Plates are common structural elements of most engineering structures, including aerospace, automotive, and civil engineering structures. The study of plates from theoretical perspective as well as experimental viewpoint is fundamental to understanding of the behavior of such structures. The dynamic characteristics of plates, such as natural vibrations, transient responses for the external forces and so on, are especially of importance in actual environments. In this paper, we conside the envelope surface created by the vibrations of a square plate on a weakly nonliner elastic foundation and analyze the stability of the uniform solution of the governing equation for the envelope surface. We derive the two-dimensional equation that governs the spatial and temporal evolution of the envelope surface on cubic nonlinear elastic foundation. The fact that the governing equation becomes the quintic nonlinear Schrödinger equation is shown. Also we obtain the stability condition of the uniform solution of the quintic nonlinear Schrödinger equation.

A novel quasi-statistical approach to improve the quality of triangular meshes is presented. The present method is based on modeling of an event of the mesh improvement. This event is modeled via modeling of a discrete random variable. The random variable is modeled in a tangent plane of each local domain of the mesh. One domain collects several elements with a common point. Values of random variable are calculated by modeling formula according to the initial sampling data of the projected elements with respect to all neighbors of the domain. Geometrical equivalent called potential form is constructed for each element of the domain with a mesh quality parameter value equal to the modeled numerical value. Such potential forms create potential centers of the domain. Averaging the coordinates of potential centers of the domain gives a new central point position. After geometrical realization over the entire mesh, the shapes of triangular elements are changed according to the normal distribution. It is shown experimentally that the mean of the final mesh is better than the initial one in most cases, so the event of the mesh improvement is likely occurred. Moreover, projection onto a local tangent plane included in the algorithm allows preservation of the model volume enclosed by the surface mesh. The implementation results are presented to demonstrate the functionality of the method. Our approach can provide a flexible tool for the development of mesh improvement algorithms, creating better-input parameters for the triangular meshes and other kinds of meshes intended to be applied in finite element analysis or computer graphics.

In many expositions of fuzzy methods, fuzzy techniques are described as an alternative to a more traditional statistical approach. In this paper, we present a class of fuzzy statistical decision process in which testing hypothesis can be naturally reformulated in terms of interval-valued statistics. We provide the definitions of fuzzy mean, fuzzy distance as well as investigation of their related properties. We also give some empirical examples to illustrate the techniques and to analyze fuzzy data. Empirical studies show that fuzzy hypothesis testing with soft computing for interval data are more realistic and reasonable in the social science research. Finally certain comments are suggested for the further studies. We hope that this reformation will make the corresponding fuzzy techniques more acceptable to researchers whose only experience is in using traditional statistical methods.

The z-transform technique is used to investigate the model for distribution of high-tax payers, which is proposed by two of the authors (K. Y and S. M) and others [12]--[14]. Our analysis shows an asymptotic power-law of this model with the exponent $-5/2$ when a total ``mass'' has a certain critical value. Below the critical value, the system exhibits an ordinary critical behavior, and scaling relations hold. Above the threshold, numerical simulations show that a power-law distribution coexists with a huge ``monopolized'' member. It is argued that these behaviors are observed universally in conserved aggregation processes, by analizing an extended model.

Let $P(z)$ be a monic univariate polynomial over $\mathbf{C}$, of degree $n$ and having roots $\zeta_1,\ldots,\zeta_n$. Given approximate roots $z_1,\ldots,z_n$, with $\zeta_i \simeq z_i$ ($i=1,\ldots,n$), we derive a very tight upper bound of $|\zeta_i - z_i|$, by assuming that $\zeta_i$ has no close root. The bound formula has a similarity with Smale's and Smith's formulas. We also derive a lower bound of $|\zeta_i - z_i|$ and a lower bound of $\min\{|\zeta_j - z_i|\mid j \neq i\}$.