Let N(n) be the number of all partitions of [n] with no singleton blocks. And let A(n) be the number of all partitions of [n] with at least one singleton block. Prove that for all n ≥ 1, N(n+1) = A(n). Hint: try to give (even an informal) bijective argument.

an LTIC system is specified by the equation (D² +4D + 4)y(t) = Dx(t) with initial condition y'(0) = -4... How do I find the other initial condition?

Write down the chromatic polynomials of
(i)the complete graph K7;
(ii)the complete bipartite graph K1,6.
In how many ways can these graphs be coloured with ten colours?

Find the solution of the system

x′=−4y, y′=3x,

where primes indicate derivatives with respect to t, that satisfies the initial condition x(0)=1, y(0)=−1.

4. Prove that for any n∈Z+, An ≤Sn.

find the appropriate series solution or solutions, using the frobenius method, about the origin (x_0 = 0)

2x(x+2)y'' + y' -xy = 0

Show by induction that if a prime p divides a product of n numbers, then it divides at least one of the numbers.

Number theory course. Please, I want a clear and neat and readable answer.

5. Another equation that has been used to model population growth is the Gompertz equation dP /dt = rP ln (K P ) , where r and K are positive constants, and P(t) > 0.

(a) Sketch a clearly labeled graph of f(P), where dP/dt = f(P) (state the main facts you used to obtain your answer).

(b) Draw the phaseline. Next to it, sketch a set of integral curves in the t − P plane.

(c) State and classify all equilibria.

(d) (Extra Credit) For 0 < P < K, find P 00(t) and determine where the graph of P(t) is concave up and where it is concave down. Enter the inflection point in your graph in (a). (Enter your answer in an attached blank page.)

For each part below, find a binary relation on the set S = {1,2,3,4} that satisfies the given combination of properties

a) reflexive, symmetric, and transitive

b) not reflexive, but symmetric and transitive

c) not symmetric, but reflexive and transitive

d) not transitive, but reflexive and symmetric

e) neither reflexive nor symmetric, but transitive

f) neither reflexive nor transitive, but symmetric

g) neither symmetric not transitive, but reflexive

h) not reflexive, not symmetric, and not transitive

A student has 10 tickets to the show but she has 20 friends who want to see the show. Find the number of ways she can choose ten to give the tickets to, where:

a. There are no restrictions. Simplify your answer to a final number.

for questions b and c Do not simplify your answers. Leave in Combinatorics format.

b. Two of the friends are twins so if you invite one you have to invite the other

c. Two of the friends do not like each other so if you invite one you cannot invite the other.

Explain if the set below is a vector space given standard operations.

The set of all even functions defined on R with addition and scalar multiplication defined as follows:

1.) (f+g)(x) = f(x) + g(x) (addition)
2.) (cf)(x) = cf(x)

Write the greatest common divisor over the given filed as a linear combination of the given polynomials. That is, given f(x) and g(x), find a(x) and b(x) so that d(x) = a(x)f(x) + b(x)g(x), where d(x) is the greatest common divisor of f(x) and g(x).

(a) x^10 − x^7 − x^5 + x^3 + x^2 − 1 and x^8 − x^5 − x^3 + 1 over Q.

(b) x^5 + x^4 + 2x^2 − x − 1 and x^3 + x^2 − x over Q.

(c) x^3 − 2x^2 + 3x + 1 and x^3 + 2x + 1 over Z5.

(d) x^5 + x^4 + 2x^2 + 4x + 4 and x^3 + x^2 + 4x over Z5.

I know the GCD's are

a. x^3-1 b. 1 c. 1 d.1

Please help me write them as a linear combination.