Q}}$. We have $\Re(c(Y_j))=r_\ell+j\frac{\beta_\ell}{2k}$. For $\lambda \in L_{h,-Nk^2,\ell}$ and $k$ not of the form $\frac{{\left\vert\Delta\right\vert} k'}{2N}$ for some $k' \in {\mathbb{Z}}_{>0}$ we have $\chi_\Delta(\lambda)=0$. So we may assume that $k=\frac{{\left\vert\Delta\right\vert} k'}{2N}$. By reordering the summation and using the $\operatorname{SO}^+(L)$-invariance of $\chi_\Delta$ we see that it remains to evaluate $$\sum_{j=0}^{2k\varepsilon_\ell-1}{\left(\frac{\Delta}{N\beta_\ell j + 2 N k r_\ell}\right)}\, e^{\frac{-2\pi i nj}{2k\varepsilon_\ell}}$$ for $n \in {\mathbb{Z}}_{<0}$. Using Lemma \[lem:gmdelta\], we obtain the statement of the proposition. Applications and Examples ========================= The twisted lift of the weight zero Eisenstein series and ${\log\lVert\Delta(z)\rVert}$ {#sec:Eisenlift} --------------------------------------------------------------------------------------- For $z\in{\mathbb{H}}$ and $s\in{\mathbb{C}}$, we let $$\mathcal{E}_0(z,s)=\frac12\zeta^{*}(2s+1) \sum\limits_{\gamma\in{\Gamma}_{\infty}\setminus {{\text {\rm SL}}}_2({\mathbb{Z}})} \left(\Im(\gamma z)\right)^{s+\frac12}.$$ Here $\zeta^*(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed Riemann Zeta function and ${\Gamma}_\infty$ is the stabilizer of the cusp $\infty$ in ${\Gamma}$. We now consider the case $N=1$, so we have the quadratic form $Q(\lambda)=\mathrm{det}(\lambda)$ and the lattice $$L=\left\{\begin{pmatrix} b&a\\c&-b\end{pmatrix}; a,b,c\in{\mathbb{Z}}\right\}.$$ Furthermore, we let $K$ be the one-dimensional lattice ${\mathbb{Z}}$ together with the negative definite bilinear form $(b,b')=-2bb'$. Then we have $L'/L \simeq K'/K$. We define a vector valued Eisenstein series $\mathcal{E}_{3/2,K}(\tau,s)$ of weight $3/2$ for the representation $\rho_K$ by $$\mathcal{E}_{3/2,K}(\tau,s)=-\frac{1}{4\pi}\left(s+\frac12\right)\zeta^{*}(2s+1)\sum\limits_{\gamma'\in{\Gamma}'_{\infty}\setminus {\Gamma}'}\left.(v^{\frac12\left(s-\frac12\right)}\mathfrak{e}_0)\right|_{3/2,K}\gamma',$$ where ${\Gamma}'_{\infty}$ and ${\Gamma}'$ are the inverse images of ${\Gamma}_{\infty}$ and ${\Gamma}$ in ${\text {\rm Mp}}_2({\mathbb{Z}})$. Note that $$\mathcal{F}(\tau,s):=\left(\mathcal{E}_{3/2,K}(4\tau,s)\right)_0+\left(\mathcal{E}_{3/2,K}(4\tau,s)\right)_1$$ evaluated at $s=\frac12$ is equal to Zagier’s Eisenstein series as in [@HiZa] and [@Zagierclass]. Similarly to [@BrFu06], Section 7.1, one can show the following theorem. Assume that $\Delta > 0$. We have $$I_\Delta(\tau,\mathcal{E}_0(z,s))=\Lambda\left(\varepsilon_\Delta, s+\frac12\right)\mathcal{E}_{3/2,K}(\tau,s).$$ Here $\varepsilon_\Delta(n)=\left(\frac{\Delta}{n}\right)$ and $\Lambda\left(\varepsilon_\Delta, s+\frac12\right)$ denotes the completed Dirichlet $L$-series associated with $\varepsilon_\Delta$. Note that for $\Delta<0$ the lift $I_\Delta$ vanishes. Using this, we obtain $$I_\Delta(\tau,1)= \begin{cases} 0, &\text{if } \Delta \neq 1,\\ 2\mathcal{E}_{3/2,K}\left(\tau,\frac12\right), &\text{if } \Delta = 1. \end{cases}$$ By $\Delta(z)=q\prod_{n=1}^\infty(1-q^n)^{24}$ we denote the Delta function. Following Bruinier and Funke we normalize the Petersson metric of $\Delta(z)$ such that $$\lVert \Delta(z)\rVert =e^{-6C} {\left\vert\Delta(z)(4\pi y)^6\right\vert},$$ where $C=\frac12(\gamma+\log 4\pi)$. We have $$-\frac{1}{12} I_\Delta(\tau,\log \lVert\Delta(z)\rVert) = \begin{cases} \Lambda(\varepsilon_\Delta,1)\mathcal{E}_{3/2,K}\left(\tau,\frac12\right) &\text{if } \Delta > 1,\\ \mathcal{E}'_{3/2,K}\left(\tau,\frac12\right) &\text{if } \Delta=1.\\ \end{cases}$$ In terms of arithmetic geometry we obtain the following interpretation of this result (for notation and background information, we refer to the survey article by Yang [@Yang]). By $\mathcal{M}$ we denote the Deligne-Rapoport compactification of the moduli stack over ${\mathbb{Z}}$ of elliptic curves. Kudla, Rapoport and Yang [@KRY; @Yang] construct cycles $\hat{\mathcal{Z}}(D,v)$ in the extended arithmetic Chow group of $\mathcal{M}$ for $D\in{\mathbb{Z}}$ and $v>0$. Then the Gillet-Soulé intersection pairing of $\hat{\mathcal{Z}}(D,v)$ with the normalized metrized Hodge bundle $\hat{\omega}$ on $\mathcal{M}$ is given in terms of the derivative of Zagier’s Eisenstein series [@Yang; @BrFu06]. Similarly, one can define twisted cycles $\hat{\mathcal{Z}}_\Delta(D,v)$. In contrast to the untwisted case, for $\Delta > 1$ the intersection pairing is given in terms of the value of the Eisenstein series at $s = 1/2$ (also note that the degree of the twisted divisor is $0$). We have $$\sum\limits_{D \in {\mathbb{Z}}} \langle\hat{\mathcal{Z}}_\Delta(D,v), \hat{\omega} \rangle q^D = \log(u_\Delta)\; h(\Delta)\; \mathcal{F}\left(\tau,\frac12\right),$$ where $u_\Delta$ denotes the fundamental unit and $h(\Delta)$ the class number of the real quadratic number field of discriminant $\Delta$. The twisted lift of Maass cusp forms {#sec:liftMC} ------------------------------------ As indicated in the remark in Section \[theta\] the construction of the twisted theta function directly yields a twisted version of other theta lifts. As an example we briefly consider the lift of Maass cusp forms analogously to Bruinier and Funke [@BrFu06 Section 7.2]. It was first considered by Maass ([@Maass], [@Duke], [@KS]). The underlying theta kernel is now given by the Siegel theta series for $\Delta L$ with the negative quadratic form $-Q_\Delta$; namely, $$\Theta_{\delta}(\tau,z,\varphi_{2,1})=\sum\limits_{\lambda\in \delta+\Delta L}\varphi_{2,1}(\lambda,\tau,z),$$ where $\varphi_{2,1}(\lambda,\tau,z)=v^{1/2}e^{\frac{\pi i}{{\left\vert\Delta\right\vert}}(-u(\lambda,\lambda)+iv(\lambda,\lambda)_z) }$. Following the construction in Section \[theta\] we obtain a twisted lift $I^M_\Delta(\tau,f)$. Using that $$\xi_{1/2}\varphi_{2,1}(\lambda,\tau,z)\omega=-\pi\varphi(\lambda,\tau,z),$$ a straightforward calculation yields We have $$\xi_{1/2}I^M_{\Delta}(\tau,f)=-\pi I_{\Delta}(\tau,f)$$ and for an eigenfunction $f$ of $\Omega$ with eigenvalue $\alpha$ $$\xi_{3/2}I_{\Delta}(\tau,f)=-\frac{\alpha}{4\pi} I^M_{\Delta}(\tau,f).$$ The example ${\Gamma}_0(p)$ --------------------------- We begin by explaining how to obtain the example in the Introduction. Let $N=p$ be a prime and $\Delta > 1$ a fundamental discriminant with $(\Delta,2p)=1$ such that there exists an $r \in {\mathbb{Z}}$ with $\Delta \equiv r^2 \bmod 4p$. Let $f \in M_0^!(\Gamma_0(p))$ be invariant under the Fricke involution with Fourier expansion $f(\tau)=\sum_n a(n) q^n$. The group ${\Gamma}_0(p)$ only has the two cusps $\infty, 0$ which are represented by the isotropic lines $\ell_\infty=\text{span}\left(\begin{smallmatrix} 0& 1\\0&0\end{smallmatrix}\right)$ and $\ell_0=\text{span}\left(\begin{smallmatrix} 0& 0\\-1&0\end{smallmatrix}\right)$. We then obtain $\alpha_{\ell_\infty}=1, \beta_{\ell_\infty}=1/p, \epsilon_{\ell_\infty}=p$ and $\alpha_{\ell_0}=p, \beta_{\ell_0}=1, \epsilon_{\ell_0}=p$. The Fricke involution interchanges the two cusps. The space $M^!_{3/2,\rho}$ is isomorphic to $M^{!,+}_{3/2}(\Gamma_0(4p))$, the subspace of $M^{!}_{3/2}(\Gamma_0(4p))$ containing only forms whose $n$-th Fourier coefficient is zero unless $n$ is a square modulo $4p$. The isomorphism takes $\sum_{h \in {\mathcal{D}}} f_h {\mathfrak{e}}_h$ to $\sum_{h \in {\mathcal{D}}} f_h$ [@EZ Theorem 5.6]. The assumption ${(\Delta,2p)=1}$ guarantees that we can choose the parameter $r \in {\mathbb{Z}}$ as a unit in ${\mathbb{Z}}/4p{\mathbb{Z}}$. Thus the sum $\sum_{h \in {\mathcal{D}}} I_{\Delta,r,h}$ does not depend on $r$. As described in Section \[sec:twisting\], we can identify lattice elements with integral binary quadratic forms. The action of ${\Gamma}_0(p)$ on both spaces is compatible. This way we obtain the interpretation of the coefficients of positive index of the holomorphic part. However, notice that we have to consider positive and negative definite quadratic forms. For positive $\Delta$ we have $\chi_\Delta(-Q)=\chi_\Delta(Q)$, which for $m=d/4p>0$ yields $$\sum_{h \in {\mathcal{D}}} \sum\limits_{\lambda\in {\Gamma}\setminus L_{rh,{\left\vert\Delta\right\vert}m}} \frac{\chi_{\Delta}(\lambda)}{\left|\overline{\Gamma}_{\lambda}\right|} f(D_{\lambda}) = 2 \sum\limits_{Q\in{\Gamma}\backslash\mathcal{Q}_{-d{\left\vert\Delta\right\vert},N}} \frac{\chi_{\Delta}(Q)}{{\left\vert\overline{\Gamma}_Q\right\vert}}f(\alpha_Q).$$ We use the explicit formula in Proposition \[prop:tf-\] to determine the coefficients of negative index. For every $k \in {\mathbb{Z}}_{>0}$ with $k \equiv h \bmod 2p$, we have that $\left(\begin{smallmatrix} -k/2p& 0\\0&k/2p\end{smallmatrix}\right) \in L_{-h,-k^2/4p,\ell_\infty}$ and $\left(\begin{smallmatrix} k/2p& 0\\0&-k/2p\end{smallmatrix}\right) \in L_{h,-k^2/4p,\ell_0}$. And if $h \neq 0$ we have $L_{h,-k^2/4p,\ell_\infty} = \emptyset$ and $L_{-h,-k^2/4p,\ell_0} = \emptyset$. In particular, this implies that $r_\ell=r'_{\ell}=0$ in Proposition \[prop:tf-\] and we get $$\sum_{h \in {\mathcal{D}}} {\mathbf{t}}_{\Delta,r}(f;h,m) = - 2 \sum_{\ell\in{\Gamma}\backslash\operatorname{Iso}(V)} \sum_{n \in {\mathbb{Z}}_{<0}} a^+_\ell\left(\frac{k}{{\left\vert\Delta\right\vert}p\, \beta_\ell} n\right) \sum_{h \in {\mathcal{D}}} \mu_\ell\left(rh,m,\frac{k}{{\left\vert\Delta\right\vert}}\, n\right).$$ Moreover, Proposition \[prop:tf-\] implies that $k={\left\vert\Delta\right\vert}k'$ for some $k' \in {\mathbb{Z}}_{>0}$. By our considerations above, for every given $k'$ and $\ell$ there is exactly one $h$ such that $\mu_{\ell}(rh,-{\left\vert\Delta\right\vert}{k'}^{2}/4p, k' n) \neq 0$. Using the explicit formula given in Remark \[rem:explform\], we obtain $$\sum_{h \in {\mathcal{D}}} \mu_\ell\left(rh,-\frac{{\left\vert\Delta\right\vert}{k'}^{2}}{4p}, k' n\right) = \sqrt{{\left\vert\Delta\right\vert}} k' {\left(\frac{\Delta}{n}\right)}.$$ Since $f$ is invariant under the Fricke involution, this yields the formula for the principal part in the Introduction. Note that for $\Delta \neq 1$, we do not obtain a non-holomorphic part in this case, which follows from the formula given in Theorem \[thm:main\] and the fact that $$\sum_{\lambda \in \Gamma_0(p)\backslash L_{h,m}} \chi_\Delta(\lambda)=0$$ for all $h \in {\mathcal{D}}$ and all $m \in {\mathbb{Q}}$. The computation above is also valid for $p=1$, except that we only have to consider the cusp at $\infty$ and we do not have to assume that $(\Delta,2)$=1. Computations ------------ We consider the genus 1 modular curve $X_0(11)$ and the weakly holomorphic modular function $f(z)=J(11z)$. Let $\Delta=5$ and $r=7$. Consider the case $d=8$. The class number of ${\mathbb{Q}}(\sqrt{-40})$ is 2 and a set of representatives for ${\Gamma}_0(11)\backslash L_{2/22, 40/44}$ is given by the integral binary quadratic forms $[1001, 200, 10]$, $[-1001, 200, -10]$, $[407, 90, 5]$, $[-407, 90, -5]$. For ${\Gamma}_0(11)\backslash L_{20/22, 40/44}$ a set of representatives is given by the negatives of the above forms and for all $h \neq \pm 2/22$ the set $L_{h, 40/44}$ is empty. Using the Galois-theoretic interpretation of the twisted traces, we can calculate them “by hand”: the CM point given by the form $[407, 90, 5]$ is $z_0=\frac{-90+\sqrt{-40}}{814}$ and the CM value $f(z_0) \approx 20641.38121$ is an algebraic integer of degree 2. It is in fact a root of the polynomial $x^2 - 425691312x + 8786430582336$ and is contained in the Hilbert class field $H_{-40}$ of ${\mathbb{Q}}(\sqrt{-40})$. So $f(z_0) = \frac{1}{2}\left(425691312 - 190356480\sqrt{5}\right)$. Therefore, we have for the twisted trace (for positive definite forms with $b \equiv 2 \bmod 11$) $$\frac{1}{\sqrt{5}} \sum_{Q \in \Gamma_0(11) \backslash \mathcal{Q}_{-40,11,2}} \chi_{5}(Q)f(\alpha_Q)= 190356480.$$ Since our definition of the trace includes both, positive and negative definite quadratic forms and $\chi_{5}([-a,b,-c])=\chi_{5}([a,b,c])$, we obtain ${\mathbf{t}}_{5,7}(f; \pm 6/22, 40/44) = 380712960$. Some other examples for $\Delta=5, r=7$ are $$\begin{aligned} {\mathbf{t}}_{5,7}(f; \pm 13/22, 5\cdot7/44) &= -105512960, \\ {\mathbf{t}}_{5,7}(f; \pm 5/22, 5\cdot19/44) &= -17776273511920, \\ {\mathbf{t}}_{5,7}(f; \pm 14/22, 5\cdot24/44) &= 789839951523840, \\ {\mathbf{t}}_{5,7}(f; \pm 4/22, 5\cdot28/44) &= 12446972332605440, \\ {\mathbf{t}}_{5,7}(f; \pm 12/22, 5\cdot32/44) &= 162066199437803520, \\ {\mathbf{t}}_{5,7}(f; \pm 3/22, 5\cdot35/44) &= -1001261756125748754,\\ {\mathbf{t}}_{5,7}(f; \pm 7/22, 5\cdot39/44) &= -10093084485445877760. \end{aligned}$$ It is quite amusing to explicitly construct the modular form corresponding to our theorem, similar to the forms $g_D$ given by Zagier and the Jacobi forms in §8 of [@Zagier]. The space $M_{3/2,\rho}^!$ is isomorphic to the space $J^!_{2,N}$ of weakly holomorphic Jacobi forms of weight $2$ and index $N=11$. Thus, we can as well construct it in the latter space. It is contained in the even part of the ring of weakly holomorphic Jacobi forms, which is the free polynomial algebra over $M_*^!({{\text {\rm SL}}}_2({\mathbb{Z}}))={\mathbb{C}}[E_4,E_6,\Delta(\tau)^{-1}]/(E_4^3-E_6^2 = 1728\Delta(\tau))$ in two generators $a \in \tilde{J}_{-2,1}, b \in \tilde{J}_{0,1}$. Here $E_4$ and $E_6$ are the normalized Eisenstein series of weight 4 and 6 for ${{\text {\rm SL}}}_2({\mathbb{Z}})$. We refer to [@EZ], Chapter 9 and [@Zagier], §8 for details. The Fourier developments of $a$ and $b$ begin $$\begin{aligned} a(\tau,z) &= (\zeta^{-1} -2 +\zeta)+(-2 \zeta^{-2} + 8 \zeta^{-1} - 12 +8 \zeta^{2}-2\zeta^{2})q + \ldots, \\ b(\tau,z) &= (\zeta^{-1} + 10 + \zeta)+(10 \zeta^{-2}-64\zeta^{-1}+108-64\zeta+10\zeta^{2})q + \ldots, \end{aligned}$$ where $\zeta=e(z)$ and $q=e(\tau)$, as usual (we slightly abuse notation by now using $z \in {\mathbb{C}}$ as the elliptic variable for Jacobi forms). For $\Delta=1$ we thus obtain by Theorem \[thm:main\] a weakly holomorphic Jacobi form $\phi^{(11)}_{1}(f;\tau,z)$ having the traces of $f$ as its Fourier coefficients. The Fourier expansion begins $$\begin{gathered} -\frac{1}{2}\phi^{(11)}_{1}(f;\tau,z) = (11\zeta^{-11} + \zeta^{-1} - 24 + \zeta + 11\zeta^{11})\\ + (-7256\zeta^{-6} + 885480\zeta^{-5} - 16576512\zeta^{-4} + 117966288\zeta^{-3} \\ - 425691312\zeta^{-2} + 884736744\zeta^{-1} - 1122626864 + \ldots)q + \ldots. \end{gathered}$$ The twisted traces for $\Delta=5$ as above are the coefficients of $\phi^{(11)}_{5}(f;\tau,z)$ given by $$\begin{gathered} -\frac{1}{2}\phi^{(11)}_{5}(f;\tau,z) = (11\zeta^{-11} + 11\zeta^{11})q^{-11} + (\zeta^{-7} - 190356480\zeta^{-6} + 8888136755960\zeta^{-5} \\ - 6223486166302720\zeta^{-4} + 500630878062874377\zeta^{-3} - 8824913060318164992\zeta^{-2} \\ + 45310559791371053140\zeta^{-1} - 77176788074781143040 + \ldots)q + \ldots. \end{gathered}$$ It can be obtained as $\sum_{j=0}^{11} f_j a^j b^{11-j}$, where for each $j$ the function $f_j \in M^!_{2j+2}$ has a principal part starting with $a_j(-11)q^{-11}$. The Fourier expansions of these forms and their developments in terms of $E_4$, $E_6$ and $\Delta(\tau)^{-1}$, as well as some more numerical examples, can be downloaded from the second author’s homepage[^1]. Finally, we consider the more general situation when $\Delta > 1$ is a fundamental discriminant, $N=p$ a prime and $f(z)=j(pz)$. By Theorem \[thm:main\] together with Proposition \[prop:tf-\] the corresponding Jacobi form $\phi^{(p)}_{\Delta}(f;\tau,z)=\sum_{n,r}c(4pn-r^2)q^n\zeta^r$ has the property that the coefficients only depend on the discriminant $4pn-r^2$ and all coefficients of negative discriminant vanish except for $c(-\Delta)=-2$ and $c(-p^2\Delta)=-2p$. [^1]: <http://www.mathematik.tu-darmstadt.de/~ehlen>

ArXiv

--- abstract: 'In this paper, we address the problem of dense 3D reconstruction from multiple view images subject to strong lighting variations. In this regard, a new piecewise framework is proposed to explicitly take into account the change of illumination across several wide-baseline images. Unlike multi-view stereo and multi-view photometric stereo methods, this pipeline deals with wide-baseline images that are uncalibrated, in terms of both camera parameters and lighting conditions. Such a scenario is meant to avoid use of any specific imaging setup and provide a tool for normal users without any expertise. To the best of our knowledge, this paper presents the first work that deals with such unconstrained setting. We propose a coarse-to-fine approach, in which a coarse mesh is first created using a set of geometric constraints and, then, fine details are recovered by exploiting photometric properties of the scene. Augmenting the fine details on the coarse mesh is done via a final optimization step. Note that the method does not provide a generic solution for multi-view photometric stereo problem but it relaxes several common assumptions of this problem. The approach scales very well in size given its piecewise nature, dealing with large scale optimization and with severe missing data. Experiments on a benchmark dataset *Robot data-set* show the method performance against 3D ground truth.' author: - | Reza Sabzevari$^1$, Vittori Murino$^2$, and Alessio Del Bue$^2$\ \ $^1$ Robotics and Perception Group, University of Zurich, Switzerland\ $^2$ Pattern Analysis and Computer Vision (PAVIS),\ Italian Institute of Technology, Genova, Italy bibliography: - 'pimps\_ref.bib' title: 'PiMPeR: Piecewise Dense 3D Reconstruction from Multi-View and Multi-Illumination Images' --- Conclusions {#sec:conclusions} =========== We have presented a novel photo-geometric method for dense reconstruction from multi-view with arbitrary lighting condition. The approach is able to cope with wide-baselines images from uncalibrated cameras and explicitly utilizes varying lighting directions in the image sequence. This means that, only a few images taken by the end user are enough to be able to recover the dense 3D surfaces. The piecewise approach is highly scalable since solving for image patches is computationally easier than considering whole images. This enables the pipeline to run on commodity PCs. Future work will be dedicated to studying approaches that can partition the image into a mesh while taking into account the photometric and geometric properties of the shape. For instance, the method in [@Sunkavalli:etal:2010] could be used to partition the image in more consistent patches allocated to different subspaces. In addition, more complex photometric models could be used to extract more realistic photometric attributes for the surface. Acknowledgement {#acknowledgement .unnumbered} =============== This work is supported by the Doctoral fellowship awarded by the Italian Government and the Italian Institute of Technology. The authors would like to thank Prof. Davide Scaramuzza for inspiring discussions and his valuable comments.

ArXiv

--- abstract: 'Calculations are performed for energies of isobaric analog states with isospins T=2 and T=3 in regions where they have been found experimentally e.g. f-p shell, and regions where they have not yet been found e.g. g$_{9/2}$ near Z=50,N=50. We consider two approaches–one using binding energy formulas and Coulomb energies contained therein and the other using shell model calculations. It is noted that some (but not all) calculations yield very low excitation energies for the J=0$^{+}$T=2 isobaric analog state in $^{96}$Ag.' author: - 'L. Zamick' - 'A. Escuderos' - 'S.J.Q. Robinson' - 'Y. Y. Sharon' - 'M.W. Kirson' title: 'Isobaric analog state in the f$_{7/2}$ and g$_{9/2}$ shells' --- If there were no violation of charge independence, the binding energy of the $^{96}$Pd ground state ($J=0^{+}$, $T=2$) would be identical to the binding energy of the analog state, also $J=0^{+}$, $T=2$, in $^{96}$Ag. But, since that is not the case in real life, the excitation energy of the $J=0^{+}$, $T=2$ state in $^{96}$Ag is given by $$E^{*}(J=0^{+},T=2)=BE(^{96}\text{Ag})-BE(^{96}\text{Pd})+V_{C}\,,\label{eq:exc}$$ where the $BE$s are the binding energies and $V_{C}$ includes all charge-independence violating effects. The binding energies can be obtained from the latest mass evaluation [@am11] and we assume that $V_{C}$ arises from the Coulomb interaction, which must be estimated. We use the classical form of the Coulomb energy $$E_{C}=\alpha_{C}Z^{2}/A^{1/3}\,,\label{eq:ecd}$$ supplemented by an exchange Coulomb term $$E_{xC}=\alpha_{xC}Z^{4/3}/A^{1/3}\,,\label{eq:ecx}$$ where $\alpha_{C}$ and $\alpha_{xC}$ are coefficients to be obtained from appropriate data. Several sources were compared. The simplest is the Bethe-Weizsäcker semi-empirical mass formula [@key-3; @mwk], which produces $\alpha_{C}=0.691$ MeV, $\alpha_{xC}=0$ from a fit of a four-term semi-empirical mass formula to the measured masses. An extended, ten-term mass formula [@mwk] produces $\alpha_{C}=0.774$ MeV and $\alpha_{xC}=-2.22$ MeV from a similar fit. The best mass formulation currently available is the Duflo-Zuker approach [@key-4; @mwdz] with up to 33 parameters fitted to the mass data. It includes a unified Coulomb term $$E_{C}^{DZ}=\alpha_{C}\frac{Z(Z-1)-0.76[Z(Z-1)]^{2/3}}{A^{1/3}\left[1-\frac{(N-Z)^{2}}{4A^{2}}\right]}\label{eq:dzc}$$ and the best fits to the data have $\alpha_{C}=0.700$ MeV. Binding energy differences of mirror nuclei, together with Coulomb displacement energies, can be fitted to differences of $E_{C}$ and $E_{xC}$ (eqs.(\[eq:ecd\]),(\[eq:ecx\])), from which $\alpha_{C}=0.717$ MeV and $\alpha_{xC}=-0.928$ MeV [@mwk]. The formula of Anderson et al. [@awm65]: $$V_{C}=E_{1}\overline{Z}/A^{1/3}+E_{2}\,,\label{eq:vc}$$ where $\overline{Z}=(Z_{1}+Z_{2})/2$, is a semi-empirical representation of the same data, as far as it was known at the time. Anderson et al. [@awm65] list several sets of values of $E_{1}$ and $E_{2}$. We here use the average values $E_{1}=1.441$ MeV and $E_{2}=-1.06$ MeV. Table \[tab:coule\] compares the Coulomb energy estimates, using the different prescriptions presented above, for a number of nuclei of interest for this discussion of analog state excitation energy. Though estimates of the total Coulomb energy can vary strongly between prescriptions, the differences which are relevant to the analog states show much less variability. In particular, estimates which are based on fits to mirror nuclei and Coulomb displacement energies agree very closely among themselves. The Anderson et al fit has stood the test of time remarkably well. Nucleus Bethe-Weizsäcker Ten-term Duflo-Zuker Mirror/CDE Anderson et al ----------- ------------------ ---------- ------------- ------------ ---------------- $^{44}$Sc 86.318 60.253 74.872 74.336 $^{44}$Ca 78.293 53.558 67.586 66.968 $A=44$ 8.025 6.694 7.285 7.368 7.308 $^{46}$Sc 85.048 59.366 73.872 73.242 $^{46}$Ca 77.141 52.771 66.738 65.983 $A=46$ 7.907 6.596 7.134 7.259 7.185 $^{52}$Mn 115.706 86.126 102.432 101.885 $^{52}$Cr 106.635 78.268 94.079 93.435 $A=52$ 9.071 7.858 8.353 8.450 8.399 $^{60}$Cu 148.442 115.748 133.491 132.907 $^{60}$Ni 138.381 106.788 124.048 123.433 $A=60$ 10.061 8.960 9.362 9.474 9.430 $^{94}$Rh 307.747 266.563 286.532 286.658 $^{94}$Ru 294.221 253.719 273.683 273.588 $A=94$ 13.526 12.843 12.849 13.070 13.043 $^{96}$Ag 333.362 291.169 311.153 311.530 $^{96}$Pd 319.328 277.773 297.738 297.939 $A=96$ 14.035 13.396 13.415 13.591 13.574 : \[tab:coule\]Coulomb energy estimates for some nuclei, in MeV. Lines labeled $A=$ give the differences of the two preceding lines. With relatively stable Coulomb energy differences in hand and with experimental binding energies, we are able to compute, using eq.(\[eq:exc\]), predicted excitation energies of analog states, and can compare the results with measured excitation energies, where they exist. We show in Table \[tab:exc\] results for various nuclei, some for which the excitation energy of the analog state is known and some for which it is not. The binding energy differences are taken from Ref. [@am11], the Coulomb energy differences from the Anderson et al semi-empirical fit [@awm65]. NUCLEUS Binding Energy Difference Coulomb Energy Excitation Energy Single $j$ Large space Experiment ----------- --------------------------- ---------------- ------------------- ------------ ------------- ------------ $^{44}$Sc 4.435 7.308 2.873 3.047 3.418 2.779 $^{46}$Sc 2.160 7.185 5.024 4.949 5.250 5.022 $^{52}$Mn 5.494 8.399 2.905 2.774 2.7307 2.926 $^{60}$Cu 6.910 9.430 2.520 2.235 2.726 2.536 $^{94}$Rh 10.458 13.043 2.585 1.990 3.266 2.048 2.879 $^{96}$Ag 12.453 13.574 1.121 0.900 1.9167 0.842 1.640 : \[tab:exc\]Excitation energies of isobaric analog states in MeV. In all four cases where the excitation energy of the analog state is known, our prediction agrees with the experimental value within 100 keV, and for three of them, within 25 keV. The fact that the analog state and Coulomb arguments work well in known cases gives us confidence that we can use these for the unknown case of $^{96}$Ag, where we predict an excitation energy just slightly above 1 MeV. Turning things around, if the isobaric analog state were found, then we might have a better constraint on what the binding energy is. We can compare our predicted excitation energies with selected calculations in the literature (included in Table \[tab:exc\]). We look at shell-model calculations of two basic kinds — single-$j$ or large–space — and with various effective interactions. For $^{44}$Sc and $^{46}$Sc, single-$j$-shell results ($f_{7/2}$) [@ezb05] are respectively 3.047 and 4.949 MeV, as compared with Table \[tab:exc\]’s excitation energies of 2.873 and 5.024 MeV. The large space results are also shown. In$^{52}$Mn there is reasonable agreement between predicted, single-$j$, large space and experiment. In the small space for $^{60}$Cu (p$_{3/2}$) we can use a particle-hole transformation to get the spectrum of this nucleus from the spectrum of $^{58}$Cu since three p$_{3/2}$ neutrons can be regarded as a single neutron hole. This gives a value of 2.235 MeV as compared with the experimental value of 2.536 MeV. For $^{96}$Ag single-$j$-shell results [@ze12] are 0.900 MeV with INTd and 0.842 MeV with the CCGI interaction [@ze12; @ccgi12]. These are lower than the excitation energy in Table \[tab:exc\] of 1.121 MeV. There are also large scale calculations with the jj44b [@bl-un] interaction for $^{96}$Ag—the result is 1.917 MeV, significantly larger than the predicted value. In $^{94}$Rh the jj44b interaction yields 3.266 MeV, larger than Table \[tab:exc\]’s predicted value of 2.585 MeV. The large space calculations with JUN45 are qualitatively similar.The single-$j$ INTd and CCGI results are lower, 1.990 MeV and 2.048 MeV respectively. Although it is clearly preferable to base predictions of the excitation energy of the analog state on experimentally measured binding energies, it may become necessary to use binding energies derived from mass formulas where data is unavailable. To this end, we check how susceptible these predictions are to various mass formulas. We tested the mass formulas used above to obtain Coulomb energy differences — a 5–term Bethe-Weizsäcker formula (the standard four terms, supplemented with a pairing term), its 10–term extension, and the 33-parameter Duflo–Zuker mass formulation. The results for the analog states are presented in Table \[tab:msfrm\]. In all cases, the Coulomb energy differences were obtained from the respective binding energy formulas. ----------- ---------------- --------------- ---------------- --------------- ---------------- --------------- NUCLEUS 5-term 5-term 10-term 10-term Duflo-Zuker Duflo-Zuker Binding Energy Analog Energy Binding Energy Analog Energy Binding Energy Analog Energy $^{44}$Sc 5.499 2.526 4.768 1.926 4.934 2.351 $^{46}$Sc 1.650 6.257 2.035 4.624 2.440 4.694 $^{52}$Mn 7.200 1.871 6.460 1.398 6.472 1.881 $^{60}$Cu 8.653 1.408 7.968 0.992 7.015 2.347 $^{94}$Rh 11.208 2.318 11.361 1.482 10.837 2.012 $^{96}$Ag 13.668 0.367 13.453 -0.057 12.885 0.530 ----------- ---------------- --------------- ---------------- --------------- ---------------- --------------- : \[tab:msfrm\]Excitation energies of isobaric analog states (in MeV) based on mass formulas, for comparison with predicted analog state energy in Table \[tab:exc\]. The Duflo-Zuker results are closet to the predictions based on the atomic mass evaluation (fourth column of Table \[tab:exc\]),.As might have been expected, mass formulas with smaller rms deviations from the measured data are better predictors of the analog state excitation energy. Even so the calculated value for $^{52}$Mn (1.881 MeV) is considerably lower than the experimental value (2.926 MeV). Why study isobaric analog states? One reason has to do with the strange dualism in nuclear structure that has emerged over the years. For the most part calculations of the excited states of nuclei have been performed with little mind to the binding energies or saturation properties. On the other hand binding energies and nuclear densities are addressed in Hartree-Fock calculations with interactions for which it makes no sense to calculate nuclear spectra. With isobaric analog state energies we have an in your face confrontation of these two approaches. As shown in Eq. (\[eq:exc\]) one needs good binding energies and good Coulomb energies to correctly predict the excitation energies of these states. As has been noted in ref [@ze12] with the shell model one can get very impressive fits for many energy levels in say $^{96}$Ag but no one has even tried to calculate the energy of the isobaric analog state here until now. Hopefully our work will stimulate trying to get a unified approach in which both the spectra and bulk properties of nuclei are treated in a unified manner. Another point of interest is the possibility that in some region of the periodic table the T=2 isobaric analog state would become the ground state. In Table \[tab:msfrm\] the 10 point formula yields such a result. If this were indeed the case there would be a drastic difference in the decay mode of the nucleus in question. Instead of the usual decay mode – electron capture – one would now have an allowed Fermi transition. This would lead to a much shorter lifetime and could influence how the elements evolve. This does not occur in $^{96}$Ag where it is known that the decay mode is electron capture but it might occur in heavier nuclei. For example Z=66, A=132 closes the h$_{11/2}$ shell. Consider a very proton rich nucleus with A=128 and Z=63, a nucleus with 3proton holes and one neutron hole. The excttation energies for 5- term, 10-term and Duflo-Zuker are respectively -0.009, 1.251 and 0.366 MeV. Before leaving we would briefly like to defend out inclusion of small space calculations in Table \[tab:exc\]. There is a precedent for this in the work of Talmi and collaborators [@key-1]. They obtain excellent agreement with binding energies in several regions, e.g. the Ca isotopes, using a single j-shell formula but with phenomenological parameters.. We here adopt the same philosophy of obtaining two body matrix elements from experiment. Matrix elements form experiment implicitly contain not all but many nuclear correlations. In view of the differing results of shell model calculations and mass formulas it would be of great interest to measure the excitation energies of isobaric analog states in the $g_{9/2}$ region. We hope that this work will encourage experimentalists to look not only for the surprisingly neglected $J=0^{+}$ isobaric analog states in $^{94}$Rh and $^{96}$Ag, but also for other such states throughout this region. We thank Klaus Blaum for his help. [References]{} G. Audi, F.G.Kondev, M.Wang, B.Pfeiffer, X.Sun, J.Blanchot,and M. MacCormick, Chinese Physics (HEP and NP), **36**(12), 1157 (2012). H.A. Bethe and R.F. Bacher, Rev. Mod. Phys.**82**, 8 (1936). M.W.Kirson, Nucl.Phys. A **798**, 29 (2008). J.Duflo,A.P. Zuker ,Phys. Rev C **52**, R23 (1995). M.W.Kirson, Nucl.Phys. A **893**, 27 (2012). J.D. Anderson, C. Wong, and J.W. McClure, Phys. Rev. **138**, B615 (1965). A. Escuderos, L. Zamick, and B.F. Bayman, *Wave functions in the $f_{7/2}$ shell, for educational purposes and ideas*, <http://arxiv.org/abs/nucl-th/0506050> (2005). M. Honma, T. Otsuka, B.A. Brown, and T. Mizusaki, Phys. Rev. C **69**, 034335 (2004). L. Zamick and A. Escuderos, Nucl. Phys. A **889**, 8 (2012). B.A. Brown, A.F. Lisetskiy, unpublished. The jj44b Hamiltonian was obtained from a fit to about 600 binding energies and excitation energies with a method similar to that used for the JUN45 Hamiltonian [@cetal10] L. Coraggio, A. Covello, A. Gargano, and N. Itaco, Phys. Rev. C **85**, 034335 (2012). M. Honma, T. Otsuka, T. Mizusaki and M. Hjorth-Jensen, Phys Rev C **80**, 064323 (2009). B. Cheal, et al., Phys. Rev. Lett. **104**, 252502 (2010). I. Talmi, Simple Models of Complex Nuclei, Harwood Academic Publishers (New York, 1993).

