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-NA0T4oBgHgl3EQfPP8X/content/tmp_files/2301.02171v1.pdf.txt
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|
1 |
+
Strong Convergence of Peaks Over a Threshold
|
2 |
+
S. A. Padoan
|
3 |
+
Department of Decision Sciences, Bocconi University, Italy
|
4 |
+
and
|
5 |
+
S. Rizzelli
|
6 |
+
Department of Statistical Sciences, Catholic University, Italy
|
7 |
+
January 6, 2023
|
8 |
+
Abstract
|
9 |
+
Extreme Value Theory plays an important role to provide approximation re-
|
10 |
+
sults for the extremes of a sequence of independent random variable when their
|
11 |
+
distribution is unknown. An important one is given by the Generalised Pareto dis-
|
12 |
+
tribution HΞ³(x) as an approximation of the distribution Ft(s(t)x) of the excesses
|
13 |
+
over a threshold t, where s(t) is a suitable norming function. In this paper we
|
14 |
+
study the rate of convergence of Ft(s(t)Β·) to HΞ³ in variational and Hellinger dis-
|
15 |
+
tances and translate it into that regarding the Kullback-Leibler divergence between
|
16 |
+
the respective densities. We discuss the utility of these results in the statistical ο¬eld
|
17 |
+
by showing that the derivation of consistency and rate of convergence of estimators
|
18 |
+
of the tail index or tail probabilities can be obtained thorough an alternative and
|
19 |
+
relatively simpliο¬ed approach, if compared to usual asymptotic techniques.
|
20 |
+
Keywords: Contraction Rate, Consistency, Exceedances, Extreme Quantile, Gener-
|
21 |
+
alised Pareto, Tail Index.
|
22 |
+
2020 Mathematics Subject Classiο¬cation: Primary 60G70; secondary 62F12, 62G20
|
23 |
+
1
|
24 |
+
Introduction
|
25 |
+
Extreme Value Theory (EVT) develops probabilistic models and methods for describ-
|
26 |
+
ing the random behaviour of extreme observations that rarely occur. These theoretical
|
27 |
+
foundations are very important for studying practical problems in environmental, cli-
|
28 |
+
mate, insurance and ο¬nancial ο¬elds (e.g., Embrechts et al., 2013; Dey and Yan, 2016),
|
29 |
+
to name a few.
|
30 |
+
In the univariate setting, the most popular approaches for statistical analysis are the
|
31 |
+
so-called Block Maxima (BM) and Peaks Over Threshold (POT) (see e.g. B¨ucher and
|
32 |
+
Zhou, 2021, for a review). Let X1, . . . , Xn be independent and identically distributed
|
33 |
+
(i.i.d.) random variables according to a common distribution F. The ο¬rst approach
|
34 |
+
concerns the modelling of k sample maxima derived over blocks of a certain size m, i.e.
|
35 |
+
Mm,i = max(X(iβ1)m+1, . . . , Xim), i β {1, . . . , k}. In this case, under some regularity
|
36 |
+
conditions (e.g. de Haan and Ferreira, 2006, Ch. 1), the weak limit theory establishes
|
37 |
+
that F m(amx+bm) converges pointwise to GΞ³(x) as m β β, for every continuity point
|
38 |
+
x of GΞ³, where GΞ³ is the Generalised Extreme Value (GEV) distribution, am > 0 and
|
39 |
+
bm are suitable norming constants for each m = 1, 2, . . . and Ξ³ β R is the so-called
|
40 |
+
tail index, which describes the tail heaviness of F (e.g. de Haan and Ferreira, 2006,
|
41 |
+
Ch. 1). The second method concerns the modelling of k random variables out of the
|
42 |
+
n available that exceed a high threshold t, or, equivalently, of k threshold excesses Yj,
|
43 |
+
1
|
44 |
+
arXiv:2301.02171v1 [math.PR] 5 Jan 2023
|
45 |
+
|
46 |
+
j = 1, . . . , k, which are i.i.d. copies of Y = X βt|X > t. In this context, the Generalised
|
47 |
+
Pareto (GP) distribution, say HΞ³, appears as weak limit law of appropriately normalised
|
48 |
+
high threshold exceedances, i.e. for all x > 0, Ft(s(t)x) converges pointwise to HΞ³(x)
|
49 |
+
as t β xβ, for all the continuity points x of HΞ³(x), where Ft(x) = P(Y β€ x) and
|
50 |
+
s(t) > 0 is a suitable scaling function for any t β€ xβ, with xβ = inf(x : F(x) <
|
51 |
+
β). This result motivates the POT approach, which was introduced decades ago by
|
52 |
+
the seminal paper Balkema and de Haan (1974). Since then, few other convergence
|
53 |
+
results emerged.
|
54 |
+
For instance, the uniform convergence of Ft(s(t) Β· ) to HΞ³ and the
|
55 |
+
coresponding convergence rate have been derived by Pickands III (1975) and Raoult
|
56 |
+
and Worms (2003), respectively. Similar results but in Wasserstein distance have been
|
57 |
+
recently established by Bobbia et al. (2021). As for the GEV distribution, more results
|
58 |
+
are available.
|
59 |
+
In particular, there are suο¬cient conditions to ensure, in addition to
|
60 |
+
weak convergence, that F m(am Β· +bm) converges to GΞ³ for example uniformly and
|
61 |
+
in variational distance and the density of F m(am Β· +bm) converges pointwise, locally
|
62 |
+
uniformly and uniformly to that of GΞ³ (e.g. Falk et al., 2010, Ch. 2; Resnick, 2007, Ch.
|
63 |
+
2).
|
64 |
+
The main contribution of this article is to provide new convergence results that can
|
65 |
+
be useful in practical problems for the POT approach. Motivated by the utility in the
|
66 |
+
statistical ο¬eld to asses the asymptotic accuracy of estimation procedures, we study
|
67 |
+
stronger forms of convergence than the pointwise one, as limtβxβ D(Ft(s(t) Β· ), HΞ³) = 0,
|
68 |
+
where D( Β· ; Β· ) is either the variational distance, the Hellinger distance or the Kullback-
|
69 |
+
Leibler divergence. In particular, we provide upper bounds for the rate of convergence
|
70 |
+
to zero of D(Ft(s(t) Β· ); HΞ³) in the case that D( Β· ; Β· ) is the variational and Hellinger dis-
|
71 |
+
tance, and further translate them into bounds on Kullback-Leibler divergence between
|
72 |
+
the densities of Ft(s(t)Β·) and HΞ³, respectively.
|
73 |
+
Estimators of the tail index Ξ³ (and other related quantities) are typically deο¬ned
|
74 |
+
as functionals of the random variables (Y1, . . . , Yk), as for instance the popular Hill
|
75 |
+
(Hill, 1975), Moment (Dekkers et al., 1989), Pickands (Pickands III, 1975), Maximum
|
76 |
+
Likelihood (ML, Jenkinson, 1969), Generalised Probability Weighted Moment (GPWM,
|
77 |
+
Hosking et al., 1985) estimators, to name a few. In real applications, the distribution
|
78 |
+
F is typically unknown and so is F(s(t) Β· ). Although, for large t, HΞ³ provides a model
|
79 |
+
approximation for Ft(s(t) Β· ), when one wants to derive asymptotic properties as the
|
80 |
+
consistency and especially the rate of convergence of the tail index estimators (or other
|
81 |
+
related quantities), still the fact that (after rescaling) the random variables (Y1, . . . , Yk)
|
82 |
+
are actually distributed according to Ft(s(t) Β· ) needs to be taken into account, which
|
83 |
+
makes asymptotic derivations quite burdensome. These are even more complicated if t
|
84 |
+
is determined on the basis of the (k + 1)-th largest order statistic of the original sample
|
85 |
+
X1, . . . , Xn, which is the most common situation in practical applications. In this case,
|
86 |
+
the threshold is in fact random and, up to rescaling, Ft(s(t) Β· ) only gives a conditional
|
87 |
+
model for the variables Yj given a ο¬xed value t of the chosen statistic. Asymptotic
|
88 |
+
properties for POT methods have been studied in the last ο¬fty years, see for example
|
89 |
+
Hall and Welsh (1984), Drees (1998), Dekkers and de Haan (1993) and the reference
|
90 |
+
therein.
|
91 |
+
Leveraging on our strong convergence results we can show that, for random sequences
|
92 |
+
(such as sequences of estimators) convergence results in probability that hold under the
|
93 |
+
limit model HΞ³, are also valid for a rescaled sample of excesses over a large order statistic.
|
94 |
+
Precisely, we show that the distribution of the latter, up to rescaling and reordering,
|
95 |
+
is contiguous to that of an ordered i.i.d. sample from HΞ³ (e.g., van der Vaart, 2000,
|
96 |
+
Ch. 6.2). As a by product of this result, one can derive the consistency and rate of
|
97 |
+
convergence of a tail index estimator (or an estimator of a related quantity) by deο¬ning
|
98 |
+
it as a functional of the random sequence (Z1, . . . , Zk) which is distributed according the
|
99 |
+
2
|
100 |
+
|
101 |
+
limit model HΞ³, and, if density of Ft(s(t)Β·) satisο¬es some regularity conditions, then the
|
102 |
+
same asymptotic results hold even when such estimator is deο¬ned through the sequence
|
103 |
+
of excesses. This approach simpliο¬es a lot the computations as asymptotic properties
|
104 |
+
are easily derivable under the limit model.
|
105 |
+
The article is organised as follows, Section 2 of the paper provides a brief summary
|
106 |
+
of the probabilistic context on which our results are based. Section 3 provides our new
|
107 |
+
results on strong convergence to a Pareto model. Section 4 explains in what applications
|
108 |
+
concerning statistical estimation our results are useful. Section 5 provides the proofs of
|
109 |
+
the main results.
|
110 |
+
2
|
111 |
+
Background
|
112 |
+
Let X be a random variable with a distribution function F that is in the domain of
|
113 |
+
attraction of the GEV distribution GΞ³, shortly denoted as F β D(GΞ³). This means that
|
114 |
+
there are norming constants am > 0 and bm β R for m = 1, 2, . . . such that
|
115 |
+
lim
|
116 |
+
mββ F m(amx + bm) = exp
|
117 |
+
οΏ½
|
118 |
+
β (1 + Ξ³x)β1/Ξ³οΏ½
|
119 |
+
=: GΞ³(x),
|
120 |
+
(2.1)
|
121 |
+
for all x β R such that 1 + Ξ³x > 0, where Ξ³ β R, and this is true if only if there is a
|
122 |
+
scaling function s(t) > 0 with t < xβ such that
|
123 |
+
lim
|
124 |
+
tβxβ Ft(s(t)x) = 1 β (1 + Ξ³x)β1/Ξ³ =: HΞ³(x),
|
125 |
+
(2.2)
|
126 |
+
e.g., de Haan and Ferreira (2006, Theorem 1.1.6). The densities of HΞ³ and GΞ³ are
|
127 |
+
hΞ³(x) = (1 + Ξ³x)β(1/Ξ³+1)
|
128 |
+
and
|
129 |
+
gΞ³(x) = GΞ³(x)hΞ³(x),
|
130 |
+
respectively. Let U(v) := F β(1 β 1/v), for v β₯ 1, where F β is the left-continuous
|
131 |
+
inverse function of F and Gβ(exp(β1/x)) = (xΞ³ β 1)/Ξ³.
|
132 |
+
Then, we recall that the
|
133 |
+
ο¬rst-order condition in formula (2.1) is equivalent to the limit result
|
134 |
+
lim
|
135 |
+
vββ
|
136 |
+
U(vx) β U(v)
|
137 |
+
a(v)
|
138 |
+
= xΞ³ β 1
|
139 |
+
Ξ³
|
140 |
+
,
|
141 |
+
(2.3)
|
142 |
+
for all x > 0, where a(v) > 0 is a suitable scaling function. In particular, we have that
|
143 |
+
s(t) = a(1/(1 β F(t))), see de Haan and Ferreira (2006, Ch. 1) for possible selections of
|
144 |
+
the function a.
|
145 |
+
A stronger convergence form than that in formula (2.2) is the uniform one, i.e.
|
146 |
+
sup
|
147 |
+
xβ[0, xββt
|
148 |
+
s(t) )
|
149 |
+
|Ft(s(t)x) β HΞ³(x)| β 0,
|
150 |
+
t β xβ.
|
151 |
+
To establish the speed at which Ft(s(t)x) converges uniformly to HΞ³(x), Raoult and
|
152 |
+
Worms (2003) relied on a speciο¬c formulation of the well-known second-order condi-
|
153 |
+
tion. In its general form, the second order condition requires the existence of a posi-
|
154 |
+
tive function a and a positive or negative function A, named rate function, such that
|
155 |
+
limvββ |A(v)| = 0 and
|
156 |
+
lim
|
157 |
+
vββ
|
158 |
+
U(vx)βU(v)
|
159 |
+
a(v)
|
160 |
+
β xΞ³β1
|
161 |
+
Ξ³
|
162 |
+
A(v)
|
163 |
+
= D(x),
|
164 |
+
x > 0,
|
165 |
+
3
|
166 |
+
|
167 |
+
where D is a non-null function which is not a multiple of (xΞ³ β 1)/Ξ³, see de Haan and
|
168 |
+
Ferreira (2006, Deο¬nition 2.3.1). The rate function A is necessarily regularly varying at
|
169 |
+
inο¬nity with index Ο β€ 0, named second-order parameter (de Haan and Ferreira, 2006,
|
170 |
+
Theorem 2.3.3). In the sequel, we use the same speciο¬c form of second order condition
|
171 |
+
of Raoult and Worms (2003) to obtain decay rates for stronger metrics than uniform
|
172 |
+
distance between distribution functions.
|
173 |
+
3
|
174 |
+
Strong results for POT
|
175 |
+
In this section, we discuss strong forms of convergence for the distribution of rescaled
|
176 |
+
exceedances over a threshold. First, in Section 3.1, we discuss convergence to a GP
|
177 |
+
distribution in variational and Hellinger distance, drawing a connection with known
|
178 |
+
results for density convergence of normalized maxima. In Section 3.2 we quantify the
|
179 |
+
speed of convergence in variational and Hellinger distance. Finally, in Section 3.3, we
|
180 |
+
show how these can be used to also bound Kullback-Leibler divergences. Throughout,
|
181 |
+
for a twice diο¬erentiable function W(x) on R, we denote with W β²(x) = (β/βx)W(x)
|
182 |
+
and W β²β²(x) = (β2/βx2)W(x) the ο¬rst and second order derivatives, respectively.
|
183 |
+
3.1
|
184 |
+
Strong convergence under classical assumptions
|
185 |
+
Let the distribution function F be twice diο¬erentiable. In the sequel, we denote f = F β²,
|
186 |
+
gm = (F m(am Β· +bm))β² and ft = F β²
|
187 |
+
t.
|
188 |
+
Under the following classical von Mises-type
|
189 |
+
conditions
|
190 |
+
lim
|
191 |
+
xββ
|
192 |
+
xf(x)
|
193 |
+
1 β F(x) = 1
|
194 |
+
Ξ³ ,
|
195 |
+
Ξ³ > 0,
|
196 |
+
lim
|
197 |
+
xβxβ
|
198 |
+
(xβ β x)f(x)
|
199 |
+
1 β F(x)
|
200 |
+
= β1
|
201 |
+
Ξ³ ,
|
202 |
+
Ξ³ < 0,
|
203 |
+
(3.1)
|
204 |
+
lim
|
205 |
+
xβxβ
|
206 |
+
f(x)
|
207 |
+
οΏ½ xβ
|
208 |
+
x (1 β F(v)dv)
|
209 |
+
(1 β F(x))2
|
210 |
+
= 0,
|
211 |
+
Ξ³ = 0,
|
212 |
+
we know that the ο¬rst-order condition in formula (2.3) is satisο¬ed and it holds that
|
213 |
+
lim
|
214 |
+
vββ va(v)f(a(v)x + U(v)) = (1 + Ξ³x)β1/Ξ³β1
|
215 |
+
(3.2)
|
216 |
+
locally uniformly for (1 + Ξ³x) > 0. Since the equality gm(x) = F mβ1(amx + bm)hm(x)
|
217 |
+
holds true, with bm = U(m), am = a(m) and hm(x) = mamf(amx + bm), and since
|
218 |
+
F mβ1(amx+bm) converges to GΞ³(x) locally uniformly as m β β, the convergence result
|
219 |
+
in formula (3.2) thus implies that gm(x) converges to gΞ³(x) locally uniformly (Resnick,
|
220 |
+
2007, Ch. 2.2).
|
221 |
+
On the other hand, the density pertaining to Ft(s(t)x) is
|
222 |
+
lt(x) := ft(s(t)x)s(t) = s(t)f(s(t)x + t)
|
223 |
+
1 β F(t)
|
224 |
+
and, setting v = 1/(1 β F(t)), we have a(v) = s(t) and v β β as t β xβ. Therefore,
|
225 |
+
a further implication of the convergence result in formula (3.2) is that lt(x) converges
|
226 |
+
to hΞ³(x) locally uniformly for x > 0, if Ξ³ β₯ 0, or x β (0, β1/Ξ³), if Ξ³ < 0. In turn, by
|
227 |
+
Scheο¬eβs lemma we have
|
228 |
+
lim
|
229 |
+
tβxβ V (Pt, P) = 0,
|
230 |
+
where
|
231 |
+
V (Pt, P) = sup
|
232 |
+
BβB
|
233 |
+
|Pt(B) β P(B)|
|
234 |
+
4
|
235 |
+
|
236 |
+
is the total variation distance between the probability measures
|
237 |
+
Pt(B) := P
|
238 |
+
οΏ½X β t
|
239 |
+
s(t)
|
240 |
+
β B
|
241 |
+
οΏ½οΏ½οΏ½οΏ½X > t
|
242 |
+
οΏ½
|
243 |
+
and P(B) := P(Z β B),
|
244 |
+
and where Z is a random variable with distribution HΞ³ and B is a set in the Borel
|
245 |
+
Ο-ο¬eld of R, denoted by B. Let
|
246 |
+
H 2(lt; hΞ³) :=
|
247 |
+
οΏ½ οΏ½οΏ½
|
248 |
+
lt(x) β
|
249 |
+
οΏ½
|
250 |
+
hΞ³(x)
|
251 |
+
οΏ½2
|
252 |
+
dx
|
253 |
+
be the square of the Hellinger distance. It is well know that the Hellinger and total
|
254 |
+
variation distances are related as
|
255 |
+
H 2(lt; hΞ³) β€ 2V (Pt, P) β€ 2H (lt; hΞ³),
|
256 |
+
(3.3)
|
257 |
+
see e.g. Ghosal and van der Vaart (2017, Appendix B). Therefore, the conditions in
|
258 |
+
formula (3.1) ultimately entail that also the Hellinger distance between the density of
|
259 |
+
rescaled peaks over a threshold lt and the GP density hΞ³ converges to zero as t β xβ.
|
260 |
+
In the next subsection we introduce a stronger assumption, allowing us to also quantify
|
261 |
+
the speed of such convergence.
|
262 |
+
3.2
|
263 |
+
Convergence rates
|
264 |
+
As in Raoult and Worms (2003) we rely on the following assumption, in order to derive
|
265 |
+
the convergence rate for the variational and Hellinger distance.
|
266 |
+
Condition 3.1. Assume that F is twice diο¬erentiable. Moreover, assume that there
|
267 |
+
exists Ο β€ 0 such that
|
268 |
+
A(v) := vUβ²β²(v)
|
269 |
+
U β²(v) + 1 β Ξ³
|
270 |
+
deο¬nes a function of constant sign near inο¬nity, whose absolute value |A(v)| is regularly
|
271 |
+
varying as v β β with index of variation Ο.
|
272 |
+
When Condition 3.1 holds then the classical von-Mises conditions in formula (3.1)
|
273 |
+
are also satisο¬ed for the cases where Ξ³ is positive, negative or equal to zero, respec-
|
274 |
+
tively. Furthermore, Condition 3.1 implies that an appropriate scaling function for the
|
275 |
+
exceedances of a high threshold t < xβ, which complies with the equivalent ο¬rst-order
|
276 |
+
condition (2.2), is deο¬ned as
|
277 |
+
s(t) = (1 β F(t))/f(t).
|
278 |
+
With such a choice of the scaling function s, we establish the following results.
|
279 |
+
Theorem 3.2. Assume Condition 3.1 is satisο¬ed with Ξ³ > β1/2. Then, there exist
|
280 |
+
constants ci > 0 with i = 1, 2, Ξ±j > 0 with j = 1, ..., 4, K > 0 and t0 < xβ such that
|
281 |
+
H 2(lt; hΞ³)
|
282 |
+
K|A(v)|2 β€ S(v)
|
283 |
+
(3.4)
|
284 |
+
for all t β₯ t0, where v = 1/(1 β F(t)) and
|
285 |
+
S(v) :=
|
286 |
+
οΏ½
|
287 |
+
1 β |A(v)|Ξ±1 + 4 exp (c1|A(v)|Ξ±2) ,
|
288 |
+
if Ξ³ β₯ 0
|
289 |
+
1 β |A(v)|Ξ±3 + 4 exp (c2|A(v)|Ξ±4) ,
|
290 |
+
if Ξ³ < 0
|
291 |
+
.
|
292 |
+
5
|
293 |
+
|
294 |
+
Given the relationship between the total variation and Hellinger distances in (3.3),
|
295 |
+
the following result is a direct consequence of Theorem (3.2).
|
296 |
+
Corollary 3.3. Under the assumptions of Theorem 3.2, for all t β₯ t0
|
297 |
+
V (Pt, P) β€ |A(v)|
|
298 |
+
οΏ½
|
299 |
+
KS(v).
|
300 |
+
Theorem 3.2 implies that the Hellinger and variational distances of the probability
|
301 |
+
density and measure of rescaled exceedances from their GP distribution counterparts are
|
302 |
+
bounded from above by C|A(v)|, for a positive constant C, as the threshold t approaches
|
303 |
+
the end-point xβ. Since for a ο¬xed x β β©tβ₯t0(0, xββt
|
304 |
+
s(t) ) it holds that
|
305 |
+
|Ft(s(t)x) β HΞ³(x)| β€ V (Pt, P)
|
306 |
+
and since Raoult and Worms (2003, Theorem 2(i)) implies that |Ft(s(t)x)βHΞ³(x)|/|A(v)|
|
307 |
+
converges to a positive constant, there also exists c > 0 such that, for all large t, c|A(v)|
|
308 |
+
is a lower bound for variational and Hellinger distances. Therefore, since
|
309 |
+
c|A(v)| β€ V (Pt, P) β€ H (lt; hΞ³) β€ C|A(v)|,
|
310 |
+
the decay rate of variational and Hellinger distances is precisely |A(v)| as t β xβ.
|
311 |
+
Diο¬erently from the result on uniform convergence in Raoult and Worms (2003),
|
312 |
+
our results on convergence rates in the stronger total variation and Hellinger topologies
|
313 |
+
are given for Ξ³ > β1/2. Although the bound in formula (3.4) remains mathematically
|
314 |
+
valid also for tail indices below β1/2, the restriction Ξ³ > β1/2 is imposed to guarantee
|
315 |
+
that constants Ξ±3, Ξ±4 in the deο¬nition of S(v) are positive, so that S(v) is positive and
|
316 |
+
bounded as t approaches xβ. Note that such a behaviour of S is essential to deduce
|
317 |
+
from the bound in formula (3.4) that the rate of convergence is |A(v)|.
|
318 |
+
3.3
|
319 |
+
Kullback-Leibler divergences
|
320 |
+
A further implication of Theorem 3.2 concerns the speed of convergence to zero of the
|
321 |
+
Kullback-Leibler divergence
|
322 |
+
K (Λlt; hΞ³) :=
|
323 |
+
οΏ½
|
324 |
+
ln
|
325 |
+
οΏ½
|
326 |
+
Λlt(x)/hΞ³(x)
|
327 |
+
οΏ½
|
328 |
+
Λlt(x)dx,
|
329 |
+
and the divergences of higher order p β₯ 2
|
330 |
+
Dp(Λlt; hΞ³) :=
|
331 |
+
οΏ½ οΏ½οΏ½οΏ½ln
|
332 |
+
οΏ½
|
333 |
+
Λlt(x)/hΞ³(x)
|
334 |
+
οΏ½οΏ½οΏ½οΏ½
|
335 |
+
p Λlt(x)dx,
|
336 |
+
where Λlt = (Ft(Λs(t) Β· ))β² and Λs(t) is a scaling function possibly diο¬erent from s(t), which
|
337 |
+
ensures that the support of the conditional distribution Ft(Λs(t)x) is contained in that
|
338 |
+
of the GP distribution HΞ³ when Ξ³ < 0, i.e. x β R : x < (xβ β t)/Λs(t) < β1/Ξ³. We recall
|
339 |
+
indeed that, when Ξ³ is negative, the end-point (xβ β t)/s(t) of lt converges to β1/Ξ³ as
|
340 |
+
t approaches xβ. Nevertheless, for t < xβ it can be that (xβ β t)/s(t) > β1/Ξ³, entailing
|
341 |
+
that K (lt; hΞ³) = Dp(lt; hΞ³) = β. The introduction of a more ο¬exible scaling function
|
342 |
+
Λs is thus meant to rule out this uninteresting situation. In order to exploit Theorem
|
343 |
+
3.2 to give bounds on Kullback-Leibler and higher order divergences, we ο¬rst introduce
|
344 |
+
by the next two lemmas a uniform bound on density ratios and a Lipschitz continuity
|
345 |
+
result.
|
346 |
+
Lemma 3.4. Under the assumptions of Theorem 3.2, if Ο < 0 and Ξ³ ΜΈ= 0, and if
|
347 |
+
Λs(t)/s(t) β 1 as t β xβ, then there exist a t1 < xβ and a constant M β (0, β) such
|
348 |
+
that
|
349 |
+
sup
|
350 |
+
tβ₯t1
|
351 |
+
sup
|
352 |
+
0<x< xββt
|
353 |
+
Λs(t)
|
354 |
+
Λlt(x)
|
355 |
+
hΞ³(x) < M.
|
356 |
+
6
|
357 |
+
|
358 |
+
Lemma 3.5. Let Ξ³ > β1/2. Then, there exists Ο΅ > 0 and L > 0 such that
|
359 |
+
H 2(hΞ³; hΞ³β²(Ο Β· )Ο) < L2(|Ξ³ β Ξ³β²|2 + |1 β Ο|2)
|
360 |
+
whenever |Ξ³ β Ξ³β²|2 + |1 β Ο|2 < Ο΅2.
|
361 |
+
Next, using the uniform bound on density ratio provided in Lemma 3.4 and the
|
362 |
+
Lipschitz continuity property established in Lemma 3.5, we are able to translate the
|
363 |
+
upper bounds on the squared Hellinger distance H 2(lt, hΞ³) into upper bounds on the
|
364 |
+
Kullback-Leibler divergence K (Λlt; hΞ³) and higher order divergences Dp(Λlt; hΞ³).
|
365 |
+
Corollary 3.6. Under the assumptions of Theorem 3.2 with in particular Ο < 0 and
|
366 |
+
Ξ³ ΜΈ= 0, if there also exists B > 0 such that, for all large t < xβ,
|
367 |
+
|s(t)/Λs(t) β 1| β€ B|A(v)|,
|
368 |
+
then there exists a t2 < xβ such that, for all t β₯ t2
|
369 |
+
(a) K (Λlt; hΞ³) β€ 2M(
|
370 |
+
οΏ½
|
371 |
+
KS(v) + BL)2|A(v)|2
|
372 |
+
(b) Dp(Λlt; hΞ³) β€ 2p!M(
|
373 |
+
οΏ½
|
374 |
+
KS(v) + BL)2|A(v)|2, with p β₯ 2.
|
375 |
+
To extend the general results in Lemma 3.4 and Corollary 3.6 to the case of Ξ³ = 0
|
376 |
+
seems to be technically over complicated.
|
377 |
+
Nevertheless, there are speciο¬c examples
|
378 |
+
where the properties listed in such lemmas are satisο¬ed, such as the following one.
|
379 |
+
Example 3.7. Let F(x) = exp(β exp(βx)), x β R, be the Gumbel distribution function.
|
380 |
+
In this case, Condition 3.1 is satisο¬ed with Ξ³ = 0 and Ο = β1, so that Theorem 3.2
|
381 |
+
applies to this example, and for an arbitrarily small Ο΅ > 0 we have
|
382 |
+
lt(x)/h0(x) β€ exp(exp(βt)) < 1 + Ο΅
|
383 |
+
for all x > 0 and suitably large t. Hence, the bounded density ratio property is satisο¬ed
|
384 |
+
and it is still possible to conclude that Dp(lt; h0)/|A(v)|2 and K (lt; h0)/|A(v)|2 can be
|
385 |
+
bounded from above as in Corollary 3.6.
|
386 |
+
4
|
387 |
+
Implications
|
388 |
+
From a statistical stand point, the results introduced in Sections 3 can be used to study
|
389 |
+
consistency and rate of contraction of estimators of the true value for a quantity of
|
390 |
+
interest relative to the distribution of threshold exceedances within a POT approach.
|
391 |
+
First, in Section 4.1, we illustrate an application to a density estimation problem.
|
392 |
+
Second, in Section 4.2, we discuss the problem of studying estimatorsβ asymptotic ac-
|
393 |
+
curacy in more general terms. A by product of our theory in Section 3 is that the
|
394 |
+
consistency of estimators of the GP distribution parameters or related quantities can
|
395 |
+
be easily derived by means of a contiguity result (e.g. van der Vaart, 2000, Ch. 6),
|
396 |
+
provided that appropriate regularity conditions are satisο¬ed, avoiding complicated and
|
397 |
+
long calculations, typically required for example by popular estimators of the tail index
|
398 |
+
Ξ³ (Hall and Welsh, 1984; Drees, 1998; Dekkers and de Haan, 1993).
|
399 |
+
4.1
|
400 |
+
Density estimation
|
401 |
+
Accurate density estimation for threshold excesses is a crucial problem for probabilis-
|
402 |
+
tic foresting of extremes, and, in particular, for the construction of reliable predictive
|
403 |
+
regions for future large observations. When a sample X1, . . . , Xn of i.i.d. random vari-
|
404 |
+
ables, with a common distribution F, is available, a simple method to estimate the
|
405 |
+
7
|
406 |
+
|
407 |
+
density ft of (approximately) a small fraction k/n of exceedances, with k β N, over a
|
408 |
+
large quantile t = U(n/k), is as follows. Let X(nβk) < . . . < X(n) denote the k + 1
|
409 |
+
largest order statistics of the sample. Then, for measurable functions Tk,i, i = 1, 2, let
|
410 |
+
οΏ½Ξ³k = Tk,1(X(nβk), ..., X(n))
|
411 |
+
be a generic estimator of the tail index Ξ³ and
|
412 |
+
οΏ½sk = Tk,2(X(nβk), ..., X(n))
|
413 |
+
be a generic estimator of the scaling function s(U(n/k)). Since under Condition 3.1 it
|
414 |
+
holds that
|
415 |
+
ft(x) β hΞ³
|
416 |
+
οΏ½
|
417 |
+
x
|
418 |
+
s(U(n/k))
|
419 |
+
οΏ½
|
420 |
+
1
|
421 |
+
s(U(n/k)),
|
422 |
+
then a plug-in estimator of ft(x) exploiting its GP approximation is given by
|
423 |
+
οΏ½hk(x) := hοΏ½Ξ³k(x/οΏ½sk)(1/οΏ½sk).
|
424 |
+
By means of Theorem 3.2 the accuracy of the above estimator can be assessed by
|
425 |
+
quantifying its rate of contraction to the true density ft in Hellinger distance. This is
|
426 |
+
formally stated by the next result.
|
427 |
+
Proposition 4.1. Under the assumptions of Theorem 3.2 and assuming further that,
|
428 |
+
for t = U(n/k) and k β‘ k(n), the following conditions are satisο¬ed as n β β:
|
429 |
+
(a) k β β and k/n β 0,
|
430 |
+
(b)
|
431 |
+
β
|
432 |
+
k|A(n/k)| β Ξ» β (0, β),
|
433 |
+
(c) |οΏ½Ξ³k β Ξ³| = Op(1/
|
434 |
+
β
|
435 |
+
k) and |οΏ½sk/s(U(n/k)) β 1| = Op(1/
|
436 |
+
β
|
437 |
+
k),
|
438 |
+
it then holds that
|
439 |
+
H (ft;οΏ½hk) = Op(1/
|
440 |
+
β
|
441 |
+
k).
|
442 |
+
For some speciο¬c choices of the estimators οΏ½Ξ³k and οΏ½sk proposed in the literature
|
443 |
+
on POT methods (e.g. de Haan and Ferreira, 2006, Ch.
|
444 |
+
3β5), assumptions (a)β(b)
|
445 |
+
of Proposition 4.1 have been used along with the second order condition to establish
|
446 |
+
asymptotic normality of the sequence
|
447 |
+
β
|
448 |
+
k
|
449 |
+
οΏ½
|
450 |
+
οΏ½Ξ³k β Ξ³,
|
451 |
+
οΏ½sk
|
452 |
+
s(U(n/k)) β 1
|
453 |
+
οΏ½
|
454 |
+
.
|
455 |
+
Such estimators thus comply with assumption (c) of Proposition 4.1, whose statement
|
456 |
+
allows to readily obtain the rate of contraction of οΏ½hk to ft in Hellinger distance. We
|
457 |
+
provide next two examples.
|
458 |
+
Example 4.2. Under the assumptions of Theorem 3.2 and conditions (a)β(b) of Propo-
|
459 |
+
sition 4.1, there exists a sequence of ML estimators of Ξ³ and s(U(n/k)) given by
|
460 |
+
(οΏ½Ξ³k, οΏ½sk) β arg max
|
461 |
+
(Ξ³,Ο)βD
|
462 |
+
k
|
463 |
+
οΏ½
|
464 |
+
i=1
|
465 |
+
hΞ³
|
466 |
+
οΏ½X(nβk+i) β X(nβk)
|
467 |
+
Ο
|
468 |
+
οΏ½ 1
|
469 |
+
Ο
|
470 |
+
where D = (β1/2, β) Γ (0, β), satisfying condition (c) of Proposition 4.1, see Drees
|
471 |
+
et al. (2004) and Zhou (2009).
|
472 |
+
8
|
473 |
+
|
474 |
+
Example 4.3. The GPWM estimators of Ξ³ and s(U(n/k)) are deο¬ned as
|
475 |
+
οΏ½Ξ³k = 1 β
|
476 |
+
οΏ½ Pk
|
477 |
+
2Qk
|
478 |
+
β 1
|
479 |
+
οΏ½β1
|
480 |
+
,
|
481 |
+
οΏ½sk = Pk
|
482 |
+
οΏ½ Pk
|
483 |
+
2Qk
|
484 |
+
β 1
|
485 |
+
οΏ½β1
|
486 |
+
,
|
487 |
+
where
|
488 |
+
Pk = 1
|
489 |
+
k
|
490 |
+
kβ1
|
491 |
+
οΏ½
|
492 |
+
i=0
|
493 |
+
οΏ½
|
494 |
+
X(nβi) β X(nβk)
|
495 |
+
οΏ½
|
496 |
+
,
|
497 |
+
Qk = 1
|
498 |
+
k
|
499 |
+
kβ1
|
500 |
+
οΏ½
|
501 |
+
i=0
|
502 |
+
i
|
503 |
+
k
|
504 |
+
οΏ½
|
505 |
+
X(nβi) β X(nβk)
|
506 |
+
οΏ½
|
507 |
+
.
|
508 |
+
Under the assumptions of Theorem 3.2 and conditions (a)β(b) of Proposition 4.1, and
|
509 |
+
assuming further that Ξ³ < 1/2, such estimators satisfy condition (c) of Proposition 4.1,
|
510 |
+
see e.g. Theorem 3.6.1 in de Haan and Ferreira (2006).
|
511 |
+
4.2
|
512 |
+
Estimation consistency
|
513 |
+
Popular estimators of the tail index Ξ³ as for example the Hill, Moment, Pickands, ML,
|
514 |
+
GPWM (Hill, 1975; Dekkers et al., 1989; Pickands III, 1975; Jenkinson, 1969; Hosking
|
515 |
+
et al., 1985), or estimators of other related quantities, are typically deο¬ned as suitable
|
516 |
+
functionals of peaks/excesses over a large order statistic X(nβk), deο¬ned though the k
|
517 |
+
larger statistics in a sample as
|
518 |
+
Yk := (X(nβk+1) β X(nβk), . . . , X(n) β X(nβk)).
|
519 |
+
Informally speaking, the random variable X(nβk) plays the role of a high threshold t
|
520 |
+
and the sequence (X(nβk+i) β X(nβk)) with i = 1, . . . , k (up to rescaling) is seen as
|
521 |
+
approximately distributed according to HΞ³.
|
522 |
+
Let Z1, . . . , Zk be a sample of i.i.d.
|
523 |
+
random variables with GP distribution HΞ³
|
524 |
+
and let Zk = (Z(1), . . . , Z(k)) be the corresponding order statistics. In this section we
|
525 |
+
establish the important statistical result that the distribution of the suitably rescaled
|
526 |
+
sequence Yk is contiguous to that of the sequence Zk. To this aim, we ο¬rst recall the
|
527 |
+
notion of contiguity, see van der Vaart (e.g., 2000, Ch. 6.2) for more details.
|
528 |
+
Deο¬nition 4.4. Let Pk and Qk be two sequence of probability measures. Qk is said to
|
529 |
+
be contiguous with respect to Pk, in symbols Pk β·Qk, if for all measurable set sequences
|
530 |
+
Ek for which Pk(Ek) = o(1) we also have Qk(Ek) = o(1).
|
531 |
+
As in Proposition 4.1, in the sequel we assume k β‘ k(n) and k β β as n β β.
|
532 |
+
Proposition 4.5. Let Pk and Qk be the probability measures relative to the random
|
533 |
+
sequences Zk and Yk/Λs(X(nβk)), respectively. Then, under the assumptions of Corollary
|
534 |
+
3.6 and assumptions (a)β(b) of Proposition 4.1, we have that Pk β· Qk.
|
535 |
+
In statistical problems where the aim is to estimate a functional of the limiting
|
536 |
+
GP distribution, say ΞΈ := Ο(HΞ³), the contiguity result in Proposition 4.5 can be used
|
537 |
+
to show that a suitable estimator Tk(Yk) of the parameter ΞΈ is consistent, or formally
|
538 |
+
speaking D(Tk(Yk), ΞΈ) = op(1), for a suitable metric D of interest. The next result and
|
539 |
+
the subsequent discussion illustrate this point.
|
540 |
+
Corollary 4.6. Under the assumption of Proposition 4.5, if Tk is a scale invariant
|
541 |
+
measurable function on (0, β)k and Tk(Zk) is consistent estimator of ΞΈ as n β β, then
|
542 |
+
also Tk(Yk) is a consistent estimator of ΞΈ as n β β.
|
543 |
+
In real applications the distribution F of the original sample (X1, . . . , Xn) is typically
|
544 |
+
unknown and as a result also the distribution of Yk is unknown.
|
545 |
+
For this reason,
|
546 |
+
9
|
547 |
+
|
548 |
+
proving consistency of an estimator of the form Tk(Yk) for the parameter ΞΈ can be quite
|
549 |
+
burdensome, and this is especially true for the derivation of its rate of contraction. We
|
550 |
+
recall that quantifying the speed of convergence, or contraction rate, of an estimator
|
551 |
+
Tk(Yk) of a parameter ΞΈ concerns the derivation of a positive sequences Ο΅k such that
|
552 |
+
Ο΅k β 0 and D(Tk(Yk), ΞΈ) = Op(Ο΅k) as k β β, for a suitable metric D.
|
553 |
+
On the contrary, to establish the consistency of an estimator of the form Tk(Zk)
|
554 |
+
for estimating ΞΈ and its contraction rate is much easier, and these preliminary results
|
555 |
+
can be readily extended to the more demanding estimator Tk(Yk) by our Corollary 4.6,
|
556 |
+
therefore establishing its consistency and the associated speed of convergence.
|
557 |
+
We conclude the section with the following remark. It should be noted that within
|
558 |
+
the POT approach it is common to use estimators deο¬ned on the basis of scale invariant
|
559 |
+
functionals Tk. This is the case for many estimators of the tail index Ξ³ as those afore-
|
560 |
+
mentioned. Nevertheless, the result of Proposition 4.5 extends also to estimators which
|
561 |
+
are not invariant to rescaling of the data, provided that the discrepancy D(Tk(Yk), ΞΈ)
|
562 |
+
can be suitably decomposed into several terms that depends on Yk/Λs(X(nβk)) up to an
|
563 |
+
op(1) reminder.
|
564 |
+
5
|
565 |
+
Proofs
|
566 |
+
5.1
|
567 |
+
Additional notation
|
568 |
+
For y > 0, we denote T(y) = U(ey) and, for t < xβ, we deο¬ne the functions
|
569 |
+
pt(y) =
|
570 |
+
οΏ½ T(y+T β1(t))βt
|
571 |
+
s(t)
|
572 |
+
β eΞ³yβ1
|
573 |
+
Ξ³
|
574 |
+
,
|
575 |
+
Ξ³ ΜΈ= 0
|
576 |
+
T(y+T β1(t))βt
|
577 |
+
s(t)
|
578 |
+
β y,
|
579 |
+
Ξ³ = 0
|
580 |
+
,
|
581 |
+
with s(t) = (1 β F(t))/f(t), and
|
582 |
+
qt(y) =
|
583 |
+
οΏ½
|
584 |
+
1
|
585 |
+
Ξ³ ln [1 + Ξ³eβΞ³ypt(y)] ,
|
586 |
+
Ξ³ ΜΈ= 0
|
587 |
+
pt(y),
|
588 |
+
Ξ³ = 0
|
589 |
+
.
|
590 |
+
Moreover, for x β (0, xβ β t), we let Οt(x) = T β1(x + t) β T β1(t). Finally, for x β R,
|
591 |
+
Ξ³ β R, Ο β€ 0 and Ο > 0, we set
|
592 |
+
IΞ³,Ο(x) =
|
593 |
+
οΏ½ x
|
594 |
+
0
|
595 |
+
eΞ³s
|
596 |
+
οΏ½ s
|
597 |
+
0
|
598 |
+
eΟzdzds
|
599 |
+
and Οx,Ξ³ = Ξ½x/Ο(Ξ³)
|
600 |
+
οΏ½
|
601 |
+
1/Ο, with
|
602 |
+
Ξ½x(Ξ³) =
|
603 |
+
οΏ½οΏ½
|
604 |
+
hΞ³(x),
|
605 |
+
1 + Ξ³x > 0
|
606 |
+
0,
|
607 |
+
otherwise
|
608 |
+
.
|
609 |
+
5.2
|
610 |
+
Auxiliary results
|
611 |
+
In this section we provide some results which are auxiliary to the proofs of the main ones,
|
612 |
+
presented in Section 3. Throughout, for Lemmas 5.1β5.6, Condition 3.1 is implicitly
|
613 |
+
assumed to hold true.
|
614 |
+
Lemma 5.1. For every Ξ΅ > 0 and every Ξ± > 0, if Ξ³ β₯ 0, or Ξ± β (0, β1/Ξ³), if Ξ³ < 0,
|
615 |
+
there exist x1 < xβ and ΞΊ1 > 0 such that, for all t β₯ x1 and y β (0, βΞ± ln |A(eT β1(t))|)
|
616 |
+
(a) if Ξ³ β₯ 0, then
|
617 |
+
eqt(y) β
|
618 |
+
οΏ½
|
619 |
+
eΒ±ΞΊ1|A(eT β1(t))|e2Ξ΅yοΏ½
|
620 |
+
;
|
621 |
+
10
|
622 |
+
|
623 |
+
(b) if Ξ³ < 0, then
|
624 |
+
eqt(y) β
|
625 |
+
οΏ½
|
626 |
+
eΒ±ΞΊ1|A(eT β1(t))|e(Ξ³βΞ΅)yοΏ½
|
627 |
+
.
|
628 |
+
Proof. By Lemma 5 in Raoult and Worms (2003), for all Ξ΅ > 0 there exists x0 such that
|
629 |
+
for all t β (x0, xβ) and y > 0,
|
630 |
+
eβΞ³x|pt(y)| β€ (1 + Ξ΅)|A(eT β1(t))|IΞ³,Ο(y)e(Ξ³βΞ΅)y.
|
631 |
+
Moreover, for a positive constant Ο1
|
632 |
+
IΞ³,Ο(y)e(Ξ³βΞ΅)y β€
|
633 |
+
οΏ½
|
634 |
+
Ο1e2Ξ΅y,
|
635 |
+
Ξ³ β₯ 0
|
636 |
+
Ο1e(Ξ³βΞ΅)y,
|
637 |
+
Ξ³ < 0
|
638 |
+
.
|
639 |
+
Combining these two inequalities, we deduce that
|
640 |
+
eβΞ³y|pt(y)| β€
|
641 |
+
οΏ½
|
642 |
+
(1 + Ξ΅)|A(eT β1(t))|Ο1e2Ξ΅y,
|
643 |
+
Ξ³ β₯ 0
|
644 |
+
(1 + Ξ΅)|A(eT β1(t))|Ο1e(Ξ³βΞ΅)y,
|
645 |
+
Ξ³ < 0
|
646 |
+
.
|
647 |
+
(5.1)
|
648 |
+
As a consequence, if Ξ³ β₯ 0, for any Ξ± > 0 there exists a constant Ο2 such that
|
649 |
+
sup
|
650 |
+
yβ(0,βΞ± ln |A(eT β1(t))|)
|
651 |
+
eβΞ³y|pt(y)| β€ Ο2|A(eT β1(t))|1β2Ρα
|
652 |
+
(5.2)
|
653 |
+
while, if Ξ³ < 0, for any Ξ± β (0, β1/Ξ³) there exists a constant Ο3 such that
|
654 |
+
sup
|
655 |
+
yβ(0,βΞ± ln |A(eT β1(t))|)
|
656 |
+
eβΞ³y|pt(y)| β€ Ο3|A(eT β1(t))|1β(Ξ΅βΞ³)Ξ±.
|
657 |
+
(5.3)
|
658 |
+
Therefore, choosing Ξ΅ suο¬ciently small, eβΞ³y|pt(y)| converges to zero uniformly over the
|
659 |
+
interval (0, βΞ± ln |A(eT β1(t))|) as t β xβ.
|
660 |
+
It now follows that, if y β (0, βΞ± ln |A(eT β1(t))|) and t > x1 for a suο¬ciently large
|
661 |
+
value x1 < xβ, when Ξ³ ΜΈ= 0 a ο¬rst-order Taylor expansion of the logarithm at 1 yields
|
662 |
+
|qt(y)| =
|
663 |
+
οΏ½οΏ½οΏ½οΏ½
|
664 |
+
1
|
665 |
+
Ξ³
|
666 |
+
Ξ³eβΞ³ypt(y)
|
667 |
+
1 + Ο(t, y)Ξ³eβΞ³ypt(y)
|
668 |
+
οΏ½οΏ½οΏ½οΏ½
|
669 |
+
β€
|
670 |
+
οΏ½
|
671 |
+
Ο4|A(eT β1(t))|e2Ξ΅y,
|
672 |
+
Ξ³ > 0
|
673 |
+
Ο5|A(eT β1(t))|e(Ξ³βΞ΅)y,
|
674 |
+
Ξ³ < 0
|
675 |
+
,
|
676 |
+
where Ο(t, y) β (0, 1) and Ο4, Ο5 are positive constants, while when Ξ³ = 0 it holds that
|
677 |
+
|qt(y)| = eΞ³yeβΞ³y|pt(y)|
|
678 |
+
β€ Ο6|A(eT β1(t))|e2Ξ΅y,
|
679 |
+
where Ο6 is a positive constant. The two results in the statement are a direct consequence
|
680 |
+
of the last two inequalities.
|
681 |
+
Lemma 5.2. For every Ξ΅ > 0 and every Ξ± > 0, if Ξ³ β₯ 0, or Ξ± β (0, β1/Ξ³), if Ξ³ < 0,
|
682 |
+
there exist x2 < xβ and ΞΊ2 > 0 such that, for all t β₯ x2 and y β (0, βΞ± ln |A(eT β1(t))|)
|
683 |
+
(a) if Ξ³ β₯ 0, then
|
684 |
+
1 + qβ²
|
685 |
+
t(y) β
|
686 |
+
οΏ½
|
687 |
+
eΒ±ΞΊ2|A(eT β1(t))|e2Ξ΅yοΏ½
|
688 |
+
;
|
689 |
+
11
|
690 |
+
|
691 |
+
(b) if Ξ³ < 0, then
|
692 |
+
1 + qβ²
|
693 |
+
t(y) β
|
694 |
+
οΏ½οΏ½
|
695 |
+
eΒ±ΞΊ2|A(eT β1(t))|e(Ξ³βΞ΅)yοΏ½
|
696 |
+
.
|
697 |
+
Proof. If Ξ³ ΜΈ= 0
|
698 |
+
1 + qβ²
|
699 |
+
t(y) =
|
700 |
+
exp
|
701 |
+
οΏ½οΏ½ ey+T β1(t)
|
702 |
+
eT β1(t)
|
703 |
+
A(u)
|
704 |
+
u du
|
705 |
+
οΏ½
|
706 |
+
1 + Ξ³eβΞ³ypt(y)
|
707 |
+
,
|
708 |
+
while if Ξ³ = 0
|
709 |
+
1 + qβ²
|
710 |
+
t(y) = exp
|
711 |
+
οΏ½οΏ½ ey+T β1(t)
|
712 |
+
eT β1(t)
|
713 |
+
A(u)
|
714 |
+
u
|
715 |
+
du
|
716 |
+
οΏ½
|
717 |
+
.
|
718 |
+
Therefore, if y β (0, βΞ± ln |A(eT β1(t))|) and t > x2 for a suο¬ciently large value x2 < xβ,
|
719 |
+
using the bounds in formulas (5.1)β(5.3) and choosing a suitably small Ξ΅ we deduce
|
720 |
+
1 + qβ²
|
721 |
+
t(y) β€
|
722 |
+
exp
|
723 |
+
οΏ½οΏ½ ey+T β1(t)
|
724 |
+
eT β1(t)
|
725 |
+
A(u)
|
726 |
+
u du
|
727 |
+
οΏ½
|
728 |
+
1 β 1(Ξ³ ΜΈ= 0)|Ξ³|eβΞ³y|pt(y)|
|
729 |
+
β€ exp
|
730 |
+
οΏ½
|
731 |
+
y|A(eT β1(t))|
|
732 |
+
οΏ½
|
733 |
+
Γ
|
734 |
+
οΏ½
|
735 |
+
οΏ½
|
736 |
+
οΏ½
|
737 |
+
1
|
738 |
+
1βΟ1|A(eT β1(t))|e2Ξ΅y ,
|
739 |
+
Ξ³ β₯ 0
|
740 |
+
1
|
741 |
+
1βΟ2|A(eT β1(t))|e(Ξ³βΞ΅)y ,
|
742 |
+
Ξ³ < 0
|
743 |
+
β€
|
744 |
+
οΏ½
|
745 |
+
οΏ½
|
746 |
+
οΏ½
|
747 |
+
exp
|
748 |
+
οΏ½
|
749 |
+
Ο3|A(eT β1(t))|e2Ξ΅yοΏ½
|
750 |
+
,
|
751 |
+
Ξ³ β₯ 0
|
752 |
+
exp
|
753 |
+
οΏ½
|
754 |
+
Ο4|A(eT β1(t))|e(Ξ³βΞ΅)yοΏ½
|
755 |
+
,
|
756 |
+
Ξ³ < 0
|
757 |
+
for positive constants Οi, i = 1, . . . , 4. Similarly,
|
758 |
+
1 + qβ²
|
759 |
+
t(y) β₯
|
760 |
+
exp
|
761 |
+
οΏ½οΏ½ ey+T β1(t)
|
762 |
+
eT β1(t)
|
763 |
+
A(u)
|
764 |
+
u du
|
765 |
+
οΏ½
|
766 |
+
1 + 1(Ξ³ ΜΈ= 0)|Ξ³|eβΞ³y|pt(y)|
|
767 |
+
β₯ exp
|
768 |
+
οΏ½
|
769 |
+
βy|A(eT β1(t))|
|
770 |
+
οΏ½
|
771 |
+
Γ
|
772 |
+
οΏ½
|
773 |
+
οΏ½
|
774 |
+
οΏ½
|
775 |
+
1
|
776 |
+
1+Ο5|A(eT β1(t))|e2Ξ΅y ,
|
777 |
+
Ξ³ β₯ 0
|
778 |
+
1
|
779 |
+
1+Ο6|A(eT β1(t))|e(Ξ³βΞ΅)y ,
|
780 |
+
Ξ³ < 0
|
781 |
+
β₯
|
782 |
+
οΏ½
|
783 |
+
οΏ½
|
784 |
+
οΏ½
|
785 |
+
exp
|
786 |
+
οΏ½
|
787 |
+
βΟ7|A(eT β1(t))|e2Ξ΅yοΏ½
|
788 |
+
,
|
789 |
+
Ξ³ β₯ 0
|
790 |
+
exp
|
791 |
+
οΏ½
|
792 |
+
βΟ8|A(eT β1(t))|e(Ξ³βΞ΅)yοΏ½
|
793 |
+
,
|
794 |
+
Ξ³ < 0
|
795 |
+
for positive constants Οi, i = 5, . . . , 8. The result now follows.
|
796 |
+
Lemma 5.3. If Ξ³ > 0 and Ο < 0, there exists a regularly varying function R with
|
797 |
+
negative index Ο± such that, deο¬ning the function
|
798 |
+
Ξ·(t) := (1 + Ξ³t)f(t)
|
799 |
+
1 β F(t)
|
800 |
+
β 1,
|
801 |
+
as v β β, Ξ·(U(v)) = O(R(v)).
|
802 |
+
Proof. Let v0 > 0 satisfy U(v0) ΜΈ= 0 and U β²(v0) ΜΈ= 0. Then, for v > v0 it holds that
|
803 |
+
Ξ·(U(v)) = 1 + Ξ³U(v)
|
804 |
+
vUβ²(v)
|
805 |
+
β 1
|
806 |
+
= 1 + Ξ³U(v0)
|
807 |
+
vUβ²(v)
|
808 |
+
+ Ξ³
|
809 |
+
οΏ½ v
|
810 |
+
v0
|
811 |
+
U β²(r)
|
812 |
+
vUβ²(v)dr β 1.
|
813 |
+
12
|
814 |
+
|
815 |
+
Moreover, by deο¬nition of A, we have the identity
|
816 |
+
Ξ³
|
817 |
+
οΏ½ v
|
818 |
+
v0
|
819 |
+
U β²(r)
|
820 |
+
vUβ²(v)dr β 1 =
|
821 |
+
οΏ½ 1
|
822 |
+
v0/v
|
823 |
+
U β²(zv)
|
824 |
+
U β²(v) dz β 1
|
825 |
+
=
|
826 |
+
οΏ½ 1
|
827 |
+
v0/v
|
828 |
+
Ξ³zΞ³β1
|
829 |
+
οΏ½
|
830 |
+
exp
|
831 |
+
οΏ½
|
832 |
+
β
|
833 |
+
οΏ½ 1
|
834 |
+
z
|
835 |
+
A(vu)
|
836 |
+
u
|
837 |
+
du
|
838 |
+
οΏ½
|
839 |
+
β 1
|
840 |
+
οΏ½
|
841 |
+
dz β
|
842 |
+
οΏ½v0
|
843 |
+
v
|
844 |
+
οΏ½Ξ³
|
845 |
+
.
|
846 |
+
Therefore, denoting by R2(v) the ο¬rst term on the right-hand side and setting
|
847 |
+
R1(v) = 1 + Ξ³U(v0)
|
848 |
+
vUβ²(v)
|
849 |
+
β
|
850 |
+
οΏ½v0
|
851 |
+
v
|
852 |
+
οΏ½Ξ³
|
853 |
+
,
|
854 |
+
we have Ξ·(U(v)) = R1(v) + R2(v).
|
855 |
+
On one hand, the function R1(v) is regularly
|
856 |
+
varying of order βΞ³. On the other hand, for any Ξ² β (0, 1), the function R2(v) can be
|
857 |
+
decomposed as follows
|
858 |
+
R2(v) =
|
859 |
+
οΏ½ vβ(1βΞ²)
|
860 |
+
v0/v
|
861 |
+
+
|
862 |
+
οΏ½ 1
|
863 |
+
vβ(1βΞ²) Ξ³zΞ³β1
|
864 |
+
οΏ½
|
865 |
+
exp
|
866 |
+
οΏ½
|
867 |
+
β
|
868 |
+
οΏ½ 1
|
869 |
+
z
|
870 |
+
A(vu)
|
871 |
+
u
|
872 |
+
du
|
873 |
+
οΏ½
|
874 |
+
β 1
|
875 |
+
οΏ½
|
876 |
+
dz
|
877 |
+
=: R2,1(v) + R2,2(v).
|
878 |
+
Assuming that A is ultimately positive and selecting v0 suitably large, we have
|
879 |
+
|R2,1(v)| β€
|
880 |
+
οΏ½ vβ(1βΞ²)
|
881 |
+
v0/v
|
882 |
+
Ξ³zΞ³β1
|
883 |
+
οΏ½
|
884 |
+
1 β exp
|
885 |
+
οΏ½
|
886 |
+
βA(vz)
|
887 |
+
z
|
888 |
+
οΏ½οΏ½
|
889 |
+
dz
|
890 |
+
= O(vβΞ³(1βΞ²))
|
891 |
+
and
|
892 |
+
|R2,2(v)| β€
|
893 |
+
οΏ½ 1
|
894 |
+
vβ(1βΞ²) Ξ³zΞ³β1 οΏ½
|
895 |
+
1 β zA(vΞ²)οΏ½
|
896 |
+
dz
|
897 |
+
= O(vβΞ³(1βΞ²) β¨ A(vΞ²)).
|
898 |
+
Consequently, there exists a regularly varying function R of index Ο± = Ξ³(Ξ² β 1) β¨ ΟΞ²
|
899 |
+
complying with the property in the statement as v β β.
|
900 |
+
Similarly, if A is ultimately negative, choosing Ξ² such that Ξ² < 2Ξ³ and v0 suitably
|
901 |
+
large, we have
|
902 |
+
|R2,1(v)| β€
|
903 |
+
οΏ½ vβ(1βΞ²)
|
904 |
+
v0/v
|
905 |
+
Ξ³zΞ³β1 οΏ½
|
906 |
+
uA(v0) β 1
|
907 |
+
οΏ½
|
908 |
+
dz
|
909 |
+
= O(vβ(Ξ³βΞ²/2)(1βΞ²))
|
910 |
+
and
|
911 |
+
|R2,2(v)| β€
|
912 |
+
οΏ½ 1
|
913 |
+
vβ(1βΞ²) Ξ³zΞ³β1 οΏ½
|
914 |
+
zA(vΞ²) β 1
|
915 |
+
οΏ½
|
916 |
+
dz
|
917 |
+
= O(vβ(Ξ³βΞ²/2)(1βΞ²) β¨ |A(vΞ²)|)
|
918 |
+
as v β β. Hence, there exists a regularly varying function R of index Ο± = (Ξ² β 1)(Ξ³ β
|
919 |
+
Ξ²/2)β¨ΟΞ² complying with the property in the statement. The proof is now complete.
|
920 |
+
Lemma 5.4. If Ξ³ > 0 and Ο < 0, there exists x3 β (0, β) and Ξ΄ > 0 such that, for all
|
921 |
+
x β₯ x3,
|
922 |
+
f(x) = hΞ³(x)
|
923 |
+
οΏ½
|
924 |
+
1 + O({1 β HΞ³(x)}Ξ΄)
|
925 |
+
οΏ½
|
926 |
+
.
|
927 |
+
13
|
928 |
+
|
929 |
+
Proof. Let Rβ(t) := R(1/(1 β F(t))), where R is as in Lemma 5.3. Then Rβ(t) is regu-
|
930 |
+
larly varying of index Ο±/Ξ³ (Resnick, 2007, Proposition 0.8(iv)). In turn, by Karamataβs
|
931 |
+
theorem (e.g, Resnick, 2007, Proposition 0.6(a)) we have that for a large tβ
|
932 |
+
οΏ½ β
|
933 |
+
tβ
|
934 |
+
|Ξ·(t)|
|
935 |
+
1 + Ξ³tdt < β
|
936 |
+
and thus, by Proposition 2.1.4 in Falk et al. (2010), we conclude that
|
937 |
+
Ο := lim
|
938 |
+
tββ
|
939 |
+
1 β F(t)
|
940 |
+
1 β HΞ³(t) β (0, β).
|
941 |
+
(5.4)
|
942 |
+
As a consequence, for any Ξ΄ β (0, βΟ±), as t β β
|
943 |
+
Rβ(t) βΌ R
|
944 |
+
οΏ½
|
945 |
+
1
|
946 |
+
Ο(1 β HΞ³(t))
|
947 |
+
οΏ½
|
948 |
+
= O({1 β HΞ³(t)}Ξ΄).
|
949 |
+
The conclusion now follows by Proposition 2.1.5 in Falk et al. (2010).
|
950 |
+
Lemma 5.5. If Ξ³ < 0 and Ο < 0, there exists a a regularly varying function ΛR with
|
951 |
+
negative index ΛΟ± = (β1) β¨ (βΟ/Ξ³) such that, deο¬ning the function
|
952 |
+
ΛΞ·(y) := (1 β Ξ³y)f(xβ β 1/y)
|
953 |
+
[1 β F(xβ β 1/y)]y2 β 1,
|
954 |
+
as y β β, ΛΞ·(y) = O( ΛR(y)).
|
955 |
+
Proof. By deο¬nition,
|
956 |
+
ΛΞ· (y) =
|
957 |
+
f(xβ β 1/y)
|
958 |
+
[1 β F(xβ β 1/y)]y2 β Ξ³
|
959 |
+
οΏ½
|
960 |
+
f(xβ β 1/y)
|
961 |
+
y(1 β F(xβ β 1/y)) + 1
|
962 |
+
Ξ³
|
963 |
+
οΏ½
|
964 |
+
=: ΛΞ·1 (y) + ΛΞ·2 (y) .
|
965 |
+
On one hand, we have that, as y β β
|
966 |
+
ΛΞ·1 (y) = O(1/y).
|
967 |
+
On the other hand, for v > 1 we have the identity
|
968 |
+
ΛΞ·2
|
969 |
+
οΏ½
|
970 |
+
1
|
971 |
+
xβ β U(v)
|
972 |
+
οΏ½
|
973 |
+
=
|
974 |
+
οΏ½ β
|
975 |
+
1
|
976 |
+
Ξ³zΞ³β1
|
977 |
+
οΏ½
|
978 |
+
1 β exp
|
979 |
+
οΏ½οΏ½ z
|
980 |
+
1
|
981 |
+
A(uv)
|
982 |
+
u
|
983 |
+
du
|
984 |
+
οΏ½οΏ½
|
985 |
+
dz.
|
986 |
+
Hence, if A is ultimately positive,
|
987 |
+
ΛΞ·2
|
988 |
+
οΏ½
|
989 |
+
1
|
990 |
+
xβ β U(v)
|
991 |
+
οΏ½
|
992 |
+
β€ βΞ³
|
993 |
+
οΏ½ β
|
994 |
+
1
|
995 |
+
zΞ³β1(zA(v) β 1)dz
|
996 |
+
= O(A(v))
|
997 |
+
while, if A is ultimately negative,
|
998 |
+
οΏ½οΏ½οΏ½οΏ½ΛΞ·2
|
999 |
+
οΏ½
|
1000 |
+
1
|
1001 |
+
xβ β U(v)
|
1002 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ³A(v)
|
1003 |
+
οΏ½ β
|
1004 |
+
1
|
1005 |
+
zΞ³β1 ln zdz
|
1006 |
+
= O(|A(v)|).
|
1007 |
+
As a result of the two above inequalities, as v β β
|
1008 |
+
ΛΞ·2(t) = O
|
1009 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½A
|
1010 |
+
οΏ½
|
1011 |
+
1
|
1012 |
+
1 β F(xβ β 1/y)
|
1013 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½
|
1014 |
+
οΏ½
|
1015 |
+
,
|
1016 |
+
Therefore, by regular variation of 1/(1 β F(xβ β 1/y)) with index β1/Ξ³, ΛΞ·2(y) is even-
|
1017 |
+
tually dominated by a regularly varing function of index βΟ/Ξ³. The ο¬nal result now
|
1018 |
+
follows.
|
1019 |
+
14
|
1020 |
+
|
1021 |
+
Lemma 5.6. If Ξ³ < 0 and Ο < 0, there exist ΛΞ΄ > 0 such that, as y β β,
|
1022 |
+
f(xβ β 1/y)
|
1023 |
+
y2
|
1024 |
+
= (1 β Ξ³y)1/Ξ³β1 οΏ½
|
1025 |
+
1 + O({1 β HβΞ³(y)}
|
1026 |
+
ΛΞ΄)
|
1027 |
+
οΏ½
|
1028 |
+
Proof. The function Λf(y) := f(xβ β 1/y)yβ2 is the density of the distribution function
|
1029 |
+
ΛF(y) := F(xββ1/y), which is in the domain of attraction of GΛΞ³, with ΛΞ³ = βΞ³. Moreover,
|
1030 |
+
ΛΞ·(y) = (1 + ΛΞ³y) Λf(y)
|
1031 |
+
1 β ΛF(y)
|
1032 |
+
β 1.
|
1033 |
+
By Lemma 5.5 and regular variation of 1 β HΛΞ³ with index β1/ΛΞ³, we have
|
1034 |
+
ΛΞ·(y) = O({1 β HΛΞ³(y)}
|
1035 |
+
ΛΞ΄)
|
1036 |
+
for any ΛΞ΄ > 0 such that βΛΞ΄/ΛΞ³ > ΛΟ±. Therefore, by Proposition 2.1.5 in Falk et al. (2010),
|
1037 |
+
as y β β it holds that
|
1038 |
+
Λf(y) = hΛΞ³(y)[1 + O({1 β HΛΞ³(y)}
|
1039 |
+
ΛΞ΄)],
|
1040 |
+
which is the result.
|
1041 |
+
Lemma 5.7. Let Ξ½β²
|
1042 |
+
x(Ξ³) = (β/βΞ³)Ξ½x(Ξ³).
|
1043 |
+
(a) If ΞΎ : R οΏ½β (0, β), then it holds that
|
1044 |
+
οΏ½οΏ½ β
|
1045 |
+
0
|
1046 |
+
[Ξ½β²x(ΞΎ(x))]2 dx β€ 1
|
1047 |
+
2
|
1048 |
+
οΏ½οΏ½ β
|
1049 |
+
0
|
1050 |
+
(1 + xΞΎ(x))β3β
|
1051 |
+
1
|
1052 |
+
ΞΎ(x)
|
1053 |
+
οΏ½x ln(1 + xΞΎ(x))
|
1054 |
+
ΞΎ(x)
|
1055 |
+
οΏ½2
|
1056 |
+
dx
|
1057 |
+
+ 1
|
1058 |
+
2
|
1059 |
+
οΏ½οΏ½ β
|
1060 |
+
0
|
1061 |
+
(1 + xΞΎ(x))β3β
|
1062 |
+
1
|
1063 |
+
ΞΎ(x) x2dx.
|
1064 |
+
(b) If instead ΞΎ : R οΏ½β (Ξ³β, 0), then
|
1065 |
+
οΏ½οΏ½ β1/Ξ³β
|
1066 |
+
0
|
1067 |
+
[Ξ½β²x(ΞΎ(x))]2 dx β€ 1
|
1068 |
+
2
|
1069 |
+
οΏ½οΏ½ β1/Ξ³β
|
1070 |
+
0
|
1071 |
+
(1 + xΞΎ(x))β3β
|
1072 |
+
1
|
1073 |
+
ΞΎ(x) x4dx
|
1074 |
+
+ 1
|
1075 |
+
2
|
1076 |
+
οΏ½οΏ½ β1/Ξ³β
|
1077 |
+
0
|
1078 |
+
(1 + xΞΎ(x))β3β
|
1079 |
+
1
|
1080 |
+
ΞΎ(x) x2dx.
|
1081 |
+
Proof. Let Οx(Ξ³) := (β/βΞ³) ln(1 β HΞ³(x)). Then, for any x > 0, if ΞΎ(Β·) > 0, or x β
|
1082 |
+
(0, β1/Ξ³β), if ΞΎ(Β·) β (Ξ³β, 0) we have
|
1083 |
+
Ξ½β²
|
1084 |
+
x(ΞΎ(x)) = 1
|
1085 |
+
2(1 + xΞΎ(x))β 1
|
1086 |
+
2 β
|
1087 |
+
1
|
1088 |
+
2ΞΎ(x) Οx(ΞΎ(x)) β 1
|
1089 |
+
2(1 + xΞΎ(x))β 3
|
1090 |
+
2 β
|
1091 |
+
1
|
1092 |
+
2ΞΎ(x) x.
|
1093 |
+
If ΞΎ(Β·) > 0, by Minkowski inequality
|
1094 |
+
οΏ½οΏ½ β
|
1095 |
+
0
|
1096 |
+
[Ξ½β²x(ΞΎ(x))]2 dx β€ 1
|
1097 |
+
2
|
1098 |
+
οΏ½οΏ½ β
|
1099 |
+
0
|
1100 |
+
(1 + xΞΎ(x))β1β
|
1101 |
+
1
|
1102 |
+
ΞΎ(x) [Οx(ΞΎ(x))]2 dx
|
1103 |
+
+ 1
|
1104 |
+
2
|
1105 |
+
οΏ½οΏ½ β
|
1106 |
+
0
|
1107 |
+
(1 + xΞΎ(x))β3β
|
1108 |
+
1
|
1109 |
+
ΞΎ(x) x2dx.
|
1110 |
+
15
|
1111 |
+
|
1112 |
+
The result at point (a) now follows from the above inequality and the fact that, by
|
1113 |
+
equations (B.5)-(B.6) in B¨ucher and Segers (2017),
|
1114 |
+
0 β€ Οx(ΞΎ(x)) β€ x ln(1 + xΞΎ(x))
|
1115 |
+
ΞΎ(x)(1 + xΞΎ(x)).
|
1116 |
+
If ΞΎ(Β·) β (Ξ³β, 0), inequality (B.8) in BΒ¨ucher and Segers (2017) implies that for any
|
1117 |
+
x β (0, β1/Ξ³β)
|
1118 |
+
0 β€ Οx(ΞΎ(x)) β€
|
1119 |
+
x2
|
1120 |
+
1 + xΞΎ(x).
|
1121 |
+
This inequality and an argument by Minkowsi inequality, analogous to the previous one,
|
1122 |
+
now lead to the result at point (b).
|
1123 |
+
Lemma 5.8. Set Οβ²
|
1124 |
+
Ξ³,x(Ο) = (β/βΟ)ΟΞ³,x(Ο).
|
1125 |
+
(a) If Ο : R οΏ½β (1 Β± Ο΅), with Ο΅ β (0, 1), and if Ξ³ > 0
|
1126 |
+
οΏ½οΏ½ β
|
1127 |
+
0
|
1128 |
+
οΏ½
|
1129 |
+
Οβ²Ξ³,x(Ο(x))
|
1130 |
+
οΏ½2 dx β€
|
1131 |
+
οΏ½1
|
1132 |
+
Ξ³ +
|
1133 |
+
οΏ½
|
1134 |
+
1
|
1135 |
+
2Ξ³ + 1
|
1136 |
+
οΏ½ οΏ½1 + Ο΅
|
1137 |
+
1 β Ο΅
|
1138 |
+
οΏ½5/2
|
1139 |
+
.
|
1140 |
+
(b) If Ξ³ β (β1/2, 0) and Ο(x) β (Οβ, 1), with Οβ β (0, 1), there is a constant ΞΆ > 0 such
|
1141 |
+
that
|
1142 |
+
οΏ½οΏ½ β Οβ
|
1143 |
+
Ξ³
|
1144 |
+
0
|
1145 |
+
οΏ½
|
1146 |
+
Οβ²Ξ³,x(Ο(x))
|
1147 |
+
οΏ½2 dx β€
|
1148 |
+
1
|
1149 |
+
Οβ
|
1150 |
+
ββΞ³ΞΆ
|
1151 |
+
οΏ½ 1
|
1152 |
+
Ξ³2 + 1
|
1153 |
+
οΏ½
|
1154 |
+
.
|
1155 |
+
If instead Ο(x) > 1,
|
1156 |
+
οΏ½οΏ½ β 1
|
1157 |
+
Ξ³
|
1158 |
+
0
|
1159 |
+
οΏ½
|
1160 |
+
Οβ²Ξ³,x(Ο(x))
|
1161 |
+
οΏ½2 dx β€
|
1162 |
+
1
|
1163 |
+
ββΞ³ΞΆ
|
1164 |
+
οΏ½ 1
|
1165 |
+
Ξ³2 + 1
|
1166 |
+
οΏ½
|
1167 |
+
.
|
1168 |
+
Proof. Note that for x such that 1 + Ξ³x/Ο > 0
|
1169 |
+
Οβ²
|
1170 |
+
Ξ³,x(Ο) = (1 + Ξ³x/Ο)β 1
|
1171 |
+
2Ξ³ β 3
|
1172 |
+
2
|
1173 |
+
Ο5/2
|
1174 |
+
x
|
1175 |
+
Ξ³ + (1 + Ξ³x/Ο)β 1
|
1176 |
+
2Ξ³ β 3
|
1177 |
+
2
|
1178 |
+
Ο3/2
|
1179 |
+
.
|
1180 |
+
Consequently, if Ο : R οΏ½β (1 Β± Ο΅) and Ξ³ > 0, by Minkowski inequality
|
1181 |
+
οΏ½οΏ½ β
|
1182 |
+
0
|
1183 |
+
οΏ½
|
1184 |
+
Οβ²Ξ³,x(Ο(x))
|
1185 |
+
οΏ½2 dx
|
1186 |
+
β€
|
1187 |
+
οΏ½οΏ½ β
|
1188 |
+
0
|
1189 |
+
(1 + Ξ³x/Ο(x))β 1
|
1190 |
+
Ξ³ β3
|
1191 |
+
Ο5(x)
|
1192 |
+
οΏ½x
|
1193 |
+
Ξ³
|
1194 |
+
οΏ½2
|
1195 |
+
dx +
|
1196 |
+
οΏ½οΏ½ β
|
1197 |
+
0
|
1198 |
+
(1 + Ξ³x/Ο(x))β 1
|
1199 |
+
Ξ³ β3
|
1200 |
+
Ο3(x)
|
1201 |
+
dx
|
1202 |
+
β€ (1 + Ο΅)
|
1203 |
+
3
|
1204 |
+
2
|
1205 |
+
(1 β Ο΅)
|
1206 |
+
5
|
1207 |
+
2
|
1208 |
+
1
|
1209 |
+
Ξ³ + (1 + Ο΅)
|
1210 |
+
1
|
1211 |
+
2
|
1212 |
+
(1 β Ο΅)
|
1213 |
+
3
|
1214 |
+
2
|
1215 |
+
οΏ½
|
1216 |
+
1
|
1217 |
+
2Ξ³ + 1
|
1218 |
+
οΏ½ 1
|
1219 |
+
2
|
1220 |
+
and the result at point (a) follows.
|
1221 |
+
16
|
1222 |
+
|
1223 |
+
If instead, Ο(Β·) β (Οβ, 1), for some Οβ β (0, 1), and Ξ³ β (β1/2, 0), there is a constant
|
1224 |
+
ΞΆ > 0 such that
|
1225 |
+
οΏ½οΏ½ β Οβ
|
1226 |
+
Ξ³
|
1227 |
+
0
|
1228 |
+
οΏ½
|
1229 |
+
Οβ²Ξ³,x(Ο(x))
|
1230 |
+
οΏ½2 dx
|
1231 |
+
β€
|
1232 |
+
οΏ½οΏ½ β Οβ
|
1233 |
+
Ξ³
|
1234 |
+
0
|
1235 |
+
(1 + Ξ³x/Ο(x))β 1
|
1236 |
+
Ξ³ β3
|
1237 |
+
Ο5(x)
|
1238 |
+
οΏ½x
|
1239 |
+
Ξ³
|
1240 |
+
οΏ½2
|
1241 |
+
dx +
|
1242 |
+
οΏ½οΏ½ β Οβ
|
1243 |
+
Ξ³
|
1244 |
+
0
|
1245 |
+
(1 + Ξ³x/Ο(x))β 1
|
1246 |
+
Ξ³ β3
|
1247 |
+
Ο3(x)
|
1248 |
+
dx
|
1249 |
+
β€
|
1250 |
+
οΏ½οΏ½ β Οβ
|
1251 |
+
Ξ³
|
1252 |
+
0
|
1253 |
+
(1 + Ξ³x/Οβ)β1+ΞΆ
|
1254 |
+
Ο3β
|
1255 |
+
1
|
1256 |
+
Ξ³4 dx +
|
1257 |
+
οΏ½οΏ½ β Οβ
|
1258 |
+
Ξ³
|
1259 |
+
0
|
1260 |
+
(1 + Ξ³x/Οβ)β1+ΞΆ
|
1261 |
+
Ο3β
|
1262 |
+
dx
|
1263 |
+
=
|
1264 |
+
οΏ½
|
1265 |
+
1
|
1266 |
+
ΞΆΟ2β(βΞ³)5 +
|
1267 |
+
οΏ½
|
1268 |
+
1
|
1269 |
+
ΞΆΟ2β(βΞ³).
|
1270 |
+
The ο¬rst half of the statement at point (b) is now established. The second half of the
|
1271 |
+
statement can be proved analogously.
|
1272 |
+
5.3
|
1273 |
+
Proof of Theorem 3.2
|
1274 |
+
For every xt > 0, it holds that
|
1275 |
+
H 2(lt; hΞ³) =
|
1276 |
+
οΏ½ xt
|
1277 |
+
0
|
1278 |
+
+
|
1279 |
+
οΏ½ β
|
1280 |
+
xt
|
1281 |
+
οΏ½οΏ½
|
1282 |
+
ft(x) β
|
1283 |
+
οΏ½
|
1284 |
+
hΞ³(x/s(t))/s(t)
|
1285 |
+
οΏ½2
|
1286 |
+
dx
|
1287 |
+
β€
|
1288 |
+
οΏ½ Οt(xt)
|
1289 |
+
0
|
1290 |
+
eβy
|
1291 |
+
οΏ½
|
1292 |
+
1 β
|
1293 |
+
οΏ½
|
1294 |
+
eqt(y)(1 + qβ²
|
1295 |
+
t(y))
|
1296 |
+
οΏ½2
|
1297 |
+
dy
|
1298 |
+
+
|
1299 |
+
οΏ½οΏ½
|
1300 |
+
1 β Ft(xt) +
|
1301 |
+
οΏ½
|
1302 |
+
1 β HΞ³(xt/s(t))
|
1303 |
+
οΏ½2
|
1304 |
+
=: I1(t) + I2(t).
|
1305 |
+
Let xt be such that the following equality holds
|
1306 |
+
Οt(xt) = βΞ± ln |A(eT β1(t))|,
|
1307 |
+
for a positive constant Ξ± to be speciο¬ed later. Then, by Lemmas 5.1-5.2, for a suitably
|
1308 |
+
small Ξ΅ > 0 there exist constants ΞΊ3, ΞΊ4 > 0 such that for all suο¬ciently large t
|
1309 |
+
I1(t) β€
|
1310 |
+
οΏ½
|
1311 |
+
οΏ½
|
1312 |
+
οΏ½
|
1313 |
+
οΏ½ βΞ± ln |A(eT β1(t))|
|
1314 |
+
0
|
1315 |
+
ΞΊ3|A(eT β1(t))|2e(4Ξ΅β1)ydy,
|
1316 |
+
Ξ³ β₯ 0
|
1317 |
+
οΏ½ βΞ± ln |A(eT β1(t))|
|
1318 |
+
0
|
1319 |
+
ΞΊ4|A(eT β1(t))|2e(Ξ³βΞ΅β1)ydy,
|
1320 |
+
Ξ³ < 0
|
1321 |
+
β€
|
1322 |
+
οΏ½
|
1323 |
+
οΏ½
|
1324 |
+
οΏ½
|
1325 |
+
ΞΊ3|A(eT β1(t))|2 οΏ½
|
1326 |
+
1 β |A(eT β1(t))|Ξ±1
|
1327 |
+
οΏ½
|
1328 |
+
,
|
1329 |
+
Ξ³ β₯ 0
|
1330 |
+
ΞΊ4|A(eT β1(t))|2 οΏ½
|
1331 |
+
1 β |A(eT β1(t))|Ξ±3
|
1332 |
+
οΏ½
|
1333 |
+
,
|
1334 |
+
Ξ³ < 0
|
1335 |
+
,
|
1336 |
+
where Ξ±1 := Ξ±(1 β 4Ξ΅) and Ξ±3 := Ξ±(1 β 2(Ξ΅ β Ξ³)) are positive. Moreover, on one hand
|
1337 |
+
we have the identity
|
1338 |
+
1 β Ft(xt) = |A(eT β1(t))|Ξ±.
|
1339 |
+
On the other hand, for some constants ΞΊ5, ΞΊ6 > 0 we have the inequality
|
1340 |
+
1 β HΞ³(xt/s(t)) = |A(eT β1(t))|Ξ± exp
|
1341 |
+
οΏ½
|
1342 |
+
βqt
|
1343 |
+
οΏ½
|
1344 |
+
βΞ± ln |A(eT β1(t))|
|
1345 |
+
οΏ½οΏ½
|
1346 |
+
β€
|
1347 |
+
οΏ½
|
1348 |
+
οΏ½
|
1349 |
+
οΏ½
|
1350 |
+
|A(eT β1(t))|Ξ± exp
|
1351 |
+
οΏ½
|
1352 |
+
ΞΊ5|A(eT β1(t))|1β2Ρα�
|
1353 |
+
,
|
1354 |
+
Ξ³ β₯ 0
|
1355 |
+
|A(eT β1(t))|Ξ± exp
|
1356 |
+
οΏ½
|
1357 |
+
ΞΊ6|A(eT β1(t))|1β(Ξ΅βΞ³)Ξ±οΏ½
|
1358 |
+
,
|
1359 |
+
Ξ³ < 0
|
1360 |
+
.
|
1361 |
+
17
|
1362 |
+
|
1363 |
+
Consequently,
|
1364 |
+
I2(t) β€
|
1365 |
+
οΏ½
|
1366 |
+
οΏ½
|
1367 |
+
οΏ½
|
1368 |
+
|A(eT β1(t))|Ξ± οΏ½
|
1369 |
+
1 + exp
|
1370 |
+
οΏ½
|
1371 |
+
ΞΊ5
|
1372 |
+
2 |A(eT β1(t))|1β2Ρα��
|
1373 |
+
,
|
1374 |
+
Ξ³ β₯ 0
|
1375 |
+
|A(eT β1(t))|Ξ± οΏ½
|
1376 |
+
1 + exp
|
1377 |
+
οΏ½
|
1378 |
+
ΞΊ6
|
1379 |
+
2 |A(eT β1(t))|1β(Ξ΅βΞ³)Ξ±οΏ½οΏ½
|
1380 |
+
,
|
1381 |
+
Ξ³ < 0
|
1382 |
+
.
|
1383 |
+
Now, if Ξ³ β₯ 0, we can choose Ξ± > 2 and Ξ΅ small enough, so that
|
1384 |
+
|A(eT β1(t))|Ξ± < |A(eT β1(t))|2
|
1385 |
+
and Ξ±2 := 1 β 2Ρα > 0. If instead Ξ³ β (β1/2, 0), we can choose Ξ± slightly larger than
|
1386 |
+
2 and Ξ΅ small enough, so that the inequality in the above display is still satisο¬ed and
|
1387 |
+
Ξ±4 := 1βΞ±(Ξ΅βΞ³) > 0. The conclusion then follows noting that T β1(t) = β ln(1βF(t))
|
1388 |
+
and, in turn,
|
1389 |
+
|A(eT β1(t))| = |A(v)|.
|
1390 |
+
5.4
|
1391 |
+
Proof of Lemma 3.4
|
1392 |
+
We analyse the cases where Ξ³ > 0 and Ξ³ < 0 separately.
|
1393 |
+
Case 1: Ξ³ > 0. In this case, Λs(t) = s(t) = (1 β F(t))/f(t) and Λlt = lt. By Lemma
|
1394 |
+
5.4, there are positive constants ΞΊ, Ξ΄ and Ο΅ such that, for all large t and all x > 0
|
1395 |
+
lt(x)
|
1396 |
+
hΞ³(x) β€ hΞ³(s(t)x + t)
|
1397 |
+
hΞ³(x)
|
1398 |
+
s(t)
|
1399 |
+
1 β F(t)
|
1400 |
+
οΏ½
|
1401 |
+
1 + ΞΊ {1 β HΞ³(s(t)x + t)}Ξ΄οΏ½
|
1402 |
+
β€
|
1403 |
+
οΏ½
|
1404 |
+
1 + Ξ³x
|
1405 |
+
(1 + Ξ³t)/s(t) + Ξ³x
|
1406 |
+
οΏ½1+1/Ξ³
|
1407 |
+
1 + Ο΅
|
1408 |
+
(s(t))1/Ξ³(1 β F(t)).
|
1409 |
+
Moreover, by Lemma 5.3 it holds that as t β β
|
1410 |
+
1 + Ξ³t
|
1411 |
+
s(t)
|
1412 |
+
= 1 + Ξ·(t) = 1 + o(1)
|
1413 |
+
and, in turn, (s(t))1/Ξ³ βΌ (1+Ξ³t)1/Ξ³. These two facts, combined with the tail equivalence
|
1414 |
+
relation in formula (5.4), imply that for all suο¬ciently large t and all x > 0
|
1415 |
+
lt(x)
|
1416 |
+
hΞ³(x) β€
|
1417 |
+
οΏ½
|
1418 |
+
1 + Ξ³x
|
1419 |
+
1 β Ο΅ + Ξ³x
|
1420 |
+
οΏ½1+1/Ξ³
|
1421 |
+
1 + Ο΅
|
1422 |
+
(1 β Ο΅)Ο
|
1423 |
+
β€
|
1424 |
+
οΏ½
|
1425 |
+
1
|
1426 |
+
1 β Ο΅
|
1427 |
+
οΏ½1+1/Ξ³
|
1428 |
+
1 + Ο΅
|
1429 |
+
(1 β Ο΅)Ο .
|
1430 |
+
The result now follows.
|
1431 |
+
Case 2: Ξ³ < 0. In this case, for any x β (0, (xβ β t)/Λs(t))
|
1432 |
+
Λlt(x) = f
|
1433 |
+
οΏ½
|
1434 |
+
xβ β 1
|
1435 |
+
y
|
1436 |
+
οΏ½ 1
|
1437 |
+
y2
|
1438 |
+
y2Λs(t)
|
1439 |
+
1 β F(t)
|
1440 |
+
where
|
1441 |
+
y β‘ y(x, t) :=
|
1442 |
+
1
|
1443 |
+
Λs(t)
|
1444 |
+
οΏ½xβ β t
|
1445 |
+
Λs(t)
|
1446 |
+
β x
|
1447 |
+
οΏ½β1
|
1448 |
+
Note that y is bounded from below by 1/(xβ β t), which converges to β as t β xβ.
|
1449 |
+
Thus, by Lemma 5.6 there are positive constants ΛΞ΄, Ο΅ and ΛΞΊ such that
|
1450 |
+
Λlt(x) β€ (1 β Ξ³y)1/Ξ³β1[1 + ΛΞΊ{1 β HβΞ³(y)}
|
1451 |
+
ΛΞ΄] y2Λs(t)
|
1452 |
+
1 β F(t)
|
1453 |
+
β€
|
1454 |
+
οΏ½
|
1455 |
+
1 + Ξ³ Λs(t)
|
1456 |
+
xβ β t
|
1457 |
+
οΏ½
|
1458 |
+
β1
|
1459 |
+
Ξ³
|
1460 |
+
οΏ½
|
1461 |
+
x
|
1462 |
+
οΏ½β 1
|
1463 |
+
Ξ³ β1 οΏ½
|
1464 |
+
(xβ β t)
|
1465 |
+
οΏ½
|
1466 |
+
1 β Λs(t)x
|
1467 |
+
xβ β t
|
1468 |
+
οΏ½
|
1469 |
+
β Ξ³
|
1470 |
+
οΏ½ 1
|
1471 |
+
Ξ³ β1 Λs(t)(1 + Ο΅)
|
1472 |
+
1 β F(t) (xβ β t)β 1
|
1473 |
+
Ξ³ β1.
|
1474 |
+
18
|
1475 |
+
|
1476 |
+
By hypothesis, it holds that
|
1477 |
+
xβ β t
|
1478 |
+
Λs(t)
|
1479 |
+
β€ β1
|
1480 |
+
Ξ³ ,
|
1481 |
+
thus
|
1482 |
+
οΏ½
|
1483 |
+
1 + Ξ³ Λs(t)
|
1484 |
+
xβ β t
|
1485 |
+
οΏ½
|
1486 |
+
β1
|
1487 |
+
Ξ³
|
1488 |
+
οΏ½
|
1489 |
+
x
|
1490 |
+
οΏ½β 1
|
1491 |
+
Ξ³ β1
|
1492 |
+
β€ (1 + Ξ³x)β1/Ξ³β1.
|
1493 |
+
Moreover, it holds that
|
1494 |
+
1 β Λs(t)x
|
1495 |
+
xβ β t > 0,
|
1496 |
+
thus
|
1497 |
+
οΏ½
|
1498 |
+
(xβ β t)
|
1499 |
+
οΏ½
|
1500 |
+
1 β Λs(t)x
|
1501 |
+
xβ β t
|
1502 |
+
οΏ½
|
1503 |
+
β Ξ³
|
1504 |
+
οΏ½ 1
|
1505 |
+
Ξ³ β1
|
1506 |
+
β€ (βΞ³)
|
1507 |
+
1
|
1508 |
+
Ξ³ β1.
|
1509 |
+
Finally, for all large t,
|
1510 |
+
Λs(t)
|
1511 |
+
xβ β t β€ β(1 + Ο΅)Ξ³.
|
1512 |
+
Combining all the above inequalities we can now conclude that, for all large t and for
|
1513 |
+
any x β (0, (xβ β t)/Λs(t)),
|
1514 |
+
Λlt(x)
|
1515 |
+
hΞ³(x) β€ (1 + Ο΅)2(βΞ³)
|
1516 |
+
1
|
1517 |
+
Ξ³ (xβ β t)β 1
|
1518 |
+
Ξ³
|
1519 |
+
1 β F(t) .
|
1520 |
+
Now, setting t = U(v), we have that v β β if and only if t β xβ and, by Theorem
|
1521 |
+
2.3.6 in de Haan and Ferreira (2006), there is a constant Ο > 0 such that for all large t
|
1522 |
+
(xβ β t)β 1
|
1523 |
+
Ξ³
|
1524 |
+
1 β F(t)
|
1525 |
+
β€ v[(1 + Ο΅)ΟvΞ³]β 1
|
1526 |
+
Ξ³ = [(1 + Ο΅)Ο]β 1
|
1527 |
+
Ξ³
|
1528 |
+
The result now follows.
|
1529 |
+
5.5
|
1530 |
+
Proof of Lemma 3.5
|
1531 |
+
Note that for any Ξ³β² > β1/2 and Ο > 0
|
1532 |
+
H (hΞ³; hΞ³β²(Ο Β· )Ο) β€
|
1533 |
+
οΏ½οΏ½
|
1534 |
+
R
|
1535 |
+
[Ξ½x(Ξ³) β Ξ½x(Ξ³β²)]2 dx +
|
1536 |
+
οΏ½οΏ½
|
1537 |
+
R
|
1538 |
+
[ΟΞ³,x(Ο) β ΟΞ³,x(1)]2 dx
|
1539 |
+
In what follows, we bound the two terms on the right-hand side for Ξ³β² β (Ξ³ Β± Ο΅) and
|
1540 |
+
Ο β (1 Β± Ο΅), for a suitably small Ο΅ > 0. We study the the cases where Ξ³ > 0, Ξ³ < 0 and
|
1541 |
+
Ξ³ = 0 separately.
|
1542 |
+
Case 1: Ξ³ > 0. An application of the mean-value theorem and Lemma 5.7(a) yields
|
1543 |
+
that, for a function ΞΎ(x) β (Ξ³ β§ Ξ³β², Ξ³ β¨ Ξ³β²),
|
1544 |
+
οΏ½οΏ½
|
1545 |
+
R
|
1546 |
+
[Ξ½x(Ξ³) β Ξ½x(Ξ³β²)]2 dx = |Ξ³ β Ξ³β²|
|
1547 |
+
οΏ½οΏ½ β
|
1548 |
+
0
|
1549 |
+
[Ξ½β²x(ΞΎ(x))]2 dx
|
1550 |
+
β€ |Ξ³ β Ξ³β²|
|
1551 |
+
2
|
1552 |
+
οΏ½οΏ½ β
|
1553 |
+
0
|
1554 |
+
(1 + xΞΎ(x))β3β
|
1555 |
+
1
|
1556 |
+
ΞΎ(x)
|
1557 |
+
οΏ½x ln(1 + xΞΎ(x))
|
1558 |
+
ΞΎ(x)
|
1559 |
+
οΏ½2
|
1560 |
+
dx
|
1561 |
+
+ |Ξ³ β Ξ³β²|
|
1562 |
+
2
|
1563 |
+
οΏ½οΏ½ β
|
1564 |
+
0
|
1565 |
+
(1 + xΞΎ(x))β3β
|
1566 |
+
1
|
1567 |
+
ΞΎ(x) x2dx.
|
1568 |
+
19
|
1569 |
+
|
1570 |
+
On one hand, it holds that
|
1571 |
+
οΏ½ β
|
1572 |
+
0
|
1573 |
+
(1 + xΞΎ(x))β3β
|
1574 |
+
1
|
1575 |
+
ΞΎ(x)
|
1576 |
+
οΏ½x ln(1 + xΞΎ(x))
|
1577 |
+
ΞΎ(x)
|
1578 |
+
οΏ½2
|
1579 |
+
dx
|
1580 |
+
β€ 4
|
1581 |
+
οΏ½ β
|
1582 |
+
0
|
1583 |
+
(1 + x(Ξ³ β Ο΅))β1β
|
1584 |
+
1
|
1585 |
+
Ξ³+Ο΅
|
1586 |
+
οΏ½ln(1 + x(Ξ³ β Ο΅))
|
1587 |
+
(Ξ³ β Ο΅)2
|
1588 |
+
οΏ½2
|
1589 |
+
dx
|
1590 |
+
β€ 8(Ξ³ + Ο΅)3
|
1591 |
+
(Ξ³ β Ο΅)5 .
|
1592 |
+
On the other hand, it holds that
|
1593 |
+
οΏ½ β
|
1594 |
+
0
|
1595 |
+
(1 + ΞΎ(x))β3β
|
1596 |
+
1
|
1597 |
+
ΞΎ(x) x2dx
|
1598 |
+
οΏ½οΏ½οΏ½
|
1599 |
+
οΏ½ β
|
1600 |
+
0
|
1601 |
+
(1 + x(Ξ³ β Ο΅))β1β
|
1602 |
+
1
|
1603 |
+
Ξ³+Ο΅
|
1604 |
+
1
|
1605 |
+
(Ξ³ β Ο΅)2 dx
|
1606 |
+
β€ (Ξ³ + Ο΅)
|
1607 |
+
(Ξ³ β Ο΅)3 .
|
1608 |
+
While, an application of the mean-value theorem and Lemma 5.8(a) yields that, for a
|
1609 |
+
function Ο(x) β (1 β§ Ο, 1 β¨ Ο),
|
1610 |
+
οΏ½
|
1611 |
+
R
|
1612 |
+
οΏ½
|
1613 |
+
ΟΞ³β²,x(Ο) β ΟΞ³,x(1)
|
1614 |
+
οΏ½2 dx =
|
1615 |
+
οΏ½ β
|
1616 |
+
0
|
1617 |
+
οΏ½
|
1618 |
+
Οβ²
|
1619 |
+
Ξ³,x(Ο(x))
|
1620 |
+
οΏ½2 dx
|
1621 |
+
β€
|
1622 |
+
οΏ½ 1
|
1623 |
+
Ξ³2 +
|
1624 |
+
οΏ½
|
1625 |
+
1
|
1626 |
+
2Ξ³ + 1
|
1627 |
+
οΏ½2 οΏ½1 + Ο΅
|
1628 |
+
1 β Ο΅
|
1629 |
+
οΏ½5
|
1630 |
+
.
|
1631 |
+
The result now follows.
|
1632 |
+
Case 2: Ξ³ < 0. Assume that Ξ³ < Ξ³β², then an application of the mean-value theorem
|
1633 |
+
and Lemma 5.7(b) yields that, for a function ΞΎ(x) β (Ξ³, Ξ³β²),
|
1634 |
+
οΏ½
|
1635 |
+
R
|
1636 |
+
οΏ½
|
1637 |
+
Ξ½x(Ξ³) β Ξ½x(Ξ³β²)
|
1638 |
+
οΏ½2 dx = |Ξ³ β Ξ³β²|2
|
1639 |
+
οΏ½ β1/Ξ³
|
1640 |
+
0
|
1641 |
+
οΏ½
|
1642 |
+
Ξ½β²
|
1643 |
+
x(ΞΎ(x))
|
1644 |
+
οΏ½2 dx + 1 β HΞ³β²(β1/Ξ³)
|
1645 |
+
β€ |Ξ³ β Ξ³β²|2
|
1646 |
+
4
|
1647 |
+
οΏ½
|
1648 |
+
οΏ½
|
1649 |
+
οΏ½οΏ½ β1/Ξ³
|
1650 |
+
0
|
1651 |
+
(1 + xΞΎ(x))β3β
|
1652 |
+
1
|
1653 |
+
ΞΎ(x) x4dx
|
1654 |
+
+
|
1655 |
+
οΏ½οΏ½ β1/Ξ³
|
1656 |
+
0
|
1657 |
+
(1 + xΞΎ(x))β3β
|
1658 |
+
1
|
1659 |
+
ΞΎ(x) x2dx
|
1660 |
+
οΏ½
|
1661 |
+
οΏ½
|
1662 |
+
2
|
1663 |
+
+ 1 β HΞ³β²(β1/Ξ³).
|
1664 |
+
First, for a constant Ξ² satisfying 0 < Ξ² < 1/(Ο΅ β Ξ³) β 2, we have that
|
1665 |
+
οΏ½ β1/Ξ³
|
1666 |
+
0
|
1667 |
+
(1 + xΞΎ(x))β3β
|
1668 |
+
1
|
1669 |
+
ΞΎ(x) x4dx β€ 1
|
1670 |
+
Ξ³4
|
1671 |
+
οΏ½ β1/Ξ³
|
1672 |
+
0
|
1673 |
+
(1 + Ξ³x)β1+Ξ²dx
|
1674 |
+
β€
|
1675 |
+
1
|
1676 |
+
(βΞ³)5
|
1677 |
+
1
|
1678 |
+
Ξ² .
|
1679 |
+
Similarly,
|
1680 |
+
οΏ½ β1/Ξ³
|
1681 |
+
0
|
1682 |
+
(1 + xΞΎ(x))β3β
|
1683 |
+
1
|
1684 |
+
ΞΎ(x) x2dx β€ 1
|
1685 |
+
Ξ³2
|
1686 |
+
οΏ½ β1/Ξ³
|
1687 |
+
0
|
1688 |
+
(1 + Ξ³x)β1+Ξ²dx
|
1689 |
+
β€
|
1690 |
+
1
|
1691 |
+
(βΞ³)3
|
1692 |
+
1
|
1693 |
+
Ξ² .
|
1694 |
+
20
|
1695 |
+
|
1696 |
+
Finally, if Ο΅ is small enough, 1 β HΞ³β²(β1/Ξ³) β€ (1 β Ξ³β²/Ξ³)2. Thus, we can conclude that
|
1697 |
+
οΏ½
|
1698 |
+
R
|
1699 |
+
οΏ½
|
1700 |
+
Ξ½x(Ξ³) β Ξ½x(Ξ³β²)
|
1701 |
+
οΏ½2 dx β€ |Ξ³ β Ξ³β²|2 1 + 1/2Ξ²
|
1702 |
+
(βΞ³)5 .
|
1703 |
+
A similar reasoning when Ξ³ > Ξ³β² yields that
|
1704 |
+
οΏ½
|
1705 |
+
R
|
1706 |
+
οΏ½
|
1707 |
+
Ξ½x(Ξ³) β Ξ½x(Ξ³β²)
|
1708 |
+
οΏ½2 dx β€ |Ξ³ β Ξ³β²|2 1 + 2/Ξ²
|
1709 |
+
(βΞ³β²)5
|
1710 |
+
β€ |Ξ³ β Ξ³β²|2 1 + 2/Ξ²
|
1711 |
+
(βΞ³ β Ο΅)5 .
|
1712 |
+
Next, assuming that Ο < 1, an application of the mean-value theorem and the ο¬rst
|
1713 |
+
half of Lemma 5.8(b) yields that for a function Ο(x) β (1 β Ο΅, 1) and a constant ΞΆ > 0
|
1714 |
+
οΏ½
|
1715 |
+
R
|
1716 |
+
[ΟΞ³,x(Ο) β ΟΞ³,x(1)]2 dx = (1 β Ο)2
|
1717 |
+
οΏ½ βΟ/Ξ³
|
1718 |
+
0
|
1719 |
+
οΏ½
|
1720 |
+
Οβ²
|
1721 |
+
Ξ³,x(Ο(x))
|
1722 |
+
οΏ½2 dx + (1 β Ο)β1/Ξ³
|
1723 |
+
β€ (1 β Ο)2
|
1724 |
+
οΏ½
|
1725 |
+
1
|
1726 |
+
ΞΆ(1 β Ο΅)2(βΞ³)
|
1727 |
+
οΏ½ 1
|
1728 |
+
Ξ³2 + 1
|
1729 |
+
οΏ½2
|
1730 |
+
+ 1
|
1731 |
+
οΏ½
|
1732 |
+
.
|
1733 |
+
While, if Ο > 1, for a function Ο(x) β (1, 1 + Ο΅)
|
1734 |
+
οΏ½
|
1735 |
+
R
|
1736 |
+
[ΟΞ³,x(Ο) β ΟΞ³,x(1)]2 dx = (1 β Ο)2
|
1737 |
+
οΏ½ β1/Ξ³
|
1738 |
+
0
|
1739 |
+
οΏ½
|
1740 |
+
Οβ²
|
1741 |
+
Ξ³,x(Ο(x))
|
1742 |
+
οΏ½2 dx + (1 β 1/Ο)β1/Ξ³
|
1743 |
+
β€ (1 β Ο)2
|
1744 |
+
οΏ½
|
1745 |
+
1
|
1746 |
+
ΞΆ(βΞ³)
|
1747 |
+
οΏ½ 1
|
1748 |
+
Ξ³2 + 1
|
1749 |
+
οΏ½2
|
1750 |
+
+ 1
|
1751 |
+
οΏ½
|
1752 |
+
.
|
1753 |
+
The result now follows.
|
1754 |
+
Case 3: Ξ³ = 0. Assume that Ξ³β² > 0, then an application of the mean-value theorem
|
1755 |
+
and Lemma 5.7(a) yields that, for a function ΞΎ(x) β (0, Ξ³β²),
|
1756 |
+
οΏ½
|
1757 |
+
R
|
1758 |
+
οΏ½
|
1759 |
+
Ξ½x(Ξ³) β Ξ½x(Ξ³β²)
|
1760 |
+
οΏ½2 dx = |Ξ³ β Ξ³β²|2
|
1761 |
+
οΏ½ β
|
1762 |
+
0
|
1763 |
+
οΏ½
|
1764 |
+
Ξ½β²
|
1765 |
+
x(ΞΎ(x))
|
1766 |
+
οΏ½2 dx
|
1767 |
+
β€ |Ξ³ β Ξ³β²|2
|
1768 |
+
4
|
1769 |
+
οΏ½
|
1770 |
+
οΏ½
|
1771 |
+
οΏ½οΏ½ β
|
1772 |
+
0
|
1773 |
+
(1 + xΞΎ(x))β3β
|
1774 |
+
1
|
1775 |
+
ΞΎ(x)
|
1776 |
+
οΏ½x ln(1 + xΞΎ(x))
|
1777 |
+
ΞΎ(x)
|
1778 |
+
οΏ½2
|
1779 |
+
dx
|
1780 |
+
+
|
1781 |
+
οΏ½οΏ½ ββ
|
1782 |
+
0
|
1783 |
+
(1 + xΞΎ(x))β3β
|
1784 |
+
1
|
1785 |
+
ΞΎ(x) x2dx
|
1786 |
+
οΏ½
|
1787 |
+
οΏ½
|
1788 |
+
2
|
1789 |
+
.
|
1790 |
+
On one hand, we have
|
1791 |
+
οΏ½ β
|
1792 |
+
0
|
1793 |
+
(1 + xΞΎ(x))β3β
|
1794 |
+
1
|
1795 |
+
ΞΎ(x)
|
1796 |
+
οΏ½x ln(1 + xΞΎ(x))
|
1797 |
+
ΞΎ(x)
|
1798 |
+
οΏ½2
|
1799 |
+
dx β€
|
1800 |
+
οΏ½ β
|
1801 |
+
0
|
1802 |
+
(1 + xΞΎ(x))β3β
|
1803 |
+
1
|
1804 |
+
ΞΎ(x) x4dx
|
1805 |
+
β€
|
1806 |
+
οΏ½ β
|
1807 |
+
0
|
1808 |
+
(1 + xΞ³β²)β3β 1
|
1809 |
+
Ξ³β² x4dx +
|
1810 |
+
οΏ½ β
|
1811 |
+
0
|
1812 |
+
eβxx4dx
|
1813 |
+
β€ 36 + Ξ(5).
|
1814 |
+
On the other hand, we have
|
1815 |
+
οΏ½ β
|
1816 |
+
0
|
1817 |
+
(1 + xΞΎ(x))β3β
|
1818 |
+
1
|
1819 |
+
ΞΎ(x) x2dx β€
|
1820 |
+
οΏ½ β
|
1821 |
+
0
|
1822 |
+
(1 + xΞ³β²)β3β 1
|
1823 |
+
Ξ³β² x2dx +
|
1824 |
+
οΏ½ β
|
1825 |
+
0
|
1826 |
+
eβxx2dx
|
1827 |
+
β€ 3 + Ξ(3).
|
1828 |
+
21
|
1829 |
+
|
1830 |
+
Assume next that Ξ³β² < 0, then an application of the mean-value theorem and Lemma
|
1831 |
+
5.7(b) yields that, for a function ΞΎ(x) β (βΟ΅, 0),
|
1832 |
+
οΏ½
|
1833 |
+
R
|
1834 |
+
οΏ½
|
1835 |
+
Ξ½x(Ξ³) β Ξ½x(Ξ³β²)
|
1836 |
+
οΏ½2 dx = |Ξ³ β Ξ³β²|2
|
1837 |
+
οΏ½ β1/Ξ³β²
|
1838 |
+
0
|
1839 |
+
οΏ½
|
1840 |
+
Ξ½β²
|
1841 |
+
x(ΞΎ(x))
|
1842 |
+
οΏ½2 dx + e1/Ξ³β²
|
1843 |
+
β€ |Ξ³ β Ξ³β²|2
|
1844 |
+
4
|
1845 |
+
οΏ½
|
1846 |
+
οΏ½
|
1847 |
+
οΏ½οΏ½ β1/Ξ³β²
|
1848 |
+
0
|
1849 |
+
(1 + xΞΎ(x))β3β
|
1850 |
+
1
|
1851 |
+
ΞΎ(x) x4dx
|
1852 |
+
+
|
1853 |
+
οΏ½οΏ½ β1/Ξ³β²
|
1854 |
+
0
|
1855 |
+
(1 + xΞΎ(x))β3β
|
1856 |
+
1
|
1857 |
+
ΞΎ(x) x2dx
|
1858 |
+
οΏ½
|
1859 |
+
οΏ½
|
1860 |
+
2
|
1861 |
+
+ e1/Ξ³β².
|
1862 |
+
On one hand, for Ο΅ suο¬ciently small we have
|
1863 |
+
οΏ½ β1/Ξ³β²
|
1864 |
+
0
|
1865 |
+
(1 + xΞΎ(x))β3β
|
1866 |
+
1
|
1867 |
+
ΞΎ(x) x4dx β€
|
1868 |
+
οΏ½ β1/Ξ³β²
|
1869 |
+
0
|
1870 |
+
(1 + xΞ³β²)β3β 1
|
1871 |
+
Ξ³β² x4dx +
|
1872 |
+
οΏ½ β1/Ξ³β²
|
1873 |
+
0
|
1874 |
+
eβxx4dx
|
1875 |
+
β€ 13
|
1876 |
+
2 Ξ(5)
|
1877 |
+
and
|
1878 |
+
οΏ½ β1/Ξ³β²
|
1879 |
+
0
|
1880 |
+
(1 + xΞΎ(x))β3β
|
1881 |
+
1
|
1882 |
+
ΞΎ(x) x2dx β€
|
1883 |
+
οΏ½ β1/Ξ³β²
|
1884 |
+
0
|
1885 |
+
(1 + xΞ³β²)β3β 1
|
1886 |
+
Ξ³β² x4dx +
|
1887 |
+
οΏ½ β1/Ξ³β²
|
1888 |
+
0
|
1889 |
+
eβxx2dx
|
1890 |
+
β€ 3
|
1891 |
+
2Ξ(3).
|
1892 |
+
On the other hand, for Ο΅ suο¬ciently small we have e1/Ξ³β² β€ |Ξ³β² β Ξ³|2.
|
1893 |
+
Finally, some algebraic manipulations yield
|
1894 |
+
οΏ½
|
1895 |
+
R
|
1896 |
+
[ΟΞ³,x(Ο) β ΟΞ³,x(1)]2 dx =
|
1897 |
+
οΏ½ β
|
1898 |
+
0
|
1899 |
+
οΏ½οΏ½
|
1900 |
+
eβx/Ο 1
|
1901 |
+
Ο β
|
1902 |
+
β
|
1903 |
+
eβx
|
1904 |
+
οΏ½2
|
1905 |
+
dx
|
1906 |
+
β€ (1 β Ο)2
|
1907 |
+
(1 β Ο΅)2
|
1908 |
+
οΏ½
|
1909 |
+
1 + 1
|
1910 |
+
2
|
1911 |
+
οΏ½1 + Ο΅
|
1912 |
+
1 β Ο΅
|
1913 |
+
οΏ½3/2οΏ½2
|
1914 |
+
.
|
1915 |
+
The proof is now complete.
|
1916 |
+
5.6
|
1917 |
+
Proof of Corollary 3.6
|
1918 |
+
By Lemma 8.2 in Ghosal et al. (2000)
|
1919 |
+
K (Λlt; hΞ³) β€ 2
|
1920 |
+
οΏ½
|
1921 |
+
οΏ½
|
1922 |
+
sup
|
1923 |
+
0<x< xββt
|
1924 |
+
Λs(t)
|
1925 |
+
Λlt(x)
|
1926 |
+
hΞ³(x)
|
1927 |
+
οΏ½
|
1928 |
+
οΏ½ H 2(Λlt; hΞ³).
|
1929 |
+
Moreover, by Lemma B.3 in Ghosal and van der Vaart (2017), for p β₯ 2
|
1930 |
+
Dp(Λlt; hΞ³) β€ 2p!
|
1931 |
+
οΏ½
|
1932 |
+
οΏ½
|
1933 |
+
sup
|
1934 |
+
0<x< xββt
|
1935 |
+
Λs(t)
|
1936 |
+
Λlt(x)
|
1937 |
+
hΞ³(x)
|
1938 |
+
οΏ½
|
1939 |
+
οΏ½ H 2(Λlt; hΞ³).
|
1940 |
+
Furthermore, by triangular inequality and Lemma 3.5, for all large t
|
1941 |
+
H (Λlt; hΞ³) = H
|
1942 |
+
οΏ½
|
1943 |
+
lt; hΞ³
|
1944 |
+
οΏ½
|
1945 |
+
Β· s(t)
|
1946 |
+
Λs(t)
|
1947 |
+
οΏ½ s(t)
|
1948 |
+
Λs(t)
|
1949 |
+
οΏ½
|
1950 |
+
β€ H (lt; hΞ³) + H
|
1951 |
+
οΏ½
|
1952 |
+
hΞ³; hΞ³
|
1953 |
+
οΏ½
|
1954 |
+
Β· s(t)
|
1955 |
+
Λs(t)
|
1956 |
+
οΏ½ s(t)
|
1957 |
+
Λs(t)
|
1958 |
+
οΏ½
|
1959 |
+
β€ H (lt; hΞ³) + L|s(t)/Λs(t) β 1|
|
1960 |
+
β€ H (lt; hΞ³) + LB|A(v)|.
|
1961 |
+
22
|
1962 |
+
|
1963 |
+
The conclusion now follows by combining the above inequalities and applying Theorem
|
1964 |
+
3.2 and Lemma 3.4.
|
1965 |
+
5.7
|
1966 |
+
Proof of Proposition 4.1
|
1967 |
+
By invariance of Hellinger distance under rescaling and triangle inequality
|
1968 |
+
H (ft;οΏ½hk) = H
|
1969 |
+
οΏ½
|
1970 |
+
lt; hοΏ½Ξ³k
|
1971 |
+
οΏ½
|
1972 |
+
Β· s(t)
|
1973 |
+
οΏ½sk(t)
|
1974 |
+
οΏ½ s(t)
|
1975 |
+
οΏ½sk(t)
|
1976 |
+
οΏ½
|
1977 |
+
β€ H (lt; hΞ³) + H
|
1978 |
+
οΏ½
|
1979 |
+
hΞ³; hοΏ½Ξ³k
|
1980 |
+
οΏ½
|
1981 |
+
Β· s(t)
|
1982 |
+
οΏ½sk(t)
|
1983 |
+
οΏ½ s(t)
|
1984 |
+
οΏ½sk(t)
|
1985 |
+
οΏ½
|
1986 |
+
.
|
1987 |
+
On one hand, by Theorem 3.2 and assumption (b), as n β β
|
1988 |
+
H (lt; hΞ³) = O(|A(n/k)|) = O(1/
|
1989 |
+
β
|
1990 |
+
k).
|
1991 |
+
Moreover, by Lemma 3.5 and assumption (c), as n β β
|
1992 |
+
H
|
1993 |
+
οΏ½
|
1994 |
+
hΞ³; hοΏ½Ξ³k
|
1995 |
+
οΏ½
|
1996 |
+
Β· s(t)
|
1997 |
+
οΏ½sk(t)
|
1998 |
+
οΏ½ s(t)
|
1999 |
+
οΏ½sk(t)
|
2000 |
+
οΏ½
|
2001 |
+
= Op
|
2002 |
+
οΏ½
|
2003 |
+
οΏ½
|
2004 |
+
οΏ½
|
2005 |
+
|Ξ³ β οΏ½Ξ³k|2 +
|
2006 |
+
οΏ½οΏ½οΏ½οΏ½1 β s(t)
|
2007 |
+
οΏ½sk(t)
|
2008 |
+
οΏ½οΏ½οΏ½οΏ½
|
2009 |
+
2
|
2010 |
+
οΏ½
|
2011 |
+
οΏ½
|
2012 |
+
= Op(1/
|
2013 |
+
β
|
2014 |
+
k).
|
2015 |
+
The result now follows.
|
2016 |
+
5.8
|
2017 |
+
Proof of Proposition 4.5
|
2018 |
+
Let Qk denote the probability measure relative to the random sequence
|
2019 |
+
(Yk/Λs(X(nβk)), X(nβk)).
|
2020 |
+
Let Zk be the order statistics of an iid sample from HΞ³, independent from X1, X2 . . .,
|
2021 |
+
and denote by Pk the probability measure relative to the random sequence (Zk, X(nβk)).
|
2022 |
+
In what follows, we prove that Pk β· Qk, which implies the result in the statement.
|
2023 |
+
We start by recalling that, as n β β,
|
2024 |
+
1/(1 β F(X(nβk)))
|
2025 |
+
n/k
|
2026 |
+
= 1 + op(1),
|
2027 |
+
see e.g. Lemma 2.2.3 in de Haan and Ferreira (2006). Hence, deο¬ning the set Bk :=
|
2028 |
+
(U((1 Β± Ο΅))n/k), for a small Ο΅ > 0, we have that for any measurable set sequence Ek
|
2029 |
+
Pk(Ek) = Pk(Ek|X(nβk) β Bk)(1 + o(1)) + o(1)
|
2030 |
+
and
|
2031 |
+
Qk(Ek) = Pk(Ek|X(nβk) β Bk)(1 + o(1)) + o(1)
|
2032 |
+
as n β β. Therefore, it suο¬ces to prove that
|
2033 |
+
Pk( Β· |X(nβk) β Bk) β· Qk( Β· |X(nβk) β Bk).
|
2034 |
+
To do it, we denote by Οk and Οk the (Lebesgue) densities pertaining to the two condi-
|
2035 |
+
tional probability measures in the formula above and prove that
|
2036 |
+
lim sup
|
2037 |
+
nββ K (Οk; Οk) < β.
|
2038 |
+
(5.5)
|
2039 |
+
23
|
2040 |
+
|
2041 |
+
Clearly, it holds that for almost every (y, t) β Rk+2
|
2042 |
+
Οk(y, t) = fYk/Λs(X(nβk))(y|X(nβk) = t)
|
2043 |
+
fX(nβk)(t)1(t β Bk)
|
2044 |
+
P(X(nβk) β Bk)
|
2045 |
+
,
|
2046 |
+
where fYk/Λs(X(nβk))(y|X(nβk) = t) and fX(nβk)(t) are the conditional density of Yk/Λs(X(nβk))
|
2047 |
+
given X(nβk) = t and the marginal density of X(nβk), respectively. Moreover,
|
2048 |
+
Οk(y, t) = hZk(y)
|
2049 |
+
fX(nβk)(t)1(t β Bk)
|
2050 |
+
P(X(nβk) β Bk)
|
2051 |
+
,
|
2052 |
+
where hZk(y) is the density of Zk. As a consequence,
|
2053 |
+
K (Οk; Οk) =
|
2054 |
+
οΏ½
|
2055 |
+
Bk
|
2056 |
+
K (fYk/Λs(X(nβk))( Β· |X(nβk) = t); hZk)
|
2057 |
+
fX(nβk)(t)
|
2058 |
+
P(X(nβk) β Bk)dt.
|
2059 |
+
By Lemma B.11 in Ghosal and van der Vaart (2017) and Lemma 3.4.1 in de Haan and
|
2060 |
+
Ferreira (2006)
|
2061 |
+
K (fYk/Λs(X(nβk))( Β· |X(nβk) = t); hZk) β€ kK (Λlt; hΞ³).
|
2062 |
+
Moreover, by Corollary 3.6, there is a constant Ξ > 0 such that for all large n
|
2063 |
+
sup
|
2064 |
+
tβBk
|
2065 |
+
K (Λlt; hΞ³) β€ Ξ
|
2066 |
+
οΏ½οΏ½οΏ½A
|
2067 |
+
οΏ½
|
2068 |
+
(1 β Ο΅)n
|
2069 |
+
k
|
2070 |
+
οΏ½οΏ½οΏ½οΏ½
|
2071 |
+
2
|
2072 |
+
β€ Ξ(1 β Ο΅)Ο(1 + Ο΅)
|
2073 |
+
οΏ½οΏ½οΏ½A
|
2074 |
+
οΏ½n
|
2075 |
+
k
|
2076 |
+
οΏ½οΏ½οΏ½οΏ½
|
2077 |
+
2
|
2078 |
+
.
|
2079 |
+
Combining the above inequalities we obtain that
|
2080 |
+
K (Οk; Οk) β€ Ξ(1 β Ο΅)Ο(1 + Ο΅)k
|
2081 |
+
οΏ½οΏ½οΏ½A
|
2082 |
+
οΏ½n
|
2083 |
+
k
|
2084 |
+
οΏ½οΏ½οΏ½οΏ½
|
2085 |
+
2
|
2086 |
+
β Ξ(1 β Ο΅)Ο(1 + Ο΅)Ξ»2
|
2087 |
+
as n β β, where the convergence result in the second line follows from assumption (b).
|
2088 |
+
The result in formula (5.5) is now established and the proof is complete.
|
2089 |
+
Acknowledgements
|
2090 |
+
Simone Padoan is supported by the Bocconi Institute for Data Science and Analytics
|
2091 |
+
(BIDSA), Italy.
|
2092 |
+
References
|
2093 |
+
Balkema, A. A. and L. de Haan (1974). Residual life time at great age. The Annals of
|
2094 |
+
probability 2, 792β804.
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2095 |
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Bobbia, B., C. Dombry, and D. Varron (2021). The coupling method in extreme value
|
2096 |
+
theory. Bernoulli 27, 1824β1850.
|
2097 |
+
B¨ucher, A. and J. Segers (2017). On the maximum likelihood estimator for the Gener-
|
2098 |
+
alized Extreme-Value distribution. Extremes 20, 839β872.
|
2099 |
+
B¨ucher, A. and C. Zhou (2021). A Horse Race between the Block Maxima Method and
|
2100 |
+
the PeakβoverβThreshold Approach. Statistical Science 36, 360β378.
|
2101 |
+
24
|
2102 |
+
|
2103 |
+
de Haan, L. and A. Ferreira (2006). Extreme Value Theory: An Introduction. Springer.
|
2104 |
+
Dekkers, A. L. and L. de Haan (1993). Optimal choice of sample fraction in extreme-
|
2105 |
+
value estimation. Journal of Multivariate Analysis 47, 173β195.
|
2106 |
+
Dekkers, A. L., J. H. Einmahl, and L. de Haan (1989). A moment estimator for the
|
2107 |
+
index of an extreme-value distribution. The Annals of Statistics 17, 1833β1855.
|
2108 |
+
Dey, D. K. and J. Yan (2016). Extreme value modeling and risk analysis: methods and
|
2109 |
+
applications. CRC Press.
|
2110 |
+
Drees, H. (1998). Optimal rates of convergence for estimates of the extreme value index.
|
2111 |
+
Annals of Statistics 26, 434β448.
|
2112 |
+
Drees, H., A. Ferreira, and L. de Haan (2004). On maximum likelihood estimation of
|
2113 |
+
the extreme value index. The Annals of Applied Probability 14, 1179β1201.
|
2114 |
+
Embrechts, P., C. Kl¨uppelberg, and T. Mikosch (2013). Modelling extremal events: for
|
2115 |
+
insurance and ο¬nance, Volume 33. Springer Science & Business Media.
|
2116 |
+
Falk, M., J. H¨usler, and R.-D. Reiss (2010). Laws of small numbers: extremes and rare
|
2117 |
+
events. Springer Science & Business Media.
|
2118 |
+
Ghosal, S., J. K. Ghosh, and A. W. van der Vaart (2000). Convergence rates of posterior
|
2119 |
+
distributions. The Annals of Statistics 28, 500β531.
|
2120 |
+
Ghosal, S. and A. van der Vaart (2017).
|
2121 |
+
Fundamentals of Nonparametric Bayesian
|
2122 |
+
Inference. Cambridge University Press.
|
2123 |
+
Hall, P. and A. H. Welsh (1984). Best attainable rates of convergence for estimates of
|
2124 |
+
parameters of regular variation. The Annals of Statistics 12, 1079β1084.
|
2125 |
+
Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution.
|
2126 |
+
Annals of statistics 3, 1163β1174.
|
2127 |
+
Hosking, J. R. M., J. R. Wallis, and E. F. Wood (1985). Estimation of the General-
|
2128 |
+
ized Extreme-Value Distribution by the Method of Probability-Weighted Moments.
|
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+
Technometrics 27, 251β261.
|
2130 |
+
Jenkinson, A. (1969). Statistics of extemes. In Estimation of maximum ο¬oods, WMO
|
2131 |
+
Tech. Note 98, pp. 183β228.
|
2132 |
+
Pickands III, J. (1975). Statistical inference using extreme order statistics. The Annals
|
2133 |
+
of Statistics 3, 119β131.
|
2134 |
+
Raoult, J.-P. and R. Worms (2003). Rate of convergence for the generalized pareto
|
2135 |
+
approximation of the excesses. Advances in Applied Probability 35, 1007β1027.
|
2136 |
+
Resnick, S. I. (2007). Extreme Values, Regular Variation, and Point Processes, Vol-
|
2137 |
+
ume 4. Springer Science & Business Media.
|
2138 |
+
van der Vaart, A. (2000). Asymptotic Statistics. Cambridge University Press.
|
2139 |
+
Zhou, C. (2009). Journal of Multivariate Analysis 100, 794β815.
|
2140 |
+
25
|
2141 |
+
|
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|
1 |
+
Single-material MoS2 thermoelectric junction enabled by substrate
|
2 |
+
engineering
|
3 |
+
Mohammadali Razeghi1, Jean Spiece3, OΔuzhan OΔuz1, Doruk PehlivanoΔlu2, Yubin Huang3, Ali
|
4 |
+
Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan KasΔ±rga1,2*
|
5 |
+
1 Bilkent University UNAM β Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara,
|
6 |
+
Turkey
|
7 |
+
2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey
|
8 |
+
3 IMCN/NAPS, UniversitΓ© Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium
|
9 |
+
4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.K
|
10 |
+
*Corresponding Author: [email protected]
|
11 |
+
Abstract
|
12 |
+
To realize a thermoelectric power generator, typically a junction between two materials with different
|
13 |
+
Seebeck coefficient needs to be fabricated. Such difference in Seebeck coefficients can be induced by
|
14 |
+
doping, which renders difficult when working with two-dimensional (2d) materials. Here, we employ
|
15 |
+
substrate effects to form a thermoelectric junction in ultra-thin few-layer MoS2 films. We investigated
|
16 |
+
the junctions with a combination of scanning photocurrent microscopy and scanning thermal
|
17 |
+
microscopy. This allows us to reveal that thermoelectric junctions form across the substrate-
|
18 |
+
engineered parts. We attribute this to a gating effect induced by interfacial charges in combination
|
19 |
+
with alterations in the electron-phonon scattering mechanisms. This work demonstrates that
|
20 |
+
substrate engineering is a promising strategy to develop future compact thin-film thermoelectric
|
21 |
+
power generators.
|
22 |
+
Main Text
|
23 |
+
In ultra-thin materials with large surface-to-bulk ratio, interactions with the substrate can have strong
|
24 |
+
impact on the materials properties 1β6. It is therefore important to understand this so-called substrate-
|
25 |
+
effect, especially in order to optimize the reliability of future devices based on two-dimensional (2d)
|
26 |
+
semiconducting materials. As an example, the choice of substrate for mono- and few-layer MoS2 has
|
27 |
+
been shown to strongly affect its Raman modes and photoluminescence (PL)7, electronic8, and thermal
|
28 |
+
transport9 properties. In this work, we employ the substrate effect to enable completely new
|
29 |
+
functionalities in a 2d semiconductor device. To this end, we engineer the substrate that atomically
|
30 |
+
thin MoS2 is deposited on. Using a combination of scanning photocurrent microscopy (SPCM) along
|
31 |
+
with scanning thermal microscopy (SThM) we demonstrate that substrate engineering is a powerful
|
32 |
+
way to build a thermoelectric junction.
|
33 |
+
|
34 |
+
|
35 |
+
Figure 1 a. Schematic of a substrate-engineered device: a MoS2 flake is suspended over a circular hole
|
36 |
+
drilled in the substrate. Metal contacts are used for scanning photocurrent microscopy (SPCM),
|
37 |
+
scanning thermal gate microscopy (SThGM) and I-V measurements. The inset shows a magnification
|
38 |
+
of the area indicated by the dashed yellow square, where Seebeck coefficients of supported and
|
39 |
+
suspended parts are labelled with π1 and π2, respectively. b. Optical microscope image of a multi-
|
40 |
+
layered device over circular holes with indium contacts, marked with grey overlays. Scale bar: 10 Β΅m.
|
41 |
+
c. SPCM reflection map and the corresponding open-circuit photocurrent map acquired from the
|
42 |
+
yellow dashed rectangle in b with 532 nm laser. {πΌπππ, πΌπππ₯} = {β0.5, 0.5} nA. d. Photocurrent map
|
43 |
+
from the red dashed rectangle region in c. Black circle is the position of the hole determined from the
|
44 |
+
reflection image. Right panel shows the photocurrent, πΌππΆ vs bias taken from point 1 (red dots) and
|
45 |
+
point 2 (blue dots) over the suspended part of the crystal marked on the left panel. Lower graph is the
|
46 |
+
derived photoconductance, πΊππΆ vs. bias.
|
47 |
+
In the following we predict that a thermoelectric junction with a Seebeck coefficient difference of tens
|
48 |
+
of Β΅V/K can be fabricated when connecting regions of suspended MoS2 to supported regions. We
|
49 |
+
assume that the Seebeck coefficient π in thermal equilibrium is composed of contributions from the
|
50 |
+
energy-dependent diffusion (ππ), scattering (πΟ) and the phonon-drag (πππ), so that π = ππ + ππ +
|
51 |
+
πππ 9,10. Here, ππ and πΟ terms can be written from the Mott relation assuming that MoS2 is in the
|
52 |
+
highly conductive state and electrons are the majority carriers:
|
53 |
+
πΟ = β
|
54 |
+
π2ππ΅
|
55 |
+
2π
|
56 |
+
3π
|
57 |
+
πlnπ
|
58 |
+
ππΈ | πΈ=πΈπΉ and ππ = Β±
|
59 |
+
ππ΅
|
60 |
+
π [
|
61 |
+
πΈπΉβπΈπΆ
|
62 |
+
ππ΅π β
|
63 |
+
(π+2)πΉπ+1(π)
|
64 |
+
(π+1)πΉπ(π) ]
|
65 |
+
where π is the temperature, ππ΅ is the Boltzmann constant, π is the electronβs charge, π is the relaxation
|
66 |
+
time, πΈπΉ is the Fermi energy, πΈπΆ is the conduction band edge energy, π is scattering parameter and πΈ
|
67 |
+
is the energy. πΉπ(π) is the m-th order Fermi integral11. In the 2d limit, π is energy independent, thus
|
68 |
+
ππ is zero. πππ term can be estimated from the theory of phonon-drag in semiconductors in the first
|
69 |
+
order as πππ = β
|
70 |
+
π½π£πππ
|
71 |
+
ππ where, π£π and ππ are the group velocity and the mean free path of a phonon,
|
72 |
+
π½ is a parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and π is
|
73 |
+
the electron mobility, respectively10. Importantly, ππ and π are heavily affected by the presence of a
|
74 |
+
substrate12 which implies that the πππ term gets strongly modified when the MoS2 flake is suspended.
|
75 |
+
|
76 |
+
pended
|
77 |
+
ported
|
78 |
+
Reflection Map
|
79 |
+
0mV
|
80 |
+
S1
|
81 |
+
noelectric
|
82 |
+
Photocurrent Map
|
83 |
+
0mVIndeed, we find that for suspended MoS2 at room temperature πππ β β100 Β΅V/K and for MoS2 on
|
84 |
+
SiO2 at room temperature πππ β β230 Β΅V/K. Similarly, ππ is heavily influenced by the presence or
|
85 |
+
absence of the substrate as electron density depends on the interfacial Coulomb impurities and short-
|
86 |
+
ranged defects11β17. We estimate that that for MoS2, ππ ranges from -400 Β΅V/K to -200 Β΅V/K for carrier
|
87 |
+
concentrations ranging from 1012 cm-2 (suspended few layer MoS2) to 3 x 1013 cm-2 (SiO2 supported
|
88 |
+
few layer MoS2).18β20 As a result, a substrate engineered thermoelectric junction with a Seebeck
|
89 |
+
coefficient difference of Ξπ β 70 Β΅V/K can be formed along the MoS2 flake (see Figure 1a and
|
90 |
+
Supporting Information).
|
91 |
+
To test this hypothesis, we fabricated substrate-engineered MoS2 devices by mechanical exfoliation
|
92 |
+
and dry transfer21 of atomically thin MoS2 flakes on substrates (sapphire or oxidized silicon) with pre-
|
93 |
+
patterned trenches/holes formed by focused ion beam (FIB). We contacted the flakes with Indium
|
94 |
+
needles22β24 which are suitable for achieving Ohmic contacts to MoS225,26 (gold-contacted device
|
95 |
+
measurements are shown in Supporting Information). A typical device is shown in Figure 1b. We then
|
96 |
+
used scanning photocurrent microscopy, to locally heat up the junction with a focused laser beam and
|
97 |
+
to measure the photothermoelectric current that is generated (see Methods for experimental details).
|
98 |
+
Figure 1c shows the greyscale reflection intensity map and the corresponding photocurrent
|
99 |
+
distribution over the device. For the few-layer suspended MoS2 devices we observe a bipolar
|
100 |
+
photoresponse at the junctions between the supported and the suspended part of the crystal. The
|
101 |
+
spatial distribution of the signal agrees well with the finite element analysis simulations, given in the
|
102 |
+
supporting information, and suggests the formation of a thermoelectric junction. When applying a
|
103 |
+
voltage bias π to the junction, the photocurrent, πΌππΆ changes linearly with bias, while the
|
104 |
+
photoconductance, πΊππΆ =
|
105 |
+
πΌππΆ
|
106 |
+
π βπΌππΆ
|
107 |
+
0
|
108 |
+
π
|
109 |
+
(πΌππΆ
|
110 |
+
π , πΌππΆ
|
111 |
+
0 : photocurrent under π and 0 mV bias, respectively) stays
|
112 |
+
constant (Figure 1d). Such bias-independent photoconductance is typically an indication for an
|
113 |
+
photothermoelectric nature of the observed signal22,24,27β29. Although we propose that the
|
114 |
+
photocurrent in substrate-engineered MoS2 devices is dominated by the photothermal effect
|
115 |
+
(PTE)30,31, other possible mechanisms have been reported that may lead to a photovoltaic response.
|
116 |
+
These include (1) strain related effects such as strain modulation of materials properties and flexo-
|
117 |
+
photovoltaic effect13, and (2) substrate proximity related effects that forms a built-in electric field32.
|
118 |
+
Next, we present experimental evidence for a thermoelectric origin of the observed photocurrent. To
|
119 |
+
this end, we employed scanning thermal gate microscopy (SThGM), where a hot AFM tip heats up the
|
120 |
+
junction locally while the resulting voltage build-up on the devices is recorded (see Methods). Since
|
121 |
+
no laser-illumination of the sample is required in this method, it can be used to ultimately exclude
|
122 |
+
photovoltaic effects. Figure 2 compares SPCM and SThGM maps of the same holes. We observed the
|
123 |
+
same bipolar signals in the suspended regions with both experimental methods. Thanks to its sub-100
|
124 |
+
nm lateral resolution, SThGM further allows us to observe local variations of the thermovoltage in
|
125 |
+
supported MoS2 that can be attributed to charge puddles induced by local doping via the substrate33β
|
126 |
+
35. We confirmed that the SThGM signal disappears when no power is dissipated in the probe heater,
|
127 |
+
which rules out parasitic effects induced by the laser used for AFM feedback. Furthermore, SThGM
|
128 |
+
allows us to estimate the magnitude of the local Seebeck coefficient variations. Using the probe-
|
129 |
+
calibration data we obtain a value of Ξπ = 72 Β± 10 Β΅V/K (See supporting information). Despite the
|
130 |
+
uncertainties regarding the real sample temperature, the obtained Ξπ value is very close to the
|
131 |
+
theoretically predicted value.
|
132 |
+
|
133 |
+
|
134 |
+
Figure 2 a. SPCM reflection map and b. photocurrent map of the device shown in the inset of panel a.
|
135 |
+
Scale bar: 10 Β΅m. The yellow rectangle indicates the region that was investigated by SThGM in c (AFM
|
136 |
+
height map) and d (SThGM thermovoltage map). e. SPCM map of the same region excerpted from the
|
137 |
+
map given in b. Color scale is the same as in panel b. Scale bars in c, d and e: 3 Β΅m.
|
138 |
+
To understand why suspending MoS2 alters its Seebeck coefficient, we first would like to discuss the
|
139 |
+
possibility of strain induced changes in the materials properties. MoS2, like graphene, is nominally
|
140 |
+
compressed when deposited on a substrate36β39. Upon suspending the crystals, the free-standing part
|
141 |
+
either adheres to the sidewalls of the hole and dimples or, bulges. As a result, strain might be present
|
142 |
+
in the free-standing part of the crystal. Strain can affect both the bandgap and the Seebeck coefficient
|
143 |
+
of MoS2. The indirect optical gap is modulated by -110 meV/%-strain for a trilayer MoS236,40. Ab initio
|
144 |
+
studies show a ~10% decrease in the Seebeck coefficient of monolayer MoS2 per 1% tensile strain 41.
|
145 |
+
To estimate the biaxial strain, we performed atomic force microscopy (AFM) height trace mapping on
|
146 |
+
the samples. Most samples, regardless of the geometry of the hole exhibit slight bulging of a few
|
147 |
+
nanometers. For the MoS2 flakes suspended on the circular holes in the device shown in Figure 3a,
|
148 |
+
the bulge height is πΏπ‘ β 25 nm. Similar πΏπ‘ values were measured for other devices. The biaxial strain
|
149 |
+
can then be calculated using an uniformly loaded circular membrane model, and is as low as 0.0025%
|
150 |
+
42. Such a small strain on MoS2 is not sufficient to induce a significant change in bandgap or Seebeck
|
151 |
+
coefficient 43β45.
|
152 |
+
Next, we consider the substrate induced changes on the material properties. The presence or the
|
153 |
+
absence of the substrate can cause enhanced or diminished optical absorption due to the screening
|
154 |
+
effects, Fermi level pinning46 and charges donated by the substrate7,47. More significantly, the doping
|
155 |
+
effect due to the trapped charges at the interface with the substrate can locally gate the MoS2 and
|
156 |
+
modify the number of charge carriers48 and thus its Seebeck coefficient. To investigate the
|
157 |
+
electrostatic impact of the substrate on the MoS2 membrane, we investigated the surface potential
|
158 |
+
difference (SPD) on devices using Kelvin Probe Force Microscopy (KPFM). SPD can provide an insight
|
159 |
+
on the band bending of the MoS2 due to the substrate effects49. Figure 3b-d shows the AFM height
|
160 |
+
trace map and the uncalibrated SPD map of the sample. SPD across the supported and suspended part
|
161 |
+
of the flake is on the order of 50 mV. This shift in the SPD value hints that there is a slight change in
|
162 |
+
the Fermi level of the suspended part with respect to the supported part of the crystal. The same type
|
163 |
+
of charge carriers is dominant on both sides of the junction formed by the suspended and supported
|
164 |
+
parts of the crystal. The band structure formed by such a junction in zero bias cannot be used in
|
165 |
+
separation of photoinduced carriers50, however, it can lead to the formation of a thermoelectric
|
166 |
+
junction11,51. This is in line with the SThGM measurements.
|
167 |
+
|
168 |
+
|
169 |
+
Figure 3 a. AFM height trace map of a device suspended over circular holes show a bulge of πΏπ‘ β 25
|
170 |
+
nm. The line trace is overlayed on the map. Scale bar: 4 Β΅m. b. AFM height trace map of the sample
|
171 |
+
shows the bulged and dimpled parts of the flake. Scale bar: 4 Β΅m. c. KPFM map of the sample shows
|
172 |
+
the variation in the surface potential. Scale bar: 4 Β΅m. d. Line traces taken along the numbered lines
|
173 |
+
in c. Direction of the arrows in c indicates the direction of the line plot.
|
174 |
+
In the remainder of the paper, we aim at controlling the electrostatics that are responsible for the
|
175 |
+
formation of a thermoelectric junction. Charge transport in MoS2 is dominated by electrons due to
|
176 |
+
unintentional doping52,53. Modulating the density and the type of free charge carriers can be done by
|
177 |
+
applying a gate voltage ππ to the junction54. This significantly modifies the magnitude and the sign of
|
178 |
+
the Seebeck coefficient as demonstrated in previous studies16,30,31,55. The Mott relation56 can be used
|
179 |
+
to model the Seebeck coefficient as a function of ππ:
|
180 |
+
π =
|
181 |
+
π2ππ΅
|
182 |
+
2π
|
183 |
+
3π
|
184 |
+
1
|
185 |
+
π
|
186 |
+
ππ
|
187 |
+
πππ
|
188 |
+
πππ
|
189 |
+
ππΈ | πΈ=πΈπΉ eq.(1)
|
190 |
+
Here, π is the temperature, ππ΅ is the Boltzmann constant, π is the electronβs charge, π
is the device
|
191 |
+
resistance, πΈπΉ is the Fermi energy and πΈ is the energy.
|
192 |
+
Since hole transport is limited due to substrate induced Fermi level pinning on SiO2 supported MoS2
|
193 |
+
field-effect devices,46 to observe the sign inversion of the Seebeck coefficient (see the Supporting
|
194 |
+
Information for measurements on device fabricated on SiO2 and Al2O3 coated SiO2) we followed an
|
195 |
+
alternative approach to emulate suspension: we fabricated heterostructure devices where the crystal
|
196 |
+
is partially supported by hexagonal boron nitride (h-BN). h-BN is commonly used to encapsulate two-
|
197 |
+
dimensional materials thanks to its hydrophobic and atomically smooth surface. This leads to less
|
198 |
+
unintentional doping due to the interfacial charge trapping and reduced electron scattering7,57,58. A
|
199 |
+
~10 ML MoS2 is placed over a 10 nm thick h-BN crystal to form a double-junction device (see
|
200 |
+
supporting information for a single-junction device formed by a MoS2 flake which is partially placed
|
201 |
+
over a h-BN flake) and indium contacts are placed over the MoS2. The device is on 1 Β΅m thick oxide
|
202 |
+
coated Si substrate where Si is used as the back-gate electrode. Figure 4a shows the optical
|
203 |
+
micrograph of the device and its schematic. The presence of h-BN modifies the SPD by 80 mV β a value
|
204 |
+
very similar to the values we find for suspended devices (see SI) β which is consistent with the relative
|
205 |
+
n-doping by the h-BN substrate32,57. We therefore attribute this difference to the Fermi level shift due
|
206 |
+
to the difference in interfacial charge doping by the different substrates.
|
207 |
+
|
208 |
+
ot
|
209 |
+
0 1234ΞΌm
|
210 |
+
Figure 4 a. Optical micrograph of a Si back-gated MoS2 device partially placed over h-BN. Its cross-
|
211 |
+
sectional schematic is shown in the lower panel. Scale bar: 10 Β΅m. b. SPCM reflection map and the
|
212 |
+
photocurrent map of the device shown in a. πΌπππ₯ = 3 nA and πΌπππ = β3 nA. Scale bar: 10 Β΅m. c.
|
213 |
+
Current-Voltage graph versus ππΊ from -40 to 40 V. Inset shows the resistance versus ππΊ. d. πΌππΆ vs. ππΊ
|
214 |
+
recorded at the points marked in the SPCM map in b.
|
215 |
+
Figure 4b shows the SPCM map under zero gate voltage. We observe a bipolar photocurrent signal
|
216 |
+
from the junctions between h-BN and SiO2 supported MoS2. Raman mapping (see the Supporting
|
217 |
+
Information) reveals slight intensity decrease and a small shift of the A1π peak over the h-BN
|
218 |
+
supported part of the MoS2. This is consistent with the stiffening of the Raman mode due to the higher
|
219 |
+
degree of charged impurities in SiO2 as compared to h-BN7. By applying a gate voltage to the device,
|
220 |
+
its resistance can be tuned significantly as free charges are depleted (Figure 4c). Under large positive
|
221 |
+
gate voltages, the I-V characteristic becomes asymmetric. To investigate the dependence of the
|
222 |
+
photocurrent on carrier type and concentration, the laser is held at specific positions on the device as
|
223 |
+
marked in Figure 4d, and the gate is swept from positive to negative voltages with respect to the
|
224 |
+
ground terminal. For positive gate voltages, the magnitude of the photoresponse from both junctions,
|
225 |
+
between h-BN and SiO2 supported MoS2, (points 2 and 3) decrease. When a negative gate voltage is
|
226 |
+
applied, the magnitude of the photoresponse at both junctions increases by almost a factor of two at
|
227 |
+
ππΊ = β21.5 V. Once this maximum is reached, the amplitude of the photocurrent at both points
|
228 |
+
decreases and has the same value as the photocurrent generated over the MoS2 (point 4) at ππΊ =
|
229 |
+
β34.5 V.
|
230 |
+
These observations can be qualitatively explained as follows: at a gate voltage of ππΊ = β34.5 π, the
|
231 |
+
majority charge carrier type in the h-BN supported part changes from electrons to holes. As a
|
232 |
+
consequence, the Seebeck coefficients of MoS2 resting on h-BN and SiO2, respectively, become similar,
|
233 |
+
which leads to βπ β 0, and curves 2,3 and 4 in Figure 4d cross. The photocurrent signal recorded near
|
234 |
+
the indium contacts (points 1 and 5) decreases non-monotonically with decreasing ππΊ and reaches
|
235 |
+
|
236 |
+
SiO2
|
237 |
+
Si
|
238 |
+
Imax
|
239 |
+
Iminzero at ππΊ = β40 π. At this voltage the Seebeck coefficient of MoS2 on SiO2 reaches that of Indium
|
240 |
+
(SIn = + 1.7 Β΅V/K)59.
|
241 |
+
In conclusion we demonstrated that substrate engineering can be used to generate a thermoelectric
|
242 |
+
junction in atomically thin MoS2 devices. Similar strategies can be employed in other low dimensional
|
243 |
+
materials that exhibit large and tunable Seebeck coefficients. This might in particular be promising at
|
244 |
+
low temperature where effects like band-hybridization and Kondo scattering can produce a very
|
245 |
+
strong photothermoelectric effect9.
|
246 |
+
Author Contributions
|
247 |
+
T.S.K. designed and conceived the experiments, T.S.K. and P.G. prepared the manuscript. M.R.
|
248 |
+
fabricated devices, performed the experiment and analyzed the results. D.P. prepared the substrates,
|
249 |
+
performed simulations, and helped with the experiments. O.O. performed the AFM and KPFM
|
250 |
+
measurements and A.S. performed some of the earlier measurements. J.S., Y.H. and P.G. performed
|
251 |
+
the SThGM measurements and analyzed the results. P.S.D and J.M.R.W contributed discussions on the
|
252 |
+
implementation of VITA-DM-GLA-1 SThM probes. All authors discussed the results and reviewed the
|
253 |
+
final version of the manuscript.
|
254 |
+
Competing Interests
|
255 |
+
The Authors declare no Competing Financial or Non-Financial Interests.
|
256 |
+
Methods
|
257 |
+
SPCM setup is a commercially available setup from LST Scientific Instruments Ltd. which offers a
|
258 |
+
compact scanning head with easily interchangeable lasers. Two SR-830 Lock-in amplifiers are
|
259 |
+
employed, one for the reflection map and the other for the photocurrent/voltage measurements. In
|
260 |
+
the main text we reported the photocurrent (a measurement of the photovoltage is given in Figure
|
261 |
+
S2). The incident laser beam is chopped at a certain frequency and focused onto the sample through
|
262 |
+
a 40x objective. The electrical response is collected through gold probes pressed on the electrical
|
263 |
+
contacts of the devices and the signal is amplified by a lock-in amplifier set to the chopping frequency
|
264 |
+
of the laser beam. Although various wavelengths (406, 532, 633 nm) are employed for the
|
265 |
+
measurements, unless otherwise stated we used 532 nm in the experiments reported in the main text
|
266 |
+
(see Figure S3 for SPCM measurements with different wavelengths). All the excitation energies are
|
267 |
+
above the indirect bandgap of the few layer MoS2.
|
268 |
+
Scanning Thermal Microscopy measurements were performed with a Dimension Icon (Bruker) AFM
|
269 |
+
under ambient conditions. The probe used in the experiments is VITA-DM-GLA-1 made of a palladium
|
270 |
+
heater on a silicon nitride cantilever and tip. The radius is typically in the order of 25-40 nm. The heater
|
271 |
+
is part of a modified Wheatstone bridge and is driven by a combined 91 kHz AC and DC bias, as
|
272 |
+
reported elsewhere. The signal is detected via a SR830 lock-in amplifier and fed in the AFM controller.
|
273 |
+
This signal monitors the probe temperature and thus allows to locally map the thermal conductance
|
274 |
+
of the sample. In this work, the power supplied to the probe gives rise to a 45K excess temperature.
|
275 |
+
While the probe is scanning the sample, we measure the voltage drop across the device using a low
|
276 |
+
noise preamplifier (SR 560). This voltage is created by the local heating induced by the hot SThM tip.
|
277 |
+
It is then fed also to the AFM controller and recorded simultaneously. In this study, the thermovoltage
|
278 |
+
measurements were performed without modulating the heater power. We note that it is also possible
|
279 |
+
to generate similar maps by varying the heater temperature and detecting thermovoltage via lock-in
|
280 |
+
detection.
|
281 |
+
|
282 |
+
Data Availability
|
283 |
+
Source data available from the corresponding authors upon request.
|
284 |
+
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Dominated Photocurrent in Graphene. Nat. Nanotechnol. 2011 72 2012, 7 (2), 114β118.
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https://doi.org/10.1038/nnano.2011.243.
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Lee, C.; Hong, J.; Lee, W. R.; Kim, D. Y.; Shim, J. H. Density Functional Theory Investigation of
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502 |
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the Electronic Structure and Thermoelectric Properties of Layered MoS2, MoSe2 and Their
|
503 |
+
Mixed-Layer Compound. J. Solid State Chem. 2014, 211, 113β119.
|
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+
https://doi.org/10.1016/j.jssc.2013.12.012.
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(52)
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+
Siao, M. D.; Shen, W. C.; Chen, R. S.; Chang, Z. W.; Shih, M. C.; Chiu, Y. P.; Cheng, C. M. Two-
|
507 |
+
Dimensional Electronic Transport and Surface Electron Accumulation in MoS2. Nat. Commun.
|
508 |
+
2018, 9 (1), 1β12. https://doi.org/10.1038/s41467-018-03824-6.
|
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+
(53)
|
510 |
+
Dagan, R.; Vaknin, Y.; Henning, A.; Shang, J. Y.; Lauhon, L. J.; Rosenwaks, Y. Two-Dimensional
|
511 |
+
Charge Carrier Distribution in MoS2 Monolayer and Multilayers. Appl. Phys. Lett. 2019, 114
|
512 |
+
(10), 101602. https://doi.org/10.1063/1.5078711.
|
513 |
+
|
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(54)
|
515 |
+
Jahangir, I.; Koley, G.; Chandrashekhar, M. V. S. Back Gated FETs Fabricated by Large-Area,
|
516 |
+
Transfer-Free Growth of a Few Layer MoS2 with High Electron Mobility. Appl. Phys. Lett.
|
517 |
+
2017, 110 (18), 182108. https://doi.org/10.1063/1.4982595.
|
518 |
+
(55)
|
519 |
+
Dobusch, L.; Furchi, M. M.; Pospischil, A.; Mueller, T.; Bertagnolli, E.; Lugstein, A. Electric Field
|
520 |
+
Modulation of Thermovoltage in Single-Layer MoS 2. Appl. Phys. Lett. 2014, 105 (25), 253103.
|
521 |
+
https://doi.org/10.1063/1.4905014.
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|
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Ashcroft, N.; Mermin, D. Solid State Physics; Thomson Learning Inc.: London, 1976.
|
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(57)
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+
Joo, M. K.; Moon, B. H.; Ji, H.; Han, G. H.; Kim, H.; Lee, G.; Lim, S. C.; Suh, D.; Lee, Y. H.
|
526 |
+
Electron Excess Doping and Effective Schottky Barrier Reduction on the MoS2/h-BN
|
527 |
+
Heterostructure. Nano Lett. 2016, 16 (10), 6383β6389.
|
528 |
+
https://doi.org/10.1021/acs.nanolett.6b02788.
|
529 |
+
(58)
|
530 |
+
Li, L.; Lee, I.; Lim, D.; Kang, M.; Kim, G. H.; Aoki, N.; Ochiai, Y.; Watanabe, K.; Taniguchi, T.
|
531 |
+
Raman Shift and Electrical Properties of MoS2 Bilayer on Boron Nitride Substrate.
|
532 |
+
Nanotechnology 2015, 26 (29), 295702. https://doi.org/10.1088/0957-4484/26/29/295702.
|
533 |
+
(59)
|
534 |
+
Shakouri, A.; Li, S. Thermoelectric Power Factor for Electrically Conductive Polymers. In
|
535 |
+
International Conference on Thermoelectrics, ICT, Proceedings; IEEE, 1999; pp 402β406.
|
536 |
+
https://doi.org/10.1109/ict.1999.843415.
|
537 |
+
|
538 |
+
|
539 |
+
|
540 |
+
|
541 |
+
Supporting Information: Single-material MoS2 thermoelectric junction
|
542 |
+
enabled by substrate engineering
|
543 |
+
Mohammadali Razeghi1, Jean Spiece3, OΔuzhan OΔuz1, Doruk PehlivanoΔlu2, Yubin Huang3, Ali
|
544 |
+
Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan KasΔ±rga1,2*
|
545 |
+
1 Bilkent University UNAM β Institute of Materials Science and Nanotehcnology, Bilkent 06800 Ankara,
|
546 |
+
Turkey
|
547 |
+
2 Department of Physics, Bilkent University, Bilkent 06800 Ankara, Turkey
|
548 |
+
3 IMCN/NAPS, UniversitΓ© Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium
|
549 |
+
4 James Watt School of Engineering, University of Glasgow, Glasgow G12 8LT, U.K
|
550 |
+
*Corresponding Author: [email protected]
|
551 |
+
|
552 |
+
1. Theoretical prediction of the substrate-effect induced Seebeck coefficient difference in MoS2
|
553 |
+
As discussed in the main text, we assume that the Seebeck coefficient S in thermal equilibrium is
|
554 |
+
composed of contributions from the energy-dependent diffusion (ππ), scattering (πΟ) and the phonon-
|
555 |
+
drag (πππ), so that π = ππ + ππ + πππ. Here, ππ and πΟ terms can be written from the Mott relation
|
556 |
+
assuming that MoS2 is in the highly conductive state and electrons are the majority carriers:
|
557 |
+
πΟ = β
|
558 |
+
π2ππ΅
|
559 |
+
2π
|
560 |
+
3π
|
561 |
+
πlnπ
|
562 |
+
ππΈ | πΈ=πΈπΉ and ππ = Β±
|
563 |
+
ππ΅
|
564 |
+
π [
|
565 |
+
πΈπΉβπΈπΆ
|
566 |
+
ππ΅π β
|
567 |
+
(π+2)πΉπ+1(π)
|
568 |
+
(π+1)πΉπ(π) ]
|
569 |
+
As mentioned in the main text, πΟ is zero as π is energy independent in the 2d limit. ππ term is
|
570 |
+
composed of constants related to material properties, scattering parameter π and the Fermi integral
|
571 |
+
of the π-th order: πΉπ = β«
|
572 |
+
[
|
573 |
+
π₯π
|
574 |
+
ππ₯βπ + 1]ππ₯
|
575 |
+
β
|
576 |
+
0
|
577 |
+
. The scattering parameters of 2d materials are listed in Table
|
578 |
+
1.1,2 Here, as discussed in detail in Ref. 1, π = 0 adequately accounts for the acoustic phonon scattering
|
579 |
+
and small deviations of experimental data from the calculated values is due to the other scattering
|
580 |
+
mechanisms. As a result, at the room temperature ππ for suspended MoS2 (1012 cm-2) is about -400
|
581 |
+
Β΅V/K and for SiO2 supported MoS2 (1013 cm-2) is about -200 Β΅V/K.
|
582 |
+
Table 1. Scattering parameters π of 2d materials.
|
583 |
+
Scattering mechanism
|
584 |
+
π
|
585 |
+
Charged Impurity Scattering
|
586 |
+
3/2
|
587 |
+
Acoustic Phonon Scattering
|
588 |
+
0
|
589 |
+
Intervalley Scattering
|
590 |
+
0
|
591 |
+
Strongly Screened Coulomb Scattering
|
592 |
+
-1/2
|
593 |
+
|
594 |
+
πππ term can be estimated from the theory of phonon-drag in semiconductors in the first order as
|
595 |
+
πππ = β
|
596 |
+
π½π£πππ
|
597 |
+
ππ where, π£π and ππ are the group velocity and the mean free path of a phonon, π½ is a
|
598 |
+
parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and π is the
|
599 |
+
electron mobility, respectively. As the dominant charge carriers are electrons, πππ term has a negative
|
600 |
+
sign. We use the parameters given in Table 2. Based on the values given in the table we obtain πππ
|
601 |
+
πππ2 =
|
602 |
+
β230 Β΅V/K and οΏ½οΏ½οΏ½ππ
|
603 |
+
ππ’π = β100 Β΅V/K.
|
604 |
+
|
605 |
+
The total π = ππ + πππ for suspended and SiO2 supported parts can be calculated by adding both
|
606 |
+
contributions. πππ’π = β500 Β΅V/K and ππππ2 = β430 Β΅V/K. Of course, we consider this to be a rough
|
607 |
+
estimate as we ignore charged impurity scattering and strongly screeded Coulomb scattering. Also
|
608 |
+
there are certain errors associated with the measurement of the parameters used for the calculation
|
609 |
+
of the Seebeck coefficients. However, overall, this calculation shows that the substrate induced effect
|
610 |
+
must be present under right experimental conditions.
|
611 |
+
Table 2. Parameters used for πΊππ
calculation
|
612 |
+
Parameter
|
613 |
+
On SiO2 (Ref. 3)
|
614 |
+
Suspended (Ref. 3)
|
615 |
+
π£π
|
616 |
+
7 105 cm/s
|
617 |
+
7 105 cm/s
|
618 |
+
ππ
|
619 |
+
5 nm
|
620 |
+
20 nm
|
621 |
+
π
|
622 |
+
5 cm2/V.s
|
623 |
+
50 cm2/V.s
|
624 |
+
|
625 |
+
2. SPCM map on a gold electrode substrate engineered MoS2 device
|
626 |
+
Throughout the study we used indium contacted devices thanks to their rapid fabrication. To compare
|
627 |
+
our indium device results, we fabricated gold contacted devices. Figure S1 shows the optical
|
628 |
+
microscope images and corresponding SPCM reflection and photocurrent maps. There is no qualitative
|
629 |
+
difference between the indium contacted devices and gold contacted device in the substrate-
|
630 |
+
engineered photocurrent. Despite IV measurement is collected from 0.25 to -0.25 V its rectifying
|
631 |
+
behaviour can be observed. Power dependence of the photocurrent from the substrate engineered
|
632 |
+
junction is also comparable to the one reported in indium contacted devices.
|
633 |
+
|
634 |
+
Figure S1 a. Optical microscope micrograph of a gold contacted substrate engineered MoS2 device is
|
635 |
+
shown. Scale bar is 10 Β΅m. b. SPCM reflection map and c. photocurrent map. d. IV curve shows signs
|
636 |
+
of rectifying nature of the contacts. e. Power dependence of the photocurrent at one of the side of
|
637 |
+
the junction is plotted in a log-log graph and the exponent is about 0.63.
|
638 |
+
|
639 |
+
b
|
640 |
+
a
|
641 |
+
3. Scanning photovoltage microscopy, AFM and KPFM measurements on a parallel trench
|
642 |
+
device
|
643 |
+
Figure S2 shows an MoS2 device fabricated on trenches drilled on sapphire with different depths. We
|
644 |
+
performed SPVM, AFM and KPFM Measurements. First, AFM measurements show that the crystal is
|
645 |
+
stuck to the bottom of the 100 nm deep trench (Figure S2b). For the rest of the trenches the flake
|
646 |
+
bulges about 10 nm above the surface (Figure S2c). AFM height trace map also reveals a peculiar
|
647 |
+
wrinkle formation over the suspended part of the flake.
|
648 |
+
In this measurement we operated the scanning microscope at photovoltage mode. Figure S2d shows
|
649 |
+
the reflection map and the corresponding photovoltage map. The bipolar response is evident with
|
650 |
+
slightly lower positive signal in some of the trenches. This asymmetry can be explained by lower
|
651 |
+
heating of one side of the samples due to the scan direction. One important observation that agrees
|
652 |
+
well with the photothermoelectric photoresponse is that the 100 nm trench shows very small
|
653 |
+
photovoltage as compared to other trenches.
|
654 |
+
Figure S2e shows the KPFM and AFM profiles. The suspended part of the crystal has 60 meV lower
|
655 |
+
surface potential difference. This is consistent with other KPFM measurements. The lower panel shows
|
656 |
+
the variation in the height over the wrinkles. The workfunction is calculated with a calibrated tip and
|
657 |
+
it follows the wrinkles of the sample. However, the difference in the SPD is not due to the variations in
|
658 |
+
the height profile of the crystal. The change in the workfunciton is a good indication of the changes in
|
659 |
+
the electronic landscape of the device upon suspension. Small variations along the wrinkles are also
|
660 |
+
expected due to formation of varying stress regions along the crystal.
|
661 |
+
|
662 |
+
Figure S2 a. Optical microscope image of the device with trench depths labelled next to it. b. AFM
|
663 |
+
height trace map and c. line trace taken along the height trace. The bulge of the crystal over the
|
664 |
+
trenches is clear. d. Reflection and photovoltage maps obtained by operating the scanning microscope
|
665 |
+
in photovoltage mode. Scale bar is 5 Β΅m. e. Left panel shows the workfunction and height taken over
|
666 |
+
|
667 |
+
Profile1
|
668 |
+
103nm
|
669 |
+
Onmthe red lines marked on the maps given in the right panel. The variation of the workfunction along the
|
670 |
+
trench is very small and correlated with the wrinkles of the crystal.
|
671 |
+
4. SPCM maps taken at different laser wavelengths and incidence polarization
|
672 |
+
We used three different wavelengths, 406, 532 and 633 nm, in our experiments all of them which are
|
673 |
+
at an energy larger than the band gap of MoS2. Figure S3 shows the SPCM results collected with
|
674 |
+
different laser wavelengths. Also, polarization dependence of the photocurrent measured at each end
|
675 |
+
of the trench as well as a point over the contact is given in Figure S3f. There is no polarization
|
676 |
+
dependence of the photocurrent. This shows that The effect is not due to built in polarization fields.
|
677 |
+
|
678 |
+
Figure S3 a. Optical microscope image of a two-terminal substrate-engineered MoS2 device with
|
679 |
+
different trench widths. Scale bar is 10 Β΅m. b. SPCM reflection map of the region marked with yellow
|
680 |
+
rectangle in a. c, d and e show photocurrent map taken at different wavelengths. At each run laser
|
681 |
+
power is set to ~40 Β΅W. The measured signal in all three measurements are very close and the overall
|
682 |
+
photocurrent features are the same. f. Incident polarization of the 633 nm laser is rotated and πΌππΆ is
|
683 |
+
measured at three different points marked by colored arrows on d, black dots- near contact, red dots-
|
684 |
+
at the positive side and blue dots- at the negative side of the trench. There is no polarization
|
685 |
+
dependence of the measured photocurrent at the three points where photocurrent is measured.
|
686 |
+
5. Finite element simulation of a substrate modified thermoelectric junction
|
687 |
+
To understand how a substrate modified thermoelectric junction would behave depending on how the
|
688 |
+
contacts are configured, we performed finite element analysis simulations using COMSOL
|
689 |
+
Multiphysics. An irregularly shaped crystal is modelled over a substrate with a hole and voltage at a
|
690 |
+
floating terminal is measured with respect to different laser positions. The observed pattern agrees
|
691 |
+
with our measurements. Figure S4 shows the thermoelectric emf generated and temperature
|
692 |
+
distribution maps.
|
693 |
+
|
694 |
+
406 nm
|
695 |
+
532 nm
|
696 |
+
633 nm
|
697 |
+
Figure S4 a. An arbitrary crystal is modelled over a SiO2 substrate with a hole. The outline drawn over
|
698 |
+
the voltage distribution map shows the outline of the crystal and the outline of the hole. The red line
|
699 |
+
indicates the ground terminal and the blue line indicates the floating voltage terminal where the
|
700 |
+
photothermoelectric emf is measured from like in the experiments. b. For comparison, another
|
701 |
+
terminal is simulated as the floating terminal. c. Temperature distribution vs. laser position is shown.
|
702 |
+
As clear, the maximum temperature rise is achieved at the center of the hole.
|
703 |
+
6. KPFM on h-BN supported and suspended MoS2
|
704 |
+
|
705 |
+
Figure S5 a. Optical micrograph of an MoS2 crystal partially suspended over a trench and partially
|
706 |
+
supported on h-BN (outlined by blacked dashed lines). White dashed square shows the AFM region. b.
|
707 |
+
AFM height trace map and c. corresponding SPD map is given. d. SPD line traces from the colored lines
|
708 |
+
in c are plotted. The difference between the SPD in h-BN supported, suspended and SiO2 supported
|
709 |
+
parts are evident. Scale bars are 2 Β΅m.
|
710 |
+
7. Gate dependent measurements
|
711 |
+
We performed gate dependent SPCM measurements both on suspended and h-BN supported MoS2
|
712 |
+
devices. In both cases, we used 1 Β΅m SiO2 coated Si wafers. Si is used as the back gate in both device
|
713 |
+
configurations. We reported the h-BN supported junctions in the main text as devices over holes
|
714 |
+
showed significant change upon application of negative gate bias. Figure S5 shows the degradation of
|
715 |
+
the suspended device. After application of a few volts the device irreversibly shows a contrast change
|
716 |
+
|
717 |
+
hBN
|
718 |
+
0.12
|
719 |
+
0.10
|
720 |
+
hBNSupported
|
721 |
+
58mV
|
722 |
+
Suspended
|
723 |
+
:35mV
|
724 |
+
Sio,Supported
|
725 |
+
0.06
|
726 |
+
0.04
|
727 |
+
0.0
|
728 |
+
0.5
|
729 |
+
1.0
|
730 |
+
1.5
|
731 |
+
2.0
|
732 |
+
lenght [um]starting from the edges of the hole. We fabricated a long trench with open ends to see if the trapped
|
733 |
+
air within the hole is causing the observed contrast change. However, same contrast change is
|
734 |
+
observed after applying negative gate voltages. We observe that the contrast change starts from near
|
735 |
+
the hole and expands from there. At the moment we are not fully aware of the reasons leading this
|
736 |
+
contrast change. We consider that the release of the adsorbed molecules on the surface of the
|
737 |
+
substrate under large negative gate voltages lead to such degradation.
|
738 |
+
|
739 |
+
Figure S6 a. Optical microscope micrograph of indium contacted MoS2 on SiO2/Si with stair-like holes
|
740 |
+
before and after application of gate voltages down to ππΊ = β20 π. Lower panel shows a clear contrast
|
741 |
+
change around the holes extending to the indium contacts. b. SPCM maps taken at ππΊ = 0 π with 532
|
742 |
+
nm of 86 Β΅W on sample: (i) before gating, (ii) after ππΊ = β15 π scan and (iii) after the scan in (iii).
|
743 |
+
πΌπππ₯ = 6.5 nA and πΌπππ = β6.5 nA. Scan starts from top left corner to the bottom left corner with
|
744 |
+
progressing to the right in raster scan pattern. Scale bars are 5 Β΅m.
|
745 |
+
To prevent the sample degradation problem under large negative gate voltages, we coated the
|
746 |
+
substrate surface with 5 nm thick Al2O3 using atomic layer deposition (ALD) method after milling the
|
747 |
+
holes with FIB. Then, the device is fabricated over the ALD coated surface. The device didnβt show any
|
748 |
+
sign of degradation and produced pronounced photoresponse. Measurements from the device is given
|
749 |
+
in Figure S6. Although the device exhibits the expected gate dependent response, as discussed in the
|
750 |
+
main text, there is no carrier inversion induced reduction in the photovoltage due to the Fermi level
|
751 |
+
pinning.
|
752 |
+
|
753 |
+
ii
|
754 |
+
ili
|
755 |
+
BeforeV
|
756 |
+
After VG
|
757 |
+
Figure S7 a. Schematic of the device along with the optical image is shown. The sample is coated with
|
758 |
+
30 nm thick Al2O3 to passivate the SiO2 surface and to minimize the pinholes. Scale bar is 10 Β΅m. b.
|
759 |
+
Photovoltage map collected in DC mode without the Lock-in amplifier and chopper. i is the reflection
|
760 |
+
map, and photovoltage maps at ii is the ππΊ =-60 V, iii ππΊ = 0 V, iv ππΊ =60 V. Here, ππππ₯ = 20 mV and
|
761 |
+
ππππ = -20 mV. c. Photovoltage line trace taken along the dashed arrow given in b-ii. Large signal
|
762 |
+
corresponds to the more negative gate voltages. d. Photovoltage data collected from points indicated
|
763 |
+
on b-ii. This sample showed no Seebeck coefficient inversion due to possible Fermi level pinning
|
764 |
+
induced by the substrate as discussed in the main text.
|
765 |
+
H-BN supported devices performed better and showed no sign of such a contrast change. Figure S7
|
766 |
+
shows the reflection and the photocurrent maps reported in the main text and the photocurrent from
|
767 |
+
point 2 and 3 subtracted from point 4, marked on the photocurrent map. Both junctions of the h-BN
|
768 |
+
show almost identical response under gate voltage (point 3 data is multiplied by -1 for viewing
|
769 |
+
convenience).
|
770 |
+
|
771 |
+
|
772 |
+
ii
|
773 |
+
MoS2
|
774 |
+
B
|
775 |
+
in
|
776 |
+
In
|
777 |
+
A/2O3
|
778 |
+
SiO.
|
779 |
+
Si
|
780 |
+
ili
|
781 |
+
IV
|
782 |
+
Point A
|
783 |
+
Point B
|
784 |
+
Point C
|
785 |
+
Figure S8 a. Same figure from the main text is copied here for convenience. b. Raman intensity map
|
786 |
+
and the π΄1π peak shift map is given. c. Gate dependent signal from point 4 is subtracted from the
|
787 |
+
gate dependent data from point 2 (red curve) and point 3 (blue curve). Blue curve is multiplied by -1
|
788 |
+
for viewing convenience.
|
789 |
+
|
790 |
+
8. Scanning Thermal Microscope Calibration and Seebeck variation estimation
|
791 |
+
The Scanning Thermal Microscope (SThM) measurements were performed on a commercial Bruker
|
792 |
+
Icon instrument with a VITA-GLA-DM-1 probe. The probe, consisting of the silicon nitride lever with a
|
793 |
+
Pd heater/thermometer has been calibrated on a hot plate to relate the temperature to its electrical
|
794 |
+
resistance. The calibration curves are shown on figure S9.
|
795 |
+
|
796 |
+
Figure S9 a. SThM probe calibration of the electrical resistance with the supplied power. b.
|
797 |
+
Temperature as a function of electrical resistance
|
798 |
+
As described elsewhere4,5, the probe is part of a modified Wheatstone bridge which is balanced at low
|
799 |
+
voltage. During the measurements, we applied a combined AC (91 kHz) and DC bias on the bridge
|
800 |
+
which heats the probe and creates a bridge offset that directly measures the probe heater
|
801 |
+
temperature. For most experiments, we applied 1mW on the probe creating a Ξπ of 50 Β± 2 K, when
|
802 |
+
the probe was far away from the sample.
|
803 |
+
When the SThM tip is brought into contact with the devices, it locally heats the materials below its
|
804 |
+
apex. While the probe scans the surface, the device open circuit voltage is recorded and amplified via
|
805 |
+
a SR830 voltage preamplifier. This voltage is referred to as the thermovoltage. We excluded any
|
806 |
+
|
807 |
+
E
|
808 |
+
Imax
|
809 |
+
3(a)
|
810 |
+
(b)
|
811 |
+
368
|
812 |
+
100
|
813 |
+
80
|
814 |
+
367
|
815 |
+
(Ohms)
|
816 |
+
60
|
817 |
+
366
|
818 |
+
40
|
819 |
+
365
|
820 |
+
R
|
821 |
+
= 363.68 + 12.73 P
|
822 |
+
20
|
823 |
+
,= -1607.39 + 4.48 R
|
824 |
+
applied
|
825 |
+
probe
|
826 |
+
probe
|
827 |
+
364
|
828 |
+
0.05
|
829 |
+
0.10
|
830 |
+
0.15
|
831 |
+
0.20
|
832 |
+
0.25
|
833 |
+
0.30
|
834 |
+
0.35
|
835 |
+
364
|
836 |
+
368
|
837 |
+
372
|
838 |
+
376
|
839 |
+
380
|
840 |
+
384
|
841 |
+
P
|
842 |
+
applied (mW)
|
843 |
+
R.
|
844 |
+
Rprobe (Ohms)shortcut between the probe and the device as no leakage current could be measured between the
|
845 |
+
probe and both contacts.
|
846 |
+
The thermovoltage can be written analytically as6,7,
|
847 |
+
ππ‘β(π₯) = β β« π(π₯) ππ
|
848 |
+
ππ₯ (π₯)ππ₯
|
849 |
+
π΅
|
850 |
+
π΄
|
851 |
+
|
852 |
+
where π(π₯) is the position dependent Seebeck coefficient and
|
853 |
+
ππ
|
854 |
+
ππ₯ (π₯) is the position dependent
|
855 |
+
temperature gradient. Both are integrated over the whole device length from A to B.
|
856 |
+
As shown elsewhere6,7, it is possible to deconvolute the Seebeck coefficient from the temperature
|
857 |
+
gradient. This however requires a precise estimation of the temperature gradient and thus the sample
|
858 |
+
temperature rise under the tip, Ξππ πππππ.
|
859 |
+
As we know the probe temperature far away from the sample (50 Β± 2 K) and we monitor its
|
860 |
+
temperature via the Wheatstone bridge, we know that the probe temperature in contact with the
|
861 |
+
sample is 43.8 Β± 4 K. The probe cooling occurs because of several heat transfer mechanisms4,5 (solid-
|
862 |
+
solid conduction, air conduction, water meniscus, β¦).
|
863 |
+
For those probes, the Pd heater is however distributed over the whole triangular shaped silicon nitride
|
864 |
+
tip4,5. This implies that the tip temperature and probe temperature are different. We turned to finite
|
865 |
+
element modelling (COMSOL Multiphysics) to estimate the tip temperature over the MoS2 suspended
|
866 |
+
and supported sample. Figure S10 shows the overall simulated probe and sample.
|
867 |
+
We used reported values for the in-plane and out-of-plane MoS2 thermal conductivity as well as for
|
868 |
+
the MoS2-glass interface conductance. Reported values vary greatly in literature8β16. However, to the
|
869 |
+
best of our knowledge, for a thick sample (>10 layers), the values are on the order of 30 Wm-1K-1 for
|
870 |
+
the supported in-plane, 60 Wm-1K-1 for the suspended in-plane and 3 Wm-1K-1 for the cross-plane
|
871 |
+
conductivities. For the substrate interface conductance, we used 1 MWm-2K-1.
|
872 |
+
|
873 |
+
|
874 |
+
Figure S10 a. Finite element model for the SThM probe on a MoS2 suspended sample. b. Zoomed-in
|
875 |
+
view of the model where the temperature gradient is visible on the sample surface.
|
876 |
+
Using those material parameters, we estimated a ratio between the probe temperature and the tip
|
877 |
+
apex temperature of 4.9. The model also accounts for the tip-sample thermal resistance. This method
|
878 |
+
|
879 |
+
(a)
|
880 |
+
(b)and model were experimentally confirmed elsewhere4,5,17. Taking these into consideration, we obtain
|
881 |
+
a sample temperature rise Ξππ πππππ of 7.4 Β± 0.7 K. This gives a Seebeck variation of 72Β±10 Β΅VK-1.
|
882 |
+
|
883 |
+
References
|
884 |
+
(1)
|
885 |
+
Ng, H. K.; Chi, D.; Hippalgaonkar, K. Effect of Dimensionality on Thermoelectric Powerfactor of
|
886 |
+
Molybdenum Disulfide. J Appl Phys 2017, 121 (20), 204303.
|
887 |
+
https://doi.org/10.1063/1.4984138.
|
888 |
+
(2)
|
889 |
+
Wu, J.; Liu, Y.; Liu, Y.; Liu, Y.; Cai, Y.; Zhao, Y.; Ng, H. K.; Watanabe, K.; Taniguchi, T.; Zhang, G.;
|
890 |
+
Qiu, C. W.; Chi, D.; Neto, A. H. C.; Thong, J. T. L.; Loh, K. P.; Hippalgaonkar, K. Large
|
891 |
+
Enhancement of Thermoelectric Performance in MoS2/h-BN Heterostructure Due to Vacancy-
|
892 |
+
Induced Band Hybridization. Proc Natl Acad Sci U S A 2020, 117 (25), 13929β13936.
|
893 |
+
https://doi.org/10.1073/pnas.2007495117.
|
894 |
+
(3)
|
895 |
+
Cui, X.; Lee, G. H.; Kim, Y. D.; Arefe, G.; Huang, P. Y.; Lee, C. H.; Chenet, D. A.; Zhang, X.; Wang,
|
896 |
+
L.; Ye, F.; Pizzocchero, F.; Jessen, B. S.; Watanabe, K.; Taniguchi, T.; Muller, D. A.; Low, T.; Kim,
|
897 |
+
P.; Hone, J. Multi-Terminal Transport Measurements of MoS2 Using a van Der Waals
|
898 |
+
Heterostructure Device Platform. Nature Nanotechnology 2015 10:6 2015, 10 (6), 534β540.
|
899 |
+
https://doi.org/10.1038/nnano.2015.70.
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900 |
+
(4)
|
901 |
+
Tovee, P.; Pumarol, M.; Zeze, D.; Kjoller, K.; Kolosov, O. Nanoscale Spatial Resolution Probes
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902 |
+
for Scanning Thermal Microscopy of Solid State Materials. J Appl Phys 2012, 112 (11).
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903 |
+
https://doi.org/10.1063/1.4767923.
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(5)
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905 |
+
Spiece, J.; Evangeli, C.; Lulla, K.; Robson, A.; Robinson, B.; Kolosov, O. Improving Accuracy of
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906 |
+
Nanothermal Measurements via Spatially Distributed Scanning Thermal Microscope Probes. J
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907 |
+
Appl Phys 2018, 124 (1), 015101. https://doi.org/10.1063/1.5031085.
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908 |
+
(6)
|
909 |
+
Harzheim, A.; Spiece, J.; Evangeli, C.; McCann, E.; Falko, V.; Sheng, Y.; Warner, J. H.; Briggs, G.
|
910 |
+
A. D.; Mol, J. A.; Gehring, P.; Kolosov, O. v. Geometrically Enhanced Thermoelectric Effects in
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911 |
+
Graphene Nanoconstrictions. Nano Lett 2018, 18 (12), 7719β7725.
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912 |
+
https://doi.org/10.1021/ACS.NANOLETT.8B03406/ASSET/IMAGES/MEDIUM/NL-2018-
|
913 |
+
03406E_M006.GIF.
|
914 |
+
(7)
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915 |
+
Harzheim, A.; Evangeli, C.; Kolosov, O. v.; Gehring, P. Direct Mapping of Local Seebeck
|
916 |
+
Coefficient in 2D Material Nanostructures via Scanning Thermal Gate Microscopy. 2d Mater
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917 |
+
2020, 7 (4), 041004. https://doi.org/10.1088/2053-1583/ABA333.
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918 |
+
(8)
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919 |
+
Frausto-Avila, C. M.; Arellano-Arreola, V. M.; Yañez Limon, J. M.; de Luna-Bugallo, A.; Gomès,
|
920 |
+
S.; Chapuis, P. O. Thermal Boundary Conductance of CVD-Grown MoS2 Monolayer-on-Silica
|
921 |
+
Substrate Determined by Scanning Thermal Microscopy. Appl Phys Lett 2022, 120 (26),
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922 |
+
262202. https://doi.org/10.1063/5.0092553.
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923 |
+
(9)
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924 |
+
Taube, A.; Judek, J.; ΕapiΕska, A.; Zdrojek, M. Temperature-Dependent Thermal Properties of
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925 |
+
Supported MoS2 Monolayers. ACS Appl Mater Interfaces 2015, 7 (9), 5061β5065.
|
926 |
+
https://doi.org/10.1021/ACSAMI.5B00690/SUPPL_FILE/AM5B00690_SI_001.PDF.
|
927 |
+
(10)
|
928 |
+
Yue, X. F.; Wang, Y. Y.; Zhao, Y.; Jiang, J.; Yu, K.; Liang, Y.; Zhong, B.; Ren, S. T.; Gao, R. X.; Zou,
|
929 |
+
M. Q. Measurement of Interfacial Thermal Conductance of Few-Layer MoS2 Supported on
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930 |
+
|
931 |
+
Different Substrates Using Raman Spectroscopy. J Appl Phys 2020, 127 (10), 104301.
|
932 |
+
https://doi.org/10.1063/1.5128613.
|
933 |
+
(11)
|
934 |
+
Gabourie, A. J.; Suryavanshi, S. v.; Farimani, A. B.; Pop, E. Reduced Thermal Conductivity of
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935 |
+
Supported and Encased Monolayer and Bilayer MoS2. 2d Mater 2020, 8 (1), 011001.
|
936 |
+
https://doi.org/10.1088/2053-1583/ABA4ED.
|
937 |
+
(12)
|
938 |
+
Zhang, X.; Sun, D.; Li, Y.; Lee, G. H.; Cui, X.; Chenet, D.; You, Y.; Heinz, T. F.; Hone, J. C.
|
939 |
+
Measurement of Lateral and Interfacial Thermal Conductivity of Single- and Bilayer MoS2 and
|
940 |
+
MoSe2 Using Refined Optothermal Raman Technique. ACS Appl Mater Interfaces 2015, 7 (46),
|
941 |
+
25923β25929. https://doi.org/10.1021/ACSAMI.5B08580/ASSET/IMAGES/LARGE/AM-2015-
|
942 |
+
085805_0003.JPEG.
|
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+
(13)
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944 |
+
Jo, I.; Pettes, M. T.; Ou, E.; Wu, W.; Shi, L. Basal-Plane Thermal Conductivity of Few-Layer
|
945 |
+
Molybdenum Disulfide. Appl Phys Lett 2014, 104 (20), 201902.
|
946 |
+
https://doi.org/10.1063/1.4876965.
|
947 |
+
(14)
|
948 |
+
Yuan, P.; Wang, R.; Wang, T.; Wang, X.; Chemistry, Y. X.-P.; 2018, undefined. Nonmonotonic
|
949 |
+
Thickness-Dependence of in-Plane Thermal Conductivity of Few-Layered MoS 2: 2.4 to 37.8
|
950 |
+
Nm. pubs.rsc.org.
|
951 |
+
(15)
|
952 |
+
Bae, J. J.; Jeong, H. Y.; Han, G. H.; Kim, J.; Kim, H.; Kim, M. S.; Moon, B. H.; Lim, S. C.; Lee, Y. H.
|
953 |
+
Thickness-Dependent in-Plane Thermal Conductivity of Suspended MoS2 Grown by Chemical
|
954 |
+
Vapor Deposition. Nanoscale 2017, 9 (7), 2541β2547. https://doi.org/10.1039/C6NR09484H.
|
955 |
+
(16)
|
956 |
+
Meng, X.; Pandey, T.; Jeong, J.; Fu, S.; Yang, J.; Chen, K.; Singh, A.; He, F.; Xu, X.; Zhou, J.;
|
957 |
+
Hsieh, W. P.; Singh, A. K.; Lin, J. F.; Wang, Y. Thermal Conductivity Enhancement in MoS2
|
958 |
+
under Extreme Strain. Phys Rev Lett 2019, 122 (15), 155901.
|
959 |
+
https://doi.org/10.1103/PHYSREVLETT.122.155901/FIGURES/3/MEDIUM.
|
960 |
+
(17)
|
961 |
+
Gehring, P.; Harzheim, A.; Spièce, J.; Sheng, Y.; Rogers, G.; Evangeli, C.; Mishra, A.; Robinson,
|
962 |
+
B. J.; Porfyrakis, K.; Warner, J. H.; Kolosov, O. v.; Briggs, G. A. D.; Mol, J. A. Field-Effect Control
|
963 |
+
of Graphene-Fullerene Thermoelectric Nanodevices. Nano Lett 2017, 17 (11), 7055β7061.
|
964 |
+
https://doi.org/10.1021/ACS.NANOLETT.7B03736/ASSET/IMAGES/NL-2017-
|
965 |
+
03736Q_M017.GIF.
|
966 |
+
|
967 |
+
|
09AzT4oBgHgl3EQfDPrL/content/tmp_files/load_file.txt
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|
1 |
+
arXiv:2301.00744v1 [math.CO] 2 Jan 2023
|
2 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A
|
3 |
+
GARSIDE MONOID
|
4 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
5 |
+
Abstract. We study two families of lattices whose number of elements are given by
|
6 |
+
the numbers in even (respectively odd) positions in the Fibonacci sequence. The even
|
7 |
+
Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially
|
8 |
+
ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even
|
9 |
+
one. We give a combinatorial proof of the lattice property, relying on a description of
|
10 |
+
words for the Garside element in terms of SchrΓΆder trees, and on a recursive description
|
11 |
+
of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins
|
12 |
+
in the lattice. As a byproduct we also obtain that the number of words for the Garside
|
13 |
+
element is given by a little SchrΓΆder number.
|
14 |
+
Contents
|
15 |
+
1.
|
16 |
+
Introduction
|
17 |
+
1
|
18 |
+
2.
|
19 |
+
Deο¬nition and structure of the poset
|
20 |
+
3
|
21 |
+
2.1.
|
22 |
+
Deο¬nition of the poset
|
23 |
+
3
|
24 |
+
2.2.
|
25 |
+
Lattice property
|
26 |
+
4
|
27 |
+
3.
|
28 |
+
SchrΓΆder trees and words for the Garside element
|
29 |
+
7
|
30 |
+
3.1.
|
31 |
+
labelling of SchrΓΆder trees
|
32 |
+
7
|
33 |
+
3.2.
|
34 |
+
Words for the Garside element in terms of SchrΓΆder trees
|
35 |
+
10
|
36 |
+
4.
|
37 |
+
Enumerative results
|
38 |
+
17
|
39 |
+
4.1.
|
40 |
+
Number of simple elements
|
41 |
+
17
|
42 |
+
4.2.
|
43 |
+
Number of left-divisors of the lcm of the atoms and odd Fibonacci lattice
|
44 |
+
17
|
45 |
+
4.3.
|
46 |
+
Number of words for the divisors of the Garside element
|
47 |
+
18
|
48 |
+
References
|
49 |
+
20
|
50 |
+
1. Introduction
|
51 |
+
Several algebraic structures naturally yield examples of lattices: as elementary examples,
|
52 |
+
one can cite the lattice of subsets of a given set ordered by inclusion, or the lattice of
|
53 |
+
subgroups of a given group.
|
54 |
+
One can then study which properties are satisο¬ed by the
|
55 |
+
obtained lattices, or conversely, starting from a known lattice, wondering for instance if
|
56 |
+
it can be realized in a given algebraic framework, or if a property of the lattice implies
|
57 |
+
properties of the attached algebraic structure(s) and vice-versa.
|
58 |
+
The aim of this paper is to give a combinatorial description of a ο¬nite lattice that
|
59 |
+
appeared in the framework of Garside theory. We will not recall results and principles
|
60 |
+
of Garside theory as they will not be used in this paper, but the interested reader can
|
61 |
+
look at [5, 4] for more on the topic.
|
62 |
+
This is a branch of combinatorial group theory
|
63 |
+
which aims at establishing properties of families of inο¬nite groups such as the solvability
|
64 |
+
of the word problem, the conjugacy problem, the structure of the center, etc. Roughly
|
65 |
+
speaking, a Garside group is a group of fraction of a monoid (called a Garside monoid) with
|
66 |
+
|
67 |
+
2
|
68 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
69 |
+
particularly nice divisibility properties, which ensures that the above-mentioned problems
|
70 |
+
can be solved. Such a monoid M has no nontrivial invertible element, and comes equipped
|
71 |
+
with a distinguished element β (called a Garside element) whose left- and right-divisors are
|
72 |
+
ο¬nite, coincide, generate the monoid, and form a lattice under left- and right-divisibility.
|
73 |
+
The left- or right-divisors of β are called the simples.
|
74 |
+
The fundamental example of a Garside group is the n-strand Artin braid group [7].
|
75 |
+
It admits several non-equivalent Garside structures (i.e., nonisomorphic Garside monoids
|
76 |
+
whose group of fractions are isomorphic to the n-strand braid group), and the lattice of
|
77 |
+
simples in the ο¬rst discovered such Garside structure is isomorphic to the weak Bruhat
|
78 |
+
order on the symmetric group. Several widely studied lattices can be realized as lattices
|
79 |
+
of simples of a Garside monoid: this includes the lattices of left and right weak Bruhat
|
80 |
+
order on any ο¬nite Coxeter group [3, 6], the lattice of (generalized) noncrossing partitions
|
81 |
+
attached to a ο¬nite Coxeter group [1, 2], etc. (see also [12] for many other examples). This
|
82 |
+
suggests the following question:
|
83 |
+
Question. Which lattices can appear as lattices of simples of Garside monoids ?
|
84 |
+
The aim of this paper is to study a family Pn of lattices arising as simples of a family Mn,
|
85 |
+
n β₯ 2 of Garside monoids introduced by the ο¬rst author [8]. For n = 2, the corresponding
|
86 |
+
Garside group is isomorphic to the 3-strand braid group B3, while in general it is isomorphic
|
87 |
+
to the (n, n + 1)-torus knot group, which for n > 3 is a (strict) extension of the (n + 1)-
|
88 |
+
strand braid group Bn+1. The lattice property of Pn follows from the fact proven in op.
|
89 |
+
cit. that Mn is a Garside monoid, but it gives very little information about the structure
|
90 |
+
and properties of the lattice. For instance, one does not have a formula enumerating the
|
91 |
+
number of simples, and only an algorithm to calculate meet and joins in the lattice.
|
92 |
+
In Section 2 we give a new proof of the lattice property of Pn (Theorem 2.8) by exhibiting
|
93 |
+
the recursive structure of the poset. Every lattice Pn turns out to contain the lattices Pi,
|
94 |
+
i < n as sublattices. Note that an ingredient of the proof of Theorem 2.8 is proven later on
|
95 |
+
in the paper, as it relies on a combinatorial description for the set of words for the Garside
|
96 |
+
element in terms of SchrΓΆder trees.
|
97 |
+
More precisely, in Section 3 we establish a simple
|
98 |
+
bijection between the set of words for βn and the set of SchrΓΆder trees on n+1 leaves, in such
|
99 |
+
a way that applying a deο¬ning relation of Mn to a word amounts to applying what we call a
|
100 |
+
"local move" on the corresponding SchrΓΆder tree (Theorem 3.12 and Corollary 3.13). These
|
101 |
+
local moves are given by speciο¬c edge contraction and are related to the notion of reο¬nement
|
102 |
+
considered in [10]. This allows us to establish in Proposition 3.16 an isomorphism of posets
|
103 |
+
between subposets of Pn and Pi, i < n, required in the proof of Theorem 2.8.
|
104 |
+
Finally, the obtained recursive description of Pn together with the description of words
|
105 |
+
for βn in terms of SchrΓΆder trees allows us to derive a few enumerative results. This is
|
106 |
+
done in Section 4. The ο¬rst one is that the number of elements of Pn is given by F2n, where
|
107 |
+
Fi is the i-th Fibonacci number (Lemma 4.1). We thus call Pn the even Fibonacci lattice.
|
108 |
+
The atoms of Mn turn out to have the same left- and right-lcm, which is strictly less than
|
109 |
+
βn. We also show that the sublattice of Pn deο¬ned as the order ideal of this lcm has F2nβ1
|
110 |
+
elements (Lemma 4.3), and thus call it the odd Fibonacci lattice. Other enumerative results
|
111 |
+
include the determination of the number of words for the Garside elements (Corollary 3.14),
|
112 |
+
and the number of words for the whole set of simples (Theorem 4.7).
|
113 |
+
Recall that the Garside monoid Mn under study in this paper has group of fractions
|
114 |
+
isomorphic to the (n, n + 1)-torus knot group. This Garside structure was generalized to
|
115 |
+
all torus knot groups in [9]. It would be interesting to have a description of the lattices of
|
116 |
+
simples of this bigger family of Garside monoids.
|
117 |
+
|
118 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
119 |
+
3
|
120 |
+
1
|
121 |
+
Ο1
|
122 |
+
Ο2
|
123 |
+
Ο2Ο1
|
124 |
+
Ο3
|
125 |
+
Ο1Ο3
|
126 |
+
Ο3Ο1
|
127 |
+
Ο1Ο3Ο1
|
128 |
+
Ο3Ο2
|
129 |
+
Ο2Ο1Ο3
|
130 |
+
Ο3Ο2Ο1
|
131 |
+
Ο2
|
132 |
+
3
|
133 |
+
Ο3Ο1Ο3
|
134 |
+
Ο2
|
135 |
+
3Ο1
|
136 |
+
(Ο1Ο3)2
|
137 |
+
(Ο3Ο1)2
|
138 |
+
Ο3
|
139 |
+
3
|
140 |
+
Ο3Ο2Ο1Ο3
|
141 |
+
Ο2
|
142 |
+
3Ο1Ο3
|
143 |
+
(Ο3Ο1)2Ο3
|
144 |
+
Ο4
|
145 |
+
3
|
146 |
+
Figure 1. The even Fibonacci lattice for n = 3 and (in blue) the odd Fibonacci
|
147 |
+
lattice inside it.
|
148 |
+
2. Definition and structure of the poset
|
149 |
+
2.1. Deο¬nition of the poset. The beginning of this section is devoted to explaining how
|
150 |
+
the poset under study is deο¬ned. We recall the deο¬nition of the monoid from which it is
|
151 |
+
built, as well as a few properties of this monoid (all of which are proven in [8]).
|
152 |
+
Let M be a monoid and a, b β M. We say that a is a left divisor of b (or that b is a
|
153 |
+
right multiple of a) if there is c β M such that ac = b. We similarly deο¬ne right divisors
|
154 |
+
and left multiples.
|
155 |
+
Let M0 be the trivial monoid and for n β₯ 1, let Mn be the monoid deο¬ned by the
|
156 |
+
presentation
|
157 |
+
(2.1)
|
158 |
+
οΏ½
|
159 |
+
Ο1, Ο2, . . . , Οn
|
160 |
+
οΏ½οΏ½οΏ½οΏ½ Ο1ΟnΟi = Οi+1Οn for all 1 β€ i β€ n β 1
|
161 |
+
οΏ½
|
162 |
+
.
|
163 |
+
We denote by S the set of generators {Ο1, Ο2, . . . , Οn}, and by R the deο¬ning relations of
|
164 |
+
Mn. This monoid was introduced by the ο¬rst author in [8, Deο¬nition 4.1]. Note that this
|
165 |
+
monoid is equipped with a length function Ξ» : Mn ββ Zβ₯0 given by the multiplicative
|
166 |
+
extension of Ξ»(Οi) = i for all i = 1, . . . , n, which is possible since the deο¬ning relations
|
167 |
+
do not change the length of a word. As a corollary, the only invertible element in Mn is
|
168 |
+
|
169 |
+
4
|
170 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
171 |
+
the identity, and the left- and right-divisibility relations are partial orders on Mn. We will
|
172 |
+
write a β€L b or simply a β€ b if a left-divides b, and a β€R b if a right-divides b.
|
173 |
+
This monoid was shown to be a so-called Garside monoid (see [8, Theorem 4.18]), with
|
174 |
+
corresponding Garside group (which has the same presentation as Mn) isomorphic to the
|
175 |
+
(n, n + 1)-torus knot group, that is, the fundamental group of the complement of the
|
176 |
+
torus knot Tn,n+1 in S3.
|
177 |
+
Garside monoids have several important properties.
|
178 |
+
Among
|
179 |
+
them, the left- and right-divisibility relations equip Mn with two lattice structures, and
|
180 |
+
Mn comes equipped with a distinguished element βn, called a Garside element, which has
|
181 |
+
the following two properties
|
182 |
+
(1) The set of left divisors of βn coincides with its set of right divisors, and forms a
|
183 |
+
ο¬nite set.
|
184 |
+
(2) The set of left (or right) divisors of βn generates Mn.
|
185 |
+
This Garside element is given by βn = Οn+1
|
186 |
+
n
|
187 |
+
. In particular, as any Garside monoid is a
|
188 |
+
lattice for both left- and right-divisibility, the set Div(βn) of left (or right) divisors of βn
|
189 |
+
is a ο¬nite lattice if equipped by the order relation given by the restriction of left- (or right-)
|
190 |
+
divisibility on Mn. The set Div(βn) is the set of simple elements or simples of Mn. In
|
191 |
+
general (Div(βn), β€L) and (Div(βn), β€R) will not be isomorphic as posets. But we always
|
192 |
+
have
|
193 |
+
(Div(βn), β€L) βΌ= (Div(βn), β€R)op
|
194 |
+
(see for instance [8, Lemma 2.19]; such a property holds in any Garside monoid).
|
195 |
+
We will give a new proof that (Div(βn), β€) (and hence (Div(βn), β€R) is a lattice, in
|
196 |
+
a way which will exhibit a recursive structure of the poset. To this end, we will require
|
197 |
+
(sometimes without mentioning it) a few basic results on the monoid Mn which are either
|
198 |
+
explained above or proven in [8]:
|
199 |
+
(1) The left- and right-divisibility relations on Mn are partial orders.
|
200 |
+
(2) The monoid Mn is both left- and right-cancellative, i.e., for a, b, c β Mn, we have
|
201 |
+
that ab = ac β b = c, and ba = ca β b = c (see [8, Propositions 4.9 and 4.12]),
|
202 |
+
(3) The set of left- and right-divisors of βn coincide. In fact, the element βn is central
|
203 |
+
in Mn, hence as Mn is cancellative, for a, b β Mn such that ab = βn, we have
|
204 |
+
ab = ba (see [8, Proposition 4.15])
|
205 |
+
2.2. Lattice property. The aim of this subsection is to prove a few properties of simple
|
206 |
+
elements of Mn, and to derive a new algebraic proof that Div(βn) is a lattice.
|
207 |
+
Proposition 2.1. Let x1x2 Β· Β· Β· xk be a word for βn, with xi β S for all i = 1, . . . , k. There
|
208 |
+
are i1 = 1 < i2 < Β· Β· Β· < iβ β€ k such that
|
209 |
+
β’ For all j = 1, . . . , β, the word yj := xijxij+1 Β· Β· Β· xij+1β1 (with the convention that
|
210 |
+
iβ+1 = k + 1) is a word for a power of Οn,
|
211 |
+
β’ The decomposition y1|y2| Β· Β· Β· |yβ of the word x1x2 Β· Β· Β· xk is maximal in the sense that
|
212 |
+
no word among the yj can be decomposed as a product of two nonempty words which
|
213 |
+
are words for powers of Οn.
|
214 |
+
Morever, a decomposition with the above properties is unique.
|
215 |
+
Proof. The existence of the decomposition is clear using the fact that Mn is cancellative:
|
216 |
+
given the word x1x2 Β· Β· Β· xk, consider the smallest i β {1, 2, . . . , k} such that x1x2 Β· Β· Β· xk is
|
217 |
+
a word for a power of Οn. Such an i has to exist, as x1x2 Β· Β· Β· xk is a word for a power of
|
218 |
+
Οn. Then set i2 := i + 1. By cancellativity in Mn, since x1 Β· Β· Β· xi and x1 Β· Β· Β· xk are both
|
219 |
+
words for a power of Οn, the word xi+1 Β· Β· Β· xk must also be a word for a power of Οn. Hence
|
220 |
+
one can go on, arguing the same with the word xi+1 Β· Β· Β· xk. Again by cancellativity, this
|
221 |
+
decomposition must be maximal.
|
222 |
+
|
223 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
224 |
+
5
|
225 |
+
Now assume that the decomposition is not unique, that is, assume that y1|y2| Β· Β· Β· |yβ
|
226 |
+
and z1|z2| Β· Β· Β· |zββ² are two decompositions of the word x1x2 Β· Β· Β· xk satisfying the properties
|
227 |
+
of the statement. As both y1 and z1 are words for a power of Οn, if y1 ΜΈ= z1, then one
|
228 |
+
word must be strict preο¬x of the other, say z1 is a strict preο¬x of y1. But this contradicts
|
229 |
+
the maximality of the decomposition y1|y2| Β· Β· Β· |yβ: indeed if y1 = x1x2 Β· Β· Β· xi2β1 and z1 =
|
230 |
+
x1x2 Β· Β· Β· xp with p < i2 β 1, we can decompose y1 nontrivially as x1x2 Β· Β· Β· xp|xp+1 Β· Β· Β· xi2β1,
|
231 |
+
and by cancellativity both x1 Β· Β· Β· xp and xp+1 Β· Β· Β· xi2β1 are words for powers of Οn.
|
232 |
+
β‘
|
233 |
+
Example 2.2. Consider the word Ο3Ο1Ο7Ο1Ο7Ο5Ο4Ο7Ο7Ο1Ο7Ο6 in M7. We claim that this is a
|
234 |
+
word for the Garside element Ο8
|
235 |
+
7 of M7. Indeed, using the deο¬ning relation Ο1Ο7Οi = Οi+1Ο7
|
236 |
+
with i = 5 and 6, we get that
|
237 |
+
Ο3(Ο1Ο7Ο1Ο7Ο5)Ο4Ο7Ο7(Ο1Ο7Ο6) = Ο3Ο3
|
238 |
+
7Ο4Ο4
|
239 |
+
7,
|
240 |
+
and we observe also applying deο¬ning relations that
|
241 |
+
Ο3Ο3
|
242 |
+
7Ο4Ο4
|
243 |
+
7 = Ο1Ο7Ο2Ο2
|
244 |
+
7Ο4 = Ο1Ο7Ο1Ο7Ο1Ο7Ο4 = Ο1Ο7Ο1Ο7Ο5Ο7 = Ο1Ο7Ο6Ο2
|
245 |
+
7 = Ο4
|
246 |
+
7.
|
247 |
+
The decomposition according to Proposition 2.1 is given by
|
248 |
+
Ο3Ο1Ο7Ο1Ο7Ο5Ο4
|
249 |
+
οΏ½
|
250 |
+
οΏ½οΏ½
|
251 |
+
οΏ½
|
252 |
+
:=y1
|
253 |
+
| Ο7
|
254 |
+
οΏ½οΏ½οΏ½οΏ½
|
255 |
+
:=y2
|
256 |
+
| Ο7
|
257 |
+
οΏ½οΏ½οΏ½οΏ½
|
258 |
+
:=y3
|
259 |
+
| Ο1Ο7Ο6
|
260 |
+
οΏ½ οΏ½οΏ½ οΏ½
|
261 |
+
:=y4
|
262 |
+
.
|
263 |
+
It is indeed clear by considering Ξ»(u) for u preο¬xes of y1 or y4 that whenever u is a proper
|
264 |
+
preο¬x, we do not have Ξ»(u) equal to a multiple of 7, which is a necessary condition for a
|
265 |
+
word to represent a power of Ο7.
|
266 |
+
Lemma 2.3. Let 1 β€ k β€ n. Then
|
267 |
+
S β© {x β Div(βn) | x β€ ΟkΟk
|
268 |
+
n} = {Ο1, Ο2, . . . , Οk}.
|
269 |
+
Proof. We argue by induction on k. The result is clear for k = 1, as no deο¬ning relation of
|
270 |
+
Mn can be applied to the word Ο1Οn. Now let k > 1. Observe that
|
271 |
+
ΟkΟk
|
272 |
+
n = (Ο1Οn)k = (Ο1Οn)(Ο1Οn)kβ1.
|
273 |
+
In particular we have Ο1 β€ ΟkΟk
|
274 |
+
n and by induction, we get Ο1ΟnΟi β€ ΟkΟk
|
275 |
+
n for all 1 β€ i β€ kβ1.
|
276 |
+
As Ο1ΟnΟi = Οi+1Οn we get that {Ο1, Ο2, . . . , Οk} β S β© {x β Div(βn) | x β€ ΟkΟk
|
277 |
+
n}.
|
278 |
+
It remains to show that no other Οi can be a left-divisor of ΟkΟk
|
279 |
+
n. Hence assume that
|
280 |
+
i > k and Οi β€ ΟkΟk
|
281 |
+
n. Hence there is a word x1x2 Β· Β· Β· xp for ΟkΟk
|
282 |
+
n, where xi β S for all i,
|
283 |
+
such that x1 = Οi. As the words x1x2 Β· Β· Β· xp and ΟkΟk
|
284 |
+
n represent the same element, they
|
285 |
+
can be related by a ο¬nite sequence of words w0 = x1x2 Β· Β· Β· xp, w1, . . . , wq = ΟkΟk
|
286 |
+
n, where
|
287 |
+
each wi is a word with letters in S and wi+1 is obtained from wi by applying a single
|
288 |
+
relation somewhere in the word. As the ο¬rst letter of w0 diο¬ers from the ο¬rst letter of
|
289 |
+
wq, there must exist some 0 β€ β < q such that wβ begins by Οi but wβ+1 does not. It
|
290 |
+
follows that the relation allowing one to pass from wβ to wβ+1 has to be applied at the
|
291 |
+
beginning of the word wβ. But the only possible relation with one side beginning by Οi
|
292 |
+
is ΟiΟn = Ο1ΟnΟiβ1. It follows that Ο1ΟnΟiβ1 β€ ΟkΟk
|
293 |
+
n = (Ο1Οn)k. By cancellativity, we get
|
294 |
+
that Οiβ1 β€ (Ο1Οn)kβ1 = Οkβ1Οkβ1
|
295 |
+
n
|
296 |
+
. By induction this forces one to have i β 1 β€ k β 1,
|
297 |
+
contradicting our assumption that i > k.
|
298 |
+
β‘
|
299 |
+
Similarly, we have
|
300 |
+
Lemma 2.4. Let 1 β€ k β€ n. Then
|
301 |
+
S β© {x β Div(βn) | x β€R Οk
|
302 |
+
n} = {Οn, Οnβ1, . . . , Οnβk+1}.
|
303 |
+
|
304 |
+
6
|
305 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
306 |
+
Proof. As for Lemma 2.3, we argue by induction on k. The result is clear for k = 1. Hence
|
307 |
+
assume that k > 1. As (Ο1Οn)nβjΟj = Οnβj+1
|
308 |
+
n
|
309 |
+
, we get that Οj β€R Οk
|
310 |
+
n for all j such that
|
311 |
+
n β j + 1 β€ k, that is, for all j β₯ n β k + 1. It remains to show that no other Οj can
|
312 |
+
right-divide Οk
|
313 |
+
n. Hence assume that Οj β€R Οk
|
314 |
+
n, where j < n β k + 1. Arguing as in the
|
315 |
+
proof of Lemma 2.3, we see that Ο1ΟnΟj = Οj+1Οn must be a right-divisor of Οk
|
316 |
+
n, hence by
|
317 |
+
cancellativity that Οj+1 β€R Οkβ1
|
318 |
+
n
|
319 |
+
. By induction this forces j + 1 β₯ n β k + 2, contradicting
|
320 |
+
our assumtion that j < n β k + 1.
|
321 |
+
β‘
|
322 |
+
For x β Div(βn), let d(x) := max{k β₯ 0 | Οk
|
323 |
+
n β€ x}. Let 0 β€ i β€ n + 1 and let
|
324 |
+
Di
|
325 |
+
n := {x β Div(βn) | d(x) = i}.
|
326 |
+
Note that
|
327 |
+
Div(βn) =
|
328 |
+
οΏ½
|
329 |
+
0β€iβ€n+1
|
330 |
+
Di
|
331 |
+
n.
|
332 |
+
We have Dn
|
333 |
+
n = {Οn
|
334 |
+
n}, Dn+1
|
335 |
+
n
|
336 |
+
= {βn}.
|
337 |
+
Lemma 2.5. Let x β Div(βn) and i = d(x). Let xβ² β Mn such that x = Οi
|
338 |
+
nxβ². Note that
|
339 |
+
xβ² β D0
|
340 |
+
n. Let x1x2 Β· Β· Β· xk be a word for x, where xi β S for all i = 1, . . . , k. Then there is
|
341 |
+
1 β€ β β€ k such that x1x2 Β· Β· Β· xβ is a word for Οi
|
342 |
+
n (and hence xβ+1 Β· Β· Β· xk is a word for xβ² by
|
343 |
+
cancellativity). In other words, any word for x has a preο¬x which is a word for Οi
|
344 |
+
n.
|
345 |
+
Proof. It suο¬ces to show that if z1z2 Β· Β· Β· zp is an expression for x such that z1z2 Β· Β· Β· zq is
|
346 |
+
an expression for Οi
|
347 |
+
n (q β€ p, then one cannot apply a deο¬ning relation of Mn on the word
|
348 |
+
z1z2 Β· Β· Β· zp simultaneously involving letters of the word z1z2 Β· Β· Β· zq and letters of the word
|
349 |
+
zq+1 Β· Β· Β· zp. Let us consider the three possible cases where this could occur: one could have
|
350 |
+
Ο1Οn|Οj, Ο1|ΟnΟj, or Οj+1|Οn (1 β€ j < n), where the | separates the letters zq and zq+1.
|
351 |
+
The last two cases cannot happen, since one would have zq+1 = Οn, hence zq+1 Β· Β· Β· zp would
|
352 |
+
be a word for xβ² beginning by Οn, contradicting the fact that xβ² β D0
|
353 |
+
n. It remains to show
|
354 |
+
that the case Ο1Οn|Οj cannot happen. Hence assume that zqβ1 = Ο1, zq = Οn, zq+1 = Οj.
|
355 |
+
By cancellativity, as z1z2 Β· Β· Β· zq is a word for Οi
|
356 |
+
n, it implies that Ο1 β€R Οiβ1
|
357 |
+
n
|
358 |
+
. By lemma 2.4,
|
359 |
+
this implies that n β (i β 1) + 1 = 1, hence that i = n + 1.
|
360 |
+
Since x β Div(βn) and
|
361 |
+
x = Οn+1
|
362 |
+
n
|
363 |
+
xβ² = βnxβ², we get xβ² = 1, contradicting the fact that zq+1 = Οj.
|
364 |
+
β‘
|
365 |
+
Lemma 2.6. Let i, j β {0, 1, . . . , n + 1}, with i ΜΈ= j. Let x β Di
|
366 |
+
n, y β Dj
|
367 |
+
n. Assume that
|
368 |
+
x β€ y. Then i < j and x < Οj
|
369 |
+
n β€ y.
|
370 |
+
Proof. It is clear that i < j, since Οi
|
371 |
+
n β€ y as Οi
|
372 |
+
n β€ x, hence j < i would contradict y β Dj
|
373 |
+
n.
|
374 |
+
In particular x < y. Let xβ², yβ² such that x = Οi
|
375 |
+
nxβ² and y = Οj
|
376 |
+
nyβ². Note that xβ², yβ² both lie
|
377 |
+
in D0
|
378 |
+
n. Since x β€ y and Mn is cancellative, we get that xβ² < Οjβi
|
379 |
+
n yβ². It implies that there
|
380 |
+
exists a word x1x2 Β· Β· Β· xk for Οjβi
|
381 |
+
n
|
382 |
+
yβ² (xi β S) and 1 β€ β < k such that x1x2 Β· Β· Β· xβ is a word
|
383 |
+
for xβ². Now by lemma 2.5, there is 0 β€ ββ² β€ k such that x1x2 Β· Β· Β· xββ² is a word for Οjβi
|
384 |
+
n . If
|
385 |
+
ββ² β€ β, then Οjβi
|
386 |
+
n
|
387 |
+
β€ xβ², contradicting the fact that xβ² β D0
|
388 |
+
n. Hence ββ² > β, and xβ² < Οjβi
|
389 |
+
n .
|
390 |
+
Multiplying by Οi
|
391 |
+
n on the left we get x < Οj
|
392 |
+
n.
|
393 |
+
β‘
|
394 |
+
Lemma 2.7. Let z1, z2 β Di
|
395 |
+
n. Let 1 β€ k1 < k2 β€ n and assume that there are two cover
|
396 |
+
relations z1 β€Β· Οk1
|
397 |
+
n , z2 β€Β· Οk2
|
398 |
+
n in (Div(βn), β€). Then z1 < z2.
|
399 |
+
Proof. As z1 β€ Οk1
|
400 |
+
n , z2 β€ Οk2
|
401 |
+
n
|
402 |
+
are cover relations, there are 1 β€ j1, j2 β€ n such that
|
403 |
+
z1Οj1 = Οk1
|
404 |
+
n , z2Οj2 = Οk2
|
405 |
+
n . By lemma 2.4, for β β {1, 2} we have jβ β {n β kβ + 1, . . . , n} and
|
406 |
+
Οkβ
|
407 |
+
n = Οjβ+kββ1βn
|
408 |
+
n
|
409 |
+
(Ο1Οn)nβjβΟjβ.
|
410 |
+
In particular, we have
|
411 |
+
zβ = Οjβ+kββ1βn
|
412 |
+
n
|
413 |
+
(Ο1Οn)nβjβ
|
414 |
+
|
415 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
416 |
+
7
|
417 |
+
and as (Ο1Οn)nβjβ = ΟnβjβΟnβjβ
|
418 |
+
n
|
419 |
+
, by lemma 2.3 we see that Οn cannot be a left divisor of
|
420 |
+
(Ο1Οn)nβjβ, and hence that d(zβ) = jβ + kβ β 1 β n. But d(zβ) = i for β β {1, 2}, and since
|
421 |
+
k1 < k2 we deduce that j1 > j2. Since zβ = Οi
|
422 |
+
n(Ο1Οn)nβjβ, we get that z1 < z2, which
|
423 |
+
concludes the proof.
|
424 |
+
β‘
|
425 |
+
Theorem 2.8. The poset (Div(βn), β€) is a lattice. Given i, j β {0, 1, . . . , n+1} with i β€ j
|
426 |
+
and x β Di
|
427 |
+
n, y β Dj
|
428 |
+
n, we have
|
429 |
+
x β§ y = x β§i
|
430 |
+
οΏ½οΏ½
|
431 |
+
i
|
432 |
+
{z β Di
|
433 |
+
n | z β€ y}
|
434 |
+
οΏ½
|
435 |
+
,
|
436 |
+
where β¨i and β§i denote the meet and join on the restriction of the left-divisibility order on
|
437 |
+
Di
|
438 |
+
n, which itself forms a lattice. Note that if i = j we simply get x β§ y = x β§i y.
|
439 |
+
Proof. The proof is by induction on n. We have Div(β0) = {β’}, and Div(β1) = {1, Ο1, Ο2
|
440 |
+
1},
|
441 |
+
which is a lattice. Hence assume that n β₯ 2. By Proposition 3.16 below, the restriction of
|
442 |
+
the left-divisibility to Di
|
443 |
+
n yields an isomorphism of poset with Div(βnβi) if i ΜΈ= 0, n+1, while
|
444 |
+
the restriction to D0
|
445 |
+
n yields an isomorphism of poset with Div(βnβ1), and the restriction to
|
446 |
+
Dn+1
|
447 |
+
n
|
448 |
+
an isomorphism of posets with Div(β0) = {β’}. In particular, by induction, all these
|
449 |
+
posets are lattices. As the poset (Div(βn), β€) is ο¬nite and admits a maximal element, it
|
450 |
+
suο¬ces to show that x β§ y as deο¬ned by the formula above is indeed the join of x and y.
|
451 |
+
It is clear that xβ§y β€ x. Let us show that xβ§y β€ y. If i = j this is clear, hence assume
|
452 |
+
that i < j. By lemma 2.6 we see that
|
453 |
+
οΏ½
|
454 |
+
i
|
455 |
+
{z β Di
|
456 |
+
n | z β€ y} =
|
457 |
+
οΏ½
|
458 |
+
i
|
459 |
+
{z β Di
|
460 |
+
n | z β€ Οj
|
461 |
+
n}.
|
462 |
+
It suο¬ces to check that οΏ½
|
463 |
+
i{z β Di
|
464 |
+
n | z β€ Οj
|
465 |
+
n} β€ Οj
|
466 |
+
n. Note that
|
467 |
+
οΏ½
|
468 |
+
i
|
469 |
+
{z β Di
|
470 |
+
n | z β€ Οj
|
471 |
+
n} =
|
472 |
+
οΏ½
|
473 |
+
i
|
474 |
+
{z β Di
|
475 |
+
n | z β€ Οj
|
476 |
+
n and (z β€Β· x β€ Οj
|
477 |
+
n β x /β Di
|
478 |
+
n)}.
|
479 |
+
Now by lemma 2.6, if z β Di
|
480 |
+
n and x is any element such that z β€Β· x β€ Οj
|
481 |
+
n and x /β Di
|
482 |
+
n, then
|
483 |
+
x = Οk
|
484 |
+
n for some k (necessarily smaller than or equal to j). It implies that
|
485 |
+
οΏ½
|
486 |
+
i
|
487 |
+
{z β Di
|
488 |
+
n | z β€ Οj
|
489 |
+
n} =
|
490 |
+
οΏ½
|
491 |
+
i
|
492 |
+
{z β Di
|
493 |
+
n | z β€ Οj
|
494 |
+
n and z β€Β· Οk
|
495 |
+
n for some k β€ j}.
|
496 |
+
By lemma 2.7, we have that
|
497 |
+
οΏ½
|
498 |
+
i
|
499 |
+
{z β Di
|
500 |
+
n | z β€ Οj
|
501 |
+
n and z β€Β· Οk
|
502 |
+
n for some k β€ j}
|
503 |
+
has to be an element of the set {z β Di
|
504 |
+
n | z β€ Οj
|
505 |
+
n and z β€Β· Οk
|
506 |
+
n for some k β€ j}, hence that
|
507 |
+
it is in particular a left-divisor of Οj
|
508 |
+
n (and hence of y).
|
509 |
+
Now assume that u β€ x, y. We can assume that u β Di
|
510 |
+
n, otherwise by lemma 2.6 we
|
511 |
+
have u < Οi
|
512 |
+
n β€ x β§ y. As u β€ y, we have that u β€ οΏ½
|
513 |
+
i{z β Di
|
514 |
+
n | z β€ y}. And hence, that
|
515 |
+
u β€ x β§i
|
516 |
+
οΏ½οΏ½
|
517 |
+
i{z β Di
|
518 |
+
n | z β€ y}
|
519 |
+
οΏ½
|
520 |
+
= x β§ y.
|
521 |
+
β‘
|
522 |
+
3. SchrΓΆder trees and words for the Garside element
|
523 |
+
3.1. labelling of SchrΓΆder trees. A rooted plane tree is a tree embedded in the plane
|
524 |
+
with one distinguished vertex called the root. The vertices of degree 1 are called the leaves
|
525 |
+
of the tree and the other vertices are called inner vertices. One can consider rooted trees
|
526 |
+
as directed graphs by orienting the edges from the root toward the leaves. If there is an
|
527 |
+
oriented edge from a vertex v to a vertex w, we say that v is the parent of w and w is a
|
528 |
+
child of v. As can be seen in Figure 2, we draw the trees with their root on the top and the
|
529 |
+
|
530 |
+
8
|
531 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
532 |
+
leaves on the bottom. The planar embedding induces a total ordering (from left to right)
|
533 |
+
on the children of each vertex, hence we can speak about the leftmost child of a vertex.
|
534 |
+
Alternatively one has a useful recursive deο¬nition of a rooted plane tree: it is either the
|
535 |
+
empty tree with no inner vertex and a single leaf or a tuple T = (r, Tr) where r is the root
|
536 |
+
vertex and Tr is an ordered list of rooted plane trees. If T is a tree with the ο¬rst deο¬nition,
|
537 |
+
the vertex r is its root and the list Tr is the list of subtrees, ordered from left to right,
|
538 |
+
obtained by removing the root r and all the edges adjacent to r in T.
|
539 |
+
Deο¬nition 3.1.
|
540 |
+
(1) A SchrΓΆder tree is a rooted plane tree in which each inner vertex has at least two
|
541 |
+
children.
|
542 |
+
(2) A binary tree is a rooted plane tree in which each inner vertex has exactly two
|
543 |
+
children.
|
544 |
+
(3) The size of a tree is its number of leaves.
|
545 |
+
(4) The height of a tree is the number of vertices in a maximal chain of descendants.
|
546 |
+
(5) The SchrΓΆder tree on n leaves in which every child of the root is a leaf is called
|
547 |
+
the SchrΓΆder bush. We denote it by Ξ΄n.
|
548 |
+
(6) The SchrΓΆder tree given by the binary tree in which every right child (resp. every
|
549 |
+
left child) is a leaf is called a left comb (resp. a right comb).
|
550 |
+
Β· Β· Β·
|
551 |
+
Figure 2. From left to right: the unique SchrΓΆder tree with 1 leaf, the unique
|
552 |
+
SchrΓΆder tree with two leaves, the three SchrΓΆder trees with 3 leaves.
|
553 |
+
Then the
|
554 |
+
SchrΓΆder bush and on its right a left comb.
|
555 |
+
The SchrΓΆder trees are counted by the so-called little SchrΓΆder numbers. The sequence
|
556 |
+
starts with 1, 1, 3, 11, 45, 197, 903, 4279, 20793, ... and is referred as A001003 in [11].
|
557 |
+
We will label (and read the labels of) the vertices and the leaves of our trees using the
|
558 |
+
so-called post-order traversal. This is a recursive algorithm that visits each vertex and leaf
|
559 |
+
of the tree exactly once. Concretely, if T =
|
560 |
+
οΏ½
|
561 |
+
r, (T1, . . . , Tk)
|
562 |
+
οΏ½
|
563 |
+
is a rooted planar tree, then
|
564 |
+
we recursively apply the algorithm to T1, T2 until Tk and ο¬nally we visit the root r. When
|
565 |
+
the algorithm meets an empty tree it visits its leaf and then, the recursion stops and it
|
566 |
+
goes up one level in the recursive process. The ο¬rst vertex visited by the algorithm is the
|
567 |
+
leftmost leaf of T, then the algorithm moves to its parent v (but does not visit v) and visits
|
568 |
+
the second subtree of v starting with the leftmost leaf and so on. We refer to Figure 3 for
|
569 |
+
an illustration where the ο¬rst vertex visited by the algorithm is labeled by 1, the second
|
570 |
+
by 2 and so on. The last vertex visited by the algorithm is always the root of T. Let m, n
|
571 |
+
be two integers such that m β₯ n β 1. We then label a SchrΓΆder tree T with n β₯ 2 leaves
|
572 |
+
by labelling its vertices one after the other with respect to the total order deο¬ned by the
|
573 |
+
post-order traversal, using the following rules:
|
574 |
+
(1) Let v be the leftmost child of a vertex w. Then w is the root of a SchrΓΆder tree
|
575 |
+
οΏ½
|
576 |
+
w, (T1, Β· Β· Β· , Tk)
|
577 |
+
οΏ½
|
578 |
+
and v is the root of T1. The label Ξ»(v) of v is equal to the number
|
579 |
+
of leaves of the forest consisting of all the trees T2, Β· Β· Β· , Tk.
|
580 |
+
|
581 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
582 |
+
9
|
583 |
+
20
|
584 |
+
11
|
585 |
+
3
|
586 |
+
1
|
587 |
+
2
|
588 |
+
6
|
589 |
+
4
|
590 |
+
5
|
591 |
+
10
|
592 |
+
7
|
593 |
+
8
|
594 |
+
9
|
595 |
+
12
|
596 |
+
19
|
597 |
+
15
|
598 |
+
13 14
|
599 |
+
18
|
600 |
+
16 17
|
601 |
+
Figure 3. Post-order traversal of a SchrΓΆder tree of size 12.
|
602 |
+
(2) If v is not the leftmost child of a vertex of T, we consider LD(v) the set of its
|
603 |
+
leftmost descendants consisting of the leftmost child of v and its leftmost child and
|
604 |
+
so one. Then the label of v is m β οΏ½
|
605 |
+
wβLD(v) Ξ»(w). Note that using the post-order
|
606 |
+
traversal, the label of the leftmost descendants of a vertex v are already determined
|
607 |
+
when we visit v.
|
608 |
+
The result is a labelled SchrΓΆder tree that we denote by Lm(T).
|
609 |
+
This procedure is
|
610 |
+
illustrated in Figure 4.
|
611 |
+
Deο¬nition 3.2. Let Lm(T) be a labelled SchrΓΆder tree with n leaves labelled by m β₯ nβ1.
|
612 |
+
The sum of the labels of the vertices of T is called its weight (with respect to m).
|
613 |
+
Lemma 3.3. Let T be a SchrΓΆder tree with n leaves and m β₯ n β 1. Then the integers
|
614 |
+
labelling Lm(T) are strictly nonnegative with the exception of the root which may be labelled
|
615 |
+
by 0.
|
616 |
+
Proof. If a vertex is a leftmost child, then its label is a number of leaves, hence it is positive.
|
617 |
+
If v is not a leftmost child, then it is labelled by m β οΏ½
|
618 |
+
wβLD(v) Ξ»(w). Each Ξ»(w) is equal
|
619 |
+
to a certain number of leaves of T and the set of leaves associated to distinct vertices of
|
620 |
+
LD(v) do not intersect. Moreover, exactly one element of LD(v) is a leaf and this leaf is
|
621 |
+
not counted in οΏ½
|
622 |
+
wβLD(v) Ξ»(w). We therefore have
|
623 |
+
(3.1)
|
624 |
+

