MATH 4111 HOMEWORK ASSIGNMENTS, FALL '07
 

INSTRUCTOR'S POLICY:  TRY HARD TO HAND IN ASSIGNMENTS IN CLASS ON THE INDICATED DUE DATE AND, IF YOU'RE UNABLE TO ATTEND, GIVE YOUR COMPLETED ASSIGNMENT TO A FRIEND TO HAND IN.  PAPERS SLIPPED UNDER MY OFFICE DOOR WITHIN AN HOUR OR TWO AFTER CLASS ON THE DUE DATE HAVE  GOOD ODDS  OF GETTING INTO THE GRADER'S HANDS AND BEING GRADED.      IN THE EVENT OF A SPECIAL PROBLEM (ILLNESS, ETC.) CONTACT ME BY E-MAIL. UNLESS I'VE MADE A SPECIAL ALLOWANCE, PAPERS RECEIVED AFTER I'VE PASSED ON ALL OTHER PAPERS TO THE GRADER (USUALLY DONE BY MID AFTERNOON ON THE DUE DATE) WILL NOT BE GRADED AND WILL NOT RECEIVE ANY CREDIT.  DON'T PUT ANY PAPERS IN MY MAILSLOT SINCE I CHECK IT ONLY SPORADICALLY.
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ASSIGNMENT #1.  DUE TUESDAY,  SEPTEMBER 11

     (i)    Read Chapter 1 in Rosenlicht's book and get started on reading Chapter 2.
    (ii)  Do the following problems from Chapter 1 (pages 12-13):   4 and 5 (all parts), 7(b), (c), (e), (f) , 8, and 9.
    (iii) Use mathematical induction to prove that, for each positive integer n, the sum of the third powers (cubes) of the first n positive integers is [n(n+1)/2]^2.  Thus, for n = 3, 1 +8+27=36=6^2 with 6 = 3(4)/2.   Done efficiently, the induction proof takes only a few lines.
     (iv) Use mathematical induction to prove the binomial theorem (look it up if you’ve forgotten the statement).  Then go on to prove by induction a trinomial theorem for expansion of the nth power of a sum a+b+c in terms of products of powers of a, b, and c with certain “trinomial” coefficients all involving n! in the numerator and appropriate products of three factorials in the denominators.

      (v) For extra credit,  hypothesize a statement of a general multinomial theorem for expansion of the nth power of a sum of m variables  and go on to prove your hypothesis either by induction on m for each fixed n or induction on n for each fixed m (one way is easier than the other!)

Example of an inductive proof:  For each positive integer n, we assert that
          (Hn)   1 + 2 + ...+n = n(n+1)/2     [arithmetic progression formula]
The assertion (H1) is true since 1=1(2)/2.   If (Hn) is true, then the sum of the first n+1 integers being the sum of the first n plus n+1, we infer that the sum of the first n+1 integers is equal to
n(n+1)/2 + (n+1) =(n+1)(n/2 +1) = (n+1)(n+2)/2.  Since n+2 = (n+1)+1, this means (H(n+1)) is true.
By the principle of mathematical induction, (Hn) is true for every positive integer n.   [Using a few shorthand symbols to eliminate the words I used above, the same proof could be written down in two lines.]

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ASSIGNMENT #2.        DUE TUESDAY, SEPTEMBER 18
          Do the following problems from Chapter 2 (pages 29-31):  4 (b)  [try to avoid brute force division/multiplication via algebraic trickery], 5 (c),(d) [be careful with signs], 6, 7, 10(b),(c), 13, 14, 15, 16.  Feel free to ask in class for hints on some of these problems.    

ASSIGNMENT #3        DUE THURSDAY, SEPTEMBER 27

            NOTE:  THE DUE DATE HAS BEEN PUSHED UP TO THE 27TH BECAUSE THERE WON’T BE ANY HOMEWORK DUE THE FOLLOWING WEEK IN VIEW OF THE EXAM TO BE GIVEN ON THURSDAY, OCT. 4.  THE HOMEWORK PROBLEMS ON THIS ASSIGNMENT WILL BE GOOD PRACTICE FOR THE EXAM ALTHOUGH THE EXAM PROBLEMS WON’T BE AS HARD AS SOME OF THESE HOMEWORK PROBLEMS.

