User adel ahmadyan - MathOverflow

karhunen-Loeve expansion of Poisson process

2012-05-04T01:12:09Z

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 $X_t-\lambda t$?

The auto-correlation function of poisson process is $R(s,t)=\lambda^2 st + \lambda \min(s,t)$. By definition, KL expansion should satisfy $\int_0^T R(s,t) \phi_n(t) dt = \lambda_n \phi_n(s)$.

I've problems figuring out how to solve the integrated equation.

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.

The bound on derivatives of a function

2012-12-08T00:19:50Z

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

The $n^{th}$ derivative of $f$ in the interval $[a, b]$. So $| \frac{dn}{dx^n} f | \leq M$
All derivatives of $f$ in interval $[a,b]$ so $1\leq n < \infty : | \frac{dn}{dx^n} f | \leq M$.

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].

An algorithm for checking if a nonlinear function f is always positive

2012-05-16T19:45:41Z

Is there an algorithm to check if a given (possibly nonlinear) function f is always positive?

The idea that I currently have is to find the roots of the function (using newton-raphson algorithm or similar techniques, see http://en.wikipedia.org/wiki/Root-finding_algorithm) 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.

For example, in Maple, function verify can do this, but I need to implement it in my own program. Maple Help on verify: http://www.maplesoft.com/support/help/Maple/view.aspx?path=verify/function_shells Maple example: assume(x,'real'); verify(x^2+1,0,'greater_than' ); --> returns true, since for every x we have x^2+1 > 0

Mirror question on stack-exchange: http://stackoverflow.com/questions/10625585/an-algorithm-for-checking-if-a-nonlinear-function-f-is-always-positive

[edit] 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).

$x$ 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.

On Direction of a Vector Field w.r.t. a Given Hyperplane

2012-02-25T01:40:26Z

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 $L= \{ {\bf x} | A {\bf x} = b \}$.

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.

I appreciate any hint or directions toward the answer.

For example, in $\mathbb{R}^2$, for a system described by

$x' = -2x + 2y $

$y' = -2x - 2y$

and for a line $y=x$, the direction of the vector field changes at point $(0,0)$.

Comment by Adel Ahmadyan

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