11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0<x <c,0 11 (c, 1) = 0, u(x, 0) = f(x), where u is continuous for0sxc,0 and where n is a positive integer. Answer: u(x, 1) Σ A,Jn(gjx) exp (-α,1), where a", and A, are the constants j-1
11. Solve this boundary value problem for u(x, t): n2 xu,-(x14),--11 (0
If ū +3+w -0, show that u u xe - w xu
Prove that each of the metrics (4.1) , (4.2) , and (4.3) has
property (4.4) .
211/2 4.2) (4.3) A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of component entries {XkU)}-i converges to x, in Xk. (4.4)
211/2
4.2) (4.3)
A sequence {xu) = (xe)-1 converges to x = (지,. . .x,) in X if and only if for each k the sequence of...
Pro The more resource in the show is there are con that the best het vlees who the - Vand - Calculate the power (a) to the currenthe Show Work is REQUIRED for this questions on Show Work
If E [exp(aX)] exists for a given constant a, then show that for t > 0 (a) exp(−at)P (X > t) < E [exp(aX )] , if a > 0. (b) exp(−at)P (X < t) < E [exp(aX )] , if a < 0.
Let XU(a, b) be a uniformly distributed random variable. Use the definition of mean and variance to show that: (a) E(X)t (b) Var(x)2
a) Show that the wave function y(x) = N exp( – x²/(2a?)) with a? = () is a solution of the Schrödinger equation for harmonic oscillator with potential V(x) = k x2/2. (10 pt) b) What is the energy of harmonic oscillator with the wave function y(x) in terms of k and m? (5 pt) c) Sketch the potential energy of harmonic oscillator, the energy level corresponding to y(x), the wave function (x), and the probability density associated with y(x)...
4 Linearize the following ODE around Xo 2T,u,-1 0 0 x 2 sin(x) + xu + u2
1. Suppose that the joint density of X and Y is given by exp(-y) (1- exp(-x)), if 0 S y,0 syS oo exp(-x) (1- exp(-y)), if 0SyS ,0 oo (e,y)exp(-y) Then . The marginal density of X (and also that of Y), ·The conditional density of Y given X = x and vice versa, Cov(X, Y) . Are X and Y independent? Explain with proper justification.
(a) Show the expression of the Hamiltonian operator for the single electron in Het and Write out the exact wavefunction expression for an electron in He* in the quantum state of (3,1,0). (b) Determine the values for the angular momentum and energy for the electron in the quantum state of (3,1,0).