Related papers: Conley-Morse-Forman theory for combinatorial multi…
We conduct a topological-numerical analysis of global dynamics in a discrete-time two-gene Andrecut-Kauffman model. This model describes gene expression regulation through nonlinear interactions. We use rigorous numerical methods to…
We investigate the collapsibility of systolic finite simplicial complexes of arbitrary dimension. The main tool we use in the proof is discrete Morse theory. We shall consider a convex subcomplex of the complex and project any simplex of…
The identification of dynamics from time series data is a problem of general interest. It is well established that dynamics on the level of invariant sets, the primary objects of interest in the classical theory of dynamical systems, is not…
In this paper we study travelling front solutions for nonlocal equations of the type \begin{equation} \partial_t u = N * S(u) + \nabla F(u), \qquad u(t,x) \in \mathbf{R}^d. \end{equation} Here $N *$ denotes a convolution-type operator in…
For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…
Multisymplectic geometry is an adequate formalism to geometrically describe first order classical field theories. The De Donder-Weyl equations are treated in the framework of multisymplectic geometry, solutions are identified as integral…
The modeling framework of port-Hamiltonian systems is systematically extended to constrained dynamical systems (descriptor systems, differential-algebraic equations). A new algebraically and geometrically defined system structure is…
The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a…
The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Convenient parameterizations of matrices in terms of vectors transform (certain classes of) matrix equations into covariant (hence rotation-invariant) vector equations. Certain recently introduced such parameterizations are tersely…
We study the combinatorial properties of 2-dimensional tropical complexes. In particular, we prove tropical analogues of the Hodge index theorem and Noether's formula. In addition, we introduce algebraic equivalence for divisors on tropical…
We outline the theory of reflections for prederivators, derivators and stable derivators. In order to parallel the classical theory valid for categories, we outline how reflections can be equivalently described as categories of fractions,…
This paper refined and introduced some notations (namely attractors, physical attractors, proper attractors, topologically exact and topologically mixing) within the context of relations. We establish necessary and sufficient conditions,…
We introduce combinatorial objects named matricubes that provide a generalization of the theory of matroids. As matroids provide a combinatorial axiomatization of hyperplane arrangements, matricubes provide a combinatorial axiomatization of…
Let $\xi$ be an analytic vector field in $\mathbb{R}^3$ with an isolated singularity at the origin and having only hyperbolic singular points after a reduction of singularities $\pi:M\to\mathbb{R}^3$. Assuming certain conditions to be…
The purpose of this note is to survey a methodology to solve systems of polynomial equations and inequalities. The techniques we discuss use the algebra of multivariate polynomials with coefficients over a field to create large-scale linear…
We discuss the notions of indices of vector fields and 1-forms and their generalizations to singular varieties and varieties with actions of finite groups, as well as indices of collections of vector fields and 1-forms.
We develop method that allows to derive reductions and solutions to hyperbolic systems of partial differential equations. The method is based on using functions that are constant in the direction of characteristics of the system. These…
This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined…
In this short review we first recall combinatorial or ($0-$dimensional) quantum field theory (QFT). We then give the main idea of a standard QFT method, called the intermediate field method, and we review how to apply this method to a…