EXOTIC SYMMETRIC SPACES OF HIGHER LEVEL: SPRINGER CORRESPONDENCE FOR COMPLEX REFLECTION GROUPS

Let G?=?GL(V) for a 2n-dimensional vector space V, and θ an involutive automorphism of G such that H?=?G ^θ ???Sp(V). Let Open image in new window be the set of unipotent elements g ∈ G such that θ(g)?=?g ^?1. For any integer r?≥?2, we consider the variety Open image in new window , on which H acts diagonally. Let Open image in new window be a complex reflection group. In this paper, generalizing the known result for r?=?2, we show that there exists a natural bijective correspondence (Springer correspondence) between the set of irreducible representations of W _n,r and a certain set of H-equivariant simple perverse sheaves on Open image in new window . We also consider a similar problem for Open image in new window , on which G acts diagonally, where G?=?GL(V) for a finite-dimensional vector space V.     

An analog of Rudin’s theorem for continuous radial positive definite functions of several variables

Let Open image in new window be the class of radial real-valued functions of m variables with support in the unit ball \(\mathbb{B}\) of the space ? ^m that are continuous on the whole space ? ^m and have a nonnegative Fourier transform. For m ≥ 3, it is proved that a function f from the class Open image in new window can be presented as the sum \(\sum {f_k \tilde *f_k } \) of at most countably many self-convolutions of real-valued functions f _k with support in the ball of radius 1/2. This result generalizes the theorem proved by Rudin under the assumptions that the function f is infinitely differentiable and the functions f _k are complex-valued.     

A weird relation between two cardinals

For a set M, let \({\text {seq}}(M)\) denote the set of all finite sequences which can be formed with elements of M, and let \([M]^2\) denote the set of all 2-element subsets of M. Furthermore, for a set A, let Open image in new window denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory \(\textsf {ZF}\): There exists a set M such that Open image in new window and no function Open image in new window is finite-to-one.     

Asymptotic Normality of Some Graph Sequences

For a simple finite graph G denote by Open image in new window the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If \(E_n\) is the graph on n vertices with no edges then Open image in new window coincides with Open image in new window , the ordinary Stirling number of the second kind, and so we refer to Open image in new window as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of \(E_n\), is asymptotically normal—essentially, as n grows, the histogram of Open image in new window , suitably normalized, approaches the density function of the standard normal distribution. In light of Harper’s result, it is natural to ask for which sequences \((G_n)_{n \ge 0}\) of graphs is there asymptotic normality of Open image in new window . Thanh and Galvin conjectured that if for each n, \(G_n\) is acyclic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that \(G_n\) has no more than \(o(\sqrt{n/\log n})\) components. Here we settle Thanh and Galvin’s conjecture in the affirmative, and significantly extend it, replacing “acyclic” in their conjecture with “co-chromatic with a quasi-threshold graph, and with negligible chromatic number”. Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in the Weyl algebra, and work of Kahn on the matching polynomial of a graph.     

The Parseval equality for multiple Fourier-Stieltjes series with respect to the Haar system

Let a function f : \(\Pi ^{ * ^m } \) → ? be Lebesgue integrable on \(\Pi ^{ * ^m } \) and Riemann-Stieltjes integrable with respect to a function G : \(\Pi ^{ * ^m } \) → ? on \(\Pi ^{ * ^m } \). Then the Parseval equality

Open image in new window

holds, where

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(k) = (f, χk) = (L)

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f(x)χk(x) dx and \(\widehat{dG}\)(k) =

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χk(x) dG(x) are Fourier coefficients of the function f and Fourier-Stieltjes coefficients of the function G with respect to the Haar system, respectively; the integrals in the equality and in the definition of the coefficients of the function G are the Riemann-Stieltjes integrals; the series in the right-hand side of the equality converges in the sense of rectangular partial sums; and the overline indicates the complex conjugation. If f : Π ^m → ? is a complex-valued Lebesgue integrable function, G is a complex-valued function of bounded variation on Π ^m ,

Open image in new window

are Fourier-Lebesgue coefficients of the function f _x(t) = f(x ⊕ t), where ⊕ is the group addition, then the Parseval equality

Open image in new window

holds for almost all x ∈ \(\Pi ^{ * ^m } \) in the sense of any summation method with respect to which the Fourier series of Lebesgue integrable functions are summable to these functions almost everywhere (the integral here is interpreted in the sense of Lebesgue-Stieltjes).

