User physics monkey - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:37:11Z http://mathoverflow.net/feeds/user/23953 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/111222/is-there-a-lattice-model-of-e8-manifold Is there a lattice model of E8 manifold? Physics Monkey 2012-11-01T22:46:12Z 2012-11-03T08:42:23Z <p><strong>Background</strong></p> <p>I'm using physics terminology because I'm not sure what the right mathematical terminology is, perhaps a simplicial complex?</p> <p>I'm interested, for various physics reasons, in four manifolds and specifically in their intersection forms. I'm especially interested in the E8 manifold, but if I understand the situation correctly, this manifold is not smooth. I have even read statements that it cannot be triangulated in a certain sense. Obviously I can read the various definitions, but I don't have any intuition for them and I suspect getting the intuition would take me way too far afield. For my physics purposes I would have really liked this manifold to have a nice differential form cohomology, so I'm trying to understand what I can use instead.</p> <p><strong>Main question</strong></p> <p>As an example of a simpler structure that I could use, I would be happy with a lattice model of the E8 manifold. By "lattice model" I mean something like the way a large square lattice with periodic boundary conditions is a model of a torus. There is a discrete notion of points, links, and plaquettes so that the various topological properties are correctly captured. For example, I have in essence non-contractible loops and so forth.</p> <p>Does something like this exist for the E8 manifold i.e. a discrete structure with the right intersection form, or is this impossible?</p> http://mathoverflow.net/questions/100008/error-bounds-for-truncating-a-probability-distribution-based-on-the-entropy/106671#106671 Answer by Physics Monkey for Error bounds for truncating a probability distribution based on the entropy? Physics Monkey 2012-09-08T13:49:04Z 2012-09-08T13:49:04Z <p>I was able to answer my own question with a little help from my fellow physicists :)</p> <hr> <p><strong>Similar result for Renyi entropy</strong></p> <p>I stumbled upon a weaker version of this result before finding Lemma 2 of <a href="http://arxiv.org/abs/cond-mat/0505140" rel="nofollow">http://arxiv.org/abs/cond-mat/0505140</a> which also gives a simple proof using majorization.</p> <p>Let $p_\chi$ denote the probability distribution truncated to its $\chi$ largest values (see above), and let the error be $\epsilon = ||p-p_\chi||_1$.</p> <p>"Lemma 2": If $S_\alpha = \frac{1}{1-\alpha} \ln{\left(\sum_n (p(n))^\alpha\right)}$ is the Renyi entropy then we have $\ln{(\epsilon)} \leq \frac{\alpha}{1-\alpha} \left(S_\alpha - \ln{\left(\frac{\chi}{1-\alpha}\right)}\right)$ for $\alpha &lt;1$.</p> <p>This gives a partial answer to my question in that keeping roughly $e^{S_\alpha}$ states for $\alpha &lt;1$ is guaranteed to lead to small error.</p> <hr> <p><strong>Case of $\alpha =1$</strong></p> <p>This leaves open the question of $\alpha=1$. I came up with a trivial and very pathological counterexample showing that no simple theorem of the type I was asking about can exist.</p> <p>Consider a probability distribution over $n$-bit strings given by $p(0...0) = p + \frac{1-p}{2^n}$ and $p(other) = \frac{1-p}{2^n}$. For $\alpha &lt;1$ then $n\rightarrow \infty$ we have $S_\alpha = n \ln{2}$ while for $\alpha=1$ then $n\rightarrow \infty$ we have $S_1 = (1-p) n \ln{2}$. The Renyi entropy for $\alpha > 1$ doesn't even scale with $n$.</p> <p>Keeping $\chi = e^{S_1}$ states leads to an error of $\epsilon = \frac{(2^n - 2^{(1-p)n})(1-p)}{2^n} \sim (1-p)$. Thus we can make $\epsilon$ as close to one as we like by taking $p$ to zero.</p> http://mathoverflow.net/questions/100008/error-bounds-for-truncating-a-probability-distribution-based-on-the-entropy Error bounds for truncating a probability distribution based on the entropy? Physics Monkey 2012-06-19T14:23:01Z 2012-09-08T13:49:04Z <p><strong>Heuristic Background</strong></p> <p>Consider a set of states labeled $n=1,2,...$ in order of non-increasing probability $p(n)$. The standard Shannon argument gives meaning to the entropy $S$ of $p$ in terms of the number of states needed to encode the distribution with small error in the limit of many iid copies.</p> <p>Roughly speaking I want to know how badly this can fail in the one-shot setting. My primary motivation is to understand how non-trivial it is to be able to show that $e^S$ states suffice to give small error for a given probability distribution. The distributions I have in mind have a system size-like parameter (like the number of iid copies) but the "copies" are correlated in a complicated way.</p> <p>Given the subject matter this may be a very elementary question, but I cannot seem to find much information on it.</p> <hr> <p><strong>Question</strong></p> <p>Fix a large number $S$. Let $p$ be a probability distribution with $p(n) \geq p(n+1)$ and entropy $S$, and let $p_S$ be the probability truncated to its first $e^S$ states i.e. $p_S(n \leq e^S) = p(n)$ and $p_S(n > e^S) = 0$. Is there a bound on the error $||p - p_S ||_1$ (varying $p$ with fixed $S$) or can I make this as close to one as I want?</p> <p>Thanks.</p> http://mathoverflow.net/questions/98783/approximating-fractal-curves Approximating fractal curves Physics Monkey 2012-06-04T17:05:09Z 2012-06-28T05:18:16Z <p>Is there a known algorithm for approximating a fractal curve, say as specified by some iterative procedure e.g. a Koch snowflake, in terms of $f^{-1}(0)$ for some "simple" function $f$?</p> <p>Specifically, consider the set $\mathcal{F} = \{(x,y)|f(x,y) = 0 \}$ where $f(x,y) = \sum_{n,m=-N}^N t_{nm} e^{i n x + i m y}$ and $f$ is real. I would like a procedure to determine the parameters $t_{nm}$ such that the set $\mathcal{F}$ is close to the actual fractal curve. Presumably the number $N$ will grow as the required error decreases.</p> <p>I have been trying to approach this problem by truncating the iteration procedure for the fractal and approximating that piecewise linear curve, but I realized that I don't know a good way to do this either.</p> <p>This is my first time posting, so my apologies if the question is too elementary.</p> http://mathoverflow.net/questions/98783/approximating-fractal-curves Comment by Physics Monkey Physics Monkey 2012-06-19T14:31:51Z 2012-06-19T14:31:51Z This kind of function arises very naturally as the energy spectrum in what is called a quantum tight binding model. One considers a particle on some graph, say a square lattice, and defines a Hamiltonian by specifying an amplitude for the particle to hop from one site to another. If the lattice has a translation symmetry then one can diagonalize the Hamiltonian in terms of plane waves giving the function above with $t_{nm}$ the amplitude to hop n sites horizontally and m sites vertically. The shape of this energy spectrum is very important to the physics of fermions like electrons.