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+Local Volatility in Interest Rate Models.
+V.M. Belyaev
+U.S. Bank, Minneapolis, MN, USA
+February 1, 2023
+Abstract
+A new approach to Local Volatility implementation in the interest
+rate model is presented. The major tool of this approach is a small
+volatility approximation. This approximation works very well and it
+can be used to calibrate all ATM swaptions. It works fast and accu-
+rate. In order to reproduce all available swaption prices we need to
+take into account the dependence of forward volatility on the current
+swap rate. Here we assume that forward volatility is a deterministic
+function on strike, tenor, and expiration at every point on the grid.
+We determine these functions and apply them in Monte-Carlo calcu-
+lations.
+It was demonstrated that this approach works well. However, in
+the case of short term and low tenor swaptions we observed errors in
+swaption pricing. To fix this problem we need to modify the scenario
+generation process.
+1
+Introduction
+Local Volatility Model was presented by Dupire [1] in 1994. According to this
+model forward volatility is a deterministic function of a current underlying
+S(t) price and time. Model dynamics in this case has the following form:
+dS(t) = µ(t)S(t)dt + σL(S(t), t)S(t)dW(t);
+(1)
+where µ(T) = r(t)−y(t) is a arbitrage free drift; r(t) and y(t) are risk free rate
+and dividend yield; dW(t) is Brownian motion; σL(S(t), t) is a deterministic
+Local Volatility.
+1
+arXiv:2301.13595v1  [q-fin.PR]  31 Jan 2023
+
+Representing the option price as a function of forward one C(F(T), X, T)
+where F(T) = S(0)e
+� T
+0 (r(t)−y(t))dt we have
+σ2
+L(X, t) =
+∂C
+∂T
+1
+2X2 ∂2C
+∂X2
+.
+(2)
+where C(F(T), X, T) is a call option price; X is a strike.
+Having arbitrage-free interpolated/extrapolated volatility surface we can
+calculate local volatility by eq.(2) and generate scenarios which are perfectly
+calibrated to all available option prices. This procedure is deterministic and
+fast.
+Later Gatheral [2] derived the following formula for local volatility which
+is expressed in terms of implied volatility itself:
+σ2
+L(X, t) =
+∂w
+∂T
+1 − y
+w
+∂w
+∂y + 1
+4
+�
+− 1
+4 − 1
+w + y2
+w2
+� �
+∂w
+∂y
+�2 + 1
+2
+∂2w
+∂y2
+;
+(3)
+where w(y, T) is an implied variance of the option with strike X and time
+to expiration T; y = ln(X/F(T)). The formula makes it easier to calculate
+Local Volatility.
+In the case of normal volatilities models where
+dS(t) = µ(t)dt + σ(S(t), t)dW(t);
+(4)
+µ(T) = ∂F(T)
+∂T , formula for Local Volatility is also known [3]:
+σ2
+L(X, t) =
+dw
+dT
+1 − y
+w
+∂w
+∂y + 1
+4
+�
+− 1
+w + y2
+w2
+� �
+∂w
+∂y
+�2 + 1
+2
+∂2w
+∂y2
+.
+(5)
+Here we use the following notation: y = X − F(T).
+To calibrate interest rate model we will use small volatility approximation
+[4]. This approximation works very well and it means that bond price dy-
+namic approximately is a normal one. So, Local Volatility can be calculated
+according to eq.(5).
+In Section 2 we describe a Small Volatility approximation and demon-
+strate its accuracy. This approximation can be used to calibrate the model
+for selected OTM strikes as well. In Section 3 we apply this approximation in
+order to to calculate deterministic sensitivities to strikes at every point of the
+2
+
+grid. In Section 4 fixed tenor dynamics and forward volatility calculation are
+discussed. In Section 5 we use these sensitivities to calculate current forward
+volatilities according to eq.(5) to generate interest rate scenarios.
+All calculations are completed for March 29, 2022 SOFR rate and swap-
+tions market data. Even though the fact that the approach works well we
+observe relatively big errors in case of short-term and low-tenor swaptions.
+2
+Small Volatility Approximation
+Here we consider HJM interest rate model. HJM model [5] has the following
+dynamics:
+df(t, T) = α(t, T)dt + σ(t, T)dW(t);
+(6)
+where f(t, T) is a forward rate:
+B(t, T) = e−� T
+t f(t,τ)dτ;
+(7)
+B(t, T) is a zero coupon risk-free bond; σ(t, T) is a normal volatility; dW(t, T)
+is a Brownian motion; and
+α(t, T) = σ(t, T)
+� T
+t
+σ(t, τ)dτ
+(8)
+is a drift.
+The drift is chosen to satisfy martingale condition on bond prices
+B(0, T) =
+�
+e−� t
+0 r(τ)dτB(t, T)
+�
+; ∀t ∈ [0, T].
+(9)
+Distribution of discounted bond prices at time T can be presented in the
+following form:
+e−� T
+0 r(τ)dτB(T, T1) =
+= B(0, T1)e−� T
+0 dτ � T1
+τ
+α(τ,t)dt−� T
+0 dW(τ)� T1
+τ
+σ(τ,t)dt =
+= B(0, T1)
+�
+1 −
+� T
+0 dW(τ)
+� T1
+τ
+σ(τ, t)dt + o(σ)
+�
+.
+(10)
+3
+
+It means that the distribution of SOFR swap present value is:
+PV (T) = e−� T
+0 r(t)dt
+N
+�
+n=1
+B(T, Tn)
+�
+rs + 1 − B(T, Tn−1)
+B(T, Tn)
+�
+=
+= e−� T
+0 r(t)dt
+�
+rs
+N
+�
+n=1
+B(T, Tn) − B(T, T) + B(T, TN)
+�
+≃
+≃ (rs − rATM)
+N
+�
+n=1
+B(0, Tn) + Σ(T, N)ξ
+√
+T;
+(11)
+where Tn are times of payments; rs and rATM = B(0,T)−B(0,TN)
+�
+n=1NB(0,Tn) are swap rate
+and ATM rate; T0 = T.
+Volatility Σ(T, N) is calculated according to the following formulas:
+Σ2(T, N)T =
+� T
+0 v2(t, Ndt)dt;
+v(t, N) = rs
+N
+�
+n=1
+B(0, Tn)
+� Tn
+t
+σ(t, τ)dτ −
+−B(0.T)
+� T
+t
+σ(t, τ)dτ + B(0, TN)
+� TN
+t
+σ(t, τ)dτ.
+(12)
+In order to calibrate the interest rate model we can use interpolated
+volatilities or to assume that all unknown volatilities are equal to each other.
+Here we use the first approach.
+In the case of the grid with 3 month time step the first swaption is a tenor
+1 expired in 3 months. According to (12) we have:
+v(dt, 1) = rsB(0, 5dt)
+4
+�
+k=0
+(k + 1)σ(0, k)dt −
+−B(0, dt)σ(0, 0)dt +
++B(0, 5dt)
+4
+�
+k=0
+(k + 1)σ(0, k)dt;
+(13)
+where dt = 0.25.
+Assuming that all unknown volatilities are equal in (13)
+σ(0, k) = σ(0, 0); ∀k < 5;
+(14)
+we can calculate volatilities in (13).
+4
+
+We can apply this procedure to other expirations and tenors, taking into
+account already defined volatility values. For every available next tenor and
+time to expiration we obtain the following equation:
+Σ2(i, j) = Aσ2 + Bσ + C;
+(15)
+where A, B, C are factors that can be determined by using bond prices and
+already calculated volatilities; σ is an unknown forward volatility.
+Schematically calibration process can be represented in the following way:
+0
+0.25
+0.5
+0.75
+1.0
+0
+0.25
+0.5
+0.75
+-
+Te + Tenor
+6
+Te
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+��
+f
+f
+f
+f
+f
+f
+f
+f
+f
+f
+f
+v
+f
+f
+f
+f
+f
+f
+v
+f
+f
+f
+Σ(1, 5)
+Σ(2, 6)
+σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 5) σ(0, 5)
+σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1)
+where Σ(1, 5) and Σ(2, 6) are input volatilities of 1-year swaptions with
+3- and 6-months to expiration.
+This procedure leads to a good calibrated prices for all ATM swaptions,
+see Fig.1 where we compare input ATM volatilities with volatilities calculated
+from Monte-Carlo prices.
+Forward volatility surface is shown on Fig.2.
+3
+Sensitivity Calculation.
+In the previous section we describe how to calibrate forward volatilities to
+reproduce all available ATM swaptions prices. This procedure can be applied
+to find forward volatilities calibrated to all swaption prices with shifted strikes
+from their ATM values.
+5
+
+Sswaption quotes are available within ±2% shifts. The calibration process
+is similar to the ATM swaption calibration. The only difference is a change of
+ATM volatility to the shifted one for selected strike. The calibration quality
+is good (see Fig.3).
+In the selected data set for SOFR swaption volatility smile, we have
+strikes up to ±2% rate shift from ATM rate. This smile can be fitted by
+quadratic polynomial as it is demonstrated in Fig.4. Splines can also be used
+but obtained results are very similar. It means that we can use quadratic
+interpolation.
+However, we need to take care of extrapolation, due to the swap rates can
+go out of [−2%, 2%] range rate shifts. This extrapolation need to be contin-
+uously differentiable and it would be very helpful if we can use extrapolation
+with limited volatility, in order to to use small volatility approximation.
+We choose the following quadratic extrapolation:
+v(x) = αu + βu(x − x0) + γu(x − x0)2;
+x0 < x < xu;
+(16)
+v(x) = αd + βd(x − (−x0)) + γd(x − (−x0))2;
+xd < x < −x0;
+where x0 = 2% is the current maximal available rate shift in input data; final
+points of extrapolation are chosen as xd = −10% and xu = 10%.
+To obtain continuously differentiable function, we need to have
+αu = v(x0);
+αd = v(−x0);
+βu = dv(x)
+dx
+|x=x0
+;
+βd = dv(x)
+dx
+|x=−x0
+.
+(17)
+To get zero slope at the end points xu,d we have
+γu(xu − x0) = −1
+2βu;
+γd(xd − (−x0)) = −1
+2βd;
+(18)
+Above end points we assume that
+v(x) =
+�
+v(xd);
+if x < xd;
+v(xu);
+if x > xu;
+(19)
+It leads to the smooth volatility curve depicted in Fig.5.
+6
+
+4
+Fixed Tenor Dynamics
+Before going to Scenario Generation Process let us consider fixed tenor dy-
+namics.
+In the case of selected tenor N we consider the following ATM
+swaption values distribution which present value is:
+PV (S(T, N)) = e−� T
+0 r(τ)dτ
+�
+rATM(T, N)
+N
+�
+n=1
+B(T, Tn) − 1 + B(T, TN)
+�
+=
+= e−� T
+0 r(τ)dτ
+�B(0.T) − B(0, TN)
+�N
+n=1 B(0, Tn)
+N
+�
+n=1
+B(T, Tn) − 1 + B(T, TN)
+�
+.(20)
+In small volatility approximation Eq.(20) has the following form:
+PV (S(T, N)) = Σ(T, N)ξ
+√
+T.
+(21)
+In the case of non-zero constant difference between current swaption rate and
+ATM rate
+δ = r(T, N) − rATM(T, N);
+(22)
+swap present value distribution is
+PV (S(T, N, δ)) = δ
+N
+�
+n=1
+B(0, Tn) + Σ(T, N, δ)ξ
+√
+T =
+= A(T, N)
+�
+δ + Σ(T, N, δ)
+A(T, N) ξ
+�
+;
+(23)
+where Σ(T, N, δ) is implied volatility of fixed tenor underlying for selected
+rate shift;
+A(T, N) =
+N
+�
+n=1
+B(0, tn).
+(24)
+As we can see from (23) we can use (5) to determine forward volatility
+assuming deterministic dependence of local forward volatility for selected
+point on the grid.
+7
+
+5
+Local Volatility Scenarios
+In the scenario generation process we can use formula for Local Volatility
+of normal volatility model. Here we assume that local forward volatility on
+every point on the grid can be defined deterministically from eq.(5). Below
+we describe this approach.
+In addition to forward volatility, the formula depends on total variance
+w.
+Deterministic variance can be calculated in the following form:
+w(−2%, n, k) = v2
+f(−2%, n − 1, k)dt + w(−2%, n − 1, k);
+w(ATM, n, k) = v2
+f(ATM, n − 1, k)dt + w(ATM, n − 1, k);
+w(2%, n, k) = v2
+f(2%, n − 1, k)dt + w(2%, n − 1, k);
+(25)
+where vf(shift, n, k) is a forward volatility for selected time step n and point
+on the grid k.
+Then we use interpolation-extrapolation formulas to get function w(x, n, k)
+for every point on the grid.
+This procedure gives us all deterministic functions needed to calculate
+forward volatility according to equation (5).
+To generate scenarios we need to calculate current swap rate which is a
+difference between current and initial ATM swap rates.
+Initial ATM swap rate is:
+rS(0, ti, tenor) = B(0, ti) − B(0, ti + tenor)
+�tenor
+n=1 B(0, ti + n)
+.
+(26)
+Observed swap rate at time ti is:
+rS(ti, ti, tenor) = 1 − B(ti, ti + tenor)
+�tenor
+n=1 B(ti + n)
+.
+(27)
+So, the current rate strike is
+X(ti, ti, tenor) = rS(ti, ti, tenor) − rS(0, ti, tenor).
+(28)
+In the case of tenor = 1 we assume that this strike is the same for all
+time steps between 0 and tenor = 1. For tenor = 2 we use strike for all
+times between tenor = 1 and tenor = 2 only etc.. As was mentioned in the
+8
+
+previous section this procedure works. We use it because we already have
+calculated volatility sensitivities on swap rates.
+In calculations we apply
+calculated variance for selected strike and point on the grid (25).
+We apply this procedure and generated 100,000 scenarios. ATM swaption
+prices for tenors 1, 5, 10 and 30 are shown in Fig.6.
+1 Year Tenor 1 smile looks good (see Fig.7). All Tenor 1 expirations also
+in a good agreement with maximal errors for 5-year expiration, see Fig.8.
+All other expirations of Tenor 1 swaptions are in better agreement with the
+input prices, Fig.9.
+Errors in higher tenors swaptions are smaller. You can see it in case of 5
+Year Tenor 2 swaption, Fig.10.
+Note, that using 1 month time step improve the quality of calibrated
+scenario set, see Fig.11.
+6
+Conclusion
+Implementation of Local Volatility Model in interest rate model is presented.
+Calibration is deterministic, it works fast and is accurate. Observed short
+term and low tenor swaption errors can be improved by modifying scenario
+generation process.
+Approach and results were presented on QuantMinds International Con-
+ference 2022, Barcelona, Spain [6].
+9
+
+References
+[1] Dupire, B. (1994). ”Pricing With a Smile.” Risk 7, pp. 18-20.
+[2] Gatheral, J. (2006). ”The Volatility Surface: A Practitioner´ıs Guide.”
+New York, NY: John Wiley & Sons.
+[3] Costeanu, V. & Pirjol D. ”Asymptotic expansion for the normal im-
+plied volatility in local volatility models” , arXiv:1105.3359v1, q-fin.CP,
+(2011);
+[4] V.M. Belyaev : “Swaption Prices in HJM Model. Nonparametric Fit”,
+arXiv:1697.01619, [ q-fin.PR], (2016); QuantMinds International Con-
+ferences (2017-2021);
+[5] Heath, D., R. Jarrow, and A. Morton (1990):
+”Bond Pricing and
+the Term Structure of Interest Rates: A Discrete Time Approxima-
+tion”.Journal of Financial and Quantitative Analysis, 25: 419−440.
+[6] V.M. Belyaev : “Local Volatility in Interest Rate Models”, QuantMinds
+International Conference 2022, Barcelona, Spain.
+10
+
+FIGURES
+Figure 1: ATM Swaption Normal Volatilities.
+11
+
+Tenor 1
+1.6%
+1.4%
+1.2%
+1.0%
+0.8%
+●Vol
+0.6%
+OMC
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Time to ExpirationTenor 5
+1.4%
+1.2%
+1.0%
+0.8%
+0.6%
+●Vol
+0.4%
+OMC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Time to ExpirationTenor10
+1.2%
+1.0%
+0.8%
+0.6%
+.Vol
+0.4%
+OMC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Time to ExpirationTenor30
+1.2%
+1.0%
+0.8%
+0.6%
+.Vol
+0.4%
+OMC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Time to ExpirationFigure 2: ATM Forward Volatility Surface.
+12
+
+0.01
+0.008
+Forward Volatility
+0.006
+0.004
+0.002
+30
+07
+25
+30
+20
+25
+20
+15
+15
+10
+10
+5
+5
+Timeto Exp
+Tenor
+0
+0Figure 3: OTM Swaption Normal Volatilities.
+13
+
+Tenor 10, strike shift-2%
+1.6%
+1.4%
+1.2%
+1.0%
+0.8%
+●Vol
+0.6%
+0.4%
+●MC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Timeto ExpirationTenor 10, strike shift 2%
+1.80%
+1.60%
+1.40%
+1.20%
+1.00%
+0.80%
+Vol
+0.60%
+OMC
+0.40%
+0.20%
+0.00%
+0
+5
+10
+15
+20
+25
+30
+35
+Time to ExpirationTenor 30, strike shift-2%
+1.6%
+1.4%
+1.2%
+1.0%
+0.8%
+●Vol
+0.6%
+0.4%
+●MC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Timeto ExpirationTenor30,strike shift 2%
+1.60%
+1.40%
+1.20%
+1.00%
+0.80%
+Vol
+0.60%
+0.40%
+OMC
+0.20%
+0.00%
+0
+5
+10
+15
+20
+25
+30
+35
+Time to ExpirationTenor 1, strike shift-2%
+1.8%
+1.6%
+1.4%
+1.2%
+1.0%
+0.8%
+.Vol
+0.6%
+●MC
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Timeto ExpirationTenor 1, strike shift 2%
+2.00%
+1.80%
+1.60%
+1.40%
+1.20%
+1.00%
+0.80%
+.Vol
+0.60%
+OMC
+0.40%
+0.20%
+0.00%
+0
+5
+10
+15
+20
+25
+30
+35
+Time to ExpirationTenor 5, strike shift -2%
+1.6%
+1.4%
+1.2%
+1.0%
+0.8%
+0.6%
+·Vol
+0.4%
+●MC
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30
+Timeto ExpirationTenor 5, strike shift 2%
+2.00%
+1.80%
+1.60%
+1.40%
+1.20%
+1.00%
+0.80%
+.Vol
+0.60%
+OMC
+0.40%
+0.20%
+0.00%
+0
+5
+10
+15
+20
+25
+30
+35
+Time to ExpirationFigure 4: 1 Year Tenor 1. Swaption volatility smile.
+14
+
+2.0%
+1.9%
+y= 5.668x*+ 0.0831x + 0.0148
+R²=0.9971
+1.8%
+1.7%
+1.6%
+Input
+....... Inter polation.
+1.5%
+1.4%
+1.3%
+1.2%
+-2.5%
+-2.0%
+-1.5%
+1.0%
+0.5%
+0.0%
+0.5%
+1.0%
+1.5%
+2.0%
+2.5%Figure 5: 1 Year Tenor 1 Volatility Extrapolation.
+Figure 6: ATM Swaption Normal Volatilities with smile.
+15
+
+3.5%
+3.0%
+2.5%
+2.0%
+Input
+Model
+1.0%
+0.5%
+0.0%
+-10%
+-5%
+%0
+5%
+10%Tenor 1
+1.8%
+1.6%
+1.4%
+1.2%
+1.0%
+oInput
+0.8%
+eMcVol
+0.6%
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30Tenor 5
+1.4%
+1.2%
+1.0%
+0.8%
+oInput
+0.6%
+eMcVol
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30Tenor 10
+1.2%
+1.0%
+0.8%
+0.6%
+oInput
+o McVol
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30Tenor 30
+1.2%
+1.0%
+0.8%
+0.6%
+oInput
+o McVol
+0.4%
+0.2%
+0.0%
+0
+5
+10
+15
+20
+25
+30Figure 7:
+Swaption volatility smile.
+16
+
+1 Year, Tenor 1
+2.0%
+1.8%
+1.6%
+.
+1.2%
+1.0%
+oInput
+0.8%
+.McVol
+0.6%
+0.4%
+0.2%
+0.0%
+-2.5%
+1.5%
+-0.5%
+0.5%
+1.5%
+2.5%Figure 8:
+Swaption volatility smile.
+17
+
+5 Years, Tenor 1
+1.4%
+.
+1.2%
+:
+1.0%
+0.8%
+.Vol
+0.6%
+OMC
+0.4%
+0.2%
+0.0%
+-0.03
+-0.02
+-0.01
+0
+0.01
+0.02
+0.03Figure 9:
+Swaption volatility smile.
+18
+
+10Years,Tenor1
+1.2%
+1.0%
+. 8.8%
+0.6%
+.Vol
+OMC
+0.4%
+0.2%
+0.0%
+-0.03
+-0.02
+-0.01
+0
+0.01
+0.02
+0.03Figure 10:
+Swaption volatility smile.
+Figure 11: Swaption volatility smiles. 1 month time steps.
+19
+
+5 Years, Tenor 2
+1.4%
+1.2%
+0.8%
+.Vol
+0.6%
+OMC
+0.4%
+0.2%
+0.0%
+-0.03
+0.02
+-0.01
+0
+0.01
+0.02
+0.033 Years, Tenor 1
+1.8%
+1.6%
+1.4%
+.
+1.2%
+1.0%
+eInput
+0.8%
+●MC
+0.6%
+0.4%
+0.2%
+0.0%
+-2.5%
+1.5%
+%5'0-
+0.5%
+1.5%
+2.5%5 Years, Tenor 1
+1.4%
+1.2%
+1.0%
+0.8%
+Input
+0.6%
+●MC
+0.4%
+0.2%
+0.0%
+-2.5%
+-1.5%
+%5'0-
+0.5%
+1.5%
+2.5%10 Years, Tenor 1
+1.2%
+1.0%
+0.6%
+oInput
+●MC
+0.4%
+0.2%
+0.0%
+-2.5%
+1.5%
+-0.5%
+0.5%
+1.5%
+2.5%30 Years, Tenor 1
+0.8%
+0.7%
+0.6%
+0.5%
+0.4%
+oInput
+0.3%
+●MC
+0.2%
+0.1%
+0.0%
+-2.5%
+1.5%
+%5'0-
+0.5%
+1.5%
+2.5%

0NFRT4oBgHgl3EQfkTex/content/tmp_files/load_file.txt

@@ -0,0 +1,519 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf,len=518
+page_content='Local Volatility in Interest Rate Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Belyaev  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Bank, Minneapolis, MN, USA  February 1, 2023  Abstract  A new approach to Local Volatility implementation in the interest  rate model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' The major tool of this approach is a small  volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This approximation works very well and it  can be used to calibrate all ATM swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' It works fast and accu-  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' In order to reproduce all available swaption prices we need to  take into account the dependence of forward volatility on the current  swap rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Here we assume that forward volatility is a deterministic  function on strike, tenor, and expiration at every point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  We determine these functions and apply them in Monte-Carlo calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  It was demonstrated that this approach works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' However, in  the case of short term and low tenor swaptions we observed errors in  swaption pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' To fix this problem we need to modify the scenario  generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  1  Introduction  Local Volatility Model was presented by Dupire [1] in 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' According to this  model forward volatility is a deterministic function of a current underlying  S(t) price and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Model dynamics in this case has the following form:  dS(t) = µ(t)S(t)dt + σL(S(t), t)S(t)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (1)  where µ(T) = r(t)−y(t) is a arbitrage free drift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' r(t) and y(t) are risk free rate  and dividend yield;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' dW(t) is Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' σL(S(t), t) is a deterministic  Local Volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='13595v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='PR]  31 Jan 2023  Representing the option price as a function of forward one C(F(T), X, T)  where F(T) = S(0)e  � T  0 (r(t)−y(t))dt we have  σ2  L(X, t) =  ∂C  ∂T  1  2X2 ∂2C  ∂X2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (2)  where C(F(T), X, T) is a call option price;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' X is a strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Having arbitrage-free interpolated/extrapolated volatility surface we can  calculate local volatility by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (2) and generate scenarios which are perfectly  calibrated to all available option prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This procedure is deterministic and  fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Later Gatheral [2] derived the following formula for local volatility which  is expressed in terms of implied volatility itself:  σ2  L(X, t) =  ∂w  ∂T  1 − y  w  ∂w  ∂y + 1  4  �  − 1  4 − 1  w + y2  w2  � �  ∂w  ∂y  �2 + 1  2  ∂2w  ∂y2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (3)  where w(y, T) is an implied variance of the option with strike X and time  to expiration T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' y = ln(X/F(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' The formula makes it easier to calculate  Local Volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In the case of normal volatilities models where  dS(t) = µ(t)dt + σ(S(t), t)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (4)  µ(T) = ∂F(T)  ∂T , formula for Local Volatility is also known [3]:  σ2  L(X, t) =  dw  dT  1 − y  w  ∂w  ∂y + 1  4  �  − 1  w + y2  w2  � �  ∂w  ∂y  �2 + 1  2  ∂2w  ∂y2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (5)  Here we use the following notation: y = X − F(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  To calibrate interest rate model we will use small volatility approximation  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This approximation works very well and it means that bond price dy-  namic approximately is a normal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' So, Local Volatility can be calculated  according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In Section 2 we describe a Small Volatility approximation and demon-  strate its accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This approximation can be used to calibrate the model  for selected OTM strikes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' In Section 3 we apply this approximation in  order to to calculate deterministic sensitivities to strikes at every point of the  2  grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' In Section 4 fixed tenor dynamics and forward volatility calculation are  discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' In Section 5 we use these sensitivities to calculate current forward  volatilities according to eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (5) to generate interest rate scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  All calculations are completed for March 29, 2022 SOFR rate and swap-  tions market data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Even though the fact that the approach works well we  observe relatively big errors in case of short-term and low-tenor swaptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  2  Small Volatility Approximation  Here we consider HJM interest rate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' HJM model [5] has the following  dynamics:  df(t, T) = α(t, T)dt + σ(t, T)dW(t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (6)  where f(t, T) is a forward rate:  B(t, T) = e−� T  t f(t,τ)dτ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (7)  B(t, T) is a zero coupon risk-free bond;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' σ(t, T) is a normal volatility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' dW(t, T)  is a Brownian motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' and  α(t, T) = σ(t, T)  � T  t  σ(t, τ)dτ  (8)  is a drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  The drift is chosen to satisfy martingale condition on bond prices  B(0, T) =  �  e−� t  0 r(τ)dτB(t, T)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (9)  Distribution of discounted bond prices at time T can be presented in the  following form:  e−� T  0 r(τ)dτB(T, T1) =  = B(0, T1)e−� T  0 dτ � T1  τ  α(τ,t)dt−� T  0 dW(τ)� T1  τ  σ(τ,t)dt =  = B(0, T1)  �  1 −  � T  0 dW(τ)  � T1  τ  σ(τ, t)dt + o(σ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (10)  3  It means that the distribution of SOFR swap present value is:  PV (T) = e−� T  0 r(t)dt  N  �  n=1  B(T, Tn)  �  rs + 1 − B(T, Tn−1)  B(T, Tn)  �  =  = e−� T  0 r(t)dt  �  rs  N  �  n=1  B(T, Tn) − B(T, T) + B(T, TN)  �  ≃  ≃ (rs − rATM)  N  �  n=1  B(0, Tn) + Σ(T, N)ξ  √  T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (11)  where Tn are times of payments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' rs and rATM = B(0,T)−B(0,TN)  �  n=1NB(0,Tn) are swap rate  and ATM rate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' T0 = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Volatility Σ(T, N) is calculated according to the following formulas:  Σ2(T, N)T =  � T  0 v2(t, Ndt)dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  v(t, N) = rs  N  �  n=1  B(0, Tn)  � Tn  t  σ(t, τ)dτ −  −B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='T)  � T  t  σ(t, τ)dτ + B(0, TN)  � TN  t  σ(t, τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (12)  In order to calibrate the interest rate model we can use interpolated  volatilities or to assume that all unknown volatilities are equal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Here we use the first approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In the case of the grid with 3 month time step the first swaption is a tenor  1 expired in 3 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' According to (12) we have:  v(dt, 1) = rsB(0, 5dt)  4  �  k=0  (k + 1)σ(0, k)dt −  −B(0, dt)σ(0, 0)dt +  +B(0, 5dt)  4  �  k=0  (k + 1)σ(0, k)dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (13)  where dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Assuming that all unknown volatilities are equal in (13)  σ(0, k) = σ(0, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ∀k < 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (14)  we can calculate volatilities in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  4  We can apply this procedure to other expirations and tenors, taking into  account already defined volatility values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' For every available next tenor and  time to expiration we obtain the following equation:  Σ2(i, j) = Aσ2 + Bσ + C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (15)  where A, B, C are factors that can be determined by using bond prices and  already calculated volatilities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' σ is an unknown forward volatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Schematically calibration process can be represented in the following way:  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='75 Te + Tenor  6  Te  �  �  �  �  �  �  �  �  �  �  �  �  �  ��  f  f  f  f  f  f  f  f  f  f  f  v  f  f  f  f  f  f  v  f  f  f  Σ(1, 5)  Σ(2, 6)  σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 0) σ(0, 5) σ(0, 5)  σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1) σ(1, 1)  where Σ(1, 5) and Σ(2, 6) are input volatilities of 1-year swaptions with  3- and 6-months to expiration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  This procedure leads to a good calibrated prices for all ATM swaptions,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='1 where we compare input ATM volatilities with volatilities calculated  from Monte-Carlo prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Forward volatility surface is shown on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  3  Sensitivity Calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In the previous section we describe how to calibrate forward volatilities to  reproduce all available ATM swaptions prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This procedure can be applied  to find forward volatilities calibrated to all swaption prices with shifted strikes  from their ATM values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  5  Sswaption quotes are available within ±2% shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' The calibration process  is similar to the ATM swaption calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' The only difference is a change of  ATM volatility to the shifted one for selected strike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' The calibration quality  is good (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In the selected data set for SOFR swaption volatility smile, we have  strikes up to ±2% rate shift from ATM rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This smile can be fitted by  quadratic polynomial as it is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Splines can also be used  but obtained results are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' It means that we can use quadratic  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  However, we need to take care of extrapolation, due to the swap rates can  go out of [−2%, 2%] range rate shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' This extrapolation need to be contin-  uously differentiable and it would be very helpful if we can use extrapolation  with limited volatility, in order to to use small volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  We choose the following quadratic extrapolation:  v(x) = αu + βu(x − x0) + γu(x − x0)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  x0 < x < xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (16)  v(x) = αd + βd(x − (−x0)) + γd(x − (−x0))2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  xd < x < −x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  where x0 = 2% is the current maximal available rate shift in input data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' final  points of extrapolation are chosen as xd = −10% and xu = 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  To obtain continuously differentiable function, we need to have  αu = v(x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  αd = v(−x0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  βu = dv(x)  dx  |x=x0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  βd = dv(x)  dx  |x=−x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (17)  To get zero slope at the end points xu,d we have  γu(xu − x0) = −1  2βu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  γd(xd − (−x0)) = −1  2βd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (18)  Above end points we assume that  v(x) =  �  v(xd);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  if x < xd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  v(xu);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  if x > xu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (19)  It leads to the smooth volatility curve depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  6  4  Fixed Tenor Dynamics  Before going to Scenario Generation Process let us consider fixed tenor dy-  namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In the case of selected tenor N we consider the following ATM  swaption values distribution which present value is:  PV (S(T, N)) = e−� T  0 r(τ)dτ  �  rATM(T, N)  N  �  n=1  B(T, Tn) − 1 + B(T, TN)  �  =  = e−� T  0 r(τ)dτ  �B(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='T) − B(0, TN)  �N  n=1 B(0, Tn)  N  �  n=1  B(T, Tn) − 1 + B(T, TN)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (20)  In small volatility approximation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (20) has the following form:  PV (S(T, N)) = Σ(T, N)ξ  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (21)  In the case of non-zero constant difference between current swaption rate and  ATM rate  δ = r(T, N) − rATM(T, N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (22)  swap present value distribution is  PV (S(T, N, δ)) = δ  N  �  n=1  B(0, Tn) + Σ(T, N, δ)ξ  √  T =  = A(T, N)  �  δ + Σ(T, N, δ)  A(T, N) ξ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (23)  where Σ(T, N, δ) is implied volatility of fixed tenor underlying for selected  rate shift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  A(T, N) =  N  �  n=1  B(0, tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (24)  As we can see from (23) we can use (5) to determine forward volatility  assuming deterministic dependence of local forward volatility for selected  point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  7  5  Local Volatility Scenarios  In the scenario generation process we can use formula for Local Volatility  of normal volatility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Here we assume that local forward volatility on  every point on the grid can be defined deterministically from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Below  we describe this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In addition to forward volatility, the formula depends on total variance  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Deterministic variance can be calculated in the following form:  w(−2%, n, k) = v2  f(−2%, n − 1, k)dt + w(−2%, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  w(ATM, n, k) = v2  f(ATM, n − 1, k)dt + w(ATM, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  w(2%, n, k) = v2  f(2%, n − 1, k)dt + w(2%, n − 1, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (25)  where vf(shift, n, k) is a forward volatility for selected time step n and point  on the grid k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Then we use interpolation-extrapolation formulas to get function w(x, n, k)  for every point on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  This procedure gives us all deterministic functions needed to calculate  forward volatility according to equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  To generate scenarios we need to calculate current swap rate which is a  difference between current and initial ATM swap rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Initial ATM swap rate is:  rS(0, ti, tenor) = B(0, ti) − B(0, ti + tenor)  �tenor  n=1 B(0, ti + n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (26)  Observed swap rate at time ti is:  rS(ti, ti, tenor) = 1 − B(ti, ti + tenor)  �tenor  n=1 B(ti + n)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (27)  So, the current rate strike is  X(ti, ti, tenor) = rS(ti, ti, tenor) − rS(0, ti, tenor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  (28)  In the case of tenor = 1 we assume that this strike is the same for all  time steps between 0 and tenor = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' For tenor = 2 we use strike for all  times between tenor = 1 and tenor = 2 only etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='. As was mentioned in the  8  previous section this procedure works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' We use it because we already have  calculated volatility sensitivities on swap rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  In calculations we apply  calculated variance for selected strike and point on the grid (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  We apply this procedure and generated 100,000 scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ATM swaption  prices for tenors 1, 5, 10 and 30 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  1 Year Tenor 1 smile looks good (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' All Tenor 1 expirations also  in a good agreement with maximal errors for 5-year expiration, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  All other expirations of Tenor 1 swaptions are in better agreement with the  input prices, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Errors in higher tenors swaptions are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' You can see it in case of 5  Year Tenor 2 swaption, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Note, that using 1 month time step improve the quality of calibrated  scenario set, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  6  Conclusion  Implementation of Local Volatility Model in interest rate model is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Calibration is deterministic, it works fast and is accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Observed short  term and low tenor swaption errors can be improved by modifying scenario  generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  Approach and results were presented on QuantMinds International Con-  ference 2022, Barcelona, Spain [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  9  References  [1] Dupire, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ”Pricing With a Smile.” Risk 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' 18-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  [2] Gatheral, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ”The Volatility Surface: A Practitioner´ıs Guide.”  New York, NY: John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  [3] Costeanu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' & Pirjol D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' ”Asymptotic expansion for the normal im-  plied volatility in local volatility models” , arXiv:1105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='3359v1, q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='CP,  (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Belyaev : “Swaption Prices in HJM Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Nonparametric Fit”,  arXiv:1697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='01619, [ q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='PR], (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' QuantMinds International Con-  ferences (2017-2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  [5] Heath, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=', R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Jarrow, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Morton (1990):  ”Bond Pricing and  the Term Structure of Interest Rates: A Discrete Time Approxima-  tion”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='Journal of Financial and Quantitative Analysis, 25: 419−440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  [6] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content=' Belyaev : “Local Volatility in Interest Rate Models”, QuantMinds  International Conference 2022, Barcelona, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
+page_content='  10  FIGURES  Figure 1: ATM Swaption Normal Volatilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFRT4oBgHgl3EQfkTex/content/2301.13595v1.pdf'}
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+arXiv:2301.01688v1  [math.OC]  4 Jan 2023
+Feedback Stabilization of Tank-Liquid
+System with Robustness to Surface Tension
+Iasson Karafyllis, Filippos Vokos
+Department of Mathematics,
+National Technical University of Athens,
+Zografou Campus, 15780, Athens, Greece,
+emails: [email protected], fi[email protected]
+Miroslav Krstic
+Department of Mechanical and Aerospace Eng.,
+University of California, La Jolla, San Diego,
+CA 92093-0411, USA email: [email protected]
+January 5, 2023
+Abstract
+We construct a robust stabilizing feedback law for the viscous Saint-
+Venant system of Partial Differential Equations (PDEs) with surface tension
+and without wall friction. The Saint-Venant system describes the move-
+ment of a tank which contains a viscous liquid. We assume constant con-
+tact angles between the liquid and the walls of the tank and we achieve
+a spill-free exponential stabilization with robustness to surface tension by
+using a Control Lyapunov Functional (CLF). The proposed CLF provides a
+parameterized family of sets which approximate the state space from the
+interior. Based on the CLF, we construct a nonlinear stabilizing feedback
+law which ensures that the closed-loop system converges exponentially to
+the desired equilibrium point in the sense of an appropriate norm.
+1
+Introduction
+The Saint-Venant model, which was derived in [2], constitutes a significant and
+very influential mathematical model in fluid mechanics. It is also referred in
+literature as the shallow water model. Recent extensions of the Saint-Venant
+model take into account various types of forces such as viscous stresses, surface
+tension and friction forces (see [10,19,29,33,41,43]).
+The feedback stabilization problem of the Saint-Venant model is a challeng-
+ing problem. The dominant cases studied in the litterature include the inviscid
+model - which ignores forces such as viscous stresses and surface tension - and
+1
+
+the linearized model (see [1,3–5,11–14,16–18,30,35,36]). In [11,12,14,35,36]
+the problem of the movement of an 1-D tank which contains a fluid is stud-
+ied. More specifically, [11,12,14] provide controllability results for the Saint-
+Venant model without viscosity, without friction and without surface tension,
+while [35] suggests a new variational formulation of Saint-Venant equations
+and proves the steady-state controllability of the linear approximations of sev-
+eral control configurations. In [36] the inviscid Saint-Venant model is studied
+and appropriate stabilizing full-state feedback and output feedback control
+laws are constructed. In [3–5, 12, 13, 16–18, 30] the movement of a fluid in an
+open channel is studied. Stabilization results are provided in [1,3,4,13,17,18].
+In [3, 4, 17, 18, 30] the linearized Saint-Venant model is being used while [1]
+deals with a general linear hyperbolic system which appears in Saint-Venant
+equations among other linear hyperbolic laws. The works [5, 12, 13, 16] study
+the nonlinear Saint-Venant model. In [5] the feedforward control problem of
+general nonlinear hyperbolic systems is studied and an application using the
+Saint-Venant model with friction is provided. In [12, 13] local convergence of
+the state of hyperbolic systems of conservation laws is guaranteed using a strict
+Lypaunov function which exploits Riemann invariants. An application to the
+inviscid, frictonless Saint-Venant model is provided as well. The paper [16]
+achieves regulation of the water flow and level in water-ways using the invis-
+cid Saint-Venant model without friction and without surface tension.
+Very few studies in the literature deal with the nonlinear viscous Saint-
+Venant model that is used for the description of the movement of a tank which
+contains an incompressible, Newtonian fluid. The first work that studied the
+nonlinear viscous Saint-Venant model without wall friction and without sur-
+face tension was [23]. In [23] an appropriate nonlinear feedback law is con-
+structed which provides semiglobal stabilization results by following a CLF
+methodology. The work [25] extends the results obtained in [23] in the case
+where wall friction forces are taken into account. In [25] both the case of a
+velocity independent friction coefficient and the general case of friction co-
+efficient are studied. A robust with respect to wall friction stabilizing feed-
+back law is constructed. Another study which deals with the nonlinear viscous
+Saint-Venant model is [24]. In [24] a stabilizing output-feedback control law for
+the viscous Saint-Venant PDE system without wall friction and without surface
+tension is constructed. The output-feedback control law is utilized through a
+functional-observer methodology and a CLF methodology.
+The study of the movement of a fluid which interacts with a gas bound-
+ary and a solid boundary is inevitably intertwined with the notion of the sur-
+face tension and the notions of contact angle and wettability (see [27, 34]).
+Surface tension is crucial as it acts in the interface between liquid and gas.
+From a mathematical point of view surface tension is very important because
+it changes the order of the PDEs (it is expressed by a third order term). Con-
+tact angle is the angle at which the fluid surface intersects with a solid bound-
+ary as stated in [34], and it is a measure of wettability of the solid surface.
+There is a wide literature concerning the topic of contact angles (see for in-
+stance [20,21,27,28,37,38,42,43]). The concept of contact angle is significant
+2
+
+in our study because it provides an additional boundary condition.
+In this paper we solve the feedback stabilization problem for a tank con-
+taining a liquid modeled by the viscous Saint-Venant system of PDEs with sur-
+face tension and without wall friction. We consider the case of constant contact
+angles between the liquid and the walls of the tank, as in [37,38]. We utilize a
+specific form of the feedback law initially presented in [25], which constitutes
+a more general form of the feedback law in [23] with robustness to surface
+tension. Indeed, we saw that the proposed feedback law guarantees stabiliza-
+tion no matter what the value of the surface tension coefficient is. Therefore, the
+knowledge of the surface tension coefficient is not necessary and the feedback
+law is independent of the surface tension coefficient. We achieve a spill-free
+exponential stabilization, with robustness to surface tension. As in [23–25] we
+follow a CLF methodology and we design the feedback law based on an appro-
+priate functional, which is the CLF. The CLF determines a specific parameter-
+ized set which approximates the state space of the control problem from the
+interior.
+Although this work presents enough technical similarities with [25], there
+are some crucial differences. Firstly, in contrast with [25], the system of PDEs
+contains an extra term due to surface tension and does not contain a friction
+term. Moreover, in order for the model to be complete and for the problem
+to be well-posed, an additional boundary condition is used. The additional
+boundary condition is provided by the assumption of a constant contact angle.
+Here we use only one CLF while in [25] two different functionals are proposed.
+As a consequence this work does not provide a bound for the sup-norm of
+the fluid velocity, as in [25], due to the absence of an appropriate functional.
+Here the CLF is different from the corresponding one in [25], as it contains an
+additional potential energy term due to the effect of the surface tension.
+This paper is organized as follows. In Section 2 the control problem is
+described as well as its main objective. In Section 3 we provide the intuitive
+ideas and the statements of the results of this work along with some auxiliary
+lemmas. Section 4 includes all the proofs of the results presented in Section 3.
+Finally, Section 5 points out the conclusions of this work and suggests topics
+for future research.
+Notation
+∗ R+ = [0,+∞) is the set of non-negative real numbers.
+∗ Let S ⊆ Rn be an open set and let A ⊆ Rn be a set such that S ⊆ A ⊆ cl(S).
+By C0(A;Ω), we denote the class of continuous functions on A, which
+take values in Ω ⊆ Rm. By Ck(A;Ω), where k ≥ 1 is an integer, we denote
+the class of functions on A ⊆ Rn, which takes values in Ω ⊆ Rm and has
+continuous derivatives of order k. In other words, the functions of class
+Ck(A;Ω) are the functions which have continuous derivatives of order k
+in S = int(A) that can be continued continuously to all points in ∂S ∩ A.
+When Ω = R then we write C0(A) or Ck(A). When I ⊆ R is an interval and
+3
+
+G ∈ C1(I) is a function of a single variable, G′(h) denotes the derivative
+with respect to h ∈ I.
+∗ Let I ⊆ R be an interval, let a < b be given constants and let u : I ×[a,b] →
+R be a given function. We utilize the notation u[t] to denote the profile
+at certain t ∈ I, i.e., (u[t])(x) = u(t,x) for all x ∈ [a,b]. When u(t,x) is three
+times differentiable with respect to x ∈ [a,b], we use the notation ux(t,x),
+uxx(t,x) and uxxx(t,x) for the first, second and third derivative of u with
+respect to x ∈ [a,b] respectively, i.e.,
+ux(t,x) = ∂u
+∂x (t,x),uxx(t,x) = ∂2 u
+∂x2 (t,x) and uxxx(t,x) = ∂3 u
+∂x3 (t,x)
+When u(t,x) is differentiable with respect to t, we use the notation ut(t,x)
+for the derivative of u with respect to t, i.e.,
+ut(t,x)= ∂u
+∂t (t,x)
+∗ Given a set U ⊆ Rn, χU denotes the characteristic function of U defined
+by
+χU(x) :=
+�
+1
+for all x ∈ U
+0
+for all x � U
+The sign function sgn : R → R is the function defined by
+sgn(x) :=
+
+1
+for x > 0
+0
+for x = 0
+−1
+for x < 0
+∗ Consider given constants a,b such that a < b . For p ∈ [1,+∞), Lp(a,b)
+denotes the set of equivalence classes of Lebesgue measurable functions
+u : (a,b) → R with
+∥u∥p :=
+�� b
+a
+|u(x)|p dx
+�1/p
+< +∞.
+L∞(a,b) denotes the set of equivalence classes of Lebesgue measurable
+functions u : (a,b) → R with
+∥u∥∞ := esssup
+x��(a,b)
+(|u(x)|) < +∞.
+For an integer k ≥ 1, Hk(a,b) denotes the Sobolev space of functions in
+L2(a,b) with all its weak derivatives up to order k ≥ 1 in L2(a,b).
+4
+
+2
+The Control Problem
+We want to manipulate the motion of a tank which contains a viscous, Newto-
+nian, incompressible liquid. Viscosity is utilized as a gain in the controller on
+the difference between the boundary liquid levels and to settle a region of at-
+traction. The tank is subject to an acceleration which we consider as the control
+input and obeys Newton’s second law. The problem is described by the viscous
+Saint-Venant equations. We restrict our study to the one-dimensional (1-D)
+case of the model. Moreover, contrary to prior works, in this work we do not
+neglect the surface tension that acts on the free surface (liquid-gas interface)
+but we neglect friction with the tank walls.
+We intend to drive asymptotically the tank to a specified position. The
+aforementioned goal must be achieved without liquid spillage and by having
+both the tank and the liquid within the tank at rest. The equations describ-
+ing the motion of the liquid in the tank can be derived by performing mass
+and momentum balances (from first principles assuming that the liquid pres-
+sure is the combination of hydrostatic pressure and capillary pressure given
+by the Young-Laplace equation (see [15]) and by ignoring friction with the
+tank walls). The equations can also be derived by using approximations of the
+Navier-Stokes equations for the incompressible fluid (see [6–8, 28, 32, 37, 38];
+but see also [21,29] for fluid equations involving capillary phenomena).
+We denote by a(t) the position of the left side of the tank at time t ≥ 0 and
+we consider the length of the tank to be L > 0 (a constant). The evolution of the
+liquid level and of the liquid velocity is described by the following equations
+Ht + (Hu)z = 0, for t > 0, z ∈ [a(t),a(t) + L]
+(1)
+(Hu)t +
+�
+Hu2 + 1
+2gH2�
+z
+− σH
+
+
+Hzz
+�
+1 + H2z
+�3/2
+
+
+z
+= µ(Huz)z
+for t > 0, z ∈ (a(t),a(t) + L)
+(2)
+where H(t,z) > 0, u(t,z) ∈ R are the liquid level and the liquid velocity, respec-
+tively, at time t ≥ 0 and position z ∈ [a(t),a(t) + L], while g,µ,σ > 0 (constants)
+are the acceleration of gravity, the kinematic viscosity of the liquid and the ra-
+tio of the surface tension and liquid density, respectively. In some papers the
+term
+
+
+Hzz
+�
+1 + H2z
+�3/2
+
+
+z
+is replaced by Hzzz (see [6–8,32], but here we prefer a more
+accurate description of the surface tension.
+The liquid velocities at the walls of the tank are equal with the tank velocity.
+Consequently:
+u(t,a(t)) = u(t,a(t) + L) = w(t), for t ≥ 0
+(3)
+where w(t) = ˙a(t) is the velocity of the tank at time t ≥ 0. Moreover, we get for
+the tank
+¨a(t) = −f (t), for t > 0
+(4)
+5
+
+where −f (t), the control input to the problem, is the tank acceleration. Defin-
+ing the quantities
+v(t,x) := u(t,a(t) + x) − w(t)
+(5)
+h(t,x) := H(t,a(t) + x)
+(6)
+ξ(t) := a(t) − a∗
+(7)
+where a∗ ∈ R is the position (a constant) which we want the left side of the tank
+to reach, we get the model:
+˙ξ = w, for t ≥ 0
+(8)
+˙w = −f , for t ≥ 0
+(9)
+ht + (hv)x = 0, for t > 0, x ∈ [0,L]
+(10)
+(hv)t +
+�
+hv2 + 1
+2gh2�
+x
+− σh
+
+
+hxx
+�
+1 + h2x
+�3/2
+
+
+x
+= µ(hvx)x + hf ,
+for t > 0, x ∈ (0,L)
+(11)
+v(t,0) = v(t,L) = 0, for t ≥ 0
+(12)
+Equations (10) and (12) imply that every classical solution of (8)-(12) satisfies
+the following
+d
+d t
+�� L
+0
+h(t,x)dx
+�
+= 0 for all t > 0
+(13)
+Consequently, the total mass of the liquid m > 0 is constant, and without loss of
+generality we can assume that every solution of (8)-(12) satisfies the equation
+� L
+0
+h(t,x)dx ≡ m
+(14)
+Due to the nature of our problem it is important to mention that the liquid
+level h(t,x) must be positive for all times, i.e., we must have:
+min
+x∈[0,L](h(t,x)) > 0, for t ≥ 0
+(15)
+Contrary to prior works, model (8)-(12), (14) is not a complete mathematical
+description of the system. This can be seen directly by studying the lineariza-
+tion of model (8)-(12), (14) but also can be seen by studying the literature
+(see [27, 37, 38, 42, 43] and references therein). For a complete mathematical
+model of the system we need two additional boundary conditions that describe
+the interaction between the liquid and the solid walls of the tank. There are
+many ways to describe the evolution of the angle of contact of a liquid with a
+solid boundary (see the detailed presentation in [27]). In [37, 38], Schweizer
+suggested (based on energy arguments and the fact that there might be a dis-
+crepancy between the actual microscopic and the apparent macroscopic con-
+tact angle) the use of a constant contact angle. Moreover, the assumption of
+6
+
+a constant contact angle allows the well-posedness of the overall problem (at
+least for small data; see [37,38,42]). The constant contact angle approach has
+been used extensively in the literature (see for instance [20,42,43]).
+In this work, we adopt the constant contact angle approach by imposing a
+contact angle equal to π/2. Therefore, the model (8)-(12), (14) is accompanied
+by the following boundary conditions:
+hx(t,0) = hx(t,L) = 0, for t ≥ 0
+(16)
+In order to avoid liquid spillage the following condition must be satisfied:
+max
+x∈[0,L](h(t,x)) < Hmax, for t ≥ 0
+(17)
+where Hmax > 0 is the height of the tank walls. We consider classical solutions
+for the system (8)-(12), (14), (16), i.e., we consider
+ξ ∈ C2 (R+), w ∈ C1 (R+), h ∈ C1 ([0,+∞) × [0,L]; (0,+∞)) ∩C3 ((0,+∞) ×(0,L)),
+v ∈ C0([0,+∞) × [0,L]) ∩C1 ((0,+∞) ×[0,L]) with v[t] ∈ C2 ((0,L)) for each t > 0
+that satisfy equations (8)-(12), (14), (16) for a given input f ∈ C0 (R+).
+For the system (8)-(12), (14), (16) with f (t) ≡ 0 (which is the open loop
+system), there exists a continuum of equilibrium points, i.e., the points
+h(x) ≡ h∗,v(x) ≡ 0, for x ∈ [0,L]
+(18)
+ξ ∈ R,w = 0
+(19)
+where h∗ = m/L. We assume that the equilibrium points satisfy the condition
+(17), i.e., h∗ < Hmax.
+We intend to construct a robust with respect to surface tension control law
+of the form
+f (t) = F (h[t],v[t],ξ(t),w(t)), for t > 0,
+(20)
+which stabilizes the equilibrium point with ξ = 0. In addition to that we im-
+pose the condition (17).
+It follows from (18), (19) that the desired equilibrium point is not asymp-
+totically stable for the open-loop system. Consequently the described control
+problem is not at all trivial.
+3
+The feedback law
+3.1
+The Control Lyapunov Functional (CLF)
+We define the set S ⊂ R2 ×
+�
+C0 ([0,L])
+�2 as follows:
+(ξ,w,h,v) ∈ S ⇔
+
+h ∈ C0 ([0,L];(0,+∞)) ∩ H1(0,L)
+v ∈ C0 ([0,L])
+� L
+0
+h(x)dx = m
+(ξ,w) ∈ R2,v(0) = v(L) = 0
+(21)
+7
+
+The above definition guarantees that every (ξ,w,h,v) ∈ S satisfies (12) and (14).
+In addition to that, we define the following functionals for all (ξ,w,h,v) ∈ S:
+V (ξ,w,h,v) := δE(h,v) + W(h,v) + qk2
+2 ξ2 + q
+2 (w + kξ)2
+(22)
+E(h,v) := 1
+2
+� L
+0
+h(x)v2(x)dx + g
+2
+���h − h∗χ[0,L]
+���2
+2
++σ
+� L
+0
+� �
+1 + (h′(x))2 − 1
+�
+dx
+(23)
+W(h,v) := 1
+2
+� L
+0
+h−1(x)(h(x)v(x) + µh′(x))2 dx + g
+2
+���h − h∗χ[0,L]
+���2
+2
++σ
+� L
+0
+
+
+�
+1 + (h′(x))2 − 1
+
+dx
+(24)
+where k,q > 0 are position error and velocity gains (to be selected) respectively,
+δ > 0 and h∗ = m/L. In particular:
+• the functional E is the mechanical energy of the liquid within the tank as
+it is the sum of the potential energy
+g
+2
+���h − h∗χ[0,L]
+���2
+2 + σ
+� L
+0
+� �
+1 + (h′(x))2 − 1
+�
+dx
+and the kinetic energy
+1
+2
+� L
+0
+h(x)v2(x)dx
+of the liquid. It should be noticed that there is no contribution to the
+mechanical energy of the tank-liquid interface which allows to give the
+interpretation that the boundary condition (16) (a constant contact angle)
+is a result of the absence of interaction between liquid and solid.
+• the functional W is a kind of mechanical energy of the liquid within
+the tank and has been used extensively in the literature of isentropic,
+compressible liquid flow (see [22,31,39,40]) as well as in [23–25].
+The functional V (ξ,w,h,v) defined by (22) will be utilized as a CLF for the
+system, and for the derivation of useful bounds for the function h as guaran-
+teed by the following lemma.
+8
+
+Lemma 1. Let constants q,k,δ > 0 be given, and define the increasing function
+G ∈ C0(R) ∩ C1((−∞,0) ∪ (0,+∞)) as follows
+G(h) :=
+
+sgn(h − h∗)
+�2
+3h
+√
+h − 2h∗ √
+h + 4
+3h∗ √
+h∗
+�
+for h > 0
+−4
+3h∗ √
+h∗ + h
+for h ≤ 0
+(25)
+Denote by G−1 : R → R the inverse function of G and define the constant
+c :=
+1
+µ
+�
+δg
+(26)
+Then for every (ξ,w,h,v) ∈ S, the following inequality holds:
+Q1 (V (ξ,w,h,v)) ≤ h(x) ≤ Q2 (V (ξ,w,h,v)), for all x ∈ [0,L],
+(27)
+where the functions Qi : R+ → R (i = 1,2) are defined as follows for all s ≥ 0:
+Q1(s) := max
+�
+G−1 (−cs),N1(s),N2(s)
+�
+(28)
+Q2(s) := min
+�
+G−1 (cs),P1(s),P2(s)
+�
+(29)
+with the functions Ni : R+ → R (i = 1,2) and Pi : R+ → R (i = 1,2) defined by the
+following expressions for all s ≥ 0:
+N1(s) := h∗ −
+�
+2m(1 + δ)s
+δµ2
+,
+(30)
+N2(s) := h∗ −
+��
+s
+σ(δ + 1) + L
+�2
+− L2,
+(31)
+P1(s) := h∗ +
+�
+2m(1 + δ)s
+δµ2
+,
+(32)
+P2(s) := h∗ +
+��
+s
+σ(δ + 1) + L
+�2
+− L2
+(33)
+Remark 1. It follows from (25), (26), (28) and the fact that h∗ = m/L that
+Q1 (V (ξ,w,h,v)) > 0 when
+V (ξ,w,h,v) < max(θ1,θ2,θ3)
+(34)
+with
+θ1 := 4
+3µh∗ �
+δgh∗, θ2 :=
+µ2h∗δ
+2L(1 + δ)
+and
+θ3 := σ (δ + 1)
+� �
+(h∗)2 + L2 − L
+�
+9
+
+Definitions (28) and (29) imply that Q2 : R+ → R is an increasing function
+while Q1 : R+ → R is a decreasing function.
+It is important to mention that Lemma 1 is more general than Lemma 1
+in [23] and Lemma 1 in [25]. Lemma 1 in [23] can be applied only for the case
+δ = 1 and σ = 0, while Lemma 1 in [25] can be applied only for the case σ = 0.
+Here Lemma 1 can be applied for all δ > 0 and σ ≥ 0.
+3.2
+The state space
+As in [23–25] the state space will be appropriately defined in order to exclude
+states of the set S defined by (21) that violate the condition (17), i.e, the states
+that cause liquid spillage. We define the following
+X :=
+�
+(ξ,w,h,v) ∈ S : max
+x∈[0,L](h(x)) < Hmax
+�
+(35)
+R := 2µ
+�
+δgh∗
+3
+(Hmax − h∗)min(ζ1,ζ2)
+(36)
+where
+ζ1 := max(Γ1,Γ2,Γ3) and
+(37)
+ζ2 :=
+h∗
+Hmax − h∗ max(2,∆1,∆2)
+(38)
+with Γ1,Γ2,Γ3,∆1 and ∆2 defined as follows:
+Γ1 :=
+�
+Hmax
+h∗
+−
+2
+√
+h∗
+√Hmax +
+√
+h∗ ,
+(39)
+Γ2 := 3µ
+√
+δ(Hmax − h∗)
+4m(1 + δ)
+�
+gh∗ ,
+(40)
+Γ3 :=
+3σ(δ + 1)
+� �
+L2 + (Hmax − h∗)2 − L
+�
+2µ
+�
+δgh∗ (Hmax − h∗)
+,
+(41)
+∆1 :=
+3µ
+√
+δ
+4L
+�
+gh∗ (1 + δ)
+,
+(42)
+∆2 :=
+3σ(δ + 1)
+√
+h∗
+2µ
+�
+δg
+� �
+(h∗)2 + L2 + L
+�
+(43)
+The aforementioned definition (36), the fact that h∗ < Hmax and Lemma 1
+imply for all (ξ,w,h,v) ∈ S with V (ξ,w,h,v) < R the following
+0 < Q1 (V (ξ,w,h,v)) ≤ h(x) ≤ Q2 (V (ξ,w,h,v)) < Hmax, for all x ∈ [0,L]
+(44)
+10
+
+Consequently, the conditions (17) for avoiding liquid spillage are satisfied when
+(ξ,w,h,v) ∈ S with V (ξ,w,h,v) < R.
+The set X defined by (35) is the state space of system (8)-(12), (14), (16).
+In particular, we consider as state space the metric space X ⊂ R2 × H1 (0,L) ×
+L2 (0,L) with metric induced by the norm of the underlying normed linear
+space R2 × H1 (0,L) × L2 (0,L), i.e.,
+∥(ξ,w,h,v)∥X =
+�
+ξ2 + w2 + ∥h∥2
+2 +
+���h′���2
+2 + ∥v∥2
+2
+�1/2
+(45)
+However, we need to approximate the state space from its interior by using
+certain parameterized sets that allow us to obtain useful estimates. We define
+XV (r) := {(ξ,w,h,v) ∈ S : V (ξ,w,h,v) ≤ r }, for r ≥ 0
+(46)
+Inequalities (44) imply that
+XV (r) ⊆ X, for all r ∈ [0,R)
+(47)
+As indicated by the following proposition the set XV (r) for r > 0 contains a
+neighborhood of
+�
+0,0,h∗χ[0,L],0
+�
+(in the topology of X with metric induced by
+the norm ∥ ∥X defined by (45)).
+Proposition 1. Let constants q,k,δ > 0 be given. Then for every (ξ,w,h,v) ∈ S
+satisfying the inequality
+���(0,w,h − h∗χ[0,L],v)
+���X ≤ ε
+(48)
+for some ε > 0 with
+ε < min(h∗,Hmax − h∗)/
+√
+L,
+(49)
+the following inequality holds:
+V (ξ,w,h,v) ≤ C1
+���(ξ,w,h − h∗χ[0,L],v)
+���2
+X + C2
+���(ξ,w,h − h∗χ[0,L],v)
+���X
+(50)
+where
+C1 := max
+
+
+µ2
+h∗ − ε
+√
+L
+, δ + 1
+2
+g, (δ + 2)Hmax
+2
+,q, 3qk2
+2
+
+,
+(51)
+C2 := σ(δ + 1)
+√
+L
+(52)
+and ∥·∥X is defined by (45).
+3.3
+Stabilization results
+The following theorem guarantees exponential stabilization of the state of the
+system (8)-(12), (14), (16) by means of the nonlinear feedback law (55).
+11
+
+Theorem 1 (Stabilization of the Tank-Liquid System).
+Let arbitrary constants ω,k,q,δ > 0 be given and define R by means of (36). Let
+arbitrary r ∈ [0,R) be given and assume that
+k < qθ(r)
+(53)
+where
+θ(r) :=
+ωgµδπ2Q1(r)
+gµδπ2Q1(r) + 2ωL(mgLHmax(δ + 1)2 + 2µ2δπ2Q1(r))
+(54)
+where Q1 is defined by (28). Then there exist constants M,λ > 0 with the following
+property:
+(P) Every classical solution of the system (8)-(12), (14), (16) and
+f (t) = −ω
+�
+(δ + 1)
+� L
+0
+h(t,x)v(t,x)dx + µ(h(t,L) − h(t,0)) − q(w(t) + kξ(t))
+�
+,
+for t > 0
+(55)
+with (ξ(0),w(0),h[0],v[0]) ∈ XV (r), satisfies (ξ(t),w(t),h[t], v[t]) ∈ XV (r) and the
+following estimate for t ≥ 0:
+����
+�
+ξ(t),w(t),h[t] − h∗χ[0,L],v[t]
+�����X
+≤ M exp(−λt)
+����
+�
+ξ(0),w(0),h[0] − h∗χ[0,1],v[0]
+�����X
+(56)
+Remarks on Theorem 1.
+1) The arbitrary quantities ω,k,q,δ > 0 are the control parameters. We should
+point out that the ratio k/q must be sufficiently small due to (53), and this is
+the only restriction for the control parameters.
+2) The set XV (r) is the set for which exponential stabilization is achieved. As
+indicated by Proposition 1, the set XV (r) for r > 0 contains a neighborhood
+of
+�
+0,0,h∗χ[0,L],0
+�
+(in the topology of X with metric induced by the norm ∥ ∥X
+defined by (45)). The size of the set XV (r) depends on r ∈ [0,R) and on δ,q,k
+(recall (36) and (22)). It is straightforward to see that the larger the parameter
+q (or k) the smaller the set XV (r). However, the dependence of XV (r) on δ
+(through the dependence of R on δ) is not clear. On the contrary it is a very
+complicated, non-monotonic dependence.
+3) The feedback law (55) only requires the measurement of the four following
+quantities:
+• the position of the tank denoted by ξ(t), and the velocity of the tank
+denoted by w(t),
+• the total momentum of the liquid, i.e., the quantity
+� L
+0
+h(t,x)v(t,x)dx,
+and
+12
+
+• the difference the liquid level at the tank walls, i.e., the quantity h(t,L) −
+h(t,0).
+It should be emphasized that the feedback law (55) does not require the mea-
+surement of the whole liquid level and liquid velocity profile whereas it is
+completely independent of the surface tension coefficient.
+4) The feedback law (55) is the same feedback law that was used in [23, 25].
+When the results in [23,25] and Theorem 1 are taken into account then it fol-
+lows that the feedback law (55) guarantees robustness with respect to surface
+tension as well as robustness with respect to wall friction forces. From a con-
+trol point of view, this is an ideal situation: the feedback law (55) is robust
+with respect to all possible perturbations of the basic model, its measurement
+requirements are minimal and guarantees exponential stabilization of the cor-
+responding closed-loop (nonlinear; not the linearized) system.
+5) In contrast with [25], Theorem 1 does not provide an estimate for the norm
+∥vx[t]∥2, and consequently an estimate for the sup-norm of the fluid velocity.
+A topic for future research is the contruction of an appropriate CLF based on
+which an estimate for the norm ∥vx[t]∥2 can be obtained.
+4
+Proofs
+Proof of Lemma 1. The proof is exactly the same with the proof of Lemma 1
+in [25]. The only difference is that here we can obtain an additional estimate for
+���h − h∗χ[0,L]
+���∞. Indeed, due to the fact that the function ϕ : R+ → R+defined
+by
+ϕ(s) =
+√
+s2 + 1 − 1, for s ≥ 0
+(57)
+is increasing and convex, we can use Jensen’s inequality (see page 120 in [9])
+and get for all h ∈ C0 ([0,L];(0,+∞)) ∩ H1(0,L) with
+� L
+0
+h(x)dx = m:
+ϕ
+�1
+L
+���h′���1
+�
+= ϕ
+�1
+L
+� L
+0
+���h′(x)
+���dx
+�
+≤ 1
+L
+� L
+0
+ϕ
+����h′(x)
+���
+�
+dx = 1
+L
+� L
+0
+� �
+(h′(x))2 + 1 − 1
+�
+dx
+(58)
+Using (58), the inequality
+���h − h∗χ[0,L]
+���∞ ≤ ∥h′∥1 (which is a direct consequence
+of the fact that there exists x∗ ∈ [0,L] such that h(x∗) = h∗; a consequence of
+continuity of h, the mean value theorem and the facts that
+� L
+0
+h(x)dx = m, h∗ =
+m/L), the fact that the function ϕ−1 : R+ → R+ (the inverse function of ϕ) is
+increasing with ϕ−1(s) =
+�
+(s + 1)2 − 1 for s ≥ 0 and the inequality
+� L
+0
+� �
+(h′(x))2 + 1 − 1
+�
+dx ≤ V (ξ,w,h,v)
+σ(δ + 1)
+(59)
+13
+
+which is a direct consequence of definitions (22), (23), (24), we get for all
+(ξ,w,h,v) ∈ S:
+���h − h∗χ[0,L]
+���∞ ≤
+��
+L + V (ξ,w,h,v)
+σ(δ + 1)
+�2
+− L2
+(60)
+Using the additional estimate (60) in conjunction with the estimates shown in
+the proof of Lemma 1 in [25] and definitions (26), (28) and (29) we get (27) .
+The proof is complete.
+□
+Proof of Proposition 1. Consider arbitrary (ξ, w,h,v) ∈ S satisfying (48) and (49).
+Definitions (22), (23), (24) and the inequalities
+(h(x)v(x) + µh′(x))2 ≤ 2h2(x)v2(x) + 2µ2 (h′(x))2 ,
+(61)
+(w + kξ)2 ≤ 2w2 + 2k2ξ2,
+(62)
+�
+1 + (h′(x))2 − 1 ≤
+���h′(x)
+���
+(63)
+imply:
+V (ξ,w,h,v) ≤ δ + 2
+2
+� L
+0
+h(x)v2(x)dx + µ2
+� L
+0
+h−1(x)(h′(x))2 dx
++δ + 1
+2
+g
+���h − h∗χ[0,L]
+���2
+2 + 3qk2
+2
+ξ2 + qw2 + σ(δ + 1)
+���h′���1
+(64)
+Following the arguments of the proof of Proposition 2.5 in [25] we obtain from
+(64) the following:
+V (ξ,w,h,v) ≤ δ + 2
+2
+Hmax ∥v∥2
+2 + qw2 + 3qk2
+2
+ξ2
++µ2 �
+h∗ − ε
+√
+L
+�−1 ���h′���2
+2 + δ + 1
+2
+g
+���h − h∗χ[0,L]
+���2
+2 + σ(δ + 1)
+√
+L
+���h′���2
+(65)
+Inequality (50) is a direct consequence of (65) and definition (45). The proof is
+complete.
+□
+In order to give the proof of the main result of this study, we need to provide
+some preliminary lemmas along with their proofs.
+Lemma 2. Every classical solution of the system (8)-(12), (14), (16) satisfies the
+following equations for all t > 0:
+d
+dt E(h[t],v[t]) = −µ
+� L
+0
+h(t,x)v2
+x(t,x)dx + f (t)
+� L
+0
+h(t,x)v(t,x)dx
+(66)
+14
+
+d
+dt W(h[t],v[t]) = −µg ∥hx[t]∥2
+2 − µσ
+� L
+0
+h2xx(t,x)dx
+�
+1 + h2x(t,x)
+�3/2
++f (t)
+� L
+0
+(h(t,x)v(t,x) + µhx(t,x))dx
+(67)
+where E,W are defined by (23), (24), respectively.
+Proof. Due to (10) and (11) we get for t > 0, x ∈ (0,L):
+vt(t,x) + v(t,x)vx(t,x) + ghx(t,x)
+= σh−1(t,x)
+
+
+1 + h2x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
++µh−1(t,x)(h(t,x)vx(t,x))x + f (t)
+(68)
+Combining definition (23), (10) and (68) we get for all t > 0 the following ex-
+pression for the time derivative of the functional (23) :
+d
+dt E(h[t],v[t]) = −1
+2
+� L
+0
+(h(t,x)v(t,x))xv2(t,x)dx
+−
+� L
+0
+h(t,x)v2(t,x)vx(t,x)dx − g
+� L
+0
+h(t,x)v(t,x)hx(t,x)dx
++σ
+� L
+0
+v(t,x)
+
+
+1 + h2
+x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+dx
++µ
+� L
+0
+v(t,x)(h(t,x)vx(t,x))x dx + f (t)
+� L
+0
+h(t,x)v(t,x)dx
+−g
+� L
+0
+(h(t,x)v(t,x))x(h(t,x) − h∗)dx
+−σ
+� L
+0
+hx(t,x)
+�
+1 + h2x(t,x)
+(h(t,x)v(t,x))xxdx
+(69)
+Using (69), integration by parts as in the proof of Lemma 2.11 in [25], (12),
+15
+
+(16) and the fact that for all t > 0
+σ
+� L
+0
+v(t,x)
+
+
+1 + h2x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+dx
+= −σ
+� L
+0
+vx(t,x)1 + h2
+x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+dx
+(70)
+− σ
+� L
+0
+hx(t,x)
+�
+1 + h2x(t,x)
+(h(t,x)v(t,x))xxdx
+= σ
+� L
+0
+vx(t,x)1 + h2x(t,x) + hxx(t,x)h(t,x)
+�
+1 + h2x(t,x)
+�3/2
+dx
+(71)
+as a consequence of integration by parts as well, we obtain equation (66).
+Next we define for all t ≥ 0 and x ∈ [0,L]:
+ϕ(t,x) := h(t,x)v(t,x) + µhx(t,x)
+(72)
+Definition (72), (10) and (11) imply for all t > 0 and x ∈ (0,L):
+ϕt(t,x) = −
+
+v(t,x)ϕ(t,x) + 1
+2gh2(t,x) − σ 1 + h2
+x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
++h(t,x)f (t)
+(73)
+Using definition (24) along with (73) and (10), we get for all t > 0 :
+d
+dt W(h[t],v[t]) = 1
+2
+� L
+0
+h−2(t,x)ϕ2(t,x)(h(t,x)v(t,x))xdx
+−
+� L
+0
+h−1(t,x)ϕ(t,x)
+�
+ϕ(t,x)v(t,x) + 1
+2gh2(t,x)
+�
+x
+dx
++σ
+� L
+0
+h−1(t,x)ϕ(t,x)
+
+
+1 + h2x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+dx
++f (t)
+� L
+0
+ϕ(t,x)dx − g
+� L
+0
+(h(t,x) − h∗)(h(t,x)v(t,x))xdx
+−σ
+� L
+0
+hx(t,x)(h(t,x)v(t,x))xx
+�
+1 + h2x(t,x)
+dx
+(74)
+Using (12) and integration by parts as in proof of Lemma 2.11 in [25], we obtain
+16
+
+from (74) and definition (72) for all t > 0:
+d
+dt W(h[t],v[t]) = −µg ∥hx[t]∥2
+2 + f (t)
+� L
+0
+(h(t,x)v(t,x) + µhx(t,x))dx
++σ
+� L
+0
+v(t,x)
+
+
+1 + h2x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+dx
++µσ
+� L
+0
+h−1(t,x)hx(t,x)
+
+
+1 + h2x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+dx
+−σ
+� L
+0
+hx(t,x)(h(t,x)v(t,x))xx
+�
+1 + h2x(t,x)
+dx
+(75)
+Using (16), (70), (71) and the fact that
+h(t,x)
+
+
+hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+=
+
+
+1 + h2
+x(t,x) + h(t,x)hxx(t,x)
+�
+1 + h2x(t,x)
+�3/2
+
+
+x
+(76)
+we obtain from (75) equation (67) for all t > 0. The proof is complete.
+□
+Lemma 3. Let constants q,k,δ > 0 be given. Then there exists a non-decreasing
+function Λ : [0,R) → (0,+∞), where R > 0 is defined by (36) such that for every
+(ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) < R, the following
+inequality holds:
+V (ξ,w,h,v)
+Λ(V (ξ,w,h,v)) ≤
+���h′���2
+2 +
+� L
+0
+(h′′(x))2
+�
+1 + (h′(x))2�3/2 dx
++
+� L
+0
+h(x)(v′(x))2 dx + ξ2 + (w + kξ)2
+(77)
+Proof. Let arbitrary (ξ,w,h,v) ∈ X with v ∈ H1(0,L), h ∈ H2(0,L) and V (ξ,w,h,v) <
+R be given. Using the same arguments as in the proof of Lemma 2.12 in [25]
+and the fact that
+� L
+0
+� �
+1 + (h′(x))2 − 1
+�
+dx ≤
+���h′���2
+2
+(78)
+17
+
+we obtain the following estimate:
+V (ξ,w,h,v) ≤ L2 (δ + 2)Q2(V (ξ,w,h,v))
+2π2Q1(V (ξ,w,h,v))
+� L
+0
+h(x)(v′(x))2 dx
++
+
+
+(δ + 1)
+�
+gL2 + 2σ
+�
+2
++
+µ2
+Q1 (V (ξ,w,h,v))
+
+
+���h′���2
+2 + qk2
+2 ξ2 + q
+2 (w + kξ)2
+≤ Λ(V (ξ,w,h,v))
+×
+����h′���2
+2 +
+� L
+0
+(h′′(x))2
+�
+1 + (h′(x))2�3/2 dx +
+� L
+0
+h(x)(v′(x))2 dx + ξ2 + (w + kξ)2
+�
+(79)
+where
+Λ(s) := 1
+2 max
+�
+κ1 + 2µ2
+Q1 (s), κ2Q2(s)
+Q1(s) ,κ3
+�
+, for s ∈ [0,R)
+(80)
+with κ1 := (δ + 1)
+�
+gL2 + 2σ
+�
+, κ2 := L2 (δ + 2)/π2 and κ3 := qmax(1,k2). Defini-
+tion (80) and the fact that Q2 : R+ → R is an increasing function and Q1 : R+ →
+R is a decreasing function imply that Λ : [0,R) → (0,+∞) is a non-decreasing
+function. Inequality (77) holds as a direct consequence of (79). The proof is
+complete.
+□
+Lemma 4. Let constants q,k,δ > 0 be given. Then there exist non-decreasing func-
+tions Gi : [0,R) → (0,+∞), i = 1,2, where R > 0 is defined by (36), such that for
+every (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R, the following inequalities hold:
+���(ξ,w,h − h∗χ[0,L],v)
+���2
+X ≤ V (ξ,w,h,v)G1 (V (ξ,w,h,v))
+(81)
+V (ξ,w,h,v)
+G2 (V (ξ,w,h,v)) ≤
+���(ξ,w,h �� h∗χ[0,L],v)
+���2
+X
+(82)
+where ∥·∥X is defined by (45).
+Proof. Let arbitrary (ξ,w,h,v) ∈ X with V (ξ,w,h,v) < R be given. Using defini-
+tions (22), (23), (24), inequalities (61), (62), the inequality
+�
+1 + (h′(x))2 ≤ 1 + (h′(x))2
+(83)
+and (44) we obtain
+V (ξ,w,h,v) ≤ δ + 2
+2
+Hmax ∥v∥2
+2 + δ + 1
+2
+g
+���h − h∗χ[0,L]
+���2
+2
++
+�
+µ2
+Q1 (V (ξ,w,h,v)) + σ (δ + 1)
+����h′���2
+2 + 3qk2
+2
+ξ2 + qw2
+(84)
+18
+
+Inequality (84) implies inequality (82) with
+G2 (s) := max
+�δ + 2
+2
+Hmax, δ + 1
+2
+g,
+µ2
+Q1 (s) + σ (δ + 1), 3qk2
+2
+,q
+�
+,
+for s ∈ [0,R)
+(85)
+The fact that Q1 : R+ → R is a decreasing function and the above definition
+imply that G2 : [0,R) → (0,+∞) is a non-decreasing function.
+The proof of inequality (81) is exactly the same with the proof of Lemma 4
+in [25]. The proof is complete.
+□
+Lemma 5. Let constants ω,k,q,δ > 0 and r ∈ [0,R) be given, where R > 0 is defined
+by (36). Then every classical solution of the system (8)-(12), (14), (16) and (55)
+satisfies the following inequality for all t > 0 for which V (ξ(t),w(t),h[t],v[t]) < R:
+d
+dt V (ξ(t),w(t),h[t],v[t]) ≤ −3µg
+4 ∥hx[t]∥2
+2 − qk3ξ2(t)
+−
+µδ
+2Hmax
+�
+2Hmax − Q1(r)Q2 (V (t))
+Q1 (V (t))
+�� L
+0
+h(t,x)v2
+x(t,x)dx
+−µσ
+� L
+0
+h2xx(t,x)
+�
+1 + h2x(t,x)
+�3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2
+(86)
+where V (t) = V (ξ(t),w(t),h[t],v[t]), θ(r) is defined by (54) and Qi : R+ → R (i =
+1,2) are the functions defined by (28) and (29).
+Proof. Let ω,k,q,δ > 0 be given constants and let r ∈ [0,R) be a constant, where
+R > 0 is defined by (36). In addition to that we consider a classical solution of
+the system (8)-(12), (14), (16) and (55) at a time t > 0 for which V (ξ(t),w(t),h[t],v[t]) <
+R. Using Lemma 2, (66), (67) and definition (22) and by following the same
+procedure as in the proof of Lemma 2.14 in [25] by assuming zero friction
+coefficient, we establish the following inequality:
+d
+dt V (ξ(t),w(t),h[t],v[t]) ≤ −3µg
+4
+∥hx[t]∥2
+2 − µδ
+� L
+0
+h(t,x)v2
+x(t,x)dx
+−µσ
+� L
+0
+h2xx(t,x)
+�
+1 + h2x(t,x)
+�3/2 dx − q(qθ(r) − k)(w(t) + kξ(t))2 − qk3ξ2(t)
++µδπ2Q1(r)
+2L2Hmax
+� L
+0
+h(t,x)v2(t,x)dx
+(87)
+Since v(t,0) = v(t,L) = 0 (recall (12)), by virtue of Wirtinger’s inequality and
+(44), we get:
+∥v[t]∥2
+2 ≤ L2
+π2 ∥vx[t]∥2
+2 ≤
+L2
+π2Q1 (V (t))
+� L
+0
+h(t,x)v2
+x(t,x)dx
+(88)
+Combining (44), (87) and (88), we obtain (86). The proof is complete.
+□
+19
+
+We can now present the proof of Theorem 1.
+Proof of Theorem 1. Let constants ω,q,k,δ > 0 satisfying (53). Let constant r ∈
+[0,R) be given. Consider a classical solution of the system (8)-(12), (14), (16)
+with (55) that satisfies V (ξ(0),w(0),h[0],v[0]) ≤ r. Let r ∈ (r,R) be a constant
+that satisfies:
+Q2 (r)
+Q1 (r) < 2Hmax
+Q1(r)
+(89)
+The existence of ¯r ∈ (r,R) is a direct consequence of the continuity of the func-
+tions involved in (89).
+Due to (53), Lemma 5, (86) and (89) the following implication is true:
+If t > 0 and V (ξ(t),w(t),h[t],v[t]) ≤ r then d
+d t V (ξ(t),w(t),h[t],v[t]) ≤ 0
+(90)
+A contradiction argument as in the proof of Theorem 2.6 in [25] implies that
+V (ξ(t),w(t), h[t],v[t]) ≤ r for all t ≥ 0.
+Implication (90) and the fact V (ξ(t),w(t),h[t],v[t]) ≤ r for all t ≥ 0 imply
+that
+d
+d t V (ξ(t),w(t),h[t],v[t]) ≤ 0 for all t > 0
+(91)
+Due to the above and the continuity of the mapping t → V (ξ(t),w(t),h[t], v[t]),
+we get that
+V (ξ(t),w(t),h[t],v[t]) ≤ V (ξ(0),w(0),h[0],v[0]) ≤ r < R,for all t ≥ 0
+(92)
+Consequently, (ξ(t),w(t),h[t],v[t]) ∈ XV (r) for all t ≥ 0 (recall (46)). Using (92)
+and Lemma 5, we conclude that (86) holds for all t > 0. Using (92), (86) and
+the fact that Q2 : R+ → R is an increasing function while Q1 : R+ → R is a
+decreasing function, we obtain the following estimate for t > 0
+d
+dt V (ξ(t),w(t),h[t],v[t])
+≤ −β(r)
+�
+∥hx[t]∥2
+2 +
+� L
+0
+h(t,x)v2
+x(t,x)dx +
+� L
+0
+h2xx(t,x)
+�
+1 + h2x(t,x)
+�3/2 dx
++ξ2(t) + (w(t) + kξ(t))2
+�
+(93)
+where
+β(r) := min
+�3µg
+4 , µδ(2Hmax − Q2 (r))
+2Hmax
+,qk3,q(qθ(r) − k),µσ
+�
+(94)
+Notice that (53) and the fact that r ∈ [0,R) in conjunction with definitions (29),
+(36), (93) imply that β(r) > 0. It follows from Lemma 3, (77), the continuity
+of the mapping t → V (ξ(t),w(t),h[t],v[t]), (recall that v ∈ C0 (R+ ;H1 (0,L)
+�
+,
+20
+
+h ∈ C1 (R+ × [0,L];(0,+∞)) and v ∈ C0 (R+ × [0,L])), estimates (92), (93), Lemma
+4, (81) and (82) that the following estimate holds for all t ≥ 0:
+���(ξ(t),w(t),h[t] − h∗χ[0,L],v[t])
+���2
+X
+≤ Ω(r)exp
+�
+−β(r)t
+Λ(r)
+����(ξ(0),w(0),h[0] − h∗χ[0,L],v[0])
+���2
+X
+(95)
+with
+Ω(r) := G1 (r)G2 (r)
+(96)
+where Λ is the non-decreasing function involved in (77) and Gi : [0,R) →
+(0,+∞) (i = 1,2) are the non-decreasing functions involved in (81), (82). Es-
+timate (56) with M =
+�
+Ω(r) and λ = β(r)
+2Λ(r) is a consequence of estimate (95).
+The proof is complete.
+□
+5
+Concluding Remarks
+In this work we managed to show that the robust with respect to wall friction
+nonlinear feedback law proposed in [25] provides also robust stabilization re-
+sults with respect to surface tension. This shows even more the significance of
+the CLFs as stabilizing tools for the infinite-dimensional case of systems de-
+scribed by PDEs and illustrates the fact that robustness is inherent in the CLF
+methodology.
+The present study deals with the case of viscous Saint-Venant system with
+surface tension and without wall friction. It is of interest to study the more
+challenging problem of the viscous Saint-Venant system with surface tension
+and with wall friction as well as the construction of an additional functional
+which provides a bound for the sup-norm of the fluid velocity. In addition to
+that, other topics for future research are the study of existence and unique-
+ness of the solutions for the closed-loop system, the study of the problem with
+non constant (dynamic) contact angles, the study of the output feedback stabi-
+lization problem, the construction of appropriate numerical schemes and the
+derivation of stability estimates in stronger spatial norms. Concerning the out-
+put feedback stabilization problem there are many interesting studies in the
+literature that may contribute, such as [26] which suggests a finite-dimensional
+observer control of the (1-D) heat equation under Neumann actuation.
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+24
+