        Do the following 4 problems from Chapter 3 (pages 61-63):    11, 19, 21,23.  Also do the following 3 additional problems:

        (5) Let f(x) = 1/(x+2) for x a real number not equal to -2.  Obviously, f is continuous on the set of real numbers not equal to -2.  We can use f to inductively define sequences by choosing a starting value x0 and defining xn+1 = f(xn).    Check that f has two equilibrium values (solutions of f(x) =x), one positive and one negative.  If x0 > -2, show that the sequence converges to the positive equilibrium value.
As mentioned in #13, the key thing is to check that the subsequences with n even and n odd are each monotonic, one monotone increasing and the other monotone decreasing.  For this, you'll want to study f(f(x))= (x+2)/(2x+5).
            For extra credit, discuss what happens when x0 < -2.  WARNING:  this gets tricky so, if your time is limited, don't consider it. Among other things, you have to worry about initial values x0

for which there is some value of n such that xn =-2 with xn+1  then not defined.   For this, it’s easiest to calculate the inverse function g of f, i.e. solve y=f(x) for x=g(y)  and look at the sequence defined by g with initial value -2.

        (6)  Check that the arc length metric on the unit sphere in R^3 is equivalent to the chord length metric.   Thus, for each two points on the unit sphere, there is a unique great circle passing through these two points and the arc length metric is the arc length of the shorter of the two arcs of the great circle while the chord length metric is the Euclidean distance between the two points (= restriction to the unit sphere of the Euclidean/Cartesian metric on R^3).   Why is it enough to make your comparisons for two points on the equator of the sphere?  Note that this reduces you to a comparison of arc lengths and chord lengths for circles in R^2 and you can use polar coordinates to simplify the calculations.

(7) In class, we’ll eventually show that, for 1<p<(infinity),  there is a norm || . ||p on R^n defined by

||(x1 , x2, …,xn ||p = 1/p power of the sum of |xi|p from i=1 to n.

 For p not equal to q,  show directly that || . ||p and || . ||q are equivalent.  If you can, use Lagrange multipliers to determine the best constants relating these two norms, i.e., maximize/minimize one norm subject to the constraint that the other norm has value 1.

 

 

ASSIGNMENT#4, DUE THURSDAY, OCTOBER 11

1.      Using the definitions of open, closed, interior, closure, boundary given in class for subsets of a metric space,  show the following for any subset S of a metric space (E, D):

(i)                     The closure of S is closed (thus, every limit point of the closure of S belongs to the closure);

(ii)                   The interior of S is open (thus, any point in the interior of the interior of S is in the interior of S);

(iii)                 E is the disjoint union of the interior of S, the interior of the complement of S, and the boundary of S;

(iv)                 The interior of S is the complement of the closure of the complement of S;

(v)                   The closure of S is the disjoint union of the interior of S and the boundary of S (also the non-disjoint union of S and the boundary of S);

(vi)                 The boundary of S (and its complement) is the intersection of the closure of S and the closure of the complement of S.

           Hints:  Draw the picture we put on the board  in class.  Each of (i)-(vi) should only take a couple of lines.

 

2.      Let f(x) = x/(1+|x|).  Check that f maps the real line 1-1onto the open interval (-1,1) with both f and its inverse function g(y)=y/(1-|y|) being continuous.  Extend f to a function from the extended real numbers (= union of R and {+infinity, -infinity}) onto [-1,1] by defining f(infinity) =1, f(-infinity)=-1. Then check that d(x,y) =|f(x)-f(y)| defines a metric on the extended real numbers in which the extended real numbers are the completion of the restriction of d to R.  Justify the statement that (R, |  |) is locally equivalent to (R,d) but not globally equivalent.

 

3.      Identify R^(n+1) with the Cartesian product of R^n and R and identify each x in R^n with the pair (x,0) in R^(n+1).  Let ||.|| be the Euclidean norm on either R^n or R^(n+1).  Call q=(0,1) the “north pole” in R^(n+1) and let S be the unit sphere about 0 in R^(n+1), i.e. all points (u,t) for which ||u||^2 +t^2=1. 

(i)                     For x in R^n, check that F(x) =(2x/(1+||x||^2), (||x||^2 -1)/(||x||^2 +1)) defines a point p in S\{q} with p on the line joining x and q.  

(ii)                   Check that F is 1-1 from R^n onto S\{q} with inverse G(u,t) =u/(1-t).  G is called stereographic projection from S\{q} onto R^n.

(iii)                 Show that d(x,y) = ||F(x)-F(y)|| defines a metric on R^n which is locally equivalent to the Euclidean metric but not globally equivalent.  In what sense is S the completion of (R^n, d)?

