On Value Distribution of Δ~nf

In this paper, we mainly study zeros and poles of the forward differences Δnf(z), where f(z) is a finite order meromorphic function with two Borel exceptional values.     

General Regular Variation of n-th Order and the 2nd Order Edgeworth Expansion of the Extreme Value Distribution （Ⅰ）

In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.     

General Regular Variation of the n-th Order and 2nd Order Edgeworth Expansions of the Extreme Value Distribution （Ⅱ）

In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.     

On the Mean Value of the Infinite Order Legendre Series

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Distribution of Primitive λ-Roots of Composite Moduli II

We improve estimates for the distribution of primitiveλ-roots of a composite modulus q yielding an asymptotic formula for the number of primitiveλ-roots in any interval I of length |I| >> q1/2 c. Similar results are obtained for the distribution of ordered pairs (x,x-1) with x a primitive A-root, and for the number of primitiveλ-roots satisfying inequalities such as |x - x-1| < B.     

Existence of Positive Solutions to Semipositone Singular Dirichlet Boundary Value Problems

The paper presents the conditions which guarantee that for some positive value of μ there are positive solutions of the differential equation （Ф（x＇））＇＋μQ（t, x, x＇） = 0 satisfying the Dirichlet boundary conditions x（0） = x（T） = 0. Here Q is a continuous function on the set [0, T] × （0, ∞） ~ （R ／ {0}） of the semipositone type and Q is singular at the value zero of its phase variables.     

On the Periodic Zeta‐Function. II

We obtain a more precise asymptotic formula than in [Lith. Math. J., 41(2), 168–177 (2001)] for the mean square of the periodic zetafunction.     

Further Extension on a Theorem of the Value Distribution of f＇f^n

By using small function method, the following result is obtained. If f(z) is transcendental meromorphic and that Ψ(z) is non-zero meromorphic and that T(r, Ψ) =S(r, f), then (n 1)T(r,f)≤(-N)(r,1/f'fn-Ψ) 2(-N)(r,1/f) (-N)(r,f) S(r,f).     

Further Extension on a Theorem of the Value Distribution of f′f~n

By using small function method,the following result is obtained.If f(z) is transcendental meromorphic and thatΨ(z) is non-zero meromorphic and that T(r,Ψ)= S(r,f),then (n 1)T(r,f)≤■(r,(1/f′f~n-Ψ)) 2■(r,(1/f)) ■(r,f) S(r,f).     

On the Mean Value of the Complete Trigonometric Sums with Dirichlet Characters

The main purpose of this paper is, using the analytic method, to study the mean value properties of the complete trigonometric sums with Dirichlet characters, and give an exact calculating formula for its fourth power mean.     

On Eigenvalue Intervals and Eigenfunctions of Nonresonance Singular Dirichlet Boundary Value Problems

In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ>0 is a parameter, p>0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].     

Series Expansions for the First Passage Distribution of Wong–Pearson Jump-Diffusions

Abstract

We explore the Erlang series approach for the first-time passage problem for a particular class of jump-diffusions with polynomial state-dependent coefficients. This approach may be viewed as a discrete analog of the Laplace transform, which replaces the differential equations with polynomial coefficients satisfied by this function by algebraic recurrences. We identify cases in which the expansion is finite and in which the recurrence is of second order, and thus more easily solved.     

Limiting Empirical Singular Value Distribution of?Restrictions of?Discrete Fourier Transform Matrices

We determine the limiting empirical singular value distribution for discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed.     

On Scattering of Poles for Schr?dinger Operator from the Point of View of Dirichlet Series

In this article, we are concerned with the scattering problem of Schr?dinger operators with compactly supported potentials on the real line. We aim at combining the theory of Dirichlet series with scattering theory. New estimate on the number of poles is obtained under the situation that the growth of power series which is related to the potential is not too fast by using a classical result of Littlewood. We propose a new approach of Dirichlet series such that significant upper bounds and lower bounds on the number of poles are obtained. The results obtained in this paper improve and extend some related conclusions on this topic.     

On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds

This paper continues the discussion, begun in J. Schwartz and M. Sharir [Comm. Pure Appl. Math., in press], of the following problem, which arises in robotics: Given a collection of bodies B, which may be hinged, i.e., may allow internal motion around various joints, and given a region bounded by a collection of polyhedral or other simple walls, decide whether or not there exists a continuous motion connecting two given positions and orientations of the whole collection of bodies. We show that this problem can be handled by appropriate refinements of methods introduced by A. Tarski [“A Decision Method for Elementary Algebra and Geometry,” 2nd ed., Univ. of Calif. Press, Berkeley, 1951] and G. Collins [in “Second GI Conference on Automata Theory and Formal Languages,” Lecture Notes in Computer Science, Vol. 33, pp. 134–183, Springer-Verlag, Berlin, 1975], which lead to algorithms for this problem which are polynomial in the geometric complexity of the problem for each fixed number of degrees of freedom (but exponential in the number of degrees of freedom). Our method, which is also related to a technique outlined by J. Reif [in “Proceedings, 20th Symposium on the Foundations of Computer Science,” pp. 421–427, 1979], also gives a general (but not polynomial time) procedure for calculating all of the homology groups of an arbitrary real algebraic variety. Various algorithmic issues concerning computations with algebraic numbers, which are required in the algorithms presented in this paper, are also reviewed.     

Type II Hermite�CPad�� Approximations of Generalized Hypergeometric Series

We shall present short proofs for type II (simultaneous) Hermite–Padé approximations of the generalized hypergeometric and q-hypergeometric series

Pre-dual of the Function Algebra A ?��(D) and Representation of Functions in Dirichlet Series

In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A ^−∞(D) (D being either a bounded C ²-smooth convex domain in ${\mathbb{C}^N (N > 1)}${\mathbb{C}^N (N > 1)} , or a bounded convex domain in \mathbbC{\mathbb{C}}) as a space of entire functions with certain growth. A possibility of representation of functions from the pre-dual space in a form of Dirichlet series with frequencies from D, is also studied.     

On the Distribution of the Limit of Products of I.I.D. 2×2 Random Stochastic Matrices

This article gives sufficient conditions for the limit distribution of products of i.i.d. 2×2 random stochastic matrices to be continuous singular, when the support of the distribution of the individual random matrices is finite.     

General Term of Sequence

In the study of number sequences,we learn to uncover how the sequence grows.Then we make a generalisation about the number sequence by stating the general term in algebraic form. Below is an activity using matches(or toothpicks,etc)to build and expand a number pattern. Given the pattern in the figures as follows: