Related papers: Quantitative Monoidal Algebra: Axiomatising Distan…
Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched…
An algebraic formalism for the study of a system of charged particles interacting with an external quantum field is developed. The notion of monoidal categories with duality is used for the description of composite systems and corresponding…
We define a diagrammatic monoidal category, together with a full and essentially surjective monoidal functor from this category to the category of modules over the exceptional Lie algebra of type $F_4$. In this way, we obtain a set of…
The framework of quantitative equational logic has been successfully applied to reason about algebras whose carriers are metric spaces and operations are nonexpansive. We extend this framework in two orthogonal directions: algebras endowed…
Diagrammatically speaking, grammatical calculi such as pregroups provide wires between words in order to elucidate their interactions, and this enables one to verify grammatical correctness of phrases and sentences. In this paper we also…
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…
This is mainly a brief review of some key achievements in a `hot'' area of theoretical and mathematical physics. The principal aim is to outline the basic structures underlying {\em integrable} quantum field theory models with {\em…
The quantum symmetry of a rational quantum field theory is a finite- dimensional multi-matrix algebra. Its representation category, which determines the fusion rules and braid group representations of superselection sectors, is a braided…
The machine learning community has recently put effort into quantized or low-precision arithmetics to scale large models. This paper proposes performing probabilistic inference in the quantized, discrete parameter space created by these…
When designing plans in engineering, it is often necessary to consider attributes associated to objects, e.g. the location of a robot. Our aim in this paper is to incorporate attributes into existing categorical formalisms for planning,…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
In this sequel to my previous paper, "Is String Theory in Knots?" I explore ways of constructing symmetries through an algebraic stepping process using knotted graphs. The hope is that this may lead to an algebraic formulation of string…
String theory is a quantum theory that reproduces the results of General Relativity at long distances but is completely different at short distances. Mathematically, string theory is based on a very new -- and little understood -- framework…
We present a diagrammatic approach to quantum dynamics based on the categorical algebraic structure of strongly complementary observables. We provide physical semantics to our approach in terms of quantum clocks and quantisation of time. We…
We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the…
This paper contains two new results: 1. We amend the notion of abstract basis in a dagger symmetric monoidal category, as well as its corresponding graphical representation, in order to accommodate non-self-dual dagger compact structures;…
The connection between contextuality and graph theory has led to many developments in the field. In particular, the sets of probability distributions in many contextuality scenarios can be described using well known convex sets from graph…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
A well-defined distance on the parameter space is key to evaluating estimators, ensuring consistency, and building confidence sets. While there are typically standard distances to adopt in a continuous space, this is not the case for…