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@@ -0,0 +1,2578 @@
+Mon. Not. R. Astron. Soc. 000, 000–000 (0000)
+Printed 10 January 2023
+(MN LATEX style file v2.2)
+Thermal hysteresis and front propagation in dense
+planetary rings
+R´emy Larue1,2,3, Henrik Latter1⋆, Hanno Rein4,5
+1DAMTP, University of Cambridge, CMS, Wilberforce Road, Cambridge CB3 0WA, UK
+2ENS Paris-Saclay, 4 avenue des Sciences 91190 Gif-sur-Yvette, France
+3Laboratoire de Physique Subatomique et de Cosmologie, Universit´e Grenoble-Alpes, CNRS/IN2P3, Grenoble INP, 38000 Grenoble, France
+4Department of Physical and Environmental Sciences, University of Toronto at Scarborough, Toronto, Ontario M1C 1A4, Canada
+5David A. Dunlap Department of Astronomy and Astrophysics, University of Toronto, Toronto, Ontario, M5S 3H4, Canada
+ABSTRACT
+Saturn’s rings are composed of icy grains, most in the mm to m size ranges, under-
+going several collisions per orbit. Their collective behaviour generates a remarkable
+array of structure over many orders of magnitude, much of it not well understood.
+On the other hand, the collisional properties and parameters of individual ring par-
+ticles are poorly constrained; usually N-body simulations and kinetic theory employ
+hard-sphere models with a coefficient of restitution ϵ that is constant or a decreasing
+function of impact speed. Due to plastic deformation of surface regolith, however, it
+is likely that ϵ will be more complicated, at the very least a non-monotonic function.
+We undertake N-body simulations with the REBOUND code with non-monotonic ϵ
+laws to approximate surfaces that are friable but not sticking. Our simulations reveal
+that such ring models can support two thermally stable steady states for the same
+(dynamical) optical depth: a cold and a warm state. If the ring breaks up into radial
+bands of one or the other state, we find that warmer states tend to migrate into the
+colder states via a coherent travelling front. We also find stationary ‘viscous’ fronts,
+which connect states of different optical depth, but the same angular momentum flux.
+We discuss these preliminary results and speculate on their implications for structure
+formation in Saturn’s B and C-rings, especially with respect to structures that appear
+in Cassini images but not in occultations.
+Key words: instabilities – waves – planets and satellites: rings
+1
+INTRODUCTION
+Saturn’s rings flaunt an extraordinary array of axisymmetric
+structure, both quasi-regular and chaotic, ranging over some
+four orders of magnitude in length - from 10 m to 100 km
+(Colwell et al. 2009, Cuzzi et al. 2018). Yet despite several
+decades of theoretical effort, their origins are only partially
+understood (Schmidt et al. 2009, Estrada et al. 2018, Salo et
+al. 2018). In particular, the disjunct bands of high and low
+optical depth in the B-ring (Horn and Cuzzi 1996, Colwell et
+al. 2007), the plateaus in the C-ring (Tiscareno et al. 2019),
+and the irregular intermediate scale striations in the A and
+B-rings (Porco et al. 2005) are presently without plausible
+explanations. Simply put, there is too much observed struc-
+ture and too few suitable instabilities (or related processes)
+in our theoretical models. Perhaps it is time to re-assess
+⋆ E-mail: [email protected]
+some of our fundamental assumptions and explore a wider
+range of alternative scenarios.
+It is probable, though not assured, that much of the
+ring’s unexplained structure arises spontaneously due to
+its peculiar granular flow. Since the 1980s researchers have
+turned to kinetic theory or N-body simulations to model
+this flow, initially calculating the thermal balances under-
+lying ring equilibria, and then the (viscous) instabilities
+that might generate structure (e.g., H¨ameen-Anttila 1982,
+Araki & Tremaine 1986, Wisdom & Tremaine 1988, Salo
+1991, H¨ameen-Anttila & Salo 1993, Salo et al. 2001, Lat-
+ter & Ogilve 2006, 2008). These studies have made several
+strong assumptions, especially regarding the nature of the
+ring particles and their collisional behaviour, for instance
+rarely deviating from a hard-sphere model with either a con-
+stant coefficient of restitution ϵ or a ‘Bridges law’ (Bridges
+et al. 1984), whereby collisions below some critical impact
+speed are perfectly elastic. In reality, ring particles are likely
+to be irregularly shaped and coated in a regolith of small par-
+© 0000 RAS
+arXiv:2301.03289v1  [astro-ph.EP]  9 Jan 2023
+
+2
+Larue, Latter, Rein
+ticles ≲ 1 cm (e.g. Doyle et al.1989, Nicholson et al. 2008,
+Morishima et al. 2012; Deau 2015) and, being irregular and
+fluffy, their surfaces should produce an enhanced inelasticity
+at low impact speeds, and indeed possible particle adhesion.
+In light of this, the adoption of a constant ϵ, or a Bridges law,
+may significantly misrepresent some of the ring’s collective
+collisional dynamics. Our paper tests this idea by exploring
+other, physically motivated, prescriptions for ϵ. We find, in
+fact, that even very simple changes to the collision law can
+give remarkably different outcomes.
+Continuum mechanical models of viscoelastic collisions
+that account for fluffy and/or sticky surfaces demonstrate
+that ϵ is a non-monotonic function of impact speed vcoll.
+Beneath some critical speed we have ϵ = 0, but on in-
+creasing vcoll, ϵ rises, plateaus, and then decreases again
+(Gorkavyi 1985, Hertzsch 2002, Albers & Spahn 2006, Bril-
+liantov et al. 2007). Laboratory experiments appear to con-
+firm this picture (Gorkavyi 1989, Hatzes et al. 1991, Bridges
+et al. 1996). We implement collision laws of this basic form
+in our paper and term them ‘regolith laws’. In addition,
+at or below the critical speed colliding particles may stick,
+but we neglect this important effect in order to avoid the
+vexed and complicated issue of size-distribution dynamics
+(e.g. Brilliantov et al. 2015). Our approach is mainly nu-
+merical, via N-body simulations of monodisperse, spherical,
+indestructible particles with the code REBOUND; but we
+also employ a dense gas kinetic theory, where appropriate.
+Note that we do not include self-gravity and thus our sim-
+ulations fail to exhibit wakes, nor do they support viscous
+overstability, both important phenomena we hope to test
+in the future. Our study is distinct but complementary to
+recent N-body simulations that explicitly test the role of
+adhesion, especially on instabilities (Ballouz et al. 2017, Lu
+et al. 2018; see also Section 16.7.1.7 in Salo et al. 2018). Our
+main focus, in contrast, will be on disk thermodynamics.
+Our first main result is that regolith laws permit a dense
+ring to fall into one of two thermally stable states at the
+same optical depth: (a) a very dense state with filling fac-
+tors ∼ 0.3 and low temperatures, c ≲ aΩ (where c is velocity
+dispersion, a is particle radius, and Ω is orbital frequency)
+and (b) a moderately dense state with lower filling factors
+(≲ 0.1) and a slightly warmer temperature, c ≳ 4aΩ. This
+bistability generally favours optical depths less than 1, but
+can be pushed up to higher values if we broaden our parame-
+ter range. We also find in certain circumstance that the cold
+state at low optical depth is metastable: shot noise permits
+the ring to spontaneously jump into the hot state.
+Our second set of results explores what happens when
+different thermal states spatially adjoin. If two states of
+the same optical depth but different temperature connect, a
+travelling ‘thermal front’ develops that can reach speeds of
+≲ aΩ, while maintaining a steady spatial structure. If the
+front is too slow, the disparity in the angular momentum
+flux between the two states reorganises the front profile so
+that the flux is uniform but the optical depth undergoes a
+jump, what we term a static ‘viscous front’. Some of the
+latter behavior mirrors that witnessed by Salo and Schmidt
+(2010) in their simulations of viscous instability.
+The plan of the paper is as follows. The next section
+begins with a review of the extant literature on low-impact
+collisions between regolith covered and/or sticky particles,
+moving on to a presentation of the model collision laws we
+use, and then our numerical methods. Subsequently, we de-
+tail out results: the calculation of thermal equilibria and hys-
+teresis in smallish boxes (Section 3), potential metastability
+(Section 4), and finally results on spatially adjoining states,
+i.e. thermal and viscous front (Section 5). We conclude in
+Section 6.
+2
+BACKGROUND AND METHODS
+This section presents the physical set-up and numerical
+model by which we attack the thermal equilibria of rings
+composed of regolith-coated particles. We first devote some
+space to set the scene, by reviewing the theoretical and ex-
+perimental literature and explaining the key ideas and pa-
+rameters that underlie work in this area. The model collision
+laws we adopt are then exhibited, followed by the details of
+the N-body simulations with REBOUND we conduct.
+2.1
+Collisional physics and the coefficient of
+restitution
+We aim to describe the collisional dynamics of many ring
+particles in a local patch of a planetary ring. From the outset
+we make several strong assumptions that we concede may
+distort our results: the particles are taken to be identical,
+spherical, and frictionless. Most of the ring mass is in metre-
+sized particles, and thus it is that population that we track.
+Only binary collisions are considered, and these are deemed
+inelastic, so that g′ · k = −ϵ(g · k), where g is the relative
+velocity of two colliding particles before the collision and g′
+afterwards, k is the unit vector pointing between the two
+particles centres at the moment of collision, and ϵ is the
+coefficient of restitution. This coefficient lies between 0 and
+1 and is usually a function of the impact speed vcoll = |g ·
+k|. We neglect the possibility of two particles sticking and
+assume that all the specifics of the particle surfaces can be
+encapsulated in the functional behaviour of ϵ. Because we
+find the ring dynamics are so sensitive to ϵ, we now spend
+some time discussing this important physical input.
+2.1.1
+Theoretical and experimental background
+Research exploring the collisional behaviour of regolith-
+covered particles can be separated into analytical calcula-
+tions, drawing on continuum mechanics, and laboratory ex-
+periments, approximating Saturnian conditions. We attempt
+to review and synthesise this body of work.
+The seminal experiments in this area were described
+in Bridges et al. (1984) and collided smooth ice spheres
+with an ice block at temperatures ∼ 170K. This work pro-
+duced the collision law ϵ = min
+�
+1, (vcoll/vcrit)−0.234�
+, for
+vcrit = 0.008 cm s−1, a defining feature of which is perfect
+elasticity at sufficiently low collision speeds (vcoll < vcrit).
+This collision law became the standard for subsequent N-
+body simulations and other theoretical work. Subsequently,
+broken power laws of this type were shown to arise naturally
+in generalisations of the Hertz theory to viscoelastic solids
+(Dilley 1993, Hertzsch et al. 1995, Brilliantov et al. 1996,
+Thornton 1997). However, such theoretical work must posit
+that the surfaces of the colliding spheres are smooth and
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+3
+that irreversible energy losses arise solely from viscoelastic
+deformations inside the spheres.
+Shortly after the Bridges experiments, two neglected
+but insightful papers by Gorkavyi (1985, 1989) highlighted
+the importance of regolith and argued against perfectly elas-
+tic restitution at low impact speed. Gorkavyi emphasised
+that ϵ can be dramatically altered at small vcoll because (a)
+impact energy can be used up when reshaping a soft fri-
+able surface (leaving nothing left over for elastic rebound)
+and/or (b) rebounding motion can be countered by surface
+stickiness. Using energy arguments, the 1985 paper sketches
+out three regimes: (a) at sufficiently low vcoll, there is total
+energy loss and thus ϵ = 0 (sticking/adhesion is not consid-
+ered); (b) at slightly larger vcoll, ϵ increases with vcoll; and
+then (c) after a turning point, ϵ decreases with vcoll (tradi-
+tional restitution). The collision law is hence non-monotonic.
+Gorkavyi (1989) followed this up with simple experiments
+using powders, metals, and marble at room temperature and
+pressure, which agree with earlier lab work by Hartmann
+(1978, 1985), in a different context, using rocks.
+Subsequent papers from the Bridges research group ex-
+amined how the state of the particle surface influenced colli-
+sions, with a particular focus on the adhesive effect of frost,
+a thin layer of microscopic structure that might behave simi-
+larly to the thicker regolith layer expected on larger ring par-
+ticles. Hatzes et al. (1991) showed frosty particles can stick
+at impact speeds below some critical level (a few mm s−1),
+but did not examine explicitly how it changed the form of
+ϵ. Bridges et al. (1996) conducted a large set of experiments
+for different kinds of ices and vcoll at relevant temperatures,
+which further strengthened the case for sticking, and also
+showed that ϵ exhibited the three main features predicted
+by Gorkavyi.
+On the theoretical side, the 2000s witnessed various ex-
+tensions of Hertz contact mechanics, accounting for both
+viscoelasticity and particle adhesion via JRK theory (Albers
+and Spahn 2006, Brilliantov et al. 2007; see also Thornton
+and Ning 1998, and Chokshi et al. 1993, the latter in the
+context of ISM grains). Notable is the work by Hertzsch
+(2002) who modelled the two effects of sticking and of pas-
+sive regolith deformation, as discussed by Gorkavyi, treating
+the passive regolith as a deformable viscous non-sticky ‘soft
+layer’. Both physical effects appear to influence the form of
+ϵ similarly. In all cases, non-monotonic ϵ laws were mathe-
+matically derived.
+Brilliantov et al. (2007) provides estimates for solid
+water-ice particles of various sizes that, despite several
+strong assumptions, help with Saturnian applications. For
+metre-sized water-ice impactors, the theory predicts that the
+maximum value ϵ takes is relatively large, potentially above
+0.7. For cm sized particles, it drops to ≈ 0.3. On the other
+hand, the critical vcoll for sticking is roughly 10−2 cm s−1 for
+metre-sized ice impactors, and this rises to greater than 0.1
+cm s−1 for cm-sized particles. Because of the model assump-
+tions care must be taken, however, when applying these es-
+timates, and in fact the quoted critical collision speeds are
+probably gross lower limits. The theory omits the energy
+dissipation channel associated with irreversible regolith de-
+formation (as well as internal fracture) by treating the par-
+ticles as solid-ice non-spinning viscoelastic spheres. It also
+sets the unknown dissipative constant A by fitting a (non-
+sticking) viscoelastic model (Brilliantov et al. 1996) to the
+(non-sticking) experimental data of Bridges et al. Nonethe-
+less, the Brilliantov results provide a useful starting point
+for our study.
+Before moving on, we flag additional physics not yet
+discussed. In applying the above ideas and prescriptions to
+an ensemble of colliding particles, one must acknowledge
+that, by virtue of the collisions themselves, particles’ surface
+properties will evolve. Repeated collisions will presumably
+‘compactify’ particle regolith and hence reduce the mean
+critical sticking speed. On the other hand, bombardment by
+micrometeoroids will disturb the surfaces and there will be
+accretion of very small floating particles, processes that will
+rejuvenate regolith. It follows that, in addition to the size
+distribution dynamics (e.g. Longaretti 1989, Bodrova et al.
+2012, Brilliantov et al. 2015), there will take place related
+dynamics controlling the mean surface properties. We do
+not attempt to construct a model for this interesting process
+here.
+2.1.2
+Important scales
+This subsection briefly outlines the key velocity scales rel-
+evant for our problem. We assume that there is a single
+critical sticking speed vstick below which two impactors will
+adhere. We also assume a second critical impact speed vcrit
+below which ϵ = 0. It may be that these two speeds are the
+same, though in general we expect vstick < vcrit, i.e. it is
+possible for all the energy of the impact to be used up re-
+shaping the surface and resisting the adhesive attraction of
+the regolith, thereby allowing the impactors to roll clear of
+each other. Particle spin and tidal shear may facilitate such
+non-sticking ϵ = 0 encounters.
+A third key speed is the velocity dispersion c, as impact
+speeds will be distributed around it. Thus the relative size
+of c relative to vcrit will determine which collisional regime
+(sticking, non-sticking, etc.) the particles are in. Partly con-
+trolling c is the orbital shear speed across a particle, aΩ
+(recall a is particle radius and Ω the orbital frequency). The
+importance of this scale issues from the fact that dense cold
+rings adopt a velocity dispersion c ∼ aΩ, in the absence of
+gravity wakes, and c ≲ 5aΩ, when gravity wakes are present
+(e.g., Araki & Tremaine 1986, Salo et al. 2018)1. It follows
+that if c ∼ aΩ ≫ vcrit then the regolith is not going to fea-
+ture much in the mean thermal dynamics, and hence the
+determination of c. On the other hand, if c ∼ aΩ ≪ vcrit
+then the surface properties are going to be important. Com-
+plicating this picture, of course, is the size dependence of
+both aΩ and vcrit. In a polydisperse ring, however, the ve-
+locity dispersion of smaller particles will be similar to the
+metre-sized particles (Salo et al. 2018). We now obtain some
+bounds on the important parameter vcrit/(aΩ).
+First we situate ourselves at a representative location
+in the C-ring, in which gravity wakes are likely absent, and
+set Ω ≈ 10−4 s−1. If a = 1 m, the most dynamically im-
+portant size, aΩ is roughly 0.01 cm/s. Next, applying the
+estimates from Brilliantov et al. (2007) (cf. Section 2.1.1)
+1 The second estimate can be obtained by assuming a gravita-
+tionally unstable ring settles into a state where the Toomre Q is
+∼ 1, and then taking typical values for the surface density (e.g.
+Hedman & Nicholson 2013, 2016)
+© 0000 RAS, MNRAS 000, 000–000
+
+4
+Larue, Latter, Rein
+and setting vcrit = vstick, we obtain vcrit/(aΩ) ∼ 1. For cm
+sizes, vcrit/(aΩ) ≳ 10 (noting that the velocity dispersion of
+this population is set by the metre sizes). As argued earlier,
+the Brilliantov estimates for vcrit only provide lower bounds,
+and hence we conclude that it is likely that the C-ring is in
+a regime where surface regolith properties will matter.
+At a representative location in the A or B-ring, we must
+take into account gravity wakes. Thus we find ourselves in
+a more ambiguous situation: the Brilliantov estimates yield
+vcrit/c ≳ 0.1 for metre-sized particles, and vcrit/c ≳ 1 for
+cm-sized particles. Depending on how badly the Brilliantov
+results underestimate vcrit, we could be in a marginal regime
+or in a regolith-dominated regime. Certainly, further work
+on the collisional dynamics of ice would help decide on this
+point. As we do not simulate self-gravity, for now we just
+assume that aΩ < vcrit, and leave open its importance to
+future work.
+2.1.3
+Model coefficients of restitution
+This section presents the two classes of non-monotonic ‘re-
+golith’ ϵ-law we use in this paper. We have attempted to
+paramaterise these laws in two readily understandable quan-
+tities: vcrit, the impact speed at which collisions are perfectly
+inelastic (cf. Section 2.1.2); and ϵmax, the turning point value
+of ϵ (i.e., its maximum).
+A broken power law (BPL) for ϵ, though somewhat
+crude has the benefits that it has few input parameters and
+some headway can be made with it using kinetic theory. We
+define the law in the following way:
+ϵ(vcoll) =
+�
+ϵ0,
+if vcoll < vcrit.
+ϵmax (vcoll/vcrit)−p ,
+if vcoll ⩾ vcrit.
+(1)
+We set the exponent p
+=
+0.234, following Bridges et
+al. (1984), though it could take other values. The quantity
+ϵ0 we set equal to either 1, to obtain the Bridges et al. law
+itself, or equal to 0, to get the opposite perfectly inelastic
+law. The Bridges BPL is plotted in Fig. 1 in blue.
+A more realistic non-monotonic ϵ law that is smoother
+and exhibits something of a plateau near its maximum can
+be defined in several ways. We choose the following:
+ϵ(vcoll) =
+�
+0,
+if vcoll < vcrit
+1.625 ϵmax ζ/(1 + ζ1.234),
+vcoll ⩾ vcrit,
+(2)
+where ζ = (vcoll−vcrit)/b and b is the plateau ‘width’, usually
+set to aΩ. Constants have been chosen so that ϵ approaches
+the Bridges law for large vcoll. To facilitate the discussion
+later, when we compare the different models, we refer to
+Eq. (2) as a ‘realistic’ law (though it is yet to be determined
+how realistic it is). We plot it in Fig. 1 in red.
+2.2
+The potential for bistability
+Before presenting our numerical methods and the results
+that ensue, we briefly explain why a non-monotonic colli-
+sion law, such as given in Eq. (2) and displayed in Fig. 1,
+potentially yields two stable states for the same parameters.
+At lower optical depths, N-body simulations and ki-
+netic theory show that the Bridges law yields equilibria with
+c > aΩ, and thus most collisions sample the power-law de-
+creasing segment of the ϵ curve (Salo 1991, Latter & Ogilvie
+0
+1
+2
+3
+4
+5
+6
+vcoll / vcrit
+0
+0.2
+0.4
+0.6
+0.8
+1
+Figure 1. Two forms of the coefficient of restitution ϵ as a func-
+tion of impact speed vcoll. The solid blue curve is the Bridges
+law, see Eq. (1), with ϵ0 = 1. The red solid curve is the ‘regolith’
+law, Eq. (2), with b = (1/4)vcrit and ϵmax = 0.75. In addition,
+we have sketched two velocity distribution functions with black
+dotted curves; see discussion in Section 2.2.
+2008). As mentioned above, the realistic regolith law we
+adopt approaches the Bridges law for impact speeds larger
+than the turning point in ϵ, and is a reasonable approxi-
+mation near the turning point. One might then expect that
+collisions employing the regolith law would sample similar
+values of ϵ and the resulting thermal equilibria will resemble
+the Bridges equilibria, giving us a ‘warm’ ring. In Fig. 1 we
+superimpose a mock impact velocity distribution at larger
+vcoll to indicate such a state.
+On the other hand, when ϵ is a constant and taken to
+be equal to zero the thermal equilibria are especially cold,
+with c ∼ aΩ (e.g. Araki & Tremaine 1986). It follows that
+our regolith law might be capable of supporting these very
+cold equilibria as well. This should certainly be the case
+if vcrit is much larger than aΩ. In this circumstance, most
+impact speeds will fall below vcrit and thus yield perfectly
+inelastic collisions with ϵ = 0, never sampling the non-zero
+segment of the ϵ curve. Fig. 1 indicates a schematic velocity
+distribution for this state, centred on a value less than vcrit.
+Both the warm state and the cold state are thermally
+stable, as has been shown separately in N-body simulations.
+And thus a non-monotonic law may yield bistability. The
+disk may fall into either the cold or the warm homogeneous
+state for exactly the same parameters (most notably optical
+depth τ)2. Which is chosen depends on the initial conditions.
+Moreover, it follows there must also be an intermediate ther-
+mally unstable state separating the two stable states, though
+this will not normally be observed. The argument for bista-
+bility is strongest in a regime where vcrit ≫ aΩ. A question
+then is: what is the minimum value of vcrit that yields bista-
+2 This bistability is different to the ‘phase transitions’ associ-
+ated with viscous instability, which drives the system to a non-
+homogeneous state characterised by abutting radial regions of
+high and low optical depth (e.g., Lukkari 1981, H¨ameen-Anttila
+1982, Salo & Schmidt 2010).
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+5
+bility? Our simulations results in Section 3 aim to answer
+this and other questions.
+2.3
+N-body simulations
+In this subsection we further outline the physical model we
+adopt and the numerical methods used to calculate its non-
+trivial thermal dynamics. We seek to determine the evolu-
+tion of a large number of inelastically colliding particles, and
+thus our main tool will be local N-body simulations.
+2.3.1
+Equations of motion
+We solve the equations of motion in the Hill approximation
+(Hill 1878), a local coordinate system that is co-rotating
+with a particle on a circular orbit. The gravity from the cen-
+tral object is linearized in local coordinates and the orbital
+frequency is a constant. This allows, but does not restrict,
+us to use shear-periodic boundary conditions. In that case,
+the Hill approximation is also referred to as the shearing
+sheet. In our notation, the x, y, and z coordinates point in
+the radial, azimuthal and vertical direction, respectively.
+Treating the central object, Saturn, as a point source,
+the equations of motion for a test particle can be written as
+¨x = 2Ω ˙y + 3Ω2x + F coll
+x
+,
+(3)
+¨y = −2Ω ˙x + F coll
+y
+,
+(4)
+¨z = Ω2z + F coll
+z
+,
+(5)
+where Fcoll is the (intermittent) acceleration exerted on a
+particle during a collision. In the absence of collisions, the
+solution to these equations can be written as epicycles (e.g.
+Rein & Tremaine 2011).
+The particles move within a finite-size numerical do-
+main/box. We denote the radial length of the box by Lx
+and the azimuthal length by Ly. In all our experiments, the
+vertical length of the box Lz has been chosen to be large
+enough so that no particle ever crosses the vertical bound-
+aries. Otherwise, the box is periodic in y and shear-periodic
+in x.
+The only further ingredients needed are the finite par-
+ticle radius a and a collision model. We treat particles as
+hard spheres (they are not permitted to overlap) and the
+outcome of a collision is described using a normal coefficient
+of restitution, as described in Section 2.1.3. The particles
+have no spin.
+2.3.2
+Numerical method
+We use the freely available N-body code REBOUND (Rein
+& Liu 2012) to perform all of the simulations presented in
+the paper. To evolve the equations of motion forward in
+time, we use the Symplectic Epicycle Integrator (SEI, Rein
+& Tremaine 2011) which is well suited for simulations of
+particle motion within the Hill approximation.
+Collisions are detected using a nearest neighbour tree
+search. We randomize the order in which collisions are re-
+solved after each timestep. We found that this removes spu-
+rious correlations which might otherwise be introduced when
+choosing a specific order in which collisions are resolved (i.e.
+resolving them from left to right, by a numerical particle
+identifier, or by the position in memory).
+2.3.3
+Diagnostics
+In order to probe the collective behaviour of the granular
+flow, we require a number of averaged quantities. We define
+the mean normal geometrical optical depth τ as the total
+projected area of the particles on the (x, y) plane divided by
+the total area of the (x, y) plane. In other words,
+τ = Nπa2/(LxLy),
+(6)
+where N is the number of particles. Thus, τ is stipulated
+at the beginning of each run and does not change. We also
+define the radially and temporally varying optical depths,
+by subdividing the radial domain into thin strips of radial
+length LS:
+τ(xi, t) = Ni(t)πa2/(LSLy),
+(7)
+where xi is the radial location of, and Ni(t) is the number
+of particles in, the i’th strip at time t.
+The filling factor is defined as the proportion of volume
+taken up by the particles. For spherical particles it can be
+defined as FF = (4π/3)na3, where n is volumetric number
+density. Particularly useful is the filling factor at the mid-
+plane FF0, which requires the calculation of the number
+density at z = 0.
+The mean velocity dispersion tensor is computed via
+Wij = ⟨ ˙xi ˙xj⟩
+(8)
+where ( ˙x1, ˙x2, ˙x3) = ( ˙x, ˙y + 3
+2Ωx, ˙z) is the velocity relative
+to the shear and the angle brackets indicate a suitable aver-
+age over the particles and possibly over time. The velocity
+dispersion c2 is then Wii/3. Note that this definition is only
+correct if there are no mean flows additional to the Keple-
+rian shear. If such flows are slow (as in viscous instability),
+the error will be small, however.
+The translational (local) component of the kinematic
+viscosity is
+νtrans = (2/3)Wxy/Ω.
+(9)
+The collisional (non-local) component of the viscosity is
+νcoll =
+2
+3ΩNδt
+�
+(x⟩ − x⟨)∆py
+(10)
+where the sum is taken over all binary collisions that occur
+in a time interval δt. Here M is the total mass of all ring
+particles, ∆py is the transfer of specific y momentum from
+the inner to the outer particle in each collision, and x⟩ and
+x⟨ are the radial locations of the two impacting particles
+(Wisdom & Tremaine 1988; Daisaka, Tanaka & Ida 2001).
+As we neglect self-gravity, there is no gravitational or wake
+contribution to the overall momentum transport. The total
+viscosity is hence νtot = νtrans + νcoll.
+To determine the thermal conductivity of a given equi-
+librium state, we follow the method of Salo et al. (2001) and
+create a steady non-uniform temperature T profile in the
+radial (x) direction, where T = c2. In our cold-state simula-
+tions, we achieve this by making vcrit radially dependent in
+the collision law. In our hot-state simulations, we vary ϵmax
+by a small amount in the radial direction. In either case,
+we end up with a steady-state sinusoidal radial temperature
+profile, though some experimentation is required to find the
+right amplitude for the variations in vcrit and ϵmax. The goal
+is to keep the perturbations in the temperature ∆T small,
+but not too small so that they are dominated by shot noise.
+© 0000 RAS, MNRAS 000, 000–000
+
+6
+Larue, Latter, Rein
+We typically use a simulation with Lx = Ly = 200a and run
+it for at least 1000 orbits.
+After setting up the nonuniform temperature profiles,
+we then measure specific translational (local) and collisional
+(non-local) heat fluxes,
+qtrans
+i
+= 1
+2σ⟨c2ci⟩
+(11)
+qcoll
+i
+= σ � ∆xiδEs
+Nδt
+(12)
+where σ = N/(LxLy) is the number surface density, δxi is
+the absolute difference of the i-coordinates of the two parti-
+cles involved in a collisions, and δEs is the change in trans-
+ported energy (as opposed to dissipated energy) during the
+collision for the particle with the larger xi coordinate. Fi-
+nally, we assume the heat flux is linearly dependent on the
+temperature gradient,
+q = −κ∇T.
+(13)
+We can then correlate the measured qx and ∂xT and retrieve
+the conductivity κ using a least squares fit. Finally, to verify
+our set up was working properly, we successfully reproduced
+Fig. 8 in Salo et al. (2001), though omit these results for the
+sake of space.
+2.3.4
+Parameters and initial conditions
+In all our N-body simulations, we adopt units so that a = 1
+and Ω = 1, though in what follows a and Ω reappear oc-
+casionally in order to make a point. As a consequence, the
+main physically relevant input is the collision law. Specifi-
+cally, we have some combination of vcrit/(aΩ), b, and ϵmax
+for non-constant collision laws. We also have the sizes of the
+numerical domain Lx and Ly and a constant dimensionless
+time-step Ωdt.
+We use initial conditions where particles are arranged
+uniformly in the plane with a uniform optical depth τ.
+Therefore an important initial input is particle number N
+while keeping the computational domain fixed. Particles are
+normally distributed in the z-direction. The initial velocities
+are also normally distributed with an initial velocity disper-
+sion c0. In most cases we initialize the particles close to the
+thermal equilibrium we believe to be present.
+We present convergence tests in Appendix A. These
+tests shows that our simulations are converged as we vary
+numerical parameters for both extremely high and low opti-
+cal depth, as well as hot and cold equilibria. For the regimes
+that we are interested in, we found that a dimensionless
+timestep of 10−3 and a box size of 10s to 100s particle radii
+are sufficient. The large box sizes are needed only for very
+hot and dilute rings.
+2.4
+Kinetic theory
+Though not the focus of this paper, it is useful to have some
+kinetic theoretical results, especially as they reveal the exis-
+tence of the additional (thermally unstable) middle branch
+of equilibrium solutions. The formalism adopted is Latter
+and Ogilvie’s (2008) reformulation of Araki and Tremaine
+(1986), which does not attempt to solve the Boltzmann-
+Enskog equation but rather a truncated moment hierarchy
+of continuum equations.
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+100
+101
+C
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+10-2
+100
+total
+1
+10
+5
+20
+0
+Figure 2. Velocity dispersion and total angular momentum flux
+τνtot versus optical depth τ for various hard-sphere ϵ laws, cal-
+culated from N-body simulations. In the top panel the appended
+numbers ‘1-20’ describe the values of vcrit/(aΩ) when using the
+standard Bridges law, whereas ‘0’ indicates runs with a constant
+ϵ = 0. In the bottom panel, the ordering of the curves is retained.
+The green symbols indicate that the viscous flux is decreasing and
+the disk viscously unstable.
+In previous deployments of this approach, the depen-
+dence of ϵ on the impact speed was only approximately in-
+corporated via a ‘pre-averaging’ procedure (see Section 2.2.7
+in Latter and Ogilvie 2008). Though convenient, this in-
+troduces unacceptable errors when using complicated non-
+monotonic laws as in Section 2.1.3. Thus the complete for-
+malism is adopted. This does require completing three (in-
+stead of two) integrations in the collision term. The other
+main approximations adopted are ‘vertical locality’ and a
+triaxial Gaussian for the velocity ellipsoid (see Araki and
+Tremaine 1986 and Latter and Ogilvie 2008 for more de-
+tails).
+3
+HOMOGENEOUS STEADY STATES
+In this section we simulate various thermodynamic equilibria
+and demonstrate that a non-monotonic epsilon law supports
+up to two equilibria for a given optical depth. We charac-
+terise these several states with respect to not only their ve-
+locity dispersion, but also their packing fraction FF0 and
+transport properties, especially with respect to angular mo-
+mentum and heat.
+We begin by reproducing previous results in the litera-
+ture with both a constant and monotonic epsilon law so as
+to verify that our code is working properly. Moreover, as ar-
+gued in Section 2.2, some of the equilibria obtained are lim-
+iting cases of those appearing in the bistable circumstances
+explored later and are thus useful in setting the scene.
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+7
+0
+1
+2
+3
+0
+5
+10
+15
+0
+1
+2
+3
+0
+5
+10
+0
+1
+2
+3
+0
+0.2
+0.4
+0.6
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+5
+10
+C
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.1
+0.2
+0.3
+0.4
+FF0
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.5
+1
+1.5
+2
+ tot
+Realistic
+max=0.75
+Realistic
+max=0.923
+BPL
+max=0.8
+Figure 3. Selected equilibrium properties as functions of τ for three regolith ϵ-laws (the three columns). The leftmost column shows
+equilibria computed with the broken-power law model (BPL) with ϵ0 = 0 and ϵmax = 0.8, whereas the other two columns show the
+realistic model with ϵmax = 0.75 and 0.923. In all cases vcrit = 5. In the top row the joined circles denote the velocity dispersion calculated
+by N-body simulations, with the colours indicating hot (red) or cold (blue) branches. The second and third rows show the filling factor
+and total angular momentum flux respectively. The dashed curve indicates equivalent solutions obtained from the kinetic theory (in the
+BPL case only). In the bottom row, a green symbol indicates expected viscous instability.
+Figure 4. The distribution of impact velocities in simulations
+using the ‘realistic’ law at τ = 1 with parameters vcrit = 5, b = 1,
+and ϵmax = 0.923. The left panel shows the system in the cold
+state. The right panel shows the system in the hot state. The red
+line corresponds to the ϵ law adopted.
+3.1
+Comparison with previous calculations
+Our reference cases include the simulations of Salo (1991),
+who employed a Bridges law but with a variable scale ve-
+locity, i.e. Eq.(1) with ϵ0 = 1 and vcrit = 1, 5, 10, 20 (see
+Section 2.1), and also simulations with a constant ϵ = 0,
+which brings about a very cold state. The results of our cal-
+culations are plotted in Fig. 2, in which we show the velocity
+dispersion c and angular momentum flux τνtot versus optical
+depth τ. The simulations were run until they were collision-
+ally relaxed, and then continued for the same length of time
+to obtain averaged quantities. When τ was low, and colli-
+sions relatively infrequent, the total run time was > 1000
+Ω−1; but at higher τ (∼ 2) runs could be as a short as 50-80
+Ω−1.
+Direct comparison of Fig. 2 with the numerical results
+of Salo (1991; cf. his Figs 3-5) shows good agreement, and
+also consistency with the kinetic theory of Latter & Ogilvie
+(2008) (note that both these works denote vcrit by vb). An
+interesting feature of the ‘warmer’ solution branches is the
+decreasing viscosity with τ. In fact, the hottest case, vcrit =
+20, is viscously unstable because the gradient of the angular
+momentum flux τν is negative in an interval of τ (green
+markers).
+By inflating vcrit in the Bridges law the velocity disper-
+sion of the system can be controlled and, in particular, set
+to ‘warm’ values greater than aΩ and, consequently, greater
+than the temperature of the very cold ϵ = 0 states. These
+warm and cold states help illustrate the arguments presented
+in Section 2.2. If we take one of the two non-monotonic col-
+lision laws and set vcrit ten or more times aΩ, then start
+the simulation with a hot initial condition, we might expect
+the subsequent spread of impact speeds to be sufficiently
+far from ϵ’s turning point (cf. Fig. 1) so that the system
+settles into a warm ‘Bridges equilibrium’, similar to those
+plotted in Fig. 2. On the other hand, if we begin the same
+simulation but with very cold initial velocities (≪ vcrit), the
+subsequent spread of impact speeds will remain less than
+vcrit and ϵ will almost always take the value of 0; the system
+© 0000 RAS, MNRAS 000, 000–000
+
+LD
+LD
+0.B
+0.B
+0.6
+0.6
+0.4
+t0
+0.2
+0.2
+20
+0.0
+-0.0
+0
+0
+24
+mpact velociby y8
+Larue, Latter, Rein
+will then converge to the appropriate constant ϵ = 0 state in
+Fig. 2. Note that the Bridges law produces a velocity disper-
+sion c that decreases with τ and we may then expect that
+for sufficiently large τ the upper ‘hot state’ will be too close
+to the ‘cold state’ and bistability may disappear.
+3.2
+Non-monotonic collision laws
+In this section we calculate equilibria for ‘regolith’ epsilon
+laws that are non-monotonic: either the broken power law
+(BPL) with ϵ0 = 0 or the realistic law (2). The parameters
+are ϵmax = 0.75, 0.8 or 0.923, vcrit = 5aΩ, and b = 1, though
+we examine a broader spread of values in Section 3.2.2. We
+first examine in some detail the thermal properties of the
+states, then their transport of angular momentum and heat.
+3.2.1
+Thermal hysteresis
+Figure 3 constitutes the first main results of the paper. Here
+we plot the equilibrium velocity dispersions (top row), filling
+factors (middle row), and total radial angular momentum
+fluxes (τνtot; bottom row) obtained in a sequence of simu-
+lations at different optical depths and for different ϵ mod-
+els and parameters. Each circular marker corresponds to a
+different simulation. These values are obtained by time av-
+eraging a quantity once the system has become collisionally
+mature, as earlier. For example, τ = 0.1 runs were run for
+1600Ω−1 and averaged for the last 800Ω−1, while at τ = 2
+the total run length was 80Ω−1, with the averaging taking
+place over the last 40Ω−1.
+As is clear, in the three models presented, two steady
+state branches (distinguished by red and blue) are possi-
+ble within a certain range of optical depth. Which of the
+two the system selects depends on the initial condition: a
+‘cold start’ (low initial c) usually (but not always) takes the
+system to the nearby cold state, whereas a ‘hot start’ (ini-
+tial c sufficiently high) settles on the hot state. Typically,
+runs starting with c = 0.5aΩ converged to the nearby cold
+state, while runs beginning with c = 10aΩ migrated to the
+hot state, if one was available, even if that state’s veloc-
+ity dispersion was significantly larger than the initial c. The
+direction of migration is discussed further in Section 4.
+The apparent bistability extends over a range of small
+to intermediate optical depths. Beyond a special τ the hot
+state disappears, and all hot start simulations landed on the
+cold branch. At small τ we never found that the cold state
+disappeared, except in the case of the realistic model with
+ϵmax = 0.923 and τ = 0.1; this equilibrium was metastable
+(explored in more detail in Section 4). The bistable regime’s
+width (in τ) depends on the parameters. From Fig. 3, in-
+creasing the ϵmax in the realistic model from 0.75 to 0.923
+moved the special τ from roughly 0.5 to 1.6 (cf. middle and
+right columns).
+The cold equilibria take c values very much in agree-
+ment with the constant ϵ = 0 states simulated in the previ-
+ous subsection, while the hot state resembles a Bridges law,
+with c decreasing with τ. In fact, the hot simulations of the
+realistic model with ϵmax = 0.923 take a similar c as the
+Bridges vcrit = 10 runs, while those with ϵmax = 0.75 resem-
+ble a Bridges law with (roughly) vcrit = 5. These similarities
+bolster our interpretation of the two states as ‘separated’ by
+Figure 5. Grids of simulations undertaken with different vcrit
+and ϵmax using the realistic regolith law with widths b = 1 (top)
+and b = 2 (bottom). Colours correspond to values of |chot −ccold|
+(see text). The contour is a conservative boundary between cases
+that support bistability (to the right and above) and those that
+do not.
+the turning point of the ϵ curve: only a minority of collisions
+in the hot state occur with the low impact speeds that would
+trigger ϵ = 0, while collisions in the cold state rarely occur
+with impact speeds sufficiently large to trigger larger ϵ. To
+flesh out this point further we plot in Fig. 4 the distribution
+function of impact speed for a hot state (right panel) and
+a cold state (left panel) for the same τ = 1 (and other pa-
+rameters). Superimposed in red is the ϵ law used. As the left
+panel indicates, cold state collisions are almost completely
+inelastic; the narrow spread in impact speeds barely overlaps
+the portion of the curve for which ϵ ̸= 0. In contrast, the hot
+state (shown in the right panel) is much broader and thus
+samples a wide range of ϵ, but importantly peaks at speeds
+which yield collisions with a small dissipation of energy.
+The filling factors in the middle row of Fig. 3 reveal that
+the hot branches are far less dense than the cold branches.
+For example, in the realistic model with ϵ = 0.923, at τ = 1
+the hot state possesses a filling factor of 0.08, while the cold
+state has 0.35. The difference, of course, is not due to the
+surface number density (which is the same) but because the
+disk semi-thickness is so different between these two states:
+in the hot state it is ≈ 6a, compared to ∼ a in the cold state.
+The ratio of the two filling factors should scale roughly with
+the ratio of semi-thicknesses and that is indeed what we see.
+The hot state branch terminates when its velocity dis-
+persion approaches a critical value ∼ 3. In reality the sys-
+tem here encounters a saddle-node bifurcation and the solu-
+tion curve bends ‘backwards’ thus forming an intermediate
+branch of thermally unstable solutions. Because these solu-
+tions are unstable they cannot manifest in N-body simula-
+tions3, but they can be calculated by kinetic theory. Kinetic
+3 See Salo et al. (1988) for a numerical exploration of a thermally
+unstable state.
+© 0000 RAS, MNRAS 000, 000–000
+
+b=1
+30
+0.9
+25
+0.8
+20
+max
+15
+0.7
+10
+0.6
+5
+0.5
+0
+0
+1
+2
+3
+4
+5
+9
+7
+8
+9
+b=2
+50
+0.9
+40
+0.8
+max
+30
+0.7
+20
+0.6
+10
+0.5
+0
+0
+1
+2
+3
+4
+5
+9
+7
+8
+9
+critThermal hysteresis in rings
+9
+theoretical equilibria are plotted in the leftmost column with
+a dashed black curve; the top and middle panels show clearly
+an intermediate cool, semi-dense branch. The agreement be-
+tween theory and simulations is qualitative good, with the
+biggest deviation in the translational viscosity in the hot
+state, a discrepancy that has been noted in previous com-
+parisons (Latter and Ogilvie 2008, Rein and Latter 2013).4
+3.2.2
+Parameter survey
+In the preceding subsection we examined only three param-
+eter sets/models; in this subsection we adopt the realistic ϵ
+law and scan through vcrit and ϵmax for two different widths
+b. Our aim is to determine how representative the thermal
+hysteresis explored in the previous subsection really is. Of
+particular interest are the lowest values of vcrit and ϵmax that
+yield bistability.
+In Fig. 5 we present ‘bistability plots’ for b = 1 and
+2. Each square in the grid corresponds to a parameter pair
+(vcrit, ϵmax), and for each square we conduct two simulations
+with τ = 0.1, one with a hot initial condition and the other
+with a cold initial condition. Each simulation has been run
+until thermal equilibrium has been obtained, and the dif-
+ference in final velocity dispersion calculated, |chot − ccold|.
+Finally, the square is coloured accordingly (cf. the colour
+bar). If the difference in final c is between 0 and 5, we in-
+terpret that the two simulations are converging on to the
+same (cold) equilibrium. Values larger than 5 (admittedly, a
+rather large value, given Fig. 3) we assume correspond to a
+bistable situation: the two simulations are settling on differ-
+ent thermal states. In both panels we have superimposed the
+contour of |chot − ccold| = 5. The reader should then assign
+bistability to regions of the parameter plane above and/or
+to the right of this curve.
+The plots indicate, as expected, that bistability is
+favoured by larger values of vcrit and ϵmax. Increasing both
+parameters helps to separate the typical impact speeds of
+the hot state from those of the cold state. Interestingly,
+the bistable region is quite rectangular. Thus when b = 1,
+bistability is guaranteed (roughly) if both vcrit > 4 and
+ϵmax > 0.7. We expect that these parameter restrictions
+should hold roughly for other non-monotonic laws. Finally,
+the range of bistability is also sensitive to the width of the
+epsilon law, as the b = 2 plot demonstrates. Increasing the
+width also helps separate out the two states. In the b = 2
+case bistability occurs when vcrit > 3 and ϵmax > 0.65.
+3.2.3
+Viscous properties
+The equilibrium states discussed in the previous subsection
+support a viscous stress that, by acting on the background
+orbital shear, transports angular momentum radially across
+the numerical domain. The viscous properties of the flow are
+important thermodynamically because the stress extracts
+free energy from the shear, thus providing the heating source
+in the thermal balances undergirding these states. But the
+viscous stress is also important dynamically because it can
+4 Unfortunately, numerical difficulties prevented us calculating
+kinetic solutions for the realistic model.
+τ
+κL (C)
+κNL (C)
+κL (H)
+κNL (H)
+0.1
+4.75
+0.42
+77.94
+0.72
+0.2
+5.92
+0.62
+119.26
+2.01
+0.3
+7.42
+1.15
+111.88
+3.39
+0.4
+6.83
+1.61
+89.95
+3.59
+Table 1. Calculated translational (local) thermal conductivities
+κL and collisional (non-local) thermal conductivities κNL in cold
+(C) and hot (H) equilibria at various optical depths τ. A realistic
+collision law is adopted with ϵmax = 0.75, vcrit = 5, and b = 1.
+beget instabilities, such as the viscous overstability and in-
+stability (Schmidt et al. 2009). In particular, if d(τνtot)/dτ
+is negative then viscous instability occurs (Lin and Boden-
+heimer 1981, Lukkari 1981, Ward 1981).
+The angular momentum flux is plotted in the bottom
+row of Fig. 3. Note that a subset of hot states possess a
+decreasing flux and are thus viscously unstable; these are
+marked in green. In the BPL model, the unstable interval
+encompasses τ of 0.4 and 0.5, whereas in the realistic model
+only the ϵmax = 0.923 case yields instability and then for
+τ between approximately 0.8 and 1.6. Instability here is as-
+sociated with a dominant translational viscosity, which can
+decline at sufficiently large τ. Growing modes do not appear
+in these simulations, however, because the numerical do-
+main size is smaller than the shortest unstable wavelength;
+in Section 5.2.2 we simulate larger domains and recover the
+instability.
+3.2.4
+Thermal conductivity
+Anticipating later sections which explore different thermal
+states that spatially adjoin, we compute the radial flux of
+thermal energy. In the absence of any mean spatial gradients,
+such as in the homogeneous equilibria calculated, the flux
+must be zero. But if two states connect in radius the flux
+must control, in part, how their interface evolves.
+As explained in Section 2.3.3, we adopt the approach of
+Salo et al (2001) and impose a radial sinusoidal temperature
+structure upon the box, through the parameters vcrit and
+ϵmax. In Fig. 7 we show calculations of the radial thermal
+flux qx and the thermal conductivity κ for a fixed set of
+parameters (ϵmax = 0.75, vcrit = 5, b = 1) and for the same
+optical depth τ = 0.2. The left four panels correspond to the
+cold state (c ≈ 1), and the right to the hot state (c ≈ 6).
+The top left panel in each case describes the temperature
+profile across the box, while the top right panel shows the
+temperature gradient (solid blue), the translational (local,
+‘L’) heat flux (dashed gold), and the collisional (nonlocal,
+‘NL’) heat flux (dotted green). The latter two are plotted
+separately as functions of the temperature gradient in the
+bottom panels; a best-fit line extracts the conductivities.
+In both the hot and cold cases, the translational heat
+flux dominates the collisional flux. This means that the heat
+flux in the two states differs significantly, despite possessing
+the same τ. In Table I we list κ for a range of τ and otherwise
+with the same parameters as in Fig. 6.
+© 0000 RAS, MNRAS 000, 000–000
+
+10
+Larue, Latter, Rein
+100
+0
+100
+x
+0.75
+0.80
+0.85
+0.90
+0.95
+T
+100
+0
+100
+x
+0.02
+0.00
+0.02
+dT/dx
+qx (L)
+qx (NL)
+100
+0
+100
+x
+40
+45
+50
+T
+100
+0
+100
+x
+20
+10
+0
+10
+20
+0.003
+0.000
+0.003
+- dT/dx
+0.02
+0.00
+0.02
+qx (L)
+L=5.92
+0.003
+0.000
+0.003
+- dT/dx
+0.004
+0.002
+0.000
+0.002
+0.004
+qx (NL)
+NL=0.62
+0.2
+0.0
+0.2
+- dT/dx
+20
+10
+0
+10
+20
+30
+qx (L)
+L=119.26
+0.2
+0.0
+0.2
+- dT/dx
+0.4
+0.2
+0.0
+0.2
+0.4
+qx (NL)
+NL=2.01
+COLD, max = 0.75, vc = 5, b = 1,
+= 0.2
+HOT, max = 0.75, vc = 5, b = 1,
+= 0.2
+Figure 6. Thermal diffusivity measurements for τ = 0.2 in the cold state (left panels) and hot state (right panels) for the realistic model
+with ϵmax = 0.75, vcrit = 5, and b = 1.
+Figure 7. Velocity dispersion as a function of time for runs with τ = 0.1 (top panel) and τ = 0.2 (bottom panel). The realistic model is
+adopted with ϵmax = 0.923, vcrit = 5, and b = 1.
+4
+METASTABILITY
+In the last section we calculated steady states that appear to
+be thermally stable, at least linearly according to a contin-
+uum interpretation. However, N-body systems are replete
+with small but finite amplitude shot noise that continually
+tests the nonlinear stability of any steady state. If the basin
+of attraction of a linearly stable state is small relative to the
+amplitude of these fluctuations, the system can potentially
+jump out of the state and migrate elsewhere. Many phys-
+ical and biological systems offer similar examples of noise
+destabilising what should be linearly stable fixed points (e.g.
+Mel’nikov 1991, May 1973, De Swart and Grasman 1987,
+Majda, Timofeyev and Vanden-Eijinden 1999, 2003). In this
+section we investigate this possibility.
+Our focus will be on cold states of low-optical depth
+and on the hot states near the saddle node bifurcation. The
+reason is that these states are close to the unstable mid-
+dle branch which can serve as the boundary of the basin
+of attraction in each case. We find that, for the parameters
+and models we employ, metastability is relatively uncom-
+mon, only occurring in certain dilute and cold states. In
+particular, states near the saddle node are generally stable
+to shot noise perturbations.
+Before presenting our results we emphasise that we
+only explore the effect of intrinsic shot noise, but in real
+rings there are several other sources of finite amplitude
+disturbances that may work similarly, e.g. meteoroid bom-
+bardment, embedded moonlets, density waves, and gravity
+wakes.
+© 0000 RAS, MNRAS 000, 000–000
+
+20
+15
+10
+5
+0
+0
+100
+200
+300
+400
+500
+time [orbits]1.1
+1.0
+0.9
+0.8
+0
+25
+50
+75
+100
+125
+150
+175
+200
+time [orbits]Thermal hysteresis in rings
+11
+Figure 8. Velocity dispersion as a function of time for runs of different initial conditions with τ = 1.61 (top panel) and τ = 1.64 (bottom
+panel). The realistic model is adopted with ϵmax = 0.923, vcrit = 5, and b = 1.
+4.1
+Cold to hot transitions
+We find spontaneous transitions from the cold lower branch
+to the hot upper branch in only a few low τ cases when
+adopting a realistic collision law and ϵmax = 0.923. Specif-
+ically, when τ = 0.1 the system can hover about the cold
+steady state for several hundred orbits before jumping to
+the hot state.
+To probe this behaviour we ran 24 runs with slightly
+different initial conditions (varying both particles’ locations
+and velocities) but all starting with the same low c. To make
+doubly certain that the system is as close to the cold equi-
+librium as possible, and that any future transition is not
+the result of a wayward initial condition, we force ϵ = 0 (a
+constant) for several orbits at the start.
+The evolution of these runs are plotted in the top panel
+of Fig. 7, with the shaded region indicating when ϵ = 0.
+As is clear from the figure, all but three runs jumped to
+the hot state by 500 orbits (roughly > 25 collision times),
+though there was a wide spread of transition times, indica-
+tive that the process is stochastic and issues from the noise:
+ultimately, after some period, an overenthusiastic collision,
+dissipating insufficient velocity dispersion, seeds a patch of
+more energetic particles, which then spreads spatially and
+takes over the system.
+Of course, this is only part of the story, because en-
+ergetic events must happen at slightly larger τ but do not
+appear to instigate runaway heating. Indeed, we undertake
+a similar experiment at τ = 0.2, plotted in the lower panel
+of Fig. 7, and witness no transitions at all. What is key is
+the overall basin of attraction of the cold state; as shown by
+the kinetic curves in the top left panel of Fig. 3, the middle
+unstable branch and the cold lower branch become closest
+at low τ. The middle branch acts as the boundary of the
+lower state’s basin of attraction (at least in this simple phase
+space projection); thus at low τ it becomes more likely that
+a finite amplitude perturbation can tip the system over this
+boundary. That said, it is not straightforward to firmly con-
+nect microphysical fluctuations (shot noise) to such a mean
+finite-amplitude perturbation in this phase space.
+4.2
+Hot to cold transitions
+We now check if it is possible to obtain spontaneous hot to
+cold transitions. We focus on states near the tip of the saddle
+node, i.e. the termination of the hot branch (see top row in
+Fig. 3), and examine a range of τ between 1.61 to 1.65 in
+the realistic model with ϵmax = 0.923. We simulate several
+runs with slightly different initial conditions, as before, and
+plot the results in Fig. 8, top and bottom panels. As in the
+previous subsection, to ensure that we start the simulations
+in a hot state we set vcrit to a very small value initially. Over
+several orbits (indicated by the shaded area in the figures),
+we slowly increase vcrit to the nominal value.
+Unlike cold to hot transitions, the systems either imme-
+diately drop to the cold state or relax into the hot state on
+a timescale of 10 orbits or so (a handful of collision times).
+At τ = 1.65 all the simulations ended up in the cold state,
+while at 1.64, some stayed in the hot state, while at lower
+tau again (1.61) most stay in the hot state. Putting aside
+the percentages in one or the other, the system transitions
+promptly or not at all. We attribute this more to the initial
+condition at the end of the blue phase, rather than having
+to wait for a more sluggish group of collisions that lead to a
+‘chain reaction’ and a switching of states.
+The difference with the low τ runs explored earlier may
+partially be explained by the separation between the middle
+and hot branches, which is relatively large, even near the
+tip of the saddle node (see kinetic theory curves in top left
+panel of Fig. 3). Once a system settles on to the hot state,
+and its initial conditions mostly forgotten, its intrinsic shot
+noise is insufficient to tip it out of its basin of attraction and
+into the cold state.
+© 0000 RAS, MNRAS 000, 000–000
+
+5
+4
+3
+2
+1
+0
+50
+100
+0
+150
+200
+250
+time [orbits]5
+4
+3
+2
+1
+0
+100
+0
+50
+150
+200
+250
+time [orbits]12
+Larue, Latter, Rein
+600
+400
+200
+0
+200
+400
+600
+x
+20
+10
+0
+10
+20
+z
+t=0 orbits
+Figure 9. Initial condition for the fiducial thermal-front simula-
+tion described in Section 5.1.1 in the form of an (x, z) projection
+of the particle positions.
+5
+THERMAL AND VISCOUS FRONTS
+Having computed several homogeneous states, we now ex-
+plore the dynamics when different states spatially adjoin. If
+a ring region is bistable, then it is likely that such situa-
+tions occur, given the varying dynamical histories at differ-
+ent radii. Our main focus is on the structure and evolution
+of the transition (or front) between two states. We will con-
+sider two cases: (a) thermal fronts, which join two states
+of the same τ but different c, and (b) viscous fronts, which
+connect two states of the same angular momentum flux τν,
+but different τ and c
+Thermal fronts involve a hot and a cold state, with the
+pair joined by a vertical line in the top panels of Fig. 3.
+Though sharing the same optical depth, they possess dis-
+tinct vertical thicknesses that may produce a photometric
+variation, and thus observable structure (e.g. Salo and Kar-
+jalainen 2003). However, the two states will support different
+angular momentum fluxes τν, and thus mass may pile up or
+evacuate near the thermal front, potentially leading to non-
+steadiness and a complete break down of the structure. We
+find that this is avoided if the front itself moves sufficiently
+fast.
+One might expect radial mass redistribution is negated
+if two adjoining states possess the same angular momentum
+flux, with the pair joined by a horizontal line in the bot-
+tom panels of Fig. 3. In fact, similar structures have already
+been witnessed in simulations of the viscous instability with
+monotonic ϵ laws (Salo and Schmidt 2010). We find, how-
+ever, that the finite width of the front itself spoils the ex-
+act matching of fluxes and makes the establishment of such
+fronts more complicated.
+5.1
+Thermal fronts
+In order to explore the structure and dynamics of fronts
+connecting equilibria of different temperatures but the same
+surface density, we concentrate on a single parameter set.
+The behaviour obtained is then interpreted using a simple
+continuum model, before other parameters are trialled.
+5.1.1
+Fiducial case
+Our fiducial run employs a realistic ϵ law with the following
+parameters: ϵmax = 0.75, vcrit = 5, and b = 1. We examine
+a hot and cold state of the same τ = 0.2, with the former
+possessing c = 6.7 and the latter c = 0.87. We adopt a wide
+box of radial size 1000a and insert a strip of particles from
+the (previously computed) hot state in the centre (with ra-
+dial extent 100a), while distributing particles from the cold
+state throughout the rest of the numerical domain. Figure
+9 plots this initial condition as a projection of the particle
+locations in the (x, z) plane. Away from the borders of the
+hot/cold zones, the ring is in thermal equilibrium.
+The subsequent evolution of the ring is shown in Fig. 10,
+which presents four snapshots at different times on each row.
+The left panels describe the (x, z) projections of the parti-
+cles, while the right panels plot the radial variation of τ
+(blue) and c (red). As is clear, the two fronts move radially
+into the cold state, until the hot state takes over the box
+entirely. Meanwhile, τ remain roughly constant throughout,
+except for some minor deviations around the front itself.
+The front speed is constant until the moment that the
+cold state evaporates. This is demonstrated in Figure 11,
+which plots the location of the rightmost front as a func-
+tion of time. A c intermediate between the c in the hot and
+cold states was selected (here c = 4) and its x location was
+determined at each time-step, which provided a means to
+capture the movement of the front as a whole. The front
+speed is 0.685aΩ, thus slightly less than c in the cold state.
+Generally, in bistable systems, the conductivity controls
+the structure of fronts; a small conductivity yields a narrow
+transition, while a large conductivity gives a more diffuse
+transition (e.g. Latter and Balbus 2012). In our granular gas,
+the thermal conductivity κ depends on c, and thus jumps
+by at least an order of magnitude as we go from the cold
+to the hot state (see Table I). This explains why the front
+structure is sharp near the cold state (though always longer
+than the ‘granularity scale’, a), while broader and smoother
+near the hot state. The overall width of the front (≳ 100a)
+is hence determined approximately by κ in the hot phase.
+5.1.2
+Physics of front motion; a simple continuum model
+The basic mechanism driving the movement of a thermal
+front relies on the finite-amplitude perturbations arising
+from the proximity of the different states. These perturba-
+tions can only be communicated via thermal diffusion. For
+example, near a front, the cold state will receive thermal en-
+ergy (via diffusion) from the adjacent hot state. If the energy
+received is sufficient to push the cold ring material out of the
+cold state’s basin of attraction, then one might expect it to
+heat up and settle on the hot state; as a consequence, the
+front advances into the cold phase. But, by the same token,
+on the other side of the front, material in the hot state will
+also be perturbed by the heat flux and will cool down. If this
+cooled material is pushed beyond the hot state’s basin of at-
+traction, then it will undergo a runaway cooling and then
+we might expect the front to advance into the hot state.
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+13
+600
+400
+200
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+t=191 orbits
+Figure 10. Snapshots of a thermal front at t = 0.8, 8, 80 and 191 orbits. Panels on the left describe a projection of ring particles on to
+the (x, z) plane. Panels on the right depict the x-dependent velocity dispersion c (red) and optical depth τ (blue).
+© 0000 RAS, MNRAS 000, 000–000
+
+14
+Larue, Latter, Rein
+0
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+t 
+0
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+x-coordinate of front
+Figure 11. Outer front radial location as a function of time in
+the simulation shown in Fig. 10.
+Which thermal runaway is favoured on average depends on
+the relative sizes of the hot and cold state’s basins of attrac-
+tion, which can be approximated (roughly) by how close the
+intermediate unstable state is to either state (see discussion
+in the section on metastability, and also Latter and Balbus
+2012).
+These ideas can be illustrated by a continuum model.
+The energy equation of the gas may be written as
+∂tE = Λ(E) + ∂x(k∂xE),
+where E = (3/2)c2, Λ combines viscous heating and col-
+lisional cooling, and k is thermal diffusivity (= 2κ/(3σ)).
+Thus Λ = 0 when E is equal to the stable hot, cold, and
+unstable intermediate steady states, EH, EC, and EI, re-
+spectively. Moreover, dΛ/dE < 0 when E = EH or EC. We
+assume a steady front, moving at speed vf, with the hot
+state to the right and the cold state to the left, and thus
+introduce the comoving variable ξ = x − vft, which trans-
+forms the energy equation into a type of Stefan problem for
+the front shape E(ξ) and speed vf,
+∂ξ(k∂ξE) + vf∂ξE + Λ(E) = 0.
+(14)
+The boundary conditions are E → EH as ξ → ∞ and
+E → EC as ξ → −∞ (hot to the right and cold to the left).
+This is a nonlinear eigenvalue problem that, after specify-
+ing the functional forms of Λ(E) and k(E), would normally
+require a numerical solution. In Appendix B we adopt sim-
+ple prescriptions for these functions and solve the equation,
+thereby illustrating some of the main features discussed be-
+low and qualitatively reproducing our N-body results.
+An illuminating expression for the speed vf can be ob-
+tained by multiplying Eq. (14) by dE/dξ and integrating
+between −∞ and ∞. After some manipulation, one gets
+vf = −
+� EH
+EC Λ dE
+� ∞
+−∞(dE/dξ)2dξ −
+� ∞
+−∞(dk/dE)(dE/dξ)3dξ
+2
+� ∞
+−∞(dE/dξ)2dξ
+. (15)
+If the thermal diffusivity is a constant, the second term is
+zero. In this case, the sign of vf is determined solely by the
+integral of the heating/cooling term Λ. Because Λ(EH) =
+Λ(EI) = Λ(EC) = 0, the integral can be subdivided into (a)
+a positive part (between EC and EI) that measures the ‘size’
+of the cold state’s basin of attraction, and (b) a negative
+part (between EI and EH) that measures the hot state’s
+basin of attraction. The proximity of EI to either EC or
+EH indicates the basins’ relative sizes. If EI is closer to EC,
+then the integral is dominated by the positive area, vf < 0,
+and the front moves into the cold state. Physically, cold ring
+material near a front finds it easier to undergo a heating
+runaway, when perturbed by the front, than hot material
+finds a cooling runaway; thus, the front advances into the
+cold material. If EI is closer to EH, then the converse holds
+and the front moves into the hot state. Turning now to the
+top row of Fig. 3 (first panel especially), one naively expects
+that at low τ fronts initially move into the cold state, but at
+higher τ fronts are slower and then at some critical τ may
+reverse direction.
+If k depends on E then things are more complicated.
+The second term in Eq. (15) is a weighted average of dk/dE,
+and shows that a non-uniformity in the transport of heat
+moderates the effect discussed above. If the front shape is
+monotonic in ξ, then dE/dξ > 0 throughout and the sign of
+the second term is determined by dk/dE. As demonstrated
+in Section 3.2.3 and Table I, dk/dE > 0, and so the second
+term in Eq. (15) is always positive, thus biasing the front’s
+movement into the cold state. The underlying mechanism
+here rests not on the system’s bistability but on exacerbating
+the imbalance in the heat flux throughout the front struc-
+ture: at any given point more heat is arriving from the hot
+state than is being evacuated.
+The discussion above suggests that the sharp region at
+the foot of the front controls the front speed. Taking an
+order of magnitude approach and equating the three terms
+in Eq. (14) yields the estimate vf ∼
+�
+kC/tth, where the
+thermal timescale is defined as tth = E/Λ ∼ c2/(νΩ2), and
+kC is the diffusivity evaluated in the cold state. Putting in
+values for the cold state gives us vf ∼ aΩ, which is consistent
+with the value calculated numerically. The width λ of the
+front extending through the hot phase can then be estimated
+by balancing the first two terms in Eq. (14); we find λ ∼
+kH/vf ≳ 500a, which is also consistent with the simulation.
+5.1.3
+Front stability
+We conducted a short survey of fronts at different τ and
+calculated their speeds. When τ = 0.1 we found vf = 0.518,
+and when τ = 0.3, vf = 0.591. While no clear trend could be
+observed between τ = 0.1 − 0.3, we expected at larger τ, as
+we approached the saddle node, that the front speed should
+decrease. In fact, what we found for τ = 0.4 or larger is that
+the front would slow to a halt and then viscously reshape;
+i.e. τ would evolve away from a uniform profile. Ultimately,
+the system moves to a state of constant angular momentum
+flux τν, and the thermal front dissolves.
+As mentioned earlier, the issue here is that across a
+thermal front τ is constant, but τν is not. As a consequence,
+mass can potentially build-up/evacuate. If the front moves
+faster than τ can be viscously redistributed, then we expect
+the front to remain coherent and to travel unimpeded. If the
+front speed is too slow, then it will be viscously reshaped and
+will collapse. For the model chosen, τ ⩽ 0.3 corresponds to
+the first case, and τ > 0.3 to the latter.
+A rough criterion for the ‘stability’ of the front to vis-
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+15
+cous redistribution would tension the relative sizes of the
+front speed vf and the viscous diffusion speed. To deter-
+mine an estimate on the latter, we employ the lengthscale
+of the abrupt transition at the foot of the structure and
+thus estimate the diffusion speed as ∼ (νC/κC)vf. A sim-
+ple criterion for front dissolution requires that this speed
+is greater than vf, and hence depends solely on the size of
+the Prandtl number Pr = ν/κ in the cold state: when Pr is
+greater than a critical value Prc, we expect the front to dis-
+solve. Indeed, Pr increases monotonically between τ = 0.1
+and 0.4, though takes relatively small values. At τ = 0.4, we
+find that Pr ∼ 0.04, which must be near Prc.
+5.2
+Viscous fronts and viscous instability
+Given the issue of the unbalanced angular momentum in
+thermal fronts, it is natural to explore fronts that join states
+with the same viscous transport properties, specifically ντ.
+We present simulations of such joined states in this subsec-
+tion, in addition to a short treatment of viscous instability.
+A simple continuum model can guide our expectations.
+In the shearing sheet, the one-dimensional diffusion equation
+for viscous Keplerian disks is
+∂tτ = 3∂2
+x(ντ)
+(e.g. Lynden-Bell and Pringle 1973). Suppose a viscous front
+moves with speed vf with τ → τA as x → −∞ and τ → τB as
+x → ∞. As earlier, we adopt a comoving variable ξ = x−vft,
+which permits the complete integration of the problem. We
+find that vf = 0 (the structure must be stationary) and
+ντ (= νAτA = νBτB) is a constant throughout the entirety
+of the front. The last constraint is a potential difficulty: while
+it is possible to find two homogeneous steady states of the
+same ντ (cf. panels in the bottom row of Fig. 3), a realis-
+tic front will have a finite width in which τ will vary and
+thus ντ will deviate from the required constant value. Our
+simulations show, in fact, that the system can overcome this
+problem by settling on a front structure in which the average
+ντ equals νAτA = νBτB.
+5.2.1
+Fronts
+We present a fiducial simulation with the realistic law, and
+parameters b = 1, ϵmax = 0.923, and vcrit = 5. To construct a
+suitable initial condition that might produce a viscous front,
+we select two thermally and viscously stable states with the
+same ντ from the bottom right panel of Fig. 3. Such pairs
+are joined by horizontal lines. We select two states of the
+same angular momentum flux ντ ≈ 2, with optical depths
+τ = 1.5 and τ = 0.16. The numerical domain is chosen to
+be sufficiently large (L = 800) to accommodate relatively
+undisturbed expanses of the two states, in addition to the
+front itself; the low τ state is placed between x = −100 and
+100, with the high τ state taking up the remainder of the
+box.
+Figure 12 shows eight snapshots of the resulting simu-
+lation at different times. In each panel we plot τ (red) and
+τν (blue). At t = 0, the angular momentum flux τν is a
+constant, but τ undergoes two jumps (at x = ±100a). As
+the system evolves, the two jumps/fronts relax and exhibit a
+characteristic width, with τ taking values between those of
+the two steady states. An immediate consequence is that the
+angular momentum flux within the fronts begins to deviate
+from the fixed value ≈ 2. In fact, the first four panels show
+that it takes significantly larger values than 2, in agreement
+with the bottom right panel of Fig. 3, which shows that
+states with τ between 0.16 and 1.5 exhibit ντ > 2. Because
+of the enhanced flux in the fronts, mass is being transported
+out of the fronts, which then appear to move as the system
+evolves far way from the initial condition.
+Ultimately, we find that the system redistributes the
+mass throughout the numerical domain so that τν is roughly
+constant (≈ 7), but still allows for strong variations in τ.
+This outcome is not a constant τ state, but consists of
+two static viscous fronts joining two homogeneous states
+of τ ≈ 0.4 and 2.7, which according to Fig. 3 possess the
+same angular momentum flux (∼ 7). Evidently, the front
+that joins the two states also possesses a similar approxi-
+mate flux, though this is difficult to determine from Fig. 3.
+A similar final state was found by Salo and Schmidt (2010)
+when simulating the viscous instability directly (see next
+subsection).
+This static structure is an interesting outcome for the
+system, but we stress that it is possible only because of
+the periodicity of the numerical domain. Owing to those
+boundary conditions, mass in the whole domain can be re-
+distributed until the desired constant ντ state can be found.
+In a more realistic setting, the system is unlikely to come
+to steady state and the front will continue to move until it
+encounters large-scale variations in background disk proper-
+ties, etc.
+5.2.2
+Viscous instability
+In the previous subsection we explored two adjoined vis-
+cously stable states, but the lower right panel of Fig. 3 in-
+dicates that there is a branch of viscously unstable states
+of intermediate τ between roughly 0.8 and 1.6. An obvious
+question is: to where does the system evolve if started from
+one of these states? We thus present a simulation with the
+same collisional parameters as earlier, but with a homoge-
+neous τ of 1.4. According to Fig. 3, this state is viscously
+unstable. Figure 13 shows 5 snapshots of the system’s evo-
+lution.
+Despite possessing a constant τν, the system moves
+slowly away from this state and begins to develop grow-
+ing patches of high and low τ. Unlike the previous subsec-
+tion, where the evolution is being driven by large-scale flux
+imbalances, here there is an instability mechanism, in which
+small-scale fluctuations in the flux self-reinforce (Lin and Bo-
+denheimer 1981, Lukkari 1981, Ward 1981). Ultimately, the
+system settles on a sequence of distinct high-τ islands sur-
+rounded by relatively dilute regions, but both with roughly
+the same flux (≈ 6, in this case), as is necessary for a steady
+state.
+These results are very similar to those predicted by
+H¨ameen-Anttila (1982) and witnessed in Lukkari (1981) and
+Salo and Schmidt (2010), though they use a monotonic col-
+lision law. A key difference is that in the monotonic ϵ simu-
+lations, the final outcome joins states from the same branch,
+while in our non-monotonic simulations states from different
+branches adjoin. An interesting consequence of this is that it
+is still possible for the system to separate into a sequence of
+© 0000 RAS, MNRAS 000, 000–000
+
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+t=50 orbits
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+2.0
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+4
+6
+8
+10
+12
+t=2000 orbits
+Figure 12. Snapshots of an example viscous front, showing optical depth and angular momentum flux as a function of x. The initial
+condition connects two states of different τ but the same angular momentum flux τν. Despite this balance, the system evolves, redis-
+tributing mass and angular momentum until a steady state is achieved characterised by a different constant τν. The collision law employs
+the realistic model with vcrit = 5, ϵmax = 0.923, b = 1. Snapshots are at t = 5, 20, 30, 50, 100, 500, 1000, and 2000 orbits.
+high and low τ states (of the same ντ), even when there is no
+intermediate viscously unstable state. In particular, this ap-
+pears achievable for the parameters of the middle column in
+Fig. 3. More generally, systems with non-monotonic collision
+laws have more freedom to exhibit viscous phase-separation
+in radius.
+6
+DISCUSSION AND CONCLUSION
+Most previous work describing the local collisional dynam-
+ics of Saturn’s rings uses relatively simple collision models.
+Given the poorly constrained nature of the collisions, and
+the numerical challenges involved, this is understandable,
+and indeed some success has been achieved in certain appli-
+cations (e.g. self-gravity wakes, viscous overstability). How-
+ever, current models still fail to describe much (if not most)
+of the irregular axisymmetric structure exhibited in Saturn’s
+B and C rings. This invites us to experiment with other more
+complicated collision laws, in particular those that account
+(in a basic way) for surface regolith on ring particles, which
+is deemed to be present and important (e.g. Nicholson et
+al. 2008, Morishima et al. 2012, Deau 2015).
+We conduct N-body simulations with the REBOUND
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+17
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+t=50 orbits
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+t=750 orbits
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+10
+12
+t=1000 orbits
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+200
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+400
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+10
+12
+t=1050 orbits
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+200
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+1.5
+2.0
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+0
+2
+4
+6
+8
+10
+12
+t=2000 orbits
+Figure 13. Snapshots showing the progress of viscous instability starting from an unstable state of τ = 1.4. The collisional parameters
+are vcrit = 5, ϵmax = 0.923, b = 1. The panels describe the x dependent optical depth τ (red) and the angular momentum flux τν (blue).
+Snapshots are at 50, 750, 1000, 1050, and 2000 orbits.
+code of a local patch of Saturn’s rings in which particles un-
+dergo collisions with a prescribed coefficient of restitution
+ϵ depending on impact speed. The main novelty of our ap-
+proach is to employ an ϵ that is a non-monotonic function
+of impact speed, as is suggested by theoretical and experi-
+mental studies of regolith-coated particles (cf. Section 2.1).
+Below a critical impact speed we set ϵ = 0, though neglect
+particle sticking. This relatively minor change in the phys-
+ical set-up immediately introduces major thermodynamical
+changes. For the same optical depth, the rings yield two
+thermally stable steady states, a hot c ≳ 4aΩ and a cold
+c < aΩ state. Which is selected depends on the local ther-
+mal and/or dynamical history, and thus different ring radii
+might fall into one or the other.
+An obvious follow up question is to ask what happens
+at the boundaries of two adjoining different states? We run
+additional simulations in larger domains and find that in
+general the hot state will engulf the cold state, with the
+transition front moving at a speed ≈ 0.5aΩ. Slower mov-
+ing fronts break down because of the imbalance in angular
+momentum flux across the transition. Stationary ‘viscous
+fronts’ are also simulated which join states of different opti-
+cal depth and c but the same angular momentum flux. Note
+that it need not necessarily be the case that hot states always
+take over: smooth variations in the ring’s background prop-
+erties may change propagation, and large amplitude pertur-
+bations (meteoroids, density waves, gravity wakes, etc.) will
+also complicate the picture.
+Our simulation results are exploratory, and should be
+taken as a demonstration of what happens when one relaxes
+the strong modelling assumptions of previous work. They are
+perhaps not yet ready for direct application to structure for-
+mation in Saturn’s rings, not least because of the parameters
+in our regolith laws are poorly constrained. Nonetheless, it is
+irresistible to speculate. We anticipate that a thermal front,
+connecting a warm and cold state of the same dynamical
+optical depth, gives rise to photometric variation (which the
+Cassini cameras may have picked up) but no variation de-
+tectable by occultation experiments. This is precisely the sit-
+uation in the C-ring plateaus (Hedman and Nicholson 2013),
+and indeed, there is evidence of size segregation across these
+structures which may tie in to the greater chance of sticking
+in the colder phase (Marouf et al. 2013, Colwell et al. 2018).
+It may also be relevant for the 10km striations shown by
+Cassini’s cameras in the A and B-rings (cf. Figs 5A and 5B
+in Porco et al. 2005). On the other hand, the steady viscous
+fronts our simulations support, which connect states of high
+and moderate optical depth, bear some resemblance to the
+disjunct bands in the middle B-ring (Colwell et al. 2009).
+A great deal more theoretical work and modelling is needed
+before these associations can be made secure. In particu-
+lar, applications to ring regions exhibiting self-gravity wakes
+must remain tentative until we produce better constrained
+estimates on typical sticking speeds.
+Other areas of future work could explore the interplay
+between the hysteresis and self-gravity wakes, on one hand,
+and viscous overstability, on the other. For example, we
+might anticipate wakes appear only in the cold state, chang-
+ing its viscous properties, and providing energy to jump into
+the hot state. More generally wake activity will produce en-
+hanced heating and thus a change in the thermodynamic
+balances calculated in this paper. Viscous overstability gen-
+erates nonlinear travelling wavetrains which may also favour
+the cold phase; these waves will reflect off the boundaries be-
+tween states, hence complicating the nonlinear saturation of
+the wave turbulence. Simulations including realistic photom-
+etry of thermal fronts might help establish if they might cor-
+respond to any observable structure (Salo and Karjalainen
+2003). Finally, the robustness of bistability must be estab-
+lished when particle sticking is permitted, as in recent sim-
+ulations by Ballouz et al. (2017) and Lu et al. (2018).
+© 0000 RAS, MNRAS 000, 000–000
+
+18
+Larue, Latter, Rein
+ACKNOWLEDGMENTS
+The authors thank the reviewer Heikki Salo and Juergen
+Schmidt, who generously provided a set of helpful and thor-
+ough comments that markedly improved the paper.
+DATA AVAILABILITY
+The data underlying this article will be shared on reasonable
+request to the corresponding author.
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+APPENDIX A: CONVERGENCE TESTS
+We present some results showing the behaviour of a subset
+of our equilibrium solutions as the numerical parameters are
+varied. In particular, we explore their dependence on the size
+of the time-step dt and the numerical domain, showing that
+convergence is achieved when the former is sufficiently small
+and the latter sufficiently large. To simplify the study, we
+adopt a standard Bridges law for two different vcrit (yielding
+hot and warm equilibria) and also a constant ϵ = 0 (yielding
+cold equilibria. We examine very dilute cases τ = 0.1 and
+very dense cases τ = 2.5, thereby determining the numerical
+requirements at the physical ‘boundaries’ of our main set of
+results, and thus for the main results themselves.
+Our convergence results are plotted in Figs A1 and A2,
+the former showing the velocity dispersion c as a function of
+dt, the latter c as a function of box size. Time steps of 10−2
+© 0000 RAS, MNRAS 000, 000–000
+
+Thermal hysteresis in rings
+19
+10-3
+10-2
+10-1
+dt
+100
+101
+102
+C
+Bridges laws
+vcrit=20,  = 0.1
+vcrit=20,  = 2.5
+vcrit=1,  = 0.1
+vcrit=1,  = 2.5
+10-4
+10-3
+10-2
+10-1
+dt
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+C
+ = 0
+ = 0.1
+ = 2.5
+Figure A1. Convergence tests in time step for several set-ups
+spanning dilute and cold, dense and hot, etc.
+or less and a box size of 30 or greater appear to be sufficient
+in most cases. In our main equilibrium runs in Section 3, we
+use dt = 10−3 and a box size of 100.
+APPENDIX B: ILLUSTRATIVE TOY FRONTS
+In this appendix we calculate thermal fronts using the simple
+continuum model of Section 5.1.2 with prescribed functions
+for Λ and k. Noting the bistability at low τ, we adopt a lo-
+gistic reaction term and a linear diffusivity, which in suitable
+units take the form
+Λ = (E − EC)(E − EI)(E − EH),
+k = αE,
+where EC < EI < EH are constant parameters denoting the
+cold, intermediate, and hot states (respectively), and α is an
+additional constant. Both EC and EH are thermally stable,
+but EI is unstable. The basins of attraction of EC and EH,
+however, are controlled by their proximity to EI.
+These functional choices simplify the integrals in the nu-
+merator of (15). The integral of Λ becomes simply −(EC −
+EH)3(EC −2EI +EH)/12, and is negative when the interme-
+diate state is less than the arithmetic mean of the hot and
+cold states, EI < (EC + EH)/2, and positive otherwise. In
+other words, the front will tend to move into the cold state
+when the intermediate state is closer to the cold state, i.e.
+when its basin of attraction is smaller. Similarly, the front
+will tend to move into the hot state when EI is closer to
+EH. If the three thermal states are equidistant and k is a
+101
+102
+Box size
+100
+101
+C
+Bridges laws
+vcrit=20,  = 0.1
+vcrit=20,  = 2.5
+vcrit=1,  = 0.1
+vcrit=1,  = 2.5
+101
+102
+Box size
+0.5
+1
+1.5
+2
+C
+ = 0
+ = 0.1
+ = 2.5
+Figure A2. Convergence tests in box size for several set-ups
+spanning dilute and cold, dense and hot, etc.
+constant, then c = 0 and the front profile can be expressed
+in terms of elliptic integrals.
+The second term in (15) cannot be evaluated without
+knowledge of the front profile. It nonetheless simplifies to
+−α
+� ∞
+−∞(dE/dξ)3dξ, which is clearly negative for monotonic
+front profiles. Thus the linear k law favours the front’s move-
+ment into the cold state, as discussed in Section 5.1.2.
+Finally, we numerically solved (14) using a relaxation
+method (Press et al. 1986), due to the problem’s charac-
+teristic stiffness. We plot a representative front solution in
+Fig. B1. As in Fig. 10, the front is sharp at the cold tran-
+sition, where k is small, and diffuse at the hot transition,
+where it is an order of magnitude larger.
+© 0000 RAS, MNRAS 000, 000–000
+
+20
+Larue, Latter, Rein
+-2
+-1
+0
+1
+2
+3
+4
+0
+2
+4
+6
+8
+10
+12
+E
+Figure B1. Illustrative front calculated numerically, with pa-
+rameters EC = 1, EI = 1.5, EH = 12, and α = 0.5. The front
+moves to the left into the cold state with a speed c = −12.3728.
+© 0000 RAS, MNRAS 000, 000–000
+