ArXiv

--- abstract: 'This is the first paper in a series. We develop a general deformation theory of objects in homotopy and derived categories of DG categories. Namely, for a DG module $E$ over a DG category we define four deformation functors ${\operatorname{Def}}^{\h}(E)$, ${\operatorname{coDef}}^{\h}(E)$, ${\operatorname{Def}}(E)$, ${\operatorname{coDef}}(E)$. The first two functors describe the deformations (and co-deformations) of $E$ in the homotopy category, and the last two - in the derived category. We study their properties and relations. These functors are defined on the category of artinian (not necessarily commutative) DG algebras.' address: - 'Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia and Independent Univeroty of Moscow' - 'Department of Mathematics, Indiana University, Bloomington, IN 47405, USA' - 'Algebra Section, Steklov Mathematical Institute, 8 Gubkina str., Moscow, 119991 Russia' author: - 'Alexander I. Efimov' - 'Valery A. Lunts' - 'Dmitri O. Orlov' title: 'Deformation theory of objects in homotopy and derived categories I: general theory' --- Introduction ============ It is well known (see for example [@De1], [@De2], [@Dr2], [@Ge1], [@Ge2], [@Hi]) that for many mathematical objects $X$ (defined over a field of characteristic zero) the formal deformation theory of $X$ is controlled by a DG Lie algebra $\mathfrak{g}=\mathfrak{g}(X)$ of (derived) infinitesimal automorphisms of $X$. This is so in case $X$ is an algebra, a compact complex manifold, a principal $G$-bundle, etc.. Let ${{\mathcal M}}(X)$ denote the base of the universal deformation of $X$ and $o\in {{\mathcal M}}(X)$ be the point corresponding to $X$. Then (under some conditions on $\mathfrak{g}$) the completion of the local ring $\hat{{{\mathcal O}}}_{{{\mathcal M}}(X),o}$ is naturally isomorphic to the linear dual of the homology space $H_0(\mathfrak{g})$. The space $H_0(\mathfrak{g})$ is a co-commutative coalgebra, hence its dual is a commutative algebra. The homology $H_0(\mathfrak{g})$ is the zero cohomology group of $B\mathfrak{g}$ – the bar construction of $\mathfrak{g}$, which is a co-commutative DG coalgebra. It is therefore natural to consider the DG “formal moduli space” ${{\mathcal M}}^{DG}(X)$, so that the corresponding completion $\hat{{{\mathcal O}}}_{{{\mathcal M}}^{DG}(X),o}$ of the “local ring” is the linear dual $(B\mathfrak{g})^*$, which is a commutative DG algebra. The space ${{\mathcal M}}^{DG}(X)$ is thus the “true” universal deformation space of $X$; it coincides with ${{\mathcal M}}(X)$ in case $H^i(B\mathfrak{g})=0$ for $i\neq 0$. In particular, it appears that the primary object is not the DG algebra $(B\mathfrak{g})^*$, but rather the DG coalgebra $B\mathfrak{g}$ (this is the point of view in [@Hi]). In any case, the corresponding deformation functor is naturally defined on the category of commutative artinian DG algebras (see [@Hi]). Note that the passage from a DG Lie algebra $\mathfrak{g}$ to the commutative DG algebra $(B\mathfrak{g})^*$ is an example of the Koszul duality for operads [@GiKa]. Indeed, the operad of DG Lie algebras is Koszul dual to that of commutative DG algebras. Some examples of DG algebraic geometry are discussed in [@Ka], [@Ci-FoKa1], [@Ci-FoKa2]. This paper (and the following papers [@LOII], [@LOIII]) is concerned with a general deformation theory in a slightly different context. Namely, we consider deformations of “linear” objects $E$, such as objects in a homotopy or a derived category. More precisely, $E$ is a right DG module over a DG category ${{\mathcal A}}$. In this case the deformation theory of $E$ is controlled by ${{\mathcal B}}=\End (E)$ which is a DG [*algebra*]{} (and not a DG Lie algebra). (This works equally well in positive characteristic.) Then the DG formal deformation space of $E$ is the “Spec” of the (noncommutative!) DG algebra $(B{{\mathcal B}})^*$ – the linear dual of the bar construction $B{{\mathcal B}}$ which is a DG coalgebra. Again this is in agreement with the Koszul duality for operads, since the operad of DG algebras is self-dual. (All this was already anticipated in [@Dr2].) More precisely, let ${\operatorname{dgart}}$ be the category of local artinian (not necessarily commutative) DG algebras and $\bf{Gpd}$ be the 2-category of groupoids. For a right DG module $E$ over a DG category ${{\mathcal A}}$ we define four pseudo-functors $${\operatorname{Def}}^{\h}(E), {\operatorname{coDef}}^{\h}(E), {\operatorname{Def}}(E), {\operatorname{coDef}}(E):{\operatorname{dgart}}\to {\bf Gpd}.$$ The first two are the [*homotopy*]{} deformation and co-deformation pseudo-functors, i.e. they describe deformations (and co-deformations) of $E$ in the homotopy category of DG ${{\mathcal A}}^{op}$-modules; and the last two are their [*derived*]{} analogues. We prove that the pseudo-functors ${\operatorname{Def}}^{\h}(E)$, ${\operatorname{coDef}}^{\h}(E)$ are equivalent and depend only on the quasi-isomorphism class of the DG algebra $\End (E)$. The derived pseudo-functors ${\operatorname{Def}}(E)$, ${\operatorname{coDef}}(E)$ need some boundedness conditions to give the “right” answer and in that case they are equivalent to ${\operatorname{Def}}^{\h}(F)$ and $ {\operatorname{coDef}}^{\h}(F)$ respectively for an appropriately chosen h-projective or h-injective DG module $F$ which is quasi-isomorphic to $E$ (one also needs to restrict the pseudo-functors to the category ${\operatorname{dgart}}_-$ of negative artinian DG algebras). This first paper is devoted to the study of general properties of the above four pseudo-functors and relations between them. Part 1 of the paper is a rather lengthy review of basics of DG categories and DG modules over them with some minor additions that we did not find in the literature. The reader who is familiar with basic DG categories is suggested to go directly to Part 2, except for looking up the definition of the DG functors $i^*$ and $i^!$. In the second paper [@LOII] we study the pro-representability of these pseudo-functors. Recall that “classically” one defines representability only for functors with values in the category of sets (since the collection of morphisms between two objects in a category is a set). For example, given a moduli problem in the form of a pseudo-functor with values in the 2-category of groupoids one then composes it with the functor $\pi _0$ to get a set valued functor, which one then tries to (pro-) represent. This is certainly a loss of information. But in order to represent the original pseudo-functor one needs the source category to be a bicategory. It turns out that there is a natural bicategory $2\text{-}{\operatorname{adgalg}}$ of augmented DG algebras. (Actually we consider two versions of this bicategory, $2\text{-}{\operatorname{adgalg}}$ and $2^\prime\text{-}{\operatorname{adgalg}}$, but then show that they are equivalent). We consider its full subcategory $2\text{-}{\operatorname{dgart}}_-$ whose objects are negative artinian DG algebras, and show that the derived deformation functors can be naturally extended to pseudo-functors $${\operatorname{coDEF}}_-(E):2\text{-}{\operatorname{dgart}}_- \to {\bf Gpd},\quad {\operatorname{DEF}}_-(E):2^\prime\text{-}{\operatorname{dgart}}_- \to {\bf Gpd}.$$ Then (under some finiteness conditions on the cohomology algebra $H({{\mathcal C}})$ of the DG algebra ${{\mathcal C}}=\bR \Hom (E,E)$) we prove pro-representability of these pseudo-functors by some local complete DG algebra described by means of $A_{\infty}$-structure on $H({{\mathcal C}})$. This pro-representability appears to be more “natural” for the pseudo-functor ${\operatorname{coDEF}}_-$, because there exists a “universal co-deformation” of the DG ${{\mathcal C}}^{op}$-module ${{\mathcal C}}$. The pro-representability of the pseudo-functor ${\operatorname{DEF}}_-$ may then be formally deduced from that of ${\operatorname{coDEF}}_-$. In the third paper [@LOIII] we show how to apply our deformation theory of DG modules to deformations of complexes over abelian categories. We also discuss examples from algebraic geometry. We note that the noncommutative deformations (i.e. over noncommutative artinian rings) of modules were already considered by Laudal in [@La]. The basic difference between our work and [@La] (besides the fact that our noncommutative artinian algebras are DG algebras) is that we work in the derived context. That is we only deform the differential in a suitably chosen complex and keep the module structure constant. It is our pleasure to thank A.Bondal, P.Deligne, M.Mandell, M.Larsen and P.Bressler for useful discussions. We especially appreciate the generous help of B.Keller. We also thank W.Goldman and V.Schechtman for sending us copies of letters [@De1] and [@Dr2] respectively and W.Lowen for sending us the preprint [@Lo]. We also thank J.Stasheff for his useful comments on the first version of this paper. Artinian DG algebras ==================== We fix a field $k$. All algebras are assumed to be ${{\mathbb Z}}$ graded $k$-algebras with unit and all categories are $k$-linear. Unless mentioned otherwise $\otimes $ means $\otimes _k$. For a homogeneous element $a$ we denote its degree by $\bar{a}$. A [*module*]{} always means a (left) graded module. A DG algebra ${{\mathcal B}}=({{\mathcal B}},d_{{{\mathcal B}}})$ is a (graded) algebra with a map $d=d_{{{\mathcal B}}}:{{\mathcal B}}\to {{\mathcal B}}$ of degree 1 such that $d^2=0$, $d(1)=0$ and $$d(ab)=d(a)b+(-1)^{\bar{a}}ad(b).$$ Given a DG algebra ${{\mathcal B}}$ its opposite is the DG algebra ${{\mathcal B}}^{op}$ which has the same differential as ${{\mathcal B}}$ and multiplication $$a\cdot b=(-1)^{\bar{a}\bar{b}}ba,$$ where $ba$ is the product in ${{\mathcal B}}$. When there is a danger of confusion of the opposite DG algebra ${{\mathcal B}}^{op}$ with the degree zero part of ${{\mathcal B}}$ we will add a comment. We denote by ${\operatorname{dgalg}}$ the category of DG algebras. A (left) DG module over a DG algebra ${{\mathcal B}}$ is called a DG ${{\mathcal B}}$-module or, simply a ${{\mathcal B}}$-module. A [*right*]{} ${{\mathcal B}}$-module is a DG module over ${{\mathcal B}}^{op}$. If ${{\mathcal B}}$ is a DG algebra and $M$ is a usual (not DG) module over the algebra ${{\mathcal B}}$, then we say that $M^{\gr}$ is a ${{\mathcal B}}^{\gr}$-module. An [*augmentation*]{} of a DG algebra ${{\mathcal B}}$ is a (surjective) homomorphism of DG algebras ${{\mathcal B}}\to k$. Its kernel is a DG ideal (i.e. an ideal closed under the differential) of ${{\mathcal B}}$. Denote by ${\operatorname{adgalg}}$ the category of augmented DG algebras (morphisms commute with the augmentation). Let $R$ be an algebra. We call $R$ [*artinian*]{}, if it is finite dimensional and has a (graded) nilpotent two-sided (maximal) ideal $m\subset R$, such that $R/m=k$. Let ${{\mathcal R}}$ be an augmented DG algebra. We call ${{\mathcal R}}$ [*artinian*]{} if ${{\mathcal R}}$ is artinian as an algebra and the maximal ideal $m\subset R$ is a DG ideal, i.e. the quotient map $R\to R/m$ is an augmentation of the DG algebra ${{\mathcal R}}$. Note that a homomorphism of artinian DG algebras automatically commutes with the augmentations. Denote by ${\operatorname{dgart}}$ the category of artinian DG algebras. An artinian DG algebra ${{\mathcal R}}$ is called positive (resp. negative) if negative (resp. positive) degree components of ${{\mathcal R}}$ are zero. Denote by ${\operatorname{dgart}}_+$ and ${\operatorname{dgart}}_-$ the corresponding full subcategories of ${\operatorname{dgart}}$. Let ${\operatorname{art}}:={\operatorname{dgart}}_-\cap {\operatorname{dgart}}_+$ be the full subcategory of ${\operatorname{dgart}}$ consisting of (not necessarily commutative) artinian algebras concentrated in degree zero. Denote by ${\operatorname{cart}}\subset {\operatorname{art}}$ the full subcategory of commutative artinian algebras. Given a DG algebra ${{\mathcal B}}$ one studies the category ${{\mathcal B}}\text{-mod}$ and the corresponding homotopy and derived categories. A homomorphism of DG algebras induces various functors between these categories. We will recall these categories and functors in the more general context of DG categories in the next section. DG categories ============= In this section we recall some basic facts about DG categories which will be needed in this paper. Our main references here are [@BoKa], [@Dr], [@Ke]. A DG category is a $k$-linear category ${{\mathcal A}}$ in which the sets $\Hom (A,B)$, $A,B\in Ob{{\mathcal A}}$, are provided with a structure of a ${{\mathbb Z}}$-graded $k$-module and a differential $d:\Hom(A,B)\to \Hom (A,B)$ of degree 1, so that for every $A,B,C\in {{\mathcal A}}$ the composition $\Hom (A,B)\times\Hom (B,C)\to \Hom (A,C)$ comes from a morphism of complexes $\Hom (A,B)\otimes \Hom (B,C)\to \Hom (A,C)$. The identity morphism $1_A\in \Hom (A,A)$ is closed of degree zero. The simplest example of a DG category is the category $DG(k)$ of complexes of $k$-vector spaces, or DG $k$-modules. Note also that a DG algebra is simply a DG category with one object. Using the supercommutativity isomorphism $S\otimes T\simeq T\otimes S$ in the category of DG $k$-modules one defines for every DG category ${{\mathcal A}}$ the opposite DG category ${{\mathcal A}}^{op}$ with $Ob{{\mathcal A}}^{op}=Ob{{\mathcal A}}$, $\Hom_{{{\mathcal A}}^{op}}(A,B)=\Hom _{{{\mathcal A}}}(B,A)$. We denote by ${{\mathcal A}}^{\gr}$ the [*graded* ]{} category which is obtained from ${{\mathcal A}}$ by forgetting the differentials on $\Hom $’s. The tensor product of DG-categories ${{\mathcal A}}$ and ${{\mathcal B}}$ is defined as follows: \(i) $Ob({{\mathcal A}}\otimes {{\mathcal B}}):=Ob{{\mathcal A}}\times Ob{{\mathcal B}}$; for $A\in Ob{{\mathcal A}}$ and $B\in Ob{{\mathcal B}}$ the corresponding object is denoted by $A\otimes B$; \(ii) $\Hom(A\otimes B,A^\prime \otimes B^\prime):=\Hom (A,A^\prime)\otimes \Hom (B,B^\prime)$ and the composition map is defined by $(f_1\otimes g_1)(f_2\otimes g_2):= (-1)^{\bar{g_1}\bar{f_2}}f_1f_2\otimes g_1g_2.$ Note that the DG categories ${{\mathcal A}}\otimes {{\mathcal B}}$ and ${{\mathcal B}}\otimes {{\mathcal A}}$ are canonically isomorphic. In the above notation the isomorphism DG functor $\phi$ is $$\phi (A\otimes B)=(B\otimes A), \quad \phi(f\otimes g)=(-1)^{\bar{f}\bar{g}}(g\otimes f).$$ Given a DG category ${{\mathcal A}}$ one defines the graded category ${\operatorname{Ho}}^\bullet ({{\mathcal A}})$ with $Ob{\operatorname{Ho}}^\bullet ({{\mathcal A}})=Ob{{\mathcal A}}$ by replacing each $\Hom$ complex by the direct sum of its cohomology groups. We call ${\operatorname{Ho}}^\bullet ({{\mathcal A}})$ the [*graded homotopy category*]{} of ${{\mathcal A}}$. Restricting ourselves to the 0-th cohomology of the $\Hom $ complexes we get the [*homotopy category*]{} ${\operatorname{Ho}}({{\mathcal A}})$. Two objects $A,B\in Ob{{\mathcal A}}$ are called DG [*isomorphic*]{} (or, simply, isomorphic) if there exists an invertible degree zero morphism $f\in \Hom(A,B)$. We say that $A,B$ are [*homotopy equivalent*]{} if they are isomorphic in ${\operatorname{Ho}}({{\mathcal A}})$. A DG-functor between DG-categories $F:{{\mathcal A}}\to {{\mathcal B}}$ is said to be a [*quasi-equivalence*]{} if ${\operatorname{Ho}}^\bullet(F):{\operatorname{Ho}}^\bullet({{\mathcal A}})\to {\operatorname{Ho}}^\bullet({{\mathcal B}})$ is an equivalence of graded categories. We say that $F$ is a DG [*equivalence*]{} if it is fully faithful and every object of ${{\mathcal B}}$ is DG isomorphic to an object of $F({{\mathcal A}})$. Certainly, a DG equivalence is a quasi-equivalence. DG categories ${{\mathcal C}}$ and ${{\mathcal D}}$ are called [*quasi-equivalent*]{} if there exist DG categories ${{\mathcal A}}_1,...,{{\mathcal A}}_n$ and a chain of quasi-equivalences $${{\mathcal C}}\leftarrow {{\mathcal A}}_1 \rightarrow ...\leftarrow {{\mathcal A}}_n \rightarrow {{\mathcal D}}.$$ Given DG categories ${{\mathcal A}}$ and ${{\mathcal B}}$ the collection of covariant DG functors ${{\mathcal A}}\to {{\mathcal B}}$ is itself the collection of objects of a DG category, which we denote by ${\operatorname{Fun}}_{{\operatorname{DG}}}({{\mathcal A}},{{\mathcal B}})$. Namely, let $\Phi $ and $\Psi$ be two DG functors. Put $\Hom ^k(\Phi ,\Psi)$ equal to the set of natural transformations $t:\Phi ^{\gr} \to \Psi ^{\gr}[k]$ of graded functors from ${{\mathcal A}}^{\gr}$ to ${{\mathcal B}}^{\gr}$. This means that for any morphism $f \in \Hom_{{{\mathcal A}}}^s(A,B)$ one has $$\Psi (f )\cdot t(A)=(-1)^{ks}t(B)\cdot \Phi (f).$$ On each $A\in {{\mathcal A}}$ the differential of the transformation $t$ is equal to $d(t(A))$ (one easily checks that this is well defined). Thus, the closed transformations of degree 0 are the DG transformations of DG functors. A similar definition gives us the DG-category consisting of the contravariant DG functors ${\operatorname{Fun}}_{{\operatorname{DG}}}({{\mathcal A}}^{op} ,{{\mathcal B}})={\operatorname{Fun}}_{{\operatorname{DG}}}({{\mathcal A}},{{\mathcal B}}^{op})$ from ${{\mathcal A}}$ to ${{\mathcal B}}$. DG modules over DG categories ----------------------------- We denote the DG category ${\operatorname{Fun}}_{{\operatorname{DG}}}({{\mathcal A}},DG(k))$ by ${{\mathcal A}}\text{-mod}$ and call it the category of DG ${{\mathcal A}}$-modules. There is a natural covariant DG functor $h:{{\mathcal A}}\to {{\mathcal A}}^{op}\text{-mod}$ (the Yoneda embedding) defined by $h^A(B):=\Hom _{{{\mathcal A}}}(B,A)$. As in the “classical” case one verifies that the functor $h$ is fully faithful, i.e. there is a natural isomorphism of complexes $$\Hom _{{{\mathcal A}}}(A,A^\prime)=\Hom_{{{\mathcal A}}^{op}\text{-mod}}(h^A,h^{A^\prime}).$$ Moreover, for any $M\in {{\mathcal A}}^{op}\text{-mod}$, $A\in {{\mathcal A}}$ $$\Hom _{{{\mathcal A}}^{op}\text{-mod}}(h^A,M)=M(A).$$ The DG ${{\mathcal A}}^{op}$-modules $h^A$, $A\in {{\mathcal A}}$ are called [*free*]{}. For $A\in {{\mathcal A}}$ one may consider also the covariant DG functor $h_A(B):=\Hom _{{{\mathcal A}}}(A,B)$ and the contravariant DG functor $h^*_A(B):=\Hom _k(h_A(B),k)$. For any $M\in {{\mathcal A}}^{op}\text{-mod}$ we have $$\Hom _{{{\mathcal A}}^{op}\text{-mod}}(M,h^*_A)=\Hom _k(M(A),k).$$ A DG ${{\mathcal A}}^{op}$-module $M$ is called acyclic, if the complex $M(A)$ is acyclic for all $A\in {{\mathcal A}}$. Let $D({{\mathcal A}}^{op})$ denote the [*derived category*]{} of DG ${{\mathcal A}}^{op}$-modules, i.e. $D({{\mathcal A}}^{op})$ is the Verdier quotient of the homotopy category ${\operatorname{Ho}}({{\mathcal A}}^{op}\text{-mod})$ by the subcategory of acyclic DG-modules. This is a triangulated category. A DG ${{\mathcal A}}^{op}$-module $P$ is called h-[*projective*]{} if for any acyclic DG ${{\mathcal A}}^{op}$-module $N$ the complex $\Hom (P,N)$ is acyclic. A free DG module is h-projective. Denote by $\P({{\mathcal A}}^{op})$ the full DG subcategory of ${{\mathcal A}}^{op}\text{-mod}$ consisting of h-projective DG modules. Similarly, a DG ${{\mathcal A}}^{op}$-module $I$ is called h-[*injective*]{} if for any acyclic DG ${{\mathcal A}}^{op}$-module $N$ the complex $\Hom (N,I)$ is acyclic. For any $A\in {{\mathcal A}}$ the DG ${{\mathcal A}}^{op}$-module $h^*_A$ is h-injective. Denote by ${{\mathcal I}}({{\mathcal A}}^{op})$ the full DG subcategory of ${{\mathcal A}}^{op}\text{-mod}$ consisting of h-injective DG modules. For any DG category ${{\mathcal A}}$ the DG categories ${{\mathcal A}}^{op}\text{-mod}$, $\P({{\mathcal A}}^{op})$, ${{\mathcal I}}({{\mathcal A}}^{op})$ are (strongly) pre-triangulated ([@Dr; @BoKa], also see subsection 3.5 below). Hence the homotopy categories ${\operatorname{Ho}}({{\mathcal A}}^{op}\text{-mod})$, ${\operatorname{Ho}}(\P({{\mathcal A}}^{op}))$, ${\operatorname{Ho}}({{\mathcal I}}({{\mathcal A}}^{op}))$ are triangulated. The following theorem was proved in [@Ke]. The inclusion functors $\P({{\mathcal A}}^{op} )\hookrightarrow {{\mathcal A}}^{op}\text{-mod}$, ${{\mathcal I}}({{\mathcal A}}^{op})\hookrightarrow {{\mathcal A}}^{op}\text{-mod}$ induce equivalences of triangulated categories ${\operatorname{Ho}}(\P({{\mathcal A}}^{op}))\simeq D({{\mathcal A}}^{op})$ and ${\operatorname{Ho}}({{\mathcal I}}({{\mathcal A}}^{op}))\simeq D({{\mathcal A}}^{op})$. Actually, it will be convenient for us to use some more precise results from [@Ke]. Let us recall the relevant definitions. A DG ${{\mathcal A}}^{op}$-module $M$ is called relatively projective if $M$ is a direct summand of a direct sum of DG ${{\mathcal A}}^{op}$-modules of the form $h^A[n]$, $A\in {{\mathcal A}}$, $n\in {{\mathbb Z}}$. A DG ${{\mathcal A}}^{op}$-module $P$ is said to have property (P) if it admits a filtration $$0=F_{-1}\subset F_0\subset F_1\subset ... P$$ such that (F1) $\cup_iF_i=P$; (F2) the inclusion $F_i\hookrightarrow F_{i+1}$ splits as a morphism of graded modules; (F3) each quotient $F_{i+1}/F_i$ is a relatively projective DG ${{\mathcal A}}^{op}$-module. A DG ${{\mathcal A}}^{op}$-module $M$ is called relatively injective if $M$ is a direct summand of a direct product of DG ${{\mathcal A}}^{op}$-modules of the form $h_A^*[n]$, $A\in {{\mathcal A}}$, $n\in {{\mathbb Z}}$. A DG ${{\mathcal A}}^{op}$-module $I$ is said to have property (I) if it admits a filtration $$I=F_{0}\supset F_1\supset ...$$ such that (F1’) the canonical morphism $$I\to \lim_{\leftarrow}I/F_i$$ is an isomorphism; (F2’) the inclusion $F_{i+1}\hookrightarrow F_i$ splits as a morphism of graded modules; (F3’) each quotient $F_{i}/F_{i+1}$ is a relatively injective DG ${{\mathcal A}}^{op}$-module. ([@Ke]) a) A DG ${{\mathcal A}}^{op}$-module with property (P) is $h$-projective. b\) For any $M\in {{\mathcal A}}^{op}\text{-mod}$ there exists a quasi-isomorphism $P\to M$, such that the DG ${{\mathcal A}}^{op}$-module $P$ has property (P). c\) A DG ${{\mathcal A}}^{op}$-module with property (I) is $h$-injective. d\) For any $M\in {{\mathcal A}}^{op}\text{-mod}$ there exists a quasi-isomorphism $M\to I$, such that the DG ${{\mathcal A}}^{op}$-module $I$ has property (I). a\) Assume that a DG ${{\mathcal A}}^{op}$-module $M$ has an increasing filtration $M_1\subset M_2\subset ...$ such that $\cup M_i=M$, each inclusion $M_i\hookrightarrow M_{i+1}$ splits as a morphism of graded modules, and each subquotient $M_{i+1}/M_i$ is $h$-projective. Then $M$ is h-projective. b) Assume that a DG ${{\mathcal A}}^{op}$-module $N$ has a decreasing filtration $N=N_1\supset N_2\supset ...$ such that $\cap N_i=0$, each inclusion $N_{i+1}\hookrightarrow N_i$ splits as a morphism of graded modules, each subquotient $N_i/N_{i+1}$ is h-injective (hence $N/N_i$ is h-injective for each $i$) and the natural map $$N\to \lim_{\leftarrow}N/N_i$$ is an isomorphism. Then $N$ is h-injective. Some DG functors ---------------- Let ${{\mathcal B}}$ be a small DG category. The complex $${\operatorname{Alg}}_{{{\mathcal B}}}:=\bigoplus _{A,B\in Ob {{\mathcal B}}}\Hom(A,B)$$ has a natural structure of a DG algebra possibly without a unit. It has the following property: every finite subset of ${\operatorname{Alg}}_{{{\mathcal B}}}$ is contained in $e{\operatorname{Alg}}_{{{\mathcal B}}} e$ for some idempotent $e$ such that $de=0$ and $\bar{e}=0$. We say that a DG module $M$ over ${\operatorname{Alg}}_{{{\mathcal B}}}$ is [*quasi-unital*]{} if every element of $M$ belongs to $eM$ for some idempotent $e\in {\operatorname{Alg}}_{{{\mathcal B}}}$ (which may be assumed closed of degree $0$ without loss of generality). If $\Phi $ is a DG ${{\mathcal B}}$-module then $$M_{\Phi}:=\oplus _{A\in Ob {{\mathcal B}}}\Phi (A)$$ is a quasi-unital DG module over ${\operatorname{Alg}}_{{{\mathcal B}}}$. This way we get a DG equivalence between DG category of DG ${{\mathcal B}}$-modules and that of quasi-unital DG modules over ${\operatorname{Alg}}_{{{\mathcal B}}}$. Recall that a homomorphism of (unital) DG algebras $\phi :{{\mathcal A}}\to {{\mathcal B}}$ induces functors $$\phi _*:{{\mathcal B}}^{op}\text{-mod}\to {{\mathcal A}}^{op}\text{-mod},$$ $$\phi ^*:{{\mathcal A}}^{op}\text{-mod}\to {{\mathcal B}}^{op} \text{-mod}$$ $$\phi ^!:{{\mathcal A}}^{op}\text{-mod}\to {{\mathcal B}}^{op} \text{-mod}$$ where $\phi _*$ is the restriction of scalars, $\phi ^*(M)=M \otimes _{{{\mathcal A}}}{{\mathcal B}}$ and $\phi ^!(M)=\Hom _{{{\mathcal A}}^{op}}({{\mathcal B}},M)$. The DG functors $(\phi ^*,\phi _*)$ and $(\phi _*,\phi ^!)$ are adjoint: for $M\in {{\mathcal A}}^{op}\text{-mod}$ and $N\in {{\mathcal B}}^{op}\text{-mod}$ there exist functorial isomorphisms of complexes $$\Hom (\phi ^*M,N)=\Hom (M,\phi _*N),\quad \Hom (\phi _*N,M)=\Hom (N,\phi ^!M).$$ This generalizes to a DG functor $F:{{\mathcal A}}\to {{\mathcal B}}$ between DG categories. We obtain DG functors $$F _*:{{\mathcal B}}^{op}\text{-mod}\to {{\mathcal A}}^{op}\text{-mod},$$ $$F ^*:{{\mathcal A}}^{op}\text{-mod}\to {{\mathcal B}}^{op}\text{-mod}.$$ $$F ^!:{{\mathcal A}}^{op}\text{-mod}\to {{\mathcal B}}^{op}\text{-mod}.$$ Namely, the DG functor $F$ induces a homomorphism of DG algebras $F:{\operatorname{Alg}}_{{{\mathcal A}}}\to {\operatorname{Alg}}_{{{\mathcal B}}}$ and hence defines functors $F_*$, $F^*$ between quasi-unital DG modules as above. (These functors $F_*$ and $F^*$ are denoted in [@Dr] by ${\operatorname{Res}}_F$ and ${\operatorname{Ind}}_F$ respectively.) The functor $F^!$ is defined as follows: for a quasi-unital ${\operatorname{Alg}}_{{{\mathcal A}}}^{op}$-module $M$ put $$F^!(M)=\Hom _{{\operatorname{Alg}}_{{{\mathcal A}}}^{op}}({\operatorname{Alg}}_{{{\mathcal B}}},M)^{{\operatorname{qu}}},$$ where $N^{{\operatorname{qu}}}\subset N$ is the [*quasi-unital*]{} part of a ${\operatorname{Alg}}_{{{\mathcal B}}}^{op}$-module $N$ defined by $$N^{{\operatorname{qu}}}:={\operatorname{Im}}(N\otimes _k {\operatorname{Alg}}_{{{\mathcal B}}}\to N).$$ The DG functors $(F ^*,F _*)$ and $(F_*,F^!)$ are adjoint. Let $F:{{\mathcal A}}\to {{\mathcal B}}$ be a DG functor. Then a\) $F_*$ preserves acyclic DG modules; b\) $F^*$ preserves h-projective DG modules; c\) $F^!$ preserves h-injective DG modules. The first assertion is obvious and the other two follow by adjunction. By Theorem 3.1 above the DG subcategories $\P({{\mathcal A}}^{op})$ and ${{\mathcal I}}({{\mathcal A}}^{op})$ of ${{\mathcal A}}^{op}\text{-mod}$ allow us to define (left and right) derived functors of DG functors $G:{{\mathcal A}}^{op}\text{-mod}\to {{\mathcal B}}^{op}\text{-mod}$ in the usual way. Namely for a DG ${{\mathcal A}}^{op}$-module $M$ choose quasi-isomorphisms $P\to M$ and $M\to I$ with $P\in \P({{\mathcal A}}^{op})$ and $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Put $$\bL G(M):=G(P),\quad \quad \bR G(M):=G(I).$$ In particular for a DG functor $F:{{\mathcal A}}\to {{\mathcal B}}$ we will consider derived functors $\bL F^*:D({{\mathcal A}}^{op})\to D({{\mathcal B}}^{op})$, $\bR F^!:D({{\mathcal A}}^{op})\to D({{\mathcal B}}^{op})$. We also have the obvious functor $F_*:D({{\mathcal B}}^{op})\to D({{\mathcal A}}^{op})$. The functors $(\bL F^*,F_*)$ and $(F_*, \bR F^!)$ are adjoint. Assume that the DG functor $F:{{\mathcal A}}\to {{\mathcal B}}$ is a quasi-equivalence. Then a\) $F^*:\P({{\mathcal A}}^{op})\to \P({{\mathcal B}}^{op})$ is a quasi-equivalence; b\) $\bL F ^*:D({{\mathcal A}}^{op})\to D({{\mathcal B}}^{op})$ is an equivalence; c\) $F_*:D({{\mathcal B}}^{op})\to D({{\mathcal A}}^{op})$ is an equivalence. d\) $\bR F^!:D({{\mathcal A}}^{op})\to D({{\mathcal B}}^{op})$ is an equivalence. e\) $F^!:{{\mathcal I}}({{\mathcal A}}^{op})\to {{\mathcal I}}({{\mathcal B}}^{op})$ is a quasi-equivalence. a\) is proved in [@Ke] and it implies b) by Theorem 3.1. c) (resp. d)) follows from b) (resp. c) by adjunction. Finally, e) follows from d) by Theorem 3.1. Given DG ${{\mathcal A}}^{op}$-modules $M,N$ we denote by ${\operatorname{Ext}}^n(M,N)$ the group of morphisms $\Hom ^n _{D({{\mathcal A}})}(M,N)$. DG category ${{\mathcal A}}_{{{\mathcal R}}}$ --------------------------------------------- Let ${{\mathcal R}}$ be a DG algebra. We may and will consider ${{\mathcal R}}$ as a DG category with one object whose endomorphism DG algebra is ${{\mathcal R}}$. We denote this DG category again by ${{\mathcal R}}$. Note that the DG category ${{\mathcal R}}^{op}\text{-mod}$ is just the category of right DG modules over the DG algebra ${{\mathcal R}}$. For a DG category ${{\mathcal A}}$ we denote the DG category ${{\mathcal A}}\otimes {{\mathcal R}}$ by ${{\mathcal A}}_{{{\mathcal R}}}$. Note that the collections of objects of ${{\mathcal A}}$ and ${{\mathcal A}}_{{{\mathcal R}}}$ are naturally identified. A homomorphism of DG algebras $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ induces the obvious DG functor $\phi={\operatorname{id}}\otimes \phi :{{\mathcal A}}_{{{\mathcal R}}}\to {{\mathcal A}}_{{{\mathcal Q}}}$ (which is the identity on objects), whence the DG functors $\phi _*$, $ \phi ^*$, $\phi ^!$ between the DG categories ${{\mathcal A}}^{op} _{{{\mathcal R}}}\text{-mod}$ and ${{\mathcal A}}^{op} _{{{\mathcal Q}}}\text{-mod}$. For $M \in {{\mathcal A}}_{{{\mathcal R}}}^{op} \text{-mod}$ we have $$\phi ^*(M)=M\otimes _{{{\mathcal R}}}{{{\mathcal Q}}}.$$ In case ${{\mathcal Q}}^{\gr}$ is a finitely generated ${{\mathcal R}}^{\gr}$-module we have $$\phi ^!(M)=\Hom _{{{\mathcal R}}^{op}}({{\mathcal Q}},M).$$ In particular, if ${{\mathcal R}}$ is augmented then the canonical homomorphisms of DG algebras $p:k\to {{\mathcal R}}$ and $i:{{\mathcal R}}\to k$ induce functors $$p:{{\mathcal A}}\to {{\mathcal A}}_{{{\mathcal R}}},\quad i:{{\mathcal A}}_{{{\mathcal R}}}\to {{\mathcal A}},$$ such that $i\cdot p={\operatorname{Id}}_{{{\mathcal A}}}$. So for $S\in {{\mathcal A}}^{op}\text{-mod}$ and $T\in {{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$ we have $$p^*(S)=S\otimes _k{{\mathcal R}}, \quad i^*(T)=T\otimes _{{{\mathcal R}}}k, \quad i^!(T)=\Hom _{{{\mathcal R}}^{op}}(k,T).$$ For an artinian DG algebra ${{\mathcal R}}$ we denote by ${{\mathcal R}}^*$ the DG ${{\mathcal R}}^{op}$-module $\Hom _k({{\mathcal R}},k)$. This is a left ${{\mathcal R}}$-module by the formula $$rf(q):=(-1)^{(\bar{f}+\bar{q})\bar{r}}f(qr)$$ and a right ${{\mathcal R}}$-module by the formula $$fr(p):=f(rp)$$ for $r,p\in {{\mathcal R}}$ and $f\in {{\mathcal R}}^*$. The augmentation map ${{\mathcal R}}\to k$ defines the canonical (left and right) ${{\mathcal R}}$-submodule $k\subset {{\mathcal R}}^*$. Moreover, the embedding $k\hookrightarrow {{\mathcal R}}^*$ induces an isomorphism $k\to \Hom _{{{\mathcal R}}}(k,{{\mathcal R}}^*)$. Let ${{\mathcal R}}$ be an artinian DG algebra. A DG ${{\mathcal A}}^{op} _{{{\mathcal R}}}$-module $M$ is called graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree) if there exists a DG ${{\mathcal A}}^{op}$-module $K$ such that $M^{\gr}\simeq (K\otimes {{\mathcal R}})^{\gr}$ (resp. $M^{\gr}\simeq (K\otimes {{\mathcal R}}^*)^{\gr}$). Note that for such $M$ one may take $K=i^*M$ (resp. $K=i^!M$). Let ${{\mathcal R}}$ be an artinian DG algebra. a\) The full DG subcategories of DG ${{\mathcal A}}_{{{\mathcal R}}}^{op}$-modules consisting of graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree) modules are DG isomorphic. Namely, if $M\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ is graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree) then $M\otimes _{{{\mathcal R}}}{{\mathcal R}}^*$ (resp. $\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,M)$) is graded ${{\mathcal R}}$-cofree (resp. graded ${{\mathcal R}}$-free). b\) Let $M$ be a graded ${{\mathcal R}}$-free module. There is a natural isomorphism of DG ${{\mathcal A}}^{op}$-modules $$i^*M\stackrel{\sim}{\to}i^!(M\otimes _{{{\mathcal R}}}{{\mathcal R}}^*).$$ a\) If $M$ is graded ${{\mathcal R}}$-free, then obviously $M\otimes _{{{\mathcal R}}}{{\mathcal R}}^*$ is graded ${{\mathcal R}}$-cofree. Assume that $N$ is graded ${{\mathcal R}}$-cofree, i.e. $N^{\gr}=(K\otimes {{\mathcal R}}^*)^{\gr}$. Then $$(\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,N))^{\gr}=(K\otimes \Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,{{\mathcal R}}^*))^{\gr},$$ since $\dim _k{{\mathcal R}}<\infty$. On the other hand $$\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,{{\mathcal R}}^*)=\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,\Hom _k({{\mathcal R}},k))=\Hom _k({{\mathcal R}}^*\otimes _{{{\mathcal R}}}{{\mathcal R}},k)={{\mathcal R}},$$ so $(\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,N))^{\gr}=(K\otimes {{\mathcal R}})^{\gr}$. b\) For an arbitrary DG ${{\mathcal A}}_{{{\mathcal R}}}^{op}$-module $M$ we have a natural (closed degree zero) morphism of DG ${{\mathcal A}}^{op}$-modules $$i^*M\to i^!(M\otimes _{{{\mathcal R}}}{{\mathcal R}}^*),\quad m\otimes 1\mapsto (1\mapsto m\otimes i),$$ where $i:{{\mathcal R}}\to k$ is the augmentation map. If $M$ is graded ${{\mathcal R}}$-free this map is an isomorphism. Let ${{\mathcal R}}$ be an artinian DG algebra. Assume that a DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module $M$ satisfies property (P) (resp. property (I)). Then $M$ is graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree). Notice that the collection of graded ${{\mathcal R}}$-free objects in ${{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$ is closed under taking direct sums, direct summands (since the maximal ideal $m\subset {{\mathcal R}}$ is nilpotent) and direct products (since ${{\mathcal R}}$ is finite dimensional). Similarly for graded ${{\mathcal R}}$-cofree objects since the DG functors in Lemma 3.9 a) preserve direct sums and products. Also notice that for any $A \in {{\mathcal A}}_{{{\mathcal R}}}$ the DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module $h^A$ (resp. $h_A^*$) is graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree). Now the proposition follows since a DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module $P$ (resp. $I$) with property (P) (resp. property (I)) as a graded module is a direct sum of relatively projective DG modules (resp. a direct product of relatively injective DG modules). Let ${{\mathcal R}}$ be an artinian DG algebra. Then for any DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module $M$ there exist quasi-isomorphisms $P\to M$ and $M\to I$ such that $P\in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$, $I\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $P$ is graded ${{\mathcal R}}$-free, $I$ is graded ${{\mathcal R}}$-cofree. Indeed, this follows from Theorem 3.4 and Proposition 3.10 above. Let ${{\mathcal R}}$ be an artinian DG algebra and $S,T\in {{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$ be graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree). a\) There is an isomorphism of graded vector spaces $\Hom(S,T)=\Hom(i^*S,i^*T) \otimes {{\mathcal R}}$, (resp. $\Hom(S,T)=\Hom(i^!S,i^!T) \otimes {{\mathcal R}}$), which is an isomorphism of algebras if $S=T.$ In particular, the map $i ^*:\Hom (S,T)\to \Hom (i^*S,i^*T)$ (resp. $i ^!:\Hom(S,T)\to \Hom(i^!S,i^!T)$) is surjective. b\) The DG module $S$ has a finite filtration with subquotients isomorphic to $i^*S$ as DG ${{\mathcal A}}^{op}$-modules (resp. to $i^!S$ as DG ${{\mathcal A}}^{op}$-modules). c\) The DG algebra $\End(S)$ has a finite filtration by DG ideals with subquotients isomorphic to $\End (i^*S)$ (resp. $\End(i^!S)$). d\) If $f\in \Hom (S,T)$ is a closed morphism of degree zero such that $i^*f$ (resp. $i^!f$) is an isomorphism or a homotopy equivalence or a quasi-isomorphism, then $f$ is also such. Because of Lemma 3.9 above it suffices to prove the proposition for graded ${{\mathcal R}}$-free modules. So assume that $S$, $T$ are graded ${{\mathcal R}}$-free. a\) This holds because ${{\mathcal R}}$ is finite dimensional. b\) We can refine the filtration of ${{\mathcal R}}$ by powers of the maximal ideal to get a filtration $F_i{{\mathcal R}}$ by ideals with 1-dimensional subquotients (and zero differential). Then the filtration $F_iS:=S\cdot F_i{{\mathcal R}}$ satisfies the desired properties. c\) Again the filtration $F_i\End (S):=\End(S)\cdot F_i{{\mathcal R}}$ has the desired properties. d\) If $i^*f$ is an isomorphism, then $f$ is surjective by the Nakayama lemma for ${{\mathcal R}}$. Also $f$ is injective since $T$ is graded ${{\mathcal R}}$-free. Assume that $i^*f$ is a homotopy equivalence. Let $C(f)\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ be the cone of $f$. (It is also graded ${{\mathcal R}}$-free.) Then $i^*C(f)\in {{\mathcal A}}^{op}\text{-mod}$ is the cone $C(i^*f)$ of the morphism $i^*f$. By assumption the DG algebra $\End (C(i^*f))$ is acyclic. But by part c) the complex $\End (C(f))$ has a finite filtration with subquotients isomorphic to the complex $\End (C(i^*f))$. Hence $\End (C(f))$ is also acyclic, i.e. the DG module $C(f)$ is null-homotopic, i.e. $f$ is a homotopy equivalence. Assume that $i^*f$ is a quasi-isomorphism. Then in the above notation $C(i^*f)$ is acyclic. Since by part b) $C(f)$ has a finite filtration with subquotients isomorphic to $C(i^*f)$, it is also acyclic. Thus $f$ is a quasi-isomorphism. More DG functors ---------------- So far we considered DG functors $F_*$, $F^*$, $F^!$ between the DG categories ${{\mathcal A}}^{op}$-mod and ${{\mathcal B}}^{op}$-mod which came from a DG functor $F:{{\mathcal A}}\to {{\mathcal B}}$. We will also need to consider a different type of DG functors. For an artinian DG algebra ${{\mathcal R}}$ and a small DG category ${{\mathcal A}}$ we will consider two types of “restriction of scalars” DG functors $\pi _*,\pi _!:{{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}\to {{\mathcal R}}^{op}\text{-mod}$. Namely, for $M\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ put $$\pi _*M:=\prod_{A\in Ob{{\mathcal A}}_{{{\mathcal R}}}}M(A),\quad \pi _!M:=\bigoplus_{A\in Ob{{\mathcal A}}_{{{\mathcal R}}}}M(A).$$ We will also consider the two “extension of scalars” functors $\pi ^*,\pi ^!:{{\mathcal R}}^{op}\text{-mod}\to {{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$ defined by $$\pi ^*(N)(A):=N\otimes \bigoplus_{B\in Ob{{\mathcal A}}}\Hom_{{{\mathcal A}}}(A,B), \quad \pi ^!(N)(A):=\Hom _k(\bigoplus_{B\in Ob{{\mathcal A}}}\Hom_{{{\mathcal A}}}(B,A),N)$$ for $A\in Ob{{\mathcal A}}_{{{\mathcal R}}}$. Notice that the DG functors $(\pi ^*,\pi _*)$ and $(\pi _!,\pi ^!)$ are adjoint, that is for $M\in {{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$ and $N\in {{\mathcal R}}^{op}\text{-mod}$ there is a functorial isomorphism of complexes $$\Hom (\pi ^*N,M)=\Hom (N,\pi_*M), \quad \Hom (\pi _!M,N)=\Hom (M,\pi^!N).$$ The DG functors $\pi^*,\pi ^!$ preserve acyclic DG modules, hence $\pi _*$ preserves h-injectives and $\pi _!$ preserves h-projectives. We have the following commutative functorial diagrams $$\begin{array}{ccc} {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod} & \stackrel{i^*}{\longrightarrow} & {{\mathcal A}}^{op}\text{-mod}\\ \pi _!\downarrow & & \pi _!\downarrow \\ {{\mathcal R}}^{op}\text{-mod} & \stackrel{i^*}{\longrightarrow} & DG(k), \end{array}$$ $$\begin{array}{ccc} {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod} & \stackrel{i^!}{\longrightarrow} & {{\mathcal A}}^{op}\text{-mod}\\ \pi _*\downarrow & & \pi _*\downarrow \\ {{\mathcal R}}^{op}\text{-mod} & \stackrel{i^!}{\longrightarrow} & DG(k). \end{array}$$ Fix $E\in {{\mathcal A}}^{op}\text{-mod}$ and put ${{\mathcal B}}=\End(E)$. Consider the DG functor $$\Sigma =\Sigma ^E:{{\mathcal B}}^{op}\text{-mod}\to {{\mathcal A}}^{op}\text{-mod}$$ defined by $\Sigma (M)=M\otimes _{{{\mathcal B}}}E$. Clearly, $\Sigma ({{\mathcal B}})=E$. This DG functor gives rise to the functor $$\bL \Sigma :D({{\mathcal B}}^{op})\to D({{\mathcal A}}^{op}),\quad \quad \bL \Sigma (M)=M\stackrel{\bL}{\otimes }_{{{\mathcal B}}}E.$$ Pre-triangulated DG categories ------------------------------ For any DG category ${{\mathcal A}}$ there exists a DG category ${{\mathcal A}}^{pre-tr}$ and a canonical full and faithful DG functor $F:{{\mathcal A}}\to {{\mathcal A}}^{\pre-tr}$ (see [@BoKa; @Dr]). The homotopy category ${\operatorname{Ho}}({{\mathcal A}}^{\pre-tr})$ is canonically triangulated. The DG category ${{\mathcal A}}$ is called [*pre-triangulated*]{} if the DG functor $F$ is a quasi-equivalence. The DG category ${{\mathcal A}}^{\pre-tr}$ is pre-triangulated. Let ${{\mathcal B}}$ be another DG category and $G:{{\mathcal A}}\to {{\mathcal B}}$ be a quasi-equivalence. Then $G^{\pre-tr}:{{\mathcal A}}^{\pre-tr}\to {{\mathcal B}}^{\pre-tr}$ is also a quasi-equivalence. The DG functor $F$ induces a DG isomorphism of DG categories $F_*:({{\mathcal A}}^{\pre-tr})^{op}\text{-mod} \to {{\mathcal A}}^{op}\text{-mod}$. Hence the functors $F_*:D(({{\mathcal A}}^{\pre-tr})^{op})\to D({{\mathcal A}}^{op})$ and $\bL F^*:D({{\mathcal A}}^{op})\to D(({{\mathcal A}}^{\pre-tr})^{op})$ are equivalences. We obtain the following corollary. Assume that a DG functor $G_1:{{\mathcal A}}\to {{\mathcal B}}$ induces a quasi-equivalence $G_1^{\pre-tr}:{{\mathcal A}}^{\pre-tr}\to {{\mathcal B}}^{\pre-tr}$. Let ${{\mathcal C}}$ be another DG category and consider the DG functor $G:=G_1\otimes {\operatorname{id}}:{{\mathcal A}}\otimes {{\mathcal C}}\to {{\mathcal B}}\otimes {{\mathcal C}}$. Then the functors $G_*,\bL G^*, \bR G^!$ between the derived categories $D(({{\mathcal A}}\otimes {{\mathcal C}})^{op})$ and $D(({{\mathcal B}}\otimes {{\mathcal C}})^{op})$ are equivalences. The DG functor $G$ induces the quasi-equivalence $G ^{\pre-tr}:({{\mathcal A}}\otimes {{\mathcal C}})^{\pre-tr}\to ({{\mathcal B}}\otimes {{\mathcal C}})^{\pre-tr}$. Hence the corollary follows from the above discussion and Proposition 3.6. Suppose ${{\mathcal B}}$ is a pre-triangulated DG category. Let $G_1:{{\mathcal A}}\hookrightarrow {{\mathcal B}}$ be an embedding of a full DG subcategory so that the triangulated category ${\operatorname{Ho}}({{\mathcal B}})$ is generated by the collection of objects $G_1(Ob{{\mathcal A}})$. Then the assumptions of the previous corollary hold. A few lemmas ------------ Let ${{\mathcal R}}$, ${{\mathcal Q}}$ be DG algebras and $M$ be a DG ${{\mathcal Q}}\otimes {{\mathcal R}}^{op}$-module. a\) For any DG modules $N$, $S$ over the DG algebras ${{\mathcal Q}}^{op}$ and ${{\mathcal R}}^{op}$ respectively there is a natural isomorphism of complexes $$\Hom _{{{\mathcal R}}^{op}}(N\otimes _{{{\mathcal Q}}}M,S)\stackrel{\sim}{\to}\Hom _{{{\mathcal Q}}^{op}}(N,\Hom _{{{\mathcal R}}^{op}}(M,S)).$$ b\) There is a natural quasi-isomorphism of complexes $$\bR \Hom _{{{\mathcal R}}^{op}}(N\stackrel{\bL}{\otimes }_{{{\mathcal Q}}}M,S)\stackrel{\sim}{\to}\bR \Hom _{{{\mathcal Q}}^{op}}(N,\bR \Hom _{{{\mathcal R}}^{op}}(M,S)).$$ a\) Indeed, for $f\in \Hom _{{{\mathcal R}}^{op}}(N\otimes _{{{\mathcal Q}}}M,S)$ define $\alpha (f)\in \Hom _{{{\mathcal Q}}}(N,\Hom _{{{\mathcal R}}^{op}}(M,S))$ by the formula $\alpha (f)(n)(m)=f(n\otimes m)$. Conversely, for $g\in \Hom _{{{\mathcal Q}}}(N,\Hom _{{{\mathcal R}}^{op}}(M,S))$ define $\beta (g)\in \Hom _{{{\mathcal R}}^{op}}(N\otimes _{{{\mathcal Q}}}M,S)$ by the formula $\beta(g)(n\otimes m)=g(n)(m)$. Then $\alpha $ and $\beta$ are mutually inverse isomorphisms of complexes. b\) Choose quasi-isomorphisms $P\to N$ and $S\to I$, where $P\in {{\mathcal P}}({{\mathcal Q}}^{op})$ and $I\in {{\mathcal I}}({{\mathcal R}}^{op})$ and apply a). Let ${{\mathcal R}}$ be an artinian DG algebra. Then in the DG category ${{\mathcal R}}^{op}\text{-mod}$ a direct sum of copies of ${{\mathcal R}}^*$ is h-injective. Let $V$ be a graded vector space, $M=V\otimes {{\mathcal R}}^*\in {{\mathcal R}}^{op}\text{-mod}$ and $C$ an acyclic DG ${{\mathcal R}}^{op}$-module. Notice that $M=\Hom _k({{\mathcal R}},V)$ since $\dim {{\mathcal R}}<\infty$. Hence the complex $$\Hom _{{{\mathcal R}}^{op}}(C,M)=\Hom _{{{\mathcal R}}^{op}}(C,\Hom _k({{\mathcal R}},V))=\Hom _k(C\otimes _{{{\mathcal R}}}{{\mathcal R}},V)=\Hom _k(C,V)$$ is acyclic. Let ${{\mathcal B}}$ be a DG algebra, such that ${{\mathcal B}}^i=0$ for $i>0$. Then the category $D({{\mathcal B}}^{op})$ has truncation functors: for any DG ${{\mathcal B}}$-module $M$ there exists a short exact sequence in the abelian category $Z^0({{\mathcal B}}\text{-mod})$ $$\tau _{<0}M\to M\to \tau _{\geq 0}M,$$ where $H^i(\tau _{<0}M)=0$ if $i\geq 0$ and $H^i(\tau _{\geq 0}M)=0$ for $i<0$. Indeed, put $\tau _{<0}M:=\oplus _{i<0}M^i\oplus d(M^{-1})$. Let ${{\mathcal B}}$ be a DG algebra, s.t. ${{\mathcal B}}^i=0$ for $i>0$ and $\dim {{\mathcal B}}^i<\infty$ for all $i$. Let $N$ be a DG ${{\mathcal B}}$-module with finite dimensional cohomology. Then there exists an h-projective DG ${{\mathcal B}}$-module $P$ and a quasi-isomorphism $P\to N$, where $P$ in addition satisfies the following conditions a\) $P^i=0$ for $i>>0$, b\) $\dim P^i<\infty$ for all $i$. First assume that $N$ is concentrated in one degree, say $N^i=0$ for $i\neq 0$. Consider $N$ as a $k$-module and put $P_0:={{\mathcal B}}\otimes N$. We have a natural surjective map of DG ${{\mathcal B}}$-modules $\epsilon :P_0\to N$ which is also surjective on the cohomology. Let $K:={\operatorname{Ker}}\epsilon$. Then $K^i=0$ for $i>0$ and $\dim K^i<\infty$ for all $i$. Consider $K$ as a DG $k$-module and put $P_{-1}:={{\mathcal B}}\otimes K$. Again we have a surjective map of DG ${{\mathcal B}}$-modules $P_{-1}\to K$ which is surjective and surjective on cohomology. And so on. This way we obtain an exact sequence of DG ${{\mathcal B}}$-modules $$...\to P_{-1}\to P_0\stackrel{\epsilon}{\to}N\to 0,$$ where $P_{-j}^i=0$ for $i>0$ and $\dim P_{-j}^i<\infty$ for all $j$. Let $P:=\oplus _jP_{-j}[j]$ be the “total” DG ${{\mathcal B}}$-module of the complex $...\to P_{-1} \to P_0\to 0$. Then $\epsilon :P\to N$ is a quasi-isomorphism. Since each DG ${{\mathcal B}}$-module $P_{-j}$ has the property (P), the module $P$ is h-projective by Remark 3.5a). Also $P^i=0$ for $i>0$ and $\dim P^i<\infty $ for all $i$. How consider the general case. Let $H^s(N)=0$ and $H^i(N)=0$ for all $i<s$. Replacing $N$ by $\tau _{\geq s}N$ (Lemma 3.19) we may and will assume that $N^i=0$ for $i<s$. Then $M:=({\operatorname{Ker}}d_N)\cap N^s$ is a DG ${{\mathcal B}}$-submodule of $N$ which is not zero. If the embedding $M\hookrightarrow N$ is a quasi-isomorphism, then we may replace $N$ by $M$ and so we are done by the previous argument. Otherwise we have a short exact sequence of DG ${{\mathcal B}}$-modules $$o\to M\to N\to N/M\to 0$$ with $\dim H(M), \dim H(N/M)<\dim H(N)$. By the induction on $\dim H(N)$ we may assume that the lemma holds for $M$ and $N/M$. But then it also holds for $N$. Let ${{\mathcal B}}$ be a DG algebra, s.t. ${{\mathcal B}}^i=0$ for $i>0$, $\dim {{\mathcal B}}^i<\infty$ for all $i$ and the algebra $H^0(\B)$ is local. Let $N$ be a DG ${{\mathcal B}}$-module with finite dimensional cohomology. Then $N$ is quasi-isomorphic to a finite dimensional DG ${{\mathcal B}}$-module. By Lemma 3.20 there exists a bounded above and locally finite DG ${{\mathcal B}}$-module $P$ which is quasi-isomorphic to $N$. It remains to apply the appropriate truncation functor to $P$ (Lemma 3.19). Let ${{\mathcal B}}$ be an augmented DG algebra, s.t. ${{\mathcal B}}^i=0$ for $i>0$, $\dim {{\mathcal B}}^i<\infty$ for all $i$ and the algebra $H^0({{\mathcal B}})$ is local. Denote by $\langle k\rangle \subset D({{\mathcal B}})$ the triangulated envelope of the DG ${{\mathcal B}}$-module $k$. Let $N$ be a DG ${{\mathcal B}}$-module with finite dimensional cohomology. Then $N\in \langle k\rangle$. By the previous corollary we may assume that $N$ is finite dimensional. But then an easy applying of the Nakayama lemma for $H^0({{\mathcal B}})$ shows that $N$ has a filtration by DG ${{\mathcal B}}$-modules with subquotients isomorphic to $k$. Let ${{\mathcal B}}$ and ${{\mathcal C}}$ be DG algebras. Consider the DG algebra ${{\mathcal B}}\otimes {{\mathcal C}}$ and a homomorphism of DG algebras $F:{{\mathcal B}}\to {{\mathcal B}}\otimes {{\mathcal C}}$, $F(b)=b\otimes 1$. Let $N$ be an h-projective (resp. h-injective) DG ${{\mathcal B}}\otimes {{\mathcal C}}$-module. Then the DG ${{\mathcal B}}$-module $F_*N$ is also h-projective (resp. h-injective). The assertions follow from the fact that the DG functor $F_*:{{\mathcal B}}\otimes {{\mathcal C}}\text{-mod}\to {{\mathcal B}}\text{-mod}$ has a left adjoint DG functor $F^*$ (resp. right adjoint DG functor $F^!$) which preserves acyclic DG modules. Indeed, $$F^*(M)={{\mathcal C}}\otimes _{k}M,\quad \quad F^!(M)=\Hom _k({{\mathcal C}},M).$$ The homotopy deformation and co-deformation pseudo-functors =========================================================== Denote by ${\bf Gpd}$ the 2-category of groupoids. Let ${{\mathcal E}}$ be a category and $F,G:{{\mathcal E}}\to {\bf Gpd}$ two pseudo-functors. A morphism $\epsilon :F\to G$ is called full and faithful (resp. an equivalence) if for every $X\in Ob{{\mathcal E}}$ the functor $\epsilon _X:F(X)\to G(X)$ is full and faithful (resp. an equivalence). We call $F$ and $G$ equivalent if there exists an equivalence $F\to G$. It the rest of this paper we will usually denote by ${{\mathcal A}}$ a fixed DG category and by $E$ a DG ${{\mathcal A}}^{op}$-module. Let us define the homotopy deformation pseudo-functor ${\operatorname{Def}}^{\h} (E):{\operatorname{dgart}}\to {\bf Gpd}$. This functor describes “infinitesimal” (i.e. along artinian DG algebras) deformations of $E$ in the homotopy category of DG ${{\mathcal A}}^{op}$-modules. Let ${{\mathcal R}}$ be an artinian DG algebra. An object in the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}^{\h} (E)$ is a pair $(S,\sigma)$, where $S\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ and $\sigma :i^*S\to E$ is an isomorphism of DG ${{\mathcal A}}^{op}$-modules such that the following holds: there exists an isomorphism of graded ${{\mathcal A}}^{op} _{{{\mathcal R}}}$-modules $\eta :(E\otimes {{\mathcal R}})^{\gr} \to S^{\gr}$ so that the composition $$E= i^*(E\otimes {{\mathcal R}}) \stackrel{i^*(\eta)}{\to} i^*S\stackrel{\sigma}{\to}E$$ is the identity. Given objects $(S,\sigma),(S^\prime ,\sigma ^\prime)\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E)$ a map $f:(S,\sigma)\to (S^\prime,\sigma ^\prime)$ is an isomorphism $f:S\to S^\prime$ such that $\sigma ^\prime \cdot i^*f=\sigma$. An allowable homotopy between maps $f,g$ is a homotopy $h:f\to g$ such that $i^*(h)=0$. We define morphisms in ${\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E)$ to be classes of maps modulo allowable homotopies. Note that a homomorphism of artinian DG algebras $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ induces the functor $\phi ^*:{\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E)\to {\operatorname{Def}}_{{{\mathcal Q}}}^{\h}(E)$. This defines the pseudo-functor $${\operatorname{Def}}{^h}(E):{\operatorname{dgart}}\to {\bf Gpd}.$$ We refer to objects of ${\operatorname{Def}}_{{{\mathcal R}}}^{\h} (E)$ as homotopy ${{\mathcal R}}$-deformations of $E$. The term “homotopy” in the above definition is used to distinguish the pseudo-functor ${\operatorname{Def}}^{\h}$ from the pseudo-functor ${\operatorname{Def}}$ of [*derived deformations*]{} (Definition 10.1). It may be justified by the fact that ${\operatorname{Def}}^{\h}(E)$ depends (up to equivalence) only on the isomorphism class of $E$ in ${\operatorname{Ho}}({{\mathcal A}}^{op}\text{-mod})$ (Corollary 8.4 a)). We call $(p^*E,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E)$ the trivial ${{\mathcal R}}$-deformation of $E$. Denote by ${\operatorname{Def}}_+^{\h}(E)$, ${\operatorname{Def}}_-^{\h}(E)$, ${\operatorname{Def}}_0^{\h}(E)$, ${\operatorname{Def}}^{\h}_{{\operatorname{cl}}}(E)$ the restrictions of the pseudo-functor ${\operatorname{Def}}^{\h}(E)$ to subcategories ${\operatorname{dgart}}_+$, ${\operatorname{dgart}}_-$, ${\operatorname{art}}$, ${\operatorname{cart}}$ respectively. Let us give an alternative description of the same deformation problem. We will define the homotopy [*co-deformation*]{} pseudo-functor ${\operatorname{coDef}}^{\h}(E)$ and show that it is equivalent to ${\operatorname{Def}}^{\h}(E)$. The point is that in practice one should use ${\operatorname{Def}}^{\h}(E)$ for a h-projective $E$ and ${\operatorname{coDef}}^{\h}(E)$ for a h-injective $E$ (see Section 11). For an artinian DG algebra ${{\mathcal R}}$ recall the ${{\mathcal R}}^{op}$-module ${{\mathcal R}}^* =\Hom _k({{\mathcal R}},k)$. Let ${{\mathcal R}}$ be an artinian DG algebra. An object in the groupoid ${\operatorname{coDef}}^{\h}_{{{\mathcal R}}}(E)$ is a pair $(T, \tau)$, where $T$ is a DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module and $\tau :E\to i^!T$ is an isomorphism of DG ${{\mathcal A}}^{op}$-modules so that the following holds: there exists an isomorphism of graded ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-modules $\delta :T^{\gr}\to (E\otimes {{\mathcal R}}^*)^{\gr}$ such that the composition $$E \stackrel{\tau}{\to}i^!T \stackrel{i^!(\delta)}{\to} i^!(E\otimes {{\mathcal R}}^*) =E$$ is the identity. Given objects $(T,\tau)$ and $(T^\prime,\tau ^\prime)\in {\operatorname{coDef}}^h_{{{\mathcal R}}}(E)$ a map $g:(T,\tau)\to (T^\prime ,\tau ^\prime)$ is an isomorphism $f:T\to T^\prime$ such that $i^!f \cdot \tau =\tau ^\prime$. An allowable homotopy between maps $f,g$ is a homotopy $h:f\to g$ such that $i^!(h)=0$. We define morphisms in ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(E)$ to be classes of maps modulo allowable homotopies. Note that a homomorphism of DG algebras $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ induces the functor $\phi ^!:{\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(E)\to {\operatorname{coDef}}_{{{\mathcal Q}}}^{\h}(E)$. This defines the pseudo-functor $${\operatorname{coDef}}^{\h}(E):{\operatorname{dgart}}\to {\bf Gpd}.$$ We refer to objects of ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h} (E)$ as homotopy ${{\mathcal R}}$-co-deformations of $E$. For example we can take $T=E\otimes {{\mathcal R}}^*$ with the differential $d_{E,R^*}:=d_E\otimes 1+1\otimes d_{R^*}$ (and $\tau ={\operatorname{id}}$). This we consider as the [*trivial*]{} ${{\mathcal R}}$-co-deformation of $E$. Denote by ${\operatorname{coDef}}_+^{\h}(E)$, ${\operatorname{coDef}}_-^{\h}(E)$, ${\operatorname{coDef}}_0^{\h}(E)$, ${\operatorname{coDef}}_{{\operatorname{cl}}}^{\h}(E)$ the restrictions of the pseudo-functor ${\operatorname{coDef}}^{\h}(E)$ to subcategories ${\operatorname{dgart}}_+$, ${\operatorname{dgart}}_-$, ${\operatorname{art}}$, ${\operatorname{cart}}$ respectively. There exists a natural equivalence of pseudo-functors $$\delta =\delta ^E:{\operatorname{Def}}^{\h} (E)\to {\operatorname{coDef}}^{\h}(E).$$ We use Lemma 3.9 above. Namely, let $S$ be an ${{\mathcal R}}$-deformation of $E$. Then $S\otimes _{{{\mathcal R}}}{{\mathcal R}}^*$ is an ${{\mathcal R}}$-co-deformation of $E$. Conversely, given an ${{\mathcal R}}$-co-deformation $T$ of $E$ the DG ${{\mathcal A}}^{op}_{{{\mathcal R}}}$-module $\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^*,T)$ is an ${{\mathcal R}}$-deformation of $E$. This defines mutually inverse equivalences $\delta _{{{\mathcal R}}}$ and $\delta _{{{\mathcal R}}}^{-1}$ between the groupoids ${\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E)$ and ${\operatorname{coDef}}^{\h} _{{{\mathcal R}}}(E)$, which extend to morphisms between pseudo-functors ${\operatorname{Def}}^{\h} (E)$ and ${\operatorname{coDef}}^{\h}(E)$. Let us be a little more explicit. Let $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ be a homomorphism of artinian DG algebras and $S\in {\operatorname{Def}}^{\h}(E)$. Then $$\delta _{{{\mathcal Q}}} \cdot \phi ^*(S)=S\otimes _{{{\mathcal R}}}Q\otimes _{{{\mathcal Q}}}{{\mathcal Q}}^*=S\otimes _{{{\mathcal R}}}{{\mathcal Q}}^*,\quad \quad \phi ^!\cdot \delta _{{{\mathcal R}}}(S)=\Hom _{{{\mathcal R}}^{op}}({{\mathcal Q}},S\otimes _{{{\mathcal R}}}{{\mathcal R}}^*).$$ The isomorphism $\alpha _{\phi}$ of these DG ${{\mathcal A}}_{{{\mathcal Q}}}^{op}$-modules is defined by $\alpha _{\phi}(s\otimes f)(q)(r):=sf(q\phi (r))$ for $s\in S$, $f\in {{\mathcal Q}}^*$, $q\in {{\mathcal Q}}$, $r\in {{\mathcal R}}$. Given another homomorphism $\psi :{{\mathcal Q}}\to {{\mathcal Q}}^\prime$ of DG algebras one checks the cocycle condition $\alpha _{\psi \phi}=\psi ^!(\alpha _{\phi})\cdot \alpha _{\psi}$ (under the natural isomorphisms $(\psi \phi)^*=\psi ^* \phi ^*$, $(\psi \phi)^!=\psi ^! \phi ^!$). Maurer-Cartan pseudo-functor ============================= For a DG algebra ${{\mathcal C}}$ with the differential $d$ consider the (inhomogeneous) quadratic map $$Q:{{\mathcal C}}^1 \to {{\mathcal C}}^2; \quad Q(\alpha )=d\alpha +\alpha ^2.$$ We denote by $MC({{\mathcal C}})$ the (usual) Maurer-Cartan cone $$MC({{\mathcal C}})=\{ \alpha \in {{\mathcal C}}^1\vert Q(\alpha )=0\}.$$ Note that $\alpha \in MC({{\mathcal C}})$ is equivalent to the operator $d+\alpha :{{\mathcal C}}\to {{\mathcal C}}$ having square zero. Thus the set $MC({{\mathcal C}})$ describes the space of “internal” deformations of the differential in the complex ${{\mathcal C}}$. Let ${{\mathcal B}}$ be a DG algebra with the differential $d$ and a nilpotent DG ideal ${{\mathcal I}}\subset {{\mathcal B}}$. We define the Maurer-Cartan groupoid ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})$ as follows. The set of objects of ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})$ is the cone $MC({{\mathcal I}})$. Maps between objects are defined by means of the gauge group $G({{\mathcal B}},{{\mathcal I}}):=1+{{\mathcal I}}^0$ (${{\mathcal I}}^0$ is the degree zero component of ${{\mathcal I}}$) acting on ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})$ by the formula $$g:\alpha \mapsto g\alpha g^{-1}+gd(g^{-1}),$$ where $g\in G({{\mathcal B}},{{\mathcal I}})$, $\alpha \in MC({{\mathcal I}})$. (This comes from the conjugation action on the space of differentials $g:d+\alpha \mapsto g(d+\alpha )g^{-1}$.) So if $g(\alpha)=\beta$, we call $g$ a map from $\alpha $ to $\beta$. Denote by $G(\alpha ,\beta)$ the collection of such maps. We define the set $\Hom (\alpha , \beta)$ in the category ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})$ to consist of homotopy classes of maps, where the homotopy relation is defined as follows. There is an action of the group ${{\mathcal I}}^{-1}$ on the set $G(\alpha ,\beta)$: $$h:g\mapsto g+d(h)+\beta h+h\alpha,$$ for $h\in {{\mathcal I}}^{-1}, g\in G(\alpha ,\beta)$. We call two maps [*homotopic*]{}, if they lie in the same ${{\mathcal I}}^{-1}$-orbit. To make the category ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})$ well defined we need to prove a lemma. Let $\alpha _1, \alpha _2, \alpha _3, \alpha _4\in MC({{\mathcal I}})$ and $g_1\in G(\alpha _1,\alpha _2)$, $g_1,g_3\in G(\alpha _2,\alpha _3)$, $g_4\in G(\alpha _3 ,\alpha _4)$. If $g_2$ and $g_3$ are homotopic, then so are $g_2g_1$ and $g_3g_1$ (resp. $g_4g_2$ and $g_4g_3$). Omit. Let ${{\mathcal C}}$ be another DG algebra with a nilpotent DG ideal ${{\mathcal J}}\subset {{\mathcal C}}$. A homomorphism of DG algebras $\psi :{{\mathcal B}}\to {{\mathcal C}}$ such that $\psi ({{\mathcal I}})\subset {{\mathcal J}}$ induces the functor $$\psi ^*:{{\mathcal M}}{{\mathcal C}}({{\mathcal B}},{{\mathcal I}})\to {{\mathcal M}}{{\mathcal C}}({{\mathcal C}},{{\mathcal J}}).$$ Let ${{\mathcal B}}$ be a DG algebra and ${{\mathcal R}}$ be an artinian DG algebra with the maximal ideal $m\subset {{\mathcal R}}$. Denote by ${{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$ the Maurer-Cartan groupoid ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}}\otimes {{\mathcal R}},{{\mathcal B}}\otimes m)$. A homomorphism of artinian DG algebras $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ induces the functor $\phi ^*:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {{\mathcal M}}{{\mathcal C}}_{{{\mathcal Q}}}({{\mathcal B}})$. Thus we obtain the Maurer-Cartan pseudo-functor $${{\mathcal M}}{{\mathcal C}}({{\mathcal B}}):{\operatorname{dgart}}\to {\bf Gpd}.$$ We denote by ${{\mathcal M}}{{\mathcal C}}_+({{\mathcal B}})$, ${{\mathcal M}}{{\mathcal C}}_-({{\mathcal B}})$, ${{\mathcal M}}{{\mathcal C}}_0({{\mathcal B}})$, ${{\mathcal M}}{{\mathcal C}}_{{\operatorname{cl}}}({{\mathcal B}})$ the restrictions of the pseudo-functor ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}})$ to subcategories ${\operatorname{dgart}}_+$, ${\operatorname{dgart}}_-$, ${\operatorname{art}}$, ${\operatorname{cart}}$. A homomorphism of DG algebras $\psi:{{\mathcal C}}\to {{\mathcal B}}$ induces a morphism of pseudo-functors $$\psi ^*:{{\mathcal M}}{{\mathcal C}}({{\mathcal C}})\to {{\mathcal M}}{{\mathcal C}}({{\mathcal B}}).$$ Description of pseudo-functors ${\operatorname{Def}}^{\h}(E)$ and ${\operatorname{coDef}}^{\h}(E)$ ================================================================================================== We are going to give a description of the pseudo-functor ${\operatorname{Def}}^{\h}$ and hence also of the pseudo-functor ${\operatorname{coDef}}^{\h}$ via the Maurer-Cartan pseudo-functor ${{\mathcal M}}{{\mathcal C}}$. Let ${{\mathcal A}}$ be a DG category and $E\in {{\mathcal A}}^{op}\text{-mod}$. Denote by ${{\mathcal B}}$ the DG algebra $\End(E)$. Then there exists an equivalence of pseudo-functors $\theta =\theta ^E: {{\mathcal M}}{{\mathcal C}}({{\mathcal B}})\to {\operatorname{Def}}^{\h}(E)$. (Hence also ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}})$ and ${\operatorname{coDef}}^{\h}(E)$ are equivalent.) Fix an artinian DG algebra ${{\mathcal R}}$ with the maximal ideal $m$. Let us define an equivalence of groupoids $$\theta _{{{\mathcal R}}}:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E).$$ Denote by $S_0=p^*E\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ the trivial ${{\mathcal R}}$-deformation of $E$ with the differential $d_{E,{{\mathcal R}}}=d_E\otimes 1+1\otimes d_{{{\mathcal R}}}$. There is a natural isomorphism of DG algebras $\End(S_0)={{\mathcal B}}\otimes {{\mathcal R}}$. Let $\alpha \in {{\mathcal M}}{{\mathcal C}}({{\mathcal B}}\otimes m)={{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$. Then in particular $\alpha \in \End ^1(S_0)$. Hence $d_{\alpha}:=d_{E,{{\mathcal R}}}+\alpha$ is an endomorphism of degree 1 of the graded module $S_0^{\gr}$. The Maurer-Cartan condition on $\alpha$ is equivalent to $d_{\alpha}^2=0$. Thus we obtain an object $S_{\alpha}\in {{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$. Clearly $i^*S_{\alpha}=E$, so that $$\theta _{{{\mathcal R}}}(\alpha):=(S_{\alpha},{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(E).$$ One checks directly that this map on objects extends naturally to a functor $\theta _{{{\mathcal R}}}:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E)$. Indeed, maps between Maurer-Cartan objects induce isomorphisms of the corresponding deformations; also homotopies between such maps become allowable homotopies between the corresponding isomorphisms. It is clear that the functors $\theta _{{{\mathcal R}}}$ are compatible with the functors $\phi ^*$ induced by morphisms of DG algebras $\phi :{{\mathcal R}}\to {{\mathcal Q}}$. So we obtain a morphism of pseudo-functors $$\theta :{{\mathcal M}}{{\mathcal C}}({{\mathcal B}})\to {\operatorname{Def}}^{\h}(E).$$ It suffices to prove that $\theta _{{{\mathcal R}}}$ is an equivalence for each ${{\mathcal R}}$. [**Surjective.**]{} Let $(T,\tau )\in {\operatorname{Def}}^{h}_{{{\mathcal R}}}(E)$. We may and will assume that $T^{gr}=S_0^{gr}$ and $\tau ={\operatorname{id}}$. Then $\alpha _T:=d_T-d_{{{\mathcal R}},E}\in \End ^1(S_0)=({{\mathcal B}}\otimes {{\mathcal R}})^1$ is an element in $MC({{\mathcal B}}\otimes {{\mathcal R}})$. Since $i^*\alpha _T=0$ it follows that $\alpha _T\in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$. Thus $(T,\tau )=\theta _{{{\mathcal R}}}(\alpha _T)$. [**Full.**]{} Let $\alpha, \beta \in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$. An isomorphism between the corresponding objects $\theta _{{{\mathcal R}}}(\alpha)$ and $\theta _{{{\mathcal R}}}(\beta)$ is defined by an element $f\in \End (S_0)=({{\mathcal B}}\otimes {{\mathcal R}})$ of degree zero. The condition $i^*f={\operatorname{id}}_Z$ means that $f\in 1+({{\mathcal B}}\otimes m)^0$. Thus $f\in G(\alpha ,\beta)$. [**Faithful.**]{} Let $\alpha, \beta \in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$ and $f,g\in G(\alpha ,\beta)$. One checks directly that $f$ and $g$ are homotopic (i.e. define the same morphism in ${{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$) if and only if there exists an allowable homotopy between $\theta _{{{\mathcal R}}}(f)$ and $\theta _{{{\mathcal R}}}(g)$. This proves the proposition. For $E\in {{\mathcal A}}^{op}\text{-mod}$ the pseudo-functors ${\operatorname{Def}}^{\h}(E)$ and ${\operatorname{coDef}}^{\h}(E)$ depend (up to equivalence) only on the DG algebra $\End(E)$. We will prove a stronger result in Corollary 8.2 below. Let $E\in {{\mathcal A}}^{op}\text{-mod}$ and denote ${{\mathcal B}}=\End(E)$. Consider ${{\mathcal B}}$ as a (free) right ${{\mathcal B}}$-module, i.e. ${{\mathcal B}}\in {{\mathcal B}}^{op}\text{-mod}$. Then ${\operatorname{Def}}^h({{\mathcal B}})\simeq {\operatorname{Def}}^h(E)$ ($\simeq {\operatorname{coDef}}^h({{\mathcal B}})\simeq {\operatorname{coDef}}^h(E)$) because $\End ({{\mathcal B}})=\End (E)={{\mathcal B}}$. We will describe this equivalence directly in Section 9 below. Obstruction Theory ================== It is convenient to describe the obstruction theory for our (equivalent) deformation pseudo-functors ${\operatorname{Def}}^h$ and ${\operatorname{coDef}}^h$ using the Maurer-Cartan pseudo-functor ${{\mathcal M}}{{\mathcal C}}({{\mathcal B}})$ for a fixed DG algebra ${{\mathcal B}}$. Let ${{\mathcal R}}$ be an artinian DG algebra with a maximal ideal $m$, such that $m^{n+1}=0$. Put $I=m^n$, $\overline{{{\mathcal R}}}={{\mathcal R}}/I$ and $\pi :{{\mathcal R}}\to \overline{{{\mathcal R}}}$ the projection morphism. We have $mI=Im=0$. Note that the kernel of the homomorphism $1 \otimes \pi:{{\mathcal B}}\otimes {{\mathcal R}}\to {{\mathcal B}}\otimes \overline{{{\mathcal R}}}$ is the (DG) ideal ${{\mathcal B}}\otimes I$. The next proposition describes the obstruction theory for lifting objects and morphisms along the functor $$\pi^*:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}}).$$ It is close to [@GoMi]. Note however a difference in part 3) and part 4) since we do not assume that out DG algebras live in nonnegative dimensions (and of course we work with DG algebras and not with DG Lie algebras). 1). There exists a map $o_2:Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}})\to H^2({{\mathcal B}}\otimes I)$ such that $\alpha \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}})$ is in the image of $\pi ^*$ if and only if $o_2(\alpha)=0$. Furthermore if $\alpha ,\beta \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}})$ are isomorphic, then $o_2(\alpha)=0$ if and only if $o_2(\beta)=0$. 2). Let $\xi \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}})$. Assume that the fiber $(\pi ^*)^{-1}(\xi)$ is not empty. Then there exists a simply transitive action of the group $Z^1({{\mathcal B}}\otimes I)$ on the set $Ob(\pi ^*)^{-1}(\xi)$. Moreover the composition of the difference map $$Ob(\pi ^*) ^{-1}(\xi)\times Ob(\pi ^*) ^{-1}(\xi)\to Z^1({{\mathcal B}}\otimes I)$$ with the projection $$Z^1({{\mathcal B}}\otimes I)\to H^1({{\mathcal B}}\otimes I)$$ which we denote by $$o_1:Ob(\pi ^*)^{-1}(\xi)\times Ob(\pi ^*)^{-1}(\xi)\to H^1({{\mathcal B}}\otimes I)$$ has the following property: for $\alpha ,\beta \in Ob(\pi ^*)^{-1}(\xi)$ there exists a morphism $\gamma :\alpha \to \beta$ s.t. $\pi ^*(\gamma)={\operatorname{id}}_{\xi}$ if and only if $o_1(\alpha ,\beta)=0$. 3). Let $\tilde{\alpha },\tilde{\beta}\in Ob{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$ be isomorphic objects and let $f:\alpha \to \beta$ be a morphism from $\alpha =\pi ^*(\tilde{\alpha})$ to $\beta =\pi ^*(\tilde{\beta})$. Then there is a transitive action of the group $H^0({{\mathcal B}}\otimes I)$ on the set $(\pi ^*)^{-1}(f)$ of morphisms $\tilde{f}:\tilde{\alpha}\to \tilde{\beta}$ such that $\pi ^*(\tilde{f})=f$. 4). In the notation of 3) suppose that the fiber $(\pi^*)^{-1}(f)$ is non-empty. Then the kernel of the above action coincides with the kernel of the map $$\label{kernel} H^0({{\mathcal B}}\otimes I)\to H^0({{\mathcal B}}\otimes m, d^{\alpha,\beta}),$$ where $d^{\alpha,\beta}$ is a differential on the graded vector space ${{\mathcal B}}\otimes m$ given by the formula $$d^{\alpha,\beta}(x)=dx+\beta x-(-1)^{\bar{x}}x\alpha.$$ In particular the difference map $$o_0:(\pi ^*)^{-1}(f)\times (\pi ^*)^{-1}(f)\to {\operatorname{Im}}(H^0({{\mathcal B}}\otimes I)\to H^0({{\mathcal B}}\otimes m,d^{\alpha,\beta}))$$ has the property: if $\tilde{f},\tilde{f}^\prime\in (\pi ^*)^{-1}(f)$, then $\tilde{f}=\tilde{f}^\prime$ if and only if $o_0(\tilde{f},\tilde{f}^\prime)=0$. 1\) Let $\alpha \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}})=MC({{\mathcal B}}\otimes (m/I))$. Choose $\tilde{\alpha}\in ({{\mathcal B}}\otimes m)^1$ such that $\pi (\tilde{\alpha})=\alpha$. Consider the element $$Q(\tilde{\alpha})=d\tilde{\alpha}+\tilde{\alpha}^2\in ({{\mathcal B}}\otimes m)^2.$$ Since $Q(\alpha)=0$ we have $Q(\tilde{\alpha})\in ({{\mathcal B}}\otimes I)^2$. We claim that $dQ(\tilde{\alpha})=0$. Indeed, $$dQ(\tilde{\alpha})=d(\tilde{\alpha}^2)=d(\tilde{\alpha})\tilde{\alpha}-\tilde{\alpha}d(\tilde{\alpha}).$$ We have $d(\tilde{\alpha})\equiv \tilde{\alpha}^2(mod({{\mathcal B}}\otimes I)).$ Hence $dQ(\tilde{\alpha})=-\tilde{\alpha}^3+\tilde{\alpha}^3=0$ (since $I\cdot m=0$). Furthermore suppose that $\tilde{\alpha}^\prime\in ({{\mathcal B}}\otimes m)^1$ is another lift of $\alpha$, i.e. $\tilde{\alpha}^\prime -\tilde{\alpha}\in ({{\mathcal B}}\otimes I)^1$. Then $$Q(\tilde{\alpha}^\prime)-Q(\tilde{\alpha})=d(\tilde{\alpha}^\prime-\tilde{\alpha})+ (\tilde{\alpha}^\prime -\tilde{\alpha})(\tilde{\alpha}^\prime +\tilde{\alpha})=d(\tilde{\alpha}^\prime -\tilde{\alpha}).$$ Thus the cohomology class of the cocycle $Q(\tilde{\alpha})$ is independent of the lift $\tilde{\alpha}$. We denote this class by $o_2(\alpha)\in H^2({{\mathcal B}}\otimes I)$. If $\alpha =\pi ^*(\tilde{\alpha})$ for some $\tilde{\alpha}\in Ob{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$, then clearly $o_2(\alpha)=0$. Conversely, suppose $o_2(\alpha)=0$ and let $\tilde{\alpha}$ be as above. Then $dQ(\tilde{\alpha})=d\tau$ for some $\tau \in ({{\mathcal B}}\otimes I)^1$. Put $\tilde{\alpha}^\prime=\tilde{\alpha}-\tau$. Then $$Q(\tilde{\alpha}^\prime)=d\tilde{\alpha}-d\tau +\tilde{\alpha}^2-\tilde{\alpha}\tau -\tau \tilde{\alpha}+\tau ^2=Q(\tilde{\alpha})-d\tau=0.$$ Let us prove the last assertion in 1). Assume that $\pi ^*(\tilde{\alpha})=\alpha$ and $\beta =g(\alpha)$ for some $g\in 1+({{\mathcal B}}\otimes m/I)^0$. Choose a lift $\tilde{g}\in 1+({{\mathcal B}}\otimes m)^0$ of $g$ and put $\tilde{\beta}:=\tilde{g}(\tilde{\alpha})$. Then $\pi ^*(\tilde{\beta})=\beta$. This proves 1). 2). Let $\alpha \in Ob(\pi ^*)^{-1}(\xi)$ and $\eta \in Z^1({{\mathcal B}}\otimes I)$. Then $$Q(\alpha +\eta)=d\alpha +d\eta +\alpha ^2 +\alpha \eta +\eta \alpha +\eta ^2=Q(\alpha)+d\eta =0.$$ So $\alpha +\eta \in Ob(\pi ^*)^{-1}(\xi)$. This defines the action of the group $Z^1({{\mathcal B}}\otimes I)$ on the set $Ob(\pi ^*)^{-1}(\xi)$. Let $\alpha ,\beta \in Ob(\pi ^*)^{-1}(\xi)$. Then $\alpha -\beta \in ({{\mathcal B}}\otimes I)^1$ and $$d(\alpha -\beta)=d\alpha -d\beta +\beta (\alpha -\beta)+(\alpha -\beta)\beta +(\alpha -\beta )^2=Q(\alpha )-Q(\beta )=0.$$ Thus $Z^1({{\mathcal B}}\otimes I)$ acts simply transitively on $Ob(\pi ^*)^{-1}(\xi)$. Now let $o_1(\alpha ,\beta)\in H^1({{\mathcal B}}\otimes I)$ be the cohomology class of $\alpha -\beta$. We claim that there exists a morphism $\gamma :\alpha \to \beta$ covering ${\operatorname{id}}_{\xi}$ if and only if $o_1(\alpha ,\beta)=0$. Indeed, let $\gamma$ be such a morphism. Then by definition the morphisms $\pi ^*(\gamma)$ and ${\operatorname{id}}_{\xi}$ are homotopic. That is there exists $h\in ({{\mathcal B}}\otimes (m/I))^{-1}$ such that $${\operatorname{id}}_{\xi}=\pi ^*(\gamma)+d(h)+\xi h+h\xi.$$ Choose a lifting $\tilde{h}\in ({{\mathcal B}}\otimes m)^{-1}$ on $h$ and replace the morphism $\gamma$ by the homotopical one $$\delta =\gamma +d(\tilde{h})+\beta \tilde{h}+\tilde{h}\alpha.$$ Thus $\delta =1+u$, where $u\in ({{\mathcal B}}\otimes I)^0$. But then $$\beta =\delta \alpha \delta ^{-1}+\delta d(\delta ^{-1})=\alpha -du,$$ so that $o_1(\alpha ,\beta)=0$. Conversely, let $\alpha -\beta=du$ for some $u\in ({{\mathcal B}}\otimes I)^0$. Then $\delta =1+u$ is a morphism from $\alpha $ to $\beta$ and $\pi ^*(\delta)={\operatorname{id}}_{\xi}$. This proves 2). 3). Let us define the action of the group $Z^0({{\mathcal B}}\otimes I)$ on the set $(\pi ^*)^{-1}(f)$. Let $\tilde{f}:\tilde{\alpha}\to \tilde{\beta}$ be a lift of $f$, and $v\in Z^0({{\mathcal B}}\otimes I)$. Then $\tilde{f}+v$ also belongs to $(\pi ^*)^{-1}(f)$. If $v=du$ for $u\in ({{\mathcal B}}\otimes I )^{-1}$, then $$\tilde{f}+v=\tilde{f}+du+\tilde{\beta} u+u \tilde{\alpha}$$ and hence morphisms $\tilde{f}$ and $\tilde{f}+v$ are homotopic. This induces the action of $H^0({{\mathcal B}}\otimes I)$ on the set $(\pi ^*)^{-1}(f)$. To show that this action is transitive let $\tilde{f}^\prime :\tilde{\alpha }\to \tilde{\beta}$ be another morphism in $(\pi ^*)^{-1}(f)$. This means by definition that there exists $h\in ({{\mathcal B}}\otimes (m/I))^{-1}$ such that $$f=\pi ^*(\tilde{f}^\prime)+dh+\beta h+h\alpha.$$ Choose a lifting $\tilde{h}\in ({{\mathcal B}}\otimes m)^{-1}$ of $h$ and replace $\tilde{f}^\prime$ by the homotopical morphism $$\tilde{g}=\tilde{f}^\prime+d\tilde{h}+\tilde{\beta}\tilde{h}+ \tilde{h}\tilde{\alpha}.$$ Then $\tilde{g}=\tilde{f}+v$ for $v\in ({{\mathcal B}}\otimes I)^0$. Since $\tilde{f}, \tilde{g}:\tilde{\alpha}\to \tilde{\beta}$ we must have that $v\in Z^0({{\mathcal B}}\otimes I)$. This shows the transitivity and proves 3). 4). Suppose that for some $v\in Z^0({{\mathcal B}}\otimes I)$ and for some $\tilde{f}\in (\pi^*)^{-1}(f)$ we have that $\tilde{f}+v=\tilde{f}$. This means, by definition, that there exists an element $h\in ({{\mathcal B}}\otimes m)^{-1}$ such that $d^{\alpha,\beta}(h)=v$. In other words, the class $[v]\in H^0({{\mathcal B}}\otimes I)$ lies in the kernel of the map (\[kernel\]). This proves 4). Invariance theorem and its implications ======================================= Let $\phi :{{\mathcal B}}\to {{\mathcal C}}$ be a quasi-isomorphism of DG algebras. Then the induced morphism of pseudo-functors $$\phi ^*:{{\mathcal M}}{{\mathcal C}}({{\mathcal B}})\to {{\mathcal M}}{{\mathcal C}}({{\mathcal C}})$$ is an equivalence. The proof is almost the same as that of Theorem 2.4 in [@GoMi]. We present it for reader’s convenience and also because of the slight difference in language: in [@GoMi] they work with DG Lie algebras as opposed to DG algebras. Fix an artinian DG algebra ${{\mathcal R}}$ with the maximal ideal $m\subset {{\mathcal R}}$, such that $m^{n+1}=0$. We prove that $$\phi ^*:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})$$ is an equivalence by induction on $n$. If $n=o$, then both groupoids contain one object and one morphism, so are equivalent. Let $n>0$. Put $I=m^n$ with the projection $\pi :{{\mathcal R}}\to {{\mathcal R}}/I=\overline{{{\mathcal R}}}$. We have the commutative functorial diagram $$\begin{array}{ccc} {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}}) & \stackrel{\phi ^*}{\rightarrow} & {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})\\ \pi ^*\downarrow & & \downarrow \pi ^*\\ {{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal B}}) & \stackrel{\phi ^*}{\rightarrow} & {{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal C}}). \end{array}$$ By induction we may assume that the bottom functor is an equivalence. To prove the same about the top one we need to analyze the fibers of the functor $\pi ^*$. This has been done by the obstruction theory. We will prove that the functor $$\phi ^*:{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})\to {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})$$ is surjective on the isomorphism classes of objects, is full and is faithful. [**Surjective on isomorphism classes.**]{} Let $\beta \in Ob{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})$. Then $\pi ^*\beta \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}} }({{\mathcal C}})$. By the induction hypothesis there exists $\alpha ^\prime \in Ob{{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal C}})$ and an isomorphism $g: \phi ^*\alpha ^\prime \to \pi ^* \beta$. Now $$H^2(\phi)o_2(\alpha ^\prime)=o_2(\phi ^*\alpha ^\prime )= o_2(\pi ^*\beta )=0.$$ Hence $o_2(\alpha ^\prime)=0$, so there exists $\tilde{\alpha }\in Ob{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$ such that $\pi ^*\tilde{\alpha}=\alpha ^\prime$, and hence $$\phi ^*\pi ^*\tilde{\alpha}=\pi ^*\phi ^*\tilde{\alpha}=\phi ^*\alpha ^\prime.$$ Choose a lift $\tilde{g}\in 1+({{\mathcal C}}\otimes m)^0$ of $g$ and put $\tilde{\beta}=\tilde{g}^{-1}(\beta)$. Then $$\pi ^*(\tilde{\beta})=\pi ^*(\tilde{g}^{-1}(\beta))=g^{-1}\pi ^*\beta=\phi ^*\alpha ^\prime.$$ The obstruction to the existence of an isomorphism $\phi ^*\tilde{\alpha}\to \tilde{\beta}$ covering ${\operatorname{id}}_{\pi^*(\alpha ^\prime)}$ is an element $o_1(\phi ^*(\tilde{\alpha}), \tilde{\beta})\in H^1({{\mathcal C}}\otimes I)$. Since $H^1(\phi)$ is surjective there exists a cocycle $u\in Z^1({{\mathcal B}}\otimes I)$ such that $H^1(\phi)[u]=o_1(\phi ^*(\tilde{\alpha}),\tilde{\beta})$. Put $\alpha =\tilde{\alpha}-u\in Ob{{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$. Then $$\begin{array}{rcl}o_1(\phi^*\alpha ,\tilde{\beta}) & = & o_1(\phi ^*\alpha ,\phi ^*\tilde{\alpha})+o_1(\phi ^*\tilde{\alpha},\tilde{\beta})\\ & = & H^1(\phi)o_1(\alpha ,\tilde{\alpha})+o_1(\phi^*\tilde{\alpha},\tilde{\beta})\\ & = & -H^1(\phi)[u]+o_1(\phi ^*\tilde{\alpha},\beta)=0 \end{array}$$ This proves the surjectivity of $\phi ^*$ on isomorphism classes. [**Full.**]{} Let $f:\phi ^*\alpha _1\to \phi ^*\alpha _2$ be a morphism in ${{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})$. Then $\pi ^*f$ is a morphism in ${{\mathcal M}}{{\mathcal C}}_{\overline{{{\mathcal R}}}}({{\mathcal C}})$: $$\pi ^*(f):\phi^*\pi^*\alpha _1\to \phi ^*\pi ^*\alpha _2.$$ By induction hypothesis there exists $g:\pi ^*\alpha _1\to \pi ^*\alpha _2$ such that $\phi ^*(g)=\pi^*(f)$. Let $\tilde{g}\in 1+({{\mathcal C}}\otimes m)^0$ be any lift of $g$. Then $\pi ^*(\tilde{g}\alpha _1)=\pi ^*\alpha _2$. The obstruction to the existence of a morphism $\gamma :\tilde{g}\alpha _1\to \alpha _2$ covering ${\operatorname{id}}_{\pi ^*\alpha _2}$ is an element $o_1(\tilde{g}\alpha _1,\alpha _2)\in H^1({{\mathcal B}}\otimes I)$. By assumption $H^1(\phi)$ is an isomorphism and we know that $$H^1(\phi)(o_1(\tilde{g}\alpha _1,\alpha _2))=o_1(\phi^*\tilde{g}\alpha _1,\phi ^*\alpha _2)=0,$$ since the morphism $f\cdot (\phi ^*\tilde{g})^{-1}$ is covering the identity morphism ${\operatorname{id}}_{\pi ^*\phi ^*\alpha _2}$. Thus $o_1(\tilde{g}\alpha _1,\alpha _2)=0$ and $\gamma $ exists. Then $\gamma \cdot \tilde{g}:\alpha _1\to \alpha _2$ is covering $g:\pi ^*\alpha _1\to \pi^*\alpha _2$. Hence both morphisms $\phi ^*(\gamma \cdot \tilde{g})$ and $f$ are covering $\pi ^*(f)$. The obstruction to their equality is an element $o_0(\phi ^*(\gamma \cdot \tilde{g}),f)\in {\operatorname{Im}}(H^0({{\mathcal C}}\otimes I)\to H^0({{\mathcal C}}\otimes m))$. Let $v\in H^0({{\mathcal C}}\otimes I)$ be a representative of this element and $u\in Z^0({{\mathcal B}}\otimes I)$ be a representative of the inverse image of $v$ under $H^0(\phi)$. Then $\phi ^*(\gamma \cdot \tilde{g}+u)=f$. [**Faithful.**]{} Let $\gamma _1,\gamma _2:\alpha _1\to \alpha _2$ be morphisms in ${{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal B}})$ with $\phi^*\gamma _1=\phi^*\gamma _2$. Then $\phi ^*\pi ^*\gamma _1=\phi ^*\pi ^*\gamma _2$. By the induction hypothesis $\pi ^*\gamma _1=\pi ^*\gamma _2$, so the obstruction $o_0(\gamma _1,\gamma _2)\in {\operatorname{Im}}(H^0({{\mathcal B}}\otimes I)\to H^0({{\mathcal B}}\otimes m,d^{\alpha_1,\alpha_2}))$ is defined. Now the image of $o_0(\gamma _1,\gamma _2)$ under the map$$\label{iso}{\operatorname{Im}}(H^0({{\mathcal B}}\otimes I)\to H^0({{\mathcal B}}\otimes m,d^{\alpha_1,\alpha_2})) \to {\operatorname{Im}}(H^0({{\mathcal C}}\otimes I)\to H^0({{\mathcal C}}\otimes m,d^{\phi^*\alpha_1,\phi^*\alpha_2}))$$ equals to $o_0(\phi^*\gamma _1,\phi^*\gamma _2)=0$. So it remains to prove that the map (\[iso\]) is an isomorphism. Clearly, it is sufficient to prove that the morphism of complexes $$\phi_{{{\mathcal R}}}^{\alpha_1,\alpha_2}:({{\mathcal B}}\otimes m,d^{\alpha_1,\alpha_2}))\to ({{\mathcal C}}\otimes m,d^{\phi^*\alpha_1,\phi^*\alpha_2}))$$ is a quasi-isomorphism. Note that these complexes have finite filtrations by subcomplexes ${{\mathcal B}}\otimes m^i$ and ${{\mathcal C}}\otimes m^i$ respectively. The morphism $\phi_{{{\mathcal R}}}^{\alpha_1,\alpha_2}$ is compatible with these filtrations and induces quasi-isomorphisms on the subquotients. Hence $\phi_{{{\mathcal R}}}^{\alpha_1,\alpha_2}$ is a quasi-isomorphism. This proves the theorem. The homotopy (co-) deformation pseudo-functor of $E\in {{\mathcal A}}^{op}\text{-mod}$ depends (up to equivalence) only on the quasi-isomorphism class of the DG algebra $\End (E)$. This follows from Theorem 8.1 and Proposition 6.1. The next proposition provides two examples of this situation. It was communicated to us by Bernhard Keller. (Keller) a) Assume that $E^\prime \in {{\mathcal A}}^{op}\text{-mod}$ is homotopy equivalent to $E$. Then the DG algebras $\End (E)$ and $\End(E^\prime )$ are canonically quasi-isomorphic. b\) Let $P\in {{\mathcal P}}({{\mathcal A}}^{op})$ and $I\in {{\mathcal I}}({{\mathcal A}}^{op})$ be quasi-isomorphic. Then the DG algebras $\End(P)$ and $\End(I)$ are canonically quasi-isomorphic. a\) Let $g:E\to E^\prime$ be a homotopy equivalence. Consider its cone $C(g)\in {{\mathcal A}}^{op}\text{-mod}$. Let ${{\mathcal C}}\subset \End (C(g))$ be the DG subalgebra consisting of endomorphisms which leave $E^\prime $ stable. There are natural projections $p:{{\mathcal C}}\to \End(E^\prime)$ and $q:{{\mathcal C}}\to \End (E)$. We claim that $p$ and $q$ are quasi-isomorphisms. Indeed, ${\operatorname{Ker}}(p)$ (resp. ${\operatorname{Ker}}(q)$) is the complex $\Hom (E[1],C(g))$ (resp. $\Hom (C(g),E^\prime)$). These complexes are acyclic, since $g$ is a homotopy equivalence. b\) The proof is similar. Let $f:P\to I$ be a quasi-isomorphism. Then the cone $C(f)$ is acyclic. We consider the DG subalgebra $\D\subset \End (C(f))$ which leaves $I$ stable. Then $\D$ is quasi-isomorphic to $\End(I)$ and $\End(P)$ because the complexes $\Hom (P[1],C(f))$ and $\Hom (C(f),I)$ are acyclic. a\) If DG ${{\mathcal A}}^{op}$-modules $E$ and $E^\prime$ are homotopy equivalent then the pseudo-functors ${\operatorname{Def}}^{\h}(E)$, ${\operatorname{coDef}}^{\h}(E)$, ${\operatorname{Def}}^{\h}(E^\prime)$, ${\operatorname{coDef}}^{\h}(E^\prime)$ are canonically equivalent. b\) Let $P\to I$ be a quasi-isomorphism between $P\in {{\mathcal P}}({{\mathcal A}}^{op})$ and $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Then the pseudo-functors ${\operatorname{Def}}^{\h}(P)$, ${\operatorname{coDef}}^{\h}(P)$, ${\operatorname{Def}}^{\h}(I)$, ${\operatorname{coDef}}^{\h}(I)$ are canonically equivalent. Indeed, this follows from Proposition 8.3 and Corollary 8.2. Actually, one can prove a more precise statement. Fix an artinian DG algebra ${{\mathcal R}}$. a\) Let $g:E\to E^\prime $ be a homotopy equivalence of DG ${{\mathcal A}}^{op}$-modules. Assume that $(V,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E)$ and $(V^\prime,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E^\prime)$ are objects that correspond to each other via the equivalence ${\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E)\simeq {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E^\prime)$ of Corollary 8.4. Then there exists a homotopy equivalence $\tilde{g}:V\to V^\prime$ which extends $g$, i.e. $i^*\tilde{g}=g$. Similarly for the objects of ${\operatorname{coDef}}^{\h}_{{{\mathcal R}}}$ with $i^!$ instead of $i^*$. b\) Let $f:P\to I$ be a quasi-isomorphism with $P\in {{\mathcal P}}({{\mathcal A}}^{op})$, $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Assume that $(S,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)$ and $(T,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(I)$ are objects that correspond to each other via the equivalence ${\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)\simeq {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(I)$ of Corollary 8.4. Then there exists a quasi-isomorphism $\tilde{f}:S\to T$ which extends $f$, i.e. $i^*\tilde{f}=f$. Similarly for the objects of ${\operatorname{coDef}}^{\h}_{{{\mathcal R}}}$ with $i^!$ instead of $i^*$. a\) Consider the DG algebra $${{\mathcal C}}\subset \End(C(g))$$ as in the proof of Proposition 8.3. We proved there that the natural projections $\End (E)\leftarrow {{\mathcal C}}\rightarrow \End(E^\prime)$ are quasi-isomorphisms. Hence the induced functors between groupoids ${{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}(\End (E))\leftarrow {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})\rightarrow {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}(\End (E^\prime))$ are equivalences by Theorem 8.1. Using Proposition 6.1 we may and will assume that deformations $(V, {\operatorname{id}})$, $(V^\prime ,{\operatorname{id}})$ correspond to elements $\alpha _E\in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}(\End (E))$, $\alpha _{E^\prime}\in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}(\End (E^\prime))$ which come from the same element $\alpha \in {{\mathcal M}}{{\mathcal C}}_{{{\mathcal R}}}({{\mathcal C}})$. Consider the DG modules $E\otimes {{\mathcal R}}$, $E^\prime \otimes {{\mathcal R}}$ with the differentials $d_E\otimes 1+1\otimes d_{{{\mathcal R}}}$ and $d_{E^\prime}\otimes 1+1\otimes d_{{{\mathcal R}}}$ respectively and the morphism $g\otimes 1:E\otimes {{\mathcal R}}\to E^\prime \otimes {{\mathcal R}}$. Then $${{\mathcal C}}\otimes {{\mathcal R}}=\left( \begin{array}{cc} \End(E^\prime\otimes {{\mathcal R}}) & \Hom (E[1]\otimes {{\mathcal R}},E^\prime \otimes {{\mathcal R}}) \\ 0 & \End(E\otimes {{\mathcal R}}) \end{array} \right) \subset \End(C(g\otimes 1)),$$ and $$\alpha =\left( \begin{array}{cc} \alpha _{E^\prime} & t\\ 0 & \alpha _{E} \end{array}\right).$$ Recall that the differential in the DG module $C(g\otimes 1)$ is of the form $(d_{E^\prime}\otimes 1, d_E[1]\otimes 1+g[1]\otimes 1)$. The element $\alpha$ defines a new differential $d_{\alpha}$ on $C(g\otimes 1)$ which is $(d_{E^\prime}\otimes 1+\alpha _{E^\prime}, (d_E[1]\otimes 1+\alpha _E)+(g[1]\otimes 1+t))$. The fact that $d_{\alpha }^2=0$ implies that $\tilde{g}:=g\otimes 1+t[-1]:V\to V^\prime$ is a closed morphism of degree zero and hence the DG module $C(g\otimes 1)$ with the differential $d_\alpha$ is the cone $C(\tilde{g})$ of this morphism. Clearly, $i^*\tilde{g}=g$ and it remains to prove that $\tilde{g}$ is a homotopy equivalence. This in turn is equivalent to the acyclicity of the DG algebra $\End(C(\tilde{g}))$. But recall that the differential in $\End(C(\tilde{g}))$ is an “${{\mathcal R}}$-deformation” of the differential in the DG algebra $\End(C(g))$ which is acyclic, since $g$ is a homotopy equivalence. Therefore $\End(C(\tilde{g}))$ is also acyclic. This proves the first statement in a). The last statement follows by the equivalence of groupoids ${\operatorname{Def}}^{\h}_{{{\mathcal R}}}\simeq {\operatorname{coDef}}^{\h}_{{{\mathcal R}}}$ (Proposition 4.7). The proof of b) is similar: exactly in the same way we construct a closed morphism of degree zero $\tilde{f}:S\to T$ which extends $f$. Then $\tilde{f}$ is a quasi-isomorphism, because $f$ is such. Fix an artinian DG algebra ${{\mathcal R}}$. a\) Let $g:E\to E^\prime $ be a homotopy equivalence as in Proposition 8.5a). Let $(V,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E)$ and $(V^\prime,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E^\prime)$ be objects corresponding to each other under the equivalence ${\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E)\simeq {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(E^\prime)$. Then $i^*V=\bL i^*V$ if and only if $i^*V^\prime=\bL i^*V^\prime$. Similarly for the objects of ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}$ with $i^!$ and $\bR i^!$ instead of $i^*$ and $\bL i^*$. b\) Let $f:P\to I$ be a quasi-isomorphism as in Proposition 8.5b). Let $(S,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(P)$ and $(T,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(I)$ be objects which correspond to each other under the equivalence ${\operatorname{Def}}_{{{\mathcal R}}}^{\h}(P)\simeq {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(I).$ Then $i^*S=\bL i^*S$ if and only if $i^*T =\bL i^*T$. Similarly for the objects of ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}$ with $i^!$ and $\bR i^!$ instead of $i^*$ and $\bL i^*$. This follows immediately from Proposition 8.5. Let $F:{{\mathcal A}}\to {{\mathcal C}}$ be a DG functor which induces an equivalence of derived categories $\bL F^*:D({{\mathcal A}}^{op})\to D({{\mathcal C}}^{op})$. (For example, this is the case if $F$ induces a quasi-equivalence $F^{\pre-tr}:{{\mathcal A}}^{\pre-tr}\to {{\mathcal C}}^{\pre-tr}$ (Corollary 3.15)). a\) Let $P\in {{\mathcal P}}({{\mathcal A}}^{op})$. Then the map of DG algebras $F^*:\End (P)\to \End (F^*(P))$ is a quasi-isomorphism. Hence the deformation pseudo-functors ${\operatorname{Def}}^{\h}$ and ${\operatorname{coDef}}^{\h}$ of $P$ and $F^*(P)$ are equivalent. b\) Let $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Then the map of DG algebras $F^!:\End(I)\to \End (F^!(I))$ is a quasi-isomorphism. Hence the deformation pseudo-functors ${\operatorname{Def}}^{\h}$ and ${\operatorname{coDef}}^{\h}$ of $I$ and $F^!(I)$ are equivalent. a\) By Lemma 3.6 we have $F^*(P)\in {{\mathcal P}}({{\mathcal C}}^{op})$. Hence the assertion follows from Theorems 3.1 and 8.1. b\) The functor $\bR F^!:D({{\mathcal A}}^{op})\to D({{\mathcal C}}^{op})$ is also an equivalence because of adjunctions $(F_*,\bR F^!),(\bL F^* ,F_*)$. Also $F^!(I)\in {{\mathcal I}}({{\mathcal C}}^{op})$ (Lemma 3.6). Hence the assertion follows from Theorems 3.1 and 8.1. Direct relation between pseudo-functors ${\operatorname{Def}}^{\h}(F)$ and ${\operatorname{Def}}^{\h}({{\mathcal B}})$ (${\operatorname{coDef}}^{\h}(F)$ and ${\operatorname{coDef}}^{\h}({{\mathcal B}})$) =========================================================================================================================================================================================================== DG functor $\Sigma$ ------------------- Let $F\in {{\mathcal A}}^{op}\text{-mod}$ and put ${{\mathcal B}}=\End(F)$. Recall the DG functor from Example 3.14 $$\Sigma =\Sigma ^F:{{\mathcal B}}^{op}\text{-mod}\to {{\mathcal A}}^{op}\text{-mod},\quad \Sigma (M)=M\otimes _{{{\mathcal B}}}F.$$ For each artinian DG algebra ${{\mathcal R}}$ we obtain the corresponding DG functor $$\Sigma _{{{\mathcal R}}}:({{\mathcal B}}\otimes {{\mathcal R}}) ^{op}\text{-mod}\to {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod},\quad \Sigma _{{{\mathcal R}}}(M)=M\otimes _{{{\mathcal B}}}F.$$ The DG functors $\Sigma _{{{\mathcal R}}}$ have the following properties. a\) If a DG $({{\mathcal B}}\otimes {{\mathcal R}}) ^{op}$-module $M$ is graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree), then so is the DG ${{\mathcal A}}_{{{\mathcal R}}} ^{op}$-module $\Sigma _{{{\mathcal R}}}(M)$. b\) Let $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ be a homomorphism of artinian DG algebras. Then there are natural isomorphisms of DG functors $$\Sigma _{{{\mathcal Q}}}\cdot \phi ^*=\phi ^*\cdot \Sigma _{{{\mathcal R}}}, \quad \Sigma _{{{\mathcal R}}}\cdot \phi _*=\phi _*\cdot \Sigma _{{{\mathcal Q}}}.$$ In particular, $$\Sigma \cdot i ^*=i ^*\cdot \Sigma _{{{\mathcal R}}}.$$ c\) There is a natural isomorphism of DG functors $$\Sigma _{{{\mathcal Q}}}\cdot \phi ^!=\phi ^!\cdot \Sigma _{{{\mathcal R}}}$$ on the full DG subcategory of DG $({{\mathcal B}}\otimes {{\mathcal R}})^{op}$-modules $M$ such that $M^{\gr}\simeq M_1^{\gr}\otimes M_2^{\gr}$ for a ${{\mathcal B}}^{op}$-module $M_1$ and an ${{\mathcal R}}^{op}$-module $M_2$. (This subcategory includes in particular graded ${{\mathcal R}}$-cofree modules.) Therefore $$\Sigma \cdot i ^!=i ^!\cdot \Sigma _{{{\mathcal R}}}$$ on this subcategory. d\) For a graded ${{\mathcal R}}$-free DG $({{\mathcal B}}\otimes {{\mathcal R}}) ^{op}$-module $M$ there is a functorial isomorphism $$\Sigma _{{{\mathcal R}}}(M\otimes _{{{\mathcal R}}} {{\mathcal R}}^*)=\Sigma _{{{\mathcal R}}}(M)\otimes _{{{\mathcal R}}}{{\mathcal R}}^*$$ The only nontrivial assertion is c). For any DG $({{\mathcal B}}\otimes {{\mathcal R}})^{op}$-module $M$ there is a natural closed morphism of degree zero of DG ${{\mathcal A}}^{op}_{{{\mathcal Q}}}$-modules $$\gamma _M:\Hom _{{{\mathcal R}}^{op}}(Q, M)\otimes _{{{\mathcal B}}}F\to \Hom _{{{\mathcal R}}^{op}}(Q, M\otimes _{{{\mathcal B}}}F), \quad \gamma(g\otimes f)(q)=(-1)^{\bar{f}\bar{q}}g(q)\otimes f.$$ Since ${{\mathcal Q}}$ is a finite ${{\mathcal R}}^{op}$-module $\gamma _M$ is an isomorphism if $M^{\gr}\simeq M_1^{\gr}\otimes M_2^{\gr}$ for a ${{\mathcal B}}^{op}$-module $M_1$ and an ${{\mathcal R}}^{op}$-module $M_2$. a\) For each artinian DG algebra ${{\mathcal R}}$ the DG functor $\Sigma _{{{\mathcal R}}}$ induces functors between groupoids $${\operatorname{Def}}^{\h}(\Sigma _{{{\mathcal R}}}):{\operatorname{Def}}^{\h}_{{{\mathcal R}}}({{\mathcal B}})\to {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(F),$$ $${\operatorname{coDef}}^{\h}(\Sigma _{{{\mathcal R}}}):{\operatorname{coDef}}^{\h}_{{{\mathcal R}}}({{\mathcal B}})\to {\operatorname{coDef}}^{\h}_{{{\mathcal R}}}(F),$$ b\) The collection of DG functors $\{\Sigma _{{{\mathcal R}}}\}_{{{\mathcal R}}}$ defines morphisms of pseudo-functors $${\operatorname{Def}}^{\h}(\Sigma ):{\operatorname{Def}}^{\h}({{\mathcal B}})\to {\operatorname{Def}}^{\h}(F),$$ $${\operatorname{coDef}}^{\h}(\Sigma ):{\operatorname{Def}}^{\h}({{\mathcal B}})\to {\operatorname{Def}}^{\h}(F).$$ c\) The morphism ${\operatorname{Def}}^{\h}(\Sigma )$ is compatible with the equivalence $\theta $ of Proposition 6.1. That is the functorial diagram $$\begin{array}{ccc} {{\mathcal M}}{{\mathcal C}}({{\mathcal B}}) & = & {{\mathcal M}}{{\mathcal C}}({{\mathcal B}})\\ \theta ^{{{\mathcal B}}} \downarrow & & \downarrow \theta ^{F}\\ {\operatorname{Def}}^{\h}({{\mathcal B}}) & \stackrel{{\operatorname{Def}}^{\h}(\Sigma)}{\rightarrow} & {\operatorname{Def}}^{\h}(F) \end{array}$$ is commutative. d\) The morphisms ${\operatorname{Def}}^{\h}(\Sigma )$ and ${\operatorname{coDef}}^{\h}(\Sigma )$ are compatible with the equivalence $\delta$ of Proposition 4.7. That is the functorial diagram $$\begin{array}{ccc} {\operatorname{Def}}^{\h}({{\mathcal B}}) & \stackrel{{\operatorname{Def}}^{\h}(\Sigma)}{\rightarrow} & {\operatorname{Def}}^{\h}(F)\\ \delta ^{{{\mathcal B}}} \downarrow & & \downarrow \delta ^{F}\\ {\operatorname{coDef}}^{\h}({{\mathcal B}}) & \stackrel{{\operatorname{coDef}}^{\h}(\Sigma)}{\rightarrow} & {\operatorname{coDef}}^{\h}(F) \end{array}$$ is commutative. e\) The morphisms ${\operatorname{Def}}^{\h}(\Sigma )$ and ${\operatorname{coDef}}^{\h}(\Sigma )$ are equivalences, i.e. for each ${{\mathcal R}}$ the functors ${\operatorname{Def}}^{\h}(\Sigma _{{{\mathcal R}}})$ and ${\operatorname{coDef}}^{\h}(\Sigma _{{{\mathcal R}}})$ are equivalences. a\) and b) follow from parts a),b),c) of Lemma 9.1; c) is obvious; d) follows from part d) of Lemma 9.1; e) follows from c) and d). DG functor $\psi^*$ ------------------- Let $\psi :{{\mathcal C}}\to {{\mathcal B}}$ be a homomorphism of DG algebras. Recall the corresponding DG functor $$\psi ^*:{{\mathcal C}}^{op}\text{-mod}\to {{\mathcal B}}^{op}\text{-mod},\quad \psi ^*(M)=M\otimes _{{{\mathcal C}}}{{\mathcal B}}.$$ For each artinian DG algebra ${{\mathcal R}}$ we obtain a similar DG functor $$\psi ^*_{{{\mathcal R}}}:({{\mathcal C}}\otimes {{\mathcal R}}) ^{op} \text{-mod}\to ({{\mathcal B}}\otimes {{\mathcal R}}) ^{op}\text{-mod},\quad \psi ^*(M)=M\otimes _{{{\mathcal C}}}{{\mathcal B}}.$$ The next lemma and proposition are complete analogues of Lemma 9.1 and Proposition 9.2. The DG functors $\psi ^* _{{{\mathcal R}}}$ have the following properties. a\) If a DG $({{\mathcal C}}\otimes {{\mathcal R}}) ^{op}$-module $M$ is graded ${{\mathcal R}}$-free (resp. graded ${{\mathcal R}}$-cofree), then so is the DG $({{\mathcal B}}\otimes {{\mathcal R}}) ^{op}$-module $\psi ^*_{{{\mathcal R}}}(M)$. b\) Let $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ be a homomorphism of artinian DG algebras. Then there are natural isomorphisms of DG functors $$\psi^* _{{{\mathcal Q}}}\cdot \phi ^*=\phi ^*\cdot \psi^* _{{{\mathcal R}}}, \quad \psi^* _{{{\mathcal R}}}\cdot \phi _*=\phi _*\cdot \psi ^* _{{{\mathcal Q}}}.$$ In particular, $$\psi^* \cdot i ^*=i ^*\cdot \psi^* _{{{\mathcal R}}}.$$ c\) There is a natural isomorphism of DG functors $$\psi^* _{{{\mathcal Q}}}\cdot \phi ^!=\phi ^!\cdot \psi^* _{{{\mathcal R}}}$$ on the full DG subcategory of DG $({{\mathcal C}}\otimes {{\mathcal R}})^{op}$-modules $M$ such that $M^{\gr}\simeq M_1^{\gr}\otimes M_2^{\gr}$ for a ${{\mathcal C}}^{op}$-module $M_1$ and an ${{\mathcal R}}^{op}$-module $M_2$. (This subcategory includes in particular graded ${{\mathcal R}}$-cofree modules.) Therefore $$\psi^* \cdot i ^!=i ^!\cdot \psi^* _{{{\mathcal R}}}$$ on this subcategory. d\) For a graded ${{\mathcal R}}$-free DG $({{\mathcal C}}\otimes {{\mathcal R}}) ^{op}$-module $M$ there is a functorial isomorphism $$\psi^* _{{{\mathcal R}}}(M\otimes _{{{\mathcal R}}} {{\mathcal R}}^*)=\psi^* _{{{\mathcal R}}}(M)\otimes _{{{\mathcal R}}}{{\mathcal R}}^*$$ As in Lemma 9.1, the only nontrivial assertion is c). For any DG $({{\mathcal C}}\otimes {{\mathcal R}})^{op}$-module $M$ there is a natural closed morphism of degree zero of DG $A_{{{\mathcal Q}}}^{op}$-modules $$\eta_M:\Hom _{{{\mathcal R}}^{op}}(Q, M)\otimes _{{{\mathcal C}}}{{\mathcal B}}\to \Hom _{{{\mathcal R}}^{op}}(Q, M\otimes _{{{\mathcal C}}}{{\mathcal B}}), \quad \gamma(g\otimes f)(q)=(-1)^{\bar{f}\bar{q}}g(q)\otimes f.$$ Since ${{\mathcal Q}}$ is a finite ${{\mathcal R}}^{op}$-module $\eta_M$ is an isomorphism if $M^{\gr}\simeq M_1^{\gr}\otimes M_2^{\gr}$ for a ${{\mathcal B}}^{op}$-module $M_1$ and an ${{\mathcal R}}^{op}$-module $M_2$. a\) For each artinian DG algebra ${{\mathcal R}}$ the DG functor $\psi^* _{{{\mathcal R}}}$ induces functors between groupoids $${\operatorname{Def}}^{\h}(\psi^* _{{{\mathcal R}}}):{\operatorname{Def}}^{\h}_{{{\mathcal R}}}({{\mathcal C}})\to {\operatorname{Def}}^{\h}_{{{\mathcal R}}}({{\mathcal B}}),$$ $${\operatorname{coDef}}^{\h}(\psi^* _{{{\mathcal R}}}):{\operatorname{coDef}}^{\h}_{{{\mathcal R}}}({{\mathcal C}})\to {\operatorname{coDef}}^{\h}_{{{\mathcal R}}}({{\mathcal B}}),$$ b\) The collection of DG functors $\{\psi^* _{{{\mathcal R}}}\}_{{{\mathcal R}}}$ defines morphisms $${\operatorname{Def}}^{\h}(\psi^* ):{\operatorname{Def}}^{\h}({{\mathcal C}})\to {\operatorname{Def}}^{\h}({{\mathcal B}}),$$ $${\operatorname{coDef}}^{\h}(\psi^* ):{\operatorname{Def}}^{\h}({{\mathcal C}})\to {\operatorname{Def}}^{\h}({{\mathcal B}}).$$ c\) The morphism ${\operatorname{Def}}^{\h}(\psi^* )$ is compatible with the equivalence $\theta $ of Proposition 6.1. That is the functorial diagram $$\begin{array}{ccc} {{\mathcal M}}{{\mathcal C}}({{\mathcal C}}) & \stackrel{\psi^*}{\rightarrow} & {{\mathcal M}}{{\mathcal C}}({{\mathcal B}})\\ \theta ^{{{\mathcal C}}} \downarrow & & \downarrow \theta ^{{{\mathcal B}}}\\ {\operatorname{Def}}^{\h}({{\mathcal C}}) & \stackrel{{\operatorname{Def}}^{\h}(\psi^*)}{\rightarrow} & {\operatorname{Def}}^{\h}({{\mathcal B}}) \end{array}$$ is commutative. d\) The morphisms ${\operatorname{Def}}^{\h}(\psi^* )$ and ${\operatorname{coDef}}^{\h}(\psi^* )$ are compatible with the equivalence $\delta$ of Proposition 4.7. That is the functorial diagram $$\begin{array}{ccc} {\operatorname{Def}}^{\h}({{\mathcal C}}) & \stackrel{{\operatorname{Def}}^{\h}(\psi^*)}{\rightarrow} & {\operatorname{Def}}^{\h}({{\mathcal B}})\\ \delta ^{{{\mathcal C}}} \downarrow & & \downarrow \delta ^{{{\mathcal B}}}\\ {\operatorname{coDef}}^{\h}({{\mathcal C}}) & \stackrel{{\operatorname{coDef}}^{\h}(\psi^*)}{\rightarrow} & {\operatorname{coDef}}^{\h}({{\mathcal B}}) \end{array}$$ is commutative. e\) Assume that $\psi$ is a quasi-isomorphism. Then the morphisms ${\operatorname{Def}}^{\h}(\psi^* )$ and ${\operatorname{coDef}}^{\h}(\psi^* )$ are equivalences, i.e. for each ${{\mathcal R}}$ the functors ${\operatorname{Def}}^{\h}(\psi^* _{{{\mathcal R}}})$ and ${\operatorname{coDef}}^{\h}(\psi^* _{{{\mathcal R}}})$ are equivalences. a\) and b) follow from parts a),b),c) of Lemma 9.3; c) is obvious; d) follows from part d) of Lemma 9.3; e) follows from c),d) and Theorem 8.1. Later we will be especially interested in the following example. (Keller). a) Assume that the DG algebra ${{\mathcal B}}$ satisfies the following conditions: $H^i({{\mathcal B}})=0$ for $i<0$, $H^0({{\mathcal B}})=k$ (resp. $H^0({{\mathcal B}})=k$). Then there exists a DG subalgebra ${{\mathcal C}}\subset {{\mathcal B}}$ with the properties: ${{\mathcal C}}^i=0$ for $i<0$, ${{\mathcal C}}^0=k$, and the embedding $\psi:{{\mathcal C}}\hookrightarrow {{\mathcal B}}$ is a quasi-isomorphism (resp. the induced map $H^i(\psi):H^i({{\mathcal C}})\to H^i({{\mathcal B}})$ is an isomorphism for $i\geq 0$). Indeed, put ${{\mathcal C}}^0=k$, ${{\mathcal C}}^1=K\oplus L$, where $d(K)=0$ and $K$ projects isomorphically to $H^1({{\mathcal B}})$, and $d:L\stackrel{\sim}{\to}d({{\mathcal B}}^1)\subset {{\mathcal B}}^2$. Then take ${{\mathcal C}}^i={{\mathcal B}}^i$ for $i\geq 2$ and ${{\mathcal C}}^i=0$ for $i<0$. The derived deformation and co-deformation pseudo-functors ========================================================== The pseudo-functor ${\operatorname{Def}}(E)$ -------------------------------------------- Fix a DG category ${{\mathcal A}}$ and an object $E\in {{\mathcal A}}^{op}\text{-mod}$. We are going to define a pseudo-functor ${\operatorname{Def}}(E)$ from the category ${\operatorname{dgart}}$ to the category ${\bf Gpd}$ of groupoids. This pseudo-functor assigns to a DG algebra ${{\mathcal R}}$ the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(E)$ of ${{\mathcal R}}$-deformations of $E$ in the [*derived*]{} category $D({{\mathcal A}}^{op})$. Fix an artinian DG algebra ${{\mathcal R}}$. An object of the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(E)$ is a pair $(S,\sigma)$, where $S\in D({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $\sigma$ is an isomorphism (in $D({{\mathcal A}}^{op})$) $$\sigma :\bL i^*S\to E.$$ A morphism $f:(S,\sigma)\to (T,\tau)$ between two ${{\mathcal R}}$-deformations of $E$ is an isomorphism (in $D({{\mathcal A}}_{{{\mathcal R}}} ^{op})$) $f:S\to T$, such that $$\tau \cdot \bL i^*(f)=\sigma.$$ This defines the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(E)$. A homomorphism of artinian DG algebras $\phi:{{\mathcal R}}\to {{\mathcal Q}}$ induces the functor $$\bL\phi ^*:{\operatorname{Def}}_{{{\mathcal R}}}(E)\to {\operatorname{Def}}_{{{\mathcal Q}}}(E).$$ Thus we obtain a pseudo-functor $${\operatorname{Def}}(E):{\operatorname{dgart}}\to {\bf Gpd}.$$ We call ${\operatorname{Def}}(E)$ the pseudo-functor of derived deformations of $E$. A quasi-isomorphism $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ of artinian DG algebras induces an equivalence of groupoids $$\bL\phi ^*:{\operatorname{Def}}_{{{\mathcal R}}}(E)\to {\operatorname{Def}}_{{{\mathcal Q}}}(E).$$ Indeed, $\bL\phi ^*:D({{\mathcal A}}_{{{\mathcal R}}} ^{op})\to D({{\mathcal A}}_{{{\mathcal Q}}} ^{op})$ is an equivalence of categories (Proposition 3.7) which commutes with the functor $\bL i^*$. A quasi-isomorphism $\delta:E_1\to E_2$ of DG ${{\mathcal A}}^{op}$-modules induces an equivalence of pseudo-functors $$\delta _*:{\operatorname{Def}}(E_1)\to {\operatorname{Def}}(E_2)$$ by the formula $\delta _*(S,\sigma)=(S,\delta \cdot \sigma)$. Let $F:{{\mathcal A}}\to {{\mathcal A}}^\prime$ be a DG functor which induces a quasi-equivalence $F^{\pre-tr}:{{\mathcal A}}^{\pre-tr}\to {{\mathcal A}}^{\prime \pre-tr}$ (this happens for example if $F$ is a quasi-equivalence). Then for any $E\in D({{\mathcal A}}^{op})$ the deformation pseudo-functors ${\operatorname{Def}}(E)$ and ${\operatorname{Def}}(\bL F^*(E))$ are canonically equivalent. (Hence also ${\operatorname{Def}}(F_*(E^\prime))$ and ${\operatorname{Def}}(E^\prime)$ are equivalent for any $E^\prime \in D({{\mathcal A}}^{\prime 0})$). For any artinian DG algebra ${{\mathcal R}}$ the functor $F$ induces a commutative functorial diagram $$\begin{array}{ccc} D({{\mathcal A}}_{{{\mathcal R}}} ^{op}) & \stackrel{\bL (F \otimes {\operatorname{id}})^*}{\longrightarrow} & D({{\mathcal A}}_{{{\mathcal R}}}^{\prime 0})\\ \downarrow \bL i^* & & \downarrow \bL i^*\\ D({{\mathcal A}}^{op}) & \stackrel{\bL F^*}{\longrightarrow} & D({{\mathcal A}}^{\prime 0}) \end{array}$$ where $\bL F^*$ and $\bL (F\otimes {\operatorname{id}})^*$ are equivalences by Corollary 3.15. The horizontal arrows define a functor $F^*_{{{\mathcal R}}}:{\operatorname{Def}}_{{{\mathcal R}}}(E)\to {\operatorname{Def}}_{{{\mathcal R}}}(\bL F^*(E))$. Moreover these functors are compatible with the functors $\bL \phi ^*:{\operatorname{Def}}_{{{\mathcal R}}}\to {\operatorname{Def}}_{{{\mathcal Q}}}$ induced by morphisms $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ of artinian DG algebras. So we get the morphism $F^*:{\operatorname{Def}}(E)\to {\operatorname{Def}}(\bL F^*(E))$ of pseudo-functors. It is clear that for each ${{\mathcal R}}$ the functor $F^*_{{{\mathcal R}}}$ is an equivalence. Thus $F^*$ is also such. Suppose that ${{\mathcal A}}^\prime$ is a pre-triangulated DG category (so that the homotopy category ${\operatorname{Ho}}({{\mathcal A}}^\prime)$ is triangulated). Let $F:{{\mathcal A}}\hookrightarrow {{\mathcal A}}^\prime$ be an embedding of a full DG subcategory so that the triangulated category ${\operatorname{Ho}}({{\mathcal A}}^\prime)$ is generated by the collection of objects $F(Ob{{\mathcal A}})$. Then the assumption of the previous proposition holds. In the definition of the pseudo-functor ${\operatorname{Def}}(E)$ we could work with the homotopy category of h-projective DG modules instead of the derived category. Indeed, the functors $i^*$ and $\phi ^*$ preserve h-projective DG modules. Denote by ${\operatorname{Def}}_+(E)$, ${\operatorname{Def}}_-(E)$, ${\operatorname{Def}}_0(E)$, ${\operatorname{Def}}_{{\operatorname{cl}}}(E)$ the restrictions of the pseudo-functor ${\operatorname{Def}}(E)$ to subcategories ${\operatorname{dgart}}_+$, ${\operatorname{dgart}}_-$, ${\operatorname{art}}$, ${\operatorname{cart}}$ respectively. The pseudo-functor ${\operatorname{coDef}}(E)$ ---------------------------------------------- Now we define the pseudo-functor ${\operatorname{coDef}}(E)$ of [*derived co-deformations*]{} in a similar way replacing everywhere the functors $(\cdot )^*$ by $(\cdot )^!$. Fix an artinian DG algebra ${{\mathcal R}}$. An object of the groupoid ${\operatorname{coDef}}_{{{\mathcal R}}}(E)$ is a pair $(S,\sigma)$, where $S\in D({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $\sigma$ is an isomorphism (in $D({{\mathcal A}}^{op})$) $$\sigma :E\to \bR i^!S.$$ A morphism $f:(S,\sigma)\to (T,\tau)$ between two ${{\mathcal R}}$-deformations of $E$ is an isomorphism (in $D({{\mathcal A}}_{{{\mathcal R}}}^{op})$) $f:S\to T$, such that $$\bR i^!(f)\cdot \sigma=\tau.$$ This defines the groupoid ${\operatorname{coDef}}_{{{\mathcal R}}}(E)$. A homomorphism of artinian DG algebras $\phi:{{\mathcal R}}\to {{\mathcal Q}}$ induces the functor $$\bR\phi ^!:{\operatorname{coDef}}_{{{\mathcal R}}}(E)\to {\operatorname{coDef}}_{{{\mathcal Q}}}(E).$$ Thus we obtain a pseudo-functor $${\operatorname{coDef}}(E):{\operatorname{dgart}}\to {\bf Gpd}.$$ We call ${\operatorname{coDef}}(E)$ the functor of derived co-deformations of $E$. A quasi-isomorphism $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ of artinian DG algebras induces an equivalence of groupoids $$\bR\phi ^!:{\operatorname{coDef}}_{{{\mathcal R}}}(E)\to {\operatorname{coDef}}_{{{\mathcal Q}}}(E).$$ Indeed, $\bR\phi ^!:D({{\mathcal A}}_{{{\mathcal R}}}^{op})\to D({{\mathcal A}}_{{{\mathcal Q}}}^{op})$ is an equivalence of categories (Proposition 3.7) which commutes with the functor $\bR i^!$. A quasi-isomorphism $\delta:E_1\to E_2$ of ${{\mathcal A}}$-DG-modules induces an equivalence of pseudo-functors $$\delta ^*:{\operatorname{coDef}}(E_2)\to {\operatorname{coDef}}(E_1)$$ by the formula $\delta ^*(S,\sigma)=(S,\sigma \cdot \delta)$. Let $F:{{\mathcal A}}\to {{\mathcal A}}^\prime$ be a DG functor as in Proposition 10.4 above. Consider the induced equivalence of derived categories $\bR F^!:D({{\mathcal A}}^{op})\to D({{\mathcal A}}^{\prime 0})$ (Corollary 3.15). Then for any $E\in D({{\mathcal A}}^{op})$ the deformation pseudo-functors ${\operatorname{coDef}}(E)$ and ${\operatorname{coDef}}(\bR F^!(E))$ are canonically equivalent. (Hence also ${\operatorname{coDef}}(F_*(E^\prime))$ and ${\operatorname{coDef}}(E^\prime)$ are equivalent for any $E^\prime \in D({{\mathcal A}}^{\prime 0})$). For any artinian DG algebra ${{\mathcal R}}$ the functor $F$ induces a commutative functorial diagram $$\begin{array}{ccc} D({{\mathcal A}}_{{{\mathcal R}}}^{op}) & \stackrel{\bR ((F \otimes {\operatorname{id}})^!)}{\longrightarrow} & D({{\mathcal A}}_{{{\mathcal R}}}^{\prime 0})\\ \downarrow \bR i^! & & \downarrow \bR i^!\\ D({{\mathcal A}}^{op}) & \stackrel{\bR F^!}{\longrightarrow} & D({{\mathcal A}}^{\prime 0}), \end{array}$$ where $\bR (F \otimes {\operatorname{id}})^!$ is an equivalence by Corollary 3.15. The horizontal arrows define a functor $ F^!_{{{\mathcal R}}}:{\operatorname{coDef}}_{{{\mathcal R}}}(E)\to {\operatorname{coDef}}_{{{\mathcal R}}}(\bR F^!(E))$. Moreover these functors are compatible with the functors $\bR \phi ^!:{\operatorname{coDef}}_{{{\mathcal R}}}\to {\operatorname{coDef}}_{{{\mathcal Q}}}$ induced by morphisms $\phi :{{\mathcal R}}\to {{\mathcal Q}}$ of artinian DG algebras. So we get the morphism $F^!:{\operatorname{coDef}}(E)\to {\operatorname{coDef}}(\bR F^!(E))$. It is clear that for each ${{\mathcal R}}$ the functor $F^!_{{{\mathcal R}}}$ is an equivalence. Thus $F^!$ is also such. Let $F:{{\mathcal A}}^\prime \to {{\mathcal A}}$ be as in Example 10.5 above. Then the assumption of the previous proposition holds. In the definition of the pseudo-functor ${\operatorname{coDef}}(E)$ we could work with the homotopy category of h-injective DG modules instead of the derived category. Indeed, the functors $i^!$ and $\phi ^!$ preserve h-injective DG modules. Denote by ${\operatorname{coDef}}_+(E)$, ${\operatorname{coDef}}_-(E)$, ${\operatorname{coDef}}_0(E)$, ${\operatorname{coDef}}_{{\operatorname{cl}}}(E)$ the restrictions of the pseudo-functor ${\operatorname{coDef}}(E)$ to subcategories ${\operatorname{dgart}}_+$, ${\operatorname{dgart}}_-$, ${\operatorname{art}}$, ${\operatorname{cart}}$ respectively. The pseudo-functors ${\operatorname{Def}}(E)$ and ${\operatorname{coDef}}(E)$ are not always equivalent (unlike their homotopy counterparts ${\operatorname{Def}}^{\h}(E)$ and ${\operatorname{coDef}}^{\h}(E)$). In fact we expect that pseudo-functors ${\operatorname{Def}}$ and ${\operatorname{coDef}}$ are the “right ones” only in case they can be expressed in terms of the pseudo-functors ${\operatorname{Def}}^{\h}$ and ${\operatorname{coDef}}^{\h}$ respectively. (See the next section). Relation between pseudo-functors ${\operatorname{Def}}$ and ${\operatorname{Def}}^h$ (resp. ${\operatorname{coDef}}$ and ${\operatorname{coDef}}^h$) ==================================================================================================================================================== The ideal scheme that should relate these deformation pseudo-functors is the following. Let ${{\mathcal A}}$ be a DG category, $E\in {{\mathcal A}}^{op}\text{-mod}$. Choose quasi-isomorphisms $P\to E$ and $E\to I$, where $P\in \P({{\mathcal A}}^{op})$ and $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Then there should exist natural equivalences $${\operatorname{Def}}(E)\simeq {\operatorname{Def}}^h (P),\quad\quad {\operatorname{coDef}}(E)\simeq {\operatorname{coDef}}^h(I).$$ Unfortunately, this does not always work. Let ${{\mathcal A}}$ be just a graded algebra $A=k[t]$, i.e. ${{\mathcal A}}$ contains a single object with the endomorphism algebra $k[t]$, $\deg (t)=1$ (the differential is zero). Take the artinian DG algebra ${{\mathcal R}}$ to be ${{\mathcal R}}=k[\epsilon]/(\epsilon ^2)$, $\deg(\epsilon)=0$. Let $E=A$ and consider a DG ${{\mathcal A}}_{{{\mathcal R}}}^{op}$-module $M=E\otimes {{\mathcal R}}$ with the differential $d_M$ which is the multiplication by $t\otimes \epsilon$. Clearly, $M$ defines an object in ${\operatorname{Def}}^h_{{{\mathcal R}}}(E)$ which is not isomorphic to the trivial deformation. However, one can check (Proposition 11.18) that $\bL i^*M$ is not quasi-isomorphic to $E$ (although $i^*M=E$), thus $M$ does not define an object in ${\operatorname{Def}}_{{{\mathcal R}}}(E)$. This fact and the next proposition show that the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(E)$ is connected (contains only the trivial deformation), so it is not the “right” one. Assume that ${\operatorname{Ext}}^{-1}(E,E)=0$. 1\) Fix a quasi-isomorphism $P\to E$, $P\in {{\mathcal P}}({{\mathcal A}}^{op})$. Let ${{\mathcal R}}$ be an artinian DG algebra and $(S, {\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)$. The following conditions are equivalent: a\) $S\in \P({{\mathcal A}}_{{{\mathcal R}}}^{op})$, b\) $i^*S=\bL i^*S$, c\) $(S,{\operatorname{id}})$ defines an object in the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(E)$. The pseudo-functor ${\operatorname{Def}}(E)$ is equivalent to the full pseudo-subfunctor of ${\operatorname{Def}}^{\h}(P)$ consisting of objects $(S,{\operatorname{id}}) \in {\operatorname{Def}}^{\h}(P)$, where $S$ satisfies a) (or b)) above. 2\) Fix a quasi-isomorphism $E\to I$ with $I\in {{\mathcal I}}({{\mathcal A}}^{op})$. Let ${{\mathcal R}}$ be an artinian DG algebra and $(T, {\operatorname{id}})\in {\operatorname{coDef}}^{\h} _{{{\mathcal R}}}(I)$. The following conditions are equivalent: a’) $T\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$, b’) $i^!T=\bR i^!T$, c’) $(T,{\operatorname{id}})$ defines an object in the groupoid ${\operatorname{coDef}}_{{{\mathcal R}}}(E)$. The pseudo-functor ${\operatorname{coDef}}(E)$ is equivalent to the full pseudo-subfunctor of ${\operatorname{coDef}}^{\h}(I)$ consisting of objects $(T,{\operatorname{id}}) \in {\operatorname{coDef}}^{\h}(I)$, where $T$ satisfies a’) (or b’)) above. 1\) It is clear that a) implies b) and b) implies c). We will prove that c) implies a). We may and will replace the pseudo-functor ${\operatorname{Def}}(E)$ by an equivalent pseudo-functor ${\operatorname{Def}}(P)$ (Remark 10.3). Since $(S, {\operatorname{id}})$ defines an object in ${\operatorname{Def}}_{{{\mathcal R}}}(P)$ there exists a quasi-isomorphism $g:\tilde{S}\to S$ where $\tilde{S}$ has property (P) (hence $\tilde{S}\in \P({{\mathcal A}}_{{{\mathcal R}}}^{op})$), such that $i^*g: i^*\tilde{S}\to i^*S=P$ is also a quasi-isomorphism. Denote $Z=i^*\tilde{S}$. Then $Z\in \P({{\mathcal A}}^{op})$ and hence $i^*g$ is a homotopy equivalence. Since both $\tilde{S}$ and $S$ are graded ${{\mathcal R}}$-free, the map $g$ is also a homotopy equivalence (Proposition 3.12d)). Thus $S\in \P({{\mathcal A}}_{{{\mathcal R}}}^{op})$. Let us prove the last assertion in 1). Fix an object $(\overline{S} ,\tau) \in {\operatorname{Def}}_{{{\mathcal R}}}(P)$. Replacing $(\overline{S}, \tau)$ by an isomorphic object we may and will assume that $\overline{S}$ satisfies property (P). In particular, $\overline{S}\in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $\overline{S}$ is graded ${{\mathcal R}}$-free. This implies that $(\overline{S},{\operatorname{id}})\in {\operatorname{Def}}^{\h} _{{{\mathcal R}}}(W)$ where $W=i^*\overline{S}$. We have $W\in \P({{\mathcal A}}^{op})$. The quasi-isomorphism $\tau :W\to P$ is therefore a homotopy equivalence. By Corollary 8.4a) and Proposition 8.5a) there exists an object $(S^\prime,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)$ and a homotopy equivalence $\tau ^\prime :\overline{S}\to S^\prime$ such that $i^*(\tau ^\prime)=\tau$. This shows that $(\overline{S}, \tau)$ is isomorphic (in ${\operatorname{Def}}_{{{\mathcal R}}}(P)$) to an object $(S^\prime ,{\operatorname{id}})\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)$, where $S^\prime \in \P({{\mathcal A}}_{{{\mathcal R}}}^{op})$. Let $(S ,{\operatorname{id}}),(S^\prime, {\operatorname{id}}) \in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(P)$ be two objects such that $S,S^\prime \in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$. Consider the obvious map $$\delta :\Hom _{{\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)}((S,{\operatorname{id}}),(S^\prime,{\operatorname{id}}))\to \Hom _{{\operatorname{Def}}_{{{\mathcal R}}}(P)}((S,{\operatorname{id}}),(S^\prime,{\operatorname{id}})).$$ It suffices to show that $\delta$ is bijective. Let $f:(S,{\operatorname{id}}) \to (S^\prime, {\operatorname{id}})$ be an isomorphism in ${\operatorname{Def}}_{{{\mathcal R}}}(P)$. Since $S,S^\prime\in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $P \in {{\mathcal P}}({{\mathcal A}}^{op})$ this isomorphism $f$ is a homotopy equivalence $f:S\to S^\prime$ such that $i^*f$ is homotopic to ${\operatorname{id}}_P$. Let $h:i^*f\to {\operatorname{id}}$ be a homotopy. Since $S$, $S^\prime$ are graded ${{\mathcal R}}$-free the map $i^*:\Hom (S, S^\prime )\to \Hom (P,P)$ is surjective (Proposition 3.12a)). Choose a lift $\tilde{h}:S\to S^\prime[1]$ of $h$ and replace $f$ by $\tilde{f}=f-d\tilde{h}$. Then $i^*\tilde{f}=id$. Since $S$ and $S^\prime$ are graded ${{\mathcal R}}$-free $\tilde{f}$ is an isomorphism (Proposition 3.12d)). This shows that $\delta$ is surjective. Let $g_1,g_2:S\to S^\prime$ be two isomorphisms (in ${{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$) such that $i^*g_1=i^*g_2={\operatorname{id}}_P$. That is $g_1,g_2$ represent morphisms in ${\operatorname{Def}}_{{{\mathcal R}}}^{\h}(P)$. Assume that $\delta (g_1)=\delta (g_2)$, i.e. there exists a homotopy $s:g_1\to g_2$. Then $d(i^*s)=i^*(ds)=0$. Since by our assumption $H^{-1}\Hom (P,P)=0$ there exists $t\in \Hom ^{-2}(P,P)$ with $dt=i^*s$. Choose a lift $\tilde{t}\in \Hom ^{-2}(S,S^\prime)$ of $t$. Then $\tilde{s}:=s-d\tilde{t}$ is an allowable homotopy between $g_1$ and $g_2$. This proves that $\delta $ is injective and finishes the proof of 1). The proof of 2) is very similar, but we present it for completeness. Again it is clear that a’) implies b’) and b’) implies c’). We will prove that c’) implies a’) We may and will replace the functor ${\operatorname{coDef}}(E)$ by an equivalent functor ${\operatorname{coDef}}(I)$ (Remark 10.10). Since $(T,{\operatorname{id}})$ defines an object in ${\operatorname{coDef}}_{{{\mathcal R}}}(I)$, there exists a quasi-isomorphism $g:T\to \tilde{T}$ where $\tilde{T}$ has property (I) (hence $\tilde{T} \in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$), such that $i^!g:I=i^!T\to i^!\tilde{T}$ is also a quasi-isomorphism. Denote $K=i^!\tilde{T}$. Then $K\in {{\mathcal I}}({{\mathcal A}}^{op})$ and hence $i^!g$ is a homotopy equivalence. Since both $T$ and $\tilde{T}$ are graded ${{\mathcal R}}$-cofree, the map $g$ is also a homotopy equivalence (Proposition 3.12d)). Thus $T\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}} ^{op})$. Let us prove the last assertion in 2). Fix an object $(\overline{T} ,\tau) \in {\operatorname{coDef}}_{{{\mathcal R}}}(I)$. Replacing $(\overline{T}, \tau)$ by an isomorphic object we may and will assume that $\overline{T}$ satisfies property (I). In particular, $\overline{T}\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}} ^{op})$ and $\overline{T}$ is graded ${{\mathcal R}}$-cofree. This implies that $(\overline{T},{\operatorname{id}})\in {\operatorname{coDef}}^{\h} _{{{\mathcal R}}}(L)$ where $L=i^!\overline{T}$. We have $L\in {{\mathcal I}}({{\mathcal A}}^{op})$ and hence the quasi-isomorphism $\tau :I\to L$ is a homotopy equivalence. By Corollary 8.4a) and Proposition 8.5a) there exist an object $(T^\prime,{\operatorname{id}})\in {\operatorname{coDef}}^{\h}_{{{\mathcal R}}}(I)$ and a homotopy equivalence $\tau ^\prime : T^\prime\to \overline{T}$ such that $i^!\tau ^\prime =\tau$. In particular, $T^\prime\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}} ^{op})$. This shows that $(\overline{T},\tau)$ is isomorphic (in ${\operatorname{coDef}}_{{{\mathcal R}}}(I)$) to an object $(T^\prime ,{\operatorname{id}}) \in {\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(I)$ where $T^\prime \in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$. Let $(T ,{\operatorname{id}}),(T^\prime, {\operatorname{id}}) \in {\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(I)$ be two objects such that $T,T^\prime \in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$. Consider the obvious map $$\delta :\Hom _{{\operatorname{coDef}}^{\h}_{{{\mathcal R}}}(I)}((T,{\operatorname{id}}),(T^\prime,{\operatorname{id}}))\to \Hom _{{\operatorname{coDef}}_{{{\mathcal R}}}(I)}((T,{\operatorname{id}}),(T^\prime,{\operatorname{id}})).$$ It suffices to show that $\delta$ is bijective. Let $f:(T,{\operatorname{id}}) \to (T^\prime, {\operatorname{id}})$ be an isomorphism in ${\operatorname{coDef}}_{{{\mathcal R}}}(I)$. Since $T,T^\prime\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$ and $I \in {{\mathcal I}}({{\mathcal A}}^{op})$ this isomorphism $f$ is a homotopy equivalence $f:T\to T^\prime$ such that $i^!f$ is homotopic to ${\operatorname{id}}_I$. Let $h:i^!f\to {\operatorname{id}}$ be a homotopy. Since $T$, $T^\prime$ are graded ${{\mathcal R}}$-cofree the map $i^!:\Hom (T, T^\prime )\to \Hom (I,I)$ is surjective (Proposition 3.12a)). Choose a lift $\tilde{h}:T\to T^\prime[1]$ of $h$ and replace $f$ by $\tilde{f}=f-d\tilde{h}$. Then $i^!\tilde{f}=id$. Since $T$ and $T^\prime$ are graded ${{\mathcal R}}$-cofree $\tilde{f}$ is an isomorphism (Proposition 3.12d)). This shows that $\delta$ is surjective. Let $g_1,g_2:T\to T^\prime$ be two isomorphisms (in ${{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$) such that $i^!g_1=i^!g_2={\operatorname{id}}_I$. That is $g_1,g_2$ represent morphisms in ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(I)$. Assume that $\delta (g_1)=\delta (g_2)$, i.e. there exists a homotopy $s:g_1\to g_2$. Then $d(i^!s)=i^!(ds)=0$. Since by our assumption $H^{-1}\Hom (I,I)=0$ there exists $t\in \Hom ^{-2}(I,I)$ with $dt=i^!s$. Choose a lift $\tilde{t}\in \Hom ^{-2}(T,T^\prime)$ of $t$. Then $\tilde{s}:=s-d\tilde{t}$ is an allowable homotopy between $g_1$ and $g_2$. This proves that $\delta $ is injective. In the situation of Proposition 11.2 using Corollary 8.4b) also obtain full and faithful morphisms of pseudo-functors $ {\operatorname{Def}}(E)$, ${\operatorname{coDef}}(E)$ to each of the equivalent pseudo-functors ${\operatorname{Def}}^{\h}(P)$, ${\operatorname{coDef}}^{\h}(P)$, ${\operatorname{Def}}^{\h}(I)$, ${\operatorname{coDef}}^{\h}(I)$. Assume that ${\operatorname{Ext}}^{-1}(E,E)=0$. Let $F\in {{\mathcal A}}^{op}\text{-mod}$ be an h-projective or an h-injective quasi-isomorphic to $E$. a\) The pseudo-functor ${\operatorname{Def}}(E)$ ($\simeq {\operatorname{Def}}(F)$) is equivalent to the full pseudo-subfunctor of ${\operatorname{Def}}^{\h}(F)$ which consists of objects $(S,{\operatorname{id}})$ such that $i^*S=\bL i^*S$. b\) The pseudo-functor ${\operatorname{coDef}}(E)$ ($\simeq {\operatorname{coDef}}(F))$ is equivalent to the full pseudo-subfunctor of ${\operatorname{coDef}}^{\h}(F)$ which consists of objects $(T,{\operatorname{id}})$ such that $i^!T=\bR i^!T$. a). In case $F$ is h-projective this is Proposition 11.2 1). Assume that $F$ is h-injective. Choose a quasi-isomorphism $P\to F$ where $P$ is h-projective. Again by Proposition 11.2 1) the assertion holds for $P$ instead of $F$. But then it also holds for $F$ by Corollary 8.6 b). b). In case $F$ is h-injective this is Proposition 11.2 2). Assume that $F$ is h-projective. Choose a quasi-isomorphism $F\to I$ where $I$ is h-injective. Then again by Proposition 11.2 2) the assertion holds for $I$ instead of $F$. But then it also holds for $F$ by Corollary 8.6 b). The next theorem provides an example when the pseudo-functors ${\operatorname{Def}}_-$ and ${\operatorname{Def}}^{\h}_-$ (resp. ${\operatorname{coDef}}_- $ and ${\operatorname{coDef}}^{\h}_-$) are equivalent. An object $M\in {{\mathcal A}}^{op}\text{-mod}$ is called bounded above (resp. below) if there exists $i$ such that $M(A)^j=0$ for all $A\in {{\mathcal A}}$ and all $j\geq i$ (resp. $j\leq i$). Assume that ${\operatorname{Ext}}^{-1}(E,E)=0$. a\) Suppose that there exists an h-projective or an h-injective $P\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded above and quasi-isomorphic to $E$. Then the pseudo-functors ${\operatorname{Def}}_-(E)$ and ${\operatorname{Def}}_-^{\h}(P)$ are equivalent. b\) Suppose that there exists an h-projective or an h-injective $I\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded below and quasi-isomorphic to $E$. Then the pseudo-functors ${\operatorname{coDef}}_-(E)$ and ${\operatorname{coDef}}_-^{\h}(I)$ are equivalent. Fix ${{\mathcal R}}\in {\operatorname{dgart}}_-$. In both cases it suffices to show that the embedding of groupoids ${\operatorname{Def}}_{{{\mathcal R}}} (E)\simeq {\operatorname{Def}}_{{{\mathcal R}}}(P) \subset {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(P)$ (resp. ${\operatorname{coDef}}_{{{\mathcal R}}} (E)\simeq {\operatorname{coDef}}_{{{\mathcal R}}}(I) \subset {\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(I)$) in Corollary 11.4 is essentially surjective. a\) It suffices to prove the following lemma. Let $M\in {{\mathcal A}}^{op}\text{-mod}$ be bounded above and $(S,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(M)$. The DG ${{\mathcal A}}_{{{\mathcal R}}}^{op}$-module $S$ is acyclic for the functor $i^*$, i.e. $\bL i^*S=i^*S$. Indeed, in case $M=P$ the lemma implies that $S$ defines an object in ${\operatorname{Def}}_{{{\mathcal R}}}(P)$ (Corollary 11.4 a)). Choose a quasi-isomorphism $f:Q\to S$ where $Q\in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}} ^{op})$. We need to prove that $i^*f$ is a quasi-isomorphism. It suffices to prove that $\pi _!i^*f$ is a quasi-isomorphism (Example 3.13). Recall that $\pi _!i^*=i^*\pi _!$. Thus it suffices to prove that $\pi _!f$ is a homotopy equivalence. Clearly $\pi _!f$ is a quasi-isomorphism. The DG ${{\mathcal R}}^{op}$-module $\pi _!Q$ is h-projective (Example 3.13). We claim that the DG ${{\mathcal R}}^{op}$-module $\pi _!S $ is also h-projective. Since the direct sum of h-projective DG modules is again h-projective, it suffices to prove that for each object $A\in {{\mathcal A}}$ the DG ${{\mathcal R}}^{op}$-module $S(A)$ is h-projective. Take some object $A\in {{\mathcal A}}$. We have that $S(A)$ is bounded above and since ${{\mathcal R}}\in {\operatorname{dgart}}_-$ this DG ${{\mathcal R}}^{op}$-module has an increasing filtration with subquotients being free DG ${{\mathcal R}}^{op}$-modules. Thus $S(A)$ satisfies property (P) and hence is h-projective. It follows that the quasi-isomorphism $\pi _!f:\pi_! Q\to \pi _!S$ is a homotopy equivalence. Hence $i^*\pi _!f=\pi _!i^*f$ is also such. b\) The following lemma implies (by Corollary 11.4 b)) that an object in ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(I)$ is also an object in ${\operatorname{coDef}}_{{{\mathcal R}}}(I)$, which proves the theorem. Let $T\in {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$ be graded cofree and bounded below. Then $T$ is acyclic for the functor $i^!$, i.e. $\bR i ^!T=i ^!T$. Denote $N=i^!T\in {{\mathcal A}}^{op}\text{-mod}$. Choose a quasi-isomorphism $g:T\to J$ where $J\in {{\mathcal I}}({{\mathcal A}}_{{{\mathcal R}}} ^{op})$. We need to prove that $i^!g$ is a quasi-isomorphism. It suffices to show that $\pi _* i^!g$ is a quasi-isomorphism. Recall that $\pi _*i^!=i^!\pi _*$. Thus it suffices to prove that $\pi _*g$ is a homotopy equivalence. Clearly it is a quasi-isomorphism. Recall that the DG ${{\mathcal R}}^{op}$-module $\pi _*J$ is h-injective (Example 3.13) We claim that $\pi _*T$ is also such. Since the direct product of h-injective DG modules is again h-injective, it suffices to prove that for each object $A\in {{\mathcal A}}$ the DG ${{\mathcal R}}^{op}$-module $T(A)$ is h-injective. Take some object $A\in {{\mathcal A}}$. Since ${{\mathcal R}}\in {\operatorname{dgart}}_-$ the DG ${{\mathcal R}}^{op}$-module $T(A)$ has a decreasing filtration $$G^0\supset G^1\supset G^2\supset ...,$$ with $$\gr T(A)=\oplus _{j}(T(A))^j \otimes {{\mathcal R}}^*.$$ A direct sum of shifted copies of the DG ${{\mathcal R}}^{op}$-module ${{\mathcal R}}^*$ is h-injective (Lemma 3.18). Thus each $(T(A))^j \otimes {{\mathcal R}}^*$ is h-injective and hence each quotient $T(A)/G^j$ is h-injective. Also $$T(A)=\lim_{\leftarrow}T(A)/G^j.$$ Therefore $T(A)$ is h-injective by Remark 3.5. It follows that $\pi _*g$ is a homotopy equivalence, hence also $i^!\pi _*g$ is such. The last theorem allows us to compare the functors ${\operatorname{Def}}_-$ and ${\operatorname{coDef}}_-$ in some important special cases. Namely we have the following corollary. Assume that a\) ${\operatorname{Ext}}^{-1}(E,E)=0$; b\) there exists an h-projective or an h-injective $P\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded above and quasi-isomorphic to $E$; c\) there exists an h-projective or an h-injective $I\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded below and quasi-isomorphic to $E$; Then the pseudo-functors ${\operatorname{Def}}_-(E)$ and ${\operatorname{coDef}}_-(E)$ are equivalent. We have a quasi-isomorphism $P\to I$. Hence by Proposition 8.3 the DG algebras $\End (P)$ and $\End (I)$ are quasi-isomoprhic. Therefore, in particular, the pseudo-functors ${\operatorname{Def}}_-^{\h}(P)$ and ${\operatorname{coDef}}_-^{\h}(I)$ are equivalent (Corollary 8.4b)). It remains to apply the last theorem. In practice in order to find the required bounded resolutions one might need to pass to a “smaller” DG category. So it is useful to have the following stronger corollary. Let $F:{{\mathcal A}}\to {{\mathcal A}}^\prime$ be a DG functor which induces a quasi-equivalence $F^{\pre-tr}:{{\mathcal A}}^{\pre-tr}\to {{\mathcal A}}^{\prime \pre-tr}$. Consider the corresponding equivalence $F_*:D({{\mathcal A}}^{\prime 0})\to D({{\mathcal A}}^{op})$ (Corollary 3.15). Let $E\in {{\mathcal A}}^{\prime 0}\text{-mod}$ be such that a\) ${\operatorname{Ext}}^{-1}(E,E)=0$; b\) there exists an h-projective or an h-injective $P\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded above and quasi-isomorphic to $F_*(E)$; c\) there exists an h-projective or an h-injective $P\in {{\mathcal A}}^{op}\text{-mod}$ which is bounded below and quasi-isomorphic to $F_*(E)$; Then the pseudo-functors ${\operatorname{Def}}_-(E)$ and ${\operatorname{coDef}}_-(E)$ are equivalent. By the above corollary the pseudo-functors ${\operatorname{Def}}_-(F_*(E))$ and ${\operatorname{coDef}}_-(F_*(E))$ are equivalent. By Proposition 10.4 the pseudo-functors ${\operatorname{Def}}_-(E)$ and ${\operatorname{Def}}_-(F_*(E))$ are equivalent. Since the functor $\bR F^!:D({{\mathcal A}}^{op})\to D({{\mathcal A}}^{\prime 0})$ is also an equivalence, we conclude that the pseudo-functors ${\operatorname{coDef}}_-(E)$ and ${\operatorname{coDef}}_-(F_*(E))$ are equivalent by Proposition 10.11. If in the above corollary the DG category ${{\mathcal A}}^\prime$ is pre-triangulated, then one can take for ${{\mathcal A}}$ a full DG subcategory of ${{\mathcal A}}^\prime$ such that ${\operatorname{Ho}}({{\mathcal A}}^\prime)$ is generated as a triangulated category by the subcategory ${\operatorname{Ho}}({{\mathcal A}})$. One can often choose ${{\mathcal A}}$ to have one object. Let ${{\mathcal C}}$ be a bounded DG algebra, i.e. ${{\mathcal C}}^i=0$ for $|i|>>0$ and also $H^{-1}({{\mathcal C}})=0$. Then by Theorem 11.6 and Proposition 4.7 $${\operatorname{coDef}}_-({{\mathcal C}})\simeq {\operatorname{coDef}}^{\h}_-({{\mathcal C}})\simeq {\operatorname{Def}}^{\h}_-({{\mathcal C}})\simeq {\operatorname{Def}}_-({{\mathcal C}}).$$ The following theorem makes the equivalence of Corollary 11.9 more explicit. Let us first introduce some notation. For an artinian DG algebra ${{\mathcal R}}$ consider the DG functors $$\eta _{{{\mathcal R}}}, \epsilon _{{{\mathcal R}}}: {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod} \to {{\mathcal A}}_{{{\mathcal R}}}^{op}\text{-mod}$$ defined by $$\epsilon _{{{\mathcal R}}} (M)=M\otimes _{{{\mathcal R}}}{{\mathcal R}}^*, \quad \eta _{{{\mathcal R}}}(N)=\Hom _{{{\mathcal R}}^{op}}({{\mathcal R}}^* ,N).$$ They induce the corresponding functors $$\bR \eta _{{{\mathcal R}}}, \bL \epsilon _{{{\mathcal R}}}:D({{\mathcal A}}^{op}_{{{\mathcal R}}})\to D({{\mathcal A}}^{op}_{{{\mathcal R}}}).$$ Let $E \in {{\mathcal A}}^{op}\text{-mod}$ satisfy the assumptions a), b), c) of Corollary 11.9. Fix ${{\mathcal R}}\in {\operatorname{dgart}}_-$. Then the following holds. 1\) Let $F\in {{\mathcal A}}^{op}\text{-mod}$ be h-projective or h-injective quasi-isomorphic to $E$. a\) For any $(S, \sigma )\in {\operatorname{Def}}^{\h}_{{{\mathcal R}}}(F)$ we have $i^*S=\bL i^*S$. b\) For any $(T,\tau )\in {\operatorname{coDef}}^{\h}_{{{\mathcal R}}}(F)$ we have $i^!T=\bR i^!T$. 2\) There are natural equivalences of pseudo-functors ${\operatorname{Def}}^{\h}_-(F)\simeq {\operatorname{Def}}_-(E)$, ${\operatorname{coDef}}_-^{\h}(F)\simeq {\operatorname{coDef}}_-(E)$. 3\) The functors $\bL \epsilon _{{{\mathcal R}}}$ and $\bR \eta _{{{\mathcal R}}}$ induce mutually inverse equivalences $$\bL \epsilon _{{{\mathcal R}}}:{\operatorname{Def}}_{{{\mathcal R}}}(E)\to {\operatorname{coDef}}_{{{\mathcal R}}}(E),$$ $$\bR \eta _{{{\mathcal R}}}:{\operatorname{coDef}}_{{{\mathcal R}}}(E)\to {\operatorname{Def}}_{{{\mathcal R}}}(E).$$ 1a). We may and will assume that $\sigma ={\operatorname{id}}$. Choose a bounded above h-projective or h-injective $P\in {{\mathcal A}}^{op}\text{-mod}$, which is quasi-isomorphic to $E$. Then there exists a quasi-isomorphism $P\to F$ (or $F\to P$). The pseudo-functors ${\operatorname{Def}}_-^{\h}(P)$ and ${\operatorname{Def}}_-^{\h}(F)$ are equivalent by Corollary 8.4 (a) or b)). By Theorem 11.6 a) ${\operatorname{Def}}_-^{\h}(P)\simeq {\operatorname{Def}}_-(P)$. Hence by Corollary 11.4 a) for each $(S^\prime ,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}(P)$ we have $i^*S^\prime =\bL i^*S^\prime $. Now Corollary 8.6 (a) or b)) implies that $i^*S=\bL i^*S$. This proves 1a). 1b). We may and will assume that $\tau ={\operatorname{id}}$. The proof is similar to that of 1a). Namely, choose a bounded below h-projective or h-injective $I\in {{\mathcal A}}^{op}\text{-mod}$ quasi-isomorphic to $E$. Then there exists a quasi-isomorphism $F\to I$ (or $I\to F$). The pseudo-functors ${\operatorname{coDef}}_-^{\h}(I)$ and ${\operatorname{coDef}}_-^{\h}(F)$ are equivalent and by Corollary 8.4 (a) or b)). By Theorem 11.6 a) ${\operatorname{coDef}}_-^{\h}(I)\simeq {\operatorname{coDef}}_-(I)$. Hence by Corollary 11.4 b) for each $(T^\prime ,{\operatorname{id}})\in {\operatorname{coDef}}^{\h}(I)$ we have $i^!T^\prime =\bR i^!T^\prime$. Now Corollary 8.6 (a) or b)) implies that $i^!T=\bR i^!T$. 2\) This follows from 1), Corollary 11.4 a), b). 3\) This follows from 2) and the fact that $\epsilon _{{{\mathcal R}}}$ and $\eta _{{{\mathcal R}}}$ induce inverse equivalences between ${\operatorname{Def}}_{{{\mathcal R}}}^{\h}(F)$ and ${\operatorname{coDef}}_{{{\mathcal R}}}^{\h}(F)$ (Proposition 4.7). Let DG algebras ${{\mathcal B}}$ and ${{\mathcal C}}$ be quasi-isomorphic and $H^{-1}({{\mathcal B}})=0$ ($=H^{-1}({{\mathcal C}})$). Suppose that the pseudo-functors ${\operatorname{Def}}({{\mathcal B}})$ and ${\operatorname{Def}}^{\h}({{\mathcal B}})$ (resp. ${\operatorname{coDef}}({{\mathcal B}})$ and ${\operatorname{coDef}}^{\h}({{\mathcal B}})$) are equivalent. Then the same is true for ${{\mathcal C}}$. Similar results hold for the pseudo-functors ${\operatorname{Def}}_-, {\operatorname{Def}}^{\h} _-, {\operatorname{coDef}}_-, ...$. We may and will assume that there exists a morphism of DG algebras $\psi :{{\mathcal B}}\to {{\mathcal C}}$ which is a quasi-isomorphism. By Proposition 8.6 a) the pseudo-functors ${\operatorname{Def}}^{\h}({{\mathcal B}})$ and ${\operatorname{Def}}^{\h}({{\mathcal C}})$ are equivalent. By Proposition 10.4 the pseudo-functors ${\operatorname{Def}}({{\mathcal B}})$ are ${\operatorname{Def}}({{\mathcal C}})$ are equivalent. By Proposition 11.2 a) ${\operatorname{Def}}({{\mathcal B}})$ (resp. ${\operatorname{Def}}({{\mathcal C}})$) is a full pseudo-subfunctor of ${\operatorname{Def}}^{\h}({{\mathcal B}})$ (resp. ${\operatorname{Def}}^{\h}({{\mathcal C}})$). Thus is ${\operatorname{Def}}({{\mathcal B}})\simeq {\operatorname{Def}}^{\h}({{\mathcal B}})$, then also ${\operatorname{Def}}({{\mathcal C}})\simeq {\operatorname{Def}}^{\h}({{\mathcal C}})$. The proof for ${\operatorname{coDef}}$ and ${\operatorname{coDef}}^{\h}$ is similar using Proposition 8.6 a), Proposition 10.11 and Proposition 11.2 b). Let ${{\mathcal B}}$ be a DG algebra such that $H^{-1}({{\mathcal B}})=0$. Assume that ${{\mathcal B}}$ is quasi-isomorphic to a DG algebra ${{\mathcal C}}$ such that ${{\mathcal C}}$ is bounded above (resp. bounded below). Then the pseudo-functors ${\operatorname{Def}}_-({{\mathcal B}})$ and ${\operatorname{Def}}^{\h}_-({{\mathcal B}})$ are equivalent (resp. ${\operatorname{coDef}}_-({{\mathcal B}})$ and ${\operatorname{coDef}}^{\h}_-({{\mathcal B}})$ are equivalent). By Theorem 11.6 a) we have that ${\operatorname{Def}}_-({{\mathcal C}})$ and ${\operatorname{Def}}_-^{\h}({{\mathcal C}})$ are equivalent (resp. ${\operatorname{coDef}}_-({{\mathcal C}})$ and ${\operatorname{coDef}}_-^{\h}({{\mathcal C}})$ are equivalent). It remains to apply Proposition 11.14. Relation between pseudo-functors ${\operatorname{Def}}_-(E)$, ${\operatorname{coDef}}_-(E)$ and ${\operatorname{Def}}_-({{\mathcal C}})$, ${\operatorname{coDef}}_-({{\mathcal C}})$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ The next proposition follows immediately from our previous results. Let ${{\mathcal A}}$ be a DG category and $E\in {{\mathcal A}}^{op}\text{-mod}$. Assume that a\) ${\operatorname{Ext}}^{-1}(E,E)=0$; b\) there exists a bounded above (resp. bounded below) h-projective or h-injective $F\in {{\mathcal A}}^{op}\text{-mod}$ which is quasi-isomorphic to $E$; c\) there exists a bounded above (resp. bounded below) DG algebra ${{\mathcal C}}$ which is quasi-isomorphic to $\End (F)$. Then the pseudo-functors ${\operatorname{Def}}_-(E)$ and ${\operatorname{Def}}_-({{\mathcal C}})$ (resp. ${\operatorname{coDef}}_-(E)$ and ${\operatorname{coDef}}_- ({{\mathcal C}})$) are equivalent. Assume that $F$ and ${{\mathcal C}}$ are bounded above. Then ${\operatorname{Def}}_-(E)\simeq {\operatorname{Def}}^{\h}_-(F)$ and ${\operatorname{Def}}_-({{\mathcal C}})\simeq {\operatorname{Def}}^{\h}_-({{\mathcal C}})$ by Theorem 11.6 a). Also ${\operatorname{Def}}^{\h}_-(F)\simeq {\operatorname{Def}}^{\h}_-({{\mathcal C}})$ by Proposition 6.1 and Theorem 8.1. Assume that $F$ and ${{\mathcal C}}$ are bounded below. Then ${\operatorname{coDef}}_-(E)\simeq {\operatorname{coDef}}^{\h}_-(F)$ and ${\operatorname{coDef}}_-({{\mathcal C}})\simeq {\operatorname{coDef}}^{\h}_-({{\mathcal C}})$ by Theorem 11.6 b). Also ${\operatorname{coDef}}^{\h}_-(F)\simeq {\operatorname{coDef}}^{\h}_-({{\mathcal C}})$ by Proposition 6.1 and Theorem 8.1. The equivalences of pseudo-functors ${\operatorname{Def}}^{\h}_-({{\mathcal C}})\simeq {\operatorname{Def}}^{\h}_-(F)$, ${\operatorname{coDef}}^{\h}_-({{\mathcal C}})\simeq {\operatorname{coDef}}^{\h}_-(F)$ in the proof of last proposition can be made explicit. Put ${{\mathcal B}}=\End(F)$. Assume, for example, that $\psi :{{\mathcal C}}\to {{\mathcal B}}$ is a homomorphism of DG algebras which is a quasi-isomorphism. Then the composition of DG functors (Propositions 9.2, 9.4) $$\Sigma ^F\cdot \psi ^*:{{\mathcal C}}^{op}\text{-mod}\to {{\mathcal A}}^{op}\text{-mod}$$ induces equivalences of pseudo-functors $${\operatorname{Def}}^{\h} (\Sigma ^F \cdot \psi ^*): {\operatorname{Def}}^{\h}({{\mathcal C}})\simeq {\operatorname{Def}}^{\h}(F)$$ $${\operatorname{coDef}}^{\h} (\Sigma ^F \cdot \psi ^*):{\operatorname{coDef}}^{\h}({{\mathcal C}})\simeq {\operatorname{coDef}}^{\h}(F)$$ by Propositions 9.2e) and 9.4f). Pseudo-functors ${\operatorname{Def}}(E)$, ${\operatorname{coDef}}(E)$ are not determined by the DG algebra $\bR \Hom (E,E)$ ---------------------------------------------------------------------------------------------------------------------------- One might expect that the derived deformation and co-deformation pseudo-functors ${\operatorname{Def}}_-(E)$, ${\operatorname{coDef}}_-(E)$ depend only on the (quasi-isomorphism class of the) DG algebra $\bR \Hom (E,E)$. This would be an analogue of Theorem 8.1 for the derived deformation theory. Unfortunately this is not true as is shown in the next proposition (even for the “classical” pseudo-functors ${\operatorname{Def}}_{{\operatorname{cl}}}$, ${\operatorname{coDef}}_{{\operatorname{cl}}}$). This is why all our comparison results for the pseudo-functors ${\operatorname{Def}}_-$ and ${\operatorname{coDef}}_-$ such as Theorems 11.6, 11.13, Corollaries 11.9, 11.15, Proposition 11.16 need some boundedness assumptions. Consider the DG algebra $A=k[x]$ with the zero differential and $\deg (x)=1$. Let ${{\mathcal A}}$ be the DG category with one object whose endomorphism DG algebra is $A$. Then ${{\mathcal A}}^{op}\text{-mod}$ is the DG category of DG modules over the DG algebra $A^{op}=A$. Denote by abuse of notation the unique object of ${{\mathcal A}}$ also by $A$ and consider the DG ${{\mathcal A}}^{op}$-modules $P=h^A$ and $I=h_A^*$. The first one is h-projective and bounded below while the second one is h-injective and bounded above (they are the graded dual of each other). Note that the DG algebras $\End (P)$ and $\End (I)$ are isomorphic: $$\End (P)=A, \quad \End (I)=A^{**}=A.$$ Let ${{\mathcal R}}=k[\epsilon]/(\epsilon ^2)$ be the (commutative) artinian DG algebra with the zero differential and $\deg(\epsilon)=0$. In the above notation the following holds: a\) The groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(P)$ is connected. b\) The groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(I)$ is not connected. c\) The groupoid ${\operatorname{coDef}}_{{{\mathcal R}}}(I)$ is connected. d\) The groupoid ${\operatorname{coDef}}_{{{\mathcal R}}}(P)$ is not connected. Let $(S,{\operatorname{id}})\in {\operatorname{Def}}_{{{\mathcal R}}}^{\h}(I)$. Then $S=I\otimes _k{{\mathcal R}}$ as a graded $(A\otimes {{\mathcal R}})^{op}$-module and the differential in $S$ is equal to “multiplication by $\lambda (x\otimes \epsilon)$” for some $\lambda \in k$. We denote this differential $d_{\lambda}$ and the deformation $S$ by $S _{\lambda}$. By Lemma 11.7 each $(S_{\lambda}, {\operatorname{id}})$ is also an object in the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(I)$. Notice that for $\lambda \neq 0$ we have $H(S_{\lambda})=k$ and if $\lambda =0$ then $H(S_{\lambda})=A\otimes {{\mathcal R}}$. This shows for example that $(S_1,{\operatorname{id}})$ and $(S_0,{\operatorname{id}})$ are non-isomorphic objects in ${\operatorname{Def}}_{{{\mathcal R}}}(I)$ and proves b). The proof of d) is similar using Lemma 11.8. Let us prove a). By Proposition 11.2, 1) the groupoid ${\operatorname{Def}}_{{{\mathcal R}}}(P)$ is equivalent to the full subcategory of ${\operatorname{Def}}^{\h}_{{{\mathcal R}}}(P)$ consisting of objects $(S,{\operatorname{id}})$ such that $S\in {{\mathcal P}}({{\mathcal A}}_{{{\mathcal R}}}^{op})$ or, equivalently, $i^*S=\bL i^*S$. As in the proof of b) above we have $S=P\otimes {{\mathcal R}}$ as a graded $(A\otimes {{\mathcal R}})^{op}$-module and the differential in $S$ is equal to “multiplication by $\lambda (x\otimes \epsilon)$” for some $\lambda \in k$. Again we denote the corresponding $S$ by $S_{\lambda}$. It is clear that the trivial homotopy deformation $S_0$ is h-projective in ${{\mathcal A}}^{op}_{{{\mathcal R}}}\text{-mod}$, hence it is also an object in ${\operatorname{Def}}_{{{\mathcal R}}}(P)$. It remains to prove that for $\lambda \neq 0$ the DG ${{\mathcal A}}_{{{\mathcal R}}}^{op}$-module $S_{\lambda}$ is not h-projective. Since the DG functor $\pi _!$ preserves h-projectives (Example 3.13) it suffices to show that $S_{\lambda }$ considered as a DG ${{\mathcal R}}$-module is not h-projective. We have $$\pi _!S_{\lambda}=\bigoplus_{n\geq 0}{{\mathcal R}}[-n]$$ with the differential $\lambda \epsilon :{{\mathcal R}}[-n]\to {{\mathcal R}}[-n-1]$. Consider the DG ${{\mathcal R}}$-module $$N=\bigoplus_{n=-\infty}^{\infty}{{\mathcal R}}[-n]$$ with the same differential $\lambda \epsilon :{{\mathcal R}}[-n]\to {{\mathcal R}}[-n-1].$ Note that $N$ is acyclic (since $\lambda \neq 0$) and the obvious embedding of DG ${{\mathcal R}}$-modules $\pi _!S_{\lambda}\hookrightarrow N$ is not homotopic to zero. Hence $\pi _!S_{\lambda}$ is not h-projective. This proves a). The proof of c) is similar using Proposition 11.2, 2) and the DG functor $\pi_*$ from Example 3.13. [ELO3]{} J. Benabou, Introduction to bicategories, Reports of the Midwest Category Seminar, LNM 47 (1967). 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W. Goldman, J. Milson, The deformation theory of representations of fundamental groups of compact Kahler manifolds, [*Publ. Math. IHES*]{}, [**67**]{} (1988), 43-96. V. Hinich, DG coalgebras as formal stacks, arXiv:math.AG/9812034. D. Husemoller, J. Moore, J. Stasheff, Differential homological algebra and homogeneous spaces, Journal of Algebra 5 (1974), 113-185. M. M. Kapranov, Injective resolutions of BG and derived moduli spaces of local systems, alg-geom/9704009. B. Keller, Deriving DG categories, [*Ann. scient. Éc. Norm. Sup.*]{}, $4^e$ série, t. 27, (1994), 63-102. O. A.  Laudal, Noncommutative deformations of modules, Homology, Homotopy and Applications, Vol. 4 (2) (2002) 357-396. A. Efimov, V. A. Lunts, D. Orlov, Deformation theory of objects in homotopy and derived categories II: pro-representability of the deformation functor (preprint). A. Efimov, V. A. Lunts, D. Orlov, Deformation theory of objects in homotopy and derived categories III: abelian categories (preprint). 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ArXiv

