# Ngô Quốc Anh

## November 18, 2008

### MA5233 – Computational Mathematics

Filed under: Linh Tinh — Ngô Quốc Anh @ 19:25

Lecturer: Weizhu Bao

http://www.math.nus.edu.sg/~bao/teach/ma5233/ma5233.html

Detailed Course Outline

• Introduction (lectures 1-2)
• History and milestone algorithms lecture 1
• Basic concepts:
• Numerical error
• Convergence and order of accuracy
• Stability and conditioning number
• Basic techniques lecture 2
• Asymptotic error expansion
• Richardson extrapolation
• Perturbation analysis
• Some techniques to reduce round-off errors
• Iterative methods for linear systems (lectures 3-4)
• Introduction lecture 3
• Classic iterative methods
• Jacobi Method
• Gauss-Seidel method
• Successive Overrelaxation method (SOR)
• Convergence analysis
• Krylov subspace methods lecture 4
• Steepest descent method
• Conjugate gradient (CG) method
• Preconditioning
• GMRES for nonsymmetric matrix
• Methods for nonlinear systems (lecture 5)
• Introduction lecture 5
• Newton’s method
• Quasi-Newton method
• Convergence analysis
• Numerical methods for ODEs (lectures 6-8)
• Introduction lecture 6
• Single-step method
• Euler methods
• Runge-Kutta methods
• Stability and convergence analysis
• Multi-step methods lecture 7
• Extension to first order systems and high order equations lecture 8
• Methods for stiff systems
• Numerical methods for PDEs (lectures 9-12)

My notes: Runge-Kutta Methods, Multi-step Methods

## November 17, 2008

### Illustration of some well-known PDEs via Matlab

Filed under: Linh Tinh, Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 17:11

In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces. It is particularly famous as the prototypical example of an exactly solvable model, that is, a non-linear partial differential equation whose solutions can be exactly and precisely specified. The solutions in turn include prototypical examples of solitons. KdV can be solved by means of the inverse scattering transform. The mathematical theory behind the KdV equation is rich and interesting, and, in the broad sense, is a topic of active mathematical research. The equation is named for Diederik Korteweg and Gustav de Vries who studied it in (Korteweg & de Vries 1895), though the equation first appears in (Boussinesq 1877, p. 360).

## November 6, 2008

### Differentiability via weird argument

Filed under: Các Bài Tập Nhỏ, Giải Tích 1, Giải Tích 2 — Ngô Quốc Anh @ 16:37

Question. Suppose that $f:\mathbb R \to \mathbb R$  is such that $f$ is continuous and $|f|$ is differentiable. Should $f$ also be differentiable?

Proof. Let $g=|f|$. Since $y=f(x)$ satisfies the differentiable equation

$y^2=g(x)^2$,

$f$ is differentiable at all points with $f(x) \neq 0$ by the implicit function theorem and

$2yy'=2gg'$,

and

$\displaystyle f'(x)=\frac{g(x)g'(x)}{f(x)}$.

Where $f(x)=0$, we note that $g$ has a minimum and $g'(x)=0$. This gives

$\displaystyle\lim_{t\to 0}\left|\frac{f(x+t)}{t}\right|=0$

and $f$ is differentiable by the definition.

Comment. The only points of interests are the zeros of $f$ since $f$ has the same sign in some neighborhood of points which are not roots. So there goes half the work. Geometrically it doesn’t make sense for $|f|$ to have anything but derivative 0 at roots of $f$ and this can be verified by taking left hand and right hand limits of the quotient.

## November 5, 2008

### Elementary Analysis through Examples and Exercises (Texts in the Mathematical Sciences)

Filed under: Sách Hay — Ngô Quốc Anh @ 14:03

This volume on mathematical analysis offers a comprehensive set of both traditional and new examples and exercises with detailed solutions. It includes many topics important in current research that are not found in other basic analysis books.

It begins with a comparison of real numbers viewed as a totally ordered field or, alternatively, constructed using the Dedekind cut method. Properties surrounding real numbers are explored and many interesting relationships are proven using mathematical induction. Functions are then developed with special emphasis on topics such as asymptotics, n-levels of composition and periodicity of certain functions. Sequences and series for both the discrete and continuous case are concurrently developed showing contrast wherever possible. The order of growth for sequences diverging to infinity is incorporated with its counterpart given for functions. The usual properties of functions, together with their limit theory including the differential calculus are compared carefully with examples illustrating their fundamental properties. The graphs of these functions are then closely studied giving as much detail as possible for a wide variety of functions.

Audience: Undergraduate and graduate students in all areas of mathematical science and its applications.

• Hardcover: 332 pages
• Publisher: Springer; 1 edition (June 30, 1995)
• Language: English
• ISBN-10: 079233597X
• ISBN-13: 978-0792335979
• Product Dimensions: 9.2 x 6.1 x 0.8 inches

Tôi có bản copy của cuốn sách này..

### Complex Analysis through Examples and Exercises (Texts in the Mathematical Sciences)

Filed under: Sách Hay — Ngô Quốc Anh @ 14:00

This volume on complex analysis offers an exposition of the theory of complex analysis via a comprehensive set of examples and exercises. The book is self-contained and the exposition of new notions and methods is introduced step by step. A minimal amount of expository theory is included at the beginning of each section in the Preliminaries, with maximum effort placed on well-selected examples and exercises capturing the essence of the material.

The examples contain complete solutions and serve as a model for solving similar problems given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are given. Special sections contain so-called Composite Examples which consist of combinations of different types of examples explaining some problems completely and giving the reader an opportunity to check all his previously accepted knowledge.

Audience: This volume is intended for undergraduate and graduate students in mathematics, physics, technology and economics interested in complex analysis.

• Hardcover: 352 pages
• Publisher: Springer; 1 edition (July 31, 1999)
• Language: English
• ISBN-10: 0792357876
• ISBN-13: 978-0792357872
• Product Dimensions: 9.4 x 6.1 x 1 inches

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### Exercises in Functional Analysis (Texts in the Mathematical Sciences)

Filed under: Sách Hay — Ngô Quốc Anh @ 13:58

This book of exercises in Functional Analysis contains almost 450 exercises (all with complete solutions), providing supplementary examples, counter-examples and applications for the basic notions usually presented in an introductory course in Functional Analysis.

It contains three parts. The first one contains exercises on the general properties for sets in normed spaces, linear bounded operators on normed spaces, reflexivity, compactness in normed spaces, and on the basic principles in Functional Analysis: the Hahn-Banach theorem, the Uniform Boundedness Principle, the Open Mapping and the Closed Graph theorems.

The second one contains exercises on the general theory of Hilbert spaces, the Riesz representation theorem, orthogonality in Hilbert spaces, the projection theorem and linear bounded operators on Hilbert spaces. The third one deals with linear topological spaces, and includes a large number of exercises on the weak topologies.

• Hardcover: 451 pages
• Publisher: Springer; 1 edition (September 30, 2003)
• Language: English
• ISBN-10: 1402015607
• ISBN-13: 978-1402015601
• Product Dimensions: 9.2 x 6.3 x 1.3 inches

Tôi có bản photocopy (không phải eBook) của sách này…