4.      Do #6, #10, 26, 30 on pp. 61-64.

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ASSIGNMENT #5 DUE TUESDAY, OCTOBER 23

            Do the following problems from Chapter 4, pp. 90-95: 1(b), 1(d), 4, 9(b), 10(b), 14,15, 17, 21,22.  Also prove the following variation on #21: with E and E’ as in #21 and with S a subset of E whose closure K in E is compact, a continuous function f from S into E’ extends to a continuous function from K into E’ if and only if f is uniformly continuous on S.  

 

Comments and hints:

  1(b) is easy--use continuity of F(x) =x to get continuity of f.  But 1(d) is not easy--you need to separately investigate continuity at rational and irrational points and use the properties of rational numbers we discussed earlier. 

 The key point in #4 is the assumption that f is onto--without this, the conclusions wouldn't follow. 

 #9(b) and 10(b) are "old chestnuts" which likely appeared in your Calculus III textbook--just a line or two ought to suffice. 

 #14 and #15 are "intuitively clear" but you'll need to think through how to concisely justify the indicated conclusions.

For #17, you use the same trick as in 9(b), namely that when a and b  are positive, a - b  is the product of the sum and difference of the postive square roots of a and b. 

 #21 and the variation on #21 aren’t particularly hard but are very useful in applications.  The idea in both problems is to use uniform continuity of the original function to get continuity of the extended function. 

 #22  may cause you trouble, especially if your linear algebra background is weak.  But it’s very important in multivariable calculus.  

 

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ASSIGNMENT #6.  DUE TUESDAY, OCT. 30

             Do the following problems from Chapter 4, pp. 90-95:  30, 32, 34, 37, 39, 41, 44, 45.

Also do the following additional problem:  In class, we gave two examples of subsets of R^2 which are connected but not path connected.  If we take the Cartesian product of  one of these sets with an interval, we get a subset of R^3 which is connected but not path connected.  Give an example of a subset of R^3 which is connected, not path connected, and NOT the Cartesian product of a subset of R^2 and an interval.  For these, you might want to generalize the graph of sin(1/x) example by the graph of something like sin(1/(|x| +|y|) or alternative generalize the spiral curve example with a spiral surface spiraling outward to the unit Euclidean sphere in R^3.

For your amusement (?), you might want to read over #31.  The idea of a continuous function on [0,1] having the solid cube in R^3 as its image set is decidedly counter-intuitive.   Such space-filling curves are often used in geometry and topology for counter-examples to various assertions and go a long way toward establishing the principle that continuity is not enough to get nice geometric properties.

Comments and hints:

#30.  When Rosenlicht says "closed interval in E^2", he means "closed rectangle in R^2", i.e. the Cartesian product S = I x J of two closed intervals I=[a,b] and J=[c,d] in R.  If f is a continuous R-valued function on S, then, for each y in J, g(x) = f(x,y) is continuous from I to R.  Use the intermediate value theorem for each such g to show f can't be one-to-one.

#32.  Note that we are dealing with a sequence of functions defined inductively by initial function =
square root of x, (n+1)st  function at x = square root of (x+(nth function at x)).  This is the same kind of thing we encountered earlier with dynamical systems problems so the same techniques apply.  Thus, if the sequence is going to converge at some x, the limit F(x) must be >= 0 and satisfy the equation  F(x)^2 = x + F(x).  The point of the problem is to show that the sequence does converge to F(x).  Is the convergence uniform or only pointwise?

#34.  These are a little tricky.  In each case, it's not hard to calculate pointwise limits but determining whether the converge is or isn't uniform takes a little work.

#37.   It's easy to see that the sum functions converge uniformly but not so clear what happens for the product functions.  First consider the case where the limit functions for each of the two sequences are bounded and use the triangle inequality to see that the product sequence converges uniformly to the product of the limits.  Then take E = R (real numbers) and look at examples of situations where one or both of the limit functions isn't bounded.

#39.   There are many ways to go about constructing a sequence of functions with the stated properties. You may want to try to find a pointwise convergent sequence of continuous functions where the limit function is 1 at the points 1, 1/2, 1/3, 1/4,..., 1/n,... and 0 elsewhere.  Think of functions whose graphs are sharply peaked “tents” with a small base.  

#41.  This one is admittedly not easy but it isn't as bad as you may initially think.  It's important that the functions are defined on a compact metric space.  Try to come up with an argument first showing that any point p lies in an open set where the sequence converges uniformly, then use compactness to obtain E as the union of finitely many such open sets and deduce uniform convergence on all of E.