--- abstract: 'Producing a large amount of annotated speech data for training ASR systems remains difficult for more than 95% of languages all over the world which are low-resourced. However, we note human babies start to learn the language by the sounds (or phonetic structures) of a small number of exemplar words, and “generalize" such knowledge to other words without hearing a large amount of data. We initiate some preliminary work in this direction. Audio Word2Vec is used to learn the phonetic structures from spoken words (signal segments), while another autoencoder is used to learn the phonetic structures from text words. The relationships among the above two can be learned jointly, or separately after the above two are well trained. This relationship can be used in speech recognition with very low resource. In the initial experiments on the TIMIT dataset, only 2.1 hours of speech data (in which 2500 spoken words were annotated and the rest unlabeled) gave a word error rate of 44.6%, and this number can be reduced to 34.2% if 4.1 hr of speech data (in which 20000 spoken words were annotated) were given. These results are not satisfactory, but a good starting point.' address: 'National Taiwan University, Taiwan' bibliography: - 'mybib.bib' - 'IR\_bib.bib' - 'ref\_dis.bib' - 'segment.bib' - 'transfer.bib' - 'INTERSPEECH16.bib' - 'ICASSP13.bib' - 'refs.bib' title: 'From Semi-supervised to Almost-unsupervised Speech Recognition with Very-low Resource by Jointly Learning Phonetic Structures from Audio and Text Embeddings' --- **Index Terms**: automatic speech recognition, semi-supervised Introduction {#sec:intro} ============ Automatic speech recognition (ASR) has achieved remarkable success in many applications [@bahdanau2016end; @amodei2016deep; @zhang2017very]. However, with existing technologies, machines have to learn from a huge amount of annotated data to achieve acceptable accuracy, which makes the development of such technologies for new languages with low resource challenging. Collecting a large amount of speech data is expensive, not to mention having the data annotated. This remains true for at least 95% of languages all over the world. Substantial effort has been reported on semi-supervised ASR [@vesely2013semi; @dikici2016semi; @thomas2013deep; @grezl2014combination; @vesely2017semi; @chen2018almost; @karita2018semi; @drexler2018combining]. However, in most cases a large amount of speech data including a good portion annotated were still needed. So training ASR systems with relatively little data, most of which are not annotated, remains to be an important but unsolved problem. Speech recognition under such “very-low" resource conditions is the target task of this paper. We note human babies start to learn the language by the sounds of a small number of exemplar words without hearing a large amount of data. They more or less learn those words by “how they sound", or the phonetic structures for the words. These exemplar words and their phonetic structures then seem to “generalize" to other words and sentences they learn later on. It is certainly highly desired if machines can do that too. In this paper we initiate some preliminary work in this direction. Audio Word2Vec was proposed to transform spoken words (signal segments for words without knowing the underlying words they represent) to vectors of fixed dimensionality [@DBLP:conf/interspeech/ChungWSLL16] carrying information about the phonetic structures of the spoken words. Segmental Audio Word2Vec was further proposed to jointly segment an utterance into a sequence of spoken words and transform them into a sequence of vectors [@SSAE]. Substantial effort has been made to try to align such audio embeddings with word embeddings [@chung2018unsupervised], which was one way to teach machines to learn the words jointly with their sounds or phonetic structures. Approaches of semi-supervised end-to-end speech recognition approaches along similar directions were also reported recently [@karita2018semi; @drexler2018combining]. But all these works still used relatively large amount of training data. On the other hand, unsupervised phoneme recognition and almost-unsupervised word recognition were recently achieved to some extent using zero or close-to-zero aligned audio and text data [@DBLP:conf/interspeech/LiuCLL18; @chen2018almost], primarily by mapping the audio embeddings with text tokens, whose “very-low" resource setting is the goal of this paper. In this work, we let the machines learn the phonetic structures of words from the embedding spaces of respective spoken and text words, as well as the relationships between the two. All these can be learned jointly, or separately for spoken and text words individually followed by learning the relationships between the two. It was found the former is better, and reasonable speech recognition was achievable with very low resource. In the initial experiments on the TIMIT dataset, only 2.1 hours of total speech data (in which 2500 spoken words were annotated and the rest unlabeled) gave a word error rate of 44.6%, and this number can be reduced to 34.2% if 4.1 hr of speech data (in which 20000 spoken words were annotated) were given. These results are not satisfactory, but a good starting point. ![The architecture of the proposed approach.[]{data-label="fig:overview"}](Overview.png){width="\linewidth"} ![Embedding alignment (red dotted block in middle of Figure \[fig:overview\]) realized by transformation between two latent spaces.[]{data-label="fig:alignment"}](alignment.png){width="\linewidth"} Proposed Approach {#sec:proposed} ================= For clarity, we denote the speech corpus as $\mathbf{X} = {\{\mathbf{x}_{i}\}}_{i=1}^{M}$, which consists of $M$ spoken words, each represented as $\mathbf{x}_i=(\mathbf{x}_{i_1}, \mathbf{x}_{i_2}, ..., \mathbf{x}_{i_T})$, where $\mathbf{x}_{i_t}$ is the acoustic feature vector at time t and $T$ is the length of the spoken word. Each spoken word $\mathbf{x}_i$ corresponds to a text word in $\mathbf{W} = {\{w_{k}\}}_{k=1}^{N}$, where $N$ is the number of distinct text words in the corpus. We can represent each text word as a sequence of subword units, like phonemes or characters, and denote it as $\mathbf{y}_i=(\mathbf{y}_{i_1}, \mathbf{y}_{i_2}, ..., \mathbf{y}_{i_L})$, where $\mathbf{y}_{i_l}$ is the one-hot vector for the $l$^th^ subword and $L$ is the number of subwords in the word. A small set of known paired data is denoted as $\mathbf{Z} = {\{(\mathbf{x}_{j}, \mathbf{y}_{j})\}}$, where $(\mathbf{x}_{j}, \mathbf{y}_{j})$ corresponds to the same text word. In the initial work here we focus on the joint learning of words in audio and text forms, so we assume all training spoken words have been properly segmented with good boundaries. Many existing approaches can be used to segment utterances into spoken words automatically [@DBLP:conf/naacl/TranTBGLO18; @DBLP:journals/jstsp/TangLKGLDSR17; @kamper2017embedded; @kamper2017segmental; @SRAILICASSP15; @WordEmbedIS14; @QbyELSTMICASSP15; @settle2017query; @SemanticRepresentationICASSP18], including the Segmental Audio Word2Vec [@SSAE] mentioned above. Extension to entire utterance input without segmentation is left for future work. A text word corresponds to many different spoken words with varying acoustic factors such as speaker or microphone characteristics, and noise. We jointly refer to all such acoustic factors as speaker characteristics below for simplicity. Intra-domain Unsupervised Autoencoder Architecture {#subsec:unsupervised} -------------------------------------------------- There are three encoders and two decoders in the architecture of the proposed approach in Figure \[fig:overview\]. We use two encoders $E_p$ and $E_s$ to encode the phonetic structures and speaker characteristics of a spoken word $\mathbf{x}_i$ into an audio phonetic vector $\mathbf{v_{p_a}}$ and a speaker vector $\mathbf{v_s}$ respectively. Meanwhile, we use another encoder $E_t$ to encode the phonetic structure of a text word $\mathbf{y}_i$ into a text phonetic vector $\mathbf{v_{p_t}}$, where text words $\mathbf{y}_i$ are represented as sequences of one-hot vectors for subwords. The audio decoder $D_a$ takes the pair ($\mathbf{v_{p_a}}$, $\mathbf{v_s}$) as input and reconstruct the original spoken word features $\mathbf{x}_i'$. The text decoder $D_t$ takes $\mathbf{v_{p_t}}$ as input and reconstruct the original text word features $\mathbf{y}_i'$. Two intra-domain losses are used for unsupervised training: 1. Intra-domain audio reconstruction loss, which is the mean-square-error between the audio original and the reconstructed features: $$\begin{aligned} \label{in_a_r_loss} L_{in\_a\_r} &= \sum_{i} \| \mathbf{x}_i - D_a(E_p(\mathbf{x}_i), E_s(\mathbf{x}_i)) \|_2^2. \end{aligned}$$ 2. Intra-domain text reconstruction loss, which is the negative log-likelihood for the text vector sequences to be reconstructed: $$\begin{aligned} \label{in_t_r_loss} L_{in\_t\_r} &= - \sum_{i} logPr(\mathbf{y}_i | D_t(E_t(\mathbf{y}_i))). \end{aligned}$$ Cross-domain Reconstruction with Paired Data {#subsec:cross-domain} -------------------------------------------- When the latent spaces for the phonetic structures for spoken words $\mathbf{x}_i$ and text words $\mathbf{y}_i$ are individually learned based on the intra-domain reconstruction losses (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\]), they can be very different, since the former are continuous signals with varying length and behavior, while the latter are sequences of discrete symbols with given length. So here we try to learn them jointly using relatively small number of known pairs of spoken words $\mathbf{x}_j$ and the corresponding text words $\mathbf{y}_j$, $\mathbf{Z} = {\{(\mathbf{x}_{j}, \mathbf{y}_{j})\}}$. Hopefully the two latent spaces can be twisted somehow and end up with a single common latent space, in which both phonetic vectors for audio and text, $\mathbf{v_{p_a}}$ and $\mathbf{v_{p_t}}$, can be properly represented. So two cross-domain losses below are used: 1. Cross-domain audio reconstruction loss: $$\begin{aligned} L_{cr\_a\_r} &= \sum_{(\mathbf{x}_j, \mathbf{y}_j) \in \mathbf{Z}} \| \mathbf{x}_j - D_a(E_t(\mathbf{y}_j), E_s(\mathbf{x}_j)) \|_2^2. \label{cr_a_r_loss} \end{aligned}$$ 2. Cross-domain text reconstruction loss: $$\begin{aligned} L_{cr\_t\_r} &= - \sum_{(\mathbf{x}_j, \mathbf{y}_j) \in \mathbf{Z}} logPr(\mathbf{y}_j | D_t(E_p(\mathbf{x}_j))). \label{cr_t_r_loss} \end{aligned}$$ By minimizing the reconstruction loss for the audio/text features obtained with the phonetic vectors computed from input sequences in the other domain as in (\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\]), the phonetic vectors of spoken and text words can be somehow aligned to carry some consistent information about the phonetic structures. Cross-domain Alignment of Phonetic Vectors with Paired Data {#subsec:emb_align} ----------------------------------------------------------- On top of the cross-domain reconstruction losses (\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\]), the two latent spaces can be further aligned by a hinge loss for all known pairs of spoken and text words $(\mathbf{x}_j, \mathbf{y}_j)$: 1. Cross-domain embedding loss: $$\begin{aligned} L_{cr\_emb} &= \sum_{(\mathbf{x}_j, \mathbf{y}_j) \in \mathbf{Z}} \| E_p(\mathbf{x}_j) - E_t(\mathbf{y}_j) \|_2^2 \\ &+ \sum_{(\mathbf{x}_i, \mathbf{y}_j) \notin \mathbf{Z}} \max(0, \lambda - \| E_p(\mathbf{x}_i) - E_t(\mathbf{y}_j) \|_2^2). \label{cr_emb_loss} \end{aligned}$$ In the second term of (\[cr\_emb\_loss\]), for each text word $\mathbf{y}_j$, we randomly sample $\mathbf{x}_i$ such that $(\mathbf{x}_i, \mathbf{y}_j) \notin \mathbf{Z}$ to serve as a negative sample. In this way, the phonetic vectors corresponding to different text words can be kept far enough apart. Here in (\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\])(\[cr\_emb\_loss\]) the small number of paired spoken and text words $\{(\mathbf{x}_j$, $\mathbf{y}_j)\} \in \mathbf{Z}$ serve just as the small number of exemplar words and their sounds when human babies start to learn the language. The reconstruction and alignment across the two spaces is somehow to try to “generalize" the phonetic structures of these exemplar words to other words in the language as human babies do. Joint Learning and Inference {#subsec:inference} ---------------------------- The total loss function $L$ to be minimized during training is the weighted sum of the above five losses: $$\begin{aligned} L &= \alpha_1 L_{in\_a\_r} + \alpha_2 L_{in\_t\_r} \\ &+ \alpha_3 L_{cr\_a\_r} + \alpha_4 L_{cr\_t\_r} + \alpha_5 L_{cr\_emb} \label{loss} \end{aligned}$$ During inference, for each distinct text word $w_k$ in training data, we compute its text phonetic vector $(\mathbf{v_{p_t}})_{k}$, k = 1, ..., N. Then for each spoken word $\mathbf{x}_i$ in testing data, we apply softmax on the negative distance between its audio phonetic vector $\mathbf{v_{p_a}}$ and each text phonetic vector $(\mathbf{v_{p_t}})_{k}$ to get the posterior probability for each text word $Pr_{a}(w_k|\mathbf{x}_i)$: $$\begin{aligned} Pr_{a}(w_k|\mathbf{x}_i) &= \frac{\exp(-\| E_p(\mathbf{x}_i) - (\mathbf{v_{p_t}})_{k} \|_2^2)}{\sum_{j=1}^N \exp(-\| E_p(\mathbf{x}_i) - (\mathbf{v_{p_t}})_{j} \|_2^2)}. \label{pr_a} \end{aligned}$$ When a large amount of unpaired text data is available, a language model can be trained and integrated into the inference. Suppose the spoken word $\mathbf{x}_i$ is the $t$-th spoken word in an utterance $\mathbf{u}$ and its corresponding text word is $u_t$. The log probability for recognition is then: $$\begin{aligned} \log Pr(u_t=w_k|\mathbf{x}_i) &= \log Pr_{a}(w_k|\mathbf{x}_i) \\ &+ \beta \log Pr_{LM}(u_t=w_k), \label{pr} \end{aligned}$$ where the first term is as in (\[pr\_a\]), and $Pr_{LM}(\cdot)$ is the language model score. All $\alpha_i$ and $\beta$ above are hyperparameters. Cycle Consistency Regularization {#subsec:cycle} -------------------------------- We can further add a cycle-consistency loss to the original loss function (\[loss\]): $$\begin{aligned} L_{cycle} &= \sum_{(\mathbf{x}_j, \mathbf{y}_j) \in \mathbf{Z}} \| \mathbf{x}_j - D_a(E_t(D_t(E_p(\mathbf{x}_j))), E_s(\mathbf{x}_j)) \|_2^2 \\ &+ \| \mathbf{y}_j - D_t(E_p(D_a(E_t(\mathbf{y}_j), E_s(\mathbf{x}_j)))) \|_2^2. \label{cycle_loss} \end{aligned}$$ Part of the first term was shown by the dotted purple cycle in the right of Figure \[fig:overview\], while part of the second term was shown by the dotted blue loop in the left of the figure. [|c||c|c|c|c|c|c|]{} &\ & & & & & &\ 39809 & - & - & - & - & - & 32.9\ & - & - & - & - & 36.6 &\ 10000 & - & - & - & 42.3 & 38.4 & 36.4\ 5000 & - & - & 48.2 & 44.8 & 41.0 & 38.9\ & - & 55.3 & 50.3 & 47.1 & & 42.5\ 1000 & 65.0 & 57.8 & 54.2 & 51.5 & 50.2 & 48.2\ 600 & 65.2 & 61.7 & 57.9 & 56.5 & 55.4 & 54.7\ 200 & 69.7 & 69.1 & 67.6 & 68.9 & 67.4 & 67.6\ 100 & 77.2 & 76.3 & 78.0 & 78.3 & 82.8 & 78.7\ 50 & 82.8 & 81.8 & 80.5 & 80.0 & 82.1 & 85.4\ Separate Learning then Transformation {#subsec:dis_spk} ------------------------------------- Because the continuous signals of spoken words and the discrete symbol sequences of text words are very different, the alignment between the two latent spaces as mentioned above may not be smooth. For example, during the joint learning in (\[loss\]) the cross-domain losses (\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\])(\[cr\_emb\_loss\]) inevitably disturb the intra-domain losses (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\]) and produce distortions on the phonetic structures for the individual audio and text domains. Of course there exist a different option, i.e., training the intra-domain phonetic structures separately for spoken and text words first, and then learn a transformation between them. This concept can be understood by replacing the red dotted block in the middle right of Figure \[fig:overview\] denoted by “Embedding Alignment" by that shown in Figure \[fig:alignment\]. In this way Figure \[fig:overview\] becomes two independent autoencoders, for spoken and text words on the left and right, respectively trained with intra-domain reconstruction losses (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\]) only, plus a set of alignment transformations $\mathbf{M_{at}}$ and $\mathbf{M_{ta}}$ between the two latent spaces. In this way the phonetic structures over the spoken and text words may be better learned separately in different spaces. In the left part of Figure \[fig:overview\], however, a set of GAN-based [@chen2018almost; @WGANGP] criteria is needed to disentangle the speaker characteristics from phonetic structures (not shown in Figure \[fig:overview\]), while in the original Figure \[fig:overview\] this disentanglement can be achieved with the help from the text word autoencoder. ![The 2-dim display of the WER (%) performance spectrum of Table \[table:unpair\_num\] for different training data sizes (hrs), and different N (number of paired words). The black dots are the real experimental results. The contours are produced based on linear interpolation among black dots.[]{data-label="fig:contour"}](contour.png){width="\linewidth"} The phonetic vectors $\mathbf{v_{p_a}}$ and $\mathbf{v_{p_t}}$ separately trained are first normalized in all dimensions and projected onto their lower dimensional space by PCA. The projected vectors in the principal component spaces are respectively denoted as $\mathbf{A} = \{\mathbf{a}_i\}_{i=1}^M$ for audio and $\mathbf{T} = \{\mathbf{t}_i\}_{i=1}^N$ for text. The paired spoken and text words, $\mathbf{Z} = \{(\mathbf{x}_j, \mathbf{y}_j)\}$, are denoted here as $\mathbf{Z} = \{(\mathbf{a}_j, \mathbf{t}_j)\}$, in which $\mathbf{a}_j$ and $\mathbf{t}_j$ correspond to the same word. Then a pair of transformation matrices, $\mathbf{M_{at}}$ and $\mathbf{M_{ta}}$ are learned, where $\mathbf{M_{at}}$ maps a vector $\mathbf{a}$ in $\mathbf{A}$ to the space of $\mathbf{T}$, that is, $\mathbf{t} = \mathbf{M_{at}}\mathbf{a}$, while $\mathbf{M_{ta}}$ maps a vector $\mathbf{t}$ in $\mathbf{T}$ to the space of $\mathbf{A}$. $\mathbf{M_{at}}$ and $\mathbf{M_{ta}}$ are initialized as identity matrices and then learned iteratively with gradient descent minimizing the objective function: $$\begin{aligned} L_t = \sum_{(\mathbf{a}_j, \mathbf{t}_j) \in \mathbf{Z}} \|\mathbf{t}_j - \mathbf{M_{at}}\mathbf{a}_j \|_2^2 + \sum_{(\mathbf{a}_j, \mathbf{t}_j) \in \mathbf{Z}} \|\mathbf{a}_j - \mathbf{M_{ta}}\mathbf{t}_j \|_2^2 \\ + \lambda^\prime \sum_{(\mathbf{a}_j, \mathbf{t}_j) \in \mathbf{Z}} (\| \mathbf{a}_j - \mathbf{M_{ta}}\mathbf{M_{at}}\mathbf{a}_j \|_2^2 + \| \mathbf{t}_j - \mathbf{M_{at}}\mathbf{M_{ta}}\mathbf{t}_j \|_2^2). \end{aligned} \label{eq:align}$$ In the first two terms, we want the transformation of $\mathbf{a}_j$ by $\mathbf{M_{at}}$ to be close to $\mathbf{t}_j$ and vice versa. The last two terms are for cycle consistency, i.e., after transforming $\mathbf{a}_j$ to the space of $\mathbf{T}$ by $\mathbf{M_{at}}$ and then transforming back by $\mathbf{M_{ta}}$, it should end up with the original $\mathbf{a}_j$, and vice versa. $\lambda^\prime$ is a hyper-parameter. [|c||c|c|c||c|c|c||c|]{} & &\ & & &\ & & & & & & &\ 39809 & 32.9 & 31.7 & **31.3** & 68.6 & 67.0 & 65.6 & 38.6\ 1000 & 48.2 & **47.4** & 47.5 & 71.8 & 72.5 & 74.6 & 60.5\ -- ------- ---------- ------ ------ ------ ------ ------ 39809 **32.9** 33.1 33.9 33.2 44.8 50.4 1000 **48.2** 51.4 49.3 48.6 57.0 69.5 -- ------- ---------- ------ ------ ------ ------ ------ : Ablation studies for the proposed approach of joint learning in (\[loss\]) when removing a loss term in (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\])(\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\])(\[cr\_emb\_loss\]) with 4.1 hrs of data.[]{data-label="table:ablation"} ------------------------------ ---------- ---------- ---------- ---------- ---------- (\[loss\]) **32.9** **48.2** 67.6 78.7 85.4 Plus cycle (\[cycle\_loss\]) 41.4 51.5 **66.4** **74.7** **82.3** ------------------------------ ---------- ---------- ---------- ---------- ---------- : Contributions by the cycle-consistency in (\[cycle\_loss\]) of Subsection \[subsec:cycle\] for 4.1 hrs of data and different N.[]{data-label="table:cycle"} Experimental Setup {#sec:exp} ================== Dataset {#subsec:dataset} ------- TIMIT [@garofolo1993darpa] dataset was used here. Its training set contains only 4620 utterances (4.1 hours) with a total of 39809 words, or 4893 distinct words. So this dataset is close to the “very-low" resource setting considered here. We followed the standard Kaldi recipe [@povey2011kaldi] to extract the MFCCs of 39-dim with utterance-wise cepstral mean and variance normalization (CMVN) applied as the acoustic features. Model Implementation {#subsec:implementation} -------------------- The three encoders $E_p$, $E_s$ and $E_t$ in Figure \[fig:overview\] were all Bi-GRUs with hidden layer size 256. The decoders $D_a$ and $D_t$ were GRUs with hidden layer size 512 and 256 respectively. The value of threshold $\lambda$ in (\[cr\_emb\_loss\]) was set to 0.01. Hyperparameters $(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5, \beta)$ were set to $(0.2, 1.0, 0.2, 1.0, 5.0, 0.01)$. We trained a tri-gram language model on the transcriptions of TIMIT data and performed beam search with beam size 10 during inference in (\[pr\]) to obtain the recognition results. Adam optimizer [@DBLP:journals/corr/KingmaB14] was used and the initial learning rate was $10^{-4}$. The mini-batch size was 32. In realizing the embedding alignment in Figure \[fig:alignment\], the discriminator used in the audio embedding for disentangling the speaker vector was a two-layer fully-connected network with hidden size 256, and the mapping functions $\mathbf{M_{at}}$ and $\mathbf{M_{ta}}$ were linear matrices, following the setting of the previous work [@chen2018almost]. Experiments {#sec:exp} =========== Performance Spectrum for Different Training Data Sizes and Different Number of Paired Words {#subsec:exp_data_amount} ------------------------------------------------------------------------------------------- First we wish to see the achievable performance in word error rates (WER) (%) over the testing set for the joint learning approach of (\[loss\]) in Subsection \[subsec:inference\] when the training data size and the numbers of paired words (N) are respectively reduced to as small as possible. All the encoders and decoders are single-layer GRUs. The results are listed in Table \[table:unpair\_num\] (blank for upper left corner because only smaller number of words can be labeled and made paired for smaller data size). A 2-dim display of this performance spectrum is shown in Figure \[fig:contour\], where the black dots are the real results in Table \[table:unpair\_num\], while the contours are produced based on linear interpolation among black dots. We can see from Table \[table:unpair\_num\] only 2.1 hr of total data (in which 2500 spoken words were labeled and the rest unlabeled) gave an WER of 44.6% (in red), and this number can be reduced to 34.2% if 4.1 hr of data (in which 20000 words labeled) were available (in blue). We can also see how the WER varied when the total data size was changed for a fixed value of N (e.g. N=2500, the horizontal dotted red line in Figure \[fig:contour\]) or N was changed for a fixed data size (e.g. 2.1 hr, the vertical orange line in Figure \[fig:contour\]). Right now these numbers are still relatively high (specially for $N \leq 1000$ or less than 1.0 hr of data), but the smooth, continuous performance spectrum may imply the proposed approach is a good direction and better performance may be achievable in the future. For example, the aligned phonetic structures for the N paired words seemed to “generalize" to more words not labeled. Also, it can be found that in the lower half of Figure \[fig:contour\] the contours are more horizontal, implying for small N (e.g. $N \leq 600$) the help offered by larger data size may be limited. In the upper half of the figure \[fig:contour\], however, the contours go up on the left implying for larger N (e.g. $N \geq 600$) larger data size gave lower WER. Different Learning Strategies and Model Parameters {#subsec:exp_approaches} -------------------------------------------------- Table \[table:unpair\_num\] and Figure \[fig:contour\] are for the joint learning strategy in (\[loss\]) of Subsection \[subsec:inference\] and single-layer GRUs. Here we wish to see the performance for the strategy of separate learning plus a transformation afterwards in (\[eq:align\]) of Subsection \[subsec:dis\_spk\]. The results are in the left section (Joint) and middle section (Separate) of Table \[table:jnt\_sep\], for 4.1 hrs of data and N=39809, 1000. Results for 2 and 3 layers of GRUs in encoders/decoders (L=2, 3) are also listed. The results in Table \[table:jnt\_sep\] empirically showed joint learning the phonetic structures from spoken and text words together with the alignment between them outperformed the strategy of separate learning then transformation. Very probably the phonetic structures of subword unit sequences of given length are very different from those of signal segments of different length, so aligning and warping them during joint learning gives smoother mapping relationships, while a forced transformation between two separately trained structures may be too rigid. Also, the model with L=2 achieved slightly better results in comparison with L=1 in the case of 4.1 hrs of data here, while overfitting happened with L=3 when N was small. All the above results are for phonemes taken as the subword units. The right column of Table \[table:jnt\_sep\] are the results for characters instead with joint learning and L=1. We see characters worked much worse than phonemes. Clearly the phoneme sequences described the phonetic structures much better than character sequences. Ablation Studies and Cycle-consistency Regularization {#subsec:exp_ablation} ----------------------------------------------------- In Table \[table:ablation\], we performed ablation studies for joint learning in (\[loss\]) of Subsection \[subsec:inference\] and 4.1 hrs data and N=39809 and 1000 by removing a loss term in (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\])(\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\])(\[cr\_emb\_loss\]) in Subsection \[subsec:unsupervised\], \[subsec:cross-domain\] and \[subsec:emb\_align\]. We see all reconstruction losses in (\[in\_a\_r\_loss\])(\[in\_t\_r\_loss\])(\[cr\_a\_r\_loss\])(\[cr\_t\_r\_loss\]) were useful, but the cross-domain text reconstruction loss in (\[cr\_t\_r\_loss\]) was specially important, obviously because the phoneme sequences described the phonetic structures most precisely, and the cross-domain reconstruction offered good mapping relationships. On the other hand, the cross-domain embedding loss learning from paired spoken and text words in (\[cr\_emb\_loss\]) made the most significant contribution here. The knowledge learned here from paired data “generalize" to other unlabeled words. We also tested the cycle-consistency mentioned in (\[cycle\_loss\]) of Subsection \[subsec:cycle\] for 4.1 hrs of data and different N. The results in Table \[table:ablation\] showed that the cycle consistency may not help for larger N, but became useful for smaller N (e.g. $N \leq 200$) when too few number of such paired words or “anchor points" were inadequate for constructing the mapping relationships. This is because the cycle-consistency condition required every paired spoken and text word to go through all encoders and decoders. Discussion and Conclusion {#sec:end} ========================= In this work, we investigate the possibility of performing speech recognition with very low resource (small data size with small number of paired labeled words) by joint learning the phonetic structures from audio and text embeddings. Smooth and continuous WER performance spectrum when the data size and number of paired words were respectively reduced to as small as possible was obtained. The achieved WERs were still relatively high, but implied a good direction for future work.

ArXiv

--- abstract: | We present a study on differentiating direct production mechanisms of the newly discovered Higgs-like boson at the LHC based on several inclusive observables. The ratios introduced reveal the parton constituents or initial state radiations involved in the production mechanisms, and are directly sensitive to fractions of contributions from different channels. We select three benchmark models, including the SM Higgs boson, to illustrate how the theoretical predictions of the above ratios are different for the $gg$, $b\bar b(c\bar c)$, and $q\bar q$ (flavor universal) initial states in the direct production. We study implications of current Tevatron and LHC measurements. We also show expectations from further LHC measurements with high luminosities. **Keywrords**: Higgs Physics, Beyond Standard Model author: - Jun Gao title: 'Differentiating the production mechanisms of the Higgs-like resonance using inclusive observables at hadron colliders' --- SMU-HEP-13-22\ Aug 22, 2013 Introduction\[sec:intro\] ========================= Recently, a new resonance with a mass around 126 GeV has been discovered by the ATLAS [@Aad:2012tfa] and CMS [@Chatrchyan:2012ufa]. It is considered to be a highly Standard Model (SM) Higgs-like particle with measured production rate consistent with the SM Higgs boson through $\gamma\gamma$, $ZZ^*$, $WW^*$, and $\tau\tau$ channels [@Aad:2012tfa; @Chatrchyan:2012ufa]. Although further efforts are required in order to determine the features of the new resonance, like the spin, couplings with SM particles, and self-couplings. The spin-1 hypothesis is excluded by the observation of the $\gamma\gamma$ decay mode according to the Landau-Yang theorem [@Landau:1948kw; @Yang:1950rg]. Many proposals have been suggested to distinguish between the spin-0 and spin-2 hypotheses mainly focusing on the kinematic distributions, e.g, angular distributions [@Choi:2002jk; @Gao:2010qx; @DeRujula:2010ys; @Englert:2010ud; @Ellis:2012wg; @Bolognesi:2012mm; @Choi:2012yg; @Ellis:2012jv; @Englert:2012xt; @Banerjee:2012ez; @Modak:2013sb; @Boer:2013fca; @Frank:2013gca], event shapes [@Englert:2013opa] and other observables [@Boughezal:2012tz; @Ellis:2012xd; @Alves:2012fb; @Geng:2012hy; @Djouadi:2013yb]. Recent measurements [@ATLAS:2013xla; @ATLAS:2013mla; @Aad:2013xqa; @CMS:xwa] show a favor of spin-0 over specific spin-2 scenarios. As for the couplings, the current direct information or constraints are for the relative strength between different observed channels, i.e., $\gamma\gamma$, $ZZ^*$, $WW^*$, and $\tau\tau$ [@Aad:2012tfa; @Chatrchyan:2012ufa]. Without knowing the total decay width and rates from other unobserved channels it is difficult to determine the absolute strength of the couplings of the new resonance at the LHC. Or later we can further measure the couplings through a combined analysis after the observation of the associated production modes with the SM $W$ and $Z$ bosons or the vector-boson fusion (VBF) production mode [@Plehn:2001nj; @Giardino:2012ww; @Rauch:2012wa; @Azatov:2012rd; @Low:2012rj; @Carmi:2012in; @Plehn:2012iz; @Djouadi:2012rh]. Among all the couplings of the new resonance, the ones with gluons or quarks are important but difficult to be measured at the LHC since the corresponding decay modes consist of two jets, which suffer from huge QCD backgrounds at the LHC even for the heavy-quark (charm or bottom quark) jets. Moreover, it is extremely hard to discriminate the couplings with gluons and light-quarks from the resonance decay. This relates to the answer to a more essential question, i.e., the direct production of the new resonance is dominated by the gluon fusion or quark annihilation. In the SM, the loop-induced gluon fusion is dominant while the heavy-quark annihilation only contributes at a percent level. As for other hypotheses, like in the two Higgs doublet models, the heavy-quark contributions can be largely enhanced [@Djouadi:2005gj; @Meng:2012uj], or in the graviton-like cases [@ArkaniHamed:1998rs; @Randall:1999ee], the light-quark contributions are important as well. Similar as in the determination of the spin of the new resonance, we can use the angular distributions of the observed decay products, like $\gamma\gamma$, $ZZ^*$, and $WW^*$, to differentiate the $gg$ and $q\bar q$ production mechanisms as in [@ATLAS:2013xla; @ATLAS:2013mla; @Aad:2013xqa; @CMS:xwa]. But the analyses are highly model-dependent, i.e., the angular distributions are sensitive to the spin of the resonance as well as the structures of the couplings with the decay products [@Gao:2010qx]. On another hand, since these two production mechanisms depend on different flavor constituents of the parton distribution functions (PDFs), they may show distinguishable behaviors by looking at the ratios of the event rate at different colliders or center-of-mass energies as previously shown in [@Mangano:2012mh] for various SM processes at the LHC including for the SM Higgs boson, or the rapidity distribution of the resonance. Even more ambitious, we may look at the production of the resonance in association with an additional photon or jet from initial state radiations which are presumably to be different for the gluon and quark initial states. Unlike the case of the angular distributions, all these observables are insensitive to details of couplings with the decay products. Thus they may serve as good discriminators of the direct production mechanisms of the new resonance. Based on the above ideas we present a study of using inclusive observables to discriminate the mechanisms of the direct production of the new resonance, including both the theoretical predictions and the experimental feasibilities. In Section \[sec:set\], we describe the benchmark models of the production mechanisms studied in this paper, and introduce several inclusive observables that are used in our study. Section \[sec:ben\] compares the theoretical predictions of the observables from different models. In Section \[sec:exp\] we discuss the applications on current experimental data from the Tevatron and LHC, and also future measurements at the LHC. Section \[sec:con\] is a brief conclusion. Model setups and inclusive observables\[sec:set\] ================================================= We select three benchmark models in the study, including the pure SM case, an alternative spin-0 resonance with enhanced couplings to the charm and bottom quarks, and a spin-2 resonance with universal couplings to all the quarks. As explained in the introduction, our analyses mainly rely on the fractions of the $gg$ and $q\bar q$ contributions in the production mechanism and are insensitive to details of couplings with the decay products. More precisely, the relevant effective couplings for the spin-0 cases are given by $${\mathcal L}_{spin-0}={g_1^{(0)}\over v}HG^{\mu\nu}G_{\mu\nu}+{g_2^{(0)}\over v} (m_cH\bar{\Psi}_c\Psi_c+m_bH\bar{\Psi}_b\Psi_b),$$ and for the spin-2 case by [@Englert:2012xt] $${\mathcal L}_{spin-2}=g_1^{(2)}Y_{\mu\nu}T_{G}^{\mu\nu}+g_2^{(2)}Y_{\mu\nu}T_{q} ^{\mu\nu},$$ with $H$ being the scalar particle, $Y_{\mu\nu}$ the general spin-2 fields [@spin2a; @spin2b], $G_{\mu\nu}$ the field strength of QCD, and $\Psi_{c,b}$ the charm and bottom quarks. We choose graviton-inspired couplings for the spin-2 case with $T_{G}^{\mu\nu}$ and $T_{q}^{\mu\nu}$ being the energy-momentum tensors of the gluon and quarks (flavor universal) as can be found in [@Hagiwara:2008jb]. Here we suppress all other couplings of the resonance with the $W$, $Z$ bosons, photon, and $\tau$ lepton, which are adjusted to satisfy the corresponding decay branching ratios observed [@Aad:2012tfa; @Chatrchyan:2012ufa], especially the couplings with photons should be suppressed in order to be consistent with the experimental measurements. We work under an effective Lagrangian approach and will not discuss about the possible UV completion of the theory. For model A, the pure SM, we have $$v=246\, {\rm GeV},\,\, g_1^{(0)}={\alpha_s\over 12\pi},\,\, g_2^{(0)}=1,\,\, m_{c(b)}=0.634(2.79)\, {\rm GeV},$$ where $g_1^{(0)}$ are evaluated at the LO in the infinite top quark mass limit, and the heavy-quark masses are $\overline{\rm MS}$ running mass at the resonance mass $m_X=126\,{\rm GeV}$ [@Beringer:1900zz; @Chetyrkin:2000yt]. From a phenomenological point of view, we introduce model B, the heavy-quark dominant case with $g_1^{(0)}=0$. Note that $g_1^{(0)}$ always receives non-zero contributions from the heavy-quark loops proportional to $g_2^{(0)}$. However, in global analyses of the Higgs couplings [@Carmi:2012in; @Djouadi:2012rh], it is always treated as another free parameter that could in principle vanish, since its actual value depends on details of the underlying new physics. Thus model B is a phenomenological simplification of models with heavy-quark annihilation dominant in the production, e.g., supersymmetric models with large $\tan \beta$ [@Djouadi:2005gj]. The absolute value of $g_2^{(0)}$ is irrelevant for the study here. Similar for model C, the spin-2 case, we set $g_1^{(2)}=0$ with the production dominated by the light quarks. It is shown that a spin-2 model with minimal couplings [@Gao:2010qx] to the vector bosons has been ruled out by both the ATLAS and CMS despite of the production mechanism [@Aad:2013xqa; @CMS:xwa]. The measurements utilize angular distributions of final states from decay vector bosons. As shown in [@Gao:2010qx], these angular distributions are sensitive to detailed structures of the couplings to the vector bosons. Thus the exclusion could not be applied to a general spin-2 model involving much more free parameters in the vector boson couplings [@Gao:2010qx]. In contrast the observables introduced below are independent of the couplings to the decay vector bosons. The inclusive observables we studied can be divided into three categories. First one is the ratio of the inclusive cross sections of the direct production, $R^{1}$, including the cross sections at the Tevatron, and at the LHC with different center-of-mass energies. The second one is the ratio of the direct production cross sections in the inner and full rapidity region of the produced resonance, $R^{2}$. These two observables probe the production mechanisms through the differences of the relevant PDFs. The third observable, $R^{3}$, is the ratio of the production cross section of the resonance in association with a photon to the one of the direct production. It differentiates the production channels by measuring the initial state radiations. For the calculation of $R^{3}$ we neglect the small explicit couplings of the new resonance with photons in the production. Other observables that might be sensitive to the production mechanisms are related to the initial state QCD radiations, like the $p_T$ spectrum or jet-bin cross sections [@Dittmaier:2012vm; @Wiesemann:2012ij] of the resonance, which are again different for the $gg$ and $q\bar q$ initial states. But that will be even more challenging in both the theory predictions and experimental measurements. Benchmark comparisons\[sec:ben\] ================================ Ratios of the total cross section\[sec:ben1\] --------------------------------------------- Here we calculate the total cross sections of the direct production of the new resonance at the Tevatron and LHC with $\sqrt s=7$, 8, and 14 TeV. At the leading order (LO), they are related to the following parton-parton luminosities, $$\begin{aligned} \label{eq:lum} &L_{gg}(\tau)=\int_{\tau}^1\frac{dx_1}{x_1}\int_{\tau/x_1}^1\frac{dx_2}{x_2}\tau^2 f_{g/h_1}(x_1, \mu_f)f_{g/h_2}(x_2, \mu_f)\delta(x_1x_2-\tau), \nonumber \\ &L_{c\bar c(b \bar b)}(\tau)=\int_{\tau}^1\frac{dx_1}{x_1}\int_{\tau/x_1}^1\frac{dx_2}{x_2}\tau^2 [f_{c(b)/h_1}(x_1, \mu_f)f_{\bar c(\bar b)/h_2}(x_2, \mu_f)+h_1\leftrightarrow h_2]\delta(x_1x_2-\tau), \nonumber\\ &L_{q\bar q}(\tau)=\sum_{q}\int^1_{\tau} \frac{dx_1}{x_1} \int^1_{\tau/x_1} \frac{dx_2}{x_2} \tau^2 [f_{q/h_1}(x_1, \mu_f)f_{\bar q/h_2}(x_2, \mu_f)+h_1\leftrightarrow h_2]\delta(x_1x_2-\tau),\end{aligned}$$ where $\tau=m_X^2/s$, $x_{1,2}$ are the momentum fractions. $\mu_f$ is the factorization scale and set to $m_X$ in our calculations. $f_{i/h}(x)$ are the PDFs, and the sum in $L_{q\bar q}$ runs over all the 5 active quark flavors. Thus the typical Bjorken $x\sim m_X/\sqrt s$ are about 0.06, 0.018, 0.016, and 0.009 at the Tevatron, LHC 7, 8, and 14 TeV. While beyond LO, there are also contributions from other flavor combinations subject to different $x_1-x_2$ constraints. We select 5 ratios from all the cross sections, $R^1_{L7/T}=\sigma({\rm LHC}\,7\, {\rm TeV})/\sigma({\rm Tevatron})$, similar for $R^1_{L8/T}$, $R^1_{L14/T}$, $R^1_{L14/L7}$, and $R^1_{L14/L8}$. The cross sections for models A and B can be calculated up to next-to-next-to-leading order (NNLO) in QCD using the numerical code iHixs1.3 [@Anastasiou:2011pi]. While it is only calculated at the LO for the model C. Note that the ratios $R^1$ at the LO are totally determined by the behaviors of the parton-parton luminosities in Eq. (\[eq:lum\]) and are independent of the detailed structures of the couplings, while at higher orders they may show slight dependence on the couplings. We set the renormalization scale to $m_X=126\,{\rm GeV}$ as well, and use the most recent NNLO PDFs including CT10 [@Gao:2013xoa], MSTW 2008 [@Martin:2009iq], and NNPDF2.3 [@Ball:2012cx]. The PDF and $\alpha_s$ uncertainties are calculated and combined using the prescription in [@Ball:2012wy]. In Table. \[tab:r1a\] we show the predicted ratios $R^1$ for the SM Higgs boson from different PDF groups. It can be seen that the current uptodate NNLO PDFs give pretty close results for the ratios. The combined PDF+$\alpha_s$ uncertainties are about 7% for the ratios of the NNLO cross sections at the LHC over Tevatron due to the relatively large uncertainties of the gluon PDF at the large $x$ region. While the uncertainties are reduced to a level of about 2% for the ratios at the LHC. Theoretical uncertainties due to the missing higher order QCD corrections can be estimated by looking at the differences of the results at different orders, which are smaller compared to the combined PDF+$\alpha_s$ uncertainties and are not considered in our analysis. Tables. \[tab:r1b\] and \[tab:r1c\] show similar results for the model B and C. The heavy quark PDFs are mostly generated through the evolution of the gluon PDF. Thus the results of the model B are close to the SM case. The model C predicts very different results compared to the SM or model B, for the ratios of the NNLO cross sections at the LHC over Tevatron, and also shows smaller uncertainties, since the cross sections are dominated by the light quark scattering. ---------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Combined LO NLO NNLO LO NLO NNLO LO NLO NNLO NNLO $R^1_{L7/T}$ $17.9^{+0.8}_{-1.0}$ $17.5^{+0.8}_{-0.9}$ $17.0^{+0.7}_{-0.9}$ $18.1^{+0.5}_{-0.5}$ $17.7^{+0.5}_{-0.5}$ $17.2^{+0.5}_{-0.5}$ $18.6^{+0.6}_{-0.6}$ $18.1^{+0.6}_{-0.5}$ $17.5^{+0.5}_{-0.5}$ $17.1^{+1.1}_{-1.1}$ $R^1_{L8/T}$ $22.9^{+1.1}_{-1.3}$ $22.4^{+1.0}_{-1.2}$ $21.7^{+1.0}_{-1.2}$ $23.2^{+0.7}_{-0.7}$ $22.6^{+0.7}_{-0.7}$ $21.9^{+0.7}_{-0.7}$ $23.9^{+0.8}_{-0.8}$ $23.2^{+0.7}_{-0.7}$ $22.4^{+0.7}_{-0.7}$ $21.8^{+1.5}_{-1.5}$ $R^1_{L14/T}$ $59.9^{+3.4}_{-4.1}$ $58.5^{+3.1}_{-3.8}$ $56.3^{+3.0}_{-3.6}$ $60.7^{+2.3}_{-2.2}$ $59.3^{+2.2}_{-2.1}$ $57.0^{+2.1}_{-2.0}$ $62.2^{+2.4}_{-2.3}$ $60.6^{+2.2}_{-2.1}$ $58.1^{+2.1}_{-2.0}$ $56.6^{+4.3}_{-4.3}$ $R^1_{L14/L7}$ $3.34^{+0.04}_{-0.05}$ $3.35^{+0.04}_{-0.05}$ $3.32^{+0.04}_{-0.05}$ $3.35^{+0.03}_{-0.03}$ $3.35^{+0.03}_{-0.03}$ $3.32^{+0.03}_{-0.03}$ $3.34^{+0.03}_{-0.03}$ $3.34^{+0.03}_{-0.02}$ $3.31^{+0.02}_{-0.02}$ $3.31^{+0.05}_{-0.05}$ $R^1_{L14/L8}$ $2.61^{+0.02}_{-0.03}$ $2.62^{+0.02}_{-0.03}$ $2.60^{+0.02}_{-0.03}$ $2.62^{+0.02}_{-0.02}$ $2.62^{+0.02}_{-0.02}$ $2.60^{+0.02}_{-0.02}$ $2.61^{+0.02}_{-0.02}$ $2.61^{+0.02}_{-0.01}$ $2.59^{+0.01}_{-0.01}$ $2.59^{+0.03}_{-0.03}$ ---------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ : \[tab:r1a\]Predicted ratios $R^1$ at different orders from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for the case of pure SM. ---------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ Combined LO NLO NNLO LO NLO NNLO LO NLO NNLO NNLO $R^1_{L7/T}$ $23.0^{+1.5}_{-1.8}$ $22.7^{+1.6}_{-1.9}$ $23.4^{+1.7}_{-2.0}$ $23.5^{+1.0}_{-1.0}$ $23.2^{+1.1}_{-1.1}$ $24.0^{+1.2}_{-1.2}$ $24.6^{+1.2}_{-1.2}$ $24.4^{+1.3}_{-1.2}$ $25.3^{+1.5}_{-1.4}$ $24.2^{+3.2}_{-3.2}$ $R^1_{L8/T}$ $29.8^{+2.1}_{-2.5}$ $29.4^{+2.2}_{-2.6}$ $30.4^{+2.4}_{-2.8}$ $30.5^{+1.4}_{-1.4}$ $30.0^{+1.5}_{-1.5}$ $31.2^{+1.7}_{-1.7}$ $32.0^{+1.7}_{-1.6}$ $31.6^{+1.8}_{-1.7}$ $33.0^{+2.0}_{-1.9}$ $31.4^{+4.3}_{-4.3}$ $R^1_{L14/T}$ $81.2^{+6.6}_{-7.8}$ $79.4^{+6.8}_{-7.9}$ $82.8^{+7.5}_{-8.6}$ $83.1^{+4.7}_{-4.6}$ $81.6^{+5.0}_{-4.8}$ $85.3^{+5.6}_{-5.4}$ $87.4^{+5.2}_{-4.9}$ $85.8^{+5.5}_{-5.1}$ $90.0^{+6.2}_{-5.7}$ $85.6^{+13.1}_{-13.1}$ $R^1_{L14/L7}$ $3.53^{+0.06}_{-0.07}$ $3.50^{+0.06}_{-0.07}$ $3.54^{+0.06}_{-0.08}$ $3.54^{+0.04}_{-0.04}$ $3.52^{+0.04}_{-0.04}$ $3.55^{+0.05}_{-0.04}$ $3.54^{+0.04}_{-0.04}$ $3.52^{+0.04}_{-0.04}$ $3.55^{+0.04}_{-0.04}$ $3.53^{+0.09}_{-0.09}$ $R^1_{L14/L8}$ $2.72^{+0.03}_{-0.04}$ $2.70^{+0.04}_{-0.04}$ $2.72^{+0.04}_{-0.04}$ $2.73^{+0.02}_{-0.02}$ $2.71^{+0.03}_{-0.02}$ $2.73^{+0.03}_{-0.03}$ $2.73^{+0.02}_{-0.02}$ $2.71^{+0.02}_{-0.02}$ $2.73^{+0.02}_{-0.02}$ $2.72^{+0.05}_{-0.05}$ ---------------- ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ ------------------------ : \[tab:r1b\]Predicted ratios $R^1$ at different orders from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model B. ---------------- ------------------------ ------------------------ ------------------------ ------------------------ Combined LO LO LO LO $R^1_{L7/T}$ $3.96^{+0.07}_{-0.06}$ $4.00^{+0.04}_{-0.06}$ $3.95^{+0.06}_{-0.05}$ $3.98^{+0.10}_{-0.10}$ $R^1_{L8/T}$ $4.68^{+0.08}_{-0.08}$ $4.72^{+0.05}_{-0.07}$ $4.67^{+0.07}_{-0.06}$ $4.70^{+0.12}_{-0.12}$ $R^1_{L14/T}$ $9.17^{+0.20}_{-0.20}$ $9.19^{+0.13}_{-0.16}$ $9.10^{+0.14}_{-0.12}$ $9.18^{+0.25}_{-0.25}$ $R^1_{L14/L7}$ $2.32^{+0.02}_{-0.02}$ $2.30^{+0.01}_{-0.01}$ $2.30^{+0.01}_{-0.01}$ $2.31^{+0.02}_{-0.02}$ $R^1_{L14/L8}$ $1.96^{+0.01}_{-0.01}$ $1.94^{+0.01}_{-0.01}$ $1.95^{+0.01}_{-0.01}$ $1.96^{+0.02}_{-0.02}$ ---------------- ------------------------ ------------------------ ------------------------ ------------------------ : \[tab:r1c\]Predicted ratios $R^1$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model C. Centrality ratio\[sec:ben2\] ---------------------------- At the LO, the rapidity of the produced resonance in the lab frame is given by, $y=\ln(x_1/x_2)/2$, or equivalently $y=\ln((1+\beta)/(1-\beta))/2$, where $\beta$ is the boost of the resonance. We define the centrality $R^2$ as the ratio of the production cross section in the central region (with $|y|<1$) to the one in the full rapidity region, which are related to the corresponding ratio of the parton-parton luminosities at the LO, $L(\tau, |y|<1)/L(\tau)$. For illustration purpose, we show the above luminosity ratio as functions of the rapidity cutoff in Fig. \[fig:r2\] for different parton combinations shown in Eq. (\[eq:lum\]). ![\[fig:r2\] Luminosity fractions as a function of the rapidity cutoff at the LHC with different center-of-mass energies.](hr2.eps){width="80.00000%"} The calculated centrality ratios for the models A, B, and C are listed in Tables. \[tab:r2a\]-\[tab:r2c\] for different PDFs at the LHC. Again for the pure SM case, the predictions are at the NNLO in QCD from HNNLO1.3 code [@Catani:2007vq]. Others are only calculated at the LO. Here we simply choose the central region of $|y|<1$ for the definition of $R^2$. In principle one can find the optimized value that gives largest distinctions of the three models. Similar to the case of $R^1$, the models A and B give close results of $R^2$ but with larger uncertainties compared to $R^1$. The differences of the predictions from the model C with the ones from the model A or B are still significant. ------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined LO NLO NNLO LO NLO NNLO LO NLO NNLO NNLO $R^2_{L7}$ $0.536^{+0.009}_{-0.013}$ $0.536^{+0.009}_{-0.013}$ $0.533^{+0.009}_{-0.013}$ $0.538^{+0.005}_{-0.007}$ $0.537^{+0.005}_{-0.007}$ $0.538^{+0.005}_{-0.007}$ $0.548^{+0.008}_{-0.008}$ $0.546^{+0.008}_{-0.008}$ $0.547^{+0.008}_{-0.008}$ $0.539^{+0.018}_{-0.018}$ $R^2_{L8}$ $0.518^{+0.009}_{-0.012}$ $0.519^{+0.009}_{-0.012}$ $0.526^{+0.009}_{-0.012}$ $0.518^{+0.009}_{-0.003}$ $0.522^{+0.009}_{-0.003}$ $0.532^{+0.009}_{-0.003}$ $0.529^{+0.008}_{-0.008}$ $0.530^{+0.008}_{-0.008}$ $0.538^{+0.008}_{-0.008}$ $0.531^{+0.017}_{-0.017}$ $R^2_{L14}$ $0.453^{+0.007}_{-0.008}$ $0.453^{+0.007}_{-0.008}$ $0.450^{+0.007}_{-0.008}$ $0.454^{+0.004}_{-0.004}$ $0.454^{+0.004}_{-0.004}$ $0.452^{+0.004}_{-0.004}$ $0.461^{+0.005}_{-0.005}$ $0.460^{+0.005}_{-0.005}$ $0.458^{+0.005}_{-0.005}$ $0.453^{+0.012}_{-0.012}$ ------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r2a\]Predicted ratios $R^2$ at different orders from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for the case of pure SM. ------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined LO LO LO LO $R^2_{L7}$ $0.575^{+0.012}_{-0.017}$ $0.578^{+0.006}_{-0.008}$ $0.592^{+0.010}_{-0.010}$ $0.580^{+0.023}_{-0.023}$ $R^2_{L8}$ $0.555^{+0.012}_{-0.015}$ $0.556^{+0.014}_{-0.004}$ $0.571^{+0.010}_{-0.010}$ $0.561^{+0.022}_{-0.022}$ $R^2_{L14}$ $0.487^{+0.009}_{-0.011}$ $0.489^{+0.005}_{-0.006}$ $0.498^{+0.007}_{-0.007}$ $0.490^{+0.016}_{-0.016}$ ------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r2b\]Predicted ratios $R^2$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model B. ------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined LO LO LO LO $R^2_{L7}$ $0.364^{+0.004}_{-0.005}$ $0.358^{+0.005}_{-0.002}$ $0.361^{+0.002}_{-0.002}$ $0.362^{+0.007}_{-0.007}$ $R^2_{L8}$ $0.351^{+0.004}_{-0.005}$ $0.345^{+0.002}_{-0.003}$ $0.348^{+0.002}_{-0.002}$ $0.349^{+0.008}_{-0.008}$ $R^2_{L14}$ $0.309^{+0.004}_{-0.005}$ $0.303^{+0.002}_{-0.003}$ $0.309^{+0.002}_{-0.002}$ $0.306^{+0.007}_{-0.007}$ ------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r2c\]Predicted ratios $R^2$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model C. Associated production\[sec:ben3\] --------------------------------- Here we consider the ratios of the cross sections for the resonance production in association with a photon to the ones of the direct production, $R^3\equiv\sigma_{X+\gamma}/\sigma_{X}$. The advantage is that for the case of the SM, this associated production mode is largely suppressed with main contributions from the $b\bar b$ annihilation at the LHC [@Abbasabadi:1997zr]. While for models B and C, the associated production is only suppressed by the QED couplings even though the statistics are low at the LHC. The calculations for the associated production are performed at the LO. Thus, for consistency we use the LO cross sections of the direct production as well. Moreover, for the model C, we apply a form factor [@Frank:2013gca] $$F=\left(\frac{\Lambda^2}{\hat s+\Lambda^2}\right)^5,$$ to the associated production by multiplying it with the squared amplitudes since the effective operator there violates unitarity above a certain energy scale. Here $\hat s$ is the square of the partonic center-of-mass energy, and we choose the cutoff scale $\Lambda$ to be $800\,{\rm GeV}$. We select the events from the associated production with a rapidity cut of $|y_{\gamma}|<2$ and a transverse momentum cut $p_{T,\gamma}>15\, {\rm GeV}$ on the photon. Here we adopt a relatively lower $p_T$ cut on the photon in order to maximize the statistics of the associated production. Fig. \[fig:r3\] shows ratios of the cross sections of associated production to the ones of the direct production as functions of the $p_T$ cut of the photon at the LHC with different center-of-mass energies. It can be seen that for the SM case, the cross sections of the associated production are negligible, less then $10^{-4}$ times the cross sections of the direct production. While for models B and C the ratios are larger by an order of magnitude comparing to the SM, and the associated production may be observable at the LHC. For lower $p_T$ cutoff the ratios from models B and C are close. At moderate or high $p_T$ cutoff the ratios from model C are larger due to the power enhancement from high dimension operators, and are sensitive to the form factor applied and the UV completion of the theory. The central values and the PDF+$\alpha_s$ uncertainties of $R^3$ predicted in different models are listed in Tables. \[tab:r3a\]-\[tab:r3c\]. ![\[fig:r3\] Ratios of the cross sections of the associated production to the ones of the direct production as functions of the $p_T$ cut of the photon, at the LHC with different center-of-mass energies.](hr3.eps){width="80.00000%"} We may also utilize production of the resonance in association with a jet, and study the effects on observables like jet-bin (jet-veto) fractions, $p_T$ distribution of the resonance as recently measured in [@TheATLAScollaboration:2013eia]. The cross sections of associated production with a jet are much larger compared to the case of a photon due to the strong couplings as well as opening of new partonic channels. Especially for the SM case, $gg$ channel now contributes and dominates over all others. Similarly, we consider a ratio of the one-jet inclusive cross sections to the total inclusive ones, $\sigma_{X+jet}/\sigma_{X}$. For example, using both LO cross sections, we obtain the ratio as 0.355 (0.128) for model A (B) at the LHC 8 TeV. Here we require a jet to have $|y|<2$ and $p_T>30\, {\rm GeV}$. We can see the ratio is larger for the SM case, in contrary with the case of a photon, because of the stronger radiations from gluon initial states and the high dimension effective operators. Thus this ratio may have some discrimination powers on different production mechanisms. At the same time it also has larger theoretical and experimental uncertainties associated with the jet. The resummed $p_T$ spectrums of the resonance produced through $gg$ and $b\bar b$ initial states have been predicted in [@Glosser:2002gm; @Bozzi:2003jy; @Field:2004tt] and [@Field:2004nc; @Belyaev:2005bs] respectively. Shapes of the two distributions are very similar with both peak located around $10\sim 20$ GeV at the LHC for a resonance mass of about 120 GeV. Note that experimentally the jet may fake a photon with a rate depending on both the kinematics and photon isolation criteria. For the SM case, this may induce non-negligible contributions to the photon associated production. We will not discuss these possibilities in the analysis since they are highly dependent on details of the experiments. ----------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined $\times10^{-3}$ LO LO LO LO $R^3_{L7}$ $0.077^{+0.003}_{-0.003}$ $0.075^{+0.002}_{-0.002}$ $0.077^{+0.002}_{-0.002}$ $0.077^{+0.004}_{-0.004}$ $R^3_{L8}$ $0.079^{+0.003}_{-0.002}$ $0.077^{+0.002}_{-0.002}$ $0.080^{+0.002}_{-0.002}$ $0.079^{+0.004}_{-0.004}$ $R^3_{L14}$ $0.085^{+0.002}_{-0.002}$ $0.083^{+0.002}_{-0.002}$ $0.086^{+0.002}_{-0.002}$ $0.085^{+0.004}_{-0.004}$ ----------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r3a\]Predicted ratios $R^3$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for the case of pure SM. ----------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined $\times10^{-3}$ LO LO LO LO $R^3_{L7}$ $1.407^{+0.014}_{-0.014}$ $1.398^{+0.008}_{-0.006}$ $1.405^{+0.007}_{-0.007}$ $1.408^{+0.017}_{-0.017}$ $R^3_{L8}$ $1.424^{+0.014}_{-0.013}$ $1.417^{+0.007}_{-0.006}$ $1.426^{+0.007}_{-0.008}$ $1.425^{+0.016}_{-0.016}$ $R^3_{L14}$ $1.467^{+0.013}_{-0.015}$ $1.464^{+0.003}_{-0.007}$ $1.478^{+0.008}_{-0.008}$ $1.470^{+0.017}_{-0.017}$ ----------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r3b\]Predicted ratios $R^3$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model B. ----------------- --------------------------- --------------------------- --------------------------- --------------------------- Combined $\times10^{-3}$ LO LO LO LO $R^3_{L7}$ $3.291^{+0.057}_{-0.066}$ $3.365^{+0.030}_{-0.019}$ $3.344^{+0.034}_{-0.036}$ $3.302^{+0.089}_{-0.084}$ $R^3_{L8}$ $3.364^{+0.058}_{-0.066}$ $3.438^{+0.025}_{-0.023}$ $3.420^{+0.035}_{-0.037}$ $3.376^{+0.085}_{-0.085}$ $R^3_{L14}$ $3.458^{+0.057}_{-0.061}$ $3.523^{+0.026}_{-0.022}$ $3.521^{+0.037}_{-0.039}$ $3.474^{+0.084}_{-0.084}$ ----------------- --------------------------- --------------------------- --------------------------- --------------------------- : \[tab:r3c\]Predicted ratios $R^3$ at the LO from various PDFs with the PDF+$\alpha_s$ uncertainties at 68% C.L. for model C. Experimental Implications\[sec:exp\] ==================================== Total cross section measurement at the Tevatron and LHC\[sec:exp1\] ------------------------------------------------------------------- The ratios $R^1$, especially the ratios of the total cross sections from the LHC to Tevatron, show a large distinction between gluon or heavy-quark initiated cases (model A or B) and the light-quark case (model C). For example, the central predictions for $R^1_{L7/T}$ are 17.1, 24.2, and 4.0 for the three models respectively according to Tables. \[tab:r1a\]-\[tab:r1c\]. With the full data sample, the combined Tevatron measurements of the inclusive cross sections of the new resonance are summarized in Ref. [@Aaltonen:2013kxa]. Corresponding measurements from the LHC at 7 and 8 TeV can be found in [@Aad:2012tfa; @Chatrchyan:2012ufa]. We show all the measured cross sections from different decay channels in Table. \[tab:back\], which are normalized to the predictions of the SM Higgs boson. Note that for the $\tau\tau$ channel we show the recent updated results instead [@atau; @ctau]. The ATLAS and CMS results are combined here by taking a weighted average with weights of one over square of the corresponding experimental errors. Thus correlations of systematic uncertainties in the two experiments are simply neglected, resulting in optimistic estimations of the combined uncertainties. Most of the results shown are for the inclusive productions, which also receive contributions from the Higgs-strahlung or VBF final states. Presumably they are only a small fraction compared to the ones from the direct production in the experimental analyses. It can be seen that the experimental errors, especially the ones from the Tevatron are far above the theoretical ones shown in Tables. \[tab:r1a\]-\[tab:r1c\]. Thus from Tables. \[tab:r1a\]-\[tab:r1c\] and neglecting the theoretical errors, we obtain the theoretical predictions for $R^1_{L7(8)/T}$ as 1(1), 1.42(1.44), and 0.23(0.22) for models A, B, and C respectively, using the relative strength (all cross sections normalized to the corresponding predictions of the SM Higgs boson). Without knowing the precise probability distribution of the experimental measurements we simply assume they are Gaussian distributed with the errors symmetrized. Based on the two data points ($\gamma\gamma$ and $WW^*$ channels) we calculate the $\chi^2$ values as 1.9, 2.4 and 3.3 for the models A, B, and C, respectively. Thus all three models agree well with the current data. The predictive power of $R^1_{L7(8)/T}$ is mostly limited by the large experimental errors from Tevatron. However, further precise measurements from the LHC may show improvements on discriminations of the three models. For example, assuming the central measurements to be exactly the same as the SM predictions and the fractional errors reduced to 20% for both the $\gamma\gamma$ and $WW^*$ channels, the $\chi^2$ for model C would be 8.4, corresponding to an exclusion at 98.5% C.L. We can also look at the ratios $R^1$ at the LHC with different energies. But they are not so distinguishable among different initial states since the light quarks there are mostly sea-like for the corresponding energies. For the model C, using the relative strength the predictions for $R^1_{L14/L7(8)}$ are 0.70(0.76), which require a high experimental precision in order to distinguish them with the SM predictions with values 1(1). $\gamma\gamma$ $ZZ^*$ $WW^*$ $\tau\tau$ combined ----------- --------------------- ---------------------- ------------------------ --------------------- ------------------------ Tevatron $6.0^{+3.4}_{-3.1}$ – $0.94^{+0.85}_{-0.83}$ – – ATLAS $1.8^{+0.5}_{-0.5}$ $1.2^{+0.6}_{-0.6}$ $1.3^{+0.5}_{-0.5}$ $1.4^{+0.5}_{-0.4}$ $1.4^{+0.3}_{-0.3}$ CMS $1.4^{+0.6}_{-0.6}$ $0.7^{+0.5}_{-0.4}$ $0.7^{+0.5}_{-0.5}$ $1.1^{+0.4}_{-0.4}$ $0.87^{+0.23}_{-0.23}$ ATLAS+CMS $1.6^{+0.4}_{-0.4}$ $0.9^{+0.4}_{-0.4} $ $1.0^{+0.4}_{-0.4}$ $1.2^{+0.3}_{-0.3}$ $1.1^{+0.2}_{-0.2}$ : \[tab:back\]Measured production cross sections of the new resonance through different decay channels at the Tevatron and LHC (7 and 8 TeV combined). All values are normalized to the corresponding cross sections of the SM Higgs boson production. The ATLAS and CMS results are combined by taking a weighted average neglecting correlations. Expectations from the centrality ratios\[sec:exp2\] --------------------------------------------------- The centrality ratios $R^2$ at the LHC also display moderate differences between the model A or B and the model C. To measure the rapidity of the resonance we need to fully reconstruct the final state kinematics. Thus the most promising decay channels for measuring $R^2$ are $\gamma\gamma$ and $ZZ^*$. As shown in Tables. \[tab:r2a\]-\[tab:r2c\], the theoretical errors for the predictions of $R^2$ are a few percents and are rather small compared to the experimental ones. The central predictions for $R^2_{L14}$ are 0.45 and 0.31 for the SM and model C. For both of the two decay channels the experimental errors of $R^2$ are expected to be dominated by the statistical errors whether due to the low event rate or large backgrounds. At the LHC 7 TeV (5.1 $fb^{-1}$), for the diphoton channel after all the selection cuts, the CMS measurement expects about 77 signal events and 311 events per GeV (invariant mass window) from the backgrounds for the case of the SM Higgs boson [@Chatrchyan:2012ufa]. If we assume a 100 $fb^{-1}$ data sample at 14 TeV from each of the CMS and ATLAS experiments, and assume the same event selection efficiencies, the expected event numbers within a mass window of 4 GeV will be about $1.0\times 10^4$ for the SM Higgs boson and $1.1\times 10^5$ for the backgrounds.[^1] Then the expected measurement of $R^2_{L14}$ is about $0.45\pm 0.024$ including only the statistical error.[^2] Thus for this case we may exclude the model C (with $R^2_{L14}$=0.31) at $5\sigma$ C.L.. The $ZZ^*\rightarrow 4l$ channel is almost background free and the observed event number at the CMS is 9 for 7 and 8 TeV combined [@Chatrchyan:2012ufa]. With the same assumptions as the $\gamma\gamma$ channel, the expected event rate is about 513, and the measurement of $R^2_{L14}$ is $0.45\pm 0.036$ for $ZZ^*$ channel. The statistical error is larger than the one of the diphoton channel but the measurement is free of the systematic errors from the background estimations. A more comprehensive study on $R^2$ should be done by the experimentalist to further examine the backgrounds and all the systematic errors, which may change the conclusions here. Observability of the associated production\[sec:exp3\] ------------------------------------------------------ The associated production of the SM Higgs boson with photon is almost unobservable at the LHC with a rate of less than $10^{-4}$ of the direct production rate. While for models B and C the rates are an order of magnitude higher. Even though they may be still difficult to be observed. In order to suppress the backgrounds and obtain sufficient statistics, the diphoton decay channel is the only realistic solution. Thus we need to look at the tri-photon final state. As a quick estimation for the background, we can calculate the ratios of the cross sections of the SM direct tri-photon production (intrinsic backgrounds) to the ones of diphoton production. The selection cuts for the two or three photon events ($p_T$ ordered) are as below $$\begin{aligned} &&|\eta_{\gamma}|<2,\, \Delta R_{\gamma\gamma}>0.4,\, p_{T,1}>30\,{\rm GeV},\, p_{T,2}>20\,{\rm GeV},\,\nonumber\\ &&p_{T,3}>15\,{\rm GeV},\, 124<m_{12}<128\,{\rm GeV}.\end{aligned}$$ Here both the cross sections of the diphoton and tri-photon productions are calculated at the LO using Madgraph 4 [@Alwall:2007st]. Contributions from quark fragmentations and gluon-initiated loop diagrams are not included. We plot the cross section $\sigma_{3\gamma}$ as well as the ratio $\sigma_{3\gamma}/\sigma_{2\gamma}$ as functions of the $p_T$ threshold of the softest photon in the tri-photon production in Fig. \[fig:back\]. The ratios are similar to the results of the models B and C shown in Fig. \[fig:r3\], with $\sigma_{3\gamma}/\sigma_{2\gamma} \sim 0.0022$ for $p_{T,3}>15\,{\rm GeV}$. ![\[fig:back\] (a), Cross sections of the SM tri-photon production at the LHC; (b), ratios of the cross sections of the SM tri-photon production to the ones of diphoton production. The selection cuts are applied to both the tri-photon and diphoton events.](sm-tri1.eps "fig:"){width="38.00000%"} ![\[fig:back\] (a), Cross sections of the SM tri-photon production at the LHC; (b), ratios of the cross sections of the SM tri-photon production to the ones of diphoton production. The selection cuts are applied to both the tri-photon and diphoton events.](sm-tri2.eps "fig:"){width="40.00000%"} By the same assumptions as in Section \[sec:exp2\], the expected background event rate is about $1.1\times 10^5$ for the diphoton channel at the LHC of 14 TeV and $\mathcal{L}=100\, fb^{-1}$. Simply multiplying it with the ratio $\sigma_{3\gamma}/\sigma_{2\gamma}$, we estimate a background rate of about 242 for the tri-photon final state. Similarly, using the numbers in Tables. \[tab:r3a\]-\[tab:r3c\], the expected signal event rates are about $0.8$, $16$ and $24$ for the SM, models B and C respectively. We can see that the signal rates of the models B and C are of the similar size as the $1\sigma$ statistical fluctuation of the background. Thus although the associated production mode shows a large distinction between the SM and the alternative models, i.e., models B and C, but it requires a high luminosity for the experimental measurements, e.g., around 900 (400) $fb^{-1}$ in order to discriminate the SM with the model B (C) at $3\sigma$ C.L. Also note that for a variation of the model B where the charm quark coupling is dominant instead of the bottom quark, the associated production rate can be further enhanced by about a factor of 4 from the electric charge. Conclusions\[sec:con\] ====================== We performed a study on differentiating the direct production mechanisms of the newly discovered Higgs-like boson at the LHC based on several inclusive observables introduced, including the ratios of the production rates at different colliders and energies, the centrality ratios of the resonance, and the ratios of the rates of associated production with a photon to the ones of direct production. Above ratios reveal neither the parton constituents nor initial state radiations involved in the production mechanisms, and are independent of the couplings to the decay products. We select three benchmark models, including the SM Higgs boson, to illustrate how the theoretical predictions of the above ratios are different for the $gg$, $b\bar b(c\bar c)$, and $q\bar q$ (flavor universal) initial states in the direct production. The theoretical uncertainties of the predictions are also discussed. All three models are found to be in good agreement with inclusive rate ratios from current measurements at the Tevatron and LHC. Moreover, we show expectations from further LHC measurements with high luminosities. The centrality ratio measurements are supposed to be able to separate the $gg$ or $b\bar b(c\bar c)$ initial states with $q\bar q$. 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ArXiv

