User mathphysicist - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:13:37Z http://mathoverflow.net/feeds/user/2149 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109222/jacobi-method-on-first-order-partial-differential-equations/109247#109247 Answer by mathphysicist for Jacobi method on first order partial differential equations mathphysicist 2012-10-09T18:24:48Z 2012-10-09T18:24:48Z <p>For starters see e.g. chapter IX of Forsyth's classical <a href="http://books.google.com/books?id=hQgZ94-coocC" rel="nofollow">Treatise on Differential Equations</a>.</p> http://mathoverflow.net/questions/94840/any-applications-integrable-systems-pde-ode-q-to-math-biology-pharmakin/94877#94877 Answer by mathphysicist for Any applications integrable systems (pde,ode, q-,...) to math. biology (pharmakinetics, pharmadynamics) ? mathphysicist 2012-04-22T20:13:57Z 2012-04-22T20:13:57Z <p>There are some integrable Lotka--Volterra systems, see e.g. the paper <em>O.I. Bogoyavlenskij</em>, <a href="http://www.springerlink.com/content/w8377477754n406r/" rel="nofollow">Integrable Lotka--Volterra systems</a>, <em>Regular and Chaotic Dynamics</em>, 2008, Vol. 13, No. 6, pp. 543–556. </p> http://mathoverflow.net/questions/87072/eliminating-1st-order-terms-in-elliptic-partial-differential-equation/91512#91512 Answer by mathphysicist for Eliminating 1st order terms in elliptic partial differential equation mathphysicist 2012-03-18T07:04:53Z 2012-03-18T15:20:10Z <p>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 <a href="http://www.atlantis-press.com/php/download_paper.php?id=952" rel="nofollow">this paper</a> 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".</p> http://mathoverflow.net/questions/78037/video-lectures-for-algebraic-geometry/90672#90672 Answer by mathphysicist for Video lectures for Algebraic Geometry mathphysicist 2012-03-09T08:44:29Z 2012-03-09T08:44:29Z <p>There are also three nice <a href="http://www.theorie.physik.uni-muenchen.de/activities/schools/archiv/2011_asc_school/videos_asc_school_2011/videos_bruzzo/index.html" rel="nofollow">videos of lectures</a> by Ugo Bruzzo entitled <em>Algebraic geometry for physicists</em>.</p> http://mathoverflow.net/questions/84255/how-about-the-lie-algebra-over-commutative-ring/84265#84265 Answer by mathphysicist for How about the Lie algebra over commutative ring? mathphysicist 2011-12-25T12:32:19Z 2011-12-25T12:32:19Z <p>Actually, it is possible to go even further and define Lie algebras over <em>noncommutative</em> rings, see the paper <a href="http://math.uoregon.edu/~arkadiy/noncomloopalg.pdf" rel="nofollow">Lie algebras and Lie groups over noncommutative rings</a> by Berenstein and Retakh.</p> http://mathoverflow.net/questions/82887/extending-lie-algebra-homomorphisms/82903#82903 Answer by mathphysicist for Extending Lie algebra homomorphisms. mathphysicist 2011-12-07T19:45:09Z 2011-12-08T00:17:08Z <p>For finite-dimensional Lie algebras, see e.g. the papers <a href="http://dx.doi.org/10.2307/2372443" rel="nofollow">Extensions of Representations of Lie Groups and Lie Algebras, I</a> by G. Hochschild and G.D. Mostow (<a href="http://dx.doi.org/10.2307/2372788" rel="nofollow">part II</a> by Mostow is mostly on Lie groups), <a href="http://dx.doi.org/10.2307/2373191" rel="nofollow">Extensions of representations of algebraic linear groups</a> by A. Białynicki-Birula and the two preceding authors, and <a href="http://projecteuclid.org/euclid.pjm/1102699676" rel="nofollow">Extensions of representations of Lie algebras</a> 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 <a href="http://www.jstor.org/stable/2374946" rel="nofollow">Extensions of modules over loop algebras</a> by Fialowski and Malikov and in <a href="http://arxiv.org/abs/1103.4367" rel="nofollow">this recent preprint</a> by E. Neher and A. Savage.