User bruce arnold - MathOverflow

The factorial of -1, -2, -3, ...

2009-12-30T11:32:10Z

Well, n! is for integer n < 0 not defined -- as yet.

So the question is: How could a sensible generalization of the factorial for negative integers look like?

Clearly a good generalization should have a clear combinatorial meaning which combines well with the nonnegative case.

Answer by Bruce Arnold for What would you want to see at the Museum of Mathematics?

2010-12-26T14:49:09Z

Look at the 'surfer video' which among other things shows how visualizations of algebraic geometry can be presented in real-time in an exhibition.

Who invented the gamma function?

2009-12-25T14:36:50Z

Who was the first person who solved the problem of extending the factorial to non-integer arguments?

Detlef Gronau writes [1]: "As a matter of fact, it was Daniel Bernoulli who gave in 1729 the first representation of an interpolating function of the factorials in form of an infinite product, later known as gamma function."

On the other hand many other places say it was Leonhard Euler. Will the real inventor please stand up?

[1] "Why is the gamma function so as it is?" by Detlef Gronau, Teaching Mathematics and Computer Science, 1/1 (2003), 43-53.

Answer by Bruce Arnold for Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?

2010-09-28T20:22:12Z

Let me answer by rephrasing your question. How will Andrew Wiles' proof of Fermat's Last Theorem be seen in the year 2260? I think it will be acknowledged then as much as it is acknowledged today that the proof was a major step in the history of mathematics. However, I seriously doubt that his proof will be considered as 'rigorous' by the standards of the year 2260. A well-respected math journal will then require a proof formally verified by a symbolic engine. (See the Notices of AMS 2008, vol. 55, issue 11).

Answer by Bruce Arnold for Consequences of the Riemann hypothesis

2010-03-05T22:48:54Z

(Jeffrey C. Lagarias) The following is equivalent to RH. Let $H_n = \sum\limits_{j=1}^n \frac{1}{j}$ be the $n$-th harmonic number. For each $n \ge 1$ $$\sum\limits_{d|n} d \le H_n + \exp (H_n) \log (H_n),$$ with equality only for $n = 1.$ (An Elementary Problem Equivalent to the Riemann Hypothesis. See also OEIS A057641.)

Answer by Bruce Arnold for Which mathematical ideas have done most to change history?

2010-02-01T18:19:58Z

The introduction of order. Orders are ubiquitous in life. And orders are ubiquitous mathematics. These are the concepts of reflexivity, antisymmetry and transitivity and those of maximality.

Answer by Bruce Arnold for Computer Algebra Errors

2010-01-12T12:11:16Z

There are too many to be listed on the margins of MO.

Look at the archives of the newsgroups comp.soft-sys.math.maple, comp.soft-sys.matlab, sci.math.symbolic, comp.soft-sys.math.mathematica. There you can find hundreds of bugs reported.

There is a notorious CAS bug hunter who once maintained a bug list for Maple and shows more than 5000 disturbing observations. (Press the Go! button.) Or go to MapleSoft and search Maple Primes.

Please don't shoot the messenger.

Answer by Bruce Arnold for What programming languages do mathematicians use?

2010-01-08T12:18:44Z

For mathematicians which do scientific computing in the sens of numerical analysis a very good choice are the XSC languages (C-XSC and Pascal-XSC) which provide tools to solve numerical problems with a verification of the results. See U. Kulisch et alia, Springer Series in Computational Mathematics, vol. 21, for an introduction to Verified Computing and this link for the software.

An inequality relating the factorial to the primorial.

2010-01-02T17:34:17Z

Let [a,b] = {k integer | a < k <= b}. Further let

Comp[a,b] = product_{c in [a,b]} c composite;
Fact[a,b] = product_{k in [a,b]} k integer;
Prim[a,b] = product_{p in [a,b]} p prime.

Question: For n > 2 and n not in {10,15,27,39} is it true that

$$ \text{Comp}[{\left\lfloor n /2 \right\rfloor}, n] < \text{Fact}[1, {\left\lfloor n /2 \right\rfloor}] \ \text{Prim}[{\left\lfloor n /2 \right\rfloor}, n] \ ? $$

Update: The state of affairs: Gjergji Zaimi showed that for large enough n the inequality is true. In my answer I affirm that the inequality is true in the range 40 <= n <= 10^5. It remains open whether 10^5 is 'large enough' in the sense of Gjergji's analysis.