--- abstract: 'We review the current status of compact object simulations that are based on the Smooth Particle Hydrodynamics (SPH) method. The first main part of this review is dedicated to SPH as a numerical method. We begin by discussing relevant kernel approximation techniques and discuss the performance of different kernel functions. Subsequently, we review a number of different SPH formulations of Newtonian, special- and general relativistic ideal fluid dynamics. We particularly point out recent developments that increase the accuracy of SPH with respect to commonly used techniques. The second main part of the review is dedicated to the application of SPH in compact object simulations. We discuss encounters between two white dwarfs, between two neutron stars and between a neutron star and a stellar-mass black hole. For each type of system, the main focus is on the more common, gravitational wave-driven binary mergers, but we also discuss dynamical collisions as they occur in dense stellar systems such as cores of globular clusters.' author: - bibliography: - 'refs.bib' title: SPH Methods in the Modelling of Compact Objects --- =1 Introduction {#sec:intro} ============ Relevance of compact object encounters -------------------------------------- The vast majority of stars in the Universe will finally become a compact stellar object, either a white dwarf, a neutron star or a black hole. Our Galaxy therefore harbors large numbers of them, probably $\sim 10^{10}$ white dwarfs, several $10^8$ neutron stars and tens of millions of stellar-mass black holes. These objects stretch the physics that is known from terrestrial laboratories to extreme limits. For example, the structure of white dwarfs is governed by electron degeneracy pressure, they are therefore Earth-sized manifestations of the quantum mechanical Pauli-principle. Neutron stars, on the other hand, reach in their cores multiples of the nuclear saturation density ($2.6 \times 10^{14}$ ) which makes them excellent probes for nuclear matter theories. The dimensionless compactness parameter $\mathcal{C}= (G/c^2) (M/R)= R_{\rm S}/(2R)$, where $M$,$R$ and $R_{\rm S}$ are mass, radius and Schwarzschild radius of the object, can be considered as a measure of the strength of a gravitational field. It is proportional to the Newtonian gravitational potential and directly related to the gravitational redshift. For black holes, the parameter takes values of 0.5 at the Schwarzschild radius and for neutron stars it is only moderately smaller, $\mathcal{C}\approx 0.2$, both are gigantic in comparison to the solar value of $\sim 10^{-6}$. General relativity has so far passed all tests to high accuracy [@will14a], but most of them have been performed in the limit of weak gravitational fields. Neutron stars and black holes, in contrast, offer the possibility to test gravity in the strong field regime [@psaltis08]. Compact objects that had time to settle into an equilibrium state possess a high degree of symmetry and are essentially perfectly spherically symmetric. Moreover, they are cold enough to be excellently described in a $T=0$ approximation (since for all involved species $i$ the thermal energies are much smaller than the involved Fermi-energies, $k T_i \ll E_{{\rm F},i}$) and they are in chemical equilibrium. For such systems a number of interesting results can be obtained by (semi-) analytical methods. It is precisely because they are in a “minimum energy state” that such systems are hardly detectable in isolation, and certainly not out to large distances. Compact objects, however, still possess – at least in principle – very large energy reservoirs and in cases where these reservoirs can be tapped, they can produce electromagnetic emission that is so bright that it can serve as cosmic beacons. For example, each nucleon inside of a carbon-oxygen white dwarf can potentially still release $\approx$ 0.9 MeV via nuclear burning to the most stable elements, or approximately $1.7 \times 10^{51} $ erg per solar mass. The gravitational binding energy of a neutron star or black hole is even larger, $E_{\rm grav} \sim G M^2/R = \mathcal{C} M c^2 = 3.6 \times 10^{53} \; {\rm erg} \; (\mathcal{C}/0.2) (M/$). Tapping these gigantic energy reservoirs, however, usually requires special, often catastrophic circumstances, such as the collision or merger with yet another compact object. For example, the merger of a neutron star with another neutron star or with a black hole likely powers the sub-class of short Gamma-ray bursts (GRBs) and mergers of two white dwarfs are thought to trigger type Ia supernovae. Such events are highly dynamic and do not possess enough symmetries to be accurately described by (semi-) analytical methods. They require a careful, three-dimensional numerical modelling of gravity, the hydrodynamics and the relevant “microphysical” ingredients such as neutrino processes, nuclear burning or a nuclear equation of state. When/why SPH? ------------- Many of such modelling efforts have involved the Smoothed Particle Hydrodynamics method (SPH), originally suggested by [@gingold77] and by [@lucy77]. There are a number of excellent numerical methods to deal with problems of ideal fluid dynamics, but each numerical method has its particular merits and challenges, and it is usually pivotal in terms of work efficiency to choose the best method for the problem at hand. For this reason, we want to collect here the major strengths of, but also point out challenges for the SPH method. The probably most outstanding feature of SPH is that it allows in a straight forward way to *exactly conserve* mass, energy, momentum and angular momentum *by construction*, i.e., independent of the numerical resolution. This property is ensured by appropriate symmetries in the SPH particle indices in the evolution equations, see Section \[chap:SPH\], together with gradient estimates (usually kernel gradients, but see Section \[sec:SPH\_with\_integral\_gradients\]) that are anti-symmetric with respect to the exchange of two particle indices. For example, as will be illustrated in Section \[sec:appl\_WDWD\], the mass transfer between two stars and the resulting orbital dynamics are extremely sensitive to the accurate conservation of angular momentum. Eulerian methods are usually seriously challenged in accurately simulating the orbital dynamics in the presence of mass transfer, but this can be achieved when special measures are taken, see, for example, [@dsouza06] and [@motl07]. Another benefit that comes essentially for free is the natural adaptivity of SPH. Since the particles move with the local fluid velocity, they naturally trace the flow motion. As a corollary, simulations are not bound, like usually in Eulerian simulations, to a pre-defined “simulation volume”, but instead they can follow whatever the astrophysical system decides to do, be it a collapse or a rapid expansions or both in different parts of the flow. This automatic “refinement on density” is also closely related to the fact that vacuum does not pose a problem: such regions are simply devoid of SPH particles and no computational resources are wasted on regions without matter. In Eulerian methods, in contrast, vacuum is usually treated as a “background fluid” in which the “fluid of interest” moves and special attention needs to be payed to avoid artefacts caused by the interaction of these two fluids. For example, the neutron star surface of a binary neutron star system close to merger moves with an orbital velocity of $\sim 0.1c$ against the “vacuum” background medium. This can cause strong shocks and it may become challenging to disentangle, say, physical neutrino emission from the one that is entirely due to the artificial interaction with the background medium. There are, however, also hydrodynamic formulations that would in principle allow to avoid such an artificial “vacuum” [@duez02]. On the other hand, SPH’s “natural tendency to ignore vacuum” may also become a disadvantage in cases where the major interest of the investigation is a tenuous medium close to a dense one, say, for gas that is accreted onto the surface of a compact star. Such cases are probably more efficiently handled with an adaptive Eulerian method that can refine on physical quantities that are different from density. Several examples of hybrid approaches between SPH and Eulerian methods are discussed in Section \[sec:WDWD\_SNIa\]. SPH is Galilean invariant and thus independent of the chosen computing frame. The lack of this property can cause serious artefacts for Eulerian schemes if the simulation is performed in an inappropriate reference frame, see [@springel10b] for a number of examples. Particular examples related to binary mergers have been discussed in [@new97] and [@swesty00]: simulating the orbital motion of a binary system in a space-fixed frame can lead to an entirely spurious inspiral and merger, while simulations in a corotating frame may yield accurate results. For SPH, in contrast, it does not matter in which frame the calculation is performed. Another strong asset of SPH is its exact advection of fluid properties: an attribute attached to an SPH particle is simply carried along as the particle moves. This makes it particularly easy to, say, post-process the nucleosynthesis from a particle trajectory, without any need for additional “tracer particles”. In Eulerian methods high velocities with respect to the computational grid can seriously compromise the accuracy of a simulation. For SPH, this is essentially a “free lunch”, see for example Figure \[fig:advection\], where a high-density wedge is advected with a velocity of 0.9999 c through the computational domain. The particle nature of SPH also allows for a natural transition to n-body methods. For example, if ejected material from a stellar encounter becomes ballistic so that hydrodynamic forces are negligible, one may decide to simply follow the long-term evolution of point particles in a gravitational potential instead of a fluid. Such a treatment can make time scales accessible that cannot be reached within a hydrodynamic approach [@faber06b; @rosswog07a; @lee07; @ramirezruiz09; @lee10a]. SPH can straight forwardly be combined with highly flexible and accurate gravity solvers such as tree methods. Such approaches are extremely powerful for problems in which a fragmentation with complicated geometry due to the interplay of gas dynamics and self-gravity occurs, say in star or planet formation. Many successful examples of couplings of SPH with trees exist in the literature. Maybe the first one was the use the Barnes-Hut oct-tree algorithm [@barnes86] within the TREESPH code [@hernquist89], closely followed by the implementation of a mutual nearest neighbour tree due to Press for the simulation of white dwarf encounters [@benz90b]. By now a number of very fast and flexible tree variants are routinely used in SPH (e.g., [@dave97; @carraro98a; @springel01a; @wadsley04; @springel05a; @wetzstein09; @nelson09]) and for a long time SPH-tree combinations were at the leading edge ahead of Eulerian approaches that only have caught up later, see for example [@kravtsov99; @tessier02]. More recently, also ideas from the fast multipole method have been borrowed [@dehnen00; @dehnen02; @gafton11; @dehnen14] that allow for a scaling with the particle number $N$ that is close to $O(N)$ or even below, rather than the $O(N \log N)$ scaling that is obtained for traditional tree algorithms. But like any other numerical method, SPH also has to face some challenges. One particular example that had received a lot of attention in recent years, was the struggle of standard SPH formulations to properly deal with some fluid instabilities [@thacker00; @agertz07; @springel10a; @read10]. As will be discussed in Section \[sec:volume\_elements\], the problem is caused by surface tension forces that can emerge near contact discontinuities. This insight triggered a flurry of suggestions on how to improve this aspect of SPH [@price08a; @cha10; @read10; @hess10; @valcke10; @junk10; @abel11; @gaburov11; @murante11; @read12; @hopkins13; @saitoh13]. Arguably the most robust way to deal with this is the reformulation of SPH in terms of different volume elements as discussed in Section \[sec:volume\_elements\], an example is shown in Section \[sec:KH\]. Another notoriously difficult field is the inclusion of magnetic fields into SPH. This is closely related to preserving the $\nabla \cdot \vec{B}=0$ constraint during the MHD evolution, but also here there has been substantial progress in recent years [@boerve01; @boerve04; @price04a; @price04b; @price05; @price06; @boerve06; @rosswog07a; @dolag09; @buerzle11a; @buerzle11b; @tricco12; @tricco13]. Moreover, magnetic fields may be very important in regions of low density and due to SPH’s “tendency to ignore vacuum”, such regions are poorly sampled. For a detailed discussion of the current state of SPH and magnetic fields we refer to the recent review of [@price12a]. Artificial dissipation is also often considered as a major drawback. However, if dissipation is steered properly, see Section \[sec:Newtonian\_shocks\], the performance should be very similar to the one of approximate Riemann solver approaches. A Riemann solver approach may from an aesthetical point of view be more appealing, though, and a number of such approaches have been successfully implemented and tested [@inutsuka02; @cha03; @cha10; @murante11; @puri14]. Contrary to what was often claimed in the early literature, however, SPH is not necessarily a very efficient method. It is true that if only the bulk matter distribution is of interest, one can often obtain robust results already with an astonishingly small number of SPH particles. To obtain accurate results for the thermodynamics of the flow, however, still usually requires large particle numbers. In large SPH simulations it becomes a serious challenge to maintain cache-coherence since particles that were initially close in space and computer memory can later on become completely scattered throughout different (mostly slow) parts of the memory. This can be improved by using cache-friendly variables (aggregate data that are frequently used together into common variables, even if they have not much of a physical connection) and/or by various sorting techniques to re-order particles in memory according to their spatial location. This can be done –in the simplest case– by using a uniform hash grid, but in many astrophysical applications hierarchical structures such as trees are highly preferred for sorting the particles, see e.g., [@warren95; @springel05a; @nelson09; @gafton11]. While such measures can easily improve the performance by factors of a few, they come with some bookkeeping overhead which, together with, say, individual time steps and particle sinks or sources, can make codes rather unreadable and error-prone. Finally, we will briefly discuss in Section \[sec:IC\] the construction of accurate initial conditions where the particles are in a true (and stable) numerical equilibrium. This is yet another a non-trivial SPH issue. Roadmap through this review --------------------------- This text is organized as follows: - In Section \[chap:kernel\_approx\] we discuss those kernel approximation techniques that are needed for the discussed SPH discretizations. We begin with the basics and then focus on issues that are needed to appreciate some of the recent improvements of the SPH method. Some of these issues are by their very nature rather technical. Readers that are familiar with basic kernel interpolation techniques could skip this section at first reading. - In Section \[chap:SPH\] we discuss SPH discretizations of ideal fluid dynamics for both the Newtonian and the relativistic case. Since several reviews have appeared in the last years, we only concisely summarize the more traditional formulations and focus on recent improvements of the method. - The last section, Section \[sec:appl\], is dedicated to astrophysical applications. We begin with double white dwarf encounters (Section \[sec:appl\_WDWD\]) and then turn to encounters between two neutron stars and between a neutron star and a stellar-mass black hole (Section \[sec:appl\_NSNS\_NSBH\]). In each of these cases, our emphasis is on the more common, gravitational wave-driven binary systems, but we also discuss dynamical collisions as they may occur, for example, in a globular cluster. Sections \[chap:kernel\_approx\] and \[chap:SPH\] provide the basis for an understanding of SPH as a numerical method and they should pave the way to the most recent developments. The less numerically inclined reader may, however, just catch the most basic SPH ideas from Section \[sec:Newt\_vanilla\] and then jump to his/her preferred astrophysical topic in Section \[sec:appl\]. The modular structure of sections \[chap:kernel\_approx\] and \[chap:SPH\] should allow, whenever needed, for a selective consultation on the more technical issues of SPH. Kernel Approximation {#chap:kernel_approx} ==================== SPH uses kernel approximation techniques to calculate densities and pressure gradients from a discrete set of freely moving particles, see Section \[chap:SPH\]. Since kernel smoothing is one of the key ideas of the SPH technique, we will begin by collecting a number of basic relations that are necessary to understand how the method works and to judge the accuracy of different approaches. The discussed approximation techniques will be applied in Section \[chap:SPH\], both for the traditional formulations of SPH and for the recent improvements of the method. Function interpolation ---------------------- A kernel-smoothed representation of a function $A$ can be obtained via A ()= A(’) W( - ’,h) dr’, \[eq:integ\_approx\] where $W$ is a smoothing kernel. The width of the kernel function is determined by the quantity $h$, which, in an SPH context, is usually referred to as the “smoothing length”. Obviously, the kernel has the dimension of an inverse volume, it should be normalized to unity, $\int W(\vec{r}',h) \, dr'= 1$, and it should possess the $\delta$-property, \_[h 0]{} A ()= A(), so that the original function is reproduced exactly in the limit of a vanishing kernel support. The choice of kernel functions is discussed in Section \[sec:kernel\_choice\] in more detail, for now we only assume that the kernels are radial, i.e., $W(\vec{r})= W(|\vec{r}|)$, which leads to the very useful property \_a W(|\_a - \_b|,h)= - \_b W(|\_a - \_b|,h) \[eq:anti\_sym\] that allows for a straight forward conservation of Nature’s conservation laws. This topic is laid out in much detail in Section 2.4 of [@rosswog09b] and we refer the interested reader to this text for a further discussion and technical details. There are plausible arguments that suggest, for example, spheroidal kernels [@fulbright95; @shapiro96], but such approaches do usually violate angular momentum conservation. In the following we will only discuss radial kernels. Figure \[fig:interaction\_sketch\] sketches how a particle $a$ interacts with its neighbour particles $b$ via a radial kernel of support radius $Q h_a$. Let us assume that a function $A$ is known at the positions $\vec{r}_b$, $A_b= A(\vec{r}_b)$. One can then approximate Eq. (\[eq:integ\_approx\]) as A ()\_b V\_b A\_b W( - \_b,h), \[eq:std\_SPH\_interpolant\] where $V_b$ is the volume associated with the fluid parcel (“particle”) at $\vec{r}_b$. In nearly all SPH formulations that are used in astrophysics the particle volume is estimated as $V_b= m_b/\rho_b$ with $m_b$ being the (fixed) particle mass and $\rho_b$ its mass density. While being a straight forward choice, this is by no means the only option. Recently, SPH formulations have been proposed [@saitoh13; @hopkins13; @rosswog15b] that are based on different volume elements and that can yield substantial improvements. In the following, we will therefore keep a general volume element $V_b$ in most equations and only occasionally we give the equations for specific volume element choices. The assessment of the kernel approximation quality in a practical SPH simulation is actually a rather non-trivial problem. Such an analysis is straight forward for the continuum kernel approximation [@benz90a; @monaghan92], or, in discrete form, for particles that are distributed on a regular lattice [@monaghan05], but both of these estimates have limited validity for practical SPH simulations. In practice a discrete kernel approximation is used and particles are neither located on a regular grid nor are they randomly distributed, they are “disordered, but in an orderly way” [@monaghan05]. The exact particle distribution depends on the dynamics of the flow and on the kernel that is used. An example of a particle distribution that arises in a practical simulation (a Kelvin–Helmholtz instability) is shown in Figure \[fig:particle\_dist\]. Note in particular that the particles sample space in a regular way and no “clumping” occurs as would be the case for random particle positions. Why the particles arrange in such a regular way is discussed in Section \[sec:self\_regularization\]. Apart from the particle distribution, the accuracy depends, of course, on the chosen kernel function $W$ and this is discussed further in Section \[sec:kernel\_choice\] (see also Figure \[fig:particle\_dist\_2\]). Since the kernel function determines the distribution into which the particles settle, the kernel function and the particle distribution cannot be considered as independent entities. This makes the analytical accuracy assessment of SPH very difficult and also suggests to interpret results based on a prescribed, fix particle distribution with some caution. In any case, results should be further scrutinized in practical benchmark tests with known solutions. Initial conditions are another very important issue for the accuracy of SPH simulations, they are further discussed in Section \[sec:IC\]. To obtain indicators for the quality of a function interpolation, it is instructive to assume that the function $A$ is smooth in the neighborhood of the particle of interest, $a$, and that it can be expanded in a Taylor-series around $\vec{r}_a$: A\_b= A\_a + (A)\_a (\_[b]{}-\_a) + (\_i \_j A)\_a (\_[b]{}-\_a)\^i (\_[b]{}-\_a)\^j + O(r\_[ab]{}\^3), \[eq:Taylor\_A\] where $r_{ab}= |\vec{r}_a - \vec{r}_b|$ and the Einstein summation convention has been used. Inserting the expansion into the discrete approximation Eq. (\[eq:std\_SPH\_interpolant\]) and requiring that $\langle A \rangle_a$ should be a close approximation of $A_a$, one finds the “interpolation quality indicators” \_1: \_b V\_b W\_[ab]{}(h\_a) 1 \_2: \_b V\_b (\_[b]{}-\_a) W\_[ab]{}(h\_a) 0, \[eq:quality\_int\] where $W_{ab}(h_a)= W(|\vec{r}_{a}-\vec{r}_b|/h_a)$. The first relation simply states that the particles should provide a good approximation to a partition of unity. Function derivatives {#sec:SPH_derivs} -------------------- We restrict the discussion here to the first-order derivatives that we will need in Section \[chap:SPH\], for higher-order derivatives that are required for some applications we refer the interested reader to the literature [@espagnol03; @monaghan05; @price12a]. There are various ways to calculate gradients for function values that are known at given particle positions. Obviously, the accuracy of the gradient estimate is of concern, but also the symmetry in the particle indices since it can allow to enforce exact numerical conservation of physically conserved quantities. An accurate gradient estimate without built-in conservation can be less useful in practice than a less accurate estimate that exactly obeys Nature’s conservation laws, see Section 5 in [@price12a] for a striking example of how a seemingly good gradient without built-in conservation can lead to pathological particle distributions. The challenge is to combine exact conservation with an accurate gradient estimate. ### Direct gradient of the approximant {#sec:direct_gradient} The straight forward (though not most accurate) gradient estimate is the direct gradient of the interpolant Eq. (\[eq:std\_SPH\_interpolant\]) (A)\^[(0)]{} ()= \_b V\_b A\_b W( - \_b,h). \[eq:std\_SPH\_gradient\] Proceeding as above, we can insert again Eq. (\[eq:Taylor\_A\]) into Eq. (\[eq:std\_SPH\_gradient\]), (A)\_a\^[(0)]{} &=& A\_a \_b V\_b \_a W\_[ab]{}(h\_a) + \_b V\_b (A (\_[b]{}-\_[a]{}) ) \_a W\_[ab]{}(h\_a) + …,\[eq:gradient\_Taylor\_expansion\] which delivers the “gradient quality indicators” \_3: \_b V\_b \_a W\_[ab]{}(h\_a) 0 \_4: \_b V\_b (\_[b]{}-\_[a]{})\^i (\_a W\_[ab]{}(h\_a))\^j \^[ij]{} \[eq:gradient\_quality\] from the requirement that the estimate closely approximates the gradient of $A$. $\mathcal{Q}_3$ is simply the gradient of the quality indicator $\mathcal{Q}_1$ and therefore again an expression of the partition of unity requirement. Note, however, that even when all function values are the same, $A_b= A_0$, the gradient estimate does not necessarily vanish exactly. This is a direct consequence of Eq. (\[eq:quality\_int\]) not being an exact partition of unity. Note that therefore the reproduction of even constant functions is not enforced and this is often referred to as lack of zeroth order consistency. There are, however, benefits from a particle distribution noticing its imperfections, since, together with exact conservation, this provides an automatic “re-meshing” mechanism that drives the particles into a regular distribution [@price12a]. Numerical experiments show that the longterm evolution of a particle distribution with such a re-meshing mechanism built in leads to much more accurate results than seemingly more accurate gradient estimates without such a mechanism. The reason is that in the first case the particles arrange themselves in a regular configuration (such as in Figure \[fig:particle\_dist\]) where the quality indicators are fulfilled to a high degree, while in the latter case pathological particle distributions can develop that sample the fluid only very poorly. ### Constant-exact gradients An immediate way to improve the gradient estimate is to simply subtract the gradient of the residual in the approximate partition of unity, see Eq. (\[eq:quality\_int\]), or, equivalently, the first error term in Eq. (\[eq:gradient\_Taylor\_expansion\]). The resulting gradient estimate (A)\_a\^[(1)]{}= \_b V\_b (A\_b - A\_a) \_a W\_[ab]{}(h\_a), \[eq:const\_exact\_gradient\] now manifestly vanishes for a constant function $A$, i.e., if all $A_k$ are the same, independent of the particle distribution. ### Linear-exact gradients Exact gradients of linear functions can be constructed in the following way [@price04c]. Start with the RHS of Eq. (\[eq:std\_SPH\_gradient\]) at $\vec{r}_a$ and again insert the Taylor expansion of $A_b$ around $\vec{r}_a$ \_b V\_b A\_b \_a W\_[ab]{}= \_b V\_b { A\_a + (A)\_a (\_[b]{}-\_[a]{}), + …} \_a W\_[ab]{} which can be re-arranged into \_b V\_b (A\_b - A\_a) (\_a W\_[ab]{})\^i= (A)\_a\^k M\^[ki]{}. Here the sum over the common index $k$ is implied and the matrix is given by M\^[ki]{}= \_b V\_b (\_b - \_a)\^k (\_a W\_[ab]{})\^i. \[eq:MIK\] Solving for the gradient component then yields (A)\_a\^k= (M\^[ki]{})\^[-1]{} \_b V\_b (A\_b-A\_a) (\_a W\_[ab]{})\^i. \[eq:lin\_exact\_gradient\] Note that the sum is the same as in the constant-exact gradient estimate Eq. (\[eq:const\_exact\_gradient\]), corrected by the matrix $M^{-1}$ which depends on the properties of the particle distribution. It is straight forward to double-check that this exactly reproduces the gradient of a linear function. Assume that $A(\vec{r})= A_0 + (\nabla A)_0 \cdot (\vec{r}-\vec{r}_0)$ so that A\_b - A\_a= (A)\_0 (-\_a). If this is inserted into Eq. (\[eq:lin\_exact\_gradient\]) one finds, as expected, (A)\_a\^k= (M\^[ki]{})\^[-1]{} \_b (A)\_0\^l (-\_a)\^l (\_a W\_[ab]{})\^i= (A)\_0\^l (M\^[ki]{})\^[-1]{} M\^[li]{}= (A)\_0\^k. Obviously, the linear-exact gradient comes at the price of inverting a $D \times D$ matrix in $D$ dimensions, however, since the inversion of this small matrix can be done analytically, this does not represent a major computational burden. ### Integral-based gradients {#sec:integral_gradients} Integral-based higher-order derivatives [@brookshaw85; @monaghan05] have for a long time been appreciated for their insensitivity to particle noise. Surprisingly, integral-based estimates for first-order derivatives have only recently been explored [@garcia_senz12; @cabezon12a]. Start from the vector \_A ()= ( - ’) W(| - ’|, h) dV’ and, similar to above, insert a first-order Taylor expansion of $A(\vec{r}')$ around $\vec{r}$ (sum over $k$) \_A\^i ()= ( - ’)\^i W(| - ’|, h) dV’, so that the gradient component representation (exact for linear functions) is given by (A)\^k\_= \^[ki]{}() \_A\^i(), where the matrix $\tilde{C}^{ki}$ is the inverse of \^[ki]{}()= \^[ik]{}()= ( - ’)\^k ( - ’)\^i W(| - ’|, h) dV’. The matrix $\tilde{T}^{ki}$ only depends on the positions while $\tilde{\vec{I}}_A$ contains the function to be differentiated. If we replace the integral by summations (and drop the tilde), the integral-based gradient estimate reads (sum over $d$) (A)\^k\_= C\^[kd]{}() I\^d\_A(), \[eq:full\_IA\_gradient\] where $(C^{kl})= (T^{kl})^{-1}$ and T\^[kl]{}() &=& \_b V\_b W(| - \_b|, h) ( - \_b)\^k ( - \_b)\^l\ I\^l\_A() &=& \_b V\_b W(| - \_b|, h) ( - \_b)\^l. \[eq:matrices\] It is worth mentioning that for a radial kernel its gradient can be written as \_a W\_[ab]{}(h\_a)= (\_b - \_a) Y\_[ab]{}(h\_a), \[eq:kernel\_gradient\] where $Y$ is also a valid, positively definite and compactly supported kernel function. Therefore, if Eq. (\[eq:kernel\_gradient\]) is inserted in Eqs. (\[eq:MIK\]) and (\[eq:lin\_exact\_gradient\]), one recovers Eq. (\[eq:full\_IA\_gradient\]), i.e., the linear-exact and the integral-based gradient are actually equivalent for radial kernels. For a good interpolation, where the quality indicator $\mathcal{Q}_2$ in Eq. (\[eq:quality\_int\]) vanishes to a good approximation, one can drop the term containing $A(\vec{r})$ I\^l\_A() - \_b V\_b W(| - \_b|, h) A\_b ( - \_b)\^l , so that the gradient estimate in integral approximation (IA) explicitly reads (sum over $d$) (A)\^k\_[IA]{}= \_b V\_b A\_b C\^[kd]{} (\_b - )\^d W(| - \_b|, h) \_b V\_b A\_b G\^k\_b(). \[eq:IA\_gradient\] Comparison of Eq. (\[eq:IA\_gradient\]) with Eq. (\[eq:std\_SPH\_gradient\]) suggests that $\vec{G}_b(\vec{r})$ takes over the role that is usually played by $\nabla W(\vec{r} - \vec{r}_b,h)$: W( - \_b,h) \_b(). \[eq:nabla\_W\_to\_G\] While being slightly less accurate than Eqs.(\[eq:full\_IA\_gradient\])–(\[eq:matrices\]), see Section \[sec:accuracy\], the approximation Eq. (\[eq:IA\_gradient\]) has the major advantage that $\vec{G}$ is anti-symmetric with respect to the exchange of $\vec{r}$ and $\vec{r}_b$, just as the direct gradient of the radial kernel, see Eq. (\[eq:anti\_sym\]). Therefore it allows in a straight-forward way to enforce exact momentum conservation, though with a substantially more accurate gradient estimate. This type of gradient has in a large number of benchmark tests turned out to be superior to the traditional SPH approach. ### Accuracy assessment of different gradient prescriptions {#sec:accuracy} We perform a numerical experiment to measure the accuracy of different gradient prescriptions for a regular particle distribution. Since, as outlined above, kernel function and particle distribution are not independent entities, such tests at fixed particle distribution are a useful accuracy indicator, but they should be backed up by tests in which the particles can evolve dynamically. We place SPH particles in a 2D hexagonal lattice configuration in $[-1,1] \times [-1,1]$. Each particle is assigned a pressure value according $P(x,y)= 2 + x$ and we use the different prescriptions discussed in Sections \[sec:direct\_gradient\] –\[sec:integral\_gradients\] to numerically measure the pressure gradient. The relative average errors, $\epsilon= N^{-1}\sum_{b=1}^{N} \epsilon_b$ with $\epsilon_b= |(\p_x P)_b - 1|$, as a function of the kernel support are shown in Figure \[fig:gradient\_prescriptions\]. The quantity $\eta$ determines the smoothing length via $h_b= \eta \left(m_b/\rho_b\right)^{1/D}$, while $D$ denotes the number of spatial dimensions. For this numerical experiment the standard cubic spline kernel M$_4$, see Section \[sec:kernel\_choice\], has been used. We apply the “standard” SPH-gradient, Eq. (\[eq:std\_SPH\_gradient\]), the approximate integral-based gradient (“IA gradient”), Eq. (\[eq:IA\_gradient\]), the full integral-based gradient (“fIA gradient”), Eq.(\[eq:full\_IA\_gradient\]), and the linearly exact-gradient (“LE gradient”), Eq. (\[eq:lin\_exact\_gradient\]). Note that for this regular particle distribution, the constant-exact gradient, Eq. (\[eq:const\_exact\_gradient\]), is practically indistinguishable from the standard prescription, since it differs only by a term that is proportional to the first “gradient quality indicator”, Eq. (\[eq:gradient\_quality\]), which vanishes here to a very good approximation. Clearly, the more sophisticated gradient prescriptions yield vast improvements of the gradient accuracy. Both the LE and fIA gradients reproduce the exact value to within machine precision, the IA gradient which has the desired anti-symmetry property, improves the gradient accuracy of the standard SPH estimate by approximately 10 orders of magnitude. As expected from the term that was neglected in Eq. (\[eq:matrices\]), the accuracy of the “IA gradient” deteriorates substantially (to an accuracy level comparable to the standard prescription) for a less regular particle distribution, see [@rosswog15b]. Therefore, the usefulness of the IA gradient depends on how regularly the SPH particles are distributed in a practical simulation. The original work by [@garcia_senz12] and our own extensive numerical experiments [@rosswog15b], however, show that the IA gradient is also in practical numerical tests highly superior to the standard kernel gradient prescription. Which kernel function to choose? {#sec:kernel_choice} -------------------------------- In the following, we briefly explore the properties of a selection of kernel functions. We write normalized SPH kernels in the form W(|-’|,h)= w(q), \[eq:normalized\_SPH\_kernel\] where $h$ is the smoothing length that determines the support of $W$, $q= |\vec{r}-\vec{r}'|/h$ and $D$ is the number of spatial dimensions. The normalizations are obtained from \^[-1]{}= { [ll]{} 2 \_0\^[Q]{} w(q) dq & [in 1D]{}\ \ 2 \_0\^[Q]{} w(q) q dq & [in 2D]{}\ \ 4 \_0\^[Q]{} w(q) q\^2 dq & [in 3D]{}, .\ \[eq:normalization\] where $Q$ is the kernel support (= 2 for most of the following kernels). In the following we will give the kernels in the form that is usually found in the literature. For a fair comparison in terms of computational effort/neighbor number we stretch the kernels in all plots and experiments to a support of 2$h$. So if a kernel has a normalization $\sigma_{lh}$ for a support of $l h$, it has normalization $\sigma_{kh}= (l/k)^D \sigma_{lh}$ if it is stretched to a support of $k h$. ### Kernels with vanishing central derivatives “Bell-shaped” kernels with their vanishing derivatives at the origin are rather insensitive to the exact positions of nearby particles and therefore they are good density estimators [@monaghan92]. For this reason they have been widely used in SPH simulations. The kernels that we discuss here and their derivatives are plotted in Figure \[fig:kernels\]. More recently, kernels with non-vanishing central derivatives have been (re-)suggested, some of them are discussed in Section \[sec:peaked\_kernels\]. ### B-spline functions: M$_4$ and M$_6$ kernels {#b-spline-functions-m_4-and-m_6-kernels .unnumbered} The most commonly used SPH kernels are the so-called B-spline functions [@schoenberg46], $M_n$, which are generated as Fourier transforms: M\_n(x,h)= \_[-]{}\^\^n (kx) dk. The smoothness of the $M_n$ functions increases with $n$ and they are continuous up to the $(n-2)$-th derivative. Since SPH requires at the very least the continuity in the first and second derivative, the cubic spline kernel M$_4$ w\_4(q)= { [ll]{} (2-q)\^3 - (1-q)\^3 & 0 q &lt; 1\ (2-q)\^3 & 1 q &lt; 2\ 0 & [else]{} . is the lowest-order SPH kernel that is a viable option, it is often considered the “standard choice” in SPH. As we will show below, much more accurate choices are available at a moderate extra cost. The normalization factor $\sigma$ of M$_4$ has values of $[2/3,10/(7 \pi),1/\pi]$ in 1, 2 and 3 dimensions. The quintic spline kernel M$_6$ has also occasionally been used in SPH simulations. It reads (truncated at support $Q= 3$) w\_6(q)= { [ll]{} (3-q)\^5 - 6(2-q)\^5 + 15(1-q)\^5 & 0 q &lt; 1\ (3-q)\^5 - 6(2-q)\^5 & 1 q &lt; 2\ (3-q)\^5 & 2 q &lt; 3\ 0 & [else]{} . with normalizations of $[1/120, 7/(478 \pi), 1/(120 \pi)]$ in 1, 2 and 3 dimensions. Although this is the commonly used form, we re-scale the 6 kernel in all plots to a support of $Q= 2$ to enable a fair and easy comparison with other kernels in terms of computational effort (i.e., neighbor number). ### A parametrized family of Kernels {#a-parametrized-family-of-kernels .unnumbered} More recently, a one parameter family of kernels has been suggested [@cabezon08] W\_[[H]{},n]{}= { [ll]{} 1 & q = 0\ ( )\^n & 0 &lt; q 2\ 0 & [else,]{} . where $n$ determines the smoothness and the shape of the kernel, see Figure \[fig:kernels\]. The normalization of this kernel family, $\sigma_{{\rm H},n}$, for integer $n$ from 3 to 9 is given in Table \[tab:kernel\_params\]. In [@cabezon08], their Table 2, a fifth order polynomial is given that provides the normalization for continuous $n$ between 3 and 7. The $W_{{\rm H},3}$ kernel is very similar to the M$_4$ (not shown), while $W_{{\rm H},5}$ is a close approximation of M$_6$, see Figure \[fig:kernels\], provided they have the same support. [c|r]{}\ \ $q_c$ & $0.75929848$\ A & $11.01753798$\ B & $-38.11192354$\ C & $-16.61958320$\ D & $69.78576728$\ $\sigma_{1D}$ & $8.24554795 \times10^{-3}$\ $\sigma_{2D}$ & $4.64964683 \times10^{-3}$\ $\sigma_{3D}$ & $2.65083908 \times10^{-3}$\ ### Wendland kernels {#wendland-kernels .unnumbered} An interesting class of kernels with compact support and positive Fourier transforms are the so-called Wendland functions [@wendland95]. In several areas of applied mathematics Wendland functions have been well appreciated for their good interpolation properties, but they have not received much attention as SPH kernels and have only recently been explored in more detail [@dehnen12; @hu14a; @rosswog15b]. [@dehnen12] have pointed out in particular that these kernels are not prone to the pairing instability, see Section \[sec:pairing\], despite having a vanishing central derivative. These kernels have been explored in a large number of benchmark tests [@rosswog15b] and they are high appreciated for their “cold fluid properties”: the particle distribution remains highly ordered even in dynamical simulations (e.g., Figure \[fig:particle\_dist\_2\], upper right panel) and only allows for very little noise. Here we only experiment with one particular example, the $C^6$ smooth W\_[3,3]{}= (1 - q )\_[+]{}\^8 ( 32 q\^3 + 25 q\^2 + 8 q + 1 ) \[eq:wend33\], see e.g., [@schaback06], where the symbol $(.)_+$ denotes the cutoff function $\max (.,0)$. The normalization $\sigma_W$ is 78/(7$\pi$) and 1365/(64 $\pi$) in 2 and 3 dimensions. ### Kernels with non-vanishing central derivatives {#sec:peaked_kernels} A number of kernels have been suggested in the literature whose derivatives remain finite in the centre so that the repulsive forces between particles never vanish. The major motivation behind such kernels is to achieve a very regular particle distribution and in particular to avoid the pairing instability, see Section \[sec:pairing\]. However, as recently pointed out by [@dehnen12], the pairing instability is not necessarily related to a vanishing central derivative, instead, non-negative Fourier transforms have been found as a necessary condition for stability against pairing. Also kernels with vanishing central derivatives can possess this properties. We explore here only two peaked kernel functions, one, the Linear Quartic core (LIQ) kernel, that has been suggested as a cure to improve SPH’s performance in Kelvin–Helmholtz instabilities and another one, the Quartic Core M$_6$ (QCM$_6$) kernel, mainly for pedagogical reasons to illustrate how an even very subtle change in the central part of the kernel can seriously deteriorate the approximation quality. For a more extensive account on kernels with non-vanishing central derivatives we refer to the literature [@thomas92; @fulk96; @valcke10; @read10]. ### Linear quartic kernel {#linear-quartic-kernel .unnumbered} The centrally peaked “Linear Quartic” (LIQ) kernel [@valcke10] reads W\_[LIQ]{}(q)= { [ l l l]{} F - q q x\_s\ A q\^4 + B q\^3 + C q\^2 + D q + E x\_s &lt; q 1\ 0 .with $x_s=0.3, A=-500/343, B=1300/343, C=-900/343, D=-100/343, E=200/343$ and $F= 13/20$. The normalization constant $\sigma_{\rm LIQ}$ is 1000/447, 3750/403 $\pi$ and 30000/2419$\pi$ in one, two and three dimensions. ### Quartic core M$_6$ kernel {#quartic-core-m_6-kernel .unnumbered} We also explore a very subtle modification of the well-known and appreciated M$_6$ kernel so that it exhibits a small, but non-zero derivative in the centre. This “Quartic core $M_6$ kernel” (QCM$_6$) is constructed by replacing the second derivative of the $M_6$ kernel for $q<q_c$ by a parabola whose parameters have been chosen so that the kernel fits smoothly and differentiably the $M_6$ kernel at the transition radius defined by $d^2 w_6/dq^2(q_c)= 0$ (numerical value $q_c= 0.75929848$). The QCM$_6$-kernel then reads: W\_[QCM\_6]{}(q)= { [ll]{} A q\^4 + B q\^2 + C q + D& 0 q &lt; q\_c\ (3-q)\^5 - 6(2-q)\^5 + 15(1-q)\^5& q\_c q &lt; 1\ (3-q)\^5 - 6(2-q)\^5 & 1 q &lt; 2\ (3-q)\^5 & 2 q &lt; 3\ 0 & [else.]{} . The coefficients $A, B, C$ and $D$ are determined by the conditions $w_{\rm QCM_6}(q_c)= w_6(q_c)$, $w'_{\rm QCM_6}(q_c)= w'_6(q_c)$, $w''_{\rm QCM_6}(q_c)= w''_6(q_c)$ and $w'''_{\rm QCM_6}(q_c)= w'''_6(q_c)$, where the primes indicate the derivatives with respect to $q$. The resulting numerical coefficients are given in Table \[tab:kernel\_params\]. Note that QCM$_6$ is continuous everywhere up to the third derivative. As can be seen in Figure \[fig:kernels\], the QCM$_6$ kernel (violet dots) deviates only subtly from M$_6$ (solid red line), but even this very subtle modification does already seriously compromise the accuracy of the kernel. In a recent study [@rosswog15b], it was found, however, that this kernel has the property of producing only very little noise, particles placed initially on a quadratic or hexagonal lattice (in pressure equilibrium) remain on this lattice configuration. The different kernels and their derivatives are summarized in Figure \[fig:kernels\]. As mentioned above, the M$_6$, QCM$_6$ and W$_{3,3}$ kernels have been rescaled to a support of $Q=2$ to allow for an easy comparison. Note how the kernels become more centrally peaked with increasing order, for example, the W$_{\rm H,9}$ kernel only deviates noticeably from zero inside of $q=1$, so that it is very insensitive to particles entering or leaving its support near $q=2$. ### Accuracy assessment of different kernels ### Density estimates {#density-estimates .unnumbered} We assess the density estimation accuracy of the different kernels in a numerical experiment. The particles are placed in a 2D hexagonal lattice configuration in $[-1,1] \times [-1,1]$. This configuration corresponds to the closest packing of spheres with radius $r_{\rm s}$ where each particle possesses an effective area of $A_{\rm eff}= 2 \sqrt{3} r_{\rm s}^2$. Each particle is now assigned the same mass $m_b= \rho_0 A_{\rm eff}$ to obtain the uniform density $\rho_0$. Subsequently the densities at the particle locations, $\rho_b$, are calculated via the standard SPH expression for the density, Eq. (\[eq:dens\_sum\]), and the average error $\epsilon= N^{-1}\sum_{b=1}^{N} \epsilon_b$ with $\epsilon_b= |\rho_b-\rho_0|/\rho_0$ is determined. Figure \[fig:density\_errors\], upper panel, shows the error as a function of $\eta$ which parameterizes the kernel support size via $h_b= \eta \left(m_b/\rho_b\right)^{1/D}$, $D$ being again the number of spatial dimensions. Interestingly, the “standard” cubic spline kernel (“CS”, solid black line in the figure) actually does not perform well. At typical values of $\eta$ near 1.3 the relative density error is a few times $10^{-3}$. Just replacing it by the quintic spline kernel (“6”, solid red line in the figure) improves the density estimate for similar values of $\eta$ already by two orders of magnitude. The Wendland kernel (“$W_{3,3}$”, dashed blue line in the figure) continuously decreases the error with increasing $\eta$ and therefore does not show the pairing instability at any value of $\eta$ [@dehnen12]. It maintains a very regular particle distribution with very little noise [@rosswog15b], see also the Gresho–Chan vortex test that is shown in Section \[sec:gresho\]. In these fixed particle distribution tests the -kernels perform particularly well. At large smoothing length they deliver exquisite density estimates. For example, the W$_{\rm H,9}$ kernel is at $\eta > 1.9$ more than two orders of magnitude more accurate than 6. The centrally peaked kernels turn out to be rather poor density estimators. The shown LIQ kernel (filled black circles) needs a large $\eta$ beyond 2 to achieve a density estimate better than 1%. Even the subtle modification of the central core in the QCM$_6$ kernel, substantially deteriorates the density estimate (violet dots) in comparison to the original M$_6$ kernel. One needs a value of $\eta>1.8$ for a density accuracy that is better than 1%, at this large $\eta$ the 7 kernel is approximately four orders of magnitude more accurate. After extensive numerical experiments a recent study [@rosswog15b] still gave overall preference to the Wendland kernel Eq. (\[eq:wend33\]). It gave less accurate results on “frozen” particle distributions than, say, the $W_{\rm H,9}$-kernel, but in dynamical experiments where particles were allowed to move, the Wendland kernel maintained a substantially lower noise level. ### Gradient estimates {#gradient-estimates .unnumbered} We perform a similar experiment to measure the gradient accuracy. The particles are set up as before and each particle is assigned a pressure $P(x,y)= 2 + x$. We use the straight forward SPH estimate (P)\_a= \_b P\_b \_a W\_[ab]{}(h\_a) \[eq:press\_grad\] to calculate the pressure gradient. The average error, $\epsilon= N^{-1}\sum_{b=1}^{N} \epsilon_b$ with $\epsilon_b \equiv |\nabla P - (\nabla P)_b|/ |\nabla P|$ is shown in Figure \[fig:density\_errors\], right panel, for different kernels as a function of $\eta$. Again, the “standard” cubic spline kernel (solid black) does not perform particularly well, only for $\eta>1.8$ reaches the gradient estimate an accuracy better than 1%. In a dynamical situation, however, the particle distribution would at such a large kernel support already fall prey to the pairing instability. For moderately small supports, say $\eta<1.6$, the 6 kernel is substantially more accurate. Again, the accuracy of the Wendland kernel increases monotonically with increasing $\eta$, and the three -kernels perform best in this static test. As in the case of the density estimates, the peaked kernels perform rather poorly and only achieve a 1% accuracy for extremely large values of $\eta$. ### Kernel choice and pairing instability {#sec:pairing} Figure \[fig:density\_errors\] suggests to increase the kernel support to achieve greater accuracy in density and gradient estimates (though at the price of reduced spatial resolution). For many bell-shaped kernels, however, the support cannot be increased arbitrarily since for large smoothing lengths the so-called “pairing instability” sets in where particles start to form pairs. In the most extreme case, two particles can merge into effectively one. Nevertheless, this is a rather benign instability. An example is shown in Figure \[fig:particle\_dist\_2\], where for the interaction of a blast wave with a low-density bubble we have once chosen (left panels) a kernel-smoothing length combination (M$_4$ kernel with $\eta= 2.2$) that leads to strong particle pairing and once (right panels) a combination (Wendland kernel with $\eta=2.2$) that produces a very regular particle distribution. Note that despite the strong pairing in the left column, the continuum properties (the lower row shows the density as an example) are still reasonably well reproduced. The pairing simply leads to a loss of resolution, though at the original computational cost. This instability has traditionally been explained by means of the inflection point in the kernel derivatives of bell-shaped kernels, see Figure \[fig:kernels\], that leads to decreasing and finally vanishing repelling forces once the inflection point has been crossed. [@dehnen12] in contrast argue that the stability of the Wendland kernels with respect to pairing is due to the density error being monotonically declining, see Figure \[fig:density\_errors\], and non-negative kernel Fourier transforms. They investigated in particular a number of bell-shaped Wendland kernels with strictly positive Fourier transforms and they did not find any sign of the instability despite the vanishing central derivatives. This is, of course, a desirable feature for convergence studies since the neighbor number can be increased without limit. Summary: kernel approximation ----------------------------- The improvement of the involved kernel approximation techniques is one obvious way how to further enhance the accuracy of SPH. One should strive, however, to preserve SPH’s most important asset, its exact numerical conservation. The simplest possible, yet effective, improvement is to just replace the kernel function, see Section \[sec:kernel\_choice\]. We have briefly discussed a number of kernels and assessed their accuracy in estimating a uniform density and the gradient of a linear function for the case where the SPH particles are placed on a hexagonal lattice. The most widely used kernel function, the cubic spline M$_4$, does actually not perform particularly well, neither for the density nor the gradient estimate. At moderate extra cost, however, one can use substantially more accurate kernels, for example the quintic spline kernel, M$_6$, or the higher-order members of the W$_{H,n}$ family. Another, very promising kernel family are the Wendland functions. They are not prone to the pairing instability and therefore show much better convergence properties than kernels that start forming pairs beyond a critical support size. Moreover, the Wendland kernel that we explored in detail [@rosswog15b] is very reluctant to allow for particle motion on a sub-resolution scale and it maintains a very regular particle distribution, even in highly dynamical tests. The explored peaked kernels, in contrast, performed rather poorly in both estimating densities and gradients. SPH Formulations of Ideal Fluid Dynamics {#chap:SPH} ======================================== Smoothed Particle Hydrodynamics (SPH) is a completely mesh-free, Lagrangian method that was originally suggested in an astrophysical context [@gingold77; @lucy77], but by now it has also found many applications in the engineering world, see [@monaghan12a] for a starting point. Since a number of detailed reviews exists, from the “classics” [@benz90a; @monaghan92] to more recent ones [@monaghan05; @rosswog09b; @springel10a; @price12a], we want to avoid becoming too repetitive about SPH basics and therefore put the emphasis here on recent developments. Many of them have very good potential, but have not yet fully made their way into practical simulations. Our emphasis here is also meant as a motivation for computational astrophysicists to keep their simulation tools up-to-date in terms of methodology. A very explicit account on the derivation of various SPH aspects has been provided in [@rosswog09b], therefore we will sometimes refer the interested reader to this text for more technical details. The basic idea of SPH is to represent a fluid by freely moving interpolation points – the particles – whose evolution is governed Nature’s conservation laws. These particles move with the local fluid velocity and their densities and gradients are determined by the kernel approximation techniques discussed in Section \[chap:kernel\_approx\], see Figure \[fig:sph\_flow\]. The corresponding evolution equations can be formulated in such a way that mass, energy, momentum and angular momentum are conserved by construction, i.e., they are fulfilled independent of the numerical resolution. In the following we use the convention that the considered particle is labeled with “a”, its neighbors with “b” and a general particle with “k”, see also the sketch in Figure \[fig:interaction\_sketch\]. Moreover, the difference between two vectors is denoted as $\vec{A}_{ab}= \vec{A}_a - \vec{A}_b$ and the symbol $B_{ab}$ refers to the arithmetic average $B_{ab}= (B_a+B_b)/2$ of two scalar functions. Choice of the SPH volume element {#sec:volume_elements} -------------------------------- Up to now the choice of the volume element in the kernel approximation, see Section \[chap:kernel\_approx\], has been kept open. The traditional choice is simply $V_b= m_b/\rho_b$, where $\rho$ is calculated as a kernel-weighted sum over nearby masses, see Eq. (\[eq:dens\_sum\]) and $m$ is the SPH particle mass. As will be discussed in Sections \[sec:SR\_SPH\] and \[sec:GR\_SPH\], this translates in a straight forward manner to relativistic SPH. In the latter case the particle mass $m$ is replaced by the SPH particle’s baryon number $\nu$ and the mass density $\rho$ is replaced by the baryon number density as calculated in the “computing frame”, see below.\ It has recently been realized [@saitoh13] that this “standard” choice of volume element is responsible for the spurious surface tension forces that can occur in many SPH implementations near contact discontinuities. At such discontinuities the pressure is continuous, but density $\rho$ and internal energy $u$ exhibit a jump. For a polytropic equation of state, $P= (\Gamma-1) u \rho$, the product of density and internal energy must be the same on both sides to ensure a single value of $P$ at the discontinuity, i.e., $\rho_1 u_1= \rho_2 u_2$, where the subscripts label the two sides of the discontinuity. This means in particular, that the jumps in $u$ and $\rho$ need to be consistent with each other, otherwise the mismatch can cause spurious forces that have an effect like a surface tension and can suppress weakly triggered fluid instabilities, see for example [@thacker00; @agertz07; @springel10a; @read10]. In the “standard” SPH formulation such a mismatch can occur because the density estimate is smooth, but the internal energy enters the SPH equations as an un-smoothed quantity. One may question, however, whether an unresolvably sharp transition in $u$ is a viable numerical initial condition in the first place. The problem can be alleviated if also the internal energy is smoothed by applying some artificial thermal conductivity and this has been shown to work well for Kelvin–Helmholtz instabilities [@price08a]. But, it is actually a non-trivial problem to design appropriate triggers that supply conductivity exclusively where needed and not elsewhere. Artificial conductivity applied where it is undesired can have catastrophic consequences, for example by removing physically required energy/pressure gradients for a star in hydrostatic equilibrium. An alternative cure comes from using different volume elements in the SPH discretization process. The first step in this direction was probably taken by [@ritchie01] who realized that by using the internal energy as a weight in the SPH density estimate a much sharper density transition could be achieved than by the standard SPH density sum where each particle is, apart from the kernel, only weighted by its mass. In [@saitoh13] it was pointed out that SPH formulations that do not include density explicitly in the equations of motion do avoid the pressure becoming multi-valued at contact discontinuities. Since the density usually enters the equation of motion via the choice of the volume element, a different choice can possibly avoid the problem altogether. This observation is consistent with the findings of [@hess10] who used a particle hydrodynamics method, but calculated the volumes via a Voronoi tessellation rather than via smooth density sum. In their approach no spurious surface tension effects have been observed. Closer to the original SPH spirit is the class of kernel-based particle volume estimates that have recently been suggested by [@hopkins13] as a generalization of the approach from [@saitoh13]. Recently, such volume elements have been generalized for the use in special-relativistic studies [@rosswog15b]. Assume that we have an (exact or approximate) partition of unity so that \_b \_b()= 1 \[eq:PU\] and a function $U$ can be approximated as () \_b U\_b \_b(), \[eq:PU\_approx\] where $U_b$ are the known function values at positions $\vec{r}_b$. This can be used to split up space (total volume $V$) into volume elements $V_b$ V = dV= (\_b \_b()) dV= \_b V\_b, where Eq. (\[eq:PU\]) has been inserted and the particle volume V\_b= \_b() dV \[eq:def\_volume\] has been introduced. One may use, for example, a Shepard-type partition of unity [@shepard68] \_b()= , with $W_b(\vec{r})= W(|\vec{r}-\vec{r}_b|)$ being a smoothing kernel so that upon using Eq. (\[eq:def\_volume\]) the particle volume becomes V\_b= dV. Making use of the approximate $\delta$-property of the kernel, this yields V\_b= , where $W_{kb}= W(|\vec{r}_k-\vec{r}_b|)$. This, of course, has the simple physical interpretation of locally sampling the particle number density (keep in mind that the kernel has the dimension of an inverse volume) and taking its inverse as the particle volume. While this is a straight forward and plausible approach, one can in principle generalize this volume estimate by weighting the kernel with any scalar property $X$ [@hopkins13] of the particles, so that the volume becomes V\_b\^[(X)]{}= \[eq:gen\_vol\_element\] and one can choose to calculate the density via \_b\^[(X)]{}= = \_[X, b]{}, \[eq:density\] respectively. If the smoothing length is adjusted to a multiple of the typical particle separation, h\_b= (V\_b\^[(X)]{})\^[1/D]{}, \[eq:eta\] $D$ being the number of spatial dimensions, the derivatives of the quantity $\kappa_{X,b}$ become \_a \_[X,b]{}= \_k X\_k \_a W\_[bk]{}(h\_b) = \_k X\_k \_[bk]{} \_b W\_[bk]{}(h\_b), \[eq:kappa\_derivs\] with the generalized “grad-h terms” being \_b= 1 - \_k X\_k . For the SPH discretization process one needs the derivatives of the volume elements \_a V\_b = - \_k X\_k \_a W\_[bk]{}(h\_b) \[eq:nabla\_V\] and = - \_k X\_k \_[bk]{} \_b W\_[bk]{}(h\_b). \[eq:ddt\_V\] Newtonian SPH {#sec:Newt_SPH} ------------- In its most basic form, the task is simply to solve the Lagrangian conservation equations for mass, energy and momentum of an ideal fluid [@landau59]: &=& -,\ &=& () ,\ &=& -,\[eq:Newt\_Euler\] where $\rho$ is the mass density, $d/dt= \p_t + \vec{v} \cdot \nabla$ the Lagrangian time derivative, $\vec{v}$ the fluid velocity, $u$ the specific thermal energy and $P$ the pressure. Like in other numerical methods, many different discrete approximations to the continuum equations can be used and they may differ substantially in their accuracy. In the following, we will summarize commonly used SPH discretizations that have been used for simulations of compact objects. They differ in their derivation strategy, the resulting symmetries in the particle indices, the volume elements, the manner how gradients are calculated and in the way they deal with shocks. ### “Vanilla ice” SPH {#sec:Newt_vanilla} We will begin with the simplest, but fully conservative, SPH formulation (“vanilla ice”) that is still used in many astrophysical simulations. A detailed step-by-step derivation with particular attention to conservation issues can be found in Section 2.3 of [@rosswog09b]. By using the volume element $V_b= m_b/\rho_b$ and the derivative prescription Eq. (\[eq:const\_exact\_gradient\]) one finds the discrete form of the Lagrangian continuity equation = - = \_b m\_b \_[ab]{} \_a W\_[ab]{} \[eq:Newt\_drho\_dt\], where $\vec{v}_{ab}= \vec{v}_a - \vec{v}_b$ and $W_{ab}= W(|\vec{r}_a-\vec{r}_b|,h)$. As an alternative to this “density-by-integration” approach one can also estimate the density at a particle position $\vec{r}_a$ as a weighted sum over contributing particles (“density-by-summation”) \_a= \_b m\_b W\_[ab]{}(h\_a). \[eq:dens\_sum\] In practice, there is little difference between the two. Note that this density estimate corresponds to the choice $X= m$ in Eq. (\[eq:gen\_vol\_element\]). Usually the particle masses are kept constant so that exact mass conservation is enforced automatically. Most often, the specific energy $u$ is evolved in time, its evolution equation follows directly from Eq. (\[eq:Newt\_drho\_dt\]) and the adiabatic first law of thermodynamics, u\_b/\_b = P\_b/\_b\^2 \[eq:first\_law\] as = () = \_b m\_b \_[ab]{} \_a W\_[ab]{}. \[eq:Newt\_du\_dt\] A discrete form of the momentum equation that enforces exact conservation can be found by using the identity = + () \[eq:basic:nabla\_P\_rho\] in the Lagrangian momentum equation = - = - \_b m\_b ( + ) \_a W\_[ab]{}. \[eq:Newt\_momentum\_equation\] If the kernel $W_{ab}$ in Eqs. (\[eq:Newt\_du\_dt\]) and (\[eq:Newt\_momentum\_equation\]) is evaluated with a symmetric combination of smoothing lengths, say with $h_{ab}= (h_a+h_b)/2$, then Eqs. (\[eq:dens\_sum\]), (\[eq:Newt\_du\_dt\]) and (\[eq:Newt\_momentum\_equation\]) form, together with an equation of state, a closed set of equations that enforces the exact conservation of mass, energy, linear and angular momentum by construction. For practical simulations that may possibly involve shocks, Eqs. (\[eq:Newt\_du\_dt\]) and (\[eq:Newt\_momentum\_equation\]) need to be augmented by extra measures (e.g., by artificial viscosity) to ensure that entropy is produced in a shock and that kinetic energy is properly transformed into thermal energy, see Section \[sec:Newtonian\_shocks\]. ### SPH from a variational principle {#sec:SPH_from_variational_principle} More elegantly, a set of SPH equations can be obtained by starting from the (discretized) Lagrangian of an ideal fluid [@gingold82; @speith98; @monaghan01; @springel02; @rosswog09b; @price12a]. This ensures that conservation of mass, energy, momentum and angular momentum is, by construction, built into the discretized form of the resulting fluid equations. Below, we use the generalized volume element Eq. (\[eq:gen\_vol\_element\]) without specifying the choice of the weight $X$ so that a whole class of SPH equations is produced [@hopkins13]. In this derivation the volume of an SPH particle takes over the fundamental role that is usually played by the density sum. Therefore we will generally express the SPH equations in terms of volumes rather than densities, we only make use of the latter for comparison with known equation sets. The Lagrangian of an ideal fluid can be written as [@eckart60] L= (x) ( - u(x)) dx, and on using Eq. (\[eq:PU\_approx\]) it can be discretized into L { \_b \_b ( - u\_b) \_b(x) }dx = \_b \_b V\_b ( - u\_b ), where we have used the definition of the particle volume Eq. (\[eq:def\_volume\]). For the choice $m_b= \rho_b V_b$ the standard SPH-Lagrangian L= \_b m\_b ( - u\_b ) is recovered. The Euler–Lagrange equations - = 0 then yield, for a fixed particle mass $m_a$, &=& = - \_b m\_b = \_b P\_b , \[eq:momentum\_from\_Lagrangian\] where Eq. (\[eq:first\_law\]) was used. The energy equation follows directly from the first law of thermodynamics as = - . With Eqs. (\[eq:nabla\_V\]), (\[eq:ddt\_V\]), $\nabla_b W_{ab}= -\nabla_a W_{ab}$ and $\nabla_a W_{bk}= \nabla_b W_{bk}(\delta_{ba} - \delta_{ka})$ the SPH equations for a general volume element of the form given in Eq. (\[eq:gen\_vol\_element\]) become V\_b &=& \[eq:gen\_vol\_N\]\ m\_a &=& - \_[b]{} X\_a X\_b { \_a W\_[ab]{}(h\_a) + \_a W\_[ab]{}(h\_b) }\[eq:gen\_mom\_N\]\ m\_a &=& \_b X\_b \_[ab]{} \_a W\_[ab]{}(h\_a).\[eq:gen\_en\_N\] For the choice $X=m$ this reduces to the commonly used equation set [@monaghan01; @price04c; @rosswog07c]. Explicit forms of the equations, also for other choices of $X$, are given in Tab. \[tab:explicit\_forms\_Newt\_SPH\]. ### The GADGET equations {#the-gadget-equations .unnumbered} The arguably most wide-spread SPH simulation code, <span style="font-variant:small-caps;">Gadget</span> [@springel01a; @springel05a], uses an entropy formulation of SPH. It uses the entropic function $A(s)$ occurring in a polytropic equation of state, $P= A(s) \rho^\Gamma$. If no entropy is created, the quantity $A$ is simply advected by the fluid element and only the momentum equation needs to be solved explicitly. In <span style="font-variant:small-caps;">Gadget</span> the smoothing lengths are adapted so that h\_a\^3 \_a = N\_[SPH]{} |[m]{} \[eq:gadget\_h\] is fulfilled, where $N_{\rm SPH}$ is the typical neighbor number and $\bar{m}$ is an average particle mass. The Euler–Lagrange equations then yield = - \_b m\_b { \_a W\_[ab]{}(h\_a) + \_a W\_[ab]{}(h\_b) }, \[eq:momentum\_gadget\] where the $f_k$ are the grad-h terms that read f\_k= (1 + )\^[-1]{}. Obviously, in this formulation artificial dissipation terms need to be applied also to $A$ to ensure that entropy is produced at the right amounts in shocks. ### Self-regularization in SPH {#sec:self_regularization} SPH possesses a built-in “self-regularization” mechanism, i.e. SPH particles feel, in addition to pressure gradients, a force that aims at driving them towards an optimal particle distribution. This corresponds to (usually ad hoc introduced) “re-meshing” steps that are used in Lagrangian mesh methods. The ability to automatically re-mesh is closely related the lack of zeroth order consistency of SPH that was briefly described in Section \[sec:direct\_gradient\]: the particles “realize” that their distribution is imperfect and they have to adjust accordingly. Particle methods without such a re-meshing mechanism can quickly evolve into rather pathological particle configurations that yield, in long-term, very poor results, see [@price12a] for an explicit numerical example. To understand this mechanism better, it is instructive to expand $P_b$ in Eq. (\[eq:momentum\_from\_Lagrangian\]) around $P_a$ (sum over $k$) m\_a ()\^i&=& \_b { P\_a + (P)\_a\^k (\_b - \_a )\^k + …} ()\^i\ && P\_a ( \_b V\_b)\^i + (P)\_a\^k \_b (\_b - \_a )\^k ()\^i\ && P\_a e\_0\^i - (P)\_a\^k V\_a D\^[ki]{}\ && f\_[reg]{}\^i + f\_[hyd]{}\^i. The first term is the “regularization force” that is responsible for driving particles into a good distribution. It only vanishes, $\vec{e}_0=0$, if the sum of the volumes is constant. The second term, $f_{\rm hyd}^i$, is the approximation to the hydrodynamic force. For a good particle distribution, $e_0^i \rightarrow 0$ and $D^{ki} \rightarrow \delta^{ki}$ and thus (using $\rho_a= m_a/V_a$) the Euler equation (\[eq:Newt\_Euler\]) is reproduced. ### SPH with integral-based derivatives {#sec:SPH_with_integral_gradients} One can find SPH equations based on the more accurate integral approximation derivatives by using the formal replacement Eq. (\[eq:nabla\_W\_to\_G\]). By replacing the kernel gradients, one finds &=& - \_[b]{} X\_a X\_b { \_a + \_b }\ &=& \_b X\_b \_[ab]{} \_a, with (\_[a]{})\^k= C\^[kd]{}(\_a,h\_a) (\_b - \_a)\^d W\_[ab]{}(h\_a) and (\_[b]{})\^k= C\^[kd]{}(\_b,h\_b) (\_b - \_a)\^d W\_[ab]{}(h\_b), where a summation over $d$ is implied and the density can be calculated via Eq. (\[eq:gen\_vol\_N\]). For the conventional choice $X= m$ this reproduces (up to the grad-h terms) the original equation set derived in [@garcia_senz12] from a Lagrangian. The experiments in [@garcia_senz12; @cabezon12a; @rosswog15b] clearly show that the use of this gradient prescription substantially improves the accuracy of SPH. ### Treatment of shocks {#sec:Newtonian_shocks} The equations of gas dynamics allow for discontinuities to emerge even from perfectly smooth initial conditions [@landau59]. At discontinuities, the differential form of the fluid equations is no longer valid, and their integral form needs to be used, which, at shocks, translates into the Rankine–Hugoniot conditions, which relate the upstream and downstream properties of the flow. They show in particular, that the entropy increases in shocks, i.e., that dissipation occurs inside the shock front. For this reason the inviscid SPH equations need to be augmented by further measures that become active near shocks. Another line of reasoning that suggests using artificial viscosity is the following. One can think of the SPH particles as (macroscopic) fluid elements that follow streamlines, so in this sense SPH is a method of characteristics. Problems can occur when particle trajectories cross since in such a case fluid properties at the crossing point can become multi-valued. The term linear in the velocities in Eq. (\[eq:basic:PI\_AV\]) was originally also introduced as a measure to avoid particle interpenetration where it should not occur [@hernquist89] and to damp particle noise. [@read12] designed special dissipation switches to avoid particle properties becoming multi-valued at trajectory crossings. ### Common form of the dissipative equations {#common-form-of-the-dissipative-equations .unnumbered} There are a number of successful Riemann solver implementations (see for example [@inutsuka02; @cha03; @cha10; @murante11; @iwasaki11; @puri14]), but the most widespread approach is the use “artificial” dissipation terms in SPH. Such terms are not necessarily meant to mimic physical dissipation, their only purpose is to ensure that the Rankine–Hugoniot relations are fulfilled, though on a numerically resolvable rather than a microscopic scale. The most commonly chosen approach is to add the following terms ( )\_[diss]{} = - \_b m\_b \_[ab]{} \_a W\_[ab]{} ( )\_[diss]{}= \_b m\_b \_[ab]{} \_[ab]{} \_a W\_[ab]{} \[eq:basic\_AV\] to the momentum and energy equation, where $\Pi_{ab}$ is the artificial viscosity tensor. As long as $\Pi_{ab}$ is symmetric in $a$ and $b$, the conservation of energy, linear and angular momentum is ensured by the form of the equation and the anti-symmetry of $\nabla_a W_{ab}$ with respect to the exchange of $a$ and $b$. There is some freedom in choosing $\Pi_{ab}$, but the most commonly used form is [@monaghan83] \_[ab]{} = { [cl]{} \_[ab]{} \_[ab]{} &lt; 0\ 0\ . ,[where]{} \_[ab]{}= , \[eq:basic:PI\_AV\] where the barred quantities are arithmetic averages of particle $a$ and $b$ and $\vec{v}_{ab}= \vec{v}_a - \vec{v}_b$. This is a SPH-translation of a bulk and a von-Neumann–Richtmyer viscosity, with viscous pressures of $P_B \propto - \rho l \nabla \cdot \vec{v}$ and $P_{\rm NR} \propto \rho l^2 (\nabla \cdot \vec{v})^2$, respectively, where $l$ is the local resolution length. This form has been chosen because it is Galilean invariant, vanishes for rigid rotation and ensures exact conservation. The parameters $\alpha$ and $\beta$ set the strength of the dissipative terms and are usually chosen so that good results are obtained in standard benchmark tests. Comparison with Riemann solver approaches [@monaghan97] suggests an alternative form that involves signal velocities and jumps in variables across characteristics. The main idea of these “discontinuity capturing terms” is that for any conserved scalar variable $A$ with $\sum_a m_a dA_a/dt=0$ a dissipative term of the form ( )\_[diss]{}= \_b m\_b (A\_a-A\_b)\_[ab]{} \_a W\_[ab]{} should be added, where the parameter $\alpha_{A,b}$ determines the exact amount of dissipation and $v_{\rm sig}$ is a signal velocity between particle $a$ and $b$. Applied to the velocity and the thermokinetic energy $e= u + v^2/2$, this yields ( )\_[diss]{}&=& \_b m\_b \_a W\_[ab]{}\[basic:eq:v\_diss\]\ ( )\_[diss]{}&=& \_b m\_b \_[ab]{}\_a W\_[ab]{},\[basic:eq:e\_diss\] where, following [@price08a], the energy $e^\ast$ includes velocity components along the line joining particles $a$ and $b$, $e^\ast_a= \frac{1}{2} \alpha v_{\rm sig} (\vec{v}_a\cdot\hat{e}_{ab})^2 + \alpha_u v_{\rm sig}^u u_a$. Note that in this equation different signal velocities and dissipation parameters can be used for the velocities and the thermal energy terms. Using $du_a/dt= d\hat{e}_a/dt -\vec{v}_a \cdot d\vec{v}_a/dt$, this translates into ( )\_[diss]{}= - \_b \_[ab]{} \_a W\_[ab]{} for the thermal energy equation. The first term in this equation bears similarities with the “standard” artificial viscosity prescription, see Eq. (\[eq:basic:PI\_AV\]). The second one expresses the exchange of thermal energy between particles and therefore represents an artificial thermal conductivity which smoothes discontinuities in the specific energy. Such artificial conductivity had been suggested earlier to cure the so-called “wall heating problem” [@noh87]. Tests have shown that artificial conductivity substantially improves SPH’s performance in simulating Sedov blast waves [@rosswog07c] and in the treatment of Kelvin–Helmholtz instabilities [@price08a]. Note that the general strategy suggests to use dissipative terms also for the continuity equation so that particles can exchange mass in a conservative way. This has, so far, been rarely applied, but [@read12] find good results with this strategy in multi-particle-mass simulations. ### Time dependent dissipation parameters {#time-dependent-dissipation-parameters .unnumbered} The choice of the dissipation parameters is crucial and simply using fixed parameters applies dissipation also where it is not needed and actually unwanted. As a result, one simulates some type of viscous fluid rather than the intended inviscid medium. Therefore, [@morris97] suggested an approach with time-dependent parameters for each particle and with triggers that indicate where dissipation is needed. Using $\beta_a= 2\alpha_a$ in Eq. (\[eq:basic:PI\_AV\]) they evolved $\alpha_a$ according to = \_a\^+ - \_a\^- \_a\^+= (-()\_a,0) \_a\^- = , \[eq:alpha\_steering\_MM\] where $\alpha_{\min}$ represents a minimum, “floor” value for the viscosity parameter and $\tau_a\sim h_a/c_{{\rm s},a}$ is the individual decay time scale. This approach (or slight modifications of it) has been shown to substantially reduce unwanted effects in practical simulations [@rosswog00; @dolag05; @wetzstein09] but, as already realized in the original publication, triggering on the velocity divergence also raises the viscosity in a slow compression with $(\nabla\cdot\vec{v})=$ const where it is actually not needed. ### New shock triggers {#new-shock-triggers .unnumbered} More recently, some improvements of the basic Morris and Monaghan idea were suggested by [@cullen10]. First, the authors argued that the floor value can be safely set to zero, provided that $\alpha$ can grow fast enough. To ensure the latter, the current value of $\alpha_a$ is compared to values indicated by triggers and, if necessary, $\alpha_a$ is increased instantaneously rather than by solving Eq. (\[eq:alpha\_steering\_MM\]). Second, instead of using $(\nabla\cdot\vec{v})_a$ one triggers on its time derivative to find the locally desired dissipation parameter: A\_a= \_a \_[a, des]{}= \_ , \[eq:trigger\_CD\] where $c_a$ is a signal velocity, $\xi_a$ a limiter, see below, and $\alpha_{\max}$ a maximally admissible dissipation value. If $\alpha_{a, \rm des} > \alpha_{\rm a}$ then $\alpha_{\rm a}$ is raised immediately to $\alpha_{a, \rm des}$, otherwise it decays according to the $\mathcal{A}_a^-$-term in Eq. (\[eq:alpha\_steering\_MM\]). Apart from overall substantially reducing dissipation, this approach possesses the additional virtue that $\alpha$ peaks $\sim 2$ smoothing lengths ahead of a shock front and decays immediately after. Tests show that this produces much less unwanted dissipation than previous approaches. [@read12] suggest a similar strategy, but trigger on the *spatial* change of the compression A\_a= \_a h\_a\^2 |()|\_a \_[a,des]{}= \_ , \[eq:trigger\_RH\] where $\nabla \cdot \vec{v} < 0$ and $\alpha_{\rm des}= 0$ otherwise. The major idea is to detect convergence *before* it actually occurs. Particular care is taken to ensure that all fluid properties remain single valued as particles approach each other and higher-order gradient estimators are used. With this approach they find good results with only very little numerical noise. ### Noise triggers {#noise-triggers .unnumbered} The time scale $\tau_a$ on which dissipation is allowed to decay is usually chosen in a trial-and-error approach. If chosen too large, the dissipation may be higher than desired, if decaying too quickly, velocity noise may build up again after a shock has passed. Therefore, a safer strategy is allow for a fast decay, but devise additional triggers that switch on if noise is detected. The main idea for identifying noise is to monitor sign fluctuations of $\nabla \cdot \vec{v}$ in the vicinity of a particle. A particle is considered “noisy” if some of its neighbours are being compressed while others expand.\ For example, the ratio \[eq:noise\_trigg\] can be used to construct a noise trigger. It deviates from $\pm 1$ in a noisy region since contributions of different sign are added up in $S_{1,a}$ and therefore such deviations can be used as a noise indicator: \_a\^[(1)]{}= | - 1|, \[eq:N\_trigg\_1\] where the quantity \_[1,a]{}= { [ll]{} -S\_[1,a]{} & [if]{} ()\_[a]{} &lt; 0\ S\_[1,a]{} & [else]{} . . If all particles in the neighborhood are either compressed or expanding, $\mathcal{N}_a^{(1)}$ vanishes. This trigger only reacts on sign changes, but does not account for the strengths of the (de-)compressions. A second noise trigger that accounts for this uses \^+\_a = \_[b, \_b&gt;0]{}\^[N\^+]{} \_b \^-\_a = - \_[b, \_b&lt;0]{}\^[N\^-]{} \_b, where $N^+/N^-$ is number of neighbor particles with positive/negative $\divv$. The trigger then reads \_a\^[(2)]{}= . \[eq:N\_trigg\_2\] If there are sign fluctuations in $\nabla \cdot \vec{v}$, but they are small compared to $c_s/h$, the product is very small, if we have a uniform expansion or compression one of the factors will be zero. So only for sign changes [*and*]{} significantly large compressions/expansions will the product have a substantial value. Like in Eq. (\[eq:trigger\_CD\]) the noise triggers $\mathcal{N}_a^{(1)}$ and $\mathcal{N}_a^{(2)}$ can be compared against reference values to steer the dissipation parameters. For more details on noise triggers see [@rosswog15b].\ Note that in all these triggers [@cullen10; @read12; @rosswog15b] the gradients can straight forwardly be calculated from accurate expressions such as Eq. (\[eq:full\_IA\_gradient\]) or (\[eq:lin\_exact\_gradient\]) rather than from kernel gradients. ### Limiters {#limiters .unnumbered} Complementary to these triggers one can also try to actively suppress dissipation where it is unwanted (with ideally working triggers this should, of course, not be necessary). For example, [@balsara95] suggested to distinguish between shock and shear motion based on the ratio \_a\^[B]{}= . \[eq:Balsara\] Dissipation is suppressed, $\xi_{a}^{\rm B} \rightarrow 0$, where $|\nabla \times \vec{v}|_a \gg |\nabla \cdot \vec{v}|_a$, whereas $\xi_{a}^{\rm B}$ tends to unity in the opposite limit. If symmetrized limiters are applied, $\Pi_{ab}'= \Pi_{ab} \; \bar{\xi}_{ab}$, exact conservation is ensured. The Balsara limiter has been found very useful in many applications [@steinmetz96; @navarro97; @rosswog00], but it can be challenged if shocks occur in a shearing environment like an accretion disk [@owen04]. Part of this challenge comes from the finite accuracy of the standard SPH-derivatives. It is, however, straight forward [@cullen10; @read12; @rosswog15b] to use more accurate derivatives such as, for example, Eq. (\[eq:lin\_exact\_gradient\]), in the limiters. Suppression of dissipation in shear flows can then be obtained by simply replacing the SPH-gradient operators in Eq. (\[eq:Balsara\]) by more accurate expressions, or by simply adding terms proportional to (accurate estimates for) $|\nabla \times \vec{v}|$ in the denominators of Eqs. (\[eq:trigger\_CD\]) and (\[eq:trigger\_RH\]) (instead of multiplying $\alpha_{\rm des}$ with a limiter). Another alternative is the limiter proposed in [@cullen10] that also makes use of more accurate gradient estimates: \_[a]{}\^[C]{}= , where, in our notation and for generalized volume elements, R\_a= \_b [sign]{}()\_b X\_b W\_[ab]{}. Note that $R_a$ is simply the ratio of a density summation where each term is weighted by the sign of $\divv$ and the normal density, Eq. (\[eq:density\]). Therefore, near a shock $R_a \rightarrow -1$. The matrix ${\bf S}$ is the traceless symmetric part of the velocity gradient matrix $(\p_i v_j)$ and a measure for the local shear. Similar to the Balsara factor, $\xi^{\rm C}$ approaches unity if compression clearly dominates over shear and it vanishes in the opposite limit. Special-relativistic SPH {#sec:SR_SPH} ------------------------ The special-relativistic SPH equations can –like the Newtonian ones– be elegantly derived from a variational principle [@monaghan01; @rosswog09b; @rosswog10a; @rosswog10b]. We discuss here a formulation [@rosswog15b] that uses generalized volume elements, see Eq. (\[eq:gen\_vol\_element\]). It is assumed that space-time is flat, that the metric, $\eta_{\mu \nu}$, has the signature (-,+,+,+) and units in which the speed of light is equal to unity, $c=1$, are adopted. We use the Einstein summation convention and reserve Greek letters for space-time indices from 0…3 with 0 being the temporal component, while $i$ and $j$ refer to spatial components and SPH particles are labeled by $a,b$ and $k$. The Lagrangian of a special-relativistic perfect fluid can then be written as [@fock64] L= - T\^ U\_U\_dV, where the energy-momentum tensor of an ideal fluid reads T\^= (e + P) U\^ U\^ + P \^ with $e$ being the energy density, $P$ the pressure and $U^\lambda$ the four-velocity of a fluid element. One can write the energy density as a sum of a contribution from rest mass density and one from the internal energy e= \_[rest]{} c\^2 + u \_[rest]{}= n m\_0 c\^2 (1 + u/c\^2), \[eq:energy\_density\] where the speed of light was, for clarity, shown explicitly. The baryon number density $n$ is measured in the local fluid rest frame and the average baryon mass is denoted by $m_0$. With the conventions that all energies are written in units of $m_0c^2$ and $c=1$ we can use the normalization of the four-velocity, $U_\mu U^\mu= - 1$, to simplify the Lagrangian to L= - n(1+u) dV. The calculation will be performed in an a priori chosen “computing frame” (CF) and –due to length contraction– the baryon number density measured in this frame, $N$, is increased by a Lorentz factor $\gamma$ with respect to the local fluid rest frame N= n \[eq:N\_vs\_n\]. Therefore, the Lagrangian can be written as L= - dV N () L- \_b V\_b N\_b = - \_b \_b ,\[eq:SR\_Lagrangian\] where the baryon number carried by particle $b$, $\nu_b$, has been introduced. If volume elements of the form Eq. (\[eq:gen\_vol\_element\]) are introduced, one can calculate CF baryon number densities as in the Newtonian case (see Eq. (\[eq:density\])) N\_b= = \_k X\_k W\_[bk]{}(h\_b), \[eq:CF\_dens\] which reduces for the choice $X=\nu$ to the equivalent of the standard SPH sum, Eq. (\[eq:dens\_sum\]), but with $\rho/m$ being replaced by $N/\nu$. To obtain the equations of motion, $\nabla_a N_b$ and $d N_a/dt$ are needed, which follow from Eq. (\[eq:gen\_vol\_element\]) as \_a N\_b= \_k X\_k \_a W\_[bk]{}(h\_b) = \_k X\_k \_[bk]{} \_b W\_[bk]{}(h\_b)\[eq:dNdt\_SR\] and which contain the generalized “grad-h” terms \_b= 1 - \_k X\_k . It turns out to be beneficial to use the canonical momentum per baryon \_a = \_a \_a ( 1+u\_a + ) \[eq:S\_a\] as a numerical variable. Its evolution equation follows directly from the Euler–Lagrange equations as [@rosswog15b] = - \_[b]{} { \_a W\_[ab]{}(h\_a) + \_a W\_[ab]{}(h\_b) }. \[eq:gen\_mom\_SR\] For the choice $V_k= \nu_k/N_k$, this reduces to the momentum equation given in [@rosswog10b]. The canonical energy E\_a \_a - L = \_a \_a \_a, suggests to use \_a = \_a \_a + = \_a (1 + u\_a + ) - = \_a \_a - , \[eq:en\_a\] as numerical energy variable. Here the specific, relativistic enthalpy was abbreviated as \_a= 1 + u\_a + . \[eq:enthalpy\] The subsequent derivation is identical to the one in [@rosswog09b] up to their Eq. (165), = \_a + , which, upon using Eqs. (\[eq:dNdt\_SR\]) and (\[eq:gen\_mom\_SR\]), yields the special-relativistic energy equation = - \_b { \_b \_a W\_[ab]{}(h\_a) + \_a \_a W\_[ab]{}(h\_b)}. \[eq:gen\_ener\_SR\] Again, for $V_k= \nu_k/N_k$, this reduces to the energy equation given in [@rosswog10b]. Of course, the set of equations needs to be closed by an equation of state which in the simplest case can be a polytrope. Note that by choosing the canonical energy and momentum as numerical variables, one avoids complications such as time derivatives of Lorentz factors, that have plagued earlier SPH formulations [@laguna93a]. The price one has to pay is that the physical variables (such as $u$ and $\vec{v}$) need to be recovered at every time step from $N$, $\epsilon$ and $\vec{S}$ by solving a non-linear equation. For this task approaches very similar to what is used in Eulerian relativistic hydrodynamics can be applied [@chow97; @rosswog10b]. As in the non-relativistic case, these equations need to be augmented by extra measures to deal with shocks. The dissipative terms [@chow97] ()\_[diss]{}= - \_b \_b \_[ab]{} \_[ab]{}= - (\_a\^-\_b\^) \_[ab]{} \[eq:diss\_mom\] ()\_[diss]{}= - \_b \_b \_[ab]{} \_[ab]{} = - (\_a\^-\_b\^)\_[ab]{} \[eq:diss\_en\] with symmetrized kernel gradients = , \_k\^= , \_k\^= \^\_k (1+u\_k+) \_k \_k\^= \^\_k (1+u\_k+) - , where the asterisk denotes the projection to the line connecting two particles, give good results, even in very challenging shock tests [@rosswog10b]. A good choice for $v_{\rm sig}$ is [@rosswog10b] v\_[sig,ab]{}= (\_a,\_b),\[eq:vsig\] where \_k\^= (0,\^\_k) with $\lambda^\pm_k$ being the extreme local eigenvalues of the Euler equations, see e.g., [@marti03], \^\_k= and $c_{{\rm s},k}$ is the relativistic sound velocity of particle $k$, $c_{s,k}= \sqrt{\frac{(\Gamma-1) ({\mathcal{E}}-1)}{{\mathcal{E}}}}$. In 1 D, this simply reduces to the usual velocity addition law, $\lambda^\pm_k= (v_k\pm c_{{\rm s},k})/(1\pm v_k c_{{\rm s},k}) $. As in the non-relativistic case, the challenge lies in designing triggers that switch on where needed, but not otherwise. The strategies that can be applied here are straight forward translations of those described in Section \[sec:Newtonian\_shocks\]. We refer to [@rosswog15b] for more details on this topic. ### Integral approximation-based special-relativistic SPH {#integral-approximation-based-special-relativistic-sph .unnumbered} Like in the Newtonian case, alternative SPH equations can be obtained [@rosswog15b] by replacing the kernel gradients by the functions $\vec{G}$, see Eq. (\[eq:IA\_gradient\]): = - \_b { P\_a V\_a\^2 \_[a]{} + P\_b V\_b\^2 \_[b]{} } \[eq:momentum\_eq\_no\_diss\_integral\] and = - \_b { P\_a V\_a\^2 \_b\_[a]{} + P\_b V\_b\^2 \_a\_[b]{} }, \[eq:ener\_eq\_no\_diss\_integral\] where (sum over $d$) (\_[a]{})\^k= C\^[kd]{}(\_a,h\_a) (\_b - \_a)\^d W\_[ab]{}(h\_a) and (\_[b]{})\^k= C\^[kd]{}(\_b,h\_b) (\_b - \_a)\^d W\_[ab]{}(h\_b). The density calculation remains unchanged from Eq. (\[eq:CF\_dens\]). The same dissipative terms as for the kernel-gradient-based approach can be used, but it is important to replace $\overline{\nabla_a W_{ab}}$ by \_[ab]{}= , since otherwise numerical instabilities can occur. This form has been extensively tested [@rosswog15b] and shown to deliver significantly more accurate results than the kernel-gradient-based approach. General-relativistic SPH {#sec:GR_SPH} ------------------------ For binaries that contain a neutron or a black hole general-relativistic effects are important. The first relativistic SPH formulations were developed by [@kheyfets90] and [@mann91; @mann93]. Shortly after, [@laguna93a] developed a 3D, general-relativistic SPH code that was subsequently applied to the tidal disruption of stars by massive black holes [@laguna93b]. Their SPH formulation is complicated by several issues: the continuity equation contains a gravitational source term that requires SPH kernels for curved space-times. Moreover, owing to their choice of variables, the equations contain time derivatives of Lorentz factors that are treated by finite difference approximations and restrict the ability to handle shocks to only moderate Lorentz factors. The Laguna et al. formulation has been extended by [@rantsiou08] and applied to neutron star black hole binaries, see Section \[sec:appl\_NSBH\]. We focus here on SPH in a fixed background metric, approximate GR treatments are briefly discussed in Section \[sec:appl\_NSNS\_NSBH\]. In smooth continuation of the Newtonian and special-relativistic approaches, we focus here on the derivation from a Lagrangian. To this end, we assume that a prescribed metric $g_{\mu \nu}$ is known as a function of the coordinates and that the perturbations that the fluid induces to the space-time geometry can be safely neglected. Again, $c=1$ and signature ($-,+,+,+$) are assumed and the same index conventions as in the special relativistic section are adopted. Contravariant spatial indices of a vector quantity $w$ at particle $a$ are denoted as $w^i_a$, while covariant ones will be written as $(w_i)_a$. The line element and proper time are given by $ds^2= g_{\mu \nu} \, dx^\mu \, dx^\nu$ and $d\tau^2= - ds^2$ and the proper time is related to a coordinate time $t$ by d= dt, where a generalization of the Lorentz-factor v\^= was introduced. This relates to the four-velocity $U^\nu$, normalized to $U^\mu U_\mu= -1$, by v\^= = = = . \[eq:v\_mu\] The Lagrangian of an ideal relativistic fluid can be written as [@fock64] L= - T\^ U\_U\_ dV, where $g= {\rm det}(g_{\mu \nu})$ and $T^{\mu\nu}$ denotes the energy-momentum tensor of an ideal fluid without viscosity and conductivity T\^= (e+P)U\^U\^+ P g\^ with the energy density $e$ given in Eq. (\[eq:energy\_density\]). With these conventions the Lagrangian can be written, similar to the special-relativistic case, as L= - n(1+u) dV\[eq:Lagrangian\_cont\]. As before, for practical simulations we give up general covariance and choose a particular “computing frame”. To find a SPH discretization in terms of a suitable density variable one can express local baryon number conservation, $(U^\mu n);_\mu= 0$, as [@siegler00a] \_( U\^n)= 0, or, more explicitly, as \_t (N) + \_i(N v\^i)= 0, \[eq:continuity\_N\] where Eq. (\[eq:v\_mu\]) was used and the computing frame baryon number density N= n\[eq:N\_n\] was introduced. The total conserved baryon number can then be expressed as a sum over fluid parcels with volume $\Delta V_b$ located at $\vec{r}_b$, where each parcel carries a baryon number $\nu_b$ = N dV \_b N\_b V\_b = \_b \_b.\[eq:parcel\_volumes\] Eq. (\[eq:continuity\_N\]) looks like the Newtonian continuity equation which suggests to use it in the SPH discretization process of a quantity $f$: () \_b f\_b W(-\_b,h),\[eq:SPH\_discretization\] where the subscript $b$ indicates that a quantity is evaluated at a position $\vec{r}_b$ and $W$ is the smoothing kernel. If all $\nu_b$ are kept constant in time, exact baryon number conservation is guaranteed and no continuity equation needs to be solved (this can be done, though, if desired). If Eq. (\[eq:SPH\_discretization\]) is applied to the baryon number density $N$ at the position of particle $a$, one finds N\_a= N(\_a)= \_b \_b W(\_a-\_b,h\_a)\[eq:N\_r\]. Note that the locally smoothed quantities are evaluated with flat-space kernels which assumes that the local space-time curvature radius is large in comparison to the local fluid resolution length. Such an approach is very convenient, but (more involved) alternatives to this approach exist [@kheyfets90; @laguna93a]. It is usually desirable to adjust the smoothing length locally to fully exploit the natural adaptivity of a particle method. As before, one can adopt the smoothing length according to $ h_a= \eta (\nu_a/N_a)^{1/3}. $ Due to the mutual dependence an iteration is required at each time step to obtain consistent values for both density and smoothing length, similar to the Newtonian and special-relativistic case. Motivated by Eqs. (\[eq:N\_n\]) and (\[eq:parcel\_volumes\]), one can re-write the fluid Lagrangian, Eq. (\[eq:Lagrangian\_cont\]), in terms of our computing frame number density $N$, L= - N dV, or, in discretized form, L\_[SPH]{}= - \_b \_b ( )\_b. This Lagrangian has the same form as in the special-relativistic case, see Eq. (\[eq:SR\_Lagrangian\]), with the Lorentz factor $\gamma$ being replaced with its generalized form $\Theta$. Using the Euler–Lagrange equations and taking care of the velocity dependence of both the generalized Lorentz-factor $\Theta$ and the internal energy, $\frac{\p u_b}{\p v^i_a}= \frac{\p u_b}{\p n_b} \frac{\p n_b}{\p v^i_a}$, one finds the canonical momentum per baryon (S\_i)\_a = \_a (1+u\_a+ ) (g\_[i]{} v\^)\_a= (1+u\_a+ ) (U\_i)\_a. which generalizes Eq. (\[eq:S\_a\]). The Euler–Lagrange equations [@rosswog10a] deliver &=& - \_b \_b { + } + ( T\^ )\_a, \[eq:GR\_momentum\_evolution\] again very similar to the special-relativistic case. Like before, the canonical energy E\_a v\^i\_a - L= \_a \_a ( v\^i\_a (S\_i)\_a + ) suggests to use e\_a v\^i\_a (S\_i)\_a + as numerical variable whose evolution equation follows from straight forward differentiation [@rosswog10a] as &=& - \_b \_b { + }- ( T\^ \_t g\_)\_a. \[eq:GR\_energy\_evolution\] Together with an equation of state, the equations (\[eq:N\_r\]), (\[eq:GR\_momentum\_evolution\]) and (\[eq:GR\_energy\_evolution\]) represent a complete and self-consistently derived set of SPH equations. The gravitational terms are identical to those of [@siegler00a] and [@monaghan01], but the hydrodynamic terms differ in both the particle symmetrization and the presence of the grad-h terms \_b= 1- \_k \_k . Note that the only choices in our above derivation were the $h$-dependence in Eq. (\[eq:N\_r\]) and how to adapt the smoothing length. The subsequent calculation contained no arbitrariness concerning the symmetry in particle indices, everything followed stringently from the first law of thermodynamics and the Euler–Lagrange equations. Another important point to note is that the derived energy equation, Eq. (\[eq:GR\_energy\_evolution\]), does not contain destabilizing time derivatives of Lorentz factors [@norman86] on the RHS – in contrast to earlier SPH formulations [@laguna93a]. The above SPH equations can also be recast in the language of the 3+1 formalism, this can be found in [@tejeda12a]. ### Limiting cases It is a straight forward exercise to show that, in the limit of vanishing hydrodynamic terms (i.e., $u$ and $P$ = 0), the evolution equations (\[eq:GR\_momentum\_evolution\]) and (\[eq:GR\_energy\_evolution\]) reduce to the equation of geodesic motion [@tejeda12a]. If, in the opposite limit, we are neglecting the gravitational terms in Eqs. (\[eq:GR\_momentum\_evolution\]) and (\[eq:GR\_energy\_evolution\]) and assume flat space-time with Cartesian coordinates one has $\sqrt{-g} \rightarrow 1$ and $\Theta \rightarrow \gamma$, and Eq. (\[eq:N\_n\]) becomes $N= \gamma n$. The momentum and energy equations reduce in this limit to the previous equations (\[eq:gen\_mom\_SR\]) and (\[eq:gen\_ener\_SR\]) (for $X= \nu$). Frequently used SPH codes ------------------------- A number of SPH codes are regularly used throughout various areas of astrophysics and it is beyond the scope of this review to give a detailed account of each of them. In Table \[tab:SPH\_codes\] we briefly summarize some of the basic features of Newtonian SPH codes that have been used in the context of compact object simulations and that will be referred to in the subsequent sections of this review. For a more detailed code description we refer to the original papers. SPH approaches that use GR or various approximations to it are further discussed in Section \[sec:appl\_NSNS\_NSBH\]. [ &gt;p[3.8cm]{}lllllllll ]{} reference & name/group & SPH equations & self-gravity & AV steering & EOS, burning &\ \ [@springel01a; @springel05a] & <span style="font-variant:small-caps;">Gadget</span> & entropy equation, Eq. (\[eq:momentum\_gadget\]) & Oct-tree & fixed parameters & polytrope\ [@starcrash] & <span style="font-variant:small-caps;">StarCrash</span> & “vanilla ice”& FFT on grid & fixed parameters & polytrope\ [@rosswog02a] & Leicester & “vanilla ice” & binary tree [@benz90b] & time-dep., Balsara-switch & Shen\ [@rosswog08b] & Bremen & “vanilla ice” & binary tree [@benz90b] & time-dep., Balsara-switch & Helmholtz + network\ [@rosswog07c] & <span style="font-variant:small-caps;">Magma</span> & hydro & self-gravity & binary tree & time-dep., Balsara-switch, & Shen &\ & & from Lagrangian & & conductivity &\ [@guerrero04] & Barcelona & “vanilla ice” & oct-tree & fixed parameters,& ions, electrons, photons&\ & & & & Balsara-switch & + network &\ [@fryer06] & <span style="font-variant:small-caps;">SNSPH</span> & “vanilla ice” & oct-tree & fixed parameters & polytrope,\ & & & & & Helmholtz + network\ [@wadsley04] & <span style="font-variant:small-caps;">Gasoline</span> & “vanilla ice” & K-D tree & fixed parameters, & polytrope &\ & & & & Balsara-switch\ “Shen” EOS: [@shen98a; @shen98b], “Helmholtz” EOS: [@timmes99] Importance of initial conditions {#sec:IC} -------------------------------- It is not always sufficiently appreciated how important the initial conditions are for the accuracy of SPH simulations. Unfortunately, it can become rather non-trivial to construct them. One obvious requirement is the regularity of a particle distribution so that the quality indicators $\mathcal{Q}_1 - \mathcal{Q}_4$, Eqs. (\[eq:quality\_int\]) and (\[eq:gradient\_quality\]), are fulfilled with high accuracy. This suggests placing the particles initially on some type of lattice. However, these lattices should ideally possess at least two more properties: a) they should not contain preferred directions b) they should represent a *stable* minimum energy configuration for the applied SPH formulation. Condition a) is desirable since otherwise shocks propagating along a symmetry axis of a lattice will continuously “collect” particles in this direction and this can lead to “lattice ringing effects”. Condition b) is important since a regular lattice does by no means guarantee that the configuration actually represents an equilibrium for the SPH particles. As briefly outlined in Section \[sec:SPH\_from\_variational\_principle\], the momentum equation derived from a Lagrangian also contains a “regularization force” contribution that strives to achieve a good particle distribution. If the initial lattice does not represent such a configuration, the particles will start to move “off the lattice” and this will introduce noise. In Figure \[fig:noise\] we show the result of a numerical experiment. 10K particles are placed either on a quadratic or hexagonal lattice in the domain $[-1,1] \times [-1,1]$ with particles at the edges being fixed. If the configurations are allowed to evolve without dissipation (here we use the common cubic spline kernel), the particles in the quadratic lattice case soon begin to move off the grid and gain average velocities of $\langle v \rangle \approx 0.015 \; c_{\rm s}$, while the remain in the initial configuration when initially placed on a hexagonal lattice ($\langle v \rangle \approx 10^{-4} \; c_{\rm s}$). This is discussed in more detail in [@rosswog15b]. From a theoretical point of view, the relation between kernel, particle configuration and stability/noise is to date only incompletely understood and must be addressed in careful future studies. In experiments [@rosswog15b] different kernels show a very different noise behavior and this seems to be uncorrelated with the accuracy properties of the kernels: kernels that are rather poor density and gradient estimators may be excellent in producing very little noise (see for example the QCM$_6$ kernel described in [@rosswog15b]). On the other hand, very accurate kernels (like W$_{\rm H,9}$) may be still allow for a fair amount of noise in a dynamical simulation. The family of Wendland kernels [@wendland95] has a number of interesting properties, among them the stability against particle pairing despite having a vanishing central derivative [@dehnen12] and their reluctance to tolerate sub-resolution particle motion [@rosswog15b], in other words noise. In those experiments where particle noise is relevant for the overall accuracy, for example in the Gresho–Chan vortex test, see Section \[sec:gresho\], the Wendland kernel, Eq. (\[eq:wend33\]), performs substantially better than any other of the explored kernels. A heuristic approach to construct good, low-noise initial conditions for the actually used kernel function is to apply a velocity-dependent damping term, $\vec{f}_{\rm damp} \propto - \vec{v}_{\rm damp,a}/\tau_a$, with a suitably chosen damping time scale $\tau_a$ to the momentum equation. This “relaxation process” can be applied until some convergence criterion is met (say, the kinetic energy or some density oscillation amplitude has dropped below some threshold). This procedure becomes, however, difficult to apply for more complicated initial conditions and it can become as time consuming as the subsequent production simulation. For an interesting recent suggestion on how to construct more general initial conditions see [@diehl12]. The performance of SPH ---------------------- Here we cannot give an exhaustive overview over the performance of SPH in general. We do want to address, however, a number of issues that are of particular relevance in astrophysics. Thereby we put particular emphasis on the new improvements of SPH that were discussed in the previous sections. Most astrophysical simulations, however, are still carried out with older methodologies. This is natural since, on the one hand, SPH is still a relatively young numerical method and improvements are constantly being suggested and, on the other hand, writing a well-tested and robust production code is usually a rather laborious endeavor. Nevertheless, efforts should be taken to ensure that latest developments find their way into production codes. In this sense the below shown examples are also meant as a motivation for computational astrophysicists to keep their simulation tools up-to-date. The accuracy of SPH with respect to commonly used techniques can be enhanced by using - higher-order kernels, for example the $W_{\rm H,9}$ or the Wendland kernel $W_{3,3}$, see Section \[sec:kernel\_choice\] - different volume elements, see Section \[sec:volume\_elements\] - more accurate integral-based derivatives, see Section \[sec:integral\_gradients\] - modern dissipation triggers, see Section \[sec:Newtonian\_shocks\]. In the examples shown here, we use a special-relativistic SPH code (“SPHINCS\_SR”, [@rosswog15b]) to demonstrate SPH’s performance in a few astrophysically relevant examples. Below, we refer several times to the “$\mathcal{F}_1$ formulation”: it consists of baryon number density calculated via Eq. (\[eq:CF\_dens\]) with volume weight $X= P^{0.05}$, the integral approximation-based form of the SPH equations, see Eqs. (\[eq:momentum\_eq\_no\_diss\_integral\]) and (\[eq:ener\_eq\_no\_diss\_integral\]), and shock trigger Eq. (\[eq:trigger\_CD\]) and noise triggers, see [@rosswog15b] for more details. ### Advection {#sec:advection_test} SPH is essentially perfect in terms of advection: if a particle carries a certain property, say some electron fraction, it simply transports this property with it (unless the property is changed by physical processes, of course). The advection accuracy is, contrary to Eulerian schemes, independent of the numerical resolution and just governed by the integration accuracy of the involved ordinary differential equations. We briefly want to illustrate the advection accuracy in an example that is a very serious challenge for Eulerian schemes, but essentially a “free lunch” for SPH: the highly supersonic advection of a density pattern through the computational domain with periodic boundary conditions. To this end, we set up 7K SPH particles inside the domain $[-1,1] \times [-1,1]$. Each fluid element is assigned a velocity in the x-direction of $v_x= 0.9999$ corresponding to a Lorentz factor of $\gamma\approx 70.7$ and periodic boundary conditions are applied. The result of this experiment is shown in Figure \[fig:advection\]: after crossing the computational domain ten times (more than 23 000 time steps; right panel) the shape of the triangle has not changed in any noticeable way with respect to the initial condition (left panel). ### Shocks {#sec:shock_test} Since shocks play a crucial role in many areas of astrophysics we want to briefly discuss SPH’s performance in shocks. As an illustration, we show the result of a 2D relativistic shock tube test where the left state is given by $[P,v_x,v_y,N]_L= [40/3,0,0,10]$ and the right state by $[P,v_x,v_y,N]_R= [10^{-6},0,0,1]$. The polytropic exponent is $\gamma = 5/3$ and the results are shown at $t= 0.25$. In the experiment 80K particles were initially placed on a hexagonal lattice in region $[-0.4,0.4] \times [-0.02,0.02]$. Overall, the exact solution (red solid line) is well captured, see Figure \[fig:shock\]. The discontinuities, however, are less sharp than those obtained with modern High Resolution Shock Capturing methods with the same resolution. Characteristic for SPH is that the particles that were initially located on a lattice get compressed in the shock and, in the post-shock region, they need to “re-grid” themselves into a new particle distribution. This also means that the particles have to pass each other and therefore also acquire a small velocity component in y-direction. As a result, there is some unavoidable “re-meshing noise” behind the shock front. This can be tamed by using smoother kernels and, in general, the details of the results depend on the exactly chosen dissipation parameters and initial conditions. Note that the noise trigger that is used in the shown simulation applies dissipation in the re-gridding region behind the shock, therefore the contact discontinuity has been somewhat broadened. Reducing the amount of dissipation in this region (i.e., raising the tolerable reference value for noise) would sharpen the contact discontinuity, but at the price of increased re-meshing noise in the velocity. ### Fluid instabilities {#sec:KH} The standard formulation of SPH has recently been criticized [@thacker00; @agertz07; @springel10a; @read10] for its performance in handling of fluid instabilities. As discussed in Section \[sec:volume\_elements\], this is related to the different smoothness of density and internal energy which can lead to spurious pressure forces acting like a surface tension. We want to briefly illustrate that a modern formulation can capture Kelvin–Helmholtz instabilities accurately. Three stripes of hexagonal lattices are placed in the domain $[-1,1] \times [-1,1]$ so that a density of $N=2$ is realized in the middle stripe ($|y|<0.3$) and $N=1$ in the surrounding stripes. The high-density stripe moves with $v_x=0.2$, the other two with $v_x=-0.2$, the pressure is $P_0=10$ everywhere and the polytropic exponent is $\Gamma=5/3$. No perturbation mode is triggered explicitly, we wait until small perturbations that occur as particles at the interface pass each other grow into a healthy Kelvin–Helmholtz instability. Figure \[fig:KH\] shows snapshots at $t=$ 2, 3.5 and 5.0. Note that “standard methods” (fixed, high dissipation parameters, volume element $\nu/N$ or $m/\rho$ and direct gradients of the M$_4$ kernel) do, for this setup, not lead to a noticeable instability on the shown time scale [@rosswog15b]. ### The Gresho–Chan vortex {#sec:gresho} The Gresho–Chan vortex [@gresho90] is considered as a particularly difficult test, in general and in particular for SPH. As shown in [@springel10a], standard SPH shows very poor convergence in this test. The test deals with a stationary vortex that should be in stable equilibrium. Since centrifugal forces and pressure gradients balance exactly, any deviation from the initial configuration that develops over time is spurious and of purely numerical origin. The azimuthal component of the velocity in this test rises linearly up to a maximum value of $v_0$ which is reached at $r= R_1$ and subsequently decreases linearly back to zero at 2$R_1$ v\_(r)= v\_0 { [ l l l]{} u u 1\ 2 - u 1 &lt; u 2,\ 0 u &gt; 2\ . where $u= r/R_1$. We require that centrifugal and pressure accelerations balance, therefore the pressure becomes P(r)= P\_0 + { [ l l l]{} v\_0\^2 u\^2 u 1\ 4 v\_0\^2 ( - u + + 1 ) 1 &lt; u 2\ 4 v\_0\^2 (2 - ) u &gt; 2.