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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 field
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 simplified 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 Classification: 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 financial fields (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 first approach
34
+ concerns the modelling of k sample maxima derived over blocks of a certain size m, i.e.
35
+ Mm,i = max(X(iβˆ’1)m+1, . . . , Xim), i ∈ {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†’ ∞, 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 γ ∈ 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 βˆ’t|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†’ xβˆ—, for all the continuity points x of HΞ³(x), where Ft(x) = P(Y ≀ x) and
50
+ s(t) > 0 is a suitable scaling function for any t ≀ xβˆ—, with xβˆ— = inf(x : F(x) <
51
+ ∞). 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 sufficient 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 field to asses the asymptotic accuracy of estimation procedures, we study
67
+ stronger forms of convergence than the pointwise one, as limtβ†’xβˆ— 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 γ (and other related quantities) are typically defined
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 fixed value t of the chosen statistic. Asymptotic
88
+ properties for POT methods have been studied in the last fifty 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 defining
98
+ it as a functional of the random sequence (Z1, . . . , Zk) which is distributed according the
99
+ 2
100
+
101
+ limit model Hγ, and, if density of Ft(s(t)·) satisfies some regularity conditions, then the
102
+ same asymptotic results hold even when such estimator is defined through the sequence
103
+ of excesses. This approach simplifies 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 Gγ, shortly denoted as F ∈ D(Gγ). This means that
114
+ there are norming constants am > 0 and bm ∈ R for m = 1, 2, . . . such that
115
+ lim
116
+ mβ†’βˆž F m(amx + bm) = exp
117
+ οΏ½
118
+ βˆ’ (1 + Ξ³x)βˆ’1/Ξ³οΏ½
119
+ =: GΞ³(x),
120
+ (2.1)
121
+ for all x ∈ R such that 1 + γx > 0, where γ ∈ R, and this is true if only if there is a
122
+ scaling function s(t) > 0 with t < xβˆ— such that
123
+ lim
124
+ tβ†’xβˆ— Ft(s(t)x) = 1 βˆ’ (1 + Ξ³x)βˆ’1/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)βˆ’(1/Ξ³+1)
128
+ and
129
+ gΞ³(x) = GΞ³(x)hΞ³(x),
130
+ respectively. Let U(v) := F ←(1 βˆ’ 1/v), for v β‰₯ 1, where F ← is the left-continuous
131
+ inverse function of F and G←(exp(βˆ’1/x)) = (xΞ³ βˆ’ 1)/Ξ³.
132
+ Then, we recall that the
133
+ first-order condition in formula (2.1) is equivalent to the limit result
134
+ lim
135
+ vβ†’βˆž
136
+ U(vx) βˆ’ U(v)
137
+ a(v)
138
+ = xΞ³ βˆ’ 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 βˆ’ 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
+ x∈[0, xβˆ—βˆ’t
148
+ s(t) )
149
+ |Ft(s(t)x) βˆ’ HΞ³(x)| β†’ 0,
150
+ t β†’ xβˆ—.
151
+ To establish the speed at which Ft(s(t)x) converges uniformly to HΞ³(x), Raoult and
152
+ Worms (2003) relied on a specific 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
+ limvβ†’βˆž |A(v)| = 0 and
156
+ lim
157
+ vβ†’βˆž
158
+ U(vx)βˆ’U(v)
159
+ a(v)
160
+ βˆ’ xΞ³βˆ’1
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 βˆ’ 1)/Ξ³, see de Haan and
168
+ Ferreira (2006, Definition 2.3.1). The rate function A is necessarily regularly varying at
169
+ infinity with index ρ ≀ 0, named second-order parameter (de Haan and Ferreira, 2006,
170
+ Theorem 2.3.3). In the sequel, we use the same specific 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 differentiable function W(x) on R, we denote with W β€²(x) = (βˆ‚/βˆ‚x)W(x)
182
+ and W β€²β€²(x) = (βˆ‚2/βˆ‚x2)W(x) the first and second order derivatives, respectively.
183
+ 3.1
184
+ Strong convergence under classical assumptions
185
+ Let the distribution function F be twice differentiable. In the sequel, we denote f = F β€²,
186
+ gm = (F m(am Β· +bm))β€² and ft = F β€²
187
+ t.
188
+ Under the following classical von Mises-type
189
+ conditions
190
+ lim
191
+ xβ†’βˆž
192
+ xf(x)
193
+ 1 βˆ’ F(x) = 1
194
+ Ξ³ ,
195
+ Ξ³ > 0,
196
+ lim
197
+ xβ†’xβˆ—
198
+ (xβˆ— βˆ’ x)f(x)
199
+ 1 βˆ’ F(x)
200
+ = βˆ’1
201
+ Ξ³ ,
202
+ Ξ³ < 0,
203
+ (3.1)
204
+ lim
205
+ xβ†’xβˆ—
206
+ f(x)
207
+ οΏ½ xβˆ—
208
+ x (1 βˆ’ F(v)dv)
209
+ (1 βˆ’ F(x))2
210
+ = 0,
211
+ Ξ³ = 0,
212
+ we know that the first-order condition in formula (2.3) is satisfied and it holds that
213
+ lim
214
+ vβ†’βˆž va(v)f(a(v)x + U(v)) = (1 + Ξ³x)βˆ’1/Ξ³βˆ’1
215
+ (3.2)
216
+ locally uniformly for (1 + Ξ³x) > 0. Since the equality gm(x) = F mβˆ’1(amx + bm)hm(x)
217
+ holds true, with bm = U(m), am = a(m) and hm(x) = mamf(amx + bm), and since
218
+ F mβˆ’1(amx+bm) converges to GΞ³(x) locally uniformly as m 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 βˆ’ F(t)
224
+ and, setting v = 1/(1 βˆ’ F(t)), we have a(v) = s(t) and v β†’ ∞ as t β†’ xβˆ—. 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 ∈ (0, βˆ’1/Ξ³), if Ξ³ < 0. In turn, by
227
+ Scheffe’s lemma we have
228
+ lim
229
+ tβ†’xβˆ— V (Pt, P) = 0,
230
+ where
231
+ V (Pt, P) = sup
232
+ B∈B
233
+ |Pt(B) βˆ’ P(B)|
234
+ 4
235
+
236
+ is the total variation distance between the probability measures
237
+ Pt(B) := P
238
+ οΏ½X βˆ’ t
239
+ s(t)
240
+ ∈ B
241
+ οΏ½οΏ½οΏ½οΏ½X > t
242
+ οΏ½
243
+ and P(B) := P(Z ∈ B),
244
+ and where Z is a random variable with distribution HΞ³ and B is a set in the Borel
245
+ Οƒ-field of R, denoted by B. Let
246
+ H 2(lt; hΞ³) :=
247
+ οΏ½ οΏ½οΏ½
248
+ lt(x) βˆ’
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) ≀ 2V (Pt, P) ≀ 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†’ xβˆ—.
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 differentiable. Moreover, assume that there
267
+ exists ρ ≀ 0 such that
268
+ A(v) := vUβ€²β€²(v)
269
+ U β€²(v) + 1 βˆ’ Ξ³
270
+ defines a function of constant sign near infinity, whose absolute value |A(v)| is regularly
271
+ varying as v β†’ ∞ with index of variation ρ.
272
+ When Condition 3.1 holds then the classical von-Mises conditions in formula (3.1)
273
+ are also satisfied for the cases where γ 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 < xβˆ—, which complies with the equivalent first-order
276
+ condition (2.2), is defined as
277
+ s(t) = (1 βˆ’ 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 satisfied with Ξ³ > βˆ’1/2. Then, there exist
280
+ constants ci > 0 with i = 1, 2, Ξ±j > 0 with j = 1, ..., 4, K > 0 and t0 < xβˆ— such that
281
+ H 2(lt; hΞ³)
282
+ K|A(v)|2 ≀ S(v)
283
+ (3.4)
284
+ for all t β‰₯ t0, where v = 1/(1 βˆ’ F(t)) and
285
+ S(v) :=
286
+ οΏ½
287
+ 1 βˆ’ |A(v)|Ξ±1 + 4 exp (c1|A(v)|Ξ±2) ,
288
+ if Ξ³ β‰₯ 0
289
+ 1 βˆ’ |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) ≀ |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 xβˆ—. Since for a fixed x ∈ ∩tβ‰₯t0(0, xβˆ—βˆ’t
304
+ s(t) ) it holds that
305
+ |Ft(s(t)x) βˆ’ HΞ³(x)| ≀ V (Pt, P)
306
+ and since Raoult and Worms (2003, Theorem 2(i)) implies that |Ft(s(t)x)βˆ’HΞ³(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)| ≀ V (Pt, P) ≀ H (lt; hΞ³) ≀ C|A(v)|,
310
+ the decay rate of variational and Hellinger distances is precisely |A(v)| as t β†’ xβˆ—.
311
+ Differently 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 > βˆ’1/2. Although the bound in formula (3.4) remains mathematically
314
+ valid also for tail indices below βˆ’1/2, the restriction Ξ³ > βˆ’1/2 is imposed to guarantee
315
+ that constants α3, α4 in the definition of S(v) are positive, so that S(v) is positive and
316
+ bounded as t approaches xβˆ—. 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 (˜lt; hγ) :=
323
+ οΏ½
324
+ ln
325
+ οΏ½
326
+ ˜lt(x)/hγ(x)
327
+ οΏ½
328
+ ˜lt(x)dx,
329
+ and the divergences of higher order p β‰₯ 2
330
+ Dp(˜lt; hγ) :=
331
+ οΏ½ οΏ½οΏ½οΏ½ln
332
+ οΏ½
333
+ ˜lt(x)/hγ(x)
334
+ οΏ½οΏ½οΏ½οΏ½
335
+ p ˜lt(x)dx,
336
+ where ˜lt = (Ft(˜s(t) Β· ))β€² and ˜s(t) is a scaling function possibly different from s(t), which
337
+ ensures that the support of the conditional distribution Ft(˜s(t)x) is contained in that
338
+ of the GP distribution HΞ³ when Ξ³ < 0, i.e. x ∈ R : x < (xβˆ— βˆ’ t)/˜s(t) < βˆ’1/Ξ³. We recall
339
+ indeed that, when Ξ³ is negative, the end-point (xβˆ— βˆ’ t)/s(t) of lt converges to βˆ’1/Ξ³ as
340
+ t approaches xβˆ—. Nevertheless, for t < xβˆ— it can be that (xβˆ— βˆ’ t)/s(t) > βˆ’1/Ξ³, entailing
341
+ that K (lt; hΞ³) = Dp(lt; hΞ³) = ∞. The introduction of a more flexible scaling function
342
+ ˜s 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 first 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 ρ < 0 and γ ̸= 0, and if
347
+ ˜s(t)/s(t) β†’ 1 as t β†’ xβˆ—, then there exist a t1 < xβˆ— and a constant M ∈ (0, ∞) such
348
+ that
349
+ sup
350
+ tβ‰₯t1
351
+ sup
352
+ 0<x< xβˆ—βˆ’t
353
+ ˜s(t)
354
+ ˜lt(x)
355
+ hΞ³(x) < M.
356
+ 6
357
+
358
+ Lemma 3.5. Let Ξ³ > βˆ’1/2. Then, there exists Ο΅ > 0 and L > 0 such that
359
+ H 2(hΞ³; hΞ³β€²(Οƒ Β· )Οƒ) < L2(|Ξ³ βˆ’ Ξ³β€²|2 + |1 βˆ’ Οƒ|2)
360
+ whenever |Ξ³ βˆ’ Ξ³β€²|2 + |1 βˆ’ Οƒ|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 (˜lt; hγ) and higher order divergences Dp(˜lt; hγ).
365
+ Corollary 3.6. Under the assumptions of Theorem 3.2 with in particular ρ < 0 and
366
+ Ξ³ ΜΈ= 0, if there also exists B > 0 such that, for all large t < xβˆ—,
367
+ |s(t)/˜s(t) βˆ’ 1| ≀ B|A(v)|,
368
+ then there exists a t2 < xβˆ— such that, for all t β‰₯ t2
369
+ (a) K (˜lt; hΞ³) ≀ 2M(
370
+ οΏ½
371
+ KS(v) + BL)2|A(v)|2
372
+ (b) Dp(˜lt; hΞ³) ≀ 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 specific examples
378
+ where the properties listed in such lemmas are satisfied, such as the following one.
379
+ Example 3.7. Let F(x) = exp(βˆ’ exp(βˆ’x)), x ∈ R, be the Gumbel distribution function.
380
+ In this case, Condition 3.1 is satisfied with Ξ³ = 0 and ρ = βˆ’1, so that Theorem 3.2
381
+ applies to this example, and for an arbitrarily small Ο΅ > 0 we have
382
+ lt(x)/h0(x) ≀ exp(exp(βˆ’t)) < 1 + Ο΅
383
+ for all x > 0 and suitably large t. Hence, the bounded density ratio property is satisfied
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 estimators’ 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 satisfied, 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 ∈ N, over a
408
+ large quantile t = U(n/k), is as follows. Let X(nβˆ’k) < . . . < 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(nβˆ’k), ..., X(n))
411
+ be a generic estimator of the tail index Ξ³ and
412
+ οΏ½sk = Tk,2(X(nβˆ’k), ..., 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‰ˆ 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 ≑ k(n), the following conditions are satisfied as n β†’ ∞:
429
+ (a) k β†’ ∞ and k/n β†’ 0,
430
+ (b)
431
+ √
432
+ k|A(n/k)| β†’ Ξ» ∈ (0, ∞),
433
+ (c) |οΏ½Ξ³k βˆ’ Ξ³| = Op(1/
434
+ √
435
+ k) and |οΏ½sk/s(U(n/k)) βˆ’ 1| = Op(1/
436
+ √
437
+ k),
438
+ it then holds that
439
+ H (ft;οΏ½hk) = Op(1/
440
+ √
441
+ k).
442
+ For some specific choices of the estimators �γk and �sk proposed in the literature
443
+ on POT methods (e.g. de Haan and Ferreira, 2006, Ch.
444
+ 3–5), assumptions (a)–(b)
445
+ of Proposition 4.1 have been used along with the second order condition to establish
446
+ asymptotic normality of the sequence
447
+ √
448
+ k
449
+ οΏ½
450
+ οΏ½Ξ³k βˆ’ Ξ³,
451
+ οΏ½sk
452
+ s(U(n/k)) βˆ’ 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)–(b) of Propo-
459
+ sition 4.1, there exists a sequence of ML estimators of Ξ³ and s(U(n/k)) given by
460
+ (�γk, �sk) ∈ arg max
461
+ (Ξ³,Οƒ)∈D
462
+ k
463
+ οΏ½
464
+ i=1
465
+ hΞ³
466
+ οΏ½X(nβˆ’k+i) βˆ’ X(nβˆ’k)
467
+ Οƒ
468
+ οΏ½ 1
469
+ Οƒ
470
+ where D = (βˆ’1/2, ∞) Γ— (0, ∞), 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 γ and s(U(n/k)) are defined as
475
+ οΏ½Ξ³k = 1 βˆ’
476
+ οΏ½ Pk
477
+ 2Qk
478
+ βˆ’ 1
479
+ οΏ½βˆ’1
480
+ ,
481
+ οΏ½sk = Pk
482
+ οΏ½ Pk
483
+ 2Qk
484
+ βˆ’ 1
485
+ οΏ½βˆ’1
486
+ ,
487
+ where
488
+ Pk = 1
489
+ k
490
+ kβˆ’1
491
+ οΏ½
492
+ i=0
493
+ οΏ½
494
+ X(nβˆ’i) βˆ’ X(nβˆ’k)
495
+ οΏ½
496
+ ,
497
+ Qk = 1
498
+ k
499
+ kβˆ’1
500
+ οΏ½
501
+ i=0
502
+ i
503
+ k
504
+ οΏ½
505
+ X(nβˆ’i) βˆ’ X(nβˆ’k)
506
+ οΏ½
507
+ .
508
+ Under the assumptions of Theorem 3.2 and conditions (a)–(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 defined as suitable
516
+ functionals of peaks/excesses over a large order statistic X(nβˆ’k), defined though the k
517
+ larger statistics in a sample as
518
+ Yk := (X(nβˆ’k+1) βˆ’ X(nβˆ’k), . . . , X(n) βˆ’ X(nβˆ’k)).
519
+ Informally speaking, the random variable X(nβˆ’k) plays the role of a high threshold t
520
+ and the sequence (X(nβˆ’k+i) βˆ’ X(nβˆ’k)) 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 first recall the
527
+ notion of contiguity, see van der Vaart (e.g., 2000, Ch. 6.2) for more details.
528
+ Definition 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 ≑ k(n) and k β†’ ∞ as n β†’ ∞.
532
+ Proposition 4.5. Let Pk and Qk be the probability measures relative to the random
533
+ sequences Zk and Yk/˜s(X(nβˆ’k)), respectively. Then, under the assumptions of Corollary
534
+ 3.6 and assumptions (a)–(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, ∞)k and Tk(Zk) is consistent estimator of ΞΈ as n β†’ ∞, then
542
+ also Tk(Yk) is a consistent estimator of ΞΈ as n 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 ↓ 0 and D(Tk(Yk), ΞΈ) = Op(Ο΅k) as k 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 defined 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/˜s(X(nβˆ’k)) 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 < xβˆ—, we define the functions
569
+ pt(y) =
570
+ οΏ½ T(y+T βˆ’1(t))βˆ’t
571
+ s(t)
572
+ βˆ’ eΞ³yβˆ’1
573
+ Ξ³
574
+ ,
575
+ Ξ³ ΜΈ= 0
576
+ T(y+T βˆ’1(t))βˆ’t
577
+ s(t)
578
+ βˆ’ y,
579
+ Ξ³ = 0
580
+ ,
581
+ with s(t) = (1 βˆ’ F(t))/f(t), and
582
+ qt(y) =
583
+ οΏ½
584
+ 1
585
+ Ξ³ ln [1 + Ξ³eβˆ’Ξ³ypt(y)] ,
586
+ Ξ³ ΜΈ= 0
587
+ pt(y),
588
+ Ξ³ = 0
589
+ .
590
+ Moreover, for x ∈ (0, xβˆ— βˆ’ t), we let Ο†t(x) = T βˆ’1(x + t) βˆ’ T βˆ’1(t). Finally, for x ∈ R,
591
+ Ξ³ ∈ R, ρ ≀ 0 and Οƒ > 0, we set
592
+ Iγ,ρ(x) =
593
+ οΏ½ x
594
+ 0
595
+ eΞ³s
596
+ οΏ½ s
597
+ 0
598
+ eρzdzds
599
+ and ψx,Ξ³ = Ξ½x/Οƒ(Ξ³)
600
+ οΏ½
601
+ 1/Οƒ, 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 5.1–5.6, 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± ∈ (0, βˆ’1/Ξ³), if Ξ³ < 0,
615
+ there exist x1 < xβˆ— and ΞΊ1 > 0 such that, for all t β‰₯ x1 and y ∈ (0, βˆ’Ξ± ln |A(eT βˆ’1(t))|)
616
+ (a) if Ξ³ β‰₯ 0, then
617
+ eqt(y) ∈
618
+ οΏ½
619
+ eΒ±ΞΊ1|A(eT βˆ’1(t))|e2Ξ΅yοΏ½
620
+ ;
621
+ 10
622
+
623
+ (b) if Ξ³ < 0, then
624
+ eqt(y) ∈
625
+ οΏ½
626
+ eΒ±ΞΊ1|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)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 ∈ (x0, xβˆ—) and y > 0,
630
+ eβˆ’Ξ³x|pt(y)| ≀ (1 + Ξ΅)|A(eT βˆ’1(t))|IΞ³,ρ(y)e(Ξ³βˆ’Ξ΅)y.
631
+ Moreover, for a positive constant Ο‘1
632
+ IΞ³,ρ(y)e(Ξ³βˆ’Ξ΅)y ≀
633
+ οΏ½
634
+ Ο‘1e2Ξ΅y,
635
+ Ξ³ β‰₯ 0
636
+ Ο‘1e(Ξ³βˆ’Ξ΅)y,
637
+ Ξ³ < 0
638
+ .
639
+ Combining these two inequalities, we deduce that
640
+ eβˆ’Ξ³y|pt(y)| ≀
641
+ οΏ½
642
+ (1 + Ξ΅)|A(eT βˆ’1(t))|Ο‘1e2Ξ΅y,
643
+ Ξ³ β‰₯ 0
644
+ (1 + Ξ΅)|A(eT βˆ’1(t))|Ο‘1e(Ξ³βˆ’Ξ΅)y,
645
+ Ξ³ < 0
646
+ .
647
+ (5.1)
648
+ As a consequence, if Ξ³ β‰₯ 0, for any Ξ± > 0 there exists a constant Ο‘2 such that
649
+ sup
650
+ y∈(0,βˆ’Ξ± ln |A(eT βˆ’1(t))|)
651
+ eβˆ’Ξ³y|pt(y)| ≀ Ο‘2|A(eT βˆ’1(t))|1βˆ’2Ρα
652
+ (5.2)
653
+ while, if Ξ³ < 0, for any Ξ± ∈ (0, βˆ’1/Ξ³) there exists a constant Ο‘3 such that
654
+ sup
655
+ y∈(0,βˆ’Ξ± ln |A(eT βˆ’1(t))|)
656
+ eβˆ’Ξ³y|pt(y)| ≀ Ο‘3|A(eT βˆ’1(t))|1βˆ’(Ξ΅βˆ’Ξ³)Ξ±.
657
+ (5.3)
658
+ Therefore, choosing Ξ΅ sufficiently small, eβˆ’Ξ³y|pt(y)| converges to zero uniformly over the
659
+ interval (0, βˆ’Ξ± ln |A(eT βˆ’1(t))|) as t β†’ xβˆ—.
660
+ It now follows that, if y ∈ (0, βˆ’Ξ± ln |A(eT βˆ’1(t))|) and t > x1 for a sufficiently large
661
+ value x1 < xβˆ—, when Ξ³ ΜΈ= 0 a first-order Taylor expansion of the logarithm at 1 yields
662
+ |qt(y)| =
663
+ οΏ½οΏ½οΏ½οΏ½
664
+ 1
665
+ Ξ³
666
+ Ξ³eβˆ’Ξ³ypt(y)
667
+ 1 + Ο‘(t, y)Ξ³eβˆ’Ξ³ypt(y)
668
+ οΏ½οΏ½οΏ½οΏ½
669
+ ≀
670
+ οΏ½
671
+ Ο‘4|A(eT βˆ’1(t))|e2Ξ΅y,
672
+ Ξ³ > 0
673
+ Ο‘5|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)y,
674
+ Ξ³ < 0
675
+ ,
676
+ where Ο‘(t, y) ∈ (0, 1) and Ο‘4, Ο‘5 are positive constants, while when Ξ³ = 0 it holds that
677
+ |qt(y)| = eΞ³yeβˆ’Ξ³y|pt(y)|
678
+ ≀ Ο‘6|A(eT βˆ’1(t))|e2Ξ΅y,
679
+ where Ο‘6 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± ∈ (0, βˆ’1/Ξ³), if Ξ³ < 0,
682
+ there exist x2 < xβˆ— and ΞΊ2 > 0 such that, for all t β‰₯ x2 and y ∈ (0, βˆ’Ξ± ln |A(eT βˆ’1(t))|)
683
+ (a) if Ξ³ β‰₯ 0, then
684
+ 1 + qβ€²
685
+ t(y) ∈
686
+ οΏ½
687
+ eΒ±ΞΊ2|A(eT βˆ’1(t))|e2Ξ΅yοΏ½
688
+ ;
689
+ 11
690
+
691
+ (b) if Ξ³ < 0, then
692
+ 1 + qβ€²
693
+ t(y) ∈
694
+ οΏ½οΏ½
695
+ eΒ±ΞΊ2|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)yοΏ½
696
+ .
697
+ Proof. If Ξ³ ΜΈ= 0
698
+ 1 + qβ€²
699
+ t(y) =
700
+ exp
701
+ οΏ½οΏ½ ey+T βˆ’1(t)
702
+ eT βˆ’1(t)
703
+ A(u)
704
+ u du
705
+ οΏ½
706
+ 1 + Ξ³eβˆ’Ξ³ypt(y)
707
+ ,
708
+ while if Ξ³ = 0
709
+ 1 + qβ€²
710
+ t(y) = exp
711
+ οΏ½οΏ½ ey+T βˆ’1(t)
712
+ eT βˆ’1(t)
713
+ A(u)
714
+ u
715
+ du
716
+ οΏ½
717
+ .
718
+ Therefore, if y ∈ (0, βˆ’Ξ± ln |A(eT βˆ’1(t))|) and t > x2 for a sufficiently large value x2 < xβˆ—,
719
+ using the bounds in formulas (5.1)–(5.3) and choosing a suitably small Ξ΅ we deduce
720
+ 1 + qβ€²
721
+ t(y) ≀
722
+ exp
723
+ οΏ½οΏ½ ey+T βˆ’1(t)
724
+ eT βˆ’1(t)
725
+ A(u)
726
+ u du
727
+ οΏ½
728
+ 1 βˆ’ 1(Ξ³ ΜΈ= 0)|Ξ³|eβˆ’Ξ³y|pt(y)|
729
+ ≀ exp
730
+ οΏ½
731
+ y|A(eT βˆ’1(t))|
732
+ οΏ½
733
+ Γ—
734
+ οΏ½
735
+ οΏ½
736
+ οΏ½
737
+ 1
738
+ 1βˆ’Ο‰1|A(eT βˆ’1(t))|e2Ξ΅y ,
739
+ Ξ³ β‰₯ 0
740
+ 1
741
+ 1βˆ’Ο‰2|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)y ,
742
+ Ξ³ < 0
743
+ ≀
744
+ οΏ½
745
+ οΏ½
746
+ οΏ½
747
+ exp
748
+ οΏ½
749
+ Ο‰3|A(eT βˆ’1(t))|e2Ξ΅yοΏ½
750
+ ,
751
+ Ξ³ β‰₯ 0
752
+ exp
753
+ οΏ½
754
+ Ο‰4|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)yοΏ½
755
+ ,
756
+ Ξ³ < 0
757
+ for positive constants Ο‰i, i = 1, . . . , 4. Similarly,
758
+ 1 + qβ€²
759
+ t(y) β‰₯
760
+ exp
761
+ οΏ½οΏ½ ey+T βˆ’1(t)
762
+ eT βˆ’1(t)
763
+ A(u)
764
+ u du
765
+ οΏ½
766
+ 1 + 1(Ξ³ ΜΈ= 0)|Ξ³|eβˆ’Ξ³y|pt(y)|
767
+ β‰₯ exp
768
+ οΏ½
769
+ βˆ’y|A(eT βˆ’1(t))|
770
+ οΏ½
771
+ Γ—
772
+ οΏ½
773
+ οΏ½
774
+ οΏ½
775
+ 1
776
+ 1+Ο‰5|A(eT βˆ’1(t))|e2Ξ΅y ,
777
+ Ξ³ β‰₯ 0
778
+ 1
779
+ 1+Ο‰6|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)y ,
780
+ Ξ³ < 0
781
+ β‰₯
782
+ οΏ½
783
+ οΏ½
784
+ οΏ½
785
+ exp
786
+ οΏ½
787
+ βˆ’Ο‰7|A(eT βˆ’1(t))|e2Ξ΅yοΏ½
788
+ ,
789
+ Ξ³ β‰₯ 0
790
+ exp
791
+ οΏ½
792
+ βˆ’Ο‰8|A(eT βˆ’1(t))|e(Ξ³βˆ’Ξ΅)yοΏ½
793
+ ,
794
+ Ξ³ < 0
795
+ for positive constants Ο‰i, i = 5, . . . , 8. The result now follows.
796
+ Lemma 5.3. If γ > 0 and ρ < 0, there exists a regularly varying function R with
797
+ negative index ϱ such that, defining the function
798
+ Ξ·(t) := (1 + Ξ³t)f(t)
799
+ 1 βˆ’ F(t)
800
+ βˆ’ 1,
801
+ as v β†’ ∞, Ξ·(U(v)) = O(R(v)).
802
+ Proof. Let v0 > 0 satisfy U(v0) ΜΈ= 0 and U β€²(v0) ΜΈ= 0. Then, for v > v0 it holds that
803
+ Ξ·(U(v)) = 1 + Ξ³U(v)
804
+ vUβ€²(v)
805
+ βˆ’ 1
806
+ = 1 + Ξ³U(v0)
807
+ vUβ€²(v)
808
+ + Ξ³
809
+ οΏ½ v
810
+ v0
811
+ U β€²(r)
812
+ vUβ€²(v)dr βˆ’ 1.
813
+ 12
814
+
815
+ Moreover, by definition of A, we have the identity
816
+ Ξ³
817
+ οΏ½ v
818
+ v0
819
+ U β€²(r)
820
+ vUβ€²(v)dr βˆ’ 1 =
821
+ οΏ½ 1
822
+ v0/v
823
+ U β€²(zv)
824
+ U β€²(v) dz βˆ’ 1
825
+ =
826
+ οΏ½ 1
827
+ v0/v
828
+ Ξ³zΞ³βˆ’1
829
+ οΏ½
830
+ exp
831
+ οΏ½
832
+ βˆ’
833
+ οΏ½ 1
834
+ z
835
+ A(vu)
836
+ u
837
+ du
838
+ οΏ½
839
+ βˆ’ 1
840
+ οΏ½
841
+ dz βˆ’
842
+ οΏ½v0
843
+ v
844
+ οΏ½Ξ³
845
+ .
846
+ Therefore, denoting by R2(v) the first term on the right-hand side and setting
847
+ R1(v) = 1 + Ξ³U(v0)
848
+ vUβ€²(v)
849
+ βˆ’
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 βˆ’Ξ³. On the other hand, for any Ξ² ∈ (0, 1), the function R2(v) can be
857
+ decomposed as follows
858
+ R2(v) =
859
+ οΏ½ vβˆ’(1βˆ’Ξ²)
860
+ v0/v
861
+ +
862
+ οΏ½ 1
863
+ vβˆ’(1βˆ’Ξ²) Ξ³zΞ³βˆ’1
864
+ οΏ½
865
+ exp
866
+ οΏ½
867
+ βˆ’
868
+ οΏ½ 1
869
+ z
870
+ A(vu)
871
+ u
872
+ du
873
+ οΏ½
874
+ βˆ’ 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)| ≀
880
+ οΏ½ vβˆ’(1βˆ’Ξ²)
881
+ v0/v
882
+ Ξ³zΞ³βˆ’1
883
+ οΏ½
884
+ 1 βˆ’ exp
885
+ οΏ½
886
+ βˆ’A(vz)
887
+ z
888
+ οΏ½οΏ½
889
+ dz
890
+ = O(vβˆ’Ξ³(1βˆ’Ξ²))
891
+ and
892
+ |R2,2(v)| ≀
893
+ οΏ½ 1
894
+ vβˆ’(1βˆ’Ξ²) Ξ³zΞ³βˆ’1 οΏ½
895
+ 1 βˆ’ zA(vΞ²)οΏ½
896
+ dz
897
+ = O(vβˆ’Ξ³(1βˆ’Ξ²) ∨ A(vΞ²)).
898
+ Consequently, there exists a regularly varying function R of index Ο± = Ξ³(Ξ² βˆ’ 1) ∨ ρβ
899
+ complying with the property in the statement as v 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)| ≀
903
+ οΏ½ vβˆ’(1βˆ’Ξ²)
904
+ v0/v
905
+ Ξ³zΞ³βˆ’1 οΏ½
906
+ uA(v0) βˆ’ 1
907
+ οΏ½
908
+ dz
909
+ = O(vβˆ’(Ξ³βˆ’Ξ²/2)(1βˆ’Ξ²))
910
+ and
911
+ |R2,2(v)| ≀
912
+ οΏ½ 1
913
+ vβˆ’(1βˆ’Ξ²) Ξ³zΞ³βˆ’1 οΏ½
914
+ zA(vΞ²) βˆ’ 1
915
+ οΏ½
916
+ dz
917
+ = O(vβˆ’(Ξ³βˆ’Ξ²/2)(1βˆ’Ξ²) ∨ |A(vΞ²)|)
918
+ as v β†’ ∞. Hence, there exists a regularly varying function R of index Ο± = (Ξ² βˆ’ 1)(Ξ³ βˆ’
919
+ β/2)∨ρβ complying with the property in the statement. The proof is now complete.
920
+ Lemma 5.4. If γ > 0 and ρ < 0, there exists x3 ∈ (0, ∞) and δ > 0 such that, for all
921
+ x β‰₯ x3,
922
+ f(x) = hΞ³(x)
923
+ οΏ½
924
+ 1 + O({1 βˆ’ HΞ³(x)}Ξ΄)
925
+ οΏ½
926
+ .
927
+ 13
928
+
929
+ Proof. Let Rβˆ—(t) := R(1/(1 βˆ’ F(t))), where R is as in Lemma 5.3. Then Rβˆ—(t) is regu-
930
+ larly varying of index Ο±/Ξ³ (Resnick, 2007, Proposition 0.8(iv)). In turn, by Karamata’s
931
+ theorem (e.g, Resnick, 2007, Proposition 0.6(a)) we have that for a large tβˆ—
932
+ � ∞
933
+ tβˆ—
934
+ |Ξ·(t)|
935
+ 1 + γtdt < ∞
936
+ and thus, by Proposition 2.1.4 in Falk et al. (2010), we conclude that
937
+ Ο„ := lim
938
+ tβ†’βˆž
939
+ 1 βˆ’ F(t)
940
+ 1 βˆ’ HΞ³(t) ∈ (0, ∞).
941
+ (5.4)
942
+ As a consequence, for any Ξ΄ ∈ (0, βˆ’Ο±), as t β†’ ∞
943
+ Rβˆ—(t) ∼ R
944
+ οΏ½
945
+ 1
946
+ Ο„(1 βˆ’ HΞ³(t))
947
+ οΏ½
948
+ = O({1 βˆ’ HΞ³(t)}Ξ΄).
949
+ The conclusion now follows by Proposition 2.1.5 in Falk et al. (2010).
950
+ Lemma 5.5. If γ < 0 and ρ < 0, there exists a a regularly varying function ˜R with
951
+ negative index ˜ϱ = (βˆ’1) ∨ (βˆ’Ο/Ξ³) such that, defining the function
952
+ ˜η(y) := (1 βˆ’ Ξ³y)f(xβˆ— βˆ’ 1/y)
953
+ [1 βˆ’ F(xβˆ— βˆ’ 1/y)]y2 βˆ’ 1,
954
+ as y β†’ ∞, ˜η(y) = O( ˜R(y)).
955
+ Proof. By definition,
956
+ ˜η (y) =
957
+ f(xβˆ— βˆ’ 1/y)
958
+ [1 βˆ’ F(xβˆ— βˆ’ 1/y)]y2 βˆ’ Ξ³
959
+ οΏ½
960
+ f(xβˆ— βˆ’ 1/y)
961
+ y(1 βˆ’ F(xβˆ— βˆ’ 1/y)) + 1
962
+ Ξ³
963
+ οΏ½
964
+ =: ˜η1 (y) + ˜η2 (y) .
965
+ On one hand, we have that, as y β†’ ∞
966
+ ˜η1 (y) = O(1/y).
967
+ On the other hand, for v > 1 we have the identity
968
+ ˜η2
969
+ οΏ½
970
+ 1
971
+ xβˆ— βˆ’ U(v)
972
+ οΏ½
973
+ =
974
+ � ∞
975
+ 1
976
+ Ξ³zΞ³βˆ’1
977
+ οΏ½
978
+ 1 βˆ’ 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
+ ˜η2
988
+ οΏ½
989
+ 1
990
+ xβˆ— βˆ’ U(v)
991
+ οΏ½
992
+ ≀ βˆ’Ξ³
993
+ � ∞
994
+ 1
995
+ zΞ³βˆ’1(zA(v) βˆ’ 1)dz
996
+ = O(A(v))
997
+ while, if A is ultimately negative,
998
+ ����˜η2
999
+ οΏ½
1000
+ 1
1001
+ xβˆ— βˆ’ U(v)
1002
+ οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ³A(v)
1003
+ � ∞
1004
+ 1
1005
+ zΞ³βˆ’1 ln zdz
1006
+ = O(|A(v)|).
1007
+ As a result of the two above inequalities, as v β†’ ∞
1008
+ ˜η2(t) = O
1009
+ οΏ½οΏ½οΏ½οΏ½οΏ½A
1010
+ οΏ½
1011
+ 1
1012
+ 1 βˆ’ F(xβˆ— βˆ’ 1/y)
1013
+ οΏ½οΏ½οΏ½οΏ½οΏ½
1014
+ οΏ½
1015
+ ,
1016
+ Therefore, by regular variation of 1/(1 βˆ’ F(xβˆ— βˆ’ 1/y)) with index βˆ’1/Ξ³, ˜η2(y) is even-
1017
+ tually dominated by a regularly varing function of index βˆ’Ο/Ξ³. The final result now
1018
+ follows.
1019
+ 14
1020
+
1021
+ Lemma 5.6. If Ξ³ < 0 and ρ < 0, there exist ˜δ > 0 such that, as y β†’ ∞,
1022
+ f(xβˆ— βˆ’ 1/y)
1023
+ y2
1024
+ = (1 βˆ’ Ξ³y)1/Ξ³βˆ’1 οΏ½
1025
+ 1 + O({1 βˆ’ Hβˆ’Ξ³(y)}
1026
+ ˜δ)
1027
+ οΏ½
1028
+ Proof. The function ˜f(y) := f(xβˆ— βˆ’ 1/y)yβˆ’2 is the density of the distribution function
1029
+ ˜F(y) := F(xβˆ—βˆ’1/y), which is in the domain of attraction of G˜γ, with ˜γ = βˆ’Ξ³. Moreover,
1030
+ ˜η(y) = (1 + ˜γy) ˜f(y)
1031
+ 1 βˆ’ ˜F(y)
1032
+ βˆ’ 1.
1033
+ By Lemma 5.5 and regular variation of 1 βˆ’ H˜γ with index βˆ’1/˜γ, we have
1034
+ ˜η(y) = O({1 βˆ’ H˜γ(y)}
1035
+ ˜δ)
1036
+ for any ˜δ > 0 such that βˆ’ΛœΞ΄/˜γ > ˜ϱ. Therefore, by Proposition 2.1.5 in Falk et al. (2010),
1037
+ as y β†’ ∞ it holds that
1038
+ ˜f(y) = h˜γ(y)[1 + O({1 βˆ’ H˜γ(y)}
1039
+ ˜δ)],
1040
+ which is the result.
1041
+ Lemma 5.7. Let Ξ½β€²
1042
+ x(Ξ³) = (βˆ‚/βˆ‚Ξ³)Ξ½x(Ξ³).
1043
+ (a) If ΞΎ : R οΏ½β†’ (0, ∞), then it holds that
1044
+ �� ∞
1045
+ 0
1046
+ [Ξ½β€²x(ΞΎ(x))]2 dx ≀ 1
1047
+ 2
1048
+ �� ∞
1049
+ 0
1050
+ (1 + xΞΎ(x))βˆ’3βˆ’
1051
+ 1
1052
+ ΞΎ(x)
1053
+ οΏ½x ln(1 + xΞΎ(x))
1054
+ ΞΎ(x)
1055
+ οΏ½2
1056
+ dx
1057
+ + 1
1058
+ 2
1059
+ �� ∞
1060
+ 0
1061
+ (1 + xΞΎ(x))βˆ’3βˆ’
1062
+ 1
1063
+ ΞΎ(x) x2dx.
1064
+ (b) If instead ΞΎ : R οΏ½β†’ (Ξ³βˆ—, 0), then
1065
+ οΏ½οΏ½ βˆ’1/Ξ³βˆ—
1066
+ 0
1067
+ [Ξ½β€²x(ΞΎ(x))]2 dx ≀ 1
1068
+ 2
1069
+ οΏ½οΏ½ βˆ’1/Ξ³βˆ—
1070
+ 0
1071
+ (1 + xΞΎ(x))βˆ’3βˆ’
1072
+ 1
1073
+ ΞΎ(x) x4dx
1074
+ + 1
1075
+ 2
1076
+ οΏ½οΏ½ βˆ’1/Ξ³βˆ—
1077
+ 0
1078
+ (1 + xΞΎ(x))βˆ’3βˆ’
1079
+ 1
1080
+ ΞΎ(x) x2dx.
1081
+ Proof. Let Ο•x(Ξ³) := (βˆ‚/βˆ‚Ξ³) ln(1 βˆ’ HΞ³(x)). Then, for any x > 0, if ΞΎ(Β·) > 0, or x ∈
1082
+ (0, βˆ’1/Ξ³βˆ—), if ΞΎ(Β·) ∈ (Ξ³βˆ—, 0) we have
1083
+ Ξ½β€²
1084
+ x(ΞΎ(x)) = 1
1085
+ 2(1 + xΞΎ(x))βˆ’ 1
1086
+ 2 βˆ’
1087
+ 1
1088
+ 2ΞΎ(x) Ο•x(ΞΎ(x)) βˆ’ 1
1089
+ 2(1 + xΞΎ(x))βˆ’ 3
1090
+ 2 βˆ’
1091
+ 1
1092
+ 2ΞΎ(x) x.
1093
+ If ΞΎ(Β·) > 0, by Minkowski inequality
1094
+ �� ∞
1095
+ 0
1096
+ [Ξ½β€²x(ΞΎ(x))]2 dx ≀ 1
1097
+ 2
1098
+ �� ∞
1099
+ 0
1100
+ (1 + xΞΎ(x))βˆ’1βˆ’
1101
+ 1
1102
+ ΞΎ(x) [Ο•x(ΞΎ(x))]2 dx
1103
+ + 1
1104
+ 2
1105
+ �� ∞
1106
+ 0
1107
+ (1 + xΞΎ(x))βˆ’3βˆ’
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 ≀ Ο•x(ΞΎ(x)) ≀ x ln(1 + xΞΎ(x))
1115
+ ΞΎ(x)(1 + xΞΎ(x)).
1116
+ If ΞΎ(Β·) ∈ (Ξ³βˆ—, 0), inequality (B.8) in BΒ¨ucher and Segers (2017) implies that for any
1117
+ x ∈ (0, βˆ’1/Ξ³βˆ—)
1118
+ 0 ≀ Ο•x(ΞΎ(x)) ≀
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 Οˆβ€²
1124
+ Ξ³,x(Οƒ) = (βˆ‚/βˆ‚Οƒ)ψγ,x(Οƒ).
1125
+ (a) If Ο‚ : R οΏ½β†’ (1 Β± Ο΅), with Ο΅ ∈ (0, 1), and if Ξ³ > 0
1126
+ �� ∞
1127
+ 0
1128
+ οΏ½
1129
+ Οˆβ€²Ξ³,x(Ο‚(x))
1130
+ οΏ½2 dx ≀
1131
+ οΏ½1
1132
+ Ξ³ +
1133
+ οΏ½
1134
+ 1
1135
+ 2Ξ³ + 1
1136
+ οΏ½ οΏ½1 + Ο΅
1137
+ 1 βˆ’ Ο΅
1138
+ οΏ½5/2
1139
+ .
1140
+ (b) If Ξ³ ∈ (βˆ’1/2, 0) and Ο‚(x) ∈ (Οƒβˆ—, 1), with Οƒβˆ— ∈ (0, 1), there is a constant ΞΆ > 0 such
1141
+ that
1142
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1143
+ Ξ³
1144
+ 0
1145
+ οΏ½
1146
+ Οˆβ€²Ξ³,x(Ο‚(x))
1147
+ οΏ½2 dx ≀
1148
+ 1
1149
+ Οƒβˆ—
1150
+ βˆšβˆ’Ξ³ΞΆ
1151
+ οΏ½ 1
1152
+ Ξ³2 + 1
1153
+ οΏ½
1154
+ .
1155
+ If instead Ο‚(x) > 1,
1156
+ οΏ½οΏ½ βˆ’ 1
1157
+ Ξ³
1158
+ 0
1159
+ οΏ½
1160
+ Οˆβ€²Ξ³,x(Ο‚(x))
1161
+ οΏ½2 dx ≀
1162
+ 1
1163
+ βˆšβˆ’Ξ³ΞΆ
1164
+ οΏ½ 1
1165
+ Ξ³2 + 1
1166
+ οΏ½
1167
+ .
1168
+ Proof. Note that for x such that 1 + Ξ³x/Οƒ > 0
1169
+ Οˆβ€²
1170
+ Ξ³,x(Οƒ) = (1 + Ξ³x/Οƒ)βˆ’ 1
1171
+ 2Ξ³ βˆ’ 3
1172
+ 2
1173
+ Οƒ5/2
1174
+ x
1175
+ Ξ³ + (1 + Ξ³x/Οƒ)βˆ’ 1
1176
+ 2Ξ³ βˆ’ 3
1177
+ 2
1178
+ Οƒ3/2
1179
+ .
1180
+ Consequently, if Ο‚ : R οΏ½β†’ (1 Β± Ο΅) and Ξ³ > 0, by Minkowski inequality
1181
+ �� ∞
1182
+ 0
1183
+ οΏ½
1184
+ Οˆβ€²Ξ³,x(Ο‚(x))
1185
+ οΏ½2 dx
1186
+ ≀
1187
+ �� ∞
1188
+ 0
1189
+ (1 + Ξ³x/Ο‚(x))βˆ’ 1
1190
+ Ξ³ βˆ’3
1191
+ Ο‚5(x)
1192
+ οΏ½x
1193
+ Ξ³
1194
+ οΏ½2
1195
+ dx +
1196
+ �� ∞
1197
+ 0
1198
+ (1 + Ξ³x/Ο‚(x))βˆ’ 1
1199
+ Ξ³ βˆ’3
1200
+ Ο‚3(x)
1201
+ dx
1202
+ ≀ (1 + Ο΅)
1203
+ 3
1204
+ 2
1205
+ (1 βˆ’ Ο΅)
1206
+ 5
1207
+ 2
1208
+ 1
1209
+ Ξ³ + (1 + Ο΅)
1210
+ 1
1211
+ 2
1212
+ (1 βˆ’ 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·) ∈ (Οƒβˆ—, 1), for some Οƒβˆ— ∈ (0, 1), and Ξ³ ∈ (βˆ’1/2, 0), there is a constant
1224
+ ΞΆ > 0 such that
1225
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1226
+ Ξ³
1227
+ 0
1228
+ οΏ½
1229
+ Οˆβ€²Ξ³,x(Ο‚(x))
1230
+ οΏ½2 dx
1231
+ ≀
1232
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1233
+ Ξ³
1234
+ 0
1235
+ (1 + Ξ³x/Ο‚(x))βˆ’ 1
1236
+ Ξ³ βˆ’3
1237
+ Ο‚5(x)
1238
+ οΏ½x
1239
+ Ξ³
1240
+ οΏ½2
1241
+ dx +
1242
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1243
+ Ξ³
1244
+ 0
1245
+ (1 + Ξ³x/Ο‚(x))βˆ’ 1
1246
+ Ξ³ βˆ’3
1247
+ Ο‚3(x)
1248
+ dx
1249
+ ≀
1250
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1251
+ Ξ³
1252
+ 0
1253
+ (1 + Ξ³x/Οƒβˆ—)βˆ’1+ΞΆ
1254
+ Οƒ3βˆ—
1255
+ 1
1256
+ Ξ³4 dx +
1257
+ οΏ½οΏ½ βˆ’ Οƒβˆ—
1258
+ Ξ³
1259
+ 0
1260
+ (1 + Ξ³x/Οƒβˆ—)βˆ’1+ΞΆ
1261
+ Οƒ3βˆ—
1262
+ dx
1263
+ =
1264
+ οΏ½
1265
+ 1
1266
+ ΞΆΟƒ2βˆ—(βˆ’Ξ³)5 +
1267
+ οΏ½
1268
+ 1
1269
+ ΞΆΟƒ2βˆ—(βˆ’Ξ³).
1270
+ The first 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
+ � ∞
1280
+ xt
1281
+ οΏ½οΏ½
1282
+ ft(x) βˆ’
1283
+ οΏ½
1284
+ hΞ³(x/s(t))/s(t)
1285
+ οΏ½2
1286
+ dx
1287
+ ≀
1288
+ οΏ½ Ο†t(xt)
1289
+ 0
1290
+ eβˆ’y
1291
+ οΏ½
1292
+ 1 βˆ’
1293
+ οΏ½
1294
+ eqt(y)(1 + qβ€²
1295
+ t(y))
1296
+ οΏ½2
1297
+ dy
1298
+ +
1299
+ οΏ½οΏ½
1300
+ 1 βˆ’ Ft(xt) +
1301
+ οΏ½
1302
+ 1 βˆ’ HΞ³(xt/s(t))
1303
+ οΏ½2
1304
+ =: I1(t) + I2(t).
1305
+ Let xt be such that the following equality holds
1306
+ Ο†t(xt) = βˆ’Ξ± ln |A(eT βˆ’1(t))|,
1307
+ for a positive constant α to be specified 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 sufficiently large t
1309
+ I1(t) ≀
1310
+ οΏ½
1311
+ οΏ½
1312
+ οΏ½
1313
+ οΏ½ βˆ’Ξ± ln |A(eT βˆ’1(t))|
1314
+ 0
1315
+ ΞΊ3|A(eT βˆ’1(t))|2e(4Ξ΅βˆ’1)ydy,
1316
+ Ξ³ β‰₯ 0
1317
+ οΏ½ βˆ’Ξ± ln |A(eT βˆ’1(t))|
1318
+ 0
1319
+ ΞΊ4|A(eT βˆ’1(t))|2e(Ξ³βˆ’Ξ΅βˆ’1)ydy,
1320
+ Ξ³ < 0
1321
+ ≀
1322
+ οΏ½
1323
+ οΏ½
1324
+ οΏ½
1325
+ ΞΊ3|A(eT βˆ’1(t))|2 οΏ½
1326
+ 1 βˆ’ |A(eT βˆ’1(t))|Ξ±1
1327
+ οΏ½
1328
+ ,
1329
+ Ξ³ β‰₯ 0
1330
+ ΞΊ4|A(eT βˆ’1(t))|2 οΏ½
1331
+ 1 βˆ’ |A(eT βˆ’1(t))|Ξ±3
1332
+ οΏ½
1333
+ ,
1334
+ Ξ³ < 0
1335
+ ,
1336
+ where Ξ±1 := Ξ±(1 βˆ’ 4Ξ΅) and Ξ±3 := Ξ±(1 βˆ’ 2(Ξ΅ βˆ’ Ξ³)) are positive. Moreover, on one hand
1337
+ we have the identity
1338
+ 1 βˆ’ Ft(xt) = |A(eT βˆ’1(t))|Ξ±.
1339
+ On the other hand, for some constants ΞΊ5, ΞΊ6 > 0 we have the inequality
1340
+ 1 βˆ’ HΞ³(xt/s(t)) = |A(eT βˆ’1(t))|Ξ± exp
1341
+ οΏ½
1342
+ βˆ’qt
1343
+ οΏ½
1344
+ βˆ’Ξ± ln |A(eT βˆ’1(t))|
1345
+ οΏ½οΏ½
1346
+ ≀
1347
+ οΏ½
1348
+ οΏ½
1349
+ οΏ½
1350
+ |A(eT βˆ’1(t))|Ξ± exp
1351
+ οΏ½
1352
+ ΞΊ5|A(eT βˆ’1(t))|1βˆ’2Ρα�
1353
+ ,
1354
+ Ξ³ β‰₯ 0
1355
+ |A(eT βˆ’1(t))|Ξ± exp
1356
+ οΏ½
1357
+ ΞΊ6|A(eT βˆ’1(t))|1βˆ’(Ξ΅βˆ’Ξ³)Ξ±οΏ½
1358
+ ,
1359
+ Ξ³ < 0
1360
+ .
1361
+ 17
1362
+
1363
+ Consequently,
1364
+ I2(t) ≀
1365
+ οΏ½
1366
+ οΏ½
1367
+ οΏ½
1368
+ |A(eT βˆ’1(t))|Ξ± οΏ½
1369
+ 1 + exp
1370
+ οΏ½
1371
+ ΞΊ5
1372
+ 2 |A(eT βˆ’1(t))|1βˆ’2Ρα��
1373
+ ,
1374
+ Ξ³ β‰₯ 0
1375
+ |A(eT βˆ’1(t))|Ξ± οΏ½
1376
+ 1 + exp
1377
+ οΏ½
1378
+ ΞΊ6
1379
+ 2 |A(eT βˆ’1(t))|1βˆ’(Ξ΅βˆ’Ξ³)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 βˆ’1(t))|Ξ± < |A(eT βˆ’1(t))|2
1385
+ and Ξ±2 := 1 βˆ’ 2Ρα > 0. If instead Ξ³ ∈ (βˆ’1/2, 0), we can choose Ξ± slightly larger than
1386
+ 2 and Ρ small enough, so that the inequality in the above display is still satisfied and
1387
+ Ξ±4 := 1βˆ’Ξ±(Ξ΅βˆ’Ξ³) > 0. The conclusion then follows noting that T βˆ’1(t) = βˆ’ ln(1βˆ’F(t))
1388
+ and, in turn,
1389
+ |A(eT βˆ’1(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, ˜s(t) = s(t) = (1 βˆ’ F(t))/f(t) and ˜lt = 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) ≀ hΞ³(s(t)x + t)
1397
+ hΞ³(x)
1398
+ s(t)
1399
+ 1 βˆ’ F(t)
1400
+ οΏ½
1401
+ 1 + ΞΊ {1 βˆ’ HΞ³(s(t)x + t)}Ξ΄οΏ½
1402
+ ≀
1403
+ οΏ½
1404
+ 1 + Ξ³x
1405
+ (1 + Ξ³t)/s(t) + Ξ³x
1406
+ οΏ½1+1/Ξ³
1407
+ 1 + Ο΅
1408
+ (s(t))1/Ξ³(1 βˆ’ F(t)).
1409
+ Moreover, by Lemma 5.3 it holds that as t β†’ ∞
1410
+ 1 + Ξ³t
1411
+ s(t)
1412
+ = 1 + Ξ·(t) = 1 + o(1)
1413
+ and, in turn, (s(t))1/γ ∼ (1+γt)1/γ. These two facts, combined with the tail equivalence
1414
+ relation in formula (5.4), imply that for all sufficiently large t and all x > 0
1415
+ lt(x)
1416
+ hΞ³(x) ≀
1417
+ οΏ½
1418
+ 1 + Ξ³x
1419
+ 1 βˆ’ Ο΅ + Ξ³x
1420
+ οΏ½1+1/Ξ³
1421
+ 1 + Ο΅
1422
+ (1 βˆ’ Ο΅)Ο„
1423
+ ≀
1424
+ οΏ½
1425
+ 1
1426
+ 1 βˆ’ Ο΅
1427
+ οΏ½1+1/Ξ³
1428
+ 1 + Ο΅
1429
+ (1 βˆ’ Ο΅)Ο„ .
1430
+ The result now follows.
1431
+ Case 2: Ξ³ < 0. In this case, for any x ∈ (0, (xβˆ— βˆ’ t)/˜s(t))
1432
+ ˜lt(x) = f
1433
+ οΏ½
1434
+ xβˆ— βˆ’ 1
1435
+ y
1436
+ οΏ½ 1
1437
+ y2
1438
+ y2˜s(t)
1439
+ 1 βˆ’ F(t)
1440
+ where
1441
+ y ≑ y(x, t) :=
1442
+ 1
1443
+ ˜s(t)
1444
+ οΏ½xβˆ— βˆ’ t
1445
+ ˜s(t)
1446
+ βˆ’ x
1447
+ οΏ½βˆ’1
1448
+ Note that y is bounded from below by 1/(xβˆ— βˆ’ t), which converges to ∞ as t β†’ xβˆ—.
1449
+ Thus, by Lemma 5.6 there are positive constants ˜δ, ϡ and ˜κ such that
1450
+ ˜lt(x) ≀ (1 βˆ’ Ξ³y)1/Ξ³βˆ’1[1 + ˜κ{1 βˆ’ Hβˆ’Ξ³(y)}
1451
+ ˜δ] y2˜s(t)
1452
+ 1 βˆ’ F(t)
1453
+ ≀
1454
+ οΏ½
1455
+ 1 + γ ˜s(t)
1456
+ xβˆ— βˆ’ t
1457
+ οΏ½
1458
+ βˆ’1
1459
+ Ξ³
1460
+ οΏ½
1461
+ x
1462
+ οΏ½βˆ’ 1
1463
+ Ξ³ βˆ’1 οΏ½
1464
+ (xβˆ— βˆ’ t)
1465
+ οΏ½
1466
+ 1 βˆ’ ˜s(t)x
1467
+ xβˆ— βˆ’ t
1468
+ οΏ½
1469
+ βˆ’ Ξ³
1470
+ οΏ½ 1
1471
+ Ξ³ βˆ’1 ˜s(t)(1 + Ο΅)
1472
+ 1 βˆ’ F(t) (xβˆ— βˆ’ t)βˆ’ 1
1473
+ Ξ³ βˆ’1.
1474
+ 18
1475
+
1476
+ By hypothesis, it holds that
1477
+ xβˆ— βˆ’ t
1478
+ ˜s(t)
1479
+ ≀ βˆ’1
1480
+ Ξ³ ,
1481
+ thus
1482
+ οΏ½
1483
+ 1 + γ ˜s(t)
1484
+ xβˆ— βˆ’ t
1485
+ οΏ½
1486
+ βˆ’1
1487
+ Ξ³
1488
+ οΏ½
1489
+ x
1490
+ οΏ½βˆ’ 1
1491
+ Ξ³ βˆ’1
1492
+ ≀ (1 + Ξ³x)βˆ’1/Ξ³βˆ’1.
1493
+ Moreover, it holds that
1494
+ 1 βˆ’ ˜s(t)x
1495
+ xβˆ— βˆ’ t > 0,
1496
+ thus
1497
+ οΏ½
1498
+ (xβˆ— βˆ’ t)
1499
+ οΏ½
1500
+ 1 βˆ’ ˜s(t)x
1501
+ xβˆ— βˆ’ t
1502
+ οΏ½
1503
+ βˆ’ Ξ³
1504
+ οΏ½ 1
1505
+ Ξ³ βˆ’1
1506
+ ≀ (βˆ’Ξ³)
1507
+ 1
1508
+ Ξ³ βˆ’1.
1509
+ Finally, for all large t,
1510
+ ˜s(t)
1511
+ xβˆ— βˆ’ t ≀ βˆ’(1 + Ο΅)Ξ³.
1512
+ Combining all the above inequalities we can now conclude that, for all large t and for
1513
+ any x ∈ (0, (xβˆ— βˆ’ t)/˜s(t)),
1514
+ ˜lt(x)
1515
+ hΞ³(x) ≀ (1 + Ο΅)2(βˆ’Ξ³)
1516
+ 1
1517
+ Ξ³ (xβˆ— βˆ’ t)βˆ’ 1
1518
+ Ξ³
1519
+ 1 βˆ’ F(t) .
1520
+ Now, setting t = U(v), we have that v β†’ ∞ if and only if t β†’ xβˆ— 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
+ (xβˆ— βˆ’ t)βˆ’ 1
1523
+ Ξ³
1524
+ 1 βˆ’ F(t)
1525
+ ≀ v[(1 + Ο΅)Ο–vΞ³]βˆ’ 1
1526
+ Ξ³ = [(1 + Ο΅)Ο–]βˆ’ 1
1527
+ Ξ³
1528
+ The result now follows.
1529
+ 5.5
1530
+ Proof of Lemma 3.5
1531
+ Note that for any Ξ³β€² > βˆ’1/2 and Οƒ > 0
1532
+ H (hΞ³; hΞ³β€²(Οƒ Β· )Οƒ) ≀
1533
+ οΏ½οΏ½
1534
+ R
1535
+ [Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)]2 dx +
1536
+ οΏ½οΏ½
1537
+ R
1538
+ [ψγ,x(Οƒ) βˆ’ ψγ,x(1)]2 dx
1539
+ In what follows, we bound the two terms on the right-hand side for Ξ³β€² ∈ (Ξ³ Β± Ο΅) and
1540
+ Οƒ ∈ (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) ∈ (Ξ³ ∧ Ξ³β€², Ξ³ ∨ Ξ³β€²),
1544
+ οΏ½οΏ½
1545
+ R
1546
+ [Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)]2 dx = |Ξ³ βˆ’ Ξ³β€²|
1547
+ �� ∞
1548
+ 0
1549
+ [Ξ½β€²x(ΞΎ(x))]2 dx
1550
+ ≀ |Ξ³ βˆ’ Ξ³β€²|
1551
+ 2
1552
+ �� ∞
1553
+ 0
1554
+ (1 + xΞΎ(x))βˆ’3βˆ’
1555
+ 1
1556
+ ΞΎ(x)
1557
+ οΏ½x ln(1 + xΞΎ(x))
1558
+ ΞΎ(x)
1559
+ οΏ½2
1560
+ dx
1561
+ + |Ξ³ βˆ’ Ξ³β€²|
1562
+ 2
1563
+ �� ∞
1564
+ 0
1565
+ (1 + xΞΎ(x))βˆ’3βˆ’
1566
+ 1
1567
+ ΞΎ(x) x2dx.
1568
+ 19
1569
+
1570
+ On one hand, it holds that
1571
+ � ∞
1572
+ 0
1573
+ (1 + xΞΎ(x))βˆ’3βˆ’
1574
+ 1
1575
+ ΞΎ(x)
1576
+ οΏ½x ln(1 + xΞΎ(x))
1577
+ ΞΎ(x)
1578
+ οΏ½2
1579
+ dx
1580
+ ≀ 4
1581
+ � ∞
1582
+ 0
1583
+ (1 + x(Ξ³ βˆ’ Ο΅))βˆ’1βˆ’
1584
+ 1
1585
+ Ξ³+Ο΅
1586
+ οΏ½ln(1 + x(Ξ³ βˆ’ Ο΅))
1587
+ (Ξ³ βˆ’ Ο΅)2
1588
+ οΏ½2
1589
+ dx
1590
+ ≀ 8(Ξ³ + Ο΅)3
1591
+ (Ξ³ βˆ’ Ο΅)5 .
1592
+ On the other hand, it holds that
1593
+ � ∞
1594
+ 0
1595
+ (1 + ΞΎ(x))βˆ’3βˆ’
1596
+ 1
1597
+ ΞΎ(x) x2dx
1598
+ οΏ½οΏ½οΏ½
1599
+ � ∞
1600
+ 0
1601
+ (1 + x(Ξ³ βˆ’ Ο΅))βˆ’1βˆ’
1602
+ 1
1603
+ Ξ³+Ο΅
1604
+ 1
1605
+ (Ξ³ βˆ’ Ο΅)2 dx
1606
+ ≀ (Ξ³ + Ο΅)
1607
+ (Ξ³ βˆ’ Ο΅)3 .
1608
+ While, an application of the mean-value theorem and Lemma 5.8(a) yields that, for a
1609
+ function Ο‚(x) ∈ (1 ∧ Οƒ, 1 ∨ Οƒ),
1610
+ οΏ½
1611
+ R
1612
+ οΏ½
1613
+ ΟˆΞ³β€²,x(Οƒ) βˆ’ ψγ,x(1)
1614
+ οΏ½2 dx =
1615
+ � ∞
1616
+ 0
1617
+ οΏ½
1618
+ Οˆβ€²
1619
+ Ξ³,x(Ο‚(x))
1620
+ οΏ½2 dx
1621
+ ≀
1622
+ οΏ½ 1
1623
+ Ξ³2 +
1624
+ οΏ½
1625
+ 1
1626
+ 2Ξ³ + 1
1627
+ οΏ½2 οΏ½1 + Ο΅
1628
+ 1 βˆ’ Ο΅
1629
+ οΏ½5
1630
+ .
1631
+ The result now follows.
1632
+ Case 2: Ξ³ < 0. Assume that Ξ³ < Ξ³β€², then an application of the mean-value theorem
1633
+ and Lemma 5.7(b) yields that, for a function ΞΎ(x) ∈ (Ξ³, Ξ³β€²),
1634
+ οΏ½
1635
+ R
1636
+ οΏ½
1637
+ Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)
1638
+ οΏ½2 dx = |Ξ³ βˆ’ Ξ³β€²|2
1639
+ οΏ½ βˆ’1/Ξ³
1640
+ 0
1641
+ οΏ½
1642
+ Ξ½β€²
1643
+ x(ΞΎ(x))
1644
+ οΏ½2 dx + 1 βˆ’ HΞ³β€²(βˆ’1/Ξ³)
1645
+ ≀ |Ξ³ βˆ’ Ξ³β€²|2
1646
+ 4
1647
+ οΏ½
1648
+ οΏ½
1649
+ οΏ½οΏ½ βˆ’1/Ξ³
1650
+ 0
1651
+ (1 + xΞΎ(x))βˆ’3βˆ’
1652
+ 1
1653
+ ΞΎ(x) x4dx
1654
+ +
1655
+ οΏ½οΏ½ βˆ’1/Ξ³
1656
+ 0
1657
+ (1 + xΞΎ(x))βˆ’3βˆ’
1658
+ 1
1659
+ ΞΎ(x) x2dx
1660
+ οΏ½
1661
+ οΏ½
1662
+ 2
1663
+ + 1 βˆ’ HΞ³β€²(βˆ’1/Ξ³).
1664
+ First, for a constant Ξ² satisfying 0 < Ξ² < 1/(Ο΅ βˆ’ Ξ³) βˆ’ 2, we have that
1665
+ οΏ½ βˆ’1/Ξ³
1666
+ 0
1667
+ (1 + xΞΎ(x))βˆ’3βˆ’
1668
+ 1
1669
+ ΞΎ(x) x4dx ≀ 1
1670
+ Ξ³4
1671
+ οΏ½ βˆ’1/Ξ³
1672
+ 0
1673
+ (1 + Ξ³x)βˆ’1+Ξ²dx
1674
+ ≀
1675
+ 1
1676
+ (βˆ’Ξ³)5
1677
+ 1
1678
+ Ξ² .
1679
+ Similarly,
1680
+ οΏ½ βˆ’1/Ξ³
1681
+ 0
1682
+ (1 + xΞΎ(x))βˆ’3βˆ’
1683
+ 1
1684
+ ΞΎ(x) x2dx ≀ 1
1685
+ Ξ³2
1686
+ οΏ½ βˆ’1/Ξ³
1687
+ 0
1688
+ (1 + Ξ³x)βˆ’1+Ξ²dx
1689
+ ≀
1690
+ 1
1691
+ (βˆ’Ξ³)3
1692
+ 1
1693
+ Ξ² .
1694
+ 20
1695
+
1696
+ Finally, if Ο΅ is small enough, 1 βˆ’ HΞ³β€²(βˆ’1/Ξ³) ≀ (1 βˆ’ Ξ³β€²/Ξ³)2. Thus, we can conclude that
1697
+ οΏ½
1698
+ R
1699
+ οΏ½
1700
+ Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)
1701
+ οΏ½2 dx ≀ |Ξ³ βˆ’ Ξ³β€²|2 1 + 1/2Ξ²
1702
+ (βˆ’Ξ³)5 .
1703
+ A similar reasoning when Ξ³ > Ξ³β€² yields that
1704
+ οΏ½
1705
+ R
1706
+ οΏ½
1707
+ Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)
1708
+ οΏ½2 dx ≀ |Ξ³ βˆ’ Ξ³β€²|2 1 + 2/Ξ²
1709
+ (βˆ’Ξ³β€²)5
1710
+ ≀ |Ξ³ βˆ’ Ξ³β€²|2 1 + 2/Ξ²
1711
+ (βˆ’Ξ³ βˆ’ Ο΅)5 .
1712
+ Next, assuming that Οƒ < 1, an application of the mean-value theorem and the first
1713
+ half of Lemma 5.8(b) yields that for a function Ο‚(x) ∈ (1 βˆ’ Ο΅, 1) and a constant ΞΆ > 0
1714
+ οΏ½
1715
+ R
1716
+ [ψγ,x(Οƒ) βˆ’ ψγ,x(1)]2 dx = (1 βˆ’ Οƒ)2
1717
+ οΏ½ βˆ’Οƒ/Ξ³
1718
+ 0
1719
+ οΏ½
1720
+ Οˆβ€²
1721
+ Ξ³,x(Ο‚(x))
1722
+ οΏ½2 dx + (1 βˆ’ Οƒ)βˆ’1/Ξ³
1723
+ ≀ (1 βˆ’ Οƒ)2
1724
+ οΏ½
1725
+ 1
1726
+ ΞΆ(1 βˆ’ Ο΅)2(βˆ’Ξ³)
1727
+ οΏ½ 1
1728
+ Ξ³2 + 1
1729
+ οΏ½2
1730
+ + 1
1731
+ οΏ½
1732
+ .
1733
+ While, if Οƒ > 1, for a function Ο‚(x) ∈ (1, 1 + Ο΅)
1734
+ οΏ½
1735
+ R
1736
+ [ψγ,x(Οƒ) βˆ’ ψγ,x(1)]2 dx = (1 βˆ’ Οƒ)2
1737
+ οΏ½ βˆ’1/Ξ³
1738
+ 0
1739
+ οΏ½
1740
+ Οˆβ€²
1741
+ Ξ³,x(Ο‚(x))
1742
+ οΏ½2 dx + (1 βˆ’ 1/Οƒ)βˆ’1/Ξ³
1743
+ ≀ (1 βˆ’ Οƒ)2
1744
+ οΏ½
1745
+ 1
1746
+ ΞΆ(βˆ’Ξ³)
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 Ξ³β€² > 0, then an application of the mean-value theorem
1755
+ and Lemma 5.7(a) yields that, for a function ΞΎ(x) ∈ (0, Ξ³β€²),
1756
+ οΏ½
1757
+ R
1758
+ οΏ½
1759
+ Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)
1760
+ οΏ½2 dx = |Ξ³ βˆ’ Ξ³β€²|2
1761
+ � ∞
1762
+ 0
1763
+ οΏ½
1764
+ Ξ½β€²
1765
+ x(ΞΎ(x))
1766
+ οΏ½2 dx
1767
+ ≀ |Ξ³ βˆ’ Ξ³β€²|2
1768
+ 4
1769
+ οΏ½
1770
+ οΏ½
1771
+ �� ∞
1772
+ 0
1773
+ (1 + xΞΎ(x))βˆ’3βˆ’
1774
+ 1
1775
+ ΞΎ(x)
1776
+ οΏ½x ln(1 + xΞΎ(x))
1777
+ ΞΎ(x)
1778
+ οΏ½2
1779
+ dx
1780
+ +
1781
+ οΏ½οΏ½ βˆ’βˆž
1782
+ 0
1783
+ (1 + xΞΎ(x))βˆ’3βˆ’
1784
+ 1
1785
+ ΞΎ(x) x2dx
1786
+ οΏ½
1787
+ οΏ½
1788
+ 2
1789
+ .
1790
+ On one hand, we have
1791
+ � ∞
1792
+ 0
1793
+ (1 + xΞΎ(x))βˆ’3βˆ’
1794
+ 1
1795
+ ΞΎ(x)
1796
+ οΏ½x ln(1 + xΞΎ(x))
1797
+ ΞΎ(x)
1798
+ οΏ½2
1799
+ dx ≀
1800
+ � ∞
1801
+ 0
1802
+ (1 + xΞΎ(x))βˆ’3βˆ’
1803
+ 1
1804
+ ΞΎ(x) x4dx
1805
+ ≀
1806
+ � ∞
1807
+ 0
1808
+ (1 + xΞ³β€²)βˆ’3βˆ’ 1
1809
+ Ξ³β€² x4dx +
1810
+ � ∞
1811
+ 0
1812
+ eβˆ’xx4dx
1813
+ ≀ 36 + Ξ“(5).
1814
+ On the other hand, we have
1815
+ � ∞
1816
+ 0
1817
+ (1 + xΞΎ(x))βˆ’3βˆ’
1818
+ 1
1819
+ ΞΎ(x) x2dx ≀
1820
+ � ∞
1821
+ 0
1822
+ (1 + xΞ³β€²)βˆ’3βˆ’ 1
1823
+ Ξ³β€² x2dx +
1824
+ � ∞
1825
+ 0
1826
+ eβˆ’xx2dx
1827
+ ≀ 3 + Ξ“(3).
