User jason mraz - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:47:01Z http://mathoverflow.net/feeds/user/25080 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference Automorphism Group of a p-group : Looking for a Reference Jason Mraz 2012-11-23T07:31:58Z 2012-11-26T10:55:07Z <p>In the following post by DavidLHarden : <a href="http://mathoverflow.net/questions/68109/orders-of-automorphism-groups-of-p-groups" rel="nofollow">See Here</a></p> <p>He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n$ , then $|AutG|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$ " . </p> <p>I can't find any reference for this theorem , </p> <p>Does someone know of any reference for this fact?</p> <p>Thanks in advance</p> http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group Number of Normal subgroups In a p-Group Jason Mraz 2012-10-01T23:22:33Z 2012-10-03T00:41:04Z <p>Dear all,</p> <p>Does someone know of any paper/method that enables us counting/estimating the number of normal subgroups of some p-group of order $p ^n$ ($n$ is some natural number ? ) .</p> <p>Is there anyway we can count the maximal subgroups it has (i.e.- the groups of order $p^{n-1}$ ? ) ?</p> <p>Thanks in advance</p> http://mathoverflow.net/questions/102563/simply-connected-regions-and-phragmen-lindelof-theorem Simply-Connected Regions and Phragmen-Lindelöf Theorem Jason Mraz 2012-07-18T17:00:16Z 2012-09-15T14:43:39Z <p>It's easy to see that the Phargmen-Lindelöf theorem from complex analysis can be generalized to non-simply-connected regions. Namely to regions $G$ with the property that for each $z \in \partial_\infty G$ there is a sphere $V$ in $\mathbb{C}_\infty$ centered at $z$ such that $V \cap G$ is simply connected. </p> <p>The problem is that I can't think about any example of non-simply connected regions that have this property and of simply-connected regions that don't have this property...</p> <p>Does the exterior of a unit ball can be considered as an example of non-simply connected region that don't satisfy the property above? What about regions that do satisfy this property?</p> <p>BTW - $\partial _ \infty$ is the boundary of $G$ with the boundary at infinity (and $\mathbb{C}_\infty$ is the Riemann-Sphere ) .</p> <p>Thanks in advance </p> http://mathoverflow.net/questions/105303/linear-independence-group-theory Linear Independence & Group Theory Jason Mraz 2012-08-23T07:04:39Z 2012-09-01T05:43:35Z <p>Given an elementary abelian $p$-group $G$, it's well known that it can be seen as a vector space over $\mathbb{Z}_p$. </p> <p>But, does someone have an idea about possible sources where I can find proofs that use this fact? Examples: linear independence of elements in a group $G$ modulo $\phi(G)$ ($\phi(G)$ is the Frattini group of $G$ ), etc. ...</p> <p>Does someone have an idea?</p> <p>Thanks! </p> http://mathoverflow.net/questions/102061/different-measures-on-r2 Different Measures On R2 Jason Mraz 2012-07-12T18:34:40Z 2012-07-25T08:31:19Z <p>Dear all,</p> <p>Is there any possible way to construct a set $A \subseteq \mathbb{R}^n$ for which $H^{n-1} (\partial A) > Leb^ + (A )$?</p> <p>Where $H^{n-1} (\partial A)$ is the Hausdorff measure of the boundary of $A$ and: $Leb^{+} (A) = \lim_{\epsilon \to 0 } \frac{ Leb(A_ \epsilon) - Leb(A) }{\epsilon}$ , $A_\epsilon := { x \in \mathbb{R} ^n | d(x,A) \leq \epsilon }$ =Minkowski's content with respect to Lebesgue measure. </p> <p>Thanks in advance ! </p> http://mathoverflow.net/questions/102744/commutator-length-in-free-groups Commutator Length In Free Groups Jason Mraz 2012-07-20T15:00:46Z 2012-07-20T15:00:46Z <p>An interesting question: Given a group $G$ , we define the commutator-length of one of its elements $g$ (it's a known notion) to be the minimal number of commutators in the group such that their product equals $g$ (i.e.- $cl(g) = min (n \qquad | \qquad g= [a_1 , b_1 ] [a_2,b_2 ] ...[a_n,b_n] )$ . </p> <p>Now, given a free-group $F= &lt; x_1 ,..., x_n >$ . Is it known that $cl (x_i x_j ) >1$ ? (i.e.