On a question of T Mizokami

In the short note, we shall give a negative answer of a question which israised in[1].     

On a Problem of D. H. Lehmer

The main purpose of this paper is to use the Fourier expansion for character sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the difference between a D. H. Lehmer number and its inverse modulo p （an odd prime）, A interesting mean square value formula is also given.     

On a question of Koll��r

We show: if a bounded domain in a Stein space covers a compact complex space, it must be smooth. This give a negative answer to a question of Kollár.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

On a conjecture of the Randić index and the minimum degree of graphs

The Randi? index $R\left(G\right)$ of a graph $G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where $d\left(u\right)$ is the degree of a vertex $u$ and the summation extends over all edges $uv$ of $G$. Delorme et al. (2002)  [6] put forward a conjecture concerning the minimum Randi? index among all$n$-vertex connected graphs with the minimum degree at least $k$. In this work, we show that the conjecture is true given the graph contains $k$ vertices of degree $n?1$. Further, it is true among $k$-trees.     

On a Conjecture of S.Halperin

If X is a finite simply connected CW complex, then H_*(X,Q) is finitedimensional, let n_X = max{i|H_i(X,Q)≠0}. On the other hand,π_i(X) is the directsum of finitely many copies of Z and finite Abelian group. We call an interval[k,l] a torsion gap for X if π_k(X) and π_l (X) both coutain copies of Z, andπ_i (X)(k相似文献   

On a Problem of A.C. Woods

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On a question of Sárközy on gaps of product sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b ₁ < b ₂ < …} of all products a _i a _j with a _i , a _j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b _n+1 ? b _n ? α ^?3 and that for t ≥ 2 there are infinitely many t-gaps b _n+t ? b _n ? t ² α ^?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a _i /a _j , a _i , a _j ∈ A} when A ? {1, …, N} and |A| = αN.     

On a theorem of M. Satô

In 1953 Nash [7] introduced the class of functions Φ. In this paper the behaviour of generalized Cesàro (C,α _n )-means (α _n ∈(−1,0)) of trigonometric Fourier series of the classes H ^ω ∩Φ in the space of continuous functions is studied. The sharpness of the results obtained is shown.     

On a homoclinic origin of Hénon-like maps

We review bifurcations of homoclinic tangencies leading to Hénon-like maps of various kinds.     

On a question of Brézis and Nirenberg concerning the degree of circle maps

One considers maps F of the unit circle into itself. For maps of class C ¹ or in the Sobolev class H?, the degree or winding number is equal to the sum of the series in terms of the Fourier coefficients c _n of F (e ^it ) = f (t). Here one studies the possible relation between this series (in symmetric arrangement) and the degree for arbitrary continuous maps F. It is shown that for such maps, the series may fail to be convergent or Abel summable. If the series does converge, the sum may have any value different from the degree of F.     

On a question of Külshammer for representations of finite groups in reductive groups

Let G be a simple algebraic group of type G₂ over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ₂ and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ₂ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H¹(Γ, V) → H¹(Γ_p, V) induced by restriction of 1-cocycles has an infinite fibre.