Evaluation of Series Involving the Product of the Tail of $${\zeta(k)}$$ and $${\zeta(k+1)}$$ζ(k+1)

We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function values

$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$

where \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.

     

Asymptotic Representation of L<Subscript>p</Subscript>-Minimal Polynomials, 1 < p < ∞

While the theory of asymptotics for L₂-minimal polynomials is highly developed, so far not much is known about L_p-minimal polynomials, Indeed, Bernstein gave asymptotics for the minimum deviation, Fekete and Walsh gave nth root asymptotics and, recently, Lubinsky and Saff came up with asymptotics outside the support [-1,1]. But the main point of interest, the asymptotic representation on the support, still remains open. Here we settle it for weight functions of the form where w is positive and on [-1,1] with and 相似文献   

Обобшенные проиэводные и граничные своиства проиэводных интеграла Пуассона для полупространства R + k+1 (k>1)

Реэуме. Вводятся понятия обобшеннои частнои проиэводнои и обобшенного оератора Лапласа для функции многих переменных. Докаэана теоремы типа теоремы Фату о граничных своиствах частхых проиэводных интегралал Пуассона для полупространства, когда плотностя интеграла имеет обобшеннуу частнуу проиэводнуу и обобшенныи оператор Лапласа.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

Thompson’s conjecture for the alternating group of degree 2<Emphasis Type="Italic">p</Emphasis> and 2<Emphasis Type="Italic">p</Emphasis>+1

For a finite group G denote by N(G) the set of conjugacy class sizes of G. In 1980s, J.G.Thompson posed the following conjecture: If L is a finite nonabelian simple group, G is a finite group with trivial center and N(G) = N(L), then G ? L. We prove this conjecture for an infinite class of simple groups. Let p be an odd prime. We show that every finite group G with the property Z(G) = 1 and N(G) = N(A _i ) is necessarily isomorphic to A _i , where i ∈ {2p, 2p + 1}.     

Convergence Theorems for ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Norm Minimizers with Respect to <Emphasis Type="Italic">p</Emphasis>

Let C be a fixed compact convex subset of and let x_p be the unique minimal _p-norm element in C for any . In this paper, we study the convergence of x_p as p or , respectively. We characterize also the limit point as the minimal element of C with respect to the lexical minimax order relation or the lexical minitotal order relation, respectively.Communicated by D. G. LuenbergerThe author thanks Professor H. Komiya for valuable advice and an anonymous referee for kind comments.     

Remarks on Hausdorff measure and stability for the <Emphasis Type="Italic">p</Emphasis>-obstacle problem (1 < <Emphasis Type="Italic">p</Emphasis> < 2)

In this paper, we consider the obstacle problem for the inhomogeneous p-Laplace equation

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

Algebraic independence over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> of values of analytic functions at points from ℂ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

General theorems on the algebraic independence over ℚ _p of the values of analytic functions at points from ℂ _p and their applications to particular examples are presented in the paper.     

On the Fučík Spectrum of the <Emphasis Type="Italic">p</Emphasis>-Laplacian

We construct and variationally characterize by a min-max procedure an unbounded sequence of continuous and strictly decreasing curves in the Fuík spectrum of the p-Laplacian. Applications to quasilinear elliptic boundary value problems are given.     

ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Diagonally Dominant Symmetric Operators

The existence of _p-universal diagonally dominant symmetric liner operators on separable Hilbert spaces is discussed. Also, properties of these operators and their characterization in terms of their spectra are investigated.     

On Perturbations of ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-localized Frames

In this paper, we give some sufficient conditions under which perturbations preserve ? _p -localized frames. Using an arbitrary given sequence, we provide a simple way for constructing ? _p -localized sequences.     

Jackson-type inequalities and exact values of widths of classes of functions in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>, 1 ≤ <Emphasis Type="Italic">p</Emphasis> < ∞

In the spaces S^p, 1 p < , introduced by Stepanets, we obtain exact Jackson-type inequalities and compute the exact values of widths of classes of functions determined by averaged moduli of continuity of order m.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 595–605, May, 2004.     

Borel equivalence relations between <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">ℓ</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we show that, for each p 〉 1, there are continuum many Borel equivalence relations between Rω/l1 and Rω/p ordered by ≤B which are pairwise Borel incomparable.     

On existence of a universal function for <Emphasis Type="Italic">L</Emphasis> <Superscript> <Emphasis Type="Italic">p</Emphasis> </Superscript>[0, 1] with <Emphasis Type="Italic">p</Emphasis>∈(0, 1)

We show that, for every number p ∈ (0, 1), there is g ∈ L¹[0, 1] (a universal function) that has monotone coefficients c_k(g) and the Fourier–Walsh series convergent to g (in the norm of L¹[0, 1]) such that, for every f ∈ L^p[0, 1], there are numbers δ_k = ±1, 0 and an increasing sequence of positive integers N_q such that the series ∑ _k=0^+∞δ_kc_k(g)W_k (with {W_k} theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (?1, 0), of its Cesáro means converge to f in the metric of L^p[0, 1].     

$$I_0$$ and combinatorics at $$\lambda ^+$$λ+

We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\).     

Rational approximations of functions of Markov-Stieltjes type in Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>, 0 < <Emphasis Type="Italic">p</Emphasis> ≤ ∞

Orders of the best approximations for functions of Markov-Stieltjes type in the space H ^p are considered in the case of intersection of the measure support and the boundary of a unit circle. The results of J.-E. Andersson for functions with a singularity at a single point of a unit circumference are extended to the case of two singularities on that circumference.