|
625 |
+
ο£
|
626 |
+
οΏ½
|
627 |
+
wβLD(v)
|
628 |
+
Ξ»(w)
|
629 |
+
ο£Ά
|
630 |
+
ο£Έ + 1 β€ n,
|
631 |
+
hence m β οΏ½
|
632 |
+
wβLD(v) Ξ»(w) β₯ 0. Moreover if οΏ½
|
633 |
+
wβLD(v) Ξ»(w) = m, then by (3.1) we must
|
634 |
+
have m = n β 1. It follows that v has n descendants since the leftmost leaf which is a
|
635 |
+
descendant of v is not counted, hence v is the root of T.
|
636 |
+
β‘
|
637 |
+
This labelling is almost determined by the recursive structure of the tree, as shown by
|
638 |
+
the following result.
|
639 |
+
Lemma 3.4. Let T =
|
640 |
+
οΏ½
|
641 |
+
r, (T1, . . . , Tk)
|
642 |
+
οΏ½
|
643 |
+
be a SchrΓΆder tree and v be a vertex of Ti for
|
644 |
+
i β {1, . . . , k}. Then,
|
645 |
+
(1) If v is not the root of T1, then its label in Lm(T) is equal to its label in Lm(Ti).
|
646 |
+
(2) If v is the root of T1, then its label in Lm(T1) is equal to the sum of the labels of v
|
647 |
+
and of the root of T in Lm(T).
|
648 |
+
Proof. Let v be a vertex of Ti. If v is a leftmost child in T which is not the root of T1,
|
649 |
+
then its label is a number of leaves of a certain forest which is contained in Ti. Hence this
|
650 |
+
number is the same in the big tree T or in the extracted tree Ti. If v is not a leftmost
|
651 |
+
child, then its label is determined by the labels of its leftmost descendants, hence it is the
|
652 |
+
same in the tree T as in the extracted tree Ti since we have just shown that the labels of
|
653 |
+
leftmost descendants which are not the root of T1 agree. The root of T1 has a diο¬erent
|
654 |
+
behaviour since in T it is a leftmost child and this is not the case in T1. Hence if v is the
|
655 |
+
root of T1, denoting by Ξ»1 the label of v in T1, we have Ξ»1(v) = m β οΏ½
|
656 |
+
wβLD(v) Ξ»1(w). The
|
657 |
+
|
658 |
+
10
|
659 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
660 |
+
labels of the descendants of v are the same in T and in T1, that is, we have Ξ»1(w) = Ξ»(w)
|
661 |
+
for all w β LD(v). In T, the label of the root r is given by
|
662 |
+
Ξ»(r) = m β Ξ»(v) β
|
663 |
+
οΏ½
|
664 |
+
wβLD(v)
|
665 |
+
Ξ»(w) = m β Ξ»(v) β
|
666 |
+
οΏ½
|
667 |
+
wβLD(v)
|
668 |
+
Ξ»1(w).
|
669 |
+
Hence we have Ξ»(r) + Ξ»(v) = Ξ»1(v).
|
670 |
+
β‘
|
671 |
+
3.2. Words for the Garside element in terms of SchrΓΆder trees. Reading the la-
|
672 |
+
belled tree Lm(T) using the post-order traversal and associating the generator Οi to the
|
673 |
+
letter i with the convention that Ο0 = e, gives a map Ξ¦m from the set of SchrΓΆder trees
|
674 |
+
labelled by m to the set Sβ of words for the elements of the monoid Mm. We refer to
|
675 |
+
Figure 4 for an illustration.
|
676 |
+
0
|
677 |
+
5
|
678 |
+
5
|
679 |
+
1
|
680 |
+
11
|
681 |
+
10
|
682 |
+
1
|
683 |
+
11
|
684 |
+
9
|
685 |
+
2
|
686 |
+
11 11
|
687 |
+
11
|
688 |
+
8
|
689 |
+
2
|
690 |
+
1
|
691 |
+
11
|
692 |
+
10
|
693 |
+
1
|
694 |
+
11
|
695 |
+
Figure
|
696 |
+
4. Example
|
697 |
+
of
|
698 |
+
the
|
699 |
+
labelling
|
700 |
+
of
|
701 |
+
a
|
702 |
+
SchrΓΆder
|
703 |
+
tree
|
704 |
+
of
|
705 |
+
size
|
706 |
+
12
|
707 |
+
with
|
708 |
+
m
|
709 |
+
=
|
710 |
+
11.
|
711 |
+
The
|
712 |
+
corresponding
|
713 |
+
element
|
714 |
+
in
|
715 |
+
the
|
716 |
+
monoid
|
717 |
+
M11
|
718 |
+
is
|
719 |
+
Ο1Ο11Ο5Ο1Ο11Ο10Ο2Ο11Ο11Ο9Ο5Ο11Ο1Ο11Ο2Ο1Ο11Ο10Ο8.
|
720 |
+
Deο¬nition 3.5. Let T be a non-empty SchrΓΆder tree. If T has a subtree T1 satisfying the
|
721 |
+
three following properties:
|
722 |
+
(1) The root r1 of T1 is not the root of T, hence it has a parent r0 which has at least
|
723 |
+
two children,
|
724 |
+
(2) The root r1 has exactly two children,
|
725 |
+
(3) The right subtree of T1 is the empty tree with only one leave.
|
726 |
+
Then, we can construct another tree οΏ½T by contracting the edge r0 β r1, in other words by
|
727 |
+
removing the root r1 of T1 and attaching the two subtrees of T1 to r0. See Figure 5 for an
|
728 |
+
illustration. We call such a transformation, or the inverse transformation, a local move.
|
729 |
+
Note that, since r0 has at least two children in the conο¬guration described above (see also
|
730 |
+
the left picture in Figure 5), we get that r0 has at least three children in the conο¬guration
|
731 |
+
obtained after applying the local move. In particular, to apply a local move in the other
|
732 |
+
direction, we need to have a SchrΓΆder tree οΏ½T with a subtree T1 satisfying :
|
733 |
+
(1) The parent r0 of T1 (which is allowed to be the root of T) has at least three children,
|
734 |
+
(2) The tree T1 is not the last child of r0, and is directly followed by an empty tree
|
735 |
+
with only one leaf.
|
736 |
+
r0
|
737 |
+
Sk
|
738 |
+
r1
|
739 |
+
r2
|
740 |
+
A1
|
741 |
+
Sk+2
|
742 |
+
ββ
|
743 |
+
r0
|
744 |
+
Sk
|
745 |
+
r2
|
746 |
+
A1
|
747 |
+
Sk+2
|
748 |
+
Figure 5. Local move.
|
749 |
+
|
750 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
751 |
+
11
|
752 |
+
Formally, if the subtree of T with root r0 is S =
|
753 |
+
οΏ½
|
754 |
+
r0, (S1, Β· Β· Β· , Sk, T1, Sk+2, Β· Β· Β· , Sr)
|
755 |
+
οΏ½
|
756 |
+
and
|
757 |
+
the subtree T1 is
|
758 |
+
οΏ½
|
759 |
+
r1, (A1, A2)
|
760 |
+
οΏ½
|
761 |
+
, then we obtain the tree οΏ½T by replacing S by
|
762 |
+
οΏ½
|
763 |
+
r0, (S1, Β· Β· Β· , Sk, A1, A2, Sk+2, Β· Β· Β· , Sr)
|
764 |
+
οΏ½
|
765 |
+
.
|
766 |
+
Lemma 3.6. Let T and S be two SchrΓΆder trees with n leaves. Then one can pass from
|
767 |
+
the tree T to the tree S by applying a sequence of local moves.
|
768 |
+
Proof. It is enough to show that T can be transformed into the SchrΓΆder bush Ξ΄nβrecall
|
769 |
+
that this is the SchrΓΆder tree in which every child of the root is a leafβby a sequence of
|
770 |
+
local moves. The SchrΓΆder tree S can then be transformed as well into Ξ΄n, and hence T
|
771 |
+
can be transformed into S. We argue by induction on the number of leaves. For n = 1 and
|
772 |
+
n = 2 there is nothing to prove. If T has a subtree S which is not of the form Ξ΄k, then we
|
773 |
+
can transform S into Ξ΄k for some k by applying the induction hypothesis to S. Hence we
|
774 |
+
can assume that T =
|
775 |
+
οΏ½
|
776 |
+
r, (Ξ΄n1, Β· Β· Β· , Ξ΄nk)
|
777 |
+
οΏ½
|
778 |
+
with οΏ½ ni = n. If T is not equal to Ξ΄n, then it has
|
779 |
+
at least a non-empty subtree S. If S has only two leaves, then we can apply a local move to
|
780 |
+
remove its root and to attach the two leaves to the root of T. If it has more than 3 leaves,
|
781 |
+
by induction there is a sequence of local moves from S to a left comb. Then, by repeatedly
|
782 |
+
applying a local move at the root of the left comb, we remove all the inner vertices of the
|
783 |
+
left comb and attach all its leaves to the root of T. Applying this to all subtrees S of T
|
784 |
+
wich are not empty, we end up getting Ξ΄n.
|
785 |
+
β‘
|
786 |
+
Lemma 3.7. Let T be a SchrΓΆder tree with n leaves and m β₯ n β 1. Then Ξ¦m(T) is a
|
787 |
+
word for Οnβ1(Οm)nβ1Οmβn+1 in Mm. In particular if m = n β 1, then it is a word for the
|
788 |
+
Garside element of Mnβ1.
|
789 |
+
Proof. If T = Ξ΄n is the SchrΓΆder tree with only one root and n leaves, then Ξ¦m(T) =
|
790 |
+
Οnβ1(Οm)nβ1Οmβn+1. If T is another SchrΓΆder tree, then by Lemma 3.6 there is a sequence
|
791 |
+
of local moves from T to Ξ΄n. To ο¬nish the proof it is enough to show that applying a local
|
792 |
+
move to a SchrΓΆder tree T amounts to applying a relation of the monoid Mm to Ξ¦m(T).
|
793 |
+
This is easily obtained by staring at Figure 5.
|
794 |
+
Indeed, if T is the tree at the left of Figure 5, then the label of r2 is 1, the label of the
|
795 |
+
leaf on its right is m and the label of r1 is a certain integer β. Since r1 is not the root of
|
796 |
+
T, we have 1 β€ β. Moreover, since r1 is not a leaf of T, we have β < m. Hence in Ξ¦m(T)
|
797 |
+
we have the factor Ο1ΟmΟβ with 1 β€ β β€ m β 1.
|
798 |
+
If οΏ½T denotes the right tree of Figure 5, then the label of r2 is β + 1. Indeed r2 is a
|
799 |
+
leftmost child in οΏ½T if and only if r1 is a leftmost child in T. In this case its label is the
|
800 |
+
number of leaves of the forest in its right and in οΏ½T there is precisely one more leaf in this
|
801 |
+
forest than in T. In the other case, the label of r2 in οΏ½T is m β οΏ½
|
802 |
+
wβLD(r2) Ξ»(w). The label
|
803 |
+
of r1 is β = m β 1 β οΏ½
|
804 |
+
wβLD(r2) Ξ»(w). So the label of r2 is β + 1. The leaf on the right of r2
|
805 |
+
in οΏ½T is labelled by m, hence Ξ¦m( οΏ½T) is obtained by replacing Ο1ΟmΟβ in Ξ¦m(T) by Οβ+1Οm,
|
806 |
+
and vice-versa.
|
807 |
+
β‘
|
808 |
+
Proposition 3.8. For m = nβ1, the map Ξ¦m from the set of SchrΓΆder trees with n leaves
|
809 |
+
to the set of words for Οn
|
810 |
+
nβ1 in Mnβ1 is surjective.
|
811 |
+
Proof. We have to show that to each word y for Οn
|
812 |
+
nβ1 β Mnβ1, we can attach a SchrΓΆder
|
813 |
+
tree T with n leaves, in such a way that Ξ¦m(T) = y. The word y and the word Οn
|
814 |
+
nβ1
|
815 |
+
can be transformed into each other by applying a sequence of deο¬ning relations of Mm.
|
816 |
+
We already know that the word Οn
|
817 |
+
nβ1 is in the image of Ξ¦m since it is the image of the
|
818 |
+
SchrΓΆder bush. To conclude the proof, we therefore need to show the following claim: given
|
819 |
+
a SchrΓΆder tree S, if the corresponding labelling has a substring of the form 1mβ (resp.
|
820 |
+
(β + 1)m) with 1 β€ β β€ m β 1, then we are necessarily in the conο¬guration of the left
|
821 |
+
|
822 |
+
12
|
823 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
824 |
+
picture in Figure 5 (resp. the right picture), and hence we can apply a local move. Indeed,
|
825 |
+
as one can pass from the word Οn
|
826 |
+
nβ1 to the word y by a sequence of deο¬ning relations
|
827 |
+
let y0 = Οn
|
828 |
+
nβ1, y1, . . . , yk = y be expressions of Οn
|
829 |
+
nβ1 such that yi is obtained from yiβ1
|
830 |
+
by applying a single relation in Mm.
|
831 |
+
Applying the relation on y0 = Ξ¦m(T) to get y1
|
832 |
+
corresponds to applying a local move on T to get a SchrΓΆder tree T1 and as seen in the
|
833 |
+
proof of lemma 3.7, we get Ξ¦m(T1) = y1.
|
834 |
+
To show the claim, assume that S is a SchrΓΆder tree with labelling having a substring
|
835 |
+
of the form 1mβ with 1 β€ β β€ m β 1. Note that m can only be the label of a leaf. Let v be
|
836 |
+
the parent of that leaf. It is a root of a family of trees, say (v, T1, . . . , Tk) and our leaf with
|
837 |
+
label m corresponds to one of the trees Ti (which has to be empty). It is clear that such a
|
838 |
+
tree cannot be T1: indeed, as T1 is the leftmost child of v, in that case m = n β 1 would
|
839 |
+
be the number of leafs in the forest T2, . . . , Tk, which is at most n β 1. As m = n β 1,
|
840 |
+
the only possibility would be that v is the root of S, hence m would be the ο¬rst label and
|
841 |
+
therefore could not be preceded by a label 1. Hence m labels one of the trees T2, . . . , Tk,
|
842 |
+
say Ti. It follows that the label 1 preceding m is the label of the root of Tiβ1. If i = 2
|
843 |
+
then k = 2 as the label 1 is then the label of the leftmost child of v, meaning that there is
|
844 |
+
only one leaf in the forest T2, . . . , Tk. In that case, it only remains to show that v cannot
|
845 |
+
be the root of S to match the conο¬guration in the left picture of Figure 5. But this is
|
846 |
+
clear for if v was the root of S, the last label would be m corresponding to T2, hence
|
847 |
+
no β could appear. Hence v is not the root of S, and its label is β. Now if i ΜΈ= 2, then
|
848 |
+
i β 1 ΜΈ= 1. The root vβ² of Tiβ1 is labelled by 1 and as vβ² is not the leftmost child of v, we
|
849 |
+
have 1 = Ξ»(vβ²) = mβοΏ½
|
850 |
+
wβLD(vβ²) Ξ»(w), yielding οΏ½
|
851 |
+
wβLD(vβ²) Ξ»(w) = mβ1. This means that
|
852 |
+
there are m leaves in Tiβ1, and as there is one leaf in Ti and m = nβ1, the only possibility
|
853 |
+
is that i β 1 = 1 and k = 2, contradicting i ΜΈ= 2.
|
854 |
+
Now, assume that S is a SchrΓΆder tree with labelling having a substring of the form
|
855 |
+
(β + 1)m with 1 β€ β β€ m β 1. Again, m can only label a leaf. Let v be the parent of that
|
856 |
+
leaf as above, which is a root of a family T1, . . . , Tk of trees with Ti corresponding to our
|
857 |
+
leaf for some i. We need to show that i ΜΈ= 1 and k β₯ 3. In the previous case we have seen
|
858 |
+
that if i = 1, then m = n β 1 is the number of leaves in T2, . . . , Tk, forcing v to be the root
|
859 |
+
of S and m to be the ο¬rst label in S. Hence i β₯ 2. If k = 2 (hence i = 2), then the root of
|
860 |
+
T1 is labelled by 1 = β + 1, contradicting 1 β€ β. Hence k β₯ 2.
|
861 |
+
β‘
|
862 |
+
Lemma 3.9.
|
863 |
+
(1) Let T be a SchrΓΆder tree with n leaves labelled by m β₯ n β 1. Then,
|
864 |
+
the weight of T is nm.
|
865 |
+
(2) Let w be a vertex of T which is not a leaf and v its leftmost child, that is w is the
|
866 |
+
root of a SchrΓΆder tree
|
867 |
+
οΏ½
|
868 |
+
w, (T1, Β· Β· Β· , Tk)
|
869 |
+
οΏ½
|
870 |
+
and v is the root of T1. Then the weight of
|
871 |
+
the forest F = (T2, Β· Β· Β· , Tk) attached to w is Ξ»(v)m, and the labelling of of a vertex
|
872 |
+
in a tree Ti for i β₯ 2 is the same as its labelling inside T.
|
873 |
+
Proof. The ο¬rst result is proved by induction on the number of leaves. If the tree has one
|
874 |
+
leaf the result holds by deο¬nition of our labelling. Let T = (r, T1, Β· Β· Β· , Tk) be a SchrΓΆder
|
875 |
+
tree, where Ti has ni leaves. By induction, the tree Ti has weight mni for i β₯ 1. Us-
|
876 |
+
ing Lemma 3.4, the sum of the labels of the vertices of the tree Ti (in T) is equal to mni
|
877 |
+
for i β₯ 2 and the sum of the labels of the vertices of T1 and of the root of T is equal to
|
878 |
+
mn1. Hence, the tree T has weight οΏ½k
|
879 |
+
i=1 mni = mn. For the second point, the number of
|
880 |
+
leaves of the forest F is equal to Ξ»(v). Hence by the ο¬rst point, the forest F has weight
|
881 |
+
Ξ»(v)m.
|
882 |
+
β‘
|
883 |
+
Proposition 3.10. Let m β₯ n β 1. Then the map Ξ¦m from the set of SchrΓΆder trees with
|
884 |
+
n leaves to the set of words for the element Οnβ1(Οm)nβ1Οmβn+1 β Mm is injective.
|
885 |
+
|
886 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
887 |
+
13
|
888 |
+
Proof. Let T = (r, T1, Β· Β· Β· , Tk) be a SchrΓΆder tree with n leaves labelled by m. This proof is
|
889 |
+
purely combinatorial and it only involves the word W in N obtained by reading the labels
|
890 |
+
of the tree in post-order. The ο¬rst step of the proof is to remark that one can recover the
|
891 |
+
decomposition βroot and list of subtreesβ of a SchrΓΆder tree just by looking at W. We will
|
892 |
+
illustrate the algorithm in Example 3.11 below. Precisely we want to split the word W
|
893 |
+
into a certain number of factors W = W1 Β· Β· Β· Wk such that each subword Wi is equal to the
|
894 |
+
word obtained by reading the labels of Lm(Ti) in post-order.
|
895 |
+
The ο¬rst letter w1 of W is the label of the leftmost leaf r1 of T and by induction we will
|
896 |
+
ο¬nd the letters w2, w3, Β· Β· Β· , wi corresponding to the ancestors r2, r3, Β· Β· Β· , ri of r1. Since the
|
897 |
+
labels of these vertices count a number of leaves of T, when οΏ½i
|
898 |
+
j=1 wj = n β 1, then all the
|
899 |
+
leaves of T have been counted so ri is the root of T1 and we stop the induction.
|
900 |
+
If we have found the letter wk corresponding to rk ΜΈ= r, then wk is the number of leaves
|
901 |
+
of the right forest attached to the parent rk+1 of rk. By Lemma 3.9, the weight of F is
|
902 |
+
m Β· wk, hence the word obtained by reading the vertices of F is wk+1 Β· Β· Β· wi where i is the
|
903 |
+
smallest integer such that οΏ½i
|
904 |
+
j=k+1 wj = mwk. All these letters correspond to the vertices
|
905 |
+
of F, hence the next letter is the label of the vertex read after F in the post-order traversal,
|
906 |
+
which is the vertex rk+1.
|
907 |
+
Since the word W only contains strictly non-negative integers (except possibly the label
|
908 |
+
of the root of T), at each step of the induction the value w1 +Β· Β· Β·+wi strictly increases and
|
909 |
+
the induction stops. If wn1 is the letter corresponding to the leftmost child of the root of T,
|
910 |
+
then the word w1 Β· Β· Β· wn1 is the word obtained by reading all the vertices of the subtree T1.
|
911 |
+
By Lemma 3.4, this is almost the word obtained by reading Lm(T1) we just need to βcorrectβ
|
912 |
+
the label of the root of T1 by adding the label of the root of T which is the last letter wl of
|
913 |
+
W. To conclude the word consisting of the labels of T1 is WT1 = w1 Β· Β· Β· wn1β1(wn1 + wl).
|
914 |
+
Let οΏ½
|
915 |
+
W be the word obtained by removing the letters w1, Β· Β· Β· , wn1 and wl. We use the
|
916 |
+
same procedure to extract the subwords corresponding to the other subtrees of T. Due
|
917 |
+
to the asymmetry of Lemma 3.4, there is a slight diο¬erence. We have found all the labels
|
918 |
+
w1, w2, Β· Β· Β· , wt of the vertices r1, r2, Β· Β· Β· , rt of the left branch of Ti when οΏ½t
|
919 |
+
j=1 wj = m and
|
920 |
+
there is no need to βcorrectβ the word as above.
|
921 |
+
We are now ready to prove that Ξ¦m is injective.
|
922 |
+
If the words of two trees T =
|
923 |
+
(r, (T1, Β· Β· Β· , Tk)) and S = (s, (S1, Β· Β· Β· , Sl)) obtained by reading the labels of their vertices
|
924 |
+
in post-order are equal, then by the discussion above we have k = l and for i β {1, Β· Β· Β· , k},
|
925 |
+
the words obtained by reading the vertices of the subtrees Lm(Ti) and Lm(Si) are equal.
|
926 |
+
By induction on the number of leaves, we have Si = Ti for i = 1, Β· Β· Β· , k and we get that
|
927 |
+
T = S.
|
928 |
+
β‘
|
929 |
+
Example 3.11. We illustrate the decomposition involved in the proof of Proposition 3.10
|
930 |
+
with the example of Figure 4. We consider the leftmost subtree T1 of T with n = 7 leaves
|
931 |
+
and which is labelled by m = 11. We have Ξ¦11(T1) = Ο1Ο11Ο5Ο1Ο11Ο10Ο2Ο11Ο11Ο9Ο5. The
|
932 |
+
ο¬rst letter 1 tels us that the forest on the right of the leftmost leaf r1 has 1 vertex. Its
|
933 |
+
weight is m = 11. Hence Ο11 labels the only vertex of the forest and the next letter 5
|
934 |
+
corresponds to the parent r2 of r1. Since 1 + 5 = 6 we know that it is the leftmost child
|
935 |
+
of the root. Hence the word Ο1Ο11Ο5 is obtained by reading the vertices of the leftmost
|
936 |
+
subtree S of T1. We apply the βcorrectionβ and we get Ο1Ο11Ο10 = Ξ¦11(S). The rest of the
|
937 |
+
word Ο1Ο11Ο10Ο2Ο11Ο11Ο9 corresponds to the other subtrees of T1 and it splits as Ο1Ο11Ο10
|
938 |
+
and Ο2Ο11Ο11Ο9.
|
939 |
+
Combining Proposition 3.8 and Proposition 3.10 we get our main result of the section:
|
940 |
+
Theorem 3.12. For m = n β 1, the map Ξ¦m from the set of SchrΓΆder trees with n leaves
|
941 |
+
to the set of words for Οn
|
942 |
+
nβ1 in Mnβ1 is bijective.
|
943 |
+
|
944 |
+
14
|
945 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
946 |
+
Corollary 3.13. The following two graphs are isomorphic under Ξ¦nβ1:
|
947 |
+
(1) The graph of words for Οn
|
948 |
+
nβ1 in Mnβ1, where vertices are given by expressions of
|
949 |
+
Οn
|
950 |
+
nβ1 and there is an edge between two expressions whenever they diο¬er by applica-
|
951 |
+
tion of a single relation,
|
952 |
+
(2) The graph of SchrΓΆder trees with n leaves, where vertices are given by SchrΓΆder trees
|
953 |
+
and there is an edge between two trees whenever they diο¬er by application of a local
|
954 |
+
move.
|
955 |
+
Proof. The previous theorem gives the bijection between the sets of vertices. The proof of
|
956 |
+
Lemma 3.7 shows that whenever one can apply a local move, one can apply a relation on
|
957 |
+
the corresponding words. The proof of Proposition 3.8 shows that whenever one can apply
|
958 |
+
a relation on words, a local move can be applied on the corresponding trees.
|
959 |
+
β‘
|
960 |
+
We illustrate the situation for M3 in Figure 6 below.
|
961 |
+
Corollary 3.14. The number of words for the Garside element of Mn is a little SchrΓΆder
|
962 |
+
number A001003 [11].
|
963 |
+
Lemma 3.15. Let T = (r, S1, Β· Β· Β· , Sk) be a SchrΓΆder tree with n leaves labelled by m = nβ1.
|
964 |
+
Then, the word obtained by reading all the labels of a subtree Sj is a word for Οljn where lj
|
965 |
+
is the number of leaves of Sj.
|
966 |
+
Proof. Let us assume that the tree Sj has s + 1 leaves. By Lemma 3.7, the labels of the
|
967 |
+
subtree Sj is a word for ΟsΟs
|
968 |
+
nβ1Οnβ1βs. If s = 0, then we have a word for Οnβ1. Otherwise,
|
969 |
+
we can apply the relations [8, Lemma 4.5] with i = s and j = nβ1. Alternatively, using the
|
970 |
+
SchrΓΆder trees, it is easy to see that these relations comes from the following modiο¬cations
|
971 |
+
of the trees.
|
972 |
+
The word ΟsΟs
|
973 |
+
nβ1Οnβ1βs correspond to the case where the tree Sj is the
|
974 |
+
SchrΓΆder bush with s + 1 leaves. Using our local moves, we can modify it to the left comb.
|
975 |
+
The corresponding word is now (Ο1Οs)sΟnβ1βs. Now we can inductively apply the local
|
976 |
+
move to contract the edge between the root of T and the root left comb. The result is s+1
|
977 |
+
empty trees attached to the root of T and the corresponding word is Οs+1
|
978 |
+
nβ1.
|
979 |
+
β‘
|
980 |
+
Proposition 3.16. Let n β₯ 1. We have the following isomorphisms of posets:
|
981 |
+
(1) D0
|
982 |
+
n βΌ= Div(βnβ1), Dn+1
|
983 |
+
n
|
984 |
+
βΌ= Div(β0) = {β’},
|
985 |
+
(2) For all 1 β€ i β€ n, Di
|
986 |
+
n βΌ= Div(βnβi),
|
987 |
+
where every set is ordered by the restriction of the left-divisibility order in the monoid Mk
|
988 |
+
for suitable k.
|
989 |
+
Proof. We begin by proving the second statement. An element x of Di
|
990 |
+
n can be written in the
|
991 |
+
form Οi
|
992 |
+
nxβ², where xβ² is uniquely determined by cancellativity, and such that Οn is not a left-
|
993 |
+
divisor of xβ². In particular, there is y a divisor of βn such that Οi
|
994 |
+
nxβ²y = Οn+1
|
995 |
+
n
|
996 |
+
, and y ΜΈ= 1. We
|
997 |
+
associate a tree (or rather a family of trees) to x as follows. Write xβ² as a product a1a2 Β· Β· Β· aj
|
998 |
+
of elements of S. Complete the word Οi
|
999 |
+
na1a2 Β· Β· Β· aj to a word Οi
|
1000 |
+
na1a2 Β· Β· Β· ajb1b2 Β· Β· Β· bβ for βn,
|
1001 |
+
i.e., choose a word b1b2 Β· Β· Β· bβ for y.
|
1002 |
+
There are several possibilitΓ©s for the biβs, but the
|
1003 |
+
condition that x β Di
|
1004 |
+
n ensures that, writing the corresponding SchrΓΆder tree in the form
|
1005 |
+
(r, T1, T2, . . . , Ti, S1, S2, Β· Β· Β· Sd), where the i ο¬rst trees are empty trees with a single leaf,
|
1006 |
+
then a1a2 Β· Β· Β· aj has all its labels inside S1. Indeed, the labelling a1a2 Β· Β· Β· aj begins at the
|
1007 |
+
beginning (in the post-order convention) of the tree S1 since the trees T1, T2, . . . , Ti yield
|
1008 |
+
the label Οi
|
1009 |
+
n, and if another tree among S2, . . . , Sd was partly labelled by the aiβs, then a
|
1010 |
+
power of Οn would left-divide xβ², since the word obtained from S1 is a power of Οn (lemma
|
1011 |
+
3.15). It is then possible to reduce all the trees S1, S2, . . . , Sd to a single tree S still having
|
1012 |
+
the labelling a1, a2, . . . , aj at the beginning, by ο¬rst reducing S2, . . . , Sd to a set of empty
|
1013 |
+
|
1014 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
1015 |
+
15
|
1016 |
+
Ο2Ο1Ο3Ο2Ο1Ο3
|
1017 |
+
Ο3Ο1Ο3Ο2Ο3
|
1018 |
+
Ο3Ο1Ο3Ο1Ο3Ο1
|
1019 |
+
Ο3Ο2Ο3Ο3Ο1
|
1020 |
+
Ο2Ο3Ο3Ο1Ο3
|
1021 |
+
Ο3Ο3Ο3Ο3
|
1022 |
+
Ο3Ο3Ο1Ο3Ο2
|
1023 |
+
Ο3Ο2Ο1Ο3Ο2Ο1
|
1024 |
+
Ο1Ο3Ο1Ο3Ο1Ο3
|
1025 |
+
Ο1Ο3Ο2Ο3Ο3
|
1026 |
+
Ο1Ο3Ο2Ο1Ο3Ο2
|
1027 |
+
1
|
1028 |
+
2
|
1029 |
+
2
|
1030 |
+
1
|
1031 |
+
3
|
1032 |
+
3
|
1033 |
+
3
|
1034 |
+
2
|
1035 |
+
1
|
1036 |
+
3
|
1037 |
+
3
|
1038 |
+
3
|
1039 |
+
1
|
1040 |
+
1
|
1041 |
+
1
|
1042 |
+
3
|
1043 |
+
3
|
1044 |
+
3
|
1045 |
+
1
|
1046 |
+
2
|
1047 |
+
3
|
1048 |
+
3
|
1049 |
+
1
|
1050 |
+
2
|
1051 |
+
3
|
1052 |
+
3
|
1053 |
+
3
|
1054 |
+
3
|
1055 |
+
3
|
1056 |
+
3
|
1057 |
+
3
|
1058 |
+
3
|
1059 |
+
3
|
1060 |
+
2
|
1061 |
+
1
|
1062 |
+
3
|
1063 |
+
3
|
1064 |
+
1
|
1065 |
+
2
|
1066 |
+
2
|
1067 |
+
1
|
1068 |
+
3
|
1069 |
+
1
|
1070 |
+
1
|
1071 |
+
1
|
1072 |
+
3
|
1073 |
+
3
|
1074 |
+
3
|
1075 |
+
2
|
1076 |
+
1
|
1077 |
+
3
|
1078 |
+
3
|
1079 |
+
3
|
1080 |
+
2
|
1081 |
+
1
|
1082 |
+
3
|
1083 |
+
2
|
1084 |
+
1
|
1085 |
+
3
|
1086 |
+
Figure 6. Illustration of Corollary 3.13 for n = 4: the graph of reduced words for
|
1087 |
+
β3 and the isomorphic graph of SchrΓΆder trees on 4 = 3 + 1 leaves.
|
1088 |
+
trees οΏ½T2, . . . , οΏ½Tdβ² with single leafs, and then merging S1 and οΏ½T2 using a local move, then
|
1089 |
+
merging the resulting tree with οΏ½T3, and so on (see Figure 7 for an illustration). In this way
|
1090 |
+
we associate to x a SchrΓΆder tree of the form (r, T1, T2, . . . , Ti, S), where the Tkβs are empty
|
1091 |
+
trees with a single leaf, and the labelling corresponding to the chosen word a1a2 Β· Β· Β· aj is an
|
1092 |
+
initial section of the tree S (in fact, in algebraic terms, what we did is modify the word for
|
1093 |
+
y to get a suitable one yielding a unique tree after the empty trees). Note that by initial
|
1094 |
+
section we mean a preο¬x of the word obtained from the labelling of S read in post-order,
|
1095 |
+
where we exclude the label of the root, i.e., if the root has a label, then the preο¬x is strict.
|
1096 |
+
We denote by Sn,i the set of such SchrΓΆder trees, that is, those SchrΓΆder trees on n leaves
|
1097 |
+
with i + 1 child of the root, and such that the i ο¬rst child are leafs. Note that the tree
|
1098 |
+
that we attached to x depends on a choice of word for x, but applying a deο¬ning relation
|
1099 |
+
|
1100 |
+
16
|
1101 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
1102 |
+
in the word x corresponds to applying a local move in the tree S, and this cannot make S
|
1103 |
+
split into several trees since the root of S is frozen (its label corresponds to the last letter
|
1104 |
+
of y ΜΈ= 1). Hence we can apply all local moves with all labels in the (strict) initial section
|
1105 |
+
corresponding to a word for x, and we keep a SchrΓΆder tree on nβi+1 leaves. In this way,
|
1106 |
+
forgetting the i ο¬rst empty trees, what we attached to x is an equivalence class of a (strict)
|
1107 |
+
initial section of a SchrΓΆder tree on n β i + 1 leaves under local moves, that is, a divisor
|
1108 |
+
of βnβi. This mapping is injective since one can recover a word for x from the obtained
|
1109 |
+
SchrΓΆder tree on n β i + 1 leaves easily by mapping S to (r, T1, . . . , Ti, S), labelling such a
|
1110 |
+
tree, and reading the word obtained by reading the i ο¬rst empty trees and then the initial
|
1111 |
+
section.
|
1112 |
+
It remains to show that it is surjective. Hence consider an initial section of a SchrΓΆder
|
1113 |
+
tree S on n β i leaves. We must show that, in the tree (r, T1, T2, . . . , Ti, S), the initial
|
1114 |
+
section of S is a word a1a2 Β· Β· Β· aj which labels an element xβ² of D0
|
1115 |
+
n. Assume that Οn is a
|
1116 |
+
left-divisor of xβ². Then, using local moves only involving those labels in the initial section
|
1117 |
+
of S corresponding to a word for xβ², one can transform (r, T1, T2, . . . , Ti, S) into a tree of the
|
1118 |
+
form (r, T1, T2, . . . , Ti, Ti+1, Sβ²
|
1119 |
+
1, . . . , Sβ²
|
1120 |
+
e), i.e., S can be split into several trees, the ο¬rst one
|
1121 |
+
(corresponding to Οn) being an empty tree. This is a contradiction: to split S into several
|
1122 |
+
trees, one would need to apply a local move involving the root of S, which is frozen since
|
1123 |
+
the initial section does not cover the root. Hence xβ² β D0
|
1124 |
+
n, and our mapping is surjective.
|
1125 |
+
This completes the proof of the second point, as it is clear that our mappings preserve
|
1126 |
+
left-divisibility.
|
1127 |
+
For the ο¬rst point, we have Dn+1
|
1128 |
+
n
|
1129 |
+
= {Οn+1
|
1130 |
+
n
|
1131 |
+
}, hence there is nothing to prove. To show
|
1132 |
+
that D0
|
1133 |
+
n βΌ= Div(βnβ1), one proceeds in a similar way as in the proof of point 1. Let x β D0
|
1134 |
+
n
|
1135 |
+
and let y such that xy = βn. Choose words for x and y, and consider the corresponding
|
1136 |
+
SchrΓΆder tree T = (r, T1, . . . , Tk). Since x β D0
|
1137 |
+
n, the initial section of T corresponding to
|
1138 |
+
the word for x must be a proper initial section of T1. Using local moves on T2, . . . , Tk (which
|
1139 |
+
amounts to changing the word for y), we can ο¬nd a SchrΓΆder tree that is equivalent to T
|
1140 |
+
under local moves, and that is of the form (r, οΏ½T1, οΏ½T2), where οΏ½T1 still has the chosen word
|
1141 |
+
for x as a proper initial section, and οΏ½T2 is the empty tree with only one leaf. In particular
|
1142 |
+
οΏ½T1 is a SchrΓΆder tree on n leaf. Applying deο¬ning relations to words for x amounts to
|
1143 |
+
applying local moves inside the ο¬rst tree, and arguing as in the ο¬rst point this establishes
|
1144 |
+
the isomorphism of posets between D0
|
1145 |
+
n and Div(βnβ1).
|
1146 |
+
β‘
|
1147 |
+
r
|
1148 |
+
Ti
|
1149 |
+
S1
|
1150 |
+
οΏ½T1
|
1151 |
+
οΏ½T2
|
1152 |
+
ββ
|
1153 |
+
r
|
1154 |
+
Ti
|
1155 |
+
r1
|
1156 |
+
S1
|
1157 |
+
οΏ½T1
|
1158 |
+
οΏ½T2
|
1159 |
+
ββ
|
1160 |
+
r
|
1161 |
+
Ti
|
1162 |
+
r2
|
1163 |
+
r1
|
1164 |
+
S1
|
1165 |
+
οΏ½T1
|
1166 |
+
οΏ½T2
|
1167 |
+
=
|
1168 |
+
r
|
1169 |
+
Ti
|
1170 |
+
S
|
1171 |
+
Figure 7. Illustration for the proof of Proposition 3.16.
|
1172 |
+
|
1173 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
1174 |
+
17
|
1175 |
+
4. Enumerative results
|
1176 |
+
We have already seen (Theorem 3.12) that the words for Οn+1
|
1177 |
+
n
|
1178 |
+
are in bijection with
|
1179 |
+
SchrΓΆder trees on n+1 leaves. In this section, we give some additional enumerative results
|
1180 |
+
for several families of particular elements of Mn.
|
1181 |
+
4.1. Number of simple elements.
|
1182 |
+
Corollary 4.1. Let n β₯ 2, and let An := |Div(βn)|. Then
|
1183 |
+
An = 2A0 + 2Anβ1 +
|
1184 |
+
nβ2
|
1185 |
+
οΏ½
|
1186 |
+
i=1
|
1187 |
+
Ai.
|
1188 |
+
(4.1)
|
1189 |
+
It follows that An = F2n, where F0, F1, F2, . . . denotes the Fibonacci sequence 1, 2, 3, 5, 8, ...
|
1190 |
+
inductively deο¬ned by F0 = 1, F1 = 2, and Fi = Fiβ1 + Fiβ2 for all i β₯ 2. The sequence of
|
1191 |
+
the Ans is referred as A001906 in [11].
|
1192 |
+
Proof. The equality (4.1) follows immediately from the disjoint union Div(βn) = οΏ½
|
1193 |
+
0β€iβ€n+1 Di
|
1194 |
+
n
|
1195 |
+
and Proposition 3.16. We have A0 = F0, A1 = 3 = F2, and it is elementary to check that
|
1196 |
+
the inductive formula given by 4.1 is also satisο¬ed by the sequence F2n. This shows that
|
1197 |
+
An = F2n for all n β₯ 0.
|
1198 |
+
β‘
|
1199 |
+
Deο¬nition 4.2. We call the lattice (Div(βn), β€) the even Fibonacci lattice.
|
1200 |
+
4.2. Number of left-divisors of the lcm of the atoms and odd Fibonacci lat-
|
1201 |
+
tice. The set DivL(Οn
|
1202 |
+
n) of left-divisors of Οn
|
1203 |
+
n also forms a lattice under the restriction
|
1204 |
+
of left-divisibility, since it is an order ideal in the lattice (Div(βn), β€). In terms of the
|
1205 |
+
Garside monoid Mn, the element Οn
|
1206 |
+
n is both the left- and right-lcm of the generators
|
1207 |
+
S = {Ο1, Ο2, . . . , Οn} (see [8, Corollary 4.17]). For n β₯ 1 we set Bn := |DivL(Οn
|
1208 |
+
n)|.
|
1209 |
+
Lemma 4.3. We have Bn = F2nβ1 for all n β₯ 1. The sequence of the Bns is referred as
|
1210 |
+
A001519 in [11].
|
1211 |
+
Proof. Let x β Div(βn). We claim that x β DivL(Οn
|
1212 |
+
n) if and only if Οnx β Div(βn). Indeed,
|
1213 |
+
if x β€ Οn
|
1214 |
+
n, there is y β Mn such that xy = Οn
|
1215 |
+
n. We then have Οnxy = Οn+1
|
1216 |
+
n
|
1217 |
+
= βn, hence
|
1218 |
+
Οnx is a left-divisor of βn. Conversely, assume that Οnx β Div(βn). It follows that there
|
1219 |
+
is y β Div(βn) such that Οnxy = βn = Οn+1
|
1220 |
+
n
|
1221 |
+
. By cancellativity we get that xy = Οn
|
1222 |
+
n, hence
|
1223 |
+
x β DivL(Οn
|
1224 |
+
n).
|
1225 |
+
It follows that DivL(Οn
|
1226 |
+
n) is in bijection with the set
|
1227 |
+
{Οnx | x β Div(βn)} β© Div(βn).
|
1228 |
+
But this set is nothing but οΏ½
|
1229 |
+
1β€iβ€n+1 Di
|
1230 |
+
n. It follows that
|
1231 |
+
Bn = |Div(βn)| β |D0
|
1232 |
+
n| = |Div(βn)| β |Div(βnβ1)|,
|
1233 |
+
where the last equality follows from point (1) of Proposition 3.16. By Corollary 4.1 we
|
1234 |
+
thus get that
|
1235 |
+
Bn = An β Anβ1 = F2n β F2nβ2 = F2nβ1,
|
1236 |
+
which concludes the proof.
|
1237 |
+
β‘
|
1238 |
+
Deο¬nition 4.4. We call the lattice (DivL(Οn
|
1239 |
+
n), β€) the odd Fibonacci lattice.
|
1240 |
+
Both lattices for M3 are depicted in Figure 1.
|
1241 |
+
Remark 4.5. Note that the set of right-divisors of Οn
|
1242 |
+
n also has cardinality Bn: in fact, the
|
1243 |
+
two posets (DivL(Οn
|
1244 |
+
n), β€L) and (DivR(Οn
|
1245 |
+
n), β€R) are anti-isomorphic via x οΏ½β x, where x is
|
1246 |
+
the element of Mn such that xx = Οn
|
1247 |
+
n (this element is unique by right-cancellativity).
|
1248 |
+
|
1249 |
+
18
|
1250 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
1251 |
+
4.3. Number of words for the divisors of the Garside element.
|
1252 |
+
Lemma 4.6. Let T1 and T2 be two SchrΓΆder trees with n leaves labelled by m β₯ n β 1, and
|
1253 |
+
denote by m1 and m2 the corresponding words obtained by reading the labels in post-order.
|
1254 |
+
If the words m1 and m2 have a common preο¬x x1x2 Β· Β· Β· xl, then xi labels a leftmost child in
|
1255 |
+
T1 if and only if it labels a leftmost child in T2.
|
1256 |
+
Proof. We prove the result by induction on the number of leaves. If x1 Β· Β· Β· xl is obtained
|
1257 |
+
by reading all the vertices of T1 = (r, S1, Β· Β· Β· , Sk), then m1 = x1 Β· Β· Β· xl = m2 and by
|
1258 |
+
Proposition 3.10, we have T1 = T2, hence there is nothing to prove. Otherwise, let Sj be
|
1259 |
+
the ο¬rst subtree of T1 which is not covered by the word x1 Β· Β· Β· xl, similarly let Uk the ο¬rst
|
1260 |
+
subtree of T2 = (r, U1, Β· Β· Β· Uv) which is not covered by x1 Β· Β· Β· xl. Looking at the proof of
|
1261 |
+
Proposition 3.10, we see that the ο¬rst subtrees S1, Β· Β· Β· , Sjβ1 are completely determined by
|
1262 |
+
the word x1 Β· Β· Β· xl, hence we have j = k and Si = Ui for all i < k. Let xs be the letter of
|
1263 |
+
x1 Β· Β· Β· xl labelling the ο¬rst vertex of Sj. Let mβ²
|
1264 |
+
1 be the subword of m1 and mβ²
|
1265 |
+
2 the subword
|
1266 |
+
of m2 starting at the xs. As explained in the proof of Proposition 3.10, we can determine
|
1267 |
+
the subword mj
|
1268 |
+
1 of mβ²
|
1269 |
+
1 which correspond to Sj. The trees Uj and Sj do not need to have the
|
1270 |
+
same number of leaves. If one of the trees, say Uj, has less leaves, then one can apply local
|
1271 |
+
moves in the trees Uj, Uj+1, Β· Β· Β· Uv as in the proof of Proposition 3.16 in order to obtain
|
1272 |
+
a tree ΛUj with the same number of leaves as Sj. This will modify the word mβ²
|
1273 |
+
2, but not
|
1274 |
+
the preο¬x xs Β· Β· Β· xl, and xi labels a leftmost child in Uj if and only if it labels a leftmost
|
1275 |
+
child in ΛUj (see Figure 7 for an illustration). After doing the modiο¬cation, we consider the
|
1276 |
+
subword mj
|
1277 |
+
2 corresponding to the tree ΛUj and apply the induction hypothesis to mj
|
1278 |
+
1 and
|
1279 |
+
mj
|
1280 |
+
2.
|
1281 |
+
β‘
|
1282 |
+
Theorem 4.7. The set of words for the left-divisors of Οn+1
|
1283 |
+
n
|
1284 |
+
is in bijection with the set of
|
1285 |
+
SchrΓΆder trees with n + 2 leaves.
|
1286 |
+
Proof. Let us denote by sk the number of SchrΓΆder trees with k + 1 leaves, and dk the
|
1287 |
+
number of words for the divisors of Οk+1
|
1288 |
+
k
|
1289 |
+
.
|
1290 |
+
Recall that Div(βn) = οΏ½
|
1291 |
+
0β€iβ€n+1 Di
|
1292 |
+
n, and let di
|
1293 |
+
n be the number of words for the elements
|
1294 |
+
of Di
|
1295 |
+
n. If i = n + 1, then Οn+1
|
1296 |
+
n
|
1297 |
+
is the only element of Di
|
1298 |
+
n and by Theorem 3.12, there are
|
1299 |
+
sn words for this element, hence we have dn+1
|
1300 |
+
n
|
1301 |
+
= sn.
|
1302 |
+
Let 0 β€ i β€ n and w = x1 Β· Β· Β· xl be a word for an element of Di
|
1303 |
+
n. The word w is a strict
|
1304 |
+
preο¬x of a SchrΓΆder tree T = (r, S1, Β· Β· Β· , Sk). By Lemma 2.5, w = w1w2 where w1 is a word
|
1305 |
+
for Οi
|
1306 |
+
n and Οn is not a left divisor of w2 (when i = 0 the word w1 is empty). Let Sj be the
|
1307 |
+
last subtree of T which has a vertex labelled by a letter of w1. We can apply a succession
|
1308 |
+
of deο¬ning relations to w1 in order to obtain Οi
|
1309 |
+
n. These relations correspond to local move
|
1310 |
+
in the trees S1, Β· Β· Β· , Sj which collapse all the trees S1, Β· Β· Β· Sj to empty trees. In order to
|
1311 |
+
reduce Sj to a list of empty trees we must use its root. Since the root is always the last
|
1312 |
+
label of the tree in post-order, the word w1 covers all the ο¬rst j trees which have in total
|
1313 |
+
i leaves. Since Οn does not divide w2, we see that w2 is a (possibly empty) strict preο¬x
|
1314 |
+
of Sj+1. It is also possible to modify the trees Sj+2, Β· Β· Β· , Sk without changing the ο¬rst j
|
1315 |
+
trees. Indeed, as in the proof of Proposition 3.16 we can reduce the trees Sj+2, Β· Β· Β· , Sk to
|
1316 |
+
empty trees and then merge them (until we can) to Sj+1.
|
1317 |
+
β’ When i = 0, after modiο¬cation we obtain a tree ΛT = (r, ΛS, L) where L the empty
|
1318 |
+
tree, ΛS is a tree with n leaves and w = w2 is a strict preο¬x of ΛS.
|
1319 |
+
β’ When 1 β€ i β€ n, we obtain a tree ΛT = (r, S1, Β· Β· Β· , Sj, οΏ½Sj+1) and w2 is a strict preο¬x
|
1320 |
+
of the tree οΏ½Sj+1 with n + 1 β i leaves.
|
1321 |
+
In both cases, the tree οΏ½Sj+1 is obtained by possibly introducing new vertices to Sj+1,
|
1322 |
+
and as Figure 7 shows, these new vertices occur after the vertices of Sj+1, in post-order,
|
1323 |
+
|
1324 |
+
ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
|
1325 |
+
19
|
1326 |
+
hence w2 is still a strict preο¬x of οΏ½Sj+1. Hence, we see that in the decomposition w = w1w2
|
1327 |
+
of Lemma 2.5, the word w1 is obtained by reading all the vertices of a SchrΓΆder tree with
|
1328 |
+
i leaves and w2 is a strict preο¬x of a SchrΓΆder tree, denoted by ΛS, with li + 1 leaves where
|
1329 |
+
li = n β i leaves if i ΜΈ= 0 and li = n β 1 if i = 0.
|
1330 |
+
Let w = w1w2 be a word of an element of Di
|
1331 |
+
n with w2 having t letters. Let ΛS be a
|
1332 |
+
SchrΓΆder tree with li + 1 leaves having w2 as a strict preο¬x. Then, we construct a word
|
1333 |
+
Ξ³(w) by ο¬rst extracting ΛS, then labelling it accordingly to its number of leaves (i.e., with
|
1334 |
+
m = li) and ο¬nally taking its ο¬rst t letters in post-order. Algebraically, it is easy to see
|
1335 |
+
how the word Ξ³(w) is obtained from w2: if wi is the label of a leftmost child in ΛS, we have
|
1336 |
+
Ξ³(w)i = wi. Otherwise, since the tree ΛSj has li + 1 leaves, we have Ξ³(w)i = wi β n + (li).
|
1337 |
+
A priori Ξ³(w) depends on the choice of a tree ΛS, but Lemma 4.6 tells us that Ξ³(w) only
|
1338 |
+
depends on w2. The word Ξ³(w) is a preο¬x of a SchrΓΆder tree with li + 1 leaves, hence it
|
1339 |
+
is a word for a divisor of βli. We have obtained a map Ξ³ from the set of words for the
|
1340 |
+
elements of Di
|
1341 |
+
n to the set of words for the divisors of βli.
|
1342 |
+
Conversely, if z is a word of length k for a divisor of βli, it is a preο¬x (strict since the
|
1343 |
+
root is not contributing) of a SchrΓΆder tree S with li +1 leaves. We can view S as a subtree
|
1344 |
+
of a SchrΓΆder tree with n + 1 leaves by considering:
|
1345 |
+
β’ T = (r, S, L) when i = 0;
|
1346 |
+
β’ T = (r, Ξ΄i, S) when i β₯ 1.
|
1347 |
+
Reading up to the ο¬rst k letters of the subtree S produces a word w = w1w2 of an
|
1348 |
+
element of Di
|
1349 |
+
n such that Ξ³(w) = z. Hence Ξ³ is surjective and we set Η«(z) = w2. As before
|
1350 |
+
Η«(z) only depends on z, not on the tree having z as a preο¬x.
|
1351 |
+
When i = 0, the map Ξ³ is injective, indeed if w and z are two words such that Ξ³(w) =
|
1352 |
+
Ξ³(z), then by Lemma 4.6 the labels of the leftmost child in Ξ³(w) and Ξ³(z) are the same,
|
1353 |
+
hence w and z are equal. This proves that d0
|
1354 |
+
n = dnβ1.
|
1355 |
+
When i β₯ 1, then Ξ³ is far from being injective, since it forgets the ο¬rst part of the tree.
|
1356 |
+
The set of words for the elements of Di
|
1357 |
+
n is the disjoint union of two sets E1 and E2 where
|
1358 |
+
E1 is the set of words w = w1w2 where w1 covers exactly one tree S1 and E2 is the set
|
1359 |
+
of words where w1 covers at least two trees. Note that when i = 1, the set E2 is empty
|
1360 |
+
otherwise both sets are non-empty. Indeed E2 contains at least all the words of the form
|
1361 |
+
Οi
|
1362 |
+
nw2 and E1 contains at least the words of the form Οiβ1Οiβ1
|
1363 |
+
n
|
1364 |
+
Οnβiβ1w2 which correspond
|
1365 |
+
to the SchrΓΆder bush Ξ΄i attached as the leftmost subtree of a SchrΓΆder tree.
|
1366 |
+
If z is a word for a divisor of βli, we compute the cardinality of the preimage of z by Ξ³
|
1367 |
+
by looking at Ξ³β1(z) β© E1 and Ξ³β1(z) β© E2. If i = 1, we obviously only consider the ο¬rst
|
1368 |
+
case. The elements of Ξ³β1(z) β© E1 are obtained by concatenation of the word of a single
|
1369 |
+
SchrΓΆder tree with i leaves and Η«(z), and the elements of Ξ³β1(z) β© E2 are concatenation
|
1370 |
+
of the words of a forest with i leaves made of at least two SchrΓΆder tree and Η«(z). Such a
|
1371 |
+
forest is nothing but a SchrΓΆder tree with i-leaves from which the root has been removed.
|
1372 |
+
So we have
|
1373 |
+
|Ξ³β1(z) β© E1| = siβ1 = |Ξ³β1(z) β© E2|.
|
1374 |
+
Taking the sum on all possible words z, we have d1
|
1375 |
+
n = s0 Β· dnβ1 and di
|
1376 |
+
n = 2 Β· siβ1 Β· dnβi
|
1377 |
+
when n β₯ i β₯ 2.
|
1378 |
+
We have obtained:
|
1379 |
+
d0
|
1380 |
+
n = dnβ1;
|
1381 |
+
d1
|
1382 |
+
n = s0 Β· dnβ1 = dnβ1;
|
1383 |
+
and
|
1384 |
+
di
|
1385 |
+
n = 2 Β· siβ1 Β· dnβi when n β₯ i β₯ 2 and dn+1
|
1386 |
+
n
|
1387 |
+
= sn.
|
1388 |
+
|
1389 |
+
20
|
1390 |
+
THOMAS GOBET AND BAPTISTE ROGNERUD
|
1391 |
+
By induction on the number of leaves, we have di = si+1, for every i β€ n β 1, and
|
1392 |
+
dn = 2sn + 2
|
1393 |
+
n
|
1394 |
+
οΏ½
|
1395 |
+
i=2
|
1396 |
+
siβ1snβi+1 + sn
|
1397 |
+
= 3sn + 2
|
1398 |
+
nβ1
|
1399 |
+
οΏ½
|
1400 |
+
i=1
|
1401 |
+
sisnβi.
|
1402 |
+
Using generating functions, it is not diο¬cult to check that this implies that dn = sn+1, see
|
1403 |
+
for example [13, Theorem 5].
|
1404 |
+
β‘
|
1405 |
+
References
|
1406 |
+
[1] D. Bessis, The dual braid monoid, Ann. Sci. Γcole Norm. Sup. 36 (2003), 647-683.
|
1407 |
+
[2] J. Birman, K.H. Ko, and S.J. Lee, A New Approach to the Word and Conjugacy Problems in the
|
1408 |
+
Braid Groups, Adv. in Math. 139 (1998), 322β353.
|
1409 |
+
[3] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245β271.
|
1410 |
+
[4] P. Dehornoy, F. Digne, D. Krammer, E. Godelle, and J. Michel. Foundations of Garside theory, Tracts
|
1411 |
+
in Mathematics 22, Europ. Math. Soc. (2015).
|
1412 |
+
[5] P. Dehornoy and L. Paris, Gaussian groups and Garside groups, two generalisations of Artin groups,
|
1413 |
+
Proc. London Math. Soc. (3) 79 (1999), no. 3, 569-604.
|
1414 |
+
[6] P. Deligne, Les immeubles des groupes de tresses gΓ©nΓ©ralisΓ©s, Invent. Math. 17 (1972), 273-302.
|
1415 |
+
[7] F.A. Garside, The braid group and other groups, Quart. J. Math. Oxford Ser. 20 (1969), no. 2,
|
1416 |
+
235β254.
|
1417 |
+
[8] T. Gobet, On some torus knot groups and submonoids of the braid groups, J. Algebra 607 (2022),
|
1418 |
+
Part B, 260-289.
|
1419 |
+
[9] T. Gobet, A new Garside structure on torus knot groups and some complex braid groups, preprint
|
1420 |
+
(2022), https://arxiv.org/abs/2209.02291.
|
1421 |
+
[10] J-L. Loday, Realization of the Stasheο¬ polytope, Arch. Math., 83 (2004), 267-278.
|
1422 |
+
[11] OEIS Foundation Inc, The On-Line Encyclopedia of Integer Sequences, Published electronically at
|
1423 |
+
https://oeis.org
|
1424 |
+
[12] M. Picantin, Petits groupes gaussiens, PhD Thesis, UniversitΓ© de Caen, 2000.
|
1425 |
+
[13] F. Qi, B. Guo Some explicit and recursive formulas of the large and little SchrΓΆder numbers, Arab
|
1426 |
+
Journal of Mathematical Sciences Vol: 23, Issue: 2, Page: 141-147 (2017).
|
1427 |
+
Institut Denis Poisson, CNRS UMR 7350, FacultΓ© des Sciences et Techniques, UniversitΓ©
|
1428 |
+
de Tours, Parc de Grandmont, 37200 TOURS, France
|
1429 |
+
Email address: [email protected]
|
1430 |
+
Institut de MathΓ©matiques de Jussieu, Paris Rive Gauche (IMJ-PRG), Campus des Grands
|
1431 |
+
Moulins, UniversitΓ© de Paris - Boite Courrier 7012, 8 Place AurΓ©lie Nemours, 75205 PARIS
|
1432 |
+
Cedex 13, France
|
1433 |
+
Email address: [email protected]
|
1434 |
+
|
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39E2T4oBgHgl3EQfjgft/content/tmp_files/2301.03970v1.pdf.txt
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|
1 |
+
arXiv:2301.03970v1 [math.GR] 10 Jan 2023
|
2 |
+
Ulam stability of lamplighters and Thompson groups
|
3 |
+
Francesco Fournier-Facio and Bharatram Rangarajan
|
4 |
+
January 11, 2023
|
5 |
+
Abstract
|
6 |
+
We show that a large family of groups is uniformly stable relative to unitary groups
|
7 |
+
equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten
|
8 |
+
p-norms. These include lamplighters Ξ β Ξ where Ξ is inο¬nite and amenable, as well as
|
9 |
+
several groups of dynamical origin such as the classical Thompson groups F, F β², T and
|
10 |
+
V . We prove this by means of vanishing results in asymptotic cohomology, a theory
|
11 |
+
introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable
|
12 |
+
for studying uniform stability.
|
13 |
+
Along the way, we prove some foundational results
|
14 |
+
in asymptotic cohomology, and use them to prove some hereditary features of Ulam
|
15 |
+
stability. We further discuss metric approximation properties of such groups, taking
|
16 |
+
values in unitary or symmetric groups.
|
17 |
+
1
|
18 |
+
Introduction
|
19 |
+
Let Ξ be a countable discrete group, and let U be a family of ο¬nite-dimensional unitary
|
20 |
+
groups. The problem of stability asks whether every almost-homomorphism Ξ β U β U
|
21 |
+
is close to a homomorphism. To formalize this we need to choose a norm, and a way to
|
22 |
+
interpret these approximate notions. We focus on the classical setting of uniform defects and
|
23 |
+
distances, with respect to submultiplicative norms.
|
24 |
+
Let U := {(U(k), β₯Β·β₯)} be a family of ο¬nite-dimensional unitary groups equipped with bi-
|
25 |
+
invariant submultiplicative norms β₯Β·β₯ (we allow U(k) to appear multiple times with diο¬erent
|
26 |
+
norms). For instance β₯ Β· β₯ could be the operator norm - the most classical case - or more
|
27 |
+
generally a Schatten p-norm. Given a map Ο : Ξ β U(k), we deο¬ne its defect to be
|
28 |
+
def(Ο) := sup
|
29 |
+
g,hβΞ
|
30 |
+
β₯Ο(gh) β Ο(g)Ο(h)β₯.
|
31 |
+
Given another map Ο : Ξ β U(k), we deο¬ne the distance between them to be
|
32 |
+
dist(Ο, Ο) := sup
|
33 |
+
gβΞ
|
34 |
+
β₯Ο(g) β Ο(g)β₯.
|
35 |
+
Deο¬nition 1.1. A uniform asymptotic homomorphism is a sequence of maps Οn : Ξ β U(kn)
|
36 |
+
such that def(Οn) β 0. We denote this simply by Ο : Ξ β U. We say that Ο, Ο : Ξ β U are
|
37 |
+
uniformly asymptotically close if they have the same range degrees and dist(Οn, Οn) β 0.
|
38 |
+
The group Ξ is uniformly U-stable if every uniform asymptotic homomorphism is uni-
|
39 |
+
formly asymptotically close to a sequence of homomorphisms.
|
40 |
+
1
|
41 |
+
|
42 |
+
We can also talk quantitatively about stability, by asking how close a homomorphism we
|
43 |
+
can choose, in terms of the defect. This leads to the notion of stability with a linear estimate,
|
44 |
+
which will be relevant for us and which we deο¬ne precisely in Section 2.1.
|
45 |
+
Early mentions of similar problems can be found in the works of von Neumann [vN29]
|
46 |
+
and Turing [Tur38]. In [Ula60, Chapter 6] Ulam discussed more general versions of stability,
|
47 |
+
which has since inspired a large body of work. Uniform U-stability has been studied mostly
|
48 |
+
when U is the family of unitary groups equipped with the operator norm, for which the
|
49 |
+
notion is typically referred to as Ulam stability. In this contest, Kazhdan proved stability of
|
50 |
+
amenable groups [Kaz82], while Burger, Ozawa and Thom proved stability of certain special
|
51 |
+
linear groups over S-integers, and instability of groups admitting non-trivial quasimorphisms
|
52 |
+
[BOT13].
|
53 |
+
More recently, the second author, Glebsky, Lubotzky and Monod proved Ulam stabil-
|
54 |
+
ity of certain lattices in higher rank Lie groups, with respect to arbitrary submultiplicative
|
55 |
+
norms [GLMR23]. For the proof, they introduce a new cohomology theory, called asymptotic
|
56 |
+
cohomology, and prove that stability is implied by the vanishing of certain asymptotic co-
|
57 |
+
homology classes Ξ± β H2
|
58 |
+
a(Ξ, V). We refer the reader to Section 2.2 for the relevant deο¬nitions.
|
59 |
+
The goal of this paper is to further the understanding of asymptotic cohomology, and
|
60 |
+
apply this to prove new stability results.
|
61 |
+
The main one is the stability of the classical
|
62 |
+
Thompson groups:
|
63 |
+
Theorem 1.2 (Section 5). Thompsonβs groups F, F β², T and V are uniformly U-stable, with
|
64 |
+
a linear estimate.
|
65 |
+
As remarked by Arzhantseva and PΛaunescu [AP15, Open problem], the analogous state-
|
66 |
+
ment for pointwise stability in permutation of F would imply that F is not soο¬c, thus proving
|
67 |
+
at once the existence of a non-soο¬c group and the non-amenability of F: two of the most
|
68 |
+
remarkable open problems in modern group theory. We will discuss these problems and their
|
69 |
+
relation to our results in Section 7.
|
70 |
+
Theorem 1.2 for F and F β² will follow from a stability result for certain lamplighters.
|
71 |
+
Given groups Ξ, Ξ, the corresponding lamplighter (or restricted wreath product) is the group
|
72 |
+
Ξ β Ξ = (βΞΞ) β Ξ, where Ξ acts by shifting the coordinates.
|
73 |
+
Theorem 1.3. Let Ξ, Ξ be two countable groups, where Ξ is inο¬nite and amenable. Then
|
74 |
+
Ξ β Ξ is uniformly U-stable, with a linear estimate.
|
75 |
+
By itself, Theorem 1.3 provides a plethora of examples of uniformly U-stable groups, to a
|
76 |
+
degree of ο¬exibility that was not previously available. For instance, using classical embedding
|
77 |
+
results [HNN49] it immediately implies the following:
|
78 |
+
Corollary 1.4. Every countable group embeds into a 3-generated group which is uniformly
|
79 |
+
U-stable, with a linear estimate.
|
80 |
+
In particular, this gives a proof that there exist uncountably many ο¬nitely generated
|
81 |
+
uniformly U-stable groups, a fact which could also be obtained by applying Kazhdanβs The-
|
82 |
+
orem [Kaz82] to an inο¬nite family of ο¬nitely generated amenable groups, such as the one
|
83 |
+
constructed by B. H. Neumann [Neu37].
|
84 |
+
2
|
85 |
+
|
86 |
+
In order to obtain stability of F and F β² from Theorem 1.3, we exploit coamenability.
|
87 |
+
Recall that a subgroup Ξ β€ Ξ is coamenable if the coset space Ξ/Ξ admits a Ξ-invariant
|
88 |
+
mean. It is well known that F β² and F contain a coamenable lamplighter F β Z. Therefore the
|
89 |
+
stability of F and F β² (Corollary 5.8) follows from Theorem 1.3, and the following result:
|
90 |
+
Proposition 1.5. Let Ξ β€ Ξ be coamenable. If Ξ is uniformly U-stable with a linear estimate,
|
91 |
+
then so is Ξ.
|
92 |
+
This can be seen as a relative version of the celebrated result of Kazhdan, stating that
|
93 |
+
amenable groups are uniformly U-stable [Kaz82]. To complete the picture, we also prove
|
94 |
+
another relative version of Kazhdanβs Theorem, which is sort of dual to Proposition 1.5:
|
95 |
+
Proposition 1.6. Let N β€ Ξ be an amenable normal subgroup. If Ξ is uniformly U-stable
|
96 |
+
with a linear estimate, then so is Ξ/N.
|
97 |
+
The fact that Theorem 1.2 follows from Theorem 1.3 and Proposition 1.5 is not special
|
98 |
+
to Thompsonβs group F: this phenomenon is typical of several groups of piecewise linear and
|
99 |
+
piecewise projective homeomorphisms, which enjoy some kind of self-similarity properties
|
100 |
+
(Theorem 5.1 and Corollary 5.2). Stability of T and V then follow from these results, to-
|
101 |
+
gether with a bounded generation argument analogous to the one from [BOT13] (Corollaries
|
102 |
+
5.11 and 5.12).
|
103 |
+
As we mentioned above, the tool underlying the proofs of Theorem 1.3 and Proposition
|
104 |
+
1.5 is asymptotic cohomology, in particular the vanishing of certain classes in degree 2. In
|
105 |
+
this framework, Theorem 1.3 takes the following form:
|
106 |
+
Theorem 1.7. Let Ξ, Ξ be two countable groups, where Ξ is inο¬nite and amenable. Then
|
107 |
+
Hn
|
108 |
+
a(Ξ β Ξ, V) = 0 for all n β₯ 1 and all ο¬nitary dual asymptotic Banach βΞ-modules V.
|
109 |
+
Here the word ο¬nitary refers to the fact that these modules arise from stability problems
|
110 |
+
with respect to ο¬nite-dimensional unitary representations. This hypothesis is crucial: see
|
111 |
+
Remark 6.1. Propositions 1.5 and 1.6 also follow from results in asymptotic cohomology,
|
112 |
+
that this time does not need the ο¬nitary assumption:
|
113 |
+
Proposition 1.8. Let Ξ β€ Ξ be coamenable. Then the restriction map Hn
|
114 |
+
a(Ξ, V) β Hn
|
115 |
+
a(Ξ, V)
|
116 |
+
is injective, for all n β₯ 0 and all dual asymptotic Banach βΞ-modules V.
|
117 |
+
Proposition 1.9. Let N β€ Ξ be an amenable normal subgroup. Then the pullback map
|
118 |
+
Hn
|
119 |
+
a(Ξ/N, V) β Hn
|
120 |
+
a(Ξ, V) is an isomorphism, for all n β₯ 0 and all dual asymptotic Banach
|
121 |
+
β(Ξ/N)-modules V.
|
122 |
+
Despite the lack of a general theorem connecting the two theories, asymptotic cohomology
|
123 |
+
seems to be closely connected to bounded cohomology, a well-established cohomology theory
|
124 |
+
[Joh72, Gro82, Iva85, Mon01, Fri17] that has become a fundamental tool in rigidity theory.
|
125 |
+
The vanishing result for asymptotic cohomology of lattices leading to stability [GLMR23]
|
126 |
+
follows closely the vanishing result for bounded cohomology of high-rank lattices [BM99,
|
127 |
+
BM02, MS04]. Similarly, our proofs of Theorem 1.7 and Propositions 1.8 and 1.9 follow
|
128 |
+
closely the corresponding bounded-cohomological results: for Theorem 1.7 this was recently
|
129 |
+
proven by Monod [Mon22], while for Proposition 1.8 this is a foundational result in bounded
|
130 |
+
cohomology [Mon01, 8.6] (see also [MP03]), and Proposition 1.9 is an analogue of Gromovβs
|
131 |
+
3
|
132 |
+
|
133 |
+
Mapping Theorem [Gro82]. Note that the bounded cohomology of T and V has also been
|
134 |
+
recently computed [FFLM21, MN21, And22], but only with trivial real coeο¬cients, and our
|
135 |
+
proofs are of a diο¬erent nature.
|
136 |
+
We thus hope that the steps we undertake to prove our main results will be useful to
|
137 |
+
produce more computations in asymptotic cohomology, and therefore more examples of uni-
|
138 |
+
formly U-stable, and in particular Ulam stable, groups.
|
139 |
+
Our results have applications to the study of approximating properties of groups. While
|
140 |
+
questions on pointwise approximation, such as soο¬city, hyperlinearity, and matricial ο¬nite-
|
141 |
+
ness, are in some sense disjoint from the content of this paper, our stability results imply
|
142 |
+
that some of the groups considered in this paper are not uniformly approximable with respect
|
143 |
+
to the relevant families U (Corollary 7.6). We are also able to treat the case of symmetric
|
144 |
+
groups endowed with the Hamming distance, by a more direct argument (Proposition 7.7).
|
145 |
+
We end this introduction by proposing a question. There is a notion of strong Ulam stabil-
|
146 |
+
ity, where the approximations take values in unitary groups of possibly inο¬nite-dimensional
|
147 |
+
Hilbert spaces, with the operator norm. It is a well-known open question whether strong
|
148 |
+
Ulam stability coincides with amenability. In this direction it is known that strong Ulam
|
149 |
+
stable groups have no non-abelian free subgroups [BOT13, Theorem 1.2], but there exist
|
150 |
+
groups without non-abelian free subgroups that are not strong Ulam stable [Alp20].
|
151 |
+
On the other hand, our results also prove uniform U-stability stability of the piecewise
|
152 |
+
projective groups of Monod [Mon13] and LodhaβMoore [LM16], which are nonamenable and
|
153 |
+
without free subgroups (see Section 5.2). Therefore we ask the following:
|
154 |
+
Question 1.10. Let Ξ be a countable group without non-abelian free subgroups.
|
155 |
+
Is Ξ
|
156 |
+
uniformly U-stable (with a linear estimate)? Or at least Ulam stable?
|
157 |
+
In particular, are all countable torsion groups Ulam stable?
|
158 |
+
In other words: if Ξ is not Ulam stable, must Ξ contain a non-abelian free subgroup? To
|
159 |
+
our knowledge it is not every known if groups admitting non-trivial quasimorphisms must
|
160 |
+
contain non-abelian free subgroups: see [Man05] and [Cal10] for partial results in this direc-
|
161 |
+
tion.
|
162 |
+
Conventions: All groups are assumed to be discrete and countable. The set of natu-
|
163 |
+
ral numbers N starts at 0. A non-principal ultraο¬lter Ο on N is ο¬xed for the rest of the paper.
|
164 |
+
Outline: We start in Section 2 by reviewing the framework of asymptotic cohomology
|
165 |
+
and its applications to stability, as developed in [GLMR23]. In Section 3 we discuss hered-
|
166 |
+
itary properties for Ulam stability, and prove Propositions 1.5 and 1.6. We then move to
|
167 |
+
lamplighters and prove Theorem 1.3 in Section 4, then to Thompson groups proving Theorem
|
168 |
+
1.2 in Section 5. In Section 6 we provide examples showing that some of our results and some
|
169 |
+
of the results from [GLMR23] are sharp, and conclude in Section 7 by discussing applications
|
170 |
+
to the study of metric approximations of groups.
|
171 |
+
Acknowledgements: The authors are indebted to Alon Dogon, Lev Glebsky, Alexander
|
172 |
+
Lubotzky and Nicolas Monod for useful conversations.
|
173 |
+
4
|
174 |
+
|
175 |
+
2
|
176 |
+
Uniform stability and asymptotic cohomology
|
177 |
+
In this section, we shall brieο¬y summarize the notion of defect diminishing that allows us
|
178 |
+
to formulate the stability problem as a problem of lifting of homomorphisms with abelian
|
179 |
+
kernel, which in turn motivates the connection to second cohomology. For a more detailed
|
180 |
+
description, refer to Section 2 in [GLMR23].
|
181 |
+
2.1
|
182 |
+
Uniform stability and defect diminishing
|
183 |
+
We begin by reviewing some basic notions of ultraproducts and non-standard analysis, be-
|
184 |
+
fore formulating the stability problem as a homomorphism lifting problem. For this, it is
|
185 |
+
convenient to describe a uniform asymptotic homomorphism (which is a sequence of maps)
|
186 |
+
as one map of ultraproducts. This in turn allows us to perform a soft analysis to obtain
|
187 |
+
a (true) homomorphism to a quotient group. Recall that Ο is a ο¬xed non-principal ultra-
|
188 |
+
ο¬lter on N. The algebraic ultraproduct οΏ½
|
189 |
+
Ο Xn of an indexed collection {Xn}nβN of sets is
|
190 |
+
deο¬ned to be οΏ½
|
191 |
+
Ο Xn := οΏ½
|
192 |
+
nβN Xn/ βΌ where for {xn}nβN, {yn}nβN β οΏ½
|
193 |
+
nβN Xn, we deο¬ne
|
194 |
+
{xn}nβN βΌ {yn}nβN if {n : xn = yn} β Ο. Ultraproducts can be made to inherit algebraic
|
195 |
+
structures of their building blocks. For instance, for a group Ξ, the ultraproduct οΏ½
|
196 |
+
Ο Ξ, called
|
197 |
+
the ultrapower and denoted βΞ, is itself a group. Another important example we will use is
|
198 |
+
the ο¬eld of hyperreals βR, the ultrapower of R.
|
199 |
+
Objects (sets, functions, etc.) that arise as ultraproducts of standard objects are referred
|
200 |
+
to as internal. Important examples of non-internal objects are the subsets βRb of bounded
|
201 |
+
hyperreals, consisting of elements {xn}Ο β βR for which there exists S β Ο and C β Rβ₯0 such
|
202 |
+
that |xn| β€ C for every n β S, and the subset βRinf of inο¬nitesimals, consisting of elements
|
203 |
+
{xn}Ο β βR such that for every real Ξ΅ > 0, there exists S β Ο such that |xn| < Ξ΅ for every
|
204 |
+
n β S.
|
205 |
+
For x, y β βR, write x = OΟ(y) if x/y β βRb, and write x = oΟ(y) if x/y β βRinf. In
|
206 |
+
particular, x β βRb is equivalent to x = OΟ(1) while Ξ΅ β βRinf is equivalent to Ξ΅ = oΟ(1). The
|
207 |
+
subset βRb forms a valuation ring with βRinf being the unique maximal ideal, with quotient
|
208 |
+
βRb/βRinf βΌ= R. The quotient map st : βRb β R is known as the standard part map or limit
|
209 |
+
along the ultraο¬lter Ο. The previous construction can also be replicated for Banach spaces.
|
210 |
+
Let {Wn}nβN be a family of Banach spaces. Then W = οΏ½
|
211 |
+
Ο Wn can be given the structure
|
212 |
+
of a βR-vector space. In fact, it also comes equipped with a βR-valued norm, allowing us to
|
213 |
+
deο¬ne the external subsets Wb and Winf. The quotient Λ
|
214 |
+
W := Wb/Winf is a real Banach space.
|
215 |
+
Given a uniform asymptotic homomorphism {Οn : Ξ β U(kn)}nβN with def(Οn) =: Ξ΅n β
|
216 |
+
0, construct the internal map Ο : βΞ β οΏ½
|
217 |
+
Ο U(kn) where Ο := οΏ½
|
218 |
+
Ο Οn, with (hyperreal) defect
|
219 |
+
Ξ΅ := {Ξ΅n}Ο β βRinf. Then the question of uniform stability with a linear estimate can be
|
220 |
+
rephrased as asking whether there exists an internal homomorphism Ο : βΞ β οΏ½
|
221 |
+
Ο U(kn) such
|
222 |
+
that their (hyperreal) distance satisο¬es dist(Ο, Ο) := {dist(Οn, Οn)}Ο = OΟ(Ξ΅).
|
223 |
+
The advantage of rephrasing the question in terms of internal maps is that an internal
|
224 |
+
map Ο : βΞ β οΏ½
|
225 |
+
Ο U(kn) with defect Ξ΅ β βRinf induces a true homomorphism ΛΟ : βΞ β
|
226 |
+
οΏ½
|
227 |
+
Ο U(kn)/B(Ξ΅) where B(Ξ΅) is the (external) normal subgroup of οΏ½
|
228 |
+
Ο U(kn) comprising ele-
|
229 |
+
ments that are at a distance OΟ(Ξ΅) from the identity. In particular, the question of uniform
|
230 |
+
stability with a linear estimate can equivalently be rephrased as asking whether given such
|
231 |
+
an internal map Ο, can the homomorphism ΛΟ : βΞ β οΏ½
|
232 |
+
Ο U(kn)/B(Ξ΅) be lifted to an internal
|
233 |
+
5
|
234 |
+
|
235 |
+
homomorphism Ο : βΞ β οΏ½
|
236 |
+
Ο U(kn).
|
237 |
+
Reinterpreting uniform stability with a linear estimate as a homomorphism lifting problem
|
238 |
+
motivates a cohomological approach to capturing the obstruction. However, the obstacle here
|
239 |
+
is that the kernel B(Ξ΅) of the lifting problem is not abelian. This can be handled by lifting
|
240 |
+
in smaller steps so that each step involves an abelian kernel. Deο¬ne a normal subgroup I(Ξ΅)
|
241 |
+
of B(Ξ΅) comprising elements that are at a distance of oΟ(Ξ΅) from the identity. Then we can
|
242 |
+
attempt to lift ΛΟ : βΞ β οΏ½
|
243 |
+
Ο U(kn)/B(Ξ΅) to an internal map Ο : βΞ β οΏ½
|
244 |
+
Ο U(kn) that is a
|
245 |
+
homomorphism modulo I(Ξ΅). The problem is simpler from the cohomological point of view:
|
246 |
+
since the norms are submultiplicative, the kernel B(Ξ΅)/I(Ξ΅) of this lifting problem is abelian.
|
247 |
+
The group Ξ is said to have the defect diminishing property with respect to U if such a lift
|
248 |
+
exists; more explicitly, Ξ has the defect diminishing property if for every uniform asymptotic
|
249 |
+
homomorphism Ο : Ξ β U there exists a uniform asymptotic homomorphism Ο with the
|
250 |
+
same range such that dist(Ο, Ο) = OΟ(def(Ο)) and def(Ο) = oΟ(def(Ο)).
|
251 |
+
Theorem 2.1 ([GLMR23, Theorem 2.3.11]). Ξ has the defect diminishing property with
|
252 |
+
respect to U if and only if Ξ is uniformly U-stable with a linear estimate.
|
253 |
+
The obstruction to such a homomorphism lifting, with an abelian kernel B(Ξ΅)/I(Ξ΅), can
|
254 |
+
be carefully modeled using a cohomology Hβ’
|
255 |
+
a(Ξ, W) so that H2
|
256 |
+
a(Ξ, W) = 0 implies the defect
|
257 |
+
diminishing property (and consequently, uniform stability with a linear estimate).
|
258 |
+
Here
|
259 |
+
W = οΏ½
|
260 |
+
Ο u(kn) is an internal Lie algebra of οΏ½
|
261 |
+
Ο U(kn) equipped with an asymptotic action of
|
262 |
+
the ultrapower βΞ constructed from the uniform asymptotic homomorphism Ο that we start
|
263 |
+
out with. The logarithm of the defect map
|
264 |
+
βΞ Γ βΞ β
|
265 |
+
οΏ½
|
266 |
+
Ο
|
267 |
+
U(kn) : (g1, g2) οΏ½β Ο(g1)Ο(g2)Ο(g1g2)β1
|
268 |
+
would correspond to an asymptotic 2-cocycle in H2
|
269 |
+
a(Ξ, W). Such a cocycle is a coboundary
|
270 |
+
in this setting (that is, it represents the zero class in H2
|
271 |
+
a(Ξ, W)), if and only if the defect
|
272 |
+
diminishing property holds for the asymptotic homomorphism Ο.
|
273 |
+
2.2
|
274 |
+
Asymptotic cohomology
|
275 |
+
The reduction to a lifting problem with abelian kernel motivates a cohomology theory of Ξ
|
276 |
+
with coeο¬cients in the internal Lie algebra W = οΏ½
|
277 |
+
Ο u(kn) of οΏ½
|
278 |
+
Ο U(kn), equipped with an
|
279 |
+
asymptotic conjugation action of Ξ. In this section we review the formal deο¬nition of this
|
280 |
+
cohomology, and state some results from [GLMR23] that we shall need to work with it.
|
281 |
+
Let (Vn)nβ₯1 be a sequence of Banach spaces, and let V := οΏ½
|
282 |
+
Ο
|
283 |
+
Vn be their algebraic ultra-
|
284 |
+
product: we refer to such V as an internal Banach space. For v β V we denote by β₯vβ₯ the
|
285 |
+
hyperreal (β₯vnβ₯)Ο β βR. We then denote by
|
286 |
+
Vb := {v β V : β₯vβ₯ β βRb};
|
287 |
+
Vinf := {v β V : β₯vβ₯ β βRinf}.
|
288 |
+
Then the quotient ΛV := Vb/Vinf is a real Banach space, whose norm is induced by the
|
289 |
+
ultralimit of β₯ Β· β₯ on Vb. For each Vn denote by V #
|
290 |
+
n its continuous dual, and let V# be the
|
291 |
+
corresponding algebraic ultraproduct. The pairing β¨Β·, Β·β©n : V #
|
292 |
+
n Γ Vn β R induces a pairing
|
293 |
+
V# Γ V β βR which descends to ΛV# Γ ΛV β R. We call V# the internal dual of V.
|
294 |
+
6
|
295 |
+
|
296 |
+
Now let Ξ be a countable discrete group, and let Ο : βΞΓV β V be an internal map which
|
297 |
+
preserves β₯ Β· β₯ and induces an isometric linear action ΛΟ : βΞ Γ ΛV β ΛV of βΞ. Such a map Ο
|
298 |
+
is referred to as an asymptotic βΞ-action on V. We then call (Ο, V), or V, if Ο is understood
|
299 |
+
from context, an asymptotic Banach βΞ-module. Given an internal Banach βΞ-module (Ο, V),
|
300 |
+
the contragradient on each coordinate induces an internal map Ο# : βΞ Γ V# β V# mak-
|
301 |
+
ing (Ο#, V#) into an asymptotic Banach βΞ-module. We call a module V a dual asymptotic
|
302 |
+
Banach βΞ-module if V is the dual of some asymptotic βΞ-module denoted Vβ. We decorate
|
303 |
+
these deο¬nitions with the adjective ο¬nitary if each Vn is ο¬nite-dimensional.
|
304 |
+
Now for each m β₯ 0 deο¬ne the internal Banach space Lβ((βΞ)m, V) := οΏ½
|
305 |
+
Ο
|
306 |
+
ββ(Ξm, Vn)
|
307 |
+
(note that m is ο¬xed and n runs through the natural numbers with respect to the ultraο¬lter
|
308 |
+
Ο). Similarly to before, for f β Lβ((βΞ)m, V) we denote β₯fβ₯ := (β₯fnβ₯)Ο β βR and
|
309 |
+
Lβ
|
310 |
+
b ((βΞ)m, V) := {f β Lβ((βΞ)m, V) : β₯fβ₯ β βRb};
|
311 |
+
Lβ
|
312 |
+
inf((βΞ)m, V) := {f β Lβ((βΞ)m, V) : β₯fβ₯ β βRinf}.
|
313 |
+
Given an asymptotic βΞ-action Ο on V, we can construct a natural asymptotic βΞ-action
|
314 |
+
Οm : βΞ Γ Lβ((βΞ)m, V) β Lβ((βΞ)m, V) given by
|
315 |
+
(Οm(g)(f))(g1, g2, . . . , gm) := ΟG(g)f(gβ1g1, . . . , gβ1gm)
|
316 |
+
(1)
|
317 |
+
Then the quotient
|
318 |
+
ΛLβ((βΞ)m, V) := Lβ
|
319 |
+
b ((βΞ)m, V)/Lβ
|
320 |
+
inf((βΞ)m, V)
|
321 |
+
is again a real Banach space equipped with an isometric βΞ-action induced coordinate-wise
|
322 |
+
by Οm, which deο¬nes the invariant subspaces ΛLβ((βΞ)m, V)
|
323 |
+
βΞ.
|
324 |
+
Now deο¬ne the internal coboundary map
|
325 |
+
dm : Lβ((βΞ)m, V) β Lβ((βΞ)m+1, V);
|
326 |
+
dm(f)(g0, . . . , gm) :=
|
327 |
+
m
|
328 |
+
οΏ½
|
329 |
+
j=0
|
330 |
+
(β1)jf(g0, . . . , Λgj, . . . , gm),
|
331 |
+
(2)
|
332 |
+
which descends to coboundary maps
|
333 |
+
Λ
|
334 |
+
dm : ΛLβ((βΞ)m, V) β ΛLβ((βΞ)m+1, V).
|
335 |
+
Since Λ
|
336 |
+
dm is βΞ-equivariant, it deο¬nes the cochain complex:
|
337 |
+
0
|
338 |
+
Λ
|
339 |
+
d0
|
340 |
+
ββ ΛLβ(βΞ, V)
|
341 |
+
βΞ
|
342 |
+
Λ
|
343 |
+
d1
|
344 |
+
ββ ΛLβ((βΞ)2, V)
|
345 |
+
βΞ
|
346 |
+
Λd2
|
347 |
+
ββ ΛLβ((βΞ)3, V)
|
348 |
+
βΞ
|
349 |
+
Λ
|
350 |
+
d3
|
351 |
+
ββ Β· Β· Β·
|
352 |
+
Deο¬nition 2.2 ([GLMR23, Deο¬nition 4.2.2]). The m-th asymptotic cohomology of Ξ with
|
353 |
+
coeο¬cients in V is
|
354 |
+
Hm
|
355 |
+
a (Ξ, V) := ker(
|
356 |
+
Λ
|
357 |
+
dm+1)/ im( Λ
|
358 |
+
dm).
|
359 |
+
7
|
360 |
+
|
361 |
+
Other resolutions may also be used to compute asymptotic cohomology. Recall ([Mon01,
|
362 |
+
5.3.2]) that a regular Ξ-space S is said to be a Zimmer-amenable Ξ-space if there exists a Ξ-
|
363 |
+
equivariant conditional expectation m : Lβ(ΞΓS) β Lβ(S). Let S be a regular Ξ-space with
|
364 |
+
a Zimmer-amenable action of Ξ, and let Lβ((βS)m, V) := οΏ½
|
365 |
+
Ο
|
366 |
+
Lβ
|
367 |
+
wβ(Sm, Vn) (where Lβ
|
368 |
+
wβ(Sm, Vn)
|
369 |
+
is the space of bounded weak-β measurable function classes from Sm to Vn). Again, the
|
370 |
+
asymptotic βΞ-action on V gives rise to a natural asymptotic βΞ-action on Lβ((βΞ)m, V) as
|
371 |
+
in (1), making Lβ((βS)m, V) an asymptotic Banach βΞ-module. The coboundary maps too
|
372 |
+
can be deο¬ned just as in (2), to construct the cochain complex, and we have:
|
373 |
+
Theorem 2.3 ([GLMR23, Theorem 4.3.3]). Let S be a Zimmer-amenable Ξ-space, and V
|
374 |
+
be a dual asymptotic Banach βΞ-module. Then Hβ’
|
375 |
+
a(Ξ, V) can be computed as the asymptotic
|
376 |
+
cohomology of the cochain complex
|
377 |
+
0
|
378 |
+
Λ
|
379 |
+
d0
|
380 |
+
ββ ΛLβ(βS, V)
|
381 |
+
βΞ
|
382 |
+
Λ
|
383 |
+
d1
|
384 |
+
ββ ΛLβ((βS)2, V)
|
385 |
+
βΞ
|
386 |
+
Λ
|
387 |
+
d2
|
388 |
+
ββ ΛLβ((βS)3, V)
|
389 |
+
βΞ
|
390 |
+
Λ
|
391 |
+
d3
|
392 |
+
ββ Β· Β· Β·
|
393 |
+
In the context of uniform U-stability, the relevant asymptotic Banach βΞ-module we shall
|
394 |
+
be interested is the ultraproduct W = οΏ½
|
395 |
+
Ο u(kn), where u(kn) is the Lie algebra of U(kn). Note
|
396 |
+
that we are only considering ο¬nite-dimensional unitary groups, so such a module is ο¬nitary.
|
397 |
+
Given a uniform asymptotic homomorphism Ο : βΞ β οΏ½
|
398 |
+
Ο U(n) with defect def(Ο) β€Ο Ξ΅ β
|
399 |
+
βRinf, this can be used to construct the asymptotic action Ο : βΞ Γ W β W given by
|
400 |
+
Ο(g)v = Ο(g)vΟ(g)β1, making W an asymptotic Banach βΞ-module. We call such a module
|
401 |
+
an Ulam βΞ-module supported on U.
|
402 |
+
Also, consider the map Ξ± : βΞ Γ βΞ β W given by
|
403 |
+
Ξ±(g1, g2) = 1
|
404 |
+
Ξ΅ log(Ο(g1)Ο(g2)Ο(g1g2)β1).
|
405 |
+
(3)
|
406 |
+
This map Ξ± induces an inhomogeneous 2-cocycle ΛΞ± : βΞ Γ βΞ β
|
407 |
+
Λ
|
408 |
+
W, and thus deο¬nes
|
409 |
+
a class in H2
|
410 |
+
a(Ξ, W), under the usual correspondence between inhomogeneous cochains and
|
411 |
+
invariant homogeneous cochains [GLMR23, Theorem 4.2.4]. We call such a class an Ulam
|
412 |
+
class supported on U. This class vanishes, i.e. ΛΞ± is a coboundary, precisely when Ο has the
|
413 |
+
defect diminishing property. Thus Theorem 2.1 yields:
|
414 |
+
Theorem 2.4 ([GLMR23, Theorem 4.2.4]). Ξ is uniformly U-stable with respect to U if and
|
415 |
+
only if all Ulam classes supported on U vanish. In particular, if H2
|
416 |
+
a(Ξ, W) = 0 for every Ulam
|
417 |
+
βΞ-module supported on U, then Ξ is uniformly U-stable, with a linear estimate.
|
418 |
+
3
|
419 |
+
Hereditary properties
|
420 |
+
In this section, we ο¬rst prove Proposition 1.8 and deduce Proposition 1.5 from it; then
|
421 |
+
analogously we prove Proposition 1.9 and deduce Proposition 1.6 from it. Both stability
|
422 |
+
statements are not symmetric, and in fact we will see in Section 6 that the converses do not
|
423 |
+
hold.
|
424 |
+
3.1
|
425 |
+
More on Zimmer-amenability
|
426 |
+
For the proofs of Propositions 1.8 and 1.9, we will need a more precise version of Theorem 2.3
|
427 |
+
in a special case. A regular Ξ-space S is said to be discrete if it is a countable set equipped
|
428 |
+
8
|
429 |
+
|
430 |
+
with the counting measure. It follows from the equivalent characterizations in [AEG94] that
|
431 |
+
a discrete Ξ-space is Zimmer-amenable precisely when each point stabilizer is amenable. In
|
432 |
+
particular:
|
433 |
+
1. If Ξ β€ Ξ is a subgroup, then the action of Ξ on Ξ by left multiplication is free, so Ξ is
|
434 |
+
a discrete Zimmer-amenable Ξ-space.
|
435 |
+
2. If N β€ Ξ is an amenable subgroup, then the action of Ξ on the coset space Ξ/N has
|
436 |
+
stabilizers equal to conjugates of N, so Ξ/N is a discrete Zimmer-amenable Ξ-space.
|
437 |
+
For such spaces, we can provide an explicit chain map that implements the isomorphism
|
438 |
+
in cohomology from Theorem 2.3. Indeed, the proof of Theorem 2.3 works by starting with
|
439 |
+
a Ξ-homotopy equivalence between the two complexes:
|
440 |
+
0 β Lβ(Ξ) β Lβ(Ξ2) β Lβ(Ξ3) β Β· Β· Β·
|
441 |
+
0 β Lβ(S) β Lβ(S2) β Lβ(S3) β Β· Β· οΏ½οΏ½
|
442 |
+
which is then extended internally to the asymptotic version of these complexes. The case
|
443 |
+
of dual asymptotic coeο¬cients follows via some suitable identiο¬cations of the corresponding
|
444 |
+
complexes (see the paragraph preceding [GLMR23, Theorem 4.20]). In the case of a discrete
|
445 |
+
group Ξ and a discrete Zimmer-amenable Ξ-space, the homotopy equivalence above can be
|
446 |
+
chosen to be the orbit map
|
447 |
+
om
|
448 |
+
b : Lβ(Sm) ββ Lβ(Ξm)
|
449 |
+
om
|
450 |
+
b (f)(g1, . . . , gm) = f(g1b, . . . , gmb);
|
451 |
+
where b β S is some choice of basepoint [Fri17, Section 4.9]. Therefore in this case we obtain
|
452 |
+
the following more explicit version of Theorem 2.3:
|
453 |
+
Theorem 3.1. Let S be a discrete Zimmer-amenable Ξ-space, with a basepoint b β S, and
|
454 |
+
let V be a dual asymptotic Banach βΞ-module. Then the orbit map
|
455 |
+
om
|
456 |
+
b : Lβ((βS)m, V) ββ Lβ((βΞ)m, V)
|
457 |
+
om
|
458 |
+
b (f)(g1, . . . , gm) = f(g1b, . . . , gmb);
|
459 |
+
induces induces an isomorphism between Hβ’
|
460 |
+
a(Ξ, V) and the cohomology of the complex
|
461 |
+
0
|
462 |
+
Λ
|
463 |
+
d0
|
464 |
+
ββ ΛLβ(βS, V)
|
465 |
+
βΞ
|
466 |
+
Λ
|
467 |
+
d1
|
468 |
+
ββ ΛLβ((βS)2, V)
|
469 |
+
βΞ
|
470 |
+
Λ
|
471 |
+
d2
|
472 |
+
ββ ΛLβ((βS)3, V)
|
473 |
+
βΞ
|
474 |
+
Λ
|
475 |
+
d3
|
476 |
+
ββ Β· Β· Β·
|
477 |
+
In the two basic examples of discrete Zimmer-amenable spaces from above, we obtain:
|
478 |
+
Corollary 3.2. Let Ξ β€ Ξ be a subgroup, and let V be a dual asymptotic Banach βΞ-
|
479 |
+
module, which restricts to a dual asymptotic Banach βΞ-module.
|
480 |
+
Then the restriction of
|
481 |
+
cochains Lβ((βΞ)m, V) β Lβ((βΞ)m, V) induces an isomorphism between Hβ’
|
482 |
+
a(Ξ, V) and the
|
483 |
+
cohomology of the complex:
|
484 |
+
0
|
485 |
+
Λ
|
486 |
+
d0
|
487 |
+
ββ ΛLβ(βΞ, V)
|
488 |
+
βΞ
|
489 |
+
Λ
|
490 |
+
d1
|
491 |
+
ββ ΛLβ((βΞ)2, V)
|
492 |
+
βΞ
|
493 |
+
Λ
|
494 |
+
d2
|
495 |
+
ββ ΛLβ((βΞ)3, V)
|
496 |
+
βΞ
|
497 |
+
Λ
|
498 |
+
d3
|
499 |
+
ββ Β· Β· Β·
|
500 |
+
Proof. Seeing Ξ as a discrete Zimmer-amenable Ξ-space, with basepoint 1 β Ξ, the orbit map
|
501 |
+
is nothing but the restriction of cochains, and we conclude by Theorem 3.1.
|
502 |
+
9
|
503 |
+
|
504 |
+
Corollary 3.3. Let N β€ Ξ be an amenable normal subgroup, and let V be a dual asymptotic
|
505 |
+
Banach β(Ξ/N)-module, which pulls back to a dual asymptotic Banach βΞ-module. Then the
|
506 |
+
pullback of cochains Lβ((β(Ξ/N))m, V) β Lβ((βΞ)m, V) induces an isomorphism between
|
507 |
+
Hβ’
|
508 |
+
a(Ξ, V) and Hβ’
|
509 |
+
a(Ξ/N, V).
|
510 |
+
Proof. Seeing Ξ/N as a discrete Zimmer-amenable Ξ-space, with basepoint the coset N, the
|
511 |
+
orbit map is nothing but the pullback of cochains. So Theorem 3.1 yields an isomorphism
|
512 |
+
between Hβ’
|
513 |
+
a(Ξ, V) and the cohomology of the complex:
|
514 |
+
0
|
515 |
+
Λ
|
516 |
+
d0
|
517 |
+
ββ ΛLβ(β(Ξ/N), V)
|
518 |
+
βΞ
|
519 |
+
Λ
|
520 |
+
d1
|
521 |
+
ββ ΛLβ((β(Ξ/N))2, V)
|
522 |
+
βΞ
|
523 |
+
Λ
|
524 |
+
d2
|
525 |
+
ββ ΛLβ((β(Ξ/N))3, V)
|
526 |
+
βΞ
|
527 |
+
Λd3
|
528 |
+
ββ Β· Β· Β·
|
529 |
+
But since the action of βΞ on both β(Ξ/N) and V factors through β(Ξ/N), the above complex
|
530 |
+
coincides with the standard one computing Hβ’
|
531 |
+
a(Ξ/N, V).
|
532 |
+
We will use these explicit isomorphisms in this section. Later, for the proof of Theorem
|
533 |
+
1.7, non-discrete Zimmer-amenable spaces will also appear, but in that case we will only need
|
534 |
+
the existence of an abstract isomorphism as in Theorem 2.3.
|
535 |
+
3.2
|
536 |
+
Restrictions and coamenability
|
537 |
+
Let Ξ β€ Ξ be a (not necessarily coamenable) subgroup, and V be a dual asymptotic Ba-
|
538 |
+
nach βΞ-module, which restricts to a dual asymptotic Banach βΞ-module. The restriction
|
539 |
+
ΛLβ((βΞ)β’, V)
|
540 |
+
βΞ β ΛLβ((βΞ)β’, V)
|
541 |
+
βΞ induces a map in cohomology, called the restriction map,
|
542 |
+
and denoted
|
543 |
+
resβ’ : Hβ’
|
544 |
+
a(Ξ, V) β Hβ’
|
545 |
+
a(Ξ, V).
|
546 |
+
This map behaves well with respect to Ulam classes:
|
547 |
+
Lemma 3.4. Let W be an Ulam βΞ-module supported on U. Then W is also an Ulam βΞ-
|
548 |
+
module supported on U, and the restriction map res2 : H2
|
549 |
+
a(Ξ, W) β H2
|
550 |
+
a(Ξ, W) sends Ulam
|
551 |
+
classes to Ulam classes.
|
552 |
+
Proof. Let Ο : Ξ β U be a uniform asymptotic homomorphism, and let W be the corre-
|
553 |
+
sponding Ulam βΞ-module. Then restricting Οn to Ξ for each n yields a uniform asymptotic
|
554 |
+
homomorphism Ο|Ξ : Ξ β U, with def(Ο|Ξ) β€Ο def(Ο) and endows W with an asymptotic
|
555 |
+
βΞ-action making it into an Ulam βΞ-module supported on U. The cocycle corresponding to
|
556 |
+
Ο is deο¬ned via the map
|
557 |
+
Ξ± : βΞ Γ βΞ β W : (g1, g2) οΏ½β 1
|
558 |
+
Ξ΅ log(Ο(g1)Ο(g2)Ο(g1g2)β1).
|
559 |
+
Since def(Ο|Ξ) β€Ο Ξ΅, restricting Ξ± to βΞ Γ βΞ yields a valid cocycle associated to the
|
560 |
+
uniform asymptotic homomorphism ΟΞ. It follows that the chain map ΛLβ((βΞ)β’, V)
|
561 |
+
βΞ β
|
562 |
+
ΛLβ((βΞ)β’, V)
|
563 |
+
βΞ preserves the set of cocycles deο¬ned via uniform asymptotic homomorphisms,
|
564 |
+
and therefore preserves Ulam classes.
|
565 |
+
Now suppose that Ξ β€ Ξ is coamenable. This means, by deο¬nition, that there exists a
|
566 |
+
Ξ-invariant mean on Ξ/Ξ; that is, there exists a linear functional m : ββ(Ξ/Ξ) β R such
|
567 |
+
that
|
568 |
+
1. m(1Ξ/Ξ) = 1, where 1Ξ/Ξ denotes the constant function.
|
569 |
+
10
|
570 |
+
|
571 |
+
2. |m(f)| β€ β₯fβ₯ for all f β ββ(Ξ/Ξ).
|
572 |
+
3. m(g Β· f) = m(f) for all g β Ξ and all f β ββ(Ξ/Ξ).
|
573 |
+
As with the absolute case [GLMR23, Lemma 3.20], we have the following:
|
574 |
+
Lemma 3.5. Suppose that Ξ β€ Ξ is coamenable, and let V be a dual asymptotic Banach
|
575 |
+
βΞ-module. Then there exists an internal map m : Lβ(βΞ/βΞ, V) β V which induces a map
|
576 |
+
Λm : ΛLβ(βΞ/βΞ, V) β ΛV with the following properties:
|
577 |
+
1. If Λf is the constant function equal to Λv β ΛV, then Λm( Λf) = Λv.
|
578 |
+
2. β₯ Λm( Λf)β₯ β€ β₯ Λfβ₯ for all Λf β ΛLβ(βΞ/βΞ, V).
|
579 |
+
3. Λm(g Β· Λf) = Λm( Λf) for all g β βΞ and all Λf β ΛLβ(βΞ/βΞ, V).
|
580 |
+
Proof. Consider f = {fn}Ο β Lβ(βΞ/βΞ, V). Since V is a dual asymptotic βΞ-module with
|
581 |
+
predual Vβ, for each Ξ» β Vβ, we get an internal map
|
582 |
+
f Ξ» : βΞ/βΞ β βR : x οΏ½β f(x)(Ξ»).
|
583 |
+
Note that f Ξ» being internal, it is of the form {f Ξ»
|
584 |
+
n}Ο where f Ξ»
|
585 |
+
n β ββ(Ξ/Ξ). This allows us to
|
586 |
+
construct the internal map mΞ»
|
587 |
+
in : Lβ(βΞ/βΞ, V) β βR as
|
588 |
+
mΞ»
|
589 |
+
in(f) = {m
|
590 |
+
οΏ½
|
591 |
+
f Ξ»
|
592 |
+
n
|
593 |
+
οΏ½
|
594 |
+
}Ο
|
595 |
+
and ο¬nally min : Lβ(βΞ/βΞ, V) β V as
|
596 |
+
min(f)(Ξ») = mΞ»
|
597 |
+
in(f)
|
598 |
+
It is straightforward to check that min as deο¬ned induces a linear map Λm : ΛLβ(βΞ/βΞ, V) β ΛV.
|
599 |
+
As for βΞ-equivariance, this follows from the observation that (g Β· f)Ξ»(x) = Ο(g)f(gβ1x)(Ξ»)
|
600 |
+
while (gΒ·f Ξ»)(x) = f(gβ1x)(Ξ»). The conditions on Λm follow from the deο¬nition and properties
|
601 |
+
of the Ξ-invariant mean m on ββ(Ξ/Ξ).
|
602 |
+
We are now ready to prove Proposition 1.8. The proof goes along the lines of [Mon01,
|
603 |
+
Proposition 8.6.2].
|
604 |
+
Proposition (Proposition 1.8). Let Ξ β€ Ξ be coamenable. Then the restriction map Hn
|
605 |
+
a(Ξ, V) β
|
606 |
+
Hn
|
607 |
+
a(Ξ, V) is injective, for all n β₯ 0 and all dual asymptotic Banach βΞ-modules V.
|
608 |
+
Proof. We implement the asymptotic cohomology of Ξ using the complex ΛLβ((βΞ)β’, V)
|
609 |
+
βΞ
|
610 |
+
from Corollary 3.2. Since the chain map that deο¬nes the restriction map factors through
|
611 |
+
this complex, and the chain map ΛLβ((βΞ)β’, V)
|
612 |
+
βΞ β ΛLβ((βΞ)β’, V)
|
613 |
+
βΞ induces an isomorphism
|
614 |
+
in cohomology (Corollary 3.2), it suο¬ces to show that the chain inclusion ΛLβ((βΞ)β’, V)
|
615 |
+
βΞ β
|
616 |
+
ΛLβ((βΞ)β’, V)
|
617 |
+
βΞ induces an injective map in cohomology. Henceforth, we will refer to this as
|
618 |
+
the restriction map.
|
619 |
+
Our goal is construct a transfer map, that is a linear map transβ’ : Hβ’
|
620 |
+
a(Ξ, V) β Hβ’
|
621 |
+
a(Ξ, V)
|
622 |
+
such that transβ’ β¦ resβ’ is the identity on Hβ’
|
623 |
+
a(Ξ, V). Then it follows at once that resβ’ must be
|
624 |
+
injective. By the above paragraph, we may do this by constructing an internal chain map
|
625 |
+
οΏ½
|
626 |
+
trans
|
627 |
+
β’ : ΛLβ((βΞ)β’, V)
|
628 |
+
βΞ β ΛLβ((βΞ)β’, V)
|
629 |
+
βΞ that restricts to the identity on ΛLβ((βΞ)β’, V)
|
630 |
+
βΞ.
|
631 |
+
11
|
632 |
+
|
633 |
+
Let f β Lβ((βΞ)k, V) be such that Λf β ΛLβ((βΞ)k, V)
|
634 |
+
βΞ. For each x β (βΞ)k, deο¬ne
|
635 |
+
fx : βΞ β V
|
636 |
+
fx(g) := Ο(g)f(gβ1x)
|
637 |
+
In other words, fx(g) is just (Ο1(g)f)(x) as in (1). Since Λf β ΛLβ((βΞ)k, V)
|
638 |
+
βΞ, for any Ξ³ β βΞ
|
639 |
+
and g β βΞ,
|
640 |
+
fx(gΞ³) β fx(g) β Vinf
|
641 |
+
Let us choose representatives of left βΞ-cosets in βΞ and restrict fx to this set of repre-
|
642 |
+
sentatives so that we can regard fx as an internal map fx : βΞ/βΞ β V. Moreover, since
|
643 |
+
fx β Lβ(βΞ/βΞ, V), we can apply the mean m constructed in Lemma 3.5 to deο¬ne the internal
|
644 |
+
map transk(f) : Lβ((βΞ)k, V) β Lβ((βΞ)k, V) by
|
645 |
+
transk(f)(x) = m(fx)
|
646 |
+
Since Λm is βΞ-invariant, this means that for g β βΞ, m(fgx) β Ο(g)m(fx) β Vinf, and implies
|
647 |
+
that
|
648 |
+
(transk(f))(gx) β Ο(g) transk(f)(x) β Vinf.
|
649 |
+
This establishes that for f β Lβ((βΞ)k, V) with Λf β ΛLβ((βΞ)k, V)
|
650 |
+
βΞ, we have
|
651 |
+
οΏ½
|
652 |
+
transk(f) β
|
653 |
+
ΛLβ((βΞ)k, V)
|
654 |
+
βΞ. Therefore transβ’ induces a chain map
|
655 |
+
Λ
|
656 |
+
trans
|
657 |
+
β’ : ΛLβ((βΞ)β’, V)
|
658 |
+
βΞ β ΛLβ((βΞ)β’, V)
|
659 |
+
βΞ.
|
660 |
+
Finally, if Λf is already βΞ-invariant, then fx is constant up to inο¬nitesimals, and thus m(fx)
|
661 |
+
is equal, up to an inο¬nitesimal, to the value of that constant, which is f(x). This shows that
|
662 |
+
οΏ½
|
663 |
+
trans
|
664 |
+
k is the identity when restricted to ΛLβ((βΞ)β’, V)
|
665 |
+
βΞ, and concludes the proof.
|
666 |
+
Proposition 1.5 is now an easy consequence.
|
667 |
+
Proposition (Proposition 1.5). Let Ξ β€ Ξ be coamenable. If Ξ is uniformly U-stable with a
|
668 |
+
linear estimate, then so is Ξ.
|
669 |
+
Proof. Suppose that Ξ is uniformly U-stable with a linear estimate, and let Ξ be a coamenable
|
670 |
+
supergroup of Ξ. We aim to show that Ξ is also uniformly U-stable with a linear estimate.
|
671 |
+
By Theorem 2.4, it suο¬ces to show that all Ulam classes supported on U vanish in H2
|
672 |
+
a(Ξ, W),
|
673 |
+
where W is an Ulam βΞ-module. Now by Proposition 1.8, it suο¬ces to show that the images
|
674 |
+
of such classes under the restriction map res2 : H2
|
675 |
+
a(Ξ, W) β H2
|
676 |
+
a(Ξ, W) vanish, since the latter
|
677 |
+
is injective. By Lemma 3.4 these are Ulam classes of Ξ. But since Ξ is uniformly U-stable
|
678 |
+
with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2
|
679 |
+
a(Ξ, W) vanish, and we
|
680 |
+
conclude.
|
681 |
+
3.3
|
682 |
+
Pullbacks and amenable kernels
|
683 |
+
Let N β€ Ξ be an amenable normal subgroup, and let V be a dual asymptotic Banach β(Ξ/N)-
|
684 |
+
module, which pulls back to a dual asymptotic Banach βΞ-module. Precomposing cochains by
|
685 |
+
the projection βΞ β β(Ξ/N) deο¬nes the pullback pβ’ : Hβ’
|
686 |
+
a(Ξ/N, V) β Hβ’
|
687 |
+
a(Ξ, V). The following
|
688 |
+
can be proven via a similar argument as in Lemma 3.4:
|
689 |
+
Lemma 3.6. Let W be an Ulam β(Ξ/N)-module. Then W is also an Ulam βΞ-module, and
|
690 |
+
the pullback p2 : H2
|
691 |
+
a(Ξ/N, W) β H2
|
692 |
+
a(Ξ, W) sends Ulam classes to Ulam classes.
|
693 |
+
12
|
694 |
+
|
695 |
+
With this language, Proposition 1.9 is just a reformulation of Corollary 3.3:
|
696 |
+
Proposition (Proposition 1.9). Let N β€ Ξ be an amenable normal subgroup. Then the
|
697 |
+
pullback Hn
|
698 |
+
a(Ξ/N, V) β Hn
|
699 |
+
a(Ξ, V) is an isomorphism, for all n β₯ 0 and all dual asymptotic
|
700 |
+
Banach βΞ-modules V.
|
701 |
+
And we deduce Proposition 1.6 analogously:
|
702 |
+
Proposition (Proposition 1.6). Let N β€ Ξ be an amenable normal subgroup. If Ξ is uni-
|
703 |
+
formly U-stable with a linear estimate, then so is Ξ/N.
|
704 |
+
Proof. Suppose that Ξ is uniformly U-stable with a linear estimate, and let N be an amenable
|
705 |
+
normal subgroup of Ξ. We aim to show that Ξ/N is also uniformly U-stable with a linear
|
706 |
+
estimate. By Theorem 2.4, it suο¬ces to show that all Ulam classes supported on U vanish
|
707 |
+
in H2
|
708 |
+
a(Ξ/N, W), where W is an Ulam β(Ξ/N)-module. Now by Proposition 1.9, it suο¬ces
|
709 |
+
to show that the pullback of such classes under H2
|
710 |
+
a(Ξ/N, W) β H2
|
711 |
+
a(Ξ, W) vanish, since the
|
712 |
+
latter is an isomorphism. By Lemma 3.6 these are Ulam classes of Ξ. But since Ξ is uniformly
|
713 |
+
U-stable with a linear estimate, by Theorem 2.4 again, all Ulam classes in H2
|
714 |
+
a(Ξ, W) vanish,
|
715 |
+
and we conclude.
|
716 |
+
4
|
717 |
+
Asymptotic cohomology of lamplighters
|
718 |
+
In this section we prove Theorem 1.7, which we recall for the readerβs convenience:
|
719 |
+
Theorem. Let Ξ, Ξ be two countable groups, where Ξ is inο¬nite and amenable.
|
720 |
+
Then
|
721 |
+
Hn
|
722 |
+
a(Ξ β Ξ, V) = 0 for all n β₯ 1 and all ο¬nitary dual asymptotic Banach βΞ-modules V.
|
723 |
+
Remark 4.1. In fact, the theorem will hold for a larger class of coeο¬cients, obtained as
|
724 |
+
ultraproducts of separable Banach spaces. This does not however lead to a stronger stability
|
725 |
+
result: see Remark 6.1.
|
726 |
+
We start by ο¬nding a suitable Zimmer-amenable Ξ-space:
|
727 |
+
Lemma 4.2 ([Mon22, Corollary 8, Proposition 9]). Let Ξ, Ξ be two countable groups, where
|
728 |
+
Ξ is amenable. Let Β΅0 be a distribution of full support on Ξ, and let Β΅ be the product measure
|
729 |
+
on S := ΞΞ. Then S is a Zimmer-amenable (Ξ β Ξ)-space.
|
730 |
+
The reason why this space is useful for computations is that it is highly ergodic. Recall that
|
731 |
+
a Ξ-space S is ergodic if every Ξ-invariant function S β R is essentially constant. When S is
|
732 |
+
doubly ergodic, that is the diagonal action of Ξ on S ΓS is ergodic, we even obtain ergodicity
|
733 |
+
with separable coeο¬cients, meaning that for every Ξ-module E, every Ξ-equivariant map
|
734 |
+
S β E is essentially constant [Mon22, 2.A, 4.B].
|
735 |
+
Lemma 4.3 (Kolmogorov [Mon22, 2.A, 4.B]). Let Ξ, Ξ be two countable groups, where Ξ is
|
736 |
+
inο¬nite, and let S be as in Lemma 4.2. Then Sm is an ergodic (ΞβΞ)-space, for every m β₯ 1.
|
737 |
+
For our purposes, we will need an approximate version of ergodicity (namely, almost
|
738 |
+
invariant functions are almost constant) and also the module E will only be endowed with
|
739 |
+
an approximate action of Ξ. The ergodicity assumption still suο¬ces to obtain this:
|
740 |
+
13
|
741 |
+
|
742 |
+
Lemma 4.4. Let S be a probability measure Ξ-space, and suppose that the action of Ξ on S
|
743 |
+
is ergodic. Then whenever f : S β R is a measurable function such that β₯g Β· f β fβ₯ < Ξ΅ for
|
744 |
+
all g β Ξ, there exists a constant c β R such that |f(s) β c| < Ξ΅ for almost every s β S.
|
745 |
+
Proof. We deο¬ne F : S β R : s οΏ½β esssupgβΞf(gβ1s). By construction, F is Ξ-invariant,
|
746 |
+
and moreover β₯F β fβ₯ < Ξ΅. By ergodicity, F is essentially equal to a constant c, and thus
|
747 |
+
|f(s) β c| < Ξ΅ for a.e. s β S.
|
748 |
+
Lemma 4.5. Let S be a probability measure Ξ-space, and suppose that the action of Ξ on
|
749 |
+
S Γ S is ergodic. Suppose moreover, that E is a separable Banach space endowed with a map
|
750 |
+
Ξ Γ E β E : v οΏ½β g Β· v such that β₯g Β· vβ₯ = β₯vβ₯ for all g β Ξ, v β E.
|
751 |
+
Then whenever f : S β E is a measurable function such that β₯g Β· f β fβ₯ < Ξ΅ for all
|
752 |
+
g β Ξ, where (g Β· f)(s) = g Β· f(gβ1s), there exists a vector v β E such that β₯f(s) β vβ₯ < 3Ξ΅
|
753 |
+
for almost every s β S.
|
754 |
+
Proof. We deο¬ne F : S Γ S β R : (s, t) οΏ½β β₯f(s) β f(t)β₯. Then
|
755 |
+
β₯g Β· F β Fβ₯ = ess sup| β₯g Β· f(gβ1s) β g Β· f(gβ1t)β₯ β β₯f(s) β f(t)β₯ |
|
756 |
+
β€ ess supβ₯g Β· f(gβ1s) β g Β· f(gβ1t) β (f(s) β f(t))β₯ β€ 2β₯g Β· f β fβ₯ < 2Ξ΅.
|
757 |
+
By the previous lemma, there exists a constant c such that |F(s, t) β c| < 2Ξ΅ for all Ξ΅ > 0. If
|
758 |
+
c < Ξ΅, then |f(s) β f(t)| < 3Ξ΅ for a.e. s, t β S, which implies the statement.
|
759 |
+
Otherwise, β₯f(s) β f(t)β₯ > Ξ΅ for a.e. s, t β S. Let D β E be a countable dense subset.
|
760 |
+
Then for each d β D the set f β1(BΞ΅/2(d)) is a measurable subset of S, and the union of
|
761 |
+
such sets covers S. Since D is countable, there must exist d β D such that f β1(BΞ΅/2(d)) has
|
762 |
+
positive measure. But for all s, t in this set, β₯f(s) β f(t)β₯ < Ξ΅, a contradiction.
|
763 |
+
We thus obtain:
|
764 |
+
Proposition 4.6. Let S be a doubly ergodic Ξ-space. Let (Vn)nβ₯1 be a sequence of separable
|
765 |
+
dual Banach spaces such that V = οΏ½
|
766 |
+
Ο
|
767 |
+
Vn has the structure of a dual asymptotic Banach
|
768 |
+
Ξ-module be the corresponding asymptotic βΞ-module. Then the natural inclusion ΛV
|
769 |
+
βΞ β
|
770 |
+
ΛLβ(βS, V)
|
771 |
+
βΞ is an isomorphism.
|
772 |
+
Proof. Let f β Lβ
|
773 |
+
b (βS, V) = οΏ½
|
774 |
+
Ο
|
775 |
+
Lβ(S, Vn) be a lift of an element Λf β ΛLβ(βS, V)
|
776 |
+
βΞ. We write
|
777 |
+
f = (fn)Ο. Then fact that Λf is βΞ-invariant means that for every sequence (gn)nβN β Ξ it holds
|
778 |
+
(gnΒ·fnβfn)Ο β Lβ
|
779 |
+
inf(βS, V). Since this holds for every sequence (gn)nβN, a diagonal argument
|
780 |
+
implies that there exists Ξ΅ β βRinf such that for every g β Ξ it holds (g Β· fn βfn)Ο β βRinf. It
|
781 |
+
then follows from Lemma 4.5 that there exist (vn)Ο β V such that (fn β 1vn) β Lβ
|
782 |
+
inf(βS, V).
|
783 |
+
Therefore f represents the same element of ΛLβ(βS, V) as the image of an element of V. Since
|
784 |
+
Λf is βΞ-invariant, the corresponding element is actually in ΛV
|
785 |
+
βΞ.
|
786 |
+
We are ο¬nally ready to prove Theorem 1.7:
|
787 |
+
Proof of Theorem 1.7. Let Ξ, Ξ be countable groups, where Ξ is inο¬nite and amenable. By
|
788 |
+
Lemma 4.2, using the same notation, S is a Zimmer-amenable (Ξ β Ξ)-space. Therefore we
|
789 |
+
can apply Theorem 2.3, and obtain that the following complex computes Hβ
|
790 |
+
a(Ξ β Ξ; V):
|
791 |
+
0
|
792 |
+
Λ
|
793 |
+
d0
|
794 |
+
ββ ΛLβ(βS, V)
|
795 |
+
βΞ
|
796 |
+
Λ
|
797 |
+
d1
|
798 |
+
ββ ΛLβ((βS)2, V)
|
799 |
+
βΞ
|
800 |
+
Λ
|
801 |
+
d2
|
802 |
+
ββ ΛLβ((βS)3, V)
|
803 |
+
βΞ
|
804 |
+
Λ
|
805 |
+
d3
|
806 |
+
ββ Β· Β· Β·
|
807 |
+
14
|
808 |
+
|
809 |
+
Now by Lemma 4.3, Sm is a doubly ergodic (Ξ β Ξ)-space, for every m β₯ 1. Thus Proposition
|
810 |
+
4.6 applies, and the natural inclusion ΛV
|
811 |
+
βΞ β ΛLβ((βS)m, V)
|
812 |
+
βΞ is an isomorphism for every
|
813 |
+
m β₯ 1. Thus the above complex is isomorphic to
|
814 |
+
0
|
815 |
+
Λ
|
816 |
+
d0
|
817 |
+
ββ ΛV
|
818 |
+
βΞ
|
819 |
+
Λ
|
820 |
+
d1
|
821 |
+
ββ ΛV
|
822 |
+
βΞ
|
823 |
+
Λ
|
824 |
+
d2
|
825 |
+
ββ ΛV
|
826 |
+
βΞ
|
827 |
+
Λd3
|
828 |
+
ββ Β· Β· Β·
|
829 |
+
Each diο¬erential Λ
|
830 |
+
dm is an alternating sum of (m+1) terms all equal to each other. Therefore
|
831 |
+
Λ
|
832 |
+
dm is the identity whenever m is even, and it vanishes whenever m is odd. The conclusion
|
833 |
+
follows.
|
834 |
+
5
|
835 |
+
Thompson groups
|
836 |
+
In this section we prove Theorem 1.2. The statement for F β² will be a special case of a more
|
837 |
+
general result for a large family of self-similar groups. The most general statement is the
|
838 |
+
following:
|
839 |
+
Theorem 5.1. Let Ξ be a group, Ξ0 a subgroup with the following properties:
|
840 |
+
1. There exists g β Ξ such that the groups {giΞ0gβi : i β Z} pairwise commute;
|
841 |
+
2. Every ο¬nite subset of Ξ is contained in some conjugate of Ξ0.
|
842 |
+
Then Hn
|
843 |
+
a(Ξ, V) = 0 for all n β₯ 1 and all ο¬nitary dual asymptotic Banach βΞ-modules V. In
|
844 |
+
particular, Ξ is uniformly U-stable, with a linear estimate.
|
845 |
+
The theorem applies to the following large family of groups of homeomorphisms of the
|
846 |
+
real line:
|
847 |
+
Corollary 5.2. Let Ξ be a proximal, boundedly supported group of orientation-preserving
|
848 |
+
homeomorphisms of the line. Then Hn
|
849 |
+
a(Ξ, V) = 0 for all n β₯ 1 and all ο¬nitary dual asymptotic
|
850 |
+
Banach βΞ-modules V. In particular, Ξ is uniformly U-stable, with a linear estimate.
|
851 |
+
Remark 5.3. The fact that such groups have no quasimorphisms is well-known: see e.g.
|
852 |
+
[GG17, FFL21, Mon22].
|
853 |
+
We refer the reader to Section 5.2 for the relevant deο¬nitions. In Corollary 5.8 we will
|
854 |
+
apply Corollary 5.2 to Thompsonβs group F β²; the result for Thompsonβs group F will follow
|
855 |
+
from Proposition 1.5. We deduce the stability of Thompsonβs group T and V from these
|
856 |
+
general criteria in Section 5.3.
|
857 |
+
5.1
|
858 |
+
Self-similar groups
|
859 |
+
In this section we prove Theorem 5.1. This will be done in a series of lemmas:
|
860 |
+
Lemma 5.4. Let Ξ be a group, and suppose that there exists g β Ξ and Ξ0 β€ Ξ such that
|
861 |
+
{giΞ0gβi : i β Z} pairwise commute. Then there exists an epimorphism Ξ0 β Z β β¨Ξ0, gβ© with
|
862 |
+
amenable (in fact, metabelian) kernel.
|
863 |
+
This is well-known and stated without proof in [Mon22]. We include a proof for com-
|
864 |
+
pleteness.
|
865 |
+
15
|
866 |
+
|
867 |
+
Proof. To make a clear distinction, we denote by H the abstract group Ξ0, and by Ξ0 the
|
868 |
+
subgroup of Ξ. So we want to construct an epimorphism H βZ β β¨Ξ0, gβ© β€ Ξ with metabelian
|
869 |
+
kernel. We deο¬ne naturally
|
870 |
+
Ο((gi)iβZ, p) =
|
871 |
+
οΏ½οΏ½
|
872 |
+
iβZ
|
873 |
+
tigitβi
|
874 |
+
οΏ½
|
875 |
+
tp.
|
876 |
+
Note that this product is well-deο¬ned since there are only ο¬nitely many non-identity terms,
|
877 |
+
and the order does not matter since diο¬erent conjugates commute. By construction Ο is
|
878 |
+
injective on Hi, that is the copy of H supported on the i-th coordinate in H β Z.
|
879 |
+
Let
|
880 |
+
K := ker Ο β© οΏ½
|
881 |
+
i Hi, and note that K is the kernel of the retraction H β Z β Z restricted to
|
882 |
+
ker Ο. So it suο¬ces to show that K is abelian.
|
883 |
+
Let g, h β K and write them as (gi)iβZ and (hi)iβZ (we omit the Z-coordinate since it is
|
884 |
+
always 0). We need to show that g and h commute. We have
|
885 |
+
1Ξ = Ο(g) =
|
886 |
+
οΏ½
|
887 |
+
iβZ
|
888 |
+
tigitβi
|
889 |
+
and thus
|
890 |
+
g0 =
|
891 |
+
οΏ½
|
892 |
+
iΜΈ=0
|
893 |
+
tigitβi β Ξ.
|
894 |
+
But now g0 belongs to a group generated by conjugates of Ξ0 in Ξ that commute with it. In
|
895 |
+
particular this implies that g0 and h0 commute in Ξ. Since Ο|H0 is injective, this shows that
|
896 |
+
g0 and h0 commute in H0. Running the same argument on the other coordinates, we obtain
|
897 |
+
that gi and hi commute in Hi, for all i β Z, and thus g and h commute.
|
898 |
+
The next facts are all contained in the literature:
|
899 |
+
Lemma 5.5 ([Mon22, Proposition 10]). Suppose that Ξ0 β€ Ξ is such that every ο¬nite subset
|
900 |
+
of Ξ is contained in some Ξ-conjugate of Ξ0. Then Ξ0 is coamenable in Ξ.
|
901 |
+
Lemma 5.6 ([MP03]). Let K β€ H β€ Ξ.
|
902 |
+
1. If K is coamenable in Ξ, then H is coamenable in Ξ;
|
903 |
+
2. If K is coamenable in H and H is coamenable in Ξ, then K is coamenable in Ξ.
|
904 |
+
Remark 5.7. We warn the reader that if K is coamenable in Ξ, then K need not be
|
905 |
+
coamenable in H [MP03].
|
906 |
+
We are now ready to prove Theorem 5.1:
|
907 |
+
Proof of Theorem 5.1. Let Ξ, Ξ0 and g be as in the statement. By Lemma 5.4, there exists
|
908 |
+
a map Ξ0 β Z β β¨Ξ0, gβ© with metabelian kernel. By Theorem 1.7 and Proposition 1.9, we
|
909 |
+
have Hn
|
910 |
+
a(β¨Ξ0, gβ©, V) for all n β₯ 1 and all ο¬nitary dual asymptotic Banach βΞ-modules V. Now
|
911 |
+
by Lemma 5.5, Ξ0 is coamenable in Ξ. Finally, by Lemma 5.6, β¨Ξ0, gβ© is coameanble in Ξ.
|
912 |
+
Proposition 1.8 allows to conclude.
|
913 |
+
5.2
|
914 |
+
Groups of homeomorphisms of the line
|
915 |
+
Let Ξ be a group acting by homeomorphisms on the real line. We say that the action is
|
916 |
+
proximal if for all reals a < b and c < d there exists g β Ξ such that g Β· a < c < d < g Β· b.
|
917 |
+
The support of g β Ξ is the set {x β R : g Β· x ΜΈ= x}. We say that Ξ is boundedly supported if
|
918 |
+
every element has bounded support. Note that boundedly supported homeomorphisms are
|
919 |
+
automatically orientation-preserving.
|
920 |
+
16
|
921 |
+
|
922 |
+
Proof of Corollary 5.2. Let Ξ be as in the statement. Let Ξ0 be the subgroup of elements
|
923 |
+
whose support is contained in [0, 1]. Let g β Ξ be such that g(0) > 1: such an element exists
|
924 |
+
because the action of Ξ is proximal. Then it follows by induction, and the fact that Ξ is
|
925 |
+
orientation-preserving, that the intervals {gi[0, 1] : i β Z} are pairwise disjoint. Therefore
|
926 |
+
the conjugates giΞ0gβi pairwise commute.
|
927 |
+
Since Ξ is boundedly supported, for every ο¬nite subset A β Ξ there exists n such that
|
928 |
+
the support of each element of A is contained in [βn, n]. By proximality, there exists h β Ξ
|
929 |
+
such that h(0) < βn and h(1) > n. Then hΞ0hβ1 is the subgroup of elements whose support
|
930 |
+
is contained in [βn, n], in particular it contains A.
|
931 |
+
Thus Theorem 5.1 applies and we conclude.
|
932 |
+
Let us now show how to obtain the statements on F and F β² from Theorem 1.2 from
|
933 |
+
Corollary 5.2 and Proposition 1.5.
|
934 |
+
We refer the reader to [CFP96] for more details on
|
935 |
+
Thompsonβs groups.
|
936 |
+
Thompsonβs group F is the group of orientation-preserving piecewise linear homeomor-
|
937 |
+
phisms of the interval, with breakpoints in Z[1/2] and slopes in 2Z. The derived subgroup F β²
|
938 |
+
coincides with the subgroup of boundedly supported elements. The action of F β² (and thus
|
939 |
+
F) on [0, 1] preserves Z[1/2] β© (0, 1), and acts highly transitively on it; that is, for every pair
|
940 |
+
of ordered n-tuples in Z[1/2] β© (0, 1) there exists an element of F β² sending one to the other.
|
941 |
+
Corollary 5.8. Thompsonβs groups F and F β² are uniformly U-stable, with a linear estimate.
|
942 |
+
Proof. We identify (0, 1) with the real line. The group F β² is boundedly supported, and it is
|
943 |
+
proximal, since it acts transitively on ordered pairs of a dense set. Therefore Corollary 5.2
|
944 |
+
applies and F β² is uniformly U-stable, with a linear estimate.
|
945 |
+
Since the quotient F/F β² is abelian, thus amenable, we see that F β² is coamenable in F, and
|
946 |
+
thus conclude from Proposition 1.5 that F β² is uniformly U-stable, with a linear estimate.
|
947 |
+
Remark 5.9. We could also deduce the stability of F from the stability of F β² more directly,
|
948 |
+
without appealing to Proposition 1.5. Indeed, since F β² is uniformly U-stable, simple, and
|
949 |
+
not linear, every homomorphism F β² β U(n) is trivial - something we will come back to in
|
950 |
+
the next section. Therefore uniform U-stability of F β² implies that every uniform asymptotic
|
951 |
+
homomorphism F β² β U is uniformly close to the trivial one. It follows that every uniform
|
952 |
+
asymptotic homomorphism F β U is uniformly asymptotically close to one that factors
|
953 |
+
through Z2. We conclude by the stability of amenable groups [Kaz82, GLMR23].
|
954 |
+
Other groups to which these criteria apply include more piecewise linear groups [BS16],
|
955 |
+
such as the SteinβThompson groups [Ste92], or the golden ratio Thompson group of Cleary
|
956 |
+
[Cle00, BNR21]. In such generality some more care is needed, since the commutator subgroup
|
957 |
+
is sometimes a proper subgroup of the boundedly supported subgroup. The criteria also apply
|
958 |
+
for the piecewise proejective groups of Monod [Mon13] and LodhaβMoore [LM16]. In this
|
959 |
+
case, further care is needed, since the role of the commutator subgroup in the proofs above
|
960 |
+
has to be taken by the double commutator subgroup [BLR18]. This ties back to Question
|
961 |
+
1.10 from the introduction.
|
962 |
+
5.3
|
963 |
+
T and V
|
964 |
+
In this section, we show how our previous results allow to prove stability of groups of home-
|
965 |
+
omorphisms of the circle and of the Cantor set as well. For simplicity of the exposition, we
|
966 |
+
17
|
967 |
+
|
968 |
+
only focus on Thompsonβs groups T and V , but the proofs generalize to some analogously
|
969 |
+
deο¬ned groups, with the appropriate modiο¬cations. Our proof will involve a bounded gen-
|
970 |
+
eration argument for stability that was pioneered in [BOT13]. We will only use it a simple
|
971 |
+
version thereof, closer to the one from [BC20]. Recall that Ξ is said to be boundedly generated
|
972 |
+
by the collection of subgroups H if there exists k β₯ 1 such that the sets {H1 Β· Β· Β· Hk : Hi β H}
|
973 |
+
cover Ξ.
|
974 |
+
Lemma 5.10. Let Ξ be a discrete group. Suppose that there exists a subgroup H β€ Ξ with
|
975 |
+
the following properties:
|
976 |
+
1. Every homomorphism H β U(n) is trivial;
|
977 |
+
2. H is uniformly U-stable (with a linear estimate);
|
978 |
+
3. Ξ is boundedly generated by the conjugates of H.
|
979 |
+
Then Ξ is uniformly U-stable (with a linear estimate).
|
980 |
+
Proof. Let Οn : Ξ β U(dn) be a uniform asymptotic homomorphism with def(Οn) =: Ξ΅n.
|
981 |
+
Then Οn|H : H β U(dn) is a uniform asymptotic homomorphism of H, therefore it is Ξ΄n-
|
982 |
+
close to a homomorphism, where Ξ΄n β 0. But by assumption such a homomorphism must
|
983 |
+
be trivial, so β₯Οn(h) β Iknβ₯ β€ Ξ΄n for all n. The same holds for all conjugates of H, up to
|
984 |
+
replacing Ξ΄n by Ξ΄n + 2Ξ΅n.
|
985 |
+
By bounded generation, there exists k β₯ 1 such that each g β Ξ can be written as
|
986 |
+
g = h1 Β· Β· Β· hk, where each hi belongs to a conjugate of H. We estimate:
|
987 |
+
β₯Οn(g) β Idnβ₯ =
|
988 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½Οn
|
989 |
+
οΏ½ k
|
990 |
+
οΏ½
|
991 |
+
i=1
|
992 |
+
hi
|
993 |
+
οΏ½
|
994 |
+
β Idn
|
995 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½ β€
|
996 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½Οn
|
997 |
+
οΏ½kβ1
|
998 |
+
οΏ½
|
999 |
+
i=1
|
1000 |
+
hi
|
1001 |
+
οΏ½
|
1002 |
+
Οn(hk) β Idn
|
1003 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½ + Ξ΅n
|
1004 |
+
=
|
1005 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½Οn
|
1006 |
+
οΏ½kβ1
|
1007 |
+
οΏ½
|
1008 |
+
i=1
|
1009 |
+
hi
|
1010 |
+
οΏ½
|
1011 |
+
β Idn
|
1012 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½ + β₯Οn(hk) β Idnβ₯ + Ξ΅n β€ Β· Β· Β·
|
1013 |
+
Β· Β· Β· β€
|
1014 |
+
k
|
1015 |
+
οΏ½
|
1016 |
+
i=1
|
1017 |
+
β₯Οn(hi) β Idnβ₯ + kΞ΅n β€ k(Ξ΄n + Ξ΅n).
|
1018 |
+
Therefore Οn is k(Ξ΄n + Ξ΅n)-close to the trivial homomorphism, and we conclude.
|
1019 |
+
Thompsonβs group T is the group of orientation-preserving piecewise linear homeomor-
|
1020 |
+
phisms of the circle R/Z preserving Z[1/2]/Z, with breakpoints in Z[1/2]/Z, and slopes in
|
1021 |
+
2Z. Given x β Z[1/2]/Z, the stabilizer of x is naturally isomorphic to F. Moreover, the germ
|
1022 |
+
stabilizer T(x) (i.e. the group consisting of elements that ο¬x pointwise some neighbourhood
|
1023 |
+
of x) is isomorphic to F β².
|
1024 |
+
Corollary 5.11. Thompsonβs group T is uniformly U-stable with a linear estimate.
|
1025 |
+
Proof. We claim that Lemma 5.10 applies with H = T(0) βΌ= F β². Item 1. follows from the
|
1026 |
+
fact F β² does not embed into U(n) (for instance because it contains F as a subgroup, which
|
1027 |
+
is ο¬nitely generated and not residually ο¬nite, and so cannot be linear by Malβcevβs Theorem
|
1028 |
+
[Mal40]), and F β² is simple [CFP96]. Also, F β² is uniformly U-stable with a linear estimate,
|
1029 |
+
by Corollary 5.8. Therefore we are left to show the bounded generation statement. We will
|
1030 |
+
18
|
1031 |
+
|
1032 |
+
show that for every g β T there exist x, y β Z[1/2]/Z such that g β T(x)T(y). This suο¬ces
|
1033 |
+
because T acts transitively on Z[1/2]/Z, so T(x) and T(y) are both conjugate to H = T(0).
|
1034 |
+
Let 1 ΜΈ= g β T, and choose x ΜΈ= y β Z[1/2]/Z such that g(y) /β {x, y}. Let I be a small
|
1035 |
+
dyadic arc around y such that x /β I and x, y /β g(I). Choose an element f β T(x) such
|
1036 |
+
that f(I) = g(I). Let h be an element supported on I such that h|I = f β1g|I. Since x /β I,
|
1037 |
+
we also have h β T(x). Moreover hβ1f β1g|I = id|I, so hβ1f β1g β G(y). We conclude that
|
1038 |
+
g = fh Β· hβ1f β1g β T(x)T(y).
|
1039 |
+
Thompsonβs group V can be described as a group of homeomorphisms of the dyadic Cantor
|
1040 |
+
set X := 2N. A dyadic brick is a clopen subset of the form XΟ := Ο Γ 2N>k, for some Ο β 2k,
|
1041 |
+
and every two dyadic bricks are canonically homeomorphic via XΟ β XΟ : Ο Γ x οΏ½β Ο Γ x.
|
1042 |
+
An element g β V is deο¬ned by two ο¬nite partitions of V of the same size into dyadic bricks,
|
1043 |
+
that are sent to each other via canonical homeomorphisms.
|
1044 |
+
Corollary 5.12. Thompsonβs group V is uniformly U-stable, with a linear estimate.
|
1045 |
+
The proof is very similar to the proof for T, so we only sketch it:
|
1046 |
+
Sketch of proof. Let x β 2N be a dyadic point, that is a sequence that is eventually all 0,
|
1047 |
+
and let V (x) denote the subgroup of V consisting of elements that ο¬x a neighbourhood of x
|
1048 |
+
pointwise. The same argument as in the proof of Corollary 5.11 shows that V is boundedly
|
1049 |
+
generated by conjugates of V (x).
|
1050 |
+
Now V (x) is isomorphic to a directed union of copies of V , which is ο¬nitely generated
|
1051 |
+
and simple [CFP96], so by Malβcevβs Theorem every homomorphism V (x) β U(n) is trivial.
|
1052 |
+
Finally, V (x) contains a copy V0 of V such that the pair (V (x), V0) satisο¬es the hypotheses of
|
1053 |
+
Theorem 5.1 (see [And22, Proposition 4.3.4] and its proof). We conclude by Lemma 5.10.
|
1054 |
+
6
|
1055 |
+
Sharpness of our results
|
1056 |
+
In this section we point out certain ways in which our results are sharp, by providing explicit
|
1057 |
+
counterexamples to generalizations and converses.
|
1058 |
+
Remark 6.1. There is a notion of strong Ulam stability, where one takes U to include unitary
|
1059 |
+
groups of inο¬nite-dimensional Hilbert spaces as well, typically equipped with the operator
|
1060 |
+
norm. It is shown in [BOT13] that a subgroup of a strongly Ulam stable group is Ulam
|
1061 |
+
stable. Therefore it is clear that Theorem 1.3 does not hold for strong Ulam stability. Even
|
1062 |
+
restricting to separable Hilbert spaces does not help: it follows from the construction in
|
1063 |
+
[BOT13] that if a countable group contains a free subgroup, then separable Hilbert spaces
|
1064 |
+
already witness the failure of strong Ulam stability.
|
1065 |
+
The framework of stability via asymptotic cohomology can be developed in this general
|
1066 |
+
setting as well, with dual asymptotic Banach modules that are not ο¬nitary. Therefore the
|
1067 |
+
counterexample above shows that Theorem 1.7 really needs the ο¬nitary assumption. The fact
|
1068 |
+
that we could obtain dual asymptotic Banach modules obtained as ultraproducts of separable
|
1069 |
+
spaces, analogously to [Mon22], does not help, since the dual asymptotic Banach modules
|
1070 |
+
arising from a stability problem over inο¬nite-dimensional Hilbert spaces are not of this form,
|
1071 |
+
even when the Hilbert space are separable.
|
1072 |
+
19
|
1073 |
+
|
1074 |
+
Remark 6.2. We proved in Proposition 1.5 that if Ξ is coamenable in Ξ and Ξ is uniformly
|
1075 |
+
U-stable with a linear estimate, then so is Ξ. The converse does not hold. Let Fn be a free
|
1076 |
+
group of rank n β₯ 2. Then Ξ := οΏ½
|
1077 |
+
nβ₯1 Fn admits a non-trivial quasimorphism, so it is not
|
1078 |
+
uniformly U(1)-stable [BOT13], in particular it is not uniformly U-stable. However, Ξ is
|
1079 |
+
coamenable in Fn β Z, which is uniformly U-stable with a linear estimate by Theorem 1.3.
|
1080 |
+
On the other hand, if we replace βcoamenableβ by βο¬nite indexβ, then the converse does
|
1081 |
+
hold. This follows from the induction procedure in [BOT13] for Ulam stability, as detailed
|
1082 |
+
in [Gam11, Lemma II.22]; the same proof can be generalized to all submultiplicative norms
|
1083 |
+
[GLMR23, Lemma 4.3.6].
|
1084 |
+
Remark 6.3. We proved in Proposition 1.6 that if N is an amenable normal subgroup of Ξ,
|
1085 |
+
and Ξ is uniformly U-stable with a linear estimate, then so is Ξ/N. The converse does not
|
1086 |
+
hold. Let Ξ be the lift of Thompsonβs group T, that is, the group of orientation-preserving
|
1087 |
+
homeomorphisms of R that commute with the group Z of integer translations and induce T
|
1088 |
+
on the quotient R/Z. These groups ο¬t into a central extension
|
1089 |
+
1 β Z β Ξ β T β 1.
|
1090 |
+
Now T is uniformly U-stable with a linear estimate, by Corollary 5.11, however Ξ is not:
|
1091 |
+
it is not even uniformly U(1)-stable, by [BOT13], since it has a non-trivial quasimorphism
|
1092 |
+
[GS87].
|
1093 |
+
The next two remarks show that some results from [GLMR23] are also sharp.
|
1094 |
+
Remark 6.4. The fundamental result of [GLMR23] is that the vanishing of asymptotic
|
1095 |
+
cohomology implies uniform U-stability. The converse does not hold. Indeed, since u(1) βΌ= R
|
1096 |
+
with trivial adjoint action (because U(1) is abelian), it follows that the implication of Theorem
|
1097 |
+
2.4 specializes to: If H2
|
1098 |
+
a(Ξ, βR) = 0, then Ξ is uniformly U(1)-stable, where βR is seen as a
|
1099 |
+
dual asymptotic βΞ-module with a trivial βΞ action.
|
1100 |
+
Now, let again Ξ be the lift of Thompsonβs group T, so that Ξ contains a central subgroup
|
1101 |
+
Z with Ξ/Z βΌ= T. The fact that Ξ is not uniformly U(1)-stable implies that H2
|
1102 |
+
a(Ξ, βR) ΜΈ= 0.
|
1103 |
+
But Proposition 1.9 then shows that H2
|
1104 |
+
a(T, βR) ΜΈ= 0 either. However, T is uniformly U-stable
|
1105 |
+
with a linear estimate, by Corollary 5.11. Morally, this is due to the fact that H2
|
1106 |
+
b(Ξ, R) βΌ=
|
1107 |
+
H2
|
1108 |
+
b(T, R) βΌ= R, but the former is spanned by a quasimorphisms, while the latter is not (see
|
1109 |
+
e.g. [Cal09, Chapter 5]).
|
1110 |
+
Remark 6.5. In [BOT13] it is shown that groups admitting non-trivial quasimorphisms are
|
1111 |
+
not uniformly U(1)-stable. In [GLMR23, Proposition 1.0.6] this result is sharpened: the
|
1112 |
+
authors show that Ξ is uniformly U(1)-stable if and only if the non-zero element in the image
|
1113 |
+
of H2
|
1114 |
+
b(Ξ, Z) in H2
|
1115 |
+
b(Ξ, R) have Gromov norm β₯Β·β₯ bounded away from 0. They use this to show
|
1116 |
+
that a ο¬nitely presented group is uniformly U(1)-stable if and only if it admits no non-trivial
|
1117 |
+
quasimorphism [GLMR23, Corollary 1.0.10].
|
1118 |
+
The hypothesis of ο¬nite presentability is necessary. Let Ξn denote the lift of Thompsonβs
|
1119 |
+
group T to R/nZ. That is, Ξn is the group of orientation-preserving homeomorphisms of the
|
1120 |
+
topological circle R/nZ, which commute with the cyclic group of rotations Z/nZ and induce
|
1121 |
+
T on the quotient R/Z. Now T has no unbounded quasimorphisms (see e.g. [Cal09, Chapter
|
1122 |
+
5]), and so Ξn also has no unbounded quasimorphisms (this follows from the left exactness
|
1123 |
+
of the quasimorphism functor [Cal09, Remark 2.90]). Therefore the group Ξ := οΏ½
|
1124 |
+
nβ₯2 Ξn has
|
1125 |
+
no unbounded quasimorphisms.
|
1126 |
+
20
|
1127 |
+
|
1128 |
+
However, we claim that Ξ is not uniformly U(1)-stable. By [GLMR23, Proposition 1.0.6],
|
1129 |
+
it suο¬ces to show that there exist bounded cohomology classes 0 ΜΈ= Οn β im(H2
|
1130 |
+
b(Ξ, Z) β
|
1131 |
+
H2
|
1132 |
+
b(Ξ, R) such that β₯Οnβ₯ β 0. We let Οn be the Euler class of the representation Ξ β Ξn β
|
1133 |
+
Homeo+(R/nZ), which admits an integral representative and so lies in the image of H2
|
1134 |
+
b(Ξ, Z)
|
1135 |
+
(see [Ghy01] for more information about Euler classes of circle actions). Moreover, using the
|
1136 |
+
terminology of [Bur11], the representation is minimal, unbounded, and has a centralizer of
|
1137 |
+
order n. Therefore β₯Οnβ₯ = 1/2n by [Bur11, Corollary 1.6], and we conclude.
|
1138 |
+
Note that Ξ is countable but inο¬nitely generated. It would be interesting to produce a
|
1139 |
+
ο¬nitely generated example (which would necessarily be inο¬nitely presented).
|
1140 |
+
7
|
1141 |
+
Approximation properties
|
1142 |
+
In this section we discuss open problems about approximation properties of the groups treated
|
1143 |
+
in this paper, and their relation to our results. We recall the following notions:
|
1144 |
+
Deο¬nition 7.1. Let G be a family of metric groups. We say that Ξ is (pointwise, uniformly)
|
1145 |
+
G-approximable if there exists a (pointwise, uniform) asymptotic homomorphism Οn : Ξ β
|
1146 |
+
Gn β G that is moreover asymptotically injective, meaning that for all g β Ξ, g ΜΈ= 1 it holds
|
1147 |
+
lim inf
|
1148 |
+
nββ Οn(g) > 0.
|
1149 |
+
The above terminology is not standard: most of the literature only deals with the point-
|
1150 |
+
wise notion, and refers to that as G-approximability. The notion of uniform approximability
|
1151 |
+
appeared in [FF21] with the name of strong G-approximability.
|
1152 |
+
Example 7.2. If G is the family of symmetric groups equipped with the normalized Hamming
|
1153 |
+
distance, then pointwise G-approximable groups are called soο¬c [Gro99, Wei00].
|
1154 |
+
If G is the family of unitary groups equipped with the HilbertβSchmidt distance, then
|
1155 |
+
pointwise G-approximable groups are called hyperlinear [RΛ08].
|
1156 |
+
All amenable and residually ο¬nite groups are soο¬c, and all soο¬c groups are hyperlinear.
|
1157 |
+
It is a major open question to determine whether there exists a non-soο¬c group.
|
1158 |
+
In our context of submultiplicative norms on unitary groups, the following two notions of
|
1159 |
+
approximability have been studied:
|
1160 |
+
Example 7.3. Let G be the family of unitary groups equipped with the operator norm.
|
1161 |
+
Then pointwise G-approximable groups are called MF [CDE13]. All amenable groups are
|
1162 |
+
MF [TWW17]. It is an open problem to determine whether there exists a non-MF group.
|
1163 |
+
Let G be the family of unitary groups equipped with the Frobenius norm, or more generally
|
1164 |
+
with a Schatten p-norm, for 1 < p < β. Groups that are not pointwise G-approximable have
|
1165 |
+
been constructed in [DCGLT20, LO20]. This is one of the very few cases in which a non-
|
1166 |
+
example for pointwise approximability is known.
|
1167 |
+
The following observation is well-known, and due to Glebsky and Rivera [GR09] and
|
1168 |
+
Arzhantseva and PΛaunescu in the pointwise symmetric case [AP15]. We give a general proof
|
1169 |
+
for reference:
|
1170 |
+
Proposition 7.4. Let G be a family of metric groups that are locally residually ο¬nite, and
|
1171 |
+
let Ξ be a ο¬nitely generated group. Suppose that Ξ is (pointwise, uniformly) G-stable and
|
1172 |
+
(pointwise, uniformly) G-approximable. Then Ξ is residually ο¬nite.
|
1173 |
+
21
|
1174 |
+
|
1175 |
+
The hypothesis on G covers all cases above. When the groups in G are ο¬nite, this is clear,
|
1176 |
+
and when they are linear, this follows from Malβcevβs Theorem [Mal40].
|
1177 |
+
Proof. We proceed with the proof without specifying the type of asymptotic homomorphisms,
|
1178 |
+
closeness, and approximability: the reader should read everything as pointwise, or everything
|
1179 |
+
as uniform.
|
1180 |
+
Let Ο : Ξ β G be an asymptotically injective asymptotic homomorphism. By stability,
|
1181 |
+
there exists a sequence of homomorphisms Ο : Ξ β G which is asymptotically close to Ο.
|
1182 |
+
Since Ο is asymptotically injective, for each g β Ξ there exists N such that Οn(g) β₯ Ο for all
|
1183 |
+
n β₯ N and some Ο = Ο(g) > 0. Up to taking a larger N, we also have that Οn(g) β₯ Ο/2, in
|
1184 |
+
particular Οn(g) ΜΈ= 1. Since Οn(Ξ) is a ο¬nitely generated subgroup of Gn β G, it is residually
|
1185 |
+
ο¬nite by hypothesis, and so Οn(g) survives in some ο¬nite quotient of Οn(Ξ). Since this is also
|
1186 |
+
a ο¬nite quotient of Ξ, we conclude that Ξ is residually ο¬nite.
|
1187 |
+
In the special case of pointwise stability and Thompsonβs group F, we obtain the following
|
1188 |
+
more general version of a remark of Arzhantseva and Paunescu [AP15, Open problem]:
|
1189 |
+
Corollary 7.5. Let G be the family of symmetric groups with the normalized Hamming
|
1190 |
+
distance, the family of unitary groups with the HilbertβSchmidt norm, or the family of unitary
|
1191 |
+
groups with the operatorn norm. If Thompsonβs group F is pointwise G-stable, then it is not
|
1192 |
+
pointwise G-approximable, and in particular it is non-amenable.
|
1193 |
+
As we mentioned in the introduction, the amenability of Thompsonβs group F is one of
|
1194 |
+
the most outstanding open problems in modern group theory.
|
1195 |
+
Proof. Thompsonβs group F is not residually ο¬nite [CFP96]. So it follows from Proposition
|
1196 |
+
7.4 that it cannot be simultaneously pointwise G-stable and pointwise G-approximable. The
|
1197 |
+
last statement follows from the fact that amenable groups are soο¬c, hyperlinear, and MF.
|
1198 |
+
On the other hand, our results allow to settle the uniform approximability of Thompsonβs
|
1199 |
+
groups, with respect to unitary groups and submultiplicative norms:
|
1200 |
+
Corollary 7.6. As usual, let U be the family of unitary groups equipped with submultiplicative
|
1201 |
+
norms. Then Thompsonβs groups F, F β², T and V are not uniformly U-approximable. The
|
1202 |
+
same holds for Ξ β Ξ, whenever Ξ is inο¬nite and amenable, and Ξ is non-abelian.
|
1203 |
+
We remark that Thompsonβs groups T and V are generally regarded as good candidates
|
1204 |
+
for counterexamples to approximability problems.
|
1205 |
+
Proof. The statement for F, T and V follows from Theorem 1.2 and Proposition 7.4, together
|
1206 |
+
with the fact that they are not residually ο¬nite, and the statement for F β² (which is not ο¬nitely
|
1207 |
+
generated) follows from the fact that F β² contains a copy of F [CFP96]. The lamplighter case
|
1208 |
+
follows from Theorem 1.3 and Proposition 7.4, together with the fact that such lamplighters
|
1209 |
+
are not residually ο¬nite [Gru57].
|
1210 |
+
We do not know whether Thompsonβs groups are uniformly G-approximable, when G is
|
1211 |
+
the family of unitary groups equipped with the HilbertβSchmidt norm, and we conjecture
|
1212 |
+
that this is not the case. In the next section, we examine the case of symmetric groups via
|
1213 |
+
a more direct argument.
|
1214 |
+
22
|
1215 |
+
|
1216 |
+
7.1
|
1217 |
+
Approximations by symmetric groups
|
1218 |
+
We end by proving, by a cohomology-free argument, that some of the groups studied in this
|
1219 |
+
paper are not uniformly approximable by symmetric groups, in a strong sense. For the rest
|
1220 |
+
of this section, we denote by S the family of symmetric groups equipped with the normalized
|
1221 |
+
Hamming distance. Our main result is an analogue of Corollary 5.2 for this approximating
|
1222 |
+
family (see Section 5.2 for the relevant deο¬nitions):
|
1223 |
+
Proposition 7.7. Let Ξ be a proximal, boundedly supported group of orientation-preserving
|
1224 |
+
homeomorphisms of the line. Then every uniform asymptotic homomorphism Οn : Ξβ² β
|
1225 |
+
Skn β S is uniformly asymptotically close to the trivial one. In particular, Ξβ² is uniformly
|
1226 |
+
S-stable, and not uniformly S-approximable.
|
1227 |
+
The non-approximability follows from the fact that Ξβ² is non-trivial (see Lemma 7.8).
|
1228 |
+
Note that for Ξ as in the statement, Ξβ² is simple [GG17, Theorem 1.1], so in particular every
|
1229 |
+
homomorphism Ξβ² β Skn is trivial.
|
1230 |
+
The proof relies on known results on the ο¬exible uniform stability of amenable groups
|
1231 |
+
[BC20] and uniform perfection of groups with proximal actions [GG17]. The ο¬niteness of the
|
1232 |
+
groups in S will play a crucial role. We start with the following lemma:
|
1233 |
+
Lemma 7.8. Let Ξ be as in Proposition 7.7. Then Ξβ² is non-trivial, and the action of Ξβ² on
|
1234 |
+
the line has no global ο¬xpoints.
|
1235 |
+
Proof. If Ξβ² is trivial, then Ξ is abelian. This contradicts that the action is proximal and
|
1236 |
+
boundedly supported. Indeed, given g β Ξ, since g is centralized, the action of Ξ on R must
|
1237 |
+
preserve the support of g, which is a proper subset of R. But then the action cannot be
|
1238 |
+
proximal.
|
1239 |
+
Now the set of global ο¬xpoints of Ξβ² is a closed subset X β R. Since Ξβ² is normal in Ξ,
|
1240 |
+
the action of Ξ preserves X. But the action of Ξ on R is proximal, in particular every orbit
|
1241 |
+
is dense, and since X is closed we obtain X = R. That is, Ξβ² acts trivially on R. Since Ξ
|
1242 |
+
is a subgroup of Homeo+(R), this implies that Ξβ² is trivial, which contradicts the previous
|
1243 |
+
paragraph.
|
1244 |
+
We proceed with the proof:
|
1245 |
+
Proof of Proposition 7.7. It follows from [GG17, Theorem 1.1] that Ξβ² is 2-uniformly perfect;
|
1246 |
+
that is, every element of Ξβ² may be written as the product of at most 2 commutators (this
|
1247 |
+
uses the proximality hypothesis). Therefore it suο¬ces to show that there exists a constant C
|
1248 |
+
such that for all g, h β Ξβ² it holds dkn(Οn([g, h]), idkn) β€ CΞ΅n, where dkn denotes the Hamming
|
1249 |
+
distance on Skn and Ξ΅n := def(Οn). We drop the subscript n on Ο and Ξ΅ for clarity.
|
1250 |
+
Now let g, h β Ξβ², and let I, J β R be bounded intervals such that g is supported on I
|
1251 |
+
and h is supported on J. Since Ξβ² acts without global ο¬xpoints by Lemma 7.8, there exists
|
1252 |
+
t β Ξβ² such that t Β· inf(J) > sup(I). Since Ξβ² is orientation-preserving, the same holds for
|
1253 |
+
all powers of t. In particular [g, tihtβi] = 1 for all i β₯ 1. Next, we apply [BC20, Theorem
|
1254 |
+
1.2] to the amenable group β¨tβ©, to obtain an integer N such that kn β€ N β€ (1 + 1218Ξ΅)kn
|
1255 |
+
and a permutation Ο in SN such that dN(Ο(t)i, Ο i) β€ 2039Ξ΅ for all i β Z. Here dN denotes
|
1256 |
+
the normalized Hamming distance on the symmetric group SN, and Ο is extended to a map
|
1257 |
+
23
|
1258 |
+
|
1259 |
+
Ο : Ξβ² β SN with every Ο(g) ο¬xing each point in {kn + 1, . . . , N}.
|
1260 |
+
We compute (using
|
1261 |
+
Ο N! = idN):
|
1262 |
+
dkn(Ο([g, h]), idkn) β€ dN(Ο([g, h]), idN) β€ dN([Ο(g), Ο(h)], idN) + O(Ξ΅)
|
1263 |
+
= dN([Ο(g), Ο N!Ο(h)Ο βN!], idN) + O(Ξ΅)
|
1264 |
+
β€ dN([Ο(g), Ο(tN!)Ο(h)Ο(tβN!)], idN) + O(Ξ΅)
|
1265 |
+
β€ dN(Ο([g, tN!htβN!]), idN) + O(Ξ΅)
|
1266 |
+
= dN(Ο(1), idN) + O(Ξ΅) β€ O(Ξ΅).
|
1267 |
+
Thus, there exists a constant C independent of g and h (C = 20000 suο¬ces) such that
|
1268 |
+
dkn(Ο([g, h]), idkn) β€ CΞ΅, which concludes the proof.
|
1269 |
+
Corollary 7.9. Consider the Thompson groups F β², F, T.
|
1270 |
+
1. Every asymptotic homomorphism Οn : F β² β Skn β S is uniformly asymptotically close
|
1271 |
+
to the trivial one.
|
1272 |
+
2. Every asymptotic homomorphism Οn : F β Skn β S is uniformly asymptotically close
|
1273 |
+
to one that factors through the abelianization.
|
1274 |
+
3. Every asymptotic homomorphism Οn : T β Skn β S is uniformly asymptotically close
|
1275 |
+
to the trivial one.
|
1276 |
+
Proof. Item 1. is an instance of Proposition 7.7: indeed F β² satisο¬es the hypotheses for Ξ,
|
1277 |
+
and F β²β² = F β² since F β² is simple. For Item 2., pick a section Ο : Ab(F) β F, and deο¬ne
|
1278 |
+
Οn(g) := Οn(Ο(Ab(g))). Using that Οn|F β² is uniformly asymptotically close to the sequence
|
1279 |
+
of trivial maps, we obtain that Οn and Οn are uniformly asymptotically close, and Οn factors
|
1280 |
+
as F β Ab(F)
|
1281 |
+
Οnβ¦Ο
|
1282 |
+
ββββ Skn. Finally, Item 3. follows again from Item 1. and the fact that every
|
1283 |
+
element of T can be written as a product of two elements in isomorphic copies of F β² (see the
|
1284 |
+
proof of Corollary 5.11).
|
1285 |
+
The corollary immediately implies that F, F β² and T are not uniformly S-approximable,
|
1286 |
+
and that F β² and T are uniformly S-stable. Since F has inο¬nite abelianization, it follows from
|
1287 |
+
[BC20, Theorem 1.4] that it is not uniformly S-stable. However the corollary together with
|
1288 |
+
[BC20, Theorem 1.2] implies that it is ο¬exibly uniformly S-stable; that is, every uniform
|
1289 |
+
asymptotic homomorphism is uniformly close to a sequence of homomorphisms taking values
|
1290 |
+
in a symmetric group of slightly larger degree. The case of Thompsonβs group V can also be
|
1291 |
+
treated analogously (see the sketch of proof of Corollary 5.12).
|
1292 |
+
References
|
1293 |
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27
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|
1 |
+
arXiv:2301.08615v1 [hep-ph] 20 Jan 2023
|
2 |
+
Photo-production of lowest Ξ£β
|
3 |
+
1/2β state within the Regge-eο¬ective Lagrangian approach
|
4 |
+
Yun-He Lyu,1 Han Zhang,1 Neng-Chang Wei,2 Bai-Cian Ke,1 En Wang,1 and Ju-Jun Xie3, 2, 4
|
5 |
+
1School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan 450001, China
|
6 |
+
2School of Nuclear Sciences and Technology, University of Chinese Academy of Sciences, Beijing 101408, China
|
7 |
+
3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
|
8 |
+
4Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics,
|
9 |
+
Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China
|
10 |
+
(Dated: January 23, 2023)
|
11 |
+
Since the lowest Ξ£β state, with quantum numbers spin-parity JP = 1/2β, is far from estab-
|
12 |
+
lished experimentally and theoretically, we have performed a theoretical study on the Ξ£β
|
13 |
+
1/2β photo-
|
14 |
+
production within the Regge-eο¬ective Lagrangian approach. Taking into account that the Ξ£β
|
15 |
+
1/2β
|
16 |
+
couples to the Β―KN channel, we have considered the contributions from the t-channel K exchange
|
17 |
+
diagram. Moreover, these contributions from t-channel Kβ exchange, s-channel nucleon pole, u-
|
18 |
+
channel Ξ£ exchange, and the contact term, are considered. The diο¬erential and total cross sections
|
19 |
+
of the process Ξ³n β K+Ξ£ββ
|
20 |
+
1/2β are predicted with our model parameters. The results should be
|
21 |
+
helpful to search for the Ξ£β
|
22 |
+
1/2β state experimentally in future.
|
23 |
+
PACS numbers:
|
24 |
+
I.
|
25 |
+
INTRODUCTION
|
26 |
+
The study of the low-lying excited Ξβ and Ξ£β hyperon
|
27 |
+
resonances is one of the most important issues in hadron
|
28 |
+
physics.
|
29 |
+
Especially, since the Ξ(1405) was discovered
|
30 |
+
experimentally [1, 2], its nature has called many atten-
|
31 |
+
tions [3β8], and one explanation for Ξ(1405) is that it
|
32 |
+
is a Β―KN hadronic molecular state [9β14]. In addition,
|
33 |
+
the isospin I = 1 partner of the Ξ(1405), the lowest
|
34 |
+
Ξ£β
|
35 |
+
1/2β is crucial to understand the light baryon spec-
|
36 |
+
tra. At present, there is a Ξ£β(1620) with JP = 1/2β
|
37 |
+
listed in the latest version of Review of Particle Physics
