Tong-type identity and the mean square of the error term for an extended Selberg class

In 1956, Tong established an asymptotic formula for the mean square of the error term of the summatory function of the Piltz divisor function d3(n). The aim of this paper is to generalize Tong's method to a class of Dirichlet series L(s) which satisfies a functional equation. Let a(n) be an arithmetical function related to a Dirichlet series L(s), and let E(x) be the error term of ′n xa(n). In this paper, after introducing a class of Diriclet series with a general functional equation(which contains the well-known Selberg class), we establish a Tong-type identity and a Tong-type truncated formula for the error term of the Riesz mean of the coefficients of this Dirichlet series L(s). This kind of Tong-type truncated formula could be used to study the mean square of E(x) under a certain assumption. In other words, we reduce the mean square of E(x) to the problem of finding a suitable constant σ*which is related to the mean square estimate of L(s). We shall represent some results of functions in the Selberg class of degrees 2–4.     

关于平行三角形集簇直线横截的Helly数

We prove two results about the problem of finding the Helly number for line transversals to a family of parallel triangles in the plane: (1) If each three triangles of a family of parallel right triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all the triangles of the family; (2) If each three triangles of a family of parallel obtuse triangles are intersected by an ascending (or a descending) line, then there is an ascending (or a descending) line that intersects all the triangles of the family. We also obtain that the Helly number of a family of parallel right or obtuse triangles is 3.     

ANISOTROPIC (p,q)-EQUATIONS WITH COMPETITION PHENOMENA

We consider a nonlinear Robin problem driven by the anisotropic(p,q)-Laplacian and with a reaction exhibiting the competing effects of a parametric sublinear(concave) term and of a superlinear(convex) term.We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies.We also prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.     

The local Hlder exponent for the entropy of real unimodal maps In Memory of Lei Tan

We consider the topological entropy h(θ) of real unimodal maps as a function of the kneading parameter θ(equivalently, as a function of the external angle in the Mandelbrot set). We prove that this function is locally Hlder continuous where h(θ) 0, and more precisely for any θ which does not lie in a plateau the local Hlder exponent equals exactly, up to a factor log 2, the value of the function at that point.This confirms a conjecture of Isola and Politi(1990), and extends a similar result for the dimension of invariant subsets of the circle.     

A direct product decomposition of QMV algebras

We study the direct product decomposition of quantum many-valued algebras (QMV algebras) which generalizes the decomposition theorem of ortholattices (orthomodular lattices).In detail,for an idempo- tent element of a given QMV algebra,if it commutes with every element of the QMV algebra,it can induce a direct product decomposition of the QMV algebra.At the same time,we introduce the commutant C(S) of a set S in a QMV algebra,and prove that when S consists of idempotent elements,C(S) is a subalgebra of the QMV algebra.This also generalizes the cases of orthomodular lattices.     

A NEW APPROACH TO SOLVE SYSTEMS OF LINEAR EQUATIONS

1. IntroductionThe new aPProaCh is based on the analysis of the motion of a damped harmonic oscillatorin the gravitational field 11]. The associated equation of motion ismXtt + oXt + aX = b (1)where X = X(t), is the one dimensions di8PlaCement of Of a mass m under a dissipation(o > 0), a ~nic potential (a > 0) and a constant acceleration (b, gravitational field). Thetotal energy variation is given by the equationwhereThe solution of the motion equation (1) is given by the sum of two contr…     

Vector subdivision schemes in (L_p(R~5))~r(1≤p≤∞) spaces

The purpose of this paper is to investigate the refinement equations of the formwhere the vector of functions = (1, … ,r)T is in (LP(R8))T,1 ≤ p ≤∞, α(α),α ∈ Z5, is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s x a integer matrix such that limn→ ∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions (0 ∈ (LP(R8))r and use the iteration schemes fn := Qan0,n = 1,2,…, where Qa is the linear operator defined on (Lp(R8))r given byThis iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group Z8/MZ8 containing 0.     

