1)In your own words describe what is the Dirac delta function. (Keep in mind that it is actually not a function in the traditional sense.

2)Explain why Dirac delta function is not a function

3)Write down the Laplace transform of Dirac delta function and explain how we compute this Laplace transform.

4)In your own words, explain the physical justification of the use of Dirac delta function.

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

Find a power series solution of the given differential equation in powers of x.

x²y''+x(x-3)y'+4y=0

Let X be a set and A a σ-algebra of subsets of X. (a) What does it mean for a function f : X → R to be measurable? [2%] (b) If f and g are measurable and α, β ∈ R show that the function αf + βg is also measurable. [7%] (c) (i) Suppose that f is a measurable function. Is |f| measurable? (Give a proof or a counterexample.) [3%] (ii) Suppose that |f| is a measurable function. Is f measurable? (Give a proof or a counterexample.) [3%] (iii) Let X = R and let f(x) = 3 if x is rational and f(x) = 1 if x is not. What is the smallest σ-algebra of subsets of R with respect to which f is measurable?

Is it possible to have 60 coins, all of which are pennies, dimes or quarters, with a total worth $3. List all possibilities.

Please fully explain each step.

general solution :

5?2?′′+23??′+16,2=0

Euler-Cauchy :

?2?′′+3??′+?=0 , ?(1)=3,6 , ?′(1)=0,4

Show that projective plane is homeomorphic to sphere modulo from vector x to - vector x

An environmentalist wants to determine the relationship between the number of fires, in thousands, and the number of acres burned, in hundreds of thousands. Based on this data, decide if the correlation is significant at alpha = 0.05. Number of fires x 73 74 58 48 80 65 54 49 Number of acres burned y 64 46 22 23 51 12 29 10

5. When x = 61 what is y and what does it mean I this context.

6. Determine r^2.

7. Determine Sset.

8. Find the 95% prediction interval when x = 61.

prove that L^∞ is complete

Consider the function ?: (−1,1) × (−1,1) → ℝ given by ?(?, ?) = sin(?^? + ?? + ?² ).

1. Find a bound for the directional derivative of ? in any direction, i.e. find a constant ? such that |?_??(?, ?)| ≤ ? for all (?, ?) ∈ (−1,1) × (−1,1) and ? ∈ ℝ ² with |?| = 1.

Suppose K is a nonempty compact subset of a metric space X and x ∈ X.

(i) Give an example of an x ∈ X for which there exists distinct points p, r ∈ K such that, for all q ∈ K, d(p, x) = d(r, x) ≤ d(q, x).

(ii) Show, there is a point p ∈ K such that, for all other q ∈ K, d(p, x) ≤ d(q, x).

[Suggestion: As a start, let S = {d(x, y) : y ∈ K} and show there is a sequence (qn) from K such that the numerical sequence (d(x, qn)) converges to inf(S).] 63

(iii) Let X = R \ {0} and K = (0, 1]. Is there a point x ∈ X with no closest point in K? Is K closed, bounded, complete, compact?

(iv) Let E = {e0, e1, . . . } be a countable set. Define a metric d on E by d(ej , ek) = 1 for j not equal k and j, k not equal 0; d(ej , ej ) = 0 and d(e0, ej ) = 1 + 1/j for j not equal 0. Show d is a metric on E. Let K = {e1, e2, . . . } and x = e0. Is there a closest point in K to x? Is K closed, bounded, complete, compact?

3. For each of the following relations on the set Z of integers, determine if it is reflexive, symmetric, antisymmetric, or transitive. On the basis of these properties, state whether or not it is an equivalence relation or a partial order.

(a) R = {(a, b) ∈ Z 2 ∶ a 2 = b 2 }.

(b) S = {(a, b) ∈ Z 2 ∶ ∣a − b∣ ≤ 1}.

proof: L t^(n+1)*f(t)=(-1)^(n+1)*(d^(n+1)/ds^(n+1))*F(s)

Consider the group homomorphism φ : S₃ × S₅→ S₅ and φ((σ, τ )) = τ .

(a) Determine the kernel of φ. Prove your answer. Call K the kernel.

(b) What are all the left cosets of K in S₃× S₅ using set builder notation.

(c) What are all the right cosets of K in S₃ × S₅ using set builder notation.

(d) What is the preimage of an element σ ∈ S₅ under φ?

(e) Compare your answers in parts (b)-(d). What do you notice?