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

1. Calculate the Transversality Conditions

2. and then solve the ff:

a.)  Ux +Uy =1-U

U (X, X+X^2) = SIN X

b) XUx + YUy= 4U

with initial condition u=1 on the unit circle x^2 +y^2 =1

( you will need to parameterize the circle in terms of parameter r).

What is Monte Carlo simulation? What principles underlie its use, and what steps are followed in applying it? Be detailed and give an example.

1. From historical data, Harry’s Car Wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every 5 minutes. One car at a time is cleaned in this example of a single-channel waiting line. Assuming Poisson arrivals and exponential service times, find the
   1. average number of cars inline.
   2. average time a car waits before it is washed.
   3. average time a car spends in the service system.
   4. utilization rate of the car wash.
   5. probability that no cars are in the system.

Please be detailed and show your work.

If we rename y=x^2, f(x)=x^2, could we rename the inverse f^-1(x)? Explain your answer

The Discounted Cash Flow methodology. The difference between interest rates and different compounding methods. The concepts of present value and future value as well as discounting. How to determine a firm's cash flow in the context of required rates of return and compensation for risk.Please give us one example from your research, work, or personal life of an application of the above concepts.

Two events E1 and E2 are said to be independent if p(E1∩E2) =p(E1)p(E2). Two (fair, standard) dice are rolled, and their sum is found.

(a) Are the events “the sum is even” and “the sum is greater than 7” independent?

(b) Are the events “the sum is a multiple of 3” and “the sum is greater than 7”independent?

Suppose we assume the logic has just 5 statement letters (i.e. Boolean variables): A, B, C, D, and E.

E1. Show a model that is a counterexample to (A ^ B) |= (B --> (not C))?

E2. Expand the definition of entailment in the following two statements, explaining without using "entails" what they mean:

a. P |= (Q v R)

b. either (P |= Q) or (P |= R) or both Here, P, Q, and R are arbitrary formulas.

E3. Prove that 2a and 2b are not necessarily the same by finding formulas P, Q, and R for which one is true and the other is false.

Prove that for n ⩾ 2 there are exactly two n-vertex graphs with n − 1 distinct degrees (up to isomorphism). The other answers on the website are incorrect.