Some evaluation of q-analogues of Euler sums

In this paper, we discuss the analytic representations of q-Euler sums which involve q-harmonic numbers through q-polylogarithms, either linearly or nonlinearly, and give explicit formulae for several classes of q-Euler sums in terms of q-polylogarithms and q-special functions. Furthermore, we develop new closed form representations of sums of quadratic and cubic parametric q-Euler sums. Finally, we can find that the q-Euler sums are reducible to the classical Euler sums when q approaches 1.     

Equivalence of the Metaplectic Representation with Sums of Wavelet Representations for a Class of Subgroups of the Symplectic Group

Equivalence of the metaplectic representation with a sum of affine representations is discussed for a class of subgroups of the symplectic group. The paper begins with a study of sums of wavelet transforms, and it is shown that the usual admissibility conditions for the wavelet transform apply to transforms by sums of affine representations as well. A construction of admissible vectors, bandlimited admissible vectors and frames for sums of affine representations by means of transversals is given. A class of subgroups of the symplectic group which are compact extensions affine groups are identified, and it is shown how subrepresentations of the metaplectic representation can be equivalent to affine representations. This equivalence is illuminated by several examples.     

Summation formula involving harmonic numbers

Some identities of sums associated with harmonic numbers and binomial coefficients are developed. Integral representations and closed form identities of these sums are also given.     

The representation of the generalized linear Euler sums with parameters

In this paper, by using residue method, we obtain the representations of some basic linear generalized Euler sums with parameters. Based on the linear generalized Euler sums with parameters, some new Euler sums are obtained and expressed in the closed forms. When the parameters of new Euler sums take special values, we can get some usual expressions of Euler sums. Moreover, the integrals of many special functions can be expressed as the Euler sums given in this paper.     

Binary terms in polynomial representations of Boolean functions

Polynomial representations of Boolean functions by binary terms are considered. The construction of terms involves variables and residual functions. Special cases of such representations are the decomposition of a function with respect to variables, Zhegalkin polynomials, and representations of functions as sums of conjunctions of residual functions.     

Some alternating sums of Lucas numbers

We consider alternating sums of squares of odd and even terms of the Lucas sequence and alternating sums of their products. These alternating sums have nice representations as products of appropriate Fibonacci and Lucas numbers.     

Identities for Alternating Inverse Squared Binomial and Harmonic Number Sums

We develop new families of closed-form representations of sums of alternating harmonic numbers and reciprocal squared binomial coefficients including integral representations.     

Harmonic numbers at half integer values

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series.     

Some evaluation of harmonic number sums

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.     

Representations of ordered random variables in the terms of the sums or products of independent variables

This is a survey of the available representations capable of expressing dependent ordered variables like order statistics and records as sums of independent summands or as products of independent random multipliers. Some new similar representations are obtained that in a similar manner express the common distributions of representatives of these two types of ordered random variables.     

Sharp bounds for the partition function of integer sequences

By means of a new criterion for higher multiplicities of additive representations of integers as sums of distinct terms of a fixed integer sequence sharp bounds are determined for the partition function of many sequences with the aid of a computer.     

Toward an Infinite-Component Field Theory with a Double Symmetry: Free Fields

We begin a study of possibilities of describing hadrons in terms of monolocal fields that transform under proper Lorentz group representations that are infinite direct sums of finite-dimensional irreducible representations. The additional requirement that the free-field Lagrangians be invariant under the secondary symmetry transformations generated by the polar or the axial four-vector representation of the orthochronous Lorentz group provides an effective mechanism for selecting the class of representations considered and eliminating an infinite number of arbitrary parameters allowed by the relativistic invariance of the Lagrangians.     

Representations,decompositions and sample function continuity of random fields with independent increments

Standard fare in the study of representations and decompositions of processes with independent increments is pursued in the somewhat more complex setting of vector-valued random fields having independent increments over disjoint sets. Such processes are first constructed as almost surely uniformly convergent sums of Poisson type summands, that immediately yield information on sample function properties of versions. The constructions employed, which include a generalized version of the Ferguson-Klass construction with uniform convergence, are new even in the simpler setting of processes in one-dimensional time.Following these constructions, or representations, an analogue of the Lévy-Ito decomposition for Lévy processes is developed, which then enables a number of simple sample function properties of these processes to be read off from the Lévy measure in their characteristic functionals.The paper concludes with a study of general centred additive random fields and an appendix incorporating a brief survey of the theory of centred sums of independent random variables.     

Duality respecting representations and compatible complexity measures for gammoids

We show that every gammoid has special digraph representations, such that a representation of the dual of the gammoid may be easily obtained by reversing all arcs. In an informal sense, the duality notion of a poset applied to the digraph of a special representation of a gammoid commutes with the operation of forming the dual of that gammoid. We use these special representations in order to define a complexity measure for gammoids, such that the classes of gammoids with bounded complexity are closed under duality, minors, and direct sums.     

Locally Quadratic Modules and Minuscule Representations

We give a new, geometric proof of a theorem by Timmesfeld showing that for simple Chevalley groups, abstract modules where all roots act quadratically are direct sums of minuscule representations. Our proof is uniform, treats finite and infinite fields on an equal footing, and includes Lie rings.     

Sums of fractions and finiteness of monodromy

We solve an elementary number theory problem on sums of fractional parts. We apply our result to deduce the finiteness of the image of certain monodromy representations.     

Asymptotic values for degrees associated with strips of young diagrams

The sums S^(β)_l(n) occur in the representations of the symmetric and the general linear groups, in combinatorics and in PI algebras. We give asymptotic values for these sums.     

Asymptotic degeneration of representations of quivers

We define asymptotic degeneration of nilpotent representations of an arbitrary finite quiver, using large tensor powers and small direct sums, and characterize this notion by a simple and effective criterion.     

Determination of the Direct Sums of Rational Groups by H-Representations of the Endomorphism Rings up to Equality

The problem of determination of Abelian groups (up to isomorphism) by their rings of endomorphisms in the class of completely decomposable torsion-free Abelian groups has been solved earlier. For the class of direct sums of rational groups, one can speak of determination of Abelian groups by rational representations of their endomorphism rings up to equality. In this paper, we consider this problem for the class of finite direct sums of rational groups and for some subclasses.