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  1. M

    A Boundary Value Problem

    I have a BVP of the form u" + f(x)u = g(x) , u(0)=u(1)= 0 where f(x) and g(x) are positive functions. I suspect that u(x) < 0 in the domain 0 < x < 1. How do I go proving this. I have try proving by contradiction. Assuming first u > 0 but I can't deduce that u" > 0 which contradict that u has...
  2. M

    Runge-Kutta Method - Need help with the calculus

    When deriving the Runge-Kutta Method to solve y'=f(x) we need to use Taylor expansion. Hence we need to differentiate the function many times. y'(x)=f(y(x)) y''(x)=f'(y(x))y'(x) = f'(y(x))f(y(x)) y''' = f''(y(x))(f(y(x)),y'(x)) + f'(y(x))f'(y(x))y'(x) I can understand the second...
  3. M

    MATLAB Matlab: Suppressing the output

    I get this example from >>help eval and add a semicolon. for n = 1:12 eval(['M' num2str(n) ' = magic(n)']); end The above commands display all 12 magic square. How do I suppress the output? I only want matlab to assign the variables not display them.
  4. M

    Solution for Sylvester Equation

    Given n by n matrices A, B, C. I know how to solve the Sylvester equation AX + XB + C = 0 using the matlab command >> X=lyap(A,B,C) But how do we solve the extended Sylvester equation AX + XB + CXD + E = 0 ? Either numerical or analytical method I'm willing to learn.
  5. M

    MATLAB Matlab - can't even do a simple arithmatic

    This really surprised me. >> format rat >> 1/3-1/2+1/6 ans = -1/36028797018963968 Even school student knows that the answer is 0. Even format short does not give a correct answer. >> format short >> 1/3-1/2+1/6 ans = -2.7756e-017
  6. M

    Linear pde of order one

    I want to determine whether u=-x^3_1-x_1-\sqrt{3}x_2 is a stabilizing control for the system \begin{array}{cc}\dot{x}_1=x_2\\ \dot{x}_2=x^3_1+u\end{array} with cost functional \frac{1}{2}\int^{\infty}_0 x^2_1 +x^2_2+u^2 \ dt. After looking at some examples, I understand that I have to find...
  7. M

    Optimal Control Problem - LQR

    I'm trying to pick up optimal control by self study. At the moment I'm working on linear quadratic regulator and trying to reproduce the result publish in this paper. Curtis and Beard, Successive collocation: An approximation to optimal nonlinear control, Proceedings of the American Control...
  8. M

    Cancellation Property

    Let A,B be mxn matrices and C be nxk matrix. What is the necessary or sufficient condition such that AC=BC implies A=B ? In my work, A and B are m by m matrices and C is just a column vector m by 1. In this specialized case, what are the condition imposed on the elements of C such that AC=BC...
  9. M

    Laplace Inversion

    When trying to solve a pde using Laplace transform, I need to invert an expression of the form \frac{\exp{(-as-b\sqrt{s})}}{s^2} A friend told me that Mathematica cannot invert such expression. I try using convolution but a bit loss when trying to evaluate the integral of Erfc(.)...
  10. M

    Nth roots of a matrix

    Matlab help state that the square root of X = \begin{pmatrix} 7 & 10 \\ 15 & 22 \end{pmatrix} are A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} , B = \begin{pmatrix} 1.5667 & 1.7408 \\ 2.6112 & 4.1779 \end{pmatrix} , C=-A and D=-B . When I used the matlab command...
  11. M

    A Quotation Again

    Carl Friedrich Gauss is known as the "prince of mathematicians" in mathematics literature. But I think something is not right with this quotation from him. "Mathematics is the Queen of the Sciences". Is maths gender female? Is maths a science?
  12. M

    Series expansion

    I'm trying to work something on inverse Laplace transform. I need to express a transfer function F(s) to the form F(s)=\frac{s^{-1} (a_0 + a_1s^{-1} + a_2s^{-2}+ ... }{b_0 + b_1s^{-1} + b_2s^{-2}+ ... } I can easily do it for rational function e.g. \frac{s^3+2s^2+3s+1}{s+4}= \frac{s^{-1}...
  13. M

    Delta function

    I'm working on discretizing pde with boundary and initial conditions. The assumption of the method is that functions must be at least twice differentiable. I have an intial condition given by the following function u(x,0)=f(x)=\left\{\begin{array}{cc}x,&\mbox{ } 0 \leq x <1\\2-x, & \mbox{ } 1...
  14. M

