Extra Question – Class Exercise 6.3Continuous Time Markov ChainsSuppose that one particle (created by a chain reaction) enters a space andrandomly produces with rate a (a new particle has the same reproductionrate). Let K, > 0 be the number (integer) of particles created at time tstarting from Ko = 1.Write down the master equation for P<(t) = P(K = k) and find the so-lution of the master equation starting from P(0) = dk.1. In other words,solve the equation for k = 1 and then for k = 2.Keep particle creation and annihilation in mind.

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MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The eigenvalues of the initial value problem y' +Ây=0, y(0)=0, y'(7)=0 are a) , = vn, for n=1,2,3,... b) 1, =n², for n=1,2,3,... c) 2, = 2n, for n=1,2,3,... d) , =, for n=1,2,3,... 2'
The behaviour of an unforced mechanical system is governed by the differential equation 5 2 -1 0 x(1) = 3 6-9 x(t), x(0) = 1 1 1 1 0 (a) Show that the eigenvalues of the system matrix are 6, 3, 3 and that there is only one linearly independent eigenvector corresponding to the eigenvalue 3. Obtain the eigenvectors corresponding to the eigenvalues 6 and 3 and a further generalized eigenvector for the eigenvalue 3. (b) Write down a generalized modal matrix M and confirm that AM = MJ for an appropriate Jordan matrix J. x(t) = Me M-¹x(0) obtain the solution to the given differential equation. (c) Using the result
Suppose you are told that the maximum eigenvalue of matrix A is dmax=3. Write a java program to compute each node's eigenvector centrality based on the equation c,(v;) =E; AjiC, (v;) Adjacency matrix ( for A) Adjacency Matrix 3 4 1 1 1 1 1 2 3 1 1 1 4 1 1 1 6 1 1 7 1 1 1. 1. 1. 00
Consider a dynamic system with three states, s_1s1, s_2s2, and s_3s3. Let A = \begin{pmatrix} 0.7&0.3 &0.3 \\0.2& 0.6 & 0.2 \\ 0.1 & 0.1 &0.5 \end{pmatrix}A=⎝⎛0.70.20.10.30.60.10.30.20.5⎠⎞ be the transition matrix such that a_{ij}aij is the probability of moving from s_jsj to s_isi. Let \overrightarrow{x} = \begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}x=⎝⎛x1x2x3⎠⎞ be a stationery distribution of AA with sum of elements x_1+x_2+x_3=120x1+x2+x3=120. Find x_2x2.
Suppose a system of first-order linear differential equations has eigenvalues A₁ = -3, A2=-2.3. The equilibrium point (origin) can be classified as: O Flux Point O Saddle Point O Origin Point O Stable Node (Sink) O Unstable Node (Source)
Consider a birth and death process with states {0,1}. Let the birth and death rates be Ao = a > 0, A₁ = 0, μo = 0, and µ₁ = b > 0. = ko = A1 (a) Show that the infinitesimal generator matrix A = [ a a satisfies the property A² = - 2 -(a + b)A. b Thus, for n = 1, 2, 3,..., A = (a - b)-1A holds. n (b) Using the result from part (a) to compute the matrix exponential et in closed form and thus find the transition probability matrix P(t) = e¹A, t≥ 0. (c) Compute lim →∞ P(t) and use it to find the stationary distribution.
In Sl units, the position x of a spring-loaded electromagnetic valve varies over timet in the manner described by the 2nd order ODE * + ax +(B+ K)x = K The term on the RHS is designed so that, if x is disturbed from its rest position, then it automatically returns, ideally, after a short period of time with no overshoot. 1 the rest position of the system is x = When a = 1, B= 1 and K = 4' a) Suppose that this system is at rest for the period 0 < t < 3, but at t = 3 the system is disturbed. Solve the ODE 'by hand' with initial conditions x(3)= and x(3)= 2; and
Linear regression aims to fit the parameters based on the training set D = {(x(i),y(i)), i = 1,2,..., m} so that the hypothesis function ho (x) 6T.x = 00+ 01x₁ + 0₂x₂+......+ Onxn can better predict the output y of a new input vector x. Please derive the stochastic gradient descent update rule which can update o repeatedly to minimize the least squares cost function (). =
determine the eigenvalues and eigenvectors if the eigenvalues are real (or use results from exercises from Section 4.2. if you have covered those exercises). Also classify the system (state whether stable or unstable node, stable or unstable spiral, center, saddle point) and in all cases sketch the phase plane of the linear system. (As a hint, problems with * have complex eigenvalues.) When checking your answers with those in the back of the book, keep in mind that any nonzero multiple of the given eigenvector may be used.

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