User glougloubarbaki - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T18:00:03Z http://mathoverflow.net/feeds/user/19189 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/121462/applications-of-discrete-time-dynamics Applications of discrete-time dynamics glougloubarbaki 2013-02-11T11:10:25Z 2013-03-20T18:15:17Z <p>Hello,</p> <p>I am a graduate student in the field of discrete-time dynamics. I am wondering about applications of this field outside of mathematics. More precisely, I would like to know if there are "real life" situations where dynamical notions provide a significant insight, or even better, a power of prediction. </p> <p>For example, is there a situation which is naturally modelized by discrete-time dynamics in which chaos is observed (I know about Lorenz attractor and meteorology in the continuous-time case) ? Or a situation in which estimations of the radius of an attractor is helpful (let's say outside of algorithms to find numerical roots), or structural stability, Lyapunov exponents, entropy, etc. play a concrete role ?</p> <p>Sorry if this question is a bit too general.</p> http://mathoverflow.net/questions/112181/about-principal-values-and-wirtinger-derivative About principal values and Wirtinger derivative glougloubarbaki 2012-11-12T15:08:11Z 2012-11-12T15:47:53Z <p>Let $K$ be a compact of the plane of Lebesgues measure 0 and $\Omega$ a domain containing $K$. Denote by $E$ the vector space of functions that are holomorphic on $\Omega - K$. </p> <p>I'm interested in knowing whether we can define a $\overline{\partial}$ operator on $E$ in the sense of distributions . The problem is, elements of $E$ do not define a priori distributions since they need not be locally integrable. So the problem amounts to finding a standard way to make them into distributions. </p> <p>In the case where $f \in E$ is a rational fraction with poles in $K$, the decomposition theorem asserts that it is a sum of $\frac{a_i}{(z-y_i)^{n_i}}$, and those can be made into distributions by taking principal values. Is there a way to generalize this to any $f \in E$ ?</p> <p>To rephrase more precisely my initial question : is there an operator $L : E \rightarrow D'(\Omega)$ such that :</p> <ul> <li><p>$L = \overline{\partial}$ on $E \cap D'(\Omega)$</p></li> <li><p>$L$ is continuous on $\mathbb{C}(z)$ with respect to its topology</p></li> </ul> <p>For example, $\frac{1}{z(z-\epsilon)} \rightarrow \frac{1}{z^2}$, and $\overline{\partial} \frac{1}{z(z-\epsilon)} \rightarrow \frac{1}{\pi}\delta'$ in the sense of distributions, so $L\frac{1}{z^2}$ should be $\frac{1}{\pi}\delta'$, and more generally $L\frac{1}{z^n}$ should be $\frac{1}{\pi}\delta^{(n)}(-1)^{n-1}$.</p> http://mathoverflow.net/questions/80708/is-there-an-axiomatic-approach-of-the-notion-of-dimension Is there an axiomatic approach of the notion of dimension ? glougloubarbaki 2011-11-11T21:15:53Z 2012-05-21T09:17:08Z <p>There are many notions of dimension : algebraic, topological, Hausdorff, Minkowski... (and others). While the topological one generalize the algebraic one, the last three need not coincide for every sets. Yet it is generally acknowledged that the Hausdorff dimension has "nice enough" properties to work with (the interest of the Minkowski dimension lies mainly in the fact that it's easier to compute).</p> <p>So my main question is this : is there an axiomatic approach that would tidy up this mess ? For example, is there a result of the form : if you ask these axioms then the only map from "reasonnable sets" to the set of positive real integers is the Hausdorff dimension ? (or another one ?). If so what are they ?</p> <p>Are there also a clearly identified list of properties that you would ask from any notion of dimension ? I give the following as an example : </p> <ul> <li>it should coincide with the algebraic dimension for finite dimensional vector spaces</li> <li>dim A $\leq$ dim B if $A \subset B$</li> <li>some sort of nice behaviour for cartesian products (at least for reasonnable sets)</li> <li>some sort of nice behaviour for infinite increasing unions and/or decreasing intersections</li> </ul> http://mathoverflow.net/questions/83274/picture-of-the-set-of-discontinuity-of-degree-2-rational-julia-sets Picture of the set of discontinuity of degree 2 rational Julia sets glougloubarbaki 2011-12-12T20:04:26Z 2011-12-12T22:02:34Z <p>Let $Rat_d$ be the set of all rational fraction of degree $d$ and $X_d \subset Rat_d$ be the bifurcation locus of rational fractions of degree $d$, i.e. the closure of the set of discontinuity of the map $f \mapsto J(f)$ defined on $Rat_d$, and associating to $f$ its Julia set (with the topology of Hausdorff distance).