#44 and #45.   In contrast to the last three problems, these two are very easy.  Just unravel the definitions.  So, if you're having trouble with some of the earlier problems, you may want to do these problems first.

 

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ASSIGNMENT #7.   DUE TUESDAY, NOV. 6.  Do the following problems from Chapter 5, pp. 108-110:  # 4, 6, 7, 12.  Also do the following problems from Chapter 9, pp. 212-213:  #1,3,4.

Comments:  The four Chapter 5 problems plus #3 and #4 from Chapter 9 rest on the Mean Value Theorem. The wording in the hint in #7 from Chapter 5 is very confusing.  What’s meant is that the subset of R^2 defined in the hint is connected and the function defined on the set is continuous so its image in the real line is connected, i.e. an interval.                                                        For #12 from Chap. 5, you'll need to use the Mean Value Theorem several times for f and functions closely related to f.  Unraveling the somewhat confusing wording in the problem, for any a and b in the interval I on which f is defined with a<b and any points c and x in (a,b), there are 3 (x,y) points be considered:  the point (x,y_chord(x)) on the chord line through (a,f(a)) and (b,f(b)); the point (x,f(x) on the graph of f; and the point (x,y_tan(x)) on the line tangent to the graph of f at (c,f(c)).  Convexity of f means that  y_tan(x) <= f(x) <= y_chord(x) for all a,b,c, and x.  As stated in the text,  Problem #12 asks  you to show that, when f happens to be twice differentiable at each point in I, then f is convex on I if and only if the 2nd derivative of f is everywhere non-negative.  In fact, the hypothesis of non-negative 2nd derivatives serves only to guarantee that f ′ is monotonically increasing;  using just the assumption that f is differentiable on I plus using the mean value property for derivatives established in #7, show that f is convex on I if and only if f ′ is increasing.  All you'll need is the point-slope formula for straight lines and the Mean Value Theorem.

                     For #1 in Chapter 9, you'll want to first show that the indicated function is differentiable at points (a,b) where both a and b are non-zero, then go on to investigate whether or not the function is differentiable at (a,b) when either a or b or both are zero. Note that if f is differentiable at (a,b), then at (a,b) the differential df must be the linear map taking (x-a,y-b) to     A(x-a) +B(y-b) where A and B are the partial derivatives of f at (a,b) with respect to x and y.    So, to investigate differentiability at (a,b), first see whether or not A and B exist, then go on to see whether the error function  goes to zero as (x-a,y-b) approaches (0,0).  We'll prove next week that if all partial derivatives of a function exist and are continuous on some open set, then the function is differentiable at each point of the open set.  You can just quote this result as a shortcut to proving that the error function goes to zero provided you can show that the partial derivatives exist and are continuous in an open ball around your chosen point.

                     Note that Rosenlicht uses a highly non-standard notation for partial derivatives, namely f prime sub i for the partial derivative ∂f/∂xi  of f with respect to the ith variable.

 

ASSIGNMENT #8  Due Tuesday, Nov. 27

              Do the following problems from Chapter 9, pp. 212-214: #   #  9,  11, 13, 14, 16, 20.

Also read over Chapter 6 and start browsing through Chapter 7.

Hints:  On #9, you can use without proof the differentiation under the integral formula of #6.

On #11, you may want to warm up with the hint in the book for the cases n=2 and n=3.

Note that #13 generalizes our discussion in class of 2nd order differences to 3rd order differences.

Aside from Rosenlicht’s horrible notation for partial derivatives, #14 is a straightforward computation.

For #16, recall that an n-by-n matrix A is symmetric if A is equal to its transpose (equivalently, the i,j entry of A coincides with the j,i entry for all i and j) and A is then said to be positive definite if

<Ax, x>  is > 0 for each non-zero column vector x.  Here < . , .>  stands for the standard inner product (dot product) on R^n.  For positive semi-definite we replace > by >= .  Similarly, A is negative definite if  <Ax,x>  is < 0  for each non-zero x, relaxing < to <= for negative semi-definite.

