How a minimal surface leaves a thin obstacle

We prove the optimal regularity and a detailed analysis of the free boundary of the solutions to the thin obstacle problem for nonparametric minimal surfaces with flat obstacles.     

MOP/GP models for machine learning

Techniques for machine learning have been extensively studied in recent years as effective tools in data mining. Although there have been several approaches to machine learning, we focus on the mathematical programming (in particular, multi-objective and goal programming; MOP/GP) approaches in this paper. Among them, Support Vector Machine (SVM) is gaining much popularity recently. In pattern classification problems with two class sets, its idea is to find a maximal margin separating hyperplane which gives the greatest separation between the classes in a high dimensional feature space. This task is performed by solving a quadratic programming problem in a traditional formulation, and can be reduced to solving a linear programming in another formulation. However, the idea of maximal margin separation is not quite new: in the 1960s the multi-surface method (MSM) was suggested by Mangasarian. In the 1980s, linear classifiers using goal programming were developed extensively.This paper presents an overview on how effectively MOP/GP techniques can be applied to machine learning such as SVM, and discusses their problems.     

Convergence in L space for the homogenization problems of elliptic and parabolic equations in the plane

We study the convergence rate of an asymptotic expansion for the elliptic and parabolic operators with rapidly oscillating coefficients. First we propose homogenized expansions which are convolution forms of Green function and given force term of elliptic equation. Then, using local L^p-theory, the growth rate of the perturbation of Green function is found. From the representation of elliptic solution by Green function, we estimate the convergence rate in L^p space of the homogenized expansions to the exact solution. Finally, we consider L2(0,T:H1(Ω)) or L∞(Ω×(0,T)) convergence rate of the first order approximation for parabolic homogenization problems.     

Direct estimates on intersection probabilities of random walks

A study is made of intersection properties of independent random walks ind-dimensional lattice spaceZ ^d . A simple method is developed which makes it possible to estimate intersection probalities of two random walks with killing ratem directly. It is expected that the method can be generalized and extended to other issues.     

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.     

Dynamics of solitary waves over an erodible surface

In this paper, we propose a new approach in the study of the dynamics of solitary waves over an erodible surface consisting of non-cohesive particles transported by a constant unidirectional fluid. Using the multiple scale perturbation method, we show that the system can be described by the Kadomtsev–Petviashvili equation. The solution presented here shows that this mathematical model can be used to explain the dynamical behaviour of sand dunes in the desert.     

On the Uniqueness in the Inverse Conductivity Problem

Let be a smooth domain in R² containing a polygon D. The inverse conductivity problem to the the elliptic equation is considered. We show that D is uniquely determined from boundary measurements corresponding two appropriately chosen Neumann datas.     

On the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank

We show that the Denjoy rank and the Zalcwasser rank are incomparable. We construct for any countable ordinal differentiable functions and such that the Zalcwasser rank and the Kechris-Woodin rank of are but the Denjoy rank of is 2 and the Denjoy rank and the Kechris-Woodin rank of are but the Zalcwasser rank of is 1. We then derive a theorem that shows the surprising behavior of the Denjoy rank, the Kechris-Woodin rank and the Zalcwasser rank.

     

Bounds on exponentials of local number operators in quantum statistical mechanics

We consider the quantum systems of interacting Bose particles confined to a bounded region of the configuration spaces ^v . For a class of superstable interactions we obtain bounds on exponentials of local number operators for any temperature and activity. The method we use is the Wiener integral formalism in statistical mechanics. As a consequence any thermodynamic limit states are entire analytic and locally normal in the CCR algebra. In some cases these are modular states.Research supported in part by a grant from Korean Science Foundation     

A perfectly matched layer approach to the nonlinear Schrödinger wave equations

Absorbing boundary conditions (ABCs) are generally required for simulating waves in unbounded domains. As one of those approaches for designing ABCs, perfectly matched layer (PML) has achieved great success for both linear and nonlinear wave equations. In this paper we apply PML to the nonlinear Schrödinger wave equations. The idea involved is stimulated by the good performance of PML for the linear Schrödinger equation with constant potentials, together with the time-transverse invariant property held by the nonlinear Schrödinger wave equations. Numerical tests demonstrate the effectiveness of our PML approach for both nonlinear Schrödinger equations and some Schrödinger-coupled systems in each spatial dimension.