</p> http://mathoverflow.net/questions/77279/movies-about-mathematics-mathematicians/77300#77300 Answer by mathphysicist for Movies about mathematics/mathematicians mathphysicist 2011-10-05T21:21:18Z 2011-10-05T21:21:18Z <p>There is also a Russian movie <a href="http://en.wikipedia.org/wiki/Sofia_Kovalevskaya_%28film%29" rel="nofollow">Sofia Kovalevskaya</a>.</p> http://mathoverflow.net/questions/70108/non-polynomial-integrals-of-motion-for-polynomial-dynamical-systems/76504#76504 Answer by mathphysicist for Non-polynomial integrals of motion for polynomial dynamical systems mathphysicist 2011-09-27T12:28:45Z 2011-09-27T12:28:45Z <p>There are such examples (with transcendental integrals of motion) already on $\mathbb{R}^{4}$, see the <a href="http://link.aps.org/doi/10.1103/PhysRevLett.52.1057" rel="nofollow">paper</a> of Hietarinta. </p> <p>The Hamiltonian is $$H=p_x^2/2+p_y^2/2+2y p_x p_y-x,$$ the desired integral is $$I_1=p_y \exp(p_x^2).$$</p> http://mathoverflow.net/questions/75238/how-to-locate-an-obscure-paper/75251#75251 Answer by mathphysicist for How to locate an obscure paper? mathphysicist 2011-09-12T19:22:48Z 2011-09-12T19:28:05Z <p>A minor addition to the excellent answer by Dmitri Pavlov: <a href="http://icyb.kiev.ua/m/104/en/kovalenko_i.n..html?id=104" rel="nofollow">this</a> appears to be the author's (other?) web page, and his e-mail address is given there. </p> http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Answer by mathphysicist for real roots for polynomials mathphysicist 2011-07-18T11:51:11Z 2011-07-18T11:51:11Z <p>This is just a bit too long for a comment :)</p> <p>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 <em>real</em> coefficients, which reduces your question to finding criteria for a <em>real-coefficients</em> polynomial to have a real root, and these are discussed <a href="http://mathoverflow.net/questions/20946/criteria-to-determine-whether-a-real-coefficient-polynomial-has-real-root" rel="nofollow">here</a>.</p> http://mathoverflow.net/questions/70602/vladimir-voevodskys-2002-icm-lecture/70604#70604 Answer by mathphysicist for Vladimir Voevodskys 2002 ICM Lecture. mathphysicist 2011-07-18T09:13:23Z 2011-07-18T10:11:30Z <p>Some of Voevodsky videos are here:</p> <p><a href="http://video.ias.edu/taxonomy/term/42" rel="nofollow">http://video.ias.edu/taxonomy/term/42</a></p> <p><a href="http://www.mathnet.ru/PresentFiles/425/425.flv" rel="nofollow">http://www.mathnet.ru/PresentFiles/425/425.flv</a></p> <p><a href="http://www.mathunion.org/Videos/ICM98/ICMs/vladimir_voevodsky.html" rel="nofollow">http://www.mathunion.org/Videos/ICM98/ICMs/vladimir_voevodsky.html</a></p> <p><a href="http://claymath.msri.org/voevodsky2002.mov" rel="nofollow">http://claymath.msri.org/voevodsky2002.mov</a></p> <p>The last two talks, <em>Algebraic Cycles and Motives</em> and <em>An Intuitive Introduction to Motivic Homotopy Theory</em>, are perhaps the closest to what you look for. As for the last talk, the notes from it are available here: </p> <p><a href="http://www.cwru.edu/artsci/phil/Voevodsky.pdf" rel="nofollow">http://www.cwru.edu/artsci/phil/Voevodsky.pdf</a></p> http://mathoverflow.net/questions/68527/a-name-for-pde-systems-which-are-neither-under-nor-overdetermined A name for PDE systems which are neither under- nor overdetermined? mathphysicist 2011-06-22T15:45:59Z 2011-06-22T18:53:01Z <p>The concepts of <a href="http://eom.springer.de/O/o070660.htm" rel="nofollow">overdetermined</a> and <a href="http://eom.springer.de/u/u095150.htm" rel="nofollow">underdetermined</a> 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 <em>neither</em> overdetermined <em>nor</em> 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.