     

A BKR Operation for Events Occurring for Disjoint Reasons with High Probability

Given events A and B on a product space \(S={\prod }_{i = 1}^{n} S_{i}\), the set \(A \Box B\) consists of all vectors x = (x₁,…,x_n) ∈ S for which there exist disjoint coordinate subsets K and L of {1,…,n} such that given the coordinates x_i,i ∈ K one has that x ∈ A regardless of the values of x on the remaining coordinates, and likewise that x ∈ B given the coordinates x_j,j ∈ L. For a finite product of discrete spaces endowed with a product measure, the BKR inequality

$$ P(A \Box B) \le P(A)P(B) $$

(1)

was conjectured by van den Berg and Kesten (J Appl Probab 22:556–569, 1985) and proved by Reimer (Combin Probab Comput 9:27–32, 2000). In Goldstein and Rinott (J Theor Probab 20:275–293, 2007) inequality Eq. 1 was extended to general product probability spaces, replacing \(A \Box B\) by the set Open image in new window consisting of those outcomes x for which one can only assure with probability one that x ∈ A and x ∈ B based only on the revealed coordinates in K and L as above. A strengthening of the original BKR inequality Eq. 1 results, due to the fact that Open image in new window . In particular, it may be the case that \(A \Box B\) is empty, while Open image in new window is not. We propose the further extension Open image in new window depending on probability thresholds s and t, where Open image in new window is the special case where both s and t take the value one. The outcomes Open image in new window are those for which disjoint sets of coordinates K and L exist such that given the values of x on the revealed set of coordinates K, the probability that A occurs is at least s, and given the coordinates of x in L, the probability of B is at least t. We provide simple examples that illustrate the utility of these extensions.

     

Local lagrange interpolation with cubic splines on tetrahedral partitions

We describe an algorithm for constructing a Lagrange interpolation pair based on C¹ cubic splines defined on tetrahedral partitions. In particular, given a set of points , we construct a set P containing and a spline space based on a tetrahedral partition whose set of vertices include such that interpolation at the points of P is well-defined and unique. Earlier results are extended in two ways: (1) here we allow arbitrary sets , and (2) the method provides optimal approximation order of smooth functions.     

A comparison of several enumerative algorithms for Sudoku

Sudoku is a puzzle played of an n × n grid Open image in new window where n is the square of a positive integer m. The most common size is n=9. The grid is partitioned into n subgrids of size m × m. The player must place exactly one number from the set N={1, …, n} in each row and each column of Open image in new window as well as in each subgrid. A grid is provided with some numbers already in place, called givens. In this paper, some relationships between Sudoku and several operations research problems are presented. We model the problem by means of two mathematical programming formulations. The first one consists of an integer linear programming model, while the second one is a tighter non-linear integer programming formulation. We then describe several enumerative algorithms to solve the puzzle and compare their relative efficiencies. Two basic backtracking algorithms are first described for the general Sudoku. We then solve both formulations by means of constraint programming. Computational experiments are performed to compare the efficiency and effectiveness of the proposed algorithms. Our implementation of a backtracking algorithm can solve most benchmark instances of size 9 within 0.02?s, while no such instance was solved within that time by any other method. Our implementation is also much faster than an existing alternative algorithm.     

