User mohsen - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T15:15:23Z http://mathoverflow.net/feeds/user/8725 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/122566/generators-of-symmetric-group Generators of symmetric group Mohsen 2013-02-21T19:11:56Z 2013-02-22T02:13:54Z <p>Hi</p> <p>Let $n$ be an odd natural number (sufficiently large), and $1\leq t,k &lt; n$. Let $A= { a_{1},a_{2},\ldots,a_{t} }$, $B= { b_{1},b_ {2},\ldots,b_{n-t} }$, $C= { c_{1},c_{2} \ldots,c_{k} }$ and $D= { d_{1},d_{2},\ldots,d_{n-k} }$ be subsets of ${ 1,2,\ldots,n }$ such that $A\cup B=C\cup D= { 1,2,\ldots,n }$ and $A\cap B=C\cap D=\phi$ and $A\neq C$ and $A\neq D$. What can we say about the subgroup generated by $(a_{1},a_{2},\ldots,a_{t})(b_{1},b_{2},\ldots,b_{n-t})$ and $(c_{1},c_{2},\ldots,c_{k})(d_{1},d_{2},\ldots,d_{n-k})$. I think that this subgroup is $S_{n}$. Is it true? Maybe there is a simple counterexample but I've not found it.</p> <p>Clearly, $n$ should be odd. Otherwise, this group is a subgroup of $A_{n}$.</p> http://mathoverflow.net/questions/116914/diophantic-equation-from-finite-semisimple-rings Diophantic equation from finite semisimple rings Mohsen 2012-12-20T20:50:59Z 2012-12-24T01:09:01Z <p>Let $k$ and $k'$ and $n_{1},\ldots,n_{k}$ and $m_{1},\ldots,m_{k'}$ be natural numbers. Let $f_{1}\leq \ldots \leq f_{k}$ and $e_{1} \leq \ldots \leq e_{k'}$ be power primes, such that the following equations hold:</p> <p>$\prod_{j=1}^{j=k} \prod_{i=0}^{n_{j}-1}(f_{j}^{n_{j}}-f_{j}^{i})=\prod_{j=1}^{j=k'} \prod_{i=0}^{m_{j}-1}(e_{j}^{m_{j}}-e_{j}^{i})$</p> <p>and</p> <p>$\prod_{j=1}^{j=k}f_{j}^{n_{j}^{2}}=\prod_{j=1}^{j=k'}e_{j}^{m_{j}^{2}}.$</p> <p>Is it achievable that $k=k'$ and for each $i=1,\ldots,k$ we have $f_{i}=e_{i}$ and $n_{i}=m_{i}$?</p> <p>Actually, this question arises from the question that, if two finite semisimple ring $R$ and $R'$ have the same number of members and the same number of unit members, then can we say that these rings are isomorphic together?</p> <p>If there is some counterexample what additional conditions should be employed to obtain that $R\simeq R'$?</p> http://mathoverflow.net/questions/116962/strongly-regular-cayley-graphs Strongly regular cayley graphs Mohsen 2012-12-21T10:40:18Z 2012-12-21T13:50:10Z <p>Let $G$ and $Cay(A,S)$ be strongly regular graphs with the same parameters. Is it true that $G$ is a cayley graph?</p> http://mathoverflow.net/questions/116220/what-graph-parameters-are-determined-by-parameters-for-strongly-regular-graph/116959#116959 Answer by Mohsen for What graph parameters are determined by parameters for strongly regular graph Mohsen 2012-12-21T10:28:12Z 2012-12-21T10:28:12Z <p>The diameter, energy and number of closed walks could be determined by parameters.</p> http://mathoverflow.net/questions/112357/groups-of-exponent-4 Groups of exponent 4 Mohsen 2012-11-14T06:54:14Z 2012-11-14T11:00:04Z <p>Is there a classification of finite nonabelian 2-groups of exponent 4?</p> <p>What about, finite nonabelian 3-groups of exponent 3?</p> http://mathoverflow.net/questions/109977/1-or-1-as-an-eigenvalue-of-graph 1 or -1 as an eigenvalue of graph Mohsen 2012-10-18T05:36:07Z 2012-11-02T08:52:02Z <p>I have a regular and arc transitive graph which I think that either 1 or -1 is an eigenvalues of adjacency matrix of this graph. How can I prove it? Is there any classification of graphs which have 1 or -1 as an eigenvalue? Is there any paper related to this problem?