</p> <p><strong>EDIT:</strong> It was suggested by Igor Khavkine and Robert Bryant that one should consider <em>formally integrable</em> 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 <em>completely integrable</em> (via the inverse scattering transform), and this might confuse the non-expert readers. Is there any sensible way out of this conundrum?</p> <p>Thanks in advance!</p> http://mathoverflow.net/questions/68331/info-on-symplectic-orthogonal-groups-of-gln-r-r-a-ring-not-necessarily-divisi/68345#68345 Answer by mathphysicist for Info on Symplectic/Orthogonal groups of Gl(n,R); R a ring, not necessarily division ring. mathphysicist 2011-06-21T06:39:57Z 2011-06-21T06:39:57Z <p>Your goal appears to require a somewhat different approach, at least for noncommutative rings; see the paper <a href="http://pages.uoregon.edu/arkadiy/noncomloopalg.pdf" rel="nofollow">Lie algebras and Lie groups over noncommutative rings</a> 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.</p> http://mathoverflow.net/questions/66046/which-nonlinear-pdes-are-of-interest-to-algebraic-geometers-and-why Which nonlinear PDEs are of interest to algebraic geometers and why? mathphysicist 2011-05-26T11:44:02Z 2011-06-13T20:21:30Z <blockquote> <p><strong>Motivation</strong> </p> </blockquote> <p>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. <a href="http://books.google.com/books?id=Pz8P_t48XAUC" rel="nofollow">this book</a>) 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 <a href="http://books.google.com/books?id=n5yb3HlW-wgC" rel="nofollow">two</a> <a href="http://www.ams.org/bookstore-getitem/item=MMONO-182" rel="nofollow">books</a>.</p> <p>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 <a href="http://eom.springer.de/s/s083380.htm" rel="nofollow">Schottky problem</a> and the applications of the Monge-Ampere equations in Kahler geometry.</p> <blockquote> <p><strong>Question</strong></p> </blockquote> <p>Which are the <em>other</em> applications of the nonlinear PDEs in (broadly understood) algebraic geometry? In other words, which nonlinear PDEs are of interest to algebraic geometers and why? </p> <p><strong>EDIT:</strong> It should be obvious, but to play it safe I would like to spell this out loud and clear: please feel free to share <strong>not only</strong> the <em>already known</em> 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. </p> http://mathoverflow.net/questions/66281/names-of-certain-surfaces/66672#66672 Answer by mathphysicist for Names of certain surfaces mathphysicist 2011-06-01T16:57:27Z 2011-06-01T16:57:27Z <p>You may wish to check whether your surfaces are equivalent to any from <a href="http://www1-c703.uibk.ac.at/mathematik/project/bildergalerie/gallery.html" rel="nofollow">this list</a>.</p> http://mathoverflow.net/questions/66172/homogeneous-linear-differential-equation-system-with-simple-periodical-coefficien/66222#66222 Answer by mathphysicist for Homogeneous linear differential equation system with simple periodical coefficient matrix mathphysicist 2011-05-27T20:32:53Z 2011-05-27T20:45:12Z <p>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 $\delta=\Phi_b-\Phi_a$.</p> <p>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.