User peter cudmore - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T13:54:00Z http://mathoverflow.net/feeds/user/15811 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128282/calculate-the-inverse-of-a-matrix/128299#128299 Answer by Peter Cudmore for Calculate the inverse of a matrix Peter Cudmore 2013-04-22T00:54:25Z 2013-04-22T00:54:25Z <p>I think this is more appropriate for mathSE, however;</p> <p>Recall that a linear differential equation can have fixed points as nodes, saddles and centers.</p> <p>Your question requires the fixed point to be a node of some variety, so the real parts of the eigenvalues of $A$ must be the same sign.</p> <p>If they all have negative real parts, the node is stable and you will get the solution you are after.</p> <p>If they all have positive real parts, simply reverse the direction of time and solve as you figured in your comment above.</p> <p>If they are of alternating sign or have zero real part, this approach will no longer work.</p> http://mathoverflow.net/questions/122593/lower-bounds-of-laplace-transform-of-characteristic-functions Lower bounds of laplace transform of characteristic functions Peter Cudmore 2013-02-22T02:39:13Z 2013-02-26T01:26:13Z <p>Cross-posted on <a href="http://math.stackexchange.com/questions/310813/lower-bounds-of-laplace-transform-of-characteristic-functions" rel="nofollow">maths.stackexchange</a></p> <p>I have the following integral: $$f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt$$ where $\varphi_X(t)$ is the characteristic function of some undetermined probability distribution and $\mu$ is some complex variable with strictly positive real part $\mu_r>0$.</p> <p>It is easy enough to prove that this function is bounded above, $|f(\mu)| \le \frac{1}{\mu_r}$.</p> <p>However, numerics suggest that for 'nice' distributions (symmetric, and non-increasing on $[0,\infty)$), the integral is bounded below by $|f(\mu_r)|\le|f(\mu)|$.</p> <p><strong>So my questions are:</strong></p> <blockquote> <p>1) Are there established results on finding lower bounds of the Laplace transform of a characteristic function. (My trawling of google scholar and the like haven't produced anything. It seems like such an elementary problem that surely someone has considered it before.)</p> <p>2) If not what general techniques are used to look for lower bounds on integrals such as this?</p> </blockquote> <p>Background: The integral comes from investigating how non-identical frequency distribution of linear oscillators a particular collective behavior problem. This has occurred before in such problems and has been treated asymptotically for strictly real $\mu$ (ref. 1) and commented on for complex $\mu$ (appendix ref. 2). Due to the parameter regions we are interested in (around $\Re{\mu} =1$) the approximations made previously no longer hold.</p> <p>Many Thanks, Pete.</p> <p>References:</p> <p>Ref. 1 - RE. Mirollo, SH Strogatz. <em>Amplitude Death in Limit Cycle Oscillators</em>. <a href="http://www.springerlink.com/index/ln051rp502550471.pdf" rel="nofollow">http://www.springerlink.com/index/ln051rp502550471.pdf</a></p> <p>Ref. 2 - PC. Matthews, RE. Mirollo, SH Strogatz. <em>Dynamics of a large system of coupled nonlinear oscillators</em>. <a href="http://www.sciencedirect.com/science/article/pii/016727899190129W" rel="nofollow">http://www.sciencedirect.com/science/article/pii/016727899190129W</a></p> <p><em>edit: (typo) corrected to non-increasing</em></p> http://mathoverflow.net/questions/69752/synchronization-frequency-of-the-kuramoto-model-for-coupling-matrix-with-constant/70149#70149 Answer by Peter Cudmore for Synchronization frequency of the Kuramoto model for coupling matrix with constant rows Peter Cudmore 2011-07-12T16:59:14Z 2011-07-12T16:59:14Z <p>As far as literature goes:</p> <p>"From Kuramoto to Crawford..." - S Strogatz: <a href="http://omnis.if.ufrj.br/~monica/ACMSM2008/StrogatzPD2000.pdf" rel="nofollow">http://omnis.if.ufrj.br/~monica/ACMSM2008/StrogatzPD2000.pdf</a></p> <p>"Synchronization in Complex Networks" - Arenas et al <a href="http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.2976v3.pdf" rel="nofollow">http://arxiv.org/PS_cache/arxiv/pdf/0805/0805.2976v3.pdf</a></p> <p>However,</p> <p>The special case you mention can be reduced to the mean field kuramoto model by introducing a complex order parameter. $re^{i \Psi} = \frac{1}{n}\sum\limits_{j=1}^n e^{i\theta_j}$</p> <p>So $\dot\theta_i = \omega_i + K_ir\sin(\Psi - \theta_i)$</p> <p>Also, if you let $\omega_i \rightarrow \Omega + \Delta\omega_i$ and $\theta_i \rightarrow \theta_i + \Omega t$ where $\Omega$ is your mean frequency</p> <p>you get</p> <p>$\dot\theta_i = \Delta\omega_i + K_ir\sin(\Psi - \theta_i)$</p> <p>with the mean of all $\Delta\omega_i$ are zero and most importantly, $\dot\theta_i = 0$ now corresponds to that oscillator running at your mean frequency.</p> <p>Now if $K_i >\frac{2}{\pi g(0)} \forall i$ with $g$ the p.d.f for your $\Delta\omega_i$'s, then you'll definately get a large proportion of oscillators synching to your mean frequency. (Strogatz' paper covers this derivation for constant $K$)</p> <p>How many is a tough question, depending on your p.d.f. For general $K_i$ i think Arenas' paper should provide a starting point. </p> http://mathoverflow.net/questions/67924/eigenvalues-of-the-sum-of-a-diagonal-and-a-unit-matrix Eigenvalues of the sum of a diagonal and a unit matrix Peter Cudmore 2011-06-16T10:03:40Z 2011-06-16T12:39:12Z <p>I'm trying to find information on the eigenvalues of an $n \times n$ matrix A such that </p> <p><code>$A = D + J$</code></p> <p>Where $D$ is some complex valued diagonal matrix, and $J$ is an matrix consisting of all $1$'s.<br> When $D$ has identical values, the problem is equivalent to finding the eigenvalues of $J$.</p> <p>So my question is this:<br> <em>If $D$ has non-identical values (specifically, non-identical imaginary components), is there an elementary way to compute the eigenvalues of $A$ ?</em> </p> <p>The problem comes from linearising about the origin of a system of $n$ near identical coupled resonators. $D$ relates to the behaviour of each resonator, $J$ relates to the coupling process. </p> http://mathoverflow.net/questions/125020/partial-linearization-near-a-hyperbolic-fixed-point-classical-scattering Comment by Peter Cudmore Peter Cudmore 2013-03-20T00:14:15Z 2013-03-20T00:14:15Z This looks strikingly familiar to the Poincare Linearization theorem and Normal Forms in the theory of ODEs. http://mathoverflow.net/questions/67924/eigenvalues-of-the-sum-of-a-diagonal-and-a-unit-matrix/67935#67935 Comment by Peter Cudmore Peter Cudmore 2011-06-16T13:11:05Z 2011-06-16T13:11:05Z Thanks very much, this was exactly what i was looking for.