Crossover

October 31, 2014

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287 New working groups

October 22, 2014

(This post is specifically for Math 287 students.)

Starting Monday, you guys should be organized in new groups. No group can have three members that are together in the current groups. When I arrive on Monday, the groups should already be formed. You guys should start working on the laboratory on Polyhedra, Chapter 7. Make sure to bring whatever materials may be needed for this.

There have been complaints about not everybody contributing their share to their respective groups. This is not acceptable, but it is sadly the main reason for the reorganization. So: If I receive two complaints about a member not contributing as required, and there are no reasonable extenuating circumstances, that person will be dropped out of their group (receiving a zero in their current project as a result). If the issue is not expected to be resolved for the next report, a new reorganization of groups will be triggered as a result.

Also, I am unhappy with the level of some of the reports. It seems the peer reviewing of other groups’ drafts is not being taken as seriously as needed. So: Now, on the dates drafts are due, each group should bring three copies of their current draft. When you review another group’s draft, write the members of your group on the copy you are reviewing. I’ll collect the copies, with your comments, copy them for my records and return. If I identify something that a group should have noticed and mentioned, but did not, that group will be penalized (since this means the group did not take their refereeing role seriously). Conversely, if a group mentions something that should be addressed, but I do not see the issue resolved in the final report, the group that failed to address the given comments will be penalized.

Finally, this being a mathematics course, I expect your projects to include proofs. If a project lacks proofs it will receive a failing grade.

Feel free to contact me be email if any of the above needs clarifying.

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Two model theory meetings

October 19, 2014

José Iovino has asked me to help advertise the following:

At the 2015 Joint mathematics meetings (JMM), January 10-13, in San Antonio, TX, there will be two special sessions on model theory.

The first is Beyond First Order Model Theory, a special session of the ASL and the AMS. See here for the schedule, list of speakers, and abstracts.

The second is Model Theory and Applications. See here for schedule, speakers, and abstracts.

The first session will be preceded by a conference (with the same title) at the University of Texas, San Antonio. Additional details can be found here, the speakers are Will Boney (University of Illinois at Chicago), H. Jerome Keisler (University of Wisconsin – Madison), Michael Makkai (McGill),  Maryanthe Malliaris (University of Chicago), Paul Larson (Miami University), Chris Laskowski (University of Maryland), and Sebastien Vasey (Carnegie Mellon).

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414/514 References on continuous nowhere differentiable functions

October 19, 2014

Just as the last two times I have taught 414/514, I am assigning a final project on the topic of continuous nowhere differentiable functions (see here and here for the previous times).

The project requires that you choose an example of a continuous nowhere differentiable function, and to write a report describing the function, indicating who first introduced it, and presenting complete proofs of its continuity and nowhere differentiability. Additional information relevant for context is highly encouraged.

I am including links to two encyclopedic references on the subject. Feel free to follow the arguments there closely if needed, or to consult other sources, but make sure that what you turn in is your own version of the details of the argument, and that full details (rather than a sketch) are provided.

  1. Johan Thim’s Master thesis (Continuous nowhere differentiable functions), written under the supervision of Lech Maligranda.
  2. A.N. Singh’s short book on The theory and construction of non-differentiable functions. (See here for a short review.)

As I mentioned before,

Please take this project very seriously (in particular, do not copy details from books or papers, I want to see your own version of the details as you work through the arguments). Feel free to ask for feedback as you work on it; in fact, asking for feedback is a good idea. Do not wait until the last minute.

The project should be typeset and is due Wednesday, December 17 (though I strongly encourage you to turn it in earlier).

Please contact me by email as soon as you have chosen the topic you are going to cover, and I’ll list it here, to avoid repetitions.

  • Stephanie Potter: Wen’s function.
  • Jeremy Siegert: Orlicz functions.
  • Stuart Nygard: Besicovitch’s function.
  • Monica Agana: Koch’s snowflake.

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414/514 Homework 2 – Monotone and Baire one functions

October 10, 2014

This set is due in three weeks, on Monday, November 3, at the beginning of lecture.

1. Let f:[a,b]\to\mathbb R be increasing. We know that f(x-) and f(x+) exist for all x\in[a,b], and that f has at most countably many points of discontinuity, say t_1,t_2,\dots For each i let I_i,J_i be the intervals (f(t_i-),f(t_i)) and (f(t_i),f(t_i+)). Some of these intervals may be empty, but for each i at least one of them is not. (Here we follow the convention that f(a-)=f(a) and f(b+)=f(b).) Let \mathrm{lh}(I) denote the length of the interval I, and say that an interval (\alpha,\beta) precedes a point t iff \beta\le t.

Verify that \sum_i(\mathrm{lh}(I_i)+\mathrm{lh}(J_i))<+\infty and, more generally, for any x,

s(x):=\sum\{\mathrm{lh}(I_i)\mid I_i precedes f(x)\} +\sum\{\mathrm{lh}(J_i)\mid J_i precedes f(x)\}<+\infty.

Define a function f_0:[a,b]\to\mathbb R by setting f_0(x)=f(x)-s(x). Show that f_0 is increasing and continuous.

