Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}.

First prove that (S,*) is a group.

Then, what can you say about the bijection μ?

A contractor builds two types of homes. The Carolina requires one lot, $160,000 capital, and 160 worker-days of labor, whereas the Savannah requires one lot, $240,000 capital, and 160 worker-days of labor. The contractor owns 300 lots and has $48,000,000 available capital and 43,200 worker-days of labor. The profit on the Carolina is $60,000 and the profit on the Savannah is $70,000.

1)Find how many of each type of home should be built to maximize profit.

aCarolina homes

bSavannah homes

2Find the maximum possible profit.

Miles driven by millennials. In 2009, the number of miles driven per year by persons aged 16–34 was 7,900. Assume that the number of miles driven was decreasing by 300 miles per year.

a. Model this information with a linear equation.

b. Use this model to predict how many miles persons aged 16–34 will drive per year in 2019.

c. Explain why this model would not be expected to hold in 2030.

y''−2xy' + λy = 0, −∞ < x < ∞ where λ is a constant, is known as the Hermite equation, named after the famous mathematician Charles Hermite. This equation is an important equation in mathematical physics.

• Find the first four terms in each of two solutions about x = 0 and show that they form a fundamental set of solutions

• Observe that if λ is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial. Find the polynomial solutions for λ = 0,2,4,6,8,10. Note that each polynomial is determined only up to a multiplicative constant.

• The Hermite polynomial, Hn(x), is defined as the polynomial solution of the Hermite equation with λ = 2n for which the coefficient of xn is 2n. Find H0(x),...,H5(x).

• Give some plots of these polynomials

A wing-body model is tested in a wind tunnel with a flow of 100 m/s at standard sea-level conditions.

The wing area is 1.5 m^2 and the mean aerodynamic chord length is 0.45 m. Measurements of lift force, L, and moment about the center of gravity, MCG are made using the wind tunnel force balance.

The lead mass is removed and a horizontal tail without an elevator is added to the model. The distance from the airplane center of gravity to the aerodynamic center of the tail is 1.0 m.

The area of the tail is 0.4 m2 and the tail-setting angle is -2.0°. The tail has a lift-curve slope of 0.12 per degree. Experimental measurements give the downwash in terms of angle of attack, ε = α 0.42 , i.e. the downwash is zero for zero angle of attack, 0
ε = 0

For the case where the angle of attack, α = 5.0°, and the lift, L= 4134 N:

(i) Calculate MCG
(ii) Determine whether the model possesses longitudinal static stability. Calculate Mo
(iii) Calculate the location of the neutral point and the static margin if h = 0.26.
(iv) An elevator is added to the horizontal tail with the property / 0.04 L E T
∂ ∂ = C δ per degree.

In two ways calculate the elevator deflection angle required to trim the aircraft at an angle of attack of 8.0°. Answer should be the same.

Construct your own valid DEDUCTIVE arguments by applying the FIVE argument forms (rules) on the worksheet below. You will need to insert your own example for each rule, following the form of the argument. A TRANSLATION KEY MUST BE PROVIDED FOR EACH EXERCISE (see above for an example).

FORMS/RULES:

Modus Ponens

1) If p, then q.

2) p.

-------------------

3) Thus, q.

Modus Tollens

1) If p, then q.

2) Not q.

-------------------

3) Thus, not p.

Hypothetical Syllogism

1) If p, then q.

2) If q, then r.

---------------------------

3) Thus, if p, then r.

Disjunctive Syllogism

1) p or q.

2) Not p.

---------------

3) Thus, q.

Dilemma

1) p or q.

2) If p, then r.

3) If q, then s.

-------------------

4) Thus, r or s.

Show that the number of labelled simple graphs with n vertices is 2^n(n-1)/2. (By induction way!)

Use the annihilator method to write the form of the particular solution but do not solve the differential equation

1. y”+y=e^(-x)+x^2

2. y”’+y=xe^(2x)cosx+sinx

Find the general solution

1. y”-2y’-8y=4e^(2x)-21e^(-3x)

Find the general solution of the given differential equation. Then find the solution that passes through the given initial solution

1. y”+6y+10y=3xe^(-3x)-2e^(3x)cosx, y(0)=1, y’(0)=-2

Mathematics real analysis(please follow the comment)

Theorem: There exists a unique positive real number a is in R, satisfying a^2=2

Please proof it. Hint(you need to prove Uniqueness and Existence. For Existence(you need to let a=supA and prove contradicts a^2<2, a^2>2 also use the Archimedean property) Show every steps. Use contradiction proof

LINEAR ALGEBRA Prove or disapprove (no counterexamples):

Given that sets A = {u₁....u_n} and B = {v₁....v_n} are subsets vectors of a vector space V.

If the vectors in A and B, respectively, are independent and dim(Span(u₁....u_n)) = dim(Span(v₁....v_n))

then Span(u₁....u_n) = Span(v₁....v_n)

(a) Suppose K is a subgroup of H, and H is a subgroup of G.
If |K|= 20 and |G| = 600, what are the possible values for |H|?
(b) Determine the number of elements of order 15 in Z30 Z24.

Transformational, Authentic, Adaptive and Servant Leadership In the prior questions you shared your opinions regarding your behavior as a leader and your leadership style. . Discuss some of the leadership behavior you believe marketing leaders should demonstrate and explain why it is important marketing leaders demonstrate those specific behaviors. 100 word minimum

choose an application of multivariable calculus to your field of study (or another interest) and make an infographic about that application. Can you help me find a topic about multivariable calc?

Consider the matrix A = [2, -1, 1, 2; 0, 2, 1, 1; 0, 0, 2, 2; 0, 0, 0, 1]. Find P, so that P^(-1) A P is in Jordan normal form.