Vladimir Protasov

Refinement Equations with Nonnegative Coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σk∈ℤPk Ф(nx−k), where n≥2 is an integer, the coefficients {Pk} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.

Joint Spectral Characteristics of Matrices: A Conic Programming Approach

We propose a new method to compute the joint spectral radius and the joint spectral subradius of a set of matrices. We first restrict our attention to matrices that leave a cone invariant. The accuracy of our algorithm, depending on geometric properties of the invariant cone, is estimated. We then extend our method to arbitrary sets of matrices by a lifting procedure, and we demonstrate the efficiency of the new algorithm by applying it to several problems in combinatorics, number theory, and discrete mathematics.

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

Some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed. The maximal number of n-bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n; the coefficient of the exponent is the capacity of the forbidden patterns. In this paper, new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy. The computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given. Subsequently, a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, the existence of extremal norms is proved for any set of matrices arising in the capacity computation. Based on this result, a second capacity approximating algorithm is proposed. The usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately

The Geometric Approach for Computing the Joint Spectral Radius

This research was supported by the grant RFBR 05-01-00066 and by the grant 304.2003.1 supporting the leading scientific schools. In this paper we describe the geometric approach for computing the joint spectral radius of a finite family of linear operators acting in finite-dimensional Eucledian space. The main idea is to use the invariant sets of of these operators. It is shown that any irreducible family of operators possesses a centrally-symmetric invariant compact set, not necessarily unique. The Minkowski norm generated by the convex hull of an invariant set (invariant body) possesses special extremal properties that can be put to good use in exploring the joint spectral radius. In particular, approximation of the invariant bodies by polytopes gives an algorithm for computing the joint spectral radius with a prescribed relative deviation ε. This algorithm is polynomial with respect to 1/ε if the dimension is fixed. Another direction of our research is the asymptotic behavior of the orbit of an arbitrary point under the action of all products of given operators. We observe some relations between the constants of the asymptotic estimations and the sizes of the invariant bodies. In the last section we give a short overview on the extension of geometric approach to the Lp-spectral radius.

Optimizing the Spectral Radius

We suggest a new approach to finding the maximal and the minimal spectral radii of linear operators from a given compact family of operators, which share a common invariant cone (e.g., family of no...

Multivariate Wavelet Frames

We proved that for any matrix dilation and for any positive integer

Weak stability of switching dynamical systems and fast computation of the p-radius of matrices

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Invariant Polytopes of Sets of Matrices with Application to Regularity of Wavelets and Subdivisions

, there exists a compactly supported tight wavelet frame with approximation order

Rank-one corrections of nonnegative matrices, with an application to matrix population models

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Polytope Lyapunov Functions for Stable and for Stabilizable LSS

. Explicit methods for construction of dual and tight wavelet frames with a given number of vanishing moments are suggested.