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We characterize the set of planar locally finite Cayley graphs, and give a finite representation of these graphs by a special kind of finite state automata called labeling schemes. As a result, we are able to enumerate and describe all…
The $\mu$-neutral linear fractional multi-delayed differential nonhomogeneous system with noncommutative coefficient matrices is introduced. The novel $\mu$-neutral multi-delayed perturbation of Mittag-Leffler type matrix function is…
This Ph.D. thesis contains original contributions to several areas within the disciplines of disordered systems, numerical linear algebra, and scientific computing: (1) Theoretical and numerical study of the errors caused by using certain…
A periodic perturbation generates a complicated dynamics close to separatrices and saddle points. We construct an asymptotic solution which is close to the separatrix for the unperturbed Duffing's oscillator over a long time. This solution…
We consider the structured-output prediction problem through probabilistic approaches and generalize the "perturb-and-MAP" framework to more challenging weighted Hamming losses, which are crucial in applications. While in principle our…
Recent results in control systems and numerical integration literature utilize invariant set theory to lift dynamical systems evolving on nonlinear manifolds to those evolving on vector spaces. We leverage this technique to propose an…
This short course offers a new perspective on randomized algorithms for matrix computations. It explores the distinct ways in which probability can be used to design algorithms for numerical linear algebra. Each design template is…
For a singularly perturbed system of reaction--diffusion equations, assuming that the 0th order solutions in regular and singular regions are all stable, we construct matched asymptotic expansions for formal solutions to any desired order…
Cylindrical algebraic decomposition (CAD) is an important tool for the investigation of semi-algebraic sets, with applications within algebraic geometry and beyond. We recently reported on a new implementation of CAD in Maple which…
In this paper, we propose an incremental algorithm for computing cylindrical algebraic decompositions. The algorithm consists of two parts: computing a complex cylindrical tree and refining this complex tree into a cylindrical tree in real…
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as the method for deriving such recurrence…
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
Simple semitoric systems were classified about ten years ago in terms of a collection of invariants, essentially given by a convex polygon with some marked points corresponding to focus-focus singularities. Each marked point is endowed with…
Topological invariants allow to characterize Hamiltonians, predicting the existence of topologically protected in-gap modes. Those invariants can be computed by tracing the evolution of the occupied wavefunctions under twisted boundary…
The parameter-uniform convergence of a fitted operator method for a singularly perturbed differential equation is normally available only for uniform meshes. Here we establish the parameter-uniform convergence of a fitted operator method on…
We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we…
The dynamics of a linear dynamical system over a finite field can be described by using the elementary divisors of the corresponding matrix. It is natural to extend the investigation to a general finite commutative ring. In a previous…
The present paper explores substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit, and render an effective method to compute…
Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear…
Linear systems are the bedrock of virtually all numerical computation. Machine learning poses specific challenges for the solution of such systems due to their scale, characteristic structure, stochasticity and the central role of…