|
38 |
+
(RPP) [15]. It should be stressed that the Ξ£β(1620) state
|
39 |
+
is a one-star baryon resonance, and many studies indicate
|
40 |
+
that the lowest Ξ£β
|
41 |
+
1/2β resonance is still far from estab-
|
42 |
+
lished, and its mass was predicted to lie in the range of
|
43 |
+
1380 βΌ 1500 MeV [13, 16β19]. Thus, searching for the
|
44 |
+
lowest Ξ£β
|
45 |
+
1/2β is helpful to understand the low-lying ex-
|
46 |
+
cited baryons with JP = 1/2β and the light ο¬avor baryon
|
47 |
+
spectra.
|
48 |
+
The analyses of the relevant data of the process
|
49 |
+
Kβp β ΞΟ+Οβ suggest that there may exist a Ξ£β
|
50 |
+
1/2β
|
51 |
+
resonance with mass about 1380 MeV [16, 17], which
|
52 |
+
is consistent with the predictions of the unquenched
|
53 |
+
quark models [20].
|
54 |
+
The analyses of the KβΞ£ photo-
|
55 |
+
production also indicate that the Ξ£β
|
56 |
+
1/2β is possibly buried
|
57 |
+
under the Ξ£β(1385) peak with mass of 1380 MeV [21],
|
58 |
+
and it is proposed to search for the Ξ£β
|
59 |
+
1/2β in the pro-
|
60 |
+
cess Ξc β Ξ·Ο+Ξ [22]. A more delicate analysis of the
|
61 |
+
CLAS data on the process Ξ³p β KΞ£Ο [23] suggests that
|
62 |
+
the Ξ£β
|
63 |
+
1/2β peak should be around 1430 MeV [13].
|
64 |
+
In
|
65 |
+
Refs. [24, 25], we suggest to search for such state in the
|
66 |
+
processes of Οc0(1P) β Β―Ξ£Ξ£Ο and Οc0(1P) β Β―ΞΞ£Ο. In
|
67 |
+
addition, Ref. [26] has found one Ξ£β
|
68 |
+
1/2β state with mass
|
69 |
+
around 1400 MeV by solving coupled channel scattering
|
70 |
+
equations, and Ref. [27] suggests to search for this state
|
71 |
+
in the photo-production process Ξ³p β K+Ξ£β0
|
72 |
+
1/2β.
|
73 |
+
Itβs worth mentioning that a Ξ£β(1480) resonance with
|
74 |
+
JP = 1/2β has been listed on the previous version of
|
75 |
+
RPP [28].
|
76 |
+
As early as 1970, the Ξ£β(1480) resonance
|
77 |
+
was reported in the ΞΟ+, Ξ£Ο, and p Β―K0 channels of the
|
78 |
+
Ο+p scattering in the Princeton-Pennsylvania Accelera-
|
79 |
+
tor 15-in.βΌhydrogen bubble chamber [29, 30]. In 2004,
|
80 |
+
a bump structure around 1480 MeV was observed in the
|
81 |
+
K0
|
82 |
+
Sp(Β―p) invariant mass spectrum of the inclusive deep
|
83 |
+
inelastic ep scattering by the ZEUS Collaboration [31].
|
84 |
+
Furthermore, a signal for a resonance at 1480 Β± 15 MeV
|
85 |
+
with width of 60 Β± 15 MeV was observed in the process
|
86 |
+
pp β K+pY β0 [32]. Theoretically, the Ξ£β(1480) was in-
|
87 |
+
vestigated within diο¬erent models [33β36]. In Ref. [36],
|
88 |
+
the S-wave meson-baryon interactions with strangeness
|
89 |
+
S = β1 were studied within the unitary chiral approach,
|
90 |
+
and one narrow pole with pole position of 1468βi 13 MeV
|
91 |
+
was found in the second Riemann sheet, which could be
|
92 |
+
associated with the Ξ£β(1480) resonance. However, the
|
93 |
+
Ξ£β(1480) signals are insigniο¬cant, and the existence of
|
94 |
+
this state still needs to be conο¬rmed within more precise
|
95 |
+
experimental measurements.
|
96 |
+
As we known, the photo-production reactions have
|
97 |
+
been used to study the excited hyperon states Ξ£β and Ξβ,
|
98 |
+
and the Crystal Ball [37β39], LEPS [40], and CLAS [23]
|
99 |
+
Collaborations have accumulated lots of relevant exper-
|
100 |
+
imental data.
|
101 |
+
For instance, with these data, we have
|
102 |
+
analyzed the process Ξ³p β KΞβ(1405) to deepen the un-
|
103 |
+
derstanding of the Ξβ(1405) nature in Ref. [41]. In order
|
104 |
+
to conο¬rm the existence of the Ξ£β(1480), we propose to
|
105 |
+
investigate the process Ξ³N β KΞ£β(1480) 1 within the
|
106 |
+
1 Here after, we denote Ξ£β(1480) as the lowest Ξ£β
|
107 |
+
1/2β state unless
|
108 |
+
otherwise stated.
|
109 |
+
|
110 |
+
2
|
111 |
+
Regge-eο¬ective Lagrange approach.
|
112 |
+
Considering the Ξ£β(1480) signal was ο¬rst observed
|
113 |
+
in the Ο+Ξ invariant mass distribution of the process
|
114 |
+
Ο+p β Ο+K+Ξ, and the signiο¬cance is about 3 βΌ
|
115 |
+
4Ο [30], we search for the charged Ξ£β(1480) in the process
|
116 |
+
Ξ³n β K+Ξ£ββ
|
117 |
+
1/2β, which could also avoid the contribu-
|
118 |
+
tions of possible excited Ξβ states. We will consider the
|
119 |
+
t-, s-, u-channels diagrams in the Born approximation
|
120 |
+
by employing the eο¬ective Lagrangian approach, and the
|
121 |
+
t-channel K/Kβ exchanges terms within Regge model.
|
122 |
+
Then we will calculate the diο¬erential and total cross
|
123 |
+
sections of the process Ξ³n β K+Ξ£ββ
|
124 |
+
1/2β reaction, which
|
125 |
+
are helpful to search for Ξ£β
|
126 |
+
1/2β experimentally.
|
127 |
+
This paper is organized as follows. In Sec. II, the the-
|
128 |
+
oretical formalism for studying the Ξ³n β K+Ξ£ββ(1480)
|
129 |
+
reactions are presented. The numerical results of total
|
130 |
+
and diο¬erential cross sections and discussion are shown
|
131 |
+
in Sec. III. Finally, a brief summary is given in the last
|
132 |
+
section.
|
133 |
+
II.
|
134 |
+
FORMALISM
|
135 |
+
The reaction mechanisms of the Ξ£β(1480) (β‘ Ξ£β)
|
136 |
+
photo-production process are depicted in the Fig. 1,
|
137 |
+
where we have taken into account the contributions from
|
138 |
+
the t-channel K and Kβ exchange term, s-channel nu-
|
139 |
+
cleon pole term, u-channel Ξ£ exchange term, and the
|
140 |
+
contact term, respectively.
|
141 |
+
Ξ³(k1)
|
142 |
+
K(k2)
|
143 |
+
N(p1)
|
144 |
+
Ξ£β(p2)
|
145 |
+
K, Kβ
|
146 |
+
Ξ³
|
147 |
+
K
|
148 |
+
N
|
149 |
+
Ξ£β
|
150 |
+
N
|
151 |
+
Ξ³
|
152 |
+
Ξ£β
|
153 |
+
N
|
154 |
+
K
|
155 |
+
Ξ³
|
156 |
+
K
|
157 |
+
N
|
158 |
+
Ξ£β
|
159 |
+
Ξ£
|
160 |
+
(a)
|
161 |
+
(b)
|
162 |
+
(c)
|
163 |
+
(d)
|
164 |
+
FIG. 1: The mechanisms of the Ξ³n β K+Ξ£ββ
|
165 |
+
1/2β process. (a)
|
166 |
+
t-channel K/Kβ exchange terms, (b) s-channel nuclear term,
|
167 |
+
(c) u-channel Ξ£ exchange term, and (d) contact term. The
|
168 |
+
k1, k2, p1, and p2 stand for the four-momenta of the initial
|
169 |
+
photon, kaon, neutron, and Ξ£β(1480), respectively.
|
170 |
+
To compute the scattering amplitudes of the Feynman
|
171 |
+
diagrams shown in Fig. 1 within the eο¬ective Lagrangian
|
172 |
+
approach, we use the Lagrangian densities for the elec-
|
173 |
+
tromagnetic and strong interaction vertices as used in
|
174 |
+
Refs. [27, 42β46]
|
175 |
+
LΞ³KK = βie
|
176 |
+
οΏ½
|
177 |
+
Kβ (βΒ΅K) β
|
178 |
+
οΏ½
|
179 |
+
βΒ΅Kβ οΏ½
|
180 |
+
K
|
181 |
+
οΏ½
|
182 |
+
AΒ΅,
|
183 |
+
(1)
|
184 |
+
LΞ³KKβ = gΞ³KKβǫ¡ναββΒ΅AΞ½βΞ±Kβ
|
185 |
+
Ξ²K,
|
186 |
+
(2)
|
187 |
+
LΞ³NN = βe Β―N
|
188 |
+
οΏ½
|
189 |
+
Ξ³Β΅Λe β
|
190 |
+
ΛΞΊN
|
191 |
+
2MN
|
192 |
+
Ο¡νβΞ½
|
193 |
+
οΏ½
|
194 |
+
AΒ΅N,
|
195 |
+
(3)
|
196 |
+
LγΣΣβ = e¡ΣΣβ
|
197 |
+
2MN
|
198 |
+
¯Σγ5Ο¡νβΞ½A¡Σβ + h.c.,
|
199 |
+
(4)
|
200 |
+
LKNΞ£ = βigKNΞ£ Β―NΞ³5Ξ£K + h.c.,
|
201 |
+
(5)
|
202 |
+
LKβNΞ£β = igKβNΞ£β
|
203 |
+
β
|
204 |
+
3
|
205 |
+
Β―KβΒ΅ Β―Ξ£βγ¡γ5N + h.c.
|
206 |
+
(6)
|
207 |
+
LKNΞ£β = gKNΞ£β Β―K Β―
|
208 |
+
Ξ£βN + h.c.,
|
209 |
+
(7)
|
210 |
+
where e(=
|
211 |
+
β
|
212 |
+
4ΟΞ±) is the elementary charge unit, AΒ΅ is the
|
213 |
+
photon ο¬led, and Λe β‘ (1+Ο3)/2 denotes the charge opera-
|
214 |
+
tor acting on the nucleon ο¬eld. ΛΞΊN β‘ ΞΊpΛe+ΞΊn(1βΛe) is the
|
215 |
+
anomalous magnetic moment, and we take ΞΊn = β1.913
|
216 |
+
for neutron [15]. MN and MΞ£ denote the masses of nu-
|
217 |
+
cleon and the ground-state of Ξ£ hyperon, respectively.
|
218 |
+
The strong coupling gKNΞ£ is taken to be 4.09 from
|
219 |
+
Ref. [47].
|
220 |
+
The gΞ³KKβ = 0.254 GeVβ1 is determined
|
221 |
+
from the experimental data of ΞKββK+Ξ³ [15] and the
|
222 |
+
value of gKβNΞ£β = β3.26 β i0.06 is taken from Ref [26].
|
223 |
+
In addition, the coupling gKNΞ£β = 8.74 GeV is taken
|
224 |
+
from Ref. [36], and the transition magnetic moment
|
225 |
+
¡ΣΣβ = 1.28 is taken from Ref. [27]
|
226 |
+
With the eο¬ective interaction Lagrangian densities
|
227 |
+
given above, the invariant scattering amplitudes are de-
|
228 |
+
ο¬ned as
|
229 |
+
M = Β―uΞ£β(p2, sΞ£β)MΒ΅
|
230 |
+
huN(k2, sp)Η«Β΅(k1, Ξ»),
|
231 |
+
(8)
|
232 |
+
where uΞ£β and uN stand for the Dirac spinors, respec-
|
233 |
+
tively, while Η«Β΅(k1, Ξ») is the photon polarization vector
|
234 |
+
and the sub-indice h corresponds to diο¬erent diagrams
|
235 |
+
of Fig. 1. The reduced amplitudes MΒ΅
|
236 |
+
h are written as
|
237 |
+
MΒ΅
|
238 |
+
Kβ =
|
239 |
+
egΞ³KKβgKβNΞ£β
|
240 |
+
β
|
241 |
+
3MKβ(t β M 2
|
242 |
+
Kβ)ǫαβ¡νk1Ξ±k2Ξ²Ξ³Ξ½Ξ³5,
|
243 |
+
(9)
|
244 |
+
MΒ΅
|
245 |
+
Kβ = β2iegKNΞ£β
|
246 |
+
t β M 2
|
247 |
+
K
|
248 |
+
kΒ΅
|
249 |
+
2 ,
|
250 |
+
(10)
|
251 |
+
MΒ΅
|
252 |
+
Ξ£β = βi
|
253 |
+
e¡ΣΣβgKNΞ£
|
254 |
+
2Mn(u β M 2
|
255 |
+
Ξ£β)(q/u β MΞ£)Ο¡νk1Ξ½, (11)
|
256 |
+
MΒ΅
|
257 |
+
n =
|
258 |
+
ΞΊngKNΞ£β
|
259 |
+
2Mn(s β M 2n)Ο¡νk1Ξ½(q/s + Mn).
|
260 |
+
(12)
|
261 |
+
In order to keep the full photoproduction amplitudes
|
262 |
+
considered here gauge invariant, we adopt the amplitude
|
263 |
+
of the contact term
|
264 |
+
MΒ΅
|
265 |
+
c = βiegKNΞ£β
|
266 |
+
pΒ΅
|
267 |
+
2
|
268 |
+
p2 Β· k1
|
269 |
+
,
|
270 |
+
(13)
|
271 |
+
for Ξ³n β K+Ξ£ββ
|
272 |
+
1/2β.
|
273 |
+
|
274 |
+
3
|
275 |
+
It is known that the Reggeon exchange mechanism
|
276 |
+
plays a crucial role at high energies and forward an-
|
277 |
+
gles [48β51], thus we will adopt Regge model for mod-
|
278 |
+
eling the t-channel K and Kβ contributions by replacing
|
279 |
+
the usual pole-like Feynman propagator with the corre-
|
280 |
+
sponding Regge propagators as follows,
|
281 |
+
1
|
282 |
+
t β M 2
|
283 |
+
K
|
284 |
+
β FRegge
|
285 |
+
K
|
286 |
+
=
|
287 |
+
οΏ½ s
|
288 |
+
sK
|
289 |
+
0
|
290 |
+
οΏ½Ξ±K(t)
|
291 |
+
ΟΞ±β²
|
292 |
+
K
|
293 |
+
sin(ΟΞ±K(t))Ξ(1 + Ξ±K(t)),(14)
|
294 |
+
1
|
295 |
+
t β M 2
|
296 |
+
Kβ
|
297 |
+
β FRegge
|
298 |
+
Kβ
|
299 |
+
=
|
300 |
+
οΏ½ s
|
301 |
+
sKβ
|
302 |
+
0
|
303 |
+
οΏ½Ξ±Kβ(t)
|
304 |
+
ΟΞ±β²
|
305 |
+
Kβ
|
306 |
+
sin(ΟΞ±Kβ(t))Ξ(Ξ±Kβ(t)),(15)
|
307 |
+
with Ξ±K(t) = 0.7 GeVβ2 Γ (t β M 2
|
308 |
+
K) and Ξ±Kβ(t) = 1 +
|
309 |
+
0.83 Gevβ2 Γ (t β M 2
|
310 |
+
Kβ) the linear Reggeon trajectory.
|
311 |
+
The constants sK
|
312 |
+
0 and sKβ
|
313 |
+
0
|
314 |
+
are determined to be 3.0 GeV2
|
315 |
+
and 1.5 GeV2, respectively [52]. Here, the Ξ±β²
|
316 |
+
K and Ξ±β²
|
317 |
+
Kβ
|
318 |
+
are the Regge-slopes.
|
319 |
+
Then, the full photo-production amplitudes for Ξ³n β
|
320 |
+
K+Ξ£ββ
|
321 |
+
1/2β reaction can be expressed as
|
322 |
+
MΒ΅ =
|
323 |
+
οΏ½
|
324 |
+
MΒ΅
|
325 |
+
Kβ + MΒ΅
|
326 |
+
c
|
327 |
+
οΏ½ οΏ½
|
328 |
+
t β M 2
|
329 |
+
Kβ
|
330 |
+
οΏ½
|
331 |
+
FRegge
|
332 |
+
K
|
333 |
+
+ MΒ΅
|
334 |
+
Ξ£βfu
|
335 |
+
+ MΒ΅
|
336 |
+
Kβ
|
337 |
+
οΏ½
|
338 |
+
t β M 2
|
339 |
+
Kβ
|
340 |
+
οΏ½
|
341 |
+
FRegge
|
342 |
+
Kβ
|
343 |
+
+ MΒ΅
|
344 |
+
nfs,
|
345 |
+
(16)
|
346 |
+
While FRegge
|
347 |
+
K
|
348 |
+
and FRegge
|
349 |
+
Kβ
|
350 |
+
stand for the Regge propaga-
|
351 |
+
tors. The form factors fs and fu are included to suppress
|
352 |
+
the large momentum transfer of the intermediate par-
|
353 |
+
ticles and describe their oο¬-shell behavior, because the
|
354 |
+
intermediate hadrons are not point-like particles.
|
355 |
+
For
|
356 |
+
s-channel and u-channel baryon exchanges, we use the
|
357 |
+
following form factors [42, 53]
|
358 |
+
fi(q2
|
359 |
+
i ) =
|
360 |
+
οΏ½
|
361 |
+
Ξ4
|
362 |
+
i
|
363 |
+
Ξ4
|
364 |
+
i + (q2
|
365 |
+
i β M 2
|
366 |
+
i )2
|
367 |
+
οΏ½2
|
368 |
+
, i = s, u
|
369 |
+
(17)
|
370 |
+
with Mi and qi being the masses and four-momenta of
|
371 |
+
the intermediate baryons, and the Ξi is the cut-oο¬ values
|
372 |
+
for baryon exchange diagrams.
|
373 |
+
In this work, we take
|
374 |
+
Ξs = Ξu = 1.5 GeV, and will discuss the results with
|
375 |
+
diο¬erent cut-oο¬.
|
376 |
+
Finally, the unpolarized diο¬erential cross section in the
|
377 |
+
center of mass (c.m.) frame for the Ξ³n β KΞ£ββ
|
378 |
+
1/2β reac-
|
379 |
+
tion reads
|
380 |
+
dΟ
|
381 |
+
dβ¦ = MNMΞ£β|βkc.m.
|
382 |
+
1
|
383 |
+
||βpc.m.
|
384 |
+
1
|
385 |
+
|
|
386 |
+
8Ο2(s β M 2
|
387 |
+
N)2
|
388 |
+
οΏ½
|
389 |
+
Ξ»,sp,sΞ£β
|
390 |
+
|M|2,
|
391 |
+
(18)
|
392 |
+
where s denotes the invariant mass square of the center
|
393 |
+
of mass (c.m.) frame for Ξ£β
|
394 |
+
1/2β photo-production. Here
|
395 |
+
βkc.m.
|
396 |
+
1
|
397 |
+
and βpc.m.
|
398 |
+
1
|
399 |
+
are the three-momenta of the photon and
|
400 |
+
K meson in the c.m.
|
401 |
+
frame, while dβ¦ = 2ΟdcosΞΈc.m.,
|
402 |
+
with ΞΈc.m. the polar outgoing K scattering angle.
|
403 |
+
III.
|
404 |
+
NUMERICAL RESULTS AND
|
405 |
+
DISCUSSIONS
|
406 |
+
In this section, we show our numerical results of the dif-
|
407 |
+
ferential and total cross sections for the Ξ³n β K+Ξ£ββ
|
408 |
+
1/2β
|
409 |
+
reaction.
|
410 |
+
The masses of the mesons and baryons are
|
411 |
+
taken from RPP [15], as given in Table I. In addition, the
|
412 |
+
mass and width of the Ξ£(1480) are M = 1480 Β± 15 GeV
|
413 |
+
and Ξ = 60 Β± 15 GeV, respectively [28].
|
414 |
+
TABLE I: Particle masses used in this work.
|
415 |
+
Particle
|
416 |
+
Mass (MeV)
|
417 |
+
n
|
418 |
+
939.565
|
419 |
+
Ξ£β
|
420 |
+
1197.449
|
421 |
+
K+
|
422 |
+
493.677
|
423 |
+
Kβ
|
424 |
+
493.677
|
425 |
+
Kβ
|
426 |
+
891.66
|
427 |
+
First we show the angle dependence of the diο¬erential
|
428 |
+
cross sections for the Ξ³n β K+Ξ£ββ
|
429 |
+
1/2β reaction in Fig. 2,
|
430 |
+
where the the center-of-mass energies W = βs varies
|
431 |
+
from 2.0 to 2.8 GeV. The black curves labeled as βTotalβ
|
432 |
+
show the results of all the contributions from the t-, s-,
|
433 |
+
u-channels, and contact term. The blue-dot curves and
|
434 |
+
red-dashed curves stand for the contributions from the
|
435 |
+
u-channel Ξ£ exchange and t-channel K exchange mecha-
|
436 |
+
nism, respectively. The magenta-dot-dashed curves and
|
437 |
+
the green-dot curves correspond to the contributions
|
438 |
+
from the s-channel and t-channel Kβ exchange diagrams,
|
439 |
+
respectively, while the cyan-dot-dashed curves represent
|
440 |
+
the contribution from the contact term. According to the
|
441 |
+
diο¬erential cross sections, one can ο¬nd that the t-channel
|
442 |
+
K meson exchange term plays an important role at for-
|
443 |
+
ward angles for the process Ξ³n β K+Ξ£ββ
|
444 |
+
1/2β, mainly due
|
445 |
+
to the Regge eο¬ects of the t-change K exchange. The
|
446 |
+
K-Reggeon exchange shows steadily increasing behavior
|
447 |
+
with cosΞΈc.m. and falls oο¬ drastically at very forward an-
|
448 |
+
gles. In addition, the u-channel Ξ£ exchange term mainly
|
449 |
+
contribute to the backward angles for both processes.
|
450 |
+
It should be stressed that the contribution from the t-
|
451 |
+
channel Kβ exchange term is very small and could be
|
452 |
+
safely neglected for the process Ξ³n β K+Ξ£ββ
|
453 |
+
1/2β, which
|
454 |
+
is consistent with the results of Ref. [27].
|
455 |
+
In addition to the the diο¬erential cross sections, we
|
456 |
+
have also calculated the total cross section of the Ξ³n β
|
457 |
+
K+Ξ£ββ
|
458 |
+
1/2β reaction as a function of the initial photon en-
|
459 |
+
ergy. The results are shown in Fig. 3. The black curve
|
460 |
+
labeled as βTotalβ shows the results of all the contribu-
|
461 |
+
tions, including t-, s-, u- channels and contact term. The
|
462 |
+
blue-dot and red-dashed curves stand for the contribu-
|
463 |
+
tions from the u- channel Ξ£ exchange and t- channel
|
464 |
+
K exchange mechanism, respectively. The magenta-dot-
|
465 |
+
dashed and the green-dot curves show the contribution of
|
466 |
+
s-channel and t-channel Kβ exchange diagrams, respec-
|
467 |
+
tively, while the cyan-dot-dashed curve represents the
|
468 |
+
|
469 |
+
4
|
470 |
+
0
|
471 |
+
0.5
|
472 |
+
1
|
473 |
+
1.5
|
474 |
+
2
|
475 |
+
2.5
|
476 |
+
3
|
477 |
+
3.5
|
478 |
+
4
|
479 |
+
4.5
|
480 |
+
dΟ/dcosΞΈc.m. (Β΅b)
|
481 |
+
cosΞΈc.m.
|
482 |
+
W=2.0 GeV
|
483 |
+
K-t
|
484 |
+
K*-t
|
485 |
+
s-channel
|
486 |
+
u-channel
|
487 |
+
contact term
|
488 |
+
Total
|
489 |
+
W=2.1 GeV
|
490 |
+
W=2.2 GeV
|
491 |
+
0
|
492 |
+
0.5
|
493 |
+
1
|
494 |
+
1.5
|
495 |
+
2
|
496 |
+
2.5
|
497 |
+
3
|
498 |
+
3.5
|
499 |
+
4
|
500 |
+
4.5
|
501 |
+
W=2.3 GeV
|
502 |
+
W=2.4 GeV
|
503 |
+
W=2.5 GeV
|
504 |
+
0
|
505 |
+
0.5
|
506 |
+
1
|
507 |
+
1.5
|
508 |
+
2
|
509 |
+
2.5
|
510 |
+
3
|
511 |
+
3.5
|
512 |
+
4
|
513 |
+
4.5
|
514 |
+
-1
|
515 |
+
-0.5
|
516 |
+
0
|
517 |
+
0.5
|
518 |
+
1
|
519 |
+
W=2.6 GeV
|
520 |
+
-1
|
521 |
+
-0.5
|
522 |
+
0
|
523 |
+
0.5
|
524 |
+
1
|
525 |
+
W=2.7 GeV
|
526 |
+
-1
|
527 |
+
-0.5
|
528 |
+
0
|
529 |
+
0.5
|
530 |
+
1
|
531 |
+
W=2.8 GeV
|
532 |
+
FIG. 2: (Color online) Ξ³n β K+Ξ£ββ
|
533 |
+
1/2β diο¬erential cross sections as a function of cosΞΈc.m. are plotted for Ξ³n-invariant mass
|
534 |
+
intervals (in GeV units). The black curve labeled as βTotalβ shows the results of all the contributions, including t-, s-, u- channels
|
535 |
+
and contact term. The blue-dot and red-dashed curves stand for the contributions from the eο¬ective Lagrangian approach u-
|
536 |
+
channel Ξ£ exchange and t- channel K exchange mechanism, respectively. The magenta-dot-dashed and the green-dot-dashed
|
537 |
+
curves show the contribution of s-channel and t-channel Kβ exchange diagrams, respectively, while the cyan-dot-dashed curve
|
538 |
+
represent the contribution of the contact term.
|
539 |
+
contribution of the contact term. For the Ξ³n β K+Ξ£ββ
|
540 |
+
1/2β
|
541 |
+
reaction its total cross section attains a maximum value
|
542 |
+
of about 4.3 Β΅b at EΞ³ = 2.3 GeV. It is expected that the
|
543 |
+
Ξ£β(1480) could be observed by future experiments in the
|
544 |
+
process Ξ³n β K+Ξ£ββ (1480) β Ξ£βΟ0/Ξ£0Οβ/Ξ£βΞ³.
|
545 |
+
Finally, we also show the total cross section for Ξ³n β
|
546 |
+
K+Ξ£ββ
|
547 |
+
1/2β with the cut-oο¬ Ξs/u = 1.2, 1.5, and 1.8 GeV
|
548 |
+
in Fig. 4, where one can ο¬nd the total cross sections are
|
549 |
+
weakly dependence on the value of the cut-oο¬. Since the
|
550 |
+
precise couplings of the Ξ£(1480) are still unknown, the
|
551 |
+
|
552 |
+
5
|
553 |
+
0
|
554 |
+
0.5
|
555 |
+
1
|
556 |
+
1.5
|
557 |
+
2
|
558 |
+
2.5
|
559 |
+
3
|
560 |
+
3.5
|
561 |
+
4
|
562 |
+
4.5
|
563 |
+
5
|
564 |
+
5.5
|
565 |
+
1.5
|
566 |
+
2
|
567 |
+
2.5
|
568 |
+
3
|
569 |
+
3.5
|
570 |
+
4
|
571 |
+
Ο (Β΅b)
|
572 |
+
EΞ³ (GeV)
|
573 |
+
K-t
|
574 |
+
K*-t
|
575 |
+
s-channel
|
576 |
+
u-channel
|
577 |
+
contact term
|
578 |
+
Total
|
579 |
+
FIG. 3:
|
580 |
+
(Color online) Total cross section for Ξ³n
|
581 |
+
β
|
582 |
+
K+Ξ£β
|
583 |
+
1/2β is plotted as a function of the lab energy EΞ³. The
|
584 |
+
black curve labeled as βTotalβ shows the results of all the con-
|
585 |
+
tributions, including t-,s-,u- channels and contact term. The
|
586 |
+
blue-dot and red-dashed curves stand for the contributions
|
587 |
+
from the eο¬ective Lagrangian approach u- channel Ξ£ exchange
|
588 |
+
and t- channel K exchange mechanism, respectively.
|
589 |
+
The
|
590 |
+
magenta-dot-dashed and the green-dot curves show the con-
|
591 |
+
tribution of s-channel and t-channel Kβ exchange diagrams,
|
592 |
+
respectively, while the cyan-dot-dashed curve represents the
|
593 |
+
contribution of the contact term.
|
594 |
+
0
|
595 |
+
0.5
|
596 |
+
1
|
597 |
+
1.5
|
598 |
+
2
|
599 |
+
2.5
|
600 |
+
3
|
601 |
+
3.5
|
602 |
+
4
|
603 |
+
4.5
|
604 |
+
5
|
605 |
+
5.5
|
606 |
+
1.5
|
607 |
+
2
|
608 |
+
2.5
|
609 |
+
3
|
610 |
+
3.5
|
611 |
+
4
|
612 |
+
Ο (Β΅b)
|
613 |
+
EΞ³ (GeV)
|
614 |
+
Ξs,u = 1.2 GeV
|
615 |
+
Ξs,u = 1.5 GeV
|
616 |
+
Ξs,u = 1.8 GeV
|
617 |
+
FIG. 4:
|
618 |
+
(Color online) Total cross section for Ξ³n
|
619 |
+
β
|
620 |
+
K+Ξ£β
|
621 |
+
1/2β with the cut-oο¬ Ξs/u = 1.2, 1.5, and 1.8 GeV.
|
622 |
+
future experiment would be helpful to constrain these
|
623 |
+
couplings if the state Ξ£(1480) is conο¬rmed.
|
624 |
+
IV.
|
625 |
+
SUMMARY
|
626 |
+
The lowest Ξ£ββ
|
627 |
+
1/2β is far from established, and its ex-
|
628 |
+
istence is important to understand the low-lying excited
|
629 |
+
baryon with JP = 1/2β. There are many experimen-
|
630 |
+
tal hints of the Ξ£β(1480), which has been listed in the
|
631 |
+
previous version of the Review of Particle Physics. We
|
632 |
+
propose to search for this state in the photoproduction
|
633 |
+
process to conο¬rm its existence.
|
634 |
+
Assuming that the JP
|
635 |
+
=
|
636 |
+
1/2β low lying state
|
637 |
+
Ξ£β (1480) has a sizeable coupling to the Β―KN according
|
638 |
+
the study of Ref. [36], we have phenomenologically inves-
|
639 |
+
tigated the Ξ³n β K+Ξ£ββ
|
640 |
+
1/2β reaction by considering the
|
641 |
+
contributions from the t-channel K/Kβ exchange term,
|
642 |
+
s-channel nucleon term, u-channel Ξ£ exchange term, and
|
643 |
+
contact term within the Regge-eο¬ective Lagrange ap-
|
644 |
+
proach.
|
645 |
+
The diο¬erential cross sections and total cross
|
646 |
+
sections for these processes are calculated with our model
|
647 |
+
parameters. The total cross section of Ξ³n β K+Ξ£ββ
|
648 |
+
1/2β
|
649 |
+
is about 4.3 Β΅b around EΞ³ = 2.3 GeV. We encourage
|
650 |
+
our experimental colleagues to measure Ξ³n β K+Ξ£ββ
|
651 |
+
1/2β
|
652 |
+
process.
|
653 |
+
Acknowledgements
|
654 |
+
This
|
655 |
+
work
|
656 |
+
is
|
657 |
+
supported
|
658 |
+
by
|
659 |
+
the
|
660 |
+
National
|
661 |
+
Natu-
|
662 |
+
ral Science Foundation of China under Grant Nos.
|
663 |
+
12192263, 12075288, 11735003, and 11961141012, the
|
664 |
+
Natural Science Foundation of Henan under Grand No.
|
665 |
+
222300420554.
|
666 |
+
It is also supported by the Project of
|
667 |
+
Youth Backbone Teachers of Colleges and Universities
|
668 |
+
of Henan Province (2020GGJS017), the Youth Talent
|
669 |
+
Support Project of Henan (2021HYTP002), the Open
|
670 |
+
Project of Guangxi Key Laboratory of Nuclear Physics
|
671 |
+
and Nuclear Technology, No.NLK2021-08, the Youth In-
|
672 |
+
novation Promotion Association CAS.
|
673 |
+
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2
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7
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|
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version https://git-lfs.github.com/spec/v1
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1 |
+
filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf,len=495
|
2 |
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page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
3 |
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page_content='03520v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
4 |
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page_content='FA] 9 Jan 2023 CLASSIFYING WEAK PHASE RETRIEVAL P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
5 |
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
6 |
+
page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
7 |
+
page_content=' AKRAMI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
8 |
+
page_content=' We will give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
9 |
+
page_content=' These results give a complete understanding of the dif- ference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
10 |
+
page_content=' We also answer two longstanding open problems on weak phase retrieval: (1) We show that the families of weak phase retrievable frames {xi}m i=1 in Rn are not dense in the family of m-element sets of vectors in Rn for all m β₯ 2n β 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
11 |
+
page_content=' (2) We show that any frame {xi}2nβ2 i=1 containing one or more canonical basis vectors in Rn cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
12 |
+
page_content=' We provide numerous examples to show that the obtained results are best possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
13 |
+
page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
14 |
+
page_content=' Introduction The concept of frames in a separable Hilbert space was originally introduced by Duο¬n and Schaeο¬er in the context of non-harmonic Fourier series [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
15 |
+
page_content=' Frames are a more ο¬exible tool than bases because of the redundancy property that make them more applicable than bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
16 |
+
page_content=' Phase retrieval is an old problem of recovering a signal from the absolute value of linear measurement coeο¬cients called intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
17 |
+
page_content=' Phase retrieval and norm retrieval have become very active areas of research in applied mathematics, computer science, engineering, and more today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
18 |
+
page_content=' Phase retrieval has been deο¬ned for both vectors and subspaces (projections) in all separable Hilbert spaces, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
19 |
+
page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
20 |
+
page_content=', [3], [4], [5], [6], [9], [10] and [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
21 |
+
page_content=' The concept of weak phase retrieval weakened the notion of phase retrieval and it has been ο¬rst deο¬ned for vectors in ([8] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
22 |
+
page_content=' The rest of the paper is organized as follows: In Section 2, we give the basic deο¬nitions and certain preliminary results to be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
23 |
+
page_content=' Weak phase retrieval by vectors is introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
24 |
+
page_content=' In section 4 we show that any family of vectors {xi}2nβ2 i=1 doing weak phase retrieval cannot contain a unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
25 |
+
page_content=' In section 5, we show that the weak phase retrievable frames are not dense in all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
26 |
+
page_content=' And in section 6 we give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
27 |
+
page_content=' These results give a complete understanding of the diο¬erence between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
28 |
+
page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
29 |
+
page_content=' preliminaries First we give the background material needed for the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
30 |
+
page_content=' Let H be a ο¬nite or inο¬nite dimensional real Hilbert space and B(H) the class of all bounded linear operators deο¬ned on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
31 |
+
page_content=' The natural numbers and real numbers are denoted by βNβ and βRβ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
32 |
+
page_content=' We use [m] instead of the set {1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
33 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
34 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
35 |
+
page_content=', m} and use [{xi}iβI] instead of span{xi}iβI, where I is a ο¬nite or countable subset of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
36 |
+
page_content=' We 2010 Mathematics Subject Classiο¬cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
37 |
+
page_content=' 42C15, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
38 |
+
page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
39 |
+
page_content=' Real Hilbert frames, Full spark, Phase retrieval, Weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
40 |
+
page_content=' The ο¬rst author was supported by NSF DMS 1609760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
41 |
+
page_content=' 1 2 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
42 |
+
page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
43 |
+
page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
44 |
+
page_content=' AKRAMI denote by Rn a n dimensional real Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
45 |
+
page_content=' We start with the deο¬nition of a real Hilbert space frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
46 |
+
page_content=' Deο¬nition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
47 |
+
page_content=' A family of vectors {xi}iβI in a ο¬nite or inο¬nite dimensional separable real Hilbert space H is a frame if there are constants 0 < A β€ B < β so that Aβ₯xβ₯2 β€ οΏ½ iβI |β¨x, xiβ©|2 β€ Bβ₯xβ₯2, for all f β H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
48 |
+
page_content=' The constants A and B are called the lower and upper frame bounds for {xi}iβI, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
49 |
+
page_content=' If an upper frame bound exists, then {xi}iβI is called a B-Bessel seqiemce or simply Bessel when the constant is implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
50 |
+
page_content=' If A = B, it is called an A-tight frame and in case A = B = 1, it is called a Parseval frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
51 |
+
page_content=' The values {β¨x, xiβ©}β i=1 are called the frame coeο¬cients of the vector x β H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
52 |
+
page_content=' It is immediate that a frame must span the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
53 |
+
page_content=' We will need to work with Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
54 |
+
page_content=' Deο¬nition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
|
55 |
+
page_content=' A family X = {xi}iβI in a ο¬nite or inο¬nite dimensional real Hilbert space H is a Riesz sequence if there are constants 0 < A β€ B < β satisfying A οΏ½ iβI |ci|2 β€ β₯ οΏ½ iβI cixiβ₯2 β€ B οΏ½ iβI |ci|2 for all sequences of scalars {ci}iβI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If it is complete in H, we call X a Riesz basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For an introduction to frame theory we recommend [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Throughout the paper the orthogonal projection or simply projection will be a self- adjoint positive projection and {ei}β i=1 will be used to denote the canonical basis for the real space Rn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', a basis for which β¨ei, ejβ© = Ξ΄i,j = οΏ½ 1 if i = j, 0 if i ΜΈ= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A family of vectors {xi}iβI in a real Hilbert space H does phase (norm) retrieval if whenever x, y β H, satisfy |β¨x, xiβ©| = |β¨y, xiβ©| for all i β I, then x = Β±y (β₯xβ₯ = β₯yβ₯).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Phase retrieval was introduced in reference [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' See reference [1] for an introduc- tion to norm retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Note that if {xi}iβI does phase (norm) retrieval, then so does {aixi}iβI for any 0 < ai < β for all i β I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' But in the case where |I| = β, we have to be careful to maintain frame bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' This always works if 0 < infiβI ai β€ supiβIai < β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' But this is not necessary in general [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The complement property is an essential issue here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A family of vectors {xi}iβI in a ο¬nite or inο¬nite dimensional real Hilbert space H has the complement property if for any subset J β I, either span{xi}iβJ = H or span{xi}iβJc = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Fundamental to this area is the following for which the ο¬nite dimensional case appeared in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' WEAK PHASE RETRIEVAL 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A family of vectors {xi}iβI does phase retrieval in Rn if and only if it has the complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We recall: Deο¬nition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A family of vectors {xi}m i=1 in Rn is full spark if for every I β [m] with |I| = n , {xi}iβI is linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If {xi}m i=1 does phase retrieval in Rn, then m β₯ 2nβ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If m = 2nβ 1, {xi}m i=1 does phase retrieval if and only if it is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We rely heavily on a signiο¬cant result from [2]: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If {xi}2nβ2 i=1 does weak phase retrieval in Rn then for every I β [2nβ2], if x β₯ span{xi}iβI and y β₯ {xi}iβIc then x β₯xβ₯ + y β₯yβ₯ and x β₯xβ₯ β y β₯yβ₯ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' In particular, if β₯xβ₯ = β₯yβ₯ = 1, then x + y and x β y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Hence, if x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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85 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , an) then y = (Η«1a1, Η«2a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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88 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , Η«nan), where Η«i = Β±1 for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The above theorem may fail if β₯xβ₯ ΜΈ= β₯yβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For example, consider the weak phase retrievable frame in R3: \uf8ee \uf8ef\uf8ef\uf8f0 1 1 1 β1 1 1 1 β1 1 1 1 β1 \uf8f9 \uf8fa\uf8fa\uf8fb Also, x = (0, 1, β1) is perpendicular to rows 1 and 2 and y = (0, 1 2, 1 2) is orthogonal to rows 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' But x + y = (0, 3 2, 1 2) and x β y = (0, β1 2 , β3 2 ) and these are not disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' But if we let them have the same norm we get x = (0, 1, β1) and y = (0, 1, 1) so x + y = (0, 1, 0) and x β y = (0, 0, 1) and these are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Weak phase retrieval The notion of βWeak phase retrieval by vectorsβ in Rn was introduced in [8] and was developed further in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' One limitation of current methods used for retrieving the phase of a signal is computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Recall that a generic family of (2n β 1)- vectors in Rn satisο¬es phaseless reconstruction, however no set of (2n β 2)-vectors can (See [7] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By generic we are referring to an open dense set in the set of (2n β 1)-element frames in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Two vectors x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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106 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , bn) in Rn weakly have the same phase if there is a |ΞΈ| = 1 so that phase(ai) = ΞΈphase(bi) for all i β [n], for which ai ΜΈ= 0 ΜΈ= bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If ΞΈ = 1, we say x and y weakly have the same signs and if ΞΈ = β1, they weakly have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Therefore with above deο¬nition the zero vector in Rn weakly has the same phase with all vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For x β R, sgn(x) = 1 if x > 0 and sgn(x) = β1 if x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A family of vectors {xi}m i=1 does weak phase retrieval in Rn if for any x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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121 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , bn) in Rn with |β¨x, xiβ©| = |β¨y, xiβ©| for all i β [m], then x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A fundamental result here is 4 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' AKRAMI Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' [8] Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , bn) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The following are equivalent: (1) We have sgn(aiaj) = sgn(bibj), for all 1 β€ i ΜΈ= j β€ n (2) Either x, y have weakly the same sign or they have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It is clear that if {xi}m i=1 does weak phase retrieval in Rn, then {cixi}m i=1 does weak phase retrieval as long as ci > 0 for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The following appears in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If X = {xi}m i=1 does weak phase retrieval in Rn, then m β₯ 2n β 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Finally, we have: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' [7] If a frame X = {xi}2nβ2 i=1 does weak phase retrieval in Rn, then X is a full spark frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Clearly the converse of above theorem is not hold, for example {(1, 0), (0, 1)} is full spark frame that fails weak phase retrieval in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If {xi}iβI does phase retrieval and R is an invertible operator on the space then {Rxi}iβI does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' This follows easily since |β¨x, Rxiβ©| = |β¨y, Rxiβ©| implies |β¨Rβx, xiβ©| = |β¨Rβy, xiβ©|, and so Rβx = ΞΈRβy for |ΞΈ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since R is invertible, x = ΞΈy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' This result fails badly for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For example, let e1 = (1, 0), e2 = (0, 1), x1 = ( 1 β 2, 1 β 2, x2 = ( 1 β 2, β1 β 2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then {e1, e2} fails weak phase retrieval, {x1, x2} does weak phase retrieval and Uei = xi is a unitary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Frames Containing Unit Vectors Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Any frame {xi}2nβ2 i=1 whith one or more canonical basis vectors in Rn cannot do weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We proceed by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Recall that {xi}2nβ2 i=1 must be full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {ei}n i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Assume I β {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', 2nβ2} with |I| = n β 1 and assume x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , bn) with β₯xβ₯ = β₯yβ₯ = 1 and x β₯ X = span{xi}iβI and y β₯ span{xi}2nβ2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' After reindexing {ei}n i=1 and {xi}2nβ2 i=1 }, we assume x1 = e1, I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', nβ1 and Ic = {n, n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , 2nβ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since β¨x, x1β© = a1 = 0, by Theorem 2, b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let P be the projection on span{ei}n i=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So {Pxi}2nβ2 i=n is (n β 1)-vectors in an (n β 1)-dimensional space and y is orthogonal to all these vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So there exist {ci}2nβ2 i=n not all zero so that 2nβ2 οΏ½ i=n ciPxi = 0 and so 2nβ1 οΏ½ i=n cixi(1)x1 β 2nβ2 οΏ½ i=n cixi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' That is, our vectors are not full spark, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The fact that there are (2nβ 2) vectors in the theorem is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For example, e1, e2, e1 + e2 is full spark in R2, so it does phase retrieval - and hence weak phase retrieval - despite the fact that it contains both basis vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The converse of Theorem 5 is not true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, β2, 3), (1, β2, β3)} in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Every set of its two same coordinates, {(1, 2), (0, 1), (0, β2), (1, β2)}, {(1, 3), (0, 0), (0, 3), (1, β3)}, and WEAK PHASE RETRIEVAL 5 {(2, 3), (1, 0), (β2, 3), (β2, β3)} do weak phase retrieval in R2, but by Theorem 5, X cannot do weak phase retrieval in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Weak Phase Retrievable Frames are not Dense in all Frames If m β₯ 2n β 1 and {xi}m i=1 is full spark then it has complement property and hence does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since the full spark frames are dense in all frames, it follows that the frames doing phase retrieval are dense in all frames with β₯ 2n β 1 vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We will now show that this result fails for weak phase retrievable frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The easiest way to get very general frames failing weak phase retrieval is: Choose x, y β Rn so that x + y, x β y do not have the same or opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let X1 = xβ₯ and Y1 = yβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then span{X1, X2} = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Choose {xi}nβ1 i=1 vectors spanning X1 and {xi}2nβ2 i=n be vectors spanning X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then {xi}2nβ2 i=1 is a frame for Rn with x β₯ xi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', n β 1 and y β₯ xi, for all i = n, n + 1, , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , 2n β 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It follows that |β¨x + y, xiβ©| = |β¨x β y, xiβ©|, for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , n, but x, y do not have the same or opposite signs and so {xi}2nβ2 i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If X is a subspace of Rn, we deο¬ne the sphere of X as SX = {x β X : β₯xβ₯ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Deο¬nition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If X, Y are subspaces of Rn, we deο¬ne the distance between X and Y as d(X, Y ) = supxβSXinfyβSY β₯x β yβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It follows that if d(X, Y ) < Η« then for any x β X there is a z β SY so that β₯ x β₯xβ₯ β zβ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Letting y = β₯xβ₯z we have that β₯yβ₯ = β₯xβ₯ and β₯x β yβ₯ < Η«β₯xβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let X, Y be hyperplanes in Rn and unit vectors x β₯ X, y β₯ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If d(X, Y ) < Η« then min{β₯x β yβ₯, β₯x + yβ₯} < 6Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since span{y, Y } = Rn, x = ay + z for some z β Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By replacing y by βy if necessary, we may assume 0 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By assumption, there is some w β X with β₯wβ₯ = β₯zβ₯ so that β₯w β zβ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Now a = aβ₯yβ₯ = β₯ayβ₯ = β₯x β zβ₯ β₯ β₯x β wβ₯ β β₯w β zβ₯ β₯ β₯xβ₯ β Η« = 1 β Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So, 1 β a < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Also, 1 = β₯xβ₯2 = a2 + β₯wβ₯2 implies a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 0 < 1 β a < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 1 = β₯xβ₯2 = β₯ay + zβ₯2 = a2β₯yβ₯2 + β₯zβ₯2 = a2 + β₯zβ₯2 β₯ (1 β Η«)2 + β₯zβ₯2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So β₯zβ₯2 β€ 1 β (1 β Η«)2 = 2Η« β Η«2 β€ 2Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Finally, β₯x β yβ₯2 = β₯(ay + z) β yβ₯2 β€ (β₯(1 β a)yβ₯ + β₯zβ₯)2 β€ (1 β a)2β₯yβ₯2 + β₯zβ₯2 + 2(1 β a)β₯yβ₯β₯zβ₯ < Η«2 + 2Η« + 2 β 2Η«2 < 6Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' AKRAMI β‘ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let X, Y be hyperplanes in Rn, {xi}nβ1 i=1 be a unit norm basis for X and {yi}nβ1 i=1 be a unit norm basis for Y with basis bounds B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If οΏ½nβ1 i=1 β₯xi β yiβ₯ < Η« then d(X, Y ) < 2Η«B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let 0 < A β€ B < β be upper and lower basis bounds for the two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Given a unit vector x = οΏ½nβ1 i=1 aixi β X, let y = οΏ½nβ1 i=1 aiyi β Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We have that sup1β€iβ€nβ1|ai| β€ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We compute: β₯x β yβ₯ = β₯ nβ1 οΏ½ i=1 ai(xi β yi)β₯ β€ nβ1 οΏ½ i=1 |ai|β₯xi β yiβ₯ β€ (sup1β€iβ€nβ1|ai|) nβ1 οΏ½ i=1 β₯xi β yiβ₯ β€ BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So β₯yβ₯ β₯ β₯xβ₯ β β₯x β yβ₯ β₯ 1 β BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' οΏ½οΏ½οΏ½οΏ½x β y β₯yβ₯ οΏ½οΏ½οΏ½οΏ½ β€ β₯x β yβ₯ + οΏ½οΏ½οΏ½οΏ½y β y β₯yβ₯ οΏ½οΏ½οΏ½οΏ½ β€ BΗ« + 1 β₯yβ₯β₯(1 β β₯yβ₯)yβ₯ = BΗ« + (1 β β₯yβ₯) β€ 2BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It follows that d(X, Y ) < 2BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {xi}n i=1 be a basis for Rn with unconditional basis constant B and assume yi β Rn satisο¬es οΏ½n i=1 β₯xi β yiβ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then {yi}n i=1 is a basis for Rn which is 1 + Η«B-equivalent to {xi}n i=1 and has unconditional basis constant B(1 + Η«B)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Fix {ai}n i=1 and compute β₯ n οΏ½ i=1 aiyiβ₯ β€ β₯ n οΏ½ i=1 aixiβ₯ + β₯ n οΏ½ i=1 |ai|(xi β yi)β₯ β€ β₯ n οΏ½ i=1 aixiβ₯ + (sup1β€iβ€n|ai|) n οΏ½ i=1 β₯xi β yiβ₯ β€ β₯ n οΏ½ i=1 aixiβ₯ + (sup1β€iβ€n|ai|)Η« β€ β₯ n οΏ½ i=1 aixiβ₯ + Η«Bβ₯ n οΏ½ i=1 aixiβ₯ = (1 + Η«B)β₯ n οΏ½ i=1 aixiβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' WEAK PHASE RETRIEVAL 7 Similarly, β₯ n οΏ½ i=1 |ai|yiβ₯ β₯ (1 β Η«B)β₯ n οΏ½ i=1 aixiβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So {xi}n i=1 is (1 + Η«B)-equivalent to {yi}n i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For Η«i = Β±1, β₯ n οΏ½ i=1 Η«iaiyiβ₯ β€ (1 + Η«B)β₯ n οΏ½ i=1 Η«iaixiβ₯ β€ B(1 + Η«B)β₯ n οΏ½ i=1 aixiβ₯ β€ B(1 + Η«B)2β₯ n οΏ½ i=1 aiyiβ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' and so {yi}n i=1 is a B(1 + Η«B) unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The family of m-element weak phase retrieval frames are not dense in the set of m-element frames in Rn for all m β₯ 2n β 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We may assume m = 2nβ2 since for larger m we just repeat the (2n-2) vec- tors over and over until we get m vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {ei}n i=1 be the canonical orthonormal basis for Rn and let xi = ei for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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257 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By [10], there is an orthonormal sequence {xi}2nβ2 i=n+1 so that {xi}2nβ2 i=1 is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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262 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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263 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', n β 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let X = span{xi}nβ1 i=1 and Y = span{xi}2nβ2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then x = en β₯ X and there is a β₯yβ₯ = 1 with y β₯ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Note that β¨x β y, enβ© ΜΈ= 0 ΜΈ= β¨x + y, enβ©, for otherwise, x = Β±y β₯ span{xi}iΜΈ=n, contradicting the fact that the vectors are full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So there is a j = n and a Ξ΄ > 0 so that |(x + y)(j)|, |(x β y)(j)| β₯ Ξ΄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We will show that there exists an 0 < Η« so that whenever {yi}2nβ2 i=1 are vectors in Rn satisfying οΏ½n i=1 β₯xi β yiβ₯ < Η«, then {yi}n i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Fix 0 < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Assume {yi}2nβ2 i=1 are vectors so that οΏ½2nβ2 i=1 β₯xiβyiβ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Choose unit vectors xβ² β₯ span{yi}iβI, yβ² β₯ span{yi}iβIc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By Proposition 2 and Lemma 1, we may choose Η« small enough (and change signs if necessary) so that β₯xβxβ²β₯, β₯yβyβ²β₯ < Ξ΄ 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Hence, since the unconditional basis constant is B, |[(x + y) β (xβ² + yβ²)](j)| β€ |(x β xβ²)j| + |(y β yβ²)(j)| < Bβ₯x β xβ²β₯ + Bβ₯y β yβ²β₯ β€ 2B Ξ΄ 4B = Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It follows that |(xβ² + yβ²)(j)| β₯ |(x + y)(j)| β |[(x + y) β (xβ² + yβ²)](j)| β₯ Ξ΄ β 1 2Ξ΄ = Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Similarly, |(xβ² β yβ²)(j)| > Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So xβ² + yβ², xβ² β yβ² are not disjointly supported and so {yi}2nβ2 i=1 fails weak phase retrieval by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Classifying Weak Phase Retrieval In this section we will give several surprising equivalences and consequences of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' These results give a complete understanding of the diο¬erence between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Now we give a surprising and very strong classiο¬cation of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' AKRAMI Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {xi}2nβ2 i=1 be non-zero vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The following are equivalent: (1) The family {xi}2nβ2 i=1 does weak phase retrieval in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (2) If x, y β Rn and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='1) |β¨x, xiβ©| = |β¨y, xiβ©| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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290 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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291 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , 2n β 2, then one of the following holds: (a) x = Β±y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (b) x and y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (1) β (2): Given the assumption in the theorem, assume (a) fails and we will show that (b) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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297 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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298 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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300 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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301 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since {xi}2nβ2 i=1 does weak phase retrieval, replacing y by βy if necessary, Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='1 implies aj = bj whenever aj ΜΈ= 0 ΜΈ= bj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let I = {1 β€ i β€ 2n β 2 : β¨x, xiβ© = β¨y, yiβ©.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then x + y β₯ xi for all i β Ic and x β y β₯ xi for all i β I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' By Theorem 2, x + y β₯x + y + x β y β₯x β yβ₯ and x + y β₯x + yβ₯ β x β y β₯x β yβ₯ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Assume there is a 1 β€ j β€ n with aj = bj ΜΈ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then (x + y)(j) β₯x + yβ₯ + (x β y)(j) β₯x β yβ₯ = 2aj β₯x + yβ₯ and (x + y)(j) β₯x + yβ₯ β (x β y)(j) β₯x β yβ₯ = 2aj β₯x + yβ₯, Contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (2) β (1): This is immediate since (a) and (b) give the conditions for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Phase retrieval is when (a) in the theorem holds for every x, y β Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So this the- orem shows clearly the diο¬erence between weak phase retrieval and phase retrieval: namely when (b) holds at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If {xi}2nβ2 i=1 does weak phase retrieval in Rn, then there are disjointly supported non-zero vectors x, y β Rn satisfying: |β¨x, xiβ©| = |β¨y, xiβ©| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , 2n β 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since {xi}2nβ2 i=1 must fail phase retrieval, (b) of Theorem 7 must hold at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Deο¬nition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {ei}n i=1 be the canonical orthonormal basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If J β [n], we deο¬ne PJ as the projection onto span{ei}iβJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {xi}m i=1 be unit vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The following are equivalent: (1) Whenever I β [2n β 2] and 0 ΜΈ= x β₯ xi for i β I and 0 ΜΈ= y β₯ xi for i β Ic, there is no j β [n] so that β¨x, ejβ© = 0 = β¨y, ejβ©.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (2) For every J β [n] with |J| = n β 1, {Pjxi}2nβ2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (3) For every J β [n] with |J| < n, {PJxi}2nβ2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' WEAK PHASE RETRIEVAL 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (1) β (2): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So assume (2) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then choose J β [n] with |J| = n β 1, J = [n] \\ {j}, and {PJxi}2nβ2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' In particular, it fails complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' That is, there exists I β [2nβ 2] and span {PJxi}iβI ΜΈ= PJRn and span {Pjxi}iβIc ΜΈ= PJRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So there exists norm one vectors x, y in PJRn with PJx = x β₯ PJxi for all i β I and PJy = y β₯ PJxi for all i β Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Extend x, y to all of Rn by setting x(j) = y(j) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Hence, x β₯ xi for i β I and y β₯ xi for i β Ic, proving (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (2) β (3): This follows from the fact that every projection of a set of vectors doing phase retrieval onto a subset of the basis also does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (3) β (2): This is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (3) β (1): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So assume (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then there is a I β [2nβ 2] and 0 ΜΈ= x β₯ xi for i β I and 0 ΜΈ= y β₯ xi for i β Ic and a j β [n] so that β¨x, ejβ© = β¨y, ejβ© = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' It follows that x = PJx, y = PJy are non zero and x β₯ Pjxi for all i β I and y β₯ Pjxi for i β Ic, so {PJxi}2nβ2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' The assumptions in the theorem are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' That is, in general, {xi}m i=1 can do weak phase retrieval and {PJxi}m i=1 may fail phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For example, in R3 consider the row vectors {xi}4 i=1 of: \uf8ee \uf8ef\uf8ef\uf8f0 1 1 1 β1 1 1 1 β1 1 1 1 β1 \uf8f9 \uf8fa\uf8fa\uf8fb This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, β1) β₯ PJxi for i = 1, 2 and y = (0, 1, 1) β₯ xi for i = 3, 4 and {PJxi}4 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Assume {xi}2nβ2 i=1 does weak phase retrieval in Rn and for every J β [n] {PJxi}2nβ2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then if x, y β Rn and |β¨x, xiβ©| = |β¨y, xiβ©| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' , 2n β 2, then there is a J β [n] so that x(j) = οΏ½ aj ΜΈ= 0 for j β J 0 for j β Jc y(j) = οΏ½ 0 for j β J bj ΜΈ= 0 for j β Jc Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let {ei}n i=1 be the unit vector basis of Rn and for I β [n], let PI be the projection onto XI = span{ei}iβI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For every m β₯ 1, there are vectors {xi}m i=1 so that for every I β [1, n], {PIxi}m i=1 is full spark in XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We do this by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' For m=1, let x1 = (1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' This satisο¬es the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' So assume the theorem holds for {xi}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Choose I β [1, n] with |I| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Choose J β I with |J| = k β 1 and let XJ = span{xi}iβJ βͺ {xi}iβIc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Then XJ is a hyperplane in Rn for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since there only exist ο¬nitely many such Jβ²s there is a vector xm+1 /β XJ for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' We will show that {xi}m+1 i=1 satisο¬es the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Let I β [1, n] and J β I with |J| = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' If PIxm+1 /β XJ, then {PIxi}iβJ is linearly independent by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' On the other hand, if m + 1 β J then xm+1 /β XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' But, if PIxm+1 β span{PIxi}iβJ\\m+1, since (I β PI)xm+1 β span{ei}iβIc, it follows that xm+1 β XJ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' β‘ 10 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' AKRAMI Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' In the above proposition, none of the xi can have a zero coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Since if it does, projecting the vectors onto that coordinate produces a zero vector and so is not full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' References [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Akrami, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Casazza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Hasankhani Fard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Rahimi, A note on norm retrievable real Hilbert space frames, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' (517)2, (2023) 126620.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Casazza, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Akrami, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Rahimi, fundamental results on weak phase retrieval, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Anal, arXiv: 2110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='06868, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Bahmanpour, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Cahill, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Casazza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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page_content=' Jasper, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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9dE1T4oBgHgl3EQfoAQF/vector_store/index.pkl
ADDED
@@ -0,0 +1,3 @@
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version https://git-lfs.github.com/spec/v1
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size 103720
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A9AyT4oBgHgl3EQf3_rL/content/tmp_files/2301.00780v1.pdf.txt
ADDED
The diff for this file is too large to render.
See raw diff
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A9AyT4oBgHgl3EQf3_rL/content/tmp_files/load_file.txt
ADDED
The diff for this file is too large to render.
See raw diff
|
|
AtFLT4oBgHgl3EQfFC_E/vector_store/index.faiss
ADDED
@@ -0,0 +1,3 @@
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1 |
+
version https://git-lfs.github.com/spec/v1
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size 9371693
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BdAzT4oBgHgl3EQfh_0J/content/tmp_files/2301.01491v1.pdf.txt
ADDED
@@ -0,0 +1,3401 @@
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|
1 |
+
Higher order Bernstein-BΒ΄ezier and NΒ΄edΒ΄elec ο¬nite elements for the
|
2 |
+
relaxed micromorphic model
|
3 |
+
Adam Sky1,
|
4 |
+
Ingo Muench2,
|
5 |
+
Gianluca Rizzi3
|
6 |
+
and
|
7 |
+
Patrizio Neο¬4
|
8 |
+
January 5, 2023
|
9 |
+
Abstract
|
10 |
+
The relaxed micromorphic model is a generalized continuum model that is well-posed in the space
|
11 |
+
X = [H 1]3 Γ [H (curl)]3. Consequently, ο¬nite element formulations of the model rely on H 1-conforming
|
12 |
+
subspaces and NΒ΄edΒ΄elec elements for discrete solutions of the corresponding variational problem. This work
|
13 |
+
applies the recently introduced polytopal template methodology for the construction of NΒ΄edΒ΄elec elements.
|
14 |
+
This is done in conjunction with Bernstein-BΒ΄ezier polynomials and dual numbers in order to compute hp-
|
15 |
+
FEM solutions of the model. Bernstein-BΒ΄ezier polynomials allow for optimal complexity in the assembly
|
16 |
+
procedure due to their natural factorization into univariate Bernstein base functions. In this work, this
|
17 |
+
characteristic is further augmented by the use of dual numbers in order to compute their values and their
|
18 |
+
derivatives simultaneously. The application of the polytopal template methodology for the construction of
|
19 |
+
the NΒ΄edΒ΄elec base functions allows them to directly inherit the optimal complexity of the underlying Bernstein-
|
20 |
+
BΒ΄ezier basis. We introduce the Bernstein-BΒ΄ezier basis along with its factorization to univariate Bernstein
|
21 |
+
base functions, the principle of automatic diο¬erentiation via dual numbers and a detailed construction of
|
22 |
+
NΒ΄edΒ΄elec elements based on Bernstein-BΒ΄ezier polynomials with the polytopal template methodology. This is
|
23 |
+
complemented with a corresponding technique to embed Dirichlet boundary conditions, with emphasis on
|
24 |
+
the consistent coupling condition. The performance of the elements is shown in examples of the relaxed
|
25 |
+
micromorphic model.
|
26 |
+
Key words: NΒ΄edΒ΄elec elements, Bernstein-BΒ΄ezier elements, relaxed micromorphic model, dual numbers, au-
|
27 |
+
tomatic diο¬erentiation, hp-FEM, generalized continua.
|
28 |
+
1
|
29 |
+
Introduction
|
30 |
+
One challenge that arises in the computation of materials with a pronounced micro-structure is the necessity of
|
31 |
+
modelling the complex geometry of the domain as a whole, in order to correctly capture its intricate kinematics.
|
32 |
+
In other words, unit-cell geometries in metamaterials or various hole-shapes in porous media have to be accounted
|
33 |
+
for in order to assert the viability of the model. Naturally, this correlates with the resolution of the discretization
|
34 |
+
in ο¬nite element simulations, resulting in longer computation times.
|
35 |
+
The relaxed micromorphic model [35] oο¬ers an alternative approach by introducing a continuum model with
|
36 |
+
enriched kinematics, accounting for the independent distortion arising from the micro-structure. As such, for
|
37 |
+
each material point, the model introduces the microdistortion ο¬eld P in addition to the standard displacement
|
38 |
+
ο¬eld u. Consequently, each material point is endowed with twelve degrees of freedom, eο¬ectively turning into
|
39 |
+
an aο¬ne-deformable micro-body with its own orientation. In contrast to the classical micromorphic model [17]
|
40 |
+
1Corresponding author: Adam Sky, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund,
|
41 |
+
August-Schmidt-Str. 8, 44227 Dortmund, Germany, email: [email protected]
|
42 |
+
2Ingo Muench, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-Str.
|
43 |
+
8, 44227 Dortmund, Germany, email: [email protected]
|
44 |
+
3Gianluca Rizzi, Institute of Structural Mechanics, Statics and Dynamics, Technische Universit¨at Dortmund, August-Schmidt-
|
45 |
+
Str. 8, 44227 Dortmund, Germany, email: [email protected]
|
46 |
+
4Patrizio Neο¬,
|
47 |
+
Chair for Nonlinear Analysis and Modelling, Faculty of Mathematics, Universit¨at Duisburg-Essen, Thea-
|
48 |
+
Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neο¬@uni-due.de
|
49 |
+
1
|
50 |
+
arXiv:2301.01491v1 [math.NA] 4 Jan 2023
|
51 |
+
|
52 |
+
by Eringen [15] and Mindlin [29], the relaxed micromorphic model does not employ the full gradient of the
|
53 |
+
microdistortion DP in its energy functional but rather its skew-symmetric part Curl P , designated as the
|
54 |
+
micro-dislocation. Therefore, the micro-dislocation Curl P remains a second-order tensor, whereas DP is a
|
55 |
+
third-order tensor.
|
56 |
+
Further, the model allows the transition between materials with a pronounced micro-
|
57 |
+
structure and homogeneous materials using the characteristic length scale parameter Lc, which governs the
|
58 |
+
inο¬uence of the micro-structure. In highly homogeneous materials the characteristic length scale parameter
|
59 |
+
approaches zero Lc β 0, and for materials with a pronounced micro-structure its value is related to the size of
|
60 |
+
the underlying unit-cell geometry. Recent works demonstrate the eο¬ectiveness of the model in the simulation
|
61 |
+
of band-gap metamaterials [7, 10, 13, 27, 28] and shielding against elastic waves [4, 40, 41, 46].
|
62 |
+
Furthermore,
|
63 |
+
analytical solutions are already available for bending [43], torsion [42], shear [44], and extension [45] kinematics.
|
64 |
+
We note that the usage of the curl operator in the free energy functional directly inο¬uences the appropriate
|
65 |
+
Hilbert spaces for existence and uniqueness of the related variational problem. Namely, the relaxed micromorphic
|
66 |
+
model is well-posed in {u, P } β X = [H 1]3 Γ [H (curl)]3 [18,34], although the regularity of the microdistortion
|
67 |
+
can be improved to P β [H 1]3Γ3 for certain smoothness of the data [22, 38]. As shown in [52], the X -space
|
68 |
+
asserts well-posedness according to the Lax-Milgram theorem, such that H 1-conforming subspaces and NΒ΄edΒ΄elec
|
69 |
+
elements [9,30,31] inherit the well-posedness property as well.
|
70 |
+
In this work we apply the polytopal template methodology introduced in [50] in order to construct higher
|
71 |
+
order NΒ΄edΒ΄elec elements based on Bernstein polynomials [23] and apply the formulation to the relaxed micro-
|
72 |
+
morphic model. Bernstein polynomials are chosen due to their optimal complexity property in the assembly
|
73 |
+
procedure [1]. We further enhance this feature by employing dual numbers [16] in order to compute the values of
|
74 |
+
the base functions and their derivatives simultaneously. The polytopal template methodology allows to extend
|
75 |
+
this property to the assembly of the NΒ΄edΒ΄elec base functions, resulting in fast computations. Alternatively, the
|
76 |
+
formulation of higher order elements on the basis of Legendre polynomials can be found in [48, 54, 58]. The
|
77 |
+
construction of low order NΒ΄edΒ΄elec elements can be found in [5, 51] and speciο¬cally in the context of the the
|
78 |
+
relaxed micromorphic model in [47,49,52,53].
|
79 |
+
This paper is structured as follows. First, we introduce the relaxed micromorphic model and its limit cases
|
80 |
+
with respect to the characteristic length scale parameter Lc, after which we reduce it to a model of antiplane
|
81 |
+
shear [55]. Next, we shortly discuss Bernstein polynomials and dual numbers for automatic diο¬erentiation. The
|
82 |
+
BΒ΄ezier polynomial basis for triangles and tetrahedra is introduced, along with its factorization, highlighting
|
83 |
+
its compatibility with dual numbers. We consider a numerical example in antiplane shear for two-dimensional
|
84 |
+
elements, a three-dimensional example for convergence of cylindrical bending, and a benchmark for the behaviour
|
85 |
+
of the model with respect to the characteristic length scale parameter Lc. Lastly, we present our conclusions
|
86 |
+
and outlook.
|
87 |
+
The following deο¬nitions are employed throughout this work:
|
88 |
+
β’ vectors are indicated by bold letters. Non-bold letters represent scalars;
|
89 |
+
β’ in general, formulas are deο¬ned using the Cartesian basis, where the base vectors are denoted by e1, e2
|
90 |
+
and e3;
|
91 |
+
β’ three-dimensional domains in the physical space are denoted with V β R3. The corresponding reference
|
92 |
+
domain is given by β¦;
|
93 |
+
β’ analogously, in two dimensions we employ A β R2 for the physical domain and Ξ for the reference domain;
|
94 |
+
β’ curves on the physical domain are denoted by s, whereas curves in the reference domain by Β΅;
|
95 |
+
β’ the tangent and normal vectors in the physical domain are given by t and n, respectively. Their counter-
|
96 |
+
parts in the reference domain are Ο for tangent vectors and Ξ½ for normal vectors.
|
97 |
+
2
|
98 |
+
|
99 |
+
2
|
100 |
+
The relaxed micromorphic model
|
101 |
+
The relaxed micromorphic model [35] is governed by a free energy functional, incorporating the gradient of the
|
102 |
+
displacement ο¬eld Du, the microdistortion P and the Curl of the microdistortion
|
103 |
+
I(u, P ) = 1
|
104 |
+
2
|
105 |
+
οΏ½
|
106 |
+
V
|
107 |
+
β¨sym(Du β P ), Ce sym(Du β P )β© + β¨sym P , Cmicro sym P β©
|
108 |
+
+ β¨skew(Du β P ), Cc skew(Du β P )β© + Β΅macroL2
|
109 |
+
cβ¨Curl P , L Curl P β© dV
|
110 |
+
β
|
111 |
+
οΏ½
|
112 |
+
V
|
113 |
+
β¨u, fβ© + β¨P , Mβ© dV β min
|
114 |
+
w.r.t.
|
115 |
+
{u, P } ,
|
116 |
+
(2.1)
|
117 |
+
where the Curl operator for second order tensors is deο¬ned row-wise as
|
118 |
+
Curl P =
|
119 |
+
οΏ½
|
120 |
+
οΏ½
|
121 |
+
curl(
|
122 |
+
οΏ½P11
|
123 |
+
P12
|
124 |
+
P13
|
125 |
+
οΏ½
|
126 |
+
)
|
127 |
+
curl(
|
128 |
+
οΏ½P21
|
129 |
+
P22
|
130 |
+
P23
|
131 |
+
οΏ½
|
132 |
+
)
|
133 |
+
curl(
|
134 |
+
οΏ½
|
135 |
+
P31
|
136 |
+
P32
|
137 |
+
P33
|
138 |
+
οΏ½
|
139 |
+
)
|
140 |
+
οΏ½
|
141 |
+
οΏ½ =
|
142 |
+
οΏ½
|
143 |
+
οΏ½
|
144 |
+
P13,y β P12,z
|
145 |
+
P11,z β P13,x
|
146 |
+
P12,x β P11,y
|
147 |
+
P23,y β P22,z
|
148 |
+
P21,z β P23,x
|
149 |
+
P22,x β P21,y
|
150 |
+
P33,y β P32,z
|
151 |
+
P31,z β P33,x
|
152 |
+
P32,x β P31,y
|
153 |
+
οΏ½
|
154 |
+
οΏ½ ,
|
155 |
+
curl p = β Γ p ,
|
156 |
+
p : V β R3 β R3 ,
|
157 |
+
(2.2)
|
158 |
+
and curl(Β·) is the vectorial curl operator. The displacement ο¬eld and the microdistortion ο¬eld are functions of
|
159 |
+
the reference domain
|
160 |
+
u : V β R3 β R3 ,
|
161 |
+
P : V β R3 β R3Γ3 .
|
162 |
+
(2.3)
|
163 |
+
The tensors Ce, Cmicro, L β R3Γ3Γ3Γ3 are standard positive deο¬nite fourth order elasticity tensors. For isotropic
|
164 |
+
materials they take the form
|
165 |
+
Ce = Ξ»e1 β 1 + 2Β΅e J ,
|
166 |
+
Cmicro = Ξ»micro1 β 1 + 2Β΅micro J .
|
167 |
+
(2.4)
|
168 |
+
where 1 is the second order identity tensor and J is the fourth order identity tensor. The fourth order tensor
|
169 |
+
Cc β R3Γ3Γ3Γ3 is a positive semi-deο¬nite material tensor related to Cosserat micro-polar continua and accounts
|
170 |
+
for inο¬nitesimal rotations Cc : so(3) β so(3), where so(3) is the space of skew-symmetric matrices.
|
171 |
+
For isotropic materials there holds Cc = 2Β΅c J, where Β΅c β₯ 0 is called the Cosserat couple modulus. Further,
|
172 |
+
for simplicity, we assume L = J in the following. The macroscopic shear modulus is denoted by Β΅macro and
|
173 |
+
Lc represents the characteristic length scale motivated by the geometry of the microstructure. The forces and
|
174 |
+
micro-moments are given by f and M, respectively.
|
175 |
+
Equilibrium is found at minima of the energy functional, which is strictly convex (also for Cc β‘ 0). As such,
|
176 |
+
we consider variations with respect to its parameters, namely the displacement and the microdistortion. Taking
|
177 |
+
variations of the energy functional with respect to the displacement ο¬eld u yields
|
178 |
+
Ξ΄uI =
|
179 |
+
οΏ½
|
180 |
+
V
|
181 |
+
β¨sym DΞ΄u, Ce sym(Du β P )β© + β¨skew DΞ΄u, Cc skew(Du β P )β© β β¨Ξ΄u, fβ© dV = 0 .
|
182 |
+
(2.5)
|
183 |
+
The variation with respect to the microdistortion P results in
|
184 |
+
Ξ΄P I =
|
185 |
+
οΏ½
|
186 |
+
V
|
187 |
+
β¨sym Ξ΄P , Ce sym(Du β P )β© + β¨skew Ξ΄P , Cc skew(Du β P )β©
|
188 |
+
β β¨sym Ξ΄P , Cmicro sym P β© β Β΅macroL2
|
189 |
+
cβ¨Curl Ξ΄P , Curl P β© + β¨Ξ΄P , Mβ© dV = 0 .
|
190 |
+
(2.6)
|
191 |
+
From the total variation we extract the bilinear form
|
192 |
+
a({Ξ΄u, Ξ΄P }, {u, P }) =
|
193 |
+
οΏ½
|
194 |
+
V
|
195 |
+
β¨sym(DΞ΄u β Ξ΄P ), Ce sym(Du β P )β© + β¨sym Ξ΄P , Cmicro sym P β©
|
196 |
+
+ β¨skew(DΞ΄u β Ξ΄P ), Cc skew(Du β P )β© + Β΅macroL2
|
197 |
+
cβ¨Curl Ξ΄P , Curl P β© dV ,
|
198 |
+
(2.7)
|
199 |
+
and linear form of the loads
|
200 |
+
l({Ξ΄u, Ξ΄P }) =
|
201 |
+
οΏ½
|
202 |
+
V
|
203 |
+
β¨Ξ΄u, fβ© + β¨Ξ΄P , Mβ© dV .
|
204 |
+
(2.8)
|
205 |
+
3
|
206 |
+
|
207 |
+
Applying integration by parts to Eq. (2.5) yields
|
208 |
+
οΏ½
|
209 |
+
βV
|
210 |
+
β¨Ξ΄u , [Ce sym(Du β P ) + Cc skew(Du β P )] nβ© dA
|
211 |
+
β
|
212 |
+
οΏ½
|
213 |
+
V
|
214 |
+
β¨Ξ΄u , Div[Ce sym(Du β P ) + Cc skew(Du β P )] β fβ© dV = 0 .
|
215 |
+
(2.9)
|
216 |
+
Likewise, integration by parts of Eq. (2.6) results in
|
217 |
+
οΏ½
|
218 |
+
V
|
219 |
+
β¨Ξ΄P , Ce sym(Du β P ) + Cc skew(Du β P ) β Cmicro sym P β Β΅macroL2
|
220 |
+
c Curl Curl P + Mβ© dV
|
221 |
+
β Β΅macroL2
|
222 |
+
c
|
223 |
+
οΏ½
|
224 |
+
βV
|
225 |
+
β¨Ξ΄P , Curl P Γ nβ© dA = 0 .
|
226 |
+
(2.10)
|
227 |
+
The strong form is extracted from Eq. (2.9) and Eq. (2.10) by splitting the boundary
|
228 |
+
A = AD βͺ AN ,
|
229 |
+
AD β© AN = β
,
|
230 |
+
(2.11)
|
231 |
+
into a Dirichlet boundary with embedded boundary conditions and a Neumann boundary with natural boundary
|
232 |
+
conditions, such that no tractions are imposed on the Neumann boundary
|
233 |
+
β Div[Ce sym(Du β P ) + Cc skew(Du β P )] = f
|
234 |
+
in
|
235 |
+
V ,
|
236 |
+
(2.12a)
|
237 |
+
βCe sym(Du β P ) β Cc skew(Du β P ) + Cmicro sym P + Β΅macro L2
|
238 |
+
c Curl Curl P = M
|
239 |
+
in
|
240 |
+
V ,
|
241 |
+
(2.12b)
|
242 |
+
u = οΏ½u
|
243 |
+
on
|
244 |
+
Au
|
245 |
+
D ,
|
246 |
+
(2.12c)
|
247 |
+
P Γ n = οΏ½P Γ n
|
248 |
+
on
|
249 |
+
AP
|
250 |
+
D , (2.12d)
|
251 |
+
[Ce sym(Du β P ) + Cc skew(Du β P )] n = 0
|
252 |
+
on
|
253 |
+
Au
|
254 |
+
N ,
|
255 |
+
(2.12e)
|
256 |
+
Curl P Γ n = 0
|
257 |
+
on
|
258 |
+
AP
|
259 |
+
N .
|
260 |
+
(2.12f)
|
261 |
+
The force stress tensor οΏ½Ο := Ce sym(Du β P ) + Cc skew(Du β P ) is symmetric if and only if Cc β‘ 0, a case
|
262 |
+
which is permitted. Problem. 2.12 represents a tensorial Maxwell-problem coupled to linear elasticity. We
|
263 |
+
observe that the Dirichlet boundary condition for the microdistortion controls only its tangential components.
|
264 |
+
It is unclear, how to control the micro-movements of a material point without also aο¬ecting the displacement.
|
265 |
+
Therefore, the relaxed micromorphic model introduces the so called consistent coupling condition [11]
|
266 |
+
P Γ n = DοΏ½u Γ n
|
267 |
+
on
|
268 |
+
AP
|
269 |
+
D ,
|
270 |
+
(2.13)
|
271 |
+
where the prescribed displacement on the Dirichlet boundary οΏ½u automatically dictates the tangential component
|
272 |
+
of the microdistortion on that same boundary. Consequently, the consistent coupling condition enforces the
|
273 |
+
deο¬nitions AD = Au
|
274 |
+
D = AP
|
275 |
+
D and AN = Au
|
276 |
+
N = AP
|
277 |
+
N (see Fig. 2.1). Further, the consistent coupling condition
|
278 |
+
substitutes Eq. (2.12d).
|
279 |
+
The set of equations in Problem. 2.12 remains well-posed for Cc β‘ 0 due to the
|
280 |
+
generalized Korn inequality for incompatible tensor ο¬elds [24β26,36]. The inequality relies on a non-vanishing
|
281 |
+
Dirichlet boundary for the microdistortion ο¬eld AP
|
282 |
+
D ΜΈ= β
, which the consistent coupling condition guarantees.
|
283 |
+
2.1
|
284 |
+
Limits of the characteristic length scale parameter - a true two scale model
|
285 |
+
In the relaxed micromorphic model the characteristic length Lc takes the role of a scaling parameter between
|
286 |
+
the well-deο¬ned macro and the micro scales. This property, unique to the relaxed micromorphic model, allows
|
287 |
+
the theory to interpolate between materials with a pronounced micro-structure and homogeneous materials,
|
288 |
+
thus relating the characteristic length scale parameter Lc to the size of the micro-structure in metamaterials.
|
289 |
+
In the lower limit Lc β 0 the continuum is treated as homogeneous and the solution of the classical Cauchy
|
290 |
+
continuum theory is retrieved [3,32]. This can be observed by reconsidering Eq. (2.12b) for Lc = 0,
|
291 |
+
βCe sym(Du β P ) β Cc skew(Du β P ) + Cmicro sym P = M ,
|
292 |
+
(2.14)
|
293 |
+
which can now be used to express the microdistortion P algebraically
|
294 |
+
sym P = (Ce + Cmicro)β1(sym M + Ce sym Du) ,
|
295 |
+
skew P = Cβ1
|
296 |
+
c
|
297 |
+
skew M + skew Du .
|
298 |
+
(2.15)
|
299 |
+
4
|
300 |
+
|
301 |
+
x
|
302 |
+
y
|
303 |
+
n
|
304 |
+
f
|
305 |
+
M
|
306 |
+
V
|
307 |
+
AD = Au
|
308 |
+
D = AP
|
309 |
+
D
|
310 |
+
AN = Au
|
311 |
+
N = AP
|
312 |
+
N
|
313 |
+
Figure 2.1: The domain in the relaxed micromorphic model with Dirichlet and Neumann boundaries under
|
314 |
+
internal forces and micro-moments. The Dirichlet boundary of the microdistortion is given by the consistent
|
315 |
+
coupling condition. The model can capture the complex kinematics of an underlying micro-structure.
|
316 |
+
Setting M = 0 corresponds to Cauchy continua, where micro-moments are not accounted for. Thus, one ο¬nds
|
317 |
+
Cc skew(Du β P ) = 0 ,
|
318 |
+
Ce sym(Du β P ) = Cmicro sym P ,
|
319 |
+
sym P = (Ce + Cmicro)β1Ce sym Du .
|
320 |
+
(2.16)
|
321 |
+
Applying the former results to Eq. (2.12a) yields
|
322 |
+
β Div[Ce sym(Du β P )] = β Div[Cmicro(Ce + Cmicro)β1Ce sym Du] = β Div[Cmacro sym Du] = f ,
|
323 |
+
(2.17)
|
324 |
+
where the deο¬nition
|
325 |
+
Cmacro = Cmicro(Ce + Cmicro)β1Ce
|
326 |
+
(2.18)
|
327 |
+
relates the meso- and micro-elasticity tensors to the classical macro-elasticity tensor of the Cauchy continuum.
|
328 |
+
In fact, Cmacro contains the material constants that arise from standard homogenization for large periodic
|
329 |
+
structures [3,32]. For isotropic materials one can directly express the macro parameters [33]
|
330 |
+
Β΅macro =
|
331 |
+
Β΅e Β΅micro
|
332 |
+
Β΅e + Β΅micro
|
333 |
+
,
|
334 |
+
2Β΅macro + 3Ξ»macro =
|
335 |
+
(2Β΅e + 3Ξ»e)(2Β΅micro + 3Ξ»micro)
|
336 |
+
(2Β΅e + 3Ξ»e) + (2Β΅micro + 3Ξ»micro)
|
337 |
+
(2.19)
|
338 |
+
in terms of the parameters of the relaxed micromorphic model.
|
339 |
+
In the upper limit Lc β +β, the stiο¬ness of the micro-body becomes dominant. As the characteristic
|
340 |
+
length Lc can be viewed as a zoom-factor into the microstructure, the state Lc β +β can be interpreted as the
|
341 |
+
entire domain being the micro-body itself. However, this is only theoretically possible as in practice, the limit is
|
342 |
+
given by the size of one unit cell. Since the energy functional being minimized contains Β΅macroL2
|
343 |
+
cβ₯ Curl P β₯2, on
|
344 |
+
contractible domains and bounded energy this implies the reduction of the microdistortion to a gradient ο¬eld
|
345 |
+
P β Dv due to the classical identity
|
346 |
+
Curl Dv = 0
|
347 |
+
β v β [C β(V )]3 ,
|
348 |
+
(2.20)
|
349 |
+
thus asserting ο¬nite energies of the relaxed micromorphic model for arbitrarily large characteristic length values
|
350 |
+
Lc. The corresponding energy functional in terms of the reduced kinematics {u, v} : V β R3 now reads
|
351 |
+
I(u, v) = 1
|
352 |
+
2
|
353 |
+
οΏ½
|
354 |
+
V
|
355 |
+
β¨sym(Du β Dv), Ce sym(Du β Dv)β© + β¨sym Dv, Cmicro sym Dvβ©
|
356 |
+
+ β¨skew(Du β Dv), Cc skew(Du β Dv)β© dV β
|
357 |
+
οΏ½
|
358 |
+
V
|
359 |
+
β¨u, fβ© + β¨Dv, Mβ© dV ,
|
360 |
+
(2.21)
|
361 |
+
such that variation with respect to the two vector ο¬elds u and v leads to
|
362 |
+
Ξ΄uI =
|
363 |
+
οΏ½
|
364 |
+
V
|
365 |
+
β¨sym DΞ΄u, Ce sym(Du β Dv)β© + β¨skew DΞ΄u, Cc skew(Du β Dv)β© β β¨Ξ΄u, fβ© dV = 0 ,
|
366 |
+
(2.22a)
|
367 |
+
Ξ΄vI =
|
368 |
+
οΏ½
|
369 |
+
V
|
370 |
+
β¨sym DΞ΄v, Ce sym(Du β Dv)β© + β¨skew DΞ΄v, Cc skew(Du β Dv)β©
|
371 |
+
β β¨sym DΞ΄v, Cmicro sym Dvβ© + β¨DΞ΄v, Mβ© dV = 0 .
|
372 |
+
(2.22b)
|
373 |
+
5
|
374 |
+
|
375 |
+
The resulting bilinear form is given by
|
376 |
+
a({Ξ΄u, Ξ΄v}, {u, v}) =
|
377 |
+
οΏ½
|
378 |
+
V
|
379 |
+
β¨sym(DΞ΄u β DΞ΄v), Ce sym(Du β Dv)β© + β¨sym DΞ΄v, Cmicro sym Dvβ©
|
380 |
+
+ β¨skew(DΞ΄u β DΞ΄v), Cc skew(Du β Dv)β© dV .
|
381 |
+
(2.23)
|
382 |
+
By partial integration of Eq. (2.22a) and Eq. (2.22b) one ο¬nds the equilibrium equations
|
383 |
+
β Div[Ce sym(Du β Dv) + Cc skew(Du β Dv)] = f
|
384 |
+
in
|
385 |
+
V ,
|
386 |
+
(2.24a)
|
387 |
+
β Div[Ce sym(Du β Dv) + Cc skew(Du β Dv)] + Div[Cmicro sym Dv] = Div M
|
388 |
+
in
|
389 |
+
V .
|
390 |
+
(2.24b)
|
391 |
+
We can now substitute the right-hand side of Eq. (2.24a) into Eq. (2.24b) to ο¬nd
|
392 |
+
β Div(Cmicro sym Dv) = f β Div M .
|
393 |
+
(2.25)
|
394 |
+
Clearly, setting v = u satisο¬es both local equilibrium equations Eq. (2.24a) and Eq. (2.24b) for f = 0. Further,
|
395 |
+
the consistent coupling condition Eq. (2.13) is also automatically satisο¬ed, asserting the equivalence of the
|
396 |
+
tangential projections of both ο¬elds on the boundary of the domain.
|
397 |
+
Since, as shown in [32, 52] using the
|
398 |
+
extended Brezzi theorem, the case Lc β +β is well-posed (including Cc β‘ 0), the solution v = u is the unique
|
399 |
+
solution to the bilinear form Eq. (2.23) with the right-hand side
|
400 |
+
l({Ξ΄u, Ξ΄v}) = β¨DΞ΄v, Mβ© dV .
|
401 |
+
(2.26)
|
402 |
+
Eο¬ectively, equation Eq. (2.25) implies that the limit Lc β +β deο¬nes a classical Cauchy continuum with a
|
403 |
+
ο¬nite stiο¬ness governed by Cmicro, representing the upper limit of the stiο¬ness for the relaxed micromorphic
|
404 |
+
continuum [32], where the corresponding forces read m = Div M. We emphasize that this interpretation of
|
405 |
+
Cmicro is impossible in the classical micromorphic model since there the limit Lc β +β results in a constant
|
406 |
+
microdistortion ο¬eld P : V β R3Γ3 as its full gradient DP is incorporated via Β΅macroL2
|
407 |
+
cβ₯DP β₯2 into the energy
|
408 |
+
functional [6].
|
409 |
+
2.2
|
410 |
+
Antiplane shear
|
411 |
+
We introduce the relaxed micromorphic model of antiplane shear1 [55] by reducing the displacement ο¬eld to
|
412 |
+
u =
|
413 |
+
οΏ½0,
|
414 |
+
0,
|
415 |
+
uοΏ½T ,
|
416 |
+
(2.27)
|
417 |
+
such that u = u(x, y) is a function of the x β y-plane. Consequently, its gradient reads
|
418 |
+
Du =
|
419 |
+
οΏ½
|
420 |
+
οΏ½
|
421 |
+
0
|
422 |
+
0
|
423 |
+
0
|
424 |
+
0
|
425 |
+
0
|
426 |
+
0
|
427 |
+
u,x
|
428 |
+
u,y
|
429 |
+
0
|
430 |
+
οΏ½
|
431 |
+
οΏ½ .
|
432 |
+
(2.28)
|
433 |
+
The structure of the microdistortion tensor is chosen accordingly
|
434 |
+
P =
|
435 |
+
οΏ½
|
436 |
+
οΏ½
|
437 |
+
0
|
438 |
+
0
|
439 |
+
0
|
440 |
+
0
|
441 |
+
0
|
442 |
+
0
|
443 |
+
p1
|
444 |
+
p2
|
445 |
+
0
|
446 |
+
οΏ½
|
447 |
+
οΏ½ ,
|
448 |
+
Curl P =
|
449 |
+
οΏ½
|
450 |
+
οΏ½
|
451 |
+
0
|
452 |
+
0
|
453 |
+
0
|
454 |
+
0
|
455 |
+
0
|
456 |
+
0
|
457 |
+
0
|
458 |
+
0
|
459 |
+
p2,x β p1,y
|
460 |
+
οΏ½
|
461 |
+
οΏ½ =
|
462 |
+
οΏ½
|
463 |
+
οΏ½
|
464 |
+
0
|
465 |
+
0
|
466 |
+
0
|
467 |
+
0
|
468 |
+
0
|
469 |
+
0
|
470 |
+
0
|
471 |
+
0
|
472 |
+
curl2Dp
|
473 |
+
οΏ½
|
474 |
+
οΏ½ .
|
475 |
+
(2.29)
|
476 |
+
1Note that the antiplane shear model encompasses 1 + 2 = 3 degrees of freedom and is the simplest non-trivial active version
|
477 |
+
for the relaxed micromorphic model, as the one-dimensional elongation ansatz features only 1 + 1 = 2 degrees of freedom and
|
478 |
+
eliminates the curl operator
|
479 |
+
I(u, p) = 1
|
480 |
+
2
|
481 |
+
οΏ½
|
482 |
+
s
|
483 |
+
(Ξ»e + 2Β΅e)|uβ² β p|2 + (Ξ»micro + 2Β΅micro)|p|2 ds β
|
484 |
+
οΏ½
|
485 |
+
s
|
486 |
+
u f + p m ds β min
|
487 |
+
w.r.t.
|
488 |
+
{u, p} ,
|
489 |
+
since Du = uβ² e1 β e1 and P = p e1 β e1, such that skew(Du β P ) = 0 and Curl P = 0. This is not to be confused with uniaxial
|
490 |
+
extension, which entails 1 + 3 = 4 degrees of freedom [45].
|
491 |
+
6
|
492 |
+
|
493 |
+
Analogously to the displacement ο¬eld u, the microdistortion P is also set to be a function of the {x, y}-variables
|
494 |
+
P = P (x, y). We observe the following sym-skew decompositions of the gradient and microdistortion tensors
|
495 |
+
sym P = 1
|
496 |
+
2
|
497 |
+
οΏ½
|
498 |
+
οΏ½
|
499 |
+
0
|
500 |
+
0
|
501 |
+
p1
|
502 |
+
0
|
503 |
+
0
|
504 |
+
p2
|
505 |
+
p1
|
506 |
+
p2
|
507 |
+
0
|
508 |
+
οΏ½
|
509 |
+
οΏ½ ,
|
510 |
+
sym(Du β P ) = 1
|
511 |
+
2
|
512 |
+
οΏ½
|
513 |
+
οΏ½
|
514 |
+
0
|
515 |
+
0
|
516 |
+
u,x β p1
|
517 |
+
0
|
518 |
+
0
|
519 |
+
u,y β p2
|
520 |
+
u,x β p1
|
521 |
+
u,y β p2
|
522 |
+
0
|
523 |
+
οΏ½
|
524 |
+
οΏ½ ,
|
525 |
+
skew(Du β P ) = 1
|
526 |
+
2
|
527 |
+
οΏ½
|
528 |
+
οΏ½
|
529 |
+
0
|
530 |
+
0
|
531 |
+
p1 β u,x
|
532 |
+
0
|
533 |
+
0
|
534 |
+
p2 β u,y
|
535 |
+
u,x β p1
|
536 |
+
u,y β p2
|
537 |
+
0
|
538 |
+
οΏ½
|
539 |
+
οΏ½ .
|
540 |
+
(2.30)
|
541 |
+
Clearly, there holds
|
542 |
+
tr[sym P ] = tr[sym(Du β P )] = tr[skew(Du β P )] = 0 ,
|
543 |
+
(2.31)
|
544 |
+
such that the contraction with the material tensors reduces to
|
545 |
+
Ce sym(Du β P ) = 2Β΅e sym(Du β P ) ,
|
546 |
+
Cmicro sym(Du β P ) = 2Β΅micro sym P ,
|
547 |
+
Cc skew(Du β P ) = 2Β΅c skew(Du β P ) .
|
548 |
+
(2.32)
|
549 |
+
As such, the quadratic forms of the energy functional are given by
|
550 |
+
β¨sym(Du β P ), Ce sym(Du β P )β© = Β΅eβ₯βu β pβ₯2 ,
|
551 |
+
(2.33a)
|
552 |
+
β¨skew(Du β P ), Cc skew(Du β P )β© = Β΅cβ₯βu β pβ₯2 ,
|
553 |
+
(2.33b)
|
554 |
+
β¨sym P , Cmicro sym P β© = Β΅microβ₯pβ₯2 ,
|
555 |
+
(2.33c)
|
556 |
+
with the deο¬nitions
|
557 |
+
βu =
|
558 |
+
οΏ½u,x
|
559 |
+
u,y
|
560 |
+
οΏ½
|
561 |
+
,
|
562 |
+
p =
|
563 |
+
οΏ½p1
|
564 |
+
p2
|
565 |
+
οΏ½
|
566 |
+
.
|
567 |
+
(2.34)
|
568 |
+
The resulting energy functional for antiplane shear reads therefore
|
569 |
+
I(u, p) = 1
|
570 |
+
2
|
571 |
+
οΏ½
|
572 |
+
A
|
573 |
+
(Β΅e + Β΅c)β₯βu β pβ₯2 + Β΅microβ₯pβ₯2 + Β΅macroL2
|
574 |
+
cβ₯curl2Dpβ₯2 dA β
|
575 |
+
οΏ½
|
576 |
+
A
|
577 |
+
u f + β¨p, mβ© dA .
|
578 |
+
(2.35)
|
579 |
+
In order to maintain consistency with the three-dimensional model we must choose Β΅c = 0. The reasoning for
|
580 |
+
this choice is explained upon in Remark 2.1 (see also Fig. 2.2). Consequently, the energy functional is given by
|
581 |
+
I(u, p) = 1
|
582 |
+
2
|
583 |
+
οΏ½
|
584 |
+
A
|
585 |
+
Β΅eβ₯βu β pβ₯2 + Β΅microβ₯pβ₯2 + Β΅macroL2
|
586 |
+
cβ₯curl2Dpβ₯2 dA
|
587 |
+
β
|
588 |
+
οΏ½
|
589 |
+
A
|
590 |
+
u f + β¨p, mβ© dA β min
|
591 |
+
w.r.t.
|
592 |
+
{u, p} .
|
593 |
+
(2.36)
|
594 |
+
Note that on two-dimensional domains the diο¬erential operators are reduced to
|
595 |
+
βu =
|
596 |
+
οΏ½u,x
|
597 |
+
u,y
|
598 |
+
οΏ½
|
599 |
+
,
|
600 |
+
Rβu =
|
601 |
+
οΏ½ u,y
|
602 |
+
βu,x
|
603 |
+
οΏ½
|
604 |
+
,
|
605 |
+
R =
|
606 |
+
οΏ½
|
607 |
+
0
|
608 |
+
1
|
609 |
+
β1
|
610 |
+
0
|
611 |
+
οΏ½
|
612 |
+
,
|
613 |
+
curl2Dp = div(Rp) = p2,x β p1,y ,
|
614 |
+
(2.37)
|
615 |
+
where we note that curl2D is just a rotated divergence. Taking variations of the energy functional with respect
|
616 |
+
to the displacement ο¬eld results in
|
617 |
+
Ξ΄uI =
|
618 |
+
οΏ½
|
619 |
+
A
|
620 |
+
Β΅eβ¨βΞ΄u, βu β pβ© β Ξ΄u f dA = 0 ,
|
621 |
+
(2.38)
|
622 |
+
and variation with respect to the microdistortion yields
|
623 |
+
Ξ΄pI =
|
624 |
+
οΏ½
|
625 |
+
A
|
626 |
+
Β΅eβ¨Ξ΄p, βu β pβ© β Β΅microβ¨Ξ΄p, pβ© β Β΅macroL2
|
627 |
+
c(curl2DΞ΄p)curl2Dp + β¨Ξ΄p, mβ© dA = 0 .
|
628 |
+
(2.39)
|
629 |
+
7
|
630 |
+
|
631 |
+
Consequently, one ο¬nds the bilinear and linear forms
|
632 |
+
a({Ξ΄u, Ξ΄p}, {u, p}) =
|
633 |
+
οΏ½
|
634 |
+
A
|
635 |
+
Β΅eβ¨βΞ΄u β Ξ΄p, βu β pβ© + Β΅microβ¨Ξ΄p, pβ© + Β΅macroL2
|
636 |
+
c(curl2DΞ΄p)curl2Dp dA ,
|
637 |
+
(2.40a)
|
638 |
+
l({Ξ΄u, Ξ΄p}) =
|
639 |
+
οΏ½
|
640 |
+
A
|
641 |
+
Ξ΄u f + β¨Ξ΄p, mβ© dA .
|
642 |
+
(2.40b)
|
643 |
+
Partial integration of Eq. (2.38) results in
|
644 |
+
οΏ½
|
645 |
+
βA
|
646 |
+
Ξ΄u β¨Β΅e(βu β p), nβ© ds β
|
647 |
+
οΏ½
|
648 |
+
A
|
649 |
+
Ξ΄u [Β΅e div(βu β p) + f] dA = 0 ,
|
650 |
+
(2.41)
|
651 |
+
and analogously for Eq. (2.39), yielding
|
652 |
+
οΏ½
|
653 |
+
A
|
654 |
+
β¨Ξ΄p, Β΅e(βu β p) β Β΅micro p β Β΅macroL2
|
655 |
+
cRβcurl2Dp + mβ© dA β
|
656 |
+
οΏ½
|
657 |
+
βA
|
658 |
+
β¨Ξ΄p, Β΅macroL2
|
659 |
+
c(curl2Dp) tβ© ds = 0 . (2.42)
|
660 |
+
Consequently, the strong form reads
|
661 |
+
βΒ΅e div(βu β p) = f
|
662 |
+
in
|
663 |
+
A ,
|
664 |
+
(2.43a)
|
665 |
+
βΒ΅e(βu β p) + Β΅micro p + Β΅macroL2
|
666 |
+
cRβcurl2Dp = m
|
667 |
+
in
|
668 |
+
A ,
|
669 |
+
(2.43b)
|
670 |
+
u = οΏ½u
|
671 |
+
on
|
672 |
+
su
|
673 |
+
D ,
|
674 |
+
(2.43c)
|
675 |
+
β¨p, tβ© = β¨οΏ½p, tβ©
|
676 |
+
on
|
677 |
+
sP
|
678 |
+
D ,
|
679 |
+
(2.43d)
|
680 |
+
β¨βu, nβ© = β¨p, nβ©
|
681 |
+
on
|
682 |
+
su
|
683 |
+
N ,
|
684 |
+
(2.43e)
|
685 |
+
curl2Dp = 0
|
686 |
+
on
|
687 |
+
sP
|
688 |
+
N .
|
689 |
+
(2.43f)
|
690 |
+
The consistent coupling condition accordingly reduces to
|
691 |
+
β¨p, tβ© = β¨βοΏ½u, tβ©
|
692 |
+
on
|
693 |
+
sD = sP
|
694 |
+
D = su
|
695 |
+
D .
|
696 |
+
(2.44)
|
697 |
+
Remark 2.1
|
698 |
+
Note that without setting Β΅c = 0 in the antiplane shear model, the analogous result to Eq. (2.17) in the limit
|
699 |
+
Lc β 0 would read
|
700 |
+
β
|
701 |
+
οΏ½ Β΅micro [Β΅e + Β΅c]
|
702 |
+
Β΅e + Β΅c + Β΅micro
|
703 |
+
οΏ½
|
704 |
+
οΏ½
|
705 |
+
οΏ½οΏ½
|
706 |
+
οΏ½
|
707 |
+
ΜΈ=Β΅macro
|
708 |
+
βu = f ,
|
709 |
+
(2.45)
|
710 |
+
where the relation to the macro parameter Β΅macro in Eq. (2.19) is lost. Further, the limit deο¬ned in Eq. (2.16)
|
711 |
+
with M = 0 yields the contradiction
|
712 |
+
sym P = (Ce + Cmicro)β1Ce sym Du ,
|
713 |
+
Cc skew P = Cc skew Du ,
|
714 |
+
(2.46)
|
715 |
+
since the equations degenerate to
|
716 |
+
p =
|
717 |
+
Β΅e
|
718 |
+
Β΅e + Β΅micro
|
719 |
+
βu ,
|
720 |
+
Β΅cp = Β΅cβu ,
|
721 |
+
(2.47)
|
722 |
+
due to the equivalent three-dimensional forms for antiplane shear. Choosing Β΅micro = 0 leads to a loss of structure
|
723 |
+
in the strong form Problem. 2.43, while satisfying Eq. (2.47). As such, we must set the Cosserat couple modulus
|
724 |
+
Β΅c = 0 to preserve the structure of the equations and satisfy both Eq. (2.19) and Eq. (2.47).
|
725 |
+
Although the relaxed micromorphic model includes the Cosserat model as a singular limit for Cmicro β +β
|
726 |
+
(Β΅micro β +β), it is impossible to deduce the Cosserat model of antiplane shear as a limit of the antiplane
|
727 |
+
relaxed micromorphic model, since one needs to satisfy Eq. (2.47) for Β΅c > 0 and Β΅micro β +β, which is
|
728 |
+
impossible.
|
729 |
+
8
|
730 |
+
|
731 |
+
The kinematic reduction of the relaxed micromorphic model to antiplane shear and its behaviour in the limit
|
732 |
+
cases of its material parameters is depicted in Fig. 2.2.
|
733 |
+
relaxed micromorphic
|
734 |
+
Cosserat elasticity
|
735 |
+
linear elasticity
|
736 |
+
with Cmacro
|
737 |
+
antiplane relaxed
|
738 |
+
micromorphic
|
739 |
+
antiplane Cosserat
|
740 |
+
elasticity
|
741 |
+
antiplane linear
|
742 |
+
elasticity
|
743 |
+
with Β΅macro
|
744 |
+
Lc β 0
|
745 |
+
Cmicro β +β ,
|
746 |
+
Β΅c > 0
|
747 |
+
Lc β 0 ,
|
748 |
+
Β΅c οΏ½οΏ½οΏ½ 0
|
749 |
+
Β΅micro β +β ,
|
750 |
+
Β΅c > 0
|
751 |
+
(contradiction)
|
752 |
+
antiplane
|
753 |
+
shear
|
754 |
+
antiplane
|
755 |
+
shear
|
756 |
+
antiplane
|
757 |
+
shear
|
758 |
+
antiplane linear
|
759 |
+
elasticity
|
760 |
+
with Β΅micro
|
761 |
+
linear elasticity
|
762 |
+
with Cmicro
|
763 |
+
Lc β +β
|
764 |
+
Lc β +β
|
765 |
+
two-scale
|
766 |
+
model
|
767 |
+
two-scale
|
768 |
+
model
|
769 |
+
non-
|
770 |
+
commutative
|
771 |
+
Figure 2.2: Kinematic reduction of the relaxed micromorphic model to antiplane shear and consistency at limit
|
772 |
+
cases according to Remark 2.1 and Section 2.1. The two-scale nature of the relaxed micromorphic model can
|
773 |
+
be clearly observed.
|
774 |
+
3
|
775 |
+
Polynomial basis
|
776 |
+
In this section we brieο¬y introduce Bernstein polynomials and dual numbers. Bernstein polynomials are used
|
777 |
+
to construct both the H 1-conforming subspace and, in conjunction with the polytopal template methodology,
|
778 |
+
the NΒ΄edΒ΄elec elements. The computation of derivatives of the Bernstein base functions is achieved by employing
|
779 |
+
dual numbers, thus enabling the calculation of the value and the derivative of a base function simultaneously.
|
780 |
+
3.1
|
781 |
+
Bernstein polynomials
|
782 |
+
Bernstein polynomials of order p are given by the binomial expansion of the barycentric representation of the
|
783 |
+
unit line
|
784 |
+
1 = (Ξ»1 + Ξ»2)p = ((1 β ΞΎ) + ΞΎ)p =
|
785 |
+
p
|
786 |
+
οΏ½
|
787 |
+
i=0
|
788 |
+
οΏ½p
|
789 |
+
i
|
790 |
+
οΏ½
|
791 |
+
ΞΎi(1 β ΞΎ)pβi =
|
792 |
+
p
|
793 |
+
οΏ½
|
794 |
+
i=0
|
795 |
+
p!
|
796 |
+
i!(p β i)!ΞΎi(1 β ΞΎ)pβi ,
|
797 |
+
(3.1)
|
798 |
+
9
|
799 |
+
|
800 |
+
b4
|
801 |
+
0(ΞΎ)
|
802 |
+
b4
|
803 |
+
1(ΞΎ)
|
804 |
+
b4
|
805 |
+
2(ΞΎ)
|
806 |
+
b4
|
807 |
+
3(ΞΎ)
|
808 |
+
b4
|
809 |
+
4(ΞΎ)
|
810 |
+
ΞΎ
|
811 |
+
1
|
812 |
+
1
|
813 |
+
0
|
814 |
+
1
|
815 |
+
1/2
|
816 |
+
1/2
|
817 |
+
Figure 3.1: Bernstein base functions of degree p = 4 on the unit domain. Their sum forms a partition of unity.
|
818 |
+
The base functions are symmetric for ΞΎ = 0.5 with respect to their indices and always positive.
|
819 |
+
where ΞΎ β [0, 1]. The Bernstein polynomial reads
|
820 |
+
bp
|
821 |
+
i (ΞΎ) =
|
822 |
+
οΏ½p
|
823 |
+
i
|
824 |
+
οΏ½
|
825 |
+
ΞΎi(1 β ΞΎ)pβi .
|
826 |
+
(3.2)
|
827 |
+
A direct result of the binomial expansion is that Bernstein polynomials form a partition of unity, see also Fig. 3.1
|
828 |
+
p
|
829 |
+
οΏ½
|
830 |
+
i=0
|
831 |
+
bp
|
832 |
+
i (ΞΎ) = 1 .
|
833 |
+
(3.3)
|
834 |
+
Another consequence is that Bernstein polynomials are non-negative and less than or equal to 1
|
835 |
+
0 β€ bp
|
836 |
+
i (ΞΎ) β€ 1 ,
|
837 |
+
ΞΎ β [0, 1] .
|
838 |
+
(3.4)
|
839 |
+
A necessary condition for the use of Bernstein polynomials in ο¬nite element approximations is for them to span
|
840 |
+
the entire polynomial space.
|
841 |
+
Theorem 3.1 (Span of Bernstein polynomials)
|
842 |
+
The span of Bernstein polynomials forms a basis of the one-dimensional polynomial space
|
843 |
+
Pp(ΞΎ) = span{bp
|
844 |
+
i } ,
|
845 |
+
ΞΎ β R .
|
846 |
+
(3.5)
|
847 |
+
Proof. First we observe
|
848 |
+
dim(span{bp
|
849 |
+
i }) = dim Pp(ΞΎ) = p + 1 .
|
850 |
+
(3.6)
|
851 |
+
The proof of linear independence is achieved by contradiction. Let the set span{bp
|
852 |
+
i } with 0 < i β€ p be linearly
|
853 |
+
dependent, then there exists some combination with at least one non-zero constant ci ΜΈ= 0 such that
|
854 |
+
p
|
855 |
+
οΏ½
|
856 |
+
i=1
|
857 |
+
cibp
|
858 |
+
i (ΞΎ) = 0 ,
|
859 |
+
d
|
860 |
+
dΞΎ
|
861 |
+
p
|
862 |
+
οΏ½
|
863 |
+
i=1
|
864 |
+
cibp
|
865 |
+
i (ΞΎ) = 0 .
|
866 |
+
(3.7)
|
867 |
+
However, by the partition of unity property Eq. (3.3), only the full combination (0 β€ i β€ p) generates a constant
|
868 |
+
and by the exact sequence property the kernel of the diο¬erentiation operator is exactly the space of constants
|
869 |
+
ker(β) = R. The linear independence of the full span also follows from the partition of unity property, since
|
870 |
+
constants cannot be constructed otherwise.
|
871 |
+
10
|
872 |
+
|
873 |
+
Bernstein polynomials can be evaluated eο¬ciently using the recursive formula
|
874 |
+
bp
|
875 |
+
0(ΞΎ) = (1 β ΞΎ)p ,
|
876 |
+
bp
|
877 |
+
i+1(ΞΎ) =
|
878 |
+
(p β i)ΞΎ
|
879 |
+
(p + 1)(1 β ΞΎ)bp
|
880 |
+
i (ΞΎ) ,
|
881 |
+
i β {0, 1, ..., p β 1} ,
|
882 |
+
(3.8)
|
883 |
+
which allows for fast evaluation of the base functions.
|
884 |
+
Remark 3.1
|
885 |
+
Note that the formula Eq. (3.8) implies limΞΎβ1 bp
|
886 |
+
i+1(ΞΎ) = β. As such, evaluations using the formula are required
|
887 |
+
to use ΞΎ < 1 preferably with additional tolerance. The limit case ΞΎ = 1 is zero for all Bernstein base functions
|
888 |
+
aside from the last function belonging to the vertex, which simply returns one
|
889 |
+
bp
|
890 |
+
i (1) = 0
|
891 |
+
β i ΜΈ= p ,
|
892 |
+
bp
|
893 |
+
p(1) = 1 .
|
894 |
+
(3.9)
|
895 |
+
3.2
|
896 |
+
Dual numbers
|
897 |
+
Dual numbers [16] can be used to deο¬ne deο¬ne an augmented algebra, where the derivative of a function can
|
898 |
+
be computed simultaneously with the evaluation of the function. This enhancement is also commonly used
|
899 |
+
in forward automatic diο¬erentiation [8, 37], not to be confused with numerical diο¬erentiation, since unlike in
|
900 |
+
numerical diο¬erentiation, automatic diο¬erentiation is no approximation and yields the exact derivative. The
|
901 |
+
latter represents an alternative method to ο¬nding the derivatives of base functions, as opposed to explicit
|
902 |
+
formulas or approximations. Dual numbers augment the classical numbers by adding a non-zero number Ξ΅ with
|
903 |
+
a zero square Ξ΅2 = 0.
|
904 |
+
Deο¬nition 3.1 (Dual number)
|
905 |
+
The dual number is deο¬ned by
|
906 |
+
x + xβ²Ξ΅ ,
|
907 |
+
Ξ΅ βͺ 1 ,
|
908 |
+
(3.10)
|
909 |
+
where xβ² is the derivative (only in automatic diο¬erentiation), Ξ΅ is an abstract number (inο¬nitesimal) and formally
|
910 |
+
Ξ΅2 = 0.
|
911 |
+
The augmented algebra results automatically from the deο¬nition of the dual number.
|
912 |
+
Deο¬nition 3.2 (Augmented dual algebra)
|
913 |
+
The standard algebraic operations take the following form for dual numbers
|
914 |
+
1. Addition and subtraction
|
915 |
+
(x + xβ²Ξ΅) Β± (y + yβ²Ξ΅) = x Β± y + (xβ² Β± yβ²)Ξ΅ .
|
916 |
+
(3.11)
|
917 |
+
2. Multiplication
|
918 |
+
(x + xβ²Ξ΅)(y + yβ²Ξ΅) = xy + (xyβ² + xβ²y)Ξ΅ ,
|
919 |
+
(3.12)
|
920 |
+
since formally Ξ΅2 = 0.
|
921 |
+
3. Division is achieved by ο¬rst deο¬ning the inverse element
|
922 |
+
(x + xβ²Ξ΅)(y + yβ²Ξ΅) = 1
|
923 |
+
ββ
|
924 |
+
y = 1
|
925 |
+
x,
|
926 |
+
yβ² = β xβ²
|
927 |
+
x2 ,
|
928 |
+
(3.13)
|
929 |
+
such that
|
930 |
+
(x + xβ²Ξ΅)/(y + yβ²Ξ΅) = x/y + (xβ²/y β xyβ²/y2)Ξ΅ .
|
931 |
+
(3.14)
|
932 |
+
Application of the above deο¬nitions to polynomials
|
933 |
+
p(x + Ξ΅) =
|
934 |
+
β
|
935 |
+
οΏ½
|
936 |
+
i=0
|
937 |
+
ci(x + Ξ΅)i =
|
938 |
+
β
|
939 |
+
οΏ½
|
940 |
+
i=0
|
941 |
+
1
|
942 |
+
οΏ½
|
943 |
+
j=0
|
944 |
+
ci
|
945 |
+
οΏ½i
|
946 |
+
j
|
947 |
+
οΏ½
|
948 |
+
xiβjΞ΅j =
|
949 |
+
β
|
950 |
+
οΏ½
|
951 |
+
i=0
|
952 |
+
cixi + Ξ΅
|
953 |
+
β
|
954 |
+
οΏ½
|
955 |
+
i=1
|
956 |
+
i cixiβ1 = p(x) + pβ²(x)Ξ΅ ,
|
957 |
+
(3.15)
|
958 |
+
allows the extension to various types of analytical functions with a power-series representation (such as trigono-
|
959 |
+
metric or hyperbolic).
|
960 |
+
11
|
961 |
+
|
962 |
+
v1
|
963 |
+
v3
|
964 |
+
v2
|
965 |
+
Ξ
|
966 |
+
Ξ½
|
967 |
+
Ο
|
968 |
+
ΞΎ
|
969 |
+
Ξ·
|
970 |
+
x1
|
971 |
+
x3
|
972 |
+
x2
|
973 |
+
Ae
|
974 |
+
t
|
975 |
+
n
|
976 |
+
x
|
977 |
+
y
|
978 |
+
x : Ξ β Ae
|
979 |
+
Figure 4.1: Barycentric mapping of the reference triangle to an element in the physical domain.
|
980 |
+
Deο¬nition 3.3 (General dual numbers function)
|
981 |
+
A function of a dual number is deο¬ned in general by
|
982 |
+
f(x + Ξ΅) = f(x) + f β²(x)Ξ΅ ,
|
983 |
+
(3.16)
|
984 |
+
being a fundamental formula for forward automatic diο¬erentiation.
|
985 |
+
The deο¬nition of dual numbers makes them directly applicable to the general rules of diο¬erentiation, such as
|
986 |
+
the chain rule or product rule, in which case the derivative is simply the composition of previous computations
|
987 |
+
with Ξ΅. The logic of dual numbers can be understood intuitively by the directional derivative
|
988 |
+
d
|
989 |
+
dxf(x) = βxβ²f(x) = d
|
990 |
+
dΞ΅f(x + xβ²Ξ΅)
|
991 |
+
οΏ½οΏ½οΏ½οΏ½
|
992 |
+
Ξ΅=0
|
993 |
+
= lim
|
994 |
+
Ξ΅β0
|
995 |
+
f(x + xβ²Ξ΅) β f(x)
|
996 |
+
Ξ΅
|
997 |
+
,
|
998 |
+
(3.17)
|
999 |
+
where dividing by Ξ΅ and setting Ξ΅ = 0 are deferred to the last step of the computation, being the extraction of
|
1000 |
+
the derivative and equivalent to the operation f(x + Ξ΅) β f(x) with the augmented algebra of dual numbers.
|
1001 |
+
In this work we apply dual numbers for the computation of Bernstein polynomials using the recursive formula
|
1002 |
+
Eq. (3.8), thus allowing to iteratively compute each base function simultaneously with its derivative.
|
1003 |
+
4
|
1004 |
+
Triangular elements
|
1005 |
+
The triangle elements are mapped from the reference element Ξ to the physical domain Ae via barycentric
|
1006 |
+
coordinates
|
1007 |
+
x(ΞΎ, Ξ·) = (1 β ΞΎ β Ξ·)x1 + Ξ· x2 + ΞΎ x3 ,
|
1008 |
+
x : Ξ β Ae ,
|
1009 |
+
Ξ = {(ΞΎ, Ξ·) β [0, 1]2 | ΞΎ + Ξ· β€ 1} ,
|
1010 |
+
(4.1)
|
1011 |
+
where xi represent the coordinates of the vertices of one triangle in the physical domain, see Fig. 4.1. The
|
1012 |
+
corresponding Jacobi matrix reads
|
1013 |
+
J = Dx =
|
1014 |
+
οΏ½x3 β x1,
|
1015 |
+
x2 β x1
|
1016 |
+
οΏ½
|
1017 |
+
β R2Γ2 .
|
1018 |
+
(4.2)
|
1019 |
+
4.1
|
1020 |
+
The Bernstein-BΒ΄ezier basis for triangles
|
1021 |
+
The base functions on the triangle reference element are deο¬ned using the binomial expansion of the barycentric
|
1022 |
+
coordinates on the domain Ξ
|
1023 |
+
1 = (Ξ»1 + Ξ»2 + Ξ»3)p = ([1 β ΞΎ β Ξ·] + Ξ· + ΞΎ)p .
|
1024 |
+
(4.3)
|
1025 |
+
As such, the BΒ΄ezier base functions read
|
1026 |
+
bp
|
1027 |
+
ij(Ξ»1, Ξ»2, Ξ»3) =
|
1028 |
+
οΏ½
|
1029 |
+
p
|
1030 |
+
i
|
1031 |
+
οΏ½ οΏ½
|
1032 |
+
p β i
|
1033 |
+
j
|
1034 |
+
οΏ½
|
1035 |
+
Ξ»pβiβj
|
1036 |
+
1
|
1037 |
+
Ξ»j
|
1038 |
+
2Ξ»i
|
1039 |
+
3 ,
|
1040 |
+
(4.4)
|
1041 |
+
12
|
1042 |
+
|
1043 |
+
(a)
|
1044 |
+
(b)
|
1045 |
+
(c)
|
1046 |
+
Figure 4.2: Cubic vertex (a), edge (b) and cell (c) BΒ΄ezier base functions on the reference triangle.
|
1047 |
+
(0,0)
|
1048 |
+
(1,0)
|
1049 |
+
(1,1)
|
1050 |
+
(0,1)
|
1051 |
+
Ξ±
|
1052 |
+
Ξ²
|
1053 |
+
Ξ
|
1054 |
+
(0,0)
|
1055 |
+
(1,0)
|
1056 |
+
(0,1)
|
1057 |
+
ΞΎ
|
1058 |
+
Ξ·
|
1059 |
+
ΞΎ : Ξ± β Ξ
|
1060 |
+
Figure 4.3: Duο¬y transformation from a quadrilateral to a triangle by collapse of the coordinate system.
|
1061 |
+
with the equivalent bivariate form
|
1062 |
+
bp
|
1063 |
+
ij(ΞΎ, Ξ·) =
|
1064 |
+
οΏ½p
|
1065 |
+
i
|
1066 |
+
οΏ½ οΏ½p β i
|
1067 |
+
j
|
1068 |
+
οΏ½
|
1069 |
+
(1 β ΞΎ β Ξ·)pβiβjΞ·jΞΎi ,
|
1070 |
+
(4.5)
|
1071 |
+
of which some examples are depicted in Fig. 4.2. The Duο¬y transformation
|
1072 |
+
ΞΎ : [0, 1]2 β Ξ ,
|
1073 |
+
{Ξ±, Ξ²} οΏ½β {ΞΎ, Ξ·} ,
|
1074 |
+
(4.6)
|
1075 |
+
given by the relations
|
1076 |
+
ΞΎ = Ξ± ,
|
1077 |
+
Ξ± = ΞΎ ,
|
1078 |
+
Ξ· = (1 β Ξ±)Ξ² ,
|
1079 |
+
Ξ² =
|
1080 |
+
Ξ·
|
1081 |
+
1 β ΞΎ ,
|
1082 |
+
(4.7)
|
1083 |
+
allows to view the triangle as a collapsed quadrilateral, see Fig. 4.3. Inserting the Duο¬y map into the deο¬nition
|
1084 |
+
of the BΒ΄ezier base function yields the split
|
1085 |
+
bp
|
1086 |
+
ij(ΞΎ, Ξ·) =
|
1087 |
+
οΏ½p
|
1088 |
+
i
|
1089 |
+
οΏ½ οΏ½p β i
|
1090 |
+
j
|
1091 |
+
οΏ½
|
1092 |
+
(1 β ΞΎ β Ξ·)pβiβjΞ·jΞΎi
|
1093 |
+
=
|
1094 |
+
οΏ½p
|
1095 |
+
i
|
1096 |
+
οΏ½ οΏ½p β i
|
1097 |
+
j
|
1098 |
+
οΏ½
|
1099 |
+
(1 β Ξ± β [1 β Ξ±]Ξ²)pβiβj(1 β Ξ±)jΞ²jΞ±i
|
1100 |
+
=
|
1101 |
+
οΏ½p
|
1102 |
+
i
|
1103 |
+
οΏ½ οΏ½p β i
|
1104 |
+
j
|
1105 |
+
οΏ½
|
1106 |
+
(1 β Ξ±)pβiβj(1 β Ξ²)pβiβj(1 β Ξ±)jΞ²jΞ±i
|
1107 |
+
(4.8)
|
1108 |
+
=
|
1109 |
+
οΏ½p
|
1110 |
+
i
|
1111 |
+
οΏ½
|
1112 |
+
(1 β Ξ±)pβiΞ±i
|
1113 |
+
οΏ½p β i
|
1114 |
+
j
|
1115 |
+
οΏ½
|
1116 |
+
(1 β Ξ²)pβiβjΞ²j
|
1117 |
+
= bp
|
1118 |
+
i (Ξ±) bpβi
|
1119 |
+
j
|
1120 |
+
(Ξ²) .
|
1121 |
+
In other words, the Duο¬y transformation results in a natural factorization of the BΒ΄ezier triangle into Bernstein
|
1122 |
+
base functions [1]. The latter allows for fast evaluation using sum factorization. Further, it is now clear that
|
1123 |
+
BΒ΄ezier triangles are given by the interpolation of BΒ΄ezier curves, where the degree of the polynomial decreases
|
1124 |
+
13
|
1125 |
+
|
1126 |
+
ΞΎ
|
1127 |
+
Ξ·
|
1128 |
+
outer BΒ΄ezier curve with p = 3
|
1129 |
+
inner BΒ΄ezier curves with p < 3
|
1130 |
+
control polygon of Ξ·-curves
|
1131 |
+
outer BΒ΄ezier curves with p = 3
|
1132 |
+
inner BΒ΄ezier curves with p = 3
|
1133 |
+
Figure 4.4: BΒ΄ezier triangle built by interpolating BΒ΄ezier curves with an ever decreasing polynomial degree.
|
1134 |
+
v1
|
1135 |
+
v3
|
1136 |
+
v2
|
1137 |
+
ΞΎ
|
1138 |
+
Ξ·
|
1139 |
+
Figure 4.5: Traversal order of base functions. The purple lines represent the order in which the base functions
|
1140 |
+
are constructed by the factorized evaluation. Note that the traversal order on each edge is intrinsically from
|
1141 |
+
the lower to the higher vertex index.
|
1142 |
+
between each curve, see Fig. 4.4. In order to compute gradients on the reference domain one applies the chain
|
1143 |
+
rule
|
1144 |
+
βΞΎbp
|
1145 |
+
ij = (DΞ±ΞΎ)βT βΞ±bp
|
1146 |
+
ij ,
|
1147 |
+
DΞ±ΞΎ =
|
1148 |
+
οΏ½ 1
|
1149 |
+
0
|
1150 |
+
βΞ²
|
1151 |
+
1 β Ξ±
|
1152 |
+
οΏ½
|
1153 |
+
,
|
1154 |
+
(DΞ±ΞΎ)βT =
|
1155 |
+
1
|
1156 |
+
1 β Ξ±
|
1157 |
+
οΏ½1 β Ξ±
|
1158 |
+
Ξ²
|
1159 |
+
0
|
1160 |
+
1
|
1161 |
+
οΏ½
|
1162 |
+
.
|
1163 |
+
(4.9)
|
1164 |
+
The factorization is naturally suited for the use of dual numbers since the Ξ±-gradient of a base function reads
|
1165 |
+
βΞ±bp
|
1166 |
+
ij(Ξ±, Ξ²) =
|
1167 |
+
οΏ½
|
1168 |
+
οΏ½οΏ½οΏ½
|
1169 |
+
bpβi
|
1170 |
+
j
|
1171 |
+
d
|
1172 |
+
dΞ±bp
|
1173 |
+
i
|
1174 |
+
bp
|
1175 |
+
i
|
1176 |
+
d
|
1177 |
+
dΞ² bpβi
|
1178 |
+
j
|
1179 |
+
οΏ½
|
1180 |
+
οΏ½οΏ½οΏ½ ,
|
1181 |
+
(4.10)
|
1182 |
+
such that only the derivatives of the Bernstein base functions with respect to their parameter are required.
|
1183 |
+
The Duο¬y transformation induces an intrinsic optimal order of traversal of the base functions, compare
|
1184 |
+
Fig. 4.5, namely
|
1185 |
+
(i, j) = (0, 0) β (0, 1) β ... β (2, 2) β ... β (i, p β i) β ... β (p, 0) ,
|
1186 |
+
(4.11)
|
1187 |
+
which respects a clockwise orientation of the element, compare [52]. Thus, the order of the sequence of discrete
|
1188 |
+
values on common edges is determined by the global orientation. In order to relate a base function to a polytopal
|
1189 |
+
piece of the element, one observes the following result.
|
1190 |
+
Observation 4.1 (Triangle base functions)
|
1191 |
+
The polytope of each base function bp
|
1192 |
+
ij(ΞΎ, Ξ·) can be determined as follows:
|
1193 |
+
14
|
1194 |
+
|
1195 |
+
β’ The indices (0, 0), (0, p) and (p, 0) represent the ο¬rst, second and last vertex base functions, respectively.
|
1196 |
+
β’ The indices (0, j) with 0 < j < p and (i, 0) with 0 < i < p represent the ο¬rst and second edge base
|
1197 |
+
functions, respectively. Base functions of the slanted edge are given by (i, p β i) with 0 < i < p.
|
1198 |
+
β’ The remaining index combinations are cell base functions.
|
1199 |
+
With the latter observation, the construction of vertex-, edge- and cell base functions follows the intrinsic
|
1200 |
+
traversal order induced by the Duο¬y transformation and relates to a speciο¬c polytope via index-pairs.
|
1201 |
+
4.2
|
1202 |
+
NΒ΄edΒ΄elec elements of the second type
|
1203 |
+
We construct the base functions for the NΒ΄edΒ΄elec element of the second type using the polytopal template
|
1204 |
+
methodology introduced in [50]. The template sets read
|
1205 |
+
T1 = {e2, e1} ,
|
1206 |
+
T2 = {e1 + e2, e1} ,
|
1207 |
+
T3 = {e1 + e2, βe2} ,
|
1208 |
+
T12 = {e2, βe1} ,
|
1209 |
+
T13 = {e1, e2} ,
|
1210 |
+
T23 = {(1/2)(e1 β e2), e1 + e2} ,
|
1211 |
+
T123 = {e1, e2} .
|
1212 |
+
(4.12)
|
1213 |
+
The space of BΒ΄ezier polynomials is split across the polytopes of the reference triangle into
|
1214 |
+
Bp(Ξ) =
|
1215 |
+
οΏ½ 3
|
1216 |
+
οΏ½
|
1217 |
+
i=1
|
1218 |
+
Vp
|
1219 |
+
i (Ξ)
|
1220 |
+
οΏ½
|
1221 |
+
β
|
1222 |
+
οΏ½
|
1223 |
+
οΏ½
|
1224 |
+
οΏ½
|
1225 |
+
οΏ½
|
1226 |
+
jβJ
|
1227 |
+
Ep
|
1228 |
+
j (Ξ)
|
1229 |
+
οΏ½
|
1230 |
+
οΏ½
|
1231 |
+
οΏ½ β Cp
|
1232 |
+
123(Ξ) ,
|
1233 |
+
J = {(1, 2), (1, 3), (2, 3)} ,
|
1234 |
+
(4.13)
|
1235 |
+
where Vp
|
1236 |
+
i are the sets of the vertex base functions, Ep
|
1237 |
+
j are the sets of edge base functions, Cp
|
1238 |
+
123 is the set of cell
|
1239 |
+
base functions, and the β indicates summation over non-overlapping spaces. Consequently, the NΒ΄edΒ΄elec basis
|
1240 |
+
is given by
|
1241 |
+
N p
|
1242 |
+
II =
|
1243 |
+
οΏ½ 3
|
1244 |
+
οΏ½
|
1245 |
+
i=1
|
1246 |
+
Vp
|
1247 |
+
i β Ti
|
1248 |
+
οΏ½
|
1249 |
+
β
|
1250 |
+
οΏ½
|
1251 |
+
οΏ½
|
1252 |
+
οΏ½
|
1253 |
+
οΏ½
|
1254 |
+
jβJ
|
1255 |
+
Ep
|
1256 |
+
j β Tj
|
1257 |
+
οΏ½
|
1258 |
+
οΏ½
|
1259 |
+
οΏ½ β {Cp
|
1260 |
+
123 β T123} ,
|
1261 |
+
J = {(1, 2), (1, 3), (2, 3)} .
|
1262 |
+
(4.14)
|
1263 |
+
Using the BΒ΄ezier basis one ο¬nds the following base functions, which inherit the optimal complexity of the
|
1264 |
+
underlying basis.
|
1265 |
+
Deο¬nition 4.1 (BΒ΄ezier-NΒ΄edΒ΄elec II triangle basis)
|
1266 |
+
The following base functions are deο¬ned on the reference triangle.
|
1267 |
+
β’ On the edges the base function reads
|
1268 |
+
e12 :
|
1269 |
+
Ο(ΞΎ, Ξ·) = bp
|
1270 |
+
00e2 ,
|
1271 |
+
Ο(ΞΎ, Ξ·) = bp
|
1272 |
+
0p(e1 + e2) ,
|
1273 |
+
Ο(ΞΎ, Ξ·) = bp
|
1274 |
+
0je2 ,
|
1275 |
+
0 < j < p ,
|
1276 |
+
e13 :
|
1277 |
+
Ο(ΞΎ, Ξ·) = bp
|
1278 |
+
00e1 ,
|
1279 |
+
Ο(ΞΎ, Ξ·) = bp
|
1280 |
+
p0(e1 + e2) ,
|
1281 |
+
Ο(ΞΎ, Ξ·) = bp
|
1282 |
+
i0e1 ,
|
1283 |
+
0 < i < p ,
|
1284 |
+
e23 :
|
1285 |
+
Ο(ΞΎ, Ξ·) = bp
|
1286 |
+
0pe1 ,
|
1287 |
+
Ο(ΞΎ, Ξ·) = βbp
|
1288 |
+
p0e2 ,
|
1289 |
+
Ο(ΞΎ, Ξ·) = (1/2) bp
|
1290 |
+
i,pβi(e1 β e2) ,
|
1291 |
+
0 < i < p ,
|
1292 |
+
(4.15)
|
1293 |
+
where the ο¬rst two base functions for each edge are the vertex-edge base functions and the third equation
|
1294 |
+
generates pure edge base functions.
|
1295 |
+
β’ The cell base functions read
|
1296 |
+
c123 :
|
1297 |
+
Ο(ΞΎ, Ξ·) = βbp
|
1298 |
+
0je1 ,
|
1299 |
+
0 < j < p ,
|
1300 |
+
Ο(ΞΎ, Ξ·) = bp
|
1301 |
+
i0e2 ,
|
1302 |
+
0 < i < p ,
|
1303 |
+
Ο(ΞΎ, Ξ·) = bp
|
1304 |
+
i,pβi(e1 + e2) ,
|
1305 |
+
0 < i < p ,
|
1306 |
+
Ο(ΞΎ, Ξ·) = bp
|
1307 |
+
ije2 ,
|
1308 |
+
0 < i < p ,
|
1309 |
+
0 < j < p β i ,
|
1310 |
+
Ο(ΞΎ, Ξ·) = bp
|
1311 |
+
ije1 ,
|
1312 |
+
0 < i < p ,
|
1313 |
+
0 < j < p β i ,
|
1314 |
+
(4.16)
|
1315 |
+
15
|
1316 |
+
|
1317 |
+
where the ο¬rst three are the respective edge-cell base functions. The remaining two are pure cell base
|
1318 |
+
functions.
|
1319 |
+
4.3
|
1320 |
+
NΒ΄edΒ΄elec elements of the ο¬rst type
|
1321 |
+
In order to construct the NΒ΄edΒ΄elec element of the ο¬rst type we rely on the construction of the kernel introduced
|
1322 |
+
in [58] via the exact de Rham sequence and the polytopal template for the non-kernel base functions following
|
1323 |
+
[50]. The complete NΒ΄edΒ΄elec space reads
|
1324 |
+
N p
|
1325 |
+
I = N 0
|
1326 |
+
I β
|
1327 |
+
οΏ½
|
1328 |
+
οΏ½
|
1329 |
+
οΏ½
|
1330 |
+
οΏ½
|
1331 |
+
jβJ
|
1332 |
+
βEp+1
|
1333 |
+
j
|
1334 |
+
οΏ½
|
1335 |
+
οΏ½
|
1336 |
+
οΏ½ β βCp+1
|
1337 |
+
123 β
|
1338 |
+
οΏ½ 2
|
1339 |
+
οΏ½
|
1340 |
+
i=1
|
1341 |
+
Vp
|
1342 |
+
i β Ti
|
1343 |
+
οΏ½
|
1344 |
+
β
|
1345 |
+
οΏ½
|
1346 |
+
οΏ½
|
1347 |
+
οΏ½
|
1348 |
+
οΏ½
|
1349 |
+
jβJ
|
1350 |
+
Ep
|
1351 |
+
j β Tj
|
1352 |
+
οΏ½
|
1353 |
+
οΏ½
|
1354 |
+
οΏ½ β {Cp
|
1355 |
+
123 β T123} ,
|
1356 |
+
J = {(1, 2), (1, 3), (2, 3)} ,
|
1357 |
+
(4.17)
|
1358 |
+
where we relied on the decomposition Eq. (4.14). Applying the construction to the BΒ΄ezier basis yields the
|
1359 |
+
following base functions.
|
1360 |
+
Deο¬nition 4.2 (BΒ΄ezier-NΒ΄edΒ΄elec I triangle basis)
|
1361 |
+
We deο¬ne the base functions on the reference triangle.
|
1362 |
+
β’ On the edges we employ the lowest order NΒ΄edΒ΄elec base functions and the edge gradients
|
1363 |
+
e12 :
|
1364 |
+
Ο(ΞΎ, Ξ·) = ΟI
|
1365 |
+
1 ,
|
1366 |
+
Ο(ΞΎ, Ξ·) = βΞΎbp+1
|
1367 |
+
0j
|
1368 |
+
,
|
1369 |
+
0 < j < p + 1 ,
|
1370 |
+
e13 :
|
1371 |
+
Ο(ΞΎ, Ξ·) = ΟI
|
1372 |
+
2 ,
|
1373 |
+
Ο(ΞΎ, Ξ·) = βΞΎbp+1
|
1374 |
+
i0
|
1375 |
+
,
|
1376 |
+
0 < i < p + 1 ,
|
1377 |
+
e23 :
|
1378 |
+
Ο(ΞΎ, Ξ·) = ΟI
|
1379 |
+
3 ,
|
1380 |
+
Ο(ΞΎ, Ξ·) = βΞΎbp+1
|
1381 |
+
i,p+1βi ,
|
1382 |
+
0 < i < p + 1 .
|
1383 |
+
(4.18)
|
1384 |
+
β’ The cell functions read
|
1385 |
+
c123 :
|
1386 |
+
Ο(ΞΎ, Ξ·) = bp
|
1387 |
+
00ΟI
|
1388 |
+
3 ,
|
1389 |
+
Ο(ΞΎ, Ξ·) = bp
|
1390 |
+
0pΟI
|
1391 |
+
2 ,
|
1392 |
+
Ο(ΞΎ, Ξ·) = bp
|
1393 |
+
0j(ΟI
|
1394 |
+
3 β ΟI
|
1395 |
+
2) ,
|
1396 |
+
0 < j < p ,
|
1397 |
+
Ο(ΞΎ, Ξ·) = bp
|
1398 |
+
i0(ΟI
|
1399 |
+
1 + ΟI
|
1400 |
+
3) ,
|
1401 |
+
0 < i < p ,
|
1402 |
+
Ο(ΞΎ, Ξ·) = bp
|
1403 |
+
i,pβi(ΟI
|
1404 |
+
1 β ΟI
|
1405 |
+
2) ,
|
1406 |
+
0 < i < p ,
|
1407 |
+
Ο(ΞΎ, Ξ·) = bp
|
1408 |
+
ij(ΟI
|
1409 |
+
1 β ΟI
|
1410 |
+
2 + ΟI
|
1411 |
+
3) ,
|
1412 |
+
0 < i < p ,
|
1413 |
+
0 < j < p β i ,
|
1414 |
+
Ο(ΞΎ, Ξ·) = βΞΎbp+1
|
1415 |
+
ij
|
1416 |
+
,
|
1417 |
+
0 < i < p + 1 ,
|
1418 |
+
0 < j < p + 1 β i ,
|
1419 |
+
(4.19)
|
1420 |
+
where the last formula gives the cell gradients and the remaining base functions are non-gradients.
|
1421 |
+
The deο¬nition relies on the base functions of the lowest order NΒ΄edΒ΄elec element of the ο¬rst type [5,50]
|
1422 |
+
ΟI
|
1423 |
+
1(ΞΎ, Ξ·) =
|
1424 |
+
οΏ½
|
1425 |
+
Ξ·
|
1426 |
+
1 β ΞΎ
|
1427 |
+
οΏ½
|
1428 |
+
,
|
1429 |
+
ΟI
|
1430 |
+
2(ΞΎ, Ξ·) =
|
1431 |
+
οΏ½1 β Ξ·
|
1432 |
+
ΞΎ
|
1433 |
+
οΏ½
|
1434 |
+
,
|
1435 |
+
ΟI
|
1436 |
+
3(ΞΎ, Ξ·) =
|
1437 |
+
οΏ½ Ξ·
|
1438 |
+
βΞΎ
|
1439 |
+
οΏ½
|
1440 |
+
.
|
1441 |
+
(4.20)
|
1442 |
+
5
|
1443 |
+
Tetrahedral elements
|
1444 |
+
The tetrahedral elements are mapped from the reference tetrahedron β¦ by the three-dimensional barycentric
|
1445 |
+
coordinates onto the physical domain Ve, see Fig. 5.1
|
1446 |
+
x(ΞΎ, Ξ·, ΞΆ) = (1 β ΞΎ β Ξ· β ΞΆ)x1 + ΞΆ x2 + Ξ· x3 + ΞΎ x4 ,
|
1447 |
+
x : β¦ β Ve ,
|
1448 |
+
β¦ = {(ΞΎ, Ξ·, ΞΆ) β [0, 1]3 | ΞΎ + Ξ· + ΞΆ β€ 1} .
|
1449 |
+
(5.1)
|
1450 |
+
16
|
1451 |
+
|
1452 |
+
ΞΎ
|
1453 |
+
Ξ·
|
1454 |
+
ΞΆ
|
1455 |
+
β¦
|
1456 |
+
v1
|
1457 |
+
v4
|
1458 |
+
v3
|
1459 |
+
v2
|
1460 |
+
Ο
|
1461 |
+
Ξ½
|
1462 |
+
Ve
|
1463 |
+
x2
|
1464 |
+
x1
|
1465 |
+
x3
|
1466 |
+
x4
|
1467 |
+
x
|
1468 |
+
y
|
1469 |
+
z
|
1470 |
+
t
|
1471 |
+
n
|
1472 |
+
x : β¦ β Ve
|
1473 |
+
Figure 5.1: Barycentric mapping of the reference tetrahedron to an element in the physical domain.
|
1474 |
+
The corresponding Jacobi matrix reads
|
1475 |
+
J = Dx =
|
1476 |
+
οΏ½x4 β x1,
|
1477 |
+
x3 β x1,
|
1478 |
+
x2 β x1
|
1479 |
+
οΏ½
|
1480 |
+
β R3Γ3 .
|
1481 |
+
(5.2)
|
1482 |
+
5.1
|
1483 |
+
The Bernstein-BΒ΄ezier basis for tetrahedra
|
1484 |
+
Analogously to triangle elements, the BΒ΄ezier tetrahedra on the unit tetrahedron β¦ are deο¬ned using the barycen-
|
1485 |
+
tric coordinates by expanding the coeο¬cients of
|
1486 |
+
(Ξ»1 + Ξ»2 + Ξ»3 + Ξ»4)p = ([1 β ΞΎ β Ξ· β ΞΆ] + ΞΆ + Ξ· + ΞΎ)p = 1 ,
|
1487 |
+
(5.3)
|
1488 |
+
thus ο¬nding
|
1489 |
+
bp
|
1490 |
+
ijk(Ξ»1, Ξ»2, Ξ»3, Ξ»4) =
|
1491 |
+
οΏ½p
|
1492 |
+
i
|
1493 |
+
οΏ½ οΏ½p β i
|
1494 |
+
j
|
1495 |
+
οΏ½ οΏ½p β i β j
|
1496 |
+
k
|
1497 |
+
οΏ½
|
1498 |
+
Ξ»pβiβjβk
|
1499 |
+
1
|
1500 |
+
Ξ»k
|
1501 |
+
2Ξ»j
|
1502 |
+
3Ξ»k
|
1503 |
+
4 ,
|
1504 |
+
(5.4)
|
1505 |
+
with the equivalent trivariate form
|
1506 |
+
bp
|
1507 |
+
ijk(ΞΎ, Ξ·, ΞΆ) =
|
1508 |
+
οΏ½
|
1509 |
+
p
|
1510 |
+
i
|
1511 |
+
οΏ½ οΏ½
|
1512 |
+
p β i
|
1513 |
+
j
|
1514 |
+
οΏ½ οΏ½p β i β j
|
1515 |
+
k
|
1516 |
+
οΏ½
|
1517 |
+
(1 β ΞΎ β Ξ· β ΞΆ)pβiβjβkΞΆkΞ·jΞΎi .
|
1518 |
+
(5.5)
|
1519 |
+
We construct the Duο¬y transformation by mapping the unit tetrahedron as a collapsed hexahedron
|
1520 |
+
ΞΎ : [0, 1]3 β β¦ ,
|
1521 |
+
{Ξ±, Ξ², Ξ³} οΏ½β {ΞΎ, Ξ·, ΞΆ} ,
|
1522 |
+
(5.6)
|
1523 |
+
using the relations
|
1524 |
+
ΞΎ = Ξ± ,
|
1525 |
+
Ξ· = (1 β Ξ±)Ξ² ,
|
1526 |
+
ΞΆ = (1 β Ξ±)(1 β Ξ²)Ξ³ ,
|
1527 |
+
Ξ± = ΞΎ ,
|
1528 |
+
Ξ² =
|
1529 |
+
Ξ·
|
1530 |
+
1 β ΞΎ ,
|
1531 |
+
Ξ³ =
|
1532 |
+
ΞΆ
|
1533 |
+
1 β ΞΎ β Ξ· ,
|
1534 |
+
(5.7)
|
1535 |
+
as depicted in Fig. 5.2. Applying the Duο¬y transformation to BΒ΄ezier tetrahedra
|
1536 |
+
bp
|
1537 |
+
ijk(ΞΎ, Ξ·, ΞΆ) =
|
1538 |
+
οΏ½p
|
1539 |
+
i
|
1540 |
+
οΏ½ οΏ½p β i
|
1541 |
+
j
|
1542 |
+
οΏ½ οΏ½p β i β j
|
1543 |
+
k
|
1544 |
+
οΏ½
|
1545 |
+
(1 β ΞΎ β Ξ· β ΞΆ)pβiβjβkΞΆkΞ·jΞΎi
|
1546 |
+
=
|
1547 |
+
οΏ½
|
1548 |
+
p
|
1549 |
+
i
|
1550 |
+
οΏ½ οΏ½
|
1551 |
+
p β i
|
1552 |
+
j
|
1553 |
+
οΏ½ οΏ½
|
1554 |
+
p β i β j
|
1555 |
+
k
|
1556 |
+
οΏ½
|
1557 |
+
(1 β Ξ± β (1 β Ξ±)Ξ² β (1 β Ξ±)(1 β Ξ²)Ξ³)pβiβjβk
|
1558 |
+
Β· (1 β Ξ±)k(1 β Ξ²)kΞ³k(1 β Ξ±)jΞ²jΞ±i
|
1559 |
+
=
|
1560 |
+
οΏ½p
|
1561 |
+
i
|
1562 |
+
οΏ½ οΏ½p β i
|
1563 |
+
j
|
1564 |
+
οΏ½ οΏ½p β i β j
|
1565 |
+
k
|
1566 |
+
οΏ½
|
1567 |
+
(1 β Ξ±)pβiβjβk(1 β Ξ²)pβiβjβk(1 β Ξ³)pβiβjβk
|
1568 |
+
(5.8)
|
1569 |
+
Β· (1 β Ξ±)k(1 β Ξ²)kΞ³k(1 β Ξ±)jΞ²jΞ±i
|
1570 |
+
=
|
1571 |
+
οΏ½p
|
1572 |
+
i
|
1573 |
+
οΏ½
|
1574 |
+
(1 β Ξ±)pβiΞ±i
|
1575 |
+
οΏ½p β i
|
1576 |
+
j
|
1577 |
+
οΏ½
|
1578 |
+
(1 β Ξ²)pβiβjΞ²j
|
1579 |
+
οΏ½p β i β j
|
1580 |
+
k
|
1581 |
+
οΏ½
|
1582 |
+
(1 β Ξ³)pβiβjβkΞ³k
|
1583 |
+
= bp
|
1584 |
+
i (Ξ±)bpβi
|
1585 |
+
j
|
1586 |
+
(Ξ²)bpβiβj
|
1587 |
+
k
|
1588 |
+
(Ξ³) ,
|
1589 |
+
17
|
1590 |
+
|
1591 |
+
Ξ±
|
1592 |
+
Ξ²
|
1593 |
+
Ξ³
|
1594 |
+
(0,0,0)
|
1595 |
+
(1,0,0)
|
1596 |
+
(0,0,1)
|
1597 |
+
(1,1,0)
|
1598 |
+
(1,1,1)
|
1599 |
+
(0,1,1)
|
1600 |
+
ΞΎ
|
1601 |
+
Ξ·
|
1602 |
+
ΞΆ
|
1603 |
+
β¦
|
1604 |
+
(0,0,0)
|
1605 |
+
(1,0,0)
|
1606 |
+
(0,1,0)
|
1607 |
+
(0,0,1)
|
1608 |
+
ΞΎ : Ξ± β β¦
|
1609 |
+
Figure 5.2: Duο¬y mapping of the unit hexahedron to the unit tetrahedron.
|
1610 |
+
leads to an intrinsic factorization via univariate Bernstein base functions, which allow for fast evaluations
|
1611 |
+
using sum factorization [1]. Further, since the pair bpβi
|
1612 |
+
j
|
1613 |
+
(Ξ²)bpβiβj
|
1614 |
+
k
|
1615 |
+
(Ξ³) spans a BΒ΄ezier triangle, it is clear that
|
1616 |
+
the multiplication with bp
|
1617 |
+
i (Ξ±) interpolates between that triangle and a point in space, eο¬ectively spanning a
|
1618 |
+
tetrahedron. In order to compute gradients the chain rule is employed with respect to the Duο¬y transformation
|
1619 |
+
βΞΎbp
|
1620 |
+
ijk = (DΞ±ΞΎ)βT βΞ±bp
|
1621 |
+
ijk ,
|
1622 |
+
DΞ±ΞΎ =
|
1623 |
+
οΏ½
|
1624 |
+
οΏ½
|
1625 |
+
1
|
1626 |
+
0
|
1627 |
+
0
|
1628 |
+
βΞ²
|
1629 |
+
1 β Ξ±
|
1630 |
+
0
|
1631 |
+
(Ξ² β 1)Ξ³
|
1632 |
+
(Ξ± β 1)Ξ³
|
1633 |
+
(1 β Ξ±)(1 β Ξ²)
|
1634 |
+
οΏ½
|
1635 |
+
οΏ½ ,
|
1636 |
+
(DΞ±ΞΎ)βT =
|
1637 |
+
1
|
1638 |
+
(1 β Ξ±)(1 β Ξ²)
|
1639 |
+
οΏ½
|
1640 |
+
οΏ½
|
1641 |
+
(1 β Ξ±)(1 β Ξ²)
|
1642 |
+
(1 β Ξ²)Ξ²
|
1643 |
+
Ξ³
|
1644 |
+
0
|
1645 |
+
1 β Ξ²
|
1646 |
+
Ξ³
|
1647 |
+
0
|
1648 |
+
0
|
1649 |
+
1
|
1650 |
+
οΏ½
|
1651 |
+
οΏ½ .
|
1652 |
+
(5.9)
|
1653 |
+
We use dual numbers to compute the derivative of each Bernstein base function and construct the Ξ±-gradient
|
1654 |
+
βΞ±bp
|
1655 |
+
ijk(Ξ±, Ξ², Ξ³) =
|
1656 |
+
οΏ½
|
1657 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
1658 |
+
bpβi
|
1659 |
+
j
|
1660 |
+
bpβiβj
|
1661 |
+
k
|
1662 |
+
d
|
1663 |
+
dΞ±bp
|
1664 |
+
i
|
1665 |
+
bp
|
1666 |
+
i bpβiβj
|
1667 |
+
k
|
1668 |
+
d
|
1669 |
+
dΞ² bpβi
|
1670 |
+
j
|
1671 |
+
bp
|
1672 |
+
i bpβi
|
1673 |
+
j
|
1674 |
+
d
|
1675 |
+
dΞ³ bpβiβj
|
1676 |
+
k
|
1677 |
+
οΏ½
|
1678 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
1679 |
+
.
|
1680 |
+
(5.10)
|
1681 |
+
The Duο¬y transformation results in the optimal order of traversal of the base functions depicted in Fig. 5.3.
|
1682 |
+
Note that the traversal order agrees with the oriental deο¬nitions introduced in [52] and each oriented face has
|
1683 |
+
the same order of traversal as the triangle Fig. 4.5. We relate the base functions to their respective polytopes
|
1684 |
+
using the index triplets.
|
1685 |
+
Observation 5.1 (Tetrahedron base functions)
|
1686 |
+
The polytope of each base function bp
|
1687 |
+
ijk(ΞΎ, Ξ·, ΞΆ) is determined as follows.
|
1688 |
+
β’ the indices (0, 0, 0), (0, 0, p), (0, p, 0) and (p, 0, 0) represent the respective vertex base functions;
|
1689 |
+
β’ the ο¬rst edge is associated with the triplet (0, 0, k) where 0 < k < p, the second with (0, j, 0) where 0 < j < p
|
1690 |
+
and the third with (i, 0, 0) where 0 < i < p. The slated edges are given by (0, j, p β j) with 0 < j < p,
|
1691 |
+
(i, 0, p β i) with 0 < i < p and (i, p β i, 0) with 0 < i < p, respectively;
|
1692 |
+
β’ the base functions of the ο¬rst face are given by (0, j, k) with 0 < j < p and 0 < k < p β j. The second face
|
1693 |
+
is associated with the base functions given by the triplets (i, 0, k) with 0 < i < p and 0 < k < p β i. The
|
1694 |
+
base functions of the third face are related to the indices (i, j, 0) with 0 < i < p and 0 < j < p β i. Lastly,
|
1695 |
+
the base functions of the slated face are given by (i, j, p β i β j) with 0 < i < p and 0 < j < p β i;
|
1696 |
+
β’ the remaining indices correspond to the cell base functions.
|
1697 |
+
Examples of BΒ΄ezier base functions on their respective polytopes are depicted in Fig. 5.4.
|
1698 |
+
18
|
1699 |
+
|
1700 |
+
ΞΎ
|
1701 |
+
Ξ·
|
1702 |
+
ΞΆ
|
1703 |
+
v1
|
1704 |
+
v4
|
1705 |
+
v3
|
1706 |
+
v2
|
1707 |
+
Figure 5.3: Order of traversal of tetrahedral BΒ΄ezier base functions on the unit tetrahedron. The traversal order
|
1708 |
+
on each face agrees with an orientation of the vertices fijk = {vi, vj, vk} such that i < j < k. The traversal
|
1709 |
+
order on each edge is from the lower index vertex to the higher index vertex.
|
1710 |
+
(a)
|
1711 |
+
(b)
|
1712 |
+
(c)
|
1713 |
+
(d)
|
1714 |
+
Figure 5.4: Quartic BΒ΄ezier vertex (a), edge (b), face (c), and cell (c) base functions on the reference tetrahedron.
|
1715 |
+
19
|
1716 |
+
|
1717 |
+
5.2
|
1718 |
+
NΒ΄edΒ΄elec elements of the second type
|
1719 |
+
The BΒ΄ezier polynomial space is split according to the polytopes of the reference tetrahedron
|
1720 |
+
Bp(β¦) =
|
1721 |
+
οΏ½ 4
|
1722 |
+
οΏ½
|
1723 |
+
i=1
|
1724 |
+
Vp
|
1725 |
+
i (β¦)
|
1726 |
+
οΏ½
|
1727 |
+
β
|
1728 |
+
οΏ½
|
1729 |
+
οΏ½
|
1730 |
+
οΏ½
|
1731 |
+
οΏ½
|
1732 |
+
jβJ
|
1733 |
+
Ep
|
1734 |
+
j (β¦)
|
1735 |
+
οΏ½
|
1736 |
+
οΏ½
|
1737 |
+
οΏ½ β
|
1738 |
+
οΏ½οΏ½
|
1739 |
+
kβK
|
1740 |
+
Fp
|
1741 |
+
k(β¦)
|
1742 |
+
οΏ½
|
1743 |
+
β Cp
|
1744 |
+
1234(β¦) ,
|
1745 |
+
J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
|
1746 |
+
K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} ,
|
1747 |
+
(5.11)
|
1748 |
+
where Vp
|
1749 |
+
i are the sets of vertex base functions, Ep
|
1750 |
+
j are the sets of edge base functions, Fp
|
1751 |
+
k are the sets of face
|
1752 |
+
base functions and Cp
|
1753 |
+
1234 is the set of cell base functions. We apply the template sets from [50]
|
1754 |
+
T1 = {e3, e2, e1} ,
|
1755 |
+
T2 = {e1 + e2 + e3, e2, e1} ,
|
1756 |
+
T3 = {e1 + e2 + e3, βe3, e1} ,
|
1757 |
+
T4 = {e1 + e2 + e3, βe3, βe2} ,
|
1758 |
+
T12 = {e3, βe2, βe1} ,
|
1759 |
+
T13 = {e2, e3, βe1} ,
|
1760 |
+
T14 = {e1, e3, e2} ,
|
1761 |
+
T23 = {e2, e1 + e2 + e3, βe1} ,
|
1762 |
+
T24 = {e1, e1 + e2 + e3, e2} ,
|
1763 |
+
T34 = {e1, e1 + e2 + e3, βe3} ,
|
1764 |
+
T123 = {e3, e2, βe1} ,
|
1765 |
+
T124 = {e3, e1, e2} ,
|
1766 |
+
T134 = {e2, e1, βe3} ,
|
1767 |
+
T234 = {e2, e1, e1 + e2 + e3} ,
|
1768 |
+
T1234 = {e3, e2, e1} ,
|
1769 |
+
(5.12)
|
1770 |
+
to span the NΒ΄edΒ΄elec element of the second type
|
1771 |
+
N p
|
1772 |
+
II =
|
1773 |
+
οΏ½ 4
|
1774 |
+
οΏ½
|
1775 |
+
i=1
|
1776 |
+
Vp
|
1777 |
+
i β Ti
|
1778 |
+
οΏ½
|
1779 |
+
β
|
1780 |
+
οΏ½
|
1781 |
+
οΏ½
|
1782 |
+
οΏ½
|
1783 |
+
οΏ½
|
1784 |
+
jβJ
|
1785 |
+
Ep
|
1786 |
+
j β Tj
|
1787 |
+
οΏ½
|
1788 |
+
οΏ½
|
1789 |
+
οΏ½ β
|
1790 |
+
οΏ½οΏ½
|
1791 |
+
kβK
|
1792 |
+
Fp
|
1793 |
+
k β Tk
|
1794 |
+
οΏ½
|
1795 |
+
β {Cp
|
1796 |
+
1234 β T1234} ,
|
1797 |
+
J = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
|
1798 |
+
K = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)} .
|
1799 |
+
(5.13)
|
1800 |
+
We can now deο¬ne the BΒ΄ezier-NΒ΄edΒ΄elec element of the second type for arbitrary powers while inheriting optimal
|
1801 |
+
complexity.
|
1802 |
+
Deο¬nition 5.1 (BΒ΄ezier-NΒ΄edΒ΄elec II tetrahedral basis)
|
1803 |
+
We deο¬ne the base functions on the reference tetrahedron:
|
1804 |
+
β’ on the edges the base functions read
|
1805 |
+
e12 :
|
1806 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1807 |
+
000e3 ,
|
1808 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1809 |
+
00p(e1 + e2 + e3) ,
|
1810 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1811 |
+
00ke3 ,
|
1812 |
+
0 < k < p ,
|
1813 |
+
e13 :
|
1814 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1815 |
+
000e2 ,
|
1816 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1817 |
+
0p0(e1 + e2 + e3) ,
|
1818 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1819 |
+
0j0e2 ,
|
1820 |
+
0 < j < p ,
|
1821 |
+
e14 :
|
1822 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1823 |
+
000e1 ,
|
1824 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1825 |
+
p00(e1 + e2 + e3) ,
|
1826 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1827 |
+
i00e1 ,
|
1828 |
+
0 < i < p ,
|
1829 |
+
e23 :
|
1830 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1831 |
+
00pe2 ,
|
1832 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1833 |
+
0p0e3 ,
|
1834 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1835 |
+
0j,pβje2 ,
|
1836 |
+
0 < j < p ,
|
1837 |
+
e24 :
|
1838 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1839 |
+
00pe1 ,
|
1840 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1841 |
+
p00e3 ,
|
1842 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1843 |
+
i0,pβie1 ,
|
1844 |
+
0 < i < p ,
|
1845 |
+
e34 :
|
1846 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1847 |
+
0p0e1 ,
|
1848 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1849 |
+
p00e2 ,
|
1850 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1851 |
+
i,pβi,0e1 ,
|
1852 |
+
0 < i < p ,
|
1853 |
+
(5.14)
|
1854 |
+
where the ο¬rst two base functions on each edge are the vertex-edge base functions;
|
1855 |
+
20
|
1856 |
+
|
1857 |
+
β’ the face base functions are given by
|
1858 |
+
f123 :
|
1859 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1860 |
+
00ke2 ,
|
1861 |
+
0 < k < p ,
|
1862 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1863 |
+
0j0e3 ,
|
1864 |
+
0 < j < p ,
|
1865 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1866 |
+
0j,pβj(e1 + e2 + e3) ,
|
1867 |
+
0 < j < p ,
|
1868 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1869 |
+
0jke3 ,
|
1870 |
+
0 < j < p ,
|
1871 |
+
0 < k < p β j ,
|
1872 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1873 |
+
0jke2 ,
|
1874 |
+
0 < j < p ,
|
1875 |
+
0 < k < p β j ,
|
1876 |
+
f124 :
|
1877 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1878 |
+
00ke1 ,
|
1879 |
+
0 < k < p ,
|
1880 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1881 |
+
i00e3 ,
|
1882 |
+
0 < i < p ,
|
1883 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1884 |
+
i0,pβi(e1 + e2 + e3) ,
|
1885 |
+
0 < i < p ,
|
1886 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1887 |
+
i0ke3 ,
|
1888 |
+
0 < i < p ,
|
1889 |
+
0 < k < p β i ,
|
1890 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1891 |
+
i0ke1 ,
|
1892 |
+
0 < i < p ,
|
1893 |
+
0 < k < p β i ,
|
1894 |
+
f134 :
|
1895 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1896 |
+
0j0e1 ,
|
1897 |
+
0 < j < p ,
|
1898 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1899 |
+
i00e2 ,
|
1900 |
+
0 < i < p ,
|
1901 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1902 |
+
i,pβi,0(e1 + e2 + e3) ,
|
1903 |
+
0 < i < p ,
|
1904 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1905 |
+
ij0e2 ,
|
1906 |
+
0 < i < p ,
|
1907 |
+
0 < j < p β i ,
|
1908 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1909 |
+
ij0e1 ,
|
1910 |
+
0 < i < p ,
|
1911 |
+
0 < j < p β i ,
|
1912 |
+
f234 :
|
1913 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1914 |
+
0j,pβje1 ,
|
1915 |
+
0 < j < p ,
|
1916 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1917 |
+
i0,pβie2 ,
|
1918 |
+
0 < i < p ,
|
1919 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1920 |
+
i,pβi,0e3 ,
|
1921 |
+
0 < i < p ,
|
1922 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1923 |
+
ij,pβiβje2 ,
|
1924 |
+
0 < i < p ,
|
1925 |
+
0 < j < p β i ,
|
1926 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1927 |
+
ij,pβiβje1 ,
|
1928 |
+
0 < i < p ,
|
1929 |
+
0 < j < p β i ,
|
1930 |
+
(5.15)
|
1931 |
+
where the ο¬rst three formulas for each face are the edge-face base functions;
|
1932 |
+
β’ ο¬nally, the cell base functions read
|
1933 |
+
c1234 :
|
1934 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1935 |
+
0jke1 ,
|
1936 |
+
0 < j < p ,
|
1937 |
+
0 < k < p β j ,
|
1938 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1939 |
+
i0ke2 ,
|
1940 |
+
0 < i < p ,
|
1941 |
+
0 < k < p β i ,
|
1942 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
1943 |
+
ij0e3 ,
|
1944 |
+
0 < i < p ,
|
1945 |
+
0 < j < p β i ,
|
1946 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1947 |
+
ij,pβiβj(e1 + e2 + e3) ,
|
1948 |
+
0 < i < p ,
|
1949 |
+
0 < j < p β i ,
|
1950 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1951 |
