User mathphysicist - MathOverflow

Answer by mathphysicist for Jacobi method on first order partial differential equations

2012-10-09T18:24:48Z

For starters see e.g. chapter IX of Forsyth's classical Treatise on Differential Equations.

Answer by mathphysicist for Any applications integrable systems (pde,ode, q-,...) to math. biology (pharmakinetics, pharmadynamics) ?

2012-04-22T20:13:57Z

There are some integrable Lotka--Volterra systems, see e.g. the paper O.I. Bogoyavlenskij, Integrable Lotka--Volterra systems, Regular and Chaotic Dynamics, 2008, Vol. 13, No. 6, pp. 543–556.

Answer by mathphysicist for Eliminating 1st order terms in elliptic partial differential equation

2012-03-18T07:04:53Z

The necessary and sufficient conditions for transforming one second-order differential operator of the type you are interested in into another are given by the so-called Cotton theorem, see Theorem 1 in this paper by Finkel and Kamran. In your case, you want the linear part of the transformed operator to vanish ($\tilde{\mathcal{A}}=0$ in the notation of the above paper), whence you can readily extract the conditions you ask for. Roughly speaking, in order to have no first-order terms in your transformed operator, the linear part of your original operator should be "pure gauge up to a change of independent variables".

Answer by mathphysicist for Video lectures for Algebraic Geometry

2012-03-09T08:44:29Z

There are also three nice videos of lectures by Ugo Bruzzo entitled Algebraic geometry for physicists.

Answer by mathphysicist for How about the Lie algebra over commutative ring?

2011-12-25T12:32:19Z

Actually, it is possible to go even further and define Lie algebras over noncommutative rings, see the paper Lie algebras and Lie groups over noncommutative rings by Berenstein and Retakh.

Answer by mathphysicist for Extending Lie algebra homomorphisms.

2011-12-07T19:45:09Z

For finite-dimensional Lie algebras, see e.g. the papers Extensions of Representations of Lie Groups and Lie Algebras, I by G. Hochschild and G.D. Mostow (part II by Mostow is mostly on Lie groups), Extensions of representations of algebraic linear groups by A. Białynicki-Birula and the two preceding authors, and Extensions of representations of Lie algebras by J.G. Ryan. The key words 'extension(s) of representations' will land you a number of other references upon searching in Google or MathSciNet. For loop algebras, which you are interested in, some results can be found in the paper Extensions of modules over loop algebras by Fialowski and Malikov and in this recent preprint by E. Neher and A. Savage.

Answer by mathphysicist for Movies about mathematics/mathematicians

2011-10-05T21:21:18Z

There is also a Russian movie Sofia Kovalevskaya.

Answer by mathphysicist for Non-polynomial integrals of motion for polynomial dynamical systems

2011-09-27T12:28:45Z

There are such examples (with transcendental integrals of motion) already on $\mathbb{R}^{4}$, see the paper of Hietarinta.

The Hamiltonian is $$H=p_x^2/2+p_y^2/2+2y p_x p_y-x,$$ the desired integral is $$I_1=p_y \exp(p_x^2).$$

Answer by mathphysicist for How to locate an obscure paper?

2011-09-12T19:22:48Z

A minor addition to the excellent answer by Dmitri Pavlov: this appears to be the author's (other?) web page, and his e-mail address is given there.

Answer by mathphysicist for real roots for polynomials

2011-07-18T11:51:11Z

This is just a bit too long for a comment :)

Let $P$ be your polynomial and $x$ its real root. Obviously, $P(x)=0$ if and only if $$Q(x)\equiv (\mathrm{Re} P(x))^2+(\mathrm{Im} P(x))^2=0.$$ Now, $Q(x)$ is a polynomial with real coefficients, which reduces your question to finding criteria for a real-coefficients polynomial to have a real root, and these are discussed here.

Answer by mathphysicist for Vladimir Voevodskys 2002 ICM Lecture.

2011-07-18T09:13:23Z

Some of Voevodsky videos are here:

http://video.ias.edu/taxonomy/term/42

http://www.mathnet.ru/PresentFiles/425/425.flv

http://www.mathunion.org/Videos/ICM98/ICMs/vladimir_voevodsky.html

http://claymath.msri.org/voevodsky2002.mov

The last two talks, Algebraic Cycles and Motives and An Intuitive Introduction to Motivic Homotopy Theory, are perhaps the closest to what you look for. As for the last talk, the notes from it are available here:

http://www.cwru.edu/artsci/phil/Voevodsky.pdf

A name for PDE systems which are neither under- nor overdetermined?

