Let A/B = {w| wx ∈ A for some x ∈ B}. Show that if A is context free and B is regular, then A/B is context free.
Use the pumping lemma to show that the following languages are not context free. a. {0n1n0n1n| n ≥ 0}
In: Advanced Math
Let A and B be sets. Then we denote the set of functions with domain A and codomain B as B^A. In other words, an element f∈B^A is a function f:A→B.
Prove: Let ?,?∈?^? (that is, ? and ? are real-valued functions with domain ?) and define a relation ≡ on ?^? by ?≡?⟺?(0)=?(0). (That is, ? and ? are equivalent if and only if they share the same value at ?=0) Then ≡ is an equivalence relation on ?^?.
Prove: Suppose that ?:?→? f:A→B and ?,?⊆? Then ?(?∩?)⊆?(?)∩?(?)
Prove: Suppose that ?:?→? f:A→B and E,F⊆B. Then ?^(−1)(?)∪?(−1)(?)⊆?(?∪?)
Prove: Suppose that {Ai} is a partititon of B. Then the relation x∼y for defined x,y∈A by x∼y⟺∃Ai(x,y∈Ai) is an equivalence relation.
In: Advanced Math
Write down every permutation in S3 as a product of 2-cycles in the most efficient way you can find (i.e., use the fewest possible transpositions). Now, write every permutation in S3 as a product of adjacent 2-cycles, but don’t worry about whether your decomposition are efficient. Any observations about the number of transpositions you used in each case? Think about even versus odd.
In: Advanced Math
Find a particular solution to the equation
2y''+0.1y'+2y = cos(t) + 4cos(5t)
In: Advanced Math
In: Advanced Math
Solve the following logic problems. Remember, everyone you meet is either a knight or a knave, knights make true statements, and knaves make false statements. Give your reasoning for each problem.
a) You meet two residents, Alex and Bill. They say the following: Alex: I’m a knight. Bill: Alex is a knight, but I’m a knave. Is Alex a knight or a knave? Is Bill a knight or a knave?
b) You meet Clara and Davis, who are all like: Clara: One of us is a knight and the other is a knave. Davis: Clara is a knave. Is Clara a knight or a knave? Is Davis a knight or a knave?
c) You meet Edith and Frank, though only Edith speaks. Edith: Both Frank and I are knaves. Is Edith a knight or a knave? Is Frank a knight or a knave? (Note: Frank’s silence gives no indication of his type, but you can figure out from Edith’s statement.)
d) You meet Gina, Herbert, and Ichabod. Gina: Ichabod is a knave, if and only if I’m a knight. Herbert: Ichabod is a knight, if and only if I’m a knave. Ichabod: I like pudding. Does Ichabod like pudding?
In: Advanced Math
In: Advanced Math
1. [9 marks] Consider the boundary-value problem,
y′′ +2εy1/2 = 0, y(0) = 1,y(1) = 3/2.
Letting y = y0 + εy1 + ε2y2 + . . . , find y0 and y1 and hence y
with error O(ε2).
In: Advanced Math
1. Write two examples of media or types of communication in each section. Note that each section is described by the column header AND the row title. Leave a section blank if, by definition, no example is possible.
Mediated Communication |
Not Mediated Communication |
|
Mass Communication |
||
Not mass communication |
2. What specifically distinguishes mass communication from earlier means of communication?
In: Advanced Math
In: Advanced Math
Consider the differential equation,
L[y] = y'' + p(t)y' + q(t)y = 0, (1)
whose coefficients p and q are continuous on some open interval I. Choose some point t0 in I. Let y1 be the solution of equation (1) that also satisfies the initial conditions
y(t0) = 1,
y'(t0) = 0,
and let y2 be the solution of equation (1) that satisfies the initial conditions
y(t0) = 0,
y'(t0) = 1.
Then y1 and y2 form a
fundamental set of solutions of equation (1).
Find the fundamental set of solutions specified by the theorem
above for the given differential equation and initial point.
y'' + 8y' − 9y = 0,
t0 = 0
y1(t) | = | |
y2(t) | = |
In: Advanced Math
Find the solution of the given initial value problem y'' + 4y = t^2 + 3e^t + e^2t cost, y(0) = 0, y'(0) = 2,
using method of undetermined coefficients
In: Advanced Math
In control systems analysis, transfer functions are developed that mathematically relate the dynamics of a system’s input to its output. A transfer function for a robotic positioning system is given by ?(?) = ?(?) ?(?) = ?3+12.5?2+50.5?+66 ?4+19?3+122?2+296?+192 . Where, ?(?) = system gain, ?(?) = system output, ?(?) = system input, and ? = Laplace transform complex frequency. Now, use Bairstow’s method to determine the roots of the numerator and denominator and factor these into the form ?(?) = (?+?1)(?+?2)(?+?3) (?+?1)(?+?2)(?+?3)(?+?4) . Where, ?? and ?? = the roots of the numerator and denominator, respectively. [Hints: To perform the evaluation of complex roots, Bairstow’s method divides the polynomial by a quadratic factor ?2 − ?? − ?. Use initial guesses of ? = ? = −1, for determining the roots of both numerator and denominator, and perform up to four iteration.]
In: Advanced Math
Matrix A2= [1 2 3; 4 5 6; 7 8 9; 3 2 4; 6 5 4; 9 8 7]
Note: TA2 is defined to be a linear transformation that maps any vector x to A2* x. That is TA2 = A2*x. Also the range of the Linear transformation represented by A2 is the same as the column space of A2.
l) Find a basis for the null(TA2).
m) Find nullity of A2, TA2 and A2tA2.
n) Find rank(A2), rank(A2t), rank(TA2) and rank(A2tA2).
I don't need any explanation, a concise answer is good enough. Please only help if you can give a full and correct answer, thank you!
In: Advanced Math