Show that the set of rigid motions E(3) forms a group.

Define the probability density functions (PDF): Binomial, Uniform, and Normal distributions. Provide one example of each of them, and their graphs.

What is the main difference between Deterministic Models and Stochastic Models? Provide two examples for each type of models.

Find the general solution of the equations:

a) y'' + 6y' +5y = 0
b) 16y" - 8y' + 145y = 0
c) 4y" - 4y' + y = 0

Due October 25. Let R denote the set of complex numbers of the form a + b √ 3i, with a, b ∈ Z. Define N : R → Z≥0, by N(a + b √ 3i) = a 2 + 3b 2 . Prove: (i) R is closed under addition and multiplication. Conclude R is a ring and also an integral domain. (ii) Prove N(xy) = N(x)N(y), for all x, y ∈ R. (ii) Prove that 1, −1 are the only units in R.

Cardano's rule gives one root for the solution of a depressed cubic.

a. How does this solve the problem of finding all the roots of the depressed cubic equation?

b. Obtain the depressed cubic that results from the following cubic equation:

x³ + 3x² + x + 1 = 0 .

c. Now use Cardano's rule to solve your resulting depressed cubic. Don’t expect a pretty answer!

Find the linear space of eigenfunctions for the problem with periodic boundary conditions

u′′(x) = λu(x)

u(0) = u(2π)

u′(0) = u′(2π)

for (a) λ = −1 (b) λ = 0 (c) λ = 1.

Note that you should look for nontrivial eigenfunctions

y″+9y′=162sin(9t)+324cos(9t)

y(0)=7

y'(0)=7

Use the one solution given below to find the general solution of the differential equation below by reduction of order method:

(1 - 2x) y'' + 2y' + (2x - 3) y = 0

One solution: y₁ = e^x

Let V be the 3-dimensional vector space of all polynomials of order less than or equal to 2 with real coefficients.
(a) Show that the function B: V ×V →R given by B(f,g) = f(−1)g(−1) + f(0)g(0) + f(1)g(1) is an inner product and write out its Gram matrix with respect to the basis (1,t,t²).

DO NOT COPY YOUR SOLUTION FROM OTHER SOLUTIONS

How do I draw a complete subgroup diagram? The question asks me to give a complete subgroup diagram for Z₂₅^x

A Trigonometric Polynomial of order n is a function of the form: ?(?) = ?0 + ?1 cos ? + ?1 sin ? + ?3 cos(2?) + ?2sin(2?) + ⋯ + ?ncos(??) + ?nsin (??)

1) Show that the set {1, cos ? , sin ? , cos(2?) , sin(2?)} is a basis for the vector space

?2 = {?(?) | ?(?)?? ? ????????????? ?????????? ?? ????? ≤ 2}

< ?, ? > = ∫ ?(?)?(?)?? defines an inner-product on T

2) Use Gram-Schmidt to show an ONB for T is:

?0 = 1 √2? , ?1 = 1/ √? cos(?) , ?2 = 1 /√? cos(2?) ?3 = 1/√? sin(?) , ?4 = 1/ √? sin(2?)

Given any ?(?) ∈ ?2

????r? = < ?, ?0 > ?0+ < ?, ?1 > ?1+ . . + < ?, ?4 > ?4

3) Show that in ?m

????r ? = ?0 + ∑ [?n cos(??) + ?nsin (??)] from n to m, when  n=1

Where: ?0 = 1/2pi ∫ ?(?)?? from 0 to 2pi

?n = 1/pi∫ ?(?) cos(??) ?? from 0 to 2pi , ? ≥ 1

?n = 1/pi ∫ ?(?) sin(??) ?? from 0 to 2pi , ? ≥ 1

We call the ?/? and ?/? the Fourier Coefficients of ?(?)

We call the Projection the Fourier Series for ?(?)

4) Compute the ?2 series for ?(?) = ?^2 using − pi/2 < ? < pi/2. Plot your series and f(x) together

5) Compute the series for ?(?) = ?^3 using − pi/2 < ? < pi/2 . Plot your series and g(x) together

Prove the case involving ¬E of the inductive step of the (strong) soundness theorem for natural deduction in classical propositional logic.