Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:

c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.

d) N/f(M) satisfies the Universal Property of Cokernels.

A 1-kilogram mass is attached to a spring whose constant is 16 N/m, and the entire system is then submerged in a liquid that imparts a damping force numerically equal to 10 times the instantaneous velocity. The mass is initially released from rest from a point 1 meter below the equilibrium position.

1. Show the Diagram, given and required only
( No need to solve, Thumbs up will only be given if 1 is followed.)

let α ∈ C be a zero of the polynomial t^3 − 4t + 2 = 0 and let R = {a1 + bα + cα^2 : a,b,c ∈ Z}. Show that R is a integral domain and

Show that α − 1 and 2α − 1 are units in R. [Hint: what if x = t + 1?

1) Solve each of the following differential equations.

a)16y"-8y'+y=0

b) (d^4y)/(dx^4)-13((d^2y)/(dx^2))+36y=0

2) use Variation of Parameters to solve y"+16y=(1/3)csc4t

3) use undetermined coefficients to solve y"-5y'+4y=3e^(3t)-5e^(2t) with y'(0)=-1 and y(0)=1

4) Explain why the product (A+B)(A-B) not equal A^2-B^2 fro two NXN matrices A and B. what is the product of (A+B)(A-B)?

Suppose h, k, r, s, t∈Z.Set a=3rt(2s+t) and b=3rs(s+2t).Prove the cubic polynomial

f (x) = (x − h)(x − a − h)(x − b − h) + k
passes through the point (h,k), has integer roots, has local extrema with integer coordinates, and has an inflection

point with integer coordinates.

f(x) = ((x − 1)^2) e^x

How easy would it be to apply the Bisection Method compared to Newton's method and modified Newton's method to the function f(x)? Explain.

Let a and b be integers which are not both zero.

(a) If c is an integer such that there exist integers x and y with ax+by = c, prove that gcd(a, b) | c.

(b) If there exist integers x and y such that ax + by = 1, explain why gcd(a, b) = 1.

(c) Let d = gcd(a,b), and write a = da′ and b = db′ for some a′,b′ ∈ Z. Prove that gcd(a′,b′) = 1.

In Matlab (diary on), do the following: 

1. Generate N+1=11 equi-spaced nodes Xi in the interval [−5,5]:
         X = [-5:1:5];    (to see values, omit the ;)
   or    X = linspace(-5,5,11);
   and evaluate f(x) at these nodes:
         Y = 1./(1+X.^2);
   The N+1 points (Xi, Yi) are the data points to be interpolated by various methods. 
   Plot them to see where they are:
         plot(X,Y,'o')
         title('N+1 = 11 equi-spaced points of Runge function')
         pause 

   Also generate lots of points xi at which to evaluate f(x) and the interpolants for plotting:
         x = [-5:0.1:5];   (this is a lower case x, not X)
   Evaluate f(x) at these xi's and plot y=f(x) and the data points:
         plot(x,y,'-', X,Y,'o')
         title('Runge f(x) and data pts')
         pause 

Now, we use the data points (Xi, Yi) to construct various interpolants.
A good interpolant should "reproduce" the function f(x) as close as possible. 
Let's try a few.


2. Nth degree interpolating polynomial:
   Use Matlab's polyinterp to construct (the coefficients of) the Nth degree interpolating polynomial (here N=10):
         pN = polyfit( X,Y, N); 
   Now this can be evaluated anywhere in the interval [-5,5] with polyval, e.g. at the xi's:
         v = polyval( pN, x); 
   Find the Inf-norm error ∥y-v∥∞:
         err = norm(y-v, inf) 
   and plot both f(x) and pN(x) on the same plot:
         plot(x,y,'-b', x,v,'*-r')
         title('f(x) and pN(x) at plotting pts')
         pause 

   Is this a good interpolant ?  Why ?
        

   
3. Interpolation at Chebychev nodes:
   Generate N+1=11 Chebychev points (Xchebi, Ychebi) in [a,b]:
         fprintf('------ chebychev nodes ------\n')
         K = N+1;
         a=−5;  b=5;
         for i=1:K
             Xcheb(i)=(a+b)/2 + (b−a)/2 *cos( (i−0.5)*pi/K );
         end
         Ycheb = 1./(1+Xcheb.^2);

   Follow the steps in 2. to produce the Nth degree interpolating polynomial pNcheb 
   based on the Chebychev nodes, its values vcheb at the xi's, the error ∥y − vcheb∥∞,
   and plot both f(x) and pNcheb(x) on the same plot.

   Compare the error and plot with those from 2.
   Which one is better ?  why ?
        


4. Piecewise linear interpolation:
   Use Matlab's interp1 to construct the linear interpolant:
         lin = interp1(X,Y, x, 'linear'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        

5. Piecewise cubic interpolation:
   Use Matlab's interp1 to construct the cubic interpolant:
         cub = interp1(X,Y, x, 'cubic'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


6. Cubic spline interpolation:
   Use Matlab's interp1 to construct the spline interpolant:
         spl = interp1(X,Y, x, 'spline'); 
   Repeat the steps of 2. 

   Compare errors and plots.
        


7. To see that the error gets worse for bigger N for equi-spaced nodes but not for Chebychev nodes 
   (for this f(x) at least), repeat 2. and 3. with N = 20.

Let U = {A ∈ Mat(2; ℚ) : AB = BA for all B ∈ Mat(2; ℚ)}.

(i) Show that  U is a subspace of Mat(2; ℚ).

(ii) Show that E ∈ Mat(2; ℚ) is a basis of U. (E: identity matrix)

(iii) Find the complement for U

5. Show that if R is a division ring,then Mn(R) has no nontrivial two-sided ideals.

Need the detailed calculation process about why at the same BER condition, BPSK is 3dB more power efficient that BFSK

Suppose U is a subspace of a finite dimensional vector space V. Prove there exists an operator T on V such that the null space of T is U.