In this talk, a reduced-order modeling methodology based on centroidal Voronoi tessellations (CVT's)is introduced. CVT's are special Voronoi tessellations for which the generators of the Voronoi diagram are also the centers of mass (means) of the corresponding Voronoi cells. The discrete data sets, CVT's are closely related to the h-means clustering techniques. Even with the use of good mesh generators, discretization schemes, and solution algorithms, the computational simulation of complex, turbulent, or chaotic systems still remains a formidable endeavor. For example, typical finite element codes may require many thousands of degrees of freedom for the accurate simulation of fluid flows. The situation is even worse for optimization problems for which multiple solutions of the complex state system are usually required or in feedback control problems for which real-time solutions of the complex state system are needed. There hava been many studies devoted to the development, testing, and use of reduced-order models for complex systems such as unsteady fluid flows. The types of reduced-ordered models that we study are those attempt to determine accurate approximate solutions of a complex system using very few degrees of freedom. To do so, such models have to use basis functions that are in some way intimately connected to the problem being approximated. Once a very low-dimensional reduced basis has been determined, one can employ it to solve the complex system by applying, e.g., a Galerkin method. In general, reduced bases are globally supported so that the discrete systems are dense; however, if the reduced basis is of very low dimension, one does not care about the lack of sparsity in the discrete system. A discussion of reduced-ordering modeling for complex systems such as fluid flows is given to provide a context for the application of reduced-order bases. Then, detailed descriptions of CVT-based reduced-order bases and how they can be constructed of complex systems are given. Subsequently, some concrete incompressible flow examples are used to illustrate the construction and use of CVT-based reduced-order bases. The CVT-based reduced-order modeling methodology is shown to be effective for these examples and is also shown to be inexpensive to apply compared to other reduced-order methods.

In this work we consider the mathematical formulation and numerical resolution of the linear feedback control problem for Boussinesq equations. The controlled Boussinesq equations is given by $$\frac{{\partial}u}{{\partial}t}-{\nu}{\Delta}u+(u{\cdot}{\nabla}u+{\nabla}p={\beta}{\theta}g+f+F\;\;in\;(0,\;T){\times}\;{\Omega}$$, $${\nabla}{\cdot}u=0\;\;in\;(0,\;T){\times}{\Omega}$$, $$u|_{{\partial}{\Omega}=0,\;u(0,x)=\;u_0(x)$$ $$\frac{{\partial}{\theta}}{{\partial}t}-k{\Delta}{\theta}+(u{\cdot}){\theta}={\tau}+T,\;\;in(0,\;T){\times}{\Omega}$$ $${\theta}|_{{\partial}{\Omega}=0,\;\;{\theta}(0,X)={\theta}_0(X)$$, where $\Omega$ is a bounded open set in $R^{n}$, n=2 or 3 with a $C^{\infty}$ boundary ${\partial}{\Omega}$. The control is achieved by means of a linear feedback law relating the body forces to the velocity and temperature field, i.e., $$f=-{\gamma}_1(u-U),\;\;{\tau}=-{\gamma}_2({\theta}-{\Theta}}$$ where (U,$\Theta$) are target velocity and temperature. We show that the unsteady solutions to Boussinesq equations are stabilizable by internal controllers with exponential decaying property. In order to compute (approximations to) solution, semi discrete-in-time and full space-time discrete approximations are also studied. We prove that the difference between the solution of the discrete problem and the target solution decay to zero exponentially for sufficiently small time step.

We studied closed set-valued Choquet integrals in two papers(1997, 2000) and convergence theorems under some sufficient conditions in two papers(2003), for examples : (i) convergence theorems for monotone convergent sequences of Choquet integrably bounded closed set-valued functions, (ii) covergence theorems for the upper limit and the lower limit of a sequence of Choquet integrably bounded closed set-valued functions. In this presentation, we consider fuzzy number-valued functions and define Choquet integrals of fuzzy number-valued functions. But these concepts of fuzzy number-valued Choquet inetgrals are all based on the corresponding results of interval-valued Choquet integrals. We also discuss their properties which are positively homogeneous and monotonicity of fuzzy number-valued Choquet integrals. Furthermore, we will prove convergence theorems for fuzzy number-valued Choquet integrals. They will be used in the following applications : (1) Subjectively probability and expectation utility without additivity associated with fuzzy events as in Choquet integrable fuzzy number-valued functions, (2) Capacity measure which are presented by comonotonically additive fuzzy number-valued functionals, and (3) Ambiguity measure related with fuzzy number-valued fuzzy inference.

