January 27, 2011
Recall that we defined Nim addition and Nim multiplication on the natural numbers by setting:
- is the result of writing in binary and adding without carrying.
- is the unique binary operation on that is commutative, associative, distributive over and satisfies:
- whenever . Here, for all , call these numbers “Fermat’s 2-powers”.
The first problem is that it is not quite clear that is even well-defined (i.e., are there any functions at all that behave as required of ? Is there really only one such function?). Begin by checking this, by showing that the rules above give us that is the result of writing as sums of (multiplications by powers of 2) of Fermat 2-powers, and expanding using associativity, distributivity, and the two rules about how to multiply with Fermat 2-powers. (Verify that any can indeed be written as such a sum in a unique way, and that this indeed shows that is completely characterized by our description.)
Show that if we set , then and each are fields (over , with addition and multiplication given by .
In lecture I erroneously mentioned that is algebraically closed. This is not the case. For example, show that the equation has no solutions in when we interpret “ is a solution” to mean that .
Nim addition and multiplication were introduced by John Conway. A bit of online searching will give you references for this exercise, but please abstain from looking for them. I will provide references and some additional details once the homework has been turned in.
This set is due February 11 at the beginning of lecture.
January 21, 2011
For Part III, see here.
(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list. If you have corrections/updates, please email me. Sorry for the delay with posting this.)
January 20, 2011
For Part II, see here.
(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list.)
January 15, 2011
Set theory and its applications. L. Babinkostova, A. E. Caicedo, S. Geschke, M. Scheepers, eds. Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. ISBN-10: 0-8218-4812-7 ISBN-13: 978-0-8218-4812-8
Here is a link to the AMS page for it, a link to its table of contents, and the preface:
The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small, locally funded conference dedicated to Set Theory and its Applications. A number of years after its inception BEST started being funded by the National Science Foundation. Without this funding it would not have been possible to maintain the conference. The conference remained relatively small with many opportunities for its participants to meet informally. We like to think that during these years BEST has made it possible for the numerous set theorists who have participated in it to absorb, besides the new developments featured in the conference talks, also part of the folklore and traditions of the ﬁeld of set theory and its relatives. An explicit effort was made to bring together role models from various career stages in set theory as well as the new generation to support some notion of continuity in the ﬁeld.
This volume has a similar purpose. In it the reader will ﬁnd valuable papers ranging from surveys that put in print here set theoretic knowledge that has been around for several decades as unpublished lore, to hybrid survey-research papers, to pure research papers. Readers can be assured of the authority of each paper since each has been carefully refereed. The reader will also ﬁnd that the subjects treated in these papers range over several of the historically strongly represented areas of set theory and its relatives. Rather than expounding the virtues of each paper individually here, we invite the reader to learn from the authors.
Bringing to publication such a collection of papers is not possible without the generous dedication of authors and referees and the services of a publisher. We would like to thank all authors and referees for their selﬂess contributions to this volume. And we particularly would like to thank the publisher, Contemporary Mathematics, and Christine Thivierge, for the guidance they provided during this process.
January 6, 2011
Today at lunch we got a trifecta:
The effort have the potential to pay off handsomely today
That’s just ok. But then:
People try thing, because they just don’t want it enough.
Photographic memory! Remember to put in film, or its digital!?