• Consider the group Z/mZ ⊕ Z/nZ, for any positive integers m and n that are divisible by 4. How many elements of order 4 does G have, and why?

4. Use the Euler Characteristic Theorem to prove that if G is drawn in the plane with n connected components, then IVI - |E| +|F| = n +1

Give examples of derangements of {1,2,3,4,5,6,7,8,9} of order 3 and of order 20. (A permutation in Sn is a derangement if it has no fixed points).

AllElectronics carries 1000 products, P1, . . . , P1000.

Consider customers Ada, Bob, and Cathy such that Ada and Bob purchase three products in common, P1, P2, and P3. For the other 997 products, Ada and Bob independently purchase seven of them randomly.

Cathy purchases 10 products, randomly selected from the 1000 products.

- In Euclidean distance, what is the probability that dist(Ada,Bob) > dist(Ada,Cathy)?

- What if Jaccard similarity is used?

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t

y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3

For the IVPs above, make a log-log plot of the error of Backward Euler and Implicit Trapezoidal Method, at t = 1 as a function of hwithh=0.1×2−k for0≤k≤5.

The spread of a highly contagious virus in a high school can be described by the logistic function

g(x)=43/1+6.40e^−0.36x

where x is the number of days after the virus is identified in the school and​ g(x) is the total number of people who are infected by the virus.

​(a) How many students had the virus when it was first​ discovered?

​(b) Graph the function for

0≤x≤50.

​(c) Determine when the virus will infect

24 students.

​(d) According to this​ model, will the virus infection level off at any point in the​ future?

Consider a mass-spring system with an iron ball (weight 16 pound force) that stretches 8/9 ft with undamped motion. The spring is initially displaced 6 inches upwards from its equilibrium position and given an initial velocity of 1 ft/s downward. Assume the relation mg=hk.

a. Find the displacement at any time t.

b. Find the natural frequency of the mass-spring (i.e. iron ball-spring) system.

c. How many cycles per minutes will the system execute?

d. What would be the amplitude and phase shift?

e. What will its motion be at t= 10 seconds if we pull the ball down from rest by 1.5ft and let it start with zero initial velocity?

d. Now examine two cases with a damping factor of 12 lb/sec and 75 lb/sec with same initial conditions.

Number Theory:

Let p be an odd number. Recall that a primitive root, mod p, is an integer g such that g^p-1 = 1 mod p, and no smaller power of g is congruent to 1 mod p. Some results in this chapter can be proved via the existence of a primitive root(Theorem 6.26)

(c) Given a primitive root g, and an integer a such that a is not congruent to 0 mod p, prove that a is a square modulo p if and only if a = g^e for an even number e. Use this to prove Euler's criterion: a is a square mod p if and only if a^(p-1)/2 = 1 mod p.

i)prove that a cylindrical co-ordinate system is orthogonal.                                     iii) Express the velocity v and acceleration of a particle in cylindrical co-ordinates.                                                   iii) find the square of the element of arc length in cylindrical co-ordinates and determine the corresponding scale factors.                                                           iv) The transformation from retangular to cylindrical co-ordinates is defined by the transformation:.                                   x=pcos#,y=psin#, z=z                                    find the Jacobian of the transformation

Plot π(x), Li(x),x/ln(x) in Mathematica for each of the following ranges: 2≤x≤10,000; 10,000≤x≤20,000; and,100,000≤x≤110,000.

Suppose G is a group and H and H are both subgroups of G.

Let HK={hk, h∈H and k ∈K}

a.give a example such that |HK| not equal to |H| |K|

b. give a example to show f :HK →H ⨯K given by f(hk) = (h,k) may not be well defined.

Hello!

Is someone able to solve the integral of the following function?
f(x) = x sin(pix/2)/(x^4+4).
The boundaries are -inf to +inf.

Thank you!

Prove that the range of a matrix A is equal to the number of singular non-null values of the matrix and Explain how the condition number of a matrix A relates to its singular values.