    MATLAB Matlab function

    I normally create anonymous function in order to avoid creating extra m-file if the function is simple enough. For example if I want to integrate the function f(x)=x2+x, I just write a simple matlab script like myfun=@(x) x.^2 + x; quad(myfun,0,1) But how do we create an anonymous...
  15. M

    Solve XA=Y

    I have a system of linear equations which can be expressed as XA=Y where X and Y are row vectors. The vector Y and the matrix A are given. I need to solve for X. I can rephrase the same equation as AtXt=Yt but the answer will still be the same. I try using matlab to solve for X using the...
  16. M

    Dean Schlicter who

    The following quotation from Dean Schlicter is an inspiration "Go down deep enough into anything and you will find mathematics." But who is Dean Schlicter. I don't see much information about him on the internet except for that quotation. Do someone have knowledge about him?
  17. M

    System of linear second order differential equations

    I think I know how to solve \frac{d\vec{x}}{dt}= A \vec{x} where A is a constant nXn matrix. We just compute the eigenvalues and the corresponding eigenvectors. But how do we solve \frac{d^2\vec{x}}{dt^2}= A \vec{x} Can we say straight away that the solution is...
  18. M

    How to read a gz file?

    I just downloaded the file and it is quite a large file about 75Mb. But I do not know how to read the file. Please tell me what to do.:cry: I'm using window XP and I know how to unzip dot zip or dot rar. However my computer doesn't recognized the new file format. Why it is...
  19. M

    Critic of heat equation

    One critic of the Fourier heat equation \frac{\partial T}{\partial t}=k\nabla^2 T that I recently came across is that it gives rise to infinite speed of heat propagation. I understand that the speed cannot be infinite because it contradict special relativity that no speed should be...
  20. M

    Infinite dimensional representation of su(2)

    I'm trying to understand this paper which the author claimed that he had constructed an infinite dimensional representation of the su(2) algebra. The hermitian generators are given by J_x=\frac{i}{2}(\sqrt{N+1}a-a^\dagger\sqrt{N+1} ) J_y=-\frac{i}{2}(\sqrt{N+1}a+a^\dagger\sqrt{N+1} )...
  21. M

    Character of a representation

    I'm trying to understand this paper on the representation of SU(2). I know these definitions: A representation of a group G is a homomorphism from G to a group of operator on a vector space V. The dimension of the representation is the dimension of the vector V. If D(g) is a matrix...
  22. M

    Frobenius Method again!

    Assume that x=0 is a regular singular for x2y" + xp(x)y' + q(x)y = 0 and the indicial equation has equal roots \lambda = \lambda_1 = \lambda_2 The first solution is alway known to be of the form y_1(x) = x^{\lambda_1}\sum a_n x^n Although tedious, I know how to obtain the second...
  23. M

    Matrix Exponential

    Theorem Let A be a square matrix nXn then exp(At) can be written as exp(At)=\alpha_{n-1}A^{n-1}t^{n-1} + \alpha_{n-2}A^{n-2}t^{n-2} + ... + \alpha_1At + \alpha_0 I where \alpha_0 , \alpha_1 , ... , \alpha_{n-1} are functions of t. Let define r(\lambda)=\alpha_{n-1}\lambda^{n-1} +...
  24. M

    Integral Equation

    This forum discuss Differential Equations - ODE, PDE, DDE, SDE, DAE I know the abbreviation ODE and PDE. But what are DDE and DAE ? SDE must stand for system of DE. Do you also discuss Integral Equation in this forum?
  25. M

    Symplectic geometry. What's this?

    My training is in mathematics. But during my free time I also try to understand fundamental physics. Recently I came across a material which said that the geometry of classical mechanics is symplectic. I'm not sure of the meaning. It was relating to the Hamiltonian which I'm also not...
  26. M

    Projectile Problem again

    I thought I have already master the topic on projectile. Now I'm not that sure. I fail to solve the Problem 3.4 in the book An Introduction to Mechanics by Daniel Kleppner and Robert J. Kolenkow. The verbatim problem is as follows: "An instrument-carrying projectile accidentally explodes at...
  27. M

    A sliding block - what's wrong with my arguement

    I'm trying to solve the". I have done the following, using Newton's Second Law \frac{N}{\sqrt{2}}i + \frac{N}{\sqrt{2}}j - mgj = m(\ddot{x} i + \ddot{y}j) where N is the normal force on the block m. Comparing the coefficients where...
  28. M

    Why do we get different answers ?

    We were trying to solve the problem 28.32 on page 289 of the Schaum's Series Differential Equations by Richard Bronson and Gabriel Costa. The DE is 4 x^2 y'' + (4 x + 2 x^2) y' + (3 x - 1) y = 0 We use the Frobenius method to solve this equation since x=0 is a regular singular point. The...