1828
+ 21
1829
+
1830
+ Assume next that Ξ³β€² < 0, then an application of the mean-value theorem and Lemma
1831
+ 5.7(b) yields that, for a function ΞΎ(x) ∈ (βˆ’Ο΅, 0),
1832
+ οΏ½
1833
+ R
1834
+ οΏ½
1835
+ Ξ½x(Ξ³) βˆ’ Ξ½x(Ξ³β€²)
1836
+ οΏ½2 dx = |Ξ³ βˆ’ Ξ³β€²|2
1837
+ οΏ½ βˆ’1/Ξ³β€²
1838
+ 0
1839
+ οΏ½
1840
+ Ξ½β€²
1841
+ x(ΞΎ(x))
1842
+ οΏ½2 dx + e1/Ξ³β€²
1843
+ ≀ |Ξ³ βˆ’ Ξ³β€²|2
1844
+ 4
1845
+ οΏ½
1846
+ οΏ½
1847
+ οΏ½οΏ½ βˆ’1/Ξ³β€²
1848
+ 0
1849
+ (1 + xΞΎ(x))βˆ’3βˆ’
1850
+ 1
1851
+ ΞΎ(x) x4dx
1852
+ +
1853
+ οΏ½οΏ½ βˆ’1/Ξ³β€²
1854
+ 0
1855
+ (1 + xΞΎ(x))βˆ’3βˆ’
1856
+ 1
1857
+ ΞΎ(x) x2dx
1858
+ οΏ½
1859
+ οΏ½
1860
+ 2
1861
+ + e1/Ξ³β€².
1862
+ On one hand, for Ο΅ sufficiently small we have
1863
+ οΏ½ βˆ’1/Ξ³β€²
1864
+ 0
1865
+ (1 + xΞΎ(x))βˆ’3βˆ’
1866
+ 1
1867
+ ΞΎ(x) x4dx ≀
1868
+ οΏ½ βˆ’1/Ξ³β€²
1869
+ 0
1870
+ (1 + xΞ³β€²)βˆ’3βˆ’ 1
1871
+ Ξ³β€² x4dx +
1872
+ οΏ½ βˆ’1/Ξ³β€²
1873
+ 0
1874
+ eβˆ’xx4dx
1875
+ ≀ 13
1876
+ 2 Ξ“(5)
1877
+ and
1878
+ οΏ½ βˆ’1/Ξ³β€²
1879
+ 0
1880
+ (1 + xΞΎ(x))βˆ’3βˆ’
1881
+ 1
1882
+ ΞΎ(x) x2dx ≀
1883
+ οΏ½ βˆ’1/Ξ³β€²
1884
+ 0
1885
+ (1 + xΞ³β€²)βˆ’3βˆ’ 1
1886
+ Ξ³β€² x4dx +
1887
+ οΏ½ βˆ’1/Ξ³β€²
1888
+ 0
1889
+ eβˆ’xx2dx
1890
+ ≀ 3
1891
+ 2Ξ“(3).
1892
+ On the other hand, for Ο΅ sufficiently small we have e1/Ξ³β€² ≀ |Ξ³β€² βˆ’ Ξ³|2.
1893
+ Finally, some algebraic manipulations yield
1894
+ οΏ½
1895
+ R
1896
+ [ψγ,x(Οƒ) βˆ’ ψγ,x(1)]2 dx =
1897
+ � ∞
1898
+ 0
1899
+ οΏ½οΏ½
1900
+ eβˆ’x/Οƒ 1
1901
+ Οƒ βˆ’
1902
+ √
1903
+ eβˆ’x
1904
+ οΏ½2
1905
+ dx
1906
+ ≀ (1 βˆ’ Οƒ)2
1907
+ (1 βˆ’ Ο΅)2
1908
+ οΏ½
1909
+ 1 + 1
1910
+ 2
1911
+ οΏ½1 + Ο΅
1912
+ 1 βˆ’ 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 (˜lt; hΞ³) ≀ 2
1920
+ οΏ½
1921
+ οΏ½
1922
+ sup
1923
+ 0<x< xβˆ—βˆ’t
1924
+ ˜s(t)
1925
+ ˜lt(x)
1926
+ hΞ³(x)
1927
+ οΏ½
1928
+ � H 2(˜lt; hγ).
1929
+ Moreover, by Lemma B.3 in Ghosal and van der Vaart (2017), for p β‰₯ 2
1930
+ Dp(˜lt; hΞ³) ≀ 2p!
1931
+ οΏ½
1932
+ οΏ½
1933
+ sup
1934
+ 0<x< xβˆ—βˆ’t
1935
+ ˜s(t)
1936
+ ˜lt(x)
1937
+ hΞ³(x)
1938
+ οΏ½
1939
+ � H 2(˜lt; hγ).
1940
+ Furthermore, by triangular inequality and Lemma 3.5, for all large t
1941
+ H (˜lt; hγ) = H
1942
+ οΏ½
1943
+ lt; hΞ³
1944
+ οΏ½
1945
+ Β· s(t)
1946
+ ˜s(t)
1947
+ οΏ½ s(t)
1948
+ ˜s(t)
1949
+ οΏ½
1950
+ ≀ H (lt; hΞ³) + H
1951
+ οΏ½
1952
+ hΞ³; hΞ³
1953
+ οΏ½
1954
+ Β· s(t)
1955
+ ˜s(t)
1956
+ οΏ½ s(t)
1957
+ ˜s(t)
1958
+ οΏ½
1959
+ ≀ H (lt; hΞ³) + L|s(t)/˜s(t) βˆ’ 1|
1960
+ ≀ 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
+ ≀ 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†’ ∞
1988
+ H (lt; hΞ³) = O(|A(n/k)|) = O(1/
1989
+ √
1990
+ k).
1991
+ Moreover, by Lemma 3.5 and assumption (c), as n 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 βˆ’ οΏ½Ξ³k|2 +
2006
+ οΏ½οΏ½οΏ½οΏ½1 βˆ’ s(t)
2007
+ οΏ½sk(t)
2008
+ οΏ½οΏ½οΏ½οΏ½
2009
+ 2
2010
+ οΏ½
2011
+ οΏ½
2012
+ = Op(1/
2013
+ √
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/˜s(X(nβˆ’k)), X(nβˆ’k)).
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(nβˆ’k)).
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†’ ∞,
2024
+ 1/(1 βˆ’ F(X(nβˆ’k)))
2025
+ n/k
2026
+ = 1 + op(1),
2027
+ see e.g. Lemma 2.2.3 in de Haan and Ferreira (2006). Hence, defining 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(nβˆ’k) ∈ Bk)(1 + o(1)) + o(1)
2030
+ and
2031
+ Qk(Ek) = Pk(Ek|X(nβˆ’k) ∈ Bk)(1 + o(1)) + o(1)
2032
+ as n β†’ ∞. Therefore, it suffices to prove that
2033
+ Pk( Β· |X(nβˆ’k) ∈ Bk) β–· Qk( Β· |X(nβˆ’k) ∈ Bk).
2034
+ To do it, we denote by Ο€k and Ο‡k the (Lebesgue) densities pertaining to the two condi-
2035
+ tional probability measures in the formula above and prove that
2036
+ lim sup
2037
+ nβ†’βˆž K (Ο‡k; Ο€k) < ∞.
2038
+ (5.5)
2039
+ 23
2040
+
2041
+ Clearly, it holds that for almost every (y, t) ∈ Rk+2
2042
+ Ο‡k(y, t) = fYk/˜s(X(nβˆ’k))(y|X(nβˆ’k) = t)
2043
+ fX(nβˆ’k)(t)1(t ∈ Bk)
2044
+ P(X(nβˆ’k) ∈ Bk)
2045
+ ,
2046
+ where fYk/˜s(X(nβˆ’k))(y|X(nβˆ’k) = t) and fX(nβˆ’k)(t) are the conditional density of Yk/˜s(X(nβˆ’k))
2047
+ given X(nβˆ’k) = t and the marginal density of X(nβˆ’k), respectively. Moreover,
2048
+ Ο€k(y, t) = hZk(y)
2049
+ fX(nβˆ’k)(t)1(t ∈ Bk)
2050
+ P(X(nβˆ’k) ∈ Bk)
2051
+ ,
2052
+ where hZk(y) is the density of Zk. As a consequence,
2053
+ K (Ο‡k; Ο€k) =
2054
+ οΏ½
2055
+ Bk
2056
+ K (fYk/˜s(X(nβˆ’k))( Β· |X(nβˆ’k) = t); hZk)
2057
+ fX(nβˆ’k)(t)
2058
+ P(X(nβˆ’k) ∈ 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/˜s(X(nβˆ’k))( Β· |X(nβˆ’k) = t); hZk) ≀ kK (˜lt; hΞ³).
2062
+ Moreover, by Corollary 3.6, there is a constant Ξ› > 0 such that for all large n
2063
+ sup
2064
+ t∈Bk
2065
+ K (˜lt; hΞ³) ≀ Ξ›
2066
+ οΏ½οΏ½οΏ½A
2067
+ οΏ½
2068
+ (1 βˆ’ Ο΅)n
2069
+ k
2070
+ οΏ½οΏ½οΏ½οΏ½
2071
+ 2
2072
+ ≀ Ξ›(1 βˆ’ Ο΅)ρ(1 + Ο΅)
2073
+ οΏ½οΏ½οΏ½A
2074
+ οΏ½n
2075
+ k
2076
+ οΏ½οΏ½οΏ½οΏ½
2077
+ 2
2078
+ .
2079
+ Combining the above inequalities we obtain that
2080
+ K (Ο‡k; Ο€k) ≀ Ξ›(1 βˆ’ Ο΅)ρ(1 + Ο΅)k
2081
+ οΏ½οΏ½οΏ½A
2082
+ οΏ½n
2083
+ k
2084
+ οΏ½οΏ½οΏ½οΏ½
2085
+ 2
2086
+ β†’ Ξ›(1 βˆ’ Ο΅)ρ(1 + Ο΅)Ξ»2
2087
+ as n 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, 792–804.
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+ Bobbia, B., C. Dombry, and D. Varron (2021). The coupling method in extreme value
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+ theory. Bernoulli 27, 1824–1850.
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+ B¨ucher, A. and J. Segers (2017). On the maximum likelihood estimator for the Gener-
2098
+ alized Extreme-Value distribution. Extremes 20, 839–872.
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+ B¨ucher, A. and C. Zhou (2021). A Horse Race between the Block Maxima Method and
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+ the Peak–over–Threshold Approach. Statistical Science 36, 360–378.
2101
+ 24
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+
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+ de Haan, L. and A. Ferreira (2006). Extreme Value Theory: An Introduction. Springer.
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+ Dekkers, A. L. and L. de Haan (1993). Optimal choice of sample fraction in extreme-
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+ value estimation. Journal of Multivariate Analysis 47, 173–195.
2106
+ Dekkers, A. L., J. H. Einmahl, and L. de Haan (1989). A moment estimator for the
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+ index of an extreme-value distribution. The Annals of Statistics 17, 1833–1855.
2108
+ Dey, D. K. and J. Yan (2016). Extreme value modeling and risk analysis: methods and
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+ applications. CRC Press.
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+ Drees, H. (1998). Optimal rates of convergence for estimates of the extreme value index.
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+ Annals of Statistics 26, 434–448.
2112
+ Drees, H., A. Ferreira, and L. de Haan (2004). On maximum likelihood estimation of
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+ the extreme value index. The Annals of Applied Probability 14, 1179–1201.
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+ Embrechts, P., C. Kl¨uppelberg, and T. Mikosch (2013). Modelling extremal events: for
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+ insurance and finance, Volume 33. Springer Science & Business Media.
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+ Falk, M., J. H¨usler, and R.-D. Reiss (2010). Laws of small numbers: extremes and rare
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+ events. Springer Science & Business Media.
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+ Ghosal, S., J. K. Ghosh, and A. W. van der Vaart (2000). Convergence rates of posterior
2119
+ distributions. The Annals of Statistics 28, 500–531.
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, 1079–1084.
2125
+ Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution.
2126
+ Annals of statistics 3, 1163–1174.
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.
2129
+ Technometrics 27, 251–261.
2130
+ Jenkinson, A. (1969). Statistics of extemes. In Estimation of maximum floods, WMO
2131
+ Tech. Note 98, pp. 183–228.
2132
+ Pickands III, J. (1975). Statistical inference using extreme order statistics. The Annals
2133
+ of Statistics 3, 119–131.
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, 1007–1027.
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, 794–815.
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+ 25
2141
+
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1
+ Single-material MoS2 thermoelectric junction enabled by substrate
2
+ engineering
3
+ Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali
4
+ Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan KasΔ±rga1,2*
5
+ 1 Bilkent University UNAM – 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 1–6. 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 𝑆1 and 𝑆2, 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. {πΌπ‘šπ‘–π‘›, πΌπ‘šπ‘Žπ‘₯} = {βˆ’0.5, 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, 𝐼𝑃𝐢 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, 𝐺𝑃𝐢 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 𝑆 in thermal equilibrium is composed of contributions from the
50
+ energy-dependent diffusion (𝑆𝑁), scattering (𝑆τ) and the phonon-drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + π‘†πœ +
51
+ 𝑆𝑝𝑑 9,10. Here, 𝑆𝑁 and 𝑆τ 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
+ 𝑆τ = βˆ’
54
+ πœ‹2π‘˜π΅
55
+ 2𝑇
56
+ 3𝑒
57
+ πœ•ln𝜏
58
+ πœ•πΈ | 𝐸=𝐸𝐹 and 𝑆𝑁 = Β±
59
+ π‘˜π΅
60
+ 𝑒 [
61
+ πΈπΉβˆ’πΈπΆ
62
+ π‘˜π΅π‘‡ βˆ’
63
+ (π‘Ÿ+2)πΉπ‘Ÿ+1(πœ‚)
64
+ (π‘Ÿ+1)πΉπ‘Ÿ(πœ‚) ]
65
+ where 𝑇 is the temperature, π‘˜π΅ is the Boltzmann constant, 𝑒 is the electron’s charge, 𝜏 is the relaxation
66
+ time, 𝐸𝐹 is the Fermi energy, 𝐸𝐢 is the conduction band edge energy, π‘Ÿ is scattering parameter and 𝐸
67
+ is the energy. πΉπ‘š(πœ‚) is the m-th order Fermi integral11. In the 2d limit, 𝜏 is energy independent, thus
68
+ π‘†πœ is zero. 𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first
69
+ order as 𝑆𝑝𝑑 = βˆ’
70
+ 𝛽𝑣𝑝𝑙𝑝
71
+ πœ‡π‘‡ where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon,
72
+ 𝛽 is a parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and πœ‡ is
73
+ the electron mobility, respectively10. Importantly, 𝑙𝑝 and πœ‡ are heavily affected by the presence of a
74
+ substrate12 which implies that the 𝑆𝑝𝑑 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 𝑆𝑝𝑑 β‰ˆ βˆ’100 Β΅V/K and for MoS2 on
84
+ SiO2 at room temperature 𝑆𝑝𝑑 β‰ˆ βˆ’230 Β΅V/K. Similarly, 𝑆𝑁 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 defects11–17. We estimate that that for MoS2, 𝑆𝑁 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).18–20 As a result, a substrate engineered thermoelectric junction with a Seebeck
89
+ coefficient difference of Δ𝑆 β‰ˆ 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
+ needles22–24 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 𝑉 to the junction, the photocurrent, 𝐼𝑃𝐢 changes linearly with bias, while the
104
+ photoconductance, 𝐺𝑃𝐢 =
105
+ 𝐼𝑃𝐢
106
+ 𝑉 βˆ’πΌπ‘ƒπΆ
107
+ 0
108
+ 𝑉
109
+ (𝐼𝑃𝐢
110
+ 𝑉 , 𝐼𝑃𝐢
111
+ 0 : photocurrent under 𝑉 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 signal22,24,27–29. 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 substrate33–
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 Δ𝑆 = 72 Β± 10 Β΅V/K (See supporting information). Despite the
130
+ uncertainties regarding the real sample temperature, the obtained Δ𝑆 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 substrate36–39. 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 𝛿𝑑 β‰ˆ 25 nm. Similar 𝛿𝑑 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 43–45.
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 𝛿𝑑 β‰ˆ 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 𝑉𝑔 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 𝑉𝑔:
180
+ 𝑆 =
181
+ πœ‹2π‘˜π΅
182
+ 2𝑇
183
+ 3𝑒
184
+ 1
185
+ 𝑅
186
+ 𝑑𝑅
187
+ 𝑑𝑉𝑔
188
+ 𝑑𝑉𝑔
189
+ 𝑑𝐸 | 𝐸=𝐸𝐹 eq.(1)
190
+ Here, 𝑇 is the temperature, π‘˜π΅ is the Boltzmann constant, 𝑒 is the electron’s charge, 𝑅 is the device
191
+ resistance, 𝐸𝐹 is the Fermi energy and 𝐸 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 – a value
204
+ very similar to the values we find for suspended devices (see SI) – 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. πΌπ‘šπ‘Žπ‘₯ = 3 nA and πΌπ‘šπ‘–π‘› = βˆ’3 nA. Scale bar: 10 Β΅m. c.
213
+ Current-Voltage graph versus 𝑉𝐺 from -40 to 40 V. Inset shows the resistance versus 𝑉𝐺. d. 𝐼𝑃𝐢 vs. 𝑉𝐺
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 A1𝑔 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
+ 𝑉𝐺 = βˆ’21.5 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 𝑉𝐺 =
229
+ βˆ’34.5 V.
230
+ These observations can be qualitatively explained as follows: at a gate voltage of 𝑉𝐺 = βˆ’34.5 𝑉, 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 βˆ†π‘† β‰ˆ 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 𝑉𝐺 and reaches
235
+
236
+ SiO2
237
+ Si
238
+ Imax
239
+ Iminzero at 𝑉𝐺 = βˆ’40 𝑉. 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
+ References
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537
+
538
+
539
+
540
+
541
+ Supporting Information: Single-material MoS2 thermoelectric junction
542
+ enabled by substrate engineering
543
+ Mohammadali Razeghi1, Jean Spiece3, Oğuzhan Oğuz1, Doruk Pehlivanoğlu2, Yubin Huang3, Ali
544
+ Sheraz2, Phillip S. Dobson4, Jonathan M. R. Weaver4, Pascal Gehring3, T. Serkan KasΔ±rga1,2*
545
+ 1 Bilkent University UNAM – 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 (𝑆𝑁), scattering (𝑆τ) and the phonon-
555
+ drag (𝑆𝑝𝑑), so that 𝑆 = 𝑆𝑁 + π‘†πœ + 𝑆𝑝𝑑. Here, 𝑆𝑁 and 𝑆τ 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
+ 𝑆τ = βˆ’
558
+ πœ‹2π‘˜π΅
559
+ 2𝑇
560
+ 3𝑒
561
+ πœ•ln𝜏
562
+ πœ•πΈ | 𝐸=𝐸𝐹 and 𝑆𝑁 = Β±
563
+ π‘˜π΅
564
+ 𝑒 [
565
+ πΈπΉβˆ’πΈπΆ
566
+ π‘˜π΅π‘‡ βˆ’
567
+ (π‘Ÿ+2)πΉπ‘Ÿ+1(πœ‚)
568
+ (π‘Ÿ+1)πΉπ‘Ÿ(πœ‚) ]
569
+ As mentioned in the main text, 𝑆τ is zero as 𝜏 is energy independent in the 2d limit. 𝑆𝑁 term is
570
+ composed of constants related to material properties, scattering parameter π‘Ÿ and the Fermi integral
571
+ of the π‘Ÿ-th order: πΉπ‘Ÿ = ∫
572
+ [
573
+ π‘₯π‘š
574
+ 𝑒π‘₯βˆ’πœ‚ + 1]𝑑π‘₯
575
+ ∞
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‘Ÿ = 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 𝑆𝑁 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 𝒓 of 2d materials.
583
+ Scattering mechanism
584
+ 𝒓
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
+ 𝑆𝑝𝑑 term can be estimated from the theory of phonon-drag in semiconductors in the first order as
595
+ 𝑆𝑝𝑑 = βˆ’
596
+ 𝛽𝑣𝑝𝑙𝑝
597
+ πœ‡π‘‡ where, 𝑣𝑝 and 𝑙𝑝 are the group velocity and the mean free path of a phonon, 𝛽 is a
598
+ parameter to modify the electron-phonon interaction strength and ranges from 0 to 1, and πœ‡ is the
599
+ electron mobility, respectively. As the dominant charge carriers are electrons, 𝑆𝑝𝑑 term has a negative
600
+ sign. We use the parameters given in Table 2. Based on the values given in the table we obtain 𝑆𝑝𝑑
601
+ 𝑆𝑖𝑂2 =
602
+ βˆ’230 Β΅V/K and ���𝑝𝑑
603
+ 𝑆𝑒𝑠 = βˆ’100 Β΅V/K.
604
+
605
+ The total 𝑆 = 𝑆𝑁 + 𝑆𝑝𝑑 for suspended and SiO2 supported parts can be calculated by adding both
606
+ contributions. 𝑆𝑆𝑒𝑠 = βˆ’500 Β΅V/K and 𝑆𝑆𝑖𝑂2 = βˆ’430 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 𝑺𝒑𝒅 calculation
612
+ Parameter
613
+ On SiO2 (Ref. 3)
614
+ Suspended (Ref. 3)
615
+ 𝑣𝑝
616
+ 7 105 cm/s
617
+ 7 105 cm/s
618
+ 𝑙𝑝
619
+ 5 nm
620
+ 20 nm
621
+ πœ‡
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 𝐼𝑃𝐢 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 𝑉𝐺 = βˆ’20 𝑉. Lower panel shows a clear contrast
741
+ change around the holes extending to the indium contacts. b. SPCM maps taken at 𝑉𝐺 = 0 𝑉 with 532
742
+ nm of 86 Β΅W on sample: (i) before gating, (ii) after 𝑉𝐺 = βˆ’15 𝑉 scan and (iii) after the scan in (iii).
743
+ πΌπ‘šπ‘Žπ‘₯ = 6.5 nA and πΌπ‘šπ‘–π‘› = βˆ’6.5 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 didn’t 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 𝑉𝐺 =-60 V, iii 𝑉𝐺 = 0 V, iv 𝑉𝐺 =60 V. Here, π‘‰π‘šπ‘Žπ‘₯ = 20 mV and
761
+ π‘‰π‘šπ‘–π‘› = -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 𝐴1𝑔 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 Δ𝑇 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
+ π‘‰π‘‘β„Ž(π‘₯) = βˆ’ ∫ 𝑆(π‘₯) πœ•π‘‡
848
+ πœ•π‘₯ (π‘₯)𝑑π‘₯
849
+ 𝐡
850
+ 𝐴
851
+
852
+ where 𝑆(π‘₯) is the position dependent Seebeck coefficient and
853
+ πœ•π‘‡
854
+ πœ•π‘₯ (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, Ξ”π‘‡π‘ π‘Žπ‘šπ‘π‘™π‘’.
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, …).
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 literature8–16. 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 Ξ”π‘‡π‘ π‘Žπ‘šπ‘π‘™π‘’ 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), 13929–13936.
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), 534–540.
899
+ https://doi.org/10.1038/nnano.2015.70.
900
+ (4)
901
+ Tovee, P.; Pumarol, M.; Zeze, D.; Kjoller, K.; Kolosov, O. Nanoscale Spatial Resolution Probes
902
+ for Scanning Thermal Microscopy of Solid State Materials. J Appl Phys 2012, 112 (11).
903
+ https://doi.org/10.1063/1.4767923.
904
+ (5)
905
+ Spiece, J.; Evangeli, C.; Lulla, K.; Robson, A.; Robinson, B.; Kolosov, O. Improving Accuracy of
906
+ Nanothermal Measurements via Spatially Distributed Scanning Thermal Microscope Probes. J
907
+ Appl Phys 2018, 124 (1), 015101. https://doi.org/10.1063/1.5031085.
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
911
+ Graphene Nanoconstrictions. Nano Lett 2018, 18 (12), 7719–7725.
912
+ https://doi.org/10.1021/ACS.NANOLETT.8B03406/ASSET/IMAGES/MEDIUM/NL-2018-
913
+ 03406E_M006.GIF.
914
+ (7)
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
917
+ 2020, 7 (4), 041004. https://doi.org/10.1088/2053-1583/ABA333.
918
+ (8)
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),
922
+ 262202. https://doi.org/10.1063/5.0092553.
923
+ (9)
924
+ Taube, A.; Judek, J.; ŁapiΕ„ska, A.; Zdrojek, M. Temperature-Dependent Thermal Properties of
925
+ Supported MoS2 Monolayers. ACS Appl Mater Interfaces 2015, 7 (9), 5061–5065.
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
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
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
+ 25923–25929. https://doi.org/10.1021/ACSAMI.5B08580/ASSET/IMAGES/LARGE/AM-2015-
942
+ 085805_0003.JPEG.
943
+ (13)
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), 2541–2547. 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), 7055–7061.
964
+ https://doi.org/10.1021/ACS.NANOLETT.7B03736/ASSET/IMAGES/NL-2017-
965
+ 03736Q_M017.GIF.
966
+
967
+
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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
+ Definition and structure of the poset
20
+ 3
21
+ 2.1.
22
+ Definition 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 satisfied 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 finite 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 infinite 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 βˆ† (called a Garside element) whose left- and right-divisors are
72
+ finite, coincide, generate the monoid, and form a lattice under left- and right-divisibility.
73
+ The left- or right-divisors of βˆ† 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 first 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 finite Coxeter group [3, 6], the lattice of (generalized) noncrossing partitions
81
+ attached to a finite 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 first 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 βˆ†n and the set of SchrΓΆder trees on n+1 leaves, in such
99
+ a way that applying a defining 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 specific edge contraction and are related to the notion of refinement
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 βˆ†n in terms of SchrΓΆder trees allows us to derive a few enumerative results. This is
106
+ done in Section 4. The first 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
+ βˆ†n. We also show that the sublattice of Pn defined as the order ideal of this lcm has F2nβˆ’1
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
+ ρ1
122
+ ρ2
123
+ ρ2ρ1
124
+ ρ3
125
+ ρ1ρ3
126
+ ρ3ρ1
127
+ ρ1ρ3ρ1
128
+ ρ3ρ2
129
+ ρ2ρ1ρ3
130
+ ρ3ρ2ρ1
131
+ ρ2
132
+ 3
133
+ ρ3ρ1ρ3
134
+ ρ2
135
+ 3ρ1
136
+ (ρ1ρ3)2
137
+ (ρ3ρ1)2
138
+ ρ3
139
+ 3
140
+ ρ3ρ2ρ1ρ3
141
+ ρ2
142
+ 3ρ1ρ3
143
+ (ρ3ρ1)2ρ3
144
+ ρ4
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. Definition of the poset. The beginning of this section is devoted to explaining how
150
+ the poset under study is defined. We recall the definition 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 ∈ 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 ∈ M such that ac = b. We similarly define right divisors
154
+ and left multiples.
155
+ Let M0 be the trivial monoid and for n β‰₯ 1, let Mn be the monoid defined by the
156
+ presentation
157
+ (2.1)
158
+ οΏ½
159
+ ρ1, ρ2, . . . , ρn
160
+ οΏ½οΏ½οΏ½οΏ½ ρ1ρnρi = ρi+1ρn for all 1 ≀ i ≀ n βˆ’ 1
161
+ οΏ½
162
+ .
163
+ We denote by S the set of generators {ρ1, ρ2, . . . , ρn}, and by R the defining relations of
164
+ Mn. This monoid was introduced by the first author in [8, Definition 4.1]. Note that this
165
+ monoid is equipped with a length function Ξ» : Mn βˆ’β†’ Zβ‰₯0 given by the multiplicative
166
+ extension of λ(ρi) = i for all i = 1, . . . , n, which is possible since the defining 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 ≀L b or simply a ≀ b if a left-divides b, and a ≀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 βˆ†n, called a Garside element, which has
181
+ the following two properties
182
+ (1) The set of left divisors of βˆ†n coincides with its set of right divisors, and forms a
183
+ finite set.
184
+ (2) The set of left (or right) divisors of βˆ†n generates Mn.
185
+ This Garside element is given by βˆ†n = ρn+1
186
+ n
187
+ . In particular, as any Garside monoid is a
188
+ lattice for both left- and right-divisibility, the set Div(βˆ†n) of left (or right) divisors of βˆ†n
189
+ is a finite lattice if equipped by the order relation given by the restriction of left- (or right-)
190
+ divisibility on Mn. The set Div(βˆ†n) is the set of simple elements or simples of Mn. In
191
+ general (Div(βˆ†n), ≀L) and (Div(βˆ†n), ≀R) will not be isomorphic as posets. But we always
192
+ have
193
+ (Div(βˆ†n), ≀L) ∼= (Div(βˆ†n), ≀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(βˆ†n), ≀) (and hence (Div(βˆ†n), ≀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 ∈ 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 βˆ†n coincide. In fact, the element βˆ†n is central
203
+ in Mn, hence as Mn is cancellative, for a, b ∈ Mn such that ab = βˆ†n, 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(βˆ†n) is a lattice.
207
+ Proposition 2.1. Let x1x2 Β· Β· Β· xk be a word for βˆ†n, with xi ∈ S for all i = 1, . . . , k. There
208
+ are i1 = 1 < i2 < Β· Β· Β· < iβ„“ ≀ k such that
209
+ β€’ For all j = 1, . . . , β„“, the word yj := xijxij+1 Β· Β· Β· xij+1βˆ’1 (with the convention that
210
+ iβ„“+1 = k + 1) is a word for a power of ρn,
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 ρn.
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 · · · xk, consider the smallest i ∈ {1, 2, . . . , k} such that x1x2 · · · xk is
217
+ a word for a power of ρn. Such an i has to exist, as x1x2 · · · xk is a word for a power of
218
+ ρn. Then set i2 := i + 1. By cancellativity in Mn, since x1 · · · xi and x1 · · · xk are both
219
+ words for a power of ρn, the word xi+1 · · · xk must also be a word for a power of ρn. 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· |zβ„“β€² 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 ρn, if y1 ̸= z1, then one
228
+ word must be strict prefix of the other, say z1 is a strict prefix of y1. But this contradicts
229
+ the maximality of the decomposition y1|y2| Β· Β· Β· |yβ„“: indeed if y1 = x1x2 Β· Β· Β· xi2βˆ’1 and z1 =
230
+ x1x2 Β· Β· Β· xp with p < i2 βˆ’ 1, we can decompose y1 nontrivially as x1x2 Β· Β· Β· xp|xp+1 Β· Β· Β· xi2βˆ’1,
231
+ and by cancellativity both x1 Β· Β· Β· xp and xp+1 Β· Β· Β· xi2βˆ’1 are words for powers of ρn.
232
+ β–‘
233
+ Example 2.2. Consider the word ρ3ρ1ρ7ρ1ρ7ρ5ρ4ρ7ρ7ρ1ρ7ρ6 in M7. We claim that this is a
234
+ word for the Garside element ρ8
235
+ 7 of M7. Indeed, using the defining relation ρ1ρ7ρi = ρi+1ρ7
236
+ with i = 5 and 6, we get that
237
+ ρ3(ρ1ρ7ρ1ρ7ρ5)ρ4ρ7ρ7(ρ1ρ7ρ6) = ρ3ρ3
238
+ 7ρ4ρ4
239
+ 7,
240
+ and we observe also applying defining relations that
241
+ ρ3ρ3
242
+ 7ρ4ρ4
243
+ 7 = ρ1ρ7ρ2ρ2
244
+ 7ρ4 = ρ1ρ7ρ1ρ7ρ1ρ7ρ4 = ρ1ρ7ρ1ρ7ρ5ρ7 = ρ1ρ7ρ6ρ2
245
+ 7 = ρ4
246
+ 7.
247
+ The decomposition according to Proposition 2.1 is given by
248
+ ρ3ρ1ρ7ρ1ρ7ρ5ρ4
249
+ οΏ½
250
+ οΏ½οΏ½
251
+ οΏ½
252
+ :=y1
253
+ | ρ7
254
+ οΏ½οΏ½οΏ½οΏ½
255
+ :=y2
256
+ | ρ7
257
+ οΏ½οΏ½οΏ½οΏ½
258
+ :=y3
259
+ | ρ1ρ7ρ6
260
+ οΏ½ οΏ½οΏ½ οΏ½
261
+ :=y4
262
+ .
263
+ It is indeed clear by considering λ(u) for u prefixes of y1 or y4 that whenever u is a proper
264
+ prefix, we do not have λ(u) equal to a multiple of 7, which is a necessary condition for a
265
+ word to represent a power of ρ7.
266
+ Lemma 2.3. Let 1 ≀ k ≀ n. Then
267
+ S ∩ {x ∈ Div(βˆ†n) | x ≀ ρkρk
268
+ n} = {ρ1, ρ2, . . . , ρk}.
269
+ Proof. We argue by induction on k. The result is clear for k = 1, as no defining relation of
270
+ Mn can be applied to the word ρ1ρn. Now let k > 1. Observe that
271
+ ρkρk
272
+ n = (ρ1ρn)k = (ρ1ρn)(ρ1ρn)kβˆ’1.
273
+ In particular we have ρ1 ≀ ρkρk
274
+ n and by induction, we get ρ1ρnρi ≀ ρkρk
275
+ n for all 1 ≀ i ≀ kβˆ’1.
276
+ As ρ1ρnρi = ρi+1ρn we get that {ρ1, ρ2, . . . , ρk} βŠ† S ∩ {x ∈ Div(βˆ†n) | x ≀ ρkρk
277
+ n}.
278
+ It remains to show that no other ρi can be a left-divisor of ρkρk
279
+ n. Hence assume that
280
+ i > k and ρi ≀ ρkρk
281
+ n. Hence there is a word x1x2 · · · xp for ρkρk
282
+ n, where xi ∈ S for all i,
283
+ such that x1 = ρi. As the words x1x2 · · · xp and ρkρk
284
+ n represent the same element, they
285
+ can be related by a finite sequence of words w0 = x1x2 · · · xp, w1, . . . , wq = ρkρk
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 first letter of w0 differs from the first letter of
289
+ wq, there must exist some 0 ≀ β„“ < q such that wβ„“ begins by ρi 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 ρi
292
+ is ρiρn = ρ1ρnρiβˆ’1. It follows that ρ1ρnρiβˆ’1 ≀ ρkρk
293
+ n = (ρ1ρn)k. By cancellativity, we get
294
+ that ρiβˆ’1 ≀ (ρ1ρn)kβˆ’1 = ρkβˆ’1ρkβˆ’1
295
+ n
296
+ . By induction this forces one to have i βˆ’ 1 ≀ k βˆ’ 1,
297
+ contradicting our assumption that i > k.
298
+ β–‘
299
+ Similarly, we have
300
+ Lemma 2.4. Let 1 ≀ k ≀ n. Then
301
+ S ∩ {x ∈ Div(βˆ†n) | x ≀R ρk
302
+ n} = {ρn, ρnβˆ’1, . . . , ρnβˆ’k+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 (ρ1ρn)nβˆ’jρj = ρnβˆ’j+1
308
+ n
309
+ , we get that ρj ≀R ρk
310
+ n for all j such that
311
+ n βˆ’ j + 1 ≀ k, that is, for all j β‰₯ n βˆ’ k + 1. It remains to show that no other ρj can
312
+ right-divide ρk
313
+ n. Hence assume that ρj ≀R ρk
314
+ n, where j < n βˆ’ k + 1. Arguing as in the
315
+ proof of Lemma 2.3, we see that ρ1ρnρj = ρj+1ρn must be a right-divisor of ρk
316
+ n, hence by
317
+ cancellativity that ρj+1 ≀R ρkβˆ’1
318
+ n
319
+ . By induction this forces j + 1 β‰₯ n βˆ’ k + 2, contradicting
320
+ our assumtion that j < n βˆ’ k + 1.
321
+ β–‘
322
+ For x ∈ Div(βˆ†n), let d(x) := max{k β‰₯ 0 | ρk
323
+ n ≀ x}. Let 0 ≀ i ≀ n + 1 and let
324
+ Di
325
+ n := {x ∈ Div(βˆ†n) | d(x) = i}.
326
+ Note that
327
+ Div(βˆ†n) =
328
+ οΏ½
329
+ 0≀i≀n+1
330
+ Di
331
+ n.
332
+ We have Dn
333
+ n = {ρn
334
+ n}, Dn+1
335
+ n
336
+ = {βˆ†n}.
337
+ Lemma 2.5. Let x ∈ Div(βˆ†n) and i = d(x). Let xβ€² ∈ Mn such that x = ρi
338
+ nxβ€². Note that
339
+ xβ€² ∈ D0
340
+ n. Let x1x2 · · · xk be a word for x, where xi ∈ S for all i = 1, . . . , k. Then there is
341
+ 1 ≀ β„“ ≀ k such that x1x2 Β· Β· Β· xβ„“ is a word for ρi
342
+ n (and hence xβ„“+1 Β· Β· Β· xk is a word for xβ€² by
343
+ cancellativity). In other words, any word for x has a prefix which is a word for ρi
344
+ n.
345
+ Proof. It suffices 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 ρi
347
+ n (q ≀ p, then one cannot apply a defining 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
+ ρ1ρn|ρj, ρ1|ρnρj, or ρj+1|ρn (1 ≀ j < n), where the | separates the letters zq and zq+1.
351
+ The last two cases cannot happen, since one would have zq+1 = ρn, hence zq+1 · · · zp would
352
+ be a word for xβ€² beginning by ρn, contradicting the fact that xβ€² ∈ D0
353
+ n. It remains to show
354
+ that the case ρ1ρn|ρj cannot happen. Hence assume that zqβˆ’1 = ρ1, zq = ρn, zq+1 = ρj.
355
+ By cancellativity, as z1z2 · · · zq is a word for ρi
356
+ n, it implies that ρ1 ≀R ρiβˆ’1
357
+ n
358
+ . By lemma 2.4,
359
+ this implies that n βˆ’ (i βˆ’ 1) + 1 = 1, hence that i = n + 1.
360
+ Since x ∈ Div(βˆ†n) and
361
+ x = ρn+1
362
+ n
363
+ xβ€² = βˆ†nxβ€², we get xβ€² = 1, contradicting the fact that zq+1 = ρj.
364
+ β–‘
365
+ Lemma 2.6. Let i, j ∈ {0, 1, . . . , n + 1}, with i ̸= j. Let x ∈ Di
366
+ n, y ∈ Dj
367
+ n. Assume that
368
+ x ≀ y. Then i < j and x < ρj
369
+ n ≀ y.
370
+ Proof. It is clear that i < j, since ρi
371
+ n ≀ y as ρi
372
+ n ≀ x, hence j < i would contradict y ∈ Dj
373
+ n.
374
+ In particular x < y. Let xβ€², yβ€² such that x = ρi
375
+ nxβ€² and y = ρj
376
+ nyβ€². Note that xβ€², yβ€² both lie
377
+ in D0
378
+ n. Since x ≀ y and Mn is cancellative, we get that xβ€² < ρjβˆ’i
379
+ n yβ€². It implies that there
380
+ exists a word x1x2 Β· Β· Β· xk for ρjβˆ’i
381
+ n
382
+ yβ€² (xi ∈ S) and 1 ≀ β„“ < k such that x1x2 Β· Β· Β· xβ„“ is a word
383
+ for xβ€². Now by lemma 2.5, there is 0 ≀ β„“β€² ≀ k such that x1x2 Β· Β· Β· xβ„“β€² is a word for ρjβˆ’i
384
+ n . If
385
+ β„“β€² ≀ β„“, then ρjβˆ’i
386
+ n
387
+ ≀ xβ€², contradicting the fact that xβ€² ∈ D0
388
+ n. Hence β„“β€² > β„“, and xβ€² < ρjβˆ’i
389
+ n .
390
+ Multiplying by ρi
391
+ n on the left we get x < ρj
392
+ n.
393
+ β–‘
394
+ Lemma 2.7. Let z1, z2 ∈ Di
395
+ n. Let 1 ≀ k1 < k2 ≀ n and assume that there are two cover
396
+ relations z1 ≀· ρk1
397
+ n , z2 ≀· ρk2
398
+ n in (Div(βˆ†n), ≀). Then z1 < z2.
399
+ Proof. As z1 ≀ ρk1
400
+ n , z2 ≀ ρk2
401
+ n
402
+ are cover relations, there are 1 ≀ j1, j2 ≀ n such that
403
+ z1ρj1 = ρk1
404
+ n , z2ρj2 = ρk2
405
+ n . By lemma 2.4, for β„“ ∈ {1, 2} we have jβ„“ ∈ {n βˆ’ kβ„“ + 1, . . . , n} and
406
+ ρkβ„“
407
+ n = ρjβ„“+kβ„“βˆ’1βˆ’n
408
+ n
409
+ (ρ1ρn)nβˆ’jℓρjβ„“.
410
+ In particular, we have
411
+ zβ„“ = ρjβ„“+kβ„“βˆ’1βˆ’n
412
+ n
413
+ (ρ1ρn)nβˆ’jβ„“
414
+
415
+ ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
416
+ 7
417
+ and as (ρ1ρn)nβˆ’jβ„“ = ρnβˆ’jℓρnβˆ’jβ„“
418
+ n
419
+ , by lemma 2.3 we see that ρn cannot be a left divisor of
420
+ (ρ1ρn)nβˆ’jβ„“, and hence that d(zβ„“) = jβ„“ + kβ„“ βˆ’ 1 βˆ’ n. But d(zβ„“) = i for β„“ ∈ {1, 2}, and since
421
+ k1 < k2 we deduce that j1 > j2. Since zβ„“ = ρi
422
+ n(ρ1ρn)nβˆ’jβ„“, we get that z1 < z2, which
423
+ concludes the proof.
424
+ β–‘
425
+ Theorem 2.8. The poset (Div(βˆ†n), ≀) is a lattice. Given i, j ∈ {0, 1, . . . , n+1} with i ≀ j
426
+ and x ∈ Di
427
+ n, y ∈ Dj
428
+ n, we have
429
+ x ∧ y = x ∧i
430
+ οΏ½οΏ½
431
+ i
432
+ {z ∈ Di
433
+ n | z ≀ y}
434
+ οΏ½
435
+ ,
436
+ where ∨i and ∧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 ∧ y = x ∧i y.
439
+ Proof. The proof is by induction on n. We have Div(βˆ†0) = {β€’}, and Div(βˆ†1) = {1, ρ1, ρ2
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(βˆ†nβˆ’i) if i ΜΈ= 0, n+1, while
444
+ the restriction to D0
445
+ n yields an isomorphism of poset with Div(βˆ†nβˆ’1), and the restriction to
446
+ Dn+1
447
+ n
448
+ an isomorphism of posets with Div(βˆ†0) = {β€’}. In particular, by induction, all these
449
+ posets are lattices. As the poset (Div(βˆ†n), ≀) is finite and admits a maximal element, it
450
+ suffices to show that x ∧ y as defined by the formula above is indeed the join of x and y.
451
+ It is clear that x∧y ≀ x. Let us show that x∧y ≀ 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 ∈ Di
456
+ n | z ≀ y} =
457
+ οΏ½
458
+ i
459
+ {z ∈ Di
460
+ n | z ≀ ρj
461
+ n}.
462
+ It suffices to check that οΏ½
463
+ i{z ∈ Di
464
+ n | z ≀ ρj
465
+ n} ≀ ρj
466
+ n. Note that
467
+ οΏ½
468
+ i
469
+ {z ∈ Di
470
+ n | z ≀ ρj
471
+ n} =
472
+ οΏ½
473
+ i
474
+ {z ∈ Di
475
+ n | z ≀ ρj
476
+ n and (z ≀· x ≀ ρj
477
+ n β‡’ x /∈ Di
478
+ n)}.
479
+ Now by lemma 2.6, if z ∈ Di
480
+ n and x is any element such that z ≀· x ≀ ρj
481
+ n and x /∈ Di
482
+ n, then
483
+ x = ρk
484
+ n for some k (necessarily smaller than or equal to j). It implies that
485
+ οΏ½
486
+ i
487
+ {z ∈ Di
488
+ n | z ≀ ρj
489
+ n} =
490
+ οΏ½
491
+ i
492
+ {z ∈ Di
493
+ n | z ≀ ρj
494
+ n and z ≀· ρk
495
+ n for some k ≀ j}.
496
+ By lemma 2.7, we have that
497
+ οΏ½
498
+ i
499
+ {z ∈ Di
500
+ n | z ≀ ρj
501
+ n and z ≀· ρk
502
+ n for some k ≀ j}
503
+ has to be an element of the set {z ∈ Di
504
+ n | z ≀ ρj
505
+ n and z ≀· ρk
506
+ n for some k ≀ j}, hence that
507
+ it is in particular a left-divisor of ρj
508
+ n (and hence of y).
509
+ Now assume that u ≀ x, y. We can assume that u ∈ Di
510
+ n, otherwise by lemma 2.6 we
511
+ have u < ρi
512
+ n ≀ x ∧ y. As u ≀ y, we have that u ≀ οΏ½
513
+ i{z ∈ Di
514
+ n | z ≀ y}. And hence, that
515
+ u ≀ x ∧i
516
+ οΏ½οΏ½
517
+ i{z ∈ Di
518
+ n | z ≀ y}
519
+ οΏ½
520
+ = x ∧ 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 definition 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 first definition,
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
+ Definition 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 finally 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 first 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 first 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 βˆ’ 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 defined 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 βˆ’ οΏ½
605
+ w∈LD(v) λ(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
+ Definition 3.2. Let Lm(T) be a labelled SchrΓΆder tree with n leaves labelled by m β‰₯ nβˆ’1.
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 βˆ’ 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 βˆ’ οΏ½
618
+ w∈LD(v) λ(w). Each λ(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
+ w∈LD(v) λ(w). We therefore have
623
+ (3.1)
624
+ 
625
+ ο£­
626
+ οΏ½
627
+ w∈LD(v)
628
+ Ξ»(w)
629
+ ο£Ά
630
+ ο£Έ + 1 ≀ n,
631
+ hence m βˆ’ οΏ½
632
+ w∈LD(v) Ξ»(w) β‰₯ 0. Moreover if οΏ½
633
+ w∈LD(v) λ(w) = m, then by (3.1) we must
634
+ have m = n βˆ’ 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 ∈ {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 different
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 βˆ’ οΏ½
656
+ w∈LD(v) λ1(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 ∈ LD(v). In T, the label of the root r is given by
662
+ Ξ»(r) = m βˆ’ Ξ»(v) βˆ’
663
+ οΏ½
664
+ w∈LD(v)
665
+ Ξ»(w) = m βˆ’ Ξ»(v) βˆ’
666
+ οΏ½
667
+ w∈LD(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 ρi to the
673
+ letter i with the convention that ρ0 = e, gives a map Φm from the set of Schrâder trees
674
+ labelled by m to the set S⋆ 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
+ ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5ρ11ρ1ρ11ρ2ρ1ρ11ρ10ρ8.
720
+ Definition 3.5. Let T be a non-empty Schrâder 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 βˆ’ 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 configuration described above (see also
730
+ the left picture in Figure 5), we get that r0 has at least three children in the configuration
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
+ ←→
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΄n–recall
769
+ that this is the SchrΓΆder tree in which every child of the root is a leaf–by 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 βˆ’ 1. Then Ξ¦m(T) is a
787
+ word for ρnβˆ’1(ρm)nβˆ’1ρmβˆ’n+1 in Mm. In particular if m = n βˆ’ 1, then it is a word for the
788
+ Garside element of Mnβˆ’1.
789
+ Proof. If T = Ξ΄n is the SchrΓΆder tree with only one root and n leaves, then Ξ¦m(T) =
790
+ ρnβˆ’1(ρm)nβˆ’1ρmβˆ’n+1. If T is another SchrΓΆder tree, then by Lemma 3.6 there is a sequence
791
+ of local moves from T to δn. To finish 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„“. Moreover, since r1 is not a leaf of T, we have β„“ < m. Hence in Ξ¦m(T)
797
+ we have the factor ρ1ρmρℓ with 1 ≀ β„“ ≀ m βˆ’ 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 βˆ’ οΏ½
802
+ w∈LD(r2) λ(w). The label
803
+ of r1 is β„“ = m βˆ’ 1 βˆ’ οΏ½
804
+ w∈LD(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 ρ1ρmρℓ in Ξ¦m(T) by ρℓ+1ρm,
806
+ and vice-versa.
807
+ β–‘
808
+ Proposition 3.8. For m = nβˆ’1, the map Ξ¦m from the set of SchrΓΆder trees with n leaves
809
+ to the set of words for ρn
810
+ nβˆ’1 in Mnβˆ’1 is surjective.
811
+ Proof. We have to show that to each word y for ρn
812
+ nβˆ’1 ∈ Mnβˆ’1, we can attach a SchrΓΆder
813
+ tree T with n leaves, in such a way that Φm(T) = y. The word y and the word ρn
814
+ nβˆ’1
815
+ can be transformed into each other by applying a sequence of defining relations of Mm.
816
+ We already know that the word ρn
817
+ nβˆ’1 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„“ ≀ m βˆ’ 1, then we are necessarily in the configuration 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 ρn
826
+ nβˆ’1 to the word y by a sequence of defining relations
827
+ let y0 = ρn
828
+ nβˆ’1, y1, . . . , yk = y be expressions of ρn
829
+ nβˆ’1 such that yi is obtained from yiβˆ’1
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„“ ≀ m βˆ’ 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 βˆ’ 1 would
839
+ be the number of leafs in the forest T2, . . . , Tk, which is at most n βˆ’ 1. As m = n βˆ’ 1,
840
+ the only possibility would be that v is the root of S, hence m would be the first 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 Tiβˆ’1. 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 configuration 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 βˆ’ 1 ΜΈ= 1. The root vβ€² of Tiβˆ’1 is labelled by 1 and as vβ€² is not the leftmost child of v, we
849
+ have 1 = Ξ»(vβ€²) = mβˆ’οΏ½
850
+ w∈LD(vβ€²) Ξ»(w), yielding οΏ½
851
+ w∈LD(vβ€²) Ξ»(w) = mβˆ’1. This means that
852
+ there are m leaves in Tiβˆ’1, and as there is one leaf in Ti and m = nβˆ’1, the only possibility
853
+ is that i βˆ’ 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„“ ≀ m βˆ’ 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 βˆ’ 1 is the number of leaves in T2, . . . , Tk, forcing v to be the root
859
+ of S and m to be the first 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„“. 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 βˆ’ 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 first result is proved by induction on the number of leaves. If the tree has one
874
+ leaf the result holds by definition of our labelling. Let T = (r, T1, · · · , Tk) be a Schrâder
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 λ(v). Hence by the first point, the forest F has weight
881
+ Ξ»(v)m.
882
+ β–‘
883
+ Proposition 3.10. Let m β‰₯ n βˆ’ 1. Then the map Ξ¦m from the set of SchrΓΆder trees with
884
+ n leaves to the set of words for the element ρnβˆ’1(ρm)nβˆ’1ρmβˆ’n+1 ∈ 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 first step of the proof is to remark that one can recover the
891
+ decomposition β€˜root and list of subtrees’ 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 first letter w1 of W is the label of the leftmost leaf r1 of T and by induction we will
896
+ find the letters w2, w3, · · · , wi corresponding to the ancestors r2, r3, · · · , ri of r1. Since the
897
+ labels of these vertices count a number of leaves of T, when οΏ½i
898
+ j=1 wj = n βˆ’ 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 β€˜correct’
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· wn1βˆ’1(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 difference. 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 β€˜correct’ 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 ∈ {1, · · · , 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 Φ11(T1) = ρ1ρ11ρ5ρ1ρ11ρ10ρ2ρ11ρ11ρ9ρ5. The
932
+ first 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 ρ11 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 ρ1ρ11ρ5 is obtained by reading the vertices of the leftmost
936
+ subtree S of T1. We apply the β€˜correction’ and we get ρ1ρ11ρ10 = Ξ¦11(S). The rest of the
937
+ word ρ1ρ11ρ10ρ2ρ11ρ11ρ9 corresponds to the other subtrees of T1 and it splits as ρ1ρ11ρ10
938
+ and ρ2ρ11ρ11ρ9.
939
+ Combining Proposition 3.8 and Proposition 3.10 we get our main result of the section:
940
+ Theorem 3.12. For m = n βˆ’ 1, the map Ξ¦m from the set of SchrΓΆder trees with n leaves
941
+ to the set of words for ρn
942
+ nβˆ’1 in Mnβˆ’1 is bijective.
943
+
944
+ 14
945
+ THOMAS GOBET AND BAPTISTE ROGNERUD
946
+ Corollary 3.13. The following two graphs are isomorphic under Ξ¦nβˆ’1:
947
+ (1) The graph of words for ρn
948
+ nβˆ’1 in Mnβˆ’1, where vertices are given by expressions of
949
+ ρn
950
+ nβˆ’1 and there is an edge between two expressions whenever they differ 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 differ 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 = nβˆ’1.
964
+ Then, the word obtained by reading all the labels of a subtree Sj is a word for ρljn 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 ρsρs
968
+ nβˆ’1ρnβˆ’1βˆ’s. If s = 0, then we have a word for ρnβˆ’1. Otherwise,
969
+ we can apply the relations [8, Lemma 4.5] with i = s and j = nβˆ’1. Alternatively, using the
970
+ Schrâder trees, it is easy to see that these relations comes from the following modifications
971
+ of the trees.
972
+ The word ρsρs
973
+ nβˆ’1ρnβˆ’1βˆ’s 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 (ρ1ρs)sρnβˆ’1βˆ’s. 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 ρs+1
978
+ nβˆ’1.
979
+ β–‘
980
+ Proposition 3.16. Let n β‰₯ 1. We have the following isomorphisms of posets:
981
+ (1) D0
982
+ n ∼= Div(βˆ†nβˆ’1), Dn+1
983
+ n
984
+ ∼= Div(βˆ†0) = {β€’},
985
+ (2) For all 1 ≀ i ≀ n, Di
986
+ n ∼= Div(βˆ†nβˆ’i),
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 ρi
992
+ nxβ€², where xβ€² is uniquely determined by cancellativity, and such that ρn is not a left-
993
+ divisor of xβ€². In particular, there is y a divisor of βˆ†n such that ρi
994
+ nxβ€²y = ρn+1
995
+ n
996
+ , and y ΜΈ= 1. We
997
+ associate a tree (or rather a family of trees) to x as follows. Write xβ€² as a product a1a2 Β· Β· Β· aj
998
+ of elements of S. Complete the word ρi
999
+ na1a2 · · · aj to a word ρi
1000
+ na1a2 Β· Β· Β· ajb1b2 Β· Β· Β· bβ„“ for βˆ†n,
1001
+ i.e., choose a word b1b2 Β· Β· Β· bβ„“ for y.
1002
+ There are several possibilitΓ©s for the bi’s, but the
1003
+ condition that x ∈ Di
1004
+ n ensures that, writing the corresponding SchrΓΆder tree in the form
1005
+ (r, T1, T2, . . . , Ti, S1, S2, · · · Sd), where the i first 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 ρi
1009
+ n, and if another tree among S2, . . . , Sd was partly labelled by the ai’s, then a
1010
+ power of ρn would left-divide xβ€², since the word obtained from S1 is a power of ρn (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 first reducing S2, . . . , Sd to a set of empty
1013
+
1014
+ ODD AND EVEN FIBONACCI LATTICES ARISING FROM A GARSIDE MONOID
1015
+ 15
1016
+ ρ2ρ1ρ3ρ2ρ1ρ3
1017
+ ρ3ρ1ρ3ρ2ρ3
1018
+ ρ3ρ1ρ3ρ1ρ3ρ1
1019
+ ρ3ρ2ρ3ρ3ρ1
1020
+ ρ2ρ3ρ3ρ1ρ3
1021
+ ρ3ρ3ρ3ρ3
1022
+ ρ3ρ3ρ1ρ3ρ2
1023
+ ρ3ρ2ρ1ρ3ρ2ρ1
1024
+ ρ1ρ3ρ1ρ3ρ1ρ3
1025
+ ρ1ρ3ρ2ρ3ρ3
1026
+ ρ1ρ3ρ2ρ1ρ3ρ2
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
+ βˆ†3 and the isomorphic graph of SchrΓΆder trees on 4 = 3 + 1 leaves.
1088
+ trees οΏ½T2, . . . , οΏ½Tdβ€² 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 Tk’s 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 prefix 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 prefix 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 first child are leafs. Note that the tree
1098
+ that we attached to x depends on a choice of word for x, but applying a defining 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 nβˆ’i+1 leaves. In this way,
1106
+ forgetting the i first empty trees, what we attached to x is an equivalence class of a (strict)
1107
+ initial section of a SchrΓΆder tree on n βˆ’ i + 1 leaves under local moves, that is, a divisor
1108
+ of βˆ†nβˆ’i. This mapping is injective since one can recover a word for x from the obtained
1109
+ SchrΓΆder tree on n βˆ’ 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 first 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 βˆ’ 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 xβ€² of D0
1115
+ n. Assume that ρn is a
1116
+ left-divisor of xβ€². Then, using local moves only involving those labels in the initial section
1117
+ of S corresponding to a word for xβ€², one can transform (r, T1, T2, . . . , Ti, S) into a tree of the
1118
+ form (r, T1, T2, . . . , Ti, Ti+1, Sβ€²
1119
+ 1, . . . , Sβ€²
1120
+ e), i.e., S can be split into several trees, the first one
1121
+ (corresponding to ρn) 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 xβ€² ∈ 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 first point, we have Dn+1
1128
+ n
1129
+ = {ρn+1
1130
+ n
1131
+ }, hence there is nothing to prove. To show
1132
+ that D0
1133
+ n ∼= Div(βˆ†nβˆ’1), one proceeds in a similar way as in the proof of point 1. Let x ∈ D0
1134
+ n
1135
+ and let y such that xy = βˆ†n. Choose words for x and y, and consider the corresponding
1136
+ Schrâder tree T = (r, T1, . . . , Tk). Since x ∈ 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 find a Schrâder 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
+ �T1 is a Schrâder tree on n leaf. Applying defining relations to words for x amounts to
1143
+ applying local moves inside the first tree, and arguing as in the first point this establishes
1144
+ the isomorphism of posets between D0
1145
+ n and Div(βˆ†nβˆ’1).
1146
+ β–‘
1147
+ r
1148
+ Ti
1149
+ S1
1150
+ οΏ½T1
1151
+ οΏ½T2
1152
+ βˆ’β†’
1153
+ r
1154
+ Ti
1155
+ r1
1156
+ S1
1157
+ οΏ½T1
1158
+ οΏ½T2
1159
+ βˆ’β†’
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 ρn+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(βˆ†n)|. Then
1183
+ An = 2A0 + 2Anβˆ’1 +
1184
+ nβˆ’2
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 defined by F0 = 1, F1 = 2, and Fi = Fiβˆ’1 + Fiβˆ’2 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(βˆ†n) = οΏ½
1193
+ 0≀i≀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 satisfied by the sequence F2n. This shows that
1197
+ An = F2n for all n β‰₯ 0.
1198
+ β–‘
1199
+ Definition 4.2. We call the lattice (Div(βˆ†n), ≀) 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(ρn
1202
+ n) of left-divisors of ρn
1203
+ n also forms a lattice under the restriction
1204
+ of left-divisibility, since it is an order ideal in the lattice (Div(βˆ†n), ≀). In terms of the
1205
+ Garside monoid Mn, the element ρn
1206
+ n is both the left- and right-lcm of the generators
1207
+ S = {ρ1, ρ2, . . . , ρn} (see [8, Corollary 4.17]). For n β‰₯ 1 we set Bn := |DivL(ρn
1208
+ n)|.
1209
+ Lemma 4.3. We have Bn = F2nβˆ’1 for all n β‰₯ 1. The sequence of the Bns is referred as
1210
+ A001519 in [11].
1211
+ Proof. Let x ∈ Div(βˆ†n). We claim that x ∈ DivL(ρn
1212
+ n) if and only if ρnx ∈ Div(βˆ†n). Indeed,
1213
+ if x ≀ ρn
1214
+ n, there is y ∈ Mn such that xy = ρn
1215
+ n. We then have ρnxy = ρn+1
1216
+ n
1217
+ = βˆ†n, hence
1218
+ ρnx is a left-divisor of βˆ†n. Conversely, assume that ρnx ∈ Div(βˆ†n). It follows that there
1219
+ is y ∈ Div(βˆ†n) such that ρnxy = βˆ†n = ρn+1
1220
+ n
1221
+ . By cancellativity we get that xy = ρn
1222
+ n, hence
1223
+ x ∈ DivL(ρn
1224
+ n).
1225
+ It follows that DivL(ρn
1226
+ n) is in bijection with the set
1227
+ {ρnx | x ∈ Div(βˆ†n)} ∩ Div(βˆ†n).
1228
+ But this set is nothing but οΏ½
1229
+ 1≀i≀n+1 Di
1230
+ n. It follows that
1231
+ Bn = |Div(βˆ†n)| βˆ’ |D0
1232
+ n| = |Div(βˆ†n)| βˆ’ |Div(βˆ†nβˆ’1)|,
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 βˆ’ Anβˆ’1 = F2n βˆ’ F2nβˆ’2 = F2nβˆ’1,
1236
+ which concludes the proof.
1237
+ β–‘
1238
+ Definition 4.4. We call the lattice (DivL(ρn
1239
+ n), ≀) 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 ρn
1242
+ n also has cardinality Bn: in fact, the
1243
+ two posets (DivL(ρn
1244
+ n), ≀L) and (DivR(ρn
1245
+ n), ≀R) are anti-isomorphic via x οΏ½β†’ x, where x is
1246
+ the element of Mn such that xx = ρn
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 βˆ’ 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 prefix x1x2 · · · 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 first subtree of T1 which is not covered by the word x1 · · · xl, similarly let Uk the first
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 first subtrees S1, Β· Β· Β· , Sjβˆ’1 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 first vertex of Sj. Let mβ€²
1264
+ 1 be the subword of m1 and mβ€²
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 mβ€²
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 ˜Uj with the same number of leaves as Sj. This will modify the word mβ€²
1273
+ 2, but not
1274
+ the prefix xs · · · xl, and xi labels a leftmost child in Uj if and only if it labels a leftmost
1275
+ child in ˜Uj (see Figure 7 for an illustration). After doing the modification, we consider the
1276
+ subword mj
1277
+ 2 corresponding to the tree ˜Uj 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 ρn+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 ρk+1
1288
+ k
1289
+ .
1290
+ Recall that Div(βˆ†n) = οΏ½
1291
+ 0≀i≀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 ρn+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 ≀ i ≀ 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
+ prefix of a Schrâder tree T = (r, S1, · · · , Sk). By Lemma 2.5, w = w1w2 where w1 is a word
1305
+ for ρi
1306
+ n and ρn 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 defining relations to w1 in order to obtain ρi
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 first j trees which have in total
1313
+ i leaves. Since ρn does not divide w2, we see that w2 is a (possibly empty) strict prefix
1314
+ of Sj+1. It is also possible to modify the trees Sj+2, · · · , Sk without changing the first 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 modification we obtain a tree ˜T = (r, ˜S, L) where L the empty
1318
+ tree, ˜S is a tree with n leaves and w = w2 is a strict prefix of ˜S.
1319
+ β€’ When 1 ≀ i ≀ n, we obtain a tree ˜T = (r, S1, Β· Β· Β· , Sj, οΏ½Sj+1) and w2 is a strict prefix
1320
+ of the tree οΏ½Sj+1 with n + 1 βˆ’ 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 prefix of �Sj+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 prefix of a Schrâder tree, denoted by ˜S, with li + 1 leaves where
1329
+ li = n βˆ’ i leaves if i ΜΈ= 0 and li = n βˆ’ 1 if i = 0.
1330
+ Let w = w1w2 be a word of an element of Di
1331
+ n with w2 having t letters. Let ˜S be a
1332
+ Schrâder tree with li + 1 leaves having w2 as a strict prefix. Then, we construct a word
1333
+ γ(w) by first extracting ˜S, then labelling it accordingly to its number of leaves (i.e., with
1334
+ m = li) and finally taking its first t letters in post-order. Algebraically, it is easy to see
1335
+ how the word γ(w) is obtained from w2: if wi is the label of a leftmost child in ˜S, we have
1336
+ Ξ³(w)i = wi. Otherwise, since the tree ˜Sj has li + 1 leaves, we have Ξ³(w)i = wi βˆ’ n + (li).
1337
+ A priori γ(w) depends on the choice of a tree ˜S, but Lemma 4.6 tells us that γ(w) only
1338
+ depends on w2. The word γ(w) is a prefix of a Schrâder tree with li + 1 leaves, hence it
1339
+ is a word for a divisor of βˆ†li. 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 βˆ†li.
1342
+ Conversely, if z is a word of length k for a divisor of βˆ†li, it is a prefix (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 first 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
+ ǫ(z) only depends on z, not on the tree having z as a prefix.
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 = dnβˆ’1.
1355
+ When i β‰₯ 1, then Ξ³ is far from being injective, since it forgets the first 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
+ ρi
1362
+ nw2 and E1 contains at least the words of the form ρiβˆ’1ρiβˆ’1
1363
+ n
1364
+ ρnβˆ’iβˆ’1w2 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 βˆ†li, we compute the cardinality of the preimage of z by Ξ³
1367
+ by looking at Ξ³βˆ’1(z) ∩ E1 and Ξ³βˆ’1(z) ∩ E2. If i = 1, we obviously only consider the first
1368
+ case. The elements of Ξ³βˆ’1(z) ∩ E1 are obtained by concatenation of the word of a single
1369
+ SchrΓΆder tree with i leaves and Η«(z), and the elements of Ξ³βˆ’1(z) ∩ 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
+ |Ξ³βˆ’1(z) ∩ E1| = siβˆ’1 = |Ξ³βˆ’1(z) ∩ E2|.
1374
+ Taking the sum on all possible words z, we have d1
1375
+ n = s0 Β· dnβˆ’1 and di
1376
+ n = 2 Β· siβˆ’1 Β· dnβˆ’i
1377
+ when n β‰₯ i β‰₯ 2.
1378
+ We have obtained:
1379
+ d0
1380
+ n = dnβˆ’1;
1381
+ d1
1382
+ n = s0 Β· dnβˆ’1 = dnβˆ’1;
1383
+ and
1384
+ di
1385
+ n = 2 Β· siβˆ’1 Β· dnβˆ’i 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 ≀ n βˆ’ 1, and
1392
+ dn = 2sn + 2
1393
+ n
1394
+ οΏ½
1395
+ i=2
1396
+ siβˆ’1snβˆ’i+1 + sn
1397
+ = 3sn + 2
1398
+ nβˆ’1
1399
+ οΏ½
1400
+ i=1
1401
+ sisnβˆ’i.
1402
+ Using generating functions, it is not difficult 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. Γ‰cole 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), 322–353.
1409
+ [3] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.
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
+ 235–254.
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 Stasheff 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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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“ ≀ Ξ› where Ξ› is infinite and amenable, as well as
9
+ several groups of dynamical origin such as the classical Thompson groups F, F β€², 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 finite-dimensional unitary
20
+ groups. The problem of stability asks whether every almost-homomorphism Ξ“ β†’ U ∈ 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), βˆ₯Β·βˆ₯)} be a family of finite-dimensional unitary groups equipped with bi-
25
+ invariant submultiplicative norms βˆ₯Β·βˆ₯ (we allow U(k) to appear multiple times with different
26
+ norms). For instance βˆ₯ Β· βˆ₯ 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 define its defect to be
28
+ def(Ο†) := sup
29
+ g,hβˆˆΞ“
30
+ βˆ₯Ο†(gh) βˆ’ Ο†(g)Ο†(h)βˆ₯.
31
+ Given another map ψ : Ξ“ β†’ U(k), we define the distance between them to be
32
+ dist(Ο†, ψ) := sup
33
+ gβˆˆΞ“
34
+ βˆ₯Ο†(g) βˆ’ ψ(g)βˆ₯.
35
+ Definition 1.1. A uniform asymptotic homomorphism is a sequence of maps Ο†n : Ξ“ β†’ U(kn)
36
+ such that def(Ο†n) β†’ 0. We denote this simply by Ο† : Ξ“ β†’ U. We say that Ο†, ψ : Ξ“ β†’ U are
37
+ uniformly asymptotically close if they have the same range degrees and dist(Ο†n, ψn) 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 define 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 α ∈ H2
58
+ a(Ξ“, V). We refer the reader to Section 2.2 for the relevant definitions.
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). Thompson’s groups F, F β€², T and V are uniformly U-stable, with
64
+ a linear estimate.
65
+ As remarked by Arzhantseva and P˘aunescu [AP15, Open problem], the analogous state-
66
+ ment for pointwise stability in permutation of F would imply that F is not sofic, thus proving
67
+ at once the existence of a non-sofic 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 β€² 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“ ≀ Ξ› = (βŠ•Ξ›Ξ“) β‹Š Ξ›, where Ξ› acts by shifting the coordinates.
73
+ Theorem 1.3. Let Ξ“, Ξ› be two countable groups, where Ξ› is infinite and amenable. Then
74
+ Ξ“ ≀ 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 flexibility 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 finitely generated
81
+ uniformly U-stable groups, a fact which could also be obtained by applying Kazhdan’s The-
82
+ orem [Kaz82] to an infinite family of finitely 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 β€² from Theorem 1.3, we exploit coamenability.
87
+ Recall that a subgroup Ξ› ≀ Ξ“ is coamenable if the coset space Ξ“/Ξ› admits a Ξ“-invariant
88
+ mean. It is well known that F β€² and F contain a coamenable lamplighter F ≀ Z. Therefore the
89
+ stability of F and F β€² (Corollary 5.8) follows from Theorem 1.3, and the following result:
90
+ Proposition 1.5. Let Ξ› ≀ 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 Kazhdan’s Theorem, which is sort of dual to Proposition 1.5:
95
+ Proposition 1.6. Let N ≀ 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 Thompson’s 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 infinite and amenable. Then
107
+ Hn
108
+ a(Ξ“ ≀ Ξ›, V) = 0 for all n β‰₯ 1 and all finitary dual asymptotic Banach βˆ—Ξ“-modules V.
109
+ Here the word finitary refers to the fact that these modules arise from stability problems
110
+ with respect to finite-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 finitary assumption:
113
+ Proposition 1.8. Let Ξ› ≀ 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 βˆ—Ξ“-modules V.
117
+ Proposition 1.9. Let N ≀ 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
+ βˆ—(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 Gromov’s
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 coefficients, and our
135
+ proofs are of a different 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 soficity, hyperlinearity, and matricial finite-
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 infinite-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 Lodha–Moore [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 ultrafilter Ο‰ on N is fixed 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 briefly 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 fixed non-principal ultra-
188
+ filter on N. The algebraic ultraproduct �
189
+ Ο‰ Xn of an indexed collection {Xn}n∈N of sets is
190
+ defined to be �
191
+ Ο‰ Xn := οΏ½
192
+ n∈N Xn/ ∼ where for {xn}n∈N, {yn}n∈N ∈ �
193
+ n∈N Xn, we define
194
+ {xn}n∈N ∼ {yn}n∈N if {n : xn = yn} ∈ 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 βˆ—Ξ“, is itself a group. Another important example we will use is
198
+ the field of hyperreals βˆ—R, 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 βˆ—Rb of bounded
201
+ hyperreals, consisting of elements {xn}Ο‰ ∈ βˆ—R for which there exists S ∈ Ο‰ and C ∈ Rβ‰₯0 such
202
+ that |xn| ≀ C for every n ∈ S, and the subset βˆ—Rinf of infinitesimals, consisting of elements
203
+ {xn}Ο‰ ∈ βˆ—R such that for every real Ξ΅ > 0, there exists S ∈ Ο‰ such that |xn| < Ξ΅ for every
204
+ n ∈ S.
205
+ For x, y ∈ βˆ—R, write x = OΟ‰(y) if x/y ∈ βˆ—Rb, and write x = oΟ‰(y) if x/y ∈ βˆ—Rinf. In
206
+ particular, x ∈ βˆ—Rb is equivalent to x = OΟ‰(1) while Ξ΅ ∈ βˆ—Rinf is equivalent to Ξ΅ = oΟ‰(1). The
207
+ subset βˆ—Rb forms a valuation ring with βˆ—Rinf being the unique maximal ideal, with quotient
208
+ βˆ—Rb/βˆ—Rinf ∼= R. The quotient map st : βˆ—Rb β†’ R is known as the standard part map or limit
209
+ along the ultrafilter Ο‰. The previous construction can also be replicated for Banach spaces.
210
+ Let {Wn}n∈N be a family of Banach spaces. Then W = �
211
+ Ο‰ Wn can be given the structure
212
+ of a βˆ—R-vector space. In fact, it also comes equipped with a βˆ—R-valued norm, allowing us to
213
+ define the external subsets Wb and Winf. The quotient ˜
214
+ W := Wb/Winf is a real Banach space.
215
+ Given a uniform asymptotic homomorphism {Ο†n : Ξ“ β†’ U(kn)}n∈N with def(Ο†n) =: Ξ΅n β†’
216
+ 0, construct the internal map Ο† : βˆ—Ξ“ β†’ οΏ½
217
+ Ο‰ U(kn) where Ο† := οΏ½
218
+ Ο‰ Ο†n, with (hyperreal) defect
219
+ Ξ΅ := {Ξ΅n}Ο‰ ∈ βˆ—Rinf. Then the question of uniform stability with a linear estimate can be
220
+ rephrased as asking whether there exists an internal homomorphism ψ : βˆ—Ξ“ β†’ οΏ½
221
+ Ο‰ U(kn) such
222
+ that their (hyperreal) distance satisfies dist(Ο†, ψ) := {dist(Ο†n, ψn)}Ο‰ = OΟ‰(Ξ΅).
