User ben weiss - MathOverflow

How to select a journal?

2009-11-30T15:58:44Z

What are good criteria for selecting a journal to submit a paper to?

One criterion per answer, please.

It is easy to group journals by subject and prestige, but is there a thought-process that you use to determine which journal is good for your specific paper?

For example, if you have a clearly top paper, it is not hard to find a journal that will accept it (the only question is which top journal to submit it to), but do you have favorite journals for specific types of projects, papers that are not earth-shattering, or other categories of papers?

Twin Prime Conjecture Reference

2009-12-03T05:17:24Z

I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but according to Goldston

http://www.math.sjsu.edu/~goldston/twinprimes.pdf

it was stated nowhere until de Polignac. I'm hoping to resolve this issue by accessing either primary historical documents, or other reputable secondary sources (Goldston being one such example). I have looked at de Polignac's work, and he does indeed make a conjecture, but have been unable to find anything definitive (besides Goldston's statements) that there was no conjecture earlier. If this is too specific for MO, I'll remove the question. Thank you.

Equivalence of Finiteness of Class number to Property of Ideals of Algebraic Integers

2010-03-22T16:13:51Z

I was reminded of this problem by a recent question of Regenbogen. It is not too difficult to prove that since every number field has finite class group, that every ideal in the ring of algebraic integers is either principal or infinitely generated. I've been unable to prove that these statements are actually equivalent, and this is what I'd be interested to know.

In particular, if they are equivalent, then this may provide a different way of proving finiteness of class number.

Answer by Ben Weiss for What are the worst notations, in your opinion ?

2010-03-18T15:54:35Z

As Trevor Wooley used to always say in class, ``Vinogradov's notation sucks....the constants away."

For those who don't know, Vinogradov's notation in this context is $f(x)\ll g(x)$ meaning $f(x) = O(g(x)).$ (if you prefer big-O notation, that is).

Answer by Ben Weiss for Books you would like to see translated into English.

2010-03-12T04:33:57Z

Bombieri's "Le Grand Crible dans la Théorie Analytique des Nombres"

Answer by Ben Weiss for Alternative proof of unique factorization for ideals in a Dedekind ring

2010-03-09T01:20:03Z

I believe that the proof in Marcus's "Number Fields" contains a proof which does not rely on integral closure except to say that if an element satisfies a monic polynomial, it is in the domain. I'll summarize the lemmas he uses:

1) For any ideal $I$, there is an ideal $J$ so that $IJ$ is principle.

2) For any proper ideal $I$, there is an element $x$ in the field of fractions and not in the Dedekind domain so that $xI$ is still in the Dedekind domain.

To prove the second lemma he uses integrally closed, but only to show that an element of the field of fractions satisfies a monic polynomial, and is thus in the Dedekind domain.

3) The ideal classes form a group. This is a quick consequence of the previous lemmas.

4) Some group results about the ideals. (the google books view which I am using is missing the last page).

I think after that there is no more use of integrally closed, but as I said I'm missing the last page of the proof. Hope this helps.

Answer by Ben Weiss for Factorials in Pascals Triangle

2010-03-04T09:31:54Z

This is only a start, but maybe it'll suggest to others how to proceed. If $p$ is a prime number, then there is no integer $1 \le n < p$ and integer $m$ with $${p\choose{n}} = m!$$ This is because the LHS is divisible by $p$ but is less than $p!$ and the right hand side cannot be both divisible by $p$ and less than $p!$

I'm not sure how to generalize this, but I hope it helps.

Answer by Ben Weiss for Set of vectors separated by at least a specified angle

2010-03-03T01:40:16Z

The subject name you are looking for is spherical codes. A good reference for this subject is Conway and Sloane's "Sphere Packings, Lattices, and Groups." In chapter 9 they give the details of the proof for the best bounds (I believe it is due to Levenstein, but don't have the book with me).

This ends up being related to density of sphere packings. There's a very elegant proof in the book which relates the answer to your question in dimension $n+1$ to the maximal density of sphere packing in dimension $n.$

Sorry I don't have my references with me, but this is all in chapter 9 of the book.