</p> <p>Now denote by $M_d$ the quotient of $Rat_d$ under conjugation by homographies, and by $Y_d$ the projection of $X_d$ onto $M_d$. (if you consider polynomials only instead of rational fractions, then $M_2=\mathbb{C}$ and $Y_2$ is the boundary of the Mandelbrot set).</p> <p>A result of Milnor gives an explicit diffeomorphism from $M_2$ to $\mathbb{C}^2$.</p> <p>Question : is there somewhere a computer picture of $Y_2$ in the coordinates defined by Milnor ? (like a projection, or a family of cuts). Judging by the aspect of the Mandelbrot's set boundary, one would expect quite a complicated and interesting set.</p> http://mathoverflow.net/questions/80687/is-it-realistic-to-want-to-classify-minimal-sub-systems-in-small-dimension Is it realistic to want to classify minimal sub-systems (in small dimension) ? glougloubarbaki 2011-11-11T14:56:23Z 2011-11-12T14:39:48Z <p>Let $f$ be a homeomorphism of a topological space onto itself. We recall that a minimal subset for $f$ is a closed invariant subset $F$ such that there is no closed invariant subsets of $F$ under $f$. Equivalently, $F$ is a closed subset for which every orbit is dense. Using Zorn's Lemma it can be shown that such sets always exist.</p> <p>It is easy to see that $R$ cannot be a minimal subset, or that irrationnal rotations on the circle are. </p> <p>Is it possible to classify such sets (and/or the associated homeomorphisms), say, in dimension less than two ? More precisely, given a homeomorphism of the plane, is it possible to classify its possible minimal subsets ? </p> <p>If it turns out that this is too ambitious, are there results of this type for a suitably well-behaved (but still large) class of dynamical systems ? </p> http://mathoverflow.net/questions/121462/applications-of-discrete-time-dynamics/121471#121471 Comment by glougloubarbaki glougloubarbaki 2013-02-13T13:56:39Z 2013-02-13T13:56:39Z thank you, very nice answer ! http://mathoverflow.net/questions/121462/applications-of-discrete-time-dynamics Comment by glougloubarbaki glougloubarbaki 2013-02-11T16:22:39Z 2013-02-11T16:22:39Z @vaughn : yes, but my question is precisely : is such chaos observed in his experiment ? equivalently, is the model relevant ? http://mathoverflow.net/questions/121462/applications-of-discrete-time-dynamics Comment by glougloubarbaki glougloubarbaki 2013-02-11T13:09:09Z 2013-02-11T13:09:09Z to my knowledge, the logistic equation modelizes population growth (among other things) and chaos is not observed in such problems http://mathoverflow.net/questions/112181/about-principal-values-and-wirtinger-derivative/112187#112187 Comment by glougloubarbaki glougloubarbaki 2012-11-13T18:25:21Z 2012-11-13T18:25:21Z Thank you, this helps a lot ! This brings another question to my mind : is there a notion of order of polar part for elements of $E$ ? if not, one might be tempted to define the order of a polar part as the order (plus one) in the sense of distributions when the corresponding hyperfunction is actually a distribution. would it be reasonable ? http://mathoverflow.net/questions/83274/picture-of-the-set-of-discontinuity-of-degree-2-rational-julia-sets Comment by glougloubarbaki glougloubarbaki 2011-12-12T20:51:10Z 2011-12-12T20:51:10Z (that's supposed to be a $\sigma_1$ above) http://mathoverflow.net/questions/83274/picture-of-the-set-of-discontinuity-of-degree-2-rational-julia-sets Comment by glougloubarbaki glougloubarbaki 2011-12-12T20:50:28Z 2011-12-12T20:50:28Z if $\lambda$, $\mu$, $\nu$ are the three fixed points of a quadratic rational map, and $\sigma_i$ are the symetric elementary functions, the map is $\bar{f} \mapsto (\sigma_(\lambda,\mu,\nu), \sigma_2(\lambda,\mu,\nu)$ http://mathoverflow.net/questions/80687/is-it-realistic-to-want-to-classify-minimal-sub-systems-in-small-dimension/80742#80742 Comment by glougloubarbaki glougloubarbaki 2011-11-12T15:48:24Z 2011-11-12T15:48:24Z thank you for your answer... you are right, my question was poorly formulated (I don't claim that minimal subsystems always exist in the non-compact case). I didn't know about that notion of universal minima system, I'll look into it. http://mathoverflow.net/questions/80687/is-it-realistic-to-want-to-classify-minimal-sub-systems-in-small-dimension/80754#80754 Comment by glougloubarbaki glougloubarbaki 2011-11-12T15:45:02Z 2011-11-12T15:45:02Z thank you very much for your answer ! I think it answers completely my question... http://mathoverflow.net/questions/80708/is-there-an-axiomatic-approach-of-the-notion-of-dimension Comment by glougloubarbaki glougloubarbaki 2011-11-12T10:10:05Z 2011-11-12T10:10:05Z my bad, I meant to write positive real numbers of course http://mathoverflow.net/questions/80699/constructing-an-l2-space-with-a-given-orthonormal-basis/80705#80705 Comment by glougloubarbaki glougloubarbaki 2011-11-11T21:19:39Z 2011-11-11T21:19:39Z Oh ok that makes sense.