Alexander Basyrov | Searcharxiv

arXiv2018

Sums of squares in pseudoconvex hypersurfaces and torsion phenomena for Catlin's boundary systems

Given a pseudoconvex hypersurface in C^n and an arbitrary weight, we show the existence of local coordinates in which the polynomial model contains a particularly simple sum of squares of monomials. Our second main result provides a normalization of a part of any Catlin boundary system. We illustrate by an example that this normalization cannot be extended to the rest of the boundary system due to the existence of what we refer to as torsion.

math.CV↗

arXiv2012

Error Bounds on Derivatives during Simulations

The methods commonly used for numerical differentiation, such as the "center-difference formula" and "four-points formula" are unusable in simulations or real-time data analysis because they require knowledge of the future. In Bard'11, an algorithm was shown that generates formulas that require knowledge only of the past and present values of $f(t)$ to estimate $f'(t)$. Furthermore, the algorithm can handle irregularly spaced data and higher-order derivatives. That work did not include a rigorous proof of correctness nor the error bounds. In this paper, the correctness and error bounds of that algorithm are proven, explicit forms are given for the coefficients, and several interesting corollaries are proven.

math.NA↗