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+arXiv:2301.11769v1  [astro-ph.EP]  27 Jan 2023
+MNRAS 000, 1–13 (2023)
+Preprint 30 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Formation of polar circumstellar discs in binary star systems
+Jeremy L. Smallwood,1,2★ Rebecca G. Martin2 and Stephen H. Lubow3
+1Institute of Astronomy and Astrophysics, Academia Sinica, Taipei 10617, Taiwan
+2Department of Physics and Astronomy, University of Nevada, Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA
+3Space Telescope Science Institute, Baltimore, MD 21218, USA
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We investigate the flow of material from highly misaligned and polar circumbinary discs that feed the formation of circumstellar
+discs around each binary component. With three-dimensional hydrodynamic simulations we consider equal mass binaries with
+low eccentricity. We also simulate inclined test particles and highly-misaligned circumstellar discs around one binary component
+for comparison.During Kozai-Lidov (KL) cycles, the circumstellar disc structure is altered through exchangesof disc eccentricity
+with disc tilt. Highly inclined circumstellar discs and test particles around individual binary components can experience very
+strong KL oscillations. The continuous accretion of highly misaligned material from the circumbinary disc allows the KL
+oscillations of circumstellar discs to be long-lived. In this process, the circumbinary material is continuously delivered with a
+high inclination to the lower inclination circumstellar discs. We find that the simulation resolution is important for modeling
+the longevity of the KL oscillations. An initially polar circumbinary disc forms nearly polar, circumstellar discs that undergo
+KL cycles. The gas steams accreting onto the polar circumstellar discs vary in tilt during each binary orbital period, which
+determines how much material is accreted onto the discs. The long-lived KL cycles in polar circumstellar discs may lead to the
+formation of polar S-type planets in binary star systems.
+Key words: binaries: general – circumstellar matter– accretion, accretion discs
+1 INTRODUCTION
+The majority of stars born in dense stellar clusters are part
+of binary star systems (Duquennoy & Mayor 1991; Ghez et al.
+1993; Duchêne & Kraus 2013). The observed orbital eccentrici-
+ties of binaries vary with orbital separation (Raghavan et al. 2010;
+Tokovinin & Kiyaeva 2016). For tight binaries, the eccentricities are
+small, which implies that there has been circularization of the bi-
+nary orbit caused by stellar tidal dissipation (Zahn 1977). More
+widely-separated binaries have observed eccentricities ranging from
+푒b = 0.39 to 0.59, with a considerable number of highly eccentric
+systems with 푒b > 0.8. The interactions of the binary with sur-
+rounding gas may be responsible for the present-day observed binary
+eccentricities (Goldreich & Tremaine 1980; Artymowicz et al. 1991;
+Artymowicz 1992; Armitage & Natarajan 2005; Cuadra et al. 2009;
+Roedig et al. 2011; Muñoz et al. 2019; Zrake et al. 2021). Circumbi-
+nary discs of gas and dust are sometimes observed to be responsible
+to be providing accreting material onto the binary (e.g., Alves et al.
+2019). The gas flow dynamics from the circumbinary disc onto the
+binary components has significant implications for planet formation
+scenarios in binary systems.
+Circumbinary discs are commonly observed to be moderately to
+highly misaligned to the binary orbital plane. For example, the pre-
+main sequence binary KH 15D has a circumbinary disc inclined
+by 5 − 16◦ (Chiang & Murray-Clay 2004; Smallwood et al. 2019;
+Poon et al. 2021). The radial extent of the disc is narrow and pre-
+★ E-mail: [email protected]
+sumed to be rigidly precessing to explain the unique periodic light
+curve. A ∼ 60◦ inclined circumbinary disc is found around the
+main-sequence binary IRS 43 (Brinch et al. 2016), along with mis-
+aligned circumstellar discs around each binary component. There is
+an observed misalignment of about 70◦ between the circumbinary
+disc and the circumprimary disc in HD 142527 (Marino et al. 2015;
+Owen & Lai 2017). Another young binary, HD 98800 BaBb, has the
+only observed polar (inclined by ∼ 90◦) gaseous circumbinary disc
+(Kennedy et al. 2019). The 6–10 Gyr old binary system, 99 Herculis,
+has a nearly polar (about 87◦) debris ring (Kennedy et al. 2012;
+Smallwood et al. 2020). Apart from binaries, stars may also form
+in higher-order systems (Tokovinin 2014a,b). The circumtriple disc
+around the hierarchical triple star system, GW Ori, is tilted by about
+38◦ (Bi et al. 2020; Kraus et al. 2020; Smallwood et al. 2021a).
+The observations of inclined circumbinary discs have implications
+on planet formation models. Observations from space and ground-
+based telescopes reveal that ∼ 50 per cent of the confirmed exoplan-
+ets reside in binary systems (Horch et al. 2014; Deacon et al. 2016;
+Ziegler et al. 2018). For example, the binary system 훾 Cep AB hosts a
+giant planet around the primary star, 훾 Cep Ab (Hatzes et al. 2003). It
+is crucial to study the structure and evolution of protoplanetary discs
+since these are the sites for planet formation (D’Angelo & Lissauer
+2018). A forming planet’s orbital properties are directly related to
+the orientation of the protoplanetary disc. For example, the observed
+young binary system XZ Tau shows both the circumprimary and
+circumsecondary discs are misaligned to the binary orbital plane
+(Ichikawa et al. 2021). The binary system HD 142527 shows the
+presence of a misaligned inner disc around one of the stellar com-
+© 2023 The Authors
+
+2
+Smallwood et al.
+ponents, presumably fed from the circumbinary disc (Price et al.
+2018b). Furthermore, IRAS 04158+2805 is a binary system where
+the two circumstellar discs and the circumbinary discs have been
+observed to be misaligned (Ragusa et al. 2021). Therefore, highly-
+inclined circumstellar discs may give birth to planets on highly-tilted
+orbits.
+Due to viscous dissipation, a misaligned circumbinary disc un-
+dergoes nodal precession and evolves towards either a coplanar or
+polar alignment. For an initially low-inclination circumbinary disc,
+the disc precesses about the angular momentum vector of the bi-
+nary and eventually evolves to be coplanar to the binary orbital
+plane (Facchini et al. 2013; Foucart & Lai 2014). Slightly misaligned
+discs around an eccentric binary undergo tilt oscillations as they
+align, due to the nonaxisymmetric potential produced by the ec-
+centric binary (Smallwood et al. 2019, 2020). For highly inclined
+discs around eccentric orbit binaries, the angular momentum vec-
+tor of the disc precesses about the eccentricity vector of the bi-
+nary (e.g. Aly et al. 2015), which leads the disc to align perpen-
+dicular (i.e., polar) to the binary orbital plane (Martin & Lubow
+2017; Lubow & Martin 2018; Zanazzi & Lai 2018; Martin & Lubow
+2018; Cuello & Giuppone 2019). A massive circumbinary disc that
+is undergoing polar alignment aligns to a generalized polar state
+which is less than 90◦ (Zanazzi & Lai 2018; Martin & Lubow 2019;
+Chen et al. 2019).
+Circumbinary gas discs contain a central cavity around the bi-
+nary where little material is present. The cavity size is determined
+by where the tidal torque is balanced with the viscous torque
+(Artymowicz & Lubow 1994; Lubow et al. 2015; Miranda & Lai
+2015; Franchini et al. 2019b; Hirsh et al. 2020; Ragusa et al. 2020).
+The strength of the binary torque on the disc is dependent on
+the tilt of the circumbinary disc and binary eccentricity. The
+tidal torque at a given radius is zero when the circumbinary
+disc is polar and the binary eccentricity approaches 푒b
+= 1
+(Lubow & Martin 2018) or if the disc is retrograde (e.g., Nixon et al.
+2013). In the simplest models, the production of an outward
+forcing torque by the binary can prevent circumbinary material
+from flowing through the cavity (Lynden-Bell & Pringle 1974;
+Pringle 1991). However, material from the circumbinary disc flows
+through the binary cavity in the form of gaseous streams (e.g.
+Artymowicz & Lubow 1996; Günther & Kley 2002; Nixon & King
+2012; Shi et al. 2012; D’Orazio et al. 2013; Farris et al. 2014;
+Muñoz et al. 2019; Alves et al. 2019). These streams are respon-
+sible for forming and replenishing circumstellar discs around each
+binary component. The accretion of material onto the circumstellar
+discs may aid in the formation of 푆–type planets, those that orbit one
+component of a binary. Accretion of material onto the central binary
+may be suppressed for small disc aspect ratios.
+The structure of a circumstellar disc around one star is
+strongly affected by the tidal field of the binary compan-
+ion
+(Papaloizou & Pringle
+1977;
+Artymowicz & Lubow
+1994;
+Pichardo et al. 2005; Jang-Condell 2015). Circumstellar discs around
+each binary component undergo tidal truncation. A circumstellar disc
+in a circular orbit binary is typically truncated to about one-third to
+one-half of the binary orbital separation The tidal truncation radius
+is expected to decrease with increasing binary eccentricity.
+Kozai-Lidov (KL) oscillations (Kozai 1962; Lidov 1962) have
+been studied extensively to analyze several astronomical pro-
+cesses involving bodies that orbit a member of a binary sys-
+tem that begin on highly misaligned orbits. During KL os-
+cillations, the object’s inclination is exchanged for eccentricity,
+and vice versa. These processes include asteroids and irregular
+satellites (Kozai 1962; Nesvorný et al. 2003), artificial satellites
+(Lidov 1962), tidal disruption events (Chen et al. 2011), forma-
+tion of Type Ia supernovae (Kushnir et al. 2013), triple star systems
+(Eggleton & Kiseleva-Eggleton 2001; Fabrycky & Tremaine 2007),
+planet formation with inclined stellar companions (Wu & Murray
+2003; Takeda & Rasio 2005), giant outbursts in Be/X-ray bi-
+naries (Martin et al. 2014a; Martin & Franchini 2019), inclined
+planetary companions (Nagasawa et al. 2008), mergers of bina-
+ries in galactic nuclei (Blaes et al. 2002; Antonini & Perets 2012;
+Hamers et al. 2018; Hoang et al. 2018; Fragione et al. 2019a,b), stel-
+lar compact objects (Thompson 2011), and blue straggler stars
+(Perets & Fabrycky 2009).
+A highly misaligned initially circular disc around one compo-
+nent of a binary undergoes KL cycles in which its inclination is
+exchanged for eccentricity, and vice versa (Martin et al. 2014a). Due
+to disc dissipation by viscosity and shocks, these oscillations are
+typically significantly damped after a few oscillations. KL oscilla-
+tions can occur in a fluid disc with a wide variety of disc and binary
+parameters (Fu et al. 2015a). When the disc becomes eccentric, it
+overflows its Roche lobe and transfers material to the companion
+star (Franchini et al. 2019a). Self-gravity of a disc can suppress disc
+KL oscillations if the disc is close to being gravitationally unstable
+(Fu et al. 2015b). KL oscillations in a circumstellar disc may have
+significant consequences for planet formation since strong shocks in
+the gas are produced during high eccentricity phases (Fu et al. 2017).
+A misaligned circumbinary disc may form misaligned circumstel-
+lar discs around the individual binary components (e.g., Nixon et al.
+2013; Smallwood et al. 2021b). A highly misaligned disc around
+one component of a binary may be unstable to the Kozai-Lidov (KL)
+mechanism (Martin et al. 2014a). Smallwood et al. (2021b) simu-
+lated the flow of gas originating from an initially misaligned cir-
+cumbinary disc by 60◦. The misaligned gas streams that flow into the
+binary cavity result in formation of highly tilted circumstellar discs
+around each binary component. The inclined circumstellar discs in
+turn undergo KL oscillations. However, the KL oscillations are long-
+lived, due to the continuous accretion of inclined material from the
+circumbinary disc. Long-lived KL cycles have important implica-
+tions for planet formation in binary systems.
+In this work, we extend the previous study Smallwood et al.
+(2021b) and consider more highly inclined circumbinary discs. We
+first revisit the dynamics of highly inclined test particle orbits around
+one component of a binary in Section 2. In Section 3, we describe
+the setup for our hydrodynamical simulations. In Section 4, we dis-
+cuss the results of our circumprimary disc simulations. We simulate
+a highly inclined circumprimary disc in a binary to explore the dy-
+namics of the KL cycles. Previous studies have only dealt with cir-
+cumprimary disc inclinations ≲ 60◦, while we consider higher tilts,
+including a polar circumprimary disc. In Section 5, we show the re-
+sults of our hydrodynamical simulations with an initial circumbinary
+disc, where we consider the flow of material from discs with various
+initial misalignments, including a polar circumbinary disc. Finally, a
+summary is given in Section 6.
+2 KOZAI-LIDOV OSCILLATIONS OF TEST PARTICLES
+Before considering discs, we consider the properties of test parti-
+cle orbits that undergo KL oscillations. As a consequence of the
+conservation of the component of the angular momentum that is
+perpendicular to the binary orbital plane, the test particle’s inclina-
+tion is recurrently exchanged for eccentricity. This conservation is
+MNRAS 000, 1–13 (2023)
+
+Formation of polar circumstellar discs
+3
+Figure 1. Eccentricity (upper panel) and inclination (lower panel) evolution
+of circumprimary test particles under the influence of a circular binary for
+initially circular orbit particles. We vary the initial particle orbital tilt, 푖0,
+beginning with 30◦ (black), 45◦ (blue), 60◦ (red), 75◦ (green), 80◦ (yellow),
+85◦ (purple), and 90◦ (pink). The initial orbital radius of the particle is set at
+푟0 = 0.06푎, where 푎 is the separation of the binary. The time is in units of
+binary orbital period 푃orb.
+expressed as
+�
+1 − 푒2p cos 푖p ≈ const,
+(1)
+where 푖p is the particle inclination with respect to the binary orbital
+plane and 푒p is the eccentricity of the test particle. A initially circular
+orbit particle initially gains eccentricity while reducing its orbital tilt
+(i.e. going towards alignment which means higher values of | cos 푖p|)
+and then circularizes while gaining orbital tilt back to its original
+inclination. For an initially circular orbit particle, KL oscillations
+only occur if the initial tilt of the test particle 푖p0 satisfies cos2 푖p0 <
+cos2 푖cr = 3/5 (Innanen et al. 1997), which requires that 39◦ ≲ 푖p0 ≲
+141◦. From Eq. (1), an initially circular particle orbit can achieve a
+maximum eccentricity given by
+푒max =
+�
+1 − 5
+3 cos2 푖p0.
+(2)
+The increase in a circular particle’s eccentricity can be quite signif-
+icant. For example, if the particle’s initial orbit is tilted by 60◦, the
+maximum eccentricity reached during a KL cycle is about 0.75.
+For eccentric binaries, stronger effects from KL oscillations
+have been found to exist (Ford et al. 2000; Lithwick & Naoz 2011;
+Naoz et al. 2011, 2013a,b; Teyssandier et al. 2013; Li et al. 2014;
+Liu et al. 2015). The KL oscillation period for a particle in the po-
+tential of an eccentric binary is approximately given by
+휏KL
+푃b
+≈ 푀1 + 푀2
+푀2
+푃b
+푃 (1 − 푒2
+b)3/2
+(3)
+(Holman et al. 1997; Innanen et al. 1997; Kiseleva et al. 1998),
+where 푀1 and 푀2 are the masses of the primary and secondary
+components of the binary, respectively, 푃 = 2휋/
+�
+퐺푀1/푎3p is the
+orbital period of the particle with semimajor axis 푎p, 푃b = 2휋/Ωb
+is the orbital period of the binary, 푒b is the binary eccentricity, and
+Ωb =
+�
+퐺(푀1 + 푀2)/푎3
+b is the binary orbital frequency for binary
+semimajor axis 푎b.
+To simulate an inclined circumprimary test particle in a binary,
+we use the 푁–body integrator, MERCURY (Chambers 1999). The
+test particle is orbiting the primary companion with an initial tilt
+푖0 relative to the binary orbital plane. The binary components have
+equal mass so that 푀1 = 푀2 = 푀/2, where 푀 is the total mass
+of the binary. Fu et al. (2015b) ran numerous test particle orbits
+showing the effects the particle and binary parameters have on the
+induced KL oscillations. Following their work, we model an eccentric
+inclined particle around one component of an eccentric binary, more
+applicable to binary systems.
+We first simulate an inclined particle in a circular binary to match
+previous results. Fig. 1 shows the eccentricity and inclination of a
+circumprimary particle as a function of time that begins on a cir-
+cular orbit. The analytic solution for these test particle orbits in the
+quadrupole approximation is given in Lubow (2021). We consider
+various initial tilts of the test particle orbit. The critical inclination
+that the test particle orbit must have to induce KL cycles is ∼ 39◦.
+Thus, a particle tilt of 30◦ (black line) does not undergo KL oscil-
+lations. As the initial inclination of the particle increases, the KL
+oscillations become more frequent, and the growth in the eccentric-
+ity becomes more prominent (in agreement with Fig. 1 in Fu et al.
+(2015b)). The trough in the inclination profile of a test particle be-
+comes narrower with initial inclination. An initial particle orbit tilt
+of 90◦ becomes unstable and collides with the primary star during
+the first KL oscillation because the particles eccentricity exceeds
+1.0. The eccentricity of the polar particle increases almost up to its
+maximum eccentricity before the tilt begins to change.
+Next, we set the initial particle tilt to 60◦ around a slightly eccen-
+tric binary with 푒b = 0.1, as we will consider in the disc simulations.
+We model various initial test particle eccentricities ranging from 0.0
+to 0.5. Figure 2 shows the eccentricity and inclination of eccentric
+circumprimary particles as a function of time in binary orbital pe-
+riods. An inclined circular test particle within an eccentric binary
+has an increased frequency in KL oscillations when compared to a
+particle orbiting one component of a circular binary, as expected by
+equation (3). From Figure 2, when the particle eccentricity is in-
+creased, the maximum eccentricity reached during a KL oscillation
+also increases. However, the difference between the initial eccentric-
+ity to the maximum eccentricity of the particle decreases as the initial
+particle eccentricity increases.
+Lastly, we examine the KL mechanism for a nearly polar particle.
+From Fig. 1, an initially circular orbit particle with an initial orbital
+tilt of 85◦ is unstable to KL oscillations but is otherwise stable.
+We consider a nearly polar orbit particle with an initial orbital tilt
+푖0 = 85◦ around a binary with eccentricity 푒b = 0.1. In Fig. 3 we
+show the particle eccentricity and inclination as a function of time
+in binary orbital periods. The various lines correspond to different
+initial particle eccentricities ranging from 0.0 to 0.5. For all values
+of the initial particle eccentricity we consider, the particle proceeds
+through KL cycles in a periodic fashion. Unlike the particle beginning
+at a tilt of 60◦, a nearly polar particle exhibits similar maximum
+eccentricity close to unity during a KL oscillation regardless of initial
+particle eccentricity. The minimum inclination reached during each
+KL oscillation is roughly independent of particle initial eccentricity.
+MNRAS 000, 1–13 (2023)
+
+100.8
+0.6
+0.4
+0.2
+0
+80
+60
+40
+0
+100
+200
+300
+400
+50
+t/Porb4
+Smallwood et al.
+Figure 2. Eccentricity (upper panel) and inclination (lower panel) evolution
+of circumprimary test particles under the influence of binary with eccentricity
+푒b = 0.1. The initial tilt of the particle orbit is set to 60◦. We vary the initial
+particle eccentricity 푒0 beginning with 푒0 = 0 (black), 0.1 (blue), 0.2 (red),
+0.3 (green), 0.4 (yellow), 0.5 (purple). The initial orbital radius of the particle
+is set at 푟0 = 0.06푎, where 푎 is the separation of the binary. The time is in
+units of binary orbital period 푃orb.
+3 HYDRODYNAMICAL-SIMULATION SETUP
+We use the smoothed particle hydrodynamics (SPH) code phantom
+(Price et al. 2018a) to model gaseous circumbinary and circumstellar
+discs. phantom has been tested extensively for modeling misaligned
+circumbinary discs (Nixon 2012; Nixon et al. 2013; Nixon & Lubow
+2015; Facchini et al. 2018; Smallwood et al. 2019; Poblete et al.
+2019; Smallwood et al. 2020; Aly & Lodato 2020; Hirsh et al. 2020;
+Smallwood et al. 2021b), as well as misaligned circumstellar discs
+around individual binary components (e.g. Martin et al. 2014b;
+Doğan et al. 2015; Franchini et al. 2020). The suite of simulations
+is summarised in Table 1. In this section we describe the setup for
+the binary star, circumprimary disc, and circumbinary disc in further
+detail.
+3.1 Binary star setup
+We model the binary star system as a pair of sink particles, with an
+initial binary separation 푎. The binary is not static but rather evolves
+freely in time. Each sink particle is given an initial mass with 푀1
+being the primary mass and 푀2 being the secondary mass. The total
+binary mass is thereby 푀 = 푀1 + 푀2. All of our simulations assume
+an equal-mass binary (푀1 = 푀2). In Cartesian coordinates, the orbit
+of the binary lies in the 푥-푦 plane initially. The binary begins ini-
+tially at apastron along the 푥-axis. The massive sink particles have
+a hard accretion boundary, meaning that when particles penetrate
+the sink accretion radius, the particle’s mass and angular momentum
+are deposited onto the star (e.g., Bate et al. 1995). A large accretion
+radius is often used to reduce the computation time significantly by
+neglecting to resolve close-in particle orbits. In this work, however,
+we are interested in resolving the formation and evolution of the cir-
+Figure 3. Same as Fig. 2 but for nearly polar test particles with an initial
+orbital tilt 푖0 = 85◦.
+cumstellar material. Therefore, we adopt a relatively small accretion
+radius of 0.05푎 for simulations that begin with a circumbinary disc
+and an accretion radius of 0.025푎 for simulations that begin with
+a circumprimary disc. Using a smaller accretion radius for the cir-
+cumprimary disc simulations ensures that the disc lifetime is longer,
+along with higher disc resolution. The more eccentric the binary,
+the smaller the outer truncation radius for the circumstellar discs
+(Artymowicz & Lubow 1994). Having a small binary eccentricity
+helps with the resolution of the circumstellar discs. On the other
+hand, to have a stable polar circumbinary disc, the binary eccentric-
+ity needs to be a non-zero value. The initial binary eccentricity is set
+to 푒b = 0.1, with the binary eccentricity vector along the positive
+푥–axis. With this value of binary eccentricity, the critical tilt of the
+circumbinary disc to remain nearly polar is ∼ 77◦ (see eq. 33 in
+Martin & Lubow 2019).
+3.2 Circumprimary disc setup
+To model a circumprimary disc, we follow the methods of
+Martin et al. (2014b). Runs 1-5 in Table 1 simulate initially a circum-
+primary disc. The inner and outer disc radii are set at 푟in = 0.025푎
+and 푟out = 0.25푎, respectively, with a initial total disc mass
+푀CPD = 10−3푀. The circumprimary disc consists of 750, 000 equal-
+mass Lagrangian particles. We neglect any effects of self-gravity. The
+disc surface density profile is initially a power law distribution given
+by
+Σ(푟) = Σ0
+� 푟
+푟in
+�−푝
+,
+(4)
+where we set 푝 = 3/2. We adopt a locally isothermal disc with
+sound speed 푐s ∝ 푅−3/4, 퐻/푟 = 0.035 at 푟 = 푟in, and 퐻/푟 = 0.02
+at 푟 = 푟out. With this prescription, the viscosity parameter 훼 and
+⟨ℎ⟩/퐻 are effectively constant over the radial extend of the disc
+(Lodato & Pringle 2007). For the circumprimary disc simulations,
+we take the Shakura & Sunyaev (1973) 훼 parameter to be 0.01. To
+MNRAS 000, 1–13 (2023)
+
+0.8
+0.6
+0.4
+0.2
+0
+90
+80
+70
+60
+50
+40
+30
+0
+20
+40
+60
+80
+10
+t/Porb10.8
+0.6
+0.4
+0.2
+0
+60
+50
+40
+30
+0
+20
+40
+60
+80
+10
+t/Porb0Formation of polar circumstellar discs
+5
+Table 1. The setup of the SPH simulations that includes an initial circumprimary disc (CPD) or circumbinary disc (CBD). The table lists the initial parameters
+beginning with the disc tilt 푖0, inner disc radius 푟in, outer disc radius 푟out, 훼 viscosity parameter, disc aspect ratio at inner disc radius 퐻/푟in, disc aspect ratio at
+outer disc radius 퐻/푟out, the number of particles, and whether or not the circumstellar discs undergo the Kozai-Lidov (KL) instability.
+Model
+Disc Setup
+푖0/◦
+푟in/푎
+푟out/푎
+훼
+퐻/푟in
+퐻/푟out
+# Particles
+KL unstable?
+run1
+CPD
+60
+0.025
+0.25
+0.01
+0.035
+0.02
+750, 000
+Yes
+run2
+CPD
+70
+0.025
+0.25
+0.01
+0.035
+0.02
+750, 000
+Yes
+run3
+CPD
+80
+0.025
+0.25
+0.01
+0.035
+0.02
+750, 000
+Yes
+run4
+CPD
+90
+0.025
+0.25
+0.01
+0.035
+0.02
+750, 000
+Yes
+run5
+CPD
+100
+0.025
+0.25
+0.01
+0.035
+0.02
+750, 000
+Yes
+run6∗
+CBD
+60
+1.6
+2.6
+0.1
+0.1
+0.088
+1.5 × 106
+Yes
+run7
+CBD
+60
+1.6
+2.6
+0.1
+0.1
+0.088
+750, 000
+Yes
+run8
+CBD
+90
+1.6
+2.6
+0.1
+0.1
+0.088
+1.5 × 106
+Yes
+∗ Simulation from Smallwood et al. (2021b)
+accomplish this, the SPH artificial viscosity coefficients are set as
+훼AV = 0.18 and 훽AV = 2.0. The disc is resolved with shell-averaged
+smoothing length per scale height ⟨ℎ⟩/퐻 ≈ 0.55.
+3.3 Circumbinary disc setup
+To model an initially flat but tilted gaseous circumbinary disc, we
+follow the methods of Smallwood et al. (2021b). Runs 6, 7, and 8
+in Table 1 describe the simulations of a circumbinary disc. The disc
+initially consists of 1.5 × 106 equal-mass Lagrangian SPH particles.
+We also model a 750, 000 particle simulation for a resolution study.
+The simulations run for 45 푃orb, where 푃orb is the orbital period of
+the binary. This is sufficient time for the forming circumstellar discs
+to reach a quasi-steady state. We simulate initially highly misaligned
+disc inclinations of 푖0 = 60◦, 90◦. A disc with 푖0 = 90◦ is in a polar
+configuration, where the angular momentum vector of the disc is
+aligned to the eccentricity vector of the binary. At the beginning
+of our simulations, we select an initial inner disc radius, 푟in, and
+outer disc radius, 푟out, where the initial total disc mass, 푀CBD, is
+confined. All of the simulations model a low-mass circumbinary disc
+such that 푀CBD = 10−3푀. We choose the circumbinary disc to be
+radially very narrow and close to the binary orbit. This is done to
+maximise the accretion rate onto the binary and hence the resolution
+of the circumstellar discs (e.g., Smallwood et al. 2021b). For our
+simulations, we take 푟in = 1.6푎 and 푟out = 2.6푎. The tidal torque
+is weaker at a given radius for a more highly misaligned disc which
+allows the inner disc radius to lie closer to the binary than a coplanar
+disc (e.g., Lubow et al. 2015; Miranda & Lai 2015; Lubow & Martin
+2018). The inner truncation radius of a polar circumbinary disc is
+around 1.6 푎 (Franchini et al. 2019b), much smaller than the 2 − 3 푎
+expected for coplanar discs (Artymowicz & Lubow 1994).
+The disc surface density profile follows from Equation (4). The
+physical disc viscosity is incorporated by using artificial viscosity
+훼av, which is detailed in Lodato & Price (2010). By using our sur-
+face density profile and a disc aspect ratio 퐻/푟 = 0.1 at 푟in, the
+shell-averaged smoothing length per scale height ⟨ℎ⟩/퐻 and the disc
+viscosity parameter 훼 are constant over the radial extent of the disc
+(Lodato & Pringle 2007). The circumbinary disc is initially resolved
+with ⟨ℎ⟩/퐻 ≈ 0.11. The parameters for the simulations require a
+high viscosity in order to maximize the accretion rate on to the
+circumstellar discs and provide better resolution. We consider a
+relatively high value for the Shakura & Sunyaev (1973) 훼SS of 0.1.
+In a more realistic system, the disc viscosity may be lower.
+In order to more accurately simulate the formation and develop-
+ment of circumstellar discs, we adopt the locally isothermal equation
+of state of Farris et al. (2014) and set the sound speed 푐s to be
+푐s = F 푐s0
+�
+푎
+푀1 + 푀2
+�푞 � 푀1
+푟1
++ 푀2
+푟2
+�푞
+,
+(5)
+where 푟1 and 푟2 are the radial distances from the primary and sec-
+ondary stars, respectively, and 푐s0 is a constant with dimensions of
+velocity. 푞 is set to 3/4. F is a dimensionless function of position
+that we define below. This sound speed prescription guarantees that
+the temperature profiles in the circumprimary and circumsecondary
+discs are set by the primary and secondary stars, respectively. For
+푟1, 푟2 ≫ 푎, 푐s is set by the distance from the binary centre of mass.
+To increase the resolution of the circumstellar discs, we include a
+function F in Equation (5) as detailed in Smallwood et al. (2021b).
+The purpose of F is to modify the sound speed around each binary
+component so that the viscous timescale is longer. This increases
+the mass (and hence the resolution) in the steady-state circumstellar
+discs. We take
+F =
+�√
+0.001,
+if 푟1 or 푟2 < 푟c,
+1,
+otherwise,
+(6)
+where 푟c is the cutoff radius. We set a cutoff radius of 푟c = 0.35푎
+from each binary component (e.g., Smallwood et al. 2021b). Using
+the prescription mentioned above ensures that the disc aspect ratio
+of the circumstellar discs at radius 푟 = 0.1푎 is 퐻/푟 ∼ 0.01, which
+is one-tenth of the disc aspect ratio at the initial inner circumbinary
+disc radius.
+3.4 Analysis routine
+We analyse the disc and binary parameters as a function of time. The
+parameters include tilt, eccentricity, the longitude of the ascending
+node, mass, and mass accretion rate. To probe the circumprimary
+disc simulations, we average over particles in the radial range from
+0.025푎 to a distance of 0.30푎. For the circumbinary disc simulations,
+we average over particles in the radial range from 1.4푎 to a distance
+of 10푎. For the forming circumstellar discs, we average over all
+particles bound to each binary component (i.e., the specific energies,
+kinetic plus potential, of the particles are negative, neglecting the
+thermal energy). The tilt, 푖, is defined as the angle between the initial
+angular momentum vector of the binary (the 푧-axis) and the angular
+momentum vector of the disc. The longitude of the ascending node,
+휙, is measured relative to the 푥-axis (the initial binary eccentricity
+vector).
+MNRAS 000, 1–13 (2023)
+
+6
+Smallwood et al.
+20
+40
+60
+80
+100
+120
+0
+0.5
+1
+-100
+0
+100
+0
+0.5
+1
+0
+5
+10
+15
+10-5
+2
+3
+4
+5
+Panel 1
+Figure 4. Evolution of a KL unstable circumprimary disc as a function of time
+in units of the binary orbital period 푃orb. We simulate five different initial disc
+inclinations, which are 60◦ (run1 from Table 1, black), 70◦ (run2, blue), 80◦
+(run3, red), 90◦ (run4, green), and 100◦ (run5, yellow). The disc parameters
+are tilt 푖 (panel 1), eccentricity 푒 (panel 2), longitude of the ascending node
+휙 (panel 3), and disc mass 푀d (panel 4). The mass accretion rate �푀 onto the
+primary star is shown in panel 5.
+4 HYDRODYNAMICAL RESULTS WITH A
+CIRCUMPRIMARY DISC
+This section considers the evolution of a circumprimary disc in the
+absence of accretion from a circumbinary disc. This enables us to
+disentangle the effect of accretion onto the circumstellar discs. We
+focus on large circumprimary disc misalignments in an eccentric
+binary star system. We consider five different initial disc tilts, 60◦
+(run1 from Table 1), 70◦ (run2), 80◦ (run3), 90◦ (run4), and 100◦
+(run5). Figure 4 shows the disc tilt, eccentricity, the longitude of
+the ascending node, the mass of the circumprimary disc, and the
+accretion rate onto the primary star as a function of time in binary
+orbital periods. The disc exhibits KL cycles for each initial tilt, where
+the disc eccentricity and inclination are exchanged. For a disc with
+an initial tilt of 60◦, Martin et al. (2014a) found that the first KL
+oscillation occurred around 10 Porb for a circular binary. In our case,
+the disc with the same initial tilt undergoes the first KL oscillation
+much sooner due to the binary having a slightly eccentric orbit (see
+Fig.12 in Fu et al. 2015a). Due to viscous dissipation and the lack
+of circumbinary material, the KL oscillations damp quickly in time.
+For higher initial inclinations, 70◦, 88◦, 90◦ and 100◦, the discs do
+not survive after one KL oscillation for our given sink size. The
+x-z plane      t = 0 Porb
+0.2a
+y-z plane      t = 0 Porb
+0.2a
+x-z plane      t = 10 Porb
+0.2a
+y-z plane      t = 10 Porb
+0.2a
+x-z plane      t = 15 Porb
+0.2a
+y-z plane      t = 15 Porb
+0.2a
+Figure 5. The evolution of polar circumprimary disc (run4 from Table 1).
+The white circles denote the eccentric orbit binary components with an initial
+binary separation of 푎. The top row shows the initial disc setup. The middle
+and bottom rows show the disc evolution at 푡 = 10 푃orb and 푡 = 15 푃orb,
+respectively, where 푃orb is the binary orbital period. The color denotes the
+gas surface density, with the orange regions being about three orders of mag-
+nitude larger than the purple regions. The left column shows the 푥–푧 plane,
+and the right column shows the 푦–푧 plane. At 푡 = 10 푃orb, the circumpri-
+mary disc is highly eccentric due to the Kozai-Lidov instability. Also, at this
+time, a circumsecondary disc is being formed from material flowing close to
+the secondary binary component from the eccentric circumprimary disc. At
+푡 = 15 푃orb, the circumprimary disc has completely dissipated from being
+accreted onto the primary star and transferring material to the secondary star.
+At this time, there is more material in the newly formed circumsecondary
+disc.
+discs become very eccentric, which leads to the majority of the disc
+material being accreted by the primary star. Increasing the resolution
+of these simulations does not lengthen the disc lifetime. However,
+if we were to use a smaller sink size, then the disc could survive
+through the KL oscillations. A smaller sink size would ensure that
+a larger portion of the disc could survive. An accretion radius of
+∼ 0.01 au is comparable to the size of the star, but we simulate
+a larger sink size for computational reasons and to compare with
+the circumbinary disc simulations detailed in the next Section. The
+initially polar disc’s tilt does not change much from polar before the
+majority of the disc is accreted. This is likely a consequence of the
+high disc eccentricities that are developed which is consistent with
+the results for test particle orbits (see Fig. 1). In the retrograde case,
+MNRAS 000, 1–13 (2023)
+
+Formation of polar circumstellar discs
+7
+60
+70
+80
+0
+0.2
+0.4
+0.6
+-100
+0
+100
+10-6
+10-4
+10
+20
+30
+40
+10-6
+10-4
+2
+3
+4
+5
+Panel 1
+Figure 6. Resolution study for a circumbinary disc that is initially misaligned
+by 60◦. The blue curves represent the simulation with initially 1.5 × 106
+particles in the circumbinary disc, while the red curves denotes the simulation
+with initially 750, 000 particles. The first four panels show the disc parameters
+for the newly forming circumprimary disc as a function of time in units of the
+binary orbital period, 푃orb. The disc parameters are tilt 푖 (panel 1), eccentricity
+푒 (panel 2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d
+(panel 4). The black dotted curve in the third panel denotes the circumbinary
+disc. The lower panel shows the mass accretion rate onto the primary star
+�푀pri (panel 5).
+푖0 = 100◦, as the disc eccentricity increases, the inclination also
+increases, opposite to the prograde cases.
+Highly inclined particle orbits experience a large (nearly 180◦)
+shift in 휙 within a small time interval centered about the eccentricity
+maximum (see the plot for Ω(푡) in Figure 1 of Lubow (2021)). This
+large shift does not appear in Figure 4 or in any of our other phase
+results. We are not sure why this is the case. Perhaps the disc is
+unable to respond to such a large shift within a short time.
+We further examine the evolution of the polar (푖0 = 90◦) circumpri-
+mary disc. In Fig. 5, we show the polar circumprimary disc structure
+at three different times, 푡 = 0 푃orb, 10 푃orb, and 15 푃orb. Initially, the
+polar disc around the primary star (left white dot) is edge-on in the
+푥-푧 plane and face-on 푦-푧 plane. At 푡 = 10 푃orb, the disc is at peak ec-
+centricity growth from the KL instability. Also, at this time, streams
+of material from the circumprimary disc flow around the secondary
+star (right white dot) and begin forming a circumsecondary disc. At
+푡 = 15 푃orb, the circumprimary disc has dissipated due to accretion
+onto the primary star and transporting material to the circumsec-
+ondary disc. The newly formed circumsecondary disc is at a lower
+tilt, below the threshold, to induce the KL cycles.
+5 HYDRODYNAMICAL RESULTS WITH A
+CIRCUMBINARY DISC
+In this section we examine how misaligned and polar circumbinary
+material flows through the binary cavity and forms circumstellar
+discs around each binary component. We first conduct a resolution
+study of our earlier work from Smallwood et al. (2021b), modeling
+an initially 60◦ misaligned circumbinary disc. We then focus on the
+polar circumbinary disc case.
+5.1 Resolution Study
+We examine a circumbinary disc with an initial misalignment of
+푖0 = 60◦ with two different initial numbers of particles, 1.5 × 106
+(run6) and 750, 000 (run7). The upper four panels in Figure 6 show
+the circumprimary disc parameters as a function of time. The bot-
+tom panel shows the mass accretion rate onto the primary star. The
+blue curves represent the 1.5 × 106 particle simulation, while the red
+curves represent the 750, 000 particle simulation. Panels 1 and 2 show
+the evolution of disc eccentricity and inclination where the forming
+circumprimary disc undergoes KL oscillations from the continuous
+accretion of material from the circumbinary disc. The oscillations
+damp in time at both resolutions, with the lower resolution simu-
+lation damping more quickly. Therefore, the oscillations are likely
+limited by resolution. If the accretion timescale is long compared
+to the KL timescale, we expect the KL oscillations to damp over
+time, similar to the circumprimary disc simulations without accre-
+tion shown in the previous Section. If the accretion timescale is short
+compared to the KL timescale, there should be no KL oscillations
+present. In this case, the material moves through the disc faster than
+it becomes unstable to KL oscillations. We expect the optimal oscil-
+lations when the timescales are comparable because the disc refills
+mass on the timescale that the oscillations take place. For the simula-
+tion with a 60◦ tilted circumbinary disc, the accretion timescales for
+the primary and secondary are ∼ 1.5 Porb, whereas the KL timescale
+for this simulation is ∼ 5 Porb. The simulation is in the regime where
+the accretion timescale is shorter than the KL oscillation timescale
+because when the disc becomes eccentric during the KL oscillations,
+a large amount of disc material is accreted, reducing the accretion
+timescale. However, the accretion timescale is dependent on the disc
+viscosity. In our hydrodynamical simulations, we use an artificial
+viscosity to model an expected Shakura & Sunyaev (1973) viscos-
+ity coefficient. The number of Lagrangian particles determines how
+close the artificial viscosity is to the actual value. Thus, the 훼 is
+artificially higher at lower resolutions, leading to a shorter accre-
+tion timescale. For our higher-resolution simulation, the 훼 is lower,
+leading to a longer accretion timescale.
+Panel 3 in Fig. 6 shows the longitude of the ascending node as
+a function of time. The precession rate of the circumprimary disc
+is only slightly faster than the circumbinary disc on average. In the
+absence of the effects of KL oscillations, the nodal precession rate
+of the primary disc, assuming constant surface density Σ out to disc
+radius 푟 from the primary, is given by
+휔pr = − 15푀2푟3
+32푀1푎3
+b
+cos (푖) Ω(푟),
+(7)
+where 푖 is inclination angle of the primary disc relative to the binary
+orbital plane and Ω =
+�
+퐺푀1/푟3 is the angular velocity in the disc
+MNRAS 000, 1–13 (2023)
+
+8
+Smallwood et al.
+x-z plane      	t = 0 Porb
+a
+y-z plane      	t = 0 Porb
+a
+x-z plane      	t = 25 Porb
+a
+y-z plane      	t = 25 Porb
+a
+Figure 7. The formation of polar circumstellar discs from an initially low-
+mass polar circumbinary disc (run8). The white circles denote the eccentric
+orbit binary components with an initial binary separation of 푎. The upper
+panels denote the initial disc setup, while the bottom panels show the disc
+evolution at 푡 = 25 푃orb, where 푃orb is the binary orbital period. At this time,
+nearly polar circumstellar discs are forming around each binary component.
+The color denotes the gas density using a weighted density interpolation,
+which gives a mass-weighted line of sight average. The yellow regions are
+about three orders of magnitude larger than the purple. The left column shows
+the 푥–푧 plane, and the right column shows the 푦–푧 plane.
+(Larwood et al. 1996). With 푟 = 0.35 푎b, we find 휔pr = 6◦/푃orb
+with a revolution period of ∼ 56 Porb. Therefore, the circumstellar
+discs should have nodally precessed 75 per cent of a revolution in
+45 Porb. In panel 3 we see that the circumstellar discs have only
+completed roughly 30 per cent of a nodal revolution. It is possible
+that the circumprimary phase is affected by the phase of accreted
+gas from the circumbinary disc that undergoes relatively slow nodal
+precession. As discussed in Section 4, KL oscillations modify the
+nodal precession rate of a test particle in a way that we do not
+see in the disc simulations. Lastly, the mass in the circumprimary
+discs oscillates in time, with the troughs corresponding with each
+high eccentricity period. During each high eccentricity phase, the
+accretion rate peaks as seen in panel 5.
+5.2 Polar discs
+In this section, we present a hydrodynamical simulation of the flow of
+material from a polar circumbinary disc onto the binary components
+(run8). The top row of Fig. 7 shows the initial configuration of
+the polar circumbinary disc around an eccentric binary. The bottom
+row shows the disc structure at 푡 = 25 푃orb. The circumbinary disc
+remains nearly polar (∼ 90◦) as shown in the 푥-푧 plane. Material flows
+from the polar circumbinary disc and forms nearly polar circumstellar
+discs around each binary component. The cavity size is smaller in
+the polar disc compared to a coplanar disc simulation as expected
+(Lubow et al. 2015; Miranda & Lai 2015).
+The upper four panels in Fig. 8 show the inclination, eccentricity,
+the longitude of the ascending node, and disc mass for the three
+60
+80
+100
+120
+0
+0.2
+0.4
+0.6
+-100
+0
+100
+10-6
+10-4
+10-2
+10
+20
+30
+40
+10-4
+2
+3
+4
+5
+Panel 1
+Figure 8. Simulation results for run8 for an initially polar circumbinary disc.
+The disc parameters are shown for the circumprimary, circumsecondary, and
+circumbinary discs as a function of time in units of the binary orbital period,
+푃orb. The upper four panels show the disc tilt 푖 (panel 1), eccentricity 푒 (panel
+2), longitude of the ascending node 휙 (panel 3), and disc mass 푀d (panel 4)
+for the three discs. The lower panel shows the mass accretion rate onto the
+sinks �푀 (panel 5).
+discs as a function of time in binary orbital periods. The lower panel
+shows the mass accretion rate onto the sinks. The circumstellar discs
+form at a time of ∼ 10 Porb, later than in the simulation with a lower
+level of circumbinary disc misalignment. The circumbinary disc tilt
+evolves in time. Since we model a disc with a non-zero mass, it
+will align to a generalised polar state with an inclination that is <
+90◦ (e.g., Martin & Lubow 2019; Chen et al. 2019). In this case, the
+circumstellar discs form slightly retrograde, with a tilt just above 90◦.
+The primary and secondary discs form with an eccentricity of ∼ 0.25.
+However, the polar circumstellar discs undergo the KL instability,
+which forces the disc eccentricity and tilt to oscillate in time. Looking
+at panels 1 and 2, we see that as the disc eccentricity increases, the
+disc tilt also increases, the opposite of the conventional KL case
+involving prograde orbits. However, this result is consistent with the
+KL mechanism for retrograde orbits. Panel 3 shows the evolution of
+the longitude of the ascending node in time. Since the circumprimary
+and circumsecondary discs are nearly polar, they exhibit very little
+precession (see equation 7 and discussion below it). The mass of the
+polar circumstellar discs oscillates in time (panel 4), likely due to
+the oscillating disc eccentricity. The polar circumbinary disc has lost
+∼ 25 per cent of its initial mass.
+MNRAS 000, 1–13 (2023)
+
+0Formation of polar circumstellar discs
+9
+a
+Figure 9. Edge-on view (푥–푧 plane) of a polar circumbinary disc (run8) at
+a time 푡 = 5 푃orb. We ignore the main portions of the disc confined within
+푟 < 0.45푎b, where 푎b is the separation of the binary. The binary components
+are shown as the green dots. The colours denote the disc surface density,
+with the orange regions being about three orders of magnitude larger than the
+purple regions. We overlay the velocity vectors shown by the black arrows.
+The length of the arrow is proportional to the velocities of the particles. We
+see two asymmetric lobes of material that are produced by the binary. Several
+of the velocity vectors are directed away from the plane of the circumbinary
+disc; however, the material then falls back onto the disc gap.
+The KL oscillations from Fig. 8 damp in time. However, from our
+resolution study, the damping is primarily due to the initial number of
+particles. The accretion timescale for this simulation is ∼ 15 Porb, and
+the KL timescale in this case is ∼ 10 Porb. The accretion timescale
+is longer in the polar simulation than in the 60◦ simulation because
+the polar circumstellar discs become less eccentric during each KL
+cycle, accreting less disc material. For a higher resolution, we expect
+the KL oscillations to be long-lived even for polar circumstellar discs.
+On the bottom-left panel in Fig. 7, we see that some material is
+flung out of the disc plane on both sides of the polar circumbinary
+disc. This material forms two lobes on both sides of the disc. Figure 9
+shows the edge-on view of the disc surface density, along with the
+velocity vectors. The material is being flung outwards but remains
+bound to the binary. Therefore, the material then falls back into the
+gap region of the circumbinary disc. Throughout the simulation, the
+material is periodically flung out every 0.5 푃orb when the binary
+components pass through the polar circumbinary disc plane.
+We further examine the flow of polar circumbinary material onto
+the forming circumstellar discs. First, we investigate the tilt of the
+gaseous streams that accrete onto the circumstellar discs as a function
+of time. Figure 10 shows the circumbinary disc tilt as a function of
+disc radius. The inner edge of the disc lies roughly at 1.6푎. The
+curves that are shown at radii < 1.6푎 map the tilt of the streams.
+We show the disc tilt for a full binary orbital period from 20 Porb
+to 21 Porb in increments of 0.1 Porb. At every 0.5 Porb, the tilt of the
+streams are low at ∼ 80◦. When the binary orbital period is not at
+half increments, the tilt of the streams increases beyond 90◦. For
+example, at times 20.2 − 20.3 Porb and 20.6 − 20.7 Porb, the streams
+are highly tilted. Recall that the forming circumstellar discs initially
+1
+1.5
+2
+2.5
+75
+80
+85
+90
+95
+100
+20.0
+20.1
+20.2
+20.3
+20.4
+20.5
+20.6
+20.7
+20.8
+20.9
+21.0
+Figure 10. Circumbinary disc tilt, 푖, as a function of radius, 푟, for the polar
+circumbinary disc. The color corresponds to the time in binary orbital periods,
+Porb.
+form at a high disc tilt, > 90◦. Therefore, whenever the gaseous
+streams are highly tilted, there is an increased accretion of material
+onto the circumstellar discs from the circumbinary disc. When the
+streams are less inclined, every 0.5 Porb, there will be less material
+accreted onto the polar circumstellar discs. This phenomenon is also
+consistent with Fig. 9, where material is flung out of the plane of the
+circumbinary disc every 0.5 Porb. We test this by further visualizing
+the inflow of material. Figure 11 shows snapshots of zoomed-in views
+in the 푥–푧 and 푦–푧 planes of the disc surface density, showing the
+gaseous streams accreting onto the nearly polar circumstellar discs.
+The snapshots show the flow of material over 20 Porb to 20.9 Porb
+in increments of 0.1 Porb. Higher density streams occur at times
+20.3 Porb and 20.7 Porb. The flow of material decreases every 0.5 Porb
+during the orbit. At these times, the steams are less dense, leading to
+less material accreting onto the circumstellar discs.
+We relate the flow of material from Fig. 11 to the mass of the
+circumstellar discs. Figure 12 shows the mass of the circumprimary
+disc from 20 Porb to 25 Porb folded on top of one another for each
+orbital period. The vertical dashed-lines denote the times when the
+binary is aligned with the circumbinary disc plane, which is assumed
+when the stars are both aligned with 푥–푧 plane. Each time the binary
+aligns to the plane of the disc, the masses of the circumstellar discs
+increase. The mass of the disc decreases every 0.5 Porb. This be-
+haviour repeats every orbital period. Overall, the disc mass deceases
+in time due to the KL mechanism.
+6 SUMMARY
+In this work, we investigated the flow of material from a circumbi-
+nary disc that results in the formation circumstellar discs around each
+binary component. We simulated an initially highly misaligned and
+polar circumbinary disc using three-dimensional SPH. We consid-
+ered cases of low initial binary eccentricity (typically 푒b = 0.1) and
+binary mass ratio of unity. We also simulated cases of test particles
+MNRAS 000, 1–13 (2023)
+
+10
+Smallwood et al.
+x-z plane      	t = 20.0 Porb
+0.25a
+y-z plane      	t = 20.0 Porb
+0.25a
+x-z plane      	t = 20.5 Porb
+0.25a
+y-z plane      	t = 20.5 Porb
+0.25a
+x-z plane      	t = 20.1 Porb
+0.25a
+y-z plane      	t = 20.1 Porb
+0.25a
+x-z plane      	t = 20.6 Porb
+0.25a
+y-z plane      	t = 20.6 Porb
+0.25a
+x-z plane      	t = 20.2 Porb
+0.25a
+y-z plane      	t = 20.2 Porb
+0.25a
+x-z plane      	t = 20.7 Porb
+0.25a
+y-z plane      	t = 20.7 Porb
+0.25a
+x-z plane      	t = 20.3 Porb
+0.25a
+y-z plane      	t = 20.3 Porb
+0.25a
+x-z plane      	t = 20.8 Porb
+0.25a
+y-z plane      	t = 20.8 Porb
+0.25a
+x-z plane      	t = 20.4 Porb
+0.25a
+y-z plane      	t = 20.4 Porb
+0.25a
+x-z plane      	t = 20.9 Porb
+0.25a
+y-z plane      	t = 20.9 Porb
+0.25a
+Figure 11. Zoomed-in snapshots of the disc surface density showing the flow of material from a polar circumbinary disc onto the nearly polar circumstellar
+discs. The white circles denote the eccentric orbit binary components with an initial binary separation of 푎. The color denotes the gas density using a weighted
+density interpolation, which gives a mass-weighted line of sight average. The yellow regions are about three orders of magnitude larger than the purple. We view
+the orbit of the binary in the 푥–푧 and 푦–푧 planes. The snapshots show a period from 20 Porb to 20.9 Porb in increments of 0.1 Porb, where 푃orb is time in binary
+orbital periods.
+MNRAS 000, 1–13 (2023)
+
+00001100Formation of polar circumstellar discs
+11
+0.2
+0.4
+0.6
+0.8
+2.5
+3
+3.5
+4
+4.5
+5
+10-6
+20-21 Porb
+21-22 Porb
+22-23 Porb
+23-24 Porb
+24-25 Porb
+Figure 12. The circumprimary disc mass evolution during one binary orbital
+period, Porb, at times 20 − 21 Porb (blue), 21 − 22 Porb (red), 22 − 23 Porb
+(yellow), 23−24 Porb (purple), and 24−25 Porb (green). The mass of the disc
+decreases every 0.5 Porb. The vertical dashed-lines denote the times when
+the binary is aligned with the circumbinary disc plane during 20 − 21 Porb.
+An increased flow of material onto the circumstellar discs occurs when the
+binary is aligned with the circumbinary disc plane.
+around the primary star and cases of circumprimary discs only (i.e.,
+no circumbinary or circumsecondary discs) for comparison.
+In order to carry out these simulations in a reasonable amount of
+time, we made some compromises on our choice of parameters. In
+particular, we introduced a higher viscosity parameter for the cir-
+cumbinary disc than is likely to occur and a lower temperature of
+the gas in the gap region. These choices were made to improve the
+resolution of the simulations. Even with these parameters, the reso-
+lution is still playing a role in our results (see Fig. 6). While we have
+chosen the disc parameters (훼 and 퐻/푅) in our simulations to max-
+imise the accretion rate on to the binary components and therefore
+the simulation resolution, we expect the general behaviour to persist
+for more realistic parameters applicable to protoplanetary discs. The
+mass of the circumstellar disc scales with the infall accretion rate.
+If the resolution of the circumstellar disc is too poor, then the disc
+artificially accretes rapidly due to the artificially enhanced effects of
+viscosity at low density in the SPH code.
+We first examined the behavior of initially highly inclined circum-
+stellar discs that are not supplied with material from a circumbinary
+disc. A polar test particle in orbit around a primary star reaches an
+eccentricity of nearly unity during the first KL cycle, forcing the
+particle to become unbound or hit the central star. Similarly, initially
+highly inclined circumstellar discs around individual binary compo-
+nents can experience very strong KL oscillations. For an equal mass
+binary containing only a single circumstellar disc at high inclina-
+tion between 70◦ and 100◦, the disc undergoes only a single KL
+oscillation before losing nearly all its mass for our given sink size.
+Some of the disc mass is transferred to the companion star to form
+a low inclination disc that does not undergo KL oscillations. These
+results suggests that such high inclinations of discs are short-lived
+due to enhanced dissipation from shocks that leads to tilt evolution
+on short timescales. In contrast, discs that are highly inclined but are
+not subject to KL oscillations would undergo much slower evolution.
+In particular, a polar disc would not precess (see e.g., equation (7))
+and therefore not warp. The disc would then not be subject to torques
+that act to change its inclination.
+In this work, and from Smallwood et al. (2021b), we showed that
+the continuous accretion of material from the circumbinary disc
+allows the effects of KL oscillations on circumstellar discs to be
+much longer-lived. In this process, the circumbinary material is con-
+tinuously delivered with a high inclination to the lower inclination
+circumstellar discs. We found that the simulation resolution is impor-
+tant for modeling the longevity of the KL oscillations. We find longer
+lived KL oscillations that show signs of mild weakening in time, pos-
+sibly due to the resolution (e.g., Figure 6). The balance between the
+accretion timescale and the KL timescale determines whether the
+oscillations are sustained or damp in time. If the circumstellar disc
+material were to accrete on a much shorter timescale than the KL os-
+cillation period, we would not expect the KL oscillations to operate.
+We found that with increasing resolution, the accretion timescale
+becomes comparable to the KL timescale, favoring sustained KL
+oscillations.
+Planet formation is thought to still occur in non-zero eccentric-
+ity discs (Silsbee & Rafikov 2021). In the case of S-type planets
+(planets orbiting one of the stellar companions in a binary), gravita-
+tional perturbations from an eccentric orbit stellar companion and an
+eccentric disc increase planetesimal eccentricities, leading to colli-
+sional fragmentation, rather than growth, of planetesimals. However,
+Rafikov & Silsbee (2015) analyzed the planetesimal motion in ec-
+centric protoplanetary discs when the planetesimals were affected by
+gas drag and disc gravity. They found that the planetesimals could
+withstand collisional fragmentation and erosion, thereby providing
+a pathway to forming planetary cores by coagulation in a binary. It
+is not clear how those results carry over to the case of highly ec-
+centric discs undergoing KL oscillations. However, the formation of
+nearly polar circumstellar discs from this work may give rise to the
+formation of nearly polar planets that become Kozai-unstable. Planet
+formation in a polar circumstellar disc requires the disc to last for a
+sufficiently long time. We speculate that this is possible provided that
+the disc is continuously accreting material in a polar configuration.
+Observations of misaligned planetary systems show a prefer-
+ence for nearly polar orbits with true obliquities 휓 in the range
+휓 = 80◦ − 125◦ (Albrecht et al. 2021; Dawson & Albrecht 2021).
+For example, two observed ultra-short-period hot Jupiters in po-
+lar orbits around an A-type star are Kelt-9b (Ahlers et al. 2020a)
+and MASCARA-4b (Ahlers et al. 2020b). The majority of planets
+studied by Albrecht et al. (2021) were hot Jupiters, since the mea-
+surements for these types of planets are more precise. However,
+a few warm-Neptunes with polar orbits were observed, including
+HAT-P-11b (Sanchis-Ojeda & Winn 2011), GJ 436b (Bourrier et al.
+2018, 2022), HD 3167c (Dalal et al. 2019; Bourrier et al. 2021), and
+WASP-107b (Dai & Winn 2017; Rubenzahl et al. 2021). A more re-
+cent warm Neptune, GJ 3470b, is also observed to be on a polar orbit
+(Stefànsson et al. 2022).
+ACKNOWLEDGEMENTS
+We thank the anonymous reviewer for helpful suggestions that pos-
+itively impacted the work. We thank Daniel Price for providing the
+phantom code for SPH simulations and acknowledge the use of
+SPLASH (Price 2007) for the rendering of the figures. Computer
+support was provided by UNLV’s National Supercomputing Center.
+MNRAS 000, 1–13 (2023)
+
+12
+Smallwood et al.
+We acknowledge support from NASA XRP grants 80NSSC19K0443
+and 80NSSC21K0395. This research was supported in part by the
+National Science Foundation under Grant No. NSF PHY-1748958.
+SHL thanks the Simons Foundation for support during a visit to the
+Flatiron Institute.
+DATA AVAILABILITY
+The data supporting the plots within this article are available
+on reasonable request to the corresponding author. A pub-
+lic version of the phantom, splash, and mercury codes are
+available
+at
+https://github.com/danieljprice/phantom,
+http://users.monash.edu.au/~dprice/splash/download.html,
+and https://github.com/4xxi/mercury, respectively.
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+

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+Topological superconductor candidates PdBi2Te4 and PdBi2Te5 from a generic ab
+initio strategy
+Aiyun Luo,1, ∗ Ying Li,1, ∗ Yi Qin,1, ∗ Jingnan Hu,1 Biao Lian,2 and Gang Xu1, 3, 4, †
+1Wuhan National High Magnetic Field Center & School of Physics,
+Huazhong University of Science and Technology, Wuhan 430074, China
+2Department of Physics, Princeton University, Princeton, NJ 08544, United States of America
+3Institute for Quantum Science and Engineering,
+Huazhong University of Science and Technology, Wuhan 430074, China
+4Wuhan Institute of Quantum Technology, Wuhan 430074, China
+Superconducting topological metals (SCTMs) have recently emerged as a promising platform of
+topological superconductivity (TSC) and Majorana zero modes(MZMs) for quantum computation.
+Despite their importance in both fundamental research and applications, SCTMs are very rare in
+nature. In addition, some superconductors with topological electronic structures have been reported
+recently, but a feasible program to determine their TSC properties is still lacking. Here, we propose
+a new strategy to design SCTMs by intercalating the superconducting units into the topological
+insulators.
+A program that characterizes the superconducting BdG Chern number of 2D BdG
+Hamiltonian from ab initio calculations is also developed. Following this strategy, PdBi2Te5 and
+PdBi2Te4 are found to be experimentally synthesizable and ideal SCTMs. Chiral TSC could be
+realized in such SCTMs by incorporating topological surface states with Zeeman effect, which can
+be realized by an external magnetic field or in proximity to ferromagnetic (FM) insulator. Our
+strategy provides a new method for identifying the SCTMs and TSC candidates, and the program
+makes it possible to design and modulate the TSC candidates from ab initio calculations.
+MAIN
+As one of the most important systems in both fundamental physics and topological quantum computation, topo-
+logical superconductors (TSCs) have attracted increasing interest for their ability to support Majorana fermions
+and anyons with non-Abelian statistics [1–17]. Currently, the search for TSCs candidates has been focused on two
+∗ These authors made equal contributions to this work.
+† e-mail address: [email protected]
+arXiv:2301.02844v1  [cond-mat.mtrl-sci]  7 Jan 2023
+
+2
+experimental schemes. One is the architectures by the combination of conventional superconductors with topological
+insulators (TIs) [18–20] or 1D nanowires [21, 22], but this approach brings high requirements for sample fabrication
+and interface engineering. The other route is to achieve TSCs in superconducting topological metals (SCTMs) that
+host both topological electronic structures at the Fermi level and superconductivity in one compound [23–36], in which
+the topological surface states are gapped by the “self-proximity effect” of bulk superconductivity, thus avoiding the
+complications of interface engineering. This approach has successfully predicted the SCTM FeTe0.55Se0.45 [24–26] and
+similar compounds of iron-based superconductors [27–30], owing to the favorable SC gap and non-trivial band topology.
+Beside the MZMs, 1D helical/chiral Majorana states have also been reported in domain walls of FeTe0.55Se0.45 [37]
+and the magnetism-superconductor heterostructures [38–45]. It is also proposed that the propagating chiral Majorana
+states can be applied to realize non-Abelian quantum gate operations, which could be 103 faster than the currently
+existing quantum computation schemes [46].
+Encouraged by the success of Fe(Se,Te) [24–26], many topological materials that host both superconductivity and
+topological electronic structures are proposed [47–52]. However, very rare experimental progress of TSC has been
+made in such SCTMs. This is because, on the one hand, all of them are not the ideal SCTMs, whose band structures
+are too complicated, the topological surface states are usually buried in the bulk states and difficult to form the
+pairing required by TSC. On the other hand, lacking a direct characterization of the TSC properties from ab initio
+calculations also hinders the effective experimental search in such materials. Therefore, a general program that could
+calculate the TSC invariant from first-principles calculations is highly desirable.
+In this work, we develop a program to characterize the superconducting topological invariant of 2D system from ab
+initio calculations. Besides, we also propose a new strategy to design ideal SCTMs by intercalating superconducting
+units into topological insulators.
+Following this strategy, PdBi2Te5 and PdBi2Te4 are found to be ideal SCTMs
+that host topological surface states at the Fermi level and superconductivity at 0.57 K and 3.11 K respectively. By
+performing the superconducting energy spectrum and topological invariant calculations, we identify that chiral TSC
+could be realized in the slab of such SCTMs by introducing considerable Zeeman splitting on the topological surface
+states, which can be realized by an external magnetic field or in proximity to FM insulators. Our strategy provides
+a new framework to enrich SCTMs and TSC candidates, and the program makes it possible to design and modulate
+the TSC system from ab initio calculations, which can also be extended to study the TSC properties in other system,
+such as magnetic TI/SC heterostructure, SC/FM heterostructure and SC/TI/SC heterostructure.
+Inspired by the construction of magnetic TI MnBi2Te4 [53, 54], we propose that the SCTMs can be designed by
+
+3
+intercalating the SC units into the TIs, as illustrated by the schematic of Fig. 1(a). As an ideal SCTM, the target
+crystal should be relatively stable in both energy and structure. More importantly, it must inherit the topological
+electronic structures of the parent TI near the Fermi level, and also the superconductivity of the parent SC as shown in
+Fig. 1(b). However, the combination of topological electronic structures and SC does not result in TSC eventually. The
+realization of TSC generally requires a delicate modulation of many parameters, such as SC pairing, Zeeman splitting
+and chemical potential, et al [18–26, 38–45]. Thus, the ability to characterize the TSC invariant and determine the
+required parameters in real materials from the ab initio calculations is not only of theoretical significance, but also
+highly desirable in experiment.
+Here we develop a program to simulate the superconducting properties and characterize its topological invariant in
+2D slab system from ab initio calculations, in which the necessary ingredients to realize chiral TSC based on SCTM
+are included, such as bulk band structures, SC pairing, Zeeman splitting, Rashba spin-orbit coupling and chemical
+potential. The workflow of this program is shown in Fig. 2. First, one should calculate the electronic structures of
+SCTM materials, and construct the localized Wannier functions that capture all electronic features from the first-
+principles calculations, referred as ˆHbulk. The next step is to construct the slab Hamiltonian ˆHslab with open boundary
+condition along a certain direction [55]. In general, the spin-orbit coupling (SOC) and surface effect can be included
+automatically in ˆHslab through the first-principles calculations with SOC. So that the topological properties, such as
+the surface states and spin-texture, can be directly studied by using ˆHslab. On the other hand, one can also construct
+a slab Hamiltonian ˆHnsoc
+slab that excluded SOC from the non-SOC first-principles calculations, and add ˆHSOC and ˆHsurf
+manually to simulate the variable SOC and surface effect in the topological electronic states and TSC. In this work,
+we will adopt the former type of ˆHslab, in which only the intrinsic SOC of the real material is included. With adopting
+particle-hole transformation, the ˆHslab can be extended to BdG Hamiltonian ˆHBdG
+slab by adding SC pairing ˆHsc and
+Zeeman splitting ˆHz. In the Nambu basis Φk = (ck,j,α,↑, ck,j,α,↓, c†
+−k,j,α,↑, c†
+−k,j,α,↓), where the cj,α,σ is the fermion
+operator denotes an electron at j layer with orbital α and spin σ(↑, ↓), the BdG Hamiltonian is formulated as:
+HBdG
+slab (k) =
+�
+�
+�
+Hslab(k) − µ
+∆(k)
+Ơ(k)
+−H∗
+slab(−k) + µ
+�
+�
+� + Mzτz.
+(1)
+In Eq. 1, µ is the chemical potential, which can be used to simulate the carriers doping. ∆(k) denotes the SC pairing
+matrices, which could be both singlet and triplet pairing form. For the conventional s-wave SC, ∆(k) is expressed as:
+∆(k) = ∆s × Islab ⊗ (iσy ⊗ Iorb),
+(2)
+
+4
+where ∆s is the magnitude of intrinsic bulk s-wave pairing, σy is the Pauli matrix in spin space, Islab (Iorb) is an
+Nslab × Nslab (Norb × Norb) identity matrix that represents the number of slab layers (Wannier orbitals). τz is the
+Pauli matrix in particle-hole space, Mz is the Zeeman splitting energy, and Hz = Mzτz is used to simulate the
+influence of the external magnetic field or the proximity effect of the FM insulator. Thus, Hz can be chosen to be
+applied for the whole slab or just few surface layers, depending on the slab thickness, strength of magnetic field, the
+type of the SC et al. In principle, chiral TSC can be achieved by modulating the SC pairing, Zeeman splitting and
+chemical potential [38–45], which can be further revealed by calculating the superconducting energy spectrum and
+the superconducting topological invariant.
+In the gaped 2D superconducting system, the topological superconductors are classified by BdG Chern number in
+the absence of time-reversal symmetry [3]. Such superconducting topological invariants can be characterized by the
+evolution of Wilson loop [56–58]. For the occupied quasiparticle states |uBdG
+n,k1,k2⟩, where k1 and k2 are momenta along
+two primitive vectors of the Brillouin zone (BZ), the Berry phase of the Wilson loop along k2 at a fixed k1 can be
+expressed as:
+W(k1) = −Im ln
+�
+i
+det M (i)
+k1 ,
+(3)
+with the overlap matrix M (i)
+k1,mn = ⟨uBdG
+m,k1,k(i)
+2 |uBdG
+n,k1,k(i+1)
+2
+��, where k(i)
+2
+is the i-th discretized momenta along k2 direction.
+The winding number of W(k1) with respect to k1 is equal to the superconducting BdG Chern number CBdG.
+Next, we take TI Bi2Te3 [59, 60], SC PdTe [61, 62] and SC PdTe2 [50–52] as parent compounds to demonstrate that
+our SCTMs strategy is feasible. Experimentally, Bi2Te3 (space group R¯3m, a = 4.35 ˚A, c = 30.36 ˚A), PdTe (space
+group space group P63/mmc, a = 4.152 ˚A, c = 5.671 ˚A, Tc = 2.3 K) and PdTe2 (space group P¯3m1, a = 4.03 ˚A,
+c = 5.12 ˚A, Tc = 1.64 K) all adopt the triangle lattice and have very similar in-plane lattice constants, which makes
+it much easier to integrate them together to form a new compound. According to our calculations, the stable unit of
+PdBi2Te5 and PdBi2Te4 adopt octuple-layer (OL) structure and septuple-layer (SL) structure respectively, as shown
+in Fig. 3(a)(also Fig. S1) and Fig. S2 of Supplementary Material(SM) [63]. They both favor the ABC stacking along
+c-direction, and form the rhombohedral unit cell as shown in Fig. 3(a), which is 73 meV/f.u.
+(73 meV/f.u.
+for
+PdBi2Te4) and 46 meV/f.u. (12 meV/f.u. PdBi2Te4) lower than the AA and AB stacking structures. The detailed
+crystal parameters and total energy of different stacking PbBi2Te5 and PbBi2Te4 are tabulated in the Table. S1 and
+Table. S2, respectively [63].
+The formation energy of PdBi2Te5 and PdBi2Te4 are calculated to study their thermodynamic stability by using
+EPdmBinTel
+f
+= EPdmBinTel − mEPd − nEBi − lETe, with Ei(i=PdmBinTel, Pd, Bi and Te) means the calculated
+
+5
+total energy per formula in the ground state. The calculated EP dBi2T e5
+f
+and EP dBi2T e4
+f
+are −3.184 eV/f.u. and
+−2.476 eV/f.u., which means that 3.184 eV and 2.476 eV can be released during their synthesis processes from the
+constituent elements. To further manifest their thermodynamic stability, we construct the convex hull diagram in
+Fig. 3(b) with all of the synthesized Pd-Bi-Te compounds, whose crystal parameters and the calculated formation
+energy have been tabulated in Table. S3 and Table. S4, respectively [63]. Fig. 3(b) shows that PdBi2Te5 and PdBi2Te4
+are 13 meV/atom and 61 meV/atom above the convex hull respectively.
+Moreover, considering that metastable
+PdBi2Te3, 52 meV and 3 meV higher than PdBi2Te5 and PdBi2Te4 as shown in Fig. 3(b), has been synthesized
+in experiments [64, 65], we thus conclude that PdBi2Te5 and PdBi2Te4 could be synthesized in experiments. For
+PdBi2Te5, we propose a synthetic route through the growth of Bi2Te3 and PdTe2 layer by layer. Our calculated
+results reveal that bulk PdBi2Te5 is 59 meV/f.u. lower than the total energy of free standing Bi2Te3 and PdTe2 layers,
+which strongly suggest that PdTe2 layer tends to deposit on Bi2Te3 to form new PdBi2Te5 crystal. To investigate
+their dynamical stability, we calculate the phonon dispersion of PdBi2Te5 and PdBi2Te4, and plot them in Fig. 3(c)
+and Fig. S3(a) [63]. There are 24 (21) phonon modes with fully real positive frequencies for PdBi2Te5 (PdBi2Te4),
+which indicates that the rhombohedral unit cells are dynamically stable. Based on these results, we conclude that
+PdBi2Te5 and PdBi2Te4 are relatively thermodynamically and dynamically stability in the rhombohedral structure,
+and further experimental investigation is called for.
+Then we study the electronic structures and topological properties of PdBi2Te5 and PdBi2Te4. Since PdBi2Te5
+and PbBi2Te4 exhibit similar electronic structures and non-trivial band topology, we only show the detailed density
+of states (DOS), band structures, and topological surface states of PdBi2Te5 as an example in the main text, one
+can check the results of PdBi2Te4 in Section III and Figs. S3 of the SM [63]. In Fig. 3(d), we plot the total and
+projected DOS of PdBi2Te5, which gives rise to DOS(0 eV) = 1.91 states/eV at Fermi level, indicating its metallic
+nature and the possibility of superconductivity. The projected DOS demonstrates that the states between −1 eV
+and 1 eV are dominated by the p-orbitals of Te hybridized with d-orbitals from Pd and p-orbitals from Bi. The
+hybridization is also manifested by the projected band structures shown in Fig. 3(e), which shows that two bands
+with p-orbital components from Te or Bi cross the Fermi level and form several Fermi surfaces. Further detailed
+orbital components analysis demonstrates that a continuous band gap (yellow region in Fig. 3(e)) and band inversion
+exists between the nominal valence band and conduction band around the Fermi level, which implies that PdBi2Te5
+inherits the topological electronic nature of Bi2Te3 successfully. The nontrivial band topology can be confirmed by
+calculating the Z2 topological invariant of time-reversal invariant insulators [66]. Given that rhombohedral PdBi2Te5
+
+6
+possesses inversion symmetry and a continuous band gap, the Z2 topological invariant νTI = (1 − P)/2 is determined
+by the product P of the parity of the wave function at the TRIM points [66]. Our calculated results give Z2 index
+νTI = 1, confirming PdBi2Te5 is a Z2 topological metal. To visualize the bulk–boundary correspondence, we calculate
+and plot the topological surface states on the (001) surface in Fig. 3(f). The surface states are similar to that of
+Bi2Te3 [59, 60], the Dirac cone at the Γ point manifest approximately −6.3 meV below the Fermi level (the dashed
+line in Fig. 3(f)).
+To investigate the superconducting property of PdBi2Te5, we perform the electron-phonon calculations based on
+density functional perturbation theory [67]. The calculated electron-phonon coupling constant λ = 0.43 and loga-
+rithmic average phonon frequency ωlog = 97 cm−1, as tabulated in Table. S5 [63]. Furthermore, the superconducting
+transition temperature (Tc) is estimated by using the reduced Allen-Dynes formula [68, 69]:
+Tc = ωlog
+1.20 exp
+�
+−
+1.04(1 + λ)
+λ − µ∗(1 + 0.62λ)
+�
+,
+(4)
+where µ∗ is the effective Coulomb potential. By adopting a typical µ∗ = 0.1, the Tc of PdBi2Te5 is estimated as
+0.57 K. As comparison, the calculated λ and ωlog in PdTe2 is 0.52 and 112 cm−1, respectively. Accordingly, the
+estimated Tc in PdTe2 is 1.59 K, which agrees well with the experimental Tc of 1.64 K [50–52]. These results clearly
+demonstrate that the SC in PdTe2 is well inherited into the PdBi2Te5.
+We now study the TSC property of the PdBi2Te5 slab by introducing the SC pairing and Zeeman splitting into the
+topological surface states. Usually, the Zeeman splitting is applied by external magnetic field or in proximity to a FM
+insulator, as illustrated in Fig. 4(a). As a concrete example, we use a 2D slab consisting of 10-OL PdBi2Te5, which is
+thick enough to avoid the hybridization between top layer and bottom layer (Fig. 3(f)). Since PdBi2Te5 is an intrinsic
+SC, the estimated s-wave superconducting gap ∆s = 1.0 meV is introduced globally for all 10-OLs. The out-of-plane
+Zeeman splitting is applied only in the bottom layer consisting of one Bi2Te3 and one PdTe2, by assuming PdBi2Te5
+is the conventional SC from the parent type-I SC PdTe2 [50]. The chemical potential µ is set at the energy of surface
+Dirac cone at the Γ point (about −6.3 meV below the Fermi level). In Fig. 4(b), we show the low energy spectrum
+of HBdG at Γ point as a function of Zeeman splitting energy Mz, which manifest that the superconducting spectrum
+is fully gaped with an energy gap of ∆ at Mz = 0. As Mz increases, the superconducting gap at the Γ point closes
+and reopens. This behavior indicates that a topological phase transition happens at critical point Mz/∆ = 3.1, and
+this 2D slab enters chiral TSC phase characterized by a nonzero BdG Chern number and chiral Majorana edge states
+according to previous model simulations [38–40].
+To firmly verify its topological property and visualize the low energy physics in the TSC phase, we calculate the
+
+7
+superconducting energy spectrum at Mz=5 meV and ∆=1 meV in Fig. 4(c). The corresponding Wilson loop evolutions
+for the occupied states are ploted in Fig. 4(d). The zoom-in image of Fig. 4(c) reveals that a full superconducting gap
+is opened in the whole BZ, indicating that the system is a well defined chiral TSC. The Wilson loop evolution exhibits
+a nontrivial chiral winding number 1, which directly confirms the superconducting BdG Chern number CBdG = 1.
+Given that the experimental accessible magnetization energy usually reaches a few tens of meV, our results provide
+a feasible guideline for discovery the chiral TSC phase in PdBi2Te5.
+Finally, we would like to point out that the chiral TSC phase could also be realized in PdBi2Te4 as shown in
+Fig. S4 [63], which exhibits a similar superconducting spectrum gap closing behavior with respect to Mz/∆ as in
+PdBi2Te5.
+In addition, we emphasize that our material design strategy can also be applied to search for other
+SCTM candidates. For example, our calculated results demonstrate that AuBi2Te5 formed by SC AuTe2 interacting
+into Bi2Te3 is also an ideal SCTM, whose detailed crystal structures, dynamic stability, electronic structures, and
+topological surface states are discussed in Section V and Fig. S5 of SM [63].
+Therefore, we expect that SCTM
+AuBi2Te5 could also be a TSC candidate. Last, we would like to point out that the program can be extended to study
+many 2D topological superconducting heterostructure systems, such as magnetic TI/SC heterostructure, SC/FM
+heterostructure and SC/TI/SC heterostructure. This will make it possible to determine the accurate parameters of
+the TSC phase and simulate their TSC property in such systems from first-principles calculations. We expect our
+program to be also useful for optimizing the experimental setup, stimulating the field of TSC study.
+ACKNOWLEDGMENTS
+This work is supported by the National Key Research and Development Program of China (2018YFA0307000),
+and the National Natural Science Foundation of China (12274154, 11874022). B.L. is supported by the Alfred P.
+Sloan Foundation, the National Science Foundation through Princeton University’s Materials Research Science and
+Engineering Center DMR-2011750, and the National Science Foundation under award DMR-2141966.
+METHOD
+The first-principles calculations based on density functional theory are performed by the Vienna ab initio simulation
+package [70, 71] with treating Perdew–Burke–Ernzerhof type of generalized gradient approximation as the exchange-
+correlation potential [72]. The cutoff energy for wave function expansion is set as 450 eV, k-points grid 13×13×13
+
+8
+is used for sampling the first BZ. All crystal structures are fully optimized until the force on each atom is less than
+0.01 eV/˚A, and the SOC is included self-consistently. The electron-phonon coupling calculations with van der Waals
+correction [73] are carried out in Quantum Espresso [74] based on the perturbation theory. A Hermite-Gaussian
+smearing of 0.0025 Ryd is used for the electronic integration. The 8×8×8 k-mesh is used for the electron-phonon
+coupling strength λ calculations, and the dynamical matrices are calculated on a 4×4×4 phonon-momentum grid.
+Besides, a 2×2×2 supercell is built to calculate the phonon dispersion by using PHONOPY [75]. For the surface
+calculation, the Wannier functions of Pd-d, Bi-p and Te-p orbitals are constructed by using WANNIER90 [76]. A
+slab consisting of 10-OL PdBi2Te5 layers with a bottom surface terminated as the Bi2Te3 layer is implemented in
+WannierTools [55], which is further used to calculate the electronic surface states, the superconducting spectrum, and
+the superconducting BdG Chern number.
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+wannier90:
+A tool for obtaining maximally-localised wannier functions.
+Computer Physics
+Communications 178, 685–699 (2008).
+
+13
+FIG. 1. The strategy to design idea SCTMs by intercalating the SC units into the TI.
+FIG. 2. The generic flow chart to characterize the TSC properties from ab initio calculations.
+
+(a)
+TI
+SC
+SCTM
+(b)SCTM structure
+Stability
+Band Topology
+Superconductivity
+DFT + Wannier90 Hbulk
+V
+Slab Hamiltonian Hslab
+Zeeman coupling H
+SC pairing Hsc
+BdG Hamiltonian HBdG
+Energy dispersion
+Wilson loop
+TSC candidate14
+FIG. 3. (a) The side view of the crystal structures of PdBi2Te5, in which the octuple-layer (OL) unit of PdBi2Te5 formed by
+the Bi2Te3 quintuple-layer and PdTe2 triple-layer is marked by a grey dashed rectangle. (b) Convex hull diagram for Pd-Bi-Te
+system, the energy above convex hull is displayed by color-bar. (c) The phonon dispersion of PdBi2Te5. (d) The total DOS
+and projected DOS of the Pd, Te, Bi atoms in PdBi2Te5, the zoom-in image shows the projected DOS near the Fermi level.
+(e) The orbital-projected band structures of PdBi2Te5, where a continuous band gap around the Fermi level is marked by the
+yellow shade. (f) The topological surface states on (001) surface of PdBi2Te5.
+
+(a)
+b
+(e)
+Bi
+Pd
+(meV/atom)
+Bi-p
+Pd
+60
+.
+Te
+e
+0.5
+Bi
+40
+PdBi2
+RdBi
+Bi4Te3
+PdBi2Te3
+Energy
+20
+e
+Te
+BiTe
+Energy
+PdBi2Te4
+BizTe3
+PdBi2Tes
+OL
+PdTe2 PdTe
+c
+-0.5
+F
+04
+IZ
+F
+d)
+20
+0.2
+Total
+4
+- Pd d
+5
+Te_p
+0
+Bi p
+0.0
+5
+-0.18
+4
+2
+0
+2
+4
+k
+Energy (eV)
+M15
+FIG. 4. (a) The schematic to realize chiral TSC in PdBi2T5 slab. (b) The low energy spectrum at the Γ point as the function of
+Zeeman splitting energy Mz. (c) The superconducting spectrum along high symmetry paths with Mz = 5 meV and ∆ = 1 meV,
+the zoom-in image shows the full gap in the whole BZ. (d) The Wilson loop spectrum for all occupied states of 4(c), which
+manifest the superconducting BdG Chern number CBdG = 1 clearly.
+
+(b) 3
+(a)
+CBdG=0
+CBdG=1
+2
+chiral Majorana states
+(n-OL PdBizTes)
+V
+SCTM
+0
+M
+-1
+FM insulator
+-2
+-3
+0
+2
+M/△
+8
+10
+(c)
+(d)
+40
+0.5
+20
+(meV)
+0
+0(2元)
+20
+Energy
+40
+V
+0
+-0.5
+K
+M
+-元
+元Supplementary Material: Topological superconductor candidates PdBi2Te4 and
+PdBi2Te5 from a generic ab initio strategy
+Aiyun Luo,1, ∗ Ying Li,1, ∗ Yi Qin,1, ∗ Jingnan Hu,1 Biao Lian,2 and Gang Xu1, 3, 4, †
+1Wuhan National High Magnetic Field Center & School of Physics,
+Huazhong University of Science and Technology, Wuhan 430074, China
+2Department of Physics, Princeton University, Princeton, NJ 08544, United States of America
+3Institute for Quantum Science and Engineering,
+Huazhong University of Science and Technology, Wuhan 430074, China
+4Wuhan Institute of Quantum Technology, Wuhan 430074, China
+I.
+The crystal structures of PdBi2Te5 and PdBi2Te4
+The crystal structures of PdBi2Te5 are shown in Fig. S1. Since both Bi2Te3(Fig. S1(a)) and PdTe2(Fig. S1(b))
+are Van der Waals materials, a naturally stable unit of PdBi2Te5 would be an octuple-layer (OL) block as shown in
+Fig. S1(e). With different stacking sequences along c-direction, the crystal structures of AA-stacking, AB-stacking
+and ABC-stacking are shown in Fig. S1(c), (d) and (e), respectively. As a result, the relaxed lattice parameters and
+total energies of these structures are tabulated in Table. S1. Our calculated results demonstrate that the most stable
+structure of PdBi2Te5 is the ABC-stacking rhombohedral unit cell as shown in Fig. S1(e), which is 73 meV/f.u. and
+46 meV/f.u. lower than the AA and AB stacking structures.
+The crystal structures of PdBi2Te4 are shown in Fig. S2, which is formed by the integration of Bi2Te3(Fig. S2(a))
+and PdTe(Fig. S2(b)). Like the construction of MnBi2Te4 [1, 2], a naturally stable unit of PdBi2Te4 would be a
+septuple-layer (SL) block as shown by the grey dash rectangle in Fig. S2(c)-(e). Similar to the case of PdBi2Te5, the
+relaxed lattice parameters and total energies of AA-stacking, AB-stacking, and ABC-stacking sequences of PdBi2Te4
+are tabulated in Table. S2. Our calculations demonstrate that SL-PdBi2Te4 favor the ABC stacking rhombohedral unit
+cell as shown in Fig. S2(e), which is 73 meV/f.u. and 12 meV/f.u. lower than the AA(Fig. S2(c)) and AB(Fig. S2(d))
+stacking structures.
+∗ These authors made equal contributions to this work.
+† e-mail address: [email protected]
+arXiv:2301.02844v1  [cond-mat.mtrl-sci]  7 Jan 2023
+
+2
+FIG. S1. The crystal structures of different stacking sequences of PdBi2Te5. (a) A side view of the crystal structures of Bi2Te3,
+the quintuple-layer block is marked by the green dashed rectangle. (b) PdTe2, the triple-layer block is marked by the purple
+dashed rectangle. PdBi2Te5 for (c) AA stacking, (d) AB stacking, and (e) ABC stacking, respectively. A octuple-layer(OL)
+block of PdBi2Te5 is marked by the grey dashed rectangle.
+
+(c)
+(e)
+a
+OL
+(d)
+(b)
+Te
+Bi
+Pd3
+FIG. S2. The crystal structures of different stacking sequences of PdBi2Te4. (a) Bi2Te3. (b) PdTe. PdBi2Te4 for (c) AA
+stacking, (d) AB stacking, and (e) ABC stacking, respectively. A SL block of PdBi2Te4 is marked by the grey dashed rectangle.
+
+(c)
+a
+(e)
+SL
+(d)
+(b)
+Te
+Bi
+Pd4
+TABLE S1. The crystal parameters and total energies of different stacking sequences of PdBi2Te5 from our relaxed results.
+Compound
+Stacking
+Space group
+Lattice (˚A)
+Atom
+Wyckoff
+Coordinate
+Energy(eV/f.u.)
+PdBi2Te5
+AA
+P3m1
+a = b = 4.287
+Pd
+1c
+(0.667, 0.333, 0.013)
+-30.385
+c = 16.325
+Bi1
+1a
+0, 0, 0.363
+Bi2
+1a
+0, 0, 0.623
+Te1
+1b
+(0.333, 0.667, 0.090)
+Te2
+1b
+(0.333, 0.667, 0.492)
+Te3
+1c
+(0.667, 0.333, 0.251)
+Te4
+1c
+(0.667, 0.333, 0.936)
+Te5
+1a
+(0, 0, 0.733)
+PdBi2Te5
+AB
+P3m1
+a = b = 4.315
+Pd1
+1c
+(0.667, 0.333, 0.483)
+-30.412
+c = 30.987
+Pd2
+1a
+(0, 0, 0)
+Bi1
+1b
+(0.333, 0.667, 0.792)
+Bi2
+1c
+(0.667, 0.333, 0.658)
+Bi3
+1c
+(0.667, 0.333, 0.308)
+Bi4
+1a
+(0, 0, 0.175)
+Te1
+1b
+(0.333, 0.667, 0.046)
+Te2
+1b
+(0.333, 0.667, 0.437)
+Te3
+1b
+(0.333, 0.667, 0.600)
+Te4
+1b
+(0.333, 0.667, 0.241)
+Te5
+1c
+(0.667, 0.333, 0.956)
+Te6
+1c
+(0.667, 0.333, 0.115)
+Te7
+1c
+(0.667, 0.333, 0.851)
+Te8
+1a
+(0, 0, 0.528)
+Te9
+1a
+(0, 0, 0.726)
+Te10
+1a
+(0, 0, 0.367)
+PdBi2Te5
+ABC
+R¯3m
+a = b = 4.304
+Pd
+3b
+(0, 0, 0.5)
+-30.458
+c = 46.391
+Bi
+6c
+(0, 0, 0.3788)
+Te1
+3a
+(0, 0, 0)
+Te2
+6c
+(0, 0, 0.1396)
+Te3
+6c
+(0, 0, 0.2488)
+
+5
+TABLE S2. The crystal parameters and total energies of different stacking sequences of PdBi2Te4 from our relaxed results.
+Compound
+Stacking
+Space group
+Lattice (˚A)
+Atom
+Wyckoff
+Coordinate
+Energy(eV/f.u.)
+PdBi2Te4
+AA
+P¯3m1
+a = b = 4.275
+Pd
+1b
+(0, 0, 0.5)
+-26.776
+c = 13.035
+Bi
+2d
+(0.667, 0.333, 0.2325)
+Te1
+2d
+(0.333, 0.667, 0.397)
+Te2
+2c
+(0, 0, 0.09)
+PdBi2Te4
+AB
+P¯3m1
+a = b = 4.275
+Pd
+2d
+(0.333, 0.667, 0.75025)
+-26.837
+c = 26.070
+Bi1
+2c
+(0, 0, 0.61675)
+Bi2
+2d
+(0.667, 0.333, 0.88375)
+Te1
+2d
+(0.333, 0.667, 0.95375)
+Te2
+2d
+(0.333, 0.667, 0.54675)
+Te3
+2c
+(0, 0, 0.80175)
+Te4
+2d
+(0.667, 0.333, 0.88375)
+PdBi2Te4
+ABC
+R¯3m
+a = b =4.296
+Pd
+3a
+(0, 0, 0)
+-26.849
+c = 40.546
+Bi
+6c
+(0, 0, 0.4215)
+Te1
+6c
+(0, 0, 0.3000)
+Te2
+6c
+(0, 0, 0.8668)
+
+6
+II.
+The detailed crystal parameters and formation energies of Pd-Bi-Te compounds for convex hull
+The fundamental thermodynamic nature of Pd-Bi-Te compounds can be fully characterized in terms of the convex
+hull diagram [3, 4], which is defined as formation energy versus composition diagram with the ground state at special
+compositions. For all of the known Pd-Bi-Te compounds as tabulated in Table. S3, we calculate their formation
+energy and tabular the results in Table. S4. Based on these results, we construct the convex hull diagram of Pd-Bi-Te
+compounds in Fig.3(b) of the main text. The ternary compounds PdBi2Te3, PdBi2Te4 and Pd2Te5 are 65 meV/atom,
+59 meV/atom and 13 meV/atom above the convex hull respectively, indicating that they are metastable phases with
+respect to decomposition into the energetically favorable binary phases. Significantly, PdBi2Te3 above the convex
+hull has already been synthesized in experiments, which is higher 52 meV/atom and 7 meV/atom than PdBi2Te5 and
+PdBi2Te4. Therefore, we expect that PdBi2Te5 and PdBi2Te4 could be synthesized in the future.
+
+7
+TABLE S3. The space group, lattice constants and atomic coordinates of Pd-Bi-Te compounds used in the convex hull.
+Compound
+Space group
+Lattice (˚A)
+Atom
+Wyckoff
+Coordinate
+Pd
+Fm¯3m
+a = b = c = 3.889
+Pd
+4a
+(0, 0, 0)
+Bi
+R¯3m
+a = 4.523, c = 11.800
+Bi
+3b
+(0, 0, 0.227)
+Te
+P3121
+a = 4.445, c = 5.91
+Te
+3a
+(0.225, 0.0, 0.0)
+PdBi
+Cmc21
+a = 8.707
+Pd1
+8b
+(0.274, 0.125, 0.053)
+b = 7.203
+Pd2
+4a
+(0, 0.108, 0.225)
+c = 10.662
+Pd3
+4a
+(0, 0.65, 0.225)
+Bi1
+8b
+(0.226, 0.375, 0.278)
+Bi2
+4a
+(0, 0.108, 0.5)
+Bi3
+4a
+(0, 0.35, 0)
+PdBi2
+I4/mmm
+a = b = 3.362
+Pd
+2a
+(0, 0, 0)
+c = 12.983
+Bi
+4e
+(0, 0, 0.363)
+PdTe
+P63/mmc
+a = b = 4.152
+Pd
+2a
+(0, 0, 0)
+c = 5.671
+Te
+2c
+(0.333, 0.667, 0.25)
+PdTe2
+P¯3m1
+a = b = 4.028
+Pd
+1a
+(0, 0, 0)
+c = 5.118
+Te
+2d
+(0.333, 0.667, 0.25)
+BiTe
+P¯3m1
+a = b = 4.400
+Bi1
+2d
+(0.333, 0.667, 0.291)
+c = 24.000
+Bi2
+2d
+(0.333, 0.667, 0.541)
+Bi3
+2c
+(0, 0, 0.126)
+Te1
+2d
+(0.333, 0.667, 0.056)
+Te2
+2d
+(0.333, 0.667, 0.789)
+Te3
+2c
+(0, 0, 0.362)
+Bi2Te3
+R¯3m
+a = b = 4.35
+Bi
+6c
+(0, 0, 0.600)
+c = 30.36
+Te1
+6c
+(0, 0, 0.790)
+Te2
+3a
+(0, 0, 0)
+Bi4Te3
+R¯3m
+a = b = 4.451
+Bi1
+6c
+(0, 0, 0.146)
+c = 41.888
+Bi2
+6c
+(0, 0, 0.283)
+Te1
+6c
+(0, 0, 0.426)
+Te2
+3a
+(0, 0, 0)
+PdBi2Te3
+R¯3m
+a = b = 4.421
+Pd
+3b
+(0, 0, 0.5)
+c = 30.337
+Bi
+6c
+(0, 0, 0.561)
+Te1
+3a
+(0, 0, 0)
+Te2
+6c
+(0, 0, 0.796)
+
+8
+TABLE S4. The total energy and formation energy of Pd-Bi-Te compounds used in the convex hull.
+Compound
+Energy(eV/f.u.)
+Formation energy(eV/atom)
+Pd
+-5.161
+–
+Bi
+-3.804
+–
+Te
+-2.901
+–
+PdTe
+-9.022
+-0.480
+PdTe2
+-1.349
+-0.450
+PdBi
+-9.565
+-0.3
+PdBi2
+-13.568
+-0.266
+BiTe
+-7.346
+-0.320
+Bi2Te3
+-18.258
+-0.389
+Bi4Te3
+-25.979
+-0.294
+PdBi2Te3
+-23.038
+-0.261
+PdBi2Te4
+-26.849
+-0.354
+PdBi2Te5
+-30.458
+-0.398
+
+9
+III.
+The phonon spectrum, electronic structures and superconducting properties of PdBi2Te4
+In Fig. S3(a), we calculate and plot the phonon dispersion of PdBi2Te4. It clearly shows that there are 21 phonon
+modes with fully real positive frequencies, indicating that the rhombohedral PdBi2Te4 are dynamically stable. Based
+on the stable rhombohedral structure, we carry out the calculations with SOC of electronic properties of PdBi2Te4.
+The total and projected density of states (DOS) are shown in Fig. S3(b), which shows that the total DOS at the Fermi
+level is about of 3.95 states/eV, indicating its metallic nature and the probability of superconductivity. Fig. S3(c)
+shows the projected band structures of PdBi2Te4, from which we can see that two bands with p-orbital components
+from Te or Bi cross the Fermi level and a continuous direct gap (the yellow region in Fig. S3(c)) exists around the
+Fermi level. Further detailed orbital components analysis demonstrates that band inversion is present between the
+nominal valence band and conduction band, indicating the non-trivial band topology of bulk PdBi2Te4. In Fig. S3(d),
+we calculate and plot the topological surface states on the (001) surface of PdBi2Te4, in which the Dirac surface states
+at the Γ point manifest approximately −62 meV below the Fermi level, confirming PdBi2Te4 is a Z2 topological metal.
+To investigate the superconducting properties of PdBi2Te4 and PdBi2Te5, we calculate their electron-phonon
+coupling constant λ and logarithmic average phonon frequency ωlog as tabulated in Table. S5. As a comparison, the
+λ and ωlog of parent SC PdTe and PdTe2 are also calculated. Moreover, the superconducting transition temperature
+(Tc) is estimated by using the reduced Allen-Dynes formula as Eq.(4) in the main text. By adopting a typical µ∗ =
+0.1, the Tc of PdBi2Te4 and PdBi2Te5 is estimated as 3.11 K and 0.57 K, respectively. Furthermore, the Tc of PdTe
+and PdTe2 is estimated as 2.55 K and 1.59 K, which are consistent with the experimental results very well [5–9].
+These results clearly demonstrate that the SC in PdTe and PdTe2 is well inherited into the bulk of PdBi2Te4 and
+PdBi2Te5.
+TABLE S5. The calculated electron-phonon coupling strength λ, logarithmic average phonon frequency ωlog and the estimated
+Tc (µ∗ = 0.1) of PdTe, PdTe2, PdBi2Te4 and PdBi2Te5.
+Compound
+λ
+ωlog(cm−1)
+Tc(K)
+Texp
+c
+(K)
+PdTe
+0.61
+106
+2.55
+2.3 [5], 4.5 [6]
+PdTe2
+0.52
+112
+1.59
+1.64 [7–9]
+PdBi2Te4
+0.70
+89
+3.11
+–
+PdBi2Te5
+0.43
+97
+0.57
+–
+
+10
+FIG. S3. The phonon dispersion and electronic properties of PdBi2Te4. (a) The phonon dispersion of PdBi2Te4. (b) The total
+DOS and projected DOS of the Pd, Te, Bi atoms in PdBi2Te4, the zoom-in image shows the projected DOS near the Fermi
+level. (c) The orbital-projected band structures of PdBi2Te4, a continuous band gap around the Fermi level is marked by the
+yellow shade. (d) The topological surface states on (001) surface of PdBi2Te4.
+IV.
+The chiral TSC phase in PdBi2Te4
+In this section, we study the chiral TSC phase in 20-SL PdBi2Te4 by incorporating its topological and supercon-
+ducting properties with Zeeman splitting. In this case, the chemical potential µ is set at the energy of surface Dirac
+cone at the Γ point (about −62 meV below the Fermi level), the estimated s-wave superconducting gap ∆s = 1.0 meV
+is introduced globally for all 20-SLs, and the out-of-plane Zeeman splitting is applied only in the bottom layer. In
+Fig. S4(a), we show the low energy spectrum of HBdG at Γ point as a function of Zeeman splitting energy Mz, which
+manifests that two superconducting bands cross each other at critical point Mz/∆ = 5. This behavior indicates that
+
+a
+b
+20
+Total
+4
+(states/eV
+-Pd d
+Frequency (THz)
+15
+-Te p
+-Bi p
+10
+Density
+0
+F
+-2
+0
+2
+4
+Energy (eV)
+C
+Bi-p
+Te-p
+0.5
+0.0
+Energy (eV)
+-0.1
+-0.5
+-0.2
+F
+M11
+a topological phase transition happens at the critical point. Fig. S4(b) shows the superconducting energy dispersion
+with Mz=10 meV and ∆=1 meV. Its zoomed-in image, calculated in the whole BZ, directly reveals the full gap
+characters of superconducting spectrum, indicating that the system is a well defined chiral TSC. To further verify
+its nontrivial topological properties, the superconducting BdG Chern number for all occupied states is calculated by
+the Wilson loop method as shown in Fig. S4(c). The spectrum of Wilson loop exhibits a non-trivial chiral winding
+number 1, which directly confirms the superconducting BdG Chern number CBdG = 1.
+FIG. S4. The chiral TSC phase in slab PdBi2T4. (a) The low energy spectrum at the Γ point as the function of Zeeman
+splitting energy Mz, where ∆ = 1 meV. (b) The superconducting spectrum along high symmetry paths with Mz = 10 meV
+and ∆ = 1 meV, the zoom-in image shows the full gap in the whole BZ. (c) The Wilson loop spectrum of all occupied states
+in (b), which manifest the superconducting BdG Chern number CBdG = 1 clearly.
+V.
+The electronic structures and topological properties of AuBi2Te5
+The crystal structure of rhombohedral AuBi2Te5 is shown in Fig. S5(a), which is formed by the layered intercalation
+of Bi2Te3 and AuTe2. In Fig. S5(b), we show the calculated phonon dispersion along the high symmetry lines of
+AuBi2Te5, in which the most important observation is that all phonon modes have positive frequency throughout the
+BZ, indicating the dynamical stability of the rhombohedral structure. In Fig. S5(c), we calculate and plot the total
+and projected DOS of AuBi2Te5, which gives rise to DOS(0 eV)=5.41 states/eV at Fermi level, indicating its metallic
+nature and the possibility of superconductivity. Fig. S5(d) shows the orbital-projected band structures of AuBi2Te5,
+in which a continuous band gap around the Fermi level is marked by the yellow shades. Further detailed orbital
+components analysis demonstrates that a band inversion exists between the nominal valence band and conduction
+
+(a)
+(b)
+(c)
+40
+0.5
+CBdG=0
+CBdG=1
+2
+20
+(meV)
+2元）
+0
+A
+-1
+-2
+0
+-3
+群
+V
+-0.5
+0
+5 Mz/△ 10
+15
+K
+M
+Ky
+-元
+元12
+band, which implies that AuBi2Te5 is a topological metal that inherits from parent Bi2Te3. In Fig. S5(e), we show
+the calculated surface states of AuBi2Te5 in the (001) surface. It clearly shows that the topological surface states are
+presented in the bulk gap, confirming AuBi2Te5 is a Z2 topological metal.
+FIG. S5. The crystal structure, phonon dispersion, and electronic properties of AuBi2Te5. (a) The side view of the crystal
+structure. (b) The phonon dispersion along high symmetry path. (d) The total DOS and projected DOS of the Au, Te, Bi
+atoms. (e) The orbital-projected band structures, a continuous band gap around the Fermi level is marked by the yellow shade.
+(f) The topological surface states on (001) surface.
+[1] Lee, D. S. et al. Crystal structure, properties and nanostructuring of a new layered chalcogenide semiconductor, Bi2MnTe4.
+CrystEngComm 15, 5532–5538 (2013).
+[2] Zhang, D. et al. Topological axion states in the magnetic insulator MnBi2Te4 with the quantized magnetoelectric effect.
+Phys. Rev. Lett. 122, 206401 (2019).
+
+(a)
+(b)
+(c)
+10
+- Total
+Au
+Density (states/eV)
+Au d
+8
+Te p
+Bi
+Bi p
+6
+Te
+OL
+F
+0
+(e)
+Energy (ev)
+(p)
+0.2
+Bi-p
+te
+0.1
+(eV)
+0.0
+Energy (
+Energy
+0.1
+-0.2
+-0.3
+F
+7
+K
+M13
+[3] Sun, Y., Lv, J., Xie, Y., Liu, H. & Ma, Y. Route to a superconducting phase above room temperature in electron-doped
+hydride compounds under high pressure. Phys. Rev. Lett. 123, 097001 (2019).
+[4] Sharan, A. & Lany, S. Computational discovery of stable and metastable ternary oxynitrides. The Journal of Chemical
+Physics 154, 234706 (2021).
+[5] Matthias, B. T. Superconducting compounds of nonsuperconducting elements. Phys. Rev. 90, 487–487 (1953).
+[6] Karki, A. B., Browne, D. A., Stadler, S., Li, J. & Jin, R. PdTe: a strongly coupled superconductor. Journal of Physics:
+Condensed Matter 24, 055701 (2012).
+[7] Kudo, K., Ishii, H. & Nohara, M. Composition-induced structural instability and strong-coupling superconductivity in
+Au1−xPdxTe2. Phys. Rev. B 93, 140505 (2016).
+[8] Leng, H., Paulsen, C., Huang, Y. K. & de Visser, A. Type-I superconductivity in the Dirac semimetal PdTe2. Phys. Rev.
+B 96, 220506 (2017).
+[9] Das, S. et al. Conventional superconductivity in the type-II Dirac semimetal PdTe2. Phys. Rev. B 97, 014523 (2018).
+

