1. Prove that h(t) * (t) = (t) *h(t) 2. A system has an impulse function h(t) = sinº (3t)u(t). Find the unit step (NOTE: an integral table is posted on D2L.) 3. Consider a system with input (t) and output y(t). Let r(t) y(t) = 1 + x(t-1) Is this system linear? Is it causal? Is it BIBO stable? Justify your answer
Prove the following: Using Convolution, determine y(t) when x(t) = 4u(t) and h(t) = e-2t u(t) for t > 0 answer: y(t) = 2[1-e-2t]
14. If f(a) and g(x) are polynomials over the field F, and h(x)-f(x) t gx), prove that h(c)-f(c) + g(c) for all c in F. 15. If f(x) and g(x) are polynomials over the field F, and p(x)fx)g(x), prove that p(c) -f(c)g(c) for all c in F
Question #2 Prove the entropy chain rules a) b) H(X, Y) = H(X|Y) + H(Y) 1(X: Y) = H(X)-H(XIY )
(e) For subsets {A,Jael, prove that2 I) Evaluate (g) Prove that XAAB (XA-X) (h) Use characteristic functions to prove the distributive law: AU(BnC) (AUB)n (AUC) Hint: start with the right-hand side. 1In this problem, the product of two functions and g is defined by (Jg)(x)-f() and the sum is defined by (f +g)(x) :-f(x) + g(x), as usua 2Here, Π denotes the product of an indexed set of numbers. For example: rL TL TL i n! i-1 -1
(e) For...
(8) Prove that dt= 1-t n=1 for x e [-a, a],0< a< 1 and deduce from there a power series expansion for -In(1-x)
(8) Prove that dt= 1-t n=1 for x e [-a, a],0
1. For differentiable vector functions u and v, prove: u'(t) X v(t)+ u(t) X v'(t) lu(t) X v(t)] 2. For the differentiable vector function u and real-valued function f, prove: lu(f(t)))= f(t)u' (f (t))
1. For differentiable vector functions u and v, prove: u'(t) X v(t)+ u(t) X v'(t) lu(t) X v(t)] 2. For the differentiable vector function u and real-valued function f, prove: lu(f(t)))= f(t)u' (f (t))
1. Prove that the function f: X → Y is injective if and only if it satisfies the following condition: For any set T and functions g: T → X and h : T → X, o g = f o h implies g = h.
4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r).
4. Let F be a field. Prove that for all polynonials f(x), g(x), h (z) є FI2], if f(x) divides g(x) and f(z) divides h(r), then for all polynomials s(r),t() E Fr, f() divides s()g(r) +t(x)h(r).
Given that H is a set and that x is an interior point of the set R∖H, prove that x is not close to H. So, I have an example where they are opposite of the question, I just don't know how to spin it around. Suppose that S is a set of real numbers and that x is a given number. Then the following two conditions are equivalent to one another: 1.The number x is close to the set...