Answer by Ben Weiss for How do you become a good listener?

2010-02-20T19:23:53Z

The best advice I ever received was from Jordan Ellenberg, I hope he doesn't mind my rephrasing of it here. When sitting in a seminar, try to figure out an interesting question to ask the speaker. This will ensure that you listen to enough of the talk that you learn something interesting (it doesn't have to be the main part of the talk, but anything you enjoy in the talk), and spend time focusing your thoughts around the subject.

Answer by Ben Weiss for Haar measure on a quotient, References for.

2010-02-05T15:39:00Z

"Fourier Analysis on Number Fields" by Ramakrishnan and Valenza deals with many of the same topics, but starts in chapter 1 with exactly this material and works up to Tate's thesis in chapter 7. I hope this helps.

Answer by Ben Weiss for Prime divisors of numbers 2^n + 3

2010-01-30T07:22:33Z

(Edited as the comments below suggest)

The ABC conjecture seemed to me like it would play a roll, however it comes up a little short:

"Are there infinitely many primes $p$ so that for each $p$ there is some integer $n$ with $p^2|2^n + 3?"$

If the ABC conjecture is true, then this answer to this question is almost "no", but still there is a problem at the end of the argument.

The ABC conjecture states that for any $\epsilon > 0$ there is a constant $K_\epsilon$ so that for any co-prime triple $A < B < C$ with $A+B = C$ then $$C \le K_\epsilon\prod_{p|ABC}p^{1 + \epsilon}.$$

So, if there is such an infinite collection of primes, then for the corresponding infinite $n$ where this is true then $2^n + 3 = p^2C$ then $$p^2C \le K_\epsilon(6Cp)^{1+\epsilon}.$$

(Edited: The following sentence is incorrect "But this will clearly run into problems for sufficiently large $p.$" But I wanted to leave it so Kevin's comment makes sense.)

Note that as $C = C(p)$ is a function of $p$ then the $C^\epsilon$ (when $C$ is square-free, or nearly square-free) term may still allow this inequality to work.

Answer by Ben Weiss for Applications of the Chinese remainder theorem

2010-01-29T05:31:26Z

One answer I don't see here: Lagrange interpolation. If one takes, for example, the ring $\mathbb{Q}[x]$, and realizes CRT as a statement about rings and direct sums of $R/P$ over a set of co-prime $P,$ then one can construct polynomials which have cycles of arbitrary length in the rationals (or any number of cycles of arbitrary length). Lagrange interpolation has other applications, but the proof is CRT.

Answer by Ben Weiss for Cool problems to impress students with group theory

2010-01-29T04:50:49Z

This is related to many of the answers already here, but a little different. When I was an undergrad, I got fixated on the following problem:

Given a deck of $n$ cards, how many perfect shuffles does it take to get back to the starting position. Does it matter if this preserves the top and bottom card (what I called an "out shuffle") or mixed them in ("an in shuffle")? This leads to the discrete logarithm problem and other stuff with cyclic groups.

Answer by Ben Weiss for probability in number theory

2010-01-26T05:12:42Z

The one I learned from is Tenenbaum.

My personal favorite application is to derive heuristics for the twin prime conjecture (and more general Hardy-Littlewood conjecture). For an excellent exposition on this, see Soundararajan's article.

Answer by Ben Weiss for non-Dedekind Domain in which every ideal is generated by at most two elements

2010-01-25T21:46:33Z

[Edited to restrict to the case of quadratic orders. --PLC]

Take any non-maximal order of a quadratic number field. This is not Dedekind because it fails to be integrally closed in its field of fractions. Every ideal is a free abelian subgroup of rank at most $2$.

For example: $\mathbb{Z}[\sqrt{-3}].$

I hope this answers your question. For further reading on Dedekind domains, and non-maximal Orders, I highly recommend the chapter on it in Neukirch's Algebraic Number Theory.

Answer by Ben Weiss for Are there any books that take a 'theorems as problems' approach?

2010-01-23T02:03:49Z

My favorite such book is Problems in Analytic Number Theory by Ram Murty. There could not be enough good things said about it.

Answer by Ben Weiss for Recovering $\Phi(n)$ from a multiple?

