An individual possesses 5 umbrellas which he employs in going from his home to the office, and vice versa. If he is at home at the beginning of a day and it is raining, then he will take an umbrella with him to the office provided there is one to be taken. Similarly, if he is at the office and at the end of a day it is raining, he will take one to go home (provided there is one to be taken at the office). If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning or at the end of a day with probability 0.35.

(a)Define a Markov chain for this system by the construction of the one-step transition matrix (Hint: Define the states of the chain as the number of umbrellas the individual has in the place he is at (home or office). Assume that there is a transition each time he changes places (from home to the office or vice versa)

(b)Find the steady state probabilities, by the formulation of the steady state equations.

(c) What fraction of time does the man get wet? Justify your answer.

Suppose ?⃗ (?,?)=−??⃗ +??⃗ and ? is the line segment from point ?=(2,0) to ?=(0,3).

(a) Find a vector parametric equation ?⃗ (?) for the line segment ? so that points ? and ? correspond to ?=0 and ?=1, respectively. ?⃗ (?)=

(b) Using the parametrization in part (a), the line integral of ?⃗ along ? is ∫??⃗ ⋅??⃗ =∫???⃗ (?⃗ (?))⋅?⃗ ′(?)??=∫?? ?? with limits of integration ?= and ?=

(c) Evaluate the line integral in part (b).

(d) What is the line integral of ?⃗ around the clockwise-oriented triangle with corners at the origin, ?, and ?? Hint: Sketch the vector field and the triangle.

Use the Gauss–Jordan method to determine whether each of the following linear systems has no solution, a unique solution, or an infinite number of solutions. Indicate the solutions (if any exist).

i.     x₁+ x₂ +x₄ = 3

             x₂ + x₃ = 4

       x₁ + 2x₂ + x₃ + x₄ = 8

ii.    x₁ + 2x₂ + x₃ = 4

       x₁ + 2x₂ = 6

iii.   x₁ + x₂ =1

     2x₁ + x₂=3

     3x₁ + 2x= 4

let R = Z x Z. P be the prime ideal {0} x Z and S = R - P. Prove that S^-1R is isomorphic to Q.

A sequence is just an infinite list of numbers (say real numbers, we often denote these by a0,a1,a2,a3,a4,.....,ak,..... so that ak denotes the k-th term in the sequence. It is not hard to see that the set of all sequences, which we will call S, is a vector space.

a) Consider the subset, F, of all sequences, S, which satisfy: ∀k ≥ 2,a(sub)k = a(sub)k−1 + a(sub)k−2. Prove that F is a vector subspace of S.

b) Prove that if 10,a1,a2,a3,.... is a sequence if F for which a0=a1=0 then the sequence is the zero sequence, that is ∀k ≥ 0,a(sub)k = 0

c) Prove that the vector space F has dimension at most 2.

d) Prove that the sequences given by x(sub)k = ((1+root(5))/2)^k and y(sub)k = ((1-root(5))/2)^k are both elements in F and are linearly independent.

e) Consider the sequence defined recursively by a0=0, a1=1 ∀k > 1; ak = ak−1 + ak−2 , express this sequence an as a linear combination of xn and yn.

Show by induction that for all n natural numbers 0+1+4+9+16+...+ n^2 = n(n+1)(2n+1)/6.

Assume the reader understands derivatives, and knows the definition of instantaneous velocity (dx/dt), and knows how to calculate integrals but is struggling to understand them. Use students’ prior knowledge to provide an explanation that includes the concept and physical meaning of the integral of velocity with respect to time.

Reminder: The user is comfortable with the calculations, but is struggling with the concept. To fully address the prompt, emphasize the written explanation in English over the calculation.*

A biologist must make a medium to grow a type of bacteria. The percentage of salt in the medium is given by S=0.01(x^2)(y^2)z , where S is the percentage expressed as a decimal. And where x, y, and z are the amounts in liters of 3 different nutrients mixed together to create the medium. The ideal salt percentage for this type of bacteria is 35.7%. The costs of the x, y, and z nutrient solutions are respectively 10 , 7, and 8 dollars per liter. Determine the minimum cost that can be achieved.

(Round your answers to the nearest 4 decimal places.)

In the study of population dynamics one of the most famous models for a growing but bounded population is the logistic equation dP/dt = P(a − bP), where a and b are positive constants. First find the general solution, then given a particular solution with p(0)=1. Then plot the solution in MATLAB with two sets of (a,b) values that you choose.

A metropolitan area is facing a serious problem with disposing of its waste. Its current landfill is almost full and it is looking for other sites that can fulfill its likely future needs. A landfill must not only be large enough to handle the weekly needs of the region, but has to be as environmentally benign as possible. This means that the types of materials that are placed in the landfill must not exceed certain threshold limits. And of course, officials are mainly concerned with satisfying the needs for disposal in the least costly manner possible. To simplify the problem of analyzing one particular site, the metropolitan planners have divided the region into three major parts and estimated the amount of waste (in tons) that can be transported from each part of the region (due to public opposition to the number of trucks on the local roads) per week. In addition, the amount of nonorganic material per ton deposited in the landfill must be kept at a minimum in order for the landfill to provide the maximum capacity over its useful life. It is expected that the absolute limit of non-organic material allowed per week in the landfill will be a composite 900 pounds per ton. The relevant data is shown in the following table.

a) Write out the set of linear equations needed for the problem assuming that planners expect the landfill to handle 1,850 tons per week. That is, what equation do you want to optimize and what equations would you use as constraints?

b) If planners are expecting the landfill to handle 1,850 tons per week, what is the optimal distribution of waste delivery from the three locations in the region? (Hand in the results from Excel: linear program input sheet, the Answer Report, Sensitivity Report, and Limits Report) How much will the city pay per week for the landfill service?

c) Suppose you want to do a sensitivity analysis on your analysis. In particular, you are interested in answering the following questions. How would the optimal cost change if you were able to obtain 500 tons per week from location 3 instead of the current 850 tons? Show how you would calculate this answer by referencing your sensitivity analysis form.

d) Suppose a trucking firm comes to you and says that they could lower the cost per ton of transporting waste from location 3 from $105 per ton to $95 per ton for a nominal fee (they're anxious to get the business). Using your results and sensitivity analysis from part b, what effect will this change have on (i) the optimal cost? (ii) If the firm will charge the equivalent of a weekly flat fee of $1,000, should metro officials accept this offer?

Use eulers Method with step size h=.01 to approximate the solution to the initial value problem y'=2x-y^2, y(6)=0 at the points x=6.1, 6.2, 6.3, 6.4, 6.5

Let A be a commutative ring and F a field. Show that A is an algebra over F if and only if A contains (an isomorphic copy of) F as a subring.

1. Give exact solutions for each of the recurrence relations. Find the equilibrium values. Are the equilibrium values stable?

a. x(n+1) = 1.5x(n) x(0) = 20

b. x(n+1) = -0.75x(n) + 5 x(0) = 10

c. x(n+1) = 1.2x(n) - 5 x(0) = 2