Now, for each n>0, define f_n:[a,b]\to\mathbb R so that f_n\upharpoonright[a,t_n)=f_{n-1}\upharpoonright[a,t_n), f_n(t_n)=f_{n-1}(t_n)+\mathrm{lh}(I_n), and f_n(x)=f_{n-1}(x)+\mathrm{lh}(I_n)+\mathrm{lh}(J_n) for all x\in(t_n,b]. Show that each f_n is increasing, and its only discontinuity points are t_1,\dots,t_n.

Prove that f_n\to f uniformly.

Use this to provide a (new) proof that increasing functions are in Baire class one.

2. Solve exercise 3.Q in the van Rooij-Schikhof book: If f:[a,b]\to\mathbb R is such that for all x, we have that f(x-) and f(x+) exist, then f is the uniform limit of a sequence of step functions. The approach suggested in the book is the following:

Show that it suffices to argue that for every \epsilon>0 there is a step function s such that |f(x)-s(x)|<\epsilon for all x.

To do this, consider the set A=\{x\in[a,b]\mid there is a step function s on [a,x] such that |f(t)-s(t)|<\epsilon for all t\in[a,x]\}.

Show that A is non-empty. Show that if a\le y\le x and x\in A, then also y\in A. This shows that A is an interval {}[a,\alpha) or {}[a,\alpha], with \alpha\le b. Show that in fact the second possibility occurs, that is, \alpha\in A. For this, the assumption that f(\alpha-) exists is useful. Finally, show that \alpha=b. For this, use now the assumption that f(\alpha+) exists.

3. (This problem is optional.) Find a counterexample to the following statement: If f:[a,b]\to\mathbb R is the pointwise limit of a sequence of functions f_1,f_2,\dots, then there is a dense subset X\subseteq [a,b] where the convergence is in fact uniform. What if f and the functions f_n are continuous?  Can you find a (reasonable) weakening of the statement that is true?

4. (This is example 1.1 in Andrew Bruckner’s Differentiation of real functions, CRM monograph series, AMS, 1994. MR1274044 (94m:26001).) We want to define a function f:[0,1]\to\mathbb R. Let C be the Cantor set in {}[0,1]. Whenever (a,b) is one of the components of the complement of C, we define f(x)=(2(x-a)/(b-a))-1 for x\in[a,b]. For x not covered by this case, we define f(x)=0. Verify that f is a Darboux continuous function, and that it is discontinuous at every point of C.

Verify that f is not of Baire class one, but that there is a Baire class one function that coincides with f except at (some of) the endpoints of intervals [a,b] as above.

Verify that f is in Baire class two.

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414/514 Simple examples of Baire class one functions

October 6, 2014

Recall that a real-valued function f defined on an interval I is (in) Baire class one (\mathcal B_1) iff it is the pointwise limit of continuous functions.

Examples are continuous functions, of course, but functions in \mathcal B_1 do not need to be continuous. An easy example is the function f:[0,1]\to\mathbb R given by f(x)=0 if x\ne 1 and f(x)=1 if x=1. This is the pointwise limit of the functions f_n(x)=x^n. By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in \mathcal B_1.

Step functions are another source of examples. Suppose that a=x_0<x_1<\dots<x_{n-1}<x_n=b and that s:[a,b]\to\mathbb R is constant on each (x_i,x_{i+1}). Then s is the pointwise limit of the functions s_k, defined as follows: Fix a decreasing sequence \epsilon_k converging to 0, with \epsilon_k\le 1/k and 2\epsilon_k<x_{i+1}-x_i for all i. Now define \hat s_k as the restriction of f to

\displaystyle \{x_0,x_1,\dots,x_n\}\cup\bigcup_{i=0}^{n-1}[x_i+\epsilon_k,x_{i+1}-\epsilon_k],

and let s_k:[a,b]\to\mathbb R extend \hat s_k by joining consecutive endpoints of the components of its domain with straight segments.

An important source of additional examples is the class of derivatives. Suppose f:\mathbb R\to\mathbb R and f(x)=g'(x) for all x. This is the pointwise limit of the functions f_n(x) given by

f_n(x)=\displaystyle\frac{g\left(x+\frac1n\right)-g(x)}{\frac1n}.

This simple construction does not quite work if f is defined on a bounded interval (as x+1/n may fall outside the interval for some values of x). We can modify this easily by using straight segments as in the case of step functions: Say f:[a,b]\to\mathbb R. For n large enough so 1/n<b-a, define f_n(x) as above for x\in[a,b-1/n], and now set f_n(b)=f(b) and extend f_n linearly in the interval {}[b-1/n,b].

Additional examples can be obtained by observing, first, that \mathcal B_1 is a real vector space, and second, that it is closed under uniform limits (the latter is not quite obvious). This gives us, for instance, that all monotone functions are in \mathcal B_1, since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing f:[a,b]\to\mathbb R, let f_n(x)=\lfloor nf(x)\rfloor/n. It follows that all functions of bounded variation are in \mathcal B_1, since any such function is the difference of two increasing functions.

Another interesting source of examples is characteristic functions. Given X\subseteq\mathbb R, the function \chi_X is in \mathcal B_1 iff X is both an \mathbf F_\sigma and a \mathbf G_\delta set.

On the other hand, \chi_{\mathbb Q} is not in \mathcal B_1, since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.

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