Let $\Omega$ be a bounded, open, and polygonal domain in $R^2$ with re-entrant corners. We consider the following Partial Differential Equations: $$(I-\nabla\nabla\cdot+\nabla^{\bot}\nabla\times)u\;=\;f\;in\;\Omega$$, $$n\cdotu\;0\;0\;on\;{\Gamma}_{N}$$, $${\nabla}{\times}u\;=\;0\;on\;{\Gamma}_{N}$$, $$\tau{\cdot}u\;=\;0\;on\;{\Gamma}_{D}$$, $$\nabla{\cdot}u\;=\;0\;on\;{\Gamma}_{D}$$ where the symbol $\nabla\cdot$ and $\nabla$ stand for the divergence and gradient operators, respectively; $f{\in}L^2(\Omega)^2$ is a given vector function, $\partial\Omega=\Gamma_{D}\cup\Gamma_{N}$ is the partition of the boundary of $\Omega$; nis the outward unit vector normal to the boundary and $\tau$represents the unit vector tangent to the boundary oriented counterclockwise. For simplicity, assume that both $\Gamma_{D}$ and $\Gamma_{N}$ are nonempty. Denote the curl operator in $R^2$ by $$\nabla\times\;=\;(-{\partial}_2,{\partial}_1$$ and its formal adjoint by $${\nabla}^{\bot}\;=\;({-{\partial}_1}^{{\partial}_2}$$ Consider a weak formulation(WF): Find $u\;\in\;V$ such that $$a(u,v):=(u,v)+(\nabla{\cdot}u,\nabla{\cdot}v)+(\nabla{\times}u,\nabla{\times}V)=(f,v),\;A\;v{\in}V$$. (2) We assume there is only one singular corner. There are many methods to deal with the domain singularities. We introduce them shortly and we suggest a new Finite Element Methods by using Singular representation for the solution.

Option pricing theory developed by Black and Sholes depends on an arbitrage opportunity argument. An investor can exactly replicate the returns to any option on that stock by continuously adjusting a portfolio consisting of a stock and a riskless bond. The value of the option equal the value of the replicating portfolio. However, transactions costs invalidate the Black-Sholes arbitrage argument for option pricing, since continuous revision implies infinite trading, Discrete revision using Black-Sholes deltas generates errors which are correlated with the market, and do not approach zero with more frequent revision when transactions costs are included. Stochastic calculus serves as a fundamental tool in the mathematical finance. We closely look at the utility maximization theory which is one of the main option valuation methods. We also see that how the stochastic optimal control problems and their solution methods are applied to the theory.

오늘날 단일 슈퍼컴퓨터로는 처리가 불가능한 거대한 문제들의 해법이 시도되고 있는데, 이들은 지리적으로 분산된 슈퍼컴퓨터, 데이터베이스, 과학장비 및 디스플레이 장치 등을 초고속 통신망으로 연결한 GRID 환경에서 효과적으로 실행시킬 수 있다. GRID는 1990년대 중반 과학 및 공학용 분산 컴퓨팅의 연구 과정에서 등장한 것으로, 점차 응용분야가 넓어지고 있다. 그러나 GRID 같은 분산 환경은 기존의 단일 병렬 시스템과는 많은 점에서 다르며 이전의 기술들을 그대로 적용하기에는 무리가 있다. 기존 병렬 시스템에서는 주로 동기 알고리즘(synchronous algorithm)이 사용되는데, 직렬 연산과 같은 결과를 얻기 위해 동기화(synchronization)가 필요하며, 부하 균형이 필수적이다. 그러나 부하 균형은 이질 클러스터(heterogeneous cluster)처럼 프로세서들의 성능이 서로 다르거나, 지리적으로 분산된 계산자원을 사용하는 GRID 환경에서는 이기종의 문제뿐 아니라 네트워크를 통한 메시지의 전송 지연 등으로 유휴시간이 길어질 수밖에 없다. 이처럼 동기화의 필요성에 의한 연산의 지연을 해결하는 하나의 방안으로 비동기 반복법(asynchronous iteration)이 나왔으며, 지금도 활발히 연구되고 있다. 이는 알고리즘의 동기점을 가능한 한 제거함으로써 빠른 프로세서의 유휴 시간을 줄이는 것이 목적이다. 즉 비동기 알고리즘에서는, 각 프로세서는 다른 프로세서로부터 갱신된 데이터가 올 때까지 기다리지 않고 계속 다음 작업을 수행해 나간다. 따라서 동시에 갱신된 데이터를 교환한 후 다음 단계로 진행하는 동기 알고리즘에 비해, 미처 갱신되지 않은 데이터를 사용하는 경우가 많으므로 전체적으로는 연산량 대비의 수렴 속도는 느릴 수 있다 그러나 각 프로세서는 거의 유휴 시간이 없이 연산을 수행하므로 wall clock time은 동기 알고리즘보다 적게 걸리며, 때로는 50%까지 빠른 결과도 보고되고 있다 그러나 현재까지의 연구는 모두 어떤 수렴조건을 만족하는 선형 시스템의 해법에 국한되어 있으며 비교적 구현하기 쉬운 공유 메모리 시스템에서의 연구만 보고되어 있다. 본 연구에서는 행렬의 주요 고유쌍을 구하는 데 있어 비동기 반복법의 적용 가능성을 타진하기 위해 우선 이론적으로 단순한 멱승법을 사용하여 실험하였고 그 결과 순수한 비동기 반복법은 수렴하기 어렵다는 결론을 얻었다 그리하여 동기 알고리즘에 비동기적 요소를 추가한 혼합 병렬 알고리즘을 제안하고, MPI(Message Passing Interface)를 사용하여 수원대학교의 Hydra cluster에서 구현하였다. 그 결과 특정 노드의 성능이 다른 것에 비해 현저하게 떨어질 때 전체적인 알고리즘의 수렴 속도가 떨어지는 것을 상당히 완화할 수 있음이 밝혀졌다.