223
+ The advantage of rephrasing the question in terms of internal maps is that an internal
224
+ map Ο† : βˆ—Ξ“ β†’ οΏ½
225
+ Ο‰ U(kn) with defect Ξ΅ ∈ βˆ—Rinf induces a true homomorphism ΛœΟ† : βˆ—Ξ“ 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 Oω(Ρ) 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 ΛœΟ† : βˆ—Ξ“ β†’ οΏ½
232
+ Ο‰ U(kn)/B(Ξ΅) be lifted to an internal
233
+ 5
234
+
235
+ homomorphism ψ : βˆ—Ξ“ 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. Define a normal subgroup I(Ρ)
241
+ of B(Ρ) comprising elements that are at a distance of oω(Ρ) from the identity. Then we can
242
+ attempt to lift ΛœΟ† : βˆ—Ξ“ β†’ οΏ½
243
+ Ο‰ U(kn)/B(Ξ΅) to an internal map ψ : βˆ—Ξ“ 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 ψ with the
250
+ same range such that dist(Ο†, ψ) = OΟ‰(def(Ο†)) and def(ψ) = 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 βˆ—Ξ“ constructed from the uniform asymptotic homomorphism Ο† that we start
263
+ out with. The logarithm of the defect map
264
+ βˆ—Ξ“ Γ— βˆ—Ξ“ β†’
265
+ οΏ½
266
+ Ο‰
267
+ U(kn) : (g1, g2) οΏ½β†’ Ο†(g1)Ο†(g2)Ο†(g1g2)βˆ’1
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 coefficients 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 definition 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 ∈ V we denote by βˆ₯vβˆ₯ the
285
+ hyperreal (βˆ₯vnβˆ₯)Ο‰ ∈ βˆ—R. We then denote by
286
+ Vb := {v ∈ V : βˆ₯vβˆ₯ ∈ βˆ—Rb};
287
+ Vinf := {v ∈ V : βˆ₯vβˆ₯ ∈ βˆ—Rinf}.
288
+ Then the quotient ˜V := Vb/Vinf is a real Banach space, whose norm is induced by the
289
+ ultralimit of βˆ₯ Β· βˆ₯ on Vb. For each Vn denote by V #
290
+ n its continuous dual, and let V# be the
291
+ corresponding algebraic ultraproduct. The pairing ⟨·, ·⟩n : V #
292
+ n Γ— Vn β†’ R induces a pairing
293
+ V# Γ— V β†’ βˆ—R which descends to ˜V# Γ— ˜V 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€ : βˆ—Ξ“Γ—V β†’ V be an internal map which
297
+ preserves βˆ₯ Β· βˆ₯ and induces an isometric linear action ΛœΟ€ : βˆ—Ξ“ Γ— ˜V β†’ ˜V of βˆ—Ξ“. Such a map Ο€
298
+ is referred to as an asymptotic βˆ—Ξ“-action on V. We then call (Ο€, V), or V, if Ο€ is understood
299
+ from context, an asymptotic Banach βˆ—Ξ“-module. Given an internal Banach βˆ—Ξ“-module (Ο€, V),
300
+ the contragradient on each coordinate induces an internal map Ο€# : βˆ—Ξ“ Γ— V# β†’ V# mak-
301
+ ing (Ο€#, V#) into an asymptotic Banach βˆ—Ξ“-module. We call a module V a dual asymptotic
302
+ Banach βˆ—Ξ“-module if V is the dual of some asymptotic βˆ—Ξ“-module denoted Vβ™­. We decorate
303
+ these definitions with the adjective finitary if each Vn is finite-dimensional.
304
+ Now for each m β‰₯ 0 define the internal Banach space L∞((βˆ—Ξ“)m, V) := οΏ½
305
+ Ο‰
306
+ β„“βˆž(Ξ“m, Vn)
307
+ (note that m is fixed and n runs through the natural numbers with respect to the ultrafilter
308
+ Ο‰). Similarly to before, for f ∈ L∞((βˆ—Ξ“)m, V) we denote βˆ₯fβˆ₯ := (βˆ₯fnβˆ₯)Ο‰ ∈ βˆ—R and
309
+ L∞
310
+ b ((βˆ—Ξ“)m, V) := {f ∈ L∞((βˆ—Ξ“)m, V) : βˆ₯fβˆ₯ ∈ βˆ—Rb};
311
+ L∞
312
+ inf((βˆ—Ξ“)m, V) := {f ∈ L∞((βˆ—Ξ“)m, V) : βˆ₯fβˆ₯ ∈ βˆ—Rinf}.
313
+ Given an asymptotic βˆ—Ξ“-action Ο€ on V, we can construct a natural asymptotic βˆ—Ξ“-action
314
+ ρm : βˆ—Ξ“ Γ— L∞((βˆ—Ξ“)m, V) β†’ L∞((βˆ—Ξ“)m, V) given by
315
+ (ρm(g)(f))(g1, g2, . . . , gm) := Ο€G(g)f(gβˆ’1g1, . . . , gβˆ’1gm)
316
+ (1)
317
+ Then the quotient
318
+ ˜L∞((βˆ—Ξ“)m, V) := L∞
319
+ b ((βˆ—Ξ“)m, V)/L∞
320
+ inf((βˆ—Ξ“)m, V)
321
+ is again a real Banach space equipped with an isometric βˆ—Ξ“-action induced coordinate-wise
322
+ by ρm, which defines the invariant subspaces ˜L∞((βˆ—Ξ“)m, V)
323
+ βˆ—Ξ“.
324
+ Now define the internal coboundary map
325
+ dm : L∞((βˆ—Ξ“)m, V) β†’ L∞((βˆ—Ξ“)m+1, V);
326
+ dm(f)(g0, . . . , gm) :=
327
+ m
328
+ οΏ½
329
+ j=0
330
+ (βˆ’1)jf(g0, . . . , Λ†gj, . . . , gm),
331
+ (2)
332
+ which descends to coboundary maps
333
+ ˜
334
+ dm : ˜L∞((βˆ—Ξ“)m, V) β†’ ˜L∞((βˆ—Ξ“)m+1, V).
335
+ Since ˜
336
+ dm is βˆ—Ξ“-equivariant, it defines the cochain complex:
337
+ 0
338
+ ˜
339
+ d0
340
+ βˆ’β†’ ˜L∞(βˆ—Ξ“, V)
341
+ βˆ—Ξ“
342
+ ˜
343
+ d1
344
+ βˆ’β†’ ˜L∞((βˆ—Ξ“)2, V)
345
+ βˆ—Ξ“
346
+ ˜d2
347
+ βˆ’β†’ ˜L∞((βˆ—Ξ“)3, V)
348
+ βˆ—Ξ“
349
+ ˜
350
+ d3
351
+ βˆ’β†’ Β· Β· Β·
352
+ Definition 2.2 ([GLMR23, Definition 4.2.2]). The m-th asymptotic cohomology of Ξ“ with
353
+ coefficients in V is
354
+ Hm
355
+ a (Ξ“, V) := ker(
356
+ ˜
357
+ dm+1)/ im( ˜
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 : L∞(Γ×S) β†’ L∞(S). Let S be a regular Ξ“-space with
364
+ a Zimmer-amenable action of Ξ“, and let L∞((βˆ—S)m, V) := οΏ½
365
+ Ο‰
366
+ L∞
367
+ wβˆ—(Sm, Vn) (where L∞
368
+ wβˆ—(Sm, Vn)
369
+ is the space of bounded weak-βˆ— measurable function classes from Sm to Vn). Again, the
370
+ asymptotic βˆ—Ξ“-action on V gives rise to a natural asymptotic βˆ—Ξ“-action on L∞((βˆ—Ξ“)m, V) as
371
+ in (1), making L∞((βˆ—S)m, V) an asymptotic Banach βˆ—Ξ“-module. The coboundary maps too
372
+ can be defined 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 βˆ—Ξ“-module. Then Hβ€’
375
+ a(Ξ“, V) can be computed as the asymptotic
376
+ cohomology of the cochain complex
377
+ 0
378
+ ˜
379
+ d0
380
+ βˆ’β†’ ˜L∞(βˆ—S, V)
381
+ βˆ—Ξ“
382
+ ˜
383
+ d1
384
+ βˆ’β†’ ˜L∞((βˆ—S)2, V)
385
+ βˆ—Ξ“
386
+ ˜
387
+ d2
388
+ βˆ’β†’ ˜L∞((βˆ—S)3, V)
389
+ βˆ—Ξ“
390
+ ˜
391
+ d3
392
+ βˆ’β†’ Β· Β· Β·
393
+ In the context of uniform U-stability, the relevant asymptotic Banach βˆ—Ξ“-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 finite-dimensional unitary groups, so such a module is finitary.
397
+ Given a uniform asymptotic homomorphism Ο† : βˆ—Ξ“ β†’ οΏ½
398
+ Ο‰ U(n) with defect def(Ο†) ≀ω Ξ΅ ∈
399
+ βˆ—Rinf, this can be used to construct the asymptotic action Ο€ : βˆ—Ξ“ Γ— W β†’ W given by
400
+ Ο€(g)v = Ο†(g)vΟ†(g)βˆ’1, making W an asymptotic Banach βˆ—Ξ“-module. We call such a module
401
+ an Ulam βˆ—Ξ“-module supported on U.
402
+ Also, consider the map Ξ± : βˆ—Ξ“ Γ— βˆ—Ξ“ β†’ W given by
403
+ Ξ±(g1, g2) = 1
404
+ Ξ΅ log(Ο†(g1)Ο†(g2)Ο†(g1g2)βˆ’1).
405
+ (3)
406
+ This map Ξ± induces an inhomogeneous 2-cocycle ˜α : βˆ—Ξ“ Γ— βˆ—Ξ“ β†’
407
+ ˜
408
+ W, and thus defines
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. ˜α 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
+ βˆ—Ξ“-module supported on U, then Ξ“ is uniformly U-stable, with a linear estimate.
418
+ 3
419
+ Hereditary properties
420
+ In this section, we first 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› ≀ 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 ≀ 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†’ L∞(Ξ“) β†’ L∞(Ξ“2) β†’ L∞(Ξ“3) β†’ Β· Β· Β·
441
+ 0 β†’ L∞(S) β†’ L∞(S2) β†’ L∞(S3) β†’ Β· Β· οΏ½οΏ½
442
+ which is then extended internally to the asymptotic version of these complexes. The case
443
+ of dual asymptotic coefficients follows via some suitable identifications 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 : L∞(Sm) βˆ’β†’ L∞(Ξ“m)
449
+ om
450
+ b (f)(g1, . . . , gm) = f(g1b, . . . , gmb);
451
+ where b ∈ 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 ∈ S, and
454
+ let V be a dual asymptotic Banach βˆ—Ξ“-module. Then the orbit map
455
+ om
456
+ b : L∞((βˆ—S)m, V) βˆ’β†’ L∞((βˆ—Ξ“)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
+ ˜
463
+ d0
464
+ βˆ’β†’ ˜L∞(βˆ—S, V)
465
+ βˆ—Ξ“
466
+ ˜
467
+ d1
468
+ βˆ’β†’ ˜L∞((βˆ—S)2, V)
469
+ βˆ—Ξ“
470
+ ˜
471
+ d2
472
+ βˆ’β†’ ˜L∞((βˆ—S)3, V)
473
+ βˆ—Ξ“
474
+ ˜
475
+ d3
476
+ βˆ’β†’ Β· Β· Β·
477
+ In the two basic examples of discrete Zimmer-amenable spaces from above, we obtain:
478
+ Corollary 3.2. Let Ξ› ≀ Ξ“ be a subgroup, and let V be a dual asymptotic Banach βˆ—Ξ“-
479
+ module, which restricts to a dual asymptotic Banach βˆ—Ξ›-module.
480
+ Then the restriction of
481
+ cochains L∞((βˆ—Ξ“)m, V) β†’ L∞((βˆ—Ξ›)m, V) induces an isomorphism between Hβ€’
482
+ a(Ξ›, V) and the
483
+ cohomology of the complex:
484
+ 0
485
+ ˜
486
+ d0
487
+ βˆ’β†’ ˜L∞(βˆ—Ξ“, V)
488
+ βˆ—Ξ›
489
+ ˜
490
+ d1
491
+ βˆ’β†’ ˜L∞((βˆ—Ξ“)2, V)
492
+ βˆ—Ξ›
493
+ ˜
494
+ d2
495
+ βˆ’β†’ ˜L∞((βˆ—Ξ“)3, V)
496
+ βˆ—Ξ›
497
+ ˜
498
+ d3
499
+ βˆ’β†’ Β· Β· Β·
500
+ Proof. Seeing Ξ“ as a discrete Zimmer-amenable Ξ›-space, with basepoint 1 ∈ 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 ≀ Ξ“ be an amenable normal subgroup, and let V be a dual asymptotic
505
+ Banach βˆ—(Ξ“/N)-module, which pulls back to a dual asymptotic Banach βˆ—Ξ“-module. Then the
506
+ pullback of cochains L∞((βˆ—(Ξ“/N))m, V) β†’ L∞((βˆ—Ξ“)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
+ ˜
516
+ d0
517
+ βˆ’β†’ ˜L∞(βˆ—(Ξ“/N), V)
518
+ βˆ—Ξ“
519
+ ˜
520
+ d1
521
+ βˆ’β†’ ˜L∞((βˆ—(Ξ“/N))2, V)
522
+ βˆ—Ξ“
523
+ ˜
524
+ d2
525
+ βˆ’β†’ ˜L∞((βˆ—(Ξ“/N))3, V)
526
+ βˆ—Ξ“
527
+ ˜d3
528
+ βˆ’β†’ Β· Β· Β·
529
+ But since the action of βˆ—Ξ“ on both βˆ—(Ξ“/N) and V factors through βˆ—(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› ≀ Ξ“ be a (not necessarily coamenable) subgroup, and V be a dual asymptotic Ba-
538
+ nach βˆ—Ξ“-module, which restricts to a dual asymptotic Banach βˆ—Ξ›-module. The restriction
539
+ ˜L∞((βˆ—Ξ“)β€’, V)
540
+ βˆ—Ξ“ β†’ ˜L∞((βˆ—Ξ›)β€’, V)
541
+ βˆ—Ξ› 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 βˆ—Ξ“-module supported on U. Then W is also an Ulam βˆ—Ξ›-
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 βˆ—Ξ“-module. Then restricting Ο†n to Ξ› for each n yields a uniform asymptotic
554
+ homomorphism Ο†|Ξ› : Ξ› β†’ U, with def(Ο†|Ξ›) ≀ω def(Ο†) and endows W with an asymptotic
555
+ βˆ—Ξ›-action making it into an Ulam βˆ—Ξ›-module supported on U. The cocycle corresponding to
556
+ Ο† is defined via the map
557
+ Ξ± : βˆ—Ξ“ Γ— βˆ—Ξ“ β†’ W : (g1, g2) οΏ½β†’ 1
558
+ Ξ΅ log(Ο†(g1)Ο†(g2)Ο†(g1g2)βˆ’1).
559
+ Since def(Ο†|Ξ›) ≀ω Ξ΅, restricting Ξ± to βˆ—Ξ› Γ— βˆ—Ξ› yields a valid cocycle associated to the
560
+ uniform asymptotic homomorphism φΛ. It follows that the chain map ˜L∞((βˆ—Ξ“)β€’, V)
561
+ βˆ—Ξ“ β†’
562
+ ˜L∞((βˆ—Ξ›)β€’, V)
563
+ βˆ—Ξ› preserves the set of cocycles defined via uniform asymptotic homomorphisms,
564
+ and therefore preserves Ulam classes.
565
+ Now suppose that Ξ› ≀ Ξ“ is coamenable. This means, by definition, that there exists a
566
+ Ξ“-invariant mean on Ξ“/Ξ›; that is, there exists a linear functional m : β„“βˆž(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)| ≀ βˆ₯fβˆ₯ for all f ∈ β„“βˆž(Ξ“/Ξ›).
572
+ 3. m(g Β· f) = m(f) for all g ∈ Ξ“ and all f ∈ β„“βˆž(Ξ“/Ξ›).
573
+ As with the absolute case [GLMR23, Lemma 3.20], we have the following:
574
+ Lemma 3.5. Suppose that Ξ› ≀ Ξ“ is coamenable, and let V be a dual asymptotic Banach
575
+ βˆ—Ξ“-module. Then there exists an internal map m : L∞(βˆ—Ξ“/βˆ—Ξ›, V) β†’ V which induces a map
576
+ ˜m : ˜L∞(βˆ—Ξ“/βˆ—Ξ›, V) β†’ ˜V with the following properties:
577
+ 1. If ˜f is the constant function equal to ˜v ∈ ˜V, then ˜m( ˜f) = ˜v.
578
+ 2. βˆ₯ ˜m( ˜f)βˆ₯ ≀ βˆ₯ ˜fβˆ₯ for all ˜f ∈ ˜L∞(βˆ—Ξ“/βˆ—Ξ›, V).
579
+ 3. ˜m(g Β· ˜f) = ˜m( ˜f) for all g ∈ βˆ—Ξ“ and all ˜f ∈ ˜L∞(βˆ—Ξ“/βˆ—Ξ›, V).
580
+ Proof. Consider f = {fn}Ο‰ ∈ L∞(βˆ—Ξ“/βˆ—Ξ›, V). Since V is a dual asymptotic βˆ—Ξ“-module with
581
+ predual Vβ™­, for each Ξ» ∈ Vβ™­, we get an internal map
582
+ f Ξ» : βˆ—Ξ“/βˆ—Ξ› β†’ βˆ—R : 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„“βˆž(Ξ“/Ξ›). This allows us to
586
+ construct the internal map mΞ»
587
+ in : L∞(βˆ—Ξ“/βˆ—Ξ›, V) β†’ βˆ—R as
588
+ mΞ»
589
+ in(f) = {m
590
+ οΏ½
591
+ f Ξ»
592
+ n
593
+ οΏ½
594
+ }Ο‰
595
+ and finally min : L∞(βˆ—Ξ“/βˆ—Ξ›, V) β†’ V as
596
+ min(f)(Ξ») = mΞ»
597
+ in(f)
598
+ It is straightforward to check that min as defined induces a linear map ˜m : ˜L∞(βˆ—Ξ“/βˆ—Ξ›, V) β†’ ˜V.
599
+ As for βˆ—Ξ“-equivariance, this follows from the observation that (g Β· f)Ξ»(x) = Ο€(g)f(gβˆ’1x)(Ξ»)
600
+ while (gΒ·f Ξ»)(x) = f(gβˆ’1x)(Ξ»). The conditions on ˜m follow from the definition and properties
601
+ of the Ξ“-invariant mean m on β„“βˆž(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› ≀ 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 βˆ—Ξ“-modules V.
608
+ Proof. We implement the asymptotic cohomology of Ξ› using the complex ˜L∞((βˆ—Ξ“)β€’, V)
609
+ βˆ—Ξ›
610
+ from Corollary 3.2. Since the chain map that defines the restriction map factors through
611
+ this complex, and the chain map ˜L∞((βˆ—Ξ“)β€’, V)
612
+ βˆ—Ξ› β†’ ˜L∞((βˆ—Ξ›)β€’, V)
613
+ βˆ—Ξ› induces an isomorphism
614
+ in cohomology (Corollary 3.2), it suffices to show that the chain inclusion ˜L∞((βˆ—Ξ“)β€’, V)
615
+ βˆ—Ξ“ β†’
616
+ ˜L∞((βˆ—Ξ“)β€’, V)
617
+ βˆ—Ξ› 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€’ : ˜L∞((βˆ—Ξ“)β€’, V)
628
+ βˆ—Ξ› β†’ ˜L∞((βˆ—Ξ›)β€’, V)
629
+ βˆ—Ξ“ that restricts to the identity on ˜L∞((βˆ—Ξ›)β€’, V)
630
+ βˆ—Ξ“.
631
+ 11
632
+
633
+ Let f ∈ L∞((βˆ—Ξ“)k, V) be such that ˜f ∈ ˜L∞((βˆ—Ξ“)k, V)
634
+ βˆ—Ξ›. For each x ∈ (βˆ—Ξ“)k, define
635
+ fx : βˆ—Ξ“ β†’ V
636
+ fx(g) := Ο€(g)f(gβˆ’1x)
637
+ In other words, fx(g) is just (ρ1(g)f)(x) as in (1). Since ˜f ∈ ˜L∞((βˆ—Ξ“)k, V)
638
+ βˆ—Ξ›, for any Ξ³ ∈ βˆ—Ξ›
639
+ and g ∈ βˆ—Ξ“,
640
+ fx(gΞ³) βˆ’ fx(g) ∈ Vinf
641
+ Let us choose representatives of left βˆ—Ξ›-cosets in βˆ—Ξ“ and restrict fx to this set of repre-
642
+ sentatives so that we can regard fx as an internal map fx : βˆ—Ξ“/βˆ—Ξ› β†’ V. Moreover, since
643
+ fx ∈ L∞(βˆ—Ξ“/βˆ—Ξ›, V), we can apply the mean m constructed in Lemma 3.5 to define the internal
644
+ map transk(f) : L∞((βˆ—Ξ“)k, V) β†’ L∞((βˆ—Ξ“)k, V) by
645
+ transk(f)(x) = m(fx)
646
+ Since ˜m is βˆ—Ξ“-invariant, this means that for g ∈ βˆ—Ξ“, m(fgx) βˆ’ Ο€(g)m(fx) ∈ Vinf, and implies
647
+ that
648
+ (transk(f))(gx) βˆ’ Ο€(g) transk(f)(x) ∈ Vinf.
649
+ This establishes that for f ∈ L∞((βˆ—Ξ“)k, V) with ˜f ∈ ˜L∞((βˆ—Ξ“)k, V)
650
+ βˆ—Ξ›, we have
651
+ οΏ½
652
+ transk(f) ∈
653
+ ˜L∞((βˆ—Ξ“)k, V)
654
+ βˆ—Ξ“. Therefore transβ€’ induces a chain map
655
+ ˜
656
+ trans
657
+ β€’ : ˜L∞((βˆ—Ξ“)β€’, V)
658
+ βˆ—Ξ› β†’ ˜L∞((βˆ—Ξ›)β€’, V)
659
+ βˆ—Ξ“.
660
+ Finally, if ˜f is already βˆ—Ξ“-invariant, then fx is constant up to infinitesimals, and thus m(fx)
661
+ is equal, up to an infinitesimal, 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 ˜L∞((βˆ—Ξ›)β€’, V)
665
+ βˆ—Ξ“, and concludes the proof.
666
+ Proposition 1.5 is now an easy consequence.
667
+ Proposition (Proposition 1.5). Let Ξ› ≀ 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 suffices to show that all Ulam classes supported on U vanish in H2
672
+ a(Ξ“, W),
673
+ where W is an Ulam βˆ—Ξ“-module. Now by Proposition 1.8, it suffices 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 ≀ Ξ“ be an amenable normal subgroup, and let V be a dual asymptotic Banach βˆ—(Ξ“/N)-
684
+ module, which pulls back to a dual asymptotic Banach βˆ—Ξ“-module. Precomposing cochains by
685
+ the projection βˆ—Ξ“ β†’ βˆ—(Ξ“/N) defines 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 βˆ—(Ξ“/N)-module. Then W is also an Ulam βˆ—Ξ“-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 ≀ 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 βˆ—Ξ“-modules V.
701
+ And we deduce Proposition 1.6 analogously:
702
+ Proposition (Proposition 1.6). Let N ≀ 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 suffices to show that all Ulam classes supported on U vanish
707
+ in H2
708
+ a(Ξ“/N, W), where W is an Ulam βˆ—(Ξ“/N)-module. Now by Proposition 1.9, it suffices
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 reader’s convenience:
719
+ Theorem. Let Ξ“, Ξ› be two countable groups, where Ξ› is infinite and amenable.
720
+ Then
721
+ Hn
722
+ a(Ξ“ ≀ Ξ›, V) = 0 for all n β‰₯ 1 and all finitary dual asymptotic Banach βˆ—Ξ“-modules V.
723
+ Remark 4.1. In fact, the theorem will hold for a larger class of coefficients, 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 finding 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 := ΓΛ. Then S is a Zimmer-amenable (Ξ“ ≀ 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 Γ—S is ergodic, we even obtain ergodicity
733
+ with separable coefficients, 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
+ infinite, and let S be as in Lemma 4.2. Then Sm is an ergodic (Γ≀Λ)-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 suffices 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 βˆ₯g Β· f βˆ’ fβˆ₯ < Ξ΅ for
744
+ all g ∈ Ξ“, there exists a constant c ∈ R such that |f(s) βˆ’ c| < Ξ΅ for almost every s ∈ S.
745
+ Proof. We define F : S β†’ R : s οΏ½β†’ esssupgβˆˆΞ“f(gβˆ’1s). By construction, F is Ξ“-invariant,
746
+ and moreover βˆ₯F βˆ’ fβˆ₯ < Ξ΅. By ergodicity, F is essentially equal to a constant c, and thus
747
+ |f(s) βˆ’ c| < Ξ΅ for a.e. s ∈ 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 βˆ₯g Β· vβˆ₯ = βˆ₯vβˆ₯ for all g ∈ Ξ“, v ∈ E.
751
+ Then whenever f : S β†’ E is a measurable function such that βˆ₯g Β· f βˆ’ fβˆ₯ < Ξ΅ for all
752
+ g ∈ Ξ“, where (g Β· f)(s) = g Β· f(gβˆ’1s), there exists a vector v ∈ E such that βˆ₯f(s) βˆ’ vβˆ₯ < 3Ξ΅
753
+ for almost every s ∈ S.
754
+ Proof. We define F : S Γ— S β†’ R : (s, t) οΏ½β†’ βˆ₯f(s) βˆ’ f(t)βˆ₯. Then
755
+ βˆ₯g Β· F βˆ’ Fβˆ₯ = ess sup| βˆ₯g Β· f(gβˆ’1s) βˆ’ g Β· f(gβˆ’1t)βˆ₯ βˆ’ βˆ₯f(s) βˆ’ f(t)βˆ₯ |
756
+ ≀ ess supβˆ₯g Β· f(gβˆ’1s) βˆ’ g Β· f(gβˆ’1t) βˆ’ (f(s) βˆ’ f(t))βˆ₯ ≀ 2βˆ₯g Β· f βˆ’ fβˆ₯ < 2Ξ΅.
757
+ By the previous lemma, there exists a constant c such that |F(s, t) βˆ’ c| < 2Ξ΅ for all Ξ΅ > 0. If
758
+ c < Ξ΅, then |f(s) βˆ’ f(t)| < 3Ξ΅ for a.e. s, t ∈ S, which implies the statement.
759
+ Otherwise, βˆ₯f(s) βˆ’ f(t)βˆ₯ > Ξ΅ for a.e. s, t ∈ S. Let D βŠ‚ E be a countable dense subset.
760
+ Then for each d ∈ D the set f βˆ’1(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 ∈ D such that f βˆ’1(BΞ΅/2(d)) has
762
+ positive measure. But for all s, t in this set, βˆ₯f(s) βˆ’ f(t)βˆ₯ < 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 βˆ—Ξ“-module. Then the natural inclusion ˜V
769
+ βˆ—Ξ“ β†’
770
+ ˜L∞(βˆ—S, V)
771
+ βˆ—Ξ“ is an isomorphism.
772
+ Proof. Let f ∈ L∞
773
+ b (βˆ—S, V) = οΏ½
774
+ Ο‰
775
+ L∞(S, Vn) be a lift of an element ˜f ∈ ˜L∞(βˆ—S, V)
776
+ βˆ—Ξ“. We write
777
+ f = (fn)Ο‰. Then fact that ˜f is βˆ—Ξ“-invariant means that for every sequence (gn)n∈N βŠ‚ Ξ“ it holds
778
+ (gnΒ·fnβˆ’fn)Ο‰ ∈ L∞
779
+ inf(βˆ—S, V). Since this holds for every sequence (gn)n∈N, a diagonal argument
780
+ implies that there exists Ξ΅ ∈ βˆ—Rinf such that for every g ∈ Ξ“ it holds (g Β· fn βˆ’fn)Ο‰ ∈ βˆ—Rinf. It
781
+ then follows from Lemma 4.5 that there exist (vn)Ο‰ ∈ V such that (fn βˆ’ 1vn) ∈ L∞
782
+ inf(βˆ—S, V).
783
+ Therefore f represents the same element of ˜L∞(βˆ—S, V) as the image of an element of V. Since
784
+ ˜f is βˆ—Ξ“-invariant, the corresponding element is actually in ˜V
785
+ βˆ—Ξ“.
786
+ We are finally ready to prove Theorem 1.7:
787
+ Proof of Theorem 1.7. Let Ξ“, Ξ› be countable groups, where Ξ› is infinite and amenable. By
788
+ Lemma 4.2, using the same notation, S is a Zimmer-amenable (Ξ“ ≀ Ξ›)-space. Therefore we
789
+ can apply Theorem 2.3, and obtain that the following complex computes Hβˆ—
790
+ a(Ξ“ ≀ Ξ›; V):
791
+ 0
792
+ ˜
793
+ d0
794
+ βˆ’β†’ ˜L∞(βˆ—S, V)
795
+ βˆ—Ξ“
796
+ ˜
797
+ d1
798
+ βˆ’β†’ ˜L∞((βˆ—S)2, V)
799
+ βˆ—Ξ“
800
+ ˜
801
+ d2
802
+ βˆ’β†’ ˜L∞((βˆ—S)3, V)
803
+ βˆ—Ξ“
804
+ ˜
805
+ d3
806
+ βˆ’β†’ Β· Β· Β·
807
+ 14
808
+
809
+ Now by Lemma 4.3, Sm is a doubly ergodic (Ξ“ ≀ Ξ›)-space, for every m β‰₯ 1. Thus Proposition
810
+ 4.6 applies, and the natural inclusion ˜V
811
+ βˆ—Ξ“ β†’ ˜L∞((βˆ—S)m, V)
812
+ βˆ—Ξ“ is an isomorphism for every
813
+ m β‰₯ 1. Thus the above complex is isomorphic to
814
+ 0
815
+ ˜
816
+ d0
817
+ βˆ’β†’ ˜V
818
+ βˆ—Ξ“
819
+ ˜
820
+ d1
821
+ βˆ’β†’ ˜V
822
+ βˆ—Ξ“
823
+ ˜
824
+ d2
825
+ βˆ’β†’ ˜V
826
+ βˆ—Ξ“
827
+ ˜d3
828
+ βˆ’β†’ Β· Β· Β·
829
+ Each differential ˜
830
+ dm is an alternating sum of (m+1) terms all equal to each other. Therefore
831
+ ˜
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 β€² 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, Ξ“0 a subgroup with the following properties:
840
+ 1. There exists g ∈ Ξ“ such that the groups {giΞ“0gβˆ’i : i ∈ Z} pairwise commute;
841
+ 2. Every finite subset of Ξ“ is contained in some conjugate of Ξ“0.
842
+ Then Hn
843
+ a(Ξ“, V) = 0 for all n β‰₯ 1 and all finitary dual asymptotic Banach βˆ—Ξ“-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 finitary dual asymptotic
850
+ Banach βˆ—Ξ“-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 definitions. In Corollary 5.8 we will
854
+ apply Corollary 5.2 to Thompson’s group F β€²; the result for Thompson’s group F will follow
855
+ from Proposition 1.5. We deduce the stability of Thompson’s 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 ∈ Ξ“ and Ξ“0 ≀ Ξ“ such that
861
+ {giΞ“0gβˆ’i : i ∈ Z} pairwise commute. Then there exists an epimorphism Ξ“0 ≀ Z β†’ βŸ¨Ξ“0, g⟩ 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 Ξ“0, and by Ξ“0 the
868
+ subgroup of Ξ“. So we want to construct an epimorphism H ≀Z β†’ βŸ¨Ξ“0, g⟩ ≀ Ξ“ with metabelian
869
+ kernel. We define naturally
870
+ Ο•((gi)i∈Z, p) =
871
+ οΏ½οΏ½
872
+ i∈Z
873
+ tigitβˆ’i
874
+ οΏ½
875
+ tp.
876
+ Note that this product is well-defined since there are only finitely many non-identity terms,
877
+ and the order does not matter since different conjugates commute. By construction Ο• is
878
+ injective on Hi, that is the copy of H supported on the i-th coordinate in H ≀ Z.
879
+ Let
880
+ K := ker Ο• ∩ οΏ½
881
+ i Hi, and note that K is the kernel of the retraction H ≀ Z β†’ Z restricted to
882
+ ker Ο•. So it suffices to show that K is abelian.
883
+ Let g, h ∈ K and write them as (gi)i∈Z and (hi)i∈Z (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
+ i∈Z
888
+ tigitβˆ’i
889
+ and thus
890
+ g0 =
891
+ οΏ½
892
+ iΜΈ=0
893
+ tigitβˆ’i ∈ Ξ“.
894
+ But now g0 belongs to a group generated by conjugates of Ξ“0 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 ∈ 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 Ξ“0 ≀ Ξ“ is such that every finite subset
900
+ of Ξ“ is contained in some Ξ“-conjugate of Ξ“0. Then Ξ“0 is coamenable in Ξ“.
901
+ Lemma 5.6 ([MP03]). Let K ≀ H ≀ 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“, Ξ“0 and g be as in the statement. By Lemma 5.4, there exists
908
+ a map Ξ“0 ≀ Z β†’ βŸ¨Ξ“0, g⟩ with metabelian kernel. By Theorem 1.7 and Proposition 1.9, we
909
+ have Hn
910
+ a(βŸ¨Ξ“0, g⟩, V) for all n β‰₯ 1 and all finitary dual asymptotic Banach βˆ—Ξ“-modules V. Now
911
+ by Lemma 5.5, Ξ“0 is coamenable in Ξ“. Finally, by Lemma 5.6, βŸ¨Ξ“0, g⟩ 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 ∈ Ξ“ such that g Β· a < c < d < g Β· b.
917
+ The support of g ∈ Ξ“ is the set {x ∈ 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 Ξ“0 be the subgroup of elements
923
+ whose support is contained in [0, 1]. Let g ∈ 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 ∈ Z} are pairwise disjoint. Therefore
926
+ the conjugates giΞ“0gβˆ’i pairwise commute.
927
+ Since Ξ“ is boundedly supported, for every finite subset A βŠ‚ Ξ“ there exists n such that
928
+ the support of each element of A is contained in [βˆ’n, n]. By proximality, there exists h ∈ Ξ“
929
+ such that h(0) < βˆ’n and h(1) > n. Then hΞ“0hβˆ’1 is the subgroup of elements whose support
930
+ is contained in [βˆ’n, 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 β€² 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
+ Thompson’s groups.
936
+ Thompson’s 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 β€²
938
+ coincides with the subgroup of boundedly supported elements. The action of F β€² (and thus
939
+ F) on [0, 1] preserves Z[1/2] ∩ (0, 1), and acts highly transitively on it; that is, for every pair
940
+ of ordered n-tuples in Z[1/2] ∩ (0, 1) there exists an element of F β€² sending one to the other.
941
+ Corollary 5.8. Thompson’s groups F and F β€² are uniformly U-stable, with a linear estimate.
942
+ Proof. We identify (0, 1) with the real line. The group F β€² 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 β€² is uniformly U-stable, with a linear estimate.
945
+ Since the quotient F/F β€² is abelian, thus amenable, we see that F β€² is coamenable in F, and
946
+ thus conclude from Proposition 1.5 that F β€² 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 β€² more directly,
948
+ without appealing to Proposition 1.5. Indeed, since F β€² is uniformly U-stable, simple, and
949
+ not linear, every homomorphism F β€² β†’ U(n) is trivial - something we will come back to in
950
+ the next section. Therefore uniform U-stability of F β€² implies that every uniform asymptotic
951
+ homomorphism F β€² 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 Stein–Thompson 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 Lodha–Moore [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 Thompson’s groups T and V , but the proofs generalize to some analogously
969
+ defined groups, with the appropriate modifications. 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 ∈ H}
973
+ cover Ξ“.
974
+ Lemma 5.10. Let Ξ“ be a discrete group. Suppose that there exists a subgroup H ≀ 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 Ο†n : Ξ“ β†’ U(dn) be a uniform asymptotic homomorphism with def(Ο†n) =: Ξ΅n.
981
+ Then Ο†n|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 βˆ₯Ο†n(h) βˆ’ Iknβˆ₯ ≀ 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 ∈ Ξ“ can be written as
986
+ g = h1 Β· Β· Β· hk, where each hi belongs to a conjugate of H. We estimate:
987
+ βˆ₯Ο†n(g) βˆ’ Idnβˆ₯ =
988
+ οΏ½οΏ½οΏ½οΏ½οΏ½Ο†n
989
+ οΏ½ k
990
+ οΏ½
991
+ i=1
992
+ hi
993
+ οΏ½
994
+ βˆ’ Idn
995
+ οΏ½οΏ½οΏ½οΏ½οΏ½ ≀
996
+ οΏ½οΏ½οΏ½οΏ½οΏ½Ο†n
997
+ οΏ½kβˆ’1
998
+ οΏ½
999
+ i=1
1000
+ hi
1001
+ οΏ½
1002
+ Ο†n(hk) βˆ’ Idn
1003
+ οΏ½οΏ½οΏ½οΏ½οΏ½ + Ξ΅n
1004
+ =
1005
+ οΏ½οΏ½οΏ½οΏ½οΏ½Ο†n
1006
+ οΏ½kβˆ’1
1007
+ οΏ½
1008
+ i=1
1009
+ hi
1010
+ οΏ½
1011
+ βˆ’ Idn
1012
+ οΏ½οΏ½οΏ½οΏ½οΏ½ + βˆ₯Ο†n(hk) βˆ’ Idnβˆ₯ + Ξ΅n ≀ Β· Β· Β·
1013
+ Β· Β· Β· ≀
1014
+ k
1015
+ οΏ½
1016
+ i=1
1017
+ βˆ₯Ο†n(hi) βˆ’ Idnβˆ₯ + kΞ΅n ≀ k(Ξ΄n + Ξ΅n).
1018
+ Therefore Ο†n is k(Ξ΄n + Ξ΅n)-close to the trivial homomorphism, and we conclude.
1019
+ Thompson’s 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 ∈ 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 fix pointwise some neighbourhood
1023
+ of x) is isomorphic to F β€².
1024
+ Corollary 5.11. Thompson’s group T is uniformly U-stable with a linear estimate.
1025
+ Proof. We claim that Lemma 5.10 applies with H = T(0) ∼= F β€². Item 1. follows from the
1026
+ fact F β€² does not embed into U(n) (for instance because it contains F as a subgroup, which
1027
+ is finitely generated and not residually finite, and so cannot be linear by Mal’cev’s Theorem
1028
+ [Mal40]), and F β€² is simple [CFP96]. Also, F β€² 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 ∈ T there exist x, y ∈ Z[1/2]/Z such that g ∈ T(x)T(y). This suffices
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 ̸= g ∈ T, and choose x ̸= y ∈ Z[1/2]/Z such that g(y) /∈ {x, y}. Let I be a small
1035
+ dyadic arc around y such that x /∈ I and x, y /∈ g(I). Choose an element f ∈ T(x) such
1036
+ that f(I) = g(I). Let h be an element supported on I such that h|I = f βˆ’1g|I. Since x /∈ I,
1037
+ we also have h ∈ T(x). Moreover hβˆ’1f βˆ’1g|I = id|I, so hβˆ’1f βˆ’1g ∈ G(y). We conclude that
1038
+ g = fh Β· hβˆ’1f βˆ’1g ∈ T(x)T(y).
1039
+ Thompson’s 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 XΟƒ := Οƒ Γ— 2N>k, for some Οƒ ∈ 2k,
1041
+ and every two dyadic bricks are canonically homeomorphic via Xσ → Xτ : σ × x �→ τ × x.
1042
+ An element g ∈ V is defined by two finite partitions of V of the same size into dyadic bricks,
1043
+ that are sent to each other via canonical homeomorphisms.
1044
+ Corollary 5.12. Thompson’s 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 ∈ 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 fix 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 finitely generated
1051
+ and simple [CFP96], so by Mal’cev’s 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) satisfies 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 infinite-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 finitary. Therefore the
1067
+ counterexample above shows that Theorem 1.7 really needs the finitary 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 infinite-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 ≀ Z, which is uniformly U-stable with a linear estimate by Theorem 1.3.
1080
+ On the other hand, if we replace β€œcoamenable” by β€œο¬nite index”, 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 Thompson’s 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 fit 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) ∼= 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“, βˆ—R) = 0, then Ξ“ is uniformly U(1)-stable, where βˆ—R is seen as a
1099
+ dual asymptotic βˆ—Ξ“-module with a trivial βˆ—Ξ“ action.
1100
+ Now, let again Ξ“ be the lift of Thompson’s group T, so that Ξ“ contains a central subgroup
1101
+ Z with Ξ“/Z ∼= T. The fact that Ξ“ is not uniformly U(1)-stable implies that H2
1102
+ a(Ξ“, βˆ—R) ΜΈ= 0.
1103
+ But Proposition 1.9 then shows that H2
1104
+ a(T, βˆ—R) 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) ∼=
1107
+ H2
1108
+ b(T, R) ∼= 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 βˆ₯Β·βˆ₯ bounded away from 0. They use this to show
1116
+ that a finitely 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 finite presentability is necessary. Let Ξ“n denote the lift of Thompson’s
1119
+ group T to R/nZ. That is, Ξ“n 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 Ξ“n 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 Ξ“n 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 suffices to show that there exist bounded cohomology classes 0 ΜΈ= ρn ∈ im(H2
1130
+ b(Ξ“, Z) β†’
1131
+ H2
1132
+ b(Ξ“, R) such that βˆ₯ρnβˆ₯ β†’ 0. We let ρn be the Euler class of the representation Ξ“ β†’ Ξ“n 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 βˆ₯ρnβˆ₯ = 1/2n by [Bur11, Corollary 1.6], and we conclude.
1138
+ Note that Ξ“ is countable but infinitely generated. It would be interesting to produce a
1139
+ finitely generated example (which would necessarily be infinitely 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
+ Definition 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 Ο†n : Ξ“ β†’
1146
+ Gn ∈ G that is moreover asymptotically injective, meaning that for all g ∈ Ξ“, g ΜΈ= 1 it holds
1147
+ lim inf
1148
+ nβ†’βˆž Ο†n(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 sofic [Gro99, Wei00].
1154
+ If G is the family of unitary groups equipped with the Hilbert–Schmidt distance, then
1155
+ pointwise G-approximable groups are called hyperlinear [R˘08].
1156
+ All amenable and residually finite groups are sofic, and all sofic groups are hyperlinear.
1157
+ It is a major open question to determine whether there exists a non-sofic 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 < ∞. 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 P˘aunescu 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 finite, and
1171
+ let Ξ“ be a finitely generated group. Suppose that Ξ“ is (pointwise, uniformly) G-stable and
1172
+ (pointwise, uniformly) G-approximable. Then Ξ“ is residually finite.
1173
+ 21
1174
+
1175
+ The hypothesis on G covers all cases above. When the groups in G are finite, this is clear,
1176
+ and when they are linear, this follows from Mal’cev’s 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 ψ : Ξ“ β†’ G which is asymptotically close to Ο†.
1182
+ Since Ο† is asymptotically injective, for each g ∈ Ξ“ there exists N such that Ο†n(g) β‰₯ ρ for all
1183
+ n β‰₯ N and some ρ = ρ(g) > 0. Up to taking a larger N, we also have that ψn(g) β‰₯ ρ/2, in
1184
+ particular ψn(g) ΜΈ= 1. Since ψn(Ξ“) is a finitely generated subgroup of Gn ∈ G, it is residually
1185
+ finite by hypothesis, and so ψn(g) survives in some finite quotient of ψn(Ξ“). Since this is also
1186
+ a finite quotient of Ξ“, we conclude that Ξ“ is residually finite.
1187
+ In the special case of pointwise stability and Thompson’s 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 Hilbert–Schmidt norm, or the family of unitary
1191
+ groups with the operatorn norm. If Thompson’s 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 Thompson’s group F is one of
1194
+ the most outstanding open problems in modern group theory.
1195
+ Proof. Thompson’s group F is not residually finite [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 sofic, hyperlinear, and MF.
1198
+ On the other hand, our results allow to settle the uniform approximability of Thompson’s
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 Thompson’s groups F, F β€², T and V are not uniformly U-approximable. The
1202
+ same holds for Ξ“ ≀ Ξ›, whenever Ξ› is infinite and amenable, and Ξ“ is non-abelian.
1203
+ We remark that Thompson’s 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 finite, and the statement for F β€² (which is not finitely
1207
+ generated) follows from the fact that F β€² 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 finite [Gru57].
1210
+ We do not know whether Thompson’s groups are uniformly G-approximable, when G is
1211
+ the family of unitary groups equipped with the Hilbert–Schmidt 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 definitions):
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 Ο†n : Ξ“β€² β†’
1225
+ Skn ∈ S is uniformly asymptotically close to the trivial one. In particular, Ξ“β€² is uniformly
1226
+ S-stable, and not uniformly S-approximable.
1227
+ The non-approximability follows from the fact that Ξ“β€² is non-trivial (see Lemma 7.8).
1228
+ Note that for Ξ“ as in the statement, Ξ“β€² is simple [GG17, Theorem 1.1], so in particular every
1229
+ homomorphism Ξ“β€² β†’ Skn is trivial.
1230
+ The proof relies on known results on the flexible uniform stability of amenable groups
1231
+ [BC20] and uniform perfection of groups with proximal actions [GG17]. The finiteness 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 Ξ“β€² is non-trivial, and the action of Ξ“β€² on
1234
+ the line has no global fixpoints.
1235
+ Proof. If Ξ“β€² is trivial, then Ξ“ is abelian. This contradicts that the action is proximal and
1236
+ boundedly supported. Indeed, given g ∈ 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 fixpoints of Ξ“β€² is a closed subset X βŠ‚ R. Since Ξ“β€² 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, Ξ“β€² acts trivially on R. Since Ξ“
1242
+ is a subgroup of Homeo+(R), this implies that Ξ“β€² 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 Ξ“β€² is 2-uniformly perfect;
1246
+ that is, every element of Ξ“β€² may be written as the product of at most 2 commutators (this
1247
+ uses the proximality hypothesis). Therefore it suffices to show that there exists a constant C
1248
+ such that for all g, h ∈ Ξ“β€² it holds dkn(Ο†n([g, h]), idkn) ≀ CΞ΅n, where dkn denotes the Hamming
1249
+ distance on Skn and Ξ΅n := def(Ο†n). We drop the subscript n on Ο† and Ξ΅ for clarity.
1250
+ Now let g, h ∈ Ξ“β€², and let I, J βŠ‚ R be bounded intervals such that g is supported on I
1251
+ and h is supported on J. Since Ξ“β€² acts without global fixpoints by Lemma 7.8, there exists
1252
+ t ∈ Ξ“β€² such that t Β· inf(J) > sup(I). Since Ξ“β€² is orientation-preserving, the same holds for
1253
+ all powers of t. In particular [g, tihtβˆ’i] = 1 for all i β‰₯ 1. Next, we apply [BC20, Theorem
1254
+ 1.2] to the amenable group ⟨t⟩, to obtain an integer N such that kn ≀ N ≀ (1 + 1218Ξ΅)kn
1255
+ and a permutation Ο„ in SN such that dN(Ο†(t)i, Ο„ i) ≀ 2039Ξ΅ for all i ∈ 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† : Ξ“β€² β†’ SN with every Ο†(g) fixing each point in {kn + 1, . . . , N}.
1260
+ We compute (using
1261
+ Ο„ N! = idN):
1262
+ dkn(Ο†([g, h]), idkn) ≀ dN(Ο†([g, h]), idN) ≀ dN([Ο†(g), Ο†(h)], idN) + O(Ξ΅)
1263
+ = dN([Ο†(g), Ο„ N!Ο†(h)Ο„ βˆ’N!], idN) + O(Ξ΅)
1264
+ ≀ dN([Ο†(g), Ο†(tN!)Ο†(h)Ο†(tβˆ’N!)], idN) + O(Ξ΅)
1265
+ ≀ dN(Ο†([g, tN!htβˆ’N!]), idN) + O(Ξ΅)
1266
+ = dN(Ο†(1), idN) + O(Ξ΅) ≀ O(Ξ΅).
1267
+ Thus, there exists a constant C independent of g and h (C = 20000 suffices) such that
1268
+ dkn(Ο†([g, h]), idkn) ≀ CΞ΅, which concludes the proof.
1269
+ Corollary 7.9. Consider the Thompson groups F β€², F, T.
1270
+ 1. Every asymptotic homomorphism Ο†n : F β€² β†’ Skn ∈ S is uniformly asymptotically close
1271
+ to the trivial one.
1272
+ 2. Every asymptotic homomorphism Ο†n : F β†’ Skn ∈ S is uniformly asymptotically close
1273
+ to one that factors through the abelianization.
1274
+ 3. Every asymptotic homomorphism Ο†n : T β†’ Skn ∈ S is uniformly asymptotically close
1275
+ to the trivial one.
1276
+ Proof. Item 1. is an instance of Proposition 7.7: indeed F β€² satisfies the hypotheses for Ξ“,
1277
+ and F β€²β€² = F β€² since F β€² is simple. For Item 2., pick a section Οƒ : Ab(F) β†’ F, and define
1278
+ ψn(g) := Ο†n(Οƒ(Ab(g))). Using that ψn|F β€² is uniformly asymptotically close to the sequence
1279
+ of trivial maps, we obtain that Ο†n and ψn are uniformly asymptotically close, and ψn factors
1280
+ as F β†’ Ab(F)
1281
+ Ο†nβ—¦Οƒ
1282
+ βˆ’βˆ’βˆ’β†’ 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 β€² (see the
1284
+ proof of Corollary 5.11).
1285
+ The corollary immediately implies that F, F β€² and T are not uniformly S-approximable,
1286
+ and that F β€² and T are uniformly S-stable. Since F has infinite 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 flexibly 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 Thompson’s group V can also be
1291
+ treated analogously (see the sketch of proof of Corollary 5.12).
1292
+ References
1293
+ [AEG94]
1294
+ S. Adams, G. A. Elliott, and T. Giordano. Amenable actions of groups. Trans. Amer. Math.
1295
+ Soc., 344(2):803–822, 1994.
1296
+ [Alp20]
1297
+ A. Alpeev. Lamplighters over non-amenable groups are not strongly Ulam stable. arXiv preprint
1298
+ arXiv:2009.11738, 2020.
1299
+ [And22]
1300
+ K. Andritsch.
1301
+ Bounded cohomology of groups acting on Cantor sets.
1302
+ arXiv preprint
1303
+ arXiv:2210.00459, 2022.
1304
+ 24
1305
+
1306
+ [AP15]
1307
+ G. Arzhantseva and L. P˘aunescu. Almost commuting permutations are near commuting per-
1308
+ mutations. J. Funct. Anal., 269(3):745–757, 2015.
1309
+ [BC20]
1310
+ O. Becker and M. Chapman. Stability of approximate group actions: uniform and probabilistic.
1311
+ J. Eur. Math. Soc. (JEMS), To appear. arXiv: 2005.06652, 2020.
1312
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1313
+ J. Burillo, Y. Lodha, and L. Reeves. Commutators in groups of piecewise projective homeomor-
1314
+ phisms. Adv. Math., 332:34–56, 2018.
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1317
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1318
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1319
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+ Funct. Anal., 265(1):135–152, 2013.
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1342
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+ nonapproximable groups. Forum Math. Sigma, 8:Paper No. e18, 37, 2020.
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+ F. Fournier-Facio.
1353
+ Ultrametric analogues of Ulam stability of groups.
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1355
+ arXiv:2105.00516, 2021.
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+ F. Fournier-Facio and Y. Lodha. Second bounded cohomology of groups acting on 1-manifolds
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+ and applications to spectrum problems. arXiv preprint arXiv:2111.07931, 2021.
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+ F. Fournier-Facio, C. L¨oh, and M. Moraschini. Bounded cohomology and binate groups. J.
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+ Aust. Math. Soc., To appear. arXiv:2111.04305, 2021.
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+ R. Frigerio. Bounded cohomology of discrete groups, volume 227 of Mathematical Surveys and
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+ Monographs. American Mathematical Society, Providence, RI, 2017.
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+ [Gam11]
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+ C. Gamm.
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+ Ξ΅-representations of groups and Ulam stability.
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+ Master’s thesis, Georg-August-
1369
+ Universit¨at G¨ottingen, 2011.
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+ [GG17]
1371
+ S. R. Gal and J. Gismatullin. Uniform simplicity of groups with proximal action. Trans. Amer.
1372
+ Math. Soc. Ser. B, 4:110–130, 2017. With an appendix by N. Lazarovich.
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+ [Ghy01]
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+ E. Ghys. Groups acting on the circle. Enseign. Math. (2), 47(3-4):329–407, 2001.
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+ [GLMR23]
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+ L. Glebsky, A. Lubotzky, N. Monod, and B. Rangarajan. Asymptotic cohomology and uniform
1377
+ stability for lattices in semisimple groups. arXiv preprint arXiv:2301.00476, 2023.
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+ L. Glebsky and L. M. Rivera. Almost solutions of equations in permutations. Taiwanese J.
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+ Math., 13(2A):493–500, 2009.
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+ M. Gromov. Volume and bounded cohomology. Inst. Hautes Β΄Etudes Sci. Publ. Math., (56):5–99
1383
+ (1983), 1982.
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+ 25
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+
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+ [Gro99]
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+ M. Gromov.
1388
+ Endomorphisms of symbolic algebraic varieties.
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+ J. Eur. Math. Soc. (JEMS),
1390
+ 1(2):109–197, 1999.
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1392
+ K. W. Gruenberg. Residual properties of infinite soluble groups. Proc. London Math. Soc. (3),
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+ 7:29–62, 1957.
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+ E. Ghys and V. Sergiescu. Sur un groupe remarquable de diff´eomorphismes du cercle. Comment.
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+ Math. Helv., 62(2):185–239, 1987.
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+ G. Higman, B. H. Neumann, and H. Neumann. Embedding theorems for groups. J. London
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+ Math. Soc., 24:247–254, 1949.
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+ [Iva85]
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+ N. V. Ivanov. Foundations of the theory of bounded cohomology. volume 143, pages 69–109,
1402
+ 177–178. 1985. Studies in topology, V.
1403
+ [Joh72]
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+ B. E. Johnson. Cohomology in Banach algebras. Memoirs of the American Mathematical Society,
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+ No. 127. American Mathematical Society, Providence, R.I., 1972.
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+ D. Kazhdan. On Ξ΅-representations. Israel J. Math., 43(4):315–323, 1982.
1408
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+ Y. Lodha and J. T. Moore. A nonamenable finitely presented group of piecewise projective
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+ homeomorphisms. Groups Geom. Dyn., 10(1):177–200, 2016.
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+ [LO20]
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+ A. Lubotzky and I. Oppenheim. Non p-norm approximated groups. J. Anal. Math., 141(1):305–
1413
+ 321, 2020.
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+ A. Malcev. On isomorphic matrix representations of infinite groups. Rec. Math. [Mat. Sbornik]
1416
+ N.S., 8 (50):405–422, 1940.
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+ [Man05]
1418
+ J. F. Manning. Geometry of pseudocharacters. Geom. Topol., 9:1147–1185, 2005.
1419
+ [MN21]
1420
+ N. Monod and S. Nariman.
1421
+ Bounded and unbounded cohomology of homeomorphism and
1422
+ diffeomorphism groups. arXiv preprint arXiv:2111.04365, 2021.
1423
+ [Mon01]
1424
+ N. Monod. Continuous bounded cohomology of locally compact groups, volume 1758 of Lecture
1425
+ Notes in Mathematics. Springer-Verlag, Berlin, 2001.
1426
+ [Mon13]
1427
+ N. Monod.
1428
+ Groups of piecewise projective homeomorphisms.
1429
+ Proc. Natl. Acad. Sci. USA,
1430
+ 110(12):4524–4527, 2013.
1431
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1432
+ N. Monod. Lamplighters and the bounded cohomology of Thompson’s group. Geom. Funct.
1433
+ Anal., 32(3):662–675, 2022.
1434
+ [MP03]
1435
+ N. Monod and S. Popa. On co-amenability for groups and von Neumann algebras. C. R. Math.
1436
+ Acad. Sci. Soc. R. Can., 25(3):82–87, 2003.
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+ [MS04]
1438
+ N. Monod and Y. Shalom. Cocycle superrigidity and bounded cohomology for negatively curved
1439
+ spaces. J. Differential Geom., 67(3):395–455, 2004.
1440
+ [Neu37]
1441
+ B. H. Neumann. Some remarks on infinite groups. Journal of the London Mathematical Society,
1442
+ 1(2):120–127, 1937.
1443
+ [R˘08]
1444
+ F. R˘adulescu.
1445
+ The von Neumann algebra of the non-residually finite Baumslag group
1446
+ ⟨a, b|ab3aβˆ’1 = b2⟩ embeds into RΟ‰. In Hot topics in operator theory, volume 9 of Theta Ser.
1447
+ Adv. Math., pages 173–185. Theta, Bucharest, 2008.
1448
+ [Ste92]
1449
+ M. Stein. Groups of piecewise linear homeomorphisms. Trans. Amer. Math. Soc., 332(2):477–
1450
+ 514, 1992.
1451
+ [Tur38]
1452
+ A. M. Turing. Finite approximations to Lie groups. Ann. of Math. (2), 39(1):105–111, 1938.
1453
+ [TWW17]
1454
+ A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear Cβˆ—-algebras. Ann. of Math.
1455
+ (2), 185(1):229–284, 2017.
1456
+ [Ula60]
1457
+ S. M. Ulam. A collection of mathematical problems. Interscience Tracts in Pure and Applied
1458
+ Mathematics, no. 8. Interscience Publishers, New York-London, 1960.
1459
+ [vN29]
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+ J. von Neumann. Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik. Z.
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+ Phys., 57(1):30–70, 1929.
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+ 26
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+
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1465
+ B. Weiss. Sofic groups and dynamical systems. SankhyΒ―a Ser. A, 62(3):350–359, 2000. Ergodic
1466
+ theory and harmonic analysis (Mumbai, 1999).
1467
+ Department of Mathematics, ETH Z¨urich, Switzerland
1468
+ E-mail address: [email protected]
1469
+ Einstein Institute of Mathematics, Hebrew University of Jerusalem, Israel
1470
+ E-mail address: [email protected]
1471
+ 27
1472
+
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1
+ arXiv:2301.08615v1 [hep-ph] 20 Jan 2023
2
+ Photo-production of lowest Ξ£βˆ—
3
+ 1/2βˆ’ state within the Regge-effective 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£βˆ— state, with quantum numbers spin-parity JP = 1/2βˆ’, is far from estab-
12
+ lished experimentally and theoretically, we have performed a theoretical study on the Ξ£βˆ—
13
+ 1/2βˆ’ photo-
14
+ production within the Regge-effective Lagrangian approach. Taking into account that the Ξ£βˆ—
15
+ 1/2βˆ’
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 Kβˆ— exchange, s-channel nucleon pole, u-
18
+ channel Ξ£ exchange, and the contact term, are considered. The differential and total cross sections
19
+ of the process Ξ³n β†’ K+Ξ£βˆ—βˆ’
20
+ 1/2βˆ’ are predicted with our model parameters. The results should be
21
+ helpful to search for the Ξ£βˆ—
22
+ 1/2βˆ’ state experimentally in future.
23
+ PACS numbers:
24
+ I.
25
+ INTRODUCTION
26
+ The study of the low-lying excited Ξ›βˆ— and Ξ£βˆ— 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 [3–8], and one explanation for Ξ›(1405) is that it
32
+ is a Β―KN hadronic molecular state [9–14]. In addition,
33
+ the isospin I = 1 partner of the Ξ›(1405), the lowest
34
+ Ξ£βˆ—
35
+ 1/2βˆ’ is crucial to understand the light baryon spec-
36
+ tra. At present, there is a Ξ£βˆ—(1620) with JP = 1/2βˆ’
37
+ listed in the latest version of Review of Particle Physics
38
+ (RPP) [15]. It should be stressed that the Ξ£βˆ—(1620) state
39
+ is a one-star baryon resonance, and many studies indicate
40
+ that the lowest Ξ£βˆ—
41
+ 1/2βˆ’ resonance is still far from estab-
42
+ lished, and its mass was predicted to lie in the range of
43
+ 1380 ∼ 1500 MeV [13, 16–19]. Thus, searching for the
44
+ lowest Ξ£βˆ—
45
+ 1/2βˆ’ is helpful to understand the low-lying ex-
46
+ cited baryons with JP = 1/2βˆ’ and the light flavor baryon
47
+ spectra.
48
+ The analyses of the relevant data of the process
49
+ Kβˆ’p β†’ Λπ+Ο€βˆ’ suggest that there may exist a Ξ£βˆ—
50
+ 1/2βˆ’
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 Kβˆ—Ξ£ photo-
55
+ production also indicate that the Ξ£βˆ—
56
+ 1/2βˆ’ is possibly buried
57
+ under the Ξ£βˆ—(1385) peak with mass of 1380 MeV [21],
58
+ and it is proposed to search for the Ξ£βˆ—
59
+ 1/2βˆ’ in the pro-
60
+ cess Ξ›c β†’ Ξ·Ο€+Ξ› [22]. A more delicate analysis of the
61
+ CLAS data on the process Ξ³p β†’ KΣπ [23] suggests that
62
+ the Ξ£βˆ—
63
+ 1/2βˆ’ 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 Ο‡c0(1P) β†’ ¯ΣΣπ and Ο‡c0(1P) β†’ ¯ΛΣπ. In
67
+ addition, Ref. [26] has found one Ξ£βˆ—
68
+ 1/2βˆ’ 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£βˆ—0
72
+ 1/2βˆ’.
73
+ It’s worth mentioning that a Ξ£βˆ—(1480) resonance with
74
+ JP = 1/2βˆ’ has been listed on the previous version of
75
+ RPP [28].
76
+ As early as 1970, the Ξ£βˆ—(1480) resonance
77
+ was reported in the Λπ+, Σπ, and p Β―K0 channels of the
78
+ Ο€+p scattering in the Princeton-Pennsylvania Accelera-
79
+ tor 15-in.∼hydrogen 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 βˆ—0 [32]. Theoretically, the Ξ£βˆ—(1480) was in-
87
+ vestigated within different models [33–36]. In Ref. [36],
88
+ the S-wave meson-baryon interactions with strangeness
89
+ S = βˆ’1 were studied within the unitary chiral approach,
90
+ and one narrow pole with pole position of 1468βˆ’i 13 MeV
91
+ was found in the second Riemann sheet, which could be
92
+ associated with the Ξ£βˆ—(1480) resonance. However, the
93
+ Ξ£βˆ—(1480) signals are insignificant, and the existence of
94
+ this state still needs to be confirmed 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£βˆ— and Ξ›βˆ—,
98
+ and the Crystal Ball [37–39], 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†’ KΞ›βˆ—(1405) to deepen the un-
103
+ derstanding of the Ξ›βˆ—(1405) nature in Ref. [41]. In order
104
+ to confirm the existence of the Ξ£βˆ—(1480), we propose to
105
+ investigate the process Ξ³N β†’ KΞ£βˆ—(1480) 1 within the
106
+ 1 Here after, we denote Ξ£βˆ—(1480) as the lowest Ξ£βˆ—
107
+ 1/2βˆ’ state unless
108
+ otherwise stated.
109
+
110
+ 2
111
+ Regge-effective Lagrange approach.
112
+ Considering the Ξ£βˆ—(1480) signal was first observed
113
+ in the Ο€+Ξ› invariant mass distribution of the process
114
+ Ο€+p β†’ Ο€+K+Ξ›, and the significance is about 3 ∼
115
+ 4Οƒ [30], we search for the charged Ξ£βˆ—(1480) in the process
116
+ Ξ³n β†’ K+Ξ£βˆ—βˆ’
117
+ 1/2βˆ’, which could also avoid the contribu-
118
+ tions of possible excited Ξ›βˆ— states. We will consider the
119
+ t-, s-, u-channels diagrams in the Born approximation
120
+ by employing the effective Lagrangian approach, and the
121
+ t-channel K/Kβˆ— exchanges terms within Regge model.
122
+ Then we will calculate the differential and total cross
123
+ sections of the process Ξ³n β†’ K+Ξ£βˆ—βˆ’
124
+ 1/2βˆ’ reaction, which
125
+ are helpful to search for Ξ£βˆ—
126
+ 1/2βˆ’ experimentally.
127
+ This paper is organized as follows. In Sec. II, the the-
128
+ oretical formalism for studying the Ξ³n β†’ K+Ξ£βˆ—βˆ’(1480)
129
+ reactions are presented. The numerical results of total
130
+ and differential 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£βˆ—(1480) (≑ Ξ£βˆ—)
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 Kβˆ— 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£βˆ—(p2)
145
+ K, Kβˆ—
146
+ Ξ³
147
+ K
148
+ N
149
+ Ξ£βˆ—
150
+ N
151
+ Ξ³
152
+ Ξ£βˆ—
153
+ N
154
+ K
155
+ Ξ³
156
+ K
157
+ N
158
+ Ξ£βˆ—
159
+ Ξ£
160
+ (a)
161
+ (b)
162
+ (c)
163
+ (d)
164
+ FIG. 1: The mechanisms of the Ξ³n β†’ K+Ξ£βˆ—βˆ’
165
+ 1/2βˆ’ process. (a)
166
+ t-channel K/Kβˆ— 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£βˆ—(1480), respectively.
170
+ To compute the scattering amplitudes of the Feynman
171
+ diagrams shown in Fig. 1 within the effective Lagrangian
172
+ approach, we use the Lagrangian densities for the elec-
173
+ tromagnetic and strong interaction vertices as used in
174
+ Refs. [27, 42–46]
175
+ LΞ³KK = βˆ’ie
176
+ οΏ½
177
+ K† (βˆ‚Β΅K) βˆ’
178
+ οΏ½
179
+ βˆ‚Β΅K†�
180
+ K
181
+ οΏ½
182
+ AΒ΅,
183
+ (1)
184
+ LΞ³KKβˆ— = gΞ³KKβˆ—Η«Β΅Ξ½Ξ±Ξ²βˆ‚Β΅AΞ½βˆ‚Ξ±Kβˆ—
185
+ Ξ²K,
186
+ (2)
187
+ LΞ³NN = βˆ’e Β―N
188
+ οΏ½
189
+ Ξ³Β΅Λ†e βˆ’
190
+ Λ†ΞΊN
191
+ 2MN
192
+ ΟƒΒ΅Ξ½βˆ‚Ξ½
193
+ οΏ½
194
+ AΒ΅N,
195
+ (3)
196
+ LΞ³Ξ£Ξ£βˆ— = eΒ΅Ξ£Ξ£βˆ—
197
+ 2MN
198
+ ¯Σγ5ΟƒΒ΅Ξ½βˆ‚Ξ½AΒ΅Ξ£βˆ— + h.c.,
199
+ (4)
200
+ LKNΞ£ = βˆ’igKNΞ£ Β―NΞ³5Ξ£K + h.c.,
201
+ (5)
202
+ LKβˆ—NΞ£βˆ— = igKβˆ—NΞ£βˆ—
203
+ √
204
+ 3
205
+ Β―Kβˆ—Β΅ Β―Ξ£βˆ—Ξ³Β΅Ξ³5N + h.c.
206
+ (6)
207
+ LKNΞ£βˆ— = gKNΞ£βˆ— Β―K Β―
208
+ Ξ£βˆ—N + h.c.,
209
+ (7)
210
+ where e(=
211
+ √
212
+ 4πα) is the elementary charge unit, AΒ΅ is the
213
+ photon filed, and Λ†e ≑ (1+Ο„3)/2 denotes the charge opera-
214
+ tor acting on the nucleon field. Λ†ΞΊN ≑ ΞΊpΛ†e+ΞΊn(1βˆ’Λ†e) is the
215
+ anomalous magnetic moment, and we take ΞΊn = βˆ’1.913
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 gΞ³KKβˆ— = 0.254 GeVβˆ’1 is determined
221
+ from the experimental data of Ξ“Kβˆ—β†’K+Ξ³ [15] and the
222
+ value of gKβˆ—NΞ£βˆ— = βˆ’3.26 βˆ’ i0.06 is taken from Ref [26].
223
+ In addition, the coupling gKNΞ£βˆ— = 8.74 GeV is taken
224
+ from Ref. [36], and the transition magnetic moment
225
+ Β΅Ξ£Ξ£βˆ— = 1.28 is taken from Ref. [27]
226
+ With the effective interaction Lagrangian densities
227
+ given above, the invariant scattering amplitudes are de-
228
+ fined as
229
+ M = Β―uΞ£βˆ—(p2, sΞ£βˆ—)MΒ΅
230
+ huN(k2, sp)Η«Β΅(k1, Ξ»),
231
+ (8)
232
+ where uΞ£βˆ— 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 different diagrams
235
+ of Fig. 1. The reduced amplitudes MΒ΅
236
+ h are written as
237
+ MΒ΅
238
+ Kβˆ— =
239
+ egΞ³KKβˆ—gKβˆ—NΞ£βˆ—
240
+ √
241
+ 3MKβˆ—(t βˆ’ M 2
242
+ Kβˆ—)ǫαβ¡νk1Ξ±k2Ξ²Ξ³Ξ½Ξ³5,
243
+ (9)
244
+ MΒ΅
245
+ Kβˆ’ = βˆ’2iegKNΞ£βˆ—
246
+ t βˆ’ M 2
247
+ K
248
+ kΒ΅
249
+ 2 ,
250
+ (10)
251
+ MΒ΅
252
+ Ξ£βˆ’ = βˆ’i
253
+ eΒ΅Ξ£Ξ£βˆ—gKNΞ£
254
+ 2Mn(u βˆ’ M 2
255
+ Ξ£βˆ—)(q/u βˆ’ MΞ£)σ¡νk1Ξ½, (11)
256
+ MΒ΅
257
+ n =
258
+ ΞΊngKNΞ£βˆ—
259
+ 2Mn(s βˆ’ M 2n)σ¡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 = βˆ’iegKNΞ£βˆ—
266
+ pΒ΅
267
+ 2
268
+ p2 Β· k1
269
+ ,
270
+ (13)
271
+ for Ξ³n β†’ K+Ξ£βˆ—βˆ’
272
+ 1/2βˆ’.
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 [48–51], thus we will adopt Regge model for mod-
278
+ eling the t-channel K and Kβˆ— contributions by replacing
279
+ the usual pole-like Feynman propagator with the corre-
280
+ sponding Regge propagators as follows,
281
+ 1
282
+ t βˆ’ M 2
283
+ K
284
+ β†’ FRegge
285
+ K
286
+ =
287
+ οΏ½ s
288
+ sK
289
+ 0
290
+ οΏ½Ξ±K(t)
291
+ πα′
292
+ K
293
+ sin(παK(t))Ξ“(1 + Ξ±K(t)),(14)
294
+ 1
295
+ t βˆ’ M 2
296
+ Kβˆ—
297
+ β†’ FRegge
298
+ Kβˆ—
299
+ =
300
+ οΏ½ s
301
+ sKβˆ—
302
+ 0
303
+ οΏ½Ξ±Kβˆ—(t)
304
+ πα′
305
+ Kβˆ—
306
+ sin(παKβˆ—(t))Ξ“(Ξ±Kβˆ—(t)),(15)
307
+ with Ξ±K(t) = 0.7 GeVβˆ’2 Γ— (t βˆ’ M 2
308
+ K) and Ξ±Kβˆ—(t) = 1 +
309
+ 0.83 Gevβˆ’2 Γ— (t βˆ’ M 2
310
+ Kβˆ—) the linear Reggeon trajectory.
311
+ The constants sK
312
+ 0 and sKβˆ—
313
+ 0
314
+ are determined to be 3.0 GeV2
315
+ and 1.5 GeV2, respectively [52]. Here, the Ξ±β€²
316
+ K and Ξ±β€²
317
+ Kβˆ—
318
+ are the Regge-slopes.
