Problem 8. A bipartite graph G = (V,E) is a graph whose vertices can be partitioned into two (disjoint) sets V₁ and V₂, such that every edge joins a vertex in V₁ with a vertex in V₂. This means no edges are within V₁ or V₂ (or symbolically: ∀u, v ∈ V₁, {u,v} ∉ E and ∀u,v ∈ V₂, {u,v} ∉ E).

8(a) Show that the complete graph K₂ is a bipartite graph.

8(b) Prove that no complete graph K_n, where n > 2, is a bipartite graph.

8(c) Prove that every rooted tree forms a bipartite graph.

Formulate the situation as a system of linear equations. Be sure to state clearly the meaning of each variable. Solve using the Gauss-Jordan method. State your final answer in terms of the original question. (If the system is inconsistent, answer INCONSISTENT. If the system is dependent, parametrize the solutions in terms of the parameters u and v.)

A large commercial farm needs 38,000 pounds of potash, 73,000 pounds of nitrogen, and 27,000 pounds of phosphoric acid. Three brands of fertilizer, GrowRite, MiracleMix, and GreatGreen, are available and contain the amounts of potash, nitrogen, and phosphoric acid per truckload listed in the table. How many truckloads of each brand should be used to provide the required potash, nitrogen, and phosphoric acid?

Find the first four nonzero terms in a power series expansion about x0 for a general solution to the given differential equation with the given value for x₀.

(x^2-8x)y''+5y=0, x₀=4

Prove that {??+?:?,?∈?} is dense in ? if and only if  r is an irrational number.

2. (Improved Euler’s Method ) Second, work out the first three steps by hand. Then approximate y(2) for each of the initial value problems using Improved Euler’s method, first with a step size of h = .1 and then with a step size of h = .05 using the Excel spreadsheet. (a) dy dx = 2xy, y(0) = 1 (b) dy dx = x − y x + 2y , y(0) = 1 (c) dy dx = y + x, y(0) = 1 ignore the excell portion

Problem 4: Consider the tabulated standard-form LP starting with X₁ and X₂ basic. Please apply the revised simplex method and solve the LP.

Note: in each iteration determine Matrix A, B, E, V, C, and the improving direction!

Amy and Lester are the owners of The Hardware Store on main street in a large town. Without consulting Amy, Lester hires an accountant to prepare a formal Balance Sheet, Income Statement, and Statement of Cash Flows for The Hardware Store. Amy understands the importance of having a Balance Sheet and Income Statement, but she does not understand the importance of having a Statement of Cash Flows. Amy reasons that examining the Cash account in the General Ledger is sufficient to know where the cash came from and where it was spent. Presume you are the accountant that was hired by Lester. It is your task to explain to Amy the purpose of the Statement of Cash Flows along with its uses and limitations. Please make your initial post your response to Amy.

Statistics Analysis question! BUS310

In a one-way ANOVA test, we conclude there are differences across population means if the

a. between variation is small relative to the within variation

b.  between variation is large relative to the within variation

c.     between variation and within variation are equal

Suppose a researcher sets up a design in which there are five different means and a total of 32 measurements in the study. For alpha = .01, the critical table F value is ____.

a. 4.07

b. 3.75

c. 3.78

d, 4.91

e. 4.1

The strength of a linear relationship in simple linear regression change if the units of the data are converted, say from feet to inches.

a. True

b. False

The contingency table in a chi-square test of independence between two variables has four rows and six columns representing the number of categories for each variable. The number of degrees of freedom associated with the test statistic is __________.

a.15

b. 24

c. 10

d. 20

Which of the following statements is true regarding the chi-square goodness-of-fit test for normality:

a. The test does depend on which and how many categories we use for the histogram.

b. The test is not very effective unless the sample size is large, say, at least 80 or 100.

c. The test tends to be too sensitive if the sample size is really large.

d. None of the above responses are true

e. Responses a, b and c are correct

The weakness of scatterplots is that they:

a. do not actually quantify the relationships between variables

b. only help identify outliers

c.can be misleading about the types of relationships they indicate

d. do not help identify linear relationships

A cost accountant is developing a regression model to predict the total cost of producing a batch of printed circuit boards as a linear function of batch size (the number of boards produced in one lot or batch). The slope of the accountant’s model is ______.

a. unit variable cost

b. total variable cost

c. fixed cost

d. batch size

e. total cost

A chi-square goodness-of-fit test can be used as an alternative to z-test for population proportions if there are only two classifications.

a. true

b. false

Thank you!

An ?-pyramid is created with ? blocks at the base, ? − 1 blocks on the next level, until the peak which has 1 block.

Pyramids are colored according to the following rules: (1) Each block can be red, gold, or black, and (2) All 3-block units composed of one block on top of two others must either all be the same color or all be different colors. Answer the following:

a) How many ways are there to color a 7-pyramid?

b) What is the maximum number of black blocks in a 25-pyramid that has at least one red block and at least one gold block?

Using the bisection method:
    Make a program to use this method using the following three functions and use it to find the root of this function f (x) = x * x * x-8.
a) A function so that the user between xlower and xupper that meets the value of the function has a different sign and if he does not ask for new values.
b) A function to find the root and call it bisection and perform a maximum of iterations (maximum approximations of the root). All estimated root values have to be printed.
c) A function to obtain the values of the function.

Given the prime factors p and​ q, the encryption exponent​ e, and the ciphertext​ C, apply the RSA algorithm to find ​(a) the decryption exponent d and ​(b) the plaintext message M.

I have to get d and M

Abstract Algebra

Let n ≥ 2. Show that S_n is generated by each of the following sets.

(a) S₁ = {(1, 2), (1, 2, 3), (1, 2, 3, 4), ..., (1, 2, 3,..., n)}

(b) S₂ = {(1, 2, 3, ..., n-1), (1, 2, 3, ..., n)}

Consider the integers from 1 to 10. Give the set of pairs (a, b) that corresponds to relation a ≡ b mod 1.

Prove Theorem 29.9 (Cantor). There are countably many algebraic numbers.In this project, you will prove this theorem.