(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.
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).
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.
- Johan Thim’s Master thesis (Continuous nowhere differentiable functions), written under the supervision of Lech Maligranda.
- 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.
This set is due in three weeks, on Monday, November 3, at the beginning of lecture.
1. Let be increasing. We know that and exist for all , and that has at most countably many points of discontinuity, say For each let be the intervals and . Some of these intervals may be empty, but for each at least one of them is not. (Here we follow the convention that and .) Let denote the length of the interval , and say that an interval precedes a point iff .
Verify that and, more generally, for any ,
precedes precedes .
Define a function by setting . Show that is increasing and continuous.
Now, for each , define so that , , and for all . Show that each is increasing, and its only discontinuity points are .
Prove that 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 is such that for all , we have that and exist, then 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 there is a step function such that for all .
To do this, consider the set there is a step function on such that for all .
Show that is non-empty. Show that if and , then also . This shows that is an interval or , with . Show that in fact the second possibility occurs, that is, . For this, the assumption that exists is useful. Finally, show that . For this, use now the assumption that exists.
3. (This problem is optional.) Find a counterexample to the following statement: If is the pointwise limit of a sequence of functions , then there is a dense subset where the convergence is in fact uniform. What if and the functions 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 . Let be the Cantor set in . Whenever is one of the components of the complement of , we define for . For not covered by this case, we define . Verify that is a Darboux continuous function, and that it is discontinuous at every point of .
Verify that is not of Baire class one, but that there is a Baire class one function that coincides with except at (some of) the endpoints of intervals as above.
Verify that is in Baire class two.
Recall that a real-valued function defined on an interval is (in) Baire class one () iff it is the pointwise limit of continuous functions.
Examples are continuous functions, of course, but functions in do not need to be continuous. An easy example is the function given by if and if . This is the pointwise limit of the functions . By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in .
Step functions are another source of examples. Suppose that and that is constant on each . Then is the pointwise limit of the functions , defined as follows: Fix a decreasing sequence converging to , with and for all . Now define as the restriction of to
and let extend 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 and for all . This is the pointwise limit of the functions given by
This simple construction does not quite work if is defined on a bounded interval (as may fall outside the interval for some values of ). We can modify this easily by using straight segments as in the case of step functions: Say . For large enough so , define as above for , and now set and extend linearly in the interval .
Additional examples can be obtained by observing, first, that 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 , since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing , let . It follows that all functions of bounded variation are in , since any such function is the difference of two increasing functions.
Another interesting source of examples is characteristic functions. Given , the function is in iff is both an and a set.
On the other hand, is not in , since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.