\ . In the literature on non-relativistic hydrodynamics [@liska03; @springel10a; @read12; @dehnen12] usually $v_0= 1$ is chosen together with $R_1= 0.2$, a uniform density $\rho= 1$ and a polytropic exponent of 5/3. Since we perform this test with the special-relativistic code SPHINCS\_SR in the Newtonian limit, we use most of these values, but choose $R_1= 2 \times 10^{-4}$ and $v_0= 10^{-3}$ to be safely in the non-relativistic regime. We use this test here to illustrate which difference modern concepts can make in this challenging test, see Figure \[fig:gresho\]. The left panel shows the result from a modern SPH formulation (formulation $\mathcal{F}_1$ from [@rosswog15b]) and one that applies “traditional” approaches (fixed, high dissipation parameters, volume element $\nu/N$ or $m/\rho$ and direct gradients of the M$_4$ kernel). The traditional approach fails completely and actually does not converge to the correct solution [@springel10a; @rosswog15b] while the new, more sophisticated approach yields very good results. Summary: SPH ------------ The most outstanding property of SPH is its exact numerical conservation. This can straight forwardly be achieved via symmetries in the particle indices of the SPH equations together with anti-symmetric gradient estimates. The most elegant and least arbitrary strategy to obtain a conservative SPH formulation is to start from a fluid Lagrangian and to derive the evolution equations via a variational principle. This approach can be applied in the Newtonian, special- and general relativistic case, see Sections \[sec:Newt\_SPH\], \[sec:SR\_SPH\] and \[sec:GR\_SPH\]. Advection of fluid properties is essentially perfect in SPH and it is in particular not dependent on the coordinate frame in which the simulation is performed. SPH robustly captures shocks, but they are, at a given resolution, not as sharp as those from state-of-the-art high-resolution shock capturing schemes. Moreover, standard SPH has been criticized for its (in)ability to resolve fluid instabilities under certain circumstances. Another issue that requires attention when performing SPH simulations are initial conditions. When not perpared carefully, they can easily lead to noisy results, since the regularization force discussed in Sec. \[sec:self\_regularization\] leads for poor particle distribution to a fair amount of particle velocity fluctuations. This issue is particular severe when poor kernel functions and/or low neighbour numbers are used, see Sec. \[chap:kernel\_approx\]. Recently, a number of improvements to SPH techniques have been suggested. These include a) more accurate gradient estimates, see Sec \[sec:SPH\_derivs\], b) new volume elements which eliminate spurious surface tension effects, see Section \[sec:volume\_elements\], c) higher-order kernels, see Section \[sec:kernel\_choice\] and d) more sophisticated dissipation switches, see Section \[sec:Newtonian\_shocks\]. As illustrated, for example by the Gresho–Chan vortex test in Section \[sec:gresho\], enhancing SPH with these improvements can substantially increase its accuracy with respect to older SPH formulations. Astrophysical Applications {#sec:appl} ========================== The remaining part of this review is dedicated to actual applications of SPH to astrophysical studies of compact objects. We focus on encounters of - two white dwarfs (Section \[sec:appl\_WDWD\]), - two neutron stars (Section \[sec:appl\_NSNS\]) and - a neutron star with a black hole (Section \[sec:appl\_NSBH\]). In each case the focus is on gravitational wave-driven binary mergers. In locations with large stellar number densities, e.g., globular clusters, dynamical collisions between stars occur frequently and encounters between two neutron stars and a neutron star with a stellar-mass black hole may yield very interesting signatures. Therefore such encounters are also briefly discussed. In each of these fields a wealth of important results have been achieved with a number of different methods. Naturally, since the scope of this review are SPH methods, we will focus our attention here to those studies that are at least partially based on SPH simulations. For further studies that are based on different methods we have to refer to the literature. Double white-dwarf encounters {#sec:appl_WDWD} ----------------------------- ### Relevance White Dwarfs (WDs) are the evolutionary end stages of most stars in the Universe, for every solar mass of stars that forms $\sim 0.22$ WDs will be produced on average. As a result, the Milky Way contains $\sim 10^{10}$ WDs [@napiwotzki09] in total and $\sim 10^8$ double WD systems [@nelemans01a]. About half of these systems have separations that are small enough (orbital periods $<10$ hrs) so that gravitational wave emission will bring them into contact within a Hubble time, making them a major target for the eLISA mission [@elisa13]. Once in contact, in almost all cases the binary system will merge, in the remaining small fraction of cases mass transfer may stabilize the orbital decay and lead to long-lived interacting binaries such as AM CVn systems [@paczynski67; @warner95; @nelemans01b; @nelemans05; @solheim10]. Those systems that merge may have a manifold of interesting possible outcomes. The merger of two He WDs may produce a low-mass He star [@webbink84; @iben86; @saio00; @han02], He-CO mergers may form hydrogen-deficient giant or R CrB stars [@webbink84; @iben96; @clayton07] and if two CO WDs merge, the outcome may be a more massive, possibly hot and high B-field WD [@bergeron91; @barstow95; @segretain97]. A good fraction of the CO-CO merger remnants probably transforms into ONeMg WDs which finally, due to electron captures on Ne and Mg, undergo an accretion-induced collapse (AIC) to a neutron star [@saio85; @nomoto91; @saio98]. Given that the nuclear binding energy that can still be released by burning to iron group elements (1.6 MeV from He, 1.1 MeV from C and 0.8 MeV from O) is large, it is not too surprising that there are also various pathways to thermonuclear explosions. The ignition of helium on the surface of a WD may lead to weak thermonuclear explosions [@bildsten07; @foley09; @perets10], sometimes called “.Ia” supernovae. The modern view is that WDWD mergers might also trigger type Ia supernovae (SN Ia) [@webbink84; @iben84] and, in some cases, even particularly bright “super-Chandrasekhar” explosions, e.g., [@howell06; @hicken07]. SN Ia are important as cosmological distance indicators, as factories for intermediate mass and iron-group nuclei, as cosmic ray accelerators, kinetic energy sources for galaxy evolution or simply in their own right as end points of binary stellar evolution. After having been the second-best option behind the “single degenerate” model for decades, it now seems entirely possible that double degenerate mergers are behind a sizeable fraction of SN Ia. It seems that with the re-discovery of double degenerates as promising type Ia progenitors an interesting time for supernovae research has begun. See [@howel11] and [@maoz14] for two excellent recent reviews on this topic. Below, we will briefly summarize the challenges in a numerical simulation of a WDWD merger (Section \[sec:challenges\_WDWD\]) and then discuss recent results concerning mass transferring systems (Section \[sec:WDWD\_MT\]) and, closely related, to the final merger of a WDWD binary and possibilities to trigger SN Ia (Section \[sec:WDWD\_SNIa\]). We will also briefly discuss dynamical collisions of WDs (Section \[sec:WDWD\_collisions\]). For SPH studies that explore the gravitational wave signatures of WDWD mergers we refer to the literature [@loren05; @dan11; @vandenbroek12]. Note that in this section we explicitly include the constants $G$ and $c$ in the equations to allow for a simple link to the astrophysical literature. ### Challenges {#sec:challenges_WDWD} WDWD merger simulations are challenging for a number of reasons not the least of which are the onset of mass transfer and the self-consistent triggering of thermonuclear explosions. While two white dwarfs revolve around their common centre of mass, gravitational wave emission reduces the separation $a$ of a circular binary orbit at a rate of [@peters63; @peters64] \_[GW]{}= - , where $m_1$ and $m_2$ are the component masses. Although it is gravitational wave emission that drives the binary towards mass transfer/merger in the first place, its dynamical impact at the merger stage is completely negligible since = = 6.6 10\^8 ( )\^[5/2]{} ( ) ( ) ( )\^[1/2]{}. Mass transfer will set in once the size of the Roche lobe of one of the stars has become comparable to the enclosed star. Due to the inverted mass-radius relationship of WDs, it is always the less massive WD (“secondary”) that fills its Roche lobe first. From the Roche lobe size and Kepler’s third law, the average density $\bar{\rho}$ of the donor star can be related to the orbital period [@paczynski71; @frank02] | , where $P_{\rm hr}$ is the orbital period measured in hours. In other words: the shorter the orbital period, the higher the density of the mass donating star. For periods below 1 hr the donor densities exceed those of main sequence stars which signals that a compact star is involved. If one uses the typical dynamical time scale of a WD, $\tau_{\rm dyn}\approx (G \bar{\rho})^{-1/2}$, one finds 10, so that a single orbit would already require $\sim 10\,000 (n_{\rm dyn}/1000)$ numerical time steps, if $n_{\rm dyn}$ denotes the number of numerical time steps per stellar dynamical time. This demonstrates that long-lived mass transfer over tens of orbital periods can become quite computationally expensive and may place limits on the numerical resolution that can be afforded in such a simulation. On the other hand, when numerically resolvable mass transfer sets in, it already has a rate of $$\label{eq:mmin} \dot M_{\rm lim}\sim \frac{1\ {\rm particle \; mass}}{{\rm orbital \; period}} \sim 2 \times 10^{-8} \frac{M_{\odot}}{\rm s}\left(\frac{10^6}{\rm npart}\right) \left(\frac{M}{1\,M_{\odot}}\right)^{3/2} \left(\frac{2\cdot 10^9\ {\rm cm}}{a_0}\right)^{3/2},$$ where ${\rm npart}$ is the total number of SPH particles, ${\rm M_{tot}}$ is the total mass of the binary and $a_{0}$ is the separation between the stars at the onset of mass transfer. This limit due to finite numerical resolution is several orders of magnitude above the Eddington limit of WDs, therefore sub-Eddington accretion rates are hardly ever resolvable within a global, 3D SPH simulation. The transferred matter comes initially from the tenuous WD surface which, in SPH, is the poorest resolved region of the star. Following this matter is also a challenge for Eulerian methods since it needs to be disentangled from the “vacuum” background and, due to the resolution-dependent angular momentum conservation, it is difficult to obtain the correct feedback on the orbital evolution. In other words: the consistent simulation of mass transfer and its feedback on the binary dynamics is a serious challenge for every numerical method. The onset of mass transfer represents a juncture in the life of WDWD binary, since now the stability of mass transfer decides whether the binary can survive or will inevitably merge. The latter depends sensitively on the internal structure of the donor star, the binary mass ratio and the angular momentum transport mechanisms [e.g., @marsh04; @gokhale07]. Due to the inverse mass-radius relationship of WDs, the secondary will expand on mass loss and therefore tendentially speed up the mass loss further. On the other hand, since the mass is transferred to the higher mass object, momentum/centre of mass conservation will tend to widen the orbit and therefore tendentially reduce mass transfer. If the circularization radius of the transferred matter is smaller than the primary radius it will directly impact on the stellar surface and tend to spin up the accreting star. In this way, orbital angular momentum is lost to the spin of the primary which, in turn, decreases the orbital separation and accelerates the mass transfer. If, on the other hand, the circularization radius is larger than the primary radius and a disk can form, angular momentum can, via the large lever arm of the disk, be fed back into the orbital motion and stabilize the binary system [@iben98; @piro11]. To make things even more complicated, if tidal interaction substantially heats up the mass donating star it may impact on its internal structure and therefore change its response to mass loss. To capture these complex angular momentum transfer mechanisms reliably in a simulation requires a very accurate numerical angular momentum conservation. We want to briefly illustrate this point with a small numerical experiment. A $0.3+0.6\,M_{\odot}$ WD binary system is prepared in a Keplerian orbit, so that mass transfer is about to set in. To mimic the effect of numerical angular momentum loss in a controllable way, we add small artificial forces similar to those emerging from gravitational wave emission [@peters63; @peters64; @davies94] and adjust the overall value so that 4% or 0.5% of angular momentum per orbit are lost. These results are compared to a simulation without artificial loss terms where the angular momentum is conserved to better than 0.01% per orbit, see Figure \[fig:conservation\_GWs\]. The effect on the mutual separation $a$ (in $10^9\,{\rm cm}$) is shown in the upper panels and the gravitational wave amplitude, $h_+$ ($r$ is the distance to the observer) as calculated in the quadrupole formalism, are shown the lower panels. Even the moderate loss of 0.5% angular momentum per orbit leads to a quick artificial merger and a mass transfer duration that is reduced by more than a factor of three. These conservation requirements make SPH a natural choice for WDWD merger simulations and it has indeed been the first method used for these type of problems. As outlined above, one of the most exciting possibilities is the triggering of thermonuclear explosions during the interaction of two WDs. Such explosions can either be triggered by a shock wave where the thermonuclear energy generation behind the shock wears down possible dissipative effects or, spontaneously, if the local conditions for burning are favorable enough so that it occurs faster than the star can react by expanding. [@seitenzahl09] have studied detonation conditions in detail via local simulations and found that critical detonation conditions can require that length scales down to centimeters are resolved which is, of course, a serious challenge for global, 3D simulations of objects with radii of $\sim 10^9$ cm. There is also a huge disparity in terms of time scales. Whenever nuclear burning is important for the dynamics of the gas flow, the nuclear time scales are many orders of magnitude shorter than the admissible hydrodynamic time steps. Therefore, nuclear networks are usually implemented via operator splitting methods, see e.g., [@benz89; @rosswog09a; @raskin10; @garcia_senz13]. Because of the exact advection in SPH the post-processing of hydrodynamic trajectories with larger nuclear networks to obtain detailed abundance patterns is straight forward. For burning processes in tenuous surface layers, however, SPH is seriously challenged since here the resolution is poorest. For such problems, hybrid approaches that combine SPH with, say, AMR methods [@guillochon10; @dan15] seem to be the best strategies. ### Dynamics and mass transfer in white dwarf binaries systems {#sec:WDWD_MT} Three-dimensional simulations of WDWD mergers were pioneered by [@benz90b]. Their major motivation was to understand the merger dynamics and the possible role of double degenerate systems as SN Ia progenitors. They used an SPH formulation as described in Section \[sec:Newt\_vanilla\] (“vanilla ice”) together with 7000 SPH particles, an equation of state for a non-degenerate ideal gas with a completely degenerate, fully relativistic electron component and they restricted themselves to the study of a 0.9 - 1.2 system. No attempts were undertaken to include nuclear burning in this study (but see [@benz89]). Each star was relaxed in isolation, see Section \[sec:IC\], and subsequently placed in a circular Keplerian orbit so that the secondary was overfilling its critical lobe by $\sim8\%$. Under these conditions the secondary star was disrupted within slightly more than two orbital periods, forming a three-component system of a rather unperturbed primary, a hot pressure supported spherical envelope and a rotationally supported outer disk. About 0.6 % of a solar mass were able to escape, the remaining $\sim 1.7$ , supported mainly by pressure gradients, showed no sign of collapse. [@rasio95] were more interested in the equilibrium and the (secular, dynamical and mass transfer) stability properties of close binary systems. They studied systems both with stiff ($\Gamma> 5/3$; as models for neutron stars) and soft ($\Gamma= 5/3$) polytropic equations of state, as approximations for the EOS of (not too massive) WDs and low-mass main sequence star binaries. They put particular emphasis on constructing accurate, *synchronized* initial conditions [@rasio94]. These were obtained by relaxing the binary system in a corotating frame where, in equilibrium, all velocities should vanish. The resulting configurations satisfied the virial theorem to an accuracy of about one part in $10^3$. With these initial conditions they found a more gradual increase in the mass transfer rate in comparison to [@benz90b], but nevertheless the binary was disrupted after only a few orbital periods. [@segretain97] focussed on the question whether particularly massive and hot WDs could be the result of binary mergers [@bergeron91]. They applied a simulation technology similar to [@benz90b] and concentrated on a binary system with non-spinning WDs of 0.6 and 0.9 . They showed, for example, that such a merger remnant would need to lose about 90% of its angular momentum in order to reproduce properties of the observed candidate WDs. Although [@rasio95] had already explored the construction of accurate initial conditions, essentially all subsequent simulations [@guerrero04; @loren05; @yoon07a; @loren09; @pakmor10; @pakmor11; @zhu13a] were carried out with rather approximate initial conditions consisting of spherical stars placed in orbits where, according to simple Roche-lobe geometry estimates, mass transfer should set in. [@marsh04] had identified in a detailed orbital stability analysis definitely stable regions (roughly for primary masses substantially larger than the companion mass), definitely unstable (mass ratios between 2/3 and 1) and an intermediate region where the stability of mass transfer is less clear. Motivated by large discrepancies in the mass transfer duration that had been observed between careful grid-based [@motl02; @dsouza06; @motl07] and earlier SPH simulations [@benz90b; @rasio95; @segretain97; @guerrero04; @yoon07a; @pakmor10] [@dan09; @dan11] focussed on the mass transfer in this unclear regime. They very carefully relaxed the binary system in a corotating frame and thereby adiabatically reduced the mutual separation until the first particle climbed up to saddle point $L_1$ in the effective potential, see Figure \[fig:effect\_potential\]. In their study such carefully constructed initial conditions were compared to the previously commonly used approximate initial conditions. Apart from the inaccuracies inherent to the analytical Roche lobe estimates, approximate initial conditions also neglect the tidal deformations of the stars and therefore seriously underestimate the initial separation at the onset of mass transfer. For this reason, such initial conditions have up to 15% too little angular momentum and, as a result, binary systems with inaccurate initial conditions merge too violently on a much too short time scale. As a result, temperatures and densities in the final remnant are over- and the size of tidal tails are underestimated. The carefully prepared binary systems all showed dozens of orbits of numerically resolvable mass transfer. Given that, due to the finite resolution, the mass transfer is already highly super-Eddington when it starts being resolvable all the results on mass transfer duration have to be considered as strict lower limits. One particular example, a 0.2 He-WD and a 0.8 CO-WD, merged with approximate initial conditions within two orbital periods (comparable to earlier SPH results), but only after painfully long 84 orbital periods when the initial conditions were prepared carefully. This particular example also illustrated the suitability of SPH for such investigations: during the orbital evolution, which corresponds to $\approx$ 17000 dynamical time scales, energy and angular momentum were conserved to better than 1%! All of the investigated (according to the Marsh et al. analysis unstable) binary systems merged in the end although only after several dozens of orbital periods. Some systems showed a systematic widening of the orbits after the onset of mass transfer. Although they were still disrupted in the end, this indicated that systems in the parameter space vicinity of this 0.2 - 0.8 system may evolve into short-period AM CVn systems. In two recent studies, [@dan12; @dan14a] systematically explored the parameter space by simulating 225 different binary systems with masses ranging from 0.2 to 1.2 . All of the initial conditions were prepared carefully as [@dan11]. Despite the only moderate resolution (40 K particles) that could be afforded in such a broad study, they found excellent agreement with the orbital evolution predicted by mass transfer stability analysis [@marsh04; @gokhale07]. ### Double white dwarf mergers and possible pathways to thermonuclear supernovae {#sec:WDWD_SNIa} The merger of two white dwarfs, the so-called “double degenerate scenario”, had already been suggested relatively early on [@webbink84; @iben84] as a promising type Ia progenitor channel. It was initially modelled as CO-rich matter being accreted from a thick disk onto a central, cold WD [@nomoto85; @saio85; @saio98; @piersanti03a; @piersanti03b; @saio04]. Since for such thick disks accretion rates close to the Eddington limit ($\dot{M} \sim 10^{-5}$ yr$^{-1}$) are expected, most studies concluded that carbon ignition would start in the envelope of the central WD and, as the burning flame propagates inwards within $\sim$ 5000 years, it would transform the WD from CO into ONeMg [@saio85; @saio98]. When approaching the Chandrasekhar mass, Ne and Mg would undergo electron captures and the final result would be an accretion-induced collapse to a neutron star rather than a SN Ia. Partially based on these studies, the double degenerate model was long regarded as only the second best model that had some good motivation (consistent rates, lack of hydrogen in SN Ia spectra, Chandrasekhar mass as motivation for the uniformity of type Ia properties), but lacked a convincing pathway to an explosion. The Barcelona group were the first to explore the effect of nuclear burning during a WD merger event [@guerrero04]. They implemented the reduced 14-isotope $\alpha$-network of [@benz89] with updated reaction rates into a “vanilla-ice” SPH code with artificial viscosity enhanced by the Balsara factor, see Sections \[sec:Newt\_vanilla\] and \[sec:Newtonian\_shocks\]. They typically used 40 K particles, approximate initial conditions as described above and explored six different combinations of masses/chemical compositions. They found an orbital dynamics similar to [@benz89; @segretain97] and, although in the outer, partially degenerate layers of the central core temperatures around $10^9$ K were encountered, no dynamically important nuclear burning was observed. Whenever it set in, the remnant had time to quench it by expansion, both for the He and CO accreting systems. Therefore they concluded that direct SN Ia explosions were unlikely, but some remnants could evolve into subdwarf B objects as suggested in [@iben90]. [@yoon07a] challenged the “classical picture” of the cold WD accreting from a thick disk as an oversimplification. Instead, they suggested that the subsequent secular evolution of the remnant would be better studied by treating the central object as a differentially rotating CO star with a central, slowly rotating, cold core engulfed by a rapidly rotating hot envelope, which, in turn, is embedded and fed by a centrifugally supported Keplerian accretion disk. The further evolution of such a system is then governed by the thermal cooling of the hot envelope and the redistribution of angular momentum inside of the central remnant and the accretion of the matter from the disk into the envelope. They based their study of the secular remnant evolution on a dynamical merger calculation of two CO WDs with 0.6 and 0.9 . To this end they used an SPH code originally developed for neutron star merger calculations [@rosswog99; @rosswog00; @rosswog02a; @rosswog03a; @rosswog08b] extended by the Helmholtz EOS [@timmes99] and a quasi-equilibrium reduced $\alpha$-network [@hix98]. Particular care was taken to avoid artefacts from the artificial viscosity treatment and time-dependent viscosity parameters [@morris97] and a Balsara-switch [@balsara95], see Section \[sec:Newtonian\_shocks\], were used in the simulation. As suggested by the work of [@segretain97], they assumed non-synchronized stars and started the simulations from approximate initial conditions, see Section \[sec:WDWD\_MT\]. Once a stationary remnant had been formed, the results were mapped into a 1D hydrodynamic stellar evolution code [@yoon04a] and its secular evolution was followed including the effects of rotation and angular momentum transport. They found that the growth of the stellar core is controlled by the neutrino cooling at the interface between the core and the envelope and that carbon ignition could be avoided provided that a) once the merger reaches a quasi-static equilibrium temperatures are below the carbon ignition threshold, b) the angular momentum loss occurs on a time scale longer than the neutrino cooling time scale and c) the mass accretion from the centrifugally supported disk is low enough ($\dot{M} \le 5 \times 10^{-6} - 10^{-5}$ yr$^{-1}$). From such remnants an explosion may be triggered $\sim 10^5$ years after the merger. Such systems, however, may need unrealistically low viscosities. A more recent study [@shen12a] started from two remnants of CO WD mergers [@dan11] and followed their viscous longterm evolution. Their conclusion was more in line with earlier studies: they expected that the long-term result would be ONe or an accretion-induced collapse to a neutron star rather than a SN Ia. In recent years, WD mergers have been extensively explored as possible pathways to SN Ia. Not too surprisingly, a number of pathways have been discovered that very likely lead to a thermonuclear explosion directly prior to or during the merger. Whether these explosions are responsible for (some fraction of) normal SNe Ia or for peculiar subtypes needs to be further explored in future work. Many of the recent studies used a number of different numerical tools to explore various aspects of WDWD mergers. We focus here on those studies where SPH simulations were involved. ### Explosions prior to merger {#explosions-prior-to-merger .unnumbered} [@dan11] had carefully studied the impact of mass transfer on the orbital dynamics. In these SPH simulations the feedback on the orbit is accurately modelled, but due to SPH’s automatic “refinement on density” the properties of the transferred matter are not well resolved. Therefore, a “best-of-both-worlds” approach was followed in [@guillochon10]/[@dan11] where the impact of the mass transfer on the orbital dynamics was simulated with SPH, while recording the orbital evolution and the mass transfer rate. This information was used in a second set of simulations that was performed with the FLASH code [@fryxell00]. This second study focussed on the detailed hydrodynamic interaction of the transferred mass with the accretor star. For He-CO binary systems where helium directly impacts on a primary of a mass $>$ 0.9 , they found that helium surface explosions can be self-consistently triggered via Kelvin–Helmholtz (KH) instabilities. These instabilities occur at the interface between the incoming helium stream and an already formed helium torus around the primary. “Knots” produced by the KH instabilities can lead to local ignition points once the triple-alpha time scale becomes shorter than the local dynamical time scale. The resulting detonations travel around the primary surface and collide on the side opposite to the ignition point. Such helium surface detonations may resemble weak type Ia SNe [@bildsten07; @foley09; @perets10] and they may drive shock waves into the CO core which concentrate in one or more focal points, similar to what was found in the 2D study of [@fink07]. This could possibly lead to an explosion via a “double-detonation” mechanism. In a subsequent large-scale parameter study [@dan12; @dan14a] found, based on a comparison between nuclear burning and hydrodynamical time scales, that a large fraction of the helium-accreting systems do produce explosions early on: all dynamically unstable systems with primary masses $< 1.1$ together with secondary masses $>0.4$ triggered helium-detonations at surface contact. A good fraction of these systems could also produce in addition KH-instability-induced detonations as described in detail in [@guillochon10]. There was no definitive evidence for explosions prior to contact for any of the studied CO-transferring systems. ### Explosions during merger {#explosions-during-merger .unnumbered} [@pakmor10] studied double degenerate mergers, but – contrary to earlier studies– they focussed on very massive WDs with masses close to 0.9 . They used the <span style="font-variant:small-caps;">Gadget</span> code [@springel05a] with some modifications [@pakmor12a], for example, the Timmes EOS [@timmes99] and a 13 isotope network were implemented for their study. In order to facilitate the network implementation, the energy equation (instead of, as usually in <span style="font-variant:small-caps;">Gadget</span>, the entropy equation) was evolved. No efforts were undertaken to reduce the constant, untriggered artificial viscosity. They placed the stars on orbit with the approximate initial conditions described above and found the secondary to be disrupted within two orbital periods. In a second step, several hot ($>2.9 \times 10^9$K) particles were identified and the remnant was artificially ignited in these hot spots. The explosion was followed with a grid-based hydrodynamics code [@fink07; @roepke07] that had been used in earlier SN Ia studies. In a third step, the nucleosynthesis was post-processed and synthetic light curves were calculated [@kromer09]. The explosion resulted in a moderate amount of [$^{56}\mathrm{Ni}$ $\;$]{}(0.1 ), large amounts (1.1 ) of intermediate mass elements and oxygen (0.5 ) and less than 0.1 of unburnt carbon. The kinetic energy of the explosion ($1.3 \times 10^{51}$ erg) was typical for a SN Ia, but the resulting velocities were relatively small, so that the explosion resembled a sub-luminous 1991bg-like supernova. An important condition for reaching ignition is a mass ratio close to unity. Some variation in total mass is expected, but cases with less than 0.9 of a primary mass would struggle to reach the ignition temperatures and – if successful – the lower densities would lead to even lower resulting [$^{56}\mathrm{Ni}$ $\;$]{}masses and therefore lower luminosities. For substantially higher masses, in contrast, the burning would proceed at larger densities and therefore result in much larger amounts of [$^{56}\mathrm{Ni}$]{}. Based on population synthesis models [@ruiter09] they estimated that mergers of this type of system could account for 2–11 % of the observed SN Ia rate. In a follow up study [@pakmor11] the sensitivity of the proposed model to the mass ratio was studied. The authors concluded that binaries with a primary mass near 0.9 ignite a detonation immediately at contact, provided that the mass ratio $q$ exceeds 0.8. Both the abundance tomography and the lower-than-standard velocities provided support for the idea of this type of merger producing sub-luminous, 1991bg-type supernovae. As a variation of the theme, [@pakmor12b] also explored the merger of a higher mass system with 0.9 and 1.1 . Using the same assumption about the triggering of detonations as before, they found a substantially larger mass of [$^{56}\mathrm{Ni}$ $\;$]{}(0.6 ), 0.5 of intermediate mass elements, 0.5 of Oxygen and about 0.15 of unburnt carbon. Due to its higher density, only the primary is able to burn [$^{56}\mathrm{Ni}$ $\;$]{}and therefore the brightness of the SN Ia would be closely related to the primary mass. The secondary is only incompletely burnt and thus provides the bulk of the intermediate mass elements. Overall, the authors concluded that such a merger reproduces the observational properties of normal SN Ia reasonably well. In [@kromer13] the results of a 0.9 and 0.76 CO-CO merger were analyzed and unburned oxygen close to the centre of the ejecta was found which produces narrow emission lines of \[O1\] in the late-time spectrum, similar to what is observed in the sub-luminous SN 2010lp [@leibundgut93]. [@fryer10] applied a sequence of computational tools to study the spectra that can be expected from a supernova triggered by a double-degenerate merger. Motivated by population synthesis calculations, they simulated a CO-CO binary of 0.9 and 1.2 with the SNSPH code [@fryer06]. They assumed that the remnant would explode at the Chandrasekhar mass limit, into a gas cloud consisting of the remaining merger debris. They found a density profile with $\rho \propto r^{-4}$, inserted an explosion [@meakin09] into such a matter distribution and calculated signatures with the radiation-hydrodynamics code <span style="font-variant:small-caps;">Rage</span> [@fryer07; @fryer09]. They found that in such “enshrouded” SNe Ia the debris extends and delays the X-ray flux from a shock breakout and produces a signal that is closer to a SN Ib/c. Also the V-band peak was extended and much broader than in a normal SN Ia with the early spectra being dominated by CO lines only. They concluded that, within their model, a CO-CO merger with a total mass $>1.5$ would not produce spectra and light curves that resemble normal type Ia supernovae. One of the insights gained from the study of [@pakmor10] was that (massive) mergers with similar masses are more likely progenitors of SNe Ia than the mergers with larger mass differences that were studied earlier. This motivated [@zhu13a] to perform a large parameter study where they systematically scanned the parameter space from 0.4-0.4 up to 1.0-1.0 . In their study, they used the <span style="font-variant:small-caps;">Gasoline</span> code [@wadsley04] together with the Helmholtz EOS [@timmes99; @timmes00a], no nuclear reactions were included. All their simulations were performed with non-spinning stars and approximate initial conditions as outlined above. Mergers with “similar” masses produced a well-mixed, hot central core while “dissimilar” masses produced a rather unaffected cold core surrounded by a hot envelope and disk, consistent with earlier studies. They found that the central density ratio of the accreting and donating star of $\rho_a/\rho_d > 0.6$ is a good criterion for those systems that produce hot cores (i.e., to define “similar” masses). [@dan14a] also performed a very broad scan of the parameter space. They studied the temperature distribution inside the remnant for different stellar spins: tidally locked initial conditions produce hot spots (which are the most likely locations for detonations to be initiated) in the outer layers of the core, while irrotational systems produce them deep inside of the core, consistent with the results of [@zhu13a]. Thus, the spin state of the WDs may possibly be decisive for the question where the ignition is triggered which may have its bearings on the resulting supernova. Dan et al. found essentially no chemical mixing between the stars for mass ratios below $q \approx 0.45$, but maximum mixing for a mass ratio of unity. Contrary to [@zhu13a], nowhere complete mixing was found, but this difference can be convincingly attributed to the different stellar spins that were investigated (tidal locking in [@dan14a], no spins in [@zhu13a]). In addition to the helium-accreting systems that likely undergo a detonation for a total mass beyond 1.1 , they also found CO binaries with total masses beyond 2.1 to be prone to a CO explosion. Such systems may be candidates for the so-called super-Chandrasekhar SN Ia explosions. They also discussed the possibility of “hybrid supernovae” where a ONeMg core with a significant helium layer collapses and forms a neutron star. While technically being a (probably weak) core-collapse supernova [@podsiadlowski04; @kitaura06], most of the explosion energy may come from helium burning. Such hybrid supernovae may be candidates for the class of “Ca-rich” SNe Ib [@perets10] as the burning conditions seem to favor the production of intermediate mass elements. [@raskin12a] explored 10 merging WDWD binary systems with total masses between 1.28 and 2.12 with the SNSPH code [@fryer06] coupled to the Helmholtz EOS and a 13-isotope nuclear network. They used a heuristic procedure to construct tidally deformed, synchronized binaries by letting the stars fall towards each other in free fall, and repeatedly set the fluid velocities to zero. They had coated their CO WDs with atmospheres of helium (smaller than 2.5% of the total mass) and found that for all cases where the primary had a mass of 1.06 , the helium detonated, no Carbon detonation was encountered, though. Given the difficulty to identify the central engine of a SNe Ia on purely theoretical grounds, [@raskin13a] explored possible observational signatures stemming from the tidal tails of a WD merger. They followed the ejecta from a SNSPH simulation [@fryer06] with n-body methods and – assuming spherical symmetry – they explored by means of a 1D Lagrangian code how a supernova would interact with such a medium. Provided the time lag between merger and supernova is short enough ($<100$ s), detectable shock emission at radio, optical, and/or X-ray wavelengths is expected. For delay times between $10^8$ s and 100 years one expects broad NaID absorption features, and, since this has not been observed to date, they concluded that if (some) type Ia supernovae are indeed caused by WDWD mergers, the delay times need to be either short ($<100$ s) or rather long ($>100$ years). If the tails can expand and cool for $\sim 10^4$ years, they produce the observable narrow NaID and Ca II K& K lines which are seen in some fraction of type Ia supernovae. An interesting study from a Santa Cruz–Berkeley collaboration [@moll14a; @raskin14a] combined again the strengths of different numerical methods. The merger process was calculated with the SNSPH code [@fryer06], the subsequent explosion with the grid-based code <span style="font-variant:small-caps;">Castro</span> [@almgren10; @zhang11] and synthetic light curves and spectra were calculated with <span style="font-variant:small-caps;">Sedona</span> [@kasen06]. For some cases without immediate explosion the viscous remnant evolution was followed further with ZEUS-MP2 [@hayes06]. The first part of the study [@moll14a] focussed on prompt detonations, while the second part explored the properties of detonations that emerge in later phases, after the secondary has been completely disrupted. For the first part, three mergers (1.20 - 1.06, 1.06 - 1.06 and 0.96 - 0.81 ) were simulated with the SPH code and subsequently mapped into <span style="font-variant:small-caps;">Castro</span>, where soon after simulation start ($<0.1$ s) detonations emerged. They found the best agreement with common SNe Ia (0.58 of [$^{56}\mathrm{Ni}$]{}) for the binary with 0.96 - 0.81 . More massive systems lead to more [$^{56}\mathrm{Ni}$ $\;$]{}and therefore unusually bright SNe Ia. The remnant asymmetry at the moment of detonation leads to large asymmetries in the elemental distributions and therefore to strong viewing angle effects for the resulting supernova. Depending on viewing angle, the peak bolometric luminosity varied by a factor of two and the flux in the ultraviolet even varied by an order of magnitude. All of the three models approximately fulfilled the width-luminosity relation (“brighter means broader”; [@phillips99]). The companion study [@raskin14a] explored cases where the secondary has become completely disrupted before a detonation sets in, so that the primary explodes into a disk-like CO structure. To initiate detonations, the SPH simulations were stopped once a stationary structure had formed, all the material with $\rho>10^6$ was burnt in a separate simulation and, subsequently, the generated energy was deposited back as thermal energy into the merger remnant and the SPH simulation was resumed. As a double-check of the robustness of this approach, two simulations were also mapped into <span style="font-variant:small-caps;">Castro</span> and detonated there as well. Overall there was good agreement both in morphology and the nucleosynthetic yields. The explosion inside the disk produced an hourglass-shaped remnant geometry with strong viewing angle effects. The disk scale height, initially set by the mass ratio and the different merger dynamics and burning processes, turned out to be an important factor for the viewing angle dependence of the later supernova. The other crucial factor was the primary mass that determines the resulting amount of [$^{56}\mathrm{Ni}$]{}. Interestingly, the location of the detonation spot, whether at the surface or in the core of the primary, had a relatively small effect compared to the presence of an accretion disk. While qualitatively in agreement with the width-luminosity relation, the lightcurves lasted much longer than standard SNe Ia. The surrounding CO disk from the secondary remained essentially unburnt, but, impeding the expansion, it lead to relatively small intermediate mass element absorption velocities. The large asymmetries in the abundance distribution could lead to a large overestimate of the involved [$^{56}\mathrm{Ni}$ $\;$]{}masses if spherical symmetry is assumed in interpreting observations. The lightcurves and spectra were peculiar with weak features from intermediate mass elements but relatively strong carbon absorption. The study also explored how longer-term viscous evolution before a detonation sets in would affect the supernova. Longer delay times were found to produce likely larger [$^{56}\mathrm{Ni}$ $\;$]{}masses and more symmetrical remnants. Such systems might be candidates for super-Chandra SNe Ia. ### Simulations of white dwarf–white dwarf collisions {#sec:WDWD_collisions} A physically interesting alternative to gravitational wave-driven binary mergers are dynamical collisions between two WDs as they are expected in globular clusters and galactic cores. They have first been explored, once more, by [@benz89], probably in one of the first hydrodynamic simulations of WDs that included a nuclear reaction network. At that time they found that nuclear burning could, in central collisions, help to undbind a substantial amount of matter, but this amount was found to be irrelevant for the chemical evolution of galaxies. More recently, the topic has been taken up again by [@rosswog09c] and, independently, by [@raskin09]. Both of these studies employed SPH simulations ($2 \times 10^6$ SPH particles in [@rosswog09c], $8 \times 10^5$ particles in [@raskin09]) coupled to small nuclear reactions networks. Both groups concluded that an amount of radioactive [$^{56}\mathrm{Ni}$ $\;$]{}could be produced that is comparable to what is observed in normal type Ia supernovae ($\sim 0.5$ ). In [@rosswog09c] one of the simulations was repeated within the <span style="font-variant:small-caps;">Flash</span> code [@fryxell00] to judge the robustness of the result and given the enormous sensitivity of the nuclear reactions to temperature overall good agreement was found, see Figure \[fig:WDWD\_collision\]. Rosswog et al. also post-processed the nucleosynthesis with larger nuclear networks and calculated synthetic light curves and spectra with the <span style="font-variant:small-caps;">Sedona</span> code [@kasen06]. Interestingly, the resulting light-curves and spectra [@rosswog09c] look like normal type Ia supernova, even the width-luminosity relationship [@phillips99] is fulfilled to good accuracy. These results have been confirmed recently in further studies [@raskin10; @hawley12; @kushnir13; @garcia_senz13]. This result is interesting for a number of reasons. First, the detonation mechanism is parameter-free and extremely robust: the free-fall velocity between WDs naturally produces relative velocities in excess of the WD sound speeds, therefore strong shocks are inevitable. Moreover, the most likely involved WD masses are near the peak of the mass distribution, $\sim 0.6$ , or, due to mass segregation effects, possibly slightly larger, but well below the Chandrasekhar mass. This has the benefit that the nuclear burning occurs at moderate densities ($\rho \sim 10^7$ ) and thus produces naturally the observed mix of $\sim 0.5$ [$^{56}\mathrm{Ni}$ $\;$]{}and intermediate-mass elements, without any fine-tuning such as the deflagration-detonation transition that is required in the single-degenerate scenario [@hillebrandt00]. However, based on simple order of magnitude estimates, the original studies [@raskin09; @rosswog09c] concluded that, while being very interesting explosions, the rates would likely be too low to make a substantial contribution to the observed supernova sample. More recently, however, there have been claims [@thompson11; @kushnir13] that the Kozai–Lidov mechanism in triple stellar systems may substantially boost the rates of WDWD collisions so that they could constitute a sizeable fraction of the SN Ia rate. Contrary to these claims, a recent study by [@hamers13] finds that the contribution from the triple-induced channels to SN Ia is small. Here further studies are needed to quantify how relevant collisions really are for explaining normal SNe Ia. ### Summary double white dwarf encounters SPH has very often been used to model mergers of, and later on also collisions between, two WDs. This is mainly since SPH is not restricted by any predefined geometry and has excellent conservation properties. As illustrated in the numerical experiment shown in Figure \[fig:conservation\_GWs\], even small (artificial) losses of angular momentum can lead to very large errors in the prediction of the mass transfer duration and the gravitational wave signal. SPH’s tendency to “follow the density” makes it ideal to predict, for example, the gravitational wave signatures of WD mergers. But it is exactly this tendency which makes it very difficult for SPH to study thermonuclear ignition processes in low-density regions that are very important, for example, for the double-detonation scenario. This suggests to apply in such cases a combination of numerical tools: SPH for bulk motion and orbital dynamics and Eulerian (Adaptive Mesh) hydrodynamics for low-density regions that need high resolution. As outlined above, there have recently been a number of successful studies that have followed such strategies. SPH simulations have played a pivotal role in “re-discovering” the importance of white dwarf mergers (and possibly, to some extent, collisions) as progenitor systems of SNe Ia. In the last few years, a number of new possible pathways to thermonuclear explosions prior to or during a WD merger have been discovered. There is, however, not yet a clear consensus whether they produce just “peculiar” SN Ia-like events or whether they may be even responsible for the bulk of “standard” SN Ia. Here, a lot of progress can be expected within the next few years, both from the modelling and the observational side. Encounters between neutron stars and black holes {#sec:appl_NSNS_NSBH} ------------------------------------------------ ### Relevance The relevance of compact binary systems consisting of two neutron stars (NS) or a neutron star and a stellar mass black hole (BH) is hard to overrate. The first observed double neutron star system, PSR 1913+16, proved, although indirectly, the existence of gravitational waves (GWs): the orbit decays in excellent agreement with the predictions from general relativity (GR) [@taylor89; @weisberg10]. By now there are 10 binary systems that are thought to consist of two neutron stars [@lorimer08], but to date no NSBH system has been identified, although both types of systems should form in similar evolutionary processes. This is usually attributed to possibly smaller numbers in comparison to NS-NS systems [@belczynski09] and to a lower detection probability in current surveys. Once formed, gravitational wave emission drives compact binary systems towards coalescence on a time scale approximately given by [@lorimer08] \_= 9.83 10\^6 ()\^[8/3]{} ( )\^[-2/3]{} ( )\^[-1]{} (1-e\^2)\^[7/2]{}, where $P_{\rm orb}$ is the current orbital period, $M$ the total and $\mu$ the reduced mass and $e$ the eccentricity. This implies that the initial orbital period must be $\leq 1$ day for a coalescence to occur within a Hubble time. As the inspiral time depends sensitively on the orbital period and eccentricity, which are set by individual evolutionary histories, one expects a large spread in inspiralling times. Near its innermost stable circular orbit (ISCO) a compact binary system emits gravitational waves at a frequency and amplitude of f && 594 ( )\^[3/2]{} ( )\[eq:GW\_frequ\]\ h && 3.6 10\^[-22]{} ( ) ( ) ( ) ( ),\[eq:GW\_ampl\] where $a$ is the binary separation, $r$ the distance to the GW source and the scalings are oriented at a NSBH system. Thus the emission from the late inspiral stages will sweep through the frequency bands of the advanced detector facilities from $\sim 10$ to $\sim 3000$ Hz [@ligo; @virgo] making compact binary mergers (CBMs) the prime targets for the ground-based gravitational wave detections. The estimated coalescence rates are rather uncertain, though, they range from 1 - 1000 Myr$^{-1}$ MWEG$^{-1}$ for NSNS binaries and from 0.05 - 100 Myr$^{-1}$ MWEG$^{-1}$ for NSBH binaries [@abadie10], where MWEG stands for “Milky Way Equivalent Galaxy”. In addition, CBMs have long been suspected to be the engines of short Gamma-ray Bursts (sGRBs) [@paczynski86; @goodman86; @eichler89; @narayan92]. Although already their projected distribution on the sky and their fluence distribution pointed to a cosmological source [@goodman86; @paczynski86; @schmidt88; @meegan92; @piran92; @schmidt01; @guetta05], their cosmological origin was only firmly established by the first afterglow observations for short bursts in 2005 [@hjorth05; @bloom06]. This established the scale for both distance and energy and proved that sGRBs occur in both early- and late-type galaxies. CBMs are natural candidates for sGRBs since accreting compact objects are very efficient converters of gravitational energy into electromagnetic radiation, they occur at rates that are consistent with those of sGRBs [@guetta05; @nakar06; @guetta06] and CBMs are expected to occur in both early and late-type galaxies. Kicks imparted at birth provide a natural explanation for the observed projected offsets of $\sim$ 5 kpc from their host galaxy [@fong13], and, with dynamical time scales of $\sim1$ ms (either orbital at the ISCO or the neutron star oscillation time scales), they naturally provide the observed short-time fluctuations. Moreover, for cases where an accretion torus forms, the expected viscous lifetime is roughly comparable with a typical sGRB duration ($\sim 0.2$ s). While this picture is certainly not without open questions, see [@piran04; @lee07; @nakar07; @gehrels09; @berger11; @berger14a] for recent reviews, it has survived the confrontation with three decades of observational results and – while competitors have emerged – it is still the most commonly accepted model for the engine of short GRBs. [@lattimer74; @lattimer76] and [@lattimer77] suggested that the decompression of initially cold neutron star matter could lead to rapid neutron capture, or “r-process”, nucleosynthesis so that CBMs may actually also be an important source of heavy elements. Although discussed convincingly in a number of subsequent publications [@symbalisty82; @eichler89], this idea kept the status of an exotic alternative to the prevailing idea that the heaviest elements are formed in core-collapse supernovae, see [@arcones13a] for a recent review. Early SPH simulations of NSNS mergers [@rosswog98c; @rosswog99] showed that of order $\sim$ 1% of neutron-rich material is ejected per merger event, enough to be a substantial or even the major source of r-process. A nucleosynthesis post-processing of these SPH results [@freiburghaus99b] confirmed that this material is indeed a natural candidate for the robust, heavy r-process [@sneden08]. Initially it was doubted [@qian00; @argast04] that CMBs as main r-process source are consistent with galactic chemical evolution, but a recent study based on a detailed chemical evolution model [@matteucci14] finds room for a substantial contribution of neutron star mergers and recent hydrodynamic galaxy formation studies [@shen14a; @vandevoort14a] even come to the conclusion that neutron star mergers are consistent with being the dominant r-process source in the Universe. Moreover, state-of-the-art supernova models seem unable to provide the conditions that are needed to produce heavy elements with nucleon numbers in excess of $A=90$ [@arcones07; @fischer10; @huedepohl10; @roberts10]. On the other hand, essentially all recent studies agree that CBMs eject enough mass to be at least a major r-process source [@oechslin07a; @bauswein13a; @rosswog14a; @hotokezaka13a; @kyutoku13] and that the resulting abundance pattern beyond $A\approx 130$ resembles the solar-system distribution [@metzger10b; @roberts11; @korobkin12a; @goriely11a; @goriely11b; @eichler15]. Recent studies [@wanajo14; @just14] suggest that compact binary mergers with their different ejection channels for neutron-rich matter could actually even be responsible for the whole range of r-process. In June 2013 the SWIFT satellite detected a relatively nearby ($z=0.356$) sGRB, GRB 130603B, [@melandri13] for which the HST [@tanvir13a; @berger13a] detected 9 days after the burst a nIR point source with properties close to what had been predicted [@kasen13a; @barnes13a; @grossman14a; @tanaka13a; @tanaka14a] by models for “macro-” or “kilonovae” [@li98; @kulkarni05; @rosswog05a; @metzger10a; @metzger10b; @roberts11], radioactively powered transients from the decay of freshly produced r-process elements. If this interpretation is correct, GRB 130603B would provide the first observational confirmation of the long-suspected link between CBMs, nucleosynthesis and Gamma-Ray Bursts. ### Differences between double neutron and neutron star black hole mergers Both NSNS and NSBH systems share the same three stages of the merger: a) the *secular inspiral* where the mutual separation is much larger than the object radii and the orbital evolution can be very accurately described by Post-Newtonian methods [@blanchet06], b) the *merger phase* where relativistic hydrodynamics is important and c) the subsequent *ringdown phase*. Although similar in their formation paths and in their relevance, there are a number of differences between NSNS and NSBH systems. For example, when the surfaces of two neutron stars come into contact the neutron star matter can heat up and radiate neutrinos. Closely related, if matter from the interaction region is dynamically ejected, it may – due to the large temperatures – have the chance to change its electron fraction due to weak interactions. Such material may have a different nucleosynthetic signature than the matter that is ejected during a NSBH merger. Moreover, the interface between two neutron stars is prone to hydrodynamic instabilities that can amplify existing neutron star magnetic fields [@price06; @anderson08b; @rezzolla11; @zrake13]. The arguably largest differences between the two types of mergers, however, are the total binary mass and its mass ratio $q$. For neutron stars, the deviation from unity $|1-q|$ is small (for masses that are known to better than 0.01 , J1807-2500B has the largest deviation with a mass ratio of $q=0.88$ [@lattimer12a]) while for NSBH systems a broad range of total masses is expected. Since the merger dynamics is very sensitive to the mass ratio, a much larger diversity is expected in the dynamical behavior of NSBH systems. The larger bh mass has also as a consequence that the plunging phase of the orbit sets in at larger separations and therefore lower GW frequencies, see Eq. (\[eq:GW\_frequ\]). Moreover, for black holes the dimensionless spin parameter can be close to unity, $a_{\mathrm{BH}}= c J/G M^2 \approx 1$, while for neutron stars it is restricted by the mass shedding limit to somewhat lower values $a_{\mathrm{NS}} < 0.7$ [@lo11], therefore spin-orbit coupling effects could potentially be larger for NSBH systems and lead to observable effects e.g., [@buonanno03; @grandclement04]. ### Challenges {#challenges} Compact binary mergers are challenging to model since in principle each of the fundamental interactions enters: the strong (equation of state), weak (composition, neutrino emission, nucleosynthesis), electromagnetic (transients, neutron star crust) and of course (strong) gravity. Huge progress has been achieved during the last decade, but so far none of the approaches “has it all” and, depending on the focus of the study, different compromises have to be accepted. The compactness parameters, $\mathcal{C}\equiv GM/R c^2$, of $\sim 0.2$ for neutron stars and $\sim 0.5$ for black holes suggest a general-relativistic treatment. Contrary to the WDWD case discussed in Section \[sec:appl\_WDWD\] now the gravitational and orbital time scales can become comparable = = 5.0 ( )\^[5/2]{} ( ) ( ) ( )\^[1/2]{}, \[eq:tau\_GW\_tau\_orb\] so that backreaction from the gravitational wave emission on the dynamical evolution can no more be safely neglected. And, unless one considers the case where the black hole completely dominates the spacetime geometry, it is difficult to make admissible approximations to dynamical GR gravity. Of comparable importance is the neutron star equation of state (EOS). Together with gravity it determines the compactness of the neutron stars which in turn impacts on peak GW frequencies. It also influences the torus masses, the amount of ejected matter, and, since it sets the $\beta$-equilibrium value of the electron fraction $Y_e$ in neutron star matter, also the resulting nucleosynthesis and the possibly emerging electromagnetic transients. Unfortunately, the EOS is not well known beyond $\sim$ 3 times nuclear density. On the other hand, since the EOS seriously impacts on a number of observable properties from a CBM, one can be optimistic and hope to conversely constrain the high-density EOS via astronomical observations. Since the bulk of a CBM remnant consists of high density matter ($\rho>10^{10}$), photons are very efficiently trapped and the only cooling agents are neutrinos. Moreover, with temperatures of order MeV, weak interactions become so fast that they change the electron fractions $Y_e$ substantially on a dynamical time scale ($\sim$ 1 ms). While such effects can be safely ignored when gravitational wave emission is the main focus, these processes are crucial for the neutrino signature, for the “engine physics” of GRBs (e.g., via $\nu\bar{\nu}$ annihilation), and for nucleosynthesis, since the nuclear reactions are very sensitive to the neutron-to-proton ratio which is set by the weak interactions. Fortunately, the treatment of neutrino interactions is not as delicate as in the core-collapse supernova case where changes on the percent level can decide between a successful and a failed explosion [@janka07]. Thus, depending on the exact focus, leakage schemes or simple transport approximations may be admissible in the case of compact binary mergers. But also approximate treatments face the challenge that the remnant is intrinsically three-dimensional, that the optical depths change from $\tau \sim 10^4$ in the hypermassive neutron star (e.g., [@rosswog03a]), to essentially zero in the outer regions of the disk and that the neutrino-nucleon interactions are highly energy-dependent. Neutron stars are naturally endowed with strong magnetic fields and a compact binary merger offers a wealth of possibilities to amplify them. They may be decisive for the fundamental mechanism to produce a GRB in the first place, but they may also determine – via transport of angular momentum – when the central object produced in a NSNS merger collapses into a black hole or how accretion disks evolve further under the transport mediated via the magneto-rotational instability (MRI) [@balbus98]. In addition to these challenges from different physical processes, there are also purely numerical challenges which are, however, closely connected to the physical ones. For example, the length scales that need to be resolved to follow the growth of the MRI can be minute in comparison to the natural length scales of the problem. Or, another example, the viscous dissipation time scale of an accretion disk, \_[visc]{} \~0.3 () ( )\^[3/2]{} ()\^[1/2]{} ( )\^2 \[eq:tau\_visc\] with $\alpha$ being the Shakura–Sunyaev viscosity parameterization [@shakura73], $M$ the central mass, $r$ a typical disk radius and $H$ the disk scale height, may be challengingly long in comparison to the hydrodynamic time step that is allowed by the CFL stability criterion [@press92], t &lt; 10\^[-5]{} () (). \[eq:CFL\] ### The current status of SPH- vs grid-based simulations {#sec:SPH_vs_Eulerian_NSNS_NSBH} A lot of progress has been achieved in recent years in simulations of CBMs. This includes many different microphysical aspects as well as the dynamic solution of the Einstein equations. The main focus of this review are SPH methods and therefore we will restrict the detailed discussion to work that makes use of SPH. Nevertheless, it is worth briefly comparing the current status of SPH-based simulations with those that have been obtained with grid-based methods. As will be explained in more detail below, SPH had from the beginning a very good track record with respect to the implementation of various microphysics ingredients. On the relativistic gravity side, however, it is lagging behind in terms of implementations that dynamically solve the Einstein equations, a task that has been achieved with Eulerian methods already more than a decade ago [@shibata99; @shibata00]. In SPH, apart from Newtonian gravity, Post-Newtonian and Conformal Flatness Approaches (CFA) exist, but up to now no coupling between SPH-hydrodynamics and a dynamic spacetime solver has been achieved. Naturally, this has implications for the types of problems that have been addressed and there are interesting questions related to nsns and nsbh mergers that have so far not yet (or only approximately) been tackled with SPH approaches. One example with far-reaching astrophysical consequences is the collapse of a hypermassive neutron star (HMNS) that temporarily forms after a binary neutron star merger. Observations now indicate a lower limit on the maximum neutron star mass of around 2.0 (1.97 $\pm$ 0.04 for PSR J1614+2230, see [@demorest10] and 2.01 $\pm$ 0.04 for PSR J0348+0432, see [@antoniadis13]) and therefore it is very likely that the hot and rapidly differentially rotating central remnant of a nsns merger is at least temporarily stabilized against a gravitational collapse to a black hole, see e.g., [@hotokezaka13b]. It appears actually entirely plausible that the low-mass end of the nsns distribution may actually leave behind a massive but stable neutron star rather than a black hole. In SPH, the question when a collapse sets in, can so far only be addressed within the Conformal Flatness Approximation (CFA), see below. Being exact for spherical symmetry, the CFA should be fairly accurate in describing the collapse itself. It is, however, less clear how accurate the CFA is during the last inspiral stages where the deviations from spherical symmetry are substantial. Therefore quantitative statements about the HMNS lifetimes need to be interpreted with care. On the other hand, differential rotation has a major impact in the stabilisation and therefore hydrodynamic resolution is also crucial for the question of the collapse time. Here, SPH with its natural tendency to refine on density should perform very well once coupled to a dynamic spacetime solver. Another question of high astrophysical significance is the amount of matter that becomes ejected into space during a nsns or nsbh merger. It is likely one of the major sources of r-process in the cosmos and thought to cause electromagnetic transients similar to the recently observed “macronova” event in the aftermath of GRB 130603B [@tanvir13a; @berger13b]. In terms of mass ejection, one could expect large differences between the results of fully relativistic, grid-based hydrodynamics results and Newtonian or approximate GR SPH approaches. The expected differences are twofold. On the one hand, Newtonian/approximate GR treatments may yield stars of a different compactness which in turn would influence the dynamics and torques and therefore the ejecta amount. But apart from gravity, SPH has a clear edge in dealing with ejecta: mass conservation is exact, advection is exact (i.e., a composition only changes due to nuclear reactions but not due to numerical effects), angular momentum conservation is exact and vacuum *really* corresponds to the absence of matter. Eulerian schemes usually face here the challenges that conservation of mass, angular momentum and the accuracy of advection are resolution-dependent and that vacuum most often is modelled as background fluid of lower density. Given this rather long list of challenges, it is actually encouraging that the results of different groups with very different methods agree reasonably well these days. For NSNS mergers Newtonian SPH simulations [@rosswog13b] find a range from $8 \times 10^{-3} - 4 \times 10^{-2}$ , approximate GR SPH calculations [@bauswein13a] find a range from $10^{-3} - 2 \times 10^{-2}$ and full GR calculations (Hotokezaka et al. 2013) find $10^{-4} - 10^{-2}$ . Even the results from Newtonian nsbh calculations agree quite well with the GR results (compare Table 1 in [@rosswog13b] and the results of [@kyutoku13]). Another advantage of SPH is that one can also decide to just focus on the ejected matter. For example, a recent SPH-based study [@rosswog14a] has followed the evolution of the dynamic merger ejecta for as long as 100 years, while Eulerian methods are usually restricted to very few tens of milliseconds. During this expansion the density was followed from supra-nuclear densities ($> 2 \times 10^{14}$ g/ccm) down to values that are below the interstellar matter density ($< 10^{-25}$ g/ccm). For many of the topics that will be addressed below, there have been parallel efforts on the Eulerian side and within the scope of this review we will not be able to do justice to all these parallel developments. As a starting point, we want to point to a number of excellent textbooks [@alcubierre08; @baumgarte10; @rezzolla13a] that deal with relativistic (mostly Eulerian) fluid dynamics and to various recent review articles [@duez10a; @shibata11; @faber12; @pfeiffer12; @lehner14a]. ### Neutron star –neutron star mergers {#sec:appl_NSNS} ### Early Newtonian calculations with polytropic equation of state {#early-newtonian-calculations-with-polytropic-equation-of-state .unnumbered} The earliest NSNS merger calculations [@rasio92; @davies94; @zhuge94; @rasio94; @rasio95; @zhuge96] were performed with Newtonian gravity and a polytropic equation of state, sometimes a simple gravitational wave backreaction force was added. While these initial studies were, of course, rather simple, they set the stage for future efforts and settled questions about the qualitative merger dynamics and some of the emerging phenomena. For example, they established the emergence of a Kelvin–Helmholtz unstable vortex sheet at the interface between the two stars, which, due to the larger shear, is more pronounced for initially non-rotating stars. Moreover, they confirmed the expectation that a relatively baryon-free funnel would form along the binary rotation axis [@davies94] (although this conclusion may need to be revisited in the light of emerging, neutrino-driven winds, see e.g., [@dessart09; @perego14b; @martin15]). These early simulations also established the basic morphological differences between tidally locked and irrotational binaries and between binaries of different mass ratios. In addition, these studies also drove technical developments that became very useful in later studies such as the relaxation techniques to construct synchronized binary systems [@rasio94] or the procedures to analyze the GW energy spectrum in the frequency band [@zhuge94; @zhuge96]. See also [@rasio99] for a review on earlier research. ### Studies with focus on microphysics {#studies-with-focus-on-microphysics .unnumbered} Studies with a focus on microphysics (in a Newtonian framework) were pioneered by [@ruffert96] who implemented a nuclear equation of state and a neutrino-leakage scheme into their Eulerian (PPM) hydrodynamics code. In SPH, the effects of a nuclear equation of state (EOS) were first explored in [@rosswog99]. The authors implemented the Lattimer–Swesty EOS [@lattimer91] and neutrino cooling in the simple free-streaming limit to bracket the effects that neutrino emission may possibly have. To avoid artefacts from excessive artificial dissipation, they also included the time-dependent viscosity parameters suggested by [@morris97], see Section \[sec:Newtonian\_shocks\]. They found torus masses between 0.1 and 0.3 , and, maybe most importantly, that between $4 \times 10^{-3}$ and $4 \times 10^{-2}$ of neutron star matter becomes ejected. A companion paper [@freiburghaus99b] post-processed trajectories from this study and found that all the matter undergoes r-process and yields an abundance pattern close to the one observed in the solar system, provided that the initial electron fraction is $Y_e\approx 0.1$. A subsequent study [@rosswog00] explored the effects of different initial stellar spins and mass ratios $q\neq 1$ on the ejecta masses. The simulation ingredients were further refined in [@rosswog02a; @rosswog03a; @rosswog03c]. Here the [@shen98a; @shen98b] EOS, extended down to very low densities, was used and a detailed multi-flavor neutrino leakage scheme was developed [@rosswog03a] that takes particular care to avoid using average neutrino energies. These studies were also performed at a substantially higher resolution (up to $10^6$ particles) than previous SPH studies of the topic. The typical neutrino luminosities turned out to be $\sim 2 \times 10^{53}$ erg/s with typical average neutrino energies of 8/15/20 MeV for $\nu_e, \bar{\nu}_e$ and the heavy lepton neutrinos. Since GRBs were a major focus of the studies, neutrino annihilation was calculated in a post-processing step and, barring possible complications from baryonic pollution, it was concluded that $\nu \bar{\nu}$ annihilation should lead to relativistic ouflows and could produce moderately energetic sGRBs. Simple estimates indicated, however, that strong neutrino-driven winds are likely to occur, that could, depending on the wind geometry, pose a possible threat for the emergence of ultra-relativistic outflow/a sGRB. A more recent, 2D study of the neutrino emission from a merger remnant [@dessart09] found indeed strong baryonic winds with mass loss rates $\dot{M} \sim 10^{-3}$ /s emerging along the binary rotation axis. Recently, studies of neutrino-driven winds have been extended to 3D [@perego14b] and the properties of the blown-off material have been studied in detail. This complex of topics, $\nu-$driven winds, baryonic pollution, collapse of the central merger remnant will for sure receive more attention in the future. Based on simple arguments, it was also argued that any initial seed magnetic fields should be amplified to values near equipartition, making magnetically launched/aided outflow likely. Subsequent studies [@price06; @anderson08b; @rezzolla11; @zrake13] found indeed a strong amplification of initial seed magnetic fields. In a recent set of studies the neutron star mass parameter space was scanned by exploring systematically mass combinations from 1.0 to 2.0 [@korobkin12a; @rosswog13a; @rosswog13b]. The main focus here was the dynamically ejected mass and its possible observational signatures. One interesting result [@korobkin12a] was that the nucleosynthetic abundance pattern is essentially identical for the dynamic ejecta of all mass combinations and even NSBH systems yield practically an identical pattern. While extremely robust to a variation of the astrophysical parameters, the pattern showed some sensitivity to the involved nuclear physics, for example to a change of the mass formula or the distribution of the fission fragments. The authors concluded that the dynamic ejecta of neutron star mergers are excellent candidates for the source of the heavy, so-called “strong r-process” that is observed in a variety of metal-poor stars and shows each time the same relative abundance pattern for nuclei beyond barium [@sneden08]. Based on these results, predictions were made for the resulting “macronovae” (or sometimes called “kilonovae”) [@li98; @kulkarni05; @rosswog05a; @metzger10a; @metzger10b; @roberts11]. The first set of models assumed spherical symmetry [@piran13a; @rosswog13a], but subsequent studies [@grossman14a] were based on the 3D remnant structure obtained by hydrodynamic simulations of the expanding ejecta. This study included the nuclear energy release during the hydrodynamic evolution [@rosswog14a]. The study substantially profited from SPH’s geometric flexibility and its treatment of vacuum as just empty (i.e., SPH particle-free) space. The ejecta expansion was followed for as many as 40 orders of magnitude in density, from nuclear matter down to the densities of interstellar matter. Since from the latter calculations the 3D remnant structure was known, also viewing angle effects for macronovae could be explored [@grossman14a]. Accounting for the very large opacities of the r-process ejecta [@barnes13a; @kasen13a], [@grossman14a] predicted that the resulting macronova should peak, depending on the binary system, between 3 and 5 days after the merger in the nIR, roughly consistent with what has been observed in the aftermath of GRB 130603B [@tanvir13a; @berger13b]. ### Studies with approximate GR gravity {#studies-with-approximate-gr-gravity .unnumbered} A natural next step beyond Newtonian gravity is the application of Post-Newtonian expansions. [@blanchet90] developed an approximate formalism for moderately relativistic, self-gravitating fluids which allows to write all the equations in a quasi-Newtonian form and casts all relativistic non-localities in terms of Poisson equations with compactly supported sources. The 1PN equations require the solution of eight Poisson equations and to account for the lowest order radiation reaction terms requires the solution of yet another Poisson equation. While – with nine Poisson equations – computationally already quite heavy, the efforts to implement the scheme into SPH by two groups [@faber00; @ayal01; @faber01; @faber02b] turned out to be not particularly useful, mainly since for realistic neutron stars with compactness $\mathcal{C}\approx 0.17$ the corrective 1PN terms are comparable to the Newtonian ones, which can lead to instabilities. As a result, one of the groups [@ayal01] decided to study “neutron stars” of small compactness ($M < 1$ , $R\approx 30$ km), while the other [@faber00; @faber01; @faber02b] artificially downsized the 1PN effects by choosing a different speed of light for the corresponding terms. While both approaches represented admissible first steps, the corresponding results are astrophysically difficult to interpret. A second, more successful approach, was the resort to the so-called conformal flatness approximation (CFA) [@isenberg08; @wilson95a; @wilson96; @mathews97; @mathews98]. Here the basic assumption is that the spatial part of the metric is conformally flat, i.e., it can be written as a multiple (the prefactor depends on space and time and absorbs the overall scale of the metric) of the Kronecker symbol $\gamma_{ij}= \Psi^4 \delta_{ij}$, and that it remains so during the subsequent evolution. The latter, however, is an assumption and by no means guaranteed. Physically this corresponds to gravitational wave-free space time. Consequently, the inspiral of a binary system has to be achieved by adding an ad hoc radiation reaction force. The CFA also cannot handle frame dragging effects. On the positive side, for spherically symmetric space times the CFA coincides exactly with GR and for small deviation from spherical symmetry, say for rapidly rotating neutron stars, it has been shown [@cook96] to deliver very accurate results. For more general cases such as a binary merger, the accuracy is difficult to assess. Nevertheless, given how complicated the overall problem is, the CFA is certainly a very useful approximation to full GR, in particular since it is computationally much more efficient than solving Einstein’s field equations. The CFA was implemented into SPH by [@oechslin02] and slightly later by [@faber04]. The major difference between the two approaches was that Oechslin et al. solved the set of six coupled, non-linear elliptic field equations by means of a multi-grid solver [@press92], while Faber et al. used spectral methods from the <span style="font-variant:small-caps;">Lorene</span> library on two spherically symmetric grids around the stars. Both studies used polytropic equations of state ([@oechslin02] used $\Gamma= 2.0, 2.6$ and 3.0; [@faber04] used $\Gamma= 2.0$) and approximative radiation reaction terms based on the Burke–Thorne potential [@burke71; @thorne69b]. Oechslin et al. used a combination of a bulk and a von-Neumann–Richtmyer artificial viscosity steered similarly as in the [@morris97] approach, while Faber et al.argued that shocks would not be important and artificial dissipation would not be needed. In a subsequent study, [@oechslin04] explored how the presence of quark matter in neutron stars would impact on a NSNS merger and its gravitational wave signal. They combined a relativistic mean field model (above $\rho = 2 \times 10^{14}$ ) with a stiff polytrope as a model for the hadronic EOS and added in an MIT bag model so that quark matter would appear at $5 \times 10^{14}$ and would completely dominate the EOS for high densities ($> 10^{15}$ ). While the impact on the GW frequencies at the ISCO remained moderate ($<10\%$), the post-merger GW signal was substantially influenced in those cases where the central object did not collapse immediately into a BH. In a subsequent study, [@oechslin06] implemented the [@shen98a; @shen98b] EOS and a range of NS mass ratios was explored, mainly with respect to the question how large resulting torus masses would be and whether such merger remnants could likely power bursts similar to GRBs 050509b, 050709, 050724, 050813. The found range of disk masses from 1–9% of the baryonic mass of the NSNS binary was considered promising and broadly consistent with CBM being the central engines of sGRB. In [@oechslin07a] the same group also explored the Lattimer–Swesty EOS [@lattimer91], the cold EOS of [@akmal98] and ideal gas equations of state with parameters fitted to nuclear EOSs. The merger outcome was rather sensitive to the nuclear matter EOS: the remnant collapsed either immediately or very soon after merger for the soft Lattimer–Swesty EOS and for all other cases it did not show signs of collapse for tens of dynamical time scales. Both ejecta and disk masses were found to increase with an increasing deviation of the mass ratio from unity. The ejecta masses were in a range between $10^{-3}$ and $10^{-2}$ , comparable, but slightly lower than the earlier, Newtonian estimates [@rosswog99]. In terms of their GW signature, it turned out that the peak in the GW wave energy spectrum that is related to the formation of the hypermassive merger remnant has a frequency that is sensitive to the nuclear EOS [@oechslin07b]. In comparison, the mass ratio and neutron star spin only had a weak impact. This line of work was subsequently continued by Bauswein et al. in a series of papers [@bauswein09; @bauswein10a; @bauswein10b; @bauswein12a; @bauswein12b; @bauswein13a; @bauswein13b]. In their first study, they explored the merger of two strange stars [@bauswein09; @bauswein10a]. If strange quark matter is really the ground state of matter as hypothesized [@bodmer71; @witten84], compact stars made of strange quark matter might exist. Such stars would differ from neutron stars in the sense that they are self-bound, they do not have an overall inverse mass-radius relationship and they can be more compact. Therefore the gravitational torques close to merger are different and it is more difficult to tidally tear apart a strange star. In their study [@bauswein10a] the quark matter EOS was modelled via the MIT bag model [@chodos74; @farhi84] for two different bag constants (60 and 80 MeV/fm$^3$) and a large number of binary systems (each time 130K SPH particles) was explored. The coalescence of two strange stars is indeed morphologically different from a neutron star merger: the result is a differentially rotating hypermassive object with sharp boundary layers surrounded by a thin and clumpy strange matter disk, see Figure \[fig:NSNS\_strange\] for an example of a strange star merger with 1.2 and 1.35 [@bauswein10a]. Moreover, due to the greater compactness, the peak GW frequencies were larger during both inspiral and the subsequent ringdown phase. If the merger of two strange stars would eject matter in the form of “strangelets” these should contribute to the cosmic ray flux. The ejected mass and therefore the contribution to the cosmic ray flux strongly depends on the chosen bag constant and for large values no mass loss could be resolved. For such values neutron stars and strange stars could coexist, since neutron stars would not be converted into strange stars by capturing cosmic strangelets. In this and another study [@bauswein10b] thermal effects (and their consistent treatment) were shown to have a substantial impact on the remnant structure. In subsequent work a very large number of microphysical EOSs was explored [@bauswein12a; @bauswein12b; @bauswein13a; @bauswein13b]. Here the authors systematically explored which imprint the nuclear EOS would have on the GW signal. They found that the peak frequency of the post-merger signal correlates well with the radii of the non-rotating neutron stars [@bauswein12a; @bauswein12b] and concluded that a GW detection would allow to constrain the ns radius within a few hundred meters. In a follow-up study [@bauswein13b] explored the threshold mass beyond which a prompt collapse to a black hole occurs. The study also showed that the ratio between this threshold mass and the maximum mass is tightly correlated with the compactness of the non-rotating maximum mass configuration. In a separate study [@bauswein13a] they used their large range of equations of state and several mass ratios to systematically explore dynamic mass ejection. According to their study, softer equations of state with correspondingly smaller radii eject a larger amount of mass. In the explored cases they found a range from $10^{-3}$ to $2 \times 10^{-2}$ to be dynamically ejected. For the arguably most likely case with 1.35 and 1.35 they found a range of ejecta masses of about one order of magnitude, determined by the equation of state. Moreover, consistent with other studies, they found a robust r-process that produces a close-to-solar abundance pattern beyond nucleon number of $A= 130$ and they discussed the implications for “macronovae” and possibly emerging radio remnants due to the ejecta. ### Neutron star–black hole mergers {#sec:appl_NSBH} A number of issues that have complicated the merger dynamics in the WDWD case, such as stability of mass transfer or the formation of a disk, see Section \[sec:WDWD\_MT\], are also very important for NSBH mergers. Here, however, they are further complicated by a poorly known high-density equation of state which determines the mass-radius relationship and therefore the reaction of the neutron star on mass loss, general-relativistic effects such as the appearance of an innermost stable circular orbit or effects from the bh spin and the fact that now the GW radiation-reaction time scale can become comparable to the dynamical time scale, see Eq. (\[eq:tau\_GW\_tau\_orb\]). Of particular relevance is the question for which binary systems sizeable accretion tori form since they are thought to be the crucial “transformation engines” that channel available energy into (relativistic) outflow. The final answer to this question requires 3D numerical simulations with the relevant physics, but a qualitative idea can be gained form simple estimates (but, based on the experience from WDWD binaries, keep in mind that even plausible approximations can yield rather poor results, see Section \[sec:WDWD\_MT\]). Mass transfer is expected to set in when the Roche volume becomes comparable to the volume of the neutron star. By applying Paczynski’s estimate for the Roche lobe radius [@paczynski71] and equating it with the ns radius, one finds that the onset of mass transfer (which we use here as a proxy for the tidal disruption radius) can be expected near a separation of a\_[MT]{}= 2.17 R\_ ( )\^[1/3]{} 26 ( ) ( )\^[1/3]{} . Since $a_{\rm MT}$ grows, in the limit $q \ll 1$, only proportional to $M_{\mathrm{BH}}^{1/3}$, but the ISCO and the event horizon grow $\propto M_{\mathrm{BH}}$, the onset of mass transfer/disruption can take place inside the ISCO for large bh masses. At the very high end of bh masses, the neutron star is swallowed as whole without being disrupted at all. A qualitative illustration (for fiducial neutron star properties, $M_{\mathrm{NS}}= 1.4$ and $R_{\mathrm{NS}}= 12$ km) is shown in Figure \[fig:BH\_radii\]. Roughly, already for black holes near $M_{\mathrm{BH}}\approx 8$ the mass transfer/disruption occurs near the ISCO which makes it potentially difficult to form a massive torus from ns debris. So, low mass black holes are clearly preferred as GRB engines. Numerical simulations [@faber06b] have shown, however, that even if the disruption occurs deep inside the ISCO this does not necessarily mean that all the matter is doomed to fall straight into the hole and a torus can still possibly form. When discussing disk formation in a GRB context, it is worth keeping in mind that even seemingly small disk masses allow, at least in principle, for the extraction of energies, E\_[extr]{} \~1.810\^[51]{} ( ) ( ), that are large enough to accommodate the isotropic gamma-ray energies, $E_{\gamma, \rm iso} \sim 10^{50}$ erg, that have been inferred for short bursts [@berger14a]. If short bursts are collimated into a half-opening angle $\theta$, their true energies are substantially lower than this number, $E_{\gamma, \rm true} = \left( E_{\gamma, \rm iso}/65 \right) \left( \theta/10^\circ\right)^2$. ### Early Newtonian calculations with polytropic equations of state {#early-newtonian-calculations-with-polytropic-equations-of-state .unnumbered} The first NSBH simulations were performed by [@lee99a; @lee99b] and [@lee00; @lee01a] using Newtonian physics and polytropic equations of state. Although simple and missing some qualitative features of black holes (such as a last stable orbit), these simulations provided insight and a qualitative understanding of the system dynamics, the impact of neutron star spin, the mass ratio and the equation of state. The first set of simulations [@lee99a] were carried out with an SPH formulation similar to the one of [@hernquist89], but an alternative kernel gradient symmetrization. In this study, the artificial viscosity tensor Eq. (\[eq:basic:PI\_AV\]) was implemented with fixed dissipation parameters $\alpha$ and $\beta$ and $\sim 10^4$ SPH particles were used. The black hole was modelled as a Newtonian point mass with an absorbing boundary condition at the Schwarzschild radius $R_{\rm S}$, no backreaction from gravitational wave emission was accounted for. In the simulations of initially tidally locked binaries with a stiff EOS ($\Gamma=3$; [@lee99a]), they found that the neutron star survived the onset of mass transfer and kept orbiting, at a reduced mass, for several orbital periods. A similar behavior had been realized in subsequent work with a stiff nuclear equation of state [@rosswog04b; @rosswog07b]. In a follow-up paper [@lee99b], a simple point-mass backreaction force was applied, and, in one case, the Paczynski–Wiita [@paczynski80] potential was used (now with absorbing boundary at $1.5\,R_{\rm S}$). But the main focus of this study was to explore the effect of a softer equation of state ($\Gamma=5/3$). In all explored cases the system dynamics was very different from the previous study, the neutron star was completely disrupted and formed a massive disk of $\sim 0.2 $ , with $\sim 0.01$ being dynamically ejected. The sensitivity to the EOS stiffness is not entirely surprising, since the solutions to the Lane–Emden equations give the mass-radius relationship = (-2)/(3-4) \[eq:LE\_MR\] for a polytropic star, so that the neutron star reacts differently on mass loss: it shrinks for $\Gamma > 2$, so mass loss is quenched, and expands for $\Gamma < 2$ and therefore further enhances mass transfer. In a second set of calculations ($\approx 80$ K SPH particles), they explored non-rotating neutron stars that were modelled as compressible triaxial ellipsoids according to the semi-analytic work of [@lai93a; @lai93b; @lai94b], both with stiff ($\Gamma=2.5$ and 3) [@lee00] and soft ($\Gamma=5/3$ and 2) [@lee01a] polytropic equations of state. They used the same simulation technology, but also applied a Balsara-limiter, see Eq. (\[eq:Balsara\]), in their artificial viscosity treatment and only purely Newtonian interaction between ns and bh was considered. For the $\Gamma=3$ case, the neutron star survived again until the end of the simulation, with $\Gamma=2.5$ it survived the first mass transfer episode but was subsequently completely disrupted and formed a disk of nearly 0.2 , about 0.03 were dynamically ejected. [@lee01b] also simulated mergers between a black hole and a strange star which was modelled with a simple quark-matter EOS. The dynamical evolution for such systems was quite different from the polytropic case: the strange star was stretched into thin matter stream that wound around the black hole and was finally swallowed. Although “starlets” of $ \approx 0.03$ formed during the disruption process, all of them were in the end swallowed by the hole within milliseconds, no mass loss could be resolved. ### Studies with focus on microphysics {#studies-with-focus-on-microphysics-1 .unnumbered} The first NSBH studies based on Newtonian gravity, but including detailed microphysics were performed by [@ruffert99] and [@janka99] using a Eulerian PPM code on a Cartesian grid. The first Newtonian-gravity-plus-microphysics SPH simulations of NSBH mergers were discussed in [@rosswog04b; @rosswog05a]. Here the black hole was simulated as a Newtonian point mass with an absorbing boundary and a simple GW backreaction force was applied. For the neutron star the [@shen98a; @shen98b] temperature-dependent nuclear EOS was used and the star was modelled with $3\times 10^5 - 10^6$ SPH particles. In addition, neutrino cooling and electron/positron captures were followed with a detailed multi-flavor leakage scheme [@rosswog03a]. The initial study focussed on systems with low mass black holes ($q=0.5 - 0.1$) since this way there are greater chances to disrupt the neutron star outside of the ISCO, see above. Moreover, both (carefully constructed) corotating and irrotational neutron stars were studied. In all cases the core of the neutron star ($0.15 - 0.68$ ) survived the initial mass transfer episodes until the end of the simulations (22 - 64 ms). If disks formed at all during the simulated time, they had only moderate masses ($\sim0.005$ ). One of the NSBH binary ($M_{\mathrm{NS}}=1.4$ , $M_{\mathrm{BH}}=3$ ) systems was followed throughout the whole mass transfer episode [@rosswog07b] which lasted for 220 ms or 47 orbital revolutions and only ended when the neutron star finally became disrupted and resulted the formation of a disk of 0.05 . A set of test calculations with a stiff ($\Gamma=3$) and a softer polytropic EOS ($\Gamma=2$) indicated that such episodic mass transfer is related to the stiffness of the ns EOS and only occurs for stiff cases, consistent with the results of [@lee00]. Subsequent studies with better approximations to relativistic gravity, e.g., [@faber06a], have seen qualitatively similar effects for stiffer EOSs, but after a few orbital periods the neutron was always disrupted. [@shibata11] discussed such episodic, long-lived mass transfer in a GR context and concluded that while possible, it has so far never been seen in fully relativistic studies. A study [@rosswog05a] with simulation tools similar to [@rosswog04b] focussed on higher mass, non-spinning black holes ($M_{\mathrm{BH}}= 14 \dots 20$ ) that were approximated by pseudo-relativistic potentials [@paczynski80]. While being very simple to implement, this approach mimics some GR effects quite well and in particular it has an innermost stable circular particle orbit at the correct location ($6 GM_{\mathrm{BH}}/c^2$), see [@tejeda13a] for quantitative assessment of various properties. In none of these high black hole mass cases was episodic mass transfer observed, the neutron star was always completely disrupted shortly after the onset of mass transfer. Although disks formed for systems below 18 , a large part of them was inside the ISCO and was falling practically radially into the hole on a dynamical time scale. As a result, they were thin and cold and not considered promising GRB engines. It was suggested, however, that even black holes at the high end of the mass distribution could possibly be GRB engines, provided they spin rapidly enough, since then both ISCO and horizon move closer to the bh. The investigated systems ejected between 0.01 and 0.2 at large velocities ($\sim 0.5$ c) and analytical estimates suggested that such systems should produce bright optical/near-infrared transients (“macronovae”) powered by the radioactive decay of the freshly produced r-process elements within the ejecta, as originally suggested by [@li98]. ### Studies with approximate GR gravity around a non-spinning black hole {#studies-with-approximate-gr-gravity-around-a-non-spinning-black-hole .unnumbered} [@faber06a; @faber06b] studied the merger of a non-rotating black hole with a polytropic neutron star in approximate GR gravity. While the hydrodynamics code was fully relativistic, the self-gravity of the neutron star was treated within the conformal flatness (CF) approximation. Since the black hole was kept at a fixed position its mass needed to be substantially larger than the one of the neutron star, therefore a mass ratio of $q=0.1$ was chosen. The neutron star matter was modelled with two (by nuclear matter standards) relatively soft polytropes ($\Gamma= 1.5$ and 2). In the first study [@faber06a] they focussed on tidally locked neutron stars and solved the five linked non-linear field equations of the CF approach by means of the <span style="font-variant:small-caps;">Lorene</span> libraries, the second study [@faber06b] used irrotational neutron stars and solved the CF equations by means of a Fast Fourier transform solver. In a first case they considered a neutron star of compactness $\mathcal{C}= 0.15$, and, to simulate a case where the disruption of the neutron star occurs near the ISCO, they also considered a second case where $\mathcal{C}$ was only 0.09. The first case turned out to be astrophysically unspectacular: the entire neutron star was swallowed as a whole without leaving matter behind. The second, “undercompact” case, however, see Figure \[fig:NSBH\_faber\], showed some very interesting behavior: the neutron star spiralled towards the black hole, became tidally stretched and although at some point 98% of the ns mass were inside of the ISCO, see panel two, via a rapid redistribution of angular momentum, a substantial fraction of the matter was ejected as a one-armed spiral. Approximately 12% of the initial neutron star formed a disk and an additional 13% of the initial neutron star were tidally ejected into unbound orbits. Such systems, they concluded, would be interesting sGRB engines. ### Studies in the fixed metric of a spinning black hole {#studies-in-the-fixed-metric-of-a-spinning-black-hole .unnumbered} [@rantsiou08] explored how the outcome of a neutron star black hole merger depends on the spin of the black hole and on the inclination angle of the binary orbit with respect to the equatorial plane of the black hole. They used the relativistic SPH code originally developed by [@laguna93a; @laguna93b] to study the tidal disruption of a main sequence star by a massive black hole. The new code version employed Kerr–Schild coordinates to avoid coordinate singularities at the horizon as they appear in the frequently used Boyer–Lindquist coordinates. Since the spacetime was kept fixed, they focused on a small mass ratio $q= 0.1$, where the impact of the neutron star on the spacetime is sub-dominant. Both the black hole mass and spin were frozen at their initial values during the simulation and the GW backreaction was implemented via the quadrupole approximation in the point mass limit, similar to the one used by [@lee99b]. The neutron star itself was modelled as a $\Gamma=2.0$ polytrope with Newtonian self-gravity, the artificial dissipation parameters were fixed to 0.2 (instead of values near unity which are needed to properly deal with shocks). Note that $\Gamma=2$ is a special choice, since a Newtonian star does not change its radius if mass is added or lost, see Eq. (\[eq:LE\_MR\]). The bulk of the simulations was calculated with $10^4$ SPH particles, in one case $10^5$ particles were used to confirm the robustness of the results. For the case of a Schwarzschild black hole ($a_{\mathrm{BH}}=0$) they found that neither a disk formed nor any material was ejected. For equatorial mergers with spinning black holes, it required a spin parameter of $a_{\mathrm{BH}} > 0.7$ for any mass to form a disk or to become ejected. For a rapidly spinning bh ($a_{\mathrm{BH}}=0.75$) an amount of matter of order 0.01 became unbound, for a close-to-maximally spinning bh ($a_{\mathrm{BH}}=0.99$) a huge amount of matter ($> 0.4$ ) was ejected. Mergers with inclination angles $> 60^\circ$ lead to the complete swallowing of the neutron star. An example of close-to-maximally spinning black hole ($a_{\mathrm{BH}}=0.99$) and a neutron star whose orbital plane is inclined by $45^\circ$ with respect to the black hole spin is shown in Figure \[fig:NSBH\_rantsiou\]. Here, as much as 25% of the neutron star, in the shape of a helix, become unbound. ### Collisions between two neutron stars and between a neutron star and a black hole {#sec:compact_collisions} Traditionally, the focus of compact object encounters have been GW-driven binary systems such as the Hulse–Taylor pulsar [@taylor89; @weisberg10]. More recently, however, also dynamical collisions/high-eccentricity encounters between two compact objects have attracted a fair amount of interest [@kocsis06a; @oleary09; @lee10a; @east12; @kocsis12; @gold12; @gold13]. Unfortunately, their rates are at least as difficult to estimate as those of GW-driven mergers. Collisions differ from gravitational wave-driven mergers in a number of ways. For example, since gravitational wave emission of eccentric binaries efficiently removes angular momentum in comparison to energy, primordial binaries will have radiated away their eccentricity and will finally merge from a nearly circular orbit. On the contrary, binaries that have formed dynamically, say in a globular cluster, start from a small orbital separation, but with large eccentricities and may not have had the time to circularize until merger. This leads to pronouncedly different gravitational wave signatures, “chirping” signals of increasing frequency and amplitude for mergers and initially well-separated, repeated GW bursts that continue from minutes to days in the case of collisions. Moreover, compact binaries are strongly gravitationally bound at the onset of the dynamical merger phase while collisions, in contrast, have total orbital energies close to zero and need to get rid of energy and angular momentum via GW emission and/or through mass shedding episodes in order to form a single remnant. Due to the strong dependence on the impact parameter and the lack of strong constraints on it, one expects a much larger variety of dynamical behavior for collisions than for mergers. [@lee10a] provided detailed rate estimates of compact object collisions and concluded that such encounters could possibly produce an interesting contribution to the observed GRB rate. They also performed the first SPH simulations of such encounters. Using the SPH code from their earlier studies [@lee99a; @lee99b; @lee00; @lee01a], they explored the dynamics and remnant structure of encounters with different strengths between all types of compact stellar objects (WD/NS/BH; typically with 100K SPH particles). Here polytropic equations of state were used and black holes were treated as Newtonian point masses with absorbing boundaries at their Schwarzschild radii. Their calculations indicated in particular that such encounters would produce interesting GRB engines with massive disks and additional external reservoirs (one tidal tail for each close encounter) where a large amounts of matter ($>0.1$ ) could be stored to possibly prolong the central engine activity, as observed in some bursts. In addition, a substantial amount of mass was dynamically ejected (0.03 for NSNS and up to 0.2 for NSBH systems). In [@rosswog13a] various signatures of gravitational wave-driven mergers and dynamical collisions were compared, both for NSNS and NSBH encounters. The study applied Newtonian SPH (with up to $8 \times 10^6$ particles) together with a nuclear equation of state [@shen98a; @shen98b] and a detailed neutrino leakage scheme [@rosswog03a]. As above, black holes were modelled as Newtonian point masses with absorbing boundaries at the $R_S$. A simulation result of a strong encounter between a 1.3 and a 1.4 neutron star (pericentre distance equal to the average of the two neutron star radii) is shown in Figure \[fig:NSNS\_collision\]. Due to the strong shear at their interface, a string of Kelvin–Helmholtz vortices forms in each of the close encounters before a final central remnant is formed. Such conditions offer plenty of opportunity for magnetic field amplification [@price06; @anderson08b; @obergaulinger10; @rezzolla11; @zrake13]. In all explored cases the neutrino luminosity was at least comparable to the merger case, $L_\nu \approx 10^{53}$ erg/s, but for the more extreme cases exceeded this value by an order of magnitude. Thus, if neutrino annihilation should be the main agent driving a GRB outflow, chances for collisions should be at least as good as in the merger case. But both scenarios also share the same caveat: neutrino-driven, baryonic pollution could prevent in at least a fraction of cases the emergence of relativistic outflows. In NSBH collisions the neutron star took usually several encounters before being completely disrupted. In some cases its core survived several encounters and was finally ejected with a mass of $\sim 0.1$ . Of course, this offers a number of interesting possibilities (production of low-mass neutron stars, explosion of the NS core at the minimal mass etc.). But first of all, such events may be very rare and it needs to be seen whether such behavior can occur at all in the general-relativistic case. Generally, both NSNS and NSBH collisions ejected large quantities of unbound debris. Collisions between neutron stars ejected a few percent (dependent on the impact strength) of a solar mass, while all investigated NSBH collisions ejected $\sim 0.15$ , consistent with the findings of [@lee10a]. Since NSBH encounters should dominate the rates [@lee10a], it was concluded in [@rosswog13a] that collisions must be (possibly much) less frequent than 10% of the NSNS merger rate to avoid a conflict with constraints from the chemical evolution of galaxies. Since here the ejecta velocities and masses are substantially larger ($v_{\rm ej}\sim 0.2$c and $m_{\rm ej}\sim 0.1$ ) than in the neutron star merger case ($v_{\rm ej} \sim 0.1$c and $m_{\rm ej}\sim 0.01$ ) simple scaling relations [@grossman14a] suggest that a resulting radioactively powered macronova should peak after t\_[P]{}= 11 ( )\^[1/2]{} with a luminosity of L\_[P]{}= 8.8 10\^[40]{} ( )\^[0.65]{} ( )\^[0.35]{}. Here $\kappa$ is the r-process material opacity [@kasen13a] and a radioactive heating rate $\dot{\epsilon} \propto t^{-1.3}$ [@metzger10b; @korobkin12a] has been assumed. ### Post-merger disk evolution SPH simulations were also applied to study the long-term evolution of accretion disks that have formed during a CBM. Lee and collaborators [@lee02; @lee04; @lee05b; @lee09] started from their NSBH merger simulations, see Section \[sec:appl\_NSBH\], and followed the fate of the resulting disks. Since the viscous disk time scale, see Eq. (\[eq:tau\_visc\]), by far exceeds the numerical time step allowed by the CFL condition, Eq. (\[eq:CFL\]), the previous results were mapped into a 2D version of their code and they followed the evolution, driven by a Shakura-Sunyaev “$\alpha$-viscosity” prescription [@shakura73], for hundreds of milliseconds. Consistent with their NSBH merger simulations, the black hole was treated as a Newtonian point mass with an absorbing boundary at $R_{\rm S}= 2 G M_{\mathrm{BH}}/c^2$, the disk self-gravity was neglected. In the first study [@lee02] the disk matter was modelled with a polytropic EOS ($\Gamma=4/3$) and locally dissipated energy was assumed to be emitted via neutrinos. Subsequent studies [@lee04; @lee05b] applied increasingly more sophisticated microphysics, the latter study accounted for pressure contributions from relativistic electrons with an arbitrary degree of degeneracy and an ideal gas of nucleons and alpha particles. These latter studies accounted for opacity-dependent neutrino cooling and also considered trapped neutrinos as a source of pressure, but no distinction between different neutrino flavors was made. It turned out that at the transition between the inner, neutrino-opaque and the outer, transparent regions an inversion of the lepton number gradient builds up, with minimum values $Y_e \approx 0.05$ close to the transition radius ($\sim 10^7$ cm), values close to 0.1 near the BH and proton-rich conditions ($Y_e>0.5$) at large radii. Such lepton number gradients drive strong convective motions that shape the inner disk regions. Overall, neutrino luminosities around $\approx 10^{53}$ erg/s were found and around $10^{52}$ erg were emitted in neutrinos over the lifetime of the disk ($\sim 0.4$ s). ### Summary encounters between neutron stars an black holes Compact binary mergers are related to a number of vibrant astrophysical topics. They are likely the first sources whose gravitational waves are detected directly, they probably power short Gamma-ray Bursts and they may be the astrophysical sites where a large fraction of the heaviest elements in the Universe are forged. With these high promises for different fields comes also the necessity to reliably include a broad range of physical processes. Here, much progress has been achieved in the last one and a half decades. The task for the future will be to bring the different facets of compact binary encounters into a coherent, bigger astrophysical picture. Very likely, our understanding will be challenged once, say, the first Gamma-ray burst is also observed as a gravitational wave event. If the interpretation of the recent nIR transient related to GRB 130603B as a “macronova” is correct, this is the first link between two suspected, but previously unproven aspects of the compact binary merger phenomenon: Gamma-ray bursts and heavy element nucleosynthesis. This may just have been one of the first heralds of the beginning era of multi-messenger astronomy. SPH has played a major role in achieving our current understanding of the astrophysics of compact binary mergers. The main reasons for its use were its geometrical flexibility, its excellent numerical conservation properties and the ease with which new physics ingredients can be implemented. A broad range of physical ingredients has been implemented into the simulations that exist to date. These include a large number of different equations of state, weak interactions/neutrino emission and magnetic fields. In terms of gravitational physics, mergers have been simulated in Newtonian, post-Newtonian and in conformal flatness approximations to GR. An important milestone, however, that at the time of writing (June 2014) still has to be achieved, is an SPH simulation where the spacetime is self-consistently evolved in dynamic GR. Acknowledgements {#sec:acknowledgements .unnumbered} ================ This work has benefited from the discussions with many teachers, students, colleagues and friends and I want to collectively thank all of them. Particular thanks goes to W. Benz, M. Dan, M.B. Davies, W. Dehnen, O. Korobkin, W.H. Lee, J.J. Monaghan, D.J. Price, E. Ramirez-Ruiz, E. Tejeda and F.K. Thielemann. This work has been supported by the Swedish Research Council (VR) under grant 621-2012-4870, the CompStar network, COST Action MP1304, and in part by the National Science Foundation Grant No. PHYS-1066293. The hospitality of the Aspen Center for Physics is gratefully acknowledged. Some of the figures in this article have been produced by the visualization software <span style="font-variant:small-caps;">Splash</span> [@price07d].