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6dE1T4oBgHgl3EQfBQKx/content/tmp_files/2301.02850v1.pdf.txt

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+Programmable wave-based analog computing
+machine: a metastructure that designs metastructures
+Dimitrios C. Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗
+1Department of Electrical and Systems Engineering,
+School of Engineering and Applied Sciences,
+University of Pennsylvania, Philadelphia, 19104, U.S.A.
+†These authors contributed equally to this work.
+‡Present address: Meta Materials Inc. (Europe),
+Ap. Pavlou 10A, Marousi, 15123, Greece.
+∗To whom correspondence should be addressed; e-mail: [email protected]
+January 10, 2023
+Abstract: The ability to perform mathematical computations using metastruc-
+tures is an emergent paradigm that carries the potential of wave-based analog
+computing to the realm of near-speed-of-light, low-loss, compact devices. We
+theoretically introduce and experimentally verify the concept of a reconfig-
+urable metastructure that performs analog complex mathematical computa-
+tions using electromagnetic waves. Reconfigurable, RF-based components en-
+dow our device with the ability to perform stationary and non-stationary iter-
+ative algorithms. After demonstrating matrix inversion (stationary problem),
+we use the machine to tackle two major non-stationary problems: root finding
+with Newton’s method and inverse design (constrained optimization) via the
+Lagrange multiplier method. The platform enables possible avenues for wave-
+1
+arXiv:2301.02850v1  [physics.app-ph]  22 Dec 2022
+
+based, analog computations for general linear algebraic problems and beyond
+in compact, ultrafast, and parallelized ways.
+One-Sentence Summary:
+A reconfigurable wave-based analog computing metastructure that
+can inverse-design a metastructure.
+Calculators of various kinds have emerged by forging numerical algorithms with corre-
+sponding technological platforms. While the algorithms describe the mathematical paths on
+how solutions to problems can be found, the platforms are responsible for the transliteration of
+this abstract path into measurable quantities. The algorithms, the platforms, and their fusion
+define such systems’ features and limitations. Following the ever-growing demand for ultrafast,
+compact, low/near-zero-power, and integrable cyber-physical devices for mathematical compu-
+tations, it is organic that significant research efforts focus on making these numerical systems
+as optimal and efficient as possible.
+This quest led to the exploration and development of unconventional analog computing sys-
+tems that exploit electromagnetic waves to deliver parallelized, ultrafast, compact, low-power
+computations (1–4). The two main categories in this domain involve systems that utilize free-
+space (scattering) elements (e.g. lenses (3)), and waveguides (e.g. photonic systems (5, 6)
+and phased arrays (7)). Sufficient free space propagation can act as dense matrix multiplica-
+tion (8). Realized with traditional optics, this results in bulky devices (3,9), while metasurfaces
+can be more compact (10–12). However, in both there can be major bottlenecks regarding
+photonic and electronic integration.
+Waveguiding systems offer more mature solutions for
+integrable and reconfigurable devices, at the expense of much larger footprints compared to
+their free-space counterparts. In all cases, their main challenge is reconfigurability since its
+implementation requires some form of a-priori mathematical calculations. For instance, meta-
+surfaces/complex media (13, 14) requires optimization, and photonic meshes require operator
+2
+
+decomposition (15–18).
+In terms of their mathematical abilities, the above examples demonstrate wave-based analog
+computing with functionalities such as integration/differentiation in space (19–22) and time
+(23), matrix-vector multiplication (24), emulating equations through physical phenomena (25,
+26), or acting as platforms for neural network functionalities (3,6,27,28). The intersection with
+the metamaterial paradigm delivered a series of remarkable analog computing devices with
+matrix multiplication (4,19,29) and ultimately equation solving (matrix inversion) capabilities
+(30). In most of the cases the matrix computations (especially the matrix inversion (30)) were
+performed through stationary algorithms (31), such as the Jacobi method, where the matrix
+(operator/kernel) does not change with the iteration count.
+The fundamental and far-reaching question we address here is whether a wave-based analog
+metastructure can be reconfigurable simply and intuitively, without needing a-priori calcula-
+tions. Most importantly, the resolution to this question endows one with the ability to implement
+stationary and non-stationary algorithms. We propose a device based on an RF waveguide archi-
+tecture with reconfigurable components. Regarding stationary problems, we use this device to
+perform matrix inversion of a statistically large number of matrices. As for non-stationary prob-
+lems, we demonstrate both root finding using Newton’s method and Inverse Design. All three
+examples are only possible due to the reconfigurability of the device and hint at the possibility
+of deeper explorations into the realm of advanced numerical algebra methods.
+A conceptual representation of the main idea is pictorially summarized in Fig. 1 (A). The
+main property of our proof-of-concept system distinctively different from all previous metas-
+tructure approaches (i.e. (30)) is its reconfigurability; the metastructure has the ability to rapidly
+take on different matrices (operators or kernels) K. To facilitate this, we employ a wave-based
+direct complex matrix (DCM) architecture, which offers an intuitive and simple implementation
+of any desired matrix (32). Using waves instead of currents and voltages, it is analogous to the
+3
+
+crossbar architecture used in electronic analog computing systems (33–35), and it can be seen as
+a generalized phased array feed (7). In this device, a collection of n × n tunable phase shifting
+and amplifying elements (which can also act as attenuating element) connect an input vector
+of n complex amplitudes on an array of transmission lines to a similar output vector through
+combiners. This architecture can be seen schematically in Fig. 1 (B) and its corresponding
+experimental implementation in Fig. 1 (C).
+The key component of the metadevice is the multiplier module (Fig. 1 (D)) so named be-
+cause given an input signal characterized by its complex amplitude at 45MHz, Vin, it will render
+a similar output Vout = zVin, where z is a complex multiplication factor. This module consists of
+two basic components: (a) a voltage-controlled phase shifter with over 360 degrees of potential
+phase rotation, and (b) a voltage-controlled amplifier with ≈47dB of dynamic range (-30dB to
++17dB). Through the use of an embedded microcontroller unit (MCU) in each multiplier, each
+device can be controlled externally through a suitable communication network and a computer
+(see supplementary material). While the design frequency is 45MHz, the module could be im-
+plemented at RF (GHz) and photonic (THz) platforms platforms, following the same principle
+of operation. The experimental DCM implementation consists of 25 multipliers to yield 5 × 5
+complex matrices. The ingress and egress stages (32) are implemented with five 1-to-5 power
+splitters (ingress stage) and five 5-to-1 signal combiners (egress stage). The multipliers are
+clustered into five groups, one for each matrix row. In Fig. 1(B) we depict the planar schematic
+of the DCM suitable for photonic implementation. However, for the RF implementation we
+stacked and routed the components vertically (Fig. 1 (C)), making the device compact for our
+particular wavelength and platform choice. Different stacking or integrated circuitry approaches
+can potentially be used to further reduce its overall footprint.
+The metadevice can be operated in one of two configurations with dramatically different
+results. When the DCM is set in an open-loop configuration (Fig 1 (E) inset), it can be used
+4
+
+for rapidly calculating parallelized matrix-vector multiplication. However, a closed-loop con-
+figuration can be created by connecting the outputs and the inputs with a feedback loop using
+properly designed couplers (Fig 1 (F) inset). When the DCM is in a closed-loop configuration,
+the metadevice can rapidly calculate parallelized matrix inversion (equation solving). This is a
+unique feature of metadevices/metastructures (30,32) that incorporate feedback loops.
+First we investigate the stationary analysis capabilities of our metadevice. For the assess-
+ment of the open and closed loop operation, we performed a series of randomized trials, one
+instance of which is presented in Fig. 1 (E) and (F). For each trial, a random passive matrix
+A ∈ C5×5 was chosen and applied to the metadevice in both its configurations. Each mea-
+surement was performed by exciting each input port in turn with all other inputs appropriately
+terminated and then observing the complex amplitudes on each of the output ports. For the
+open-loop configuration, this corresponds to performing five matrix-vector multiplications, or
+as A · I where each column of I is progressively applied (one column at a time) as separate
+vectors. The closed-loop configuration was measured similarly, but in this case we are prob-
+ing the steady-state of the metadevice which corresponds to (I − K)−1 · I = A−1 · I. While
+this measurement technique fully characterizes both configurations, in practice dense complex
+vectors will be input and read to achieve parallelized results.
+The estimated relative error ||Aexact − Ameas||2/||Aexact||2 for both cases (Fig. 1 (E) and (F))
+revealed an error about 0.001 and 0.005, respectively. Despite the component imperfections,
+misalignments, measurement noise, and other stochastic errors, the measured results are in
+excellent agreement with the theoretical values. The calibration procedure of the metadevice
+and the statistical analysis of the full trial set (100 values) are presented in the Supplementary
+Material.
+A single multiplier module has a rise time of approximately 80 ns to achieve its desired com-
+plex value. This value is approximately 4T assuming one-period duration of T = 1/45MHz ≈
+5
+
+22.2 ns. In the open loop configuration, the total response requires approximately 5T, including
+signal delays in connections and splitters. The duration of the closed-loop case is affected by
+the platform and the condition number of the inverted matrix (32), but in principle is in the
+same order of magnitude. Possible photonic implementations may further reduce this time to
+the picosecond range (28) and below (36).
+We now apply the implemented metadevice to two characteristic non-stationary problems
+that highlight its mathematical abilities: (i) root finding of a system of five equations with five
+unknowns using Newton’s iterative technique and (ii) implementing an inverse-design problem
+using the Lagrangian multiplier formalism for constrained optimization. Both cases require
+that the kernel be reprogrammed in each iteration step. Note that our approach is not restricted
+to these two problems; instead, we choose these to highlight the potential of the introduced
+metadevice.
+For the first case we construct a simple nonlinear toy problem and we apply Newton’s algo-
+rithm (37) (Fig. 2 (A)) for finding one possible root. The vector problem statement reads
+f(z) = [f1(z), f2(z), f3(z), f4(z), f5(z)]T = 0
+(1)
+where f ∈ C5×1, z = [z1, z2, z3, z4, z5]T ∈ C5×1, and 0 is the zero vector. We construct the
+vector function to have the following polynomial form
+f1(z) = (z1 − r1)(z2 − 4.2i)(z3 + 2)(z4 − 5i)(z5 − 3.5)
+(2)
+f2(z) = (z1 − 3.9)(z2 − r2)(z3 + 2.5i)(z4 − 3.2i)(z5 − 4.2)
+(3)
+f3(z) = (z1 + 5.2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.1)
+(4)
+f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i)
+(5)
+f5(z) = (z1 − 5.2i)(z2 − 4)(z3 + 4.75i)(z4 − 8)(z5 − r5)
+(6)
+where r = [r1, r2, r3, r4, r5]T are the vertices of a regular pentagon with 1/4 radius (see Fig.
+2(B)) and the other factors represent additional extraneous roots far from the starting point.
+6
+
+For the evaluation of Newton’s method we need to calculate the Jacobian matrix, i.e., Jij =
+∂fi
+∂zj or
+Jf(z) =
+�
+�
+�
+�
+�
+∂f1
+∂z1
+∂f1
+∂z2
+· · ·
+∂f1
+∂z5
+∂f2
+∂z1
+∂f2
+∂z2
+· · ·
+∂f2
+∂z5
+...
+...
+...
+...
+∂f5
+∂z1
+∂f5
+∂z2
+· · ·
+∂f5
+∂z5
+�
+�
+�
+�
+�
+(7)
+therefore the root can be estimated by the following iterative process
+zn+1 = zn − αJ−1
+f (zn)f(zn)
+(8)
+where α = 0.2 is a relaxation constant (32).
+In Fig. 2 (A), we can see the required algorithm steps that implement the iterative scheme
+described by Eq. (8). Note that the Jacobian changes value in each iteration and it is required
+that its inverse is calculated anew. This is traditionally a computationally expensive operation
+which is accelerated through the use of our metadevice. The results are then used to update the
+z. The method converges successfully after a few iterations.
+A numerical version (using MATLAB) is compared with the experimental results illustrated
+in Fig. 2 (B). We observe that for both MATLAB and the experiment, the estimation vector
+converges close to the exact roots. Moreover, the estimated vector reaches a stationary point as
+the iteration count increases. After 15 iterations the relative error is ||z − r||2/||r||2 ≈ 0.0023.
+This is similar to the accuracy achieved for the stationary trials, thus representing the accuracy
+floor of our system. A similar picture is also visible by comparing three specific iterations, as
+illustrated in Fig. 2 (C), where a comparison of the full Jacobian is presented.
+The experimental results do not precisely follow the paths indicated by the numerical im-
+plementation realized using MATLAB. This can be explained by adding random noise to the
+Jacobian on each iteration step. The added noise N ∈ C5×5 is a random complex matrix that fol-
+lows a normal distribution inside a disk with radius rN = 0.01λmax, where λmax is the maximum
+eigenvalue of the Jacobian. The noise creates many possible paths, all of which successfully
+7
+
+converge and we observe that our measured results comfortably lie within these families of
+curves. Note that some solution branches are more susceptible to this noise than others (e.g. r1
+(blue) and r3 (red) curves in Fig. 2(B)) and this is due to the details of the toy problem solved.
+Generally, the numerical accuracy of the device has a threshold that depends on both the
+implementation and the measuring apparatus (vector network analyzer (VNA)). When higher
+precision computations are required, this device can be a part of a mixed-precision computing
+system. In these systems, part of the calculations are done in a fast, low-precision estimation
+stage and then fed and further refined at a higher precision stage, similar to the in-memory
+mixed-precision approaches in electronic platforms (38).
+For the second example, we chose the case of an inverse design problem (Fig. 3 (A)). We
+assume that our design consists of a collection of m = 5 two-dimensional (2D) scatterers with
+circular cross section at fixed known locations r = [r1, ..., r5], each with an unknown bounded
+permittivity ε = [ε1, ..., ε5] ∈ C5×1. The goal is to achieve a specific user-defined scattered field
+measured at a series of n = 4 detection (objective) points, o = [o1, ..., o4]. Note that in our case
+we assume a collection of cylindrical circular scatterers (2D) excited with a monochromatic
+incident field of λw wavelength. The x-propagating incident field (kx) is a polarized in the
+z-direction (TE wave - Ez) with the ejωt convention.
+The scatterers are coupled, making this a nonlinear problem modeled using the Lippmann–
+Schwinger (39) scheme, solved with a standard discrete dipole approximation (DDA) method-
+ology (40). Each scatterer will respond to the local (self-excluded) electric field, which consists
+of the known incident electric field, einc ∈ C5×1, and the scattered field from all other scatterers,
+esca ∈ C5×1. The scatterers exhibits a complex polarization vector p = A(einc + esca) ∈ C5×1
+where A is the normalized polarizabilitiy diagonal matrix, i.e., A = diag(ε − εbackground).
+The field interaction between the scatterers are expressed via the Greens matrix G ∈
+C5×5
+(hollow symmetric matrix) such that esca = Gp.
+We may express the polarization vector
+8
+
+p = A(einc + Gp), which indicates the mutual dependence of p. Therefore, the polarization
+vector can be calculated as p = (A−1 − G)−1einc. Finally, we use the four objective points o to
+measure the scattered field vector emeas = Gprp ∈ C4×1 where Gpr ∈ C4×5 is the propagator
+Greens function. The measured field is then compared to a (user-defined) objective eobj ∈ C4×1.
+A typical constrained minimization problem (primal) can be written as (37)
+min
+x,y
+f(x, y)
+s.t.
+g(x, y) ≤ 0
+(9)
+where f(x, y) are the objectives and g(x, y) are the constraints. For such problems the La-
+grangian (dual) problem is expressed as
+max
+λ
+min
+x,y
+L(x, y, λ) = f(x, y) + λg(x, y)
+(10)
+Note that x and y may be subject to further requirements such as domains and bounds.
+For our particular example we have that x = p, y = ε, and f(p, ε) = 1/2||Gprp − eobj||2 and
+g(p, ε) = 1/2||(A(ε)−1 − G)p − einc||2. In our formulation, the objective is the scattered field at
+the observation points. The constraints comprise the self-consistency of the polarization vector
+(physics). Also, the permittivity vector is subject to specific bounds, i.e., ε ∈ R and ε ∈ [1, 5].
+Note that g(p, ε) is nonlinear with respect to p and ε and therefore requires a non-stationary
+approach.
+Following an initialization, our numerical evaluation of the above is implemented by a non-
+stationary algorithm that requires repeated application of the following three steps. First, we
+minimize with respect to ε by examining ∇εL(p, ε, λ) = 0. At this step we project the resulting
+permittivity vector to the desired domain and bounds. Second, we minimize with respect to p by
+examining ∇pL(p, ε, λ) = 0. At this stage the required stationary matrix inversion is performed
+with our metadevice. Finally we maximize for λ by using ∇λL(p, ε, λ) = 0. These steps are
+repeated until convergence is achieved, i.e.,
+E = ||eobj − emeas||2/||eobj||2 < δ
+(11)
+9
+
+(for more information see SM).
+As a numerical test case, the scatterers are assumed to be lossless with permittivity of ε =
+[ε1, ε2, ε3, ε4, ε5] = [3.5, 1.5, 1.5, 3.5, 1.5]. The objective scattered field at the detection points
+o, as depicted in (Fig. 3(A)), is eobj = [−0.0086 − 0.0078j, 0.0089 − 0.0132j, −0.0066 −
+0.0120j, 0.0043 − 0.0004j]. The values were extracted from the DDA method and verified with
+a full-wave COMSOL simulation. Note that the Fig 3(A) depicts the complex (hue/saturation)
+of the electric scattered field (Ez), i.e. the difference between the total field and the incident
+excitation.
+Figure 3 (B) depicts a set of four cases for the same algorithm. In the first case (black
+line), the idealized (noiseless, no filtering) computer evaluation of the algorithm is given - we
+observe that after only 20 iterations the error drops below 10−3. The experimental results are
+presented in Fig. 3(B) as red dots. The measured results exhibit an optimal point (minimum
+error) after 87 iterations, with an error of 0.00172. As an analog device, there is an additional
+systematic/stochastic/experimental noise to the system which affects the fidelity of the matrix
+inversion. We apply a simple averaging filtering scheme on the polarization estimation, i.e.,
+pnew = (1 − αF)p + αFpprevious, with αF = 0.25, as a way to partially mitigate this noise.
+The filter affects the convergence speed by increasing the iteration count but also significantly
+improves the accuracy/fidelity of the matrix inversion, hence the metadevice’s performance.
+This feature is illustrated in Fig. 3 (B), where the retrieved experimental results are compared
+to the idealized computer evaluation with the applied filter (blue line). We also performed a
+series of 100 randomized cases of the idealized filtered computer evaluation with added noise
+to the estimated/measured polarization vector (faint blue lines in Fig. 3(B)). The noise profile
+is similar to the one used in the first example (Newton’s method). The measured results are
+well contained within these error bounds. Note that iteration count is not equivalent of time.
+For a traditional computer evaluation, each iteration (with its required matrix-inversion) could
+10
+
+ultimately be slower than the convergence time of an optimized hardware implementation of
+the metadevice.
+Due to systematic/measurement noise, the error begins to grow after the experimental accu-
+racy floor is obtained - an indication that a termination criterion could be applied at this point.
+This result also agrees with the maximum accuracy we obtained in the previous non-stationary
+example. More sophisticated error-correcting and filtering schemes can possibly push the accu-
+racy below this threshold. For instance, αF could be adaptively tuned during the non-stationary
+evaluation to realize a mixed-precision computing system.
+At the minimum error point (iteration 87), the extracted permittivity estimation is illustrated
+in (Fig. 3(C)). Notice that the values are very close to the numerical test case objectives and
+permittivities. Finally, Fig. 3 (D) illustrates the path of the scattering vector, emeas, for these 87
+iterations. Similar to the above example, the faint paths represent the added noise effects to the
+numerical evaluation.
+For both presented non-stationary examples, it is evident that our metadevice can act ei-
+ther as an ultrafast analog computing machine and mathematics calculator with waves, or in
+a broader sense as an electromagnetic emulator for inverse design (41). It can be used for a
+plethora of realistic problems where the linear response of a system (i.e. matrix-vector mul-
+tiplication) or the solution of a system of equations (stationary problems, matrix inversion)
+is required. Moreover, the intuitive reconfigurability of this metadevice also enables the per-
+formance of constrained optimization tasks, like the ones required in non-stationary problems
+such as inverse design, where the desired response of complex media requires intensive opti-
+mization (42). In short, this metastructure can design metastructures. Finally, an adaptation
+of the above proof-of-concept metadevice in RF-IC, photonic, or hybrid platforms can make it
+an excellent candidate for on-the-fly or computation-through-propagation ultrafast, parallelized
+calculations.
+11
+
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+Acknowledgments
+The authors would like to thank Mario Junior Mencagli for useful discussions and preliminary
+experimental survey on the subject. D.C.T acknowledges Luiz F. O. Chamon and Juan Cervi˜no
+for the useful inputs and discussions regarding the constrained optimization algorithm.
+Funding
+: This work is supported in part by the Air Force Office of Scientific Research
+(AFOSR) Multidisciplinary University Research Initiative (MURI) grant numbers FA9550-17-
+1-0002 and FA9550-21-1-0312.
+Competing Interests
+: N.E. is a strategic scientific advisor/consultant to Meta Materials Inc.
+The authors have no competing interest.
+14
+
+Authors Contributions:
+N.E. conceived the idea for the reconfigurable device that solve
+equations, acquired the funds, and supervised the project. D.C.T developed further the rele-
+vant theories and analyses of the project. B.E. designed and programmed the device and the
+device’s calibration routine. D.C.T. and B.E. assembled, built, tested the components, and per-
+formed simulations and experimental measurements. D.C.T and B.E. developed the numerical
+examples. All the authors discussed the results. D.C.T wrote the first draft of the manuscript
+and D.C.T, B.E. and N.E. discussed, developed, and edited the final version of the manuscript.
+Data and materials availability:
+All data needed to evaluate the conclusions in the paper are
+present in the main text and the supplementary materials.
+Supplementary Materials
+Materials and Methods
+Supplementary Text
+Figs. S1 to S9
+References (1-15)
+15
+
+Figure 1: A reconfigurable wave-based analog computing metastructure: (A) A conceptual
+representation that describes the main objective, i.e., a reconfigurable device that can provide
+us with repeated matrix inversions of arbitrary matrices in order to achieve stationary and non-
+stationary algorithms. The device can implement any given kernel K = I − A and give the
+(I − K)−1 = A−1. (B) The central component of the design consists of a direct complex
+matrix (DCM) (32) architecture of 5x5 elements. (C) The experimental realization of this design
+for the 45 MHz operating frequency. (D) The essential element of the DCM is the multiplier
+module, consisting of both a phase-shifting and an amplification part (which can also function
+as attenuation part); controlled with an onboard microntroller unit. Finally, (E) and (F) depict
+the performance of the DCM machine in the open-loop (matrix-vector multiplication) and the
+closed-loop (matrix inversion) setups. In both we compare the experimentally obtained matrix
+to one computed conventionally to see good agreement.
+16
+
+(A)
+x = (I-K)"b
+(C)
+(E)
+(out)
+b (in)
+K
+=A
+90
+K
+n
+0.25
+in
+ino
+w
+180
+0
+-
+K
+n-2
+C
+o exact
+exp
+270
+a,
+(F)
+(B)
+a.
+12
+couplers
+2
+a.
+ino
+kernel
+(D)
+K=I-A
+90
+41
+control
+in
+out
+?2
+multiplier
+phase shift
+0.5
+52
+amp
+in.
+ino
+180
+in.
+in
+kernel
+exact
+in
+out
+phase shift
+amp
+exp
+K
+multiplier mod
+270Figure 2: Experimental verification of Newton’s root finding method with the proposed
+metadevice: (A) The algorithmic steps implemented in Newton’s root finding method. A fixed
+kernel is programmed into the DCM machine in each iteration. The measured results are used
+for calculating the next steps of the algorithm. (B) A comparison between the experimental
+results and a numerical implementation of the algorithm. The faint solid lines are cases where
+additional stochastic noise has been added to the system. We observe that the experimental
+trajectory is well contained within these simulated noisy paths. (C) A comparison between
+the numerical (left column) and measured (right column) evaluation of the inverse Jacobian at
+different iterations.
+17
+
+J-1(zn)
+(A)
+(C)
+4×103
+Algorithm 1: Newton's Method
+1: Initialize z, =[0,0,0,0,0]
+numerical
+experiment
+2: for n = 1, ... , m do
+3:
+J(z), f(zn), α
+> Jacobian, function, scaling
+4:
+K, = I - α,J(zh)
+> kernel for DCM
+n= 1
+5: 
+J-1(z.) = DCMnx(K.)
+> matrix inverse with DCM
+7:
+d,= J-1(z.)f(zn)
+8:
+Zn+1 = Zn- αd.
+9: end for
+(B)
+iterations
+numerical
+num + noise
+?n=15
+n= 5
+experiment
+0.2
+exact
+@n=5
+imag
+n= 1
+0
+ n=15
+-0.2
+-0.2
+0
+0.2
+realFigure 3: A metastructure that designs a metastructure: Numerical and experimental
+results: (A) Schematic of the numerical test case. A set of five two-dimensional (2D) scatterers
+with unknown permittivities, to be determined via our analog metadevice, ε = [ε1, ..., ε5] subject
+to a plane wave excitation. Pictured is the complex-valued scattered field. The scattered fields
+at the observations points [o1, ..., o4] are used as the benchmark values of this problem in which
+the objective fields are shown as the color in each torus. The algorithm tunes each permittivity
+in order to match the scattered field (center of each torus) to the objective fields. (B) The relative
+error E for the algorithm computed both numerically, and experimentally using the metadevice,
+under various noise and filtering scenarios. The experimental device gives a minimum relative
+error of 0.00172 at 87 iterations. (C) Objective permittivities (black rings) compared to those
+computed numerically (blue cross) and experimentally via the metadevice (yellow rhombus)
+using the described algorithm at iteration 87. (D) Evolution of scattered field vector up to
+iteration 87 in comparison to objective fields for experiment and simulation under various noise
+and filtering scenarios.
+18
+
+100
+(A)
+(B)
+experiment
+num
+num+filter
+>X
+2入
+num+noise+filter
+W
+min (87,0.0017)
+scatterers 入w/25
+OP
+10-1
+error
+3
+102
+0.025
+025
+objectivepoints
+103
+re
+0
+20
+40
+60
+80
+100
+120
+iterations
+(C)
+(D)
+X10-3
+numerical
+04
+0
+experiment
+exact
+permittivity
+3
+imag
+-5
+O
+2
+U
+-10
+1
+sca:numerical
+esca:experiment
+-15
+2
+3
+4
+5
+numberof scaterer
+-0.01
+-0.005
+0
+0.005
+0.01
+realSupporting Material
+Dimitrios C. Tzarouchis,1†‡ Brian Edwards,1† Nader Engheta1∗
+1Department of Electrical and Systems Engineering,
+School of Engineering and Applied Sciences,
+University of Pennsylvania, Philadelphia, 19104, U.S.A.
+†These authors contributed equally to this work.
+‡Present address: Meta Materials Inc. (Europe),
+Ap. Pavlou 10A, Marousi, 15123, Greece.
+∗To whom correspondence should be addressed; e-mail: [email protected]
+January 10, 2023
+1
+Details for the root finding algorithm
+In the following the lowercase quantities are vectors, the capitalized ones are matrices while Greek
+letters denote scalars - the subscripts follow a logical notation. The problem statement for the root
+finding procedure is
+f(z) = 0
+(1)
+find z ∈ Cm×1 that satisfies the above equation (roots) of f ∈ Cm×1. The above problem is solved
+using Newton’s method for finding the root of a vector polynomial function [1]. For example we have
+f(z) =
+�
+�
+�
+�
+�
+�
+f1(z1, z2, z3, z4, z5)
+f2(z1, z2, z3, z4, z5)
+f3(z1, z2, z3, z4, z5)
+f4(z1, z2, z3, z4, z5)
+f5(z1, z2, z3, z4, z5)
+�
+�
+�
+�
+�
+�
+(2)
+with z1·5 ∈ C or (equivalently)
+f1(z) = (z1 − r1)(z2 − 4.2i)(z3 + 2)(z4 − 5i)(z5 − 3.5)
+(3)
+f2(z) = (z1 − 3.9)(z2 − r2)(z3 + 2.5i)(z4 − 3.2i)(z5 − 4.2)
+(4)
+f3(z) = (z1 + 5.2i)(z2 − 4)(z3 − r3)(z4 − 4i)(z5 − 7.1)
+(5)
+f4(z) = (z1 − 3)(z2 − 7i)(z3 + 4)(z4 − r4)(z5 − 5i)
+(6)
+f5(z) = (z1 − 5.2i)(z2 − 4)(z3 + 4.75i)(z4 − 8)(z5 − r4)
+(7)
+where
+r = 1
+4 (s1 + c1i, −s1 + c1i, −s2 − c2i, s2 − c2i, 1i)T
+(8)
+with c1 = cos(2π/5), c2 = cos(π/5), s1 = sin(2π/5), and s2 = sin(4π/5). The point corresponds to
+the vertices of a regular pentagon. For the evaluation of Newton’s method we need to calculate the
+Jacobian matrix, i.e., Jij = ∂fi
+∂xj (here i and j are indexes) or
+Jf(z) =
+�
+�
+�
+�
+�
+�
+∂f1
+∂z1
+∂f1
+∂z2
+· · ·
+∂f1
+∂z5
+∂f2
+∂z1
+∂f2
+∂z2
+· · ·
+∂f2
+∂z5
+...
+...
+...
+...
+∂f5
+∂z1
+∂f5
+∂z2
+· · ·
+∂f5
+∂z5
+�
+�
+�
+�
+�
+�
+(9)
+1
+arXiv:2301.02850v1  [physics.app-ph]  22 Dec 2022
+
+therefore the root can be found as
+zn+1 = zn − αJ−1
+f (zn)f(zn)
+(10)
+where α is a relaxation constant. Here we used α = 0.2. In terms of an algorithm, we have the
+following routine
+Algorithm 1 Root finding with Newton’s method
+1: Initial guess for z1
+2: for n = 1, . . . , m do
+3:
+Jf(zn)
+4:
+αλ =
+2
+λmin+λmax
+▷ Scaling factor: λmim/max are the min/max eigenvalues of Jf(zn)
+5:
+Kn = I − αλJf(zn)
+▷ Kernel that is fed to DCM machine
+6:
+dn = J−1
+f (zn)f(zn)
+▷ Compute matrix inverse with the DCM machine
+7:
+zn+1 = zn − αdn
+8: end for
+2
+Details on the inverse design algorithm
+In this section we present the details for the inverse design algorithm implemented in text. The al-
+gorithm consist of a part of the DDA methodology for the quantification of the problem and an its
+adaptation to a Lagrange formalism for solving the require inverse scattering problem, the determina-
+tion of the permittivity of the scatterers. Note that both methods are arguably the simplest methods to
+follow, since they offer an intuitive understanding on the formulated problem and the coorresponding
+inverse-design (constraints optimization) problem.
+2.1
+Notes on the DDA method
+In this section we present a few details regarding the DDA method used in the main text. The details
+can be found also in [2, 3, 4, 5]. A similar methodological approach was also used in [6].
+We start by assuming that each 2D scatterer (assuming a point in the x-y plane) acquires its
+z-oriented dipole moment due to the local electric field, i.e.,
+p = αeloc
+(11)
+where the eloc is the vector of local z-polarized electric fields at the center of each point and α is the
+polarizability that depends on the shape and the material composition of each 2D scatterer. The local
+field is the sum of the incident field and the secondary fields generated from all the other dipoles such
+that:
+eloc = einc + Gp
+(12)
+where einc is the incident field vector, p is the induced polarization vector and G is the 2D Green’s
+function. In our case we consider a two-dimensional (2D) problem with a transverse electric (TE)
+excitation (the field is normal (z-direction) to the x-y plane). Therefore the corresponding Green’s
+function reads
+G = G(ri − rk) = −j k2
+0
+4πε0
+H(2)
+0 (k0|ri − rk|)
+(13)
+where H(2)
+0 (k0|ri − rk|) is the Hankel function of the second type (with the time harmonic convention
+e+jωt) and 0-th order and k0 = ω0√µ0ε0 is the free-space wavenumber [7]. The G is a CN×N Toeplitz
+matrix with zero diagonal entries since the |ri − rk| is treated as in (assuming a uniformly spaced
+discrete grid) [2, 3, 4, 5].
+By combining Eqs. (11) and (12) we obtain the following expression, arranged using the matrix
+formulation as follows
+p =
+�
+A−1 − G
+�−1 einc
+(14)
+2
+
+where the lowercase quantities p = [p1, p2, ..., pN]T , A = diag(α), α = Acellε0[ε1 − 1, ε2 − 1, ..., εN − 1]
+(Acell is the cross-sectional area of a cylinders) and einc = [einc
+1 , einc
+2 , ..., einc
+N ]T are CN×1 vectors, diag(·)
+is the diagonal matrix operator.
+Finally, the scattered field observed at M specified discrete detection points (in general M ̸= N)
+is given by:
+esca = Gpr p = Gpr
+�
+diag(α−1) − G
+�−1 einc
+(15)
+where Gpr ∈ CM×N is the “propagator" Green’s function matrix. This propagator function connects
+the induced dipole polarization vectors of the scatterers with the desired detection (or objective) points.
+The above matrix representation of the scattering problem allow us to have a clear inspection of the
+unknown quantities. These quantities are the ones that will be formulated as a Lagrange multiplier
+algorithm for the solution of the desired constrained optimization problem. We note here that, as seen
+in Eq. (15), the forward scattering problem requires a matrix inversion to evaluate the polarization
+density vectors induced in each scattering cell, as we have discussed in our previous work [6] in which
+we utilized the same DDA approach for the evaluation Eq. (14) and the matrix-vector operation of
+Eq. (15) for different excitation and for different scattering scenarios.
+2.2
+Notes on the Lagrange multiplier algorithm
+For this, we utilize the DDA algorithm (where we closely follow the contrast source inversion method![8])
+and the Lagrange multiplier method for applying the constraints and finding the optimal solution.
+First, in terms of the defined problem, we have that the polarization is connected with the following
+expressions
+p = A(einc + Gp)
+(16)
+and
+esca = Gprp
+(17)
+A typical constrained minimization problem (primal) can be written as [9, 1, 10]
+min
+x,y
+y∈R
+0≤y≤1
+f(x, y)
+s.t.
+g(x, y) ≤ 0
+(18)
+where f(x, y) is the objective and g(x, y) are the constraints also subject to further requirements of
+the problem such as y ∈ R and 0 ≥ y ≥ 1 For such problems the dual Lagrangian problem is expressed
+as
+max
+λ
+min
+x,y
+y∈R
+0≤y≤1
+L(x, y, λ) = f(x, y) + λg(x, y)
+(19)
+which is essentially a dual unconstrained problem (since all the constraints are encapsulated to the λ
+term). It is worth noting that the Lagrange multiplier should be positive real, λ ∈ R+. Finding an
+approximate solution to the primal inverse scattering problem is therefore reduced to finding a solution
+to the above dual problem. Notice that the Lagrange multiplier can be applied to either f(x, y) or
+g(x, y) without affecting the outcome of the overall process.
+The algorithm for solving the above dual problem is the following:
+• Step 0: initial x0 and λ0
+• Step 1: minimize yn, i.e., via ∇yL(xn−1, yn, λn−1) = 0
+• Step 2: project yn into y ∈ R and 0 ≤ y ≤ 1
+• Step 3: minimize xn, i.e., ∇xL(x, yn, λn−1) = 0
+• Step 4: maximize λn, i.e., ∇λL(xn, yn, λ) = 0
+• Step 5: Repeat steps 1-4 until the error is minimized
+3
+
+For our particular example we have that x = p, y = A = diag(ε − 1), and f(p, A) = 1/2||(A−1 −
+G)p − einc||2 and g(p) = 1/2||Gprp − eobj||2, and Lagrange function reads
+L(p, A, λ) = ||(A−1 − G)p − Aeinc||2 + λ||Gpp − eobj||2
+(20)
+The corresponding algorithmic steps are:
+• Step 0: initial p0 and λ0
+• Step 1: minimize An via ∇AL(pn−1, A, λn−1) = 0
+– We have that ∇AL(pn−1, A, λn−1) = ∇f(p, A)∗||(A−1−G)p−einc|| (∗ is complex conjugate).
+This expression lead to An = p/(Gp − einc). In practice this is a simple calculation since A
+is a diagonal matrix, i.e., A = diag(ε − 1).
+• Step 2: project An into An ∈ R and 0 ≤ A ≤ 4 (for the range ε ∈ [1, 5])
+– this is the point where essentially the required properties and bound of the permittivity can
+be implemented. These bounds or constrains can be general
+– the above projection is rather a simple projection that does not guarantee always the min-
+imum within the projection domain.
+A more accurate projection would be of the form
+An = proj[An−1 − η∇A||(A−1
+n−1 − G)pn−1 − einc||2].
+• Step 3: minimize pn, i.e., ∇pL(p, An, λn−1) = 0 (DCM metadevice).
+– pn = K−1
+n en
+L
+– Kn = (A−1
+n
+− G)∗(A−1
+n
+− G) + λn−1G∗
+prGpr
+– eL
+n = λn−1G∗
+preobj + (A−1
+n
+− G)∗einc
+– The matrix inversion pn is performed with our DCM metadevice
+– Due to noise error a simple weighted average filtering is applied, i.e., pn = (1 − αF )pn−1 +
+αF pn with αF = 0.25
+• Step 4: maximize λn, i.e., ∇λL(An, pn, λ) = 0
+– This maximization can be calculated by a simple gradient descent, i.e., λn = λn−1 +
+η (∇λL(pn, An, λ) − δ) or λn = λn−1 + η
+�
+||Gppn − eobj||2 − δ
+�
+– Notice that this is an gradient ascent since we assume η > 0, therefore we maximize the
+problem.
+• Step 5: Repeat steps 1-4 until the error is minimized. In our case we used the following error
+– ||esca − eobj||2/||eobj||2
+Note that the quantities η and δ are the step and minimal error quantities that are user determined.
+The whole process stop either when λ reaches a plateau, or when the required error criterion is met.
+The optimization goal was set as ||esca−eobj||2
+||eobj||2
+< δ, where esca = Gppm with pm = (A−1
+m − G)−1einc
+being the final m-th evaluation of the iteration.
+Notice that our approach has several similarities with the contrast source inversion method and
+other similar inverse scattering methods [11, 8, 12, 13].
+Undoubtedly this approach is only one of the available methods for approximating the inverse design
+problem. This is rather an attempt to showcase the ability of our device for performing inverse design
+with desired objectives and constraints by exposing the crucial parts of the algorithm, such as the
+matrix inversions. This part is usually implicit within commercially available FDTD or FEM software.
+Hence here we developed our own methodology so we can have deeper inspection to quantities. As
+a remark, the field of inverse design and inverse scattering is a very rich field with a plethora of
+methodologies that try to address similar problems [14].
+4
+
+Figure S1: Photograph of the experimental setup with the corresponding components.
+3
+RF Design, PCB, Device Implementation
+A photograph of the experimental setup is shown in Fig S1, where all parts are designated accordingly.
+3.1
+Measurement
+Measurements were performed using an ENA-5071C two port VNA. In order to avoid the saturation
+of the amplifiers (multiplier module) the VNA power level was set to be −20dBm for the open loop
+configuration and −10dBm for the closed loop configuration. The VNA was set to have an IF band-
+width of 10 kHz. The single frequency measurements (1601 point) at 45MHz with averaging applied
+after obtaining the measured signal from VNA.
+3.2
+Multiplier
+The schematic of the multiplier is depicted in Fig. S2. The multiplier was designed to perform multi-
+plication on the incoming complex amplitude such that a new complex amplitude is rendered at the
+output. In other words, the output is Vout = zVin. This involves changing both the amplitude and
+phase of the incoming signal. Phase change was performed using a pair of serially connected Minicir-
+cuit JSPHS-51+ Phase Shifters (PS). Each phase shifter provides slightly over 180 degrees of rotation.
+The amplitude change was performed using the Analog Devices AD603ARZ Variable Gain Amplifier
+(VGA). The Multiplier design contain the appropriate loads such that both the input and output of
+the device externally appears as 50 Ohm.
+Both of these devices are controlled using analog voltages with ranges of [−0.5V, +0.5V] and
+[0V, 12V] for the VGA and PS, respectively. In order to create a common control mechanism, op-
+amp level shifting circuits were used to put these on a common [0V, 5V] interface. The Multiplier
+5
+
+couplers
+system in (trigger)
+system out (read
+DCMcontrol
+kernel out
+kernel.in
+kernel(DCM
+VNA port 2
+VNAport
+1-6.switch
+1-5 switch
+DCMpower sourceFigure S2: Schematics for the multiplier: The PCB layout design (top figure), and the corresponding
+subcircuit (pictures from Altium®). Bottom figure represent the AWR Microwave Office® schematic
+with the realistic data
+board has a connection that allows for a daughter board. The daughter board is supplied with 0V and
++5V and is responsible for returning two control voltages in the range of [0V, 5V].
+This simple interface allows for a number of possible control schematics. At its most simple scenario,
+the control board can consist of a pair of potentiometers. However, we will present another control
+board which utilizes a microcontroller to receive UART input and render the two analog voltages.
+The VGA’s dynamic range could be shifted using an external resistor. This was set so that the
+Multipliers’s range (including load elements, PS losses, etc) was [-30dB – +17dB]. The multiplier
+effectively saturates if the input is greater than -10dBm. Therefore for all measurements the reference
+input signal that was used was -30dBm for avoiding any saturation effects.
+It should be stated that the VGA imparts a varying phase change and the PS pair imparts an
+amplitude change. This will be addressed later.
+3.3
+1-5 splitter (5-1 combiner)
+The schematic of the 1-5 splitter is depicted in Fig. S3. An ideal passive n-way splitter is comprised of
+a summation port and n feed ports. The scattering parameters are expected to be reciprocal such that
+for the ith feed port |SSi|2 = |SiS|2 = 1/n and all other elements within the matrix are zero. Due to
+losses, a real splitter will fall short of this precise definition. Our splitter was based on the Minicircuits
+AD5PS-1+, which yielded good performance at 45MHz with approximately -7.2dB split ratio for all
+outputs.Note that 1/5 ≈ −7.0dB.
+6
+
+: +15
+GND 15 
+5
+C8
+GND
+GND
+JGA
+RF
+GND
+NetU1_7
+NetU1_7
+GNDGND
+R8
+2 : GND
+ 5 : GND
+000
+R6
+00
+ _2 : GND
+1 : NetJ1_-1
+5
+R7
+1 : NetC2_2
+000
+000
+5 : GND
+2 : GND
+C2
++15
+R1
+3
+R2
+GND
+61300211121
+U2
+VGASub
+VGASub SchDoo
+ RF_In RF_Out 
+> Cont_ In
+CONMCX003.031
+CON ACX003.031
+ PSInput
+JSPHS-51+
+JSPHS-51+
+613b041142
+C6
+CNT O
+10uF
++5
+ContProc
+C7
+ContProcessor. SchDoc
+10uF
+J3
+[0.5]
+> PS_In
+PS_Out 
+[0.15]
++15
+4
+10uF
+[0.5]
+ VGA_In VGA_Out 
+[-0.5, 0.51]
+951103-8622-AR
+P3
+61300411021
+CNT
+RFIn
+RFOut
+100R1%
+.7nF5%
+>100R 1%
+100R1%
+LM6172IMX
+1k 1%
+[0.15]
+PS Out
+PS Im
+[0,5]
+VINP
+VOUT7
++5H
+VPOS
+O v[so's0-]
+0.1uF 5%
+GRES
+DINOO
+R7
+FDBK
+4
+3.5k 1%
+7.15k 1%
+AD603ARZ
+0.1uF 5%
+VGAIn
+[0.5]
+1k 1%
+Cont InSUBCKT
+TLIN
+ID=S2
+TLIN
+SUBCKT
+TLIN
+PORT1
+ID=TL1
+NET='phase BLK"
+ID=TL2
+ID=S1
+ID=TL3
+P=1
+Z0=50 Ohm
+Vph1=V/ph
+Z0=50 Ohm
+NET="AD_603"
+Z0=50 Ohm
+Z=50 Ohm
+EL=el Deg
+Vph2=Vph
+EL=el Deg
+VG=Vg_test
+EL=el Deg
+Pwr=[-30] dBm
+F0=45 MHz
+F0=45 MHz
+F0=45 MHz
+PORT
+P=2 
+Z=50 OhmFigure S3: Schematic and layout for 1-5 splitter based on the Minicircuits AD5PS-1+ (pictures from
+Altium®)
+3.4
+Feedback coupler
+Figure S4: Schematic and layout for the feedback coupler (pictures from Altium®)).
+The schematic of the feedback coupler is depicted in Fig. S4 The Feedback coupler must perform
+several tasks.
+• Provide near unity feedback
+• Introduce the input signal
+• Sample the output signal
+Therefore, the feedback coupler is a four-port device wherein the primary path has near unity trans-
+mission such that the feedback is strong.
+3.5
+Switches
+In order to replicate having a 10 port VNA, we utilized two demo boards (EV1HMC253AQS24), which
+acted as RF SP8T RF switches, i.e., an analog multiplexer. For one SP8T, we utilized five of these
+ports for illuminating the bank of five couplers. The other SP8T was used to receive signals from the
+couplers. The remaining three ports on each were used for system sanity checks. Note that the stock
+high-pass 100pF capacitors on these boards were switched to 470pF for better transmission at 45MHz.
+7
+
+CONMCX003.031
+P1
+CONMCX003.031
+U1
+NelJi_
+5 : GND
+P3
+PS
+: Net/3
+GND
+:NeJ4.1
+5 : GND
+CONMCX003.031
+2 : GND
+CONMCX003.031
+AD5PsJi+
+139-AD5PS-1
+CONMCX003.031
+2lst
+CONMCX003.031J1
+J2
+OO
+OO
+1 : NetJ1_
+1 : NetJ1_1
+OGO
+R3
+R1
+OGO
+Nei1.1
+Nei.1
+Neia.1
+NeiJ4.1
+R4
+82
+J3
+J4
+2
+OGO
+OGO
+1 : NetJ3_1
+1 : NetJ4_1
+OO
+OO
+>R3
+>R1
+1k
+1k
+GND
+GND
+GND
+R4
+R2
+GND
+50R
+50R
+GNDWhile the off ports were nominally matched to 50 Ohm from DC-2.5GHz, there was significant
+reflections.
+Deeper inspection of the datasheet indicated that the "off" ports were only matched
+above 500MHz. Measurements indicated that the off ports were approximately "open" at the design
+frequency and therefore reflections from the off ports could be significantly reduced with parallel 50Ohm
+terminations. However, this was not done as this it would have reduced power within the system on
+the "on" port. Rather, we note that any polluting signal from these "open" off ports will have crossed
+through the coupler twice. Due to the small coupling coefficient of the feedback coupler, these values
+will have become very small.
+The VNA was calibrated to the end of the switch ports. Measurements indicated that transmission
+through each of the switch ports was similar enough as to not warrant individual calibrations on each.
+Each of the switches was actuated by three digital inputs to address the 23 = 8 ports on each
+switch. These digital signals were created by a micro-controller which was programmed to respond to
+UART commands from an attached computer. Code is available at github.com/brianedw/RFMath/
+Arduino/mcu_control_V2/mcu_control_V2.ino.
+3.6
+Micro Controller Unit (MCU)
+The two analog input control voltages for each Multiplier was created by an MCU Control Board,
+which attached directly to the Multiplier. The heart of this board is a Metro-Mini MCU.
+Each control line was connected to both an 8-bit PWM DAC pin (labeled “fast”) and a 10-bit
+PWM DAC pin (labeled “slow”). While both pins connected to the control line through a high-pass
+filter, the fast DAC utilized a lower capacitance and resistance than that of the slow DAC. During a
+set operation, both pins would drive to their appropriate values, during this time, the behavior of the
+collective output would be dominated by the fast DAC and rapidly converge, but exhibit large ripples.
+After 20ms, the fast PWM DAC pin would switch to a high-impedance state, leaving the voltage to
+settle in the remaining difference utilizing the slow DAC alone. The high-pass filter was designed to
+maintain accuracy of 10bit. Since the Metro-Mini is a 5V compliant device, the generated voltages
+nicely matched to the expected inputs of the Multiplier.
+Each MCU board had two 3-pin UART input connectors. These were shorted such that one could
+be used to receive a command from "upstream" while the other would effectively passively repeat the
+signal. Additionally, each MCU board had two 3-pin UART output connector which were similarly
+shorted together, allowing it to transmit the same message to two devices. Each MCU was programmed
+with a unique identification number. Upon receiving a UART command, it would either act on that
+command or repeat the command on its output UART pins for downstream devices. This input/output
+configuration created a lot of possibilities for control topologies. However, in practice we found that
+we could use a single MCU board (no multiplier attached), as a bridge between the computer and the
+array of MCU Boards and that this array could all be connected in parallel such that the output of
+the bridge was effectively driving 25 inputs. Note that the required time complexity is of the order
+of O(n2).
+Possibly this complexity can be further reduced by implementing different connectivity
+schemes than the simple serial one that we used. Code is available at github.com/brianedw/RFMath/
+Arduino/mcu_control_V2/mcu_control_V2.ino.
+4
+Tuning/Calibration
+As stated in 3.2, the VGA has a minor effect on the phase and the PS has a minor effect on the
+amplitude.
+In other words, the phase and amplitude responses are coupled.
+Additionally, other
+systematic errors are present such as nonidealities in the level shifting circuits due to resistor tolerances.
+When connected in a network that includes RF jumper cables of varying length, there will also be phase
+shifts that naturally arise. In short, the relationship between the control voltages and the response
+of the Multiplier in situ, are repeatable, but difficult to predict without developing a more complex
+model.
+We found that an effective strategy to capture, model, and invert the relationship between control
+voltages and system response goes as follows.
+1. A collection of Multipliers are swept across their input values to map the relationship between
+control voltage and complex multiplier response.
+8
+
+2. These responses were analyzed using Principle Component Analysis (PCA) [15].
+3. The multipliers were assembled into the open-loop configuration and the response of the entire
+open-loop network was measured under many sets of input control voltages.
+4. These results were compared to a theoretical model of the network wherein the weights of the
+components could be adjusted until the theoretical results matched the experimental results.
+5. With accurate PCA weights in hand, the Multipliers can be immediately adjusted to achieve a
+desired multiplication factor by inverting the model to achieve any open-loop kernel.
+6. Additional refinement can be obtained by changing the device configuration into the closed loop,
+which now includes the feedback couplers. Again, we measure the response of the closed-loop
+network under many input conditions.
+7. We further refine the PCA weights of each multiplier to match this more demanding data set.
+This becomes our final device model for both the open- and closed-loop configurations.
+We will go into detail on each one of these items in the following sections.
+4.1
+Multiplier PCA
+A collection of 35 multipliers were each mapped using the MCU control boards, capable of 10-bit
+resolution on both control voltages. The mapping occurred with a grid of values based on [0, 11, ...,
+1012, 1023] on both controls. Ideally, the mapping of two Multipliers would yield identical responses.
+However, for all the reasons stated above, they do not. All of the mappings were compared using a
+complex domain PCA analysis. Typically, in PCA, one would examine deviations from the mean, but
+here we take another approach. Rather, the collection of mappings were analyzed directly to yield a
+set of 4 PCA components. The response of any individual Multiplier could then be found as the linear
+superposition of these components given by:
+m(dvga, dPS) =
+3
+�
+i=0
+wici(dvga, dPS)
+The term c0(dvga, dPS) is effectively the “average” response scaled by a complex factor, while the next
+several components represent likely deviations due to the systematic errors described above. Within a
+PCA analysis the final PCA components (i.e. c34(dvga, dPS), not shown) should be nearly pure noise.
+We found that only the first four terms were needed to effectively model any given Multiplier.
+Given any randomly chosen multiplier, we can find the complex valued PCA weights wi through a
+least-squares analysis. As opposed to the "deviation from the mean" approach, the above formulation
+is particularly useful for RF engineering. While the Multipliers were measured directly at their input
+and output ports and analyzed as such, the model can easily account for the addition of RF cables
+which would provide attenuation and phase rotation. These will appear as a complex scaling of all
+of the components weights and the Multiplier’s behavior (RF jumpers cables included) can still be
+captured as the simple linear superposition of the PCA components.
+In fact, any losses or phase
+rotations along the Multipliers flow path can be incorporated into these weights. Therefore, we do
+not characterize the individual multipliers, but delay this until the architecture is fully assembled, as
+described in the next section.
+Regardless, we will use least-squares to find the set of wi which characterizes the average multiplier
+response. We call these the “base weights”.
+4.2
+Open-Loop Device Fitting
+The goal of the this section is to determine the PCA weights that characterize each Multiplier in
+situ, so that the system errors can be captured and modeled. The open-loop DCM system was fully
+assembled including jumper cables, splitters, and couplers. All multipliers within the array were set
+to the same input value (dvga, dPS) pair. The transmission matrix of the system was then measured.
+This was repeated for all possible combinations of 10 evenly spaced values in the range [0, 1023] to
+9
+
+Figure S5:
+PCA Components and Average Multiplier Response.
+The first four panels represent
+c0(dvga, dPS), c1(dvga, dPS), c2(dvga, dPS), and c3(dvga, dPS) and have a maximum saturation of 0.05.
+The final image shows the response of the “average” Multiplier with a maximum saturation at 7.5
+yield 100 measured transmission matrices, Tmeas. Note that not all of these 100 transmission matrices
+represent "passive" operators.
+The same system was modeled using Scikit-RF, wherein the following assumptions were made:
+• The 5-1 splitters were ideal such that power was evenly split with no phase
+• All jumpers were zero-length
+• The coupler feedback path was ideal with no power removed
+• The multipliers were all assumed to be “average” and the their responses were assumed to be
+given by the “base weights”.
+The system was simulated using SciKit-RF for each input pair (dvga, dPS) to yield 100 measured
+transmission matrices Tsim(w), which are naturally a function of each Multipliers PCA weight. We
+can then define an error error(w) = |Tsim(w) − Tmeas|2 and optimize w until that error is minimized.
+It should be noted, that with only four PCA weights per Multiplier, in theory, only 4 transmission
+matrices are required to fully define the system. Using 100 helps guarantee that normal measurement
+noise does not unduly influence the fitting. Additionally, if a low error can be achieved across 100
+measurements using only 4 weights, then we can be confident that the model was sufficient to capture
+the entire open loop system response, K.
+4.3
+Setting the Open Loop System Response
+Given a desired open-loop system response, K, we need to calculate the necessary multiplier values for
+the DCM architecture, mi,j, gathered to form M. In this case, the simplicity of the DCM architecture
+makes this trivial. If we assume an idealized passive five port splitters such that given an input of 1W
+at the summation port, s, we will observe 1/5W on each branch port, i. Put in terms of Scattering
+Parameters, Ss,i = 1/
+√
+5 and via reciprocity Si,s = 1/
+√
+5. Since we have such splitters at the input
+and output of the Multiplier array, K = (1/
+√
+5)M(1/
+√
+5) and therefore M = 5K. Note that since we
+fitted the PCA weights of the Multipliers under the assumption of ideal components, it is appropriate
+to assume ideal components here.
+With each of the desired mi,j in hand to achieve a given K, the next step is to determine the
+required (dvga, dPS). This can be done using a number of function inversion schemes such a gradient
+descent. In practice, this could be very fast as it is likely that in many applications, each new M will
+be a small step from the previous M and therefore each multiplier will change only slightly.
+4.4
+Closed-Loop Device Fitting
+Due to the recursive nature of the closed-loop configuration (Matrix Inversion), the accuracy require-
+ments are more stringent than for the open-loop configuration. Moreover, additional degrees of freedom
+are introduced in the form of coupler coefficients. These can be considered part of w. In short, the
+devices must be fitted again.
+We will employ a similar strategy as was used in the Open-Loop Device Fitting. Using the open-
+loop calibrated device models, a sequence of randomly generated passive transmission matrices, K,
+10
+
+co(dvga, dps)
+Ci(dvga, dps)
+C2(dvga, dps)
+C3(dvga, dps)
+mave(dvga, dps)
+imag
+imag
+ima
+imag
+dps
+real
+real
+real
+real
+realare shown to the system. Note, unlike the open-loop matrices, in order to guarantee convergence,
+these matrices must be passive. We model the closed-loop system using Scikit-RF. Using the open-
+loop weights as a starting point, we optimize the multiplier weights and coupling coefficients until the
+simulated Tsim(w) matches the measured Tmeas. This represents a small, but necessary, refinement
+from the open-loop device model and can be used for both open- and closed-loop applications.
+4.5
+Setting the Closed Loop System Response
+Setting the closed-loop system response, K is identical to setting the open-loop response. In both
+cases, each desired mi,j is used to find the required (dvga, dPS) using a function inversion scheme.
+5
+System Accuracy
+We performed an open loop measurement on 100 complex-valued random matrices with (eigenvalues)
+values within the unit circle. For these, we configured the open-loop with the target (or ideal kernel)
+Ae and retrieved the measured results Am. We define as error the quantity
+||Am − Ae||2
+||Ae||2
+100%
+(21)
+In Fig. S6, we can see the difference between the two matrices for 100 random cases. We observe
+that all the results are within a 0.05 − 0.3% percent error. Similarly, we performed the same error
+analysis for the same 100 random matrices, only this time on a closed-loop setup (matrix-inversion).
+The results (Fig. S7) reveal that the error can climb up to 20%, but for most of the results, we get a
+matrix inversion with less than 2% error. Finally, we assess the matrix inversion fidelity by evaluating
+the trace of the A−1
+m Ae product. Ideally the trace of the product tr(A−1
+m Ae)/5 = 1. In Fig. S8, we
+observe that this product spans between 0.5 − 1.5. However, for the particular examples we used in
+the manuscript, this accuracy can be maintained at reasonably high levels once error-correcting and
+filtering techniques are applied. Note that for the closed-loop case, the level of the measured voltage
+is in the order of µV, very close to the noise floor of the VNA device we used. For the open loop
+operation, the measured voltage was hundreds of mV.
+0
+50
+100
+0
+0.1
+0.2
+0.3
+Figure S6: The error between the exact and the measured matrices, open loop configuration, for 100
+random complex matrices.
+11
+
+0
+50
+100
+0
+5
+10
+15
+20
+Figure S7: The error between the exact and the measured matrices, closed loop configuration (matrix
+inversion), for 100 random complex matrices.
+6
+System Transient Analysis
+6.1
+Single Multiplier
+In terms of the time response of the multiplier module the transient analysis reveal (Fig. S9) that
+the module obtained the desired value approximately within 3-4 signal periods, i.e., T = 22.2ns. The
+measurements were performed using the RIGOL DG4062 pulse generator (15 sinusoidal pulses at
+45MHz), and the measured response extracted with the RIGOL DS1104 oscilloscope.
+The open loop response is therefore assumed to be very close to the single multipliers response
+since both splitters and connecting cables introduce a small phase shift to the signal. The closed loop
+transient response is affected by both the multiplier timing and the condition number of the input
+matrix (kernel) as shown in [16].
+12
+
+0
+50
+100
+0
+0.5
+1
+1.5
+2
+Figure S8: The fidelity of the matrix inversion expressed in terms of the normalized trace of the A−1
+m Ae
+product for 100 random complex matrices.
+measurements
+in
+out
+0
+50
+100
+150
+200
+ns
+simulations
+in
+out
+Figure S9: The transient response of a single multiplier module.
+The blue curves correspond to
+the input signal, while the red curves are the measured (top) and simulated (bottom) using AWR
+Microwave Office® results. The agreement is excellent. It is evident that it takes approximately 3 to
+4 signal periods for the multiplier to obtain the desired output signal. Here we assumed small signal
+amplification (VGA voltage is +.05) and the phase shift voltage is 0V.
+13
+
+7
+De-embedding the solution
+7.1
+Open Loop
+Let us define the open-loop response as
+Vout = KVin
+Note that this includes not only the DCM architecture (multipliers, splitters, jumpers), but also the
+through channel of the input/output coupler. In other words, the open-loop is defined using all of
+the components of the closed-loop. However, the loop has been broken "open" just after the coupler
+array and measured at this point. Since in the closed configuration, these measurement planes were
+coincident, upon "closing" the loop, these measurement will then represent the complete response of
+the loop. While a minor perturbation to the results, this definition assumes that the weakly coupled
+additional ports on the coupler are properly terminated.
+Let us further define response of only the DCM architecture as K′. When the system is in a closed
+loop configuration, this relates the vector exiting the coupler array (V4) to the vector incident on the
+coupler array (V2).
+V2 = K′V4
+The coupler array introduces a small loss as the input is introduced and the output is measured. The
+near unity transmission is named α1. It is clear then that K = α1K′.
+7.2
+Closed Loop
+The closed loop response is fully defined by the open-loop response and the definition of the scattering
+parameters of the coupler.
+V2 = K′V4
+(22)
+V3 = α2V1 + βV2
+(23)
+V4 = βV1 + α1V2
+(24)
+Our goal is to solve the equations for V4, which represents the vectorial solution of the problem in
+question. For the expected solution, this should be done such that the solution depends only on the
+kernel K and the input vector (V1). For the measured solution, this should be only in terms of the
+measured results (V3) and the known input (V1).
+7.3
+Expected Solution
+We begin by applying the definitions above
+V4 = βV1 + α1V2
+V4 = βV1 + α1K′V4
+V4 = βV1 + KV4
+and then solve the final equation for the V4.
+V4 = (I − K)−1βV1
+7.4
+Measured Solution
+We begin with Eq 23:
+V3 = α2V1 + βV2
+and then solve it for V2
+V2 = 1
+β V3 − α2
+β V1
+14
+
+and then substitute the above into 24
+V4 = βV1 + α1( 1
+β V3 − α2
+β V1)
+and then simplify
+V4 = (β − α1α2
+β
+)V1 + α1
+β V3
+Note that in many real world cases, the coupler will be defined such that we can assume α2 → 0
+V4 = βV1 + α1
+β V3
+References
+[1] D. P. Bertsekas, Nonlinear programming (Athena Scientific„ Belmont, Mass. :, 1995).
+[2] E. M. Purcell, C. R. Pennypacker, Astrophys. J. 186, 705 (1973).
+[3] B. T. Draine, P. J. Flatau, J. Opt. Soc. Am. A 11, 1491 (1994).
+[4] M. A. Yurkin, A. G. Hoekstra, J. Quant. Spectrosc. Radiat. Transf. 106, 558 (2007).
+[5] S. P. Groth, A. G. Polimeridis, J. K. White, J. Quant. Spectrosc. Radiat. Transf. 240, 106689
+(2020).
+[6] V. Nikkhah, D. C. Tzarouchis, A. Hoorfar, N. Engheta, ACS Photonics (2022).
+[7] C. A. Balanis, Advanced engineering electromagnetics (John Wiley & Sons, 1999).
+[8] P. M. van den Berg, R. E. Kleinman, Inverse Probl. 13, 1607 (1997).
+[9] D. Bertsekas, Convex optimization theory, vol. 1 (Athena Scientific, 2009).
+[10] S. Boyd, S. P. Boyd, L. Vandenberghe, Convex optimization (Cambridge University Press, 2004).
+[11] R. E. Kleinman, P. den Berg, J. Comput. Appl. Math. 42, 17 (1992).
+[12] D. Colton, R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, vol. 93 of Applied
+Mathematical Sciences (Springer New York, New York, NY, 2013).
+[13] S. Boutami, S. Fan, Journal of the Optical Society of America B 36, 2378 (2019).
+[14] Z. Li, R. Pestourie, Z. Lin, S. G. Johnson, F. Capasso, ACS Photonics 9, 2178 (2022).
+[15] I. T. Jolliffe, Principal component analysis for special types of data (Springer, 2002).
+[16] D. C. Tzarouchis, M. J. Mencagli, B. Edwards, N. Engheta, Light Sci. Appl. 11, 263 (2022).
+15
+

