A note on the triangle conjecture

We prove some particular cases of the following conjecture of Perrin and Schützenberger, known as “the triangle conjecture.” Let A = {a, b} be a two-letter alphabet, d a positive integer and let B_d = {aⁱba^j| 0 ? i + j ? d}. If X ? B_d is a code, then |X| ? d + 1.     

Riesz extremal measures on the sphere for axis-supported external fields

We investigate the minimal Riesz s-energy problem for positive measures on the d-dimensional unit sphere Sd in the presence of an external field induced by a point charge, and more generally by a line charge. The model interaction is that of Riesz potentials |x−y|^−s with d−2?s<d. For a given axis-supported external field, the support and the density of the corresponding extremal measure on Sd is determined. The special case s=d−2 yields interesting phenomena, which we investigate in detail. A weak^∗ asymptotic analysis is provided as s→⁺(d−2).     

Limit sets of dynamical systems on the space of probability measures

Let m be a dynamical system on the space of probability measures $\text{M}$₁(R^d), and let Λ + (?) be the positive limit set for ? ∈ $\text{M}$₁(R^d), where ? has compact support K ?R^d. The main result of this paper states that support of Λ⁺(?) ?

,support of Λ + (δ_x), where δ_x is the Dirac measure at point x.     

On the order of an automorphism of a smooth hypersurface

In this paper we give an effective criterion as to when a positive integer q is the order of an automorphism of a smooth hypersurface of dimension n and degree d, for every d ≥ 3, n ≥ 2, (n, d) ≠ (2, 4), and gcd(q, d) = gcd(q, d ? 1) = 1. This allows us to give a complete criterion in the case where q = p is a prime number. In particular, we show the following result: If X is a smooth hypersurface of dimension n and degree d admitting an automorphism of prime order p then p < (d ? 1) ⁿ⁺¹; and if p > (d ? 1) ⁿ then X is isomorphic to the Klein hypersurface, n = 2 or n + 2 is prime, and p = Φ _n+2(1 ? d) where Φ _n+2 is the (n+2)-th cyclotomic polynomial. Finally, we provide some applications to intermediate jacobians of Klein hypersurfaces.     

The rate of approximation of Gaussian radial basis neural networks in continuous function space

There have been many studies on the dense theorem of approximation by radial basis feedforword neural networks, and some approximation problems by Gaussian radial basis feedforward neural networks(GRBFNs)in some special function space have also been investigated. This paper considers the approximation by the GRBFNs in continuous function space. It is proved that the rate of approximation by GRNFNs with n~d neurons to any continuous function f defined on a compact subset K(R~d)can be controlled by ω(f, n~(-1/2)), where ω(f, t)is the modulus of continuity of the function f .     

Translational tilings by a polytope, with multiplicity

We study the problem of covering ? ^d by overlapping translates of a convex polytope, such that almost every point of ? ^d is covered exactly k times. Such a covering of Euclidean space by a discrete set of translations is called a k-tiling. The investigation of simple tilings by translations (which we call 1-tilings in this context) began with the work of Fedorov [5] and Minkowski [15], and was later extended by Venkov and McMullen to give a complete characterization of all convex objects that 1-tile ? ^d . By contrast, for k ≥2, the collection of polytopes that k-tile is much wider than the collection of polytopes that 1-tile, and there is currently no known analogous characterization for the polytopes that k-tile. Here we first give the necessary conditions for polytopes P that k-tile, by proving that if P k-tiles ? ^d by translations, then it is centrally symmetric, and its facets are also centrally symmetric. These are the analogues of Minkowski’s conditions for 1-tiling polytopes, but it turns out that very new methods are necessary for the development of the theory. In the case that P has rational vertices, we also prove that the converse is true; that is, if P is a rational polytope, is centrally symmetric, and has centrally symmetric facets, then P must k-tile ? ^d for some positive integer k.     