ArXiv

--- abstract: 'Recent simulations of solar active regions have shown that it is possible to reproduce both the total intensity and the general morphology of the high temperature emission observed at soft X-ray wavelengths using static heating models. There is ample observational evidence, however, that the solar corona is highly variable, indicating a significant role for dynamical processes in coronal heating. Because they are computationally demanding, full hydrodynamic simulations of solar active regions have not been considered previously. In this paper we make first application of an impulsive heating model to the simulation of an entire active region, AR8156 observed on 1998 February 16. We model this region by coupling potential field extrapolations to full solutions of the time-dependent hydrodynamic loop equations. To make the problem more tractable we begin with a static heating model that reproduces the emission observed in 4 different *Yohkoh* Soft X-Ray Telescope (SXT) filters and consider dynamical heating scenarios that yield time-averaged SXT intensities that are consistent with the static case. We find that it is possible to reproduce the total observed soft X-ray emission in all of the SXT filters with a dynamical heating model, indicating that nanoflare heating is consistent with the observational properties of the high temperature solar corona.' author: - 'Harry P. Warren' - 'Amy R. Winebarger' title: 'Static and Dynamic Modeling of a Solar Active Region. I: Soft X-Ray Emission' --- Introduction ============ Understanding how the Sun’s corona is heated to high temperatures remains one of the most significant challenges in solar physics. Unfortunately, the complexity of the solar atmosphere, with its many disparate spatial and temporal scales, makes it impossible to represent with a single, all encompassing model. Instead we need to break the problem up into smaller, more manageable pieces (e.g., see the recent review by @klimchuk2006). For example, kinetic theory or generalized MHD is used to describe the microphysics of the energy release process. Ideal and resistive MHD are used to study the evolution of coronal magnetic fields and the conditions that give rise to energy release. Finally, one dimensional hydrodynamical modeling is employed to calculate the response of the solar atmosphere to the release of energy. This last step is a critical one in the process of understanding coronal emission because it links theoretical models with solar observations. Even here, however, most previous work has focused on modeling small pieces of the Sun, such as individual loops [e.g., @aschwanden2001b; @reale2000]. Though understanding the heating in individual structures is an important first step, it has been difficult to apply this information to constrain the properties of the global coronal heating mechanism. Recent advances in high performance computing have made it possible to simulate large regions of the corona, at least with static heating models. [@schrijver2004], for example, have coupled potential field source-surface models of the coronal magnetic field with parametric fits to solutions of the hydrostatic loop equations to calculate visualizations of the full Sun. Comparisons between the simulation results and full-disk solar images indicate that the energy flux ($F_H$) into a corona loop scales as $B_F/L$, where $B_F$ is the foot point field strength and $L$ is the loop length. [@schrijver2005b] also find that this form for the heating flux is consistent with the flux-luminosity relationship derived from X-ray observations of other cool dwarf stars [@schrijver2005b]. [@warren2006b] have performed similar simulations for 26 solar active regions using potential field extrapolations and full solutions to the hydrostatic loop equations. These simulation results indicate that the observed emission is consistent with a volumetric heating rate ($\epsilon_S$) that scales as $\bar{B}/L$, where $\bar{B}$ is the field strength averaged along the field line. In the sample of active regions used in that study $\bar{B}\sim B_F/L$ so that $F_H\sim \epsilon_S L \sim \bar{B}\sim B_F/L$, and this form for the volumetric heating rate is consistent with the energy flux determined by [@schrijver2004]. In these previous studies it was possible to use static heating models to reproduce the high temperature emission observed at soft X-ray wavelengths, but not the lower temperature emission typically observed in the EUV. The static models are not able to account for the EUV loops evident in the solar images. Recent work has shown that the active region loops observed at these lower temperatures are often evolving [@urra2006; @winebarger2003b]. Simulation results suggest that these loops can be understood using dynamical models where the loops are heated impulsively and are cooling [e.g., @spadaro2003; @warren2003]. Furthermore, spectrally resolved observations have indicated pervasive red shifts in active regions at upper transition region temperatures [e.g., @winebarger2002], suggesting that much of the plasma in solar active regions near 1MK has been heated to higher temperatures and is cooling. Finally, [@warren2006b] found that static heating in loops with constant cross section yields footpoint emission that is much brighter than what is observed. This suggests that static heating models may not be consistent with the observations, even in the central cores of active regions. The need for exploring dynamical heating models of the solar corona is clear, but there are a number of problems that make this difficult in practice. One problem is the many free parameters possible in parameterizations of impulsive heating models. In addition to the magnitude and spatial location of the heating, it is possible to vary the temporal envelope and repetition rate of the heating [e.g., @patsourakos2006; @testa2005]. Furthermore, dynamical solutions to the hydrodynamic loop equations are much more computationally intensive to calculate than static solutions, limiting our ability to explore parameter space. In this paper we explore the application of impulsive heating models to the high temperature emission observed in active region 8156 on 1998 February 16. To make the problem more tractable we begin with a static heating model that reproduces the emission observed in 4 different *Yohkoh* Soft X-Ray Telescope (SXT) filters and look for dynamical models that yield time-averaged SXT intensities that are in agreement with those computed from the static solutions. Relating the time-averaged intensities derived from the full dynamical solutions to the observed intensities is based on the idea that the emission from a single feature results from the superposition of even finer, dynamical structures that are in various stages of heating and cooling. This idea is similar to the nanoflare model of coronal heating [e.g., @parker1983; @cargill1994]. Other nanoflare heating scenarios are possible, such as heating events on larger scale threads that are distributed randomly in space and time, but are not considered here. We find that it is possible to construct a dynamical heating model that reproduces the total soft X-ray emission in each SXT filter. This indicates that nanoflare heating is consistent with the observational properties of the high temperature corona. Observations ============ Observations from SXT [@tsuneta1991] on *Yohkoh* form the basis for this work. The SXT, which operated from late 1991 to late 2001, was a grazing incidence telescope with a nominal spatial resolution of about 5 (25 pixels). Temperature discrimination was achieved through the use of several focal plane filters. The SXT response extended from approximately 3Å to approximately 40Å and the instrument was sensitive to plasma above about 2MK. In addition to the SXT images we use full-Sun magnetograms taken with the MDI instrument [@scherrer1995] on *SOHO* to provide information on the distribution of photospheric magnetic fields. The spatial resolution of the MDI magnetograms is comparable to the spatial resolution of EIT and SXT. In this study we use the synoptic MDI magnetograms which are taken every 96 minutes. To constrain the static heating model we require observations of an active region in multiple SXT filters. Observations in the thickest SXT analysis filters, the “thick aluminum” (Al12) and the “beryllium” (Be119), are crucial for this work. As we will show, observations in the thinner analysis filters, such as the “thin aluminum” (Al.1) and the “sandwich” (AlMg) filters, do not have the requisite temperature discrimination for this modeling. To identify candidate observations we made a list of all SXT partial-frame (as opposed to full disk) observations with observations in the Al.1, AlMg, Al12, and Be119 filters between the beginning of the *Yohkoh* mission and the end of 2001. Since the potential field extrapolation is also important to this analysis, we required that the active lie within 400 of disk center. We also use consider observations from the EIT [@delaboudiniere1995] on *SOHO*. EIT is a normal incidence telescope that takes full-Sun images in four wavelength ranges, 304Å (which is generally dominated by emission from ), 171Å ( and ), 195Å (), and 284Å (). EIT has a spatial resolution of 26. Images in all four wavelengths are typically taken 4 times a day and these synoptic data are used in this study. From a visual inspection of the available data we selected observations of AR8156 taken 1998 February 16 near 8UT. This region is shown in full-disk SXT and MDI images in Figure \[fig:1\]. The region of interest observed in SXT, EIT, and MDI is shown in Figure \[fig:2\]. These images represent the observations taken closest to the MDI magnetogram. The total intensities in the SXT partial frame images for this region during the period beginning 1998 February 15 23:30 UT and ending 1998 February 16 13:00 UT are generally within $\pm20$% of the total intensities in these SXT images, indicating an absence of major flare activity during this time. Static Modeling =============== To model the topology of this active region we use a simple potential field extrapolation of the photospheric magnetic field. For each MDI pixel with a field strength greater than 50G we calculate a field line. Some representative field lines are shown in Figure \[fig:2\]. It is clear that such a simple model does not fully reproduce the observed topology of the images. The long loops in the bottom half of the images, for example, are shifted relative to the field lines computed from the potential field extrapolation. However, as we argued previously [@warren2006b], our goal is not to reproduce the exact morphology of the active region. Rather, we are primarily interested in the more general properties of the active region emission, such as the total intensity or the distribution of intensities. The potential field extrapolation only serves to provide a realistic distribution of loop lengths. One subtlety with coupling a potential field extrapolation with solutions to the hydrostatic loop equations is the difference in boundary conditions. The field lines originate in the photosphere where the plasma temperature is approximately 4,000K. The boundary condition for the loop footpoints, however are typically set at 10,000 or 20,000K in the numerical solutions to the hydrodynamic loop equations. Furthermore, studies of the topology of the quiet Sun have shown that a significant fraction of the field lines close at heights below 2.5Mm, a typical chromospheric height [@close2003]. To avoid these small scale loops we use the portion of the field line above 2.5Mm in the modeling and exclude all field lines that do not reach this height. For each of the 1956 field lines ultimately included in the simulation we calculate a solution to the hydrostatic loop equations using a numerical code written by Aad van Ballegooijen (e.g., @hussain2002 [@schrijver2005]). Following our previous work, our volumetric heating function is assumed to be $$\epsilon_S = \epsilon_0 \left(\frac{\bar{B}}{\bar{B}_0}\right) \left(\frac{L_0}{L}\right), \label{eq:heating}$$ where $\bar{B}$ is the field strength averaged along the field line, and $L$ is the total loop length. We assume a constant cross section and a uniform distribution of heating along each loop. Note that the variation in gravity along the loop is determined from the geometry of the field line. The numerical solution to the hydrostatic loop equation provides the variation in the density, temperature, and velocity along the loop. The temperatures, densities, and loop geometry are then used to compute the expected response in the SXT and EIT filters. For our work we use the CHIANTI atomic database [e.g. @dere1997] to compute the instrumental responses and the radiative losses used in the hydrostatic code (see @brooks2006 for a discussion of the instrumental responses and radiative losses). In our previous work the value for $\epsilon_0$ was chosen to be 0.0492erg cm$^{-3}$ s$^{-1}$ so that a “typical” field line ($\bar{B}=\bar{B}_0$ and $L=L_0$) had an apex temperature of $T_0=4$MK. We also found that for this value of $\epsilon_0$ a filling factor of about 10% was needed to reproduce the SXT emission observed in the Al.1 or AlMg filters. In the absence of information from the hotter SXT filters the value for $\epsilon_0$ is poorly constrained. The values adopted for the parameters $\bar{B}_0$ and $L_0$ are 76G and 29Mm respectively. For this active region we have observations in the hotter SXT filters so we have performed active region simulations for a range of $T_0$ (equivalently $\epsilon_0$) values. The resulting total intensities in each of the 4 filters as a function of $T_0$ are shown in Figure \[fig:3\]. It is clear from this figure that the static model cannot reproduce all of the SXT intensities for a filling factor of 1. For a filling factor of 1 the value of $T_0$ needed to reproduce the Al.1 intensity yields a Be119 intensity that is too low. Similarly, the value of $T_0$ that reproduces the Be119 intensity for a filling factor of 1 yields Al.1 intensities that are too large. By doing a least squares fit of the simulation results to the observations and varying both the value of $T_0$ and the filling factor we find that we can reproduce all of the SXT intensities to within 10% for $T_0=3.8$MK and a filling factor of 7.6%, values close to what we used in our previous work. These simulation results also highlight the importance of the SXT Al12 and Be119 filters in modeling the observations. The ratio between the Al.1 and AlMg filters is simply too shallow to be of any use in constraining the magnitude of the heating. The Al.1 to Be119 ratio, in contrast, varies by almost an order of magnitude as $T_0$ is varied from 2 to 5MK. The total intensity represents the minimum level of agreement between the simulation and the observations. The distribution of the simulated intensities must also look similar to what is observed. To transform the 1D intensities into 3D intensities we assume that the intensity at any point in space is related to the intensity on the field line by $$I(x,y,z) = kI(x_0,y_0,z_0)\exp\left[ -\frac{\Delta^2}{2\sigma_r^2}\right]$$ where $\Delta^2 = (x-x_0)^2+(y-y_0)^2+(z-z_0)^2$ and $2.355\sigma_r$, the FWHM, is set equal to the assumed diameter of the flux tube. A normalization constant ($k$) is introduced so that the integrated intensity of over all space is equal to the intensity integrated along the field line. This approach for the visualization is based on the method used in [@karpen2001]. The resulting simulated SXT images are shown in Figure \[fig:4\] where they are compared with the observations. The simulations clearly do a reasonable job reproducing these data, particularly in the core of the active region. At the periphery of the active region the simulation does not match either the morphology of the emission or its absolute magnitude exactly. The general impression, however, is that the model intensities are generally similar to the observed intensities outside of the active region core even if the morphology doesn’t match exactly. One change that we have made from our previous methodology [@warren2006b] is to include all of the field lines with footpoint field strengths above 500G. These field lines have been largely excluded in previous work because sunspots are generally faint in soft X-rays images (see @golub1997 [@schrijver2004; @fludra2003]). In these observations, however, the exclusion of the field lines rooted in strong field leads to small, but noticeable differences between the simulated and observed emission. As can be inferred from Figure \[fig:2\], excluding these field lines leads to an absence of emission on either side of the bright feature in the center of the active region. This suggests that the algorithm used to select which field lines are included in the simulation needs to be studied more carefully. The histograms of the intensities offer an additional point of comparison between the simulations and the observations. As shown in Figure \[fig:4\], the distributions of the intensities are very similar in both cases, supporting the qualitative agreement between the visualizations and the actual solar images. Dynamic Modeling ================ The principal difficulties with full hydrodynamic modeling of solar active regions are the many degrees of freedom available to parameterize the heating function and the computational difficulty of calculating the numerical solutions. For this exploratory study we make several simplifying assumptions. First, we consider dynamic simulations that are closely related to the static solutions. Since the static modeling of the SXT observations presented in the previous section adequately reproduces the total intensities, the distribution of intensities, and the general morphology of the images, it seems reasonable to consider dynamical heating that would reproduce the static solutions in some limit. Second, we utilize the time-averaged properties of these solutions in computing the simulated intensities. Our assumption is that the emission from a single field line in the static model actually results from the superposition of even finer, dynamical structures that are in various stages of heating and cooling. This is similar to the nanoflare picture of coronal heating [e.g., @parker1983; @cargill1994]. Finally, we will also make use of grids of solutions where we interpolate to determine the simulated intensities rather than computing solutions for each field line individually. In the static case we have used the average magnetic field strength and loop length to infer the volumetric heating rate ($\epsilon_s$) for each field line. For the dynamic case we consider volumetric heating rates of the form $$\epsilon_D(t) = g(t)R\epsilon_S + \epsilon_B, \label{eq:heatingD}$$ where $g(t)$ is a step or boxcar function envelope on the heating, $\epsilon_S$ is the static heating rate determined from Equation \[eq:heating\], $R$ is a arbitrary scaling factor, and $\epsilon_B$ is a weak background heating rate that establishes a cool, tenuous equilibrium atmosphere in the loop. To solve the hydrodynamic loop equations numerically we use the NRL Solar Flux Tube Model (SOLFTM) code [e.g., @mariska1987; @mariska1989]. In the limit of an infinite heating window and $R=1$ the dynamic solutions would converge to the static solutions and all of the properties of the static simulation would be recovered. This is the primary motivation for our choice of the heating function given in Equation \[eq:heatingD\]. The good agreement between the observations and the static model suggest that the energetics of the static model are not far off. For $R=1$ and a finite duration to the heating we expect that the calculated SXT emission will generally be less than in the static case because it takes a finite time for chromospheric plasma to evaporate up into the loop. Thus simply truncating the heating will not produce acceptable results. If we increase the heating somewhat from the static case ($R>1$) and consider a finite duration we expect larger SXT intensities relative to the $R=1$, finite duration case since the evaporation will be faster and the temperatures will be higher with the increased heating. Since the time to fill the loop with plasma will depend on the sound crossing time ($\tau_s\sim L/c_s$, with $c_s$ the sound speed) the behavior of the dynamic solutions relative to the static solutions will also depend on loop length. For a finite duration to the heating the intensities in the dynamical simulations of the shorter loops will more closely resemble the results from the static solution. An illustrative dynamical simulation is shown in Figure \[fig:5\]. Here $R=1.5$ and the heating duration is 200s. For these parameters the apex densities are somewhat somewhat lower than the corresponding static solution. The time-averaging also reduces the SXT intensities significantly relative to their peak values. When the filling factor is included in the calculation of the SXT intensities from the static solutions, however, we see that the SXT intensities calculated from the two different simulations are very similar in all of the filters of interest. Note that the computed intensities are somewhat dependent on the interval chosen for the time averaging. We assume that each small scale thread is heated once then allowed to cool fully before being heated again. In practice, we terminate the dynamical simulation when the apex temperature falls below 0.7MK. The SOLFTM only has an adaptive mesh in the transition region and cannot resolve the formation of very cool material in the corona. Radiative losses become very large at low temperatures so the loops evolve very rapidly past this point and the time-averaged intensities should be only weakly dependent on when the dynamical simulation is stopped. While calculations such as this, which show that the SXT intensities computed from the dynamical simulation and those computed from the static simulation can be comparable, are encouraging, they represent a special case. In general, for a fixed value of $R$ the ratio of the dynamic and static intensities will be greater than 1 for shorter loops and smaller than 1 for longer loops. We would like to know what would happen if we performed dynamical simulations for all of the loops in AR8156. Would the total intensities in the dynamical simulation match the observations? The dynamical solutions we have investigated in this paper typically take about 500s to perform on an Intel Pentium 4-based workstation. For our 1956 field lines this amounts to about 11 days of cpu time. While such calculations can be done in principle, particularly on massively parallel machines with 100s of nodes, they’re too lengthy for the exploratory work we consider here. To circumvent this computational limitation we consider a grid of solutions that encompass the range of loop lengths and heating rates that are present in our static simulation of this active region. In Figure \[fig:6\] we also show a plot of total loop length ($L$) and energy flux ($\epsilon_S L$) for each field line in the static simulation of AR8156. Note that we use the energy flux instead of the volumetric heating rate because, as indicated in the plot, these variables are largely uncorrelated and we can use a simple rectangular grid. The volumetric heating rate and the loop length, in contrast, are correlated. The procedure we adopt is to calculate dynamical solutions for the $L, \epsilon_S L$ pairs on this grid, determine the total SXT intensities for these solutions, and then use interpolation to estimate the SXT intensities for each field line in the simulation. To investigate the effects of varying $R$ on the dynamical solutions we have computed $10\times10$ grids for $R=$1.0, 1.25, 1.5, 1.75, and 2.0 for a total of 500 dynamical simulations. One important difference between these grid solutions and the static solutions discussed in the previous section is the loop geometry. In the static simulation the loop geometry is determined by the field line. In the dynamic simulation the loop is assumed to be perpendicular to the solar surface. Since the density scale height for high temperature loops is large this difference has only a small effect on the simulation of the SXT emission. The effect is much more pronounced for the lower temperature loops imaged in EIT and precludes the use of the interpolation grid for these intensities. In Figure \[fig:6\] we show the resulting SXT Al12 intensities for the $R=1.5$ grid of solutions. The most intense loops in the dynamic simulations are the shortest loops with the most intense heating. These loops come the closest to reaching equilibrium parameters with the finite duration heating. The faintest loops are the longest loops with the weakest heating. In Figure \[fig:6\] we also show the ratio of the SXT intensities from the dynamic and static simulations. That is, we compare the total intensity determined from the static solution with a heating rate of $\epsilon_S$ with the total intensity determined with the time dependent heating rate $\epsilon_D$ given in Equation \[eq:heatingD\]. These ratios indicate that for a significant region of this parameter space the total intensities in the dynamic heating are close to those computed for the static simulations. The shorter field lines are somewhat more intense while the longer field lines are generally fainter. This suggests that the dynamic intensities integrated over all of the field lines should produce results similar to the static simulation. As expected, for smaller values of $R$ these ratios are systematically smaller and for larger values of $R$ these ratios are systematically larger. We have used the results from all of the dynamic simulation grids to estimate the total SXT intensity in each filter as a function of $R$. The results are shown in Figure \[fig:7\]. For $R=1$ the total intensities are smaller than what is observed by about 50%. For $R=2$ the intensities are all too large by about 100%. For the $R=1.5$ case, which we have highlighted in Figures \[fig:5\] and \[fig:6\], the simulated total intensities are within 20% of the measurements in all 4 filters, and this case come closest to reproducing the observations. The differences between the model calculations and the observations are not systematic. The calculated Al12 and Be119 intensities are very close to the observations while the Al1 and AlMg are a little too high. The duration of the heating, which we have chosen to be 200s, may explain this discrepancy, at least partially. If the heating were reduced in duration stronger heating (larger $R$) would be required to match the observed intensities. This would lead to higher temperatures and lead to somewhat different ratios among three filters. One of the primary motivations for introducing dynamic modeling is the inability of static models to account for the EUV observations at lower temperatures. Because we have used interpolation grids to infer the intensities it is not possible to compute images similar to those presented in Figure \[fig:4\]. We can, however, consider the morphology of individual loops with the static and dynamic heating scenarios. To investigate this we calculate the time-averaged intensity in SXT and EIT along the loop. As illustrated in Figure \[fig:8\], the dynamic heating is clearly moving in the right direction. The morphology of the high temperature plasma imaged with SXT is largely unchanged in the dynamic simulation while the EUV emission shows full loops. This suggests that the dynamic simulations of active regions will look much closer to the observations shown in Figure \[fig:2\] than the synthetic images calculated from static heating models. One implication of Figure \[fig:8\] is that relatively cool loops imaged in the EUV should be co-spatial with high temperature loops imaged in soft X-rays. There is some evidence that this is not observed. [@schmieder2004] and [@nitta2000], for example, argue that the EUV loops may be observed near soft X-ray loops, but that they are generally not co-spatial. It is possible, however, that the geometry of the loops changes as they cool [@winebarger2005]. [@antiochos2003] suggest that the observed EUV loop emission in an active region is not bright enough to account for the cooling of soft X-ray loops. However, the contrast between the background corona and the EUV loops is low [@cirtain2006], and it is possible that the total EUV intensity in an active region is consistent with the cooling of high temperature loops. Summary and Discussion ====================== We have investigated the use of static and dynamic heating models in the simulation of AR8156. Recent work has shown that static models can capture many of the observed properties of the high temperature soft X-ray emission from solar active regions and our results confirm this. We are able to reproduce both the total intensity and the general morphology of this active region with a static heating model. Furthermore, our results show that this agreement extends to the the hotter SXT filters which have not been considered before. The application of dynamic heating models to active region emission on this scale has not been considered previously. Only the properties of individual loops have been explored [e.g. @warren2002b]. The computational complexity of the dynamical simulations precludes the calculation of individual solutions for each field line and we have utilized interpolation grids for estimating the expected SXT intensities for each field line. We find that it is possible to reproduce the observed SXT intensities in 4 filters, including the high temperature Al12 and Be119 filters, using the dynamical model. Conceptually, the simple dynamical heating model investigated here, where we assume that the emission from a solar feature results from the superposition of many, very fine structures that are in various stages of heating and cooling, is closely related to the nanoflare model of coronal heating [e.g., @parker1983; @cargill1994]. The use of time-averaged intensities computed from the dynamical simulations implicitly assumes that the heating is very coherent, with each infinitesimal thread being heated once and then allowed to cool and drain before being heated again. Other scenarios are possible, such as heating events on larger scale threads that are distributed randomly in time. The spatial and temporal characteristics of coronal heating is likely to fall somewhere in between these extremes. The analysis of high spatial resolution (0.5) EUV images suggests that current solar instrumentation may be close to resolving individual threads in the corona [@aschwanden2005a; @aschwanden2005b], but considerable work remains to be done to determine the fundamental spatial scale for coronal heating. The geometrical properties of coronal threads is also unclear at present. We have assumed constant cross sections in our modeling, consistent with the observational results [@klimchuk1992; @watko2000]. In the static modeling of solar active regions there has been some evidence that the loops with expanding cross sections better reproduce the observations [@schrijver2004; @warren2006b]. Detailed comparisons between simulated and observed solar images are needed to resolve this issue. Despite the many limitations to our modeling the results that we have presented are encouraging and provide a framework for further exploration. The highest priority for future work is the full dynamical simulation of solar active regions without the use of interpolation grids so that synthetic soft X-ray and EUV images can be computed and compared with observations. Another priority is the comparison of active region simulations with spatially and spectrally resolved observations from the upcoming Solar-B mission. Spectral diagnostics, such as Doppler velocities and nonthermal widths, are another dimension that have not been explored in the context of this modeling. The authors would like to thank John Mariska for his helpful comments on the manuscript. Yohkoh is a mission of the Institute of Space and Astronautical Sciences (Japan), with participation from the U.S. and U.K. The EIT data are courtesy of the EIT consortium. This research was supported by NASA’s Supporting Research and Technology and Guest Investigator programs and by the Office of Naval Research. , S. K., [Karpen]{}, J. T., [DeLuca]{}, E. E., [Golub]{}, L., & [Hamilton]{}, P. 2003, , 590, 547 , M. J. 2005, , 634, L193 , M. J., & [Nightingale]{}, R. W. 2005, , 633, 499 , M. J., [Schrijver]{}, C. J., & [Alexander]{}, D. 2001, , 550, 1036 , D. H., & [Warren]{}, H. P. 2006, , 164, 202 , P. J. 1994, , 422, 381 , J., [Martens]{}, P. C. H., [Acton]{}, L. W., & [Weber]{}, M. 2006, , 235, 295 , R. M., [Parnell]{}, C. E., [Mackay]{}, D. H., & [Priest]{}, E. R. 2003, , 212, 251 Delaboudinière, J.-P., et al. 1995, , 162, 291 Dere, K. P., Landi, E., Mason, H. E., [Monsignori Fossi]{}, B. C., & Young, P. R. 1997, , 125, 149 , A., & [Ireland]{}, J. 2003, in The Future of Cool-Star Astrophysics: 12th Cambridge Workshop on Cool Stars , Stellar Systems, and the Sun (2001 July 30 - August 3), eds. A. Brown, G.M. Harper, and T.R. 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The dotted lines indicate the observed intensities. The simulation grid with $R=1.5$ best approximates the observed intensities.[]{data-label="fig:7"}](f08.eps) 

ArXiv

--- abstract: 'We consider the forced problem $-\Delta_p u - V(x)|u|^{p-2} u = f(x)$, where $\Delta_p$ is the $p$-Laplacian ($1<p<\infty$) in a domain $\Omega\subset \mathbb{R}^N$, $V\ge 0$ and $Q_V (u) := \int_\Omega |\nabla u|^p\, dx - \int_\Omega V|u|^p\,dx$ satisfies the condition (A) below. We show that this problem has a solution for all $f$ in a suitable space of distributions. Then we apply this result to some classes of functions $V$ which in particular include the Hardy potential and the potential $V(x)=\lambda_{1,p}(\Omega)$, where $\lambda_{1,p}(\Omega)$ is the Poincaré constant on an infinite strip.' address: - 'Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden' - 'Inst. de Recherche en Math. et Phys., Université catholique de Louvain, 1348 Louvain-la-Neuve, Belgium' author: - Andrzej Szulkin - Michel Willem title: | On some weakly coercive quasilinear problems\ with forcing --- Introduction {#intro} ============ Our purpose is to solve the forced problem $$-\Delta_p u - V(x)|u|^{p-2} u = f(x)$$ where $\Delta_p u := \textrm{div}\ (|\nabla u|^{p-2} \nabla u)$ and $f\in \mathcal{D}' ({\Omega})$, the space of distributions on the domain ${\Omega}$. Our assumptions are the following:\ **(A)** *$1<p<\infty$, $V\in L^r_{\emph{loc}} \ ({\Omega})$ with $r$ as in , ${\Omega}$ is a domain in $\mathbb{R}^N$, $V \geq 0$ and for all test functions $u\in \mathcal{D} ({\Omega}) \backslash \{0\}$, $$\label{eq1.1} \mathcal{Q}_V (u) := \int_{\Omega}|\nabla u|^p \,dx - \int_{\Omega}V|u|^p \,dx > 0.$$ There exists $1<q\leq p$ such that $p-1<q$, $$\label{eq1.2} r=1 \ (N<q), \quad 1<r<+\infty \ (N=q), \quad 1/r+(p-1)/q^\ast=1 \ (N>q)$$ and there exists $W \in \mathcal{C} ({\Omega})$, $W>0$, such that for all $u\in\mathcal{D}({\Omega})$, $$\label{eq1.3} {\displaystyle}\left(\int_{\Omega}(|\nabla u|^q + |u|^q)W\,dx\right)^{p/q} \leq \mathcal{Q}_V (u).$$* Let us recall that $q^\ast:=Nq/(N-q)$. Our first example of $V$ is the *quadratic Hardy potential* $(N\geq 3$, $p=2)$: $$\label{hardy1} V(x):=\left({N-2\over 2}\right)^2 |x|^{-2}.$$ The corresponding forced problem is solved in [@6] using the Brezis-Vazquez remainder term for the quadratic Hardy inequality ([@4] and [@10]). A second example is the *Hardy potential* $(1<p<N)$: $$\label{hardy2} V(x) := \left({N-p\over p}\right)^p |x|^{-p}.$$ The corresponding forced problem is partially solved in [@3] using the Abdellaoui-Colorado-Peral remainder term for the Hardy inequality [@1]. A third example is a potential $V \in L^\infty_{\textrm{loc}}({\Omega})$ such that (1.3) is satisfied with $p=q$ (see [@9]). If $p=2$, the natural energy space is the completion $H$ of $\mathcal{D}({\Omega})$ with respect to the norm $[\mathcal{Q}_V(u)]^{1/2}$, and it suffices to have $V\in L^1_{\textrm{loc}}({\Omega})$. Then $H$ is a Hilbert space with an obvious inner product. The following result is immediate: \[thma\] For each $f\in H^*$ (the dual space of $H$) the problem $$-\Delta u - V(x)u = f(x)$$ has a unique solution $u\in H$. This is an extension of Lemma 1.1$'$ in [@6] though the argument is exactly the same as there. As we shall see at the beginning of Section \[appl\], more can be said about $u$ and $H^*$ if $V$ is the Hardy potential in a bounded domain. When $p\neq 2$, one can expect no uniqueness as in Theorem \[thma\], see [@dem], pp. 11-12 and [@fhtt], Section 4. Hence the functional $[\mathcal{Q}_V(u)]^{1/p}$ is no longer convex, so it cannot serve as a norm, and the second conjugate functional $Q^{\ast\ast}_V$ was used in [@9] to define the energy space. Our goal is to generalize all the above results by using only the Hahn-Banach theorem to define an energy space and to obtain a priori bounds. We also consider the case of constant potential $$V\equiv {\lambda}_{1, p}({\Omega}):=\inf \left\{\int_{\Omega}|\nabla u |^p \,dx : u\in \mathcal{D}({\Omega}),\int_{\Omega}|u|^p \,dx=1\right\}$$ on the cylindrical domain ${\Omega}= \omega \times \mathbb{R}^M$ where $\omega\subset\r^{N-M}$ is bounded. Then we have $$\label{pi} {\lambda}_{1,p} ({\Omega})={\lambda}_{1,p}(\omega) > 0,$$ see Section \[po\]. The paper is organized as follows. In Section \[aeconv\] we prove an almost everywhere convergence result for the gradients. In Section \[approx\] we solve a sequence of approximate problems. In Section \[main\] we state and prove our main result. Section \[po\] is devoted to the proof of and to remainder terms for the Poincaré inequality. In Section \[appl\] we apply the main result to the potentials mentioned above. Almost everywhere convergence of the gradients {#aeconv} ============================================== Let us recall a classical result (see [@8]). \[lem2.1\] For every $1<p<2$ there exists $c>0$ such that for all $x,y\in\mathbb{R}^N$, $$c\, |x-y|^2\big/\left(|x|+|y|\right)^{p-2} \leq \left(|x|^{p-2} x-|y|^{p-2}y\right) \cdot (x-y).$$ For every $p\geq 2$ there exists $c>0$ such that for all $x,y\in \mathbb{R}^N$, $$c\, |x-y|^p\leq\left(|x|^{p-2} x-|y|^{p-2} y\right) \cdot (x-y).$$ We define the truncation $T$ by $$Ts:=s, \ |s|\leq 1; \quad Ts:=s/|s| , \ |s|>1.$$ The following theorem is a variant of a result which may be found e.g. in [@2], [@5], [@rak]. Here we provide a very simple argument, similar to that in [@5]. \[theo2.2\] Let $1<q\leq p$ and $(u_n) \subset W^{1,p}_{\textrm{loc}} ({\Omega})$ be such that for all $\omega \subset \subset \ {\Omega}$ 1. ${\displaystyle}\sup_n \ \|u_n\|_{W^{1,q}(\omega)} < +\infty$, 2. ${\displaystyle}\lim_{m,n \to\infty} \int_\omega \bigl(|\nabla u_m|^{p-2} \nabla u_m -\nabla u_n|^{p-2} \nabla u_n\bigr) \cdot \nabla T(u_m-u_n) \,dx = 0$. Then there exists a subsequence $(n_k)$ and $u\in W^{1,q}_{\textrm{loc}} ({\Omega})$ such that $$\label{eq2.1} u_{n_k} \to u\quad \text{and} \quad \nabla u_{n_k} \to \nabla u \quad \text{almost everywhere on }{\Omega}.$$ It suffices to prove that for all $\omega \subset \subset {\Omega}$ there exist a subsequence $(n_k)$ and $u\in W^{1,q}(\omega)$ satisfying a.e. on $\omega$ and to use a diagonal argument. Let $\omega \subset \subset {\Omega}$. By assumption (a), extracting if necessary a subsequence, we can assume that for some $u\in W^{1,q}(\omega)$, $$u_n \to u \ \textrm{in} \ L^q (\omega),\quad u_n \to u \ \textrm{a.e.} \ \textrm{on} \ \omega.$$ Let us define $$\begin{aligned} E_{m,n} & := \{x \in \omega : |u_m(x)-u_n(x)| < 1 \}, \\ e_{m,n} & := \left(|\nabla u_m|^{p-2} \nabla u_m - |\nabla u_n|^{p-2} \nabla u_n\right) \cdot \nabla (u_m-u_n). \end{aligned}$$ Then $e_{m,n}\ge 0$ and by assumption (b), $$\lim_{m,n\to\infty} \int_\omega e_{m,n} \chi_{E_{m,n}} \,dx = 0.$$ Since $\chi_{E_{m,n}}\to 1$ a.e. on $\omega$ as $m,n\to\infty$, it follows, extracting if necessary a subfamily, that $e_{m,n}\to 0$ a.e. on $\omega$ as $m,n\to\infty$. By Lemma \[lem2.1\], $|\nabla u_m -\nabla u_n|\to 0$ a.e. on $\omega$ as $m,n\to\infty$. Hence $\nabla u_n \to v$ a.e. on $\omega$. Since by assumption (a), $$\sup_n \|\nabla u_n\|_{L^q (\omega,\mathbb{R}^N)} < \infty,$$ it follows from Proposition 5.4.7 in [@11] that $$\nabla u_n \rightharpoonup v \ \textrm{in} \ L^q (\omega,\mathbb{R}^N).$$ We conclude that $v=\nabla u$. Approximate problems {#approx} ==================== In this section we assume that is satisfied, $V\in L^1_{\textrm{loc}}(\Omega)$ and $f\in W^{-1,p'} ({\Omega})$. Let $0<\varepsilon < 1$. We shall apply Ekeland’s variational principle to the functional $$\varphi(u) := {1\over p} \int_{\Omega}|\nabla u|^p \,dx -{1-\varepsilon\over p} \int_{\Omega}V|u|^p \,dx + {\varepsilon\over p} \int_{\Omega}|u|^p \,dx -\langle f,u\rangle$$ defined on $W^{1,p}_0 ({\Omega})$. By , for every $u\in W^{1,p}_0 ({\Omega})$, $V|u|^p \in L^1 ({\Omega})$ and $$\int_{\Omega}V|u|^p \,dx \leq \int_{\Omega}|\nabla u|^p \,dx.$$ Moreover, the functional $$u \mapsto \int_{\Omega}V|u|^p \,dx$$ is continuous and Gâteaux-differentiable on $W^{1,p}_0 ({\Omega})$ (see [@11], Theorem 5.4.1). It is clear that, on $W^{1,p}_0({\Omega})$, $$\label{eq3.1} -\|f\|_{W^{-1,p'}({\Omega})} \|u\|_{W^{1,p}({\Omega})} + {\varepsilon\over p} \|u\|^p_{W^{1,p}({\Omega})} \leq \varphi(u).$$ Hence $\varphi$ is bounded below and by Ekeland’s variational principle ([@wi], Theorem 2.4) there exists a sequence $(u_n)\subset W^{1,p}_0 ({\Omega})$ such that $$\label{eq3.2} \varphi (u_n) \to c := \displaystyle \inf_{W^{1,p}_0({\Omega})} \varphi \quad\text{and}\quad \varphi' (u_n)\to 0 \ \textrm{in} \ W^{-1,p'}({\Omega}).$$ We deduce from that $$\label{eq3.3} \sup_n \|u_n\|_{W^{1,p}({\Omega})} < +\infty.$$ Going if necessary to a subsequence, we can assume the existence of $u\in W^{1,p}_0({\Omega})$ such that $$\label{eq3.4} u_n \to u \quad \text{a.e. on } {\Omega}.$$ \[lem3.1\] Let $\zeta \in \mathcal{D}({\Omega})$. Then $$\lim_{m,n\to\infty} \int_{\Omega}\bigl[|\nabla u_m|^{p-2} \nabla u_m - |\nabla u_n|^{p-2} \nabla u_n\bigr] \cdot \zeta \nabla T(u_m-u_n) \,dx = 0.$$ Because of , we have that $$\label{eq3.5} -\Delta_p u_n - (1-\varepsilon) V |u_n|^{p- 2} u_n + \varepsilon |u_n|^{p- 2} u_n = f+g_n,$$ where $g_n \to 0$ in $W^{-1,p'}({\Omega})$. Testing with $\zeta T(u_n-u_m)$, we see that it suffices to prove that $$\begin{aligned} &\lim_{m,n\to\infty} \int_{\Omega}|\nabla u_m|^{p-1} |\nabla \zeta| \ |T(u_m-u_n)| \,dx = 0, \label{a1} \\ &\lim_{m,n\to\infty} \int_{\Omega}| u_m|^{p-1} | \zeta| \ |T(u_m-u_n)| \,dx = 0, \label{a2} \\ &\lim_{m,n\to\infty} \int_{\Omega}V| u_m|^{p-1} | \zeta| \ |T(u_m-u_n)| \,dx = 0. \label{a3}\end{aligned}$$ By and the fact that $|T(u_n-u_m)|\le 1$, $\lim_{m,n\to\infty}\int_{\textrm{spt}\, \zeta}|T(u_n-u_m)|\,dx = 0$. Hence $$\lim_{m,n\to\infty} \int_{\textrm{spt}\, \zeta} |T(u_m-u_n)|^{p} \,dx = 0, \quad \lim_{m,n\to\infty} \int_{\textrm{spt}\,\zeta} V|T(u_m-u_n)|^p \,dx = 0$$ and , follow from and Hölder’s inequality. Since $$\int_{\textrm{spt}\, \zeta}V|u_m|^{p-1}|T(u_m-u_n)| \,dx \le \left(\int_{\textrm{spt}\, \zeta}V|u_m|^p\,dx\right)^{(p-1)/p} \left(\int_{\textrm{spt}\, \zeta}V|T(u_m-u_n)|^p\,dx\right)^{1/p},$$ using and also follows. \[theo3.2\] There exists $u\in W^{1,p}_0({\Omega})$ such that $\varphi(u)= \displaystyle \inf_{W^{1,p}_0({\Omega})} \varphi$ and $\varphi'(u)=0$. Assumption (b) of Theorem \[theo2.2\] (with $q=p$) follows from Lemma \[lem3.1\]. Extracting a subsequence, we can assume that $$\label{eq3.6} \nabla u_n \to \nabla u \quad \text{a.e. on } {\Omega}.$$ By we have that, for every $\zeta \in \mathcal{D}({\Omega})$, $$\int_{\Omega}|\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \zeta \,dx-(1-\varepsilon) \int_{\Omega}V|u_n|^{p-2} u_n \zeta \,dx + \varepsilon \int_{\Omega}|u_n|^{p-2} u_n \zeta \,dx = \langle f+ g_n,\zeta\rangle.$$ Using , , and Proposition 5.4.7 in [@11], we obtain $$\int_{\Omega}|\nabla u|^{p-2} \nabla u \cdot \nabla \zeta \,dx-(1-\varepsilon) \int_{\Omega}V|u|^{p-2} u \zeta \,dx + \varepsilon \int_{\Omega}|u|^{p-2} u \zeta \,dx = \langle f, \zeta\rangle,$$ so that $\varphi'(u)=0$. As in [@7], the homogeneity of $\mathcal{Q}_V$ implies $$\begin{aligned} \inf_{W^{1,p}_0 ({\Omega})} \varphi &= \lim_{n\to\infty} \varphi (u_n) \\ &= \lim_{n\to\infty} \left[{1\over p} \langle \varphi' (u_n),u_n \rangle +\left({1\over p}-1\right) \langle f,u_n\rangle\right] \\ &= \left({1\over p}-1\right) \langle f,u\rangle = \varphi (u) - {1\over p} \langle \varphi'(u),u\rangle = \varphi (u).\end{aligned}$$ Main result {#main} =========== In this section we assume that (A) is satisfied and we define, on $\mathcal{D}'({\Omega})$, $$\|f\| := \sup \{\langle f,u \rangle : u\in \mathcal{D}({\Omega}), \ \mathcal{Q}_V (u)=1\}$$ so that $$\label{eq4.1} \langle f,u \rangle \leq \|f\| \ [\mathcal{Q}_V(u)]^{1/p}.$$ On the spaces $$X:=\{f\in \mathcal{D}'({\Omega}) : \|f\| < \infty \} \quad\text{and}\quad Y:=W^{1,q}_0 ({\Omega},Wdx)$$ we respectively use the norm defined above and the natural norm. Note that the space $X$ has been introduced by Takáč and Tintarev in [@9]. \[lem4.1\] Let $f\in Y^\ast$. Then $f\in X$ and $$\|f\| \leq \|f\|_{Y^\ast}.$$ Let $u\in \mathcal{D}({\Omega})$. By assumption we have $$\langle f,u \rangle \leq \|f\|_{Y^\ast} \|u\|_Y \leq \|f\|_{Y^\ast} [\mathcal{Q}_V (u)]^{1/p}.$$ \[lem4.2\] *(a)* Let $u\in W^{1,p}_0({\Omega})$. Then $$\|u\|_Y \leq \|u\|_{X^\ast} \leq [\mathcal{Q}_V (u)]^{1/p} \leq \|u\|_{W^{1,p}}({\Omega}).$$ *(b)* Let $f\in X$. Then $f\in W^{-1,p'}({\Omega})$ and $$\|f\|_{W^{-1,p'}({\Omega})} \leq \|f\|.$$ \(a) Let $u\in \mathcal{D}({\Omega})$. Using the Hahn-Banach theorem and the preceding lemma, we obtain $$\|u\|_Y = \sup_{f\in Y^\ast \atop \|f\|_{Y^\ast}\leq 1} \langle f,u \rangle \leq \sup_{f\in X \atop \|f\| \leq 1} \langle f,u \rangle = \|u\|_{X^\ast}.$$ It follows from that $$\sup_{f\in X \atop \|f\|\leq 1} \langle f,u \rangle \leq [\mathcal{Q}_V (u)]^{1/p}.$$ Since $V\geq 0$, it is clear that $$[\mathcal{Q}_V (u)]^{1/p} \leq \|u\|_{W^{1,p}({\Omega})}.$$ Now it is easy to conclude by density of $\mathcal{D}({\Omega})$. \(b) If $f\in X$ and $u\in \mathcal{D}({\Omega})$, then $$\langle f,u \rangle \leq \|f\| [\mathcal{Q}_V(u)]^{1/p} \leq \|f\| \ \|u\|_{W^{1,p}({\Omega})}.$$ Let $f\in X$ and let $(\varepsilon_n)\subset \ ]0,1[$ be such that $\varepsilon_n \downarrow 0$. Then $f\in W^{-1,p'}(\Omega)$, so by Theorem \[theo3.2\], for every $n$ there exists $u_n\in W^{1,p}_0({\Omega})$ such that $$\label{eq4.2} -\Delta_p u_n - (1-\varepsilon_n)V |u_n|^{p-2} u_n + \varepsilon_n |u_n|^{p-2} u_n = f$$ and $u_n$ minimizes the functional $$\varphi_n(v):={1\over p} \int_{\Omega}|\nabla v|^p \,dx - {1-\varepsilon_n\over p} \int_{\Omega}V|v|^p \,dx + {\varepsilon_n\over p} \int_{\Omega}|v|^p \,dx-\langle f,u \rangle.$$ on $W^{1,p}_0 ({\Omega})$. In fact below we shall not use the minimizing property of $u_n$ but only the fact that holds. \[lem4.3\] Let $f\in X$. Then $$\sup_n \|u_n\|_Y \leq \sup_n \mathcal{Q}_V (u_n) < \infty.$$ Lemma \[lem4.2\] and equation imply that $$\begin{aligned} \|u_n\|^p_{X^\ast} & \leq \mathcal{Q}_V (u_n) \leq \mathcal{Q}_V (u_n) + \varepsilon_n \int_{\Omega}(V+1) |u_n|^p \,dx \\ &= \langle f,u_n \rangle \leq \|f\|_X \|u_n\|_{X^\ast}.\end{aligned}$$ Since $p>1$, we obtain the conclusion. Going if necessary to a subsequence, we can assume the existence of $u\in Y$ such that $$\label{eq4.3} u_n \to u \quad \text{a.e. on }{\Omega}.$$ \[lem4.4\] Let $\zeta\in \mathcal{D}({\Omega})$. Then $$\lim_{m,n \to \infty} \int_{\Omega}\left[|\nabla u_m |^{p-2} u_m - |\nabla u_n|^{p-2} u_n\right] \cdot \zeta \nabla T (u_m-u_n) \,dx = 0.$$ Because of , as in the proof of Lemma \[lem3.1\] it suffices to show that $$\begin{aligned} &\lim_{m,n\to\infty} \int_{\Omega}|\nabla u_m|^{p-1} |\nabla \zeta| \ |T(u_m-u_n)|\,dx=0, \\ &\lim_{m,n\to\infty} \int_{\Omega}| u_m|^{p-1} | \zeta| \ |T(u_m-u_n)|\,dx=0, \\ &\lim_{m,n\to\infty} \int_{\Omega}V| u_m|^{p-1} |\zeta| \ |T(u_m-u_n)|\,dx=0. \end{aligned}$$ We assume that $N>q$. The other cases are similar but simpler. By Lemma \[lem4.3\], $(u_n)$ is bounded in $W^{1,q}_{\textrm{loc}} ({\Omega})$, so by the Sobolev theorem, $(u_n)$ is bounded in $L^{q^\ast}_{\textrm{loc}} ({\Omega})$. Since by , $$\lim_{m,n\to\infty} \int_{\textrm{spt} \, \zeta} |T(u_m-u_n)|^{({q\over p-1})'} \,dx=0 \quad{and}\quad \lim_{m,n\to\infty} \int_{\textrm{spt}\ \zeta} V^r |T(u_m-u_n)|^r \,dx=0,$$ it is easy to conclude using and Hölder’s inequality. Note in particular that $$\int_{\textrm{spt}\, \zeta}V|u_m|^{p-1}|T(u_m-u_n)| \,dx \le \left(\int_{\textrm{spt}\, \zeta}|u_m|^{q^*}\,dx\right)^{(p-1)/q^*} \left(\int_{\textrm{spt}\, \zeta}V^r|T(u_m-u_n)|^r\,dx\right)^{1/r}.$$ \[theo4.5\] Assume (A) is satisfied and $f\in \mathcal{D}' ({\Omega})$ is such that $$\label{cond} \sup \{ \langle f,u \rangle : u\in \mathcal{D}({\Omega}),\ \mathcal{Q}_V (u)=1\} < +\infty.$$ Then there exists $u\in W^{1,q}_0 ({\Omega},Wdx)$ such that, in $\mathcal{D}' ({\Omega})$, $$\label{eq} -\Delta_p u-V(x) |u|^{p-2} u = f(x).$$ Let $\zeta \in \mathcal{D}({\Omega})$. By we have $$\label{eq4.4} \int_{\Omega}|\nabla u_n|^{p-2} \nabla u_n \cdot \nabla \zeta \,dx-(1-\varepsilon_n) \int_{\Omega}V |u_n|^{p-2} u_n \zeta \,dx + \varepsilon_n \int_{\Omega}|u_n|^{p-2} u_n \zeta \,dx = \langle f,\zeta \rangle.$$ Let us recall that $(u_n)$ is bounded in $W^{1,q}_{\textrm{loc}}({\Omega})$ and $u_n\to u$ a.e. on ${\Omega}$. Assumption (b) of Theorem \[theo2.2\] follows from Lemma \[lem4.4\]. Extracting a subsequence, we can assume that $$\nabla u_n \to \nabla u \quad\text{a.e. on }{\Omega}.$$ We assume that $N>q$ and we choose $\omega$ such that $$\textrm{spt}\, \zeta \subset \omega \subset\subset {\Omega}.$$ Using Proposition 5.4.7 in [@11], we obtain $$|\nabla u_n|^{p-2} \nabla u_n \rightharpoonup |\nabla u|^{p-2} \nabla u \ \ \textrm{in} \ L^{{q\over p-1}} (\omega) \quad\text{and}\quad |u_n|^{p-2} u_n \rightharpoonup |u|^{p-2} u \ \ \textrm{in} \ L^{{q^\ast\over p-1}}(\omega).$$ It follows then from that $$\int_{\Omega}|\nabla u|^{p-2} \nabla u \cdot \nabla \zeta \,dx - \int_{\Omega}V|u|^{p-2} u \zeta \,dx = \langle f,\zeta \rangle.$$ The following variant of Theorem \[theo4.5\] will be needed in one of the applications in Section \[appl\], see Theorem \[thmb\]. \[cor1\] Theorem \[theo4.5\] remains valid if we replace (case $N>q$) in (A) by the conditions $$\label{eqe} \begin{aligned} & V\in L^r_{\textrm{loc}}(\Omega) \quad \text{where }1/r+(p-1)(p-q)/q = 1 \\ & \text{and} \quad \io V^{q/p}|u|^q\,dx \le C\io|\nabla u|^q\,dx \quad \text{for some }C>0. \end{aligned}$$ The argument is similar except that we must show that the third limit in the proof of Lemma \[lem4.4\] is zero also when holds and that we can pass to the limit in the second integral in . Using Lemma \[lem4.3\], Hölder’s inequality, and the fact that $r=q/[(q-p+1)p]\ge 1$, we obtain $$\begin{aligned} \lim_{m,n\to\infty} \int_{\textrm{spt} \, \zeta} V| u_m|^{p-1} |\zeta| & \, |T(u_m-u_n)|\,dx \le \lim_{m,n\to\infty}\left(\int_{\textrm{spt} \, \zeta} V^{q/p}|u_m|^q\,dx\right)^{(p-1)/q} \times \\ & \times \left(\int_{\textrm{spt} \, \zeta} V^{r}|T(u_m-u_n)|^{pr}\,dx\right)^{1/(pr)} = 0.\end{aligned}$$ Let $E\subset \textrm{spt} \, \zeta$. Similarly as above, we have $$\begin{aligned} \int_EV|u_n|^{p-1}\,dx \le \left(\int_E V^{q/p}|u_n|^q\,dx\right)^{(p-1)/q} \left(\int_E V^{r}\,dx\right)^{1/(pr)} \le D \left(\int_E V^{r}\,dx\right)^{1/(pr)}.\end{aligned}$$ Since the integrand on the right-hand side is in $L^1(\textrm{spt} \, \zeta)$, it follows that $V|u_n|^{p-1}$ are uniformly integrable and we can pass to the limit in the second integral in according to the Vitali theorem, see e.g. [@11 Theorem 3.1.9]. Note that in the case $q=p$ this result is stronger than Theorem \[theo4.5\] because $V\in L^1_{\text{loc}}(\Omega)$ is allowed for any $p$. Poincaré inequality with remainder term {#po} ======================================= Let $\Omega := \omega\times \r^M$, where $\omega$ is a domain in $\r^{N-M}$ and $N>M>p$. For $x\in \Omega$ we shall write $x=(y,z)$, where $y\in\omega$ and $z\in \rm$. Recall from the introduction that $$\lambda_{1,p}(\Omega) := \inf\left\{\io|\nabla u|^p\,dx: u\in\calD(\Omega),\ \io|u|^p\,dx = 1\right\}.$$ It is well known that $\lambda_{1,p}(\omega) = \lambda_{1,p}(\Omega)$ if $p=2$, see e.g. [@est], Lemma 3. We shall show that this is also true for general $p\in(1,\infty)$. \[lem1\] $\lambda_{1,p}(\Omega) = \lambda_{1,p}(\omega)$. First we show that $\lambda_{1,p}(\Omega) \ge \lambda_{1,p}(\omega)$. Let $u\in\calD(\Omega)$, $\|u\|_{L^p(\Omega)}=1$. Then $$\begin{aligned} \io|\nabla u|^p\,dx & \ge \io|\nabla_y u|^p\,dx = \irm dz\iom|\nabla_yu|^p\,dy \\ & \ge \irm dz \,\lambda_{1,p}(\omega)\iom|u|^p\,dy = \lambda_{1,p}(\omega)\io|u|^p\,dx = \lambda_{1,p}(\omega).\end{aligned}$$ Taking the infimum on the left-hand side we obtain the conclusion. To show the reverse inequality, let $u(x)=v(y)w(z)$, where $v\in \calD(\omega)\setminus\{0\}$ and $w\in \calD(\r^M)\setminus\{0\}$. For each $\eps>0$ there exists $C_\eps>0$ such that $$|\nabla u|^p \le (|w|\,|\nabla_yv|+|v|\,|\nabla_zw|)^p \le (1+\eps)|w|^p|\nabla_yv|^p + C_\eps |v|^p|\nabla_zw|^p.$$ Hence $$\lambda_{1,p}(\Omega) \le \frac{\io|\nabla u|^p\,dx}{\io|u|^p\,dx} \le \frac{(1+\eps)\iom|\nabla_yv|^p\,dy}{\iom|v|^p\,dy} + \frac{C_\eps\irm|\nabla_zw|^p\,dz}{\irm|w|^p\,dz}.$$ Taking the infimum with respect to $v$ and $w$, we obtain $$\lambda_{1,p}(\Omega)\le (1+\eps)\lambda_{1,p}(\omega).$$ Since $\eps$ has been chosen arbitrarily, it follows that $\lambda_{1,p}(\Omega)\le \lambda_{1,p}(\omega)$. Now we state the main result of this section. \[poincare\] For each $u\in \calD(\Omega)$ the following holds:\ *(a)* If $p\ge 2$, then $$\io(|\nabla u|^p-\lambda_{1,p}(\Omega)|u|^p)\,dx \ge \left(\frac{M-p}p\right)^p\io\frac{|u|^p}{|z|^p}\,dx.$$ *(b)* If $1<p<2$, then $$\io(|\nabla u|^p-\lambda_{1,p}(\Omega)|u|^p)\,dx \ge 2^{(p-2)/2} \left(\frac{M-p}p\right)^p\io\frac{|u|^p}{|z|^p}\,dx.$$ \(a) Let $p\ge 2$. Then $(a+b)^{p/2} \ge a^{p/2}+b^{p/2}$ for all $a,b\ge 0$, hence $$|\nabla u|^p \ge |\nabla_yu|^p+|\nabla_zu|^p.$$ Using this, Lemma \[lem1\] and Hardy’s inequality in $\rm$, we obtain $$\begin{aligned} \io(|\nabla u|^p & -\lambda_{1,p}(\Omega)|u|^p)\,dx \\ & \ge \irm dz\iom(|\nabla_yu|^p-\lambda_{1,p}(\Omega) |u|^p)\,dy + \iom dy\irm|\nabla_zu|^p\,dz \\ & \ge \iom dy\irm|\nabla_zu|^p\,dz \ge \iom dy \left(\frac{M-p}p\right)^p\irm \frac{|u|^p}{|z|^p}\,dz \\ & = \left(\frac{M-p}p\right)^p\io\frac{|u|^p}{|z|^p}\,dx.\end{aligned}$$ \(b) Let $1<p<2$. It is easy to see that $(a+b)^{p/2} \ge 2^{(p-2)/2} (a^{p/2}+b^{p/2})$ for such $p$ and all $a,b\ge 0$. Hence $$|\nabla u|^p \ge2^{(p-2)/2}\left( |\nabla_yu|^p+|\nabla_zu|^p\right)$$ and we can proceed as above. Here we have not excluded the case $ \lambda_{1,p}(\omega)=0$ but the result is only interesting if $\lambda_{1,p}(\omega)$ is positive. A sufficient condition for this is that $\omega$ has finite measure. Applications {#appl} ============ In this section we work out some applications of Theorem \[theo4.5\] for the potentials $V$ mentioned in the introduction. Let $\Omega$ be a domain in $\rn$. If $\Omega$ is bounded, $0\in\Omega$ and $V$ is the quadratic Hardy potential , then more can be said about the solution $u$ and the space $H^*$ in Theorem \[thma\]. Given $1\le q<2$, we have $$\io |\nabla u|^2\,dx - \io V(x)u^2\,dx \ge C(q,\Omega)\|u\|_{W^{1,q}(\Omega)}^2$$ for all $u\in\calD(\Omega)$ ([@10], Theorem 2.2). We see that $V\in L^r(\Omega)$ if $N/(N-1)<q<2$ ($r$ is as in ) and holds with constant $W$. So $H^*=X$ and we also have $Y=W^{1,q}_0(\Omega)$ where $X,Y$ are as in Section \[main\]. Hence by Lemma \[lem4.2\] and Theorem \[theo4.5\], $H^*\subset W^{-1,2}(\Omega)$ and the solution $u$ is in $W^{1,q}_0(\Omega)$ for any $1\le q<2$. Let $x=(y,z)\in\Omega\subset \r^k\times\r^{N-k}$, where $N\ge k>p>1$, and consider the Hardy potential $$V(x) := \left(\frac{k-p}p\right)^p|y|^{-p}.$$ \[thmb\] Let $\Omega$ be a bounded domain containing the origin and let $V$ be the Hardy potential above. Then for each $f$ satisfying and each $1\le q<p$ there exists a solution $u\in W^{1,q}_0(\Omega)$ to . Recall from Lemma \[lem4.2\] that if $f$ satisfies , then $f\in W^{-1,p'}(\Omega)$. According to Lemma 2.1 in [@3] (see also [@1], Theorem 1.1), for each $1<q<p$ there exists a constant $C(q,\Omega)$ such that $$\mathcal{Q}_V(u) \ge C(q,\Omega)\left(\io|\nabla u|^q\,dx\right)^{p/q}, \quad u\in\calD(\Omega).$$ So the Poincaré inequality implies that holds for some constant $W$. Since also holds if $q$ is sufficiently close to $p$, we obtain the conclusion using Theorem \[cor1\] ($k(p-1)/(k-1)<q<p$ is needed in order to have $V\in L^r_{\text{loc}}(\Omega)$ with $r$ as in ). This result extends Theorem 3.1 in [@3] where it was assumed that $f\in L^\gamma(\Omega)$ for some $\gamma>(p^*)'$. In our next theorem we essentially recover the main result (Theorem 4.3) of [@9]. \[thmc\] Let $\Omega$ be a domain and let $V\in L^\infty_{\emph{loc}}(\Omega)$, $V\ge 0$. Suppose that $$\label{tt} \mathcal{Q}_V(u) \ge \io\wt W|u|^p\,dx \quad \text{for all } u\in\calD(\Omega) \text{ and some } \wt W\in \mathcal{C}(\Omega), \ \wt W>0.$$ Then for each $f$ satisfying there exists a solution $u\in W^{1,p}_{\emph{loc}}(\Omega)$ to . According to Proposition 3.1 in [@9] (see also (2.6) there), implies that $\mathcal{Q}_V$ satisfies with $q=p$. Hence our Theorem \[theo4.5\] applies. \[thmd\] Let $\Omega = \omega\times \r^M$, where $\omega\subset \r^{N-M}$ is a domain such that $\lambda_{1,p}(\omega)>0$, $N>M>p$, and let $V(x) = \lambda_{1,p}(\Omega)$. Then for each $f$ satisfying there exists a solution $u\in W^{1,p}_{\emph{loc}}(\Omega)$ to . Let $x=(y,z)\in\omega\times\r^M$. It follows from Theorem \[poincare\] that $$\mathcal{Q}_V(u) = \io|\nabla u|^p\,dx-\lambda_{1,p}(\Omega)\io|u|^p\,dx \ge \io \wt W|u|^p\,dx \quad \text{for all }u\in\calD(\Omega),$$ where $\wt W(x)=C_p/(1+|z|^p)$ and $C_p$ is the constant on the right-hand side of respectively (a) and (b) of Theorem \[poincare\]. So the conclusion follows from Theorem \[thmc\]. Below we give an example showing that the solution we obtain need not be in $W^{1,p}(\Omega)$. *Let $\Omega = \omega\times \rm$, where $\omega\subset\r^{N-M}$ is such that $\lambda_1 := \lambda_{1,2}(\omega) > 0$ and put $$\mathcal{Q}(u) := \io(|\nabla u|^2-\lambda_1u^2)\,dx.$$ By the definition of $\lambda_1$, for each $n$ we can find $u_n\in \calD(\Omega)$ such that $$\left(1-\frac1n\right)\io|\nabla u_n|^2\,dx \le \lambda_1\io u_n^2\,dx.$$ Hence $$Q(u_n) \le \frac1n\io |\nabla u_n|^2\,dx.$$ By normalization, we can assume $$Q(u_n) = \frac1{n^2}.$$ Translating along $\rm$, we may assume $\text{spt\,}u_n\cap \text{spt\,}u_m =\emptyset$ if $n\ne m$. We define $$f_n := -\Delta u_n-\lambda_1u_n \quad \text{and} \quad f:= \sum_{n=1}^\infty f_n \quad \text{in }\calD'(\Omega).$$ Then $f\in X$ because $$\|f\| = \left(\sum_{n=1}^\infty \frac1{n^2}\right)^{1/2} < \infty,$$ and $u = \sum_{n=1}^\infty u_n$ is a weak solution for the equation $$-\Delta u -\lambda_1 u = f(x).$$ Moreover, $u$ is the unique solution in $H$ as follows from Theorem \[thma\]. But $$\io|\nabla u|^2\,dx = \sum_{n=1}^\infty\io|\nabla u_n|^2\,dx \ge \sum_{n=1}^\infty \frac1n = +\infty,$$ so $u\not\in W^{1,2}(\Omega)$.* [99]{} Abdellaoui, B., Colorado, E., Peral, I., “Some improved Caffarelli-Kohn-Nirenberg inequalities", *Calc. Var. Partial Differential Equations* 23 (2005), 327-345. Boccardo, L., Murat, F., “Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations", *Nonlinear Anal.* 19 (1992), 581-597. Brandolini, B., Chiacchio, F., Trombetti, C., “Some remarks on nonlinear elliptic problems involving Hardy potentials", *Houston J. Math.* 33 (2007), 617-630. Brezis, H., Vázquez, J., “Blow-up solutions of some nonlinear elliptic problems", *Rev. Mat. Univ. Complut. Madrid* 10 (1997), 443-469. de Valeriola, S., Willem, M., “On some quasilinear critical problems", *Adv. Nonlinear Stud.* 9 (2009), 825-836. del Pino, M., Elgueta, M., Manásevich, R., “A homotopic deformation along $p$ of a Leray-Schauder degree result and existence for $(|u'|^{p-2}u')'+f(t,u)=0$, $u(0)=u(T)=0$, $p>1$, *J. Diff. Eq.* 80 (1989), 1-13. Dupaigne, L., “A nonlinear elliptic PDE with the inverse square potential", *J. Anal. Math.* 86 (2002), 359-398. Esteban, M.J., “Nonlinear elliptic problems in strip-like domains: symmetry of positive vortex rings”, *Nonl. Anal.* 7 (1983), 365-379. Fleckinger-Pellé, J., Hernández, J., Taká[č]{}, P., de Thélin, F., “Uniqueness and positivity for solutions of equations with the $p$-Laplacian”, *Lecture Notes in Pure and Appl. Math.* 194, Dekker, New York, 1998, pp. 141–155. Garcia Azorero J.P., Peral Alonso, I., “Hardy inequalities and some critical elliptic and parabolic problems", *J. Diff. Eq.* 144 (1998), 441-476. Rakotoson, J.M., “Quasilinear elliptic problems with measures as data”, *Diff. Int. Eq.* 4 (1991), 449-457. Simon, J., “Régularité de la solution d’une équation non linéaire dans $\mathbb{R}^N$, *Lecture Notes in Math.* 665, Springer, Berlin, 1978, 205-227. Takáč, P., Tintarev, K., “Generalized minimizer solutions for equations with the $p$-Laplacian and a potential term", *Proc. Roy. Soc. Edinburgh Sect. A* 138 (2008), 201-221. Vazquez, J., Zuazua, E., “The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential", *J. Funct. Anal.* 173 (2000), 103-153. Willem, M., “Minimax Theorems”, Birkhäuser, Boston, 1996. Willem, M., “Functional Analysis. Fundamentals and Applications", Birkhäuser/Springer, New York, 2013.