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+Property-Based Mutation Testing
+Ezio Bartocci
+TU Wien
+Vienna, Austria
+[email protected]
+Leonardo Mariani
+University of Milano-Bicocca
+Milan, Italy
+[email protected]
+Dejan Niˇckovi´c
+Austrian Institute of Technology
+Vienna, Austria
+[email protected]
+Drishti Yadav
+TU Wien
+Vienna, Austria
+[email protected]
+©2023 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including
+reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or
+reuse of any copyrighted component of this work in other works.
+Abstract—Mutation testing is an established software quality
+assurance technique for the assessment of test suites. While it is
+well-suited to estimate the general fault-revealing capability of a
+test suite, it is not practical and informative when the software
+under test must be validated against specific requirements. This
+is often the case for embedded software, where the software is
+typically validated against rigorously-specified safety properties.
+In such a scenario (i) a mutant is relevant only if it can impact
+the satisfaction of the tested properties, and (ii) a mutant is
+meaningfully-killed with respect to a property only if it causes
+the violation of that property. To address these limitations of
+mutation testing, we introduce property-based mutation testing, a
+method for assessing the capability of a test suite to exercise
+the software with respect to a given property. We evaluate
+our property-based mutation testing framework on Simulink
+models of safety-critical Cyber-Physical Systems (CPS) from the
+automotive and avionic domains and demonstrate how property-
+based mutation testing is more informative than regular mutation
+testing. These results open new perspectives in both mutation
+testing and test case generation of CPS.
+Index Terms—Cyber-Physical Systems, Mutation Testing, Sig-
+nal Temporal Logic (STL), Simulink Models, Software Testing
+I. INTRODUCTION
+Software has a pivotal role in safety-critical applications,
+from autonomous vehicles to medical devices. Inadequate soft-
+ware quality assurance may result in potentially catastrophic
+system failures. It is thus important to thoroughly test software,
+checking that it does not violate its critical properties.
+Mutation testing (MT) is a well-established technique to
+measure the adequacy of a test suite w.r.t. a fault model [1]–
+[4]: MT first injects some artificial defects in the software-
+under-test, and then measures the thoroughness of the test suite
+as the percentage of injected faults that the test suite can reveal.
+The injection is performed through mutation operators that
+modify the software according to well-defined patterns. The
+resulting modified program is called a mutant. A test case kills
+a mutant if its execution causes observable differences in the
+behavior of the original and mutated programs. The ratio of
+killed mutants w.r.t. the mutants that are not equivalent to the
+original program is known as the mutation score. Ideally, a
+test suite should reach a mutation score equal to one.
+While MT is effective when the test suite has to be assessed
+against a wide set of faults spread in the software, it loses its
+effectiveness when the purpose of a test suite is to validate
+the software against specific requirements. This is particularly
+true in the embedded software domain, where software must
+be often validated against rigorously-defined safety properties.
+For example, the ATCS (Automatic Transmission Controller
+System) we used in the experimental evaluation is annotated
+with several safety properties expressed with Signal Temporal
+Logic (STL) [5], and test cases are designed to validate the
+software against these properties.
+When applying mutation testing to assess the capability of a
+test suite to thoroughly exercise a software w.r.t. a given prop-
+erty, there are two challenges to take into consideration: the
+relevance of the mutants and the relevance of the executions
+that kill the mutants.
+Relevance of the mutants w.r.t. a tested property. Not all
+the mutants are relevant to assess the thoroughness of a test
+suite against a property. In fact, only the mutants whose
+effects propagate in a way that ultimately causes the property
+violation are relevant. A mutant that does not impact a property
+shall also not contribute to measuring the adequacy of a test
+suite against that property. Regular MT does not distinguish
+between these mutants, and hence does not consider the
+difference between them when computing the mutation score.
+Relevance of the execution that kills a mutant. Producing
+different outputs for the original and the mutated programs is
+insufficient to kill a mutant when a test suite is assessed against
+a property. In fact, a test is thoroughly exercising the software
+w.r.t. a property only if the difference in the two outputs
+is severe and relevant enough to cause a violation of the
+property under consideration. Otherwise, the test is generating
+differences that are marginal w.r.t. the testing objective. For
+instance, in our evaluation, we assessed the test cases for the
+ATCS against the property that requires the engine speed and
+the vehicle speed to remain below certain thresholds. Several
+tests succeeded in exercising a mutant in the Transmission
+component, causing differences in the outputs, but failed to
+produce outputs that violate these properties, which is a clear
+inadequacy of the test suite. This situation is visually illus-
+trated in Fig. 1 (top), where the test is generating differences in
+the engine and vehicle speeds without exceeding the threshold.
+The mutant would be counted as killed according to regular
+mutation testing, although the test does not make the software
+to violate the property. In practice, if the fault would be present
+in the original model, the test would not reveal it. This also
+exemplifies how mutations could be easily killed according to
+regular mutation testing in data-flow models, where most of
+the components are activated in every computation and values
+easily propagate through the blocks in the model. However, the
+arXiv:2301.13615v1  [cs.SE]  31 Jan 2023
+
+propagated values often result in minor and non-significant
+output differences. Killing mutants while taking the tested
+properties under consideration is a definitely harder challenge.
+For instance, Fig. 1 (bottom) shows the case of a test that
+reveal the mutant by violating the tested properties, obtained
+in our experiments.
+0
+10
+20
+30
+Time (seconds)
+0
+20
+40
+60
+80
+100
+120
+Vehicle Speed (mph)
+Output (Vehicle Speed)
+0
+10
+20
+30
+Time (seconds)
+0
+1000
+2000
+3000
+4000
+Engine Speed (RPM)
+Output (Engine Speed)
+Original model
+Mutant
+Threshold
+0
+10
+20
+30
+Time (seconds)
+0
+20
+40
+60
+80
+100
+120
+Vehicle Speed (mph)
+Output (Vehicle Speed)
+0
+10
+20
+30
+Time (seconds)
+0
+1000
+2000
+3000
+4000
+Engine Speed (RPM)
+Output (Engine Speed)
+26
+28
+30
+116
+118
+120
+122
+Fig. 1. Output plots for the original and mutated models of ATCS: (top) for
+a test case satisfying the property on the mutant, (bottom) for a test case
+violating the property on the mutant. The portion of the output trace (vehicle
+speed) responsible for property violation is highlighted.
+In this paper, we address these challenges by defining the
+notion of Property-Based Mutation Testing (PBMT) to assess
+test suites against properties or specifications. To this end,
+we revise the key notions of mutation testing to measure the
+effectiveness of the test suites as their capability to exercise
+the software against a property. We also define a search-based
+test generation strategy for Simulink models to effectively and
+automatically identify the relevant mutants that could be killed
+with meaningful executions, from a set of injected mutants. We
+provide empirical evidence that PBMT is more informative
+than MT to assess the thoroughness of test suites, considering
+two benchmarks in the domain of safety-critical CPS with
+requirements expressed in STL formalism.
+In summary, this paper makes the following contributions:
+1) We introduce the novel notion of Property-Based Muta-
+tion Testing for testing software against properties.
+2) We define a search-based strategy to automatically iden-
+tify the mutants that contribute to PBMT experiments.
+3) We report empirical results for Simulink models, demon-
+strating that PBMT is more informative than regular MT
+when software is tested against properties.
+4) We make tools and experimental data publicly available
+for reproduction and to ease follow-up research1.
+1https://gitlab.com/DrishtiYadav/mt
+Paper Organization. Section II presents the overview of
+regular mutation testing. Section III presents property-based
+mutation testing, our proposed approach. Section IV describes
+testing CPS Simulink models against STL specifications. Sec-
+tion V presents our evaluation of two safety-critical industrial
+benchmarks. Section VI discusses threats to validity. Sec-
+tion VII describes the lessons learned. Section VIII presents
+related work. Section IX concludes the paper.
+II. MUTATION TESTING
+In this section, we present the background and fundamental
+concepts of regular mutation testing.
+Mutation testing relies on two fundamental assumptions [1],
+[2]: (1) the Competent Programmer Hypothesis that states
+that programmers create programs that differ from the correct
+one mostly by small syntactic errors, and (2) the Coupling
+Effect that asserts that “complex faults are coupled to simple
+faults in such a way that a test data set that detects all
+simple faults in a program will detect a high percentage
+of the complex faults” [6]. Several studies investigate these
+hypotheses demonstrating that results obtained with mutation
+testing can reliably predict the results obtained for the vast
+majority of high-priority real bugs [7]–[10]. Although not
+every bug couples with mutants, mutation testing can still be
+considered a good tool to measure test suite quality.
+We now introduce the key concepts of mutation testing.
+Definition II.1 (Mutation operator). A mutation operator is
+a source-code transformation that introduces a modification in
+the program-under-test. More rigorously, given a program P,
+a mutation operator op is a function that takes as inputs P and
+a location k inside P and creates a syntactic alteration of P
+at location k, if the location can be mutated with op.
+Definition II.2 (Mutant). For a given program P and a set of
+mutation operators O = {op1, op2, ..., opn}, a mutant p is the
+result of the application of a mutation operator op ∈ O to P
+at a specified location k. A mutant created by the application
+of only one mutation operator to P is known as First Order
+Mutant (FOM). The application of multiple mutation operators
+to P results in a Higher Order Mutant (HOM) [11].
+Given a test suite T , and a test t ∈ T , we write t |= p when
+the test passes on p and t ̸|= p when the test fails on p. We
+denote with O(t, p) the output generated by p with t and with
+T p
+U the (universal) set of every possible valid test case for p.
+Definition II.3 (Killed Mutant). A mutant p is said to be killed
+by T if at least one test case t in T fails when exercising p,
+i.e., ∃t ∈ T : t ̸|= p.
+Definition II.4 (Live Mutant). Mutants that do not lead to the
+failure of any test case t ∈ T are said to be live or survived.
+Formally, p is said to be live if ∀t ∈ T , t |= p.
+Definition II.5 (Equivalent Mutant). A mutant p is equivalent
+to the original program P if they both generate the same
+output for any possible input. Formally, p is equivalent to
+P if ∀t ∈ T P
+U , O(t, p) = O(t, P). In other words, no test
+
+case can distinguish an equivalent mutant from the original
+program [12]. Note that the detection of equivalent mutants is
+undecidable.
+Definition II.6 (Invalid Mutant). A mutant p is considered
+invalid if it cannot be compiled [13]. Such a mutant is not
+included in the mutation coverage.
+Definition II.7 (Mutation coverage). The adequacy of a
+test suite T can be measured using the mutation coverage
+(hereafter, mutation score MS): the ratio of mutants killed
+w.r.t. the total number of non-equivalent and valid mutants:
+Mutation coverage =
+#killed mutants
+#valid mutants − #equivalent mutants
+T is said to achieve 100% mutation test adequacy if it kills all
+non-equivalent valid mutants. Full mutation coverage ensures
+that T is (i) robust against the modeled mutation types, and
+(ii) sensitive to small changes in the program-under-test (P).
+Definition II.8 (Redundant Mutant). Redundant mutants are
+not beneficial as they consume resources without contributing
+to the test process as they are killed whenever other mutants
+are killed. This redundancy can be expressed by duplicate
+and subsumed mutants [14]. Duplicate mutants are equivalent
+with each other but not equivalent to the original program [3].
+Subsumed mutants are not equivalent with each other but are
+killed by the same test cases. The subsumption relation is
+defined as follows [15]: We say that pi subsumes pj, denoted
+pi → pj, iff the following two properties hold:
+1) ∃t ∈ T P
+U : t ̸|= pi. In other words, there exists some test
+case t s.t. pi and P yield different outputs on t, i.e., pi
+is not equivalent to P.
+2) ∀t ∈ T P
+U , if t ̸|= pi, then t ̸|= pj. In other words, for
+every possible test case t on P, if pi yields a different
+output than P on t, then so does pj.
+With Regular MT, for a test case t ∈ T to kill a mutant p,
+the following three conditions must be satisfied [16], [17]:
+1) Reachability: t must reach the mutated statement in p.
+2) Necessity: t must infect the program state by causing
+different program states for p and P.
+3) Sufficiency: the incorrect program state must propagate to
+the output of p and be checked by t, i.e., there is an
+observable difference in the outputs of p and P for t.
+The above three conditions are known as the RIP model.
+The capability of a test case t ∈ T to kill a mutant p is
+governed by the observability of the program state, leading to
+following two common types of mutation testing:
+1) Weak mutation testing: A mutant p is killed by a test suite
+T if only the first two conditions of the RIP model are
+satisfied.
+2) Strong mutation testing: For a test case t ∈ T to kill a
+mutant p, all three conditions of the RIP model must be
+met.
+Tests, in particular automated tests, usually include an
+explicit comparison of the observed program behavior to the
+expected behavior using an oracle. Thus, automated tests
+usually examine specific portions of the output state. However,
+the oracle will fail to identify the failure if it does not check the
+specific part of the output state which contains the erroneous
+value. Therefore, the oracle should also reveal the failure [18],
+as proposed in the RIPR model. This paper further elaborates
+this concept defining how mutation testing can be designed
+to validate and measure the quality of a test suite w.r.t. a
+requirement, in our case taking the form of a rigorously
+defined STL property for a MathWorks® Simulink model.
+III. PROPERTY-BASED MUTATION TESTING
+In this section, we present PBMT, a mutation testing ap-
+proach designed to validate test suites against programs and
+properties. We assume that we have a program P expressed in
+a language L as the software-under-test (SUT), a property φ
+of the SUT, a test suite T and a set of mutation operators O.
+PBMT measures how thoroughly the test suite T validates P
+against the property φ, studying the capability of T to reveal
+faults —of type defined in O—that may impact φ.
+Definition III.1 (φ-killed mutant). A mutant p is said to be
+φ-killed by a test suite T ⊂ T P
+U iff ∃ a test case t ∈ T such
+that the following conditions hold:
+1) O(t, P) |= φ, i.e., t satisfies φ when executed on the
+original program P, and
+2) O(t, p) ̸|= φ, i.e., t violates φ when executed on the
+mutant p. It follows that t exercises the mutation/fault
+in p in such a way that its effect is propagated to the
+output up to the violation of the property φ.
+The above two conditions collectively guarantee that the
+execution of p against t yields an output strong enough to
+violate φ (i.e., O(t, p) ̸|= φ), while still passing in the original
+program (i.e., O(t, P) |= φ). This implies that the test is
+specifically good in exercising the software so that the fault,
+if present, is propagated to the output, producing significant
+behavioral differences up to the point of violating φ.
+Similar to the concept of equivalent mutants in regular MT,
+we introduce a refined version of equivalent mutants which
+we call: φ-trivially different mutants. The intuition is that in
+this context, a mutant is irrelevant not only if it is equivalent
+(i.e., it shows no behavioral differences w.r.t. the original
+program), but also if the introduced behavioral differences are
+not relevant w.r.t. the property φ, that is, no test case t ∈ T P
+U
+can distinguish between p and P.
+Definition III.2 (φ-trivially different mutant). A mutant p is
+φ-trivially different from P iff ∄t ∈ T P
+U
+: O(t, P) |= φ ∧
+O(t, p) ̸|= φ.
+The set of the φ-trivially different mutants include equiva-
+lent mutants. The identification of φ-trivially different mutants
+is undecidable.
+Definition III.3 (φ-adequate test suite). A test suite T is φ-
+adequate w.r.t. a set of mutation operators O if it kills all the
+non φ-trivially different mutants that can be generated by O.
+
+Definition III.4 (Mutation score). If KDφ denotes the φ-
+killed mutants and NTDφ denotes the non φ-trivially different
+mutants, the mutation score assigned with a test suite T for a
+program P and a set of mutation operators O is
+MSφ = |KDφ|
+|NTDφ|
+(1)
+The objective of implementing test suites that are adequate
+according to PBMT results in the Mutant killing problem. That
+is, given a program P, a mutant of P denoted by p and a
+property φ, the mutant killing problem is the problem of
+finding a test case t such that O(t, P) |= φ, and O(t, p) ̸|= φ.
+PBMT is usually more challenging than regular MT since:
+• Higher risks of introducing φ-trivially different mutants:
+PBMT can potentially generate more irrelevant mutations
+than mutation testing since, in addition to equivalent
+mutants, there might be mutants that are not equivalent
+but introduce irrelevant differences w.r.t. a property φ.
+• Harder to kill mutants: The faults must be exercised in
+such a way that it does not only propagate to the output
+but also leads to the violation of φ.
+IV. MUTATION TESTING OF SIMULINK CPS PROGRAMS
+We instantiate PBMT in the context of safety-critical CPS
+Simulink (data-flow) models where the system safety prop-
+erties are expressed using STL. While extensive details of
+Simulink models [19]–[21] and STL [5], [22], [23] are avail-
+able elsewhere, we introduce below the key concepts to make
+the paper self-contained. We conclude by presenting a novel
+technique to automatically determine the mutants that could
+be φ-killed by test suites.
+A. Simulink models
+The MathWorks® Simulink environment is widely used for
+CPS model-based development [24], [25]. Simulink allows
+non-software engineers to design complex systems, compile
+them to low-level code, and simulate the designed models to
+observe their behavior against some test inputs. In general,
+a Simulink model is the block diagram representation of a
+system using blocks and lines (aka connections) as in Fig. 2.
+A block receives data via its input ports and performs a defined
+operation on its input data depending on its functionality. After
+processing the input data, a block transmits the output data
+via its output ports, along (directed) lines. Each line in the
+model can be uniquely identified using (1) the source block
+and its associated output port, and (2) the target block and its
+associated input port. The model receives its inputs from a set
+of input blocks and emits the output through a set of output
+blocks. Usually, a block can be either atomic (i.e., it does
+not include any other block within it) or hierarchical (i.e., it
+includes other blocks within it).
+When creating a model, a tester can either use standard
+blocks from built-in libraries or create new custom blocks
+from scratch. After designing the model, a tester compiles and
+simulates the model using a suitable solver and simulation
+Fig. 2. A Simulink model with hierarchical blocks (b4, b8) and atomic blocks
+(remaining), input ports (black nodes), output ports (white nodes), inputs (In1
+and In2) and outputs (Out1, Out2 and Out3).
+mode. Simulink allows to execute the model using user-
+specified sample times (either fixed-length or variable-length).
+A Simulink model M when simulated against a test case t
+yields the model simulation output as the set of traces of all
+input-internal-output signals. We denote the model simulation
+output with O(t, M). A Simulink model can have multiple
+outputs (such as Fig. 2’s Out1, Out2 and Out3).
+B. Signal Temporal Logic (STL)
+In recent years, for the verification of safety-critical CPS,
+researchers have used temporal logic formalisms to express
+safety properties. Signal Temporal Logic (STL) [5] is a well-
+known specification formalism used to express temporal prop-
+erties of dense-time real-valued behaviors of hybrid (i.e., both
+continuous and discrete dynamic) systems, including safety-
+critical CPS. The syntax of STL is formally defined as follows:
+Φ := f(x(j)) > 0 | ¬Φ | Φ1 ∧ Φ2 | □IΦ | ♦IΦ | Φ1UIΦ2
+Here, the formula of the form f(x(j)) > 0 represents a signal
+predicate, where x(j) is the value of a signal x at time instant
+j, and f is a function from signal domain D to R. I ⊆ R≥0 is
+an arbitrary time-interval. The propositional logic operators ¬
+and ∧ follow the obvious logical semantics, i.e., ¬ indicates
+logical negation and ∧ indicates logical conjunction. Other
+temporal operators are as follows:
+• □IΦ (always operator) indicates that Φ must be true for
+all samples in I.
+• ♦IΦ (eventually operator) indicates that Φ must be true
+at least once for samples in I.
+• Φ1UIΦ2 means that Φ1 must be true in I until Φ2
+becomes true. UI refers to as until operator.
+The Boolean satisfaction semantics aka qualitative seman-
+tics of STL offers a boolean witness of the property Φ. The
+Boolean satisfaction of the signal predicate is simply ⊤ if it
+is satisfied; otherwise ⊥. We use the operators U, ♦, and □
+to denote UI, ♦I, and □I with I = [0, ∞).
+Besides the qualitative semantics, STL also offers quanti-
+tative semantics [23] that allows to compute the degree of
+satisfaction of Φ by the traces generated by a system after
+executing it against a test input. The degree of satisfaction of
+
+b1
+In1
+I1nO
+Dut2
+In2
+Out3Φ for a trace q is measured using a robust satisfaction function
+ρ(q, Φ) that computes a real value that indicates the distance
+of the trace q from satisfying (|=s) the property Φ. Formally,
+ρ(q, Φ) > 0 ⇒ q |=s Φ, and ρ(q, Φ) < 0 ⇒ q ̸|=s Φ.
+C. Mutations in Simulink
+From a conceptual perspective, mutations are simply mod-
+ifications to the behavior of the Simulink model. Usually,
+alterations can be made in a Simulink model in two ways:
+1) Line mutations: changing the behavior of the signals that
+propagate through lines from one block to another block
+(see ‘Fault in line’ in Fig. 3), or
+2) Block mutations: changing the behavior of a block (see
+‘Fault in block’ in Fig. 3), for instance, by making changes
+in its functionality.
+Fig. 3. Mutations in a SUT (the seeded fault blocks F are highlighted in red).
+A, B and C are blocks of original SUT. Internal signals s and s′ provide
+knowledge of the fault location.
+D. Robustness Measure
+The notion of robustness function ρ becomes useful when
+we need to search for a test t that passes the execution of the
+model M w.r.t. an STL requirement φ. We use the following
+notations [23]:
+1) ρ(O(t, M), φ) < ϵ ⇒ O(t, M) ̸|= φ, i.e., t fails on M
+with respect to the specification φ
+2) ρ(O(t, M), φ) > ϵ ⇒ O(t, M) |= φ, i.e., t passes on M
+with respect to the specification φ.
+Here, the parameter ϵ represents the degree of violation of
+the property as assessed by the robustness function ρ. The
+standard choice is ϵ = 0 which implies that the identification
+of passing or failing test case i.e., satisfaction or violation is
+based on even a small (non-zero) deviation in the observed
+behavior of M from the expected behavior w.r.t. φ.
+E. Search-based generation of mutation adequate test cases
+A key challenge in mutation testing, including PBMT, is
+accurately computing the mutation score, due to the undecid-
+able problem of identifying the equivalent mutants. In PBMT,
+this problem is even harder due to the need of identifying
+the φ-trivially different mutants, which include but are not
+limited to the equivalent mutants. To address this challenge,
+we defined a search-based test generation strategy that exploits
+the knowledge of the mutants and their locations to generate
+targeted executions that demonstrate if a mutant can be φ-
+killed. Although nothing could be said about the mutants not
+killed according to this procedure, the experimental results
+show that assuming this procedure can identify every φ-
+killable mutant may give an accurate approximation of the
+mutation score.
+Note that the proposed test strategy cannot be used to gen-
+erate tests in a real situation, since it exploits the knowledge of
+the fault location that is normally unknown when a software is
+tested. However, the proposed test generation strategy is useful
+in the context of PBMT to collect accurate empirical data.
+In particular, we formulate the ‘Property-based test search
+problem’, an optimization problem of finding a φ-adequate
+test case as:
+Property-based test search problem
+INPUT: a Simulink model M, a first-order mutant M′
+(with signal s changed into signal s′ or a block b with
+output s changed into a block b′ with output s′), and
+a property φ.
+PROBLEM:
+Find
+t
+s.t.
+ρ(O(t, M), φ)
+>
+0,
+ρ(O(t, M′), φ) < 0 and D(s, s′) is maximum.
+The proposed ‘Property-based test search problem’ com-
+bines three key features, two deriving from the definition of
+φ-killed mutant and one guiding the search toward the mutant,
+and toward producing an execution that exploits the mutant to
+significantly alter the state of the system:
+• ρ(O(t, M), φ) > 0 requires finding a test that passes on
+the original program,
+• ρ(O(t, M′), φ) < 0 requires finding a test that violates
+φ in the modified program, and
+• D(s, s′) is maximum requires the mutation to impact on
+the internal signal as much as possible.
+We choose the Euclidean distance (aka L2 norm) as the
+metric to compute the distance between s and s′. Since CPS
+models involve continuous real-valued variables, Euclidean
+distance, a prominent metric for real vector spaces, is a
+good candidate for computing the distance. More rigorously,
+given two finite-length signals s = (s1, · · · , sk) and s′ =
+(s′
+1, · · · , s′
+k), each with k samples, the Euclidean distance
+between s and s′ is mathematically expressed as:
+D(s, s′) = ||s − s′||2 =
+�
+�
+�
+�
+k
+�
+i=1
+(si − s′
+i)2
+The optimization task is to maximize D(s, s′) subject to
+the constraints ρ(O(t, M), φ) > 0 and ρ(O(t, M′), φ) < 0.
+To solve the formulated test search problem, we exploit
+BCA [26], a recently developed global optimizer as outlined
+in Algorithm 1. We chose BCA over other available optimizers
+on account of its superior convergence and speed. While being
+a global search with BCA in essence, Algorithm 1 introduces
+two differences w.r.t. standard BCA: (1) The initial population
+(Line 2) is a set of test cases randomly generated in their
+valid numerical input domain. (2) Fitness (Line 3) corresponds
+to the value of the test objective function for the given
+population of test cases. The test objective function is obtained
+
+Original SUT
+Mutated
+(Faulty)
+SUT
+B
+Fault in line
+B
+Faults in block
+(block mutation)by converting the constrained optimization problem into an
+unconstrained problem using the scalar penalty constraint
+handling method [27]. The algorithm updates the test cases
+(Line 6-8) and finds the best solution for the new population
+depending on their fitness values (Lines 9-10). The candidate
+fittest amongst all others in the population is accepted as the
+new global best solution (Lines 11-14). The algorithm returns
+the best solution if all the constraints are satisfied. Algorithm 1
+terminates (loop at Line 5) if either a test case satisfying the
+optimization constraints is found, or the budget is exhausted
+(time budget or the maximum number of iterations).
+Algorithm 1: Search-based test generation.
+Input : M : A Simulink model.
+M′ : A mutant of M.
+φ : An STL specification.
+Output: tbest : A test case that φ-kills M′.
+1 Initialize optimizer parameters
+2 IP ← GENERATEINITIALPOPULATION()
+3 FP ← Fitness(IP, M, M′, φ)
+4 tbest, Fbest ← BestFound(FP)
+5 while TimeOut() do
+6
+for each candidate k ∈ IP do
+7
+knew ← Update(k)
+8
+end for
+9
+FP ← Fitness(IP, M, M′, φ)
+10
+tnew, F ← BestFound(FP)
+11
+if F > Fbest then
+12
+Fbest ← F ;
+// update best fitness
+13
+tbest ← tnew ;
+// update best test
+14
+end if
+15 end while
+16 return tbest
+For each mutant, we solve the formulated ‘Property-based
+test search problem’ to find a test case that φ-kills it. The
+resulting test suite is a fault-directed test suite that is likely to
+reveal all the non φ-trivially different mutants.
+F. Test suite reduction
+To maintain a small and practical fault-directed test suite,
+we reduce its size automatically. We consider a test case tr
+φ-redundant w.r.t. a fault-directed test suite T if the set of
+φ-killed mutants by T remains unchanged after the inclusion
+of tr in T , i.e., |KDφ|T = |KDφ|T ∪ tr.
+A φ-non-redundant test suite does not contain φ-redundant
+test cases. Usually, a test suite can contain redundant test cases
+while retaining the same testing power in the sense that they
+are capable of killing the same mutants w.r.t. φ. In other words,
+a single test case can cover more than one mutation.
+In our experiments, we use the greedy algorithm similar
+to the one proposed in [28] for test suite reduction. In the
+worst-case scenario, p test cases are required to cover all p
+non φ-trivially different mutations. In practice, fewer tests are
+usually necessary.
+V. EVALUATION
+Our evaluation aims to study Property-Based Mutation
+Testing (PBMT) for testing CPS Simulink models against STL
+properties, also w.r.t. regular Mutation Testing (MT).
+A. Research Questions
+Our experiments address the following research questions:
+RQ1. Does PBMT measure the adequacy of a test suite
+better than MT when a safety property is targeted? To answer
+this research question, we assess the adequacy of multiple
+test suites using both PBMT and MT, and discuss how the
+resulting scores reflect the intrinsic capability of the test cases
+to exercise the software based on the target property.
+RQ2. Are mutation operators equally contributing in
+PBMT? To answer this research question, we study the impact
+of different mutation operators on the mutation score, aiming
+at discovering operators that tend to generate mutants that are
+either trivial or particularly hard to detect.
+B. Experimental Setup
+We performed our experiments on a MacBook Pro with
+Apple M1 chip, 16 GB RAM, macOS Monterey with MAT-
+LAB™ R2018b. For our evaluation, we developed a prototype
+implementation of both PBMT and MT with CPS Simulink
+models in MATLAB. We used the RTAMT library [29] for
+offline evaluation of STL properties.
+We limit the scope of the evaluation to FOMs. Moreover, we
+use a fixed-length sampling when running Simulink models
+with faults active from the beginning to the end of the
+simulation. In the following, we describe our experimental
+subjects, mutants and test suites.
+1) Experimental subjects: We evaluate PBMT on Simulink
+models of two industrial benchmarks across the safety-critical
+domain, each one publicly available in the Simulink/Stateflow
+online documentation of MathWorks® [30], [31]: ATCS, an
+Automatic Transmission Controller System, and AECS, an
+Aircraft Elevator Control System.
+ATCS is a typical automotive drivetrain with the two inputs
+throttle and brake governing the vehicle speed v (mph) and
+the engine speed ω (RPM). Both user inputs are in the range
+[0, 100] for all time instants. As one of the safety properties,
+ATCS requires that v and ω must always remain below their
+thresholds ¯v and ¯ω, respectively. This is represented in STL
+in Table I where ¯v = 120 mph and ¯ω = 4500 RPM.
+AECS from the avionics-aerospace domain controls the
+positions of the left and right elevators of an aircraft using
+the pilot command. In general, the elevator position should
+maintain a constant value if the aircraft is flying at the desired
+level. Among the safety requirements, the AECS requires that
+whenever the Pilot Command cmd goes beyond a threshold
+m, the measured elevator position pos must stabilize (should
+not exceed cmd by more than n units) within T + a time
+units. This is formally expressed with the STL specification
+in Table I where m = 0.09, T = 2, a = 1 and n = 0.02.
+
+TABLE I
+DETAILS OF SIMULINK MODELS OF OUR CASE STUDIES.
+Model
+Ref.
+#Blocks
+#Lines
+φ (STL specification)
+qT
+Sample time
+#Samples
+ATCS
+[32]
+65
+92
+□((v ≤ ¯v) ∧ (ω ≤ ¯ω))
+30
+0.04
+751
+AECS
+[33]
+825
+577
+□(↑ (cmd ≥ m) → ♦[0,T ]□[0,a](|cmd − pos| ≤ n))
+10
+0.01
+1001
+2) Fault seeding and mutant generation: For each experi-
+mental subject, we generated mutants using the FIM prototype
+tool [34] that supports the following mutation operators for
+Simulink models: Negate, Stuck-at, Absolute, Noise,
+Bias/Offset, Time Delay, Package Drop, ROR (Re-
+lational Operator Replacement), LOR (Logical Operator Re-
+placement), S2P (Sum to Product mutation), P2S (Product to
+Sum mutation) and ASR (Arithmetic Sign Replacement). The
+detailed description of these operators can be found in [34].
+Since FIM does not support the injection of faults in look-
+up tables (LUTs), we extended the tool implementing two
+additional operators: (1) Stuck-at 0 fault in any one entry, and
+(2) swapped entries (from two randomly chosen neighbors).
+Table II reports, for each subject, the number of mutants
+generated for the specific mutation operator. Table III indicates
+the total number of mutants generated for every subject and
+their generation time. Mutant generation is fast: On an average
+(across ATCS and AECS), the generation of a mutant takes
+1.74 seconds.
+TABLE II
+NUMBER OF MUTANTS OF OUR EXPERIMENTAL SUBJECTS.
+Type
+# Mutants
+ATCS
+AECS
+Noise
+13
+17
+Bias/Offset
+13
+17
+Negate
+13
+17
+Absolute
+13
+17
+ROR
+0
+10
+S2P
+1
+3
+P2S
+2
+6
+ASR
+3
+8
+LUT
+2
+5
+TABLE III
+INFORMATION OF GENERATED MUTANTS.
+Subject
+Mutants generated
+Mutant generation time (seconds)
+ATCS
+60
+68.76
+AECS
+100
+261.64
+3) Test Suite: To compare PBMT to MT, we assess test
+suites generated according to two different strategies: Adap-
+tive Random Testing (ART) [35] and Falsification Testing
+(FT) [36], [37]. ART is a baseline strategy that generates
+evenly distributed test cases (within valid input ranges),
+thereby ensuring adequate diversity in the test inputs. On the
+other hand, FT generates counterexamples i.e., test cases that
+violate a property for a given model [38], [39]. Note that ART
+and FT work in radically complementary ways. ART quickly
+generates many test inputs, considering diversity, but ignoring
+the property under test. On the contrary, FT specifically targets
+the generation of a test that violates the property under test.
+In particular, for each mutant M′, FT attempts to generate a
+test case t such that O(t, M′) ̸|= φ. The hypothesis is that
+ART could obtain higher MS, but smaller MSφ since the
+generated tests do not depend on φ. On the contrary, FT should
+kill fewer mutants in general, but more mutants relevant to φ,
+and thus obtain higher MSφ.
+In our evaluation, we generated 30 and 50 test cases
+with ART for ATCS and AECS, respectively. FT generates
+a property-violating test per mutant, if successful.
+For collecting data to address our research questions, we
+have executed all the test cases in the test suite for every
+subject and every generated mutant. To perform our exper-
+iments, we executed multiple simulations in parallel using
+the Parallel Computing Toolbox™ in the MATLAB/Simulink®
+environment. Table IV provides, for each subject, the total
+number of test cases executed (including both test suites) and
+the total execution time.
+TABLE IV
+SCALE OF EXPERIMENTS.
+Subject
+Total test cases executed
+Total execution time (seconds)
+ATCS
+90
+2,490
+AECS
+150
+25,912
+C. Results
+RQ1 studies the extent to which PBMT-based testing can
+better capture the thoroughness of a test suite w.r.t. a safety
+property that the software-under-test must fulfil. To this end,
+we apply both MT and PBMT to our experimental subjects
+and compute the mutation scores MS and MSφ. Note that we
+use exactly the same mutants to compute both scores. Table V
+reports the results.
+We report the results for regular mutation testing (MT)
+and Property-Based Mutation Testing (PBMT) in two different
+rows, while columns ATCS and AECS correspond to the two
+subject systems. For each subject system, we indicate the
+scores achieved by the test suites generated with Adaptive
+Random Testing (TART ) and Falsification Testing (TF T ). In
+details, we report the number of mutants that have been
+generated, the number of killable and φ-killable mutants, the
+number of mutants killed by each test suite according to MT
+and PBMT, and finally the mutation scores MS and MSφ.
+
+TABLE V
+RESULTS OF MUTATION TESTING.
+Approach
+ATCS
+AECS
+TART
+TF T
+TART
+TF T
+MT
+# Mutants
+60
+60
+100
+100
+# Killable mutants
+47
+47
+83
+83
+# Killed mutants
+47
+46
+74
+70
+Mutation Score MS (in %)
+100%
+97.87%
+89.15%
+84.33%
+PBMT
+# Mutants
+60
+60
+100
+100
+# φ-killable mutants
+47
+47
+83
+83
+# φ-killed mutants
+25
+27
+39
+35
+MSφ (in %)
+53.19%
+57.44%
+46.98%
+42.16%
+To identify the killable mutants, we had to identify the
+equivalent ones. To this end, we inspected the non-killed
+mutants to determine if a mutation generated a variant that
+cannot be distinguished from the original program. We could
+identify every equivalent mutant with high-confidence. In fact,
+the 13 equivalent mutants in the ATCS model all belong
+to the Absolute fault type injected in the ‘Transmission’
+component and all try to change into positive values some
+signals that could not be negative. The exact same situation
+happened for the 17 equivalent mutants found in the AECS
+model. To determine the φ-killable mutants, we used the
+Search-based test generation (SBTG) technique presented in
+Algorithm 1. Note that the SBTG strategy is more computa-
+tionally expensive than ART and FT due to the optimization
+constraints. Our procedure automatically identified every φ-
+killable mutant with thirty independent runs of our search
+algorithm and a maximum number of iterations (set to 1000)
+as the stopping criterion. The remaining φ-trivially different
+mutants are all equivalent mutants that cannot be killed. This
+result provides confidence on the capability of our approach
+to support fully automated experiments with Simulink models
+by assuming that the mutants not killed with our strategy are
+φ-trivially different mutants that do not need to be killed, and
+thus can be excluded from the computation of MSφ.
+By comparing the results obtained for MT to the results
+obtained with PBMT, we can notice the mutation score ob-
+tained with MT is significantly higher than the mutation score
+obtained with PBMT. In fact, the value of MS ranges between
+84.33% and 100% for the four test suites and the two subject
+systems. On the other hand, the value of MSφ ranges between
+42.16% and 57.44%. This is also due to the intrinsic nature of
+both Simulink models and data-flow computations, where it is
+generally easy to activate every component (i.e., to generate a
+sequence of inputs that exercise every element in a program),
+but it is definitely harder to activate these components while
+guaranteeing they contribute to the computation propagating
+the fault to the output, finally causing observable issues. That
+is, it is relatively easy to reach faults, but it is still hard
+to meaningfully propagate and detect faults. This result is
+confirmed across the test suites generated with two alternative
+strategies.
+These results demonstrate that MT may mislead testers
+when there are important properties to be validated. For in-
+stance, referring to Fig. 1 (top), the test case can kill the mutant
+but cannot φ-kill it. In fact, the test suites generated with ART
+and FT achieve high mutation score (MS), possibly inducing
+testers to believe the test suites are thoroughly exercising
+software. On the contrary, it turns out that the test cases are not
+good enough to guarantee that even the simple faults (e.g., like
+the ones we injected) that may affect the property are actually
+detected.
+It is also interesting that FT, which targets the falsification of
+the property, in comparison to ART, which addresses diversity
+neglecting the existence of the property, does not kill more
+mutants. Combined with the evidence that almost half of the
+killable mutants have not been φ-killed, this suggests that more
+research is needed to exercise software thoroughly w.r.t. a
+target property, at least for Simulink programs.
+We finally checked for the capability of the generated
+tests to kill and φ-kill mutants. Interestingly, there is often
+high redundancy across tests, that is, each test can kill many
+mutants. For instance, all the mutants that have been killed
+with ART could be killed by a single test. This reinforces the
+idea that there are some surface faults that are easy to reveal,
+but at the same time there are other faults that, even if simple
+in structure, require more sophisticated tests to be revealed.
+On the other hand, we found that four test cases, derived
+with our SBTG technique are needed to reveal all 47 φ-killable
+mutations of ATCS. Likewise, all 83 φ-killable mutations of
+AECS could be revealed with 12 test cases. This suggests
+that compact but effective test suites could be designed to
+reveal faults according to PBMT. Yet, PBMT requires a higher
+number of tests than regular MT to φ-kill and kill mutants,
+respectively.
+RQ2 assesses the contribution of individual mutation oper-
+ators in PBMT. The goal is to identify the operators that tend
+to generate easy-to-kill mutants (simple mutants), which do
+not contribute much to measuring the adequacy of a test suite,
+and the operators that tend to generate hard-to-kill mutants
+(stubborn mutants), which can contribute more in measuring
+the thoroughness of a test suite.
+Table VI reports the following results for each mutation
+operator: (1) the number of mutants generated, (2) the number
+(and percentage) of φ-trivially different mutants, (3) the num-
+
+ber (and percentage) of NTDφ (i.e., non φ-trivially different
+mutants), (4) mutation score achieved by ART, (5) mutation
+score achieved by FT, and (6) number (and percentage) of
+NTDφ mutants not killed by any test generation technique
+(neither ART nor FT). Note that Table VI reports the combined
+results for our two experimental subjects (ATCS and AECS).
+At least half of the mutations generated by the Negate,
+ROR, S2P and ASR operators have been killed neither by
+ART nor by FT. This may suggest that these operators might
+be more useful than others for PBMT because they tend to
+generate faults that are not easy to propagate to the output.
+For instance, all the mutants of AECS with the Negate
+operator were generated by alterations in the Right Outer
+Hydraulic Actuator component. The available test cases can
+easily infect the execution (e.g., they change the output of the
+‘Line resistance’ block), but fail to propagate the infection due
+to the presence of an intermediate signal (e.g., ‘Piston Force’)
+that masks changes if differences are not large enough.
+None of the mutants generated by ROR has been detected
+by TART and TF T . In particular, we observe that for all
+available test cases t ∈ TART ∪ TF T , with the execution of
+the ROR mutations, the robustness value evaluated for the STL
+property for every mutant is the same as that obtained for the
+original model. However, there exist test cases that produce
+visible differences in the outputs and φ-kill the mutants as
+demonstrated by the tests obtained with our SBTG technique.
+Mutations generated by S2P have been also hard to φ-
+kill. Besides, some mutations with ASR operator could not be
+detected by test cases in TART and TF T . Though these mu-
+tants alter the internal signal, the data-flow computations and
+propagation of signals do not affect the property. For instance,
+the ASR mutation in the ‘Hydraulic Actuator’ component of
+Right Inner Hydraulic Actuator unit of AECS (−+ replaced
+by +−) creates significant variations in the local signal but is
+not strong enough to φ-kill the mutant.
+On the other hand, two operators have not been particularly
+useful. The Absolute operator only generated equivalent
+mutants. This suggests that this operator must be used care-
+fully, only with systems known to process negative values, and
+possibly controlling the locations where the fault is injected.
+This case is quite infrequent in CPS. In fact, we have not
+observed any useful mutation in our two subjects. All the
+mutations generated by LUT were easy to φ-kill, with only
+one exception, which generates values hard to propagate to the
+output (but still feasible to propagate as demonstrated by the
+test suites generated with our SBTG approach). Although this
+operator is the only one targeting look-up tables, testers might
+consider skipping it when there are strong time constraints on
+the testing process.
+VI. THREATS TO VALIDITY
+We now discuss the threats to validity centered around the
+following perspectives of validity and threats:
+External validity. The main threat to external validity
+concerns with the generalization of our results. Indeed, the
+reported evidence may not generalize to every software
+system. In fact, we experimented in the domain of data-
+flow oriented computations (i.e., Simulink models), and our
+observations may not hold in other contexts (e.g., object-
+oriented programs). However, results are already quite clear
+and explainable in the domain of safety-critical CPS Simulink
+programs, where testing software against safety properties is
+particularly relevant. Moreover, the size of the experiment
+made affordable the manual analysis of mutations to identify
+equivalent mutants.
+Another threat to validity is the representativeness of the
+injected faults. The results reported in this study are based on
+typical mutation operators for Simulink models. In particular,
+we used the FIM tool [34] and its mutation operators, extended
+with additional mutation operator to address lookup tables.
+Internal
+validity. In our experiments, we considered
+only FOMs, i.e., faulty Simulink models with only one
+fault/mutation. Models can have multiple faults/mutations that
+may influence each other. Hence, the results might differ when
+tested with multi-fault Simulink models. Nevertheless, since
+most of the existing research on mutation testing focuses
+on FOMs of software artifacts [40], [41], we assessed our
+technique with single-fault models, leaving the study of HOMs
+for future work.
+Conclusion validity. Random variations is the main threat
+to conclusion validity. We mitigate this threat by making thirty
+independent runs of the test generation algorithms.
+VII. LESSONS LEARNED
+We now discuss the lessons learned from our experiments.
+Lesson 1 - It is challenging to generate PBMT-adequate
+test suites. Our study shows how none of the two state-of-
+the-art test generation strategies for Simulink programs we
+experimented with achieved high mutation score with PBMT.
+Indeed, PBMT is more laborious than regular MT: a test
+case that can kill a mutant might not φ-kill the same mutant.
+The embedded software industry heavily relies on properties
+for verification and validation activities, and it is important
+to design testing tools that thoroughly exercise the software.
+The definition of PBMT is a relevant advance to the state-of-
+the-practice that may influence and guide the design of more
+sophisticated and effective test generation strategies.
+Lesson 2 - MT does not capture well the thoroughness
+of a test suite. MT can still be applied to Simulink programs.
+However, test generation techniques could easily kill mutants
+as long as properties are not considered. This reveals that it is
+important to not only design executions that cover mutants, but
+that also propagate the errors produced by mutants, amplifying
+its visibility on the outputs. These characteristics of a test are
+not well assessed with MT.
+Lesson 3 - PBMT-driven test case generation can result
+in effective test cases. We defined a SBTG technique to find
+test cases that demonstrate that mutants could be φ-killed.
+Such a strategy has been highly effective in φ-killing mutants
+and could be the basis for the design of a mutation-based test
+case generation strategy.
+
+TABLE VI
+SUMMARY OF RESULTS OF PBMT FOR INDIVIDUAL OPERATORS.
+Noise
+Negate
+Bias
+Absolute
+ROR
+S2P
+P2S
+ASR
+LUT
+# Mutants generated
+30
+30
+30
+30
+10
+4
+8
+11
+7
+# (%) of φ-trivially different mutants
+0 (0%)
+0 (0%)
+0 (0%)
+30 (100%)
+0 (0%)
+0 (0%)
+0 (0%)
+0 (0%)
+0 (0%)
+# (%) NTDφ
+30 (100%)
+30 (100%)
+30 (100%)
+0 (0%)
+10 (100%)
+4 (100%)
+8 (100%)
+11 (100%)
+7 (100%)
+MSφART (in %)
+66.67%
+43.33%
+46.66%
+0%
+0%
+25%
+62.5%
+45.45%
+85.71%
+MSφF T (in %)
+70%
+43.33%
+50%
+0%
+0%
+25%
+62.5%
+45.45%
+28.57%
+# (%) NTDφ not killed by ART+FT
+9 (30%)
+17 (56.66%)
+15 (50%)
+0 (0%)
+10 (100%)
+3 (75%)
+3 (37.5%)
+6 (54.54%)
+1 (14.28%)
+Lesson 4 - Not all mutations are equally useful to
+test CPS Simulink models. Based on our results, we might
+deduce that some operators are more likely to generate φ-
+trivially different mutants. For instance, the Absolute oper-
+ator always generated equivalent mutants. On the other hand,
+some operators (e.g., Negate, ROR, and ASR) generated
+mutants that were hard to φ-kill, calling for test case generation
+techniques that exercise the software in non-trivial ways.
+VIII. RELATED WORK
+Mutation Testing. From the software engineering perspec-
+tive, mutation analysis is one of the powerful software testing
+techniques that can evaluate the test suite quality [1], [2]. The
+mutation testing and analysis literature includes a large number
+of theoretical studies and empirical investigations of various
+kinds of software artifacts [42], [43].
+The work in [44] combines symbolic execution, concolic
+execution, and evolutionary testing to automate the test gener-
+ation for weak mutation testing of programs. Along a similar
+line of research, the work in [45] proposes a path selection
+strategy to pick up test cases capable of killing the mutants.
+Related research on test suite minimization include techniques
+based on Integer Linear Programming (ILP) [46], Greedy
+algorithms [28], [47], formal concept analysis [48], etc.
+The most prominent works concerning the applicability of
+mutation testing to safety-critical industrial systems include
+the empirical investigations reported in [3], [49]–[51]. Al-
+though the work in [3] proposes a well-defined mutation
+analysis pipeline for test suite quality assessment of embedded
+software, it misses to address the importance of properties
+associated with the software and the ways to handle them
+during mutation testing. Contrary to the existing research on
+regular MT, we use properties—which allow us to express
+software requirements and specifications—to formalize the
+notion of killing the mutants.
+Mutations with Simulink models. Mutation mainly relies
+on alterations in the Simulink model by seeding defects using
+mutation operators [52]. Researchers have proposed several
+tools for creating mutants: SIMULTATE [53], MODIFI [54],
+ErrorSim [55], FIBlock [56], and FIM [34]. We also mention
+SLforge [19], a tool for automatically generating random valid
+Simulink models for differential testing. In our experiments,
+we used FIM since it provides a higher degree of automation
+compared to the other tools.
+Mutation-based test case generation. With regular MT,
+the mutation-based test case generation approaches exploit the
+mutants to generate test cases that can pick up the errors and
+discover the mutants. Some approaches considered generating
+tests that can reveal mutations introduced in the specification
+(e.g., in UML models) [57]–[62]. PBMT is different in many
+ways: it does not target mutations in the specification and it
+introduces a novel notion of mutation testing.
+The approaches designed to address Simulink models focus
+on targeted test-data generation either using search-based test-
+ing [63], [64] or behavioral analysis approaches (for instance,
+bounded reachability) [65], [66]. In essence, the main objective
+of these techniques is to generate a mutation-adequate test
+suite that achieves full mutation coverage based on the RIP
+model. Inspired by these techniques, we designed our search
+strategy to automatically φ-kill mutants. Further, PBMT intro-
+duces a novel instance of mutation testing that assesses the
+mutation adequacy of test suites w.r.t. properties, which has
+not been considered in mutation-based testing so far.
+IX. CONCLUSION
+We presented Property-based Mutation Testing (PBMT),
+a novel approach to mutation testing that promises efficient
+evaluation of test suites concerning software properties. Our
+formalization of mutant killability concerns with the satisfac-
+tion (and violation) of a property for the original program (and
+its mutated version). We provide rigorous semantics for PBMT
+and its associated mutant killing problem, enabling search-
+based generation of test cases using a global optimizer. We
+used different test generation strategies for creating test suites
+and observed their impact on mutant killability.
+We studied PBMT on two Simulink models across the
+safety-critical CPS domain, providing evidence that testing
+software against properties is more challenging and relevant
+than opting for regular MT, in which mutants can be easily
+killed. Finally, our evaluation shows that state-of-the-art Adap-
+tive Random Testing and Falsification Testing techniques are
+still weak in terms of their capability of generating test suites
+that can effectively kill mutants when tested against properties.
+Future work concerns adapting PBMT to closely related
+CPS modeling languages, including Simulink models inte-
+grated with Stateflow Charts. We further plan to conduct
+additional investigations with HOMs.
+ACKNOWLEDGMENT
+This work has been supported by the Doctoral College
+Resilient Embedded Systems, which is run jointly by the TU
+Wien’s Faculty of Informatics and the UAS Technikum Wien.
+
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+Reduced Reference Quality Assessment for Point
+Cloud Compression
+Yipeng Liu
+Cooperative MediaNet Innovation Center
+Shanghai Jiao Tong University
+Shanghai, China
+[email protected]
+Qi Yang
+Cooperative MediaNet Innovation Center
+Shanghai Jiao Tong University
+Shanghai, China
+yang [email protected]
+Yiling Xu
+Cooperative MediaNet Innovation Center
+Shanghai Jiao Tong University
+Shanghai, China
+[email protected]
+Abstract—In this paper, we propose a reduced reference (RR)
+point cloud quality assessment (PCQA) model named R-PCQA
+to quantify the distortions introduced by the lossy compression.
+Specifically, we use the attribute and geometry quantization
+steps of different compression methods (i.e., V-PCC, G-PCC and
+AVS) to infer the point cloud quality, assuming that the point
+clouds have no other distortions before compression. First, we
+analyze the compression distortion of point clouds under separate
+attribute compression and geometry compression to avoid their
+mutual masking, for which we consider 5 point clouds as
+references to generate a compression dataset (PCCQA) containing
+independent attribute compression and geometry compression
+samples. Then, we develop the proposed R-PCQA via fitting the
+relationship between the quantization steps and the perceptual
+quality. We evaluate the performance of R-PCQA on both the
+established dataset and another independent dataset. The results
+demonstrate that the proposed R-PCQA can exhibit reliable
+performance and high generalization ability.
+I.
+INTRODUCTION
+Recently, point cloud has emerged as a promising represen-
+tation format in prevalent 3D applications (e.g., autonomous
+driving [1] and augmented reality [2]), for which the point
+cloud compression (PCC) is of great interest for providing
+efficient service in practices. Currently, the Moving Picture
+Experts Group (MPEG) has applied the separable measure-
+ments of geometry and attribute distortion in the course of
+lossy PCC. For the geometry distortion, MPEG proposes to use
+the point-to-point (p2point) [3], or point-to-plane (p2plane) [4]
+to quantify the spatial perturbation; while for the attribute
+part, the PSNRyuv is proposed to measure the differences
+between corresponding color channels. Besides these metrics
+which have already been applied in MPEG PCC standardiza-
+tion, some other metrics which consider more human visual
+characteristics and present better performance on public PCQA
+databases are also developed, such as [5]–[15]. However, they
+are full reference (FR) metrics which require both the reference
+and distorted samples and have high computational complexity
+for real-time quality prediction.
+In many practical cases, e.g., transmission, the timely
+feedback is expected, and only the compressed samples and
+the meta data are available, in which the reduced reference
+(RR) PCQA metrics are indispensable. Only a few researches
+explore the RR methods for PCQA. [16] uses the statistical in-
+formation (e.g., the luminance histogram) as the substitute for
+the complete samples, but still requires the backend processing.
+[17] applies the quantization parameters in V-PCC to estimate
+the quality of compressed samples and guide rate control,
+but other prevalent compression strategies (e.g., G-PCC) are
+ignored. Therefore, in this paper, we propose a general RR
+PCQA model for compression distortions named R-PCQA
+which only takes the attribute and geometry quantization steps
+of compression schemes (including V-PCC [18], G-PCC [19]
+and AVS [20], V-PCC and G-PCC are provided by MPEG
+while AVS is recommended by China Audio-Video Coding
+Standard) as variables, since the quality of compressed point
+clouds is highly related to the compression parameters. To fully
+study the relationship between the compression parameters
+and perceptual quality, we first establish a complete subjective
+database for PCC, named PCC quality assessment (PCCQA)
+database.
+The reason why we establish PCCQA while many PCQA
+datasets [7], [21]–[24] have been proposed is that current
+databases only consider the superimposed compression distor-
+tion, i.e. lossy-geometry (G)-lossy-attribute (A) compression,
+which is recommended in the Common Test Conditions (CTC)
+[25]–[27]. The separate compression strategies, i.e. lossless-
+G-lossy-A and lossy-G-lossless-A compression, which are
+not included in the CTC are often ignored. Considering the
+mutual masking between geometry and attribute distortions
+[28], they are useful for exploring the relationship between the
+perceptual quality and compression parameters. In PCCQA,
+the reference point clouds are compressed by V-PCC, G-PCC
+and AVS under lossless-G-lossy-A condition, lossy-G-lossless-
+A condition, and lossy-G-lossy-A condition respectively.
+To model the relationship between the perceptual quality
+and the compression parameters, we first convert all the
+compression parameters to the quantization steps. Then, we
+model the relationship between the perceptual quality and the
+attribute/geometry quantization steps respectively via using the
+least square fitting. Finally, the proposed R-PCQA combines
+the attribute compression model and geometry compression
+model to predict the final quality scores.
+The rest of this paper is organized as follows: section II
+introduces the established PCCQA dataset; section III presents
+the proposed R-PCQA; section IV illustrates the experiment
+results; the conclusion is summarized in section V.
+978-1-6654-7592-1/22/$31.00 © 2022 IEEE
+arXiv:2301.01009v1  [eess.IV]  3 Jan 2023
+
+II.
+PCCQA DATABASE
+To better explore the relationship between the perceptual
+quality and the attribute/geometry compression parameters,
+we first establish a database called PCCQA under several
+compression conditions.
+Five reference point clouds are selected from MPEG and
+AVS point cloud datasets. These reference point clouds are
+ensured to have no holes and other distortions under 1080P
+presentation with size-2 primitives. The reference point clouds
+are then distorted with 3 compression algorithms, i.e. V-
+PCC [18], G-PCC [19] and AVS [20]. Each compression is
+conducted under 3 conditions, i.e. lossless-G-lossy-A, lossy-G-
+lossless-A, and lossy-G-lossy-A. The compression parameters
+are shown in Table I. In total, 225 compressed point clouds
+are generated.
+TABLE I: Compression parameters used for distorted point
+cloud generation.
+Conditions
+Parameters
+GPCC lossy-G-lossless-A
+VPCC lossy-G-lossless-A
+AVS lossy-G-lossless-A
+(positionQuantizationScale)
+0.75 0.5 0.25 0.125 0.0625
+(geomQP)
+22 32 37 42 51
+(geom quant step)
+2 4 8 12 16
+GPCC lossless-G-lossy-A
+VPCC lossless-G-lossy-A
+AVS lossless-G-lossy-A
+(qp)
+35 39 43 47 51
+(textureQP)
+32 37 42 47 51
+(attr quant param)
+24 32 40 44 48
+GPCC lossy-G-lossy-A
+VPCC lossy-G-lossy-A
+AVS lossy-G-lossy-A
+(positionQuantizationScale, qp)
+0.75,35 0.5,39 0.25,43 0.125,47 0.0625,51
+(geomQP, textureQP)
+24,32 28,37 32,42 36,47 40,51
+(geom quant step, attr quant param)
+2,24 4,32 8,40 12,44 16,48
+To annotate the compressed point clouds, a subjective
+experiment is organized to collect the Mean Opinion Scores
+(MOS). We adopt the double stimulus method since it can
+obtain more stable results for minor impairments. The ex-
+periment process and environment setting strictly follow the
+ITU-R Recommendation BT. 500 [29]. Such a method for
+collecting subjective MOS is also adopted in other researches,
+such as [23], [30]–[32].
+III.
+PROPOSED QUALITY ASSESSMENT MODEL R-PCQA
+A. Unifying the Compression Parameters
+The compression parameters with different meanings are
+used in V-PCC, G-PCC and AVS, but the used compression
+parameters can all be converted to the quantization steps.
+Thus, to better explore the relationship between the perceptual
+quality and the quantization, we first convert these compres-
+sion parameters to quantization steps, denoted as Qs, before
+proposing the R-PCQA.
+In V-PCC, the parameters textureQP and geomQP, de-
+noted as QP, are used to control the attribute compression
+and geometry compression respectively, which apply
+Qs = round(2
+QP −4
+6
+),
+(1)
+where round(·) means converting a number to the nearest
+integer.
+In attribute compression of G-PCC, the compression pa-
+rameter qp has the same meaning as QP in V-PCC, following
+(a)
+(b)
+Fig. 1: Variation of Qs as function of MOS for different
+samples. (a) under V-PCC lossless-G-lossy-A condition; (b)
+under V-PCC lossy-G-lossless-A condition.
+the same conversion formula in Eq. (1). The parameter posi-
+tionQuantizationScale, denoted as S, is used to control the
+geometry quantization, which can be converted to Qs by
+Qs = 1
+S .
+(2)
+In AVS, the parameter attr quant param, denoted as
+QPa, is used to control the attribute quantization, which can
+use the following formulation to convert it to Qs
+Qs = 2
+QPa
+8 .
+(3)
+For the geometry compression of AVS, the parameter
+geom quant step shares the same meaning with the quan-
+tization step Qs.
+B. Overall Quality Model
+We use the average MOS in PCCQA to fit the mathematical
+model for quality prediction. The relationships between MOS
+and Qs under V-PCC lossless-G-lossy-A condition and V-PCC
+lossy-G-lossless-A condition are shown in Fig. 1. We can see
+that under the same compression condition, different samples
+share the fitting model with basically the same shape but
+are added to different additive factors. Therefore, we assume
+MOS and Qs satisfy the following relationship under a certain
+compression condition:
+MOS = F(Qs) + c(pc),
+(4)
+
+5
+4.5
+4
+3.5
+SO
+3
+2.5
+2
+→ Sample 1 Sample 2 ^ Sample 3
+1.5
+Sample 4 X Sample 5
+1
+50
+100
+150
+200
+250
+05
+4.5
+4
+3.5
+OS
+3
+2.5
+2
+1.5
+← Sample 1
+ Sample 2 ^ Sample 3
+Sample 4 × Sample 5
+1
+50
+100
+150
+200
+250
+0
+Oswhere F denotes the fitting function which is related to the
+quantization step Qs. c denotes the additive factor which is
+related to the intrinsic characteristics of the point cloud pc.
+On the whole, different samples share the same relationship
+model under a certain compression condition, but they are
+added to an additive sample factor. To deal with the addi-
+tive sample factor, we use Qs and average MOS which is
+denoted as MOS to build up the relationship model for each
+compression condition:
+MOSf = MOS = F(Qs) + c(pc),
+(5)
+where MOSf is the final predicted quality score and c denotes
+the average value of additive factors.
+C. Modeling the Attribute Compression
+The relationships between MOS and Qs are illustrated in
+Fig. 2. For the attribute compression of all V-PCC, G-PCC and
+AVS compression algorithms, the relationship between MOS
+and Qs follows the linear model, i.e.,
+MOSa = c1,a ∗ Qsa + c2,a,
+(6)
+where Qsa denotes the quantization steps for attribute com-
+pression. c1,a and c2,a are the model parameters, whose fitting
+values are shown in Table II.
+TABLE II: Fitting parameters in the attribute compression
+model.
+V-PCC
+G-PCC
+AVS
+c1,a
+-0.0089
+-0.01
+-0.0519
+c2,a
+4.4862
+5.3515
+5.1337
+D. Modeling the Geometry Compression
+For geometry compression of V-PCC, the relationship
+between MOS and Qs follows the natural logarithm function,
+i.e.,
+MOSg,V −P CC = c1,g ∗ lnQsg + c2,g,
+(7)
+where Qsg denotes the quantization steps for geometry com-
+pression. c1,g and c2,g denote the model parameters.
+For geometry compression of G-PCC and AVS compres-
+sion algorithms, the relationship between MOS and Qs fol-
+lows the linear model, i.e.,
+MOSg,G−P CC,AV S = c1,g ∗ Qsg + c2,g.
+(8)
+The fitted parameters in the geometry compression models
+are shown in Table III:
+TABLE III: Fitting parameters in the geometry compression
+model.
+V-PCC
+G-PCC
+AVS
+c1,g
+-0.559
+-0.2381
+-0.273
+c2,g
+5.4165
+5.3818
+5.5034
+E. Combining the Attribute Model and Geometry Model
+The point clouds are often compressed in both attribute
+and geometry, and the attribute degradation and geometry
+degradation are superimposed on the point clouds at the same
+time. As explored in Section IV-B, the linear combination
+of the attribute model and geometry model can accurately
+estimate the quality. We take the weighted summation of
+MOSa and MOSg to predict the final quality scores.
+For V-PCC, the established model is
+MOSf = p1,a ∗ Qsa + p1,g ∗ lnQsg + P.
+(9)
+For G-PCC and AVS, the established model is
+MOSf = p1,a ∗ Qsa + p1,g ∗ Qsg + P,
+(10)
+where MOSf is the predicted quality scores, Qsa is the
+quantization steps for attribute compression, and Qsg is the
+quantization steps for geometry compression. p1,a = 1
+2 ∗ c1,a,
+p1,g = 1
+2 ∗ c1,g, and P = 1
+2 ∗ (c2,a + c2,g) to cast the predicted
+quality scores under the same range of subjective scores.
+F. Analysis
+Some findings can be made in the experiment: i) Eq. 6
+and Eq. 7 demonstrate that for the V-PCC distortion, the
+geometry distortion is more annoying compared with the
+attribute distortion, but the human eyes are more sensitive to
+the quantization change in the attribute compression; ii) for
+the geometry compression, the fitting curves of V-PCC and G-
+PCC are different, which derives from that the quantization of
+V-PCC is conducted on the projection while the quantization
+of G-PCC and AVS is conducted on octree; iii) for the attribute
+compression, all the three compression algorithms follow the
+linear model, since their quantization is all conducted on RGB,
+resulting in the similar perceptual pattern.
+A potential concern is whether the obtained relation func-
+tion is generic for different datasets. As discussed in Section
+III-B, the difference of reference samples will only affect
+the additive factors, as P in Eq. 9 and Eq. 10 which is a
+predefined constant. Thus, the obtained relation function can
+still accurately predict the quality rank of samples in other
+datasets, which is demonstrated by the cross-dataset evaluation
+in Section IV-C.
+IV.
+EXPERIMENTS
+In this section, we evaluate the performance of the pro-
+posed R-PCQA on the established PCCQA and WPC [24]
+dataset. Specifically, we use PCCQA to fit the model parame-
+ters and evaluate the fitting errors. Then, we evaluate on WPC
+dataset as cross check to verify the performance of R-PCQA
+and its generalization ability.
+A. Error Analysis
+The proposed PCCQA consists of three parts, part 1:
+lossless-G-lossy-A, part 2: lossy-G-lossless-A and part 3:
+lossy-G-lossy-A. The proposed R-PCQA is fitted on the
+lossless-G-lossy-A and lossy-G-lossless-A parts, and we use
+the remaining lossy-G-lossy-A part to evaluate the perfor-
+mance. Especially, we note the former two parts as the training
+
+(a)
+(b)
+(c)
+(d)
+(e)
+(f)
+Fig. 2: Variation of Qs as function of average MOS for each compression condition. The top row is under lossless-G-lossy-A,
+and the bottom row is under lossy-G-lossless-A. (a) (d) is for V-PCC, (b) (e) is for G-PCC, and (c) (f) is for AVS.
+set and the latter part as the testing set. The mean, standard
+deviation and 95% quantile of fitting errors MOS − MOSf
+on the testing set are shown in Table IV. The correlation
+performance on the testing set is shown in Table V.
+TABLE IV: Mean, standard deviation and 95% quantile of the
+fitting errors on the testing set.
+Mean
+Standard deviation
+95% quantile
+V-PCC
+0.0378
+0.5885
+0.7561
+G-PCC
+-0.5794
+0.5113
+0.0969
+AVS
+-0.1356
+0.2845
+0.2531
+We can see from Table IV and Table V that the proposed
+model can not only fit the dataset accurately, but also conforms
+to the characteristics of human visual system.
+B. Combination Analysis
+The correlation performance of four combination schemes
+of the attribute model and geometry model on the testing set
+is shown in Table V.
+TABLE V: Correlation performance of four combination
+schemes on the testing set.
+PLCC
+SROCC
+RMSE
+PLCC
+SROCC
+RMSE
+Linear Combination
+Multiplicative Combination
+V-PCC
+0.8360
+0.8554
+0.5070
+0.8360
+0.8554
+0.5070
+G-PCC
+0.9854
+0.9582
+0.2098
+0.9853
+0.9582
+0.2100
+AVS
+0.9917
+0.9854
+0.1650
+0.9913
+0.9854
+0.1691
+GA Combination
+AG Combination
+V-PCC
+0.8351
+0.8554
+0.5082
+0.8356
+0.8554
+0.5075
+G-PCC
+0.9444
+0.9582
+0.4046
+0.9767
+0.9582
+0.2644
+AVS
+0.9881
+0.9854
+0.1978
+0.9862
+0.9854
+0.2127
+We can see from Table V that: i) the linear combination
+is determined due to its slightly better performance and sim-
+pler calculation for two relationship model mixing; ii) the
+combination schemes hardly affect the performance, which
+indicates that the obtained relationship models for attribute
+and geometry are independent. Due to the removal of mutual
+masking, it is not necessary to consider the interaction of
+attribute and geometry components in the mixing.
+C. Cross-dataset Evaluation
+After the model is established on the proposed dataset, we
+evaluate its generalization performance on another independent
+dataset, the V-PCC part of WPC [24], which contains 400
+distorted samples derived from 16 reference point clouds with
+25 different quantization parameters. The results are shown in
+Table VI.
+TABLE VI: Cross-dataset performance on WPC dataset.
+PLCC
+SROCC
+PLCC
+SROCC
+M-p2po (FR) [3]
+0.61
+0.58
+H-PSNRyuv (FR) [33]
+0.29
+0.23
+M-p2pl (FR) [34]
+0.63
+0.59
+PCQM (FR) [12]
+0.74
+0.75
+H-p2po (FR) [3]
+0.51
+0.46
+GraphSIM (FR) [13]
+0.74
+0.75
+H-p2pl (FR) [34]
+0.55
+0.48
+MPED (FR) [35]
+0.60
+0.59
+PSNRyuv (FR) [33]
+0.46
+0.47
+PCM RR (RR) [16]
+0.42
+0.38
+R-PCQA (RR)
+0.88
+0.88
+We can see from Table VI that: i) for the compression
+distortions, the proposed R-PCQA exhibits the state-of-the-
+art performance which only needs the assistance of two
+compression parameters, even compared with the existing FR
+metrics; ii) the model parameters derived from PCCQA still
+exhibit robust performance on another independent dataset,
+which demonstrates the generalization ability of the proposed
+RR metric R-PCQA; iii) the massive increase in points after
+reconstruction may interfere with the measurement of point-
+wise FR metrics, resulting in the poor performance of some
+FR metrics.
+V.
+CONCLUSION
+In this paper, we analyze the compression distortions of
+point clouds under separate attribute compression and geom-
+etry compression to avoid their mutual masking. Then by
+fitting the relationship between the quantization steps and the
+
+5
+4.5
+SO
+4
+3.5
+3
+2.5
+2
+1.5
+1
+0
+100
+200
+300
+s5
+4.5
+SO)
+4
+3.5
+3
+2.5
+2
+1.5
+1
+0
+100
+200
+300
+Os5
+4.5
+SOI
+4
+3.5
+Average
+3
+2.5
+2
+1.5
+1
+0
+20
+40
+60
+80
+Qs5
+4.5
+so
+4
+7
+3.5
+3
+2.5
+2
+1.5
+1
+0
+100
+200
+300
+Os5
+4.5
+SOI
+4
+M
+3.5
+Average
+3
+2.5
+2
+1.5
+1
+0
+5
+10
+15
+20
+Qs5
+4.5
+SOI
+4
+3.5
+Average
+3
+2.5
+2
+1.5
+1
+0
+5
+10
+15
+20
+Qsperceptual quality, we propose a RR PCQA model, called R-
+PCQA, for evaluating V-PCC, G-PCC and AVS distortions.
+The experiment results have demonstrated that the proposed
+R-PCQA exhibits reliable and robust performance.
+VI.
+ACKNOWLEDGEMENT
+This paper is supported in part by National Key R&D
+Program of China (2018YFE0206700), National Natural Sci-
+ence Foundation of China (61971282, U20A20185). The cor-
+responding author is Yiling Xu (e-mail: [email protected]).
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+

