On the 2<Emphasis Type="Italic">m</Emphasis>-th power mean of Dirichlet <Emphasis Type="Italic">L</Emphasis>-functions with the weight of trigonometric sums

Let p be a prime, χ denote the Dirichlet character modulo p, f (x) = a ₀ + a ₁ x + ... + a _k x ^k is a k-degree polynomial with integral coefficients such that (p, a ₀, a ₁, ..., a _k ) = 1, for any integer m, we study the asymptotic property of

Approximate <Emphasis Type="Italic">C</Emphasis>*-ternary ring homomorphisms

In this paper, we establish the generalized Hyers–Ulam–Rassias stability of C*-ternary ring homomorphisms associated to the Trif functional equation

Structure theorems of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">n</Emphasis>)-Azumaya algebras

Let k be a field and E(n) be the 2 ⁿ⁺¹-dimensional pointed Hopf algebra over k constructed by Beattie, Dăscălescu and Grünenfelder [J. Algebra, 2000, 225: 743–770]. E(n) is a triangular Hopf algebra with a family of triangular structures R _M parameterized by symmetric matrices M in M _n (k). In this paper, we study the Azumaya algebras in the braided monoidal category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } and obtain the structure theorems for Azumaya algebras in the category $ E_{(n)} \mathcal{M}^{R_M } $ E_{(n)} \mathcal{M}^{R_M } , where M is any symmetric n×n matrix over k.     

A note on some relations among special sums of reciprocals modulo <Emphasis Type="Italic">p</Emphasis>

In this note the sums s(k, N) of reciprocals are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers. Published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan MSM 0021630518 “Simulation modeling of mechatronic systems”.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

The signed <Emphasis Type="Italic">k</Emphasis>-domination number of directed graphs

Let k ≥ 1 be an integer, and let D = (V; A) be a finite simple digraph, for which d _D ⁻ ≥ k − 1 for all v ɛ V. A function f: V → {−1; 1} is called a signed k-dominating function (SkDF) if f(N ⁻[v]) ≥ k for each vertex v ɛ V. The weight w(f) of f is defined by $ \sum\nolimits_{v \in V} {f(v)} $ \sum\nolimits_{v \in V} {f(v)} . The signed k-domination number for a digraph D is γ _kS (D) = min {w(f|f) is an SkDF of D. In this paper, we initiate the study of signed k-domination in digraphs. In particular, we present some sharp lower bounds for γ _kS (D) in terms of the order, the maximum and minimum outdegree and indegree, and the chromatic number. Some of our results are extensions of well-known lower bounds of the classical signed domination numbers of graphs and digraphs.     

Compact embedded rotation hypersurfaces of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis>+1</Superscript>

In this paper, we prove that and round geodesic spheres are the only n-dimensional compact embedded rotation hypersurfaces with H_m = 0 (1 ≤ m ≤ n − 1) in a unit sphere Sⁿ⁺¹(1). When m = 1, our result reduces to the result of T. Otsuki [O1], [O2], Brito and Leite [BL]. The project is supported by the grant No. 10531090 of NSFC.     

Inequalities for derivatives of functions in the spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

  We obtain a new sharp inequality for the local norms of functions x ∈ L _{∞, ∞} ^r (R), namely,

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

A bound on the <Emphasis Type="Italic">k</Emphasis>-domination number of a graph

Let G be a graph with vertex set V(G), and let k ⩾ 1 be an integer. A subset D ⊆ V(G) is called a k-dominating set if every vertex υ ∈ V(G)-D has at least k neighbors in D. The k-domination number γ _k (G) of G is the minimum cardinality of a k-dominating set in G. If G is a graph with minimum degree δ(G) ⩾ k + 1, then we prove that

Generalized (<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">d</Emphasis>)-ray systems of points and inequalities for nonoverlapping domains and open sets

