A Common Generalization to Theorems on Set Systems with <Emphasis Type="Italic">L</Emphasis>-intersections

In this paper, we provide a common generalization to the well-known Erdös–Ko–Rado Theorem, Frankl–Wilson Theorem, Alon–Babai–Suzuki Theorem, and Snevily Theorem on set systems with L-intersections. As a consequence, we derive a result which strengthens substantially the well-known theorem on set systems with k-wise L-intersections by Füredi and Sudakov [J. Combin. Theory, Ser. A, 105, 143–159 (2004)]. We will also derive similar results on L-intersecting families of subspaces of an n-dimensional vector space over a finite field F _q , where q is a prime power.     

F-空间中的一个基本定理及其在数值分析中的应用

本文针对F-空间中闭算子方程的一般逼近格式,研究其相容性、收敛性和稳定性之间的关系.所得的主要结果是：这种一般逼近格式在相容性条件下,其收敛性与稳定性是等价的.此定理可以看作是对Lax等价原理的推广,是求解第一类闭算子方程的一般逼近格式的基本定理.为得到这一主要结果,本文还给出了F-空间中的一条基本定理,众所周知的一致有界原理,闭图像定理和开映像定理是其简单推论.     

A game-theoretic equivalence to the Hahn-Banach theorem

Let Γ_X() = X, A (X), υ be a cooperative von Neumann game with side payments, where X is a nonempty set of arbitrary cardinality, A(X) the Boolean ring generated from P(X) with the operations Δ and ∩ for addition and multiplication, respectively, such that S² =S for all S ε A (X), and with ;() = 0. The Shapley-Bondareva-Schmeidler Theorem, which states that a game of the form Γ_X() = X, A (X), is weak if and only if the core of Γ_X(),ζ(Γ_X()), is normal, may be regarded as the fundamental theorem for weak cooperative games with side-payments. In this paper we use an ultrapower construction on the reals, , to summarize a common mathematical theme employed in various constructions used to establish the Shapley-Bondareva-Schmeidler Theorem in the literature (Dalbaen, 1974; Kannai, 1969; Schmeidler, 1967, 1972). This common mathematical theme is that the space L, comprised of finite, real linear combinations of the collection of functions, {χ_a : a ε A (X)}, possesses a certain extension property that is intimately related to the Hahn-Banach Theorem of functional analysis. A close inspection of the extension property reveals that the Shapley-Bondareva-Schmeidler Theorem is in fact equivalent to the Hahn-Banach Theorem.     

Structure Theorem for (d, g, h)-Maps

The (3x + 1)-Map, T, acts on the set, Π, of positive integers not divisible by 2 or 3. It is defined by , where k is the largest integer for which T (x) is an integer. The (3x + 1)-Conjecture asks if for every x ε Π there exists an integer, n, such that T ⁿ (x) = 1. The Statistical (3x + 1)-Conjecture asks the same question, except for a subset of Π of density 1. The Structure Theorem proven in [𝕊] shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x + 1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x + 1)-Map, and expand on the consequences derived in [𝕊]. The generalizations we consider are determined by positive coprime integers, d and g, with g > d ≥ 2, and a periodic function, h (x). The map T is defined by the formula , where k is again the largest integer for which T (x) is an integer. We prove an analogous Structure Theorem for (d, g, h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift and positive diffusion constant. This shows that it is natural to expect that typical trajectories return to the origin if and escape to infinity otherwise. Received: 18 April 2002     

Maximum Matchings in the n-Dimensional Cube

The problem of efficient computation of maximum matchings in the n-dimensional cube, which is applied in coding theory, is solved. For an odd n, such a matching can be found by the method given in our Theorem 2. This method is based on the explicit construction (Theorem 1) of the maps of the vertex set that induce largest matchings in any bipartite subgraph of the n-dimensional cube for any n.     

The Three Gap Theorem and Riemannian geometry

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.      

Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds,or better

This paper will do the following: (1) Establish a (better than) Thue-Siegel-Roth-Schmidt theorem bounding the approximation of solutions of linear differential equations over valued differential fields; (2) establish an effective better than Thue-Siegel-Roth-Schmidt theorem bounding the approximation of irrational algebraic functions (of one variable over a constant field of characteristic zero) by rational functions; (3) extend Nevanlinna's Three Small Function Theorem to an n small function theorem (for each positve integer n), by removing Chuang's dependence of the bound upon the relative “number” of poles and zeros of an auxiliary function; (4) extend this n Small Function Theorem to the case in which the n small functions are algebroid (a case which has applications in functional equations); (5) solidly connect Thue-Siegel-Roth-Schmidt approximation theory for functions with many of the Nevanlinna theories. The method of proof is (ultimately) based upon using a Thue-Siegel-Roth-Schmidt type auxiliary polynomial to construct an auxiliary differential polynomial.     