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+page_content='  Specifically, we use the attribute and geometry quantization  steps of different compression methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=', V-PCC, G-PCC and  AVS) to infer the point cloud quality, assuming that the point  clouds have no other distortions before compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' First, we  analyze the compression distortion of point clouds under separate  attribute compression and geometry compression to avoid their  mutual masking, for which we consider 5 point clouds as  references to generate a compression dataset (PCCQA) containing  independent attribute compression and geometry compression  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Then, we develop the proposed R-PCQA via fitting the  relationship between the quantization steps and the perceptual  quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' We evaluate the performance of R-PCQA on both the  established dataset and another independent dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The results  demonstrate that the proposed R-PCQA can exhibit reliable  performance and high generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  INTRODUCTION  Recently, point cloud has emerged as a promising represen-  tation format in prevalent 3D applications (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=', autonomous  driving [1] and augmented reality [2]), for which the point  cloud compression (PCC) is of great interest for providing  efficient service in practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Currently, the Moving Picture  Experts Group (MPEG) has applied the separable measure-  ments of geometry and attribute distortion in the course of  lossy PCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' For the geometry distortion, MPEG proposes to use  the point-to-point (p2point) [3], or point-to-plane (p2plane) [4]  to quantify the spatial perturbation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' while for the attribute  part, the PSNRyuv is proposed to measure the differences  between corresponding color channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Besides these metrics  which have already been applied in MPEG PCC standardiza-  tion, some other metrics which consider more human visual  characteristics and present better performance on public PCQA  databases are also developed, such as [5]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' However, they  are full reference (FR) metrics which require both the reference  and distorted samples and have high computational complexity  for real-time quality prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  In many practical cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=', transmission, the timely  feedback is expected, and only the compressed samples and  the meta data are available, in which the reduced reference  (RR) PCQA metrics are indispensable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Only a few researches  explore the RR methods for PCQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' [16] uses the statistical in-  formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=', the luminance histogram) as the substitute for  the complete samples, but still requires the backend processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  [17] applies the quantization parameters in V-PCC to estimate  the quality of compressed samples and guide rate control,  but other prevalent compression strategies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=', G-PCC) are  ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Therefore, in this paper, we propose a general RR  PCQA model for compression distortions named R-PCQA  which only takes the attribute and geometry quantization steps  of compression schemes (including V-PCC [18], G-PCC [19]  and AVS [20], V-PCC and G-PCC are provided by MPEG  while AVS is recommended by China Audio-Video Coding  Standard) as variables, since the quality of compressed point  clouds is highly related to the compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' To fully  study the relationship between the compression parameters  and perceptual quality, we first establish a complete subjective  database for PCC, named PCC quality assessment (PCCQA)  database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  The reason why we establish PCCQA while many PCQA  datasets [7], [21]–[24] have been proposed is that current  databases only consider the superimposed compression distor-  tion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' lossy-geometry (G)-lossy-attribute (A) compression,  which is recommended in the Common Test Conditions (CTC)  [25]–[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The separate compression strategies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' lossless-  G-lossy-A and lossy-G-lossless-A compression, which are  not included in the CTC are often ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Considering the  mutual masking between geometry and attribute distortions  [28], they are useful for exploring the relationship between the  perceptual quality and compression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' In PCCQA,  the reference point clouds are compressed by V-PCC, G-PCC  and AVS under lossless-G-lossy-A condition, lossy-G-lossless-  A condition, and lossy-G-lossy-A condition respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  To model the relationship between the perceptual quality  and the compression parameters, we first convert all the  compression parameters to the quantization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Then, we  model the relationship between the perceptual quality and the  attribute/geometry quantization steps respectively via using the  least square fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Finally, the proposed R-PCQA combines  the attribute compression model and geometry compression  model to predict the final quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  The rest of this paper is organized as follows: section II  introduces the established PCCQA dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' section III presents  the proposed R-PCQA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' section IV illustrates the experiment  results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' the conclusion is summarized in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  978-1-6654-7592-1/22/$31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='00 © 2022 IEEE  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='01009v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='IV]  3 Jan 2023  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  PCCQA DATABASE  To better explore the relationship between the perceptual  quality and the attribute/geometry compression parameters,  we first establish a database called PCCQA under several  compression conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  Five reference point clouds are selected from MPEG and  AVS point cloud datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' These reference point clouds are  ensured to have no holes and other distortions under 1080P  presentation with size-2 primitives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The reference point clouds  are then distorted with 3 compression algorithms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' V-  PCC [18], G-PCC [19] and AVS [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Each compression is  conducted under 3 conditions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' lossless-G-lossy-A, lossy-G-  lossless-A, and lossy-G-lossy-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The compression parameters  are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' In total, 225 compressed point clouds  are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  TABLE I: Compression parameters used for distorted point  cloud generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  Conditions  Parameters  GPCC lossy-G-lossless-A  VPCC lossy-G-lossless-A  AVS lossy-G-lossless-A  (positionQuantizationScale)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0625  (geomQP)  22 32 37 42 51  (geom quant step)  2 4 8 12 16  GPCC lossless-G-lossy-A  VPCC lossless-G-lossy-A  AVS lossless-G-lossy-A  (qp)  35 39 43 47 51  (textureQP)  32 37 42 47 51  (attr quant param)  24 32 40 44 48  GPCC lossy-G-lossy-A  VPCC lossy-G-lossy-A  AVS lossy-G-lossy-A  (positionQuantizationScale, qp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='75,35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5,39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='25,43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='125,47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0625,51  (geomQP, textureQP)  24,32 28,37 32,42 36,47 40,51  (geom quant step, attr quant param)  2,24 4,32 8,40 12,44 16,48  To annotate the compressed point clouds, a subjective  experiment is organized to collect the Mean Opinion Scores  (MOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' We adopt the double stimulus method since it can  obtain more stable results for minor impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The ex-  periment process and environment setting strictly follow the  ITU-R Recommendation BT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 500 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Such a method for  collecting subjective MOS is also adopted in other researches,  such as [23], [30]–[32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  PROPOSED QUALITY ASSESSMENT MODEL R-PCQA  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Unifying the Compression Parameters  The compression parameters with different meanings are  used in V-PCC, G-PCC and AVS, but the used compression  parameters can all be converted to the quantization steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  Thus, to better explore the relationship between the perceptual  quality and the quantization, we first convert these compres-  sion parameters to quantization steps, denoted as Qs, before  proposing the R-PCQA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  In V-PCC, the parameters textureQP and geomQP, de-  noted as QP, are used to control the attribute compression  and geometry compression respectively, which apply  Qs = round(2  QP −4  6  ),  (1)  where round(·) means converting a number to the nearest  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  In attribute compression of G-PCC, the compression pa-  rameter qp has the same meaning as QP in V-PCC, following  (a)  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 1: Variation of Qs as function of MOS for different  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' (a) under V-PCC lossless-G-lossy-A condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' (b)  under V-PCC lossy-G-lossless-A condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  the same conversion formula in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The parameter posi-  tionQuantizationScale, denoted as S, is used to control the  geometry quantization, which can be converted to Qs by  Qs = 1  S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  (2)  In AVS, the parameter attr quant param, denoted as  QPa, is used to control the attribute quantization, which can  use the following formulation to convert it to Qs  Qs = 2  QPa  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  (3)  For the geometry compression of AVS, the parameter  geom quant step shares the same meaning with the quan-  tization step Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Overall Quality Model  We use the average MOS in PCCQA to fit the mathematical  model for quality prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The relationships between MOS  and Qs under V-PCC lossless-G-lossy-A condition and V-PCC  lossy-G-lossless-A condition are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' We can see  that under the same compression condition, different samples  share the fitting model with basically the same shape but  are added to different additive factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Therefore, we assume  MOS and Qs satisfy the following relationship under a certain  compression condition:  MOS = F(Qs) + c(pc),  (4)  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SO  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  → Sample 1 Sample 2 ^ Sample 3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  Sample 4 X Sample 5  1  50  100  150  200  250  05  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  OS  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  ← Sample 1  Sample 2 ^ Sample 3  Sample 4 × Sample 5  1  50  100  150  200  250  0  Oswhere F denotes the fitting function which is related to the  quantization step Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' c denotes the additive factor which is  related to the intrinsic characteristics of the point cloud pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  On the whole, different samples share the same relationship  model under a certain compression condition, but they are  added to an additive sample factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' To deal with the addi-  tive sample factor, we use Qs and average MOS which is  denoted as MOS to build up the relationship model for each  compression condition:  MOSf = MOS = F(Qs) + c(pc),  (5)  where MOSf is the final predicted quality score and c denotes  the average value of additive factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Modeling the Attribute Compression  The relationships between MOS and Qs are illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' For the attribute compression of all V-PCC, G-PCC and  AVS compression algorithms, the relationship between MOS  and Qs follows the linear model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=',  MOSa = c1,a ∗ Qsa + c2,a,  (6)  where Qsa denotes the quantization steps for attribute com-  pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' c1,a and c2,a are the model parameters, whose fitting  values are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  TABLE II: Fitting parameters in the attribute compression  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  V-PCC  G-PCC  AVS  c1,a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0089  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0519  c2,a  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='4862  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='3515  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='1337  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Modeling the Geometry Compression  For geometry compression of V-PCC, the relationship  between MOS and Qs follows the natural logarithm function,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=',  MOSg,V −P CC = c1,g ∗ lnQsg + c2,g,  (7)  where Qsg denotes the quantization steps for geometry com-  pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' c1,g and c2,g denote the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  For geometry compression of G-PCC and AVS compres-  sion algorithms, the relationship between MOS and Qs fol-  lows the linear model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=',  MOSg,G−P CC,AV S = c1,g ∗ Qsg + c2,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  (8)  The fitted parameters in the geometry compression models  are shown in Table III:  TABLE III: Fitting parameters in the geometry compression  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  V-PCC  G-PCC  AVS  c1,g  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='559  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='2381  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='273  c2,g  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='4165  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='3818  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5034  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Combining the Attribute Model and Geometry Model  The point clouds are often compressed in both attribute  and geometry, and the attribute degradation and geometry  degradation are superimposed on the point clouds at the same  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' As explored in Section IV-B, the linear combination  of the attribute model and geometry model can accurately  estimate the quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' We take the weighted summation of  MOSa and MOSg to predict the final quality scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  For V-PCC, the established model is  MOSf = p1,a ∗ Qsa + p1,g ∗ lnQsg + P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  (9)  For G-PCC and AVS, the established model is  MOSf = p1,a ∗ Qsa + p1,g ∗ Qsg + P,  (10)  where MOSf is the predicted quality scores, Qsa is the  quantization steps for attribute compression, and Qsg is the  quantization steps for geometry compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' p1,a = 1  2 ∗ c1,a,  p1,g = 1  2 ∗ c1,g, and P = 1  2 ∗ (c2,a + c2,g) to cast the predicted  quality scores under the same range of subjective scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Analysis  Some findings can be made in the experiment: i) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 6  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 7 demonstrate that for the V-PCC distortion, the  geometry distortion is more annoying compared with the  attribute distortion, but the human eyes are more sensitive to  the quantization change in the attribute compression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' ii) for  the geometry compression, the fitting curves of V-PCC and G-  PCC are different, which derives from that the quantization of  V-PCC is conducted on the projection while the quantization  of G-PCC and AVS is conducted on octree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' iii) for the attribute  compression, all the three compression algorithms follow the  linear model, since their quantization is all conducted on RGB,  resulting in the similar perceptual pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  A potential concern is whether the obtained relation func-  tion is generic for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' As discussed in Section  III-B, the difference of reference samples will only affect  the additive factors, as P in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 9 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 10 which is a  predefined constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Thus, the obtained relation function can  still accurately predict the quality rank of samples in other  datasets, which is demonstrated by the cross-dataset evaluation  in Section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  EXPERIMENTS  In this section, we evaluate the performance of the pro-  posed R-PCQA on the established PCCQA and WPC [24]  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Specifically, we use PCCQA to fit the model parame-  ters and evaluate the fitting errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Then, we evaluate on WPC  dataset as cross check to verify the performance of R-PCQA  and its generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Error Analysis  The proposed PCCQA consists of three parts, part 1:  lossless-G-lossy-A, part 2: lossy-G-lossless-A and part 3:  lossy-G-lossy-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The proposed R-PCQA is fitted on the  lossless-G-lossy-A and lossy-G-lossless-A parts, and we use  the remaining lossy-G-lossy-A part to evaluate the perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Especially, we note the former two parts as the training  (a)  (b)  (c)  (d)  (e)  (f)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' 2: Variation of Qs as function of average MOS for each compression condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The top row is under lossless-G-lossy-A,  and the bottom row is under lossy-G-lossless-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' (a) (d) is for V-PCC, (b) (e) is for G-PCC, and (c) (f) is for AVS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  set and the latter part as the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The mean, standard  deviation and 95% quantile of fitting errors MOS − MOSf  on the testing set are shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The correlation  performance on the testing set is shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  TABLE IV: Mean, standard deviation and 95% quantile of the  fitting errors on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  Mean  Standard deviation  95% quantile  V-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0378  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5885  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='7561  G-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5794  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5113  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='0969  AVS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='1356  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='2845  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='2531  We can see from Table IV and Table V that the proposed  model can not only fit the dataset accurately, but also conforms  to the characteristics of human visual system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Combination Analysis  The correlation performance of four combination schemes  of the attribute model and geometry model on the testing set  is shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  TABLE V: Correlation performance of four combination  schemes on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  PLCC  SROCC  RMSE  PLCC  SROCC  RMSE  Linear Combination  Multiplicative Combination  V-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='8360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='8554  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5070  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='8360  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='8554  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5070  G-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9854  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9582  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='2098  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9853  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9582  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='2100  AVS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9917  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9854  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='1650  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='9913  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
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+page_content='1691  GA Combination  AG Combination  V-PCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
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+page_content='2644  AVS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
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+page_content='2127  We can see from Table V that: i) the linear combination  is determined due to its slightly better performance and sim-  pler calculation for two relationship model mixing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' ii) the  combination schemes hardly affect the performance, which  indicates that the obtained relationship models for attribute  and geometry are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Due to the removal of mutual  masking, it is not necessary to consider the interaction of  attribute and geometry components in the mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Cross-dataset Evaluation  After the model is established on the proposed dataset, we  evaluate its generalization performance on another independent  dataset, the V-PCC part of WPC [24], which contains 400  distorted samples derived from 16 reference point clouds with  25 different quantization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The results are shown in  Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  TABLE VI: Cross-dataset performance on WPC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  PLCC  SROCC  PLCC  SROCC  M-p2po (FR) [3]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='61  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='58  H-PSNRyuv (FR) [33]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='29  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='23  M-p2pl (FR) [34]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='59  PCQM (FR) [12]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='75  H-p2po (FR) [3]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='51  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='46  GraphSIM (FR) [13]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='75  H-p2pl (FR) [34]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='48  MPED (FR) [35]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='59  PSNRyuv (FR) [33]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='47  PCM RR (RR) [16]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='42  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='38  R-PCQA (RR)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='88  We can see from Table VI that: i) for the compression  distortions, the proposed R-PCQA exhibits the state-of-the-  art performance which only needs the assistance of two  compression parameters, even compared with the existing FR  metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' ii) the model parameters derived from PCCQA still  exhibit robust performance on another independent dataset,  which demonstrates the generalization ability of the proposed  RR metric R-PCQA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' iii) the massive increase in points after  reconstruction may interfere with the measurement of point-  wise FR metrics, resulting in the poor performance of some  FR metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  CONCLUSION  In this paper, we analyze the compression distortions of  point clouds under separate attribute compression and geom-  etry compression to avoid their mutual masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' Then by  fitting the relationship between the quantization steps and the  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SO  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  100  200  300  s5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SO)  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  100  200  300  Os5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SOI  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  Average  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  20  40  60  80  Qs5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  so  4  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  100  200  300  Os5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SOI  4  M  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  Average  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  5  10  15  20  Qs5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  SOI  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  Average  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='5  1  0  5  10  15  20  Qsperceptual quality, we propose a RR PCQA model, called R-  PCQA, for evaluating V-PCC, G-PCC and AVS distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  The experiment results have demonstrated that the proposed  R-PCQA exhibits reliable and robust performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  ACKNOWLEDGEMENT  This paper is supported in part by National Key R&D  Program of China (2018YFE0206700), National Natural Sci-  ence Foundation of China (61971282, U20A20185).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content=' The cor-  responding author is Yiling Xu (e-mail: yl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='xu@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
+page_content='  REFERENCES  [1]  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQfFPrO/content/2301.01009v1.pdf'}
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@@ -0,0 +1,3817 @@
+arXiv:2301.01022v1  [math.AP]  3 Jan 2023
+DECAY OF SOLUTIONS OF ISENTROPIC GAS DYNAMICS
+FOR LARGE DATA
+NAOKI TSUGE
+Abstract. In this paper, we are concerned with the Cauchy problem for
+isentropic gas dynamics. Through the contribution of many researchers such
+as Lax, P. D., Glimm, J., DiPerna, R. J. and Liu, T. P., the decay of solutions
+was established. They treated with initial data with the small total variation.
+On the other hand, the decay for large initial data has been open for half a
+century. Our goal is to provide a new method to analyze this problem. We
+prove the existence of a global attractor, which yields a decay of solutions
+for large data. To construct approximate solutions, we introduce a modified
+Godunov scheme.
+1. Introduction
+The present paper is concerned with isentropic gas dynamics
+
+
+
+ρt + mx = 0,
+mt +
+�m2
+ρ + p(ρ)
+�
+x
+= 0,
+x ∈ R,
+t ∈ R+,
+(1.1)
+where ρ, m and p are the density, the momentum and the pressure of the gas,
+respectively. If ρ > 0, v = m/ρ represents the velocity of the gas. For a barotropic
+gas, p(ρ) = ργ/γ, where γ ∈ (1, 5/3] is the adiabatic exponent for usual gases.
+We consider the initial value problem (1.1) with the initial data
+(ρ, m)|t=0 = (ρ0(x), m0(x)),
+(1.2)
+where
+(ρ0(x), m0(x)) = (¯ρ, 0) outside a finite interval
+(1.3)
+and ¯ρ is a positive constant. The above problem (1.1)–(1.2) can be written in the
+following form
+�
+ut + f(u)x = 0,
+x ∈ R,
+t ∈ R+,
+u|t=0 = u0(x),
+(1.4)
+by using u = t(ρ, m), f(u) = t
+�
+m, m2
+ρ + p(ρ)
+�
+.
+We recollect the known results of the above problem. DiPerna [5] proved the
+global existence of solutions to (1.1)–(1.2) by the vanishing viscosity method and
+2020 Mathematics Subject Classification. Primary 35L03, 35L65, 35Q31, 76N10, 76N15; Sec-
+ondary 35A01, 35B35, 35B50, 35L60, 76H05, 76M20.
+Key words and phrases. The Compressible Euler Equation, isentropic gas dyanmics, the com-
+pensated compactness, the Godunov scheme, global attractor, decay estimates.
+N. Tsuge’s research is partially supported by Grant-in-Aid for Scientific Research (C)
+17K05315, Japan.
+1
+
+2
+NAOKI TSUGE
+a compensated compactness argument. DiPerna first applied the method to (1.1)
+for the special case where γ = 1 + 2/n and n is an odd integer. We notice that this
+result can treat with the arbitrary L∞ data. Subsequently, Ding, Chen and Luo [6]
+and Chen [1] and [2] extended his analysis to any γ in (1, 5/3].
+On the other hand, the existence of solutions to conservation laws including
+(1.1) was established by Glimm [8]. Glimm treatd the Cauchy problem with initial
+data having small total variation.
+The theory of decay for genuinely nonlinear
+2×2 systems of conservation laws was constructed by Glimm-Lax [9]. Glimm-Lax
+showed that if initial data are constant outside a finite interval and have locally
+bounded total variation and small oscillation, then the tonal variation of the solution
+of [8] decays to zero at the rate t−1/2. The Glimm-Lax theory had been further
+developed by DiPerna and Liu: general conservation laws with a convex entropy
+function [4], general conservation laws with small initial data in total variation [10].
+However, the decay for large initial data has been open for half a century. Our
+goal in the present paper is to provide a new method to analyze this problem and
+investigate the decay structure of (1.1). We first introduce a modified Godunov
+scheme to construct approximate solutions. We next prove the existence of a global
+attractor, which yields decay estimates of solutions for large initial data.
+To state our main theorem, we define the Riemann invariants w, z, which play
+important roles in this paper, as
+Definition 1.1.
+w(u) := m
+ρ + ρθ
+θ = v + ρθ
+θ ,
+z(u) := m
+ρ − ρθ
+θ = v − ρθ
+θ
+�
+θ = γ − 1
+2
+�
+.
+These Riemann invariants satisfy the following.
+Remark 1.1.
+|w| ≥ |z|, w ≥ 0, when v ≥ 0.
+|w| ≤ |z|, z ≤ 0, when v ≤ 0.
+v = w + z
+2
+, ρ =
+�θ(w − z)
+2
+�1/θ
+, m = ρv.
+(1.5)
+From the above, the lower bound of z and the upper bound of w yield the bound of
+ρ and |v|.
+We next introduce the mechanical energy as η∗(u) = 1
+2
+m2
+ρ +
+1
+γ(γ − 1)ργ and set
+J(u) = η∗(u) − (¯ρ)γ−1
+γ − 1 ρ + (¯ρ)γ
+γ .
+(1.6)
+Remark 1.2. From the convexity of η∗, we have
+J(u) = η∗(u) − η∗(¯ρ, 0) − ∂η∗
+∂ρ (¯ρ, 0)(ρ − ¯ρ) − ∂η∗
+∂m (¯ρ, 0)m ≥ 0.
+(1.7)
+From the conservation of mass and the energy inequality, we have
+0 ≤
+� x
+−∞
+J(u(y, t))dy ≤
+� ∞
+−∞
+J(u(x, t))dx ≤
+� ∞
+−∞
+J(u0(x))dx.
+(1.8)
+Moreover, we define the entropy weak solution.
+
+THE COMPRESSIBLE EULER EQUATIONS
+3
+Definition 1.2. A measurable function u(x, t) is called a global entropy weak so-
+lution of the Cauchy problems (1.4) if
+� ∞
+−∞
+� ∞
+0
+uφt + f(u)φxdxdt +
+� ∞
+−∞
+u0(x)φ(x, 0)dx = 0
+holds for any test function φ ∈ C1
+0(R × R+) and
+� ∞
+−∞
+� ∞
+0
+η(u)ψt + q(u)ψx +
+� ∞
+−∞
+η(u0(x))ψ(x, 0)dx ≥ 0
+holds for any non-negative test function ψ ∈ C1
+0(R × R+), where (η, q) is a pair of
+convex entropy–entropy flux of (1.1).
+Finally, we define ˜z, ˜w by
+˜z(x, t) = z(x, t) −
+� x
+−∞
+J(u(y, t))dy,
+˜w(x, t) = w(x, t) −
+� x
+−∞
+J(u(y, t))dy.
+(1.9)
+Then, our main theorem is as follows.
+Theorem 1.1. We assume that
+ρ0(x) ≥ 0
+a.e. x ∈ R,
+ρ0 ∈ L∞(R),
+m0
+ρ0
+∈ L∞(R).
+(1.10)
+Then, there exists a global entropy weak solution of the Cauchy problems (1.4).
+Moreover, for any positive constant ε, there exist positive constants t0 such that the
+solution satisfies
+− (¯ρ)θ
+θ
+− E0 − ε ≤ ˜z(x, t),
+˜w(x, t) ≤ (¯ρ)θ
+θ
++ ε,
+ρ(x, t) ≥ 0,
+a.e. (x, t) ∈ R × [t0, ∞),
+(1.11)
+where
+E0 =
+� ∞
+−∞
+J(u0(x))dx.
+(1.12)
+Remark 1.3. We remark some points for the above theorem.
+In view of (1.3), we find that −(¯ρ)θ
+θ −E0 ≥ ess infx (˜z(x, 0)) , ess supx ( ˜w(x, 0)) ≥
+(¯ρ)θ
+θ . Therefore, (1.11) means the decay estimate of ess infx (˜z(x, t)) and ess infx ( ˜w(x, t)).
+We similarly obtain −(¯ρ)θ
+θ
+≥ ess infx (z(x, 0)) , ess supx (w(x, 0)) ≥ (¯ρ)θ
+θ .
+If
+−(¯ρ)θ
+θ
+− E0 − ε > ess infx (z(x, 0)) and ess supx (w(x, 0)) > (¯ρ)θ
+θ
++ E0 + ε, (1.11)
+yields the decay estimate of ess infx (z(x, t)) and ess supx (w(x, t)). In fact, we find
+
+4
+NAOKI TSUGE
+that
+ess inf
+x (z(x, t0)) − ess inf
+x (z(x, 0))
+= ess inf
+x
+�
+˜z(x, t0) +
+� x
+−∞
+J(u(y, t))dy
+�
+− ess inf
+x (z(x, 0))
+≥ −(¯ρ)θ
+θ
+− E0 − ε − ess inf
+x (z(x, 0)) > 0,
+ess sup
+x (w(x, t0)) − ess sup
+x (w(x, 0))
+= ess sup
+x
+�
+˜w(x, t0) +
+� x
+−∞
+J(u(y, t))dy
+�
+− ess sup
+x (w(x, 0))
+≤ (¯ρ)θ
+θ
++ ε + E0 − ess sup
+x (w(x, 0)) < 0.
+1.1. Outline of the proof (formal argument). The proof of main theorem is
+a little complicated. Therefore, before proceeding to the subject, let us grasp the
+point of the main estimate by a formal argument. Although (1.1) has a discontin-
+uous solution in general, we assume that solutions are smooth and the density is
+nonnegative in this section.
+We consider the physical region ρ ≥ 0 (i.e., w ≥ z.). Recalling Remark 1.1, it
+suffices to derive the lower bound of z(u) and the upper bound of w(u) to obtain
+the bound of u. To do this, we diagonalize (1.1). If solutions are smooth, we deduce
+from (1.1)
+zt + λ1zx = 0,
+wt + λ2wx = 0,
+(1.13)
+where λ1 and λ2 are the characteristic speeds defined as follows
+λ1 = v − ρθ,
+λ2 = v + ρθ.
+(1.14)
+From the conservation of mass (1.1)1 and the conservation of energy (η∗)t +
+(q∗)x = 0, we obtain
+˜zt + λ1˜zx = g1(u),
+˜wt + λ2 ˜wx = g2(u),
+(1.15)
+where
+q∗(u) = m
+�1
+2
+m2
+ρ2 + ργ−1
+γ − 1
+�
+and
+g1(u) = − (¯ρ)γ
+γ λ1 +
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv + 1
+2ρθ+1v2 − (¯ρ)γ−1
+γ − 1 ρθ+1,
+g2(u) = − (¯ρ)γ
+γ λ2 −
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv − 1
+2ρθ+1v2 + (¯ρ)γ−1
+γ − 1 ρθ+1.
+(1.16)
+To prove Theorem 1.1, we prepare the following proposition.
+
+THE COMPRESSIBLE EULER EQUATIONS
+5
+Proposition 1.2.
+
+
+
+
+
+
+
+
+
+
+
+•
+g1(u(x, t)) > 0, when ˜z(x, t) < −(¯ρ)θ
+θ
+− E0,
+•
+g1(u(x, t)) = 0, when ˜z(x, t) = −(¯ρ)θ
+θ
+− E0,
+� x
+−∞ J(u(y, t))dy = E0
+and ρ(x, t) = ¯ρ,
+(1.17)
+
+
+
+
+
+
+
+
+
+
+
+•
+g2(u(x, t)) < 0, when ˜w(x, t) > (¯ρ)θ
+θ ,
+•
+g2(u(x, t)) = 0, when ˜w(x, t) = (¯ρ)θ
+θ ,
+� x
+−∞ J(u(y, t))dy = 0
+and ρ(x, t) = ¯ρ.
+(1.18)
+Proof. We first investigate (1.17).
+When ˜z(x, t) ≤ −(¯ρ)θ
+θ
+− E0, from (1.8), we
+observe that
+z(x, t) = ˜z(x, t) +
+� x
+−∞
+J(u(y, t))dy ≤ −(¯ρ)θ
+θ .
+(1.19)
+Then, we deduce that
+g1(u)=
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv + 1
+2ρθ+1v2 − (¯ρ)γ
+γ λ1 − (¯ρ)γ−1
+γ − 1 ρθ+1
+=
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv + 1
+2ρθ+1v2 − (¯ρ)γ
+γ
+�
+z + 3 − γ
+γ − 1ρθ
+�
+− (¯ρ)γ−1
+γ − 1 ρθ+1
+=
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv + 1
+2ρθ+1v2 − (¯ρ)γ
+γ
+�
+z + 3 − γ
+γ − 1ρθ
+�
+− (¯ρ)γ−1
+γ − 1 ρθ+1
+=
+1
+γ(γ − 1)ργ+θ + 1
+γ ργ
+�
+z + ρθ
+θ
+�
++ 1
+2ρθ+1
+�
+z + ρθ
+θ
+�2
+− (¯ρ)γ
+γ
+�
+z + 3 − γ
+γ − 1ρθ
+�
+− (¯ρ)γ−1
+γ − 1 ρθ+1
+=ρθ+1
+2
+�
+z − 1
+γ
+(¯ρ)γ
+ρθ+1 + 3γ − 1
+γ(γ − 1)ρθ
+�2
++
+γ + 1
+2γ2(γ − 1)ργ+θ
+−
+1
+γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1
+γ2
+(¯ρ)γ ρθ −
+1
+2γ2 (¯ρ)2γ
+1
+ρθ+1 .
+(1.20)
+When ρ ≤ ¯ρ, since (1.20) attains the minimum at z = −(¯ρ)θ
+θ , we deduce from
+Appendix A
+g1(u) ≥ 5γ − 3
+γ(γ − 1)2 ργ+θ − 2(3γ − 1)
+γ(γ − 1)2 (¯ρ)θ ργ +
+3 − γ
+(γ − 1)2 (¯ρ)γ−1 ρθ+1 −
+3 − γ
+γ(γ − 1) (¯ρ)γ ρθ
++
+2
+γ(γ − 1) (¯ρ)��+θ
+≥0,
+(1.21)
+where the equal sign of the second inequality can be used only when ρ = ¯ρ.
+
+6
+NAOKI TSUGE
+When ρ > ¯ρ, since (1.20) attains the minimum at z = 1
+γ
+(¯ρ)γ
+ρθ+1 − 3γ − 1
+γ(γ − 1)ρθ, we
+deduce from Appendix A
+g1(u) ≥
+γ + 1
+2γ2(γ − 1)ργ+θ −
+1
+γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1
+γ2
+(¯ρ)γ ρθ −
+1
+2γ2 (¯ρ)2γ
+1
+ρθ+1
+≥0,
+(1.22)
+where the equal sign of the second inequality can be used only when ρ = ¯ρ.
+We next investigate (1.18). When ˜w(x, t) ≥ (¯ρ)θ
+θ , from (1.8), we observe that
+w(x, t) = ˜w(x, t) +
+� x
+−∞
+J(u(y, t))dy ≥ (¯ρ)θ
+θ .
+(1.23)
+Then, we deduce that
+g2(u) = − (¯ρ)γ
+γ λ2 −
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv − 1
+2ρθ+1v2 + (¯ρ)γ−1
+γ − 1 ρθ+1
+= − (¯ρ)γ
+γ
+�
+w − 3 − γ
+γ − 1ρθ
+�
+−
+1
+γ(γ − 1)ργ+θ + 1
+γ ργv − 1
+2ρθ+1v2 + (¯ρ)γ−1
+γ − 1 ρθ+1
+= − (¯ρ)γ
+γ
+�
+w − 3 − γ
+γ − 1ρθ
+�
+−
+1
+γ(γ − 1)ργ+θ + 1
+γ ργ
+�
+w − ρθ
+θ
+�
+− 1
+2ρθ+1
+�
+w − ρθ
+θ
+�2
++ (¯ρ)γ−1
+γ − 1 ρθ+1
+= − ρθ+1
+2
+�
+w + 1
+γ
+(¯ρ)γ
+ρθ+1 − 3γ − 1
+γ(γ − 1)ρθ
+�2
+−
+γ + 1
+2γ2(γ − 1)ργ+θ
++
+1
+γ − 1 (¯ρ)γ−1 ρθ+1 − γ + 1
+γ2
+(¯ρ)γ ρθ +
+1
+2γ2 (¯ρ)2γ
+1
+ρθ+1 .
+From (1.21)–(1.22), we conclude that
+g2(u) ≤ 0,
+(1.24)
+where the equal sign can be used only when ρ = ¯ρ.
+□
+Proof of Theorem (1.11)
+From (1.17)–(1.18), for any positive constant ε, there exists a positive constant
+δ such that
+g1(u) > 2δ, when ˜z ≤ −(¯ρ)θ
+θ
+− E0 − ε
+2 and 0 ≤ ρ,
+g2(u) < −2δ, when (¯ρ)θ
+θ
++ ε
+2 ≤ ˜w and 0 ≤ ρ.
+(1.25)
+We introduce ˆz, ˆw as follows.
+ˆz(x, t) = ˜z(x, t) − δt,
+ˆw(x, t) = ˜w(x, t) + δt.
+(1.26)
+We deduce from (1.15) that
+ˆzt + λ1ˆzx = g1(u) − δ,
+ˆwt + λ2 ˆwx = g2(u) + δ.
+(1.27)
+
+THE COMPRESSIBLE EULER EQUATIONS
+7
+We set
+M0 = max
+�
+(¯ρ)θ
+θ , −ess inf
+x (˜z(x, 0)) + E0, ess inf
+x ( ˜w(x, 0))
+�
+.
+(1.28)
+Then, we notice that
+−M0 − E0 ≤ ˆz(x, 0),
+ˆw(x, 0) ≤ M0.
+Let us prove that
+ˆSinv = {(ˆz, ˆw) ∈ R2; ρ ≥ 0, ˆz ≥ −M0 − E0, ˆw ≤ M0}
+is an invariant region for the Cauchy problem of (1.27) on
+0 ≤ t ≤ max
+�
+0, M0 − (¯ρ)θ
+θ
+− ε
+δ
+�
+=: t0.
+We notice that this yields (1.11) on 0 ≤ t ≤ t0.
+To achieve this, assuming
+−M0 − E0 < ˆz(x, 0),
+ˆw(x, 0) < M0
+and there exist x∗ ∈ R, 0 < t∗ ≤ t0 such that the following (1.29) or (1.30) holds,
+− M0 − E0 < ˆz(x, t), ˆw(x, t) < M0,
+x ∈ R, 0 ≤ t < t∗
+and
+ˆz(x∗, t∗) = −M0 − E0, ˆw(x∗, t∗) ≤ M0,
+(1.29)
+− M0 − E0 < ˆz(x, t), ˆw(x, t) < M0,
+x ∈ R, 0 ≤ t < t∗
+and
+ˆz(x∗, t∗) ≥ −M0 − E0, ˆw(x∗, t∗) = M0,
+(1.30)
+we will deduce a contradiction.
+To do this, we prove
+g1(u(x∗, t∗)) − δ > 0, when (1.29) holds,
+(1.31)
+g2(u(x∗, t∗)) + δ < 0, when (1.30) holds.
+(1.32)
+Let us consider (1.31). When (1.29) and 0 ≤ t∗ ≤ t0, we notice that
+˜z(x∗, t∗) ≤ −(¯ρ)θ
+θ
+− E0 − ε.
+Therefore, from (1.25), we prove (1.31). Since ˆz attains the minimum at (x∗, t∗),
+we can deduce from (1.27)1 a contradiction. We can similarly prove (1.32).
+We notice that (˜z(x, t0), ˜w(x, t0)) is contained in
+˜Sinv =
+�
+(˜z, ˜w) ∈ R2; ρ ≥ 0, ˜z ≥ −(¯ρ)θ
+θ
+− ε − E0, ˜w ≤ (¯ρ)θ
+θ
++ ε
+�
+.
+Then, we can similarly prove that ˜Sinv is an invariant region for the Cauchy problem
+of (1.15). Therefore, we conclude (1.11).
+Although the above argument is formal, it is essential. In fact, we shall implicitly
+use this argument in the proof of Theorem 3.1. We must next justify the above
+argument.
+To do this, we introduce a modified Godunov scheme in Section 2.
+Recently, the various difference schemes are developed in [11]–[18], which consist
+of known functions. On the other hand, the present approximate solutions include
+unknown functions in the form of (1.9) with constants ˜z, ˜w (see (2.15)).
+
+8
+NAOKI TSUGE
+2. Construction of Approximate Solutions
+In this section, we construct approximate solutions. Let T be any fixed positive
+constant. In the strip 0 ≤ t ≤ T , we denote the approximate solutions by u∆(x, t) =
+(ρ∆(x, t), m∆(x, t)). We denote the space mesh lengths by ∆x. Using E0 in (1.12)
+and M0 in (1.28), we take time mesh length ∆t such that
+∆x
+∆t = 2(M0 + E0).
+(2.1)
+In addition, we set
+(j, n) ∈ 2Z × Z≥0,
+where Z≥0 = {0, 1, 2, 3, . . .}. For simplicity, we use the following terminology
+xj = j∆x, tn = n∆t, tn.5 =
+�
+n + 1
+2
+�
+∆t, tn− = n∆t − 0, tn+ = n∆t + 0. (2.2)
+First we set u∆(x, t0−) = u0(x).
+Then, for j ∈ 2Z, we define E0
+j (u) by
+E0
+j (u) =
+1
+2∆x
+� xj+1
+xj−1
+u∆(x, t0−)dx.
+Next, we assume that u∆(x, t) is defined for t < tn.
+Then, for j ∈ 2Z, we define En
+j (u) by
+En
+j (u) =
+1
+2∆x
+� xj+1
+xj−1
+u∆(x, tn−)dx.
+To determine un
+j = (ρn
+j , mn
+j ) for j ∈ 2Z, we define symbols In
+j and Ln. Let the
+approximation of
+� x
+−∞ J(u(y, t))dy be
+In
+j :=
+� xj
+−∞
+J(En(x; u))dx,
+where
+En(x; u) = En
+j (u)
+x ∈ [xj−1, xj+1)
+(2.3)
+and J is defined in (1.6).
+Let D = (x(t), t) denote a discontinuity in u∆(x, t), [η∗] and [q∗] denote the
+jump of η∗(u∆(x, t)) and q∗(u∆(x, t)) across D from left to right, respectively,
+[η∗] = η∗(u∆(x(t) + 0, t)) − η∗(u∆(x(t) − 0, t)),
+[q∗] = q∗(u∆(x(t) + 0, t)) − q∗(u∆(x(t) − 0, t)),
+where q∗(u) is the flux of η∗(u) defined by
+q∗(u) = m
+�1
+2
+m2
+ρ2 + ργ−1
+γ − 1
+�
+.
+Next, to measure the error in the entropy condition and the gap of the energy at
+tn±, we introduce a functional. To do this, we deduce from the Taylor expansion
+
+THE COMPRESSIBLE EULER EQUATIONS
+9
+that
+η∗
+�
+u∆(x, tn−)
+�
+− η∗
+�
+En
+j (u)
+�
+=∇η∗(En
+j (u))
+�
+u∆(x, tn−) − En
+j (u)
+�
++
+� 1
+0
+(1 − τ) · t �
+u∆(x, tn−) − En
+j (u)
+�
+× ∇2η∗
+�
+En
+j (u) + τ
+�
+u∆(x, tn−) − En
+j (u)
+��
+dτ
+×
+�
+u∆(x, tn−) − En
+j (u)
+�
+=∇η∗(En
+j (u))
+�
+u∆(x, tn−) − En
+j (u)
+�
++ Rn
+j (x),
+(2.4)
+where
+Rn
+j (x) =
+� 1
+0
+(1 − τ) · t �
+u∆(x, tn−) − En
+j (u)
+�
+∇2η∗
+�
+En
+j (u) + τ
+�
+u∆(x, tn−) − En
+j (u)
+��
+×
+�
+u∆(x, tn−) − En
+j (u)
+�
+dτ.
+Then, we define a functional Ln as
+Ln =
+� tn
+0
+�
+x∈R
+σ[η∗] − [q∗]dt +
+n
+�
+k=0
+� ∞
+−∞
+�
+η∗(u∆(x, tk−0)) − η∗(Ek(x; u))
+�
+dx
++
+n
+�
+k=0
+�
+j∈2Z
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+Rk
+j (y)dydx,
+(2.5)
+where the summention in �
+x∈R is taken over all discontinuities in u∆(x, t) at a
+fixed time t over x ∈ R, σ is the propagating speed of the discontinuities.
+Moreover, we set
+M1 = M0 − δ∆t,
+Mn+1 =
+
+
+
+
+
+
+
+Mn − δ∆t,
+when Mn + Ln ≥ (¯ρ)θ
+θ
++ ε,
+Mn,
+when Mn + Ln < (¯ρ)θ
+θ
++ ε,
+(2.6)
+where δ is defined in (1.25). We notice that Mn ≥ (¯ρ)θ
+θ
++ ε − δ∆t.
+Using In
+j , Ln and Mn, we define un
+j as follows.
+We choose µ such that 1 < µ < 1/(2θ). If
+En
+j (ρ) :=
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx < (∆x)µ,
+(2.7)
+we define un
+j by un
+j = (0, 0); otherwise, setting
+zn
+j := max
+�
+z(En
+j (u)), −Mn − E0 − Ln + In
+j
+�
+,
+wn
+j := min
+�
+w(En
+j (u)), Mn + Ln + In
+j
+�
+,
+(2.8)
+we define un
+j by
+un
+j := (ρn
+j , mn
+j ) := (ρn
+j , ρn
+j vn
+j ) :=
+��θ(wn
+j − zn
+j )
+2
+�1/θ
+,
+�θ(wn
+j − zn
+j )
+2
+�1/θ wn
+j + zn
+j
+2
+�
+.
+Remark 2.1. We find
+−Mn − E0 − Ln + In
+j ≤ z(un
+j ),
+w(un
+j ) ≤ Mn + Ln + In
+j .
+(2.9)
+
+10
+NAOKI TSUGE
+This implies that we cut off the parts where z(En
+j (u)) < −Mn − E0 − Ln + In
+j
+and w(En
+j (u)) > Mn + Ln + In
+j in defining z(un
+j ) and w(un
+j ). Observing (3.2), the
+order of these cut parts is o(∆x). The order is so small that we can deduce the
+compactness and convergence of our approximate solutions.
+2.1. Construction of Approximate Solutions in the Cell. We then assume
+that approximate solutions u∆(x, t) are defined in domains D1 : t < tn
+(n ∈ N)
+and D2 : x < xj−1
+(j ∈ 2Z), tn ≤ t < tn+1.
+By using un
+j defined in D1
+and u∆(x, t) defined in D2, we construct the approximate solutions in the cell
+tn ≤ t < tn+1
+(n ∈ N),
+xj−1 ≤ x < xj+1
+(j ∈ 2Z).
+We first solve a Riemann problem with initial data (un
+j−1, un
+j+1). Call constants
+uL(= un
+j−1), uM, uR(= un
+j+1) the left, middle and right states, respectively. Then
+the following four cases occur.
+• Case 1 A 1-rarefaction wave and a 2-shock arise.
+• Case 2 A 1-shock and a 2-rarefaction wave arise.
+• Case 3 A 1-rarefaction wave and a 2-rarefaction arise.
+• Case 4 A 1-shock and a 2-shock arise.
+We then construct approximate solutions u∆(x, t) by perturbing the above Riemann
+solutions.
+Let α be a constant satisfying 1/2 < α < 1. We choose a positive value β small
+enough.
+In this step, we consider Case 1 in particular. The constructions of Cases 2–4
+are similar to that of Case 1. We consider only the case in which uM is away from
+the vacuum. The other case (i.e., the case where uM is near the vacuum) is a little
+technical. Therefore, we postpone this case to Appendix C.
+Consider the case where a 1-rarefaction wave and a 2-shock arise as a Riemann
+solution with initial data (un
+j , un
+j+1). Assume that uL, uM and uM, uR are connected
+by a 1-rarefaction and a 2-shock curve, respectively.
+Step 1.
+In order to approximate a 1-rarefaction wave by a piecewise constant rarefaction
+fan, we introduce the integer
+p := max {[[(zM − zL)/(∆x)α]] + 1, 2} ,
+where zL = z(uL), zM = z(uM) and [[x]] is the greatest integer not greater than x.
+Notice that
+p = O((∆x)−α).
+(2.10)
+Define
+z∗
+1 := zL, z∗
+p := zM, w∗
+i := wL (i = 1, . . . , p),
+and
+z∗
+i := zL + (i − 1)(∆x)α (i = 1, . . . , p − 1).
+We next introduce the rays x = (j + 1/2)∆x + λ1(z∗
+i , z∗
+i+1, wL)(t − n∆t) separating
+finite constant states (z∗
+i , w∗
+i ) (i = 1, . . . , p), where
+λ1(z∗
+i , z∗
+i+1, wL) := v(z∗
+i , wL) − S(ρ(z∗
+i+1, wL), ρ(z∗
+i , wL)),
+
+THE COMPRESSIBLE EULER EQUATIONS
+11
+ρ∗
+i := ρ(z∗
+i , wL) :=
+�θ(wL − z∗
+i )
+2
+�1/θ
+,
+v∗
+i := v(z∗
+i , wL) := wL + z∗
+i
+2
+and
+S(ρ, ρ0) :=
+
+
+
+
+
+�
+ρ(p(ρ) − p(ρ0))
+ρ0(ρ − ρ0)
+,
+if ρ ̸= ρ0,
+�
+p′(ρ0),
+if ρ = ρ0.
+(2.11)
+We call this approximated 1-rarefaction wave a 1-rarefaction fan.
+Step 2.
+In this step, we replace the above constant states with functions of x and t as
+follows:
+In view of (1.9), we construct u∆
+1 (x, t).
+We first determine the approximation of ˜z, ˜w in (1.9) as follows.
+˜z∆
+1 =zL −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx, ˜w∆
+1 = wL −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx,
+where u∆
+n,0(x) is a piecewise constant function defined by
+u∆
+n,0(x) = un
+j ,
+x ∈ [xj−1, xj+1)
+(j ∈ 2Z).
+(2.12)
+We set
+ˇz∆
+1 (x, t) = ˜z∆
+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx +
+� x
+x∆
+1
+J(uL)dy
++ {g1(x, t; uL) + V (uL)} (t − tn),
+ˇw∆
+1 (x, t) = ˜w∆
+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx +
+� x
+x∆
+1
+J(uL)dy
++ {g2(x, t; uL) + V (uL)} (t − tn),
+(2.13)
+where g1 and g2 are defined in (1.16), x∆
+1 = xj−1 and
+V (u) = q∗(u) − (¯ρ)γ−1
+γ − 1 m.
+(2.14)
+From (2.13), we determine ˇu∆
+1 (x, t) by the relation (1.5), that is,
+ˇu∆
+1 (x, t) = (ˇρ∆
+1 (x, t), ˇm∆
+1 (x, t)) = (ˇρ∆
+1 (x, t), ˇρ∆
+1 (x, t)ˇv∆
+1 (x, t)),
+where
+ˇρ∆
+1 (x, t) =
+�
+θ
+�
+ˇw∆
+1 (x, t) − ˇz∆
+1 (x, t)
+�
+2
+� 1
+θ
+,
+ˇv∆
+1 (x, t) = ˇw∆
+1 (x, t) + ˇz∆
+1 (x, t)
+2
+.
+
+12
+NAOKI TSUGE
+Using ˇu∆
+1 (x, t), we next define u∆
+1 (x, t) as follows.
+z∆
+1 (x, t) = ˜z∆
+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx +
+� x
+x∆
+1
+J(ˇu∆
+1 (y, t))dy
++
+�
+g1(x, t; ˇu∆
+1 ) + V (uL)
+�
+(t − tn),
+w∆
+1 (x, t) = ˜w∆
+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx +
+� x
+x∆
+1
+J(ˇu∆
+1 (y, t))dy
++
+�
+g2(x, t; ˇu∆
+1 ) + V (uL)
+�
+(t − tn).
+(2.15)
+From (2.15), we determine u∆
+1 (x, t) by the relation (1.5).
+Remark 2.2.
+(i) We notice that approximate solutions z∆
+1 , w∆
+1 and ˜z∆
+1 , ˜w∆
+1 correspond to
+z, w and ˜z, ˜w in (1.9), respectively.
+(ii) For tn < t < tn+1, our approximate solutions will satisfy
+� xj−1
+−∞
+J(u∆(x, tn+1−))dx +
+� tn+1
+tn
+�
+x≤xj−1
+(σ[η∗] − [q∗])dt
+=
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)∆t + o(∆x).
+(2.16)
+In (2.15), we thus employ the right hand side of (2.16) instead of the left
+hand side.
+(iii) Our construction of approximate solutions uses the iteration method twice
+(see (2.13) and (2.15)) to deduce (3.12).
+First, by the implicit function theorem, we determine a propagation speed σ2
+and u2 = (ρ2, m2) such that
+(1.a) z2 := z(u2) = z∗
+2
+(1.b) the speed σ2, the left state u∆
+1 (x2, tn.5) and the right state u2 satisfy the
+Rankine–Hugoniot conditions, i.e.,
+f(u2) − f(u∆
+1 (x∆
+2 (tn.5), tn.5)) = σ2(u2 − u∆
+1 (x∆
+2 (tn.5), tn.5)),
+where x∆
+2 (t) = xj + σ2(t − tn). Then we fill up by u∆
+1 (x) the sector where tn ≤ t <
+tn+1, xj−1 ≤ x < x∆
+2 (t) (see Figure 1).
+Assume that uk, u∆
+k (x, t), a propagation speed σk and x∆
+k (t) are defined. Then
+we similarly determine σk+1 and uk+1 = (ρk+1, mk+1) such that
+(k.a) zk+1 := z(uk+1) = z∗
+k+1,
+(k.b) σk < σk+1,
+(k.c) the speed σk+1, the left state u∆
+k (x∆
+k+1(tn.5), tn.5) and the right state uk+1
+satisfy the Rankine–Hugoniot conditions,
+where x∆
+k+1(t) = xj + σk+1(t − tn). Then we fill up by u∆
+k (x, t) the sector where
+tn ≤ t < tn+1, x∆
+k (t) ≤ x < x∆
+k+1(t) (see Figure 1).
+We construct u∆
+k+1(x, t) as follows.
+
+THE COMPRESSIBLE EULER EQUATIONS
+13
+Figure 1. The approximate solution in the case where a 1-
+rarefaction and a 2-shock arise in the cell.
+We first determine
+˜z∆
+k+1 =zk+1 −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx − V (uL)∆t
+2 −
+k
+�
+l=1
+� x∆
+l+1(tn.5)
+x∆
+l (tn.5)
+J(u∆
+l (x, tn.5))dx,
+˜w∆
+k+1 =wk+1 −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx − V (uL)∆t
+2 −
+k
+�
+l=1
+� x∆
+l+1(tn.5)
+x∆
+l (tn.5)
+J(u∆
+l (x, tn.5))dx,
+where x∆
+1 (t) = xj−1, x∆
+l (t) = xj +σl(t−tn)
+(l = 2, 3, . . . , k+1) and tn.5 is defined
+in (2.2).
+We next define ˇu∆
+k+1 as follows.
+ˇz∆
+k+1(x, t) =˜z∆
+k+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+k
+�
+l=1
+� x∆
+l+1(t)
+x∆
+l (t)
+J(u∆
+l (x, t))dx
++
+� x
+x∆
+k+1(t)
+J(uk+1)dy + g1(x, t; uk+1)(t − tn.5) +
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds,
+ˇw∆
+k+1(x, t) = ˜w∆
+k+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+k
+�
+l=1
+� x∆
+l+1(t)
+x∆
+l (t)
+J(u∆
+l (x, t))dx
++
+� x
+x∆
+k+1(t)
+J(uk+1)dy + g2(x, t; uk+1)(t − tn.5) +
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds.
+From the above, we determine ˇu∆
+k+1(x, t) by the relation (1.5).
+Finally, using ˇu∆
+k+1(x, t), we define u∆
+k+1(x, t) as follows.
+
+()m(cf)mg(f)()(°)(°)十-JQSQ3b
+QQ
+G14
+NAOKI TSUGE
+z∆
+k+1(x, t) =˜z∆
+k+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+k
+�
+l=1
+� x∆
+l+1(t)
+x∆
+l (t)
+J(u∆
+l (x, t))dx
++
+� x
+x∆
+k+1(t)
+J(ˇu∆
+k+1(y, t))dy + g1(x, t; ˇu∆
+k+1)(t − tn.5)
++
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds,
+w∆
+k+1(x, t)= ˜w∆
+k+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+k
+�
+l=1
+� x∆
+l+1(t)
+x∆
+l (t)
+J(u∆
+l (x, t))dx
++
+� x
+x∆
+k+1(t)
+J(ˇu∆
+k+1(y, t))dy + g2(x, t; ˇu∆
+k+1)(t − tn.5)
++
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds.
+(2.17)
+From (2.17), we determine u∆
+k+1(x, t) by the relation (1.5).
+By induction, we define ui, u∆
+i (x, t) and σi (i = 1, . . . , p − 1).
+Finally, we
+determine a propagation speed σp and up = (ρp, mp) such that
+(p.a) zp := z(up) = z∗
+p,
+(p.b) the speed σp, and the left state u∆
+p−1(x∆
+p (tn.5), tn.5) and the right state up
+satisfy the Rankine–Hugoniot conditions,
+where x∆
+p (t) = xj + σp(t − tn). We then fill up by u∆
+p−1(x, t) and up the sector
+where tn ≤ t < tn+1, x∆
+p−1(t) ≤ x < x∆
+p (t) and the line tn ≤ t < tn+1, x = x∆
+p (t),
+respectively.
+Given uL and zM with zL ≤ zM, we denote this piecewise functions of x and t
+1-rarefaction wave by R∆
+1 (uL, zM, x, t).
+On the other hand, we construct u∆
+R(x, t) as follows.
+We first set
+˜z∆
+R = zR −
+� xj+1
+−∞
+J(u∆
+n,0(x))dx, ˜w∆
+R = wR −
+� xj+1
+−∞
+J(u∆
+n,0(x))dx.
+We next construct ˇu∆
+R
+ˇz∆
+R (x, t)=˜z∆
+R +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn)
++
+� x
+xj+1
+J(uR)dy + g1(x, t; uR)(t − tn),
+ˇw∆
+R (x, t)= ˜w∆
+R +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn)
++
+� x
+xj+1
+J(uR)dy + g2(x, t; uR)(t − tn).
+From the above, we determine ˇu∆
+R(x, t) by the relation (1.5).
+
+THE COMPRESSIBLE EULER EQUATIONS
+15
+Using ˇu∆
+R(x, t), we define u∆
+R(x, t) as follows.
+z∆
+R (x, t) =˜z∆
+R +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn)
++
+� x
+xj+1
+J(ˇuR(y, t))dy + g1(x, t; ˇuR)(t − tn),
+w∆
+R (x, t) = ˜w∆
+R +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn)
++
+� x
+xj+1
+J(ˇuR(y, t))dy + g2(x, t; ˇuR)(t − tn).
+(2.18)
+From (2.18), we determine u∆
+R(x, t) by the relation (1.5).
+Now we fix u∆
+R(x, t) and u∆
+p−1(x, t). Let σs be the propagation speed of the 2-
+shock connecting uM and uR. Choosing σ⋄
+p near to σp, σ⋄
+s near to σs and u⋄
+M near to
+uM, we fill up by u∆
+M(x, t) the gap between x = xj+σ⋄
+p(t−tn) and x = xj+σ⋄
+s(t−tn),
+such that
+(M.a) σp−1 < σ⋄
+p < σ⋄
+s,
+(M.b) the speed σ⋄
+p, the left and right states u∆
+p−1(x⋄
+p, tn.5), u∆
+M(x⋄
+p, tn.5) satisfy
+the Rankine–Hugoniot conditions,
+(M.c) the speed σ⋄
+s, the left and right states u∆
+M(x⋄
+s, tn.5), u∆
+R(x⋄
+s, tn.5) satisfy the
+Rankine–Hugoniot conditions,
+where x⋄
+p := xj + σ⋄
+p∆/2, x⋄
+s := xj + σ⋄
+s∆/2 and u∆
+M(x, t) defined as follows.
+We first set
+˜z∆
+M =z⋄
+M −
+� xj+1
+−∞
+J(u∆
+n,0 (x))dx − V (uR)∆t
+2 −
+� x∆
+R (tn.5)
+xj+1
+J(u∆
+R(x, tn.5))dx,
+˜w∆
+M =w⋄
+M −
+� xj+1
+−∞
+J(u∆
+n,0 (x))dx − V (uR)∆t
+2 −
+� x∆
+R (tn.5)
+xj+1
+J(u∆
+R(x, tn.5))dx,
+where x∆
+R(t) = xj + σ⋄
+s(t − tn).
+We construct ˇu∆
+M
+ˇz∆
+M(x, t) =˜z∆
+M +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn) +
+� x∆
+R (t)
+xj+1
+J(u∆
+R(x, t))dy
++
+� x
+x∆
+R (t)
+J(uM)dy + g1(x, t; uM)(t − tn.5)
+−
+� t
+tn.5
+�
+x≤y≤xj+1
+(σ[η∗] − [q∗])ds,
+ˇw∆
+M(x, t) = ˜w∆
+M +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn) +
+� x∆
+R (t)
+xj+1
+J(u∆
+R(x, t))dy
++
+� x
+x∆
+R (t)
+J(uM)dy + g2(x, t; uM)(t − tn.5)
+−
+� t
+tn.5
+�
+x≤y≤xj+1
+(σ[η∗] − [q∗])ds.
+
+16
+NAOKI TSUGE
+From the above, we determine ˇu∆
+M(x, t) by the relation (1.5).
+Using ˇu∆
+M(x, t), we next define u∆
+M(x, t) as follows.
+z∆
+M(x, t) =˜z∆
+M +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn) +
+� x∆
+R (t)
+xj+1
+J(u∆
+R(x, t))dy
++
+� x
+x∆
+R (t)
+J(ˇu∆
+M(y, t))dy + g1(x, t; ˇu∆
+M)(t − tn.5)
+−
+� t
+tn.5
+�
+x≤y≤xj+1
+(σ[η∗] − [q∗])ds,
+w∆
+M(x, t) = ˜w∆
+M +
+� xj+1
+−∞
+J(u∆
+n,0(x))dx + V (uR)(t − tn) +
+� x∆
+R (t)
+xj+1
+J(u∆
+R(x, t))dy
++
+� x
+x∆
+R (t)
+J(ˇu∆
+M(y, t))dy + g2(x, t; ˇu∆
+M)(t − tn.5)
+−
+� t
+tn.5
+�
+x≤y≤xj+1
+(σ[η∗] − [q∗])ds.
+(2.19)
+From (2.19), we determine u∆
+M(x, t) by the relation (1.5).
+We denote this approximate Riemann solution, which consists of (2.17), (2.18),
+(2.18) , by u∆(x, t). The validity of the above construction is demonstrated in [11,
+Appendix A].
+Remark 2.3. u∆(x, t) satisfies the Rankine–Hugoniot conditions at the middle
+time of the cell, t = tn.5.
+Remark 2.4. The approximate solution u∆(x, t) is piecewise smooth in each of the
+divided parts of the cell. Then, in the divided part, u∆(x, t) satisfies
+(u∆)t + f(u∆)x − g(x, u∆) = o(1).
+To deduce that Ln is uniformly bounded, we prove the following lemma.
+Lemma 2.1.
+0 ≤
+n
+�
+k=0
+� ∞
+−∞
+�
+η∗(u∆(x, tk−0)) − η∗(Ek(x; u))
+�
+dx +
+� tn
+0
+�
+x∈R
+(σ[η∗] − [q∗])dt
+(2.20)
+=
+n
+�
+k=0
+�
+j∈2Z
+� xj+1
+xj−1
+Rn
+j (x)dx +
+� tn
+0
+�
+x∈R
+(σ[η∗] − [q∗])dt + o(∆x)
+(2.21)
+=
+� ∞
+−∞
+�
+J
+�
+u∆(x, t0−)
+�
+− J
+�
+u∆(x, tn+)
+��
+dx + o(∆x)
+(2.22)
+≤
+� ∞
+−∞
+J(u0(x))dx + o(∆x).
+(2.23)
+where o(∆x) depends only on M0, E0 and T .
+Ln ≤ C,
+(2.24)
+where C depends only on initial data.
+
+THE COMPRESSIBLE EULER EQUATIONS
+17
+Proof. We recall that our approximate solutions are constructed in [0, T ] for any
+fixed positive constant T . From (1.3) and the finite propagation, we find that our
+approximate solutions are (¯ρ, 0) outside a finite interval.
+First, from the Jensen inequality and the entropy condition, we obtain (2.20).
+Second, from (2.4), we have (2.21).
+Finally, we consider (2.22). From the similar argument to [11, (6.10)], taking
+J(u) as η in [11], we have
+n
+�
+k=0
+� ∞
+−∞
+�
+η∗(u∆(x, tk−0)) − η∗(u∆
+n,0(x))
+�
+dx +
+� tn
+0
+�
+x∈R
+(σ[η∗] − [q∗])dt
+=
+� ∞
+−∞
+�
+J
+�
+u∆(x, t0−)
+�
+− J
+�
+u∆(x, tn+)
+��
+dx + o(∆x).
+On the other hand, (2.9) and Theorem 3.2, we find that un
+j = En
+j (u) + o(∆x).
+Recalling (2.3) and (2.12), we have
+n
+�
+k=0
+� ∞
+−∞
+�
+η∗(u∆(x, tk−0)) − η∗(u∆
+n,0(x))
+�
+dx
+=
+n
+�
+k=0
+� ∞
+−∞
+�
+η∗(u∆(x, tk−0)) − η∗(Ek(x; u))
+�
+dx + o(∆x).
+Finally, observing Rn
+j (x) ≥ 0, from (2.5) and (2.23), we have (2.24).
+□
+3. The L∞ estimate of the approximate solutions
+The aim in this section is to deduce from (2.9) the following theorem:
+Theorem 3.1. For j ∈ 2Z≥0, n ∈ Z≥0 and xj−1 ≤ x < xj+1,
+z∆(x, tn+1−) ≥ − Mn+1 − E0 − Ln +
+� x
+−∞
+J(u∆(y, tn+1−))dy − o(∆x),
+w∆(x, tn+1−)≤Mn+1 + Ln +
+� x
+−∞
+J(u∆(y, tn+1−))dy
++
+� tn+1
+tn
+�
+x<xj+1
+(σ[η∗] − [q∗])dt + o(∆x),
+(3.1)
+where tn− = n∆t − 0, Mn+1 is defined in (2.6), o(∆x) depends only on M0 and
+E0, ε and δ are found in (1.25).
+Theorem 3.2. We assume that u∆(x, t) satisfies (1.10) and (3.1).
+Then, if
+En
+j (ρ) ≥ (∆x)µ, it holds that
+− Mn − E0 − Ln + In
+j − o(∆x) ≤ z(En
+j (u)),
+w(En
+j (u)) ≤ Mn + Ln + In
+j + o(∆x),
+(3.2)
+where j ∈ 2Z and o(∆x) depends only on M0.
+(3.2) is needed to ensures (2.9).
+Throughout this paper, by the Landau symbols such as O(∆x), O((∆x)2) and
+o(∆x), we denote quantities whose moduli satisfy a uniform bound depending only
+on M0 and E0 unless we specify them.
+
+18
+NAOKI TSUGE
+Now, in the previous section, we have constructed u∆(x, t) in Case 1. When
+we consider L∞ estimates in this case, main difficulty is to obtain (3.1)2 along R∆
+1 .
+Therefore, we are concerned with (3.1)2 along R∆
+1 .
+3.1. Proof of Theorem 3.2. We first observe Theorem 3.2. For x ∈ [xj−1, xj+1],
+we set
+z∆
+† (x, tn−) =z∆(x, tn−) −
+� x
+−∞
+J
+�
+u∆(y, tn−)
+�
+dy +
+� x
+xj−1
+η∗
+�
+u∆(y, tn−)
+�
+dy
+−
+� x
+xj−1
+an
+j ρ∆(y, tn−)dy +
+� x
+xj−1
+(¯ρ)γ
+γ dy,
+w∆
+† (x, tn−) =w∆(x, tn−) −
+� x
+−∞
+J
+�
+u∆(y, tn−)
+�
+dy +
+� x
+xj−1
+η∗
+�
+u∆(y, tn−)
+�
+dy
+−
+� x
+xj−1
+an
+j ρ∆(y, tn−)dy +
+� x
+xj−1
+(¯ρ)γ
+γ dy,
+where
+an
+j = ∂η∗
+∂ρ (En
+j (u)) + ∂η∗
+∂m (En
+j (u))
+�
+En
+j (v) −
+�
+En
+j (ρ)
+�θ�
+,
+En
+j (v) := En
+j (m)
+En
+j (ρ) . (3.3)
+Then, by the relation (1.5), we define ρ∆
+† (x, tn−) and v∆
+† (x, tn−). We notice that
+ρ∆
+† (x, tn−) =ρ∆(x, tn−),
+v∆
+† (x, tn−) =v∆(x, tn−) −
+� x
+−∞
+J
+�
+u∆(y, tn−)
+�
+dy +
+� x
+xj−1
+η∗
+�
+u∆(y, tn−)
+�
+dy
+−
+� x
+xj−1
+an
+j ρ∆(y, tn−)dy +
+� x
+xj−1
+(¯ρ)γ
+γ dy.
+Since Ln is positive, (3.2)2 is more difficult than (3.2)1. We thus treat with only
+(3.2)2 in this proof.
+w(En
+j (u)) =
+1
+2∆x
+� xj+1
+xj−1
+m∆(x, tn−)dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx
+�θ
+/θ
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx
+=
+1
+2∆x
+� xj+1
+xj−1
+m∆
+† (x, tn−)dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ρ∆
+† (x, tn−)dx
+�θ
+/θ
+1
+2∆x
+� xj+1
+xj−1
+ρ∆
+† (x, tn−)dx
++
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)
+�� xj−1
+−∞
+η∗
+�
+u∆(y, tn−)
+�
+dy
+�
+dx
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx
+
+THE COMPRESSIBLE EULER EQUATIONS
+19
+−
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)
+��
+(¯ρ)γ−1
+γ − 1 − an
+j
+� � x
+xj−1
+ρ∆(y, tn−)dy
+�
+dx
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx
+− (¯ρ)γ−1
+γ − 1
+� xj−1
+−∞
+ρ∆(x, tn−)dx +
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)
+�� xj−1
+−∞
+(¯ρ)γ
+γ dy
+�
+dx
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx
+=A1 + A2 − A3 + A4.
+We denote the numerator of A3 by A31. From the integration by parts, we have
+A31=
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx ×
+�
+(¯ρ)γ−1
+γ − 1 ρ − an
+j
+� � xj+1
+xj−1
+ρ∆(y, tn−)dx
+−
+1
+2∆x
+� xj+1
+xj−1
+�� x
+xj−1
+ρ∆(y, tn−)dy
+� �
+(¯ρ)γ−1
+γ − 1 ρ − an
+j
+�
+ρ∆(x, tn−)dx
+=
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx ×
+�
+(¯ρ)γ−1
+γ − 1 ρ − an
+j
+� � xj+1
+xj−1
+ρ∆(x, tn−)dx − A31.
+We thus obtain
+A31 =1
+2 ×
+1
+2∆x
+� xj+1
+xj−1
+ρ∆(x, tn−)dx ×
+�
+(¯ρ)γ−1
+γ − 1 ρ − an
+j
+� � xj+1
+xj−1
+ρ∆(y, tn−)dx.
+Therefore, we obtain
+w(En
+j (u)) =
+1
+2∆x
+� xj+1
+xj−1
+m∆
+† (x, tn−) +
+�
+1
+2∆x
+� xj+1
+xj−1
+ρ∆
+† (x, tn−)dx
+�θ
+/θ
+1
+2∆x
+� xj+1
+xj−1
+ρ∆
+† (x, tn−)dx
++
+� xj−1
+−∞
+J
+�
+u∆(x, tn−)
+�
+dx −
+(¯ρ)γ−1
+γ−1 − an
+j
+2
+� xj+1
+xj−1
+ρ∆(x, tn−)dx.
+(3.4)
+Here we introduce the following lemma. The proof is postponed to Appendix A.
+Lemma 3.3. If
+1
+2∆x
+� xj+1
+xj−1
+ρ∆
+† (x, tn−)dx ≥ (∆x)µ
+(3.5)
+and
+w∆
+† (x, tn−) ≤Mn + Ln−1 +
+� x
+xj−1
+η∗
+�
+u∆(y, tn−)
+�
+dy −
+� x
+xj−1
+an
+j ρ∆(y, tn−)dy
++
+� x
+xj−1
+(¯ρ)γ
+γ dy +
+� tn
+tn−1
+�
+x<xj−1
+(σ[η∗] − [q∗])dt + o(∆x)
+= : A(x, tn−) + o(∆x)
+(x ∈ [xj−1, xj+1]),
+(3.6)
+
+20
+NAOKI TSUGE
+the following holds
+w(En
+j (u∆
+† )) ≤ ¯Aj(tn−) + o(∆x),
+where En
+j (u∆
+† ) =
+1
+2∆x
+� xj+1
+xj−1
+u∆
+† (x, tn−)dx,
+¯Aj(tn−) =
+1
+2∆x
+� xj+1
+xj−1
+A(x, tn−)dx,
+the definition of µ is found in (2.7).
+It follows from (3.1) and this lemma that
+w(En
+j (u)) ≤Mn + Ln−1 + In
+j +
+� tn
+tn
+�
+y<xj−1
+(σ[η∗] − [q∗])dt
++
+� xj−1
+−∞
+�
+η∗
+�
+u∆(x, tn−)
+�
+− η∗
+�
+u∆
+n,0(x)
+��
+dx
++
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+�
+η∗
+�
+u∆(y, tn−)
+�
+− η∗
+�
+En
+j (u)
+��
+dydx
+−
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+an
+j
+�
+ρ∆(y, tn−) − En
+j (ρ)
+�
+dydx + o(∆x).
+(3.7)
+To complete the proof of Theorem (3.2), we must investigate
+Γn
+j (y) =η∗(u∆(y, tn−)) − η∗(En
+j (u)) − an
+j
+�
+ρ∆(y, tn−) − En
+j (ρ)
+�
+in (3.7), where an
+j is defined in (3.3).
+We then deduce from (2.4) that
+Γn
+j (y) =∂η∗
+∂m (En
+j (u))ρ∆(y, tn−)
+�
+w(y, tn−) − w(En
+j (u))
+�
+− ∂η∗
+∂m (En
+j (u))
+� 1
+0
+(1 − τ)(θ + 1)
+�
+En
+j (ρ) + τ
+�
+ρ∆(y, tn−) − En
+j (ρ)
+��θ−1 dτ
+×
+�
+ρ∆(y, tn−) − En
+j (ρ)
+�2 + Rn
+j (y)
+=En
+j (v)ρ∆(y, tn−)
+�
+w(y, tn−) − w(En
+j (u)
+�
+− En
+j (v)
+2
+� 1
+0
+(1 − τ)(θ + 1)
+�
+En
+j (ρ) + τ
+�
+ρ∆(y, tn−) − En
+j (ρ)
+��θ−1 dτ
+×
+�
+ρ∆(y, tn−) − En
+j (ρ)
+�2 + Rn
+j (y).
+We thus obtain
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+Γn
+j (y)dydx
+=
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+En
+j (v)ρ∆(y, tn−)
+�
+w(y, tn−) − w(En
+j (u))
+�
+dydx
+−
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+En
+j (v)
+2
+� 1
+0
+(1 − τ)(θ + 1)
+�
+En
+j (ρ) + τ
+�
+ρ∆(y, tn−) − En
+j (ρ)
+��θ−1 dτ
+×
+�
+ρ∆(y, tn−) − En
+j (ρ)
+�2 dydx +
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+Rn
+j (y)dydx
+= : B1 + B2 + B3.
+(3.8)
+
+THE COMPRESSIBLE EULER EQUATIONS
+21
+If En
+j (ρ) < (∆x)µ, we find B1 = o(∆x) and B2 = o(∆x). Therefore, we devote
+to investigating the case where En
+j (ρ) ≥ (∆x)µ.
+If w(En
+j (u)) ≤ Mn + Ln + In
+j , (3.2) clearly holds.
+Otherwise, we consider the following lemma.
+Lemma 3.4. If
+w(En
+j (u)) > Mn + Ln + In
+j ,
+(3.9)
+the following holds.
+1
+2∆x
+� xj+1
+xj−1
+� x
+xj−1
+Γn
+j (y)dydx ≤
+� xj+1
+xj−1
+Rn
+j (x)dx + o(∆x).
+Proof. From Theorem 3.1, we have z(En
+j (u)) ≥ −Mn − Ln + In
+j − O(∆x). From
+(3.9), we find En
+j (v) ≥ In
+j − O (∆x) (recall the definition of En
+j (v) in (3.3) ). If
+En
+j (v) < 0, since In
+j − O (∆x) ≤ En
+j (v) ≤ 0, we have
+−En
+j (v) ≤ O (∆x) .
+(3.10)
+We first treat with B1 in (3.8). If En
+j (v) ≥ 0, we deduce from Theorem 3.1 and
+(3.9) that
+B1 ≤
+� xj+1
+xj−1
+En
+j (v)ρ∆(x, tn−)
+�
+w(x, tn−) − wn
+j
+�
+dx = o(∆x).
+If En
+j (v) < 0, from (3.10), we have B1 = o(∆x).
+We next consider B2. If En
+j (v) ≥ 0, we find that B2 ≤ 0. If En
+j (v) < 0, from
+(3.10), we have B2 = o(∆x).
+□
+From Lemma 3.4, we can complete the proof of (3.2).
+3.2. Proof of Theorem 3.1. We next prove Theorem 3.1.
+Estimates of w∆(x, t) along R∆
+1 in Case 1 In this step, we estimate w∆(x, t)
+along R∆
+1 in Case 1 of Section 2. We recall that u∆ along R∆
+1 consists of u∆
+k
+(k =
+1, 2, 3, . . ., p−1). In this case, w∆(x, t) has the following properties, which is proved
+in [11, Appendix A]:
+w∆
+k+1(x∆
+k+1(tn.5), tn.5) =wk+1 = w∆
+k (x∆
+k+1(tn.5), tn.5) + O((∆x)3α−(γ−1)β)
+(k = 1, . . . , p − 2),
+(3.11)
+where tn.5 is defined in (2.2).
+We first consider ˜w∆
+1 . We recall that
+˜w∆
+1 = wL −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx.
+From (2.9), we have ˜w∆
+1 ≤ Mn + Ln.
+Since
+ˇu∆
+1 (x, t) = u∆
+1 (x, t) + O((∆x)2),
+(3.12)
+
+22
+NAOKI TSUGE
+we have
+w∆
+1 (x, t)= ˜w∆
+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+� x
+x∆
+1
+J(ˇu∆
+1 (y, t))dy
++ g2(x, t; ˇu∆)(t − tn)
+≤Mn + Ln +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+� x
+x∆
+1
+J(u∆
+1 (y, t))dy
++ g2(x, t; u∆)(t − tn) + o(∆x).
+On the other hand, from the construction of our approximate solutions, we ob-
+serve that w∆
+1 (x, t) = w∆
+1 (x, tn−) + O(∆x)
+(xj−1 ≤ x < xj+1, tn ≤ t < tn+1).
+Separating three cases, we prove (3.1)2.
+(i) If w∆
+1 (x, tn−) < Mn + Ln + In
+j −
+√
+∆x, we obtain (3.1)2.
+(ii) If w∆
+1 (x, tn−) ≥ Mn + Ln + In
+j −
+√
+∆x and Mn + Ln ≥ (¯ρ)θ
+θ
++ ε, we
+observe that w∆
+1 (x, t) ≥ w∆
+1 (x, tn−) − O(
+√
+∆x) ≥ Mn + Ln − O(
+√
+∆x) ≥
+(¯ρ)θ
+θ
++ ε − O(
+√
+∆x) ≥ (¯ρ)θ
+θ
++ ε/2, by choosing ∆x small enough. From
+(1.25), we obtain g2(x, t; u∆
+1 ) < −2δ. From (2.16), we obtain (3.1)2.
+(iii) If w∆
+1 (x, tn−) ≥ Mn +Ln +In
+j −
+√
+∆x and Mn +Ln < (¯ρ)θ
+θ
++ε, from (2.6),
+we find that (¯ρ)θ
+θ
++ ε − δ∆t ≤ Mn + Ln. Therefore, we have w∆
+1 (x, t) ≥
+w∆
+1 (x, tn−)−O(
+√
+∆x) ≥ Mn+Ln−O(
+√
+∆x) ≥ (¯ρ)θ
+θ +ε−O(
+√
+∆x) ≥ (¯ρ)θ
+θ +
+ε/2, by choosing ∆x small enough. From (1.25), we obtain g2(x, t; u∆
+1 ) <
+−2δ < 0. Recalling (2.6), from (2.16), we obtain (3.1)2.
+Next, we assume that
+w∆
+k (x, t)≤Mn + Ln +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn)
++
+� x
+xj−1
+J(u∆(y, t))dy +
+� t
+tn.5
+�
+xj−1≤y<x
+(σ[η∗] − [q∗])dt
++ (k − 1) · O((∆x)3α−(γ−1)β) + o(∆x)
+(3.13)
+for (x, t) ∈ [xj−1, x∆
+k+1(t)) × [tn, tn+1).
+We recall that
+˜w∆
+k+1 =wk+1 −
+� xj−1
+−∞
+J(u∆
+n,0(x))dx − V (uL)∆t
+2 −
+k
+�
+l=1
+� x∆
+l+1(tn.5)
+x∆
+l (tn.5)
+J(u∆
+l (x, tn.5))dx
+and u∆(x, t) consists u∆
+l (x, t) in x∆
+l (t) ≤ x < x∆
+l+1(t), tn ≤ t < tn+1
+(l =
+1, 2, 3, . . ., k + 1).
+From (3.11) and (3.13), we have
+˜w∆
+k+1 ≤Mn + Ln + k · O((∆x)3α−(γ−1)β) + o(∆x).
+
+THE COMPRESSIBLE EULER EQUATIONS
+23
+From a similar argument to w∆
+1 , we have
+w∆
+k+1(x, t)= ˜w∆
+k+1 +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+k
+�
+l=1
+� x∆
+l+1(t)
+x∆
+l (t)
+J(u∆
+l (x, t))dx
++
+� x
+x∆
+k+1(t)
+J(ˇu∆
+k+1(y, t))dy + g2(x, t; ˇu∆
+k+1)(t − tn.5)
++
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds
+≤Mn + Ln +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)(t − tn) +
+� x
+xj−1
+J(u∆(y, t))dy
++ g2(x, t; ˇu∆
+k+1)(t − tn.5) +
+� t
+tn.5
+�
+xj−1≤y≤x
+(σ[η∗] − [q∗])ds
++ k · O((∆x)3α−(γ−1)β) + o(∆x)
+(k = 1, 2, 3, . . ., p − 1).
+From (2.10) and (2.16), since {3α − (γ − 1)β} p > 1, we conclude (3.1)2.
+4. Proof of Theorem 1.1
+Our approximate solutions satisfy the following propositions holds (these proofs
+are similar to [11]–[13].).
+Proposition 4.1. The measure sequence
+η∗(u∆)t + q(u∆)x
+lies in a compact subset of H−1
+loc (Ω) for all weak entropy pair (η∗, q), where Ω ⊂
+[0, 1] × [0, 1] is any bounded and open set.
+Proposition 4.2. Assume that the approximate solutions u∆ are bounded and
+satisfy Proposition 4.1. Then there is a convergent subsequence u∆n(x, t) in the
+approximate solutions u∆(x, t) such that
+u∆n(x, t) → u(x, t)
+a.e.,
+as n → ∞.
+The function u(x, t) is a global entropy solution of the Cauchy problem (1.4).
+Moreover, from Theorem 3.1, the above solution satisfies (1.11). Therefore, we
+can prove Theorem 1.1.
+Appendix A. Proof of (1.21) and (1.22)
+A.1. Proof of (1.21). First, when 0 ≤ ρ ≤ ¯ρ, we will prove
+5γ − 3
+γ(γ − 1)2 ργ+θ − 2(3γ − 1)
+γ(γ − 1)2 (¯ρ)θ ργ +
+3 − γ
+(γ − 1)2 (¯ρ)γ−1 ρθ+1 −
+3 − γ
+γ(γ − 1) (¯ρ)γ ρθ
++
+2
+γ(γ − 1) (¯ρ)γ+θ ≥ 0.
+To this, setting t = ρ/¯ρ, we consider
+f(t) =(5γ − 3)t3θ+1 − 2(3γ − 1)t2θ+1 + γ(3 − γ)tθ+1 − (3 − γ)(γ − 1)tθ
++ 2(γ − 1),
+0 ≤ t ≤ 1.
+
+24
+NAOKI TSUGE
+Separating 3 steps, we will deduce that f(t) ≥ 0,
+0 ≤ t ≤ 1.
+Step 1
+First, we consider the neighborhood of t = 0. We set X = tθ. Solving two
+inequalities
+(5γ − 3)t3θ+1 − 2(3γ − 1)t2θ+1 + γ(3 − γ)tθ+1
+= tθ+1 �
+(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ)
+�
+≥ 0
+and
+−(3 − γ)(γ − 1)tθ + 2γ(γ − 1) = −(3 − γ)(γ − 1)X + 2(γ − 1) ≥ 0,
+we have 0 ≤ X ≤ ξ, where ξ is the smaller solution of (5γ − 3)X2 − 2(3γ − 1)X +
+γ(3 − γ) = 0. We notice that f(t) ≥ 0 in the interval 0 ≤ X ≤ ξ.
+Step 2
+Next, we consider the neighborhood of t = 1. We find that f(1) = f ′(1) = 0.
+On the other hand, from 0 ≤ t ≤ 1, γ > 1, we have
+4f ′′(t) =3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1
++ γ(γ − 1)(γ + 1)(3 − γ)tθ−1 + (γ − 1)2(3 − γ)2tθ−2
+≥3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1
++ (5γ − 3)(γ − 1)(3 − γ)tθ−1
+≥3(5γ − 3)(3γ − 1)(γ − 1)t3θ−1 − 8γ(γ − 1)(3γ − 1)t2θ−1
++ (3γ − 1)(γ − 1)(3 − γ)tθ−1
+=(3γ − 1)(γ − 1)tθ−1 �
+3(5γ − 3)X2 − 8γX + 3 − γ
+�
+.
+We thus find that f(t) ≥ 0 in the interval η ≤ X ≤ 1, where η is the larger solution
+of 3(5γ − 3)X2 − 8γX + 3 − γ = 0.
+Step 3
+Since ξ < η, from Step 1,2, it suffices to prove f(t) ≥ 0 in the interval ξ ≤ X ≤ η.
+Observing that (5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ) ≤ 0 in this interval, we have
+f(t) =tθ+1 �
+(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ)
+�
+− (3 − γ)(γ − 1)X + 2(γ − 1)
+≥X
+�
+(5γ − 3)X2 − 2(3γ − 1)X + γ(3 − γ)
+�
+− (3 − γ)(γ − 1)X + 2(γ − 1)
+=(5γ − 3)X3 − 2(3γ − 1)X2 + (3 − γ)X + 2(γ − 1) =: g(X).
+Let α, β (α < β) be tow solutions of g′(X) = 0. Then, we find that 0 < α < ξ <
+η < β < 1. Moreover, we deduce that g(η) > 0. Therefore, we can complete the
+proof.
+A.2. Proof of (1.22). Our goal in this appendix is to prove
+γ + 1
+2γ2(γ − 1)ργ+θ −
+1
+γ − 1 (¯ρ)γ−1 ρθ+1 + γ + 1
+γ2
+(¯ρ)γ ρθ −
+1
+2γ2 (¯ρ)2γ
+1
+ρθ+1 ≥ 0,
+where ρ ≥ ¯ρ. To do this, setting t = ρ/¯ρ, we prove
+g(t) =
+γ + 1
+2γ2(γ − 1)t2γ −
+1
+γ − 1tγ+1 + γ + 1
+γ2 tγ −
+1
+2γ2 ≥ 0,
+t ≥ 1.
+First, we observe that g(1) = g′(1) = g′′(1) = 0. In addition, we find that g′′′(t) ≥
+0,
+t ≥ 1. We thus conclude that g(t) ≥ 0.
+
+THE COMPRESSIBLE EULER EQUATIONS
+25
+Appendix B. Proof of Lemma 3.3
+Proof. Due to space limitations, we denote tn− by T in this section.
+Set
+ρ∆
+† (x, T ) := ˆρ(x, T ) {A(x, T )}
+2
+γ−1 ,
+m∆
+† (x, T ) := ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 ,
+En+1
+j
+(ρ∆
+† ) :=
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx,
+En+1
+j
+(m∆
+† ) :=
+1
+2∆x
+� xj+1
+xj−1
+ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 dx.
+Then, we find that
+w(ˆu(x, T )) ≤ 1 + o(∆x).
+(B.1)
+Let us prove
+w(En+1
+j
+(ρ∆
+† ), En+1
+j
+(m∆
+† )) ≤ ¯Aj(T ) + o(∆x),
+where
+¯Aj(T ) =
+1
+2∆x
+� xj+1
+xj−1
+A(x, T )dx
+and
+w(En+1
+j
+(ρ∆
+† ), En+1
+j
+(m∆
+† ))
+= En+1
+j
+(m∆
+† )/En+1
+j
+(ρ∆
+† ) + {En+1
+j
+(ρ∆
+† )}θ/θ
+=
+1
+2∆x
+� xj+1
+xj−1
+ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ+1
+/θ
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+.
+(B.2)
+Step 1.
+We find
+En+1
+j
+(ρ∆
+† ) =
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 {A(x, T )}−1 dx
+=
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 dx
++
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 ×
+�
+{A(x, T )}−1 −
+� ¯Aj(T )
+�−1�
+dx
+=
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 dx
+−
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx + o(∆x),
+where r(x, T ) = A(x, T ) − ¯Aj(T ). Recalling (3.6), we notice that r(x, T ) = O(∆x).
+
+26
+NAOKI TSUGE
+Substituting the above equation for (B.2), we obtain
+w(En+1
+j
+(ρ∆
+† ), En+1
+j
+(m∆
+† ))
+=
+1
+2∆x
+� xj+1
+xj−1
+ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ+1
+/θ
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 dx
++
+1
+2∆x
+� xj+1
+xj−1
+ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ+1
+/θ
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�2
+×
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx + o(∆x).
+(B.3)
+Set
+ω :=
+2
+γ + 1
+1
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ
+×
+1
+2∆x
+� xj+1
+xj−1
+ˆm(x, T ) {A(x, T )}
+γ+1
+γ−1 dx +
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ+1
+/θ
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+.
+(B.4)
+Then assume that the following holds.
+(En+1
+j
+(ρ∆
+† ))θ+1 ≤
+1
+2∆x
+� xj+1
+xj−1
+(ˆρ(x, T ))θ+1 {A(x, T )}
+γ+1
+γ−1 dx
+− γ + 1
+2
+ω
+� ¯Aj(T )
+�−1
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ
+×
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
+−
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+1
+2∆x
+� xj+1
+xj−1
+r(x, T )dx
+�
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx.
+(B.5)
+
+THE COMPRESSIBLE EULER EQUATIONS
+27
+This estimate shall be proved in step 2–4. Then, substituting (B.5) for (B.3),
+we deduce from (B.1) that
+w(En+1
+j
+(¯ρ), En+1
+j
+(m∆
+† )) ≤
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1
+�
+ˆv(x, T ) + {ˆρ(x, T )}θ
+θ
+�
+dx
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+γ+1
+γ−1 dx
++ o(∆x)
+≤ ¯Aj(T ) + o(∆x).
+Therefore we must prove (B.5). Separating three steps, we derive this estimate.
+Step 2.
+From (3.5), we notice that
+|ω| ≤ C(∆x)−θδ−ε,
+where C depends only on M.
+In this step, we consider the first equation of (B.3):
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ+1
+.
+Since θδ < 1/2, we first find
+En+1
+j
+(ρ∆
+† ) =
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 {A(x, T )}−ω dx
+=
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+− ω
+� ¯Aj(T )
+�−ω−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 r(x, T )dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+:=I0 − I1 + I2.
+We next estimate I1 as follows:
+I1 = ω
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx.
+
+28
+NAOKI TSUGE
+Therefore, we have
+En+1
+j
+(ρ∆
+† ) =
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 {A(x, T )}−ω dx
+=
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+− ω
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx.
+From the above, we deduce that
+(En+1
+j
+(ρ∆
+† ))θ+1 =
+�
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+−ω
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
+�θ+1
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+=
+�
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+�θ+1
++ (θ + 1)
+�
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+�θ
+× −ω
+� ¯Aj(T )
+�−1
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+=
+�
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+�θ+1
+− γ + 1
+2
+ω
+� ¯Aj(T )
+�−1
+�
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+�θ
+×
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 r(x, T )dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx.
+(B.6)
+
+THE COMPRESSIBLE EULER EQUATIONS
+29
+Step 3
+Applying the Jensen inequality to the first term of the right-hand of (B.6), we have
+�
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+�θ+1
+=
+
+
+
+
+
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+γ+1
+γ−1 ω dx
+
+
+
+
+
+θ+1
+×
+�
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+γ+1
+γ−1 ω dx
+�θ+1
+=
+
+
+
+
+
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω+
+2
+γ−1 dx
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+γ+1
+γ−1 ω dx
+
+
+
+
+
+θ+1
+×
+�
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+γ+1
+γ−1 ω dx
+�
+×
+�
+� ¯Aj(T )
+� γ+1
+2
+ω + γ + 1
+2
+ω
+� ¯Aj(T )
+� γ+1
+2
+ω−1
+1
+2∆x
+� xj+1
+xj−1
+r(x, T )dx + o(∆x)
+�
+=
+
+
+
+
+
+� ¯Aj(T )
+�−ω
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}ω− γ+1
+γ−1 ω+
+2
+γ−1 {A(x, T )}
+γ+1
+γ−1 ω dx
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+��+1
+γ−1 ω dx
+
+
+
+
+
+θ+1
+×
+�
+1
+2∆x
+� xj+1
+xj−1
+{A(x, T )}
+γ+1
+γ−1 ω dx
+�
+� ¯Aj(T )
+� γ+1
+2
+ω
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx
+≤
+1
+2∆x
+� xj+1
+xj−1
+(ˆρ(x, T ))θ+1 {A(x, T )}
+γ+1
+γ−1 dx
++ o(∆x)
+1
+2∆x
+� xj+1
+xj−1
+ˆρ(x, T ) {A(x, T )}
+2
+γ−1 dx.
+(B.7)
+From (B.6) and (B.7), we obtain (B.5) and complete the proof of lemma 3.3.
+□
+Appendix C. Construction and L∞ estimates of approximate
+solutions near the vacuum in Case 1
+In this step, we consider the case where ρM ≤ (∆x)β, which means that uM
+is near the vacuum. Since we cannot use the implicit function theorem, we must
+construct u∆(x, t) in a different way.
+Case 1 A 1-rarefaction wave and a 2-shock arise.
+
+30
+NAOKI TSUGE
+In this case, we notice that ρR ≤ (∆x)β, zR ≥ −Mn − Ln + In
+j and wR ≤
+Mn + Ln + In
+j .
+Case 1.1 ρL > (∆x)β
+We denote u(1)
+L
+a state satisfying w(u(1)
+L ) = w(uL) and ρ(1)
+L
+= (∆x)β. Let u(2)
+L
+be a state connected to u∆
+1 (xj−1, tn+1−) on the right by R∆
+1 (uL, z(1)
+L , x, tn+1−). We
+set
+(z(3)
+L , w(3)
+L ) =
+�
+(z(2)
+L , w(2)
+L ),
+if z(2)
+L
+≥ Dn
+j ,
+(Dn
+j , w(2)
+L ),
+if z(2)
+L
+< Dn
+j ,
+where
+Dn
+j = − Mn+1 − Ln +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)∆t +
+� xj+1
+xj−1
+(¯ρ)γ
+γ dx
++
+� xj+λ1(u(2)
+L )∆t
+xj−1
+η(R∆
+1 (uL, z(1)
+L , x, tn+1−))dx.
+Then, we define u∆(x, t) as follows.
+u∆(x, t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+R∆
+1 (uL, z(1)
+L , x, t),
+if xj−1 ≤ x ≤ xj + λ1(u(2)
+L )(t − tn)
+and tn ≤ t < tn+1,
+uRw(x, t),
+if xj + λ1(u(2)
+L )(t − tn)< x ≤ xj + λ2(uM, uR)(t − tn)
+and tn ≤ t < tn+1,
+u∆
+R(x, t) defined in (2.18),
+if xj + λ2(uM, uR)(t − tn)< x ≤ xj+1
+and tn ≤ t < tn+1,
+where (a) λ2(uM, uR) is a propagation speed of 2-shock wave; (b) uRw(x, t) is a
+rarefaction wave connecting u(3)
+L and u(4)
+L ; (c) u(4)
+L is defined by z(4)
+L
+= max{z(3)
+L , zM},
+w(4)
+L
+= w(3)
+L .
+Rarefaction wave
+Figure 2. Case 1.1: The approximate solution u∆ in the cell.
+
+()m(cf)mg(f)(°)(°)十-JQSQ3THE COMPRESSIBLE EULER EQUATIONS
+31
+Case 1.2 ρL ≤ (∆x)β
+We set (z(5)
+L , w(5)
+L ) = (max{zL, Dn
+j }, min{wL, U n
+j }), where
+U n
+j =Mn+1 + Ln +
+� xj−1
+−∞
+J(u∆
+n,0(x))dx + V (uL)∆t.
+Then, we define u∆(x, t) as follows.
+u∆(x, t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+u∆
+1 (x, t) defined in (2.15),
+if xj−1 ≤ x ≤ xj + λ1(uL)(t − tn)
+and tn ≤ t < tn+1,
+uRw(x, t),
+if xj + λ1(uL)(t − tn)< x ≤ xj + λ2(uM, uR)(t − tn)
+and tn ≤ t < tn+1,
+u∆
+R(x, t) defined in (2.18),
+if xj + λ2(uM, uR)(t − tn)< x ≤ xj+1
+and tn ≤ t < tn+1,
+where (a) uRw(x, t) is a rarefaction wave connecting u(5)
+L
+and u(6)
+L ; (b) u(6)
+L
+is defined
+by z(6)
+L
+= max{z(5)
+L , zM}, w(6)
+L
+= w(5)
+L .
+Remark C.1. We notice that ρ∆(x, t) = O((∆x)β) in (1.ii), (1.iii) and (2.i)–
+(2.iii). Therefore, the followings hold in these areas.
+Although (1.ii) and (2.ii) are solutions of homogeneous isentropic gas dynamics
+(i.e., g(x, t, u)) = 0), they is also a solution of (1.4) approximately
+(u∆)t + f(u∆)x − g(x, u∆) = −g(x, u∆) = O((∆x)β).
+In addition, discontinuities separating (1.i)–(1.iii) and (2.i)–(2.iii) satisfy [11,
+Lemma 5.3].
+C.1. L∞ estimates of approximate solutions. We consider Case 1.1 in partic-
+ular. It suffices to treat with uRw(x, t) in the region where xj + λ1(u(2)
+L )(t − tn) <
+x ≤ xj + λ2(uM, uR)(t − tn) and tn ≤ t < tn+1. The other cases are similar to
+Theorem 3.1.
+In this case, since ρ∆(x, t) = O((∆x)β), we have
+η∗(u∆(x, t)) = O((∆x)β).
+(C.1)
+Moreover, we notice that
+w∆(x, tn+1−) = w(2)
+L
+= w(R∆
+1 (uL, z(1)
+L , xj + λ1(u(2)
+L )∆t, tn+1−)).
+Applying Theorem 3.1 to R∆
+1 (uL, z(1)
+L , x, tn+1−), we drive
+w∆(x, tn+1−)≤Mn+1 + Ln +
+� xj+λ1(u(2)
+L )∆t
+−∞
+J(u∆(y, tn+1−))dy
++
+� tn+1
+tn
+�
+y<xj−1
+(σ[η∗] − [q∗])dt + o(∆x)
+≤Mn+1 + Ln +
+� x
+−∞
+J(u∆(y, tn+1−))dy +
+� tn+1
+tn
+�
+y<xj−1
+(σ[η∗] − [q∗])dt
++ o(∆x),
+
+32
+NAOKI TSUGE
+which means (3.1)2.
+Next, we notice that z∆(x, t) ≥ Dn
+j . In view of (2.16) and (C.1), we obtain
+(3.1)1.
+Acknowledgements.
+N. Tsuge’s research is partially supported by Grant-in-Aid for Scientific Research
+(C) 17K05315, Japan.
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+outer force. Nonlinear Anal. Real World Appl. 27, 203–220 (2016)
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+nozzle flow for large data: Application of the generalized invariant regions and the modified
+Godunov scheme. Nonlinear Anal. Real World Appl. 37, 217–238 (2017)
+[16] Tsuge, N.: Global entropy solutions to the compressible Euler equations in the isentropic noz-
+zle flow, Hyperbolic Problems: Theory, Numerics, Applications By Alberto Bressan, Marta
+Lewicka, Dehua Wang, Yuxi Zheng (Eds.), AIMS on Applied Mathematics 10, 666–673 (2020)
+[17] Tsuge, N.: Existence of a time periodic solution for the compressible Euler equation with a
+time periodic outer force. Nonlinear Anal. Real World Appl. 53, 103080 (2020)
+[18] N. Tsuge: Remarks on the energy inequality of a global L∞ solution to the compressible
+Euler equations for the isentropic nozzle flow. Commun. Math. Sci. to appear.
+Department of Mathematics Education, Faculty of Education, Gifu University, 1-1
+Yanagido, Gifu Gifu 501-1193 Japan.
+Email address: [email protected]
+

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+Towards Open World NeRF-Based SLAM
+Daniil Lisus1 and Connor Holmes1
+Abstract— Neural Radiance Fields (NeRFs) have taken the
+machine vision and robotics perception communities by storm
+and are starting to be applied in robotics applications. NeRFs
+offer versatility and robustness in map representations for
+Simultaneous Localization and Mapping. However, computa-
+tional difficulties of multilayer perceptrons (MLP) have lead to
+reductions in robustness in the state-of-the-art of NeRF-based
+SLAM algorithms in order to meet real-time requirements.
+In this report, we seek to improve accuracy and robustness
+of NICE-SLAM, a recent NeRF-based SLAM algorithm, by
+accounting for depth measurement uncertainty and using IMU
+measurements. Additionally, extend this algorithm by providing
+a model that can represent backgrounds that are too distant to
+be modeled by NeRF.
+I. INTRODUCTION
+Starting with the landmark paper by Mildenhall et al. just
+over two years ago [1], Neural Radiance Fields (NeRFs)
+have taken the machine vision and robotics perception
+communities by storm. The central idea behind NeRF is
+to combine classical graphics rendering techniques with a
+multilayer perceptron (MLP) trained on image data to learn
+an implicit representation of a given scene. The scene can
+then be rendered from novel viewpoints (i.e., view synthesis).
+Within the context of robotics, this approach holds promise to
+address shortcomings of classical dense SLAM algorithms.
+In particular, a NeRF is well suited to estimate portions of
+the map for unobserved regions [2]. Additionally, they can
+be leveraged to view maps from perspectives that may be
+interesting to users, but which have not been directly visited
+by the robot. Since they are fundamentally based on MLPs,
+NeRF maps can also be trained to be robust to changes in
+map environment conditions such as lighting [3] or time of
+the year.
+The original formulation of NeRF, which used one large
+MLP, required hours of training, was slow to render, and
+required exact knowledge of input camera poses. However, a
+flurry of advancements since have considerably improved on
+all of these issues. Several authors have shown that encoding
+the NeRF in a spatial data structure leads to considerable
+improvements in speed and accuracy, often using smaller
+MLPs to decoding spatial features during view synthesis [4]–
+[6]. In particular, NGLOD [4] proposed the use of small
+MLPs in a volumetric grid while Plenoxels [5] used a spatial
+octree and bypassed MLP use altogether. These ideas were
+further improved upon in [6] by using spatial hash tables
+to attain NeRFs that can be trained in real-time. BARF [7]
+and NeRF- - [8] have also shown that a priori exact pose
+1Authors are associated with the University of Toronto Robotics Institute,
+University of Toronto, Canada. Authors contributed equally to this work.
+[FIRSTNAME].[LASTNAME]@mail.utoronto.ca
+knowledge is unnecessary to reproduce both accurate pose
+estimates and NeRFs.
+These advancements have opened the door for the use of
+NeRFs to represent maps for Simultaneous Localization and
+Mapping (SLAM) in robotics applications. One of the first
+papers in this area was iMAP [9], which builds an MLP-
+based NeRF in real-time using RGB-D data by cleverly
+downsampling the number of pixels used to generate the
+NeRF. NICE-SLAM improved significantly on this frame-
+work by leveraging the key insight that the entire NeRF map
+need not be updated at every iteration. By introducing a spa-
+tial, voxel-based NeRF, NICE-SLAM only updates parts of
+the NeRF that are specially relevent to a given camera view.
+At the start of this project, NICE-SLAM represented the
+current state-of-the-art in the field of NeRF-based SLAM1.
+Although NICE-SLAM produces moderately robust and
+complete dense maps of the environment, it fails to perform
+competitively in the generated pose estimates compared
+to classical SLAM approaches. Additionally, the produced
+volumetric grid can become very expensive to maintain if the
+algorithm aims to function in a large environment. Looking
+forward to potential open world deployment of NeRF-based
+SLAM, all current approaches make use of a predefined,
+finite volumetric grid, without a clear way to handle visual
+information contained far from the camera or to dynamically
+expand the area of operation.
+This project begins to tackle these problems in order
+to leverage the special mapping approach of NeRF-based
+SLAM without sacrificing the quality of the state estimates
+or requiring excessive computational resources. More specif-
+ically, this project improves on the robustness of NICE-
+SLAM by considering depth uncertainty from RGB-D im-
+ages and by including IMU information to constrain the
+camera pose estimates. Additionally, the project begins to
+expand the capabilities of NICE-SLAM towards the open
+world by splitting visual information into a foreground and
+background. The background, modelled as a sphere infinitely
+far away from the foreground, is then able to contribute to the
+colour information of the map without requiring an explicit
+grid to be generated all the way to the visual range of the
+image.
+II. BASELINE ALGORITHM
+Similar to its predecessor, iMAP, and other modern SLAM
+algorithms [11], NICE-SLAM separates the phases of SLAM
+into to two parallel threads: a tracking thread to localize the
+1NeRFs and NeRF-based SLAM are currently rapidly evolving fields.
+NeRF-SLAM [10] has further developed the concepts presented in NICE-
+SLAM and extended them to monocular SLAM
+arXiv:2301.03102v1  [cs.RO]  8 Jan 2023
+
+current camera frame against the current map and a mapping
+thread which jointly optimizes the parameters of the NeRF
+map and a set of stored keyframes.
+NICE-SLAM uses a series of 3 fixed-size, voxel grids of
+encoded features with different grid resolutions to represent
+the map. When evaluating pixels for a given RGB-D image,
+NICE-SLAM uses the same ray-casting technique as [1], but
+evaluates the rays by interpolating sample points the voxel
+grid (similar to [5]), decoding each sample feature with a
+(smaller) MLP and aggregating the result into an estimated
+pixel color and depth. Additional detail on mapping and
+tracking threads is given in the subsections below.
+A. Mapping
+The mapping thread is responsible for updating the voxel-
+grid NeRF representation of the map. It does this by con-
+tinually optimizing the voxel-grid features and the stored
+poses of relevant subset of keyframes. Similar to ORB-
+SLAM, keyframes are selected on the basis of the level of
+information gain that they provide and consist of an RGB-D
+image as well as the corresponding estimate of camera pose.
+For each map update, a set of keyframes is selected based
+on predicted overlap with the current frame and are used,
+along with the current frame, to construct a loss function.
+The loss function has both depth-based and colour-based
+components. The depth-based loss for each grid resolution
+in the mapping thread based on N pixels is given by
+Lmap
+depth =
+�
+r=f,c
+1
+N
+N
+�
+n=1
+���dn − ˆdr,n
+���
+1 ,
+(1)
+where dn is the measured pixel depth and ˆdr,n is the NeRF-
+predicted pixel depth for the fine (f) and coarse (c) grid
+resolutions.
+The color-based loss is only generated at the fine grid
+resolution and is given by
+Lmap
+colour = 1
+N
+N
+�
+n=1
+���In − ˆIn
+���
+1 ,
+(2)
+where In is the measured pixel colour and ˆIn is the NeRF-
+predicted pixel colour.
+The overall optimization for the mapping portion is given
+by:
+min
+Ci,ri,Θ
+�M
+i=1 Lmap,i
+depth + λmLmap,i
+colour,
+(3)
+where i represents the ith keyframe of M frames, Θ repre-
+sents the voxel grid features, and λm is a tuning parameter.
+B. Tracking
+The tracking thread performs a similar optimization to the
+mapping thread, but only optimizes the pose of the current
+frame (i.e. does not modify the map).
+The colour component of the loss, Ltrack
+colour is computed in
+the same way as for the mapping, but is only for the current
+frame. On the other hand, the depth-based loss in the tracking
+thread is weighted based on depth uncertainty in the network.
+For each grid resolution, the loss is computed for N pixels
+as follows:
+Ltrack
+depth =
+�
+r=f,c
+1
+N
+N
+�
+n=1
+���dn − ˆdr,n
+���
+ˆσdr,n
+,
+(4)
+where variables are defined in the same way as for (1) with
+ˆσd
+r,n corresponding to the standard deviation on ˆdn. This
+standard deviation is extracted from the NeRF voxel grid by
+computing the variance from all points that contribute to the
+final ˆdn value along a pixel ray.
+The overall tracking optimization is given by:
+min
+C,r
+Ltrack
+depth + λtLtrack
+colour,
+(5)
+where C and r are the pose parameters for the current frame
+and λt is a tuning parameter.
+III. IMPROVEMENTS
+A. Depth Uncertainty
+The standard NICE-SLAM algorithm does not consider
+depth uncertainty from the measured RGB-D depth. As a
+result, when considering N pixels, each pixel contributed
+equally to the computed depth loss in (1) and (4). While
+(4) does down-weight the contributions according to a proxy
+measure of the uncertainty on the NeRF-generated depth
+value, it still treats the RGB-D depth values as equally valid.
+This equal valuation is contrary to not only potential sensor-
+specific uncertainty fluctuations or biases, for example due
+to artifacts such as vignetting, but also to the fact that most
+depth sensors measurements increase in uncertainty as a
+function of the depth. Although this uncertainty negligable
+in relatively small and constant depth environments, it can
+become considerable in a large environment.
+In order to account for this potentially varying uncertainty,
+a modification to the depth loss in (1) and (4) is proposed
+as follows:
+Lmap
+depth = 1
+N
+N
+�
+n=1
+���dn − ˆdn
+���
+1
+σdn
+,
+(6)
+Ltrack
+depth = 1
+N
+N
+�
+n=1
+���dn − ˆdn
+���
+1
+�
+σdn + ˆσdn
+,
+(7)
+where σd
+n is the standard deviation of dn, the depth measured
+in pixel n.
+B. Including IMU Data
+Motion data can serve as a valuable source of information
+for state estimation. In this report, we include this data in
+both the tracking and mapping optimization to make use of
+temporal relationship between camera poses.
+
+1) SO(3) Preliminaries: The camera orientation in this
+report is represented by the SO(3) matrix Lie group (MLG).
+For brevity, overall MLG theory is not covered in this report,
+with a general reference to everything found in [12]. How-
+ever, this section deals with various SO(3) operators, which
+need to be defined. The log(·) and exp(·) logarithmic and
+exponential operators respectively convert SO(3) element to
+and from the Lie algebra. The (·)∨ and (·)∧ vee and wedge
+operators convert elements of the Lie algebra respectively
+to and from the R3 representation, for the 3 degrees of
+freedom of SO(3). As a shorthand, the capital logarithmic
+and exponential operators are defined as
+C = exp(ξ∧) ≜ Exp(ξ) ∈ SO(3),
+(8)
+ξ = log(C)∨ ≜ Log(C) ∈ R3.
+(9)
+Additionally, the right-invariant error definition for SO(3) is
+adopted here arbitrarily. With this definition, the difference
+between a nominal ¯C and C is computed as δC = ¯CCT.
+2) IMU Modelling: Since IMU data is not considered in
+NICE-SLAM, it is not included with any of the provided
+datasets. Therefore, we generate IMU measurements from
+the provided ground truth poses. Specifically, linear and
+angular velocities are generated based on a fixed timestamp
+and corrupted with white Gaussian noise. In this project,
+“IMU” measurements refer to linear and angular velocity
+measurements. This is done in order to simplify derivations,
+since camera velocity estimates are not included, while still
+evaluating the benefit of motion data.
+A linear velocity measurements vk at time tk is generated
+from ground truth camera poses according to
+vk = ¯vk + wv
+k = CT
+k
+�rk+1 − rk
+Tk
+�
++ wv
+k,
+(10)
+where ¯vk is the true linear velocity, Ck ∈ SO(3) is the
+camera orientation at tk, rı ∈ R3 is the camera position at
+tı, ı ∈ [k, k + 1], Tk = tk+1 − tk is the time increment, and
+wv
+k ∼ N(0, Σv
+k) is the measurement noise with covariance
+Σv
+k. Note, this model assumes that the velocity measurement
+is constant over Tk.
+An angular velocity measurement uk at time tk is gener-
+ated from ground truth camera poses according to
+uk = ¯uk + wu
+k = Log(CT
+kCk+1)
+Tk
++ wu
+k,
+(11)
+where ¯uk is the true angular velocity, wu
+k ∼ N(0, Σu
+k) is the
+measurement noise with covariance Σu
+k, and the assumption
+that the measurement is constant over Tk is again made.
+3) Tracking with IMU: We add IMU-based loss to the
+tracking optimization that provides a motion prior for the
+current camera pose based on its immediate predecessor.
+Since there is no uncertainty quantification in NICE-SLAM,
+the previous camera pose is treated as fixed, with the IMU-
+based loss being weighted by a factor corresponding to the
+IMU noise.
+Consider the current camera pose Xk = (Ck, rk) as
+being at time tk and the previous camera pose Xk−1 =
+(Ck−1, rk−1) as being at time tk−1. The IMU loss is
+composed of an orientation and position component.
+The orientation component is computed as
+eC
+IMU = Log
+�
+Exp(uk−1Tk−1)CT
+kCk−1
+�
+.
+(12)
+For the final loss, this loss is weighted by the uncertainty
+from uk−1. This weight is computed by linearizing eC
+IMU
+with respect to uk−1 and propagating Σu
+k−1 through the
+linearization as
+ΣC
+IMU =
+� ∂eC
+IMU
+∂wu
+k−1
+�
+Σu
+k−1
+� ∂eC
+IMU
+∂wu
+k−1
+�T
+,
+(13)
+where wu
+k−1 enters eC
+IMU through (11). The final Jacobian,
+with the error u = ¯u−δu and the full derivation omitted for
+brevity, is
+∂eC
+IMU
+∂uk−1
+= −J−1
+ℓ
+�
+eC
+IMU
+�
+Jℓ(wu
+k−1Tk−1)Tk−1,
+(14)
+where Jℓ is the left group Jacobian of SO(3).
+The position component is computed as
+er
+IMU = vk−1 − CT
+k−1
+�rk − rk−1
+Tk−1
+�
+,
+(15)
+with a weighting resulting from vk−1 being Σr
+IMU = Σv
+k−1.
+The final IMU tracking loss is then
+Ltrack
+IMU = eCT
+IMUΣC−1
+IMUeC
+IMU + erT
+IMUΣr−1
+IMUer
+IMU.
+(16)
+4) Mapping with IMU: In order to make use of IMU
+data in the mapping optimization, IMU preintegration is
+used. Introduced in [13], IMU preintegration allows to group
+multiple IMU measurements into a relative motion increment
+(RMI). The RMI then functions as a single “measurement”,
+defining a probabilistic motion constraint between two poses
+separated by any number of IMU measurements. Since the
+keyframe used for each mapping step are not predetermined
+a potentially different number of IMU measurements can be
+connected to each consecutive keyframe. IMU preintegration
+allows to greatly simplify the computation the corresponding
+motion constraints between these keyframes, preventing the
+need to keep track of every IMU measurement received.
+The RMIs are computed incrementally between keyframes
+and are saved whenever a new keyframe is added. Consider
+the current camera pose Xk = (Ck, rk) as being at time
+tk, the previous keyframe camera pose Xi = (Ci, ri) as
+being at time ti, and the RMI ∆Xik
+= (∆Cik, ∆rik)
+connecting the two. The orientation component ∆Cik and
+position component ∆rik are incrementally updated with the
+IMU measurement that arrived at tk−1 according to
+∆Cik = ∆Cik−1 Exp(uk−1Tk−1),
+(17)
+∆rik = ∆rik−1 + ∆Cik−1(vk−1Tk−1),
+(18)
+with full derivation omitted for brevity. Note, when a new
+keyframe is created, the RMI is reset to ∆Xik = (1, 0). The
+
+uncertainty on the RMI can also be updated incrementally
+according to
+ΣRMI
+ik
+=
+� ∂∆Xik
+∂∆Xik−1
+�
+ΣRMI
+ik−1
+� ∂∆Xik
+∂∆Xik−1
+�T
++
+�∂∆Xik
+∂wk−1
+�
+Σw
+k−1
+�∂∆Xik
+∂wk−1
+�T
+, (19)
+where wk−1 =
+�
+wu
+k−1
+T
+wv
+k−1
+T�T enters ∆Xik through
+(11) and (10), and Σw
+k−1
+= diag(Σu
+k−1, Σv
+k−1). These
+Jacobians are derived, however, as discussed in Section IV-
+A, are not used in the end. For brevity, their explicit form is
+thus omitted.
+The
+final
+computed
+RMI
+is
+defined
+between
+two
+keyframes. In the case that, during optimization, involved
+keyframes are separated by one or more other not involved
+keyframes, the total RMI can be computed by multiplying
+the individual RMIs through. For example, if keyframes 2
+and 4 are involved in the optimization but 3 is not, the
+RMI between the involved keyframes can be computed as
+X24 = X23X34. Typically, the uncertainty on the RMIs also
+need to be combined, but this was not considered for this
+project.
+The RMI loss between keyframe i and j is
+e∆C
+ij
+= Log
+�
+∆CijCT
+j Ci
+�
+,
+(20)
+e∆r
+ij = ∆vij − CT
+i (rj − ri) ,
+(21)
+eRMI
+ij
+=
+�
+e∆CT
+ij
+e∆rT
+ij
+�T
+,
+(22)
+with a corresponding weight ΣRMI
+ij
+.
+The total final RMI loss for some subset of keyframes z,
+for example z ∈ [2, 4] in the example above, is written
+Lmap
+RMI =
+len(z)
+�
+q=2
+eRMIT
+z[q−1]z[q]ΣRMI
+z[q−1]z[q]eRMI
+z[q−1]z[q].
+(23)
+C. Background Model
+In order to extend NICE-SLAM to unbounded environ-
+ments without required infinite memory resources, we en-
+code the background of given scene using a spherical grid.
+This representation is similar to the Multi-Sphere Images
+(MSI) proposed in [14] and used in [5].
+1) Modeling A Sphere at Infinity: In order to preserve
+real-time capability by keeping the representation computa-
+tionally light, we have opted to model the background with
+only one background sphere. This sphere is modeled as if it
+is infinitely far away from the central world frame and, by
+extension, any given camera frame. This allows to represent
+distant objects on the horizon, which can be very helpful for
+camera/robot orientation localization.
+This representation simplifies some of our computations,
+since the only the direction of the pixel ray is required to
+evaluate the contribution of the background model. To see
+why this is true, consider a point on the background sphere
+along a given pixel ray in the camera frame, xc = β ˆxc,
+where ˆxc is the normalized direction of the ray and β
+represents the magnitude or the ray. The normalized ray in
+the world frame can be expressed as
+ˆxw =
+xw
+∥xw∥2
+= Cwcxc + rcw
+w
+∥xc + rcw
+w ∥2
+=
+Cwcβ ˆxc + rcw
+w
+β ∥ˆxc + rcw
+w /β∥2
+.
+(24)
+Now, as β → ∞, it is clear that ˆxw ≃ Cwc ˆxc, that is, we
+only need to evaluate the sphere along the rotated camera
+frame ray.
+2) Background NeRF Model: Consider the formula for a
+pixel colour corresponding to a given ray, r(t) = o + td, as
+given in the original NeRF paper [1]:
+ˆIn =
+� tf
+tn
+T(t)σ(r(t))c(r(t))dt,
+(25)
+where T(t) = Exp
+�
+−
+� tf
+tn
+σ(r(s))ds
+�
+(26)
+where σ(x) represents the scene volume density, T(t) rep-
+resents the accumulated transmittance (probability that a ray
+travels from t to tn), c(r(t)) represents the colour at a given
+point and tn, tf, represent the near and far limits of the
+depth parameter, t, respectively. Now, suppose we allow the
+far limit to extend to infinity, tf → ∞. We can rewrite the
+integral as follows:
+ˆIn,∞ =
+� ∞
+tn
+T(t)σ(r(t))c(r(t))dt
+=
+� tf
+tn
+T(t)σ(r(t))c(r(t))dt +
+� ∞
+tf
+T(t)σ(r(t))c(r(t))dt
+= ˆIn + T(tf)c∞(r(t)).
+where ˆIn is the original NICE-SLAM foreground NeRF and
+c∞(r(t)) =
+� ∞
+tf T(t)σ(r(t))c(r(t)) represents the evalua-
+tion from tf to ∞ of a separate background NeRF. We have
+made use of the fact that
+T(c) = Exp(−
+� c
+a
+σ(r(s))ds)
+= Exp(−
+� b
+a
+σ(r(s))ds) Exp(−
+� c
+b
+σ(r(s))ds)
+= T(b)T(b, c).
+We see that we can represent the background of the scene
+by simply adding a defined background colour, c∞(r(t)),
+multiplied by the final transmittance value at the boundary
+foreground NeRF, T(tf).
+We assume that the background is mostly empty except
+for some points that are very far away. By the arguments
+presented in Section III-C.1, this allows us to represent
+the background colour only in terms of the ray direction:
+c∞(r(t)) ≃ c∞(d) = cbg(d).
+Similar to the encoded color grid for NeRF-SLAM, we
+generate a 2D grid of background features that are warped
+onto a sphere. At render time, we use the grid and a pixel
+ray, d, to interpolate a feature, then decode the feature and
+scale by the boundary transmittance of the foreground NeRF,
+T(tf)2.
+2Note that NICE-SLAM already computes the rotated rays, d, and
+boundary transmittances, T(tf), so we incur no additional cost to use them
+
+IV. RESULTS
+A. Implementation Details
+We implemented the changes outlined in this paper using
+the existing NICE-SLAM codebase [2], making modifica-
+tions to the rendering and loss functions as appropriate. To
+test our changes, we initially tested on the “Demo” dataset
+provided in [2], but used scenes from the Replica dataset
+[15] for depth uncertainty and IMU testing and TUM RGB-
+D dataset [16] for final testing for background sphere testing.
+In order to test the impact of the listed changes on
+robustness, the algorithm is tested with the full, default
+number of iterations suggested by the authors of NICE-
+SLAM and with a heavily reduced number of iterations to
+approach real-time performance. Specifically, for the Replica
+datasets the “full number of iterations” corresponds to 10 and
+60 iterations for tracking and mapping respectively, whereas
+the ”low number of iterations” corresponds to 5 and 10
+iterations for tracking and mapping respectively. In some
+circumstances, reducing the number of iterations is required
+to ensure real-time capability. Indeed, we found that this
+reduction in number of iterations reduced the overall runtime
+of the NICE-SLAM from 48 minutes to 10 minutes.
+In practice, because the tracking and mapping threads are
+run in parallel, it was found to be challenging to compute
+RMIs at non-keyframe frequency. As a result, an RMI
+connection to the most recent state, which is involved in
+mapping but not guaranteed to be a keyframe, is not included.
+B. Depth Uncertainty and IMU Results
+The main results for experiments with depth uncertainty
+and IMU data are shown in Table I and Figure 1.
+Depth uncertainty parameters were set based on the spec-
+ified depth uncertainty values given for the cameras used
+in each dataset. Typically, this uncertainty increases linearly
+with the depth measurement and depends on the stereo
+baseline.
+It was initially found that the depth uncertainty was less
+impactful on the results of NICE-SLAM. However, it turns
+out that this was only true in smaller environments for which
+the depth uncertainty does not vary greatly (“Demo” dataset
+from [2]). When larger environments were considered (i.e.,
+from Replica dataset [15]), it was found that depth uncer-
+tainty moderately improved most metrics and considerably
+improved metrics for both rooms when used in combination
+with IMU data.
+Across all experiments, the standard deviation of the
+additive noise was 0.01 rad/s and 0.01 m/s for rotation and
+translation measurements, respectively. In general, integra-
+tion of IMU data lead to drastic improvement in both tracking
+error and map reconstruction error. Most notably, for the
+Room 1 test with reduced number of iterations, an order of
+magnitude improvement can be seen in Table I for the RMSE
+and worst case error for tracking as well as the average
+accuracy (Acc.) of the reconstructed map.
+The improvement at low iterations can also be seen in
+Figure 1 (b) and (d). Note that in (b) the localization and
+room reconstruction approaches catastrophic failure when
+the number of iterations is reduced to improve computation
+time. In contrast, even with reduced number of iterations, (d)
+shows that the IMU and depth uncertainty measurements lead
+to a consistent representation. This highlights an important
+conclusion: addition of depth uncertainty and IMU data can
+reduce the number of iterations required for convergence,
+thereby increasing the runtime frequency of NICE-SLAM.
+C. Background Representation
+The background NeRF representation outlined in Section
+III-C was implemented and tested on a subset of images from
+the Freiburg2 RPY scene from the TUM RGB-D dataset.
+This scene was selected because it involved mostly only
+rotations of the camera and because the scene itself consisted
+of distinct foreground and background elements.
+To simplify this experiment, the tracking part of NICE-
+SLAM was disabled and the ground truth camera posi-
+tions were used. Additionally, the IMU measurements and
+depth uncertainty measurements were not included in this
+experiment. The bounding box of the foreground NeRF was
+intentionally set to a small value (1.5 m) to observe the
+effect of the including the background. Figure 2 shows the
+background model and resulting image reconstruction (with
+and without background) for three scenes in the dataset. By
+comparing columns c) and d) with a) it is clear that the
+background model allows the details in the background to be
+represented more accurately. For this experiment, the average
+colour loss was decreased by 18.32% when the background
+model was included.
+V. CONCLUSIONS
+We have shown how the addition of depth uncertainty
+and IMU data can improve the accuracy of NICE-SLAM,
+especially when reducing the number of allowed iterations
+for each timestep. Additionally, we have demonstrated that
+spherical background model can improve the image recon-
+struction of NICE-SLAM when the scene is too large to be
+modeled by the foreground NeRF.
+In the future, the goal would be to test the implemented
+changes in a large environment where background visual in-
+formation is significantly far away from the robot. Addition-
+ally, in order to make NeRF-based SLAM truly open world,
+it is required to remove the dependency on a predefined
+grid. Potential work in this area would involve switching to
+a dynamically expanding octree. This would allow a robot
+to explore new environments while maintaining reasonable
+memory requirements by only forming finer grid resolutions
+around objects.
+ACKNOWLEDGMENT
+Thank you Alec Krawciw for taking the time out of your
+weekend to go to UTIAS and turn on the lab computer so
+that we could continue running tests for this project.
+
+(a) NICE-SLAM: Full number of iterations Default
+(b) NICE-SLAM: Low number of iterations Default
+(c) Ours: Full number of iterations IMU + Dep.
+(d) Ours: Low number of iterations IMU + Dep.
+Fig. 1: Reconstruction results for NICE-SLAM and our proposed modifications run at a full and low number of iterations.
+Fig. 2: Experiment demonstrating the effect of the background sphere across different frames in the Freiburg2 RPY scene
+from the TUM RGB-D dataset. a) Input image; b) Background sphere only; c) NICE-SLAM with background active; d)
+Original NICE-SLAM (no background model).
+REFERENCES
+[1] B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoor-
+thi, and R. Ng, “NeRF: Representing Scenes as Neural Radiance Fields
+for View Synthesis,” Aug. 2020, arXiv:2003.08934 [cs].
+
+Set Up
+Room 0
+Room 1
+Tracking
+Mapping
+Tracking
+Mapping
+RMSE
+Max.
+Acc.
+Comp.
+C.R.
+Dep. L1
+RMSE
+Max.
+Acc.
+Comp.
+C.R.
+Dep. L1
+[cm] ↓
+[cm] ↓
+[cm] ↓
+[cm] ↓
+[%] ↑
+[cm] ↓
+[cm] ↓
+[cm] ↓
+[cm] ↓
+[cm] ↓
+[%] ↑
+[cm] ↓
+Full Iter.
+Default
+0.020
+0.072
+2.744
+3.000
+91.03
+1.922
+0.026
+0.103
+2.796
+2.376
+92.43
+1.732
+Depth
+0.018
+0.074
+2.646
+2.840
+91.25
+1.763
+0.018
+0.048
+2.430
+2.257
+93.54
+1.488
+IMU
+0.019
+0.053
+2.442
+2.756
+91.10
+1.752
+0.019
+0.061
+2.300
+2.298
+93.12
+1.493
+IMU + Dep.
+0.019
+0.057
+2.796
+2.855
+90.71
+1.700
+0.020
+0.057
+2.671
+2.293
+93.22
+1.378
+Low Iter.
+Default
+0.156
+0.627
+5.083
+5.030
+67.29
+5.677
+0.531
+1.148
+15.55
+16.67
+35.68
+24.98
+Depth
+0.211
+0.539
+5.641
+6.330
+66.13
+7.937
+0.3429
+0.8199
+18.17
+18.57
+28.79
+25.89
+IMU
+0.029
+0.079
+3.357
+3.646
+84.61
+3.258
+0.031
+0.104
+3.116
+3.360
+84.51
+3.189
+IMU + Dep.
+0.029
+0.085
+3.217
+3.595
+84.98
+3.293
+0.030
+0.089
+2.635
+2.968
+88.35
+2.373
+TABLE I: Results from two trials on “Replica” dataset. Best results (within 5% of pre-rounded values) are in bold. RMSE
+refers to Root Mean Squared Error; Max. refers to worst cases error; Comp. refers to the average distance to the nearest
+reconstructed mesh from ground truth mesh; C.R. refers to the completion ratio, or percentage of points with completion
+under 5 cm; Dep. L1 refers to sum of absolute depth errors.
+[2] Z. Zhu, S. Peng, V. Larsson, W. Xu, H. Bao, Z. Cui, M. R. Oswald,
+and M. Pollefeys, “NICE-SLAM: Neural Implicit Scalable Encoding
+for SLAM,” Apr. 2022, arXiv:2112.12130 [cs].
+[3] V. Rudnev, M. Elgharib, W. Smith, L. Liu, V. Golyanik, and
+C. Theobalt, “NeRF for Outdoor Scene Relighting,” Jul. 2022,
+arXiv:2112.05140 [cs].
+[4] T.
+Takikawa,
+J.
+Litalien,
+K.
+Yin,
+K.
+Kreis,
+C.
+Loop,
+D. Nowrouzezahrai, A. Jacobson, M. McGuire, and S. Fidler,
+“Neural Geometric Level of Detail: Real-time Rendering with
+Implicit 3D Shapes,” in 2021 IEEE/CVF Conference on Computer
+Vision and Pattern Recognition (CVPR).
+Nashville, TN, USA: IEEE,
+Jun. 2021, pp. 11 353–11 362.
+[5] A. Yu, S. Fridovich-Keil, M. Tancik, Q. Chen, B. Recht, and
+A. Kanazawa, “Plenoxels: Radiance Fields without Neural Networks,”
+Dec. 2021, arXiv:2112.05131 [cs].
+[6] T. Müller, A. Evans, C. Schied, and A. Keller, “Instant neural graphics
+primitives with a multiresolution hash encoding,” ACM Transactions
+on Graphics, vol. 41, no. 4, pp. 1–15, Jul. 2022.
+[7] C.-H. Lin, W.-C. Ma, A. Torralba, and S. Lucey, “BARF: Bundle-
+Adjusting Neural Radiance Fields,” Aug. 2021, arXiv:2104.06405 [cs].
+[8] Z. Wang, S. Wu, W. Xie, M. Chen, and V. A. Prisacariu, “NeRF–
+: Neural Radiance Fields Without Known Camera Parameters,” Apr.
+2022, arXiv:2102.07064 [cs].
+[9] E. Sucar, S. Liu, J. Ortiz, and A. J. Davison, “iMAP: Implicit Mapping
+and Positioning in Real-Time,” in 2021 IEEE/CVF International
+Conference on Computer Vision (ICCV).
+Montreal, QC, Canada:
+IEEE, Oct. 2021, pp. 6209–6218.
+[10] A. Rosinol, J. J. Leonard, and L. Carlone, “NeRF-SLAM: Real-Time
+Dense Monocular SLAM with Neural Radiance Fields,” Oct. 2022,
+arXiv:2210.13641 [cs].
+[11] R. Mur-Artal, J. M. M. Montiel, and J. D. Tardos, “ORB-SLAM: A
+Versatile and Accurate Monocular SLAM System,” IEEE Transactions
+on Robotics, vol. 31, no. 5, pp. 1147–1163, Oct. 2015. [Online].
+Available: https://ieeexplore.ieee.org/document/7219438/
+[12] J. Solà, J. Deray, and D. Atchuthan, “A Micro Lie Theory for State
+Estimation in Robotics,” arXiv:1812.01537v8, 2018.
+[13] T. Lupton and S. Sukkarieh, “Visual-Inertial-Aided Navigation for
+High-Dynamic Motion in Built Environments without Initial Condi-
+tions,” IEEE Trans. Robot., vol. 28, no. 1, pp. 61–76, 2012.
+[14] K. Zhang, G. Riegler, N. Snavely, and V. Koltun, “NeRF++: Analyzing
+and Improving Neural Radiance Fields,” Oct. 2020, arXiv:2010.07492
+[cs]. [Online]. Available: http://arxiv.org/abs/2010.07492
+[15] J. Straub, T. Whelan, L. Ma, Y. Chen, E. Wijmans, S. Green,
+J. J. Engel, R. Mur-Artal, C. Ren, S. Verma, A. Clarkson, M. Yan,
+B. Budge, Y. Yan, X. Pan, J. Yon, Y. Zou, K. Leon, N. Carter,
+J. Briales, T. Gillingham, E. Mueggler, L. Pesqueira, M. Savva,
+D. Batra, H. M. Strasdat, R. De Nardi, M. Goesele, S. Lovegrove,
+and R. Newcombe, “The replica dataset: A digital replica of indoor
+spaces,” no. arXiv:1906.05797, Jun 2019, arXiv:1906.05797 [cs, eess].
+[16] J. Sturm, N. Engelhard, F. Endres, W. Burgard, and D. Cremers, “A
+benchmark for the evaluation of rgb-d slam systems,” in Proc. of the
+International Conference on Intelligent Robot Systems (IROS), Oct.
+2012.
+VI. SUPPLEMENTARY MATERIAL
+An example of a simplified (black and white) background
+rendering can be seen in Figure 3. This was used initially to
+debug the background model, but has been included in this
+section since it is instructive.
+Fig. 3: Interpolated rendering at different angles of checker-
+board background grid for angle divisions of 0.02 (rad). Zero
+(deg) camera angle corresponds to z axis in world frame.
+
+Camera Angle:
+Camera Angle:
+0.0 deg
+90.0 deg
+Camera Angle:
+Camera Angle:
+45.0 deg
+135.0 deg