2011-06-22T15:45:59Z

The concepts of overdetermined and underdetermined PDE systems are well known. However, all sources I have so far looked into appear to avoid giving any name to PDE systems which are neither overdetermined nor underdetermined. Is there any reasonably commonly used name for such PDE systems? If possible, please provide the references where the name from your answer is used.

EDIT: It was suggested by Igor Khavkine and Robert Bryant that one should consider formally integrable systems (which are neither over- nor underdetermined and have no nontrivial compatibility conditions). I like the idea but in my case this term would appear within the discussion of systems which are completely integrable (via the inverse scattering transform), and this might confuse the non-expert readers. Is there any sensible way out of this conundrum?

Thanks in advance!

Answer by mathphysicist for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring.

2011-06-21T06:39:57Z

Your goal appears to require a somewhat different approach, at least for noncommutative rings; see the paper Lie algebras and Lie groups over noncommutative rings by Berenstein and Retakh and references therein. The said paper concentrates more on Lie algebras than on Lie groups but still seems close enough to what you want.

Which nonlinear PDEs are of interest to algebraic geometers and why?

2011-05-26T11:44:02Z

Motivation

I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a number of important achievements in the study of PDEs, suffice it to mention the construction of finite-gap solutions to integrable PDEs (see e.g. this book) and the geometric approach to PDEs developed by A.M. Vinogradov et al. which revolves around the concept of diffiety (the word itself was merged from "differential" and "variety") and which the authors themselves consider, at least to some extent, as a "translation" of ideas from algebraic geometry and commutative algebra to the realm of PDEs, see e.g. these two books.

On the other hand, it appears (as far as my googling skills allow me to tell) that the other way around, i.e., in the applications of nonlinear PDEs in algebraic geometry, the interaction is at least somewhat less intense. I was able to come up with basically just two things: the Novikov conjecture (proved by Shiota) on the relation of the KP equation to the Schottky problem and the applications of the Monge-Ampere equations in Kahler geometry.

Question

Which are the other applications of the nonlinear PDEs in (broadly understood) algebraic geometry? In other words, which nonlinear PDEs are of interest to algebraic geometers and why?

EDIT: It should be obvious, but to play it safe I would like to spell this out loud and clear: please feel free to share not only the already known cases where PDEs have helped the algebraic geometers but also the more open-problem-type cases where, say, there is a PDE that could be of use in algebraic geometry but some crucial bits of information about this PDE (for instance, about the existence of solution(s) with the desired properties) are still missing.

Answer by mathphysicist for Names of certain surfaces

2011-06-01T16:57:27Z

You may wish to check whether your surfaces are equivalent to any from this list.

Answer by mathphysicist for Homogeneous linear differential equation system with simple periodical coefficient matrix

2011-05-27T20:32:53Z

Another thing to consider: introduce new independent variable $\tau=(1/\alpha)sin(\alpha z+\Phi_a)$. Also note that we have $$cos(\alpha z+\Phi_b)=cos(x+\delta)$$ and $$ cos(x+\delta)=cos(x)cos(\delta)-sin(x)sin(\delta).$$ where $x=\alpha z+\Phi_a$ and $\delta=\Phi_b-\Phi_a$.

Then the first equation of your system becomes $$ dy_1/d\tau=B \left(cos(\delta)-\displaystyle\frac{\alpha\tau}{\sqrt{1-\alpha^2\tau^2}} sin(\delta)\right) y_2,$$ while the second one is simply $$ dy_2/d\tau=A y_1. $$ This new system might be somewhat easier to investigate, be it analytically or numerically.

Answer by mathphysicist for Solving ODEs of the form $x'(t)=F(x(t))+f(t)$

2011-04-06T19:55:29Z

This is not a complete answer either but at this page you can find many special cases of your equation for which the closed form solution is available.

Answer by mathphysicist for References on Lie Groups and Dynamical systems

2011-04-02T10:49:15Z

One of the most important connections of the two fields can be found in the theory of Hamiltonian dynamical systems with Lie groups being the symmetry groups. The interplay of these leads to many interesting concepts (including, inter alia, the classical R-matrix) and results. For starters you can try the books Applications of Lie Groups to Differential Equations by Olver and Integrable Systems of Classical Mechanics and Lie Algebras by Perelomov, and the survey paper Integrable Systems and Factorization Problems by Semenov-Tian-Shansky.

