Find the values of λ (eigenvalues) for which the given problem has a nontrivial solution. Also determine the corresponding nontrivial solutions​ (eigenfunctions).

y''+2λy=0; 0<x<π, y(0)=0, y'(π)=0

This is one question about 14-bit strings

1. How many 14-bit strings that have more 0’s than 1’s?
2. How many 14-bit strings that have even number of 0’s?
3. How many 14-bit strings that have no consecutive three 0’s in a row?

Granite Housing Association Granite Housing Association (GHA), a charitable organisation, is a Registered Social Landlord (RSL). Its operations are part funded by the Government. GHA is based in the South East of England and manages 30,000 properties. It was created from 7 other RSLs with portfolios ranging from 1,800 to 14,600 properties. Five of the original RSLs had their own maintenance workforce while the other two used external contractors. 30% of the inhouse workforces are also tenants. Each workforce has remained on its own terms and conditions of employment and operates independently of each other. Activities associated with the workforce fell into three main areas: 1. Reactive maintenance 2. Void management and refurbishment 3. Programmed capital works Reactive maintenance requests are initiated by the tenant requesting repairs for damage to property, broken locks, leaking windows, etc. Void management is triggered when a tenant leaves. Typically, the property needs to be made secure, which may involve boarding windows and doors and/or installation of alarms. While the property is vacant prior to a new tenant arriving and dependent upon the condition of the premises, they may need full or partial refurbishment which may include central heating upgrades, kitchen and bathroom replacements. Programmed capital works includes the significant upgrading of a number of properties in a locality which may, for example, include replacement double-glazed windows. Two years remain on the framework agreements for the supply only of plumbing materials and supply only of building supplies, and four years remain on framework contracts for the supply and fitting of alarms, the supply only of kitchen units and the supply only of glazing products. The accountant, James Andrews, has undertaken an analysis of a new government scheme to encourage renewable forms of energy and has proposed that a major capital programme can be undertaken to install solar panels on properties with a south facing roof. For 8,000 properties the scheme is financially viable, for a further 7,000 properties the cases further work would require additional investigation to determine economic viability. The programme will be beneficial both to the tenant and to Granite. However, the in-house workforce does not have the capacity to undertake this activity.

Tasks

You are the Procurement Manager for GSA and you have been asked to consider:

(a) Initiating the procurement of an externally managed programme of installation of solar panels

(b) Outsourcing the three categories of activities undertaken by the in-house workforce.

1. What would you need to consider in relation to both the above initiatives?

2. What contractual matters would need consideration in the relation to the outsourcing initiative and the existing frameworks?

The differential equation ay" + by' + cy = f(t) has characteristic equation aλ2 + bλ + c = 0 whose roots are given in each part below. The forcing function f(t) is also indicated. Sketch the form of the homogeneous solution to the differential equation. Indicate the algebraic form of a particular solution. Sketch the form of the general solution for y(0) = y' (0) = 1.

(a) λ = −2, −3, f(t) = e −2t

(b) λ = −3, −3 (repeated real root), f(t) = sin(3t)

(c) λ = ±3i, f(t) = sin(2t)

(d) λ = ±3i, f(t) = sin(3t)

(e) λ = −2 ± 3i, f(t) = sin(3t)

1. Let R be the rectangle formed by going along line segments from 1 to i to -1 to -i and back to 1. If f(z)=1/(z-5i) then the integral around R of f(z) has value of?

2. Let C be the circle of radius fifty centered at the origin with positive orientation. Then the integral around C of f(z) = 1/(z-4) has value of?

3. Let C be the circle of radius fifty centered at the origin with positive orientation

of F(z)=[1/(z-i)] + [1/(z-2)] then the integral around C of F(z) has value of?

4. Let C be the circle of radius six centered at the origin with positive orientation.

If G(z) = [1/(z-2)] + [1/(z-8)] then the integral around C of G(z) has value of?

5. The integral of f(z) = 1/[(z-5)(z-8)] around the circle of radius one centered at z=1 with positive orientation has value zero? True or false?

Define a relation ~ on Z x Z such that (a,b) ~ (c,d) precisely when a + b = c + d.

Let R = {[(a,b)] : (a,b) in Z x Z} (i.e. R is the set of all equivalence classes of Z x Z under the equivalence relation ~). For each of the following operations, determine whether or not the operation is well defined. Prove your answer.

[(x,y)] * [(w, z)] = [(x + w, y + z)]

[(x,y)] * [(w, z)] = [(x² + w², y² + z²)]

Discrete math : Show your work please.

Consider a set X of 10 positive integers, none of which is greater than 100. Show that it has two distinct subsets whose elements have the same sum.

In 2006​, there were 11,300 students at college​ A, with a projected enrollment increase of 800 students per year. In the same​ year, there were 30,900 students at college​ B, with a projected enrollment decline of 600 students per year. According to these​ projections, when will the colleges have the same​ enrollment? What will be the enrollment at that​ time?

For all integers n > 2, show that the number of integer partitions of n in which each part is greater than one is given by p(n)-p(n-1), where p(n) is the number of integer partitions of n.

f(x)=0 if x≤0, f(x)=x^a if x>0

For what a is f continuous at x = 0

For what a is f differentiable at x = 0

For what a is f twice differentiable at x = 0

Twenty calculus students are comparing grades on their first two quizzes of the year. The class discovers that every pair of students received the same grade on at least one of the two quizzes. Prove that the entire class received the same grade on at least one of the two quizzes.

Find a particular solution to the non-homogeneous differential equation:

(a) y'' − 4y = 4t^2

(b) y '' − 4y = sin(2t)

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

(d) y '' + y = cos(t)

Consider an n×n square board, where n is a fixed even positive integer. The board is divided into n 2 unit squares. We say that two different squares on the board are adjacent if they have a common side. N unit squares on the board are marked in such a way that every unmarked square on the board is adjacent to at least one marked square. Determine the smallest possible value of N.

4) Let ? = {2, 3, 5, 7}, ? = {3, 5, 7}, ? = {1, 7}. Answer the following questions, giving reasons for your answers.

a) Is ? ⊆ ??

b)Is ? ⊆ ??

c) Is ? ⊂ ??

d) Is ? ⊆ ??

e) Is ? ⊆ ??

5) Let ? = {1, 3, 4} and ? = {2, 3, 6}. Use set-roster notation to write each of the following sets, and indicate the number of elements in each set.

a) ? × ?

b) ? × ?

c) ? × ?

d) ? × ?

4-Consider the following problem:

max − 3x1 + 2x2 − x3 + x4

s.t.

2x1 − 3x2 − x3 + x4 ≤ 0

− x1 + 2x2 + 2x3 − 3x4 ≤ 1

− x1 + x2 − 4x3 + x4 ≤ 8

x1, x2, x3, x4 ≥ 0

Use the Simplex method to verify that the optimal objective value is unbounded. Make use of the final tableau to construct an unbounded direction..