Homework Answers
1. Let h be a hash function mapping U to indexes (0...m -1]. Assume that |U|...
1. Let h be a hash function mapping U to indexes (0...m -1]. Assume that |U| m2 Could an adversary could pick a set K of m/2 keys from U which are all mapped into the same cell? What is the value of ΣzyEKFh(x,y) in this case ? Fh(x,y) is the function defined in the slides on the section discussing Universal Families of Hash functions.
Please show detailed solution Given: Ux = 3/8 Uxx0 < x < 50,t > 0 u(0,t)...
Please show detailed solution Given: Ux = 3/8 Uxx0 < x < 50,t > 0 u(0,t) = 50, u(50,1)=100, T>0 u(x,0) = 50,0 < x < 50 1. Identify the IBVP case 2. c2= ,1 = 47)2 = To= 3. Find all the values required by the general formula , p= Ti= f(x)=_
Problem 1: Let y()- r(t+2)-r(t+1)+r(t)-r(t-1)-u(t-1)-r(t-2)+r(t-3), where r(t) is the ramp function. a) plot y(t) b) plot...
Problem 1: Let y()- r(t+2)-r(t+1)+r(t)-r(t-1)-u(t-1)-r(t-2)+r(t-3), where r(t) is the ramp function. a) plot y(t) b) plot y'() c) Plot y(2t-3) d) calculate the energy of y(t) note: r(t) = t for t 0 and 0 for t < 0 Problem 2: Let x(t)s u(t)-u(t-2) and y(t) = t[u(t)-u(t-1)] a) b) plot x(t) and y(t) evaluate graphically and plot z(t) = x(t) * y(t) Problem 3: An LTI system has the impulse response h(t) = 5e-tu(t)-16e-2tu(t) + 13e-3t u(t) The input...
2(a). Compute and plot the convolution of ytryh)x where h(t) t)-u(t-4), x(t)u(t)-u(t-1) and zero else b)....
2(a). Compute and plot the convolution of ytryh)x where h(t) t)-u(t-4), x(t)u(t)-u(t-1) and zero else b). Compute and plot the convolution y(n) h(n)*x (n) where h(n)-1, for 0Sns4, x(n) 1, n 0, 1 and zero else.
Please show all your works. Thanks. 2) Let r(t) be the continuous-time signal x(t) t(0.97)' [a(t)-u(t-100)|...
Please show all your works.
Thanks.
2) Let r(t) be the continuous-time signal x(t) t(0.97)' [a(t)-u(t-100)| where u(t) is the unit step function. We sample r(t) with a sampling period of T 0.4 over ootoo to obtain r/n]. Assume that the sample for n 0 is taken at t 0. We then use sinc interpolation to generate Tr(t) from xml a) Plot r(t) using MATLAB over the smallest range that includes all nonzero values of r(t) b) Find r[n] c)...
Given v (678) = U (,1;8%)*(x",0)de where U (,t;a") = ( 217 )* <im(8=e")°/2nt and 1...
Given v (678) = U (,1;8%)*(x",0)de where U (,t;a") = ( 217 )* <im(8=e")°/2nt and 1 V (X",0) = - wie poz' /ħe-x'2/242 (TA)1/4 e’Poz' /H Using Gaussian integrals, show that 1 1/2 -2ħt) 1+ m2 1 6 1410) Lada cara no 15. (24) (-P0t/m)2 Ų (x, t) = eimr/2ht , "ofcas. 771/2 m
(1 point) Solve the heat problem with non-homogeneous boundary conditions ∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0 u(0,t)=0, u(3,t)=2, t>0,
(1 point) Solve the heat problem with non-homogeneous boundary
conditions
∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0
u(0,t)=0, u(3,t)=2, t>0,u(0,t)=0, u(3,t)=2, t>0,
u(x,0)=23x, 0<x<3.u(x,0)=23x, 0<x<3.
Recall that we find h(x)h(x), set
v(x,t)=u(x,t)−h(x)v(x,t)=u(x,t)−h(x), solve a heat problem for
v(x,t)v(x,t) and write u(x,t)=v(x,t)+h(x)u(x,t)=v(x,t)+h(x).
Find h(x)h(x)
h(x)=h(x)=
The solution u(x,t)u(x,t) can be written as
u(x,t)=h(x)+v(x,t),u(x,t)=h(x)+v(x,t),
where
v(x,t)=∑n=1∞aneλntϕn(x)v(x,t)=∑n=1∞aneλntϕn(x)
v(x,t)=∑n=1∞v(x,t)=∑n=1∞
Finally, find
limt→∞u(x,t)=limt→∞u(x,t)=
Please show all work.
(1 point) Solve the heat problem with non-homogeneous boundary conditions au ди (x, t) at (2, t), 0<x<3, t> 0 ar2 u(0,t) = 0, u(3, t) = 2, t>0, u(t,0)...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
5. In class we saw that the function r(u, v) = (sin u, (2 + cos...
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
1. Let h be a hash function mapping U to indexes (0...m -1]. Assume that |U| m2 Could an adversary could pick a set K of m/2 keys from U which are all mapped into the same cell? What is the value of ΣzyEKFh(x,y) in this case ? Fh(x,y) is the function defined in the slides on the section discussing Universal Families of Hash functions.
