Search results for "Mathematical structure"
showing 10 items of 17 documents
M-bornologies on L-valued Sets
2017
We develop an approach to the concept of bornology in the framework of many-valued mathematical structures. It is based on the introduced concept of an M-bornology on an L-valued set (X, E), or an LM-bornology for short; here L is an iccl-monoid, M is a completely distributive lattice and \(E: X\times X \rightarrow L\) is an L-valued equality on the set X. We develop the basics of the theory of LM-bornological spaces and initiate the study of the category of LM-bornological spaces and appropriately defined bounded “mappings” of such spaces.
General decidability theorems for infinite-state systems
2002
Over the last few years there has been an increasing research effort directed towards the automatic verification of infinite state systems. This paper is concerned with identifying general mathematical structures which can serve as sufficient conditions for achieving decidability. We present decidability results for a class of systems (called well-structured systems), which consist of a finite control part operating on an infinite data domain. The results assume that the data domain is equipped with a well-ordered and well-founded preorder such that the transition relation is "monotonic" (is a simulation) with respect to the preorder. We show that the following properties are decidable for …
A Side-by-Side Single Sex Age-Structured Human Population Dynamic Model: Exact Solution and Model Validation
2008
A side-by-side single sex age-structured population dynamic model is presented in this paper. The model consists of two coupled von Foerster-McKendrick-type quasi-linear partial differential equations, two initial conditions, and two boundary conditions. The state variables of the model are male and female population densities. The solutions of these partial differential equations provide explicit time and age dependence of the variables. The initial conditions define the male and female population densities at the initial time, while the boundary conditions compute the male and female births at zero-age by using fertility rates. The assumptions of the nontime-dependence of the death and fe…
Heat transfer in conducting and radiating bodies
1997
Abstract We introduce briefly some nonlocal models for heat transfer in conducting and radiating media. The goal is to give an idea of the general mathematical structure and related existence results for such models.
Quantum Field Theory
2018
Quantum field theory (QFT) shares many of its philosophical problems with quantum mechanics. This applies in particular to the quantum measurement process and the connected interpretive problems, to which QFT contributes hardly any new aspects, let alone solutions. The question as to how the objects described by the theory are spatially embedded was already also discussed for quantum mechanics. However, the new mathematical structure of QFT promises new answers, which renders the spatiotemporal interpretation of QFT the pivotal question. In this chapter, we sketch the mathematical characteristics of QFT and show that a particle as well as a field interpretation breaks down.
Unifying vectors and matrices of different dimensions through nonlinear embeddings
2020
Complex systems may morph between structures with different dimensionality and degrees of freedom. As a tool for their modelling, nonlinear embeddings are introduced that encompass objects with different dimensionality as a continuous parameter $\kappa \in \mathbb{R}$ is being varied, thus allowing the unification of vectors, matrices and tensors in single mathematical structures. This technique is applied to construct warped models in the passage from supergravity in 10 or 11-dimensional spacetimes to 4-dimensional ones. We also show how nonlinear embeddings can be used to connect cellular automata (CAs) to coupled map lattices (CMLs) and to nonlinear partial differential equations, derivi…
Geometric measures of quantum correlations: characterization, quantification, and comparison by distances and operations
2016
We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. Thes…
Quantum Mechanics as a Semantic Problem
2018
Physics, like all science, is grown out of a desire to understand the world. However, modern physics with its mathematical form has become increasingly removed from the world of everyday experience and visual imagination. In quantum mechanics it is impossible to visualize the reality represented by the theory. All we have is a consistent mathematical structure. Although the theory works perfectly well instrumentally, the question remains, how can the mathematics of the theory impart understanding? If we look at mathematics as a language, we are faced with the semantic problem: how does the language of mathematics acquire meaning? In an attempt to answer this question, I study Derrida’s earl…
Mathematical Issues in a Fully-Constrained Formulation of Einstein Equations
2008
Bonazzola, Gourgoulhon, Grandcl\'ement, and Novak [Phys. Rev. D {\bf 70}, 104007 (2004)] proposed a new formulation for 3+1 numerical relativity. Einstein equations result, according to that formalism, in a coupled elliptic-hyperbolic system. We have carried out a preliminary analysis of the mathematical structure of that system, in particular focusing on the equations governing the evolution for the deviation of a conformal metric from a flat fiducial one. The choice of a Dirac's gauge for the spatial coordinates guarantees the mathematical characterization of that system as a (strongly) hyperbolic system of conservation laws. In the presence of boundaries, this characterization also depen…
Statistical physics: Some basic principles of fluctuation and noise theory
1983
Abstract Models have traditionally played a significant role in statistical mechanics. In view of the complexity of the system which statistical mechanics attempt to describe, this is not at all surprising. The study of simplified models has frequently revealed the underlying mathematical structure of many physical questions and in so doing the study of models has contributed directly to a clarification of several paradoxes which beset statistical mechanics. In this paper some of the models which appear to be useful for the discussion of non-equilibrium phenomena are examined in some detail. As usual these models are extremely simplified versions of the actual situations. It is, finally, as…