ArXiv

--- abstract: 'We analyze the origin of ferromagnetism as a result of carrier mediation in diluted magnetic oxide semiconductors in the light of the experimental evidence reported in the literature. We propose that a combination of percolation of magnetic polarons at lower temperature and Ruderman-Kittel-Kasuya-Yosida ferromagnetism at higher temperature may be the reason for the very high critical temperatures measured (up to $\sim 700$K).' author: - 'M.J. Calderón, S. [Das Sarma]{}' title: Theory of carrier mediated ferromagnetism in dilute magnetic oxides --- Introduction ============ Semiconductors doped with magnetic ions are being studied in an effort to develop spintronics, the new kind of electronics that seeks to exploit, in addition to the charge degree of freedom as in the usual electronics, also the spin of the carriers [@zutic-review]. The first so-called diluted magnetic semiconductors (DMS) were II-VI semiconductor alloys like Zn$_{1-x}$Mn$_x$Te and Cd$_{1-x}$Mn$_x$Te [@furdyna88] originally studied in the 1980s. These materials are either spin glasses or have very low ferromagnetic (FM) critical temperatures $T_C$ ($\sim$ few K) [@ferrand01] and are, therefore, inadequate for technological applications which would require FM order at room temperature. More recently, the Mn doped III-V semiconductors In$_{1-x}$Mn$_x$As [@munekata89; @ohno92] and Ga$_{1-x}$Mn$_x$As [@ohno96; @jungwirthRMP06] showed ferromagnetism at a much higher temperature, thanks to the development of molecular beam epitaxy (MBE)-growth techniques. The current high $T_C$ record of $173$K achieved in Mn-doped GaAs by using low temperature annealing techniques [@wang02; @edmonds02; @chiba03] is promising, but still too low for actual applications. In all these materials, ferromagnetism has been proven to be carrier mediated, a necessary property for spintronics since this enables the modification of magnetic behavior through charge manipulation. This has motivated a search for alternative spintronics materials with even higher $T_{\rm C}$ and carrier mediated FM. In this direction, dilute magnetic oxides [@spaldin-review], such as magnetically-doped TiO$_2$ [@matsumoto01], ZnO [@ueda01], and SnO$_2$ [@ogale03], could represent this alternative with reported $T_{\rm C}$s above room temperature and as high as $\sim 700$K [@shinde03]. The general formula for these oxide based dilute magnetic semiconductors (O-DMS) is $$A_{1-x}M_x O_{n-\delta} \, , \nonumber$$ where $A$ is a non-magnetic cation, $M$ is a magnetic cation, $\delta$ is the concentration of oxygen vacancies which depends on the growth conditions, and $n= 1$ or $2$. Carriers, usually electrons, are provided by the oxygen vacancies that are believed to act as shallow donors [@forro94; @tang94]. This is in contrast to III-V semiconductors, like Ga$_{1-x}$Mn$_x$As, where the carriers (holes in this case) are provided by the magnetic impurities themselves which act also as donors (or acceptors). More oxygen vacancies and, thus, more carriers, are produced at low oxygen pressure [@toyosaki04; @venkatesan04]. In the process of doping with magnetic ions, usually of different valence than the ion they substitute for, oxygen or $A$ vacancies are also introduced to maintain charge neutrality. However, these oxygen vacancies have been found not to contribute directly to the electrical conductivity of the system [@chambers03]. On the contrary, the resistivity increases by orders of magnitude upon doping [@ogale03; @shinde03]. There is currently no consensus on the origin of ferromagnetism in O-DMS, in particular, whether it is an extrinsic effect due to direct interaction between the local moments in magnetic impurity clusters (or nanoclusters) or is indeed an intrinsic property caused by exchange coupling between the spin of the carriers and the local magnetic moments. This is a very important issue because spintronics requires the carriers to be polarized and this would only be guaranteed if ferromagnetism is intrinsic. Experimental evidence for carrier-mediated FM in O-DMS is not yet conclusive. In the much studied Co-doped TiO$_2$ in the rutile phase, anomalous Hall effect (AHE) [@toyosaki04] and electric field induced modulation of magnetization (by as much as $13.5\%$) [@zhao-FE-05] have been observed, arguing for carrier-mediated FM. However, AHE was also measured in a sample with magnetic clusters, casting doubts about the conclusions that can be drawn from AHE data [@shinde-HE-04]. Nevertheless, room temperature FM has also been reported in cluster free films grown or annealed at high temperatures $\sim 900 ^{o}$C  [@shinde03] by pulsed laser deposition, and in nanocrystalline films grown in conditions that preclude the formation of metallic cobalt [@chambers04]. Another piece of evidence in favor of carrier-mediated ferromagnetism is the observation that the magnetic field dependence of ferromagnetic circular dichroism is in good agreement with those of magnetization and anomalous Hall effect [@toyosaki05]. More recently, X-ray photoemission spectroscopy measurements have suggested strong hybridization between carriers in the Ti 3d band and the localized t$_{2g}$ states of Co$^{2+}$ [@quilty06]. Many reports have raised serious doubts on the magnetism of magnetically-doped ZnO [@rao05] since the results are very sensitive to sample preparation. However, it has been pointed out [@rao05; @spaldin04] that the lack of ferromagnetism in some samples can be the result of too low a density of carriers. Indeed, films of Zn$_{0.75}$Co$_{0.25}$O prepared in low oxygen partial pressure ($< 10^{-6}$ Torr), a condition that should increase the density of electrons from oxygen vacancies, were found to be ferromagnetic at room temperature [@rode03]. These samples exhibited perpendicular magnetic anisotropy [@rode03; @dinia05] and no segretion effects were significant. In the same direction, doping Zn$_{0.98}$Co$_{0.02}$O with small amounts of Cu also enhanced the system’s ferromagnetism [@lin04]. The systematic variation of magnetism in doped ZnO as a function of the magnetic dopant has been explored in Ref. [@venkatesan04], where room-temperature ferromagnetism has been found in films doped with Sc, Ti, V, Fe, Co, or Ni but not with Cr, Mn, or Cu. The same group also reported an increase in the magnetic moment per Co ion as a function of the reduction of oxygen pressure (equivalent to an increase of the density of carriers) [@venkatesan04]. Optical magnetic circular dichroism, one of the tests proposed as a signature of diluted ferromagnetism [@zhao-FE-05], has also been measured at low temperatures [@ando01], and at room temperature [@neal06]. Very recently, by analyzing the controlled introduction and removal of interstitial Zn, carrier-mediated ferromagnetism in Co-doped ZnO has been demonstrated [@kittilstved06]. The case of magnetically-doped SnO$_2$ (with $T_C \sim 600$K [@ogale03; @coey04]) seems to be different from the previous two materials in that the parent compound is highly conducting [@chopra83] (though still transparent) and that some doped samples have shown an extremely large magnetic moment ($>7 \mu_B$) per magnetic impurity [@ogale03; @coey04]. It thus appears that different doped magnetic oxides may very well have different underlying mechanisms leading to the observed ferromagnetism – in particular, there may very well be several competing FM mechanisms in play in O-DMS, a novel theme we develop further here. In this article, we address the origin of ferromagnetism as mediated by carriers in O-DMS analyzing the existing experimental evidence for magnetic and transport properties, in particular, in Co-doped TiO$_2$. Our goal is to develop a theory for O-DMS ferromagnetism (assuming it to be mediated by band or impurity-band carriers) in analogy with the better understood III-V Mn-doped DMS materials [@jungwirthRMP06; @dietlreview02; @timm03; @dassarmaSSC03; @macdonaldNatMat]. Magnetically doped III-V semiconductors are ferromagnetic for a wide range of carrier concentrations, from the insulating to highly conducting regimes, where different mechanisms are expected to prevail. It has been proposed that for high enough values of the local exchange between the carriers and the magnetic ions, an impurity band is formed [@chatto01; @calderon02]. In a highly insulating system, the Fermi level is well below the mobility edge of the impurity band. In this regime, ferromagnetism has been explained as the result of percolation of bound magnetic polarons [@kaminski02]. This mechanism is consistent with the concave shaped magnetization versus temperature $M(T)$ curves observed, for instance, in (In,Mn)As [@ohno92]. In the more conducting samples, itinerant carriers would mediate ferromagnetism via a Ruderman-Kittel-Kasuya-Yosida (RKKY) mechanism whose sign fluctuations, and the associated frustration effects, are suppressed due to the low density of carriers [@priour04], arising possibly from heavy compensation and/or existence of defects. Given that the carriers in magnetic oxides reside in an insulating impurity band, there are essentially three kinds of carrier-mediated magnetic exchange interactions which could potentially lead to intrinsic carrier-mediated ferromagnetism: double exchange (similar to the situation in manganites) [@zener] in the impurity band, bound magnetic polarons percolation (similar to the situation in insulating diluted magnetic semiconductors, e.g. Ge$_{1-x}$Mn$_x$, In$_{1-x}$Mn$_x$As, and low-$x$ Ga$_{1-x}$Mn$_x$As)  [@kaminski02], and indirect (RKKY) exchange coupling mediated by free carriers (similar to the situation in the optimally doped $x \approx 0.05$ high-$T_{\rm C}$ metallic Ga$_{1-x}$Mn$_x$As) [@priour04]. The double exchange mechanism gives, at the low carrier density of magnetic oxides, $T_{\rm C}$ proportional to the carrier density; therefore, critical temperatures exceeding room temperature ($\sim 300 K$) are essentially impossible within this model. In general, the carrier-mediated indirect RKKY exchange mechanism applies only to free carriers in an itinerant band, and therefore may be thought to be ruled out for doped magnetic oxides which are insulating impurity band materials. But we propose here that the ’standard’ RKKY mechanism may very well be playing a role in the magnetic oxides, particularly at high temperatures, where a large number of localized impurity band carriers will be thermally activated (due to the small values of the carrier binding energies in the magnetic oxides) either within the impurity band or to the conduction band, effectively becoming itinerant free carriers (albeit thermally activated ones) which can readily participate in the indirect RKKY exchange between the localized impurity magnetic moments. If the exchange coupling between local moments and the thermally activated carriers is sufficiently large, then this mechanism could explain the very high $T_C$ observed in magnetic oxides. We emphasize that $T_C$ also depends on other parameters including the activated carrier density and the magnetic moment density. At low temperature, the RKKY mechanism must freeze out since thermal activation from the impurity band is no longer operational and one must therefore have a complementary mechanism to provide carrier-mediated ferromagnetism. We believe that the bound magnetic polaron (BMP) mechanism is the plausible low temperature magnetic ordering mechanism, but it cannot explain the high (room temperature or above) claimed $T_C$ of O-DMS unless one uses completely unphysical parameters. Our new idea in this work is, therefore, the suggestion that the intrinsic carrier-mediated ferromagnetism leading to high Curie temperatures is plausible in doped magnetic oxides only if two [*complementary*]{} magnetic mechanisms (i.e. the bound magnetic polaron percolation at low temperatures and the indirect RKKY exchange mechanism in the presence of substantial thermal activation of carriers in the impurity band) are operating in parallel. We find that no single carrier-mediated mechanism, by itself, can account for the observed high $T_C$ in doped magnetic oxides. These considerations apply mainly to TiO$_2$ as high-$T$ resistivity measurements indicate existence of substantial thermally activated carrier population [@shinde03; @toyosaki04; @higgins-HE-04]. Of course, the possibility that the observed ferromagnetism in magnetic oxides arises from completely unknown extrinsic mechanisms (e.g. clustering near the surface) or from non-carrier-mediated mechanisms cannot be ruled out at this stage. Only more experimental work, perhaps motivated by our theoretical considerations presented herein, can provide definitive evidence in favor of one or more FM mechanisms in O-DMS. We mention in this context that even for much better-understood III-V DMS systems there are still debates in the literature regarding the precise role of the RKKY mechanism [@priourPRL06]. This paper is organized as follows: in Sec. \[sec:model\], the bound magnetic polaron and the RKKY model are introduced. In Sec. \[sec:compare\], magnetic and transport properties of dilute magnetic oxide semiconductors, mainly Co-doped TiO$_2$, are summarized and analyzed in the light of our proposed combined model. Sec. \[sec:discussion\] presents a discussion of our model and some of the alternatives suggested in the literature. We conclude in Sec. \[sec:conclusion\]. Model {#sec:model} ===== The general Hamiltonian that describes the physics of diluted magnetic semiconductors is (see, for instance, Ref. [@dassarma-mag-03]) $$\begin{aligned} H&=& \sum_{\alpha} \int{ d^3 x \,\, \Psi_{\alpha}^{\dagger} (x) \left(-{\frac{\nabla^2}{2 m}} + V_L(x)+ V_r(x) \right) \Psi_{\alpha} (x)} \nonumber\\ &+&\int{ d^3 x \sum_{i\alpha} W(x-R_i) \Psi_{\alpha}^{\dagger} (x) \Psi_{\alpha} (x) }\nonumber\\ &+& \int{d^3 x \sum_{i\alpha\beta} J {\mathbf S}_i \,\, \Psi_{\alpha}^{\dagger} (x) {\boldmath\sigma}_{\alpha,\beta} \Psi_{\beta} (x) \,\,a_0^3 \,\delta^3(x-R_i) } \nonumber \\ &+&\sum_{i,j} J_d (R_i-R_j) {\mathbf S}_i {\mathbf S}_j \label{eq:hamiltonian}\end{aligned}$$ where $m$ is the relevant effective mass, $V_L$ is the periodic lattice potential, $V_r$ is a potential arising from disorder (magnetic and non-magnetic) in the lattice, $W$ is a Coulombic potential arising from the oxygen vacancies that act as shallow donors, $J$ is the local exchange (Hund’s like) between the carrier spin and the magnetic impurities moments, and $J_d$ is a direct exchange between the magnetic impurities spins (which is ferromagnetic in Co-doped TiO$_2$ [@janisch06]). ${\mathbf S}_i$ is the impurity spin located at ${\mathbf R}_i$, ${\boldmath \sigma}_{\alpha,\beta}$ represents the Pauli matrices with spin indices $\alpha$ and $\beta$, and $a_0^3$ is the unit cell volume. We are neglecting the electron-electron interaction which we expect to be very small due to low carrier density in the diluted magnetic oxides. In the following, we assume the direct exchange term to be effectively included in the local exchange term. Only at very low carrier densities and in the case of antiferromagnetic $J_d$, which is [*not*]{} the case of Co-doped TiO$_2$, the direct exchange could compete with the carrier-mediated FM and cause frustration [@kaminski04]. The Hamiltonian in Eq. (\[eq:hamiltonian\]) is extremely complex to solve exactly and, consequently, we integrate out the electronic degrees of freedom and simplify the problem by considering only the term in the Hamiltonian that dictates the interaction between the spin of the carriers and the magnetic moments $$H_l=\sum_i J a_0^3 \,\,\,{\mathbf S}_i \,{\mathbf s}({\mathbf R}_i)\,. \label{eq:hamiltonian-local}$$ We, therefore, model the problem with a minimum set of parameters that effectively include other interactions in the system. This simplified Hamiltonian is solved in two complementary cases: (i) localized carriers (bound to oxygen vacancies by the interaction $W$), and (ii) itinerant carriers. When the carriers are localized, they form bound magnetic polarons that percolate at $T_C$ [@kaminski02; @coey05]. When carriers are delocalized in the conduction (or valence) band, they can mediate ferromagnetism through the RKKY mechanism. In the following, we introduce these two approaches, which are capable of producing carrier-mediated FM provided the carrier density is much lower than the magnetic impurity density. Percolation of bound magnetic polarons -------------------------------------- Bound magnetic polarons (BMP) are the result of the combination of Coulomb and magnetic exchange interactions [@kasuya68]. Carriers are localized due to electrostatic interaction with some defect (i.e. the magnetic impurity in III-V semiconductors, and the oxygen vacacies in the magnetic oxides) with a confinement radius $a_{\rm B}= \epsilon (m/m^*) a$, with $\epsilon$ the static dielectric constant, $m^*$ the effective mass of the polaron, and $a=0.52 \,{\rm\AA}$ the Bohr radius. $a_{\rm B}$ is larger (smaller) for shallower (deeper) defect levels. We assume the bound electron (or hole) wave-function has the isotropic hydrogen-like form $\Psi(r) \sim {\frac{1}{\sqrt{a_{\rm B}^3}}}\exp(-r/a_{\rm B})$ so that the exchange \[given in Eq. (\[eq:hamiltonian-local\])\] between the magnetic impurities and the carrier decays exponentially with the distance $r$ between them as $\sim \exp(-2r/a_{\rm B})$. At a certain $T$, the radius of the polaron $R_{\rm p}$ is given by the condition $$k_{\rm B} T= |J| \left(a_0/a_{\rm B}\right )^3 S s \exp(-2R_{\rm p}/a_{\rm B}) \,,$$ where $k_{\rm B}$ is the Boltzmann constant, leading to $$R_{\rm p}(T) \equiv (a_{\rm B}/2) \ln \left(sS|J|\left(a_0/a_{\rm B}\right )^3/k_{\rm B}T\right) \,, \label{eq:pol-radius}$$ where we can see that at the high temperature disordered phase there are no polarons, which start forming at $k_B T \sim sS|J|\left(a_0/a_{\rm B}\right )^3$, and then grow as temperature is lowered. The impurity spins that are a distance $r<R_{\rm p}$ from a localized carrier tend to align with the localized carrier spin. The polarized magnetic impurities form, in turn, a trapping potential for the carrier such that a finite energy is required to flip its spin. The magnetic polarons are well defined non overlapping isolated entities only at low carrier densities and sufficiently large temperatures. The size of the polarons increases as temperature decreases, eventually overlapping with neighboring BMPs. This overlap causes the alignment of their spins, therefore forming FM clusters. The FM transition takes place when an ’infinite cluster’ (of the size of the system) is formed, i.e. when the BMP percolation occurs. This scenario has been studied in Ref. [@kaminski02] in the context of III-V semiconductors. There it is shown that the bound magnetic polaron model can be mapped onto the problem of percolation of overlapping spheres. The model proposed is valid in the low carrier density regime $n_c a_{\rm B}^3 <<1$ and when the density of magnetic impurities $n_i$ is larger than the density of carriers $n_c$. Under these conditions, each carrier couples to a large number of magnetic impurities, as shown in Fig. \[fig:percolation\]. In order to calculate the BMP percolation critical temperature $T_C^{\rm perc}$ we first need to estimate the maximum temperature at which two magnetic polarons a distance $r$ from each other are still strongly correlated $T_{2\rm p} (r)$. By estimating the number of impurities which interact with both polarons [@kaminski02], this temperature is found to be given by $$k_{\rm B} T_{2p}(r) \sim a_B \sqrt{r n_i} \,s S J \left({{a_0}\over{a_{\rm B}}} \right )^3 \exp(-r/a_{\rm B})\,. \label{eq:t2p}$$ As the temperature is lowered more and more polarons overlap with each other until a cluster of the size of the sample appears. The critical polaron radius at which this happens can be calculated as the percolation radius in the problem of randomly placed overlapping spheres. This problem has been solved numerically [@pike74] giving $$r_{\rm perc}\approx 0.86/\sqrt[3]{n_c} \,.$$ Substituting this distance in Eq. \[eq:t2p\] gives the FM transition temperature $$k_{\rm B} T_C^{\rm perc} \sim s S J \left({{a_0}\over{a_{\rm B}}} \right )^3 \left(a_{\rm B}^3 n_c \right) ^{1/3} \sqrt{{{n_i}\over{n_c}}} \,\exp \left(- {{0.86}\over {\left(a_{\rm B}^3 n_c \right) ^{1/3} }}\right). \label{eq:Tc-perc}$$ The magnetization is due to the magnetic ions (since $n_i > n_c$) in the percolating cluster. It is given by $$M(T)= S n_i \,\mathcal{V} \left( r_{\rm corr} \sqrt[3]{n_c} \right)$$ where $\mathcal{V} (y)$ is the infinite cluster’s volume in the model of the overlapping spheres and $$r_{\rm corr}(T)=\left[0.86+(a_B^3 n_c)^{1/3} \ln {\frac{T_C}{T}}\right]/\sqrt[3]{n_c}$$ is the maximum distance between correlated polarons at $T$. Therefore, the shape of the magnetization curve only depends on $a_B^3 n_c$. The volume of the infinite cluster is strongly suppressed as temperature is increased producing a concave shaped $M(T)$ curve [@kaminski02], as opposed to the Weiss mean-field like convex Brillouin function shape expected for highly conducting samples. The concave shape is less pronounced for larger $a_{\rm B}^3 n_c$ [@dassarma-mag-03], that would eventually lead to a change of shape in the itinerant carrier regime, beyond the limit of applicability of this theory. Interestingly, however, the BMP percolation theory developed in the strongly localized insulating regime smoothly extrapolates to the mean-field RKKY-Zener theory (discussed below in Sec. \[sec:model\]B) in the itinerant free-carrier metallic regime [@kaminski02; @dassarma-mag-03], giving us some additional confidence in the potential validity of the composite percolation-RKKY model we are proposing here for ferromagnetism in the magnetic oxides materials. Conduction in a bound magnetic polaron system occurs via an activated process. Thus, $\rho \sim \exp(\Delta E/k_{\rm B} T)$ where $\Delta E$ is the activation (or binding) energy. $\Delta E$ depends on the binding energy of the carrier due mainly to electrostatic interaction with donors or acceptors, and in a minor degree, on the polarization due to magnetic exchange Eq. (\[eq:hamiltonian-local\]). The effect of an applied magnetic field in this system is to align non-connected polarons in such a way that the polarization part of the binding energy gets suppressed. As a result, the magnetoresistance is negative and depends exponentially on the field [@kaminski03]. It should be emphasized that the BMP picture and the associated magnetic percolation transition is only valid in the strongly localized carrier regime where each BMP (i.e. each carrier) is immobile, and a percolation transition in the random disordered configuration makes sense. ![Schematic representation of magnetic percolation in oxide based dilute magnetic semiconductors. The solid squares represent the oxygen vacancies where an electron, represented by an arrow, is localized. The gray circles represent the extension of the electron wave-function. Magnetic impurity spins are represented by small arrows whose orientation is established by antiferromagnetic exchange coupling with the localized carrier. []{data-label="fig:percolation"}](percolation.eps){width="3.0in"} RKKY model for itinerant carriers --------------------------------- As already mentioned, carriers in magnetic oxides are donated by oxygen vacancies that act as shallow donors. The carrier binding energy has been estimated for different samples of undoped TiO$_2$ to be $\sim 4$ meV [@forro94], $\sim 14$ meV [@tang94] and $\sim 41$ meV [@shinde-un]. These are smaller than the Mn acceptor binding energy of GaAs, which is of the order of $0.1$ eV [@yakunin04]. It is then possible that a large fraction of the electrons are promoted (i.e. thermally activated) above the mobility edge in the impurity band at temperatures lower than $T_C^{\rm perc}$ and that the localization picture just described cannot explain ferromagnetism by itself. This thermal activation of carriers will be extremely effective at higher temperatures, particularly since $k_{\rm B} T_C$ in this system could be substantially higher than the electron binding energy or, equivalently, the activation energy. The assumption of strict BMP localization fails in this situation as the carriers get activated into highly mobile states at higher temperatures, as manifested by the experimentally observed enhancement in the conductivity at higher temperatures. The BMP formation will be suppressed in this situation since the thermally activated band carriers are effectively free or itinerant at elevated temperatures. Itinerant carriers coupled to local moments by Eq. (\[eq:hamiltonian-local\]) lead to the well-known Zener-RKKY mechanism of indirect magnetic interaction between the magnetic impurities. This model gives an effective exchange $J_{\rm eff}$ between magnetic moments in the lattice of the form [@ruderman54; @yosida57] $$J_{\rm eff} \propto {{2 k_F R_{ij} \cos(2 k_F R_{ij})- \sin(2 k_F R_{ij})}\over {R_{ij}^4}} \label{eq:RKKY-osc}$$ where $R_{ij}=R_i-R_j$ (i.e. the distance between magnetic impurities) and $k_F$ is the Fermi momentum. $J_{\rm eff}$ oscillates in sign with distance leading in general to complicated magnetic configurations [@mattis62], frustration and glassy behavior. However, in the limit of very low density of carriers relevant here ($n_c << n_i$), $k_F r \rightarrow 0$ and $J_{\rm eff} <0$, namely, the RKKY interaction is always ferromagnetic. The RKKY ferromagnetism phase diagram has recently been calculated [@priourPRL06]. We study the RKKY interaction in the limit of non-degenerate carriers within mean-field theory (therefore, all disorder in the lattice is neglected). The interaction between carrier spins and local moments can be then described as a self-consistent process in which carrier spins see an effective magnetic field produced by the local moments (which are considered classical) $$B_{\rm eff}^{(c)}=\frac{Ja_0^3n_i \langle S_z \rangle}{g_c \mu_B}\,,$$ and the local moments see an effective field produced by the carrier spins $$B_{\rm eff}^{(i)}=\frac{Ja_0^3n_c \langle s_z \rangle}{g_i \mu_B}\,.$$ The response of the impurity spin, which follows Boltzmann statistics, to $B_{\rm eff}^{(i)}$ is [@ashcroft] $$\langle S_z \rangle = S\, \mathcal{B}_S \left({{g_i \mu_B B_{\rm eff}^{(i)}}\over{k_B T}}\right)=S\, \mathcal{B}_S \left({{S J a_o^3 n_c \langle s_z \rangle}\over{k_{\rm B} T}} \right) \label{eq:Sz}$$ where $$\mathcal{B}_s(y) \equiv {\frac{2s+1}{2s}} \coth {\frac{2s+1}{2s}} y -{\frac{1}{2s}} \coth {\frac{1}{2s}} y$$ is the Brillouin function. In the nondegenerate limit, the carrier spin distribution is not affected by the Pauli exclusion principle and, therefore, it is determined also by Boltzmann statistics rendering $$\langle s_z \rangle =s \,\mathcal{B}_s \left({{g_c \mu_B B_{\rm eff}^{(c)}}\over{k_B T}}\right)= s \, \mathcal{B}_s \left({{s J a_o^3 n_i \langle S_z \rangle}\over{k_{\rm B} T}} \right) \label{eq:sz}$$ Combining Eqs. (\[eq:Sz\]) and  (\[eq:sz\]), we get a self-consistent equation for the impurity spin $$\label{eq:RKKY} \langle S_z \rangle = S\,\mathcal{B}_S \left[{\frac{J a_0^3}{k_B T}} s \mathcal{B}_s \left({\frac{J a_0^3 n_i \langle S_z \rangle}{k_B T}}\right)\right]\, . \label{eq:RKKY-mag}$$ When the effective magnetic fields are small, as it is close to the magnetic transition, we can perform the expansion of $\mathcal{B}_s(y)$ for small values of $y$ $$\mathcal{B}_s(y) \approx {\frac{s+1}{3s}}y+O(y^3) \, ,$$ which, applied to Eq. \[eq:RKKY\], gives the critical temperature [@coey-comment] $$k_{\rm B} T_C^{RKKY} ={{1}\over{3}} \,J a_0^3 \,\sqrt{n_c n_i}\, \sqrt{(S+1) (s+1)}. \label{eq:Tc-nondeg}$$ The magnetization is mainly due to the local magnetic moments $\langle S_z \rangle$ (because $n_i>n_c$) and is given by the self-consistent Eq. \[eq:RKKY-mag\]. The resultant dependence with temperature is concave for low values of $n_c/n_i$ and convex mean-field-like for $n_c/n_i \ge 0.2$ [@dassarma-mag-03]. For higher carrier density, the carriers form a degenerate gas and the procedure followed in Ref. [@priour04] would be more appropriate. In this reference, the RKKY model is solved taking into account the disorder in the lattice. The conductivity is included via the mean free path (MFP), which produces a cutoff in the RKKY coupling reach. $T_C$ has been found to depend very strongly on the relation between $n_c$, $n_i$ and the MFP. MFP can be made larger by improving sample quality. In contrast with mean field treatments of the same model (which neglect disorder) that predict $T_C \propto n_c^{1/3}$ [@dietl01], $T_C$ is enhanced and later suppressed by increasing $n_c$. This $T_C$ suppression is due to the sign oscillations of the RKKY interaction at larger carrier densities \[see Eq. (\[eq:RKKY-osc\])\] that lead to magnetic frustration and spin glassy behavior. It is then proposed [@dassarma04] that $T_C$ improvement can arise both from increasing $n_c$ and the MFP. The magnetization curves $M(T)$ are convex in the highly conducting limit (large MFP) and concave for more insulating systems (small MFP) [@priour04; @dassarma-mag-03] appropriate for dilute magnetic oxides. This is qualitatively the same result as given by the non-degenerate approach and the BMP model. In this way, the localized and itinerant carrier models predict the same behavior for $M(T)$ in both the high and low carrier density limits, leading to correct qualitative predictions even beyond the applicability range of the two models. Calculations of dc-resistivity in diluted systems within DMFT approximation have rendered a negative magnetoresistance MR, that peaks at $T_C$ but is appreciably smaller than the MR of magnetic ordered lattices such as manganites [@hwang-condmat]. Double exchange model --------------------- In the double exchange model [@zener], a large $J$ forces the spin of the carrier to be parallel to the local magnetic moment in such a way that the kinetic energy of the carriers hopping between magnetic sites is minimized when the magnetic ions are ferromagnetically ordered. This model was proposed for manganites, where ferromagnetism and metallicity usually come together. The ferromagnetic critical temperature in the double exchange model is proportional to the density of carriers and the bandwidth [@calderon-MC98], with very small values in the low density limit appropriate to O-DMS. Therefore, double exchange is not a suitable mechanism to explain the high critical temperature of dilute magnetic oxides, and is hence not considered further in this work. However, in the strong coupling regime, where the effective exchange coupling $J$ is large ($J>>t, \, E_F$ where $t$ is the carrier bandwidth), a double exchange mechanism may be appropriate for low-$T_C$ DMS materials as discussed in Refs. [@chatto01; @akaiPRL98]. Comparison to experiment {#sec:compare} ======================== ![Activation energies (closed symbols) and estimated localization radius $a_{\rm B}$ (open symbols) for undoped and Co-doped TiO$_2$. For these estimations, the static dielectric constant $\epsilon=31$ has been used. Circles are results from Ref. [@shinde-un], squares from Ref. [@tang94], and diamonds from Ref. [@forro94]. []{data-label="fig:activation"}](conc-activation-aB.eps){width="3.0in"} We argue here that, at high enough temperatures compared to the carrier binding energy, carriers are thermally excited to conduction (or valence) bands from the impurity band becoming itinerant and mediating ferromagnetism by an effective RKKY mechanism. The carrier binding energy $\Delta E$ can be estimated by fitting the resistivity curves to $\rho= \rho_0 \exp(\Delta E/k_{\rm B} T)$. The result is unfortunately sample and composition dependent. For the undoped TiO$_2$, estimations go from $\sim 4$ meV [@forro94] to $\sim 41$ meV [@shinde-un]. In Ref. [@tang94], $a_{\rm B}$ in TiO$_2$ is estimated to be $15\, {\rm \AA}$ from the observation of an insulator to metal transition upon doping at $n_c \sim 5\times 10^{18}$ cm$^{-3}$. This corresponds to $\Delta E= 14$ meV using the static dielectric constant $\epsilon=31$ [@tang94] and $m^*=m$. We show these results in Fig. \[fig:activation\], together with the activation energy values corresponding to Co-doped TiO$_2$ films [@shinde-un]. When doped with magnetic ions, the resistivity increases by orders of magnitude proportionally to the density of impurities, its slope is dramatically enhanced, and does not show the insulator to metal transition [@ogale03; @shinde03; @toyosaki04] observed in the undoped samples. The overall increase of the resistivity is due to the scattering from charged impurities, as they produce strong disorder in the system. However, the resistivity due to the impurity scattering is not temperature dependent on the exponential scale of carrier activation. Therefore, $\Delta E$ can be calculated the same way as for undoped samples. The results for a particular series of Co-doped TiO$_2$ [@shinde-un] show that the activation energy increases upon doping (see Fig. \[fig:activation\]). We expect these values to vary widely from sample to sample, as observed for the undoped compound. (This is also consistent with the current experimental situation where the observed $T_C$ in various nominally similar Co-TiO$_2$ samples varies widely from sample to sample.) As the density of free carriers depends exponentially on the impurity energy level (or the activation energy), a significant part of the bound impurity band electrons can be promoted above the mobility edge at temperatures lower than $T_C$. This is particularly true in doped magnetic oxides, where the claimed $T_C$ ($\approx 400-1000$K) is so high that the $k_{\rm B} T_C >> \Delta E$ regime is easily reached producing a high density of thermally activated mobile band carriers at $k_{\rm B} T \gtrsim \Delta E$. The estimation of the critical temperature depends on many variables: (i) the density of carriers $n_c$, (ii) the density of magnetic ions $n_i$, (iii) the exchange coupling $J$ and, (iv) in the bound magnetic polaron theory, the localization radius $a_{\rm B}$. These parameters are either not precisely known or present a huge sample-to-sample variation: (i) The density of carriers $n_c$ has been measured by Hall effect giving results ranging from $10^{18}$ to $10^{22}$ cm$^{-3}$ [@higgins-HE-04], as the density of oxygen vacancies varies greatly with the growth conditions. (ii) The absolute density of magnetic ions is known ($x=0.1$ corresponds to $n_i \sim 3 \times 10^{21}$ cm$^{-3}$ in TiO$_2$) but we cannot estimate how many of those are magnetically active and, therefore, relevant to long-range carrier-mediated ferromagnetism. In fact, the magnetic moment per magnetic ion has a strong dependence on sample characteristics [@coey05]. This could be due to different effective values of magnetically active impurities. In particular, for Co-doped TiO$_2$, $S$ is usually close to the low spin state $S=1/2$. (iii) $J$ is not known in general though there are some estimates which place its value above $1$ eV for ZnO [@coey05]. (iv) $a_{\rm B}$ calculated from the activation energies are shown in Fig. \[fig:activation\] (right axes). Note the strong variation of $a_{\rm B}$ from sample to sample in the undoped case. This dramatically affects the estimates of the critical temperature within the polaron percolation model as shown in Fig. \[fig:temp-nc\], as $T_C^{\rm perc}$ depends exponentially on $a_{\rm B}^3 n_c$. Estimations of $T_C$ from Eqs. (\[eq:Tc-perc\]) and (\[eq:Tc-nondeg\]) depend strongly on all these unknown parameters and, consequently, $T_C$ can vary from tens to hundreds of Kelvin upon tuning their values. This is illustrated in Fig. \[fig:temp-nc\]. The local moment is taken to be $S=1$, however, the moment per magnetic ion is another quantity that is very sample dependent. A value of $n_i=3 \, \times \, 10^{21}$ cm$^{-3}$ has been used. $T_C^{\rm perc}$ \[Eq. (\[eq:Tc-perc\])\] increases with the magnetic impurity density as $\sqrt{n_i}$ while $T_C^{\rm RKKY}$ \[Eq. (\[eq:Tc-nondeg\])\] only depends on the product $\sqrt{n_i n_c}$. Note that both estimates of $T_C$ are the same order of magnitude and close to experimental data. However, due to the strong dependence of $T_C$ on unknown parameters, we should not use the calculated value of $T_C$ as the sole criterion to elucidate the applicability of a particular model. Rather, we should look at other evidence given by experiment such as trends in magnetization curves and magnetoresistance. We note, however, that we cannot rule out the possibility that the strong variation in $T_C$ between different experimental groups (or even from sample to sample in the same group) arises precisely from the variation of the sample parameters $n_i$, $n_c$, $a_{\rm B}$, etc. which will indeed lead to large $T_C$ variation! Further experiments are clearly needed to settle this important issue. ![(a) Estimation of ferromagnetic $T_C$ for the theory of bound magnetic polarons Eq. (\[eq:Tc-perc\]) and (b) for the RKKY model for itinerant carriers Eq. (\[eq:Tc-nondeg\]). $T_C$ on the right y-scale has been calculated using $J=1$ eV. $S=1$, $a_o=3.23$ Å [@simpson04], and $n_i=3 \,\times \, 10^{21} $ cm$^{-3}$ (equivalent to $x=0.1$ for doped TiO$_2$). []{data-label="fig:temp-nc"}](temp-nc-all.eps){width="3.0in"} The measured magnetization versus temperature curves $M(T)$ usually present a very constant signature within a wide range at low temperatures [@matsumoto01; @ogale03; @shinde03; @chambers01], though in some cases, a concave shape is observed [@rode03; @wang05]. As explained in Sec. \[sec:model\], this concave shape is expected for insulating systems within the polaron percolation approach and for low carrier density systems within the RKKY approach. Magnetoresistance is very sample and composition dependent and there are not many reports in the literature. In general, it is only significant at temperatures much lower than $T_{C}$ [@matsumoto01; @ogale03; @jin01]. For doped ZnO, MR shows different signs for different dopants and temperatures [@jin01], though these results may not be relevant as these samples were not ferromagnetic. Co-SnO$_2$ displays positive MR that becomes negative with increasing $T$ [@ogale03]. Anatase Co-TiO$_2$ has positive MR for $T \le 5$ K ($60\%$ at $T=2$K and $H=8$T) [@matsumoto01], while rutile Co-TiO$_2$ shows negative MR up to almost room temperature (maximum $\sim -0.4\%$ at $T=100$K and $H=8$T) [@toyosaki04]. The latter also presents an increase of MR for the least resistive samples, the ones with higher carrier density. As explained above, the polaron percolation model can explain a negative MR, whose value depends on the change of $\langle S_z \rangle$ with the applied magnetic field. At low temperatures, $\langle S_z \rangle$ should already be saturated, therefore, MR is bound to be small, consistent with results in O-DMS. On the other hand, both the percolation and RKKY models imply a maximum MR at $T_C$ when spin fluctuations and susceptibility are very large and a small magnetic field can affect magnetization dramatically. However, this behavior has not been found in O-DMS. Discussion {#sec:discussion} ========== In this work we are trying to understand the physical origin of ferromagnetism in magnetically doped oxide materials (e.g. Co-TiO$_2$) by assuming the underlying magnetic mechanism to be carrier-mediated, qualitatively similar to that operational in the widely studied DMS systems such as Ga$_{1-x}$Mn$_x$As. We use an effective model of a local exchange coupling between the localized magnetic moments (i.e. Co) and the localized carriers (since the system, e.g. Co-TiO$_2$, is an insulator at $T=0$). We first show that the very high $T_C$ (well above the room temperature) reported in the literature cannot be understood entirely within the BMP percolation picture [*by itself*]{} without fine-tuning the effective parameters (e.g. exchange coupling, carrier and magnetic moment densities) to an unreasonable degree. We therefore disagree with the recent attempts [@coey05] in the literature to attribute FM in O-DMS entirely to BMP percolation. In general, the BMP percolation theory, which is very natural for the insulating magnetic oxides and has already been invoked in the literature [@venkatesan04; @coey04; @coey05], leads to low values of $T_C$ for any reasonable assumptions about the system parameters. The observed convex shape of the magnetization curves also argues against an entirely BMP percolation model for magnetic oxide ferromagnetism. In addition, for TiO$_2$, the high temperature resistivity measurements indicate the existence of a substantial thermally activated carrier population consistent with RKKY. We therefore suggest a composite scenario by taking into account the relatively small carrier binding energy $\Delta E$ in these materials. We propose that the low temperature BMPs in the system (in the ferromagnetic state) become mobile (instead of going into a disordered paramagnetic state through the percolation transition), and these mobile carriers then produce long-range ferromagnetic coupling among the magnetic impurities through the standard RKKY-Zener mechanism, which could explain the experimentally observed high-temperature ferromagnetism. Wide variations in effective carrier ($n_c$) and magnetic moment ($n_i$) densities due to different growth conditions could easily accomodate the observed large variation in $T_C$ among different groups. We emphasize that our model of combined polaron percolation (at low temperatures, where the carriers are still bound) and RKKY-Zener (at high temperatures, where the carriers are activated into mobile states) mechanism seems to be the only reasonable possibility for the ferromagnetism with high $T_C$ of the doped magnetic oxides assuming that the ferromagnetism [*is both intrinsic and carrier-mediated*]{}. This combined model applies mainly to TiO$_2$ where high temperature resistivity is activated [@shinde03; @toyosaki04; @higgins-HE-04]. On the other hand, recent measurements on Co-doped ZnO seem to rule out RKKY in favor of an exclusive BMP mechanism [@kittilstved06]. There are obviously other possibilities for the origin of ferromagnetism, which are less interesting, but [*not*]{} necessarily less plausible. First, the oxide ferromagnetism (in Co-TiO$_2$, Co-ZnO, etc.) could be entirely extrinsic (and, some of it most likely is), arising from contaminants (as happened, for example, in the ferromagnetism of calcium hexaborides) or from magnetic nanoclustering [@shinde03]. The Co nanoclusters in Co-TiO$_2$, for example, could produce long-range ferromagnetic order through dipolar coupling. A second intrinsic possibility for oxide ferromagnetism, increasingly gaining ground on the basis of recent first principles band structure calculations, is that the ordering in insulating samples or regions arises from the standard superexchange interaction between the localized magnetic impurities (e.g. Co in Co-TiO$_2$) [@janisch06]. These local interactions would still coexist with long-range carrier-mediated ferromagnetism. Other calculations [@ye06] find a $p-d$ hopping mechanism, not related to free carriers but to clustering of the magnetic ions. These possibilities certainly cannot be ruled out, particularly in ordered magnetic oxides where such band structure supercell calculations would, in principle, be applicable. But the first principles calculations completely neglect disorder, and cannot really explain the wide variation in magnetic properties (e.g. $T_C$) seen in different laboratories. In addition, none of these posibilities (superexchange, p-d hopping mechanisms, or extrinsic nanocluster ferromagnetism) are carrier-mediated in the sense of DMS materials, and as such are out of scope for our interest. What we have discussed in this work is that, within the assumption of an intrinsic carrier-mediated ferromagnetic mechanism (related to the mechanisms operational in DMS materials), the most likely origin for ferromagnetism in DMS oxides is a combination of BMP percolation at low temperatures and RKKY-Zener coupling through activated carriers at high temperatures. Only detailed experimental work can establish the origin of ferromagnetism in magnetically doped oxides — theory can only suggest interesting possibilities without ruling out alternative mechanisms such as extrinsic ferromagnetism or direct superexchange. Among the various experimental evidence supporting the model of carrier-mediated ferromagnetism in magnetically doped oxides are the observation of AHE [@toyosaki04], electric field induced modulation of magnetization [@zhao-FE-05], and optical magnetic circular dichroism [@ando01]. It is clear that much more systematic experimental data showing the magnetic properties as a function of carrier density (and carrier properties) will be needed before the issue of definitively establishing the origin of ferromagnetism in doped magnetic oxides can be settled. Conclusion {#sec:conclusion} ========== Before concluding, we emphasize that the new idea in this paper is that high-temperature RKKY ferromagnetism may be mediated in a semiconductor doped with magnetic impurities by thermally excited carriers in an otherwise-empty itinerant (conduction or valence) semiconductor band, in contrast to the usual band-carrier mediated RKKY ferromagnetism often discussed [@priour04; @dassarma04] in the context of ferromagnetic (Ga,Mn)As where the valence band holes are thought to mediate the ferromagnetic RKKY coupling between the localized Mn magnetic moments. We believe that this RKKY coupling mediated by thermally excited carriers may be playing a role in the high observed $T_C$ in Co-doped TiO$_2$ where the experimentally measured conductivity is always activated in the ferromagnetic phase indicating the presence of substantial thermally activated free carriers. Obviously, the necessary condition for such a thermally activated RKKY ferromagnetism is that electrical conduction in the system must be activated (and therefore insulating) in nature in contrast to metallic temperature-independent conductivity. If the observed conductivity in the system is temperature-independent, then the novel thermally-excited RKKY mechanism proposed by us simply does not apply. Our motivation for suggesting this rather unusual thermally-activated RKKY DMS ferromagnetism has been the reported existence of very high transition temperatures in Co-doped TiO$_2$ which simply cannot be explained quantitatively by the bound magnetic polaron percolation picture of Kaminski and Das Sarma [@kaminski02; @dassarma-mag-03; @kaminski03] although it is certainly possible that some of the oxide ferromagnetism arises purely from the BMP mechanism as has recently been argued [@kittilstved06]. The thermally activated RKKY ferromagnetic mechanism, while being necessary for high $T_C$, cannot be sufficient since at low temperatures the thermally activated carriers freeze out leading to an exponential suppression of the thermally activated RKKY ferromagnetism. We therefore propose that BMP ferromagnetism [@kaminski02; @dassarma-mag-03; @kaminski03] mediated by strongly localized carriers take over at low temperatures, as already proposed by Coey and collaborators [@coey05] supplementing and complementing the high-temperature RKKY mechanism. The two mechanisms coexist in the same sample with the high values of $T_C$ being controlled by the thermally activated RKKY mechanism and ferromagnetism persisting to $T=0$ due to the bound magnetic polaron percolation mechanism. We note that in a single sample the two mechanisms will give rise to a unique $T_C$ depending on all the details of the system since the two mechanisms coexist whereas there could be considerable sample to sample $T_C$ variation due to the coexistence of the two mechanisms. In our picture the high-temperature thermally activated RKKY mechanism smoothly interpolates to the low-temperature BMP ferromagnetism with a single transition temperature. This picture of the coexistence of two complementary ferromagnetic mechanisms in oxides is essentially forced on us by our consideration of the possible transition temperatures achievable within the BMP model, which are just too low to explain the observations in (at least) the Co-doped TiO$_2$. Very recently, we have shown [@re-entrant] that the confluence of two competing FM mechanisms, namely the BMP percolation in the impurity band at ’low’ temperatures and the activated RKKY interaction in the conduction band at ’high’ temperatures, could lead to an intriguing and highly non-trivial re-entrant FM transition in O-DMS where lowering temperature at first leads to a non-FM phase which gives way to a lower temperature second FM phase. A direct observation of our predicted [@re-entrant] re-entrant FM would go a long way in validating the dual FM mechanism model introduced in this article. To summarize, we have analyzed different proposed models for carrier-mediated ferromagnetism in dilute magnetic oxides such as TiO$_2$, ZnO, andSnO$_2$. Due to the insulating character of these compounds, a model based on the formation of bound magnetic polarons is proposed. 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ArXiv

--- abstract: 'We report photonic quantum circuits created using an ultrafast laser processing technique that is rapid, requires no lithographic mask and can be used to create three-dimensional networks of waveguide devices. We have characterized directional couplers—the key functional elements of photonic quantum circuits—and found that they [perform as well as]{} lithographically produced waveguide devices. We further demonstrate high-performance interferometers and an important multi-photon quantum interference phenomenon for the first time in integrated optics. This direct-write approach will enable the rapid development of sophisticated quantum optical circuits and their scaling into three-dimensions.' address: - '$^1$Centre for Ultrahigh bandwidth Devices for Optical Systems (CUDOS), MQ Photonics Research Centre, Department of Physics, Macquarie University, NSW 2109, Australia' - '$^2$Centre for Quantum Photonics, H. H. Wills Physics Laboratory & Department of Electrical and Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol, BS8 1UB, UK' - '$^\star$These authors contributed equally to this work' author: - 'Graham D. Marshall,$^{1,\dagger,\star}$ Alberto Politi,$^{2,\star}$ Jonathan C. F. Matthews,$^{2,\star}$ Peter Dekker,$^1$ Martin Ams,$^1$ Michael J. Withford,$^1$ and Jeremy L. O’Brien$^{2,\ddagger}$' title: Laser written waveguide photonic quantum circuits --- [10]{} url \#1[`#1`]{}urlprefix\[2\]\[\][[\#2](#2)]{} M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information* (Cambridge University Press, 2000). V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-Enhanced Measurements: Beating the Standard Quantum Limit,” Science **306**, 1330–1336 (2004). A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. 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Introduction ============ Quantum information science promises exponential improvement and new functionality for particular tasks in computation [@nielsen], metrology [@gi-sci-306-1330], lithography [@bo-prl-85-2733] and communication [@gi-rmp-74-145; @gi-nphot-1-165]. Photonics appears destined for a central role owing to the wide compatibility, low-noise and high-speed transmission properties of photons [@gi-nphot-1-165; @ob-sci-318-1567]. However, future quantum technologies and fundamental science will require integrated optical circuits that offer high-fidelity and stability whilst enabling scalability. Silica-on-silicon waveguide photonic quantum circuits [@po-sci-320-646] are an important step, however, conventional lithography is costly, time consuming, and limited to two-dimensions. Here we demonstrate an alternative fabrication technique based on ultrafast laser processing [@da-ol-21-1729; @no-apa-77-109] that overcomes all of these issues. Quantum technologies rely on transmitting and processing information encoded in physical systems—typically two-state *qubits*—exhibiting uniquely quantum mechanical properties [@nielsen]. Photons hold great promise as qubits given their light-speed transmission, ease of manipulation at the single qubit level, low noise (or *decoherence*) properties and the multiple degrees of freedom available for encoding qubits or higher-level systems (including polarization, optical mode, path and time-bin). The problem of realizing interactions between single photonic [qubits]{} was theoretically solved in a breakthrough scheme for implementing optical non-linear interactions using only single photon sources, linear optical elements and single photon detectors [@kn-nat-409-46]. Remarkable progress in the development of single photon sources [@sh-nphot-1-215] and detectors [@ta-nphot-1-343; @ga-nphot-1-585], make a photonic approach to quantum technologies very promising. Indeed, there have been a number of important proof-of-principal demonstrations of quantum optical circuits for communication [@pr-prl-94-220406; @ya-nphot-2-488], computing [@ob-sci-318-1567], metrology [@mi-nat-429-161; @na-sci-316-726; @re-prl-98-223601], and lithography [@da-prl-87-013602]. However, the use of large-scale (bulk) optics to create these circuits places extremely stringent requirements on the alignment and stability in position of the optical components, thus making such an approach inherently unscalable. Successful operation of a quantum optical circuit requires that the individual photons be brought together at precisely the same position and with the correct phase on a succession of optical components in order to realize the high-fidelity classical and quantum interference that lies at the heart of single photon interactions [@kn-nat-409-46]. Integrated optical quantum circuits based on chip-scale waveguide networks will likely find important applications in future quantum information science along side optical fibre photonic quantum circuits that have been demonstrated in quantum key distribution [@gi-nphot-1-165] and quantum logic gate applications [@clark-2008]. Ultrafast lasers are a powerful tool not only for machining [@ga-nphot-2-219] but also for the subtle optical modification of materials. In particular, the direct-write femtosecond laser technique for creating optical waveguides in dielectric media [@da-ol-21-1729] is an alternative waveguide manufacturing technique that allows the production of low-volume complex three-dimensional optical *circuits* (Fig. \[schematic\](a)). This process has been applied to a wide range of passive and active media to create integrated devices such as microfluidic sensors [@os-apl-90-231118], waveguide-Bragg gratings [@ma-ol-31-2690] and miniature lasers [@ma-ol-33-956]. Because there is no lithography step in this procedure it enables a waveguide circuit to be taken rapidly from concept to a completed device. However, as with all waveguide fabrication processes, the devices are subject to manufacturing imperfections and there has been no previous demonstration that the use of the laser-writing technique can produce waveguides that can operate on single photons without deleterious effects on phase, spatial mode or polarization. [In this paper we report the first application of laser written waveguide circuits to photonic quantum technologies and, using single- and multi-photon interference experiments, we show that such circuits perform as well as lithographically fabricated devices and are an ideal platform for scalable quantum information science.]{} Experimental techniques ======================= Waveguide fabrication --------------------- In conventional lithographically fabricated integrated optical devices, light is guided in waveguides consisting of a core and a slightly lower refractive-index cladding or buffer layers (in a manner analogous to an optical fibre). In the commonly used flame hydrolysis deposition (FHD) fabrication method these structures are lithographically described on top of a semiconductor wafer [@po-sci-320-646]. By careful choice of core and cladding dimensions and refractive index difference it is possible to design such waveguides to support only a single transverse mode for a given wavelength range. Coupling between waveguides, to realize beamsplitter-like operation, can be achieved when two waveguides are brought sufficiently close together that the evanescent fields overlap; this architecture is known as the directional coupler. By carefully selecting the separation between the waveguides and the length of the interaction region the amount of light coupled from one waveguide into the other (the coupling ratio $1-\eta$, where $\eta$ is equivalent to beamsplitter reflectivity) and its dependance on wavelength can be tuned. A similar approach can be taken in the case of directly written waveguides, where the waveguide core is formed by local modification of silica [@chan2003a; @li-oe-16-20029] (or other materials). However, unlike the lithographic approach, direct-write circuits can be straightforwardly written in 3D. We fabricated two chips with a number of direct-write quantum circuits (DWQCs) composed of 2$\times$2 directional couplers (Fig. \[schematic\](c)) and Mach-Zehnder interferometers. The circuits were written inside high purity fused silica using a tightly focused [1 kHz repetition rate, 800 nm, 120 fs laser]{} and motion control system similar to that previously reported [@ams2005; @ams2006]. The writing laser beam was circularly polarised and passed through a 520 $\mu$m slit before being focused 170 $\mu$m below the surface of the glass using a 40$\times$ 0.6 numerical aperture microscope objective that was corrected for spherical aberrations at this depth (Fig. \[schematic\](a)). The writing process created approximately [Gaussian profile waveguides which were characterised using a near-field refractive index profilometer (from Rinck Elektronik) and a magnifying beam profiler. The *measured* refractive index profile of a typical waveguide is displayed in Fig. \[schematic\](b) and shows a peak refractive index change of 4.55$\times10^{-3}$, an average $1/e^2$ width of 5.5 $\mu$m and a $x/z$ width ratio of 0.93. At their design wavelength of 806 nm these waveguides supported a single transverse mode with orthogonal (intensity) $1/e^2$ widths of 6.1 $\mu$m $\times$ 6.1 $\mu$m (the small amount of measured RI shape-asymmetry not being evident in the guided mode)]{}. The design of the directional couplers was functionally identical except for the length of the central [interaction region that was varied from 400 to 2000 $\mu$m]{} to achieve different coupling ratios. The curved regions of the waveguides were of raised-sine form and connected the input and output waveguide pitch of 250 $\mu$m down to the closely spaced evanescent coupling region of the devices. [The Mach-Zehnder interferometers’ design comprised two 50:50 directional couplers separated by identical 1500 $\mu$m long arms. The purpose of these devices was to test the stability of both the completed waveguide circuits and the laser writing system (which was required to remain stable for the several hours required for fabrication of the directional couplers and interferometers).]{} (a) ![Fabrication and measurement of laser direct-write photonic quantum circuits. (a) A schematic of the femtosecond-laser direct-write process. [(b) A refractive index (RI) profile of a typical waveguide where the writing laser was incident from the left. The white overlayed plots show the cross section of the RI profile at the peak. The distortion to the measured $x$-axis RI profile to the right of the peak is an artifact of the measurement method.]{} (c) A schematic of an array of directional couplers fabricated by *fs* direct writing in a single fused silica chip and an optical micro-graph showing the central coupling region where the waveguides are separated by 10 $\mu$m. (d) Measurement setup based on spontaneous parametric down conversion of a continuous wave 402 nm laser diode. []{data-label="schematic"}](DWschematic2.eps){width="100.00000%"} (b) ![Fabrication and measurement of laser direct-write photonic quantum circuits. (a) A schematic of the femtosecond-laser direct-write process. [(b) A refractive index (RI) profile of a typical waveguide where the writing laser was incident from the left. The white overlayed plots show the cross section of the RI profile at the peak. The distortion to the measured $x$-axis RI profile to the right of the peak is an artifact of the measurement method.]{} (c) A schematic of an array of directional couplers fabricated by *fs* direct writing in a single fused silica chip and an optical micro-graph showing the central coupling region where the waveguides are separated by 10 $\mu$m. (d) Measurement setup based on spontaneous parametric down conversion of a continuous wave 402 nm laser diode. []{data-label="schematic"}](rip.eps){width="100.00000%"} (c) ![Fabrication and measurement of laser direct-write photonic quantum circuits. (a) A schematic of the femtosecond-laser direct-write process. [(b) A refractive index (RI) profile of a typical waveguide where the writing laser was incident from the left. The white overlayed plots show the cross section of the RI profile at the peak. The distortion to the measured $x$-axis RI profile to the right of the peak is an artifact of the measurement method.]{} (c) A schematic of an array of directional couplers fabricated by *fs* direct writing in a single fused silica chip and an optical micro-graph showing the central coupling region where the waveguides are separated by 10 $\mu$m. (d) Measurement setup based on spontaneous parametric down conversion of a continuous wave 402 nm laser diode. []{data-label="schematic"}](Fig1bnew.eps){width="100.00000%"} (d) ![Fabrication and measurement of laser direct-write photonic quantum circuits. (a) A schematic of the femtosecond-laser direct-write process. [(b) A refractive index (RI) profile of a typical waveguide where the writing laser was incident from the left. The white overlayed plots show the cross section of the RI profile at the peak. The distortion to the measured $x$-axis RI profile to the right of the peak is an artifact of the measurement method.]{} (c) A schematic of an array of directional couplers fabricated by *fs* direct writing in a single fused silica chip and an optical micro-graph showing the central coupling region where the waveguides are separated by 10 $\mu$m. (d) Measurement setup based on spontaneous parametric down conversion of a continuous wave 402 nm laser diode. []{data-label="schematic"}](1-1ExperimentalSetup_2.eps){width="100.00000%"} Single and 2-photon quantum interference ---------------------------------------- The quantum circuits were studied using [photons]{} obtained using spontaneous parametric down conversion (SPDC) sources. For the two-photon interference work the output from a continuous wave (CW) 402 nm laser diode was down converted into two unentangled 804 nm photons in a type-I phase matched BiBO crystal (Fig. \[schematic\](d)). The photon pairs were passed through 2 nm bandpass interference filters to improve indistinguishability before being coupled into two polarization maintaining single mode optical fibres (PMFs). The path length difference could be varied with the $\mu$m actuator which adjusted the arrival time of the photons with respect to each other at the waveguide chip. Photons were collected from the chip in single mode fibres (SMFs) and coupled to avalanche photo diodes (APDs) which were in turn connected to a photon counting and coincidence logic circuit. Index matching oil was used between the fibres and the device under test to reduce Fresnel reflections that contribute to coupling losses. The circuit devices had typical transmission efficiencies of 50% (which includes coupling and propagation losses). In the case of an ideal $\eta=\frac{1}{2}$ directional coupler, if two non-degenerate (*i.e.* entirely distinguishable) photons are coupled into waveguides $A$ and $B$ (Fig. \[schematic\](d)) the photons have a 50% probability of both being transmitted or reflected inside the coupler—*i.e.*, they behave like “classical" particles—such that there is a 50% probability of detecting them simultaneously at the two different APDs. In contrast, if two degenerate photons are input to $A$ and $B$ the pair will be transformed into an entangled superposition of two photons in output waveguide $C$ and two photons in output waveguide $D$: $$\label{hom} |11\rangle_{AB}\rightarrow\frac{|20\rangle_{CD}-|02\rangle_{CD}}{\sqrt{2}}.$$ This *quantum interference* ideally yields no simultaneous photon detection events at the separate APDs when the difference in arrival time of the photons at the coupler is zero, giving rise to the characteristic Hong-Ou-Mandel [@ho-prl-59-2044] (HOM) “dip" in the coincident detection rate as a function of delay (see Fig. \[dip\]). [In our setup, the relative arrival time of the two photons was the free parameter in photon distinguishability and allowed the level of quantum interference to be directly controlled. More generally the visibility of the HOM dip is also limited by other degrees of freedom including polarization, frequency, spatial mode, and beam coupling ratio (or reflectivity) of the directional coupler. While the reflectivity parameter is inherent to the optical circuitry, the remaining degrees of freedom can be attributed to both the source of photons and individual manipulation of these parameters within the circuit.]{} [Considering the device separately to the source of photons,]{} the *ideal* HOM dip visibility, $V\equiv (max-min)/max$, is a function of the equivalent beamsplitter reflectivity $\eta$: $$\label{visibility} V_{ideal}=\frac{2\eta(1-\eta)}{1 - 2 \eta + 2 \eta^2}$$ [which is a maximum for $\eta=\frac{1}{2}$]{}. Imperfections in the waveguides [that perturb the state of a photon in any degree of freedom]{} will degrade the degeneracy of the photon pairs and reduce the measured $V$ below $V_{ideal}$. [Assuming a SPDC source prepares identical photon pairs the relative visibility $V_{rel}\equiv V/V_{ideal}$ of the HOM dip provides the operational fidelity of a directional coupler and thus $V_{rel}$ is]{} the key quantifier of the performance of a photonic quantum circuit [in preserving photon degeneracy. In this work we have not corrected for imperfections in either the sources’ degeneracies or the measurement setups, hence our measurements of $V_{rel}$ are lower bounds for the fidelities of the devices.]{} In addition to high-fidelity quantum interference (as quantified by $V_{rel}$), quantum circuits require high-visibility, stable classical interference. [Using the 2-photon SPDC source as a convenient source of 804 nm photons we measured the effective reflectivity of the Mach-Zehnder interferometers by blocking one arm of the SPDC source to input one photon at a time into the arms of the interferometer. In the case of an ideal null-phase difference interferometer with 50:50 beam splitters the effective reflectivity of the device, $\eta_{MZ}$, should be 1. Values for $\eta_{MZ}<1$ or instabilities under changing environmental conditions can therefore be used as a device performance metric.]{} 3-photon quantum interference ----------------------------- quantum logic gates [@kn-nat-409-46; @sa-prl-92-017902], quantum metrology [@li-pra-77-023815], photon number filters [@sa-prl-96-083601; @re-prl-98-203602], entanglement filters [@ho-prl-88-147901; @ok-sci-323-483], and biphoton qutrit unitaries [@la-prl-100-060504]. When two photons are input into $A$ and one in $B$ an ideal $\eta=2/3$ reflectivity coupler will generate the three-photon entangled state: $$\label{21hom} |21\rangle_{AB}\rightarrow\frac{2}{3}|30\rangle_{CD}-\frac{\sqrt{3}}{3}|12\rangle_{CD}-\frac{\sqrt{2}}{3}|03\rangle_{CD},$$ where quantum interference results in no $|21\rangle_{CD}$ term. An analogue of the HOM dip can therefore be observed by measuring the rate of detecting two photons in C and one in D as a function of the delay time between the photon in $B$ and the two photons in $A$ [@sa-prl-96-083601]. To observe a $|21\rangle_{CD}$ HOM dip, as described by Eq. (\[21hom\]), we used a pulsed laser system (Fig. \[21dip\](a)) to generate four photons in 2 modes at 780 nm where the DWQCs are also single moded. The output of a $\sim$150 fs, 80 MHz repetition rate 780 nm Ti:Sapphire laser was frequency doubled to 390 nm and then down converted into pairs of 780 nm photons in a type-I phase matched BiBO crystal. The photon pairs passed through 3 nm bandpass interference filters before being coupled into two polarization maintaining single mode optical fibres. In all other respects the setup is similar to the CW one shown in Fig. \[schematic\](d). By using a fused PMF splitter and single photon APD we were able to probabilistically prepare the $|1\rangle_B$ state at input $B$ (Fig. \[21dip\](a)). Using a SMF fibre coupler we were able to probabilistically detect two photons at output $C$. [The low probability of preparing the required 4-photon state at the source and the $\frac{1}{4}$ success rate of experimental setup significantly reduced the measurement count rate from that of the 2-photon experiment. By testing circuit stability over long durations, interference measurements of this form are a crucial trial of these circuits for applications in advanced, multi-photon quantum optics. Each set of measurements with this setup took $\sim$60 hours to complete and are the first demonstration of interference between more than 2-photons in an integrated waveguide-chip platform.]{} ![Quantum interference in a laser direct-write directional coupler. The number of coincident detections are shown as a function of the arrival delay between the two interfering photons. Error bars from Poissonian statistics are smaller than the point size and the fit is a Gaussian plus linear.[]{data-label="dip"}](Fig2.eps){width="45.00000%"} Results ======= We measured the performance of the DWQC devices using the 2-photon setup shown schematically in Fig. \[schematic\](d). Figure \[dip\] shows the raw data for a HOM dip in a coupler with $\eta=0.5128\pm 0.0007$ (maximum theoretical visibility $V_{ideal}=0.9987\pm 0.0001$). The measured visibility is $0.958\pm 0.005$. Figure \[vis\] shows the measured visibility $V$ as a function of the equivalent reflectivity $\eta$ for eight couplers on the two chips. The curve is a fit of Eq. (\[visibility\]), modified to include a single parameter to account for mode mismatch [@ro-pra-72-032306]. The average relative visibility for these eight couplers is $\overline{V_{rel}}=0.952\pm 0.005$, demonstrating high performance across all couplers on both chips. ![Quantum interference visibility as a function of coupling ratio. Hong–Ou–Mandel interference visibility for couplers of various reflectivities $\eta$. Error bars are determined from fits such as those in Fig. \[dip\] and are comparable to the point size.[]{data-label="vis"}](Fig3.eps){width="45.00000%"} [Using single-photons, we measured the effective reflectivity of a Mach-Zehnder interferometer and found that the reflectivity of the device was $\eta_{MZ}=0.960\pm 0.001$. This indicates that the error in written phase shift]{} in the interferometer was very close to zero [(of the order 10 nm).]{} [Using the 4-photon source shown in Fig. \[21dip\](a) we were able to generate the three-photon entangled state described in Eqn. (\[21hom\]).]{} Figure \[21dip\](b) shows the generalized HOM dip observed in a $\eta=0.659$ reflectivity DWQC coupler. The visibility of this dip is $V_{rel}=0.84\pm0.03$ [which surpasses the value of $V=0.78\pm0.05$ previously observed in a bulk optical implementation [@sa-prl-96-083601])]{}. We believe our visibility to be limited by a small amount of temporal distinguishability of the photons produced in the source, nominally in the $|2\rangle_A$ state, and not by the waveguide device. [These results demonstrate the enhanced stability afforded by guided-wave circuits and are the first demonstration of multi-photon interactions in an integrated optics platform.]{} (a) ![Generalized quantum interference with three photons. (a) Measurement setup consisting of a SPDC source based on a frequency doubled Ti:Sapphire *fs* oscillator. (b) Generalized HOM dip for three photons. The number of coincident detections are shown as a function of the arrival delay between the two interfering photons.[]{data-label="21dip"}](2-1ExperimentalSetup_2.eps){width="100.00000%"} (b) ![Generalized quantum interference with three photons. (a) Measurement setup consisting of a SPDC source based on a frequency doubled Ti:Sapphire *fs* oscillator. (b) Generalized HOM dip for three photons. The number of coincident detections are shown as a function of the arrival delay between the two interfering photons.[]{data-label="21dip"}](Fig4.eps){width="100.00000%"} Conclusions =========== DWQCs overcome many of the limitations of standard lithographic approaches: while the devices described here were written in 2D, extension to a 3D architecture is straightforward; single devices can be made with short turnaround for rapid prototyping; and the direct-write technique easily produces devices with circular mode profiles, the size and elipticity of which can be adjusted by changing the laser focusing conditions. This ability to tailor the guided mode will enable production of waveguides that better match fibre modes and reduce photon losses, and could be combined with the ongoing development of waveguide [@fu-oe-15-12769; @zh-oe-15-10288] or fibre [@fu-prl-99-120501] based photon sources. Fabrication of sophisticated integrated quantum photonic circuits can now be achieved with only an ultrafast laser system rather than state-of-the-art semiconductor processing facilities. [Quantum circuits for photons, regardless of their application, comprise multi-path and multi-photon interferometers involving generalized quantum interference at a beam splitter (or directional coupler). High visibility interference such as that demonstrated here is therefore crucial to realizing future quantum technologies and]{} the next generation of fundamental quantum optics experiments. **Acknowledgments**\ This work was supported by the Australian Research Council through their centres of excellence program, the UK EPSRC, QIP IRC, the [Macquarie University Research Innovation Fund]{} and the US IARPA.

ArXiv

--- abstract: 'Gravity is the weakest of all four known forces in the universe. Quantum states of an elementary particle due to such a weak field is certainly very shallow and would therefore be an experimental challenge to detect. Recently an experimental attempt was made by V. V. Nesvizhevsky et al., Nature ${\bf 415}$, 297 (2002), to measure the quantum states of a neutron, which shows that ground state and few excited states are $\sim 10^{-12}$ eV. We show that the energy of the ground state of a neutron confined above Earth’s surface should be $\sim 10^{-37}$eV. The experimentally observed energy levels are $10^{25}$ times deeper than the actual energy levels it should be and thus certainly not due to gravitational effect of Earth. Therefore the correct interpretation for the painstaking experimental results of Ref. [@nes1] is due to the confinement potential of a one dimensional box of length $L \sim 50\mu$m, generated from the experimental setup as commented before [@hansoon]. Our results thus creates a new challenge to the experimentalist to resolve the shallow energy levels of the neutron in Earth’s gravitational field in future.' author: - Pulak Ranjan Giri title: 'Quantization of neutron in Earth’s gravity' --- The investigation of quantum phenomenon in gravitational field is certainly interesting and challenging [@nes1; @nes3; @peters; @ber] due its weakness of strength. To get an idea of the weakness of gravitational force over other forces a quantitative estimation may be helpful. The the gravitational attraction of two neutrons separated by a distance $r$ is $\sim 10^{-36}$ times weaker [@hartle] than the Coulomb repulsion between the two electrons separated by the same distance. One therefore needs to be very careful while investigating the quantum effects of gravity. Neutron is a possible candidate on which quantum effects of gravity can be investigated because charge neutrality will eliminate electromagnetic force from our considerations. The nature of the gravitational force $F$ of Earth (except the strength) experienced by a neutron is same (long range and proportional to the inverse of the distance between the two agents) as that of the Coulomb force experienced by an electron in a Hydrogen atom. It is therefore expected that the nature of energy states of a neutron in the Earth’s gravitational force will be similar to that of a Hydrogen atom with an infinite hard sphere core [@care; @meyer]. We need to keep in mind that the neutron is above the Earth’s surface, so we assume that the wave-function within the Earth is zero, i.e., $\psi(r)=0$ for $r\leq R_{\oplus}$, where $R_{\oplus}$ is the Earth’s radius (it is assumed that Earth is completely spherical). Since the neutron of mass $m$ can not penetrate within the Earth, it will put an upper bound to the absolute value of the energy $E_n$ of the discrete quantum states, which is $|E_n|\leq (\hbar^2/2m)R_{\oplus}^{-2}\approx 5.08\times 10^{-37}$eV [@care; @meyer]. Note that the states are $\sim10^{-25}$ times less deeper than that obtained in the recent experiment by V. V. Nesvizhevsky et al., Nature ${\bf 415}$, 297 (2002). Then the question arises that what is the reason of getting quantum states $\sim 10^{-12}$eV in the experiment? The correct interpretation for observing $\sim$peV ($1$peV= $10^{-12}$eV) states in the experiment is the following. The experimental set up consists of a bottom mirror and a top absorber with a gap of approximate $50\mu$m in between them. This can be considered as a problem of a particle in a one dimensional box [@landau] of length $L= 50\mu$m. In fact it has been commented before in Ref. [@hansoon], see the corresponding reply [@nes4] also. For the present purpose we may neglect the dynamics of the neutron in the transverse direction. The energy levels for the neutron in the potential created by the box is $E_n= -(\hbar^2\pi^2n^2/2m)L^{-2}$. The first few states for $L= 50\mu$m are respectively given by $E_1\approx 0.082$peV, $E_2\approx 0.3272$peV, $E_3\approx 0.7362$peV, $E_4\approx 1.309$peV, $E_5\approx 2.0451$peV, $E_6 \approx2.945$peV, $E_7\approx 4.01$peV. Note that the experimentally obtained first four energy levels in Ref. [@nes1] are comparable with the above obtained theoretical levels $E_4$, $E_5$, $E_6$ and $E_7$ respectively. We then need to answer what is wrong with the previous theoretical prediction of Ref. [@nes2], which shows that the discrete quantum levels due to Earth’s gravitational force is $\sim$ peV? In fact it agrees with the experimental results [@nes1]. The answer could be found partly in the potential $U(z)= mgz$ considered for the neutron above the Earth’s surface. The other drawback is that the spherical symmetry of the problem due to central force has been completely ignored and thus the dynamics of the neutron in the $z$ direction has be decoupled by assuming that in the transverse directions the particle is free. It is of course true that the potential $U(z)$ is approximately valid for $z\ll R_{\oplus}$. But for the neutron above the Earth the wave-function in principle would extend from earth’s surface to infinity. Thus $U(z)$ is useless in this situation and instead should be replaced by spherically symmetric Newton’s potential $V(r)= -GM_{\oplus}m/r$ [@penrose], where $G$ is the universal gravitational constant, $M_{\oplus}$ is the mass of the Earth and $r$ is the distance of the neutron from the Earth’s center. The potential within the Earth is as usual infinity because of the assumption that the probability of finding the neutron within the Earth is zero. The ground state of the neutron on the Earth’s surface in gravitational field is thus $E_{\mbox{g.s}}\approx-(\hbar^2/2m)R_{\oplus}^{-2}\approx -5.08\times 10^{-37}$eV [@care; @meyer], since it is the maximum deep state of the neutron. The analytical calculation for all the excited states of the neutron will be in line with Ref. [@meyer]. However analytical solutions for excited states are not important for our present purpose because they all will be even less than the ground state energy. The point that the deepest bound state (which is ground state) is $\sim 10^{-37}$eV is the most important message here. However one needs to think about the validity of the basic assumption in reality that the neutron wave-function within the Earth is zero. Because, the penetration of the quantum particle probability within the Earth will change the bound on the quantum energy levels of the neutron. But this is an issue which can be best resolved by experimental observation. We however have considered this assumption based on the experiment [@nes1] The next immediate challenge to the experimentalist is to detect the quantum states due to Earth’s gravity. Our theoretical observation does not rule out the experimental detection of $\sim 10^{-12}$eV quantum states of a neutron but rather it gives a correct interpretation for the existence of peV states. The one dimensional box potential generated from the lower mirror and top absorber of length $\sim 50\mu$m dominates the gravitational potential on the Earth’s surface. Gravitational force on the Earth is so weak that the quantum states of a neutron due to such force is $\sim 10^{-37}$eV, based on the assumption that neutron wave-function inside the Earth is zero. The experimental resolution power should be much higher than the present we have [@nes1] in order to detect such quantum states. [99]{} V. V. Nesvizhevsky et al., Nature [**415**]{}, 297 (2002). V. V. Nesvizhevsky et al., Phys. Rev. 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ArXiv

