Asymptotical analysis of Cramer‘s transforms for extrema

Asymptotic of normalized extrema of independent identically distributed random variables is analyzed. Normalization – Cramer‘s transforms [2].

Dynamics of cholera epidemic models in fluctuating environments

Based on our deterministic models for cholera epidemics, we propose a stochastic model for cholera epidemics to incorporate environmental fluctuations which is a nonlinear system of Itô stochastic differential equations. We conduct an asymptotical analysis of dynamical behaviors for the model. The basic stochastic reproduction value [Formula: see text] is defined in terms of the basic reproduction number [Formula: see text] for the corresponding deterministic model and noise intensities. The basic stochastic reproduction value determines the dynamical patterns of the stochastic model. When [Formula: see text], the cholera infection will extinct within finite periods of time almost surely. When [Formula: see text], the cholera infection will persist most of time, and there exists a unique stationary ergodic distribution to which all solutions of the stochastic model will approach almost surely as noise intensities are bounded. When the basic reproduction number [Formula: see text] for the corresponding deterministic model is greater than 1, and the noise intensities are large enough such that [Formula: see text], the cholera infection is suppressed by environmental noises. We carry out numerical simulations to illustrate our analysis, and to compare with the corresponding deterministic model. Biological implications are pointed out.

Numerical Approaches for Linear Left-invariant Diffusions onSE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

Left-invariant PDE-evolutions on the roto-translation groupSE(2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent toSE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

PIECEWISE UNIFORM PRODUCT POLAR QUANTIZATION

In this paper, simple and complete asymptotical analysis is given for a piecewise uniform product polar quantizer (PUPPQ) with respect to mean-square error (MSE) i.e. distortion (D). PUPPQ is based on uniform product polar quantizers. PUPPQ consists of L different uniform product polar quantizers. Uniform product polar quantizer conditions for optimality and all main equations for number of phase divisions and optimal number of levels for each partition are presented. These systems, although not optimal, may have asymptotic performance close to the optimum. Furthermore, the analysis and implementation can be simpler than those of optimal systems. PUPPQ has implementation complexity between optimal nonuniform polar quantization (NPQ) and uniform product polar quantization (UPPQ). The gain of PUPPQ over optimum uniform product polar quantization is also obtained (2 dB for the rate of 8 bits/sample).