A General Version of Price’s Theorem

Journal of Theoretical Probability - Assume that $$X_{Sigma } in mathbb {R}^{n}$$ is a centered random vector following a multivariate normal distribution with positive definite covariance...     

A Bicategorical Version of Masuoka’s Theorem

We give a bicategorical version of the main result of Masuoka (Tsukuba J Math 13:353–362, 1989) which proposes a non-commutative version of the fact that for a faithfully flat extension of commutative rings R í SR \subseteq S, the relative Picard group Pic(S/R) is isomorphic to the Amitsur 1–cohomology group H ¹(S/R,U) with coefficients in the units functor U.     

Some generalizations of Backʼs Theorem

The problem of the existence of jointly continuous utility functions is studied. A continuous representation theorem of Back [1] gives the existence of a continuous map from the space of total preorders topologized by closed convergence (Fell topology) to the space of utility functions with different choice sets (partial maps) endowed with a generalization of the compact-open topology. The commodity space is locally compact and second countable. Our results generalize Back?s Theorem to non-metrizable commodity spaces with a family of not necessarily total preorders. Precisely, we consider regular commodity spaces having a weaker locally compact second countable topology.     

A Finitary Version of Gromov’s Polynomial Growth Theorem

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is some R ₀ > exp(exp(Cd ^C )) for which the number of elements in a ball of radius R ₀ in a Cayley graph of G is bounded by R₀^d{R_0^d} , then G has a finite-index subgroup which is nilpotent (of step < C ^d ). An effective bound on the finite index is provided if “nilpotent” is replaced by “polycyclic”, thus yielding a non-trivial result for finite groups as well.     

A Tree Version of Kőnig's Theorem

Kőnig's theorem states that the covering number and the matching number of a bipartite graph are equal. We prove a generalization, in which the point in one fixed side of the graph of each edge is replaced by a subtree of a given tree. The proof uses a recent extension of Hall's theorem to families of hypergraphs, by the first author and P. Haxell [2]. As an application we prove a special case (that of chordal graphs) of a conjecture of B. Reed. Received January 27, 2000/Revised November 2, 2000 RID=" " ID=" " The research of the first author was supported by grants from the Israel Science Foundation, the M. & M.L Bank Mathematics Research Fund and the fund for the promotion of research at the Technion.     

A Partitioned Version of the Erdös–Szekeres Theorem for Quadrilaterals

We prove a partitioned version of the Erdös–Szekeres theorem for the case $k = 4$: any finite set $X \subset \bbbr^2$ of points in general position can be partitioned into sets $X_0, X_{ij}$ where $i=1,2,3,4$ and $j=1,\ldots,26$, so that $|X_{1j}|=|X_{2j}|=|X_{3j}|=|X_{4j}|$, $|X_0|\leq 4$ and for all $j$ every transversal $\{x_1,x_2,x_3,x_4\}$, $x_1 \in X_{1j}, x_2 \in X_{2j},x_3 \in X_{3j}, x_4 \in X_{4j}$, is in convex position. In order to prove this, we show another theorem, the partitioned version of the same type lemma, which was proved by Bárány and Valtr.     

A Non-Autonomous Version of the Denjoy–Wolff Theorem

The aim of this work is to establish the celebrated Denjoy–Wolff Theorem in the context of generalized Loewner chains. In contrast with the classical situation where essentially convergence to a certain point in the closed unit disk is the unique possibility, several new dynamical phenomena appear in this framework. Indeed, ω-limits formed by suitable closed arcs of circumferences appear now as natural possibilities of asymptotic dynamical behavior.     

A Generalized Version of the Baker–Pixley Theorem

A famous theorem by Baker and Pixley states that the term functions of a finite algebra with a (d + 1)-ary near-unanimity term are precisely the functions under which all subalgebras of the algebra's dth power are closed. In this paper, we generalize the theorem to a completely abstract level. Indeed, we obtain a version of the theorem that is stated in purely category-theoretic terms, making it applicable in any concrete or abstract category. To motivate this rather abstract result, we also discuss some of its concrete applications.     

A generalization of Menonʼs identity

In this note we give a generalization of the well-known Menon?s identity. This is based on applying the Burnside?s lemma to a certain group action.     

A Different Version of Flett＇s Mean Value Theorem and an Associated Functional Equation

In this paper we present a mean value theorem derived from Flett‘s mean value theorem. It turns out that cubic polynomials have the midpoint of the interval as their mean value point. To answer what class of functions have this property, we consider a functional equation associated with this mean value theorem. This equation is then solved in a general setting on abelian groups.     

A Generalisation of Tverberg’s Theorem

We will prove the following generalisation of Tverberg’s Theorem: given a set S⊂ℝ ^d of (r+1)(k−1)(d+1)+1 points, there is a partition of S in k sets A ₁,A ₂,…,A _k such that for any C⊂S of at most r points, the convex hulls of A ₁\C,A ₂\C,…,A _k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).     

Ramanujanʼs Master Theorem for Riemannian symmetric spaces

Ramanujan?s Master Theorem states that, under suitable conditions, the Mellin transform of a power series provides an interpolation formula for the coefficients of this series. Based on the duality of compact and noncompact reductive Riemannian symmetric spaces inside a common complexification, we prove an analogue of Ramanujan?s Master Theorem for the spherical Fourier transform of a spherical Fourier series. This extends the results proven by Bertram for Riemannian symmetric spaces of rank-one.     

A generalization of Menger’s Theorem

This paper generalizes one of the celebrated results in Graph Theory due to Karl. A. Menger (1927), which plays a crucial role in many areas of flow and network theory. This paper also introduces and characterizes strength reducing sets of nodes and arcs in weighted graphs.     

An Integral Version of ?iri?’s Fixed Point Theorem

We establish a new fixed point theorem for mappings satisfying a general contractive condition of integral type. The presented theorem generalizes the well known Ćirić’s fixed point theorem [Lj. B. Ćirić, Generalized contractions and fixed point theorems, Publ. Inst. Math. 12 (26) (1971) 19-26]. Some examples and applications are given.     

A weak variant of Hindman’s Theorem stronger than Hilbert’s Theorem

Hirst investigated a natural restriction of Hindman’s Finite Sums Theorem—called Hilbert’s Theorem—and proved it equivalent over \(\mathbf {RCA}_0\) to the Infinite Pigeonhole Principle for all colors. This gave the first example of a natural restriction of Hindman’s Theorem provably much weaker than Hindman’s Theorem itself. We here introduce another natural restriction of Hindman’s Theorem—which we name the Adjacent Hindman’s Theorem with apartness—and prove it to be provable from Ramsey’s Theorem for pairs and strictly stronger than Hirst’s Hilbert’s Theorem. The lower bound is obtained by a direct combinatorial implication from the Adjacent Hindman’s Theorem with apartness to the Increasing Polarized Ramsey’s Theorem for pairs introduced by Dzhafarov and Hirst. In the Adjacent Hindman’s Theorem homogeneity is required only for finite sums of adjacent elements.     

A Complex Analogue of Toda’s Theorem

Toda (SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P ^#P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines (Blum et al. in Bull. Am. Math. Soc. 21(1):1–46, 1989) was proved in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010). Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) is topological in nature; the properties of the topological join are used in a fundamental way. However, the constructions used in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) were semi-algebraic—they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010) to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence, we obtain a complex analogue of Toda’s theorem. The results of this paper, combined with those in Basu and Zell (Found. Comput. Math. 10(4):429–454, 2010), illustrate the central role of the Poincaré polynomial in algorithmic algebraic geometry, as well as in computational complexity theory over the complex and real numbers: the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.