+
ijke3 ,
|
1952 |
+
0 < i < p ,
|
1953 |
+
0 < j < p β i ,
|
1954 |
+
0 < k < p β i β j ,
|
1955 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1956 |
+
ijke2 ,
|
1957 |
+
0 < i < p ,
|
1958 |
+
0 < j < p β i ,
|
1959 |
+
0 < k < p β i β j ,
|
1960 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
1961 |
+
ijke1 ,
|
1962 |
+
0 < i < p ,
|
1963 |
+
0 < j < p β i ,
|
1964 |
+
0 < k < p β i β j ,
|
1965 |
+
(5.16)
|
1966 |
+
where the ο¬rst four formulas are the face-cell base functions.
|
1967 |
+
5.3
|
1968 |
+
NΒ΄edΒ΄elec elements of the ο¬rst type
|
1969 |
+
In order to construct the NΒ΄edΒ΄elec element of ο¬rst type on tetrahedra we introduce the template sets
|
1970 |
+
T1 = {ΟI
|
1971 |
+
4, ΟI
|
1972 |
+
5, ΟI
|
1973 |
+
6} ,
|
1974 |
+
T2 = {βΟI
|
1975 |
+
2, βΟI
|
1976 |
+
3, ΟI
|
1977 |
+
6} ,
|
1978 |
+
T3 = {βΟI
|
1979 |
+
3, βΟI
|
1980 |
+
5} ,
|
1981 |
+
T12 = {ΟI
|
1982 |
+
4 β ΟI
|
1983 |
+
2, ΟI
|
1984 |
+
5 β ΟI
|
1985 |
+
3} ,
|
1986 |
+
T13 = {ΟI
|
1987 |
+
1 + ΟI
|
1988 |
+
4, ΟI
|
1989 |
+
6 β ΟI
|
1990 |
+
3} ,
|
1991 |
+
T14 = {ΟI
|
1992 |
+
1 + ΟI
|
1993 |
+
5, ΟI
|
1994 |
+
2 + ΟI
|
1995 |
+
6} ,
|
1996 |
+
T23 = {ΟI
|
1997 |
+
1 β ΟI
|
1998 |
+
2, ΟI
|
1999 |
+
6 β ΟI
|
2000 |
+
5} ,
|
2001 |
+
T24 = {ΟI
|
2002 |
+
1 β ΟI
|
2003 |
+
3, ΟI
|
2004 |
+
4 + ΟI
|
2005 |
+
6} ,
|
2006 |
+
T34 = {ΟI
|
2007 |
+
2 β ΟI
|
2008 |
+
3, ΟI
|
2009 |
+
4 β ΟI
|
2010 |
+
5} ,
|
2011 |
+
T123 = {ΟI
|
2012 |
+
1 β ΟI
|
2013 |
+
2 + ΟI
|
2014 |
+
4} ,
|
2015 |
+
T124 = {ΟI
|
2016 |
+
1 β ΟI
|
2017 |
+
3 + ΟI
|
2018 |
+
5} ,
|
2019 |
+
T134 = {ΟI
|
2020 |
+
2 β ΟI
|
2021 |
+
3 + ΟI
|
2022 |
+
6} ,
|
2023 |
+
T234 = {ΟI
|
2024 |
+
4 β ΟI
|
2025 |
+
5 + ΟI
|
2026 |
+
6} ,
|
2027 |
+
(5.17)
|
2028 |
+
21
|
2029 |
+
|
2030 |
+
which are based on the lowest order NΒ΄edΒ΄elec base functions on the unit tetrahedron
|
2031 |
+
Ο1(ΞΎ, Ξ·, ΞΆ) =
|
2032 |
+
οΏ½
|
2033 |
+
οΏ½
|
2034 |
+
ΞΆ
|
2035 |
+
ΞΆ
|
2036 |
+
1 β ΞΎ β Ξ·
|
2037 |
+
οΏ½
|
2038 |
+
οΏ½ ,
|
2039 |
+
Ο2(ΞΎ, Ξ·, ΞΆ) =
|
2040 |
+
οΏ½
|
2041 |
+
οΏ½
|
2042 |
+
Ξ·
|
2043 |
+
1 β ΞΎ β ΞΆ
|
2044 |
+
Ξ·
|
2045 |
+
οΏ½
|
2046 |
+
οΏ½ ,
|
2047 |
+
Ο3(ΞΎ, Ξ·, ΞΆ) =
|
2048 |
+
οΏ½
|
2049 |
+
οΏ½
|
2050 |
+
1 β Ξ· β ΞΆ
|
2051 |
+
ΞΎ
|
2052 |
+
ΞΎ
|
2053 |
+
οΏ½
|
2054 |
+
οΏ½ ,
|
2055 |
+
Ο4(ΞΎ, Ξ·, ΞΆ) =
|
2056 |
+
οΏ½
|
2057 |
+
οΏ½
|
2058 |
+
0
|
2059 |
+
ΞΆ
|
2060 |
+
βΞ·
|
2061 |
+
οΏ½
|
2062 |
+
οΏ½ ,
|
2063 |
+
Ο5(ΞΎ, Ξ·, ΞΆ) =
|
2064 |
+
οΏ½
|
2065 |
+
οΏ½
|
2066 |
+
ΞΆ
|
2067 |
+
0
|
2068 |
+
βΞΎ
|
2069 |
+
οΏ½
|
2070 |
+
οΏ½ ,
|
2071 |
+
Ο6(ΞΎ, Ξ·, ΞΆ) =
|
2072 |
+
οΏ½
|
2073 |
+
οΏ½
|
2074 |
+
Ξ·
|
2075 |
+
βΞΎ
|
2076 |
+
0
|
2077 |
+
οΏ½
|
2078 |
+
οΏ½ .
|
2079 |
+
(5.18)
|
2080 |
+
For the non-gradient cell functions we use the construction introduced in [2]
|
2081 |
+
Rp =
|
2082 |
+
οΏ½
|
2083 |
+
(p + 1)bp
|
2084 |
+
iβejβΞ»j β
|
2085 |
+
ij
|
2086 |
+
p + 1βΞΎbp+1
|
2087 |
+
i
|
2088 |
+
| i β Io
|
2089 |
+
οΏ½
|
2090 |
+
,
|
2091 |
+
(5.19)
|
2092 |
+
where Io is the set of multi-indices of cell functions, ej is the unit multi-index with the value one at position
|
2093 |
+
j and ij is the value of the i-multi-index at position j. Note that only the ο¬rst term in the cell functions is
|
2094 |
+
required to span the next space in the sequence due to
|
2095 |
+
curl
|
2096 |
+
οΏ½
|
2097 |
+
[p + 1]bp
|
2098 |
+
iβejβΞΎΞ»j β
|
2099 |
+
ij
|
2100 |
+
p + 1βΞΎbp+1
|
2101 |
+
i
|
2102 |
+
οΏ½
|
2103 |
+
= curl([p + 1]bp
|
2104 |
+
iβejβΞΎΞ»j) .
|
2105 |
+
(5.20)
|
2106 |
+
However, without the added gradient the function would not belong to [Pp]3 βΞΎ Γ[οΏ½P]3 and consequently, would
|
2107 |
+
not be part of the NΒ΄edΒ΄elec space. By limiting Rp to Rp
|
2108 |
+
β such that Rp
|
2109 |
+
β contains only the surface permutations
|
2110 |
+
with βΞ»j = ej and the cell permutations with j β {1, 2}, one retrieves the necessary base functions. The
|
2111 |
+
sum of the lowest order NΒ΄edΒ΄elec base functions, the template base functions, gradient base functions, and the
|
2112 |
+
non-gradient cell base functions yields exactly (p+4)(p+3)(p+1)/2, thus satisfying the required dimensionality
|
2113 |
+
of the NΒ΄edΒ΄elec space. The complete space reads
|
2114 |
+
N p
|
2115 |
+
I = N 0
|
2116 |
+
I β
|
2117 |
+
οΏ½οΏ½
|
2118 |
+
iβI
|
2119 |
+
βEp+1
|
2120 |
+
i
|
2121 |
+
οΏ½
|
2122 |
+
β
|
2123 |
+
οΏ½
|
2124 |
+
οΏ½
|
2125 |
+
οΏ½
|
2126 |
+
οΏ½
|
2127 |
+
jβJ
|
2128 |
+
βFp+1
|
2129 |
+
j
|
2130 |
+
οΏ½
|
2131 |
+
οΏ½
|
2132 |
+
οΏ½ β βCp+1
|
2133 |
+
1234 β
|
2134 |
+
οΏ½ 3
|
2135 |
+
οΏ½
|
2136 |
+
k=1
|
2137 |
+
Vp
|
2138 |
+
k β Tk
|
2139 |
+
οΏ½
|
2140 |
+
β
|
2141 |
+
οΏ½οΏ½
|
2142 |
+
iβI
|
2143 |
+
Ep
|
2144 |
+
i β Ti
|
2145 |
+
οΏ½
|
2146 |
+
β
|
2147 |
+
οΏ½
|
2148 |
+
οΏ½
|
2149 |
+
οΏ½
|
2150 |
+
οΏ½
|
2151 |
+
jβJ
|
2152 |
+
Fp
|
2153 |
+
j β Tj
|
2154 |
+
οΏ½
|
2155 |
+
οΏ½
|
2156 |
+
οΏ½ β Rp+1
|
2157 |
+
β
|
2158 |
+
,
|
2159 |
+
I = {(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)} ,
|
2160 |
+
J = {(1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4)}
|
2161 |
+
.
|
2162 |
+
(5.21)
|
2163 |
+
Here, the BΒ΄ezier basis is used to construct the higher order NΒ΄edΒ΄elec base functions of the ο¬rst type.
|
2164 |
+
Deο¬nition 5.2 (BΒ΄ezier-NΒ΄edΒ΄elec I tetrahedral basis)
|
2165 |
+
The base functions are deο¬ned on the reference tetrahedron:
|
2166 |
+
β’ for the edges we use the lowest order base functions from Eq. (5.18). The remaining edge base functions
|
2167 |
+
are given by the gradients
|
2168 |
+
e12 :
|
2169 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2170 |
+
1 ,
|
2171 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2172 |
+
00k ,
|
2173 |
+
0 < k < p + 1 ,
|
2174 |
+
e13 :
|
2175 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2176 |
+
2 ,
|
2177 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2178 |
+
0j0 ,
|
2179 |
+
0 < j < p + 1 ,
|
2180 |
+
e14 :
|
2181 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2182 |
+
3 ,
|
2183 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2184 |
+
i00 ,
|
2185 |
+
0 < i < p + 1 ,
|
2186 |
+
e23 :
|
2187 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2188 |
+
4 ,
|
2189 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2190 |
+
0j,p+1βj ,
|
2191 |
+
0 < j < p + 1 ,
|
2192 |
+
e24 :
|
2193 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2194 |
+
5 ,
|
2195 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2196 |
+
i0,p+1βi ,
|
2197 |
+
0 < i < p + 1 ,
|
2198 |
+
e34 :
|
2199 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = ΟI
|
2200 |
+
6 ,
|
2201 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2202 |
+
00k ,
|
2203 |
+
0 < i < p + 1 ;
|
2204 |
+
(5.22)
|
2205 |
+
22
|
2206 |
+
|
2207 |
+
β’ on faces we employ both template base functions and gradients
|
2208 |
+
f123 :
|
2209 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2210 |
+
000ΟI
|
2211 |
+
4 ,
|
2212 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
2213 |
+
00pΟI
|
2214 |
+
2 ,
|
2215 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2216 |
+
00k(ΟI
|
2217 |
+
4 β ΟI
|
2218 |
+
2) ,
|
2219 |
+
0 < k < p ,
|
2220 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2221 |
+
0j0(ΟI
|
2222 |
+
1 + ΟI
|
2223 |
+
4) ,
|
2224 |
+
0 < j < p ,
|
2225 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2226 |
+
0j,pβj(ΟI
|
2227 |
+
1 β ΟI
|
2228 |
+
2) ,
|
2229 |
+
0 < j < p ,
|
2230 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2231 |
+
0jk(ΟI
|
2232 |
+
1 β ΟI
|
2233 |
+
2 + ΟI
|
2234 |
+
4) ,
|
2235 |
+
0 < j < p ,
|
2236 |
+
0 < k < p β j ,
|
2237 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2238 |
+
0jk ,
|
2239 |
+
0 < j < p + 1 ,
|
2240 |
+
0 < k < p + 1 β j ,
|
2241 |
+
f124 :
|
2242 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2243 |
+
000ΟI
|
2244 |
+
5 ,
|
2245 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
2246 |
+
00pΟI
|
2247 |
+
3 ,
|
2248 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2249 |
+
00k(ΟI
|
2250 |
+
5 β ΟI
|
2251 |
+
3) ,
|
2252 |
+
0 < k < p ,
|
2253 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2254 |
+
i00(ΟI
|
2255 |
+
1 + ΟI
|
2256 |
+
5) ,
|
2257 |
+
0 < i < p ,
|
2258 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2259 |
+
i0,pβi(ΟI
|
2260 |
+
1 β ΟI
|
2261 |
+
3) ,
|
2262 |
+
0 < i < p ,
|
2263 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2264 |
+
i0k(ΟI
|
2265 |
+
1 β ΟI
|
2266 |
+
3 + ΟI
|
2267 |
+
5) ,
|
2268 |
+
0 < i < p ,
|
2269 |
+
0 < k < p β i ,
|
2270 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2271 |
+
i0k ,
|
2272 |
+
0 < i < p + 1 ,
|
2273 |
+
0 < k < p + 1 β i ,
|
2274 |
+
f134 :
|
2275 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2276 |
+
000ΟI
|
2277 |
+
6 ,
|
2278 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
2279 |
+
0p0ΟI
|
2280 |
+
3 ,
|
2281 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2282 |
+
0j0(ΟI
|
2283 |
+
6 β ΟI
|
2284 |
+
3) ,
|
2285 |
+
0 < j < p ,
|
2286 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2287 |
+
i00(ΟI
|
2288 |
+
2 + ΟI
|
2289 |
+
6) ,
|
2290 |
+
0 < i < p ,
|
2291 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2292 |
+
i,pβi,0(ΟI
|
2293 |
+
2 β ΟI
|
2294 |
+
3) ,
|
2295 |
+
0 < i < p ,
|
2296 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2297 |
+
ij0(ΟI
|
2298 |
+
2 β ΟI
|
2299 |
+
3 + ΟI
|
2300 |
+
6) ,
|
2301 |
+
0 < i < p ,
|
2302 |
+
0 < j < p β i ,
|
2303 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2304 |
+
ij0 ,
|
2305 |
+
0 < i < p + 1 ,
|
2306 |
+
0 < j < p + 1 β i ,
|
2307 |
+
f234 :
|
2308 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2309 |
+
00pΟI
|
2310 |
+
6 ,
|
2311 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βbp
|
2312 |
+
0p0ΟI
|
2313 |
+
5 ,
|
2314 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2315 |
+
0j,pβj(ΟI
|
2316 |
+
6 β ΟI
|
2317 |
+
5) ,
|
2318 |
+
0 < j < p ,
|
2319 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2320 |
+
i0,pβi(ΟI
|
2321 |
+
4 + ΟI
|
2322 |
+
6) ,
|
2323 |
+
0 < i < p ,
|
2324 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2325 |
+
i,pβi,0(ΟI
|
2326 |
+
4 β ΟI
|
2327 |
+
5) ,
|
2328 |
+
0 < i < p ,
|
2329 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = bp
|
2330 |
+
ij,pβiβj(ΟI
|
2331 |
+
4 β ΟI
|
2332 |
+
5 + ΟI
|
2333 |
+
6) ,
|
2334 |
+
0 < i < p ,
|
2335 |
+
0 < j < p β i ,
|
2336 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2337 |
+
ij,pβi.j ,
|
2338 |
+
0 < i < p + 1 ,
|
2339 |
+
0 < j < p + 1 β i ;
|
2340 |
+
(5.23)
|
2341 |
+
β’ the cell base functions read
|
2342 |
+
c1234 :
|
2343 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
|
2344 |
+
iβ1,jke1 β
|
2345 |
+
i
|
2346 |
+
p + 2βΞΎbp+2
|
2347 |
+
ijk ,
|
2348 |
+
0 < i < p + 2 ,
|
2349 |
+
0 < j < p + 2 β i ,
|
2350 |
+
0 < k < p + 2 β i β j
|
2351 |
+
,
|
2352 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
|
2353 |
+
i,jβ1,ke2 β
|
2354 |
+
j
|
2355 |
+
p + 2βΞΎbp+2
|
2356 |
+
ijk ,
|
2357 |
+
0 < i < p + 2 ,
|
2358 |
+
0 < j < p + 2 β i ,
|
2359 |
+
0 < k < p + 2 β i β j
|
2360 |
+
,
|
2361 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
|
2362 |
+
ij0 e3 β
|
2363 |
+
1
|
2364 |
+
p + 2βΞΎbp+2
|
2365 |
+
ij1 ,
|
2366 |
+
0 < i < p + 2 ,
|
2367 |
+
0 < j < p + 2 β i ,
|
2368 |
+
Ο(ΞΎ, Ξ·, ΞΆ) = βΞΎbp+1
|
2369 |
+
ijk ,
|
2370 |
+
0 < i < p + 1 ,
|
2371 |
+
0 < j < p + 1 β i ,
|
2372 |
+
0 < k < p + 1 β i β j
|
2373 |
+
.
|
2374 |
+
(5.24)
|
2375 |
+
23
|
2376 |
+
|
2377 |
+
6
|
2378 |
+
Numerical quadrature
|
2379 |
+
Although the base functions are expressed using (Ξ±, Ξ², Ξ³) the domain is either the reference triangle or the
|
2380 |
+
reference tetrahedron, which require fewer quadrature points than their counterparts given by the Duο¬y trans-
|
2381 |
+
formation (quad or hexahedron). As such, we employ a mixture of the eο¬cient quadrature points introduced
|
2382 |
+
in [14,19,39,56,57] for triangles and tetrahedra, where we avoid quadrature schemes with points on the edges
|
2383 |
+
or faces of the reference domain due to the recursion formula of the Bernstein polynomials Eq. (3.8). The
|
2384 |
+
quadrature points are mapped to their equivalent expression in (Ξ±, Ξ², Ξ³). Consequently, the integration over the
|
2385 |
+
reference triangle or tetrahedron reads
|
2386 |
+
οΏ½
|
2387 |
+
Ae
|
2388 |
+
f(x, y) dA =
|
2389 |
+
οΏ½
|
2390 |
+
Ξ
|
2391 |
+
(f β¦ (ΞΎ, Ξ·))(Ξ±, Ξ²) | det J| dΞ ,
|
2392 |
+
οΏ½
|
2393 |
+
Ve
|
2394 |
+
f(x, y, z) dV =
|
2395 |
+
οΏ½
|
2396 |
+
β¦
|
2397 |
+
(f ⦠(ξ, η, ΢))(α, β, γ) | det J| d⦠.
|
2398 |
+
(6.1)
|
2399 |
+
For the lower order elements we use the Lagrangian-NΒ΄edΒ΄elec construction from [52,53].
|
2400 |
+
7
|
2401 |
+
Boundary conditions
|
2402 |
+
The degrees of freedom in [12] commute between the continuous and discrete spaces.
|
2403 |
+
As such, they allow
|
2404 |
+
to exactly satisfy the consistent coupling condition [11].
|
2405 |
+
We note that the functionals can be viewed as a
|
2406 |
+
hierarchical system of Dirichlet boundary problems. In the case of hierarchical base functions [58], they can
|
2407 |
+
be solved independently. However, here the boundary value of each polytope is required in advance due to the
|
2408 |
+
non-hierarchical nature of Bernstein polynomials. In other words, one must ο¬rst solve the problem for vertices,
|
2409 |
+
then for edges, afterwards for faces, and ο¬nally for the cell. In our case the degrees of freedom for the cell are
|
2410 |
+
irrelevant since a cell is never part of the boundary.
|
2411 |
+
7.1
|
2412 |
+
Boundary vertices
|
2413 |
+
The ο¬nite element mesh identiο¬es each vertex with a tuple of coordinates. It suο¬ces to evaluate the displacement
|
2414 |
+
ο¬eld at the vertex
|
2415 |
+
ud
|
2416 |
+
i = οΏ½u
|
2417 |
+
οΏ½οΏ½οΏ½οΏ½
|
2418 |
+
xi
|
2419 |
+
.
|
2420 |
+
(7.1)
|
2421 |
+
If the ο¬eld is vectorial, each component is evaluated at the designated vertex. The boundary conditions of the
|
2422 |
+
microdistortion ο¬eld are associated with tangential projections and as such do not have vertex-type degrees of
|
2423 |
+
freedom. This is the case since a vertex does not deο¬ne a unique tangential plane.
|
2424 |
+
7.2
|
2425 |
+
Boundary edges
|
2426 |
+
The edge functionals from [12] for the H 1-conforming subspace
|
2427 |
+
lij(u) =
|
2428 |
+
οΏ½
|
2429 |
+
si
|
2430 |
+
βqj
|
2431 |
+
βs
|
2432 |
+
βu
|
2433 |
+
βs ds ,
|
2434 |
+
q β Pp(s) ,
|
2435 |
+
(7.2)
|
2436 |
+
can be reformulated for a reference edge on a unit domain Ξ± β [0, 1]. We parametrize the edge via
|
2437 |
+
x(Ξ±) = (1 β Ξ±)x1 + Ξ±x2 .
|
2438 |
+
(7.3)
|
2439 |
+
As such, the following relation exists between the unit parameter and the arc-length parameter
|
2440 |
+
t = d
|
2441 |
+
dΞ±x = x2 β x1 ,
|
2442 |
+
ds = β₯dxβ₯ = β₯x2 β x1β₯dΞ± = β₯tβ₯dΞ± .
|
2443 |
+
(7.4)
|
2444 |
+
24
|
2445 |
+
|
2446 |
+
Ξ±
|
2447 |
+
0
|
2448 |
+
1
|
2449 |
+
ΞΎ : Ξ± β Ξ
|
2450 |
+
ΞΎ2
|
2451 |
+
ΞΎ1
|
2452 |
+
Ξ
|
2453 |
+
Ο
|
2454 |
+
ΞΎ
|
2455 |
+
Ξ·
|
2456 |
+
x2
|
2457 |
+
x1
|
2458 |
+
A
|
2459 |
+
t
|
2460 |
+
x
|
2461 |
+
y
|
2462 |
+
x : Ξ β A
|
2463 |
+
Figure 7.1: Barycentric mapping of edges from the unit domain to the reference triangle and onto the physical
|
2464 |
+
domain.
|
2465 |
+
By the chain rule we ο¬nd
|
2466 |
+
du
|
2467 |
+
ds = du
|
2468 |
+
dΞ±
|
2469 |
+
dΞ±
|
2470 |
+
ds = β₯tβ₯β1 du
|
2471 |
+
dΞ± ,
|
2472 |
+
(7.5)
|
2473 |
+
for some function u. On edges, the test and trial functions are Bernstein polynomials parametrized by the unit
|
2474 |
+
domain. The function representing the boundary condition οΏ½u(x) however, is parametrized by the Cartesian
|
2475 |
+
coordinates of the physical space. We ο¬nd its derivative with respect to the arc-length parameter by observing
|
2476 |
+
d
|
2477 |
+
ds οΏ½u = β¨ d
|
2478 |
+
dsx, βxοΏ½uβ© .
|
2479 |
+
(7.6)
|
2480 |
+
The derivative of the coordinates with respect to the arc-length is simply the normed tangent vector
|
2481 |
+
d
|
2482 |
+
dsx = dx
|
2483 |
+
dΞ±
|
2484 |
+
dΞ±
|
2485 |
+
ds = β₯tβ₯β1t .
|
2486 |
+
(7.7)
|
2487 |
+
Consequently, the edge boundary condition is given by
|
2488 |
+
οΏ½
|
2489 |
+
si
|
2490 |
+
βqj
|
2491 |
+
βs
|
2492 |
+
βu
|
2493 |
+
βs ds =
|
2494 |
+
οΏ½ 1
|
2495 |
+
0
|
2496 |
+
οΏ½
|
2497 |
+
β₯tβ₯β1 dqj
|
2498 |
+
dΞ±
|
2499 |
+
οΏ½ οΏ½
|
2500 |
+
β₯tβ₯β1 du
|
2501 |
+
dΞ±
|
2502 |
+
οΏ½
|
2503 |
+
β₯tβ₯ dΞ±
|
2504 |
+
=
|
2505 |
+
οΏ½ 1
|
2506 |
+
0
|
2507 |
+
οΏ½
|
2508 |
+
β₯tβ₯β1 dqj
|
2509 |
+
dΞ±
|
2510 |
+
οΏ½
|
2511 |
+
β¨β₯tβ₯β1t, βxοΏ½uβ©β₯tβ₯ dΞ± =
|
2512 |
+
οΏ½
|
2513 |
+
si
|
2514 |
+
βqj
|
2515 |
+
βs
|
2516 |
+
βοΏ½u
|
2517 |
+
βs ds
|
2518 |
+
β qj β Pp(Ξ±) ,
|
2519 |
+
(7.8)
|
2520 |
+
and can be solved by assembling the stiο¬ness matrix of the edge and the load vector induced by the prescribed
|
2521 |
+
displacement ο¬eld οΏ½u, representing volume forces
|
2522 |
+
kij =
|
2523 |
+
οΏ½ 1
|
2524 |
+
0
|
2525 |
+
οΏ½
|
2526 |
+
β₯tβ₯β1 dni
|
2527 |
+
dΞ±
|
2528 |
+
οΏ½ οΏ½
|
2529 |
+
β₯tβ₯β1 dnj
|
2530 |
+
dΞ±
|
2531 |
+
οΏ½
|
2532 |
+
β₯tβ₯ dΞ± ,
|
2533 |
+
fi =
|
2534 |
+
οΏ½ 1
|
2535 |
+
0
|
2536 |
+
β¨β₯tβ₯β1t, βxοΏ½uβ©
|
2537 |
+
οΏ½
|
2538 |
+
β₯tβ₯β1 dni
|
2539 |
+
dΞ±
|
2540 |
+
οΏ½
|
2541 |
+
β₯tβ₯ dΞ± .
|
2542 |
+
(7.9)
|
2543 |
+
Next we consider the Dirichlet boundary conditions for the microdistortion with the NΒ΄edΒ΄elec space of the
|
2544 |
+
second type NII. The problem reads
|
2545 |
+
οΏ½
|
2546 |
+
si
|
2547 |
+
qjβ¨t, pβ© ds =
|
2548 |
+
οΏ½
|
2549 |
+
si
|
2550 |
+
qjβ¨t, βxοΏ½uβ© ds
|
2551 |
+
β qj β Pp(si) .
|
2552 |
+
(7.10)
|
2553 |
+
Observe that on the edge the test functions qj are chosen to be the Bernstein polynomials. Further, by the
|
2554 |
+
polytopal template construction of the NII-space there holds β¨t, ΞΈiβ©|s = ni(Ξ±). Therefore, the components of
|
2555 |
+
the corresponding stiο¬ness matrix and load vectors read
|
2556 |
+
kij =
|
2557 |
+
οΏ½ 1
|
2558 |
+
0
|
2559 |
+
ni njβ₯tβ₯ dΞ± ,
|
2560 |
+
fi =
|
2561 |
+
οΏ½ 1
|
2562 |
+
0
|
2563 |
+
niβ¨t, βxοΏ½uβ©β₯tβ₯ dΞ± .
|
2564 |
+
(7.11)
|
2565 |
+
25
|
2566 |
+
|
2567 |
+
Note that in order to maintain the exactness property, the degree of the NΒ΄edΒ΄elec spaces N p
|
2568 |
+
I , N p
|
2569 |
+
II is always one
|
2570 |
+
less than the degree of the subspace Bp+1.
|
2571 |
+
Lastly, we consider the NΒ΄edΒ΄elec element of the ο¬rst type. The problem is given by
|
2572 |
+
οΏ½
|
2573 |
+
si
|
2574 |
+
qjβ¨t, pβ© ds =
|
2575 |
+
οΏ½
|
2576 |
+
si
|
2577 |
+
qjβ¨t, βxοΏ½uβ© ds
|
2578 |
+
β qj β Pp(si) .
|
2579 |
+
(7.12)
|
2580 |
+
We deο¬ne
|
2581 |
+
qi = d
|
2582 |
+
dΞ±np+1
|
2583 |
+
i
|
2584 |
+
,
|
2585 |
+
(7.13)
|
2586 |
+
and observe that on the edges the NΒ΄edΒ΄elec base functions yield
|
2587 |
+
β¨t, ΞΈjβ© = β¨t, βxnp+1
|
2588 |
+
j
|
2589 |
+
β© = d
|
2590 |
+
dΞ±np+1
|
2591 |
+
j
|
2592 |
+
.
|
2593 |
+
(7.14)
|
2594 |
+
Therefore, the components of the stiο¬ness matrix and the load vector result in
|
2595 |
+
kij =
|
2596 |
+
οΏ½ 1
|
2597 |
+
0
|
2598 |
+
dnp+1
|
2599 |
+
i
|
2600 |
+
dΞ±
|
2601 |
+
dnp+1
|
2602 |
+
j
|
2603 |
+
dΞ±
|
2604 |
+
β₯tβ₯ dΞ± ,
|
2605 |
+
fi =
|
2606 |
+
οΏ½ 1
|
2607 |
+
0
|
2608 |
+
dnp+1
|
2609 |
+
i
|
2610 |
+
dΞ±
|
2611 |
+
β¨t, βxοΏ½uβ©β₯tβ₯ dΞ± .
|
2612 |
+
(7.15)
|
2613 |
+
7.3
|
2614 |
+
Boundary faces
|
2615 |
+
We start with the face boundary condition for the H 1-conforming subspace. The problem reads
|
2616 |
+
οΏ½
|
2617 |
+
Ai
|
2618 |
+
β¨βfqj, βfuβ© dA =
|
2619 |
+
οΏ½
|
2620 |
+
Ai
|
2621 |
+
β¨βfqj, βf οΏ½uβ© dA
|
2622 |
+
β qj β Pp(Ai) .
|
2623 |
+
(7.16)
|
2624 |
+
The surface is parameterized by the barycentric mapping from the unit triangle Ξ = {(ΞΎ, Ξ·) β [0, 1]2 | ΞΎ +Ξ· β€ 1}.
|
2625 |
+
The surface gradient is given by
|
2626 |
+
βf οΏ½u = βxοΏ½u β
|
2627 |
+
1
|
2628 |
+
β₯nβ₯2 β¨βxοΏ½u, nβ©n ,
|
2629 |
+
(7.17)
|
2630 |
+
where n is the surface normal. The surface gradient can also be expressed via
|
2631 |
+
βfu = eiβx
|
2632 |
+
i u = gΞ²βΞΎ
|
2633 |
+
Ξ²u ,
|
2634 |
+
Ξ² β {1, 2} ,
|
2635 |
+
(7.18)
|
2636 |
+
where βx
|
2637 |
+
Ξ² are partial derivates with respect to the physical coordinates, βΞΎ
|
2638 |
+
Ξ² are partial derivatives with respect
|
2639 |
+
to the reference domain and gΞ² are the contravariant base vectors. The Einstein summation convention over
|
2640 |
+
corresponding indices is implied. The covariant base vectors are given by
|
2641 |
+
gΞ² = βx
|
2642 |
+
βΞΎΞ² .
|
2643 |
+
(7.19)
|
2644 |
+
One can ο¬nd the contravariant vector orthogonal to the surface by
|
2645 |
+
g3 = n = g1 Γ g2 .
|
2646 |
+
(7.20)
|
2647 |
+
We deο¬ne the mixed transformation matrix
|
2648 |
+
T =
|
2649 |
+
οΏ½
|
2650 |
+
g1 , g2 , g3οΏ½
|
2651 |
+
.
|
2652 |
+
(7.21)
|
2653 |
+
Due to the orthogonality relation β¨gi, gjβ© = Ξ΄ j
|
2654 |
+
i the transposed inverse of T is clearly
|
2655 |
+
T βT =
|
2656 |
+
οΏ½
|
2657 |
+
g1 , g2 , g3
|
2658 |
+
οΏ½
|
2659 |
+
.
|
2660 |
+
(7.22)
|
2661 |
+
Thus, we can compute the surface gradient of functions parametrized by the reference triangle via
|
2662 |
+
βfu =
|
2663 |
+
οΏ½
|
2664 |
+
g1 , g2οΏ½
|
2665 |
+
βΞΎu = T βT
|
2666 |
+
β
|
2667 |
+
βΞΎu ,
|
2668 |
+
T βT
|
2669 |
+
β
|
2670 |
+
=
|
2671 |
+
οΏ½
|
2672 |
+
g1 , g2οΏ½
|
2673 |
+
.
|
2674 |
+
(7.23)
|
2675 |
+
26
|
2676 |
+
|
2677 |
+
Further, there holds the following relation between the physical surface and the reference surface
|
2678 |
+
dA = β₯nβ₯dΞ = β₯g3β₯dΞ =
|
2679 |
+
οΏ½
|
2680 |
+
β¨g1 Γ g2, g3β© dΞ =
|
2681 |
+
β
|
2682 |
+
det T dΞ .
|
2683 |
+
(7.24)
|
2684 |
+
Consequently, we can write the components of the stiο¬ness matrix and load vector as
|
2685 |
+
kij =
|
2686 |
+
οΏ½
|
2687 |
+
Ξ
|
2688 |
+
β¨T βT
|
2689 |
+
β
|
2690 |
+
βΞΎni, T βT
|
2691 |
+
β
|
2692 |
+
βΞΎnjβ©
|
2693 |
+
β
|
2694 |
+
det T dΞ ,
|
2695 |
+
fi =
|
2696 |
+
οΏ½
|
2697 |
+
Ξ
|
2698 |
+
β¨T βT
|
2699 |
+
β
|
2700 |
+
βΞΎni, βxοΏ½u β (det T )β1β¨βxοΏ½u, nβ©nβ©
|
2701 |
+
β
|
2702 |
+
det T dΞ =
|
2703 |
+
οΏ½
|
2704 |
+
Ξ
|
2705 |
+
β¨T βT
|
2706 |
+
β
|
2707 |
+
βΞΎni, βxοΏ½uβ©
|
2708 |
+
β
|
2709 |
+
det T dΞ ,
|
2710 |
+
(7.25)
|
2711 |
+
with the orthogonality β¨gΞ², nβ© = 0 for Ξ² β {1, 2}.
|
2712 |
+
In order to embed the consistent coupling boundary condition to the microdistortion we deviate from the
|
2713 |
+
degrees of freedom deο¬ned in [12] and apply the simpler H (divR)-projection
|
2714 |
+
β¨qi, p, β©H(divR) = β¨qi, βf οΏ½uβ©H(divR)
|
2715 |
+
β qi β N p
|
2716 |
+
I (A)
|
2717 |
+
or
|
2718 |
+
β qi β N p
|
2719 |
+
II(A) .
|
2720 |
+
(7.26)
|
2721 |
+
Due to ker(curl) = βH 1 the problem reduces to
|
2722 |
+
οΏ½
|
2723 |
+
Ai
|
2724 |
+
β¨qj, pβ© + β¨curl2Dqj, curl2Dpβ© dA =
|
2725 |
+
οΏ½
|
2726 |
+
Ai
|
2727 |
+
β¨qj, βf οΏ½uβ© dA
|
2728 |
+
β qj β N p
|
2729 |
+
I (A)
|
2730 |
+
or
|
2731 |
+
β qj β N p
|
2732 |
+
II(A) .
|
2733 |
+
(7.27)
|
2734 |
+
We express the co- and contravariant Piola transformation from the two-dimensional reference domain to the
|
2735 |
+
three-dimensional physical domain using
|
2736 |
+
ΞΈi = T βT
|
2737 |
+
β
|
2738 |
+
Οi ,
|
2739 |
+
divx R ΞΈi =
|
2740 |
+
1
|
2741 |
+
β
|
2742 |
+
det T
|
2743 |
+
divΞΎ R Οi .
|
2744 |
+
(7.28)
|
2745 |
+
Thus, the stiο¬ness matrix components and load vector components read
|
2746 |
+
kij =
|
2747 |
+
οΏ½
|
2748 |
+
Ξ
|
2749 |
+
β¨T βT
|
2750 |
+
β
|
2751 |
+
Οi, T βT
|
2752 |
+
β
|
2753 |
+
Οjβ© + β¨(det T )β1/2 divΞΎ R Οi, (det T )β1/2 divΞΎ R Οjβ©
|
2754 |
+
β
|
2755 |
+
det T dΞ ,
|
2756 |
+
fi =
|
2757 |
+
οΏ½
|
2758 |
+
Ξ
|
2759 |
+
β¨T βT
|
2760 |
+
β
|
2761 |
+
Οi, βxοΏ½u β (det T )β1β¨βxοΏ½u, nβ©nβ©
|
2762 |
+
β
|
2763 |
+
det T dΞ =
|
2764 |
+
οΏ½
|
2765 |
+
Ξ
|
2766 |
+
οΏ½οΏ½T βT
|
2767 |
+
β
|
2768 |
+
Οi, βxοΏ½uβ©
|
2769 |
+
β
|
2770 |
+
det T dΞ ,
|
2771 |
+
(7.29)
|
2772 |
+
where we again make use of the orthogonality between the surface tangent vectors and its normal vector.
|
2773 |
+
8
|
2774 |
+
Numerical examples
|
2775 |
+
In the following we test the ο¬nite element formulations with an artiο¬cial analytical solution in the antiplane shear
|
2776 |
+
model and with an analytical solution for an inο¬nite plane under cylindrical bending in the three dimensional
|
2777 |
+
model. Finally, we benchmark the ability of the ο¬nite element formulations to correctly interpolate between
|
2778 |
+
micro Cmicro and macro Cmacro stiο¬nesses as described by the characteristic length scale parameter Lc. The
|
2779 |
+
majority of convergence results are presented by measuring the error in the Lebesgue norm over the domain
|
2780 |
+
β₯οΏ½u β uhβ₯L2 =
|
2781 |
+
οΏ½οΏ½
|
2782 |
+
V
|
2783 |
+
β₯οΏ½u β uhβ₯2 dV ,
|
2784 |
+
β₯ οΏ½P β P hβ₯L2 =
|
2785 |
+
οΏ½οΏ½
|
2786 |
+
V
|
2787 |
+
β₯ οΏ½P β P hβ₯2 dV ,
|
2788 |
+
(8.1)
|
2789 |
+
in which context {οΏ½u, οΏ½P } and {uh, P h} are the analytical and approximate subspace solutions, respectively.
|
2790 |
+
8.1
|
2791 |
+
Compatible microdistortion
|
2792 |
+
In [53] we explored the conditions for which the microdistortion p reduces to a gradient ο¬eld, i.e. p is compatible.
|
2793 |
+
By deο¬ning the micro-moment with a scalar potential
|
2794 |
+
m = β100 β x2 β y2
|
2795 |
+
10
|
2796 |
+
= β1
|
2797 |
+
5
|
2798 |
+
οΏ½x
|
2799 |
+
y
|
2800 |
+
οΏ½
|
2801 |
+
,
|
2802 |
+
(8.2)
|
2803 |
+
27
|
2804 |
+
|
2805 |
+
and constructing an analytical solution for the displacement ο¬eld
|
2806 |
+
οΏ½u = sin
|
2807 |
+
οΏ½x2 + y2
|
2808 |
+
5
|
2809 |
+
οΏ½
|
2810 |
+
,
|
2811 |
+
(8.3)
|
2812 |
+
we can recover the analytical solution of the microdistortion
|
2813 |
+
p =
|
2814 |
+
1
|
2815 |
+
Β΅e + Β΅micro
|
2816 |
+
(m + Β΅eβοΏ½u) = 1
|
2817 |
+
2
|
2818 |
+
οΏ½
|
2819 |
+
β1
|
2820 |
+
5
|
2821 |
+
οΏ½x
|
2822 |
+
y
|
2823 |
+
οΏ½
|
2824 |
+
+ 2
|
2825 |
+
5
|
2826 |
+
οΏ½x cos([x2 + y2]/5)
|
2827 |
+
y cos([x2 + y2]/5)
|
2828 |
+
οΏ½οΏ½
|
2829 |
+
= 1
|
2830 |
+
5
|
2831 |
+
οΏ½x cos([x2 + y2]/5)
|
2832 |
+
y cos([x2 + y2]/5)
|
2833 |
+
οΏ½
|
2834 |
+
β 1
|
2835 |
+
10
|
2836 |
+
οΏ½x
|
2837 |
+
y
|
2838 |
+
οΏ½
|
2839 |
+
,
|
2840 |
+
(8.4)
|
2841 |
+
where for simplicity we set all material constants to one. Since m is a gradient ο¬eld, the microdistortion p is
|
2842 |
+
also reduced to a gradient ο¬eld and curl2Dp = 0, see [53]. Note that this result is speciο¬c to antiplane shear
|
2843 |
+
and does not generalize to the full three-dimensional model, compare [52]. We note that the microdistortion
|
2844 |
+
is not equal to the gradient of the displacement ο¬eld and as such, their tangential projections on an arbitrary
|
2845 |
+
boundary are not automatically the same. However, for both the gradient of the displacement ο¬eld and the
|
2846 |
+
micro-moment is the tangential projection on the boundary of the circular domain A = {x β R2 | β₯xβ₯ β€ 10}
|
2847 |
+
equal to zero
|
2848 |
+
β¨βt, οΏ½uβ©
|
2849 |
+
οΏ½οΏ½οΏ½οΏ½
|
2850 |
+
βA
|
2851 |
+
= β¨t, mβ©
|
2852 |
+
οΏ½οΏ½οΏ½οΏ½
|
2853 |
+
βA
|
2854 |
+
= 0 ,
|
2855 |
+
(8.5)
|
2856 |
+
and as such the microdistortion belongs to p β H0(curl, A).
|
2857 |
+
Consequently, we can set sD = βA and the
|
2858 |
+
consistent coupling condition remains compatible.
|
2859 |
+
With the displacement and the microdistortion ο¬elds at
|
2860 |
+
hand we derive the corresponding forces
|
2861 |
+
f = 1
|
2862 |
+
25
|
2863 |
+
οΏ½
|
2864 |
+
2x2 sin
|
2865 |
+
οΏ½x2 + y2
|
2866 |
+
5
|
2867 |
+
οΏ½
|
2868 |
+
+ 2y2 sin
|
2869 |
+
οΏ½x2 + y2
|
2870 |
+
5
|
2871 |
+
οΏ½
|
2872 |
+
β 10 cos
|
2873 |
+
οΏ½x2 + y2
|
2874 |
+
5
|
2875 |
+
οΏ½
|
2876 |
+
β 5
|
2877 |
+
οΏ½
|
2878 |
+
.
|
2879 |
+
(8.6)
|
2880 |
+
The approximation of the displacement and microdistortion ο¬elds using linear and higher order elements is
|
2881 |
+
shown in Fig. 8.1. We note that even with almost 3000 ο¬nite elements and 6000 degrees of freedom the linear
|
2882 |
+
formulation is incapable of ο¬nding an adequate approximation. On the other side of the spectrum, the higher
|
2883 |
+
order approximation (degree 7) with 57 elements and 4097 degrees of freedom yields very accurate results in
|
2884 |
+
the interior of the domain. However, the exterior of the domain is captured rather poorly. This is the case since
|
2885 |
+
the geometry of the circular domain is being approximated by linear triangles. Thus, in this setting, a ο¬ner
|
2886 |
+
mesh captures the geometry in a more precise manner. The eο¬ects of the geometry on the approximation of the
|
2887 |
+
solution are also clearly visible in the convergence graphs in Fig. 8.2; only after a certain accuracy in the domain
|
2888 |
+
description is achieved do the ο¬nite elements retrieve their predicted convergence rates, compare [52,53]. This
|
2889 |
+
is clearly observable when comparing the convergence curves of the linear and seventh order elements. The
|
2890 |
+
linear element generates quadratic convergence p + 1 = 1 + 1 = 2, whereas the seventh-order element yields
|
2891 |
+
the convergence slope 7 (where 8 is expected).
|
2892 |
+
Although the seventh-order formulation encompasses more
|
2893 |
+
degrees of freedom, it employs a coarser mesh and as such, generates higher errors at the boundary. The errors
|
2894 |
+
themselves can be traced back to the consistent coupling condition since, for a non-perfect circle the gradient
|
2895 |
+
of the displacement ο¬eld induces tangential projections on the imperfect boundary. The inο¬uence of the latter
|
2896 |
+
eο¬ect is even more apparent in the convergence of the microdistortion, where the higher order formulations are
|
2897 |
+
unable to perform optimally on coarse meshes.
|
2898 |
+
8.2
|
2899 |
+
Cylindrical bending
|
2900 |
+
In order to test the capability of the ο¬nite element formulations to capture the intrinsic behaviour of the relaxed
|
2901 |
+
micromorphic model, we compare with analytical solutions of boundary-value problems. The ο¬rst example
|
2902 |
+
considers the displacement and microdistortion ο¬elds under cylindrical bending [43] for inο¬nitely extended
|
2903 |
+
plates. Let the plates be deο¬ned as V = (ββ, β)2 Γ [β1/2, 1/2], than the analytical solution for cylindrical
|
2904 |
+
bending reads
|
2905 |
+
u = ΞΊ
|
2906 |
+
οΏ½
|
2907 |
+
οΏ½
|
2908 |
+
βxz
|
2909 |
+
0
|
2910 |
+
x2/2
|
2911 |
+
οΏ½
|
2912 |
+
οΏ½ ,
|
2913 |
+
P = βΞΊ
|
2914 |
+
οΏ½
|
2915 |
+
οΏ½
|
2916 |
+
[41z + 20
|
2917 |
+
β
|
2918 |
+
82 sech(
|
2919 |
+
οΏ½
|
2920 |
+
41/2) sinh(
|
2921 |
+
β
|
2922 |
+
82z)]/1681
|
2923 |
+
0
|
2924 |
+
x
|
2925 |
+
0
|
2926 |
+
0
|
2927 |
+
0
|
2928 |
+
βx
|
2929 |
+
0
|
2930 |
+
0
|
2931 |
+
οΏ½
|
2932 |
+
οΏ½ ,
|
2933 |
+
(8.7)
|
2934 |
+
28
|
2935 |
+
|
2936 |
+
(a)
|
2937 |
+
(b)
|
2938 |
+
(c)
|
2939 |
+
(d)
|
2940 |
+
(e)
|
2941 |
+
(f)
|
2942 |
+
(g)
|
2943 |
+
(h)
|
2944 |
+
(i)
|
2945 |
+
(j)
|
2946 |
+
(k)
|
2947 |
+
(l)
|
2948 |
+
Figure 8.1: Depiction of the displacement ο¬eld (a)-(c) and the microdistortion ο¬eld (d)-(f) for the antiplane
|
2949 |
+
shear problem, for the linear element under h-reο¬nement with 225, 763 and 2966 elements, corresponding to
|
2950 |
+
485, 1591 and 6060 degrees of freedom. The p-reο¬nement of the displacement ο¬eld on the coarsest mesh of 57
|
2951 |
+
elements is visualized in (g)-(l) with p β {3, 5, 7}, corresponding to 731, 2072 and 4097 degrees of freedom.
|
2952 |
+
29
|
2953 |
+
|
2954 |
+
11
|
2955 |
+
NA144
|
2956 |
+
44103
|
2957 |
+
104
|
2958 |
+
105
|
2959 |
+
10β3
|
2960 |
+
10β1
|
2961 |
+
101
|
2962 |
+
degrees of freedom
|
2963 |
+
β₯οΏ½u β uhβ₯L2
|
2964 |
+
L1 Γ N 0
|
2965 |
+
I
|
2966 |
+
L2 Γ N 1
|
2967 |
+
II
|
2968 |
+
B3 Γ N 2
|
2969 |
+
II
|
2970 |
+
B5 Γ N 4
|
2971 |
+
II
|
2972 |
+
B7 Γ N 6
|
2973 |
+
II
|
2974 |
+
O(h2)
|
2975 |
+
O(h7)
|
2976 |
+
(a)
|
2977 |
+
103
|
2978 |
+
104
|
2979 |
+
105
|
2980 |
+
10β3
|
2981 |
+
10β1
|
2982 |
+
101
|
2983 |
+
degrees of freedom
|
2984 |
+
β₯οΏ½p β phβ₯L2
|
2985 |
+
L1 Γ N 0
|
2986 |
+
I
|
2987 |
+
L2 Γ N 1
|
2988 |
+
II
|
2989 |
+
B3 Γ N 2
|
2990 |
+
II
|
2991 |
+
B5 Γ N 4
|
2992 |
+
II
|
2993 |
+
B7 Γ N 6
|
2994 |
+
II
|
2995 |
+
O(h)
|
2996 |
+
O(h2)
|
2997 |
+
(b)
|
2998 |
+
Figure 8.2: Convergence of displacement (a) and the microdistortion (b) under h-reο¬nement for multiple poly-
|
2999 |
+
nomial degrees for the antiplane shear problem.
|
3000 |
+
where sech(x) = 1/ cosh(x), and for the following values of material constants
|
3001 |
+
Ξ»e = Ξ»micro = 0 ,
|
3002 |
+
Β΅e = Β΅macro = 1/2 ,
|
3003 |
+
Β΅c = 0 ,
|
3004 |
+
Lc = 1 ,
|
3005 |
+
Β΅micro = 20 .
|
3006 |
+
(8.8)
|
3007 |
+
The intensity of the curvature parameter ΞΊ of the plate is chosen to be ΞΊ = 14/200.
|
3008 |
+
Remark 8.1
|
3009 |
+
The particular case of the cylindrical bending for which Ξ»e = Ξ»micro = 0 (equivalent to a zero micro-Poissonβs
|
3010 |
+
ratio) has been solved, along with its more general case (Ξ»e ΜΈ= Ξ»micro ΜΈ= 0), in [43]. The advantage of considering
|
3011 |
+
this particular case is that a cut out ο¬nite plate of the inο¬nite domain automatically exhibits the consistent
|
3012 |
+
coupling boundary conditions on its side surfaces.
|
3013 |
+
Remark 8.2
|
3014 |
+
Note that the general analytical solution for cylindrical bending does not depend on Β΅c, so we can set Β΅c = 0
|
3015 |
+
without loss of generality, compare [43].
|
3016 |
+
We deο¬ne the ο¬nite domain V = [β10, 10]2 Γ [β1/2, 1/2] and the boundaries
|
3017 |
+
AD1 = {β10} Γ [β10, 10] Γ [β1/2, 1/2] ,
|
3018 |
+
AD2 = {10} Γ [β10, 10] Γ [β1/2, 1/2] ,
|
3019 |
+
AN = βV \ {AD1 β AD2} .
|
3020 |
+
(8.9)
|
3021 |
+
Additionally, on the Dirichlet boundary we impose the translated analytical solution οΏ½u = u β
|
3022 |
+
οΏ½0
|
3023 |
+
0
|
3024 |
+
3.5οΏ½T .
|
3025 |
+
The displacement ο¬eld and the last row of the microdistortion are depicted in Fig. 8.3. The displacement
|
3026 |
+
ο¬eld is dominated by its quadratic term and captured correctly.
|
3027 |
+
The last row of the microdistortion is a
|
3028 |
+
linear function and easily approximated even with linear elements. On the contrary, the P11 component of
|
3029 |
+
the microdistortion is a hyperbolic function of the z-axis.
|
3030 |
+
The results of its approximation at x = y = 0
|
3031 |
+
(the centre of the plane) are given in Fig. 8.4. We observe that even increasing the number of linear ο¬nite
|
3032 |
+
elements to the extreme only results in better oscillations around the analytical solution. In comparison, higher
|
3033 |
+
order formulations converge towards the expected hyperbolic behaviour. The approximation of the quadratic
|
3034 |
+
NΒ΄edΒ΄elec element of the ο¬rst type is nearly perfect, whereas its second type counterpart clearly deviates from
|
3035 |
+
the analytical solution at z β β0.25. Taking the cubic second type element yields the desired result. This
|
3036 |
+
phenomenon is an evident indicator of the prominent role of the Curl of the microdistortion in this type of
|
3037 |
+
problems. Firstly, the microdistortion is a non-gradient ο¬eld. Secondly, the Curl of the analytical solution
|
3038 |
+
induces an hyperbolic sine term. Such functions are often approximated using at least cubic terms in power
|
3039 |
+
series, thus explaining the necessity of such high order elements for correct computations.
|
3040 |
+
30
|
3041 |
+
|
3042 |
+
(a)
|
3043 |
+
(b)
|
3044 |
+
Figure 8.3: Displacement (a) and last row of the microdistortion (b) for the quadratic formulation using the
|
3045 |
+
NΒ΄edΒ΄elec element of the ο¬rst type.
|
3046 |
+
β0.5
|
3047 |
+
0
|
3048 |
+
0.5
|
3049 |
+
β1
|
3050 |
+
0
|
3051 |
+
1
|
3052 |
+
Β·10β2
|
3053 |
+
z-axis
|
3054 |
+
P11(z)
|
3055 |
+
ne = 5640
|
3056 |
+
ne = 44592
|
3057 |
+
ne = 354720
|
3058 |
+
(a)
|
3059 |
+
β0.5
|
3060 |
+
0
|
3061 |
+
0.5
|
3062 |
+
β1
|
3063 |
+
0
|
3064 |
+
1
|
3065 |
+
Β·10β2
|
3066 |
+
z-axis
|
3067 |
+
P11(z)
|
3068 |
+
B2 Γ N 1
|
3069 |
+
I
|
3070 |
+
B3 Γ N 2
|
3071 |
+
I
|
3072 |
+
L2 Γ N 1
|
3073 |
+
II
|
3074 |
+
B3 Γ N 2
|
3075 |
+
II
|
3076 |
+
B4 Γ N 3
|
3077 |
+
II
|
3078 |
+
(b)
|
3079 |
+
Figure 8.4: Convergence of the lowest order formulation under h-reο¬nement with 732, 5640 and 44592 elements
|
3080 |
+
(a) and of the higher order formulations under p-reο¬nement using 732 elements(b) towards the analytical solution
|
3081 |
+
(dashed curve) of the P11(z) component at x = y = 0.
|
3082 |
+
31
|
3083 |
+
|
3084 |
+
8.3
|
3085 |
+
Bounded stiο¬ness property
|
3086 |
+
The characteristic length scale parameter Lc allows the relaxed micromorphic model to capture the transition
|
3087 |
+
from highly homogeneous materials to materials with a pronounced micro-structure by governing the inο¬uence
|
3088 |
+
of the micro-structure on the overall behaviour of the model. We demonstrate this property of the model with
|
3089 |
+
an example, where we vary Lc and measure the resulting energy.
|
3090 |
+
Let the domain be given by the axis-symmetric cube V = [β1, 1]3 with a total Dirichlet boundary
|
3091 |
+
AD1 = {(x, y, z) β [β1, 1]3 | x = Β±1} ,
|
3092 |
+
AD2 = {(x, y, z) β [β1, 1]3 | y = Β±1} ,
|
3093 |
+
AD3 = {(x, y, z) β [β1, 1]3 | z = Β±1} ,
|
3094 |
+
(8.10)
|
3095 |
+
we embed the periodic boundary conditions
|
3096 |
+
οΏ½u
|
3097 |
+
οΏ½οΏ½οΏ½οΏ½
|
3098 |
+
AD1
|
3099 |
+
=
|
3100 |
+
οΏ½
|
3101 |
+
οΏ½
|
3102 |
+
(1 β y2) sin(Ο[1 β z2])/10
|
3103 |
+
0
|
3104 |
+
0
|
3105 |
+
οΏ½
|
3106 |
+
οΏ½ ,
|
3107 |
+
οΏ½u
|
3108 |
+
οΏ½οΏ½οΏ½οΏ½
|
3109 |
+
AD2
|
3110 |
+
=
|
3111 |
+
οΏ½
|
3112 |
+
οΏ½
|
3113 |
+
0
|
3114 |
+
(1 β x2) sin(Ο[1 β z2])/10
|
3115 |
+
0
|
3116 |
+
οΏ½
|
3117 |
+
οΏ½ ,
|
3118 |
+
οΏ½u
|
3119 |
+
οΏ½οΏ½οΏ½οΏ½
|
3120 |
+
AD3
|
3121 |
+
=
|
3122 |
+
οΏ½
|
3123 |
+
οΏ½
|
3124 |
+
0
|
3125 |
+
0
|
3126 |
+
(1 β y2) sin(Ο[1 β x2])/10
|
3127 |
+
οΏ½
|
3128 |
+
οΏ½ .
|
3129 |
+
(8.11)
|
3130 |
+
The material parameters are chosen as
|
3131 |
+
Ξ»macro = 2 ,
|
3132 |
+
Β΅macro = 1 ,
|
3133 |
+
Ξ»micro = 10 ,
|
3134 |
+
Β΅micro = 5 ,
|
3135 |
+
Β΅c = 1 ,
|
3136 |
+
(8.12)
|
3137 |
+
thus giving rise to the following meso-parameters via Eq. (2.19)
|
3138 |
+
Ξ»e = 2.5 ,
|
3139 |
+
Β΅e = 1.25 .
|
3140 |
+
(8.13)
|
3141 |
+
The displacement ο¬eld as well as some examples of the employed meshes are shown in Fig. 8.5. In order to
|
3142 |
+
compute the upper and lower bound on the energy we utilize the equivalent Cauchy model formulation with
|
3143 |
+
the micro- and macro elasticity parameters. In order to assert the high accuracy of the solution of the bounds
|
3144 |
+
we employ tenth order ο¬nite elements. The progression of the energy in dependence of the characteristic length
|
3145 |
+
parameter Lc is given in Fig. 8.6. We observe the high mesh dependency of the lower order formulations, where
|
3146 |
+
the energy is clearly overestimated. The higher order formulations all capture the upper bound correctly but
|
3147 |
+
diverge with respect to the result of the lower bound. Notably, the approximation using the NΒ΄edΒ΄elec element
|
3148 |
+
of the ο¬rst type is more accurate than the equivalent formulation with the NΒ΄edΒ΄elec element of the second type,
|
3149 |
+
thus indicating the non-negligible involvement of the micro-dislocation in the energy. Using standard mesh
|
3150 |
+
coarseness the cubic element formulation with NΒ΄edΒ΄elec elements of the ο¬rst type yields satisfactory results. In
|
3151 |
+
order to achieve the same on highly coarse meshes, one needs to employ seventh order elements.
|
3152 |
+
9
|
3153 |
+
Conclusions and outlook
|
3154 |
+
The intrinsic behaviour of the relaxed micromorphic model is revealed by the analytical solutions to boundary
|
3155 |
+
value problems. Clearly, the continuum exhibits hyperbolic and trigonometric solutions, which are not easily
|
3156 |
+
approximated by low order ο¬nite elements. The example provided in Section 8.2 demonstrates that cubic and
|
3157 |
+
higher order ο¬nite elements yield excellent results in approximate solutions of the model.
|
3158 |
+
The polytopal template methodology introduced in [50] allows to easily and ο¬exibly construct H (curl)-
|
3159 |
+
conforming vectorial ο¬nite elements that inherit many of the characteristics of an underlying H 1-conforming
|
3160 |
+
basis, which can be chosen independently. In this work, we made use of Bernstein-BΒ΄ezier polynomials. The
|
3161 |
+
latter boast optimal complexity properties manifesting in the form of sum factorization. The natural decom-
|
3162 |
+
position of their multi-variate versions into multiplications of univariate Bernstein base functions via the Duο¬y
|
3163 |
+
transformation allows to construct optimal iterators for their evaluation using recursion formulas. Further, this
|
3164 |
+
characteristic makes the use of dual numbers in the computation of their derivatives ideal. Finally, the intrinsic
|
3165 |
+
order of traversal induced by the factorization is exploited optimally by the choice of clock-wise orientation
|
3166 |
+
of the reference element. The consequence of these combined features is a high-performance hp-ο¬nite element
|
3167 |
+
program.
|
3168 |
+
32
|
3169 |
+
|
3170 |
+
(a)
|
3171 |
+
(b)
|
3172 |
+
(c)
|
3173 |
+
Figure 8.5: Displacement ο¬eld of the Cauchy model on the coarsest mesh of 48 ο¬nite elements of the tenth order
|
3174 |
+
(a) and depictions of the meshes with 384 (b) and 3072 (c) elements, respectively.
|
3175 |
+
The ability of the relaxed micromorphic model to interpolate between the energies of homogeneous materials
|
3176 |
+
and materials with an underlying micro-structure using the characteristic length scale parameter Lc is demon-
|
3177 |
+
strated in Section 8.3. It is also shown that in order to correctly capture the span of energies for the values of
|
3178 |
+
Lc either ο¬ne-discretizations or higher order elements are required.
|
3179 |
+
The excellent performance of the proposed higher order ο¬nite elements in the linear static case is a precur-
|
3180 |
+
sor for their application in the dynamic setting, which is important since the relaxed micromorphic model is
|
3181 |
+
often employed in the computation of elastic waves (e.g., for acoustic metamaterials), where solutions for high
|
3182 |
+
frequency ranges are commonly needed.
|
3183 |
+
The proposed computational scheme is lacking in its description of curved geometries. Due to the consistent
|
3184 |
+
coupling condition, this can easily lead to errors emanating from the boundary. Consequently, a topic for future
|
3185 |
+
works would be the investigation of curved ο¬nite elements [20,21] and their behaviour with respect to the model.
|
3186 |
+
Acknowledgements
|
3187 |
+
Angela Madeo and Gianluca Rizzi acknowledge support from the European Commission through the funding
|
3188 |
+
of the ERC Consolidator Grant META-LEGO, N⦠101001759.00
|
3189 |
+
Patrizio Neο¬ acknowledges support in the framework of the DFG-Priority Programme 2256 βVariational
|
3190 |
+
Methods for Predicting Complex Phenomena in Engineering Structures and Materialsβ, Neο¬ 902/10-1, Project-
|
3191 |
+
No. 440935806.
|
3192 |
+
10
|
3193 |
+
References
|
3194 |
+
[1] Ainsworth, M., Andriamaro, G., Davydov, O.: BernsteinβBΒ΄ezier ο¬nite elements of arbitrary order and optimal assembly
|
3195 |
+
procedures. SIAM Journal on Scientiο¬c Computing 33(6), 3087β3109 (2011)
|
3196 |
+
[2] Ainsworth, M., Fu, G.: BernsteinβBΒ΄ezier bases for tetrahedral ο¬nite elements. Computer Methods in Applied Mechanics and
|
3197 |
+
Engineering 340, 178β201 (2018)
|
3198 |
+
[3] Aivaliotis, A., Tallarico, D., dβAgostino, M.V., Daouadji, A., Neο¬, P., Madeo, A.: Frequency- and angle-dependent scattering
|
3199 |
+
of a ο¬nite-sized meta-structure via the relaxed micromorphic model. Archive of Applied Mechanics 90(5), 1073β1096 (2020)
|
3200 |
+
[4] Alberdi, R., Robbins, J., Walsh, T., Dingreville, R.: Exploring wave propagation in heterogeneous metastructures using the
|
3201 |
+
relaxed micromorphic model. Journal of the Mechanics and Physics of Solids 155, 104540 (2021)
|
3202 |
+
[5] Anjam, I., Valdman, J.: Fast MATLAB assembly of FEM matrices in 2d and 3d: Edge elements. Applied Mathematics and
|
3203 |
+
Computation 267, 252β263 (2015)
|
3204 |
+
[6] Barbagallo, G., Madeo, A., dβAgostino, M.V., Abreu, R., Ghiba, I.D., Neο¬, P.:
|
3205 |
+
Transparent anisotropy for the relaxed
|
3206 |
+
micromorphic model: Macroscopic consistency conditions and long wave length asymptotics. International Journal of Solids
|
3207 |
+
and Structures 120, 7β30 (2017)
|
3208 |
+
33
|
3209 |
+
|
3210 |
+
10β3
|
3211 |
+
100
|
3212 |
+
103
|
3213 |
+
0.1
|
3214 |
+
0.5
|
3215 |
+
0.9
|
3216 |
+
Lc
|
3217 |
+
I
|
3218 |
+
ne = 384
|
3219 |
+
ne = 3072
|
3220 |
+
ne = 24576
|
3221 |
+
ne = 48000
|
3222 |
+
Cmacro
|
3223 |
+
Cmicro
|
3224 |
+
(a)
|
3225 |
+
10β3
|
3226 |
+
100
|
3227 |
+
103
|
3228 |
+
0.1
|
3229 |
+
0.5
|
3230 |
+
0.9
|
3231 |
+
Lc
|
3232 |
+
I
|
3233 |
+
ne = 384
|
3234 |
+
ne = 3072
|
3235 |
+
ne = 24576
|
3236 |
+
Cmacro
|
3237 |
+
Cmicro
|
3238 |
+
(b)
|
3239 |
+
10β3
|
3240 |
+
100
|
3241 |
+
103
|
3242 |
+
0.1
|
3243 |
+
0.5
|
3244 |
+
0.9
|
3245 |
+
Lc
|
3246 |
+
I
|
3247 |
+
ne = 384 , B3 Γ N 2
|
3248 |
+
I
|
3249 |
+
ne = 3072 , B3 Γ N 2
|
3250 |
+
I
|
3251 |
+
ne = 384 , B3 Γ N 2
|
3252 |
+
II
|
3253 |
+
ne = 3072 , B3 Γ N 2
|
3254 |
+
II
|
3255 |
+
Cmacro
|
3256 |
+
Cmicro
|
3257 |
+
(c)
|
3258 |
+
10β3
|
3259 |
+
100
|
3260 |
+
103
|
3261 |
+
0.1
|
3262 |
+
0.5
|
3263 |
+
0.9
|
3264 |
+
Lc
|
3265 |
+
I
|
3266 |
+
B5 Γ N 4
|
3267 |
+
I
|
3268 |
+
B7 Γ N 6
|
3269 |
+
I
|
3270 |
+
B5 Γ N 4
|
3271 |
+
II
|
3272 |
+
B7 Γ N 6
|
3273 |
+
II
|
3274 |
+
Cmacro
|
3275 |
+
Cmicro
|
3276 |
+
(d)
|
3277 |
+
Figure 8.6: Energy progression of the relaxed micromorphic model with respect to Lc using the linear (a),
|
3278 |
+
quadratic (b) and cubic (c) ο¬nite element formulations. The energy computed with the coarsest mesh of 48
|
3279 |
+
elements is depicted in (d) for various polynomial powers.
|
3280 |
+
34
|
3281 |
+
|
3282 |
+
[7] Barbagallo, G., Tallarico, D., DβAgostino, M.V., Aivaliotis, A., Neο¬, P., Madeo, A.: Relaxed micromorphic model of transient
|
3283 |
+
wave propagation in anisotropic band-gap metastructures. International Journal of Solids and Structures 162, 148β163 (2019)
|
3284 |
+
[8] Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic diο¬erentiation in machine learning: a survey. Journal
|
3285 |
+
of Machine Learning Research 18, 1β43 (2018)
|
3286 |
+
[9] Bergot, M., Lacoste, P.: Generation of higher-order polynomial basis of NΒ΄edΒ΄elec H(curl) ο¬nite elements for Maxwellβs equa-
|
3287 |
+
tions. Journal of Computational and Applied Mathematics 234(6), 1937β1944 (2010). Eighth International Conference on
|
3288 |
+
Mathematical and Numerical Aspects of Waves (Waves 2007)
|
3289 |
+
[10] dβAgostino, M.V., Barbagallo, G., Ghiba, I.D., Eidel, B., Neο¬, P., Madeo, A.: Eο¬ective description of anisotropic wave
|
3290 |
+
dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. Journal of Elasticity 139(2), 299β329
|
3291 |
+
(2020)
|
3292 |
+
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|
3293 |
+
for the classical micromorphic model: existence, uniqueness and interpretation of parameters.
|
3294 |
+
Continuum Mechanics and
|
3295 |
+
Thermodynamics (2022)
|
3296 |
+
[12] Demkowicz, L., Monk, P., Vardapetyan, L., Rachowicz, W.: De Rham diagram for hp-ο¬nite element spaces. Computers and
|
3297 |
+
Mathematics with Applications 39(7), 29β38 (2000)
|
3298 |
+
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|
3299 |
+
modeling of large-scale acoustic meta-structures. Journal of the Mechanics and Physics of Solids 168, 104995 (2022)
|
3300 |
+
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|
3301 |
+
International Journal for
|
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+
Numerical Methods in Engineering 21(6), 1129β1148 (1985)
|
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+
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|
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+
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|
3305 |
+
P. Hovland, E. Phipps, J. Utke, A. Walther (eds.) Recent Advances in Algorithmic Diο¬erentiation, pp. 163β173. Springer
|
3306 |
+
Berlin Heidelberg, Berlin, Heidelberg (2012)
|
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+
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|
3308 |
+
tions of the Royal Society A 378(20190169) (2020)
|
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+
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|
3310 |
+
and continuous dependence in dynamics. Mathematics and Mechanics of Solids 20(10), 1171β1197 (2015)
|
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+
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|
3312 |
+
High-order cubature rules for tetrahedra.
|
3313 |
+
International Journal for Numerical Methods in
|
3314 |
+
Engineering 121(11), 2418β2436 (2020)
|
3315 |
+
[20] Johnen, A., Remacle, J.F., Geuzaine, C.: Geometrical validity of curvilinear ο¬nite elements. Journal of Computational Physics
|
3316 |
+
233, 359β372 (2013)
|
3317 |
+
[21] Johnen, A., Remacle, J.F., Geuzaine, C.: Geometrical validity of high-order triangular ο¬nite elements.
|
3318 |
+
Engineering with
|
3319 |
+
Computers 30(3), 375β382 (2014)
|
3320 |
+
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|
3321 |
+
Journal of Mathematical Analysis and Applications 519(2), 126806 (2023)
|
3322 |
+
[23] Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press (2007)
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3323 |
+
[24] Lewintan, P., MΒ¨uller, S., Neο¬, P.: Korn inequalities for incompatible tensor ο¬elds in three space dimensions with conformally
|
3324 |
+
invariant dislocation energy. Calculus of Variations and Partial Diο¬erential Equations 60(4), 150 (2021)
|
3325 |
+
[25] Lewintan, P., Neο¬, P.: Lp-versions of generalized Korn inequalities for incompatible tensor ο¬elds in arbitrary dimensions with
|
3326 |
+
p-integrable exterior derivative. Comptes Rendus MathΒ΄ematique 359(6), 749β755 (2021)
|
3327 |
+
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|
3328 |
+
ο¬elds. Mathematical Methods in the Applied Sciences 44(14), 11392β11403 (2021)
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+
[27] Madeo, A., Barbagallo, G., Collet, M., dβAgostino, M.V., Miniaci, M., Neο¬, P.: Relaxed micromorphic modeling of the interface
|
3330 |
+
between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design. Mathematics
|
3331 |
+
and Mechanics of Solids 23(12), 1485β1506 (2018)
|
3332 |
+
[28] Madeo, A., Neο¬, P., Ghiba, I.D., Rosi, G.: Reο¬ection and transmission of elastic waves in non-local band-gap metamaterials:
|
3333 |
+
A comprehensive study via the relaxed micromorphic model. Journal of the Mechanics and Physics of Solids 95, 441β479
|
3334 |
+
(2016)
|
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+
[29] Mindlin, R.: Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 51β78 (1964)
|
3336 |
+
[30] Nedelec, J.C.: Mixed ο¬nite elements in R3. Numerische Mathematik 35(3), 315β341 (1980)
|
3337 |
+
[31] NΒ΄edΒ΄elec, J.C.: A new family of mixed ο¬nite elements in R3. Numerische Mathematik 50(1), 57β81 (1986)
|
3338 |
+
[32] Neο¬, P., Eidel, B., dβAgostino, M.V., Madeo, A.: Identiο¬cation of scale-independent material parameters in the relaxed
|
3339 |
+
micromorphic model through model-adapted ο¬rst order homogenization. Journal of Elasticity 139(2), 269β298 (2020)
|
3340 |
+
[33] Neο¬, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for aο¬ne microstructure.
|
3341 |
+
modelling, existence of minimizers, identiο¬cation of moduli and computational results. Journal of Elasticity 87(2), 239β276
|
3342 |
+
(2007)
|
3343 |
+
[34] Neο¬, P., Ghiba, I.D., Lazar, M., Madeo, A.: The relaxed linear micromorphic continuum: well-posedness of the static problem
|
3344 |
+
and relations to the gauge theory of dislocations. The Quarterly Journal of Mechanics and Applied Mathematics 68(1), 53β84
|
3345 |
+
(2015)
|
3346 |
+
35
|
3347 |
+
|
3348 |
+
[35] Neο¬, P., Ghiba, I.D., Madeo, A., Placidi, L., Rosi, G.: A unifying perspective: the relaxed linear micromorphic continuum.
|
3349 |
+
Continuum Mechanics and Thermodynamics 26(5), 639β681 (2014)
|
3350 |
+
[36] Neο¬, P., Pauly, D., Witsch, K.J.: Maxwell meets Korn: A new coercive inequality for tensor ο¬elds with square-integrable
|
3351 |
+
exterior derivative. Mathematical Methods in the Applied Sciences 35(1), 65β71 (2012)
|
3352 |
+
[37] Neidinger, R.D.: Introduction to automatic diο¬erentiation and MATLAB object-oriented programming. SIAM Review 52(3),
|
3353 |
+
545β563 (2010)
|
3354 |
+
[38] Owczarek, S., Ghiba, I.D., Neο¬, P.: A note on local higher regularity in the dynamic linear relaxed micromorphic model.
|
3355 |
+
Mathematical Methods in the Applied Sciences 44(18), 13855β13865 (2021)
|
3356 |
+
[39] Papanicolopulos, S.A.: Eο¬cient computation of cubature rules with application to new asymmetric rules on the triangle. J.
|
3357 |
+
Comput. Appl. Math. 304, 73β83 (2016)
|
3358 |
+
[40] Perez-Ramirez, L.A., Rizzi, G., Madeo, A.: Multi-element metamaterialβs design through the relaxed micromorphic model.
|
3359 |
+
arXiv:2210.14697 (2022)
|
3360 |
+
[41] Rizzi, G., dβAgostino, M.V., Neο¬, P., Madeo, A.: Boundary and interface conditions in the relaxed micromorphic model:
|
3361 |
+
Exploring ο¬nite-size metastructures for elastic wave control.
|
3362 |
+
Mathematics and Mechanics of Solids p. 10812865211048923
|
3363 |
+
(2021)
|
3364 |
+
[42] Rizzi, G., HΒ¨utter, G., Khan, H., Ghiba, I.D., Madeo, A., Neο¬, P.: Analytical solution of the cylindrical torsion problem for
|
3365 |
+
the relaxed micromorphic continuum and other generalized continua (including full derivations). Mathematics and Mechanics
|
3366 |
+
of Solids p. 10812865211023530 (2021)
|
3367 |
+
[43] Rizzi, G., HΒ¨utter, G., Madeo, A., Neο¬, P.: Analytical solutions of the cylindrical bending problem for the relaxed micromorphic
|
3368 |
+
continuum and other generalized continua. Continuum Mechanics and Thermodynamics 33(4), 1505β1539 (2021)
|
3369 |
+
[44] Rizzi, G., HΒ¨utter, G., Madeo, A., Neο¬, P.: Analytical solutions of the simple shear problem for micromorphic models and
|
3370 |
+
other generalized continua. Archive of Applied Mechanics 91(5), 2237β2254 (2021)
|
3371 |
+
[45] Rizzi, G., Khan, H., Ghiba, I.D., Madeo, A., Neο¬, P.: Analytical solution of the uniaxial extension problem for the relaxed
|
3372 |
+
micromorphic continuum and other generalized continua (including full derivations). Archive of Applied Mechanics (2021)
|
3373 |
+
[46] Rizzi, G., Neο¬, P., Madeo, A.: Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic
|
3374 |
+
approach. Philosophical Transactions of the Royal Society A 380(2231) (2022)
|
3375 |
+
[47] Sarhil, M., Scheunemann, L., SchrΒ¨oder, J., Neο¬, P.: Size-eο¬ects of metamaterial beams subjected to pure bending: on boundary
|
3376 |
+
conditions and parameter identiο¬cation in the relaxed micromorphic model. arXiv:2210.17117 (2022)
|
3377 |
+
[48] Sch¨oberl, J., Zaglmayr, S.: High order N´ed´elec elements with local complete sequence properties. COMPEL - The international
|
3378 |
+
Journal for Computation and Mathematics in Electrical and Electronic Engineering 24(2), 374β384 (2005)
|
3379 |
+
[49] SchrΒ¨oder, J., Sarhil, M., Scheunemann, L., Neο¬, P.: Lagrange and H (curl, B) based ο¬nite element formulations for the relaxed
|
3380 |
+
micromorphic model. Computational Mechanics (2022)
|
3381 |
+
[50] Sky, A., Muench, I.: Polytopal templates for the formulation of semi-continuous vectorial ο¬nite elements of arbitrary order.
|
3382 |
+
arXiv:2210.03525 (2022)
|
3383 |
+
[51] Sky, A., Muench, I., Neο¬, P.: On [H 1]3Γ3, [H (curl)]3 and H (symCurl) ο¬nite elements for matrix-valued Curl problems. Journal
|
3384 |
+
of Engineering Mathematics 136(1), 5 (2022)
|
3385 |
+
[52] Sky, A., Neunteufel, M., Muench, I., SchΒ¨oberl, J., Neο¬, P.: Primal and mixed ο¬nite element formulations for the relaxed
|
3386 |
+
micromorphic model. Computer Methods in Applied Mechanics and Engineering 399, 115298 (2022)
|
3387 |
+
[53] Sky, A., Neunteufel, M., MΒ¨unch, I., SchΒ¨oberl, J., Neο¬, P.: A hybrid H 1 Γ H (curl) ο¬nite element formulation for a relaxed
|
3388 |
+
micromorphic continuum model of antiplane shear. Computational Mechanics 68(1), 1β24 (2021)
|
3389 |
+
[54] Solin, P., Segeth, K., Dolezel, I.: Higher-Order Finite Element Methods (1st ed.). Chapman and Hall/CRC (2003)
|
3390 |
+
[55] Voss, J., Baaser, H., Martin, R.J., Neο¬, P.: More on anti-plane shear. Journal of Optimization Theory and Applications
|
3391 |
+
184(1), 226β249 (2020)
|
3392 |
+
[56] Witherden, F., Vincent, P.: On the identiο¬cation of symmetric quadrature rules for ο¬nite element methods. Computers &
|
3393 |
+
Mathematics with Applications 69(10), 1232β1241 (2015)
|
3394 |
+
[57] Xiao, H., Gimbutas, Z.: A numerical algorithm for the construction of eο¬cient quadrature rules in two and higher dimensions.
|
3395 |
+
Computers & Mathematics with Applications 59(2), 663β676 (2010)
|
3396 |
+
[58] Zaglmayr, S.:
|
3397 |
+
High order ο¬nite element methods for electromagnetic ο¬eld computation.
|
3398 |
+
Ph.D. thesis, Johannes Kepler
|
3399 |
+
Universit¨at Linz (2006). URL https://www.numerik.math.tugraz.at/~zaglmayr/pub/szthesis.pdf
|
3400 |
+
36
|
3401 |
+
|
BdAzT4oBgHgl3EQfh_0J/content/tmp_files/load_file.txt
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CNAzT4oBgHgl3EQfTvwq/content/tmp_files/2301.01253v1.pdf.txt
ADDED
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|
1 |
+
Deep Learning for bias-correcting comprehensive
|
2 |
+
high-resolution Earth system models
|
3 |
+
Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3
|
4 |
+
1Earth System Modelling, School of Engineering & Design, Technical University of Munich,
|
5 |
+
Munich, Germany
|
6 |
+
2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
|
7 |
+
3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK
|
8 |
+
Key Points:
|
9 |
+
β’ A generative adversarial network is shown to improve daily precipitation ο¬elds from
|
10 |
+
a state-of-the-art Earth system model.
|
11 |
+
β’ Biases in long-term temporal distributions are strongly reduced by the generative
|
12 |
+
adversarial network.
|
13 |
+
β’ Our network-based approach can be complemented with quantile mapping to fur-
|
14 |
+
ther improve precipitation ο¬elds.
|
15 |
+
β1β
|
16 |
+
arXiv:2301.01253v1 [physics.ao-ph] 16 Dec 2022
|
17 |
+
|
18 |
+
Abstract
|
19 |
+
The accurate representation of precipitation in Earth system models (ESMs) is crucial for
|
20 |
+
reliable projections of the ecological and socioeconomic impacts in response to anthropogenic
|
21 |
+
global warming. The complex cross-scale interactions of processes that produces precipi-
|
22 |
+
tation are challenging to model, however, inducing potentially strong biases in ESM ο¬elds,
|
23 |
+
especially regarding extremes. State-of-the-art bias correction methods only address errors
|
24 |
+
in the simulated frequency distributions locally, at every individual grid cell. Improving
|
25 |
+
unrealistic spatial patterns of the ESM output, which would require spatial context, has
|
26 |
+
not been possible so far. Here, we show that a post-processing method based on physically
|
27 |
+
constrained generative adversarial networks (GANs) can correct biases of a state-of-the-art,
|
28 |
+
CMIP6-class ESM both in local frequency distributions and in the spatial patterns at once.
|
29 |
+
While our method improves local frequency distributions equally well as gold-standard bias-
|
30 |
+
adjustment frameworks it strongly outperforms any existing methods in the correction of
|
31 |
+
spatial patterns, especially in terms of the characteristic spatial intermittency of precipita-
|
32 |
+
tion extremes.
|
33 |
+
1 Introduction
|
34 |
+
Precipitation is a crucial climate variable and changing amounts, frequencies, or spatial
|
35 |
+
distributions have potentially severe ecological and socioeconomic impacts.
|
36 |
+
With global
|
37 |
+
warming projected to continue in the coming decades, assessing the impacts of changes
|
38 |
+
in precipitation characteristics is an urgent challenge (Wilcox & Donner, 2007; Boyle &
|
39 |
+
Klein, 2010; IPCC, 2021). Climate impact models are designed to assess the impacts of
|
40 |
+
global warming on, for example, ecosystems, crop yields, vegetation and other land-surface
|
41 |
+
characteristics, infrastructure, water resources, or the economy in general (Kotz et al., 2022),
|
42 |
+
using the output of climate or Earth system models (ESMs) as input. Especially for reliable
|
43 |
+
assessments of the ecological and socioeconomic impacts, accurate ESM precipitation ο¬elds
|
44 |
+
to feed the impact models are therefore crucial.
|
45 |
+
ESMs are integrated on spatial grids with ο¬nite resolution. The resolution is limited
|
46 |
+
by the computational resources that are necessary to perform simulations on decadal to
|
47 |
+
centennial time scales. Current state-of-the-art ESMs have a horizontal resolution on the
|
48 |
+
order of 100km, in exceptional cases going down to 50km. Smaller-scale physical processes
|
49 |
+
that are relevant for the generation of precipitation operate on scales below the size of
|
50 |
+
individual grid cells. These can therefore not be resolved explicitly in ESMs and have to
|
51 |
+
included as parameterizations of the resolved prognostic variables. These include droplet
|
52 |
+
interactions, turbulence, and phase transitions in clouds that play a central role in the
|
53 |
+
generation of precipitation.
|
54 |
+
The limited grid resolution hence introduces errors in the simulated precipitation ο¬elds,
|
55 |
+
leading to biases in short-term spatial patterns and long-term summary statistics. These
|
56 |
+
biases need to be addressed prior to passing the ESM precipitation ο¬elds to impact mod-
|
57 |
+
els. In particular, climate impact models are often developed and calibrated with input
|
58 |
+
data from reanalysis data rather than ESM simulations. These reanalyses are created with
|
59 |
+
data assimilation routines and combine various observations with high-resolution weather
|
60 |
+
models. They hence provide a much more realistic input than the ESM simulations and
|
61 |
+
statistical bias correction methods are necessary to remove biases in the ESM simulations
|
62 |
+
output and to make them more similar to the reanalysis data for which the impact models
|
63 |
+
are calibrated. Quantile mapping (QM) is a standard technique to correct systematic errors
|
64 |
+
in ESM simulations. QM estimates a mapping between distributions from historical sim-
|
65 |
+
ulations and observations that can thereafter be applied to future simulations in order to
|
66 |
+
provide more accurate simulated precipitation ο¬elds to impact models (DΒ΄equΒ΄e, 2007; Tong
|
67 |
+
et al., 2021; Gudmundsson et al., 2012; Cannon et al., 2015).
|
68 |
+
State-of-the-art bias correction methods such as QM are, however, conο¬ned to address
|
69 |
+
errors in the simulated frequency distributions locally, i.e., at every grid cell individually.
|
70 |
+
β2β
|
71 |
+
|
72 |
+
Unrealistic spatial patterns of the ESM output, which would require spatial context, have
|
73 |
+
therefore so far not been addressed by postprocessing methods. For precipitation this is
|
74 |
+
particularly important because it has characteristic high intermittency not only in time,
|
75 |
+
but also in its spatial patterns. Mulitvariate bias correction approaches have recently been
|
76 |
+
developed, aiming to improve spatial dependencies (Vrac, 2018; Cannon, 2018). However,
|
77 |
+
these approaches are typically only employed in regional studies, as the dimension of the
|
78 |
+
input becomes too large for global high-resolution ESM simulations. Moreover, such meth-
|
79 |
+
ods have been reported to suο¬er from instabilities and overο¬tting, while diο¬erences in their
|
80 |
+
applicability and assumptions make them challenging to use (FranΒΈcois et al., 2020).
|
81 |
+
Here, we employ a recently introduced postprocessing method (Hess et al., 2022) based
|
82 |
+
on a cycle-consistent adversarial network (CycleGAN) to consistently improve both local
|
83 |
+
frequency distributions and spatial patterns of state-of-art high-resolution ESM precipita-
|
84 |
+
tion ο¬elds. Artiο¬cial neural networks from computer vision and image processing have been
|
85 |
+
successfully applied to various tasks in Earth system science, ranging from weather forecast-
|
86 |
+
ing (Weyn et al., 2020; Rasp & Thuerey, 2021) to post-processing (Gr¨onquist et al., 2021;
|
87 |
+
Price & Rasp, 2022), by extracting spatial features with convolutional layers (LeCun et al.,
|
88 |
+
2015). Generative adversarial networks (Goodfellow et al., 2014) in particular have emerged
|
89 |
+
as a promising architecture that produces sharp images that are necessary to capture the
|
90 |
+
high-frequency variability of precipitation (Ravuri et al., 2021; Price & Rasp, 2022; Harris et
|
91 |
+
al., 2022). GANs have been speciο¬cally developed to be trained on unpaired image datasets
|
92 |
+
(Zhu et al., 2017). This makes them a natural choice for post-processing the output of cli-
|
93 |
+
mate projections, which β unlike weather forecasts β are not nudged to follow the trajectory
|
94 |
+
of observations; due to the chaotic nature of the atmosphere small deviations in the initial
|
95 |
+
conditions or parameters lead to exponentially diverging trajectories (Lorenz, 1996). As a
|
96 |
+
result, numerical weather forecasts lose their deterministic forecast skill after approximately
|
97 |
+
two weeks at most and century-scale climate simulations do not agree with observed daily
|
98 |
+
weather records. Indeed the task of climate models is rather to produce accurate long-term
|
99 |
+
statistics that to agree with observations.
|
100 |
+
We apply our CycleGAN approach to correct global high-resolution precipitation simu-
|
101 |
+
lations of the GFDL-ESM4 model (Krasting et al., 2018) as a representative ESM from the
|
102 |
+
Climate Model Intercomparison Project phase 6 (CMIP6). So far, GANs-based approaches
|
103 |
+
have only been applied to postprocess ESM simulations either in a regional context (FranΒΈcois
|
104 |
+
et al., 2021), or to a very-low-resolution global ESM (Hess et al., 2022). We show here that
|
105 |
+
a suitably designed CycleGAN is capable of improving even the distributions and spatial
|
106 |
+
patterns of precipitation ο¬elds from a state-of-the-art comprehensive ESM, namely GFDL-
|
107 |
+
ESM4. In particular, in contrast to rather speciο¬c existing methods for postprocessing ESM
|
108 |
+
output for climate impact modelling, we will show that the CycleGAN is general and can
|
109 |
+
readily be applied to diο¬erent ESMs and observational datasets used as ground truth.
|
110 |
+
In order to assure that physical conservation laws are not violated by the GAN-based
|
111 |
+
postprocessing, we include a suitable physical constraint, enforcing that the overall global
|
112 |
+
sum of daily precipitation values is not changed by the GAN-based transformations; es-
|
113 |
+
sentially, this assures that precipitation is only spatially redistributed (see Methods). By
|
114 |
+
framing bias correction as an image-to-image translation task, our approach corrects both
|
115 |
+
spatial patterns of daily precipitation ο¬elds on short time scales and temporal distributions
|
116 |
+
aggregated over decadal time scales. We evaluate the skill to improve spatial patterns and
|
117 |
+
temporal distributions against the gold-standard ISIMIP3BASD framework (Lange, 2019),
|
118 |
+
which relies strongly on QM.
|
119 |
+
Quantifying the βrealisticnessβ of spatial precipitation patterns is a key problem in
|
120 |
+
current research (Ravuri et al., 2021). We use spatial spectral densities and the fractal
|
121 |
+
dimension of spatial patterns as a measure to quantify the similarity of intermittent and un-
|
122 |
+
paired precipitation ο¬elds. We will show that our CycleGAN is indeed spatial context-aware
|
123 |
+
and strongly improves the characteristic intermittency in spatial precipitation patterns. We
|
124 |
+
β3β
|
125 |
+
|
126 |
+
will also show that our CycleGAN combined with a subseqeunt application of ISIMIP3BASD
|
127 |
+
routine leads to the best overall performance.
|
128 |
+
2 Results
|
129 |
+
We evaluate our CycleGAN method on two diο¬erent tasks and time scales. First, the
|
130 |
+
correction of daily rainfall frequency distributions at each grid cell locally, aggregated from
|
131 |
+
decade-long time series. Second, we quantify the ability to improve spatial patterns on daily
|
132 |
+
time scales. Our GAN approach is compared to the raw GFDL-ESM4 model output, as well
|
133 |
+
as to the ISIMIP3BASD methodology applied to the GFDL-ESM4 output.
|
134 |
+
2.1 Temporal distributions
|
135 |
+
10
|
136 |
+
6
|
137 |
+
10
|
138 |
+
5
|
139 |
+
10
|
140 |
+
4
|
141 |
+
10
|
142 |
+
3
|
143 |
+
10
|
144 |
+
2
|
145 |
+
10
|
146 |
+
1
|
147 |
+
100
|
148 |
+
Histogram
|
149 |
+
a
|
150 |
+
0
|
151 |
+
98.4
|
152 |
+
99.7
|
153 |
+
99.94
|
154 |
+
99.98
|
155 |
+
99.993
|
156 |
+
99.997
|
157 |
+
W5E5v2 precipitation percentiles
|
158 |
+
W5E5v2
|
159 |
+
GFDL-ESM4
|
160 |
+
ISIMIP3BASD
|
161 |
+
GAN
|
162 |
+
GAN (unconstrained)
|
163 |
+
GAN-ISIMIP3BASD
|
164 |
+
0
|
165 |
+
25
|
166 |
+
50
|
167 |
+
75
|
168 |
+
100
|
169 |
+
125
|
170 |
+
150
|
171 |
+
Precipitation [mm/d]
|
172 |
+
10
|
173 |
+
8
|
174 |
+
10
|
175 |
+
7
|
176 |
+
10
|
177 |
+
6
|
178 |
+
10
|
179 |
+
5
|
180 |
+
10
|
181 |
+
4
|
182 |
+
10
|
183 |
+
3
|
184 |
+
10
|
185 |
+
2
|
186 |
+
10
|
187 |
+
1
|
188 |
+
Absolute error
|
189 |
+
b
|
190 |
+
Figure
|
191 |
+
1: Histograms of relative precipitation frequencies over the entire globe and test
|
192 |
+
period (2004-2014). (a) The histograms are shown for the W5E5v2 ground truth (black),
|
193 |
+
GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), unconstrained GAN (orange),
|
194 |
+
and the constrained-GAN-ISIMIP3BASD combination (blue).
|
195 |
+
(b) Distances of the his-
|
196 |
+
tograms to the W5E5v2 ground truth are shown for the same models as in (a). Percentiles
|
197 |
+
corresponding to the W5E5v2 precipitation values are given on the second x-axis at the
|
198 |
+
top. Note that GFDL-ESM4 overestimates the frequencies of strong and extreme rainfall
|
199 |
+
events. All compared methods show similar performance in correcting the local frequency
|
200 |
+
distributions.
|
201 |
+
β4β
|
202 |
+
|
203 |
+
We compute global histograms of relative precipitation frequencies using daily time
|
204 |
+
series (Fig. 1a). The GFDL-ESM4 model overestimates frequencies in the tail, namely for
|
205 |
+
events above 50 mm/day (i.e., the 99.7th percentile). Our GAN-based method as well as
|
206 |
+
ISIMIP3BASD and the GAN-ISIMIP3BASD combination correct the histogram to match
|
207 |
+
the W5E5v2 ground truth equally well, as can be also seen in the absolute error of the
|
208 |
+
histograms (Fig. 1b).
|
209 |
+
Comparing the diο¬erences in long-term averages of precipitation per grid cell (Fig. 2
|
210 |
+
and Methods), large biases are apparent in the GFDL-ESM4 model output, especially in
|
211 |
+
the tropics. The double-peaked Intertropical Convergence Zone (ITCZ) bias is visible. The
|
212 |
+
double-ITCZ bias can also be inferred from the latitudinal proο¬le of the precipitation mean
|
213 |
+
in Fig. 3.
|
214 |
+
Table 1 summarizes the annual biases shown in Fig. 2 as absolute averages, and addi-
|
215 |
+
tionally for the four seasons. The GAN alone reduces the annual bias of the GFDL-ESM4
|
216 |
+
model by 38.7%. The unconstrained GAN performs better than the physically constrained
|
217 |
+
one, with bias reductions of 50.5%. As expected, the ISIMIP3BASD gives even better results
|
218 |
+
for correcting the local mean, since it is speciο¬cally designed to accurately transform the
|
219 |
+
local frequency distributions. It is therefore remarkable that applying the ISIMIP3BASD
|
220 |
+
procedure on the constrained GAN output improves the post-processing further, leading to
|
221 |
+
a local bias reduction of the mean by 63.6%, compared to ISIMIP3BASD with 59.4%. For
|
222 |
+
seasonal time series the order in which the methods perform is the same as for the annual
|
223 |
+
data.
|
224 |
+
Besides the error in the mean, we also compute diο¬erences in the 95th percentile for each
|
225 |
+
grid cell, shown in Fig. S1 and as mean absolute errors in Table 1. Also in this case of heavy
|
226 |
+
precipitation values we ο¬nd that ISIMIP3BASD outperforms the GAN, but that combining
|
227 |
+
GAN and ISIMIP3BASD leads to best agreement of the locally computed quantiles.
|
228 |
+
Table 1: The globally averaged absolute value of the grid cell-wise diο¬erence in the long-
|
229 |
+
term precipitation average, as well as the 95th percentile, between the W5E5v2 ground truth
|
230 |
+
and GFDL-ESM4, ISIMIP3BASD, GAN, unconstrained GAN, and the GAN-ISIMIP3BASD
|
231 |
+
combination for annual and seasonal time series (in [mm/day]). The relative improvement
|
232 |
+
over the raw GFDL-ESM4 climate model output is shown as percentages for each method.
|
233 |
+
Season
|
234 |
+
Percentile
|
235 |
+
GFDL-
|
236 |
+
ESM4
|
237 |
+
ISIMIP3-
|
238 |
+
BASD
|
239 |
+
%
|
240 |
+
GAN
|
241 |
+
%
|
242 |
+
GAN
|
243 |
+
(unconst.)
|
244 |
+
%
|
245 |
+
GAN-
|
246 |
+
ISIMIP3-
|
247 |
+
BASD
|
248 |
+
%
|
249 |
+
Annual
|
250 |
+
-
|
251 |
+
0.535
|
252 |
+
0.217
|
253 |
+
59.4
|
254 |
+
0.328
|
255 |
+
38.7
|
256 |
+
0.265
|
257 |
+
50.5
|
258 |
+
0.195
|
259 |
+
63.6
|
260 |
+
DJF
|
261 |
+
-
|
262 |
+
0.634
|
263 |
+
0.321
|
264 |
+
49.4
|
265 |
+
0.395
|
266 |
+
37.7
|
267 |
+
0.371
|
268 |
+
41.5
|
269 |
+
0.308
|
270 |
+
51.4
|
271 |
+
MAM
|
272 |
+
-
|
273 |
+
0.722
|
274 |
+
0.314
|
275 |
+
56.5
|
276 |
+
0.419
|
277 |
+
42.0
|
278 |
+
0.378
|
279 |
+
47.6
|
280 |
+
0.285
|
281 |
+
60.5
|
282 |
+
JJA
|
283 |
+
-
|
284 |
+
0.743
|
285 |
+
0.289
|
286 |
+
61.1
|
287 |
+
0.451
|
288 |
+
39.3
|
289 |
+
0.357
|
290 |
+
52.0
|
291 |
+
0.280
|
292 |
+
62.3
|
293 |
+
SON
|
294 |
+
-
|
295 |
+
0.643
|
296 |
+
0.327
|
297 |
+
49.1
|
298 |
+
0.409
|
299 |
+
36.4
|
300 |
+
0.362
|
301 |
+
43.7
|
302 |
+
0.306
|
303 |
+
52.4
|
304 |
+
Annual
|
305 |
+
95th
|
306 |
+
2.264
|
307 |
+
1.073
|
308 |
+
52.6
|
309 |
+
1.415
|
310 |
+
37.5
|
311 |
+
1.213
|
312 |
+
46.4
|
313 |
+
0.945
|
314 |
+
58.3
|
315 |
+
DJF
|
316 |
+
95th
|
317 |
+
2.782
|
318 |
+
1.496
|
319 |
+
46.2
|
320 |
+
1.725
|
321 |
+
38.0
|
322 |
+
1.655
|
323 |
+
40.5
|
324 |
+
1.432
|
325 |
+
48.5
|
326 |
+
MAM
|
327 |
+
95th
|
328 |
+
2.948
|
329 |
+
1.482
|
330 |
+
49.7
|
331 |
+
1.805
|
332 |
+
38.8
|
333 |
+
1.661
|
334 |
+
43.7
|
335 |
+
1.337
|
336 |
+
54.6
|
337 |
+
JJA
|
338 |
+
95th
|
339 |
+
2.944
|
340 |
+
1.366
|
341 |
+
53.6
|
342 |
+
1.852
|
343 |
+
37.1
|
344 |
+
1.532
|
345 |
+
48.0
|
346 |
+
1.247
|
347 |
+
57.6
|
348 |
+
SON
|
349 |
+
95th
|
350 |
+
2.689
|
351 |
+
1.495
|
352 |
+
44.4
|
353 |
+
1.741
|
354 |
+
35.3
|
355 |
+
1.592
|
356 |
+
40.8
|
357 |
+
1.366
|
358 |
+
49.2
|
359 |
+
β5β
|
360 |
+
|
361 |
+
Figure
|
362 |
+
2: Bias in the long-term average precipitation over the entire test set between
|
363 |
+
the W5E5v2 ground truth (a) and GFDL-ESM4 (b), ISIMIP3BASD (c), GAN (d), uncon-
|
364 |
+
strained GAN (e) and the GAN-ISIMIP3BASD combination (f).
|
365 |
+
2.2 Spatial patterns
|
366 |
+
We compare the ability of the GAN to improve spatial patterns based on the W5E5v2
|
367 |
+
ground truth, against the GFDL-ESM4 simulations and the ISIMIP3BASD method applied
|
368 |
+
to the GFDL-ESM4 simulations. To model realistic precipitation ο¬elds, the characteristic
|
369 |
+
spatial intermittency needs to be captured accurately.
|
370 |
+
We compute the spatial power spectral density (PSD) of global precipitation ο¬elds,
|
371 |
+
averaged over the test set for each method. GFDL-ESM4 shows noticeable deviations from
|
372 |
+
W5E5v2 in the PSD (Fig. 4). Our GAN can correct these over the entire range of wave-
|
373 |
+
β6β
|
374 |
+
|
375 |
+
W5E5v2 mean [mm/d]
|
376 |
+
GFDL-ESM4
|
377 |
+
a
|
378 |
+
b
|
379 |
+
N.09
|
380 |
+
0Β°
|
381 |
+
S.09
|
382 |
+
0
|
383 |
+
ISIMIP3BASD
|
384 |
+
GAN
|
385 |
+
N.09
|
386 |
+
0Β°
|
387 |
+
60Β°S
|
388 |
+
GAN (unconstrained)
|
389 |
+
GAN-ISIMIP3BASD
|
390 |
+
e
|
391 |
+
f
|
392 |
+
N.09
|
393 |
+
0Β°
|
394 |
+
S.09
|
395 |
+
120Β°W
|
396 |
+
60Β°W
|
397 |
+
0
|
398 |
+
60Β°E
|
399 |
+
120Β°E
|
400 |
+
120Β°W
|
401 |
+
60Β°W
|
402 |
+
0Β°
|
403 |
+
60Β°E
|
404 |
+
120Β°E
|
405 |
+
7.5
|
406 |
+
-7.5 -5.0 -2.5
|
407 |
+
0.0
|
408 |
+
2.5
|
409 |
+
5.0
|
410 |
+
Bias [mm/d]80 S
|
411 |
+
60 S
|
412 |
+
40 S
|
413 |
+
20 S
|
414 |
+
0
|
415 |
+
20 N
|
416 |
+
40 N
|
417 |
+
60 N
|
418 |
+
80 N
|
419 |
+
Latitude
|
420 |
+
0
|
421 |
+
1
|
422 |
+
2
|
423 |
+
3
|
424 |
+
4
|
425 |
+
5
|
426 |
+
6
|
427 |
+
7
|
428 |
+
Mean precipitation [mm/d]
|
429 |
+
W5E5v2
|
430 |
+
GFDL-ESM4: MAE = 0.241
|
431 |
+
ISIMIP3BASD: MAE = 0.120
|
432 |
+
GAN: MAE = 0.226
|
433 |
+
GAN (unconstrained): MAE = 0.102
|
434 |
+
GAN-ISIMIP3BASD: MAE = 0.068
|
435 |
+
Figure
|
436 |
+
3: Precipitation averaged over longitudes and the entire test set period from the
|
437 |
+
W5E5v2 ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN
|
438 |
+
(cyan), unconstrained GAN (orange) and the GAN-ISIMIP3BASD combination (blue). To
|
439 |
+
quantify the diο¬erences between the shown lines, we show their mean absolute error w.r.t
|
440 |
+
the W5E5v2 ground truth in the legend. These values are diο¬erent from the ones shown in
|
441 |
+
Table 1 as the average is taken here over the longitudes without their absolute value. The
|
442 |
+
GAN-ISIMIP3BASD approach shows the lowest error.
|
443 |
+
lengths, closely matching the W5E5v2 ground truth. Improvements over ISIMIP3BASD
|
444 |
+
are especially pronounced in the range of high frequencies (low wavelengths), which are
|
445 |
+
responsible for the intermittent spatial variability of daily precipitation ο¬elds. Adding the
|
446 |
+
physical constraint to the GAN does not aο¬ect the ability to produce realistic PSD distribu-
|
447 |
+
tions. After applying ISIMIP3BASD to the GAN-processed ο¬elds, most of the improvements
|
448 |
+
generated by the GAN are retained, as shown by the GAN-ISIMIP3BASD results.
|
449 |
+
For a second way to quantifying how realistic the simulated and post-processed pre-
|
450 |
+
cipitation ο¬elds are, with a focus on high-frequency spatial intermittency, we investigate
|
451 |
+
the fractal dimension (Edgar & Edgar, 2008) of the lines separating grid cells with daily
|
452 |
+
rainfall sums above and below a given quantile threshold (see Methods). For a sample and
|
453 |
+
qualitative comparison of precipitation ο¬elds over the South American continent see Fig. S2.
|
454 |
+
The daily spatial precipitation ο¬elds are ο¬rst converted to binary images using a quantile
|
455 |
+
threshold. The respective quantiles are determined from the precipitation distribution over
|
456 |
+
the entire test set period and globe. The mean of the fractal dimension computed with box-
|
457 |
+
counting (see Methods) (Lovejoy et al., 1987; Meisel et al., 1992; Husain et al., 2021) for each
|
458 |
+
time slice is then investigated (Fig. 5). Both the GFDL-ESM4 simulations themselves and
|
459 |
+
the results of applying the ISIMIP3BASD post-processing to them exhibit spatial patterns
|
460 |
+
with a lower fractal dimension than the W5E5v2 ground truth, implying too low spatial
|
461 |
+
intermittency. In contrast, the GAN translates spatial ο¬elds simulated by GFDL-ESM4 in
|
462 |
+
a way that results in closely matching fractal dimensions over the entire range of quantiles.
|
463 |
+
3 Discussion
|
464 |
+
Postprocessing climate projections is a fundamentally diο¬erent task from postprocessing
|
465 |
+
weather forecast simulations (Hess et al., 2022). In the latter case, data-driven postprocess-
|
466 |
+
ing methods, e.g. based on deep learning, to minimize diο¬erences between paired samples
|
467 |
+
β7β
|
468 |
+
|
469 |
+
128
|
470 |
+
256
|
471 |
+
512
|
472 |
+
1024
|
473 |
+
2048
|
474 |
+
4096
|
475 |
+
8192
|
476 |
+
Wavelength [km]
|
477 |
+
10
|
478 |
+
6
|
479 |
+
10
|
480 |
+
5
|
481 |
+
10
|
482 |
+
4
|
483 |
+
10
|
484 |
+
3
|
485 |
+
10
|
486 |
+
2
|
487 |
+
PSD [a.u]
|
488 |
+
W5E5v2
|
489 |
+
GFDL-ESM4
|
490 |
+
ISIMIP3BASD
|
491 |
+
GAN
|
492 |
+
GAN (unconstrained)
|
493 |
+
GAN-ISIMIP3BASD
|
494 |
+
Figure 4: The power spectral density (PSD) of the spatial precipitation ο¬elds is shown as
|
495 |
+
an average over all samples in the test set for the W5E5v2 ground truth (black) and GFDL-
|
496 |
+
ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan, dashed), unconstrained GAN (orange,
|
497 |
+
dashed-dotted) and the constrained-GAN-ISIMIP3BASD combination (blue, dotted). The
|
498 |
+
GANs and W5E5v2 ground truth agree so closely that they are indistinguishable. In contrast
|
499 |
+
to ISIMIP3BASD, the GAN can correct the intermittent spectrum accurately over the entire
|
500 |
+
range down to the smallest wavelengths.
|
501 |
+
of variables such as spatial precipitation ο¬elds (Hess & Boers, 2022). Beyond time scales of
|
502 |
+
a few days, however, the chaotic nature of the atmosphere leads to exponentially diverging
|
503 |
+
trajectories, and for climate or Earth system model output there is no observation-based
|
504 |
+
ground truth to directly compare to. We therefore frame the post-processing of ESM projec-
|
505 |
+
tions, with applications for subsequent 195 impact modelling in mind, as an image-to-image
|
506 |
+
translation task with unpaired samples.
|
507 |
+
To this end we apply a recently developed postprocessing method based on physically
|
508 |
+
constrained CycleGANs to global simulations of a state-of-the-art, high-resolution ESM
|
509 |
+
from the CMIP6 model ensemble, namely the GFDL-ESM4 (Krasting et al., 2018; O' Neill
|
510 |
+
et al., 2016). We evaluate our method against the gold-standard bias correction framework
|
511 |
+
ISIMIP3BASD. Our model can be trained on unpaired samples that are characteristic for
|
512 |
+
climate simulations. It is able to correct the ESM simulations in two regards: temporal
|
513 |
+
distributions over long time scales, including extremes in the distrivutionsβ tails, as well
|
514 |
+
as spatial patterns of individual global snap shots of the model output. The latter is not
|
515 |
+
possible with established methods.
|
516 |
+
Our GAN-based approach is designed as a general
|
517 |
+
framework that can be readily applied to diο¬erent ESMs and observational target datasets.
|
518 |
+
This is in contrast to existing bias-adjustment methods that are often tailored to speciο¬c
|
519 |
+
applications.
|
520 |
+
We chose to correct precipitation because it is arguably one of the hardest variables
|
521 |
+
to represent accurately in ESMs. So far, GANs have only been applied to regional studies
|
522 |
+
or low-resolution global ESMs (FranΒΈcois et al., 2021; Hess et al., 2022). The GFDL-ESM4
|
523 |
+
model simulations are hence chosen in order to test if our CycleGAN approach would lead
|
524 |
+
β8β
|
525 |
+
|
526 |
+
0.4
|
527 |
+
0.5
|
528 |
+
0.6
|
529 |
+
0.7
|
530 |
+
0.8
|
531 |
+
0.9
|
532 |
+
Quantile
|
533 |
+
1.3
|
534 |
+
1.4
|
535 |
+
1.5
|
536 |
+
1.6
|
537 |
+
1.7
|
538 |
+
Fractal dimension
|
539 |
+
W5E5v2
|
540 |
+
GFDL-ESM4: MAE = 0.048
|
541 |
+
ISIMIP3BASD: MAE = 0.037
|
542 |
+
GAN: MAE = 0.002
|
543 |
+
GAN (unconstrained): MAE = 0.002
|
544 |
+
GAN-ISIMIP3BASD: MAE = 0.004
|
545 |
+
Figure 5: The fractal dimension (see Methods) of binary global precipitation ο¬elds is com-
|
546 |
+
pared as averages for diο¬erent quantile thresholds.
|
547 |
+
Results are shown for the W5E5v2
|
548 |
+
ground truth (black) and GFDL-ESM4 (red), ISIMIP3BASD (magenta), GAN (cyan), un-
|
549 |
+
constrained GAN (orange, dashed), and the GAN-ISIMIP3BASD combination (blue). The
|
550 |
+
GAN can accurately reproduce the fractal dimension of the W5E5v2 ground truth spatial
|
551 |
+
precipitation ο¬elds over all quantile thresholds, clearly outperforming the ISIMIP3BASD
|
552 |
+
basline.
|
553 |
+
to improvements even when postprocessing global high-resolution simulations of one of the
|
554 |
+
most complex and sophisticated ESMs to date. In the same spirit, we evaluate our ap-
|
555 |
+
proach against a very strong baseline given by the state-of-the-art bias correction framework
|
556 |
+
ISIMIP3BASD, which is based on a trend-preserving QM method (Lange, 2019).
|
557 |
+
Comparing long-term summary statistics, our method yields histograms of relative pre-
|
558 |
+
cipitation frequencies that very closely agree with corresponding histograms from reanalysis
|
559 |
+
data (Fig. 1). The means that the extremes in the far end of the tail are accurately cap-
|
560 |
+
tured, with similar skill to the ISIMIP3BASD baseline that is mainly designed for this task.
|
561 |
+
Diο¬erences in the grid cell-wise long-term average show that the GAN skillfully reduces bi-
|
562 |
+
ases (Fig. 2); in particular, the often reported double-peaked ITCZ bias of the GFDL-ESM4
|
563 |
+
simulations, which is a common feature of most climate models (Tian & Dong, 2020), is
|
564 |
+
strongly reduced (Fig. 3). The ISIMIP3BASD method - being speciο¬cally designed for this
|
565 |
+
- produces slightly lower biases for grid-cell-wise averages than the GAN; we show that
|
566 |
+
combining both methods by ο¬rst applying the GAN and then the ISIMIP3BASD procedure
|
567 |
+
leads to the overall best performance.
|
568 |
+
Regarding the correction of spatial patterns of the modelled precipitation ο¬elds, we
|
569 |
+
compare the spectral density and fractal dimensions of the spatial precipitation ο¬elds. Our
|
570 |
+
results show that indeed only the GAN can capture the characteristic spatial intermittency
|
571 |
+
of precipitation closely (Figs. 4 and 5). We believe that the measure of fractal dimension
|
572 |
+
is also relevant for other ο¬elds such as nowcasting and medium-range weather forecasting,
|
573 |
+
where blurriness in deep learning-based predictions is often reported (Ravuri et al., 2021)
|
574 |
+
and needs to be further quantiο¬ed.
|
575 |
+
β9β
|
576 |
+
|
577 |
+
Post-processing methods for climate projections have to be able to preserve the trends
|
578 |
+
that result from the non-stationary dynamics of the Earth system on long-time scales. We
|
579 |
+
have therefore introduced the architecture constraint of preserving the global precipitation
|
580 |
+
amount on every day in the climate model output (Hess et al., 2022). We ο¬nd that this does
|
581 |
+
not aο¬ect the quality of the spatial patterns that are produced by our CycleGAN method.
|
582 |
+
However, the skill of correcting mean error biases is slightly reduced by the constraint. This
|
583 |
+
can be expected in part as the constraint is constructed to follow the global mean of the
|
584 |
+
ESM. Hence, biases in the global ESM mean can inο¬uence the constrained GAN. This also
|
585 |
+
motivates our choice to demonstrate the combination of the constrained GAN with the QM-
|
586 |
+
based ISIMIP3BASD procedure, since it can be applied to future climate scenarios, making
|
587 |
+
it more suitable for actual applications than the unconstrained architecture.
|
588 |
+
There are several directions to further develop or approach. The architecture employed
|
589 |
+
here has been built for equally spaced two-dimensional images. Extending the CycleGAN
|
590 |
+
architecture to perform convolutions on the spherical surface, e.g. using graph neural net-
|
591 |
+
works, might lead to more eο¬cient and accurate models. Moreover, GANs are comparably
|
592 |
+
diο¬cult to train, which could make it challenging to identify suitable network architectures.
|
593 |
+
Using large ensembles of climate simulations could provide additional training data that
|
594 |
+
could further improve the performance. Another straightforward extension of our method
|
595 |
+
would be the inclusion of further input variables or the prediction additional high-impact
|
596 |
+
physical variables, such as near-surface temperatures that are also important for regional
|
597 |
+
impact models.
|
598 |
+
4 Methods
|
599 |
+
4.1 Training data
|
600 |
+
We use global ο¬elds of daily precipitation with a horizontal resolution of 1β¦ from the
|
601 |
+
GFDL-ESM4 Earth system model (Krasting et al., 2018) and the W5E5v2 reanalysis prod-
|
602 |
+
uct (Cucchi et al., 2020; WFDE5 over land merged with ERA5 over the ocean (W5E5 v2.0),
|
603 |
+
2021) as observation-based ground truth.
|
604 |
+
The W5E5v2 dataset is based on the ERA5
|
605 |
+
(Hersbach et al., 2020) reanalysis and has been bias-adjusted using the Global Precipitation
|
606 |
+
Climatology Centre (GPCC) full data monthly product v2020 (Schneider et al., 2011) over
|
607 |
+
land and the Global Precipitation Climatology Project (GPCP) v2.3 dataset (Huο¬man et
|
608 |
+
al., 1997) over the ocean. Both datasets have been regridded to the same 1β¦ horizontal
|
609 |
+
resolution using bilinear interpolation following (Beck et al., 2019). We split the dataset
|
610 |
+
into three periods for training (1950-2000), validation (2001-2003), and testing (2004-2014).
|
611 |
+
This corresponds to 8030 samples for training, 1095 for validation, and 4015 for testing.
|
612 |
+
During pre-processing, the training data is log-transformed with Λx = log(x+Ο΅)βlog(Ο΅) with
|
613 |
+
Ο΅ = 0.0001, following Rasp and Thuerey (2021), to account for zeros in the transform. The
|
614 |
+
data is then normalized to the interval [β1, 1] following (Zhu et al., 2017).
|
615 |
+
4.2 Cycle-consistent generative adversarial networks
|
616 |
+
This section gives a brief overview of the CycleGAN used in this study. We refer to
|
617 |
+
(Zhu et al., 2017; Hess et al., 2022) for a more comprehensive description and discussion.
|
618 |
+
Generative adversarial networks learn to generate images that are nearly indistinguishable
|
619 |
+
from real-world examples through a two-player game (Goodfellow et al., 2014).
|
620 |
+
In this
|
621 |
+
set-up, a ο¬rst network G, the so-called generator, produces images with the objective to
|
622 |
+
fool a second network D, the discriminator, which has to classify whether a given sample
|
623 |
+
is generated (βfakeβ) or drawn from a real-world dataset (βrealβ). Mathematically this can
|
624 |
+
be formalized as
|
625 |
+
Gβ = min
|
626 |
+
G
|
627 |
+
max
|
628 |
+
D
|
629 |
+
LGAN(D, G),
|
630 |
+
(1)
|
631 |
+
β10β
|
632 |
+
|
633 |
+
with Gβ being the optimal generator network. The loss function LGAN(D, G) can be deο¬ned
|
634 |
+
as
|
635 |
+
LGAN(D, G) = EyβΌpy(y)[log(D(y))] + ExβΌpx(x)[log(1 β D(G(x)))],
|
636 |
+
(2)
|
637 |
+
where py(y) is the distribution of the real-world target data and samples from px(x) are
|
638 |
+
used as inputs by G to produce realistic images. The CycleGAN (Zhu et al., 2017) consists
|
639 |
+
of two generator-discriminator pairs, where the generators G and F learn inverse mappings
|
640 |
+
between two domains X and Y . This allows to deο¬ne an additional cycle-consistency loss
|
641 |
+
that constraints the training of the networks, i.e.
|
642 |
+
Lcycle(G, F) = ExβΌpx(x)[||F(G(x)) β x||1]
|
643 |
+
(3)
|
644 |
+
+ EyβΌpy(y)[||G(F(y)) β y||1].
|
645 |
+
It measures the error caused by a translation cycle of an image to the other domain and
|
646 |
+
back. Further, an additional loss term is introduced to regularize the networks to be close
|
647 |
+
to an identity mapping with,
|
648 |
+
Lident(G, F) = ExβΌpx(x)[||G(x) β x||1]
|
649 |
+
(4)
|
650 |
+
+ EyβΌpy(y)[||F(y) β y||1].
|
651 |
+
In practice, the log-likelihood loss can be replaced by a mean squared error loss to facilitate
|
652 |
+
a more stable training.
|
653 |
+
Further, the generator loss is reformulated to be minimized by
|
654 |
+
inverting the labels, i.e.
|
655 |
+
LGenerator = ExβΌpx(x)[(DX(G(x)) β 1)2]
|
656 |
+
+ EyβΌpy(y)[(DY (F(y)) β 1)2]
|
657 |
+
(5)
|
658 |
+
+ Ξ»Lcycle(G, F) + ΛΞ»Lident(G, F),
|
659 |
+
where Ξ» and ΛΞ» are set to 10 and 5 respectively following (Zhu et al., 2017). The corresponding
|
660 |
+
loss term for the discriminator networks is given by
|
661 |
+
LDiscriminator = EyβΌpy(y)[(DY (y) β 1)2] + ExβΌpx(x)[(DX(G(x)))2]
|
662 |
+
(6)
|
663 |
+
+ ExβΌpx(x)[(DX(x) β 1)2] + EyβΌpy(y)[(DY (F(y)))2].
|
664 |
+
(7)
|
665 |
+
The weights of the generator and discriminator networks are then optimized with the ADAM
|
666 |
+
(Kingma & Ba, 2014) optimizer using a learning rate of 2eβ4 and updated in an alternating
|
667 |
+
fashion. We train the network for 350 epochs and a batch size of 1, saving model checkpoints
|
668 |
+
every other epoch. We evaluate the checkpoints on the validation dataset to determine the
|
669 |
+
best model instance.
|
670 |
+
4.3 Network Architectures
|
671 |
+
Both the generator and discriminator have fully convolutional architectures. The gen-
|
672 |
+
erator uses ReLU activation functions, instance normalization, and reο¬ection padding. The
|
673 |
+
discriminator uses leaky ReLU activations with slope 0.2 instead, together with instance
|
674 |
+
normalization. For a more detailed description, we refer to our previous study (Hess et al.,
|
675 |
+
2022). The network architectures in this study are the same, only with a change in the
|
676 |
+
number of residual layers in the generator network from 6 to 7.
|
677 |
+
The ο¬nal layer of the generator can be constrained to preserve the global sum of the
|
678 |
+
input, i.e. by rescaling
|
679 |
+
Λyi = yi
|
680 |
+
οΏ½Ngrid
|
681 |
+
i
|
682 |
+
xi
|
683 |
+
οΏ½Ngrid
|
684 |
+
i
|
685 |
+
yi
|
686 |
+
,
|
687 |
+
(8)
|
688 |
+
β11β
|
689 |
+
|
690 |
+
where xi and yi are grid cell values of the generator input and output respectively and
|
691 |
+
Ngrid is the number of grid cells. The generator without this constraint will be referred
|
692 |
+
to as unconstrained in this study. The global physical constraint enforces that the global
|
693 |
+
daily precipitation sum is not aο¬ected by the CycleGAN postprocessing and hence remains
|
694 |
+
identical to the original value from the GFDL-ESM4 simualtions. This is motivated by the
|
695 |
+
observation that large-scale average trends in precipitation follow the Clausius-Clapeyron
|
696 |
+
relation (Traxl et al., 2021), which is based on thermodynamic relations and hence can be
|
697 |
+
expected to be modelled well in GFDL-ESM4.
|
698 |
+
4.4 Quantile mapping-based bias adjustment
|
699 |
+
We compare the performance of our GAN-based method to the bias adjustment method
|
700 |
+
ISMIP3BASD v3.0.1 (Lange, 2019, 2022) that has been developed for phase 3 of the Inter-
|
701 |
+
Sectoral Impact Model Intercomparison Project (Warszawski et al., 2014; Frieler et al.,
|
702 |
+
2017). This state-of-the-art bias-adjustment method is based on a trend-preserving quantile
|
703 |
+
mapping (QM) framework. It represents a very strong baseline for comparison as it has
|
704 |
+
been developed prior to this study and used not only in ISIMIP3 but also to prepare many
|
705 |
+
of the climate projections that went into the Interactive Atlas produced as part of the 6th
|
706 |
+
assessment report of working group 1 of the Intergovernmental Panel on Climate Change
|
707 |
+
(IPCC, https://interactive-atlas.ipcc.ch/). In QM, a transformation between the cumulative
|
708 |
+
distribution functions (CDFs) of the historical simulation and observations is ο¬tted and then
|
709 |
+
applied to future simulations. The CDFs can either be empirical or parametric, the latter
|
710 |
+
being a Bernoulli-gamma distribution for the precipitation in this study. The CFDs are
|
711 |
+
ο¬tted and mapped for each grid cell and day of the year separately. For bias-adjusting the
|
712 |
+
GFDL-ESM4 simulation, parametric QM was found to give the best results, while empirical
|
713 |
+
CDFs are used in combination with the GAN.
|
714 |
+
To evaluate the methods in this study we deο¬ne the grid cell-wise bias as the diο¬erence
|
715 |
+
in long-term averages as,
|
716 |
+
Bias(Λy, y) = 1
|
717 |
+
T
|
718 |
+
T
|
719 |
+
οΏ½
|
720 |
+
t=1
|
721 |
+
Λyt β 1
|
722 |
+
T
|
723 |
+
T
|
724 |
+
οΏ½
|
725 |
+
t=1
|
726 |
+
yt,
|
727 |
+
(9)
|
728 |
+
where T is the number of time steps, Λyt and Λyt the modelled and observed precipitation
|
729 |
+
respectively at time step t.
|
730 |
+
4.5 Evaluating spatial patterns
|
731 |
+
Quantifying how realistic spatial precipitation ο¬elds are is an ongoing research question
|
732 |
+
in itself, which has become more important with the application of deep learning to weather
|
733 |
+
forecasting and post-processing. In these applications, neural networks often achieve error
|
734 |
+
statistics and skill scores competitive with physical models, while the output ο¬elds can
|
735 |
+
at the same time show unphysical characteristics, such as blurring or excessive smoothing.
|
736 |
+
Ravuri et al. (2021) compare the spatial intermittency, which is characteristic of precipitation
|
737 |
+
ο¬elds, using the power spectral density (PSD) computed from the spatial ο¬elds; in the latter
|
738 |
+
study, the PSD-based quantiο¬cation was complemented by interviews with a large number
|
739 |
+
of meteorological experts. We propose the fractal dimension of binary precipitation ο¬elds
|
740 |
+
as an alternative to quantify how realistic the patterns are.
|
741 |
+
We compute the fractal dimension via the box-counting algorithm (Lovejoy et al., 1987;
|
742 |
+
Meisel et al., 1992). It quantiο¬es how spatial patterns, for example coastlines (Husain et
|
743 |
+
al., 2021), change with the scale of measurement. The box-counting algorithm divides the
|
744 |
+
image into squares and counts the number of squares that cover the binary pattern of
|
745 |
+
interest, Nsquares. The size of the squares, i.e. the scale of measurement, is then reduced
|
746 |
+
iteratively by a factor s. The fractal dimension Dfractal can then be determined from the
|
747 |
+
slope of the resulting log-log scaling, i.e.,
|
748 |
+
β12β
|
749 |
+
|
750 |
+
Dfractal = log(Nsquares)
|
751 |
+
log(s)
|
752 |
+
.
|
753 |
+
(10)
|
754 |
+
Competing interests
|
755 |
+
The authors declare no competing interests.
|
756 |
+
Data availability
|
757 |
+
The W5E5 data is available for download at https://doi.org/10.48364/ISIMIP.342217.
|
758 |
+
The GFDL-ESM4 data can be downloaded at https://esgf-node.llnl.gov/projects/
|
759 |
+
cmip6/.
|
760 |
+
Code availability
|
761 |
+
The Python code for processing and analysing the data, together with the PyTorch
|
762 |
+
Lightning (Falcon et al., 2019) code is available at https://github.com/p-hss/earth
|
763 |
+
system model gan bias correction.git. The ISIMIP3BASD code in (Lange, 2022) is
|
764 |
+
used for this study.
|
765 |
+
Acknowledgments
|
766 |
+
NB and PH acknowledge funding by the Volkswagen Foundation, as well as the European
|
767 |
+
Regional Development Fund (ERDF), the German Federal Ministry of Education and Re-
|
768 |
+
search and the Land Brandenburg for supporting this project by providing resources on the
|
769 |
+
high performance computer system at the Potsdam Institute for Climate Impact Research.
|
770 |
+
N.B. acknowledges funding by the European Unionβs Horizon 2020 research and innovation
|
771 |
+
programme under grant agreement No 820970 and under the Marie Sklodowska-Curie grant
|
772 |
+
agreement No. 956170, as well as from the Federal Ministry of Education and Research
|
773 |
+
under grant No. 01LS2001A. SL acknowledges funding from the European Unionβs Horizon
|
774 |
+
2022 research and innovation programme under grant agreement no. 101081193 Optimal
|
775 |
+
High Resolution Earth System Models for Exploring Future Climate Changes (OptimESM).
|
776 |
+
References
|
777 |
+
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788 |
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|
789 |
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|
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Cucchi, M., Weedon, G. P., Amici, A., Bellouin, N., Lange, S., M¨uller Schmied, H.,
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. . . Buontempo, C. (2020). Wfde5: bias-adjusted era5 reanalysis data for impact
|
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|
793 |
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essd.copernicus.org/articles/12/2097/2020/ doi: 10.5194/essd-12-2097-2020
|
794 |
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β13β
|
795 |
+
|
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Supporting Information for βDeep Learning for
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bias-correcting comprehensive high-resolution Earth
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system modelsβ
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Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3
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1Earth System Modelling, School of Engineering & Design, Technical University of Munich, Munich, Germany
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2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
|
937 |
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3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK
|
938 |
+
Contents of this ο¬le
|
939 |
+
1. Figure S1 to S2
|
940 |
+
January 4, 2023, 1:28am
|
941 |
+
arXiv:2301.01253v1 [physics.ao-ph] 16 Dec 2022
|
942 |
+
|
943 |
+
X - 2
|
944 |
+
:
|
945 |
+
Figure S1.
|
946 |
+
Bias maps as in Fig.
|
947 |
+
2 but with the 95th percentile instead of the mean.
|
948 |
+
Global mean absolute errors (MAEs) are given in the respective titles. Combining the GAN with
|
949 |
+
ISIMIP3BASD achieves the lowest error compared to the other methods.
|
950 |
+
January 4, 2023, 1:28am
|
951 |
+
|
952 |
+
W5E5v2 95th percentile [mm/d]
|
953 |
+
GFDL-ESM4: MAE = 2.264
|
954 |
+
b
|
955 |
+
a
|
956 |
+
N.09
|
957 |
+
0Β°
|
958 |
+
S.09
|
959 |
+
0
|
960 |
+
25
|
961 |
+
50
|
962 |
+
120Β°W
|
963 |
+
60Β°W
|
964 |
+
0Β°
|
965 |
+
60Β°E
|
966 |
+
120Β°E
|
967 |
+
120Β°W
|
968 |
+
60Β°W
|
969 |
+
0Β°
|
970 |
+
60Β°E
|
971 |
+
120Β°E
|
972 |
+
ISIMIP3BASD: MAE = 1.073
|
973 |
+
GAN: MAE =
|
974 |
+
1.415
|
975 |
+
d
|
976 |
+
N.09
|
977 |
+
0Β°
|
978 |
+
S.09
|
979 |
+
120Β°W
|
980 |
+
60Β°W
|
981 |
+
.0
|
982 |
+
60Β°E
|
983 |
+
120Β°E
|
984 |
+
60Β°W
|
985 |
+
0Β°
|
986 |
+
60Β°E
|
987 |
+
120Β°W
|
988 |
+
120Β°E
|
989 |
+
GAN (unconstrained): MAE
|
990 |
+
1.213
|
991 |
+
GAN-ISIMIP3BASD: MAE
|
992 |
+
0.945
|
993 |
+
=
|
994 |
+
e
|
995 |
+
2
|
996 |
+
120Β°W
|
997 |
+
60Β°W
|
998 |
+
0Β°
|
999 |
+
60Β°E
|
1000 |
+
120Β°E
|
1001 |
+
120Β°W
|
1002 |
+
60Β°W
|
1003 |
+
0Β°
|
1004 |
+
60Β°E
|
1005 |
+
120Β°E
|
1006 |
+
-20 -i5 -i0 -5γ0
|
1007 |
+
5
|
1008 |
+
10
|
1009 |
+
15
|
1010 |
+
20
|
1011 |
+
Differences in the 95th percentile [mm/d]:
|
1012 |
+
X - 3
|
1013 |
+
a
|
1014 |
+
50Β°S
|
1015 |
+
25Β°S
|
1016 |
+
0Β°
|
1017 |
+
100Β°W
|
1018 |
+
75Β°W
|
1019 |
+
50Β°W
|
1020 |
+
25Β°W
|
1021 |
+
W5E5v2
|
1022 |
+
c
|
1023 |
+
50Β°S
|
1024 |
+
25Β°S
|
1025 |
+
0Β°
|
1026 |
+
100Β°W
|
1027 |
+
75Β°W
|
1028 |
+
50Β°W
|
1029 |
+
25Β°W
|
1030 |
+
ISIMIP3BASD
|
1031 |
+
b
|
1032 |
+
50Β°S
|
1033 |
+
25Β°S
|
1034 |
+
0Β°
|
1035 |
+
100Β°W
|
1036 |
+
75Β°W
|
1037 |
+
50Β°W
|
1038 |
+
25Β°W
|
1039 |
+
GFDL-ESM4
|
1040 |
+
d
|
1041 |
+
50Β°S
|
1042 |
+
25Β°S
|
1043 |
+
0Β°
|
1044 |
+
100Β°W
|
1045 |
+
75Β°W
|
1046 |
+
50Β°W
|
1047 |
+
25Β°W
|
1048 |
+
GAN-ISIMIP3BASD
|
1049 |
+
5
|
1050 |
+
10
|
1051 |
+
15
|
1052 |
+
20
|
1053 |
+
25
|
1054 |
+
30
|
1055 |
+
35
|
1056 |
+
Precipitation [mm/d]
|
1057 |
+
Figure S2.
|
1058 |
+
Qualitative comparison of precipitation ο¬elds at the same date (December 21st
|
1059 |
+
2014) over the South American continent. The region is used for a comparison of the fractal
|
1060 |
+
dimension in binary precipitation patterns.
|
1061 |
+
January 4, 2023, 1:28am
|
1062 |
+
|
CNAzT4oBgHgl3EQfTvwq/content/tmp_files/load_file.txt
ADDED
The diff for this file is too large to render.
See raw diff
|
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ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt
ADDED
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1 |
+
arXiv:2301.00085v1 [math.CO] 31 Dec 2022
|
2 |
+
On the chromatic number of random regular
|
3 |
+
hypergraphs
|
4 |
+
Patrick Bennettβ
|
5 |
+
Department of Mathematics,
|
6 |
+
Western Michigan University
|
7 |
+
Kalamazoo MI 49008
|
8 |
+
Alan Friezeβ
|
9 |
+
Department of Mathematical Sciences,
|
10 |
+
Carnegie Mellon University,
|
11 |
+
Pittsburgh PA 15213.
|
12 |
+
Abstract
|
13 |
+
We estimate the likely values of the chromatic and independence numbers of the
|
14 |
+
random r-uniform d-regular hypergraph on n vertices for ο¬xed r, large ο¬xed d, and
|
15 |
+
n β β.
|
16 |
+
1
|
17 |
+
Introduction
|
18 |
+
The study of the chromatic number of random graphs has a long history. It begins with the
|
19 |
+
work of BollobΒ΄as and ErdΛos [6] and Grimmett and McDiarmid [13] who determined Ο(Gn,p),
|
20 |
+
p constant to within a factor 2, w.h.p. Matula [17] reduced this to a factor of 3/2. Then
|
21 |
+
we have the discovery of martingale concentration inequalities by Shamir and Spencer [18]
|
22 |
+
leading to the breakthrough by BollobΒ΄as [5] who determined Ο(Gn,p) asymptotically for p
|
23 |
+
constant.
|
24 |
+
The case of p β 0 proved a little more tricky, but οΏ½Luczak [15] using ideas from Frieze [10]
|
25 |
+
and [17] determined Ο(Gn,p), p = c/n asymptotically for large c. οΏ½Luczak [16] showed that
|
26 |
+
w.h.p. Ο(Gn,p), p = c/n took one of two values. It was then that the surprising power of
|
27 |
+
the second moment method was unleashed by Achlioptas and Naor [3]. Since then there has
|
28 |
+
been much work tightening our estimates for the k-colorability threshold, k β₯ 3 constant.
|
29 |
+
See for example Coja-Oghlan [7].
|
30 |
+
Random regular graphs of low degree were studied algorithmically by several authors e.g.
|
31 |
+
Achlioptas and Molloy [2] and by Shi and Wormald [19]. Frieze and οΏ½Luczak [12] introduced
|
32 |
+
βResearch supported in part by Simons Foundation Grant #426894.
|
33 |
+
β Research supported in part by NSF Grant DMS1661063
|
34 |
+
1
|
35 |
+
|
36 |
+
a way of using our knowledge of Ο(Gn,p), p = c/n to tackle Ο(Gn,r) where Gn,r denotes a
|
37 |
+
random r-regular graph and where p = r/n. Subsequently Achlioptas and Moore [2] showed
|
38 |
+
via the second moment method that w.h.p. Ο(Gn,r) was one of 3 values. This was tightened
|
39 |
+
basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8].
|
40 |
+
For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic
|
41 |
+
number for Ο(Hr(n, p) for
|
42 |
+
οΏ½nβ1
|
43 |
+
rβ1
|
44 |
+
οΏ½
|
45 |
+
p suο¬ciently large. Here Hr(n, p) is the binomial r-uniform
|
46 |
+
hypergraph where each of the
|
47 |
+
οΏ½n
|
48 |
+
r
|
49 |
+
οΏ½
|
50 |
+
possible edges is included with probability p. There are
|
51 |
+
several possibilities of a proper coloring of the vertices of a hypergraph. Here we concentrate
|
52 |
+
on the case where a vertex coloring is proper if no edge contains vertices of all the same color.
|
53 |
+
Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that
|
54 |
+
w.h.p. Ο(Hr(n, p) took one or two values. When it comes to what ew denote by Ο(Hr(n, d),
|
55 |
+
a random d-regular, r-uniform hypergraph, we are not aware of any results at all. In this
|
56 |
+
paper we extend the approach of [12] to this case:
|
57 |
+
Theorem 1. For all ο¬xed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such that for any ο¬xed
|
58 |
+
d β₯ d0 we have that w.h.p.
|
59 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
60 |
+
Ο(Hr(n, d)) β
|
61 |
+
οΏ½
|
62 |
+
(rβ1)d
|
63 |
+
r log d
|
64 |
+
οΏ½
|
65 |
+
1
|
66 |
+
rβ1
|
67 |
+
οΏ½
|
68 |
+
(rβ1)d
|
69 |
+
r log d
|
70 |
+
οΏ½
|
71 |
+
1
|
72 |
+
rβ1
|
73 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
74 |
+
β€ Ξ΅,
|
75 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
76 |
+
Ξ±(Hr(n, d)) β
|
77 |
+
οΏ½
|
78 |
+
r log d
|
79 |
+
(rβ1)d
|
80 |
+
οΏ½
|
81 |
+
1
|
82 |
+
rβ1 n
|
83 |
+
οΏ½
|
84 |
+
r log d
|
85 |
+
(rβ1)d
|
86 |
+
οΏ½
|
87 |
+
1
|
88 |
+
rβ1 n
|
89 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
90 |
+
β€ Ξ΅
|
91 |
+
(1)
|
92 |
+
Here Ξ± refers to the independence number of a hypergraph.
|
93 |
+
2
|
94 |
+
Preliminaries
|
95 |
+
2.1
|
96 |
+
Tools
|
97 |
+
We will be using the following forms of Chernoο¬βs bound (see, e.g., [11]).
|
98 |
+
Lemma 2 (Chernoο¬ bound). Let X βΌ Bin(n, p). Then for all 0 < Ξ» < np
|
99 |
+
P(|X β np| β₯ Ξ») β€ 2 exp
|
100 |
+
οΏ½
|
101 |
+
β Ξ»2
|
102 |
+
3np
|
103 |
+
οΏ½
|
104 |
+
.
|
105 |
+
(2)
|
106 |
+
Lemma 3 (McDiarmidβs inequality). Let X = f(βZ) where βZ = (Z1, . . . Zt) and the Zi are
|
107 |
+
independent random variables. Assume the function f has the property that whenever βz, βw
|
108 |
+
diο¬er in only one coordinate we have |f(βz) β f(βw)| β€ c. Then for all Ξ» > 0 we have
|
109 |
+
P(|X β E[X]| β₯ Ξ») β€ 2 exp
|
110 |
+
οΏ½
|
111 |
+
β Ξ»2
|
112 |
+
2c2t
|
113 |
+
οΏ½
|
114 |
+
.
|
115 |
+
(3)
|
116 |
+
Bal and the ο¬rst author [4] showed the following.
|
117 |
+
2
|
118 |
+
|
119 |
+
Theorem 4 (Claim 4.2 in [4]). Fix r β₯ 3, d β₯ 2, and 0 < c < rβ1
|
120 |
+
r . Let z2 be the unique
|
121 |
+
positive number such that
|
122 |
+
z2
|
123 |
+
οΏ½
|
124 |
+
(z2 + 1)rβ1 β zrβ1
|
125 |
+
2
|
126 |
+
οΏ½
|
127 |
+
(z2 + 1)r β zr
|
128 |
+
2
|
129 |
+
= c
|
130 |
+
(4)
|
131 |
+
and let
|
132 |
+
z1 =
|
133 |
+
d
|
134 |
+
r [(z2 + 1)r β zr
|
135 |
+
2].
|
136 |
+
(5)
|
137 |
+
Let h(x) = x log x. If it is the case that
|
138 |
+
h
|
139 |
+
οΏ½d
|
140 |
+
r
|
141 |
+
οΏ½
|
142 |
+
+ h(dc) + h(d(1 β c)) β h(c) β h(1 β c) β h(d) β d
|
143 |
+
r log z1 β dc log z2 < 0
|
144 |
+
(6)
|
145 |
+
then w.h.p. Ξ±(Hr(n, d)) < cn.
|
146 |
+
Krivelevich and Sudakov [14] proved the following.
|
147 |
+
Theorem 5 (Theorem 5.1 in [14]). For all ο¬xed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such
|
148 |
+
that whenever D = D(p) :=
|
149 |
+
οΏ½nβ1
|
150 |
+
rβ1
|
151 |
+
οΏ½
|
152 |
+
p β₯ d0 we have that
|
153 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
154 |
+
Ο(Hr(n, p)) β
|
155 |
+
οΏ½
|
156 |
+
(rβ1)D
|
157 |
+
r log D
|
158 |
+
οΏ½
|
159 |
+
1
|
160 |
+
rβ1
|
161 |
+
οΏ½
|
162 |
+
(rβ1)D
|
163 |
+
r log D
|
164 |
+
οΏ½
|
165 |
+
1
|
166 |
+
rβ1
|
167 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
168 |
+
β€ Ξ΅,
|
169 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
170 |
+
Ξ±(Hr(n, p)) β
|
171 |
+
οΏ½
|
172 |
+
r log D
|
173 |
+
(rβ1)D
|
174 |
+
οΏ½
|
175 |
+
1
|
176 |
+
rβ1 n
|
177 |
+
οΏ½
|
178 |
+
r log D
|
179 |
+
(rβ1)D
|
180 |
+
οΏ½
|
181 |
+
1
|
182 |
+
rβ1 n
|
183 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
184 |
+
β€ Ξ΅
|
185 |
+
with probability at least 1 οΏ½οΏ½οΏ½ o(1/n).
|
186 |
+
3
|
187 |
+
Proof
|
188 |
+
In this section we prove Theorem 1. First we give an overview. We show in Subsection 3.1
|
189 |
+
that the upper bound on Ξ± follows from Theorem 4 and some straightforward calculations.
|
190 |
+
Then the lower bound on Ο follows as well. Thus we will be done once we prove the upper
|
191 |
+
bound on Ο (since that proves the lower bound on Ξ±). This will be in Subsection 3.2. For
|
192 |
+
that we follow the methods of Frieze and οΏ½Luczak [12].
|
193 |
+
We will assume r β₯ 3 since Frieze and οΏ½Luczak [12] covered the graph case. We will use
|
194 |
+
standard asymptotic notation, and we will use big-O notation to suppress any constants
|
195 |
+
depending on r but not d. Thus, for example we will write r = O(1) and dβ1 = O(1) but
|
196 |
+
not d = O(1). This is convenient for us because even though our theorem is for ο¬xed d, it
|
197 |
+
requires d to be suο¬ciently large.
|
198 |
+
3
|
199 |
+
|
200 |
+
3.1
|
201 |
+
Upper bound on the independence number
|
202 |
+
We will apply Theorem 4 to show an upper bound on Ξ±(Hr(n, d)). Fix Ξ΅, r (but not d) and
|
203 |
+
let c = c(d) := (1 + Ξ΅)
|
204 |
+
οΏ½
|
205 |
+
r log d
|
206 |
+
(rβ1)d
|
207 |
+
οΏ½
|
208 |
+
1
|
209 |
+
rβ1. Let z2 be as deο¬ned in (4) and z1 be as deο¬ned in (5).
|
210 |
+
We see that
|
211 |
+
Lemma 6.
|
212 |
+
z2 =
|
213 |
+
c
|
214 |
+
1 β c + O (cr)
|
215 |
+
Proof. After some algebra, we re-write (4) as
|
216 |
+
z2 β
|
217 |
+
zr
|
218 |
+
2
|
219 |
+
(1 + z2)rβ1 =
|
220 |
+
c
|
221 |
+
1 β c.
|
222 |
+
and the claim follows.
|
223 |
+
Now we check (6).
|
224 |
+
h
|
225 |
+
οΏ½d
|
226 |
+
r
|
227 |
+
οΏ½
|
228 |
+
+ h(dc) + h(d(1 β c)) β h(c) β h(1 β c) β h(d) β d
|
229 |
+
r log z1 β dc log z2
|
230 |
+
=d
|
231 |
+
r log
|
232 |
+
οΏ½d
|
233 |
+
r
|
234 |
+
οΏ½
|
235 |
+
+ dc log(dc) + d(1 β c) log(d(1 β c)) β c log c β (1 β c) log(1 β c)
|
236 |
+
β d log d β d
|
237 |
+
r log z1 β dc log z2
|
238 |
+
=dc log
|
239 |
+
οΏ½
|
240 |
+
c
|
241 |
+
(1 β c)z2
|
242 |
+
οΏ½
|
243 |
+
+ d
|
244 |
+
r log [(z2 + 1)r β zr
|
245 |
+
2] + d log(1 β c) β c log c β (1 β c) log(1 β c). (7)
|
246 |
+
Now note that the ο¬rst term of (7) is
|
247 |
+
dc log
|
248 |
+
οΏ½
|
249 |
+
c
|
250 |
+
(1 β c)z2
|
251 |
+
οΏ½
|
252 |
+
= dc log
|
253 |
+
οΏ½
|
254 |
+
c
|
255 |
+
(1 β c)
|
256 |
+
οΏ½
|
257 |
+
c
|
258 |
+
1βc + O (cr+1)
|
259 |
+
οΏ½
|
260 |
+
οΏ½
|
261 |
+
= dc log
|
262 |
+
οΏ½
|
263 |
+
1
|
264 |
+
1 + O (cr)
|
265 |
+
οΏ½
|
266 |
+
= O
|
267 |
+
οΏ½
|
268 |
+
dcr+1οΏ½
|
269 |
+
.
|
270 |
+
The second term of (7) is
|
271 |
+
d
|
272 |
+
r log [(z2 + 1)r β zr
|
273 |
+
2] = d
|
274 |
+
r log
|
275 |
+
οΏ½οΏ½
|
276 |
+
1
|
277 |
+
1 β c + O
|
278 |
+
οΏ½
|
279 |
+
cr+1οΏ½οΏ½r
|
280 |
+
β
|
281 |
+
οΏ½
|
282 |
+
c
|
283 |
+
1 β c + O
|
284 |
+
οΏ½
|
285 |
+
cr+1οΏ½οΏ½rοΏ½
|
286 |
+
= d
|
287 |
+
r log
|
288 |
+
οΏ½οΏ½
|
289 |
+
1
|
290 |
+
1 β c
|
291 |
+
οΏ½r οΏ½
|
292 |
+
1 β cr + O
|
293 |
+
οΏ½
|
294 |
+
cr+1οΏ½οΏ½οΏ½
|
295 |
+
= d
|
296 |
+
r log
|
297 |
+
οΏ½
|
298 |
+
1
|
299 |
+
1 β c
|
300 |
+
οΏ½r
|
301 |
+
+ d
|
302 |
+
r log
|
303 |
+
οΏ½
|
304 |
+
1 β cr + O
|
305 |
+
οΏ½
|
306 |
+
cr+1οΏ½οΏ½
|
307 |
+
= βd log(1 β c) β d
|
308 |
+
r cr + O
|
309 |
+
οΏ½
|
310 |
+
dcr+1οΏ½
|
311 |
+
.
|
312 |
+
4
|
313 |
+
|
314 |
+
The last term of (7) is
|
315 |
+
(1 β c) log(1 β c) = O(c).
|
316 |
+
Therefore (7) becomes
|
317 |
+
β d
|
318 |
+
r cr β c log c + O
|
319 |
+
οΏ½
|
320 |
+
c + dcr+1οΏ½
|
321 |
+
= β c
|
322 |
+
οΏ½d
|
323 |
+
rcrβ1 + log c
|
324 |
+
οΏ½
|
325 |
+
+ O
|
326 |
+
οΏ½
|
327 |
+
c + dcr+1οΏ½
|
328 |
+
= β c
|
329 |
+
οΏ½
|
330 |
+
d
|
331 |
+
r (1 + Ξ΅)rβ1 r log d
|
332 |
+
(r β 1)d + log
|
333 |
+
οΏ½
|
334 |
+
(1 + Ξ΅)
|
335 |
+
οΏ½ r log d
|
336 |
+
(r β 1)d
|
337 |
+
οΏ½
|
338 |
+
1
|
339 |
+
rβ1οΏ½οΏ½
|
340 |
+
+ O
|
341 |
+
οΏ½
|
342 |
+
c + dcr+1οΏ½
|
343 |
+
= β c
|
344 |
+
οΏ½
|
345 |
+
(1 + Ξ΅)rβ1 log d
|
346 |
+
r β 1 β log d
|
347 |
+
r β 1 + O(log log d)
|
348 |
+
οΏ½
|
349 |
+
+ O
|
350 |
+
οΏ½
|
351 |
+
c + dcr+1οΏ½
|
352 |
+
= β β¦ (c log d) .
|
353 |
+
It follows from Theorem 4 that w.h.p.
|
354 |
+
Ξ±(Hr(n, d)) β€ (1 + Ξ΅)
|
355 |
+
οΏ½ r log d
|
356 |
+
(r β 1)d
|
357 |
+
οΏ½
|
358 |
+
1
|
359 |
+
rβ1
|
360 |
+
.
|
361 |
+
(8)
|
362 |
+
3.2
|
363 |
+
Upper bound on the chromatic number
|
364 |
+
Our proof of the upper bound uses the method of Frieze and οΏ½Luczak [12]. We will generate
|
365 |
+
Hr(n, d) in a somewhat complicated way. The way we generate it will allow us to use known
|
366 |
+
results on Hr(n, p) due to Krivelevich and Sudakov [14].
|
367 |
+
Set
|
368 |
+
m :=
|
369 |
+
οΏ½d β d1/2 log d
|
370 |
+
r
|
371 |
+
οΏ½
|
372 |
+
n.
|
373 |
+
(9)
|
374 |
+
Let Hβ
|
375 |
+
r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists
|
376 |
+
of r independent uniformly random vertices chosen with replacement.
|
377 |
+
We will generate
|
378 |
+
Hβ
|
379 |
+
r(n, m) as follows. We have n sets (βbucketsβ ) V1, . . . Vn and a set of rm points P :=
|
380 |
+
{p1, . . . prm}. We put each point pi into a uniform random bucket VΟ(i) independently. We
|
381 |
+
let R = {R1, . . . , Rm} be a uniform random partition of P into sets of size r. Of course, the
|
382 |
+
idea here is that the buckets Vi represent vertices and the parts of the partition R represent
|
383 |
+
edges. Thus Ri deο¬nes a hyper-edge {Ο(j) : j β Ri} for i = 1, 2, . . . , m. We denote the
|
384 |
+
hypergraph deο¬ned by R by HR.
|
385 |
+
Note that since r β₯ 3 the expected number of pairs of multi-edges in Hβ
|
386 |
+
r(n, m) is at most
|
387 |
+
οΏ½n
|
388 |
+
r
|
389 |
+
οΏ½οΏ½m
|
390 |
+
2
|
391 |
+
οΏ½ οΏ½
|
392 |
+
1
|
393 |
+
οΏ½n
|
394 |
+
r
|
395 |
+
οΏ½
|
396 |
+
οΏ½2
|
397 |
+
= O
|
398 |
+
οΏ½m2
|
399 |
+
nr
|
400 |
+
οΏ½
|
401 |
+
= O(nβ1).
|
402 |
+
5
|
403 |
+
|
404 |
+
Thus, w.h.p. there are no multi-edges. Now the expected number of βloopsβ (edges containing
|
405 |
+
the same vertex twice) is at most
|
406 |
+
nm
|
407 |
+
οΏ½r
|
408 |
+
2
|
409 |
+
οΏ½ οΏ½1
|
410 |
+
n
|
411 |
+
οΏ½2
|
412 |
+
= O(1).
|
413 |
+
Thus w.h.p. there are at most log n loops. We now remove all multi-edges and loops, and
|
414 |
+
say that M is the (random) number of edges remaining, where m β log n β€ M β€ m. The
|
415 |
+
remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges
|
416 |
+
chosen uniformly at random without replacement. Next we estimate the chromatic number
|
417 |
+
of Hr(n, M).
|
418 |
+
Claim 1. W.h.p. we have
|
419 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
420 |
+
Ο(Hr(n, M)) β
|
421 |
+
οΏ½
|
422 |
+
(rβ1)d
|
423 |
+
r log d
|
424 |
+
οΏ½
|
425 |
+
1
|
426 |
+
rβ1
|
427 |
+
οΏ½
|
428 |
+
(rβ1)d
|
429 |
+
r log d
|
430 |
+
οΏ½
|
431 |
+
1
|
432 |
+
rβ1
|
433 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
434 |
+
β€ Ξ΅
|
435 |
+
2,
|
436 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
437 |
+
Ξ±(Hr(n, M)) β
|
438 |
+
οΏ½
|
439 |
+
r log D
|
440 |
+
(rβ1)D
|
441 |
+
οΏ½
|
442 |
+
1
|
443 |
+
rβ1 n
|
444 |
+
οΏ½
|
445 |
+
r log D
|
446 |
+
(rβ1)D
|
447 |
+
οΏ½
|
448 |
+
1
|
449 |
+
rβ1 n
|
450 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
451 |
+
β€ Ξ΅
|
452 |
+
2.
|
453 |
+
Proof. We will use Theorem 5 together with a standard argument for comparing Hr(n, p)
|
454 |
+
with Hr(n, m). Set p := m/
|
455 |
+
οΏ½n
|
456 |
+
r
|
457 |
+
οΏ½
|
458 |
+
and apply Theorem 5 with Ξ΅ replaced with Ξ΅/4 so we get
|
459 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
460 |
+
Ο(Hr(n, p)) β
|
461 |
+
οΏ½
|
462 |
+
(rβ1)D
|
463 |
+
r log D
|
464 |
+
οΏ½
|
465 |
+
1
|
466 |
+
rβ1
|
467 |
+
οΏ½
|
468 |
+
(rβ1)D
|
469 |
+
r log D
|
470 |
+
οΏ½
|
471 |
+
1
|
472 |
+
rβ1
|
473 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
474 |
+
β€ Ξ΅
|
475 |
+
4
|
476 |
+
(10)
|
477 |
+
with probability at least 1 β o(1/n). Note that here
|
478 |
+
D =
|
479 |
+
οΏ½n β 1
|
480 |
+
r β 1
|
481 |
+
οΏ½
|
482 |
+
p =
|
483 |
+
οΏ½n β 1
|
484 |
+
r β 1
|
485 |
+
οΏ½
|
486 |
+
m/
|
487 |
+
οΏ½n
|
488 |
+
r
|
489 |
+
οΏ½
|
490 |
+
= rm/n = d β d1/2 log d.
|
491 |
+
Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can
|
492 |
+
replace D with d in (10) without changing the left hand side by more than Ξ΅/4 to obtain
|
493 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
494 |
+
Ο(Hr(n, p)) β
|
495 |
+
οΏ½
|
496 |
+
(rβ1)d
|
497 |
+
r log d
|
498 |
+
οΏ½
|
499 |
+
1
|
500 |
+
rβ1
|
501 |
+
οΏ½
|
502 |
+
(rβ1)d
|
503 |
+
r log d
|
504 |
+
οΏ½
|
505 |
+
1
|
506 |
+
rβ1
|
507 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
508 |
+
β€ Ξ΅
|
509 |
+
2
|
510 |
+
(11)
|
511 |
+
with probability at least 1βo(1/n). But now note that with probability β¦(nβ1/2) the number
|
512 |
+
of edges in Hr(n, p) is precisely M. Thus we have that
|
513 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
514 |
+
Ο(Hr(n, M)) β
|
515 |
+
οΏ½
|
516 |
+
(rβ1)d
|
517 |
+
r log d
|
518 |
+
οΏ½
|
519 |
+
1
|
520 |
+
rβ1
|
521 |
+
οΏ½
|
522 |
+
(rβ1)d
|
523 |
+
r log d
|
524 |
+
οΏ½
|
525 |
+
1
|
526 |
+
rβ1
|
527 |
+
οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
|
528 |
+
β€ Ξ΅
|
529 |
+
2
|
530 |
+
with probability at least 1 β o(nβ1/2). This proves the ο¬rst inequality, and the second one
|
531 |
+
follows similarly.
|
532 |
+
6
|
533 |
+
|
534 |
+
Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d). This
|
535 |
+
transformation will involve ο¬rst removing some edges from vertices of degree larger than d,
|
536 |
+
and then adding some edges to vertices of degree less than d. We deο¬ne the rank of a point
|
537 |
+
pi β Vj, to be the number of points piβ² β Vj such that iβ² β€ i. We form a new set of points
|
538 |
+
P β² β P and a partition Rβ² of P β² as follows. For any Rk β R containing a point with rank
|
539 |
+
more than d, we delete Rk from R and delete all points of Rk from P. Note that each bucket
|
540 |
+
contains at most r points of P β². Note also that Rβ² is a uniform random partition of P β². We
|
541 |
+
let HRβ² be the natural hypergraph associated with Rβ².
|
542 |
+
Now we would like to put some more points into the buckets until each bucket has exactly d
|
543 |
+
points, arriving at some set of points P β²β² β P β². We would also like a uniform partition Rβ²β² of
|
544 |
+
P β²β² into sets of size r, and we would like Rβ²β² to have many of the same parts as Rβ². We will
|
545 |
+
accomplish this by constructing a sequence P β²
|
546 |
+
1 := P β² β P β²
|
547 |
+
2 β . . . β P β²
|
548 |
+
β =: P β²β² of point sets
|
549 |
+
and a sequence Rβ²
|
550 |
+
1 := Rβ², Rβ²
|
551 |
+
2, . . . , Rβ²
|
552 |
+
β =: Rβ²β² where Rβ²
|
553 |
+
j is a uniform random partition of P β²
|
554 |
+
j.
|
555 |
+
We construct P β²
|
556 |
+
j+1, Rβ²
|
557 |
+
j+1 from P β²
|
558 |
+
j, Rβ²
|
559 |
+
j as follows. Suppose |Rβ²
|
560 |
+
j| = a (in other words Rβ²
|
561 |
+
j has
|
562 |
+
a parts), so |P β²
|
563 |
+
j| = ra. P β²
|
564 |
+
j+1 will simply be P β²
|
565 |
+
j plus r new points. Now we will choose a
|
566 |
+
random value K β {1, . . . , r} using the distribution P[K = k] = qk(a), where qk(a) is deο¬ned
|
567 |
+
as follows.
|
568 |
+
Deο¬nition 1. Consider a random partition of ra+r points into a+1 parts of size r, and ο¬x
|
569 |
+
some set Q of r points. Then for 1 β€ k β€ r, the number qk(a) is deο¬ned to be the probability
|
570 |
+
that Q meets exactly k parts of the partition.
|
571 |
+
We will then remove a uniform random set of K β 1 parts from Rβ²
|
572 |
+
j, leaving Kr points in
|
573 |
+
P β²
|
574 |
+
j+1 which are not in any remaining part of Rβ²
|
575 |
+
j. We partition those points into K parts of
|
576 |
+
size r such that each part contains at least one new point (each such partition being equally
|
577 |
+
likely), arriving at our partition Rβ²
|
578 |
+
j+1.
|
579 |
+
We claim that Rβ²
|
580 |
+
j+1 is a uniform random partition of P β²
|
581 |
+
j+1 into parts of size r. Indeed, ο¬rst
|
582 |
+
consider the r new points that are in P β²
|
583 |
+
j+1 which were not in P β²
|
584 |
+
j. The probability that a
|
585 |
+
uniform random partition of P β²
|
586 |
+
j+1 would have exactly k parts containing at least one new
|
587 |
+
point is qk. So we can generate such a random partition as follows: ο¬rst choose a random
|
588 |
+
value K with P[K = k] = qk; next we choose a uniform random set of (K β 1)r points from
|
589 |
+
P β²
|
590 |
+
j; next we choose a partition of the set of points consisting of P β²
|
591 |
+
j+1 \ P β²
|
592 |
+
j together with the
|
593 |
+
points from P β²
|
594 |
+
j we chose in the last step, where the partition we choose is uniformly random
|
595 |
+
from among all partitions such that each part contains at least one point of P β²
|
596 |
+
j+1 \P β²
|
597 |
+
j; ο¬nally,
|
598 |
+
we choose a uniform partition of the rest of the points. In our case this partition of the rest
|
599 |
+
of the points comprises the current partition of the βunusedβ (a β K + 1)r points. At the
|
600 |
+
end of this process we have that HRβ²β² is distributed as Hr(n, d).
|
601 |
+
7
|
602 |
+
|
603 |
+
3.2.1
|
604 |
+
Bounding the number of low degree vertices in HRβ²
|
605 |
+
We deο¬ne some sets of buckets. We show that w.h.p. there are few small buckets i.e few
|
606 |
+
vertices of low degree in the hypregraph HRβ². Let S0 be the buckets with at most dβ3d1/2 log d
|
607 |
+
points of P β², and let S1 be the buckets with at most dβ2d1/2 log d points of P. Let S2 be the
|
608 |
+
set of buckets that, when we remove points from P β² to get P, have at least d1/2 log d points
|
609 |
+
removed. Then S0 β S1 βͺ S2. Our goal is to bound the probability that S0 is too large.
|
610 |
+
Fix a bucket Vj and let X βΌ Bin
|
611 |
+
οΏ½
|
612 |
+
rm, 1
|
613 |
+
n
|
614 |
+
οΏ½
|
615 |
+
be the number of points of P in Vj. Then the
|
616 |
+
probability that Vj is in S1 satisο¬es
|
617 |
+
P[Vj β S1] = P
|
618 |
+
οΏ½
|
619 |
+
X β€ d β 2d1/2 log d
|
620 |
+
οΏ½
|
621 |
+
= P
|
622 |
+
οΏ½
|
623 |
+
X β rm
|
624 |
+
n β€ βd1/2 log d
|
625 |
+
οΏ½
|
626 |
+
β€ exp
|
627 |
+
οΏ½
|
628 |
+
β
|
629 |
+
d log2 d
|
630 |
+
3(d β d1/2 log d)
|
631 |
+
οΏ½
|
632 |
+
= exp
|
633 |
+
οΏ½
|
634 |
+
ββ¦
|
635 |
+
οΏ½
|
636 |
+
log2 d
|
637 |
+
οΏ½οΏ½
|
638 |
+
,
|
639 |
+
where for our inequality we have used the Chernoο¬ bound (Lemma 2). Therefore E[|S0|] β€
|
640 |
+
exp
|
641 |
+
οΏ½
|
642 |
+
ββ¦
|
643 |
+
οΏ½
|
644 |
+
log2 d
|
645 |
+
οΏ½οΏ½
|
646 |
+
n. Now we argue that |S1| is concentrated using McDiarmidβs inequality
|
647 |
+
(Lemma 3). For our application we let X = |S1| which is a function (say f) of the vector
|
648 |
+
(Z1, . . . Zrm) where Zi tells us which bucket the ith point of P went into. Moving a point
|
649 |
+
from one bucket to another can only change |S1| by at most 1 so we use c = 1. Thus we get
|
650 |
+
the bound
|
651 |
+
P(|X β E[X]| β₯ n2/3) β€ 2 exp
|
652 |
+
οΏ½
|
653 |
+
β n4/3
|
654 |
+
2rm
|
655 |
+
οΏ½
|
656 |
+
= o(1).
|
657 |
+
(12)
|
658 |
+
Now we handle S2. For 1 β€ j β€ n let Yj be the number of parts Rk β R such that Rk
|
659 |
+
contains a point in the bucket Vj as well as a point in some bucket Vjβ² where |Vjβ²| > d. Note
|
660 |
+
that if Vj β S2 then Yj β₯ d1/2 log d. We view Rk as a set of r points, say {q1, . . . , qr} each
|
661 |
+
going into a uniform random bucket. Say qi goes to bucket Vji. The probability that Rk is
|
662 |
+
counted by Yj is at most
|
663 |
+
rP[j1 = j and |Vj1| > d] + r(r β 1)P[j1 = j and |Vj2| > d]
|
664 |
+
= r
|
665 |
+
nP[|Vj1| > d
|
666 |
+
οΏ½οΏ½j1 = j] + r(r β 1)
|
667 |
+
n
|
668 |
+
P[|Vj2| > d
|
669 |
+
οΏ½οΏ½j1 = j]
|
670 |
+
β€ r2
|
671 |
+
n P[|Vj1| > d
|
672 |
+
οΏ½οΏ½j1 = j]
|
673 |
+
β€ r2
|
674 |
+
n P[Bin(rm β 1, 1/n) β₯ d] = r2
|
675 |
+
n exp
|
676 |
+
οΏ½
|
677 |
+
ββ¦
|
678 |
+
οΏ½
|
679 |
+
log2 d
|
680 |
+
οΏ½οΏ½
|
681 |
+
.
|
682 |
+
Thus we have
|
683 |
+
E[Yj] = m Β· r2
|
684 |
+
n exp
|
685 |
+
οΏ½
|
686 |
+
ββ¦
|
687 |
+
οΏ½
|
688 |
+
log2 d
|
689 |
+
οΏ½οΏ½
|
690 |
+
β€ rd exp
|
691 |
+
οΏ½
|
692 |
+
ββ¦
|
693 |
+
οΏ½
|
694 |
+
log2 d
|
695 |
+
οΏ½οΏ½
|
696 |
+
= rd1/2 exp
|
697 |
+
οΏ½
|
698 |
+
ββ¦
|
699 |
+
οΏ½
|
700 |
+
log2 d
|
701 |
+
οΏ½οΏ½
|
702 |
+
and so Markovβs inequality gives us
|
703 |
+
P
|
704 |
+
οΏ½
|
705 |
+
Yj β₯ d1/2 log d
|
706 |
+
οΏ½
|
707 |
+
β€ rd exp
|
708 |
+
οΏ½
|
709 |
+
ββ¦
|
710 |
+
οΏ½
|
711 |
+
log2 d
|
712 |
+
οΏ½οΏ½
|
713 |
+
d1/2 log d
|
714 |
+
= exp
|
715 |
+
οΏ½
|
716 |
+
ββ¦
|
717 |
+
οΏ½
|
718 |
+
log2 d
|
719 |
+
οΏ½οΏ½
|
720 |
+
8
|
721 |
+
|
722 |
+
and so E[|S2|] = n exp
|
723 |
+
οΏ½
|
724 |
+
ββ¦
|
725 |
+
οΏ½
|
726 |
+
log2 d
|
727 |
+
οΏ½οΏ½
|
728 |
+
. We use McDiarmidβs inequality once more, this time
|
729 |
+
with X = |S2|.
|
730 |
+
A change in choice of bucket changes |S2| by at most one and so (12)
|
731 |
+
continues to hold. Thus
|
732 |
+
|S0| = n exp
|
733 |
+
οΏ½
|
734 |
+
ββ¦
|
735 |
+
οΏ½
|
736 |
+
log2 d
|
737 |
+
οΏ½οΏ½
|
738 |
+
.
|
739 |
+
w.h.p.
|
740 |
+
3.2.2
|
741 |
+
A property of independent subsets of Hr(n, m)
|
742 |
+
Fix 1 β€ j β€ r β 1. Set
|
743 |
+
a :=
|
744 |
+
οΏ½
|
745 |
+
1 + Ξ΅
|
746 |
+
2
|
747 |
+
οΏ½ οΏ½ r log d
|
748 |
+
(r β 1)d
|
749 |
+
οΏ½
|
750 |
+
1
|
751 |
+
rβ1
|
752 |
+
,
|
753 |
+
ΞΊj := 10d
|
754 |
+
r
|
755 |
+
οΏ½r
|
756 |
+
j
|
757 |
+
οΏ½
|
758 |
+
aj,
|
759 |
+
p := d(r β 1)!
|
760 |
+
nrβ1
|
761 |
+
.
|
762 |
+
The expected number of independent sets A in Hr(n, p) of size at most an such that there
|
763 |
+
are ΞΊjn edges each having j vertices in A is at most
|
764 |
+
an
|
765 |
+
οΏ½
|
766 |
+
s=1
|
767 |
+
οΏ½n
|
768 |
+
s
|
769 |
+
οΏ½
|
770 |
+
(1 β p)(s
|
771 |
+
r)
|
772 |
+
οΏ½οΏ½s
|
773 |
+
j
|
774 |
+
οΏ½οΏ½ n
|
775 |
+
rβj
|
776 |
+
οΏ½
|
777 |
+
ΞΊjn
|
778 |
+
οΏ½
|
779 |
+
pΞΊjn
|
780 |
+
β€
|
781 |
+
an
|
782 |
+
οΏ½
|
783 |
+
s=1
|
784 |
+
exp
|
785 |
+
ο£±
|
786 |
+
ο£²
|
787 |
+
ο£³s log
|
788 |
+
οΏ½en
|
789 |
+
s
|
790 |
+
οΏ½
|
791 |
+
β
|
792 |
+
οΏ½s
|
793 |
+
r
|
794 |
+
οΏ½
|
795 |
+
p + ΞΊjn log
|
796 |
+

