Analysis Integral confused concept. Please clear writing and follow the comment

1. upper sum with respect to P is U(f,P)= sum of Mk(xk-xk-1). Does that equal to U(f)=inf{u(f,P): p is a element of Q} (Q is the collection of all possible partition})

Also, is Mk=sup{f(x):x is a element of {xk-1,xk}?????

2. If Q is a refinement of P, then U(f, Q)<=U(f, P) and L(f, P),<=L(f,Q). I don't understand the geometric meaning behind this because Q has more element than P and contains all of the elements from P, However, why U(f, Q)<=U(f, P)????

This Q is not the Q from question1

1)  Recall, a truth table for a proposition involving propositional symbols p and q uses four rows for the cases p true, q true, p true, q false, p false, q true and p false, q false (in that order). For example  the outcome for p v ¬q  is  T, T, F, T  since the expression is only false when q is true but p is false. Of course, we have the same outcome for any logically equivalent proposition including ¬(¬p ∧ q), (¬p ∧ q) → false  and q → p. Of these, q → p clearly reduces the number of symbols to a bare minimum. Find "minimal expressions" for the other 15 possible outcomes, which are listed below:

a) FFFF

b) FFFT

c) FFTF

d) FFTT

e) FTFF

f) FTFT

g) FTTF

h) FTTT

i) TFFF

j) TFFT

k) TFTF

l) TFTT

m) TTFF

n) TTTF

o) TTTT

You may only use symbols from the set p, q, → , ∧ , ∨, (,  ),  ↔ , false,  true, ¬ .  Each of those count 1 toward the length of the expression. (Note:  falseand true still count as single symbols, even though they have multiple letters.) In some of the answers, your expression might have just p in it and not q or vice versa. Of course, there are also two answers that don't have either p or q in them!

(a) Find a 3×3 matrix A such that 0 is the only eigenvalue of A, and the space of eigenvectors of 0 has dimension 1. (Hint: upper triangular matrices are your friend!)

(b) Find the general solution to x' = Ax.

PLEASE SHOW YOUR WORK CLEARLY.

Show that the Legendre polynomials P1 and P2 are orthogonal by explicit integration. Also show that when (P2)^ 2 is integrated over the full range of integration, the result is 2 /(2l+1) , where l is the order of the polynomial.

Let f: A ->B and g:B -> A be functions. Prove that if fog is one-to-one and gof is onto, then f is a bijection.

Suppose there are two lakes located on a stream. Clean water flows into the first
lake, then the water from the first lake flows into the second lake, and then water from the second
lake flows further downstream. The in and out flow from each lake is 500 liters per hour. The first
lake contains 100 thousand liters of water and the second lake contains 200 thousand liters of water.
A truck with 500 kg of toxic substance crashes into the first lake. Assume that the water is being
continually mixed perfectly by the stream.

a) Find the concentration of toxic substance as a function of time in both lakes.
b) When will the concentration in the first lake be below 0.001 kg per liter?
c) When will the concentration in the second lake be maximal?

(a) Find all positive values of λ for which the following boundary value problem has a nonzero solution. What are the corresponding eigenfunctions? X′′ + 4Xʹ + (λ + 4) X = 0, X′(0) = 0 and X′(1) = 0. Hint: the roots of its auxiliary equation are –2 ± σi, where λ = σ².

(b) Is λ = 0 an eigenvalue of this boundary value problem? Why or why not?

How could I mathematically prove these statements?

1. The sum of the first n positive odd numbers is square.

2. Two positive numbers have the same set of common divisors as do the smallest of them and their absolute difference.

3. For every prime p > 3, 12|(p 2 − 1).

a) Let σ = (1 2 3 4 5 6) ∈ S6, find the cycle decomposition of σ i for i = 1, 2, . . . , 6.

(b) Let σ1, . . . , σm ∈ Sn be disjoint cycles. For 1 ≤ i ≤ m, let ki be the length of σi . Determine o(σ1σ2 · · · σm)

A circuit consisting of a resistor, capacitor and power supply is called an RC circuit. Physics and Kirchoff’s laws imply that if Q is the charge on the capacitor, R is the resistance and E is the power supply, then R(dQ/dt) + (1/C)Q = E. Let R = 20, C = .1 and E = 100e −.1t. If there is no charge on the capacitor at time t=0, find the charge Q at any time after that.

In an RL circuit, as described in class, R = .1, L = 1 and E = 10(1 − e −.05t ). If there is no current in the circuit at time t = 0, find the current at any time after that.

Solve the initial value problem dy/dt = 1 − 5y/(150 − 2t) , y(0) = 5. NOTE: Since we only care about what is going on up to when the tank is empty (i.e., t < 75 minutes), you can assume 150 − 2t > 0.

Prove that for an nth order differential equation whose auxiliary equation has a repeated complex root a+bi of multiplicity k then its conjugate is also a root of multiplicity k and that the general solution of the corresponding differential equation contains a linear combination of the 2k linearly independent solutions

e^(ax)cos(bx), xe^(ax)cos(bx),  x^2e^(ax)cos(bx),...,  x^(k-1)e^(ax)cos(bx)

e^(ax)sin(bx), xe^(ax)sin(bx), x^2e^(ax)sin(bx),..., x^(k-1)e^(ax)sin(bx)

Using extended euclidean algorithm find f(x) and g(x) in: f(x)(x^5 + 4x^4 + 6x^3 + x^2 + 4x + 6) + g(x)(x^5 + 5x^4 + 10x^3 + x^2 + 5x + 10) = x^3+1

Suppose (X, dX) and (Y, dY ) are metric spaces. Define d : (X ×Y )×(X × Y ) → R by d((x, y),(a, b)) = dX(x, a) + dY (y, b). Prove d is a metric on X × Y .

The campaign manager for a politician who is running for reelection to a political office is planning the campaign. Four ways to advertise have been selected: TV ads, radio ads, billboards, and social media advertising buys. The costs of these are $900 for each TV ad, $500 for each radio ad, $600 for a billboard for 1 month, and $180 for each buy on social media (approximately 40,000 unique impressions). The audience reached by each type of advertising has been estimated to be 40,000 for each TV ad, 32,000 for each radio ad, 34,000 for each billboard, and 17,000 for each social media buy. The total monthly advertising budget is $16,000. The following goals have been established and ranked:

1. The number of people reached should be at least 1,500,000.
2. The total monthly advertising budget should not be exceeded.
3. Together, the number of ads on either TV or radio should be at least 6.
4. No more than 10 ads/buys of any one type should be used.
   1. Formulate this as a goal programming problem.
   2. Solve this using computer software.
   3. Which goals are exactly met and which are not?

Determine if each of the following sets of vectors U is a subspace of the specified vector space, and if so, describe the set geometrically:

1. (a) U ⊆ R2, where U = {〈x1,x2〉 : x1 = 0}

2. (b) U ⊆ R2, where U = {〈x1,x2〉 : x1x2 = 0}

3. (c) U⊆R3,whereU={〈x1,x2,x3〉:〈1,2,3〉·〈x1,x2,x3〉=0}

4. (d) U ⊆ R3, where U = {〈x1,x2,x3〉 : 〈1,2,2〉 · 〈x1,x2,x3〉 = 0 and

   〈1, 3, 0〉 · 〈x1, x2, x3〉 = 0}