</p> http://mathoverflow.net/questions/60744/solving-odes-of-the-form-xtfxtft/60848#60848 Answer by mathphysicist for Solving ODEs of the form $x'(t)=F(x(t))+f(t)$ mathphysicist 2011-04-06T19:55:29Z 2011-04-06T19:55:29Z <p>This is not a complete answer either but at <a href="http://eqworld.ipmnet.ru/en/solutions/ode/ode-toc1.htm" rel="nofollow">this page</a> you can find many special cases of your equation for which the closed form solution <em>is</em> available.</p> http://mathoverflow.net/questions/60322/references-on-lie-groups-and-dynamical-systems/60352#60352 Answer by mathphysicist for References on Lie Groups and Dynamical systems mathphysicist 2011-04-02T10:49:15Z 2011-04-02T10:49:15Z <p>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 <a href="http://books.google.com/books?id=sI2bAxgLMXYC" rel="nofollow">Applications of Lie Groups to Differential Equations</a> by Olver and <a href="http://www.amazon.com/gp/product/0817623361" rel="nofollow">Integrable Systems of Classical Mechanics and Lie Algebras</a> by Perelomov, and the survey paper <a href="http://arxiv.org/abs/nlin/0209057" rel="nofollow">Integrable Systems and Factorization Problems</a> by Semenov-Tian-Shansky.</p> http://mathoverflow.net/questions/57476/is-there-a-closed-formula-for-this-heat-equation-with-time-dependent-term/57481#57481 Answer by mathphysicist for Is there a closed formula for this heat equation with time dependent term. mathphysicist 2011-03-05T18:10:21Z 2011-03-11T19:33:14Z <p>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 $x=X-\int f(T)dT$, $t=T$. </p> <p>Of course, all of this works for an arbitrary smooth $f$ (not necessarily a polynomial).</p> http://mathoverflow.net/questions/56677/what-notions-are-used-but-not-clearly-defined-in-modern-mathematics/56740#56740 Answer by mathphysicist for What notions are used but not clearly defined in modern mathematics? mathphysicist 2011-02-26T15:10:46Z 2011-02-26T15:10:46Z <p>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 <a href="http://mathoverflow.net/questions/6379/what-is-an-integrable-system/" rel="nofollow">integrable system</a> yet.</p> http://mathoverflow.net/questions/54646/greens-functions/54705#54705 Answer by mathphysicist for Green's Functions mathphysicist 2011-02-07T22:33:15Z 2011-02-07T22:33:15Z <p>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. <a href="https://webspace.utexas.edu/dj955/www/notes/GreenFn.pdf" rel="nofollow">here</a> to construct the Green function. </p> http://mathoverflow.net/questions/37101/computing-the-index-of-a-lie-algebra-what-is-known-beyond-the-reductive-case Computing the index of a Lie algebra: what is known beyond the reductive case? mathphysicist 2010-08-29T23:59:41Z 2011-01-05T22:54:41Z <p>Recall that an index of a Lie algebra $\mathfrak{g}$ is <code>$\mathrm{ind}\ \mathfrak{g} := \min\limits_{\xi \in \mathfrak{g}^*} \dim \mathrm{Ann}_{\xi}$</code> where <code>$\mathrm{Ann}_{\xi}=\{h\in\mathfrak{g}| \mathrm{ad}_h^*(\xi)=0\}$</code> is the annihilator (also known as the stabilizer) of $\xi$ with respect to the co-adjoint representation. The relevant <a href="http://en.wikipedia.org/wiki/Index_of_a_Lie_algebra" rel="nofollow">Wikipedia article</a> just says that if $\mathfrak{g}$ is <a href="http://en.wikipedia.org/wiki/Reductive_Lie_algebra" rel="nofollow">reductive</a> then <code>$\mathrm{ind}\ \mathfrak{g}=\mathrm{rank}\ \mathfrak{g}$</code> 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:</p> <ol> <li><p>Can one say anything about the index of a <em>solvable</em> Lie algebra?</p></li> <li><p>What about the index of a <em>semidirect</em> sum (rather than the direct sum which <a href="http://mathoverflow.net/questions/23085/on-the-full-reducibility-of-representations-of-reductive-lie-algebras/23093#23093" rel="nofollow">occurs</a> 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.