A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.     

关于''抽象的极小极大定理''的注记

本文指出林壮鹏2000年发表的一个抽象的极大极小定理一文中主要结果的证明需要修正,然后改进了该文的结果,同时给出了一个简单的证明。     

A Shortcoming in the Geometrically Non‐Linear Shakedown Theorem

The original static shakedown theorem of Melan [1], valid for geometrically linear theory, was extended for geometrically non‐linear theory e.g. by Polizzotto and Borino [2], who presented a proof for large rotations with small strains. However, a counterexample to this extended Melan's theorem has been found. The reason of the failure is investigated and is corrected by an additional condition in the theorem. The outline of the proof is given.     

Stability regions of linear canonical systems with periodic coefficients

A new definition of the index of the stability region of a canonical system of linear differential equations with periodic coefficients is proposed. A simple proof of the Gel'fand-Lidskii theorem [1] on the structure of stability regions is given and a theorem on the directional convexity of such regions is proved. It follows from this theorem, in particular, that stability regions of parametric oscillations in a system with a sign-definite Hamiltonian are convex with respect to the frequency of parametric perturbation.     

论鲁津条件

提出几种变形的鲁津条件，并阐述它们之间的关系，利用这些结果，我们把经典的巴拿哈-查列茨基定理（[1]P．279）转化成容易证明的定理．     

A new short proof of a theorem of Ahlswede and Khachatrian

Ahlswede and Khachatrian [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] proved the following theorem, which answered a question of Frankl and Füredi [P. Frankl, Z. Füredi, Nontrivial intersecting families, J. Combin. Theory Ser. A 41 (1986) 150-153]. Let 2?t+1?k?2t+1 and n?(t+1)(k−t+1). Suppose that F is a family of k-subsets of an n-set, every two of which have at least t common elements. If |_?F∈FF|<t, then , and this is best possible. We give a new, short proof of this result. The proof in [R. Ahlswede, L.H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A 76 (1996) 121-138] requires the entire machinery of the proof of the complete intersection theorem, while our proof uses only ordinary compression and an earlier result of Wilson [R.M. Wilson, The exact bound in the Erd?s-Ko-Rado theorem, Combinatorica 4 (1984) 247-257].     

On varieties of n-solvable groups

A new and shorter proof is given for the fundamental theorem of Kaluzhnin's recent note [1]: A variety of n-Abelian groups is the sum of a variety of Abelian groups and the respective Burnside varieties of groups of exponent n and 1–n. This theorem is extended to varieties of n-solvable groups.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 629–634, December, 1968.     

From weak to strong convergence inL 1-spaces viaK-Convergence

Summary A generalized version of Komlós' theorem [6], combined with a useful property of denting points in the style of [17,22], gives a new, very efficient proof of Visintin's theorem and its generalizations [24,2,10,21,7], on equivalence of weak and strong convergence in L₁-space under denting point conditions.     

Pure strategy Nash equilibrium points and the Lefschetz fixed point theorem

A pure strategy Nash equilibrium point existence theorem is established for a class ofn-person games with possibly nonacyclic (e.g. disconnected) strategy sets. The principal tool used in the proof is a Lefschetz fixed point theorem for multivalued maps, due to Eilenberg and Montgomery, which extends their better known. Eilenberg-Montgomery fixed point theorem (EMT) [Eilenberg/Montgomery, Theorem 1, p. 215] to nonacyclic spaces. Special cases of the existence theorem are also discussed.     

矩阵对策鞍点定理的归纳法证明

Many proofs have been published for the minimax theorem, and all the published inductive proofs have been indirect ones. It has been pointed out that a direct inductive proof is needed, especially for instructional purposes, since indirect proofs are more or less implicit in nature. Such a direct proof is given in [4]. Now the minimax theorem can be stated equivalently in terms of saddle point; And it is the object of the present paper to give a direct inductive proof for the saddle point version of this theorem.     

