14.6. Say that a string x overlaps a string y if there exist strings p, q, r such that x = pq and y = qr, with q ̸= λ. For example, abcde overlaps cdefg, but does not overlap bcd or cdab.

(a) Draw the overlap relation for the four strings of length 2 over the alphabet {a, b}.

(b) Is overlap reflexive? Why or why not?

(c) Is overlap symmetric? Why or why not?

(d) Is overlap transitive? Why or why not?

1. In 1995, the enrollment at a certain university was 2400 students and increases by 11% per year.

a. What will the enrollment be in the year 2008?

b. In what year will the enrollment be 15,000?

2. The average price of a home in a certain town was $78,000 in 1990, but home prices have been falling by 4% per year. How much will the average home be worth in 2000?

3. Thermonuclear weapons use tritium, a radioactive form of hydrogen, for their nuclear reactions. Tritium decays at a rate of 5.613% per year. Safeway Labs stocks 300 pounds of Tritium.

a. How much Tritium will they have in 20 years?

b. In how many years will they have 100 pounds of Tritium?

4. A population of 200 deer is introduced to an area. If the population is growing at a rate of 4.2% per year, how many years will it take for the population to double? 25 Exponential

Please, show me all your work.

Let LaTeX: GG be an abelian group. Let LaTeX: H = { g \in G \mid g^3 = e }H = { g ∈ G ∣ g 3 = e }. Prove or disprove: LaTeX: H \leq GH ≤ G.

(a) Apply the chain rule to express ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ using ∂/∂x, ∂/∂y, and ∂/∂z.

(b) Solve algebraically for ∂/∂x, ∂/∂y, and ∂/∂z with ∂/∂ρ, ∂/∂ϕ, and ∂/∂θ when ρ does NOT equal  0 and sin ϕ does NOT equal 0. (Hint: you can use method of elimination to reduce the number of variables.)

(c) Express ∂2/∂x2 with ρ, ϕ, θ, and their partials.

a.)Find the length of the spiral r=θ for 0 ≤ θ ≤ 2

b.)Find the exact length of the polar curve r=3sin(θ), 0 ≤ θ ≤ π/3

c.)Write each equation in polar coordinates. Express as a function of t. Assume that r>0.

- y=(−9)
r=

- x^2+y^2=8
r=

- x^2 + y^2 − 6x=0

r=

-    x^2(x^2+y^2)=2y^2

r=

Suppose that a community contains 15,000 people who are susceptible to a contagious disease. If N(t) represents the number of people who have become infected in thousands, where t is time in days, and N′(t) is proportional to the product of the numbers of those who have caught the disease and of those who have not. The following logistic model can be used to model the spread of the disease.

dN/dt= 1N(15−N) dt 100

(a) Sketch the phase line (portrait) and classify all of the critical (equilibrium) points. Use arrows to indicated the flow on the phase line (away or towards a critical point).

(b) Next to your phase line, sketch a typical solution curve for the differential equation in each of the regions of the tN-plane determined by the graph(s) of the equilibrium solution(s).

(c) Solve the initial-value problem dN/dt = (1/100) N (15 − N ) , N (0) = 10 with Separation of Variables. You may leave

your solution in implicit form.

For each of the statements, begin a proof by contraposition and a proof by contradiction. This will include rewriting the statement, writing the assumptions, and writing what needs to be shown. From there, pick one of the two methods and finish the proof.

a) For all integers m and n, if m + n is even the m and n are both even or m and n are both odd.

b) For all integers a, b, and c, if a - bc then a - b. (Recall that the symbol - means “does not divide.”)

c) For all x ∈ Z, if x 2 − 6x + 5 is even, then x is odd.

2) Prove the following statement by contradiction: If a, b, and c are integers and a 2 + b 2 = c 2 , then at least one of a and b is even.

An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Each subcontractor will have enough time to work on up to two projects during the day. Each project should be completed by exactly two subcontractors. Given below are the distances between the subcontractors and the projects Projects Subcontractor A B C Westside 2 3 4 Federated 6 1 5 Goliath 5 6 6 Universal 11 2 3

a. Draw a network to represent the possible subcontractor-project assignments.

b. Develop a linear model which would be used to minimize total mileage costs.

Find a study in a professional journal which uses correlation and/or regression as at least one of the statistical processes. Write one paragraph about the purpose of the study, another paragraph about the conclusions from the study and a description of how correlation and/or regression was used in the study. Please include numerical values or the equation(s) found in the study. State complete reference information in APA format.

A particle of mass 1 is attached to a spring dashpot mechanism. The stiffness constant of the spring is 3 N/m and the drag force exerted on the particle by the dashpot mechanism is 4 times its velocity. At time t=0, the particle is stretched 1/4 m from its equilibrium position. At time t=3 seconds, an impulsive force of very short duration is applied to the system. This force imparts an impulse of 2 N*s to the particle. Find the displacement of the particle from its equilibrium position.

This question is from the dirac delta function chapter in my diff. equations book. Thanks!!

Q1-Find all possible time domain signals corresponding to the following z-transform:

X(z) = (z³ + z² + 3/2 z + 1/2 ) / (z³ + 3/2 z² + 1/2 z)

Q2-A digital linear time invariant filter has the following transfer function:

H(z) = (5 + 5z^-1) / (1 - 3/8 z^-1 + 1/16 z^-2)

a) Find the impulse response if the filter is causal.

​​​​​​Variable Identification: how do i know if they have 1 or 2 variables?

1. A study was conducted to investigate the possible link between vaccination in children and autism.

Individual: a single child

Variable 1:

QN or CL :   

Variable 2 :

QN or CL:

2. An employee at a supermarket samples shoppers to determine if a majority of them are wearing face masks.

Individual: a single shopper

Variable 1:

QN or CL :   

Variable 2 :

QN or CL:

3. Does the average movie gross more than $10,000,000?

Individual: a single movie

Variable 1:

QN or CL :

Variable 2 :

QN or CL:

4. You want to conduct a study to compare how quickly pain relief is felt for 2 different brands of aspirin.

Individual: a person with pain

Variable 1:

QN or CL :

Variable 2 :

QN or CL:

5. A realtor wants to determine if property values in a particular area have decreased since last year.

Individual: a piece of property

Variable 1:  

QN or CL :   

Variable 2 :

QN or CL:

The price of a new car is $16,000. Assume that an individual makes a down payment of 25% toward the purchase of the car and secures financing for the balance at the rate of 7%/year compounded monthly. (Round your answers to the nearest cent.)

(a) What monthly payment will she be required to make if the car is financed over a period of 36 months? Over a period of 72 months?

(b) What will the interest charges be if she elects the 36-month plan? The 72-month plan?

• Sketch a drawing that matches the values given
• Determine how to begin (Law of Sines/Law of Cosines)
• Solve for the missing values of the triangle (if possible)

1. a=4.2 in, b=2.5 in, c=5.8 in

2. b=3.9 cm, β=94.7°, α=30.6°

3. α=39.6°, a=3.7 m, c=18.4 m

4. α=41.2°, a=8.1 ft, b=10.6 ft

5. Sketch a triangle that fits the description below. Then find the area.

1. a=3 in, b=5 in, c=7 in

2. a=12.9 m, b= 6.4 m, ?=13.7°