Please show detailed solution Given: Ux = 3/8 Uxx0 < x < 50,t > 0 u(0,t) = 50, u(50,1)=100, T>0 u(x,0) = 50,0 < x < 50 1. Identify the IBVP case 2. c2= ,1 = 47)2 = To= 3. Find all the values required by the general formula , p= Ti= f(x)=_
Problem 1: Let y()- r(t+2)-r(t+1)+r(t)-r(t-1)-u(t-1)-r(t-2)+r(t-3), where r(t) is the ramp function. a) plot y(t) b) plot y'() c) Plot y(2t-3) d) calculate the energy of y(t) note: r(t) = t for t 0 and 0 for t < 0 Problem 2: Let x(t)s u(t)-u(t-2) and y(t) = t[u(t)-u(t-1)] a) b) plot x(t) and y(t) evaluate graphically and plot z(t) = x(t) * y(t) Problem 3: An LTI system has the impulse response h(t) = 5e-tu(t)-16e-2tu(t) + 13e-3t u(t) The input...
2(a). Compute and plot the convolution of ytryh)x where h(t) t)-u(t-4), x(t)u(t)-u(t-1) and zero else b). Compute and plot the convolution y(n) h(n)*x (n) where h(n)-1, for 0Sns4, x(n) 1, n 0, 1 and zero else.
Please show all your works.
Thanks.
2) Let r(t) be the continuous-time signal x(t) t(0.97)' [a(t)-u(t-100)| where u(t) is the unit step function. We sample r(t) with a sampling period of T 0.4 over ootoo to obtain r/n]. Assume that the sample for n 0 is taken at t 0. We then use sinc interpolation to generate Tr(t) from xml a) Plot r(t) using MATLAB over the smallest range that includes all nonzero values of r(t) b) Find r[n] c)...
Given v (678) = U (,1;8%)*(x",0)de where U (,t;a") = ( 217 )* <im(8=e")°/2nt and 1 V (X",0) = - wie poz' /ħe-x'2/242 (TA)1/4 e’Poz' /H Using Gaussian integrals, show that 1 1/2 -2ħt) 1+ m2 1 6 1410) Lada cara no 15. (24) (-P0t/m)2 Ų (x, t) = eimr/2ht , "ofcas. 771/2 m
(1 point) Solve the heat problem with non-homogeneous boundary
conditions
∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0∂u∂t(x,t)=∂2u∂x2(x,t), 0<x<3, t>0
u(0,t)=0, u(3,t)=2, t>0,u(0,t)=0, u(3,t)=2, t>0,
u(x,0)=23x, 0<x<3.u(x,0)=23x, 0<x<3.
Recall that we find h(x)h(x), set
v(x,t)=u(x,t)−h(x)v(x,t)=u(x,t)−h(x), solve a heat problem for
v(x,t)v(x,t) and write u(x,t)=v(x,t)+h(x)u(x,t)=v(x,t)+h(x).
Find h(x)h(x)
h(x)=h(x)=
The solution u(x,t)u(x,t) can be written as
u(x,t)=h(x)+v(x,t),u(x,t)=h(x)+v(x,t),
where
v(x,t)=∑n=1∞aneλntϕn(x)v(x,t)=∑n=1∞aneλntϕn(x)
v(x,t)=∑n=1∞v(x,t)=∑n=1∞
Finally, find
limt→∞u(x,t)=limt→∞u(x,t)=
Please show all work.
(1 point) Solve the heat problem with non-homogeneous boundary conditions au ди (x, t) at (2, t), 0<x<3, t> 0 ar2 u(0,t) = 0, u(3, t) = 2, t>0, u(t,0)...
1. (a) Derive the solution u(x, y) of Laplace's equation in the rectangle 0 < x <a, 0 <y <b, that satisfies the boundary conditions u(0,y) = 0, u(a, y) = 0, u(x,0) = 0, u(x,b) = g(x), 0 0 0 < a. (b) Find the solution if a = 4, b = 2, and g(x) = 0 <r <a/2, a-r, a/2 < x <a.
5. In class we saw that the function r(u, v) = (sin u, (2 + cos u) cos v, (2 + cos u) sin v), 0<u<27, 050521 parametrizes a torus T, which is depicted below. (a) Calculate ||ru x rull. (b) Show that T is smooth. (c) Find the equation of the tangent plane to T at (0,). (d) Find the surface area of T (e) Earlier in the semester, we observed that a torus can be built out of...
Problem 1 (20 points) Consider the PDE for the function u(x, t) e 0<x<T, t> 0 with the boundary conditions n(0, t) 0, u(T, t) 0, t> 0 and the initial condition 0 u(x, 0) 1+cos(2a), (a) Give a one-sentence physical interpretation of this problem. (b) Find the solution u(x, t) using a Fourier cosine series representation An (t) cos(nax) u(x,t)= Ao(t) + n=1
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