</p> http://mathoverflow.net/questions/110104/finite-commutative-local-ring Finite commutative local ring Mohsen 2012-10-19T16:43:59Z 2012-10-19T18:09:50Z <p>We know that, if $p$ is a prime number and $k$ is a natural number, then there is just one finite field $F$, such that $|F|=p^{k}$.</p> <p>How about finite commutative local ring?</p> <p>Let $R$ and $S$ be local rings with maximal ideal $m_{1}$ and $m_{2}$, respectively, such that $|R|=|S|$ and $|m_{1}|=|m_{2}|$. Is it true that $R\simeq S$?</p> <p>In the other words, let $p$ be a prime number and $k,n$ ($k\geq n$) be natural numbers. Can we conclude that there exists just one finite commutative local ring of order $p^{k}$, with maximal ideal of order $p^{n}$?</p> http://mathoverflow.net/questions/84898/cayley-graphs-and-its-subgraphs/110040#110040 Answer by Mohsen for Cayley graphs and its subgraphs Mohsen 2012-10-18T19:55:26Z 2012-10-18T19:55:26Z <p>To answer your first question, take a look at [P. Erdos and A. B. Evans. Representations of graphs and orthogonal Latin square graphs. J. Graph Theory 13 (1989), no. 5, 593-595.] Actually, It was shown that every graph $G$ is the induced subgraph of a circulant graph (a cayley graph on a cyclic group).</p> http://mathoverflow.net/questions/36431/eigenvalues-of-edge-regular-graphs eigenvalues of edge regular graphs Mohsen 2010-08-23T10:17:34Z 2012-04-20T09:16:59Z <p>In graph theory, an edge regular graph is defined as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be edge regular if there is also integer λ such that:</p> <p>Every two adjacent vertices have λ common neighbors.</p> <p>A graph of this kind is sometimes said to be an er(v,k,λ).</p> <p>I want know about eigenvalues of edge regular graph, how can we find eigenvalue of this graph?</p> http://mathoverflow.net/questions/63828/ideal-membership Ideal membership Mohsen 2011-05-03T17:19:43Z 2011-05-07T15:04:57Z <p>Let $n=2t$ be an even number. Let $F$ denote a finite field where $|F|=q$. Let $A_{1}, A_{2},\ldots, A_{t}$ and $B_{1},B_{2},\ldots,B_{t}$ be distinct matrices in $M_{n}(F)$. Let <code>$$X = \begin{pmatrix} x_{11} &amp; x_{12} &amp; \cdots &amp; x_{1n} \\ x_{21} &amp; x_{22} &amp; \cdots &amp; x_{2n} \\ \vdots &amp; \vdots &amp; \ddots &amp; \vdots \\ x_{n1} &amp; x_{n2} &amp; \cdots &amp; x_{nn} \\ \end{pmatrix}.$$</code> Consider the ideal <code>$$I=\langle x_{11}^{q-1}-x_{11},x_{12}^{q-1}-x_{12},\ldots,x_{nn}^{q-1}-x_{nn}\rangle$$</code> and the polynomial $$f(X)=f(x_{11},x_{12},\ldots,x_{nn})= \prod_{i=1}^{t}\det(X-A_{i})\prod_{i=1}^{t}(\det(X-B_{i})^{q-1}-1).$$ I'm looking for some condition on $F$ such that $f(X) \notin I$. Actually I think that $f(X) \notin I$ if $|F|$ is sufficiently large. In fact I know that, if $|F|>n^{2}$, then $\prod_{i=1}^{t}\det(X-A_{i})\notin I$ and $\prod_{i=1}^{t}(\det(X-B_{i})^{q-1}-1) \notin I$, but I can't find similar result about $f$.</p> http://mathoverflow.net/questions/61189/determinant-of-matrices Determinant of matrices Mohsen 2011-04-10T06:06:05Z 2011-04-10T06:06:05Z <p>Let $A$, $B$ and $C$ be matrices in $M_{11}(\mathbb{Z}_{97})$. Is there exists a matrix $D$ such that $det(A-D)=0$, $det(B-D)=0$ but $det(C-D)=-1$?