Related papers: A computational framework for connection matrix th…
We present two hypermatrix formulations of the Cayley Hamilton theorem. One of the proposed formulation naturally extends to hypermatrices the combinatorial interpretations of the classical Cayley Hamilton theorem. We conclude by discussing…
A linking theory explains how verbs' semantic arguments are mapped to their syntactic arguments---the inverse of the Semantic Role Labeling task from the shallow semantic parsing literature. In this paper, we develop the Computational…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
Abstraction plays a key role in concept learning and knowledge discovery; this paper is concerned with computational abstraction. In particular, we study the nature of abstraction through a group-theoretic approach, formalizing it as…
We develop the relative Morse index theory for linear self-adjoint operator equation without compactness assumption and give the relationship between the index defined in [44] and [45]. Then we generalize the method of saddle point…
A combinatorial framework for dynamical systems provides an avenue for connecting classical dynamics with data-oriented, algorithmic methods. Combinatorial vector fields introduced by Forman and their recent generalization to multivector…
The aim of the paper is to build a connection between two approaches towards categorical language theory: the coalgebraic and algebraic language theory for monads. For a pair of monads modelling the branching and the linear type we defined…
On a smooth manifold, we associate to any closed differential form a mapping cone complex. The cohomology of this mapping cone complex can vary with the de Rham cohomology class of the closed form. We present a novel Morse theoretical…
We propose a novel approach to iterated sparse matrix dense matrix multiplication, a fundamental computational kernel in scientific computing and graph neural network training. In cases where matrix sizes exceed the memory of a single…
Inspired by Morse theory, we introduce a topological stack Broken, which we refer to as the moduli stack of broken lines. We show that Broken can be presented as a Lie groupoid with corners and provide a combinatorial description of sheaves…
This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…
Topological data analysis can reveal higher-order structure beyond pairwise connections between vertices in complex networks. We present a new method based on discrete Morse theory to study topological properties of unweighted and…
We present module theory and linear maps as a powerful generalised and computationally efficient framework for the relational data model, which underpins today's relational database systems. Based on universal constructions of modules we…
Retrieving graphs from a large corpus, that contain a subgraph isomorphic to a given query graph, is a core operation in many real-world applications. While recent multi-vector graph representations and scores based on set alignment and…
We describe correlations functions of topological quantum mechanics (TQM) in terms of Morse theory. We review the basics of topological field theories and discuss geometric and algebraic interpretations of TQM. We prove that correlators in…
We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…
We study index theory on homogeneous spaces associated to an almost connected Lie group in terms of the topological aspect and the analytic aspect. On the topological aspect, we obtain a topological formula as a result of the Riemann-Roch…
The study of complex systems has captured widespread attention in recent years, emphasizing the exploration of interactions and emergent properties among system units. Network analysis based on graph theory has emerged as a powerful…
This paper investigates some issues arising in categorical models of reversible logic and computation. Our claim is that the structural (coherence) isomorphisms of these categorical models, although generally overlooked, have decidedly…
Network topology matrices are algebraic representations of graphs that are widely used in modeling and analysis of various applications including electrical circuits, communication networks and transportation systems. In this paper, we…