2010-01-22T21:12:50Z

One thought that is only one step worse than what you are suggesting is that if the message gets sent 3 times: $k- t\phi(n)$, $k-u\phi(n)$, and $k - s\phi(n)$, then one can recover 3 different differences, and easily perform the euclidean algorithm on the pairs. This well always result in a multiple of $\phi(n)$, and I expect that unless $s$, $t$, and $u$ have a lot of congruence relationships between themselves, then this will result in $\phi(n)$ itself.

Maximal Ellipsoid

2010-01-20T21:36:53Z

John's Theorem can be stated as "To every compact, convex body, there is a unique inscribed ellipsoid, whose volume is maximal among all inscribed ellipsoids." It goes on to classify this maximal ellipsoid. By using this theorem, one can prove that the ellipsoid of maximal volume which is contained in a square is a circle.

This strikes me as a problem which was probably studied well before Fritz John, and yet I have been unable to prove the statement about squares and circles in an elegant, but low-brow manner. Any thoughts?

Correlation of Statistical Tests

2010-01-20T15:57:32Z

Suppose I have a sequence ${x_i}_{i=1}^\infty$ of zeros and ones. I want to test if they are randomly generated according to a conjectured scheme (the example to keep in mind is that they are conjectured to be generated by independent coin flips). I can use a statistical test --say to see if the sequence 11 occurs as much as it should up to some bound $x$, or I can use a different test --say to see if $1r1$, where the $r$ is anything--occurs as much as it should up to the same bound (in general I'll be interested as $x \rightarrow \infty$).

The question is, does knowing that one test suggests that this is a randomly generated set guarantee (or somehow influence) the other test's results? Are they strongly correlated? Are there tests which would be correlated? I believe this study falls under the subject of 0-1 laws, but do not know where to begin looking. Any references, books, papers, answers would be appreciated--I'm still trying to find my way around the field.

Also please retag if you know of better tags to use.

Answer by Ben Weiss for Generalization of primitive roots

2010-01-11T00:09:47Z

There are good results related to what you are asking. Hans Roskam stated and proved a quadratic analogue of Artin's conjecture (and I think his paper is much easier to read than Hooley's). He studies the density of primes in which a fundamental unit in a real quadratic field has a maximal order.

The paper is titled, "A quadratic Analogue of Artin's Conjecture on Primitive Roots" Journal of Number Theory 81, 93-109.

You can find his papers here and this is also good.

Hope this helps.

Answer by Ben Weiss for Gaps in nx (mod 1)

2010-01-10T07:10:27Z

This reminds me of the following cute statement (which I believe has been recently generalized to arbitrary manifolds, but I couldn't find the paper): Given the sequence of points $k\alpha (\mod 1)$, there are at most 3 different distances between nearest neighbors.

Actually, I first heard it stated as $\alpha^k$ on the unit circle. To prove it, notice that the first cycle of points around the circle all have the same length between them, and then there is the remainder at the end.

When the wrap occurs, this is the remainder just shifted (because the distance between two consecutive points is always the same), and it keeps appearing between previous points in the same way at each step (and then there are the remaining points it has yet to appear between). This still leaves only 3 lengths (the two from the "cut" it produced in the previous length, and the previous lengths it has yet to cut).

I hope this helps.

Answer by Ben Weiss for English reference for a result of Kronecker?

2010-01-06T15:43:31Z

Bombieri and Gluber's recent book "Heights in Diophantine Geometry" has a proof of this in chapter 1.

Answer by Ben Weiss for Magic trick based on deep mathematics

2010-01-04T04:37:45Z

This was fascinating for me. Somehow the man takes a bagel and with one cut arrives with two pieces that are interlocked. Whether this qualifies as "magic" I dunno (it's hard to say once the trick's been explained), but it sure seems like it to me.

It doesn't hurt that I love bagels, and have the opportunity to perform this with friends/family/non-math people and can teach a little about problems/topology/counter-intuitive facts about the universe.

