Related papers: Hierarchical Structure of Uncertainty
An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some…
We demonstrate that final-state uncertainty is ubiquitous in multistable systems of coupled neuronal maps, meaning that predicting whether one such system will eventually be chaotic or nonchaotic is often nearly impossible. We propose a…
In the theory of open quantum systems, divisibility of the system dynamical maps is related to memory effects in the dynamics. By decomposing the system Hilbert space as a direct sum of several Hilbert spaces, we study the relationship…
Despite the impressive success of quantum structures to model long-standing human judgement and decision puzzles, the {\it quantum cognition research programme} still faces challenges about its explanatory power. Indeed, quantum models…
Unstable particles, together with their stable decay products, constitute probability collectives which are defined as Hilbert spaces with dimension higher than one, nondecomposable in a particle basis. Their structure is considered in the…
We introduce structured decompositions, category-theoretic structures which simultaneously generalize notions from graph theory (including treewidth, layered treewidth, co-treewidth, graph decomposition width, tree independence number,…
Ranks estimated from data are uncertain and this poses a challenge in many applications. However, estimated ranks are deterministic functions of estimated parameters, so the uncertainty in the ranks must be determined by the uncertainty in…
WWe define the notion of a random metric space and prove that with probability one such a space is isometricto the Urysohn universal metric space. The main technique is the study of universal and random distance matrices; we relate the…
In this paper we explore the concept of hierarchy as a quantifiable descriptor of ordered structures, departing from the definition of three conditions to be satisfied for a hierarchical structure: {\em order}, {\em predictability} and {\em…
In this paper we explore the mathematical structure of hierarchical organization in smooth dynamical systems. We start by making precise what we mean by a level in a hierarchy, and how the higher le vels need to respect the dynamics on the…
In e-commerce system, category prediction is to automatically predict categories of given texts. Different from traditional classification where there are no relations between classes, category prediction is reckoned as a standard…
It is well-known that biological phenomena are emergent. Emergent phenomena are quite interesting and amazing. However, they are difficult to be understood. Due to this difficulty, we propose a theory to describe emergence based on a…
We present a concept of uniform encodability of theories and develop tools related to this concept. As an application we obtain general undecidability results which are uniform for large families of structures. In the way, we define…
In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are…
As large language models (LLMs) continue to evolve, understanding and quantifying the uncertainty in their predictions is critical for enhancing application credibility. However, the existing literature relevant to LLM uncertainty…
We introduce the notion of a hierarchical quandle, which is a generalisation of diquandles and multi-quandles. Using hierarchical quandle colourings, we construct a cocycle invariants for links coloured by quandles.
In this work we propose a novel concept of a hierarchical confusion matrix, opening the door for popular confusion matrix based (flat) evaluation measures from binary classification problems, while considering the peculiarities of…
The notion of rough set captures indiscernibility of elements in a set. But, in many real life situations, an information system establishes the relation between different universes. This gave the extension of rough set on single universal…
We introduce a category Prob of probability spaces whose objects are all probability spaces and arrows are corresponding to measurable functions satisfying an absolutely continuous requirement. We can consider a Prob-arrow as an evolving…
We investigate the problem of defining group or loop structures on spheres, where by ''sphere'' we mean the level set q(x) = c of a general K-valued quadratic form q, for an invertible scalar c. When K is a field and q non-degenerate, then…