Probability and Fourier Duality for Affine Iterated Function Systems

Let d be a positive integer, and let μ be a finite measure on ? ^d . In this paper we ask when it is possible to find a subset Λ in ? ^d such that the corresponding complex exponential functions e _λ indexed by Λ are orthogonal and total in L ²(μ). If this happens, we say that (μ,Λ) is a spectral pair. This is a Fourier duality, and the x-variable for the L ²(μ)-functions is one side in the duality, while the points in Λ is the other. Stated this way, the framework is too wide, and we shall restrict attention to measures μ which come with an intrinsic scaling symmetry built in and specified by a finite and prescribed system of contractive affine mappings in ? ^d ; an affine iterated function system (IFS). This setting allows us to generate candidates for spectral pairs in such a way that the sets on both sides of the Fourier duality are generated by suitably chosen affine IFSs. For a given affine setup, we spell out the appropriate duality conditions that the two dual IFS-systems must have. Our condition is stated in terms of certain complex Hadamard matrices. Our main results give two ways of building higher dimensional spectral pairs from combinatorial algebra and spectral theory applied to lower dimensional systems.     

Über ein Turánsches Problem für ℓ-1 radiale, positiv definite Funktionen

Turán’s problem is to determine the greatest possible value of the integral ∫_? ^df(x)dx/ f (0) for positive definite functions f (x), x ∈ ?^d, supported in a given convex centrally symmetric body D ? ?^d. In this note we consider the 2-dimensional Turán problem for positive definite functions of the form f(x) = φ (∥x∥1), x ∈ ?², with φ supported in [0,π].     

On the Borsuk and Erdös—Hadwiger numbers

Two classical problems of combinatorial geometry, the Borsuk problem about splitting sets into parts of smaller diameter and the Erdös—Hadwiger problem about coloring Euclidean space, are studied. New asymptotic estimates are obtained for the quantities f(d) (the minimal number of parts of smaller diameter into which any bounded set in ?^d can be decomposed) and χ(?^d) (the minimal number of colors required to color all points ?^d so that any points at distance 1 from each other have different colors), which are the main objects of study in these problems.     

Invariance of Picard Dimensions Under Basic Perturbations

The Picard dimension \(\dim \mu\) of a signed Radon measure μ on the punctured closed unit ball 0?x|?≦?1 in the d-dimensional euclidean space with d?≧?2 is the cardinal number of the set of extremal rays of the cone of positive continuous distributional solutions u of the Schrödinger equation (???Δ?+?μ)u?=?0 on the punctured open unit ball 0?x|?x|?=?1. If the Green function of the above equation on 0?x|?Δ?+?μ)u?=?δ _y , the Dirac measure supported by the point y, exists for every y in 0?x|?μ is referred to as being hyperbolic on 0?x|?γ is a radial Radon measure which is both positive and absolutely continuous with respect to the d-dimensional Lebesgue measure dx whose Radon–Nikodym density dγ(x)/dx is bounded by a positive constant multiple of |x|^???2. The purpose of this paper is to show that the Picard dimensions of hyperbolic radial Radon measures μ are invariant under basic perturbations \(\gamma: \dim(\mu+\gamma)=\dim\mu\). Three applications of this invariance are also given.     

An Extension of a Theorem of Yao and Yao

In this paper we study N _d (k), the smallest positive integer such that any nice measure μ in $\mathbb{R}^{d}$ can be partitioned into N _d (k) convex parts of equal measure so that every hyperplane avoids at least k of them. A theorem of Yao and Yao states that N _d (1)≤2 ^d . Among other results, we obtain the bounds N _d (2)≤3?2 ^d?1 and N _d (1)≥C?2 ^d/2 for some constant C. We then apply these results to a problem on the separation of points and hyperplanes.     