|
797 |
+
ο£e(an)j
|
798 |
+
j!
|
799 |
+
nrβj
|
800 |
+
(rβj)!p
|
801 |
+
ΞΊjn
|
802 |
+
ο£Ά
|
803 |
+
ο£Έ
|
804 |
+
ο£Ό
|
805 |
+
ο£½
|
806 |
+
ο£Ύ
|
807 |
+
=
|
808 |
+
an
|
809 |
+
οΏ½
|
810 |
+
s=1
|
811 |
+
exp
|
812 |
+
οΏ½
|
813 |
+
s log
|
814 |
+
οΏ½en
|
815 |
+
s
|
816 |
+
οΏ½
|
817 |
+
β
|
818 |
+
οΏ½s
|
819 |
+
r
|
820 |
+
οΏ½
|
821 |
+
p + ΞΊjn log
|
822 |
+
οΏ½eaj
|
823 |
+
10
|
824 |
+
οΏ½οΏ½
|
825 |
+
β€ an Β· exp
|
826 |
+
οΏ½οΏ½
|
827 |
+
log
|
828 |
+
οΏ½e
|
829 |
+
a
|
830 |
+
οΏ½
|
831 |
+
β 10d
|
832 |
+
r
|
833 |
+
οΏ½r
|
834 |
+
j
|
835 |
+
οΏ½
|
836 |
+
ajβ1 log
|
837 |
+
οΏ½10
|
838 |
+
e
|
839 |
+
οΏ½οΏ½
|
840 |
+
an
|
841 |
+
οΏ½
|
842 |
+
= o(1/n)
|
843 |
+
where the last line follows since as d β β we have
|
844 |
+
log
|
845 |
+
οΏ½e
|
846 |
+
a
|
847 |
+
οΏ½
|
848 |
+
βΌ
|
849 |
+
1
|
850 |
+
r β 1 log d
|
851 |
+
and
|
852 |
+
10d
|
853 |
+
r
|
854 |
+
οΏ½r
|
855 |
+
j
|
856 |
+
οΏ½
|
857 |
+
ajβ1 log
|
858 |
+
οΏ½10
|
859 |
+
e
|
860 |
+
οΏ½
|
861 |
+
= β¦
|
862 |
+
οΏ½
|
863 |
+
d
|
864 |
+
rβj
|
865 |
+
jβ1 logβ jβ1
|
866 |
+
rβ1 d
|
867 |
+
οΏ½
|
868 |
+
β« log d.
|
869 |
+
Thus with probability 1 β o(1/n), Hr(n, p) has a coloring using (1 + Ξ΅/2)
|
870 |
+
οΏ½
|
871 |
+
(rβ1)d
|
872 |
+
r log d
|
873 |
+
οΏ½
|
874 |
+
1
|
875 |
+
rβ1 colors
|
876 |
+
such that for each color class A and for each 1 β€ j β€ r β 1 there are at most ΞΊjn edges with
|
877 |
+
j vertices in A. The hypergraph Hr(n, m), m =
|
878 |
+
οΏ½n
|
879 |
+
r
|
880 |
+
οΏ½
|
881 |
+
p will have this property w.h.p..
|
882 |
+
3.2.3
|
883 |
+
Transforming HRβ² into Hr(n, d)
|
884 |
+
Now we will complete the transformation to the random regular hypergraph Hr(n, d). We
|
885 |
+
are open to the possibility that doing so will render our coloring no longer proper, since this
|
886 |
+
9
|
887 |
+
|
888 |
+
process will involve changing some edges which might then be contained in a color class. We
|
889 |
+
will keep track of how many such βbadβ edges there are and then repair our coloring at the
|
890 |
+
end.
|
891 |
+
We have to add at most (3d1/2 log d + d exp
|
892 |
+
οΏ½
|
893 |
+
ββ¦
|
894 |
+
οΏ½
|
895 |
+
log2 d
|
896 |
+
οΏ½οΏ½
|
897 |
+
)n < (4d1/2 log d)n points, which
|
898 |
+
takes at most as many steps.
|
899 |
+
For each color class A of HRβ² deο¬ne XA,j = XA,j(i) to
|
900 |
+
be the number of edges with j vertices in A at step i. We have already established that
|
901 |
+
XA,j(0) β€ ΞΊjn. This follows from Section 3.2.2 and the fact that we have removed edges
|
902 |
+
from H(n, m) to obtain HRβ². Let Ei be the event that at step i we have that for each color
|
903 |
+
class A and for each 1 β€ j β€ r β 1 we have XA,j(i) β€ 2ΞΊjn. Then, assuming Ei holds, the
|
904 |
+
probability that XA,j increases at step i is at most
|
905 |
+
οΏ½
|
906 |
+
1β€kβ€r, jββ₯1
|
907 |
+
j1+Β·Β·Β·+jk=j
|
908 |
+
οΏ½
|
909 |
+
1β€ββ€k
|
910 |
+
2ΞΊjβn
|
911 |
+
nd/r =
|
912 |
+
οΏ½
|
913 |
+
1β€kβ€r, jββ₯1
|
914 |
+
j1+Β·Β·Β·+jk=j
|
915 |
+
οΏ½
|
916 |
+
1β€ββ€k
|
917 |
+
20
|
918 |
+
οΏ½ r
|
919 |
+
jk
|
920 |
+
οΏ½
|
921 |
+
ajk β€
|
922 |
+
οΏ½
|
923 |
+
1β€kβ€r, jββ₯1
|
924 |
+
j1+Β·Β·Β·+jk=j
|
925 |
+
20r2r2aj β€ 40r2r2aj.
|
926 |
+
Also, the largest possible increase in XA,j in one step is r. Thus, the ο¬nal value of XA,j
|
927 |
+
after at most (4d1/2 log d)n steps is stochastically dominated by ΞΊjn + rY where Y
|
928 |
+
βΌ
|
929 |
+
Bin
|
930 |
+
οΏ½
|
931 |
+
(4d1/2 log d)n, 40r2r2ajοΏ½
|
932 |
+
. An easy application of the Chernoο¬ bound tells us
|
933 |
+
P (Y > 2E[Y ]) β€ exp(ββ¦(n)).
|
934 |
+
(13)
|
935 |
+
Note that here
|
936 |
+
2E[Y ]
|
937 |
+
ΞΊjn
|
938 |
+
= 8d1/2 log d Β· 40r2r2ajn
|
939 |
+
10d
|
940 |
+
οΏ½r
|
941 |
+
j
|
942 |
+
οΏ½
|
943 |
+
ajn/r
|
944 |
+
= O(dβ1/2 log d) < 1
|
945 |
+
for suο¬ciently large d. Thus, using (13) and the union bound over all color classes A, we
|
946 |
+
have w.h.p. the ο¬nal value of XA,j is at most ΞΊjn + 2E[Y ] β€ 2ΞΊjn for all 1 β€ j β€ r β 1.
|
947 |
+
Now we address βbadβ edges, i.e. edges contained in a color class. Assuming Ei holds, the ex-
|
948 |
+
pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar =
|
949 |
+
O
|
950 |
+
οΏ½οΏ½ log d
|
951 |
+
d
|
952 |
+
οΏ½
|
953 |
+
r
|
954 |
+
rβ1οΏ½
|
955 |
+
(because it would have to be one of the colors of one of the vertices we are
|
956 |
+
adding points to). Thus the expected number of bad edges created in (4d1/2 log d)n steps
|
957 |
+
is stochastically dominated by Z βΌ r Β· Bin
|
958 |
+
οΏ½
|
959 |
+
(4d1/2 log d)n, O
|
960 |
+
οΏ½οΏ½log d
|
961 |
+
d
|
962 |
+
οΏ½
|
963 |
+
r
|
964 |
+
rβ1οΏ½ οΏ½
|
965 |
+
.
|
966 |
+
Another easy
|
967 |
+
application of Chernoο¬ shows that w.h.p. Z β€ 2E[Z] = O(dβ1/2n).
|
968 |
+
We repair the coloring as follows. First we uncolor one vertex from each bad edge, and let
|
969 |
+
the set of uncolored vertices be U where |U| = u = O
|
970 |
+
οΏ½
|
971 |
+
dβ1/2n
|
972 |
+
οΏ½
|
973 |
+
. Let
|
974 |
+
Ξ΄ := Ξ΅
|
975 |
+
2
|
976 |
+
οΏ½(r β 1)d
|
977 |
+
r log d
|
978 |
+
οΏ½
|
979 |
+
1
|
980 |
+
rβ1
|
981 |
+
.
|
982 |
+
We claim that for every S β U, |S| = s, the hypergraph induced on S has at most Ξ΄s/r
|
983 |
+
edges. This will complete our proof since it implies that the minimum degree is at most Ξ΄
|
984 |
+
and so U can be recolored using a fresh set of Ξ΄ colors, yielding a coloring of Hr(n, d) using
|
985 |
+
10
|
986 |
+
|
987 |
+
at most
|
988 |
+
Ο(Hr(n, M)) + Ξ΄ β€
|
989 |
+
οΏ½
|
990 |
+
1 + Ξ΅
|
991 |
+
2
|
992 |
+
οΏ½ οΏ½(r β 1)d
|
993 |
+
r log d
|
994 |
+
οΏ½
|
995 |
+
1
|
996 |
+
rβ1
|
997 |
+
+ Ξ΅
|
998 |
+
2
|
999 |
+
οΏ½(r β 1)d
|
1000 |
+
r log d
|
1001 |
+
οΏ½
|
1002 |
+
1
|
1003 |
+
rβ1
|
1004 |
+
= (1 + Ξ΅)
|
1005 |
+
οΏ½(r β 1)d
|
1006 |
+
r log d
|
1007 |
+
οΏ½
|
1008 |
+
1
|
1009 |
+
rβ1
|
1010 |
+
colors. The expected number of sets S with more than Ξ΄s/r edges is at most
|
1011 |
+
οΏ½
|
1012 |
+
1β€sβ€u
|
1013 |
+
οΏ½n
|
1014 |
+
s
|
1015 |
+
οΏ½οΏ½οΏ½ds
|
1016 |
+
r
|
1017 |
+
οΏ½
|
1018 |
+
Ξ΄s/r
|
1019 |
+
οΏ½
|
1020 |
+
1
|
1021 |
+
οΏ½dn
|
1022 |
+
r
|
1023 |
+
οΏ½οΏ½dnβr
|
1024 |
+
r
|
1025 |
+
οΏ½
|
1026 |
+
. . .
|
1027 |
+
οΏ½dnβΞ΄s+r
|
1028 |
+
r
|
1029 |
+
οΏ½
|
1030 |
+
β€
|
1031 |
+
οΏ½
|
1032 |
+
1β€sβ€u
|
1033 |
+
οΏ½ne
|
1034 |
+
s
|
1035 |
+
οΏ½s οΏ½(dse/r)re
|
1036 |
+
Ξ΄s/r
|
1037 |
+
οΏ½Ξ΄s/r
|
1038 |
+
(r!)Ξ΄s/r
|
1039 |
+
(dn β Ξ΄s)Ξ΄s
|
1040 |
+
β€
|
1041 |
+
οΏ½
|
1042 |
+
1β€sβ€u
|
1043 |
+
οΏ½
|
1044 |
+
ne
|
1045 |
+
s
|
1046 |
+
οΏ½
|
1047 |
+
dse
|
1048 |
+
(dn β Ξ΄s)r
|
1049 |
+
οΏ½Ξ΄ οΏ½er Β· r!
|
1050 |
+
Ξ΄s
|
1051 |
+
οΏ½Ξ΄/rοΏ½s
|
1052 |
+
.
|
1053 |
+
(14)
|
1054 |
+
Now for 1 β€ s β€ βn the term in (14) is at most
|
1055 |
+
οΏ½
|
1056 |
+
O(n) Β·
|
1057 |
+
οΏ½
|
1058 |
+
O(nβ1/2)
|
1059 |
+
οΏ½Ξ΄ Β· O(1)
|
1060 |
+
οΏ½s
|
1061 |
+
= o(1/n)
|
1062 |
+
since Ξ΄ can be made arbitrarily large by choosing d large. Meanwhile for βn β€ s β€ u we
|
1063 |
+
have that the term in (14) is at most
|
1064 |
+
οΏ½
|
1065 |
+
O(n1/2) Β· O(1) Β·
|
1066 |
+
οΏ½
|
1067 |
+
O(nβ1/2)
|
1068 |
+
οΏ½Ξ΄/rοΏ½s
|
1069 |
+
= o(1/n).
|
1070 |
+
Now since (14) has O(n) terms the whole sum is o(1) and we are done. This completes the
|
1071 |
+
proof of Theorem 1.
|
1072 |
+
4
|
1073 |
+
Summary
|
1074 |
+
We have asymptotically computed the chromatic number of random r-uniform, d-regular
|
1075 |
+
hypergraphs when proper colorings mean that no edge is mono-chromatic. It would seem
|
1076 |
+
likely that the approach we took would extend to other deο¬nitions of proper coloring. We
|
1077 |
+
have not attempted to use second moment calculations to further narrow our estimates.
|
1078 |
+
These would seem to be two natural lines of further research.
|
1079 |
+
References
|
1080 |
+
[1] P. Ayre, A. Coja-Oghlan and C. Greenhill, Hypergraph coloring up to condensation,
|
1081 |
+
Random Structures and Algorithms 54 (2019) 615 - 652.
|
1082 |
+
11
|
1083 |
+
|
1084 |
+
[2] D. Achlioptas and C. Moore, The Chromatic Number of Random Regular Graphs,
|
1085 |
+
In Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Random-
|
1086 |
+
ization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM AP-
|
1087 |
+
PROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Hei-
|
1088 |
+
delberg. Approximation, Randomization, and Combinatorial Optimization. Algorithms
|
1089 |
+
and Techniques (2004) 219β228.
|
1090 |
+
[3] D. Achlioptas and A. Naor, The two possible values of the chromatic number of a
|
1091 |
+
random graph, Annals of Mathematics 162 (2005) 1335-1351.
|
1092 |
+
[4] D. Bal and P. Bennett, The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs.
|
1093 |
+
[5] B. BollobΒ΄as, The chromatic number of random graphs, Combinatorica 8 (1988) 49-55.
|
1094 |
+
[6] B. BollobΒ΄as and P. ErdΛos, Cliques in random graphs, Mathematical Proceedings of the
|
1095 |
+
Cambridge Philosophical Society 80 (1976) 419-427.
|
1096 |
+
[7] A. Coja-Oghlan, Upper-Bounding the k-Colorability Threshold by Counting Covers,
|
1097 |
+
Electronic Journal of Combinatorics 20 (2013).
|
1098 |
+
[8] A. Coja-Oghlan, C. Efthymiou and S. Hetterich, On the chromatic number of random
|
1099 |
+
regular graphs, Journal of Combinatorial Theory B 116 (2016) 367-439.
|
1100 |
+
[9] M. Dyer, A.M. Frieze and C. Greenhill, On the chromatic number of a random hyper-
|
1101 |
+
graph, Journal of Combinatorial Theorey B 113 (2015) 68-122.
|
1102 |
+
[10] A.M. Frieze, On the independence number of random graphs, Discrete Mathematics 81
|
1103 |
+
(1990) 171-176.
|
1104 |
+
[11] A.M. Frieze and M. KaroΒ΄nski, Introduction to Random Graphs, Cambridge University
|
1105 |
+
Press, 2015.
|
1106 |
+
[12] A.M. Frieze and T. οΏ½Luczak, On the independence and chromatic numbers of random
|
1107 |
+
regular graphs, Journal of Combinatorial Theory. Series B 54 (1992) 123-132.
|
1108 |
+
[13] G. Grimmett and C. McDiarmid, On colouring random graphs, Mathematical Proceed-
|
1109 |
+
ings of the Cambridge Philosophical Society 77 (1975) 313-324.
|
1110 |
+
[14] M. Krivelevich and B. Sudakov, The chromatic numbers of random hypergraphs, Ran-
|
1111 |
+
dom Structures Algorithms 12 (1998) 381-403.
|
1112 |
+
[15] T. οΏ½Luczak, The chromatic number of random graphs, Combinatorica 11 (19990) 45-54.
|
1113 |
+
[16] T. οΏ½Luczak, A note on the sharp concentration of the chromatic number of random
|
1114 |
+
graphs, Combinatorica 11 (1991) 295-297.
|
1115 |
+
[17] D. Matula, Expose-and-Merge Exploration and the Chromatic Number of a Random
|
1116 |
+
Graph, Combinatorica 7 (1987) 275-284.
|
1117 |
+
12
|
1118 |
+
|
1119 |
+
[18] E. Shamir and J. Spencer, Sharp concentration of the chromatic number od random
|
1120 |
+
graphs Gn,p, Combinatorica 7 (1987) 121-129.
|
1121 |
+
[19] L. Shi and N. Wormald, Coloring random regular graphs, Combinatorics, Probability
|
1122 |
+
and Computing 16 (2007) 459-494.
|
1123 |
+
13
|
1124 |
+
|
ENAyT4oBgHgl3EQfSPf9/content/tmp_files/load_file.txt
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1 |
+
filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf,len=361
|
2 |
+
page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
3 |
+
page_content='00085v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
4 |
+
page_content='CO] 31 Dec 2022 On the chromatic number of random regular hypergraphs Patrick Bennettβ Department of Mathematics, Western Michigan University Kalamazoo MI 49008 Alan Friezeβ Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
5 |
+
page_content=' Abstract We estimate the likely values of the chromatic and independence numbers of the random r-uniform d-regular hypergraph on n vertices for ο¬xed r, large ο¬xed d, and n β β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
6 |
+
page_content=' 1 Introduction The study of the chromatic number of random graphs has a long history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
7 |
+
page_content=' It begins with the work of BollobΒ΄as and ErdΛos [6] and Grimmett and McDiarmid [13] who determined Ο(Gn,p), p constant to within a factor 2, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
8 |
+
page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
9 |
+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
10 |
+
page_content=' Matula [17] reduced this to a factor of 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
11 |
+
page_content=' Then we have the discovery of martingale concentration inequalities by Shamir and Spencer [18] leading to the breakthrough by BollobΒ΄as [5] who determined Ο(Gn,p) asymptotically for p constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
12 |
+
page_content=' The case of p β 0 proved a little more tricky, but οΏ½Luczak [15] using ideas from Frieze [10] and [17] determined Ο(Gn,p), p = c/n asymptotically for large c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
13 |
+
page_content=' οΏ½Luczak [16] showed that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
14 |
+
page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
15 |
+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
16 |
+
page_content=' Ο(Gn,p), p = c/n took one of two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
17 |
+
page_content=' It was then that the surprising power of the second moment method was unleashed by Achlioptas and Naor [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
18 |
+
page_content=' Since then there has been much work tightening our estimates for the k-colorability threshold, k β₯ 3 constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
19 |
+
page_content=' See for example Coja-Oghlan [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Random regular graphs of low degree were studied algorithmically by several authors e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Achlioptas and Molloy [2] and by Shi and Wormald [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Frieze and οΏ½Luczak [12] introduced βResearch supported in part by Simons Foundation Grant #426894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' β Research supported in part by NSF Grant DMS1661063 1 a way of using our knowledge of Ο(Gn,p), p = c/n to tackle Ο(Gn,r) where Gn,r denotes a random r-regular graph and where p = r/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Subsequently Achlioptas and Moore [2] showed via the second moment method that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ο(Gn,r) was one of 3 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This was tightened basically to one value by Coja-Oghlan, Efthymiou and Hetterich [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic number for Ο(Hr(n, p) for οΏ½nβ1 rβ1 οΏ½ p suο¬ciently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Here Hr(n, p) is the binomial r-uniform hypergraph where each of the οΏ½n r οΏ½ possible edges is included with probability p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' There are several possibilities of a proper coloring of the vertices of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Here we concentrate on the case where a vertex coloring is proper if no edge contains vertices of all the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Dyer, Frieze and Greehill [9] and Ayre, Coja-Oghlan and Greehill [1] established showed that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ο(Hr(n, p) took one or two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' When it comes to what ew denote by Ο(Hr(n, d), a random d-regular, r-uniform hypergraph, we are not aware of any results at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' In this paper we extend the approach of [12] to this case: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For all ο¬xed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such that for any ο¬xed d β₯ d0 we have that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, d)) β οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, d)) β οΏ½ r log d (rβ1)d οΏ½ 1 rβ1 n οΏ½ r log d (rβ1)d οΏ½ 1 rβ1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ (1) Here Ξ± refers to the independence number of a hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 2 Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='1 Tools We will be using the following forms of Chernoο¬βs bound (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=', [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Lemma 2 (Chernoο¬ bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let X βΌ Bin(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then for all 0 < Ξ» < np P(|X β np| β₯ Ξ») β€ 2 exp οΏ½ β Ξ»2 3np οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (2) Lemma 3 (McDiarmidβs inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let X = f(βZ) where βZ = (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Zt) and the Zi are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Assume the function f has the property that whenever βz, βw diο¬er in only one coordinate we have |f(βz) β f(βw)| β€ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then for all Ξ» > 0 we have P(|X β E[X]| β₯ Ξ») β€ 2 exp οΏ½ β Ξ»2 2c2t οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (3) Bal and the ο¬rst author [4] showed the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 2 Theorem 4 (Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Fix r β₯ 3, d β₯ 2, and 0 < c < rβ1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let z2 be the unique positive number such that z2 οΏ½ (z2 + 1)rβ1 β zrβ1 2 οΏ½ (z2 + 1)r β zr 2 = c (4) and let z1 = d r [(z2 + 1)r β zr 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (5) Let h(x) = x log x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' If it is the case that h οΏ½d r οΏ½ + h(dc) + h(d(1 β c)) β h(c) β h(1 β c) β h(d) β d r log z1 β dc log z2 < 0 (6) then w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ξ±(Hr(n, d)) < cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Krivelevich and Sudakov [14] proved the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Theorem 5 (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='1 in [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For all ο¬xed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such that whenever D = D(p) := οΏ½nβ1 rβ1 οΏ½ p β₯ d0 we have that οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, p)) β οΏ½ (rβ1)D r log D οΏ½ 1 rβ1 οΏ½ (rβ1)D r log D οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, p)) β οΏ½ r log D (rβ1)D οΏ½ 1 rβ1 n οΏ½ r log D (rβ1)D οΏ½ 1 rβ1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ with probability at least 1 β o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 3 Proof In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' First we give an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We show in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='1 that the upper bound on Ξ± follows from Theorem 4 and some straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then the lower bound on Ο follows as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus we will be done once we prove the upper bound on Ο (since that proves the lower bound on Ξ±).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This will be in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For that we follow the methods of Frieze and οΏ½Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will assume r β₯ 3 since Frieze and οΏ½Luczak [12] covered the graph case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will use standard asymptotic notation, and we will use big-O notation to suppress any constants depending on r but not d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus, for example we will write r = O(1) and dβ1 = O(1) but not d = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This is convenient for us because even though our theorem is for ο¬xed d, it requires d to be suο¬ciently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='1 Upper bound on the independence number We will apply Theorem 4 to show an upper bound on Ξ±(Hr(n, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Fix Ξ΅, r (but not d) and let c = c(d) := (1 + Ξ΅) οΏ½ r log d (rβ1)d οΏ½ 1 rβ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let z2 be as deο¬ned in (4) and z1 be as deο¬ned in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We see that Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' z2 = c 1 β c + O (cr) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' After some algebra, we re-write (4) as z2 β zr 2 (1 + z2)rβ1 = c 1 β c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now we check (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' h οΏ½d r οΏ½ + h(dc) + h(d(1 β c)) β h(c) β h(1 β c) β h(d) β d r log z1 β dc log z2 =d r log οΏ½d r οΏ½ + dc log(dc) + d(1 β c) log(d(1 β c)) β c log c β (1 β c) log(1 β c) β d log d β d r log z1 β dc log z2 =dc log οΏ½ c (1 β c)z2 οΏ½ + d r log [(z2 + 1)r β zr 2] + d log(1 β c) β c log c β (1 β c) log(1 β c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (7) Now note that the ο¬rst term of (7) is dc log οΏ½ c (1 β c)z2 οΏ½ = dc log οΏ½ c (1 β c) οΏ½ c 1βc + O (cr+1) οΏ½ οΏ½ = dc log οΏ½ 1 1 + O (cr) οΏ½ = O οΏ½ dcr+1οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The second term of (7) is d r log [(z2 + 1)r β zr 2] = d r log οΏ½οΏ½ 1 1 β c + O οΏ½ cr+1οΏ½οΏ½r β οΏ½ c 1 β c + O οΏ½ cr+1οΏ½οΏ½rοΏ½ = d r log οΏ½οΏ½ 1 1 β c οΏ½r οΏ½ 1 β cr + O οΏ½ cr+1οΏ½οΏ½οΏ½ = d r log οΏ½ 1 1 β c οΏ½r + d r log οΏ½ 1 β cr + O οΏ½ cr+1οΏ½οΏ½ = βd log(1 β c) β d r cr + O οΏ½ dcr+1οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 4 The last term of (7) is (1 β c) log(1 β c) = O(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Therefore (7) becomes β d r cr β c log c + O οΏ½ c + dcr+1οΏ½οΏ½ = β c οΏ½d rcrβ1 + log c οΏ½ + O οΏ½ c + dcr+1οΏ½ = β c οΏ½ d r (1 + Ξ΅)rβ1 r log d (r β 1)d + log οΏ½ (1 + Ξ΅) οΏ½ r log d (r β 1)d οΏ½ 1 rβ1οΏ½οΏ½ + O οΏ½ c + dcr+1οΏ½ = β c οΏ½ (1 + Ξ΅)rβ1 log d r β 1 β log d r β 1 + O(log log d) οΏ½ + O οΏ½ c + dcr+1οΏ½ = β β¦ (c log d) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' It follows from Theorem 4 that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ξ±(Hr(n, d)) β€ (1 + Ξ΅) οΏ½ r log d (r β 1)d οΏ½ 1 rβ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2 Upper bound on the chromatic number Our proof of the upper bound uses the method of Frieze and οΏ½Luczak [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will generate Hr(n, d) in a somewhat complicated way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The way we generate it will allow us to use known results on Hr(n, p) due to Krivelevich and Sudakov [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Set m := οΏ½d β d1/2 log d r οΏ½ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (9) Let Hβ r(n, m) be an r-uniform multi-hypergraph with m edges, where each multi-edge consists of r independent uniformly random vertices chosen with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will generate Hβ r(n, m) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We have n sets (βbucketsβ ) V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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111 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
112 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Vn and a set of rm points P := {p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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114 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
115 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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116 |
+
page_content=' prm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We put each point pi into a uniform random bucket VΟ(i) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We let R = {R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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119 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
120 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' , Rm} be a uniform random partition of P into sets of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Of course, the idea here is that the buckets Vi represent vertices and the parts of the partition R represent edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus Ri deο¬nes a hyper-edge {Ο(j) : j β Ri} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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125 |
+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We denote the hypergraph deο¬ned by R by HR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Note that since r β₯ 3 the expected number of pairs of multi-edges in Hβ r(n, m) is at most οΏ½n r οΏ½οΏ½m 2 οΏ½ οΏ½ 1 οΏ½n r οΏ½ οΏ½2 = O οΏ½m2 nr οΏ½ = O(nβ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 5 Thus, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
131 |
+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' there are no multi-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now the expected number of βloopsβ (edges containing the same vertex twice) is at most nm οΏ½r 2 οΏ½ οΏ½1 n οΏ½2 = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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135 |
+
page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
136 |
+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' there are at most log n loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We now remove all multi-edges and loops, and say that M is the (random) number of edges remaining, where m β log n β€ M β€ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The remaining hypergraph is distributed as H(n, M), the random hypergraph with M edges chosen uniformly at random without replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Next we estimate the chromatic number of Hr(n, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
144 |
+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' we have οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, M)) β οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ 2, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, M)) β οΏ½ r log D (rβ1)D οΏ½ 1 rβ1 n οΏ½ r log D (rβ1)D οΏ½ 1 rβ1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will use Theorem 5 together with a standard argument for comparing Hr(n, p) with Hr(n, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Set p := m/ οΏ½n r οΏ½ and apply Theorem 5 with Ξ΅ replaced with Ξ΅/4 so we get οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, p)) β οΏ½ (rβ1)D r log D οΏ½ 1 rβ1 οΏ½ (rβ1)D r log D οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ 4 (10) with probability at least 1 β o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Note that here D = οΏ½n β 1 r β 1 οΏ½ p = οΏ½n β 1 r β 1 οΏ½ m/ οΏ½n r οΏ½ = rm/n = d β d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now since d, D can be chosen to be arbitrarily large and d = D + O(D1/2 log D) we can replace D with d in (10) without changing the left hand side by more than Ξ΅/4 to obtain οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, p)) β οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ 2 (11) with probability at least 1βo(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' But now note that with probability β¦(nβ1/2) the number of edges in Hr(n, p) is precisely M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus we have that οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο(Hr(n, M)) β οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ β€ Ξ΅ 2 with probability at least 1 β o(nβ1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This proves the ο¬rst inequality, and the second one follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 6 Now we will start to transform Hr(n, m) to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This transformation will involve ο¬rst removing some edges from vertices of degree larger than d, and then adding some edges to vertices of degree less than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We deο¬ne the rank of a point pi β Vj, to be the number of points piβ² β Vj such that iβ² β€ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We form a new set of points P β² β P and a partition Rβ² of P β² as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For any Rk β R containing a point with rank more than d, we delete Rk from R and delete all points of Rk from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Note that each bucket contains at most r points of P β².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Note also that Rβ² is a uniform random partition of P β².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We let HRβ² be the natural hypergraph associated with Rβ².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now we would like to put some more points into the buckets until each bucket has exactly d points, arriving at some set of points P β²β² β P β².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We would also like a uniform partition Rβ²β² of P β²β² into sets of size r, and we would like Rβ²β² to have many of the same parts as Rβ².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will accomplish this by constructing a sequence P β² 1 := P β² β P β² 2 β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' β P β² β =: P β²β² of point sets and a sequence Rβ² 1 := Rβ², Rβ² 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' , Rβ² β =: Rβ²β² where Rβ² j is a uniform random partition of P β² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We construct P β² j+1, Rβ² j+1 from P β² j, Rβ² j as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Suppose |Rβ² j| = a (in other words Rβ² j has a parts), so |P β² j| = ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' P β² j+1 will simply be P β² j plus r new points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now we will choose a random value K β {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' , r} using the distribution P[K = k] = qk(a), where qk(a) is deο¬ned as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Deο¬nition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Consider a random partition of ra+r points into a+1 parts of size r, and ο¬x some set Q of r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then for 1 β€ k β€ r, the number qk(a) is deο¬ned to be the probability that Q meets exactly k parts of the partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will then remove a uniform random set of K β 1 parts from Rβ² j, leaving Kr points in P β² j+1 which are not in any remaining part of Rβ² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We partition those points into K parts of size r such that each part contains at least one new point (each such partition being equally likely), arriving at our partition Rβ² j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We claim that Rβ² j+1 is a uniform random partition of P β² j+1 into parts of size r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Indeed, ο¬rst consider the r new points that are in P β² j+1 which were not in P β² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The probability that a uniform random partition of P β² j+1 would have exactly k parts containing at least one new point is qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' So we can generate such a random partition as follows: ο¬rst choose a random value K with P[K = k] = qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' next we choose a uniform random set of (K β 1)r points from P β² j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' next we choose a partition of the set of points consisting of P β² j+1 \\ P β² j together with the points from P β² j we chose in the last step, where the partition we choose is uniformly random from among all partitions such that each part contains at least one point of P β² j+1 \\P β² j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' ο¬nally, we choose a uniform partition of the rest of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' In our case this partition of the rest of the points comprises the current partition of the βunusedβ (a β K + 1)r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' At the end of this process we have that HRβ²β² is distributed as Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='1 Bounding the number of low degree vertices in HRβ² We deο¬ne some sets of buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We show that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' there are few small buckets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='e few vertices of low degree in the hypregraph HRβ².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let S0 be the buckets with at most dβ3d1/2 log d points of P β², and let S1 be the buckets with at most dβ2d1/2 log d points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let S2 be the set of buckets that, when we remove points from P β² to get P, have at least d1/2 log d points removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then S0 β S1 βͺ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Our goal is to bound the probability that S0 is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Fix a bucket Vj and let X βΌ Bin οΏ½ rm, 1 n οΏ½ be the number of points of P in Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then the probability that Vj is in S1 satisο¬es P[Vj β S1] = P οΏ½ X β€ d β 2d1/2 log d οΏ½ = P οΏ½ X β rm n β€ βd1/2 log d οΏ½ β€ exp οΏ½ β d log2 d 3(d β d1/2 log d) οΏ½ = exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ , where for our inequality we have used the Chernoο¬ bound (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Therefore E[|S0|] β€ exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now we argue that |S1| is concentrated using McDiarmidβs inequality (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For our application we let X = |S1| which is a function (say f) of the vector (Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Zrm) where Zi tells us which bucket the ith point of P went into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Moving a point from one bucket to another can only change |S1| by at most 1 so we use c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus we get the bound P(|X β E[X]| β₯ n2/3) β€ 2 exp οΏ½ β n4/3 2rm οΏ½ = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (12) Now we handle S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For 1 β€ j β€ n let Yj be the number of parts Rk β R such that Rk contains a point in the bucket Vj as well as a point in some bucket Vjβ² where |Vjβ²| > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Note that if Vj β S2 then Yj β₯ d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We view Rk as a set of r points, say {q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' , qr} each going into a uniform random bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Say qi goes to bucket Vji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The probability that Rk is counted by Yj is at most rP[j1 = j and |Vj1| > d] + r(r β 1)P[j1 = j and |Vj2| > d] = r nP[|Vj1| > d οΏ½οΏ½j1 = j] + r(r β 1) n P[|Vj2| > d οΏ½οΏ½j1 = j] β€ r2 n P[|Vj1| > d οΏ½οΏ½j1 = j] β€ r2 n P[Bin(rm β 1, 1/n) β₯ d] = r2 n exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus we have E[Yj] = m Β· r2 n exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ β€ rd exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ = rd1/2 exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ and so Markovβs inequality gives us P οΏ½ Yj β₯ d1/2 log d οΏ½ β€ rd exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ d1/2 log d = exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ 8 and so E[|S2|] = n exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We use McDiarmidβs inequality once more, this time with X = |S2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' A change in choice of bucket changes |S2| by at most one and so (12) continues to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus |S0| = n exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2 A property of independent subsets of Hr(n, m) Fix 1 β€ j β€ r β 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Set a := οΏ½ 1 + Ξ΅ 2 οΏ½ οΏ½ r log d (r β 1)d οΏ½ 1 rβ1 , ΞΊj := 10d r οΏ½r j οΏ½ aj, p := d(r β 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' nrβ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The expected number of independent sets A in Hr(n, p) of size at most an such that there are ΞΊjn edges each having j vertices in A is at most an οΏ½ s=1 οΏ½n s οΏ½ (1 β p)(s r) οΏ½οΏ½s j οΏ½οΏ½ n rβj οΏ½ ΞΊjn οΏ½ pΞΊjn β€ an οΏ½ s=1 exp \uf8f1 \uf8f2 \uf8f3s log οΏ½en s οΏ½ β οΏ½s r οΏ½ p + ΞΊjn log \uf8eb \uf8ede(an)j j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' nrβj (rβj)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p ΞΊjn \uf8f6 \uf8f8 \uf8fc \uf8fd \uf8fe = an οΏ½ s=1 exp οΏ½ s log οΏ½en s οΏ½ β οΏ½s r οΏ½ p + ΞΊjn log οΏ½eaj 10 οΏ½οΏ½ β€ an Β· exp οΏ½οΏ½ log οΏ½e a οΏ½ β 10d r οΏ½r j οΏ½ ajβ1 log οΏ½10 e οΏ½οΏ½ an οΏ½ = o(1/n) where the last line follows since as d β β we have log οΏ½e a οΏ½ βΌ 1 r β 1 log d and 10d r οΏ½r j οΏ½ ajβ1 log οΏ½10 e οΏ½ = β¦ οΏ½ d rβj jβ1 logβ jβ1 rβ1 d οΏ½ β« log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus with probability 1 β o(1/n), Hr(n, p) has a coloring using (1 + Ξ΅/2) οΏ½ (rβ1)d r log d οΏ½ 1 rβ1 colors such that for each color class A and for each 1 β€ j β€ r β 1 there are at most ΞΊjn edges with j vertices in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The hypergraph Hr(n, m), m = οΏ½n r οΏ½ p will have this property w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='. 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='3 Transforming HRβ² into Hr(n, d) Now we will complete the transformation to the random regular hypergraph Hr(n, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We are open to the possibility that doing so will render our coloring no longer proper, since this 9 process will involve changing some edges which might then be contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We will keep track of how many such βbadβ edges there are and then repair our coloring at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We have to add at most (3d1/2 log d + d exp οΏ½ ββ¦ οΏ½ log2 d οΏ½οΏ½ )n < (4d1/2 log d)n points, which takes at most as many steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' For each color class A of HRβ² deο¬ne XA,j = XA,j(i) to be the number of edges with j vertices in A at step i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We have already established that XA,j(0) β€ ΞΊjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This follows from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='2 and the fact that we have removed edges from H(n, m) to obtain HRβ².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let Ei be the event that at step i we have that for each color class A and for each 1 β€ j β€ r β 1 we have XA,j(i) β€ 2ΞΊjn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Then, assuming Ei holds, the probability that XA,j increases at step i is at most οΏ½ 1β€kβ€r, jββ₯1 j1+Β·Β·Β·+jk=j οΏ½ 1β€ββ€k 2ΞΊjβn nd/r = οΏ½ 1β€kβ€r, jββ₯1 j1+Β·Β·Β·+jk=j οΏ½ 1β€ββ€k 20 οΏ½ r jk οΏ½ ajk β€ οΏ½ 1β€kβ€r, jββ₯1 j1+Β·Β·Β·+jk=j 20r2r2aj β€ 40r2r2aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Also, the largest possible increase in XA,j in one step is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus, the ο¬nal value of XA,j after at most (4d1/2 log d)n steps is stochastically dominated by ΞΊjn + rY where Y βΌ Bin οΏ½ (4d1/2 log d)n, 40r2r2ajοΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' An easy application of the Chernoο¬ bound tells us P (Y > 2E[Y ]) β€ exp(ββ¦(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (13) Note that here 2E[Y ] ΞΊjn = 8d1/2 log d Β· 40r2r2ajn 10d οΏ½r j οΏ½ ajn/r = O(dβ1/2 log d) < 1 for suο¬ciently large d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' Thus, using (13) and the union bound over all color classes A, we have w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+
page_content=' the ο¬nal value of XA,j is at most ΞΊjn + 2E[Y ] β€ 2ΞΊjn for all 1 β€ j β€ r β 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now we address βbadβ edges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' edges contained in a color class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Assuming Ei holds, the ex- pected number of new edges contained in any color class at step i is at most r(40)r2r2+2rar = O οΏ½οΏ½ log d d οΏ½ r rβ1οΏ½ (because it would have to be one of the colors of one of the vertices we are adding points to).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Thus the expected number of bad edges created in (4d1/2 log d)n steps is stochastically dominated by Z βΌ r Β· Bin οΏ½ (4d1/2 log d)n, O οΏ½οΏ½log d d οΏ½ r rβ1οΏ½ οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Another easy application of Chernoο¬ shows that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Z β€ 2E[Z] = O(dβ1/2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We repair the coloring as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' First we uncolor one vertex from each bad edge, and let the set of uncolored vertices be U where |U| = u = O οΏ½ dβ1/2n οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Let Ξ΄ := Ξ΅ 2 οΏ½(r β 1)d r log d οΏ½ 1 rβ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We claim that for every S β U, |S| = s, the hypergraph induced on S has at most Ξ΄s/r edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This will complete our proof since it implies that the minimum degree is at most Ξ΄ and so U can be recolored using a fresh set of Ξ΄ colors, yielding a coloring of Hr(n, d) using 10 at most Ο(Hr(n, M)) + Ξ΄ β€ οΏ½ 1 + Ξ΅ 2 οΏ½ οΏ½(r β 1)d r log d οΏ½ 1 rβ1 + Ξ΅ 2 οΏ½(r β 1)d r log d οΏ½ 1 rβ1 = (1 + Ξ΅) οΏ½(r β 1)d r log d οΏ½ 1 rβ1 colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' The expected number of sets S with more than Ξ΄s/r edges is at most οΏ½ 1β€sβ€u οΏ½n s οΏ½οΏ½οΏ½ds r οΏ½ Ξ΄s/r οΏ½ 1 οΏ½dn r οΏ½οΏ½dnβr r οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' οΏ½dnβΞ΄s+r r οΏ½ β€ οΏ½ 1β€sβ€u οΏ½ne s οΏ½s οΏ½(dse/r)re Ξ΄s/r οΏ½Ξ΄s/r (r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' )Ξ΄s/r (dn β Ξ΄s)Ξ΄s β€ οΏ½ 1β€sβ€u οΏ½ ne s οΏ½ dse (dn β Ξ΄s)r οΏ½Ξ΄ οΏ½er Β· r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ξ΄s οΏ½Ξ΄/rοΏ½s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (14) Now for 1 β€ s β€ βn the term in (14) is at most οΏ½ O(n) Β· οΏ½ O(nβ1/2) οΏ½Ξ΄ Β· O(1) οΏ½s = o(1/n) since Ξ΄ can be made arbitrarily large by choosing d large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Meanwhile for βn β€ s β€ u we have that the term in (14) is at most οΏ½ O(n1/2) Β· O(1) Β· οΏ½ O(nβ1/2) οΏ½Ξ΄/rοΏ½s = o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Now since (14) has O(n) terms the whole sum is o(1) and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 4 Summary We have asymptotically computed the chromatic number of random r-uniform, d-regular hypergraphs when proper colorings mean that no edge is mono-chromatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' It would seem likely that the approach we took would extend to other deο¬nitions of proper coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' We have not attempted to use second moment calculations to further narrow our estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' These would seem to be two natural lines of further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Ayre, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Coja-Oghlan and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Greenhill, Hypergraph coloring up to condensation, Random Structures and Algorithms 54 (2019) 615 - 652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 11 [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Achlioptas and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Moore, The Chromatic Number of Random Regular Graphs, In Jansen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=', Khanna, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=', Rolim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=', Ron, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' (eds) Approximation, Random- ization, and Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Algorithms and Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' RANDOM AP- PROX 2004 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Lecture Notes in Computer Science, vol 3122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Springer, Berlin, Hei- delberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Approximation, Randomization, and Combinatorial Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Algorithms and Techniques (2004) 219β228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Achlioptas and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Naor, The two possible values of the chromatic number of a random graph, Annals of Mathematics 162 (2005) 1335-1351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Bennett, The Matching Process and Independent Process in Random Regular Graphs and Hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' BollobΒ΄as, The chromatic number of random graphs, Combinatorica 8 (1988) 49-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' BollobΒ΄as and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' ErdΛos, Cliques in random graphs, Mathematical Proceedings of the Cambridge Philosophical Society 80 (1976) 419-427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Coja-Oghlan, Upper-Bounding the k-Colorability Threshold by Counting Covers, Electronic Journal of Combinatorics 20 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Coja-Oghlan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Efthymiou and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Hetterich, On the chromatic number of random regular graphs, Journal of Combinatorial Theory B 116 (2016) 367-439.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Dyer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Frieze and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Greenhill, On the chromatic number of a random hyper- graph, Journal of Combinatorial Theorey B 113 (2015) 68-122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Frieze, On the independence number of random graphs, Discrete Mathematics 81 (1990) 171-176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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337 |
+
page_content=' Frieze and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' KaroΒ΄nski, Introduction to Random Graphs, Cambridge University Press, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [12] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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340 |
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page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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341 |
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page_content=' Frieze and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' οΏ½Luczak, On the independence and chromatic numbers of random regular graphs, Journal of Combinatorial Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Series B 54 (1992) 123-132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [13] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Grimmett and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' McDiarmid, On colouring random graphs, Mathematical Proceed- ings of the Cambridge Philosophical Society 77 (1975) 313-324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Krivelevich and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Sudakov, The chromatic numbers of random hypergraphs, Ran- dom Structures Algorithms 12 (1998) 381-403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' οΏ½Luczak, The chromatic number of random graphs, Combinatorica 11 (19990) 45-54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [16] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' οΏ½Luczak, A note on the sharp concentration of the chromatic number of random graphs, Combinatorica 11 (1991) 295-297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' [17] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Matula, Expose-and-Merge Exploration and the Chromatic Number of a Random Graph, Combinatorica 7 (1987) 275-284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' 12 [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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page_content=' Shamir and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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358 |
+
page_content=' Spencer, Sharp concentration of the chromatic number od random graphs Gn,p, Combinatorica 7 (1987) 121-129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
359 |
+
page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
360 |
+
page_content=' Shi and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
361 |
+
page_content=' Wormald, Coloring random regular graphs, Combinatorics, Probability and Computing 16 (2007) 459-494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
362 |
+
page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
|
GtAzT4oBgHgl3EQfUvxQ/content/tmp_files/2301.01271v1.pdf.txt
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1 |
+
arXiv:2301.01271v1 [econ.GN] 15 Dec 2022
|
2 |
+
On the notion of measurable utility on a
|
3 |
+
connected and separable topological space:
|
4 |
+
an order isomorphism theorem.β
|
5 |
+
Gianmarco Caldini
|
6 |
+
7 February 2020
|
7 |
+
Abstract
|
8 |
+
The aim of this article is to deο¬ne a notion of cardinal utility function called
|
9 |
+
measurable utility and to deο¬ne it on a connected and separable subset of a weakly
|
10 |
+
ordered topological space. The deο¬nition is equivalent to the ones given by Frisch
|
11 |
+
in 1926 and by Shapley in 1975 and postulates axioms on a set of alternatives that
|
12 |
+
allow both to ordinally rank alternatives and to compare their utility diο¬erences.
|
13 |
+
After a brief review of the philosophy of utilitarianism and the history of utility
|
14 |
+
theory, the paper introduces the mathematical framework to represent intensity
|
15 |
+
comparisons of utility and proves a list of topological lemmas that will be used in
|
16 |
+
the main result. Finally, the article states and proves a representation theorem,
|
17 |
+
see Theorem 5, for a measurable utility function deο¬ned on a connected and sep-
|
18 |
+
arable subset of a weakly ordered topological space equipped with another weak
|
19 |
+
order on its cartesian product. Under some assumptions on the order relations,
|
20 |
+
Theorem 5 proves existence and uniqueness, up to positive aο¬ne transformations,
|
21 |
+
of an order isomorphism with the real line.
|
22 |
+
βI am grateful to Professor Massimo Marinacci for letting me know about the open problem.
|
23 |
+
1
|
24 |
+
|
25 |
+
Introduction
|
26 |
+
Together with notions such as value, money, market and economic agents, utility has
|
27 |
+
been one of the most controversial concepts in the whole history of economic theory.
|
28 |
+
The most important debate can be considered the one around the question whether it
|
29 |
+
is possible to deο¬ne a clear and rigorous concept of utility and an appropriate notion
|
30 |
+
of unit of measurement for utility, seen as a quantity like the physical ones. In the ο¬rst
|
31 |
+
chapter we will give a short introduction to the evolution of the concept of utility from
|
32 |
+
both a philosophical and a historical point of view. Our treatment is far from being
|
33 |
+
exhaustive. For an extensive treatment of history of utility and utility measurements,
|
34 |
+
we refer the interested reader to Stigler [39], Majumdar [26], Adams [1], Luce and
|
35 |
+
Suppes [24], Fishburn [15], [16], [17] and Moscati [29].
|
36 |
+
The second chapter will shift from the descriptive part to more formal concepts
|
37 |
+
and will be used to introduce the usual mathematical framework of decision theory.
|
38 |
+
Moreover, we will introduce deο¬nitions and axioms that will enable us to represent
|
39 |
+
comparisons of the intensity that a decision maker feels about the desirability of diο¬erent
|
40 |
+
alternatives. For this aim, we will follow the construction of Suppes and Winet [40]
|
41 |
+
and Shapley [36].
|
42 |
+
The third and last chapter will be entirely devoted to the proof of Shapleyβs theo-
|
43 |
+
rem, extending the domain of alternatives X from a convex subset of R to a connected
|
44 |
+
and separable subset of a topological space, hence providing a generalization of his
|
45 |
+
theorem. Our intended goal is to deο¬ne a rigorous notion of a speciο¬c kind of cardinal
|
46 |
+
utility function, not only able to rank alternatives, but also to compare utility diο¬er-
|
47 |
+
ences. In particular, we deο¬ne a βtwofoldβ utility function in line with the primordial
|
48 |
+
axiomatization of Frisch [18], calling it a measurable utility function. In mathematical
|
49 |
+
terms, we will prove a speciο¬c order-isomorphism theorem between a totally ordered,
|
50 |
+
connected and separable subset of a topological space and the real line.
|
51 |
+
1
|
52 |
+
Philosophy and history of utility theory
|
53 |
+
Theory of felicity, theory of justice, theory of morality, theory of virtue and theory of
|
54 |
+
utility are among the most important theories of moral philosophy and, as such, they
|
55 |
+
are constantly sources of questions that often do not ο¬nd an immediate answer. When
|
56 |
+
a human being acts, or when she makes a decision, she is, at the same time, looking for
|
57 |
+
1
|
58 |
+
|
59 |
+
justiο¬cations, either positive or normative, for the decision she has just made. We, as
|
60 |
+
human beings, are constantly trying to prove that what we did was the best thing to
|
61 |
+
do, in some well-deο¬ned sense, or, at least, the less harmful. These justiο¬cations take
|
62 |
+
into account the means, the ends and all the possible paths we have to reach our goals.
|
63 |
+
Moral philosophy is the science that comes into place when we formulate questions
|
64 |
+
about the ends, the means and the possible ways to achieve them.
|
65 |
+
Moral philosophy is essentially composed by principles, also called norms, on what
|
66 |
+
is good and what is bad. They allow to deο¬ne and to judge human actions, means and
|
67 |
+
ends. Sometimes, norms take the form of universal laws to which all human beings
|
68 |
+
are subjected.
|
69 |
+
Nevertheless, the formulation of moral laws or rules that prescribe
|
70 |
+
what a single agent should do or not do are intrinsically tied with history. Historical
|
71 |
+
experiences determine our vision of the world. Our moral philosophy is the result of
|
72 |
+
diο¬erent heritages that formed a common culture in which values like human respect, an
|
73 |
+
idea of equality between human beings and impartiality are among the most important.
|
74 |
+
Together with this general deο¬nition of morality, there exist the similar concepts of
|
75 |
+
ethics and of role morality - a speciο¬c form of professional morality. It was Jeremy Ben-
|
76 |
+
tham, in an unο¬nished manuscript which was posthumously published in 1834, to deο¬ne
|
77 |
+
the neologism deontology in the title of his book Deontology or the Science of Morality.
|
78 |
+
The manuscript stated, for the ο¬rst and only time, the particular aspects of Benthamβs
|
79 |
+
utilitarian theory as moral philosophy. This passage is clearly mentioned in SΓΈrensen
|
80 |
+
[37]:
|
81 |
+
[...] pointing out to each man on each occasion what course of conduct promises to be in
|
82 |
+
the highest degree conducive to his happiness: to his own happiness, ο¬rst and last; to the
|
83 |
+
happiness of others, no farther than in so far as his happiness is promoted by promoting
|
84 |
+
theirs, than his interest coincides with theirs (p. 5).
|
85 |
+
In this passage we can see how Bentham considered deontology to be primarily
|
86 |
+
aimed at oneβs own private felicity. Nevertheless, this does not bring any selο¬sh concern.
|
87 |
+
Benthamβs goal can be identiο¬ed with the objective study and measurement of passions
|
88 |
+
and feelings, pleasures and pains, will and action. Among these particular pleasures are
|
89 |
+
those stemming from sympathy - in Adam Smithβs sense - and they include the genuine
|
90 |
+
pleasure being happy for the good of others.
|
91 |
+
In this light, Bentham spent his life in search of the cardinal principle of ethics
|
92 |
+
and he found it in Epicurean ethics of hedonism. Hedonism comes from Greek ῾ηδον´η,
|
93 |
+
which means pleasure. Thus, classic utilitarianism, founded on hedonism, started from
|
94 |
+
2
|
95 |
+
|
96 |
+
the principle that pleasure is an intrinsic positive value and sorrow is an intrinsic neg-
|
97 |
+
ative value. It is, for this reason, somehow curious that Bentham conception, founded
|
98 |
+
on pleasure, had been called utilitarianism, from the simple observation that what is
|
99 |
+
useful is not necessarily pleasant or providing pleasure. We need always to take into
|
100 |
+
account that the term utility is intended in a functional sense; what gives utility is what
|
101 |
+
contributes the most to the individual, or universal, pleasure.
|
102 |
+
Classical utilitarian philosophers considered utilitarianism well-founded and realistic
|
103 |
+
thanks to the fact that it is based on pleasure. It is well-founded as its norms are jus-
|
104 |
+
tiο¬ed by an intrinsic, absolute value, that does not need any further justiο¬cation. It is
|
105 |
+
realistic because they thought human being to ultimately seek the maximum pleasure
|
106 |
+
and the minimum sorrow. More speciο¬cally, human beings try to choose the action
|
107 |
+
that will provide the maximum excess of pleasure against grief.
|
108 |
+
For Bentham, what really matters is the total amount of pleasure, intended as
|
109 |
+
the total excess of pleasure against sorrow: the only reasons for human actions are the
|
110 |
+
quests for pleasure, avoiding sorrow: they are the sources of our ideas, our judgments
|
111 |
+
and our determinations. Human moral judgments become statements on happiness;
|
112 |
+
pleasure (or felicity) is good and sorrow is bad. Utilitarian moral can be considered as
|
113 |
+
a βcalculated hedonismβ, that carefully evaluates the characteristics of pleasure. Wise
|
114 |
+
is the man that is able to restrain from an immediate pleasure for a future good that,
|
115 |
+
in comparison, will be more beneο¬cial. On the other side, being able to evaluate the
|
116 |
+
positive or negative consequences of an action without making mistakes is fundamental.
|
117 |
+
Hence, the correct utilitarian person should reach some kind of βmoral arithmeticβ that
|
118 |
+
allows the correct calculations to be carried out. Far from being a unanimously accepted
|
119 |
+
doctrine, we cannot forget to mention that Alessandro Manzoni wrote an essay [27] in
|
120 |
+
which he strongly criticized Benthamβs utilitarianism, saying that it is utterly wrong to
|
121 |
+
think that human beings build their moral values judgment of their actions on utility.1
|
122 |
+
From this explanation of utilitarianism, Benthamβs evaluation criterion of actions
|
123 |
+
follows as an immediate corollary: the maximum happiness for the maximum number
|
124 |
+
of people. Again, happiness is intended as state of pleasure, or absence of grief. Hence,
|
125 |
+
individual pleasure becomes no more the ultimate goal: it is the universal pleasure to
|
126 |
+
be hegemonic.2
|
127 |
+
1Manzoni [27] wrote: βNon ci vuol molto a scoprir qui un falso ragionamento fondato sullβalterazione
|
128 |
+
dβun fatto. Altro `e che lβutilit`a sia un motivo, cio`e uno deβ motivi per cui gli uomini si determinano
|
129 |
+
nella scelta dellβazioni, altro `e che sia, per tutti gli uomini, il motivo per eccellenza, lβunico motivo
|
130 |
+
delle loro determinazioni (p.775).
|
131 |
+
2This tension between individual pleasure and universal pleasure is one of the principal diο¬culties
|
132 |
+
3
|
133 |
+
|
134 |
+
This view of utilitarianism admits, at least in our minds, the conception of the
|
135 |
+
existence of a scale of pleasure in which pleasure and sorrow can be added and sub-
|
136 |
+
tracted. In other words, the idea of a calculus of felicity and grief is not completely
|
137 |
+
absurd, both in intrapersonal and interpersonal compensations.
|
138 |
+
1.1
|
139 |
+
Brief history of utilitarianism
|
140 |
+
Although it is possible to ο¬nd utilitarian reasonings in Aristoteleβs works, it is com-
|
141 |
+
monly agreed that the beginning of the history of utility can be identiο¬ed with 18th
|
142 |
+
century moral philosophy. To be even more speciο¬c, Benthamβs ideas were not isolated,
|
143 |
+
since they were already present in works by his illuministic predecessors like Richard
|
144 |
+
Cumberland, Francis Hutcheson and Cesare Beccaria. Especially Hutcheson [20] had
|
145 |
+
already deο¬ned good as pleasure and good objects as objects that create pleasure. The
|
146 |
+
novelty of Bentham was to treat pleasure as a measurable quantity, thus making the
|
147 |
+
utilitarian doctrine directly applicable to issues like tax policies and legislation. In-
|
148 |
+
deed, not only did Bentham argue that individual pleasure was measurable, but also
|
149 |
+
that happiness of diο¬erent people could be compared. Stark [38] cited in his article
|
150 |
+
Benthamβs writings in the following way:
|
151 |
+
Fortunes unequal: by a particle of wealth, if added to him who has least, more happiness
|
152 |
+
will be produced, than if added to the wealth of him who has most (vol. 1, p. 103).
|
153 |
+
Stark [38] continues:
|
154 |
+
The quantity of happiness produced by a particle of wealth (each particle being the same
|
155 |
+
magnitude) will be less and less every particle (vol. 1, p. 113).
|
156 |
+
It is easy to see how this last concept and the well-known idea of decreasing
|
157 |
+
marginal utility are related.
|
158 |
+
In the pioneering work of Jevons [21], utility functions were the primitive mathe-
|
159 |
+
matical notion to formalize and quantify Benthamβs calculus of pleasure. Utility func-
|
160 |
+
tions were tools to measure and scale the amount of well-being of human beings. It
|
161 |
+
seems clear, at this point, how the starting role of utility functions was cardinal,3 in the
|
162 |
+
sense that utility or, better, pleasure diο¬erences were well-founded and realistic notions
|
163 |
+
with a strong moral philosophy justiο¬cation.
|
164 |
+
of utilitarian moral philosophy.
|
165 |
+
3Note that before the work of Hicks and Allen [19], economists spoke about measurable utility and
|
166 |
+
not of cardinal utility.
|
167 |
+
4
|
168 |
+
|
169 |
+
Summing up, in the beginning, utility functions were designed for the mere purpose
|
170 |
+
of a calculus of pleasure and sorrow. However, even if the philosophical concept made
|
171 |
+
sense, the diο¬culties in the quantiο¬cation of any experimental measurement of pleasure
|
172 |
+
led cardinal utility theory to be seen more just like a thought process rather than a
|
173 |
+
science.
|
174 |
+
However, utility theory did not rise from philosophy alone, but it was object of
|
175 |
+
study of other sciences such as statistics, with the so-called St. Petersburg paradox,
|
176 |
+
and psychophysics, the study of physical stimuli and their relation to sensory reactions.
|
177 |
+
These two phenomena can be considered the starting point of the law of decreasing
|
178 |
+
marginal utility. It was Nicolas Bernoulli that, originally, invented what is now called
|
179 |
+
the St. Petersburg puzzle, which oο¬ered the theoretical explanation for the law of de-
|
180 |
+
creasing marginal utility of wealth. The standard version of the puzzle is the following:
|
181 |
+
a fair coin is tossed until it lands βheadβ on the ground. At that point, the player wins
|
182 |
+
2n dollars, where n is the number of times the coin was ο¬ipped. How much should one
|
183 |
+
be willing to pay for playing this game? In other words, what is the expected value of
|
184 |
+
the game, given the probability of βheadβ being 0.5? The mathematical answer is
|
185 |
+
β
|
186 |
+
οΏ½
|
187 |
+
i=1
|
188 |
+
1
|
189 |
+
2i Β· 2i = 1 + 1 + Β· Β· Β· = β.
|
190 |
+
The only rationale for this conundrum is that, if it makes sense to maximize expected
|
191 |
+
utility and if people are willing to participate to the St. Petersburg game for only a
|
192 |
+
ο¬nite amount of money, then their marginal utility as a function of wealth must be,
|
193 |
+
somewhere, decreasing.
|
194 |
+
Neither Bentham nor Bernoulli thought as decreasing marginal utility as a phe-
|
195 |
+
nomenon in need of scientiο¬c justiο¬cations. Nevertheless, this came as an immediate
|
196 |
+
consequence from the psychophysical theories discovered by Weber [42] and generalized
|
197 |
+
by Fechner [13]. One of the most important questions posed by psychophysics is what is
|
198 |
+
the functional link between diο¬erent degrees of a given stimulus and a given sensation.
|
199 |
+
What Weber did was an in-depth study to try to measure the smallest detectable change
|
200 |
+
(also called βjust noticeable diο¬erenceβ or βminimum perceptible thresholdβ) in stimuli
|
201 |
+
like heat, weight and pitch. Moreover, Fechner took this βjust noticeable diο¬erenceβ
|
202 |
+
as a unit of measurement, constructing a scale for subjective sensations. From their
|
203 |
+
studies we now have the so-called Weberβs law and Fechnerβs laws: the former states
|
204 |
+
that the relative increase of a stimulus needed to produce a just noticeable change is
|
205 |
+
5
|
206 |
+
|
207 |
+
constant and, the latter, that the magnitude of sensation is a logarithmic function of
|
208 |
+
the stimulus.
|
209 |
+
In conclusion, if wealth is a stimulus, then Benthamian utility must be the cor-
|
210 |
+
responding sensation.
|
211 |
+
In this light, St.
|
212 |
+
Petersburg puzzle can be seen as just one
|
213 |
+
materialization of these laws.
|
214 |
+
At the end of 19th century, the marginalist revolution paved the way for an ordinal
|
215 |
+
approach to the notion of utility. In fact, this was because one of the main economic
|
216 |
+
problems of late 19th century was the need of a theory of demand. One of the leading
|
217 |
+
ο¬gures that founded neoclassical theory with scientiο¬c and analytic rigor was Vilfredo
|
218 |
+
Pareto. Pareto is considered the father of the so-called ordinal approach. It was a notion
|
219 |
+
of utility that was purely comparative and it left out from the theory the initial idea for
|
220 |
+
which utility theory was developed: the existence of psychophysical and physiological
|
221 |
+
substrates. Paretoβs theory was so successful that was considered a revolution in the
|
222 |
+
notion of utility. The ordinal approach was extremely successful because it solved the
|
223 |
+
classic consumer problem based on indiο¬erence curves, and the notion of utility had a
|
224 |
+
central role in its construction. The key aspect was the replacement of marginal utility
|
225 |
+
- a notion that was meaningless in an ordinal approach - with the trick of marginal rate
|
226 |
+
of substitutions along indiο¬erence curves.
|
227 |
+
Interesting are the writings of Francis Edgeworth [10] and Pareto [31], starting
|
228 |
+
from very diο¬erent assumptions and arriving at diο¬erent conclusions.
|
229 |
+
Edgeworthβs
|
230 |
+
main contribution can be summarized in the synthesis of Benthamβs utilitarianism and
|
231 |
+
Fechnerβs psychophysics: his ideas were based on the unit of utility seen as a just
|
232 |
+
perceivable increment of pleasure. Moreover, he was interested also in an inter-personal
|
233 |
+
unit of utility to be able to carry out welfare comparisons among people. Edgeworth
|
234 |
+
was completely aware of the impossibility of testing these implications, but he was a
|
235 |
+
strong supporter of the idea of possible comparisons of happiness among people.
|
236 |
+
Pareto, on the other hand, denied Edgeworthβs intuition of comparisons of utility.
|
237 |
+
Instead, Pareto [31] reckoned the theoretical possibility of a cardinal notion of utility,
|
238 |
+
seen as the limit of the purely comparative notion he developed. Nevertheless, he also
|
239 |
+
argued that such a notion of perfect precision is not attainable and that pleasure is only
|
240 |
+
imperfectly measurable.4 Summing up, Edgeworthβs and Paretoβs ways of conceiving
|
241 |
+
measurable utility must be diο¬erentiated and utility theory is still today based on the
|
242 |
+
Paretian notion mainly because of its use in the theory of demand and in the general
|
243 |
+
4However, Pareto [32] writes: βThere is no reason for not accepting it [cardinal utility], with the
|
244 |
+
reservation that it must be veriο¬ed by the results deduced from it (p. 73).β
|
245 |
+
6
|
246 |
+
|
247 |
+
equilibrium theory.
|
248 |
+
In 1950, ordinalism was the well-established mainstream ideology in utility theory
|
249 |
+
and the cardinal notion of utility was almost completely abandoned. Nevertheless, the
|
250 |
+
purely comparative approach was not convincing everyone, mainly because peopleβs
|
251 |
+
introspection suggested the existence something more. One of the main supporters of
|
252 |
+
cardinalism was Maurice Allais who explicitly wrote in [4]:
|
253 |
+
The concept of cardinal utility [...] has almost been rejected the literature for half a cen-
|
254 |
+
tury. This rejection, based on totally unjustiο¬ed prejudices, deprived economic analysis
|
255 |
+
of an indispensable tool (p. 1).
|
256 |
+
Allais [4] admits that the theory of general economic equilibrium can be fully de-
|
257 |
+
scribed in an ordinal world, but he immediately lists a series of theories that cannot
|
258 |
+
be adequately developed without a rigorous and well-deο¬ned concept of cardinal utility
|
259 |
+
and interpersonal comparisons. Some examples are the theory of dynamic evolution of
|
260 |
+
the economy, the theory of ο¬scal policy, of income transfers, of collective preferences,
|
261 |
+
social welfare analysis and political choices, of risk, of insurance and the theory of
|
262 |
+
cooperative games. Then, Allais goes even further in his defense of cardinal utility,
|
263 |
+
arguing that even the theory of demand could become more intuitive - and with a
|
264 |
+
simpler exposition - if we could appeal to a notion of intensity of preferences. In any
|
265 |
+
case, as long as the conclusions of price theory do not change signiο¬cantly using the
|
266 |
+
ordinal and the cardinal approach, we should prefer the purely comparative approach
|
267 |
+
by Occamβs razor. But the problems with group decision making, social choice theory
|
268 |
+
and cooperative game theory still cannot be solved. Indeed, while classical economists
|
269 |
+
considered distributional problems as a fundamental part of economic science, the ordi-
|
270 |
+
nalist approach to utility theory refused completely to deal with questions that involved
|
271 |
+
interpersonal comparisons of welfare. Economists became more interested in positive
|
272 |
+
statements, rather than normative ones and the accent was put on eο¬ciency, rather
|
273 |
+
than equity. This was the case of the optimum allocations in the sense of Pareto. For
|
274 |
+
a complete overview of the main issues of welfare economics, the main problems with
|
275 |
+
an ordinal approach and the main literature, we refer the interested reader to Sen [35].
|
276 |
+
Hence, one of the main problems, still in the 21st century, is how is it possi-
|
277 |
+
ble to understand the intuitive tool of introspection to develop a rigorous theory that
|
278 |
+
economists can apply in their models and explain economic phenomena. The solution
|
279 |
+
does not exists yet. In the last years, the issue started getting the attention of few deci-
|
280 |
+
sion theorists mainly because of the powerful developments in the ο¬eld of neuroscience
|
281 |
+
7
|
282 |
+
|
283 |
+
and the new discipline of neuroeconomics. These ο¬elds of cognitive sciences are going
|
284 |
+
into the direction of overcoming the main diο¬culty of the founders of utilitarianism:
|
285 |
+
the diο¬culty of carrying out experiments on pleasure and pain and the construction of
|
286 |
+
a rigorous and well-deο¬ned scale of pleasure. Nevertheless, nothing is clear yet, mainly
|
287 |
+
because also the theoretical concept of cardinal utility is still vague. Cardinal utility
|
288 |
+
is still used as a name for a large number of formally distinct concepts and it misses a
|
289 |
+
precise and well-established deο¬nition that can be applied in decision-theoretic models.
|
290 |
+
During the 20th century a lot of methodologies to try to deο¬ne a concept of
|
291 |
+
measurement of human sensation have been deο¬ned.
|
292 |
+
Deο¬nition 1. A scale is a rule for the assignment of numbers to aspects5 of objects or
|
293 |
+
events.
|
294 |
+
The result was the development of a full taxonomy of scales, with scales that diο¬er
|
295 |
+
in terms of higher precision of measurement. For an extensive treatment of the theory
|
296 |
+
of measurement we refer the interested reader to Krantz et alii [22].
|
297 |
+
The issue of having a rigorous deο¬nition for cardinal utility was not solved by the
|
298 |
+
theory of measurement.
|
299 |
+
It was just translated in a diο¬erent language: what is the
|
300 |
+
suitable scale for measuring a given aspect? The deο¬nition of a unit of measurement
|
301 |
+
for utility was not an easy task to solve. Even in physics, where experiments can be
|
302 |
+
carried out with relatively high precision, the way a unit of measurement is deο¬ned
|
303 |
+
is not perfect. One meter was originated as the 1/10-millionth of the distance from
|
304 |
+
the equator to the north pole along a meridian through Paris. Then, the International
|
305 |
+
Bureau of Weights and Measures, founded in 1875, deο¬ned the meter as the distance
|
306 |
+
of a particular bar made by platinum and iridium kept in S`evres, near Paris. More
|
307 |
+
recently, in 1983, the Geneva Conference on Weights and Measures deο¬ned the meter
|
308 |
+
as the distance light travels, in a vacuum, in 1/299,792,458 seconds with time measured
|
309 |
+
by a cesium-133 atomic clock which emits pulses of radiation at very rapid and regular
|
310 |
+
intervals.
|
311 |
+
Increases in science allow the unit of measurement to be duplicated with a better
|
312 |
+
and better level of precision. The comparison with the unit of measurement of the
|
313 |
+
quantity utility can be carried out with the philosophical question whether it is, for
|
314 |
+
some esoteric reason, intrinsically impossible to measure human beingsβ pleasure or
|
315 |
+
whether economic science and neuroeconomics are so underdeveloped that we still have
|
316 |
+
5For example: hardness, length, volume, density, . . .
|
317 |
+
8
|
318 |
+
|
319 |
+
very poor precision in measuring human felicity.6
|
320 |
+
The same comparison can be done with light (or heat, color and wave lengths, as
|
321 |
+
it is mentioned by von Neumann and Morgenstern in [41]). For example, temperature
|
322 |
+
was, in the original concept, an ordinal quantity as long as the concept warmer was
|
323 |
+
known. Then, the ο¬rst transition can be identiο¬ed with the development of a more pre-
|
324 |
+
cise science of measurement: thermometry. With thermometry, a scale of temperature
|
325 |
+
that was unique up to linear transformations was constructed. The main feature was
|
326 |
+
the association of diο¬erent temperatures with diο¬erent classes of systems in thermal
|
327 |
+
equilibrium. Classes like these were called ο¬xed points for the scale of temperatures.
|
328 |
+
Then, the second transition can be associated with the development of thermodynam-
|
329 |
+
ics, where the absolute zero was ο¬xed, deο¬ning a reference point for the whole scale.
|
330 |
+
In physics, these phenomena had to be measured and the individual had to be able to
|
331 |
+
replicate results of such measurements every time. The same may apply to decision
|
332 |
+
theory and the notion of utility, someday. At the moment, the issue remains unclear,
|
333 |
+
even if even Pareto was not completely skeptical about the ο¬rst transition from an
|
334 |
+
ordinal purely comparative approach to that of an equality relation for utility diο¬er-
|
335 |
+
ences. Von Neumann and Morgenstern point out in [41] that the previous concept is
|
336 |
+
based on the same idea used by Euclid to describe the position on a line: the ordinal
|
337 |
+
utility concept of preference corresponds to Euclidβs notion of lying to the right of and
|
338 |
+
the derived concept of equality of utility diο¬erences with the geometrical congruence of
|
339 |
+
intervals.
|
340 |
+
Hence, the main question becomes whether the derived order relation on utility
|
341 |
+
diο¬erences can be observed and reproduced. Nobody can, at the moment, answer this
|
342 |
+
question.
|
343 |
+
1.2
|
344 |
+
Axiomatization of utility theories
|
345 |
+
In 1900, at the International Congress of Mathematicians in Paris, David Hilbert an-
|
346 |
+
nounced that he was ο¬rmly convinced that the foundation of mathematics was almost
|
347 |
+
complete. Then, he listed 23 problems to be solved and to give full consistency to
|
348 |
+
mathematics. All the rest was considered, by him, just details. Some of the problems
|
349 |
+
6Some authors, like Ellingsen [12], are certain, instead, that the philosophical question of whether
|
350 |
+
utility is intrinsically measurable or not is a spurious one, mainly because they see the issue of βmea-
|
351 |
+
surementβ as a concept that is always invented and never discovered. In this light, our question can
|
352 |
+
be rephrased as whether it is possible to deο¬ne a correct notion of measurement that allows some kind
|
353 |
+
of intrapersonal and interpersonal utility comparisons.
|
354 |
+
9
|
355 |
+
|
356 |
+
consisted in the axiomatizations of some ο¬elds of mathematics. Indeed, at the begin-
|
357 |
+
ning of the 20th century, the idea of being able to solve every mathematical problem
|
358 |
+
led mathematicians to try develop all mathematical theory from a ο¬nite set of axioms.
|
359 |
+
The main advantage of the axiomatic method was to give a clean order and to remove
|
360 |
+
ambiguity to the theory as a whole. Axioms are the fundamental truths by which it
|
361 |
+
is possible to start modeling a theory. The careful deο¬nition of them is critical in the
|
362 |
+
development of a theory that does not contain contradictions.
|
363 |
+
As a result, almost all ο¬elds of science started a process of axiomatization, utility
|
364 |
+
theory as well.
|
365 |
+
The ordinal Paretian revolution was the fertile environment where
|
366 |
+
preferences started to be seen as primitive notions. Preference relations began to be
|
367 |
+
formalized as mathematical order relations on a set of alternatives X and became the
|
368 |
+
starting point of the whole theory of choice.
|
369 |
+
As a result, utility functions became
|
370 |
+
the derived object from the preference relations. The mainstream notion of ordinal
|
371 |
+
(Paretian) utility reached its maturity with the representation theorems by Eilenberg
|
372 |
+
[11] and Debreu [8], [9]. Subsequent work in decision theory shifted from decision theory
|
373 |
+
under certainty to choice problems under uncertainty, with the pioneering article of
|
374 |
+
Ramsey [33] on the βlogic of partial belief.β In short, Ramsey [33] stated the necessity
|
375 |
+
of the development of a purely psychological method of measuring both probability
|
376 |
+
and beliefs, in strong contradiction with Keynesβ probability theory. Some years after,
|
377 |
+
the milestone works of von Neumann and Morgenstern [41] and Savage [34] gave full
|
378 |
+
authority to decision theory under uncertainty.
|
379 |
+
One of the ο¬rst treatments of preference relations as a primitive notion can be
|
380 |
+
identiο¬ed with Frisch [18], in his 1926 paper.
|
381 |
+
Ragnar Frisch was also the ο¬rst to
|
382 |
+
formulate an axiomatic notion of utility diο¬erence. Hence, two kinds of axioms were
|
383 |
+
postulated by him: the ο¬rst ones - called βaxioms of the ο¬rst kindβ - regarded the
|
384 |
+
relation able to rank alternatives in a purely comparative way, while the second axioms
|
385 |
+
- named βaxioms of the second kindβ - reο¬ected a notion of intensity of preference and
|
386 |
+
allowed utility diο¬erences to be compared.
|
387 |
+
So, in parallel to the axiomatization of
|
388 |
+
ordinal utility, also cardinal utility axiomatizations started to grow.
|
389 |
+
Frischβs article did not have the deserved impact in the academic arena, mainly
|
390 |
+
because his article was written in French and published in a Norwegian mathematical
|
391 |
+
journal. Hence, the full mathematical formalization of these two notions of preference
|
392 |
+
axioms resulted almost ten years later from the 1930s debate by Oskar Lange [23] and
|
393 |
+
Franz Alt [5]. Lange [23] deο¬ned an order relation β» on the set of alternatives X with
|
394 |
+
the meaning that, for any two alternatives x, y β X, x β» y reads βx is strictly preferred
|
395 |
+
10
|
396 |
+
|
397 |
+
to y.β Then, a corresponding relation P on ordering diο¬erences is assumed with the
|
398 |
+
meaning that, for any x, y, z, w β X, xyPzw reads βa change from y to x is strictly
|
399 |
+
preferred than a change from w to z.
|
400 |
+
More formally:
|
401 |
+
x β» y ββ u(x) > u(y) for all x, y β X
|
402 |
+
(1)
|
403 |
+
xyPzw ββ u(x) β u(y) > u(z) β u(w) for all x, y, z, w β X
|
404 |
+
(2)
|
405 |
+
The main theorem of Lange [23] can be stated as follows:
|
406 |
+
Theorem 1. If there exists a diο¬erentiable utility function u : R β R such that (1)
|
407 |
+
and (2) hold, then only positive aο¬ne transformations of that utility function represent
|
408 |
+
the given preferences β» and P.
|
409 |
+
It is immediate to see that Lange [23] provides only necessary conditions for a
|
410 |
+
utility function representation of preference relation. Moreover, it is relatively easy to
|
411 |
+
see that the assumption of diο¬erentiability of u can be largely relaxed. Hence, the issue
|
412 |
+
becomes whether it is possible to ο¬nd suο¬cient conditions on the preference relations
|
413 |
+
under which Langeβs utility function - a cardinal utility function - exists. This was
|
414 |
+
done by Franz Alt in his 1936 article [5]. Alt postulated seven axioms that guaranteed
|
415 |
+
suο¬cient and necessary conditions for the existence of a continuous utility function
|
416 |
+
- unique up to positive aο¬ne transformations - based on a preference relation and a
|
417 |
+
utility-diο¬erence ordering relation. In his set of axioms, Alt deο¬ned a notion that can
|
418 |
+
be understood as the set of alternatives X to be connected.
|
419 |
+
With Frischβs pioneering work of 1926 and 1930s debate by Lange and Alt, the
|
420 |
+
modern ingredients of cardinal utility axiomatization such as equations (1) and (2) and
|
421 |
+
connectedness of the domain of alternatives X started to be formalized. In those years,
|
422 |
+
a lot of diο¬erent axiomatic models were studied, till the article of the famous philoso-
|
423 |
+
pher of science Patrick Suppes and his doctoral student Muriel Winet [40]. In their
|
424 |
+
1955 paper, Suppes and Winet developed an abstract algebraic structure of axioms for
|
425 |
+
cardinal utility, called a diο¬erence structure, in line with old Frischβs ideas and Langeβs
|
426 |
+
formalization: not only are individuals able to ordinally rank diο¬erent alternatives, but
|
427 |
+
they are also able to compare and rank utility diο¬erences of alternatives. Indeed, Sup-
|
428 |
+
pes and Winet cited the work of Oskar Lange on the notion of utility diο¬erences and
|
429 |
+
stood in favor of the intuitive notion of introspection, elevating it to not just a mere
|
430 |
+
11
|
431 |
+
|
432 |
+
intuition, but as a solid base where to build a notion of utility diο¬erences. Suppes and
|
433 |
+
Winet continued their article saying that, up to 1950s, no adequate axiomatization for
|
434 |
+
intensity comparison had been given. Hence, as Moscati [29] nicely highlights, they
|
435 |
+
were probably unaware of Altβs representation theorem and this was probably due to
|
436 |
+
the fact that Alt [5] was published in German in a German journal. Suppes and Winet
|
437 |
+
postulated 11 axioms in total, some on the set of alternatives X and others on the
|
438 |
+
two order relations,7 providing suο¬cient and necessary condition for a cardinal utility
|
439 |
+
representation, unique up to positive aο¬ne transformations. Another approach to the
|
440 |
+
ο¬eld of axiomatization of cardinal utility was taken twenty years later by Lloyd Shap-
|
441 |
+
ley. While axiomatizations `a la Suppes and Winet started developing a set-theoretic
|
442 |
+
abstract structure, Shapley substituted the usual long list of postulates with strong
|
443 |
+
topological conditions both on the domain of alternatives X and on the topology in-
|
444 |
+
duced by the order relations. Shapley [36] constructed a cardinal utility function u
|
445 |
+
satisfying some consistency axioms between the orders and assuming the domain of u
|
446 |
+
to be a convex subset of the real line. We will enter into the details of Shapley [36] in
|
447 |
+
the next chapters.
|
448 |
+
In conclusion, the notion of cardinal utility has always suο¬ered a lack of conceptual
|
449 |
+
precision in its whole history and, for some authors like Ellingsen [12], it can be even
|
450 |
+
considered the main reason why scientists have disagreed over whether pleasure can be
|
451 |
+
measured or not.8 What is certain is that the history of cardinal utility, a part from some
|
452 |
+
sporadic articles, has been a persistent failure, mostly in its applications to economic
|
453 |
+
theory. While the main reason can be probably identiο¬ed with the almost total absence
|
454 |
+
of any rigorous and proven experimental measurement of pleasure, it is fair to observe
|
455 |
+
that part of its failure must be given to the strong reluctant opinion of the mainstream
|
456 |
+
ordinal βparty.β In fact, a large class of economists classify as βmeaninglessβ even the
|
457 |
+
mere introspective idea of a comparison of utility diο¬erence, and not just the concept
|
458 |
+
itself, when formalized in a purely comparative environment. This position is shown to
|
459 |
+
be, with a gentle expression, βepistemological laziness.β We should always remember
|
460 |
+
that no real progress in economic science can be derived from purely abstract reasoning,
|
461 |
+
but only from the combined eο¬ort of empirical measurements with theoretical analysis,
|
462 |
+
always under the wise guide of the compass of history and philosophy.
|
463 |
+
7The conditions these axioms impose are analogous to the conditions deο¬ned by Alt [5]: com-
|
464 |
+
pleteness, transitivity, continuity, and some form of additivity for the two order relations, and an
|
465 |
+
Archimedean property on the quaternary relation.
|
466 |
+
8Ellingsen [12] writes about a βfallacy of identityβ and βfallacy of unrelatedness.β
|
467 |
+
12
|
468 |
+
|
469 |
+
2
|
470 |
+
Preliminary results
|
471 |
+
The aim of this chapter is twofold. On one side, we introduce the mathematical frame-
|
472 |
+
work that enable us to represent intensity comparisons that a decision maker feels about
|
473 |
+
the desirability of diο¬erent alternatives. For this aim, we follow the construction of Sup-
|
474 |
+
pes and Winet [40] and Shapley [36]. On the other side, we state and prove a list of
|
475 |
+
lemmas that will be used in Theorem 5 and that allow us to generalize Shapleyβs proof
|
476 |
+
to a connected and separable subset of a topological space.
|
477 |
+
2.1
|
478 |
+
Basic deο¬nitions
|
479 |
+
Deο¬nition 2. A relation on a set X is a subset βΏ of the cartesian product X Γ X,
|
480 |
+
where x βΏ y means (x, y) β βΏ.
|
481 |
+
In decision theory, βΏ is usually called a preference relation, with the interpretation
|
482 |
+
that, for any two elements x, y β X, we write x βΏ y if a decision maker either strictly
|
483 |
+
prefers x to y or is indiο¬erent between the two.
|
484 |
+
Deο¬nition 3. An equivalence relation on a set X is a relation R on X that satisο¬es
|
485 |
+
1) Reο¬exivity: for all x β X, we have xRx.
|
486 |
+
2) Symmetry: for any two elements x, y β X, if xRy, then yRx.
|
487 |
+
3) Transitivity: for any three elements x, y and z β X, if xRy and yRz, then xRz.
|
488 |
+
Deο¬nition 4. A relation βΏ on a set X is called a total order relation (or a simple order,
|
489 |
+
or a linear order) if it has the following properties:
|
490 |
+
1) Completeness: for any two elements x, y β X, either x βΏ y or y βΏ x or both.
|
491 |
+
2) Antisymmetry: for any two elements x, y β X, if x βΏ y and y βΏ x, then x = y.
|
492 |
+
3) Transitivity: for any three elements x, y and z β X, if x βΏ y and y βΏ z, then
|
493 |
+
x βΏ z.
|
494 |
+
Note that if βΏ is complete, then it is also reο¬exive. The relation βΏ induces, in
|
495 |
+
turns, two other relations. Speciο¬cally, for any two elements x, y β X we write:
|
496 |
+
(i) x β» y if x βΏ y but not y βΏ x.
|
497 |
+
13
|
498 |
+
|
499 |
+
(ii) x βΌ y if x βΏ y and y βΏ x.
|
500 |
+
It is easy to see, indeed, that if βΏ is reο¬exive and transitive, then βΌ is an equivalence
|
501 |
+
relation. Given an equivalence relation βΌ on a set X and an element x β X, we deο¬ne
|
502 |
+
a subset E of X, called the equivalence class determined by x, by the equation
|
503 |
+
E := {y β X : y βΌ x}
|
504 |
+
Note that the equivalence class E determined by x contains x, since x βΌ x, hence
|
505 |
+
E is usually denoted as [x].
|
506 |
+
We will denote X/βΌ the collection {[x] : x β X} of
|
507 |
+
all equivalence classes, which is a partition of X: each x β X belongs to one, and
|
508 |
+
only one, equivalence class. In decision theory, an equivalence class is often called an
|
509 |
+
indiο¬erence curve.
|
510 |
+
Deο¬nition 5. A relation βΏ on a set X is called a weak order if it is complete and
|
511 |
+
transitive.
|
512 |
+
The problem of ο¬nding a numerical representation for a preference relation βΏ, i.e.
|
513 |
+
an order isomorphism between a generic set X and R, has been widely studied by math-
|
514 |
+
ematicians and is a familiar and well-understood concept. Such an order isomorphism
|
515 |
+
is called, in decision theory, a utility function. More formally:
|
516 |
+
Deο¬nition 6. A real-valued function u : X β R is a (Paretian) utility function for βΏ
|
517 |
+
if for all x, y β X we have
|
518 |
+
x βΏ y ββ u(x) β₯ u(y)
|
519 |
+
Utility functions βshiftβ the pairwise comparisons that characterize the order rela-
|
520 |
+
tion βΏ and its properties in the more analytically convenient space of the real numbers.
|
521 |
+
Nevertheless, as a result, the only thing that is preserved is the order, and the real
|
522 |
+
numbers that are images of the utility function cannot be interpreted as a scale where
|
523 |
+
the decision maker can compare diο¬erent intensities about the single desirability of any
|
524 |
+
two alternatives x, y β X. What is important is the ranking given by the real num-
|
525 |
+
bers, according to the usual order of the ordered ο¬eld (R, β₯). Indeed, one can easily
|
526 |
+
prove that every strictly increasing transformation of a utility function is again a utility
|
527 |
+
function. For this reason, utility functions are called ordinal and their study belong
|
528 |
+
to what is called ordinal utility theory. The main problem of ordinal utility theory is
|
529 |
+
to study suο¬cient and necessary conditions under which a relation βΏ admits a utility
|
530 |
+
representation. The original reference can be identiο¬ed with Cantor [7], but the result
|
531 |
+
has been adapted by Debreu [8].
|
532 |
+
14
|
533 |
+
|
534 |
+
In addition, to be able to solve optimization problems, one of the properties that
|
535 |
+
is desirable to have is continuity of the utility function. Debreu [8] is the ο¬rst to state
|
536 |
+
the theorem in the way we are going to. Nevertheless, he proved it making explicit
|
537 |
+
reference to Eilenberg [11]. We state here a version of this very well-known theorem.
|
538 |
+
Deο¬nition 7. A weak order βΏ on a set X is said to be continuous if, for every y β X,
|
539 |
+
the sets {x β X : x β» y} and {x β X : x βΊ y} are open.9
|
540 |
+
Theorem 2 (Eilenberg). Let βΏ be a complete and transitive relation on a connected
|
541 |
+
and separable topological space X. The following conditions are equivalent:
|
542 |
+
(i) βΏ is continuous.
|
543 |
+
(ii) βΏ admits a continuous utility function u : X β R.
|
544 |
+
One of the biggest theoretical problems of ordinal utility theory is that the expres-
|
545 |
+
sion
|
546 |
+
u(x) β u(y)
|
547 |
+
is a well-deο¬ned real number thanks to the algebraic properties of R, but it is meaning-
|
548 |
+
less in term of the interpretation of a diο¬erence of utility of two alternatives x, y β X.
|
549 |
+
In other words, a Paretian utility function does not have an intrinsic introspective psy-
|
550 |
+
chological notion of intensity of the preferences. An immediate corollary of this remark
|
551 |
+
is that the concept of marginal utility (and what is known under the Gossenβs law of
|
552 |
+
decreasing marginal utility), based on the notion of diο¬erent quotient, is meaningless.
|
553 |
+
More formally, the expression
|
554 |
+
du(x)
|
555 |
+
dx
|
556 |
+
= lim
|
557 |
+
hβ0
|
558 |
+
u(x + h) β u(x)
|
559 |
+
h
|
560 |
+
has no meaning in this setting. Nevertheless, the concept of marginal utility has been a
|
561 |
+
milestone in economic theory, proving that this notion deserves an adequate theoretical
|
562 |
+
foundation.
|
563 |
+
2.2
|
564 |
+
An overview on measurable utility theory
|
565 |
+
Let X be a set of alternatives. Pairs of alternatives (x, y) β X Γ X are intended to
|
566 |
+
represent the prospect of replacing alternative y by alternative x, that can be read as
|
567 |
+
9Note that this is the usual order topology on X.
|
568 |
+
15
|
569 |
+
|
570 |
+
βx in lieu of yβ. Deο¬ne the binary relation β½ on X ΓX called intensity preference with
|
571 |
+
the following interpretation: for any two pairs (x, y) and (z, w) in X Γ X,
|
572 |
+
(x, y) β½ (z, w)
|
573 |
+
is intended to mean that getting x over y gives at least as much added utility as getting
|
574 |
+
z over w or (if y βΏ x) at most as much added sadness. As a result, our decision maker
|
575 |
+
is endowed with a weak order preference relation βΏ on alternatives and an intensity
|
576 |
+
preference relation β½ on pairs of alternatives.
|
577 |
+
Shapley [36] proves his theorem assuming X to be a convex subset of R. As a
|
578 |
+
result, the proof exploits the full algebraic power of the ordered ο¬eld and the topological
|
579 |
+
properties of the linear continuum. Our aim is to generalize the set of alternatives X
|
580 |
+
to a connected and separable subset of a topological space, ordered with the binary
|
581 |
+
relations βΏ and β½ and with the order topology induced by the weak order βΏ.
|
582 |
+
We assume the following axioms for βΏ and β½, as in Shapley [36].
|
583 |
+
Axiom 1. For all x, y, z β X we have (x, z) β½ (y, z) ββ x βΏ y.
|
584 |
+
Axiom 1 (henceforth A1) is an assumption of consistency between the two order-
|
585 |
+
ings because it implies that the decision maker prefers to exchange z with x instead of
|
586 |
+
z with y if and only if she prefers x to y. Together with A1 we can formulate a dual
|
587 |
+
version of consistency, A1β², that can be derived from the whole set of axioms we are
|
588 |
+
going to assume later.10
|
589 |
+
Axiom 1β². For all x, y, z β X we have (z, x) β½ (z, y) ββ y βΏ x.
|
590 |
+
We now introduce the main object of this thesis: a joint real-valued representation
|
591 |
+
for the two orders βΏ and β½.
|
592 |
+
Deο¬nition 8. A real-valued function u : X β R is a measurable utility function for
|
593 |
+
(βΏ, β½) if for each pair x, y β X
|
594 |
+
x βΏ y ββ u(x) β₯ u(y)
|
595 |
+
(3)
|
596 |
+
and if, for each quadruple x, y, z, w β X
|
597 |
+
(x, y) β½ (z, w) ββ u(x) β u(y) β₯ u(z) β u(w).
|
598 |
+
(4)
|
599 |
+
The measurable terminology has nothing to do with measure theory, but it refers to
|
600 |
+
what is known as measurement theory, i.e. the ο¬eld of science that established the for-
|
601 |
+
mal foundation of quantitative measurement and the assignment of numbers to objects
|
602 |
+
10We mention A1β² as a form of axiom only because in this way we can refer to it in the proof of
|
603 |
+
Theorem 5, but we never assume it formally. A proof of it will be formulated forward with Lemma 13.
|
604 |
+
16
|
605 |
+
|
606 |
+
in their structural correspondence. Indeed, not only is a measurable utility function
|
607 |
+
able to rank pairs of alternatives according to a preference relation, but it also repre-
|
608 |
+
sents the idea of magnitude and intensity of the preference relation among alternatives.
|
609 |
+
Therefore, the numerical value u(x) that a measurable utility function assigns to the
|
610 |
+
alternative x is assuming the role of a particular unit of measurement for pleasure, that
|
611 |
+
we call util.
|
612 |
+
Recall that an ordinal utility function u is unique up to strictly monotone trans-
|
613 |
+
formations f : Im(u) β R. Hence, a measurable utility function is not ordinal. Never-
|
614 |
+
theless, it is unique up to positive aο¬ne transformations. Recall that a positive aο¬ne
|
615 |
+
transformation is a special case of a strictly monotone transformation of the follow-
|
616 |
+
ing form f(x) = Ξ±x + Ξ², with Ξ± > 0 and Ξ² β R. Positive aο¬ne transformations are
|
617 |
+
order-preserving thanks to Ξ± > 0.
|
618 |
+
Proposition 1. A measurable utility function u : X β R for (βΏ, β½) is unique up to
|
619 |
+
positive aο¬ne transformations.
|
620 |
+
Proof. If u(x) = Ξ±u(x) + Ξ² then we have
|
621 |
+
x βΏ y ββ u(x) β₯ u(y) ββ u(x) = Ξ±u(x) + Ξ² β₯ Ξ±u(y) + Ξ² = u(y)
|
622 |
+
and
|
623 |
+
(x, y) β½ (z, w) ββ u(x) β u(y) β₯ u(z) β u(w)
|
624 |
+
ββ u(x) β u(y) = Ξ±[u(x) β u(y)] β₯ Ξ±[u(z) β u(w)] = u(z) β u(w).
|
625 |
+
As a result, u and u are two utility representations for (βΏ, β½).
|
626 |
+
The whole class
|
627 |
+
of utility functions that are unique up to positive aο¬ne transformations are called
|
628 |
+
cardinal. Measurable utility functions are, therefore, cardinal and pertain to the so-
|
629 |
+
called cardinal utility theory.
|
630 |
+
Other two axioms (A2, A3) we need to introduce are the following:
|
631 |
+
Axiom 2. For all x, y, z, w β X we have (x, y) βΌ (z, w) ββ (x, z) βΌ (y, w).
|
632 |
+
Axiom 3. For all x, y, z, w β X the set
|
633 |
+
{(x, y, z, w) β X Γ X Γ X Γ X : (x, y) β½ (z, w)}
|
634 |
+
is closed in the product topology.
|
635 |
+
Axiom 2 is a βcrossoverβ property that characterizes diο¬erence comparisons of util-
|
636 |
+
ity, while Axiom 3 is a technical assumption deο¬ning the order relation β½ as continuous.
|
637 |
+
17
|
638 |
+
|
639 |
+
Shapley [36] proves his theorem on a domain of alternative outcomes that is a
|
640 |
+
nonempty, convex subset D of the real line where the preference order coincides with
|
641 |
+
the total order of (R, β₯). Moreover, β½ is assumed to be a weak order on D Γ D such
|
642 |
+
that A1, A2 and A3 are satisο¬ed.
|
643 |
+
Theorem 3 (Shapley). There exist a utility function u : D β R β R such that
|
644 |
+
x β₯ y ββ u(x) β₯ u(y)
|
645 |
+
(5)
|
646 |
+
and
|
647 |
+
(x, y) β½ (z, w) ββ u(x) β u(y) β₯ u(z) β u(w)
|
648 |
+
(6)
|
649 |
+
for all x, y, z, w β D. Moreover, this function is unique up to a positive aο¬ne transfor-
|
650 |
+
mation.
|
651 |
+
The theorem is stated as a suο¬cient condition, which is the most diο¬cult part to
|
652 |
+
prove. The necessary condition of the theorem is easily proved and we state it here as
|
653 |
+
a proposition.
|
654 |
+
Proposition 2. If the pair (β₯, β½) has a continuous measurable utility function u : D β
|
655 |
+
R β R, then β₯ is complete and transitive, β½ is complete, transitive, continuous (A3)
|
656 |
+
and satisο¬es the crossover axiom (A2), and jointly β₯ and β½ satisfy the consistency
|
657 |
+
axiom (A1).
|
658 |
+
Shapleyβs construction of the measurable utility function of Theorem 3 is extremely
|
659 |
+
elegant, but has the drawback of being too speciο¬c as u is deο¬ned on a convex subset
|
660 |
+
of R. On the other side of the spectrum, as mentioned in the ο¬rst chapter, the ο¬eld
|
661 |
+
of utility axiomatization has been proliο¬c in the 20th century and a copious number
|
662 |
+
of cardinal-utility derivations from preference-intensity axiomatizations were published.
|
663 |
+
One of the most important papers on this issue was the one published in 1955 by Patrick
|
664 |
+
Suppes and Muriel Winet. Recalling what described before, Suppes and Winet [40]
|
665 |
+
advanced an axiomatization of cardinal utility based on the assumption that individuals
|
666 |
+
are not only able to rank the utility of diο¬erent alternatives, as is assumed in the ordinal
|
667 |
+
approach to utility, but are also capable of ranking the diο¬erences between the utilities
|
668 |
+
of commodities. Nevertheless, their 11 axioms on an abstract algebraic structure were
|
669 |
+
not fully satisfactory in terms of generality: it was too general. Indeed, some of their
|
670 |
+
axioms can be derived in Shapley [36], thanks to the topological properties of R.
|
671 |
+
The aim of this research is to settle somewhere in between, ο¬nding a representation
|
672 |
+
theorem for cardinal utility function (in particular, a measurable one) keeping the
|
673 |
+
18
|
674 |
+
|
675 |
+
elegance of Shapleyβs proof and generalizing the domain of alternatives into the direction
|
676 |
+
of Suppes and Winet [40]. We will state and prove a representation theorem for a
|
677 |
+
measurable utility function u : X β R where X is a connected and separable subset of
|
678 |
+
a topological space, βΏ and β½ are weak orders and they satisfy (A1), (A2) and (A3).
|
679 |
+
Before doing this, we need to state and prove some topological preliminary results that
|
680 |
+
will be used in Theorem 5.11
|
681 |
+
2.3
|
682 |
+
A few basic lemmas
|
683 |
+
Deο¬nition 9. Let X be a topological space. X is connected if it cannot be separated
|
684 |
+
into the union of two disjoint nonempty open subsets. Otherwise, such a pair of open
|
685 |
+
sets is called a separation of X.
|
686 |
+
Deο¬nition 10. Let X be a topological space. X is separable if there exists a countable
|
687 |
+
dense subset. A dense subset D of a space X is a subset such that its closure equals the
|
688 |
+
whole space, i.e. D = X.
|
689 |
+
Deο¬nition 11. A totally ordered set (L, βΏ) having more than one element is called a
|
690 |
+
linear continuum if the following hold:
|
691 |
+
(a)
|
692 |
+
L has the least upper bound property.
|
693 |
+
(b)
|
694 |
+
If x β» y, there exists z such that x β» z β» y
|
695 |
+
We recall that a ray is a set of the following type (ββ, a) = {x β L : x βΊ a}
|
696 |
+
and (ββ, a] = {x β L : x βΎ a} in the case L does not have a minimum. In the
|
697 |
+
case L does have a minimum we write [xm, a) = {x β L : xm βΎ x βΊ a} and [xm, a] =
|
698 |
+
{x β L : xm βΎ x βΎ a}. Analogously for the sets (a, +β), [a, +β), (a, xM] , [a, xM], where
|
699 |
+
xM is the maximum of L in the case it existed.12
|
700 |
+
Given A β X, an element y β X is an upper bound for a set A if y βΏ x for all
|
701 |
+
x β A. It is a least upper bound for A if, in addition, it is the minimum of the set of all
|
702 |
+
upper bounds of A, that is if yβ² βΏ x for all x β A then yβ² βΏ y. If βΏ is antisymmetric, the
|
703 |
+
least upper bound is unique and is denoted sup A. The greatest lower bound is deο¬ned
|
704 |
+
analogously and denoted inf A.
|
705 |
+
11We thank Dr. Hendrik S. Brandsma for providing a feedback and insightful comments.
|
706 |
+
12Note that in decision theory, rays of a set X equipped with a reο¬exive and transitive binary
|
707 |
+
relation βΏ are usually denoted with the following notation L(a, βΏ) := (ββ, a] = {x β X : x βΎ a} and
|
708 |
+
U(a, βΏ) := [a, +β) = {x β X : x βΏ a}, L(a, β») := (ββ, a) and U(a, β») := (a, +β).
|
709 |
+
19
|
710 |
+
|
711 |
+
Lemma 1. Let βΏ be a total order on a connected set X. Then, X is a linear continuum
|
712 |
+
in the order topology.13
|
713 |
+
Proof. Suppose that a and b are two arbitrary but ο¬xed elements of X such that a βΊ b.
|
714 |
+
If there is no element c β X such that a βΊ c βΊ b, then X is the union of the open
|
715 |
+
rays (ββ, b) = {x β X : x βΊ b} and (a, +β) = {x β X : a βΊ x} both of which are
|
716 |
+
open sets in the order topology and are also nonempty, as the ο¬rst contains a, while
|
717 |
+
the second contains b. But this contradicts the fact that X is connected, so there must
|
718 |
+
exists an element c β X such that a βΊ c βΊ b.
|
719 |
+
Now, to show the least upper bound property, let A be a nonempty subset of X
|
720 |
+
such that A is bounded above in X. Let B be the set of all the upper bounds in X of
|
721 |
+
set A, i.e.
|
722 |
+
B := {b β X : b βΏ a for every a β A}
|
723 |
+
which is nonempty. All we need to show is that B has the least element. If B has a
|
724 |
+
smallest element (or A has a largest element, which would then be the smallest element
|
725 |
+
of B), then that element is the least upper bound of A.
|
726 |
+
Let us assume, instead, that B has no smallest element. Then, for any element
|
727 |
+
b β B, there exists an element bβ² β B such that bβ² βΊ b, and so b β (bβ², +β) β B with
|
728 |
+
(bβ², +β) being an open set in X. This shows that B is a nonempty open subset of X.
|
729 |
+
Therefore, B can be closed only in the case when B = X. But we know that B β X,
|
730 |
+
since A β X\B and A ΜΈ= β
, so it cannot be the case that B = X. Therefore, B has a
|
731 |
+
limit point b0 that does not belong to B. Then b0 is not an upper bound of set A, which
|
732 |
+
implies the existence of an element a β A such that b0 βΊ a, we can also conclude that
|
733 |
+
b0 β (ββ, a) β X\B, with (ββ, a) being an open set. This contradicts our choice of
|
734 |
+
b0 as a limit point of set B. Therefore, the set B of all the upper bounds in X of set A
|
735 |
+
must have a smallest element, and that element is the least upper bound of A.
|
736 |
+
Given A β X, we denote A or ClA the topological closure of A, that is deο¬ned as
|
737 |
+
the intersection of all closed sets containing A.
|
738 |
+
From now on denote X as a subset of a topological space (X, Ο), unless otherwise
|
739 |
+
stated.
|
740 |
+
Lemma 2. Let βΏ be a complete, transitive and continuous order on a connected set X.
|
741 |
+
13Note that the converse holds as well: βΏ is a total order on a connected set X if and only if X is a
|
742 |
+
linear continuum in the order topology.
|
743 |
+
20
|
744 |
+
|
745 |
+
Given any x, y β X, with x β» y, we have
|
746 |
+
x βΏ z βΏ y β z β X
|
747 |
+
for all z β (X, Ο)
|
748 |
+
Proof. Suppose by contradiction that there exists z β X\X such that x β» z β» y.
|
749 |
+
By the continuity of βΏ, we can partition X into two nonempty disjoint open sets
|
750 |
+
{x β X : x βΊ z} and {x β X : x β» z}, which contradicts the connectedness of X.
|
751 |
+
Lemma 3. Suppose that jointly βΏ and β½ satisfy A1. If β½ is continuous , then βΏ is
|
752 |
+
continuous.
|
753 |
+
Proof. For all arbitrary but ο¬xed y, z β X, by A1 we have {x β X : (x, z) β½ (y, z)} =