319
+ Then, the full photo-production amplitudes for Ξ³n β†’
320
+ K+Ξ£βˆ—βˆ’
321
+ 1/2βˆ’ reaction can be expressed as
322
+ MΒ΅ =
323
+ οΏ½
324
+ MΒ΅
325
+ Kβˆ’ + MΒ΅
326
+ c
327
+ οΏ½ οΏ½
328
+ t βˆ’ M 2
329
+ Kβˆ’
330
+ οΏ½
331
+ FRegge
332
+ K
333
+ + MΒ΅
334
+ Ξ£βˆ’fu
335
+ + MΒ΅
336
+ Kβˆ—
337
+ οΏ½
338
+ t βˆ’ M 2
339
+ Kβˆ—
340
+ οΏ½
341
+ FRegge
342
+ Kβˆ—
343
+ + MΒ΅
344
+ nfs,
345
+ (16)
346
+ While FRegge
347
+ K
348
+ and FRegge
349
+ Kβˆ—
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 off-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
+ Ξ›4
362
+ i
363
+ Ξ›4
364
+ i + (q2
365
+ i βˆ’ 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 Ξ›i is the cut-off values
372
+ for baryon exchange diagrams.
373
+ In this work, we take
374
+ Ξ›s = Ξ›u = 1.5 GeV, and will discuss the results with
375
+ different cut-off.
376
+ Finally, the unpolarized differential cross section in the
377
+ center of mass (c.m.) frame for the Ξ³n β†’ KΞ£βˆ—βˆ’
378
+ 1/2βˆ’ reac-
379
+ tion reads
380
+ dσ
381
+ dΩ = MNMΞ£βˆ—|βƒ—kc.m.
382
+ 1
383
+ ||βƒ—pc.m.
384
+ 1
385
+ |
386
+ 8Ο€2(s βˆ’ M 2
387
+ N)2
388
+ οΏ½
389
+ Ξ»,sp,sΞ£βˆ—
390
+ |M|2,
391
+ (18)
392
+ where s denotes the invariant mass square of the center
393
+ of mass (c.m.) frame for Ξ£βˆ—
394
+ 1/2βˆ’ photo-production. Here
395
+ βƒ—kc.m.
396
+ 1
397
+ and βƒ—pc.m.
398
+ 1
399
+ are the three-momenta of the photon and
400
+ K meson in the c.m.
401
+ frame, while dΩ = 2Ο€dcosΞΈc.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£βˆ—βˆ’
408
+ 1/2βˆ’
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£βˆ’
420
+ 1197.449
421
+ K+
422
+ 493.677
423
+ Kβˆ’
424
+ 493.677
425
+ Kβˆ—
426
+ 891.66
427
+ First we show the angle dependence of the differential
428
+ cross sections for the Ξ³n β†’ K+Ξ£βˆ—βˆ’
429
+ 1/2βˆ’ reaction in Fig. 2,
430
+ where the the center-of-mass energies W = √s varies
431
+ from 2.0 to 2.8 GeV. The black curves labeled as β€˜Total’
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 Kβˆ— exchange diagrams,
439
+ respectively, while the cyan-dot-dashed curves represent
440
+ the contribution from the contact term. According to the
441
+ differential cross sections, one can find 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£βˆ—βˆ’
444
+ 1/2βˆ’, mainly due
445
+ to the Regge effects of the t-change K exchange. The
446
+ K-Reggeon exchange shows steadily increasing behavior
447
+ with cosΞΈc.m. and falls off 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 Kβˆ— exchange term is very small and could be
452
+ safely neglected for the process Ξ³n β†’ K+Ξ£βˆ—βˆ’
453
+ 1/2βˆ’, which
454
+ is consistent with the results of Ref. [27].
455
+ In addition to the the differential cross sections, we
456
+ have also calculated the total cross section of the Ξ³n β†’
457
+ K+Ξ£βˆ—βˆ’
458
+ 1/2βˆ’ reaction as a function of the initial photon en-
459
+ ergy. The results are shown in Fig. 3. The black curve
460
+ labeled as β€˜Total’ 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 Kβˆ— 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
+ dσ/dcosθc.m. (¡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£βˆ—βˆ’
533
+ 1/2βˆ’ differential cross sections as a function of cosΞΈc.m. are plotted for Ξ³n-invariant mass
534
+ intervals (in GeV units). The black curve labeled as β€˜Total’ 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 effective 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 Kβˆ— 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£βˆ—βˆ’
540
+ 1/2βˆ’
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£βˆ—(1480) could be observed by future experiments in the
544
+ process Ξ³n β†’ K+Ξ£βˆ—βˆ’ (1480) β†’ Ξ£βˆ’Ο€0/Ξ£0Ο€βˆ’/Ξ£βˆ’Ξ³.
545
+ Finally, we also show the total cross section for Ξ³n β†’
546
+ K+Ξ£βˆ—βˆ’
547
+ 1/2βˆ’ with the cut-off Ξ›s/u = 1.2, 1.5, and 1.8 GeV
548
+ in Fig. 4, where one can find the total cross sections are
549
+ weakly dependence on the value of the cut-off. 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
+ Οƒ (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£βˆ—
583
+ 1/2βˆ’ is plotted as a function of the lab energy EΞ³. The
584
+ black curve labeled as β€˜Total’ 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 effective 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 Kβˆ— 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
+ Οƒ (Β΅b)
613
+ EΞ³ (GeV)
614
+ Ξ›s,u = 1.2 GeV
615
+ Ξ›s,u = 1.5 GeV
616
+ Ξ›s,u = 1.8 GeV
617
+ FIG. 4:
618
+ (Color online) Total cross section for Ξ³n
619
+ β†’
620
+ K+Ξ£βˆ—
621
+ 1/2βˆ’ with the cut-off Ξ›s/u = 1.2, 1.5, and 1.8 GeV.
622
+ future experiment would be helpful to constrain these
623
+ couplings if the state Σ(1480) is confirmed.
624
+ IV.
625
+ SUMMARY
626
+ The lowest Ξ£βˆ—βˆ’
627
+ 1/2βˆ’ is far from established, and its ex-
628
+ istence is important to understand the low-lying excited
629
+ baryon with JP = 1/2βˆ’. There are many experimen-
630
+ tal hints of the Ξ£βˆ—(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 confirm its existence.
634
+ Assuming that the JP
635
+ =
636
+ 1/2βˆ’ low lying state
637
+ Ξ£βˆ— (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£βˆ—βˆ’
640
+ 1/2βˆ’ reaction by considering the
641
+ contributions from the t-channel K/Kβˆ— exchange term,
642
+ s-channel nucleon term, u-channel Ξ£ exchange term, and
643
+ contact term within the Regge-effective Lagrange ap-
644
+ proach.
645
+ The differential 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£βˆ—βˆ’
648
+ 1/2βˆ’
649
+ is about 4.3 Β΅b around EΞ³ = 2.3 GeV. We encourage
650
+ our experimental colleagues to measure Ξ³n β†’ K+Ξ£βˆ—βˆ’
651
+ 1/2βˆ’
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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+ 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'}
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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 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ 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'}
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+ 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 βˆ’ 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' (2) We show that any frame {xi}2nβˆ’2 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'}
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+ 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'}
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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=' Introduction The concept of frames in a separable Hilbert space was originally introduced by Duffin and Schaeffer in the context of non-harmonic Fourier series [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Frames are a more flexible 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'}
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+ page_content=' Phase retrieval is an old problem of recovering a signal from the absolute value of linear measurement coefficients called intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' Phase retrieval has been defined 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'}
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+ page_content=' The concept of weak phase retrieval weakened the notion of phase retrieval and it has been first defined for vectors in ([8] and [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' The rest of the paper is organized as follows: In Section 2, we give the basic definitions and certain preliminary results to be used in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' In section 4 we show that any family of vectors {xi}2nβˆ’2 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'}
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+ 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'}
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+ 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'}
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+ page_content=' These results give a complete understanding of the difference 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=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' Let H be a finite or infinite dimensional real Hilbert space and B(H) the class of all bounded linear operators defined on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' The natural numbers and real numbers are denoted by β€œN” and β€œR”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ 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}i∈I] instead of span{xi}i∈I, where I is a finite or countable subset of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' We 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' 42C15, 42C40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' The first author was supported by NSF DMS 1609760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' 1 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 AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' We start with the definition of a real Hilbert space frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Definition 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}i∈I in a finite or infinite dimensional separable real Hilbert space H is a frame if there are constants 0 < A ≀ B < ∞ so that Aβˆ₯xβˆ₯2 ≀ οΏ½ i∈I |⟨x, xi⟩|2 ≀ Bβˆ₯xβˆ₯2, for all f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' The constants A and B are called the lower and upper frame bounds for {xi}i∈I, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' If an upper frame bound exists, then {xi}i∈I 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'}
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+ 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'}
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+ page_content=' The values {⟨x, xi⟩}∞ i=1 are called the frame coefficients of the vector x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ 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'}
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+ page_content=' We will need to work with Riesz sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' A family X = {xi}i∈I in a finite or infinite dimensional real Hilbert space H is a Riesz sequence if there are constants 0 < A ≀ B < ∞ satisfying A οΏ½ i∈I |ci|2 ≀ βˆ₯ οΏ½ i∈I cixiβˆ₯2 ≀ B οΏ½ i∈I |ci|2 for all sequences of scalars {ci}i∈I.' 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}∞ 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 ⟨ei, ej⟩ = δi,j = � 1 if i = j, 0 if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Definition 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}i∈I in a real Hilbert space H does phase (norm) retrieval if whenever x, y ∈ H, satisfy |⟨x, xi⟩| = |⟨y, xi⟩| for all i ∈ I, then x = Β±y (βˆ₯xβˆ₯ = βˆ₯yβˆ₯).' 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}i∈I does phase (norm) retrieval, then so does {aixi}i∈I for any 0 < ai < ∞ for all i ∈ 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| = ∞, 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 < infi∈I ai ≀ supi∈Iai < ∞.' 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=' Definition 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}i∈I in a finite or infinite dimensional real Hilbert space H has the complement property if for any subset J βŠ‚ I, either span{xi}i∈J = H or span{xi}i∈Jc = 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 finite 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}i∈I 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: Definition 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 βŠ‚ [m] with |I| = n , {xi}i∈I 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‰₯ 2nβˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' If m = 2nβˆ’ 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 significant 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}2nβˆ’2 i=1 does weak phase retrieval in Rn then for every I βŠ‚ [2nβˆ’2], if x βŠ₯ span{xi}i∈I and y βŠ₯ {xi}i∈Ic then x βˆ₯xβˆ₯ + y βˆ₯yβˆ₯ and x βˆ₯xβˆ₯ βˆ’ y βˆ₯yβˆ₯ 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 βˆ₯xβˆ₯ = βˆ₯yβˆ₯ = 1, then x + y and x βˆ’ 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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+ 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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+ 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 βˆ₯xβˆ₯ ΜΈ= βˆ₯yβˆ₯.' 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 βˆ’1 1 1 1 βˆ’1 1 1 1 βˆ’1 \uf8f9 \uf8fa\uf8fa\uf8fb Also, x = (0, 1, βˆ’1) 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 βˆ’ y = (0, βˆ’1 2 , βˆ’3 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, βˆ’1) and y = (0, 1, 1) so x + y = (0, 1, 0) and x βˆ’ 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 β€œWeak phase retrieval by vectors” 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 βˆ’ 1)- vectors in Rn satisfies phaseless reconstruction, however no set of (2n βˆ’ 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 βˆ’ 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=' Definition 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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+ 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 |θ| = 1 so that phase(ai) = θphase(bi) for all i ∈ [n], for which ai ̸= 0 ̸= 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 = βˆ’1, they weakly have the opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
113
+ page_content=' Therefore with above definition 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 ∈ R, sgn(x) = 1 if x > 0 and sgn(x) = βˆ’1 if x < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
116
+ 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'}
117
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
118
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
119
+ page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
120
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
121
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
122
+ page_content=' , bn) in Rn with |⟨x, xi⟩| = |⟨y, xi⟩| for all i ∈ [m], then x and y weakly have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
123
+ 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'}
125
+ page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
126
+ page_content=' AKRAMI Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
127
+ page_content=' [8] Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
128
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
129
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
130
+ page_content=' , an) and y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
131
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
132
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
133
+ page_content=' , bn) in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
134
+ page_content=' The following are equivalent: (1) We have sgn(aiaj) = sgn(bibj), for all 1 ≀ i ΜΈ= j ≀ 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'}
135
+ 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'}
136
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
137
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
138
+ page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
139
+ page_content=' The following appears in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
140
+ page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
141
+ page_content=' If X = {xi}m i=1 does weak phase retrieval in Rn, then m β‰₯ 2n βˆ’ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
142
+ page_content=' Finally, we have: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
143
+ page_content=' [7] If a frame X = {xi}2nβˆ’2 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'}
144
+ 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'}
145
+ page_content=' If {xi}i∈I does phase retrieval and R is an invertible operator on the space then {Rxi}i∈I 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 |⟨x, Rxi⟩| = |⟨y, Rxi⟩| implies |⟨Rβˆ—x, xi⟩| = |⟨Rβˆ—y, xi⟩|, and so Rβˆ—x = ΞΈRβˆ—y for |ΞΈ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
147
+ page_content=' Since R is invertible, x = ΞΈy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
148
+ page_content=' This result fails badly for weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
149
+ page_content=' For example, let e1 = (1, 0), e2 = (0, 1), x1 = ( 1 √ 2, 1 √ 2, x2 = ( 1 √ 2, βˆ’1 √ 2) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
150
+ 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'}
151
+ page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
152
+ page_content=' Frames Containing Unit Vectors Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
153
+ page_content=' Any frame {xi}2nβˆ’2 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'}
154
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
155
+ page_content=' We proceed by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
156
+ page_content=' Recall that {xi}2nβˆ’2 i=1 must be full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
157
+ 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'}
158
+ page_content=' Assume I βŠ‚ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
159
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
160
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
161
+ page_content=', 2nβˆ’2} with |I| = n βˆ’ 1 and assume x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
162
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
163
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
164
+ page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
165
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
166
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
167
+ page_content=' , bn) with βˆ₯xβˆ₯ = βˆ₯yβˆ₯ = 1 and x βŠ₯ X = span{xi}i∈I and y βŠ₯ span{xi}2nβˆ’2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
168
+ page_content=' After reindexing {ei}n i=1 and {xi}2nβˆ’2 i=1 }, we assume x1 = e1, I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
169
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
170
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
171
+ page_content=', nβˆ’1 and Ic = {n, n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
172
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
173
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
174
+ page_content=' , 2nβˆ’ 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
175
+ page_content=' Since ⟨x, x1⟩ = a1 = 0, by Theorem 2, b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
176
+ 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}2nβˆ’2 i=n is (n βˆ’ 1)-vectors in an (n βˆ’ 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}2nβˆ’2 i=n not all zero so that 2nβˆ’2 οΏ½ i=n ciPxi = 0 and so 2nβˆ’1 οΏ½ i=n cixi(1)x1 βˆ’ 2nβˆ’2 οΏ½ i=n cixi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
179
+ 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'}
180
+ 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'}
182
+ page_content=' The fact that there are (2nβˆ’ 2) vectors in the theorem is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
183
+ 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'}
185
+ page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
186
+ page_content=' Consider the full spark frame X = {(1, 2, 3), (0, 1, 0), (0, βˆ’2, 3), (1, βˆ’2, βˆ’3)} 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, βˆ’2), (1, βˆ’2)}, {(1, 3), (0, 0), (0, 3), (1, βˆ’3)}, and WEAK PHASE RETRIEVAL 5 {(2, 3), (1, 0), (βˆ’2, 3), (βˆ’2, βˆ’3)} 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 βˆ’ 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 βˆ’ 1 vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
191
+ 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 ∈ Rn so that x + y, x βˆ’ y do not have the same or opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
193
+ page_content=' Let X1 = xβŠ₯ and Y1 = yβŠ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
194
+ page_content=' Then span{X1, X2} = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
195
+ page_content=' Choose {xi}nβˆ’1 i=1 vectors spanning X1 and {xi}2nβˆ’2 i=n be vectors spanning X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
196
+ page_content=' Then {xi}2nβˆ’2 i=1 is a frame for Rn with x βŠ₯ xi, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
197
+ 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'}
199
+ page_content=', n βˆ’ 1 and y βŠ₯ 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'}
202
+ page_content=' , 2n βˆ’ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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+ page_content=' It follows that |⟨x + y, xi⟩| = |⟨x βˆ’ y, xi⟩|, 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'}
206
+ page_content=' , n, but x, y do not have the same or opposite signs and so {xi}2nβˆ’2 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=' Definition 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 define the sphere of X as SX = {x ∈ X : βˆ₯xβˆ₯ = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
209
+ page_content=' Definition 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 define the distance between X and Y as d(X, Y ) = supx∈SXinfy∈SY βˆ₯x βˆ’ yβˆ₯.' 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 ∈ X there is a z ∈ SY so that βˆ₯ x βˆ₯xβˆ₯ βˆ’ zβˆ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
212
+ page_content=' Letting y = βˆ₯xβˆ₯z we have that βˆ₯yβˆ₯ = βˆ₯xβˆ₯ and βˆ₯x βˆ’ yβˆ₯ < Η«βˆ₯xβˆ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
213
+ page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
214
+ page_content=' Let X, Y be hyperplanes in Rn and unit vectors x βŠ₯ X, y βŠ₯ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
215
+ page_content=' If d(X, Y ) < Η« then min{βˆ₯x βˆ’ yβˆ₯, βˆ₯x + yβˆ₯} < 6Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
216
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
217
+ page_content=' Since span{y, Y } = Rn, x = ay + z for some z ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
218
+ page_content=' By replacing y by βˆ’y if necessary, we may assume 0 < a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
219
+ page_content=' By assumption, there is some w ∈ X with βˆ₯wβˆ₯ = βˆ₯zβˆ₯ so that βˆ₯w βˆ’ zβˆ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
220
+ page_content=' Now a = aβˆ₯yβˆ₯ = βˆ₯ayβˆ₯ = βˆ₯x βˆ’ zβˆ₯ β‰₯ βˆ₯x βˆ’ wβˆ₯ βˆ’ βˆ₯w βˆ’ zβˆ₯ β‰₯ βˆ₯xβˆ₯ βˆ’ Η« = 1 βˆ’ Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
221
+ page_content=' So, 1 βˆ’ a < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
222
+ page_content=' Also, 1 = βˆ₯xβˆ₯2 = a2 + βˆ₯wβˆ₯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'}
225
+ page_content=' 0 < 1 βˆ’ a < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
226
+ page_content=' 1 = βˆ₯xβˆ₯2 = βˆ₯ay + zβˆ₯2 = a2βˆ₯yβˆ₯2 + βˆ₯zβˆ₯2 = a2 + βˆ₯zβˆ₯2 β‰₯ (1 βˆ’ Η«)2 + βˆ₯zβˆ₯2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
227
+ page_content=' So βˆ₯zβˆ₯2 ≀ 1 βˆ’ (1 βˆ’ Η«)2 = 2Η« βˆ’ Η«2 ≀ 2Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
228
+ page_content=' Finally, βˆ₯x βˆ’ yβˆ₯2 = βˆ₯(ay + z) βˆ’ yβˆ₯2 ≀ (βˆ₯(1 βˆ’ a)yβˆ₯ + βˆ₯zβˆ₯)2 ≀ (1 βˆ’ a)2βˆ₯yβˆ₯2 + βˆ₯zβˆ₯2 + 2(1 βˆ’ a)βˆ₯yβˆ₯βˆ₯zβˆ₯ < Η«2 + 2Η« + 2 √ 2Η«2 < 6Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
229
+ page_content=' 6 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
230
+ page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
231
+ page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
232
+ page_content=' AKRAMI β–‘ Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
233
+ page_content=' Let X, Y be hyperplanes in Rn, {xi}nβˆ’1 i=1 be a unit norm basis for X and {yi}nβˆ’1 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'}
234
+ page_content=' If οΏ½nβˆ’1 i=1 βˆ₯xi βˆ’ yiβˆ₯ < Η« then d(X, Y ) < 2Η«B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
235
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
236
+ page_content=' Let 0 < A ≀ B < ∞ be upper and lower basis bounds for the two bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
237
+ page_content=' Given a unit vector x = οΏ½nβˆ’1 i=1 aixi ∈ X, let y = οΏ½nβˆ’1 i=1 aiyi ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
238
+ page_content=' We have that sup1≀i≀nβˆ’1|ai| ≀ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
239
+ page_content=' We compute: βˆ₯x βˆ’ yβˆ₯ = βˆ₯ nβˆ’1 οΏ½ i=1 ai(xi βˆ’ yi)βˆ₯ ≀ nβˆ’1 οΏ½ i=1 |ai|βˆ₯xi βˆ’ yiβˆ₯ ≀ (sup1≀i≀nβˆ’1|ai|) nβˆ’1 οΏ½ i=1 βˆ₯xi βˆ’ yiβˆ₯ ≀ BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
240
+ page_content=' So βˆ₯yβˆ₯ β‰₯ βˆ₯xβˆ₯ βˆ’ βˆ₯x βˆ’ yβˆ₯ β‰₯ 1 βˆ’ BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
241
+ page_content=' οΏ½οΏ½οΏ½οΏ½x βˆ’ y βˆ₯yβˆ₯ οΏ½οΏ½οΏ½οΏ½ ≀ βˆ₯x βˆ’ yβˆ₯ + οΏ½οΏ½οΏ½οΏ½y βˆ’ y βˆ₯yβˆ₯ οΏ½οΏ½οΏ½οΏ½ ≀ BΗ« + 1 βˆ₯yβˆ₯βˆ₯(1 βˆ’ βˆ₯yβˆ₯)yβˆ₯ = BΗ« + (1 βˆ’ βˆ₯yβˆ₯) ≀ 2BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
242
+ page_content=' It follows that d(X, Y ) < 2BΗ«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
243
+ page_content=' β–‘ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
244
+ page_content=' Let {xi}n i=1 be a basis for Rn with unconditional basis constant B and assume yi ∈ Rn satisfies οΏ½n i=1 βˆ₯xi βˆ’ yiβˆ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
245
+ 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'}
246
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
247
+ page_content=' Fix {ai}n i=1 and compute βˆ₯ n οΏ½ i=1 aiyiβˆ₯ ≀ βˆ₯ n οΏ½ i=1 aixiβˆ₯ + βˆ₯ n οΏ½ i=1 |ai|(xi βˆ’ yi)βˆ₯ ≀ βˆ₯ n οΏ½ i=1 aixiβˆ₯ + (sup1≀i≀n|ai|) n οΏ½ i=1 βˆ₯xi βˆ’ yiβˆ₯ ≀ βˆ₯ n οΏ½ i=1 aixiβˆ₯ + (sup1≀i≀n|ai|)Η« ≀ βˆ₯ n οΏ½ i=1 aixiβˆ₯ + Η«Bβˆ₯ n οΏ½ i=1 aixiβˆ₯ = (1 + Η«B)βˆ₯ n οΏ½ i=1 aixiβˆ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
248
+ page_content=' WEAK PHASE RETRIEVAL 7 Similarly, βˆ₯ n οΏ½ i=1 |ai|yiβˆ₯ β‰₯ (1 βˆ’ Η«B)βˆ₯ n οΏ½ i=1 aixiβˆ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
249
+ 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'}
250
+ page_content=' For Η«i = Β±1, βˆ₯ n οΏ½ i=1 Η«iaiyiβˆ₯ ≀ (1 + Η«B)βˆ₯ n οΏ½ i=1 Η«iaixiβˆ₯ ≀ B(1 + Η«B)βˆ₯ n οΏ½ i=1 aixiβˆ₯ ≀ B(1 + Η«B)2βˆ₯ n οΏ½ i=1 aiyiβˆ₯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
251
+ 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'}
252
+ page_content=' β–‘ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
253
+ 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 βˆ’ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
254
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
255
+ page_content=' We may assume m = 2nβˆ’2 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'}
256
+ 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'}
257
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
258
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
259
+ page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
260
+ page_content=' By [10], there is an orthonormal sequence {xi}2nβˆ’2 i=n+1 so that {xi}2nβˆ’2 i=1 is full spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
261
+ page_content=' Let I = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
262
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
263
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
264
+ page_content=', n βˆ’ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
265
+ page_content=' Let X = span{xi}nβˆ’1 i=1 and Y = span{xi}2nβˆ’2 i=n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
266
+ page_content=' Then x = en βŠ₯ X and there is a βˆ₯yβˆ₯ = 1 with y βŠ₯ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
267
+ page_content=' Note that ⟨x βˆ’ y, en⟩ ΜΈ= 0 ΜΈ= ⟨x + y, en⟩, for otherwise, x = Β±y βŠ₯ 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'}
268
+ page_content=' So there is a j = n and a Ξ΄ > 0 so that |(x + y)(j)|, |(x βˆ’ y)(j)| β‰₯ Ξ΄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
269
+ page_content=' We will show that there exists an 0 < Η« so that whenever {yi}2nβˆ’2 i=1 are vectors in Rn satisfying οΏ½n i=1 βˆ₯xi βˆ’ yiβˆ₯ < Η«, then {yi}n i=1 fails weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
270
+ page_content=' Fix 0 < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
271
+ page_content=' Assume {yi}2nβˆ’2 i=1 are vectors so that οΏ½2nβˆ’2 i=1 βˆ₯xiβˆ’yiβˆ₯ < Η«.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
272
+ page_content=' Choose unit vectors xβ€² βŠ₯ span{yi}i∈I, yβ€² βŠ₯ span{yi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
273
+ page_content=' By Proposition 2 and Lemma 1, we may choose Η« small enough (and change signs if necessary) so that βˆ₯xβˆ’xβ€²βˆ₯, βˆ₯yβˆ’yβ€²βˆ₯ < Ξ΄ 4B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
274
+ page_content=' Hence, since the unconditional basis constant is B, |[(x + y) βˆ’ (xβ€² + yβ€²)](j)| ≀ |(x βˆ’ xβ€²)j| + |(y βˆ’ yβ€²)(j)| < Bβˆ₯x βˆ’ xβ€²βˆ₯ + Bβˆ₯y βˆ’ yβ€²βˆ₯ ≀ 2B Ξ΄ 4B = Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
275
+ page_content=' It follows that |(xβ€² + yβ€²)(j)| β‰₯ |(x + y)(j)| βˆ’ |[(x + y) βˆ’ (xβ€² + yβ€²)](j)| β‰₯ Ξ΄ βˆ’ 1 2Ξ΄ = Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
276
+ page_content=' Similarly, |(xβ€² βˆ’ yβ€²)(j)| > Ξ΄ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
277
+ page_content=' So xβ€² + yβ€², xβ€² βˆ’ yβ€² are not disjointly supported and so {yi}2nβˆ’2 i=1 fails weak phase retrieval by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
278
+ page_content=' β–‘ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
279
+ 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'}
280
+ page_content=' These results give a complete understanding of the difference between weak phase retrieval and phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
281
+ page_content=' Now we give a surprising and very strong classification of weak phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
282
+ page_content=' 8 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
283
+ page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
284
+ page_content=' CASAZZA AND F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
285
+ page_content=' AKRAMI Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
286
+ page_content=' Let {xi}2nβˆ’2 i=1 be non-zero vectors in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
287
+ page_content=' The following are equivalent: (1) The family {xi}2nβˆ’2 i=1 does weak phase retrieval in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
288
+ page_content=' (2) If x, y ∈ Rn and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
289
+ page_content='1) |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
290
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
291
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
292
+ page_content=' , 2n βˆ’ 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'}
293
+ page_content=' (b) x and y are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
294
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
295
+ 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'}
296
+ page_content=' Let x = (a1, a2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
297
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
298
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
299
+ page_content=' , an), y = (b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
300
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
301
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
302
+ page_content=' , bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
303
+ page_content=' Since {xi}2nβˆ’2 i=1 does weak phase retrieval, replacing y by βˆ’y if necessary, Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
304
+ 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'}
305
+ page_content=' Let I = {1 ≀ i ≀ 2n βˆ’ 2 : ⟨x, xi⟩ = ⟨y, yi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
306
+ page_content=' Then x + y βŠ₯ xi for all i ∈ Ic and x βˆ’ y βŠ₯ xi for all i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
307
+ page_content=' By Theorem 2, x + y βˆ₯x + y + x βˆ’ y βˆ₯x βˆ’ yβˆ₯ and x + y βˆ₯x + yβˆ₯ βˆ’ x βˆ’ y βˆ₯x βˆ’ yβˆ₯ are disjointly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
308
+ page_content=' Assume there is a 1 ≀ j ≀ n with aj = bj ΜΈ= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
309
+ page_content=' Then (x + y)(j) βˆ₯x + yβˆ₯ + (x βˆ’ y)(j) βˆ₯x βˆ’ yβˆ₯ = 2aj βˆ₯x + yβˆ₯ and (x + y)(j) βˆ₯x + yβˆ₯ βˆ’ (x βˆ’ y)(j) βˆ₯x βˆ’ yβˆ₯ = 2aj βˆ₯x + yβˆ₯, Contradicting Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
310
+ 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'}
311
+ page_content=' β–‘ Phase retrieval is when (a) in the theorem holds for every x, y ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
312
+ page_content=' So this the- orem shows clearly the difference 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'}
313
+ page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
314
+ page_content=' If {xi}2nβˆ’2 i=1 does weak phase retrieval in Rn, then there are disjointly supported non-zero vectors x, y ∈ Rn satisfying: |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
315
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
316
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
317
+ page_content=' , 2n βˆ’ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
318
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
319
+ page_content=' Since {xi}2nβˆ’2 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'}
320
+ page_content=' β–‘ Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
321
+ 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'}
322
+ page_content=' If J βŠ‚ [n], we define PJ as the projection onto span{ei}i∈J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
323
+ page_content=' Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
324
+ 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'}
325
+ page_content=' The following are equivalent: (1) Whenever I βŠ‚ [2n βˆ’ 2] and 0 ΜΈ= x βŠ₯ xi for i ∈ I and 0 ΜΈ= y βŠ₯ xi for i ∈ Ic, there is no j ∈ [n] so that ⟨x, ej⟩ = 0 = ⟨y, ej⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
326
+ page_content=' (2) For every J βŠ‚ [n] with |J| = n βˆ’ 1, {Pjxi}2nβˆ’2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
327
+ page_content=' (3) For every J βŠ‚ [n] with |J| < n, {PJxi}2nβˆ’2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
328
+ page_content=' WEAK PHASE RETRIEVAL 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
329
+ page_content=' (1) β‡’ (2): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
330
+ page_content=' So assume (2) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
331
+ page_content=' Then choose J βŠ‚ [n] with |J| = n βˆ’ 1, J = [n] \\ {j}, and {PJxi}2nβˆ’2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
332
+ page_content=' In particular, it fails complement property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
333
+ page_content=' That is, there exists I βŠ‚ [2nβˆ’ 2] and span {PJxi}i∈I ΜΈ= PJRn and span {Pjxi}i∈Ic ΜΈ= PJRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
334
+ page_content=' So there exists norm one vectors x, y in PJRn with PJx = x βŠ₯ PJxi for all i ∈ I and PJy = y βŠ₯ PJxi for all i ∈ Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
335
+ 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'}
336
+ page_content=' Hence, x βŠ₯ xi for i ∈ I and y βŠ₯ xi for i ∈ Ic, proving (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
337
+ 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'}
338
+ page_content=' (3) β‡’ (2): This is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
339
+ page_content=' (3) β‡’ (1): We prove the contrapositive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
340
+ page_content=' So assume (1) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
341
+ page_content=' Then there is a I βŠ‚ [2nβˆ’ 2] and 0 ΜΈ= x βŠ₯ xi for i ∈ I and 0 ΜΈ= y βŠ₯ xi for i ∈ Ic and a j ∈ [n] so that ⟨x, ej⟩ = ⟨y, ej⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
342
+ page_content=' It follows that x = PJx, y = PJy are non zero and x βŠ₯ Pjxi for all i ∈ I and y βŠ₯ Pjxi for i ∈ Ic, so {PJxi}2nβˆ’2 i=1 fails phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
343
+ page_content=' β–‘ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
344
+ page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
345
+ page_content=' The assumptions in the theorem are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
346
+ 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'}
347
+ page_content=' For example, in R3 consider the row vectors {xi}4 i=1 of: \uf8ee \uf8ef\uf8ef\uf8f0 1 1 1 βˆ’1 1 1 1 βˆ’1 1 1 1 βˆ’1 \uf8f9 \uf8fa\uf8fa\uf8fb This set does weak phase retrieval, but if J = {2, 3} then x = (0, 1, βˆ’1) βŠ₯ PJxi for i = 1, 2 and y = (0, 1, 1) βŠ₯ 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'}
348
+ page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
349
+ page_content=' Assume {xi}2nβˆ’2 i=1 does weak phase retrieval in Rn and for every J βŠ‚ [n] {PJxi}2nβˆ’2 i=1 does phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
350
+ page_content=' Then if x, y ∈ Rn and |⟨x, xi⟩| = |⟨y, xi⟩| for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
351
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
352
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
353
+ page_content=' , 2n βˆ’ 2, then there is a J βŠ‚ [n] so that x(j) = οΏ½ aj ΜΈ= 0 for j ∈ J 0 for j ∈ Jc y(j) = οΏ½ 0 for j ∈ J bj ΜΈ= 0 for j ∈ Jc Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
354
+ page_content=' Let {ei}n i=1 be the unit vector basis of Rn and for I βŠ‚ [n], let PI be the projection onto XI = span{ei}i∈I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
355
+ page_content=' For every m β‰₯ 1, there are vectors {xi}m i=1 so that for every I βŠ‚ [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'}
356
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
357
+ page_content=' We do this by induction on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
358
+ page_content=' For m=1, let x1 = (1, 1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
359
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
360
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
361
+ page_content=', 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
362
+ page_content=' This satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
363
+ 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'}
364
+ page_content=' Choose I βŠ‚ [1, n] with |I| = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
365
+ page_content=' Choose J βŠ‚ I with |J| = k βˆ’ 1 and let XJ = span{xi}i∈J βˆͺ {xi}i∈Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
366
+ 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'}
367
+ page_content=' Since there only exist finitely many such Jβ€²s there is a vector xm+1 /∈ XJ for every J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
368
+ page_content=' We will show that {xi}m+1 i=1 satisfies the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
369
+ page_content=' Let I βŠ‚ [1, n] and J βŠ‚ I with |J| = |I|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
370
+ page_content=' If PIxm+1 /∈ XJ, then {PIxi}i∈J is linearly independent by the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
371
+ page_content=' On the other hand, if m + 1 ∈ J then xm+1 /∈ XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
372
+ page_content=' But, if PIxm+1 ∈ span{PIxi}i∈J\\m+1, since (I βˆ’ PI)xm+1 ∈ span{ei}i∈Ic, it follows that xm+1 ∈ XJ, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
373
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+ page_content=' Schaeffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4dE1T4oBgHgl3EQf6QUq/content/2301.03520v1.pdf'}
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1
+ Higher order Bernstein-B´ezier and N´ed´elec finite elements for the
2
+ relaxed micromorphic model
3
+ Adam Sky1,
4
+ Ingo Muench2,
5
+ Gianluca Rizzi3
6
+ and
7
+ Patrizio Neff4
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, finite 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 differentiation 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 differentiation, 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 finite element simulations, resulting in longer computation times.
35
+ The relaxed micromorphic model [35] offers 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 field P in addition to the standard displacement
38
+ field u. Consequently, each material point is endowed with twelve degrees of freedom, effectively turning into
39
+ an affine-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 Neff,
47
+ Chair for Nonlinear Analysis and Modelling, Faculty of Mathematics, Universit¨at Duisburg-Essen, Thea-
48
+ Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neff@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
+ influence 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 effectiveness 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 influences 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 } ∈ X = [H 1]3 Γ— [H (curl)]3 [18,34], although the regularity of the microdistortion
67
+ can be improved to P ∈ [H 1]3Γ—3 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 N´ed´elec elements can be found in [5, 51] and specifically 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 differentiation. 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 definitions are employed throughout this work:
88
+ β€’ vectors are indicated by bold letters. Non-bold letters represent scalars;
89
+ β€’ in general, formulas are defined 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 βŠ‚ R3. The corresponding reference
92
+ domain is given by Ω;
93
+ β€’ analogously, in two dimensions we employ A βŠ‚ 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 field Du, the microdistortion P and the Curl of the microdistortion
103
+ I(u, P ) = 1
104
+ 2
105
+ οΏ½
106
+ V
107
+ ⟨sym(Du βˆ’ P ), Ce sym(Du βˆ’ P )⟩ + ⟨sym P , Cmicro sym P ⟩
108
+ + ⟨skew(Du βˆ’ P ), Cc skew(Du βˆ’ P )⟩ + Β΅macroL2
109
+ c⟨Curl P , L Curl P ⟩ dV
110
+ βˆ’
111
+ οΏ½
112
+ V
113
+ ⟨u, f⟩ + ⟨P , M⟩ dV β†’ min
114
+ w.r.t.
115
+ {u, P } ,
116
+ (2.1)
117
+ where the Curl operator for second order tensors is defined 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 βˆ’ P12,z
145
+ P11,z βˆ’ P13,x
146
+ P12,x βˆ’ P11,y
147
+ P23,y βˆ’ P22,z
148
+ P21,z βˆ’ P23,x
149
+ P22,x βˆ’ P21,y
150
+ P33,y βˆ’ P32,z
151
+ P31,z βˆ’ P33,x
152
+ P32,x βˆ’ P31,y
153
+ οΏ½
154
+ οΏ½ ,
155
+ curl p = βˆ‡ Γ— p ,
156
+ p : V βŠ‚ R3 β†’ R3 ,
157
+ (2.2)
158
+ and curl(·) is the vectorial curl operator. The displacement field and the microdistortion field are functions of
159
+ the reference domain
160
+ u : V βŠ‚ R3 β†’ R3 ,
161
+ P : V βŠ‚ R3 β†’ R3Γ—3 .
162
+ (2.3)
163
+ The tensors Ce, Cmicro, L ∈ R3Γ—3Γ—3Γ—3 are standard positive definite fourth order elasticity tensors. For isotropic
164
+ materials they take the form
165
+ Ce = Ξ»e1 βŠ— 1 + 2Β΅e J ,
166
+ Cmicro = Ξ»micro1 βŠ— 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 ∈ R3Γ—3Γ—3Γ—3 is a positive semi-definite material tensor related to Cosserat micro-polar continua and accounts
170
+ for infinitesimal 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 ≑ 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 field u yields
178
+ Ξ΄uI =
179
+ οΏ½
180
+ V
181
+ ⟨sym DΞ΄u, Ce sym(Du βˆ’ P )⟩ + ⟨skew DΞ΄u, Cc skew(Du βˆ’ P )⟩ βˆ’ ⟨δu, f⟩ 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
+ ⟨sym Ξ΄P , Ce sym(Du βˆ’ P )⟩ + ⟨skew Ξ΄P , Cc skew(Du βˆ’ P )⟩
188
+ βˆ’ ⟨sym Ξ΄P , Cmicro sym P ⟩ βˆ’ Β΅macroL2
189
+ c⟨Curl δP , Curl P ⟩ + ⟨δP , M⟩ 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
+ ⟨sym(DΞ΄u βˆ’ Ξ΄P ), Ce sym(Du βˆ’ P )⟩ + ⟨sym Ξ΄P , Cmicro sym P ⟩
196
+ + ⟨skew(DΞ΄u βˆ’ Ξ΄P ), Cc skew(Du βˆ’ P )⟩ + Β΅macroL2
197
+ c⟨Curl δP , Curl P ⟩ dV ,
198
+ (2.7)
199
+ and linear form of the loads
200
+ l({Ξ΄u, Ξ΄P }) =
201
+ οΏ½
202
+ V
203
+ ⟨δu, f⟩ + ⟨δP , M⟩ dV .
204
+ (2.8)
205
+ 3
206
+
207
+ Applying integration by parts to Eq. (2.5) yields
208
+ οΏ½
209
+ βˆ‚V
210
+ ⟨δu , [Ce sym(Du βˆ’ P ) + Cc skew(Du βˆ’ P )] n⟩ dA
211
+ βˆ’
212
+ οΏ½
213
+ V
214
+ ⟨δu , Div[Ce sym(Du βˆ’ P ) + Cc skew(Du βˆ’ P )] βˆ’ f⟩ dV = 0 .
215
+ (2.9)
216
+ Likewise, integration by parts of Eq. (2.6) results in
217
+ οΏ½
218
+ V
219
+ ⟨δP , Ce sym(Du βˆ’ P ) + Cc skew(Du βˆ’ P ) βˆ’ Cmicro sym P βˆ’ Β΅macroL2
220
+ c Curl Curl P + M⟩ dV
221
+ βˆ’ Β΅macroL2
222
+ c
223
+ οΏ½
224
+ βˆ‚V
225
+ ⟨δP , Curl P Γ— n⟩ 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 βˆͺ AN ,
229
+ AD ∩ AN = βˆ… ,
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
+ βˆ’ Div[Ce sym(Du βˆ’ P ) + Cc skew(Du βˆ’ P )] = f
234
+ in
235
+ V ,
236
+ (2.12a)
237
+ βˆ’Ce sym(Du βˆ’ P ) βˆ’ Cc skew(Du βˆ’ 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 βˆ’ P ) + Cc skew(Du βˆ’ 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 οΏ½Οƒ := Ce sym(Du βˆ’ P ) + Cc skew(Du βˆ’ P ) is symmetric if and only if Cc ≑ 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 affecting 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
+ definitions 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 ≑ 0 due to the
280
+ generalized Korn inequality for incompatible tensor fields [24–26,36]. The inequality relies on a non-vanishing
281
+ Dirichlet boundary for the microdistortion field AP
282
+ D ΜΈ= βˆ…, 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-defined 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
+ βˆ’Ce sym(Du βˆ’ P ) βˆ’ Cc skew(Du βˆ’ 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)βˆ’1(sym M + Ce sym Du) ,
295
+ skew P = Cβˆ’1
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 finds
317
+ Cc skew(Du βˆ’ P ) = 0 ,
318
+ Ce sym(Du βˆ’ P ) = Cmicro sym P ,
319
+ sym P = (Ce + Cmicro)βˆ’1Ce sym Du .
320
+ (2.16)
321
+ Applying the former results to Eq. (2.12a) yields
322
+ βˆ’ Div[Ce sym(Du βˆ’ P )] = βˆ’ Div[Cmicro(Ce + Cmicro)βˆ’1Ce sym Du] = βˆ’ Div[Cmacro sym Du] = f ,
323
+ (2.17)
324
+ where the definition
325
+ Cmacro = Cmicro(Ce + Cmicro)βˆ’1Ce
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†’ +∞, the stiffness 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†’ +∞ 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
+ cβˆ₯ Curl P βˆ₯2, on
344
+ contractible domains and bounded energy this implies the reduction of the microdistortion to a gradient field
345
+ P β†’ Dv due to the classical identity
346
+ Curl Dv = 0
347
+ βˆ€ v ∈ [C ∞(V )]3 ,
348
+ (2.20)
349
+ thus asserting finite 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
+ ⟨sym(Du βˆ’ Dv), Ce sym(Du βˆ’ Dv)⟩ + ⟨sym Dv, Cmicro sym Dv⟩
356
+ + ⟨skew(Du βˆ’ Dv), Cc skew(Du βˆ’ Dv)⟩ dV βˆ’
357
+ οΏ½
358
+ V
359
+ ⟨u, f⟩ + ⟨Dv, M⟩ dV ,
360
+ (2.21)
361
+ such that variation with respect to the two vector fields u and v leads to
362
+ Ξ΄uI =
363
+ οΏ½
364
+ V
365
+ ⟨sym DΞ΄u, Ce sym(Du βˆ’ Dv)⟩ + ⟨skew DΞ΄u, Cc skew(Du βˆ’ Dv)⟩ βˆ’ ⟨δu, f⟩ dV = 0 ,
366
+ (2.22a)
367
+ Ξ΄vI =
368
+ οΏ½
369
+ V
370
+ ⟨sym DΞ΄v, Ce sym(Du βˆ’ Dv)⟩ + ⟨skew DΞ΄v, Cc skew(Du βˆ’ Dv)⟩
371
+ βˆ’ ⟨sym DΞ΄v, Cmicro sym Dv⟩ + ⟨DΞ΄v, M⟩ 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
+ ⟨sym(DΞ΄u βˆ’ DΞ΄v), Ce sym(Du βˆ’ Dv)⟩ + ⟨sym DΞ΄v, Cmicro sym Dv⟩
380
+ + ⟨skew(DΞ΄u βˆ’ DΞ΄v), Cc skew(Du βˆ’ Dv)⟩ dV .
381
+ (2.23)
382
+ By partial integration of Eq. (2.22a) and Eq. (2.22b) one finds the equilibrium equations
383
+ βˆ’ Div[Ce sym(Du βˆ’ Dv) + Cc skew(Du βˆ’ Dv)] = f
384
+ in
385
+ V ,
386
+ (2.24a)
387
+ βˆ’ Div[Ce sym(Du βˆ’ Dv) + Cc skew(Du βˆ’ 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 find
392
+ βˆ’ Div(Cmicro sym Dv) = f βˆ’ Div M .
393
+ (2.25)
394
+ Clearly, setting v = u satisfies 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 satisfied, asserting the equivalence of the
396
+ tangential projections of both fields on the boundary of the domain.
397
+ Since, as shown in [32, 52] using the
398
+ extended Brezzi theorem, the case Lc β†’ +∞ is well-posed (including Cc ≑ 0), the solution v = u is the unique
399
+ solution to the bilinear form Eq. (2.23) with the right-hand side
400
+ l({δu, δv}) = ⟨Dδv, M⟩ dV .
401
+ (2.26)
402
+ Effectively, equation Eq. (2.25) implies that the limit Lc β†’ +∞ defines a classical Cauchy continuum with a
403
+ finite stiffness governed by Cmicro, representing the upper limit of the stiffness 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†’ +∞ results in a constant
406
+ microdistortion field P : V β†’ R3Γ—3 as its full gradient DP is incorporated via Β΅macroL2
407
+ cβˆ₯DP βˆ₯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 field 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 βˆ’ 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 βˆ’ 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)|uβ€² βˆ’ p|2 + (Ξ»micro + 2Β΅micro)|p|2 ds βˆ’
484
+ οΏ½
485
+ s
486
+ u f + p m ds β†’ min
487
+ w.r.t.
488
+ {u, p} ,
489
+ since Du = uβ€² e1 βŠ— e1 and P = p e1 βŠ— e1, such that skew(Du βˆ’ 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 field 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 βˆ’ P ) = 1
511
+ 2
512
+ οΏ½
513
+ οΏ½
514
+ 0
515
+ 0
516
+ u,x βˆ’ p1
517
+ 0
518
+ 0
519
+ u,y βˆ’ p2
520
+ u,x βˆ’ p1
521
+ u,y βˆ’ p2
522
+ 0
523
+ οΏ½
524
+ οΏ½ ,
525
+ skew(Du βˆ’ P ) = 1
526
+ 2
527
+ οΏ½
528
+ οΏ½
529
+ 0
530
+ 0
531
+ p1 βˆ’ u,x
532
+ 0
533
+ 0
534
+ p2 βˆ’ u,y
535
+ u,x βˆ’ p1
536
+ u,y βˆ’ p2
537
+ 0
538
+ οΏ½
539
+ οΏ½ .
540
+ (2.30)
541
+ Clearly, there holds
542
+ tr[sym P ] = tr[sym(Du βˆ’ P )] = tr[skew(Du βˆ’ P )] = 0 ,
543
+ (2.31)
544
+ such that the contraction with the material tensors reduces to
545
+ Ce sym(Du βˆ’ P ) = 2Β΅e sym(Du βˆ’ P ) ,
546
+ Cmicro sym(Du βˆ’ P ) = 2Β΅micro sym P ,
547
+ Cc skew(Du βˆ’ P ) = 2Β΅c skew(Du βˆ’ P ) .
548
+ (2.32)
549
+ As such, the quadratic forms of the energy functional are given by
550
+ ⟨sym(Du βˆ’ P ), Ce sym(Du βˆ’ P )⟩ = Β΅eβˆ₯βˆ‡u βˆ’ pβˆ₯2 ,
551
+ (2.33a)
552
+ ⟨skew(Du βˆ’ P ), Cc skew(Du βˆ’ P )⟩ = Β΅cβˆ₯βˆ‡u βˆ’ pβˆ₯2 ,
553
+ (2.33b)
554
+ ⟨sym P , Cmicro sym P ⟩ = Β΅microβˆ₯pβˆ₯2 ,
555
+ (2.33c)
556
+ with the definitions
557
+ βˆ‡u =
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)βˆ₯βˆ‡u βˆ’ pβˆ₯2 + Β΅microβˆ₯pβˆ₯2 + Β΅macroL2
574
+ cβˆ₯curl2Dpβˆ₯2 dA βˆ’
575
+ οΏ½
576
+ A
577
+ u f + ⟨p, m⟩ 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΅eβˆ₯βˆ‡u βˆ’ pβˆ₯2 + Β΅microβˆ₯pβˆ₯2 + Β΅macroL2
586
+ cβˆ₯curl2Dpβˆ₯2 dA
587
+ βˆ’
588
+ οΏ½
589
+ A
590
+ u f + ⟨p, m⟩ dA β†’ min
591
+ w.r.t.
592
+ {u, p} .
593
+ (2.36)
594
+ Note that on two-dimensional domains the differential operators are reduced to
595
+ βˆ‡u =
596
+ οΏ½u,x
597
+ u,y
598
+ οΏ½
599
+ ,
600
+ Rβˆ‡u =
601
+ οΏ½ u,y
602
+ βˆ’u,x
603
+ οΏ½
604
+ ,
605
+ R =
606
+ οΏ½
607
+ 0
608
+ 1
609
+ βˆ’1
610
+ 0
611
+ οΏ½
612
+ ,
613
+ curl2Dp = div(Rp) = p2,x βˆ’ 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 field results in
617
+ Ξ΄uI =
618
+ οΏ½
619
+ A
620
+ Β΅eβŸ¨βˆ‡Ξ΄u, βˆ‡u βˆ’ p⟩ βˆ’ 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΅e⟨δp, βˆ‡u βˆ’ p⟩ βˆ’ Β΅micro⟨δp, p⟩ βˆ’ Β΅macroL2
627
+ c(curl2Dδp)curl2Dp + ⟨δp, m⟩ dA = 0 .
628
+ (2.39)
629
+ 7
630
+
631
+ Consequently, one finds the bilinear and linear forms
632
+ a({Ξ΄u, Ξ΄p}, {u, p}) =
633
+ οΏ½
634
+ A
635
+ Β΅eβŸ¨βˆ‡Ξ΄u βˆ’ Ξ΄p, βˆ‡u βˆ’ p⟩ + Β΅micro⟨δp, p⟩ + Β΅macroL2
636
+ c(curl2DΞ΄p)curl2Dp dA ,
637
+ (2.40a)
638
+ l({Ξ΄u, Ξ΄p}) =
639
+ οΏ½
640
+ A
641
+ δu f + ⟨δp, m⟩ dA .
642
+ (2.40b)
643
+ Partial integration of Eq. (2.38) results in
644
+ οΏ½
645
+ βˆ‚A
646
+ Ξ΄u ⟨¡e(βˆ‡u βˆ’ p), n⟩ ds βˆ’
647
+ οΏ½
648
+ A
649
+ Ξ΄u [Β΅e div(βˆ‡u βˆ’ p) + f] dA = 0 ,
650
+ (2.41)
651
+ and analogously for Eq. (2.39), yielding
652
+ οΏ½
653
+ A
654
+ ⟨δp, Β΅e(βˆ‡u βˆ’ p) βˆ’ Β΅micro p βˆ’ Β΅macroL2
655
+ cRβˆ‡curl2Dp + m⟩ dA βˆ’
656
+ οΏ½
657
+ βˆ‚A
658
+ ⟨δp, ¡macroL2
659
+ c(curl2Dp) t⟩ ds = 0 . (2.42)
660
+ Consequently, the strong form reads
661
+ βˆ’Β΅e div(βˆ‡u βˆ’ p) = f
662
+ in
663
+ A ,
664
+ (2.43a)
665
+ βˆ’Β΅e(βˆ‡u βˆ’ p) + Β΅micro p + Β΅macroL2
666
+ cRβˆ‡curl2Dp = m
667
+ in
668
+ A ,
669
+ (2.43b)
670
+ u = οΏ½u
671
+ on
672
+ su
673
+ D ,
674
+ (2.43c)
675
+ ⟨p, t⟩ = ⟨�p, t⟩
676
+ on
677
+ sP
678
+ D ,
679
+ (2.43d)
680
+ βŸ¨βˆ‡u, n⟩ = ⟨p, n⟩
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
+ ⟨p, t⟩ = βŸ¨βˆ‡οΏ½u, t⟩
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
+ βˆ’
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
+ βˆ†u = f ,
709
+ (2.45)
710
+ where the relation to the macro parameter ¡macro in Eq. (2.19) is lost. Further, the limit defined in Eq. (2.16)
711
+ with M = 0 yields the contradiction
712
+ sym P = (Ce + Cmicro)βˆ’1Ce 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
+ βˆ‡u ,
720
+ Β΅cp = Β΅cβˆ‡u ,
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†’ +∞
726
+ (Β΅micro 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†’ +∞, 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†’ +∞ ,
746
+ Β΅c > 0
747
+ Lc β†’ 0 ,
748
+ Β΅c οΏ½οΏ½οΏ½ 0
749
+ Β΅micro 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†’ +∞
764
+ Lc 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 briefly 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 βˆ’ ΞΎ) + ΞΎ)p =
785
+ p
786
+ οΏ½
787
+ i=0
788
+ οΏ½p
789
+ i
790
+ οΏ½
791
+ ΞΎi(1 βˆ’ ΞΎ)pβˆ’i =
792
+ p
793
+ οΏ½
794
+ i=0
795
+ p!
796
+ i!(p βˆ’ i)!ΞΎi(1 βˆ’ ΞΎ)pβˆ’i ,
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 ξ ∈ [0, 1]. The Bernstein polynomial reads
820
+ bp
821
+ i (ΞΎ) =
822
+ οΏ½p
823
+ i
824
+ οΏ½
825
+ ΞΎi(1 βˆ’ ΞΎ)pβˆ’i .
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 ≀ bp
836
+ i (ΞΎ) ≀ 1 ,
837
+ ξ ∈ [0, 1] .
838
+ (3.4)
839
+ A necessary condition for the use of Bernstein polynomials in finite 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 βŠ† 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 ≀ 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 ≀ i ≀ p) generates a constant
868
+ and by the exact sequence property the kernel of the differentiation operator is exactly the space of constants
869
+ ker(βˆ‚) = 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 efficiently using the recursive formula
874
+ bp
875
+ 0(ΞΎ) = (1 βˆ’ ΞΎ)p ,
876
+ bp
877
+ i+1(ΞΎ) =
878
+ (p βˆ’ i)ΞΎ
879
+ (p + 1)(1 βˆ’ ΞΎ)bp
880
+ i (ΞΎ) ,
881
+ i ∈ {0, 1, ..., p βˆ’ 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 limΞΎβ†’1 bp
886
+ i+1(ξ) = ∞. 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
+ βˆ€ 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 define define 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 differentiation [8, 37], not to be confused with numerical differentiation, since unlike in
900
+ numerical differentiation, automatic differentiation is no approximation and yields the exact derivative. The
901
+ latter represents an alternative method to finding 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
+ Definition 3.1 (Dual number)
905
+ The dual number is defined by
906
+ x + xβ€²Ξ΅ ,
907
+ Ξ΅ β‰ͺ 1 ,
908
+ (3.10)
909
+ where xβ€² is the derivative (only in automatic differentiation), Ξ΅ is an abstract number (infinitesimal) and formally
910
+ Ξ΅2 = 0.
911
+ The augmented algebra results automatically from the definition of the dual number.
912
+ Definition 3.2 (Augmented dual algebra)
913
+ The standard algebraic operations take the following form for dual numbers
914
+ 1. Addition and subtraction
915
+ (x + xβ€²Ξ΅) Β± (y + yβ€²Ξ΅) = x Β± y + (xβ€² Β± yβ€²)Ξ΅ .
916
+ (3.11)
917
+ 2. Multiplication
918
+ (x + xβ€²Ξ΅)(y + yβ€²Ξ΅) = xy + (xyβ€² + xβ€²y)Ξ΅ ,
919
+ (3.12)
920
+ since formally Ξ΅2 = 0.
921
+ 3. Division is achieved by first defining the inverse element
922
+ (x + xβ€²Ξ΅)(y + yβ€²Ξ΅) = 1
923
+ ⇐⇒
924
+ y = 1
925
+ x,
926
+ yβ€² = βˆ’ xβ€²
927
+ x2 ,
928
+ (3.13)
929
+ such that
930
+ (x + xβ€²Ξ΅)/(y + yβ€²Ξ΅) = x/y + (xβ€²/y βˆ’ xyβ€²/y2)Ξ΅ .
931
+ (3.14)
932
+ Application of the above definitions to polynomials
933
+ p(x + Ξ΅) =
934
+ ∞
935
+ οΏ½
936
+ i=0
937
+ ci(x + Ξ΅)i =
938
+ ∞
939
+ οΏ½
940
+ i=0
941
+ 1
942
+ οΏ½
943
+ j=0
944
+ ci
945
+ οΏ½i
946
+ j
947
+ οΏ½
948
+ xiβˆ’jΞ΅j =
949
+ ∞
950
+ οΏ½
951
+ i=0
952
+ cixi + Ξ΅
953
+ ∞
954
+ οΏ½
955
+ i=1
956
+ i cixiβˆ’1 = p(x) + pβ€²(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
+ Definition 3.3 (General dual numbers function)
981
+ A function of a dual number is defined in general by
982
+ f(x + Ξ΅) = f(x) + f β€²(x)Ξ΅ ,
983
+ (3.16)
984
+ being a fundamental formula for forward automatic differentiation.
985
+ The definition of dual numbers makes them directly applicable to the general rules of differentiation, 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) = βˆ‚xβ€²f(x) = d
990
+ dΞ΅f(x + xβ€²Ξ΅)
991
+ οΏ½οΏ½οΏ½οΏ½
992
+ Ξ΅=0
993
+ = lim
994
+ Ξ΅β†’0
995
+ f(x + xβ€²Ξ΅) βˆ’ 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΅) βˆ’ 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 βˆ’ ΞΎ βˆ’ Ξ·)x1 + Ξ· x2 + ΞΎ x3 ,
1008
+ x : Ξ“ β†’ Ae ,
1009
+ Ξ“ = {(ΞΎ, Ξ·) ∈ [0, 1]2 | ΞΎ + Ξ· ≀ 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 βˆ’ x1,
1015
+ x2 βˆ’ x1
1016
+ οΏ½
1017
+ ∈ R2Γ—2 .
1018
+ (4.2)
1019
+ 4.1
1020
+ The Bernstein-BΒ΄ezier basis for triangles
1021
+ The base functions on the triangle reference element are defined using the binomial expansion of the barycentric
1022
+ coordinates on the domain Ξ“
1023
+ 1 = (Ξ»1 + Ξ»2 + Ξ»3)p = ([1 βˆ’ ΞΎ βˆ’ 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 βˆ’ i
1033
+ j
1034
+ οΏ½
1035
+ Ξ»pβˆ’iβˆ’j
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: Duffy 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 βˆ’ i
1067
+ j
1068
+ οΏ½
1069
+ (1 βˆ’ ΞΎ βˆ’ Ξ·)pβˆ’iβˆ’jΞ·jΞΎi ,
1070
+ (4.5)
1071
+ of which some examples are depicted in Fig. 4.2. The Duffy 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 βˆ’ Ξ±)Ξ² ,
1079
+ Ξ² =
1080
+ Ξ·
1081
+ 1 βˆ’ ΞΎ ,
1082
+ (4.7)
1083
+ allows to view the triangle as a collapsed quadrilateral, see Fig. 4.3. Inserting the Duffy map into the definition
1084
+ of the BΒ΄ezier base function yields the split
1085
+ bp
1086
+ ij(ΞΎ, Ξ·) =
1087
+ οΏ½p
1088
+ i
1089
+ οΏ½ οΏ½p βˆ’ i
1090
+ j
1091
+ οΏ½
1092
+ (1 βˆ’ ΞΎ βˆ’ Ξ·)pβˆ’iβˆ’jΞ·jΞΎi
1093
+ =
1094
+ οΏ½p
1095
+ i
1096
+ οΏ½ οΏ½p βˆ’ i
1097
+ j
1098
+ οΏ½
1099
+ (1 βˆ’ Ξ± βˆ’ [1 βˆ’ Ξ±]Ξ²)pβˆ’iβˆ’j(1 βˆ’ Ξ±)jΞ²jΞ±i
1100
+ =
1101
+ οΏ½p
1102
+ i
1103
+ οΏ½ οΏ½p βˆ’ i
1104
+ j
1105
+ οΏ½
1106
+ (1 βˆ’ Ξ±)pβˆ’iβˆ’j(1 βˆ’ Ξ²)pβˆ’iβˆ’j(1 βˆ’ Ξ±)jΞ²jΞ±i
1107
+ (4.8)
1108
+ =
1109
+ οΏ½p
1110
+ i
1111
+ οΏ½
1112
+ (1 βˆ’ Ξ±)pβˆ’iΞ±i
1113
+ οΏ½p βˆ’ i
1114
+ j
1115
+ οΏ½
1116
+ (1 βˆ’ Ξ²)pβˆ’iβˆ’jΞ²j
1117
+ = bp
1118
+ i (Ξ±) bpβˆ’i
1119
+ j
1120
+ (Ξ²) .
1121
+ In other words, the Duffy 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
+ βˆ‡ΞΎbp
1145
+ ij = (DΞ±ΞΎ)βˆ’T βˆ‡Ξ±bp
1146
+ ij ,
1147
+ DΞ±ΞΎ =
1148
+ οΏ½ 1
1149
+ 0
1150
+ βˆ’Ξ²
1151
+ 1 βˆ’ Ξ±
1152
+ οΏ½
1153
+ ,
1154
+ (DΞ±ΞΎ)βˆ’T =
1155
+ 1
1156
+ 1 βˆ’ Ξ±
1157
+ οΏ½1 βˆ’ 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
+ βˆ‡Ξ±bp
1166
+ ij(Ξ±, Ξ²) =
1167
+ οΏ½
1168
+ οΏ½οΏ½οΏ½
1169
+ bpβˆ’i
1170
+ j
1171
+ d
1172
+ dΞ±bp
1173
+ i
1174
+ bp
1175
+ i
1176
+ d
1177
+ dΞ² bpβˆ’i
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 Duffy 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 βˆ’ 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 first, 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 first and second edge base
1197
+ functions, respectively. Base functions of the slanted edge are given by (i, p βˆ’ 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 Duffy transformation and relates to a specific 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, βˆ’e2} ,
1208
+ T12 = {e2, βˆ’e1} ,
1209
+ T13 = {e1, e2} ,
1210
+ T23 = {(1/2)(e1 βˆ’ 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
+ βŠ•
1222
+ οΏ½
1223
+ οΏ½
1224
+ οΏ½
1225
+ οΏ½
1226
+ j∈J
1227
+ Ep
1228
+ j (Ξ“)
1229
+ οΏ½
1230
+ οΏ½
1231
+ οΏ½ βŠ• 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 βŠ• 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 βŠ— Ti
1248
+ οΏ½
1249
+ βŠ•
1250
+ οΏ½
1251
+ οΏ½
1252
+ οΏ½
1253
+ οΏ½
1254
+ j∈J
1255
+ Ep
1256
+ j βŠ— Tj
1257
+ οΏ½
1258
+ οΏ½
1259
+ οΏ½ βŠ• {Cp
1260
+ 123 βŠ— T123} ,
1261
+ J = {(1, 2), (1, 3), (2, 3)} .
1262
+ (4.14)
1263
+ Using the B´ezier basis one finds the following base functions, which inherit the optimal complexity of the
1264
+ underlying basis.
1265
+ Definition 4.1 (B´ezier-N´ed´elec II triangle basis)
1266
+ The following base functions are defined 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·) = βˆ’bp
1288
+ p0e2 ,
1289
+ Ο‘(ΞΎ, Ξ·) = (1/2) bp
1290
+ i,pβˆ’i(e1 βˆ’ e2) ,
1291
+ 0 < i < p ,
1292
+ (4.15)
1293
+ where the first 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·) = βˆ’bp
1298
+ 0je1 ,
1299
+ 0 < j < p ,
1300
+ Ο‘(ΞΎ, Ξ·) = bp
1301
+ i0e2 ,
1302
+ 0 < i < p ,
1303
+ Ο‘(ΞΎ, Ξ·) = bp
1304
+ i,pβˆ’i(e1 + e2) ,
1305
+ 0 < i < p ,
1306
+ Ο‘(ΞΎ, Ξ·) = bp
1307
+ ije2 ,
1308
+ 0 < i < p ,
1309
+ 0 < j < p βˆ’ i ,
1310
+ Ο‘(ΞΎ, Ξ·) = bp
1311
+ ije1 ,
1312
+ 0 < i < p ,
1313
+ 0 < j < p βˆ’ i ,
1314
+ (4.16)
1315
+ 15
1316
+
1317
+ where the first three are the respective edge-cell base functions. The remaining two are pure cell base
1318
+ functions.
1319
+ 4.3
1320
+ N´ed´elec elements of the first type
1321
+ In order to construct the N´ed´elec element of the first 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 βŠ•
1327
+ οΏ½
1328
+ οΏ½
1329
+ οΏ½
1330
+ οΏ½
1331
+ j∈J
1332
+ βˆ‡Ep+1
1333
+ j
1334
+ οΏ½
1335
+ οΏ½
1336
+ οΏ½ βŠ• βˆ‡Cp+1
1337
+ 123 βŠ•
1338
+ οΏ½ 2
1339
+ οΏ½
1340
+ i=1
1341
+ Vp
1342
+ i βŠ— Ti
1343
+ οΏ½
1344
+ βŠ•
1345
+ οΏ½
1346
+ οΏ½
1347
+ οΏ½
1348
+ οΏ½
1349
+ j∈J
1350
+ Ep
1351
+ j βŠ— Tj
1352
+ οΏ½
1353
+ οΏ½
1354
+ οΏ½ βŠ• {Cp
1355
+ 123 βŠ— 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
+ Definition 4.2 (B´ezier-N´ed´elec I triangle basis)
1361
+ We define 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·) = Ο‘I
1365
+ 1 ,
1366
+ Ο‘(ΞΎ, Ξ·) = βˆ‡ΞΎbp+1
1367
+ 0j
1368
+ ,
1369
+ 0 < j < p + 1 ,
1370
+ e13 :
1371
+ Ο‘(ΞΎ, Ξ·) = Ο‘I
1372
+ 2 ,
1373
+ Ο‘(ΞΎ, Ξ·) = βˆ‡ΞΎbp+1
1374
+ i0
1375
+ ,
1376
+ 0 < i < p + 1 ,
1377
+ e23 :
1378
+ Ο‘(ΞΎ, Ξ·) = Ο‘I
1379
+ 3 ,
1380
+ Ο‘(ΞΎ, Ξ·) = βˆ‡ΞΎbp+1
1381
+ i,p+1βˆ’i ,
1382
+ 0 < i < p + 1 .
1383
+ (4.18)
1384
+ β€’ The cell functions read
1385
+ c123 :
1386
+ Ο‘(ΞΎ, Ξ·) = bp
1387
+ 00Ο‘I
1388
+ 3 ,
1389
+ Ο‘(ΞΎ, Ξ·) = bp
1390
+ 0pΟ‘I
1391
+ 2 ,
1392
+ Ο‘(ΞΎ, Ξ·) = bp
1393
+ 0j(Ο‘I
1394
+ 3 βˆ’ Ο‘I
1395
+ 2) ,
1396
+ 0 < j < p ,
1397
+ Ο‘(ΞΎ, Ξ·) = bp
1398
+ i0(Ο‘I
1399
+ 1 + Ο‘I
1400
+ 3) ,
1401
+ 0 < i < p ,
1402
+ Ο‘(ΞΎ, Ξ·) = bp
1403
+ i,pβˆ’i(Ο‘I
1404
+ 1 βˆ’ Ο‘I
1405
+ 2) ,
1406
+ 0 < i < p ,
1407
+ Ο‘(ΞΎ, Ξ·) = bp
1408
+ ij(Ο‘I
1409
+ 1 βˆ’ Ο‘I
1410
+ 2 + Ο‘I
1411
+ 3) ,
1412
+ 0 < i < p ,
1413
+ 0 < j < p βˆ’ i ,
1414
+ Ο‘(ΞΎ, Ξ·) = βˆ‡ΞΎbp+1
1415
+ ij
1416
+ ,
1417
+ 0 < i < p + 1 ,
1418
+ 0 < j < p + 1 βˆ’ i ,
1419
+ (4.19)
1420
+ where the last formula gives the cell gradients and the remaining base functions are non-gradients.
1421
+ The definition relies on the base functions of the lowest order N´ed´elec element of the first type [5,50]
1422
+ Ο‘I
1423
+ 1(ΞΎ, Ξ·) =
1424
+ οΏ½
1425
+ Ξ·
1426
+ 1 βˆ’ ΞΎ
1427
+ οΏ½
1428
+ ,
1429
+ Ο‘I
1430
+ 2(ΞΎ, Ξ·) =
1431
+ οΏ½1 βˆ’ Ξ·
1432
+ ΞΎ
1433
+ οΏ½
1434
+ ,
1435
+ Ο‘I
1436
+ 3(ΞΎ, Ξ·) =
1437
+ οΏ½ Ξ·
1438
+ βˆ’ΞΎ
1439
+ οΏ½
1440
+ .
1441
+ (4.20)
1442
+ 5
1443
+ Tetrahedral elements
1444
+ The tetrahedral elements are mapped from the reference tetrahedron Ω by the three-dimensional barycentric
1445
+ coordinates onto the physical domain Ve, see Fig. 5.1
1446
+ x(ΞΎ, Ξ·, ΞΆ) = (1 βˆ’ ΞΎ βˆ’ Ξ· βˆ’ ΞΆ)x1 + ΞΆ x2 + Ξ· x3 + ΞΎ x4 ,
1447
+ x : Ω β†’ Ve ,
1448
+ Ω = {(ΞΎ, Ξ·, ΞΆ) ∈ [0, 1]3 | ΞΎ + Ξ· + ΞΆ ≀ 1} .
1449
+ (5.1)
1450
+ 16
1451
+
1452
+ ΞΎ
1453
+ Ξ·
1454
+ ΞΆ
1455
+ Ω
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 : Ω 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 βˆ’ x1,
1477
+ x3 βˆ’ x1,
1478
+ x2 βˆ’ x1
1479
+ οΏ½
1480
+ ∈ R3Γ—3 .
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 Ω are defined using the barycen-
1485
+ tric coordinates by expanding the coefficients of
1486
+ (Ξ»1 + Ξ»2 + Ξ»3 + Ξ»4)p = ([1 βˆ’ ΞΎ βˆ’ Ξ· βˆ’ ΞΆ] + ΞΆ + Ξ· + ΞΎ)p = 1 ,
1487
+ (5.3)
1488
+ thus finding
1489
+ bp
1490
+ ijk(Ξ»1, Ξ»2, Ξ»3, Ξ»4) =
1491
+ οΏ½p
1492
+ i
1493
+ οΏ½ οΏ½p βˆ’ i
1494
+ j
1495
+ οΏ½ οΏ½p βˆ’ i βˆ’ j
1496
+ k
1497
+ οΏ½
1498
+ Ξ»pβˆ’iβˆ’jβˆ’k
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 βˆ’ i
1513
+ j
1514
+ οΏ½ οΏ½p βˆ’ i βˆ’ j
1515
+ k
1516
+ οΏ½
1517
+ (1 βˆ’ ΞΎ βˆ’ Ξ· βˆ’ ΞΆ)pβˆ’iβˆ’jβˆ’kΞΆkΞ·jΞΎi .
