On the generation of high-quality white noise series

A new approach to the simulation of white-noise time-series is presented. The approach is based on the frequency-domain property of white-noise as having a flat power spectrum density (PSD). From such a PSD, a linear complex spectrum of random-phase multisinusoidal series (MS) may be generated. Next, the fast Fourier transform is applied to this linear spectrum to generate a random-phase multisinusoidal N-sample series, simulating the white-noise series. The properties of some MS series are discussed including a gaussian white noise multisinusoidal series. The idea is illustrated by a number of examples. They demonstrate that the spectral and correlation properties of such series are enhanced in comparison to the properties of the random phases used to generate them. They also demonstrate that the spectral and correlation properties of such series are better, especially for short series, in comparison with standard white noise generators.     

On the degree of homogeneous bent functions

In this paper, the degree of homogeneous bent functions is discussed. We prove that for any nonnegative integer k, there exists a positive integer N such that for n?N there exist no 2n- variable homogeneous bent functions having degree n-k or more, where N is the least integer satisfying .     

On the multiplicities of the zeros of Laguerre-Pólya functions

We show that all the zeros of the Fourier transforms of the functions , , are real and simple. Then, using this result, we show that there are infinitely many polynomials such that for each the translates of the function

generate . Finally, we discuss the problem of finding the minimum number of monomials , , which have the property that the translates of the functions , , generate , for a given .

     

On the zeros of certain polynomials

We prove that certain naturally arising polynomials have all of their roots on a vertical line.     

Detecting Features in Spatial Point Processes with Clutter via Local Indicators of Spatial Association

We consider the problem of detecting features of general shape in spatial point processes in the presence of substantial clutter. Our goal is to remove clutter from images where one or several features are present and have to be detected. We use a method based on local indicators of spatial association (LISA) functions, particularly on the development of a local version of the product density which is a second-order characteristic of spatial point processes. The classification method is built upon a stochastic version of the EM algorithm (SEM). This method can be applied without user input about the number or shapes of the regions. Our proposal, compared with the kth nearest-neighbor technique, is tested through simulated examples yielding high detection and low false-positive rates. Two real case studies of connective loose tissues in human organs and earthquakes are also presented.     

非平稳可微高斯过程的上穿过点过程的渐近分布

{X(t),0≤t≤T}为均方可微非平稳高斯过程。具有渐近中心化的均值m(t)和常数的方差, NT(·)为{X(t),0≤t≤T}上穿过水平uT的点过程,则在一定的条件下匕穿过点过程NT(·)依分布收敛到一Poisson过程．     

On the index integral transformation with Nicholson's function as the kernel

The integral transformation, which is associated with the Nicholson function as the kernel, is introduced and investigated in the paper. This transformation is an integral, where integration is with respect to an index of the sum of squares of Bessel functions of the first and second kind. Composition representations and relationships with the Meijer K-transform, the Kontorovich-Lebedev transform, the Mellin transform, and the sine Fourier transform are given. We also present boundedness properties, a Parseval type equality, and an inversion formula.     

On the simpleness of zeros of Stokes multipliers

The aim of this paper is to discuss the simpleness of zeros of Stokes multipliers associated with the differential equation -Φ^″(X)+W(X)Φ(X)=0, where W(X)=X^m+a₁X^m-1+?+a_m is a real monic polynomial. We show that, under a suitable hypothesis on the coefficients a_k, all the zeros of the Stokes multipliers are simple.     

On some integral properties of multidimensional Hilbert operators

In the present paper, we study the integral properties of multidimensional Hilbert transforms.     

Wavelets, generalized white noise and fractional integration: The synthesis of fractional Brownian motion

We provide an almost sure convergent expansion of fractional Brownian motion in wavelets which decorrelates the high frequencies. Our approach generalizes Lévy's midpoint displacement technique which is used to generate Brownian motion. The low-frequency terms in the expansion involve an independent fractional Brownian motion evaluated at discrete times or, alternatively, partial sums of a stationary fractional ARIMA time series. The wavelets fill in the gaps and provide the necessary high frequency corrections. We also obtain a way of constructing an arbitrary number of non-Gaussian continuous time processes whose second order properties are the same as those of fractional Brownian motion.     

现代信息技术中的快速收敛取样定理

在现代信息技术中,取样定理是模拟信号的量化以及复原为模拟信号的基础.模拟信号一般是频谱有限函数,彭瑞仁教授虽然得到了收敛速度比Shannon取样定理快得多的取样定理,然而,本文给出了收敛速度更快的负2N 1次幂取样定理(N为任意自然数),其结果比彭教授的结果更加优越.     

On Riesz transforms and maximal functions in the context of Gaussian Harmonic Analysis

The purpose of this paper is twofold. We introduce a general maximal function on the Gaussian setting which dominates the Ornstein-Uhlenbeck maximal operator and prove its weak type by using a covering lemma which is halfway between Besicovitch and Wiener. On the other hand, by taking as a starting point the generalized Cauchy-Riemann equations, we introduce a new class of Gaussian Riesz Transforms. We prove, using the maximal function defined in the first part of the paper, that unlike the ones already studied, these new Riesz Transforms are weak type independently of their orders.

     

Characterization of measures by potentials

Let μ be a measure in a Banach spaceE, f be an even function onR. We consider the potentialg(a)=f _E f(‖x?a‖)dμ(x). The question is as follows: For whichf does the potentialg determine μ uniquely? In this article we give answers in the cases whereE=l _∞ ⁿ and wheref(t)=|t| ^p andE is a finite dimensional Banach space with symmetric analytic norm. Calculating the Fourier transform of the functionf(‖x‖ _∞) we give a new proof of the J. Misiewicz's result that the functionf(‖x‖ _∞) is positive definite only iff is a constant function.     

On the Convergence of the P-Algorithm for One-Dimensional Global Optimization of Smooth Functions

The Wiener process is a widely used statistical model for stochastic global optimization. One of the first optimization algorithms based on a statistical model, the so-called P-algorithm, was based on the Wiener process. Despite many advantages, this process does not give a realistic model for many optimization problems, particularly from the point of view of local behavior. In the present paper, a version of the P-algorithm is constructed based on a stochastic process with smooth sampling functions. It is shown that, in such a case, the algorithm has a better convergence rate than in the case of the Wiener process. A similar convergence rate is proved for a combination of the Wiener model-based P-algorithm with quadratic fit-based local search.     

On the distribution of zeros of a sequence of entire functions approaching the Riemann zeta function

In this paper we study the distribution of zeros of each entire function of the sequence , which approaches the Riemann zeta function for Rez<−1, and is closely related to the solutions of the functional equations f(z)+f(2z)+?+f(nz)=0. We determine the density of the zeros of Gn(z) on the critical strip where they are situated by using almost-periodic functions techniques. Furthermore, by using a theorem of Kronecker, we also establish a formula for the number of zeros of Gn(z) inside certain rectangles in the critical strip.     

On the law of the iterated logarithm for Gaussian processes

We present some optimal conditions for the compact law of the iterated logarithm of a sequence of jointly Gaussian processes in different situations. We also discuss the local law of the iterated logarithm for Gaussian processes indexed by arbitrary index sets, in particular for self-similar Gaussian processes. We apply these results to obtain the law of the iterated logarithm for compositions of Gaussian processes. Research partially supported by NSF Grant DMS-93-02583.     

On the commutant of operators of multiplication by univalent functions

Let be a certain Banach space consisting of continuous functions defined on the open unit disk. Let be a univalent function defined on , and assume that denotes the operator of multiplication by . We characterize the structure of the operator such that . We show that for some function in . We also characterize the commutant of under certain conditions.