arXiv subjects

Iwo Biborski

Publications and source records attributed to Iwo Biborski.

4 recordsLinked to original sources

arXiv2026

A Constructive Cayley Representation of Orthogonal Matrices and Applications to Optimization

It is known that every real orthogonal matrix can be brought into the domain of the Cayley transform by multiplication with a suitable diagonal signature matrix. In this paper we provide a constructive and numerically efficient algorithm that, given a real orthogonal matrix $U$, computes a diagonal matrix $D$ with entries in $\{\pm1\}$ such that the Cayley transform of $DU$ is well defined. This yields a representation of $U$ in the form \[ U = D(I-S)(I+S)^{-1}, \] where $S$ is a skew-symmetric matrix. The proposed algorithm requires $O(n^{3})$ arithmetic operations and produces an explicit quantitative bound on the associated skew-symmetric generator. As an application, we show how this construction can be used to control singularities in Cayley-transform-based optimization methods on the orthogonal group.

math.OC↗

arXiv2021

Note on the spread of real symmetric matrices with entries in fixed interval

The spread of a matrix is defined as the maximum of distances between any two eigenvalues of that matrix. In this paper we investigate spread maximization as a function on compact convex subset of the set of real symmetric matrices. We provide some general results and further, we study spread maximizing problem on the set of symmetric matrices with entries restricted to the interval. In particular, we develop some results by X. Zhan, S. M. Fallat and J. J. Xing.

math.OC↗

arXiv2017

Solutions of quasianalytic equations

The article develops techniques for solving equations G(x,y)=0, where G(x,y)=G(x_1,...,x_n,y) is a function in a given quasianalytic class (for example, a quasianalytic Denjoy-Carleman class, or the class of infinitely differentiable functions definable in a polynomially-bounded o-minimal structure). We show that, if G(x,y)=0 has a formal power series solution y=H(x) at some point a, then H is the Taylor expansion at a of a quasianalytic solution y=h(x), where h(x) is allowed to have a certain controlled loss of regularity, depending on G. Several important questions on quasianalytic functions, concerning division, factorization, Weierstrass preparation, etc., fall into the framework of this problem (or are closely related), and are also discussed.

math.CV↗

arXiv2015

On the geometric and differential properties of closed sets definable in quasianalytic structures

In this paper we show that the equivalences between certain properties of closed subanalytic sets proved by E. Bierstone and P. Milman in \cite{[BM-1]} hold for closed sets definable in quasianalytic o-minimal structures. In particular we prove that uniform Chevalley estimate implies a stratification by the diagram of initial exponents and further, Zariski semicontinuity of the diagram of initial exponents. We also show that the stratification by the diagram implies Zariski semicontinuity of Hilbert-Samuel function.

math.AG↗