User adel ahmadyan - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T14:52:30Z http://mathoverflow.net/feeds/user/21201 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95941/karhunen-loeve-expansion-of-poisson-process karhunen-Loeve expansion of Poisson process Adel Ahmadyan 2012-05-04T01:12:09Z 2012-12-24T23:22:00Z <p>Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda t$?</p> <p>The auto-correlation function of poisson process is $R(s,t)=\lambda^2 st + \lambda \min(s,t)$. By definition, KL expansion should satisfy $\int_0^T R(s,t) \phi_n(t) dt = \lambda_n \phi_n(s)$.</p> <p>I've problems figuring out how to solve the integrated equation.</p> <p>For wiener process, this link (http://mathoverflow.net/questions/59337/karhunenloeve-approximation-of-brownian-motion-and-diffusions) and wikipedia article on KL expansion was useful.</p> http://mathoverflow.net/questions/115760/the-bound-on-derivatives-of-a-function The bound on derivatives of a function Adel Ahmadyan 2012-12-08T00:19:50Z 2012-12-08T00:19:50Z <p>Let $f(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Is there any way to find an upper bound (any bound, not necessary supremum) on </p> <ul> <li>The $n^{th}$ derivative of $f$ in the interval $[a, b]$. So $| \frac{dn}{dx^n} f | \leq M$</li> <li>All derivatives of $f$ in interval $[a,b]$ so $1\leq n &lt; \infty : | \frac{dn}{dx^n} f | \leq M$. </li> </ul> <p>Moreover, is there special class of functions that have these bounds? for example, $sin(x)$ is and all it's derivatives are bounded within interval [-1,1].</p> http://mathoverflow.net/questions/97152/an-algorithm-for-checking-if-a-nonlinear-function-f-is-always-positive An algorithm for checking if a nonlinear function f is always positive Adel Ahmadyan 2012-05-16T19:45:41Z 2012-05-17T00:12:09Z <p>Is there an algorithm to check if a given (possibly nonlinear) function f is always positive?</p> <p>The idea that I currently have is to find the roots of the function (using newton-raphson algorithm or similar techniques, see <a href="http://en.wikipedia.org/wiki/Root-finding_algorithm" rel="nofollow">http://en.wikipedia.org/wiki/Root-finding_algorithm</a>) and check for derivatives, or finding the minimum of f, but they don't seems to be the best solution to this problem, also there are a lot of convergence issues with root finding algorithms.</p> <p>For example, in Maple, function verify can do this, but I need to implement it in my own program. Maple Help on verify: <a href="http://www.maplesoft.com/support/help/Maple/view.aspx?path=verify/function_shells" rel="nofollow">http://www.maplesoft.com/support/help/Maple/view.aspx?path=verify/function_shells</a> Maple example: assume(x,'real'); verify(x^2+1,0,'greater_than' ); --> returns true, since for every x we have x^2+1 > 0</p> <p>Mirror question on stack-exchange: <a href="http://stackoverflow.com/questions/10625585/an-algorithm-for-checking-if-a-nonlinear-function-f-is-always-positive">http://stackoverflow.com/questions/10625585/an-algorithm-for-checking-if-a-nonlinear-function-f-is-always-positive</a></p> <p> Some background on the question: The function $f$ is the right hand-side differential nonlinear model for a circuit. A nonlinear circuit can be modeled as a set of ordinary differential equations by applying modified nodal analysis (MNA), for sake of simplicity, let's consider only systems with 1 dimension, so $x' = f(x)$ where $f$ describes the circuit, for example $f$ can be $f(x) = 10x - 100x^2 + 200x^3 - 300x^4 + 100x^5$ ( A model for nonlinear tunnel-diode) or $f=10 - 2sin(4x)+ 3x$ (A model for josephson junction). </p> <p>$x$ is bounded and $f$ is only defined in interval $[a,b] \in R$. $f$ is continuous. I can also make an assumption that $f$ is Lipschitz with Lipschitz constant L>0, but I don't want to unless I have to.</p> http://mathoverflow.net/questions/89457/on-direction-of-a-vector-field-w-r-t-a-given-hyperplane On Direction of a Vector Field w.r.t. a Given Hyperplane Adel Ahmadyan 2012-02-25T01:40:26Z 2012-02-25T09:08:56Z <p>A nonlinear system modeled as a set of ODE equations as $f( {\bf x}(t), {\bf \dot{x}}(t), {\bf u}(t))=0$. Let $L$ be a hyper-plane in $\mathbb{R}^n$, described by $L= \{ {\bf x} | A {\bf x} = b \}$. </p> <p>What is the direction of the vector field generated by the $f$ on $L$? Preferred answer should be in a form of a function $g({\bf x})$ where the output of g is positive in one direction and negative in the other case. </p> <p>I appreciate any hint or directions toward the answer.</p> <p>For example, in $\mathbb{R}^2$, for a system described by </p> <p>$x' = -2x + 2y$</p> <p>$y' = -2x - 2y$</p> <p>and for a line $y=x$, the direction of the vector field changes at point $(0,0)$. </p> http://mathoverflow.net/questions/49516/n-dimensional-voronoi-diagram/49521#49521 Comment by Adel Ahmadyan Adel Ahmadyan 2013-04-05T05:12:37Z 2013-04-05T05:12:37Z I'm afraid it is as bad as Allessandro fears. The main problem is the number of the hyperplanes that these algorithm generates. For a very simple example, I used qhull for 40 points in 15-d and the voronoi decomposition has more than 5 million hyperplanes Voronoi diagram by the convex hull of 40 points in 16-d: Number of Voronoi regions: 40 Number of Voronoi vertices: 1361237 Statistics for: rbox 40 s D15 | qvoronoi s p TO test Number of points processed: 40 Number of hyperplanes created: 5539466 Number of facets in hull: 2774645 Number of distance tests for qhull: 4729150