Answer by Bruce Arnold for An inequality relating the factorial to the primorial.

2010-01-03T15:57:22Z

A computational approach to the Compositorial-Factorial-Primorial Inequality (CFPI).

Let $u_{0}=1,u_{1}=1,u_{2}=1/2$ and for $n>2$ define $u_{n}$ by $$ {\text{if}\ n\ \ \text{odd} \ \text{then }\text{if}\ n\ \ prime\ \ \text{then } \ u_{n}=1/n\text{ else }u_{n}=n\ \text{fi}\ \text{fi};} $$ $$ {\text{if}\ n\ \text{even}\ \text{then } \text{if}\ n/2\ prime\ \text{then}\ u_{n} =n\text{ else }u_{n}=4/n\ \text{fi}\ \text{fi}.} $$ Let the sequence of partial products of $u_{n}$ given by $U_{0}=1$ and $$ U_{n}=U_{n-1}u_{n}\quad\left( n>0\right) . $$ The CFPI as stated in the question is equivalent to the statement $$ \text{numerator }U_{n}<\text{denominator }U_{n}\quad\left(n\geq40\right) .$$

Using this algorithm I checked the CFPI in the range $40 \leq n \leq 10^5$ and found no counterexamples.

Answer by Bruce Arnold for The factorial of -1, -2, -3, ...

2009-12-30T23:46:51Z

My question was intended somewhat along the line: Assume the Gamma function is not yet invented and Goldbach asks you the question: "What is (-n)! ?" You know that Goldbach expects a combinatorial answer in the domain of integer or rational numbers. What would you answer? I will give my answer in this sens.

Looking at GKP's ConMath, Table 253, the combined Stirling triangles in their dual form, we see: If we sum the columns in this triangle for k < 0 we get the factorial numbers, if we sum the rows for k > 0 we get the Bell numbers.

What about saying the Bell numbers are the factorial numbers at negative integers? Is the answer encoded in one of the most important triangles in combinatorics?

See what Knuth says about the origin of this duality (table on page 11).

{120}
......{24}
.1,.......{6}
10, .1,......{2}
35, .6, .1,.....{1}
50, 11, 3, 1,
24, .6, 2, 1, 1,....{1}
.0, .0, 0, 0, 0, 1,
.0, .0, 0, 0, 0, 0, 1,...{1}
.0, .0, 0, 0, 0, 0, 1, .1,....{2}
.0, .0, 0, 0, 0, 0, 1, .3, .1,....{5}
.0, .0, 0, 0, 0, 0, 1, .7, .6, .1,....{15}
.0, .0, 0, 0, 0, 0, 1, 15, 25, 10, 1,....{52}

A product of gamma values over the numbers coprime to n.

2009-12-27T12:38:35Z

Let φ(n) denote Euler's totient function and k $\perp$ n denote that k, n are integers and relatively prime. Let N = φ(n) + 1. If n is not a prime power $$ \prod_{\substack{0 < k < n, \ k \perp n }} \Gamma \left(\frac{k}{n}\right) = \sqrt{N}\prod_{ 0 < k < N}\Gamma \left(\frac{k}{N}\right) \quad (n \neq p^a) . $$ This identity was proved here as a corollary to some other identities, but the authors asked: ``is there some natural direct proof of this formula?´´

Answer by Bruce Arnold for Who invented the gamma function?

2009-12-26T23:00:48Z

The first person who gave a representation of the so called gamma function was Daniel Bernoulli in a letter to Goldbach from 1729-10-06. The letter can be seen here.

The formula reads in modern notation as given by Gronau in the article cited in the answer:

$ x! = \lim_{n\rightarrow \infty}\left(n+1+\frac{x}{2}\right)^{x-1} \prod_{i=1}^n\frac{i+1}{i+x} $

Gronau also observes that "Numerical experiments show that the formula of Bernoulli converges much faster to its limit than that of Euler ...", "that of Euler" refers here to a formula Euler has given in a letter to Goldbach dated 1729-10-13.

Gronau writes: "Euler who, at that time, stayed together with D. Bernoulli in St. Petersburg gave a similar representation of this interpolating function. But then, Euler did much more. He gave further representations by integrals, and formulated interesting theorems on the properties of this function."

