Related papers: On the Theory of Generalized Algebraic Transformat…
The increasing demand for Fourier transforms on geometric algebras has resulted in a large variety. Here we introduce one single straight forward definition of a general geometric Fourier transform covering most versions in the literature.…
Algebraic contraction is proposed to realize mappings between models Hamiltonians. This transformation contracts the algebra of the degrees of freedom underlying the Hamiltonian. The rigorous mapping between the anisotropic $XXZ$ Heisenberg…
Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum…
We survey the development and status quo of a subject best described as "generic representation theory of finite dimensional algebras", which started taking shape in the early 1980s. Let $\Lambda$ be a finite dimensional algebra over an…
Some sorts of generalized morphisms are defined from very basic mathematical objects such as sets, functions, and partial functions. A wide range of mathematical notions such as continuous functions between topological spaces, ring…
A new (in)finite dimensional algebra which is a fundamental dynamical symmetry of a large class of (continuum or lattice) quantum integrable models is introduced and studied in details. Finite dimensional representations are constructed and…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
Generalized Ising models, also known as cluster expansions, are an important tool in many areas of condensed-matter physics and materials science, as they are often used in the study of lattice thermodynamics, solid-solid phase transitions,…
This article is an expository account of the theory of twisted commutative algebras, which simply put, can be thought of as a theory for handling commutative algebras with large groups of linear symmetries. Examples include the coordinate…
Recently, an algebraic generalization of the Jordan-Wigner transformation was introduced and applied to one- and two-dimensional systems. This transformation is composed of the interactions $\eta_{i}$ that appear in the Hamiltonian ${\cal…
This paper is devoted to constructing and studying exactly solvable dynamical systems in discrete time obtained from some algebraic operations on matrices, to reductions of such systems leading to classical field theory models in…
In this review article, we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked…
This paper is mainly a semi-tutorial introduction to elementary algebraic topology and its applications to Ising-type models of statistical physics, using graphical models of linear and group codes. It contains new material on systematic…
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:…
Twisted modules over vertex algebras formalize the relations among twisted vertex operators and have applications to conformal field theory and representation theory. A recent generalization, called twisted logarithmic module, involves the…
We show that a recently introduced generalized scheme of quantum mechanics has connections to Li\'{e}nard and Levinson-Smith classes of nonlinear systems. For the Li\'{e}nard type, which has coefficients of odd and odd symmetry, we…
The main results of this paper are generalizations some classical theorems about transversals for families of finite sets to some cases of families of infinite sets.
This work intends to present a study on relations between a Lie algebra called dispersion operators algebra, linear canonical transformation and a phase space representation of quantum mechanics that we have introduced and studied in…
We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz…
Enhancing and essentially generalizing previous results on a class of (1+1)-dimensional nonlinear wave and elliptic equations, we apply several new techniques to classify admissible point transformations within this class up to the…