- a product of two generators can't be a commutator of two other elements) </p> <p>Thanks in advance </p> http://mathoverflow.net/questions/102464/discrete-spectrum-of-laplacian-and-the-associated-quadratic-form Discrete Spectrum of Laplacian and the associated Quadratic form Jason Mraz 2012-07-17T16:37:01Z 2012-07-17T16:37:01Z <p>Given the following inequality: $\int (|\nabla f|^2+V_+|f|^2)\le \int V_-|f|^2,$ , where $V_- , V_+$ are the negative and positive parts of the potential, does this imply that the spectrum of the laplacian on the domain of integration isn't discrete? </p> <p>If so, can someone explain me the reason for that? </p> <p>I found very similar conclusion in SE: <a href="http://math.stackexchange.com/questions/170507/proving-compactness-of-resolvent-of-an-operator" rel="nofollow">http://math.stackexchange.com/questions/170507/proving-compactness-of-resolvent-of-an-operator</a> (see (2) there), but I can't understand why this is true... </p> <p>thanks all</p> http://mathoverflow.net/questions/102144/schrodinger-operators-with-diverging-potential Schrodinger Operators with diverging Potential Jason Mraz 2012-07-13T13:56:04Z 2012-07-13T18:38:21Z <p>Is it well known that if $H = -\bar{\Delta} + V$ (which is defined over $L^2( \mathbb{R} ^n$ ) and $lim_{|x| \to \infty } = + \infty$, then $H$ has compact resolvent?</p> <p>Does someone know of any elegant way of proving this?</p> <p>Thanks in advance </p> http://mathoverflow.net/questions/102115/spectrum-of-laplacian-in-convex-region Spectrum Of Laplacian in Convex Region Jason Mraz 2012-07-13T08:12:04Z 2012-07-13T08:12:04Z <p>I'm given the following question (from Davies' book - Spectral theory of differential operators): Use the theorem (*) below to prove that if $\Omega$ is a convex region in $\mathbb{R}^2$ , then: $\frac{1}{4} \int _\Omega \frac{|f|^2}{d^2} d^2 x \leq \int _\Omega | \bigtriangledown f | ^2 d^2 x$ for all $f \in C_c ^\infty (\Omega )$ . Deduce that $Spec(H) \subseteq [\lambda, \infty )$ , where $\lambda^{-1} := 4Inradius(\Omega)^2$.</p> <p><strong>Theorem :</strong> Let $\Omega$ be a region in $\mathbb{R}^2$ and let $f \in C_c ^\infty (\Omega )$. Then : $\frac{1}{2} \int _\Omega \frac{|f|^2}{m(x)^2} d^2 x \leq \int _\Omega | \bigtriangledown f | ^2 d^2 x$ for all $f \in C_c ^\infty (\Omega )$ where the pseudodistance $m(x)$ is defined by $\frac{1}{m(x)} := \frac{1}{2\pi } \int_{-\pi}^{\pi} \frac{d \Theta }{d_\Theta (x)^2 }$ and $d_\Theta : \Omega \to (0,+\infty ]$ is defined by $d_\Theta (x) = min { |s| : x+se^{i \Theta} \notin \Omega }$ . </p> <p>It's also known that if $\Omega$ is regular ( i.e.- there exists a constant $c$ such that $d(x)\leq m(x) \leq cd(x)$ ) , then $Spec (H) \leq [ \frac{1}{2c^2 Inradius^2}, \infty )$ . </p> <p>How can I use these facts in order to answer Davies' question?</p> <p>Any help will be greatfully acknowledged ! </p> <p>Thanks in advance </p> http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference/114235#114235 Comment by Jason Mraz Jason Mraz 2012-11-26T11:02:18Z 2012-11-26T11:02:18Z Found it ! Helped indeed ! Thanks! http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference/114235#114235 Comment by Jason Mraz Jason Mraz 2012-11-26T10:55:25Z 2012-11-26T10:55:25Z Thanks ! I'll try to get the book! http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference/114235#114235 Comment by Jason Mraz Jason Mraz 2012-11-25T22:02:18Z 2012-11-25T22:02:18Z @DavidLHarden: But you don't want to prove something that is well known in a paper you are writing... By proving it, it's kind of saying that you are the one that figured this theorem out... But if you already saw this post... Have you got any idea for possible reference? In your first message I quoted, you said that it's well known, do you know where can I find it ? Thanks ! http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference/114235#114235 Comment by Jason Mraz Jason Mraz 2012-11-23T19:30:13Z 2012-11-23T19:30:13Z Thanks @JSpecter! I'll go over your proof later today... The problem is that a friend of mine needs this fact for a paper he's writing, and I don't think a proof of a known fact is something that he would want to put in his paper (due to copyrighting rights) Thanks anyway ! http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference/114229#114229 Comment by Jason Mraz Jason Mraz 2012-11-23T19:26:16Z 2012-11-23T19:26:16Z @Nick Gill: Thanks ! Sorry for my ignorance, but what is the $p'$ part of the result? (I can't understand what is $p'$ in your response) . I also have trouble getting into the link you just gave... It gives me an error... can you please fix it? Thanks ! http://mathoverflow.net/questions/114218/automorphism-group-of-a-p-group-looking-for-a-reference Comment by Jason Mraz Jason Mraz 2012-11-23T19:16:35Z 2012-11-23T19:16:35Z Geoff: I would try Mathscinet on Monday or something... I am not sure if this is due to Neumann or not... Thanks anyway! @mt: I couldn't find this theorem in the book you just mentioned... Thanks for the suggestion! http://mathoverflow.net/questions/109541/method-for-solving-exponential-recurrence-relations Comment by Jason Mraz Jason Mraz 2012-10-13T18:26:51Z 2012-10-13T18:26:51Z Yep ... Got any idea? http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group Comment by Jason Mraz Jason Mraz 2012-10-03T19:03:07Z 2012-10-03T19:03:07Z That's excatly the thing... I only need kind of &quot;simple&quot; estimates and bounds on the number of subgroups... I'll try to go over the lecture notes you sent and I might find something useful in them... Thanks a lot ! (I'll try to look for the book you mentioned) http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group/108584#108584 Comment by Jason Mraz Jason Mraz 2012-10-03T09:23:56Z 2012-10-03T09:23:56Z Dear @Alexander Gruber: 1) As far as I know, the Frattini subgroup is defined as the intersection of all the maximal subgroups. How can I use this (and the quotioent $P/\Phi(P)$ in order to verify the first bound you gave in your answer ? 2) Given an elementary abelian group of order $p^n$, the classification theorem tells us that it is isomorphic to $n$ copies of $\mathbb{Z} _p$ . If so, then I expect the number of normal subgroups to be $2^n$ ... What am I doing wrong? Thanks ! http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group/108615#108615 Comment by Jason Mraz Jason Mraz 2012-10-03T09:16:03Z 2012-10-03T09:16:03Z Dear @Nick Gill and @Alexander: Where can I find proofs for the facts you mention? I can't see this straight away... Can you give me some reference for the proof of these facts? Thanks ! http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group/108615#108615 Comment by Jason Mraz Jason Mraz 2012-10-02T12:43:12Z 2012-10-02T12:43:12Z Dear @Nick: Thanks a lot ! I 'll be glad if you'll be able to tell me what do you mean by a &quot;group of maximal class&quot; ... After verifying this little detail, I'll reread your answer in order to check again that I understand it... Thanks again! http://mathoverflow.net/questions/108581/number-of-normal-subgroups-in-a-p-group Comment by Jason Mraz Jason Mraz 2012-10-02T12:38:41Z 2012-10-02T12:38:41Z @Alexander Chevov: Thanks ! I had no idea about the Hall Algebra notion... But I'm still skeptic about it... Have you got any paper the gives some more details about it? Thanks again! http://mathoverflow.net/questions/106101/another-independence-in-group-theory-question Comment by Jason Mraz Jason Mraz 2012-09-01T14:15:13Z 2012-09-01T14:15:13Z I think that the independence notion I want is that if an element can be written as a linear combination of these elements, then this linear combination is unique... It's the independence notion that we need for a basis http://mathoverflow.net/questions/106037/on-the-independence-of-elements-in-the-frattini-group/106050#106050 Comment by Jason Mraz Jason Mraz 2012-08-31T17:02:29Z 2012-08-31T17:02:29Z OHHHHH I'm sorry , you meant the p-adic integers... Great Answer! I thought you meant $\mathbb{F} _p$ ! SORRY &amp; Thanks! Got Any idea about my second question? http://mathoverflow.net/questions/106037/on-the-independence-of-elements-in-the-frattini-group/106050#106050 Comment by Jason Mraz Jason Mraz 2012-08-31T16:50:01Z 2012-08-31T16:50:01Z Actually it is what I meant to ask (in (1) ) , is it also true for an infinite free pro-p group ? Can you help me with (2)? Got any idea? Thanks!