How many ±1-sequences of length n are there?For example, (1, −1, −1, 1) is not a happy sequence, because although 1 ≥ 0 and 1 − 1 ≥ 0,the sum 1−1−1 is negative, so the condition fails for k = 3.

Think of a scenario where you'd want to know if the graph that represents that scenario has an Euler Path, Euler Circuit or neither?

Problem 2.55. Consider the dihedral group D3 introduced in Problem 2.21. To give us a common starting point, let’s assume the triangle and hole are positioned so that one of the tips of the triangle is pointed up. Let r be rotation by 120◦ in the clockwise direction and let s be the reflection in D3 that fixes the top of the triangle.

(a) Describe the action of r −1 on the triangle and express r −1 as a word using r only.

(b) Describe the action of s −1 on the triangle and express s −1 as a word using s only.
(c) Prove that D3 = hr, si by writing every element of D3 as a word in r or s.

(d) Is {r, s} a minimal generating set for D3 ?

(e) Explain why there is no single generating set for D3 consisting of a single element. This proves that D3 is not cyclic.

It is important to point out that the fact that {r, s} is a minimal generating set for D3 does not imply that D3 is not a cyclic group. There are examples of cyclic groups that have minimal generating sets consisting of more than one element (see Problem 2.70).

a) Suppose that A = A^T can be row reduced without row swaps. If E is an elementary matrix such that EA has zero as a second entry in the first column, what can you say about EAE^T?

b) Use step a) to prove that any symmetric matrix that can be row reduced without swaps can be written as A = LDL^T

P.s: L is the lower triangular matrix whose diagonal contains only 1. D is a diagonal matrix whose diagonal contains the pivots of A. Originally, the triangular factorization of A is A = LDU (U is the upper triangular whose diagonal contains only 1), but since A is a symmetric matrix, it can be rewritten as A = LDL^T (U = L^T)

PLEASE HELP ME WITH THIS QUESTION. I HAVE BEEN SPENDING HOURS SOLVING IT AND I GOT STUCK. THANK YOU VERY MUCH FOR YOUR HELP!

Solve using judicious guessing.

y''+4y = t*sin(2t)

Use systematic elimination to solve the system

??/?? + ??/?? = −? + ? + 3

??/?? = −4? − 3? − 1

How to proof that the 2-partition problem can be transformed to 3-partition problem and the time complexity of the transformation

(i.e. the 2-partition problem can be solved by using an algorithm that solves the 3-partition problem)

Explain why an FMEA online excel template is much easier to find than a Tolerance analysis (TA) online excel template?

1) Use the simplex method to solve the linear programming problem. Maximize P = 6x + 5y subject to 3x + 4y ≤ 34 x + y ≤ 10 3x + y ≤ 28 x ≥ 0, y ≥ 0   The maximum is P = at (x, y) =

2) Use the simplex method to solve the linear programming problem. Maximize P = x + 2y + 3z subject to 2x + y + z ≤ 21 3x + 2y + 4z ≤ 36 2x + 5y − 2z ≤ 15 x ≥ 0, y ≥ 0, z ≥ 0   The maximum is P = at (x, y, z) =

Which of the following vectors does not belong to span{ (2,0,3), (3,0,2) } ?

a) (5,0,5)

b) (1,0,-1)

c) (3,2,0)

d) (0,0,5)

1. Let R be the relation on A = {1, 2, 3, 4, 5} given by R = {(1, 1),(1, 3),(2, 2),(2, 4),(2, 5),(3, 1),(3, 3),(4, 2),(4, 4),(4, 5),(5, 2),(5, 4),(5, 5)}.

(a) Draw the digraph which represents R.

(b) Give the 0 -1 matrix of R with respect to the natural ordering.

(c) Which of the five properties (reflexive, irreflexive, symmetric, antisymmetric, transitive) does R have? Give a brief reason why or why not each property holds.

2. Let A = {1, 2, 3, 4}, B = {α, β, γ}, and C = {x, y, z}. Further suppose S = {(1, γ),(2, α),(2, γ),(3, β),(3, γ)} and R = {(α, x),(α, y),(β, z)}.

(a) Compute the composition relation R ◦ S. Hint: It may be helpful to draw bipartite graphs.

(b) Is the relation R ◦ S a function from A to B? Why or why not?

Let G be a finite group and p be a prime number.Suppose that pr divides the order of G. Show that G has a proper subgroup of order pr.

Use Theorem 3.5.1 to find the general solution to each of the following systems. Then find a specific solution satisfying the given boundary condition.

a. f1′=2f1+4f2,f1(0)=0 f 2′ = 3 f 1 + 3 f 2 , f 2 ( 0 ) = 1

c. f1′= 4f2+4f3 f2′= f1+f2−2f3 f 3′ = − f 1 + f 2 + 4 f 3 f1(0) = f2(0) = f3(0) = 1

Prove the following more general version of the Chinese Remainder Theorem: Theorem. Let m1, . . . , mN ∈ N, and let M = lcm(m1, . . . , mN ) be their least common multiple. Let a1, . . . , aN ∈ Z, and consider the system of simultaneous congruence equations    x ≡ a1 mod m1 . . . x ≡ aN mod mN This system is solvable for x ∈ Z if and only if gcd(mi , mj )| ai − aj for all i 6= j, and the solutions are precisely given by one congruence class x ≡ b mod M. Hint: Regarding existence: For x ≡ ai mod mi and x ≡ aj mod mj , argue by reducing further modulo gcd(mi , mj ) that gcd(mi , mj )| ai − aj is a necessary condition for existence. To prove sufficiency of this condition, first treat the case N = 2. In that case, reduce the problem by the prime factors of m1 and m2 and thereby consolidate to a single system of congruence equations with coprime moduli to which the standard Chinese Remainder Theorem can be applied. This establishes existence for N = 2. Then proceed to treat the general case N > 2 by induction with respect to N. At some point, you will probably have to apply the identity lcm(gcd(m1, mN+1), . . . , gcd(mN , mN+1)) = gcd(lcm(m1, . . . , mN ), mN+1) which is valid in view of Problem 2 (this identity can be proved based on Problem 2 by induction, but you may just use it in your proof).