User jason mfash - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:17:34Z http://mathoverflow.net/feeds/user/20568 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/101564/question-about-harmonic-function-theory Question About Harmonic Function Theory jason mfash 2012-07-07T08:42:46Z 2012-08-06T20:52:57Z <p>Given a non-negative function $u$ defined on $\mathbb{R}^2$ , and satisfies : $\Delta u \leq 0$ . </p> <p>How can I prove that $u$ must be constant? Is there an easy way to do it ? </p> <p>Thanks ! </p> http://mathoverflow.net/questions/101746/commutator-subgroup-group-theory Commutator Subgroup - Group Theory jason mfash 2012-07-09T07:44:48Z 2012-07-10T09:50:01Z <p>Given a rank-2 group $G= &lt; a,b>$ . Is it true and trivial that $[G,G] = &lt; [a,b], [b,a] >$ ? </p> <p>Thanks ! </p> http://mathoverflow.net/questions/100956/some-functional-analysis-questions-laplace-operator-and-fourier-transform Some Functional Analysis Questions (Laplace Operator And Fourier Transform) jason mfash 2012-06-29T16:39:00Z 2012-06-29T16:58:24Z <p>Given a set of the k first eigenvalues $(\lambda_i)_ 1 ^k$ of some operator , and a set of the first k orthomormal eigenfunctions for these eigenvalues : $( \phi_i )$ . Define: $\Phi(x,y) = \sum_{i=1}^k \phi_i(x) \phi_i(y)$ and then define the fourier transform of this function: $\hat \Phi (z,y)= (2 \pi)^{-n/2} \int_{x \in \mathbb{R} ^ n } \Phi(x,y)e^{ix \cdot z} dx$.</p> <p>Can someone explain me the second equality in the following: $z_j \hat{\Phi} (z,y) = (2 \pi)^{-n/2} \int_{\mathbb{R}^n } \Phi(x,y)z_j e^{ixz} dx = (2 \pi)^{-n/2} \int_{\mathbb{R}^n } \Phi(x,y)(-i) \frac{\partial}{\partial x_j } e^{ix \cdot z } dx$ </p> <p>BTW- What does the notation $z_j$ means in this context?</p> <p>Hope someone will be able to help me</p> <p>Thanks in advance </p> http://mathoverflow.net/questions/100275/group-theory-zassenhaus-filtration-other-filtrations Group Theory- Zassenhaus Filtration & Other Filtrations jason mfash 2012-06-21T19:19:55Z 2012-06-21T19:43:51Z <p>Does someone know of any good papers/books/references of properties of the so-called "Zassenhaus Filtration" of a group $G$ ? </p> <p>I'm mainly interested in relations between this filtration and closely related ones such as the lower central series (which I actually already found ) , the derived series, etc... </p> <p>Any good reference will be greatfully acknowledged! I really need to know some properties of this filtration, but can't find any good book/paper that contains such</p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/99776/further-questions-regarding-cohomolgoy-theory-of-sheaves Further Questions Regarding Cohomolgoy Theory Of Sheaves jason mfash 2012-06-16T09:02:01Z 2012-06-16T09:02:01Z <p>In continuation to my previous post: <a href="http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof" rel="nofollow">http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof</a></p> <p>I'll be glad to receive some explanations regarding the following: 1) I know that when taking a sheaf $F$ , then the notation $f_* F$ corresponds to the direct image sheaf. In the answer Gunnar posted, he wrote: $f_* \mathbb{C} = \mathbb{C}^{ \oplus d}$ . What is the meaning of $f_* \mathbb{C}$ in this context? Does the notation $\mathbb{C}^{ \oplus d}$ means the direct sum of d times $\mathbb{C}$ ?</p> <p>2) Given the short exact sequence $0 \to f_* \mathbb{C} \to \mathbb{C}^{\oplus d} \to \mathcal{G} \to 0$ .<br> I understand from the previous calculation, the map between $f_*\mathbb{C}$ and $\mathbb{C}^{\oplus d}$ is the identity. But what is the map between $\mathbb{C}^{\oplus d}$ and the skyscraper sheaf $\mathcal{G}$ ?</p> <p>3) Given a Riemann Surface $X$ , does the algebraic Geometry of its genus is $\chi(X) := \chi(X,\mathbb{C})$ ? </p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/99386/another-help-in-cohomology-of-sheaves Another Help In Cohomology Of Sheaves jason mfash 2012-06-12T19:58:34Z 2012-06-12T20:37:25Z <p>in continuation of my previous post: <a href="http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof" rel="nofollow">http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof</a></p> <p>I need another help in understanding how to compute Cohomology of Sheaves (and right derived functors): Given a compact Riemann Surface $X$ , and the sheaf $O_X(U):= {f:U \to \mathbb{C} |f -holomorphic }$ , I want to compute the cohomology groups $H^n (X, O_X) = R^n \Gamma (O_X)$ where $\Gamma$ is the global section functor: $\Gamma(O_X) = O_X(X)$ . </p> <p>As far as I know, this computation should give: $H^0 (X,O_X) = \mathbb{C}$ , $H^1(X,O_X) = \mathbb{C} ^ {2g}$ , $H^2(X,O_X) = \mathbb{C}$ and the other cohomology groups vanish.