Let x,y ∈ R satisfy x < y. Prove that there exists a q ∈ Q such that x < q < y.

Strategy for solving the problem

1. Show that there exists an n ∈ N⁺ such that 0 < 1/n < y - x.
2. Letting A = {k : Z | k < ny}, where Z denotes the set of all integers, show that A is a non-empty subset of R with an upper bound in R. (Hint: Use the Archimedean Property to show that A ≠ ∅.)
3. By the Completeness Axiom, A has a least upper bound in R, which we shall denote by m. Show that m ∈ A. (Hint: Refer to Problem 3 of Homework Assignment 3.)
4. Finally, show that x < m/n < y. (Hint: It is immediate from Step 3 that m/n < y. To show that x < m/n, assume that m/n ≤ x and then derive a contradiction.)

In each of problems 1 through 4: (a) Find approximate values of the solution of the given initialvalue problem at t=0.1, 0.2, 0.3, and 0.4 using the Euler methodwith h=0.1 (b) Repeat part (a) with h=0.05. Compare the results with thosefound in (a) (c) Repeat part (a) with h=0.025. Compare the results with thosefound in (a) and (b). (d) Find the solution y=φ(t) of the given problem and evaluateφ(t) at t=0.1, 0.2, 0.3, and 0.4. Compare these values with the results of (a), (b), and (c). 3. dy/dt = 0.5 -t +2y, y(0) = 1 Please also provide solution using Matlab.

Consider the following 3-person encryption scheme based on RSA. L (can be trusted in this case) generates two large primes p and q, calculates both n and φ(n). L also chooses k1, k2 and k3 such that GCD(ki,n) = 1 and k1k2k3 ≡ 1 mod φ(n). Keys are securely distributed to three others as follows:

G: <n,k1,k2 >

J: < n, k2, k3 >

Z: < n, k3, k1 >

Answer the following questions.

(a) G has a message M1 for J. Give the encryption function for G as well as the decryption function for J, so that the message won’t be seen by anyone else.（Detailed steps）

(b) J has a message M2 for both G and Z. Give the encryption function for J, as well as decryption functions for both G and Z, so that the message won’t be seen by any other person.（Detailed steps）

2.2.6. Let S be a subset of a group G, and let S^-1 denote {s^-1: s ∈ S}.
Show that 〈S^-1〉 = 〈S 〉. In particular, for a ∈ G, 〈a〉 = 〈a^-1〉, so also
o(a) =o(a^-1)

Reduce the following modular arithmetic without the use of a calculator: PLEASE STATE THE THEOREMS/RULES YOU USE AND EXPLAIN HOW. Thanks!!

a) 104^5 mod 2669

b) 11^132 mod 133

c) 2208^5 mod 2669

d) 7^1000 mod 5

e) 2^247 mod 35

A school has a principal, many students, and many teachers. Each of these persons has a name, birthdate, and may borrow and return books. Teachers and the principal are both paid a salary; the principal evaluates the teachers. A school board supervises multiple schools and can hire and fire the principal for each school. A school has many playgrounds and rooms. A playground has many swings. Each room has many chairs and doors. Rooms include restrooms, classrooms, and the cafeteria. Each classroom has many computers and desks. Each desk has many rulers. Given this specification, you must enhance your class diagram as flowing specification (Course Registration System). The principal can add courses. Student can select some of the courses. To add courses, student will see all the courses as illustrated in Table 1 Table 1. Example of Courses Major Name ID CRN Teacher Max Classroom Time Computer CSII CS101 12345 MM 20 B10R101 TR2,3:20 R U Mathematics Calculus2 MATH201 12346 XX 10 B10R102 MW2,3:20 R U Physics Physics II PY301 12347 DD 15 B10R101 F10:11:20 R U Note: R and U is buttons. Student can select R (Register). If a student is registered for a course, this student can select U (unregister) for this course.

1. This exercise is based on one in Hartman (2007). A pharmaceutical company needs to use a supercomputer to run simulation models as part of its research on cures for AIDS, cancer, and other diseases. The firm expects to perform thousands of simulation runs per year for the next 3 years. The firm can purchase a supercomputer for $2.5 million; the annual operating and maintenance costs are $200,000 per year, and the supercomputer can perform 15,000 runs per year. For every simulation run above 15,000 in a year, the operating costs rise $1,000 per year to cover the needed overtime. A second alternative is to outsource the simulation runs to an IT firm that offers supercomputing services on demand. They will charge the pharmaceutical company $400 per simulation run. Consider a 3-year time horizon, and assume that the number of runs per year is the same every year. The firm is not sure how many simulation runs they will need to perform each year. What is the range of total cost if the number of simulation runs varies from 10,000 to 20,000 runs per year? For what range of activity (number of simulation runs per year) is purchasing a supercomputer the lowest cost alternative?

2. Consider the supercomputer example from Exercise 1 above. The firm is not sure about some of the relevant costs. The following probability distributions reflect their beliefs about the uncertain costs: the annual operating and maintenance costs are uniformly distributed on the range [$150,000, $250,000]; the additional operating costs for simulation runs above 15,000 per year are uniformly distributed on the range [$500, $1500] (per run per year). Use the method of moments to estimate the mean and variance of the costs if the firm purchases the supercomputer and they perform 20,000 runs per year. Use Monte Carlo sampling to estimate the distribution of costs if the firm purchases the supercomputer and they perform 20,000 runs per year.

With regard to multi variable calculus can some explain line and surface integrals in detail. Thank you. Provide an example problem and solve too thanks!

let f be the function on [0,1] given by f(x) = 1 if x is different of 1/2 and 2 if x is equal to 1/2

Prove that f is Riemann integrable and compute integral of f(x) dx from 0 to 1 Hint for each epsilon >0 find a partition P so that U_p (f) - L_p (f) <= epsilon

With regard to multi variable calculus can some explain vector fields in detail. Thank you. Provide an example problem and solve too thanks!

With regard to multi variable calculus can some explain Divergence Theorem and applications in detail. Thank you. Provide an example problem and solve too thanks!

With regard to multi variable calculus can some explain multiple integration in detail. Thank you. Provide an example problem and solve too thanks!

The demand equation for a computer desk is p = −4x + 290,  and the supply equation is p = 3x + 80.

(b) Find the equilibrium quantity x and price p. (Round your answers to one decimal place.)

(x, p) =

(c) Find the price at which the buyer stops buying.
$

(d) Find the price at which the supplier stops supplying.
$

(e) Is there a shortage or surplus when the price is $110? How much?

(f) Is there a shortage or surplus when the price is $206? How much?

A trail crew is constructing a 500-ft electric fence. Fence posts are placed every 5 feet. The location is 8 miles from the parking lot, so they must hike in everything to the site. They want to have enough posts but not too many since they will have to carry them back out, Leave No Trace. What is the minimum number of posts they will need to hike in? What other consideration might the crew want to consider? Additionally, between each post a plastic marker will be placed warning that the fence is electrified. What is the minimum number of these they should pack in?