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We develop a linear-algebraic framework for dimensional analysis in systems with constraints, particularly when variables are numerous or related by implicit relations so that direct elimination is impractical. By expressing both…
An algorithm for embedding finite dimensional Lie algebras into Lie algebras of vector fields (and Lie superalgebras into Lie superalgebras of vector fields) is offered in a way applicable over ground fields of any characteristic. The…
The paper deals with Henselian valued field with analytic structure. Actually, we are focused on separated analytic structures, but the results remain valid for strictly convergent analytic ones as well. A classical example of the latter is…
Developments in dynamical systems theory provides new support for the discretisation of \pde{}s and other microscale systems. Here we explore the methodology applied to the gap-tooth scheme in the equation-free approach of Kevrekidis in two…
The downward closure of a word language is the set of all (not necessarily contiguous) subwords of its members. It is well-known that the downward closure of any language is regular. While the downward closure appears to be a powerful…
The normal form and zero dynamics are powerful tools useful in analysis and control of both linear and nonlinear systems. There are no simple closed form solutions to the general zero dynamics problem for nonlinear systems. A few algorithms…
Many interesting and useful symbolic computation algorithms manipulate mathematical expressions in mathematically meaningful ways. Although these algorithms are commonplace in computer algebra systems, they can be surprisingly difficult to…
We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.
We show that the theory of algebraically closed fields with multiplicative circular orders has a model companion $\mathrm{ACFO}$. Using number-theoretic results on character sums over finite fields, we show that if $\mathbb{F}$ is an…
A set of algorithms is presented for efficient numerical calculation of the time evolution of classical dynamical systems. Starting with a first approximation for solving the differential equations that has a "reversible" character, we show…
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
Much algorithmic research in NLP aims to efficiently manipulate rich formal structures. An algorithm designer typically seeks to provide guarantees about their proposed algorithm -- for example, that its running time or space complexity is…
Connections between partial dynamcial systems, a generalized notion of partial dynamical systems defined by nested sequences of partial homeomorphisms, and triangular AF algebras which admit an integer-valued cocycle are established.
We introduce an effective algorithmic method for the computation of a lower bound for uniform expansion in one-dimensional dynamics. The approach employs interval arithmetic and thus provides a rigorous numerical result (computer-assisted…
We classify fields having finitely many finite non-commutative (not necessarily central) division algebras over them. In the process, we introduce the notion of anti-closure of a field and also make comments on fields having a linear…
An accurate method to compute enclosures of Abelian integrals is developed. This allows for an accurate description of the phase portraits of planar polynomial systems that are perturbations of Hamiltonian systems. As an example, it is…
We give an exposition of Natural Topology (NToP), which highlights its advantages for exact computation. The NToP-definition of the real numbers (and continuous real functions) matches recent expert recommendations for exact real…
We give algorithms for the computation of the algebraic de Rham cohomology of open and closed algebraic sets inside projective space or other smooth complex toric varieties. The methods, which are based on Gr\"obner basis computations in…
The algebra of volume-preserving vector fields is considered. The potentials for that fields are introduced, and induced algebra of potentials is considered. It is shown, that this algebra fails to satisfy the Jacoby identity. Analogy with…
Geometric Algebra and Calculus are mathematical languages encoding fundamental geometric relations that theories of physics seem to respect. We propose criteria given which statistics of expressions in geometric algebra are computable in…