6. A one-dimensional diatomic crystal contains 2 different atoms with masses ?1 and ?2, with ?1< ?2, in each of its unit cell. Assume that there is interaction between every nearest neighbor atoms that can represented by a spring with spring constant ?. Derive the phonon frequency dispersion relation ?(?) of the crystal. Indicate which one is the acoustic branch and which one is the optical branch.

2. Let ?1, ?2, ?3 be 3 independent random variables with uniform distribution on [0, 1]. Let ?? be the ?-th smallest among {?1, ?2, ?3}. Find the variance of ?2, and the covariance between the median ?2 and the sample mean ? = 1 3 (?1 + ?2 + ?3).

Consider two complementors: Firm 1 and Firm 2. Demand functions for firms 1 and 2 aregiven by

Q1¿)=1600−200P1-100P2

Q2¿)=1600−200P2-100P1

Suppose that the marginal cost functions of firms 1 and 2 are given byMC1(Q1)=MC2(Q2)=0. What price should each firm charge to maximize its profit? Showyour calculation precisely

Use the data set below to answer the question.

Find a 90% prediction interval for some value of y to be observed in the future when

x = −1.

(Round your answers to three decimal places.)

Given the following frequency distribution, the distribution is:

Answer: Which one it is?

Please answer each question in 1-2 sentences:

1). What is the beauty myth?

2). Give at least two examples of how the idea of beauty supports gender stratification.

3). Why do you think it has been especially difficult to convince men to share the housework?

A perfectly competitive firm has a production function ?(?₁, ?₂) = ?₁?₂. Input prices are given by ?₁ = 2 and ?₂ = 1.

a. Does this production function exhibit decreasing, constant, or increasing returns to scale?

b. In the short run, input factor 2 is fixed at x₂=8 and the firm chooses the optimal input quantity x^*₁ to minimize the cost of producing output y=72. Derive x₁^*.

c. Calculate the costs c_s associated with the above short-run solution.

d. In the long-run, the firm chooses the optimal input quantities x₁^* and x₂^* to minimize the cost of producing output y= 72. Write the Lagrangian function for this firm's long-run cost minimization problem.

e Derive the first order conditions for this firm's long-run cost minimization problem.

Java language.

1. write a conversion method that have 2 parameter 1. feet and 2. inches(as small as 1/8 inch)(make sure precondition, can't be smaller than 1/8 inch)

and give output as feet in decimal place.

2 Write a method to calculate the area of a rectangle that take in width and height (feet) (as small as 1/8 inch)(make sure precondition, can't be smaller than 1/8 inch) and round up to the closet feet.

Write test cases for both method in J unit (including precheck the precondition (as small as 1/8 inch))

Let S = {-3, -2, -1, 0, 1, 2, 3}. Define a relation R on S by: xRy if and only if x = y + 4n for some integer n.

a) Prove that R is an equivalence relation.

b) Find all the distinct equivalence classes of R.