|
754 |
+
{x : x βΏ y}. By A3, the set {x β X : (x, z) β½ (y, z)} is closed. Analogous is the case
|
755 |
+
for {x : y βΏ x}, derived from A1β².
|
756 |
+
Lemma 4. Fix y β X, the set Iy := {x β X : x βΌ y} is a closed set in X.
|
757 |
+
Proof. βΏ is continuous, so for every y β X we have that {x β X : x βΏ y} and
|
758 |
+
{x β X : y βΏ x} are closed. Pick a point x such that x βΏ y and y βΏ x, that is x βΌ y.
|
759 |
+
So we have {x β X : x βΌ y} = {x β X : x βΏ y} β© {x β X : y βΏ x} and the intersection
|
760 |
+
of two closed sets is closed.
|
761 |
+
Note that when βΏ is antisymmetric, the set Iy is a singleton and Lemma 4 reduces
|
762 |
+
to prove that X satisο¬es the T1 axiom of separation, that is every one-point set is closed.
|
763 |
+
Clearly, every Hausdorο¬ space satisο¬es it.
|
764 |
+
Lemma 5. Let βΏ be a continuous total order on a connected set X. If A β X is a
|
765 |
+
nonempty closed set in the order topology and A is bounded above (below), then supA
|
766 |
+
(infA) belongs to A.14
|
767 |
+
Proof. Suppose supA /β A. Then supA β X\A, which is open. By deο¬nition, there
|
768 |
+
exists a base element (a, b) such that
|
769 |
+
supA β (a, b) β X\A.
|
770 |
+
A is bounded above so, by Lemma 1, sup A exists and there is an element aβ such that
|
771 |
+
a βΊ aβ βΊ sup A, then aβ β (a, b) β X\A, so aβ is an upper bound of A smaller that
|
772 |
+
supA, reaching a contradiction. In the case X had a maximum, then consider the case
|
773 |
+
where sup A = max X. Let U := (x, sup A] be a basic neighborhood of sup A. Then, x
|
774 |
+
14The lemma holds even in the case we relaxed connectedness. Nevertheless, we always need to as-
|
775 |
+
sume sup A exists. If we do not assume the existence of the least upper bound, an easy counterexample
|
776 |
+
is N β R that is closed in the order topology, but sup N /β N.
|
777 |
+
21
|
778 |
+
|
779 |
+
cannot be an upper bound of A as x βΊ sup A. Hence, there exists an element a β A
|
780 |
+
such that x βΊ a βΎ sup A. Thus, as x was generic, it follows that U β© A ΜΈ= β
. This
|
781 |
+
means that every neighborhood of sup A intersects A, that is sup A β A. But A is
|
782 |
+
closed, hence sup A β A and we can conclude sup A = max A.
|
783 |
+
The case of inf A is specular.
|
784 |
+
Now we deο¬ne the notion of convergence in any topological space.
|
785 |
+
Deο¬nition 12. In an arbitrary topological space X, we say that a sequence x1, x2, . . .
|
786 |
+
of points of the space X converges to the point x of X provided that, corresponding to
|
787 |
+
each neighborhood U of x, there is a positive integer N such that xn β U for all n β₯ N.
|
788 |
+
Moreover, let βΏ a total order. We write xn β x if x1 βΎ x2 βΎ Β· Β· Β· βΎ xn βΎ . . . and
|
789 |
+
supnxn = x where sup is with respect to βΎ. The deο¬nition xn β x for a βΎ-decreasing
|
790 |
+
sequence is analogous. We say that (xn) converges monotonically to a limit point x
|
791 |
+
when either xn β x or xn β x.
|
792 |
+
We now prove one of the fundamental lemmas that allow us to generalize Shapleyβs
|
793 |
+
proof to a connected and separable subset of a topological space. Note that, as long
|
794 |
+
as Shapley [36] is working on R, sequences as βenoughβ to characterize the deο¬nition
|
795 |
+
of convergence.
|
796 |
+
This is due to the fact that there exists a countable collection of
|
797 |
+
neighborhoods around every point. This is not true in general, but it is for a speciο¬c
|
798 |
+
class of spaces that are said to satisfy the ο¬rst countability axiom.15 A space X is said
|
799 |
+
to have a countable basis at the point x if there is a countable collection {Un}nβN of
|
800 |
+
neighborhoods of x such that any neighborhood U of x contains at least one of the sets
|
801 |
+
Un. A space X that has a countable basis at each of its points is said to satisfy the
|
802 |
+
ο¬rst countability axiom.
|
803 |
+
In general, however, sequences are not powerful enough to capture the idea of
|
804 |
+
convergence we want to capture in a generic topological space. Indeed, there could
|
805 |
+
be uncountably many neighborhoods around every point, so the countability of the
|
806 |
+
natural number index of sequences cannot βreachβ these points. The ideal solution to
|
807 |
+
this problem is to deο¬ne a more general object than a sequence, called a net, and talk
|
808 |
+
about net-convergence. One can also deο¬ne a type of object called a ο¬lter and show
|
809 |
+
that ο¬lters also provide us a type of convergence which turns out to be equivalent to
|
810 |
+
net-convergence. With these more powerful tools in place of sequence convergence, one
|
811 |
+
can fully characterize the notion of convergence in any topological space.
|
812 |
+
15There are far more general classes of spaces in which convergence can be fully characterized by se-
|
813 |
+
quences. We refer the interested reader to the notion of FrΒ΄echet-Urysohn spaces and Sequential spaces.
|
814 |
+
22
|
815 |
+
|
816 |
+
Nevertheless, we are now going to show that every connected, separable and totally
|
817 |
+
ordered set X satisο¬es the ο¬rst countability axiom. In fact, we are going to prove even
|
818 |
+
more. We are going to show that X is metrizable, which means there exists a metric d
|
819 |
+
on the set X that induces the topology of X.16 We give other two deο¬nitions that will
|
820 |
+
be used to prove Lemma 6.
|
821 |
+
Deο¬nition 13. Suppose X is T1. Then X is said to be regular (or T3) if for each pair
|
822 |
+
consisting of a point x and a closed set B disjoint from x, there exist disjoint open sets
|
823 |
+
containing x and B, respectively.
|
824 |
+
Deο¬nition 14. If a space X has a countable basis for its topology, then X is said to
|
825 |
+
satisfy the second countability axiom, or to be second-countable.
|
826 |
+
Theorem 4 (Urysohn metrization theorem). Every regular space X with a count-
|
827 |
+
able basis is metrizable.
|
828 |
+
Lemma 6. Let βΏ be a continuous total order on a connected and separable topological
|
829 |
+
space X in the order topology and A β X. We have x β A if and only if there exists a
|
830 |
+
sequence (xn) β AN that converges monotonically to x.
|
831 |
+
The steps of the proof are the following:
|
832 |
+
(i) We show that X is regular17 and second-countable. By the Urysohn metrization
|
833 |
+
theorem, which provides suο¬cient (but not necessary) conditions for a space to
|
834 |
+
be metrizable, there exist a metric d that induces the topology of X.
|
835 |
+
(ii) Let A β X with X metrizable, then we have that x β A if and only if there exists
|
836 |
+
a sequence of points of A converging to x.
|
837 |
+
(iii) Finally, we use the fact that in every totally ordered topological space X, every
|
838 |
+
sequence admits a monotone subsequence. Then, if a sequence converges, all of
|
839 |
+
its subsequences converge to the same limit. Thus, we can extract our monotone
|
840 |
+
converging sequence.
|
841 |
+
Lemma 7. A totally ordered topological space X is regular in the order topology.
|
842 |
+
Proof. It is basic topology to prove that every totally ordered set is Hausdorο¬, hence
|
843 |
+
it is T1. Now, suppose x β X and B is a closed set, disjoint from x. So, x β X\B,
|
844 |
+
16A metrizable space always satisο¬es the ο¬rst countability axiom.
|
845 |
+
17In fact, one could prove that X is also normal.
|
846 |
+
23
|
847 |
+
|
848 |
+
which is open. Then, by deο¬nition of open set, there exists a basis element (a, b) such
|
849 |
+
that x β (a, b) and (a, b) β© B = β
. Pick any a0 β (a, x), and let U1 = (ββ, a0) , V1 =
|
850 |
+
(a0, β). If no such a0 exists (in our case it would, by connectedness of X), then let
|
851 |
+
U1 = (ββ, x), V1 = (a, β). In both cases, U1 β© V1 = β
. Similar is the case of the other
|
852 |
+
side, pick b0 β (x, b), and if that exists, denote U2 = (b0, β) , V2 = (ββ, b0) , and if
|
853 |
+
not, let U2 = (x, β), V2 = (ββ, b). Again, in both cases U2 β© V2 = β
. As a result, we
|
854 |
+
obtained that, in both cases, x β V1 β© V2 with V1 β© V2 open set and B β U1 βͺ U2, with
|
855 |
+
U1 βͺ U2 open set. As V1 β© V2 is disjoint from U1 βͺ U2, X is regular.
|
856 |
+
Lemma 8. A totally ordered, connected and separable topological space X is second-
|
857 |
+
countable.
|
858 |
+
Proof. Now we ο¬nd a countable basis for the order topology of X. As X is separable,
|
859 |
+
then let D β X be countable and dense in X, i.e. D = X. Then, deο¬ne
|
860 |
+
B := {(a, b) : a, b β D with a βΊ b}
|
861 |
+
together with, if there exists a minimal element m := min X and a maximal element
|
862 |
+
M := max X, the set {[m, a), (a, M], a β D}. In both cases, the collection B forms a
|
863 |
+
countable base for the topology of X. To prove this, we show that for each open set
|
864 |
+
(a, b) of the order topology of X and for every x β (a, b) there is an element (aβ², bβ²) β B
|
865 |
+
such that x β (aβ², bβ²) β (a, b).
|
866 |
+
Suppose x β (a, b) β X, then the open intervals (a, x) and (x, b) cannot be empty
|
867 |
+
by connectedness. Hence, there exist aβ² β (a, x) β© D and bβ² β (x, b) β© D. This follows
|
868 |
+
from the fact that D = X and x β D = X if and only if every open set containing x
|
869 |
+
intersects D. Then, it follows that x β (aβ², bβ²) β (a, b).
|
870 |
+
Now, when m exists, suppose x = m, then x β [m, a) and this set is nonempty
|
871 |
+
by connectedness. Hence, there exists an element aβ²β² β [m, a) β© D. So, it follows that
|
872 |
+
x β [m, aβ²β²) β [m, a). Analogous is the case when M exists.
|
873 |
+
By Lemma 7 and Lemma 8 , X satisο¬es all the assumptions of the Urysohn metriza-
|
874 |
+
tion theorem, hence X is metrizable (and, a fortiori, it is ο¬rst-countable).
|
875 |
+
Lemma 9. Let A β X with X metrizable, then x β A if and only if there exists a
|
876 |
+
sequence of points of A converging to x.
|
877 |
+
Proof. Suppose xn β x with xn β A. Then, every neighborhood U of x contains a
|
878 |
+
point of A, i.e. x β A. Conversely, we use the fact that X is metrizable.18 Let x β A
|
879 |
+
18Note, once again, that here we do not need the full strength of metrizability. All we really need
|
880 |
+
24
|
881 |
+
|
882 |
+
and let d be a metric that induces the order topology. For every n β N, we take the
|
883 |
+
neighborhood Bd(x, 1
|
884 |
+
n), of x of radius 1
|
885 |
+
n and we choose xn to be a point such that, for
|
886 |
+
all n, xn β Bd(x, 1
|
887 |
+
n) β© A. We show xn β x. Any open set U containing x contains an
|
888 |
+
Η«-neighborhood Bd(x, Η«) centered at x. Choosing N such that
|
889 |
+
1
|
890 |
+
N < Η«, then U contains
|
891 |
+
xn for all n β₯ N.
|
892 |
+
We can ο¬nally prove Lemma 6.
|
893 |
+
Proof. The if part comes trivially by deο¬nition. If there exists a sequence that converges
|
894 |
+
(monotonically) to x, then x β A by Lemma 9.
|
895 |
+
Conversely, if x β A, then by Lemma 9 we know that there exists a sequence in A
|
896 |
+
converging to x. Now we show that, in every totally ordered set (X, βΎ), every sequence
|
897 |
+
from N β (X, βΎ) has a monotone subsequence. Indeed, this is a property that has
|
898 |
+
nothing to do with the topology of X.
|
899 |
+
Let (xi)iβN be a sequence with values in X.
|
900 |
+
We say that xk is a peak of the
|
901 |
+
sequence if h > k β xh βΎ xk (we admit a slight abuse of notation here, as it would be
|
902 |
+
better to call peak the index of the sequence, and not its image). We distinguish two
|
903 |
+
cases: if there are inο¬nitely many peaks, then the subsequence of peaks is an inο¬nite
|
904 |
+
non-increasing sequence and we are done. If there are only ο¬nitely many peaks, then
|
905 |
+
let i1 be the index such that xi1 is the successor of the last peak. Then, xi1 is not a
|
906 |
+
peak. Again, we ο¬nd another index i2 > i1 such that xi2 βΏ xi1. Again, as xi2 is not a
|
907 |
+
peak, we can ο¬nd another index i3 > i2 such that xi3 βΏ xi2 βΏ xi1. Keeping deο¬ning the
|
908 |
+
sequence in this way, we get, inductively, a non-decreasing sequence.
|
909 |
+
In conclusion, as by assumption we have a sequence (xn) β AN converging to x,
|
910 |
+
this sequence admits a monotone subsequence. But, if a sequence converges to a point
|
911 |
+
x, then all of its subsequences converge to the same point x. Hence, there exists a
|
912 |
+
sequence that converges monotonically to x, proving Lemma 6.
|
913 |
+
Note that Lemma 6 could have been proven just using the notion of ο¬rst countabil-
|
914 |
+
ity. Nevertheless, we decided to take the longer path of Urysohn metrization theorem to
|
915 |
+
is a countable collection of neighborhoods around x. Moreover, both connectedness and separability
|
916 |
+
are not necessary conditions. We refer the interested reader to the nice two-page paper of Lutzer [25],
|
917 |
+
that proves a linearly ordered space X is metrizable in the order topology if and only if the diagonal
|
918 |
+
β := {(x, x) : x β X} is a countable intersection of open subsets of X Γ X, i.e. the diagonal is a GΞ΄
|
919 |
+
set. Furthermore, this condition can be shown to be equivalent to have a Ο-locally countable basis,
|
920 |
+
which is a condition more in the spirit of the Nagata-Smirnov metrization theorem which requires a
|
921 |
+
Ο-locally ο¬nite basis.
|
922 |
+
25
|
923 |
+
|
924 |
+
show how βwell-behavedβ a totally ordered, connected and separable topological space
|
925 |
+
can be.
|
926 |
+
Lemma 10. Let (X, βΏ) be a topological space with the order topology. Let β½ be another
|
927 |
+
order relation on X Γ X such that A1 and A3 hold,19 and suppose (xn), (yn) converge
|
928 |
+
to x and y respectively, and (wn), (zn) converge to w and z respectively. If for every
|
929 |
+
n β N we have (xn, yn) β½ (wn, zn) then (x, y) β½ (w, z).
|
930 |
+
Proof. Denote the set A := {(x, y, w, z) β X Γ X Γ X Γ X : (x, y) β½ (w, z)} and pick
|
931 |
+
a sequence of points with values in A, that is pick (xn, yn, wn, zn) β AN converging to
|
932 |
+
(x, y, w, z). By assumption, we have that xn β x, yn β y, wn β w, zn β z and this
|
933 |
+
is equivalent to (xn, yn, wn, zn) β (x, y, w, z). Indeed, a sequence in the product space
|
934 |
+
X Γ X Γ X Γ X converges to (x, y, w, z) if and only if it converges componentwise, i.e.
|
935 |
+
xn β x, yn β y, wn β w, zn β z. We now prove this fact.
|
936 |
+
Assume (xn, yn, wn, zn) β (x, y, w, z) in X ΓX ΓX ΓX. Let U1, U2, U3, U4 be open
|
937 |
+
sets containing x, y, w, z, respectively. Then U1ΓU2ΓU3 ΓU4 is a basis element (hence,
|
938 |
+
open) for the product topology containing (x, y, w, z). By deο¬nition of convergence, we
|
939 |
+
can ο¬nd n0 such that for all n β₯ n0 we have (xn, yn, wn, zn) β U1 Γ U2 Γ U3 Γ U4.
|
940 |
+
Thanks to the fact that projections are continuous functions, they preserve convergent
|
941 |
+
sequences and so for all n β₯ n0 we have xn β U1, yn β U2, wn β U3, zn β U4, i.e.
|
942 |
+
xn β x, yn β y, wn β w, zn β z.
|
943 |
+
Conversely, if xn β x, yn β y, wn β w, zn β z, let Uβ be an open subset of
|
944 |
+
XΓXΓXΓX such that (x, y, w, z) β Uβ. By deο¬nition of product topology, we can ο¬nd
|
945 |
+
U1 β X open in X, . . . , U4 β X open in X such that x β U1, y β U2, w β U3, z β U4. By
|
946 |
+
convergence, we have that for all i = 1, 2, 3, 4 there exists nki β N such that for all n β₯
|
947 |
+
nki we have xn β U1, yn β U2, wn β U3, zn β U4. Now pick N := max{nk1, nk2, nk3, nk4}
|
948 |
+
and for every n β₯ N we have (xn, yn, wn, zn) β U1 Γ U2 Γ U3 Γ U4 β Uβ. Hence, by
|
949 |
+
deο¬nition of convergence, (xn, yn, wn, zn) β (x, y, w, z).
|
950 |
+
Now we want to show (x, y, w, z) β A, with A closed in the product topology.
|
951 |
+
We now prove that every closed set in the product topology is sequentially closed.20
|
952 |
+
This means we want to show that if we pick a sequence of points (xn, yn, wn, zn) with
|
953 |
+
values in A β X that is converging to a point (x, y, w, z) β X, then (x, y, w, z) β A.
|
954 |
+
Pick a sequence (xn, yn, wn, zn) with values in A β X that is converging to a point
|
955 |
+
19Note that the order topology and A1 are redundant assumptions. The lemma follows immediately
|
956 |
+
by continuity of β½ alone.
|
957 |
+
20Note that when X is metrizable, a set C β X is closed ββ C is sequentially closed.
|
958 |
+
26
|
959 |
+
|
960 |
+
(x, y, w, z) β X. Then, let Uβ be any neighborhood of (x, y, w, z). By convergence,
|
961 |
+
there exist an n0 β N such that for all n β₯ n0 we have (xn, yn, wn, zn) β Uβ and, in
|
962 |
+
particular, (xn, yn, wn, zn) β Uβ β© A. Since Uβ was an arbitrary but ο¬xed neighborhood
|
963 |
+
of (x, y, w, z), then (x, y, w, z) is in the closure of A, i.e. (x, y, w, z) β A. But A is
|
964 |
+
closed, therefore A = A, so (x, y, w, z) β A, hence (x, y) β½ (w, z).
|
965 |
+
The proof of Theorem 5 in chapter 3, as in the original version of Shapley [36],
|
966 |
+
relies on two very interesting lemmas. Similar propositions have been taken as axioms
|
967 |
+
in environments that lack the topological assumptions on the set of alternatives X.
|
968 |
+
Lemma 11. Let (w,z) be an element of X Γ X. If xβ², xβ²β², y β X are such that:
|
969 |
+
(xβ², y) β½ (w, z) β½ (xβ²β², y)
|
970 |
+
(7)
|
971 |
+
then there exists a unique, up to indiο¬erence, xβ β X such that
|
972 |
+
(xβ, y) βΌ (w, z)
|
973 |
+
(8)
|
974 |
+
and xβ² βΏ xβ βΏ xβ²β².
|
975 |
+
Proof. Deο¬ne x0 := inf{x β X : (x, y) β½ (w, z)} and denote A := {x β X : (x, y) β½
|
976 |
+
(w, z)} this set. The set A is nonempty as xβ² β A, A is bounded below by xβ²β² as we
|
977 |
+
have (w, z) β½ (xβ²β², y) and, by transitivity and A1, we reach x βΏ xβ²β² for every x β A.
|
978 |
+
Thus, x0 is such that xβ² βΏ x0 βΏ xβ²β² and so x0 β X by Lemma 2. Analogously, we deο¬ne
|
979 |
+
x0 := sup{x β X : (w, z) β½ (x, y)} and denote B := {x β X : (w, z) β½ (x, y)} this set.
|
980 |
+
Then, B is nonempty as xβ²β² β B, B is bounded above by xβ² as we have (xβ², y) β½ (w, z)
|
981 |
+
and, by transitivity and A1, we reach xβ² βΏ x for every x β B. Thus, x0 is such that
|
982 |
+
xβ² βΏ x0 βΏ xβ²β² and so x0 β X by Lemma 2.
|
983 |
+
By A3, the sets A and B are closed and so, by Lemma 5, we have x0 β A and
|
984 |
+
x0 β B so that
|
985 |
+
(x0, y) β½ (w, z) β½ (x0, y)
|
986 |
+
By transitivity and by A1 we have x0 βΏ x0.
|
987 |
+
Assume now by contradiction that x0 β» x0. By Lemma 1 there exists xβ β X such
|
988 |
+
that x0 βΊ xβ βΊ x0. But then, comparing xβ with (w, z), (xβ, y) β½ (w, z) can hold only
|
989 |
+
if x0 βΌ xβ β» x0, so x0 βΌ xβ and therefore xβ should be the inο¬mum of A, reaching a
|
990 |
+
contradiction. Specular is the contradiction in the other case. Therefore, as there does
|
991 |
+
not exist any xβ β X such that x0 βΊ xβ βΊ x0, we must conclude that x0 βΌ x0. By
|
992 |
+
transitivity and A1 we have
|
993 |
+
(x0, y) βΌ (w, z) βΌ (x0, y)
|
994 |
+
27
|
995 |
+
|
996 |
+
This proves the existence of xβ β X for which (8) holds.
|
997 |
+
Let x β X be any other element of X for which (8) holds. By transitivity, (xβ, y) βΌ
|
998 |
+
(x, y). By A1, we have xβ βΌ x and this completes the proof.
|
999 |
+
Lemma 12. Let x, z β X such that x β» z. Then, there exists a unique, up to indiο¬er-
|
1000 |
+
ence, yβ β X such that
|
1001 |
+
(x, yβ) βΌ (yβ, z)
|
1002 |
+
and x β» yβ β» z.
|
1003 |
+
Proof. Deο¬ne y0 to be the least upper bound of the set C := {y β X : (x, y) β½ (y, z)}.
|
1004 |
+
This set is nonempty as if we pick y = z we have (x, z) β½ (z, z) that by A1 is equivalent
|
1005 |
+
to x βΏ z, that holds by assumption. C is also bounded from above by x as if we pick
|
1006 |
+
y = x we have (x, x) β½ (x, z) that by A1β² is equivalent to z βΏ x, that, by completeness,
|
1007 |
+
contradicts the assumption of x β» z showing that x is an upper bound for C. Since C
|
1008 |
+
is nonempty and bounded above by x, by Lemma 2 we have y0 β X.
|
1009 |
+
Similarly, by deο¬ning y0 to be the greatest lower bound of the set D := {y β X :
|
1010 |
+
(y, z) β½ (x, y)}. This set is nonempty as if we pick y = x we have (x, z) β½ (x, x) that
|
1011 |
+
by A1β² is if and only if x βΏ z, that holds by assumption. This set is also bounded
|
1012 |
+
from below by z as if we pick y = z we have (z, z) β½ (x, z) that by A1 is if and only
|
1013 |
+
if z βΏ x, that, by completeness, contradicts the assumption of x β» z showing that z is
|
1014 |
+
a lower bound for D. Since D is nonempty and bounded below by z, by Lemma 2 we
|
1015 |
+
have y0 β X.
|
1016 |
+
By A3 the sets C and D are closed, so by Lemma 5 we have y0 β C and y0 β D,
|
1017 |
+
that is
|
1018 |
+
(x, y0) β½ (y0, z) and (y0, z) β½ (x, y0)
|
1019 |
+
(9)
|
1020 |
+
We show now that y0 βΏ y0. Suppose, by contradiction, y0 β» y0. By Lemma 1 there
|
1021 |
+
exists yβ β X such that y0 β» yβ β» y0. Then, by deο¬nition of y0 we have (yβ, z) βΊ (x, yβ),
|
1022 |
+
while by the deο¬nition of y0 we have (x, yβ) βΊ (yβ, z). This contradiction shows that
|
1023 |
+
y0 βΏ y0. By A1 this is equivalent to
|
1024 |
+
(y0, z) β½ (y0, z) for all z β X.
|
1025 |
+
(10)
|
1026 |
+
By A1β² it is also equivalent to
|
1027 |
+
(x, y0) β½ (x, y0) for all x β X.
|
1028 |
+
(11)
|
1029 |
+
Putting together equation 9 with equations 10 and 11, we reach the loop
|
1030 |
+
(y0, z) β½ (y0, z) β½ (x, y0) β½ (x, y0) β½ (y0, z).
|
1031 |
+
28
|
1032 |
+
|
1033 |
+
By transitivity, we have (y0, z) βΌ (y0, z) and (x, y0) βΌ (x, y0). By A1, we conclude that
|
1034 |
+
y0 βΌ y0.
|
1035 |
+
We conclude proving that from A1, A2 and A3 we can derive A1β².
|
1036 |
+
Lemma 13. Let X be a connected subset of a topological space.
|
1037 |
+
If βΏ is complete
|
1038 |
+
and transitive, β½ is complete, transitive, satisο¬es A3 and A2, and jointly βΏ and β½
|
1039 |
+
satisfy A1, then A1β² holds, that is, for all x, y, z β X we have x βΏ y if and only if
|
1040 |
+
(z, y) β½ (z, x).
|
1041 |
+
Proof. By contradiction, suppose A1β² fails. Then, there exist x, y, z β X such that
|
1042 |
+
(z, y) β½ (z, x) and x βΊ y. We consider two cases: y β» z and y βΎ z.
|
1043 |
+
If y β» z then, being (z, y) β½ (z, x) by assumption, we have
|
1044 |
+
(x, x) βΌ (y, y) β» (z, y) β½ (z, x)
|
1045 |
+
by A2 and A1, respectively. We apply Lemma 11 to ο¬nd a w β X such that
|
1046 |
+
(w, x) βΌ (z, y) and x βΏ w βΏ z.
|
1047 |
+
Being y β» x, we have
|
1048 |
+
(z, z) βΌ (y, y) β» (x, y) βΌ (w, z) βΏ (z, z)
|
1049 |
+
by A2, A1, A2, A1, respectively. This implies a contradiction in the case y β» z.
|
1050 |
+
Assume now y βΎ z. Being y βΎ z and x βΊ y, by transitivity we have x βΊ z. We
|
1051 |
+
can proceed as in the previous case, interchanging the roles of x and y and reversing
|
1052 |
+
all the inequalities.
|
1053 |
+
3
|
1054 |
+
The theorem
|
1055 |
+
We can now state and prove Shapleyβs theorem in our general version.
|
1056 |
+
Theorem 5. Let X be a connected and separable subset of a topological space. If βΏ is
|
1057 |
+
complete and transitive, β½ is complete, transitive, satisο¬es A2 and A3, and jointly βΏ
|
1058 |
+
and β½ satisfy A1, then the pair (βΏ, β½) can be represented by a continuous measurable
|
1059 |
+
utility function u: X β R, that is, for each pair x, y β X,
|
1060 |
+
x βΏ y ββ u(x) β₯ u(y)
|
1061 |
+
(12)
|
1062 |
+
and for each quadruple x, y, z, w β X,
|
1063 |
+
(x, y) β½ (z, w) ββ u(x) β u(y) β₯ u(z) β u(w).
|
1064 |
+
(13)
|
1065 |
+
Moreover, u is unique up to positive aο¬ne transformations.
|
1066 |
+
29
|
1067 |
+
|
1068 |
+
Proof. We ο¬rst prove the result when βΏ is antisymmetric. In view of Lemma 1, through-
|
1069 |
+
out the proof we will consider suprema and inο¬ma of subsets of X.
|
1070 |
+
Suppose X is not a singleton, otherwise the result is trivially true. Let a0, a1 β X
|
1071 |
+
be two distinct elements of X such that, without loss of generality, a1 β» a0.
|
1072 |
+
Assign u(a0) = 0 and u(a1) = 1. Now we want to show that u has a unique
|
1073 |
+
extension on X which is a measurable utility function for (βΏ, β½). To ease notation,
|
1074 |
+
denote
|
1075 |
+
1 := (a1, a0) , 0 := (a0, a0) , β1 := (a0, a1).
|
1076 |
+
Clearly, 1, 0, β1 β X Γ X and, by A1 and A1β², 1 β» 0 β» β1. Then, by A2 we have
|
1077 |
+
(x, x) βΌ 0 for every x β X. Moreover, for every y β X we have either:
|
1078 |
+
(i) There exists a unique T1(y) β X such that (T1(y), y) βΌ 1
|
1079 |
+
or
|
1080 |
+
(ii) 1 β» (x, y) for all x β X
|
1081 |
+
Indeed, if (ii) fails, there exists xβ² β X such that (xβ², y) β½ 1. Since (xβ², y) β½ 1 β½ 0 βΌ
|
1082 |
+
(y, y), by Lemma 11 there exists an element T1(y) β X such that (T1(y), y) βΌ 1. By
|
1083 |
+
A1 and antisymmetry of βΏ , (T1(y), y) βΌ (yβ², y) implies T1(y) = yβ², so T1(y) is unique.
|
1084 |
+
In addition, note that y βΊ T1(y). Indeed, (y, y) βΌ 0 βΊ 1 βΌ (T1(y), y), and so A1
|
1085 |
+
implies y βΊ T1(y). In a similar way as before, for every y β X we have either:
|
1086 |
+
(i.bis) There exists a unique Tβ1(y) β X such that (Tβ1(y), y) βΌ β1
|
1087 |
+
or
|
1088 |
+
(ii.bis) β1 βΊ (x, y) for all x β X
|
1089 |
+
Indeed, if (ii.bis) fails, there exists xβ² β X such that (xβ², y) βΌ β1. Since (xβ², y) βΌ β1 βΌ
|
1090 |
+
0 βΌ (y, y), by Lemma 11 there exists an element Tβ1(y) β X such that (T1(y), y) βΌ β1.
|
1091 |
+
By A1 and antisymmetry of βΏ , (Tβ1(y), y) βΌ (yβ², y) implies Tβ1(y) = yβ², so Tβ1(y) is
|
1092 |
+
unique.
|
1093 |
+
In addition, note that Tβ1(y) βΊ y. Indeed, (Tβ1(y), y) βΌ β1 βΊ 0 βΌ (y, y), and so
|
1094 |
+
A1 implies Tβ1(y) βΊ y.
|
1095 |
+
Now deο¬ne a2 := T1(a1) if (i) holds for y = a1, i.e. if there exists a unique
|
1096 |
+
T1(a1) β X such that (T1(a1), a1) βΌ 1. Similarly, set a3 := T1(a2) if (i) holds for y = a2,
|
1097 |
+
and continue in this way till (if ever) occurs y = an for which (ii) holds, i.e. 1 β» (x, an)
|
1098 |
+
for every x β X. Analogously, we deο¬ne aβ1 := Tβ1(a0) if (i.bis) holds for y = a0, set
|
1099 |
+
aβ2 := Tβ1(aβ1) if (i.bis) holds for y = aβ1, and continue in this way till (if ever) occurs
|
1100 |
+
y = aβn for which (ii.bis) holds.
|
1101 |
+
30
|
1102 |
+
|
1103 |
+
Now deο¬ne A := {. . . , aβ2, aβ1, a0, a1, a2, . . . }, with
|
1104 |
+
Β· Β· Β· βΊ aβ2 βΊ aβ1 βΊ a0 βΊ a1 βΊ a2 βΊ . . .
|
1105 |
+
The set A can be ο¬nite or inο¬nite in either direction. If we consider now a sequence that
|
1106 |
+
from an index set Pa β Z maps to A, we deο¬ne the following function a : Pa β Z β A.
|
1107 |
+
Now we start to extend u to A. Deο¬ne the following:
|
1108 |
+
u(ap) = p
|
1109 |
+
for every p β Pa.
|
1110 |
+
Clearly, we have (12), i.e. x βΏ y if and only if u(x) β₯ u(y) for every x, y that are images
|
1111 |
+
of the sequence a, so (12) holds on A.
|
1112 |
+
Now we show that (13) holds whenever x, y, z, w β A β X, say x = ap, y = aq, z =
|
1113 |
+
apβd where p, q, p β d β Pa. Without loss of generality, assume d β₯ 0. We ο¬rst prove
|
1114 |
+
the βequalityβ case of (13), that is
|
1115 |
+
(x, y) βΌ (z, w) ββ u(x) β u(y) = u(z) β u(w)
|
1116 |
+
(14)
|
1117 |
+
By construction we have
|
1118 |
+
(ap, apβ1) βΌ 1 βΌ (aq, aqβ1)
|
1119 |
+
so, by transitivity and A2, we have:
|
1120 |
+
(ap, aq) βΌ (apβ1, aqβ1)
|
1121 |
+
Iterating this procedure ο¬nitely many times we reach:
|
1122 |
+
(x, y) = (ap, aq) βΌ (apβd, aqβd) = (z, aqβd)
|
1123 |
+
(15)
|
1124 |
+
By transitivity, (z, aqβd) βΌ (z, w) and so, by A1 aqβd = w, so that u(aqβd) = u(w).
|
1125 |
+
By deο¬nition of u we can write
|
1126 |
+
u(x) β u(y) = u(ap) β u(aq) = p β q = u(apβd) β u(aqβd) = u(z) β u(w)
|
1127 |
+
thus proving (14). Next we prove
|
1128 |
+
(x, y) β» (z, w) ββ u(x) β u(y) > u(z) β u(w)
|
1129 |
+
(16)
|
1130 |
+
By transitivity, (z, aqβd) β» (z, w) and so, by A1β², w β» aqβd, so that u(w) > u(aqβd).
|
1131 |
+
By deο¬nition of u, from (15) we can write
|
1132 |
+
u(x) β u(y) = u(ap) β u(aq) = p β q = u(apβd) β u(aqβd) > u(z) β u(w)
|
1133 |
+
thus proving (16).
|
1134 |
+
Summing up, both (12) and (13) hold on the terms of the set A. Using Lemma
|
1135 |
+
12, now we want to extend u to the points of X that lie between terms of the set A.
|
1136 |
+
Set b0 := a0 and since a1 β» a0, by Lemma 12 there exists b1 β X, with a1 β» b1 β» a0,
|
1137 |
+
31
|
1138 |
+
|
1139 |
+
such that
|
1140 |
+
(a1, b1) βΌ (b1, a0)
|
1141 |
+
Now build the set B := {. . . , bβ2, bβ1, b0, b1, b2, . . . }, with
|
1142 |
+
Β· Β· Β· βΊ bβ2 βΊ bβ1 βΊ b0 βΊ b1 βΊ b2 βΊ . . .
|
1143 |
+
based on b0, b1, in the same way we constructed A from a0, a1. Also here, we can deο¬ne
|
1144 |
+
a sequence that from an index set Pb β Z maps to B, that is, we deο¬ne the following
|
1145 |
+
function b : Pb β Z β B.
|
1146 |
+
By construction we have
|
1147 |
+
(b2, b1) βΌ (b1, b0)
|
1148 |
+
Together with (a1, b1) βΌ (b1, a0), by transitivity we have (b2, b1) βΌ (a1, b1). By A1,
|
1149 |
+
b2 = a1. Analogously, one can verify that
|
1150 |
+
b2p = ap for every p β Pa
|
1151 |
+
(17)
|
1152 |
+
So, the terms of the set B lie between the terms of the set A, i.e. the set B reο¬nes
|
1153 |
+
A and we can write
|
1154 |
+
A β B
|
1155 |
+
(18)
|
1156 |
+
Denote now c0 := b0 = a0 and we let c1 β X be that element provided by Lemma
|
1157 |
+
12 such that (b1, c1) βΌ (c1, b0). In the same way we constructed B from A, we can
|
1158 |
+
construct, from B, a third set C := {. . . , cβ2, cβ1, c0, c1, c2, . . . }, with
|
1159 |
+
Β· Β· Β· βΊ cβ2 βΊ cβ1 βΊ c0 βΊ c1 βΊ c2 βΊ . . .
|
1160 |
+
based on c0, c1. We can see that
|
1161 |
+
c2p = bp for every p β Pc
|
1162 |
+
where Pc β Z is the collection of indexes of the sequence c : Pc β Z β C.
|
1163 |
+
The set C reο¬nes B
|
1164 |
+
B β C
|
1165 |
+
(19)
|
1166 |
+
We keep iterating this process, constructing sets that reο¬ne one another and, for
|
1167 |
+
ease of notation, we denote them in the following way:
|
1168 |
+
A0 := A
|
1169 |
+
and
|
1170 |
+
a0
|
1171 |
+
p := ap β A0
|
1172 |
+
A1 := B
|
1173 |
+
and
|
1174 |
+
a1
|
1175 |
+
p := bp β A1
|
1176 |
+
A2 := C
|
1177 |
+
and
|
1178 |
+
a2
|
1179 |
+
p := cp β A2
|
1180 |
+
Β· Β· Β·
|
1181 |
+
32
|
1182 |
+
|
1183 |
+
These sets generalize the inclusions (18) and (19) as follows:
|
1184 |
+
A0 β A1 β A2 β Β· Β· Β· β An β . . .
|
1185 |
+
(20)
|
1186 |
+
So, in general, an
|
1187 |
+
p for p ΜΈ= 1 is obtained from the construction of (i) and (ii), applied to
|
1188 |
+
the points a0, an
|
1189 |
+
1. The term an
|
1190 |
+
1, for n > 0, is the βmidpointβ between anβ1
|
1191 |
+
1
|
1192 |
+
and a0, that
|
1193 |
+
exists by Lemma 12. By iterating the construction of (17), we have that
|
1194 |
+
p
|
1195 |
+
2n = q
|
1196 |
+
2m =β an
|
1197 |
+
p = am
|
1198 |
+
q
|
1199 |
+
In the spirit of (20), we extend u to all points in Aβ := οΏ½β
|
1200 |
+
n=1 An by:
|
1201 |
+
u(an
|
1202 |
+
p) = p
|
1203 |
+
2n
|
1204 |
+
for all an
|
1205 |
+
p β An
|
1206 |
+
Relations (12) and (13) hold in this extended domain: given x, y, z, w β οΏ½β
|
1207 |
+
n=1 An,
|
1208 |
+
just take n large enough so that they become, up to indiο¬erence, terms of the set An
|
1209 |
+
and proceed in the same exact way as we did for the set A0.
|
1210 |
+
To complete the construction of u we only remain to show Aβ is dense in X,
|
1211 |
+
that is Aβ = X. We ο¬rst show that none of the sets An has, for its sequences of
|
1212 |
+
points an, a point of accumulation in X. Indeed, ο¬x n and suppose by contradiction
|
1213 |
+
that an
|
1214 |
+
pk converges monotonically to aβ β X, where, without loss of generality, we
|
1215 |
+
assume an
|
1216 |
+
pk β aβ with aβ β X, i.e. (pk) is an increasing sequence of integers. Denote
|
1217 |
+
1n := (an
|
1218 |
+
1, a0) and we have, for every k β N,
|
1219 |
+
(an
|
1220 |
+
1+pk, an
|
1221 |
+
pk) β½ 1n β» 0
|
1222 |
+
By Lemma 10, we have (aβ, aβ) β½ 1n. So, by transitivity, we reach (aβ, aβ) β» 0, a
|
1223 |
+
contradiction. We conclude that, ο¬xed n, none of the sequences an with values in An
|
1224 |
+
has a limit point in X.
|
1225 |
+
To prove Aβ = X, the implication Aβ β X is trivial by construction. Now we
|
1226 |
+
want to show Aβ β X, that is all the elements of X belong to the closure of Aβ
|
1227 |
+
as well. Fix x β X such that, without loss of generality, x βΏ a0. For n β₯ 1, deο¬ne
|
1228 |
+
yn := sup{y β An : x βΏ y}. Note that a0 β {y β An : x βΏ y}, so this set is nonempty
|
1229 |
+
and we can write x βΏ yn βΏ a0. By Lemma 2, yn β X. Note further that, as shown
|
1230 |
+
before, An cannot have accumulation points in X so, as long as yn β X, it follows
|
1231 |
+
yn cannot be an accumulation point of An. So, yn must belong to An and we denote
|
1232 |
+
yn := an
|
1233 |
+
pn. As a result, we have:
|
1234 |
+
an
|
1235 |
+
pnβk βΎ x βΊ an
|
1236 |
+
pn+k for every k > 0
|
1237 |
+
(21)
|
1238 |
+
We also have that
|
1239 |
+
1n β» (x, yn)
|
1240 |
+
(22)
|
1241 |
+
33
|
1242 |
+
|
1243 |
+
Indeed, if (22) were not true, then (x, an
|
1244 |
+
pn) β½ 1n. We consider two cases: an
|
1245 |
+
1+pn β» x or
|
1246 |
+
an
|
1247 |
+
1+pn βΎ x. If an
|
1248 |
+
1+pn β» x, then, thanks to A1, we reach the following contradiction:
|
1249 |
+
1n βΌ (an
|
1250 |
+
1+pn, an
|
1251 |
+
pn) β» (x, an
|
1252 |
+
pn) β½ 1n
|
1253 |
+
(23)
|
1254 |
+
So an
|
1255 |
+
1+pn βΎ x, but this contradicts (21), that is, it contradicts yn to be the supremum.
|
1256 |
+
Thus, (22) holds. In particular, by A1 and A2, we can write (x, yn) β½ (yn, yn) βΌ 0,
|
1257 |
+
leading to
|
1258 |
+
1n β» (x, yn) β½ 0
|
1259 |
+
(24)
|
1260 |
+
Now, when n β β, as the sets An+1 β An β Anβ1 . . . are nested one into the
|
1261 |
+
other by (20), we can write, for every n β₯ 1, yn βΎ yn+1 βΎ x. Thus, the points yn form
|
1262 |
+
a non-decreasing sequence that is bounded from above by x. Call yβ the limit of this
|
1263 |
+
sequence, that is well-deο¬ned by Lemma 1. Since a0 βΎ yβ βΎ x, by Lemma 2 it follows
|
1264 |
+
that yβ β X. In particular, by Lemma 6 we have yβ β Aβ, because, for every ο¬xed
|
1265 |
+
n β₯ 1, yn is a term of the sets An, and so (yn) β AN
|
1266 |
+
β.
|
1267 |
+
As to the 1n terms, for n ο¬xed, we see that
|
1268 |
+
1n βΌ (an
|
1269 |
+
2, an
|
1270 |
+
1) βΌ (anβ1
|
1271 |
+
1
|
1272 |
+
, an
|
1273 |
+
1)
|
1274 |
+
We also have that, for every n β₯ 1, a0 βΎ an+1
|
1275 |
+
1
|
1276 |
+
βΎ an
|
1277 |
+
1.
|
1278 |
+
Thus, the points an
|
1279 |
+
1 form, for n β β, a non-increasing sequence that is bounded
|
1280 |
+
from below by a0. Call aβ the limit of this sequence, that is well-deο¬ned by Lemma 1.
|
1281 |
+
Since a0 βΎ aβ βΎ a1, by Lemma 2 we have aβ β X.
|
1282 |
+
Consider now (anβ1
|
1283 |
+
1
|
1284 |
+
, an
|
1285 |
+
1) and (x, yn). By Lemma 10 and from (24) it follows that
|
1286 |
+
(aβ, aβ) β½ (x, yβ) β½ 0
|
1287 |
+
Since, by A2, (aβ, aβ) βΌ 0, by transitivity (x, yβ) βΌ 0, so that x βΌ yβ, i.e. x = yβ as βΏ
|
1288 |
+
is antisymmetric.
|
1289 |
+
Since x was arbitrarily chosen in X and yβ β Aβ, we can conclude x β Aβ, so
|
1290 |
+
that Aβ = X. Therefore, we can extend u by continuity to the whole set X by setting
|
1291 |
+
u(x) = lim
|
1292 |
+
nββ u(xn)
|
1293 |
+
if (xn) β AN
|
1294 |
+
β converges monotonically to x. Note that u : X β R is well-deο¬ned.
|
1295 |
+
Indeed, to prove it is well-posed we show that if xn and yn are two sequences that
|
1296 |
+
converge to x, then limnββ u(xn) = limnββ u(yn). This follows easily by continuity of
|
1297 |
+
u.21 In light of Lemma 6, it is easy to see that u satisο¬es (12) and (13).
|
1298 |
+
As to uniqueness, observe that any other u that satisο¬es (12) and (13) can be
|
1299 |
+
21Recall that in every topological space X continuity implies sequential continuity. The converse
|
1300 |
+
holds if X is ο¬rst-countable.
|
1301 |
+
34
|
1302 |
+
|
1303 |
+
normalized so that u(a0) = 0 and u(a1) = 1. So, u must agree on u at each step of
|
1304 |
+
the constructive procedure for u just seen. Indeed, for a given u : X β R, deο¬ne the
|
1305 |
+
following positive aο¬ne transformation f : Im(u) β R such that
|
1306 |
+
f(x) :=
|
1307 |
+
x β u(a0)
|
1308 |
+
u(a1) β u(a0)
|
1309 |
+
It is immediate to see that, for the equivalent utility function οΏ½u := f β¦ u, we have
|
1310 |
+
οΏ½u(a0) = 0 and οΏ½u(a1) = 1.
|
1311 |
+
Summing up, we proved Theorem 5 if βΏ is antisymmetric.
|
1312 |
+
Now we drop this
|
1313 |
+
assumption. Let X/βΌ be the quotient space with respect to the equivalence relation
|
1314 |
+
βΌ. The set {x β X : x βΌ y} is a closed set in X by Lemma 4, so (X/βΌ, ΛβΏ) is a totally
|
1315 |
+
ordered connected and separable subset of a topological space, where ΛβΏ is the total
|
1316 |
+
order induced by the weak order βΏ.22 Therefore, the orders βΏ and β½ induce orders ΛβΏ
|
1317 |
+
and Λβ½ on the quotient set X/βΌ, by setting, for all [x], [y] β X/βΌ
|
1318 |
+
[x] ΛβΏ [y] ββ x βΏ y
|
1319 |
+
and, for all [x], [y], [z], [w] β X/βΌ
|
1320 |
+
([x], [y]) Λβ½ ([z], [w]) ββ (x, y) β½ (z, w)
|
1321 |
+
It is routine to show that the orders ΛβΏ over X/βΌ and Λβ½ over X/βΌ Γ X/βΌ inherit the
|
1322 |
+
same properties of βΏ and β½ used in the theorem. So, by what has been proved so far,
|
1323 |
+
there exists Λu : X/βΌ β R that satisο¬es (12) and (13) for (ΛβΏ, Λβ½). Let Ο : X β X/βΌ be
|
1324 |
+
the quotient map. Then, the function u : X β R deο¬ned as u = Λu β¦ Ο is a well-deο¬ned
|
1325 |
+
measurable utility function, i.e. it is easily seen to satisfy (12) and (13) for (βΏ, β½).
|
1326 |
+
To conclude, we show that u satisο¬es (12) and (13). If x βΌ y then [x] = [y] and,
|
1327 |
+
by the theorem we have just proved, Λu([x]) = Λu([y]), which is (Λu β¦ Ο)(x) = (Λu β¦ Ο)(y),
|
1328 |
+
and so u(x) = u(y). If x β» y, then [x] β» [y], which implies Λu([x]) > Λu([y]), which is
|
1329 |
+
(Λu β¦ Ο)(x) > (Λu β¦ Ο)(y), and so u(x) > u(y).
|
1330 |
+
Conversely, assume u(x) β₯ u(y) and suppose by contradiction x οΏ½ y that, by
|
1331 |
+
completeness, is y β» x. If u(x) = u(y) then Λu([x]) = Λu([y]) ββ [x] = [y] ββ x βΌ y,
|
1332 |
+
a contradiction. If u(x) > u(y) then Λu([x]) > Λu([y]) ββ [x] > [y] ββ x β» y, a
|
1333 |
+
contradiction. Hence, (12) holds for u.
|
1334 |
+
By deο¬nition, we have that ([x], [y]) β½ ([z], [w])
|
1335 |
+
ββ
|
1336 |
+
(x, y) β½ (z, w), for all
|
1337 |
+
[x],[y],[z],[w] β X/βΌ. So, we can write (x, y) β½ (z, w) ββ ([x], [y]) β½ ([z], [w]) ββ
|
1338 |
+
Λu([x])βΛu([y]) β₯ Λu([z])βΛu([w]) ββ u(x)βu(y) β₯ u(z)βu(w). Hence, also (13) holds
|
1339 |
+
for u.
|
1340 |
+
22That is, ΛβΏ := βΏ /βΌ β X/βΌ Γ X/βΌ.
|
1341 |
+
35
|
1342 |
+
|
1343 |
+
This completes the proof of Theorem 5.
|
1344 |
+
Graphically, we can build the following diagram to represent our construction.
|
1345 |
+
X
|
1346 |
+
X/βΌ
|
1347 |
+
R
|
1348 |
+
Ο
|
1349 |
+
u
|
1350 |
+
Λu
|
1351 |
+
References
|
1352 |
+
[1] Adams, E.W. 1960. βSurvey of Bernoullian utility theory.β Mathematical Thinking
|
1353 |
+
in the Measurement of Behavior, edited by Solomon, H., 151β268. Glencoe.
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+
[2] Allais, M. 1943. βA la Recherche dβune Discipline Economique. LβEconomie Pure.β
|
1355 |
+
Ateliers Industria, Paris.
|
1356 |
+
[3] Allais, M. 1979. βThe so-called Allais paradox and rational decisions under uncer-
|
1357 |
+
tainty.β Expected Utility Hypotheses and the Allais Paradox, edited by Allais, M.
|
1358 |
+
and Hagen, O., 437-681.
|
1359 |
+
[4] Allais, M. 1994. βThe fundamental cardinalist approach and its prospects.β Cardi-
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1360 |
+
nalism, edited by Allais, M. and Hagen, O., 289-306. Kluwer Academic Publishers.
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1361 |
+
[5] Alt, F. 1936. βΒ¨Uber die MΒ¨assbarkeit des Nutzens.β Zeitschrift fΒ¨ur NationalΒ¨okonomie
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[6] Banakh, T., Gutik, O., Potiatynyk, O., and Ravsky, A. 2012. βMetrizability of
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[9] Debreu, G. 1964. βContinuity properties of a Paretian utility.β International Eco-
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1376 |
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[13] Fechner, G.T. 1860. βElemente der Psychophysik.β Leipzig, Breitkopf und HΒ¨artel.
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[14] Feng, Z., and Heath, R. 2008. βMetrizability of topological semigroups on linearly
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1385 |
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[20] Hutcheson, P. 1728. βAn Essay on the Nature and Conduct of the Passions and
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1397 |
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|
1401 |
+
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1402 |
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1403 |
+
edited by Luce, R.D., Bush, R.R., and Galanter, E. Handbook of Mathematical
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1406 |
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|
1407 |
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|
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|
1434 |
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[40] Suppes, P., and Winet, M. 1955. βAn axiomatization of utility based on the notion
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of utility diο¬erences.β Management Science 1: 186-202.
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39
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|
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|
1 |
+
arXiv:2301.03137v1 [math.NT] 9 Jan 2023
|
2 |
+
Gaps on the intersection numbers of
|
3 |
+
sections on a rational elliptic surface
|
4 |
+
Renato Dias Costa
|
5 |
+
Abstract
|
6 |
+
Given a rational elliptic surface X over an algebraically closed ο¬eld, we investigate whether a
|
7 |
+
given natural number k can be the intersection number of two sections of X. If not, we say that
|
8 |
+
k a gap number. We try to answer when gap numbers exist, how they are distributed and how to
|
9 |
+
identify them. We use Mordell-Weil lattices as our main tool, which connects the investigation
|
10 |
+
to the classical problem of representing integers by positive-deο¬nite quadratic forms.
|
11 |
+
Contents
|
12 |
+
1
|
13 |
+
Introduction
|
14 |
+
2
|
15 |
+
2
|
16 |
+
Preliminaries
|
17 |
+
4
|
18 |
+
2.1
|
19 |
+
The Mordell-Weil Lattice
|
20 |
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
21 |
+
4
|
22 |
+
2.2
|
23 |
+
Gap numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
24 |
+
6
|
25 |
+
2.3
|
26 |
+
Bounds cmax, cmin for the contribution term . . . . . . . . . . . . . . . . . . . . . . .
|
27 |
+
6
|
28 |
+
2.4
|
29 |
+
The diο¬erence β = cmax β cmin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
30 |
+
8
|
31 |
+
2.5
|
32 |
+
The quadratic form QX
|
33 |
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
34 |
+
9
|
35 |
+
3
|
36 |
+
Intersection with a torsion section
|
37 |
+
10
|
38 |
+
4
|
39 |
+
Existence of a pair of sections with a given intersection number
|
40 |
+
11
|
41 |
+
4.1
|
42 |
+
Necessary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
43 |
+
11
|
44 |
+
4.2
|
45 |
+
Suο¬cient conditions when β β€ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
46 |
+
12
|
47 |
+
4.2.1
|
48 |
+
The case β < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
49 |
+
12
|
50 |
+
4.2.2
|
51 |
+
The case β = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
52 |
+
13
|
53 |
+
4.3
|
54 |
+
Necessary and suο¬cient conditions for β β€ 2
|
55 |
+
. . . . . . . . . . . . . . . . . . . . . .
|
56 |
+
14
|
57 |
+
4.4
|
58 |
+
Summary of suο¬cient conditions
|
59 |
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
60 |
+
15
|
61 |
+
5
|
62 |
+
Main Results
|
63 |
+
15
|
64 |
+
5.1
|
65 |
+
No gap numbers in rank r β₯ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
66 |
+
15
|
67 |
+
5.2
|
68 |
+
Gaps with probability 1 in rank r β€ 2
|
69 |
+
. . . . . . . . . . . . . . . . . . . . . . . . . .
|
70 |
+
17
|
71 |
+
5.3
|
72 |
+
Identiο¬cation of gaps when E(K) is torsion-free with rank r = 1
|
73 |
+
. . . . . . . . . . .
|
74 |
+
18
|
75 |
+
5.4
|
76 |
+
Surfaces with a 1-gap
|
77 |
+
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
|
78 |
+
20
|
79 |
+
6
|
80 |
+
Appendix
|
81 |
+
23
|
82 |
+
1
|
83 |
+
|
84 |
+
1
|
85 |
+
Introduction
|
86 |
+
Description of the problem. Let X be a rational elliptic surface over an algebraically closed
|
87 |
+
ο¬eld, i.e. a smooth, rational projective surface with a ο¬bration Ο : X β P1 whose general ο¬ber
|
88 |
+
is a smooth curve of genus 1. Assume also that Ο is relatively minimal, i.e. no ο¬ber contains an
|
89 |
+
exceptional curve in its support. We use E/K to denote the generic ο¬ber of Ο, which is an elliptic
|
90 |
+
curve over the function ο¬eld K := k(P1). By the Mordell-Weil theorem, the set E(K) of K-points
|
91 |
+
is a ο¬nitely generated Abelian group, whose rank we denote by r. The points on E(K) are in
|
92 |
+
bijective correspondence with the sections of Ο, as well as with the exceptional curves on X, so
|
93 |
+
we use these terms interchangeably. This paper addresses the following question: given sections
|
94 |
+
P1, P2 β E(K), what values can the intersection number P1 Β· P2 possibly attain?
|
95 |
+
Original motivation.
|
96 |
+
The problem originates from a previous investigation of conic bundles
|
97 |
+
on X, i.e. morphisms Ο : X β P1 whose general ο¬ber is a smooth curve of genus zero [Cos]. More
|
98 |
+
speciο¬cally, one of the ways to produce a conic bundle is by ο¬nding a pair of sections P1, P2 β E(K)
|
99 |
+
with P1 Β· P2 = 1, so that the linear system |P1 + P2| induces a conic bundle Ο|P1+P2| : X β P1
|
100 |
+
having P1 + P2 as a reducible ο¬ber. We may ask under which conditions such a pair exists. An
|
101 |
+
immediate necessary condition is that r β₯ 1, for if r = 0 any two distinct sections must be disjoint
|
102 |
+
[SS19, Cor. 8.30]. Conversely, given that r β₯ 1, does X admit such a pair? The ο¬rst observation
|
103 |
+
is that r β₯ 1 implies an inο¬nite number of sections, so we should expect inο¬nitely many values for
|
104 |
+
P1Β·P2 as P1, P2 run through E(K). Then the question is ultimately: what values may P1Β·P2 assume?
|
105 |
+
Mordell-Weil lattices. The computation of intersection numbers on a surface is a diο¬cult prob-
|
106 |
+
lem in general. However, as we are concerned with sections on an elliptic surface, the information
|
107 |
+
we need is considerably more accessible. The reason for this lies in the Mordell-Weil lattice, a
|
108 |
+
concept ο¬rst established in [Elk90], [Shi89], [Shi90]. It involves the deο¬nition of a Q-valued pair-
|
109 |
+
ing β¨Β·, Β·β© on E(K), called the height pairing [SS19, Section 6.5], inducing a positive-deο¬nite lattice
|
110 |
+
(E(K)/E(K)tor, β¨Β·, Β·β©), named the Mordell-Weil lattice.
|
111 |
+
A key aspect of its construction is the
|
112 |
+
connection with the NΓ©ron-Severi lattice, so that the height pairing and the intersection pairing
|
113 |
+
of sections are strongly intertwined. In the case of rational elliptic surfaces, the possibilities for
|
114 |
+
the Mordell-Weil lattice have already been classiο¬ed in [OS91], which gives us a good starting point.
|
115 |
+
Representation of integers.
|
116 |
+
The use of Mordell-Weil lattices in our investigation leads to
|
117 |
+
a classical problem in number theory, which is the representation of integers by positive-deο¬nite
|
118 |
+
quadratic forms. Indeed, the free part of E(K) is generated by r terms, so the height h(P) := β¨P, Pβ©
|
119 |
+
induces a positive-deο¬nite quadratic form on r variables with coeο¬cients in Q. If O β E(K) is the
|
120 |
+
neutral section and R is the set of reducible ο¬bers of Ο, then by the height formula (2)
|
121 |
+
h(P) = 2 + 2(P Β· O) β
|
122 |
+
οΏ½
|
123 |
+
vβR
|
124 |
+
contrv(P),
|
125 |
+
where the sum over v is a rational number which can be estimated. By clearing denominators,
|
126 |
+
we see that the possible values of P Β· O depend on a certain range of integers represented by a
|
127 |
+
positive-deο¬nite quadratic form with coeο¬cients in Z. This point of view is explored in some parts
|
128 |
+
of the paper, where we apply results such as the classical Lagrangeβs four-square theorem [HW79,
|
129 |
+
Β§20.5], the counting of integers represented by a binary quadratic form [Ber12, p. 91] and the more
|
130 |
+
recent Bhargava-Hankeβs 290-theorem on universal quadratic forms [BH, Thm. 1].
|
131 |
+
2
|
132 |
+
|
133 |
+
Statement of results. Given k β Zβ₯0 we investigate whether there is a pair of sections P1, P2 β
|
134 |
+
E(K) such that P1 Β· P2 = k. If such a pair does not exist, we say that X has a k-gap, or that k is
|
135 |
+
a gap number. Our ο¬rst result is a complete identiο¬cation of gap numbers in some cases:
|
136 |
+
Theorem 5.7. If E(K) is torsion-free with rank r = 1, we have the following characterization of
|
137 |
+
gap numbers on X according to the lattice T associated to the reducible ο¬bers of Ο.
|
138 |
+
T
|
139 |
+
k is a gap number β none of
|
140 |
+
the following are perfect squares
|
141 |
+
E7
|
142 |
+
k + 1, 4k + 1
|
143 |
+
A7
|
144 |
+
k+1
|
145 |
+
4 , 16k, ..., 16k + 9
|
146 |
+
D7
|
147 |
+
k+1
|
148 |
+
2 , 8k + 1, ..., 8k + 4
|
149 |
+
A6 β A1
|
150 |
+
k+1
|
151 |
+
7 , 28k β 3, ..., 28k + 21
|
152 |
+
E6 β A1
|
153 |
+
k+1
|
154 |
+
3 , 12k + 1, ..., 12k + 9
|
155 |
+
D5 β A2
|
156 |
+
k+1
|
157 |
+
6 , 24k + 1, ..., 24k + 16
|
158 |
+
A4 β A3
|
159 |
+
k+1
|
160 |
+
10 , 40k β 4, ..., 40k + 25
|
161 |
+
A4 β A2 β A1
|
162 |
+
k+1
|
163 |
+
15 , 60k β 11, ..., 60k + 45
|
164 |
+
We also explore the possibility of X having no gap numbers. We prove that, in fact, this is
|
165 |
+
always the case if the Mordell-Weil rank is big enough.
|
166 |
+
Theorem 5.2. If r β₯ 5, then X has no gap numbers.
|
167 |
+
On the other hand, for r β€ 2 we show that gap numbers occur with probability 1.
|
168 |
+
Theorem 5.4. If r β€ 2, then the set of gap numbers of X, i.e. G := {k β N | k is a gap number of X}
|
169 |
+
has density 1 in N, i.e.
|
170 |
+
lim
|
171 |
+
nββ
|
172 |
+
#G β© {1, ..., n}
|
173 |
+
n
|
174 |
+
= 1.
|
175 |
+
At last we answer the question from the original motivation, which consists in classifying the
|
176 |
+
rational elliptic surfaces with a 1-gap:
|
177 |
+
Theorem 5.8. X has a 1-gap if and only if r = 0 or r = 1 and Ο has a IIIβ ο¬ber.
|
178 |
+
3
|
179 |
+
|
180 |
+
Structure of the paper. The text is organized as follows. Section 2 introduces the main objects,
|
181 |
+
namely the Mordell-Weil lattice, the bounds cmax, cmin for the contribution term, the diο¬erence
|
182 |
+
β = cmax βcmin and the quadratic form QX induced by the height pairing. In Section 3 we explain
|
183 |
+
the role of torsion sections in the investigation. The key technical results are gathered in Section 4,
|
184 |
+
where we state necessary and suο¬cient conditions for having P1 Β· P2 = k for a given k. Section 5
|
185 |
+
contains the main results of the paper, namely: the description of gap numbers when E(K) is
|
186 |
+
torsion-free with r = 1 (Subsection 5.3), the absence of gap numbers for r β₯ 5 (Subsection 5.1),
|
187 |
+
density of gap numbers when r β€ 2 (Subsection 5.2) and the classiο¬cation of surfaces with a 1-gap
|
188 |
+
(Subsection 5.4). Section 6 is an appendix containing Table 8, which stores the relevant information
|
189 |
+
about the Mordell-Weil lattices of rational elliptic surfaces with r β₯ 1.
|
190 |
+
2
|
191 |
+
Preliminaries
|
192 |
+
Throughout the paper X denotes a rational elliptic surface over an algebraically closed ο¬eld
|
193 |
+
k of any characteristic. More precisely, X is a smooth rational projective surface with a ο¬bration
|
194 |
+
Ο : X β P1, with a section, whose general ο¬ber is a smooth curve of genus 1. We assume moreover
|
195 |
+
that Ο is relatively minimal (i.e. each ο¬ber has no exceptional curve in its support) [SS19, Def.
|
196 |
+
5.2]. The generic ο¬ber of Ο is an elliptic curve E/K over K := k(P1). The set E(K) of K-points is
|
197 |
+
called the Mordell-Weil group of X, whose rank is called the Mordell-Weil rank of X, denoted by
|
198 |
+
r := rank E(K).
|
199 |
+
In what follows we introduce the main objects of our investigation and stablish some notation.
|
200 |
+
2.1
|
201 |
+
The Mordell-Weil Lattice
|
202 |
+
We give a brief description of the Mordell-Weil lattice, which is the central tool used in the
|
203 |
+
paper. Although it can be deο¬ned on elliptic surfaces in general, we restrict ourselves to rational
|
204 |
+
elliptic surfaces. For more information on Mordell-Weil lattices, we refer the reader to the com-
|
205 |
+
prehensive introduction by Schuett and Shioda [SS19] in addition to the original sources, namely
|
206 |
+
[Elk90], [Shi89], [Shi90].
|
207 |
+
We begin by noting that points in E(K) can be regarded as curves on X and by deο¬ning the
|
208 |
+
lattice T and the trivial lattice Triv(X), which are needed to deο¬ne the Mordell-Weil lattice.
|
209 |
+
Sections, points on E(K) and exceptional curves. The sections of Ο are in bijective cor-
|
210 |
+
respondence with points on E(K). Moreover, since X is rational and relatively minimal, points on
|
211 |
+
E(K) also correspond to exceptional curves on X [SS10, Section 8.2]. For this reason we identify
|
212 |
+
sections of Ο, points on E(K) and exceptional curves on X.
|
213 |
+
The lattice T and the trivial lattice Triv(X). Let O β E(K) be the neutral section and
|
214 |
+
R := {v β P1 | Οβ1(v) is reducible} the set of reducible ο¬bers of Ο. The components of a ο¬ber
|
215 |
+
Οβ1(v) are denoted by Ξv,i, where Ξv,0 is the only component intersected by O. The NΓ©ron-Severi
|
216 |
+
group NS(X) together with the intersection pairing is called the NΓ©ron-Severi lattice.
|
217 |
+
4
|
218 |
+
|
219 |
+
We deο¬ne the following sublattices of NS(X), which encode the reducible ο¬bers of Ο:
|
220 |
+
Tv := Zβ¨Ξv,i | i ΜΈ= 0β© for v β R,
|
221 |
+
T :=
|
222 |
+
οΏ½
|
223 |
+
vβR
|
224 |
+
Tv.
|
225 |
+
By Kodairaβs classiο¬cation [SS19, Thm. 5.12], each Tv with v β R is represented by a Dynkin
|
226 |
+
diagram Am, Dm or Em for some m. We also deο¬ne the trivial lattice of X, namely
|
227 |
+
Triv(X) := Zβ¨O, Ξv,i | i β₯ 0, v β Rβ©.
|
228 |
+
Next we deο¬ne the Mordell-Weil lattice and present the height formula.
|
229 |
+
The Mordell-Weil lattice. In order to give E(K) a lattice structure, we cannot use the inter-
|
230 |
+
section pairing directly, which only deο¬nes a lattice on NS(X) but not on E(K). This is achieved
|
231 |
+
by deο¬ning a Q-valued pairing, called the height pairing, given by
|
232 |
+
β¨Β·, Β·β© : E(K) Γ E(K) β Q
|
233 |
+
P, Q οΏ½β βΟ(P) Β· Ο(Q),
|
234 |
+
where Ο : E(K) β NS(X) βZ Q is deο¬ned from the orthogonal projection with respect to Triv(X)
|
235 |
+
(for a detailed exposition, see [SS19, Section 6.5]). Moreover, dividing by torsion elements we get
|
236 |
+
a positive-deο¬nite lattice (E(K)/E(K)tor, β¨Β·, Β·β©) [SS19, Thm. 6.20], called the Mordell-Weil lattice.
|
237 |
+
The height formula. The height pairing can be explicitly computed by the height formula [SS19,
|
238 |
+
Thm. 6.24]. For rational elliptic surfaces, it is given by
|
239 |
+
β¨P, Qβ© = 1 + (P Β· O) + (Q Β· O) β (P Β· Q) β
|
240 |
+
οΏ½
|
241 |
+
vβR
|
242 |
+
contrv(P, Q),
|
243 |
+
(1)
|
244 |
+
h(P) := β¨P, Pβ© = 2 + 2(P Β· O) β
|
245 |
+
οΏ½
|
246 |
+
vβR
|
247 |
+
contrv(P),
|
248 |
+
(2)
|
249 |
+
where contrv(P) := contrv(P, P) and contrv(P, Q) are given by Table 1 [SS19, Table 6.1] assuming
|
250 |
+
P, Q meet Οβ1(v) at Ξv,i, Ξv,j resp. with 0 < i < j. If P or Q meets Ξv,0, then contrv(P, Q) := 0.
|
251 |
+
The minimal norm. Since E(K) is ο¬nitely generated, there is a minimal positive value for h(P)
|
252 |
+
as P runs through E(K) with h(P) > 0. It is called the minimal norm, denoted by
|
253 |
+
Β΅ := min{h(P) > 0 | P β E(K)}.
|
254 |
+
The narrow Mordell-Weil lattice. An important sublattice of E(K) is the narrow Mordell-Weil
|
255 |
+
lattice E(K)0, deο¬ned as
|
256 |
+
E(K)0 := {P β E(K) | P intersects Ξv,0 for all v β R}
|
257 |
+
= {P β E(K) | contrv(P) = 0 for all v β R}.
|
258 |
+
As a subgroup, E(K)0 is torsion-free; as a sublattice, it is a positive-deο¬nite even integral lattice
|
259 |
+
with ο¬nite index in E(K) [SS19, Thm. 6.44]. The importance of the narrow lattice can be explained
|
260 |
+
by its considerable size as a sublattice and by the easiness to compute the height pairing on it,
|
261 |
+
since all contribution terms vanish. A complete classiο¬cation of the lattices E(K) and E(K)0 on
|
262 |
+
rational elliptic surfaces is found in [OS91, Main Thm.].
|
263 |
+
5
|
264 |
+
|
265 |
+
Tv
|
266 |
+
A1
|
267 |
+
E7
|
268 |
+
A2
|
269 |
+
E6
|
270 |
+
Anβ1
|
271 |
+
Dn+4
|
272 |
+
Type of Οβ1(v)
|
273 |
+
III
|
274 |
+
IIIβ
|
275 |
+
IV
|
276 |
+
IVβ
|
277 |
+
In
|
278 |
+
Iβ
|
279 |
+
n
|
280 |
+
contrv(P)
|
281 |
+
1
|
282 |
+
2
|
283 |
+
3
|
284 |
+
2
|
285 |
+
2
|
286 |
+
3
|
287 |
+
4
|
288 |
+
3
|
289 |
+
i(nβi)
|
290 |
+
n
|
291 |
+
οΏ½
|
292 |
+
1
|
293 |
+
(i = 1)
|
294 |
+
1 + n
|
295 |
+
4
|
296 |
+
(i > 1)
|
297 |
+
contrv(P, Q)
|
298 |
+
-
|
299 |
+
-
|
300 |
+
1
|
301 |
+
3
|
302 |
+
2
|
303 |
+
3
|
304 |
+
i(nβj)
|
305 |
+
n
|
306 |
+
οΏ½ 1
|
307 |
+
2
|
308 |
+
(i = 1)
|
309 |
+
1
|
310 |
+
2 + n
|
311 |
+
4
|
312 |
+
(i > 1)
|
313 |
+
Table 1: Local contributions from reducible ο¬bers to the height pairing.
|
314 |
+
2.2
|
315 |
+
Gap numbers
|
316 |
+
We introduce some convenient terminology to express the possibility of ο¬nding a pair of sections
|
317 |
+
with a given intersection number.
|
318 |
+
Deο¬nition 2.1. If there are no sections P1, P2 β E(K) such that P1 Β· P2 = k, we say that X has
|
319 |
+
a k-gap or that k is a gap number of X.
|
320 |
+
Deο¬nition 2.2. We say that X is gap-free if for every k β Zβ₯0 there are sections P1, P2 β E(K)
|
321 |
+
such that P1 Β· P2 = k.
|
322 |
+
Remark 2.3. In case the Mordell-Weil rank is r = 0, we have E(K) = E(K)tor. In particular,
|
323 |
+
any two distinct sections are disjoint [SS19, Cor. 8.30], hence every k β₯ 1 is a gap number of X.
|
324 |
+
For positive rank, the description of gap numbers is less trivial, thus our focus on r β₯ 1.
|
325 |
+
2.3
|
326 |
+
Bounds cmax, cmin for the contribution term
|
327 |
+
We deο¬ne the estimates cmax, cmin for the contribution term οΏ½
|
328 |
+
v contrv(P) and state some
|
329 |
+
simple facts about them. We also provide an example to illustrate how they are computed.
|
330 |
+
The need for these estimates comes from the following. Suppose we are given a section P β E(K)
|
331 |
+
whose height h(P) is known and we want to determine P Β· O. In case P β E(K)0 we have a direct
|
332 |
+
answer, namely P Β· O = h(P)/2 β 1 by the height formula (2).
|
333 |
+
However if P /β E(K)0, the
|
334 |
+
computation of P Β· O depends on the contribution term cP := οΏ½
|
335 |
+
vβR contrv(P), which by Table 1
|
336 |
+
depends on how P intersects the reducible ο¬bers of Ο. Usually we do not have this intersection
|
337 |
+
data at hand, which is why we need estimates for cP not depending on P.
|
338 |
+
Deο¬nition 2.4. If the set R of reducible ο¬bers of Ο is not empty, we deο¬ne
|
339 |
+
cmax :=
|
340 |
+
οΏ½
|
341 |
+
vβR
|
342 |
+
max{contrv(P) | P β E(K)},
|
343 |
+
cmin := min {contrv(P) > 0 | P β E(K), v β R} .
|
344 |
+
Remark 2.5. The case R = β
only occurs when X has Mordell-Weil rank r = 8 (No. 1 in Table 8).
|
345 |
+
In this case E(K)0 = E(K) and οΏ½
|
346 |
+
vβR contrv(P) = 0 βP β E(K), hence we adopt the convention
|
347 |
+
cmax = cmin = 0.
|
348 |
+
6
|
349 |
+
|
350 |
+
Remark 2.6. We use cmax, cmin as bounds for cP := οΏ½
|
351 |
+
v contrv(P). For our purposes it is not
|
352 |
+
necessary to know whether cP actually attains one of these bounds for some P, so that cmax, cmin
|
353 |
+
should be understood as hypothetical values.
|
354 |
+
We state some facts about cmax, cmin.
|
355 |
+
Lemma 2.7. Let X be a rational elliptic surface with Mordell-Weil rank r β₯ 1. If Ο admits a
|
356 |
+
reducible ο¬ber, then:
|
357 |
+
i) cmin > 0.
|
358 |
+
ii) cmax < 4.
|
359 |
+
iii) cmin β€ οΏ½
|
360 |
+
vβR contrv(P) β€ cmax βP /β E(K)0. For P β E(K)0, only the second inequality holds.
|
361 |
+
iv) If οΏ½
|
362 |
+
vβR contrv(P) = cmin, then contrvβ²(P) = cmin for some vβ² and contrv(P) = 0 for v ΜΈ= vβ².
|
363 |
+
Proof. Item i) is immediate from the deο¬nition of cmin. For ii) it is enough to check the values
|
364 |
+
of cmax directly in Table 8. For iii), the second inequality follows from the deο¬nition of cmax and
|
365 |
+
clearly holds for any P β E(K). If P /β E(K)0, then cP := οΏ½
|
366 |
+
v contrv(P) > 0, so contrv0(P) > 0
|
367 |
+
for some v0. Therefore cP β₯ contrv0(P) β₯ cmin.
|
368 |
+
For iv), let οΏ½
|
369 |
+
v contrv(P) = cmin. Assume by contradiction that there are distinct v1, v2 such
|
370 |
+
that contrvi(P) > 0 for i = 1, 2. By deο¬nition of cmin we have cmin β€ contrvi(P) for i = 1, 2 so
|
371 |
+
cmin =
|
372 |
+
οΏ½
|
373 |
+
v
|
374 |
+
contrv(P) β₯ contrv1(P) + contrv2(P) β₯ 2cmin,
|
375 |
+
which is absurd because cmin > 0 by i). Therefore there is only one vβ² with contrvβ²(P) > 0, while
|
376 |
+
contrv(P) = 0 for all v ΜΈ= vβ². In particular, contrvβ²(P) = cmin. β
|
377 |
+
Explicit computation. Once we know the lattice T associated with the reducible ο¬bers of Ο
|
378 |
+
(Section 2.1), the computation of cmax, cmin is simple. For a ο¬xed v β R, the extreme values of the
|
379 |
+
local contribution contrv(P) are given in Table 2, which is derived from Table 1. We provide an
|
380 |
+
example to illustrate this computation.
|
381 |
+
Tv
|
382 |
+
max{contrv(P) | P β E(K)}
|
383 |
+
min{contrv(P) > 0 | P β E(K)}
|
384 |
+
Anβ1
|
385 |
+
β(nββ)
|
386 |
+
n
|
387 |
+
, where β :=
|
388 |
+
οΏ½n
|
389 |
+
2
|
390 |
+
οΏ½
|
391 |
+
nβ1
|
392 |
+
n
|
393 |
+
Dn+4
|
394 |
+
1 + n
|
395 |
+
4
|
396 |
+
1
|
397 |
+
E6
|
398 |
+
4
|
399 |
+
3
|
400 |
+
4
|
401 |
+
3
|
402 |
+
E7
|
403 |
+
3
|
404 |
+
2
|
405 |
+
3
|
406 |
+
2
|
407 |
+
Table 2: Extreme values of contrv(P).
|
408 |
+
7
|
409 |
+
|
410 |
+
Example: Let Ο with ο¬ber conο¬guration (I4, IV, III, I1). The reducible ο¬bers are I4, IV, III, so
|
411 |
+
T = A3 β A2 β A1.
|
412 |
+
By Table 2, the maximal contributions for A3, A2, A1 are 2Β·2
|
413 |
+
4
|
414 |
+
= 1,
|
415 |
+
2
|
416 |
+
3,
|
417 |
+
1
|
418 |
+
2
|
419 |
+
respectively. The minimal positive contributions are 1Β·3
|
420 |
+
4 = 3
|
421 |
+
4, 2
|
422 |
+
3, 1
|
423 |
+
2 respectively. Then
|
424 |
+
cmax = 1 + 2
|
425 |
+
3 + 1
|
426 |
+
2 = 13
|
427 |
+
6 ,
|
428 |
+
cmin = min
|
429 |
+
οΏ½3
|
430 |
+
4, 2
|
431 |
+
3, 1
|
432 |
+
2
|
433 |
+
οΏ½
|
434 |
+
= 1
|
435 |
+
2.
|
436 |
+
2.4
|
437 |
+
The diο¬erence β = cmax β cmin
|
438 |
+
In this section we explain why the value of β := cmax β cmin is relevant to our discussion,
|
439 |
+
specially in Subsection 4.2. We also verify that β < 2 in most cases and identify the exceptional
|
440 |
+
ones in Table 3 and Table 4.
|
441 |
+
As noted in Subsection 2.3, in case P /β E(K)0 and h(P) is known, the diο¬culty of determining
|
442 |
+
P Β·O lies in the contribution term cP := οΏ½
|
443 |
+
vβR contrv(P). In particular, the range of possible values
|
444 |
+
for cP determines the possibilities for P Β· O. This range is measured by the diο¬erence
|
445 |
+
β := cmax β cmin.
|
446 |
+
Hence a smaller β means a better control over the intersection number P Β· O, which is why β
|
447 |
+
plays an important role in determining possible intersection numbers. In Subsection 4.3 we assume
|
448 |
+
β β€ 2 and state necessary and suο¬cient conditions for having a pair P1, P2 such that P1 Β· P2 = k
|
449 |
+
for a given k β₯ 0. If however β > 2, the existence of such a pair is not guaranteed a priori, so a
|
450 |
+
case-by-case treatment is needed. Fortunately by Lemma 2.8 the case β > 2 is rare.
|
451 |
+
Lemma 2.8. Let X be a rational elliptic surface with Mordell-Weil rank r β₯ 1. The only cases
|
452 |
+
with β = 2 and β > 2 are in Table 3 and 4 respectively. In particular we have β < 2 whenever
|
453 |
+
E(K) is torsion-free.
|
454 |
+
No.
|
455 |
+
T
|
456 |
+
E(K)
|
457 |
+
cmax
|
458 |
+
cmin
|
459 |
+
24
|
460 |
+
Aβ5
|
461 |
+
1
|
462 |
+
Aβ
|
463 |
+
1
|
464 |
+
β3 β Z/2Z
|
465 |
+
5
|
466 |
+
2
|
467 |
+
1
|
468 |
+
2
|
469 |
+
38
|
470 |
+
A3 β Aβ3
|
471 |
+
1
|
472 |
+
Aβ
|
473 |
+
1 β β¨1/4β© β Z/2Z
|
474 |
+
5
|
475 |
+
2
|
476 |
+
1
|
477 |
+
2
|
478 |
+
53
|
479 |
+
A5 β Aβ2
|
480 |
+
1
|
481 |
+
β¨1/6β© β Z/2Z
|
482 |
+
5
|
483 |
+
2
|
484 |
+
1
|
485 |
+
2
|
486 |
+
57
|
487 |
+
D4 β Aβ3
|
488 |
+
1
|
489 |
+
Aβ
|
490 |
+
1 β (Z/2Z)β2
|
491 |
+
5
|
492 |
+
2
|
493 |
+
1
|
494 |
+
2
|
495 |
+
58
|
496 |
+
Aβ2
|
497 |
+
3
|
498 |
+
β A1
|
499 |
+
Aβ
|
500 |
+
1 β Z/4Z
|
501 |
+
5
|
502 |
+
2
|
503 |
+
1
|
504 |
+
2
|
505 |
+
61
|
506 |
+
Aβ3
|
507 |
+
2
|
508 |
+
β A1
|
509 |
+
β¨1/6β© β Z/3Z
|
510 |
+
5
|
511 |
+
2
|
512 |
+
1
|
513 |
+
2
|
514 |
+
Table 3: Cases with β = 2
|
515 |
+
8
|
516 |
+
|
517 |
+
No.
|
518 |
+
T
|
519 |
+
E(K)
|
520 |
+
cmax
|
521 |
+
cmin
|
522 |
+
β
|
523 |
+
41
|
524 |
+
A2 β Aβ4
|
525 |
+
1
|
526 |
+
1
|
527 |
+
6
|
528 |
+
οΏ½
|
529 |
+
2
|
530 |
+
1
|
531 |
+
1
|
532 |
+
2
|
533 |
+
οΏ½
|
534 |
+
β Z/2Z
|
535 |
+
8
|
536 |
+
3
|
537 |
+
1
|
538 |
+
2
|
539 |
+
13
|
540 |
+
6
|
541 |
+
42
|
542 |
+
Aβ6
|
543 |
+
1
|
544 |
+
Aβ
|
545 |
+
1
|
546 |
+
β2 β (Z/2Z)β2
|
547 |
+
3
|
548 |
+
1
|
549 |
+
2
|
550 |
+
5
|
551 |
+
2
|
552 |
+
59
|
553 |
+
A3 β A2 β Aβ2
|
554 |
+
1
|
555 |
+
β¨1/12β© β Z/2Z
|
556 |
+
8
|
557 |
+
3
|
558 |
+
1
|
559 |
+
2
|
560 |
+
13
|
561 |
+
6
|
562 |
+
60
|
563 |
+
A3 β Aβ4
|
564 |
+
1
|
565 |
+
β¨1/4β© β (Z/2Z)β2
|
566 |
+
3
|
567 |
+
1
|
568 |
+
2
|
569 |
+
5
|
570 |
+
2
|
571 |
+
Table 4: Cases with β > 2
|
572 |
+
Proof. By searching Table 8 for all cases with β = 2 and β > 2, we obtain Table 3 and Table 4
|
573 |
+
respectively. Notice in particular that in both tables the torsion part of E(K) is always nontrivial.
|
574 |
+
Consequently, if E(K) is torsion-free, then β < 2. β
|
575 |
+
2.5
|
576 |
+
The quadratic form QX
|
577 |
+
We deο¬ne the positive-deο¬nite quadratic form with integer coeο¬cients QX derived from the
|
578 |
+
height pairing. The relevance of QX is due to the fact that some conditions for having P1 Β· P2 = k
|
579 |
+
for some P1, P2 β E(K) can be stated in terms of what integers can be represented by QX (see
|
580 |
+
Corollary 4.2 and Proposition 4.12).
|
581 |
+
The deο¬nition of QX consists in clearing denominators of the rational quadratic form induced
|
582 |
+
by the height pairing; the only question is how to ο¬nd a scale factor that works in every case. More
|
583 |
+
precisely, if E(K) has rank r β₯ 1 and P1, ..., Pr are generators of its free part, then q(x1, ..., xr) :=
|
584 |
+
h(x1P1 + ... + xrPr) is a quadratic form with coeο¬cients in Q; we deο¬ne QX by multiplying q by
|
585 |
+
some integer d > 0 so as to produce coeο¬cients in Z. We show that d may always be chosen as the
|
586 |
+
determinant of the narrow lattice E(K)0.
|
587 |
+
Deο¬nition 2.9. Let X with r β₯ 1. Let P1, ..., Pr be generators of the free part of E(K). Deο¬ne
|
588 |
+
QX(x1, ..., xr) := (det E(K)0) Β· h(x1P1 + ... + xrPr).
|
589 |
+
We check that the matrix representing QX has entries in Z, therefore QX has coeο¬cients in Z.
|
590 |
+
Lemma 2.10. Let A be the matrix representing the quadratic form QX, i.e. Q(x1, ..., xr) = xtAx,
|
591 |
+
where x := (x1, ..., xr)t. Then A has integer entries. In particular, QX has integer coeο¬cients.
|
592 |
+
Proof. Let P1, ..., Pr be generators of the free part of E(K) and let L := E(K)0. The free part of
|
593 |
+
E(K) is isomorphic to the dual lattice Lβ [OS91, Main Thm.], so we may ο¬nd generators P 0
|
594 |
+
1 , ..., P 0
|
595 |
+
r
|
596 |
+
of L such that the Gram matrix B0 := (β¨P 0
|
597 |
+
i , P 0
|
598 |
+
j β©)i,j of L is the inverse of the Gram matrix
|
599 |
+
B := (β¨Pi, Pjβ©)i,j of Lβ.
|
600 |
+
9
|
601 |
+
|
602 |
+
We claim that QX is represented by the adjugate matrix of B0, i.e. the matrix adj(B0) such
|
603 |
+
that B0 Β· adj(B0) = (det B0) Β· Ir, where Ir is the r Γ r identity matrix. Indeed, by construction B
|
604 |
+
represents the quadratic form h(x1P1 + ... + xrPr), therefore
|
605 |
+
QX(x1, ..., xr) = (det E(K)0) Β· h(x1P1 + ... + xrPr)
|
606 |
+
= (det B0) Β· xtBx
|
607 |
+
= (det B0) Β· xt(B0)β1x
|
608 |
+
= xtadj(B0)x,
|
609 |
+
as claimed. To prove that A := adj(B0) has integer coeο¬cients, notice that the Gram matrix
|
610 |
+
B0 of L = E(K)0 has integer coeο¬cients (as E(K)0 is an even lattice), then so does A. β
|
611 |
+
We close this subsection with a simple consequence of the deο¬nition of QX.
|
612 |
+
Lemma 2.11. If h(P) = m for some P β E(K), then QX represents d Β· m, where d := det E(K)0.
|
613 |
+
Proof. Let P1, ..., Pr be generators for the free part of E(K). Let P = a1P1 + ... + arPr + Q, where
|
614 |
+
ai β Z and Q is a torsion element (possibly zero). Since torsion sections do not contribute to the
|
615 |
+
height pairing, then h(P β Q) = h(P) = m. Hence
|
616 |
+
QX(a1, ..., ar) = d Β· h(a1P1 + ... + arPr)
|
617 |
+
= d Β· h(P β Q)
|
618 |
+
= d Β· m. β
|
619 |
+
3
|
620 |
+
Intersection with a torsion section
|
621 |
+
Before dealing with more technical details in Section 4, we explain how torsion sections can be
|
622 |
+
of help in our investigation, specially in Subsection 4.2.
|
623 |
+
We ο¬rst note some general properties of torsion sections. As the height pairing is positive-
|
624 |
+
deο¬nite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q β E(K)tor we
|
625 |
+
have β¨Q, Pβ© = 0 for all P β E(K).
|
626 |
+
Moreover, in the case of rational elliptic surfaces, torsion
|
627 |
+
sections also happen to be mutually disjoint:
|
628 |
+
Theorem 3.1. [MP89, Lemma 1.1] On a rational elliptic surface, Q1 Β· Q2 = 0 for any distinct
|
629 |
+
Q1, Q2 β E(K)tor. In particular, if O is the neutral section, then QΒ·O = 0 for all Q β E(K)tor\{O}.
|
630 |
+
Remark 3.2. As stated in [MP89, Lemma 1.1], Theorem 3.1 holds for elliptic surfaces over C even
|
631 |
+
without assuming X is rational. However, for an arbitrary algebraically closed ο¬eld the rationality
|
632 |
+
hypothesis is needed, and a proof can be found in [SS19, Cor. 8.30].
|
633 |
+
By taking advantage of the properties above, we use torsion sections to help us ο¬nd P1, P2 β
|
634 |
+
E(K) such that P1 Β· P2 = k for a given k β Zβ₯0. This is particularly useful when β β₯ 2, in which
|
635 |
+
case E(K)tor is not trivial by Lemma 2.8.
|
636 |
+
The idea is as follows. Given k β Zβ₯0, suppose we can ο¬nd P β E(K)0 with height h(P) = 2k.
|
637 |
+
By the height formula (2), P Β· O = k β 1 < k, which is not yet what we need. In the next lemma
|
638 |
+
we show that replacing O with a torsion section Q ΜΈ= O gives P Β· Q = k, as desired.
|
639 |
+
10
|
640 |
+
|
641 |
+
Lemma 3.3. Let P β E(K)0 such that h(P) = 2k. Then P Β· Q = k for all Q β E(K)tor \ {O}.
|
642 |
+
Proof. Assume there is some Q β E(K)tor \ {O}. By Theorem 3.1, Q Β· O = 0 and by the height
|
643 |
+
formula (2), 2k = 2 + 2(P Β· O) β 0, hence P Β· O = k β 1. We use the height formula (1) for β¨P, Qβ©
|
644 |
+
in order to conclude that P Β· Q = k. Since P β E(K)0, it intersects the neutral component Ξv,0 of
|
645 |
+
every reducible ο¬ber Οβ1(v), so contrv(P, Q) = 0 for all v β R. Hence
|
646 |
+
0 = β¨P, Qβ©
|
647 |
+
= 1 + P Β· O + Q Β· O β P Β· Q β
|
648 |
+
οΏ½
|
649 |
+
vβR
|
650 |
+
contrv(P, Q)
|
651 |
+
= 1 + (k β 1) + 0 β P Β· Q β 0
|
652 |
+
= k β P Β· Q. β
|
653 |
+
4
|
654 |
+
Existence of a pair of sections with a given intersection number
|
655 |
+
Given k β Zβ₯0, we state necessary and (in most cases) suο¬cient conditions for having
|
656 |
+
P1 Β·P2 = k for some P1, P2 β E(K). Necessary conditions are stated in generality in Subsection 4.1,
|
657 |
+
while suο¬cient ones depend on the value of β and are treated separately in Subsection 4.2. In
|
658 |
+
Subsection 4.4, we collect all suο¬cient conditions proven in this section.
|
659 |
+
4.1
|
660 |
+
Necessary Conditions
|
661 |
+
If k β Zβ₯0, we state necessary conditions for having P1Β·P2 = k for some sections P1, P2 β E(K).
|
662 |
+
We note that the value of β is not relevant in this subsection, although it plays a decisive role for
|
663 |
+
suο¬cient conditions in Subsection 4.2.
|
664 |
+
Lemma 4.1. Let k β Zβ₯0. If P1 Β· P2 = k for some P1, P2 β E(K), then one of the following holds:
|
665 |
+
i) h(P) = 2 + 2k for some P β E(K)0.
|
666 |
+
ii) h(P) β [2 + 2k β cmax, 2 + 2k β cmin] for some P /β E(K)0.
|
667 |
+
Proof. Without loss of generality we may assume P2 is the neutral section, so that P1 Β· O = k. By
|
668 |
+
the height formula (2), h(P1) = 2 + 2k β c, where c := οΏ½
|
669 |
+
v contrv(P1). If P1 β E(K)0, then c = 0
|
670 |
+
and h(P1) = 2 + 2k, hence i) holds. If P1 /β E(K)0, then cmin β€ c β€ cmax by Lemma 2.7. But
|
671 |
+
h(P1) = 2 + 2k β c, therefore 2 + 2k β cmax β€ h(P1) β€ 2 + 2k β cmin, i.e. ii) holds. β
|
672 |
+
Corollary 4.2. Let k β Zβ₯0. If P1 Β· P2 = k for some P1, P2 β E(K), then QX represents some
|
673 |
+
integer in [d Β· (2 + 2k β cmax), d Β· (2 + 2k)], where d := det E(K)0.
|
674 |
+
Proof.
|
675 |
+
We apply Lemma 4.1 and rephrase it in terms of QX. If i) holds, then QX represents
|
676 |
+
d Β· (2 + 2k) by Lemma 2.11. But if ii) holds, then h(P) β [2 + 2k β cmax, 2 + 2k β cmin] and by
|
677 |
+
Lemma 2.11, QX represents d Β· h(P) β [d Β· (2 + 2k β cmax), d Β· (2 + 2k β cmin)]. In both i) and ii),
|
678 |
+
QX represents some integer in [d Β· (2 + 2k β cmax), d Β· (2 + 2k)]. β
|
679 |
+
11
|
680 |
+
|
681 |
+
4.2
|
682 |
+
Suο¬cient conditions when β β€ 2
|
683 |
+
In this subsection we state suο¬cient conditions for having P1 Β· P2 = k for some P1, P2 β E(K)
|
684 |
+
under the assumption that β β€ 2. By Lemma 2.8, this covers almost all cases (more precisely, all
|
685 |
+
but No. 41, 42, 59, 60 in Table 8). We treat β < 2 and β = 2 separately, as the latter needs more
|
686 |
+
attention.
|
687 |
+
4.2.1
|
688 |
+
The case β < 2
|
689 |
+
We ο¬rst prove Lemma 4.3, which gives suο¬cient conditions assuming β < 2, then Corollary 4.5,
|
690 |
+
which states suο¬cient conditions in terms of integers represented by QX.
|
691 |
+
This is followed by
|
692 |
+
Corollary 4.6, which is a simpliο¬ed version of Corollary 4.5.
|
693 |
+
Lemma 4.3. Assume β < 2 and let k β Zβ₯0. If h(P) β [2 + 2k β cmax, 2 + 2k β cmin] for some
|
694 |
+
P /β E(K)0, then P1 Β· P2 = k for some P1, P2 β E(K).
|
695 |
+
Proof. Let O β E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P Β· O) β c,
|
696 |
+
where c := οΏ½
|
697 |
+
v contrv(P). Since h(P) β [2 + 2k β cmax, 2 + 2k β cmin], then
|
698 |
+
2 + 2k β cmax β€ 2 + 2(P Β· O) β c β€ 2 + 2k β cmin
|
699 |
+
β c β cmax
|
700 |
+
2
|
701 |
+
β€ P Β· O β k β€ c β cmin
|
702 |
+
2
|
703 |
+
.
|
704 |
+
Therefore P Β· O β k is an integer in I :=
|
705 |
+
οΏ½ cβcmax
|
706 |
+
2
|
707 |
+
, cβcmin
|
708 |
+
2
|
709 |
+
οΏ½. We prove that 0 is the only integer in
|
710 |
+
I, so that P Β· O β k = 0, i.e. P Β· O = k. First notice that c ΜΈ= 0, as P /β E(K)0. By Lemma 2.7 iii),
|
711 |
+
cmin β€ c β€ cmax, consequently cβcmax
|
712 |
+
2
|
713 |
+
β€ 0 β€ cβcmin
|
714 |
+
2
|
715 |
+
, i.e. 0 β I. Moreover β < 2 implies that I has
|
716 |
+
length cmaxβcmin
|
717 |
+
2
|
718 |
+
= β
|
719 |
+
2 < 1, so I contains no integer except 0 as desired. β
|
720 |
+
Remark 4.4. Lemma 4.3 also applies when cmax = cmin, in which case the closed interval degen-
|
721 |
+
erates into a point.
|
722 |
+
The following corollary of Lemma 4.3 states a suο¬cient condition in terms of integers represented
|
723 |
+
by the quadratic form QX (Section 2.5).
|
724 |
+
Corollary 4.5. Assume β < 2 and let d := det E(K)0. If QX represents an integer not divisible
|
725 |
+
by d in the interval [d Β· (2+ 2k β cmax), d Β· (2+ 2k β cmin)], then P1 Β· P2 = k for some P1, P2 β E(K).
|
726 |
+
Proof. Let a1, ..., ar β Z such that QX(a1, ..., ar) β [d Β· (2 + 2k β cmax), d Β· (2 + 2k β cmin)] with
|
727 |
+
d β€ QX(a1, ..., ar). Let P := a1P1 + ... + arPr, where P1, ..., Pr are generators of the free part of
|
728 |
+
E(K). Then d β€ QX(a1, ..., ar) = d Β· h(P), which implies that h(P) /β Z. In particular P /β E(K)0
|
729 |
+
since E(K)0 is an integer lattice. Moreover h(P) = 1
|
730 |
+
dQX(a1, ..., ar) β [2 + 2k β cmax, 2 + 2k β cmin]
|
731 |
+
and we are done by Lemma 4.3. β
|
732 |
+
12
|
733 |
+
|
734 |
+
The next corollary, although weaker than Corollary 4.5, is more practical for concrete examples
|
735 |
+
and is frequently used in Subsection 5.4. It does not involve ο¬nding integers represented by QX,
|
736 |
+
but only ο¬nding perfect squares in an interval depending on the minimal norm Β΅ (Subsection 2.1).
|
737 |
+
Corollary 4.6. Assume β < 2. If there is a perfect square n2 β
|
738 |
+
οΏ½
|
739 |
+
2+2kβcmax
|
740 |
+
Β΅
|
741 |
+
, 2+2kβcmin
|
742 |
+
Β΅
|
743 |
+
οΏ½
|
744 |
+
such that
|
745 |
+
n2Β΅ /β Z, then P1 Β· P2 = k for some P1, P2 β E(K).
|
746 |
+
Proof. Take P β E(K) such that h(P) = Β΅. Since h(nP) = n2Β΅ /β Z, we must have nP /β E(K)0
|
747 |
+
as E(K)0 is an integer lattice. Moreover h(nP) = n2Β΅ β [2 + 2k β cmax, 2 + 2k β cmin] and we are
|
748 |
+
done by Lemma 4.3. β
|
749 |
+
4.2.2
|
750 |
+
The case β = 2
|
751 |
+
The statement of suο¬cient conditions for β = 2 is almost identical to the one for β < 2: the
|
752 |
+
only diο¬erence is that the closed interval Lemma 4.3 is substituted by a right half-open interval
|
753 |
+
in Lemma 4.8. This change, however, is associated with a technical diο¬culty in the case when a
|
754 |
+
section has minimal contribution term, thus the separate treatment for β = 2.
|
755 |
+
The results are presented in the following order. First we prove Lemma 4.7, which is a statement
|
756 |
+
about sections whose contribution term is minimal.
|
757 |
+
Next we prove Lemma 4.8, which states
|
758 |
+
suο¬cient conditions for β = 2, then Corollaries 4.9 and 4.10.
|
759 |
+
Lemma 4.7. Assume β = 2.
|
760 |
+
If there is P β E(K) such that οΏ½
|
761 |
+
vβR contrv(P) = cmin, then
|
762 |
+
P Β· Q = P Β· O + 1 for every Q β E(K)tor \ {O}.
|
763 |
+
Proof. If Q β E(K)tor \ {O}, then Q Β· O = 0 by Theorem 3.1. Moreover, by the height formula (1),
|
764 |
+
0 = β¨P, Qβ© = 1 + P Β· O + 0 β P Β· Q β
|
765 |
+
οΏ½
|
766 |
+
vβR
|
767 |
+
contrv(P, Q). (β)
|
768 |
+
Hence it suο¬ces to show that contrv(P, Q) = 0 βv β R. By Lemma 2.7 iv), contrvβ²(P) = cmin
|
769 |
+
for some vβ² and contrv(P) = 0 for all v ΜΈ= vβ². In particular P meets Ξv,0, hence contrv(P, Q) = 0
|
770 |
+
for all v ΜΈ= vβ². Thus from (β) we see that contrvβ²(P, Q) is an integer, which we prove is 0.
|
771 |
+
We claim that Tvβ² = A1, so that contrvβ²(P, Q) = 0 or 1
|
772 |
+
2 by Table 1. In this case, as contrvβ²(P, Q)
|
773 |
+
is an integer, it must be 0, and we are done. To see that Tvβ² = A1 we analyse contrvβ²(P). Since
|
774 |
+
β = 2, then cmin = 1
|
775 |
+
2 by Table 3 and contrvβ²(P) = cmin = 1
|
776 |
+
2. By Table 1, this only happens if
|
777 |
+
Tvβ² = Anβ1 and 1
|
778 |
+
2 = i(nβi)
|
779 |
+
n
|
780 |
+
for some 0 β€ i < n. The only possibility is i = 1, n = 2 and Tvβ² = A1. β
|
781 |
+
With the aid of Lemma 4.7 we are able to state suο¬cient conditions for β = 2.
|
782 |
+
13
|
783 |
+
|
784 |
+
Lemma 4.8. Assume β = 2 and let k β Zβ₯0. If h(P) β [2 + 2k β cmax, 2 + 2k β cmin) for some
|
785 |
+
P /β E(K)0, then P1 Β· P2 = k for some P1, P2 β E(K).
|
786 |
+
Proof. Let O β E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P Β· O) β c,
|
787 |
+
where c := οΏ½
|
788 |
+
v contrv(P). We repeat the arguments from Lemma 4.3, in this case with the right
|
789 |
+
half-open interval, so that the hypothesis that h(P) β [2 + 2k β cmax, 2 + 2k β cmin), implies that
|
790 |
+
P Β· O β k is an integer in Iβ² :=
|
791 |
+
οΏ½ cβcmax
|
792 |
+
2
|
793 |
+
, cβcmin
|
794 |
+
2
|
795 |
+
οΏ½.
|
796 |
+
Since Iβ² is half-open with length cmaxβcmin
|
797 |
+
2
|
798 |
+
= β
|
799 |
+
2 = 1, then Iβ² contains exactly one integer. If
|
800 |
+
0 β Iβ², then P Β· O β k = 0, i.e. P Β· O = k and we are done. Hence we assume 0 /β Iβ².
|
801 |
+
We claim that P Β·O = k β1. First, notice that if c > cmin, then the inequalities cmin < c β€ cmax
|
802 |
+
give cβcmax
|
803 |
+
2
|
804 |
+
β€ 0 < cβcmin
|
805 |
+
2
|
806 |
+
, i.e. 0 β Iβ², which is a contradiction. Hence c = cmin. Since β = 2, then
|
807 |
+
Iβ² = [β1, 0), whose only integer is β1. Thus P Β· O β k = β1, i.e. P Β· O = k β 1, as claimed.
|
808 |
+
Finally, let Q β E(K)tor \ {O}, so that P Β· Q = P Β· O + 1 = k by Lemma 4.7 and we are done.
|
809 |
+
We remark that E(K)tor is not trivial by Table 3, therefore such Q exists. β
|
810 |
+
The following corollaries are analogues to Corollary 4.5 and Corollary 4.6 adapted to β = 2.
|
811 |
+
Similarly to the case β < 2, Corollary 4.9 is stronger than Corollary 4.10, although the latter is
|
812 |
+
more practical for concrete examples. We remind the reader that Β΅ denotes the minimal norm
|
813 |
+
(Subsection 2.1).
|
814 |
+
Corollary 4.9. Assume β = 2 and let d := det E(K)0. If QX represents an integer not divisible
|
815 |
+
by d in the interval [dΒ·(2+2k βcmax), dΒ·(2+2k βcmin)), then P1 Β·P2 = k for some P1, P2 β E(K).
|
816 |
+
Proof. We repeat the arguments in Corollary 4.5, in this case with the half-open interval. β
|
817 |
+
Corollary 4.10. Assume β = 2. If there is a perfect square n2 β
|
818 |
+
οΏ½
|
819 |
+
2+2kβcmax
|
820 |
+
Β΅
|
821 |
+
, 2+2kβcmin
|
822 |
+
Β΅
|
823 |
+
οΏ½
|
824 |
+
such that
|
825 |
+
n2Β΅ /β Z, then P1 Β· P2 = k for some P1, P2 β E(K).
|
826 |
+
Proof. We repeat the arguments in Corollary 4.6, in this case with the half-open interval. β
|
827 |
+
4.3
|
828 |
+
Necessary and suο¬cient conditions for β β€ 2
|
829 |
+
For completeness, we present a uniο¬ed statement of necessary and suο¬cient conditions assuming
|
830 |
+
β β€ 2, which follows naturally from results in Subsections 4.1 and 4.2.
|
831 |
+
Lemma 4.11. Assume β β€ 2 and let k β Zβ₯0. Then P1 Β· P2 = k for some P1, P2 β E(K) if and
|
832 |
+
only if one of the following holds:
|
833 |
+
i) h(P) = 2 + 2k for some P β E(K)0.
|
834 |
+
ii) h(P) β [2 + 2k β cmax, 2 + 2k β cmin) for some P /β E(K)0.
|
835 |
+
iii) h(P) = 2 + 2k β cmin and οΏ½
|
836 |
+
vβR contrv(P) = cmin for some P β E(K).
|
837 |
+
Proof. If i) or iii) holds, then P Β· O = k directly by the height formula (2). But if ii) holds, it
|
838 |
+
suο¬ces to to apply Lemma 4.3 when β < 2 and by Lemma 4.8 when β = 2.
|
839 |
+
Conversely, let P1Β·P2 = k. Without loss of generality, we may assume P2 = O, so that P1Β·O = k.
|
840 |
+
By the height formula (2), h(P1) = 2 + 2k β c, where c := οΏ½
|
841 |
+
v contrv(P1).
|
842 |
+
If c = 0, then P1 β E(K)0 and h(P1) = 2+2k, so i) holds. Hence we let c ΜΈ= 0, i.e. P1 /β E(K)0,
|
843 |
+
so that cmin β€ c β€ cmax by Lemma 2.7. In case c = cmin, then h(P1) = 2 + 2k β cmin and iii) holds.
|
844 |
+
Otherwise cmin < c β€ cmax, which implies 2 + 2k β cmax β€ h(P1) < 2 + 2k β cmin, so ii) holds. β
|
845 |
+
14
|
846 |
+
|
847 |
+
4.4
|
848 |
+
Summary of suο¬cient conditions
|
849 |
+
For the sake of clarity, we summarize in a single proposition all suο¬cient conditions for having
|
850 |
+
P1 Β· P2 = k for some P1, P2 β E(K) proven in this section.
|
851 |
+
Proposition 4.12. Let k β Zβ₯0. If one of the following holds, then P1 Β· P2 = k for some P1, P2 β
|
852 |
+
E(K).
|
853 |
+
1) h(P) = 2 + 2k for some P β E(K)0.
|
854 |
+
2) h(P) = 2k for some P β E(K)0 and E(K)tor is not trivial.
|
855 |
+
3) β < 2 and there is a perfect square n2 β
|
856 |
+
οΏ½
|
857 |
+
2+2kβcmax
|
858 |
+
Β΅
|
859 |
+
, 2+2kβcmin
|
860 |
+
Β΅
|
861 |
+
οΏ½
|
862 |
+
with n2Β΅ /β Z, where Β΅ is the
|
863 |
+
minimal norm (Subsection 2.1). In case β = 2, consider the right half-open interval.
|
864 |
+
4) β < 2 and the quadratic form QX represents an integer not divisible by d := det E(K)0 in the
|
865 |
+
interval [d Β· (2 + 2k β cmax), d Β· (2 + 2k β cmin)]. In case β = 2, consider the right half-open
|
866 |
+
interval.
|
867 |
+
Proof. In 1) a height calculation gives 2 + 2k = h(P) = 2 + 2(P Β· O) β 0, so P Β· O = k. For
|
868 |
+
2), we apply Lemma 3.3 to conclude that P Β· Q = k for any Q β E(K)tor \ {O}. In 3) we use
|
869 |
+
Corollary 4.6 when β < 2 and Corollary 4.10 when β = 2. In 4), we apply Corollary 4.5 if β < 2
|
870 |
+
and Corollary 4.9 if β = 2. β
|
871 |
+
5
|
872 |
+
Main Results
|
873 |
+
We prove the four main theorems of this paper, which are independent applications of the results
|
874 |
+
from Section 4. The ο¬rst two are general attempts to describe when and how gap numbers occur:
|
875 |
+
Theorem 5.2 tells us that large Mordell-Weil groups prevent the existence of gaps numbers, more
|
876 |
+
precisely for Mordell-Weil rank r β₯ 5; in Theorem 5.4 we show that for small Mordell-Weil rank,
|
877 |
+
more precisely when r β€ 2, then gap numbers occur with probability 1. The last two theorems,
|
878 |
+
on the other hand, deal with explicit values of gap numbers: Theorem 5.7 provides a complete
|
879 |
+
description of gap numbers in certain cases, while Theorem 5.8 is a classiο¬cation of cases with a
|
880 |
+
1-gap.
|
881 |
+
5.1
|
882 |
+
No gap numbers in rank r β₯ 5
|
883 |
+
We show that if E(K) has rank r β₯ 5, then X is gap-free. Our strategy is to prove that for
|
884 |
+
every k β Zβ₯0 there is some P β E(K)0 such that h(P) = 2+2k, and by Proposition 4.12 1) we are
|
885 |
+
done. We accomplish this in two steps. First we show that this holds when there is an embedding
|
886 |
+
of Aβ
|
887 |
+
1 or of A4 in E(K)0 (Lemma 5.1). Second, we show that if r β₯ 5, then such embedding exists,
|
888 |
+
hence X is gap-free (Theorem 5.2).
|
889 |
+
15
|
890 |
+
|
891 |
+
Lemma 5.1. Assume E(K)0 has a sublattice isomorphic to Aβ4
|
892 |
+
1
|
893 |
+
or A4. Then for every β β Zβ₯0
|
894 |
+
there is P β E(K)0 such that h(P) = 2β.
|
895 |
+
Proof.
|
896 |
+
First assume Aβ4
|
897 |
+
1
|
898 |
+
β E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in
|
899 |
+
Aβ4
|
900 |
+
1 . Then h(Pi) = 2 and β¨Pi, Pjβ© = 0 for distinct i, j = 1, 2, 3, 4.
|
901 |
+
By Lagrangeβs four-square
|
902 |
+
theorem [HW79, Β§20.5] there are integers a1, a2, a3, a4 such that a2
|
903 |
+
1 + a2
|
904 |
+
2 + a2
|
905 |
+
3 + a2
|
906 |
+
4 = β. Deο¬ning
|
907 |
+
P := a1P1 + a2P2 + a3P3 + a4P4 β Aβ4
|
908 |
+
1
|
909 |
+
β E(K)0, we have
|
910 |
+
h(P) = 2a2
|
911 |
+
1 + 2a2
|
912 |
+
2 + 2a2
|
913 |
+
3 + 2a2
|
914 |
+
4 = 2β.
|
915 |
+
Now let A4 β E(K)0 with generators P1, P2, P3, P4.
|
916 |
+
Then h(Pi) = 2 for i = 1, 2, 3, 4 and
|
917 |
+
β¨Pi, Pi+1β© = β1 for i = 1, 2, 3. We need to ο¬nd integers x1, ..., x4 such that h(P) = 2β, where
|
918 |
+
P := x1P1 + ... + x4P4 β A4 β E(K)0. Equivalently, we need that
|
919 |
+
β = 1
|
920 |
+
2β¨P, Pβ© = x2
|
921 |
+
1 + x2
|
922 |
+
2 + x2
|
923 |
+
3 + x2
|
924 |
+
4 β x1x2 β x2x3 β x3x4.
|
925 |
+
Therefore β must be represented by q(x1, ..., x4) := x2
|
926 |
+
1 + x2
|
927 |
+
2 + x2
|
928 |
+
3 + x2
|
929 |
+
4 β x1x2 β x2x3 β x3x4. We
|
930 |
+
prove that q represents all positive integers. Notice that q is positive-deο¬nite, since it is induced
|
931 |
+
by β¨Β·, Β·β©. By Bhargava-Hankeβs 290-theorem [BH][Thm. 1], q represents all positive integers if and
|
932 |
+
only if it represents the following integers:
|
933 |
+
2, 3, 5, 6, 7, 10, 13, 14, 15, 17, 19, 21, 22, 23, 26,
|
934 |
+
29, 30, 31, 34, 35, 37, 42, 58, 93, 110, 145, 203, 290.
|
935 |
+
The representation for each of the above is found in Table 5. β
|
936 |
+
We now prove the main theorem of this section.
|
937 |
+
Theorem 5.2. If r β₯ 5, then X is gap-free.
|
938 |
+
Proof. We show that for every k β₯ 0 there is P β E(K)0 such that h(P) = 2 + 2k, so that by
|
939 |
+
Proposition 4.12 1) we are done. Using Lemma 5.1 it suο¬ces to prove that E(K)0 has a sublattice
|
940 |
+
isomorphic to Aβ4
|
941 |
+
1
|
942 |
+
or A4.
|
943 |
+
The cases with r β₯ 5 are No.
|
944 |
+
1-7 (Table 8).
|
945 |
+
In No.
|
946 |
+
1-6, E(K)0 = E8, E7, E6, D6, D5, A5
|
947 |
+
respectively. Each of these admit an A4 sublattice [Nis96, Lemmas 4.2,4.3]. In No. 7 we claim that
|
948 |
+
E(K)0 = D4 β A1 has an Aβ4
|
949 |
+
1
|
950 |
+
sublattice. This is the case because D4 admits an Aβ4
|
951 |
+
1
|
952 |
+
sublattice
|
953 |
+
[Nis96, Lemma 4.5 (iii)]. β
|
954 |
+
16
|
955 |
+
|
956 |
+
n
|
957 |
+
x1, x2, x3, x4 with x2
|
958 |
+
1 + x2
|
959 |
+
2 + x2
|
960 |
+
3 + x2
|
961 |
+
4 β x1x2 β x2x3 β x3x4 = n
|
962 |
+
1
|
963 |
+
1, 0, 0, 0
|
964 |
+
2
|
965 |
+
1, 0, 1, 0
|
966 |
+
3
|
967 |
+
1, 1, 2, 0
|
968 |
+
5
|
969 |
+
1, 0, 2, 0
|
970 |
+
6
|
971 |
+
1, 1, β2, β1
|
972 |
+
7
|
973 |
+
1, 1, β2, 0
|
974 |
+
10
|
975 |
+
1, 0, 3, 0
|
976 |
+
13
|
977 |
+
2, 0, 3, 0
|
978 |
+
14
|
979 |
+
1, 2, 5, 1
|
980 |
+
15
|
981 |
+
1, 5, 5, 2
|
982 |
+
17
|
983 |
+
1, 0, 4, 0
|
984 |
+
19
|
985 |
+
1, 5, 3, β1
|
986 |
+
21
|
987 |
+
1, 5, 0, 0
|
988 |
+
22
|
989 |
+
1, 5, 0, β1
|
990 |
+
23
|
991 |
+
1, 6, 6, 2
|
992 |
+
26
|
993 |
+
1, 0, 5, 0
|
994 |
+
29
|
995 |
+
2, 0, 5, 0
|
996 |
+
30
|
997 |
+
1, 5, 0, β3
|
998 |
+
31
|
999 |
+
1, 3, β4, β2
|
1000 |
+
34
|
1001 |
+
3, 0, 5, 0
|
1002 |
+
35
|
1003 |
+
1, 2, β2, 4
|
1004 |
+
37
|
1005 |
+
1, 0, 6, 0
|
1006 |
+
42
|
1007 |
+
1, 1, β4, 3
|
1008 |
+
58
|
1009 |
+
3, 0, 7, 0
|
1010 |
+
93
|
1011 |
+
1, 1, β10, 0
|
1012 |
+
110
|
1013 |
+
1, β2, 3, β8
|
1014 |
+
145
|
1015 |
+
1, 0, 12, 0
|
1016 |
+
203
|
1017 |
+
1, β5, β9, 8
|
1018 |
+
290
|
1019 |
+
1, 0, 17, 0
|
1020 |
+
Table 5: Representation of the critical integers in Bhargava-Hankeβs 290-theorem.
|
1021 |
+
5.2
|
1022 |
+
Gaps with probability 1 in rank r β€ 2
|
1023 |
+
Fix a rational elliptic surface Ο : X β P1 with Mordell-Weil rank r β€ 2. We prove that if k is
|
1024 |
+
a uniformly random natural number, then k is a gap number with probability 1. More precisely, if
|
1025 |
+
G := {k β N | k is a gap number of X} is the set of gap numbers, then G β N has density 1, i.e.
|
1026 |
+
d(G) := lim
|
1027 |
+
nββ
|
1028 |
+
#G β© {1, ..., n}
|
1029 |
+
n
|
1030 |
+
= 1.
|
1031 |
+
17
|
1032 |
+
|
1033 |
+
We adopt the following strategy. If k β N \ G, then P1 Β· P2 = k for some P1, P2 β E(K) and
|
1034 |
+
by Corollary 4.2 the quadratic form QX represents some integer t depending on k. This deο¬nes a
|
1035 |
+
function N\G β T, where T is the set of integers represented by QX. Since QX is a quadratic form
|
1036 |
+
on r β€ 2 variables, T has density 0 in N by Lemma 5.3. By analyzing the pre-images of N\G β T,
|
1037 |
+
in Theorem 5.4 we conclude that d(N \ G) = d(T) = 0, hence d(G) = 1 as desired.
|
1038 |
+
Lemma 5.3. Let Q be a positive-deο¬nite quadratic form on r = 1, 2 variables with integer coeο¬-
|
1039 |
+
cients. Then the set of integers represented by Q has density 0 in N.
|
1040 |
+
Proof. Let S be the set of integers represented by Q. If d is the greatest common divisor of the
|
1041 |
+
coeο¬cients of Q, let Sβ² be the set of integers representable by the primitive form Qβ² := 1
|
1042 |
+
d Β· Q. By
|
1043 |
+
construction Sβ² is a rescaling of S, so d(S) = 0 if and only if d(Sβ²) = 0.
|
1044 |
+
If r = 1, then Qβ²(x1) = x2
|
1045 |
+
1 and Sβ² is the set of perfect squares, so clearly d(Sβ²) = 0. If r = 2,
|
1046 |
+
then Qβ² is a binary quadratic form and the number of elements in Sβ² bounded from above by x > 0
|
1047 |
+
is given by C Β·
|
1048 |
+
x
|
1049 |
+
βlog x + o(x) with C > 0 a constant and limxββ
|
1050 |
+
o(x)
|
1051 |
+
x
|
1052 |
+
= 0 [Ber12, p. 91]. Thus
|
1053 |
+
d(Sβ²) = lim
|
1054 |
+
xββ
|
1055 |
+
C
|
1056 |
+
βlog x + o(x)
|
1057 |
+
x
|
1058 |
+
= 0. β
|
1059 |
+
We now prove the main result of this section.
|
1060 |
+
Theorem 5.4. Let Ο : X β P1 be a rational elliptic surface with Mordell-Weil rank r β€ 2. Then
|
1061 |
+
the set G := {k β N | k is a gap number of X} of gap numbers of X has density 1 in N.
|
1062 |
+
Proof. If r = 0, then the claim is trivial by Remark 2.3, hence we may assume r = 1, 2. We
|
1063 |
+
prove that S := N \ G has density 0.
|
1064 |
+
If S is ο¬nite, there is nothing to prove.
|
1065 |
+
Otherwise, let
|
1066 |
+
k1 < k2 < ... be the increasing sequence of all elements of S. By Corollary 4.2, for each n there is
|
1067 |
+
some tn β Jkn := [d Β· (2 + 2kn β cmax), d Β· (2 + 2kn)] represented by the quadratic form QX. Let T
|
1068 |
+
be the set of integers represented by QX and deο¬ne the function f : N \ G β T by kn οΏ½β tn. Since
|
1069 |
+
QX has r = 1, 2 variables, T has density 0 by Lemma 5.3.
|
1070 |
+
For N > 0, let SN := S β© {1, ..., N} and TN := T β© {1, ..., N}.
|
1071 |
+
Since T has density zero,
|
1072 |
+
#TN = o(N), i.e.
|
1073 |
+
#TN
|
1074 |
+
N
|
1075 |
+
β 0 when N β β and we need to prove that #SN = o(N). We analyze
|
1076 |
+
the function f restricted to SN. Notice that as tn β Jkn, then kn β€ N implies tn β€ d Β· (2 + 2kn) β€
|
1077 |
+
d Β· (2 + 2N). Hence the restriction g := f|SN can be regarded as a function g : SN β TdΒ·(2+2k).
|
1078 |
+
We claim that #gβ1(t) β€ 2 for all t β TdΒ·(2+2N), in which case #SN β€ 2 Β· #TdΒ·(2+2N) = o(N)
|
1079 |
+
and we are done. Assume by contradiction that gβ1(t) contains three distinct elements, say kβ1 <
|
1080 |
+
kβ2 < kβ3 with t = tβ1 = tβ2 = tβ3. Since tβi β Jkβi for each i = 1, 2, 3, then t β Jkβ1 β© Jkβ2 β© Jkβ3. We
|
1081 |
+
prove that Jkβ1 and Jkβ3 are disjoint, which yields a contradiction. Indeed, since kβ1 < kβ2 < kβ3,
|
1082 |
+
in particular kβ3 β kβ1 β₯ 2, therefore d Β· (2 + 2kβ1) β€ d Β· (2 + 2kβ3 β 4). But cmax < 4 by Lemma 2.7,
|
1083 |
+
so d Β· (2 + 2kβ1) < d Β· (2 + 2kβ3 β cmax), i.e. max Jkβ1 < min Jkβ3. Thus Jkβ1 β© Jkβ3 = β
, as desired. β
|
1084 |
+
5.3
|
1085 |
+
Identiο¬cation of gaps when E(K) is torsion-free with rank r = 1
|
1086 |
+
The results in Subsections 5.1 and 5.2 concern the existence and the distribution of gap num-
|
1087 |
+
bers. In the following subsections we turn our attention to ο¬nding gap numbers explicitly. In this
|
1088 |
+
subsection we give a complete description of gap numbers assuming E(K) is torsion-free with rank
|
1089 |
+
r = 1. Such descriptions are diο¬cult in the general case, but our assumption guarantees that each
|
1090 |
+
18
|
1091 |
+
|
1092 |
+
E(K), E(K)0 is generated by a single element and that β < 2 by Lemma 2.8, which makes the
|
1093 |
+
problem more accessible.
|
1094 |
+
We organize this subsection as follows. First we point out some trivial facts about generators
|
1095 |
+
of E(K), E(K)0 when r = 1 in Lemma 5.5. Next we state necessary and suο¬cient conditions for
|
1096 |
+
having P1 Β· P2 = k when E(K) is torsion-free with r = 1 in Lemma 5.6. As an application of the
|
1097 |
+
latter, we prove Theorem 5.7, which is the main result of the subsection.
|
1098 |
+
Lemma 5.5. Let X be a rational elliptic surface with Mordell-Weil rank r = 1. If P generates the
|
1099 |
+
free part of E(K), then
|
1100 |
+
a) h(P) = Β΅.
|
1101 |
+
b) 1/Β΅ is an even integer.
|
1102 |
+
c) E(K)0 is generated by P0 := (1/Β΅)P and h(P0) = 1/Β΅.
|
1103 |
+
Proof. Item a) is clear. Items b), c) follow from the fact that E(K)0 is an even lattice and that
|
1104 |
+
E(K) β Lβ β E(K)tor, where L := E(K)0 [OS91, Main Thm.]. β
|
1105 |
+
In what follows we use Lemma 5.5 and results from Section 4 to state necessary and suο¬cient
|
1106 |
+
conditions for having P1 Β· P2 = k for some P1, P2 β E(K) in case E(K) is torsion-free with r = 1.
|
1107 |
+
Lemma 5.6. Assume E(K) is torsion-free with rank r = 1. Then P1 Β· P2 = k for some P1, P2 β
|
1108 |
+
E(K) if and only if one of the following holds:
|
1109 |
+
i) Β΅ Β· (2 + 2k) is a perfect square.
|
1110 |
+
ii) There is a perfect square n2 β
|
1111 |
+
οΏ½
|
1112 |
+
2+2kβcmax
|
1113 |
+
Β΅
|
1114 |
+
, 2+2kβcmin
|
1115 |
+
Β΅
|
1116 |
+
οΏ½
|
1117 |
+
such that Β΅ Β· n /β Z.
|
1118 |
+
Proof. By Lemma 5.5, E(K) is generated by some P with h(P) = Β΅ and E(K)0 is generated by
|
1119 |
+
P0 := n0P, where n0 := 1
|
1120 |
+
Β΅ β 2Z.
|
1121 |
+
First assume that P1Β·P2 = k for some P1, P2. Without loss of generality we may assume P2 = O.
|
1122 |
+
Let P1 = nP for some n β Z. We show that P1 β E(K)0 implies i) while P1 /β E(K)0 implies ii).
|
1123 |
+
If P1 β E(K)0, then n0 | n, hence P1 = nP = mP0, where m :=
|
1124 |
+
n
|
1125 |
+
n0. By the height formula (2),
|
1126 |
+
2 + 2k = h(P1) = h(mP0) = m2 Β· 1
|
1127 |
+
Β΅. Hence Β΅ Β· (2 + 2k) = m2, i.e. i) holds.
|
1128 |
+
If P1 /β E(K)0, then n0 β€ n, hence Β΅ Β· n =
|
1129 |
+
n
|
1130 |
+
n0 /β Z. Moreover, h(P1) = n2h(P) = n2Β΅ and by
|
1131 |
+
the height formula (2), n2Β΅ = h(P) = 2 + 2k β c, where c := οΏ½
|
1132 |
+
v contrv(P1) ΜΈ= 0. The inequalities
|
1133 |
+
cmin β€ c β€ cmax then give 2+2kβcmax
|
1134 |
+
Β΅
|
1135 |
+
β€ n2 β€ 2+2kβcmin
|
1136 |
+
Β΅
|
1137 |
+
. Hence ii) holds.
|
1138 |
+
Conversely, assume i) or ii) holds. Since E(K) is torsion-free, β < 2 by Lemma 2.8, so we may
|
1139 |
+
apply Lemma 4.3. If i) holds, then Β΅ Β· (2 + 2k) = m2 for some m β Z. Since mP0 β E(K)0 and
|
1140 |
+
h(mP0) =
|
1141 |
+
m2
|
1142 |
+
Β΅
|
1143 |
+
= 2 + 2k, we are done by Lemma 4.3 i).
|
1144 |
+
If ii) holds, the condition Β΅ Β· n /β Z
|
1145 |
+
is equivalent to n0 β€ n, hence nP
|
1146 |
+
/β E(K)0.
|
1147 |
+
Moreover n2 β
|
1148 |
+
οΏ½
|
1149 |
+
2+2kβcmax
|
1150 |
+
Β΅
|
1151 |
+
, 2+2kβcmin
|
1152 |
+
Β΅
|
1153 |
+
οΏ½
|
1154 |
+
, implies
|
1155 |
+
h(nP) = n2Β΅ β [2 + 2k β cmax, 2 + 2k β cmin]. By Lemma 4.3 ii), we are done. β
|
1156 |
+
By applying Lemma 5.6 to all possible cases where E(K) is torsion-free with rank r = 1,
|
1157 |
+
we obtain the main result of this subsection.
|
1158 |
+
19
|
1159 |
+
|
1160 |
+
Theorem 5.7. If E(K) is torsion-free with rank r = 1, then all the gap numbers of X are described
|
1161 |
+
in Table 6.
|
1162 |
+
No.
|
1163 |
+
T
|
1164 |
+
k is a gap number β none of
|
1165 |
+
the following are perfect squares
|
1166 |
+
ο¬rst gap numbers
|
1167 |
+
43
|
1168 |
+
E7
|
1169 |
+
k + 1, 4k + 1
|
1170 |
+
1, 4
|
1171 |
+
45
|
1172 |
+
A7
|
1173 |
+
k+1
|
1174 |
+
4 , 16k, ..., 16k + 9
|
1175 |
+
8, 11
|
1176 |
+
46
|
1177 |
+
D7
|
1178 |
+
k+1
|
1179 |
+
2 , 8k + 1, ..., 8k + 4
|
1180 |
+
2, 5
|
1181 |
+
47
|
1182 |
+
A6 β A1
|
1183 |
+
k+1
|
1184 |
+
7 , 28k β 3, ..., 28k + 21
|
1185 |
+
12, 16
|
1186 |
+
49
|
1187 |
+
E6 β A1
|
1188 |
+
k+1
|
1189 |
+
3 , 12k + 1, ..., 12k + 9
|
1190 |
+
3, 7
|
1191 |
+
50
|
1192 |
+
D5 β A2
|
1193 |
+
k+1
|
1194 |
+
6 , 24k + 1, ..., 24k + 16
|
1195 |
+
6, 11
|
1196 |
+
55
|
1197 |
+
A4 β A3
|
1198 |
+
k+1
|
1199 |
+
10 , 40k β 4, ..., 40k + 25
|
1200 |
+
16, 20
|
1201 |
+
56
|
1202 |
+
A4 β A2 β A1
|
1203 |
+
k+1
|
1204 |
+
15 , 60k β 11, ..., 60k + 45
|
1205 |
+
22, 27
|
1206 |
+
Table 6: Description of gap numbers when E(K) is torsion-free with r = 1.
|
1207 |
+
Proof. For the sake of brevity we restrict ourselves to No. 55. The other cases are treated similarly.
|
1208 |
+
Here cmax = 2Β·3
|
1209 |
+
5 + 2Β·2
|
1210 |
+
4 = 11
|
1211 |
+
5 , cmin = min
|
1212 |
+
οΏ½
|
1213 |
+
4
|
1214 |
+
5, 3
|
1215 |
+
4
|
1216 |
+
οΏ½
|
1217 |
+
= 3
|
1218 |
+
4 and Β΅ = 1/20.
|
1219 |
+
By Lemma 5.6, k is a gap number if and only if neither i) nor ii) occurs. Condition i) is that
|
1220 |
+
2+2k
|
1221 |
+
20
|
1222 |
+
= k+1
|
1223 |
+
10
|
1224 |
+
is a perfect square. Condition ii) is that
|
1225 |
+
οΏ½
|
1226 |
+
2+2kβcmax
|
1227 |
+
Β΅
|
1228 |
+
, 2+2kβcmin
|
1229 |
+
Β΅
|
1230 |
+
οΏ½
|
1231 |
+
= [40k β 4, 40k + 25]
|
1232 |
+
contains some n2 with 20 β€ n. We check that 20 β€ n for every n such that n2 = 40k + β, with
|
1233 |
+
β = β4, ..., 25. Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2. Then 40 | (n2 β 40k) = β,
|
1234 |
+
which is absurd. β
|
1235 |
+
5.4
|
1236 |
+
Surfaces with a 1-gap
|
1237 |
+
In Subsection 5.3 we take each case in Table 6 and describe all its gap numbers.
|
1238 |
+
In this
|
1239 |
+
subsection we do the opposite, which is to ο¬x a number and describe all cases having it as a gap
|
1240 |
+
number. We remind the reader that our motivating problem (Section 1) was to determine when
|
1241 |
+
there are sections P1, P2 such that P1 Β· P2 = 1, which induce a conic bundle having P1 + P2 as a
|
1242 |
+
reducible ο¬ber. The answer for this question is the main theorem of this subsection:
|
1243 |
+
Theorem 5.8. Let X be a rational elliptic surface. Then X has a 1-gap if and only if r = 0 or
|
1244 |
+
r = 1 and Ο has a IIIβ ο¬ber.
|
1245 |
+
20
|
1246 |
+
|
1247 |
+
Our strategy for the proof is the following. We already know that a 1-gap exists whenever r = 0
|
1248 |
+
(Theorem 3.1) or when r = 1 and Ο has a IIIβ ο¬ber (Theorem 5.7, No. 43). Conversely, we need to
|
1249 |
+
ο¬nd P1, P2 with P1 Β· P2 = 1 in all cases with r β₯ 1 and T ΜΈ= E7.
|
1250 |
+
First we introduce two lemmas, which solve most cases with little computation, and leave the
|
1251 |
+
remaining ones for the proof of Theorem 5.8. In both Lemma 5.9 and Lemma 5.11 our goal is to
|
1252 |
+
analyze the narrow lattice E(K)0 and apply Proposition 4.12 to detect cases without a 1-gap.
|
1253 |
+
Lemma 5.9. If one of the following holds, then h(P) = 4 for some P β E(K)0.
|
1254 |
+
a) The Gram matrix of E(K)0 has a 4 in its main diagonal.
|
1255 |
+
b) There is an embedding of An β Am in E(K)0 for some n, m β₯ 1.
|
1256 |
+
c) There is an embedding of An, Dn or En in E(K)0 for some n β₯ 3.
|
1257 |
+
Proof. Case a) is trivial. Assuming b), we take generators P1, P2 from An, Am respectively with
|
1258 |
+
h(P1) = h(P2) = 2. Since An, Am are in direct sum, β¨P1, P2β© = 0, hence h(P1 + P2) = 4, as desired.
|
1259 |
+
If c) holds, then the fact that n β₯ 3 allows us to choose two elements P1, P2 among the generators
|
1260 |
+
of L1 = An, Dn or En such that h(P1) = h(P2) = 2 and β¨P1, P2β© = 0. Thus h(P1 + P2) = 4 as
|
1261 |
+
claimed. β
|
1262 |
+
Corollary 5.10. In the following cases, X does not have a 1-gap.
|
1263 |
+
β’ r β₯ 3 : all cases except possibly No. 20.
|
1264 |
+
β’ r = 1, 2 : cases No. 25, 26, 30, 32-36, 38, 41, 42, 46, 52, 54, 60.
|
1265 |
+
Proof. We look at column E(K)0 in Table 8 to ο¬nd which cases satisfy one of the conditions a),
|
1266 |
+
b), c) from Lemma 5.9.
|
1267 |
+
a) Applies to No. 12, 17, 19, 22, 23, 25, 30, 32, 33, 36, 38, 41, 46, 52, 54, 60.
|
1268 |
+
b) Applies to No. 10, 11, 14, 15, 18, 24, 26, 34, 35, 42.
|
1269 |
+
c) Applies to No. 1-10, 13, 16, 21.
|
1270 |
+
In particular, this covers all cases with r β₯ 3 (No. 1-24) except No. 20. By Lemma 5.9 in each
|
1271 |
+
of these cases there is P β E(K)0 with h(P) = 4 and we are done by Proposition 4.12 1). β
|
1272 |
+
In the next lemma we also analyze E(K)0 to detect surfaces without a 1-gap.
|
1273 |
+
Lemma 5.11. Assume E(K)0 β An for some n β₯ 1 and that E(K) has nontrivial torsion part.
|
1274 |
+
Then X does not have a 1-gap. This applies to cases No. 28, 39, 44, 48, 51, 57, 58 in Table 8.
|
1275 |
+
Proof. Take a generator P of E(K)0 with h(P) = 2 and apply Proposition 4.12 2). β
|
1276 |
+
21
|
1277 |
+
|
1278 |
+
We are ready to prove the main result of this subsection.
|
1279 |
+
Proof of Theorem 5.8. We need to show that in all cases where r β₯ 1 and T ΜΈ= E7 there are
|
1280 |
+
P1, P2 β E(K) such that P1 Β· P2 = 1. This corresponds to cases No. 1-61 except 43 in Table 8.
|
1281 |
+
The cases where r = 1 and E(K) is torsion-free can be solved by Theorem 5.10, namely No.
|
1282 |
+
45-47, 49, 50, 55, 56. Adding these cases to the ones treated in Corollary 5.10 and Lemma 5.11,
|
1283 |
+
we have therefore solved the following:
|
1284 |
+
No. 1-19, 21-26, 28, 30, 32-36, 38, 39, 41-52, 54-58, 60.
|
1285 |
+
For the remaining cases, we apply Proposition 4.12 3), which involves ο¬nding perfect squares
|
1286 |
+
in the interval
|
1287 |
+
οΏ½
|
1288 |
+
4βcmax
|
1289 |
+
Β΅
|
1290 |
+
, 4βcmin
|
1291 |
+
Β΅
|
1292 |
+
οΏ½
|
1293 |
+
(see Table 7), considering the half-open interval in the cases with
|
1294 |
+
β = 2 (No. 53, 61).
|
1295 |
+
No.
|
1296 |
+
T
|
1297 |
+
E(K)
|
1298 |
+
Β΅
|
1299 |
+
I
|
1300 |
+
n2 β I
|
1301 |
+
20
|
1302 |
+
Aβ2
|
1303 |
+
2
|
1304 |
+
β A1
|
1305 |
+
Aβ
|
1306 |
+
2 β β¨1/6β©
|
1307 |
+
1
|
1308 |
+
6
|
1309 |
+
[13, 23]
|
1310 |
+
42
|
1311 |
+
27
|
1312 |
+
E6
|
1313 |
+
Aβ
|
1314 |
+
2
|
1315 |
+
2
|
1316 |
+
3
|
1317 |
+
[4, 4]
|
1318 |
+
22
|
1319 |
+
29
|
1320 |
+
A5 β A1
|
1321 |
+
Aβ
|
1322 |
+
1 β β¨1/6β©
|
1323 |
+
1
|
1324 |
+
6
|
1325 |
+
[12, 21]
|
1326 |
+
42
|
1327 |
+
31
|
1328 |
+
A4 β A2
|
1329 |
+
1
|
1330 |
+
15
|
1331 |
+
οΏ½
|
1332 |
+
2
|
1333 |
+
1
|
1334 |
+
1
|
1335 |
+
8
|
1336 |
+
οΏ½
|
1337 |
+
2
|
1338 |
+
15
|
1339 |
+
[16, 21]
|
1340 |
+
42
|
1341 |
+
37
|
1342 |
+
A3 β A2 β A1
|
1343 |
+
Aβ
|
1344 |
+
1 β β¨1/12β©
|
1345 |
+
1
|
1346 |
+
12
|
1347 |
+
[22, 28]
|
1348 |
+
52
|
1349 |
+
40
|
1350 |
+
Aβ2
|
1351 |
+
2
|
1352 |
+
β Aβ2
|
1353 |
+
1
|
1354 |
+
β¨1/6β©β2
|
1355 |
+
1
|
1356 |
+
6
|
1357 |
+
[10, 21]
|
1358 |
+
42
|
1359 |
+
53
|
1360 |
+
A5 β Aβ2
|
1361 |
+
1
|
1362 |
+
β¨1/6β© β Z/2Z
|
1363 |
+
1
|
1364 |
+
6
|
1365 |
+
[9, 12]
|
1366 |
+
32
|
1367 |
+
59
|
1368 |
+
A3 β A2 β Aβ2
|
1369 |
+
1
|
1370 |
+
β¨1/12β© β Z/2Z
|
1371 |
+
1
|
1372 |
+
12
|
1373 |
+
[16, 42]
|
1374 |
+
42, 52, 62
|
1375 |
+
61
|
1376 |
+
Aβ3
|
1377 |
+
2
|
1378 |
+
β A1
|
1379 |
+
β¨1/6β© β Z/3Z
|
1380 |
+
1
|
1381 |
+
6
|
1382 |
+
[9, 12]
|
1383 |
+
32
|
1384 |
+
Table 7: Perfect squares in the interval I :=
|
1385 |
+
οΏ½
|
1386 |
+
4βcmax
|
1387 |
+
Β΅
|
1388 |
+
, 4βcmin
|
1389 |
+
Β΅
|
1390 |
+
οΏ½
|
1391 |
+
.
|
1392 |
+
In No. 59 we have β > 2, so a particular treatment is needed. Let T = Tv1 β Tv2 β Tv3 β Tv4 =
|
1393 |
+
A3 β A2 β A1 β A1. If P generates the free part of E(K) and Q generates its torsion part, then
|
1394 |
+
h(P) =
|
1395 |
+
1
|
1396 |
+
12 and 4P + Q meets the reducible ο¬bers at Ξv1,2, Ξv2,1, Ξv3,1, Ξv4,1 [Kur14][Example 1.7].
|
1397 |
+
By Table 1 and the height formula (2),
|
1398 |
+
42
|
1399 |
+
12 = h(4P + Q) = 2 + 2(4P + Q) Β· O β 2 Β· 2
|
1400 |
+
4
|
1401 |
+
β 1 Β· 2
|
1402 |
+
3
|
1403 |
+
β 1
|
1404 |
+
2 β 1
|
1405 |
+
2,
|
1406 |
+
hence (4P + Q) Β· O = 1, as desired. β
|
1407 |
+
22
|
1408 |
+
|
1409 |
+
6
|
1410 |
+
Appendix
|
1411 |
+
We reproduce part of the table in [OS91, Main Th.] with data on Mordell-Weil lattices of
|
1412 |
+
rational elliptic surfaces with Mordell-Weil rank r β₯ 1. We only add columns cmax, cmin, β.
|
1413 |
+
No.
|
1414 |
+
r
|
1415 |
+
T
|
1416 |
+
E(K)0
|
1417 |
+
E(K)
|
1418 |
+
cmax
|
1419 |
+
cmin
|
1420 |
+
β
|
1421 |
+
1
|
1422 |
+
8
|
1423 |
+
0
|
1424 |
+
E8
|
1425 |
+
E8
|
1426 |
+
0
|
1427 |
+
0
|
1428 |
+
0
|
1429 |
+
2
|
1430 |
+
7
|
1431 |
+
A1
|
1432 |
+
E7
|
1433 |
+
Eβ
|
1434 |
+
8
|
1435 |
+
1
|
1436 |
+
2
|
1437 |
+
1
|
1438 |
+
2
|
1439 |
+
0
|
1440 |
+
3
|
1441 |
+
6
|
1442 |
+
A2
|
1443 |
+
E6
|
1444 |
+
Eβ
|
1445 |
+
6
|
1446 |
+
2
|
1447 |
+
3
|
1448 |
+
2
|
1449 |
+
3
|
1450 |
+
0
|
1451 |
+
4
|
1452 |
+
Aβ2
|
1453 |
+
1
|
1454 |
+
D6
|
1455 |
+
Dβ
|
1456 |
+
6
|
1457 |
+
3
|
1458 |
+
2
|
1459 |
+
1
|
1460 |
+
1
|
1461 |
+
2
|
1462 |
+
5
|
1463 |
+
5
|
1464 |
+
A3
|
1465 |
+
D5
|
1466 |
+
Dβ
|
1467 |
+
5
|
1468 |
+
1
|
1469 |
+
3
|
1470 |
+
4
|
1471 |
+
1
|
1472 |
+
4
|
1473 |
+
6
|
1474 |
+
A2 β A1
|
1475 |
+
A5
|
1476 |
+
Aβ
|
1477 |
+
5
|
1478 |
+
7
|
1479 |
+
6
|
1480 |
+
1
|
1481 |
+
2
|
1482 |
+
2
|
1483 |
+
3
|
1484 |
+
7
|
1485 |
+
Aβ3
|
1486 |
+
1
|
1487 |
+
D4 β A1
|
1488 |
+
Dβ
|
1489 |
+
4 β Aβ
|
1490 |
+
1
|
1491 |
+
3
|
1492 |
+
2
|
1493 |
+
1
|
1494 |
+
2
|
1495 |
+
1
|
1496 |
+
8
|
1497 |
+
4
|
1498 |
+
A4
|
1499 |
+
A4
|
1500 |
+
Aβ
|
1501 |
+
4
|
1502 |
+
6
|
1503 |
+
5
|
1504 |
+
4
|
1505 |
+
5
|
1506 |
+
2
|
1507 |
+
5
|
1508 |
+
9
|
1509 |
+
D4
|
1510 |
+
D4
|
1511 |
+
Dβ
|
1512 |
+
4
|
1513 |
+
1
|
1514 |
+
1
|
1515 |
+
0
|
1516 |
+
10
|
1517 |
+
A3 β A1
|
1518 |
+
A3 β A1
|
1519 |
+
Aβ
|
1520 |
+
3 β Aβ
|
1521 |
+
1
|
1522 |
+
3
|
1523 |
+
2
|
1524 |
+
1
|
1525 |
+
2
|
1526 |
+
1
|
1527 |
+
11
|
1528 |
+
Aβ2
|
1529 |
+
2
|
1530 |
+
Aβ2
|
1531 |
+
2
|
1532 |
+
Aβ
|
1533 |
+
2
|
1534 |
+
β2
|
1535 |
+
4
|
1536 |
+
3
|
1537 |
+
2
|
1538 |
+
3
|
1539 |
+
2
|
1540 |
+
3
|
1541 |
+
12
|
1542 |
+
A2 β Aβ2
|
1543 |
+
1
|
1544 |
+

|
1545 |
+

|
1546 |
+

|
1547 |
+

|
1548 |
+
ο£
|
1549 |
+
4
|
1550 |
+
β1
|
1551 |
+
0
|
1552 |
+
1
|
1553 |
+
β1
|
1554 |
+
2
|
1555 |
+
β1
|
1556 |
+
0
|
1557 |
+
0
|
1558 |
+
β1
|
1559 |
+
2
|
1560 |
+
β1
|
1561 |
+
1
|
1562 |
+
0
|
1563 |
+
β1
|
1564 |
+
2
|
1565 |
+
ο£Ά
|
1566 |
+
ο£·
|
1567 |
+
ο£·
|
1568 |
+
ο£·
|
1569 |
+
ο£Έ
|
1570 |
+
1
|
1571 |
+
6
|
1572 |
+

|
1573 |
+

|
1574 |
+

|
1575 |
+

|
1576 |
+
ο£
|
1577 |
+
2
|
1578 |
+
1
|
1579 |
+
0
|
1580 |
+
β1
|
1581 |
+
1
|
1582 |
+
5
|
1583 |
+
3
|
1584 |
+
1
|
1585 |
+
0
|
1586 |
+
3
|
1587 |
+
6
|
1588 |
+
3
|
1589 |
+
β1
|
1590 |
+
1
|
1591 |
+
3
|
1592 |
+
5
|
1593 |
+
ο£Ά
|
1594 |
+
ο£·
|
1595 |
+
ο£·
|
1596 |
+
ο£·
|
1597 |
+
ο£Έ
|
1598 |
+
5
|
1599 |
+
3
|
1600 |
+
1
|
1601 |
+
2
|
1602 |
+
7
|
1603 |
+
6
|
1604 |
+
13
|
1605 |
+
Aβ4
|
1606 |
+
1
|
1607 |
+
D4
|
1608 |
+
Dβ
|
1609 |
+
4 β Z/2Z
|
1610 |
+
2
|
1611 |
+
1
|
1612 |
+
2
|
1613 |
+
3
|
1614 |
+
2
|
1615 |
+
14
|
1616 |
+
Aβ4
|
1617 |
+
1
|
1618 |
+
Aβ4
|
1619 |
+
1
|
1620 |
+
Aβ
|
1621 |
+
1
|
1622 |
+
β4
|
1623 |
+
2
|
1624 |
+
1
|
1625 |
+
2
|
1626 |
+
3
|
1627 |
+
2
|
1628 |
+
15
|
1629 |
+
3
|
1630 |
+
A5
|
1631 |
+
A2 β A1
|
1632 |
+
Aβ
|
1633 |
+
2 β Aβ
|
1634 |
+
1
|
1635 |
+
3
|
1636 |
+
2
|
1637 |
+
5
|
1638 |
+
6
|
1639 |
+
2
|
1640 |
+
3
|
1641 |
+
16
|
1642 |
+
D5
|
1643 |
+
A3
|
1644 |
+
Aβ
|
1645 |
+
3
|
1646 |
+
5
|
1647 |
+
4
|
1648 |
+
1
|
1649 |
+
1
|
1650 |
+
4
|
1651 |
+
17
|
1652 |
+
A4 β A1
|
1653 |
+

|
1654 |
+

|
1655 |
+
ο£
|
1656 |
+
4
|
1657 |
+
β1
|
1658 |
+
1
|
1659 |
+
β1
|
1660 |
+
2
|
1661 |
+
β1
|
1662 |
+
1
|
1663 |
+
β1
|
1664 |
+
2
|
1665 |
+
ο£Ά
|
1666 |
+
ο£·
|
1667 |
+
ο£Έ
|
1668 |
+
1
|
1669 |
+
10
|
1670 |
+

|
1671 |
+

|
1672 |
+
ο£
|
1673 |
+
3
|
1674 |
+
1
|
1675 |
+
β1
|
1676 |
+
1
|
1677 |
+
7
|
1678 |
+
3
|
1679 |
+
β1
|
1680 |
+
3
|
1681 |
+
7
|
1682 |
+
ο£Ά
|
1683 |
+
ο£·
|
1684 |
+
ο£Έ
|
1685 |
+
17
|
1686 |
+
10
|
1687 |
+
1
|
1688 |
+
2
|
1689 |
+
6
|
1690 |
+
5
|
1691 |
+
18
|
1692 |
+
D4 β A1
|
1693 |
+
Aβ3
|
1694 |
+
1
|
1695 |
+
Aβ
|
1696 |
+
1
|
1697 |
+
β3
|
1698 |
+
3
|
1699 |
+
2
|
1700 |
+
1
|
1701 |
+
2
|
1702 |
+
1
|
1703 |
+
19
|
1704 |
+
A3 β A2
|
1705 |
+

|
1706 |
+

|
1707 |
+
ο£
|
1708 |
+
2
|
1709 |
+
0
|
1710 |
+
β1
|
1711 |
+
0
|
1712 |
+
2
|
1713 |
+
β1
|
1714 |
+
β1
|
1715 |
+
β1
|
1716 |
+
4
|
1717 |
+
ο£Ά
|
1718 |
+
ο£·
|
1719 |
+
ο£Έ
|
1720 |
+
1
|
1721 |
+
12
|
1722 |
+