1518
+ (5.5)
1519
+ We construct the Duffy transformation by mapping the unit tetrahedron as a collapsed hexahedron
1520
+ ΞΎ : [0, 1]3 β†’ Ω ,
1521
+ {Ξ±, Ξ², Ξ³} οΏ½β†’ {ΞΎ, Ξ·, ΞΆ} ,
1522
+ (5.6)
1523
+ using the relations
1524
+ ΞΎ = Ξ± ,
1525
+ Ξ· = (1 βˆ’ Ξ±)Ξ² ,
1526
+ ΞΆ = (1 βˆ’ Ξ±)(1 βˆ’ Ξ²)Ξ³ ,
1527
+ Ξ± = ΞΎ ,
1528
+ Ξ² =
1529
+ Ξ·
1530
+ 1 βˆ’ ΞΎ ,
1531
+ Ξ³ =
1532
+ ΞΆ
1533
+ 1 βˆ’ ΞΎ βˆ’ Ξ· ,
1534
+ (5.7)
1535
+ as depicted in Fig. 5.2. Applying the Duffy transformation to BΒ΄ezier tetrahedra
1536
+ bp
1537
+ ijk(ΞΎ, Ξ·, ΞΆ) =
1538
+ οΏ½p
1539
+ i
1540
+ οΏ½ οΏ½p βˆ’ i
1541
+ j
1542
+ οΏ½ οΏ½p βˆ’ i βˆ’ j
1543
+ k
1544
+ οΏ½
1545
+ (1 βˆ’ ΞΎ βˆ’ Ξ· βˆ’ ΞΆ)pβˆ’iβˆ’jβˆ’kΞΆkΞ·jΞΎi
1546
+ =
1547
+ οΏ½
1548
+ p
1549
+ i
1550
+ οΏ½ οΏ½
1551
+ p βˆ’ i
1552
+ j
1553
+ οΏ½ οΏ½
1554
+ p βˆ’ i βˆ’ j
1555
+ k
1556
+ οΏ½
1557
+ (1 βˆ’ Ξ± βˆ’ (1 βˆ’ Ξ±)Ξ² βˆ’ (1 βˆ’ Ξ±)(1 βˆ’ Ξ²)Ξ³)pβˆ’iβˆ’jβˆ’k
1558
+ Β· (1 βˆ’ Ξ±)k(1 βˆ’ Ξ²)kΞ³k(1 βˆ’ Ξ±)jΞ²jΞ±i
1559
+ =
1560
+ οΏ½p
1561
+ i
1562
+ οΏ½ οΏ½p βˆ’ i
1563
+ j
1564
+ οΏ½ οΏ½p βˆ’ i βˆ’ j
1565
+ k
1566
+ οΏ½
1567
+ (1 βˆ’ Ξ±)pβˆ’iβˆ’jβˆ’k(1 βˆ’ Ξ²)pβˆ’iβˆ’jβˆ’k(1 βˆ’ Ξ³)pβˆ’iβˆ’jβˆ’k
1568
+ (5.8)
1569
+ Β· (1 βˆ’ Ξ±)k(1 βˆ’ Ξ²)kΞ³k(1 βˆ’ Ξ±)jΞ²jΞ±i
1570
+ =
1571
+ οΏ½p
1572
+ i
1573
+ οΏ½
1574
+ (1 βˆ’ Ξ±)pβˆ’iΞ±i
1575
+ οΏ½p βˆ’ i
1576
+ j
1577
+ οΏ½
1578
+ (1 βˆ’ Ξ²)pβˆ’iβˆ’jΞ²j
1579
+ οΏ½p βˆ’ i βˆ’ j
1580
+ k
1581
+ οΏ½
1582
+ (1 βˆ’ Ξ³)pβˆ’iβˆ’jβˆ’kΞ³k
1583
+ = bp
1584
+ i (Ξ±)bpβˆ’i
1585
+ j
1586
+ (Ξ²)bpβˆ’iβˆ’j
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
+ Ω
1604
+ (0,0,0)
1605
+ (1,0,0)
1606
+ (0,1,0)
1607
+ (0,0,1)
1608
+ ΞΎ : Ξ± β†’ Ω
1609
+ Figure 5.2: Duffy 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 bpβˆ’i
1612
+ j
1613
+ (Ξ²)bpβˆ’iβˆ’j
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, effectively spanning a
1618
+ tetrahedron. In order to compute gradients the chain rule is employed with respect to the Duffy transformation
1619
+ βˆ‡ΞΎbp
1620
+ ijk = (DΞ±ΞΎ)βˆ’T βˆ‡Ξ±bp
1621
+ ijk ,
1622
+ DΞ±ΞΎ =
1623
+ οΏ½
1624
+ οΏ½
1625
+ 1
1626
+ 0
1627
+ 0
1628
+ βˆ’Ξ²
1629
+ 1 βˆ’ Ξ±
1630
+ 0
1631
+ (Ξ² βˆ’ 1)Ξ³
1632
+ (Ξ± βˆ’ 1)Ξ³
1633
+ (1 βˆ’ Ξ±)(1 βˆ’ Ξ²)
1634
+ οΏ½
1635
+ οΏ½ ,
1636
+ (DΞ±ΞΎ)βˆ’T =
1637
+ 1
1638
+ (1 βˆ’ Ξ±)(1 βˆ’ Ξ²)
1639
+ οΏ½
1640
+ οΏ½
1641
+ (1 βˆ’ Ξ±)(1 βˆ’ Ξ²)
1642
+ (1 βˆ’ Ξ²)Ξ²
1643
+ Ξ³
1644
+ 0
1645
+ 1 βˆ’ 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
+ βˆ‡Ξ±bp
1655
+ ijk(Ξ±, Ξ², Ξ³) =
1656
+ οΏ½
1657
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
1658
+ bpβˆ’i
1659
+ j
1660
+ bpβˆ’iβˆ’j
1661
+ k
1662
+ d
1663
+ dΞ±bp
1664
+ i
1665
+ bp
1666
+ i bpβˆ’iβˆ’j
1667
+ k
1668
+ d
1669
+ dΞ² bpβˆ’i
1670
+ j
1671
+ bp
1672
+ i bpβˆ’i
1673
+ j
1674
+ d
1675
+ dΞ³ bpβˆ’iβˆ’j
1676
+ k
1677
+ οΏ½
1678
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
1679
+ .
1680
+ (5.10)
1681
+ The Duffy 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 definitions 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 first 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 βˆ’ j) with 0 < j < p,
1691
+ (i, 0, p βˆ’ i) with 0 < i < p and (i, p βˆ’ i, 0) with 0 < i < p, respectively;
1692
+ β€’ the base functions of the first face are given by (0, j, k) with 0 < j < p and 0 < k < p βˆ’ 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 βˆ’ 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 βˆ’ i. Lastly,
1695
+ the base functions of the slated face are given by (i, j, p βˆ’ i βˆ’ j) with 0 < i < p and 0 < j < p βˆ’ 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(Ω) =
1721
+ οΏ½ 4
1722
+ οΏ½
1723
+ i=1
1724
+ Vp
1725
+ i (Ω)
1726
+ οΏ½
1727
+ βŠ•
1728
+ οΏ½
1729
+ οΏ½
1730
+ οΏ½
1731
+ οΏ½
1732
+ j∈J
1733
+ Ep
1734
+ j (Ω)
1735
+ οΏ½
1736
+ οΏ½
1737
+ οΏ½ βŠ•
1738
+ οΏ½οΏ½
1739
+ k∈K
1740
+ Fp
1741
+ k(Ω)
1742
+ οΏ½
1743
+ βŠ• Cp
1744
+ 1234(Ω) ,
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, βˆ’e3, e1} ,
1757
+ T4 = {e1 + e2 + e3, βˆ’e3, βˆ’e2} ,
1758
+ T12 = {e3, βˆ’e2, βˆ’e1} ,
1759
+ T13 = {e2, e3, βˆ’e1} ,
1760
+ T14 = {e1, e3, e2} ,
1761
+ T23 = {e2, e1 + e2 + e3, βˆ’e1} ,
1762
+ T24 = {e1, e1 + e2 + e3, e2} ,
1763
+ T34 = {e1, e1 + e2 + e3, βˆ’e3} ,
1764
+ T123 = {e3, e2, βˆ’e1} ,
1765
+ T124 = {e3, e1, e2} ,
1766
+ T134 = {e2, e1, βˆ’e3} ,
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 βŠ— Ti
1778
+ οΏ½
1779
+ βŠ•
1780
+ οΏ½
1781
+ οΏ½
1782
+ οΏ½
1783
+ οΏ½
1784
+ j∈J
1785
+ Ep
1786
+ j βŠ— Tj
1787
+ οΏ½
1788
+ οΏ½
1789
+ οΏ½ βŠ•
1790
+ οΏ½οΏ½
1791
+ k∈K
1792
+ Fp
1793
+ k βŠ— Tk
1794
+ οΏ½
1795
+ βŠ• {Cp
1796
+ 1234 βŠ— 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 define the B´ezier-N´ed´elec element of the second type for arbitrary powers while inheriting optimal
1801
+ complexity.
1802
+ Definition 5.1 (B´ezier-N´ed´elec II tetrahedral basis)
1803
+ We define 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) = βˆ’bp
1833
+ 0p0e3 ,
1834
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1835
+ 0j,pβˆ’je2 ,
1836
+ 0 < j < p ,
1837
+ e24 :
1838
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1839
+ 00pe1 ,
1840
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1841
+ p00e3 ,
1842
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1843
+ i0,pβˆ’ie1 ,
1844
+ 0 < i < p ,
1845
+ e34 :
1846
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1847
+ 0p0e1 ,
1848
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1849
+ p00e2 ,
1850
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1851
+ i,pβˆ’i,0e1 ,
1852
+ 0 < i < p ,
1853
+ (5.14)
1854
+ where the first 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) = βˆ’bp
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,pβˆ’j(e1 + e2 + e3) ,
1867
+ 0 < j < p ,
1868
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1869
+ 0jke3 ,
1870
+ 0 < j < p ,
1871
+ 0 < k < p βˆ’ j ,
1872
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1873
+ 0jke2 ,
1874
+ 0 < j < p ,
1875
+ 0 < k < p βˆ’ j ,
1876
+ f124 :
1877
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
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,pβˆ’i(e1 + e2 + e3) ,
1885
+ 0 < i < p ,
1886
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1887
+ i0ke3 ,
1888
+ 0 < i < p ,
1889
+ 0 < k < p βˆ’ i ,
1890
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1891
+ i0ke1 ,
1892
+ 0 < i < p ,
1893
+ 0 < k < p βˆ’ i ,
1894
+ f134 :
1895
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
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,pβˆ’i,0(e1 + e2 + e3) ,
1903
+ 0 < i < p ,
1904
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1905
+ ij0e2 ,
1906
+ 0 < i < p ,
1907
+ 0 < j < p βˆ’ i ,
1908
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1909
+ ij0e1 ,
1910
+ 0 < i < p ,
1911
+ 0 < j < p βˆ’ i ,
1912
+ f234 :
1913
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1914
+ 0j,pβˆ’je1 ,
1915
+ 0 < j < p ,
1916
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1917
+ i0,pβˆ’ie2 ,
1918
+ 0 < i < p ,
1919
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1920
+ i,pβˆ’i,0e3 ,
1921
+ 0 < i < p ,
1922
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1923
+ ij,pβˆ’iβˆ’je2 ,
1924
+ 0 < i < p ,
1925
+ 0 < j < p βˆ’ i ,
1926
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1927
+ ij,pβˆ’iβˆ’je1 ,
1928
+ 0 < i < p ,
1929
+ 0 < j < p βˆ’ i ,
1930
+ (5.15)
1931
+ where the first three formulas for each face are the edge-face base functions;
1932
+ β€’ finally, the cell base functions read
1933
+ c1234 :
1934
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1935
+ 0jke1 ,
1936
+ 0 < j < p ,
1937
+ 0 < k < p βˆ’ j ,
1938
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1939
+ i0ke2 ,
1940
+ 0 < i < p ,
1941
+ 0 < k < p βˆ’ i ,
1942
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
1943
+ ij0e3 ,
1944
+ 0 < i < p ,
1945
+ 0 < j < p βˆ’ i ,
1946
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1947
+ ij,pβˆ’iβˆ’j(e1 + e2 + e3) ,
1948
+ 0 < i < p ,
1949
+ 0 < j < p βˆ’ i ,
1950
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1951
+ ijke3 ,
1952
+ 0 < i < p ,
1953
+ 0 < j < p βˆ’ i ,
1954
+ 0 < k < p βˆ’ i βˆ’ j ,
1955
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1956
+ ijke2 ,
1957
+ 0 < i < p ,
1958
+ 0 < j < p βˆ’ i ,
1959
+ 0 < k < p βˆ’ i βˆ’ j ,
1960
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
1961
+ ijke1 ,
1962
+ 0 < i < p ,
1963
+ 0 < j < p βˆ’ i ,
1964
+ 0 < k < p βˆ’ i βˆ’ j ,
1965
+ (5.16)
1966
+ where the first four formulas are the face-cell base functions.
1967
+ 5.3
1968
+ N´ed´elec elements of the first type
1969
+ In order to construct the N´ed´elec element of first type on tetrahedra we introduce the template sets
1970
+ T1 = {Ο‘I
1971
+ 4, Ο‘I
1972
+ 5, Ο‘I
1973
+ 6} ,
1974
+ T2 = {βˆ’Ο‘I
1975
+ 2, βˆ’Ο‘I
1976
+ 3, Ο‘I
1977
+ 6} ,
1978
+ T3 = {βˆ’Ο‘I
1979
+ 3, βˆ’Ο‘I
1980
+ 5} ,
1981
+ T12 = {Ο‘I
1982
+ 4 βˆ’ Ο‘I
1983
+ 2, Ο‘I
1984
+ 5 βˆ’ Ο‘I
1985
+ 3} ,
1986
+ T13 = {Ο‘I
1987
+ 1 + Ο‘I
1988
+ 4, Ο‘I
1989
+ 6 βˆ’ Ο‘I
1990
+ 3} ,
1991
+ T14 = {Ο‘I
1992
+ 1 + Ο‘I
1993
+ 5, Ο‘I
1994
+ 2 + Ο‘I
1995
+ 6} ,
1996
+ T23 = {Ο‘I
1997
+ 1 βˆ’ Ο‘I
1998
+ 2, Ο‘I
1999
+ 6 βˆ’ Ο‘I
2000
+ 5} ,
2001
+ T24 = {Ο‘I
2002
+ 1 βˆ’ Ο‘I
2003
+ 3, Ο‘I
2004
+ 4 + Ο‘I
2005
+ 6} ,
2006
+ T34 = {Ο‘I
2007
+ 2 βˆ’ Ο‘I
2008
+ 3, Ο‘I
2009
+ 4 βˆ’ Ο‘I
2010
+ 5} ,
2011
+ T123 = {Ο‘I
2012
+ 1 βˆ’ Ο‘I
2013
+ 2 + Ο‘I
2014
+ 4} ,
2015
+ T124 = {Ο‘I
2016
+ 1 βˆ’ Ο‘I
2017
+ 3 + Ο‘I
2018
+ 5} ,
2019
+ T134 = {Ο‘I
2020
+ 2 βˆ’ Ο‘I
2021
+ 3 + Ο‘I
2022
+ 6} ,
2023
+ T234 = {Ο‘I
2024
+ 4 βˆ’ Ο‘I
2025
+ 5 + Ο‘I
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
+ Ο‘1(ΞΎ, Ξ·, ΞΆ) =
2032
+ οΏ½
2033
+ οΏ½
2034
+ ΞΆ
2035
+ ΞΆ
2036
+ 1 βˆ’ ΞΎ βˆ’ Ξ·
2037
+ οΏ½
2038
+ οΏ½ ,
2039
+ Ο‘2(ΞΎ, Ξ·, ΞΆ) =
2040
+ οΏ½
2041
+ οΏ½
2042
+ Ξ·
2043
+ 1 βˆ’ ΞΎ βˆ’ ΞΆ
2044
+ Ξ·
2045
+ οΏ½
2046
+ οΏ½ ,
2047
+ Ο‘3(ΞΎ, Ξ·, ΞΆ) =
2048
+ οΏ½
2049
+ οΏ½
2050
+ 1 βˆ’ Ξ· βˆ’ ΞΆ
2051
+ ΞΎ
2052
+ ΞΎ
2053
+ οΏ½
2054
+ οΏ½ ,
2055
+ Ο‘4(ΞΎ, Ξ·, ΞΆ) =
2056
+ οΏ½
2057
+ οΏ½
2058
+ 0
2059
+ ΞΆ
2060
+ βˆ’Ξ·
2061
+ οΏ½
2062
+ οΏ½ ,
2063
+ Ο‘5(ΞΎ, Ξ·, ΞΆ) =
2064
+ οΏ½
2065
+ οΏ½
2066
+ ΞΆ
2067
+ 0
2068
+ βˆ’ΞΎ
2069
+ οΏ½
2070
+ οΏ½ ,
2071
+ Ο‘6(ΞΎ, Ξ·, ΞΆ) =
2072
+ οΏ½
2073
+ οΏ½
2074
+ Ξ·
2075
+ βˆ’ΞΎ
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
+ iβˆ’ejβˆ‡Ξ»j βˆ’
2085
+ ij
2086
+ p + 1βˆ‡ΞΎbp+1
2087
+ i
2088
+ | i ∈ 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 first 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
+ iβˆ’ejβˆ‡ΞΎΞ»j βˆ’
2099
+ ij
2100
+ p + 1βˆ‡ΞΎbp+1
2101
+ i
2102
+ οΏ½
2103
+ = curl([p + 1]bp
2104
+ iβˆ’ejβˆ‡ΞΎΞ»j) .
2105
+ (5.20)
2106
+ However, without the added gradient the function would not belong to [Pp]3 βŠ•ΞΎ Γ—[οΏ½P]3 and consequently, would
2107
+ not be part of the NΒ΄edΒ΄elec space. By limiting Rp to Rp
2108
+ βˆ— such that Rp
2109
+ βˆ— contains only the surface permutations
2110
+ with βˆ‡Ξ»j = ej and the cell permutations with j ∈ {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 βŠ•
2117
+ οΏ½οΏ½
2118
+ i∈I
2119
+ βˆ‡Ep+1
2120
+ i
2121
+ οΏ½
2122
+ βŠ•
2123
+ οΏ½
2124
+ οΏ½
2125
+ οΏ½
2126
+ οΏ½
2127
+ j∈J
2128
+ βˆ‡Fp+1
2129
+ j
2130
+ οΏ½
2131
+ οΏ½
2132
+ οΏ½ βŠ• βˆ‡Cp+1
2133
+ 1234 βŠ•
2134
+ οΏ½ 3
2135
+ οΏ½
2136
+ k=1
2137
+ Vp
2138
+ k βŠ— Tk
2139
+ οΏ½
2140
+ βŠ•
2141
+ οΏ½οΏ½
2142
+ i∈I
2143
+ Ep
2144
+ i βŠ— Ti
2145
+ οΏ½
2146
+ βŠ•
2147
+ οΏ½
2148
+ οΏ½
2149
+ οΏ½
2150
+ οΏ½
2151
+ j∈J
2152
+ Fp
2153
+ j βŠ— Tj
2154
+ οΏ½
2155
+ οΏ½
2156
+ οΏ½ βŠ• Rp+1
2157
+ βˆ—
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 B´ezier basis is used to construct the higher order N´ed´elec base functions of the first type.
2164
+ Definition 5.2 (B´ezier-N´ed´elec I tetrahedral basis)
2165
+ The base functions are defined 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) = Ο‘I
2170
+ 1 ,
2171
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2172
+ 00k ,
2173
+ 0 < k < p + 1 ,
2174
+ e13 :
2175
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = Ο‘I
2176
+ 2 ,
2177
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2178
+ 0j0 ,
2179
+ 0 < j < p + 1 ,
2180
+ e14 :
2181
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = Ο‘I
2182
+ 3 ,
2183
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2184
+ i00 ,
2185
+ 0 < i < p + 1 ,
2186
+ e23 :
2187
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = Ο‘I
2188
+ 4 ,
2189
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2190
+ 0j,p+1βˆ’j ,
2191
+ 0 < j < p + 1 ,
2192
+ e24 :
2193
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = Ο‘I
2194
+ 5 ,
2195
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2196
+ i0,p+1βˆ’i ,
2197
+ 0 < i < p + 1 ,
2198
+ e34 :
2199
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = Ο‘I
2200
+ 6 ,
2201
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+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
+ 000Ο‘I
2211
+ 4 ,
2212
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
2213
+ 00pΟ‘I
2214
+ 2 ,
2215
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2216
+ 00k(Ο‘I
2217
+ 4 βˆ’ Ο‘I
2218
+ 2) ,
2219
+ 0 < k < p ,
2220
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2221
+ 0j0(Ο‘I
2222
+ 1 + Ο‘I
2223
+ 4) ,
2224
+ 0 < j < p ,
2225
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2226
+ 0j,pβˆ’j(Ο‘I
2227
+ 1 βˆ’ Ο‘I
2228
+ 2) ,
2229
+ 0 < j < p ,
2230
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2231
+ 0jk(Ο‘I
2232
+ 1 βˆ’ Ο‘I
2233
+ 2 + Ο‘I
2234
+ 4) ,
2235
+ 0 < j < p ,
2236
+ 0 < k < p βˆ’ j ,
2237
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2238
+ 0jk ,
2239
+ 0 < j < p + 1 ,
2240
+ 0 < k < p + 1 βˆ’ j ,
2241
+ f124 :
2242
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2243
+ 000Ο‘I
2244
+ 5 ,
2245
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
2246
+ 00pΟ‘I
2247
+ 3 ,
2248
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2249
+ 00k(Ο‘I
2250
+ 5 βˆ’ Ο‘I
2251
+ 3) ,
2252
+ 0 < k < p ,
2253
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2254
+ i00(Ο‘I
2255
+ 1 + Ο‘I
2256
+ 5) ,
2257
+ 0 < i < p ,
2258
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2259
+ i0,pβˆ’i(Ο‘I
2260
+ 1 βˆ’ Ο‘I
2261
+ 3) ,
2262
+ 0 < i < p ,
2263
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2264
+ i0k(Ο‘I
2265
+ 1 βˆ’ Ο‘I
2266
+ 3 + Ο‘I
2267
+ 5) ,
2268
+ 0 < i < p ,
2269
+ 0 < k < p βˆ’ i ,
2270
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2271
+ i0k ,
2272
+ 0 < i < p + 1 ,
2273
+ 0 < k < p + 1 βˆ’ i ,
2274
+ f134 :
2275
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2276
+ 000Ο‘I
2277
+ 6 ,
2278
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
2279
+ 0p0Ο‘I
2280
+ 3 ,
2281
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2282
+ 0j0(Ο‘I
2283
+ 6 βˆ’ Ο‘I
2284
+ 3) ,
2285
+ 0 < j < p ,
2286
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2287
+ i00(Ο‘I
2288
+ 2 + Ο‘I
2289
+ 6) ,
2290
+ 0 < i < p ,
2291
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2292
+ i,pβˆ’i,0(Ο‘I
2293
+ 2 βˆ’ Ο‘I
2294
+ 3) ,
2295
+ 0 < i < p ,
2296
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2297
+ ij0(Ο‘I
2298
+ 2 βˆ’ Ο‘I
2299
+ 3 + Ο‘I
2300
+ 6) ,
2301
+ 0 < i < p ,
2302
+ 0 < j < p βˆ’ i ,
2303
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2304
+ ij0 ,
2305
+ 0 < i < p + 1 ,
2306
+ 0 < j < p + 1 βˆ’ i ,
2307
+ f234 :
2308
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2309
+ 00pΟ‘I
2310
+ 6 ,
2311
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ’bp
2312
+ 0p0Ο‘I
2313
+ 5 ,
2314
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2315
+ 0j,pβˆ’j(Ο‘I
2316
+ 6 βˆ’ Ο‘I
2317
+ 5) ,
2318
+ 0 < j < p ,
2319
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2320
+ i0,pβˆ’i(Ο‘I
2321
+ 4 + Ο‘I
2322
+ 6) ,
2323
+ 0 < i < p ,
2324
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2325
+ i,pβˆ’i,0(Ο‘I
2326
+ 4 βˆ’ Ο‘I
2327
+ 5) ,
2328
+ 0 < i < p ,
2329
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = bp
2330
+ ij,pβˆ’iβˆ’j(Ο‘I
2331
+ 4 βˆ’ Ο‘I
2332
+ 5 + Ο‘I
2333
+ 6) ,
2334
+ 0 < i < p ,
2335
+ 0 < j < p βˆ’ i ,
2336
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2337
+ ij,pβˆ’i.j ,
2338
+ 0 < i < p + 1 ,
2339
+ 0 < j < p + 1 βˆ’ i ;
2340
+ (5.23)
2341
+ β€’ the cell base functions read
2342
+ c1234 :
2343
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
2344
+ iβˆ’1,jke1 βˆ’
2345
+ i
2346
+ p + 2βˆ‡ΞΎbp+2
2347
+ ijk ,
2348
+ 0 < i < p + 2 ,
2349
+ 0 < j < p + 2 βˆ’ i ,
2350
+ 0 < k < p + 2 βˆ’ i βˆ’ j
2351
+ ,
2352
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
2353
+ i,jβˆ’1,ke2 βˆ’
2354
+ j
2355
+ p + 2βˆ‡ΞΎbp+2
2356
+ ijk ,
2357
+ 0 < i < p + 2 ,
2358
+ 0 < j < p + 2 βˆ’ i ,
2359
+ 0 < k < p + 2 βˆ’ i βˆ’ j
2360
+ ,
2361
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = (p + 2)bp+1
2362
+ ij0 e3 βˆ’
2363
+ 1
2364
+ p + 2βˆ‡ΞΎbp+2
2365
+ ij1 ,
2366
+ 0 < i < p + 2 ,
2367
+ 0 < j < p + 2 βˆ’ i ,
2368
+ Ο‘(ΞΎ, Ξ·, ΞΆ) = βˆ‡ΞΎbp+1
2369
+ ijk ,
2370
+ 0 < i < p + 1 ,
2371
+ 0 < j < p + 1 βˆ’ i ,
2372
+ 0 < k < p + 1 βˆ’ i βˆ’ 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 Duffy trans-
2381
+ formation (quad or hexahedron). As such, we employ a mixture of the efficient 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
+ Ω
2397
+ (f β—¦ (ΞΎ, Ξ·, ΞΆ))(Ξ±, Ξ², Ξ³) | det J| dΩ .
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 first solve the problem for vertices,
2409
+ then for edges, afterwards for faces, and finally 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 finite element mesh identifies each vertex with a tuple of coordinates. It suffices to evaluate the displacement
2414
+ field at the vertex
2415
+ ud
2416
+ i = οΏ½u
2417
+ οΏ½οΏ½οΏ½οΏ½
2418
+ xi
2419
+ .
2420
+ (7.1)
2421
+ If the field is vectorial, each component is evaluated at the designated vertex. The boundary conditions of the
2422
+ microdistortion field 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 define 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
+ βˆ‚qj
2431
+ βˆ‚s
2432
+ βˆ‚u
2433
+ βˆ‚s ds ,
2434
+ q ∈ Pp(s) ,
2435
+ (7.2)
2436
+ can be reformulated for a reference edge on a unit domain α ∈ [0, 1]. We parametrize the edge via
2437
+ x(Ξ±) = (1 βˆ’ 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 βˆ’ x1 ,
2442
+ ds = βˆ₯dxβˆ₯ = βˆ₯x2 βˆ’ x1βˆ₯dΞ± = βˆ₯tβˆ₯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 find
2466
+ du
2467
+ ds = du
2468
+ dΞ±
2469
+ dΞ±
2470
+ ds = βˆ₯tβˆ₯βˆ’1 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 find its derivative with respect to the arc-length parameter by observing
2476
+ d
2477
+ ds �u = ⟨ d
2478
+ dsx, βˆ‡xοΏ½u⟩ .
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 = βˆ₯tβˆ₯βˆ’1t .
2486
+ (7.7)
2487
+ Consequently, the edge boundary condition is given by
2488
+ οΏ½
2489
+ si
2490
+ βˆ‚qj
2491
+ βˆ‚s
2492
+ βˆ‚u
2493
+ βˆ‚s ds =
2494
+ οΏ½ 1
2495
+ 0
2496
+ οΏ½
2497
+ βˆ₯tβˆ₯βˆ’1 dqj
2498
+ dΞ±
2499
+ οΏ½ οΏ½
2500
+ βˆ₯tβˆ₯βˆ’1 du
2501
+ dΞ±
2502
+ οΏ½
2503
+ βˆ₯tβˆ₯ dΞ±
2504
+ =
2505
+ οΏ½ 1
2506
+ 0
2507
+ οΏ½
2508
+ βˆ₯tβˆ₯βˆ’1 dqj
2509
+ dΞ±
2510
+ οΏ½
2511
+ ⟨βˆ₯tβˆ₯βˆ’1t, βˆ‡xοΏ½u⟩βˆ₯tβˆ₯ dΞ± =
2512
+ οΏ½
2513
+ si
2514
+ βˆ‚qj
2515
+ βˆ‚s
2516
+ βˆ‚οΏ½u
2517
+ βˆ‚s ds
2518
+ βˆ€ qj ∈ Pp(Ξ±) ,
2519
+ (7.8)
2520
+ and can be solved by assembling the stiffness matrix of the edge and the load vector induced by the prescribed
2521
+ displacement field �u, representing volume forces
2522
+ kij =
2523
+ οΏ½ 1
2524
+ 0
2525
+ οΏ½
2526
+ βˆ₯tβˆ₯βˆ’1 dni
2527
+ dΞ±
2528
+ οΏ½ οΏ½
2529
+ βˆ₯tβˆ₯βˆ’1 dnj
2530
+ dΞ±
2531
+ οΏ½
2532
+ βˆ₯tβˆ₯ dΞ± ,
2533
+ fi =
2534
+ οΏ½ 1
2535
+ 0
2536
+ ⟨βˆ₯tβˆ₯βˆ’1t, βˆ‡xοΏ½u⟩
2537
+ οΏ½
2538
+ βˆ₯tβˆ₯βˆ’1 dni
2539
+ dΞ±
2540
+ οΏ½
2541
+ βˆ₯tβˆ₯ 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
+ qj⟨t, p⟩ ds =
2548
+ οΏ½
2549
+ si
2550
+ qj⟨t, βˆ‡xοΏ½u⟩ ds
2551
+ βˆ€ qj ∈ 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 ⟨t, θi⟩|s = ni(α). Therefore, the components of
2555
+ the corresponding stiffness matrix and load vectors read
2556
+ kij =
2557
+ οΏ½ 1
2558
+ 0
2559
+ ni njβˆ₯tβˆ₯ dΞ± ,
2560
+ fi =
2561
+ οΏ½ 1
2562
+ 0
2563
+ ni⟨t, βˆ‡xοΏ½u⟩βˆ₯tβˆ₯ 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 N´ed´elec element of the first type. The problem is given by
2572
+ οΏ½
2573
+ si
2574
+ qj⟨t, p⟩ ds =
2575
+ οΏ½
2576
+ si
2577
+ qj⟨t, βˆ‡xοΏ½u⟩ ds
2578
+ βˆ€ qj ∈ Pp(si) .
2579
+ (7.12)
2580
+ We define
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
+ ⟨t, ΞΈj⟩ = ⟨t, βˆ‡xnp+1
2588
+ j
2589
+ ⟩ = d
2590
+ dΞ±np+1
2591
+ j
2592
+ .
2593
+ (7.14)
2594
+ Therefore, the components of the stiffness 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
+ βˆ₯tβˆ₯ dΞ± ,
2605
+ fi =
2606
+ οΏ½ 1
2607
+ 0
2608
+ dnp+1
2609
+ i
2610
+ dΞ±
2611
+ ⟨t, βˆ‡xοΏ½u⟩βˆ₯tβˆ₯ 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
+ βŸ¨βˆ‡fqj, βˆ‡fu⟩ dA =
2619
+ οΏ½
2620
+ Ai
2621
+ βŸ¨βˆ‡fqj, βˆ‡f οΏ½u⟩ dA
2622
+ βˆ€ qj ∈ Pp(Ai) .
2623
+ (7.16)
2624
+ The surface is parameterized by the barycentric mapping from the unit triangle Ξ“ = {(ΞΎ, Ξ·) ∈ [0, 1]2 | ΞΎ +Ξ· ≀ 1}.
2625
+ The surface gradient is given by
2626
+ βˆ‡f οΏ½u = βˆ‡xοΏ½u βˆ’
2627
+ 1
2628
+ βˆ₯nβˆ₯2 βŸ¨βˆ‡xοΏ½u, n⟩n ,
2629
+ (7.17)
2630
+ where n is the surface normal. The surface gradient can also be expressed via
2631
+ βˆ‡fu = eiβˆ‚x
2632
+ i u = gΞ²βˆ‚ΞΎ
2633
+ Ξ²u ,
2634
+ β ∈ {1, 2} ,
2635
+ (7.18)
2636
+ where βˆ‚x
2637
+ Ξ² are partial derivates with respect to the physical coordinates, βˆ‚ΞΎ
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 = βˆ‚x
2642
+ βˆ‚ΞΎΞ² .
2643
+ (7.19)
2644
+ One can find the contravariant vector orthogonal to the surface by
2645
+ g3 = n = g1 Γ— g2 .
2646
+ (7.20)
2647
+ We define the mixed transformation matrix
2648
+ T =
2649
+ οΏ½
2650
+ g1 , g2 , g3οΏ½
2651
+ .
2652
+ (7.21)
2653
+ Due to the orthogonality relation ⟨gi, gj⟩ = δ j
2654
+ i the transposed inverse of T is clearly
2655
+ T βˆ’T =
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
+ βˆ‡fu =
2663
+ οΏ½
2664
+ g1 , g2οΏ½
2665
+ βˆ‡ΞΎu = T βˆ’T
2666
+ βˆ—
2667
+ βˆ‡ΞΎu ,
2668
+ T βˆ’T
2669
+ βˆ—
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 = βˆ₯nβˆ₯dΞ“ = βˆ₯g3βˆ₯dΞ“ =
2679
+ οΏ½
2680
+ ⟨g1 Γ— g2, g3⟩ dΞ“ =
2681
+ √
2682
+ det T dΞ“ .
2683
+ (7.24)
2684
+ Consequently, we can write the components of the stiffness matrix and load vector as
2685
+ kij =
2686
+ οΏ½
2687
+ Ξ“
2688
+ ⟨T βˆ’T
2689
+ βˆ—
2690
+ βˆ‡ΞΎni, T βˆ’T
2691
+ βˆ—
2692
+ βˆ‡ΞΎnj⟩
2693
+ √
2694
+ det T dΞ“ ,
2695
+ fi =
2696
+ οΏ½
2697
+ Ξ“
2698
+ ⟨T βˆ’T
2699
+ βˆ—
2700
+ βˆ‡ΞΎni, βˆ‡xοΏ½u βˆ’ (det T )βˆ’1βŸ¨βˆ‡xοΏ½u, n⟩n⟩
2701
+ √
2702
+ det T dΞ“ =
2703
+ οΏ½
2704
+ Ξ“
2705
+ ⟨T βˆ’T
2706
+ βˆ—
2707
+ βˆ‡ΞΎni, βˆ‡xοΏ½u⟩
2708
+ √
2709
+ det T dΞ“ ,
2710
+ (7.25)
2711
+ with the orthogonality ⟨gβ, n⟩ = 0 for β ∈ {1, 2}.
2712
+ In order to embed the consistent coupling boundary condition to the microdistortion we deviate from the
2713
+ degrees of freedom defined in [12] and apply the simpler H (divR)-projection
2714
+ ⟨qi, p, ⟩H(divR) = ⟨qi, βˆ‡f οΏ½u⟩H(divR)
2715
+ βˆ€ qi ∈ N p
2716
+ I (A)
2717
+ or
2718
+ βˆ€ qi ∈ N p
2719
+ II(A) .
2720
+ (7.26)
2721
+ Due to ker(curl) = βˆ‡H 1 the problem reduces to
2722
+ οΏ½
2723
+ Ai
2724
+ ⟨qj, p⟩ + ⟨curl2Dqj, curl2Dp⟩ dA =
2725
+ οΏ½
2726
+ Ai
2727
+ ⟨qj, βˆ‡f οΏ½u⟩ dA
2728
+ βˆ€ qj ∈ N p
2729
+ I (A)
2730
+ or
2731
+ βˆ€ qj ∈ 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 βˆ’T
2737
+ βˆ—
2738
+ Ο‘i ,
2739
+ divx R ΞΈi =
2740
+ 1
2741
+ √
2742
+ det T
2743
+ divΞΎ R Ο‘i .
2744
+ (7.28)
2745
+ Thus, the stiffness matrix components and load vector components read
2746
+ kij =
2747
+ οΏ½
2748
+ Ξ“
2749
+ ⟨T βˆ’T
2750
+ βˆ—
2751
+ Ο‘i, T βˆ’T
2752
+ βˆ—
2753
+ Ο‘j⟩ + ⟨(det T )βˆ’1/2 divΞΎ R Ο‘i, (det T )βˆ’1/2 divΞΎ R Ο‘j⟩
2754
+ √
2755
+ det T dΞ“ ,
2756
+ fi =
2757
+ οΏ½
2758
+ Ξ“
2759
+ ⟨T βˆ’T
2760
+ βˆ—
2761
+ Ο‘i, βˆ‡xοΏ½u βˆ’ (det T )βˆ’1βŸ¨βˆ‡xοΏ½u, n⟩n⟩
2762
+ √
2763
+ det T dΞ“ =
2764
+ οΏ½
2765
+ Ξ“
2766
+ οΏ½οΏ½T βˆ’T
2767
+ βˆ—
2768
+ Ο‘i, βˆ‡xοΏ½u⟩
2769
+ √
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 finite element formulations with an artificial analytical solution in the antiplane shear
2776
+ model and with an analytical solution for an infinite plane under cylindrical bending in the three dimensional
2777
+ model. Finally, we benchmark the ability of the finite element formulations to correctly interpolate between
2778
+ micro Cmicro and macro Cmacro stiffnesses 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
+ βˆ₯οΏ½u βˆ’ uhβˆ₯L2 =
2781
+ οΏ½οΏ½
2782
+ V
2783
+ βˆ₯οΏ½u βˆ’ uhβˆ₯2 dV ,
2784
+ βˆ₯ οΏ½P βˆ’ P hβˆ₯L2 =
2785
+ οΏ½οΏ½
2786
+ V
2787
+ βˆ₯ οΏ½P βˆ’ P hβˆ₯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 field, i.e. p is compatible.
2793
+ By defining the micro-moment with a scalar potential
2794
+ m = βˆ‡100 βˆ’ x2 βˆ’ y2
2795
+ 10
2796
+ = βˆ’1
2797
+ 5
2798
+ οΏ½x
2799
+ y
2800
+ οΏ½
2801
+ ,
2802
+ (8.2)
2803
+ 27
2804
+
2805
+ and constructing an analytical solution for the displacement field
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΅eβˆ‡οΏ½u) = 1
2817
+ 2
2818
+ οΏ½
2819
+ βˆ’1
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
+ βˆ’ 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 field, the microdistortion p is
2842
+ also reduced to a gradient field and curl2Dp = 0, see [53]. Note that this result is specific 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 field and as such, their tangential projections on an arbitrary
2845
+ boundary are not automatically the same. However, for both the gradient of the displacement field and the
2846
+ micro-moment is the tangential projection on the boundary of the circular domain A = {x ∈ R2 | βˆ₯xβˆ₯ ≀ 10}
2847
+ equal to zero
2848
+ βŸ¨βˆ‡t, οΏ½u⟩
2849
+ οΏ½οΏ½οΏ½οΏ½
2850
+ βˆ‚A
2851
+ = ⟨t, m⟩
2852
+ οΏ½οΏ½οΏ½οΏ½
2853
+ βˆ‚A
2854
+ = 0 ,
2855
+ (8.5)
2856
+ and as such the microdistortion belongs to p ∈ H0(curl, A).
2857
+ Consequently, we can set sD = βˆ‚A and the
2858
+ consistent coupling condition remains compatible.
2859
+ With the displacement and the microdistortion fields 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
+ βˆ’ 10 cos
2873
+ οΏ½x2 + y2
2874
+ 5
2875
+ οΏ½
2876
+ βˆ’ 5
2877
+ οΏ½
2878
+ .
2879
+ (8.6)
2880
+ The approximation of the displacement and microdistortion fields using linear and higher order elements is
2881
+ shown in Fig. 8.1. We note that even with almost 3000 finite elements and 6000 degrees of freedom the linear
2882
+ formulation is incapable of finding 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 finer
2886
+ mesh captures the geometry in a more precise manner. The effects 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 finite 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 field induces tangential projections on the imperfect boundary. The influence of the latter
2896
+ effect 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 finite element formulations to capture the intrinsic behaviour of the relaxed
2901
+ micromorphic model, we compare with analytical solutions of boundary-value problems. The first example
2902
+ considers the displacement and microdistortion fields under cylindrical bending [43] for infinitely extended
2903
+ plates. Let the plates be defined as V = (βˆ’βˆž, ∞)2 Γ— [βˆ’1/2, 1/2], than the analytical solution for cylindrical
2904
+ bending reads
2905
+ u = ΞΊ
2906
+ οΏ½
2907
+ οΏ½
2908
+ βˆ’xz
2909
+ 0
2910
+ x2/2
2911
+ οΏ½
2912
+ οΏ½ ,
2913
+ P = βˆ’ΞΊ
2914
+ οΏ½
2915
+ οΏ½
2916
+ [41z + 20
2917
+ √
2918
+ 82 sech(
2919
+ οΏ½
2920
+ 41/2) sinh(
2921
+ √
2922
+ 82z)]/1681
2923
+ 0
2924
+ x
2925
+ 0
2926
+ 0
2927
+ 0
2928
+ βˆ’x
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 field (a)-(c) and the microdistortion field (d)-(f) for the antiplane
2949
+ shear problem, for the linear element under h-refinement with 225, 763 and 2966 elements, corresponding to
2950
+ 485, 1591 and 6060 degrees of freedom. The p-refinement of the displacement field on the coarsest mesh of 57
2951
+ elements is visualized in (g)-(l) with p ∈ {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
+ 10βˆ’3
2960
+ 10βˆ’1
2961
+ 101
2962
+ degrees of freedom
2963
+ βˆ₯οΏ½u βˆ’ uhβˆ₯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
+ 10βˆ’3
2981
+ 10βˆ’1
2982
+ 101
2983
+ degrees of freedom
2984
+ βˆ₯οΏ½p βˆ’ phβˆ₯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-refinement 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-Poisson’s
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 finite plate of the infinite 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 define the finite domain V = [βˆ’10, 10]2 Γ— [βˆ’1/2, 1/2] and the boundaries
3017
+ AD1 = {βˆ’10} Γ— [βˆ’10, 10] Γ— [βˆ’1/2, 1/2] ,
3018
+ AD2 = {10} Γ— [βˆ’10, 10] Γ— [βˆ’1/2, 1/2] ,
3019
+ AN = βˆ‚V \ {AD1 βŠ• AD2} .
3020
+ (8.9)
3021
+ Additionally, on the Dirichlet boundary we impose the translated analytical solution οΏ½u = u βˆ’
3022
+ οΏ½0
3023
+ 0
3024
+ 3.5οΏ½T .
3025
+ The displacement field and the last row of the microdistortion are depicted in Fig. 8.3. The displacement
3026
+ field 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 finite
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
+ N´ed´elec element of the first type is nearly perfect, whereas its second type counterpart clearly deviates from
3035
+ the analytical solution at z β‰ˆ βˆ’0.25. 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 field. 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
+ N´ed´elec element of the first type.
3046
+ βˆ’0.5
3047
+ 0
3048
+ 0.5
3049
+ βˆ’1
3050
+ 0
3051
+ 1
3052
+ Β·10βˆ’2
3053
+ z-axis
3054
+ P11(z)
3055
+ ne = 5640
3056
+ ne = 44592
3057
+ ne = 354720
3058
+ (a)
3059
+ βˆ’0.5
3060
+ 0
3061
+ 0.5
3062
+ βˆ’1
3063
+ 0
3064
+ 1
3065
+ Β·10βˆ’2
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-refinement with 732, 5640 and 44592 elements
3080
+ (a) and of the higher order formulations under p-refinement 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 stiffness 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 influence
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 = [βˆ’1, 1]3 with a total Dirichlet boundary
3091
+ AD1 = {(x, y, z) ∈ [βˆ’1, 1]3 | x = Β±1} ,
3092
+ AD2 = {(x, y, z) ∈ [βˆ’1, 1]3 | y = Β±1} ,
3093
+ AD3 = {(x, y, z) ∈ [βˆ’1, 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 βˆ’ y2) sin(Ο€[1 βˆ’ 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 βˆ’ x2) sin(Ο€[1 βˆ’ 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 βˆ’ y2) sin(Ο€[1 βˆ’ 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 field 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 finite 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 first type is more accurate than the equivalent formulation with the N´ed´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 N´ed´elec elements of the first 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 finite elements. The example provided in Section 8.2 demonstrates that cubic and
3157
+ higher order finite elements yield excellent results in approximate solutions of the model.
3158
+ The polytopal template methodology introduced in [50] allows to easily and flexibly construct H (curl)-
3159
+ conforming vectorial finite 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 Duffy
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-finite element
3167
+ program.
3168
+ 32
3169
+
3170
+ (a)
3171
+ (b)
3172
+ (c)
3173
+ Figure 8.5: Displacement field of the Cauchy model on the coarsest mesh of 48 finite 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 fine-discretizations or higher order elements are required.
3179
+ The excellent performance of the proposed higher order finite 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 finite 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, Nβ—¦ 101001759.00
3189
+ Patrizio Neff acknowledges support in the framework of the DFG-Priority Programme 2256 β€œVariational
3190
+ Methods for Predicting Complex Phenomena in Engineering Structures and Materials”, Neff 902/10-1, Project-
3191
+ No. 440935806.
3192
+ 10
3193
+ References
3194
+ [1] Ainsworth, M., Andriamaro, G., Davydov, O.: Bernstein–BΒ΄ezier finite elements of arbitrary order and optimal assembly
3195
+ procedures. SIAM Journal on Scientific Computing 33(6), 3087–3109 (2011)
3196
+ [2] Ainsworth, M., Fu, G.: Bernstein–BΒ΄ezier bases for tetrahedral finite elements. Computer Methods in Applied Mechanics and
3197
+ Engineering 340, 178–201 (2018)
3198
+ [3] Aivaliotis, A., Tallarico, D., d’Agostino, M.V., Daouadji, A., Neff, P., Madeo, A.: Frequency- and angle-dependent scattering
3199
+ of a finite-sized meta-structure via the relaxed micromorphic model. Archive of Applied Mechanics 90(5), 1073–1096 (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, 252–263 (2015)
3204
+ [6] Barbagallo, G., Madeo, A., d’Agostino, M.V., Abreu, R., Ghiba, I.D., Neff, 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, 7–30 (2017)
3208
+ 33
3209
+
3210
+ 10βˆ’3
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
+ 10βˆ’3
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
+ 10βˆ’3
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
+ 10βˆ’3
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) finite 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., D’Agostino, M.V., Aivaliotis, A., Neff, P., Madeo, A.: Relaxed micromorphic model of transient
3283
+ wave propagation in anisotropic band-gap metastructures. International Journal of Solids and Structures 162, 148–163 (2019)
3284
+ [8] Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. Journal
3285
+ of Machine Learning Research 18, 1–43 (2018)
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+ [9] Bergot, M., Lacoste, P.: Generation of higher-order polynomial basis of NΒ΄edΒ΄elec H(curl) finite elements for Maxwell’s equa-
3287
+ tions. Journal of Computational and Applied Mathematics 234(6), 1937–1944 (2010). Eighth International Conference on
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+ Mathematical and Numerical Aspects of Waves (Waves 2007)
3289
+ [10] d’Agostino, M.V., Barbagallo, G., Ghiba, I.D., Eidel, B., Neff, P., Madeo, A.: Effective description of anisotropic wave
3290
+ dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. Journal of Elasticity 139(2), 299–329
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+ (2020)
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+ [11] d’Agostino, M.V., Rizzi, G., Khan, H., Lewintan, P., Madeo, A., Neff, P.: The consistent coupling boundary condition
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-finite element spaces. Computers and
3297
+ Mathematics with Applications 39(7), 29–38 (2000)
3298
+ [13] Demore, F., Rizzi, G., Collet, M., Neff, P., Madeo, A.: Unfolding engineering metamaterials design: Relaxed micromorphic
3299
+ modeling of large-scale acoustic meta-structures. Journal of the Mechanics and Physics of Solids 168, 104995 (2022)
3300
+ [14] Dunavant, D.A.: High degree efficient symmetrical Gaussian quadrature rules for the triangle.
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+ International Journal for
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+ Numerical Methods in Engineering 21(6), 1129–1148 (1985)
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+ [15] Eringen, A.: Microcontinuum Field Theories. I. Foundations and Solids. Springer-Verlag New York (1999)
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+ [16] Fike, J.A., Alonso, J.J.: Automatic differentiation through the use of hyper-dual numbers for second derivatives. In: S. Forth,
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+ P. Hovland, E. Phipps, J. Utke, A. Walther (eds.) Recent Advances in Algorithmic Differentiation, pp. 163–173. Springer
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+ Berlin Heidelberg, Berlin, Heidelberg (2012)
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+ [17] Forest, S.: Continuum thermomechanics of nonlinear micromorphic, strain and stress gradient media. Philosophical Transac-
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+ tions of the Royal Society A 378(20190169) (2020)
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+ [18] Ghiba, I.D., Neff, P., Madeo, A., Placidi, L., Rosi, G.: The relaxed linear micromorphic continuum: Existence, uniqueness
3310
+ and continuous dependence in dynamics. Mathematics and Mechanics of Solids 20(10), 1171–1197 (2015)
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+ [19] JaΒ΄skowiec, J., Sukumar, N.:
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+ High-order cubature rules for tetrahedra.
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+ International Journal for Numerical Methods in
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+ Engineering 121(11), 2418–2436 (2020)
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+ [20] Johnen, A., Remacle, J.F., Geuzaine, C.: Geometrical validity of curvilinear finite elements. Journal of Computational Physics
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+ Engineering with
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+ [22] Knees, D., Owczarek, S., Neff, P.: A local regularity result for the relaxed micromorphic model based on inner variations.
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+ [23] Lai, M.J., Schumaker, L.L.: Spline Functions on Triangulations. Cambridge University Press (2007)
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+ [24] Lewintan, P., M¨uller, S., Neff, P.: Korn inequalities for incompatible tensor fields in three space dimensions with conformally
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+ invariant dislocation energy. Calculus of Variations and Partial Differential Equations 60(4), 150 (2021)
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+ [25] Lewintan, P., Neff, P.: Lp-versions of generalized Korn inequalities for incompatible tensor fields in arbitrary dimensions with
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+ p-integrable exterior derivative. Comptes Rendus MathΒ΄ematique 359(6), 749–755 (2021)
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+ [26] Lewintan, P., Neff, P.: NeΛ‡cas–Lions lemma revisited: An Lp-version of the generalized Korn inequality for incompatible tensor
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+ fields. Mathematical Methods in the Applied Sciences 44(14), 11392–11403 (2021)
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+ [27] Madeo, A., Barbagallo, G., Collet, M., d’Agostino, M.V., Miniaci, M., Neff, P.: Relaxed micromorphic modeling of the interface
3330
+ between a homogeneous solid and a band-gap metamaterial: New perspectives towards metastructural design. Mathematics
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+ and Mechanics of Solids 23(12), 1485–1506 (2018)
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+ [28] Madeo, A., Neff, P., Ghiba, I.D., Rosi, G.: Reflection and transmission of elastic waves in non-local band-gap metamaterials:
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+ A comprehensive study via the relaxed micromorphic model. Journal of the Mechanics and Physics of Solids 95, 441–479
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+ (2016)
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+ [29] Mindlin, R.: Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis 16, 51–78 (1964)
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+ [31] NΒ΄edΒ΄elec, J.C.: A new family of mixed finite elements in R3. Numerische Mathematik 50(1), 57–81 (1986)
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+ micromorphic model through model-adapted first order homogenization. Journal of Elasticity 139(2), 269–298 (2020)
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+ [33] Neff, P., Forest, S.: A geometrically exact micromorphic model for elastic metallic foams accounting for affine microstructure.
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+ modelling, existence of minimizers, identification of moduli and computational results. Journal of Elasticity 87(2), 239–276
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+ (2007)
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+ and relations to the gauge theory of dislocations. The Quarterly Journal of Mechanics and Applied Mathematics 68(1), 53–84
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+ (2015)
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+ Continuum Mechanics and Thermodynamics 26(5), 639–681 (2014)
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+ [36] Neff, P., Pauly, D., Witsch, K.J.: Maxwell meets Korn: A new coercive inequality for tensor fields with square-integrable
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+ exterior derivative. Mathematical Methods in the Applied Sciences 35(1), 65–71 (2012)
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+ 545–563 (2010)
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+ Mathematical Methods in the Applied Sciences 44(18), 13855–13865 (2021)
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+ [39] Papanicolopulos, S.A.: Efficient computation of cubature rules with application to new asymmetric rules on the triangle. J.
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+ Comput. Appl. Math. 304, 73–83 (2016)
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+ [40] Perez-Ramirez, L.A., Rizzi, G., Madeo, A.: Multi-element metamaterial’s design through the relaxed micromorphic model.
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+ arXiv:2210.14697 (2022)
3360
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+ Exploring finite-size metastructures for elastic wave control.
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+ Mathematics and Mechanics of Solids p. 10812865211048923
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+ (2021)
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+ the relaxed micromorphic continuum and other generalized continua (including full derivations). Mathematics and Mechanics
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+ of Solids p. 10812865211023530 (2021)
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+ [43] Rizzi, G., H¨utter, G., Madeo, A., Neff, P.: Analytical solutions of the cylindrical bending problem for the relaxed micromorphic
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+ continuum and other generalized continua. Continuum Mechanics and Thermodynamics 33(4), 1505–1539 (2021)
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+ other generalized continua. Archive of Applied Mechanics 91(5), 2237–2254 (2021)
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+ micromorphic continuum and other generalized continua (including full derivations). Archive of Applied Mechanics (2021)
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+ approach. Philosophical Transactions of the Royal Society A 380(2231) (2022)
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+ [47] Sarhil, M., Scheunemann, L., Schr¨oder, J., Neff, P.: Size-effects of metamaterial beams subjected to pure bending: on boundary
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+ conditions and parameter identification in the relaxed micromorphic model. arXiv:2210.17117 (2022)
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+ High order finite element methods for electromagnetic field computation.
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3400
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+
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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 fields 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 fields.
15
+ –1–
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 fields,
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 fields
44
+ to feed the impact models are therefore crucial.
45
+ ESMs are integrated on spatial grids with finite 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 fields,
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 fields 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 fields to impact models (D´equ´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, confined to address
69
+ errors in the simulated frequency distributions locally, i.e., at every grid cell individually.
70
+ –2–
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 suffer from instabilities and overfitting, while differences 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 fields. Artificial 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 specifically 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 – unlike weather forecasts – 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 fields from a state-of-the-art comprehensive ESM, namely GFDL-
107
+ ESM4. In particular, in contrast to rather specific 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 different 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 fields 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 β€œrealisticness” 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 fields. We will show that our CycleGAN is indeed spatial context-aware
123
+ and strongly improves the characteristic intermittency in spatial precipitation patterns. We
124
+ –3–
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 different 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
+ –4–
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 differences 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 profile 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 specifically 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 differences 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 find 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 difference 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
+ –5–
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 fields, the characteristic
369
+ spatial intermittency needs to be captured accurately.
370
+ We compute the spatial power spectral density (PSD) of global precipitation fields,
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
+ –6–
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 differences between the shown lines, we show their mean absolute error w.r.t
440
+ the W5E5v2 ground truth in the legend. These values are different 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 fields. Adding the
446
+ physical constraint to the GAN does not affect the ability to produce realistic PSD distribu-
447
+ tions. After applying ISIMIP3BASD to the GAN-processed fields, 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 fields 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 fields over the South American continent see Fig. S2.
454
+ The daily spatial precipitation fields are first 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 fields 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 different 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 differences between paired samples
467
+ –7–
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 fields 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 fields (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 distrivutions’ 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 different ESMs and observational target datasets.
518
+ This is in contrast to existing bias-adjustment methods that are often tailored to specific
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
+ –8–
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 fields is com-
546
+ pared as averages for different 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 fields 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
+ Differences 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 specifically designed for this
565
+ - produces slightly lower biases for grid-cell-wise averages than the GAN; we show that
566
+ combining both methods by first 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 fields, we
569
+ compare the spectral density and fractal dimensions of the spatial precipitation fields. 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 fields 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 quantified.
575
+ –9–
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 find that this does
581
+ not affect 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 influence 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 efficient and accurate models. Moreover, GANs are comparably
592
+ difficult 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 fields 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 (Huffman 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 ˜x = log(x+Ο΅)βˆ’log(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 [βˆ’1, 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 first 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 (β€œfake”) or drawn from a real-world dataset (β€œreal”). Mathematically this can
624
+ be formalized as
625
+ Gβˆ— = min
626
+ G
627
+ max
628
+ D
629
+ LGAN(D, G),
630
+ (1)
631
+ –10–
632
+
633
+ with Gβˆ— being the optimal generator network. The loss function LGAN(D, G) can be defined
634
+ as
635
+ LGAN(D, G) = Ey∼py(y)[log(D(y))] + Ex∼px(x)[log(1 βˆ’ 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 define an additional cycle-consistency loss
641
+ that constraints the training of the networks, i.e.
642
+ Lcycle(G, F) = Ex∼px(x)[||F(G(x)) βˆ’ x||1]
643
+ (3)
644
+ + Ey∼py(y)[||G(F(y)) βˆ’ 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) = Ex∼px(x)[||G(x) βˆ’ x||1]
649
+ (4)
650
+ + Ey∼py(y)[||F(y) βˆ’ 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 = Ex∼px(x)[(DX(G(x)) βˆ’ 1)2]
656
+ + Ey∼py(y)[(DY (F(y)) βˆ’ 1)2]
657
+ (5)
658
+ + λLcycle(G, F) + ˜λLident(G, F),
659
+ where λ and ˜λ 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 = Ey∼py(y)[(DY (y) βˆ’ 1)2] + Ex∼px(x)[(DX(G(x)))2]
662
+ (6)
663
+ + Ex∼px(x)[(DX(x) βˆ’ 1)2] + Ey∼py(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 2eβˆ’4 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 reflection 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 final layer of the generator can be constrained to preserve the global sum of the
678
+ input, i.e. by rescaling
679
+ ˜yi = yi
680
+ οΏ½Ngrid
681
+ i
682
+ xi
683
+ οΏ½Ngrid
684
+ i
685
+ yi
686
+ ,
687
+ (8)
688
+ –11–
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 affected 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 fitted 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
+ fitted 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 define the grid cell-wise bias as the difference
715
+ in long-term averages as,
716
+ Bias(Λ†y, y) = 1
717
+ T
718
+ T
719
+ οΏ½
720
+ t=1
721
+ Λ†yt βˆ’ 1
722
+ T
723
+ T
724
+ οΏ½
725
+ t=1
726
+ yt,
727
+ (9)
728
+ where T is the number of time steps, Λ†yt and Λ†yt the modelled and observed precipitation
729
+ respectively at time step t.
730
+ 4.5 Evaluating spatial patterns
731
+ Quantifying how realistic spatial precipitation fields 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 fields 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
+ fields, using the power spectral density (PSD) computed from the spatial fields; in the latter
738
+ study, the PSD-based quantification was complemented by interviews with a large number
739
+ of meteorological experts. We propose the fractal dimension of binary precipitation fields
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 quantifies 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
+ –12–
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 Union’s 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 Union’s 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
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+ prediction using deep convolutional neural networks on a cubed sphere. Journal of
918
+ Advances in Modeling Earth Systems, 12(9), e2020MS002109.
919
+ Wfde5 over land merged with era5 over the ocean (w5e5 v2.0). (2021). ISIMIP Repository.
920
+ Retrieved from https://doi.org/10.48364/ISIMIP.342217
921
+ doi: 10.48364/ISIMIP
922
+ .342217
923
+ Wilcox, E. M., & Donner, L. J. (2007). The frequency of extreme rain events in satellite
924
+ rain-rate estimates and an atmospheric general circulation model. Journal of Climate,
925
+ 20(1), 53–69.
926
+ Zhu, J.-Y., Park, T., Isola, P., & Efros, A. A. (2017). Unpaired image-to-image translation
927
+ using cycle-consistent adversarial networks. In Proceedings of the IEEE international
928
+ conference on computer vision (pp. 2223–2232).
929
+ –15–
930
+
931
+ Supporting Information for ”Deep Learning for
932
+ bias-correcting comprehensive high-resolution Earth
933
+ system models”
934
+ Philipp Hess1,2, Stefan Lange2, and Niklas Boers1,2,3
935
+ 1Earth System Modelling, School of Engineering & Design, Technical University of Munich, Munich, Germany
936
+ 2Potsdam Institute for Climate Impact Research, Member of the Leibniz Association, Potsdam, Germany
937
+ 3Global Systems Institute and Department of Mathematics, University of Exeter, Exeter, UK
938
+ Contents of this file
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 -5γ€€0
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 fields 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
 
ENAyT4oBgHgl3EQfSPf9/content/tmp_files/2301.00085v1.pdf.txt ADDED
@@ -0,0 +1,1124 @@
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
1
+ arXiv:2301.00085v1 [math.CO] 31 Dec 2022
2
+ On the chromatic number of random regular
3
+ hypergraphs
4
+ Patrick Bennettβˆ—
5
+ Department of Mathematics,
6
+ Western Michigan University
7
+ Kalamazoo MI 49008
8
+ Alan Frieze†
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 fixed r, large fixed d, and
15
+ n 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 Erd˝os [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
+ βˆ—Research supported in part by Simons Foundation Grant #426894.
33
+ †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
+ οΏ½nβˆ’1
43
+ rβˆ’1
44
+ οΏ½
45
+ p sufficiently 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 fixed r and Ρ > 0 there exists d0 = d0(r, Ρ) such that for any fixed
58
+ d β‰₯ d0 we have that w.h.p.
59
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
60
+ Ο‡(Hr(n, d)) βˆ’
61
+ οΏ½
62
+ (rβˆ’1)d
63
+ r log d
64
+ οΏ½
65
+ 1
66
+ rβˆ’1
67
+ οΏ½
68
+ (rβˆ’1)d
69
+ r log d
70
+ οΏ½
71
+ 1
72
+ rβˆ’1
73
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
74
+ ≀ Ξ΅,
75
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
76
+ Ξ±(Hr(n, d)) βˆ’
77
+ οΏ½
78
+ r log d
79
+ (rβˆ’1)d
80
+ οΏ½
81
+ 1
82
+ rβˆ’1 n
83
+ οΏ½
84
+ r log d
85
+ (rβˆ’1)d
86
+ οΏ½
87
+ 1
88
+ rβˆ’1 n
89
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
90
+ ≀ 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 Chernoff’s bound (see, e.g., [11]).
98
+ Lemma 2 (Chernoff bound). Let X ∼ Bin(n, p). Then for all 0 < Ξ» < np
99
+ P(|X βˆ’ np| β‰₯ Ξ») ≀ 2 exp
100
+ οΏ½
101
+ βˆ’ Ξ»2
102
+ 3np
103
+ οΏ½
104
+ .
105
+ (2)
106
+ Lemma 3 (McDiarmid’s inequality). Let X = f(βƒ—Z) where βƒ—Z = (Z1, . . . Zt) and the Zi are
107
+ independent random variables. Assume the function f has the property that whenever βƒ—z, βƒ—w
108
+ differ in only one coordinate we have |f(βƒ—z) βˆ’ f(βƒ—w)| ≀ c. Then for all Ξ» > 0 we have
109
+ P(|X βˆ’ E[X]| β‰₯ Ξ») ≀ 2 exp
110
+ οΏ½
111
+ βˆ’ Ξ»2
112
+ 2c2t
113
+ οΏ½
114
+ .
115
+ (3)
116
+ Bal and the first 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 < rβˆ’1
120
+ r . Let z2 be the unique
121
+ positive number such that
122
+ z2
123
+ οΏ½
124
+ (z2 + 1)rβˆ’1 βˆ’ zrβˆ’1
125
+ 2
126
+ οΏ½
127
+ (z2 + 1)r βˆ’ zr
128
+ 2
129
+ = c
130
+ (4)
131
+ and let
132
+ z1 =
133
+ d
134
+ r [(z2 + 1)r βˆ’ 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 βˆ’ c)) βˆ’ h(c) βˆ’ h(1 βˆ’ c) βˆ’ h(d) βˆ’ d
143
+ r log z1 βˆ’ 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 fixed r and Ρ > 0 there exists d0 = d0(r, Ρ) such
148
+ that whenever D = D(p) :=
149
+ οΏ½nβˆ’1
150
+ rβˆ’1
151
+ οΏ½
152
+ p β‰₯ d0 we have that
153
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
154
+ Ο‡(Hr(n, p)) βˆ’
155
+ οΏ½
156
+ (rβˆ’1)D
157
+ r log D
158
+ οΏ½
159
+ 1
160
+ rβˆ’1
161
+ οΏ½
162
+ (rβˆ’1)D
163
+ r log D
164
+ οΏ½
165
+ 1
166
+ rβˆ’1
167
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
168
+ ≀ Ξ΅,
169
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
170
+ Ξ±(Hr(n, p)) βˆ’
171
+ οΏ½
172
+ r log D
173
+ (rβˆ’1)D
174
+ οΏ½
175
+ 1
176
+ rβˆ’1 n
177
+ οΏ½
178
+ r log D
179
+ (rβˆ’1)D
180
+ οΏ½
181
+ 1
182
+ rβˆ’1 n
183
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
184
+ ≀ 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 dβˆ’1 = O(1) but
196
+ not d = O(1). This is convenient for us because even though our theorem is for fixed d, it
197
+ requires d to be sufficiently 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
+ (rβˆ’1)d
207
+ οΏ½
208
+ 1
209
+ rβˆ’1. Let z2 be as defined in (4) and z1 be as defined in (5).
210
+ We see that
211
+ Lemma 6.
212
+ z2 =
213
+ c
214
+ 1 βˆ’ c + O (cr)
215
+ Proof. After some algebra, we re-write (4) as
216
+ z2 βˆ’
217
+ zr
218
+ 2
219
+ (1 + z2)rβˆ’1 =
220
+ c
221
+ 1 βˆ’ 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 βˆ’ c)) βˆ’ h(c) βˆ’ h(1 βˆ’ c) βˆ’ h(d) βˆ’ d
229
+ r log z1 βˆ’ dc log z2
230
+ =d
231
+ r log
232
+ οΏ½d
233
+ r
234
+ οΏ½
235
+ + dc log(dc) + d(1 βˆ’ c) log(d(1 βˆ’ c)) βˆ’ c log c βˆ’ (1 βˆ’ c) log(1 βˆ’ c)
236
+ βˆ’ d log d βˆ’ d
237
+ r log z1 βˆ’ dc log z2
238
+ =dc log
239
+ οΏ½
240
+ c
241
+ (1 βˆ’ c)z2
242
+ οΏ½
243
+ + d
244
+ r log [(z2 + 1)r βˆ’ zr
245
+ 2] + d log(1 βˆ’ c) βˆ’ c log c βˆ’ (1 βˆ’ c) log(1 βˆ’ c). (7)
246
+ Now note that the first term of (7) is
247
+ dc log
248
+ οΏ½
249
+ c
250
+ (1 βˆ’ c)z2
251
+ οΏ½
252
+ = dc log
253
+ οΏ½
254
+ c
255
+ (1 βˆ’ c)
256
+ οΏ½
257
+ c
258
+ 1βˆ’c + 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 βˆ’ zr
273
+ 2] = d
274
+ r log
275
+ οΏ½οΏ½
276
+ 1
277
+ 1 βˆ’ c + O
278
+ οΏ½
279
+ cr+1οΏ½οΏ½r
280
+ βˆ’
281
+ οΏ½
282
+ c
283
+ 1 βˆ’ c + O
284
+ οΏ½
285
+ cr+1οΏ½οΏ½rοΏ½
286
+ = d
287
+ r log
288
+ οΏ½οΏ½
289
+ 1
290
+ 1 βˆ’ c
291
+ οΏ½r οΏ½
292
+ 1 βˆ’ cr + O
293
+ οΏ½
294
+ cr+1οΏ½οΏ½οΏ½
295
+ = d
296
+ r log
297
+ οΏ½
298
+ 1
299
+ 1 βˆ’ c
300
+ οΏ½r
301
+ + d
302
+ r log
303
+ οΏ½
304
+ 1 βˆ’ cr + O
305
+ οΏ½
306
+ cr+1οΏ½οΏ½
307
+ = βˆ’d log(1 βˆ’ c) βˆ’ d
308
+ r cr + O
309
+ οΏ½
310
+ dcr+1οΏ½
311
+ .
312
+ 4
313
+
314
+ The last term of (7) is
315
+ (1 βˆ’ c) log(1 βˆ’ c) = O(c).
316
+ Therefore (7) becomes
317
+ βˆ’ d
318
+ r cr βˆ’ c log c + O
319
+ οΏ½
320
+ c + dcr+1οΏ½
321
+ = βˆ’ c
322
+ οΏ½d
323
+ rcrβˆ’1 + log c
324
+ οΏ½
325
+ + O
326
+ οΏ½
327
+ c + dcr+1οΏ½
328
+ = βˆ’ c
329
+ οΏ½
330
+ d
331
+ r (1 + Ξ΅)rβˆ’1 r log d
332
+ (r βˆ’ 1)d + log
333
+ οΏ½
334
+ (1 + Ξ΅)
335
+ οΏ½ r log d
336
+ (r βˆ’ 1)d
337
+ οΏ½
338
+ 1
339
+ rβˆ’1οΏ½οΏ½
340
+ + O
341
+ οΏ½
342
+ c + dcr+1οΏ½
343
+ = βˆ’ c
344
+ οΏ½
345
+ (1 + Ξ΅)rβˆ’1 log d
346
+ r βˆ’ 1 βˆ’ log d
347
+ r βˆ’ 1 + O(log log d)
348
+ οΏ½
349
+ + O
350
+ οΏ½
351
+ c + dcr+1οΏ½
352
+ = βˆ’ Ω (c log d) .
353
+ It follows from Theorem 4 that w.h.p.
354
+ Ξ±(Hr(n, d)) ≀ (1 + Ξ΅)
355
+ οΏ½ r log d
356
+ (r βˆ’ 1)d
357
+ οΏ½
358
+ 1
359
+ rβˆ’1
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 βˆ’ d1/2 log d
370
+ r
371
+ οΏ½
372
+ n.
373
+ (9)
374
+ Let Hβˆ—
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
+ Hβˆ—
379
+ r(n, m) as follows. We have n sets (β€œbuckets” ) V1, . . . Vn and a set of rm points P :=
380
+ {p1, . . . prm}. We put each point pi into a uniform random bucket Vφ(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 defines a hyper-edge {Ο†(j) : j ∈ Ri} for i = 1, 2, . . . , m. We denote the
384
+ hypergraph defined by R by HR.
385
+ Note that since r β‰₯ 3 the expected number of pairs of multi-edges in Hβˆ—
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(nβˆ’1).
402
+ 5
403
+
404
+ Thus, w.h.p. there are no multi-edges. Now the expected number of β€œloops” (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 βˆ’ log n ≀ M ≀ 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)) βˆ’
421
+ οΏ½
422
+ (rβˆ’1)d
423
+ r log d
424
+ οΏ½
425
+ 1
426
+ rβˆ’1
427
+ οΏ½
428
+ (rβˆ’1)d
429
+ r log d
430
+ οΏ½
431
+ 1
432
+ rβˆ’1
433
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
434
+ ≀ Ξ΅
435
+ 2,
436
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
437
+ Ξ±(Hr(n, M)) βˆ’
438
+ οΏ½
439
+ r log D
440
+ (rβˆ’1)D
441
+ οΏ½
442
+ 1
443
+ rβˆ’1 n
444
+ οΏ½
445
+ r log D
446
+ (rβˆ’1)D
447
+ οΏ½
448
+ 1
449
+ rβˆ’1 n
450
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
451
+ ≀ 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)) βˆ’
461
+ οΏ½
462
+ (rβˆ’1)D
463
+ r log D
464
+ οΏ½
465
+ 1
466
+ rβˆ’1
467
+ οΏ½
468
+ (rβˆ’1)D
469
+ r log D
470
+ οΏ½
471
+ 1
472
+ rβˆ’1
473
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
474
+ ≀ Ξ΅
475
+ 4
476
+ (10)
477
+ with probability at least 1 βˆ’ o(1/n). Note that here
478
+ D =
479
+ οΏ½n βˆ’ 1
480
+ r βˆ’ 1
481
+ οΏ½
482
+ p =
483
+ οΏ½n βˆ’ 1
484
+ r βˆ’ 1
485
+ οΏ½
486
+ m/
487
+ οΏ½n
488
+ r
489
+ οΏ½
490
+ = rm/n = d βˆ’ 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)) βˆ’
495
+ οΏ½
496
+ (rβˆ’1)d
497
+ r log d
498
+ οΏ½
499
+ 1
500
+ rβˆ’1
501
+ οΏ½
502
+ (rβˆ’1)d
503
+ r log d
504
+ οΏ½
505
+ 1
506
+ rβˆ’1
507
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
508
+ ≀ Ξ΅
509
+ 2
510
+ (11)
511
+ with probability at least 1βˆ’o(1/n). But now note that with probability Ω(nβˆ’1/2) the number
512
+ of edges in Hr(n, p) is precisely M. Thus we have that
513
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
514
+ Ο‡(Hr(n, M)) βˆ’
515
+ οΏ½
516
+ (rβˆ’1)d
517
+ r log d
518
+ οΏ½
519
+ 1
520
+ rβˆ’1
521
+ οΏ½
522
+ (rβˆ’1)d
523
+ r log d
524
+ οΏ½
525
+ 1
526
+ rβˆ’1
527
+ οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½
528
+ ≀ Ξ΅
529
+ 2
530
+ with probability at least 1 βˆ’ o(nβˆ’1/2). This proves the first 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 first removing some edges from vertices of degree larger than d,
536
+ and then adding some edges to vertices of degree less than d. We define the rank of a point
537
+ pi ∈ Vj, to be the number of points piβ€² ∈ Vj such that iβ€² ≀ i. We form a new set of points
538
+ P β€² βŠ† P and a partition Rβ€² of P β€² as follows. For any Rk ∈ 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 β€². Note also that Rβ€² is a uniform random partition of P β€². We
541
+ let HRβ€² be the natural hypergraph associated with Rβ€².
