Some Incidence Theorems and Integrable Discrete Equations

Several incidence theorems of planar projective geometry are considered. It is demonstrated that generalizations of the Pascal theorem due to Mobius give rise to the double cross-ratio equation and the Hietarinta equation. The construction corresponding to the double cross-ratio equation is a reduction to a conic section of some planar configuration (20₃15₄). This configuration provides a correct definition of the multi-dimensional quadrilateral lattices on the plane.     

Tverberg-Type Theorems for Separoids

Let $S$ be a $d$-dimensional separoid of $(k-1)(d+1)+1$ convex sets in some "large-dimensional" Euclidean space $\E^N$. We prove a theorem that can be interpreted as follows: if the separoid $S$ can be mapped with a monomorphism to a $d$-dimensional separoid of points $P$ in general position, then there exists a $k$-colouring $\varsigma\colon \ S\to K_k$ such that, for each pair of colours $i,j\in K_k$, the convex hulls of their preimages do intersect---they are not separated. Here, by a monomorphism we mean an injective function such that the preimage of separated sets are separated. In a sense, this result is "dual" to the Hadwiger-type theorems proved by Goodman and Pollack (1988) and Arocha et al. (2002). We also introduce $\T(k,d)$, the minimum number $n$ such that all $d$-dimensional separoids of order at least $n$ can be $k$-coloured as before. By means of examples and explicit colourings, we show that for all $k>2$ and $d>0$, \[(k-1)(d+1)+1<\T(k,d)<{k\choose2}(d+1)+1.\] Furthermore, by means of a probabilistic argument, we show that for each $d$ there exists a constant $C=C(d)$ such that for all $k$, $\T(k,d)\leq Ck\log k$.     

子流形刚性定理

该文改进了Ｓｈｉｏｈａｍａ和Ｘｕ的常曲率空间中子流形的刚性定理,得到了一个新的整体拼级常数犆狀,并且犆狀只与子流形的维数有关,而与其余维数无关．     

Converse Theorems for Final-stability

This paper deals with the converse theory for some aspects offinal-stability, for which a general qualitative theory wasfirst presented by the author in 1970. The theory of final-stabilityis concerned with the behaviour of the trajectories of a systemof differential equations with respect to two arbitrary setsand a finite interval of time.     

Representation Theorems for Quantales

In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for (non-commutative) Linear Logic using relations. Mathematics Subject Classification: 06D05, 06D10, 06D20, 03G25.     

Support Theorems for the Radon Transform and Cramér-Wold Theorems

This article presents extensions of the Cramér-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distribution theory and Fourier analysis we show that the presented injectivity results for the Radon transform lead to Cramér-Wold type results for measures. One purpose of this article is to contribute to making known to probabilists interesting results for the Radon transform that have been developed essentially during the 1980s and 1990s.     

Bernstein-Type Theorems and Uniqueness Theorems

Let be an entire function of finite type with respect to finite order and let be a subset of an open cone in a certain n-dimensional subspace (the smaller , the sparser ). We assume that this cone contains a ray 0} \right\}$$ " align="middle" border="0"> . It is shown that the radial indicator of at any point may be evaluated in terms of function values at points of the discrete subset . Moreover, if tends to zero fast enough as over , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on and , which are close to exact conditions, the function bounded on is bounded on the ray.     

Uniqueness Theorems for CR Functions

Let M be a CR manifold embedded in ?^s of arbitrary codimension. M is called generic if the complex hull of the tangent space in all points of M is the whole ?^s. M is minimal (in sense of Tumanov) in p ? M if there does not exist any CR submanifold of M passing through p with the same CR dimension as M but of smaller dimension. Let M be generic and minimal in some point p ? M and N be a generic submanifold of M passing through p. We prove that a continuous CR function on M vanishes identically in some neigbourhood of p if its restriction to N either vanishes in p faster then some function with non-integrable logarithm or it vanishes on a subset of N of positive measure.     

Jump Theorems for REA Operators

In [2], Jockusch and Shore have introduced a new hierarchy of sets and operators called the REA hierarchy. In this note we prove analogues of the Friedberg Jump Theorem and the Sacks Jump Theorem for many REA operators. MSC: 03D25, 03D55.     

线性算子的摄动定理

该文利用Mbekhta M于1987年引入的两个子空间来研究线性算子的摄动. 证明了如下结论：设X=K(T)+W, 其中K(T), W均闭, dim［K(T)∩N(T)］< ∞. 若TWW, TW闭, 且存在闭子空间N, 使W=［W∩N(T)］N, 则： 当S∈B(X)可逆, ST= TS, SWW, 且‖S‖充分小时， T-S为上半Fredholm算子. 在上条件下, 若dimN<∞, K(T′)闭, 则T-S为Fredholm算子, 且R(T-S)=X.     

点态化Fuzzy幂群的转移定理

从F uz zy点的角度来研究F uz zy幂群,建立F uz zy幂群的转移定理,然后通过转移定理将分明幂群的许多概念和结论直接转到F uz zy幂群,揭示了F uz zy幂群与分明幂群的关系。     

Canonical Theorems for Convex Sets

Let F be a family of pairwise disjoint compact convex sets in the plane such that none of them is contained in the convex hull of two others, and let r be a positive integer. We show that F has r disjoint ⌊ c _r n⌋ -membered subfamilies F _i (1 ≤ i ≤ r) such that no matter how we pick one element F _i from each F _i , they are in convex position, i.e., every F _i appears on the boundary of the convex hull of ⋃ _i=1 ^r F _i . (Here c _r is a positive constant depending only on r .) This generalizes and sharpens some results of Erdős and Szekeres, Bisztriczky and Fejes Tóth, Bárány and Valtr, and others. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p427.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received April 30, 1997, and in revised form August 5, 1997.     

有限群的某些定理

本文给出了有限群的如下结果：１）成为可解群与超可解群的条件；２）推广了Ｉｔｏ定理，３）超可解且内幂零的群结构。     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

Radius Theorems for Monotone Mappings

Set-Valued and Variational Analysis - For a Hilbert space X and a mapping $F: X\rightrightarrows X$ (potentially set-valued) that is maximal monotone locally around a pair $(\bar {x},\bar {y})$ in...     

On Commutativity Theorems for Rings

Some results on commutativity of rings are presented. These results are related with a conjecture of Searcoid and MacHale in 1986 on the commutativity of rings with additional conditions.