Solve the given differential equation by using an appropriate substitution. The DE is homogeneous.

=

If p is prime, then Z^*_{p is a group under multiplication.}  

Either give an example of sequences (sn) and (tn) satisfying the properties or explain why such sequences do not exist.

(a) (sn) converges, (tn) diverges, (sn + tn) converges.

(b) (sn) converges, (tn) diverges, (sntn) converges.

(c) (sn) diverges, (tn) diverges, (sntn) converges.

(d) (sn) is bounded, (tn) converges, (sntn) diverges.

(e) (sn) converges, (tn) converges, tn 6= 0 for all n, ( sn tn ) diverges.

Unless otherwise noted, all sets in this module are finite. Prove the following statements...

1. If A and B are sets then (a) |A ∪ B| = |A| + |B| − |A ∩ B| and (b) |A × B| = |A||B|.

2. If the function f : A→B is (a) injective then |A| ≤ |B|. (b) surjective then |A| ≥ |B|.

3. For each part below, there is a function f : R→R that is (a) injective and surjective. (b) injective but not surjective. (c) surjective but not injective. (d) neither injective nor surjective.

A donut shop has 10 types of donuts including chocolate. (a) How many ways are there to choose 6 donuts? (b) How many ways are there to choose 6 donuts, where at least one of the choices should include chocolate?

Unless otherwise noted, all sets in this module are finite. Prove the following statements...

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = r when 24|(k−r). If g : S→S is defined by (a) g(m) = f(7m) then g is injective and (b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C is injective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C is surjective.

4. There is a surjection f : A→B such that f −1 : B→A is not a function.

If P = (1,4) in the elliptic curve E13(1, 1) , then 4P is ????

PLEASE SHOW ALL STEPS IN DETAIL AND EXPLAIN EACH STEP...

State and prove the Weighted Mean Value Theorem for integrals.

State and prove the Gaussian Quadrature Formula. Explain (do not prove) in what sense this formula is optimal.

Prove uniqueness of (a) LU-factorisation, (b) of LDU-factorisation of a square matrix.

State and prove Simpson’s Formula with an error term.

State and prove Simpson’s Rule with an error term.

Question:

Your client is the curator of the local museum and as such is responsible for the way the various objects are displayed. Usually, the items are displayed inside the large, clear Poly(methyl methacrylate) (PMMA)1 display cabinets which are already installed in the museum, however, your client recently received a rare collection of butterflies which are housed in their own Polyethylene terephthalate (PET) glass case and she is considering which is the best way to display this item. Does she:

(i) Leave the butter collection in its glass case and display this as is on top of a stand?

(ii) Remove the butterfly collection from its case and place the collection inside the existing large display cabinet?

(iii) Leave the display in its glass case and place this inside the existing large display cabinet, in which case visitors would be looking at the butterfly collection through two layers of glass? In order to assist your client in making this decision, you do some thinking and quick calculations.

1. PMMA is also known as acrylic or acrylic glass.

a) In relation to visitors’ ability to see the collection under the glass, what is the significance of the two different refractive indices? Explain in detail.

b) Presuming the butterfly case is placed inside the museum display cabinet (as per Option

(iii)), what is the critical angle for light passing through the two layers of glass?

c) If the situation was reversed, and the butterfly case was made of PMMA and the museum display cabinet was made of PET, what would be the critical angle for light passing through the two layers of glass? Again, assume the butterfly case is placed inside the museum display cabinet.

d) After careful exploration of the situation and based on the above considerations, which option would you recommend your client select? Explain in detail.

Take the refractive index of the PET butterfly case to be 1.5750 and the refractive index of the existing PMMA museum display cabinets to be 1.4893.