Projective Center Point and Tverberg Theorems

We present projective versions of the center point theorem and Tverberg’s theorem, interpolating between the original and the so-called “dual” center point and Tverberg theorems. Furthermore we give a common generalization of these and many other known (transversal, constraint, dual, and colorful) Tverberg type results in a single theorem, as well as some essentially new results about partitioning measures in projective space.     

Strengthening topological colorful results for graphs

Various results ensure the existence of large complete and colorful bipartite graphs in properly colored graphs when some condition related to a topological lower bound on the chromatic number is satisfied. We generalize three theorems of this kind, respectively due to Simonyi and Tardos 2006), Simonyi et al. (2013), and Chen 2011). As a consequence of the generalization of Chen’s theorem, we get new families of graphs whose chromatic number equals their circular chromatic number and that satisfy Hedetniemi’s conjecture for the circular chromatic number.     

The colorful Helly theorem and colorful resolutions of ideals

We demonstrate that the topological Helly theorem and the algebraic Auslander-Buchsbaum theorem may be viewed as different versions of the same phenomenon. Using this correspondence we show how the colorful Helly theorem of I. Barany and its generalizations by G. Kalai and R. Meshulam translate to the algebraic side. Our main results are algebraic generalizations of these translations, which in particular give a syzygetic version of Helly’s theorem.     

A Helson-Lowdenslager-deBranges Theorem in

This paper presents a generalization of the invariant subspace theorem of Helson and Lowdenslager along the lines of de Branges' generalization of Beurling's theorem.

     

A note on the colorful fractional Helly theorem

Helly’s theorem is a classical result concerning the intersection patterns of convex sets in ${\mathbb{R}}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.     

Separating sets with relative interior in Fréchet spaces

A separation theorem, valid in infinite dimensional spaces, and involving the relative interior of the sets to be separated, will be extended to Fréchet spaces. This theorem will be elucidated by means of a few examples. The second separation theorem is a generalization of an existing separation theorem, valid in Fréchet spaces. This paper consists of two parts: part I contains the first theorem, the second part contains the second generalization.     

Combinatorial Generalizations of Jung’s Theorem

We consider combinatorial generalizations of Jung’s theorem on covering a set by a ball. We prove the “fractional” and “colorful” versions of the theorem.     

Notes About the Carathéodory Number

In this paper we give sufficient conditions for a compactum in ? ⁿ to have Carathéodory number less than n+1, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carathéodory theorem and give a Tverberg-type theorem for families of convex compacta.     

The colored Hadwiger transversal theorem in ℝ<Superscript><Emphasis Type="Italic">d</Emphasis></Superscript>

Hadwiger’s transversal theorem gives necessary and suffcient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in ? ^d in bijection with a set P of points in ? ^d?1. Assume that there is a coloring of F with suffciently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.     

A new type of fixed point theorem in metric spaces

We prove a generalization of Edelstein’s fixed point theorem. Though there are thousands of fixed point theorems in metric spaces, our theorem is a new type of theorem.     

Mazur定理的一个推广及其几何证明

建立了代数内点的若干性质；给出了Mazur定理的一个推广及其几何证明；证明了此推广的Mazur定理与基本分离定理等价；还得到一个严格分离定理．     

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincaré theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.     

Generalization of the Rödseth–Gupta Theorem on Binary Partitions

In this article, we give a new proof of the Rödseth–Gupta theorem on binary partitions and give one possible generalization of this theorem.     

A generalization of the Stein-Rosenberg theorem to Banach spaces

Summary In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.     

Stability of Functional Equations in Several Variables

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.     

On partitions of a partially ordered set

Using linear programming we prove a generalization of Greene and Kleitman's generalization of Dilworth's theorem on the decomposition of a partially ordered set into chains.     

A generalization of weyl’s inequalities with implications

This paper suggests a generalization of the additive Weyl inequalities to the case of two square matrices of different orders. As a consequence of the generalized Weyl inequalities, a theorem describing the location of eigenvalues of a Hermitian matrix in terms of the eigenvalues of an arbitrary Hermitian matrix of smaller order is derived. It is demonstrated that the latter theorem provides a generalization of Kahan’s theorem on clustered eigenvalues. It is also shown that the theorem on extended interlacing intervals is another consequence of the generalized additive Weyl inequalities suggested. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 49–59. Translated by L. Yu. Kolotilina.     

Uncertainty principles for the weinstein transform

The Weinstein transform satisfies some uncertainty principles similar to the Euclidean Fourier transform. A generalization and a variant of Cowling-Price theorem, Miyachi’s theorem, Beurling’s theorem, and Donoho-Stark’s uncertainty principle are obtained for the Weinstein transform.     

Helly-type theorems for pseudoline arrangements in P2

We prove duals of Radon's theorem, Helly's theorem, Carathéodory's theorem, and Kirchberger's theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Helly's theorem.     

A note on log etale Čech cohomolog

In this paper, we prove the comparison theorem between log etale cohomology and log etale Čech cohomology for certain log schemes, which is a generalization of a result of Artin on the comparison theorem between etale cohomology and etale Čech cohomology. It turns out that the naive generalization is not true, and we also give a counter-example for it. Received: 2 July 2000 / Revised version: 4 September 2000