User alexander - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T19:07:25Z http://mathoverflow.net/feeds/user/17361 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100861/a-series-representation A series representation Alexander 2012-06-28T13:23:27Z 2012-06-28T16:57:42Z <p>How to find the end of a series representation of the product $$\prod_{\substack{i=1...\infty\\ j=0...i\\ k=0...j}}\frac{1}{1-x^{i-j}y^{j-k}z^{k}}?$$</p> <p>For example for product $$\prod_{\substack{i=1...\infty\\ j=0...i}}\frac{1}{1-x^{i-j}y^j}$$ the ends of series is $$...+7x^5 + 12x^4y + 16x^3y^2 + 16x^2y^3 + 12xy^4 + 7y^5 + 5x^4 +\\ +7x^3y + 9x^2y^2 + 7xy^3 + 5y^4 + 3x^3 + 4x^2y + 4xy^2+\\ + 3y^3 + 2x^2 + 2xy + 2y^2 + x + y + 1$$</p> http://mathoverflow.net/questions/100457/number-of-jordan-canonical-forms-for-an-nxn-matrix Number of Jordan canonical forms for an nxn matrix Alexander 2012-06-23T13:41:52Z 2012-06-23T14:40:18Z <p>How many Jordan canonical forms may have an nxn matrix?</p> <p>In the article <a href="https://oeis.org/A000219" rel="nofollow">https://oeis.org/A000219</a> states that the number of Jordan canonical forms for an nxn matrix is the Number of planar partitions of n. But calculating the normal form of $4\times4$ matrix I'm obtained 14 distinct Jordan forms, and for $5\times5$ I'm obtained 27 distinct Jordan forms. Author of the article (https://oeis.org/A000219) states that the normal form 2 11 can't be obtained. I do not understand this. For example $$J=\begin{pmatrix} \lambda&amp;1&amp;0&amp;0\\ 0&amp;\lambda&amp;0&amp;0\\ 0&amp;0&amp;\lambda'&amp;0\\ 0&amp;0&amp;0&amp;\lambda'\ \end{pmatrix}.$$</p> <p>I think that the number of Jordan canonical forms for an nxn matrix is the number of partitions of n. (https://oeis.org/A001970). But how to prove it, I do not know.</p> <p>I would like to hear your opinion on this matter. Sorry for my english.</p> http://mathoverflow.net/questions/81834/number-of-perturbations-of-the-jordan-form Number of perturbations of the Jordan form Alexander 2011-11-24T20:23:57Z 2011-11-29T19:00:44Z <p>I am looking for information about the number of Jordan forms that can be obtained from a given Jordan form of a small perturbation.</p> <p>For example, if a Jordan form consists of a single cell 2x2 $$J=\begin{pmatrix} \lambda_0 &amp;1\\ 0&amp;\lambda_0\end{pmatrix},$$ by small perturbation $\begin{pmatrix} 0&amp;0\\ 0&amp;\varepsilon\end{pmatrix}$, we can only get the matrix $$\begin{pmatrix} \lambda_0 &amp;0\\ 0&amp;\lambda_1\end{pmatrix}$$ i.e. only two variants are possible.</p> <p>If the Jordan form consists of a single cell 3x3, there may be such cases: $$J=\begin{pmatrix} \lambda_0 &amp;1&amp;0\\ 0&amp;\lambda_0&amp;1\\ 0&amp;0&amp;\lambda_0\end{pmatrix},$$ $$J+ \begin{pmatrix} 0 &amp;0&amp;0\\ 0&amp;0&amp;0\\ 0&amp;0&amp;\varepsilon\end{pmatrix}\sim\begin{pmatrix} \lambda_0 &amp;1&amp;0\\ 0&amp;\lambda_0&amp;0\\ 0&amp;0&amp;\lambda_1\end{pmatrix},$$ $$J+ \begin{pmatrix} 0 &amp;0&amp;0\\ 0&amp;\varepsilon_1&amp;0\\ 0&amp;0&amp;\varepsilon_2\end{pmatrix}\sim\begin{pmatrix} \lambda_0 &amp;0&amp;0\\ 0&amp;\lambda_1&amp;0\\ 0&amp;0&amp;\lambda_2\end{pmatrix}.$$</p> <p>i.e. only three variants are possible.</p> <p>I think I proved that if the Jordan form consists of a single cell mxm, then the number of variants equal to $p(m)$ (see <a href="http://en.wikipedia.org/wiki/Partition_%28number_theory%29" rel="nofollow">http://en.wikipedia.org/wiki/Partition_%28number_theory%29</a>).