</p> <p>Is there exists a matrix $E$ such that $det(A-E)=0$, $det(B-E)=0$ but $det(C-E)=1$?</p> http://mathoverflow.net/questions/55132/eigenvalues-of-i-otimes-b-otimes-c-a-otimes-i-otimes-c-a-otimes-b-otimes-i eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I$ Mohsen 2011-02-11T15:19:11Z 2011-02-11T15:42:01Z <p>Let $A$, $B$ and $C$ be symmetric matrices. What can we say about eigenvalues of $I\otimes B\otimes C + A\otimes I \otimes C + A\otimes B \otimes I$?</p> http://mathoverflow.net/questions/54193/graph-containing-all-trees/54636#54636 Answer by Mohsen for Graph containing all trees? Mohsen 2011-02-07T14:07:29Z 2011-02-07T14:07:29Z <p>see <a href="http://www.units.muohio.edu/sumsri/sumj/2008/Bernsteinpaper.pdf" rel="nofollow">representation number of graph by ring</a></p> <p>it is useful.</p> http://mathoverflow.net/questions/54571/eigenvalues-of-ab eigenvalues of A⊕B Mohsen 2011-02-06T22:05:00Z 2011-02-07T01:16:49Z <p>Let <code>$A_{n\times n}=(a_{ij}),B_{n\times n}=(b_{ij}) \in M_{n}(\mathbb{R})$</code>, where <code>$a_{ij},b_{ij} \in \lbrace 0,1\rbrace$</code>. Boolean sum of $A,B$ denoted by <code>$(A \oplus B)_{n\times n}=(a_{ij}\oplus b_{ij})$</code> is the matrix in <code>$M_{n}(\mathbb{R})$</code> such that <code>$0\oplus 0 = 0$</code>, <code>$0\oplus 1 = 1$</code>, <code>$1\oplus 0=1$</code> and <code>$1\oplus 1 =1$</code>. Is there any inequality or relation between eigenvalues of $A,B$ and $A\oplus B$? (specially, when $A,B$ are symmetric)</p> http://mathoverflow.net/questions/54395/spectral-techniques-for-genus-of-a-graph Spectral techniques for genus of a graph Mohsen 2011-02-05T12:44:27Z 2011-02-05T21:18:23Z <p>A generic question: </p> <p>are there any spectral techniques to estimate the genus of a graph? I am interested in complete balance multipartite graph.</p> http://mathoverflow.net/questions/49214/ext-for-abelian-group EXT for abelian group Mohsen 2010-12-13T05:29:34Z 2010-12-13T05:29:34Z <p>Let $A$ and $B$ are abelian groups. Find $Ext_{\mathbb{Z}}^{n}(A,B)$. (For all $n$)</p> http://mathoverflow.net/questions/36483/sum-of-two-unitary-matrix-is-equal-to-every-matrix Sum of two unitary matrix is equal to every matrix? Mohsen 2010-08-23T19:21:53Z 2010-08-23T20:11:00Z <p>Let $R=M_{n}(Z_{2})$, can we write every matrices of $R$ as sum of two matrices of $GL_{n}(Z_{2})$?</p> http://mathoverflow.net/questions/109977/1-or-1-as-an-eigenvalue-of-graph Comment by Mohsen Mohsen 2012-10-18T20:10:57Z 2012-10-18T20:10:57Z Many thanks for your answers. Yes, it is cayley graph, and it is too large. http://mathoverflow.net/questions/63828/ideal-membership Comment by Mohsen Mohsen 2011-05-08T05:47:10Z 2011-05-08T05:47:10Z I mean for any distinct $A_{i},B_{i}$. http://mathoverflow.net/questions/54571/eigenvalues-of-ab/54579#54579 Comment by Mohsen Mohsen 2011-02-07T07:30:01Z 2011-02-07T07:30:01Z Thanks for your answering to my question, but i'm looking for a lower bound for all the eigenvalues of $A⊕B$. http://mathoverflow.net/questions/49214/ext-for-abelian-group Comment by Mohsen Mohsen 2010-12-13T05:39:06Z 2010-12-13T05:39:06Z No, $A$ and $B$ are not finitely generated.