</p></li> </ol> <p>Many thanks in advance!</p> http://mathoverflow.net/questions/45950/is-there-a-quantum-riemann-zeta-function/45958#45958 Answer by mathphysicist for Is there a "quantum" Riemann zeta function? mathphysicist 2010-11-13T20:06:41Z 2010-11-13T20:21:28Z <p>The paper by Cherednik <a href="http://www.springerlink.com/content/a0cw6fvunupf1da4/" rel="nofollow">On q-analogues of Riemann's zeta function</a> 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 <em>quantum zeta function</em> appears to have a somewhat different meaning, see e.g. the paper <a href="http://dx.doi.org/10.1088/0305-4470/29/21/014" rel="nofollow">On the quantum zeta function</a> by R.E. Crandall.</p> http://mathoverflow.net/questions/44326/most-memorable-titles/44644#44644 Answer by mathphysicist for Most memorable titles mathphysicist 2010-11-03T04:40:28Z 2010-11-03T04:40:28Z <p>Some nice titles from B.A. Kupersmidt:</p> <ul> <li><a href="http://www.mathjournals.org/bookstore?fn=20&amp;arg1=survseries&amp;ikey=SURV-78" rel="nofollow">KP or mKP</a> (a book)</li> <li><a href="http://staff.www.ltu.se/~norbert/home_journal/electronic/83art3.pdf" rel="nofollow">Dark equations</a> (an article)</li> </ul> http://mathoverflow.net/questions/44379/is-there-an-analogue-of-mathscinet-for-physics/44383#44383 Answer by mathphysicist for Is there an analogue of mathscinet for physics? mathphysicist 2010-10-31T21:53:11Z 2010-11-01T01:47:50Z <p>In addition to freely available <a href="http://scholar.google.com/advanced_scholar_search?hl=en&amp;num=100" rel="nofollow">Google Scholar</a> and <a href="http://www.slac.stanford.edu/spires/" rel="nofollow">SPIRES</a>, and subscription-based <a href="http://isiknowledge.com/WOS" rel="nofollow">Web of Science</a> and <a href="http://www.scopus.com/home.url" rel="nofollow">Scopus</a>, there is a free <a href="http://www.adsabs.harvard.edu/" rel="nofollow">NASA Astrophysics Data Systems database</a> 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 <a href="http://arxiv.org/" rel="nofollow">arXiv.org</a>: when looking at the abstract of any arXiv preprint, you see the link to its citations and references at NASA ADS under References &amp; Citations. This database has, <em>inter alia</em>, a specialized <a href="http://adsabs.harvard.edu/physics_service.html" rel="nofollow">physics and geophysics search engine</a>. </p> http://mathoverflow.net/questions/44125/what-is-a-good-introductory-text-for-moduli-theory/44199#44199 Answer by mathphysicist for What is a good introductory text for moduli theory? mathphysicist 2010-10-30T00:37:21Z 2010-10-30T01:57:18Z <p>As for somewhat informal (and mostly brief) introductions to moduli spaces, see</p> <ul> <li><p><a href="http://www.mathnet.or.kr/real/2010/08/MoonHanbom%280804%29.pdf" rel="nofollow">Introduction to moduli spaces</a> by Han-Bom Moon</p></li> <li><p><a href="http://www.ams.org/notices/200306/fea-vakil.pdf" rel="nofollow">The Moduli Space of Curves and Its Tautological Ring</a> by Ravi Vakil in the <em>Notices of the AMS</em></p></li> <li><p><a href="http://www.math.leidenuniv.nl/~streng/modular/maarten.pdf" rel="nofollow">An introduction to the moduli spaces of curves</a> by Maarten Hoeve </p></li> <li><p><a href="http://www.ma.utexas.edu/users/djensen/Curves%2520Over%2520Q.pdf" rel="nofollow">Rational Points on Moduli Spaces of Curves</a> by Dave Jensen </p></li> <li><p><a href="http://users.ictp.it/~pub_off/lectures/lns001/Looijenga/Looijenga.pdf" rel="nofollow">A minicourse on moduli of curves</a> by E. Looijenga</p></li> </ul> <p>See also the BAMS paper <a