</p> <p>It seems to me that these results have been obtained by someone, but I can not find them.</p> <p>25.11 We are working over $\mathbb{C}$.</p> <p>When we have a Jordan form $$J=\begin{pmatrix}\lambda_0&amp;0\\0&amp;\lambda_0\end{pmatrix}\ \ \ \mbox{denote by}\ \ \ 1(\lambda_0)+1(\lambda_0),$$ we may obtained $$2(\lambda_0) \ \mbox{ or} \ \ 1(\lambda_0)+1(\lambda_1).$$ There are <strong>three</strong> variants. </p> <p>If we have a Jordan form $2(\lambda_0)+1(\lambda_0)$ we may obtained $$3(\lambda_0), 2(\lambda_0)+1(\lambda_1), 1(\lambda_0)+1(\lambda_1)+1(\lambda_1),$$ $$1(\lambda_0)+1(\lambda_1)+1(\lambda_2).$$ There are <strong>five</strong> variants. </p> <p>If we have a Jordan form $3(\lambda_0)+1(\lambda_0)$ we may obtained</p> <p>$$4(\lambda_0), \ \ 3(\lambda_0)+1(\lambda_1), \ \ 2(\lambda_0)+2(\lambda_1),$$ $$2(\lambda_0)+1(\lambda_1)+1(\lambda_1), \ \ 2(\lambda_0)+1(\lambda_1)+1(\lambda_2),$$ $$1(\lambda_0)+2(\lambda_1)+1(\lambda_1)=2(\lambda_1)+1(\lambda_1)+1(\lambda_0),$$ $$1(\lambda_0)+1(\lambda_1)+1(\lambda_2)+1(\lambda_3).$$ There are <strong>eight</strong> variants. And so on.</p> http://mathoverflow.net/questions/81435/jordan-form-of-compact-operator Jordan form of compact operator Alexander 2011-11-20T16:17:50Z 2011-11-21T10:04:43Z <p>Let $X$ be Banach space over a field $\mathbb{C}$. Consider the Banach space $L_c$ of compact operators in $X$. Let $A^0\in L_c$ be fixed and $\lambda^0\neq 0$ his eigenvalue with algebraic multiplicity $m$ which is associated with Jordan form $J_{\overline k}(\lambda^0)$.</p> <p>In the canonical basis of the subspace that corresponding to eigenvalue $\lambda^0$, operator $A^0$ has the form $$A^0=\begin{pmatrix}J_{\overline k}(\lambda^0)&amp;0 \\ 0 &amp; A_{\textbf{2,2}}^0 \end{pmatrix}.$$</p> <p><strong>Questions.</strong></p> <ol> <li>Can I speak about canonical basis in Banach space or better use a Hilbert space?</li> <li>Is it natural to consider the Jordan form of compact operator on Banach or Hilbert space?</li> <li>Does any non-zero eigenvalue of a compact operator have finite multiplicity?</li> </ol> <hr> <p>21.11. Operator $A^0$ such that it has non-zero eigenvalues.</p> http://mathoverflow.net/questions/73554/distance-between-lattices-of-invariant-subspaces-of-matrices Distance between lattices of invariant subspaces of matrices Alexander 2011-08-24T11:02:38Z 2011-08-24T11:18:02Z <p>For a linear transformation $A: C^n \to C^n$ let $Inv(A)$ be the lattice of all $A$-invariant subspaces. In work I.~Gohberg, L.~Rodman "On the Distance between Lattices of Invariant Subspaces of Matrices" analysis the distanse between $Inv(A)$ and $Inv(B)$ defined as follows: $$dist(Inv(A),Inv(B))=\max\Big(\sup\limits_{M\in Inv(A)}\ \ \inf\limits_{L\in Inv(B)}||P_L-P_M||, \sup\limits_{M\in Inv(B)}\ \ \inf\limits_{L\in Inv(A)}||P_L-P_M||\Big),$$ where $P_N$ is orthogonal projector on the subspace $N$ in $C^n$.</p> <p>I can not quite understand this definition. Why is there symmetry? Maybe someone met geometric Interpretation of these definition.</p> <p>Thanks for your help.