Generalizations of Ramanujan's reciprocity theorem and their applications

First, we briefly survey Ramanujan's reciprocity theorem fora certain q-series related to partial theta functions and givea new proof of the theorem. Next, we derive generalizationsof the reciprocity theorem that are also generalizations ofthe ₁₁ summation formula and Jacobi triple product identityand show that these reciprocity theorems lead to generalizationsof the quintuple product identity, as well. Last, we presentsome applications of the generalized reciprocity theorems andproduct identities, including new representations for generatingfunctions for sums of six squares and those for overpartitions.     

Generalizations of the Liouville theorem

The purpose of this paper is to generalize the Liouville theorem for functions which are defined on the complete Riemannian manifolds. Then, we apply it to the isometric immersions between complete Riemannian manifolds in order to obtain an estimate for the size of the image of immersions in terms of the supremum of the length of their mean curvature vector in a quite general setting. The proofs are based on the Calabi's generalization of maximum principle for functions which are not necessarily differentiable.     

Generalizations of Schur's partition theorem

Schur's partition theorem states that the number of partitions ofn into distinct parts ≡ 1,2 (mod 3) is equal to the number of partitions ofn into parts with minimal difference 3 and no consecutive multiples of 3. A three-parameter generalization of Gleissberg's refinement of Schur's theorem is obtained by showing that is equal to the numerator of a certain continued fraction. Two proofs are presented, one completely combinatorial, and one using generating functions. This article was processed by the author using the Springer-Verlag T_EX mamath macro package 1990.     

Generalizations of Cantor's theorem in

A set x is Dedekind infinite if there is an injection from ω into x ; otherwise x is Dedekind finite. A set x is power Dedekind infinite if , the power set of x , is Dedekind infinite; otherwise x is power Dedekind finite. For a set x , let pdfin(x ) be the set of all power Dedekind finite subsets of x . In this paper, we prove in (without the axiom of choice) two generalizations of Cantor's theorem (i.e., the statement that for all sets x , there are no injections from into x ): The first one is that for all power Dedekind infinite sets x , there are no Dedekind finite to one maps from into pdfin(x ). The second one is that for all sets , if x is infinite and there is a power Dedekind finite to one map from y into x , then there are no surjections from y onto . We also obtain some related results.     

Generalizations of the Krasnoselskii fixed point theorem

We obtain new generalized forms of the Krasnoselskii theorem on fixed points for a sum of two operators and show that our new results encompass a number of previously known generalizations of the theorem.     

Generalizations of a theorem of Rolewicz

The aim of this article is to give a unified treatment for the theorems of Rolewicz and Neerven type for uniform exponential stability of evolution families. We obtain necessary and sufficient conditions for uniform exponential stability of evolution families, generalizing a stability theorem due to Rolewicz and we present a new proof for the Rolewicz theorem, based on the theory of Banach function spaces. Finally, we apply our results and we deduce a generalization for a classical stability theorem due to Przyluski and Rolewicz.     

Generalizations of the truncated pentagonal number theorem results

The Ramanujan Journal - In this paper, results associated with the truncated pentagonal number theorem will be generalized. A family of truncated series will be created that count the number of...     

Schützenberger groups ofH-Classes containing odd degree polynomials-Classes containing odd degree polynomials

In this note we determine when topological simplicity of a subsemigroup implies its simplicity. A measure theoretic approach is used. The main result is: In a locally compact completely simple semigroup whose maximal subgroups are compact, every subsemigroup which is either locally compact or of non-empty interior is completely simple. As a corollary, we have: In a compact semigroup, any topologically simple subsemigroup which is either locally compact or of non-empty interior is simple.     

Irreducible representations of odd degree

McKay’s original observation on characters of odd degrees of finite groups is reduced to almost simple groups.     

Generalizations of the Riesz convergence theorem for Lorentz spaces

Summary We obtain preservation inequalities for Lipschitz constants of higher order in simultaneous approximation processes by Bernstein type operators. From such inequalities we derive the preservation of the corresponding Lipschitz spaces.     

Multidimensional natural splines of odd degree

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 65–68, February, 1990.     

Generic cyclic polynomials of odd degree

For any cyclic group G of odd order and any field F whose characteristic does not divide the order of G it is possible to find a polynomial which parametrizes all extensions of F with group G. In this paper such a polynomial is explicitly constructed for each such group G which is valid for all such fields F.