A spring with a spring constant ? k of 100 pounds per foot is loaded with 1-pound weight and brought to equilibrium. It is then stretched an additional 1 inch and released. Find the equation of motion, the amplitude, and the period. Neglect friction. Then find y(t)

Describe the ring obtained from the product ring R×R by inverting the element (2,0).

and we already know that the ring R x R is isomorphic to R[x]/(x^2-1)

Find the first four nonzero terms in a power series expansion about x=0

(x^2+9)y''+y=0

Consider the Tragedy of the Commons game from the chapter with two shepherds, A and B, where sA and sB denote the number of sheep each grazes on the common pasture. Assume that the benefit per sheep (in terms of mutton and wool) equals 500-sA-sB implying that the total benefit from a flock of sA sheep is sA(500-sA-sB) and that the marginal benefit of an additional sheep (which is defined by the first partial derivative with respect to sA of the total benefit function) is 500-2sA-sB. Assume that the marginal cost of grazing sheep is 100.

1. Compute the flock sizes and shepherds’ total benefit in the Nash equilibrium.

2. Draw the best-response unction diagram corresponding to your solution.

3. Suppose A’s benefit per sheep drops to 450-sA-sB, which means that his marginal benefit becomes 450-2sA-sB. B’s benefit per sheep does not change. Compute the new Nash equilibrium flock sizes. Show the change from the original to the new Nash equilibrium in your best-response function diagram.

Determine if each of the following statements is true or false. If it’s true, explain why. If it’s false explain why not, or simply give an example demonstrating why it’s false. (A correct choice of “T/F” with no explanation will not receive any credit.)

(a) If a system has more equations than variables, its RREF must have a row of 0’s.

(b) If a consistent system has more variables than equations, then it must have infinitely many solutions.

(c) Consider a system with m equations, n variables, and rank r. If the system has a unique solution, then m = n.

Q. EXPLAIN COLLEY METHOD AND GIVE EXAMPLE?

Required: Formulate and solve the LP Relaxation of the problem. Solve it graphically, and round down to find a feasible solution.

Explain/show what excel parameters and cells should be entered into the excel to come up with the answer.

Consider the following all-integer linear program:

??? 10?1 + 3?2
?.?.
6?1 + 7?2 ≤ 40
3?1 + 1?2 ≤ 11
?1,?2 ≥ 0 and integer

y'=√2x+y+1

general solution of differential equation

Prove that a continuous function for sections(subrectangles) is integrable in R^N

In the proof of bolzano-weierstrass theorem in R^n on page 56 of "Mathematical Analysis" by Apostol, should the inequality be a/2^(m-2) < r/sqrt(n) or something related to n? a/2^(m-2) < r/2 seems not enough

How is truth found in philosophy and how is that different than in science?

How is truth found in math and how is that different than in science?

Please include the ideas of experimentally based vs. non-experimentally based.

Consider the following linear operator: L u = d^2u/dx^2+ 2 du/dx + u.

Consider the eigenvalue problem L u + λu = 0, x ∈ (0, π), u(0) = 0, u(π) = 0.

(a) Determine the possible eigenvalues and eigenfunctions.

(b) Use an integrating factor to put the eigenvalue problem in Sturm-Liouville form.

(c) What is the appropriate inner product for this system?

Show that, for each value of n, the graph associated with the alcohol C_nH_2n+1OH is a tree (the oxygen vertex has degree 2). Draw the tree corresponding to the molecule C₂H₅OH.