</p> http://mathoverflow.net/questions/123627/an-example-second-order-differential-equation-with-associated-function Comment by Alexander Alexander 2013-03-06T13:53:17Z 2013-03-06T13:53:17Z @Robert Bryant: Thank you for your help. http://mathoverflow.net/questions/123627/an-example-second-order-differential-equation-with-associated-function Comment by Alexander Alexander 2013-03-05T17:41:12Z 2013-03-05T17:41:12Z Does for any scalar differential operators with constant coefficients associated function do not exist? http://mathoverflow.net/questions/123627/an-example-second-order-differential-equation-with-associated-function Comment by Alexander Alexander 2013-03-05T16:57:27Z 2013-03-05T16:57:27Z This equation has two eigenfunctions. I need one eigenfunction and associated function that corresponds to it. $y(x)$ - eigenfunction, if $$y′′=\lambda y$$. $u(x)$ - associated function to eigenfunction $y(x)$, if $$u′′=\lambda u+y$$. http://mathoverflow.net/questions/123627/an-example-second-order-differential-equation-with-associated-function Comment by Alexander Alexander 2013-03-05T16:24:53Z 2013-03-05T16:24:53Z Sorry, i mean associated function. Fix the question. http://mathoverflow.net/questions/115086/example-of-a-third-order-differential-operator Comment by Alexander Alexander 2012-12-02T07:05:41Z 2012-12-02T07:05:41Z I inverstigate the submanifolds of compact operators with fixed Jordan structure of chosen eigenvalue. For this purpose i need an example of such operators. They can be constructed by a differential operators of the special form. http://mathoverflow.net/questions/115086/example-of-a-third-order-differential-operator Comment by Alexander Alexander 2012-12-01T17:53:54Z 2012-12-01T17:53:54Z We are my scientific adviser and me. Sorry for my english. http://mathoverflow.net/questions/104761/solvability-of-the-equation/104851#104851 Comment by Alexander Alexander 2012-08-17T05:22:37Z 2012-08-17T05:22:37Z In the question alredy right down that $I(0)&lt;0$. http://mathoverflow.net/questions/104761/solvability-of-the-equation/104851#104851 Comment by Alexander Alexander 2012-08-16T17:31:01Z 2012-08-16T17:31:01Z Thank you for your answer. This case has already been studied, but further progress is not possible. http://mathoverflow.net/questions/104761/solvability-of-the-equation Comment by Alexander Alexander 2012-08-15T14:43:44Z 2012-08-15T14:43:44Z Solution of the equation is the gap between the eigenvalues ​​ of the periodic problem of second order, which correspond to the eigenfunctions of the same oscillation. http://mathoverflow.net/questions/104761/solvability-of-the-equation Comment by Alexander Alexander 2012-08-15T13:54:57Z 2012-08-15T13:54:57Z It is my mistake. The function only has the property $$\theta(x+2\pi)=\theta(x)+2\pi$$ http://mathoverflow.net/questions/100861/a-series-representation/100877#100877 Comment by Alexander Alexander 2012-06-28T17:54:14Z 2012-06-28T17:54:14Z Thank you, I already figured out. http://mathoverflow.net/questions/100861/a-series-representation/100877#100877 Comment by Alexander Alexander 2012-06-28T16:58:53Z 2012-06-28T16:58:53Z Can you write youre Maple's commands? http://mathoverflow.net/questions/100861/a-series-representation Comment by Alexander Alexander 2012-06-28T16:57:26Z 2012-06-28T16:57:26Z Thank you. I am edit the quastion. http://mathoverflow.net/questions/100861/a-series-representation/100877#100877 Comment by Alexander Alexander 2012-06-28T16:47:41Z 2012-06-28T16:47:41Z Thank you. How to get this? Or where I can see more members of the series? http://mathoverflow.net/questions/100861/a-series-representation Comment by Alexander Alexander 2012-06-28T16:33:18Z 2012-06-28T16:33:18Z If y=0 this product $$\prod_{n=1}^\infty\frac{1}{1-x^n}$$ will be generating function of number of partitions of $n$ $$1+x+2x^2+3x^3+5x^4+7x^5...$$ (<a href="https://oeis.org/A000041" rel="nofollow">oeis.org/A000041</a>).