Maximizing a lower bound on the computational complexity

We solve a combinatorial problem that arises in determining by a method due to Engeler lower bounds for the computational complexity of algorithmic problems. Denote by G_d the class of permutation groups G of degree d that are iterated wreath products of symmetric groups, i.e. G = S_dh?11?1S_d0 with Π^h?1_i=0d_i = d for some natural number h and some sequence (d₀,…,d_h?1) of natural numbers greater than 1. The problem is to characterize those G = S_dh?11?1S_d0 in G_d on which k(G):=log|G|/max_0≤i≤h?1log(d_i!) assumes its maximum. Our solution consists of two necessary conditions for this, namely that d₀≤d₁≤?≤d_h and that d_h is the largest prime divisor of d. Consequently, if d is a prime power, say d = p^h with p prime, then a necessary and sufficient condition is that d_i = p, 0 ≤ i ≤ h ? 1.     

Inequalities for the Gaussian hypergeometric function

we study the monotonicity of certain combinations of the Gaussian hypergeometric functions F(-1/2,1/2;1;1- xc) and F(-1/2- δ,1/2 + δ;1;1- xd) on(0,1) for given 0 c 5d/6 ∞ andδ∈(-1/2,1/2),and find the largest value δ1 = δ1(c,d) such that inequality F(-1/2,1/2;1;1- xc) F(-1/2- δ,1/2 + δ;1;1- xd) holds for all x ∈(0,1). Besides,we also consider the Gaussian hypergeometric functions F(a- 1- δ,1- a + δ;1;1- x3) and F(a- 1,1- a;1;1- x2) for given a ∈ [1/29,1) and δ∈(a- 1,a),and obtain the analogous results.     

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere ?? ^d in the Euclidean space ? ^d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r ^s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? ^d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? ^d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.     

Ombres Convexité, régularité et sous-harmonicité

Letf:V→R be a function defined on a subsetV ofR ⁿ ×R ^d let?:x→inf{f(x t);t such that(x t)∈V} denote theshadow off and letΦ={(x t)∈V; f(x t)=?(x)} This paper deals with the characterization of some properties of ? in terms of the infinitesimal behavior off near points ζ∈Φ proving in particular a conjecture of J M Trépreau concerning the cased=1 Characterizations of this type are provided for the convexity the subharmonicity or theC ^{1 1} regularity of ? in the interior ofI={x∈ R ⁿ;εR ^d (x t)∈V} and in theC ^{1 1} case an expression forD ²? is given To some extent an answer is given to the following question: which convex function ?:I→R I interval ?R (resp which function √:I→R of classC ^{1 1}) is the shadow of aC ² functionf:I×R→R?     

Subadditive affine-invariant transformations of convex bodies

We prove two results about the continuous maps F, from the space of d-dimensional convex bodies K of ? ^d into the space of non-empty compact sets of ? ^d , which are subadditive and invariant by affine permutations. The first theorem gives properties of the images F(K). In the second one, we determine the largest map (with respect to inclusion of images) in the family of all subadditive and continuous affine invariants F such that F(P)?p+(2r-1)(P-p) where P is a given parallelotope with center P and where r is a number between d/(d+1) and 1.     

Description of random fields by means of one-point finite-conditional distributions

The aim of this note is to investigate the relationship between strictly positive random fields on a lattice ? ^ν and the conditional probability measures at one point given the values on a finite subset of the lattice ? ^ν . We exhibit necessary and sufficient conditions for a one-point finite-conditional system to correspond to a unique strictly positive probability measure. It is noteworthy that the construction of the aforementioned probability measure is done explicitly by some simple procedure. Finally, we introduce a condition on the one-point finite conditional system that is sufficient for ensuring the mixing of the underlying random field.     

Fast construction of irreducible polynomials over finite fields

We present a randomized algorithm that on inputting a finite field K with q elements and a positive integer d outputs a degree d irreducible polynomial in K[x]. The running time is d ^1+?(d)×(log q)^5+?(q) elementary operations. The function ? in this expression is a real positive function belonging to the class o(1), especially, the complexity is quasi-linear in the degree d. Once given such an irreducible polynomial of degree d, we can compute random irreducible polynomials of degree d at the expense of d ^1+?(d) × (log q)^1+?(q) elementary operations only.