57

Short version: LIGO matches their data onto waveforms calculated in numerical relativity. The mathematical study of black hole solutions plays a significant role in this; we couldn't trust our inferences if we didn't know a priori that black holes rapidly stabilize into a handful of low-parametric stationary configurations. Classical black holes are ...


50

I do not understand what the bounty on this question is for, as it seems to me that the other answers were already rather devastating. Here is a semi-reasoned technical answer. According to G. Lolli (the paper you cite) "Sergeyev is wary of the axiomatic method because he thinks that by adopting it we would be tied to the expressive power of a language in ...


34

In my paper about the Grossone, I point out that the logic of this formalism is identical (in my version) to using $1 + x + x^2 + ... + x^G$ as a FINITE SUM with $G$ a generic positive integer. One can then manipulate the series and look at the limiting behaviour in many cases. There is no need to invoke any new concepts about infinity. This point of view ...


33

No, research in numerical mathematics is still very relevant today. One of the main challenges is big data: scaling the usual algorithms up to larger dimensions. Today's linear systems may involve sparse matrices of dimensions 100k or 1M, for instance. Using traditional methods such as Gaussian elimination will take ages even on modern computers, and ...


31

My answer is: Don't put code in your paper. Do: put pseudocode in your paper, version control your code on Github, and add a link to your Github repository to your paper. The purpose of a paper is to be read; the purpose of code is to be executed by a computer. These purposes should not be mixed, so a readable representation of your code should be ...


26

This formula can actually be proved using only properties of the Gamma function already known to Gauss, with no need to invoke special values of Dirichlet series. The relevant identities are $$ \Gamma(z) \, \Gamma(1-z) = \frac\pi{\sin(\pi z)}, $$ already cited by john mangual as the "mirror formula", and the triplication formula for the Gamma function, i.e. ...


24

As Terry mentions in the comments, the reason for the $\sqrt{5}$ is that the limiting case, the golden ratio, forces it. There is a very neat explanation of all of this in the classic number theory book by Hardy and Wright, pages 209 to 212. I give a brief sketch of the ideas. Why $\phi$ is the worst case. As Hardy and Wright put it, "from the point of ...


24

There are several puzzling things about the question: Firstly of course $\theta$ must be irrational, and it is intended for $\{ x\}$ to denote the Bernoulli polynomial $x-[x]-1/2$ rather than the more usual fractional part. Secondly, where is the result of Hardy from? I did find this statement in the Cambridge ICM paper of Hardy and Littlewood where they ...


23

The numbers 36,29,62 were obtained as the best match between the received signals and the output of computer simulations. The 90% confidence intervals on these numbers are about $\pm 4$. The details (largely beyond my understanding) are in the technical paper here.


21

What I did in my paper on Sergeyev’s Grossone that has been mentioned in your discussion was to present an axiomatised theory of arithmetic in the language of Peano arithmetic augmented with a new constant for Grossone. I did it because many colleagues seemed to think that Sergeyev’s approach didn’t respect the standards of acceptable mathematical ...


20

Federico Poloni's answer is up to date as far as I know, and his explanation of the difficulty is good, but here's another way to think about it. It's less concrete than his explanation, but perhaps useful. The thing that confuses people is that matrix rank is not hard to compute. This is what we're used to, and it seems strange that computing tensor rank ...


17

There is a characterization of Schwarzian derivatives of rational maps: section 3 in the text: http://www.math.purdue.edu/~eremenko/dvi/schwarz3.pdf There is something similar also in arXiv:math/0512370, chapter 2. All these descriptions are various systems of algebraic equations. One of them, the "Bethe ansatz equations for the Gaudin model", proved to be ...


17

Let $f(x)={\mathrm{erf}}(x)-\tanh(x)$. It can be easily seen from Taylor series at $0$ and from asymptotics at $\infty$ that $f(x)>0$ for small $x$ and for large $x$. Let us prove that $f(x)>0$ by contradiction. Suppose that $f(x)$ is negative for some $x$, then $f'$ must have at least $3$ positive zeros, by Rolle's theorem. This means that the ...


