Almost Summing Polynomials

This paper introduces the class of almost summing homogeneous polynomials between Banach spaces. A characterization by means of vector‐valued sequences is proved, connections with the theory of absolutely summing polynomials are established and several examples are given. Some composition and coincidence theorems are obtained and the case of spaces with type or cotype is also investigated.     

Weakly Compact and Absolutely Summing Polynomials

This paper shows that, contrary to the case of linear operators, absolutely summing homogeneous polynomials are not always weakly compact. It is also shown that, regardless of the infinite dimensional Banach space E and the positive integer n, there exists an n-homogeneous polynomial P from E to E that plays the role of the identity operator in the sense that P is neither compact nor absolutely r-summing for any r, and P is weakly compact if and only if E is reflexive.     

Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.     

Generating Polynomials and Symmetric Tensor Decompositions

This paper studies symmetric tensor decompositions. For symmetric tensors, there exist linear relations of recursive patterns among their entries. Such a relation can be represented by a polynomial, which is called a generating polynomial. The homogenization of a generating polynomial belongs to the apolar ideal of the tensor. A symmetric tensor decomposition can be determined by a set of generating polynomials, which can be represented by a matrix. We call it a generating matrix. Generally, a symmetric tensor decomposition can be determined by a generating matrix satisfying certain conditions. We characterize the sets of such generating matrices and investigate their properties (e.g., the existence, dimensions, nondefectiveness). Using these properties, we propose methods for computing symmetric tensor decompositions. Extensive examples are shown to demonstrate the efficiency of proposed methods.     

Tensor Rank: Matching Polynomials and Schur Rings

We study the polynomial equations vanishing on tensors of a given rank. By means of polarization we reduce them to elements  $A$ of the group algebra ${\mathbb {Q}}[S_n\times S_n]$ and describe explicit linear equations on the coefficients of  $A$ to vanish on tensors of a given rank. Further, we reduce the study to the Schur ring over the group $S_n\times S_n$ that arises from the diagonal conjugacy action of  $S_n$ . More closely, we consider elements of ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and describe them in terms of triples of Young diagrams, their irreducible characters, and nonvanishing of their Kronecker coefficients. Also, we construct a family of elements in ${\mathbb {Q}}[S_n\times S_n]$ vanishing on tensors of rank $n-1$ and illustrate our approach by a sharp lower bound on the border rank of an explicitly produced tensor. Finally, we apply this construction to prove a lower bound $5n^2/4$ on the border rank of the matrix multiplication tensor (being, of course, weaker than the best known one $(2-\epsilon )\cdot n^2$ , due to Landsberg, Ottaviani).     

Distributional Transformations,Orthogonal Polynomials,and Stein Characterizations

A new class of distributional transformations is introduced, characterized by equations relating function weighted expectations of test functions on a given distribution to expectations of the transformed distribution on the test functions higher order derivatives. The class includes the size and zero bias transformations, and when specializing to weighting by polynomial functions, relates distributional families closed under independent addition, and in particular the infinitely divisible distributions, to the family of transformations induced by their associated orthogonal polynomial systems. For these families, generalizing a well known property of size biasing, sums of independent variables are transformed by replacing summands chosen according to a multivariate distribution on its index set by independent variables whose distributions are transformed by members of that same family. A variety of the transformations associated with the classical orthogonal polynomial systems have as fixed points the original distribution, or a member of the same family with different parameter.     

Constants and Darboux Polynomials for Tensor Products of Polynomial Algebras with Derivations

Abstract

Let d ₁ : k[X] → k[X] and d ₂ : k[Y] → k[Y] be k-derivations, where k[X] ? k[x ₁,…,x _n ], k[Y] ? k[y ₁,…,y _m ] are polynomial algebras over a field k of characteristic zero. Denote by d ₁ ⊕ d ₂ the unique k-derivation of k[X, Y] such that d| _k[X] = d ₁ and d| _k[Y] = d ₂. We prove that if d ₁ and d ₂ are positively homogeneous and if d ₁ has no nontrivial Darboux polynomials, then every Darboux polynomial of d ₁ ⊕ d ₂ belongs to k[Y] and is a Darboux polynomial of d ₂. We prove a similar fact for the algebra of constants of d ₁ ⊕ d ₂ and present several applications of our results.     

Characterizations of Some Free Random Variables by Properties of Conditional Moments of Third Degree Polynomials

We investigate Laha–Lukacs properties of noncommutative random variables (processes). We prove that some families of free Meixner distributions can be characterized by the conditional moments of polynomial functions of degree 3. We also show that this fact has consequences in describing some free Lévy processes. The proof relies on a combinatorial identity. At the end of this paper we show that this result can be extended to a \(q\) -Gausian variable.     

Summing Boolean Algebras

In this paper we will study some familes and subalgebras F of P (N) that let us characterize the unconditional convergence of series through the weak convergence of subseries ∑i∈A^xi, A ∈ F. As a consequence, we obtain a new version of the Orlicz-Pettis theorem, for Banach spaces. We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.     

Absolutely Summing Processes

Abstract

Absolutely summing processes are defined, which form a subclass of weakly operator harmonizable processes. When the parameter space is the set of real numbers, it is proved that an absolutely summing process is represented as an integral of operator stationary processes with respect to an appropriate probability measure. To do this, weak convergence of scalar and vector measures is considered. Then we prove compactness of the unit ball of vector measures under certain topologies, and we apply the Choquet theorem to derive an integral representation.     

p-Lattice Summing Operators

In this paper we study the spaces ∞_p(E, X) of p-lattice summing operators from a Banach space E to a Banach lattice X. The main results characterize those E and X for which Δ_p(E, X) = II_p(E, X) and we show that ∞(E, X)=Δ₂(E, X) for an infinite dimensional Banach lattice X of finite cotype if and only if E is isomorphic to a Hilbert space.     

Bernoulli多项式的积分多项式

首次研究了 Bernoulli多项式的积分多项式 .首先 ,给出这类多项式的定义和基本性质 ;其次 ,建立两类幂和多项式的相互关系 ;最后 ,介绍上述结果在求解自然数幂和公式方面的应用 .     

级数求和的八种方法

总结归纳无穷级数求和的八种方法,并借助实例加以说明.     

关于幂级数求和的教学实例分析

通过几组教学实例,阐述如何通过积分、求导、拆项等方法将复杂的幂级数转化为简单的基本型幂级数和等比级数以求其和函数.     

Bernoulli多项式和Euler多项式的关系

本文给出了 Bernoulli- Euler数之间的关系和 Bernoulli- Euler多项式之间的关系 ,从而深化和补充了有关文献中的相关结果 .     

数项级数和的几种典型求法

借助实例介绍利用级数收敛和数列极限存在的关系并结合阿贝尔变换求数项级数和的方法、利用幂级数和傅里叶级数的和函数在某点的函数值来求数项级数和的方法、利用基本初等函数的泰勒级数公式求数项级数和的方法.     

Euler Pseudoprime Polynomials and Strong Pseudoprime Polynomials

It is usual to emphasize the analogy between the integers and polynomials with coefficients in a finite field, comparing different notions in the two points of view. We introduce a particular rank one Drinfeld module to get an exponentiation for polynomials and then define the notions of Euler pseudoprimes and strong pseudoprimes for polynomials with coefficients in a finite field. As for the integers, we have Solovay–Strassen and Miller–Rabin tests for polynomials.