href="http://www.ams.org/journals/bull/2004-41-03/S0273-0979-04-01024-9/S0273-0979-04-01024-9.pdf" rel="nofollow">Perturbations, deformations, and variations (and near-misses") in geometry, physics, and number theory</a> by Barry Mazur, in particular, Sections 4 and 5. </p> http://mathoverflow.net/questions/43918/who-is-kirszbraun/43920#43920 Answer by mathphysicist for Who is Kirszbraun? mathphysicist 2010-10-28T01:29:32Z 2010-10-28T15:36:21Z <p><a href="http://banach.univ.gda.pl/zyciorys.html" rel="nofollow">This Banach biography web page in Polish</a> 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 <a href="http://www.zentralblatt-math.org/zmath/en/advanced/" rel="nofollow">Zentralblatt Math.</a> database, he wrote just one paper in German (just as Mark mentioned in the comments):</p> <p>Kirszbraun, M.D. <a href="http://matwbn.icm.edu.pl/ksiazki/fm/fm22/fm22112.pdf" rel="nofollow">Über die zusammenziehenden und Lipschitzschen Transformationen</a>. (German) Fundam. Math. 22, 77-108 (1934).</p> <p>Here is the <a href="http://www.zentralblatt-math.org/zmath/en/advanced/?q=an%3A60.0532.03&amp;format=complete" rel="nofollow">link</a> to the review of this paper by Freudental.</p> <p>Perhaps some Polish colleague(s) could provide further details. </p> http://mathoverflow.net/questions/43861/online-introduction-to-lattice-theory/43869#43869 Answer by mathphysicist for Online introduction to Lattice Theory? mathphysicist 2010-10-27T20:32:17Z 2010-10-27T20:56:36Z <p>For something brief to begin with see the <a href="http://www.rasmusen.org/GI/lattice.theory.notes.txt" rel="nofollow">notes</a> by Eric Rasmusen, the introductions to lattice theory by <a href="http://mizar.org/JFM/pdf/lattices.pdf" rel="nofollow">Zukowski</a> and <a href="http://www.math.s.chiba-u.ac.jp/~wang/research/coin/lattice.pdf" rel="nofollow">Wang</a>, and the Notices <a href="http://www.ams.org/notices/199711/comm-rota.pdf" rel="nofollow">article</a> by Giancarlo Rota.</p> <p>Also see the website of <a href="http://users.ece.utexas.edu/~garg/f03-lat.html" rel="nofollow">this course</a> which contains some notes.</p> http://mathoverflow.net/questions/43665/compute-lie-algebra-cohomology/43691#43691 Answer by mathphysicist for Compute Lie algebra cohomology mathphysicist 2010-10-26T16:55:37Z 2010-10-26T16:55:37Z <p>In the <a href="http://www.maplesoft.com/" rel="nofollow">Maple</a> computer algebra system you have the package <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=DifferentialGeometry/LieAlgebras/LieAlgebraCohomology" rel="nofollow">LieAlgebraCohomology</a> which should do what you want.</p> http://mathoverflow.net/questions/43331/quantum-group-calculations-in-mathematica/43576#43576 Answer by mathphysicist for Quantum Group Calculations in Mathematica mathphysicist 2010-10-25T20:46:27Z 2010-10-25T20:46:27Z <p>I just found the link to the <em>QuantumGroups</em> Mathematica package by Scott Morrison mentioned by Noah: </p> <p><a href="http://katlas.org/wiki/QuantumGroups%60" rel="nofollow">http://katlas.org/wiki/QuantumGroups%60</a></p> http://mathoverflow.net/questions/104202/does-the-operator-mathrmid-t-delta-or-its-greens-function-have-a-name Comment by mathphysicist mathphysicist 2012-08-17T19:13:03Z 2012-08-17T19:13:03Z In the <b>pseudo-Riemannian</b> case the physicists would probably refer to it as the <i>(massive) Klein--Gordon operator</i>. http://mathoverflow.net/questions/85562/reference-request-roger-howes-schur-lectures Comment by mathphysicist mathphysicist 2012-01-13T14:54:25Z 2012-01-13T14:54:25Z The link on Google books (unfortunately, apparently without preview): <a