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We show from a categorical point of view that probability measures on certain measurable or topological spaces arise canonically as the extension of probability distributions on countable sets. We do this by constructing probability monads…
Motivated by recent interest concerning "puncture repair" in the conformal geometry of compact Riemannian manifolds, a brief exposition on generalisation to the setting of quasiconformal mappings on certain metric measure spaces is…
We demonstrate that statistics for several types of set partitions are described by generating functions which appear in the theory of integrable equations.
We introduce combinatorial types of arrangements of convex bodies, extending order types of point sets to arrangements of convex bodies, and study their realization spaces. Our main results witness a trade-off between the combinatorial…
This paper presents two general criteria to determine spaceability results in the complements of unions of subspaces. The first criterion applies to countable unions of subspaces under specific conditions and is closely related to the…
Measurements are shown to be processes designed to return figures: they are effective. This effectivity allows for a formalization as Turing machines, which can be described employing computation theory. Inspired in the halting problem we…
The completeness properties of spaces of immersed curves equipped with reparametrization-invariant Riemannian metrics have recently been the subject of active research. This thesis studies the metric completion of spaces of immersed open…
A test space is the set of outcome-sets associated with a collection of experiments. This notion provides a simple mathematical framework for the study of probabilistic theories -- notably, quantum mechanics -- in which one is faced with…
We show that for a large class of marked point processes there exists a random measure m with the predictable representation property such that iterated integrals with respect to m span the space of square integrable random variables.
We show if a metric measure space admits a differentiable structure then porous sets have measure zero and hence the measure is pointwise doubling. We then give a construction to show if we only require an approximate differentiable…
We give an exposition of Natural Topology (NToP), which highlights its advantages for exact computation. The NToP-definition of the real numbers (and continuous real functions) matches recent expert recommendations for exact real…
We observe that the category of topological space, uniform spaces, and simplicial sets are all, in a natural way, full subcategories of the same larger category, namely the simplicial category of filters; this is, moreover, implicit in the…
Computational models pervade all branches of the exact sciences and have in recent times also started to prove to be of immense utility in some of the traditionally 'soft' sciences like ecology, sociology and politics. This volume is a…
In this paper, we study metric completions of triangulated categories in a representation-theoretic context. We provide a concrete description of completions of bounded derived categories of hereditary finite dimensional algebras of finite…
We continue our study of the mapping ideal of operator $p$-compact maps, previously introduced by the authors. Our approach embraces a more geometric perspective, delving into the interplay between operator $p$-compact mappings and matrix…
In this paper the concept of a partial cone metric space is investigated, some continuity type theorems, and fixed point theorems of contractive mappings in this generalized setting are proved as well as some theorems related to topological…
Learning Spaces are certain set systems that are applied in the mathematical modeling of education. We propose a suitable compression (without loss of information) of such set systems to facilitate their logical and statistical analysis.…
Let $T$ be a compact, metrisable and strongly countable-dimensional topological space. Let $\mathcal{M}^T$ be the set of all metrics $d$ on $T$ compatible with its topology, and equip $\mathcal{M}^T$ with the topology of uniform…
We recall the notion of abstract bornology, and connect it with topological spaces and size functions. As a generalization of measures of non-compactness, we show how every size function can be mapped to a maxitive measure.
We decompose the Hilbert space of wave functions into two subspaces, and assign to a given observable two effective representatives that act in the model space. The first serves to determine some of the eigenvalues of the full observable,…