</p> <p>I can't figure out how to get these results, and I can't find an appropriate example for such a calculation. </p> <p>Can someone help me understand how to compute such a thing? </p> <p>Thanks!</p> <p>Here is what I did for the n=0 case: So we obviously know that given a functor $F$ , we have $R^0 F(X) = F(X)$ which implies: $H^0 (X,O_X) = \Gamma(O_X) = {f:X\to \mathbb{C} | f-analytic }$ Why is is excatly $\mathbb{C}$ ? In order to continue the calculation for bigger n's, I need to know what is my exact sequence and what are the maps betweenthe elements of this sequence. I hope someone will be able to help me figure it out</p> <p>Thanks a lot again ! </p> http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof Question Regarding Riemann-Hurwitz Formula Proof jason mfash 2012-06-11T12:57:20Z 2012-06-11T22:34:25Z <p>Does someone knows of any good reference for a proof of the Riemann-Hurwitz Formula (of Riemann Surfaces) that uses Spectral-Sequences and Homology ?</p> <p>Thanks in advance ! </p> <p>[ I think I know how to start this kind of proof, but have no idea on how to finish it...If no one has a good reference, I might ask for the specialists' help in finishing my idea]</p> http://mathoverflow.net/questions/98975/question-regarding-rank-k-p-groups Question regarding rank k p-groups jason mfash 2012-06-06T18:54:36Z 2012-06-06T20:38:53Z <p>Does someone know of any works done regarding the density of rank k groups in the collection of groups of order $p^n$ ? i.e.- do we know that almost all p-groups are of rank 3 for example?</p> <p>Thanks in advance for any suggestions</p> http://mathoverflow.net/questions/98395/automorphism-group-of-a-p-group-finitely-generated Automorphism Group of a p-group (finitely generated) jason mfash 2012-05-30T16:04:26Z 2012-05-30T16:30:40Z <p>Does someone know whether the order of the automorphism group of a general p-group of order $p^n$ is bounded from above by $(p^n)^2$? (Every element can possibly be transferred to one of other $p^n$ elements)... </p> <p>If this fact is incorrect, is it possible to deduce a bound on the order of such an automorphism group if our p-group is finitely generated? </p> <p>Thanks in advance</p> http://mathoverflow.net/questions/98059/derived-series-frattini-series-of-a-p-group-pro-p-group Derived Series & Frattini Series of a p-group/pro-p group jason mfash 2012-05-26T18:55:35Z 2012-05-26T18:55:35Z <p>Does someone know of any work done regarding the connection between the derived series &amp; the frattini series of a pro-p group/p -group ? </p> <p>[ I'm aware of the general fact that the frattini-series can be regarded as the p-derived series ]</p> <p>What about some connection between central series and derived ? </p> <p>any varation on this theme will be also appreciated! </p> <p>Does someone know of any results regarding estimating number of groups in a certain class by using these series?</p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/98039/estimating-number-of-groups Estimating Number Of Groups jason mfash 2012-05-26T13:51:49Z 2012-05-26T15:17:12Z <p>I'll be delighted to get some help from the experts around here, regarding the following questions:</p> <p>1) If we take the collection of all rank 2 groups. Does this collection countable? Unctountable? Does someone know about any work concerning putting a topology on the collction of all groups or maybe on the collection of all rank 2 groups?</p> <p>2) Does someone know about works concerning estimations on the number of metabelian groups, or on their density in the collection of groups? </p> <p>3) Does someone know about any works about estimating the density of groups satisfying some property? i.e.- given a family of groups defined by a property $X$ , what is the limit of $\frac{no. X}{no. grps}$ when the order becomes infinite?</p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/97486/fundamental-lemma-of-geometric-group-theory Fundamental Lemma Of Geometric Group Theory jason mfash 2012-05-20T15:37:45Z 2012-05-21T08:29:49Z <p>I'll be delighted to get some help in understanding the proof of the first theorem here: <a href="http://www.math.utah.edu/~malone/QI/notes.pdf" rel="nofollow">http://www.math.utah.edu/~malone/QI/notes.pdf</a> "If G acts geometrically on X and Y (proper geodesic metric spaces) then X and Y are quasi-isometric."