Related papers: Graphical Symplectic Algebra
Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations…
We give generators and relations for the hypergraph props of Gaussian relations and positive affine Lagrangian relations. The former extends Gaussian probabilistic processes by completely-uninformative priors, and the latter extends…
There is a bijection between odd prime dimensional qudit pure stabilizer states modulo invertible scalars and affine Lagrangian subspaces of finite dimensional symplectic $\mathbb{F}_p$-vector spaces. In the language of the stabilizer…
We propose a symmetric graph convolutional autoencoder which produces a low-dimensional latent representation from a graph. In contrast to the existing graph autoencoders with asymmetric decoder parts, the proposed autoencoder has a newly…
Recently researchers working in the LFG framework have proposed algorithms for taking advantage of the implicit context-free components of a unification grammar [Maxwell 96]. This paper clarifies the mathematical foundations of these…
Graphical (Linear) Algebra is a family of diagrammatic languages allowing to reason about different kinds of subsets of vector spaces compositionally. It has been used to model various application domains, from signal-flow graphs to Petri…
We present an algebraic framework for interacting extended quantum systems to study complex phenomena characterized by the coexistence and competition of different states of matter. We start by showing how to connect different…
Directed acyclic graphs (DAGs) are a class of graphs commonly used in practice, with examples that include electronic circuits, Bayesian networks, and neural architectures. While many effective encoders exist for DAGs, it remains…
Aspects of the algebraic structure and representation theory of the quantum affine superalgebras with symmetrizable Cartan matrices are studied. The irreducible integrable highest weight representations are classified, and shown to be…
We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the…
We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-)self-adjoint and unitary operators by means of self-ortho-gonal subspaces. By orthogonality…
We propose representation of configurational physical quantities and microscopic structures for multicomponent system on lattice, by extending a concept of generalized Ising model (GIM) to graph theory. We construct graph Laplacian (and…
We study a class of edge-coloured graphs, including the chamber systems of buildings and other geometries such as affine planes, from which we build coherent configurations (also known as non-commutative association schemes). The condition…
Lagrangian curves in 4-space entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic…
Laplacian matrices of graphs arise in large-scale computational applications such as machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits; and elliptic…
Laplacian matrices of graphs arise in large-scale computational applications such as semi-supervised machine learning; spectral clustering of images, genetic data and web pages; transportation network flows; electrical resistor circuits;…
Coloured graphical models are Gaussian statistical models determined by an undirected coloured graph. These models can be described by linear spaces of symmetric matrices. We outline a relationship between the symmetries of the graph and…
We systematise and develop a graphical approach to the investigations of quantum integrable vertex statistical models and the corresponding quantum spin chains. The graphical forms of the unitarity and various crossing relations are…
This note provides an introduction to selected topics in algebraic graph theory, including strongly regular graphs, Steiner systems, and automorphism groups. We describe constructions and properties of notable graphs such as the Petersen…
We study isometric representations of product systems of correspondences over the semigroup $\mathbb{N}^k$ which are minimal dilations of finite dimensional, fully coisometric representations. We show the existence of a unique minimal…