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we will focus on the optimization problem TERn 2 where H Rnxn is symmetric, g ERn,...
(The integral of a Gaussian/Bell curve) Let Exercise 34: e~t2(1+z2) -dz 12 da f(t) and g(t) = e and h(t) f(t2 g(t) 1 Pr...
(The integral of a Gaussian/Bell curve) Let Exercise 34: e~t2(1+z2) -dz 12 da f(t) and g(t) = e and h(t) f(t2 g(t) 1 Problem sheet 9 Homework 29. Mai 2019 a) Compute h(0). b) Compute h'(t) for all t > 0 Remark: You have to argue why you can interchange differentiation and integration c) Compute lim4-,00 h(t) d) Use a) c) to show that 1 d 1 VT JR da and 2 Remark: The elegant proof of the integral of...
Asymptotic notation O satisfies the transitive property i.e. if f(n)=O(g(n)) and g(n)=O(h(n)), then f(n)=O(h(n)). Now we...
Asymptotic notation O satisfies the transitive property i.e. if f(n)=O(g(n)) and g(n)=O(h(n)), then f(n)=O(h(n)). Now we know that 2n =O(2n-1), 2n-1 =O(2n-2?),....... , 2i=O(2i-1?),....... So using rule of transitivity, we can write 2n =O(2i-1?).We can go extending this, so that finally 2n =O(2k?), where k is constant.So we can write 2n =O(1?). Do you agree to what has been proved?If not,where is the fallacy? 6 marks (ALGORITHM ANALYSIS AND DESIGN based problem)
Use R programming to solve Q2. A matrix operator H(G; k) on a pxp symmetric matrix G (iy)- with a positive integer parameter k (k < p) yields another p×p symmetric matrix H = (hij 1 with i=k,j...
Use R programming to solve
Q2. A matrix operator H(G; k) on a pxp symmetric matrix G (iy)- with a positive integer parameter k (k < p) yields another p×p symmetric matrix H = (hij 1 with i=k,j = k; (a) Use one single loop to construct the function H(G; k) in R (b) Generate a random matrix X of dimension 7x5, each element of which is id from N(0,1). Use the function H(G; k constructed in (a) to compute...
problems are related to symmetric groups. 1. In this problem we will give an explicit Cayley...
problems are related to symmetric groups. 1. In this problem we will give an explicit Cayley embedding of the quaterion group Qs into the symmetric group Sg. First, we fix the following order on the 8 elements in the group: {1,-1,1, -i, j, - j, k, -k}. Now, for each g EQs, write down the corresponding permutation 0, in cycle notation), where o,(h) = gh. Example, 0; = (1 3 2 4)(5 76 8). Now write down o, for the...
8. Suppose that we are given the following information about the functions f, g, h and...
8. Suppose that we are given the following information about the functions f, g, h and k and their derivatives; • f(1) = 3 • f'(1) = 2 • g(1) = 4 • g'(1) = -2 • h(1) = 9 . h'(1) = -1 k(1) = 10 • k'(1) = -3 (e) (5 points) Set F(x) = log2[f(x) + g(x)]. Compute F'(1). (f) (5 points) Set F(T) = log: [f(r)g(r)h(r)k(r)]. Compute F'(1).
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we h...
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
HW06: Problem 2 Previous Problem Problem List Next Problem (1 point) The equation бх? + Зу...
HW06: Problem 2 Previous Problem Problem List Next Problem (1 point) The equation бх? + Зу (*) ху can be written in the form y = f(y/x), i.e., it is homogeneous, so we can use the substitution u = equation with dependent variable u = u(x). yx to obtain a separable Introducing this substitution and using the fact that y xu' +u we can write (*) as = У = xu' u = f(u) where f(u) Separating variables we can...
The function h(2) = can be expressed in the form f(g(x)) where g(x) = (x +...
The function h(2) = can be expressed in the form f(g(x)) where g(x) = (x + 3) and f(x) is defined as: f(x)= Preview
Problem 2 (10 pt) Consider real functions with a general form of: f(x) = 20 +...
Problem 2 (10 pt) Consider real functions with a general form of: f(x) = 20 + a1x + a2|2|| (a) Using the inner product (f|g) = ', dxf(x)g(x), find an orthonormal basis of functions that take on the above form. (b) f(x) above is currently written in the basis (1, x, [x]). Rewrite f(x) in terms of the orthonormal basis you determined in (a). (c) Project the function h(x) = x2 onto each of the basis vectors you arrived at...