ALGORITHME DE PERTURBATION DIRECTRICE POUR LE PROBLME D''''AJUSTEMENT DES DONNEES

In this paper we propose an algorithm, called Directrive Perturbation (PD-Algorithm), which decrease the maximum value of the absolute errors of a fitting formula obtained by a Least-Squares Method. Applying the algorithm we can increase the accuracy of the uniform approximation of the fitting formula with the original data (x_i, f(x_i)) (i=1, 2, ..., m). Further, after applying a finite number of times of the perturbation, we can obtain the formuala of Tchebycheff approximation.     

Crossed Products over Weak Hopf Algebras Related to Cleft Extensions and Cohomology

The authors present the general theory of cleft extensions for a cocommutative weak Hopf algebra H.For a right H-comodule algebra,they obtain a bijective correspondence between the isomorphisms classes of H-cleft extensions AH → A,where AH is the subalgebra of coinvariants,and the equivalence classes of crossed systems for H over AH.Finally,they establish a bijection between the set of equivalence classes of crossed systems with a fixed weak H-module algebra structure and the second cohomology group HφZ(AH)2(H,Z(AH)),where Z(AH)is the center of AH.     

THE BOUNDARY VALUE PROBLEMS FOR THE HIGHER ORDER QUASILINEAR PARABOLIC AND ELLIPTIC SYSTEMS SATISFYING GENERAL BOUNDARY CONDITIONS

In this paper, the existence and uniqueness of the boundary value problems for the higher order quasilinear parabolic systems satisfying general boundary conditions and initial value condition are considered. We have concluded the problems to the following: if all the solutions of a family of problems of the same type which are derived from a substitution of τf(s, t, u, ...,Dx~(2b-1)u), τgj(y, t, u) and τ(x) (0≤τ≤1) for f, g, and φ respectively are uniformly bounded, then the original problems have a unique solution in H~(2b a,1 a/2b)((?)_T). Under assumption that the linear problems have a unique solution, we have proved the existence and uniqueness of the solution of the boundary value problems for the quasillnear elliptic systems.     

Minimal Representations of Cuntz Algebras

Suppose that O_n(2 ≤ n ≤∞) is a Cuntz algebra. There exists a number 1 ≤ k(π, Ω) ≤∞such that if the cyclic representations(H, π, Ω) and(H, π, Ω) of the Cuntz algebra O_n are unitary equivalent, k(π, Ω) = k(π, Ω). Applying the number, one can define a minimal representation of O_n and give a sufficient condition of the minimality for a representation of O_n. Moreover, the relations between minimal states and minimal representations of Cuntz algebras and the relations between the minimality and the irreducibility of the representation of O_n are investigated, respectively.     

A THEORY FOR MARSHALLING RALLWAY CARS INTO A TRAIN——An Approach to a Multiple Objective Problem

About twefity years ago, on some marshalling yards in China, an empirical method, alled the Tabulation Method, was used to make feasible and effective plans for marshalling railway cars into a train, especially, into a local train. Theoretical studies of this method developed some mathematical models of the nature of of combinatorial optimization. The main part of this paper is a survey of a mathematical theory, called the Theory of Standard Numeration. When there is one loco (there are two locos) in operation on the marshalling yard (one on each side of the yard), this theory says that every feasible plan for making up a local train can be characterized by a finite sequence of binary(ternary) numbers which are called the characterlshc numbers of the cars. If the minimum value of the number of ordered subsequences in a standard partition of the initial train is n (which can easily be obtained by the Tabulation Method)and 2~(m-2)相似文献   

本刊英文版Vol.30(2014),No.2论文摘要

<正>Subordination in the Sense of Bochner of L~p-semigroups and Associated Markov Processes Oana LUPASCU Abstract We show that the subordination induced by a convolution semigroup(subordination in the sense of Bochner)of a C_0-semigroup of sub-Markovian operators on an L~p space is actually associated to the subordination of a right(Markov)process.As a consequence,we solve the martingale problem associate with the Lp-infinitesimal generator of the subordinate semigroup.We also prove quasi continuity properties for the elements of the domain of the     