Given a sequence ｛ $X_{n}$｝ of independent and identically distributed random variables with F, a sequential procedure for the p-th quantile ξ$_{P}$= $F^{-1}$ (P), 0$\beta$-protection. Some asymptotic properties for the proposed procedure and of an involved stopping time are proved: asymptotic consistency, asymptotic efficiency and asymptotic normality. From one of the results an effect of smoothing based on kernel functions is discussed. The results are also extended to the contaminated case.e.e.

We provide some recent results of approximation algorithms for solving Markov Games and discuss their applications to problems that arise in Computer Science. We consider a receding horizon approach as an approximate solution to two-person zero-sum Markov games with an infinite horizon discounted cost criterion. We present error bounds from the optimal equilibrium value of the game when both players take “correlated” receding horizon policies that are based on exact or approximate solutions of receding finite horizon subgames. Motivated by the worst-case optimal control of queueing systems by Altman, we then analyze error bounds when the minimizer plays the (approximate) receding horizon control and the maximizer plays the worst case policy. We give two heuristic examples of the approximate receding horizon control. We extend “parallel rollout” and “hindsight optimization” into the Markov game setting within the framework of the approximate receding horizon approach and analyze their performances. From the parallel rollout approach, the minimizing player seeks to combine dynamically multiple heuristic policies in a set to improve the performances of all of the heuristic policies simultaneously under the guess that the maximizing player has chosen a fixed worst-case policy. Given $\varepsilon$>0, we give the value of the receding horizon which guarantees that the parallel rollout policy with the horizon played by the minimizer “dominates” any heuristic policy in the set by $\varepsilon$, From the hindsight optimization approach, the minimizing player makes a decision based on his expected optimal hindsight performance over a finite horizon. We finally discuss practical implementations of the receding horizon approaches via simulation and applications.

WLAN의 매체 특성상 AP beacon영역 내의 모든 STA들은 다른 STA의 송수신 데이터 내용에 접근할 수 있다. 따라서 상호 또는 그룹 간의 데이터프라이버시와 상호인증 서비스는 무선 랜의 중요한 이슈중의 하나이다. 무선랜을 통한 네트워크 접속 보안으로는 사용자와 AP 사이의 무선 접속구간 보안과 AP와 AS사이의 유선 구간 보안으로 정의되며, 상대적으로 취약한 무선 구간 보안이 초점이 된다. 현재 무선 구간 보안에는 WEP이 사용된다. 그러나 WEP 방식은 WEP 키와 IV 크기가 작고, 노출된 공유키를 사용하며, 암호 알고리즘(RC4)와 무결성 알고리즘(CRC-32)이 근본적으로 취약하다. 이러한 문제에 대한 해결 방법으로 IEEE 802.11i는 두 가지 접근 방식을 채택하였다. 하나는 WEP의 보안 문제점을 소프트웨어적으로 개선한 TKIP이고 다른 하나는 기존의 WEP과는 하드웨어적으로 상이한 AES을 기반으로 한 CCMP이다. 이 논문에서는 각 알고리즘에 대한 키의 흐름 및 그 안전성을 분석하였다. 이러한 방법을 통해 WEP 구조의 보안상의 취약점을 확인하고, TKIP이 WEP을 대체할 수 있을 만큼의 안전성을 갖는지를 검증한다. 또한 고려될 수 있는 공격 모델을 제시하고, 이에 대하여 알고리즘에 부가적으로 요구되는 보완점에 대해 논한다.

