User francis adams - MathOverflow

Image of L^1 under the Fourier Transform

2013-04-09T01:10:08Z

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is dense in $C_0$. Everything I have read/heard says that the range is "difficult to describe".

But, since $\mathcal{F}$ is injective and continuous, the image of $L^1$ must be Borel inside of $C_0$. Is anything else known about its descriptive complexity? If not, might this be an example of a natural set of high Borel rank?

This question may be relevant. Thanks for any insight/references.

Answer by Francis Adams for Favorite popular math book

2012-06-14T17:34:57Z

Title: The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity

Author: Amir D. Aczel

Short Description: Another book about the development of ideas about infinity. The central character is Cantor, of course, but it also looks at people before, like Bolzano and Galileo, and afterwards, including Godel and Cohen for their work on CH.

I read this in high school, having just a little calculus background, and got a lot out of it. It does try to work in the themes of "contemplation of infinity leads to insanity" and "infinity as religious insight" a bit, which might be drawbacks. But when I read it for the first time, I remember laughing out loud at how amazing the ideas involving infinte cardinalites and AC were, so at least those are presented well.

Is the Turing equivalence relation the orbit equiv. relation of the action of a countable group?

2012-05-25T01:17:44Z

The Turing equivalence relation on $\cal P(\mathbb{N})$ is defined by $A\equiv_T B$ iff $A\leq_T B$ and $B\leq_T A$. This is a countable Borel equivalence relation on the polish space $\cal P(\mathbb{N})$. And as stated, the answer to my question is yes since a theorem by Feldman and Moore says: for any countable Borel equivalence relation $E$ on a standard Borel space $X$, there is a countable group $G$ and a Borel action of $G$ on $X$ such that $E$ is the orbit equivalence relation of the action.

But from where I have looked, the Turing equivalence isn't the orbit e.r. of any countable group that is "found in nature". I was wondering if anything else was known about this, as well as any other instances of equiv. relations that are provably orbit equivalences, but with no known "natural" group doing the acting. It seems like for many important equivalences (like $E_0, E_{\infty}$, hyperfinite e.r's) we do in fact have a natural group that is acting, but perhaps that is the reason why we know so much about them.

Any references or other information would be great, thanks.

Good source for Effective Descriptive Set Theory

2012-05-14T03:29:53Z

I just finished a first course in Descriptive Set Theory using Kechris' "Classical Descriptive Set Theory" and was hoping to find a good source for learning some of the Effective DST. Kechris doesn't mention it at all, and after looking at Moschovakis' "Descriptive Set Theory", it doesn't seem like the easiest thing to read. If that is the source I should use and I just need to suck it up and work through it, that is fine. But I would appreciate any advice on other places to look, even as supplements to a tougher standard source.

If it helps with the suggestions, I don't have any particular application or direction in mind. I just liked my course and wanted to continue learning in the area. Thanks.

Answer by Francis Adams for Intuitive and/or philosophical explanation for set theory paradoxes

2012-04-15T15:24:29Z

A really interesting paper by Penelope Maddy called Believing the Axioms has a pretty substantial discussion about various justifications for axioms of set theory. Not only does it look at ZFC, but also higher axioms concerning large cardinals and determinacy. The paper doesn't really talk about why one might want to axiomatize set theory, but it does look at reasons why the axioms that currently exist can be justified, and how they reflect some intuition about how the set-theoretic universe "should" look.

It comes in two parts:

Part 1: http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf

Part 2: http://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms2.pdf

Even if this doesn't really answer your question, this is certainly an interesting read if you are curious about some philosophy of set theory.

Up-to-date version of Principia Mathematica?

2011-11-24T04:32:32Z

Background: I found this interesting translation of Godel's On formally undecidable propositions of Principia Mathematica and related systems I that, along with translating it into English, uses more modern and understandable symbols, includes some hyper text links throughout the paper, and, if I'm not mistaken, makes the language a little more readable. This seems like a good and noble undertaking: taking an important paper and revamping it to make it more accessible, or at least less intimidating for modern readers.

So while looking through it, I couldn't help but think of another work that could benefit from such a treatment, Russell and Whitehead's Principia Mathematica. Every review I have read about it is something like "It lead to a lot of important developments in logic, but is mostly a historical curiosity. And besides, it is incredibly difficult to read, so putting forth the necessary effort is probably not worth it."