16

Walter Van Assche gives a modern account of Pade Approximation, a variant of these were used in the proof that e is transcendental. In the case that $f(z)$ is the Cauchy transform of a compactly supported measure $\mu(x)$, \[ f(z) = \int \frac{1}{z-x} d\mu(x) \] Then $P_n(x)$ is an orthogonal polynomial with respect to $\mu(x)$ while \[ Q_{n-1}(z) = \...


16

The Petrov you are looking for is: Georgii Ivanovich Petrov (1912-1987), biographies are here and here and here. I quote from the third biography: G.I. Petrov was a prominent Russian scientist in the field of aerodynamics, gasdynamics, and space research. Even his first studies connected with the investigation of viscous laminar flows attracted the ...


16

I develop Arb, an arbitrary-precision interval library with special functions support. There is some code included for integration of complex analytic functions using Taylor series (documentation), which in principle should scale nicely to high precision. An example program is available here. The last test case in the example program is $\tfrac{1}{\pi} \...


16

At least in my field (numerical linear algebra), the current standard is that including the full source code is not mandatory for a publication. That said, there are many reasons why sharing your code is a good idea; for instance this article on SIAM news makes some very compelling arguments. Unless it's just a few lines, it is quite unusual to have code ...


14

I can recommend reading "Closed forms: what they are and why we care" by Jon Borwein and Richard Crandall, the article is to appear in Notices Amer. Math. Soc. 60 (2013). Edit (Dec 2012). The paper has just appeared in the Notices: pdf. Together with the sad news about Richard Crandall: he passed away.


14

Here's another plot (using Maple 17, and the first 21 terms) of the solution Carlo referred to.


14

Computer generated images of Julia sets tend to agree strongly with theory so we expect there must be some reason. I believe there are two main forces at work that generally mitigate the problem of round off error in this context. First, the predominate behavior under the iteration of a polynomial (or even generally a rational function) is stability. That ...


14

You are trying to compute a multi-dimensional theta function, and this question is studied in depth in this 2003 Math. Comp. article by Deconinck, Heil, Bobenko, van Hoeij,and Schmies.


14

One of the more convenient and popular approximations of the sum is $$\frac{2^{nH(\frac{k}{n})}}{\sqrt{8k(1-\frac{k}{n})}} \leq \sum_{i=0}^k\binom{n}{i} \leq 2^{nH(\frac{k}{n})}$$ for $0< k < \frac{n}{2}$, where $H$ is the binary entropy function. (The upper bound is exactly what Aryeh Kontorovich mentions.) You can find its proof in many textbooks, ...


13

An elementary use that was in the books on determinants (not linear algebra) I read as a student long ago: you can use them to write cute equations for objects in elementary (Euclidean or projective) geometry. For instance, the equation for the circle through three points in the plane is $$ \left| \begin{matrix} x^2+y^2 & x & y & 1 \cr a_1^2+a_2^...


13

looks like Somos's quadratic recurrence constant


13

Implementing the QR factorization with Householder rotations is cheaper ($2n^2m$ vs $3n^2m$ for a $m\times n$ matrix), and equally accurate in practice. See Section 19.6 of Higham's Accuracy and Stability of Numerical Algorithms, or Golub-Van Loan for more explicit algorithms. Moreover, in a Householder-based implementation there is a higher fraction of ...


13

After reading the original paper of Chowla-Selberg, I think it is a problem which can be solved with Kummer's Fourier Expansion of Gamma Function $$\log\Gamma(x)=(\frac{1}{2}-x)(\gamma+\log2)+(1-x)\log\pi-(\log\sin\pi x)/2+\sum_{m=1}^{\infty}\frac{\sin 2\pi m x}{\pi m}\log m,$$ where $0<x<1$. We do not need to worry about the first and the second ...


13

Actually, no matter what $A$ is, there will be nonzero solutions that will converge to zero, so you can't prove unboundedness, even though it may be true that the 'generic' solution is unbounded. Here is why: First, convert the system into a first order system in $\mathbb{R}^3$ by setting $x = x_0$, $\dot x = x_1$, and $\ddot x = x_2$. Then the equation ...


13

Disclaimer 1: Treating these topics properly would require a quick course in numerical analysis. Disclaimer 2: If you are using any sane computer system, it's already going to have a library function to solve linear systems implemented, which is going to be better of what you can code yourself if you don't have a solid grasp of numerical linear algebra and ...


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