</p> <p>In his proof, he fixed $a,b \in X$ and took an arbitrary $q \in Y$ . He then proved that $d_Y (g_0 q , g_n q) \leq R' (d(a,b)+1)$ , which (he claims) finishes the proof.</p> <p>Can you please explain me how does it implies that X,Y are quasi-isometric?</p> <p>The definition of Q.I I know is that if we have a map $f:X \to Y$ such that there exist some constants $L,A$ , such that for every $x_1 , x_2 \in X$ , $y \in Y$ : $\frac{1}{L} d(x_1,x_2) -A \leq d(f(x_1) , f(x_2) ) \leq Ld(x_1,x_2) +A$ and $d(y, f(X) \leq A$ . </p> <p>I'll be glad to receive an explanation .</p> <p>Thanks in advance !</p> http://mathoverflow.net/questions/96750/derived-series-of-pro-p-groups Derived Series of Pro-p groups jason mfash 2012-05-12T07:27:44Z 2012-05-12T09:56:10Z <p>Does every element of the derived series of a pro-p group is also a pro-p group?</p> <p>The problem reduces to showing that every element of the derived series is a closed subgroup...But is it always true?</p> <p>Hope you'll be able to help me</p> <p>Thanks ! </p> http://mathoverflow.net/questions/95366/decomposition-of-a-group-into-a-product Decomposition of a group into a Product jason mfash 2012-04-27T15:05:27Z 2012-05-05T14:11:05Z <p>I was wondering regarding the next question I encountered during my current research:</p> <p>Given a p-group $G$ of order $p^n$ that can be decomposed into the product $G=AB$ of a normal subgroup $A$ and an arbitrary subgroup $B$ , what can we say about the orders of $A$ and $B$ ? Lagrange's theorem tells us that $A,B$ must be also p-groups, but can we say something about the power of p that their order has? </p> <p>What if $B$ was cyclic? Does it change the answer?</p> <p>If we also know that the rank of $G$ is $k$ , will it help us to determine the rank of $A$ ? </p> <p>In another direction, given such a decomposition, how unique is it ? i.e.- have we got any way to estimate the number of ways in which we can decompose an arbitrary p-group into such a product?</p> <p>Hope you'll be able to help. Any reference will be greatfully acknowledged</p> <p>Thanks ! </p> http://mathoverflow.net/questions/91314/bieberbachs-theorem Bieberbach's Theorem jason mfash 2012-03-15T17:42:46Z 2012-03-15T18:53:34Z <p>Can someone explain me, what is the meaning of the term "Compact Fundamental Domain" in the following theorem? </p> <p>"Every discrete group of isometries acting on the n-dimensional euelidean space R^n with compact fundamental domain contains n linearly independent translations" ? </p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/85650/lie-algebras-symplectic-algebra Lie algebras-Symplectic Algebra jason mfash 2012-01-14T11:24:20Z 2012-01-14T19:09:06Z <p>How can I use the root-space-decomposition of the Symplectic Lie-algebra, as it appears here: <a href="http://www.math.toronto.edu/murnaghan/courses/mat445/sp.pdf" rel="nofollow">link text</a></p> <p>in order to prove that this Lie-algebra is simple?</p> <p>I'm mainly interested in the cases l=2,4 . </p> <p>Any help will be greatfuly acknowledged.</p> <p>Thanks</p> http://mathoverflow.net/questions/85642/combinatorics-polytopes-question Combinatorics- Polytopes Question jason mfash 2012-01-14T08:50:10Z 2012-01-14T09:22:52Z <p>Can someone help me solve the following question please? </p> <p>Let v be a vertex of a d-polytope P such that $0 \in intP$ . Prove that $P^* \cap \{ y \in \mathbb{R}^d \mid\left &lt; y, v\right>=1\ \}$ is a facet of $P^{*}$. </p> <p>The definitions are: $P^*=\{ y\in\mathbb{R}^{d}\mid\left &lt; x, y\right>\leq 1\ \forall x\in P\}$ and a face of P is the empty set, P itself, or an intersection of P with a supporting hyperplane (i.e.- a hyperplane, such that P is located in one of the halfspaces it determines). A facet is a face of maximal degree</p> <p>I tried showing that if there exists a vertex v such that this isn't a facet, then P is a convex hull of a finite set not containing v, which is a contradiction, but without success.