Related papers: Magnitudes, Scalable Monoids and Quantity Spaces
We review the basic idea behind resource theories, where we quantify quantum resources by specifying a restricted class of operations. This divides the state space into various sets, including states which are free (because they can be…
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,…
Many systems in biology, physics, and engineering are modeled by nonlinear dynamical systems where the states are usually unknown and only a subset of the state variables can be physically measured. Can we understand the full system from…
Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck's constant \hbar>0, meaning that the classical and quantum world views must actually {\it coexist}. Traditionally, canonical…
Quantum theory's irreducible empirical core is a probability calculus. While it presupposes the events to which (and on the basis of which) it serves to assign probabilities, and therefore cannot account for their occurrence, it has to be…
Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces,…
The definitions of classical and quantum singularities in general relativity are reviewed. The occurence of quantum mechanical singularities in certain spherically symmetric and cylindrically symmetric (including infinite line…
Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their structure assigns to every directed triangle in the graph a value which may be interpreted as the area of the triangle; alternatively, as the…
The magnitude of metric spaces does not appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this…
We argue that the finiteness of quantum gravity amplitudes in fully compactified theories (at least in supersymmetric cases) leads to a bottom-up prediction for the existence of non-trivial dualities. In particular, finiteness requires the…
Plausibility measures are structures for reasoning in the face of uncertainty that generalize probabilities, unifying them with weaker structures like possibility measures and comparative probability relations. So far, the theory of…
Relational particle models are useful toy models for quantum cosmology and the problem of time in quantum general relativity. This paper shows how to extend existing work on concrete examples of relational particle models in 1-d to include…
In this work it is demonstrated that, provided a theory involves a minimal length, this theory must be free from such infinitesimal quantities as infinitely small variations in surface of the holographic screen, its volume, and entropy. The…
We introduce strong congruence spaces, which are topological spaces that provide a useful concept of dimension for monoid schemes. We study their properties and show that, given a toric monoid scheme over an algebraically closed basis, its…
The role of mathematical models in physics has for longer been well established. The issue of their proper building and use appears to be less clear. Examples in this regard from relativity and quantum mechanics are mentioned. Comments…
The central idea of this work is the concept of prespace, a hypothetical structure that is postulated to underlie the fabric of space or space-time. I consider how such a structure could relate to space and space-time, and the implications…
Quantum cosmology in the presence of a fundamental minimal length is analyzed in the context of the flat isotropic and the Taub cosmological models. Such minimal scale comes out from a generalized uncertainty principle and the quantization…
The conceptual setting of quantum mechanics is subject to an ongoing debate from its beginnings until now. The consequences of the apparent differences between quantum statistics and classical statistics range from the philosophical…
Gathering data through measurements is at the basis of every experimental science. Ideally, measurements should be repeatable and, when extracting only coarse-grained data, they should allow the experimenter to retrieve the finer details at…
A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories…