Related papers: Valuative lattices and spectra
The space of constructible functions form a dense subspace of the space of generalized valuations. In this note we prove a somewhat stronger property that the sequential closure, taken sufficiently many (in fact, infinitely many) times, of…
New families of fourth-order composition methods for the numerical integration of initial value problems defined by ordinary differential equations are proposed. They are designed when the problem can be separated into three parts in such a…
Implicit computational complexity, which aims at characterizing complexity classes by machine-independent means, has traditionally been based, on the one hand, on programs and deductive formalisms for free algebras, and on the other hand on…
An age-old controversy in mathematics concerns the necessity and the possibility of constructive proofs. The controversy has been rekindled by recent advances which demonstrate the feasibility of a fully constructive mathematics. This…
Taking an algebraic perspective on the basic structures of Rough Concept Analysis as the starting point, in this paper we introduce some varieties of lattices expanded with normal modal operators which can be regarded as the natural rough…
In an impressive series of papers, Krivine showed at the edge of the last decade how classical realizability provides a surprising technique to build models for classical theories. In particular, he proved that classical realizability…
Dynamical system theory is a widely used technique in the analysis of cosmological models. Within this framework, the equations describing the dynamics of a model are recast in terms of dimensionless variables, which evolve according to a…
Modeling biological processes is a highly demanding task because not all processes are fully understood. Mathematical models allow us to test hypotheses about possible mechanisms of biological processes. The mathematical mechanisms…
We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of…
Deformational structures, in many aspects generalizing standard elasticity theory, are investigated in abstract form. Within free deformational structures we define algebra of deformations, classify them by its special properties, define…
In a series of papers the present authors and their coworkers have developed a family of algebraic techniques to solve a number of problems in the theory of discrete or continuous dynamical systems and to analyze numerical integrators.…
This paper describes valuation-based systems for representing and solving discrete optimization problems. In valuation-based systems, we represent information in an optimization problem using variables, sample spaces of variables, a set of…
Deep learning techniques have been hugely successful for traditional supervised and unsupervised machine learning problems. In large part, these techniques solve continuous optimization problems. Recently however, discrete generative deep…
Differential algebraic geometry seeks to extend the results of its algebraic counterpart to objects defined by differential equations. Many notions, such as that of a projective algebraic variety, have close differential analogues but their…
In these notes we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with $\mathbb Q$-vector lattices, we present the divisible hull as a categorical…
Local fields, and fields complete with respect to a discrete valuation, are essential objects in commutative algebra, with applications to number theory and algebraic geometry. We formalize in Lean the basic theory of discretely valued…
Starting from the hypothesis that both physics, in particular space-time and the physical vacuum, and the corresponding mathematics are discrete on the Planck scale we develop a certain framework in form of a '{\it cellular network}'…
We obtain, for the first time, a modular many-valued semantics for combined logics, which is built directly from many-valued semantics for the logics being combined, by means of suitable universal operations over partial non-deterministic…
We consider a scalar-valued implicit function of many variables, and provide two closed formulae for all of its partial derivatives. One formula is based on products of partial derivatives of the defining function, the other one involves…
We apply methods of nonstandard mathematics in order to regard analytic geometry in a very different way. For example, complex spaces are seen to be the "standard part" of certain algebraic nonstandard schemes. We construct a category of…