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Problem LA Suppose k and n are natural numbers....
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k,...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
Let A = ( a1 0 ... 0 0 a2 ... 0 ... ... ... 0...
Let
A = ( a1 0 ... 0
0 a2 ... 0
... ... ...
0 0 an)
be an n * n matrix, where a1, a2, . . . , an are
nonzero real numbers.
(a) Find the general solution to the system of equations ->
->
x' = A * x
(b) Solve the initial value problem x1(0) =
x2(0) = · · · = xn(0) = k, for some constant
k.
(c) Solve the initial value problem
(x1(0) x2(0)...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving k...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
Suppose n numbers X1, X2, . . . , Xn are chosen from a uniform distribution on [0, 10]. We say th...
Suppose n numbers X1, X2, . . . , Xn are chosen from a uniform distribution on [0, 10]. We say that there is an increase at i if Xi < Xi+1. Let I be the number of increases. Find E[I].
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility...
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility that the first k terms are all distinct is n - 1 п 2 n - (k – 1) P(n, k) = 2 п п п as we saw in class. For this problem, you will want to write a (very short) program to compute this number, but you do not need to submit it this time. (For both parts, you can submit a...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
How do I approach this question? *69. Suppose that f'(x) > 0 and that f(a) <...
How do I approach this question?
*69. Suppose that f'(x) > 0 and that f(a) < 0 while f(6) > 0, so that f(x) = 0 has a root r in the interval (a,b). Newton's method for finding r, starting at c in (a, b) is as follows: let Xo = c, and for n> 1, define In = In-1 - f(xn-1)/f'(xn-1). a) Taking f(x) = x – 23, a=-1/13, b=1/13 and xo = 1/V5 find x0, 11,x2,.... b) If...
Problem 5.10.10 Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2, …, Xn ...
Problem 5.10.10 Suppose you have n suitcases and suitcase i holds Xi dollars where X1, X2, …, Xn are iid continuous uniform (0, m) random variables. (Think of a number like one million for the symbol m.) Unfortunately, you don’t know Xi until you open suitcase i. Suppose you can open the suitcases one by one, starting with suitcase n and going down to suitcase 1. After opening suitcase i, you can either accept or reject Xi dollars. If you...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - ...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
Problem 15.10 Suppose that you need to transmit a discrete-time signal whose DTFT is shown in...
Problem 15.10 Suppose that you need to transmit a discrete-time signal whose DTFT is shown in Figure 15.16 with sample rate 1MHz. The specification is that the signal should be transmitted at the lowest possible rate, but in real time, i.e., the signal at the receiver must be exactly the same with a sample rate of 1MHz. Design a system (draw a block diagram) that meets these specs. Both before transmission and after reception, you can use upsampling-filtering denoted as...
13. (i) For each of the following equations, find all the natural numbers n that satisfy it (a) φ(n)-4 (b) o(n) 6 (c) ф(n) 8 (d) φ(n) = 10 (ii) Prove or disprove: (a) For every natural number k, there are only finitely many natural num- bers n such that ф(n)-k (b) For every integer n > 2, there are at least two distinction integers that are invertible modulo n (c) For every integers a, b,n with n > 1...
Let
A = ( a1 0 ... 0
0 a2 ... 0
... ... ...
0 0 an)
be an n * n matrix, where a1, a2, . . . , an are
nonzero real numbers.
(a) Find the general solution to the system of equations ->
->
x' = A * x
(b) Solve the initial value problem x1(0) =
x2(0) = · · · = xn(0) = k, for some constant
k.
(c) Solve the initial value problem
(x1(0) x2(0)...
6.32 Theorem. If k and n are natural numbers with (k, d(n)) =I, then there exist positive integers u and v satisfving ku=(n)u The previous theorem not only asserts that an appropriate exponent is always availahle, but it also tells us how to find it. The numbers u and are solutions lo a lincar Diophantine cquation just like those we studied in Chapter 6.33 Exercisc. Use your observations so far to find solutions to the follow ing congruences. Be sure...
Problem 1. Suppose that we take a random sequence of numbers modulo n. The proba- bility that the first k terms are all distinct is n - 1 п 2 n - (k – 1) P(n, k) = 2 п п п as we saw in class. For this problem, you will want to write a (very short) program to compute this number, but you do not need to submit it this time. (For both parts, you can submit a...
Prove the Binomial Theorem, that is Exercises 173 (vi) x+y y for all n e N C) Recall that for all 0rS L is divisible by 8 when n is an odd natural number vii))Show that 2 (vin) Prove Leibniz's Theorem for repeated differentiation of a product: If ande are functions of x, then prove that d (uv) d + +Mat0 for all n e N, where u, and d'a d/v and dy da respectively denote (You will need to...
How do I approach this question?
*69. Suppose that f'(x) > 0 and that f(a) < 0 while f(6) > 0, so that f(x) = 0 has a root r in the interval (a,b). Newton's method for finding r, starting at c in (a, b) is as follows: let Xo = c, and for n> 1, define In = In-1 - f(xn-1)/f'(xn-1). a) Taking f(x) = x – 23, a=-1/13, b=1/13 and xo = 1/V5 find x0, 11,x2,.... b) If...
For the following 2DOF linear mass-spring-damper system r2 (t) M-2kg K -18N/m C- 1.2N s/m i(t) - 5 sin 2t (N) f2(t)-t (N) l. Formulate an IVP for vibration analysis in terms of xi (t) and x2(t) in a matrix form. Assume that the 2. Solve an eigenvalue problem to find the natural frequencies and modeshape vectors of the system 3. What is the modal matrix of the system? Verify the orthogonal properties of the modal matrix, Ф, with system...
Problem 15.10 Suppose that you need to transmit a discrete-time signal whose DTFT is shown in Figure 15.16 with sample rate 1MHz. The specification is that the signal should be transmitted at the lowest possible rate, but in real time, i.e., the signal at the receiver must be exactly the same with a sample rate of 1MHz. Design a system (draw a block diagram) that meets these specs. Both before transmission and after reception, you can use upsampling-filtering denoted as...
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