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 β€²β€² βŠ‡ P β€². We would also like a uniform partition Rβ€²β€² of
544
+ P β€²β€² into sets of size r, and we would like Rβ€²β€² to have many of the same parts as Rβ€². We will
545
+ accomplish this by constructing a sequence P β€²
546
+ 1 := P β€² βŠ† P β€²
547
+ 2 βŠ† . . . βŠ† P β€²
548
+ β„“ =: P β€²β€² of point sets
549
+ and a sequence Rβ€²
550
+ 1 := Rβ€², Rβ€²
551
+ 2, . . . , Rβ€²
552
+ β„“ =: Rβ€²β€² where Rβ€²
553
+ j is a uniform random partition of P β€²
554
+ j.
555
+ We construct P β€²
556
+ j+1, Rβ€²
557
+ j+1 from P β€²
558
+ j, Rβ€²
559
+ j as follows. Suppose |Rβ€²
560
+ j| = a (in other words Rβ€²
561
+ j has
562
+ a parts), so |P β€²
563
+ j| = ra. P β€²
564
+ j+1 will simply be P β€²
565
+ j plus r new points. Now we will choose a
566
+ random value K ∈ {1, . . . , r} using the distribution P[K = k] = qk(a), where qk(a) is defined
567
+ as follows.
568
+ Definition 1. Consider a random partition of ra+r points into a+1 parts of size r, and fix
569
+ some set Q of r points. Then for 1 ≀ k ≀ r, the number qk(a) is defined 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 βˆ’ 1 parts from Rβ€²
572
+ j, leaving Kr points in
573
+ P β€²
574
+ j+1 which are not in any remaining part of Rβ€²
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 Rβ€²
578
+ j+1.
579
+ We claim that Rβ€²
580
+ j+1 is a uniform random partition of P β€²
581
+ j+1 into parts of size r. Indeed, first
582
+ consider the r new points that are in P β€²
583
+ j+1 which were not in P β€²
584
+ j. The probability that a
585
+ uniform random partition of P β€²
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: first choose a random
588
+ value K with P[K = k] = qk; next we choose a uniform random set of (K βˆ’ 1)r points from
589
+ P β€²
590
+ j; next we choose a partition of the set of points consisting of P β€²
591
+ j+1 \ P β€²
592
+ j together with the
593
+ points from P β€²
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 β€²
596
+ j+1 \P β€²
597
+ j; finally,
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 β€œunused” (a βˆ’ K + 1)r points. At the
600
+ end of this process we have that HRβ€²β€² is distributed as Hr(n, d).
601
+ 7
602
+
603
+ 3.2.1
604
+ Bounding the number of low degree vertices in HRβ€²
605
+ We define 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 HRβ€². Let S0 be the buckets with at most dβˆ’3d1/2 log d
607
+ points of P β€², and let S1 be the buckets with at most dβˆ’2d1/2 log d points of P. Let S2 be the
608
+ set of buckets that, when we remove points from P β€² to get P, have at least d1/2 log d points
609
+ removed. Then S0 βŠ† S1 βˆͺ S2. Our goal is to bound the probability that S0 is too large.
610
+ Fix a bucket Vj and let X ∼ 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 satisfies
617
+ P[Vj ∈ S1] = P
618
+ οΏ½
619
+ X ≀ d βˆ’ 2d1/2 log d
620
+ οΏ½
621
+ = P
622
+ οΏ½
623
+ X βˆ’ rm
624
+ n ≀ βˆ’d1/2 log d
625
+ οΏ½
626
+ ≀ exp
627
+ οΏ½
628
+ βˆ’
629
+ d log2 d
630
+ 3(d βˆ’ d1/2 log d)
631
+ οΏ½
632
+ = exp
633
+ οΏ½
634
+ βˆ’β„¦
635
+ οΏ½
636
+ log2 d
637
+ οΏ½οΏ½
638
+ ,
639
+ where for our inequality we have used the Chernoff bound (Lemma 2). Therefore E[|S0|] ≀
640
+ exp
641
+ οΏ½
642
+ βˆ’β„¦
643
+ οΏ½
644
+ log2 d
645
+ οΏ½οΏ½
646
+ n. Now we argue that |S1| is concentrated using McDiarmid’s 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 βˆ’ E[X]| β‰₯ n2/3) ≀ 2 exp
652
+ οΏ½
653
+ βˆ’ n4/3
654
+ 2rm
655
+ οΏ½
656
+ = o(1).
657
+ (12)
658
+ Now we handle S2. For 1 ≀ j ≀ n let Yj be the number of parts Rk ∈ R such that Rk
659
+ contains a point in the bucket Vj as well as a point in some bucket Vjβ€² where |Vjβ€²| > d. Note
660
+ that if Vj ∈ 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 βˆ’ 1)P[j1 = j and |Vj2| > d]
664
+ = r
665
+ nP[|Vj1| > d
666
+ οΏ½οΏ½j1 = j] + r(r βˆ’ 1)
667
+ n
668
+ P[|Vj2| > d
669
+ οΏ½οΏ½j1 = j]
670
+ ≀ r2
671
+ n P[|Vj1| > d
672
+ οΏ½οΏ½j1 = j]
673
+ ≀ r2
674
+ n P[Bin(rm βˆ’ 1, 1/n) β‰₯ d] = r2
675
+ n exp
676
+ οΏ½
677
+ βˆ’β„¦
678
+ οΏ½
679
+ log2 d
680
+ οΏ½οΏ½
681
+ .
682
+ Thus we have
683
+ E[Yj] = m Β· r2
684
+ n exp
685
+ οΏ½
686
+ βˆ’β„¦
687
+ οΏ½
688
+ log2 d
689
+ οΏ½οΏ½
690
+ ≀ rd exp
691
+ οΏ½
692
+ βˆ’β„¦
693
+ οΏ½
694
+ log2 d
695
+ οΏ½οΏ½
696
+ = rd1/2 exp
697
+ οΏ½
698
+ βˆ’β„¦
699
+ οΏ½
700
+ log2 d
701
+ οΏ½οΏ½
702
+ and so Markov’s inequality gives us
703
+ P
704
+ οΏ½
705
+ Yj β‰₯ d1/2 log d
706
+ οΏ½
707
+ ≀ rd exp
708
+ οΏ½
709
+ βˆ’β„¦
710
+ οΏ½
711
+ log2 d
712
+ οΏ½οΏ½
713
+ d1/2 log d
714
+ = exp
715
+ οΏ½
716
+ βˆ’β„¦
717
+ οΏ½
718
+ log2 d
719
+ οΏ½οΏ½
720
+ 8
721
+
722
+ and so E[|S2|] = n exp
723
+ οΏ½
724
+ βˆ’β„¦
725
+ οΏ½
726
+ log2 d
727
+ οΏ½οΏ½
728
+ . We use McDiarmid’s 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
+ βˆ’β„¦
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 ≀ j ≀ r βˆ’ 1. Set
743
+ a :=
744
+ οΏ½
745
+ 1 + Ξ΅
746
+ 2
747
+ οΏ½ οΏ½ r log d
748
+ (r βˆ’ 1)d
749
+ οΏ½
750
+ 1
751
+ rβˆ’1
752
+ ,
753
+ ΞΊj := 10d
754
+ r
755
+ οΏ½r
756
+ j
757
+ οΏ½
758
+ aj,
759
+ p := d(r βˆ’ 1)!
760
+ nrβˆ’1
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 βˆ’ p)(s
771
+ r)
772
+ οΏ½οΏ½s
773
+ j
774
+ οΏ½οΏ½ n
775
+ rβˆ’j
776
+ οΏ½
777
+ ΞΊjn
778
+ οΏ½
779
+ pΞΊjn
780
+ ≀
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
+ βˆ’
792
+ οΏ½s
793
+ r
794
+ οΏ½
795
+ p + ΞΊjn log
796
+ 
797
+ ο£­e(an)j
798
+ j!
799
+ nrβˆ’j
800
+ (rβˆ’j)!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
+ βˆ’
818
+ οΏ½s
819
+ r
820
+ οΏ½
821
+ p + ΞΊjn log
822
+ οΏ½eaj
823
+ 10
824
+ οΏ½οΏ½
825
+ ≀ an Β· exp
826
+ οΏ½οΏ½
827
+ log
828
+ οΏ½e
829
+ a
830
+ οΏ½
831
+ βˆ’ 10d
832
+ r
833
+ οΏ½r
834
+ j
835
+ οΏ½
836
+ ajβˆ’1 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†’ ∞ we have
844
+ log
845
+ οΏ½e
846
+ a
847
+ οΏ½
848
+ ∼
849
+ 1
850
+ r βˆ’ 1 log d
851
+ and
852
+ 10d
853
+ r
854
+ οΏ½r
855
+ j
856
+ οΏ½
857
+ ajβˆ’1 log
858
+ οΏ½10
859
+ e
860
+ οΏ½
861
+ = Ω
862
+ οΏ½
863
+ d
864
+ rβˆ’j
865
+ jβˆ’1 logβˆ’ jβˆ’1
866
+ rβˆ’1 d
867
+ οΏ½
868
+ ≫ log d.
869
+ Thus with probability 1 βˆ’ o(1/n), Hr(n, p) has a coloring using (1 + Ξ΅/2)
870
+ οΏ½
871
+ (rβˆ’1)d
872
+ r log d
873
+ οΏ½
874
+ 1
875
+ rβˆ’1 colors
876
+ such that for each color class A and for each 1 ≀ j ≀ r βˆ’ 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 HRβ€² 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 β€œbad” 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
+ βˆ’β„¦
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 HRβ€² define 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) ≀ ΞΊjn. This follows from Section 3.2.2 and the fact that we have removed edges
902
+ from H(n, m) to obtain HRβ€². Let Ei be the event that at step i we have that for each color
903
+ class A and for each 1 ≀ j ≀ r βˆ’ 1 we have XA,j(i) ≀ 2ΞΊjn. Then, assuming Ei holds, the
904
+ probability that XA,j increases at step i is at most
905
+ οΏ½
906
+ 1≀k≀r, jβ„“β‰₯1
907
+ j1+Β·Β·Β·+jk=j
908
+ οΏ½
909
+ 1≀ℓ≀k
910
+ 2ΞΊjβ„“n
911
+ nd/r =
912
+ οΏ½
913
+ 1≀k≀r, jβ„“β‰₯1
914
+ j1+Β·Β·Β·+jk=j
915
+ οΏ½
916
+ 1≀ℓ≀k
917
+ 20
918
+ οΏ½ r
919
+ jk
920
+ οΏ½
921
+ ajk ≀
922
+ οΏ½
923
+ 1≀k≀r, jβ„“β‰₯1
924
+ j1+Β·Β·Β·+jk=j
925
+ 20r2r2aj ≀ 40r2r2aj.
926
+ Also, the largest possible increase in XA,j in one step is r. Thus, the final value of XA,j
927
+ after at most (4d1/2 log d)n steps is stochastically dominated by ΞΊjn + rY where Y
928
+ ∼
929
+ Bin
930
+ οΏ½
931
+ (4d1/2 log d)n, 40r2r2ajοΏ½
932
+ . An easy application of the Chernoff bound tells us
933
+ P (Y > 2E[Y ]) ≀ exp(βˆ’β„¦(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(dβˆ’1/2 log d) < 1
945
+ for sufficiently large d. Thus, using (13) and the union bound over all color classes A, we
946
+ have w.h.p. the final value of XA,j is at most ΞΊjn + 2E[Y ] ≀ 2ΞΊjn for all 1 ≀ j ≀ r βˆ’ 1.
947
+ Now we address β€œbad” 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
+ rβˆ’1οΏ½
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 ∼ r · Bin
958
+ οΏ½
959
+ (4d1/2 log d)n, O
960
+ οΏ½οΏ½log d
961
+ d
962
+ οΏ½
963
+ r
964
+ rβˆ’1οΏ½ οΏ½
965
+ .
966
+ Another easy
967
+ application of Chernoff shows that w.h.p. Z ≀ 2E[Z] = O(dβˆ’1/2n).
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
+ dβˆ’1/2n
972
+ οΏ½
973
+ . Let
974
+ Ξ΄ := Ξ΅
975
+ 2
976
+ οΏ½(r βˆ’ 1)d
977
+ r log d
978
+ οΏ½
979
+ 1
980
+ rβˆ’1
981
+ .
982
+ We claim that for every S βŠ† 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΄ ≀
989
+ οΏ½
990
+ 1 + Ξ΅
991
+ 2
992
+ οΏ½ οΏ½(r βˆ’ 1)d
993
+ r log d
994
+ οΏ½
995
+ 1
996
+ rβˆ’1
997
+ + Ξ΅
998
+ 2
999
+ οΏ½(r βˆ’ 1)d
1000
+ r log d
1001
+ οΏ½
1002
+ 1
1003
+ rβˆ’1
1004
+ = (1 + Ξ΅)
1005
+ οΏ½(r βˆ’ 1)d
1006
+ r log d
1007
+ οΏ½
1008
+ 1
1009
+ rβˆ’1
1010
+ colors. The expected number of sets S with more than Ξ΄s/r edges is at most
1011
+ οΏ½
1012
+ 1≀s≀u
1013
+ οΏ½n
1014
+ s
1015
+ οΏ½οΏ½οΏ½ds
1016
+ r
1017
+ οΏ½
1018
+ Ξ΄s/r
1019
+ οΏ½
1020
+ 1
1021
+ οΏ½dn
1022
+ r
1023
+ οΏ½οΏ½dnβˆ’r
1024
+ r
1025
+ οΏ½
1026
+ . . .
1027
+ οΏ½dnβˆ’Ξ΄s+r
1028
+ r
1029
+ οΏ½
1030
+ ≀
1031
+ οΏ½
1032
+ 1≀s≀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 βˆ’ Ξ΄s)Ξ΄s
1040
+ ≀
1041
+ οΏ½
1042
+ 1≀s≀u
1043
+ οΏ½
1044
+ ne
1045
+ s
1046
+ οΏ½
1047
+ dse
1048
+ (dn βˆ’ Ξ΄s)r
1049
+ οΏ½Ξ΄ οΏ½er Β· r!
1050
+ Ξ΄s
1051
+ οΏ½Ξ΄/rοΏ½s
1052
+ .
1053
+ (14)
1054
+ Now for 1 ≀ s ≀ √n the term in (14) is at most
1055
+ οΏ½
1056
+ O(n) Β·
1057
+ οΏ½
1058
+ O(nβˆ’1/2)
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 √n ≀ s ≀ 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(nβˆ’1/2)
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 definitions 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) 219–228.
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. Bollob´as and P. Erd˝os, 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 ADDED
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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 Bennettβˆ— Department of Mathematics, Western Michigan University Kalamazoo MI 49008 Alan Frieze† 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 fixed r, large fixed d, and n 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 Erd˝os [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'}
20
+ 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'}
21
+ page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
22
+ 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'}
23
+ page_content=' Frieze and οΏ½Luczak [12] introduced βˆ—Research supported in part by Simons Foundation Grant #426894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
24
+ page_content=' †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'}
25
+ 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'}
26
+ page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
27
+ page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
28
+ page_content=' Ο‡(Gn,r) was one of 3 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
29
+ 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'}
30
+ page_content=' For random hypergraphs, Krivelevich and Sudakov [14] established the asymptotic chromatic number for Ο‡(Hr(n, p) for οΏ½nβˆ’1 rβˆ’1 οΏ½ p sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
31
+ 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'}
32
+ 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'}
33
+ 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'}
34
+ 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'}
35
+ page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
36
+ page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
37
+ 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'}
38
+ 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'}
39
+ 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'}
40
+ page_content=' For all fixed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such that for any fixed 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)) βˆ’ οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, d)) βˆ’ οΏ½ r log d (rβˆ’1)d οΏ½ 1 rβˆ’1 n οΏ½ r log d (rβˆ’1)d οΏ½ 1 rβˆ’1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ 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 Chernoff’s 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 (Chernoff bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
49
+ page_content=' Let X ∼ Bin(n, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
50
+ page_content=' Then for all 0 < Ξ» < np P(|X βˆ’ np| β‰₯ Ξ») ≀ 2 exp οΏ½ βˆ’ Ξ»2 3np οΏ½ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
51
+ page_content=' (2) Lemma 3 (McDiarmid’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' Let X = f(βƒ—Z) where βƒ—Z = (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'}
54
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
55
+ 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 βƒ—z, βƒ—w differ in only one coordinate we have |f(βƒ—z) βˆ’ f(βƒ—w)| ≀ 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 βˆ’ E[X]| β‰₯ Ξ») ≀ 2 exp οΏ½ βˆ’ 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 first 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 < rβˆ’1 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)rβˆ’1 βˆ’ zrβˆ’1 2 οΏ½ (z2 + 1)r βˆ’ zr 2 = c (4) and let z1 = d r [(z2 + 1)r βˆ’ 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 βˆ’ c)) βˆ’ h(c) βˆ’ h(1 βˆ’ c) βˆ’ h(d) βˆ’ d r log z1 βˆ’ 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 fixed r and Ξ΅ > 0 there exists d0 = d0(r, Ξ΅) such that whenever D = D(p) := οΏ½nβˆ’1 rβˆ’1 οΏ½ p β‰₯ d0 we have that οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο‡(Hr(n, p)) βˆ’ οΏ½ (rβˆ’1)D r log D οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)D r log D οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, p)) βˆ’ οΏ½ r log D (rβˆ’1)D οΏ½ 1 rβˆ’1 n οΏ½ r log D (rβˆ’1)D οΏ½ 1 rβˆ’1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ with probability at least 1 βˆ’ 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 dβˆ’1 = 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 fixed d, it requires d to be sufficiently 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 (rβˆ’1)d οΏ½ 1 rβˆ’1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' Let z2 be as defined in (4) and z1 be as defined 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 βˆ’ 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 βˆ’ zr 2 (1 + z2)rβˆ’1 = c 1 βˆ’ 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 βˆ’ c)) βˆ’ h(c) βˆ’ h(1 βˆ’ c) βˆ’ h(d) βˆ’ d r log z1 βˆ’ dc log z2 =d r log οΏ½d r οΏ½ + dc log(dc) + d(1 βˆ’ c) log(d(1 βˆ’ c)) βˆ’ c log c βˆ’ (1 βˆ’ c) log(1 βˆ’ c) βˆ’ d log d βˆ’ d r log z1 βˆ’ dc log z2 =dc log οΏ½ c (1 βˆ’ c)z2 οΏ½ + d r log [(z2 + 1)r βˆ’ zr 2] + d log(1 βˆ’ c) βˆ’ c log c βˆ’ (1 βˆ’ c) log(1 βˆ’ 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 first term of (7) is dc log οΏ½ c (1 βˆ’ c)z2 οΏ½ = dc log οΏ½ c (1 βˆ’ c) οΏ½ c 1βˆ’c + 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 βˆ’ zr 2] = d r log οΏ½οΏ½ 1 1 βˆ’ c + O οΏ½ cr+1οΏ½οΏ½r βˆ’ οΏ½ c 1 βˆ’ c + O οΏ½ cr+1οΏ½οΏ½rοΏ½ = d r log οΏ½οΏ½ 1 1 βˆ’ c οΏ½r οΏ½ 1 βˆ’ cr + O οΏ½ cr+1οΏ½οΏ½οΏ½ = d r log οΏ½ 1 1 βˆ’ c οΏ½r + d r log οΏ½ 1 βˆ’ cr + O οΏ½ cr+1οΏ½οΏ½ = βˆ’d log(1 βˆ’ c) βˆ’ 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 βˆ’ c) log(1 βˆ’ 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 βˆ’ d r cr βˆ’ c log c + O οΏ½ c + dcr+1οΏ½οΏ½ = βˆ’ c οΏ½d rcrβˆ’1 + log c οΏ½ + O οΏ½ c + dcr+1οΏ½ = βˆ’ c οΏ½ d r (1 + Ξ΅)rβˆ’1 r log d (r βˆ’ 1)d + log οΏ½ (1 + Ξ΅) οΏ½ r log d (r βˆ’ 1)d οΏ½ 1 rβˆ’1οΏ½οΏ½ + O οΏ½ c + dcr+1οΏ½ = βˆ’ c οΏ½ (1 + Ξ΅)rβˆ’1 log d r βˆ’ 1 βˆ’ log d r βˆ’ 1 + O(log log d) οΏ½ + O οΏ½ c + dcr+1οΏ½ = βˆ’ Ω (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)) ≀ (1 + Ξ΅) οΏ½ r log d (r βˆ’ 1)d οΏ½ 1 rβˆ’1 .' 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 βˆ’ 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 Hβˆ— 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 Hβˆ— 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 (β€œbuckets” ) V1, .' 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'}
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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+ 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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+ 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 Vφ(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'}
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 defines a hyper-edge {Ο†(j) : j ∈ Ri} for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
124
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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'}
127
+ page_content=' We denote the hypergraph defined 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 Hβˆ— r(n, m) is at most οΏ½n r οΏ½οΏ½m 2 οΏ½ οΏ½ 1 οΏ½n r οΏ½ οΏ½2 = O οΏ½m2 nr οΏ½ = O(nβˆ’1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
129
+ page_content=' 5 Thus, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
130
+ 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'}
132
+ page_content=' there are no multi-edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
133
+ page_content=' Now the expected number of β€œloops” (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'}
134
+ page_content=' Thus w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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'}
137
+ page_content=' there are at most log n loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
138
+ page_content=' We now remove all multi-edges and loops, and say that M is the (random) number of edges remaining, where m βˆ’ log n ≀ M ≀ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
139
+ 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'}
140
+ 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'}
141
+ page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
142
+ page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
143
+ 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'}
145
+ page_content=' we have οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο‡(Hr(n, M)) βˆ’ οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ 2, οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ξ±(Hr(n, M)) βˆ’ οΏ½ r log D (rβˆ’1)D οΏ½ 1 rβˆ’1 n οΏ½ r log D (rβˆ’1)D οΏ½ 1 rβˆ’1 n οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
146
+ page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
147
+ 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'}
148
+ 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)) βˆ’ οΏ½ (rβˆ’1)D r log D οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)D r log D οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ 4 (10) with probability at least 1 βˆ’ o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
149
+ page_content=' Note that here D = οΏ½n βˆ’ 1 r βˆ’ 1 οΏ½ p = οΏ½n βˆ’ 1 r βˆ’ 1 οΏ½ m/ οΏ½n r οΏ½ = rm/n = d βˆ’ d1/2 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
150
+ 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)) βˆ’ οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ 2 (11) with probability at least 1βˆ’o(1/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
151
+ page_content=' But now note that with probability Ω(nβˆ’1/2) 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'}
152
+ page_content=' Thus we have that οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ Ο‡(Hr(n, M)) βˆ’ οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½ ≀ Ξ΅ 2 with probability at least 1 βˆ’ o(nβˆ’1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
153
+ page_content=' This proves the first inequality, and the second one follows similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
154
+ 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'}
155
+ page_content=' This transformation will involve first 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'}
156
+ page_content=' We define the rank of a point pi ∈ Vj, to be the number of points piβ€² ∈ Vj such that iβ€² ≀ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
157
+ page_content=' We form a new set of points P β€² βŠ† P and a partition Rβ€² of P β€² as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
158
+ page_content=' For any Rk ∈ 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'}
159
+ page_content=' Note that each bucket contains at most r points of P β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
160
+ page_content=' Note also that Rβ€² is a uniform random partition of P β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
161
+ page_content=' We let HRβ€² be the natural hypergraph associated with Rβ€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
162
+ 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 β€²β€² βŠ‡ P β€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
163
+ page_content=' We would also like a uniform partition Rβ€²β€² of P β€²β€² into sets of size r, and we would like Rβ€²β€² to have many of the same parts as Rβ€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
164
+ page_content=' We will accomplish this by constructing a sequence P β€² 1 := P β€² βŠ† P β€² 2 βŠ† .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
165
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
166
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
167
+ page_content=' βŠ† P β€² β„“ =: P β€²β€² of point sets and a sequence Rβ€² 1 := Rβ€², Rβ€² 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
168
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
169
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
170
+ page_content=' , Rβ€² β„“ =: Rβ€²β€² where Rβ€² j is a uniform random partition of P β€² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
171
+ page_content=' We construct P β€² j+1, Rβ€² j+1 from P β€² j, Rβ€² j as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
172
+ page_content=' Suppose |Rβ€² j| = a (in other words Rβ€² j has a parts), so |P β€² j| = ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
173
+ page_content=' P β€² j+1 will simply be P β€² j plus r new points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
174
+ page_content=' Now we will choose a random value K ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
175
+ page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
176
+ 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 defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
179
+ page_content=' Consider a random partition of ra+r points into a+1 parts of size r, and fix some set Q of r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
180
+ page_content=' Then for 1 ≀ k ≀ r, the number qk(a) is defined 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'}
181
+ page_content=' We will then remove a uniform random set of K βˆ’ 1 parts from Rβ€² j, leaving Kr points in P β€² j+1 which are not in any remaining part of Rβ€² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
182
+ 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 Rβ€² j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
183
+ page_content=' We claim that Rβ€² j+1 is a uniform random partition of P β€² 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, first consider the r new points that are in P β€² j+1 which were not in P β€² j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
185
+ page_content=' The probability that a uniform random partition of P β€² 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'}
186
+ page_content=' So we can generate such a random partition as follows: first choose a random value K with P[K = k] = qk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
187
+ page_content=' next we choose a uniform random set of (K βˆ’ 1)r points from P β€² j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
188
+ page_content=' next we choose a partition of the set of points consisting of P β€² j+1 \\ P β€² j together with the points from P β€² 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 β€² j+1 \\P β€² j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
189
+ page_content=' finally, 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'}
190
+ page_content=' In our case this partition of the rest of the points comprises the current partition of the β€œunused” (a βˆ’ K + 1)r points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
191
+ page_content=' At the end of this process we have that HRβ€²β€² 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 HRβ€² We define some sets of buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
195
+ page_content=' We show that w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
196
+ page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
197
+ page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
198
+ page_content=' there are few small buckets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
199
+ page_content='e few vertices of low degree in the hypregraph HRβ€².' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
200
+ page_content=' Let S0 be the buckets with at most dβˆ’3d1/2 log d points of P β€², and let S1 be the buckets with at most dβˆ’2d1/2 log d points of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
201
+ page_content=' Let S2 be the set of buckets that, when we remove points from P β€² 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'}
202
+ page_content=' Then S0 βŠ† S1 βˆͺ S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
203
+ 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'}
204
+ page_content=' Fix a bucket Vj and let X ∼ Bin � rm, 1 n � be the number of points of P in Vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
205
+ page_content=' Then the probability that Vj is in S1 satisfies P[Vj ∈ S1] = P οΏ½ X ≀ d βˆ’ 2d1/2 log d οΏ½ = P οΏ½ X βˆ’ rm n ≀ βˆ’d1/2 log d οΏ½ ≀ exp οΏ½ βˆ’ d log2 d 3(d βˆ’ d1/2 log d) οΏ½ = exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ , where for our inequality we have used the Chernoff bound (Lemma 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
206
+ page_content=' Therefore E[|S0|] ≀ exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
207
+ page_content=' Now we argue that |S1| is concentrated using McDiarmid’s inequality (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
208
+ 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 βˆ’ E[X]| β‰₯ n2/3) ≀ 2 exp οΏ½ βˆ’ 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 ≀ j ≀ n let Yj be the number of parts Rk ∈ R such that Rk contains a point in the bucket Vj as well as a point in some bucket Vjβ€² where |Vjβ€²| > 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 ∈ 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 βˆ’ 1)P[j1 = j and |Vj2| > d] = r nP[|Vj1| > d οΏ½οΏ½j1 = j] + r(r βˆ’ 1) n P[|Vj2| > d οΏ½οΏ½j1 = j] ≀ r2 n P[|Vj1| > d οΏ½οΏ½j1 = j] ≀ r2 n P[Bin(rm βˆ’ 1, 1/n) β‰₯ d] = r2 n exp οΏ½ βˆ’β„¦ 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 βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ ≀ rd exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ = rd1/2 exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ and so Markov’s inequality gives us P οΏ½ Yj β‰₯ d1/2 log d οΏ½ ≀ rd exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ d1/2 log d = exp οΏ½ βˆ’β„¦ οΏ½ log2 d οΏ½οΏ½ 8 and so E[|S2|] = n exp οΏ½ βˆ’β„¦ 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 McDiarmid’s 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 βˆ’β„¦ 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 ≀ j ≀ r βˆ’ 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 βˆ’ 1)d οΏ½ 1 rβˆ’1 , ΞΊj := 10d r οΏ½r j οΏ½ aj, p := d(r βˆ’ 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' nrβˆ’1 .' 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 βˆ’ p)(s r) οΏ½οΏ½s j οΏ½οΏ½ n rβˆ’j οΏ½ ΞΊjn οΏ½ pΞΊjn ≀ an οΏ½ s=1 exp \uf8f1 \uf8f2 \uf8f3s log οΏ½en s οΏ½ βˆ’ 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=' nrβˆ’j (rβˆ’j)!' 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 βˆ’ οΏ½s r οΏ½ p + ΞΊjn log οΏ½eaj 10 οΏ½οΏ½ ≀ an Β· exp οΏ½οΏ½ log οΏ½e a οΏ½ βˆ’ 10d r οΏ½r j οΏ½ ajβˆ’1 log οΏ½10 e οΏ½οΏ½ an οΏ½ = o(1/n) where the last line follows since as d β†’ ∞ we have log οΏ½e a οΏ½ ∼ 1 r βˆ’ 1 log d and 10d r οΏ½r j οΏ½ ajβˆ’1 log οΏ½10 e οΏ½ = Ω οΏ½ d rβˆ’j jβˆ’1 logβˆ’ jβˆ’1 rβˆ’1 d οΏ½ ≫ 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 βˆ’ o(1/n), Hr(n, p) has a coloring using (1 + Ξ΅/2) οΏ½ (rβˆ’1)d r log d οΏ½ 1 rβˆ’1 colors such that for each color class A and for each 1 ≀ j ≀ r βˆ’ 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 HRβ€² 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 β€œbad” 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 βˆ’β„¦ 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 HRβ€² define 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) ≀ 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 HRβ€².' 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 ≀ j ≀ r βˆ’ 1 we have XA,j(i) ≀ 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 1≀k≀r, jβ„“β‰₯1 j1+Β·Β·Β·+jk=j οΏ½ 1≀ℓ≀k 2ΞΊjβ„“n nd/r = οΏ½ 1≀k≀r, jβ„“β‰₯1 j1+Β·Β·Β·+jk=j οΏ½ 1≀ℓ≀k 20 οΏ½ r jk οΏ½ ajk ≀ οΏ½ 1≀k≀r, jβ„“β‰₯1 j1+Β·Β·Β·+jk=j 20r2r2aj ≀ 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 final value of XA,j after at most (4d1/2 log d)n steps is stochastically dominated by κjn + rY where Y ∼ Bin � (4d1/2 log d)n, 40r2r2aj� .' 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 Chernoff bound tells us P (Y > 2E[Y ]) ≀ exp(βˆ’β„¦(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(dβˆ’1/2 log d) < 1 for sufficiently 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 final value of XA,j is at most ΞΊjn + 2E[Y ] ≀ 2ΞΊjn for all 1 ≀ j ≀ r βˆ’ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' Now we address β€œbad” 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 rβˆ’1οΏ½ (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 ∼ r Β· Bin οΏ½ (4d1/2 log d)n, O οΏ½οΏ½log d d οΏ½ r rβˆ’1οΏ½ 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 Chernoff 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 ≀ 2E[Z] = O(dβˆ’1/2n).' 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 dβˆ’1/2n 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 βˆ’ 1)d r log d οΏ½ 1 rβˆ’1 .' 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 βŠ† 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΄ ≀ οΏ½ 1 + Ξ΅ 2 οΏ½ οΏ½(r βˆ’ 1)d r log d οΏ½ 1 rβˆ’1 + Ξ΅ 2 οΏ½(r βˆ’ 1)d r log d οΏ½ 1 rβˆ’1 = (1 + Ξ΅) οΏ½(r βˆ’ 1)d r log d οΏ½ 1 rβˆ’1 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 1≀s≀u οΏ½n s οΏ½οΏ½οΏ½ds r οΏ½ Ξ΄s/r οΏ½ 1 οΏ½dn r οΏ½οΏ½dnβˆ’r 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=' οΏ½dnβˆ’Ξ΄s+r r οΏ½ ≀ οΏ½ 1≀s≀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 βˆ’ Ξ΄s)Ξ΄s ≀ οΏ½ 1≀s≀u οΏ½ ne s οΏ½ dse (dn βˆ’ 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 ≀ s ≀ √n the term in (14) is at most οΏ½ O(n) Β· οΏ½ O(nβˆ’1/2) 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 √n ≀ s ≀ u we have that the term in (14) is at most οΏ½ O(n1/2) Β· O(1) Β· οΏ½ O(nβˆ’1/2) 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'}
286
+ page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
287
+ 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'}
288
+ page_content=' It would seem likely that the approach we took would extend to other definitions of proper coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
289
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290
+ 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=' Frieze and C.' 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='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=' 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=' 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=' 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=' 12 [18] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ page_content=' [19] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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+ 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'}
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+ page_content=' 13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENAyT4oBgHgl3EQfSPf9/content/2301.00085v1.pdf'}
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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.βˆ—
5
+ Gianmarco Caldini
6
+ 7 February 2020
7
+ Abstract
8
+ The aim of this article is to define a notion of cardinal utility function called
9
+ measurable utility and to define it on a connected and separable subset of a weakly
10
+ ordered topological space. The definition 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 differences.
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 defined 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 affine transformations,
21
+ of an order isomorphism with the real line.
22
+ βˆ—I 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 define 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 first
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 definitions and axioms that will enable us to represent
39
+ comparisons of the intensity that a decision maker feels about the desirability of different
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 Shapley’s 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 define a rigorous notion of a specific kind of cardinal
46
+ utility function, not only able to rank alternatives, but also to compare utility differ-
47
+ ences. In particular, we define a β€œtwofold” 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 specific 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 find 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
+ justifications, 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-defined sense, or, at least, the less harmful. These justifications 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 define 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
+ different 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 definition of morality, there exist the similar concepts of
75
+ ethics and of role morality - a specific form of professional morality. It was Jeremy Ben-
76
+ tham, in an unfinished manuscript which was posthumously published in 1834, to define
77
+ the neologism deontology in the title of his book Deontology or the Science of Morality.
78
+ The manuscript stated, for the first and only time, the particular aspects of Bentham’s
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, first 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 one’s own private felicity. Nevertheless, this does not bring any selfish concern.
87
+ Bentham’s goal can be identified 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 Smith’s 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
+ tified by an intrinsic, absolute value, that does not need any further justification. It is
105
+ realistic because they thought human being to ultimately seek the maximum pleasure
106
+ and the minimum sorrow. More specifically, 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 β€œcalculated hedonism”, 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 beneficial. 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 β€œmoral arithmetic” 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 Bentham’s 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, Bentham’s 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: ”Non ci vuol molto a scoprir qui un falso ragionamento fondato sull’alterazione
128
+ d’un fatto. Altro `e che l’utilit`a sia un motivo, cio`e uno de’ motivi per cui gli uomini si determinano
129
+ nella scelta dell’azioni, altro `e che sia, per tutti gli uomini, il motivo per eccellenza, l’unico motivo
130
+ delle loro determinazioni (p.775).
131
+ 2This tension between individual pleasure and universal pleasure is one of the principal difficulties
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 find utilitarian reasonings in Aristotele’s works, it is com-
141
+ monly agreed that the beginning of the history of utility can be identified with 18th
142
+ century moral philosophy. To be even more specific, Bentham’s 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 defined 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 different people could be compared. Stark [38] cited in his article
150
+ Bentham’s 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 Bentham’s 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 differences were well-founded and realistic notions
163
+ with a strong moral philosophy justification.
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 difficulties in the quantification 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 offered 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 β€œhead” on the ground. At that point, the player wins
182
+ 2n dollars, where n is the number of times the coin was flipped. 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 β€œhead” being 0.5? The mathematical answer is
185
+ ∞
186
+ οΏ½
187
+ i=1
188
+ 1
189
+ 2i · 2i = 1 + 1 + · · · = ∞.
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
+ finite 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 scientific justifications. 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 different 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 β€œjust noticeable difference” or β€œminimum perceptible threshold”) in stimuli
201
+ like heat, weight and pitch. Moreover, Fechner took this β€œjust noticeable difference”
202
+ as a unit of measurement, constructing a scale for subjective sensations. From their
203
+ studies we now have the so-called Weber’s law and Fechner’s 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
+ figures that founded neoclassical theory with scientific 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. Pareto’s 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 indifference 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 indifference curves.
227
+ Interesting are the writings of Francis Edgeworth [10] and Pareto [31], starting
228
+ from very different assumptions and arriving at different conclusions.
229
+ Edgeworth’s
230
+ main contribution can be summarized in the synthesis of Bentham’s utilitarianism and
231
+ Fechner’s 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 Edgeworth’s 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, Edgeworth’s and Pareto’s ways of conceiving
241
+ measurable utility must be differentiated 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: β€œThere is no reason for not accepting it [cardinal utility], with the
244
+ reservation that it must be verified by the results deduced from it (p. 73).”
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 people’s
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 unjustified 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-defined concept of cardinal utility
259
+ and interpersonal comparisons. Some examples are the theory of dynamic evolution of
260
+ the economy, the theory of fiscal 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 significantly using the
266
+ ordinal and the cardinal approach, we should prefer the purely comparative approach
267
+ by Occam’s 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 efficiency, 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 field of neuroscience
281
+ 7
282
+
283
+ and the new discipline of neuroeconomics. These fields of cognitive sciences are going
284
+ into the direction of overcoming the main difficulty of the founders of utilitarianism:
285
+ the difficulty of carrying out experiments on pleasure and pain and the construction of
286
+ a rigorous and well-defined 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 definition that can be applied in decision-theoretic models.
290
+ During the 20th century a lot of methodologies to try to define a concept of
291
+ measurement of human sensation have been defined.
292
+ Definition 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 differ
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 definition for cardinal utility was not solved by the
298
+ theory of measurement.
299
+ It was just translated in a different language: what is the
300
+ suitable scale for measuring a given aspect? The definition 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 defined
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, defined 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 defined 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 beings’ 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 first transition can be identified 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 different temperatures with different classes of systems in thermal
327
+ equilibrium. Classes like these were called fixed 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 fixed, defining 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 first transition from an
334
+ ordinal purely comparative approach to that of an equality relation for utility differ-
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 Euclid’s notion of lying to the right of and
338
+ the derived concept of equality of utility differences with the geometrical congruence of
339
+ intervals.
340
+ Hence, the main question becomes whether the derived order relation on utility
341
+ differences 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 firmly 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 β€œmea-
351
+ surement” 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 define 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 fields 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 finite 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 definition of them is critical in the
362
+ development of a theory that does not contain contradictions.
363
+ As a result, almost all fields 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 β€œlogic of partial belief.” 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 Keynes’ 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 first treatments of preference relations as a primitive notion can be
380
+ identified with Frisch [18], in his 1926 paper.
381
+ Ragnar Frisch was also the first to
382
+ formulate an axiomatic notion of utility difference. Hence, two kinds of axioms were
383
+ postulated by him: the first ones - called β€œaxioms of the first kind” - regarded the
384
+ relation able to rank alternatives in a purely comparative way, while the second axioms
385
+ - named β€œaxioms of the second kind” - reflected a notion of intensity of preference and
386
+ allowed utility differences to be compared.
387
+ So, in parallel to the axiomatization of
388
+ ordinal utility, also cardinal utility axiomatizations started to grow.
389
+ Frisch’s 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] defined an order relation ≻ on the set of alternatives X with
394
+ the meaning that, for any two alternatives x, y ∈ X, x ≻ y reads β€œx is strictly preferred
395
+ 10
396
+
397
+ to y.” Then, a corresponding relation P on ordering differences is assumed with the
398
+ meaning that, for any x, y, z, w ∈ X, xyPzw reads β€œa change from y to x is strictly
399
+ preferred than a change from w to z.
400
+ More formally:
401
+ x ≻ y ⇐⇒ u(x) > u(y) for all x, y ∈ X
402
+ (1)
403
+ xyPzw ⇐⇒ u(x) βˆ’ u(y) > u(z) βˆ’ u(w) for all x, y, z, w ∈ X
404
+ (2)
405
+ The main theorem of Lange [23] can be stated as follows:
406
+ Theorem 1. If there exists a differentiable utility function u : R β†’ R such that (1)
407
+ and (2) hold, then only positive affine transformations of that utility function represent
408
+ the given preferences ≻ 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 differentiability of u can be largely relaxed. Hence, the issue
412
+ becomes whether it is possible to find sufficient conditions on the preference relations
413
+ under which Lange’s 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
+ sufficient and necessary conditions for the existence of a continuous utility function
416
+ - unique up to positive affine transformations - based on a preference relation and a
417
+ utility-difference ordering relation. In his set of axioms, Alt defined a notion that can
418
+ be understood as the set of alternatives X to be connected.
419
+ With Frisch’s 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 different 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 difference structure, in line with old Frisch’s ideas and Lange’s
426
+ formalization: not only are individuals able to ordinally rank different alternatives, but
427
+ they are also able to compare and rank utility differences of alternatives. Indeed, Sup-
428
+ pes and Winet cited the work of Oskar Lange on the notion of utility differences 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 differences. 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 Alt’s 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 sufficient and necessary condition for a cardinal utility
439
+ representation, unique up to positive affine transformations. Another approach to the
440
+ field 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 suffered 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 identified 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 β€œparty.” In fact, a large class of economists classify as β€œmeaningless” even the
457
+ mere introspective idea of a comparison of utility difference, 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, β€œepistemological laziness.” 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 effort 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 defined 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 β€œfallacy of identity” and β€œfallacy of unrelatedness.”
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 different 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 Shapley’s proof
476
+ to a connected and separable subset of a topological space.
477
+ 2.1
478
+ Basic definitions
479
+ Definition 2. A relation on a set X is a subset β‰Ώ of the cartesian product X Γ— X,
480
+ where x β‰Ώ y means (x, y) ∈ β‰Ώ.
481
+ In decision theory, β‰Ώ is usually called a preference relation, with the interpretation
482
+ that, for any two elements x, y ∈ X, we write x β‰Ώ y if a decision maker either strictly
483
+ prefers x to y or is indifferent between the two.
484
+ Definition 3. An equivalence relation on a set X is a relation R on X that satisfies
485
+ 1) Reflexivity: for all x ∈ X, we have xRx.
486
+ 2) Symmetry: for any two elements x, y ∈ X, if xRy, then yRx.
487
+ 3) Transitivity: for any three elements x, y and z ∈ X, if xRy and yRz, then xRz.
488
+ Definition 4. A relation β‰Ώ 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 ∈ X, either x β‰Ώ y or y β‰Ώ x or both.
491
+ 2) Antisymmetry: for any two elements x, y ∈ X, if x β‰Ώ y and y β‰Ώ x, then x = y.
492
+ 3) Transitivity: for any three elements x, y and z ∈ X, if x β‰Ώ y and y β‰Ώ z, then
493
+ x β‰Ώ z.
494
+ Note that if β‰Ώ is complete, then it is also reflexive. The relation β‰Ώ induces, in
495
+ turns, two other relations. Specifically, for any two elements x, y ∈ X we write:
496
+ (i) x ≻ y if x β‰Ώ y but not y β‰Ώ x.
497
+ 13
498
+
499
+ (ii) x ∼ y if x β‰Ώ y and y β‰Ώ x.
500
+ It is easy to see, indeed, that if β‰Ώ is reflexive and transitive, then ∼ is an equivalence
501
+ relation. Given an equivalence relation ∼ on a set X and an element x ∈ X, we define
502
+ a subset E of X, called the equivalence class determined by x, by the equation
503
+ E := {y ∈ X : y ∼ x}
504
+ Note that the equivalence class E determined by x contains x, since x ∼ x, hence
505
+ E is usually denoted as [x].
506
+ We will denote X/∼ the collection {[x] : x ∈ X} of
507
+ all equivalence classes, which is a partition of X: each x ∈ X belongs to one, and
508
+ only one, equivalence class. In decision theory, an equivalence class is often called an
509
+ indifference curve.
510
+ Definition 5. A relation β‰Ώ on a set X is called a weak order if it is complete and
511
+ transitive.
512
+ The problem of finding a numerical representation for a preference relation β‰Ώ, 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
+ Definition 6. A real-valued function u : X β†’ R is a (Paretian) utility function for β‰Ώ
517
+ if for all x, y ∈ X we have
518
+ x β‰Ώ y ⇐⇒ u(x) β‰₯ u(y)
519
+ Utility functions β€œshift” the pairwise comparisons that characterize the order rela-
520
+ tion β‰Ώ 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 different intensities about the single desirability of any
524
+ two alternatives x, y ∈ X. What is important is the ranking given by the real num-
525
+ bers, according to the usual order of the ordered field (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 sufficient and necessary conditions under which a relation β‰Ώ admits a utility
530
+ representation. The original reference can be identified 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 first 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
+ Definition 7. A weak order β‰Ώ on a set X is said to be continuous if, for every y ∈ X,
539
+ the sets {x ∈ X : x ≻ y} and {x ∈ X : x β‰Ί y} are open.9
540
+ Theorem 2 (Eilenberg). Let β‰Ώ be a complete and transitive relation on a connected
541
+ and separable topological space X. The following conditions are equivalent:
542
+ (i) β‰Ώ is continuous.
543
+ (ii) β‰Ώ 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) βˆ’ u(y)
547
+ is a well-defined real number thanks to the algebraic properties of R, but it is meaning-
548
+ less in term of the interpretation of a difference of utility of two alternatives x, y ∈ 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 Gossen’s law of
552
+ decreasing marginal utility), based on the notion of different quotient, is meaningless.
553
+ More formally, the expression
554
+ du(x)
555
+ dx
556
+ = lim
557
+ h→0
558
+ u(x + h) βˆ’ 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) ∈ 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
+ β€œx in lieu of y”. Define the binary relation ≽ on X Γ—X called intensity preference with
571
+ the following interpretation: for any two pairs (x, y) and (z, w) in X Γ— X,
572
+ (x, y) ≽ (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 β‰Ώ x) at most as much added sadness. As a result, our decision maker
575
+ is endowed with a weak order preference relation β‰Ώ on alternatives and an intensity
576
+ preference relation ≽ 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 field 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 β‰Ώ and ≽ and with the order topology induced by the weak order β‰Ώ.
582
+ We assume the following axioms for β‰Ώ and ≽, as in Shapley [36].
583
+ Axiom 1. For all x, y, z ∈ X we have (x, z) ≽ (y, z) ⇐⇒ x β‰Ώ 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, A1β€², that can be derived from the whole set of axioms we are
588
+ going to assume later.10
589
+ Axiom 1β€². For all x, y, z ∈ X we have (z, x) ≽ (z, y) ⇐⇒ y β‰Ώ x.
590
+ We now introduce the main object of this thesis: a joint real-valued representation
591
+ for the two orders β‰Ώ and ≽.
592
+ Definition 8. A real-valued function u : X β†’ R is a measurable utility function for
593
+ (β‰Ώ, ≽) if for each pair x, y ∈ X
594
+ x β‰Ώ y ⇐⇒ u(x) β‰₯ u(y)
595
+ (3)
596
+ and if, for each quadruple x, y, z, w ∈ X
597
+ (x, y) ≽ (z, w) ⇐⇒ u(x) βˆ’ u(y) β‰₯ u(z) βˆ’ 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 field of science that established the for-
601
+ mal foundation of quantitative measurement and the assignment of numbers to objects
602
+ 10We mention A1β€² 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 affine transformations. Recall that a positive affine
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 ∈ R. Positive affine transformations are
617
+ order-preserving thanks to Ξ± > 0.
618
+ Proposition 1. A measurable utility function u : X β†’ R for (β‰Ώ, ≽) is unique up to
619
+ positive affine transformations.
620
+ Proof. If u(x) = Ξ±u(x) + Ξ² then we have
621
+ x β‰Ώ y ⇐⇒ u(x) β‰₯ u(y) ⇐⇒ u(x) = Ξ±u(x) + Ξ² β‰₯ Ξ±u(y) + Ξ² = u(y)
622
+ and
623
+ (x, y) ≽ (z, w) ⇐⇒ u(x) βˆ’ u(y) β‰₯ u(z) βˆ’ u(w)
624
+ ⇐⇒ u(x) βˆ’ u(y) = Ξ±[u(x) βˆ’ u(y)] β‰₯ Ξ±[u(z) βˆ’ u(w)] = u(z) βˆ’ u(w).
625
+ As a result, u and u are two utility representations for (β‰Ώ, ≽).
626
+ The whole class
627
+ of utility functions that are unique up to positive affine 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 ∈ X we have (x, y) ∼ (z, w) ⇐⇒ (x, z) ∼ (y, w).
632
+ Axiom 3. For all x, y, z, w ∈ X the set
633
+ {(x, y, z, w) ∈ X Γ— X Γ— X Γ— X : (x, y) ≽ (z, w)}
634
+ is closed in the product topology.
635
+ Axiom 2 is a β€œcrossover” property that characterizes difference comparisons of util-
636
+ ity, while Axiom 3 is a technical assumption defining the order relation ≽ 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, ≽ is assumed to be a weak order on D Γ— D such
642
+ that A1, A2 and A3 are satisfied.
643
+ Theorem 3 (Shapley). There exist a utility function u : D βŠ† R β†’ R such that
644
+ x β‰₯ y ⇐⇒ u(x) β‰₯ u(y)
645
+ (5)
646
+ and
647
+ (x, y) ≽ (z, w) ⇐⇒ u(x) βˆ’ u(y) β‰₯ u(z) βˆ’ u(w)
648
+ (6)
649
+ for all x, y, z, w ∈ D. Moreover, this function is unique up to a positive affine transfor-
650
+ mation.
651
+ The theorem is stated as a sufficient condition, which is the most difficult 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‰₯, ≽) has a continuous measurable utility function u : D βŠ†
655
+ R β†’ R, then β‰₯ is complete and transitive, ≽ is complete, transitive, continuous (A3)
656
+ and satisfies the crossover axiom (A2), and jointly β‰₯ and ≽ satisfy the consistency
657
+ axiom (A1).
658
+ Shapley’s construction of the measurable utility function of Theorem 3 is extremely
659
+ elegant, but has the drawback of being too specific as u is defined on a convex subset
660
+ of R. On the other side of the spectrum, as mentioned in the first chapter, the field
661
+ of utility axiomatization has been prolific 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 different alternatives, as is assumed in the ordinal
667
+ approach to utility, but are also capable of ranking the differences 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, finding a representation
672
+ theorem for cardinal utility function (in particular, a measurable one) keeping the
673
+ 18
674
+
675
+ elegance of Shapley’s 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, β‰Ώ and ≽ 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
+ Definition 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
+ Definition 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
+ Definition 11. A totally ordered set (L, β‰Ώ) 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 ≻ y, there exists z such that x ≻ z ≻ y
695
+ We recall that a ray is a set of the following type (βˆ’βˆž, a) = {x ∈ L : x β‰Ί a}
696
+ and (βˆ’βˆž, a] = {x ∈ L : x β‰Ύ a} in the case L does not have a minimum. In the
697
+ case L does have a minimum we write [xm, a) = {x ∈ L : xm β‰Ύ x β‰Ί a} and [xm, a] =
698
+ {x ∈ L : xm β‰Ύ x β‰Ύ a}. Analogously for the sets (a, +∞), [a, +∞), (a, xM] , [a, xM], where
699
+ xM is the maximum of L in the case it existed.12
700
+ Given A βŠ† X, an element y ∈ X is an upper bound for a set A if y β‰Ώ x for all
701
+ x ∈ 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 yβ€² β‰Ώ x for all x ∈ A then yβ€² β‰Ώ y. If β‰Ώ is antisymmetric, the
703
+ least upper bound is unique and is denoted sup A. The greatest lower bound is defined
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 reflexive and transitive binary
707
+ relation β‰Ώ are usually denoted with the following notation L(a, β‰Ώ) := (βˆ’βˆž, a] = {x ∈ X : x β‰Ύ a} and
708
+ U(a, β‰Ώ) := [a, +∞) = {x ∈ X : x β‰Ώ a}, L(a, ≻) := (βˆ’βˆž, a) and U(a, ≻) := (a, +∞).
709
+ 19
710
+
711
+ Lemma 1. Let β‰Ώ 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 fixed elements of X such that a β‰Ί b.
714
+ If there is no element c ∈ X such that a β‰Ί c β‰Ί b, then X is the union of the open
715
+ rays (βˆ’βˆž, b) = {x ∈ X : x β‰Ί b} and (a, +∞) = {x ∈ X : a β‰Ί x} both of which are
716
+ open sets in the order topology and are also nonempty, as the first 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 ∈ X such that a β‰Ί c β‰Ί 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 ∈ X : b β‰Ώ a for every a ∈ 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 ∈ B, there exists an element bβ€² ∈ B such that bβ€² β‰Ί b, and so b ∈ (bβ€², +∞) βŠ† B with
728
+ (bβ€², +∞) 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 βŠ‚ X,
730
+ since A βŠ† X\B and A ΜΈ= βˆ…, 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 ∈ A such that b0 β‰Ί a, we can also conclude that
733
+ b0 ∈ (βˆ’βˆž, a) βŠ† X\B, with (βˆ’βˆž, 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 βŠ† X, we denote A or ClA the topological closure of A, that is defined 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 β‰Ώ be a complete, transitive and continuous order on a connected set X.
741
+ 13Note that the converse holds as well: β‰Ώ 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 ∈ X, with x ≻ y, we have
746
+ x β‰Ώ z β‰Ώ y β‡’ z ∈ X
747
+ for all z ∈ (X, Ο„)
748
+ Proof. Suppose by contradiction that there exists z ∈ X\X such that x ≻ z ≻ y.
749
+ By the continuity of β‰Ώ, we can partition X into two nonempty disjoint open sets
750
+ {x ∈ X : x β‰Ί z} and {x ∈ X : x ≻ z}, which contradicts the connectedness of X.
751
+ Lemma 3. Suppose that jointly β‰Ώ and ≽ satisfy A1. If ≽ is continuous , then β‰Ώ is
752
+ continuous.
753
+ Proof. For all arbitrary but fixed y, z ∈ X, by A1 we have {x ∈ X : (x, z) ≽ (y, z)} =
754
+ {x : x β‰Ώ y}. By A3, the set {x ∈ X : (x, z) ≽ (y, z)} is closed. Analogous is the case
755
+ for {x : y β‰Ώ x}, derived from A1β€².
756
+ Lemma 4. Fix y ∈ X, the set Iy := {x ∈ X : x ∼ y} is a closed set in X.
757
+ Proof. β‰Ώ is continuous, so for every y ∈ X we have that {x ∈ X : x β‰Ώ y} and
758
+ {x ∈ X : y β‰Ώ x} are closed. Pick a point x such that x β‰Ώ y and y β‰Ώ x, that is x ∼ y.
759
+ So we have {x ∈ X : x ∼ y} = {x ∈ X : x β‰Ώ y} ∩ {x ∈ X : y β‰Ώ x} and the intersection
760
+ of two closed sets is closed.
761
+ Note that when β‰Ώ is antisymmetric, the set Iy is a singleton and Lemma 4 reduces
762
+ to prove that X satisfies the T1 axiom of separation, that is every one-point set is closed.
763
+ Clearly, every Hausdorff space satisfies it.
764
+ Lemma 5. Let β‰Ώ be a continuous total order on a connected set X. If A βŠ† 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 /∈ A. Then supA ∈ X\A, which is open. By definition, there
768
+ exists a base element (a, b) such that
769
+ supA ∈ (a, b) βŠ† X\A.
770
+ A is bounded above so, by Lemma 1, sup A exists and there is an element a⋆ such that
771
+ a β‰Ί a⋆ β‰Ί sup A, then a⋆ ∈ (a, b) βŠ† X\A, so a⋆ 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 βŠ‚ R that is closed in the order topology, but sup N /∈ N.
777
+ 21
778
+
779
+ cannot be an upper bound of A as x β‰Ί sup A. Hence, there exists an element a ∈ A
780
+ such that x β‰Ί a β‰Ύ sup A. Thus, as x was generic, it follows that U ∩ A ΜΈ= βˆ…. This
781
+ means that every neighborhood of sup A intersects A, that is sup A ∈ A. But A is
782
+ closed, hence sup A ∈ A and we can conclude sup A = max A.
783
+ The case of inf A is specular.
784
+ Now we define the notion of convergence in any topological space.
785
+ Definition 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 ∈ U for all n β‰₯ N.
788
+ Moreover, let β‰Ώ a total order. We write xn ↑ x if x1 β‰Ύ x2 β‰Ύ Β· Β· Β· β‰Ύ xn β‰Ύ . . . and
789
+ supnxn = x where sup is with respect to β‰Ύ. The definition xn ↓ x for a β‰Ύ-decreasing
790
+ sequence is analogous. We say that (xn) converges monotonically to a limit point x
791
+ when either xn ↑ x or xn ↓ x.
792
+ We now prove one of the fundamental lemmas that allow us to generalize Shapley’s
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 β€œenough” to characterize the definition
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 specific
798
+ class of spaces that are said to satisfy the first 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}n∈N 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
+ first 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 β€œreach” these points. The ideal solution to
807
+ this problem is to define a more general object than a sequence, called a net, and talk
808
+ about net-convergence. One can also define a type of object called a filter and show
809
+ that filters 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 satisfies the first 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 definitions that will
820
+ be used to prove Lemma 6.
821
+ Definition 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
+ Definition 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 β‰Ώ be a continuous total order on a connected and separable topological
829
+ space X in the order topology and A βŠ† X. We have x ∈ A if and only if there exists a
830
+ sequence (xn) ∈ 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 sufficient (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 βŠ† X with X metrizable, then we have that x ∈ 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 Hausdorff, hence
843
+ it is T1. Now, suppose x ∈ X and B is a closed set, disjoint from x. So, x ∈ X\B,
844
+ 16A metrizable space always satisfies the first countability axiom.
845
+ 17In fact, one could prove that X is also normal.
846
+ 23
847
+
848
+ which is open. Then, by definition of open set, there exists a basis element (a, b) such
849
+ that x ∈ (a, b) and (a, b) ∩ B = βˆ…. Pick any a0 ∈ (a, x), and let U1 = (βˆ’βˆž, a0) , V1 =
850
+ (a0, ∞). If no such a0 exists (in our case it would, by connectedness of X), then let
851
+ U1 = (βˆ’βˆž, x), V1 = (a, ∞). In both cases, U1 ∩ V1 = βˆ…. Similar is the case of the other
852
+ side, pick b0 ∈ (x, b), and if that exists, denote U2 = (b0, ∞) , V2 = (βˆ’βˆž, b0) , and if
853
+ not, let U2 = (x, ∞), V2 = (βˆ’βˆž, b). Again, in both cases U2 ∩ V2 = βˆ…. As a result, we
854
+ obtained that, in both cases, x ∈ V1 ∩ V2 with V1 ∩ V2 open set and B βŠ† U1 βˆͺ U2, with
855
+ U1 βˆͺ U2 open set. As V1 ∩ V2 is disjoint from U1 βˆͺ U2, X is regular.
856
+ Lemma 8. A totally ordered, connected and separable topological space X is second-
857
+ countable.
858
+ Proof. Now we find a countable basis for the order topology of X. As X is separable,
859
+ then let D βŠ† X be countable and dense in X, i.e. D = X. Then, define
860
+ B := {(a, b) : a, b ∈ D with a β‰Ί 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 ∈ 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 ∈ (a, b) there is an element (aβ€², bβ€²) ∈ B
865
+ such that x ∈ (aβ€², bβ€²) βŠ† (a, b).
866
+ Suppose x ∈ (a, b) βŠ‚ X, then the open intervals (a, x) and (x, b) cannot be empty
867
+ by connectedness. Hence, there exist aβ€² ∈ (a, x) ∩ D and bβ€² ∈ (x, b) ∩ D. This follows
868
+ from the fact that D = X and x ∈ D = X if and only if every open set containing x
869
+ intersects D. Then, it follows that x ∈ (aβ€², bβ€²) βŠ† (a, b).
870
+ Now, when m exists, suppose x = m, then x ∈ [m, a) and this set is nonempty
871
+ by connectedness. Hence, there exists an element aβ€²β€² ∈ [m, a) ∩ D. So, it follows that
872
+ x ∈ [m, aβ€²β€²) βŠ† [m, a). Analogous is the case when M exists.
873
+ By Lemma 7 and Lemma 8 , X satisfies all the assumptions of the Urysohn metriza-
874
+ tion theorem, hence X is metrizable (and, a fortiori, it is first-countable).
875
+ Lemma 9. Let A βŠ† X with X metrizable, then x ∈ 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 ∈ A. Then, every neighborhood U of x contains a
878
+ point of A, i.e. x ∈ A. Conversely, we use the fact that X is metrizable.18 Let x ∈ 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 ∈ 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 ∈ Bd(x, 1
887
+ n) ∩ 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 finally prove Lemma 6.
893
+ Proof. The if part comes trivially by definition. If there exists a sequence that converges
894
+ (monotonically) to x, then x ∈ A by Lemma 9.
895
+ Conversely, if x ∈ 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, β‰Ύ), every sequence
897
+ from N β†’ (X, β‰Ύ) has a monotone subsequence. Indeed, this is a property that has
898
+ nothing to do with the topology of X.
899
+ Let (xi)i∈N be a sequence with values in X.
900
+ We say that xk is a peak of the
901
+ sequence if h > k β‡’ xh β‰Ύ 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 infinitely many peaks, then the subsequence of peaks is an infinite
904
+ non-increasing sequence and we are done. If there are only finitely 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 find another index i2 > i1 such that xi2 β‰Ώ xi1. Again, as xi2 is not a
907
+ peak, we can find another index i3 > i2 such that xi3 β‰Ώ xi2 β‰Ώ xi1. Keeping defining the
908
+ sequence in this way, we get, inductively, a non-decreasing sequence.
909
+ In conclusion, as by assumption we have a sequence (xn) ∈ 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 first 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
+ βˆ† := {(x, x) : x ∈ 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 Οƒ-locally countable basis,
920
+ which is a condition more in the spirit of the Nagata-Smirnov metrization theorem which requires a
921
+ Οƒ-locally finite basis.
922
+ 25
923
+
924
+ show how β€œwell-behaved” a totally ordered, connected and separable topological space
925
+ can be.
926
+ Lemma 10. Let (X, β‰Ώ) be a topological space with the order topology. Let ≽ 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 ∈ N we have (xn, yn) ≽ (wn, zn) then (x, y) ≽ (w, z).
930
+ Proof. Denote the set A := {(x, y, w, z) ∈ X Γ— X Γ— X Γ— X : (x, y) ≽ (w, z)} and pick
931
+ a sequence of points with values in A, that is pick (xn, yn, wn, zn) ∈ 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 Γ—X Γ—X Γ—X. Let U1, U2, U3, U4 be open
937
+ sets containing x, y, w, z, respectively. Then U1Γ—U2Γ—U3 Γ—U4 is a basis element (hence,
938
+ open) for the product topology containing (x, y, w, z). By definition of convergence, we
939
+ can find n0 such that for all n β‰₯ n0 we have (xn, yn, wn, zn) ∈ 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 ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ 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 U⋆ be an open subset of
944
+ XΓ—XΓ—XΓ—X such that (x, y, w, z) ∈ U⋆. By definition of product topology, we can find
945
+ U1 βŠ† X open in X, . . . , U4 βŠ† X open in X such that x ∈ U1, y ∈ U2, w ∈ U3, z ∈ U4. By
946
+ convergence, we have that for all i = 1, 2, 3, 4 there exists nki ∈ N such that for all n β‰₯
947
+ nki we have xn ∈ U1, yn ∈ U2, wn ∈ U3, zn ∈ U4. Now pick N := max{nk1, nk2, nk3, nk4}
948
+ and for every n β‰₯ N we have (xn, yn, wn, zn) ∈ U1 Γ— U2 Γ— U3 Γ— U4 βŠ† U⋆. Hence, by
949
+ definition of convergence, (xn, yn, wn, zn) β†’ (x, y, w, z).
950
+ Now we want to show (x, y, w, z) ∈ 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 βŠ† X that is converging to a point (x, y, w, z) ∈ X, then (x, y, w, z) ∈ A.
954
+ Pick a sequence (xn, yn, wn, zn) with values in A βŠ† 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 ≽ alone.
957
+ 20Note that when X is metrizable, a set C βŠ† X is closed ⇐⇒ C is sequentially closed.
958
+ 26
959
+
960
+ (x, y, w, z) ∈ X. Then, let U⋆ be any neighborhood of (x, y, w, z). By convergence,
961
+ there exist an n0 ∈ N such that for all n β‰₯ n0 we have (xn, yn, wn, zn) ∈ U⋆ and, in
962
+ particular, (xn, yn, wn, zn) ∈ U⋆ ∩ A. Since U⋆ was an arbitrary but fixed neighborhood
963
+ of (x, y, w, z), then (x, y, w, z) is in the closure of A, i.e. (x, y, w, z) ∈ A. But A is
964
+ closed, therefore A = A, so (x, y, w, z) ∈ A, hence (x, y) ≽ (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 xβ€², xβ€²β€², y ∈ X are such that:
969
+ (xβ€², y) ≽ (w, z) ≽ (xβ€²β€², y)
970
+ (7)
971
+ then there exists a unique, up to indifference, x⋆ ∈ X such that
972
+ (x⋆, y) ∼ (w, z)
973
+ (8)
974
+ and xβ€² β‰Ώ x⋆ β‰Ώ xβ€²β€².
975
+ Proof. Define x0 := inf{x ∈ X : (x, y) ≽ (w, z)} and denote A := {x ∈ X : (x, y) ≽
976
+ (w, z)} this set. The set A is nonempty as xβ€² ∈ A, A is bounded below by xβ€²β€² as we
977
+ have (w, z) ≽ (xβ€²β€², y) and, by transitivity and A1, we reach x β‰Ώ xβ€²β€² for every x ∈ A.
978
+ Thus, x0 is such that xβ€² β‰Ώ x0 β‰Ώ xβ€²β€² and so x0 ∈ X by Lemma 2. Analogously, we define
979
+ x0 := sup{x ∈ X : (w, z) ≽ (x, y)} and denote B := {x ∈ X : (w, z) ≽ (x, y)} this set.
980
+ Then, B is nonempty as xβ€²β€² ∈ B, B is bounded above by xβ€² as we have (xβ€², y) ≽ (w, z)
981
+ and, by transitivity and A1, we reach xβ€² β‰Ώ x for every x ∈ B. Thus, x0 is such that
982
+ xβ€² β‰Ώ x0 β‰Ώ xβ€²β€² and so x0 ∈ X by Lemma 2.
983
+ By A3, the sets A and B are closed and so, by Lemma 5, we have x0 ∈ A and
984
+ x0 ∈ B so that
985
+ (x0, y) ≽ (w, z) ≽ (x0, y)
986
+ By transitivity and by A1 we have x0 β‰Ώ x0.
987
+ Assume now by contradiction that x0 ≻ x0. By Lemma 1 there exists x⋆ ∈ X such
988
+ that x0 β‰Ί x⋆ β‰Ί x0. But then, comparing x⋆ with (w, z), (x⋆, y) ≽ (w, z) can hold only
989
+ if x0 ∼ x⋆ ≻ x0, so x0 ∼ x⋆ and therefore x⋆ should be the infimum of A, reaching a
990
+ contradiction. Specular is the contradiction in the other case. Therefore, as there does
991
+ not exist any x⋆ ∈ X such that x0 β‰Ί x⋆ β‰Ί x0, we must conclude that x0 ∼ x0. By
992
+ transitivity and A1 we have
993
+ (x0, y) ∼ (w, z) ∼ (x0, y)
994
+ 27
995
+
996
+ This proves the existence of x⋆ ∈ X for which (8) holds.
997
+ Let x ∈ X be any other element of X for which (8) holds. By transitivity, (x⋆, y) ∼
998
+ (x, y). By A1, we have x⋆ ∼ x and this completes the proof.
999
+ Lemma 12. Let x, z ∈ X such that x ≻ z. Then, there exists a unique, up to indiffer-
1000
+ ence, y⋆ ∈ X such that
1001
+ (x, y⋆) ∼ (y⋆, z)
1002
+ and x ≻ y⋆ ≻ z.
1003
+ Proof. Define y0 to be the least upper bound of the set C := {y ∈ X : (x, y) ≽ (y, z)}.
1004
+ This set is nonempty as if we pick y = z we have (x, z) ≽ (z, z) that by A1 is equivalent
1005
+ to x β‰Ώ z, that holds by assumption. C is also bounded from above by x as if we pick
1006
+ y = x we have (x, x) ≽ (x, z) that by A1β€² is equivalent to z β‰Ώ x, that, by completeness,
1007
+ contradicts the assumption of x ≻ 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 ∈ X.
1009
+ Similarly, by defining y0 to be the greatest lower bound of the set D := {y ∈ X :
1010
+ (y, z) ≽ (x, y)}. This set is nonempty as if we pick y = x we have (x, z) ≽ (x, x) that
1011
+ by A1β€² is if and only if x β‰Ώ 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) ≽ (x, z) that by A1 is if and only
1013
+ if z β‰Ώ x, that, by completeness, contradicts the assumption of x ≻ 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 ∈ X.
1016
+ By A3 the sets C and D are closed, so by Lemma 5 we have y0 ∈ C and y0 ∈ D,
1017
+ that is
1018
+ (x, y0) ≽ (y0, z) and (y0, z) ≽ (x, y0)
1019
+ (9)
1020
+ We show now that y0 β‰Ώ y0. Suppose, by contradiction, y0 ≻ y0. By Lemma 1 there
1021
+ exists y⋆ ∈ X such that y0 ≻ y⋆ ≻ y0. Then, by definition of y0 we have (y⋆, z) β‰Ί (x, y⋆),
1022
+ while by the definition of y0 we have (x, y⋆) β‰Ί (y⋆, z). This contradiction shows that
1023
+ y0 β‰Ώ y0. By A1 this is equivalent to
1024
+ (y0, z) ≽ (y0, z) for all z ∈ X.
1025
+ (10)
1026
+ By A1β€² it is also equivalent to
1027
+ (x, y0) ≽ (x, y0) for all x ∈ X.
1028
+ (11)
1029
+ Putting together equation 9 with equations 10 and 11, we reach the loop
1030
+ (y0, z) ≽ (y0, z) ≽ (x, y0) ≽ (x, y0) ≽ (y0, z).
1031
+ 28
1032
+
1033
+ By transitivity, we have (y0, z) ∼ (y0, z) and (x, y0) ∼ (x, y0). By A1, we conclude that
1034
+ y0 ∼ y0.
1035
+ We conclude proving that from A1, A2 and A3 we can derive A1β€².
1036
+ Lemma 13. Let X be a connected subset of a topological space.
1037
+ If β‰Ώ is complete
1038
+ and transitive, ≽ is complete, transitive, satisfies A3 and A2, and jointly β‰Ώ and ≽
1039
+ satisfy A1, then A1β€² holds, that is, for all x, y, z ∈ X we have x β‰Ώ y if and only if
1040
+ (z, y) ≽ (z, x).
1041
+ Proof. By contradiction, suppose A1β€² fails. Then, there exist x, y, z ∈ X such that
1042
+ (z, y) ≽ (z, x) and x β‰Ί y. We consider two cases: y ≻ z and y β‰Ύ z.
1043
+ If y ≻ z then, being (z, y) ≽ (z, x) by assumption, we have
1044
+ (x, x) ∼ (y, y) ≻ (z, y) ≽ (z, x)
1045
+ by A2 and A1, respectively. We apply Lemma 11 to find a w ∈ X such that
1046
+ (w, x) ∼ (z, y) and x β‰Ώ w β‰Ώ z.
