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.

Prove that if the integers 1, 2, 3, . . . , 65 are arranged in any order, then it is possible to look either left to right or right to left through the list and find nine numbers that are in increasing order

function[prob,flop]=Matlab(a,b)

a=[1,2,3];
b=[7,8,9];

if (length(a)~=length(b))
disp("it doesn't make sense");
end

prob=0;
flop=0;

for i=1:length(a)
prob=prob+a(i)*b(i);
flop=flop+2;
end
disp(prob);
disp(flop);

What is wrong with my code? it only run the disp(prob) and it doesn't run the disp(flop)

here is what I got

ans =

50

However, if I take % for the function line, it shows

50

6

1.- let(X1, τ1) and (X2, τ2) are two compact topological spaces. Prove that their topological product is also compact.

2.- Let f: X - → Y be a continuous transformation, where X is compact and Y is Hausdorff. Show that if f is bijective then f is a homeomorphism.

1.- Show that (R, τs) is connected. Also show that (a, b) is connected, with the subspace topology given by τs.

2. Let f: X → Y continue. We say that f is open if it sends open of X in open of Y. Show that the canonical projection

ρi: X1 × X2 → Xi
(x1, x2) −→ xi

It is continuous and open, for i = 1, 2, where (X1, τ1) and (X2, τ2) are two topological spaces and X1 × X2 has the product topology.

Question in graph theory:

1. Let (a₁,a₂,a₃,...a_n) be a sequence of integers.

Given that the sum of all integers = 2(n-1)

Write an algorithm that, starting with a sequence (a₁,a₂,a₃,...a_n) of positive
integers, either constructs a tree with this degree sequence or concludes that
none is possible.

A terrible despot governing one small country decided to check
how smart are people living in his country. He gathered 20 smartest people
and put hats on their heads. Everybody could see all hats except their own.
Then the despot said: "Some of the hats have a red stripe on them. I will
give you one minute to think and then ask who has a red stripe on their
hat? If nobody answers, then I will give one more minute and ask the same
question again. I will repeat it 100 times. If you guess somehow that you got
stripe on the hat you have to wait till I ask my question and say immediately
about that, because after somebody will gure out correctly that he or she
has a stripe on the hat, I will kill everybody else who has the stripe and did
not gure it out. If you say that you have a stripe and you don't , I will kill
you. " They know that each of them is really good at deduction (got A for
Math 311W when they were in college) and nobody wants to die.
a) Prove that if 2 people have red stripes on their hats, then after the terrible
despot will ask them second time all of them will say that they have them,
and despot won't be able to kill anybody.
b) What happens if three people have stripes?

Find all solutions of:
(a) 4? ≡ 3 ??? 7
(b) 9? ≡ 11 ??? 26
(c) 8? ≡ 6 ??? 14
(d) 8? ≡ 6 ??? 422

Which number set can you find:

a. the inverse of integers under multiplication?

b. the inverse of natural numbers under multiplication?

For the equation e^x =x+2,
(a) use the fixed point iteration method to determine its two roots to eight correct decimal places (you may need to write this equation in two different ways of x = g(x) in order to obtain these two roots);
(b) numerically calculate the convergence rates for your converged iterations; (c) compare these numerical convergence rates with the theoretical conver- gence rates we presented in class (also see Theorem 1.6 on page 38 of the textbook).