PRF(Pseudo Random Function)에 대한 랜덤성 검증은 pre-computation공격에 대해 알고리즘이 특별한 통계적 약점이 없이 적절하게 개발되었는지를 평가할 수 있다. 이 논문에서는 NIST에서 실시한 AES 후보 알고리즘 랜덤성 평가 기준을 적용하여 IEEE의 802.11i Draft에서 인증자와 요청자가 비밀키(PTK, GTK)를 생성하는데 사용되는 PRF의 랜덤성을 검증하였다. 랜덤성 테스트를 위해 표본 수는 300개, 표본 길이는 2$^{20}$ (= 1,048,576)으로 검정 표본을 생성하고, 유의 수준은 0.01로 선택하였다. 랜덤성 검증 방법으로는 NIST의 16가지 통계 테스트를 사용하였다.

We show how the minimization can be used to solve the quadratic matrix equattion. We then compare two different types of conjugate gradient method and show Polak and Ribire version converge more rapidly than Fletcher and Reeves version in several examples.

We propose a method of surface extension method using several functions. Interpolation theory is well developed in curve and surface. But extrapolation theory is not well developed because it is not unique outside the useful domain. It requires continuous, first derivative, second derivative continuous extension for matching in NC(Numerical Control) machine. In the past, we generate data outside the useful area and refit those data using least squares method. this has some problems which have some errors within the useful area. We keep the useful area and extend the unuseful area by a function

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX= Y. An interpolating operator for n-operators satisfies the equation AXi= Yi, for i = 1,2,...,n, In this article, we showed the following : Let H be a Hilbert space and let L be a subspace lattice on H. Let X and Y be operators acting on H. Assume that rangeX is dense in H. Then the following statements are equivalent : (1) There exists an operator A in AlgL such that AX = Y, A$\^$*/=A and every E in L reduces A. (2) sup｛(equation omitted) : n $\in$ N f$\sub$I/ $\in$ H and E$\sub$I/ $\in$ L｝<$\infty$ and = for all E in L and all f, g in H.

In this paper, we describe how to approximate the solutions of partial differential equations by bicubic spline functions whose interpolation errors at non-grid points are smaller in general than those by linear interpolations of the original solution at grid points. We show that the bicubic spline function can be represented exactly or approximately by a fuzzy system, and that the resulting fuzzy rule table shows the contours of the solution function. Thus, the fuzzy rule table is identified as a digital image and the contours in the rule table provide approximate contours of the solution of partial differential equations. Several illustrative examples are included.

In this paper, we study the convergence of relaxed multisplitting and relaxed nonstationary two-stage multisplitting methods associated with a multisplitting which is obtained from the ILU factorizations for solving a linear system whose coefficient matrix is an H-matrix. Also, parallel performance results of relaxed nonstaionary two-stage multisplitting method using ILU factorizations as inner splittings on the IBM p690 supercomputer are provided to analyze theoretical results.

In this paper, we are to find the condition that a two-dimensional Finsler space with Matsumoto metric satisfying L(${\alpha}$,${\beta}$)=${\alpha}$$^2$/(${\alpha}$-${\beta}$) be a Landsberg space and the differential equations of geodesics.

The boundary of a typical period-2 component in the degree-3 bifurcation set is formulated by a parametrization of its image which is the unit circle under the multiplier map, Some properties on the geometry of the boundary are investigated including the root point, the cusp, the component center and the length as well as the area bounded by the boundary curve. An algorithm drawing the boundary curve with Mathematica codes is proposed and its implementation exhibits a good agreement with the analysis presented here.

We prove that m-step walks and self-avoiding walks on the 2D triangle lattices can be uniquely characterized (canonized) with no more than m Euclidian distances. We also demonstrate that these canonical distances can be obtained with O(n) physical measurements.