Though this justifies the name 'Euler gamma function' Euler's representation was historically only second to Daniel Bernoulli's.

The correspondence between Goldbach, Daniel Bernoulli and Euler which undoubtedly gave birth to the gamma function is well documented in Paul Heinrich Fuss's „Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIeme siècle ..“, St. Pétersbourg, 1843.

Comment by Bruce Arnold

2011-08-21T09:32:59Z

The 'infini' in "éventail (infini) de variétés" does not match with a hand fan - it suggest much more the botanic meaning and a fractal nature, something like this 'magic fan' fraktalwelt.de/myhome/images/farn.gif.

Comment by Bruce Arnold

2011-05-31T19:07:47Z

See also Lieven le Bruyn's blog neverendingbooks.org/index.php/…

Comment by Bruce Arnold

2011-03-02T14:58:43Z

Only short <a href="books.google.de/…;.

Comment by Bruce Arnold

2011-01-28T15:15:32Z

Comment by Bruce Arnold

2010-12-10T17:35:45Z

Well of course, these three noble men discussed this subject for a long time. Do you think the sentence you cite sheds any new light on the matter resp. doubt that Daniel Bernoulli was the first to come up with the gamma function? Since my Latin is a little bit rusty I might not be able to extract further relevant information from this letter; on the other hand Bernoulli in his letter, which he wrote a week earlier, gives not the slightest indication that he had the "term general pour la suite 1, 1*2, 1*2*3, etc." from Euler. He would certainly have mentioned this if it had been the case.

Comment by Bruce Arnold

2010-11-30T23:07:20Z

Have a look at numerics.mathdotnet.com .

Comment by Bruce Arnold

2010-11-13T21:47:23Z

This might also interest you: Robert J Betts, Lack of divisibility of C(2N,N) by three fixed odd primes infinitely often, ... [1010.3070] at arXiv.

Comment by Bruce Arnold

2010-11-07T21:19:27Z

André Weil was the author of the first article of Nicolas Bourbaki. Bourbaki N.: Sur un théoreme de Carathéodory et la mesure dans les espaces topologiques. C. R. Acad. sci. Paris 201 (1935), 1309-1311. Zbl 0013.15503

Comment by Bruce Arnold

2010-10-22T15:46:56Z

With Knuth's proposal to write $k \perp n$ for `k relatively prime to n' we can write by the Moebius inversion formula $\sum_{1 \le k \le n} \left[ k \perp n \right] = \sum_{1 \le k \le n} \left[ k \mid n \right] \mu\left(\frac{n}{k}\right).$ Note that by the conventions of the Iverson bracket when the Iverson-bracketed statement is false, it `annihilates anything it is multiplied by - even if that other factor is undefined'.

Comment by Bruce Arnold

2010-10-21T13:50:05Z

Knuth's <i> x to the n falling </i> is $x^{\underline{n}}$, not $x_{\underline{n}}$.

Comment by Bruce Arnold

2010-10-04T20:59:34Z

Thus Paul Feyerabend's "Great scientists are methodological opportunists who use any moves that come to hand, even if they thereby violate canons of empiricist methodology" translates into "Great mathematicans are foundational opportunists who use any moves that come to hand, even if they thereby violate canons of logical methodology"?

Comment by Bruce Arnold

2010-09-17T16:40:37Z

I agree. However, the life-span of a personal webpage is much shorter than the life time of its owner. Therefore a repository for such software should be set up and sponsored. For example by using the $1 mil. prize the Clay Institute is looking to spend for the benefit of mathematics. Let me suggest: 1/3 mil. for MO, 1/3 mil. for the arXiv and 1/3 mil. for a mathematical software repository. This would "increase and disseminate mathematical knowledge" and maybe such a solution would also please Perelman as it would acknowledge the way he shared his work with the public?

Comment by Bruce Arnold

2010-07-26T07:06:20Z

I can confirm that there are at least 6 errors in way you wrote the 3 names.

Comment by Bruce Arnold

2010-07-13T19:47:21Z

Be careful! Optimal rulers are not necessarily Golomb rulers. There are 4 homometric optimal rulers of length 123 and a homometric pair of length 138 is known. Look around at the given site.

Comment by Bruce Arnold

2010-07-13T19:12:36Z

Some information on the related 'optimal rulers' can be found on this page: luschny.de/math/rulers/optimallist.html