An existence theorem for packing problems with implications for the computation of optimal machine schedules

Motivated by a job-shop problem we ask on which conditions n intervals v_i: with given lengths d_i , can be arranged non-overlapping on the real axis, so that every v_i is placed in a given frame [f_i s_i ].We prove a necessary and sufficient criterion analogous to the “marriage theorem” but with an additional monotony in the ages. Using this criterion we can accelerate branch and bound algorithms for job-shop scheduling by fixing partial sequences. For n = 2, the approved pair-combinatorics by Carlier and Pinson results. To show how the method works for other n. we derive the complete triple-combinatorics (n = 3: five sufficient criterions, which are necessary as a whole)     

Dedekind–MacNeille Completion of n-ordered Sets

A completion of an n-ordered set is defined, by analogy with the case of posets (2-ordered sets), as a pair , where Q is a complete n-lattice and is an n-order embedding. The Basic Theorem of Polyadic Concept Analysis is exploited to construct a completion of an arbitrary n-ordered set. The completion reduces to the Dedekind–MacNeille completion in the dyadic case, the case of posets. A characterization theorem is provided, analogous to the well-known dyadic one, for the case of joined n-ordered sets. The condition of joinedness is trivial in the dyadic case and, therefore, this characterization theorem generalizes the uniqueness theorem for the Dedekind–MacNeille completion of an arbitrary poset.      

A divergence theorem for Finsler metrics

By means of the general form of Stokes' theorem on manifolds a divergence theorem is derived for hypersurfaces which bound a compact region of ann-dimensional Finsler spaceF _n . In general the integrand of then-fold volume integral will depend on the covariant derivatives of an arbitrary vector field which defines the element of support; certain conditions under which this dependence may be circumvented are discussed. The scalar curvature ofF _n is expressed in terms of the divergence of a certain vector field: forn=2 this formula reduces to a particularly simple form, and its substitution into the aforementioned divergence theorem gives rise to a formula which represents a generalization of the classical Gauss-Bonnet Theorem.

     

Some New Applications of the Subspace Theorem

We present some applications of the Subspace Theorem to the investigation of the arithmetic of the values of Laurent series f(z) at S-unit points. For instance we prove that if f(q ⁿ ) is an algebraic integer for infinitely many n, then h(f(q ⁿ )) must grow faster than n. By similar principles, we also prove diophantine results about power sums and transcendency results for lacunary series; these include as very special cases classical theorems of Mahler. Our arguments often appear to be independent of previous techniques in the context.     

What is a Quotient Map with Respect to a Closure Operator?

It is shown that there is no good answer to the question of the title, even if we restrict our attention to S et-based topological categories. Although very closely related, neither the natural notion of c-finality (designed in total analogy to c-initiality) nor the notion of c-quotient (modelled after the behaviour of topological quotient maps) provide universally satisfactory concepts. More dramatically, in the category T op with its natural Kuratowski closure operator k, the class of k-final maps cannot be described as the class of c-quotient maps for any closure operator c, and the class of k-quotients cannot be described as the class of c-final maps for any c. We also discuss the behaviour of c-final maps under crossing with an identity map, as in Whitehead's Theorem. In T op, this gives a new stability theorem for hereditary quotient maps.     

Another Proof of Kasami's Theorem

We give a short direct proof for a famous theorem published by Kasami in 1971. In terms of Walsh analysis it states that for d = 2^2k - 2^k + 1 the Walsh spectrum of the Boolean function Tr(x ^d ) on GF(2 ⁿ ) consists precisely of the three values 0, ±2^(n+s)/2 if s = gcd(k, n) = gcd(2k, n).     

Lipschitz Flow-box Theorem

A generalization of the Flow-box Theorem is proven. The assumption of a C¹ vector field f is relaxed to the condition that f be locally Lipschitz continuous. The theorem holds in any Banach space.     

f-Rings and the Stone-Weierstrass Theorem

Using an appropriate notion of separating subring, it is shown that the classical Stone-Weierstrass Theorem for compact Hausdorff spaces is ultimately a result about f-rings. As an application the constructively valid Stone-Weierstrass Theorem for compact completely regular frames is obtained.     

Corrections: on a question of boyle and handelman concerning eigenvalues of nonnegative matrices

Professor Raphale Leowy, of the Technion-Israel Institute of Technology in Haifa, has informed us that the assumptions that we made in the statement of Theorem I (which appeared in Volume 36. 1993, 125–140) concerning the case when n≥5 are weaker that those than we made use of in the proof of the theorem. Thus without a change in the proof, only the following result is correctly proved in the theorem:     

On R∞ and Q∞-manifolds

We give a characterization of manifolds modeled on $\text{R}{}^{\mathrm{\infty }}\text{= dir lim}$ or $\text{R}{}^{\mathrm{n}}\text{Q}{}^{\mathrm{\infty }}\text{=dir lim Q}{}^{\mathrm{n}}$, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.     

Goldie's theorem for semigroups

In this paper, we try to look for a semigroup analogue of Goldie's theorem which characterizes the orders of the full matrix ringM _n (D) over a division ringD. The main result of this paper, Theorem 4.11, does give a characterization of the semigroups which have a classical left quotient semigroupM _n (G ⁰), whereG ⁰ is a group with zero. The author is grateful to the referee for his careful modification of this paper and helpful comments which resulted in several improvements.