A new proof of the septuple product identity

By means of Liouville’s theorem and the log-derivative method, we give a new proof of the septuple product identity.     

Applications of residues to combinatorial identities

A concrete aspect of Grothendieck Duality is used to give local cohomology proofs of combinatorial identities including MacMahon's master theorem, Grosswald identity, identity of Shoo, Tepper identity, and others.

     

Transformations of Border Strips and Schur Function Determinants

We introduce the notion of the cutting strip of an outside decomposition of a skew shape, and show that cutting strips are in one-to-one correspondence with outside decompositions for a given skew shape. Outside decompositions are introduced by Hamel and Goulden and are used to give an identity for the skew Schur function that unifies the determinantal expressions for the skew Schur functions including the Jacobi-Trudi determinant, its dual, the Giambelli determinant and the rim ribbon determinant due to Lascoux and Pragacz. Using cutting strips, one obtains a formula for the number of outside decompositions of a given skew shape. Moreover, one can define the basic transformations which we call the twist transformation among cutting strips, and derive a transformation theorem for the determinantal formula of Hamel and Goulden. The special case of the transformation theorem for the Giambelli identity and the rim ribbon identity was obtained by Lascoux and Pragacz. Our transformation theorem also applies to the supersymmetric skew Schur function.     

The homogeneous q-difference operator

We introduce a q-differential operator D_xy on functions in two variables which turns out to be suitable for dealing with the homogeneous form of the q-binomial theorem as studied by Andrews, Goldman, and Rota, Roman, Ihrig, and Ismail, et al. The homogeneous versions of the q-binomial theorem and the Cauchy identity are often useful for their specializations of the two parameters. Using this operator, we derive an equivalent form of the Goldman–Rota binomial identity and show that it is a homogeneous generalization of the q-Vandermonde identity. Moreover, the inverse identity of Goldman and Rota also follows from our unified identity. We also obtain the q-Leibniz formula for this operator. In the last section, we introduce the homogeneous Rogers–Szegö polynomials and derive their generating function by using the homogeneous q-shift operator.     

A proof of Markov's theorem for polynomials on Banach spaces

Our object is to present an independent proof of the extension of V.A. Markov's theorem to Gâteaux derivatives of arbitrary order for continuous polynomials on any real normed linear space. The statement of this theorem differs little from the classical case for the real line except that absolute values are replaced by norms. Our proof depends only on elementary computations and explicit formulas and gives a new proof of the classical theorem as a special case. Our approach makes no use of the classical polynomial inequalities usually associated with Markov's theorem. Instead, the essential ingredients are a Lagrange interpolation formula for the Chebyshev nodes and a Christoffel-Darboux identity for the corresponding bivariate Lagrange polynomials. We use these tools to extend a single variable inequality of Rogosinski to the case of two real variables. The general Markov theorem is an easy consequence of this.     

A Topological Intersection Theorem with Applications in H-Spaces

In this paper, a topological intersection theorem of a family of sets with H-convex sections is established. As an application to this theorem, the Nash equilibrium of abstract economy, the Ky-Fan minimax theorem, and an existence problem of solution for quasi-variational inequality are studied. The results presented in this paper modify and extend the corresponding results given in the literature.AMS Subject Classification (2000) 47H10This work is supported by the Foundation of Yunnan Science Technology Commission, China.     

Nonlinear Doob—Meyer Decomposition with Jumps

Concepts of g-supersolution ,g-martingale,g-supermartingale are introduced,wihch are related to BSDE with Brownian motion and Poisson Point process.A strict comparison theorem,monotonic limit theorem related to this type of BSDE are also discussed.As an application of these results,a nonlinear Doob-Meyer decomposition theorem is obtained.     

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Z _n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.     

Multiplicity theorems for scalar periodic problems at resonance with <Emphasis Type="Italic">p</Emphasis>-Laplacian-like operator

In this paper, we study the existence of multiple solutions for nonlinear scalar periodic problems at resonance with p-Laplacian-like operator. Using the Ekeland variational principle a two-solution theorem is obtained and using also a local linking theorem a three-solution theorem is proved.      

A theorem concerning a product of two general classes of polynomials and the multivariableH-function

A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.     

A Limit Theorem for Dirichlet L-Functions

We prove a limit theorem on the weak convergence of probability measures in the space of continuous functions for Dirichlet L-functions. The result generalizes a similar theorem for the Riemann zeta-function.     

The Bedrosian identity for the Hilbert transform of product functions

We investigate a necessary and sufficient condition which ensures validity of the Bedrosian identity for the Hilbert transform of a product function . Convenient sufficient conditions are presented, which cover the classical Bedrosian theorem and provide us with new insightful information.

     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

On the Extension of D-Poset Valued Measures

A variant of Alexandrov theorem is proved stating that a compact, subadditive D-poset valued mapping is continuous. Then the measure extension theorem is proved for MV-algebra valued measures.     

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables

A theorem concerning a product of a general class of polynomials and theH-function of several complex variables is given. Using this theorem certain integrals and expansion formula have been obtained. This general theorem is capable of giving a number of new, interesting and useful integrals, expansion formulae as its special cases.     

A reciprocity theorem for certain q-series found in Ramanujan’s lost notebook

Three proofs are given for a reciprocity theorem for a certain q-series found in Ramanujan’s lost notebook. The first proof uses Ramanujan’s ₁ψ₁ summation theorem, the second employs an identity of N. J. Fine, and the third is combinatorial. Next, we show that the reciprocity theorem leads to a two variable generalization of the quintuple product identity. The paper concludes with an application to sums of three squares. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D15 B. C. Berndt: Research partially supported by grant MDA904-00-1-0015 from the National Security Agency. A. J. Yee: Research partially supported by a grant from The Number Theory Foundation.     

F-空间中的一个基本定理及其在数值分析中的应用

本文针对F-空间中闭算子方程的一般逼近格式,研究其相容性、收敛性和稳定性之间的关系.所得的主要结果是：这种一般逼近格式在相容性条件下,其收敛性与稳定性是等价的.此定理可以看作是对Lax等价原理的推广,是求解第一类闭算子方程的一般逼近格式的基本定理.为得到这一主要结果,本文还给出了F-空间中的一条基本定理,众所周知的一致有界原理,闭图像定理和开映像定理是其简单推论.     

Discrete limit theorems for general Dirichlet series. III

Here we prove a limit theorem in the sense of the weak convergence of probability measures in the space of meromorphic functions for a general Dirichlet series. The explicit form of the limit measure in this theorem is given. Partially supported by Lithuanian Foundation of Studies and Science     

A Generalization of Niho’s Theorem

In this note, we generalize Niho’s theorem on cross-correlation functions of binary m-sequences. Our theorem applies both to binary and non-binary sequences. AMS Classification: 11T24, 94A55 Research supported by the Academy of Finland, grant 200213.     

Anti-lecture hall compositions and overpartitions

We show that the number of anti-lecture hall compositions of n with the first entry not exceeding k−2 equals the number of overpartitions of n with non-overlined parts not congruent to 0,±1 modulo k. This identity can be considered as a finite version of the anti-lecture hall theorem of Corteel and Savage. To prove this result, we find two Rogers-Ramanujan type identities for overpartitions which are analogous to the Rogers-Ramanujan type identities due to Andrews. When k is odd, we give another proof by using the bijections of Corteel and Savage for the anti-lecture hall theorem and the generalized Rogers-Ramanujan identity also due to Andrews.