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+CONTENTS
+CONTENTS
+Photoinduced pairing in Mott insulators
+Satoshi Ejima1,2⋆ and Holger Fehske1,3
+1 Institut für Physik, Universität Greifswald, 17489 Greifswald, Germany
+2 Institut für Softwaretechnologie, Abteilung High-Performance Computing, Deutsches Zentrum
+für Luft- und Raumfahrt (DLR), 22529 Hamburg, Germany
+3 Erlangen National High Performance Computing Center, Friedrich-Alexander-Universität
+Erlangen-Nürnberg, 91058 Erlangen, Germany
+* [email protected]
+January 12, 2023
+International Conference on Strongly Correlated Electron Systems
+(SCES 2022)
+Amsterdam, 24-29 July 2022
+doi:10.21468/SciPostPhysProc.?
+Abstract
+Utilizing time-evolution techniques in (infinite) matrix-product-state representation, we study
+the non-equilibrium dynamics of driven Mott insulators and demonstrate photoinduced η
+pairing directly in the thermodynamic limit. Analyzing the time evolution of the correspond-
+ing pairing correlations, we determine the optimal laser pump parameters for which long-
+range η-pairing becomes dominant after pulse irradiation. The time-dependent photoemis-
+sion spectra for this optimal pump parameter set show clear signatures of the photoinduced
+insulator-to-metal phase transition related to the formation of η pairs.
+Contents
+1
+Introduction
+2
+2
+Model
+2
+3
+Pairing correlations
+3
+4
+Non-equilibrium dynamics
+5
+5
+Conclusions
+6
+References
+7
+1
+arXiv:2301.04496v1  [cond-mat.str-el]  11 Jan 2023
+
+SCES
+Amsterdam
+2022
+金2
+MODEL
+1
+Introduction
+η pairing, proposed first by C. N. Yang in 1989 [1], gives rise to a pairing-density-wave-like off-
+diagonal long-range order in the Hubbard model. While it can be used to construct exact eigen-
+states of this model, η pairing is absent in the Hubbard model’s ground state, and therefore has
+attracted only specific attention, mostly from mathematical point of view. Recently, however, it
+was pointed out that the η-pairing state will be enforced by pulse irradiation [2]. The respective
+enhancement of pairing correlations emerged in time-dependent exact diagonalisations: Calcu-
+lating all eigenstates as well as pairing correlations for a small cluster and taking the selection
+rule of η pairs into account, Kaneko et al. showed that this photoinduced state is related to the
+η-pairing state [2].
+Meanwhile, as a result of on-going developments in (time-depenent) density-matrix renormal-
+isation group [(t-)DMRG] technique [3,4], optically driven systems in (quasi-)one-dimension can
+be simulated directly in the thermodynamic limit. In doing so, static correlation functions such
+as η-pair correlations can be computed by means of the infinite time-evolving block decimation
+(iTEBD) technique [5], taking advantage of translational invariance in the infinite matrix-product-
+state (iMPS) representation. Building window sites with so-called infinite boundary conditions
+(IBC) in the uniform update scheme [6] enables us to simulate non-equilibrium dynamics of ex-
+cited (quasi-)one-dimensional (1D) systems by a laser electric field [7].
+On this basis, in this study, we reexamine the time-evolution of photoinduced η-pairing, mainly
+to confirm or put in question previous small cluster results. Thereby we emphasize the impor-
+tance of using optimal pump pulse parameters. Furthermore, we reconsider the relation between
+the η-pairing correlations and the optical spectrum in the small-amplitude regime after pulse ir-
+radiation. Finally we prove the photoinduced insulator-to-metal phase transition by simulating
+time-dependent photoemission spectra of driven Mott insulators.
+2
+Model
+Let us consider the 1D half-filled Hubbard model,
+ˆH = −th
+�
+j,σ
+�
+ˆc†
+j,σˆcj+1,σ + H.c.
+�
++ U
+�
+j
+�
+ˆnj,↑ − 1/2
+��
+ˆnj,↓ − 1/2
+�
+,
+(1)
+where th is the nearest-neighbor transfer amplitude and U gives the on-site part of the Coulomb
+interaction. In Eq. (1), ˆc†
+j,σ (ˆcj,σ) creates (annihilates) a spin-σ (=↑,↓) electron at Wannier lattice
+site j, and ˆnj,σ = ˆc†
+j,σˆcj,σ. In the repulsive case (U > 0) the model realizes a Mott insulating
+ground state with a finite charge gap ∆.
+Exact eigenstates of the Hubbard model can be constructed by means of the operators ˆη+ =
+�
+j(−1)j ˆ∆†
+j,
+ˆη− = ( ˆη+)†, and ˆηz = 1
+2
+�
+j(ˆnj,↑ + ˆnj,↓ − 1), where ˆ∆†
+j = ˆc†
+j,↓ˆc†
+j,↑ denotes the singlet pair-creation
+operator [1]. These so-called η operators fulfill SU(2) commutation relations [ ˆη+, ˆη−] = 2 ˆηz and
+[ ˆηz, ˆη±] = ± ˆη±. Apparently, the Hubbard Hamiltonian (1) commutes with ˆη2 = 1
+2( ˆη+ ˆη−+ ˆη− ˆη+)+( ˆηz)2,
+i.e., 〈η2〉 is a conserved quantity. Long-ranged pairing correlations 〈 ˆη+
+j ˆη−
+ℓ 〉 develop when the ex-
+pectation value 〈 ˆη2〉 becomes finite, but such η-pairing states cannot be the ground state of the
+Hubbard model [1]. Pulse irradiation can establish η-paired states in Mott insulators however [2].
+To address this issue, we apply a pump pulse with amplitude A0, frequency ωp and width σp,
+2
+
+3
+PAIRING CORRELATIONS
+0
+5
+10
+15
+20
+0
+0.5
+1.0
+1.5
+t · th
+˜P(q = π, t)
+2nd(t)
+U/th = 8
+Figure 1: Typical time-evolution process of ˜P(q = π, t) and 2nd(t) for the photoin-
+duced η-pairing states in the strong-coupling regime of the driven Hubbard model with
+U/th = 8 and pump parameters A0=0.4, ωp/th = 7.0, σp = 2t−1
+h
+and t0 = 10t−1
+h .
+The iTEBD data are obtained for bond dimension χ = 1200, ensuring a truncation error
+smaller than 10−5. For the iTEBD calculations, we employ a second-order Suzuki-Trotter
+decomposition with time step 0.1t−1
+h
+(0.01t−1
+h ).
+centered at time t0(> 0):
+A(t) = A0e−(t−t0)2/(2σ2
+p) cos
+�
+ωp(t − t0)
+�
+.
+(2)
+The external time-dependent electric field A(t) changes the hopping amplitude by a Peierls phase [8]:
+thˆc†
+j,σˆcj+1,σ → theiA(t)ˆc†
+j,σˆcj+1,σ, i.e., ˆH → ˆH(t). As a result, the system being initially in the ground
+state, is driven out of equilibrium, |ψ(0)〉 → |ψ(t)〉.
+3
+Pairing correlations
+The η-pairing state can be detected evaluating the time evolution of the pair-correlation function
+P(r, t) = 1
+L
+�
+j
+〈ψ(t)| ˆ∆†
+j+r ˆ∆j + H.c.)|ψ(t)〉
+(3)
+and its Fourier transform ˜P(q, t) =
+�
+r eiqr P(r, t). As found in Refs. [2,9] for small clusters, ˜P(π, t)
+is enhanced after pulse irradiation, indicating the formation of an η-pairing state. By means of
+iTEBD this is confirmed directly in the thermodynamic limit which is demonstrated in Fig. 1 for
+a pump with A0 = 0.4, ωp/th = 7.0 and σp = 2t−1
+h
+centered at t0 = 10t−1
+h .
+˜P(π, t) shows
+a clear response to pulse irradiation and is strengthened as the system progresses in time until
+saturation is reached. Obviously, the nonlocal contributions have a stronger impact on ˜P(π, t)
+than the double occupancy nd(t) = (1/L)
+�
+j〈ψ(t)|ˆnj,↑ˆnj,↓|ψ(t)〉 [note that P(r = 0, t) = 2nd(t),
+where nd(0) > 0 for the finite U values considered].
+The enhancement process of η-pairing can be described as follows [2]: The initial state before
+pulse irradiation is the ground state of the Hubbard chain with |η = 0,ηz = 0〉, which is consis-
+tent with the numerical finding: ˜P(0, t = 0) ≃ 0 (see Fig. 1). Turning on the pump pulse, the
+3
+
+3
+PAIRING CORRELATIONS
+0
+0.4
+0.8
+1.2
+4
+6
+8
+10
+12
+A0
+ωp/th
+0
+0.4
+0.8
+1.2
+2
+4
+6
+8
+10
+12
+0
+0.4
+0.8
+1.2
+ω/th,
+ωp/th
+Im χJJ(ω)
+0
+2
+4
+6
+8
+10
+�
+P(π, t)/A2
+0
+A0 = 0.04
+A0 = 0.06
+A0 = 0.08
+A0 = 0.10
+Figure 2: (a): Contour plots of ˜P(q = π, t) in the ωp-A0 plane at t = 15t−1
+h . Again
+U/th = 8, and the pump is parametrized by σp = 2t−1
+h
+at t0 = 10t−1
+h . (b): ˜P(π, t) at
+t = 15t−1
+h
+in the small-A0 area enclosed by the dashed square in panel (a). Dividing by
+A2
+0, data can be rescaled to Imχ(ω) (black line), where Imχ(ω) is the imaginary part of
+the optical spectrum χJJ(ω).
+Hamiltonian does not commute with the η-operators anymore,
+[ ˆH(t),η+] = [ ˆH,η+]cos[A(t)] +
+�
+k
+F(k, t)ˆc†
+π−k,↓ˆc†
+k,↑ ,
+(4)
+where F(k, t) = 4th sin[A(t)]sin k. This alters the initial state to a state with a finite expectation
+value 〈 ˆη2〉. Even though the commutation relation is recovered for t ≫ t0, i.e., [ ˆH(t), ˆη+] → [ ˆH, ˆη+]
+[since A(t) → 0], |ψ(t)〉 now includes components of |η > 0,ηz = 0〉 leading to the enhancement
+of ˜P(π, t), see Fig. 1 for t > t0.
+Note that the enhancement of η pairing after pulse irradiation depends, however, strongly
+on the pump pulse parameters. The optimal parameter set for inducing η-pairing states can be
+determined examining the A0 and ωp dependences of ˜P(π, t) by iTEBD. Figure 2(a) shows the
+contour plot of ˜P(π, t) after pulse irradiation (t = 15t−1
+h ). We find a single maximum around
+A0 ≈ 0.4 and ωp/th ≈ 7.0 (marked by the “×" symbol), instead of the stripe structure observed in
+the finite-system (L = 14) exact diagonalisation (ED) simulations [2].
+Another notable results of previous ED calculations [2] was that the peak structure of ˜P(π, t) as
+a function of ωp for small A0 is essentially the same as those of the ground-state optical spectrum,
+χJJ(ω > 0) = − 1
+L 〈ψ0|ˆJ
+1
+E0 − ˆH + ħhω + iηL
+ˆJ|ψ0〉,
+(5)
+where |ψ0〉 is the ground state having energy E0 and Lorentzian width ηL. In (5), the Hubbard-
+model charge-current operator is ˆJ = ith
+�
+j,σ(ˆc†
+j,σˆcj+1,σ − ˆc†
+j+1,σˆcj,σ).
+Figure 2(b) compares the iTEBD data, obtained for ˜P(π, t) at various small A0 and t = 15t−1
+h ,
+with the t-DMRG results for χJJ(ω) (using ηL/th = 0.2), in dependence on ωp respectively ω.
+Most notably, ˜P(π, t) divided by A2
+0 scales to the imaginary part of the optical spectrum Imχ(ω).
+This can be understood as follows: The hopping term including the Peierls phase can be divided
+4
+
+4
+NON-EQUILIBRIUM DYNAMICS
+−π
+π −π
+π −π
+π
+0
+0.2
+A(ω; t)
+t = 5t−1
+h
+t = 10t−1
+h
+t = 15t−1
+h
+8
+−8
+−4
+0
+4
+0
+ω/th
+k
+(a) t = 5t−1
+h
+0
+k
+(b) t = 10t−1
+h
+0
+k
+(c) t = 15t−1
+h
+0.1
+(d)
+Figure 3: Snapshots of the photoemission spectra A(k,ω; t) indicating photoinduced η-
+pairing during the pump at times t = 5t−1
+h
+(a), 10t−1
+h
+(b) and 15t−1
+h
+(c). The pump is
+parametrized by A0 = 0.4, ωp/th = 7.0 [see ‘×’-symbol in Fig. 2(a)], and σp = 2t−1
+h
+at
+t0 = 10t−1
+h . The transient integrated density of states A(ω; t) obtained from the data of
+panels (a)-(c) is depicted in panel (d). All data are obtained by the (i)TEBD technique
+with IBC for the 1D half-filled Hubbard model with U/th = 8. Note that the time cutoff in
+the simulation of time-dependent correlation functions is T = 5t−1
+h , i.e., the integration
+in Eq. 7 extends only over the interval −T ≤ τ1,τ2 ≤ T. As a compromise between time
+and frequency resolutions we have chosen a probe pulse width σpr = 2t−1
+h .
+into kinetic and current operators as
+−th
+�
+j,σ
+�
+eiA(t)ˆc†
+j,σˆcj+1,σ + H.c.
+�
+= ˆK cos[A(t)] + ˆJ sin[A(t)],
+(6)
+where ˆK = −th
+�
+j,σ(ˆc†
+j,σˆcj+1,σ +H.c.). For small A0 and large t, the second term in Eq. (6) can be
+approximated by ˆJA0, yielding a significant contribution of A2
+0 to the pair correlations. Needless
+to say that the finite-size effects are eliminated by simulating the pair correlations directly in
+thermodynamic limit by iTEBD, leading to the single-peak structure in Fig. 2(b), in strong contrast
+to the multiple-peak structure observed in the ED calculations [2].
+4
+Non-equilibrium dynamics
+We now analyze the non-equilibrium photoemission spectra A(k,ω; t) =
+�
+σ=↑,↓ Aσ(k,ω; t) for
+the optimal pump parameter set marked by the “×"-symbol in Fig. 2(a). To explore the system
+dynamics in a non-equilibrium situation, time-dependent spectral functions of the form [10]
+Aσ(k,ω; t) =
+�
+r
+e−ikr
+� ∞
+−∞
+� ∞
+−∞
+dτ1dτ2 f (τ1,τ2;ω) · Cσ(r,τ1,τ2; t)
+(7)
+are of interest. Here, the non-equilibrium two-point correlator
+Cσ(r,τ1,τ2; t) = 〈φ(t)|ˆc†
+j+r,σ(τ1; t)ˆcj,σ(τ2; t)|φ(t)〉
+(8)
+5
+
+5
+CONCLUSIONS
+is defined relative to t, and
+f (τ1,τ2;ω) = eiω(τ1−τ2)g(τ1)g(τ2), g(τ) = exp[−τ2/2σ2
+pr]/
+�
+2πσpr
+(9)
+specify the shape of the probe pulse, e.g., in a time-dependent photoemission spectroscopy exper-
+iment. How numerically simulate two-time-dependent quantities such as Cσ(r,τ1,τ2; t) has been
+explained in detail in Ref. [7] [see paragraphs below Eq. (1)].
+Figure 3 displays our (i)TEBD results for the 1D half-filled Hubbard model in the strong-
+coupling regime (U/th = 8). Before pump irradiation the state is a Mott insulator with a noticable
+single-particle gap, see Fig. 3(a) for t = 5t−1
+h . In the midst of the pump (t = 10t−1
+h ), an extra
+dispersion above Fermi energy (ω > EF) appears and persists afterwards [Fig. 3(c)].
+Evaluating the integrated density of states
+A(ω; t) = 1
+L
+�
+k
+A(k,ω; t),
+(10)
+we see more clearly how the spectral weight is shifted from ω < EF to ω > EF due to the photoin-
+duced η-pairing. Figure 3(d) gives A(ω; t) for the photoinduced η-pairing state. Obviously, the
+spectral weight for ω > EF increases distinctly over time, indicating a photoinduced phase tran-
+sition from a Mott insulator to a metallic η-pairing state. This photoinduced insulator-to-metal
+transition should be observed in time- and angle-resolved photoemission spectroscopy, when the
+pure Hubbard model is realized experimentally, e.g., in optical lattices. We note that the pho-
+toinduced phase transition cannot be observed by simulating the time-dependent photoemission
+spectra with not-optimized pump-pulse parameters, see Ref. [7].
+5
+Conclusions
+To summarize, combining tensor-network algorithms with infinite time-evolving block decimation
+techniques, we revisited the problem of photoinducing η-pairing states in the one-dimensional
+Hubbard model at half band filling. This allowed us to prove the enhancement of the pairing
+correlations directly in the thermodynamic limit. We also determined the optimal pump-pulse
+parameter set that maximizes the η-pairing tendency. An η-pairing related Mott insulator to metal
+transition could be extracted from the time-dependent photoemission spectrum.
+We wish to stress that the numerical approach presented here can be applied to simulate the
+non-equilibrium dynamics of any (quasi-)one-dimensional translational-invariant system in entire
+ranges of interacting and driving parameters. For example, the photoinduced metallization of
+excitonic insulators was demonstrated quite recently in accordance with time- and angle-resolved
+photoemission spectroscopy experiments on Ta2NiSe5 [14,15].
+Acknowledgements
+The iTEBD simulations were performed using the ITensor library [16].
+Funding information
+S.E. was supported by Deutsche Forschungsgemeinschaft through project
+EJ 7/2-1.
+6
+
+REFERENCES
+REFERENCES
+References
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+63, 2144 (1989), doi:10.1103/PhysRevLett.63.2144.
+[2] T. Kaneko, T. Shirakawa, S. Sorella and S. Yunoki, Photoinduced η pairing in the Hubbard
+model, Phys. Rev. Lett. 122, 077002 (2019), doi:10.1103/PhysRevLett.122.077002.
+[3] S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett.
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+[4] U. Schollwöck, The density-matrix renormalization group in the age of matrix product states,
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+[5] G. Vidal, Classical simulation of infinite-size quantum lattice systems in one spatial dimension,
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+[6] V. Zauner, M. Ganahl, H. G. Evertz and T. Nishino,
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+window: scaling of signal fronts and magnetization plateaus after a local quench in quan-
+tum spin chains, J. Phys.: Condens. Matter 27(42), 425602 (2015), doi:10.1088/0953-
+8984/27/42/425602.
+[7] S. Ejima, F. Lange and H. Fehske, Nonequilibrium dynamics in pumped mott insulators, Phys.
+Rev. Research 4, L012012 (2022), doi:10.1103/PhysRevResearch.4.L012012.
+[8] R. Peierls, Zur Theorie des Diamagnetismus von Leitungselektronen, Z. Phys. 80(11), 763
+(1933), doi:10.1007/BF01342591.
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+Ejima,
+T.
+Kaneko,
+F.
+Lange,
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+Yunoki
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+H.
+Fehske,
+Photoinduced
+η-
+pairing in one-dimensional Mott insulators,
+JPS Conf. Proc. 30,
+011184 (2020),
+doi:10.7566/JPSCP.30.011184.
+[10] J. K. Freericks, H. R. Krishnamurthy and T. Pruschke, Theoretical description of time-resolved
+photoemission spectroscopy: Application to pump-probe experiments,
+Phys. Rev. Lett. 102,
+136401 (2009), doi:10.1103/PhysRevLett.102.136401.
+[11] G. Vidal, Efficient classical simulation of slightly entangled quantum computations, Phys. Rev.
+Lett. 91, 147902 (2003), doi:10.1103/PhysRevLett.91.147902.
+[12] F. Lange, S. Ejima and H. Fehske,
+Finite-temperature dynamic structure factor of
+the spin-1 XXZ chain with single-ion anisotropy,
+Phys. Rev. B 97, 060403 (2018),
+doi:10.1103/PhysRevB.97.060403.
+[13] S. Ejima, F. Lange and H. Fehske,
+Finite-temperature photoemission in the extended
+Falicov-Kimball model:
+a case study for Ta2NiSe5,
+SciPost Phys. 10(3), 077 (2021),
+doi:10.21468/scipostphys.10.3.077.
+[14] S. Ejima, F. Lange and H. Fehske, Photoinduced metallization of excitonic insulators, Phys.
+Rev. B 105, 245126 (2022), doi:10.1103/PhysRevB.105.245126.
+7
+
+REFERENCES
+REFERENCES
+[15] K. Okazaki, Y. Ogawa, T. Suzuki, T. Yamamoto, T. Someya, S. Michimae, M. Watanabe, Y. Lu,
+M. Nohara, H. Takagi, N. Katayama, H. Sawa et al., Photo-induced semimetallic states realised
+in electron–hole coupled insulators, Nat. Commun. 9(1), 4322 (2018), doi:10.1038/s41467-
+018-06801-1.
+[16] M. Fishman, S. R. White and E. M. Stoudenmire, The ITensor software library for tensor
+network calculations, 2007.14822.
+8
+

N9E3T4oBgHgl3EQfZQq9/content/tmp_files/load_file.txt

@@ -0,0 +1,324 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf,len=323
+page_content='CONTENTS  CONTENTS  Photoinduced pairing in Mott insulators  Satoshi Ejima1,2⋆ and Holger Fehske1,3  1 Institut für Physik, Universität Greifswald, 17489 Greifswald, Germany  2 Institut für Softwaretechnologie, Abteilung High-Performance Computing, Deutsches Zentrum  für Luft- und Raumfahrt (DLR), 22529 Hamburg, Germany  3 Erlangen National High Performance Computing Center, Friedrich-Alexander-Universität  Erlangen-Nürnberg, 91058 Erlangen, Germany  satoshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='ejima@dlr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='de  January 12, 2023  International Conference on Strongly Correlated Electron Systems  (SCES 2022)  Amsterdam, 24-29 July 2022  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='21468/SciPostPhysProc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Abstract  Utilizing time-evolution techniques in (infinite) matrix-product-state representation, we study  the non-equilibrium dynamics of driven Mott insulators and demonstrate photoinduced η  pairing directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Analyzing the time evolution of the correspond-  ing pairing correlations, we determine the optimal laser pump parameters for which long-  range η-pairing becomes dominant after pulse irradiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' The time-dependent photoemis-  sion spectra for this optimal pump parameter set show clear signatures of the photoinduced  insulator-to-metal phase transition related to the formation of η pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Contents  1  Introduction  2  2  Model  2  3  Pairing correlations  3  4  Non-equilibrium dynamics  5  5  Conclusions  6  References  7  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='04496v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='str-el]  11 Jan 2023  SCES  Amsterdam  2022  金2  MODEL  1  Introduction  η pairing, proposed first by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Yang in 1989 [1], gives rise to a pairing-density-wave-like off-  diagonal long-range order in the Hubbard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' While it can be used to construct exact eigen-  states of this model, η pairing is absent in the Hubbard model’s ground state, and therefore has  attracted only specific attention, mostly from mathematical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Recently, however, it  was pointed out that the η-pairing state will be enforced by pulse irradiation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' The respective  enhancement of pairing correlations emerged in time-dependent exact diagonalisations: Calcu-  lating all eigenstates as well as pairing correlations for a small cluster and taking the selection  rule of η pairs into account, Kaneko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' showed that this photoinduced state is related to the  η-pairing state [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Meanwhile, as a result of on-going developments in (time-depenent) density-matrix renormal-  isation group [(t-)DMRG] technique [3,4], optically driven systems in (quasi-)one-dimension can  be simulated directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' In doing so, static correlation functions such  as η-pair correlations can be computed by means of the infinite time-evolving block decimation  (iTEBD) technique [5], taking advantage of translational invariance in the infinite matrix-product-  state (iMPS) representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Building window sites with so-called infinite boundary conditions  (IBC) in the uniform update scheme [6] enables us to simulate non-equilibrium dynamics of ex-  cited (quasi-)one-dimensional (1D) systems by a laser electric field [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  On this basis, in this study, we reexamine the time-evolution of photoinduced η-pairing, mainly  to confirm or put in question previous small cluster results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Thereby we emphasize the impor-  tance of using optimal pump pulse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Furthermore, we reconsider the relation between  the η-pairing correlations and the optical spectrum in the small-amplitude regime after pulse ir-  radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Finally we prove the photoinduced insulator-to-metal phase transition by simulating  time-dependent photoemission spectra of driven Mott insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  2  Model  Let us consider the 1D half-filled Hubbard model,  ˆH = −th  �  j,σ  �  ˆc†  j,σˆcj+1,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  �  + U  �  j  �  ˆnj,↑ − 1/2  ��  ˆnj,↓ − 1/2  �  ,  (1)  where th is the nearest-neighbor transfer amplitude and U gives the on-site part of the Coulomb  interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' (1), ˆc†  j,σ (ˆcj,σ) creates (annihilates) a spin-σ (=↑,↓) electron at Wannier lattice  site j, and ˆnj,σ = ˆc†  j,σˆcj,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' In the repulsive case (U > 0) the model realizes a Mott insulating  ground state with a finite charge gap ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Exact eigenstates of the Hubbard model can be constructed by means of the operators ˆη+ =  �  j(−1)j ˆ∆†  j,  ˆη− = ( ˆη+)†, and ˆηz = 1  2  �  j(ˆnj,↑ + ˆnj,↓ − 1), where ˆ∆†  j = ˆc†  j,↓ˆc†  j,↑ denotes the singlet pair-creation  operator [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' These so-called η operators fulfill SU(2) commutation relations [ ˆη+, ˆη−] = 2 ˆηz and  [ ˆηz, ˆη±] = ± ˆη±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Apparently, the Hubbard Hamiltonian (1) commutes with ˆη2 = 1  2( ˆη+ ˆη−+ ˆη− ˆη+)+( ˆηz)2,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', 〈η2〉 is a conserved quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Long-ranged pairing correlations 〈 ˆη+  j ˆη−  ℓ 〉 develop when the ex-  pectation value 〈 ˆη2〉 becomes finite, but such η-pairing states cannot be the ground state of the  Hubbard model [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Pulse irradiation can establish η-paired states in Mott insulators however [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  To address this issue, we apply a pump pulse with amplitude A0, frequency ωp and width σp,  2  3  PAIRING CORRELATIONS  0  5  10  15  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='5  t · th  ˜P(q = π, t)  2nd(t)  U/th = 8  Figure 1: Typical time-evolution process of ˜P(q = π, t) and 2nd(t) for the photoin-  duced η-pairing states in the strong-coupling regime of the driven Hubbard model with  U/th = 8 and pump parameters A0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='0, σp = 2t−1  h  and t0 = 10t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  The iTEBD data are obtained for bond dimension χ = 1200, ensuring a truncation error  smaller than 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' For the iTEBD calculations, we employ a second-order Suzuki-Trotter  decomposition with time step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='1t−1  h  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='01t−1  h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  centered at time t0(> 0):  A(t) = A0e−(t−t0)2/(2σ2  p) cos  �  ωp(t − t0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  (2)  The external time-dependent electric field A(t) changes the hopping amplitude by a Peierls phase [8]:  thˆc†  j,σˆcj+1,σ → theiA(t)ˆc†  j,σˆcj+1,σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', ˆH → ˆH(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' As a result, the system being initially in the ground  state, is driven out of equilibrium, |ψ(0)〉 → |ψ(t)〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  3  Pairing correlations  The η-pairing state can be detected evaluating the time evolution of the pair-correlation function  P(r, t) = 1  L  �  j  〈ψ(t)| ˆ∆†  j+r ˆ∆j + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=')|ψ(t)〉  (3)  and its Fourier transform ˜P(q, t) =  �  r eiqr P(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' As found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' [2,9] for small clusters, ˜P(π, t)  is enhanced after pulse irradiation, indicating the formation of an η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' By means of  iTEBD this is confirmed directly in the thermodynamic limit which is demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 1 for  a pump with A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='0 and σp = 2t−1  h  centered at t0 = 10t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  ˜P(π, t) shows  a clear response to pulse irradiation and is strengthened as the system progresses in time until  saturation is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Obviously, the nonlocal contributions have a stronger impact on ˜P(π, t)  than the double occupancy nd(t) = (1/L)  �  j〈ψ(t)|ˆnj,↑ˆnj,↓|ψ(t)〉 [note that P(r = 0, t) = 2nd(t),  where nd(0) > 0 for the finite U values considered].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  The enhancement process of η-pairing can be described as follows [2]: The initial state before  pulse irradiation is the ground state of the Hubbard chain with |η = 0,ηz = 0〉, which is consis-  tent with the numerical finding: ˜P(0, t = 0) ≃ 0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Turning on the pump pulse, the  3  3  PAIRING CORRELATIONS  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='2  4  6  8  10  12  A0  ωp/th  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='2  2  4  6  8  10  12  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='2  ω/th,  ωp/th  Im χJJ(ω)  0  2  4  6  8  10  �  P(π, t)/A2  0  A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='04  A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='06  A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='08  A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='10  Figure 2: (a): Contour plots of ˜P(q = π, t) in the ωp-A0 plane at t = 15t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Again  U/th = 8, and the pump is parametrized by σp = 2t−1  h  at t0 = 10t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' (b): ˜P(π, t) at  t = 15t−1  h  in the small-A0 area enclosed by the dashed square in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Dividing by  A2  0, data can be rescaled to Imχ(ω) (black line), where Imχ(ω) is the imaginary part of  the optical spectrum χJJ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Hamiltonian does not commute with the η-operators anymore,  [ ˆH(t),η+] = [ ˆH,η+]cos[A(t)] +  �  k  F(k, t)ˆc†  π−k,↓ˆc†  k,↑ ,  (4)  where F(k, t) = 4th sin[A(t)]sin k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' This alters the initial state to a state with a finite expectation  value 〈 ˆη2〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Even though the commutation relation is recovered for t ≫ t0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', [ ˆH(t), ˆη+] → [ ˆH, ˆη+]  [since A(t) → 0], |ψ(t)〉 now includes components of |η > 0,ηz = 0〉 leading to the enhancement  of ˜P(π, t), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 1 for t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Note that the enhancement of η pairing after pulse irradiation depends, however, strongly  on the pump pulse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' The optimal parameter set for inducing η-pairing states can be  determined examining the A0 and ωp dependences of ˜P(π, t) by iTEBD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Figure 2(a) shows the  contour plot of ˜P(π, t) after pulse irradiation (t = 15t−1  h ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' We find a single maximum around  A0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4 and ωp/th ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='0 (marked by the “×" symbol), instead of the stripe structure observed in  the finite-system (L = 14) exact diagonalisation (ED) simulations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Another notable results of previous ED calculations [2] was that the peak structure of ˜P(π, t) as  a function of ωp for small A0 is essentially the same as those of the ground-state optical spectrum,  χJJ(ω > 0) = − 1  L 〈ψ0|ˆJ  1  E0 − ˆH + ħhω + iηL  ˆJ|ψ0〉,  (5)  where |ψ0〉 is the ground state having energy E0 and Lorentzian width ηL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' In (5), the Hubbard-  model charge-current operator is ˆJ = ith  �  j,σ(ˆc†  j,σˆcj+1,σ − ˆc†  j+1,σˆcj,σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Figure 2(b) compares the iTEBD data, obtained for ˜P(π, t) at various small A0 and t = 15t−1  h ,  with the t-DMRG results for χJJ(ω) (using ηL/th = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='2), in dependence on ωp respectively ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Most notably, ˜P(π, t) divided by A2  0 scales to the imaginary part of the optical spectrum Imχ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  This can be understood as follows: The hopping term including the Peierls phase can be divided  4  4  NON-EQUILIBRIUM DYNAMICS  −π  π −π  π −π  π  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='2  A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t)  t = 5t−1  h  t = 10t−1  h  t = 15t−1  h  8  −8  −4  0  4  0  ω/th  k  (a) t = 5t−1  h  0  k  (b) t = 10t−1  h  0  k  (c) t = 15t−1  h  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='1  (d)  Figure 3: Snapshots of the photoemission spectra A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) indicating photoinduced η-  pairing during the pump at times t = 5t−1  h  (a), 10t−1  h  (b) and 15t−1  h  (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' The pump is  parametrized by A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='4, ωp/th = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='0 [see ‘×’-symbol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 2(a)], and σp = 2t−1  h  at  t0 = 10t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' The transient integrated density of states A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) obtained from the data of  panels (a)-(c) is depicted in panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' All data are obtained by the (i)TEBD technique  with IBC for the 1D half-filled Hubbard model with U/th = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Note that the time cutoff in  the simulation of time-dependent correlation functions is T = 5t−1  h , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', the integration  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 7 extends only over the interval −T ≤ τ1,τ2 ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' As a compromise between time  and frequency resolutions we have chosen a probe pulse width σpr = 2t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  into kinetic and current operators as  −th  �  j,σ  �  eiA(t)ˆc†  j,σˆcj+1,σ + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  �  = ˆK cos[A(t)] + ˆJ sin[A(t)],  (6)  where ˆK = −th  �  j,σ(ˆc†  j,σˆcj+1,σ +H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' For small A0 and large t, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' (6) can be  approximated by ˆJA0, yielding a significant contribution of A2  0 to the pair correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Needless  to say that the finite-size effects are eliminated by simulating the pair correlations directly in  thermodynamic limit by iTEBD, leading to the single-peak structure in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 2(b), in strong contrast  to the multiple-peak structure observed in the ED calculations [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  4  Non-equilibrium dynamics  We now analyze the non-equilibrium photoemission spectra A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) =  �  σ=↑,↓ Aσ(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) for  the optimal pump parameter set marked by the “×"-symbol in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' To explore the system  dynamics in a non-equilibrium situation, time-dependent spectral functions of the form [10]  Aσ(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) =  �  r  e−ikr  � ∞  −∞  � ∞  −∞  dτ1dτ2 f (τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='ω) · Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t)  (7)  are of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Here, the non-equilibrium two-point correlator  Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) = 〈φ(t)|ˆc†  j+r,σ(τ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t)ˆcj,σ(τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t)|φ(t)〉  (8)  5  5  CONCLUSIONS  is defined relative to t, and  f (τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='ω) = eiω(τ1−τ2)g(τ1)g(τ2), g(τ) = exp[−τ2/2σ2  pr]/  �  2πσpr  (9)  specify the shape of the probe pulse, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', in a time-dependent photoemission spectroscopy exper-  iment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' How numerically simulate two-time-dependent quantities such as Cσ(r,τ1,τ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) has been  explained in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' [7] [see paragraphs below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Figure 3 displays our (i)TEBD results for the 1D half-filled Hubbard model in the strong-  coupling regime (U/th = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Before pump irradiation the state is a Mott insulator with a noticable  single-particle gap, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 3(a) for t = 5t−1  h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' In the midst of the pump (t = 10t−1  h ), an extra  dispersion above Fermi energy (ω > EF) appears and persists afterwards [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' 3(c)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Evaluating the integrated density of states  A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) = 1  L  �  k  A(k,ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t),  (10)  we see more clearly how the spectral weight is shifted from ω < EF to ω > EF due to the photoin-  duced η-pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Figure 3(d) gives A(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' t) for the photoinduced η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' Obviously, the  spectral weight for ω > EF increases distinctly over time, indicating a photoinduced phase tran-  sition from a Mott insulator to a metallic η-pairing state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' This photoinduced insulator-to-metal  transition should be observed in time- and angle-resolved photoemission spectroscopy, when the  pure Hubbard model is realized experimentally, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=', in optical lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' We note that the pho-  toinduced phase transition cannot be observed by simulating the time-dependent photoemission  spectra with not-optimized pump-pulse parameters, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  5  Conclusions  To summarize, combining tensor-network algorithms with infinite time-evolving block decimation  techniques, we revisited the problem of photoinducing η-pairing states in the one-dimensional  Hubbard model at half band filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' This allowed us to prove the enhancement of the pairing  correlations directly in the thermodynamic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' We also determined the optimal pump-pulse  parameter set that maximizes the η-pairing tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' An η-pairing related Mott insulator to metal  transition could be extracted from the time-dependent photoemission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  We wish to stress that the numerical approach presented here can be applied to simulate the  non-equilibrium dynamics of any (quasi-)one-dimensional translational-invariant system in entire  ranges of interacting and driving parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' For example, the photoinduced metallization of  excitonic insulators was demonstrated quite recently in accordance with time- and angle-resolved  photoemission spectroscopy experiments on Ta2NiSe5 [14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Acknowledgements  The iTEBD simulations were performed using the ITensor library [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='  Funding information  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
+page_content=' was supported by Deutsche Forschungsgemeinschaft through project  EJ 7/2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9E3T4oBgHgl3EQfZQq9/content/2301.04496v1.pdf'}
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+arXiv:2301.05151v1  [math.RT]  12 Jan 2023
+A local-global principle for unipotent
+characters
+Damiano Rossi
+Abstract
+We obtain an adaptation of Dade’s Conjecture and Späth’s Character Triple Conjecture to
+unipotent characters of simple, simply connected finite reductive groups of type A, B and C.
+In particular, this gives a precise formula for counting the number of unipotent characters of
+each defect d in any Brauer ℓ-block B in terms of local invariants associated to e-local struc-
+tures. This provides a geometric version of the local-global principle in representation theory
+of finite groups. A key ingredient in our proof is the construction of certain parametrisations of
+unipotent generalised Harish-Chandra series that are compatible with isomorphisms of charac-
+ter triples.
+Contents
+1
+Introduction
+2
+1.1
+Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2
+Notation and background material
+5
+2.1
+Characters and blocks of finite groups
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+2.2
+Finite reductive groups and unipotent characters . . . . . . . . . . . . . . . . . . . . . .
+7
+2.3
+e-Harish-Chandra theory for unipotent characters . . . . . . . . . . . . . . . . . . . . .
+7
+2.4
+Pseudo-unipotent characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+9
+3
+Compatibility with isomorphisms of character triples
+10
+3.1
+Equivariance and maximal extendibility . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+3.2
+Construction of GF -block isomorphisms of character triples . . . . . . . . . . . . . . .
+14
+3.3
+Proof of Theorem C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+18
+4
+Consequences of Theorem C
+18
+4.1
+Parametrisation of pseudo-unipotent characters of Levi subgroups . . . . . . . . . . .
+19
+4.2
+Above e-Harish-Chandra series
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+21
+2010 Mathematical Subject Classification: 20C20, 20C33.
+Key words and phrases: Dade’s Conjecture, Character Triple Conjecture, finite reductive groups, unipotent characters.
+This work is partially supportedby the EPSRC grant EP/T004592/1 and was written during a research visit of the author at
+the Universitá degli Studi di Firenze. The author would like to thank Silvio Dolfi and all the members of the algebra group
+in the Department of Mathematics for their hospitality and, in particular, Carolina Vallejo for some comments concerning
+the local-global principle. Moreover, the author would like to thank Lucas Ruhstorfer for some helpful conversation on
+the paper [Bro-Ruh].
+1
+
+5
+Towards Theorem A and Theorem B
+22
+5.1
+Preliminaries on e-chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+23
+5.2
+Proof of Theorem A
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+27
+5.3
+Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+1
+Introduction
+The local-global conjectures are currently some of the most interesting and challenging problems in
+representation theory of finite groups. Among others, these include the McKay Conjecture [McK72],
+the Alperin—McKay Conjecture [Alp76] and Alperin’s Weight Conjecture [Alp87] all of which can
+be deduced by a deeper statement known as Dade’s Conjecture [Dad92], [Dad94], [Dad97]. The
+latter also implies the celebrated Brauer’s Height Zero Conjecture introduced in [Bra56] and whose
+proof has recently been completed in [MNSFT22] and [Ruh22a] while relying on a combined effort
+of many other authors.
+In this paper, we are particularly interested in Dade’s Conjecture which, for every prime number
+ℓ, suggests a precise formula for counting the number of irreducible characters of a finite group,
+with a given ℓ-defect and belonging to a given Brauer ℓ-block, in terms of the ℓ-local structure of
+the group itself. This conjecture has been further extended in [Spä17] where the Character Triple
+Conjecture was formulated by introducing a compatibility with N-block isomorphisms of character
+triples, hereinafter denoted by ∼N, as defined in [Spä17, Definition 3.6]. This notion plays a funda-
+mental role in many aspects of group representation theory and, as we will see later, gives us a way
+to control the representation theory of local subgroups. Furthermore, it was exploited to reduce
+Dade’s Conjecture to finite quasi-simple groups as explained in [Spä17, Theorem 1.3].
+Our aim is to adapt and prove the two conjectures described in the previous paragraph to the case of
+unipotent characters of finite reductive groups. The approach considered here is inspired by ideas
+introduced by the author in [Ros22c] and provides further evidence for the conjectures formulated in
+that paper [Ros22c, Conjecture C and Conjecture D]. In particular, the ℓ-local structures considered
+above are replaced by more suitable e-local structures arising from the geometry of the underlying
+algebraic group that are compatible with the framework of Deligne–Lusztig theory. Therefore, our
+results also suggest the existence of an e-local-global principle for the representation theory of finite
+reductive groups.
+More precisely, let G be a simple, simply connected group of type A, B or C which is defined
+over an algebraically closed field of positive characteristic p and let F ∶ G → G be a Frobenius
+endomorphism endowing G, as a variety, with an Fq-structure for some power q of p. We denote by
+GF the finite reductive group consisting of the Fq-rational points on G. Furthermore, we fix an odd
+prime ℓ different from p and denote by e the multiplicative order of q modulo ℓ. We let Le(G,F)
+denote the set of e-chains of (G,F) of the form σ = {G = L0 > L1 > ⋅⋅⋅ > Ln} where each Li
+is an e-split Levi subgroup of (G,F). The final term of the e-chain σ is denoted by L(σ) = Ln,
+while ∣σ∣ ∶= n is the length of σ. Observe that the latter induces a partition of the set Le(G,F) into
+the sets Le(G,F)± consisting of those e-chains σ that satisfy (−1)∣σ∣ = ±1. Furthermore, notice
+that GF acts by conjugation on the set Le(G,F) and indicate by GF
+σ the stabiliser of the e-chain
+σ. It follows directly from the definition that this action preserves the length of e-chains and, in
+particular, it restricts to an action of GF on the set Le(G,F)>0 of e-chains of positive length.
+2
+
+Now, to each non-negative integer d and Brauer ℓ-block B of the finite group GF , we associate a
+set Ld
+u(B)± consisting of quadruples (σ,M,µ,ϑ) where σ is an e-chain belonging to Le(G,F)±,
+(M,µ) is a unipotent e-cuspidal pair of (L(σ),F) such that M does not coincide with G, and ϑ is an
+irreducible character of the e-chain stabiliser GF
+σ belonging to the character set Uchd(Bσ,(M,µ))
+defined by the choice of d, B, σ and (M,µ) as described in Definition 5.5. Once again, the group GF
+acts by conjugation on Ld
+u(B)± and we indicate the corresponding set of GF -orbits by Ld
+u(B)±/GF .
+Moreover, for every such orbit ω, we denote by ω● the corresponding GF -orbit of pairs (σ,ϑ) such
+that (σ,M,µ,ϑ) ∈ ω for some unipotent e-cuspidal pair (M,µ).
+With the above notation, we are now able to state our first main result. For simplicity, in the next
+theorem we assume that the prime ℓ does not divide the greatest common divisor (q ± 1,n + 1)
+whenever (G,F) is of type An(±q) and where An(−q) denotes 2An(q) as usual. Observe however
+that this assumption can be removed as explained in Remark 5.7 (see Theorem 5.9 for the more
+general statement).
+Theorem A. For every Brauer ℓ-block B of GF and every non-negative integer d, there exists an
+AutF(GF )B-equivariant bijection
+Λ ∶ Ld
+u(B)+/GF → Ld
+u(B)−/GF
+such that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GF (Xρ,χ,GF
+ρ ,χ)
+for every ω ∈ Ld
+u(B)+/GF, any (σ,ϑ) ∈ ω●, any (ρ,χ) ∈ Λ(ω)● and where X ∶= GF ⋊ AutF(GF )
+and AutF(GF ) is the group of automorphisms described in Section 3.1.
+The above theorem provides an adaptation of Späth’s Character Triple Conjecture to the framework
+of Deligne–Lusztig theory for the unipotent characters of finite reductive groups. Theorem A also
+offers further evidence for the validity of [Ros22c, Conjecture D], in fact the set Ld
+u(B)± introduced
+above is a subset of the set of quadruples Ld(B)± considered in [Ros22c, Conjecture D] which
+is identified by only selecting unipotent e-cuspidal pairs (M,µ) among those appearing in such
+quadruples.
+Next, we obtain a formula for counting the number of unipotent characters of ℓ-defect d in the
+Brauer ℓ-block B in terms of local invariants associated to e-local structures. For each e-chain σ of
+(G,F) with positive length, we define kd
+u(Bσ) to be the number of characters belonging to one of
+the character sets Uchd(Bσ,(M,µ)) for some unipotent e-cuspidal pair (M,µ) of (L(σ),F) up to
+GF
+σ -conjugation (see also (5.10)). Furthermore, let kd
+u(B) and kd
+c,u(B) be the number of irreducible
+characters with ℓ-defect d and belonging to the Brauer ℓ-block B that are unipotent and unipotent
+e-cuspidal respectively. Then, by using the bijection given by Theorem A we can determine the
+difference kd
+u(B) − kd
+c,u(B) in terms of an alternating sum involving the terms kd
+u(Bσ) arising
+from the e-local structure GF
+σ .
+3
+
+Theorem B. For every Brauer ℓ-block B of GF and every non-negative integer d, we have
+kd
+u(B) − kd
+c,u(B) = ∑
+σ
+(−1)∣σ∣+1kd
+u(Bσ)
+where σ runs over a set of representatives for the action of GF on Le(G,F)>0.
+We point out that the restriction on the prime ℓ made for simplification before Theorem A only
+concerns the condition on isomorphisms of character triples and hence does not affect Theorem B.
+As before, this result provides an adaptation of Dade’s Conjecture to the framework of Deligne–
+Lusztig theory for the unipotent characters of finite reductive groups and gives new evidence in
+favour of [Ros22c, Conjecture C]. The necessity for the introduction of the corrective term kd
+c,u(B)
+in the equality of Theorem B can be understood as an analogue to the exclusion of the case of blocks
+with central defect in the statement of Dade’s Conjecture or, depending on the formulation under
+consideration, of the case where d = 0. We refer the reader to the more detailed discussion given in
+the paragraph following Definition 5.2. Finally, we mention that Theorem B also provides evidence
+for a positive answer to a question recently posed by Broué [Bro22a].
+It is particularly interesting to notice that, to the author’s knowledge, Theorem B cannot be obtained
+directly using techniques available at the present time, but only as a consequence of the existence
+of GF -block isomorphisms of character triples as those considered in Theorem A. In fact, while
+Deligne–Lusztig theory allows us to control the representation theory of finite reductive groups, it
+is not sufficient to control the representation theory of e-chain stabilisersGF
+σ . However, observe that
+the stabiliser GF
+σ contains the finite reductive group L(σ)F as a normal subgroup. Therefore, we
+can first use Deligne–Lusztig theory to study the characters of L(σ)F and then apply Clifford theory
+via GF -block isomorphisms of character triples to control the characters of GF
+σ (see Proposition 4.5
+and Proposition 5.6 for further details).
+In order to achieve the latter step, we need to make Deligne–Lusztig theory and, more precisely, e-
+Harish-Chandra theory for unipotent characters compatible with GF -block isomorphisms of char-
+acter triples. This ideas was first suggested by the author in [Ros22c, Parametrisation B] and further
+studied in [Ros22d]. Our next result, which is a key ingredient in the proofs of Theorem A and
+Theorem B, establishes this conjectured parametrisation in the unipotent case under the assump-
+tion specified above. This can also be seen as an extension of the parametrisation introduced by
+Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] to the language of GF-block isomorphisms
+of character triples.
+Theorem C. For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)-
+equivariant bijection
+ΩG
+(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ)
+that preserves the ℓ-defect of characters and such that
+(Xχ,GF ,χ) ∼GF (NXχ(L),NG(L)F ,ΩG
+(L,λ)(χ))
+for every χ ∈ E(GF ,(L,λ)) and where X ∶= GF ⋊ AutF(GF ).
+The proof of Theorem C, and therefore of Theorem A and Theorem B, partially relies on certain
+4
+
+conditions on the extendibility of characters of e-split Levi subgroups that were first introduced to
+settle the inductive conditions for the McKay Conjecture and the Alperin–McKay Conjecture, and
+then further studied in the context of Parametrisation B of [Ros22c] (see the exact statement given
+in [Ros22d, Definition 5.2]). These conditions were obtain, under certain assumptions, for groups
+of type A, B and C in the papers [BS20], [Bro22b] and [Bro-Ruh] respectively. Nonetheless, a
+version of these results is expected to hold in general and hence we believe that the above theorems,
+obtained here for types A, B and C with respect to an odd prime ℓ, will extend to the general case
+as well.
+1.1
+Structure of the paper
+The paper is organised as follows. In Section 2 we introduce the necessary notation and recall the
+main definitions and results used throughout the paper. Furthermore, in Section 2.4 we introduce
+the notion of pseudo-unipotent character (see Definition 2.2) and prove a result on the regularity
+of blocks covering those containing such characters. Next, in Section 3 we start working towards
+a proof of Theorem C. First, in Section 3.1 we consider certain equivariance properties that can
+be established in the presence of extendibility conditions for characters of e-split Levi subgroups.
+Here, we also present a candidate for the bijection ΩG
+(L,λ) required by Theorem C. Next, in Sec-
+tion 3.2 we construct the required GF-block isomorphisms of character triples. Using these results,
+we can then prove Theorem C in Section 3.3. The following step is to extend the parametrisation
+of unipotent e-Harish-Chandra series in the group G, as given by Theorem C, to a parametrisa-
+tion of pseudo-unipotent e-Harish-Chandra series in F-stable Levi subgroups K of (G,F). This
+is done in Theorem 4.4. Once this is established, in Section 4.2 we exploit the theory of GF -block
+isomorphisms to obtain bijections above e-Harish-Chandra series that are required to control the
+representation theory of the e-chain stabilisers GF
+σ . A more detailed analysis of the characters of
+GF
+σ is carried out in Section 5.1. In particular, we obtain a parametrisation of the character sets
+Uchd(Bσ,(M,µ)) in Proposition 5.6. Finally, in Section 5.2 and Section 5.3 we apply these results
+to prove Theorem A and Theorem B respectively.
+2
+Notation and background material
+2.1
+Characters and blocks of finite groups
+We recall some standard notation from representation theory of finite groups as can be found in
+[Isa76] and [Nav98], for instance. Let Irr(G) the set of ordinary irreducible characters. If N ⊴ G
+and ϑ ∈ Irr(N), then we denote by Irr(G ∣ ϑ) the set of irreducible characters of G that lie above
+ϑ. More generally, if S is a subset of irreducible characters of N, then we denote by Irr(G ∣ S) the
+union of the sets Irr(G ∣ ϑ) for ϑ ∈ S, that is, the set of irreducible characters of G that lie above
+some character in the set S.
+Next, we denote by Gϑ the stabiliser of the irreducible character ϑ ∈ Irr(N) under the conjugacy
+action of G and say that ϑ is G-invariant if G = Gϑ. In this case, we say that (G,N,ϑ) is a character
+triple. These objects provide important information in the study of Clifford theory and play a crucial
+role in many aspects of the local-global conjectures. Of paramount importance is the introduction of
+certain binary relations on the set of character triples. We refer the reader to [Nav18, Chapter 5 and
+10] and [Spä18] for a more detailed introduction to these ideas and for the necessary background on
+5
+
+projective representations. The binary relation considered here was introduced in [Spä17, Definition
+3.6] and is known as N-block isomorphism of character triples, denoted by ∼N. This equivalence
+relation has further been studied in [Ros22a].
+In order to construct N-block isomorphisms of character triples, it is often useful to prove certain
+results on the extendibility of characters. Here, we introduce the notion of maximal extendibility (see
+[MS16, Definition 3.5]) that will be considered in the following sections. Let N ⊴ G be finite groups
+and consider S a subset of irreducible characters of N. Then, we say that maximal extendibility
+holds for the set S with respect to the inclusion N ⊴ G if every character ϑ ∈ S extends to its
+stabiliser Gϑ. More precisely, we can specify an extension map
+Λ ∶ S →
+∐
+N≤H≤G
+Irr(H)
+(2.1)
+that sends each character ϑ ∈ S to an extension Λ(ϑ) of ϑ to the stabiliser Gϑ.
+Next, we consider modular representation theory with respect to a fixed prime number ℓ. For χ ∈
+Irr(G), there exist unique non-negative integers d(χ), called the ℓ-defect of χ, such that ℓd(χ) =
+∣G∣ℓ/χ(1)ℓ and where for an integer n we denote by nℓ the largest power of ℓ that divides n. For
+any d ≥ 0, let Irrd(G) be the set of irreducible characters χ of G that satisfy d(χ) = d and denote by
+kd(G) its cardinality. Associated to the prime ℓ, we also have the set of Brauer ℓ-blocks of G. Each
+block is uniquely determined by the central functions λB (see [Nav98, p. 49]). For every χ ∈ Irr(G),
+we denote by bl(χ) the unique block that satisfies χ ∈ Irr(bl(χ)). Furthermore, if H ≤ G and b is
+a block of H, then bG denotes the block of G obtained via Brauer’s induction (when it is defined).
+If B is a block of G and d ≥ 0, then let Irrd(B) be the set of irreducible characters belonging to the
+block B and having defect d. The cardinality of Irrd(B) is denoted by kd(B).
+We conclude this introductory section with an analogue of [Isa76, Problem 5.3] for blocks that will
+be used in the sequel.
+Lemma 2.1. Let H ≤ G be finite groups and consider blocks b of H and B of G. If ζ is a linear
+character of G, then:
+(i) there are blocks b ⋅ ζH of H and B ⋅ ζ of G satisfying
+Irr(b ⋅ ζH) = {ψζH ∣ ψ ∈ Irr(b)}
+and
+Irr(B ⋅ ζ) = {χζ ∣ χ ∈ Irr(B)};
+(ii) If bG = B, then (b ⋅ ζH)G = B ⋅ ζ.
+Proof. The first point is [Riz18, Lemma 2.1]. Next, let g ∈ G and denote by ClG(g) the G-conjugacy
+class of g and by ClG(g)+ the corresponding conjugacy class sum in the group algebra. Since the
+intersection ClG(g) ∩ H is a union of H-conjugacy classes, we can find h1,... ,hn ∈ ClG(g) ∩ H
+such that
+ClG(g) ∩ H =
+n
+∐
+i=1
+ClH(hi)
+and where n is zero if ClG(g) ∩ H is empty. In particular, observe that ζ(hi) = ζ(g) since λ is a
+6
+
+class function of G. Now, using the notation of [Nav98, p.87] we obtain
+λB⋅ζ (ClG(g)+) = λB (ClG(g)+)ζ(g)
+= λG
+b (ClG(g)+)ζ(g)
+=
+n
+∑
+i=1
+λb (ClH(hi)+)ζ(g)
+=
+n
+∑
+i=1
+λb (ClH(hi)+)ζH(hi)
+=
+n
+∑
+i=1
+λb⋅ζH (ClH(hi)+) = λG
+b⋅ζH (ClG(g))
+where for every algebraic integer α of C we denote by α its reduction modulo a maximal ideal
+containing the prime ℓ (see [Nav98, Chapter 2]). This shows that B ⋅ ζ = (b ⋅ ζH)G and we are
+done.
+2.2
+Finite reductive groups and unipotent characters
+Let G be a connected reductive group defined over an algebraic closure of a field of positive char-
+acteristic p different from ℓ and consider a Frobenius endomorphism F ∶ G → G associated with
+an Fq-structure for a power q of p. The set of Fq-rational points on the variety G is denoted by GF
+and is called a finite reductive group. By abuse of notation we also refer to the pair (G,F) as a finite
+reductive group.
+Let L be a Levi subgroup of a parabolic subgroup P of G and assume that L (but not necessarily
+P) is F-stable. Using ℓ-adic cohomology, Deligne–Lusztig [DL76] and Lusztig [Lus76] defined a
+Z-linear map
+RG
+L≤P ∶ ZIrr(LF) → ZIrr(GF )
+with adjoint
+∗RG
+L≤P ∶ ZIrr(GF ) → ZIrr(LF) .
+The exact definition can be found in [CE04, Section 8.3]. These maps are known to be independent
+of the choice of the parabolic subgroup P in almost all cases (see [BM11] and [Tay18]) and, in
+particular, in those considered in this paper. Therefore, we will always omit P and denote RG
+L≤P
+simply by RG
+L . Next, using Deligne–Lusztig induction we define the unipotent characters of GF .
+These are the irreducible characters χ of GF that appear as an irreducible constituent of the virtual
+character RG
+T(1T) for some F-stable maximal torus T of G. The set of unipotent characters of GF
+is denoted by Uch(GF ) and its cardinality by ku(GF ). Similarly, if B is a block of GF and d a
+non-negative integer, then kd
+u(B) denotes the cardinality of the intersection Uch(GF ) ∩ Irrd(B).
+2.3
+e-Harish-Chandra theory for unipotent characters
+Denote by e the multiplicative order of q modulo ℓ, if ℓ is odd, or modulo 4, if ℓ = 2. In this section,
+we collect the main results of e-Harish-Chandra theory for unipotent characters. This was first in-
+troduced by Fong and Srinivasan [FS86] for classical groups and then further developed by Broué,
+Malle and Michel [BMM93] for unipotent characters. The compatibility of this theory with Brauer
+ℓ-blocks was described by Cabanes and Enguehard in [CE94] for good primes and completed by
+7
+
+Enguehard [Eng00] for bad primes. These results also provide a description of the characters be-
+longing to unipotent blocks (see [CE94, Theorem (iii)]). Another description of these characters was
+provided by the author in [Ros22c] under certain resctrictions on the prime ℓ (see also [Ros22c, Re-
+mark 4.14] for a comparison between the two descriptions). We refer the reader to the monographs
+[CE04] and [GM20] for a more complete account of this beautiful theory.
+The theory of Φe-subgroups that constitutes the foundation of e-Harish-Chandra theory was intro-
+duced in [BM92]. Following their terminology, we say that an F-stable torus S of G is a Φe-torus if
+its order polynomial is a power of the e-th cyclotomic polynomial, that is, if P(S,F ) = Φn
+e for some
+integer n and where Φe denotes the e-th cyclotomic polynomial (see [CE04, Definition 13.3]). Then,
+we say that a Levi subgroup L of G is an e-split Levi subgroup if there exists a Φe-torus S such
+that L = CG(S). More precisely, we say that L is an e-split Levi subgroup of (G,F) to emphasise
+the role of the Frobenius endomorphism F. Observe that, for any torus T, there exists a unique
+maximal Φe-torus of T denoted by TΦe (see [CE04, Proposition 13.5 3.4]). Then, it can be shown
+that an F-stable Levi subgroup L of G is e-split if and only if L = CG(Z○(L)Φe) (see, for instance,
+[GM20, Proposition 3.5.5]).
+Next, recall that (L,λ) is a unipotent e-cuspidal pair of (G,F) if L is an e-split Levi subgroup
+of (G,F) and λ ∈ Irr(LF ) satisfies ∗RL
+M(λ) = 0 for every e-split Levi subgroup M < L. A
+character λ with the property above is said to be a unipotent e-cuspidal character of LF . We denote
+by CPu(G,F) the set of unipotent e-cuspidal pairs of (G,F) and by kc,u(GF ) the number of
+unipotent e-cuspidal characters of GF . Moreover, we define the e-Harish-Chandra series associate
+to the e-cuspidal pair (L,λ) to be the set of irreducible constituents of the virtual character RG
+L (λ),
+denoted by E(GF ,(L,λ)).
+Unipotent characters where parametrised by Broué, Malle and Michel [BMM93, Theorem 3.2] by us-
+ing e-Harish-Chandra theory. Their description can be divided into two parts. First, each unipotent
+character lies in a unique e-Harish-Chandra series, that is,
+Uch (GF ) = ∐
+(L,λ)
+E (GF ,(L,λ))
+where (L,λ) runs over a set of representatives for the action of GF on the set of unipotent e-
+cuspidal pairs of (G,F) as explained in [BMM93, Theorem 3.2 (1)]. This is a well known fact
+and will be used throughout the paper without further reference. As a consequence of the partition
+above, it now remains to parametrise the unipotent e-Harish-Chandra series. If (L,λ) is a unipotent
+e-cuspidal pair, we denote by WG(L,λ)F ∶= NG(L)F
+λ /LF the corresponding relative Weyl group.
+Then, [BMM93, Theorem 3.2 (2)] parametrises the characters in an e-Harish-Chandra series in terms
+of the characters in the relative Weyl group by showing the existence of a bijection
+Irr(WG(L,λ)F ) → E(GF ,(L,λ)).
+(2.2)
+In Section 3 we reformulate (2.2) in order to obtain Theorem C.
+Unipotent e-Harish-Chandra series are also used to parametrise the so-called unipotent blocks, that
+is, those blocks that contain unipotent characters. This is the main result of [CE94]. More precisely,
+if ℓ is odd and good for G, with ℓ ≠ 3 if 3D4 is an irreducible rational component of (G,F), then
+for every ℓ-block B of GF there exists a unipotent e-cuspidal pair (L,λ), with (L,λ) unique up to
+8
+
+GF -conjugation, such that all the irreducible constituents of RG
+L (λ) belongs to the block B. In this
+case, we write B = bGF (L,λ) and we also have
+Uch(GF ) ∩ Irr(bGF (L,λ)) = E(GF ,(L,λ)).
+Moreover, [CE94, Proposition 3.3 (ii) and Proposition 4.2] imply that bl(λ)GF = B.
+2.4
+Pseudo-unipotent characters
+We denote by (G∗,F ∗) a group in duality with (G,F) with respect to a choice of an F-stable
+maximal torus T of G and an F ∗-stable maximal torus T∗ of G∗. If τ ∶ Gsc → [G,G] is a simply
+connected covering (see [GM20, Remark 1.5.13]), then there exists an isomorphisms between the
+abelian groups
+Z(G∗)F ∗ → Irr(GF /τ(Gsc))
+z ↦ ˆzG
+according to [CE04, (8.19)]. Notice that, if L is an F-stable Levi subgroup of G, then its dual L∗ is
+an F ∗-stable Levi subgroup of G∗ and we have Z(G∗)F ∗ ≤ Z(L∗)F ∗. In particular, every element
+z ∈ Z(G∗)F ∗ defines a linear characters of ˆzL and restriction of characters yields the equality
+(ˆzG)LF = ˆzL.
+In the next definition, we consider charactersthat are obtained by multiplying these linear characters
+with unipotent characters.
+Definition 2.2. Let (K,F) be a finite reductive group and consider a Levi subgroup of L ≤ K
+and an irreducible character θ ∈ Irr(LF). We say that θ is (K,F)-pseudo-unipotent if there exists
+an element z ∈ Z(K∗)F ∗ such that θˆzL is unipotent. Moreover, for every unipotent character
+λ ∈ Uch(LF ), we denote by psK(λ) the set of (K,F)-pseudo-unipotent characters of LF of the
+form λˆzL for some z ∈ Z(K∗)F ∗. Moreover, we denote by psK(LF ) the set of all (K,F)-pseudo
+unipotent characters of LF. When the group K coincides with L, we denote the set of characters
+psL(LF ) simply by ps(LF ).
+In accordance with the terminology introduced above, we say that an e-Harish-Chandra series of
+(K,F) is pseudo-unipotent if it is of the form E(KF ,(L,ν)) for some ν ∈ psK(λ) and where
+(L,λ) is a unipotent e-cuspidal pair of (K,F). In this case, we also say that (L,ν) is a pseudo-
+unipotent e-cuspidal pair. We define the union of all the series associated to characters in psK(λ)
+by E(KF ,(L,psK(λ))). Since
+RK
+L (λˆzL) = RK
+L (λ)ˆzK
+for every z ∈ Z(K∗)F ∗ by [CE04, (8.20)], we deduce that the elements of the pseudo-unipotent
+e-Harish-Chandra series E(KF ,(L,λˆz)) are exactly the irreducible characters of the form ϕˆzK
+for some unipotent character ϕ ∈ E(KF ,(L,λ)). Moreover, we point out that λ is the unique
+unipotent character in the set psK(λ) according to [CE04, Proposition 8.26]. Similarly, the unipotent
+characters in the set E(KF ,(L,psK(λ))) are those in the series E(KF ,(L,λ)).
+Our next lemma, shows that blocks covering pseudo-unipotent characters are regular as defined in
+[Nav98, p.210].
+9
+
+Lemma 2.3. Let L be an F-stable Levi subgroup of G and suppose that ℓ is odd and good for G.
+For every LF ≤ H ≤ NG(L)F and every character ϑ ∈ Irr(H) lying above some pseudo-unipotent
+character in ps(LF ), the block bl(ϑ) is LF-regular. In particular, the Brauer induced block bl(ϑ)H is
+defined and is the unique block of H covering bl(ϑ).
+Proof. Let ϕ ∈ Uch(LF ) and z ∈ Z(L∗)F ∗ such that ϕˆzL lies below the character ϑ and chose
+a unipotent e-cuspidal pair (M,µ) of L such that ϕ ∈ E(LF ,(M,µ)). In particular, bl(ϕ) =
+bLF (M,µ) according to [CE94]. If Q ∶= Z(M)F
+ℓ , then MF = CGF (Q) according to [CE94, Propo-
+sition 3.3 (ii)]. Moreover, observe that [CE94, Proposition 4.2] implies that bl(ϕ) = bLF (M,µ) =
+bl(µ)LF while [Riz18, Lemma 2.1] implies that bl(ϕ) and bl(ϕˆzL) have the same defect groups.
+Now, applying [Nav98, Lemma 4.13 and Theorem 9.26], we can find defect groups Dϑ, Dϕ and Dµ
+of bl(ϑ), bl(ϕ) and bl(µ) respectively with the property that Dµ ≤ Dϕ ≤ Dϑ. Since Q ≤ Oℓ(MF ) ≤
+Dµ by [Nav98, Theorem 4.8], we deduce that Q ≤ Dϑ and hence CH(Dϑ) ≤ CH(Q) = MF ≤ LF.
+By [Nav98, Lemma 9.20] we conclude that the block bl(ϑ) is LF -regular. The second part of the
+lemma now follows from [Nav98, Theorem 9.19].
+3
+Compatibility with isomorphisms of character triples
+The aim of this section is to show how the bijection (2.2) can be made compatible with isomorphisms
+of character triples and with the action of automorphisms. This property was first suggested by the
+author in [Ros22c, Parametrisation B] and further studied in [Ros22d]. Our Theorem C gives a
+solution of this conjectured result for unipotent e-Harish-Chandra series and groups of type A, B
+and C. Before proceeding further, we show how the parametrisation (2.2) can be reformulated in
+a more convenient form. For this, let (L,λ) be a unipotent e-cuspidal pair of (G,F) and assume
+that ̂λ is an extension of λ to the stabiliser NG(L)F
+λ . Then, by applying Gallagher’s theorem [Isa76,
+Corollary 6.17] and the Clifford correspondence [Isa76, Theorem 6.11] we obtain a bijection
+Irr(WG(L,λ)F ) → Irr(NG(L)F ∣ λ)
+η ↦ (̂λη)
+NG(L)F
+and therefore (2.2) holds if an only if there exists a bijection
+E(GF ,(L,λ)) → Irr(NG(L)F ∣ λ) .
+(3.1)
+This new reformulation will allow us to introduce the aforementioned compatibility with isomor-
+phisms of character triple isomorphisms.
+3.1
+Equivariance and maximal extendibility
+In this section, we consider some equivariance properties for the parametrisation (3.1) which are
+related to maximal extendibility (see (2.1)) of unipotent characters.
+As in the previous sections, consider a connected reductive group G with a Frobenius endomor-
+phism F ∶ G → G defining an Fq-structure on G. We denote by AutF(GF ) the set of those auto-
+morphisms of GF obtained by restricting some bijective morphism of algebraic groups σ ∶ G → G
+that commutes with F to the set of Fq-rational points GF . Notice that the restriction of such a
+10
+
+morphism σ to GF , which by abuse of notation we denote again by σ, is an automorphism of the
+finite group GF . We refer the reader to [CS13, Section 2.4] for further details. In particular, observe
+that any morphism σ with the properties above is determined by its restriction to GF up to a power
+of F and hence it follows that AutF(GF ) acts on the set of F-stable closed connected subgroups of
+G. Then, given an F-stable closed connected subgroup H of G, we can define the set AutF(GF )H
+consisting of those automorphisms σ as above that stabilise the algebraic group H.
+Now, let ℓ be a prime number not dividing q and denote by e the order of q modulo ℓ or q modulo 4
+if ℓ = 2. In order to control the action of automorphism on unipotent e-Harish-Chandra series, we
+exploit a result of Cabanes and Späth. More precisely, in [CS13, Theorem 3.4] it was shown that the
+parametrisation given by Broué, Malle and Michel in [BMM93, Theorem 3.2 (2)] commutes with the
+action of those automorphisms in the set AutF(GF ). Notice that the statement of [CS13, Theorem
+3.4] only considers unipotent e-cuspidal pairs (L,λ) where L is a minimal e-split Levi subgroups
+(which is enough for the purpose of dealing with the McKay Conjecture). However, their proof
+works for the general case as well.
+Proposition 3.1. For every unipotent e-cuspidal pair (L,λ) of (G,F) there exists an AutF(GF )(L,λ)-
+equivariant bijection
+IG
+(L,λ) ∶ Irr(WG(L,λ)F ) → E (GF ,(L,λ))
+such that
+IG
+(L,λ)(η)(1)ℓ = ∣GF ∶ NG(L,λ)F ∣ℓ ⋅ λ(1)ℓ ⋅ η(1)ℓ
+for every η ∈ Irr(WG(L,λ)F ).
+Proof. This follows from the proof of [CS13, Theorem 3.4]. See also [Ros22d, Theorem 3.4].
+As explained at the beginning of this section, if the character λ extends to the stabiliser NG(L)F
+λ ,
+then we can use the bijection (2.2) to obtain (3.1). A similar argument can be used to include the
+equivariance property described above and obtain an equivariant version of (3.1). Observe that, by
+the discussion on automorphisms above, it follows that the group AutF(GF ) acts on the set of e-
+cuspidal pairs (L,λ) and therefore we can define the stabiliser AutF(GF )(L,λ). Furthermore, recall
+that we denote by d(χ) the ℓ-defect of an irreducible character χ.
+Corollary 3.2. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an ex-
+tension λ◇ ∈ Irr(NG(L)F
+λ ) which is additionally AutF(GF )(L,λ)-invariant. Then there exists an
+AutF(GF )(L,λ)-equivariant bijection
+ΩG
+(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ)
+such that
+d(χ) = d(ΩG
+(L,λ)(χ))
+for every χ ∈ E(GF ,(L,λ)).
+11
+
+Proof. Consider the bijection IG
+(L,λ) given by Proposition 3.1 and define the map
+ΩG
+(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ)
+IG
+(L,λ)(η) ↦ (λ◇η)NG(L)F
+for every η ∈ Irr(WG(L,λ)F ) and where λ◇ is the extension of λ to NG(L)F
+λ given in the statement.
+This is a well defined bijection by the Clifford correspondence [Isa76, Theorem 6.11] and Gallagher’s
+theorem [Isa76, Corollary 6.17]. Moreover, for every α ∈ AutF(GF ) such that (L,λ)α = (L,λ) and
+every η ∈ Irr(WG(L,λ)F ) we have
+((λ◇η)NG(L)F )
+α
+= ((λ◇η)α)
+NG(L)F
+= (λ◇ηα)NG(L)F
+because α stabilises λ◇. On the other hand
+IG
+(L,λ)(η)α = IG
+(L,λ) (ηα)
+by the properties of IG
+(L,λ) and hence we conclude that ΩG
+(L,λ) is AutF(GF )(L,λ)-equivariant. Fur-
+thermore, if we consider η ∈ Irr(WG(L,λ)F ) and define the characters χ ∶= IG
+(L,λ)(η) and ψ ∶=
+(λ◇η)NG(L)F , then the degree formula from Proposition 3.1 implies that
+ℓd(χ) =
+∣GF ∣ℓ
+χ(1)ℓ
+=
+∣NG(L,λ)F ∣ℓ
+λ(1)ℓ ⋅ η(1)ℓ
+=
+∣NG(L)F ∣ℓ
+ψ(1)ℓ
+= ℓd(ψ)
+and hence we deduce that d(χ) = d(ψ) as required.
+Next, we consider a regular embedding G ≤ ̃G as defined in [CE04, (15.1)]. Then, ̃G is a connected
+reductive group with connected centre and whose derived subgroup coincides with that of G, that
+is, [̃G, ̃G] = [G,G]. In particular, observe that ̃G = Z(̃G)G, that G is normal in ̃G and that
+the quotient ̃G/G is an abelian group. Moreover, for every Levi subgroup L of G, we deduce
+that ̃L ∶= Z(̃G)L is a Levi subgroup of ̃G and that L ≤ ̃L is again a regular embedding. These
+observations will be used throughout this paper without further reference.
+We also recall that, according to [DM91, Proposition 13.20], restriction of characters yields a bi-
+jection between the unipotent characters of ̃GF and those of GF . In particular, every unipotent
+character of GF is ̃GF -invariant. Using this observation, we can compare the relative Weyl groups
+in ̃GF with those in GF .
+Lemma 3.3. Let (L,λ)be a unipotent e-cuspidal pair of (G,F), set ̃L = LZ(̃G) and consider a unipo-
+tent extension ̃λ of λ to ̃LF. Then, N ̃
+G(L)F
+λ = N ̃
+G(L)F
+̃λ and we have W ̃
+G(̃L,̃λ)F ≃ WG(L,λ)F .
+Proof. Since ̃λ extends λ, it is clear that the stabiliser N ̃
+G(L)F
+̃λ is contained in N ̃
+G(L)F
+λ . On the
+other hand, let x ∈ N ̃
+G(L)F
+λ and observe that ̃λx is a unipotent character of ̃LF that restricts to
+λx = λ. Then, [DM91, Proposition 13.20] implies that ̃λx = ̃λ and therefore x ∈ N ̃
+G(L)F
+̃λ . From this,
+we also conclude that N ̃
+G(L)F
+̃λ = ̃LFNG(L)F
+λ and therefore that W ̃
+G(̃L,̃λ)F ≃ WG(L,λ)F .
+12
+
+As a consequence of the lemma above, we show that when λ extends to its stabiliser NG(L)F
+λ ,
+then every irreducible character of NG(L) that lies above λ is N ̃
+G(L)F -invariant and extends to
+N ̃
+G(L)F .
+Corollary 3.4. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an extension
+λ◇ ∈ Irr(NG(L)F
+λ ). Then every character of NG(L)F lying above λ extends to N ̃
+G(L)F .
+Proof. To start, we fix a unipotent extension ̃λ of λ to ̃LF and recall that N ̃
+G(L)F
+λ = N ̃
+G(L)F
+̃λ
+according to Lemma 3.3. Then, applying [Spä10, Lemma 4.1 (a)] we deduce that there exists an
+extension ̃λ◇ of λ◇ to N ̃
+G(L)F
+λ that also extends ̃λ. Consider now an irreducible character ψ of
+NG(L)F lying above λ. By Gallagher’s theorem [Isa76, Corollary 6.17] and the Clifford correspon-
+dence [Isa76, Theorem 6.11], it follows that there exists an irreducible character η of the relative
+Weyl group WG(L,λ)F such that ψ is induced from the irreducible character ψ0 ∶= ηλ◇. More-
+over, by using Lemma 3.3, we have W ̃
+G(̃L,̃λ)F ≃ WG(L,λ)F . Then, η, viewed as a character of
+NG(L)F
+λ , admits an extension, say ̃η, to N ̃
+G(L)F
+λ . Now, define ̃ψ0 ∶= ̃η̃λ◇ and observe that ̃ψ0 lies
+above ̃λ. By the Clifford correspondence, it follows that the character ̃ψ of N ̃
+G(L)F induced from
+̃ψ0 is irreducible and therefore, applying [Isa76, Problem 5.2], we conclude that ̃ψ extends ψ. The
+proof is now complete.
+We can now construct a parametrisation of unipotent e-Harish-Chandra series in the group ̃GF
+which agrees with the bijection ΩG
+(L,λ) from Corollary 3.2 via restriction of characters.
+Proposition 3.5. Let (L,λ) be a unipotent e-cuspidal pair of (G,F) and suppose that λ has an
+extension λ◇ ∈ Irr(NG(L)F
+λ ) which is additionally AutF(GF )(L,λ)-invariant. If ̃λ is a unipotent
+extension of λ to ̃LF , then there exists a bijection ̃Ω ̃
+G
+(̃L,̃λ) making the following diagram commute
+E (̃GF ,(̃L,̃λ))
+Irr(N ̃
+G(L)F ∣ ̃λ)
+E (GF ,(L,λ))
+Irr(NG(L)F ∣ λ)
+̃Ω ̃
+G
+(̃L,̃λ)
+Res
+̃
+GF
+GF
+Res
+N ̃
+G(L)F
+NG(L)F
+ΩG
+(L,λ)
+and where ΩG
+(L,λ) is the bijection given by Corollary 3.2.
+Proof. First, observe that λ has an extension ̃λ to ̃LF according to [DM91, Proposition 13.20]. More-
+over, restrictions from ̃GF to GF induces a bijection from the set E(̃GF ,(̃L,̃λ)) to E(GF ,(L,λ))
+according to [CE94, Proposition 3.1]. Next, consider a character ψ ∈ Irr(NG(L)F ) lying above λ
+and observe that ψ admits an extension ̃ψ0 ∈ Irr(N ̃
+G(L)F ) by Corollary 3.4. Let ̃λ0 be an irre-
+ducible constituent of the restriction ̃ψ0,̃LF and notice that ̃
+λ0 is an extension of λ since ̃LF /LF
+is abelian. Now, Gallagher’s theorem [Isa76, Corollary 6.17] implies that there exists a linear char-
+acter ν ∈ Irr(̃LF /LF) such that ̃λ0ν = ̃λ. Since N ̃
+G(L)F /NG(L)F ≃ ̃LF/LF we can identify ν
+with its extension to N ̃
+G(L)F . Then, it follows that the character ̃ψ ∶= ̃ψ0ν is an extension of ψ to
+N ̃
+G(L)F lying above ̃λ. Then the assignment ψ ↦ ̃ψ defines a bijection between Irr(NG(L)F ∣ λ)
+13
+
+and Irr(N ̃
+G(L)F ∣ ̃λ) whose inverse is given by restriction of characters. We can now define
+̃Ω
+̃
+G
+(̃L,̃λ) (̃χ) ∶= ̃ψ
+for every ̃χ ∈ E(̃GF ,(̃L,̃λ)) and ̃ψ ∈ Irr(N ̃
+G(L)F ∣ ̃λ) whenever ΩG
+(L,λ)(̃χGF ) = ̃ψNG(L)F .
+3.2
+Construction of GF-block isomorphisms of character triples
+From now on, we assume that G is simple, simply connected and of type A, B or C. Furthermore,
+we suppose that ℓ is odd and denote by e the order of q modulo ℓ.
+We now give a more explicit construction of the group of automorphism AutF(GF ). Fix a max-
+imally split torus T0 contained in an F-stable Borel subgroup B0 of G. This choice corresponds
+to a set of graph automorphisms γ ∶ G → G and a field endomorphism F0 ∶ G → G. More pre-
+cisely, if we consider the set of simple roots ∆ ⊆ Φ(G,T0) corresponding to the choice T0 ⊆ B0,
+then we have an automorphism γ ∶ G → G given by γ(xα(t)) ∶= xγ(α)(t) for every t ∈ Ga and
+α ∈ ±∆ and where γ is a symmetry of the Dynkin diagram of ∆, while F0(xα(t)) ∶= xα(tp) for
+every t ∈ Ga and α ∈ Φ(G,T0). Here, we denote by xα ∶ Ga → G a one-parameter subgroup
+corresponding to α ∈ Φ(G,T0). We define the subgroup A of AutF(GF ) generated by the graph
+and field automorphisms described above.
+In addition, we choose our regular embedding G ≤ ̃G to be defined in such a way that the graph and
+field automorphisms extends to ̃G (see, for instance, [MS16, Section 2B]). In particular, the group
+A acts via automorphisms on ̃GF and we can form the external semidirect product ̃GF ⋊ A which
+acts on GF . It turns out that ̃GF ⋊A and AutF(GF ) induce the same set of automorphisms on the
+finite group GF (see, for instance, [GLS98, Section 2.5]).
+Throughout this section, we consider a fixed unipotent e-cuspidal pair (L,λ) of (G,F) and a unipo-
+tent extension ̃λ of λ to ̃LF (whose existence is ensured by [DM91, Proposition 13.20]) where, as
+always, we define ̃L ∶= LZ(̃G). In the next lemma, we show that the hypothesis of Corollary 3.2 is
+satisfied under our assumptions.
+Lemma 3.6. There exists an extension λ◇ of λ to NG(L)F
+λ that is (̃GF A)(L,λ)-invariant.
+Proof. Using [BS20, Theorem 4.3 (i)], [Bro22b, Theorem 1.2 (a)] and the results of [Bro-Ruh], we
+obtain an extension λ◇ of λ to NG(L)F
+λ which is (GF A)(L,λ)-invariant. Since (̃GF A)(L,λ) =
+̃L(GF A)(L,λ) it suffices to show that λ◇ is ̃LF-invariant. However, the latter assertion follows
+immediately from the fact that λ◇ extends to N ̃
+G(L)F
+λ according to Lemma 3.3 and [Spä10, Lemma
+4.1 (a)].
+As an immediate consequence of the lemma above, we deduce that every character of NG(L)F
+lying above λ extends to N ̃
+G(L)F . This can be considered as a local analogue of [DM91, Proposition
+13.20].
+Lemma 3.7. Every irreducible character of NG(L)F lying above λ extends to N ̃
+G(L)F .
+Proof. This follows from Corollary 3.4 whose hypothesis is satisfied by Lemma 3.6.
+14
+
+We point out that, under our assumptions, every irreducible character of NG(L)F lying above λ
+extends to its stabiliser in N ̃
+G(L)F because the quotient N ̃
+G(L)F /NG(L)F is cyclic according to
+[GM20, Proposition 1.7.5]. However, in the lemma above we are also showing, using independent
+methods, that each such character is N ̃
+G(L)F -invariant.
+Using Lemma 3.6, we can now define bijections Ω ∶= ΩG
+(L,λ) and ̃Ω ̃
+G
+(̃L,̃λ) as described in Corol-
+lary 3.2 and Proposition 3.5 respectively. In what follows, we consider the sets of characters G ∶=
+E(GF ,(L,λ)), L ∶= Irr(NG(L)F ∣ λ), ̃G ∶= E(̃GF ,(̃L,̃λ)) and ̃L ∶= Irr(N ̃
+G(L)F ∣ ̃λ). Our next
+aim is to show that the parametrisation Ω is compatible with GF -block isomorphisms of charac-
+ter triples. We start by checking the group theoretic properties required for the existence of such
+isomorphisms (see [Spä17, Remark 3.7 (i)]).
+Lemma 3.8. For every χ ∈ G and ψ ∶= Ω(χ) ∈ L we have (̃GF A)L,χ = (̃GF A)L,ψ and ̃GFAχ =
+GF (̃GF A)L,ψ.
+Proof. We argue as in the proof of [Ros22c, Lemma 4.2]. To start, we observe that (̃GF A)(L,λ),χ =
+(̃GF A)(L,λ),ψ since the map Ω is (̃GF A)(L,λ)-equivariant. Set U(χ) ∶= (̃GF A)L,χ and U(ψ) ∶=
+(̃GF A)L,ψ. First, consider x ∈ U(χ) and observe that according to [BMM93, Theorem 3.2 (1)] there
+exists y ∈ NG(L)F such that (L,λ)xy = (L,λ). In particular, xy ∈ (̃GF A)(L,λ),χ = (̃GF A)(L,λ),ψ
+and hence x ∈ U(ψ) since ψy = ψ. This shows that U(χ) ≤ U(ψ). On the other hand, suppose
+that x ∈ U(ψ). By Clifford’s theorem there exists y ∈ NG(L)F such that λxy = λ and so xy ∈
+(̃GF A)L,ψ = (̃GF A)L,χ. Since χy = χ we deduce that x ∈ U(χ) and hence U(χ) = U(ψ). To
+conclude, it is enough to show that ̃GFAχ = GF U(χ). First, notice that GF U(χ) ≤ ̃GF Aχ since
+χ is ̃GF -invariant. On the other hand, for x ∈ ̃GF Aχ we know that (L,λ)x is GF -conjugate to
+(L,λ) thanks to [BMM93, Theorem 3.2 (1)]. Therefore, we obtain x ∈ GF U(χ) and as explained
+above this concludes the proof.
+We now apply Lemma 3.8 to show that the map ̃Ω satisfies some useful equivariance properties.
+Before doing so, we need to introduce some notation. For this purpose, consider a pair (G∗,F ∗)
+dual to (G,F) and a pair (̃G∗,F ∗) dual to (̃G,F). Let i∗ ∶ ̃G∗ → G∗ be the surjection induced by
+duality from the inclusion G ≤ ̃G and observe that Ker(i∗) = Z(̃G∗) since G is simply connected
+(see [CE04, Section 15.1]). As shown in [CE04, (15.2)], there exists an isomorphism
+Ker(i∗)F → Irr(̃GF /GF)
+(3.2)
+z ↦ ̂z̃
+G
+Furthermore, if L is an F-stable Levi subgroup of G and z ∈ Ker(i∗), then we define ̂z̃L to be the
+restriction of ̂z̃
+G to ̃LF and ̂zN ̃
+G(L) to be the restriction of ̂z̃
+G to N ̃
+G(L)F . We set K ∶= Ker(i∗) and
+obtain an action of the group K on the characters of ̃GF , ̃LF and N ̃
+G(L)F as defined in [Ros22d,
+Definition 2.1]. Moreover, we consider the external semidirect product (̃GF A)⋉K given by defining
+zx as the unique element of K corresponding to the character (̂z̃
+G)x of the quotient ̃GF /GF via
+the isomorphism specified in (3.2), whenever x ∈ ̃GF A and z ∈ K. Then, for every F-stable Levi
+subgroup L of G, we obtain an action of (̃GF A)L ⋉ K on the irreducible characters of ̃LF and
+N ̃
+G(L)F . We denote by ((̃GF A)L ⋉ K)̃λ the stabiliser of ̃λ ∈ Irr(̃LF ). In particular, it follows
+that ((̃GF A)L ⋉ K)̃λ acts on the sets of characters ̃G and ̃L. Next, we show that the bijection ̃Ω is
+compatible with this action.
+15
+
+Lemma 3.9. The bijection ̃Ω is (N ̃
+G(L)F (̃GF A)(L,λ) ⋊ K)̃λ-equivariant.
+Proof. Let ̃χ ∈ ̃G and ̃ψ ∈ ̃L. By the definition of ̃Ω, we have ̃Ω(̃χ) = ̃ψ if and only if Ω(χ) = ψ where
+χ ∶= ̃χGF and ψ ∶= ̃ψNG(L)F . Now, if we consider g ∈ N ̃
+G(L)F , x ∈ (̃GF A)(L,λ) and z ∈ K such
+that (gx,z) stabilises ̃λ, then we obtain
+̃Ω(̃χ(gx,z)) = ̃ψ(gx,z)
+if and only if
+Ω((̃χ(gx,z))
+GF ) = (̃ψ(gx,z))
+NG(L)F .
+(3.3)
+However, since the restriction of ̃χ(gx,z) to GF coincides with χx and the restriction of ̃ψ(gx,z) to
+NG(L)F coincides with ψx, we deduce that the equality in (3.3) holds by the equivariant properties
+of Ω as described in Corollary 3.2.
+One of the main ingredients for the construction of the projective representations needed to obtain
+GF -block isomorphisms of character triples is given by the following two lemmas on maximal
+extendibility.
+Lemma 3.10. Maximal extendibility holds for G with respect to the inclusion GF ⊴ GF A, that is,
+every character χ ∈ G extends to GF Aχ.
+Proof. If G is of type B or C then the result follows from [Isa76, Corollary 11.22] since A is cyclic.
+Then, we can assume that G is of type A in which case the result follows from [CS17, Theorem 4.1]
+(see also [Mal08, Theorem 2.4]).
+The local version of the lemma above is a consequence of the results obtained in [BS20].
+Lemma 3.11. Maximal extendibility holds for L with respect to the inclusion NG(L)F ⊴ (GF A)L,
+that is, every character ψ ∈ L extends to (GF A)L,ψ.
+Proof. As in the proof of Lemma 3.10, it is enough to prove the result in the case where G is of type
+A. In fact, if G is of type B or C, then the quotient (GF A)(L,ψ)/NG(L)F is cyclic because it it a
+subquotient of A. Now, if G is of type A the result follows from [BS20, Theorem 1.2].
+Finally, we can start constructing isomorphisms of character triples for the bijection Ω. As a first
+step, we obtain a weaker isomorphism, know as GF -central isomorphism of character triples and
+denoted by ∼c
+GF , whose requirements are given by [Spä17, Remark 3.7 (i)-(iii)] and replacing the
+condition on defect groups by imposing that CG(N) ≤ H1 ∩ H2 with the notations used there. We
+refer the reader to [Ros22b, Definition 3.3.4] for a precise definition.
+Proposition 3.12. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have
+(̃GFAχ,GF ,χ) ∼c
+GF ((̃GF A)L,ψ,NG(L)F ,ψ) .
+16
+
+Proof. We start by constructing projective representations associated with χ and ψ. According to
+Proposition 3.5 we can find a unipotent extension ̃χ ∈ ̃G of χ to ̃GF . Furthermore, by Lemma 3.10
+there exists an extension χ′ of χ to GF Aχ. Let ̃Dglo be a representation of ̃GF affording ̃χ and D′
+glo
+a representation of GF Aχ affording χ′. Now, [Spä12, Lemma 2.11] implies that
+Pglo ∶ (̃GF A)χ → GLχ(1)(C)
+defined by Pglo(x1x2) ∶= ̃
+Dglo(x1)D′
+glo(x2) for every x1 ∈ ̃GF and x2 ∈ GF Aχ is a projective
+representation associated with χ. Next, observe that ̃ψ ∶= ̃Ω(̃χ) ∈ ̃L is an extension of ψ to N ̃
+G(L)F
+and consider an extension ψ′ of ψ to (GF A)L,ψ given by Lemma 3.11. Let ̃Dloc be a representation
+of N ̃
+G(L)F affording ̃ψ and D′
+loc a representation of (GF A)L,ψ affording ψ′. Once again, [Spä12,
+Lemma 2.11] shows that the map
+Ploc ∶ (̃GF A)L,ψ → GLψ(1)(C)
+given by Ploc(x1x2) ∶= ̃Dloc(x1)D′
+loc(x2) for every x1 ∈ N ̃
+G(L)F and x2 ∈ (GF A)L,ψ is a pro-
+jective representation associated with ψ. We denote by αglo and αloc the factor set of Pglo and
+Ploc respectively. As explained in the proof of [Ros22d, Theorem 4.3], in order to prove that αglo
+coincides with αloc via the isomorphism ̃GF Aχ/GF ≃ (̃GF A)L,ψ/NG(L)F , it suffices to show
+that
+(µglo
+x )N ̃
+G(L)F = µloc
+x
+(3.4)
+for every x ∈ (GF A)L,χ and where µglo
+x
+∈ Irr(̃GF /GF ) and µloc
+x
+∈ Irr(N ̃
+G(L)F /NG(L)F ) are
+determined by Gallagher’s theorem (see [Isa76, Corollary 6.17]) via the equalities ̃χ = µglo
+x ̃χx and ̃ψ =
+µloc
+x ̃ψx respectively. Because (GF A)L,χ = NG(L)F (GF A)(L,λ),χ, we may assume that x stabilises
+λ. Let z ∈ K such that µglo
+x
+= ˆz̃
+G and observe that (x,z) is an element of (GF A)(L,λ),χ ⋊ K that
+stabilises ̃χ. Then, applying [BMM93, Theorem 3.2 (1)], we deduce that ̃λ and ̃λ(x,z) are N ̃
+G(L)F -
+conjugate and we may choose g ∈ N ̃
+G(L)F such that ̃λ = (̃λ(x,z))g = ̃λ(xg,z). In other words
+(xg,z) ∈ (N ̃
+G(L)F (̃GF A)(L,λ) ⋊ K)̃λ
+and thus Lemma 3.9 implies that the equality ̃χ = ̃χ(xg,z) holds if and only if ̃ψ = ̃ψ(xg,z). From this,
+we immediately deduce the equality required in (3.4).
+Next, denote by ζglo and ζloc the scalar functions associated to Pglo and Ploc respectively. To con-
+clude the proof, it remains to show that ζglo and ζloc coincide on C(̃
+GF A)χ(GF ) = Z(̃GF ). As in the
+proof of [Ros22d, Theorem 4.3], it is enough to show that the restrictions of ̃χ and ̃ψ to Z(̃GF ) are
+multiples of a common irreducible constituent. This follows from the fact that unipotent characters
+contain the center in their kernel. In fact, on one hand, 1Z(̃
+GF ) is the unique irreducible constituent
+of ̃χZ(̃
+GF ) because ̃χ is unipotent. On the other hand, ̃ψ lies above ̃λ and, since Z(̃GF ) ≤ Z(̃LF )
+and ̃λ is unipotent, we deduce that 1Z(̃
+GF ) is the unique irreducible constituent of ̃ψZ(̃
+GF ). This
+completes the proof.
+We conclude this section by verifying the remaining condition [Spä17, Remark 3.7 (iv)] and obtain
+the required GF -block isomorphisms of character triples for the map Ω.
+17
+
+Proposition 3.13. For every χ ∈ G and ψ ∶= Ω(χ) ∈ we have
+(̃GFAχ,GF ,χ) ∼GF ((̃GF A)L,ψ,NG(L)F ,ψ) .
+Proof. By Proposition 3.12 it is enough to check the block theoretic requirement given by [Spä17,
+Remark 3.7 (ii) and (iv)]. First, observe that under our assumption [CE94, Proposition 3.3 (ii)] shows
+that LF = CGF (E) where E ∶= Z(L)F
+ℓ . In particular, NJ(L) = NJ(E) for every GF ≤ J ≤
+̃GF . Furthermore, for every block C0 of NJ(L) and every defect group D of C0 we have E ≤
+Oℓ(NJ(L)) ≤ D and hence C ̃
+GF (D) ≤ N ̃
+G(L)F . Now, [KS15, Theorem B] implies that for every
+block C of N ̃
+G(L)F covering C0, the induced blocks B ∶= C ̃GF and B0 ∶= CJ
+0 are well-defined and
+B covers B0.
+Let ̃χ ∈ ̃G be an extension of χ and set ̃ψ ∶= ̃Ω(̃χ). By Lemma 2.3 the block of ̃C of ̃ψ coincides
+with the induced block bl(̃λ)N ̃
+G(L)F . Furthermore, by [CE94, Proposition 4.2] we know that the
+block ̃B of ̃χ coincides with b ̃
+GF (̃L,̃λ) = bl(̃λ)̃
+GF . Then, by the transitivity of block induction we
+get ̃
+B = ̃C ̃
+GF . Consider now GF ≤ J ≤ ̃GF as in the previous paragraph and notice that bl(̃χJ)
+is the unique block of J covered by ̃B. Now, since bl(̃ψNJ (L)) is covered by ̃C, we deduce that
+bl(̃ψNJ (L))J is covered by ̃B and therefore
+bl(̃χJ) = bl( ̃ψNJ (L))
+J .
+(3.5)
+As explained in the proof of [Ros22d, Theorem 4.8] we can now use (3.5) together with Proposition
+3.12 to conclude the proof via an application of [Spä17, Theorem 4.1 (i)].
+3.3
+Proof of Theorem C
+Proof of Theorem C. The hypothesis of Corollary 3.2 is satisfied under our restrictions on G accord-
+ing to Lemma 3.6 and therefore we obtain an AutF(GF )(L,λ)-equivariant bijection
+ΩG
+(L,λ) ∶ E (GF ,(L,λ)) → Irr(NG(L)F ∣ λ)
+that preserves the ℓ-defect of characters. Next, observe that the groups ̃GF A and X ∶= GF ⋊
+AutF(GF ) induce the same automorphisms on GF according to the description given in [GLS98,
+Section 2.5]. Then, by applying [Spä17, Theorem 5.3] and Proposition 3.13, we conclude that
+(Xχ,GF ,χ) ∼GF (NX(L)ψ,NG(L)F ,ψ)
+for every χ ∈ E(GF ,(L,λ)) and where ψ ∶= ΩG
+(L,λ)(χ) and the proof is now complete.
+4
+Consequences of Theorem C
+In this section, we collect some consequences of Theorem C. First, we extend the parametrisation
+obtained in Theorem C from unipotent e-Harish-Chandra series of the simple group G to pseudo-
+unipotent (see Definition 2.2) e-Harish-Chandra series of the Levi subgroups of G. More precisely,
+for every F-stable Levi subgroup K of G, we construct a parametrisation of the e-Harish-Chandra
+series associated to e-cuspidal pairs of the form (L,λ) for some (K,F)-pseudo-unipotent character
+λ ∈ psK(LF ). In a second step, we construct character bijections above this parametrisation by ex-
+ploiting results on isomorphisms of character triples (see Corollary 4.6). This will allow us to control
+the characters of e-chain stabilisers lying above pseudo-unipotent characters (see Proposition 5.6).
+18
+
+4.1
+Parametrisation of pseudo-unipotent characters of Levi subgroups
+Let K be an F-stable Levi subgroup of G and set K0 ∶= [K,K]. Observe that since the group G
+is simply connected, the subgroup K0 is also simply connected according to [MT11, Proposition
+12.14]. In addition, under our assumption on the type of G, we deduce that the simple components
+of K0 can only be of some of the types A, B or C.
+Proposition 4.1. For every unipotent e-cuspidal pair (L0,λ0) of (K0,F) there exists a defect pre-
+serving AutF(KF
+0 )(L0,λ0)-equivariant bijection
+ΩK0
+(L0,λ0) ∶ E (KF
+0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0)
+such that
+(Yϑ,KF
+0 ,ϑ) ∼KF
+0 (NYϑ(L0),NK0(L0)F ,ΩK0
+(L0,λ0)(ϑ))
+for every ϑ ∈ E(KF
+0 ,(L0,λ0)) and where Y ∶= KF
+0 ⋊ AutF(KF
+0 ).
+Proof. Notice that K0 is the direct product of simple algebraic groups K1,... ,Kn and that the action
+of F permutes the simple components Ki. Denote the direct product of the simple components in
+each F-orbit by Hj for j = 1,... ,t. The (Hj,F) are the irreducible rational components of (K,F)
+and we have KF
+0 = HF
+1 × ⋯ × HF
+t . Similarly, if we define the intersections Mj ∶= L0 ∩ Hj, then
+we have a decomposition LF
+0 = MF
+1 × ⋯ × MF
+t . In particular, we can write λ0 = µ1 × ⋯ × µt
+with µj ∈ Irr(MF
+j ). In this case, notice that (Mj,µj) is a unipotent e-cuspidal pair of (Hj,F).
+Next, suppose that Hj = Hj,1 × ⋅⋅⋅ × Hj,mj and observe that HF
+j ≃ HF mj
+j,1 . By the discussion at the
+beginning of this section we know that Hj,1 is a simple, simply connected group of type A, B or C
+and hence it satisfies the assumptions of Theorem C. Then, via the isomorphism HF
+j ≃ HF mj
+j,1 , we
+obtain an AutF(HF
+j )(Mj,µj)-equivarint bijection
+ΩHj
+(Mj,µj) ∶ E (HF
+j ,(Mj,µj)) → Irr(NHj(Mj)F ∣µj)
+that preserves the defect of characters and such that
+(Yj,ϑ,HF
+j ,ϑ) ∼HF
+j (NYj,ϑ(Mj),NHj(Mj)F ,ΩHj
+(Mj,µj)(ϑ))
+(4.1)
+for every ϑ ∈ E(HF
+j ,(Mj,µj)) and where Yj ∶= HF
+j ⋊ AutF(HF
+j ). Since the characters in the sets
+E(KF
+0 ,(L0,λ0)) and Irr(NK0(L0)F ∣ λ0) are direct products of characters belonging to the sets
+E(HF
+j ,(Mj,µj)) and Irr(NHj(Mj)F ∣ µj) respectively, we obtain a bijection
+ΩK0
+(L0,λ0) ∶ E (KF
+0 ,(L0,λ0)) → Irr(NK0(L0)F ∣λ0)
+by setting
+ΩK0
+(L0,λ0) (ϑ1 × ⋅⋅⋅ × ϑt) ∶= ΩH1
+(M1,µ1)(ϑ1) × ⋅⋅⋅ × ΩHt
+(Mt,µt)(ϑt)
+for every ϑj ∈ E(HF
+j ,(Mj,µj)). Finally, arguing as in the proof of [Ros22c, Proposition 6.5], we de-
+duce that the bijection ΩK0
+(L0,λ0) preserves the defect of characters, is AutF(KF
+0 )(L0,λ0)-equivariant,
+and, using (4.1), it induces the KF
+0 -block isomorphisms of character triples required in the state-
+ment.
+19
+
+In our next result, we replace the automorphism group Y ∶= KF
+0 ⋊ AutF(KF
+0 ) with the group of
+automorphisms of GF stabilising K, that is, X ∶= (GF ⋊ AutF(GF ))K. To do so, we apply the
+so-called Butterfly theorem [Spä17, Theorem 5.3] which basically states that, for any finite group G,
+the notion of G-block isomorphism of character triples only depends on the automorphisms induced
+on G.
+Corollary 4.2. Let (L0,λ0) be a unipotent e-cuspidal pair of (K0,F). The map ΩK0
+(L0,λ0) given by
+Proposition 4.1 is AutF(GF )K,(L0,λ0)-equivariant and satisfies
+(Xϑ,KF
+0 ,ϑ) ∼KF
+0 (NXϑ(L0),NK0(L0)F ,ΩK0
+(L0,λ0)(ϑ))
+(4.2)
+for every ϑ ∈ E(KF
+0 ,(L0,λ0)) and where X ∶= (GF ⋊ AutF(GF ))K.
+Proof. First, observe that AutF(GF )K is contained in AutF(KF
+0 ) because K0 is an F-stable char-
+acteristic subgroup of K. In particular, we deduce that the map ΩK0
+(L0,λ0) is AutF(GF )K,(L0,λ0)-
+equivariant. Next, to obtain (4.2), we apply [Spä17, Lemma 3.8 and Theorem 5.3] to the isomorphism
+of character triples given by Proposition 4.1 as explained in the proof of [Ros22c, Corollary 6.8].
+Isomorphisms of character triples play a fundamental role in representation theory of finite groups
+and in the study of the local-global conjectures. One of the most important consequences of the
+existence of isomorphisms of character triples is the possibility to lift character bijections. For in-
+stance, the main result of [NS14], shows how to apply this technique to construct bijections above
+characters of height zero in the context of the Alperin–McKay Conjecture [NS14, Theorem B]. The
+main consequence of this result, which follows from an argument introduced by Murai [Mur12], is
+a reduction theorem for the celebrated Brauer’s Height Zero Conjecture [NS14, Theorem A]. This
+strategy ultimately lead to the solution of Brauer’s conjecture [Ruh22a] and [MNSFT22]. For other
+applications of isomorphisms of character triples see [Tur17], [NSV20], [Ros22a], [Ruh22b] [Ros23]
+and [MR22, Proposition 1.1].
+In our next result, we exploit this idea in order to lift the bijections given by Proposition 4.1 to
+the Levi subgroup K. Consequently, we extend the parametrisation of unipotent e-Harish-Chandra
+series given by Theorem C for the simple group G to a parametrisation of e-Harish-Chandra series
+associated to (K,F)-pseudo-unipotent characters for every F-stable Levi subgroup K of G. First,
+we need a preliminary lemma.
+Lemma 4.3. Let (L,λ) be a unipotent e-cuspidal pair of (K,F) and define X ∶= (GF ⋊AutF(GF ))K.
+If KF ≤ H ≤ NG(L)F and Q is an ℓ-radical subgroup of NH(L), then CX(Q) ≤ NX(L).
+Proof. Let E ∶= Z(L)F
+ℓ and observe that L = C○
+G(E) according to [CE94, Proposition 3.3 (ii)]. Now,
+since Oℓ(NH(L)) is the smallest ℓ-radical subgroup of NH(L) [Dad92, Proposition 1.4], we deduce
+that E ≤ Oℓ(NH(L)) ≤ Q and it follows that CX(Q) ≤ CX(E) ≤ NX(L) as wanted.
+20
+
+Theorem 4.4. For every unipotent e-cuspidal pair (L,λ) of (K,F) there exists a defect preserving
+AutF(GF )K,(L,λ)-equivariant bijection
+ΩK
+(L,λ) ∶ E (KF ,(L,psK(λ))) → Irr(NK(L)F ∣ psK(λ))
+such that
+(Xχ,KF ,χ) ∼KF (NXχ(L),NK(L)F ,ΩK
+(L,λ)(χ))
+for every χ ∈ E(KF ,(L,psK(λ))) and where X ∶= (GF ⋊ AutF(GF ))K.
+Proof. Recall that K0 = [K,K] and define L0 ∶= L ∩ K0 and λ0 the restriction of λ to LF
+0 . Observe
+that (L0,λ0) is a unipotent e-cuspidal pair of (K0,F). Let z ∈ Z(K∗)F ∗ and consider a character
+χ belonging to E(KF ,(L,λˆzL)). Since the restriction of λˆzL to LF
+0 coincides with λ0, [GM20,
+Corollary 3.3.25] implies that χ lies above some character in E(KF
+0 (L0,λ0)). On the other hand,
+suppose that χ ∈ Irr(KF ) lies above some χ0 ∈ E(KF
+0 ,(L0,λ0)). By [CE94, Proposition 3.1] the
+character χ0 has an extension χ′ ∈ E(KF ,(L,λ)) and hence, using Gallagher’s theorem [Isa76,
+Corollary 6.17] and [CE04, (8.19)], we can find z ∈ Z(K∗)F ∗ such that χ = χ′ˆzK. Since χ′ˆzK is a
+character of E(KF ,(L,λˆzL)) according to [CE04, (8.20)], we conclude that
+E (KF,(L,psK(λ))) = Irr(KF ∣ E (KF
+0 ,(L0,λ0))) .
+(4.3)
+Next, suppose that ψ ∈ Irr(NK(L)F ∣ λˆzL). In this case, ψ lies above the restriction of λˆzL to LF
+0
+which coincides with λ0. In particular, there exists some ϕ ∈ Irr(NK0(L0)F ∣ λ0) such that ψ lies
+above ϕ. On the other, if χ lies above such a character ϕ ∈ Irr(NK0(L0)F ∣ λ0), then it lies above
+λ0 and therefore we can find z ∈ Z(K∗)F ∗ such that ψ ∈ Irr(NK(L)F ∣ λˆzL). This shows that
+Irr(NK(L)F ∣ psK(λ)) = Irr(NK(L)F ∣ Irr(NK0(L0)F ∣ λ0)).
+(4.4)
+Finally, consider the map ΩK0
+(L0,λ0) given by Proposition 4.1. Then, the result follows from (4.3)
+and (4.4) by applying [Ros22c, Proposition 6.1 and Remark 6.2] as explained in the proof of [Ros22c,
+Corollary 6.10] and using the KF -block isomorphisms of character triples obtained in Corollary 4.2.
+Here, we consider A ∶= GF ⋊ AutF(GF ), A0 ∶= NA(L), K ∶= KF
+0 , K0 = NK0(L)F = NK0(L0)F ,
+G ∶= GF , X ∶= (GF ⋊ AutF(GF ))K, S ∶= E(KF
+0 ,(L0,λ0)), S0 ∶= Irr(NK0(L0)F ∣ λ0), V ∶=
+(GF ⋊ AutF(GF ))K,S and U ∶= (GF ⋊ AutF(GF ))K,L,Y0. Observe that the condition on defect
+groups required by [Ros22c, Proposition 6.1] is satisfied by Lemma 4.3.
+4.2
+Above e-Harish-Chandra series
+We now further extend Theorem C by lifting the character bijections from Theorem 4.4 with respect
+to normal inclusions.
+Proposition 4.5. Consider the setup of Theorem 4.4 and let KF ≤ H ≤ NG(K)F . Then, there exists
+a defect preserving AutF(GF )H,K,(L,λ)-equivariant bijection
+ΩK,H
+(L,λ) ∶ Irr(H ∣E (KF,(L,psK(λ)))) → Irr(NH(L)∣psK(λ))
+such that
+(NX(H)χ,H,χ) ∼H (NX(H,L)χ,NH(L),ψ)
+for every χ ∈ Irr(H ∣ E(KF ,(L,psK(λ)))) and where X ∶= (GF ⋊ AutF(GF ))K.
+21
+
+Proof. We apply [Ros22c, Proposition 6.1] to the bijection given by Theorem 4.4. We consider A ∶=
+GF ⋊ AutF(GF ), G ∶= GF, K ∶= KF , A0 ∶= NA(L), X ∶= NA(K), S ∶= E(KF ,(L,psK(λ))),
+S0 ∶= Irr(NK(L)F ∣ psK(λ)), U ∶= X0,λ, V ∶= XS and J ∶= H. Notice that the conditions (i)-(iii) of
+[Ros22c, Proposition 6.1] are satisfied by [BMM93, Theorem 3.2 (1)]. Furthermore, the requirements
+about defect groups are satisfied by Lemma 4.3. Therefore, as explained in [Ros22c, Proposition
+6.11], we obtain the claimed result by applying [Ros22c, Proposition 6.1 and Remark 6.2].
+Before proceeding further, we point out an interesting analogy with another important character
+correspondence. The Glauberman correspondence plays a fundamental role in the study of the local-
+global counting conjectures and lies at the heart of most reduction theorems. In its most basic form,
+it states that for every finite ℓ-group L acting on a finite ℓ′-group K, there exists a bijection
+fL ∶ IrrL(K) → Irr(NK(L))
+between the set of L-invariant characters of K and the characters of the normaliser NK(L) (see,
+for instance, [Nav18, Section 2.3]). A very deep result due to Dade [Dad80] and recently reproved by
+Turull [Tur08], shows that, if K and L are subgroups of a finite group G and KP ≤ H ≤ KNG(L),
+then the Glauberman correspondence fL can be lifted to a character correspondence for H, that is,
+there exists a bijection
+f H
+L ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ fL(χ))
+(4.5)
+for every χ ∈ IrrL(K). On the other hand, the parametrisation of unipotent e-Harish-Chandra
+series obtained by Broué, Malle and Michel [BMM93, Theorem 3.2] lies at the centre of the proofs
+of the local-global counting conjectures for finite reductive groups. It is interesting to note that our
+methods yield a character bijection above e-Harish-Chandra series which is analogous to (4.5) in
+the context of the Glauberman correspondence. This is an immediate consequence of Proposition
+4.5.
+Corollary 4.6. Consider the setup of Theorem 4.4 and let KF ≤ H ≤ NG(K)F . Then, there exists a
+bijection
+ΨH
+χ ∶ Irr(H ∣ χ) → Irr(NH(L) ∣ ΩK
+(L,λ)(χ))
+for every χ ∈ E(KF ,(L,psK(λ))).
+Proof. This follows immediately from the proof of Proposition 4.5 by following the construction
+made in [Ros22c, Proposition 6.1].
+5
+Towards Theorem A and Theorem B
+Finally, we apply the results obtained in the previous sections to prove Theorem A which is our
+main result. Then, we obtain Theorem B as a corollary by applying the e-Harish-Chandra theory
+for unipotent characters developed by Broué, Malle and Michel [BMM93] and by Cabanes and En-
+guehard [CE94]. Before doing so, we introduce the relevant notation and prove some preliminary
+results.
+22
+
+5.1
+Preliminaries on e-chains
+Our first aim is to define e-local structures for finite reductive groups that play a role analogue to that
+of ℓ-chains in the context of Dade’s Conjecture and the Character Triple Conjecture. The connection
+between the set of e-chains and that of ℓ-chains has already been studied in [Ros22c, Section 7.2].
+These results provide a way to obtain Dade’s Conjecture and the Character Triple Conjecture as a
+consequence of [Ros22c, Conjecture C and Conjecture D]. The possibility to use different types of
+chains is crucial in the study of Dade’s Conjecture and has been introduced by Knörr and Robinson
+[KR89]. Their results were insipred by previous studies conducted by many authors including Brown
+[Bro75] and Quillen [Qui78] who analised the homotopy theory of associated simplicial complexes.
+Definition 5.1. We denote by Le(G,F) the set of e-chains of the finite reductive group (G,F),
+that is, chains of the form
+σ = {G = L0 > L1 > ⋅⋅⋅ > Ln}
+where n is a non-negative integer and each Li is an e-split Levi subgroup of (G,F). We denote by
+∣σ∣ ∶= n the length of the e-chain σ and by L(σ) its last term. Furthermore, we define Le(G,F)>0
+to be the set of e-chains having length strictly larger than 0.
+Observe that the notion of length defined above, induces a partition of the set Le(G,F) into e-
+chains of even and odd length. More precisely, we denote by Le(G,F)± the subset of those e-chains
+σ ∈ Le(G,F) that satisfy (−1)∣σ∣ = ±1.
+In what follows, given an e-chain σ and an e-split Levi subgroup M of (L(σ),F), we denote by
+σ+M the e-chain obtainedby adding M at the end of σ. We also allow the possibility that M = L(σ),
+in which case we have σ + L(σ) = σ. Vice versa, we denote by σ − L(σ) the e-chain obtained by
+removing the last term L(σ) from σ. In this way we obtain (σ + M) − L(σ + M) = σ where as
+usual L(σ + M) denotes the final term of the e-chain σ + M. Here, we use the convention that
+σ0 − L(σ0) = σ0 = σ0 + G where σ0 = {G} is the trivial e-chain.
+Next, consider the action of GF on the set of e-chains Le(G,F) induced by conjugation: for every
+g ∈ GF and σ = {Li}i, we define
+σg ∶= {G = L0 > Lg
+1 > ⋅⋅⋅ > Lg
+n}.
+It follows from this definition that the stabiliser GF
+σ coincides with the intersection of the normalis-
+ers NG(Li)F for i = 1,... ,n. Similarly, we can define an action of AutF(GF ) on Le(G,F) and
+give an analogous description of the chains stabilisers AutF(GF )σ. In particular, notice that the last
+term of the chain satisfies L(σ)F ⊴ GF
+σ . Using this observation, we can use the results of Section
+4.2 to control the characters of GF
+σ that lie above pseudo-unipotent series of L(σ).
+Definition 5.2. For every e-chain σ ∈ Le(G,F) we denote by CPu(σ) the set of unipotent e-
+cuspidal pairs (M,µ) ∈ CPu(L(σ),F) that satisfy M < G. Furthermore, for any such pair (M,µ) ∈
+CPu(σ), we define the character set
+Uch(GF
+σ ,(M,µ)) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+Irr(GF
+σ ∣ E (L(σ)F ,(M,psL(σ)(µ))))
+L(σ) > M
+(5.1)
+Irr(GF
+σ ∣ E (L(σ)F ,(M,psL(σ−L(σ))(µ))))
+L(σ) = M
+(5.2)
+23
+
+The need to distinguish the cases (5.1) and (5.2) will become apparent in the proofs of Proposition
+5.6 and Theorem 5.9 below. Observe that in the definition above, we are excluding the degenerate
+case where G = L(σ) = M and therefore the chain σ − L(σ) in the case (5.2) is always defined. To
+understand the reason why we are excluding this case, we can consider an analogy with Dade’s Con-
+jecture. For every finite group G, recall that k(G) denotes the number of its irreducible characters
+and that, for any non-negative integer d, the symbol kd(G) denotes the number of those irreducible
+characters of ℓ-defect d. The local-global counting conjectures provide a way to determine the global
+invariants kd(G) in terms of ℓ-local structures. This idea was made precise by Isaacs and Navarro
+[IN20]. According to their definitions, the block-free version of Dade’s Conjecture can be stated by
+saying that the functions kd are chain local for every d > 0. Consequently, and because a sum of
+chain local functions is chain local, we deduce that the difference k − k0 = ∑d>0 kd is a chain local
+function. On the other hand, using the fact that groups admitting a character of ℓ-defect zero have
+trivial ℓ-core, it is easy to see that k0 is not chain local. The exclusion of the case G = L(σ) = M can
+be explained by interpreting these observations in the context of unipotent characters. Recall that
+ku(GF ) and kc,u(GF ) denote the number of unipotent characters of GF and unipotent e-cuspidal
+characters of GF respectively. If ℓ does not divide the order of Z(GF ), then [CE94] implies that the
+unipotent e-cuspidal characters of GF have defect zero. Therefore, as in the case of Dade’s Con-
+jecture, the global invariant we want to determine e-locally is the difference ku(GF ) − kc,u(GF ).
+Finally, notice that kc,u(GF ) is exactly the number of unipotent e-cuspidal pairs (M,µ) of L(σ)
+satisfying G = L(σ) = M.
+In the following lemma, we show that if the set Uch(GF
+σ ,(M,µ)) is non-empty then (M,µ) is
+uniquely defined up to GF
+σ -conjugation.
+Lemma 5.3. Let σ ∈ Le(G,F) and consider two unipotent e-cuspidal pairs (M,µ) and (K,κ)
+in CPu(σ). If the sets Uch(GF
+σ ,(M,µ)) and Uch(GF
+σ ,(K,κ)) have non-trivial intersection, then
+(M,µ) and (K,κ) are GF
+σ -conjugate.
+Proof. Suppose that ϑ is a character belonging to Uch(GF
+σ ,(M,µ)) and Uch(GF
+σ ,(K,κ)). If we
+set L ∶= L(σ), then we can find elements s,t ∈ Z(L∗)F ∗ and characters ϕ ∈ E(LF ,(M,µ)) and
+ψ ∈ E(LF ,(K,κ)) such that ϑ lies above ϕˆsL and ψˆtL. By Clifford’s theorem, we deduce that
+ϕˆsL = (ψˆtL)g for some g ∈ GF
+σ . Furthermore, since ˆs is a linear character, we obtain that ϕ =
+ψg(ˆtL)g(ˆsL)−1. Since both ϕ and ψg are unipotent characters of LF , using [CE04, Proposition
+8.26] we deduce that (ˆtL)g(ˆsL)−1 = 1L and therefore ϕ = ψg. But then, [BMM93, Theorem 3.2(1)]
+shows that (M,µ) and (K,κ)g are LF-conjugate and the result follows.
+Next, we describe the block theory associated to characters in the sets introduced in Definition 5.2.
+Lemma 5.4. Let σ ∈ Le(G,F) and consider a unipotent e-cuspidal pair (M,µ) ∈ CPu(σ) and a
+character ϑ ∈ Uch(GF
+σ ,(M,µ)). Then:
+(i) the block bl(ϑ) is L(σ)F -regular;
+(ii) if the character ϑ lies above a given ϕˆzL(σ) ∈ E(L(σ)F ,(M,µˆzM)) for some z ∈ Z(L(σ)∗)F ∗,
+then we have
+bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F
+and
+bl(ϑ) = bl(ϕˆzL(σ))GF
+σ = bl(µˆzM)GF
+σ
+24
+
+(iii) the induced block bl(ϑ)GF is defined.
+Proof. The first point follows from Lemma 2.3 by choosing L = L(σ) and H = GF
+σ . Furthermore,
+in the case of (5.2) observe that L(σ) ≤ L(σ − L(σ)) and hence Z(L(σ − L(σ))∗) ≤ Z(L(σ)∗).
+Therefore, we can always find ϕ and z as in the statement of (ii). Since ϕ is an irreducible constituent
+of the virtual character RL(σ)
+M
+(µ), it follows from [CE94, Proposition 4.2] (whose assumptions are
+satisfied by [CE94, Proposition 3.3 (ii)]) that bl(ϕ) = bL(σ)F (M,µ) = bl(µ)L(σ)F . Then, since ˆzM
+is the restriction of the linear character ˆzL(σ) to MF , we deduce from Lemma 2.1 that
+bl(ϕˆzL(σ)) = bl(µˆzM)L(σ)F .
+Now, [Nav98, Theorem 9.19] implies that
+bl(ϑ) = bl(ϕˆzL(σ))
+GF
+σ
+and the second point follows by the transitivity of block induction. Finally, set Q ∶= Z(M)F
+ℓ and
+observe that QCGF (Q) = MF ≤ NGF (Q) by [CE94, Proposition 3.3(ii)]. Then, [Nav98, Theorem
+4.14] implies that bl(µˆzM)GF is well defined and so is bl(ϑ)GF by (ii) and transitivity of block
+induction. This concludes the proof.
+Using the lemma above, we can now define the following character set. This yields the e-local object
+through which we can determine the number of unipotent characters in a given block of B of GF
+and with a given defect d ≥ 0 (see Section 5.3).
+Definition 5.5. Let B be a block of GF and d a non-negative integer. For every e-chain σ ∈
+Le(G,F) and unipotent e-cuspidal pair (L,λ) ∈ CPu(σ) we define the character set
+Uchd (Bσ,(M,µ)) ∶= {ϑ ∈ Uch (GF
+σ ,(M,µ)) ∣ d(ϑ) = d,bl(ϑ)GF = B}.
+where bl(ϑ)GF is defined according to Lemma 5.4 (iii). Furthermore, we denote the cardinality of
+this set by
+kd
+u (Bσ,(M,µ)) ∶= ∣Uchd (Bσ,(M,µ))∣.
+To conclude this section, we show that Proposition 4.5 can be used to parametrise the character sets
+from Definition 5.5.
+Proposition 5.6. Let B be a block of G and d a non-negative integer. If σ ∈ Le(G,F) and (M,µ) is
+a unipotent e-cuspidal pair in CPu(σ) then there exists an AutF(GF )B,σ,(M,µ)-equivariant bijection
+ΩB,d
+σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bσ+M,(M,µ))
+such that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GFσ (Xσ+M,ϑ,GF
+σ+M,ΩB,d
+σ,(M,µ)(ϑ))
+for every ϑ ∈ Uchd(Bσ,(M,µ)) and where X ∶= GF ⋊ AutF(GF ).
+25
+
+Proof. First, observe that if M coincides with the last term L(σ) of the chain σ, then we have
+σ + M = σ which implies Uchd(Bσ,(M,µ)) = Uchd(Bσ+M,(M,µ)). In this case the result holds
+by defining ΩB,d
+σ,(M,µ) as the identity. Therefore, we can assume that M < L(σ) and define ρ ∶= σ+M.
+Now, according to (5.1) we have
+Uch (GF
+σ ,(M,µ)) = Irr(GF
+σ ∣ E (L(σ)F ,(M,psL(σ)(µ)))) .
+(5.3)
+On the other hand, noticing that M coincides with the last term L(ρ) of the chain ρ and that
+ρ−L(ρ) = σ, we obtain the equality E(L(ρ)F ,(M,psL(ρ−L(ρ))(µ))) = psL(σ)(µ). Then, observing
+that GF
+ρ = NGFσ (M), we can apply (5.2) to obtain the equality
+Uch (GF
+ρ ,(M,µ)) = Irr(NGFσ (M) ∣ psL(σ)(µ))) .
+(5.4)
+Next, we apply Proposition 4.5 by choosing the groups in that statement to be H = GF
+σ , K =
+L(σ) and (L,λ) = (M,µ). By (5.3) and (5.4), we deduce that there exists an AutF(GF )σ,(M,µ)-
+equivariant bijection
+ΩL(σ),GF
+σ
+(M,µ)
+∶ Uch(GF
+σ ,(M,µ)) → Uch(GF
+ρ ,(M,µ)).
+(5.5)
+Moreover, using the H-block isomorphisms given by Proposition 4.5 together with [Spä17, Lemma
+3.8 (b)], we deduce that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GFσ (Xρ,ϑ,GF
+ρ ,ΩL(σ),GF
+σ
+(M,µ)
+(ϑ))
+(5.6)
+for every ϑ ∈ Uchd(GF
+σ ,(M,µ)). To conclude, observe first that ΩL(σ),GF
+σ
+(M,µ)
+sends characters of
+defect d to characters of defect d. Moreover, by the transitivity of block induction and using (5.6),
+we deduce that
+bl(ϑ)GF = bl(ΩL(σ),GF
+σ
+(M,µ)
+(ϑ))
+GF
+.
+This shows that the bijection from (5.5) sends characters in the set Uchd(Bσ,(M,µ)) to charac-
+ters in the set Uchd(Bσ+M,(M,µ)) and therefore it restricts to a bijection, denoted by ΩB,d
+σ,(M,µ),
+satisfying the properties required in the statement. This completes the proof.
+We conclude this section with a remark on the isomorphisms of character triples obtained in Propo-
+sition 5.6.
+Remark 5.7. Suppose that ℓ does not divide q ± 1 if G is of type A(±q). In this case, every e-split
+Levi subgroup L of G satisfies L = C○
+G(Z(L)F
+ℓ ) according to [CE04, Proposition 13.19]. This fact
+can be used to show that the GF
+σ -block isomorphisms of character triples given by Proposition 5.6
+can be extended to GF -block isomorphisms of character triples. First, we claim that
+CGF Xσ,ϑ(D) ≤ Xσ,ϑ
+(5.7)
+for every irreducible character ϑ of GF
+σ and every ℓ-radical subgroup D of GF
+σ+M. Define Qi ∶=
+Z○(Li)F
+ℓ for every e-split Levi subgroup Li appearing in the chain σ. Then, using the fact that D is
+26
+
+ℓ-radical, we obtain the inclusions Qi ≤ Oℓ(GF
+σ ) ≤ D. Therefore, every element x ∈ GF Xσ,ϑ that
+centralises D centralises also each Qi and hence normalises each Li. It follows that
+CGF Xσ,ϑ(D) ≤ (GF Xσ,ϑ)σ = Xσ,ϑ
+as required by (5.7). We can now apply [Ros22a, Lemma 2.11] to the GF
+σ -block isomorphisms given
+by Proposition 5.6 to show that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GF (Xσ+M,ϑ,GF
+σ+M,ΩB,d
+σ,(M,µ)(ϑ))
+for every ϑ ∈ Uchd(Bσ,(M,µ)).
+5.2
+Proof of Theorem A
+We are finally ready to prove our main theorem which provides a bijection for unipotent characters
+in the spirit of the Character Triple Conjecture [Spä17, Conjecture 6.3]. In this section, we prove a
+slightly stronger result that provides further information on the type of e-chains and isomorphisms
+of character triples. In the following definition we introduce the analogue of the set Cd(B)± con-
+sidered in the Character Triple Conjecture as defined in [Spä17, p. 1097].
+Definition 5.8. Let B be a block of GF and consider a non-negative integer d. We define the set
+Ld
+u(B)± = {(σ,M,µ,ϑ) ∣ σ ∈ Le(G,F)±,(M,µ) ∈ CPu(σ),ϑ ∈ Uchd (Bσ,(M,µ))} .
+The conjugacy action of GF induces an action of GF on Ld
+u(B)± defined by (σ,M,µ,ϑ)g ∶=
+(σg,Mg,µg,ϑg) for every element g ∈ GF and (σ,M,µ,ϑ) ∈ Ld
+u(B)±. We denote by Ld
+u(B)±/GF
+the corresponding set of GF -orbits of tuples. Moreover, for every such orbit ω, we denote by ω● the
+corresponding GF -orbit of pairs (σ,ϑ) such that (σ,M,µ,ϑ) ∈ ω for some (M,µ) ∈ CPu(σ). In
+other words, if we indicate by (σ,M,µ,ϑ) the GF-orbit of (σ,M,µ,ϑ), then (σ,M,µ,ϑ)
+● is the
+GF -orbit of the pairs (σg,ϑg).
+In a similar way, if AutF(GF )B denotes the set of those automorphisms α ∈ AutF(GF ) that sta-
+bilise B, then we can define (σ,M,µ,ϑ)α ∶= (σα,Mα,µα,ϑα) for every α ∈ AutF(GF )B and
+(σ,M,µ,ϑ) ∈ Ld
+u(B). In this way, we obtain an action of the group AutF(GF )B on the set Ld
+u(B)±
+and on the corresponding set of orbits Ld
+u(B)±/GF .
+Theorem 5.9. For every block B of GF and every non-negative integer d, there exists an AutF(GF )B-
+equivariant bijection
+Λ ∶ Ld
+u(B)+/GF → Ld
+u(B)−/GF .
+Moreover, for every ω ∈ Ld
+u(B)+/GF, any (σ,ϑ) ∈ ω● and any (ρ,χ) ∈ Λ(ω)● we have
+∣σ∣ = ∣ρ∣ ± 1
+and
+(Xσ,ϑ,GF
+σ ,ϑ) ∼J (Xρ,χ,GF
+ρ ,χ)
+with J = GF
+σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF
+ρ , if ∣σ∣ = ∣ρ∣ + 1, and where X ∶= GF ⋊ AutF(GF ).
+27
+
+Proof. Define A ∶= AutF(GF ) and observe that X = GF ⋊ A. In a first step, we construct an
+equivariant bijection between triples of the form (��,M,µ). More precisely, let S denote the set of
+such triples (σ,M,µ) with σ ∈ Le(G,F) and (M,µ) ∈ CPu(σ). We define a map
+∆ ∶ S → S
+by setting
+∆((σ,M,µ)) ∶=
+⎧⎪⎪⎨⎪⎪⎩
+(σ + M,M,µ) ,
+L(σ) > M
+(σ − M,M,µ) ,
+L(σ) = M.
+Observe that the chain σ−M is always defined since M < G by the definition of CPu(σ). Moreover,
+it is clear from the definition above that the map ∆ is A-equivariant and satisfies ∆2 = Id. Therefore,
+observing that ∣σ ± M∣ = ∣σ∣ ± 1, we conclude that ∆ restricts to an A-equivariant bijection
+∆ ∶ S+ → S−
+where S± denotes the set of those triples (σ,M,µ) of S that satisfy σ ∈ Le(G,F)±. Furthermore,
+notice once again that if ∆((σ,M,µ)) = (ρ,K,κ), then
+∣σ∣ = ∣ρ∣ ± 1.
+(5.8)
+Now, fix an AB-transversal T+ in S+ and observe that the image of T+ under the map ∆, denoted by
+T−, is an AB-transversal in S because of the equivariance property of ∆. Consider (σ,M,µ) ∈ T+
+and write ∆((σ,M,µ)) = (ρ,M,µ). In what follows, we may assume without loss of generality
+that L(σ) > M and that ρ = σ + M, otherwise we repeat the arguments verbatim by replacing
+(σ,M,µ) with (ρ,M,µ). By Proposition 5.6 we obtain an AB,σ,(M,µ)-equivariant bijection
+ΩB,d
+σ,(M,µ) ∶ Uchd (Bσ,(M,µ)) → Uchd (Bρ,(M,µ))
+such that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GFσ (Xρ,χ,GF
+ρ ,χ)
+(5.9)
+for every ϑ ∈ Uchd(Bσ,(M,µ)) and where χ is the image of ϑ. Consequently, if U(σ,M,µ)
++
+is an
+AB,(σ,M,µ)-transversal in the character set Uchd(Bσ,(M,µ)), then its image, denoted by U(ρ,M,µ)
+−
+,
+under the bijection above is an AB,(ρ,M,µ)-transversal in the character set Uchd(Bρ,(M,µ)) be-
+cause AB,(σ,M,µ) = AB,(ρ,M,µ).
+Now, by the discussion in the previous paragraph and using Lemma 5.3, we conclude that the sets
+of GF -orbits
+L+ ∶= {(σ,M,µ,ϑ) ∣ (σ,M,µ) ∈ T+,ϑ ∈ U(σ,M,µ)
++
+}
+and
+L− ∶= {(ρ,M,µ,χ) ∣ (ρ,M,µ) ∈ T−,χ ∈ U(ρ,M,µ)
+−
+}
+are AB-transversals in the sets Ld
+u(B)+/GF and Ld
+u(B)−/GF respectively. Finally, we can define
+the bijection Λ by setting
+Λ((σ,M,µ,ϑ)
+x) ∶= (ρ,M,µ,χ)x
+28
+
+for every x ∈ AB and every (σ,M,µ,ϑ) ∈ L+ and (ρ,M,µ,χ) ∈ L− satisfying ∆(σ,M,µ) =
+(ρ,M,µ) and such that
+χ =
+⎧⎪⎪⎪⎨⎪⎪⎪⎩
+ΩB,d
+σ,(M,µ)(ϑ),
+ρ = σ + M
+(ΩB,d
+ρ,(M,µ))
+−1
+(ϑ),
+ρ = σ − M.
+Using (5.8) and (5.9) together with the definition of Λ, we conclude that the properties required in
+the statement are satisfied and the proof is now complete.
+Now, as a consequence of Theorem 5.9 and Remark 5.7, we can finally prove Theorem A.
+Proof of Theorem A. Assume that ℓ does not divide q ± 1 whenever (G,F) is of type A(±q). Con-
+sider the bijection Λ from Theorem 5.9 and chose ω ∈ Ld
+u(B)+/GF, (σ,ϑ) ∈ ω● and (ρ,χ) ∈ Λ(ω)●.
+Then, we have
+(Xσ,ϑ,GF
+σ ,ϑ) ∼J (Xρ,χ,GF
+ρ ,χ)
+with J = GF
+σ , if ∣σ∣ = ∣ρ∣ − 1, or J = GF
+ρ , if ∣σ∣ = ∣ρ∣ + 1. In either cases, applying Remark 5.7, we
+deduce that
+(Xσ,ϑ,GF
+σ ,ϑ) ∼GF (Xρ,χ,GF
+ρ ,χ)
+as required by Theorem A.
+5.3
+Proof of Theorem B
+Our final goal is to obtain a counting argument for unipotent characters as a consequence of The-
+orem 5.9. Recall that Dade’s Conjecture provides a way to determine the number of characters in
+a given ℓ-block B and with a given defect d in terms of ℓ-local structures. Theorem B provides an
+adaptation of this idea to the unipotent characters of finite reductive groups by means of e-local
+structures compatible with e-Harish-Chandra theory (see Definition 5.5). For every σ ∈ Le(G,F)
+we define
+kd
+u(Bσ) ∶=
+∑
+(M,µ)
+kd
+u(Bσ,(M,µ))
+(5.10)
+where (M,µ) runs over a set of representatives for the action of GF
+σ on CPu(σ). Moreover, recall
+that kd
+u(B) and kd
+c,u(B) denote the number of irreducible characters belonging to the block B and
+with defect d that are unipotent and unipotent e-cuspidal respectively.
+Proof of Theorem B. To start, we determine the cardinality of the sets of GF-orbits Ld
+u(B)±/GF .
+By applying Lemma 5.3, we obtain
+∣Ld
+u(B)±/GF ∣ =
+∑
+σ,(M,µ)
+kd
+u(Bσ,(M,µ)) = ∑
+σ
+kd
+u(Bσ)
+(5.11)
+where σ runs over a set of representatives, say L±, for the action of GF on Le(G,F)± and (M,µ)
+runs over a set of representatives for the action of GF
+σ on CPu(σ). Next, we isolate the contribution
+given by the trivial chain σ0 ∶= {G} ∈ Le(G,F)+ to the sum in (5.11). In this case, we have
+L(σ0) = G and hence psL(σ)(µ) = {µ} for every (M,µ) ∈ CPu(σ0) because the center Z(G∗)F ∗
+29
+
+is trivial under our assumptions. Consequently, using Definition 5.2 and Definition 5.5, we deduce
+that
+kd
+u(Bσ0) =
+∑
+(M,µ)
+kd
+u(Bσ0,(M,µ))
+(5.12)
+=
+∑
+(M,µ)
+∣Irrd(B) ∩ E(GF ,(M,µ))∣
+= kd
+u(B) − kd
+c,u(B)
+where the last equality holds by [BMM93, Theorem 3.2 (1)] and recalling that every pair (M,µ) ∈
+CPu(σ0) satisfies M < G = L(σ0). Next, Theorem 5.9 implies that the sets Ld
+u(B)+/GF and
+Ld
+u(B)−/GF have the same cardinality and therefore we conclude from (5.11) and (5.12) that
+kd
+u(B) − kd
+c,u(B) + ∑
+σ∈L+
+σ≠σ0
+kd
+u(Bσ) = ∑
+σ∈L+
+kd
+u(Bσ) = ∑
+σ∈L−
+kd
+u(Bσ).
+(5.13)
+Finally, noticing that (−1)∣σ∣+1 = ∓1 for every σ ∈ L±, we can rewrite (5.13) as
+kd
+u(B) − kd
+c,u(B) =
+∑
+σ∈L−∪L+
+(−1)∣σ∣+1kd
+u(Bσ)
+which is exactly the equality in the statement of Theorem B.
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+ITALY
+Email address: [email protected]
+33
+

