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The aim of this paper is to provide mathematical foundations of a graph transformation language, called UnCAL, using categorical semantics of type theory and fixed points. About twenty years ago, Buneman et al. developed a graph database…
Recently, Retrieval-Augmented Generation (RAG) has achieved remarkable success in addressing the challenges of Large Language Models (LLMs) without necessitating retraining. By referencing an external knowledge base, RAG refines LLM…
Geometric modeling by constraints, whose applications are of interest to communities from various fields such as mechanical engineering, computer aided design, symbolic computation or molecular chemistry, is now integrated into standard…
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
We introduce QuAlg, an open-source symbolic algebra package for quantum information. QuAlg supports working with qubit-states, qudit-states, fock-states and even wave-functions in infinite dimenional Hilbert spaces. States can have…
Many statistical models are algebraic in that they are defined in terms of polynomial constraints, or in terms of polynomial or rational parametrizations. The parameter spaces of such models are typically semi-algebraic subsets of the…
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…
String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of…
We investigate a novel class of quantum error correcting codes to correct errors on both qubits and higher-state quantum systems represented as qudits. These codes arise from an original graph-theoretic representation of sets of quantum…
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…
This article is an interdisciplinary review and an on-going progress report over the last few years made by myself and collaborators in certain fundamental subjects on two major theoretic branches in mathematics and theoretical physics:…
Enumerative algebraic geometry deals with problems of counting geometric objects defined algebraically, An important class of enumerative problems is that of counting curves: given a class of curves in some projective variety defined by…
Diagrammatic techniques for reasoning about monoidal categories provide an intuitive understanding of the symmetries and connections of interacting computational processes. In the context of categorical quantum mechanics, Coecke and…
In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all $\mathbb F_{q^2}$-rational points of the curve is determined; the Feng-Rao designed minimum distance is…
Graphical languages offer intuitive and rigorous formalisms for quantum physics. They can be used to simplify expressions, derive equalities, and do computations. Yet in order to replace conventional formalisms, rigour alone is not…
Several constructive homological methods based on noncommutative Gr\"obner bases are known to compute free resolutions of associative algebras. In particular, these methods relate the Koszul property for an associative algebra to the…
Differential forms is a highly geometric formalism for physics used from field theories to General Relativity (GR) which has been a great upgrade over vector calculus with the advantages of being coordinate-free and carrying a high degree…
Visual commonsense reasoning task aims at leading the research field into solving cognition-level reasoning with the ability of predicting correct answers and meanwhile providing convincing reasoning paths, resulting in three sub-tasks…
Semantic parsing is the task of producing a structured meaning representation for natural language utterances or questions. Recent research has pointed out that the commonly-used sequence-to-sequence (seq2seq) semantic parsers struggle to…
Geographic Question Answering (GeoQA) addresses natural language queries in geographic domains to fulfill complex user demands and improve information retrieval efficiency. Traditional QA systems, however, suffer from limited comprehension,…