For p a given prime number, define the p-adic norm | * |p as follows on Q: Given q in Q, we can write it as a product q = (p^m)(a/b) with a,b integers which are not divisible by p, and m an integer which is uniquely determined by q (check that m is indeed uniquely determined by q). Then define |q|p = p^(-m).

Check that Q with distance dp(q1,q2) = |q1 - q2|p is a metric space (here q1-q2 just means the usual operation of subtraction in Q). Show moreover
that |q1 + q2|p <= max(|q1|p, |q2|p).

Finally, consider the following fixed point iteration xk+1 = g(xk) = arccos −1 1 + e 2x and show that finding a fixed point of g(x) is equivalent to finding a root of f(x) = 0. Use the code fixedpt.m to try to approximate the root using an initial guess of x0 = −3. Can you explain why your iteration behaves as it does? Hint: Plot the fixed-point function and think convergence!

Code in fixedpt.m:-

function [xfinal, niter, xlist] = fixedpt( gfunc, xguess, tol )
% FIXEDPT: Fixed point iteration for x=gfunc(x).
%
%  Sample usage:
%     [xfinal, niter, xlist] = fixedpt( gfunc, xguess, tol )
%
%  Input:
%     gfunc   - fixed point function 
%     xguess  - initial guess at the fixed point
%     tol     - convergence tolerance (OPTIONAL, defaults to 1e-6)
%
%  Output:
%     xfinal  - final estimate of the fixed point
%     niter   - number of iterations to convergence
%     xlist   - list of interates, an array of length 'niter'

% First, do some error checking on parameters.
if nargin < 2
  fprintf( 1, 'FIXEDPT: must be called with at least two arguments' );
  error( 'Usage:  [xfinal, niter, xlist] = fixedpt( gfunc, xguess, [tol] )' );
end
if nargin < 3, tol = 1e-6; end

% fcnchk(...) allows a string function to be sent as a parameter, and
% coverts it to the correct type to allow evaluation by feval().
gfunc = fcnchk(gfunc);
x = xguess;
xlist = [ x ];

niter = 0;
done  = 0;
while ~done,
  xnew  = feval(gfunc,  x);
  xlist = [ xlist; xnew ];  % create a list of x-values 
  niter = niter + 1;
  if abs(x-xnew) < tol,     % stopping tolerance for x only
    done = 1;
  end
  x = xnew;
end
xfinal = xnew;

show that if I is uncountable,then 2^I is not metrizable.

hint: Suppose I as index set.

3-How can digital signal processing affect the quality of sound?

U(C1, C2, C3, C4, C5) = C1∙C2∙C3∙C4∙C5

As a mathematical function, does U have a maximum or minimum value? What values of Ci correspond to the minimum value of U? What values of Ci correspond to the maximum value of U? Do these values of Ci make sense from an economic standpoint?

Now let us connect the idea of economic utility to actual dollar values. To keep the values more manageable, we will use household income rather than the entire state budget, and retail costs and measures rather than industrial ones. Find the Median Household Income for Mesa, AZ for the most recent year possible. Then find the dollar cost in Mesa, AZ for a Penny, a pound of Ground Beef, a pair of Jeans, fresh Orange Juice, and a Movie Ticket. (Entertainment is often used as a stand-in for Climate.) A Cost-of- Living Index is a good place to find much of this data. Record these prices as P1, P2, P3, P4, and P5 respectively.

Construct an equation using Median Income, the Ci and Pi values that illustrates how much of each resource the Median Household can afford to purchase. Given this restriction, do the maximum or minimum values of U change? Do the values of Ci that give the maximum or minimum values change? What are these new values? How should the Median Household budget its Income so as to maximize its Economic Utility?

Write up your findings in a paper that you could turn in to an employer. Be sure to show all your work. Include any appropriate references as well as any computational devices used.

* Solve the questions, make the table, and the graph.

Let K_n denote the simple graph on n vertices.
(a) Let v be some vertex of K_n and consider K _{n − v}, the graph obtained by deleting
v. Prove that K _{n − v} is isomorphic to K n−1 .
(b) Use mathematical induction on n to prove the following statement:
K _n , the complete graph on n vertices, has n(n-1)/2
edges

The town hall of a city wants to open some recreational centers. It has been analyzed 3 options. The opening cost and the capacity of each center are listed below.

The selected recreational centers must be hosting the students from 5 schools. In the table below is summarized the number of students at each school.

Each school must be assigned to only one recreational center. And the capacity of each center must be respected. What are the recreational centers that must be open in order to minimize the opening cost?

Illustrate the greedy procedure with the following data:

(Modern Algebra) Show that if G is a finite group that has at most one subgroup for each
divisor of its order then G is cyclical.

Evaluate if the followings are Cauchy sequences or not.

(a) a_n= (-1)ⁿ

(b) a_n= (-1)ⁿ/n

(c) a_n = n/(n+1)

(d) a_n = (cos n)/n

Solve using Laplace and Inverse Laplace Transforms.

Y’’’-y’’-4y’+4y=0

y(0)=1 y’(0)=9 y’’(0)=1

Question1 (50 pts): LU Factorization Code for Square Matrices without Row Exchange

I want you to write an LU decomposition program in Matlab for square matrices (n×n) where row exchange is not necessary (that is no pivot is 0). Here are some hints and requirements for your matlab code.

1. You should write comments for every procedure. Make sure that your code is well indexed (see below). Otherwise it will be hard for me to follow and bad programming practice for you.

2. This should be a Matlab Function called slu (don’t name it lu because matlab already has a program called lu):

3. Your program should be a function in Matlab. So your filename should be slu.m (.m is standard for matlab scripts or functions). In order to be a function you should start with

   function [L,U]=slu=slu(A)

   where A is the input square matrix. Always write what a program does using matlab

comment (%) so that it

4. First step is to determine the size of the matrix. Use size command to obtain the size of a matrix.

   [n,m]=size(A)

5. Then if n is not equal to m end the program. If n=m continue

6. So your code should look like this

   function [L,U]=slu=slu(A)
   %LU factorization of square matrices %with no row exchange
   [n,n]=size(A) %determine the size of A Tol=1.e-6 %tolerance level for zero

   for k=1:n %loop over pivots of A if(A(k,k) < tol) %check for pivot

   disp(‘cannot proceed without row exchange’) end %cannot proceed without row exchange L(k,k)=1;

for i=k+1:n %loop over each line to determine lik determine lik

for j=k:n start from row k+1 and col. k remove (pivot row×Lik) from each row of A

end

for j=k:n
write the pivot line to Ubecause pivot is fixed

end end

due 23/10/2019

Question2(50 pts): Solving matrix equation using LU

Next step is to solve the equation using substitution. So you will need to do the standard procedure for LU. First step is to solve Ly=b. And next is to solve Ux=y. For Ly=b you need to go from top to bottom, but for Ux=y you need to go from bottom xn to x1. This function will use slu function so you need to be on the same directory. Name your file “luslv.m”.

Function x=luslv(A,b)
%Solve Ax=b using L&U from slu(A) %No row exchanges

[L,U]=slu(A) %first decompose A to LU

First do a for loop from k=1:n and another for loop for j=1:k-1 and sum the contributions to b(k) from yk-1, yk-2...y1.

Then y(k)=b(k)-s %forward elim. To solve Ly=b

Then solve Ux=y. You need to start from n and go to 1. In matlab you can do it

for k=n:-1:1 %go backwards from n to 1
Sum up contributions from xk+1, xk+2...xn.

Remove the sum from y(k) and divide by the pivot