Let (Z, N, +, ·) be an ordered integral domain. Let {x1, x2, . . . , xn} be a subset of Z. Prove there exists an i, 1 ≤ i ≤ n such that xi ≥ xj for all 1 ≤ j ≤ n. Prove that Z is an infinite set. (Remark: How do you tell if a set is infinite??)

Find the Laplace transform of d²y/dt²

i) A set of 4 6-tuples (“sextuplets”) is linearly independent: (always), (never), (sometimes). ii) A set of 6 4-tuples (“quadruplets”) is linearly independent: (always), (never), (sometimes). iii) A set of 4 equations with 6 unknown variables which is consistent has a unique solution: (always), (never), (sometimes). iv) A set of 4 equations with 6 unknown variables is inconsistent: (always), (never), (sometimes) v) A set of homogeneous equations is inconsistent: (always), (never), (sometimes) vi) The solution to a set of homogeneous equations is unique: (always), (never), (sometimes)

For a 2 by 2 invertible matrix A, define the condition number to be cond(A) = ||A|| ⋅ ||A||^-1. Assume that the matrix norm is defined using the Euclidean vector norm.

(a) Find two 2by2 invertible matrices B and C such that cond(B + C) < cond(B) + cond(C).

(b) Find two 2by2 invertible matrices B and C such that cond(B + C) > cond(B) + cond(C).

(c) Suppose that A is a symmetric invertible 2by2 matrix. Find cond(2A) and cond(A2) in terms of cond(A).

(d)do the results from part (c) hold if A is not symmetric? You can either prove the results, or find counterexamples.

Use Laplace Tranform in solving the ff.:

After cooking for 45 minutes, when a cake is removed from an oven, its temperature is measured at 300°F. 3 minutes later, its temperature is 200°F. The oven is preheated, and so at t=0, the cake mixture is at the room temperature of 70°F. The temperature of the oven increases linearly until t=4 minutes, when the desired temperature of 300°F is attained; thereafter the oven temperature is constant 300°F for t is greater than or equal to 4 minutes.

Solve the following:

a.) devised a mathematical model for the temperature of a cake while it is inside the oven and after it is taken out of the oven.
b.) how long will it take the cake to cool of to a room temperature of 70°F?

Prove that \strongly connected" is an equivalence relation on the vertex set of a directed graph

Is 100202345X a valid ISBN number? If not, what would the correct check digit have to be ?

Solve the congruence 121x ≡ 5 mod 350.

The more compounding periods per year, the lower the effective rate of return. T or F

The stated interest rate is the real or true rate of return on an investment T or F

Compound interest yields considerably higher interest than simple interest. T or F

Interest is the rental fee charged by a lender to a business or individual for the use of money. T or F

Exact interest method uses 365 days as the time factor denominator in the simple interest formula T or F

The total payback of principal and interest is known as compound amount of a loan. T or F

Prove the following for undirected graphs:
(a) A 3-regular graph must have an even number of vertices.
(b) The average degree of a tree is strictly less than 2.

Compute growth function and VC dimension

H ={h: R -> {-1, +1} | h(x) = 1D(x)   where D is a finite set of R}

Please explain why and how the following are true.

I. p ⇒(q∧r) is equivalent to (p ⇒q)∧(p ⇒r)

II. Let p(x) and p(x) be defined on an random/arbitrary universe of discourse. Why, in words, is (∀x)[p(x) ∧ q(x)] equivalent to[(∀x)p (x)] ∧ [(∀x)q(x)]

prove every cauchy sequence converges

4. Verify that the Cartesian product V × W of two vector spaces V and W over (the same field) F can be endowed with a vector space structure over F, namely, (v, w) + (v ′ , w′ ) := (v + v ′ , w + w ′ ) and c · (v, w) := (cv, cw) for all c ∈ F, v, v′ ∈ V , and w, w′ ∈ W. This “product” vector space (V × W, +, ·) is commonly (and more appropriately) denoted as V ⊕ W, called the direct sum of V and W. The Euclidean plane R 2 ≡ R × R is in fact R ⊕ R. (Remark. This notion of direct sum can be extended to the direct sum of finitely many vector spaces V1 ⊕ V2 ⊕ · · · ⊕ Vk in a straightforward way.)

Consider the Newton-Raphson method for finding root of a nonlinear function
??+1=??−?(??)?′(??), ?≥0.
a) Prove that if ? is simple zero of ?(?), then the N-R iteration has quadratic convergence.
b) Prove that if ? is zero of multiplicity ? , then the N-R iteration has only linear convergence.