Answer by Ben Weiss for Bertrand postulate

2010-01-02T23:04:46Z

One remark to relate Bertrand's postulate to the prime number theorem: Chebyshev's work was related to bounding ratios of factorials--in particular $\frac{2n!}{n!n!}.$

His later proof that $C\frac{x}{\log x} < \pi(x) < D\frac{x}{\log x}$ made use of other ratios (in this case $\frac{30n!n!}{15n!5n!3n!}$). In theory one could try and improve the numbers used in the ratio to asymptotically prove the prime number theorem. Jonathan Bober (among others) have worked on this. He has catalogued many different combinations of ratios of factorials (this also ends up tying into G and E functions....but I'm already out of my depth of what I'm capable of explaining).

Answer by Ben Weiss for Simultaneous diagonalization

2009-12-22T15:49:20Z

Hi Arminius, For part (1) I believe you're asking that if two diagonalizable matrices commute, then they have the same eigenspaces (aka they are simultaneously diagonalizable). I believe this is true if you make all the eigenvalues distinct:

Let $D$ be diagonal with distinct entries, then if it commutes with $A$:

$AD$ = $DA$

on the LHS of the equation, all the columns of $A$ are scaled by different factors. On the RHS the columns are scaled. The only way to reconcile this is if $A$ is diagonal too.

Hope this helps.

Answer by Ben Weiss for What are some good resources for mathematical translation?

2009-12-21T04:15:49Z

I've found that Google has a translator application which is wonderful. If you type in a sentence, it does the correct translation; it has even known mathematical terms when presented in the correct context. It is my first resource for help with translations.

Answer by Ben Weiss for Analytic density of the set of primes starting with 1

2009-12-17T01:37:40Z

I think instead of posting my own explanation (which will only lose something in the translation) I'll instead refer you to two very interesting papers (thanks for posting this question, I haven't thought about this stuff in a couple years, and these papers were interesting reads to solve your problem.)

The first (among other things) proves that the density of primes with leading coefficient $k$ is $\log_{10}\left(\frac{k +1}{k}\right).$

Prime numbers and the first digit phenomenon by Daniel I. A. Cohen* and Talbot M. Katz in Journal of number theory 18, 261-268 (1984)

The second is a more general statement about first digits. It is

The first digit problem by Ralph Raimi in American Math Monthly vol 83 No 7

Hope this all helps.

Answer by Ben Weiss for Expressing field inclusions by polynomial equalities on coefficients

2009-12-08T05:24:11Z

This may ramble a bit much, but I hope it provides some help in how to think about the problem.

Let's see what your extension of fields looks like. We have 4 possible extensions (perhaps the same) So that any of them is

$\mathbb Q(z_i)$

$|$

$\mathbb Q\left(\sqrt2\right)$

$|$

$\mathbb Q$

Where $z_i$ ranges of the 4 possible roots $z_1,...,z_4.$ Then $\mathbb Q(z_1)$ is degree 4 (since the polynomial is irreducible), but this polynomial factors into a product of quadratics over $\mathbb Q\left(\sqrt2\right).$ So indeed we've reduced to having only two possible extensions, in that the two roots of the same quadratic generate the same extension over $\mathbb Q(\sqrt2).$

However, except for this restriction, I don't see anything else to lead to a relation on the coefficients. Hopefully this will help you or someone else get a start on the problem.

One further thought:

Since the roots appear in pairs (say $z_1$ and $z_2$ are conjugate over $\mathbb Q\left(\sqrt 2\right)$) then one can generate $\sqrt 2$ with either pair, and subtract them. However, I don't immediately see a way to gather that information from the symmetric polynomials of the roots (a.k.a. the coefficients $a_1, \ldots, a_4.$)

Extremal question on matrices

2009-12-02T02:12:07Z

The following question was posed to me a while ago. No one I know has a given a satisfactory (or even a complete) proof:

Suppose that $M$ is an $n$ x $n$ matrix of non-negative integers. Additionally, suppose that if a coordinate of $M$ is zero, then the sum of the entries in its row and its column is at least $n$.

What is the smallest that the sum of all the entries in $M$ can be?

The conjecture posed to me was that it was $\frac{n^2}{2}$ which is obtained by the diagonal matrix with $\frac{n}{2}$ in all diagonal entries.