href="http://books.google.com/books?id=EhEZAQAAIAAJ" rel="nofollow">books.google.com/books?id=EhEZAQAAIAAJ</a> http://mathoverflow.net/questions/85269/presenting-work-in-progress Comment by mathphysicist mathphysicist 2012-01-09T18:18:37Z 2012-01-09T18:18:37Z These two posts at the Secret Blogging Seminar could be somewhat helpful: <a href="http://sbseminar.wordpress.com/2008/07/19/followup-working-in-secret/" rel="nofollow">sbseminar.wordpress.com/2008/07/19/&hellip;</a> <a href="http://sbseminar.wordpress.com/2008/07/03/request-lis-preprint-or-on-not-being-a-crackpot/" rel="nofollow">sbseminar.wordpress.com/2008/07/03/&hellip;</a> http://mathoverflow.net/questions/81221/graduate-ode-textbook/81229#81229 Comment by mathphysicist mathphysicist 2011-11-18T10:49:56Z 2011-11-18T10:49:56Z Olver's book requires considerable <i>prior</i> knowledge of ODEs and PDEs, so it certainly can't be a <i>first</i> book to read on ODEs. http://mathoverflow.net/questions/46008/wick-rotation-and-the-riemann-zeta-function Comment by mathphysicist mathphysicist 2011-08-22T15:32:04Z 2011-08-22T15:32:04Z The Casahorran paper is available for free on arXiv.org: <a href="http://arxiv.org/pdf/quant-ph/0011059" rel="nofollow">arxiv.org/pdf/quant-ph/0011059</a> http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Comment by mathphysicist mathphysicist 2011-07-18T16:03:46Z 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. http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Comment by mathphysicist mathphysicist 2011-07-18T15:19:56Z 2011-07-18T15:19:56Z @Yemon: That's one way to look at it, yes. http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Comment by mathphysicist mathphysicist 2011-07-18T15:00:32Z 2011-07-18T15:00:32Z @Yemon: of course I meant for <i>real</i> $x$. http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Comment by mathphysicist mathphysicist 2011-07-18T14:38:52Z 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 <i>real</i> $X$ $P(x)\bar P(x)$ is clearly a polynomial in $x$. http://mathoverflow.net/questions/70609/real-roots-for-polynomials/70613#70613 Comment by mathphysicist mathphysicist 2011-07-18T13:39:12Z 2011-07-18T13:39:12Z @Yemon: $x$ is assumed to be <i>real</i>, and then my $Q(x)$ equals $P(x)\overline{P(x)}$ (cf. Denis' answer). http://mathoverflow.net/questions/69178/momentum-maps-and-matrix-poisson-brackets Comment by mathphysicist mathphysicist 2011-06-30T12:32:07Z 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 <i>entries</i> of your matrices? http://mathoverflow.net/questions/68527/a-name-for-pde-systems-which-are-neither-under-nor-overdetermined/68531#68531 Comment by mathphysicist mathphysicist 2011-06-23T18:17:49Z 2011-06-23T18:17:49Z @Robert Bryant: Thanks a lot! http://mathoverflow.net/questions/68527/a-name-for-pde-systems-which-are-neither-under-nor-overdetermined/68531#68531 Comment by mathphysicist mathphysicist 2011-06-23T06:28:48Z 2011-06-23T06:28:48Z @Ben McKay: to me, <i>involutive determined</i> doesn't sound bad at all. http://mathoverflow.net/questions/68527/a-name-for-pde-systems-which-are-neither-under-nor-overdetermined/68531#68531 Comment by mathphysicist mathphysicist 2011-06-23T06:27:25Z 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 <i>quasilinear</i> systems too. http://mathoverflow.net/questions/68527/a-name-for-pde-systems-which-are-neither-under-nor-overdetermined/68531#68531 Comment by mathphysicist mathphysicist 2011-06-22T17:30:38Z 2011-06-22T17:30:38Z A small problem that I have with the term <i>formally integrable</i> is that in the context I want to use it will appear together with the term <i>completely integrable</i> (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!