Inverse problem for upper asymptotic density

For a set of natural numbers, the structural properties are described when the upper asymptotic density of achieves the infimum of the upper asymptotic densities of all sets of the form , where the upper asymptotic density of is greater than or equal to the upper asymptotic density of . As a corollary, we prove that if the upper asymptotic density of is less than and the upper asymptotic density of achieves the infimum, then the lower asymptotic density of must be .

     

A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.     

Second order asymptotic optimality of estimators for a density with finite cusps

We consider i.i.d. samples from a continuous density with finite cusps. Then we obtain the bound for the second order asymptotic distribution of all asymptotically median unbiased estimators. Further we get the second order asymptotic distribution of a bias-adjusted maximum likelihood estimator, and we see that it is not generally second order asymptotically efficient.     

On the density of primitive sets

Classical results by Behrend, Erdös, Pillai, Szemerédi, the authors, and others are improved and extended to other concepts of densities in two directions, based on smooth and multiplicative weightings.     

Additive and multiplicative Sidon sets

Acta Mathematica Hungarica - We give a construction of a set $$A \subset \mathbb N$$ such that any subset $${A' \subset A}$$ with $$|A'| \gg |A|^{2/3}$$ is neither an additive nor...     

Additive and multiplicative Sidon sets

Summary Inspired by a paper of Sárk?zy [4] we study sets of integers and sets of residues with the property that all sums and all products are distinct.     

Asymptotics of orthogonal polynomials with asymptotic Freud-like weights

We derive uniform asymptotic expansions for polynomials orthogonal with respect to a class of weight functions that are real analytic and behave asymptotically like the Freud weight at infinity. Although the limiting zero distributions are the same as in the Freud cases, the asymptotic expansions are different due to the fact that the weight functions may have a finite or infinite number of zeros on the imaginary axis. To resolve the singularities caused by these zeros, an auxiliary function is introduced in the Riemann–Hilbert analysis. Asymptotic formulas are established in several regions covering the whole complex plane. We take the continuous dual Hahn polynomials as an example to illustrate our main results. Some numerical verifications are also given.     

Connection between sets of uniqueness for multiplicative systems of functions and sets of uniqueness for multiplicative transforms

A problem of uniqueness for series over multiplicative systems of functions and for multiplicative transforms is considered. It is shown that each set of uniqueness for a multiplicative transform is specified by a countable collection of sets of uniqueness for series over the corresponding multiplicative system of functions. Each set of uniqueness for a series over a multiplicative system of functions is a portion on [0, 1) of some set of uniqueness for the corresponding multiplicative transform.     

Optimal portfolios with asymptotic criteria

This paper is concerned with a portfolio optimization model for a long planning horizon. We first argue that in this case the asymptotic growth rate and the asymptotic variance are better measures of performance than the usual mean and variance of return. We next propose an efficient algorithm for calculating the asymptotic frontier, i.e., the efficient frontier relative to the new criteria. Finally, we illustrate our methods and compare the difference between our model and the classical mean-variance-model by using historical data based on the 1064 stocks of the Tokyo Stock Exchange.     

Estimating function with asymptotic bias and its estimator

We introduce the estimating function with asymptotic bias and investigate the asymptotic behavior of the estimator based on it by using their relationship. The estimator based on the estimating function with asymptotic bias has the asymptotic normality with asymptotic bias. We show that this theory has several interesting applications in practical statistics.     

Global asymptotic behavior of positive solutions for exponential form difference equations with three parameters

In this paper, we study a class of second order difference equations with three paremeters. With positive initial values, the asymptotic behavior of positive solutions are investigated.     

Elementary density bounds for self-similar sets and application

Falconer[1] used the relationship between upper convex density and upper spherical density to obtain elementary density bounds for s-sets at H S-almost all points of the sets. In this paper, following Falconer[1], we first provide a basic method to estimate the lower bounds of these two classes of set densities for the self-similar s-sets satisfying the open set condition （OSC）, and then obtain elementary density bounds for such fractals at all of their points. In addition, we apply the main results to the famous classical fractals and get some new density bounds.     

一个渐近非线性Dirichlet问题的对径解的存在性

利用锥上的不动点定理获得了一个渐近非线性Dirirchlet问题的对径解的存在定理。     

GLOBAL ASYMPTOTIC STABILITY OF THE PERIODIC LOTKA-VOLTERRA SYSTEM WITH TWO-PREDATORS AND ONE-PREY

The three species Lotka-Volterra periodic model with two predators and one prey is considered. A set of easily verifiable sufficient conditions is obtained. Finally, an example is given to illustrate the feasibility of these conditious.     

Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise

We study the asymptotic behavior of solutions to the stochastic sine-Gordon lattice equations with multiplicative white noise. We first prove the existence and uniqueness of solutions, and then establish the existence of tempered random bounded absorbing sets and global random attractors.