1047
+ Being y ≻ x, we have
1048
+ (z, z) ∼ (y, y) ≻ (x, y) ∼ (w, z) β‰Ώ (z, z)
1049
+ by A2, A1, A2, A1, respectively. This implies a contradiction in the case y ≻ z.
1050
+ Assume now y β‰Ύ z. Being y β‰Ύ z and x β‰Ί y, by transitivity we have x β‰Ί 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 Shapley’s theorem in our general version.
1056
+ Theorem 5. Let X be a connected and separable subset of a topological space. If β‰Ώ is
1057
+ complete and transitive, ≽ is complete, transitive, satisfies A2 and A3, and jointly β‰Ώ
1058
+ and ≽ satisfy A1, then the pair (β‰Ώ, ≽) can be represented by a continuous measurable
1059
+ utility function u: X β†’ R, that is, for each pair x, y ∈ X,
1060
+ x β‰Ώ y ⇐⇒ u(x) β‰₯ u(y)
1061
+ (12)
1062
+ and for each quadruple x, y, z, w ∈ X,
1063
+ (x, y) ≽ (z, w) ⇐⇒ u(x) βˆ’ u(y) β‰₯ u(z) βˆ’ u(w).
1064
+ (13)
1065
+ Moreover, u is unique up to positive affine transformations.
1066
+ 29
1067
+
1068
+ Proof. We first prove the result when β‰Ώ is antisymmetric. In view of Lemma 1, through-
1069
+ out the proof we will consider suprema and infima of subsets of X.
1070
+ Suppose X is not a singleton, otherwise the result is trivially true. Let a0, a1 ∈ X
1071
+ be two distinct elements of X such that, without loss of generality, a1 ≻ 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 (β‰Ώ, ≽). To ease notation,
1074
+ denote
1075
+ 1 := (a1, a0) , 0 := (a0, a0) , βˆ’1 := (a0, a1).
1076
+ Clearly, 1, 0, βˆ’1 ∈ X Γ— X and, by A1 and A1β€², 1 ≻ 0 ≻ βˆ’1. Then, by A2 we have
1077
+ (x, x) ∼ 0 for every x ∈ X. Moreover, for every y ∈ X we have either:
1078
+ (i) There exists a unique T1(y) ∈ X such that (T1(y), y) ∼ 1
1079
+ or
1080
+ (ii) 1 ≻ (x, y) for all x ∈ X
1081
+ Indeed, if (ii) fails, there exists xβ€² ∈ X such that (xβ€², y) ≽ 1. Since (xβ€², y) ≽ 1 ≽ 0 ∼
1082
+ (y, y), by Lemma 11 there exists an element T1(y) ∈ X such that (T1(y), y) ∼ 1. By
1083
+ A1 and antisymmetry of β‰Ώ , (T1(y), y) ∼ (yβ€², y) implies T1(y) = yβ€², so T1(y) is unique.
1084
+ In addition, note that y β‰Ί T1(y). Indeed, (y, y) ∼ 0 β‰Ί 1 ∼ (T1(y), y), and so A1
1085
+ implies y β‰Ί T1(y). In a similar way as before, for every y ∈ X we have either:
1086
+ (i.bis) There exists a unique Tβˆ’1(y) ∈ X such that (Tβˆ’1(y), y) ∼ βˆ’1
1087
+ or
1088
+ (ii.bis) βˆ’1 β‰Ί (x, y) for all x ∈ X
1089
+ Indeed, if (ii.bis) fails, there exists xβ€² ∈ X such that (xβ€², y) β‰Ό βˆ’1. Since (xβ€², y) β‰Ό βˆ’1 β‰Ό
1090
+ 0 ∼ (y, y), by Lemma 11 there exists an element Tβˆ’1(y) ∈ X such that (T1(y), y) ∼ βˆ’1.
1091
+ By A1 and antisymmetry of β‰Ώ , (Tβˆ’1(y), y) ∼ (yβ€², y) implies Tβˆ’1(y) = yβ€², so Tβˆ’1(y) is
1092
+ unique.
1093
+ In addition, note that Tβˆ’1(y) β‰Ί y. Indeed, (Tβˆ’1(y), y) ∼ βˆ’1 β‰Ί 0 ∼ (y, y), and so
1094
+ A1 implies Tβˆ’1(y) β‰Ί y.
1095
+ Now define a2 := T1(a1) if (i) holds for y = a1, i.e. if there exists a unique
1096
+ T1(a1) ∈ X such that (T1(a1), a1) ∼ 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 ≻ (x, an)
1098
+ for every x ∈ X. Analogously, we define aβˆ’1 := Tβˆ’1(a0) if (i.bis) holds for y = a0, set
1099
+ aβˆ’2 := Tβˆ’1(aβˆ’1) if (i.bis) holds for y = aβˆ’1, and continue in this way till (if ever) occurs
1100
+ y = aβˆ’n for which (ii.bis) holds.
1101
+ 30
1102
+
1103
+ Now define A := {. . . , aβˆ’2, aβˆ’1, a0, a1, a2, . . . }, with
1104
+ Β· Β· Β· β‰Ί aβˆ’2 β‰Ί aβˆ’1 β‰Ί a0 β‰Ί a1 β‰Ί a2 β‰Ί . . .
1105
+ The set A can be finite or infinite in either direction. If we consider now a sequence that
1106
+ from an index set Pa βŠ† Z maps to A, we define the following function a : Pa βŠ† Z β†’ A.
1107
+ Now we start to extend u to A. Define the following:
1108
+ u(ap) = p
1109
+ for every p ∈ Pa.
1110
+ Clearly, we have (12), i.e. x β‰Ώ 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 ∈ A βŠ‚ X, say x = ap, y = aq, z =
1113
+ apβˆ’d where p, q, p βˆ’ d ∈ Pa. Without loss of generality, assume d β‰₯ 0. We first prove
1114
+ the β€œequality” case of (13), that is
1115
+ (x, y) ∼ (z, w) ⇐⇒ u(x) βˆ’ u(y) = u(z) βˆ’ u(w)
1116
+ (14)
1117
+ By construction we have
1118
+ (ap, apβˆ’1) ∼ 1 ∼ (aq, aqβˆ’1)
1119
+ so, by transitivity and A2, we have:
1120
+ (ap, aq) ∼ (apβˆ’1, aqβˆ’1)
1121
+ Iterating this procedure finitely many times we reach:
1122
+ (x, y) = (ap, aq) ∼ (apβˆ’d, aqβˆ’d) = (z, aqβˆ’d)
1123
+ (15)
1124
+ By transitivity, (z, aqβˆ’d) ∼ (z, w) and so, by A1 aqβˆ’d = w, so that u(aqβˆ’d) = u(w).
1125
+ By definition of u we can write
1126
+ u(x) βˆ’ u(y) = u(ap) βˆ’ u(aq) = p βˆ’ q = u(apβˆ’d) βˆ’ u(aqβˆ’d) = u(z) βˆ’ u(w)
1127
+ thus proving (14). Next we prove
1128
+ (x, y) ≻ (z, w) ⇐⇒ u(x) βˆ’ u(y) > u(z) βˆ’ u(w)
1129
+ (16)
1130
+ By transitivity, (z, aqβˆ’d) ≻ (z, w) and so, by A1β€², w ≻ aqβˆ’d, so that u(w) > u(aqβˆ’d).
1131
+ By definition of u, from (15) we can write
1132
+ u(x) βˆ’ u(y) = u(ap) βˆ’ u(aq) = p βˆ’ q = u(apβˆ’d) βˆ’ u(aqβˆ’d) > u(z) βˆ’ 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 ≻ a0, by Lemma 12 there exists b1 ∈ X, with a1 ≻ b1 ≻ a0,
1137
+ 31
1138
+
1139
+ such that
1140
+ (a1, b1) ∼ (b1, a0)
1141
+ Now build the set B := {. . . , bβˆ’2, bβˆ’1, b0, b1, b2, . . . }, with
1142
+ Β· Β· Β· β‰Ί bβˆ’2 β‰Ί bβˆ’1 β‰Ί b0 β‰Ί b1 β‰Ί b2 β‰Ί . . .
1143
+ based on b0, b1, in the same way we constructed A from a0, a1. Also here, we can define
1144
+ a sequence that from an index set Pb βŠ† Z maps to B, that is, we define the following
1145
+ function b : Pb βŠ† Z β†’ B.
1146
+ By construction we have
1147
+ (b2, b1) ∼ (b1, b0)
1148
+ Together with (a1, b1) ∼ (b1, a0), by transitivity we have (b2, b1) ∼ (a1, b1). By A1,
1149
+ b2 = a1. Analogously, one can verify that
1150
+ b2p = ap for every p ∈ Pa
1151
+ (17)
1152
+ So, the terms of the set B lie between the terms of the set A, i.e. the set B refines
1153
+ A and we can write
1154
+ A βŠ† B
1155
+ (18)
1156
+ Denote now c0 := b0 = a0 and we let c1 ∈ X be that element provided by Lemma
1157
+ 12 such that (b1, c1) ∼ (c1, b0). In the same way we constructed B from A, we can
1158
+ construct, from B, a third set C := {. . . , cβˆ’2, cβˆ’1, c0, c1, c2, . . . }, with
1159
+ Β· Β· Β· β‰Ί cβˆ’2 β‰Ί cβˆ’1 β‰Ί c0 β‰Ί c1 β‰Ί c2 β‰Ί . . .
1160
+ based on c0, c1. We can see that
1161
+ c2p = bp for every p ∈ Pc
1162
+ where Pc βŠ† Z is the collection of indexes of the sequence c : Pc βŠ† Z β†’ C.
1163
+ The set C refines B
1164
+ B βŠ† C
1165
+ (19)
1166
+ We keep iterating this process, constructing sets that refine 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 ∈ A0
1172
+ A1 := B
1173
+ and
1174
+ a1
1175
+ p := bp ∈ A1
1176
+ A2 := C
1177
+ and
1178
+ a2
1179
+ p := cp ∈ A2
1180
+ Β· Β· Β·
1181
+ 32
1182
+
1183
+ These sets generalize the inclusions (18) and (19) as follows:
1184
+ A0 βŠ† A1 βŠ† A2 βŠ† Β· Β· Β· βŠ† An βŠ† . . .
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 β€œmidpoint” between anβˆ’1
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 A∞ := �∞
1200
+ n=1 An by:
1201
+ u(an
1202
+ p) = p
1203
+ 2n
1204
+ for all an
1205
+ p ∈ An
1206
+ Relations (12) and (13) hold in this extended domain: given x, y, z, w ∈ �∞
1207
+ n=1 An,
1208
+ just take n large enough so that they become, up to indifference, 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 A∞ is dense in X,
1211
+ that is A∞ = X. We first show that none of the sets An has, for its sequences of
1212
+ points an, a point of accumulation in X. Indeed, fix n and suppose by contradiction
1213
+ that an
1214
+ pk converges monotonically to a⋆ ∈ X, where, without loss of generality, we
1215
+ assume an
1216
+ pk ↑ a⋆ with a⋆ ∈ X, i.e. (pk) is an increasing sequence of integers. Denote
1217
+ 1n := (an
1218
+ 1, a0) and we have, for every k ∈ N,
1219
+ (an
1220
+ 1+pk, an
1221
+ pk) ≽ 1n ≻ 0
1222
+ By Lemma 10, we have (a⋆, a⋆) ≽ 1n. So, by transitivity, we reach (a⋆, a⋆) ≻ 0, a
1223
+ contradiction. We conclude that, fixed n, none of the sequences an with values in An
1224
+ has a limit point in X.
1225
+ To prove A∞ = X, the implication A∞ βŠ† X is trivial by construction. Now we
1226
+ want to show A∞ βŠ‡ X, that is all the elements of X belong to the closure of A∞
1227
+ as well. Fix x ∈ X such that, without loss of generality, x β‰Ώ a0. For n β‰₯ 1, define
1228
+ yn := sup{y ∈ An : x β‰Ώ y}. Note that a0 ∈ {y ∈ An : x β‰Ώ y}, so this set is nonempty
1229
+ and we can write x β‰Ώ yn β‰Ώ a0. By Lemma 2, yn ∈ X. Note further that, as shown
1230
+ before, An cannot have accumulation points in X so, as long as yn ∈ 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
+ pnβˆ’k β‰Ύ x β‰Ί an
1236
+ pn+k for every k > 0
1237
+ (21)
1238
+ We also have that
1239
+ 1n ≻ (x, yn)
1240
+ (22)
1241
+ 33
1242
+
1243
+ Indeed, if (22) were not true, then (x, an
1244
+ pn) ≽ 1n. We consider two cases: an
1245
+ 1+pn ≻ x or
1246
+ an
1247
+ 1+pn β‰Ύ x. If an
1248
+ 1+pn ≻ x, then, thanks to A1, we reach the following contradiction:
1249
+ 1n ∼ (an
1250
+ 1+pn, an
1251
+ pn) ≻ (x, an
1252
+ pn) ≽ 1n
1253
+ (23)
1254
+ So an
1255
+ 1+pn β‰Ύ 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) ≽ (yn, yn) ∼ 0,
1257
+ leading to
1258
+ 1n ≻ (x, yn) ≽ 0
1259
+ (24)
1260
+ Now, when n β†’ ∞, as the sets An+1 βŠ‡ An βŠ‡ Anβˆ’1 . . . are nested one into the
1261
+ other by (20), we can write, for every n β‰₯ 1, yn β‰Ύ yn+1 β‰Ύ x. Thus, the points yn form
1262
+ a non-decreasing sequence that is bounded from above by x. Call y⋆ the limit of this
1263
+ sequence, that is well-defined by Lemma 1. Since a0 β‰Ύ y⋆ β‰Ύ x, by Lemma 2 it follows
1264
+ that y⋆ ∈ X. In particular, by Lemma 6 we have y⋆ ∈ A∞, because, for every fixed
1265
+ n β‰₯ 1, yn is a term of the sets An, and so (yn) ∈ AN
1266
+ ∞.
1267
+ As to the 1n terms, for n fixed, we see that
1268
+ 1n ∼ (an
1269
+ 2, an
1270
+ 1) ∼ (anβˆ’1
1271
+ 1
1272
+ , an
1273
+ 1)
1274
+ We also have that, for every n β‰₯ 1, a0 β‰Ύ an+1
1275
+ 1
1276
+ β‰Ύ an
1277
+ 1.
1278
+ Thus, the points an
1279
+ 1 form, for n β†’ ∞, a non-increasing sequence that is bounded
1280
+ from below by a0. Call a⋆ the limit of this sequence, that is well-defined by Lemma 1.
1281
+ Since a0 β‰Ύ a⋆ β‰Ύ a1, by Lemma 2 we have a⋆ ∈ X.
1282
+ Consider now (anβˆ’1
1283
+ 1
1284
+ , an
1285
+ 1) and (x, yn). By Lemma 10 and from (24) it follows that
1286
+ (a⋆, a⋆) ≽ (x, y⋆) ≽ 0
1287
+ Since, by A2, (a⋆, a⋆) ∼ 0, by transitivity (x, y⋆) ∼ 0, so that x ∼ y⋆, i.e. x = y⋆ as β‰Ώ
1288
+ is antisymmetric.
1289
+ Since x was arbitrarily chosen in X and y⋆ ∈ A∞, we can conclude x ∈ A∞, so
1290
+ that A∞ = X. Therefore, we can extend u by continuity to the whole set X by setting
1291
+ u(x) = lim
1292
+ nβ†’βˆž u(xn)
1293
+ if (xn) ∈ AN
1294
+ ∞ converges monotonically to x. Note that u : X β†’ R is well-defined.
1295
+ Indeed, to prove it is well-posed we show that if xn and yn are two sequences that
1296
+ converge to x, then limnβ†’βˆž u(xn) = limnβ†’βˆž u(yn). This follows easily by continuity of
1297
+ u.21 In light of Lemma 6, it is easy to see that u satisfies (12) and (13).
1298
+ As to uniqueness, observe that any other u that satisfies (12) and (13) can be
1299
+ 21Recall that in every topological space X continuity implies sequential continuity. The converse
1300
+ holds if X is first-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, define the
1305
+ following positive affine transformation f : Im(u) β†’ R such that
1306
+ f(x) :=
1307
+ x βˆ’ u(a0)
1308
+ u(a1) βˆ’ 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 β‰Ώ is antisymmetric.
1312
+ Now we drop this
1313
+ assumption. Let X/∼ be the quotient space with respect to the equivalence relation
1314
+ ∼. The set {x ∈ X : x ∼ y} is a closed set in X by Lemma 4, so (X/∼, Λœβ‰Ώ) is a totally
1315
+ ordered connected and separable subset of a topological space, where Λœβ‰Ώ is the total
1316
+ order induced by the weak order β‰Ώ.22 Therefore, the orders β‰Ώ and ≽ induce orders Λœβ‰Ώ
1317
+ and Λœβ‰½ on the quotient set X/∼, by setting, for all [x], [y] ∈ X/∼
1318
+ [x] Λœβ‰Ώ [y] ⇐⇒ x β‰Ώ y
1319
+ and, for all [x], [y], [z], [w] ∈ X/∼
1320
+ ([x], [y]) Λœβ‰½ ([z], [w]) ⇐⇒ (x, y) ≽ (z, w)
1321
+ It is routine to show that the orders Λœβ‰Ώ over X/∼ and Λœβ‰½ over X/∼ Γ— X/∼ inherit the
1322
+ same properties of β‰Ώ and ≽ used in the theorem. So, by what has been proved so far,
1323
+ there exists ˜u : X/∼ β†’ R that satisfies (12) and (13) for (Λœβ‰Ώ, Λœβ‰½). Let Ο€ : X β†’ X/∼ be
1324
+ the quotient map. Then, the function u : X β†’ R defined as u = ˜u β—¦ Ο€ is a well-defined
1325
+ measurable utility function, i.e. it is easily seen to satisfy (12) and (13) for (β‰Ώ, ≽).
1326
+ To conclude, we show that u satisfies (12) and (13). If x ∼ y then [x] = [y] and,
1327
+ by the theorem we have just proved, ˜u([x]) = ˜u([y]), which is (˜u β—¦ Ο€)(x) = (˜u β—¦ Ο€)(y),
1328
+ and so u(x) = u(y). If x ≻ y, then [x] ≻ [y], which implies ˜u([x]) > ˜u([y]), which is
1329
+ (˜u β—¦ Ο€)(x) > (˜u 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 ≻ x. If u(x) = u(y) then ˜u([x]) = ˜u([y]) ⇐⇒ [x] = [y] ⇐⇒ x ∼ y,
1332
+ a contradiction. If u(x) > u(y) then ˜u([x]) > ˜u([y]) ⇐⇒ [x] > [y] ⇐⇒ x ≻ y, a
1333
+ contradiction. Hence, (12) holds for u.
1334
+ By definition, we have that ([x], [y]) ≽ ([z], [w])
1335
+ ⇐⇒
1336
+ (x, y) ≽ (z, w), for all
1337
+ [x],[y],[z],[w] ∈ X/∼. So, we can write (x, y) ≽ (z, w) ⇐⇒ ([x], [y]) ≽ ([z], [w]) ⇐⇒
1338
+ ˜u([x])βˆ’Λœu([y]) β‰₯ ˜u([z])βˆ’Λœu([w]) ⇐⇒ u(x)βˆ’u(y) β‰₯ u(z)βˆ’u(w). Hence, also (13) holds
1339
+ for u.
1340
+ 22That is, Λœβ‰Ώ := β‰Ώ /∼ βŠ† X/∼ Γ— X/∼.
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/∼
1347
+ R
1348
+ Ο€
1349
+ u
1350
+ ˜u
1351
+ References
1352
+ [1] Adams, E.W. 1960. β€œSurvey of Bernoullian utility theory.” Mathematical Thinking
1353
+ in the Measurement of Behavior, edited by Solomon, H., 151–268. Glencoe.
1354
+ [2] Allais, M. 1943. β€œA la Recherche d’une Discipline Economique. L’Economie Pure.”
1355
+ Ateliers Industria, Paris.
1356
+ [3] Allais, M. 1979. β€œThe so-called Allais paradox and rational decisions under uncer-
1357
+ tainty.” Expected Utility Hypotheses and the Allais Paradox, edited by Allais, M.
1358
+ and Hagen, O., 437-681.
1359
+ [4] Allais, M. 1994. β€œThe fundamental cardinalist approach and its prospects.” Cardi-
1360
+ nalism, edited by Allais, M. and Hagen, O., 289-306. Kluwer Academic Publishers.
1361
+ [5] Alt, F. 1936. β€œΒ¨Uber die MΒ¨assbarkeit des Nutzens.” Zeitschrift fΒ¨ur NationalΒ¨okonomie
1362
+ 7: 161-169.
1363
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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 field, 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-definite 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 difference βˆ† = cmax βˆ’ 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
+ Sufficient conditions when βˆ† ≀ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
+ 12
47
+ 4.2.1
48
+ The case βˆ† < 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
+ 12
50
+ 4.2.2
51
+ The case βˆ† = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
+ 13
53
+ 4.3
54
+ Necessary and sufficient conditions for βˆ† ≀ 2
55
+ . . . . . . . . . . . . . . . . . . . . . .
56
+ 14
57
+ 4.4
58
+ Summary of sufficient 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 ≀ 2
69
+ . . . . . . . . . . . . . . . . . . . . . . . . . .
70
+ 17
71
+ 5.3
72
+ Identification 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
+ field, i.e. a smooth, rational projective surface with a fibration Ο€ : X β†’ P1 whose general fiber
88
+ is a smooth curve of genus 1. Assume also that Ο€ is relatively minimal, i.e. no fiber contains an
89
+ exceptional curve in its support. We use E/K to denote the generic fiber of Ο€, which is an elliptic
90
+ curve over the function field K := k(P1). By the Mordell-Weil theorem, the set E(K) of K-points
91
+ is a finitely 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 ∈ E(K), what values can the intersection number P1 · 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 fiber is a smooth curve of genus zero [Cos]. More
98
+ specifically, one of the ways to produce a conic bundle is by finding a pair of sections P1, P2 ∈ 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 fiber. 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 first observation
103
+ is that r β‰₯ 1 implies an infinite number of sections, so we should expect infinitely 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 difficult 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 first established in [Elk90], [Shi89], [Shi90]. It involves the definition of a Q-valued pair-
109
+ ing ⟨·, ·⟩ on E(K), called the height pairing [SS19, Section 6.5], inducing a positive-definite lattice
110
+ (E(K)/E(K)tor, ⟨·, ·⟩), 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 classified 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-definite
118
+ quadratic forms. Indeed, the free part of E(K) is generated by r terms, so the height h(P) := ⟨P, P⟩
119
+ induces a positive-definite quadratic form on r variables with coefficients in Q. If O ∈ E(K) is the
120
+ neutral section and R is the set of reducible fibers of Ο€, then by the height formula (2)
121
+ h(P) = 2 + 2(P Β· O) βˆ’
122
+ οΏ½
123
+ v∈R
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-definite quadratic form with coefficients in Z. This point of view is explored in some parts
128
+ of the paper, where we apply results such as the classical Lagrange’s 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-Hanke’s 290-theorem on universal quadratic forms [BH, Thm. 1].
131
+ 2
132
+
133
+ Statement of results. Given k ∈ Zβ‰₯0 we investigate whether there is a pair of sections P1, P2 ∈
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 first result is a complete identification 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 fibers of Ο€.
138
+ T
139
+ k is a gap number ⇔ 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 βŠ• A1
150
+ k+1
151
+ 7 , 28k βˆ’ 3, ..., 28k + 21
152
+ E6 βŠ• A1
153
+ k+1
154
+ 3 , 12k + 1, ..., 12k + 9
155
+ D5 βŠ• A2
156
+ k+1
157
+ 6 , 24k + 1, ..., 24k + 16
158
+ A4 βŠ• A3
159
+ k+1
160
+ 10 , 40k βˆ’ 4, ..., 40k + 25
161
+ A4 βŠ• A2 βŠ• A1
162
+ k+1
163
+ 15 , 60k βˆ’ 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 ≀ 2 we show that gap numbers occur with probability 1.
168
+ Theorem 5.4. If r ≀ 2, then the set of gap numbers of X, i.e. G := {k ∈ N | k is a gap number of X}
169
+ has density 1 in N, i.e.
170
+ lim
171
+ nβ†’βˆž
172
+ #G ∩ {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 IIIβˆ— fiber.
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 difference
182
+ βˆ† = cmax βˆ’cmin 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 sufficient 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 ≀ 2 (Subsection 5.2) and the classification 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 field
193
+ k of any characteristic. More precisely, X is a smooth rational projective surface with a fibration
194
+ Ο€ : X β†’ P1, with a section, whose general fiber is a smooth curve of genus 1. We assume moreover
195
+ that Ο€ is relatively minimal (i.e. each fiber has no exceptional curve in its support) [SS19, Def.
196
+ 5.2]. The generic fiber 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 defined 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 defining the
208
+ lattice T and the trivial lattice Triv(X), which are needed to define 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 ∈ E(K) be the neutral section and
214
+ R := {v ∈ P1 | Ο€βˆ’1(v) is reducible} the set of reducible fibers of Ο€. The components of a fiber
215
+ Ο€βˆ’1(v) are denoted by Θv,i, where Θv,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 define the following sublattices of NS(X), which encode the reducible fibers of Ο€:
220
+ Tv := Z⟨Θv,i | i ̸= 0⟩ for v ∈ R,
221
+ T :=
222
+ οΏ½
223
+ v∈R
224
+ Tv.
225
+ By Kodaira’s classification [SS19, Thm. 5.12], each Tv with v ∈ R is represented by a Dynkin
226
+ diagram Am, Dm or Em for some m. We also define the trivial lattice of X, namely
227
+ Triv(X) := Z⟨O, Θv,i | i β‰₯ 0, v ∈ R⟩.
228
+ Next we define 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 defines a lattice on NS(X) but not on E(K). This is achieved
231
+ by defining a Q-valued pairing, called the height pairing, given by
232
+ ⟨·, ·⟩ : E(K) Γ— E(K) β†’ Q
233
+ P, Q οΏ½β†’ βˆ’Ο•(P) Β· Ο•(Q),
234
+ where Ο• : E(K) β†’ NS(X) βŠ—Z Q is defined 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-definite lattice (E(K)/E(K)tor, ⟨·, ·⟩) [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
+ ⟨P, Q⟩ = 1 + (P Β· O) + (Q Β· O) βˆ’ (P Β· Q) βˆ’
240
+ οΏ½
241
+ v∈R
242
+ contrv(P, Q),
243
+ (1)
244
+ h(P) := ⟨P, P⟩ = 2 + 2(P Β· O) βˆ’
245
+ οΏ½
246
+ v∈R
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 Ο€βˆ’1(v) at Θv,i, Θv,j resp. with 0 < i < j. If P or Q meets Θv,0, then contrv(P, Q) := 0.
251
+ The minimal norm. Since E(K) is finitely 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
+ ¡ := min{h(P) > 0 | P ∈ E(K)}.
254
+ The narrow Mordell-Weil lattice. An important sublattice of E(K) is the narrow Mordell-Weil
255
+ lattice E(K)0, defined as
256
+ E(K)0 := {P ∈ E(K) | P intersects Θv,0 for all v ∈ R}
257
+ = {P ∈ E(K) | contrv(P) = 0 for all v ∈ R}.
258
+ As a subgroup, E(K)0 is torsion-free; as a sublattice, it is a positive-definite even integral lattice
259
+ with finite 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 classification 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
+ Anβˆ’1
271
+ Dn+4
272
+ Type of Ο€βˆ’1(v)
273
+ III
274
+ IIIβˆ—
275
+ IV
276
+ IVβˆ—
277
+ In
278
+ Iβˆ—
279
+ n
280
+ contrv(P)
281
+ 1
282
+ 2
283
+ 3
284
+ 2
285
+ 2
286
+ 3
287
+ 4
288
+ 3
289
+ i(nβˆ’i)
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(nβˆ’j)
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 fibers to the height pairing.
314
+ 2.2
315
+ Gap numbers
316
+ We introduce some convenient terminology to express the possibility of finding a pair of sections
317
+ with a given intersection number.
318
+ Definition 2.1. If there are no sections P1, P2 ∈ E(K) such that P1 · P2 = k, we say that X has
319
+ a k-gap or that k is a gap number of X.
320
+ Definition 2.2. We say that X is gap-free if for every k ∈ Zβ‰₯0 there are sections P1, P2 ∈ 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 define the estimates cmax, cmin for the contribution term �
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 ∈ E(K)
331
+ whose height h(P) is known and we want to determine P · O. In case P ∈ E(K)0 we have a direct
332
+ answer, namely P Β· O = h(P)/2 βˆ’ 1 by the height formula (2).
333
+ However if P /∈ E(K)0, the
334
+ computation of P Β· O depends on the contribution term cP := οΏ½
335
+ v∈R contrv(P), which by Table 1
336
+ depends on how P intersects the reducible fibers 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
+ Definition 2.4. If the set R of reducible fibers of Ο€ is not empty, we define
339
+ cmax :=
340
+ οΏ½
341
+ v∈R
342
+ max{contrv(P) | P ∈ E(K)},
343
+ cmin := min {contrv(P) > 0 | P ∈ E(K), v ∈ R} .
344
+ Remark 2.5. The case R = βˆ… 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
+ v∈R contrv(P) = 0 βˆ€P ∈ 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 fiber, then:
357
+ i) cmin > 0.
358
+ ii) cmax < 4.
359
+ iii) cmin ≀ οΏ½
360
+ v∈R contrv(P) ≀ cmax βˆ€P /∈ E(K)0. For P ∈ E(K)0, only the second inequality holds.
361
+ iv) If οΏ½
362
+ v∈R contrv(P) = cmin, then contrvβ€²(P) = cmin for some vβ€² and contrv(P) = 0 for v ΜΈ= vβ€².
363
+ Proof. Item i) is immediate from the definition 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 definition of cmax and
365
+ clearly holds for any P ∈ E(K). If P /∈ E(K)0, then cP := �
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 definition of cmin we have cmin ≀ 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 vβ€² with contrvβ€²(P) > 0, while
376
+ contrv(P) = 0 for all v ΜΈ= vβ€². In particular, contrvβ€²(P) = cmin. β– 
377
+ Explicit computation. Once we know the lattice T associated with the reducible fibers of Ο€
378
+ (Section 2.1), the computation of cmax, cmin is simple. For a fixed v ∈ 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 ∈ E(K)}
383
+ min{contrv(P) > 0 | P ∈ E(K)}
384
+ Anβˆ’1
385
+ β„“(nβˆ’β„“)
386
+ n
387
+ , where β„“ :=
388
+ οΏ½n
389
+ 2
390
+ οΏ½
391
+ nβˆ’1
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 fiber configuration (I4, IV, III, I1). The reducible fibers are I4, IV, III, so
411
+ T = A3 βŠ• A2 βŠ• 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 difference βˆ† = cmax βˆ’ cmin
438
+ In this section we explain why the value of βˆ† := cmax βˆ’ cmin is relevant to our discussion,
439
+ specially in Subsection 4.2. We also verify that βˆ† < 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 /∈ E(K)0 and h(P) is known, the difficulty of determining
442
+ P Β·O lies in the contribution term cP := οΏ½
443
+ v∈R 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 difference
445
+ βˆ† := cmax βˆ’ cmin.
446
+ Hence a smaller βˆ† means a better control over the intersection number P Β· O, which is why βˆ†
447
+ plays an important role in determining possible intersection numbers. In Subsection 4.3 we assume
448
+ βˆ† ≀ 2 and state necessary and sufficient conditions for having a pair P1, P2 such that P1 Β· P2 = k
449
+ for a given k β‰₯ 0. If however βˆ† > 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 βˆ† > 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 βˆ† = 2 and βˆ† > 2 are in Table 3 and 4 respectively. In particular we have βˆ† < 2 whenever
453
+ E(K) is torsion-free.
454
+ No.
455
+ T
456
+ E(K)
457
+ cmax
458
+ cmin
459
+ 24
460
+ AβŠ•5
461
+ 1
462
+ Aβˆ—
463
+ 1
464
+ βŠ•3 βŠ• Z/2Z
465
+ 5
466
+ 2
467
+ 1
468
+ 2
469
+ 38
470
+ A3 βŠ• AβŠ•3
471
+ 1
472
+ Aβˆ—
473
+ 1 βŠ• ⟨1/4⟩ βŠ• Z/2Z
474
+ 5
475
+ 2
476
+ 1
477
+ 2
478
+ 53
479
+ A5 βŠ• AβŠ•2
480
+ 1
481
+ ⟨1/6⟩ βŠ• Z/2Z
482
+ 5
483
+ 2
484
+ 1
485
+ 2
486
+ 57
487
+ D4 βŠ• AβŠ•3
488
+ 1
489
+ Aβˆ—
490
+ 1 βŠ• (Z/2Z)βŠ•2
491
+ 5
492
+ 2
493
+ 1
494
+ 2
495
+ 58
496
+ AβŠ•2
497
+ 3
498
+ βŠ• A1
499
+ Aβˆ—
500
+ 1 βŠ• Z/4Z
501
+ 5
502
+ 2
503
+ 1
504
+ 2
505
+ 61
506
+ AβŠ•3
507
+ 2
508
+ βŠ• A1
509
+ ⟨1/6⟩ βŠ• Z/3Z
510
+ 5
511
+ 2
512
+ 1
513
+ 2
514
+ Table 3: Cases with βˆ† = 2
515
+ 8
516
+
517
+ No.
518
+ T
519
+ E(K)
520
+ cmax
521
+ cmin
522
+ βˆ†
523
+ 41
524
+ A2 βŠ• AβŠ•4
525
+ 1
526
+ 1
527
+ 6
528
+ οΏ½
529
+ 2
530
+ 1
531
+ 1
532
+ 2
533
+ οΏ½
534
+ βŠ• Z/2Z
535
+ 8
536
+ 3
537
+ 1
538
+ 2
539
+ 13
540
+ 6
541
+ 42
542
+ AβŠ•6
543
+ 1
544
+ Aβˆ—
545
+ 1
546
+ βŠ•2 βŠ• (Z/2Z)βŠ•2
547
+ 3
548
+ 1
549
+ 2
550
+ 5
551
+ 2
552
+ 59
553
+ A3 βŠ• A2 βŠ• AβŠ•2
554
+ 1
555
+ ⟨1/12⟩ βŠ• Z/2Z
556
+ 8
557
+ 3
558
+ 1
559
+ 2
560
+ 13
561
+ 6
562
+ 60
563
+ A3 βŠ• AβŠ•4
564
+ 1
565
+ ⟨1/4⟩ βŠ• (Z/2Z)βŠ•2
566
+ 3
567
+ 1
568
+ 2
569
+ 5
570
+ 2
571
+ Table 4: Cases with βˆ† > 2
572
+ Proof. By searching Table 8 for all cases with βˆ† = 2 and βˆ† > 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 βˆ† < 2. β– 
575
+ 2.5
576
+ The quadratic form QX
577
+ We define the positive-definite quadratic form with integer coefficients 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 ∈ 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 definition of QX consists in clearing denominators of the rational quadratic form induced
582
+ by the height pairing; the only question is how to find 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 coefficients in Q; we define QX by multiplying q by
585
+ some integer d > 0 so as to produce coefficients in Z. We show that d may always be chosen as the
586
+ determinant of the narrow lattice E(K)0.
587
+ Definition 2.9. Let X with r β‰₯ 1. Let P1, ..., Pr be generators of the free part of E(K). Define
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 coefficients 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 coefficients.
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 Lβˆ— [OS91, Main Thm.], so we may find generators P 0
594
+ 1 , ..., P 0
595
+ r
596
+ of L such that the Gram matrix B0 := (⟨P 0
597
+ i , P 0
598
+ j ⟩)i,j of L is the inverse of the Gram matrix
599
+ B := (⟨Pi, Pj⟩)i,j of Lβˆ—.
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)βˆ’1x
608
+ = xtadj(B0)x,
609
+ as claimed. To prove that A := adj(B0) has integer coefficients, notice that the Gram matrix
610
+ B0 of L = E(K)0 has integer coefficients (as E(K)0 is an even lattice), then so does A. β– 
611
+ We close this subsection with a simple consequence of the definition of QX.
612
+ Lemma 2.11. If h(P) = m for some P ∈ E(K), then QX represents d · 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 ∈ Z and Q is a torsion element (possibly zero). Since torsion sections do not contribute to the
615
+ height pairing, then h(P βˆ’ Q) = h(P) = m. Hence
616
+ QX(a1, ..., ar) = d Β· h(a1P1 + ... + arPr)
617
+ = d Β· h(P βˆ’ 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 first note some general properties of torsion sections. As the height pairing is positive-
624
+ definite on E(K)/E(K)tor, torsion sections are inert in the sense that for each Q ∈ E(K)tor we
625
+ have ⟨Q, P⟩ = 0 for all P ∈ 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 ∈ E(K)tor. In particular, if O is the neutral section, then Q·O = 0 for all Q ∈ 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 field 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 find P1, P2 ∈
634
+ E(K) such that P1 Β· P2 = k for a given k ∈ Zβ‰₯0. This is particularly useful when βˆ† β‰₯ 2, in which
635
+ case E(K)tor is not trivial by Lemma 2.8.
636
+ The idea is as follows. Given k ∈ Zβ‰₯0, suppose we can find P ∈ E(K)0 with height h(P) = 2k.
637
+ By the height formula (2), P Β· O = k βˆ’ 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 ∈ E(K)0 such that h(P) = 2k. Then P · Q = k for all Q ∈ E(K)tor \ {O}.
642
+ Proof. Assume there is some Q ∈ E(K)tor \ {O}. By Theorem 3.1, Q · O = 0 and by the height
643
+ formula (2), 2k = 2 + 2(P Β· O) βˆ’ 0, hence P Β· O = k βˆ’ 1. We use the height formula (1) for ⟨P, Q⟩
644
+ in order to conclude that P · Q = k. Since P ∈ E(K)0, it intersects the neutral component Θv,0 of
645
+ every reducible fiber Ο€βˆ’1(v), so contrv(P, Q) = 0 for all v ∈ R. Hence
646
+ 0 = ⟨P, Q⟩
647
+ = 1 + P Β· O + Q Β· O βˆ’ P Β· Q βˆ’
648
+ οΏ½
649
+ v∈R
650
+ contrv(P, Q)
651
+ = 1 + (k βˆ’ 1) + 0 βˆ’ P Β· Q βˆ’ 0
652
+ = k βˆ’ P Β· Q. β– 
653
+ 4
654
+ Existence of a pair of sections with a given intersection number
655
+ Given k ∈ Zβ‰₯0, we state necessary and (in most cases) sufficient conditions for having
656
+ P1 ·P2 = k for some P1, P2 ∈ E(K). Necessary conditions are stated in generality in Subsection 4.1,
657
+ while sufficient ones depend on the value of βˆ† and are treated separately in Subsection 4.2. In
658
+ Subsection 4.4, we collect all sufficient conditions proven in this section.
659
+ 4.1
660
+ Necessary Conditions
661
+ If k ∈ Zβ‰₯0, we state necessary conditions for having P1Β·P2 = k for some sections P1, P2 ∈ E(K).
662
+ We note that the value of βˆ† is not relevant in this subsection, although it plays a decisive role for
663
+ sufficient conditions in Subsection 4.2.
664
+ Lemma 4.1. Let k ∈ Zβ‰₯0. If P1 Β· P2 = k for some P1, P2 ∈ E(K), then one of the following holds:
665
+ i) h(P) = 2 + 2k for some P ∈ E(K)0.
666
+ ii) h(P) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin] for some P /∈ 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 βˆ’ c, where c := οΏ½
669
+ v contrv(P1). If P1 ∈ E(K)0, then c = 0
670
+ and h(P1) = 2 + 2k, hence i) holds. If P1 /∈ E(K)0, then cmin ≀ c ≀ cmax by Lemma 2.7. But
671
+ h(P1) = 2 + 2k βˆ’ c, therefore 2 + 2k βˆ’ cmax ≀ h(P1) ≀ 2 + 2k βˆ’ cmin, i.e. ii) holds. β– 
672
+ Corollary 4.2. Let k ∈ Zβ‰₯0. If P1 Β· P2 = k for some P1, P2 ∈ E(K), then QX represents some
673
+ integer in [d Β· (2 + 2k βˆ’ 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) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin] and by
677
+ Lemma 2.11, QX represents d Β· h(P) ∈ [d Β· (2 + 2k βˆ’ cmax), d Β· (2 + 2k βˆ’ cmin)]. In both i) and ii),
678
+ QX represents some integer in [d Β· (2 + 2k βˆ’ cmax), d Β· (2 + 2k)]. β– 
679
+ 11
680
+
681
+ 4.2
682
+ Sufficient conditions when βˆ† ≀ 2
683
+ In this subsection we state sufficient conditions for having P1 Β· P2 = k for some P1, P2 ∈ E(K)
684
+ under the assumption that βˆ† ≀ 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 βˆ† < 2 and βˆ† = 2 separately, as the latter needs more
686
+ attention.
687
+ 4.2.1
688
+ The case βˆ† < 2
689
+ We first prove Lemma 4.3, which gives sufficient conditions assuming βˆ† < 2, then Corollary 4.5,
690
+ which states sufficient conditions in terms of integers represented by QX.
691
+ This is followed by
692
+ Corollary 4.6, which is a simplified version of Corollary 4.5.
693
+ Lemma 4.3. Assume βˆ† < 2 and let k ∈ Zβ‰₯0. If h(P) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin] for some
694
+ P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).
695
+ Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P Β· O) βˆ’ c,
696
+ where c := οΏ½
697
+ v contrv(P). Since h(P) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin], then
698
+ 2 + 2k βˆ’ cmax ≀ 2 + 2(P Β· O) βˆ’ c ≀ 2 + 2k βˆ’ cmin
699
+ β‡’ c βˆ’ cmax
700
+ 2
701
+ ≀ P Β· O βˆ’ k ≀ c βˆ’ cmin
702
+ 2
703
+ .
704
+ Therefore P Β· O βˆ’ k is an integer in I :=
705
+ οΏ½ cβˆ’cmax
706
+ 2
707
+ , cβˆ’cmin
708
+ 2
709
+ οΏ½. We prove that 0 is the only integer in
710
+ I, so that P Β· O βˆ’ k = 0, i.e. P Β· O = k. First notice that c ΜΈ= 0, as P /∈ E(K)0. By Lemma 2.7 iii),
711
+ cmin ≀ c ≀ cmax, consequently cβˆ’cmax
712
+ 2
713
+ ≀ 0 ≀ cβˆ’cmin
714
+ 2
715
+ , i.e. 0 ∈ I. Moreover βˆ† < 2 implies that I has
716
+ length cmaxβˆ’cmin
717
+ 2
718
+ = βˆ†
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 sufficient condition in terms of integers represented
723
+ by the quadratic form QX (Section 2.5).
724
+ Corollary 4.5. Assume βˆ† < 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 βˆ’ cmax), d Β· (2+ 2k βˆ’ cmin)], then P1 Β· P2 = k for some P1, P2 ∈ E(K).
726
+ Proof. Let a1, ..., ar ∈ Z such that QX(a1, ..., ar) ∈ [d Β· (2 + 2k βˆ’ cmax), d Β· (2 + 2k βˆ’ cmin)] with
727
+ d ∀ QX(a1, ..., ar). Let P := a1P1 + ... + arPr, where P1, ..., Pr are generators of the free part of
728
+ E(K). Then d ∀ QX(a1, ..., ar) = d · h(P), which implies that h(P) /∈ Z. In particular P /∈ E(K)0
729
+ since E(K)0 is an integer lattice. Moreover h(P) = 1
730
+ dQX(a1, ..., ar) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ 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 finding integers represented by QX,
736
+ but only finding perfect squares in an interval depending on the minimal norm ¡ (Subsection 2.1).
737
+ Corollary 4.6. Assume βˆ† < 2. If there is a perfect square n2 ∈
738
+ οΏ½
739
+ 2+2kβˆ’cmax
740
+ Β΅
741
+ , 2+2kβˆ’cmin
742
+ Β΅
743
+ οΏ½
744
+ such that
745
+ n2¡ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ E(K).
746
+ Proof. Take P ∈ E(K) such that h(P) = ¡. Since h(nP) = n2¡ /∈ Z, we must have nP /∈ E(K)0
747
+ as E(K)0 is an integer lattice. Moreover h(nP) = n2Β΅ ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin] and we are
748
+ done by Lemma 4.3. β– 
749
+ 4.2.2
750
+ The case βˆ† = 2
751
+ The statement of sufficient conditions for βˆ† = 2 is almost identical to the one for βˆ† < 2: the
752
+ only difference 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 difficulty in the case when a
754
+ section has minimal contribution term, thus the separate treatment for βˆ† = 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
+ sufficient conditions for βˆ† = 2, then Corollaries 4.9 and 4.10.
759
+ Lemma 4.7. Assume βˆ† = 2.
760
+ If there is P ∈ E(K) such that �
761
+ v∈R contrv(P) = cmin, then
762
+ P · Q = P · O + 1 for every Q ∈ E(K)tor \ {O}.
763
+ Proof. If Q ∈ E(K)tor \ {O}, then Q · O = 0 by Theorem 3.1. Moreover, by the height formula (1),
764
+ 0 = ⟨P, Q⟩ = 1 + P Β· O + 0 βˆ’ P Β· Q βˆ’
765
+ οΏ½
766
+ v∈R
767
+ contrv(P, Q). (βˆ—)
768
+ Hence it suffices to show that contrv(P, Q) = 0 βˆ€v ∈ R. By Lemma 2.7 iv), contrvβ€²(P) = cmin
769
+ for some vβ€² and contrv(P) = 0 for all v ΜΈ= vβ€². In particular P meets Θv,0, hence contrv(P, Q) = 0
770
+ for all v ΜΈ= vβ€². Thus from (βˆ—) we see that contrvβ€²(P, Q) is an integer, which we prove is 0.
771
+ We claim that Tvβ€² = A1, so that contrvβ€²(P, Q) = 0 or 1
772
+ 2 by Table 1. In this case, as contrvβ€²(P, Q)
773
+ is an integer, it must be 0, and we are done. To see that Tvβ€² = A1 we analyse contrvβ€²(P). Since
774
+ βˆ† = 2, then cmin = 1
775
+ 2 by Table 3 and contrvβ€²(P) = cmin = 1
776
+ 2. By Table 1, this only happens if
777
+ Tvβ€² = Anβˆ’1 and 1
778
+ 2 = i(nβˆ’i)
779
+ n
780
+ for some 0 ≀ i < n. The only possibility is i = 1, n = 2 and Tvβ€² = A1. β– 
781
+ With the aid of Lemma 4.7 we are able to state sufficient conditions for βˆ† = 2.
782
+ 13
783
+
784
+ Lemma 4.8. Assume βˆ† = 2 and let k ∈ Zβ‰₯0. If h(P) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin) for some
785
+ P /∈ E(K)0, then P1 · P2 = k for some P1, P2 ∈ E(K).
786
+ Proof. Let O ∈ E(K) be the neutral section. By the height formula (2), h(P) = 2 + 2(P Β· O) βˆ’ 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) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin), implies that
790
+ P Β· O βˆ’ k is an integer in Iβ€² :=
791
+ οΏ½ cβˆ’cmax
792
+ 2
793
+ , cβˆ’cmin
794
+ 2
795
+ οΏ½.
796
+ Since Iβ€² is half-open with length cmaxβˆ’cmin
797
+ 2
798
+ = βˆ†
799
+ 2 = 1, then Iβ€² contains exactly one integer. If
800
+ 0 ∈ Iβ€², then P Β· O βˆ’ k = 0, i.e. P Β· O = k and we are done. Hence we assume 0 /∈ Iβ€².
801
+ We claim that P Β·O = k βˆ’1. First, notice that if c > cmin, then the inequalities cmin < c ≀ cmax
802
+ give cβˆ’cmax
803
+ 2
804
+ ≀ 0 < cβˆ’cmin
805
+ 2
806
+ , i.e. 0 ∈ Iβ€², which is a contradiction. Hence c = cmin. Since βˆ† = 2, then
807
+ Iβ€² = [βˆ’1, 0), whose only integer is βˆ’1. Thus P Β· O βˆ’ k = βˆ’1, i.e. P Β· O = k βˆ’ 1, as claimed.
808
+ Finally, let Q ∈ E(K)tor \ {O}, so that P · Q = P · 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 βˆ† = 2.
811
+ Similarly to the case βˆ† < 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 βˆ† = 2 and let d := det E(K)0. If QX represents an integer not divisible
815
+ by d in the interval [dΒ·(2+2k βˆ’cmax), dΒ·(2+2k βˆ’cmin)), then P1 Β·P2 = k for some P1, P2 ∈ 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 βˆ† = 2. If there is a perfect square n2 ∈
818
+ οΏ½
819
+ 2+2kβˆ’cmax
820
+ Β΅
821
+ , 2+2kβˆ’cmin
822
+ Β΅
823
+ οΏ½
824
+ such that
825
+ n2¡ /∈ Z, then P1 · P2 = k for some P1, P2 ∈ 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 sufficient conditions for βˆ† ≀ 2
829
+ For completeness, we present a unified statement of necessary and sufficient conditions assuming
830
+ βˆ† ≀ 2, which follows naturally from results in Subsections 4.1 and 4.2.
831
+ Lemma 4.11. Assume βˆ† ≀ 2 and let k ∈ Zβ‰₯0. Then P1 Β· P2 = k for some P1, P2 ∈ E(K) if and
832
+ only if one of the following holds:
833
+ i) h(P) = 2 + 2k for some P ∈ E(K)0.
834
+ ii) h(P) ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ cmin) for some P /∈ E(K)0.
835
+ iii) h(P) = 2 + 2k βˆ’ cmin and οΏ½
836
+ v∈R contrv(P) = cmin for some P ∈ 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
+ suffices to to apply Lemma 4.3 when βˆ† < 2 and by Lemma 4.8 when βˆ† = 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 βˆ’ c, where c := οΏ½
841
+ v contrv(P1).
842
+ If c = 0, then P1 ∈ E(K)0 and h(P1) = 2+2k, so i) holds. Hence we let c ̸= 0, i.e. P1 /∈ E(K)0,
843
+ so that cmin ≀ c ≀ cmax by Lemma 2.7. In case c = cmin, then h(P1) = 2 + 2k βˆ’ cmin and iii) holds.
844
+ Otherwise cmin < c ≀ cmax, which implies 2 + 2k βˆ’ cmax ≀ h(P1) < 2 + 2k βˆ’ cmin, so ii) holds. β– 
845
+ 14
846
+
847
+ 4.4
848
+ Summary of sufficient conditions
849
+ For the sake of clarity, we summarize in a single proposition all sufficient conditions for having
850
+ P1 · P2 = k for some P1, P2 ∈ E(K) proven in this section.
851
+ Proposition 4.12. Let k ∈ Zβ‰₯0. If one of the following holds, then P1 Β· P2 = k for some P1, P2 ∈
852
+ E(K).
853
+ 1) h(P) = 2 + 2k for some P ∈ E(K)0.
854
+ 2) h(P) = 2k for some P ∈ E(K)0 and E(K)tor is not trivial.
855
+ 3) βˆ† < 2 and there is a perfect square n2 ∈
856
+ οΏ½
857
+ 2+2kβˆ’cmax
858
+ Β΅
859
+ , 2+2kβˆ’cmin
860
+ Β΅
861
+ οΏ½
862
+ with n2¡ /∈ Z, where ¡ is the
863
+ minimal norm (Subsection 2.1). In case βˆ† = 2, consider the right half-open interval.
864
+ 4) βˆ† < 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 βˆ’ cmax), d Β· (2 + 2k βˆ’ cmin)]. In case βˆ† = 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) βˆ’ 0, so P Β· O = k. For
868
+ 2), we apply Lemma 3.3 to conclude that P · Q = k for any Q ∈ E(K)tor \ {O}. In 3) we use
869
+ Corollary 4.6 when βˆ† < 2 and Corollary 4.10 when βˆ† = 2. In 4), we apply Corollary 4.5 if βˆ† < 2
870
+ and Corollary 4.9 if βˆ† = 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 first 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 ≀ 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 classification 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 ∈ Zβ‰₯0 there is some P ∈ 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 AβŠ•
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 AβŠ•4
892
+ 1
893
+ or A4. Then for every β„“ ∈ Zβ‰₯0
894
+ there is P ∈ E(K)0 such that h(P) = 2β„“.
895
+ Proof.
896
+ First assume AβŠ•4
897
+ 1
898
+ βŠ‚ E(K)0 and let P1, P2, P3, P4 be generators for each factor A1 in
899
+ AβŠ•4
900
+ 1 . Then h(Pi) = 2 and ⟨Pi, Pj⟩ = 0 for distinct i, j = 1, 2, 3, 4.
901
+ By Lagrange’s 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„“. Defining
907
+ P := a1P1 + a2P2 + a3P3 + a4P4 ∈ AβŠ•4
908
+ 1
909
+ βŠ‚ 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 βŠ‚ E(K)0 with generators P1, P2, P3, P4.
916
+ Then h(Pi) = 2 for i = 1, 2, 3, 4 and
917
+ ⟨Pi, Pi+1⟩ = βˆ’1 for i = 1, 2, 3. We need to find integers x1, ..., x4 such that h(P) = 2β„“, where
918
+ P := x1P1 + ... + x4P4 ∈ A4 βŠ‚ E(K)0. Equivalently, we need that
919
+ β„“ = 1
920
+ 2⟨P, P⟩ = x2
921
+ 1 + x2
922
+ 2 + x2
923
+ 3 + x2
924
+ 4 βˆ’ x1x2 βˆ’ x2x3 βˆ’ x3x4.
925
+ Therefore β„“ must be represented by q(x1, ..., x4) := x2
926
+ 1 + x2
927
+ 2 + x2
928
+ 3 + x2
929
+ 4 βˆ’ x1x2 βˆ’ x2x3 βˆ’ x3x4. We
930
+ prove that q represents all positive integers. Notice that q is positive-definite, since it is induced
931
+ by ⟨·, ·⟩. By Bhargava-Hanke’s 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 ∈ 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 suffices to prove that E(K)0 has a sublattice
940
+ isomorphic to AβŠ•4
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 βŠ• A1 has an AβŠ•4
949
+ 1
950
+ sublattice. This is the case because D4 admits an AβŠ•4
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 βˆ’ x1x2 βˆ’ x2x3 βˆ’ 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, βˆ’2, βˆ’1
972
+ 7
973
+ 1, 1, βˆ’2, 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, βˆ’1
986
+ 21
987
+ 1, 5, 0, 0
988
+ 22
989
+ 1, 5, 0, βˆ’1
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, βˆ’3
998
+ 31
999
+ 1, 3, βˆ’4, βˆ’2
1000
+ 34
1001
+ 3, 0, 5, 0
1002
+ 35
1003
+ 1, 2, βˆ’2, 4
1004
+ 37
1005
+ 1, 0, 6, 0
1006
+ 42
1007
+ 1, 1, βˆ’4, 3
1008
+ 58
1009
+ 3, 0, 7, 0
1010
+ 93
1011
+ 1, 1, βˆ’10, 0
1012
+ 110
1013
+ 1, βˆ’2, 3, βˆ’8
1014
+ 145
1015
+ 1, 0, 12, 0
1016
+ 203
1017
+ 1, βˆ’5, βˆ’9, 8
1018
+ 290
1019
+ 1, 0, 17, 0
1020
+ Table 5: Representation of the critical integers in Bhargava-Hanke’s 290-theorem.
1021
+ 5.2
1022
+ Gaps with probability 1 in rank r ≀ 2
1023
+ Fix a rational elliptic surface Ο€ : X β†’ P1 with Mordell-Weil rank r ≀ 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 ∈ N | k is a gap number of X} is the set of gap numbers, then G βŠ‚ N has density 1, i.e.
1026
+ d(G) := lim
1027
+ nβ†’βˆž
1028
+ #G ∩ {1, ..., n}
1029
+ n
1030
+ = 1.
1031
+ 17
1032
+
1033
+ We adopt the following strategy. If k ∈ N \ G, then P1 · P2 = k for some P1, P2 ∈ E(K) and
1034
+ by Corollary 4.2 the quadratic form QX represents some integer t depending on k. This defines 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 ≀ 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-definite quadratic form on r = 1, 2 variables with integer coeffi-
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
+ coefficients of Q, let Sβ€² be the set of integers representable by the primitive form Qβ€² := 1
1042
+ d Β· Q. By
1043
+ construction Sβ€² is a rescaling of S, so d(S) = 0 if and only if d(Sβ€²) = 0.
1044
+ If r = 1, then Qβ€²(x1) = x2
1045
+ 1 and Sβ€² is the set of perfect squares, so clearly d(Sβ€²) = 0. If r = 2,
1046
+ then Qβ€² is a binary quadratic form and the number of elements in Sβ€² bounded from above by x > 0
1047
+ is given by C Β·
1048
+ x
1049
+ √log x + o(x) with C > 0 a constant and limxβ†’βˆž
1050
+ o(x)
1051
+ x
1052
+ = 0 [Ber12, p. 91]. Thus
1053
+ d(Sβ€²) = lim
1054
+ xβ†’βˆž
1055
+ C
1056
+ √log 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 ≀ 2. Then
1061
+ the set G := {k ∈ 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 finite, 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 ∈ Jkn := [d Β· (2 + 2kn βˆ’ cmax), d Β· (2 + 2kn)] represented by the quadratic form QX. Let T
1068
+ be the set of integers represented by QX and define 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 ∩ {1, ..., N} and TN := T ∩ {1, ..., N}.
1071
+ Since T has density zero,
1072
+ #TN = o(N), i.e.
1073
+ #TN
1074
+ N
1075
+ β†’ 0 when N β†’ ∞ and we need to prove that #SN = o(N). We analyze
1076
+ the function f restricted to SN. Notice that as tn ∈ Jkn, then kn ≀ N implies tn ≀ d Β· (2 + 2kn) ≀
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 #gβˆ’1(t) ≀ 2 for all t ∈ TdΒ·(2+2N), in which case #SN ≀ 2 Β· #TdΒ·(2+2N) = o(N)
1079
+ and we are done. Assume by contradiction that gβˆ’1(t) contains three distinct elements, say kβ„“1 <
1080
+ kβ„“2 < kβ„“3 with t = tβ„“1 = tβ„“2 = tβ„“3. Since tβ„“i ∈ Jkβ„“i for each i = 1, 2, 3, then t ∈ Jkβ„“1 ∩ Jkβ„“2 ∩ Jkβ„“3. We
1081
+ prove that Jkβ„“1 and Jkβ„“3 are disjoint, which yields a contradiction. Indeed, since kβ„“1 < kβ„“2 < kβ„“3,
1082
+ in particular kβ„“3 βˆ’ kβ„“1 β‰₯ 2, therefore d Β· (2 + 2kβ„“1) ≀ d Β· (2 + 2kβ„“3 βˆ’ 4). But cmax < 4 by Lemma 2.7,
1083
+ so d Β· (2 + 2kβ„“1) < d Β· (2 + 2kβ„“3 βˆ’ cmax), i.e. max Jkβ„“1 < min Jkβ„“3. Thus Jkβ„“1 ∩ Jkβ„“3 = βˆ…, as desired. β– 
1084
+ 5.3
1085
+ Identification 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 finding 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 difficult 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 βˆ† < 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 sufficient 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) ≃ Lβˆ— βŠ• 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 sufficient
1106
+ conditions for having P1 · P2 = k for some P1, P2 ∈ 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 · P2 = k for some P1, P2 ∈
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 ∈
1111
+ οΏ½
1112
+ 2+2kβˆ’cmax
1113
+ Β΅
1114
+ , 2+2kβˆ’cmin
1115
+ Β΅
1116
+ οΏ½
1117
+ such that ¡ · n /∈ 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
+ ¡ ∈ 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 ∈ Z. We show that P1 ∈ E(K)0 implies i) while P1 /∈ E(K)0 implies ii).
1123
+ If P1 ∈ 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 /∈ E(K)0, then n0 ∀ n, hence ¡ · n =
1129
+ n
1130
+ n0 /∈ Z. Moreover, h(P1) = n2h(P) = n2¡ and by
1131
+ the height formula (2), n2Β΅ = h(P) = 2 + 2k βˆ’ c, where c := οΏ½
1132
+ v contrv(P1) ΜΈ= 0. The inequalities
1133
+ cmin ≀ c ≀ cmax then give 2+2kβˆ’cmax
1134
+ Β΅
1135
+ ≀ n2 ≀ 2+2kβˆ’cmin
1136
+ Β΅
1137
+ . Hence ii) holds.
1138
+ Conversely, assume i) or ii) holds. Since E(K) is torsion-free, βˆ† < 2 by Lemma 2.8, so we may
1139
+ apply Lemma 4.3. If i) holds, then ¡ · (2 + 2k) = m2 for some m ∈ Z. Since mP0 ∈ 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 ¡ · n /∈ Z
1145
+ is equivalent to n0 ∀ n, hence nP
1146
+ /∈ E(K)0.
1147
+ Moreover n2 ∈
1148
+ οΏ½
1149
+ 2+2kβˆ’cmax
1150
+ Β΅
1151
+ , 2+2kβˆ’cmin
1152
+ Β΅
1153
+ οΏ½
1154
+ , implies
1155
+ h(nP) = n2Β΅ ∈ [2 + 2k βˆ’ cmax, 2 + 2k βˆ’ 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 ⇔ none of
1165
+ the following are perfect squares
1166
+ first 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 βŠ• A1
1183
+ k+1
1184
+ 7 , 28k βˆ’ 3, ..., 28k + 21
1185
+ 12, 16
1186
+ 49
1187
+ E6 βŠ• A1
1188
+ k+1
1189
+ 3 , 12k + 1, ..., 12k + 9
1190
+ 3, 7
1191
+ 50
1192
+ D5 βŠ• A2
1193
+ k+1
1194
+ 6 , 24k + 1, ..., 24k + 16
1195
+ 6, 11
1196
+ 55
1197
+ A4 βŠ• A3
1198
+ k+1
1199
+ 10 , 40k βˆ’ 4, ..., 40k + 25
1200
+ 16, 20
1201
+ 56
1202
+ A4 βŠ• A2 βŠ• A1
1203
+ k+1
1204
+ 15 , 60k βˆ’ 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+2kβˆ’cmax
1227
+ Β΅
1228
+ , 2+2kβˆ’cmin
1229
+ Β΅
1230
+ οΏ½
1231
+ = [40k βˆ’ 4, 40k + 25]
1232
+ contains some n2 with 20 ∀ n. We check that 20 ∀ n for every n such that n2 = 40k + β„“, with
1233
+ β„“ = βˆ’4, ..., 25. Indeed, if 20 | n, then 400 | n2 and in particular 40 | n2. Then 40 | (n2 βˆ’ 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 fix 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 fiber. 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 IIIβˆ— fiber.
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 IIIβˆ— fiber (Theorem 5.7, No. 43). Conversely, we need to
1249
+ find 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 ∈ 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 βŠ• 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, ⟨P1, P2⟩ = 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 ⟨P1, P2⟩ = 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 find 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 ∈ 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 ≃ 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 ∈ E(K) such that P1 · 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 finding perfect squares
1286
+ in the interval
1287
+ οΏ½
1288
+ 4βˆ’cmax
1289
+ Β΅
1290
+ , 4βˆ’cmin
1291
+ Β΅
1292
+ οΏ½
1293
+ (see Table 7), considering the half-open interval in the cases with
1294
+ βˆ† = 2 (No. 53, 61).
1295
+ No.
1296
+ T
1297
+ E(K)
1298
+ Β΅
1299
+ I
1300
+ n2 ∈ I
1301
+ 20
1302
+ AβŠ•2
1303
+ 2
1304
+ βŠ• A1
1305
+ Aβˆ—
1306
+ 2 βŠ• ⟨1/6⟩
1307
+ 1
1308
+ 6
1309
+ [13, 23]
1310
+ 42
1311
+ 27
1312
+ E6
1313
+ Aβˆ—
1314
+ 2
1315
+ 2
1316
+ 3
1317
+ [4, 4]
1318
+ 22
1319
+ 29
1320
+ A5 βŠ• A1
1321
+ Aβˆ—
1322
+ 1 βŠ• ⟨1/6⟩
1323
+ 1
1324
+ 6
1325
+ [12, 21]
1326
+ 42
1327
+ 31
1328
+ A4 βŠ• 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 βŠ• A2 βŠ• A1
1343
+ Aβˆ—
1344
+ 1 βŠ• ⟨1/12⟩
1345
+ 1
1346
+ 12
1347
+ [22, 28]
1348
+ 52
1349
+ 40
1350
+ AβŠ•2
1351
+ 2
1352
+ βŠ• AβŠ•2
1353
+ 1
1354
+ ⟨1/6βŸ©βŠ•2
1355
+ 1
1356
+ 6
1357
+ [10, 21]
1358
+ 42
1359
+ 53
1360
+ A5 βŠ• AβŠ•2
1361
+ 1
1362
+ ⟨1/6⟩ βŠ• Z/2Z
1363
+ 1
1364
+ 6
1365
+ [9, 12]
1366
+ 32
1367
+ 59
1368
+ A3 βŠ• A2 βŠ• AβŠ•2
1369
+ 1
1370
+ ⟨1/12⟩ βŠ• Z/2Z
1371
+ 1
1372
+ 12
1373
+ [16, 42]
1374
+ 42, 52, 62
1375
+ 61
1376
+ AβŠ•3
1377
+ 2
1378
+ βŠ• A1
1379
+ ⟨1/6⟩ βŠ• Z/3Z
1380
+ 1
1381
+ 6
1382
+ [9, 12]
1383
+ 32
1384
+ Table 7: Perfect squares in the interval I :=
1385
+ οΏ½
1386
+ 4βˆ’cmax
1387
+ Β΅
1388
+ , 4βˆ’cmin
1389
+ Β΅
1390
+ οΏ½
1391
+ .
1392
+ In No. 59 we have βˆ† > 2, so a particular treatment is needed. Let T = Tv1 βŠ• Tv2 βŠ• Tv3 βŠ• Tv4 =
1393
+ A3 βŠ• A2 βŠ• A1 βŠ• 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 fibers at Θv1,2, Θv2,1, Θv3,1, Θv4,1 [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 βˆ’ 2 Β· 2
1400
+ 4
1401
+ βˆ’ 1 Β· 2
1402
+ 3
1403
+ βˆ’ 1
1404
+ 2 βˆ’ 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, βˆ†.
1413
+ No.
