How do you mathematically prove a line moves faster than the other? The problem says there are two fast food restaurants, each containing two employees working the register. Restaurant 1 has two lines to service customers, and restaurant 2 has one line for customers to wait in. How can you mathematically prove that either restaurant 1 or restaurant 2 has a better line control system?

Side note: I was thinking that this is like which function has the better growth such as an exponential line vs. a power function line, but maybe I'm wrong. Please show me the easiest way.

Give a direct proof of the following theorem, upon which case you can use it for future proofs. (Hint: note that we’ve called it a corollary as in p.81, not just a theorem.) Corollary 4.12. Every integer is even or odd.

Prove the following two statements: If the permutation α is even then α^-1 is even. If the permutation α is odd then α^-1 is odd.

Define the Hamiltonian Cycle Problem and the Travelling Salesman Problem. Give a polynomial-time transformation from the Hamiltonian Cycle Problem to the Travelling Salesman Problem to claim that if the Hamiltonian Cycle is ”Hard” (i.e., NP-Complete) then Travelling Salesman Problem must also be hard.

If a population with harvesting rate h is modeled by

dx/do=9-x^2-h

find the bifurcation point for the equation.

Which of the following sets is an abstract simplical complex? For each, if the answer is no, explain why; and if the answer is yes, give the dimension of the complex, and sketch its geometric realization, up to homeomorphism. a. {[a],[b],[a,b,c]},

b. {[a],[b],[c],[a,b,c]},

c. {[a],[b],[c],[a,b]},

d. {[a],[b],[c],[d],[a,b],[c,d]},

e. {[a],[b],[c],[d],[a,b],[b,c],[c,d],[a,d],[a,c],[a,b,c]}.

A:=<<0,-1,1>|<4,0,-2>|<2,-1,0>|<2,1,1>>;
Matrix(3, 4, [[0, 4, 2, 2], [-1, 0, -1, 1], [1, -2, 0, 1]])

(a) Use the concept of matrix Rank to argue, without performing ANY calculation, why the columns of this matrix canNOT be linerly independent.

(b) Use Gauss-Jordan elimination method (you can use ReducedRowEchelonForm command) to identify a set B of linearly independent column vectors of A that span the column space of A. Express the column vectors of A that are not included in the set B as a linear combination of the vectors in the set B.

(c) Do the columns of matrix A span the entire Euclidean space
"real^3"
? Explain why yes or why not.

Solve the following initial value problems by the method of undetermined coefficients:

A.) y′′ + y = 4 sin t − cos t.

B.) y′′ = 5t4 − 2t.

A spring is stretched by 7in by a mass weighing 15lb. The mass is attached to a dashpot mechanism that has a damping constant of 0.1lb⋅s/ft and is acted on by an external force of 5cos(9t)lb. Determine the steady state response of this system. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units.

Solve the following differential equation:

y''+e^{-y}=0 where y is only a function of x.

1. Find the additive and multiplicative inverses for the residual sets Z13, Z14, and Z11. Identify each as being a group, ring or field

2. Write an algorithm for performing Euclid greatest common denominator for A,B in Matlab and demonstrate the results for A=9777 and B=106665. Compare this the the matlab function gcd(A,B) and the by hand.

3. Program Euclid’s Extended algorithm in Matlab and demonstrate on the residual sets in problem 1.

4. Demonstrate Fermats and Eulers Theorems with the value of a set to the Residual set Z13 and Z14 and with P equal to 13 and 14 respectively. Explain results

5. Perform the Chinese Remainder theorem as in the example in your text with m1=29, m2=47

Find the Laplace transform of the following functions:
a. p(t) = 6[u(t − 2) − u(t − 4)]
b. g(t) = (4 + 3e−2t)u(t)
c. h(t) = 2r(t) − 2r(t − 2)