[I'm guessing that there should be a "suppose that" in describing M. -- GJK]

Answer by Ben Weiss for When does the zeta function take on integer values?

2009-12-02T16:25:51Z

Let's look at $\zeta(s)$ for some large $\sigma$ (the real part of $s$). We can bound the function by $\int_1^\infty \frac{dx}{x^\sigma} + 1$, which is $\frac{\sigma}{\sigma-1}.$ So all these values in any $A_m$ ($m > 1$) can be bounded by a half-plane. This definitely doesn't pin down where they are, but does give a nice bound on where they are not.

Comment by Ben Weiss

2010-03-22T15:34:11Z

Great question, but I would like to debate the statement that "there is not much work going on in the subject of geometry of numbers". Curt Mcmullen just recently proved Minkowski's conjecture in 6 dimensions, and reduced all higher dimensions to a simpler problem. Geometry of numbers is still studied quite a bit; it is frequently a study of geodesics now, but lattices, lattice packings and geodesics (especially on hyperbolic space) are all geometry of numbers and still very much studied.

Comment by Ben Weiss

2010-03-20T02:35:53Z

Or e-mail Harris Nover...he knows all this stuff.

Comment by Ben Weiss

2010-03-14T22:41:50Z

I've heard that 7/22 is the correct holiday.

Comment by Ben Weiss

2010-03-13T23:01:48Z

I just confirmed your calculations up to n=1000, so I'm inclined to believe the statement. I'm also inclined to believe it is difficult.

Comment by Ben Weiss

2010-03-09T21:06:09Z

Im confused as to why x_0 must be a fixed point of f. Can't it be an involution (order two) point of f?

Comment by Ben Weiss

2010-03-09T01:31:41Z

The proof is on pages 56-60.

Comment by Ben Weiss

2010-03-07T06:13:04Z

No thoughts on how to answer your question, but as I think I know what brought it up let me ask: is there any ergodic reasons behind this question?

Comment by Ben Weiss

2010-03-04T07:49:06Z

@Rob In your suggestion, do you mean powers of 17? The example you give of 16! = 2!14!5! has the powers of 7 matching up.

Comment by Ben Weiss

2010-03-02T21:40:53Z

+1 post, but as an aside, we already know the Mersenne primes have zero density; there are at most log(x) of them up to x (as they are powers of 2).

Comment by Ben Weiss

2010-02-25T21:59:38Z

@unknown: If you are suggesting that the semantics you are getting at relies on some specifics of how rockets work, then I don't really think this has much to do with the mathematics of the problem, which is what MathOverflow is about.

Comment by Ben Weiss

2010-02-25T21:55:32Z

@Andrea: No worries, I'd never heard it as dogs. I think it's in Peter Winkler's puzzle book as rockets (but that's the only place I've heard this problem).

Comment by Ben Weiss

2010-02-25T21:48:18Z

It seems as if your problem is surreptitiously asking about the semantics of the question. There are several reasonable solutions to this problem, my favorite uses orthogonality of the rockets, and symmetry of the system, and then argues that it must be 20 seconds because the rockets are always traveling directly at each other. Tho, I'm not really sure what you are looking for. I'd rather the purpose and semantics question be asked up front, but I appreciate that you didn't want to do that.

Comment by Ben Weiss

2010-02-21T04:34:49Z

At the Midwest Number Theory Conference for Graduate Students a few years ago, you told us you do this (especially when you run a seminar). You also told us it is a good thing to do in a talk which is particularly bad so that someone has a question to ask the speaker at the end and make them feel welcome.

Comment by Ben Weiss

2010-02-05T16:20:51Z

@Pete: You're right! I didn't have my copy, and just misremembered. I tried skimming through google just now, and didn't see it. It may be in chapter 3 (which Google didn't let me see), but I doubt it.

Comment by Ben Weiss

2010-01-30T20:21:22Z

This sounds like a little like a "smooth" version of Artin's Conjecture for number fields. If I'm correct in understanding that, try looking up Hans Roskam's work on the subject "Quadratic Analogue of Artin's Conjecture."