关于Boutroux—Cartan定理的推广

推广了著名的Boutroux—Cartan定理。设aμ（μ=1，2，…，n）为复平面上任意的n个点，H为任意的一个正数，则在平面上同时使得n∏μ=1 ｜z-aμ｜≤（H/e）^n和n∑μ=1 1/｜z-aμ｜≥nlog（en）/H成立的点z可被含于总数不超过n，半径总和不超过2H的一组圈内。     

On Helly's Theorem: Algorithms and Extensions

This paper studies algorithmic Helly-type problems in the framework of the algorithmic theory of convex bodies developed by Gr?tschel, Lovász, and Schrijver. Various oracle-polynomial-time algorithms are presented that are complemented by -hardness results for polytopes. In addition, some new Helly-type theorems are derived. Received June 15, 1996, and in revised form August 16, 1996.     

Extensions of the Scherk-Kemperman Theorem

Let Γ=(V,E) be a reflexive relation with a transitive automorphism group. Let F be a finite subset of V containing a fixed element v. We prove that the size of Γ(F) (the image of F) is at least

|F|+|Γ(v)|−|Γ⁻(v)∩F|.     

Extensions of the Cauchy-Goursat Integral Theorem

It is natural to conjecture that if a function f is continuous on the closed region determined by a rectifiable 1-cycle Γ and complex-differentiable on the open region then Γ_∫f=0. The main result is an extension of the classical Cauchy-Goursat Theorem: the equality conjectured holds (with no boundary condition on f^′) under the additional hypothesis that the winding numbers of Γ define an Lp function and f satisfies a matching Hölder continuity condition near the image of Γ. (In particular, continuity suffices if p=∞.) The proof uses approximations of a rectifiable path by piecewise linear paths.     

The Sturm-Hurwitz Theorem and its Extensions

The following principle is well-known in Harmonic Analysis: If a real function has a spectral gap at the origin then it must have many sign changes. We obtain some sharp estimates showing that the set of positivity of such functions cannot be too small. We also extend the principle above to complex functions: If a complex function has a spectral gap at the origin then the variation of argument of this function must be large.     

Two Extensions to Finsler's Recurring Theorem

Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R ⁿ : (i) f( ξ ) >0 for all ξ≠ 0 with g( ξ ) =0; (ii) f-λ g>0 for some λ∈ R. We prove two extensions: 1. We admit a vector-valued quadratic form g: R ⁿ → R ^k , for which we show that (i) implies that f-λ . . . g>0 on an ( n-k+1 ) -dimensional subspace Y R ⁿ for some λ∈ R ^k . 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R ⁿ by a real vector space X . Accepted 22 February 1998     

G-分次环中一个定理的推广

本推广单分次摸情形下的G-分次环的一个定理．     

Extensions of Edelstein's Theorem on Contractive Mappings

The aim of this article is to provide extensions of Edelstein's theorem for a class of contractive mappings, namely cyclic contractive mappings. We prove the existence and convergence of best proximity points of a cyclic contractive map. We also discuss continuity properties of cyclic contractive maps. Finally, we give a characterization of such maps in the setting of a Hilbert space.     

Hilbert's Theorem 90 for Hopf Galois Extensions

For a faithfully flat extension A/B and a right A-module M, we give a new characterization of the set of descent data on M. Assuming that B is a simple Artinian ring and A/B is H-Galois, for a certain finite dimensional Hopf algebra H, we prove that Sweedler's noncommutative cohomology H ¹(H?, A) is trivial as a pointed set.     

Extensions of the Erdös-Ko-Rado Theorem

An antichain of subsets of 1,2,...,n has the Erdös-Ko-Rado property if |A_i|?n/2 and A_i∩A_j≠Ø(i=j). This paper contains a number of results concerning the distribution of sizes of sets in such a family, and also in families where the restriction |A_i|?n/2 is removed.     

柏龙树(THE PERRON TREE)定理的推广与应用

将平面上的柏龙树(the Perron Tree)定理推广了k维欧氏空间,由此推广了Fefferman定理.     

On Compactness with Respect to Countable Extensions of Ideals and the Generalized Banach Category Theorem

We extend a theorem of Hamlett and Jankovi by proving that if a topological space (X, ) is compact with respect to the countable extension of I, then the local function A ^*(I) of every subset A of X with respect to and I is a compact subspace with respect to the extension in A ^* (I). We also give a generalized version of the Banach category theorem.     

On Extensions of AO-Groups

The notion of an extension is important in the study of partially ordered groups. In the present paper the notion of a lexicographic extension of a partially ordered group by an AO-group is studied. A result is obtained concerning an AO-group G which is a lexicographic extension of a directed subgroup of G.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 277–281, 2003.     

Variations on the Theorem of Birkhoff – von Neumann and Extensions

 The theorem of Birkhoff – von Neumann concerns bistochastic matrices (i.e., matrices with nonnegative real entries such that all row sums and all column sums are equal to one). We consider here real matrices with entries unrestricted in sign and we extend the notion of permutation matrices (integral bistochastic matrices); some generalizations of the theorem are derived by using elementary properties of graph theory. Received: October 10, 2000 Final version received: April 11, 2002     

On a Theorem of Zarach Concerning the Global Pattern of Cardinals in Generic Extensions of Models of ZFC

Let M ZFC and let F: be a function on the ordinals of M.For which such F will there exist a model N of ZFC with thesame ordinals as M such that N is a cardinal if and only ifM () ( = _F())? A partial answer to this question was given in a theorem ofZarach. We present here a counter-example and correction tothis theorem. Our argument invokes results of Drake concerningthe collapse of weak cardinal powers in certain generic extensions. This work is supported in part by the National Research Councilof Canada under grant number A8216.     

论Whittaker-Shannon抽样定理及其一些推广

The aim of this paper is to present a survey of results concerning the Whittaker-Kotel'nikov-Raabe-Shannon-Someya sampling theorem and its various extensions obtained at Aachen since 1977. This theorem, basic in communication engineering, is often called the cardinal interpolation series theorem in mathematical circles. The interconnections of the sampling theorem (in the setting of Paley-Wiener space) with the theory of Fourier series and integrals are examined. Emphasis is placed upon error analysis, including the aliasing, round-off (or quantization), and time jitter errors. Some new error estimates are given, others are improved; many of the proofs are reduced to a common structure. Both deterministic and probabilistic methods are employed. whereas these results are worked out in detail, the paper also contains a brief discussion of some of the various generalizations.     

On Minimal Extensions of Rings

Given two rings R ? S, S is said to be a minimal ring extension of R, if R is a maximal subring of S. In this article, we study minimal extensions of an arbitrary ring R, with particular focus on those possessing nonzero ideals that intersect R trivially. We will also classify the minimal ring extensions of prime rings, generalizing results of Dobbs, Dobbs &; Shapiro, and Ferrand &; Olivier, on commutative minimal extensions.