2. Write a programmets) to solve the optimization problem where d is defined as above, by...
2. Write a programmets) to solve the optimization problem where d is defined as above, by the following methods (a) the steepest descent method, (b) any conjugate gradient method, and (c) any quasi-Newton method. Run the programme and terminate it when either g)110-6 or the number of iterations reaches 100. Then provide the following (without the programme): (d) For each method, describe the way of obtaining a) andr) (e) For each method, provide a table of results (rounded to 4...
(The integral of a Gaussian/Bell curve) Let Exercise 34: e~t2(1+z2) -dz 12 da f(t) and g(t) = e and h(t) f(t2 g(t) 1 Problem sheet 9 Homework 29. Mai 2019 a) Compute h(0). b) Compute h'(t) for all t > 0 Remark: You have to argue why you can interchange differentiation and integration c) Compute lim4-,00 h(t) d) Use a) c) to show that 1 d 1 VT JR da and 2 Remark: The elegant proof of the integral of...
Use R programming to solve
Q2. A matrix operator H(G; k) on a pxp symmetric matrix G (iy)- with a positive integer parameter k (k < p) yields another p×p symmetric matrix H = (hij 1 with i=k,j = k; (a) Use one single loop to construct the function H(G; k) in R (b) Generate a random matrix X of dimension 7x5, each element of which is id from N(0,1). Use the function H(G; k constructed in (a) to compute...
problems are related to symmetric groups. 1. In this problem we will give an explicit Cayley embedding of the quaterion group Qs into the symmetric group Sg. First, we fix the following order on the 8 elements in the group: {1,-1,1, -i, j, - j, k, -k}. Now, for each g EQs, write down the corresponding permutation 0, in cycle notation), where o,(h) = gh. Example, 0; = (1 3 2 4)(5 76 8). Now write down o, for the...
8. Suppose that we are given the following information about the functions f, g, h and k and their derivatives; • f(1) = 3 • f'(1) = 2 • g(1) = 4 • g'(1) = -2 • h(1) = 9 . h'(1) = -1 k(1) = 10 • k'(1) = -3 (e) (5 points) Set F(x) = log2[f(x) + g(x)]. Compute F'(1). (f) (5 points) Set F(T) = log: [f(r)g(r)h(r)k(r)]. Compute F'(1).
Question 2 Consider the differential equation We saw in class that one solution is the Bessel function (a) Suppose we have a solution to this ODE in the form y-Σχ0CnXntr where cn 0. By considering the first term of this series show that r must satisfy r2-4-0 (and hence that r = 2 or r =-2) (b) Show that any solution of the form y-ca:0G,2n-2 must satisfy C0 (c) From the theory about singular solutions we know that a linearly...
HW06: Problem 2 Previous Problem Problem List Next Problem (1 point) The equation бх? + Зу (*) ху can be written in the form y = f(y/x), i.e., it is homogeneous, so we can use the substitution u = equation with dependent variable u = u(x). yx to obtain a separable Introducing this substitution and using the fact that y xu' +u we can write (*) as = У = xu' u = f(u) where f(u) Separating variables we can...
The function h(2) = can be expressed in the form f(g(x)) where g(x) = (x + 3) and f(x) is defined as: f(x)= Preview
Problem 2 (10 pt) Consider real functions with a general form of: f(x) = 20 + a1x + a2|2|| (a) Using the inner product (f|g) = ', dxf(x)g(x), find an orthonormal basis of functions that take on the above form. (b) f(x) above is currently written in the basis (1, x, [x]). Rewrite f(x) in terms of the orthonormal basis you determined in (a). (c) Project the function h(x) = x2 onto each of the basis vectors you arrived at...
2. Write a programmets) to solve the optimization problem where d is defined as above, by the following methods (a) the steepest descent method, (b) any conjugate gradient method, and (c) any quasi-Newton method. Run the programme and terminate it when either g)110-6 or the number of iterations reaches 100. Then provide the following (without the programme): (d) For each method, describe the way of obtaining a) andr) (e) For each method, provide a table of results (rounded to 4...
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