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).

show that if H is a p sylow subgroup of a finite group G then for an arbitrary x in G x^-1 H x is also a p sylow subgroup of G

1. For this question, we define the following vectors: u = (1, 2), v = (−2, 3).

(a) Sketch following vectors on the same set of axes. Make sure to label your axes with a scale. i. 2u ii. −v iii. u + 2v iv. A unit vector which is parallel to v

(b) Let w be the vector satisfying u + v + w = 0 (0 is the zero vector). Draw a diagram showing the geometric relationship between the three vectors u, v and w.

2. Let P 1 and P2 be planes with general equations P1 : −2x + y − 4z = 2, P2 : x + 2y = 7.

(a) Let P3 be a plane which is orthogonal to both P1 and P2. If such a plane P3 exists, give a possible general equation for it. Otherwise, explain why it is not possible to find such a plane. (b) Let ` be a line which is orthogonal to both P1 and P2. If such a line ` exists, give a possible vector equation for it. Otherwise, explain why it is not possible to find such a line.

Problem 5. The operator T : H → H is an isometry if ||T f|| = ||f|| for all f ∈ H.

(a) Please, prove that if T is an isometry then (T f, T g) = (f, g) for all f, g ∈ H.

(b) Now prove that if T is an isometry then T^∗T = I.

(c) Now prove that if T is surjective and isometry (and thus unitary) then T T^∗ = I.

(d) Give an example of an isometry T that is not unitary. Hint: consider l₂(N) and the map which takes (a₁, a₂, . . .) to (0, a₁, a₂, . . .).

(e) Now Prove that if T^∗T is unitary then T is an isometry. Hint: Start with ||T f||² = (f, T^∗T f) and use Holders inequality to get ||T f|| ≤ ||f||. Next consider ||f|| = ||T ∗T f|| do get the opposite inequality.

Conception about Integral, lower sum, upper sum.. Clear writing please and follow the comment

What is the difference between lower sum and lower integral

by the textbook, lower sum is defined by the infimum but lower inegral is using supremum? how come? Please explain.

By the Real analysis