Related papers: PDEs and hypercomplex analytic functions
In this paper we study the representation of partial differential equations (PDEs) as abstract differential-algebraic equations (DAEs) with dissipative Hamiltonian structure (adHDAEs). We show that these systems not only arise when there…
Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $\mathbb{R}^{*}\supset\mathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions…
Partial differential equations (PDEs) are at the heart of many mathematical and scientific advances. While great progress has been made on the theory of PDEs of standard types during the last eight decades, the analysis of nonlinear PDEs of…
We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a…
We define an extension of operator-valued positive definite functions from the real or complex setting to topological algebras, and describe their associated reproducing kernel spaces. The case of entire functions is of special interest,…
The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
The paper gives some criteria for partial sums of rational number sequences to be not rational functions and to be not algebraic functions. As an application, we study partial sums of some famous rational number sequences in mathematical…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
The paper considers estimates for some sums and products of functions of prime numbers. Several assertions on this topic have been proven. We also study extremal estimates for strongly additive and strongly multiplicative arithmetic…
In this brief note, we consider p-adic unit roots or poles of L-functions of exponential sums defined over finite fields. In particular, we look at the number of unit roots or poles, and a congruence relation on the units. This raises a…
We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems.
In the article random functions in modules over the octonion algebra and Cayley-Dickson algebras are investigated. For their study transition measures with values in the octonion algebra and Cayley-Dickson algebras are used. Stochastic…
We introduce the notion of "quasi-symmetric" polynomials, which is a generalization of the notion of symmetry, and is particularly suited to the setting of polynomial rings over finite fields. The properties of this new class of functions…
We give a first-order definition of key polynomials, we show the links with previous definitions, that it is relevant to study key degrees, and to use a kind of valuations that we call partially multiplicative. We also prove or reprove…
The field of partial differential equations (PDEs) is vast in size and diversity. The basic reason for this is that essentially all fundamental laws of physics are formulated in terms of PDEs. In addition, approximations to these…
This paper deals with the solution of large classes of systems of nonlinear partial differential equations (PDEs) in spaces of generalized functions that are constructed as the completion of uniform convergence spaces. The existence result…
In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex…
We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that…
We study the degrees of selector functions related to the degrees in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such degrees can be represented as the unions of upper cones…