--- abstract: | ------------------------------------------------------------------------ On-device intelligence is gaining significant attention recently as it offers local data processing and low power consumption. In this research, an on-device training circuitry for threshold-current memristors integrated in a crossbar structure is proposed. Furthermore, alternate approaches of mapping the synaptic weights into fully-trained and semi-trained crossbars are investigated. In a semi-trained crossbar a confined subset of memristors are tuned and the remaining subset of memristors are not programmed. This translates to optimal resource utilization and power consumption, compared to a fully programmed crossbar. The semi-trained crossbar architecture is applicable to a broad class of neural networks. System level verification is performed with an extreme learning machine for binomial and multinomial classification. The total power for a single 4x4 layer network, when implemented in IBM 65nm node, is estimated to be $\approx$ 42.16$\upmu$W and the area is estimated to be 26.48$\upmu$m x 22.35$\upmu$m. author: - 'Abdullah M. Zyarah Dhireesha Kudithipudi' title: 'Semi-Trained Memristive Crossbar Computing Engine with In-Situ Learning Accelerator' --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010147.10010257.10010293.10010294&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computing methodologies Neural networks&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010583.10010786&lt;/concept\_id&gt; &lt;concept\_desc&gt;Hardware Emerging technologies&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; This material is based on research sponsored by AirForce Research Laboratory under agreement number FA8750-16-1-0108. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AirForce Research Laboratory or the U.S. Government. Authors’ addresses: A. M. Zyarah and D. Kudithipudi, Neuromorphic AI Lab, Rochester Institute of Technology, Rochester, NY; emails: {amz6011, dxkeec}@rit.edu. Introduction ============ On-device intelligence is gaining significant attention recently as it offers local data processing and low power consumption, suitable for energy constrained platforms (*e.g.* IoT). Porting neural networks on to embedded platforms to enable on-device intelligence requires high computational power and bandwidth. Conventional architectures, such as von Neumann architecture, suffer from throughput drop and high power draw when realizing neural networks. This can be attributed to the physical separation between processing and memory units, which leads to memory bottleneck [@indiveri2015memory]. Additionally, pure CMOS implementation of neural networks impose area and power constraints that hinder the deployment on to embedded platforms [@kim2012neural]. In 2008, a successful physical implementation of a synapse-like device called memristor was proposed by Strukov [@strukov2008missing]. Theoretically, the memristor was introduced by L. Chua in 1971 as a fourth fundamental electrical device that correlates the flux and charge in a non-linear relationship [@chua1971memristor]. The memristor acts like a non-volatile memory element [@borghetti2010memristive], consumes low energy [@prezioso2015training; @merkel2017current], has a small footprint compared to transistors, and can be integrated in high density crossbar structures [@jo2010nanoscale]. A key advantage of the crossbar structure is that it enables performing the most computationally intensive operations (multiply-accumulate) in neural networks concurrently while consuming small amount of power compared to conventional implementations [@snider2008spike; @taha2014memristor]. These properties make the memristor a natural choice for realizing neural networks in an efficient manner such that it meets embedded device constraints. Typically, memristive devices are used to model the bi-polar synaptic weights in neural networks. Due to the fact that a memristor exhibits properties similar to that of a resistor, memristors can represent only positive range of weights. Thus, either a hybrid CMOS-memristor [@kim2012neural; @soudry2015memristor] or two memristors [@alibart2013pattern; @hu2014memristor] are used to model the bipolar synaptic weights. Although modeling the synaptic weight with one memristor is easier to train, it demands additional circuitry to generate the bipolar weights. On the other hand, using two memristors to model the synaptic weights reduces the power consumption, but increases the hardware complexity and the training process. Several research groups have studied the realization of synaptic weights in memristive devices while enabling the on-device learning. To realize the synaptic weight with one memristor, Sah et al. proposed a memristor-based synaptic circuit which employs an H-Bridge and doublet generator to perform positive and negative input-weight multiplication. However, it was not studied in the context of a multi-level network or a crossbar architecture [@sah2012memristor]. In 2015, Soudry et al. presented a memristor crossbar that supports on-chip online gradient descent. In this architecture, two transistors and a memristor were used to implement a synapse. This makes the total number of transistors in the crossbar scale linearly with the number of memristors [@soudry2015memristor]. Adopting two memristors to model the synaptic weights is studied by Alibart et al. who proposed a memristor-based single-layer perceptron to classify synthetic pattern of the letters ’X’ and ’T’. The proposed design is trained using ex-situ and in-situ methods [@alibart2013pattern]. Hasan et al. presented an on-chip training circuit to account for device faults and variability in memristor-based deep neural networks [@hasan2017chip]. This network was trained with auto-encoders and backpropagation and simulated in MATLAB for classification application. When it comes to extreme learning machine (ELM), which is the neural network algorithm used to verify our proposed architecture, few research groups have studied the memristor-based ELM. In 2014 Merkel et al. proposed memristor-based ELM implementation, but it is not studied within the context of a crossbar structure [@merkel2014neuromemristive]. Later in 2015, OxRAM based ELM architecture was proposed by Suri et al. in which the nano-scale device variability is exploited to design ELM in an efficient manner  [@suri2015oxram]. Unfortunately, this work does not provide details about the hardware implementation and the training process. It is also important to mention here that most memristor-based neural network architectures proposed in literature use threshold-voltage memristors. To the best of our knowledge, no design has explored on-device learning for current-threshold memristors integrated in crossbar architecture. This paper proposes on-device training circuitry for current threshold memristors integrated into a crossbar structure. Moreover, the paper presents a different approach for realizing the synaptic weights into a memristive crossbar such that bipolar weights are obtained. The proposed approach is based on using semi-trained crossbar structure (a combination of trained and untrained memristors), where the trained memristor models the synaptic weights and the fixed ones are used in association with the trained memristor to generate bipolar synaptic weights. The proposed design is simulated in Cadence Spectre and verified for classification application in MATLAB using binomial (Diabetes and Australian Credit) and multinomial (Iris and MNIST) datasets  [@Lichman:2013; @lecun1998mnist]. For a single 4x4 layer network (crossbar and its associated control and training circuitries) implemented in IBM 65nm technology node, the total power is estimated to be $\approx$ 42.16$\upmu$W, while the area is 26.48$\upmu$m x 22.35$\upmu$m. The rest of the paper is organized as follows: Section 2 presents an overview about ELM. Section 3 and 4 discuss the design methodology and the hardware analysis. The experimental setup is described in Section 5. Section 6 demonstrates the experimental results and Section 7 concludes the paper. Overview of ELM =============== Extreme learning machine (ELM) is a multi-layer feed-forward neural network used in real-time regression and classification applications [@huang2004extreme]. It has roots back in the random vector functional link (RVFL) networks proposed in 1994 [@pao1994learning]. Primarily, ELM is composed of three successive fully connected layers: input, hidden, and output. The input layer is used to present the input data to the network, whereas the hidden and output layers conduct the feature extraction and data classification, respectively. When the input data is presented to the network, it gets relayed to the hidden layer, where all the relevant and important features are stochastically extracted [@auerbach2014online]. This is done via projecting the input data to high-dimensional space carried out by a large number of hidden neurons [@huang2014insight]. The features extracted by the hidden layer are further relayed to the output layer where the class label associated with the input is identified. A key feature of ELM is that the training is confined only to the output layer synaptic weights, whereas the hidden layer weights are randomly initialized and left unchanged [@huang2006extreme]. This feature speeds up the training in ELM and makes the algorithm attractive for hardware implementations as there is no need for back-propagation. [Figure \[ELM\]]{} illustrates the high-level architecture of an ELM. At runtime, each example in the input dataset is presented to the network as a pair. Each pair contains an input feature vector $X^p$ and its associated class label $t^p$, where $X^p \in \mathbb{R}^{n}$, $\forall p=1, 2, ....,L$ and $L$ is the dataset size. Using [Equation (\[ff\_eqn\])]{}, the network feed-forward output can be computed, where $t^*_i$ represents the predicted output of the $i^{th}$ output unit, $\forall i= 1,2, .... k$. $k$ and $\eta$ denote the total number of neurons in the output and hidden layers, $b$ is the bias, while $f$ and $z$ are the activation functions of the hidden and output neurons, respectively. $$\label{ff_eqn} t^*_i = z_{i}\Big(\sum\limits_{j=0}^{\eta-1} \beta_j f_j(X,b)\Big)$$ $$\label{weight_eqn} \beta = H^{-1} T$$ Adopting the normal equation, [Equation (\[weight\_eqn\])]{} (hidden layer output inverse ($H^{-1}$) multiplied by the desired output class labels ($T$)), to find the output layer weight matrix ($\beta$) is a common method in ELM [@huang2004extreme; @kasun2013representational] as it offers faster convergence compared to the numerical counterpart. However, realizing the matrix inverse in hardware is cumbersome [@perina2017exploiting]. Rather than using the normal equation, the iterative delta rule algorithm [@jacobs1988increased] is chosen. In delta rule, a weight $\beta_{i,j}$ connecting the $j^{th}$ neuron in hidden layer to $i^{th}$ neuron in output layer is updated according to [Equation (\[weight\_eqn2\])]{}, where $\alpha$ is the learning rate, $h_{i,j}$ and $(t^{*}_{i} - t^p_{i})$ refer to the input and the output error of the $i^{th}$ neuron, respectively. $$\label{weight_eqn2} \Delta \beta_{i,j} = \alpha \times h^p_{i,j} \times (t^{*}_{i} - t^p_{i})$$ Design Methodology ================== Memristive Crossbar Network --------------------------- In order to perform the matrix-vector multiplication in ELM, a memristive crossbar is used as it enables high-speed computations while maintaining low power consumption and area overhead. Unfortunately, the memristive crossbar structure offers only positive range of synaptic weights. Therefore, two memristors are used to model the synaptic weights in a bipolar manner. Typically, this is achieved either by using two crossbars or one crossbar with dual input (in this work, dual input refers to a signal and its negation) [@alibart2013pattern; @hu2014memristor; @hasan2016memristor; @chakma2018memristive]. Here, both approaches of realizing the synaptic weights in the memristive crossbar and their constraints will be discussed. Furthermore, in each section, the proposed optimization approach, called semi-trained crossbar, will be investigated. ### Two Crossbars topology In this topology, two crossbars are employed to generate positive and negative weight ranges [@alibart2013pattern; @hu2014memristor]. [Figure \[cross\_weight\]]{}-(a) illustrates the use of two crossbars in emulating the synaptic weights, where each weight value is given by [Equation (\[weight\])]{}. $R_{f}$ is the feedback resistance of the Op-Amp based subtracter, and $M^+_{i,j}$ and $M^-_{i,j}$ denote the memristor resistance at the crosspoint ($i,j$) for the left (pink) and right (green) crossbars, respectively. By applying an input voltage ($X^p$ = \[$x^p_0, x^p_1, ....x^p_{n-1}$\]) at the word-lines (crossbar rows), an output current ($T$ = \[$t^*_0, t^*_1,....t^*_{k-1}$\]) will be generated as given by [Equation (\[vout\])]{}, where $\beta$ is the synaptic weight matrix which can be calculated using [Equation (\[weight\])]{}. $$\beta_{i,j} = \frac{R_{f}}{M^+_{i,j}} - \frac{R_{f}}{M^-_{i,j}} \label{weight}$$ $$T = X^p \times \beta \label{vout}$$ It turns out that mapping the synaptic weights to two crossbar arrays overwhelm the learning process, as two crossbars need to be trained rather than one. Moreover, a consistent change in the memristors on the positive and negative crossbars must be sustained to ensure network convergence. Therefore, this research suggests a different approach (called semi-trained) to realize the synaptic weights. This approach utilizes only one crossbar associated with one fixed reference line. The reference line can be created either by using memristors or resistors. [Figure \[cross\_weight\]]{}-(b) illustrates the structure of the proposed approach. On the left side, a memristor crossbar is implemented to emulate the synaptic weights. On the right side, one column (denoted by $M^-$) of fixed memristors is used such that positive and negative weight ranges are obtained. By adopting this approach, the number of Op-Amps used at the bit-lines will be shrunk to almost half and thereby reduce hardware resources significantly. ### One Crossbar topology One crossbar is used to model the synaptic weights in this topology. However, in this crossbar array, the input needs to be negated to achieve bipolar input-weight matrix-vector multiplication. [Figure \[cross\_weight\_appr3\]]{} depicts the schematic of one crossbar structure, where the input vector $X^p$ and its negation ($\sim X^p$) are introduced to the crossbar and are multiplied by the synaptic weight matrix $\beta$, as given by [Equation (\[vout\])]{}. When it comes to training such a structure, again all the memristors in the crossbar need to be adjusted. By adopting the semi-trained approach, $M^-$ memristors are set to a fixed value, whereas $M^+$ memristors are trained to achieve the desired network convergence. ### Two Crossbars Vs. One Crossbar: In spite of the fact that both crossbar topologies are capable of performing the intended function (bipolar matrix-vector multiplication), when it comes to hardware, each approach imposes different constraints. The downside of using two crossbars to emulate the synaptic weights is that the input-weight multiplication is fractioned into two parts. One of them is accomplished via $M^+$ crossbar, whereas the second is done in $M^-$ crossbar. Due to this separation, additional constraints are imposed on the network input and its weight range. [Figure \[cross\_weight\_down\]]{}-(a) illustrates a circuit of one neuron with $n$ inputs, the output of the neuron is given by [Equation (\[neuron\_eq\])]{} and [Equation (\[neuron\_eq2\])]{}. $$t^*_i = (x_0 \frac{-R_f}{M^+_0} + ..... +x_{n-1} \frac{-R_f}{M^+_{n-1}}) + V_{x} \frac{-R_f}{R_x} \label{neuron_eq}$$ $$V_x = x_0 \frac{-R_x}{M^-_0} + ..... +x_{n-1} \frac{-R_x}{M^-_{n-1}} \label{neuron_eq2}$$ Due to the fact that $V_x$ is computed by the first Op-Amp, then its maximum value is always limited to the first Op-Amp biasing voltages ($V_{dd}$ and $V_{ss}$), i.e. [Equation (\[neuron\_eq3\])]{} must be satisfied. Consequently, the input and weight range will be limited as well as the crossbar size. $$\sum_{i=0}^{n-1} x_i \frac{-R_x}{M^-_i} \leq (V_{dd} - V_{ss}) \label{neuron_eq3}$$ In cases where only one crossbar is used to perform the input-weight matrix-vector multiplication, an additional inverter is needed to negate the input signal. However, in this network the input-weight multiplication will not be segregated. Thus, using only one Op-Amp at the output will suffice, as shown in [Figure \[cross\_weight\_down\]]{}-b. The output here is given by [Equation (\[neuron\_eq\_4\])]{}. Since $t^*_i$ is associated with one Op-Amp, the constraints that we had in [Equation (\[neuron\_eq3\])]{} are not applied here. Instead, [Equation (\[neuron\_eq\_5\])]{} must be satisfied, which infers that every single input feature multiplied by its corresponding weight is evaluated separately to be $\leq (V_{dd} - V_{ss})$. This eventually alleviates the constraints we had when using two Op-Amps. Reducing the input and weight range constraints gives more flexibility when it comes to hardware implementation. Moreover, large crossbars can be realized. $$t^*_i = (x_0 \frac{-R_f}{M^+_0} + x_0 \frac{R_f}{M^-_0}..... +x_{n-1} \frac{R_f}{M^-_{n-1}}) \label{neuron_eq_4}$$ $$x_i \frac{-R_f}{M^-_i} \leq (V_{dd} - V_{ss}) \label{neuron_eq_5}$$ [Figure \[Neuron\]]{} demonstrates the input-weight multiplication performed using the neuron circuits from [Figure \[cross\_weight\_down\]]{}, where all the input features ($X_{n}$, where n =2) are assigned to $0.3$sin($wt$), $R_f = R_x = 500k\Omega$, and $M^{+} = M^{-} = 250k\Omega$. $V_{out2}$ and $V_{out1}$ denote the output of neuron-(a) and neuron-(b), which can be computed based on [Equation (\[neuron\_eq\])]{} and [Equation (\[neuron\_eq\_4\])]{}, respectively. Although the output in both cases should be the same (=0v), neuron-(a) gives incorrect output as it violated the constraints in [Equation (\[neuron\_eq3\])]{} (notice that $V_x$ was clipped). This indicates that the neuron circuit in [Figure \[cross\_weight\_down\]]{}-(b) can handle more input-weight range compared to the former. Delta Rule Algorithm -------------------- In order to simplify the learning process on-chip, delta rule algorithm, described by [Equation (\[weight\_eqn2\])]{}, is used. However, realizing the delta rule equation in hardware still requires non-trivial resources (subtractor and multiplier). Therefore, this work adopts the simplified delta rule equation used in [@hu2014memristor] and shown in [Equation (\[delta\_sign\])]{}. $$\Delta \beta_{i,j} = \alpha \times S(h^p_{i,j}) \times S(t^{*}_{i} - t^p_{i}) \label{delta_sign}$$ $$S(i) := \left\{ \begin{array}{l l} 1 & \quad \text{$i > 0$} \hspace{2mm}\\ -1 & \quad \text{Otherwise} \label{permanence_equation2} \end{array} \right.$$ Here, the weight will essentially change according to the sign of the gradient and has a fixed learning rate. Although such procedure will slow down the convergence speed, the resources used for learning circuity will be significantly minimized. By applying the delta rule to the semi-trained crossbar structure, the iterative change in weight value ensures network convergence can be computed. Recall that each synaptic weight is emulated by fixed and tuned memristors. By using [Equation (\[weight\])]{} and [Equation (\[delta\_sign\])]{}, the net change in the memristor to achieve the desired weight value can be calculated, as in [Equation (\[delta\_weight\])]{}. $$M_{new} = \frac{M_{old} \times R_f}{R_{f} - \alpha \times S(x^p_{i,j}) \times S(t^{*}_{i} - t_{i,p})} \label{delta_weight}$$ $$M = \frac{1}{M^+} - \frac{1}{M^-}$$ System Design and Analysis ========================== The system level architecture of ELM consists of two main layers: hidden and output. In this section, the main focus will be on the output layer as it has similar structure to the hidden layer except that the output layer is integrated to the training circuit due to the need of synaptic weight adjustment. [Figure \[system\]]{} shows the architecture of the output layer which essentially has three parts: memristive crossbar, neuron circuit, and training circuit. The memristive crossbar represents a single layer of ELM in which each column corresponds to one neuron connected to $R_n$ ($R_n$ is the number of crossbar rows) number of synapses modeled by memristors. The crossbar is responsible for evaluating the input-weight matrix multiplication, whereas the neuron and training circuits are responsible for performing a non-linear transformation on crossbar bit-line outputs and adjusting the weights of the crossbar, respectively. Primarily, the proposed network runs in two phases: inference and learning. During the inference phase, the input vector ($X^p$ = \[$x^p_0, x^p_1, ....x^p_{n-1}$\]) is fetched to the network where it gets multiplied by the synaptic weight matrix ($\beta$) to generate the output vector ($T^*$). The output of the network is evaluated by comparing it to input class label and the difference is reported either as logic ’1’ denoting that $t^*_i > t_i$, or logic ’0’ otherwise. The output of the error computing unit is stored into a shift register to be used in the learning phase. In the learning phase, the memristor resistances are adjusted according to the sign of the gradient and learning rate. In this work, the tuning of memristor is done column-by-column (training each column takes two clock cycles) through a modified Ziksa training circuit [@zyarah2017ziksa], which is modeled by $+Tr$ and $-Tr$. Ziksa is used to form an H-Bridge across the memristors that require tuning and by allowing the current to flow through the device in both directions, bipolar weight change can be achieved (more details is provided section \[ziksa\_section\]). It is important to mention here that each Ziksa unit is controlled by local row and column units which in turn are controlled by a global controller. The global controller determines when to enable the inference and learning phases and is responsible for synchronization. In the following subsections, each unit in the system architecture is discussed in detail. Ziksa: Training Circuitry {#ziksa_section} ------------------------- Recall the simplified delta rule algorithm suggests a fixed adjustment in the memristive weight. In the adopted memristor model, the changes in memristive state variable are dependent on the current flowing in the device as given by [Equation (\[delta\_w\])]{} [@kvatinsky2013team] $$\frac{\Delta w}{\Delta t} = \begin{cases} k_{off}.\Big(\frac{i(t)}{i_{off}} - 1\Big)^{\alpha_{off}}.f_{off}(w),~~~~~0 < i_{off} < i \\ 0, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i_{on} < i < i_{off} \\ k_{on}.\Big(\frac{i(t)}{i_{on}} - 1\Big)^{\alpha_{on}}.f_{on}(w),~~~~~~~~~~~i <i_{on} < 0 \end{cases} \label{delta_w}$$ where $w$ is the memristor state variable. $k_{off}$, $k_{on}$, $\alpha_{off}$, and $\alpha_{on}$ are constants, $i_{off}$ and $i_{on}$ are the memristor current thresholds, and $f_{on}$ and $f_{off}$ describe the device window function. As the memristor exhibits properties similar to that of a resistor, the current flowing in the device will be limited by its resistance. Thus, to satisfy the learning rule constraints, in this work, we modified our previous design of Ziksa to accommodate this issue. [Figure \[Ziksa\]]{} illustrates the modified version of Ziksa in which two current mirrors are used to limit the amount of current flowing in the memristor which consequently ensures consistent adjustment of memristor resistance. The circuit works as follows: during the first clock cycle of learning, which involves increasing the weight values (this means decreasing the memristor resistance as $\beta_{i,j} \propto (1 / M_{i,j}$)) in a selected column, current will be provided to the memristor via $T_5$. To ensure a fixed change in the memristor value, this current will be limited to $\approx I^-$ by using a current mirror created by $T_{1-2}$ on the other terminal of the memristor. During the second cycle of training, the weight will be decremented by allowing the current, $\approx I^+$, to flow in the opposite direction. Practically, fixing the current through the memristor is a difficult condition to meet with the current technology limitations. However, there is still a possibility to limit the variation in the current through memristor while changing its state via a cascode current mirror. The variation in the current through memristor in regular and cascode current mirrors when using $I_{ref}$ = 4 $\upmu$A is depicted in [Figure \[CurrentMirror\]]{}-(a). The corner analysis evaluation while considering the fabrication process, ambient temperature, and supply voltage variations is shown in [Figure \[CurrentMirror\]]{}-(b). Column and Row Local Control Units ---------------------------------- Ziksa transistors are driven by local control units associated with each column and row. Recall that Ziksa has four transistors that form an H-Bridge sandwiching the memristors in the crossbar. Half of the H-Bridge transistors reside in $+Tr$ and the other half in $-Tr$. Each $-Tr$ is controlled by its associated column controller, shown in [Figure \[localCU\]]{}-(b). The column local control unit consists of a combinational logic circuit that drives the $T_5$ and $T_6$ transistors of $-Tr$. When the learning phase begins, the column controller receives two signals: $ColEn$ and $Polar$. The former signal determines whether a column is selected for training, whereas the latter refers to the system training cycle which can be either positive or negative. During the positive cycle of training, i.e. $Polar$ = ’0’, all the weights that need to be incremented are adjusted, while the weights required to be decremented are tuned in the negative cycle of training. When a column is enabled by $ColEn$, $T_5$ and $T_6$ transistors of $-Tr$ will be controlled in an alternating way. During the low cycles of the $Polar$ signal, $PT$ is set to low to enable transistor $T_5$, whereas $T_6$ is set to off via $NT$. This allows the current to flow towards the end-terminal of the crossbar row to increase the weight and vice versa for the high period of the $Polar$ signal. In case of $+Tr$, its transistors are controlled via the formed current mirrors with transistors $T_1$ and $T_4$. The output of $+Tr$ is connected to a tri-state gate controlled by the row controller, illustrated in [Figure \[localCU\]]{}-(a). It turns out that the row controller is more complex compared to the column controller because the gradient sign of delta rule is evaluated here via the input signal ($input$) and computed network error ($Error$). Based on the gradient sign, the memristor resistance will be either incremented or decremented. However, this is carried out when the training process is enabled via $TrEn$. According to the $Polar$ signal state, the memristor resistances are modulated during the appropriate training cycle. It is important to mention here that the $ColEn$, $Polar$, and $TrEn$ signals are provided to the local controllers via the global controller. Global Controller ----------------- All the layers in the proposed system architecture are controlled by a global controller which takes care of data flow and unit synchronization. The global controller runs in three main states. In the first state ($Read$), the global control unit enables the pass transistors to allow the input signals to propagate through the crossbar to perform input-weight matrix multiplication. Once this is done, the output of the network will be captured and the next state ($Train\_C1$) starts. In this state, the first round of training, positive cycle, is performed. Thus, the signals that control the local controllers must be generated. For the column controller, if a column is selected for training, its associated $ColEn$ is set to ’1’ whereas $Polar$ signal is set to low indicating the positive cycle of training. In case of the row controller, $TrEn$ is set to high, which in association with the input and computed error signs, determines the synaptic weights that need to be incremented. When the global controller runs into the third state ($Train\_C2$), the negative cycle of training will commence, and the same process from the previous cycles will be repeated except that the $Polar$ signal is set to high. The global control unit keeps moving between the second and third states until all the columns of the crossbar are trained. [Figure \[Global\_ASM\]]{} is an algorithm state machine chart demonstrating the transition between the states and the output of each one. Experimental Setup ================== The Verilog-A memristor model proposed by  [@kvatinsky2013team] is employed in this work. The memristor value is set based on the device parameters described in [@fan2014design; @kawahara20082] such that it meets the network and technology constraints, shown in [Table \[para\_table\]]{}. The high resistance state (HRS) of the memristor is selected to be 250 k$\Omega$, so that the voltage developed across the memristor does drive the current mirror transistor out of saturation. On the other hand, to keep the power dissipation through the crossbar network as minimum as possible, the low resistance state (LRS) is set to 100 k$\Omega$. **Parameter** **Value** --------------------------------------------- -------------------------- Memristor tuning current 4 $\upmu$A Max. memristor current during the inference 3 $\upmu$A Input voltage range $<$ $\left|0.5\right|$ v Current threshold 3.2 $\upmu$A Memristor low resistance state (LRS) 100 k$\Omega$ Memristor high resistance state (HRS) 250 k$\Omega$ : The parameters used for the ELM network simulation.[]{data-label="para_table"} Recall that the proposed system runs in two phases: inference and learning. In the inference phase no change in memristor values should occur. Since a threshold-current memristor is adopted to model the synaptic weights, the current through these devices should never exceed the device threshold to avoid undesired changes in the memristors. Unlike the inference phase, during the learning phase, the current must cross the device threshold-current to adjust the device resistance. According to the experimental setup used in this work, there is a variation in the current through the memristor during the inference phase. However, this variation is estimated to be $\approx$0.6 $\upmu$A. Thus, by using a current of 4 $\upmu$A during the learning phase and limiting the input current during the inference phase to be 3 $\upmu$A, no overlap between the two phases can occur. The following constraints must be fulfilled for the crossbar pass transistors[^1]: - Limit the input voltage such that $V_{DS} << \mid V_{GS} - V_{Th}\mid$. This ensures that the transistor is working in the triode region and an undistorted signal reaches the memristors. - Assume that $K(V_{GS} - 2V_{Th}) >> G_{mem}(w)$ such that the conductance of the transistor is higher compared to the memristor conductance. Thus, the voltage at the memristor is $\approx$ input voltage [@soudry2015memristor]. ------------------- ------------------------ ---------------------------- ----------------------- **DataSet** **ELM, $\eta$ = 1000** **VLSI ELM, $\eta$ = 120** **This work,** [@huang2012extreme] [@yao2017vlsi] **$\eta$ = variable** Diabetes 77.95% 77.09% 72.73% ($\eta$=65) Australian Credit 87.89% 87.89% 82.16% ($\eta$=40) Iris 96.04% - 84.66% ($\eta$=20) MNIST - - 93.53% ($\eta$=180) ------------------- ------------------------ ---------------------------- ----------------------- : Summary of classification accuracy for binomial and multinomial datasets[]{data-label="miss"} \[tnote:robots-r1\] Software implementation of ELM. \[tnote:robots-r2\] Non-memristive mixed signal implementation of ELM. \[tnote:robots-r3\] All MNIST images are preprocessed with HOG feature descriptor, described in [@zyarah2017extreme]. Experimental Results and Network verification ============================================= Network Verification -------------------- In order to verify the operation of the proposed design, each unit in the network is simulated independently and within a network in Cadence Spectre environment. Then, the same network is emulated in MATLAB and simulated for classification application under the same circuit constraints but with different configurations. The benchmarks employed in this work are selected from UCI library and chosen to be binomial (Diabetes and Australian Credit) and multinomial (Iris). This is added to the multi-class standard hand-written digits dataset, MNIST. [Figure \[Acc\_hist\]]{} depicts the weight distribution of the output layer when using MNIST dataset and the achieved accuracy for each dataset during the training and inference phases. Furthermore, the variation in accuracy that may occur due to the process variation of memristors[^2] and random weight initialization is shown (variation for 5 iterations, each iteration is averaged over 10 runs). [Table \[miss\]]{} shows a comparison of the achieved accuracy with previous ELM implementations for the same datasets. As can be noticed, although the proposed work offers lower classification accuracy, it has a simpler network as the number of hidden neurons is much lower. However, the performance degradation in the network primarily can be attributed to the input voltage and the weight range constraints imposed on the network. These constraints are due to the memristors limited resistance range and the neuron biasing voltage. Network Scalability ------------------- In order to estimate the resources needed to map a large neural network to the proposed design, a full custom design of small-scale (4x4) single layer network is implemented in Cadence using IBM 65nm technology node. [Figure \[scale\]]{} shows the exponential scaling of a single layer neural network from 2x2 up to 128x128, which tends to be linearly proportional to the total transistor count. Power Consumption ----------------- The total power consumption of the proposed design is estimated for a 4x4 single layer neural network. The power consumption is evaluated in Cadence Spectre environment while running the system at 100 MHz. Considering the worst case scenario, (all crossbar memristors are set to low resistance state), the power consumption is estimated to be 40 $\upmu$W for each 4x4 crossbar and 0.63 $\upmu$W for the digital circuit (the Op-Amp power is not considered). It is important to mention here that the power consumption in a memristor while keeping the device resistance unchanged is assumed to be similar to that of a resistor [@marani2015review]. [Table \[power\]]{} shows the comparison of power consumption while using different crossbar architectures and different approaches of realizing the synaptic weights. The power consumption of each architecture is achieved by covering all the input combinations and averaging the results. It can be noticed that the semi-trained two-crossbar architecture offers the minimum power consumption as it is compact and uses almost half the number of memristors compared to other designs. **Architecture** **Digital logic circuit** **Crossbar** **Total** ---------------------------- --------------------------- ---------------- ---------------- Fully-trained two-crossbar 1.45 $\upmu$W 39.53 $\upmu$W 40.98 $\upmu$W Semi-trained two-crossbar 1.22 $\upmu$W 20.85 $\upmu$W 22.07 $\upmu$W Fully-trained one-crossbar 1.52 $\upmu$W 41.01 $\upmu$W 42.53 $\upmu$W Semi-trained one-crossbar 1.15 $\upmu$W 41.01 $\upmu$W 42.16 $\upmu$W : The power consumption distribution of the proposed crossbar architectures.[]{data-label="power"} Conclusions =========== This paper investigates a new approach for realizing positive and negative synaptic weights in a crossbar structure using threshold-current memristors. The proposed approach relies on one memristive crossbar to model the weights and uses additional fixed (untrained) columns to generate bipolar weights. Moreover, the paper presents an updated version of the on-device training circuit, Ziksa, which can be used to modulate current-threshold memristors in a crossbar structure. The proposed network is tested for classification applications with binomial and multinomial datasets while considering memristor device variations. It is found that the process variations have limited effect on the network performance for large datasets. In these cases, the network has better generalization. In scenarios where power efficiency is a constraint the semi-trained network for two crossbar topology is preferred. Future work will investigate the network performance while considering crossbar resistance, noise effect, and other process variations. The authors would like to thank the members of the Neuromorphic AI research Lab at RIT for their support and critical feedback. The authors would also like to thank the reviewers for their time and extensive feedback to enhance the quality of the paper. [^1]: These constraints can be overcome by using transmission gates rather than pass transistors. [^2]: 10% variation in memristor resistance range (LRS and HRS) has been considered during the simulation.

ArXiv

--- abstract: 'Smartphones and mobile devices are rapidly becoming indispensable devices for many users. Unfortunately, they also become fertile grounds for hackers to deploy malware and to spread virus. There is an urgent need to have a “[*security analytic & forensic system*]{}” which can facilitate analysts to examine, dissect, associate and correlate large number of mobile applications. An effective analytic system needs to address the following questions: [*How to automatically collect and manage a high volume of mobile malware? How to analyze a zero-day suspicious application, and compare or associate it with existing malware families in the database? How to perform information retrieval so to reveal similar malicious logic with existing malware, and to quickly identify the new malicious code segment?* ]{} In this paper, we present the design and implementation of [*DroidAnalytics*]{}, a signature based analytic system to automatically collect, manage, analyze and extract android malware. The system facilitates analysts to retrieve, associate and reveal malicious logics at the “[*opcode level*]{}”. We demonstrate the efficacy of DroidAnalytics using 150,368 Android applications, and successfully determine 2,494 Android malware from 102 different families, with 342 of them being [*zero-day*]{} malware samples from six different families. To the best of our knowledge, this is the first reported case in showing such a large Android malware analysis/detection. The evaluation shows the DroidAnalytics is a valuable tool and is effective in analyzing malware repackaging and mutations.' author: - | Min Zheng, Mingshen Sun, John C.S. Lui\ Computer Science & Engineering Department\ The Chinese University of Hong Kong bibliography: - 'paper.bib' title: 'DroidAnalytics: A Signature Based Analytic System to Collect, Extract, Analyze and Associate Android Malware' --- [**Introduction**]{} {#section: introduction} ==================== Smartphones are becoming prevailing devices for many people. Unfortunately, malware on smartphones is also increasing at an unprecedented rate. Android OS-based systems, being the most popular platform for mobile devices, have been a popular target for malware developers. As stated in [@mcafee:mcafee], the exponential growth of mobile malware is mainly due to the ease of generating malware variants. Although there are number of works which focus on Android malware detection via permission leakage, it is equally important to design a system that can perform comprehensive [*malware analytics*]{}: analyze and dissect suspicious applications at the [*opcode level*]{} (instead at the permission level), and correlate these applications to existing malware in the database to determine whether they are mutated malware or even zero-day malware, and to discover which legitimate applications are infected. [**Challenges:**]{} To realize an effective analytic system for Android mobile applications, we need to overcome several technical hurdles. First, how to systematically [*collect*]{} malware from the wild. As indicated in [@blogspot], new malware variants are always hidden in many different third-party markets. Due to the competition of anti-virus companies and their fear of accidentally releasing malware to the public, companies are usually reluctant to share their malware database to researchers. Researchers in academic can only obtain a small number of mobile malware samples. Hence, how to [*automate a systematic process*]{} to obtain these malicious applications is the first hurdle we need to overcome. The second hurdle is how to identify [*repackaged applications*]{} (or mutated malware) from the vast ocean of applications and malware. As reported in[@dimva12], hackers can easily transform legitimate applications by injecting malicious logic or obfuscated program segments so that they have the same structure as the original application but contain malicious logic. Thus, how to determine whether an application is a repackaged or obfuscated malware, and which legitimate applications are infected is very challenging. The third hurdle is how to [*associate*]{} malware with existing malware (or application) so as to facilitate security analysis. The existing approach of using cryptographic hash or package name as an identifier is not effective because hackers can easily change the hash value or package name. Currently, security analysts need to go through a laborous process of manually reverse engineer a malware to discover malicious functions and structure. There is an urgent need to have an efficient method to associate malware with other malware in the database, so to examine their commonalities at the opcode level. [**Contributions:**]{} To address these problems mentioned above, we present the design and implementation of [*DroidAnalytics*]{}, an Andorid malware analytic system for malware collection, signature generation, information retrieval, and malware association based on similarity score. Furthermore, DroidAnalytic can efficiently detect zero-day repackaged malware. The contributions of our system are: - DroidAnalytics automates the processes of malware collection, analysis and management. We have successfully collected 150,368 Android applications, and determined 2,494 malware samples from 102 families. Among those, there are 342 zero-day malware samples from six different malware families. We also plan to release the malware database to the research community (please refer to <https://dl.dropbox.com/u/37123887/malware.pdf>). - DroidAnalytics uses a [*multi-level signature algorithm*]{} to extract the malware feature based on their semantic meaning at the [*opcode level*]{}. This is far more robust than a cryptographic hash of the entire application. We show how to use DroidAnalytics to combat against malware which uses repackaging or code obfuscation, as well as how to analyze malware with dynamic payloads (see Sec. \[sec: signature\]). - Unlike previous works which associate malware via “[*permission*]{}”, DroidAnalytics associates malware and generates signatures at the app/class/method level. Hence, we can easily track and analyze mutation, derivatives, and generation of new malware. DroidAnalytics can reveal malicious behavior at the method level so to identify repackaged malware, and perform class association among malware/applications (see Sec. \[sec: analytic\_capability\]). - We show how to use DroidAnalytics to detect [*zero-day*]{} repackaged malware. We have found 342 zero-day repackaged malware in six different families (see Sec. \[sec: zeroday\]). [**Design & Implementation of DroidAnalytics**]{} {#design} ================================================= Here, we present the design and implementation of DroidAnalytics. Our system consists of modules for automatic malware collection, signature generation, information retrieval and association, as well as similarity comparison between malware. We will also show how to use these functions to detect zero-day repackaged malware. ![The Architecture of the DroidAnalytics[]{data-label="figure: architecture"}](architecture){width="210pt"} Building Blocks of DroidAnalytics --------------------------------- Figure \[figure: architecture\] depicts the architecture of DroidAnalytics and its components. Let us explain the design of each component. [**$\bullet$ Extensible Crawler:**]{} In DroidAnalytics, we implement an application crawler based on Scrapy[@scrapy]. Users can specify official or third party market places, as well as blog sites and the crawler will perform regular mobile application download. The crawler enables us to systematically build up the mobile applications database for malware analysis and association. So far, we have collected 150,368 mobile applications and carried out detailed security analysis. [**$\bullet$ Dynamic Payloads Detector:**]{} To deal with malware which dynamically downloads malicious codes via the Internet or attachment files, we have implemented the dynamic payloads detector component, which determines malicious trigger code within malware packages and tracks the downloaded application and its behavior in virtual machine. Firstly, it scans the package to find suspicious files such as [.elf]{} or [.jar]{} file. Hackers usually camouflage malicious files by changing their file type. To overcome this, this component scans all files and identifies files using their [*magic numbers*]{} instead of file extension. Secondly, if an application has any Internet behavior (e.g., Internet permission or re-delegating other applications to download files[@redelegate]), the dynamic payloads detector will treat these files as the target, then runs the application in the emulator. The system will use the forward symbolic execution technique to trigger the download behavior[@TC:DroidTrace]. Both the suspicious files within the package and dynamically downloaded files from the Internet will be sent to the signature generator (which we will shortly describe) for further analysis. [**$\bullet$ Android App Information (AIS) Parser:**]{} AIS is a data structure within DroidAnalytics and it is used to represent [.apk]{} information structure. Using the AIS parser, analysts can reveal the cryptographic hash (or other basic signature) of an [.apk]{} file, its package name, permission information, broadcast receiver information and disassembled code, $\ldots$, etc. Our AIS parser decrypts the [AndroidManifest.xml]{} within an application and disassembles the [.dex]{} file into [.smali]{} code. Then it extracts package information from source code and retains it in AIS so analysts can easily retrieve this information. [**$\bullet$ Signature Generator:**]{} Anti-virus companies usually use cryptographic hash, e.g., MD5, to generate a signature for an application. This has two major drawbacks. Firstly, hackers can easily mutate an application and change its cryptographic hash. Secondly, the cryptographic hash does not provide sufficient flexibility for security analysis. In DroidAnalytics, we use a [*three-level*]{} signature generation scheme to identify each application. This signature scheme is based on the mobile application, classes, methods, as well as malware’s dynamic payloads (if any). Our signature generation is based on the following observation: [*For any functional application, it needs to invoke various Android API calls, and Android API calls sequence within a method is difficult to modify*]{} (unless one drastically changes the program’s logic, but we did not find any from the 150,368 applications we collected that used this obfuscation technique). Hence, we generate a method’s signature using the API call sequence, and given the signature of a method, create the signature of a class which composes of different methods. Finally, the signature of an application is composed of all signatures of its classes. We like to emphasize that our signature algorithm is not only for defense against malware obfuscation, but more importantly, facilitating malware analysis via class/method association (we will show in later sections). Let us present the detail of signature generation. [**(a) Android API calls table:**]{} Our system uses the API calls table of the Android SDK. The [android.jar]{} file is the framework package provided by the Android SDK. We use the Java reflection[@mccl98] to obtain all descriptions of the API calls. For each API, we extract both the [*class path*]{} and the [*method name*]{}. We assign each full path method a hex number as part of the ID. For the current version of DroidAnalytics, we extract 47,126 full path methods in the Android SDK 4.1 version as our API calls table. Table \[table:APItable\] depicts a [*snapshot*]{} of API calls table, e.g., [android/content/Intent;-&gt;&lt;init&gt; ]{} is assigned an ID 0x30291. [**Full Path Method**]{} [**Method ID**]{} --------------------------------------------------- ------------------- [android/accounts/Account;-&gt;&lt;init&gt; ]{} 0x00001 [android/content/Intent;-&gt;&lt;init&gt; ]{} 0x30291 [android/content/Intent;-&gt;toUri ]{} 0x30292 [android/telephony/SmsManager;-&gt;getDefault ]{} 0x39D53 [android/app/PendingIntent;-&gt;getBroadcast ]{} 0xF3E91 : Example of the Android API Calls Table and assigned IDs[]{data-label="table:APItable"} [**(b) Disassembling process:**]{} Each Android application is composed of different classes and each class is composed of different methods. To generate signatures for each class or method, DroidAnalytics first disassembles an [.apk]{} file, then takes the Dalvik opcodes of the [.dex]{} file and transforms them to methods and classes. Then DroidAnalytics uses the Android API calls table to generate signatures. [**(c) Generate Lev3 signature (or method signature):**]{} The system first generates a signature for each method and we call this the [*Lev3 signature*]{}. Based on the Android API calls table, the system extracts the API call ID sequence as a string in each method, then hashes this string value to produce the method’s signature. Figure \[Fig:lev3\] illustrates how to generate the Lev3 signature of a method which sends messages to another mobile phone. Figure \[Fig:lev3\] shows that the method contains three API calls. Using the Android API calls table (as in Table \[table:APItable\]), we determine their IDs. Signature of a method is generated by cancatenation of all these IDs. Note that DroidAnalytics will not extract the API calls which will not be executed in run time because these codes are usually generated via obfuscation. Furthermore, if a method (except the main method) will not be invoked by any other methods, signature generator will also ignore this method because this may be a defunct method generated by malware writers. ![The Process of Lev3 Signature Generation[]{data-label="Fig:lev3"}](level3){width="250pt"} [**(d) Generate Lev2 signature (either class signature or dynamic payload signature):**]{} Next, DroidAnalytics proceeds to generate the Lev2 signature for each [*class*]{}, and it is based on the Lev3 signatures of methods within that class. Malware writers may use various obfuscation or repackaging techniques to change the [*calling order*]{} of the methods table in a [.dex]{} file. To overcome this problem, our signature generation algorithm will first [*sort*]{} the level 3 signatures within that class, and then concatenate all these level 3 signatures to form the level 2 signature. Some malicious codes are dynamically downloaded from the Internet during execution. DroidAnalytics uses the [*dynamic payloads detector*]{} component to obtain the payload files. For the dynamic payloads which are [.dex]{} file or [.jar]{} file, DroidAnalytics treats them as classes within the malware. Given these files, the system checks their API call sequence and generates a Lev2 signature for each class within an application. For the dynamic payloads which contain, say, [.elf]{} file or [.so]{} file, DroidAnalytics treats them as a single class within that malware, then uses the cryptographic hash value (e.g., MD5) of the payload as its Lev2 signature. For the dynamic payloads which are [.apk]{} files, DroidAnalytics treats each as a new application and a class within the malware. DroidAnalytics first uses the cryptographic hash value (e.g., MD5) of the new [.apk]{} file as one Lev2 signature of that malware. Because the payload is a new application, DroidAnalytics will use the method we discussed to carry out a new signature generation. [**(e) Generate Lev1 signature (or application signature):**]{} The Lev1 signature is based on the level 2 signatures, e.g., signatures of all qualified classes within an application. In addition, the signature generator will ignore the class (except the main class) which will not be invoked by any other classes since these defunct classes may be generated via obfuscation. Malware writers may use some repackaging or obfuscation techniques to change the order of the classes table of the [.dex]{} file, our signature algorithm will first [*sort*]{} all Lev2 signatures, then concatenate these Lev2 signatures to generate the Lev1 signature. Figure \[Fig:SigAlg\] summarizes the framework of our signature algorithm. For example, the Lev3 signatures of [AAAE1]{} and [B23E8]{} are the two method signatures within the same class. Based on these two (sorted) signatures, we generate the Lev2 signature of the corresponding class, which is [53EB3]{}. Note that the Lev2 signature of [C3EB3]{} is generated from a [.dex]{} file which is a dynamic payload used to execute the malicious behavior. Based on all sorted Lev2 signatures of all classes, we generate the Lev1 signature, [F32DE]{}, of the application. ![Illustration of signature generation: the application (Lev1) signature, class level (Lev2) signatures and method level (Lev3) signatures.[]{data-label="Fig:SigAlg"}](new_signature){width="210pt"} For the current DroidAnalytics platform, we use a server which is of 2.80 GHz Duo CPU processor, 4GB memory and 2 TB hard disk, with two virtual machines in the server to implement the anti-virus engine. We carried out experiment to study the processing time to scanning and generating signatures. On average, it takes around 60 seconds to scan one application (includes the dynamic analysis), and around three seconds to generate all three level signatures, five seconds to generate AIS information, and one second to insert information into the database. As of November 2012, our system have downloaded 150,368 mobile applications from the following places: Google Play[@google:play], nine Android third party markets (e.g., [@russia; @appchina; @souapp]), two malware forums [@groups1; @groups2] and one mobile malware share blog[@blogspot]. The size of all downloaded application is 468GB. [**Utility & Effectiveness of Signature Based System**]{} {#sec: signature} ========================================================= Here, we illustrate how DroidAnalytics’ signatures can be used to analyze (and detect) malware repackaging, code obfuscation and malware with dynamic payloads. [**A. Analyzing Malware Repackaging**]{} Repackaging obfuscation technique is commonly used by malware writers to change the cryptographic hash value of an [.apk]{} file without modifying the opcodes of the [.dex]{} file. This technique is different from the repackaged technique which is to inject new packages into the legitimate applications. For example, using [Jarsigner]{} utility of Java SDK to re-sign an [.apk]{} file only changes the signature part of one [.apk]{} file, and generates a new [.apk]{} file which preserves the same logic and functionality as the original one. Another example is using the [Apktool]{}[@apkt12], which is a reverse engineering tool to disassemble and rebuild an [.apk]{} file without changing any assembly code. Although there is no modification on the assembly codes, the recompiler may change the classes order and methods order during the [*recompiling process*]{}. Therefore, repackaging obfuscation is often used to [*mutate*]{} an existing malware to generate a new version with a different signature. If an anti-virus system only identifies malware based on a cryptographic hash signature, then repackaging obfuscation techniques can easily evade the detection. DroidAnalytics can detect malware which is generated by repackaging obfuscation. Since there is no modification on the opcodes within the [.dex]{} file, and DroidAnalytics first sorts Lev2 and Lev3 signatures before generating the Lev1 signature. Therefore, DroidAnalytics will generate the [*same*]{} signature as the original even when one repackages the [.apk]{} file. [**Experiment.**]{} To illustrate the above claim, we carry out the following experiment and Figure \[Fig:lev1\_opfake\] illustrates the results. [Opfake]{} is a server-side polymorphism malware. The malware mutates automatically when it is downloaded. When analysts compare the cyclic redundancy codes (CRCs) of two [Opfake]{} downloads, it shows that the only meaningful change happens in the file [data.db]{} which is located in “[res/raw/]{}” folder. The modified [data.db]{} changes the signature data for the package in “[META-INF]{}” folder. By analyzing this form of malware, we find that all mutations in [Opfake]{} family happen in the same opcode (stored in [classes.dex]{}). Hence, our signature system will generate the [*same*]{} level 1 signature for all mutations in this malware family. Figure \[Fig:lev1\_kmin\] illustrates another example of using DroidAnalytics to analyze the [Kmin]{} family. We first calculate the Lev1 signatures of all 150,368 applications in our database. After the calculation, the result shows that the most frequent signature (which corresponds to the Lev1 signature [90b3d4af183e9f72f818c629b486fdec]{}) comes from 117 files and all these files have different MD5 values. This shows that conventional cryptographic hashing (i.e., MD5) cannot identify malware variants but DroidAnalytics can effectively identify them. Also, these 117 files are all variants of the [Kmin]{} family. After further analysis, we discover that the [Kmin]{} family is a wallpaper changer application, and all its variants have the same application structure and same malicious behavior. The only difference is that they have different icons and wallpaper files. [**B. Analyzing Malware which uses Code Obfuscation:**]{} A malware writer can use a disassembler (e.g., [Apktool]{}[@apkt12]) to convert a [.dex]{} file into [.smali]{} files, then injects new malware logic into the [.smali]{} code, rebuilds it back to a [.dex]{} file. Based on this rebuilt process, malware writers can apply various code obfuscation techniques while preserve the behavior as the original one in order to bypass the anti-virus detection. As shown in [@dimva12; @Andreas07], many mobile anti-virus products are not effective to detect code obfuscated variants. DroidAnalytics will not extract the API calls in methods and classes which will not be executed in run time (refer to Section \[design\]) because they are defunct and can be generated by obfuscators. In addition, the signature generation does not depend on the name of methods or classes, hence name obfuscation has no effect on our signature. Furthermore, the signature generation of DroidAnalytics is based on the analyst-defined API calls table. So one can flexibly update the table entries to defend against various code obfuscation techniques. [**Experiment.**]{} To illustrate the effectiveness of DroidAnalytics against code obfuscation, we chose 30 different malware from three different families (10 samples in each family) and Table \[table:obfs\] illustrates our results. The malware families in our study are: [Basebrid]{} (or [Basebridge]{}), [Gold-Dream]{}, and [Kungfu]{}. We use ADAM[@adam12], a system which can automatically transform an original malware sample to different variants by various repackaging and obfuscation techniques (e.g. inserting defunct methods, modifying methods name, ...etc). We generate seven different variants for each malware. Then we put these 240 new malware samples into the DroidAnalytics system and check their signatures. After our signature calculations, the result shows that for each malware, the original sample and seven mutated variants have [*distinct*]{} MD5 hash values (3 repackaging, 4 code obfuscation), but all of them have the [*same level 1 signature*]{}. This shows that DroidAnalytics’ signature system is effective in defending against code obfuscation. [**C. Analyzing Malware with Attachement Files or Dynamic Payloads:**]{} Some malware will dynamically download file which contains the malicious code from the Internet. Also, some attachment files within a package may contain malicious logic but they can be concealed as other valid documents (e.g., [.png]{} file, [.wma]{} file). DroidAnalytics will treat these files as dynamic payloads. By using both static and dynamic analysis techniques described in Section \[design\], DroidAnalytics accesses these payloads and generates different signatures. [**Experiment.**]{} We carried out the following experiment. From our malware database, we used our signature system and detected some malware contain the same file with a [.png]{} filename extension. But when we check the magic number of this file, it is actually an [.elf]{} file. Upon further analysis, we found that this file is a root exploit and this malware belongs to the [GinMaster]{} (or [GingerMaster]{}) family. Another example is the [Plankton]{} family. By using dynamic analysis, DroidAnalytics discovered that all malware in this family will download a [plankton\_v0.0.4.jar]{} (or similar [.jar]{}) when the main activity of the application starts. Further analysis revealed the [.jar]{} file contains malicious behavior, i.e., stealing browser’s history information, making screen shortcuts and botnet logic. Table \[table:dynamic\_payload\] depicts DroidAnalytics system detects some representative malware using dynamic payloads. [**Analytic Capability of DroidAnalytics**]{} {#sec: analytic_capability} ============================================= We conduct three experiments and show how analysts can study malware, carry out similarity measurement between applications, as well as perform class association among 150,368 mobile applications in the database. [**A. Detailed Analysis on Malware:**]{} Using DroidAnalytics, analysts can also discover which class or method uses suspicious API calls via the [*permission recursion*]{} technique. [**$\bullet$ Common Analytics on Malware.**]{} First, using the AIS parser, DroidAnalytics can reveal basic information of an application like the cryptographic hash (i.e., MD5 value), package name, broadcast receiver, $\ldots$, etc. This is illustrated in Figure \[Fig:app\_detail\]. In addition, DroidAnalytics has a built-in cloud-based APK scanner that supports diverse anti-virus scan results (e.g., Kaspersky and Antiy) to help analysts for reference. Our cloud-based APK scanner is [*extensible*]{} to accommodate other anti-virus scan engines. Last but not least, DroidAnalytics can disassemble the [.dex]{} file and extract class number, method number, and API calls number in each application, class or method. These functionalities are useful so analysts can quickly zoom in to the meta-information of a suspicious malware. Figure \[Fig:lev2\_signature\] shows these functionalities. [**$\bullet$ Permission Recursion.**]{} Current state-of-the-art systems examine the [AndroidManifest.xml]{} to discover permissions of an application. This is not informative enough since analysts do not know which class or which method uses these permissions for suspicious activities. In DroidAnalytics, we can discover the permission within a class or a method. Since each permission is related to some API calls [@felt:apd]. In DroidAnalytics, we tag permission to API calls in each method. We combine the method permissions within the same class as class permission, and combine all class permissions as application permission. This helps analysts to quickly discover suspicious methods or classes. [**$-$ Experiment.**]{} In this experiment, we choose a popular malware family [Kungfu]{} and examine the permissions at the application/class/method levels. Malware in the [Kungfu]{} family can obtain user’s information such as IMEI number, phone model,..., etc. It can also exploit the device and gain root privilege access. Once the malware obtained the root level access, it installs malicious application in the background as a back-door service. {width="360pt"} We use DroidAnalytics to generate all three-level signatures. Figure \[Fig:information\] shows the partial structure of the signatures with permission recursion of two [Kungfu]{} malware: [A4E39D]{} and [D2EF8A]{}, and together with a legitimate application [BEDIE3]{} (Lev1 signature). Firstly, based on the malware reports by our cloud APK scanner, we identify [A4E39D]{} and [D2EF8A]{} are malware which come from the [Kungfu]{} family with different package names, [net.atools.android.cmwrap]{} and [com.atools.netTrafficStats]{} respectively. By analyzing the Lev2 signature, we discover that [BCEED]{} is the common class which is within the two [Kungfu]{} applications. Secondly, from the Lev3 signature of [BCEED]{}, we use the permission recursion method to expose the method [6F100]{} with the [INSTALL\_PACKAGES]{} permission. Note that the [INSTALL\_PACKAGES]{} permission is a system permission which the application can install other unrelated applications. This shows how DroidAnalytics helps analysts to quickly discover the malicious code of methods and classes of the [Kungfu]{} family. [**$\bullet$ Similarity Measurement.**]{} DroidAnalytics can also calculate the similarity between two Android mobile applications, and the computation is based on the three level signatures that we discussed. By comparing the similarity of applications, we can determine whether an application is repackaged malware. Moreover, because we use Lev2 (class) signature as the basic building block, our approach not only can provide the similarity scores between two applications, but can inform analysts the common and different code segments between two applications. This great facilitates analysts to carry out detailed analysis. Our similarity score is based on the Lev2 signatures which represent all classes within an application. The similarity score is based on the Jaccard similarity coefficient. Given $f_{a}$ and $f_{b}$ as two Lev2 signature sets of two applications $a$ and $b$ respectively, $f_a \cap f_b$ refers to the same Lev2 signatures of $a$ and $b$ (or the [*common classes*]{} of these two applications), while $f_a \cup f_b$ represents the set of classes of these two applications. $S(x)$ is a function which returns the total number of API calls in the set $x$. Our similarity score equation between two applications is: $$J_{app}(f_{a}, f_{b})=\frac{S( f_{a}\cap f_{b})}{S(f_{a}\cup f_{b})}. $$ Let us how to use this similarity score to carry out analysis. First, given a legitimate application $X$, we can find all applications $\{Y_{1}, Y_{2},..., Y_{k}\}$ which are the top $p\%$ (say $p=20$) applications that are similar to $X$. Second, in the procedure of calculating similarity, we can identify the common and different Lev2 signatures. As a result, repackaged or mutated malware (i.e., $Y_i$), can be easily detected using the similarity score and the classes of malicious behavior can be easily determined. [**$-$ Experiment.**]{} We carry out the experiment using similarity measurement based on the Lev2 and Lev3 signatures. We use a benign application and calculate the similarity with other applications in our database to find all repackaged malware of this benign application. Furthermore, we can discover the differences between this benign application and repackaged malware (at the code level) to see what malicious codes are repackaged into legitimate application. We choose a legitimate application called [*“Touch alarm”*]{} which is downloaded from Google official market. We use DroidAnalytics to compute the similar scores with malware in the database, Table \[table:scores\] shows the similarity scores between [*“Touch alarm”*]{} and other applications in our database (in decreasing order of similarity). From the table, we can clearly observe that the package names of the top six applications are the same: [com.k99k.keel.touch.alert.freeze]{}. The first one in the table is the legitimate [*“Touch alarm”* ]{}application, while the following five applications are repackaged malware of [Adrd]{}. [Adrd]{} steals the personal information like IMEI, hardware information of users, and it will encrypt the stolen information and upload to some remote server. Moreover, it may dynamically download the newest version of [Adrd]{} and update itself. The table shows that the malware writer inserted malicious codes in benign [*“Touch alarm”*]{} and repackaged it to different variants of [Adrd]{}. [**MD5**]{} [**Package Name**]{} $\boldsymbol{S(f_{a}\cap f_{b})/S(f_{a}\cup f_{b})}$ [**Similarity Score**]{} [**Detection Result**]{} -------------------------------------- ---------------------------------- ------------------------------------------------------ -------------------------- -------------------------- [278859faa5906bedb81d9e204283153f]{} com.k99k.keel.touch.alert.freeze N/A 1 Not a Malware [effb70ccb47e8148b010675ad870c053]{} com.k99k.keel.touch.alert.freeze 674/878 0.76 Adrd.w [ef46ed2998ee540f96aaa1676993acca]{} com.k99k.keel.touch.alert.freeze 674/878 0.76 Adrd.w [cd6f6beff21d4fe5caa69fb9ff54b2c1]{} com.k99k.keel.touch.alert.freeze 674/878 0.76 Adrd.w [99f4111a1746940476e6eb4350d242f2]{} com.k99k.keel.touch.alert.freeze 674/884 0.77 Adrd.a [49bbfa29c9a109fff7fef1aa5405b47b]{} com.k99k.keel.touch.alert.freeze 674/884 0.77 Adrd.a [39ef06ad651c2acc290c05e4d1129d9b]{} org.nwhy.WhackAMole 332/674 0.49 Adrd.cw By using similarity measurement based on the three-level signature, DroidAnalytics reveals the difference between two applications at the code level. Table \[table:CompareSin\] shows the comparison of legitimate application [*“Touch alarm”*]{} and repackaged malware. Highlighted rows are the different level 2 signatures of the two applications, while the other rows represent same signature of common classes. This shows that DroidAnalytics can easily identify different classes of the two applications. By using permission recursion which we discussed previously, analysts can discover that the different level 2 signature [e48040acb2d761fedfa0e9786dd2f3c2]{} has READ\_PHONE\_STATE, READ\_CONTACTS and SEND\_SMS in repackaged malware. Using DroidAnalytics for further analysis, we find suspicious API calls like [android/telephony/TelephonyManager;-&gt;\ getSimSerialNumber]{}, [android/telephony/gsm/\ SmsManager;-&gt;endTextMessage]{} and [android/telephony/TelephonyManager;-&gt;\ getDeviceID]{}. Last but not least, we determine the malware writer inserted these malicious codes into legitimate application and published repackaged malware to various third party markets. [**$\bullet$ Class Association.**]{} Traditional analysis on malware only focuses on one malware but can not associate malware with other malware or applications. DroidAnalytics can associate legitimate applications and other malware in the class level and/or method level. Given a class signature (or Lev2 signature), DroidAnalytics keeps track of how many legitimate applications or malware using this particular class. Also, with the methodology of permission recursion, DroidAnalytics can indicate the permission usage of this class signature. By using class association, we can easily determine which class or method may possess malicious behavior, and which class is used for common task, say for pushing advertisement. Lastly, for class signatures which are used by many known malware but zero legitimate application, these are classes that analysts need to pay special attention to because it is very likely that they contain malicious code and are used in many repackaged or obfuscated malware. [**$-$ Experiment.**]{} Using DroidAnalytics, we carry out the class association experiment using 1,000 legitimate appliactions and 1,000 malware as reported by Kaspersky. Figure \[Fig:lev2\_details\] illustrates the results. After the class association, we discover a class (Lev2) signature [2bcb4f8940f00fb7f50731ee341003df]{} which is used by 143 malware and zero legitimate application. In addition, the 143 malware are all from the [Geinimi]{} family. Furthermore, this class has 47 API calls and uses [READ\_CONTACTS]{} and [SEND\_SMS]{} permissions. Therefore, we quickly identify this class contains malicious codes. In the signature database, we also find a class (Lev2) signature [9067f7292650ba0b5c725165111ef04e]{} which is used by 80 legitimate applications and 42 malware. Further analysis shows that this class is used by similar number of legitimate applications and malware, and this class uses an advertisement library called DOMOB[@domob]. Another class (Lev2) signature [a007d9e3754daef90ded300190903451]{} is used by 105 legitimate applications and 80 malware. Further examination shows that it is a class from the Google official library called AdMob[@admob]. This is illustrated in Figure \[Fig:lev2\_details\_ad\]. In summary, all experimented samples are presented in Table \[table:results1\], which represents the known malware in our database. The detection results is based on the cloud anti-virus engine using various detection engines (i.e., Kaspersky and Antiy(linux version)). Note that in the last column, $R$ ($G$) represents repackaged (generic) malware family. For those malware families with less than five samples, we lump them as “others” in the table. Antiy (linux version) [@antisc] is a commercial anti-virus product that we obtained from the company, and the product is known to run the same engine for detecting malware in smartphones. The rows are sorted in alphabetical order. Highlighted rows show the common malware families detected by both Kaspersky and Antiy. Others are uniquely detected by one anti-virus product. The penultimate row shows there are 1,295 common malware samples detected by these two anti-virus products. Hence, the number of unique malware samples is 2,148. [**Kaspersky**]{} [**Samples**]{} [**Antiy**]{} [**Samples**]{} ---------------------- ----------------- ------------------------- ----------------- Adrd 176 Adrd 57 AnSer 5 App2card 8 BaseBrid 611 Keji(BaseBrid) 299 CrWind 5 Crusewin(CrWind) 5 Deduction 19 DorDrae 10 emagsoftware 15 FakePlayer 6 FakePlayer 6 Fatakr 16 Fjcon 142 fjRece(Fjcon) 141 Gapev 5 gapp(Gapev) 4 Geinimi 139 Geinimi 128 GingerBreak 11 GinMaster 31 GingerMaster(GinMaster) 26 Glodream 13 GoldDream 7 Gonca 5 i22hk 5 Itfunz 10 jxtheme 13 Kmin 192 Kmin 179 KungFu 144 KungFu 78 Lightdd 7 Lotoor 113 Lotoor 15 MainService 121 MobileTx 15 tianxia(MobileTx) 14 Nyleaker 7 Opfake 8 Plangton 126 Plankton(Plangton) 1 Rooter 26 DroidDream(Rooter) 17 RootSystemTools 7 SeaWeth 11 seaweed(SeaWeth) 11 SendPay 9 go108(SendPay) 10 SerBG 23 Bgserv(SerBG) 31 Stiniter 50 Universalandroot 27 Latency 28 Visionaryplus 1 Wukong 10 Xsider 14 jSMSHider(Xsider) 8 YouBobmg 20 Yzhc 35 Z4root 47 Zft 5 Others (42 families) 71 Others (27 families) 44 Common 1295 Common 1295 All 2003 All 1440 : Detection Results of our Cloud-based Apk Scanner[]{data-label="table:results1"} [**Zero-day Malware Detection**]{} {#sec: zeroday} ================================== Here, we show a novel methodology in using DroidAnalytics to detect the zero-day repackaged malware. We analyze three zero-day malware families to illustrate the effectiveness of our system. Zero-day Malware ---------------- Zero-day malware is a new malware that current commercial anti-virus systems cannot detect. Anti-virus software usually relies on signatures to identify malware. However, signature can only be generated when samples are obtained. It is always a challenge for anti-virus companies to detect the zero-day malware, then update their malware detection engines as quickly as possible. In this paper, we define an application as a zero-day malware if it has malicious behavior and it cannot be detected by popular anti-virus software (e.g., Kaspersky, NOD32, Norton) using their latest signature database. As of November, 2012, we use DroidAnalytics and have successfully detected 342 zero-day repackaged malware in five different families: [AisRs]{}, [Aseiei]{}, [AIProvider]{}, [G3app]{}, [GSmstracker]{} and [YFontmaster]{} (please refer to Table \[table:zero-day\] for reference). In this paper, we use the name of the injected package (not the name of the repackaged applications) as the name of its malware family. Furthermore, all samples are scanned by Kaspersky, NOD32, Norton and Antiy using their latest database in November, 2012. We also uploaded these samples to the virustotal[@virustotal] for malware detection analysis. Note that none of the submitted samples was reported as a malware by these engines when we carried out our experiments. In [@Abhi:repackaged; @Gary:repackaged], authors reported that nearly 86.0% of all Android malware are actually repackaged versions of some legitimate applications. By camouflaging to some legitimate applications, repackaged malware can easily deceive users. Given the large percentage of repackaged malware, we explore the effectiveness of using DroidAnalytics to detect the [*zero-day repackaged malware*]{}. Zero-Day Malware Detection Methodology -------------------------------------- The process of detecting zero-day repackaged malware can be summarized by the following steps. [**Step 1:**]{} We first construct a white list for common and legitimate classes. For example, we add all legitimate level 2 signatures, such as those in utility libraries (e.g., Json library) or advertisement libraries (e.g., Google Admob library, Airpush library) to the white list. All level 2 signatures in the white list will not be used to calculate the similarity score between two applications. [**Step 2:**]{} We calculate the number of common API calls between two given applications in the database. This can be achieved by using the similarity score in Equation (\[equation: similarity score\]). $$S_{app}(f_{a}, f_{b})=S( f_{a}\cap f_{b}), \label{equation: similarity score}$$ where $f_{a}$ and $f_{b}$ as two level 2 signature sets of two applications $a$ and $b$ respectively, $S(x)$ is a function to indicate the total number of API calls in the set $x$. The above similarity score between two repackaged malwares focus on the [*common*]{} repackaged API calls that correspond to the malicious logic, and ignore the effect of other API calls in these two applications. [**Step 3:**]{} Assume we have $N$ applications in the database, then we start with $N$ clusters. The distance between two clusters is the similarity score we mentioned in Step 2. We select two applications which have the largest similarity score and combine them into one cluster. For this new cluster, we re-calculate the similarity score between this new cluster with other $N-2$ clusters. The new similarity score is computed by averaging all similarity scores between all applications in two different clusters. [**Step 4:**]{} Again, we combine two clusters which have the largest similarity score. We continue this step until the similarity score between any two clusters is less than a pre-defined threshold $T$ (say $T$ is 100). [**Step 5:**]{} After we finish the clustering process, we will use anti-virus engines to scan all of these $N$ applications. Each application may be classified as [*legitimate*]{} or [*malicious*]{}. [**Step 6:**]{} If a cluster has more than $n$ applications (say $n=10$) and a small fraction $f$ (say $f \leq 0.2$) is classified as malicious. This should be a [*suspicious cluster*]{} since it is very unlikely that more than $n$ applications are similar (in terms of class functionality) in the real-world, and most of them are classified as benign. Hence, the similarity comes when some of these applications are repackaged. We then extract their common classes (using our level 2 signature) and examine these classes. Once we find any malicious logic in these common classes, we discover a zero-day repackaged malware family. [**Experimental results in discovering zero-day repackaged malware:**]{} Let us present the results of using DroidAnalytics to discover three zero-day repackaged malware families. [**- AisRs family:**]{} We discover 87 samples of [AisRs]{} family in our database. All the malware are repackaged from legitimate applications (e.g., [*com.fengle.jumptiger*]{}, [*com.mine.videoplayer*]{}) and all of them have a common malicious package named “[com.ais.rs]{}”. This malware contains a number of botnet commands that can be remotely invoked. When the malware runs, it will first communicate with two remote servers (see Figure \[Fig:AisRs\_1\]). These two servers are camouflaged as software download websites(see Figure \[Fig:AisRs\_2\]). If any of these two servers is online, the malware will receive some commands like downloading other [.apk]{} files. They are not necessary malware, but they contain advertisement from the website, “http://push.aandroid.net”. It is interesting to note that the address is “[**a**]{}android.net”, not “android.net”. Also, one of the many botnet commands is to save the user’s application installation lists and system information to the [.SQLite]{} file, then upload this file to a remote server. ![Remote Servers of AisRs[]{data-label="Fig:AisRs_1"}](AisRs_1){width="45.00000%"} ![Camouflaged Software Download Websites[]{data-label="Fig:AisRs_2"}](AisRs_2){width="45.00000%"} [**- AIProvider family:**]{} We discover 51 samples of [AIProvider]{} family in our database. All the malware are repackaged from legitimate applications (e.g., [*jinho.eye\_check*]{}, [*com.otiasj.androradio*]{}) and all of them have a common malicious package named “[com.android.internal.provider]{}”. There are several interesting characteristics of this malware. Firstly, the malicious package name is disguised as a system package name. Since DroidAnalytics does not detect malware based on the package name, so our system can easily discover this repackaged malware. Secondly, this malware uses DES to encrypt all SMS information (e.g., telephone numbers, SMS content) and store them in the [DESUtils]{} class. Thirdly, the malware will start a service called [OperateService]{} in the background when it receives “[BOOT\_COMPLETED]{}” broadcast(see Figure \[Fig:AIProvider\_1\] and \[Fig:AIProvider\_2\]). This service will decrypt the SMS information in the [DESUtils]{} class and use this information to send SMS messages without any notification. ![Disassembled Repackaged Codes of AIProvider[]{data-label="Fig:AIProvider_1"}](AIProvider_1){width="45.00000%"} ![Malicious OperateService of AIProvider[]{data-label="Fig:AIProvider_2"}](AIProvider_2){width="30.00000%"} [**- G3app family:**]{} We discover 96 samples of [G3app]{} family in our database. All the malware are repackaged from legitimate applications (e.g., [*com.openg3.virtua lIncomingCall*]{}, [*com.cs.android.wc3*]{}) and all of them have a common malicious package named “[com.g3app]{}”. There are several malicious behaviors in the [G3app]{} malware family. Firstly, the malware will frequently pop up notification on the status bar and entice users to select it (see Figure \[Fig:g3app\_2\]). Secondly, the malware will inject trigger codes to every button of the legitimate application(see Figure \[Fig:g3app\_1\]). If the user presses any button in the repackaged application or the notification in the status bar, the malware will download other applications from the remote server. We believe the hackers want to use repackaged malware to publicize their applications and use these advertisements for financial gain. ![Trigger Notification of G3app[]{data-label="Fig:g3app_2"}](g3app_2){width="30.00000%"} ![Trigger Button of G3app[]{data-label="Fig:g3app_1"}](g3app_1){width="30.00000%"} [**Related Work**]{} {#sec: related} ==================== Before the rapid increase of Android malware in 2011, researchers focused on the permission and capability leaks of Android applications. E.g., David Barrera [*et al.*]{}[@barrera:mmap] propose a methodology to explore and analyze permission-based models in Android. Stowaway[@felt:apd] is a tool for detecting permission over privilege in Android, while ComDroid[@felt:com] is a tool which detects communication vulnerabilities. Woodpecker[@Michael:2012] analyzes each application on a smartphone to explore the reachability of a dangerous permission from a public, unguarded interface. William Enck [*et al.*]{} [@Enck:2009] propose a lightweight mobile phone application with a certification-based permission. In this paper, instead of malware detection, we focus on designing an analytic system that helps analysts to dissect, analyze and correlate Android malware with other Android applications in the database. We propose a novel signature system to identify malicious code segments and associate with other malware in the database. Our signature system is robust against obfuscation techniques that hackers may use. In August 2010, Kaspersky reported the first SMS Trojan, known as [FakePlayer]{} in Android systems[@Carlos:2011]. Since then, many malware and their variants have been discovered and mobile malware rapidly became a serious threat. Felt [*et al.*]{} study 18 Android malware in [@felt:smmw]. Enck [*et al.*]{} [@enck:saas] carry out a study with 1,100 android applications but no malware was found. Zhou Yang [*et al.*]{}[@jiang:oakland] study characterization and evolution of Android malware with 1,260 samples. However, they did not show how to systematically collect, analyze and correlate these samples. Yajin Zhou [*et al.*]{}[@zhou:drsa] were the first to present a systematic study for the detection of malicious applications on Android Markets. They successfully discover 211 malware. Their system, DroidRanger, needs malware samples to extract the footprint of each malware family before the known malware detection. For zero day malware, DroidRanger serves as a filtering system. After the filtering process, suspicious malware needs to be manually analyzed. DroidMOSS[@jiang:moss] is an Android system to detect repackaged applications using fuzzy hashing. As stated in [@jiang:moss], the system is not designed for general malware detection. Furthermore, the similarity score provided by DoridMoss is not helpful in malware analysis. In addition, obfuscation techniques can change the order of classes and methods execution, and this will introduce large deviation in the measure used in DroidMOSS. Michael Grace et al. develop RiskRanker[@risk] to analyze whether a particular application exhibits dangerous behavior. It uses class path as the malware family feature to detect more mutations. However, obfuscation can easily rearrange the opcode along an execution path. So using class path for malware feature is not effective under obfuscation attack. Our system overcomes these problems by using a novel signature algorithm to extract the malware feature at the opcode level so it captures the semantic meaning for signature generation. For PC based malware, a lot of research work focus on the signature based malware detection. For example, authors in [@Andreas07] discussed the limitation of using signature to detect malware. In [@okane:ohm], authors described the obfuscation techniques to hide malware. Authors in [@Christodorescu:2007] presented an automatic system to mine specifications of malicious behavior in malware families. Paolo Milani Comparetti [*et al.*]{}[@Comparetti:2010] proposed a solution to determine malicious functionalities of malware. However, mobile malware has different features as compared with PC based malware. It is difficult to transform a PC based malware detection solution for mobile devices. For example, [@dimva12] reported that many anti-virus products have poor performance in detecting Android malware mutations, although these products performed reasonably well for PC based malware. Repackaging is another characteristic of android malware. Authors in [@jiang:moss] showed there are many repackaged applications in Android third party markets and significant number of these applications is malware. Based on the above studies, it is clear that a more sophisticated methodology to detect and analyze Android malware is needed. [**Conclusion**]{} {#section:conclusion} =================== We present DroidAnalytics, an Android malware analytic system which can automatically collect malware, generate signatures for applications, identify malicious code segment (even at the opcode level), and at the same time, associate the malware under study with various malware and applications in the database. Our signature methodology provides significant advantages over traditional cryptographic hash like MD5-based signature. We show how to use DroidAnalytics to quickly retrieve, associate and reveal malicious logics. Using the permission recursion technique and class association, we show how to retrieve the permissions of methods, classes and application (rather than basic package information), and associate all applications in the opcode level. Using DroidAnalytics, one can easily discover repackaged applications via the similarity score. Last but not least, we have used DroidAnalytics to detect 2,494 malware samples from 102 families, with 342 zero-day malware samples from six different families. We have conducted extensive experiments to demonstrate the analytic and malware detection capabilities of DroidAnalytics.

ArXiv

--- abstract: 'It is a fundamental principle of quantum theory that an unknown state cannot be copied or, as a consequence, an unknown optical signal cannot be amplified deterministically and perfectly. Here we describe a protocol that provides nondeterministic quantum optical amplification in the coherent state basis with high gain, high fidelity and which does not use quantum resources. The scheme is based on two mature quantum optical technologies, coherent state comparison and photon subtraction. The method compares favourably with all previous nondeterministic amplifiers in terms of fidelity and success probability.' author: - 'Electra Eleftheriadou$^1$, Stephen M. Barnett$^2$ and John Jeffers$^{1*}$' title: Quantum Optical State Comparison Amplifier --- Signal amplification is a simple concept in classical physics. In electromagnetism there is no theoretical impediment to amplifying a time-varying electric field $E$ to form a perfectly-copied larger signal $gE$, where $g (>1)$ is the gain factor. There are, of course, practical limitations: the gain is typically saturated because only a finite amount of energy is available for the amplifier. Furthermore the utility of the device is limited in practice by the fact that the amplifier normally adds noise to the signal. A perfect quantum optical amplifier would increase the coherent amplitude of a state multiplicatively. For example, it would transform the coherent state $|\alpha\rangle$, the nearest quantum equivalent to a classical stable wave, as follows $$\begin{aligned} |\alpha \rangle \rightarrow |g\alpha \rangle.\end{aligned}$$ Quantum-level linear optical amplifiers have stringent limitations on their operation. It is impossible to amplify an unknown quantum optical signal without adding noise [@haus], the minimum value of which is a consequence of the uncertainty principle [@caves]. This extra required added noise swamps the quantum properties of a signal. Were it otherwise it would be possible to violate the no-cloning theorem [@wootters] and achieve superluminal communication [@herbert]. Ralph and Lund [@ralph1] suggested that this noise limit could be beaten by nondeterministic amplifiers: ones that work only in postselection. Such amplifiers transform the coherent state $|\alpha \rangle \rightarrow c |g\alpha \rangle$, where $c$ satisfies $|c| \leq \frac{1}{g}$. They proposed an amplifier based on the quantum scissors device [@scissors1; @scissors2] and this was later realised experimentally [@ralph2; @grangier]. The scheme has been extended to amplification of photonic polarization qubits using two such amplifiers [@swiss; @kocsis]. Unfortunately the amplifier uses single photons as a resource, its success probability is only a few percent and it works only in the $|0\rangle$ and $|1\rangle$ basis, so the condition on the output is $|g \alpha| \ll 1$. This restriction could be circumvented by operating several amplifiers in parallel [@ralph1], or using a two-photon version of the quantum scissors device [@jeffers], but the requirement of multiple coincident separately-heralded photons renders the effective probability of amplification tiny. Photon addition [@walker] and subtraction [@wenger] can also be used to form a nondeterministic amplifier with $g=\sqrt{2}$ [@addsubtract1; @addsubtract2], but with the same type of limitations as the scissors-based device. Another scheme relies on weak measurements and the cross-Kerr effect and so also has low success probability [@croke]. A protocol for state amplification without using quantum resources was suggested by Marek and Filip [@marekfilip]. Surprisingly, thermal noise is added to the signal. The state is then Bose-conditioned by photon subtraction, which performs the effective amplification. The amplification is not perfect, but it produces larger amplitude output states with phase variances smaller than those of the input [@usuga]. The scheme can be improved slightly if a standard optical amplifier is used to add the initial noise [@jj2; @laterguy]. The success probability in all cases is limited by the photon subtraction probability, of the order of a few percent. Here we describe a remarkably simple method for amplifying coherent states based on comparing the input state with a known coherent state [@andersson]. This type of comparison has already been used in an experimental realisation of a quantum digital signature scheme [@qds] and in various binary quantum receivers [@kennedy; @dolinar; @bondurant]. As an amplifier it has several advantages over earlier methods. We assume that Alice sends coherent states selected at random from a known set to Bob. Bob’s task is to amplify them, e.g. for later splitting and distribution of identical copies, or to determine the phase of the coherent state accurately. Higher amplitude coherent states have a smaller phase variance, so this amounts to the sharing of a reference frame - an important task in quantum communication [@reframe]. There are many other possibilities. Bob performs the amplification using the device shown in Fig. \[fig1\]. He mixes the unknown input state with another coherent state (the guess state) at a beam splitter and compares them. The beam splitter has transmission and reflection coefficients $t_1$ and $r_1$ which we take to be real and there is a phase change of $\pi$ on reflection from the lower arm. One output arm of the beam splitter falls upon a photodetector and the other output is postselected based on no photocounts being recorded. This system performs a state comparison between the reflected part of the guess state and the transmitted part of the input state. ![(Colour online) The state comparison amplifier. Bob attempts to null Alice’s input with his guess state at the first beam splitter. The second beam splitter and detector are used for photon subtraction. The output state is accepted if the first detector does not fire and the second one does.[]{data-label="fig1"}](fig1.png){width="0.9\columnwidth"} If Alice’s input state is $|\alpha \rangle$ and Bob chooses his guess state to be $|\beta \rangle=|t_1 \alpha/r_1 \rangle$ (correctly) the transmitted input state interferes destructively with the reflected guess state and the detector cannot fire. The output in the upper arm is then a coherent state of amplitude $\alpha/r_1$, which is larger than $\alpha$. This is the gain mechanism for the device. If Bob chooses his guess state incorrectly, coherent light leaks into the detector arm. This can also sometimes cause no counts at the detector. Thus the output state, *conditioned* on the detector not firing, is generally a mixed state, weighted by the probability that no counts are recorded. The probability is maximised when destructive interference occurs in the detector arm, for which the output coherent amplitude also reaches its maximum. Bob can improve the quality of his output at small cost to the success probability if he performs a photon subtraction on the output. Coherent states are eigenstates of the annihilation operator and so subtraction has no effect on them [@zavatta], but for a mixture of coherent states with different mean photon numbers the probabilities in the mixture are adjusted. When the detector fires, a subtraction occurs and it is more likely to have been due to a high-amplitude coherent state rather than a low amplitude one. Thus a subtraction is more likely when Bob has chosen his guess state well. If the subtraction is performed with a beam splitter of transmission coefficient $t_2$, then $g=t_2/r_1$ is the nominal gain of the composite system. For input and guess states $|\alpha \rangle$ and $| \beta \rangle$ the coherent amplitude in the nominal vacuum output is $t_1 \alpha - r_1 \beta$ and the other beam splitter output passes to the subtraction stage. The amplitude in the subtraction arm is therefore $-r_2(t_1 \beta+r_1 \alpha)$ and the output amplitude is $t_2(t_1 \beta+r_1 \alpha)$. We assume that the input and guess states are chosen from probability distributions over the coherent states $$\begin{aligned} \label{inputstates} \nonumber \hat{\rho}_{\mbox{in}} &=& \int d^2 \bar{\alpha} P(\bar{\alpha})| \bar{\alpha} \rangle \langle \bar{\alpha} |, \\ \hat{\rho}_{\mbox{g}} &=& \int d^2 \bar{\beta} Q(\bar{\beta})| \bar{\beta} \rangle \langle \bar{\beta} |,\end{aligned}$$ and calculate the output state and the fidelity based on these and the properties of the device. The fidelity is $$\begin{aligned} F=\int d^2\alpha P(\alpha) \langle g \alpha |\hat{\rho}_{\mbox{out}} | g \alpha \rangle,\end{aligned}$$ where $\hat{\rho}_{\mbox{out}}$ is the output state conditioned both on the input state distributions from eq.(\[inputstates\]) and on the successful operation of the device. This is the probability that the output state passes a measurement test comparing it to the amplified version of the input state and can be written $$\begin{aligned} \label{fidform} \nonumber F&=& P(T|S) = \frac{P(T,S)}{P(S)} \\ &=& \frac{\int d^2 \bar{\alpha} \int d^2 \bar{\beta} P(T|S,\bar{\alpha}, \bar{\beta}) P(S|\bar{\alpha}, \bar{\beta}) P(\bar{\alpha})Q(\bar{\beta})}{\int d^2 \bar{\alpha} \int d^2 \bar{\beta} P(S|\bar{\alpha}, \bar{\beta}) P(\bar{\alpha})Q(\bar{\beta})}\end{aligned}$$ where $P(T|S)$ is the probability that the output state will pass the fidelity test given that the device operates successfully. So far we have made no assumptions about the forms of the input and guess probability distributions, but it is instructive to consider two cases in order to comply with the requirements of a realistic communication system. The first scenario is one in which the set of states to be amplified is restricted to the binary alphabet $\{ |\alpha \rangle, |-\alpha \rangle \}$ and in the second each state in the set has the same mean photon number, but with completely uncertain phase (Fig. \[fig2\]). Here the mean photon number could either be agreed in advance, or determined by simply measuring the first few states. Without loss of generality we will assume from here on that $\alpha$ is real and positive. ![(Colour online) The sets of input states considered, binary (red, hatched), phase-covariant (blue shaded).[]{data-label="fig2"}](fig2.png){width="0.9\columnwidth"} Suppose that Alice chooses randomly from the binary set of states $$\begin{aligned} P( \bar{\alpha}) = \frac{1}{2}\left[ \delta^2 (\bar{\alpha} - \alpha) + \delta^2 (\bar{\alpha} + \alpha) \right].\end{aligned}$$ The best choice for Bob’s guess state is to choose randomly from the set $\{ |\pm t_1 \alpha/r_1 \rangle\}$. Suppose, without loss of generality, that he chooses + so $Q(\bar{\beta}) = \delta^2 (\bar{\beta} - t_1 \alpha / r_1)$. The probabilities which form the fidelity and success probability in eq. (\[fidform\]) are straightforwardly calculated [@kelleykleiner] and we leave details to the supplemental material. ![(Colour online) Fidelity as a function of intensity gain for the binary system. The red (bottom two) , green and blue (top two) plots correspond to $\alpha^2= 0.1, 0.5, 1$. Full (dashed) curves are for detector quantum efficiencies of $\eta =1 (0.5)$. The intensity reflection coefficient of the subtraction beam splitter is 0.1. The inset shows an expanded view of the low gain region. The fidelity for $\alpha^2 = 1, \eta = 0.5$ is very close to that for $\alpha^2 = 0.5, \eta = 1$ for gains greater than about 3.[]{data-label="fig3"}](newfig3.pdf){width="0.9\columnwidth"} Fig. \[fig3\] shows the fidelity as a function of gain. The fidelity drops as the gain is increased initially, rises to unity at intermediate gain and then decays. The increase at intermediate gain is a combined effect of the photon subtraction and comparison. If the first beam splitter is 50/50, then its conditioned output that falls upon the subtraction beam splitter has a coherent amplitude of either $\sqrt{2} \alpha$ (if Bob has chosen his state correctly) or zero (incorrectly). If it is zero then no photon subtraction can take place. Therefore, if there is a photon subtraction Bob must have chosen correctly and a perfect amplified copy of the state is made. At high gain for perfect detector efficiency the fidelity decays to $1/(1+\exp{[-4\alpha^2]})$, which approaches unity for large $\alpha$. Fidelity is not the only possible measure of output quality and in the supplemental material we show that the amplifier has a noise figure which increases with gain. ![(Colour online) Success probability as a function of intensity gain for the binary system. Parameters as for Fig. \[fig3\].[]{data-label="fig4"}](Fig4.pdf){width="0.9\columnwidth"} The success probability (Fig. \[fig4\]) is dominated at low gain by the photon subtraction probability, but for higher gains the subtraction probability approaches one because the coherent amplitude is large. It is important to note that the performance of our device is relatively insensitive to experimental detector imperfections. The fidelity is reasonably robust to nonunit quantum efficiency, as shown in Fig. \[fig3\]. At high gain a nonunit quantum efficiency reduces the fidelity to that which would be obtained by operating the device with reduced coherent amplitude input $\alpha \sqrt{\eta}$. The effects on success probability shown in Fig. \[fig4\] depend on two competing factors. There is an increased probability of obtaining no counts at the comparison detector, but a decreased probability of photon subtraction. In a realistic experimental scenario the ideal is to keep dark count rates low enough that they do not affect the results. For example, in the state comparison experiment of reference [@qds] the photon flux at the detectors is of the order of $10^7s^{-1}$ and the dark count rates for the SPAD detectors are 320$s^{-1}$, rendering the effects of dark counts insignificant. If Bob and Alice do not initially share a phase reference and all that Bob knows is the mean photon number of Alice’s input then $$\begin{aligned} P( \bar{\alpha}) = \frac{1}{2 \pi \alpha}\delta (|\bar{\alpha}| - \alpha),\end{aligned}$$ with $Q(\bar{\beta})$ as before. The relevant probability calculations are again left for the supplemental material, together with plots of the fidelities and success probabilities. It is instructive to compare the fidelity with that obtained using other amplification methods. In Fig. \[fig5\] we show the fidelity obtained using the state-comparison amplifier, using the quantum scissors [@ralph1; @ralph2; @grangier] and using the noise addition amplifier [@marekfilip; @usuga] for $\alpha^2=0.5$. The advantages of the state-comparison amplifier are obvious. We can see that for the binary alphabet the state comparison amplifier significantly outperforms the other systems. The effect of the photon subtraction ensures perfect amplification for twofold gain and no other amplifier can reach this. ![(Colour online) Fidelity as a function of intensity gain for the state comparison amplifier. Curves are from top to bottom: state comparison binary system (blue upper), state comparison phase-covariant (blue lower), noise addition (red) and scissors-based (green). Detector efficiencies and photon subtractions are assumed perfect in all cases.[]{data-label="fig5"}](Fig5main.pdf){width="0.9\columnwidth"} For the phase covariant state set, again the state comparison amplifier outperforms the other systems, although for low gain its edge over the noise addition amplifier is minimal in terms of fidelity. However, it does have another advantage over this system. When it works the state comparison amplifier provides knowledge of both the state to be amplified and the amplified state, knowledge which is not available in the noise addition amplifier. The lower fidelity associated with the scissors-based amplifier is largely due to the fact that it can only produce a superposition of zero and one photon, which is not useful for amplifying a state of mean photon number 0.5. For much lower mean input photon numbers the scissors-based amplifier has a higher fidelity than other methods. It is clear that to be useful in future quantum communication systems postselecting amplifiers must approach the ultimate limits of performance [@pandey]. Here we have described a nondeterministic amplifier that outperforms other schemes over a wide range of input amplitudes and gains. It does not require quantum resources and operates with high fidelity and high success probability. It uses two already demonstrated experimental techniques and is relatively straightforward to implement. The gain can be chosen via the reflectivity of a beam splitter. The main reason why this amplifier works well is that it uses the available information about the input states effectively. Amplification is performed by dumping energy into the system - an optical mode. This works best if we do it in the appropriate basis, so if we want to amplify coherent states we should place the energy in the coherent state basis. The amplifier is robust to realistic values of detector imperfections. It still works well for nonunit detector efficiencies and dark count rates should be low enough to render them unimportant. Losses within the other components, such as at beam splitters or in connecting fibres, will be small and will reduce fidelity and/or gain by a commensurate amount. The device works best in a limited state space, a feature common to all nondeterministic amplifiers, although the limits are different for each one. Both the scissors-based and photon addition/subtraction amplifiers have an output which cannot contain more than one photon and for the noise addition amplifier the gain is tailored to the input amplitude to maximize phase concentration. Our amplifier turns the state space limitation to an advantage, in that the amplifier can be tailored to work in the basis used in a particular communication system. However the state space that the device works on can always be widened at a cost to the success probability. Any cost to the fidelity depends on the gain chosen and the states added. We also remark that the amplifier described here provides gain that is dependent on the input state and this is the case for all postselecting amplifiers so far. It renders them effectively nonlinear, but it does not bring into question their status as amplifiers [@pandey]. 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A [**88**]{}, 033852 (2013); arxiv:1304.3901. Supplemental Material {#supplemental-material .unnumbered} ===================== Calculation of Fidelities and Success Probabilities for the Binary System {#calculation-of-fidelities-and-success-probabilities-for-the-binary-system .unnumbered} ------------------------------------------------------------------------- The fidelities and success probability are given by Eq. (4) in the main paper and its denominator. They depend upon the probability that the device operates successfully given that pure coherent states $|\alpha \rangle$ and $| \beta \rangle$ are input by Alice and Bob. This is the probability that the first detector does not fire and the second one does, and is provided by the the Kelley-Kleiner formula [@kelleykleiner] $$\begin{aligned} P(S|\alpha, \beta) &=& \mbox{Tr} \left\{ \hat{\rho} : \exp{(-\eta_1\hat{a}_1^\dagger\hat{a}_1)} \left[ 1- \exp{(-\eta_2\hat{a}_2^\dagger\hat{a}_2)} \right] : \right\},\end{aligned}$$ where $\hat{\rho}$ is the three mode output of the device, $\hat{a}_1$ is the annihilation operator for detector mode 1, and $\eta_1$ is the quantum efficiency of detector 1. As the input states are pure coherent states the outputs will also be pure coherent states, and $$\begin{aligned} \label{psab} P(S|\alpha, \beta) &=& \exp{(-\eta_1|t_1 \alpha - r_1 \beta |^2}) \left[ 1- \exp{(-\eta_2 r_2^2 |t_1 \beta + r_1 \alpha|^2) } \right].\end{aligned}$$ The output state given that the input amplitudes were $\alpha$ and $\beta$ is simply $|t_2(t_1\beta+r_1\alpha) \rangle$, and the overlap of this with the state $ |g \alpha \rangle = |t_2 \alpha/ r_1 \rangle$ is the probability that the output state passes the measurement test that it is an amplified version of the coherent input. Finally we can use these probability distributions in, together with the input probability distributions for the binary system to obtain the success probability and the fidelity $$\begin{aligned} \nonumber P(S) &=& \frac{1}{2} \left( 1- \exp{ \left[ -\eta_2 g^2 \alpha^2 \left(1/t_2^2 -1 \right) \right] } + \exp{\left[-4 \eta_1 \alpha^2 \left( 1-t_2^2/g^2 \right)\right]} \left\{ 1- \exp{\left[ -\eta_2 g^2 \alpha^2 \left(1/t_2^2 -1 \right) \left( 1- 2t_2^2/g^2 \right)^2\right]} \right\} \right) \\ & &\\ \nonumber P(T,S)&=& \frac{1}{2} \left( \rule{0mm}{4mm} 1- \exp{ \left[ -\eta_2 g^2 \alpha^2 \left(1/t_2^2 -1 \right) \right] } + \exp{\left[-4 \eta_1 \alpha^2 \left( 1-t_2^2/g^2 \right)\right]} \right. \\ &\times& \left. \left\{ 1- \exp{\left[ -\eta_2 g^2 \alpha^2 \left(1/t_2^2 -1 \right) \left( 1- 2t_2^2/g^2 \right)^2\right]} \right\} \exp{ \left[-4g^2 \alpha^2 \left( 1-t_2^2/g^2 \right)^2 \right] }\right).\end{aligned}$$ Noise Figure ------------ One of the more standard measures of amplifier quality is the signal to noise ratio (SNR) of the output, or the noise figure, which is the ratio of the output SNR to that of the input. These are quantities typically used in optical physics to quantify the quality of a device. We examine here a simple signal to noise ratio for the $x_1 = \left( \hat{a} + \hat{a}^\dagger \right)/\sqrt{2}$ quadrature, $$\begin{aligned} \mbox{SNR} =\frac { \langle x_1 \rangle} {\sqrt{(\Delta x_1)^2}}.\end{aligned}$$ This quantity is appropriate for a nominally real coherent input $\alpha$, for which it takes the value $2|\alpha|$. We examine the binary alphabet case for which a relatively straightforward set of calculations find $$\begin{aligned} \langle x_1 \rangle &=& \sqrt{2} g\alpha\left[ P(\alpha|S) + \left( 1-2t_2^2/g^2 \right) P(-\alpha|S)\right],\\ \nonumber \langle x_1^2 \rangle &=& \frac{1}{2} \left\{\rule{0mm}{4mm} 1 +4g^2\alpha^2\left[ \rule{0mm}{4mm}P(\alpha|S) + \left( 1-2t_2^2/g^2 \right)^2 P(-\alpha|S)\right] \right\}, \end{aligned}$$ where the probabilities $P(\alpha|S)$ and $P(-\alpha|S)$ are the conditional probabilities that the input coherent amplitudes were $\alpha$ and $-\alpha$ respectively, given that the device operated successfully. We plot the noise figure based on these formulae in Fig. \[nf\]. This clearly shows improvement for all gains larger than about 1.3. The noise figure is relatively insensitive to input coherent amplitude, and (not shown) to detector quantum efficiency. ![(Colour online) Noise figure (NF) as a function of intensity gain for the binary system. Curves are from bottom to top (red, green dashed, blue dotted) $\alpha^2=0.1, 0.5, 1$. The intensity reflection coefficient of the subtraction beam splitter is 0.1.[]{data-label="nf"}](noisefigure.pdf){width="0.65\columnwidth"} Phase-covariant amplifier -------------------------- The phase covariant amplifier calculations, for which Alice’s input state probability distribution is $P( \bar{\alpha}) = \frac{1}{2 \pi \alpha}\delta (|\bar{\alpha}| - \alpha)$, require a phase integral to be performed, $$\begin{aligned} \nonumber P(S) &=& \int d^2\bar{\alpha}\int d^2\bar{\beta} P(\bar{\alpha}) Q(\bar{\beta}) P(S|\bar{\alpha}, \bar{\beta}) \\ \nonumber &=& \frac{1}{2 \pi} \int d\theta \exp{[-2 \eta_1 \alpha^2 t_1^2 (1-\cos \theta)]} \left( 1-\exp{\{-\eta_2 \alpha^2 r_2^2 [1-2r_1^2(1-r_1^2)(1-\cos \theta )]/r_1^2\} } \right) \\ \nonumber &=& \exp{[-2 \eta_1 \alpha^2 (1-t_2^2/g^2)]} I_0[2 \eta_1 \alpha^2 (1-t_2^2/g^2)]\\ \nonumber &-& \exp{[-2 \eta_1 \alpha^2 (1-t_2^2/g^2) -\eta_2 \alpha^2 g^2 (1/t_2^2 -1) +2\eta_2 \alpha^2 (1-t_2^2) (1-t_2^2/g^2)]} \\ &\times& I_0[2 \eta_1 \alpha^2 (1-t_2^2/g^2) -2\eta_2 \alpha^2 (1-t_2^2) (1-t_2^2/g^2)], \end{aligned}$$ where $I_0$ is the modified Bessel Function of zero order. The numerator in the fidelity is the same function as above, but with the substitution $$\begin{aligned} \eta_1 \rightarrow \eta_1+ g^2-t_2^2.\end{aligned}$$ We plot the fidelity and success probability in Figs. ( \[fig5\]) and (\[fig6\]). ![(Colour online) Fidelity as a function of intensity gain for the phase-covariant system. Curves are from top to bottom (red, green, blue) $\alpha^2=0.1, 0.5, 1$. Full (dashed) curves are for $\eta=1 (0.5)$. The intensity reflection coefficient of the subtraction beam splitter is 0.05.[]{data-label="fig5"}](Fig5.pdf){width="0.65\columnwidth"} The fidelity decays with gain and the success probability increases with gain, as might be expected. The effect of photon subtraction is not as strong as in the binary case, but is still present. The other main difference is that the fidelity is highest for low mean photon number inputs, largely because even if the wrong state from the ring is amplified, it still has significant overlap with the nominal amplified state. This is not the case for higher amplitude input states, for which the amplified version of any state not sufficiently close to Bob’s guess state is effectively orthogonal to the nominal amplified output. ![(Colour online) Success probability as a function of intensity gain for the phase-covariant system system. Curves are from top to bottom (blue, green, red) $\alpha^2=1, 0.5, 0.1$. Full (dashed) curves are for $\eta=1 (0.5)$. The intensity reflection coefficient of the subtraction beam splitter is 0.05.[]{data-label="fig6"}](Fig6.pdf){width="0.65\columnwidth"} The success probability (Fig. \[fig6\]) is of similar magnitude to the equivalent for the binary system. It rises with gain, as increasing the gain amounts to adding more photons into the guess state beam splitter input port, and increasing the transmission of this beam splitter.

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