Determine whether the following statement about graph theory is true or false.

(1) If a graph with m vertices is connected, then there must be at least m-1 edges.

(2) If a graph with m vertices has at least m−1 edges, then the graph must be connected.

(3) A simple undirected graph must contain a cycle, if it has m vertices with at least m edges.

(4) A graph must contain at least m edges, if it has m vertices and contains a cycle.

(5) The number of proper vertex colorings for any bipartite graph is at most two.

(6) The graph has an Euler tour, if all the vertices of a graph have an even degree.

(7) In a tournament graph, there always exists a directed Hamiltonian cycle.

(8) A simple undirected graph a Hamiltonian cycle, if it has an Euler tour.

(9) A simple undirected graph has an Euler tour, if it has a Hamiltonian cycle.

Let sn be a Cauchy sequence such that ∀n > 1, n ∈ N, ∃m > 1, m ∈ N such that |sn − m| = 1/3 (this says that every term of the sequence is an integer plus or minus 1/3 ). Show that the sequence sn is eventually constant, i.e. after a point all terms of the sequence are the same

Find a conformal mapping which maps the upper half-plane onto the exterior of the semi-infinite strip |Re w|< 1, Im w > 0

An m × n grid graph has m rows of n vertices with vertices closest to each other connected by an edge. Find the greatest length of any path in such a graph, and provide a brief explanation as to why it is maximum. You can assume m, n ≥ 2. Please provide an explanation without using Hamilton Graph Theory.

let l be the linear transformation from a vector space V where ker(L)=0 if { v1,v2,v3} are linearly independent vectors on V prove {Lv1,Lv2,Lv3} are linearly independent vectors in V

16. Which of the following statements is false?

(a) Let S = {v1, v2, . . . , vm} be a subset of a vector space V with dim(V) = n. If m > n, then S is linearly dependent.

(b) If A is an m × n matrix, then dim Nul A = n.

(c) If B is a basis for some finite-dimensional vector space W, then the change of coordinates matrix PB is always invertible.

(d) dim(R17) = 17.

(e) If B1 and B2 are both bases for the same vector space, then B1 and B2 have the same number of vectors.

In this quiz, use the following touchdown data for Tom Brady:

2 (a) Find the correlation coefficient, accurate to four significant figures, between the number of touchdowns, t, and the number of passing yards, y.

(b) Find the equation of the regression line, y = n*t + k, giving the line of best fit for the number of passing yards, y, as a function of the number of touchdowns, t. What is the slope, n, of this line, accurate to two decimal places?

(c) Find the equation of the regression line, y = n*t + k, giving the line of best fit for the number of passing yards, y, as a function of the number of touchdowns, t. What is the value of k for this line, accurate to two decimal places?

(d)  Use the regression line you found in 2 (b) and 2 (c) to find the number of touchdowns expected if Brady passes for 4000 yards.Use the values stored in your calculator or spreadsheet, without rounding them, and give an answer with four significant figures.

(e) Use the regression line you found in 2 (b) and 2 (c) to find the number of passing yards expected if Brady throws 45 touchdowns.Use the values stored in your calculator or spreadsheet, without rounding them, and give an answer with two decimal places.

Sketch the graph of y = x³ − 3x + 3.

Consider the differential equation
(x
2 + 1)y
′′ − 4xy′ + 6y = 0.
(a) Determine all singular points and find a minimum value for the radius of convergence of
a power series solution about x0 = 0.
(b) Use a power series expansion y(x) = ∑∞
n=0
anx
n
about the ordinary point x0 = 0, to find
a general solution to the above differential equation, showing all necessary steps including the
following:
(i) recurrence relation;
(ii) determination of all coefficients in the power series;
(iii) final form of general solution as y(x) = c1y1 + c2y2.

If the function u (x, y) is a harmonic conjugate of v (x, y) prove that the curves u (x, y) = st. and v (x, y) = stations. are orthogonal to each other. These curves are called level curves. Now consider the function f (z) = 1 / z
defined throughout the complex plane except the beginning of the axes. Draw them
level curves for the real and imaginary part of this function
and notice that they are two families of curves perpendicular to each other.

Please justify and prove each statement (Use explicitly the four axioms)

a) Prove that a finite positive linear combination of metrics is a metric (Use explicitly the four axioms). If it is infinite, will it be metric?

b) Is the difference between two metrics a metric? (d₁ - d₂)

A := {{1, 0}, {0.21, 0.79}}

B := {{1, 0, 0}, {0.21, 0.79, 0}, {0.17, 0.35, 0.48}}

are the matrices regular? ergodic? find a limiting value if applicable

There are 8 Broadway musicals and they offer a special three-night package (Friday, Saturday, Sunday nights) where one can order one ticket that is good for three different musicals on successive nights (a sequence of three different musicals). A travel agent is going to order 30 of these tickets for a tour group of 30 people. How many ways are there to order a subset of 30 such tickets with the constraint that each of the 8 musicals appears on at least one ticket?

the answer is Test A: C(P(8,3),30) C(8,1)xC(P(7,3),30) + C(8,2)xC(P(6,3),30) .. i don't understand this answer what does it mean by C(P(8,3), 30)??