Related papers: Fraction space revisited
We propose a new constructive model of the real continuum based on the notion of fractal definability. Rather than assuming the continuum as a completed uncountable totality, we view it as the cumulative result of a vast space of stratified…
Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and…
In this short note we present several infinite dimensional theorems which generalize corresponding facts from the finite dimensional differential inclusions theory.
The arithmetic of natural numbers has a natural and simple encoding within sets, and the simplest set whose structure is not that of any natural number extends this set-theoretic representation to positive and negative integers. The…
One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are…
We consider the fragment F of first order arithmetic in which quantification is restricted to ''for all but finitely many.'' We show that the integers form an F-elementary substructure of the real numbers. Consequently, the F-theory of…
Unreplicated two-level factorial designs are often used in screening experiments to determine which factors out of a large plausible set are active. A theorem regarding the generalized word count pattern is stated and proved for…
We provide an overview of the results we have attained in the last decade on the identification of quantum structures in cognition and, more specifically, in the formalization and representation of natural concepts. We firstly discuss the…
Modern order and lattice theory provides convenient mathematical tools for pattern mining, in particular for condensed irredundant representations of pattern spaces and their efficient generation. Formal Concept Analysis (FCA) offers a…
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections…
In this study, we explore the field of physics through the lens of fractional dimensionality. We propose that space is not confined to integer dimensions alone, but can also be understood as a superposition of spaces that exist between…
Fractional calculus is an effective tool in incorporating the effects of non-locality and memory into physical models. In this regard, successful applications exist rang- ing from signal processing to anomalous diffusion and quantum…
Category theory is famous for its innovative way of thinking of concepts by their descriptions, in particular by establishing universal properties. Concepts that can be characterized in a universal way receive a certain quality seal, which…
This is a proposal of an algebra which aims at distributed array processing. The focus lies on re-arranging and distributing array data, which may be multi-dimensional. The context of the work is scientific processing; thus, the core…
This article was motivated by the discovery of a potential new foundation for mainstream mathematics. The goals are to clarify the relationships between primitives, foundations, and deductive practice; to understand how to determine what…
The paper introduces the notion of the size of countable sets that preserves the Part-Whole Principle and generalizes the notion of the cardinality of finite sets. The sizes of natural numbers, integers, rational numbers, and all their…
Complex systems (CS) are ubiquitous in nature. It is argued that fractional order (FO) calculus is more suitable to describe fractal systems. Motivated by the fractal space time theory some fractional generalizations of Scrodinger and…
The concept of complexity appears in virtually all areas of knowledge. Its intuitive meaning shares similarities across fields, but disagreements between its details hinders a general definition, leading to a plethora of proposed…
This paper is partly a historical survey of various approaches and methods in the fractional calculus, partly a description of the Kipriyanov extraordinary theory in comparison with the classical one. The significance and outstanding…
Disordered and frustrated graphical systems are ubiquitous in physics, biology, and information science. For models on complete graphs or random graphs, deep understanding has been achieved through the mean-field replica and cavity methods.…