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Graphs and Functions

Analytic function

Prove that if f(z) : H -> H is an analytic function from the upper-half plane to itself, then:|f(z) &#8722; f(z_0)|/|f(z) &#8722; (f(z_0))bar|<=|z &#8722; z_0|/|z &#8722; (z_0)bar| where z,z_0 are in H and |f'(z)|/Im(f(z))<=1/Im(z) where z is in H. When does equality hold?

Analytic function

3. Let D = {z : |z| < 1}. Suppose that f : D -> D is analytic, f(1/3) = 0 and f'(1/3) = 0. Show that |f(0)| <= 1/9.

Increasing/decreasing holomorphic functions

1. Let f(z) be a holomorphic function in the disc |z| < R1 and set M(r) = sup|f(z)|(|z|=r), A(r) = supR(f(z)) (|z|<r) where 0<=r<R_1 (a) Show that M(r) is monotonic and, in fact, strictly increasing, unless f is a constant. (b) Show that A(r) is monotonic and, in fact, strictly increasing, unless f is constant.

Parameter problem on parabolas

For a parabola y^2 = 4ax: 1. Find the equation of the tangent at P ( at^2, 2at ) on the parabola. 2. Find the point Q on the parabola so that PQ passes through the focus F ( a, 0 ) of the parabola 3. Show that the tangents at P and Q intersect on the directrix of the parabola

Slope

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through (-4,2) and perpendicular to the line whose equation is y = 1/3X + 7

Congruences, Equivalence Relations and Inverses

1. Show that a = b mod m is an equivalence relation on Z. I used = to mean "equal by definition to" and Z as integers. 2. Find the inverse of each of the following integers. r 1 2 3 4 5 6 ----------------------------------- r^-1 mod 7 3. Sh

Mappings, Injective and Surjective Functions and Cycles

1. Let f : X -> Y and g : Y -> Z be mappings. (1) Show that if f and g are both injective, then so is g o f : X -> Z (2) Show that if f and g are both surjective, then so is g o f : X -> Z. 2. Let alpha = 1 2 3 4 5 and Beta = 1 2 3 4 5 3 5 1 2 4 3 2 4 5 1 .

F(X) Positive and Negative

Consider the following graph of the function f(x) f(X) Positive and Negative. See attached file for full problem description.

How do you use linear equation in business?

In most businesses, increasing prices of their product can have a negative effect on the number of customers of the business. A bus company in a small town has an average number of riders of 1,000 per day. The bus company charges $2.00 for a ride. They conducted a survey of their customers and found that they will lose approxima

Petersen Graph Nonplanar

Show that the Petersen graph is nonplanar by a) showing that it has k3,3 as a subcontraction, and b) using the problem 1 show above part a) You don't have to solve problem 1. Can you explain about contraction. problem 1.-Let k>=3 be an integer , and let G be a plane graph of order n(>=k) and size m. a) If the length

Histograms

Require histograms of: 1 )Arrival Rate Number of Arrivals vs. Number of Occurrences This should be one consolidated histogram representing all of the observations from each data sheet. In plotting this data, the horizontal axis should be the number of arrivals during the 5-minute interval and the vertical axis should be th

Pairwise Disjoint Graphs and Clique Numbers

Let be pairwise disjoint graphs, and let . Prove that is called clique number of the graph G, is the maximum order among the complete subgraphs of G . Please see the attached file for the fully formatted problems.

Asymptotes, Tangents and Intercepts

(4) Explain why the function f (x) = =has a vertical asymptote but no vertical tangent. (5) Sketch the graph of the curve....for [0,2a). Show all special features such as vertical asymptotes, horizontal asymptotes, cusps, vertical tangents, and intercepts. See attached file for full problem description.

Horizontal and Vertical Asymptotes, Vertical Tangents and Cusps

(1) Find constants a and b that guarantee that the graph of the function defined by f(x)= will have a vertical asymptote at x = 5 and a horizontal asymptote at y = -3. (2) Find all vertical tangents and vertical cusps for each of the following functions. Justify your work. See attached file for full problem description.

Trees and Graphs : Outerplanar Graphs

What bound is given for X(G) by the theorem "for every graph G, X(G)<=1+max &(G') ,where the maximum is taken over all induced subgraphs G' of G" in the case that G is a) a tree? b) an outerplanar graph. Note: -&(G') this sign represent the minimum degree of G'. Yes it is that minimum -A graph G is outerplana

Vertex Chromatic Numbers and Betti Numbers

Prove for every graph G of order n, that n/B(G)<=X(G)<=n+1-B(G). X(G) is the minimum integer k for which a graph G is k-colorable is called the vertex chromatic number In the page 82 B(G) is defined like independent sets like you say but in the page 187 it other kind of B and it is define like Betti number and it is defi

Slopes and Intercepts

1. The x-coordinate of (-5,3) 2. The slope through the points (-2,5),(6,-3) 3. The y- intercept of 3x +7y =21 4. The slope of -10x-5y=0 5. The y-coordinate of (7,-6) 6. The x- intercept of -6x+6y=-6 7. Evaluate the function f(x)=3x-7forx=4 8. The slope through (8,6),(-4,0) 9. The y-intercept of y = 1/2x 10. The slop

Calculating Work and Slope of a Tangent

1. The natural length of a spring is 10 cm. A force of 25 N stretches it to a length of 20cm. How much work, in units of N-cm, is done in stretching it from a length of 10cm to a length of 15cm? Hooke's law for a spring is given by f=kx, where f is the force, x is the distance the spring is stretched, and k is a constant. 2.

Power Series and Holomorphic Functions

Let f(z) be holomorphic in the region |z|<=R with power series expansion f(z)=sum(n=0 to infinity) a_nz^n. Let the partial sum of the series be defined as s_N(z)=sum(n=0 to N) a_nz^n Show that for |z|less than R we have s_n(z)= 1/i2pi(integral over |w|=R of f(w)[(w^N+1 - z^N+1)/(w-z)]dw/w^N+1)

Holomorphic Functions

If f(z) is holomorphic on |z|<1, f(0)=1, and for all |z|<=1 we have R(f(z))>=0, then show that -2<=R(f'(0))<=2 keywords: holomorphisms