To be honest, I dislike  #20 since it entails a sufficient condition for contraction mappings which is more restrictive than necessary and whose proof entails accepting on faith an important theorem in Euclidean geometry.  To demystify this statement, observe that there are many candidates for useful  norms on the space of n-by-n real matrices.  We showed in class that the operator norm of a linear  transformation  R^n with matrix A relative to the norm ||.||_(infinity) on R^n is the maximum of the numbers rho_i =sum of the magnitudes of the ith row entries of A for 1<= i <=n . Call this the row sum norm of A. We went on to use the multi-variable mean value theorem to show that, if F is a continuously differentiable map on U with U containing a closed convex subset S for which F(S) is contained in S and, if there is a positive number c<1 for which the row sum norm of the Jacobian matrix JF(x) of F at x is less than or equal to c at each point x in S, then F is a contraction mapping on S.  In particular, this condition holds if each entry of JF(x) has magnitude less than or equal to c/n and we relied heavily on this fact in carrying out the proof via contraction maps of the Implicit Function Theorem.  The Hilbert-Schmidt norm of a matrix A, commonly denoted by |||A|||, is NOT an operator norm but is instead the Euclidean norm of A as a member of R^{n^2}.  Thus  |||A||| is the square root of the sums of the squares of all entries of A.  Problem #20 asks you to prove that, for F and S as above, F is a contraction mapping on S if there is a positive number c < 1 for which |||JF(x)||| <= c for each x in S.  Do this as follows:  (i) Let || . || =||.||­_2 be the Euclidean norm on R^n. Use the Cauchy-Schwartz inequality to show that ||(dF)_x (v)|| <=|||JF(x)||| ||v||  for every vector v.

                                                            (ii)  Accept without proof the famous adage that “a straight line is the shortest distance between two points”;  more precisely, for each pair of points c and d in R^n, ||d-c|| is less than or equal to the arc length distance along any smooth path C joining c to d

(iii)                 For any a and b in S, apply (ii) to c=F(a), d=F(b),

and C = {p(t) =F(a + t(-a): 0<= t <= 1}, noting that dp/dt (t) =application to b-a of the differential of F at the point a + t(b-a)  for each t.  Then use (i) to reach the desired conclusion.

        In representation theory,tensor analysis, and their generalizations to infinite dimensional Euclidean spaces, the Hilbert-Schmidt norm for linear transformations plays a very important role.

But, if we want to use the Euclidean norm || . || , the appropriate matrix norm is the operator norm of (dF)_x associated with  || . || and the calculation in (i) shows that this operator norm is  <= |||JF(x)||| for each x.  It turns out this operator norm is equal to the square root of the largest eigenvalue of the product of JF(x) times its transpose while |||JF(x)||| is the square root of the sum of these eigenvalue and could be much larger.  So |||JF(x)||| could be > 1 and F still a contraction.

This isn’t the end of the story.   It can be shown that a linear transformation is a contraction (with 0 its unique fixed point) relative to SOME norm on R^n if and only if all eigenvalues of the transformation have magnitude < 1.  So the “right approach” to determining necessary and sufficient conditions for a non-linear transformation F to be a contraction has to involve eigenvalues of (dF)_x having magnitude <= c for all x with c fixed and c<1.  This result is of large importance in signal processing via wavelets, data compression, and related topics. To summarize, the row sum criterion for contractibility we developed in class is the easiest one to apply, the necessary and sufficient criterion involves eigenvalue estimates (not easy), and the business about Hilbert-Schmidt norms in #20 is just “blowing smoke” on an important question.     

     

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ASSIGNMENT #9, DUE THURSDAY, DEC. 6
              Do the following problems from Chapter 6, pp. 133-134:  #8, 13, 16, 19, 20, 22.  Also do the following problems from Chapter 7, p. 161: #3, #4, #5.

Hints:  On #8, Chap. 6, you may want take a sequence of partitions with widths going to 0 and try to show that the Riemann sums for these partitions using the right endpoint rule form a Cauchy sequence and so do the Riemann sums using the left endpoint rule with the limits for these two types of Riemann sums being equal.  Depending on the approach you use, you may or may not need to show that every monotone function has at most a countable number of discontinuities.

             For #13, remember that the limit of nth roots of any positive number as n goes to infinity is equal to 1.

  
            #16 is very easy—just a matter of reviewing definitions.

             Do #19 by induction plus integration by parts.

              Among other things, #20 involves a proof that among all smooth curves joining two points,  the shortest arc length distance is the straight line distance.  To carry out #20, you’ll need to use the Mean Value Theorem for EACH component.                

                        In #22, log(x)  means the natural log of x or ln(x).  Recall that ln(x) is the integral of the function 1/t from t=1 to t=x for each positive x.  Using this plus a Riemann sum argument, #22 “drops out”.