FINITE DIFFERENCE METHODS FOR SOLVING DIFFERENTIAL …
The goal of this course is to provide numerical analysis background for finite difference methods for solving partial differential equations. The focuses are the stability and convergence theory. The partial differential equations to be discussed include •parabolic equations, •elliptic equations, •hyperbolic conservation laws.
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MATHEMATICAL MODELING AND ORDINARY …
www.math.ntu.edu.twMATHEMATICAL MODELING AND ORDINARY DIFFERENTIAL EQUATIONS I-Liang Chern Department of Mathematics National Taiwan University 2007, 2015 January 6, 2016
Methods in Applied Mathematics - 國立臺灣大學
www.math.ntu.edu.twChapter 1 Dimensional Analysis 1.1 The program of applied mathematics The program of applied mathematics consists of the following procedures. Observe a phenomenon, determine variables (quantities) that we are concerned with.
Lecture 2 : Basics of Probability Theory
www.math.ntu.edu.twLecture 2 : Basics of Probability Theory When an experiment is performed, the realization of the experiment is an outcome in the sample space. If the experiment is performed a number of times, different outcomes may occur each time or some outcomes may repeat. This “frequency of occurrence” of an outcome can be thought of as a probability.
Partial Differential Equation Toolbox User's Guide
www.math.ntu.edu.twFor Use with MATLAB® User’s Guide Computer Solutions Europe AB. How to Contact The MathWorks: 508-647-7000 Phone ... A nonlinear solver is available for the nonlinear elliptic PDE ... solver for Poisson’s equation on a rectangular grid is available. – ∇⋅()cu∇+au = f ...
第 9 章 無窮級數 (Infinite Series)
www.math.ntu.edu.tw第9 章無窮級數 9.1 數列 (4) 若an ‚ an+1,8n, 則稱fang 為非上升數列 (nondecreasing sequence)。 (5) fang 為上升或下降數列, 則統稱為單調 (monotonic)。 (6) 若存在 N, 使得 an < an+1,8 n > N, 則稱fang 為終極上升 (ultimately increasing) 數列 定義 9.1.19. (1) 若存在 M, 使得 an • M,8n, 則稱fang 為有上界 (bounded above), 且 M
Chapter 2. Order Statistics - 國立臺灣大學
www.math.ntu.edu.twcontinuous. Moreover, the above decomposition is unique. Let λ denote the Lebesgue measure on B, the σ-field of Borel sets in R. It follows from the Lebesgue decomposition theorem that we can write F c(x) = βF s(x)+(1−β)F ac(x) where 0 ≤ β ≤ 1, F s is singular with respect to λ, and F ac is absolutely continuous with respect to λ.
1 Order Statistics - 國立臺灣大學
www.math.ntu.edu.tw1 Order Statistics Definition The order statistics of a random sample X1,...,Xn are the sample values placed in ascending order. They are denoted by X(1),...,X(n). The order statistics are random variables that satisfy X(1) ≤ X(2) ≤ ··· ≤ X(n). The following are some statistics that are easily defined in terms of the order statistics.
3.2.5 Negative Binomial Distribution - 國立臺灣大學
www.math.ntu.edu.twThe negative binomial distribution is sometimes defined in terms of the random variable Y =number of failures before rth success. This formulation is statistically equivalent to the one given above in terms of X =trial at which the rth success occurs, since Y = X −r. The
5.5.3 Convergence in Distribution
www.math.ntu.edu.twIf Xn → X in distribution and Yn → a, a constant, in probability, then (a) YnXn → aX in distribution. (b) Xn +Yn → X +a in distribution. Example (Normal approximation with estimated variance) Suppose that √ n(X¯ n −µ) σ → N(0,1), but the value σ is unknown. We know Sn → σ in probability. By Exercise 5.32, σ/Sn → 1 in ...
4.2 Conditional Distributions and Independence
www.math.ntu.edu.tw4.2 Conditional Distributions and Independence Definition 4.2.1 Let (X,Y) be a discrete bivariate random vector with joint pmf f(x,y) and marginal pmfs fX(x) and fY (y). For any x such that P(X = x) = fX(x) > 0, the conditional pmf of Y given that X = …
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