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Graphs and Functions

Eulerian and Non-Eulerian Graphs

Let G be a connected graph that is not Eulerian. Prove that it is possible to add a single vertex to G together with some edges from this new vertex to some old vertices so that the new graph is Eulerian. Please see attachment for background and hints.

Graphs : Eulerian Trails

1. We noticed that a graph with more than two vertices of odd degree cannot have an Eulerian trail... (please see the attached file).

Graphs : Connectedness, Vertices and Edges

11. Let G be a graph with n>= 2 vertices. a) Prove that if G has at least (n-1) + 1 edges the G is connected. ( 2 ) b) Show that the result in (a) is best possible; that is, for each n>= 2, prove there is a graph with (n- 1)

Computing 'Ambiguity' Equation

1. Calculate the ambiguity function of a signal with an envelope u(t) = Bexp(-t^2)T^2). What should be the value of B that will make the signal of the unit energy? 2. Calculate the ambiguity of a signal with a complex envelope u(t) = Bexp(-t^2/T^2)exp(jpikt^2). Note that this is the same signal as above, except for the additi


Show that the 1 dimensional problem with equation of motion (FUNTION1) has a stable equilibrium point at x=1, and show that the period of small oscillations about the point is (FUNCTION2). (PLEASE SEE ATTACHMENT FOR FUNCTIONS)

Approximation of Functions

Let f(x)=invertedCOS(x) for EQUATION1 (the principal branch of EQUATION2) Find the polynomial of degree two EQUATION3 which minimizes EQUATION4. *(Please see attachment for all equations)

Proof f(x)=x

Suppose that F is a continuous function on [0,1] and f(x) is in [0,1] for each x. Prove that f(x)=x for some x.

Function : Analytically Find the Inverse

Please see the attached file for the fully formatted problems. Please, find analytically inverses of the following functions. a. f(x)= √(x-a)/(√(x-a) + √(b-x)) b. f(x)= (x-a)^2.5/((x-a)^2.5 + (b-x)^2.5) Where a and b are constants. Please, note that analytical solution does not require substitu

Vectors, graphs, and counting

Project 4A 1. Explain why vectors QR and RQ are not equivalent. 2. Explain in your own words when the elimination method for solving a system of equations is preferable to the substitution method. 1. In Washington DC, there is a large grassy area south of the White House known as the Ellipse. It is actually an ellipse wi

Shortest segment

If P is any point on the parabola y = x^2 except for the origin, let Q be the point where the normal line intersects the parabola again. Find the shortest possible length of the line segment PQ.

Continuous function

F(x)= square root of x - 2(the 2 is not a part of the square root)/ x-4 ; x=2 decide if the given function is continuous at the specified value of s.

Composite function

Obtain the composite functions f(g(x)) and g (f(x)), and find all (if any) values of x such that f(g(x)) = g (f(x)) f(x)= the square root of 2x+1, g(x)= 1-3x

Word problems, graphs and equations, etc.

1 In a class, there are 20 men and 15 women. Find the ratio of the number of men to the number of students in the class. First express the ratio as a fraction reduced to lowest terms. Then re write the ratio using a second method. 2.Determine whether each of the ordered pair is a solution of 3x-4y>7: (0,0), (3,-6), (-

Lattice Point and Distance Formula

I need help understanding the following problem set. 1. Mark a point at the intersection of two rulings. The lattice point. Seven units to the right and four units up, mark another. Use a ruler to find the distance from one point to the other, using the distance between two parallel rulings as the unit. What degree of ac

Directed Graphs and Trees

A. Is a directed graph weakly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph? b. If two trees have the same number of vertices and the same degrees, are the two trees isomorphic?

More Tree Trans

A. The length of the longest simple circuit in K5 is ???? b. If T is a tree with 999 vertices, then T has ???? edges.

Function problem

If F(x)=x/x+1 find F(x+h)-F(x)/h where h is not 0. This problem wants you to substitute F(x+h)-F(x)/h into F(x)=x/x+1.