Periods of polynomials over a Galois ring

The period of a monic polynomial over an arbitrary Galois ring GR(pe,d) is theoretically determined by using its classical factorization and Galois extensions of rings. For a polynomial f(x) the modulo p remainder of which is a power of an irreducible polynomial over the residue field of the Galois ring, the period of f(x) is characterized by the periods of the irreducible polynomial and an associated polynomial of the form (x-1)m+pg(x). Further results on the periods of such associated polynomials are obtained, in particular, their periods are proved to achieve an upper bound value in most cases. As a consequence, the period of a monic polynomial over GR(pe,d) is equal to pe-1 times the period of its modulo p remainder polynomial with a probability close to 1, and an expression of this probability is given.     

ASYMPTOTIC BEHAVIOR OF DYNAMIC EQUATIONS ON A PERIODIC TIME SCALES

We first prove some basic results on the periodic time scales, then on a special but practical periodic time scales T =∪[k(a + b),k(a + b) + a], the asymptotic behavior of x△ = px is explored with specific emphasis given to how the graininess of the time scales affects stability. At last, we analyze the asymptotic properties of the solutions of planar linear dynamic system, and obtain some sufficient conditions for the stability of the equalibium (0,0).     

AN ERGODIC THEOREM FOR GENERALIZED SIMPLE EXCLUSION PROCESSES WITH REVERSIBLE POSITIVE TRANSITION

Simple exclusion procosses are important infinite particle systems. This model was proposed by Spitzer (1)and studied extensively by Spitzer and Liggett (2). On the basis of the model, Yan Shi-jian and Chen Mu-fa proposed generalized exclusion processes and obtained the construction of the processes. Zheng Wen-qu obtained a necessary and sufficient condition of the reversibility and proved that the set of the reversible probability measures of such a process is equal to the set of it's Gibbs states (3). This paper is devoted to the ergodic theory of a generalized simple exclusion proess with the state space {0, 1, …, m} (m≥1, S is a countable set) and a reversible positive recurrent transition probability matrix P=(p(x, y))_(x,y)∈s Refering (4), the set of it's invariant probability measures is described and the ergodic properties of the process is obtained. We also prove that the set of it's reversible probability measures coinsides with the set of it's invariant probability measures. So the main re     

The Soft Point and Its Applications in Body Falling

From the mesoscopic point of view, a definition of soft point is introduced by considering the attributes of geometric profile and mass distribution. After that, this concept is used to develop the soft matching technique to simulate the chaotic behaviors of the equations. Especially, a tennis model with deformation factor a(t) is proposed to derive a generalized Newton-Stokes equation v′(t) = λ(v T-a(t)v(t)). Furthermore, a concept of duality of deformation factor a(t) and velocity v(t) with re...     

Property(ω_1) and Single Valued Extension Property

In this note we study the property(ω1),a variant of Weyl's theorem by means of the single valued extension property,and establish for a bounded linear operator defined on a Banach space the necessary and sufficient condition for which property(ω1) holds.As a consequence of the main result,the stability of property(ω1) is discussed.     

Shrinkage estimation analysis of correlated binary data with a diverging number of parameters

For analyzing correlated binary data with high-dimensional covariates,we,in this paper,propose a two-stage shrinkage approach.First,we construct a weighted least-squares(WLS) type function using a special weighting scheme on the non-conservative vector field of the generalized estimating equations(GEE) model.Second,we define a penalized WLS in the spirit of the adaptive LASSO for simultaneous variable selection and parameter estimation.The proposed procedure enjoys the oracle properties in high-dimensional framework where the number of parameters grows to infinity with the number of clusters.Moreover,we prove the consistency of the sandwich formula of the covariance matrix even when the working correlation matrix is misspecified.For the selection of tuning parameter,we develop a consistent penalized quadratic form(PQF) function criterion.The performance of the proposed method is assessed through a comparison with the existing methods and through an application to a crossover trial in a pain relief study.