Explicit zero density theorems for Dedekind zeta functions

This article studies the zeros of Dedekind zeta functions. In particular, we establish a smooth explicit formula for these zeros and we derive an effective version of the Deuring–Heilbronn phenomenon. In addition, we obtain an explicit bound for the number of zeros in a box.     

Some renewal theorems and their applications

Siberian Mathematical Journal -     

Some general theorems on generating functions and their applications

Several general classes of generating functions are established for a certain sequence of functions defined by Equation (1) below. By suitably specializing the various parameters involved, each of these main results can be applied to yield known as well as new generating functions for such familiar orthogonal polynomials as Jacobi, Laguerre, Hermite, and Bessel polynomials, and also for numerous interesting generalizations of these polynomials studied in the literature.     

Generalizations of some zero sum theorems

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weight A, denoted by D _A (G), is defined to be the least positive integer t such that, for every sequence (x ₁,..., x _t ) with x _i ?∈?G, there exists a non-empty subsequence \((x_{j_1},\ldots, x_{j_l})\) and a _i ?∈?A such that \(\sum_{i=1}^{l}a_ix_{j_i} = 0\). Similarly, for an abelian group G of order n, E _A (G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence \((x_{j_1} ,\ldots, x_{j_n})\) such that \(\sum_{i=1}^{n}a_ix_{j_i} = 0\), for some a _i ?∈?A. When G is of order n, one considers A to be a non-empty subset of {1,..., n???1 }. If G is the cyclic group \({\Bbb Z}/n{\Bbb Z}\), we denote E _A (G) and D _A (G) by E _A (n) and D _A (n) respectively.In this note, we extend some results of Adhikari et al (Integers 8 (2008) Article A52) and determine bounds for \(D_{R_n}(n)\) and \(E_{R_n}(n)\), where \(R_n = \{x^2 : x \in (\mathbb{Z}/n\mathbb{ Z})^*\}\). We follow some lines of argument from Adhikari et al (Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677–680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242–243) and Kneser’s theorem (Math. Z. 58 (1953) 459–484; 66 (1956) 88–110; 61 (1955) 429–434).     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

General existence theorems, alternative theorems and applications to minimax problems

We establish general theorems on maximal elements, coincidence points and nonempty intersections for set-valued mappings on GFC-spaces and show their equivalence. Applying them we derive equivalent forms of alternative theorems. As applications, we develop in detail general types of minimax theorems. The results obtained improve or include as special cases several recent ones in the literature.     

Some critical point theorems and their applications to periodic solution for second order Hamiltonian systems

Some critical point theorems without the compactness assumptions are obtained by the reduction method, the perturbation argument and the least action principle. As applications, some existence results of periodic solutions are obtained for nonautonomous second order Hamiltonian systems, which unify and generalize some recent corresponding results.     

Some fixed point theorems for non-self mappings of contractive type with applications to endpoint theory

In this paper, in the setting of complete metric spaces we establish some fixed point theorems for non-self mappings of contractive type satisfying either the Reich–Zaslavski property or the approximate fixed point property. As applications, we obtain some results in endpoint theory.     

Some stability theorems for an abstract equation in Hilbert space with applications to linear elastodynamics

Previous results of Knops and Payne concerning continuous data dependence in linear elastodynamics are extended to include the case where the elasticities may be time dependent. Our results, which include stability under perturbations of both the elasticities and the initial geometry, are obtained by applying a logarithmic convexity argument to the Cauchy problem associated with an abstract differential equation in Hilbert space.     

Some inequalities for strong mixing random variables with applications to density estimation

In this paper, we establish an inequality of the characteristic functions for strongly mixing random vectors, by which, an upper bound is provided for the supremum of the absolute value of the difference of two multivariate probability density functions based on strongly mixing random vectors. As its application, we consider the consistency and asymptotic normality of a kernel estimate of a density function under strong mixing. Our results generalize some known results in the literature.     

Some existence theorems in nonlinear analysis for mappings on GFC-spaces and applications

We establish a maximal element theorem, an intersection theorem and a coincidence-point theorem in product GFC-spaces. As examples of wide ranges of applications, we first deduce sufficient conditions for the solution existence of a mixed system of inclusions. Then using this we obtain existence results for systems of vector quasi-optimization problems and for multiobjective mathematical programs constrained by systems of inclusions. Our results are shown to improve and include recent ones in the literature.     

Some existence theorems in nonlinear analysis for mappings on GFC-spaces and applications

We establish a maximal element theorem, an intersection theorem and a coincidence-point theorem in product GFC-spaces. As examples of wide ranges of applications, we first deduce sufficient conditions for the solution existence of a mixed system of inclusions. Then using this we obtain existence results for systems of vector quasi-optimization problems and for multiobjective mathematical programs constrained by systems of inclusions. Our results are shown to improve and include recent ones in the literature.