Find the biggest “*prime gap” (see definition) from the prime numbers between 1 and 1,000,000.

*Prime gap = the difference between two consecutive primes.

The exercise can be completed manually or with a computer program. Whichever seems easiest.

1.) A sheet of paper is cut into 6 same-size parts. Each of the parts is then cut into 6 ​same-size parts and so on.

a. After the 8th cut, how many of the smallest pieces of paper are​ there?

b. After the nth​ cut, how many of the smallest pieces of paper are​ there?

2.) Find the sum of the sequence 5+15+25+35+...65

3.)How many terms are there in each of the following sequences?

a. 1,2,2²,2³,...2²⁹⁹

b.9,13,17,21,...329

c.32,33,34,35,...432

The laser sight Jupiter uses for surveying is a little off. The mean error is 0.29m, meaning that it tends to provide measurements that are 0.29m too long. The standard deviation of the errors is 0.35m. She decides to recalibrate the device, but she wants to test it afterward to see if she made things better or worse. She collects a random sample of 43 measurements of a 100m distance. a) Identify the population of interest. b) Identify the variable of interest. What type of variable is it? c) ​​​​​​​If she measured the 100m distance BEFORE recalibrating, what would the mean of the measurements have been? d) ​​​​​​​If she wishes to assess how far off the sight is AFTER recalibrating, what parameter should she estimate? Mean error ? Std. Error .0676? e) ​​​​​​​Are the conditions for estimating the parameter you chose in part d met? What assumptions would you need to make? f)​​​​​​ Estimate the parameter you chose in part d with 99% confidence. Does her recalibration appear to have improved this situation? g)​​​​​​​ If she wishes to assess how reliable the sight is AFTER recalibrating, what parameter should she estimate? h) ​​​​​​​Are the conditions for estimating the parameter you chose in part g met? What assumptions would you need to make? i)​​​​​Estimate the parameter you chose in part g with 99% confidence. Does her recalibration appear to have improved this situation? j)​​​​​​​ Overall, do you think her recalibration made things better or worse?

sample data:

1.) Find the 100th and the nth term for each of the following sequences.

a. 1,3,5,7,...

b.70,100,130...

c.1,3,9...

d.8,8⁴,8⁷,8¹⁰,...

e. 200+6x2³¹, 200+7x2³¹, 200+8x2³¹

2. Find the first five terms in sequences with the following nth terms.

a. 2n²+6

b.4n+1

c. 10ⁿ-5

d.2n-1

1. Consider a= 〈−3,1,−2〉, b= 〈−2,0,−1〉 and c= 〈−5,4,−3〉. Find the angles between the following vectors

2. Consider a=4i−j+5k, b=−i+4k and c=i−k Find the following scalar and vector projections

A database uses 20-character strings as record identifiers. The valid characters in these strings are upper-case letters in the English alphabet and decimal digits. (Recall there are 26 letters in the English alphabet and 10 decimal digits.) How many valid record identifiers are possible if a valid record identifier must meet all of the following criteria:

• Letter(s) from the set {A,E,I,O,U} occur in exactly three positions of the string.

• The last three characters in the string are distinct decimal digits that do not appear elsewhere in the string.

• The remaining characters of the string may be filled with any of the remaining letters or decimal digits.

1. (20 pts) For each of the following statements, please circle T (True) or F (False). You do not need to justify your answer. (a) T or F? Any eigenvector of a matrix is in the column space of the matrix. (b) T or F? The number of singular values of a matrix is also its rank. (c) T or F? If A is an m × n with m < n, then the dimension of its column space is greater than the dimension of its row space. (d) T or F? A symmetric matrix is diagonalizable. (e) T or F? The null space null(A) of a matrix A is orthogonal to the column space of AT . (f) T or F? Zero can be the eigenvalue of an elementary matrix. (g) T or F? If W is a vector space spanned by 4 vectors, them the dimension of W is 4.

(a) Show that there are, up to isomorphism, exactly 8 matroids whose underlying set has three elements. Calling the elements a,b,c, exhibit, for each of these matroids, its bases, cycles and independent sets.

(b) Consider the matroid M on the set E = {a,b,c,d}, where the bases are the subsets of E having precisely two elements. Detrmine all the cycles of M, and show that there is no graph G such that M is the cycle matroid M(G).

Determine if the following set forms a subspace in R^2.

The set is (x1,x2)^t ,in other words the column vector [x1,x2]. Can you go through each axiom and show your work?, I have a lot of difficulty with these types of questions and I want to make sure I understand.Thank you in advance.

(The “conjugation rewrite lemma”.) Let σ and τ be permutations.

1. (a) Show that if σ maps x to y then σ^τ maps τ(x) to τ(y).

2. (b) Suppose that σ is a product of disjoint cycles. Show that σ^τ has the same cycle structure as

   σ; indeed, wherever (... x y ...) occurs in σ, (... τ(x) τ(y) ...) occurs in σ^τ.

Write the following complex numbers in polar form, as ??^??.

1/3+4i

Derive the variation of parameters formula for the solution of the initial value problem for a non-homogeneous, linear system of first order, ordinary differential equations in terms of a fundamental matrix of solutions of the corresponding homogeneous problem.

Solve the given system of differential equations by systematic elimination.

(x(t), y(t)) =

Solve the differential equation by variation of parameters.

y'' + 3y' + 2y =

y(x) =

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈ X. (1) Show that [−, −] defines an inner product on X. (c) Suppose Y is a subspace of Fn,1 with the property that [v,v] < 0 for all nonzero v ∈ Y. (2) If X is a subspace with property (1), prove that X + K + Y is a direct sum, where K is defined in part (a).