Answer by mathphysicist for Is there a closed formula for this heat equation with time dependent term.

2011-03-05T18:10:21Z

Introduce new independent variables $T=t$, $X=x+\int f(t)dt$. Then your equation becomes nothing but the standard heat equation $$ P_T=P_{XX}. $$ You can take the formula for general solution of this equation and transform it back to your case using the inverse change of variables $x=X-\int f(T)dT$, $t=T$.

Of course, all of this works for an arbitrary smooth $f$ (not necessarily a polynomial).

Answer by mathphysicist for What notions are used but not clearly defined in modern mathematics?

2011-02-26T15:10:46Z

Not only is the notion of chaos not well-defined (cf. the answer of Gerry Myerson), but the same holds true for its opposite: there is no universally accepted definition of integrable system yet.

Answer by mathphysicist for Green's Functions

2011-02-07T22:33:15Z

This is indeed fairly standard. Using the two linearly independent solutions to your ODE that you already have you can just follow the procedure outlined e.g. here to construct the Green function.

Computing the index of a Lie algebra: what is known beyond the reductive case?

2010-08-29T23:59:41Z

Recall that an index of a Lie algebra $\mathfrak{g}$ is $\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$ where $\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| \mathrm{ad}_h^*(\xi)=0\}$ is the annihilator (also known as the stabilizer) of $\xi$ with respect to the co-adjoint representation. The relevant Wikipedia article just says that if $\mathfrak{g}$ is reductive then $\mathrm{ind}\ \mathfrak{g}=\mathrm{rank}\ \mathfrak{g}$ but I would like to build some intuition for the non-reductive case, and my googling hasn't brought about any relevant references so far. In particular, I would very much like to know:

Can one say anything about the index of a solvable Lie algebra?
What about the index of a semidirect sum (rather than the direct sum which occurs in the reductive case) $\mathfrak{g}=\mathfrak{h}\triangleright\mathfrak{a}$, where $\mathfrak{a}$ is abelian and $\mathfrak{h}$ is arbitrary? If something is known for semisimple $\mathfrak{h}$, that would be of interest too.

Many thanks in advance!

Answer by mathphysicist for Is there a "quantum" Riemann zeta function?

2010-11-13T20:06:41Z

The paper by Cherednik On q-analogues of Riemann's zeta function gives precisely the definition you're after: $$ \zeta_q(s)=\sum\limits_{n=1}^\infty q^{sn}/[n]_q^s $$ His paper also contains a brief discussion of the properties of this $q$-zeta function. On the other hand, the term quantum zeta function appears to have a somewhat different meaning, see e.g. the paper On the quantum zeta function by R.E. Crandall.

Answer by mathphysicist for Most memorable titles

2010-11-03T04:40:28Z

Some nice titles from B.A. Kupersmidt:

KP or mKP (a book)
Dark equations (an article)

Answer by mathphysicist for Is there an analogue of mathscinet for physics?

2010-10-31T21:53:11Z

In addition to freely available Google Scholar and SPIRES, and subscription-based Web of Science and Scopus, there is a free NASA Astrophysics Data Systems database which, contrary to its title, appears to have broader scope than SPIRES, at least as far as mathematical physics is concerned; it provides abstracts (but not reviews), citations, and, for some older papers, their full-text scanned versions. Now it is sort of integrated with arXiv.org: when looking at the abstract of any arXiv preprint, you see the link to its citations and references at NASA ADS under References & Citations. This database has, inter alia, a specialized physics and geophysics search engine.

Answer by mathphysicist for What is a good introductory text for moduli theory?

2010-10-30T00:37:21Z

As for somewhat informal (and mostly brief) introductions to moduli spaces, see

Introduction to moduli spaces by Han-Bom Moon
The Moduli Space of Curves and Its Tautological Ring by Ravi Vakil in the Notices of the AMS
An introduction to the moduli spaces of curves by Maarten Hoeve
Rational Points on Moduli Spaces of Curves by Dave Jensen
A minicourse on moduli of curves by E. Looijenga

See also the BAMS paper Perturbations, deformations, and variations (and ``near-misses") in geometry, physics, and number theory by Barry Mazur, in particular, Sections 4 and 5.

Answer by mathphysicist for Who is Kirszbraun?