1414
+ r
1415
+ T
1416
+ E(K)0
1417
+ E(K)
1418
+ cmax
1419
+ cmin
1420
+ βˆ†
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
+ Eβˆ—
1434
+ 8
1435
+ 1
1436
+ 2
1437
+ 1
1438
+ 2
1439
+ 0
1440
+ 3
1441
+ 6
1442
+ A2
1443
+ E6
1444
+ Eβˆ—
1445
+ 6
1446
+ 2
1447
+ 3
1448
+ 2
1449
+ 3
1450
+ 0
1451
+ 4
1452
+ AβŠ•2
1453
+ 1
1454
+ D6
1455
+ Dβˆ—
1456
+ 6
1457
+ 3
1458
+ 2
1459
+ 1
1460
+ 1
1461
+ 2
1462
+ 5
1463
+ 5
1464
+ A3
1465
+ D5
1466
+ Dβˆ—
1467
+ 5
1468
+ 1
1469
+ 3
1470
+ 4
1471
+ 1
1472
+ 4
1473
+ 6
1474
+ A2 βŠ• A1
1475
+ A5
1476
+ Aβˆ—
1477
+ 5
1478
+ 7
1479
+ 6
1480
+ 1
1481
+ 2
1482
+ 2
1483
+ 3
1484
+ 7
1485
+ AβŠ•3
1486
+ 1
1487
+ D4 βŠ• A1
1488
+ Dβˆ—
1489
+ 4 βŠ• Aβˆ—
1490
+ 1
1491
+ 3
1492
+ 2
1493
+ 1
1494
+ 2
1495
+ 1
1496
+ 8
1497
+ 4
1498
+ A4
1499
+ A4
1500
+ Aβˆ—
1501
+ 4
1502
+ 6
1503
+ 5
1504
+ 4
1505
+ 5
1506
+ 2
1507
+ 5
1508
+ 9
1509
+ D4
1510
+ D4
1511
+ Dβˆ—
1512
+ 4
1513
+ 1
1514
+ 1
1515
+ 0
1516
+ 10
1517
+ A3 βŠ• A1
1518
+ A3 βŠ• A1
1519
+ Aβˆ—
1520
+ 3 βŠ• Aβˆ—
1521
+ 1
1522
+ 3
1523
+ 2
1524
+ 1
1525
+ 2
1526
+ 1
1527
+ 11
1528
+ AβŠ•2
1529
+ 2
1530
+ AβŠ•2
1531
+ 2
1532
+ Aβˆ—
1533
+ 2
1534
+ βŠ•2
1535
+ 4
1536
+ 3
1537
+ 2
1538
+ 3
1539
+ 2
1540
+ 3
1541
+ 12
1542
+ A2 βŠ• AβŠ•2
1543
+ 1
1544
+ 
1545
+ 
1546
+ 
1547
+ 
1548
+ ο£­
1549
+ 4
1550
+ βˆ’1
1551
+ 0
1552
+ 1
1553
+ βˆ’1
1554
+ 2
1555
+ βˆ’1
1556
+ 0
1557
+ 0
1558
+ βˆ’1
1559
+ 2
1560
+ βˆ’1
1561
+ 1
1562
+ 0
1563
+ βˆ’1
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
+ βˆ’1
1581
+ 1
1582
+ 5
1583
+ 3
1584
+ 1
1585
+ 0
1586
+ 3
1587
+ 6
1588
+ 3
1589
+ βˆ’1
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
+ AβŠ•4
1606
+ 1
1607
+ D4
1608
+ Dβˆ—
1609
+ 4 βŠ• Z/2Z
1610
+ 2
1611
+ 1
1612
+ 2
1613
+ 3
1614
+ 2
1615
+ 14
1616
+ AβŠ•4
1617
+ 1
1618
+ AβŠ•4
1619
+ 1
1620
+ Aβˆ—
1621
+ 1
1622
+ βŠ•4
1623
+ 2
1624
+ 1
1625
+ 2
1626
+ 3
1627
+ 2
1628
+ 15
1629
+ 3
1630
+ A5
1631
+ A2 βŠ• A1
1632
+ Aβˆ—
1633
+ 2 βŠ• Aβˆ—
1634
+ 1
1635
+ 3
1636
+ 2
1637
+ 5
1638
+ 6
1639
+ 2
1640
+ 3
1641
+ 16
1642
+ D5
1643
+ A3
1644
+ Aβˆ—
1645
+ 3
1646
+ 5
1647
+ 4
1648
+ 1
1649
+ 1
1650
+ 4
1651
+ 17
1652
+ A4 βŠ• A1
1653
+ 
1654
+ 
1655
+ ο£­
1656
+ 4
1657
+ βˆ’1
1658
+ 1
1659
+ βˆ’1
1660
+ 2
1661
+ βˆ’1
1662
+ 1
1663
+ βˆ’1
1664
+ 2
1665
+ ο£Ά
1666
+ ο£·
1667
+ ο£Έ
1668
+ 1
1669
+ 10
1670
+ 
1671
+ 
1672
+ ο£­
1673
+ 3
1674
+ 1
1675
+ βˆ’1
1676
+ 1
1677
+ 7
1678
+ 3
1679
+ βˆ’1
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 βŠ• A1
1693
+ AβŠ•3
1694
+ 1
1695
+ Aβˆ—
1696
+ 1
1697
+ βŠ•3
1698
+ 3
1699
+ 2
1700
+ 1
1701
+ 2
1702
+ 1
1703
+ 19
1704
+ A3 βŠ• A2
1705
+ 
1706
+ 
1707
+ ο£­
1708
+ 2
1709
+ 0
1710
+ βˆ’1
1711
+ 0
1712
+ 2
1713
+ βˆ’1
1714
+ βˆ’1
1715
+ βˆ’1
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
+ AβŠ•2
1746
+ 2
1747
+ βŠ• A1
1748
+ A2 βŠ• ⟨6⟩
1749
+ Aβˆ—
1750
+ 2 βŠ• ⟨1/6⟩
1751
+ 11
1752
+ 6
1753
+ 1
1754
+ 2
1755
+ 4
1756
+ 3
1757
+ 21
1758
+ A3 βŠ• AβŠ•2
1759
+ 1
1760
+ A3
1761
+ Aβˆ—
1762
+ 3 βŠ• Z/2Z
1763
+ 2
1764
+ 1
1765
+ 2
1766
+ 3
1767
+ 2
1768
+ 22
1769
+ A3 βŠ• AβŠ•2
1770
+ 1
1771
+ A1 βŠ• ⟨4⟩
1772
+ Aβˆ—
1773
+ 1 βŠ• ⟨1/4⟩
1774
+ 2
1775
+ 1
1776
+ 2
1777
+ 3
1778
+ 2
1779
+ 23
1780
+ A2 βŠ• AβŠ•3
1781
+ 1
1782
+ A1 βŠ•
1783
+ οΏ½
1784
+ 4
1785
+ βˆ’2
1786
+ βˆ’2
1787
+ 4
1788
+ οΏ½
1789
+ Aβˆ—
1790
+ 1 βŠ• 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
+ AβŠ•5
1806
+ 1
1807
+ AβŠ•3
1808
+ 1
1809
+ Aβˆ—
1810
+ 1
1811
+ βŠ•3 βŠ• Z/2Z
1812
+ 5
1813
+ 2
1814
+ 1
1815
+ 2
1816
+ 2
1817
+ 25
1818
+ 2
1819
+ A6
1820
+ οΏ½
1821
+ 4
1822
+ βˆ’1
1823
+ βˆ’1
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
+ AβŠ•2
1843
+ 1
1844
+ Aβˆ—
1845
+ 1
1846
+ βŠ•2
1847
+ 3
1848
+ 2
1849
+ 1
1850
+ 1
1851
+ 2
1852
+ 27
1853
+ E6
1854
+ A2
1855
+ Aβˆ—
1856
+ 2
1857
+ 4
1858
+ 3
1859
+ 4
1860
+ 3
1861
+ 0
1862
+ 28
1863
+ A5 βŠ• A1
1864
+ A2
1865
+ Aβˆ—
1866
+ 2 βŠ• Z/2Z
1867
+ 2
1868
+ 1
1869
+ 2
1870
+ 3
1871
+ 2
1872
+ 29
1873
+ A5 βŠ• A1
1874
+ A1 βŠ• ⟨6⟩
1875
+ Aβˆ—
1876
+ 1 βŠ• ⟨1/6⟩
1877
+ 2
1878
+ 1
1879
+ 2
1880
+ 3
1881
+ 2
1882
+ 30
1883
+ D5 βŠ• A1
1884
+ A1 βŠ• ⟨4⟩
1885
+ Aβˆ—
1886
+ 1 βŠ• ⟨1/4⟩
1887
+ 7
1888
+ 4
1889
+ 1
1890
+ 2
1891
+ 5
1892
+ 4
1893
+ 31
1894
+ A4 βŠ• A2
1895
+ οΏ½
1896
+ 8
1897
+ βˆ’1
1898
+ βˆ’1
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 βŠ• A2
1917
+ οΏ½
1918
+ 4
1919
+ βˆ’2
1920
+ βˆ’2
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 βŠ• AβŠ•2
1938
+ 1
1939
+ οΏ½
1940
+ 6
1941
+ βˆ’2
1942
+ βˆ’2
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 βŠ• AβŠ•2
1961
+ 1
1962
+ AβŠ•2
1963
+ 1
1964
+ Aβˆ—
1965
+ 1
1966
+ βŠ•2
1967
+ 2
1968
+ 1
1969
+ 2
1970
+ 3
1971
+ 2
1972
+ 35
1973
+ AβŠ•2
1974
+ 3
1975
+ AβŠ•2
1976
+ 1
1977
+ Aβˆ—
1978
+ 1
1979
+ βŠ•2 βŠ• Z/2Z
1980
+ 2
1981
+ 3
1982
+ 4
1983
+ 5
1984
+ 4
1985
+ 36
1986
+ AβŠ•2
1987
+ 3
1988
+ ⟨4βŸ©βŠ•2
1989
+ ⟨1/4βŸ©βŠ•2
1990
+ 2
1991
+ 3
1992
+ 4
1993
+ 5
1994
+ 4
1995
+ 37
1996
+ A3 βŠ• A2 βŠ• A1
1997
+ A1 βŠ• ⟨12⟩
1998
+ Aβˆ—
1999
+ 1 βŠ• ⟨1/12⟩
2000
+ 13
2001
+ 6
2002
+ 1
2003
+ 2
2004
+ 5
2005
+ 3
2006
+ 38
2007
+ A3 βŠ• AβŠ•3
2008
+ 1
2009
+ A1 βŠ• ⟨4⟩
2010
+ Aβˆ—
2011
+ 1 βŠ• ⟨1/4⟩ βŠ• Z/2Z
2012
+ 5
2013
+ 2
2014
+ 1
2015
+ 2
2016
+ 2
2017
+ 39
2018
+ AβŠ•3
2019
+ 2
2020
+ A2
2021
+ Aβˆ—
2022
+ 2 βŠ• Z/3Z
2023
+ 2
2024
+ 2
2025
+ 3
2026
+ 4
2027
+ 3
2028
+ 40
2029
+ AβŠ•2
2030
+ 2
2031
+ βŠ• AβŠ•2
2032
+ 1
2033
+ ⟨6βŸ©βŠ•2
2034
+ ⟨1/6βŸ©βŠ•2
2035
+ 7
2036
+ 3
2037
+ 1
2038
+ 2
2039
+ 11
2040
+ 6
2041
+ 24
2042
+
2043
+ 41
2044
+ A2 βŠ• AβŠ•4
2045
+ 1
2046
+ οΏ½
2047
+ 4
2048
+ βˆ’2
2049
+ βˆ’2
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
+ AβŠ•6
2068
+ 1
2069
+ AβŠ•2
2070
+ 1
2071
+ Aβˆ—
2072
+ 1
2073
+ βŠ•2 βŠ• (Z/2Z)2
2074
+ 3
2075
+ 1
2076
+ 2
2077
+ 5
2078
+ 2
2079
+ 43
2080
+ 1
2081
+ E7
2082
+ A1
2083
+ Aβˆ—
2084
+ 1
2085
+ 3
2086
+ 2
2087
+ 3
2088
+ 2
2089
+ 0
2090
+ 44
2091
+ A7
2092
+ A1
2093
+ Aβˆ—
2094
+ 1 βŠ• Z/2Z
2095
+ 2
2096
+ 7
2097
+ 8
2098
+ 11
2099
+ 8
2100
+ 45
2101
+ A7
2102
+ ⟨8⟩
2103
+ ⟨1/8⟩
2104
+ 2
2105
+ 7
2106
+ 8
2107
+ 11
2108
+ 8
2109
+ 46
2110
+ D7
2111
+ ⟨4⟩
2112
+ ⟨1/4⟩
2113
+ 7
2114
+ 4
2115
+ 1
2116
+ 3
2117
+ 4
2118
+ 47
2119
+ A6 βŠ• A1
2120
+ ⟨14⟩
2121
+ ⟨1/14⟩
2122
+ 31
2123
+ 14
2124
+ 1
2125
+ 2
2126
+ 12
2127
+ 7
2128
+ 48
2129
+ D6 βŠ• A1
2130
+ A1
2131
+ Aβˆ—
2132
+ 1
2133
+ 2
2134
+ 3
2135
+ 2
2136
+ 1
2137
+ 2
2138
+ 49
2139
+ E6 βŠ• A1
2140
+ ⟨6⟩
2141
+ ⟨1/6⟩
2142
+ 11
2143
+ 6
2144
+ 1
2145
+ 2
2146
+ 4
2147
+ 3
2148
+ 50
2149
+ D5 βŠ• A2
2150
+ ⟨12⟩
2151
+ ⟨1/12⟩
2152
+ 23
2153
+ 12
2154
+ 2
2155
+ 3
2156
+ 5
2157
+ 4
2158
+ 51
2159
+ A5 βŠ• A2
2160
+ A1
2161
+ Aβˆ—
2162
+ 1 βŠ• Z/3Z
2163
+ 13
2164
+ 6
2165
+ 2
2166
+ 3
2167
+ 3
2168
+ 2
2169
+ 52
2170
+ D5 βŠ• AβŠ•2
2171
+ 1
2172
+ ⟨4⟩
2173
+ ⟨1/4⟩ βŠ• Z/2Z
2174
+ 9
2175
+ 4
2176
+ 1
2177
+ 2
2178
+ 7
2179
+ 4
2180
+ 53
2181
+ A5 βŠ• AβŠ•2
2182
+ 1
2183
+ ⟨6⟩
2184
+ ⟨1/6⟩ βŠ• Z/2Z
2185
+ 5
2186
+ 2
2187
+ 1
2188
+ 2
2189
+ 2
2190
+ 54
2191
+ D4 βŠ• A3
2192
+ ⟨4⟩
2193
+ ⟨1/4⟩ βŠ• Z/2Z
2194
+ 2
2195
+ 3
2196
+ 4
2197
+ 5
2198
+ 4
2199
+ 55
2200
+ A4 βŠ• A3
2201
+ ⟨20⟩
2202
+ ⟨1/20⟩
2203
+ 11
2204
+ 5
2205
+ 3
2206
+ 4
2207
+ 29
2208
+ 20
2209
+ 56
2210
+ A4 βŠ• A2 βŠ• A1
2211
+ ⟨30⟩
2212
+ ⟨1/30⟩
2213
+ 71
2214
+ 30
2215
+ 1
2216
+ 2
2217
+ 28
2218
+ 15
2219
+ 57
2220
+ D4 βŠ• AβŠ•3
2221
+ 1
2222
+ A1
2223
+ Aβˆ—
2224
+ 1
2225
+ 5
2226
+ 2
2227
+ 1
2228
+ 2
2229
+ 2
2230
+ 58
2231
+ AβŠ•2
2232
+ 3
2233
+ βŠ• A1
2234
+ A1
2235
+ Aβˆ—
2236
+ 1 βŠ• Z/4Z
2237
+ 5
2238
+ 2
2239
+ 1
2240
+ 2
2241
+ 2
2242
+ 59
2243
+ A3 βŠ• A2 βŠ• AβŠ•2
2244
+ 1
2245
+ ⟨12⟩
2246
+ ⟨1/12⟩ βŠ• Z/2Z
2247
+ 8
2248
+ 3
2249
+ 1
2250
+ 2
2251
+ 13
2252
+ 6
2253
+ 60
2254
+ A3 βŠ• AβŠ•4
2255
+ 1
2256
+ ⟨4⟩
2257
+ ⟨1/4⟩ βŠ• Z/2Z
2258
+ 3
2259
+ 1
2260
+ 2
2261
+ 5
2262
+ 2
2263
+ 61
2264
+ AβŠ•3
2265
+ 2
2266
+ βŠ• A1
2267
+ ⟨6⟩
2268
+ ⟨1/6⟩ βŠ• 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. Über die Darstellung von positiven, ganzen Zahlen durch die primitive, binÀren
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
+ Classification of fibers 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
+ Master’s thesis,
2297
+ Kyoto University, 2014.
2298
+ [MP89] R. Miranda and U. Persson. Torsion groups of elliptic surfaces. Compositio Mathematica,
2299
+ 72(3):249–267, 1989.
2300
+ [Nis96]
2301
+ K. Nishiyama. The Jacobian fibrations 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
+ 60:51–160, 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
+
JtE1T4oBgHgl3EQfYQS_/content/tmp_files/load_file.txt ADDED
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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, Serre’s open image theorem for abelian surfaces asserts that there are only finitely
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 finite set. The key inputs are
9
+ Mitchell’s classification of maximal subgroups of PSp4(Fβ„“), sampling of the characteristic polyno-
10
+ mials of Frobenius, and the Khare–Wintenberger 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 ∢= Gal(Q/Q), let β„“ be a prime, and let A[β„“] ∢= A(Q)[β„“] denote the β„“-torsion points
20
+ of A(Q). Let
21
+ ρA,β„“ ∢ GQ β†’ Aut(A[β„“])
22
+ denote the Galois representation on A[β„“].
23
+ By fixing 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 ρA,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 ρA,β„“ is not surjective β€” i.e., the set of primes β„“ for which ρA,β„“(GQ) is
30
+ contained in a proper subgroup of GSp4(Fβ„“) β€” is finite.
31
+ In the elliptic curve case, Serre subsequently provided a conditional upper bound in terms of the
32
+ conductor of E on this finite set [Ser81, Th´eor`eme 22]; this bound has since been made unconditional
33
+ [Coj05, Kra95]. There are also algorithms to compute the finite set of nonsurjective primes [Zyw15],
34
+ and practical implementations in Sage [CL12].
35
+ Serre’s open image theorem for typical abelian surfaces was made explicit by Dieulefait [Die02]
36
+ who described an algorithm that returns a finite set of primes containing the set of nonsurjective
37
+ primes. In a different 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 Classification. 11F80 (primary), 11G10, 11Y16 (secondary).
41
+ 1Abelian varieties with extra endomorphisms define 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 finite 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 suffi-
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 Mitchell’s 1914 classification of
59
+ maximal subgroups of PSp4(Fβ„“) [Mit14] and sampling of characteristic polynomials of Frobenius
60
+ elements. Indeed, ρA,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 ρA,β„“(GQ) and hence let
66
+ us explore the image.
67
+ Algorithm 3.1 makes use of the fact that if the image of ρA,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 ρA,β„“(GQ) to
70
+ be contained in that subgroup. Our Algorithm 3.1 combines Dieulefait’s conditions, with some
71
+ modest improvements, to produce a finite 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 different 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 sufficiently large, provably establishes nonsurjectivity. The parameter B is a cut-off for
78
+ the number of Frobenius elements that we use to sample the conjugacy classes of ρA,β„“(GQ).
79
+ As an illustration of the interplay between theory and practice, analyzing the β€œworst case” 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
+ ∏
85
+ β„“ nonsurjective
86
+ β„“ β‰ͺ exp(N1/2+Ο΅),
87
+ where the implied constant is absolute and effectively computable.
88
+ While we believe this bound to be far from asymptotically optimal, it is the first bound in the
89
+ literature expressed in terms of the (effectively computable) conductor.
90
+ Naturally one wants to find the sufficiently 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 βˆ’ 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 ≫ 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 fixed value of B = 1000 in
120
+ Algorithm 4.1. The combined algorithm then takes about 4 hours on MIT’s 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 ρA,β„“ is surjective, we expect that the outcome of these tests should be independent
130
+ for sufficiently 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 ρA,β„“ is surjective. There is an effective bound B0 such that for any B > B0, if we sample
133
+ the characteristic polynomials of Frobenius Pp(t) for n primes p ∈ [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„“ ≀
139
+ 9
140
+ 10 tending to 3
141
+ 4 as 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 file 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
+ filenames 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 classification
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 first 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 finite 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 ρA,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 β€˜at’ 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
+ β€˜Arithmetic Geometry, Number Theory, and Computation’.
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 defined 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 ∣ N.
187
+ Let β„“ be a prime. We write Tβ„“A for the β„“-adic Tate module of A:
188
+ Tβ„“A ≃ lim
189
+ ←�
190
+ n
191
+ A[β„“n].
192
+ This is a free Zβ„“-module of rank 2g.
193
+ For each prime p, we write Frobp ∈ 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 ∀ NA. If p is a good prime for A, then the trace ap of the action of Frobp on Tβ„“A 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 βŸ¨β‹…,β‹…βŸ©. A subspace W βŠ† V is called isotropic (for βŸ¨β‹…,β‹…βŸ©) if ⟨w1,w2⟩ = 0 for all w1,w2 ∈ W.
201
+ A subspace W βŠ† V is called nondegenerate (for βŸ¨β‹…,β‹…βŸ©) if βŸ¨β‹…,β‹…βŸ© restricts to a nondegenerate form on
202
+ W. The general symplectic group of (V,βŸ¨β‹…,β‹…βŸ©) is defined as
203
+ GSp(V,βŸ¨β‹…,β‹…βŸ©) ∢= {M ∈ GL(V ) ∢ βˆƒ mult(M) ∈ FΓ—
204
+ β„“ ∢ ⟨Mv,Mw⟩ = mult(M)⟨v,w⟩ βˆ€ v,w ∈ V }.
205
+ The map M ↦ mult(M) is a surjective homomorphism from GSp(V,βŸ¨β‹…,β‹…βŸ©) to FΓ—
206
+ β„“ called the similitude
207
+ character; its kernel is the symplectic group, denoted Sp(V,βŸ¨β‹…,β‹…βŸ©).
208
+ Usually the bilinear form is
209
+ understood from the context, in which case one drops βŸ¨β‹…,β‹…βŸ© from the notation; moreover, for our
210
+
211
+ COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
212
+ 5
213
+ purposes, we will have fixed a basis for V , one in which the bilinear form is represented by the
214
+ nonsingular skew-symmetric matrix
215
+ J ∢= ( 0
216
+ I2
217
+ βˆ’I2
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 β€² β†  W from a subrepresentation U β€² of U.
222
+ Since we are chiefly concerned with computing the sets LikelyNonsurjectivePrimes(C;B) and
223
+ PossiblyNonsurjectivePrimes(C) for a fixed 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 defined
228
+ over Q and let p be a prime of good reduction for A. Then there exists a monic integral polynomial
229
+ Pp(t) ∈ Z[t] of degree 2g with constant coefficient 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 Tβ„“A. 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 βˆ’
239
+ 2g
240
+ βˆ‘
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
+ Tβ„“A ≃ (Tβ„“A)∨ βŠ—Zβ„“ 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 tβˆ’p. The integral
250
+ characteristic polynomial for the action of Frobp on (Tβ„“A)∨ is the reversed polynomial
251
+ P ∨
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 verifiable 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 βˆ’ apt3 + bpt2 βˆ’ papt + p2 for some ap,bp ∈ 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 ≑ 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 ∈ 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 definition 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] classified the maximal subgroups of
278
+ PSp4(Fβ„“) in 1914. This can be used to deduce the following classification 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 βŠ• V2 of V , where V1 and V2 are jointly defined 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 ≃ 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±3 (mod 8): a group whose image G1920 in PGSp(V,Ο‰) has order 1920.
296
+ (b) When 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 classification 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 Mitchell’s original labels and ours below.
302
+ Mitchell’s 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 β€² in the direct sum decomposition
315
+ are individually defined over Fβ„“ are the analogues of normalizers of the split Cartan subgroup of
316
+ GL2(Fβ„“). When the two subspaces V,V β€² are not individually defined 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 briefly explain why the action of GL2(Fβ„“) on Sym3(F2
322
+ β„“) preserves a nondegenerate
323
+ symplectic form. It suffices to show that the restriction to SL2(Fβ„“) fixes a vector in β‹€2 Sym3(F2
324
+ β„“).
325
+ This follows by character theory. If W is the standard 2-dimensional representation of SL2, then
326
+ we have β‹€2(Sym3 W) ≃ Sym4 W βŠ• 1 as representations of SL2.
327
+ Remark 5. One can extract explicit generators of the exceptional maximal subgroups from Mitchell’s
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„“)). Mitchell’s 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 βŠ• 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 fixes 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 Mitchell’s descriptions. In the file exceptional.m publicly available
337
+ with our code, we verify that Magma’s list of conjugacy classes of maximal subgroups of GSp4(Fβ„“)
338
+ agree with those described in Lemma 2.3 for 3 ≀ β„“ ≀ 47.
339
+ Remark 6. The classification 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-defined on PGSp4(Fβ„“) (multiplication by a scalar c ∈ FΓ—
342
+ β„“
343
+ scales the similitude character by c2) it is well-defined 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±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 Mitchell’s original classification (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 coefficient 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-defined on PGSp4(Fβ„“). For
367
+ β„“ > 2 and βˆ— ∈ {720,1920,5040}, define
368
+ (2)
369
+ Cβ„“,βˆ— ∢= {( tr(T)2
370
+ mult(T), mid(T)
371
+ mult(T)) ∣ T ∈ an exceptional subgroup of GSp4(Fβ„“) of projective order βˆ—}
372
+ Lemma 2.4.
373
+ (1) In cases 2a and 2b of Lemma 2.3:
374
+ 3Mitchell’s notation for PGSp4(Fβ„“) is AΞ½(β„“) and for PSp4(Fβ„“) is A1(β„“).
375
+ 4Explicitly, the characteristic polynomial of T is therefore t4 βˆ’ tr(T)t3 + mid(T)t2 βˆ’ 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„“ βˆ– 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 βˆ— ∈ {1920,720}, the set Cβ„“,βˆ— defined in (2) equals the reduction modulo β„“ of the elements of
386
+ the set Cβˆ— below.
387
+ C1920 = {(0,βˆ’2),(0,βˆ’1),(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,βˆ’2)}
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 βŠ• V2 to elements in V2 and vice-versa), it follows that any element
395
+ in Gβ„“ βˆ– 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 βˆ’(Ξ»3 +Β΅3)T +Ξ»3Β΅3)(T 2 βˆ’(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 file exceptional statistics.sage, we generate the corresponding
406
+ finite subgroups over the appropriate quadratic number field to compute Cβˆ—. It follows that
407
+ the corresponding values for the subgroup Gβˆ— 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 classification of maximal subgroups of GSp4(Fβ„“) and the more familiar classi-
415
+ fication 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 defined over Fβ„“ is isomorphic to
420
+ {(m1,m2) ∈ GL2(Fβ„“)2 ∣ det(m1) = det(m2)}.
421
+ In particular, the order of Mβ„“ is β„“2(β„“ βˆ’ 1)(β„“2 βˆ’ 1)2.
422
+ (2) The subgroup Mβ„“ in the case (2b) when the two isotropic subspaces V1 and V2 are individually
423
+ defined over Fβ„“ is isomorphic to
424
+ {(m1,m2) ∈ GL2(Fβ„“)2 ∣ mT
425
+ 1 m2 = Ξ»I, for some Ξ» ∈ Fβˆ—
426
+ β„“ }.
427
+ In particular, the order of Mβ„“ is β„“(β„“ βˆ’ 1)2(β„“2 βˆ’ 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 defined over Fβ„“ is isomorphic to
433
+ {m ∈ GL2(Fβ„“2) ∣ det(m) ∈ Fβˆ—
434
+ β„“ }.
435
+ In particular, the order of Mβ„“ is β„“2(β„“ βˆ’ 1)(β„“4 βˆ’ 1).
436
+ (4) The subgroup Mβ„“ in the case (2b) when the two isotropic subspaces V1 and V2 are not indi-
437
+ vidually defined over Fβ„“ is isomorphic to GU2(Fβ„“2), i.e.,
438
+ {m ∈ GL2(Fβ„“2) ∣ mT ΞΉ(m) = Ξ»I, for some Ξ» ∈ Fβˆ—
439
+ β„“ },
440
+ where ΞΉ denotes the natural extension of the Galois automorphism of Fβ„“2/Fβ„“ to GL2(Fβ„“2). In
441
+ particular, the order of Mβ„“ is β„“(β„“2 βˆ’ 1)2.
442
+ Proof. Given a direct sum decomposition V1 βŠ• 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 defined 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 Witt’s theorem, in a
452
+ suitable basis for V1βŠ•V2 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 ∢=
455
+ ⎑⎒⎒⎒⎒⎒⎒⎒⎣
456
+ 0
457
+ 1
458
+ βˆ’1
459
+ 0
460
+ 0
461
+ 1
462
+ βˆ’1
463
+ 0
464
+ ⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
465
+ .
466
+ The condition that an element (m1,m2) ∈ Aut(V1) βŠ• 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 Witt’s theorem, in a suitable basis for V1 βŠ•V2
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 ∢=
473
+ ⎑⎒⎒⎒⎒⎒⎒⎒⎣
474
+ 0
475
+ 1
476
+ 1
477
+ 0
478
+ 0
479
+ βˆ’1
480
+ βˆ’1
481
+ 0
482
+ ⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦
483
+ .
484
+ The condition that an element (m1,m2) ∈ Aut(V1) βŠ• 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 defined over Fq2 but not defined over Fq, and we let W denote the
489
+ conjugate subspace and further assume that W βŠ•W 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 βŠ— 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 βŠ•W 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) ∈ 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 Mitchell’s
504
+ work described in the previous subsection to classify the maximal subgroups of GSp4(Fβ„“) that could
505
+ occur as the image of ρA,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 ψn∢Iβ„“ β†’ FΓ—
509
+ β„“n denote a (tame) fundamental character of level n. The n Galois-conjugate
510
+ fundamental characters ψn,1,...,ψn,n of level n are given by ψn,i ∢= Οˆβ„“i
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 Jordan–HΒ¨older factor of the Iβ„“-module A[β„“]. Then V
516
+ admits a 1-dimensional Fβ„“n-vector space structure such that ρA,β„“βˆ£Iβ„“ acts on V via the character
517
+ ψd1
518
+ n,1β‹―Οˆdn
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 field K. Then A has semistable reduction at p β‰  β„“ if and only if the action of Ip βŠ‚ GK on
525
+ Tβ„“A is unipotent of length 2.
526
+ Combining these two results allows one fine 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 Jordan–HΒ¨older factor of the Fβ„“[GQ]-
529
+ module A[β„“] βŠ— Fβ„“. If β„“ is a semistable prime, then
530
+ detXβ„“ ≃ Ο΅ β‹… cycx
531
+ β„“
532
+ for some character ϡ∢GQ β†’ Fβ„“ that is unramified at β„“ and some 0 ≀ x ≀ dimXβ„“. Moreover, Ο΅120 = 1.
533
+ Proof. The first 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 ∢ Q] dividing 120 by
536
+ [LV14a, Theorem 7.2], and so this follows from Theorem 2.7 since there are no nontrivial unramified
537
+ characters of GQ.
538
+ β–‘
539
+ We can now state Dieulefait’s classification of maximal subgroups of GSp4(Fβ„“) that can occur
540
+ as the image ρA,β„“(GQ) for a semistable prime β„“ > 7.
541
+ Proposition 2.9 ([Die02]). Let A be the Jacobian of a genus 2 curve defined over Q with Weil
542
+ pairing Ο‰ on A[β„“]. If β„“ > 7 is a semistable prime, then ρA,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„“ ≀ 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(PρA,β„“) β‰₯ (β„“βˆ’1)/12. Since exp(G1920) = 2exp(S6) = 120 and exp(G720) = exp(S5) = 60,
550
+ the exceptional maximal subgroups cannot occur as ρA,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 finite Galois exten-
555
+ sion with Galois group G = Gal(K/Q) and absolute discriminant dK. Let S βŠ† G be a nonempty
556
+ subset that is closed under conjugation. By the Chebotarev density theorem, we know that
557
+ (3)
558
+ lim
559
+ xβ†’βˆž
560
+ ∣{p ≀ x ∢ p is unramified in K and Frobp ∈ S}∣
561
+ ∣{p ≀ x}∣
562
+ = ∣S∣
563
+ ∣G∣.
564
+ Let p be the least prime such that p is unramified in K and Frobp ∈ S. There are effective 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 ≀ (4log dK + 2.5[K ∢ Q] + 5)2.
570
+ The present goal is to give an effective 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 unramified in K is not sufficient 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 finite Galois,
575
+ then
576
+ (5)
577
+ log dK ≀ ([K ∢ Q] βˆ’ 1)log rad(dK) + [K ∢ Q]log([K ∢ Q]),
578
+ where radn = ∏p∣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 βŠ† ρA,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 ρA,q(Frobp) ∈ S. Assuming GRH, we have
582
+ p ≀ (4[(2q11 βˆ’ 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 ∢ Q] ≀ ∣GSp4(Fq)∣ = q4(q4 βˆ’ 1)(q2 βˆ’ 1)(q βˆ’ 1) ≀ 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 ˜K ∢= K(√m)
588
+ where m ∢= rad(2NA). Note that the primes that ramify in ˜K are precisely 2, q, and the primes of
589
+ bad reduction for A. Thus rad(d ˜
590
+ K) = rad(2qNA). Moreover [ ˜K ∢ Q] ≀ 2q11 and by (5),
591
+ log(d ˜
592
+ K) ≀ (2q11 βˆ’ 1)log rad(2qNA) + 22q11 log(2q).
593
+ Applying [MW21, Corollary 6] to the field ˜K, we get that (under GRH) there exists a prime p
594
+ satisfying the claimed bound, that does not divide m, and for which ρA,q(Frobp) ∈ 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 finite list PossiblyNonsurjectivePrimes that provably includes all nonsurjective primes β„“. We also
599
+ prove Theorem 1.2.
600
+ Since our goal is to produce a finite list (from which we will later remove extraneous primes) it
601
+ is harmless to include the finitely many bad primes as well as 2,3,5,7. Using Proposition 2.9, it
602
+ suffices to find conditions on β„“ > 7 for which ρA,β„“(GQ) could be contained in one of the maximal
603
+ subgroups of type (1) and (2) in Lemma 2.3. We first find primes β„“ for which ρA,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 finite 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 ρA,β„“ βŠ— 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 ρA,β„“ βŠ—Fβ„“ 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„“] ∢= A[β„“] βŠ—Fβ„“ 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 ≀ x ≀ dimXβ„“ and 0 ≀ y ≀ 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 Jordan–HΒ¨older factors of A[β„“], Corollary 2.8
643
+ constrains its determinant. We have detXβ„“ = Ο΅cycx
644
+ β„“ for some character ϡ∢GQ β†’ Fβ„“ unramified at β„“,
645
+ and 0 ≀ x ≀ 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„“] ≃ cyc2
649
+ β„“, we have detYβ„“ ≃ Ο΅βˆ’1 cyc2βˆ’x
650
+ β„“
651
+ . Since the
652
+ conductor is multiplicative in extensions, we conclude that cond(ϡ)2 ∣ N. By class field 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 ≑ 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 find 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 coefficients 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 pα, and f ∣ 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 fβ€² = gcd(f,120). Compute an integer Modd as follows.
677
+ (1) Choose a nonempty finite set T of auxiliary good primes p ∀ N.
678
+ (2) For each p, compute
679
+ Rp ∢= P (fβ€²)
680
+ p
681
+ (1).
682
+ (3) Let Modd = gcdp∈T (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 ∈ T , Lemma 3.2 shows that Frobfβ€²
688
+ p acts on detXβ„“ by either pfβ€²
689
+ or by 1. Thus, the action of Frobfβ€²
690
+ p on A[β„“] has an eigenvalue that is congruent to pfβ€² or 1 modulo
691
+ β„“, and so P (fβ€²)
692
+ p
693
+ (t) has a root that is congruent to 1 or pfβ€² modulo β„“. Since the roots of P (fβ€²)(t)
694
+ multiply in pairs to pfβ€², we have P (fβ€²)
695
+ p
696
+ (pfβ€²) = p2fβ€²P (fβ€²)
697
+ p
698
+ (1). Hence β„“ divides p β‹… P (fβ€²)
699
+ p
700
+ (1) = pRp.
701
+ β–‘
702
+ Using Theorem 2.1, we can give a theoretical bound on the β€œworst case” 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‰  ∣Modd∣ β‰ͺ p240
708
+ where the implied constant is absolute.
709
+ Proof. This follows from the fact that the coefficient of ti in P (fβ€²)
710
+ p
711
+ (t) has magnitude on the order
712
+ of p(2βˆ’i)fβ€² and fβ€² ≀ 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 ∨
719
+ β„“ βŠ— cycβ„“ is the unique subrepresentation
720
+ of A[β„“]
721
+ ∨ βŠ— cycβ„“. The isomorphism Tβ„“A ≃ (Tβ„“A)∨ βŠ— cycβ„“ from (1) yields an isomorphism A[β„“] ≃
722
+ (A[β„“])∨ βŠ— cycβ„“ and hence Xβ„“ ≃ Y ∨
723
+ β„“ βŠ— cycβ„“. Otherwise, A[β„“] ≃ Xβ„“ βŠ• Yβ„“ and so the nondegeneracy of
724
+ the Weil pairing gives
725
+ Xβ„“ βŠ• Yβ„“ ≃ (X∨
726
+ β„“ βŠ— cycβ„“) βŠ• (Y ∨
727
+ β„“ βŠ— cycβ„“).
728
+ Therefore either:
729
+ (a) Xβ„“ ≃ Y ∨
730
+ β„“ βŠ— cycβ„“ and Yβ„“ ≃ X∨
731
+ β„“ βŠ— cycβ„“, or
732
+ (b) Xβ„“ ≃ X∨
733
+ β„“ βŠ— cycβ„“ and Yβ„“ ≃ Y ∨
734
+ β„“ βŠ— cycβ„“ and A[β„“] ≃ Xβ„“ βŠ• Yβ„“.
735
+ We call the first 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 Serre’s 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) ∢= t4βˆ’at3+bt2βˆ’pat+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„“ ≃ Y ∨
747
+ β„“ βŠ— 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 βˆ’ (b βˆ’ 2p)t3 + p(a2 βˆ’ 2b + 2p)t2 βˆ’ p2(b βˆ’ 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 fβ€² = gcd(f,120). Compute an integer Mrelated as follows.
757
+ (1) Choose a finite set T of auxiliary good primes p ∀ N;
758
+ (2) For each p, compute the product
759
+ Rp ∢= Q(fβ€²)
760
+ p
761
+ (1)Q(fβ€²)
762
+ p
763
+ (pfβ€²)
764
+ (3) Let Mrelated = gcdp∈T (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(fβ€²)
769
+ p
770
+ (1),
771
+ Q(fβ€²)
772
+ p
773
+ (pfβ€²), or Q(fβ€²)
774
+ p
775
+ (p2fβ€²) and hence β„“ divides pRp since Q(fβ€²)
776
+ p
777
+ (p2fβ€²) = p4fβ€²Q(fβ€²)
778
+ p
779
+ (1).
780
+ β–‘
781
+ A theoretical β€œworst case” 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 β‰ͺ q22 log2(qN),
785
+ where the implied constants are absolute and effectively computable. Moreover, for such a prime p,
786
+ ∣Mrelated∣ β‰ͺ p961 β‰ͺ q21142 log1922(qN),
787
+ where the implied constants are absolute.
788
+ Proof. By Serre’s 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 ≀ pRp = pQ(fβ€²)(1)Q(fβ€²)(pfβ€²) β‰ͺ p8fβ€²+1 β‰ͺ p961,
791
+ since the coefficient of ti in Q(fβ€²)(t) has magnitude on the order of p(4βˆ’i)fβ€² and fβ€² ≀ 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 βˆ’1.) Therefore, by the Khare–Wintenberger theorem (formerly Serre’s 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 ∈ Snew
802
+ ki (Ξ“1(Ni),Ο΅i) such that
803
+ Xβ„“ β‰… ρf1,β„“ and Yβ„“ β‰… ρf2,β„“.
804
+ Furthermore, by the multiplicativity of Artin conductors, we obtain the divisibility N1N2 ∣ 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 Xβ„“βˆ£Iβ„“ and Yβ„“βˆ£Iβ„“ must each be conjugate to either of the
808
+ following subgroups of GL2(Fβ„“):
809
+ (1
810
+ βˆ—
811
+ 0
812
+ cycβ„“
813
+ ) or (ψ2
814
+ 0
815
+ 0
816
+ Οˆβ„“
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 finite and flat 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 satisfies, for all p ∀ β„“N,
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 ∈ Snew
829
+ 2
830
+ (Ξ“0(Ni)) such that
831
+ (7)
832
+ A[β„“] ≃ ρf1,β„“ βŠ• ρf2,β„“.
833
+ We may assume without loss of generality that N1 ≀
834
+ √
835
+ N. Let p ∀ N be an auxiliary prime. We
836
+ obtain from equation (7) that the integral characteristic polynomial of Frobenius factors:
837
+ Pp(t) ≑ (t2 βˆ’ ap(f1)t + p)(t2 βˆ’ ap(f2)t + p)
838
+ mod β„“;
839
+ here we use the standard property that, for f a normalised eigenform with trivial Nebentypus,
840
+ ρf,β„“(Frobp) satisfies the polynomial equation t2 βˆ’ ap(f)t + p for p β‰  β„“. In particular, we have
841
+ Res(Pp(t),t2 βˆ’ ap(f1)t + p) ≑ 0
842
+ mod β„“.
843
+ This serves as the basis of the algorithm to find 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 ≀
847
+ √
848
+ N.
849
+ (2) For each d ∈ S:
850
+ (a) compute the Hecke L-polynomial
851
+ Qd(t) ∢= ∏
852
+ f
853
+ (t2 βˆ’ ap(f)t + p),
854
+ where the product is taken over the finitely many newforms in Snew
855
+ 2
856
+ (Ξ“0(d));
857
+ (b) choose a finite set T of auxiliary primes p ∀ N;
858
+ (c) for each auxiliary prime p, compute the resultant
859
+ Rp(d) ∢= 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) ∢= gcd
865
+ p∈T
866
+ (pRp(d)).
867
+ (3) Let Mself-dual ∢= ∏d∈S 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 ∈ 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 ∈ S and some newform f1 ∈ Snew
874
+ 2
875
+ (Ξ“0(N1)) such that Res(Pp(t),t2 βˆ’ apf1t + p) ≑ 0
876
+ (mod β„“) for every p ∈ T . In particular, Rp(N1) ≑ 0 (mod β„“), so β„“ divides M(N1) and Mself-dual.
877
+ β–‘
878
+ We can again do a β€œworst case” 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 β‰ͺ q22 log2(qN), where the implied constant is absolute and effectively computable. Moreover, for
883
+ such a prime p, we have
884
+ ∣Rp(d)∣ β‰ͺ (2p1/2)8 dim Snew
885
+ 2
886
+ (Ξ“0(d)) β‰ͺ (4p)(d+1)/3,
887
+ and so all together
888
+ ∣Mself-dual∣ β‰ͺ (4q)N1/2+Ο΅,
889
+ where the implied constants are absolute.
890
+ Proof. As in Proposition 3.8, we use Serre’s open image theorem and the Effective Chebotarev
891
+ Theorem. If Rp(d) is zero integrally, then in particular Rp(d) ≑ 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 differences of the roots of Pp(t) and Qd(t),
895
+ which all have complex absolute value p1/2. Hence the pairwise differences have absolute value
896
+ at most 2p1/2.
897
+ Moreover dimSnew
898
+ 2
899
+ (Ξ“0(d)) ≀ (d + 1)/12 by [Mar05, Theorem 2].
900
+ Since there
901
+ are 8dimSnew
902
+ 2
903
+ (Ξ“0(d)) such terms multiplied to give Rp(d), the bound for Rp(d) follows. Since
904
+ Mself-dual = ∏ d∣N
905
+ d≀
906
+ √
907
+ N
908
+ pRp(d), it suffices to bound
909
+ βˆ‘
910
+ d∣N
911
+ d≀
912
+ √
913
+ N
914
+ d + 4
915
+ 3
916
+ ≀
917
+ βˆ‘
918
+ d∣N
919
+ d≀
920
+ √
921
+ N
922
+ √
923
+ N + 4
924
+ 3
925
+ ≀ Οƒ0(N)
926
+ √
927
+ N + 4
928
+ 3
929
+ .
930
+ Since Οƒ0(N) β‰ͺ 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(Ξ“0(d)), which may be
934
+ computed via modular symbols computations. One may recover Qd(t) from Hd(t) by first 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
+ √
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 Ο†βˆΆ(Z/NZ)Γ— β†’ {Β±1}. Our goal in this subsection is to find all good primes β„“ governed by Ο†,
950
+ by which we mean that
951
+ tr(ρA,β„“(Frobp)) ≑ ap ≑ 0
952
+ mod β„“
953
+ whenever Ο†(p) = βˆ’1.
954
+ We will consider the set of all quadratic Dirichlet character Ο†βˆΆ(Z/NZ)Γ— β†’ {Β±1}. Using the struc-
955
+ ture theorem for finite 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) ∢= Ο‰(N) +
958
+ ⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎩
959
+ 0
960
+ ∢ v2(N) = 0
961
+ βˆ’1
962
+ ∢ v2(N) = 1
963
+ 0
964
+ ∢ v2(N) = 2
965
+ 1
966
+ ∢ 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) ≀ 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 Ο†βˆΆ(Z/NZ)Γ— β†’ {Β±1}.
972
+ (2) For each Ο† ∈ S:
973
+ (a) Choose a nonempty finite set T of β€œauxiliary” primes p ∀ N for which ap β‰  0 and Ο†(p) = βˆ’1.
974
+ (b) Take the greatest common divisor
975
+ MΟ† ∢= gcd
976
+ p∈T
977
+ (pap),
978
+ over all auxiliary primes p.
979
+ (3) Let Mquad ∢= βˆΟ†βˆˆS 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 Ο†βˆΆ(Z/NZ)Γ— β†’ {Β±1}. Then for
984
+ every good prime p β‰  β„“ for which Ο†(p) = βˆ’1, the prime β„“ must divide the integral trace of Frobenius
985
+ ap. Hence ℓ divides Mφ 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) = βˆ’1 and ap is nonzero is bounded by a function of q.
989
+ Assuming GRH, p β‰ͺ 22d(N)q22 log2(qN), where the implied constant is absolute and effectively
990
+ computable. Moreover, we have
991
+ ∏
992
+ Ο†βˆˆS
993
+ ∏
994
+ β„“ governed
995
+ by Ο†
996
+ β„“ β‰ͺ (23d(N)q33 log3(qN))2βˆ’21βˆ’d(N) β‰ͺ 26Ο‰(N)q66 log6(qN),
997
+ where the implied constant is absolute and effectively 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 unramified outside of N). Let ρV ∢GK β†’ V ∨ denote the representation sending Frobp to
1001
+ the linear functional Ο† ↦ 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 PρA,q, we conclude in the same way as [LV14b, Proof of Lemma 21] that for any α ∈ V ∨, there
1006
+ exists an XΞ± ∈ GSp4(Fq) with tr(XΞ±) β‰  0 such that (Ξ±,XΞ±) is in the image of ρV Γ— ρA,β„“.
1007
+ Apply the effective Chebotarev density theorem to the Galois extension corresponding to ρV Γ—
1008
+ ρA,q. This has degree at most 2d(N)∣GSp4(Fq)∣ and is unramified outside of qN. Therefore, assum-
1009
+ ing GRH and combining (4) and (5), there exists a prime
1010
+ pΞ± β‰ͺ 22d(N)q22 log2(qN)
1011
+ for which (Ξ±,XΞ±) = (ρV (FrobpΞ±),ρA,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± ∈ V ∨. 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
+ p∣ap∣ ≀ p3/2 β‰ͺ 23d(N)q33 log3(qN).
1017
+ Taking the product over all nonzero Ξ± in V (of which there are 2d(N) βˆ’ 1), each β„“ will show up half
1018
+ the time, so we obtain:
1019
+ βŽ›
1020
+ ⎜⎜⎜
1021
+ ⎝
1022
+ ∏
1023
+ β„“ governed
1024
+ by Ο† ∈ S
1025
+ β„“
1026
+ ⎞
1027
+ ⎟⎟⎟
1028
+ ⎠
1029
+ 2d(N)βˆ’1
1030
+ β‰ͺ (23d(N)q33 log3(qN))
1031
+ 2d(N)βˆ’1
1032
+ ,
1033
+ which implies the result by taking the (2d(N)βˆ’1)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 ρA,β„“ is nonsurjective, β„“ > 7, and β„“ ∀ N, then Proposition 2.9 implies
1037
+ that ρA,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
+ ρA,β„“(GQ) (and, hence, ρA,β„“(GQ) βŠ— 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 ρA,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„“ βŠ— Fβ„“ is always reducible by Lemma 2.4(1b).
1044
+ Hence we may assume that ρA,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
+ ρA,β„“
1048
+ οΏ½οΏ½β†’ Gβ„“ β†’ Gβ„“/Mβ„“ = {Β±1}
1049
+ is nontrivial and unramified outside of N, and so it corresponds to a quadratic Dirichlet character
1050
+ Ο†βˆΆ(Z/NZ)Γ— β†’ {Β±1}. Lemma 2.4(1a) shows that tr(g) = 0 in Fβ„“ for any g ∈ Gβ„“ βˆ– 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 Serre’s 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
+ ∏
1059
+ β„“ nonsurjective
1060
+ β„“ β‰ͺ exp(N1/2+Ο΅),
1061
+ where the implied constant is absolute and effectively 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 finite by Serre’s 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
+ ∏
1070
+ β„“ nonsurjective
1071
+ β„“ β‰ͺ qN1/2+Ο΅.
1072
+ On the other hand, since q is the smallest surjective prime by definition, 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) β‰ͺ ∏
1075
+ β„“<q
1076
+ β„“ ≀
1077
+ ∏
1078
+ β„“ nonsurjective
1079
+ β„“ β‰ͺ qN1/2+Ο΅.
1080
+ Combining the first and last terms, we have q β‰ͺ N1/2+Ο΅ log(q), whence q β‰ͺ N1/2+Ο΅. Plugging this
1081
+ back into (8) yields the claimed bound.
1082
+ β–‘
1083
+ 4. Testing surjectivity of ρA,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
+ sufficiently large (quantified 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 field
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 ρA,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 ρA,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)–(iii) and Tests 4.5(i)–(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 briefly summarize the contents of this section. In Section 4.1, we first 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 ρA,β„“. If we find 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 ρA,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 classification 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 βŠ† 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 ∈ 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 β€˜only if’ 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„“ βˆ– 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„“ βˆ– Mβ„“. We now consider two cases
1148
+ (i) If the two lines are individually defined 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 β€² 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 β€², 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) defined over
1165
+ Q. Then ρA,2 is surjective if and only if the size of the Galois group of the splitting field 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 ρA,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 finite 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 βˆ’ apt + bpt2 βˆ’ papt + p2 and β„“ β‰₯ 2,
1178
+ (i) Pp(t) passes Test 4.4 (i) if 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±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 βˆ’ apt + bpt2 βˆ’ 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 five 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. Justification for surjectivity tests. Considering Tests 4.4 and 4.5, we define
1195
+ CΞ± = {M ∈ GSp4(Fβ„“) ∢ PM(t) is irreducible}
1196
+ CΞ² = {M ∈ GSp4(Fβ„“) ∢ tr(M) β‰  0 and PM(t) has a linear factor of multiplicity 1}
1197
+ CΞ³1 = {M ∈ GSp4(Fβ„“) ∢ ( tr(M)2
1198
+ mult(M), mid(M)
1199
+ mult(M)) /∈ Cβ„“,1920 or β„“ ≑ Β±1
1200
+ (mod 8)}
1201
+ CΞ³2 = {M ∈ GSp4(Fβ„“) ∢ ( tr(M)2
1202
+ mult(M), mid(M)
1203
+ mult(M)) /∈ Cβ„“,720 or β„“ ≑ Β±1
1204
+ (mod 12)}
1205
+ CΞ³3 = {M ∈ GSp4(Fβ„“) ∢ ( tr(M)2
1206
+ mult(M), mid(M)
1207
+ mult(M)) /∈ Cβ„“,5040 or β„“ β‰  7}
1208
+ Cγ = Cγ1 ∩ Cγ2 ∩ Cγ3.
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„“ ∈ PossiblyNonsurjectivePrimes
1212
+ and not in LikelyNonsurjectivePrimes(B). If β„“ = 2, then by Proposition 4.3, ρA,2 is surjective. If
1213
+ β„“ > 2, this means that we found primes p1,p2,p3,p4,p5 ≀ B each distinct from β„“ and of good reduc-
1214
+ tion for A for which ρA,β„“(Frobp1) ∈ CΞ±, ρA,β„“(Frobp2) ∈ CΞ², ρA,β„“(Frobp3) ∈ CΞ³1, ρA,β„“(Frobp4) ∈ CΞ³2,
1215
+ and ρA,β„“(Frobp4) ∈ CΞ³3. Note that by (1), the similitude factor mult(ρA,β„“(Frobp)) is p. Therefore,
1216
+ by Lemma 2.4(3), it follows that ρA,β„“(GQ) is not contained in an exceptional maximal subgroup.
1217
+ The surjectivity of ρA,β„“ now follows from Proposition 4.2.
1218
+ Finally, we will show that if B is sufficiently large (as quantified 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 ≀ 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 β‰ͺ (log dK)2
1228
+ ∣S∣
1229
+ . Proposition
1230
+ 5.1, which follows, shows that the sets CΞ±, CΞ², and CΞ³ have size at least ∣ GSp4(Fβ„“)∣
1231
+ 10
1232
+ . This can be
1233
+ used to prove the ineffective 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 defined 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 ∈ Cα is given by
1242
+ Ξ±β„“ = 1
1243
+ 4 βˆ’
1244
+ 1
1245
+ 2(β„“2 + 1).
1246
+ (ii) The probability that M ∈ Cβ is given by
1247
+ Ξ²β„“ = 3
1248
+ 8 βˆ’
1249
+ 3
1250
+ 4(β„“ βˆ’ 1) +
1251
+ 1
1252
+ 2(β„“ βˆ’ 1)2 .
1253
+ (iii) The probability that M ∈ Cγ is
1254
+ Ξ³β„“ β‰₯ 1 βˆ’
1255
+ 3β„“
1256
+ β„“2 + 1.
1257
+ Remark 16. Magma code that directly verifies 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 different 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 ∣CGSp4(Fβ„“)(Ξ³)∣, which is the size of the centralizer
1262
+ ∣CGSp4(Fβ„“)(M)∣ of M in GSp4(Fβ„“) for any M in a conjugacy class of type Ξ³. The size ∣C(Ξ³)∣ of any
1263
+ conjugacy class of type Ξ³ can then easily be computed as
1264
+ ∣C(γ)∣ =
1265
+ ∣GSp4(Fβ„“)∣
1266
+ ∣CGSp4(Fβ„“)(Ξ³)∣
1267
+ and the probability that a uniformly chosen M ∈ GSp4(Fβ„“) has conjugacy type Ξ³ is then given by
1268
+ (9)
1269
+ N(γ)∣C(γ)∣
1270
+ ∣GSp4(Fβ„“)∣ =
1271
+ N(Ξ³)
1272
+ ∣CGSp4(Fβ„“)(Ξ³)∣.
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) = (β„“βˆ’1)(β„“2βˆ’1)
1277
+ 4
1278
+ and ∣CGSp4(Fβ„“)(K0)∣ =
1279
+ (β„“ βˆ’ 1)(β„“2 + 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 ∈ 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 ∈ GSp4(Fβ„“) has conjugacy
1296
+ type Ξ³ computed using (9).
1297
+ Type Ξ³ in [Shi82]
1298
+ N(Ξ³)
1299
+ ∣CGSp4(Fβ„“)(Ξ³)∣
1300
+ Associated Probability
1301
+ K0 (Irreducible)
1302
+ (β„“βˆ’1)(β„“2βˆ’1)
1303
+ 4
1304
+ (β„“2 + 1)(β„“ βˆ’ 1)
1305
+ 1
1306
+ 4 βˆ’
1307
+ 1
1308
+ 2(β„“2+1)
1309
+ H0 (Split)
1310
+ (β„“βˆ’1)(β„“βˆ’3)2
1311
+ 8
1312
+ (β„“ βˆ’ 1)3
1313
+ 1
1314
+ 8 βˆ’
1315
+ 1
1316
+ 2(β„“βˆ’1) +
1317
+ 1
1318
+ 2(β„“βˆ’1)2
1319
+ J0 (Two Simple Roots)
1320
+ (β„“βˆ’1)3
1321
+ 4
1322
+ (β„“ + 1)(β„“ βˆ’ 1)2
1323
+ 1
1324
+ 4 βˆ’
1325
+ 1
1326
+ 2(β„“+1)
1327
+ E0 (One Double Root)
1328
+ (β„“βˆ’1)(β„“βˆ’3)
1329
+ 2
1330
+ β„“(β„“ βˆ’ 1)2(β„“2 βˆ’ 1)
1331
+ 1
1332
+ 2β„“(β„“2βˆ’1) βˆ’
1333
+ 1
1334
+ β„“(β„“βˆ’1)(β„“2βˆ’1)
1335
+ E1 (One Double Root)
1336
+ (β„“βˆ’1)(β„“βˆ’3)
1337
+ 2
1338
+ β„“(β„“ βˆ’ 1)2
1339
+ 1
1340
+ 2β„“ βˆ’
1341
+ 1
1342
+ β„“(β„“βˆ’1)
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 β€œK0 (Irreducible)” 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 insufficient 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 ∈ 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 ∈ FΓ—
1359
+ β„“ and a ∈ Fβ„“2 βˆ–Fβ„“.
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
+ (β„“βˆ’1)(β„“βˆ’3)2
1364
+ 8
1365
+ conjugacy classes of type H0 correspond to quadruples of distinct elements in
1366
+ a1,a2,b1,b2 ∈ 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 = βˆ’a1 and b2 = βˆ’b1 or b1 = βˆ’a2 and b2 = βˆ’a1.
1370
+ This accounts for (β„“βˆ’1)(β„“βˆ’3)
1371
+ 4
1372
+ of the (β„“βˆ’1)(β„“βˆ’3)2
1373
+ 8
1374
+ conjugacy classes of type H0, leaving (β„“βˆ’1)(β„“βˆ’3)(β„“βˆ’5)
1375
+ 8
1376
+ conjugacy classes with non-zero trace. As a result, the probability that a matrix M ∈ GSp4(Fβ„“)
1377
+ chosen uniformly at random has non-zero trace and totally split characteristic polynomial is
1378
+ (10)
1379
+ (β„“ βˆ’ 1)(β„“ βˆ’ 3)(β„“ βˆ’ 5)
1380
+ 8(β„“ βˆ’ 1)3
1381
+ = 1
1382
+ 8 βˆ’
1383
+ 3
1384
+ 4(β„“ βˆ’ 1) +
1385
+ 1
1386
+ (β„“ βˆ’ 1)2 .
1387
+ To obtain part (ii), we add (10) to the entries in the rightmost column of the final three rows of
1388
+ Table 2, getting
1389
+ (1
1390
+ 8 βˆ’
1391
+ 3
1392
+ 4(β„“ βˆ’ 1) +
1393
+ 1
1394
+ (β„“ βˆ’ 1)2 ) + (1
1395
+ 4 βˆ’
1396
+ 1
1397
+ 2(β„“ + 1)) + (
1398
+ 1
1399
+ 2β„“(β„“2 βˆ’ 1) βˆ’
1400
+ 1
1401
+ β„“(β„“ βˆ’ 1)(β„“2 βˆ’ 1)) + ( 1
1402
+ 2β„“ βˆ’
1403
+ 1
1404
+ β„“(β„“ βˆ’ 1))
1405
+ = 3
1406
+ 8 βˆ’
1407
+ 3
1408
+ 4(β„“ βˆ’ 1) +
1409
+ 1
1410
+ 2(β„“ βˆ’ 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 βˆ’ at3 + bt2 βˆ’ amt + m2 ∢ a,b ∈ Fβ„“,m ∈ FΓ—
1416
+ β„“ and (a2
1417
+ m , b
1418
+ m) = (u,v)}
1419
+ is at most β„“ βˆ’ 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 ∣Cβ„“,720 βˆͺ Cβ„“,1920∣ ≀ 14, we obtain the bound
1423
+ Ξ³β„“ β‰₯ 1 βˆ’ 14(β„“ βˆ’ 1)(β„“ + 3)8
1424
+ ∣GSp4(Fβ„“)∣
1425
+ .
1426
+ For β„“ > 17, this implies the claimed bound. For 3 ≀ 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 ρA,β„“ is surjective. For any Ο΅ > 0, there exists an effective constant B0 (with B0 > β„“NA)
1431
+ such that for any B > B0 and each δ ∈ {α,β,γ}, we have
1432
+ ∣∣{p prime ∢ B ≀ p ≀ 2B and ρA,β„“(Frobp) ∈ CΞ΄}∣
1433
+ ∣{p prime ∢ B ≀ p ≀ 2B}∣
1434
+ βˆ’ Ξ΄β„“βˆ£ < Ο΅.
1435
+ Proof. Let G = Gal(Q(A[β„“])/Q) and S βŠ† G be any subset that is closed under conjugation. By
1436
+ taking B to be sufficiently large, we have that B > β„“NA and can make
1437
+ ∣∣{p prime ∢ B ≀ p ≀ 2B and Frobp ∈ S}∣
1438
+ ∣{p prime ∢ B ≀ p ≀ 2B}∣
1439
+ βˆ’ ∣S∣
1440
+ ∣G∣∣
1441
+ arbitrarily small by (3).
1442
+ Moreover, the previous statement can be made effective by using an
1443
+ effective 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 > β„“NA, 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
+ ρA,β„“(Frobpi) /∈ CΞ± for each i
1449
+ or
1450
+ ρA,β„“(Frobpi) /∈ CΞ² for each i
1451
+ or
1452
+ ρA,β„“(Frobpi) /∈ 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 effective constant B0 (with B0 > β„“NA) such that for all B > B0, we have
1455
+ P(B,n) < (1 βˆ’ Ξ±β„“)n + (1 βˆ’ Ξ²β„“)n + (1 βˆ’ Ξ³β„“)n + Ο΅.
1456
+ Proof. For Ξ΄ ∈ {Ξ±,Ξ²,Ξ³}, let XΞ΄ be the event that none of the ρA,β„“(Frobpi) are contained in CΞ΄. We
1457
+ then have
1458
+ P(XΞ± βˆͺ XΞ² βˆͺ XΞ³) ≀ 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 (1βˆ’Ξ±β„“)n, (1βˆ’Ξ²β„“)n, and (1βˆ’Ξ³β„“)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 sufficiently
1465
+ large. By Proposition 5.1, we have 1 βˆ’ Ξ±β„“ ≀ 4
1466
+ 5, 1 βˆ’ Ξ²β„“ ≀ 7
1467
+ 8, and 1 βˆ’ Ξ³β„“ ≀ 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 MIT’s Lovelace computer (see the Introduction for
1484
+ the hardware specification of this machine).
1485
+
1486
+ COMPUTING NONSURJECTIVE PRIMES IN GENUS 2
1487
+ 25
1488
+ We first 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 βˆ’ 48x3 + 85x2 βˆ’ 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 Kieffer [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 β€œconductor.discrimnant”.
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
+ –
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 file may be found in the README.md file 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 defined 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
+ defined 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 defined
1590
+ by the Legendre symbol
1591
+ Ο†βˆΆ(Z/5Z)Γ— β†’ {Β±1},
1592
+ 2 ↦ βˆ’1.
1593
+ In this case, Algorithm 3.13 fails to find an auxilliary prime p < 1000 for which ap β‰  0 and Ο†(p) = βˆ’1.
1594
+ This is consistent with the endomorphism calculation, since the trace of ρA,β„“(Frobp) is 0 for all
1595
+ primes p that do not split completely in Q(ΞΆp) and any inert prime in Q(
1596
+ √
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 ππ in Q(ΞΆ3), the
1601
+ representation J1(13)[β„“] splits as a direct sum VΟ€ βŠ•VΟ€ of two 2-dimensional subrepresentations that
1602
+ are dual to each other. (A similar statement holds for J1(13)[β„“]βŠ—Fβ„“ Fβ„“, and so this representation is
1603
+ never absolutely irreducible.) As expected, Algorithm 3.6 fails to find an auxiliary prime p < 1000
1604
+ for which Rp is nonzero.
1605
+ Example 6.3. The first (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 Ξ“0(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 fields, tame ramification and
1614
+ big Galois image. Math. Res. Lett., 20(1):1–17, 2013.
1615
+ [AK19]
1616
+ Jeoung-Hwan Ahn and Soun-Hi Kwon. An explicit upper bound for the least prime ideal in the Cheb-
1617
+ otarev density theorem. Ann. Inst. Fourier (Grenoble), 69(3):1411–1458, 2019.
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
+ Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John E. Cremona, Maarten Derickx,
1623
+ Min Lee, David Lowry-Duda, David Roe, Andrew V. Sutherland, and John Voight. Computing classical
1624
+ modular forms. In Jennifer S. Balakrishnan, Noam Elkies, Brendan Hassett, Bjorn Poonen, Andrew V.
1625
+ Sutherland, and John Voight, editors, Arithmetic Geometry, Number Theory, and Computation, pages
1626
+ 131–213, Cham, 2021. Springer International Publishing.
1627
+ [BCP97]
1628
+ Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language.
1629
+ J. Symbolic Comput., 24(3-4):235–265, 1997. Computational algebra and number theory (London, 1993).
1630
+ [BK94]
1631
+ Armand Brumer and Kenneth Kramer. The conductor of an abelian variety. Compositio Mathematica,
1632
+ 92(2):227–248, 1994.
1633
+ [BPR13]
1634
+ Yuri Bilu, Pierre Parent, and Marusia Rebolledo. Rational points on X+
1635
+ 0 (pr). Ann. Inst. Fourier (Greno-
1636
+ ble), 63(3):957–984, 2013.
1637
+ [BS96]
1638
+ Eric Bach and Jonathan Sorenson. Explicit bounds for primes in residue classes. Math. Comp.,
1639
+ 65(216):1717–1735, 1996.
1640
+ [Car56]
1641
+ Leonard Carlitz. Note on a quartic congruence. Amer. Math. Monthly, 63:569–571, 1956.
1642
+ [Cha97]
1643
+ 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):151–180, 1997.
1645
+ [CL12]
1646
+ John Cremona and Eric Larson. Galois representations for elliptic curves over number fields, 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):16–31, 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):503–512 (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
+ algΒ΄ebrique. I. Lecture Notes in Mathematics, Vol. 288. Springer-Verlag, 1972. SΒ΄eminaire de GΒ΄eomΒ΄etrie
1663
+ AlgΒ΄ebrique du Bois-Marie 1967–1969 (SGA 7 I).
1664
+ [Kha06]
1665
+ Chandrashekhar Khare. Serre’s modularity conjecture: The level one case. Duke Math. J., 134(3):557–
1666
+ 589, 09 2006.
1667
+ [KL90]
1668
+ Peter Kleidman and Martin Liebeck. The subgroup structure of the finite 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):1143–1146, 1995.
1673
+ [KW09a]
1674
+ Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre’s modularity conjecture (I). Invent. Math.,
1675
+ 178(3):485–504, 2009.
1676
+ [KW09b]
1677
+ Chandrashekhar Khare and Jean-Pierre Wintenberger. Serre’s modularity conjecture (II). Invent. Math.,
1678
+ 178(3):505–586, 2009.
1679
+ [KW22]
1680
+ Habiba Kadiri and Peng-Jie Wong. Primes in the Chebotarev density theorem for all number fields (with
1681
+ an Appendix by Andrew Fiori). J. Number Theory, 241:700–737, 2022.
1682
+ [Liu94]
1683
+ Qing Liu. Conducteur et discriminant minimal de courbes de genre 2. Compositio Mathematica, 94(1):51–
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
+ Jeffrey 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):271–296, 1979.
1691
+ [Lom16]
1692
+ Davide Lombardo. Explicit surjectivity of Galois representations for abelian surfaces and GL2-varieties.
1693
+ Journal of Algebra, 460:26–59, 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):517–559, 2014.
1697
+ [LV14b]
1698
+ Eric Larson and Dmitry Vaintrob. On the surjectivity of Galois representations associated to elliptic
1699
+ curves over number fields. Bull. Lond. Math. Soc., 46(1):197–209, 2014.
1700
+ [LV22]
1701
+ Davide Lombardo and Matteo Verzobio. On the local-global principle for isogenies of abelian surfaces,
1702
+ 2022. arXiv:2206.15240.
1703
+ [Mar05]
1704
+ Greg Martin. Dimensions of the spaces of cusp forms and newforms on Ξ“0(N) and Ξ“1(N). Journal of
1705
+ Number Theory, 112(2):298–331, 2005.
1706
+ [Mit14]
1707
+ Howard H. Mitchell. The subgroups of the quaternary abelian linear group. Trans. Amer. Math. Soc.,
1708
+ 15(4):379–396, 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., 22:41–49, 1973/74.
1714
+ [MW21]
1715
+ Jacob Mayle and Tian Wang. On the effective version of Serre’s 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
+ 102:241–280, 1974.
1724
+ [Ser72]
1725
+ Jean-Pierre Serre. PropriΒ΄etΒ΄es Galoisienne des points d’ordre fini des courbes elliptiques. Inventiones
1726
+ Mathematicae, 15:259–331, 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 l’IHΒ΄ES, 54:123–201, 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):179–
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 finite conformal symplectic group, CSp(4, q). Comm. Algebra,
1748
+ 10(13):1369–1419, 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), 88:492–517,
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 Kieffer. 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„“ ≑ 5 (mod 8)
1772
+ G1920
1773
+ b2 = βˆ’1 in Fβ„“
1774
+ βŽ›
1775
+ ⎜⎜⎜
1776
+ ⎝
1777
+ 1
1778
+ 0
1779
+ 0
1780
+ βˆ’1
1781
+ 0
1782
+ 1
1783
+ βˆ’1
1784
+ 0
1785
+ 0
1786
+ 1
1787
+ 1
1788
+ 0
1789
+ 1
1790
+ 0
1791
+ 0
1792
+ 1
1793
+ ⎞
1794
+ ⎟⎟⎟
1795
+ ⎠
1796
+ ,
1797
+ βŽ›
1798
+ ⎜⎜⎜
1799
+ ⎝
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
+ ⎞
1817
+ ⎟⎟⎟
1818
+ ⎠
1819
+ ,
1820
+ βŽ›
1821
+ ⎜⎜⎜
1822
+ ⎝
1823
+ 1
1824
+ 0
1825
+ 0
1826
+ βˆ’1
1827
+ 0
1828
+ 1
1829
+ 1
1830
+ 0
1831
+ 0
1832
+ βˆ’1
1833
+ 1
1834
+ 0
1835
+ 1
1836
+ 0
1837
+ 0
1838
+ 1
1839
+ ⎞
1840
+ ⎟⎟⎟
1841
+ ⎠
1842
+ ,
1843
+ βŽ›
1844
+ ⎜⎜⎜
1845
+ ⎝
1846
+ 1
1847
+ 0
1848
+ 1
1849
+ 0
1850
+ 0
1851
+ 1
1852
+ 0
1853
+ 1
1854
+ βˆ’1
1855
+ 0
1856
+ 1
1857
+ 0
1858
+ 0
1859
+ βˆ’1
1860
+ 0
1861
+ 1
1862
+ ⎞
1863
+ ⎟⎟⎟
1864
+ ⎠
1865
+ β„“ ≑ 3 (mod 8)
1866
+ G1920
1867
+ b2 = βˆ’2 in Fβ„“
1868
+ βŽ›
1869
+ ⎜⎜⎜
1870
+ ⎝
1871
+ 1
1872
+ 0
1873
+ 0
1874
+ βˆ’1
1875
+ 0
1876
+ 1
1877
+ βˆ’1
1878
+ 0
1879
+ 0
1880
+ 1
1881
+ 1
1882
+ 0
1883
+ 1
1884
+ 0
1885
+ 0
1886
+ 1
1887
+ ⎞
1888
+ ⎟⎟⎟
1889
+ ⎠
1890
+ ,
1891
+ βŽ›
1892
+ ⎜⎜⎜
1893
+ ⎝
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
+ ⎞
1911
+ ⎟⎟⎟
1912
+ ⎠
1913
+ ,
1914
+ βŽ›
1915
+ ⎜⎜⎜
1916
+ ⎝
1917
+ 1
1918
+ 0
1919
+ 0
1920
+ βˆ’1
1921
+ 0
1922
+ 1
1923
+ 1
1924
+ 0
1925
+ 0
1926
+ βˆ’1
1927
+ 1
1928
+ 0
1929
+ 1
1930
+ 0
1931
+ 0
1932
+ 1
1933
+ ⎞
1934
+ ⎟⎟⎟
1935
+ ⎠
1936
+ ,
1937
+ βŽ›
1938
+ ⎜⎜⎜
1939
+ ⎝
1940
+ 1
1941
+ 0
1942
+ 1
1943
+ 0
1944
+ 0
1945
+ 1
1946
+ 0
1947
+ 1
1948
+ βˆ’1
1949
+ 0
1950
+ 1
1951
+ 0
1952
+ 0
1953
+ βˆ’1
1954
+ 0
1955
+ 1
1956
+ ⎞
1957
+ ⎟⎟⎟
1958
+ ⎠
1959
+ β„“ ≑ 7 (mod 12)
1960
+ G720
1961
+ a2 + a + 1 = 0 in Fβ„“
1962
+ βŽ›
1963
+ ⎜⎜⎜
1964
+ ⎝
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
+ ⎞
1982
+ ⎟⎟⎟
1983
+ ⎠
1984
+ ,
1985
+ βŽ›
1986
+ ⎜⎜⎜
1987
+ ⎝
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
+ ⎞
2005
+ ⎟⎟⎟
2006
+ ⎠
2007
+ ,
2008
+ βŽ›
2009
+ ⎜⎜⎜
2010
+ ⎝
2011
+ a
2012
+ 0
2013
+ βˆ’a βˆ’ 1
2014
+ a + 1
2015
+ 0
2016
+ a
2017
+ βˆ’a βˆ’ 1
2018
+ βˆ’a βˆ’ 1
2019
+ βˆ’a βˆ’ 1
2020
+ βˆ’a βˆ’ 1
2021
+ βˆ’1
2022
+ 0
2023
+ a + 1
2024
+ βˆ’a βˆ’ 1
2025
+ 0
2026
+ βˆ’1
2027
+ ⎞
2028
+ ⎟⎟⎟
2029
+ ⎠
2030
+ ,
2031
+ βŽ›
2032
+ ⎜⎜⎜
2033
+ ⎝
2034
+ 0
2035
+ βˆ’1
2036
+ 0
2037
+ 0
2038
+ 1
2039
+ 0
2040
+ 0
2041
+ 0
2042
+ 0
2043
+ 0
2044
+ 0
2045
+ βˆ’1
2046
+ 0
2047
+ 0
2048
+ 1
2049
+ 0
2050
+ ⎞
2051
+ ⎟⎟⎟
2052
+ ⎠
2053
+ β„“ ≑ 5 (mod 12)
2054
+ G720
2055
+ b2 = βˆ’1 in Fβ„“
2056
+ βŽ›
2057
+ ⎜⎜⎜
2058
+ ⎝
2059
+ βˆ’1
2060
+ 0
2061
+ 0
2062
+ βˆ’1
2063
+ 0
2064
+ βˆ’1
2065
+ βˆ’1
2066
+ 0
2067
+ 0
2068
+ 1
2069
+ 0
2070
+ 0
2071
+ 1
2072
+ 0
2073
+ 0
2074
+ 0
2075
+ ⎞
2076
+ ⎟⎟⎟
2077
+ ⎠
2078
+ ,
2079
+ βŽ›
2080
+ ⎜⎜⎜
2081
+ ⎝
2082
+ 0
2083
+ 0
2084
+ 0
2085
+ 1
2086
+ 0
2087
+ βˆ’1
2088
+ βˆ’1
2089
+ 0
2090
+ 0
2091
+ 1
2092
+ 0
2093
+ 0
2094
+ βˆ’1
2095
+ 0
2096
+ 0
2097
+ βˆ’1
2098
+ ⎞
2099
+ ⎟⎟⎟
2100
+ ⎠
2101
+ ,
2102
+ βŽ›
2103
+ ⎜⎜⎜
2104
+ ⎝
2105
+ βˆ’b βˆ’ 1
2106
+ b
2107
+ 2b
2108
+ βˆ’2b + 1
2109
+ b
2110
+ b βˆ’ 1
2111
+ 2b + 1
2112
+ 2b
2113
+ b
2114
+ b βˆ’ 1
2115
+ βˆ’b βˆ’ 2
2116
+ βˆ’b
2117
+ βˆ’b βˆ’ 1
2118
+ b
2119
+ βˆ’b
2120
+ b βˆ’ 2
2121
+ ⎞
2122
+ ⎟⎟⎟
2123
+ ⎠
2124
+ ,
2125
+ βŽ›
2126
+ ⎜⎜⎜
2127
+ ⎝
2128
+ 0
2129
+ βˆ’b
2130
+ βˆ’2b
2131
+ 0
2132
+ b
2133
+ 0
2134
+ 0
2135
+ 2b
2136
+ βˆ’2b
2137
+ 0
2138
+ 0
2139
+ βˆ’b
2140
+ 0
2141
+ 2b
2142
+ b
2143
+ 0
2144
+ ⎞
2145
+ ⎟⎟⎟
2146
+ ⎠
2147
+ β„“ = 7
2148
+ G5040
2149
+ a = 2 satisfies
2150
+ a2 + a + 1 = 0
2151
+ βŽ›
2152
+ ⎜⎜⎜
2153
+ ⎝
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
+ ⎞
2171
+ ⎟⎟⎟
2172
+ ⎠
2173
+ ,
2174
+ οΏ½οΏ½οΏ½
2175
+ ⎜⎜⎜
2176
+ ⎝
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
+ ⎞
2194
+ ⎟⎟⎟
2195
+ ⎠
2196
+ ,
2197
+ βŽ›
2198
+ ⎜⎜⎜
2199
+ ⎝
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
+ ⎞
2217
+ ⎟⎟⎟
2218
+ ⎠
2219
+ ,
2220
+ βŽ›
2221
+ ⎜⎜⎜
2222
+ ⎝
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
+ ⎞
2240
+ ⎟⎟⎟
2241
+ ⎠
2242
+ ,
2243
+ βŽ›
2244
+ ⎜⎜⎜
2245
+ ⎝
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
+ ⎞
2263
+ ⎟⎟⎟
2264
+ ⎠
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 suffices.
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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