The theory of fuzzy random variables and fuzzy stochastic processes has been received much attentions in recent years. But convergence in distribution for fuzzy random variables has not established yet. In this talk, we restrict our concerns to level-wise continuous fuzzy random variables and obtain some characterizations of its tightness and convergence in distribution.

Sufficient conditions are given under which a generalized class of kernel-type estimators allows asymptotic approximation On the modulus of continuity This generalized class includes sample distribution function, kernel-type estimator of density function, and an estimator that may apply to the censored case. In addition, an application is given to asymptotic normality of recursive density estimators of density function at an unknown point.

미분 가능한 함수가 독립변수의 각 점에서 미분계수를 가지듯이 가장 일반화된 Cantor집합의 각 점에서 weak local dimension 을 갖는다. 이러한 weak local dimension 은 두 가지가 있는데 weak lower local dimension 과 weak upper local dimension 이 있다 weak lower local dimension 은 국소적인 의미로 perturbed Cantor 집합의 lower Cantor dimension 이고 Hausdorff dimension 과 관련이 있다. weak upper local dimension 은 국소적인 의미로 perturbed Cantor 집합의 upper Cantor dimension 이고 packing dimension 과 관련이 있다. 이때 각 점에 대응하는 유관한 측도는 quasi-self-similar measure 이며 그 점의 weak lower local dimension 이 s 이면 그 점의 s-차원 quasi-self-similar measure 의 lower local dimension 이 s 가 된다. 마찬가지로 그 점의 weak upper local dimension 이 s 이면 그 점의 s-차원 quasi-self-similar measure 의 upper local dimension 이 s 가 된다. 따라서 이와 같은 사실을 이용하면 가장 일반화된 Cantor집합의 각 점에서의 weak local dimension 을 이용하여 그 집합의 Hausdorff 또는 packing 차원의 정보를 얻을 수 있을 뿐 더러 weak local dimension 을 이용한 spectrum 을 또한 구할 수 있다. 한편 weak local dimension 과 유관한 quasi-self-similar measure 는 집합의 spectrum을 생성하며 이 spectrum 을 이루는 각 부분집합의 차원에 대하여 보다 좋은 정보를 주는 transformed dimension 과 또 다른 관련을 갖게 된다.

The Markovian arrival process with marked transitions (MMAP) is useful in modeling input processes of stochastic system with several types. Especially, the MMAP can be used to model the phenomena where the correlation of different types is considered. In this talk we discuss modeling issues for the queue with three types of customers; ordinary customers, negative customers and disasters which are correlated by using MMAP. We also present the recent results and further studies.

We propose some properties of Bayesian fuzzy hypotheses testing by revision for prior possibility distribution and posterior possibility distribution using weighted fuzzy hypotheses versus on with loss function.

A subset S of vertices of a graph G is independent if no two vertices of S are adjacent by an edge in G. Also we say that S is maximal independent if it is contained In no larger independent set in G. A planted plane tree is a tree that is embedded in the plane and rooted at an end-vertex. A (k＋1) -valent tree is a planted plane tree in which each vertex has degree one or (k＋1). We classify maximal independent sets of (k＋1) -valent trees into two groups, namely, type A and type B maximal independent sets and consider specific independent sets of these trees. We study relations among these three types of independent sets. Using the relations, we count the number of all maximal independent sets of (k＋1) -valent trees with n vertices of degree (k＋1).

미분 방정식의 수치적 해를 나타내는 방법 중 예측자-수정자 방법(predictor-corrector method)으로 알려진 Adams-Bashford-Moulton 방법은 다단계 방법을 이용하기 때문에 일단계 방법에 비하여 훨씬 좋은 수치적인 결과를 보여주고 있다. 이제, 이 다단계 방법에 오차제어 변수를 첨가한 새로운 형태의 예측자-수정자 방법을 제시하고 안정적인 해를 구할 수 있는 오차 제어 변수의 범위를 확인한다. 또한, 새로운 형태의 예측자-수정자 방법이 기존의 방법에 비하여 미분 방정식의 해에 대한 오차를 줄일 수 있는 방법임을 수치적인 결과를 통하여 검증한다.

We study averaging properties in Banach spaces, First, we seek an averaging property equivalent to the reflexivity in Banach spaces. Second, we investigate averaging properties in Banach space using the Brunet-Sucheston's spreading model.

The purpose of this paper is to solve the generalized Hyers-Ulam stability problem for a cubic functional equation 8f(x-y/2)＋8f(y-x/2)＋2f(x＋y)=9f(x)＋9f(y) on the basis of a direct method.