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1.
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. 相似文献
2.
In this paper, we first establish some existence theorems of systems of generalized vector equilibrium problems. From these
results, we obtain new variants of Ekeland’s variational principle in a Hausdorff t.v.s., a minimax theorem and minimization
theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem
and a generalization of Schauder’s fixed point theorem are also given. 相似文献
3.
By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational
principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by
the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement
of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev
theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization
theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other. 相似文献
4.
In this paper we introduce the notion of tolerance in connection with Helly-type theorems and prove, using the Erdős–Gallai
theorem, that any Helly-type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number
of exceptional sets or points. In particular, we analyze some of the classical Helly-type theorems, such as Caratheodory’s
and Tverberg’s theorems, as well as some other interesting ones. 相似文献
5.
The concepts of L-convex function and M-convex function have recently been introduced by Murota as generalizations of submodular
function and base polyhedron, respectively, and discrete separation theorems are established for L-convex/concave functions
and for M-convex/concave functions as generalizations of Frank’s discrete separation theorem for submodular/supermodular set
functions and Edmonds’ matroid intersection theorem. This paper shows the equivalence between Murota’s L-convex functions
and Favati and Tardella’s submodular integrally convex functions, and also gives alternative proofs of the separation theorems
that provide a geometric insight by relating them to the ordinary separation theorem in convex analysis.
Received: November 27, 1997 / Accepted: December 16, 1999?Published online May 12, 2000 相似文献
6.
In this paper, we attempt to give a unified approach to the existing several versions of Ekeland’s variational principle.
In the framework of uniform spaces, we introduce p-distances and more generally, q-distances. Then we introduce a new type
of completeness for uniform spaces, i.e., sequential completeness with respect to a q-distance (particularly, a p-distance),
which is a very extensive concept of completeness. By using q-distances and the new type of completeness, we prove a generalized
Takahashi’s nonconvex minimization theorem, a generalized Ekeland’s variational principle and a generalized Caristi’s fixed
point theorem. Moreover, we show that the above three theorems are equivalent to each other. From the generalized Ekeland’s
variational principle, we deduce a number of particular versions of Ekeland’s principle, which include many known versions
of the principle and their improvements. 相似文献
7.
S. Raghavan 《Proceedings Mathematical Sciences》1984,93(2-3):147-160
Recently, Bombieri and Vaaler obtained an interesting adelic formulation of the first and the second theorems of Minkowski
in the Geometry of Numbers and derived an effective formulation of the well-known “Siegel’s lemma” on the size of integral
solutions of linear equations. In a similar context involving linearinequalities, this paper is concerned with an analogue of a theorem of Khintchine on integral solutions for inequalities arising from
systems of linear forms and also with an analogue of a Kronecker-type theorem with regard to euclidean frames of integral
vectors. The proof of the former theorem invokes Bombieri-Vaaler’s adelic formulation of Minkowski’s theorem. 相似文献
8.
Murat Limoncu 《Archiv der Mathematik》2010,95(2):191-199
By using two modified Ricci tensors, we prove some theorems which correspond to Myers’s diameter estimate theorem and Bochner’s
vanishing theorem. 相似文献
9.
Deng Hua ZHANG Huai Xin CAO 《数学学报(英文版)》2007,23(2):321-326
We prove a generalization of Hyers' theorem on the stability of approximately additive mapping and a generalization of Badora's theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam's problem. 相似文献
10.
Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues
have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues,
namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s
conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting. 相似文献
11.
Petri Rosendahl 《Designs, Codes and Cryptography》2006,38(3):331-336
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. 相似文献
12.
In this paper, we establish several theorems for the explicit evaluations of Ramanujan-G?llnitz-Gordon continued fraction
by using some parameterizations of Ramanujan’s theta-functions.
Nayandeep Deka Baruah, Corresponding author. Research partially supported by grant SR/FTP/MA-02/2002 from DST, Govt. of India.
Authors’ address: Nayandeep Deka Baruah and Nipen Saikia, Department of Mathematical Sciences, Tezpur University, Napaam-784028,
Assam, India 相似文献
13.
Extreme properties of quermassintegrals of convex bodies 总被引:3,自引:0,他引:3
In this paper, we establish two theorems for the quermassintegrals of convex bodies, which are the generalizations of the
well-known Aleksandrov’ s projection theorem and Loomis-Whitney’ s inequality, respectively. Applying these two theorems,
we obtain a number of inequalities for the volumes of projections of convex bodies. Besides, we introduce the concept of the
perturbation element of a convex body, and prove an extreme property of it. 相似文献
14.
For compact Riemann surfaces, the collar theorem and Bers’ partition theorem are major tools for working with simple closed
geodesics. The main goal of this article is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional
orbifolds are a particular case of such surfaces. We consider all cone angles to be strictly less than π to be able to consider
partitions.
Emily B. Dryden—partially supported by the US National Science Foundation grant DMS-0306752.
Hugo Parlier—supported by the Swiss National Science Foundation grants 21-57251.99 and 20-68181.02. 相似文献
15.
Kai Rajala 《Mathematische Annalen》2007,339(2):445-460
We study the covering properties of mappings of bounded and exponentially integrable distortion on the unit ball. We extend
the results of Eremenko (Proc Am Math Soc 128:557–560, 2000) by proving Bloch-type theorems for mappings of exponentially
integrable distortion. In the case of mappings of bounded distortion, we formulate and prove Bloch’s theorem in its most natural
form.
Research supported by the Academy of Finland, and by NSF grant DMS 0244421. Part of this research was done when the author
was visiting at the University of Michigan. He wishes to thank the department for hospitality. 相似文献
16.
On the Equivalence and Generalized of Weyl Theorem Weyl Theorem 总被引:3,自引:0,他引:3
M. BERKANI 《数学学报(英文版)》2007,23(1):103-110
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. 相似文献
17.
We investigate a class of actions of real Lie groups on complex spaces. Using moment map techniques we establish the existence of a quotient and a version of Luna’s slice theorem as well as a version of the Hilbert–Mumford criterion. A global slice theorem is proved for proper actions. We give new proofs of results of Mostow on decompositions of groups and homogeneous spaces.First author partially supported by the Sonderforschungsbereich SFB/TR12 of the Deutsche Forschungsgemeinschaft and the DFG Schwerpunk program Globale Methoden in der komplexen Geometrie.Second author partially supported by NSA grant H98230–04–01–0070. 相似文献
18.
Joel Spencer 《Combinatorica》1981,1(3):303-307
We compare extremal theorems such as Turán’s theorem with their corresponding partition theorems such as Ramsey’s theorem.
We derive a general inequality involving chromatic number and independence number of symmetric hypergraphs. We give applications
to Ramsey numbers and to van der Waerden numbers. 相似文献
19.
M. Frigon 《Journal of Fixed Point Theory and Applications》2011,10(2):279-298
We give generalizations in complete gauge spaces of the following results: Bishop-Phelps’ theorem, Ekeland’s variational principle,
Caristi’s fixed point theorem, the drop theorem and the flower petal theorem. We show that our generalizations are equivalent.
We apply those results to obtain fixed point theorems for multivalued contractions defined on a closed subset of a complete
gauge space and satisfying a generalized inwardness condition. 相似文献
20.
A colorful theorem on transversal lines to plane convex sets 总被引:1,自引:0,他引:1
We prove a colorful version of Hadwiger’s transversal line theorem: if a family of colored and numbered convex sets in the
plane has the property that any three differently colored members have a transversal line that meet the sets consistently
with the numbering, then there exists a color such that all the convex sets of that color have a transversal line.
All authors are partially supported by CONACYT research grant 5040017. 相似文献