Use Euclid’s algorithm to find integers x, y and d for which 3936x + 1293y = d is the smallest possible positive integer. Using your answers to this as your starting point, do the following tasks.

(a)Find an integer s that has the property that s ≡ d mod 3936 and s ≡ 0 mod 1293.

(b) Find an integer S that has the property that S ≡ 573 mod 3936 and S ≡ 0 mod 1293.

(c) Find an integer T that has the property that T ≡ 126 mod 1293 and T ≡ 573 mod 3936.

(d) Is T the only number satisfying those two congruences; if not, which other numbers?

Prove that Desargues' configurations satisfy the principle of duality.

Use separation of variables to find a series solution of u_tt = c ²u_xx subject to u(0, t) = 0,

u_x(l, t) + u(l, t) = 0, u(x, 0) = φ(x), & u_t(x, 0) = ψ(x) over the domain 0 < x < `, t > 0. Provide an equation that identifies the eigenvalues and sketch a graph depicting this equation. Clearly identify the eigenfunctions

1. prove

s(n, k) = s(n − 1, k − 1) − (n − 1)s(n − 1, k).

2. What is ∑n k=0 s(n, k)?

Let L be a linear map between linear spaces U and V, such that L: U -> V and let l_{ij} be the matrix associated with L w.r.t bases {u_i} and {v_i}. Show l_{ij} changes w.r.t a change of bases (i.e u_i -> u'_i and v_j -> v'_j)

Use Laplace transforms to solve x''− 7x' + 6x = e^t + δ(t − 2) + δ(t − 4), x(0) = 0, x'(0) = 0.

Prove or provide a counterexample

If f is T_C−T_U continuous, then f is T_U−T_C continuous.

Where T_C is the open half-line topology and T_U is the usual topology.

show that an integer n > 4, is prime iff it is not a divisor of (n-1)!

Prove that for all integers n ≥ 2, the number p(n) − p(n − 1) is equal

to the number of partitions of n in which the two largest parts are

equal.

how does 2 = 26 *5 -32 * 4 becomes 26 * 5 + 32 * (-6) = 2

Write a Matlab function for:

1. Root Finding: Calculate the root of the equation f(x)=x^3 −5x^2 +3x−7

Calculate the accuracy of the solution to 1 × 10−10. Find the number of iterations required to achieve this accuracy. Compute the root of the equation with the bisection method.

Your program should output the following lines:

• Bisection Method: Method converged to root X after Y iterations with a relative error of Z.

Please solve this with only transformations (I.E. Translation matrices, Reflection matrices, Rotation matrices. You need to use matrices (linear algebra). Please only do this if you know what I mean when I say "transformations using matrices."

A 2-D polygon has the following vertices in (x, y): A (2, 1), B (1, 2), C (2, 3), D (3, 3), E (4, 2), and F (3, 1). The polygon is taken through a set of 3 transformations in the given order: 1. Reflection about the line x=4 2. Translation by -3 in x direction. 3. Rotation counter clockwise about the point (2.5, 2) by 180 degrees . Determine the final coordinates.

The Fibonacci sequence is an infinite sequence of numbers that have important consequences for theoretical mathematics and applications to arrangement of flower petals, population growth of rabbits, and genetics. For each natural number n ≥ 1, the nth Fibonacci number fn is defined inductively by

f1 = 1, f2 = 2, and fn+2 = fn+1 + fn

(a) Compute the first 8 Fibonacci numbers f1, · · · , f8.

(b) Show that for all natural numbers n, if α = 1+√5 and β = 1−√5, then fn = αn−βn .2 2 α−β

Hint: Note the relationship between α + 1 and α2 and likewise between β + 1 and β2.