On a conjecture of bollobás and bosák

It is shown that, for all sufficiently large k, the complete graph K_n can be decomposed into k factors of diameter 2 if and only if n ≥ 6k.     

On a conjecture of F. Móricz

F. Móricz has investigated the integrability of double lacunary sine series. His result, valid for special lacunary sequences, does not extend in the form originally conjectured, but we establish a suitably modified result.

     

On a conjecture by B. Grünbaum

Denote by p_k(M) or υ_k(M) the number of k-gonal faces or k-valent of the convex 3-polytope M, respectively. Completely solving a problem by B. Grünbaum, the following theorem is proved: Given sequences of nonnegative integers p = (p₃, p₄,…p_m), υ = (υ₃, υ₄,…,υ_n) satisfying ∑_k?3(6−k)p_k + 2∑_k?3(3−k)υ_k = 12, there exists a convex 3-polytope M with p_k(M) = p_k for all k ≠ 6 and υ_k for all k ≠ 3 if and only if for the sequences p, υ the following does not hold: ∑p_i = 0 for i odd and ∑υ_i = 1 for i ? 0 (mod 3).     

关于Smarandache 函数S(n)的一个猜想

对任意正整数n,著名的Smarandache 函数S(n)定义为最小的正整数m,使得n|m!.即就是S(n)=min{m∶n|m!,m∈N}.本文的主要目的是利用初等方法研究一个包含函数S(n)的猜想,并部分的得到解决.     

Answer to a conjecture of J.M. Cohen

The origin of Gelfand rings comes from [9] where the Jacobson topology and the weak topology are compared. The equivalence of these topologies defines a regular Banach algebra. One of the interests of these rings resides in the fact that we have an equivalence of categories between vector bundles over a compact manifold and finitely generated projective modules over C(M), the ring of continuous real functions on M [17].These rings have been studied by R. Bkouche (soft rings [3]) C.J. Mulvey (Gelfand rings [15]) and S. Teleman (harmonic rings [19]).Firstly we study these rings geometrically (by sheaves of modules (Theorem 2.5)) and then introduce the ?ech covering dimension of their maximal spectrums. This allows us to study the stable rank of such a ring A (Theorem 6.1), the nilpotence of the nilideal of K₀(A) - The Grothendieck group of the category of finitely generated projective A-modules - (Theorem 9.3), and an upper limit on the maximal number of generators of a finitely generated A-module as a function of the afore-mentioned dimension (Theorem 4.4).Moreover theorems of stability are established for the group K₀(A), depending on the stable rank (Theorems 8.1 and 8.2). They can be compared to those for vector bundles over a finite dimensional paracompact space [18].Thus there is an analogy between finitely generated projective modules over Gelfand rings and ?ech dimension, and finitely generated projective modules over noetherian rings and Krull dimension.     

The multiplier conjecture for the case n = 4n1

In this article we prove that in the case n = 4n₁ if (v,2 · 7 · 31) = 1 and v is not divisible by 15, then the Second Multiplier Theorem holds without the assumption n₁ > λ. This improves a result due to McFarland. © 1995 John Wiley & Sons, Inc.     

On a conjecture of V. Nikiforov

In this paper we will prove that

     

On a conjecture of S. Reich

Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.

     

On a conjecture of W. Veys

In this work we show a singular K3 surface S of such that producing a counter-example to a conjecture of Veys. Received: 22 October 1999     

On a conjecture of C. Johnson

Given a pair of n×n matricesA and B, one may form a polynomial P(A,B,λ) which generalizes the characteristic polynomial of BP(B,λ). In particular, when A=I (identity), P(A, B,λ) = P(B,λ), the characteristic polynomial of B. C. Johnson has conjectured [1] (among other things) that when A and B are hermitian and A is positive definite, then P(A,B,λ) has real roots. The case n=2 can be done by hand. In this paper we verify the conjecture for n=3.     

Cubic CNS polynomials, notes on a conjecture of W.J. Gilbert

A conjecture of W.J. Gilbert's on canonical number systems which are defined by cubic polynomials is partially proved, and it is shown that the conjecture is not complete. Applications to power integral bases of simplest and pure cubic number fields are given thereby extending results of S. Körmendi.     

Convergence of discrete analogues of a generalization of J.R. Haddock’s conjecture

This work focuses on the asymptotic behavior of solutions for a class of neutral delay difference equations, which are discrete analogues of a generalization of J.R. Haddock’s conjecture. It is shown that every bounded solution of the equations converges to a constant. Our results improve some known results from the literature.     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

On a conjecture by Y. Last

We prove a conjecture due to Y. Last. The new determinantal representation for transmission coefficient of Jacobi matrix is obtained.