We solve the extremal problem of finding the maximum of the functional

A note on <Emphasis Type="Italic">q</Emphasis>-multiplicative functions

We prove the following statement. Let , and let . Suppose that, for all and , the sequence satisfies the relation

On <Emphasis Type="Italic">T</Emphasis> points of algebroid functions

In this paper, we firstly give a new definition, namely, the T point of algebroid functions. Then by using Ahlfors’ theory of covering surfaces, we prove the existence of these points for any ν-valued algebroid functions in the unit disk satisfying $\mathop {\lim \sup }\limits_{r \to 1^ - } \frac{{T(r,w)}} {{\log \tfrac{1} {{1 - r}}}} = + \infty $\mathop {\lim \sup }\limits_{r \to 1^ - } \frac{{T(r,w)}} {{\log \tfrac{1} {{1 - r}}}} = + \infty . This extends the recent results of Xuan, Wu and Sun.     

The Maximum Size Of Graphs With A Unique<Emphasis Type="Italic">k</Emphasis>- Factor

For suitable positive integers n and k let m(n, k) denote the maximum number of edges in a graph of order n which has a unique k-factor. In 1964, Hetyei and in 1984, Hendry proved for even n and , respectively. Recently, Johann confirmed the following conjectures of Hendry: for and kn even and for n = 2kq, where q is a positive integer. In this paper we prove for and kn even, and we determine m(n, 3).     

On the 2 k-th power mean of $$ \frac{{L'}} {L}(1,\chi ) $$ with the weight of Gauss sums

The main purpose of this paper is to study the hybrid mean value of $ \frac{{L'}} {L}(1,\chi ) $ \frac{{L'}} {L}(1,\chi ) and Gauss sums by using the estimates for trigonometric sums as well as the analytic method. An asymptotic formula for the hybrid mean value $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } $ \sum\limits_{\chi \ne \chi _0 } {|\tau (\chi )||\frac{{L'}} {L}(1,\chi )|^{2k} } of $ \frac{{L'}} {L} $ \frac{{L'}} {L} and Gauss sums will be proved using analytic methods and estimates for trigonometric sums.     

Multivariate Refinement Equations and Convergence of Cascade Algorithms in Lp（0〈p〈1）Spaces

We consider the solutions of refinement equations written in the form

On rigidity of Clifford torus in a unit sphere

We extend the scalar curvature pinching theorems due to Peng-Terng, Wei-Xu and Suh-Yang. Let M be an n-dimensional compact minimal hypersurface in S n+1 satisfying Sf 4 f_3~2 ≤ 1/n S~3 , where S is the squared norm of the second fundamental form of M, and f_k =sum λ_i~k from i and λ_i (1 ≤ i ≤ n) are the principal curvatures of M. We prove that there exists a positive constant δ(n)(≥ n/2) depending only on n such that if n ≤ S ≤ n + δ(n), then S ≡ n, i.e., M is one of the Clifford torus S~k ((k/n)~1/2 ) ×S~...     

On the strong law of large numbers and additive functions

Let f(n) be a strongly additive complex-valued arithmetic function. Under mild conditions on f, we prove the following weighted strong law of large numbers: if X,X ₁,X ₂, … is any sequence of integrable i.i.d. random variables, then

Order equalities for some functionals and their application to the estimation of the best <Emphasis Type="Italic">n</Emphasis>-term approximations and widths

We study the behavior of functionals of the form

Problem of the first passage time for <Emphasis Type="Italic">p</Emphasis>-adic diffusion

This work is a continuation of paper [1], where was considered analog of the problem of the first return for ultrametric diffusion. The main result of this paper consists in construction and investigation of stochastic quantity $ \tau _{B_r (a)} $ \tau _{B_r (a)} (ω), which has meaning of the first passage time into domain B _r (a) by trajectories of the Markov stochastic process ζ(t, ω).Markov stochastic process is given by distribution density f(x, t), x ∈ ℚ _p , t ∈ R ₊, which is solution of the Cauchy problem