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+HIT-SCIR at MMNLU-22: Consistency Regularization for Multilingual
+Spoken Language Understanding
+Bo Zheng, Zhouyang Li, Fuxuan Wei, Qiguang Chen, Libo Qin, Wanxiang Che∗
+Harbin Institute of Technology
+{bzheng,zhouyangli,fxwei,qgchen,lbqin,car}@ir.hit.edu.cn
+Abstract
+Multilingual spoken language understanding
+(SLU) consists of two sub-tasks, namely intent
+detection and slot filling. To improve the per-
+formance of these two sub-tasks, we propose
+to use consistency regularization based on a
+hybrid data augmentation strategy. The consis-
+tency regularization enforces the predicted dis-
+tributions for an example and its semantically
+equivalent augmentation to be consistent. We
+conduct experiments on the MASSIVE dataset
+under both full-dataset and zero-shot settings.
+Experimental results demonstrate that our pro-
+posed method improves the performance on
+both intent detection and slot filling tasks. Our
+system1 ranked 1st in the MMNLU-22 compe-
+tition under the full-dataset setting.
+1
+Introduction
+The MMNLU-22 evaluation focuses on the prob-
+lem of multilingual natural language understanding.
+It is based on the MASSIVE dataset (FitzGerald
+et al., 2022), a multilingual spoken language under-
+standing (SLU) dataset with two sub-tasks, includ-
+ing intent detection and slot filling. Specifically,
+given a virtual assistant utterance in an arbitrary
+language, the model is designed to predict the cor-
+responding intent label and extract the slot results.
+An English example is illustrated in Figure 1.
+Fine-tuning pre-trained cross-lingual language
+models allows task-specific supervision to be
+shared and transferred across languages (Conneau
+and Lample, 2019; Conneau et al., 2020; Xue et al.,
+2021).
+This motivates the two setting for the
+MMNLU-22 evaluation, namely the full-dataset
+setting and the zero-shot setting. Participants are
+allowed to use training data in all languages under
+the full-dataset setting, while they can only access
+the English training data under the zero-shot setting.
+∗Email corresponding.
+1The code will be available at https://github.com/
+bozheng-hit/MMNLU-22-HIT-SCIR.
+Utterance
+Slot
+Intent
+Wake me
+up
+at
+five
+am
+Friday
+this week
+O
+O
+O
+O
+time time
+date
+date
+date
+set alarm
+Figure 1: An English example from the MASSIVE
+dataset. The slot label ‘O’ stands for the ‘Other’ label.
+The latter is also called zero-shot cross-lingual SLU
+in previous work (Qin et al., 2020, 2022).
+Cross-lingual data augmentation methods have
+been proven effective to improve cross-lingual
+transferability, e.g., code-switch substitution (Qin
+et al., 2020) and machine translation (Conneau and
+Lample, 2019; Singh et al., 2019). Most previ-
+ous work directly utilizes the data augmentations
+as additional training data for fine-tuning. How-
+ever, they ignore the inherent correlation between
+the original example and its semantically equiva-
+lent augmentation, which can be fully exploited
+with the consistency regularization (Zheng et al.,
+2021b). The consistency regularization enforces
+the model predictions to be more consistent for
+semantic-preserving augmentations.
+Motivated by this, we propose to apply consis-
+tency regularization based on a hybrid data aug-
+mentation strategy, including data augmentation of
+machine translation and subword sampling (Kudo,
+2018). We use machine translation augmentation
+to align the model predictions of the intent detec-
+tion task. Meanwhile, subword sampling augmen-
+tation is used to align the model predictions of
+both intent detection and slot filling tasks. The
+proposed method consistently improves the SLU
+performance on the MASSIVE dataset under both
+full-dataset and zero-shot settings. It is worth men-
+tioning that our system ranked 1st in the MMNLU-
+22 competition under the full-dataset setting. We
+achieved an exact match accuracy of 49.65 points,
+outperforming the 2nd system by 1.02 points.
+arXiv:2301.02010v1  [cs.CL]  5 Jan 2023
+
+2
+Background
+2.1
+Task Description
+The task of SLU is that given an utterance with
+a word sequence x = (x1, ..., xn) with length n.
+The model is required to solve two sub-tasks. The
+intent detection task can be seen as an utterance
+classification task to decide the intent label oI, and
+the slot filling task is a sequence labeling task that
+generates a slot label for each word in the utterance
+to obtain the slot sequence oS = (oS
+1 , ..., oS
+n).
+2.2
+Dataset Description
+The MASSIVE dataset is composed of realistic,
+human-created virtual assistant utterance text span-
+ning 51 languages, 60 intents, 55 slot types, and
+18 domains (FitzGerald et al., 2022). There are
+11,514 training utterances for each language. For
+the full-dataset setting, all training data can be used.
+For the zero-shot setting, only English training data
+can be used, yet we can translate them into other
+languages using commercial translators. There are
+2,033, 2,974, and 3,000 utterances for each lan-
+guage in the development, test, and evaluation set,
+respectively. The average performance in all lan-
+guages should be reported under the full-dataset
+setting. Meanwhile, the average performance in all
+languages except English should be reported under
+the zero-shot setting.
+2.3
+Related Work
+Pre-trained cross-lingual language models (Con-
+neau and Lample, 2019; Conneau et al., 2020; Chi
+et al., 2021a,b, 2022; Xue et al., 2021) encode dif-
+ferent languages into universal representations and
+significantly improve cross-lingual transferability.
+These models usually consist of a multilingual vo-
+cabulary (Conneau and Lample, 2019; Conneau
+et al., 2020; Xue et al., 2021; Zheng et al., 2021a)
+and a Transformer model (Vaswani et al., 2017).
+A simple yet effective way to improve cross-
+lingual fine-tuning is to populate the training data
+with cross-lingual data augmentation (Conneau
+et al., 2020). Singh et al. (2019) replace a segment
+of source language input text with its translation
+in another language as data augmentation. Qin
+et al. (2020) randomly replace words in the source-
+language training example with target-language
+words using the bilingual dictionaries. Then the
+model is fine-tuned on the generated code-switched
+data. Instead of directly treating cross-lingual data
+augmentation as extra training data, Zheng et al.
+(2021b) proposed to better use data augmentations
+based on consistency regularization.
+3
+Method
+Given the input utterance x = (x1, ..., xn) with
+length n and the corresponding intent label oI and
+slot labels oS = (oS
+1 , ..., ...oS
+n) from training cor-
+pus D, we define the loss for the two sub-tasks of
+SLU in our fine-tuning process as:
+LI =
+�
+(x,oI)∈D
+CE(fI(x), oI),
+LS =
+�
+(x,oS)∈D
+CE(fS(x), oS),
+where LI and LS stand for the intent detection task
+and the slot filling task, fI(·) and fS(·) denote the
+model which predicts task-specific probability dis-
+tributions for the input example x, CE(·, ·) denotes
+cross-entropy loss.
+3.1
+Consistency Regularization
+In order to make better use of data augmentations,
+we introduce the consistency regularization used
+in Zheng et al. (2021b), which encourages consis-
+tent predictions for an example and its semantically
+equivalent augmentation. We apply consistency
+regularization on intent detection and slot filling
+tasks, which is defined as follows:
+RI =
+�
+x∈D
+KLS(fI(x)∥fI(A(x, z))),
+RS =
+�
+x∈D
+KLS(fS(x)∥fS(A(x, z))),
+KLS(P∥Q) = KL(stopgrad(P)∥Q)+
+KL(stopgrad(Q)∥P)
+where KLS(·∥·) is the symmertrical Kullback-
+Leibler divergence, A(x, z) denotes the augmented
+version of input utterance x with data augmenta-
+tion strategy z. The regularizer encourages the
+predicted distributions of the original training ex-
+ample and its augmented version to agree with
+each other. The stopgrad(·) operation2 is used to
+stop back-propagating gradients, which is also em-
+ployed in (Jiang et al., 2020; Liu et al., 2020; Zheng
+et al., 2021b).
+3.2
+Data Augmentations
+We consider two types of data augmentation strate-
+gies for our consistency regularization method, in-
+cluding subword sampling and machine translation.
+2Implemented by .detach() in PyTorch.
+
+Subword
+Sampling
+Machine
+Translation
+𝑥
+Pretrained Language Model
+Slot Classifier
+Intent Classifier
+Intent
+Task
+Loss
+Intent
+Consistency
+Regularization
+Slot
+Consistency
+Regularization
+Slot
+Task
+Loss
+Hybrid Data Augmentation
+Wake me up at five am
+_Wa/ke/_me/_up/_at/_five/_am
+_Wake/_me/_up/_at/_f/ive/_am
+_Wa/ke/_me/_up/_at/_fiv/e/_am
+早上五点叫醒我
+朝5時に起こして
+Maak me om vijf uur wakker
+Subword
+Sampling
+Machine
+Translation
+Input Utterance
+Figure 2: Illustration of our fine-tuning framework. ‘MT’ denotes machine translation augmentation and ‘SS’
+denotes subword sampling augmentation.
+3.2.1
+Subword Sampling
+Subword sampling is to generate multiple subword
+sequences from the original text as data augmen-
+tation. We apply the on-the-fly subword sampling
+algorithm from the unigram language model (Kudo,
+2018) in SentencePiece (Kudo and Richardson,
+2018). The output distributions of slot labels are
+generated on the first subword of each word in the
+input utterance. Therefore, the subword sampling
+augmentation can be used to align the output dis-
+tribution of both intent detection and slot filling
+tasks.
+3.2.2
+Machine Translation
+Machine translation is a common and effective
+data augmentation strategy in the cross-lingual sce-
+nario (Conneau and Lample, 2019; Singh et al.,
+2019). Due to the difficulty of accessing ground-
+truth labels in translation examples, machine trans-
+lation can not be an available data augmentation
+strategy in the slot filling task. To improve the
+quality of our translations, we employ a variety
+of approaches (See Section 4.2). Unlike subword
+sampling, the output distributions of slot labels be-
+tween the translation pairs can not be aligned. Thus,
+we only use machine translation to align the output
+distributions of the intent detection task.
+3.3
+Consistency Regularization based on
+Hybrid Data Augmentations
+We illustrate our fine-tuning framework in Figure 2.
+We propose to use consistency regularization based
+on a hybrid data augmentation strategy, which in-
+cludes data augmentation of machine translation
+and subword sampling. During the training pro-
+cess, we perform task fine-tuning and consistency
+regularization for an input example simultaneously.
+Then the final training loss is defined as follows:
+L = LI + λ1LS + λ2RI + λ3RS
+where λ1 is the slot loss coefficient, λ2 and λ3
+are the corresponding weights of the consistency
+regularization for two tasks. We sample different
+data augmentation for the input example with the
+pre-defined distribution.
+4
+Experiments
+4.1
+Experimental Setup
+We consider two types of pre-trained cross-lingual
+language models, which are encoder-only models
+and Text-to-Text models.
+We use XLM-Align Base (Chi et al., 2021b) for
+the encoder-only model setting. We use a two-layer
+feed-forward network with a 3,072 hidden size. We
+use the first representation of sentences “<s>” for
+the intent detection task and the first subword of
+each word for the slot filling task.
+We use mT5 Base (Xue et al., 2021) for the Text-
+to-Text model setting. We follow FitzGerald et al.
+(2022) to concatenate “Annotate: ” and the unla-
+beled input utterance as the input of the encoder,
+and generate the text concatenation of the intent
+label and the slot labels as the decoder output. The
+labels are separated with white spaces and then
+tokenized into subwords.
+We select the model that performs the best on
+the development dataset to run prediction on the
+test and evaluation dataset. We mainly select the
+batch size in [32, 64, 128, 256], dropout rate in
+
+Text Type
+Text Content
+Slot Translation
+Text Translation
+Aligned or Not
+Plain Text
+Wake me up at five am Friday this week
+five am: 凌晨五点
+Friday this week: 本周周五
+本周周五凌晨五点叫我起床
+Yes
+Text with Slots in Brackets
+Wake me up at [five am] [Friday this week]
+在[凌晨五点][本周星期五]叫醒我
+No
+Plain Text
+set an alarm for two hours from now
+two hours from now:
+从现在起两小时后
+从现在开始设置两个小时的闹钟
+No
+Text with Slots in Brackets
+set an alarm for [two hours from now]
+设置[从现在起两小时后]的闹钟
+Yes
+Table 1: Examples of aligning slots into machine translations.
+Model
+Test Set
+Evaluation Set
+Intent Acc
+Slot F1
+EMA
+Intent Acc
+Slot F1
+EMA
+XLM-R Base
+85.10
+73.60
+63.69
+-
+-
+-
+XLM-Align Base
+86.16
+76.36
+66.42
+-
+-
+-
+mT5 Base Text-to-Text
+85.33
+76.77
+66.64
+-
+-
+-
+XLM-Align Base + Ours
+87.12
+77.99
+68.76
+85.00
+68.45
+48.64
+mT5 Base Text-to-Text + Ours
+87.60
+78.22
+69.60
+85.10
+69.08
+49.65
+Table 2: Test and evaluation results on the MASSIVE dataset under the full-dataset setting. Results of XLM-R
+Base and mT5 Base Text-to-Text are taken from FitzGerald et al. (2022).
+[0.05, 0.1, 0.15], and the hyper-parameters used in
+our proposed method, including slot loss coeffi-
+cient λ1 in [1, 2, 4], weights of consistency regu-
+larization λ2 and λ3 in [2, 3, 5, 10]. We select the
+learning rate in [5e−5, 8e−5, 1e−4] for Text-to-Text
+models. As for encoder-only models, we select the
+learning rate in [4e−6, 6e−6, 8e−6].
+4.2
+Data Processing
+For the full-dataset setting, we use examples with
+the same id in different languages as machine trans-
+lation augmentation in our fine-tuning framework.
+For the zero-shot setting, we translated the entire
+English training set into 50 languages using com-
+mercial translation APIs, such as DeepL translator
+and Google translator. These translations refer to
+plain text translations and can be used for intent
+detection training and consistency regularization.
+We used two methods to obtain a translated ex-
+ample that aligned at the slot level. One is based
+on the plain text translation. Each slot value in an
+English training example is translated into a target
+language. If the translation results of each slot can
+be found in the plain text translation, a slot-aligned
+translation is obtained. The other is based on the
+annotated English training examples. We translated
+the annotated English training example with brack-
+ets for slot values (without slot type in brackets).
+Using brackets explicitly allows the translator to
+align slots to consecutive spans. And we also trans-
+lated each slot value into the target language. If
+the translation result of each slot can be found in
+the annotated utterance translation, we obtain a slot
+alignment example after removing the brackets.
+In practice, slot-aligned examples based on plain
+text translations are preferred as the final result of
+the slot alignment. If no such example is avail-
+able, we use the slot-aligned results from annotated
+translations. Examples of slot alignment are shown
+in Table 1. For those plain text translations where
+we can not align the slot labels, we only use them
+for the training of the intent detection task.
+4.3
+Evaluation Metrics
+The evaluation in competition is mainly conducted
+using three metrics:
+• Exact Match Accuracy (EMA): The percent-
+age of utterance-level predictions where the
+intent and all slots are exactly correct.
+• Intent Accuracy (Intent Acc): The percentage
+of predictions in which the intent is correct.
+• Slot Micro F1 (Slot F1): The micro-averaged
+F1 score is calculated over all slots.
+4.4
+Results
+Table 2 shows our results on the MASSIVE dataset
+under the full-set setting. We tried different cross-
+lingual pre-trained language models under the base-
+line setting. Among them, XLM-Align Base per-
+forms the best on the intent detection task, while
+the mT5 Base Text-to-Text model performs the
+best on the slot filling task and exact match ac-
+curacy. When applying our consistency regular-
+ization method, the mT5 Base Text-to-Text model
+outperforms the XLM-Align Base model by 0.84
+points and 0.99 points on exact match accuracy on
+the test dataset and the evaluation set, respectively.
+Meanwhile, compared to the baseline model, us-
+ing consistency regularization achieves an absolute
+
+Model
+Test Set
+Evaluation Set
+Intent Acc
+Slot F1
+EMA
+Intent Acc
+Slot F1
+EMA
+XLM-R Base
+70.62
+50.27
+38.70
+-
+-
+-
+XLM-Align Base
+68.49
+54.69
+40.91
+-
+-
+-
+mT5 Base Text-to-Text
+62.92
+44.77
+34.72
+-
+-
+-
+XLM-Align Base + Ours
+85.12
+71.27
+62.18
+83.18
+62.84
+43.05
+XLM-Align Base + Ours + KD
+85.76
+73.55
+64.44
+83.89
+64.60
+44.84
+mT5 Base Text-to-Text + Ours
+84.58
+69.24
+60.59
+82.56
+60.00
+40.93
+Table 3: Test and evaluation results on the MASSIVE dataset under the zero-shot setting. Results of XLM-R Base
+and mT5 Base Text-to-Text are taken from FitzGerald et al. (2022).
+Model
+Intent Acc Slot F1 EMA
+XLM-Align Base + Ours
+87.12
+77.99
+68.76
+- Subword Sampling
+87.50
+76.08
+67.40
+- Consistency Regularization
+86.16
+76.32
+66.57
+Table 4: Ablation studies on the MASSIVE test dataset
+under the full-dataset setting.
+2.96-point improvement on exact match accuracy
+with the mT5 Base Text-to-Text model.
+Table 3 shows our results on the MASSIVE
+dataset under the zero-shot setting. For the base-
+line models, XLM-Align Base performs the best on
+all three metrics. Difference from the full-dataset
+setting, mT5 Base Text-to-Text models perform
+poorly under the zero-shot setting. We attribute
+it to the fact that Text-to-Text models strongly
+rely on the training data quality since most of the
+training data under the zero-shot setting are ob-
+tained with machine translation systems. When
+applying our consistency regularization method,
+the XLM-Align Base model outperforms the base-
+line model by 21.27 points. Distilled from the In-
+foXLM Large (Chi et al., 2021a) model will further
+improve the performance by an absolute 2.26-point.
+4.5
+Ablation Studies
+We conduct ablation studies on the test dataset of
+MASSIVE under the two settings. Table 4 shows
+the results under the full-dataset setting. Ablating
+subword sampling will degrade the performance
+by 1.36 points on the exact match accuracy, where
+the performance drop comes mainly from the slot
+filling task, indicating the subword sampling aug-
+mentation mainly works on slot filling. Ablating
+consistency regularization will degrade the perfor-
+mance by 2.19 points on the exact match accuracy.
+The performances on both intent detection and slot
+filling tasks are decreased.
+The zero-shot setting results are presented in Ta-
+Model
+Intent Acc Slot F1 EMA
+XLM-Align Base + Ours
+85.12
+71.27
+62.18
+- Subword Sampling
+85.14
+69.52
+60.94
+- Machine Translation
+72.27
+58.37
+45.50
+- Consistency Regularization
+83.90
+69.37
+59.95
+Table 5: Ablation studies on the MASSIVE test dataset
+under the zero-shot setting.
+ble 5. It can be observed that when machine trans-
+lation augmentation is removed, the exact match
+accuracy drops by 16.68 points, while the perfor-
+mance on intent detection and slot filling are also
+significantly worse. We also removed the subword
+sampling augmentation, and the performance is
+found to have the same trend as in the full-dataset
+setting. An absolute 1.24-point drop on the exact
+match accuracy and an absolute 1.75-point drop
+on slot micro F1 demonstrate that subword sam-
+pling is more beneficial for the slot filling task. By
+removing the consistency regularization, the per-
+formance of exact match accuracy will degrade by
+2.23 points. The performance shows a significant
+performance drop on both intent detection and slot
+filling tasks.
+5
+Conclusion
+We propose to use consistency regularization based
+on a hybrid data augmentation strategy to improve
+the performance of multilingual SLU. The pro-
+posed method is flexible and can be easily plugged
+into the fine-tuning process of both the encoder-
+only model and the Text-to-Text model. The ex-
+perimental results demonstrate the importance of
+consistency regularization and the hybrid data aug-
+mentation strategy, respectively.
+Acknowledgments
+This work was supported by the National Key R&D
+Program of China via grant 2020AAA0106501 and
+
+the National Natural Science Foundation of China
+(NSFC) via grant 62236004 and 61976072.
+References
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+

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+page_content='HIT-SCIR at MMNLU-22: Consistency Regularization for Multilingual  Spoken Language Understanding  Bo Zheng, Zhouyang Li, Fuxuan Wei, Qiguang Chen, Libo Qin, Wanxiang Che∗  Harbin Institute of Technology  {bzheng,zhouyangli,fxwei,qgchen,lbqin,car}@ir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='hit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='cn  Abstract  Multilingual spoken language understanding  (SLU) consists of two sub-tasks, namely intent  detection and slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' To improve the per-  formance of these two sub-tasks, we propose  to use consistency regularization based on a  hybrid data augmentation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The consis-  tency regularization enforces the predicted dis-  tributions for an example and its semantically  equivalent augmentation to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We  conduct experiments on the MASSIVE dataset  under both full-dataset and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Experimental results demonstrate that our pro-  posed method improves the performance on  both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Our  system1 ranked 1st in the MMNLU-22 compe-  tition under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  1  Introduction  The MMNLU-22 evaluation focuses on the prob-  lem of multilingual natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  It is based on the MASSIVE dataset (FitzGerald  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2022), a multilingual spoken language under-  standing (SLU) dataset with two sub-tasks, includ-  ing intent detection and slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Specifically,  given a virtual assistant utterance in an arbitrary  language, the model is designed to predict the cor-  responding intent label and extract the slot results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  An English example is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Fine-tuning pre-trained cross-lingual language  models allows task-specific supervision to be  shared and transferred across languages (Conneau  and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=',  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  This motivates the two setting for the  MMNLU-22 evaluation, namely the full-dataset  setting and the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Participants are  allowed to use training data in all languages under  the full-dataset setting, while they can only access  the English training data under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  ∗Email corresponding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  1The code will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='com/  bozheng-hit/MMNLU-22-HIT-SCIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Utterance  Slot  Intent  Wake me  up  at  five  am  Friday  this week  O  O  O  O  time time  date  date  date  set alarm  Figure 1: An English example from the MASSIVE  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The slot label ‘O’ stands for the ‘Other’ label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  The latter is also called zero-shot cross-lingual SLU  in previous work (Qin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Cross-lingual data augmentation methods have  been proven effective to improve cross-lingual  transferability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', code-switch substitution (Qin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020) and machine translation (Conneau and  Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Most previ-  ous work directly utilizes the data augmentations  as additional training data for fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' How-  ever, they ignore the inherent correlation between  the original example and its semantically equiva-  lent augmentation, which can be fully exploited  with the consistency regularization (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=',  2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The consistency regularization enforces  the model predictions to be more consistent for  semantic-preserving augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Motivated by this, we propose to apply consis-  tency regularization based on a hybrid data aug-  mentation strategy, including data augmentation of  machine translation and subword sampling (Kudo,  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We use machine translation augmentation  to align the model predictions of the intent detec-  tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Meanwhile, subword sampling augmen-  tation is used to align the model predictions of  both intent detection and slot filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The  proposed method consistently improves the SLU  performance on the MASSIVE dataset under both  full-dataset and zero-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' It is worth men-  tioning that our system ranked 1st in the MMNLU-  22 competition under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We  achieved an exact match accuracy of 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='65 points,  outperforming the 2nd system by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='02 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='02010v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='CL]  5 Jan 2023  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='1  Task Description  The task of SLU is that given an utterance with  a word sequence x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', xn) with length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  The model is required to solve two sub-tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The  intent detection task can be seen as an utterance  classification task to decide the intent label oI, and  the slot filling task is a sequence labeling task that  generates a slot label for each word in the utterance  to obtain the slot sequence oS = (oS  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', oS  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2  Dataset Description  The MASSIVE dataset is composed of realistic,  human-created virtual assistant utterance text span-  ning 51 languages, 60 intents, 55 slot types, and  18 domains (FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' There are  11,514 training utterances for each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' For  the full-dataset setting, all training data can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  For the zero-shot setting, only English training data  can be used, yet we can translate them into other  languages using commercial translators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' There are  2,033, 2,974, and 3,000 utterances for each lan-  guage in the development, test, and evaluation set,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The average performance in all lan-  guages should be reported under the full-dataset  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Meanwhile, the average performance in all  languages except English should be reported under  the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='3  Related Work  Pre-trained cross-lingual language models (Con-  neau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Chi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021a,b, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021) encode dif-  ferent languages into universal representations and  significantly improve cross-lingual transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  These models usually consist of a multilingual vo-  cabulary (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Conneau  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021a)  and a Transformer model (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  A simple yet effective way to improve cross-  lingual fine-tuning is to populate the training data  with cross-lingual data augmentation (Conneau  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' (2019) replace a segment  of source language input text with its translation  in another language as data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Qin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' (2020) randomly replace words in the source-  language training example with target-language  words using the bilingual dictionaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Then the  model is fine-tuned on the generated code-switched  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Instead of directly treating cross-lingual data  augmentation as extra training data, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  (2021b) proposed to better use data augmentations  based on consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3  Method  Given the input utterance x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', xn) with  length n and the corresponding intent label oI and  slot labels oS = (oS  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='oS  n) from training cor-  pus D, we define the loss for the two sub-tasks of  SLU in our fine-tuning process as:  LI =  �  (x,oI)∈D  CE(fI(x), oI),  LS =  �  (x,oS)∈D  CE(fS(x), oS),  where LI and LS stand for the intent detection task  and the slot filling task, fI(·) and fS(·) denote the  model which predicts task-specific probability dis-  tributions for the input example x, CE(·, ·) denotes  cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='1  Consistency Regularization  In order to make better use of data augmentations,  we introduce the consistency regularization used  in Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' (2021b), which encourages consis-  tent predictions for an example and its semantically  equivalent augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We apply consistency  regularization on intent detection and slot filling  tasks, which is defined as follows:  RI =  �  x∈D  KLS(fI(x)∥fI(A(x, z))),  RS =  �  x∈D  KLS(fS(x)∥fS(A(x, z))),  KLS(P∥Q) = KL(stopgrad(P)∥Q)+  KL(stopgrad(Q)∥P)  where KLS(·∥·) is the symmertrical Kullback-  Leibler divergence, A(x, z) denotes the augmented  version of input utterance x with data augmenta-  tion strategy z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The regularizer encourages the  predicted distributions of the original training ex-  ample and its augmented version to agree with  each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The stopgrad(·) operation2 is used to  stop back-propagating gradients, which is also em-  ployed in (Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Zheng  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2  Data Augmentations  We consider two types of data augmentation strate-  gies for our consistency regularization method, in-  cluding subword sampling and machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  2Implemented by .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='detach() in PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Subword  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Sampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Machine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='𝑥  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Pretrained Language Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Slot Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Intent Classifier  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Intent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Intent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Consistency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Regularization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Slot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Consistency  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Regularization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Slot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Task  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Hybrid Data Augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Wake me up at five am  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='_Wa/ke/_me/_up/_at/_five/_am  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='_Wake/_me/_up/_at/_f/ive/_am  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='_Wa/ke/_me/_up/_at/_fiv/e/_am  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='早上五点叫醒我  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='朝5時に起こして  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Maak me om vijf uur wakker  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Subword  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Sampling  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Machine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Input Utterance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Figure 2: Illustration of our fine-tuning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' ‘MT’ denotes machine translation augmentation and ‘SS’  denotes subword sampling augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='1  Subword Sampling  Subword sampling is to generate multiple subword  sequences from the original text as data augmen-  tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We apply the on-the-fly subword sampling  algorithm from the unigram language model (Kudo,  2018) in SentencePiece (Kudo and Richardson,  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The output distributions of slot labels are  generated on the first subword of each word in the  input utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Therefore, the subword sampling  augmentation can be used to align the output dis-  tribution of both intent detection and slot filling  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2  Machine Translation  Machine translation is a common and effective  data augmentation strategy in the cross-lingual sce-  nario (Conneau and Lample, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Singh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Due to the difficulty of accessing ground-  truth labels in translation examples, machine trans-  lation can not be an available data augmentation  strategy in the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' To improve the  quality of our translations, we employ a variety  of approaches (See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Unlike subword  sampling, the output distributions of slot labels be-  tween the translation pairs can not be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Thus,  we only use machine translation to align the output  distributions of the intent detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='3  Consistency Regularization based on  Hybrid Data Augmentations  We illustrate our fine-tuning framework in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  We propose to use consistency regularization based  on a hybrid data augmentation strategy, which in-  cludes data augmentation of machine translation  and subword sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' During the training pro-  cess, we perform task fine-tuning and consistency  regularization for an input example simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Then the final training loss is defined as follows:  L = LI + λ1LS + λ2RI + λ3RS  where λ1 is the slot loss coefficient, λ2 and λ3  are the corresponding weights of the consistency  regularization for two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We sample different  data augmentation for the input example with the  pre-defined distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  4  Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='1  Experimental Setup  We consider two types of pre-trained cross-lingual  language models, which are encoder-only models  and Text-to-Text models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  We use XLM-Align Base (Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021b) for  the encoder-only model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We use a two-layer  feed-forward network with a 3,072 hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We  use the first representation of sentences “<s>” for  the intent detection task and the first subword of  each word for the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  We use mT5 Base (Xue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021) for the Text-  to-Text model setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We follow FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  (2022) to concatenate “Annotate: ” and the unla-  beled input utterance as the input of the encoder,  and generate the text concatenation of the intent  label and the slot labels as the decoder output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The  labels are separated with white spaces and then  tokenized into subwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  We select the model that performs the best on  the development dataset to run prediction on the  test and evaluation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We mainly select the  batch size in [32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' 256],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' dropout rate in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Text Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Text Content  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Slot Translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Text Translation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Aligned or Not  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Plain Text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Wake me up at five am Friday this week  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='five am: 凌晨五点  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Friday this week: 本周周五  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='本周周五凌晨五点叫我起床  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Text with Slots in Brackets  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Wake me up at [five am] [Friday this week]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='在[凌晨五点][本周星期五]叫醒我  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Plain Text  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='set an alarm for two hours from now  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='two hours from now:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='从现在起两小时后  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='从现在开始设置两个小时的闹钟  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='No  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Text with Slots in Brackets  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='set an alarm for [two hours from now]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='设置[从现在起两小时后]的闹钟  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='Table 1: Examples of aligning slots into machine translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Model  Test Set  Evaluation Set  Intent Acc  Slot F1  EMA  Intent Acc  Slot F1  EMA  XLM-R Base  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='10  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='60  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='69 XLM-Align Base  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='16  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='36  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='42 mT5 Base Text-to-Text  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='33  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='77  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='64 XLM-Align Base + Ours  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='12  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='99  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='76  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='00  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='45  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='64  mT5 Base Text-to-Text + Ours  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='60  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='22  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='60  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='10  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='08  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='65  Table 2: Test and evaluation results on the MASSIVE dataset under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Results of XLM-R  Base and mT5 Base Text-to-Text are taken from FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='15], and the hyper-parameters used in  our proposed method, including slot loss coeffi-  cient λ1 in [1, 2, 4], weights of consistency regu-  larization λ2 and λ3 in [2, 3, 5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We select the  learning rate in [5e−5, 8e−5, 1e−4] for Text-to-Text  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' As for encoder-only models, we select the  learning rate in [4e−6, 6e−6, 8e−6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='2  Data Processing  For the full-dataset setting, we use examples with  the same id in different languages as machine trans-  lation augmentation in our fine-tuning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  For the zero-shot setting, we translated the entire  English training set into 50 languages using com-  mercial translation APIs, such as DeepL translator  and Google translator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' These translations refer to  plain text translations and can be used for intent  detection training and consistency regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  We used two methods to obtain a translated ex-  ample that aligned at the slot level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' One is based  on the plain text translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Each slot value in an  English training example is translated into a target  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' If the translation results of each slot can  be found in the plain text translation, a slot-aligned  translation is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The other is based on the  annotated English training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We translated  the annotated English training example with brack-  ets for slot values (without slot type in brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Using brackets explicitly allows the translator to  align slots to consecutive spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' And we also trans-  lated each slot value into the target language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' If  the translation result of each slot can be found in  the annotated utterance translation, we obtain a slot  alignment example after removing the brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  In practice, slot-aligned examples based on plain  text translations are preferred as the final result of  the slot alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' If no such example is avail-  able, we use the slot-aligned results from annotated  translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Examples of slot alignment are shown  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' For those plain text translations where  we can not align the slot labels, we only use them  for the training of the intent detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='3  Evaluation Metrics  The evaluation in competition is mainly conducted  using three metrics:  Exact Match Accuracy (EMA): The percent-  age of utterance-level predictions where the  intent and all slots are exactly correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Intent Accuracy (Intent Acc): The percentage  of predictions in which the intent is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Slot Micro F1 (Slot F1): The micro-averaged  F1 score is calculated over all slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='4  Results  Table 2 shows our results on the MASSIVE dataset  under the full-set setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We tried different cross-  lingual pre-trained language models under the base-  line setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Among them, XLM-Align Base per-  forms the best on the intent detection task, while  the mT5 Base Text-to-Text model performs the  best on the slot filling task and exact match ac-  curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' When applying our consistency regular-  ization method, the mT5 Base Text-to-Text model  outperforms the XLM-Align Base model by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='84  points and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='99 points on exact match accuracy on  the test dataset and the evaluation set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Meanwhile, compared to the baseline model, us-  ing consistency regularization achieves an absolute  Model  Test Set  Evaluation Set  Intent Acc  Slot F1  EMA  Intent Acc  Slot F1  EMA  XLM-R Base  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='62  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='27  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='70 XLM-Align Base  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='49  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='69  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='91 mT5 Base Text-to-Text  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='92  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='77  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='72 XLM-Align Base + Ours  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='12  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='27  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='18  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='18  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='84  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='05  XLM-Align Base + Ours + KD  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='76  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='55  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='44  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='89  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='60  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='84  mT5 Base Text-to-Text + Ours  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='58  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='24  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='59  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='56  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='00  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='93  Table 3: Test and evaluation results on the MASSIVE dataset under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Results of XLM-R Base  and mT5 Base Text-to-Text are taken from FitzGerald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Model  Intent Acc Slot F1 EMA  XLM-Align Base + Ours  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='12  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='99  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='76  Subword Sampling  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='50  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='08  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='40  Consistency Regularization  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='16  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='32  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='57  Table 4: Ablation studies on the MASSIVE test dataset  under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='96-point improvement on exact match accuracy  with the mT5 Base Text-to-Text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Table 3 shows our results on the MASSIVE  dataset under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' For the base-  line models, XLM-Align Base performs the best on  all three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Difference from the full-dataset  setting, mT5 Base Text-to-Text models perform  poorly under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We attribute  it to the fact that Text-to-Text models strongly  rely on the training data quality since most of the  training data under the zero-shot setting are ob-  tained with machine translation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' When  applying our consistency regularization method,  the XLM-Align Base model outperforms the base-  line model by 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='27 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Distilled from the In-  foXLM Large (Chi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=', 2021a) model will further  improve the performance by an absolute 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='26-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='5  Ablation Studies  We conduct ablation studies on the test dataset of  MASSIVE under the two settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Table 4 shows  the results under the full-dataset setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Ablating  subword sampling will degrade the performance  by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='36 points on the exact match accuracy, where  the performance drop comes mainly from the slot  filling task, indicating the subword sampling aug-  mentation mainly works on slot filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' Ablating  consistency regularization will degrade the perfor-  mance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='19 points on the exact match accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  The performances on both intent detection and slot  filling tasks are decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  The zero-shot setting results are presented in Ta-  Model  Intent Acc Slot F1 EMA  XLM-Align Base + Ours  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='12  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='27  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='18  Subword Sampling  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='14  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='52  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='94  Machine Translation  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='27  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='37  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='50  Consistency Regularization  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='90  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='37  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='95  Table 5: Ablation studies on the MASSIVE test dataset  under the zero-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' It can be observed that when machine trans-  lation augmentation is removed, the exact match  accuracy drops by 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='68 points, while the perfor-  mance on intent detection and slot filling are also  significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' We also removed the subword  sampling augmentation, and the performance is  found to have the same trend as in the full-dataset  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' An absolute 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='24-point drop on the exact  match accuracy and an absolute 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='75-point drop  on slot micro F1 demonstrate that subword sam-  pling is more beneficial for the slot filling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' By  removing the consistency regularization, the per-  formance of exact match accuracy will degrade by  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='23 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The performance shows a significant  performance drop on both intent detection and slot  filling tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  5  Conclusion  We propose to use consistency regularization based  on a hybrid data augmentation strategy to improve  the performance of multilingual SLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The pro-  posed method is flexible and can be easily plugged  into the fine-tuning process of both the encoder-  only model and the Text-to-Text model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content=' The ex-  perimental results demonstrate the importance of  consistency regularization and the hybrid data aug-  mentation strategy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
+page_content='  Acknowledgments  This work was supported by the National Key R&D  Program of China via grant 2020AAA0106501 and  the National Natural Science Foundation of China  (NSFC) via grant 62236004 and 61976072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
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+page_content='  Allocating large vocabulary capacity  for cross-lingual language model pre-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNA0T4oBgHgl3EQfD_8a/content/2301.02010v1.pdf'}
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