2010-10-28T01:29:32Z

This Banach biography web page in Polish says that Kirszbraun was born in 1903 or 1904 and died in 1942 (he is listed there among other Polish mathematicians who died in the course of WW II). Further googling revealed that his full name was Mojżesz David Kirszbraun. According to the Zentralblatt Math. database, he wrote just one paper in German (just as Mark mentioned in the comments):

Kirszbraun, M.D. Über die zusammenziehenden und Lipschitzschen Transformationen. (German) Fundam. Math. 22, 77-108 (1934).

Here is the link to the review of this paper by Freudental.

Perhaps some Polish colleague(s) could provide further details.

Answer by mathphysicist for Online introduction to Lattice Theory?

2010-10-27T20:32:17Z

For something brief to begin with see the notes by Eric Rasmusen, the introductions to lattice theory by Zukowski and Wang, and the Notices article by Giancarlo Rota.

Also see the website of this course which contains some notes.

Answer by mathphysicist for Compute Lie algebra cohomology

2010-10-26T16:55:37Z

In the Maple computer algebra system you have the package LieAlgebraCohomology which should do what you want.

Answer by mathphysicist for Quantum Group Calculations in Mathematica

2010-10-25T20:46:27Z

I just found the link to the QuantumGroups Mathematica package by Scott Morrison mentioned by Noah:

http://katlas.org/wiki/QuantumGroups%60

Comment by mathphysicist

2012-08-17T19:13:03Z

In the pseudo-Riemannian case the physicists would probably refer to it as the (massive) Klein--Gordon operator.

Comment by mathphysicist

2012-01-13T14:54:25Z

The link on Google books (unfortunately, apparently without preview): books.google.com/books?id=EhEZAQAAIAAJ

Comment by mathphysicist

2012-01-09T18:18:37Z

These two posts at the Secret Blogging Seminar could be somewhat helpful: sbseminar.wordpress.com/2008/07/19/… sbseminar.wordpress.com/2008/07/03/…

Comment by mathphysicist

2011-11-18T10:49:56Z

Olver's book requires considerable prior knowledge of ODEs and PDEs, so it certainly can't be a first book to read on ODEs.

Comment by mathphysicist

2011-08-22T15:32:04Z

The Casahorran paper is available for free on arXiv.org: arxiv.org/pdf/quant-ph/0011059

Comment by mathphysicist

2011-07-18T16:03:46Z

@Yemon: provided one considers $P$ as map from $\mathbb{R}$ to $\mathbb{C}$ (rather than from $\mathbb{C}$ to $\mathbb{C}$), of course. But in the case under study we are interested in real roots only anyway.

Comment by mathphysicist

2011-07-18T15:19:56Z

@Yemon: That's one way to look at it, yes.

Comment by mathphysicist

2011-07-18T15:00:32Z

@Yemon: of course I meant for real $x$.

Comment by mathphysicist

2011-07-18T14:38:52Z

@Yemon: In fact $Q(x)=P(x)\bar P(x)$ even for complex $x$ (sorry for a poor formulation in previous comment), but, and that's the main point, for real $X$ $P(x)\bar P(x)$ is clearly a polynomial in $x$.

Comment by mathphysicist

2011-07-18T13:39:12Z

@Yemon: $x$ is assumed to be real, and then my $Q(x)$ equals $P(x)\overline{P(x)}$ (cf. Denis' answer).

Comment by mathphysicist

2011-06-30T12:32:07Z

Perhaps there is a problem with the notation to begin with: I guess your Poisson brackets should include a tensor product somewhere. Could you write down the Poisson brackets of the entries of your matrices?

Comment by mathphysicist

2011-06-23T18:17:49Z

@Robert Bryant: Thanks a lot!

Comment by mathphysicist

2011-06-23T06:28:48Z

@Ben McKay: to me, involutive determined doesn't sound bad at all.

Comment by mathphysicist

2011-06-23T06:27:25Z

@Deane Yang: that's nice but a bit too restrictive (although this can be fixed if one requires that there exists a change of independent variables after which your property holds). On the other hand, your definition (with or without my modification) can be applied to quasilinear systems too.

Comment by mathphysicist

2011-06-22T17:30:38Z

A small problem that I have with the term formally integrable is that in the context I want to use it will appear together with the term completely integrable (through existence of a Lax pair and the inverse scattering transform) which might confuse the non-expert readers. Which is the best way to handle this? Many thanks in advance once again!