Periodic decomposition of measurable integer valued functions

We study those functions that can be written as a sum of (almost everywhere) integer valued periodic measurable functions with given periods. We show that being (almost everywhere) integer valued measurable function and having a real valued periodic decomposition with the given periods is not enough. We characterize those periods for which this condition is enough. We also get that the class of bounded measurable (almost everywhere) integer valued functions does not have the so-called decomposition property. We characterize those periods a₁,…,ak for which an almost everywhere integer valued bounded measurable function f has an almost everywhere integer valued bounded measurable (a₁,…,ak)-periodic decomposition if and only if Δa₁?Δakf=0, where Δaf(x)=f(x+a)−f(x).     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Existence and global exponential stability of anti‐periodic solutions for delayed quaternion‐valued cellular neural networks with impulsive effects

In this paper, we consider a class of delayed quaternion‐valued cellular neural networks (DQVCNNs) with impulsive effects. By using a novel continuation theorem of coincidence degree theory, the existence of anti‐periodic solutions for DQVCNNs is obtained with or without assuming that the activation functions are bounded. Furthermore, by constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of anti‐periodic solutions for DQVCNNs. Our results are new and complementary to the known results even when DQVCNNs degenerate into real‐valued or complex‐valued neural networks. Finally, an example is given to illustrate the effectiveness of the obtained results.     

Polynomials with a root close to an integer

Let us consider the set of polynomials with integer coefficients of a given degree and of bounded height. We prove that among all polynomials in this set with no integer roots the polynomialx ⁿ-H(x ⁿ⁻¹+x ⁿ⁻²+...+1) has a root closest to an integer. Partially supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2006 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 3, pp. 310–316, July–September, 1999.     

A new class of polynomials associated with Bernstein and beta polynomials

The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.     

Notes on generalization of the Bernoulli type polynomials

Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.     

New ideas for decomposing nonlinearities in differential equations

In this paper we consider the decomposition for the nonlinearity in a differential equation for the solution by decomposition. By analyzing and transforming the Taylor expansion of the nonlinearity about the initial solution component, the decomposition of the nonlinearity is converted to the partitions of the solution sets for a class of Diophantine equations. This conversion simplifies the discussion and presents a new idea for decompositions. We enumerate five types of partitions and their corresponding decomposition polynomials. Each of the last four types contains infinitely many kinds of decomposition polynomials in the form of finite sums. In Types 2, 3 and 4, there is a parameter q and each value of q corresponds to a class of decomposition polynomials. In Type 5, each positive integer sequence {c_j} satisfying 1 = c₁ ? c₂ ? ? and j ? c_j for j = 2, 3, … corresponds to a class of decomposition polynomials. Four classes of the Adomian polynomials [R. Rach, A new definition of the Adomian polynomials, Kybernetes 37 (2008) 910-955] are derived as particular cases.     

Explicit generalized zolotarev polynomials with complex coefficients

We give explicitly a class of polynomials with complex coefficients of degreen which deviate least from zero on [−1, 1] with respect to the max-norm among all polynomials which have the same,m + 1, 2m ≤n, first leading coefficients. Form=1, we obtain the polynomials discovered by Freund and Ruschewyh. Furthermore, corresponding results are obtained with respect to weight functions of the type 1/√ρl, whereρl is a polynomial positive on [−1, 1].     

Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

The Multivariate Integer Chebyshev Problem

The multivariate integer Chebyshev problem is to find polynomials with integer coefficients that minimize the supremum norm over a compact set in ℂ ^d . We study this problem on general sets but devote special attention to product sets such as cube and polydisk. We also establish a multivariate analog of the Hilbert–Fekete upper bound for the integer Chebyshev constant, which depends on the dimension of space. In the case of single-variable polynomials in the complex plane, our estimate coincides with the Hilbert–Fekete result.      

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Local bifurcation of critical periods in quadratic-like cubic systems

In this paper, we investigate quadratic-like cubic systems having a center at $O$ for the local bifurcation of critical periods. We provide an inductive algorithm to compute polynomials of periodic coefficients, find structures of solutions for systems of algebraic equations corresponding to weak centers of finite order, and derive conditions on parameters under which the considered equilibrium is a weak center of order $k$, $k=0,1,2,3,4$. Furthermore, we show that with appropriate perturbations, at most four critical periods bifurcate from the weak center of finite order, and we give conditions under which exactly $k$ critical periods bifurcate from the center $O$ for each integer $k=1,2,3,4$.     

Cohomology and asymptotic stability of 1-dimensional continua

We prove that a 1-dimensional continuum carrying a flow without singular points is homeomorphic to the unit circle if its first Čech cohomology group with integer coefficients is isomorphic to ℤ. As an application of this we obtain that an asymptotically stable invariant 1-dimensional continuum of a flow on a locally compact ANR, which does not contain singular points, must be a periodic orbit.     

Generating functions of the (<Emphasis Type="Italic">h,q</Emphasis>) extension of twisted Euler polynomials and numbers

By using p-adic q-deformed fermionic integral on ℤ _p , we construct new generating functions of the twisted (h, q)-Euler numbers and polynomials attached to a Dirichlet character χ. By applying Mellin transformation and derivative operator to these functions, we define twisted (h, q)-extension of zeta functions and l-functions, which interpolate the twisted (h, q)-extension of Euler numbers at negative integers. Moreover, we construct the partially twisted (h, q)-zeta function. We give some relations between the partially twisted (h, q)-zeta function and twisted (h, q)-extension of Euler numbers.      

Matrix Valued Orthogonal Polynomials Arising from the Complex Projective Space

The main purpose of this paper is to display new families of matrix valued orthogonal polynomials satisfying second-order differential equations, obtained from the representation theory of U(n). Given an arbitrary positive definite weight matrix W(t) one can consider the corresponding matrix valued orthogonal polynomials. These polynomials will be eigenfunctions of some symmetric second-order differential operator D only for very special choices of W(t). Starting from the theory of spherical functions associated to the pair (SU(n+1), U(n)) we obtain new families of such pairs {W,D}. These depend on enough integer parameters to obtain an immediate extension beyond these cases.     

Doubly periodic lozenge tilings of a hexagon and matrix valued orthogonal polynomials

We analyze a random lozenge tiling model of a large regular hexagon, whose underlying weight structure is periodic of period 2 in both the horizontal and vertical directions. This is a determinantal point process whose correlation kernel is expressed in terms of non‐Hermitian matrix valued orthogonal polynomials (OPs). This model belongs to a class of models for which the existing techniques for studying asymptotics cannot be applied. The novel part of our method consists of establishing a connection between matrix valued and scalar valued OPs. This allows to simplify the double contour formula for the kernel obtained by Duits and Kuijlaars by reducing the size of a Riemann–Hilbert problem. The proof relies on the fact that the matrix valued weight possesses eigenvalues that live on an underlying Riemann surface of genus 0. We consider this connection of independent interest; it is natural to expect that similar ideas can be used for other matrix valued OPs, as long as the corresponding Riemann surface is of genus 0. The rest of the method consists of two parts, and mainly follows the lines of a previous work of Charlier, Duits, Kuijlaars and Lenells. First, we perform a Deift–Zhou steepest descent analysis to obtain asymptotics for the scalar valued OPs. The main difficulty is the study of an equilibrium problem in the complex plane. Second, the asymptotics for the OPs are substituted in the double contour integral and the latter is analyzed using the saddle point method. Our main results are the limiting densities of the lozenges in the disordered flower‐shaped region. However, we stress that the method allows in principle to rigorously compute other meaningful probabilistic quantities in the model.     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.