A stability criterion

We come up with an independent proof for a corollary to the main theorem in [1]. This corollary is the degenerate case of the main theorem (with empty sets B ₀ and B ₁) and establishes a stability criterion for a Henselian valued field. Such a proof is given here based on an analysis of tame and purely wild extensions made in [2].     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Special-valued l-groups and Abelian covers

This paper presents a new and independent proof of the theorem (proven first by Kopytov and Gurchenkov [7] and again by Reilly [10]) that covers of the Abelian l-variety are either representable or are Scrimger covers. The proof in this paper is based upon the l-Cauchy constructions of Ball [1]; once these are applied to the problem, the proof becomes elementary.     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Triangulated Spheres and Colored Cliques

   Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

Triangulated Spheres and Colored Cliques

Abstract. In [1] a generalization of Hall's theorem was proved for families of hypergraphs. The proof used Sperner's lemma. In [5] Meshulam proved an extension of this result, using homology and the nerve theorem. In this paper we show how the triangulations method can be used to derive Meshulam's results. As in [1], the proof is based on results on extensions of triangulations from the sphere to the full ball. A typical result of this type is that any triangulation of the (d-1) -dimensional sphere S ^d-1 can be extended to a triangulation of the ball B ^d , by adding one point at a time, having degree at most 2d to its predecessors.     

On the Equivalence of Minimax Theorems

In this note we show that Ky Fan's minimax theorem and its several generalizations such as König's minimax theorem [6], M. Neumann's minimax theorem [8] and Fuchssteiner-König's minimax theorem [3] are equivalent. We also give a direct proof for Fuchssteiner-König's minimax theorem on the basis of Eidelheit's well-known seperation theorem.     

The octahedron theorem in Minkowski three-space a metric vector space proof of Miquel's theorem in the Laguerre plane

As is well known [1, p.480], the cycles and spears of the real Laguerre plane can be represented by the points and null planes of three-dimensional Minkowski space. Miquel's theorem in the Laguerre plane can then be expressed as an intrinsically interesting Minkowski space theorem the octahedron theorem. We outline the correspondence between the two theorems, and then give a metric vector space proof of the octahedron theorem, thereby providing an alternate proof of Miquel's theorem. We then discuss the generalization of both theorems to more general spaces.     

A Simplified Elementary Proof of Hadwiger's Volume Theorem

One of the most important results in geometric convexity is Hadwiger's characterization of quermassintergrals and intrinsic volumes. The importance lies in that Hadwiger's theorem provides straightforward proofs of numerous results in integral geometry such as the kinematic formulas [Santaló, L. A.: Integral Geometry and Geometric Probability, Addison-Wesley, 1976], the mean projection formulas for convex bodies [Schneider, R.: Convex Bodies: The Brunn—Minkowski Theory, Cambridge Univ. Press, 1993], and the characterization of totally invariant set functions of polynomial type [Chen, B. and Rota, G.-C.: Totally invariant set functions of polynomial type, Comm. Pure Appl. Math. 47 (1994), 187–197]. For a long time the only known proof of Hadwiger's theorem was his original one [Hadwiger, H.: Vorlesungen über Inhalt, Oberfläche and Isoperimetrie, Springer, Berlin, 1957] (long and not available in English), until a new proof was obtained by Klain [Klain, D. A.: A short proof of Hadwiger's characterization theorem, Mathematika 42 (1995), 329–339., Klain, D. A. and Rota, G.-C.: Introduction to Geometric Probability, Lezioni Lincee, Cambridge Univ. Press, 1997], using a result from spherical harmonics. The present paper provides a simplified and self-contained proof of Hadwiger's theorem.     

关于随机变量期望方差的两个注记

文献[1]中给出了有关条件期望与三个随机变量独立的两个充要条件,本文通过几个反例说明其充分性是不成立的.分析了文献[4]中一个定理证明存在的错误,并给出了新的证明.