Related papers: Mathematical structure of unit systems
We show that mixtures comprised of multicomponent systems typically are much more structurally complex than the sum of their parts; sometimes, infinitely more complex. We contrast this with the more familiar notion of statistical mixtures,…
The aim of the present paper is to investigate the half-spaces in the convexity structure of all quasiorders on a given set and to use them in an alternative approach to classical order dimension. The main result states that linear orders…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
We clarified the connection between measurements and partitions, and discussed the meaning of semiotics for measurements based on functions. The terms of property and relation quantity were defined by our understanding of partitions and…
In consciousness science, several promising approaches have been developed for how to represent conscious experience in terms of mathematical spaces and structures. What is missing, however, is an explicit definition of what a 'mathematical…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
We show that the theory of the partial order of computably enumerable equivalence relations (ceers) under computable reduction is 1-equivalent to true arithmetic. We show the same result for the structure comprised of the dark ceers and the…
Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason…
In this paper, we analyze and compare three of the many algebraic structures that have been used for modeling dependent type theories: categories with families, split type-categories, and representable maps of presheaves. We study these in…
Continuous first-order logic is used to apply model-theoretic analysis to analytic structures (e.g. Hilbert spaces, Banach spaces, probability spaces, etc.). Classical computable model theory is used to examine the algorithmic structure of…
In this paper, we introduce a mathematical structure called Euclidean Universe. This structure provides a basic framework for Non-Archimedean Mathematics and in particular for Nonstandard Analysis.
In the former article "Formal mathematical systems including a structural induction principle" we have presented a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the…
The physical aspect of a general perturbation theory is explored. Its role as a physical principle for understanding the interaction among matter with different levels of hierarchy is appreciated. It is shown that the generic perturbation…
This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the…
We introduce the notion of order projections using the order unit property of a positive element in an order unit space and characterize them in terms of (geometric) orthogonality. We describe order projections of the order unit space…
In scientific and engineering applications, physical quantities embodied as units of measurement (UoM) are frequently used. The loss of the Mars climate orbiter, attributed to a confusion between the metric and imperial unit systems,…
Basic concepts of quantum integrable systems (QIS) are presented stressing on the unifying structures underlying such diverse models. Variety of ultralocal and nonultralocal models is shown to be described by a few basic relations defining…
Higher-order interactions provide a nuanced understanding of the relational structure of complex systems beyond traditional pairwise interactions. However, higher-order network analyses also incur more cumbersome interpretations and greater…
The causal structure of a unitary transformation is the set of relations of possible influence between any input subsystem and any output subsystem. We study whether such causal structure can be understood in terms of compositional…
In this letter we briefly investigate the mathematical structure of space-time in the framework of discretization. It is shown that the discreteness of space-time may result in a new mechanical system which differ from the usual quantum…