Prove combinatorially: {{n \ choose x)= {{k+1 \ choose n-1)

This question is asking to prove combinatorially whether  n multi-choose k is equal to k+1 multi-choose n-1

Thank you

Derive a three-point formula with the highest possible order (of h) to approximate f '(a) and f "(a), respectively, using f(a − 2h), f(a + h), f(a + 2h).

The data is in the nonlinear regime. The first column is plastic strain values (e) and the second column is
corresponding true stress (s) values. In the nonlinear regime, the relation between e and s is:
? = ???
Where K,n are material constants which need to be determined using curve fitting.

a). Plot e vs s on a scatter plot
b). Find the constants n and K using curve fit. We have not learned how to fit a power law in class but here
is a hint: if you take the log of the equation above, it becomes a linear equation in log(e) and log(s).

Data

USE MATLAB CODE ONLY!
USE MATLAB CODE ONLY!

THANK YOU

1) a) Let k ≥  2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.)

b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).)

2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2 for n ≥  3. See if you can recognize the sequence a_1, a_2, a_3, a_4 . . .

3) Let L_n be the graph obtained from K_n by deleting one edge. Determine τ(L_n). (Hint: Use Cayley’s formula as a starting point.)

4) Let K_p,q denote the complete bipartite graph with partite sets of sizes p and q. Use the Matrix-Tree Theorem to calculate τ(K_p,q). Hint: Find an explicit basis for ℝ^(p+q) consisting of eigenvectors of the Laplacian matrix L(K_p,q).

5) Let G = (V, E) be a connected graph, let T, T' be spanning tress of G, and let e ∈ T\T'. Prove that there exists an edge e' ∈  T'\T such that both T-e+e' and T'+e+e' are spanning trees. (This is known as the “symmetric exchange law.”)

6) Let G be a connected graph with weight function w : E(G) → ℝ_(>0)

a) Suppose that C ⊆ G is a cycle and e ∈ C is an edge of maximum weight (i.e., w(e) ≥ w(e') for all e' ∈ C). Prove that G has an MST (Minimum Spanning Tree) not containing e.

b) Use (a) to show that the following algorithm produces an MST for all G and w:

Let T := G

while T contains a cycle do:

Let C be a cycle

Let e be an edge of C of maximum weight

Set T := T - e

Return T

Let m, n be natural numbers such that their greatest common divisor gcd(m, n) = 1. Prove that there is a natural number k such that n divides ((m^k) − 1).

1. Prove that it is impossible to have a group of nine people at a party such that each one knows exactly five others in the group.
2. Let G be a graph with n vertices, t of which have degree k and the others have degree k+1. Prove that t = (k+1)n - 2e, where e is the number of edges in G.
3. Let G be a k-regular graph, where k is an odd number. Prove that the number of edges in G is a multiple of k.
4. Let G be a graph with n vertices and exactly n-1 edges. Prove that G has either a vertex of degree 1 or an isolated vertex.
5. Show that the k-cube has 2^k vertices and k2^(k-1) edges and is bipartite.
6. Prove that if G is a simple graph with at least two vertices, then G has two or more vertices of the same degree.

T41(Robert Beezer) Consider the system of linear equations LS(A,b), and suppose that every element of the vector of constants b is a common multiple of the corresponding element of a certain column of A.More precisely, there is a complex numberα, and a column index j, such that [b]_i=α[A]_ij for all i. Prove that the system is consistent.

Prove

1. For all A, B ∈ Mmn and scalar a, we have
A + B, aA ∈ Mmn.
2. For all A, B ∈ Mmn, A + B = B + A.
3. For all A, B, C ∈ Mmn, (A + B) + C = A + (B + C).

4. For each A ∈ Mmn there is a B ∈ Mmn such that
A + B = 0mn.

Assume area B is 18 km from area A and lies in porous sedimentary rock. You are at area A. An iron asteroid strikes area B at 45° with a velocity of 17 km/s. How does the asteroid affect your survival if the asteroid has diameters of 1 m, 10 m, 100 m, 1,000 m, 10,000 m, and 100,000 m?

Determine whether the following is true or false:13^⁸⁵² ≡ 7^⁹⁰⁸ mod 15

Describe the image of the circle |z − 3| = 1 under the M¨obius transformation w = f(z) = (z − i)/(z − 4). Be sure to explain why your description is correct.

could someone please help me on this problem?