170 – Homework 4

March 22, 2013

Please do not forget the extra-credit project is due April 1st. This homework set is due Wednesday, April 3rd. Write the full statement of the questions. In each case, explain what you are doing (a list of numbers or equations is not an explanation.)

  • From the Whitman College Calculus (downloadable here), solve: From Exercises 5.1: 2, 10, 17. From Exercises 5.2: 7, 11, 15. From Exercises 5.3: 15, 16. From Exercises 5.4: 11, 12, 19.
  • From Spivak’s book: Chapter 11, problems: 16, 17, 21. Chapter 11, Appendix, problem 2. (Note that there is a typo in Ch.11, App., prob. 2. Rather than Figure 30, it should be Figure 31. Thanks to Justin Garrard for noticing this.)
  • From Whitman’s book: Exercises 6.1: 5, 7, 15, 25. Exercises 6.2: 11, 12.

170 – Homework 3

March 12, 2013

This is due Wednesday, March 20, at the beginning of lecture.

  • From Spivak’s book, Chapter 9, solve problems 2, 11, 14.
  • From Chapter 10, solve problems 2, 3, 8, 9.
  • Extra credit problem: Chapter 10, problem 18.

 


170 – The birth of calculus, Quiz 1 and extra credit project

March 4, 2013

The short BBC documentary “The birth of calculus”, produced by The Open University, and narrated by Jeremy Gray, can be found here.

Quiz 1 was Friday, March 1. Here it is.

As an extra credit project: Write (type) a short essay on the life and mathematical contributions of one of the mathematicians responsible for the development of calculus. Please email me your choice before you get started, to avoid repetitions. Make sure to cite all your references appropriately. Turn this in by FridayMonday, April 1.


Random series

March 2, 2013

A little while ago, a question was posted on MathOverflow on why lacunary series are “badly behaved”, in the sense that they their circle of convergence is their natural boundary. As the answers indicate, it is actually the opposite: This behavior is typical, in the sense that a random power series will have this property. I posted a short note pointing this out as an answer to a related question on Math.StackExchange. Here it is, with very minor edits:

Consider \sum_n X_n z^n, where the X_n=X_n(\omega) are independent random (complex) variables and z is complex.

First of all, the radius of convergence of the series (at a given \omega in the underlying measure space) is

 r(\omega)=(\limsup_{n\to\infty}|X_n(\omega)|^{1/n})^{-1}.

Note that r is a measurable function of \omega, and its value does not depend on the values of a finite number of the X_n. Kolmogorov’s zero-one law then gives us that r(\omega) is a constant, say R, almost surely. This R lies in {}[0,\infty], and both 0 and +\infty are possible values, depending on the distribution of the X_n, though the most interesting cases to study are perhaps when 0<R<+\infty.

There is a nice book that presents the relevant theory:

Jean-Pierre Kahane. Some random series of functions. Second edition. Cambridge Studies in Advanced Mathematics, 5. Cambridge University Press, Cambridge, 1985. MR0833073 (87m:60119).

(“Random Taylor series” is Chapter 4. What follows is based on Kahane’s presentation. Kahane’s book includes proofs of all the results below.)

Consideration of random series seems to have been first suggested by Borel, in

Emile Borel. Sur les séries de Taylor. C. r. hebd. Séanc. Acad. Sci., Paris 123, (1896), 1051-2.

(The journal is available here.)

Part of the problem was that at the time the concepts of probability theory were not quite formalized yet, so going from Borel’s remarks to actual theorems took some time.  Borel wrote

Si les coefficients sont quelconques, le cercle de convergence est une coupure.

What Borel is saying is that if the coefficients of a series \sum_n X_n z^n are “arbitrary”, then the circle of convergence is a natural boundary for the function. What this means is that there is no way to extend F(z)=\sum_n X_n z^n analytically beyond the circle of convergence (because the singular points are dense on the boundary).

The first actual result in this regard is due to Steinhaus in 1929: If the r_n are positive, and 0<\limsup_n r_n^{1/n}<\infty, and the \omega_n are independent random variables equidistributed on {}[0,1], then \sum_n r_ne^{2\pi i\omega_n}z^n has the circle of convergence as natural boundary, almost surely. A different formalization was found later by Paley and Zygmund, in 1932, in terms of Rademacher sequences.

On the other hand, Borel’s statement cannot quite be translated as “the coefficients are independent random variables”. Kahane’s example is the series

\sum_n (2^n\pm1)z^n,

which has radius of convergence 1/2, and 1/2 is the only singular point on the circle of convergence.

Kahane mentions a conjecture of Blackwell that the general situation should be that one of the two scenarios above applies: Either

  1. F(z)=\sum_n X_n z^n has the circle of convergence as natural boundary, or
  2. There is a series \sum_n c_n z^n (the c_n being constants, not random variables; Kahane calls it a sure series) that added to F results on a (random) Taylor series with a strictly larger circle of convergence which is its natural boundary.

The conjecture was proved in 1953 by Ryll-Nardzewski, see

Czesław Ryll-Nardzewski. D. Blackwell’s conjecture on power series with random coefficients. Studia Math. 13, (1953). 30–36. MR0054882 (14,994e).

Kahane also wrote a nice survey of these and related matters, in

Jean-Pierre Kahane. A century of interplay between Taylor series, Fourier series and Brownian motion. Bull. London Math. Soc. 29 (3), (1997), 257–279. MR1435557 (98a:01015).


Young Set Theory Workshop 2013 – Deadline extension

March 1, 2013

(From Matteo Viale.)

Due to the availability of further grants, the deadline for registration to the YSTW 2013 (the 6th edition of the Young Set Theory Workshops, taking place in Oropa Italy 10-14th June 2013) has been extended to the 15th of March 2013.

The full registration fee is 300 euros (including a double room with full board for all the days of the conference — 350 if a single room is requested). We dispose of at least 50 reduced fees of 150 euros, 10 reduced fees of 70 euros and 10 reduced fees of 0 euros. Priority in the attribution of reduced fees will be given to Ph.D. students in logic; however, all applicants can request these special fees.

Full details  on the registration procedure are given at the info and registration pages of the conference, whose official website is http://www2.dm.unito.it/paginepersonali/viale/YST2013/yst2013-home.html

Recall that the aims of the Young Set Theory Workshops are to bring together young researchers in the domain of set theory and give them the opportunity to learn from each other and from experts in a friendly environment. A long-term objective of this series of workshops is to create and maintain a network of young set theorists and senior researchers, so as to establish working contacts and help disseminate knowledge in the field.

These aims are reflected by the format of this year’s workshop, which consists of:

• Four tutorials by established experts: James Cummings, Sy Friedman, Su Gao, John Steel.
• Five invited talks by young researchers: Tristan Bice, Scott Cramer, Luca Motto Ros, Victor Perez Torres, Trevor Wilson.
• Open discussion sessions.