Hyper-bent functions and cyclic codes

Bent functions are those Boolean functions whose Hamming distance to the Reed-Muller code of order 1 equal 2^n-1-2^n/2-1 (where the number n of variables is even). These combinatorial objects, with fascinating properties, are rare. Few constructions are known, and it is difficult to know whether the bent functions they produce are peculiar or not, since no way of generating at random bent functions on 8 variables or more is known.The class of bent functions contains a subclass of functions whose properties are still stronger and whose elements are still rarer. Youssef and Gong have proved the existence of such hyper-bent functions, for every even n. We prove that the hyper-bent functions they exhibit are exactly those elements of the well-known PS_ap class, introduced by Dillon, up to the linear transformations x?δx, . Hyper-bent functions seem still more difficult to generate at random than bent functions; however, by showing that they all can be obtained from some codewords of an extended cyclic code H_n with small dimension, we can enumerate them for up to 10 variables. We study the non-zeroes of H_n and we deduce that the algebraic degree of hyper-bent functions is n/2. We also prove that the functions of class PS_ap are some codewords of weight 2^n-1-2^n/2-1 of a subcode of H_n and we deduce that for some n, depending on the factorization of 2ⁿ-1, the only hyper-bent functions on n variables are the elements of the class , obtained from PS_ap by composing the functions by the transformations x?δx, δ≠0, and by adding constant functions. We prove that non- hyper-bent functions exist for n=4, but it is not clear whether they exist for greater n. We also construct potentially new bent functions for n=12.     

Two-sided eigenvalue estimates for subordinate processes in domains

Let X={X_t,t?0} be a symmetric Markov process in a state space E and D an open set of E. Denote by X^D the subprocess of X killed upon leaving D. Let S={S_t,t?0} be a subordinator with Laplace exponent φ that is independent of X. The processes X^φ?{X_St,t?0} and are called the subordinate processes of X and X^D, respectively. Under some mild conditions, we show that, if {-μ_n,n?1} and {-λ_n,n?1} denote the eigenvalues of the generators of the subprocess of X^φ killed upon leaving D and of the process X^D respectively, then

     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions

Let X∼f(∥x-θ∥²) and let δ_π(X) be the generalized Bayes estimator of θ with respect to a spherically symmetric prior, π(∥θ∥²), for loss ∥δ-θ∥². We show that if π(t) is superharmonic, non-increasing, and has a non-decreasing Laplacian, then the generalized Bayes estimator is minimax and dominates the usual minimax estimator δ₀(X)=X under certain conditions on . The class of priors includes priors of the form for and hence includes the fundamental harmonic prior . The class of sampling distributions includes certain variance mixtures of normals and other functions f(t) of the form e^-αtβ and e^-αt+βφ(t) which are not mixtures of normals. The proofs do not rely on boundness or monotonicity of the function r(t) in the representation of the Bayes estimator as .     

On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the open unit ball B of Cⁿ. Let φ=(φ₁,…,φ_n) be a holomorphic self-map of B and g∈H(B) such that g(0)=0. In this paper we study the boundedness and compactness of the following integral-type operator, recently introduced by Xiangling Zhu and the second author

     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Young’s inequality and trace

Let φ be a positive linear functional on M_n(C) and f,g mutually conjugate in the sense of Young. In this note we show a necessary and sufficient condition for the inequality

     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

Le problème de Yamabe avec singularités

Let (Mn,g) be a compact riemannian manifold of dimension n?3. Under some assumptions, we prove that there exists a positive function φ solution of the Yamabe equation

     

Regularizations of products of residue and principal value currents

Let f₁ and f₂ be two functions on some complex n-manifold and let φ be a test form of bidegree (n,n−2). Assume that (f₁,f₂) defines a complete intersection. The integral of φ/(f₁f₂) on {²|f₁|=?₁,²|f₂|=?₂} is the residue integral . It is in general discontinuous at the origin. Let χ₁ and χ₂ be smooth functions on [0,∞] such that χj(0)=0 and χj(∞)=1. We prove that the regularized residue integral defined as the integral of , where χj=χj(²|fj|/?j), is Hölder continuous on the closed first quarter and that the value at zero is the Coleff-Herrera residue current acting on φ. In fact, we prove that if φ is a test form of bidegree (n,n−1) then the integral of is Hölder continuous and tends to the -potential of the Coleff-Herrera current, acting on φ. More generally, let f₁ and f₂ be sections of some vector bundles and assume that f₁⊕f₂ defines a complete intersection. There are associated principal value currents Uf and Ug and residue currents Rf and Rg. The residue currents equal the Coleff-Herrera residue currents locally. One can give meaning to formal expressions such as e.g. Uf∧Rg in such a way that formal Leibnitz rules hold. Our results generalize to products of these currents as well.     

Uniform asymptotic stability for perturbed neutral delay differential equations

One-dimensional perturbed neutral delay differential equations of the form (x(t)−P(t,x(t−τ)))′=f(t,x_t)+g(t,x_t) are considered assuming that f satisfies −v(t)M(φ)?f(t,φ)?v(t)M(−φ), where M(φ)=max{0,max_s∈[−r,0]φ(s)}. A typical result is the following: if ‖g(t,φ)‖?w(t)‖φ‖ and , then the zero solution is uniformly asymptotically stable providing that the zero solution of the corresponding equation without perturbation (x(t)−P(t,x(t−τ)))′=f(t,x_t) is uniformly asymptotically stable. Some known results associated with this equation are extended and improved.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Classification of the factorial functions of Eulerian binomial and Sheffer posets

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or ²n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n)=n! has Sheffer factorial function D(n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function B(n)=²n−1 as the doubling of an upside-down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra BX or the infinite cubical lattice . We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.     

Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u₀(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P^∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.     

A generalization of the Strassen converse

This note is devoted to a generalization of the Strassen converse. Let gn:R^∞→[0,∞], n?1 be a sequence of measurable functions such that, for every n?1, and for all x,y∈R^∞, where 0<C<∞ is a constant which is independent of n. Let be a sequence of i.i.d. random variables. Assume that there exist r?1 and a function ?:[0,∞)→[0,∞) with limt→∞?(t)=∞, depending only on the sequence such that lim supn→∞gn(X₁,X₂,…)=?(Er|X|) a.s. whenever Er|X|<∞ and EX=0. We prove the converse result, namely that lim supn→∞gn(X₁,X₂,…)<∞ a.s. implies Er|X|<∞ (and EX=0 if, in addition, lim supn→∞gn(c,c,…)=∞ for all c≠0). Some applications are provided to illustrate this result.