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We generalize the solution of linear recurrence relations from fields to central division algebras, adapting the standard tools of companion matrices and characteristic polynomials to the non-commutative setting. We then solve linear…
It is widely known that the recursion operator is a very important component of integrability. It allows one to describe in a compact form both hierarchies of the generalized symmetries and infinite series of the local conservation laws. In…
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that…
We generalize Jacobson's notion of primitive ring to the setting of quantales. We show that every primitive ring gives rise to a primitive quantale of ideals. We then prove a density theorem for strongly primitive quantales. Furthermore, we…
We axiomatize and generalize Markov's approach to the continuity problem for Type 1 computable functions, i.e. the problem of finding sufficient conditions on a computable topological space to obtain a theorem of the form "computable…
Replication of experimental results has been a challenge faced by many scientific disciplines, including the field of machine learning. Recent work on the theory of machine learning has formalized replicability as the demand that an…
The primitive equations in a 3D infinite layer domain are considered with linearly growing initial data in the horizontal direction, which illustrates the global atmospheric rotating or straining flows. On the boundaries, Dirichlet, Neumann…
A general theorem on factorization of matrices with polynomial entries is proven and it is used to reduce polynomial Darboux matrices to linear ones. Some new examples of linear Darboux matrices are discussed.
To enable the study of open sets in computational approaches to mathematics, lots of extra data and structure on these sets is assumed. For both foundational and mathematical reasons, it is then a natural question, and the subject of this…
In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path…
Intuitionistic logic extended with decidable propositional atoms combines classical properties in its propositional part and intuitionistic properties for derivable formulas not containing propositional symbols. Sequent calculus is used as…
We define a fragment of monadic infinitary second-order logic corresponding to an abstract separation property. We use this to define the concept of a separation subclass. We use model theoretic techniques and games to show that separation…
Many semantical aspects of programming languages, such as their operational semantics and their type assignment calculi, are specified by describing appropriate proof systems. Recent research has identified two proof-theoretic features that…
We present a recursive minimal polynomial theorem for finite sequences over a commutative integral domain $D$. This theorem is relative to any element of $D$. The ingredients are: the arithmetic of Laurent polynomials over $D$, a recursive…
To approximate a simple root of an equation we construct families of iterative maps of higher order of convergence. These maps are based on model functions which can be written as an inner product. The main family of maps discussed is…
We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such…
We present here algorithms for efficient computation of linear algebra problems over finite fields.
Permutation rational functions over finite fields have attracted high interest in recent years. However, only a few of them have been exhibited. This article studies a class of permutation rational functions constructed using trace maps on…
A transitive Lie algebra g of rational vector fields on a projective manifold which do not preserve any foliation determines a rational map to an algebraic homogenous space G/H which maps g to lie(G).
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…