On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

On a Chang Conjecture. II

Continuing [7], we here prove that the Chang Conjecture together with the Continuum Hypothesis, , implies that there is an inner model in which the Mitchell ordering is for some ordinal . Received April 9, 1996     

On a Conjecture of J. Weidmann

Consider the Sturm-Liouville differential expression l(y) = ?y″ +q(x)y on an interval (a,b) and assume that l is in the limit point case at b. Fix c ∈(a,b) and let L, Lb be self-adjoint realizations of l in ?²(a,b), ?²(c,b) respectively. If Lb has purely absolutely continuous spectrum in an interval J and if the spectral function ρ_b of Lb satisfies some mild growth conditions then the spectrum of L in J is shown to be purely absolutely continuous, too. Our result confirms a conjecture of J. Weidmann (1982). It had been shown by del Rio Castillo (1988) that in Weidmann’s original formulation this conjecture is false.     

On a Conjecture of A. Bezdek

The aim of this paper is to prove a conjecture of A. Bezdek. It generalizes conjectures of V. V. Proizvolov. The theorems of this paper are as follows: Suppose points {x _i } _i=1 ^p and a polytope X are given. Then we can choose p facets {F _i } _i=1 ^p of X so that the convex hulls \conv({x _i }\cup F _i ) either cover the polytope or do not intersect pairwise in interior points. Received April 16, 2001, and in revised form October 11, 2001. Online publication March 1, 2002.     

关于Veldman等人的一个猜想

本文证明了下列源于Veldman等人的一个猜想;设G是阶为n≥13的1—tough图且对所有独立集x,y,z有d(x) d(y) d(z)≥(3n-14)/2,则G是哈米顿的。     

On a Conjecture of Shapiro's

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On a Conjecture of Halperin

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On a Conjecture of Finotti

We prove a conjecture of Luis Finotti about cubic polynomials of one variable in characteristic p. He checked it by computer for primes p < 890 and uses it to define and study the minimal degree lift of the generic point of an ordinary elliptic curve in characteristic p to the canonical lift mod p ³ of the curve. Received: 5 June 2002     

On a Conjecture of Kemnitz

A classic theorem of Erdis, Ginzburg and Ziv states that in a sequence of 2n-1 integers there is a subsequence of length n whose sum is divisble by n. This result has led to several extensions and generalizations. A multi-dimensional problem from this line of research is the following. Let Z_nZ_n stand for the additive group of integers modulo n. Let s(n, d) denote the smallest integer s such that in any sequence of s elements from Z_n^dZ_n^d (the direct sum of d copies of Z_nZ_n) there is a subsequence of length n whose sum is 0 in Z_n^dZ_n^d. Kemnitz conjectured that s(n, 2) = 4n - 3. In this note we prove that s(p,2) ￡ 4p - 2s(p,2) \le 4p - 2 holds for every prime p. This implies that the value of s(p, 2) is either 4p-3 or 4p-2. For an arbitrary positive integer n it follows that s(n, 2) ￡ (41/10)ns(n, 2) \le (41/10)n. The proof uses an algebraic approach.     

On a Supercongruence Conjecture of Z.-W. Sun

In this paper, the author partly proves a supercongruence conjectured by Z.-W.Sun in 2013. Let p be an odd prime and let a ∈ Z⁺. Then, if p ≡ 1 (mod 3),k=0 6p^a 2k k/16^k ≡ 3/p^a(modp²) is obtained, where is the Jacobi symbol.     

关于高的一个猜想

In this paper, we prove s conjecture formulated by Gao on zero-sum.And also we make an improvement on a result due to Gao.     

一个积和不等式猜想

介绍一个积和不等式猜想,对任意的正整数n和α∈[0,1],有n-1∑k=0[(n-k)~(a-1)-(n-k+1)~(a-1)][(k+1)~(1-a)-k~(1-a)]≤(n+1)~(1-a)-n~(1-a).证明对于另外,证明与猜想相近的结论.对任意的正整数n和α∈[0,1],有n-1∑k=0[(n-k)~(a-1)-(n-k+1)~(a-1)][(k+1)~(1-a)-k~(1-a)]≤(n+1)~(1-a)-n~(1-a)+(a(2-2~(a-1)))/n~a-1/(n+1)成立.     

On Extensions of a Conjecture of Gallai

A graphGisk-criticalif it has chromatic numberkbut every proper subgraph ofGhas a (k−1)-coloring. We prove the following result. IfGis ak-critical graph of ordern>k3, thenGcontains fewer thann−3k/5+2 complete subgraphs of orderk−1.     

On a Criterion for Catalan's Conjecture

We give a new proof of a theorem of P. Mihailescu which states that the equation x ^p – y ^q = 1 is unsolvable with x, y integral and p, q odd primes, unless the congruences p ^q p (mod q ²) and q ^p q (mod p ²) hold.     

On a Modified Conjecture of De Giorgi

We study the Γ-convergence of functionals arising in the Van der Waals–Cahn–Hilliard theory of phase transitions. Their limit is given as the sum of the area and the Willmore functional. The problem under investigation was proposed as modification of a conjecture of De Giorgi and partial results were obtained by several authors. We prove here the modified conjecture in space dimensions n = 2,3.This work was supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00274, FRONTS-SINGULARITIES.     

On a Conjecture of Ditzian and Runovskii

Let M_θ be the mean operator on the unit sphere in ⁿ, n3, which is an analogue of the Steklov operator for functions of single variable. Denote by D the Laplace–Beltrami operator on the sphere which is an analogue of second derivative for functions of single variable. Ditzian and Runovskii have a conjecture on the norm of the operator θ²D(M_θ)^m, m2 from X=L^p (1p∞) to itself which can be expressed as

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. We give a proof of this conjecture.     

On a Conjecture of Fan and Raspaud

We show that every bridgeless cubic graph having a 2-factor with at most two odd circuits admits three perfect matchings with no common edge. This partially verifies a conjecture of Fan and Raspaud (1994) and supports Fulkerson's conjecture (1971).     

On a Conjecture of Kleene and Post

A proof is given that 0 ′ (the argest Turing degree containing a computably enumerable set) is definable in the structure of the degrees of unsolvability. This answers a long‐standing question of Kleene and Post, and has a number of corollaries including the definability of the jump operator.     

On a generalization of the Evans Conjecture

The Evans Conjecture states that a partial Latin square of order n with at most n-1 entries can be completed. In this paper we generalize the Evans Conjecture by showing that a partial r-multi Latin square of order n with at most n-1 entries can be completed. Using this generalization, we confirm a case of a conjecture of Häggkvist.