Related papers: A note on belief structures and S-approximation sp…

First, we prove that a random metric space can be isometrically embedded into a complete random normed module, as an application of which, it is easy to see that the notion of $d$-$\sigma$-stability introduced for a nonempty subset of a…

Dempster's rule of combination has been the most controversial part of the Dempster-Shafer (D-S) theory. In particular, Zadeh has reached a conjecture on the noncombinability of evidence from a relational model of the D-S theory. In this…

In this paper we will modify the Milnor--Thurston map, which maps a one dimensional mapping to a piece-wise linear of the same entropy, and study its properties. This will allow us to give a simple proof of monotonicity of topological…

The belief construction is a fundamental technique for transforming partially observable systems to fully observable ones while preserving the relevant semantics. It plays a central role in the analysis of partially observable systems, in…

The notion of entropy penetrates much of science. A key feature of the all-important notion of Boltzmann-Gibbs-Shannon entropy is its extensivity (additivity over independent subsystems). However, there is a need for other quantities. In…

Random events in space and time often exhibit a locally dependent structure. When the events are very rare and dependent structure is not too complicated, various studies in the literature have shown that Poisson and compound Poisson…

We report an inconsistency found in probability theory (also referred to as measure-theoretic probability). For probability measures induced by real-valued random variables, we deduce an "equality" such that one side of the "equality" is a…

I introduce a family of closeness functions between causal Lorentzian geometries of finite volume and arbitrary underlying topology. When points are randomly scattered in a Lorentzian manifold, with uniform density according to the volume…

We demonstrate a fundamental relation between the structures of physical space and of quantum theory: the set of quantum correlations in a rotational prepare-and-measure scenario can be derived from covariance alone, without assuming…

Spatial embodied intelligence requires agents to act to acquire information under partial observability. While multimodal foundation models excel at passive perception, their capacity for active, self-directed exploration remains…

The class of norm-dependent Random Matrix Ensembles is studied in the presence of an external field. The probability density in those ensembles depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact…

In this paper we propose two guiding principles that suggest a number of conjectures (some now proved) about various forms of rigidity for moduli spaces arising in algebraic geometry. Such conjectures have group-theoretic, topological and…

We propose a Bayesian evidence-based inference framework based on relative belief ratios and apply it to discriminating between one and two incoherent optical point sources using spatial-mode demultiplexing (SPADE). Unlike the Helstrom…

Confidence estimates are often "detection-like" - driven by positive evidence in favour of a decision. This empirical observation has been interpreted as showing that human metacognition is limited by biases or heuristics. Here, we show…

General acceptance of a mathematical proposition $P$ as a theorem requires convincing evidence that a proof of $P$ exists. But what constitutes "convincing evidence?" I will argue that, given the types of evidence that are currently…

While statistical learning methods have proved powerful tools for predictive modeling, the black-box nature of the models they produce can severely limit their interpretability and the ability to conduct formal inference. However, the…

Consistent belief functions represent collections of coherent or non-contradictory pieces of evidence, but most of all they are the counterparts of consistent knowledge bases in belief calculus. The use of consistent transformations cs[.]…

We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging…

Conditioning is crucial in applied science when inference involving time series is involved. Belief calculus is an effective way of handling such inference in the presence of epistemic uncertainty -- unfortunately, different approaches to…

The Collatz Conjecture's connection to dynamical systems opens it to a variety of techniques aimed at recurrence and density results. First, we turn to density results and strengthen the result of Terras through finding a strict rate of…