Non-Jumping Numbers for 4-Uniform Hypergraphs

Let r≥2 be an integer. A real number α ∈ [0,1) is a jump for r if for any Open image in new window >0 and any integer m, m≥r, any r-uniform graph with n>n₀( Open image in new window ,m) vertices and at least Open image in new window edges contains a subgraph with m vertices and at least Open image in new window edges, where c=c(α) does not depend on Open image in new window and m. It follows from a theorem of Erd?s, Stone and Simonovits that every α ∈ [0,1) is a jump for r=2. Erd?s asked whether the same is true for r≥3. Frankl and Rödl gave a negative answer by showing that Open image in new window is not a jump for r if r≥3 and l>2r. Following a similar approach, we give several sequences of non-jumping numbers generalizing the above result for r=4.     

Congruences between word length statistics for the finitary alternating and symmetric groups

Bacher and de la Harpe (arxiv:1603.07943, 2016) study conjugacy growth series of infinite permutation groups and their relationships with p(n), the partition function, and \(p(n)_\mathbf{e }\), a generalized partition function. They prove identities for the conjugacy growth series of the finitary symmetric group and the finitary alternating group. The group theory due to Bacher and de la Harpe (arxiv:1603.07943, 2016) also motivates an investigation into congruence relationships between the finitary symmetric group and the finitary alternating group. Using the Ramanujan congruences for the partition function p(n) and Atkin’s generalization to the k-colored partition function \(p_{k}(n)\), we prove the existence of congruence relations between these two series modulo arbitrary powers of 5 and 7, which we systematically describe. Furthermore, we prove that such relationships exist modulo powers of all primes \(\ell \ge 5\).     

The sum of digits of primes in $${\mathbb{Z}}$$[i]

We study the distribution of the complex sum-of-digits function s _q with basis q = –a±i, \({a \in \mathbb{Z}^+}\) for Gaussian primes p. Inspired by a recent result of Mauduit and Rivat (http://iml.univ-mrs.fr/~rivat/publications.html) for the real sum-of-digits function, we here get uniform distribution modulo 1 of the sequence (αs _q (p)) provided \({\alpha \in \mathbb{R} \setminus \mathbb{Q}}\) and q is prime with a ≥ 28. We also determine the order of magnitude of the number of Gaussian primes whose sum-of-digits evaluation lies in some fixed residue class mod m.     

On Jordan ideals and submodules: Algebraic and analytic aspects

Let

Open image in new window

be an algebra, and let X be an arbitrary

Open image in new window

-bimodule. A linear space Y ? X is called a Jordan

Open image in new window

-submodule if Ay + yA ∈ Y for all A ∈

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and y ∈ Y. (For X =

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, this coincides with the notion of a Jordan ideal.) We study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. We also consider symmetrically normed Jordan submodules over C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.

     

Improved Strategies for Radial basis Function Methods for Global Optimization

We propose some strategies that can be shown to improve the performance of the radial basis function (RBF) method by Gutmann [J. Global optim. 19(3), 201–227 (2001a)] (Gutmann-RBF) and the RBF method by Regis and Shoemaker [J. Global optim. 31, 153–171 (2005)] (CORS–RBF) on some test problems when they are initialized by symmetric Latin hypercube designs (SLHDs). Both methods are designed for the global optimization of computationally expensive functions with multiple local optima. We demonstrate how the original implementation of Gutmann-RBF can sometimes converge slowly to the global minimum on some test problems because of its failure to do local search. We then propose Controlled Gutmann-RBF (CG-RBF), which is a modification of Gutmann-RBF where the function evaluation point in each iteration is restricted to a subregion of the domain centered around a global minimizer of the current RBF model. By varying the size of this subregion in different iterations, we ensure a better balance between local and global search. Moreover, we propose a complete restart strategy for CG-RBF and CORS-RBF whenever the algorithm fails to make any substantial progress after some threshold number of consecutive iterations. Computational experiments on the seven Dixon and Szegö [Towards Global optimization, pp. 1–13. North-Holland, Amsterdam (1978)] test problems and on nine Schoen [J. Global optim. 3, 133–137 (1993)] test problems indicate that the proposed strategies yield significantly better performance on some problems. The results also indicate that, for some fixed setting of the restart parameters, the two modified RBF algorithms, namely CG-RBF-Restart and CORS-RBF-Restart, are comparable on the test problems considered. Finally, we examine the sensitivity of CG-RBF-Restart and CORS-RBF-Restart to the restart parameters.     

Implementing Brouwer’s database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on \(n\le 1300\) vertices. We have implemented most of the infinite families of graphs listed there in the open-source software Sagemath (The Sage Developers, http://www.sagemath.org), as well as provided constructions of the “sporadic” cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of n.     

Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

Let Open image in new window be the weighted local time of fractional Brownian motion B ^H with Hurst index 1/2?H?Open image in new window As an application, we investigate the weighted quadratic covariation \([f\big(B^H\big),B^H]^{(W)}\) defined by

$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $

where the limit is uniform in probability and t _k ?=?kt/n. We show that it exists and

Open image in new window

provided f is of bounded p-variation with \(1\leq p<\frac{2H}{1-H}\). Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

     

Preserving preference rankings under non-financial background risk

We investigate the impact of a non-financial background risk ??? on thepreference rankings between two independent financial risks Open image in new window ₁ and Open image in new window ₂ for anexpected-utility maximizer. More precisely, we provide necessary and sufficientconditions for the alternative (x₀+ Open image in new window ₁,y₀+ ???) to be preferred to(x₀+ Open image in new window ₂,y₀+ ???)whenever (x₀+ Open image in new window ₁,y₀) ispreferred to (x₀+ Open image in new window ₂,y₀). Utilityfunctions that preserve the preference rankings are fully characterized. Theirpractical relevance is discussed in light of recent results on the constraintsfor the modelling of the preference for the disaggregation of harms.     

Nonseparable closed vector subspaces of separable topological vector spaces

In 1983 P. Domański investigated the question: For which separable topological vector spaces E, does the separable space Open image in new window have a nonseparable closed vector subspace, where \(\hbox {c}\) is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space Open image in new window has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line \(\mathbb M\) the space of all continuous real-valued functions on \(\mathbb M\) endowed with the pointwise convergence topology, \(C_p(\mathbb M)\) contains a nonseparable closed vector subspace while \(C_p(\mathbb M)\) is separable.     

Sur les fonctions arithmétiques non entières

A theorem of Baker says that a function F entire on ? ^d such that F(? ^d ) ? ? and increasing slower (in a precise sense) than \(2^{z_1 + \cdots + z_d } \) is necessarily a polynomial. This is a multivariate generalisation of the celebrated theorem of Pólya (case d = 1). Using the theory of analytic functionals with non-compact carrier, Yoshino proved a general theorem dealing with the growth of arithmetic analytic functions, which implies that the conclusion of Baker’s theorem holds if F is only assumed to be holomorphic on the domain

Open image in new window

, j = 1,..., d}.

The case d = 1 was also treated in a different way by Gel’fond and Pólya by means of the characteristic function of Carlson-Nörlund. This function was introduced to bound in a nearly optimal way the growth of holomorphic functions of one variable that can be expanded in a Newton interpolation series in the half-plane

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.

In this article, we show how this characteristic function can also be used to bound the growth of multivariate functions defined on

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that can be expanded in multiple Newton series. These considerations enable us to improve Gel’fond-Pólya’s and Yoshino’s theorems, in particular, to remove or to weaken certain of their technical conditions.

     

Statistical simulation of one type of pairs of random variables with the use of fictitious jumps

Statistical simulation based on the sampling technique for a pair of random variables (T,

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), where T ∈ [0, +∞) and

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∈ ? ^d (d ≥ 1) is considered. The simultaneous distribution of the pair is specified in the form that is common for analogous problems in various fields. It has the form

Open image in new window

, where f is a function and m is a measure. The first variable T is the well-known random waiting time. A simulation method for the pair (T,

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) is constructed using a realization of an auxiliary Markov sequence of trial pairs. Applications of this method in particle transport theory and in kinetics of rarefied gases are discussed.