|
1723 |
+

|
1724 |
+
ο£
|
1725 |
+
7
|
1726 |
+
1
|
1727 |
+
2
|
1728 |
+
1
|
1729 |
+
7
|
1730 |
+
2
|
1731 |
+
2
|
1732 |
+
2
|
1733 |
+
4
|
1734 |
+
ο£Ά
|
1735 |
+
ο£·
|
1736 |
+
ο£Έ
|
1737 |
+
5
|
1738 |
+
3
|
1739 |
+
2
|
1740 |
+
3
|
1741 |
+
1
|
1742 |
+
23
|
1743 |
+
|
1744 |
+
20
|
1745 |
+
Aβ2
|
1746 |
+
2
|
1747 |
+
β A1
|
1748 |
+
A2 β β¨6β©
|
1749 |
+
Aβ
|
1750 |
+
2 β β¨1/6β©
|
1751 |
+
11
|
1752 |
+
6
|
1753 |
+
1
|
1754 |
+
2
|
1755 |
+
4
|
1756 |
+
3
|
1757 |
+
21
|
1758 |
+
A3 β Aβ2
|
1759 |
+
1
|
1760 |
+
A3
|
1761 |
+
Aβ
|
1762 |
+
3 β Z/2Z
|
1763 |
+
2
|
1764 |
+
1
|
1765 |
+
2
|
1766 |
+
3
|
1767 |
+
2
|
1768 |
+
22
|
1769 |
+
A3 β Aβ2
|
1770 |
+
1
|
1771 |
+
A1 β β¨4β©
|
1772 |
+
Aβ
|
1773 |
+
1 β β¨1/4β©
|
1774 |
+
2
|
1775 |
+
1
|
1776 |
+
2
|
1777 |
+
3
|
1778 |
+
2
|
1779 |
+
23
|
1780 |
+
A2 β Aβ3
|
1781 |
+
1
|
1782 |
+
A1 β
|
1783 |
+
οΏ½
|
1784 |
+
4
|
1785 |
+
β2
|
1786 |
+
β2
|
1787 |
+
4
|
1788 |
+
οΏ½
|
1789 |
+
Aβ
|
1790 |
+
1 β 1
|
1791 |
+
6
|
1792 |
+
οΏ½
|
1793 |
+
2
|
1794 |
+
1
|
1795 |
+
1
|
1796 |
+
2
|
1797 |
+
οΏ½
|
1798 |
+
13
|
1799 |
+
6
|
1800 |
+
1
|
1801 |
+
2
|
1802 |
+
5
|
1803 |
+
3
|
1804 |
+
24
|
1805 |
+
Aβ5
|
1806 |
+
1
|
1807 |
+
Aβ3
|
1808 |
+
1
|
1809 |
+
Aβ
|
1810 |
+
1
|
1811 |
+
β3 β Z/2Z
|
1812 |
+
5
|
1813 |
+
2
|
1814 |
+
1
|
1815 |
+
2
|
1816 |
+
2
|
1817 |
+
25
|
1818 |
+
2
|
1819 |
+
A6
|
1820 |
+
οΏ½
|
1821 |
+
4
|
1822 |
+
β1
|
1823 |
+
β1
|
1824 |
+
2
|
1825 |
+
οΏ½
|
1826 |
+
1
|
1827 |
+
7
|
1828 |
+
οΏ½
|
1829 |
+
2
|
1830 |
+
1
|
1831 |
+
1
|
1832 |
+
4
|
1833 |
+
οΏ½
|
1834 |
+
12
|
1835 |
+
7
|
1836 |
+
6
|
1837 |
+
7
|
1838 |
+
6
|
1839 |
+
7
|
1840 |
+
26
|
1841 |
+
D6
|
1842 |
+
Aβ2
|
1843 |
+
1
|
1844 |
+
Aβ
|
1845 |
+
1
|
1846 |
+
β2
|
1847 |
+
3
|
1848 |
+
2
|
1849 |
+
1
|
1850 |
+
1
|
1851 |
+
2
|
1852 |
+
27
|
1853 |
+
E6
|
1854 |
+
A2
|
1855 |
+
Aβ
|
1856 |
+
2
|
1857 |
+
4
|
1858 |
+
3
|
1859 |
+
4
|
1860 |
+
3
|
1861 |
+
0
|
1862 |
+
28
|
1863 |
+
A5 β A1
|
1864 |
+
A2
|
1865 |
+
Aβ
|
1866 |
+
2 β Z/2Z
|
1867 |
+
2
|
1868 |
+
1
|
1869 |
+
2
|
1870 |
+
3
|
1871 |
+
2
|
1872 |
+
29
|
1873 |
+
A5 β A1
|
1874 |
+
A1 β β¨6β©
|
1875 |
+
Aβ
|
1876 |
+
1 β β¨1/6β©
|
1877 |
+
2
|
1878 |
+
1
|
1879 |
+
2
|
1880 |
+
3
|
1881 |
+
2
|
1882 |
+
30
|
1883 |
+
D5 β A1
|
1884 |
+
A1 β β¨4β©
|
1885 |
+
Aβ
|
1886 |
+
1 β β¨1/4β©
|
1887 |
+
7
|
1888 |
+
4
|
1889 |
+
1
|
1890 |
+
2
|
1891 |
+
5
|
1892 |
+
4
|
1893 |
+
31
|
1894 |
+
A4 β A2
|
1895 |
+
οΏ½
|
1896 |
+
8
|
1897 |
+
β1
|
1898 |
+
β1
|
1899 |
+
2
|
1900 |
+
οΏ½
|
1901 |
+
1
|
1902 |
+
15
|
1903 |
+
οΏ½
|
1904 |
+
2
|
1905 |
+
1
|
1906 |
+
1
|
1907 |
+
8
|
1908 |
+
οΏ½
|
1909 |
+
28
|
1910 |
+
15
|
1911 |
+
2
|
1912 |
+
3
|
1913 |
+
6
|
1914 |
+
5
|
1915 |
+
32
|
1916 |
+
D4 β A2
|
1917 |
+
οΏ½
|
1918 |
+
4
|
1919 |
+
β2
|
1920 |
+
β2
|
1921 |
+
4
|
1922 |
+
οΏ½
|
1923 |
+
1
|
1924 |
+
6
|
1925 |
+
οΏ½
|
1926 |
+
2
|
1927 |
+
1
|
1928 |
+
1
|
1929 |
+
2
|
1930 |
+
οΏ½
|
1931 |
+
5
|
1932 |
+
3
|
1933 |
+
2
|
1934 |
+
3
|
1935 |
+
1
|
1936 |
+
33
|
1937 |
+
A4 β Aβ2
|
1938 |
+
1
|
1939 |
+
οΏ½
|
1940 |
+
6
|
1941 |
+
β2
|
1942 |
+
β2
|
1943 |
+
4
|
1944 |
+
οΏ½
|
1945 |
+
1
|
1946 |
+
10
|
1947 |
+
οΏ½
|
1948 |
+
2
|
1949 |
+
1
|
1950 |
+
1
|
1951 |
+
3
|
1952 |
+
οΏ½
|
1953 |
+
11
|
1954 |
+
5
|
1955 |
+
1
|
1956 |
+
2
|
1957 |
+
17
|
1958 |
+
10
|
1959 |
+
34
|
1960 |
+
D4 β Aβ2
|
1961 |
+
1
|
1962 |
+
Aβ2
|
1963 |
+
1
|
1964 |
+
Aβ
|
1965 |
+
1
|
1966 |
+
β2
|
1967 |
+
2
|
1968 |
+
1
|
1969 |
+
2
|
1970 |
+
3
|
1971 |
+
2
|
1972 |
+
35
|
1973 |
+
Aβ2
|
1974 |
+
3
|
1975 |
+
Aβ2
|
1976 |
+
1
|
1977 |
+
Aβ
|
1978 |
+
1
|
1979 |
+
β2 β Z/2Z
|
1980 |
+
2
|
1981 |
+
3
|
1982 |
+
4
|
1983 |
+
5
|
1984 |
+
4
|
1985 |
+
36
|
1986 |
+
Aβ2
|
1987 |
+
3
|
1988 |
+
β¨4β©β2
|
1989 |
+
β¨1/4β©β2
|
1990 |
+
2
|
1991 |
+
3
|
1992 |
+
4
|
1993 |
+
5
|
1994 |
+
4
|
1995 |
+
37
|
1996 |
+
A3 β A2 β A1
|
1997 |
+
A1 β β¨12β©
|
1998 |
+
Aβ
|
1999 |
+
1 β β¨1/12β©
|
2000 |
+
13
|
2001 |
+
6
|
2002 |
+
1
|
2003 |
+
2
|
2004 |
+
5
|
2005 |
+
3
|
2006 |
+
38
|
2007 |
+
A3 β Aβ3
|
2008 |
+
1
|
2009 |
+
A1 β β¨4β©
|
2010 |
+
Aβ
|
2011 |
+
1 β β¨1/4β© β Z/2Z
|
2012 |
+
5
|
2013 |
+
2
|
2014 |
+
1
|
2015 |
+
2
|
2016 |
+
2
|
2017 |
+
39
|
2018 |
+
Aβ3
|
2019 |
+
2
|
2020 |
+
A2
|
2021 |
+
Aβ
|
2022 |
+
2 β Z/3Z
|
2023 |
+
2
|
2024 |
+
2
|
2025 |
+
3
|
2026 |
+
4
|
2027 |
+
3
|
2028 |
+
40
|
2029 |
+
Aβ2
|
2030 |
+
2
|
2031 |
+
β Aβ2
|
2032 |
+
1
|
2033 |
+
β¨6β©β2
|
2034 |
+
β¨1/6β©β2
|
2035 |
+
7
|
2036 |
+
3
|
2037 |
+
1
|
2038 |
+
2
|
2039 |
+
11
|
2040 |
+
6
|
2041 |
+
24
|
2042 |
+
|
2043 |
+
41
|
2044 |
+
A2 β Aβ4
|
2045 |
+
1
|
2046 |
+
οΏ½
|
2047 |
+
4
|
2048 |
+
β2
|
2049 |
+
β2
|
2050 |
+
4
|
2051 |
+
οΏ½
|
2052 |
+
1
|
2053 |
+
6
|
2054 |
+
οΏ½
|
2055 |
+
2
|
2056 |
+
1
|
2057 |
+
1
|
2058 |
+
2
|
2059 |
+
οΏ½
|
2060 |
+
8
|
2061 |
+
3
|
2062 |
+
1
|
2063 |
+
2
|
2064 |
+
13
|
2065 |
+
6
|
2066 |
+
42
|
2067 |
+
Aβ6
|
2068 |
+
1
|
2069 |
+
Aβ2
|
2070 |
+
1
|
2071 |
+
Aβ
|
2072 |
+
1
|
2073 |
+
β2 β (Z/2Z)2
|
2074 |
+
3
|
2075 |
+
1
|
2076 |
+
2
|
2077 |
+
5
|
2078 |
+
2
|
2079 |
+
43
|
2080 |
+
1
|
2081 |
+
E7
|
2082 |
+
A1
|
2083 |
+
Aβ
|
2084 |
+
1
|
2085 |
+
3
|
2086 |
+
2
|
2087 |
+
3
|
2088 |
+
2
|
2089 |
+
0
|
2090 |
+
44
|
2091 |
+
A7
|
2092 |
+
A1
|
2093 |
+
Aβ
|
2094 |
+
1 β Z/2Z
|
2095 |
+
2
|
2096 |
+
7
|
2097 |
+
8
|
2098 |
+
11
|
2099 |
+
8
|
2100 |
+
45
|
2101 |
+
A7
|
2102 |
+
β¨8β©
|
2103 |
+
β¨1/8β©
|
2104 |
+
2
|
2105 |
+
7
|
2106 |
+
8
|
2107 |
+
11
|
2108 |
+
8
|
2109 |
+
46
|
2110 |
+
D7
|
2111 |
+
β¨4β©
|
2112 |
+
β¨1/4β©
|
2113 |
+
7
|
2114 |
+
4
|
2115 |
+
1
|
2116 |
+
3
|
2117 |
+
4
|
2118 |
+
47
|
2119 |
+
A6 β A1
|
2120 |
+
β¨14β©
|
2121 |
+
β¨1/14β©
|
2122 |
+
31
|
2123 |
+
14
|
2124 |
+
1
|
2125 |
+
2
|
2126 |
+
12
|
2127 |
+
7
|
2128 |
+
48
|
2129 |
+
D6 β A1
|
2130 |
+
A1
|
2131 |
+
Aβ
|
2132 |
+
1
|
2133 |
+
2
|
2134 |
+
3
|
2135 |
+
2
|
2136 |
+
1
|
2137 |
+
2
|
2138 |
+
49
|
2139 |
+
E6 β A1
|
2140 |
+
β¨6β©
|
2141 |
+
β¨1/6β©
|
2142 |
+
11
|
2143 |
+
6
|
2144 |
+
1
|
2145 |
+
2
|
2146 |
+
4
|
2147 |
+
3
|
2148 |
+
50
|
2149 |
+
D5 β A2
|
2150 |
+
β¨12β©
|
2151 |
+
β¨1/12β©
|
2152 |
+
23
|
2153 |
+
12
|
2154 |
+
2
|
2155 |
+
3
|
2156 |
+
5
|
2157 |
+
4
|
2158 |
+
51
|
2159 |
+
A5 β A2
|
2160 |
+
A1
|
2161 |
+
Aβ
|
2162 |
+
1 β Z/3Z
|
2163 |
+
13
|
2164 |
+
6
|
2165 |
+
2
|
2166 |
+
3
|
2167 |
+
3
|
2168 |
+
2
|
2169 |
+
52
|
2170 |
+
D5 β Aβ2
|
2171 |
+
1
|
2172 |
+
β¨4β©
|
2173 |
+
β¨1/4β© β Z/2Z
|
2174 |
+
9
|
2175 |
+
4
|
2176 |
+
1
|
2177 |
+
2
|
2178 |
+
7
|
2179 |
+
4
|
2180 |
+
53
|
2181 |
+
A5 β Aβ2
|
2182 |
+
1
|
2183 |
+
β¨6β©
|
2184 |
+
β¨1/6β© β Z/2Z
|
2185 |
+
5
|
2186 |
+
2
|
2187 |
+
1
|
2188 |
+
2
|
2189 |
+
2
|
2190 |
+
54
|
2191 |
+
D4 β A3
|
2192 |
+
β¨4β©
|
2193 |
+
β¨1/4β© β Z/2Z
|
2194 |
+
2
|
2195 |
+
3
|
2196 |
+
4
|
2197 |
+
5
|
2198 |
+
4
|
2199 |
+
55
|
2200 |
+
A4 β A3
|
2201 |
+
β¨20β©
|
2202 |
+
β¨1/20β©
|
2203 |
+
11
|
2204 |
+
5
|
2205 |
+
3
|
2206 |
+
4
|
2207 |
+
29
|
2208 |
+
20
|
2209 |
+
56
|
2210 |
+
A4 β A2 β A1
|
2211 |
+
β¨30β©
|
2212 |
+
β¨1/30β©
|
2213 |
+
71
|
2214 |
+
30
|
2215 |
+
1
|
2216 |
+
2
|
2217 |
+
28
|
2218 |
+
15
|
2219 |
+
57
|
2220 |
+
D4 β Aβ3
|
2221 |
+
1
|
2222 |
+
A1
|
2223 |
+
Aβ
|
2224 |
+
1
|
2225 |
+
5
|
2226 |
+
2
|
2227 |
+
1
|
2228 |
+
2
|
2229 |
+
2
|
2230 |
+
58
|
2231 |
+
Aβ2
|
2232 |
+
3
|
2233 |
+
β A1
|
2234 |
+
A1
|
2235 |
+
Aβ
|
2236 |
+
1 β Z/4Z
|
2237 |
+
5
|
2238 |
+
2
|
2239 |
+
1
|
2240 |
+
2
|
2241 |
+
2
|
2242 |
+
59
|
2243 |
+
A3 β A2 β Aβ2
|
2244 |
+
1
|
2245 |
+
β¨12β©
|
2246 |
+
β¨1/12β© β Z/2Z
|
2247 |
+
8
|
2248 |
+
3
|
2249 |
+
1
|
2250 |
+
2
|
2251 |
+
13
|
2252 |
+
6
|
2253 |
+
60
|
2254 |
+
A3 β Aβ4
|
2255 |
+
1
|
2256 |
+
β¨4β©
|
2257 |
+
β¨1/4β© β Z/2Z
|
2258 |
+
3
|
2259 |
+
1
|
2260 |
+
2
|
2261 |
+
5
|
2262 |
+
2
|
2263 |
+
61
|
2264 |
+
Aβ3
|
2265 |
+
2
|
2266 |
+
β A1
|
2267 |
+
β¨6β©
|
2268 |
+
β¨1/6β© β Z/3Z
|
2269 |
+
5
|
2270 |
+
2
|
2271 |
+
1
|
2272 |
+
2
|
2273 |
+
2
|
2274 |
+
Table 8:
|
2275 |
+
Mordell-Weil lattices of rational elliptic surfaces
|
2276 |
+
with Mordell-Weil rank r β₯ 1.
|
2277 |
+
25
|
2278 |
+
|
2279 |
+
References
|
2280 |
+
[Ber12] P. Bernays. Γber die Darstellung von positiven, ganzen Zahlen durch die primitive, binΓ€ren
|
2281 |
+
quadratischen Formen einer nicht-quadratischen Diskriminante. PhD thesis, GΓΆttingen,
|
2282 |
+
1912.
|
2283 |
+
[BH]
|
2284 |
+
M. Bhargava and J. Hanke. Universal quadratic forms and the 290-Theorem. Preprint at
|
2285 |
+
http://math.stanford.edu/~vakil/files/290-Theorem-preprint.pdf.
|
2286 |
+
[Cos]
|
2287 |
+
R. D. Costa.
|
2288 |
+
Classiο¬cation of ο¬bers of conic bundles on rational elliptic surfaces.
|
2289 |
+
arXiv:2206.03549.
|
2290 |
+
[Elk90]
|
2291 |
+
N. D. Elkies. The Mordell-Weil lattice of a rational elliptic surface. Arbeitstagung Bonn,
|
2292 |
+
1990.
|
2293 |
+
[HW79] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Clarendon
|
2294 |
+
Press, 1979.
|
2295 |
+
[Kur14] Y. Kurumadani. Pencil of cubic curves and rational elliptic surfaces.
|
2296 |
+
Masterβs thesis,
|
2297 |
+
Kyoto University, 2014.
|
2298 |
+
[MP89] R. Miranda and U. Persson. Torsion groups of elliptic surfaces. Compositio Mathematica,
|
2299 |
+
72(3):249β267, 1989.
|
2300 |
+
[Nis96]
|
2301 |
+
K. Nishiyama. The Jacobian ο¬brations on some K3 surfaces and their Mordell-Weil groups.
|
2302 |
+
Japanese Journal of Mathematics, 22(2), 1996.
|
2303 |
+
[OS91]
|
2304 |
+
K. Oguiso and T. Shioda. The Mordell-Weil lattice of a rational elliptic surface. Com-
|
2305 |
+
mentarii Mathematici Universitatis Sancti Pauli, 40, 1991.
|
2306 |
+
[Shi89]
|
2307 |
+
T. Shioda. The Mordell-Weil lattice and Galois representation, I, II, III. Proceedings of
|
2308 |
+
the Japan Academy, 65(7), 1989.
|
2309 |
+
[Shi90]
|
2310 |
+
T. Shioda. On the Mordell-Weil lattices. Commentarii Mathematici Universitatis Sancti
|
2311 |
+
Pauli, 39(7), 1990.
|
2312 |
+
[SS10]
|
2313 |
+
M. Schuett and T. Shioda.
|
2314 |
+
Elliptic surfaces.
|
2315 |
+
Advanced Studies in Pure Mathematics,
|
2316 |
+
60:51β160, 2010.
|
2317 |
+
[SS19]
|
2318 |
+
M. Schuett and T. Shioda. Mordell-Weil Lattices, volume 70 of Ergebnisse der Mathematik
|
2319 |
+
und ihrer Grenzgebiete. Springer, 2019.
|
2320 |
+
26
|
2321 |
+
|
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|
1 |
+
COMPUTING NONSURJECTIVE PRIMES ASSOCIATED TO GALOIS
|
2 |
+
REPRESENTATIONS OF GENUS 2 CURVES
|
3 |
+
BARINDER S. BANWAIT, ARMAND BRUMER, HYUN JONG KIM, ZEV KLAGSBRUN, JACOB MAYLE,
|
4 |
+
PADMAVATHI SRINIVASAN, AND ISABEL VOGT
|
5 |
+
Abstract. For a genus 2 curve C over Q whose Jacobian A admits only trivial geometric en-
|
6 |
+
domorphisms, Serreβs open image theorem for abelian surfaces asserts that there are only ο¬nitely
|
7 |
+
many primes β for which the Galois action on β-torsion points of A is not maximal. Building on
|
8 |
+
work of Dieulefait, we give a practical algorithm to compute this ο¬nite set. The key inputs are
|
9 |
+
Mitchellβs classiο¬cation of maximal subgroups of PSp4(Fβ), sampling of the characteristic polyno-
|
10 |
+
mials of Frobenius, and the KhareβWintenberger modularity theorem. The algorithm has been
|
11 |
+
submitted for integration into Sage, executed on all of the genus 2 curves with trivial endomor-
|
12 |
+
phism ring in the LMFDB, and the results incorporated into the homepage of each such curve on
|
13 |
+
a publicly-accessible branch of the LMFDB.
|
14 |
+
1. Introduction
|
15 |
+
Let C/Q be a smooth, projective, geometrically integral curve (referred to hereafter as a nice
|
16 |
+
curve) of genus 2, and let A be its Jacobian. We assume throughout that A admits no nontrivial
|
17 |
+
geometric endomorphisms; that is, we assume that End(AQ) = Z, and we refer to any abelian
|
18 |
+
variety satisfying this property as typical1. We also say that a nice curve is typical if its Jacobian is
|
19 |
+
typical. Let GQ βΆ= Gal(Q/Q), let β be a prime, and let A[β] βΆ= A(Q)[β] denote the β-torsion points
|
20 |
+
of A(Q). Let
|
21 |
+
ΟA,β βΆ GQ β Aut(A[β])
|
22 |
+
denote the Galois representation on A[β].
|
23 |
+
By ο¬xing a basis for A[β], and observing that A[β]
|
24 |
+
admits a nondegenerate Galois-equivariant alternating bilinear form, namely the Weil pairing, we
|
25 |
+
may identify the codomain of ΟA,β with the general symplectic group GSp4(Fβ).
|
26 |
+
In a letter to VignΒ΄eras [Ser00, Corollaire au ThΒ΄eor`eme 3], Serre proved an open image theorem
|
27 |
+
for typical abelian varieties of dimensions 2 or 6, or of odd dimension, generalizing his celebrated
|
28 |
+
open image theorem for elliptic curves [Ser72]. More precisely, the set of nonsurjective primes β for
|
29 |
+
which the representation ΟA,β is not surjective β i.e., the set of primes β for which ΟA,β(GQ) is
|
30 |
+
contained in a proper subgroup of GSp4(Fβ) β is ο¬nite.
|
31 |
+
In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the
|
32 |
+
conductor of E on this ο¬nite set [Ser81, ThΒ΄eor`eme 22]; this bound has since been made unconditional
|
33 |
+
[Coj05, Kra95]. There are also algorithms to compute the ο¬nite set of nonsurjective primes [Zyw15],
|
34 |
+
and practical implementations in Sage [CL12].
|
35 |
+
Serreβs open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02]
|
36 |
+
who described an algorithm that returns a ο¬nite set of primes containing the set of nonsurjective
|
37 |
+
primes. In a diο¬erent direction Lombardo [Lom16, Theorem 1.3] provided an upper bound on the
|
38 |
+
nonsurjective primes involving the stable Faltings height of A.
|
39 |
+
Date: January 6, 2023.
|
40 |
+
2010 Mathematics Subject Classiο¬cation. 11F80 (primary), 11G10, 11Y16 (secondary).
|
41 |
+
1Abelian varieties with extra endomorphisms deο¬ne a thin set (in the sense of Serre) in Ag and as such are not
|
42 |
+
the typically arising case.
|
43 |
+
1
|
44 |
+
arXiv:2301.02222v1 [math.NT] 5 Jan 2023
|
45 |
+
|
46 |
+
2
|
47 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
48 |
+
In this paper we develop Algorithms 3.1 and 4.1, which together allow for the exact determination
|
49 |
+
of the nonsurjective primes for C, yielding our main result as follows.
|
50 |
+
Theorem 1.1. Let C/Q be a typical genus 2 curve whose Jacobian A has conductor N.
|
51 |
+
(1) Algorithm 3.1 produces a ο¬nite list PossiblyNonsurjectivePrimes(C) that provably contains all
|
52 |
+
nonsurjective primes.
|
53 |
+
(2) For a given bound B > 0, Algorithm 4.1 produces a sublist LikelyNonsurjectivePrimes(C;B)
|
54 |
+
of PossiblyNonsurjectivePrimes(C) that contains all the nonsurjective primes.
|
55 |
+
If B is suο¬-
|
56 |
+
ciently large, then the elements of LikelyNonsurjectivePrimes(C;B) are precisely the nonsurjec-
|
57 |
+
tive primes of A.
|
58 |
+
The two common ingredients in Algorithms 3.1 and 4.1 are Mitchellβs 1914 classiο¬cation of
|
59 |
+
maximal subgroups of PSp4(Fβ) [Mit14] and sampling of characteristic polynomials of Frobenius
|
60 |
+
elements. Indeed, ΟA,β is nonsurjective precisely when its image is contained in one of the proper
|
61 |
+
maximal subgroups of GSp4(Fβ). The (integral) characteristic polynomial of Frobenius at a good
|
62 |
+
prime p is computationally accessible since it is determined by counting points on C over Fpr for
|
63 |
+
small r. The reduction of this polynomial modulo β gives the characteristic polynomial of the action
|
64 |
+
of the Frobenius element on A[β]. By the Chebotarev density theorem, the images of the Frobenius
|
65 |
+
elements for varying primes p equidistribute over the conjugacy classes of ΟA,β(GQ) and hence let
|
66 |
+
us explore the image.
|
67 |
+
Algorithm 3.1 makes use of the fact that if the image of ΟA,β is nonsurjective, then the character-
|
68 |
+
istic polynomials of Frobenius at auxiliary primes p will be constrained modulo β. Using this idea,
|
69 |
+
Dieulefait worked out the constraints imposed by each type of maximal subgroup for ΟA,β(GQ) to
|
70 |
+
be contained in that subgroup. Our Algorithm 3.1 combines Dieulefaitβs conditions, with some
|
71 |
+
modest improvements, to produce a ο¬nite list PossiblyNonsurjectivePrimes(C).
|
72 |
+
Algorithm 4.1 then weeds out the extraneous surjective primes from PossiblyNonsurjectivePrimes(C).
|
73 |
+
Equipped with the prime β, the task here is try to generate enough diο¬erent elements in the image
|
74 |
+
to rule out containment in any proper maximal subgroup. The key input is a purely group-theoretic
|
75 |
+
condition (Proposition 4.2) that guarantees that a subgroup is all of GSp4(Fβ) if it contains par-
|
76 |
+
ticular types of elements. This algorithm is probabilistic and depends on the choice of a parameter
|
77 |
+
B which, if suο¬ciently large, provably establishes nonsurjectivity. The parameter B is a cut-oο¬ for
|
78 |
+
the number of Frobenius elements that we use to sample the conjugacy classes of ΟA,β(GQ).
|
79 |
+
As an illustration of the interplay between theory and practice, analyzing the βworst caseβ run
|
80 |
+
time of each step in Algorithm 3.1 yields a new theoretical bound, conditional on the Generalized
|
81 |
+
Riemann Hypothesis (GRH), on the product of all nonsurjective primes in terms of the conductor.
|
82 |
+
Theorem 1.2. Let C/Q be a typical genus 2 curve with conductor N. Assuming the Generalized
|
83 |
+
Riemann Hypothesis (GRH), we have, for any Ο΅ > 0,
|
84 |
+
β
|
85 |
+
β nonsurjective
|
86 |
+
β βͺ exp(N1/2+Ο΅),
|
87 |
+
where the implied constant is absolute and eο¬ectively computable.
|
88 |
+
While we believe this bound to be far from asymptotically optimal, it is the ο¬rst bound in the
|
89 |
+
literature expressed in terms of the (eο¬ectively computable) conductor.
|
90 |
+
Naturally one wants to ο¬nd the suο¬ciently large value of B in Theorem 1.1(2), which the next
|
91 |
+
result gives, conditional on GRH.
|
92 |
+
Theorem 1.3. Let C/Q be a typical genus 2 curve, B be a positive integer, and q be the largest
|
93 |
+
prime in LikelyNonsurjectivePrimes(C;B). Assuming GRH, the set LikelyNonsurjectivePrimes(C;B)
|
94 |
+
is precisely the set of nonsurjective primes of C, provided that
|
95 |
+
B β₯ (4[(2q11 β 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5)
|
96 |
+
2 .
|
97 |
+
|
98 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
99 |
+
3
|
100 |
+
The proof of Theorem 1.3 involves an explicit Chebotarev bound due to Bach and Sorenson
|
101 |
+
[BS96] that is dependent on GRH. An unconditional version of Theorem 1.3 can be given using an
|
102 |
+
unconditional Chebotarev result (for instance [KW22]), though the bound for B will be exponential
|
103 |
+
in q. In addition, if we assume both GRH and the Artin Holomorphy Conjecture (AHC), then a
|
104 |
+
version of Theorem 1.3 holds with the improved asymptotic bound B β« q11 log2(qNA), but without
|
105 |
+
an explicit constant.
|
106 |
+
Unfortunately, the bound from Theorem 1.3 is prohibitively large to use in practice. By way of
|
107 |
+
illustration, consider the smallest (with respect to conductor) typical genus 2 curve, which has a
|
108 |
+
model
|
109 |
+
y2 + (x3 + 1)y = x2 + x,
|
110 |
+
and label 249.a.249.1 in the L-functions and modular forms database (LMFDB) [LMF22]. The
|
111 |
+
output of Algorithm 3.1 is the set {2,3,5,7,83}. Applying Algorithm 4.1 with B = 100 rules out
|
112 |
+
the prime 83, suggesting that 7 is the largest nonsurjective prime. Subsequently applying Theorem
|
113 |
+
1.3 with q = 7 yields the value B = 3.578 Γ 1023 for which LikelyNonsurjectivePrimes(C;B) coincides
|
114 |
+
with the set of nonsurjective primes associated with C. With this value of B, our implementation of
|
115 |
+
the algorithm was still running after 24 hours, after which we terminated it. Even if the version of
|
116 |
+
Theorem 1.3 that relies on AHC could be made explicit, the value of q11 log2(qNA) in this example
|
117 |
+
is on the order of 1011, which would still be a daunting prospect.
|
118 |
+
To execute the combined algorithm on all typical genus 2 curves in the LMFDB - which at the
|
119 |
+
time of writing constitutes 63,107 curves - we have decided to take a ο¬xed value of B = 1000 in
|
120 |
+
Algorithm 4.1. The combined algorithm then takes about 4 hours on MITβs Lovelace computer,
|
121 |
+
a machine with 2 AMD EPYC 7713 2GHz processors, each with 64 cores, and a total of 2TB of
|
122 |
+
memory. The result of this computation of nonsurjective primes for these curves is available to
|
123 |
+
view on the homepage of each curve in the LMFDB beta:
|
124 |
+
https://beta.lmfdb.org
|
125 |
+
In addition, the combined algorithm has been run on a much larger set of 1,823,592 curves
|
126 |
+
provided to us by Andrew Sutherland. See Section 6 for the results of this computation.
|
127 |
+
Algorithm 4.1 samples the characteristic polynomial of Frobenius Pp(t) for each prime p of
|
128 |
+
good reduction for the curve up to a particular bound and applies Tests 4.4 and 4.5 to Pp(t).
|
129 |
+
Assuming that ΟA,β is surjective, we expect that the outcome of these tests should be independent
|
130 |
+
for suο¬ciently large primes. More precisely,
|
131 |
+
Theorem 1.4. Let C/Q be a typical genus 2 curve with Jacobian A and suppose β is an odd prime
|
132 |
+
such that ΟA,β is surjective. There is an eο¬ective bound B0 such that for any B > B0, if we sample
|
133 |
+
the characteristic polynomials of Frobenius Pp(t) for n primes p β [B,2B] chosen uniformly and
|
134 |
+
independently at random, the probability that none of these pass Tests 4.4 or 4.5 is less than 3β
( 9
|
135 |
+
10)
|
136 |
+
n.
|
137 |
+
Remark 1. In fact, for each prime β satisfying the conditions of Theorem 1.4, there is an explicit
|
138 |
+
constant cβ β€
|
139 |
+
9
|
140 |
+
10 tending to 3
|
141 |
+
4 as β β β which may be computed using Corollary 5.3 such that
|
142 |
+
bound of 3 β
( 9
|
143 |
+
10)
|
144 |
+
n in Theorem 1.4 can be replaced by 3 β
cn
|
145 |
+
β .
|
146 |
+
The combined algorithm to probabilistically determine the nonsurjective primes of a nice genus
|
147 |
+
2 curve over Q has been implemented in Sage [The20], and it will appear in a future release of this
|
148 |
+
software2. Until then, the implementation is available at the following repository:
|
149 |
+
https://github.com/ivogt/abeliansurfaces
|
150 |
+
The README.md ο¬le contains detailed instructions on its use. This repository also contains other
|
151 |
+
scripts in both Sage and Magma [BCP97] useful for verifying some of the results of this work; any
|
152 |
+
ο¬lenames used in the sequel will refer to the above repository.
|
153 |
+
2see https://trac.sagemath.org/ticket/30837 for the ticket tracking this integration.
|
154 |
+
|
155 |
+
4
|
156 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
157 |
+
Outline of this paper. In Section 2, we begin by reviewing the properties of the characteristic
|
158 |
+
polynomial of Frobenius with a view towards computational aspects. We also recall the classiο¬cation
|
159 |
+
of maximal subgroups of GSp4(Fβ). In Section 3, we explain Algorithm 3.1 and establish Theorem
|
160 |
+
1.1(1); that is, for each of the maximal subgroups of GSp4(Fβ) listed in Section 2.4, we generate a
|
161 |
+
list of primes that provably contains all primes β for which the mod β image of Galois is contained
|
162 |
+
in this maximal subgroup. Theorem 1.2 is also proved in this section (Subsection 3.3). In Section 4,
|
163 |
+
we ο¬rst prove a group-theoretic criterion (Proposition 4.2) for a subgroup of GSp4(Fβ) to equal
|
164 |
+
GSp4(Fβ). Then, for each β in the ο¬nite list from Section 3, we ascertain whether the characteristic
|
165 |
+
polynomials of the Frobenius elements sampled satisfy the group-theoretic criterion; Theorem 1.1(2)
|
166 |
+
and Theorem 1.3 also follow from this study. In Section 5 we prove Theorem 1.4 concerning the
|
167 |
+
probability of output error, assuming that Frobenius elements distribute in ΟA,β(GQ) as they would
|
168 |
+
in a randomly chosen element of GSp4(Fβ). Finally, in Section 6, we close with remarks concerning
|
169 |
+
the execution of the algorithm on the large dataset of genus 2 curves mentioned above, and highlight
|
170 |
+
some interesting examples that arose therein.
|
171 |
+
Acknowledgements. This work was started at a workshop held remotely βatβ the Institute for
|
172 |
+
Computational and Experimental Research in Mathematics (ICERM) in Providence, RI, in May
|
173 |
+
2020, and was supported by a grant from the Simons Foundation (546235) for the collaboration
|
174 |
+
βArithmetic Geometry, Number Theory, and Computationβ.
|
175 |
+
It has also been supported by the
|
176 |
+
National Science Foundation under Grant No. DMS-1929284 while the authors were in residence
|
177 |
+
at ICERM during a Collaborate@ICERM project held in May 2022. We are grateful to Noam Elkies
|
178 |
+
for providing interesting examples of genus 2 curves in the literature, Davide Lombardo for helpful
|
179 |
+
discussions related to computing geometric endomorphism rings, and to Andrew Sutherland for
|
180 |
+
providing a dataset of Hecke characteristic polynomials that were used for executing our algorithm
|
181 |
+
on all typical genus 2 curves in the LMFDB, as well as making available the larger dataset of
|
182 |
+
approximately 2 million curves that we ran our algorithm on.
|
183 |
+
2. Preliminaries
|
184 |
+
2.1. Notation. Let A be an abelian variety of dimension g deο¬ned over Q. By conductor we mean
|
185 |
+
the Artin conductor N = NA of A. We write Nsq for the largest integer such that N2
|
186 |
+
sq β£ N.
|
187 |
+
Let β be a prime. We write TβA for the β-adic Tate module of A:
|
188 |
+
TβA β lim
|
189 |
+
βοΏ½
|
190 |
+
n
|
191 |
+
A[βn].
|
192 |
+
This is a free Zβ-module of rank 2g.
|
193 |
+
For each prime p, we write Frobp β Gal(Q/Q) for an absolute Frobenius element associated to p.
|
194 |
+
By a good prime p for an abelian variety A, we mean a prime p for which A has good reduction, or
|
195 |
+
equivalently p β€ NA. If p is a good prime for A, then the trace ap of the action of Frobp on TβA is
|
196 |
+
an integer. See Section 2.2 for a discussion of the characteristic polynomial of Frobenius.
|
197 |
+
By a typical abelian variety A, we mean an abelian variety with geometric endomorphism ring
|
198 |
+
Z. A typical genus 2 curve is a nice curve whose Jacobian is a typical abelian surface.
|
199 |
+
Let V be a 4-dimensional vector space over Fβ endowed with a nondegenerate skew-symmetric
|
200 |
+
bilinear form β¨β
,β
β©. A subspace W β V is called isotropic (for β¨β
,β
β©) if β¨w1,w2β© = 0 for all w1,w2 β W.
|
201 |
+
A subspace W β V is called nondegenerate (for β¨β
,β
β©) if β¨β
,β
β© restricts to a nondegenerate form on
|
202 |
+
W. The general symplectic group of (V,β¨β
,β
β©) is deο¬ned as
|
203 |
+
GSp(V,β¨β
,β
β©) βΆ= {M β GL(V ) βΆ β mult(M) β FΓ
|
204 |
+
β βΆ β¨Mv,Mwβ© = mult(M)β¨v,wβ© β v,w β V }.
|
205 |
+
The map M β¦ mult(M) is a surjective homomorphism from GSp(V,β¨β
,β
β©) to FΓ
|
206 |
+
β called the similitude
|
207 |
+
character; its kernel is the symplectic group, denoted Sp(V,β¨β
,β
β©).
|
208 |
+
Usually the bilinear form is
|
209 |
+
understood from the context, in which case one drops β¨β
,β
β© from the notation; moreover, for our
|
210 |
+
|
211 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
212 |
+
5
|
213 |
+
purposes, we will have ο¬xed a basis for V , one in which the bilinear form is represented by the
|
214 |
+
nonsingular skew-symmetric matrix
|
215 |
+
J βΆ= ( 0
|
216 |
+
I2
|
217 |
+
βI2
|
218 |
+
0 ),
|
219 |
+
where I2 is the 2 Γ 2 identity matrix.
|
220 |
+
By a subquotient W of a Galois module U, we mean a Galois module W that admits a surjection
|
221 |
+
U β² β W from a subrepresentation U β² of U.
|
222 |
+
Since we are chieο¬y concerned with computing the sets LikelyNonsurjectivePrimes(C;B) and
|
223 |
+
PossiblyNonsurjectivePrimes(C) for a ο¬xed curve C, we will henceforth, for ease of notation, drop
|
224 |
+
the C from the notation for these sets.
|
225 |
+
2.2. Integral characteristic polynomial of Frobenius. The theoretical result underlying the
|
226 |
+
whole approach is the following.
|
227 |
+
Theorem 2.1 (Weil, see [ST68, Theorem 3]). Let A be an abelian variety of dimension g deο¬ned
|
228 |
+
over Q and let p be a prime of good reduction for A. Then there exists a monic integral polynomial
|
229 |
+
Pp(t) β Z[t] of degree 2g with constant coeο¬cient pg such that for any β β p, the polynomial Pp(t)
|
230 |
+
modulo β is the characteristic polynomial of the action of Frobp on TβA. Furthermore, every root
|
231 |
+
of Pp(t) has complex absolute value p1/2.
|
232 |
+
The polynomials Pp(t) are computationally accessible by counting points on C over Fpr r = 1,2.
|
233 |
+
See [Poo17, Chapter 7] for more details.
|
234 |
+
In fact, Pp(t) can be accessed via the frobenius_
|
235 |
+
polynomial command in Sage. In particular, we denote the trace of Frobenius by ap. By the
|
236 |
+
Grothendieck-Lefschetz trace formula, if A = JacX, p is a prime of good reduction for X, and
|
237 |
+
Ξ»1,...,Ξ»2g are the roots of Pp(t), then
|
238 |
+
#X(Fpr) = pr + 1 β
|
239 |
+
2g
|
240 |
+
β
|
241 |
+
i=1
|
242 |
+
Ξ»r
|
243 |
+
i .
|
244 |
+
2.3. The Weil pairing and consequences on the characteristic polynomial of Frobenius.
|
245 |
+
The nondegenerate Weil pairing gives an isomorphism (of Galois modules):
|
246 |
+
(1)
|
247 |
+
TβA β (TβA)β¨ βZβ Zβ(1).
|
248 |
+
The Galois character acting on Zβ(1) is the β-adic cyclotomic character, which we denote by cycβ.
|
249 |
+
The integral characteristic polynomial for the action of Frobp on Zβ(1) is simply tβp. The integral
|
250 |
+
characteristic polynomial for the action of Frobp on (TβA)β¨ is the reversed polynomial
|
251 |
+
P β¨
|
252 |
+
p (t) = Pp(1/t) β
t2g/pg
|
253 |
+
whose roots are the inverses of the roots of Pp(t).
|
254 |
+
We now record a few easily veriο¬able consequences of the nondegeneracy of the Weil pairing
|
255 |
+
when dim(A) = 2.
|
256 |
+
Lemma 2.2.
|
257 |
+
(i) The roots of Pp(t) come in pairs that multiply out to p. In particular, Pp(t) has no root with
|
258 |
+
multiplicity 3.
|
259 |
+
(ii) Pp(t) = t4 β apt3 + bpt2 β papt + p2 for some ap,bp β Z.
|
260 |
+
(iii) If the trace of an element of GSp4(Fβ) is 0 mod β, then its characteristic polynomial is re-
|
261 |
+
ducible modulo β. In particular, this applies to Pp(t) when ap β‘ 0 (mod β).
|
262 |
+
(iv) If A[β] is a reducible GQ-module, then Pp(t) is reducible modulo β.
|
263 |
+
Proof. Parts (i) and (ii) are immediate from the fact that the non-degenerate Weil pairing allows
|
264 |
+
us to pair up the four roots of Pp(t) into two pairs that each multiply out to p.
|
265 |
+
|
266 |
+
6
|
267 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
268 |
+
For part (iii), suppose that M β GSp4(Fβ) has tr(M) = 0. Then the characteristic polynomial
|
269 |
+
PM(t) of M is of the form t4 +bt2 +c2. When the discriminant of PM is 0 modulo β, the polynomial
|
270 |
+
PM has repeated roots and is hence reducible. So assume that the discriminant of PM is nonzero
|
271 |
+
modulo β. When β β 2, the result follows from [Car56, Theorem 1]. When β = 2, a direct computation
|
272 |
+
shows that the characteristic polynomial of a trace 0 element of GSp4(F2) is either (t + 1)4 or
|
273 |
+
(t2 + t + 1)2, which are both reducible.
|
274 |
+
Part (iv) is immediate from Theorem 2.1 since Pp(t) mod β by deο¬nition is the characteristic
|
275 |
+
polynomial for the action of Frobp on A[β].
|
276 |
+
β‘
|
277 |
+
2.4. Maximal subgroups of GSp4(Fβ). Mitchell [Mit14] classiο¬ed the maximal subgroups of
|
278 |
+
PSp4(Fβ) in 1914. This can be used to deduce the following classiο¬cation of maximal subgroups of
|
279 |
+
GSp4(Fβ) with surjective similitude character.
|
280 |
+
Lemma 2.3 (Mitchell). Let V be a 4-dimensional Fβ-vector space endowed with a nondegener-
|
281 |
+
ate skew-symmetric bilinear form Ο. Then any proper subgroup G of GSp(V,Ο) with surjective
|
282 |
+
similitude character is contained in one of the following types of maximal subgroups.
|
283 |
+
(1) Reducible maximal subgroups
|
284 |
+
(a) Stabilizer of a 1-dimensional isotropic subspace for Ο.
|
285 |
+
(b) Stabilizer of a 2-dimensional isotropic subspace for Ο.
|
286 |
+
(2) Irreducible subgroups governed by a quadratic character
|
287 |
+
Normalizer Gβ of the group Mβ that preserves each summand in a direct sum decomposition
|
288 |
+
V1 β V2 of V , where V1 and V2 are jointly deο¬ned over Fβ and either:
|
289 |
+
(a) both nondegenerate for Ο; or
|
290 |
+
(b) both isotropic for Ο.
|
291 |
+
Moreover, Mβ is an index 2 subgroup of Gβ.
|
292 |
+
(3) Stabilizer of a twisted cubic
|
293 |
+
GL(W) acting on Sym3 W β V , where W is a 2-dimensional Fβ-vector space.
|
294 |
+
(4) Exceptional subgroups See Table A for explicit generators for the groups described below.
|
295 |
+
(a) When β β‘ Β±3 (mod 8): a group whose image G1920 in PGSp(V,Ο) has order 1920.
|
296 |
+
(b) When β β‘ Β±5 (mod 12) and β β 7: a group whose image G720 in PGSp(V,Ο) has order 720.
|
297 |
+
(c) When β = 7: a group whose image G5040 in PGSp(V,Ο) has order 5040.
|
298 |
+
Remark 2. We have chosen to label the maximal subgroups in the classiο¬cation using invariant
|
299 |
+
subspaces for the symplectic pairing Ο on V , following the more modern account due to Aschbacher
|
300 |
+
(see [Lom16, Section 3.1]; for a more comprehensive treatment see [KL90]). For the convenience of
|
301 |
+
the reader, we record the correspondence between Mitchellβs original labels and ours below.
|
302 |
+
Mitchellβs label
|
303 |
+
Label in Lemma 2.3
|
304 |
+
Group having an invariant point and plane
|
305 |
+
1a
|
306 |
+
Group having an invariant parabolic congruence
|
307 |
+
1b
|
308 |
+
Group having an invariant hyperbolic or elliptic congruence
|
309 |
+
2a
|
310 |
+
Group having an invariant quadric
|
311 |
+
2b
|
312 |
+
Table 1. Dictionary between maximal subgroup labels in [Die02]/[Mit14] and Lemma 2.3
|
313 |
+
Remark 3. The maximal subgroups in (1) are the analogues of the Borel subgroup of GL2(Fβ).
|
314 |
+
The maximal subgroups in (2) when the two subspaces V,V β² in the direct sum decomposition
|
315 |
+
are individually deο¬ned over Fβ are the analogues of normalizers of the split Cartan subgroup of
|
316 |
+
GL2(Fβ). When the two subspaces V,V β² are not individually deο¬ned over Fβ instead, the maximal
|
317 |
+
subgroups in (2) are analogues of the normalizers of the non-split Cartan subgroups of GL2(Fβ).
|
318 |
+
|
319 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
320 |
+
7
|
321 |
+
Remark 4. We brieο¬y explain why the action of GL2(Fβ) on Sym3(F2
|
322 |
+
β) preserves a nondegenerate
|
323 |
+
symplectic form. It suο¬ces to show that the restriction to SL2(Fβ) ο¬xes a vector in β2 Sym3(F2
|
324 |
+
β).
|
325 |
+
This follows by character theory. If W is the standard 2-dimensional representation of SL2, then
|
326 |
+
we have β2(Sym3 W) β Sym4 W β 1 as representations of SL2.
|
327 |
+
Remark 5. One can extract explicit generators of the exceptional maximal subgroups from Mitchellβs
|
328 |
+
original work3. Indeed [Mit14, the proof of Theorem 8, page 390] gives four explicit matrices that
|
329 |
+
generate a G1920 (which is unique up to conjugacy in PGSp4(Fβ)). Mitchellβs description of the
|
330 |
+
other exceptional groups is in terms of certain projective linear transformations called skew perspec-
|
331 |
+
tivities attached to a direct sum decomposition V = V1 β V2 into 2-dimensional subspaces. A skew
|
332 |
+
perspectivity of order n with axes V1 and V2 is the projective linear transformation that scales V1 by
|
333 |
+
a primitive nth root of unity and ο¬xes V2. This proof also gives the axes of the skew perspectivities
|
334 |
+
of order 2 and 3 that generate the remaining exceptional groups [Mit14, pages 390-391]. Table 5
|
335 |
+
lists generators of (one representative of the conjugacy class of) each of the exceptional maximal
|
336 |
+
subgroup extracted from Mitchellβs descriptions. In the ο¬le exceptional.m publicly available
|
337 |
+
with our code, we verify that Magmaβs list of conjugacy classes of maximal subgroups of GSp4(Fβ)
|
338 |
+
agree with those described in Lemma 2.3 for 3 β€ β β€ 47.
|
339 |
+
Remark 6. The classiο¬cation of exceptional maximal subgroups of PSp4(Fβ) is more subtle than
|
340 |
+
that of PGSp4(Fβ), because of the constraint on the similitude character of matrices in PSp4(Fβ).
|
341 |
+
While the similitude character is not well-deο¬ned on PGSp4(Fβ) (multiplication by a scalar c β FΓ
|
342 |
+
β
|
343 |
+
scales the similitude character by c2) it is well-deο¬ned modulo squares. The group PSp4(Fβ) is the
|
344 |
+
kernel of this natural map:
|
345 |
+
1 β PSp4(Fβ) β PGSp4(Fβ)
|
346 |
+
mult
|
347 |
+
οΏ½οΏ½β FΓ
|
348 |
+
β /(FΓ
|
349 |
+
β )2 β {Β±1} β 1.
|
350 |
+
An exceptional subgroup of PGSp4(Fβ) gives rise to an exceptional subgroup of PSp4(Fβ) of either
|
351 |
+
the same size or half the size depending on the image of mult restricted to that subgroup, which
|
352 |
+
in turn depends on the congruence class of β. For this reason, the maximal exceptional subgroups
|
353 |
+
of PSp4(Fβ) in Mitchellβs original classiο¬cation (also recalled in Dieulefait [Die02, Section 2.1]) can
|
354 |
+
have order 1920 or 960 and 720 or 360 depending on the congruence class of β, and 2520 (for
|
355 |
+
β = 7). Such an exceptional subgroup gives rise to a maximal exceptional subgroup of PGSp4(Fβ)
|
356 |
+
only when mult is surjective (i.e., its intersection with PSp4(Fβ) is index 2), which explains the
|
357 |
+
restricted congruence classes of β for which they arise.
|
358 |
+
We now record a lemma that directly follows from the structure of maximal subgroups described
|
359 |
+
above. This lemma will be used in Section 4 to devise a criterion for a subgroup of GSp4(Fβ) to be
|
360 |
+
the entire group. For an element T in GSp4(Fβ), let tr(T), mid(T), mult(T) denote the trace of
|
361 |
+
T, the middle coeο¬cient of the characteristic polynomial of T, and the similitude character applied
|
362 |
+
to T respectively4. For a scalar Ξ», we have
|
363 |
+
tr(Ξ»T) = Ξ»tr(T),
|
364 |
+
mid(Ξ»T) = Ξ»2 mid(T),
|
365 |
+
mult(Ξ»T) = Ξ»2 mult(T).
|
366 |
+
Hence the quantities tr(T)2/mult(T) and mid(T)/mult(T) are well-deο¬ned on PGSp4(Fβ). For
|
367 |
+
β > 2 and β β {720,1920,5040}, deο¬ne
|
368 |
+
(2)
|
369 |
+
Cβ,β βΆ= {( tr(T)2
|
370 |
+
mult(T), mid(T)
|
371 |
+
mult(T)) β£ T β an exceptional subgroup of GSp4(Fβ) of projective order β}
|
372 |
+
Lemma 2.4.
|
373 |
+
(1) In cases 2a and 2b of Lemma 2.3:
|
374 |
+
3Mitchellβs notation for PGSp4(Fβ) is AΞ½(β) and for PSp4(Fβ) is A1(β).
|
375 |
+
4Explicitly, the characteristic polynomial of T is therefore t4 β tr(T)t3 + mid(T)t2 β mult(T) tr(T)t + mult(T)2.
|
376 |
+
|
377 |
+
8
|
378 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
379 |
+
(a) every element in Gβ β Mβ has trace 0, and,
|
380 |
+
(b) the group Mβ stabilizes a non-trivial linear subspace of F
|
381 |
+
4
|
382 |
+
β.
|
383 |
+
(2) Every element that is contained in a maximal subgroup corresponding to the stabilizer of a
|
384 |
+
twisted cubic has a reducible characteristic polynomial.
|
385 |
+
(3) For β β {1920,720}, the set Cβ,β deο¬ned in (2) equals the reduction modulo β of the elements of
|
386 |
+
the set Cβ below.
|
387 |
+
C1920 = {(0,β2),(0,β1),(0,0),(0,1),(0,2),(1,1),(2,1),(2,2),(4,2),(4,3),(8,4),(16,6)}
|
388 |
+
C720 = {(0,1),(0,0),(4,3),(1,1),(16,6),(0,2),(1,0),(3,2),(0,β2)}
|
389 |
+
We also have
|
390 |
+
C7,5040 = {(0,0),(0,1),(0,2),(0,5),(0,6),(1,0),(1,1),(2,6),(3,2),(4,3),(5,3),(6,3)}.
|
391 |
+
Proof.
|
392 |
+
(1) In cases 2a and 2b of Lemma 2.3, since any element of the normalizer Gβ that is not in Mβ
|
393 |
+
switches elements in the two subspaces V1 and V2 (i.e. maps elements in the subspace V1
|
394 |
+
in the decomposition V1 β V2 to elements in V2 and vice-versa), it follows that any element
|
395 |
+
in Gβ β Mβ has trace zero.
|
396 |
+
(2) The conjugacy class of maximal subgroups corresponding to the stabilizer of a twisted cubic
|
397 |
+
comes from the embedding GL2(Fβ)
|
398 |
+
ΞΉοΏ½β GSp4(Fβ) induced by the natural action of GL2(Fβ)
|
399 |
+
on the space of monomials of degree 3 in 2 variables. If M is a matrix in GL2(Fβ) with
|
400 |
+
eigenvalues λ,¡ (possibly repeated), then the eigenvalues of ι(M) are λ3,¡3,λ2¡,λ¡2 and
|
401 |
+
hence the characteristic polynomial of ΞΉ(M) factors as (T 2 β(Ξ»3 +Β΅3)T +Ξ»3Β΅3)(T 2 β(Ξ»2Β΅+
|
402 |
+
λ¡2)T + Ξ»3Β΅3) over Fβ which is reducible over Fβ.
|
403 |
+
(3) This follows from the description of the maximal subgroups given in Table 5. Each case
|
404 |
+
(except G5040 that only occurs for β = 7) depends on a choice of a root of a quadratic
|
405 |
+
polynomial. In the ο¬le exceptional statistics.sage, we generate the corresponding
|
406 |
+
ο¬nite subgroups over the appropriate quadratic number ο¬eld to compute Cβ. It follows that
|
407 |
+
the corresponding values for the subgroup Gβ in GSp4(Fβ) can be obtained by reducing
|
408 |
+
these values modulo β. Since the group G5040 only appears for β = 7, we directly compute
|
409 |
+
the set C7,5040.
|
410 |
+
β‘
|
411 |
+
Remark 7. The condition in Lemma 2.4(3) is the analogue of the condition [Ser72, Proposition 19
|
412 |
+
(iii)] used to rule out exceptional maximal subgroups of GL2(Fβ).
|
413 |
+
We end this subsection by including the following lemma, to further highlight the similarities
|
414 |
+
between the above classiο¬cation of maximal subgroups of GSp4(Fβ) and the more familiar classi-
|
415 |
+
ο¬cation of maximal subgroups of GL2(Fβ). This lemma is not used elsewhere in the article and is
|
416 |
+
thus for expositional purposes only.
|
417 |
+
Lemma 2.5.
|
418 |
+
(1) The subgroup Mβ in the case (2a) when the two nondegenerate subspaces V1 and V2 are indi-
|
419 |
+
vidually deο¬ned over Fβ is isomorphic to
|
420 |
+
{(m1,m2) β GL2(Fβ)2 β£ det(m1) = det(m2)}.
|
421 |
+
In particular, the order of Mβ is β2(β β 1)(β2 β 1)2.
|
422 |
+
(2) The subgroup Mβ in the case (2b) when the two isotropic subspaces V1 and V2 are individually
|
423 |
+
deο¬ned over Fβ is isomorphic to
|
424 |
+
{(m1,m2) β GL2(Fβ)2 β£ mT
|
425 |
+
1 m2 = Ξ»I, for some Ξ» β Fβ
|
426 |
+
β }.
|
427 |
+
In particular, the order of Mβ is β(β β 1)2(β2 β 1).
|
428 |
+
|
429 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
430 |
+
9
|
431 |
+
(3) The subgroup Mβ in the case (2a) when the two nondegenerate subspaces V1 and V2 are not
|
432 |
+
individually deο¬ned over Fβ is isomorphic to
|
433 |
+
{m β GL2(Fβ2) β£ det(m) β Fβ
|
434 |
+
β }.
|
435 |
+
In particular, the order of Mβ is β2(β β 1)(β4 β 1).
|
436 |
+
(4) The subgroup Mβ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi-
|
437 |
+
vidually deο¬ned over Fβ is isomorphic to GU2(Fβ2), i.e.,
|
438 |
+
{m β GL2(Fβ2) β£ mT ΞΉ(m) = Ξ»I, for some Ξ» β Fβ
|
439 |
+
β },
|
440 |
+
where ΞΉ denotes the natural extension of the Galois automorphism of Fβ2/Fβ to GL2(Fβ2). In
|
441 |
+
particular, the order of Mβ is β(β2 β 1)2.
|
442 |
+
Proof. Given a direct sum decomposition V1 β V2 of a vector space V over Fq, we get a natural
|
443 |
+
embedding of Aut(V1) Γ Aut(V2) (β
GL2(Fq)2) into Aut(V ) (β
GL4(Fq)), whose image consists of
|
444 |
+
automorphisms that preserve this direct sum decomposition. We will henceforth refer to elements
|
445 |
+
of Aut(V1) Γ Aut(V2) as elements of Aut(V ) using this embedding. To understand the subgroup
|
446 |
+
Mβ of GSp4(Fq) in cases (1) and (2) where the two subspaces in the direct sum decomposition are
|
447 |
+
individually deο¬ned over Fq, we need to further impose the condition that the automorphisms in
|
448 |
+
the image of the map Aut(V1) Γ Aut(V2) β Aut(V ) preserve the symplectic form Ο on V up to a
|
449 |
+
scalar.
|
450 |
+
In (1), without any loss of generality, the two nondegenerate subspaces V1 and V2 can be chosen
|
451 |
+
to be orthogonal complements under the nondegenerate pairing Ο, and so by Wittβs theorem, in a
|
452 |
+
suitable basis for V1βV2 obtained by concatenating a basis of V1 and a basis of V2, the nondegenerate
|
453 |
+
symplectic pairing Ο has the following block-diagonal shape:
|
454 |
+
B βΆ=
|
455 |
+
β‘β’β’β’β’β’β’β’β£
|
456 |
+
0
|
457 |
+
1
|
458 |
+
β1
|
459 |
+
0
|
460 |
+
0
|
461 |
+
1
|
462 |
+
β1
|
463 |
+
0
|
464 |
+
β€β₯β₯β₯β₯β₯β₯β₯β¦
|
465 |
+
.
|
466 |
+
The condition that an element (m1,m2) β Aut(V1) β Aut(V2) preserves the symplectic pairing
|
467 |
+
up to a similitude factor of Ξ» is the condition (m1,m2)T B(m1,m2) = Ξ»B, which boils down to
|
468 |
+
det(m1) = Ξ» = det(m2).
|
469 |
+
Similarly, in (2), without any loss of generality, by Wittβs theorem, in a suitable basis for V1 βV2
|
470 |
+
obtained by concatenating a basis of the isotropic subspace V1 and a basis of the isotropic subspace
|
471 |
+
V2, the nondegenerate symplectic pairing Ο has the following block-diagonal shape.
|
472 |
+
B βΆ=
|
473 |
+
β‘β’β’β’β’β’β’β’β£
|
474 |
+
0
|
475 |
+
1
|
476 |
+
1
|
477 |
+
0
|
478 |
+
0
|
479 |
+
β1
|
480 |
+
β1
|
481 |
+
0
|
482 |
+
β€β₯β₯β₯β₯β₯β₯β₯β¦
|
483 |
+
.
|
484 |
+
The condition that an element (m1,m2) β Aut(V1) β Aut(V2) preserves the symplectic pairing
|
485 |
+
up to a similitude factor of Ξ» is the condition (m1,m2)T B(m1,m2) = Ξ»B, which again boils down
|
486 |
+
to mT
|
487 |
+
1 m2 = Ξ»I.
|
488 |
+
If we have a subspace W deο¬ned over Fq2 but not deο¬ned over Fq, and we let W denote the
|
489 |
+
conjugate subspace and further assume that W βW gives a direct sum decomposition of V , then we
|
490 |
+
get a natural embedding of Aut(W) (β
GL2(Fq2)) into Aut(V ) (β
GL4(Fq)) whose image consists
|
491 |
+
of automorphisms that commute with the natural involution of V β Fq2 induced by the Galois
|
492 |
+
automorphism of Fq2 over Fq. The proofs of cases (3) and (4) are analogous to the cases (1) and (2)
|
493 |
+
respectively, by using the direct sum decomposition W βW and letting m2 = ΞΉ(m1). The condition
|
494 |
+
that det(m1) = det(m2) in (1) becomes the condition det(m1) = det(m2) = detm1 = det(m1), or
|
495 |
+
|
496 |
+
10
|
497 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
498 |
+
equivalently, that det(m1) β Fq in (3). Similarly, the condition that mT
|
499 |
+
1 m2 = Ξ»I in (2) becomes the
|
500 |
+
condition that mT
|
501 |
+
1 ΞΉ(m1) = Ξ»I in (4).
|
502 |
+
β‘
|
503 |
+
2.5. Image of inertia and (tame) fundamental characters. Dieulefait [Die02] used Mitchellβs
|
504 |
+
work described in the previous subsection to classify the maximal subgroups of GSp4(Fβ) that could
|
505 |
+
occur as the image of ΟA,β . This was achieved via an application of a fundamental result of Serre
|
506 |
+
and Raynaud that strongly constrains the action of inertia at β, and which we now recall.
|
507 |
+
Fix a prime β > 3 that does not divide the conductor N of A. Let Iβ be an inertia subgroup
|
508 |
+
at β. Let ΟnβΆIβ β FΓ
|
509 |
+
βn denote a (tame) fundamental character of level n. The n Galois-conjugate
|
510 |
+
fundamental characters Οn,1,...,Οn,n of level n are given by Οn,i βΆ= Οβi
|
511 |
+
n . Recall that the fundamental
|
512 |
+
character of level 1 is simply the mod β cyclotomic character cycβ, and that the product of all
|
513 |
+
fundamental characters of a given level is the cyclotomic character.
|
514 |
+
Theorem 2.6 (Serre [Ser72], Raynaud [Ray74], cf. [Die02][Theorem 2.1). Let β be a semistable
|
515 |
+
prime for A. Let V /Fβ be an n-dimensional JordanβHΒ¨older factor of the Iβ-module A[β]. Then V
|
516 |
+
admits a 1-dimensional Fβn-vector space structure such that ΟA,ββ£Iβ acts on V via the character
|
517 |
+
Οd1
|
518 |
+
n,1β―Οdn
|
519 |
+
n,n
|
520 |
+
with each di equal to either 0 or 1.
|
521 |
+
On the other hand, the following fundamental result of Grothendieck constrains the action of
|
522 |
+
inertia at semistable primes p β β.
|
523 |
+
Theorem 2.7 (Grothendieck [GRR72, ExposΒ΄e IX, Prop 3.5]). Let A be an abelian variety over a
|
524 |
+
number ο¬eld K. Then A has semistable reduction at p β β if and only if the action of Ip β GK on
|
525 |
+
TβA is unipotent of length 2.
|
526 |
+
Combining these two results allows one ο¬ne control of the determinant of a subquotient of A[β];
|
527 |
+
this will be used in Section 3.
|
528 |
+
Corollary 2.8. Let A/Q be an abelian surface, and let Xβ be a JordanβHΒ¨older factor of the Fβ[GQ]-
|
529 |
+
module A[β] β Fβ. If β is a semistable prime, then
|
530 |
+
detXβ β Ο΅ β
cycx
|
531 |
+
β
|
532 |
+
for some character Ο΅βΆGQ β Fβ that is unramiο¬ed at β and some 0 β€ x β€ dimXβ. Moreover, Ο΅120 = 1.
|
533 |
+
Proof. The ο¬rst part follows immediately from Theorem 2.6.
|
534 |
+
For the fact that Ο΅120 = 1, every
|
535 |
+
abelian surface attains semistable reduction over an extension K/Q with [K βΆ Q] dividing 120 by
|
536 |
+
[LV14a, Theorem 7.2], and so this follows from Theorem 2.7 since there are no nontrivial unramiο¬ed
|
537 |
+
characters of GQ.
|
538 |
+
β‘
|
539 |
+
We can now state Dieulefaitβs classiο¬cation of maximal subgroups of GSp4(Fβ) that can occur
|
540 |
+
as the image ΟA,β(GQ) for a semistable prime β > 7.
|
541 |
+
Proposition 2.9 ([Die02]). Let A be the Jacobian of a genus 2 curve deο¬ned over Q with Weil
|
542 |
+
pairing Ο on A[β]. If β > 7 is a semistable prime, then ΟA,β(GQ) is either all of GSp(A[β],Ο) or it
|
543 |
+
is contained in one of the maximal subgroups of Types (1) or (2) in Lemma 2.3.
|
544 |
+
See also [Lom16, Proposition 3.15] for an expanded exposition of why the image of GQ cannot
|
545 |
+
be contained in maximal subgroup of Type (3) for a semistable prime β > 7.
|
546 |
+
Remark 8. However, if β is a prime of additive reduction, or if β β€ 7, then the image of GQ may also
|
547 |
+
be contained in any of the four types of maximal subgroups described in Lemma 2.3. Nevertheless,
|
548 |
+
by [LV22, Theorem 6.6], for any prime β > 24, we have that the exponent of the projective image is
|
549 |
+
bounded exp(PΟA,β) β₯ (ββ1)/12. Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60,
|
550 |
+
the exceptional maximal subgroups cannot occur as ΟA,β(GQ) for β > 1441.
|
551 |
+
|
552 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
553 |
+
11
|
554 |
+
2.6. A consequence of the Chebotarev density theorem. Let K/Q be a ο¬nite Galois exten-
|
555 |
+
sion with Galois group G = Gal(K/Q) and absolute discriminant dK. Let S β G be a nonempty
|
556 |
+
subset that is closed under conjugation. By the Chebotarev density theorem, we know that
|
557 |
+
(3)
|
558 |
+
lim
|
559 |
+
xββ
|
560 |
+
β£{p β€ x βΆ p is unramiο¬ed in K and Frobp β S}β£
|
561 |
+
β£{p β€ x}β£
|
562 |
+
= β£Sβ£
|
563 |
+
β£Gβ£.
|
564 |
+
Let p be the least prime such that p is unramiο¬ed in K and Frobp β S. There are eο¬ective versions
|
565 |
+
of the Chebotarev density theorem that give bounds on p. The best known unconditional bounds
|
566 |
+
are polynomial in dK [LMO79, AK19, KW22]. Under GRH, the best known bounds are polynomial
|
567 |
+
in log dK. In particular Bach and Sorenson [BS96] showed that under GRH,
|
568 |
+
(4)
|
569 |
+
p β€ (4log dK + 2.5[K βΆ Q] + 5)2.
|
570 |
+
The present goal is to give an eο¬ective version of the Chebotarev density theorem in the context
|
571 |
+
of abelian surfaces. We will use a corollary of (4) that is noted in [MW21] which allows for the
|
572 |
+
avoidance of a prescribed set of primes by taking a quadratic extension of K. We do this because
|
573 |
+
we will take K = Q(A[β]), and p being unramiο¬ed in K is not suο¬cient to imply that p is a prime
|
574 |
+
of good reduction for A. Lastly, we will use that by [Ser81, Proposition 6], if K/Q is ο¬nite Galois,
|
575 |
+
then
|
576 |
+
(5)
|
577 |
+
log dK β€ ([K βΆ Q] β 1)log rad(dK) + [K βΆ Q]log([K βΆ Q]),
|
578 |
+
where radn = βpβ£n p denotes the radical of an integer n.
|
579 |
+
Lemma 2.10. Let A/Q be a typical principally polarized abelian surface with conductor NA. Let q
|
580 |
+
be a prime. Let S β ΟA,q(GQ) be a nonempty subset that is closed under conjugation. Let p be the
|
581 |
+
least prime of good reduction for A such that p β q and ΟA,q(Frobp) β S. Assuming GRH, we have
|
582 |
+
p β€ (4[(2q11 β 1)log rad(2qNA) + 22q11 log(2q)] + 5q11 + 5)
|
583 |
+
2 .
|
584 |
+
Proof. Let K = Q(A[q]). Then K/Q is Galois and
|
585 |
+
[K βΆ Q] β€ β£GSp4(Fq)β£ = q4(q4 β 1)(q2 β 1)(q β 1) β€ q11.
|
586 |
+
As raddK is the product of primes that ramify in Q(A[q]), the criterion of NΒ΄eron-Ogg-Shafarevich
|
587 |
+
for abelian varieties [ST68, Theorem 1] implies that rad(dK) divides rad(qNA). Let ΛK βΆ= K(βm)
|
588 |
+
where m βΆ= rad(2NA). Note that the primes that ramify in ΛK are precisely 2, q, and the primes of
|
589 |
+
bad reduction for A. Thus rad(d Λ
|
590 |
+
K) = rad(2qNA). Moreover [ ΛK βΆ Q] β€ 2q11 and by (5),
|
591 |
+
log(d Λ
|
592 |
+
K) β€ (2q11 β 1)log rad(2qNA) + 22q11 log(2q).
|
593 |
+
Applying [MW21, Corollary 6] to the ο¬eld ΛK, we get that (under GRH) there exists a prime p
|
594 |
+
satisfying the claimed bound, that does not divide m, and for which ΟA,q(Frobp) β S.
|
595 |
+
β‘
|
596 |
+
3. Finding a finite set containing all nonsurjective primes
|
597 |
+
In this section we describe Algorithm 3.1 referenced in Theorem 1.1(1). This algorithm produces
|
598 |
+
a ο¬nite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes β. We also
|
599 |
+
prove Theorem 1.2.
|
600 |
+
Since our goal is to produce a ο¬nite list (from which we will later remove extraneous primes) it
|
601 |
+
is harmless to include the ο¬nitely many bad primes as well as 2,3,5,7. Using Proposition 2.9, it
|
602 |
+
suο¬ces to ο¬nd conditions on β > 7 for which ΟA,β(GQ) could be contained in one of the maximal
|
603 |
+
subgroups of type (1) and (2) in Lemma 2.3. We ο¬rst ο¬nd primes β for which ΟA,β has (geometrically)
|
604 |
+
reducible image (and hence is contained in a maximal subgroup in case (1) of Lemma 2.3 or in a
|
605 |
+
subgroup Mβ in case (2)). To treat the geometrically irreducible cases, we then make use of the
|
606 |
+
observation from Lemma 2.4 1a that every element outside of an index 2 subgroup has trace 0.
|
607 |
+
|
608 |
+
12
|
609 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
610 |
+
Algorithm 3.1. Given a typical genus 2 curve C/Q with conductor N and Jacobian A, compute
|
611 |
+
a ο¬nite list PossiblyNonsurjectivePrimes of primes as follows.
|
612 |
+
(1) Initialize PossiblyNonsurjectivePrimes = [2,3,5,7].
|
613 |
+
(2) Add to PossiblyNonsurjectivePrimes all primes dividing N.
|
614 |
+
(3) Add to PossiblyNonsurjectivePrimes the good primes β for which ΟA,β β Fβ could be reducible via
|
615 |
+
Algorithms 3.3, 3.6, and 3.10.
|
616 |
+
(4) Add to PossiblyNonsurjectivePrimes the good primes β for which ΟA,β βFβ could be irreducible but
|
617 |
+
nonsurjective via Algorithm 3.13.
|
618 |
+
(5) Return PossiblyNonsurjectivePrimes.
|
619 |
+
At a very high-level, each of the subalgorithms of Algorithm 3.1 makes use of a set of auxiliary
|
620 |
+
good primes p. We compute the integral characteristic polynomial of Frobenius Pp(t) and use it to
|
621 |
+
constrain those β β p for which the image could have a particular shape.
|
622 |
+
Remark 9. Even though robust methods to compute the conductor N of a genus 2 curve are not
|
623 |
+
implemented at the time of writing, the odd-part Nodd of N can be computed via genus2red
|
624 |
+
function of PARI and the genus2reduction module of SageMath, both based on an algorithm
|
625 |
+
of Liu [Liu94]. Moreover, [BK94, Theorem 6.2] bounds the 2-exponent of N above by 20 and hence
|
626 |
+
N can be bounded above by 220Nodd. While these algorithms can be run only with the bound
|
627 |
+
220Nodd, it will substantially increase the run-time of the limiting Algorithm 3.10.
|
628 |
+
We now explain each of these steps in detail.
|
629 |
+
3.1. Good primes that are not geometrically irreducible. In this section we describe the
|
630 |
+
conditions that β must satisfy for the base-extension A[β] βΆ= A[β] βFβ Fβ to be reducible. In this
|
631 |
+
case, the representation A[β] is an extension
|
632 |
+
(6)
|
633 |
+
0 β Xβ β A[β] β Yβ β 0
|
634 |
+
of a (quotient) representation Yβ by a (sub) representation Xβ. Recall that Nsq denotes the largest
|
635 |
+
square divisor of N.
|
636 |
+
Lemma 3.2. Let β be a prime of good reduction for A and suppose that A[β] sits in sequence (6).
|
637 |
+
Let p β β be a good prime for A and let f denote the order of p in (Z/NsqZ)Γ. Then there exists
|
638 |
+
0 β€ x β€ dimXβ and 0 β€ y β€ dimYβ such that Frobgcd(f,120)
|
639 |
+
p
|
640 |
+
acts on detXβ by pgcd(f,120)x, respectively
|
641 |
+
on detYβ by pgcd(f,120)y.
|
642 |
+
Proof. Since β is a good prime and Xβ is composed of JordanβHΒ¨older factors of A[β], Corollary 2.8
|
643 |
+
constrains its determinant. We have detXβ = Ο΅cycx
|
644 |
+
β for some character Ο΅βΆGQ β Fβ unramiο¬ed at β,
|
645 |
+
and 0 β€ x β€ dimXβ, and Ο΅120 = 1. Hence Frob120
|
646 |
+
p
|
647 |
+
acts on detXβ by cycβ(Frobp)120x = p120x.
|
648 |
+
In fact, we can do slightly better. Since detA[β] β cyc2
|
649 |
+
β, we have detYβ β Ο΅β1 cyc2βx
|
650 |
+
β
|
651 |
+
. Since the
|
652 |
+
conductor is multiplicative in extensions, we conclude that cond(Ο΅)2 β£ N. By class ο¬eld theory,
|
653 |
+
the character Ο΅ factors through (Z/cond(Ο΅)Z)Γ, and hence through (Z/NsqZ)Γ, sending Frobp
|
654 |
+
to p (mod Nsq). Since pf β‘ 1 (mod Nsq), we have that Ο΅(Frobp)gcd(f,120) = 1, and we see that
|
655 |
+
Frobgcd(f,120)
|
656 |
+
p
|
657 |
+
acts on detXβ by pgcd(f,120)x. Exchanging the roles of Xβ and Yβ, we deduce the
|
658 |
+
analogous statement for Yβ.
|
659 |
+
β‘
|
660 |
+
This is often enough information to ο¬nd all β for which A[β] has a nontrivial subquotient. Namely,
|
661 |
+
by Theorem 2.1, every root of Pp(t) has complex absolute value p1/2. Thus the gcd(f,120)-th power
|
662 |
+
of each root has complex absolute value pgcd(f,120)/2, and hence is never integrally equal to 1 or
|
663 |
+
pgcd(f,120). Since Lemma 3.2 guarantees that this equality must hold modulo β for any good prime
|
664 |
+
β for which A[β] is reducible with a 1-dimensional subquotient, we always get a nontrivial condition
|
665 |
+
on β. Some care must be taken to rule out β for which A[β] only has 2-dimensional subquotient(s).
|
666 |
+
|
667 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
668 |
+
13
|
669 |
+
3.1.1. Odd-dimensional subquotient. Let p be a good prime.
|
670 |
+
Given a polynomial P(t) and an
|
671 |
+
integer f, write P (f)(t) for the polynomial whose roots are the fth powers of roots of P(t).
|
672 |
+
Universal formulas for such polynomials in terms of the coeο¬cients of P(t) are easy to compute,
|
673 |
+
and are implemented in our code in the case where P is a degree 4 polynomial whose roots multiply
|
674 |
+
in pairs to pΞ±, and f β£ 120.
|
675 |
+
Algorithm 3.3. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of
|
676 |
+
p in (Z/NsqZ)Γ and write fβ² = gcd(f,120). Compute an integer Modd as follows.
|
677 |
+
(1) Choose a nonempty ο¬nite set T of auxiliary good primes p β€ N.
|
678 |
+
(2) For each p, compute
|
679 |
+
Rp βΆ= P (fβ²)
|
680 |
+
p
|
681 |
+
(1).
|
682 |
+
(3) Let Modd = gcdpβT (pRp) over all auxiliary primes.
|
683 |
+
Return the list of prime divisors β of Modd.
|
684 |
+
Proposition 3.4. Any good prime β for which A[β] has an odd-dimensional subrepresentation is
|
685 |
+
returned by Algorithm 3.3.
|
686 |
+
Proof. Since A[β] is 4-dimensional and has an odd-dimensional subrepresentation, it has a 1-
|
687 |
+
dimensional subquotient. For any p β T , Lemma 3.2 shows that Frobfβ²
|
688 |
+
p acts on detXβ by either pfβ²
|
689 |
+
or by 1. Thus, the action of Frobfβ²
|
690 |
+
p on A[β] has an eigenvalue that is congruent to pfβ² or 1 modulo
|
691 |
+
β, and so P (fβ²)
|
692 |
+
p
|
693 |
+
(t) has a root that is congruent to 1 or pfβ² modulo β. Since the roots of P (fβ²)(t)
|
694 |
+
multiply in pairs to pfβ², we have P (fβ²)
|
695 |
+
p
|
696 |
+
(pfβ²) = p2fβ²P (fβ²)
|
697 |
+
p
|
698 |
+
(1). Hence β divides p β
P (fβ²)
|
699 |
+
p
|
700 |
+
(1) = pRp.
|
701 |
+
β‘
|
702 |
+
Using Theorem 2.1, we can give a theoretical bound on the βworst caseβ of this step of the
|
703 |
+
algorithm using only one auxiliary prime p. Of course, taking the greatest common divisor over
|
704 |
+
multiple auxiliary primes will likely remove extraneous factors, and in practice this step of the
|
705 |
+
algorithm runs substantially faster than other steps.
|
706 |
+
Proposition 3.5. Algorithm 3.3 terminates. More precisely, if p is any good prime for A, then
|
707 |
+
0 β β£Moddβ£ βͺ p240
|
708 |
+
where the implied constant is absolute.
|
709 |
+
Proof. This follows from the fact that the coeο¬cient of ti in P (fβ²)
|
710 |
+
p
|
711 |
+
(t) has magnitude on the order
|
712 |
+
of p(2βi)fβ² and fβ² β€ 120.
|
713 |
+
β‘
|
714 |
+
3.1.2. Two-dimensional subquotients. We now assume that A[β] is reducible, but does not have
|
715 |
+
any odd-dimensional subquotients.
|
716 |
+
In particular, it has an irreducible subrepresentation Xβ of
|
717 |
+
dimension 2, with irreducible quotient Yβ of dimension 2. If A[β] is reducible but indecomposable,
|
718 |
+
then Xβ is the unique subrepresentation of A[β] and Y β¨
|
719 |
+
β β cycβ is the unique subrepresentation
|
720 |
+
of A[β]
|
721 |
+
β¨ β cycβ. The isomorphism TβA β (TβA)β¨ β cycβ from (1) yields an isomorphism A[β] β
|
722 |
+
(A[β])β¨ β cycβ and hence Xβ β Y β¨
|
723 |
+
β β cycβ. Otherwise, A[β] β Xβ β Yβ and so the nondegeneracy of
|
724 |
+
the Weil pairing gives
|
725 |
+
Xβ β Yβ β (Xβ¨
|
726 |
+
β β cycβ) β (Y β¨
|
727 |
+
β β cycβ).
|
728 |
+
Therefore either:
|
729 |
+
(a) Xβ β Y β¨
|
730 |
+
β β cycβ and Yβ β Xβ¨
|
731 |
+
β β cycβ, or
|
732 |
+
(b) Xβ β Xβ¨
|
733 |
+
β β cycβ and Yβ β Y β¨
|
734 |
+
β β cycβ and A[β] β Xβ β Yβ.
|
735 |
+
We call the ο¬rst case related 2-dimensional subquotients and the second case self-dual 2-dimensional
|
736 |
+
subrepresentations.
|
737 |
+
We will see that the ideas of Lemma 3.2 easily extend to treat the related
|
738 |
+
subquotient case; we will use the validity of Serreβs conjecture to treat the self-dual case. In the
|
739 |
+
|
740 |
+
14
|
741 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
742 |
+
case that A[β] is decomposable, the above two cases correspond respectively to the index 2 subgroup
|
743 |
+
Mβ in cases (2a) (the isotropic case) and (2b) (the nondegenerate case) of Lemma 2.3.
|
744 |
+
3.1.3. Related two-dimensional subquotients. Let p be a good prime. Let Pp(t) βΆ= t4βat3+bt2βpat+p2
|
745 |
+
be the characteristic polynomial of Frobp acting on A[β]. Suppose that Ξ± and Ξ² are the eigenvalues
|
746 |
+
of Frobp acting on the subrepresentation Xβ. Then, since Xβ β Y β¨
|
747 |
+
β β cycβ, the eigenvalues of the
|
748 |
+
action of Frobp on Yβ are p/Ξ± and p/Ξ². The action of Frobp on detXβ is therefore by a product of
|
749 |
+
two of the roots of Pp(t) that do not multiply to p. Note that there are four such pairs of roots of
|
750 |
+
Pp(t) that do not multiply to p. Let Qp(t) be the quartic polynomial whose roots are the products
|
751 |
+
of pairs of roots of Pp(t) that do not multiply to p. By design, the roots of Qp(t) have complex
|
752 |
+
absolute value p, but are not equal to p. (It is elementary to work out that
|
753 |
+
Qp(t) = t4 β (b β 2p)t3 + p(a2 β 2b + 2p)t2 β p2(b β 2p)t + p4
|
754 |
+
and is a quartic whose roots multiply in pairs to p2.)
|
755 |
+
Algorithm 3.6. Given a typical genus 2 Jacobian A/Q of conductor N, let f denote the order of
|
756 |
+
p in (Z/NsqZ)Γ and write fβ² = gcd(f,120). Compute an integer Mrelated as follows.
|
757 |
+
(1) Choose a ο¬nite set T of auxiliary good primes p β€ N;
|
758 |
+
(2) For each p, compute the product
|
759 |
+
Rp βΆ= Q(fβ²)
|
760 |
+
p
|
761 |
+
(1)Q(fβ²)
|
762 |
+
p
|
763 |
+
(pfβ²)
|
764 |
+
(3) Let Mrelated = gcdpβT (pRp).
|
765 |
+
Return the list of prime divisors β of Mrelated.
|
766 |
+
Proposition 3.7. Any good prime β for which A[β] has related two-dimensional subquotients is
|
767 |
+
returned by Algorithm 3.6.
|
768 |
+
Proof. Proceed similarly as in the proof of Proposition 3.4 β in particular, β divides Q(fβ²)
|
769 |
+
p
|
770 |
+
(1),
|
771 |
+
Q(fβ²)
|
772 |
+
p
|
773 |
+
(pfβ²), or Q(fβ²)
|
774 |
+
p
|
775 |
+
(p2fβ²) and hence β divides pRp since Q(fβ²)
|
776 |
+
p
|
777 |
+
(p2fβ²) = p4fβ²Q(fβ²)
|
778 |
+
p
|
779 |
+
(1).
|
780 |
+
β‘
|
781 |
+
A theoretical βworst caseβ analysis yields the following.
|
782 |
+
Proposition 3.8. Algorithm 3.6 terminates. More precisely, if q is the smallest surjective prime
|
783 |
+
for A, then a good prime p for which Rp is nonzero is bounded by a function of q. Assuming GRH,
|
784 |
+
p βͺ q22 log2(qN),
|
785 |
+
where the implied constants are absolute and eο¬ectively computable. Moreover, for such a prime p,
|
786 |
+
β£Mrelatedβ£ βͺ p961 βͺ q21142 log1922(qN),
|
787 |
+
where the implied constants are absolute.
|
788 |
+
Proof. By Serreβs open image theorem for genus 2 curves, such a prime q exists, and by Lemma
|
789 |
+
2.10, the prime p can be chosen such that Rp is nonzero modulo q. Finally,
|
790 |
+
Mrelated β€ pRp = pQ(fβ²)(1)Q(fβ²)(pfβ²) βͺ p8fβ²+1 βͺ p961,
|
791 |
+
since the coeο¬cient of ti in Q(fβ²)(t) has magnitude on the order of p(4βi)fβ² and fβ² β€ 120.
|
792 |
+
β‘
|
793 |
+
|
794 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
795 |
+
15
|
796 |
+
3.1.4. Self-dual two-dimensional subrepresentations. In this case, both subrepresentations Xβ and
|
797 |
+
Yβ are absolutely irreducible 2-dimensional Galois representations with determinant the cyclotomic
|
798 |
+
character cycβ. It follows that the representations are odd (i.e., the determinant of complex con-
|
799 |
+
jugation is β1.) Therefore, by the KhareβWintenberger theorem (formerly Serreβs conjecture on
|
800 |
+
the modularity of mod-β Galois representations) [Kha06, KW09a, KW09b], both Xβ and Yβ are
|
801 |
+
modular; that is, for i = 1,2, there exist newforms fi β Snew
|
802 |
+
ki (Ξ1(Ni),Ο΅i) such that
|
803 |
+
Xβ β
Οf1,β and Yβ β
Οf2,β.
|
804 |
+
Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 β£ N.
|
805 |
+
Lemma 3.9. Both f1 and f2 have weight two and trivial Nebentypus; that is, k1 = k2 = 2, and
|
806 |
+
Ο΅1 = Ο΅2 = 1.
|
807 |
+
Proof. From Theorem 2.6, we have that Xββ£Iβ and Yββ£Iβ must each be conjugate to either of the
|
808 |
+
following subgroups of GL2(Fβ):
|
809 |
+
(1
|
810 |
+
β
|
811 |
+
0
|
812 |
+
cycβ
|
813 |
+
) or (Ο2
|
814 |
+
0
|
815 |
+
0
|
816 |
+
Οβ
|
817 |
+
2
|
818 |
+
).
|
819 |
+
The assertion of weight 2 now follows from [Ser87, Proposition 3]. (Alternatively, one may use
|
820 |
+
Proposition 4 of loc. cit., observing that XοΏ½οΏ½οΏ½ and Yβ are ο¬nite and ο¬at as group schemes over Zβ
|
821 |
+
because β is a prime of good reduction.)
|
822 |
+
From Section 1 of loc. cit., the Nebentypus Ο΅i of fi satisο¬es, for all p β€ βN,
|
823 |
+
detXβ(Frobp) = p β
Ο΅i(p),
|
824 |
+
where this equality is viewed inside F
|
825 |
+
Γ
|
826 |
+
β . The triviality follows.
|
827 |
+
β‘
|
828 |
+
We therefore have newforms fi β Snew
|
829 |
+
2
|
830 |
+
(Ξ0(Ni)) such that
|
831 |
+
(7)
|
832 |
+
A[β] β Οf1,β β Οf2,β.
|
833 |
+
We may assume without loss of generality that N1 β€
|
834 |
+
β
|
835 |
+
N. Let p β€ N be an auxiliary prime. We
|
836 |
+
obtain from equation (7) that the integral characteristic polynomial of Frobenius factors:
|
837 |
+
Pp(t) β‘ (t2 β ap(f1)t + p)(t2 β ap(f2)t + p)
|
838 |
+
mod β;
|
839 |
+
here we use the standard property that, for f a normalised eigenform with trivial Nebentypus,
|
840 |
+
Οf,β(Frobp) satisο¬es the polynomial equation t2 β ap(f)t + p for p β β. In particular, we have
|
841 |
+
Res(Pp(t),t2 β ap(f1)t + p) β‘ 0
|
842 |
+
mod β.
|
843 |
+
This serves as the basis of the algorithm to ο¬nd all primes β in this case.
|
844 |
+
Algorithm 3.10. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer
|
845 |
+
Mself-dual as follows.
|
846 |
+
(1) Compute the set S of divisors d of N with d β€
|
847 |
+
β
|
848 |
+
N.
|
849 |
+
(2) For each d β S:
|
850 |
+
(a) compute the Hecke L-polynomial
|
851 |
+
Qd(t) βΆ= β
|
852 |
+
f
|
853 |
+
(t2 β ap(f)t + p),
|
854 |
+
where the product is taken over the ο¬nitely many newforms in Snew
|
855 |
+
2
|
856 |
+
(Ξ0(d));
|
857 |
+
(b) choose a ο¬nite set T of auxiliary primes p β€ N;
|
858 |
+
(c) for each auxiliary prime p, compute the resultant
|
859 |
+
Rp(d) βΆ= Res(Pp(t),Qd(t));
|
860 |
+
|
861 |
+
16
|
862 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
863 |
+
(d) Take the greatest common divisor
|
864 |
+
M(d) βΆ= gcd
|
865 |
+
pβT
|
866 |
+
(pRp(d)).
|
867 |
+
(3) Let Mself-dual βΆ= βdβS M(d).
|
868 |
+
Return the list of prime divisors β of Mself-dual.
|
869 |
+
Proposition 3.11. Any good prime β for which A[β] has self-dual two-dimensional subrepresenta-
|
870 |
+
tions is returned by Algorithm 3.10.
|
871 |
+
Proof. If β is in T for any d β S, then β is in the output because Mself-dual is a multiple of M(d)
|
872 |
+
which in turn is a multiple of any element of T . Otherwise, as explained before Algorithm 3.10,
|
873 |
+
there is some N1 β S and some newform f1 β Snew
|
874 |
+
2
|
875 |
+
(Ξ0(N1)) such that Res(Pp(t),t2 β apf1t + p) β‘ 0
|
876 |
+
(mod β) for every p β T . In particular, Rp(N1) β‘ 0 (mod β), so β divides M(N1) and Mself-dual.
|
877 |
+
β‘
|
878 |
+
We can again do a βworst caseβ theoretical analysis of this algorithm to conclude the following.
|
879 |
+
As this indicates, this is by far the limiting step of the algorithm.
|
880 |
+
Proposition 3.12. Algorithm 3.10 terminates. More precisely, if q is the smallest surjective prime
|
881 |
+
for A, then a good prime p for which Rp(d) is nonzero is bounded by a function of q. Assuming GRH,
|
882 |
+
p βͺ q22 log2(qN), where the implied constant is absolute and eο¬ectively computable. Moreover, for
|
883 |
+
such a prime p, we have
|
884 |
+
β£Rp(d)β£ βͺ (2p1/2)8 dim Snew
|
885 |
+
2
|
886 |
+
(Ξ0(d)) βͺ (4p)(d+1)/3,
|
887 |
+
and so all together
|
888 |
+
β£Mself-dualβ£ βͺ (4q)N1/2+Ο΅,
|
889 |
+
where the implied constants are absolute.
|
890 |
+
Proof. As in Proposition 3.8, we use Serreβs open image theorem and the Eο¬ective Chebotarev
|
891 |
+
Theorem. If Rp(d) is zero integrally, then in particular Rp(d) β‘ 0 (mod q) and Pp(t) is reducible
|
892 |
+
modulo q. Since GSp4(Fq) contains elements that do not have reducible characteristic polynomial,
|
893 |
+
Lemma 2.10 implies that such elements are the image of Frobp for p bounded as claimed.
|
894 |
+
The resultant Rp(d) is the product of the pairwise diο¬erences of the roots of Pp(t) and Qd(t),
|
895 |
+
which all have complex absolute value p1/2. Hence the pairwise diο¬erences have absolute value
|
896 |
+
at most 2p1/2.
|
897 |
+
Moreover dimSnew
|
898 |
+
2
|
899 |
+
(Ξ0(d)) β€ (d + 1)/12 by [Mar05, Theorem 2].
|
900 |
+
Since there
|
901 |
+
are 8dimSnew
|
902 |
+
2
|
903 |
+
(Ξ0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows. Since
|
904 |
+
Mself-dual = β dβ£N
|
905 |
+
dβ€
|
906 |
+
β
|
907 |
+
N
|
908 |
+
pRp(d), it suο¬ces to bound
|
909 |
+
β
|
910 |
+
dβ£N
|
911 |
+
dβ€
|
912 |
+
β
|
913 |
+
N
|
914 |
+
d + 4
|
915 |
+
3
|
916 |
+
β€
|
917 |
+
β
|
918 |
+
dβ£N
|
919 |
+
dβ€
|
920 |
+
β
|
921 |
+
N
|
922 |
+
β
|
923 |
+
N + 4
|
924 |
+
3
|
925 |
+
β€ Ο0(N)
|
926 |
+
β
|
927 |
+
N + 4
|
928 |
+
3
|
929 |
+
.
|
930 |
+
Since Ο0(N) βͺ NΟ΅ by [Apo76, (31) on page 296], we obtain the claimed bound.
|
931 |
+
β‘
|
932 |
+
Remark 10. The polynomial Qd(t) in step (2) of Algorithm 3.10 is closely related to the charac-
|
933 |
+
teristic polynomial Hd(t) of the Hecke operator Tp acting on the space S2(Ξ0(d)), which may be
|
934 |
+
computed via modular symbols computations. One may recover Qd(t) from Hd(t) by ο¬rst homoge-
|
935 |
+
nizing H with an auxiliary variable z (say) to obtain Hd(t,z), and setting t = 1+pz2 (an observation
|
936 |
+
we made in conjunction with Joseph Wetherell). In our computation of nonsurjective primes for
|
937 |
+
the database of genus 2 curves with conductor at most 220 (including those in the LMFDB), we
|
938 |
+
only needed to use polynomials Qd(t) for level up to 210 (since step (1) of the Algorithm has a
|
939 |
+
β
|
940 |
+
N term). We are grateful to Andrew Sutherland for providing us with a precomputed dataset
|
941 |
+
for these levels resulting from the creation of an extensive database of modular forms going well
|
942 |
+
beyond what was previously available [BBB+21].
|
943 |
+
|
944 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
945 |
+
17
|
946 |
+
Remark 11. Our Sage implementation uses two auxiliary primes in Step 2(b) of the above algorithm.
|
947 |
+
Increasing the number of such primes yields smaller supersets at the expense of longer runtime.
|
948 |
+
3.2. Good primes that are geometrically irreducible. Let Ο be any quadratic Dirichlet char-
|
949 |
+
acter ΟβΆ(Z/NZ)Γ β {Β±1}. Our goal in this subsection is to ο¬nd all good primes β governed by Ο,
|
950 |
+
by which we mean that
|
951 |
+
tr(ΟA,β(Frobp)) β‘ ap β‘ 0
|
952 |
+
mod β
|
953 |
+
whenever Ο(p) = β1.
|
954 |
+
We will consider the set of all quadratic Dirichlet character ΟβΆ(Z/NZ)Γ β {Β±1}. Using the struc-
|
955 |
+
ture theorem for ο¬nite abelian groups and the fact that Ο factors through (Z/NZ)Γ/((Z/NZ)Γ)2,
|
956 |
+
this set has the structure of an F2-vector space of dimension
|
957 |
+
d(N) βΆ= Ο(N) +
|
958 |
+
β§βͺβͺβͺβͺβͺβͺβͺβ¨βͺβͺβͺβͺβͺβͺβͺβ©
|
959 |
+
0
|
960 |
+
βΆ v2(N) = 0
|
961 |
+
β1
|
962 |
+
βΆ v2(N) = 1
|
963 |
+
0
|
964 |
+
βΆ v2(N) = 2
|
965 |
+
1
|
966 |
+
βΆ v2(N) β₯ 3,
|
967 |
+
where Ο(m) denotes the number of prime factors of m and v2(m) is the 2-adic valuation of m. In
|
968 |
+
particular, d(N) β€ Ο(N) + 1.
|
969 |
+
Algorithm 3.13. Given a typical genus 2 Jacobian A/Q of conductor N, compute an integer Mquad
|
970 |
+
as follows.
|
971 |
+
(1) Compute the set S of quadratic Dirichlet characters ΟβΆ(Z/NZ)Γ β {Β±1}.
|
972 |
+
(2) For each Ο β S:
|
973 |
+
(a) Choose a nonempty ο¬nite set T of βauxiliaryβ primes p β€ N for which ap β 0 and Ο(p) = β1.
|
974 |
+
(b) Take the greatest common divisor
|
975 |
+
MΟ βΆ= gcd
|
976 |
+
pβT
|
977 |
+
(pap),
|
978 |
+
over all auxiliary primes p.
|
979 |
+
(3) Let Mquad βΆ= βΟβS MΟ.
|
980 |
+
Return the list of prime divisors β of Mquad.
|
981 |
+
Proposition 3.14. Any good prime β for which A[β] is governed by a quadratic character is
|
982 |
+
returned by Algorithm 3.13.
|
983 |
+
Proof. Suppose that A[β] is governed by the quadratic character ΟβΆ(Z/NZ)Γ β {Β±1}. Then for
|
984 |
+
every good prime p β β for which Ο(p) = β1, the prime β must divide the integral trace of Frobenius
|
985 |
+
ap. Hence β divides MΟ and Mquad.
|
986 |
+
β‘
|
987 |
+
Proposition 3.15. Algorithm 3.13 terminates. More precisely, if q is the smallest surjective prime
|
988 |
+
for A, then a good prime p for which Ο(p) = β1 and ap is nonzero is bounded by a function of q.
|
989 |
+
Assuming GRH, p βͺ 22d(N)q22 log2(qN), where the implied constant is absolute and eο¬ectively
|
990 |
+
computable. Moreover, we have
|
991 |
+
β
|
992 |
+
ΟβS
|
993 |
+
β
|
994 |
+
β governed
|
995 |
+
by Ο
|
996 |
+
β βͺ (23d(N)q33 log3(qN))2β21βd(N) βͺ 26Ο(N)q66 log6(qN),
|
997 |
+
where the implied constant is absolute and eο¬ectively computable.
|
998 |
+
Proof. We imitate the proof of [LV14b, Lemma 21] in our setting. Let V be the d-dimensional
|
999 |
+
F2-vector space of quadratic Dirichlet characters of modulus N (equivalently, quadratic Galois
|
1000 |
+
characters unramiο¬ed outside of N). Let ΟV βΆGK β V β¨ denote the representation sending Frobp to
|
1001 |
+
the linear functional Ο β¦ Ο(p). Since the character for PGSp4(Fq)/PSp4(Fq) is the abelianization
|
1002 |
+
|
1003 |
+
18
|
1004 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1005 |
+
of PΟA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any Ξ± β V β¨, there
|
1006 |
+
exists an XΞ± β GSp4(Fq) with tr(XΞ±) β 0 such that (Ξ±,XΞ±) is in the image of ΟV Γ ΟA,β.
|
1007 |
+
Apply the eο¬ective Chebotarev density theorem to the Galois extension corresponding to ΟV Γ
|
1008 |
+
ΟA,q. This has degree at most 2d(N)β£GSp4(Fq)β£ and is unramiο¬ed outside of qN. Therefore, assum-
|
1009 |
+
ing GRH and combining (4) and (5), there exists a prime
|
1010 |
+
pΞ± βͺ 22d(N)q22 log2(qN)
|
1011 |
+
for which (Ξ±,XΞ±) = (ΟV (FrobpΞ±),ΟA,q(FrobpΞ±)). Let Ο be a character not in the kernel of Ξ±. Any
|
1012 |
+
exceptional prime β governed by Ο must divide pΞ±apΞ±, which is nonzero because it is nonzero modulo
|
1013 |
+
q. This proves that the algorithm terminates, since every Ο is not in the kernel of precisely half
|
1014 |
+
of all Ξ± β V β¨. We now bound the size of the product of all β governed by a character in S. If β is
|
1015 |
+
governed by Ο, then β divides the quantity
|
1016 |
+
pβ£apβ£ β€ p3/2 βͺ 23d(N)q33 log3(qN).
|
1017 |
+
Taking the product over all nonzero Ξ± in V (of which there are 2d(N) β 1), each β will show up half
|
1018 |
+
the time, so we obtain:
|
1019 |
+
β
|
1020 |
+
βββ
|
1021 |
+
β
|
1022 |
+
β
|
1023 |
+
β governed
|
1024 |
+
by Ο β S
|
1025 |
+
β
|
1026 |
+
β
|
1027 |
+
βββ
|
1028 |
+
β
|
1029 |
+
2d(N)β1
|
1030 |
+
βͺ (23d(N)q33 log3(qN))
|
1031 |
+
2d(N)β1
|
1032 |
+
,
|
1033 |
+
which implies the result by taking the (2d(N)β1)th root of both sides.
|
1034 |
+
β‘
|
1035 |
+
Putting all of these pieces together, we obtain the following.
|
1036 |
+
Proof of Theorem 1.1(1). If ΟA,β is nonsurjective, β > 7, and β β€ N, then Proposition 2.9 implies
|
1037 |
+
that ΟA,β(GQ) must be in one of the maximal subgroups of Type (1) or (2) listed in Lemma
|
1038 |
+
2.3. If it is contained in one of the reducible subgroups, i.e. the subgroups of Type (1), then
|
1039 |
+
ΟA,β(GQ) (and, hence, ΟA,β(GQ) β Fβ) is reducible, and so β is added to PossiblyNonsurjectivePrimes
|
1040 |
+
in Step (3) by Propositions 3.4, 3.7, and 3.11.
|
1041 |
+
If ΟA,β(GQ) is contained in one of the index 2
|
1042 |
+
subgroups Mβ of an irreducible subgroup of Type (2) listed in Lemma 2.3, then again β is added to
|
1043 |
+
PossiblyNonsurjectivePrimes in Step (3), since Mβ β Fβ is always reducible by Lemma 2.4(1b).
|
1044 |
+
Hence we may assume that ΟA,β(GQ) is contained in one of the irreducible maximal subgroups
|
1045 |
+
Gβ of Type (2) listed in Lemma 2.3, but not in the index 2 subgroup Mβ. The normalizer character
|
1046 |
+
GQ
|
1047 |
+
ΟA,β
|
1048 |
+
οΏ½οΏ½β Gβ β Gβ/Mβ = {Β±1}
|
1049 |
+
is nontrivial and unramiο¬ed outside of N, and so it corresponds to a quadratic Dirichlet character
|
1050 |
+
ΟβΆ(Z/NZ)Γ β {Β±1}. Lemma 2.4(1a) shows that tr(g) = 0 in Fβ for any g β Gβ β Mβ. Consequently,
|
1051 |
+
β is governed by Ο (in the language of Section 3.2), so β is added to PossiblyNonsurjectivePrimes in
|
1052 |
+
Step (4) by Proposition 3.14.
|
1053 |
+
β‘
|
1054 |
+
3.3. Bounds on Serreβs open image theorem. In this section we combine the theoretical worst
|
1055 |
+
case bounds in the Algorithms 3.3, 3.6, 3.10, and 3.13 to give a bound on the smallest surjective
|
1056 |
+
good prime q, and the product of all nonsurjective primes, thereby establishing Theorem 1.2.
|
1057 |
+
Corollary 3.16. Let A/Q be a typical genus 2 Jacobian of conductor N. Assuming GRH, we have
|
1058 |
+
β
|
1059 |
+
β nonsurjective
|
1060 |
+
β βͺ exp(N1/2+Ο΅),
|
1061 |
+
where the implied constant is absolute and eο¬ectively computable.
|
1062 |
+
|
1063 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1064 |
+
19
|
1065 |
+
Proof. Let q be the smallest surjective good prime for A, which is ο¬nite by Serreβs open image
|
1066 |
+
theorem. Multiplying the bounds in Propositions 3.5, 3.8, 3.12, and 3.15 by the conductor N, the
|
1067 |
+
product of all nonsurjective primes is bounded by a function of q and N of the following shape
|
1068 |
+
(8)
|
1069 |
+
β
|
1070 |
+
β nonsurjective
|
1071 |
+
β βͺ qN1/2+Ο΅.
|
1072 |
+
On the other hand, since q is the smallest surjective prime by deο¬nition, the product of all primes
|
1073 |
+
less than q divides the product of all nonsurjective primes. Using [Ser81, Lemme 11], we have
|
1074 |
+
exp(q) βͺ β
|
1075 |
+
β<q
|
1076 |
+
β β€
|
1077 |
+
β
|
1078 |
+
β nonsurjective
|
1079 |
+
β βͺ qN1/2+Ο΅.
|
1080 |
+
Combining the ο¬rst and last terms, we have q βͺ N1/2+Ο΅ log(q), whence q βͺ N1/2+Ο΅. Plugging this
|
1081 |
+
back into (8) yields the claimed bound.
|
1082 |
+
β‘
|
1083 |
+
4. Testing surjectivity of ΟA,β
|
1084 |
+
In this section we establish Theorem 1.1(2). The goal is to weed out any extraneous nonsur-
|
1085 |
+
jective primes in the output PossiblyNonsurjectivePrimes of Algorithm 3.1 to produce a smaller list
|
1086 |
+
LikelyNonsurjectivePrimes(B) containing all nonsurjective primes (depending on a chosen bound
|
1087 |
+
B) by testing the characteristic polynomials of Frobenius elements up to the bound B. If B is
|
1088 |
+
suο¬ciently large (quantiο¬ed in Section 5), the list LikelyNonsurjectivePrimes(B) is provably the list
|
1089 |
+
of nonsurjective primes.
|
1090 |
+
Algorithm 4.1. Given an integer B and the output PossiblyNonsurjectivePrimes of Algorithm 3.1
|
1091 |
+
run on the typical hyperelliptic genus 2 curve with equation y2 + h(x)y = f(x), output a sublist
|
1092 |
+
LikelyNonsurjectivePrimes(B) of PossiblyNonsurjectivePrimes as follows.
|
1093 |
+
(1) Initialize LikelyNonsurjectivePrimes(B) as PossiblyNonsurjectivePrimes.
|
1094 |
+
(2) Remove 2 from LikelyNonsurjectivePrimes(B) if the size of the Galois group of the splitting ο¬eld
|
1095 |
+
of 4f + h2 is 720.
|
1096 |
+
(3) For each good prime p < B, while LikelyNonsurjectivePrimes(B) is nonempty:
|
1097 |
+
(a) Compute the integral characteristic polynomial Pp(t) of Frobp.
|
1098 |
+
(b) For each prime β in LikelyNonsurjectivePrimes(B), run Tests 4.4(i), (ii), and (iii) on Pp(t)
|
1099 |
+
to rule out ΟA,β(GQ) being contained in one of the exceptional maximal subgroups.
|
1100 |
+
(c) For each prime β in LikelyNonsurjectivePrimes(B), run Tests 4.5(i) and (ii) on Pp(t) to rule
|
1101 |
+
out ΟA,β(GQ) being contained in one of the nonexceptional maximal subgroups.
|
1102 |
+
(d) For a given prime β, if each of the 5 tests Tests 4.4(i)β(iii) and Tests 4.5(i)β(ii) have
|
1103 |
+
succeeded for some prime p, remove β from LikelyNonsurjectivePrimes(B).
|
1104 |
+
(4) Return LikelyNonsurjectivePrimes(B).
|
1105 |
+
Remark 12. In our implementation of Step 3 of this algorithm, we have chosen to only use primes
|
1106 |
+
p of good reduction for the curve as auxiliary primes, which is a stronger condition than being a
|
1107 |
+
good prime for the Jacobian A. More precisely, the primes that are good for the Jacobian but bad
|
1108 |
+
for the curve are precisely the prime factors of the discriminant 4f + h2 of a minimal equation for
|
1109 |
+
the curve that do not divide the conductor NA of the Jacobian. At such a prime, the reduction
|
1110 |
+
of the curve consists of two elliptic curves E1 and E2 intersecting transversally at a single point.
|
1111 |
+
Since there are many auxiliary primes p < B to choose from, excluding bad primes for the curve is
|
1112 |
+
not a serious restriction, but allows us to access the characteristic polynomial of Frobenius directly
|
1113 |
+
by counting points on the reduction of the curve. This is not strictly necessary: one could use the
|
1114 |
+
characteristic polynomials of Frobenius for the elliptic curves E1 and E2, which can be computed
|
1115 |
+
using the genus2reduction module of SageMath.
|
1116 |
+
We brieο¬y summarize the contents of this section. In Section 4.1, we ο¬rst prove a purely group-
|
1117 |
+
theoretic criterion for a subgroup of GSp4(Fβ) to equal the whole group. Then in Section 4.2,
|
1118 |
+
|
1119 |
+
20
|
1120 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1121 |
+
we explain Test 4.4 and Test 4.5, whose validity follows immediately from Lemma 2.4(3) and
|
1122 |
+
Proposition 4.2 respectively. The main idea of these tests is to use auxiliary good primes p β β to
|
1123 |
+
generate characteristic polynomials in the image of ΟA,β. If we ο¬nd enough types of characteristic
|
1124 |
+
polynomials to rule out each proper maximal subgroup of GSp4(Fβ) (cf. Proposition 4.2), then we
|
1125 |
+
can conclude that ΟA,β is surjective. In Section 4.3, we prove Theorems 1.1(2) and 1.3 that justify
|
1126 |
+
this algorithm.
|
1127 |
+
4.1. A group-theoretic criterion. We now use the classiο¬cation of maximal subgroups of GSp4(Fβ)
|
1128 |
+
described in Section 2.4 to deduce a group-theoretic criterion for a subgroup G of GSp4(Fβ) to be
|
1129 |
+
the whole group. This is analogous to [Ser72, Proposition 19 (i)-(ii)].
|
1130 |
+
Proposition 4.2. Fix a prime β β 2 and a subgroup G β GSp4(Fβ) with surjective similitude
|
1131 |
+
character. Assume that G is not contained in one of the exceptional maximal subgroups described
|
1132 |
+
in Lemma 2.3(4). Then G = GSp4(Fβ) if and only if there exists matrices X,Y β G such that
|
1133 |
+
(a) the characteristic polynomial of X is irreducible; and
|
1134 |
+
(b) traceY β 0 and the characteristic polynomial of Y has a linear factor with multiplicity one.
|
1135 |
+
Proof. The βonly ifβ direction follows from Proposition 5.1 below, where we show that a nonzero
|
1136 |
+
proportion of elements of GSp4(Fβ) satisfy the conditions in (a) and (b).
|
1137 |
+
Now assume that the group G has elements X and Y as in the statement of the proposition. We
|
1138 |
+
have to show that G = GSp4(Fβ). By assumption, G is not a subgroup of a maximal subgroup of
|
1139 |
+
type (4). For each of the remaining types of maximal subgroups in Lemma 2.3, we will use one of
|
1140 |
+
the elements X or Y to rule out G being contained in a subgroup of that type.
|
1141 |
+
(a) By Lemma 2.2 (iv), every element of a subgroup of type (1) has a reducible characteristic
|
1142 |
+
polynomial. The same is true for elements of type (3) by Lemma 2.4 (2). This is violated by
|
1143 |
+
the element X, so G cannot be contained in a subgroup of type (1) or type (3).
|
1144 |
+
(b) Recall the notation used in the description of a type (2) maximal subgroups in Lemma 2.3.
|
1145 |
+
By Lemma 2.4 1a, every element in Gβ β Mβ has trace 0. By Lemma 2.2 (iii), an element with
|
1146 |
+
irreducible characteristic polynomial automatically has nonzero trace. Hence both X and Y
|
1147 |
+
have nonzero trace, and so cannot be contained in Gβ β Mβ. We now consider two cases
|
1148 |
+
(i) If the two lines are individually deο¬ned over Fβ, then every element in Mβ preserves a
|
1149 |
+
two-dimensional subspace and hence has a reducible characteristic polynomial. This is
|
1150 |
+
violated by the element X.
|
1151 |
+
(ii) If the two lines are permuted by GFβ, then the action of Mβ on the corresponding subspaces
|
1152 |
+
V and V β² are conjugate. Therefore, every Fβ-rational eigenvalue for the action of Frobp
|
1153 |
+
on V , also appears as an eigenvalue for the action on V β², with the same multiplicity. This
|
1154 |
+
is violated by the element Y .
|
1155 |
+
Hence G cannot be contained in a maximal subgroup of type (2).
|
1156 |
+
Since any subgroup of GSp4(Fβ) that is not contained in a proper maximal subgroup of GSp4(Fβ)
|
1157 |
+
must equal GSp4(Fβ), we are done.
|
1158 |
+
β‘
|
1159 |
+
Remark 13. [AdRK13, Corollary 2.2] gives a very similar criterion for a subgroup G of GSp4(Fβ)
|
1160 |
+
to contain Sp4(Fβ), namely that it contains a transvection, and also an element with irreducible
|
1161 |
+
characteristic polynomial (and hence automatically nonzero trace).
|
1162 |
+
4.2. Surjectivity tests.
|
1163 |
+
4.2.1. Surjectivity test for β = 2.
|
1164 |
+
Proposition 4.3. Let A be the Jacobian of the hyperelliptic curve y2 + h(x)y = f(x) deο¬ned over
|
1165 |
+
Q. Then ΟA,2 is surjective if and only if the size of the Galois group of the splitting ο¬eld of 4f + h2
|
1166 |
+
is 720.
|
1167 |
+
|
1168 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1169 |
+
21
|
1170 |
+
Proof. This follows from the fact that GSp4(F2) β
S6 which is a group of size 720, and that the
|
1171 |
+
representation ΟA,2 is the permutation action of the Galois group on the six roots of 4f + h2.
|
1172 |
+
β‘
|
1173 |
+
4.2.2. Surjectivity tests for β β 2.
|
1174 |
+
The tests to rule out the exceptional maximal subgroups rely on the existence of the ο¬nite lists
|
1175 |
+
C1920 and C720 (independent of β), and C7,5040 given in Lemma 2.4(3).
|
1176 |
+
Test 4.4 (Tests for ruling out exceptional maximal subgroups of GSp4(Fβ) for β β 2).
|
1177 |
+
Given a polynomial Pp(t) = t4 β apt + bpt2 β papt + p2 and β β₯ 2,
|
1178 |
+
(i) Pp(t) passes Test 4.4 (i) if β β‘ Β±1 (mod 8) or (a2
|
1179 |
+
p/p,bp/p) mod β lies outside of C1920 mod β.
|
1180 |
+
(ii) Pp(t) passes Test 4.4 (ii) if β β‘ Β±1 (mod 12) or (a2
|
1181 |
+
p/p,bp/p) mod β lies outside of C720 mod β.
|
1182 |
+
(iii) Pp(t) passes Test 4.4 (iii) if β β 7 or (a2
|
1183 |
+
p/p,bp/p) mod β lies outside of C7,5040.
|
1184 |
+
Test 4.5 (Tests for ruling out non-exceptional maximal subgroups for β β 2).
|
1185 |
+
Given a polynomial Pp(t) = t4 β apt + bpt2 β papt + p2 and β β₯ 2,
|
1186 |
+
(i) Pp(t) passes Test 4.5 (i) if Pp(t) modulo β is irreducible.
|
1187 |
+
(ii) Pp(t) passes Test 4.5 (ii) if Pp(t) modulo β has a linear factor of multiplicity 1 and has nonzero
|
1188 |
+
trace.
|
1189 |
+
For any one of the ο¬ve tests above, say that the test succeeds if a given polynomial Pp(t) passes
|
1190 |
+
the corresponding test.
|
1191 |
+
Remark 14. We call an auxiliary prime p a witness for a given prime β if the polynomial Pp(t)
|
1192 |
+
passes one of our tests for β. The verbose output of our code prints witnesses for each of our tests
|
1193 |
+
for each prime β in PossiblyNonsurjectivePrimes but not in LikelyNonsurjectivePrimes(B).
|
1194 |
+
4.3. Justiο¬cation for surjectivity tests. Considering Tests 4.4 and 4.5, we deο¬ne
|
1195 |
+
CΞ± = {M β GSp4(Fβ) βΆ PM(t) is irreducible}
|
1196 |
+
CΞ² = {M β GSp4(Fβ) βΆ tr(M) β 0 and PM(t) has a linear factor of multiplicity 1}
|
1197 |
+
CΞ³1 = {M β GSp4(Fβ) βΆ ( tr(M)2
|
1198 |
+
mult(M), mid(M)
|
1199 |
+
mult(M)) /β Cβ,1920 or β β‘ Β±1
|
1200 |
+
(mod 8)}
|
1201 |
+
CΞ³2 = {M β GSp4(Fβ) βΆ ( tr(M)2
|
1202 |
+
mult(M), mid(M)
|
1203 |
+
mult(M)) /β Cβ,720 or β β‘ Β±1
|
1204 |
+
(mod 12)}
|
1205 |
+
CΞ³3 = {M β GSp4(Fβ) βΆ ( tr(M)2
|
1206 |
+
mult(M), mid(M)
|
1207 |
+
mult(M)) /β Cβ,5040 or β β 7}
|
1208 |
+
CΞ³ = CΞ³1 β© CΞ³2 β© CΞ³3.
|
1209 |
+
Proof of Theorem 1.1(2) and Theorem 1.3. Let B > 0. Since LikelyNonsurjectivePrimes(B) is a sub-
|
1210 |
+
list of PossiblyNonsurjectivePrimes, which contains all nonsurjective primes by Theorem 1.1(1), any
|
1211 |
+
prime not in PossiblyNonsurjectivePrimes is surjective. Now consider β β PossiblyNonsurjectivePrimes
|
1212 |
+
and not in LikelyNonsurjectivePrimes(B). If β = 2, then by Proposition 4.3, ΟA,2 is surjective. If
|
1213 |
+
β > 2, this means that we found primes p1,p2,p3,p4,p5 β€ B each distinct from β and of good reduc-
|
1214 |
+
tion for A for which ΟA,β(Frobp1) β CΞ±, ΟA,β(Frobp2) β CΞ², ΟA,β(Frobp3) β CΞ³1, ΟA,β(Frobp4) β CΞ³2,
|
1215 |
+
and ΟA,β(Frobp4) β CΞ³3. Note that by (1), the similitude factor mult(ΟA,β(Frobp)) is p. Therefore,
|
1216 |
+
by Lemma 2.4(3), it follows that ΟA,β(GQ) is not contained in an exceptional maximal subgroup.
|
1217 |
+
The surjectivity of ΟA,β now follows from Proposition 4.2.
|
1218 |
+
Finally, we will show that if B is suο¬ciently large (as quantiο¬ed by Theorem 1.3), then any
|
1219 |
+
prime β in PossiblyNonsurjectivePrimes is nonsurjective. Since the sets CΞ±, CΞ², CΞ³1, CΞ³2 and CΞ³3
|
1220 |
+
are nonempty by Proposition 5.1 below and closed under conjugation, it follows by Lemma 2.10,
|
1221 |
+
there exist primes p1,p2,p3,p4,p5 β€ B as above.
|
1222 |
+
β‘
|
1223 |
+
|
1224 |
+
22
|
1225 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1226 |
+
Remark 15. If we assume both GRH and AHC, Ram Murty and Kumar Murty [MM97, p. 52] noted
|
1227 |
+
(see also [FJ20, Theorem 2.3]) that the bound (4) can be replaced with p βͺ (log dK)2
|
1228 |
+
β£Sβ£
|
1229 |
+
. Proposition
|
1230 |
+
5.1, which follows, shows that the sets CΞ±, CΞ², and CΞ³ have size at least β£ GSp4(Fβ)β£
|
1231 |
+
10
|
1232 |
+
. This can be
|
1233 |
+
used to prove the ineο¬ective version of Theorem 1.3 which relies on AHC noted in the introduction
|
1234 |
+
in a manner similar to the proof of Theorem 1.3.
|
1235 |
+
5. The probability of success
|
1236 |
+
In this section we prove Theorem 1.4, by studying the probability that a matrix chosen uniformly
|
1237 |
+
at random from GSp4(Fβ) is contained in each of CΞ±, CΞ², and CΞ³ deο¬ned in Section 4.3. Let Ξ±β, Ξ²β,
|
1238 |
+
and Ξ³β respectively be the probabilities that a matrix chosen uniformly at random from GSp4(Fβ)
|
1239 |
+
is contained in CΞ±, CΞ², or CΞ³.
|
1240 |
+
Proposition 5.1. Let M be a matrix chosen uniformly at random from GSp4(Fβ) with β odd. Then
|
1241 |
+
(i) The probability that M β CΞ± is given by
|
1242 |
+
Ξ±β = 1
|
1243 |
+
4 β
|
1244 |
+
1
|
1245 |
+
2(β2 + 1).
|
1246 |
+
(ii) The probability that M β CΞ² is given by
|
1247 |
+
Ξ²β = 3
|
1248 |
+
8 β
|
1249 |
+
3
|
1250 |
+
4(β β 1) +
|
1251 |
+
1
|
1252 |
+
2(β β 1)2 .
|
1253 |
+
(iii) The probability that M β CΞ³ is
|
1254 |
+
Ξ³β β₯ 1 β
|
1255 |
+
3β
|
1256 |
+
β2 + 1.
|
1257 |
+
Remark 16. Magma code that directly veriο¬es the sizes of CΞ±,CΞ²,CΞ³ (i.e. computes Ξ±β,Ξ²β,Ξ³β) for
|
1258 |
+
small β may be found in helper_scripts/SanityCheckProbability.m in the repository.
|
1259 |
+
[Shi82] characterizes all conjugacy classes of elements of GSp4(Fβ) for β odd, grouping them into
|
1260 |
+
26 diο¬erent types. For each type Ξ³, Shinoda further computes the number N(Ξ³) of conjugacy
|
1261 |
+
classes of type Ξ³ and the size of the centralizer β£CGSp4(Fβ)(Ξ³)β£, which is the size of the centralizer
|
1262 |
+
β£CGSp4(Fβ)(M)β£ of M in GSp4(Fβ) for any M in a conjugacy class of type Ξ³. The size β£C(Ξ³)β£ of any
|
1263 |
+
conjugacy class of type Ξ³ can then easily be computed as
|
1264 |
+
β£C(Ξ³)β£ =
|
1265 |
+
β£GSp4(Fβ)β£
|
1266 |
+
β£CGSp4(Fβ)(Ξ³)β£
|
1267 |
+
and the probability that a uniformly chosen M β GSp4(Fβ) has conjugacy type Ξ³ is then given by
|
1268 |
+
(9)
|
1269 |
+
N(Ξ³)β£C(Ξ³)β£
|
1270 |
+
β£GSp4(Fβ)β£ =
|
1271 |
+
N(Ξ³)
|
1272 |
+
β£CGSp4(Fβ)(Ξ³)β£.
|
1273 |
+
To prove Proposition 5.1, we will need to examine a handful of types of conjugacy classes of
|
1274 |
+
GSp4(Fβ).
|
1275 |
+
There is only a single conjugacy type Ξ³ whose characteristic polynomials are irreducible. This
|
1276 |
+
type is denoted K0 in [Shi82] where it is shown there that N(K0) = (ββ1)(β2β1)
|
1277 |
+
4
|
1278 |
+
and β£CGSp4(Fβ)(K0)β£ =
|
1279 |
+
(β β 1)(β2 + 1).
|
1280 |
+
While there is only one way for a polynomial to be irreducible, there are several ways for a
|
1281 |
+
quartic polynomial to have a root of odd order. However, only some of these can occur if f(t) is
|
1282 |
+
the characteristic polynomial of a matrix M β GSp4(Fβ) and we only need to concern ourselves
|
1283 |
+
with the following three possibilities:
|
1284 |
+
(a) f(t) splits completely over Fβ;
|
1285 |
+
(b) f(t) has two roots over Fβ, both of which occur with multiplicity one; and
|
1286 |
+
(c) f(t) has two simple roots and one double root over Fβ.
|
1287 |
+
|
1288 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1289 |
+
23
|
1290 |
+
Cases (a) and (b) correspond to the conjugacy types H0 and J0 in [Shi82] respectively.
|
1291 |
+
In
|
1292 |
+
contrast, there are two types of conjugacy classes for which f(t) has two simple roots and one
|
1293 |
+
double root, which are denoted E0 and E1 in [Shi82].
|
1294 |
+
The number of conjugacy classes and centralizer size for each of these conjugacy types is given by
|
1295 |
+
Table 2, along with the associated probability that a uniform random M β GSp4(Fβ) has conjugacy
|
1296 |
+
type Ξ³ computed using (9).
|
1297 |
+
Type Ξ³ in [Shi82]
|
1298 |
+
N(Ξ³)
|
1299 |
+
β£CGSp4(Fβ)(Ξ³)β£
|
1300 |
+
Associated Probability
|
1301 |
+
K0 (Irreducible)
|
1302 |
+
(ββ1)(β2β1)
|
1303 |
+
4
|
1304 |
+
(β2 + 1)(β β 1)
|
1305 |
+
1
|
1306 |
+
4 β
|
1307 |
+
1
|
1308 |
+
2(β2+1)
|
1309 |
+
H0 (Split)
|
1310 |
+
(ββ1)(ββ3)2
|
1311 |
+
8
|
1312 |
+
(β β 1)3
|
1313 |
+
1
|
1314 |
+
8 β
|
1315 |
+
1
|
1316 |
+
2(ββ1) +
|
1317 |
+
1
|
1318 |
+
2(ββ1)2
|
1319 |
+
J0 (Two Simple Roots)
|
1320 |
+
(ββ1)3
|
1321 |
+
4
|
1322 |
+
(β + 1)(β β 1)2
|
1323 |
+
1
|
1324 |
+
4 β
|
1325 |
+
1
|
1326 |
+
2(β+1)
|
1327 |
+
E0 (One Double Root)
|
1328 |
+
(ββ1)(ββ3)
|
1329 |
+
2
|
1330 |
+
β(β β 1)2(β2 β 1)
|
1331 |
+
1
|
1332 |
+
2β(β2β1) β
|
1333 |
+
1
|
1334 |
+
β(ββ1)(β2β1)
|
1335 |
+
E1 (One Double Root)
|
1336 |
+
(ββ1)(ββ3)
|
1337 |
+
2
|
1338 |
+
β(β β 1)2
|
1339 |
+
1
|
1340 |
+
2β β
|
1341 |
+
1
|
1342 |
+
β(ββ1)
|
1343 |
+
Table 2. Number of conjugacy classes and centralizer sizes for each conjugacy class
|
1344 |
+
type in [Shi82].
|
1345 |
+
Proof of Proposition 5.1. Part (i) is simply the entry in Table 2 in the last column corresponding
|
1346 |
+
to the βK0 (Irreducible)β type.
|
1347 |
+
We now establish part (ii). As indicated in the discussion above Table 2, the only conjugacy
|
1348 |
+
classes of matrices in GSp4(Fβ) whose characteristic polynomials have some linear factors of odd
|
1349 |
+
multiplicity are those of the types H0,J0,E0,E1. However, for part (ii) since we are only interested
|
1350 |
+
in matrices M also having non-zero trace, it is insuο¬cient to simply sum over the rightmost entries
|
1351 |
+
in the bottom four rows of Table 2. From [Shi82, Table 2], we see that the elements of E0 and E1
|
1352 |
+
have trace c(a+1)2
|
1353 |
+
a
|
1354 |
+
for some c,a β FΓ
|
1355 |
+
β with a β Β±1. In particular, it follows that elements of types E0
|
1356 |
+
and E1 have nonzero traces. The elements of J0 have trace (c+a)(c+aβ)
|
1357 |
+
c
|
1358 |
+
where c β FΓ
|
1359 |
+
β and a β Fβ2 βFβ.
|
1360 |
+
Therefore, the elements of J0 also have nonzero trace.
|
1361 |
+
It remains to analyze which conjugacy classes of Type H0 have nonzero trace. Following [Shi82],
|
1362 |
+
the
|
1363 |
+
(ββ1)(ββ3)2
|
1364 |
+
8
|
1365 |
+
conjugacy classes of type H0 correspond to quadruples of distinct elements in
|
1366 |
+
a1,a2,b1,b2 β FΓ
|
1367 |
+
β satisfying a1b1 = a2b2 modulo the action of swapping any of a1 with b1, a2 with
|
1368 |
+
b2, or a1,b1 with a2,b2. The eigenvalues of any matrix in the conjugacy class are a1, a2, b1, and b2.
|
1369 |
+
Consequently the matrix has trace zero only if either a2 = βa1 and b2 = βb1 or b1 = βa2 and b2 = βa1.
|
1370 |
+
This accounts for (ββ1)(ββ3)
|
1371 |
+
4
|
1372 |
+
of the (ββ1)(ββ3)2
|
1373 |
+
8
|
1374 |
+
conjugacy classes of type H0, leaving (ββ1)(ββ3)(ββ5)
|
1375 |
+
8
|
1376 |
+
conjugacy classes with non-zero trace. As a result, the probability that a matrix M β GSp4(Fβ)
|
1377 |
+
chosen uniformly at random has non-zero trace and totally split characteristic polynomial is
|
1378 |
+
(10)
|
1379 |
+
(β β 1)(β β 3)(β β 5)
|
1380 |
+
8(β β 1)3
|
1381 |
+
= 1
|
1382 |
+
8 β
|
1383 |
+
3
|
1384 |
+
4(β β 1) +
|
1385 |
+
1
|
1386 |
+
(β β 1)2 .
|
1387 |
+
To obtain part (ii), we add (10) to the entries in the rightmost column of the ο¬nal three rows of
|
1388 |
+
Table 2, getting
|
1389 |
+
(1
|
1390 |
+
8 β
|
1391 |
+
3
|
1392 |
+
4(β β 1) +
|
1393 |
+
1
|
1394 |
+
(β β 1)2 ) + (1
|
1395 |
+
4 β
|
1396 |
+
1
|
1397 |
+
2(β + 1)) + (
|
1398 |
+
1
|
1399 |
+
2β(β2 β 1) β
|
1400 |
+
1
|
1401 |
+
β(β β 1)(β2 β 1)) + ( 1
|
1402 |
+
2β β
|
1403 |
+
1
|
1404 |
+
β(β β 1))
|
1405 |
+
= 3
|
1406 |
+
8 β
|
1407 |
+
3
|
1408 |
+
4(β β 1) +
|
1409 |
+
1
|
1410 |
+
2(β β 1)2 .
|
1411 |
+
|
1412 |
+
24
|
1413 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1414 |
+
To prove (iii), we start by noting that for any pair (u,v), the cardinality of the set
|
1415 |
+
{t4 β at3 + bt2 β amt + m2 βΆ a,b β Fβ,m β FΓ
|
1416 |
+
β and (a2
|
1417 |
+
m , b
|
1418 |
+
m) = (u,v)}
|
1419 |
+
is at most β β 1.
|
1420 |
+
By [Cha97, Theorem 3.5], the number of matrices in GSp4(Fβ) with a given
|
1421 |
+
characteristic polynomial is at most (β+3)8. Assuming β β 7, by combining these observations, and
|
1422 |
+
noting that β£Cβ,720 βͺ Cβ,1920β£ β€ 14, we obtain the bound
|
1423 |
+
Ξ³β β₯ 1 β 14(β β 1)(β + 3)8
|
1424 |
+
β£GSp4(Fβ)β£
|
1425 |
+
.
|
1426 |
+
For β > 17, this implies the claimed bound. For 3 β€ β β€ 17, we directly check the claim using
|
1427 |
+
Magma.
|
1428 |
+
β‘
|
1429 |
+
Lemma 5.2. Let C/Q be a typical genus 2 curve with Jacobian A and suppose β is an odd prime
|
1430 |
+
such that ΟA,β is surjective. For any Ο΅ > 0, there exists an eο¬ective constant B0 (with B0 > βNA)
|
1431 |
+
such that for any B > B0 and each Ξ΄ β {Ξ±,Ξ²,Ξ³}, we have
|
1432 |
+
β£β£{p prime βΆ B β€ p β€ 2B and ΟA,β(Frobp) β CΞ΄}β£
|
1433 |
+
β£{p prime βΆ B β€ p β€ 2B}β£
|
1434 |
+
β Ξ΄ββ£ < Ο΅.
|
1435 |
+
Proof. Let G = Gal(Q(A[β])/Q) and S β G be any subset that is closed under conjugation. By
|
1436 |
+
taking B to be suο¬ciently large, we have that B > βNA and can make
|
1437 |
+
β£β£{p prime βΆ B β€ p β€ 2B and Frobp β S}β£
|
1438 |
+
β£{p prime βΆ B β€ p β€ 2B}β£
|
1439 |
+
β β£Sβ£
|
1440 |
+
β£Gβ£β£
|
1441 |
+
arbitrarily small by (3).
|
1442 |
+
Moreover, the previous statement can be made eο¬ective by using an
|
1443 |
+
eο¬ective version of the Chebotarev density theorem. The result then follows because each of the
|
1444 |
+
sets CΞ±, CΞ², and CΞ³ is closed under conjugation.
|
1445 |
+
β‘
|
1446 |
+
For positive integers n and B > βNA, let P(B,n) be the probability that n primes p1,...,pn
|
1447 |
+
(possibly non-distinct) chosen uniformly at random in the interval [B,2B] have the property that
|
1448 |
+
ΟA,β(Frobpi) /β CΞ± for each i
|
1449 |
+
or
|
1450 |
+
ΟA,β(Frobpi) /β CΞ² for each i
|
1451 |
+
or
|
1452 |
+
ΟA,β(Frobpi) /β CΞ³ for each i.
|
1453 |
+
Corollary 5.3. Suppose C and β are as in Lemma 5.2 and let n be a positive integer. For any
|
1454 |
+
Ο΅ > 0, there exists an eο¬ective constant B0 (with B0 > βNA) such that for all B > B0, we have
|
1455 |
+
P(B,n) < (1 β Ξ±β)n + (1 β Ξ²β)n + (1 β Ξ³β)n + Ο΅.
|
1456 |
+
Proof. For Ξ΄ β {Ξ±,Ξ²,Ξ³}, let XΞ΄ be the event that none of the ΟA,β(Frobpi) are contained in CΞ΄. We
|
1457 |
+
then have
|
1458 |
+
P(XΞ± βͺ XΞ² βͺ XΞ³) β€ P(XΞ±) + P(XΞ²) + P(XΞ³)
|
1459 |
+
The result then follows by Lemma 5.2, which shows that there exists a B0 such that the probabilities
|
1460 |
+
of XΞ±, XΞ², and XΞ³ can be made arbitrarily close to (1βΞ±β)n, (1βΞ²β)n, and (1βΞ³β)n respectively.
|
1461 |
+
β‘
|
1462 |
+
Proof of Theorem 1.4. The claim made by Theorem 1.4 is that P(B,n) < 3β
( 9
|
1463 |
+
10)
|
1464 |
+
n for B suο¬ciently
|
1465 |
+
large. By Proposition 5.1, we have 1 β Ξ±β β€ 4
|
1466 |
+
5, 1 β Ξ²β β€ 7
|
1467 |
+
8, and 1 β Ξ³β β€ 9
|
1468 |
+
10 for all β odd. The result
|
1469 |
+
then follows from Corollary 5.3 because (4
|
1470 |
+
5)
|
1471 |
+
n + (7
|
1472 |
+
8)
|
1473 |
+
n + ( 9
|
1474 |
+
10)
|
1475 |
+
n < 3 β
( 9
|
1476 |
+
10)
|
1477 |
+
n.
|
1478 |
+
β‘
|
1479 |
+
6. Results of computation and interesting examples
|
1480 |
+
We report on the results of running our algorithm on a dataset of 1,823,592 typical genus 2
|
1481 |
+
curves with conductor bounded by 220 that are part of a new dataset of approximately 5 million
|
1482 |
+
curves currently being prepared for addition into the LMFDB. Running our algorithm on all of
|
1483 |
+
these curves in parallel took about 45 hours on MITβs Lovelace computer (see the Introduction for
|
1484 |
+
the hardware speciο¬cation of this machine).
|
1485 |
+
|
1486 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1487 |
+
25
|
1488 |
+
We ο¬rst show in Table 3 how many of these curves were nonsurjective at particular primes,
|
1489 |
+
indicating also if this can be explained by the existence of a rational torsion point of that prime
|
1490 |
+
order. We found 31 as the largest nonsurjective prime, which occurred for the curve
|
1491 |
+
(11)
|
1492 |
+
y2 + (x + 1)y = x5 + 23x4 β 48x3 + 85x2 β 69x + 45
|
1493 |
+
of conductor 72 β
312 and discriminant 72 β
319 (the prime 2 was also nonsurjective here).
|
1494 |
+
The
|
1495 |
+
Jacobian of this curve does not admit a nontrivial rational 31-torsion point, so unlike many other
|
1496 |
+
instances of nonsurjective primes we observed, this one cannot be explained by the presence of
|
1497 |
+
rational torsion. One could ask if it might be explained by the existence of a Q-rational 31-isogeny
|
1498 |
+
(as suggested by Algorithm 3.1, since 31 is returned by Algorithm 3.6). This seems to be the case
|
1499 |
+
- see forthcoming work of van Bommel, Chidambaram, Costa, and Kieο¬er [vBCCK22] where the
|
1500 |
+
isogeny class of this curve (among others) is computed.
|
1501 |
+
nonsurj. prime
|
1502 |
+
No. of curves w/ torsion
|
1503 |
+
No. of curves w/o torsion
|
1504 |
+
Example curve
|
1505 |
+
2
|
1506 |
+
1,100,706
|
1507 |
+
462,616
|
1508 |
+
464.a.464.1
|
1509 |
+
3
|
1510 |
+
79,759
|
1511 |
+
98,750
|
1512 |
+
277.a.277.2
|
1513 |
+
5
|
1514 |
+
12,040
|
1515 |
+
10,809
|
1516 |
+
16108.b.64432.1
|
1517 |
+
7
|
1518 |
+
1,966
|
1519 |
+
2,213
|
1520 |
+
295.a.295.2
|
1521 |
+
11
|
1522 |
+
167
|
1523 |
+
210
|
1524 |
+
4288.b.548864.1
|
1525 |
+
13
|
1526 |
+
108
|
1527 |
+
310
|
1528 |
+
439587.d.439587.1
|
1529 |
+
17
|
1530 |
+
22
|
1531 |
+
61
|
1532 |
+
1996.b.510976.1
|
1533 |
+
19
|
1534 |
+
10
|
1535 |
+
20
|
1536 |
+
1468.6012928
|
1537 |
+
23
|
1538 |
+
2
|
1539 |
+
8
|
1540 |
+
1696.1736704
|
1541 |
+
29
|
1542 |
+
1
|
1543 |
+
5
|
1544 |
+
976.999424
|
1545 |
+
31
|
1546 |
+
0
|
1547 |
+
1
|
1548 |
+
47089.1295541485872879
|
1549 |
+
Table 3. Nonsurjective primes in the dataset, and whether they are explained by
|
1550 |
+
torsion, with examples from the LMFDB dataset if available, else a string of the
|
1551 |
+
form βconductor.discrimnantβ.
|
1552 |
+
We also observed (see Table 4) that the vast majority of curves had less than 3 nonsurjective
|
1553 |
+
primes.
|
1554 |
+
No. of nonsurj. primes
|
1555 |
+
No. of curves
|
1556 |
+
Example curve
|
1557 |
+
Nonsurj. primes of example
|
1558 |
+
0
|
1559 |
+
211,620
|
1560 |
+
743.a.743.1
|
1561 |
+
β
|
1562 |
+
1
|
1563 |
+
1,455,473
|
1564 |
+
1923.a.1923.1
|
1565 |
+
5 (torsion)
|
1566 |
+
2
|
1567 |
+
155,186
|
1568 |
+
976.a.999424.1
|
1569 |
+
2, 29(torsion)
|
1570 |
+
3
|
1571 |
+
1,313
|
1572 |
+
15876.a.15876.1
|
1573 |
+
2, 3, 5
|
1574 |
+
Table 4. Frequency count of nonsurjective primes in the dataset, with examples
|
1575 |
+
from the LMFDB dataset.
|
1576 |
+
Instructions for obtaining the entire results ο¬le may be found in the README.md ο¬le of the
|
1577 |
+
repository.
|
1578 |
+
Remark 17. It would be interesting to know if there is a uniform upper bound on the largest prime
|
1579 |
+
β that could occur as a nonsurjective prime for the Jacobian of a genus 2 curve deο¬ned over Q,
|
1580 |
+
|
1581 |
+
26
|
1582 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1583 |
+
analogous to the conjectural bound of 37 for the largest nonsurjective prime for elliptic curves
|
1584 |
+
deο¬ned over Q (see e.g. [BPR13, Introduction]). As the example of (11) shows, this bound - if it
|
1585 |
+
exists - would have to be at least 31.
|
1586 |
+
We conclude with a few examples that illustrate where Algorithm 3.1 fails when the abelian
|
1587 |
+
surface has extra (geometric) endomorphisms.
|
1588 |
+
Example 6.1. The Jacobian A of the genus 2 curve 3125.a.3125.1 on the LMFDB given by y2+y =
|
1589 |
+
x5 has End(AQ) = Z but End(AQ) = Z[ΞΆ5]. Let Ο be the Dirichlet character of modulus 5 deο¬ned
|
1590 |
+
by the Legendre symbol
|
1591 |
+
ΟβΆ(Z/5Z)Γ β {Β±1},
|
1592 |
+
2 β¦ β1.
|
1593 |
+
In this case, Algorithm 3.13 fails to ο¬nd an auxilliary prime p < 1000 for which ap β 0 and Ο(p) = β1.
|
1594 |
+
This is consistent with the endomorphism calculation, since the trace of ΟA,β(Frobp) is 0 for all
|
1595 |
+
primes p that do not split completely in Q(ΞΆp) and any inert prime in Q(
|
1596 |
+
β
|
1597 |
+
5) automatically does
|
1598 |
+
not split completely in Q(ΞΆ5).
|
1599 |
+
Example 6.2. The modular curve X1(13) (169.a.169.1) has genus 2 and its Jacobian J1(13) has
|
1600 |
+
CM by Z[ΞΆ3] over Q. As in [MT74, Claim 2, page 45], for any prime β that splits as ΟΟ in Q(ΞΆ3), the
|
1601 |
+
representation J1(13)[β] splits as a direct sum VΟ βVΟ of two 2-dimensional subrepresentations that
|
1602 |
+
are dual to each other. (A similar statement holds for J1(13)[β]βFβ Fβ, and so this representation is
|
1603 |
+
never absolutely irreducible.) As expected, Algorithm 3.6 fails to ο¬nd an auxiliary prime p < 1000
|
1604 |
+
for which Rp is nonzero.
|
1605 |
+
Example 6.3. The ο¬rst (ordered by conductor) curve whose Jacobian J admits real multiplication
|
1606 |
+
over Q is the curve 529.a.529.1; indeed, this Jacobian is isogenous to the Jacobian of the modular
|
1607 |
+
curve X0(23). Since there is a single Galois orbit of newforms - call it f - of level Ξ0(23) and weight
|
1608 |
+
2, we have that J is isogenous to the abelian variety Af associated to f, and thus we expect the
|
1609 |
+
integer Mself-dual output by Algorithm 3.10 to be zero for any auxiliary prime, which is indeed the
|
1610 |
+
case.
|
1611 |
+
References
|
1612 |
+
[AdRK13]
|
1613 |
+
Sara Arias-de Reyna and Christian Kappen. Abelian varieties over number ο¬elds, tame ramiο¬cation and
|
1614 |
+
big Galois image. Math. Res. Lett., 20(1):1β17, 2013.
|
1615 |
+
[AK19]
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1616 |
+
Jeoung-Hwan Ahn and Soun-Hi Kwon. An explicit upper bound for the least prime ideal in the Cheb-
|
1617 |
+
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1618 |
+
[Apo76]
|
1619 |
+
Tom M. Apostol. Introduction to analytic number theory. Undergraduate Texts in Mathematics. Springer-
|
1620 |
+
Verlag, New York-Heidelberg, 1976.
|
1621 |
+
[BBB+21]
|
1622 |
+
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1623 |
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|
1624 |
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|
1625 |
+
Sutherland, and John Voight, editors, Arithmetic Geometry, Number Theory, and Computation, pages
|
1626 |
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+
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|
1628 |
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Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language.
|
1629 |
+
J. Symbolic Comput., 24(3-4):235β265, 1997. Computational algebra and number theory (London, 1993).
|
1630 |
+
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|
1631 |
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Armand Brumer and Kenneth Kramer. The conductor of an abelian variety. Compositio Mathematica,
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1632 |
+
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|
1633 |
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[BPR13]
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1634 |
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Yuri Bilu, Pierre Parent, and Marusia Rebolledo. Rational points on X+
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1635 |
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0 (pr). Ann. Inst. Fourier (Greno-
|
1636 |
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1637 |
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|
1638 |
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Eric Bach and Jonathan Sorenson. Explicit bounds for primes in residue classes. Math. Comp.,
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1639 |
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|
1641 |
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Leonard Carlitz. Note on a quartic congruence. Amer. Math. Monthly, 63:569β571, 1956.
|
1642 |
+
[Cha97]
|
1643 |
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Nick Chavdarov. The generic irreducibility of the numerator of the zeta function in a family of curves
|
1644 |
+
with large monodromy. Duke Math. J., 87(1):151β180, 1997.
|
1645 |
+
[CL12]
|
1646 |
+
John Cremona and Eric Larson. Galois representations for elliptic curves over number ο¬elds, 2012.
|
1647 |
+
SageMath.
|
1648 |
+
|
1649 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1650 |
+
27
|
1651 |
+
[Coj05]
|
1652 |
+
Alina Carmen Cojocaru. On the surjectivity of the Galois representations associated to non-CM elliptic
|
1653 |
+
curves. Canad. Math. Bull., 48(1):16β31, 2005. With an appendix by Ernst Kani.
|
1654 |
+
[Die02]
|
1655 |
+
Luis V. Dieulefait. Explicit determination of the images of the Galois representations attached to abelian
|
1656 |
+
surfaces with End(A) = Z. Experiment. Math., 11(4):503β512 (2003), 2002.
|
1657 |
+
[FJ20]
|
1658 |
+
Daniel Fiorilli and Florent Jouve. Distribution of Frobenius elements in families of Galois extensions,
|
1659 |
+
2020.
|
1660 |
+
[GRR72]
|
1661 |
+
Alexander Grothendieck, Michel Raynaud, and Dock Sang Rim. Groupes de monodromie en gΒ΄eomΒ΄etrie
|
1662 |
+
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|
1663 |
+
AlgΒ΄ebrique du Bois-Marie 1967β1969 (SGA 7 I).
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1664 |
+
[Kha06]
|
1665 |
+
Chandrashekhar Khare. Serreβs modularity conjecture: The level one case. Duke Math. J., 134(3):557β
|
1666 |
+
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|
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+
[KL90]
|
1668 |
+
Peter Kleidman and Martin Liebeck. The subgroup structure of the ο¬nite classical groups, volume 129 of
|
1669 |
+
London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1990.
|
1670 |
+
[Kra95]
|
1671 |
+
Alain Kraus. Une remarque sur les points de torsion des courbes elliptiques. C. R. Acad. Sci. Paris SΒ΄er.
|
1672 |
+
I Math., 321(9):1143β1146, 1995.
|
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+
[KW09a]
|
1674 |
+
Chandrashekhar Khare and Jean-Pierre Wintenberger. Serreβs modularity conjecture (I). Invent. Math.,
|
1675 |
+
178(3):485β504, 2009.
|
1676 |
+
[KW09b]
|
1677 |
+
Chandrashekhar Khare and Jean-Pierre Wintenberger. Serreβs modularity conjecture (II). Invent. Math.,
|
1678 |
+
178(3):505β586, 2009.
|
1679 |
+
[KW22]
|
1680 |
+
Habiba Kadiri and Peng-Jie Wong. Primes in the Chebotarev density theorem for all number ο¬elds (with
|
1681 |
+
an Appendix by Andrew Fiori). J. Number Theory, 241:700β737, 2022.
|
1682 |
+
[Liu94]
|
1683 |
+
Qing Liu. Conducteur et discriminant minimal de courbes de genre 2. Compositio Mathematica, 94(1):51β
|
1684 |
+
79, 1994.
|
1685 |
+
[LMF22]
|
1686 |
+
The LMFDB Collaboration. The L-functions and modular forms database. http://www.lmfdb.org,
|
1687 |
+
2022. [Online; accessed 12 December 2022].
|
1688 |
+
[LMO79]
|
1689 |
+
Jeο¬rey C. Lagarias, Hugh L. Montgomery, and Andrew M. Odlyzko. A bound for the least prime ideal
|
1690 |
+
in the Chebotarev density theorem. Invent. Math., 54(3):271β296, 1979.
|
1691 |
+
[Lom16]
|
1692 |
+
Davide Lombardo. Explicit surjectivity of Galois representations for abelian surfaces and GL2-varieties.
|
1693 |
+
Journal of Algebra, 460:26β59, 2016.
|
1694 |
+
[LV14a]
|
1695 |
+
Eric Larson and Dmitry Vaintrob. Determinants of subquotients of Galois representations associated
|
1696 |
+
with abelian varieties. Journal of the Institute of Mathematics of Jussieu, 13(3):517β559, 2014.
|
1697 |
+
[LV14b]
|
1698 |
+
Eric Larson and Dmitry Vaintrob. On the surjectivity of Galois representations associated to elliptic
|
1699 |
+
curves over number ο¬elds. Bull. Lond. Math. Soc., 46(1):197β209, 2014.
|
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[LV22]
|
1701 |
+
Davide Lombardo and Matteo Verzobio. On the local-global principle for isogenies of abelian surfaces,
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1702 |
+
2022. arXiv:2206.15240.
|
1703 |
+
[Mar05]
|
1704 |
+
Greg Martin. Dimensions of the spaces of cusp forms and newforms on Ξ0(N) and Ξ1(N). Journal of
|
1705 |
+
Number Theory, 112(2):298β331, 2005.
|
1706 |
+
[Mit14]
|
1707 |
+
Howard H. Mitchell. The subgroups of the quaternary abelian linear group. Trans. Amer. Math. Soc.,
|
1708 |
+
15(4):379β396, 1914.
|
1709 |
+
[MM97]
|
1710 |
+
M. Ram Murty and V. Kumar Murty. Non-vanishing of L-functions and applications. Modern Birkh¨auser
|
1711 |
+
Classics. Birkh¨auser/Springer Basel AG, Basel, 1997. [2011 reprint of the 1997 original] [MR1482805].
|
1712 |
+
[MT74]
|
1713 |
+
Barry Mazur and John Tate. Points of order 13 on elliptic curves. Invent. Math., 22:41β49, 1973/74.
|
1714 |
+
[MW21]
|
1715 |
+
Jacob Mayle and Tian Wang. On the eο¬ective version of Serreβs open image theorem,
|
1716 |
+
2021.
|
1717 |
+
arXiv:2109.08656.
|
1718 |
+
[Poo17]
|
1719 |
+
Bjorn Poonen. Rational points on varieties, volume 186 of Graduate Studies in Mathematics. American
|
1720 |
+
Mathematical Society, Providence, RI, 2017.
|
1721 |
+
[Ray74]
|
1722 |
+
Michel Raynaud. SchΒ΄emas en groupes de type (p, . . . , p). Bulletin de la SociΒ΄etΒ΄e MathΒ΄ematique de France,
|
1723 |
+
102:241β280, 1974.
|
1724 |
+
[Ser72]
|
1725 |
+
Jean-Pierre Serre. PropriΒ΄etΒ΄es Galoisienne des points dβordre ο¬ni des courbes elliptiques. Inventiones
|
1726 |
+
Mathematicae, 15:259β331, 1972.
|
1727 |
+
[Ser81]
|
1728 |
+
Jean-Pierre
|
1729 |
+
Serre.
|
1730 |
+
Quelques
|
1731 |
+
applications
|
1732 |
+
du
|
1733 |
+
thΒ΄eor`eme
|
1734 |
+
de
|
1735 |
+
densitΒ΄e
|
1736 |
+
de
|
1737 |
+
Chebotarev.
|
1738 |
+
Publications
|
1739 |
+
MathΒ΄ematiques de lβIHΒ΄ES, 54:123β201, 1981.
|
1740 |
+
[Ser87]
|
1741 |
+
Jean-Pierre Serre. Sur les reprΒ΄esentations modulaires de degrΒ΄e 2 de Gal(Q/Q). Duke Math. J., 54(1):179β
|
1742 |
+
230, 1987.
|
1743 |
+
[Ser00]
|
1744 |
+
Jean-Pierre Serre. Lettre `a Marie-France VignΒ΄eras du 10/2/1986. In Oeuvres - Collected Papers IV.
|
1745 |
+
Springer-Verlag Berlin Heidelberg, 2000.
|
1746 |
+
[Shi82]
|
1747 |
+
Ken-ichi Shinoda. The characters of the ο¬nite conformal symplectic group, CSp(4, q). Comm. Algebra,
|
1748 |
+
10(13):1369β1419, 1982.
|
1749 |
+
|
1750 |
+
28
|
1751 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
1752 |
+
[ST68]
|
1753 |
+
Jean-Pierre Serre and John Tate. Good reduction of abelian varieties. Ann. of Math. (2), 88:492β517,
|
1754 |
+
1968.
|
1755 |
+
[The20]
|
1756 |
+
The Sage Developers. SageMath, the Sage Mathematics Software System (Version 9.2), 2020. https:
|
1757 |
+
//www.sagemath.org.
|
1758 |
+
[vBCCK22] Raymond van Bommel, Shiva Chidambaram, Edgar Costa, and Jean Kieο¬er. Computing isogeny classes
|
1759 |
+
of typical principally polarized abelian surfaces over the rationals. In preparation, 2022.
|
1760 |
+
[Zyw15]
|
1761 |
+
David Zywina. On the surjectivity of mod β representations associated to elliptic curves, 2015.
|
1762 |
+
arXiv:1508.07661.
|
1763 |
+
|
1764 |
+
COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
|
1765 |
+
29
|
1766 |
+
Appendix A. Exceptional maximal subgroups of GSp4(Fβ)
|
1767 |
+
β
|
1768 |
+
type
|
1769 |
+
choices
|
1770 |
+
generators
|
1771 |
+
β β‘ 5 (mod 8)
|
1772 |
+
G1920
|
1773 |
+
b2 = β1 in Fβ
|
1774 |
+
β
|
1775 |
+
βββ
|
1776 |
+
β
|
1777 |
+
1
|
1778 |
+
0
|
1779 |
+
0
|
1780 |
+
β1
|
1781 |
+
0
|
1782 |
+
1
|
1783 |
+
β1
|
1784 |
+
0
|
1785 |
+
0
|
1786 |
+
1
|
1787 |
+
1
|
1788 |
+
0
|
1789 |
+
1
|
1790 |
+
0
|
1791 |
+
0
|
1792 |
+
1
|
1793 |
+
β
|
1794 |
+
βββ
|
1795 |
+
β
|
1796 |
+
,
|
1797 |
+
β
|
1798 |
+
βββ
|
1799 |
+
β
|
1800 |
+
1
|
1801 |
+
0
|
1802 |
+
0
|
1803 |
+
b
|
1804 |
+
0
|
1805 |
+
1
|
1806 |
+
b
|
1807 |
+
0
|
1808 |
+
0
|
1809 |
+
b
|
1810 |
+
1
|
1811 |
+
0
|
1812 |
+
b
|
1813 |
+
0
|
1814 |
+
0
|
1815 |
+
1
|
1816 |
+
β
|
1817 |
+
βββ
|
1818 |
+
β
|
1819 |
+
,
|
1820 |
+
β
|
1821 |
+
βββ
|
1822 |
+
β
|
1823 |
+
1
|
1824 |
+
0
|
1825 |
+
0
|
1826 |
+
β1
|
1827 |
+
0
|
1828 |
+
1
|
1829 |
+
1
|
1830 |
+
0
|
1831 |
+
0
|
1832 |
+
β1
|
1833 |
+
1
|
1834 |
+
0
|
1835 |
+
1
|
1836 |
+
0
|
1837 |
+
0
|
1838 |
+
1
|
1839 |
+
β
|
1840 |
+
βββ
|
1841 |
+
β
|
1842 |
+
,
|
1843 |
+
β
|
1844 |
+
βββ
|
1845 |
+
β
|
1846 |
+
1
|
1847 |
+
0
|
1848 |
+
1
|
1849 |
+
0
|
1850 |
+
0
|
1851 |
+
1
|
1852 |
+
0
|
1853 |
+
1
|
1854 |
+
β1
|
1855 |
+
0
|
1856 |
+
1
|
1857 |
+
0
|
1858 |
+
0
|
1859 |
+
β1
|
1860 |
+
0
|
1861 |
+
1
|
1862 |
+
β
|
1863 |
+
βββ
|
1864 |
+
β
|
1865 |
+
β β‘ 3 (mod 8)
|
1866 |
+
G1920
|
1867 |
+
b2 = β2 in Fβ
|
1868 |
+
β
|
1869 |
+
βββ
|
1870 |
+
β
|
1871 |
+
1
|
1872 |
+
0
|
1873 |
+
0
|
1874 |
+
β1
|
1875 |
+
0
|
1876 |
+
1
|
1877 |
+
β1
|
1878 |
+
0
|
1879 |
+
0
|
1880 |
+
1
|
1881 |
+
1
|
1882 |
+
0
|
1883 |
+
1
|
1884 |
+
0
|
1885 |
+
0
|
1886 |
+
1
|
1887 |
+
β
|
1888 |
+
βββ
|
1889 |
+
β
|
1890 |
+
,
|
1891 |
+
β
|
1892 |
+
βββ
|
1893 |
+
β
|
1894 |
+
0
|
1895 |
+
0
|
1896 |
+
0
|
1897 |
+
b
|
1898 |
+
0
|
1899 |
+
0
|
1900 |
+
b
|
1901 |
+
0
|
1902 |
+
0
|
1903 |
+
b
|
1904 |
+
2
|
1905 |
+
0
|
1906 |
+
b
|
1907 |
+
0
|
1908 |
+
0
|
1909 |
+
2
|
1910 |
+
β
|
1911 |
+
βββ
|
1912 |
+
β
|
1913 |
+
,
|
1914 |
+
β
|
1915 |
+
βββ
|
1916 |
+
β
|
1917 |
+
1
|
1918 |
+
0
|
1919 |
+
0
|
1920 |
+
β1
|
1921 |
+
0
|
1922 |
+
1
|
1923 |
+
1
|
1924 |
+
0
|
1925 |
+
0
|
1926 |
+
β1
|
1927 |
+
1
|
1928 |
+
0
|
1929 |
+
1
|
1930 |
+
0
|
1931 |
+
0
|
1932 |
+
1
|
1933 |
+
β
|
1934 |
+
βββ
|
1935 |
+
β
|
1936 |
+
,
|
1937 |
+
β
|
1938 |
+
βββ
|
1939 |
+
β
|
1940 |
+
1
|
1941 |
+
0
|
1942 |
+
1
|
1943 |
+
0
|
1944 |
+
0
|
1945 |
+
1
|
1946 |
+
0
|
1947 |
+
1
|
1948 |
+
β1
|
1949 |
+
0
|
1950 |
+
1
|
1951 |
+
0
|
1952 |
+
0
|
1953 |
+
β1
|
1954 |
+
0
|
1955 |
+
1
|
1956 |
+
β
|
1957 |
+
βββ
|
1958 |
+
β
|
1959 |
+
β β‘ 7 (mod 12)
|
1960 |
+
G720
|
1961 |
+
a2 + a + 1 = 0 in Fβ
|
1962 |
+
β
|
1963 |
+
βββ
|
1964 |
+
β
|
1965 |
+
a
|
1966 |
+
0
|
1967 |
+
0
|
1968 |
+
0
|
1969 |
+
0
|
1970 |
+
a
|
1971 |
+
0
|
1972 |
+
0
|
1973 |
+
0
|
1974 |
+
0
|
1975 |
+
1
|
1976 |
+
0
|
1977 |
+
0
|
1978 |
+
0
|
1979 |
+
0
|
1980 |
+
1
|
1981 |
+
β
|
1982 |
+
βββ
|
1983 |
+
β
|
1984 |
+
,
|
1985 |
+
β
|
1986 |
+
βββ
|
1987 |
+
β
|
1988 |
+
a
|
1989 |
+
0
|
1990 |
+
0
|
1991 |
+
0
|
1992 |
+
0
|
1993 |
+
1
|
1994 |
+
0
|
1995 |
+
0
|
1996 |
+
0
|
1997 |
+
0
|
1998 |
+
a
|
1999 |
+
0
|
2000 |
+
0
|
2001 |
+
0
|
2002 |
+
0
|
2003 |
+
1
|
2004 |
+
β
|
2005 |
+
βββ
|
2006 |
+
β
|
2007 |
+
,
|
2008 |
+
β
|
2009 |
+
βββ
|
2010 |
+
β
|
2011 |
+
a
|
2012 |
+
0
|
2013 |
+
βa β 1
|
2014 |
+
a + 1
|
2015 |
+
0
|
2016 |
+
a
|
2017 |
+
βa β 1
|
2018 |
+
βa β 1
|
2019 |
+
βa β 1
|
2020 |
+
βa β 1
|
2021 |
+
β1
|
2022 |
+
0
|
2023 |
+
a + 1
|
2024 |
+
βa β 1
|
2025 |
+
0
|
2026 |
+
β1
|
2027 |
+
β
|
2028 |
+
βββ
|
2029 |
+
β
|
2030 |
+
,
|
2031 |
+
β
|
2032 |
+
βββ
|
2033 |
+
β
|
2034 |
+
0
|
2035 |
+
β1
|
2036 |
+
0
|
2037 |
+
0
|
2038 |
+
1
|
2039 |
+
0
|
2040 |
+
0
|
2041 |
+
0
|
2042 |
+
0
|
2043 |
+
0
|
2044 |
+
0
|
2045 |
+
β1
|
2046 |
+
0
|
2047 |
+
0
|
2048 |
+
1
|
2049 |
+
0
|
2050 |
+
β
|
2051 |
+
βββ
|
2052 |
+
β
|
2053 |
+
β β‘ 5 (mod 12)
|
2054 |
+
G720
|
2055 |
+
b2 = β1 in Fβ
|
2056 |
+
β
|
2057 |
+
βββ
|
2058 |
+
β
|
2059 |
+
β1
|
2060 |
+
0
|
2061 |
+
0
|
2062 |
+
β1
|
2063 |
+
0
|
2064 |
+
β1
|
2065 |
+
β1
|
2066 |
+
0
|
2067 |
+
0
|
2068 |
+
1
|
2069 |
+
0
|
2070 |
+
0
|
2071 |
+
1
|
2072 |
+
0
|
2073 |
+
0
|
2074 |
+
0
|
2075 |
+
β
|
2076 |
+
βββ
|
2077 |
+
β
|
2078 |
+
,
|
2079 |
+
β
|
2080 |
+
βββ
|
2081 |
+
β
|
2082 |
+
0
|
2083 |
+
0
|
2084 |
+
0
|
2085 |
+
1
|
2086 |
+
0
|
2087 |
+
β1
|
2088 |
+
β1
|
2089 |
+
0
|
2090 |
+
0
|
2091 |
+
1
|
2092 |
+
0
|
2093 |
+
0
|
2094 |
+
β1
|
2095 |
+
0
|
2096 |
+
0
|
2097 |
+
β1
|
2098 |
+
β
|
2099 |
+
βββ
|
2100 |
+
β
|
2101 |
+
,
|
2102 |
+
β
|
2103 |
+
βββ
|
2104 |
+
β
|
2105 |
+
βb β 1
|
2106 |
+
b
|
2107 |
+
2b
|
2108 |
+
β2b + 1
|
2109 |
+
b
|
2110 |
+
b β 1
|
2111 |
+
2b + 1
|
2112 |
+
2b
|
2113 |
+
b
|
2114 |
+
b β 1
|
2115 |
+
βb β 2
|
2116 |
+
βb
|
2117 |
+
βb β 1
|
2118 |
+
b
|
2119 |
+
βb
|
2120 |
+
b β 2
|
2121 |
+
β
|
2122 |
+
βββ
|
2123 |
+
β
|
2124 |
+
,
|
2125 |
+
β
|
2126 |
+
βββ
|
2127 |
+
β
|
2128 |
+
0
|
2129 |
+
βb
|
2130 |
+
β2b
|
2131 |
+
0
|
2132 |
+
b
|
2133 |
+
0
|
2134 |
+
0
|
2135 |
+
2b
|
2136 |
+
β2b
|
2137 |
+
0
|
2138 |
+
0
|
2139 |
+
βb
|
2140 |
+
0
|
2141 |
+
2b
|
2142 |
+
b
|
2143 |
+
0
|
2144 |
+
β
|
2145 |
+
βββ
|
2146 |
+
β
|
2147 |
+
β = 7
|
2148 |
+
G5040
|
2149 |
+
a = 2 satisο¬es
|
2150 |
+
a2 + a + 1 = 0
|
2151 |
+
β
|
2152 |
+
βββ
|
2153 |
+
β
|
2154 |
+
2
|
2155 |
+
0
|
2156 |
+
0
|
2157 |
+
0
|
2158 |
+
0
|
2159 |
+
2
|
2160 |
+
0
|
2161 |
+
0
|
2162 |
+
0
|
2163 |
+
0
|
2164 |
+
1
|
2165 |
+
0
|
2166 |
+
0
|
2167 |
+
0
|
2168 |
+
0
|
2169 |
+
1
|
2170 |
+
β
|
2171 |
+
βββ
|
2172 |
+
β
|
2173 |
+
,
|
2174 |
+
οΏ½οΏ½οΏ½
|
2175 |
+
βββ
|
2176 |
+
β
|
2177 |
+
2
|
2178 |
+
0
|
2179 |
+
0
|
2180 |
+
0
|
2181 |
+
0
|
2182 |
+
1
|
2183 |
+
0
|
2184 |
+
0
|
2185 |
+
0
|
2186 |
+
0
|
2187 |
+
2
|
2188 |
+
0
|
2189 |
+
0
|
2190 |
+
0
|
2191 |
+
0
|
2192 |
+
1
|
2193 |
+
β
|
2194 |
+
βββ
|
2195 |
+
β
|
2196 |
+
,
|
2197 |
+
β
|
2198 |
+
βββ
|
2199 |
+
β
|
2200 |
+
6
|
2201 |
+
0
|
2202 |
+
5
|
2203 |
+
2
|
2204 |
+
0
|
2205 |
+
6
|
2206 |
+
5
|
2207 |
+
5
|
2208 |
+
5
|
2209 |
+
5
|
2210 |
+
4
|
2211 |
+
0
|
2212 |
+
2
|
2213 |
+
5
|
2214 |
+
0
|
2215 |
+
4
|
2216 |
+
β
|
2217 |
+
βββ
|
2218 |
+
β
|
2219 |
+
,
|
2220 |
+
β
|
2221 |
+
βββ
|
2222 |
+
β
|
2223 |
+
0
|
2224 |
+
6
|
2225 |
+
0
|
2226 |
+
0
|
2227 |
+
1
|
2228 |
+
0
|
2229 |
+
0
|
2230 |
+
0
|
2231 |
+
0
|
2232 |
+
0
|
2233 |
+
0
|
2234 |
+
6
|
2235 |
+
0
|
2236 |
+
0
|
2237 |
+
1
|
2238 |
+
0
|
2239 |
+
β
|
2240 |
+
βββ
|
2241 |
+
β
|
2242 |
+
,
|
2243 |
+
β
|
2244 |
+
βββ
|
2245 |
+
β
|
2246 |
+
4
|
2247 |
+
6
|
2248 |
+
0
|
2249 |
+
0
|
2250 |
+
6
|
2251 |
+
6
|
2252 |
+
0
|
2253 |
+
0
|
2254 |
+
0
|
2255 |
+
0
|
2256 |
+
4
|
2257 |
+
1
|
2258 |
+
0
|
2259 |
+
0
|
2260 |
+
1
|
2261 |
+
6
|
2262 |
+
β
|
2263 |
+
βββ
|
2264 |
+
β
|
2265 |
+
Table 5. Explicit generators for each exceptional maximal subgroup in GSp4(Fβ)
|
2266 |
+
(up to conjugacy). The matrices described in Table 5 depend on an auxiliary choice
|
2267 |
+
of a parameter denoted either a and b in each case. In each row, any one choice of
|
2268 |
+
the corresponding a and b satisfying the equations described in the table suο¬ces.
|
2269 |
+
|
2270 |
+
30
|
2271 |
+
BANWAIT, BRUMER, KIM, KLAGSBRUN, MAYLE, SRINIVASAN, AND VOGT
|
2272 |
+
Barinder S. Banwait, Department of Mathematics & Statistics, Boston University, Boston, MA
|
2273 |
+
Email address: [email protected]
|
2274 |
+
URL: https://barinderbanwait.github.io/
|
2275 |
+
Armand Brumer, Department of Mathematics, Fordham University, New York, NY
|
2276 |
+
Email address: [email protected]
|
2277 |
+
Hyun Jong Kim, Department of Mathematics, University of Wisconsin-Madison, Madison, WI
|
2278 |
+
Email address: [email protected]
|
2279 |
+
URL: https://sites.google.com/wisc.edu/hyunjongkim
|
2280 |
+
Zev Klagsbrun, Center for Communications Research, San Diego, CA
|
2281 |
+
Email address: [email protected]
|
2282 |
+
Jacob Mayle, Department of Mathematics, Wake Forest University, Winston-Salem, NC
|
2283 |
+
Email address: [email protected]
|
2284 |
+
Padmavathi Srinivasan, ICERM, Providence, RI
|
2285 |
+
Email address: padmavathi [email protected]
|
2286 |
+
URL: https://padmask.github.io/
|
2287 |
+
Isabel Vogt, Department of Mathematics, Brown University, Providence, RI
|
2288 |
+
Email address: [email protected]
|
2289 |
+
URL: https://www.math.brown.edu/ivogt/
|
2290 |
+
|
K9E0T4oBgHgl3EQfSgAu/content/tmp_files/load_file.txt
ADDED
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|
|