</p> <p>HOpe you'll be able to help me</p> http://mathoverflow.net/questions/101746/commutator-subgroup-group-theory/101747#101747 Comment by jason mfash jason mfash 2012-07-09T08:14:08Z 2012-07-09T08:14:08Z Thanks a lot !!!! http://mathoverflow.net/questions/101564/question-about-harmonic-function-theory/101572#101572 Comment by jason mfash jason mfash 2012-07-07T14:01:21Z 2012-07-07T14:01:21Z Thanks a lot!!!! http://mathoverflow.net/questions/101564/question-about-harmonic-function-theory/101568#101568 Comment by jason mfash jason mfash 2012-07-07T10:21:51Z 2012-07-07T10:21:51Z @Wasilewki: It's probably a possible solution, but unfortunately I have no idea what Ito's lemma, Wiener process nor supermartingale are... Thanks anyway ! ( I was thinking about some PDE method or something) http://mathoverflow.net/questions/101564/question-about-harmonic-function-theory Comment by jason mfash jason mfash 2012-07-07T09:42:22Z 2012-07-07T09:42:22Z @Kofi: Your example is indeed wrong, since it is not non-negative... http://mathoverflow.net/questions/100956/some-functional-analysis-questions-laplace-operator-and-fourier-transform/100959#100959 Comment by jason mfash jason mfash 2012-06-29T18:10:29Z 2012-06-29T18:10:29Z Great ! That's excatly what I was missing. Thanks a lot ! http://mathoverflow.net/questions/100956/some-functional-analysis-questions-laplace-operator-and-fourier-transform Comment by jason mfash jason mfash 2012-06-29T18:10:18Z 2012-06-29T18:10:18Z Thanks a lot !!! http://mathoverflow.net/questions/100956/some-functional-analysis-questions-laplace-operator-and-fourier-transform/100959#100959 Comment by jason mfash jason mfash 2012-06-29T17:06:25Z 2012-06-29T17:06:25Z Thanks for your quick reply ! ! The problem is that I can't figure out how to &quot;derive&quot; the equality I mentioned... What are they doing in order to get it? Thanks again !!! http://mathoverflow.net/questions/100275/group-theory-zassenhaus-filtration-other-filtrations/100280#100280 Comment by jason mfash jason mfash 2012-06-22T06:42:20Z 2012-06-22T06:42:20Z So what you're actually saying is that the derived series is contained in the zassenhaus filtration ? What about any results about bounding the other direction ? (something like - the n'th term of the zass. filtration is contained in the 10000n's term of the derived series)? Have you got any idea? Thanks a lot again ! http://mathoverflow.net/questions/100275/group-theory-zassenhaus-filtration-other-filtrations/100280#100280 Comment by jason mfash jason mfash 2012-06-21T20:09:26Z 2012-06-21T20:09:26Z Thanks a lot Mark Sapir! Have you got any idea if one of these references contains some relation between the derived series and the Zass. Filtration? http://mathoverflow.net/questions/99776/further-questions-regarding-cohomolgoy-theory-of-sheaves Comment by jason mfash jason mfash 2012-06-16T13:41:04Z 2012-06-16T13:41:04Z Thanks again Gunnar, regarding 1- But what do you mean by $f_* \mathbb{C}$ ? It seems not suitable for the direct image sheaf case regarding 3- I know the definition of the topological Euler char of $X$. But in your previous answers, you wrote: $\chi(\mathbb{C}^{\oplus d }) = d \chi(Y)$ . How did you get this result? Thanks ! http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof/99311#99311 Comment by jason mfash jason mfash 2012-06-14T11:15:07Z 2012-06-14T11:15:07Z And thanks again for your generous help ! http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof/99311#99311 Comment by jason mfash jason mfash 2012-06-14T11:14:54Z 2012-06-14T11:14:54Z Thanks for these notes !! They really helped ! http://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof/99311#99311 Comment by jason mfash jason mfash 2012-06-13T17:42:32Z 2012-06-13T17:42:32Z Hi again Gunnar, since I can't find any detailed examples regarding such calculations, I will be glad if you'll be able to detail the following: you deduce that since the sequence degenerates at the $E_2$ - level, then $H^k (Y, f_* F) = H^k (X,F)$ . Can you please explain it to me? I think it'll finish my misunderstandings. Hope you'll be able to help me for last time Thanks! – http://mathoverflow.net/questions/99386/another-help-in-cohomology-of-sheaves Comment by jason mfash jason mfash 2012-06-13T15:52:01Z 2012-06-13T15:52:01Z Great ! Thanks a lot ! http://mathoverflow.net/questions/99386/another-help-in-cohomology-of-sheaves/99388#99388 Comment by jason mfash jason mfash 2012-06-13T06:13:54Z 2012-06-13T06:13:54Z Thanks a lot ! Can you explain me just what do you mean by a &quot;fine resolution&quot;? Thanks !