Here is the question: does such a thing exist, and would such a paper be worthwhile? Is there anything in the Principia that would be of interest to modern mathematicians. (Finding some points of interest and reproducing them seems like a better idea than trying to reproduce all three volumes.) Godel's theorem and his proof are still interesting, in spite of the fact that there are many other (better?) ways of getting to the Incompleteness theorems.

Another example of this sort of revision is The Annotated Turing.

I'd appreciate any knowledge of such a source, or any insight into why one wouldn't be too valuable, or really any thoughts on looking at Principia Mathematica at all.

Answer by Francis Adams for Which math paper maximizes the ratio (importance)/(length)?

2011-11-07T21:46:03Z

This should be a comment in regards to the answer of A One-Sentence Proof That Every Prime p≡1(mod4) Is a Sum of Two Squares D. Zagier , but I don't have the reputation. While it isn't the actual paper, there is a short but interesting note that goes through how such an involution is constructed in the first result of a google search here.

Answer by Francis Adams for What should be learned in an introductory analytic number theory course?

2011-07-28T05:00:51Z

Having recently finished my math undergrad I audited a course based on Davenport and had a reading course using Apostol, which to use depends on what skills and smaller results you want them to come away with. The things you mentioned like big-O and summation are given a pretty thorough treatment in Apostol. I certainly wasn't cheated and really appreciated having some practice with the skills. It also had enough material for you to have some flexibility.

But, it might seem like too much of an undergraduate text (it introduces the definition of a group before it talks about characters). It also gives a pretty elementary proof for Dirichlet's theorem. Which you may not want.

A book not mentioned that also has a lot of topics and is nice to learn from is Additive Number Theory by Melvyn Nathanson. The material here is very different from that of the other two, but still worthwhile and accessible.

If I had to pick one, I'd go with Apostol. It was so readable and I felt like I got a great foundation in the ideas and skills of number theory.

Answer by Francis Adams for Complete mathematics

2011-06-07T04:23:54Z

This answer is possibly unhelpful and definitely under-informed, but a good source to look into might be the book Complete Theories by Abraham Robinson. I haven't read it, I just happened to see it as I was perusing the mathematics section in my library. If you have done any research, you may have come across this book. If not, it would be worth checking out.

Answer by Francis Adams for Textbook recommendations for undergraduate proof-writing class

2011-04-28T16:31:43Z

The book I used in my 'proofs' class was "Doing Mathematics: An Introduction to Proofs and Problem Solving" by Steven Galovich, here on Amazon.

The class was called "Mathematical Structures", which is an apt name since the class wasn't solely about learning to prove things. It was learning to prove things in the context of learning about basic mathematical objects. It starts with basic logic, but after it introduces sets, relations, functions, equivlance relations and the like, it goes onto to develop the ideas of cardinality, including Cantor-Bernstein. It also has a couple other topics, like some basic combinatorics, the constructions of number systems, or looking at consequences of the field axioms.

It was a great introduction to what math is "really about" coming after some mostly computational calculus and linear algebra courses. The price is about $50, so it is a little more than you were looking for. But it is absolutely a book worth having.

Answer by Francis Adams for What are some mathematical concepts that were (pretty much) created from scratch and do not owe a debt to previous work?

2010-05-27T01:54:58Z

It seems like Dirichlet's Theorem on Primes in Arithmetic Progressions came out of nowhere, or at least his methods of proof. While the complex analysis may not have been new, his application of it, through the Dirichlet characters and the series he made from them, to number theory was pretty novel.

Comment by Francis Adams

2012-06-22T12:51:24Z

Sequences are taught before real analysis, usually in Calc 2 along with infinite series. And the more basic material is suitable for high school, even a decent precalculus class. These are only sequences of reals so it isn't very general, and while they are taught, students might not really "understand" them until later.

Comment by Francis Adams

2012-05-25T02:52:23Z

This website is for research level mathematics. This is a website that is more suited for homework help. math.stackexchange.com

Comment by Francis Adams

2012-04-26T19:20:49Z

For homework type problems, try this site: math.stackexchange.com This site is for research-level mathematics.

Comment by Francis Adams

2011-12-01T16:08:26Z

I'm sure Igor understands what the words mean and what the question says. But maybe he wants to know why you are asking it. Do you already know one direction? Do you have any evidence for thinking this is true? What were you doing when you thought this question? Things like that would help.