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Decoherence is the phenomenon of non-unitary dynamics that arises as a consequence of coupling between a system and its environment. It has important harmful implications for quantum information processing, and various solutions to the…
The tridiagonal representation approach is an algebraic method for solving second order differential wave equations. Using this approach in the solution of quantum mechanical problems, we encounter two new classes of orthogonal polynomials…
The tetrahedron equation arises as a generalization of the famous Yang--Baxter equation to the 2+1-dimensional quantum field theory and the 3-dimensional statistical mechanics. Very little is still known about its solutions. Here a…
We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. For this, we introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group…
The aim of this note is: (a) to propose a generalization of tetrahedron equations from \cite{S} and of their solutions. Due to appearance of a larger number of parameters the $R$-matrices from \cite{S} will be replaced by…
This note considers linear recurrences (also called linear difference equations) in unknowns indexed by the integers. We characterize a unique \emph{reduced} linear recurrence with the same solutions as a given linear recurrence, and…
In this paper, we present formula solutions of a family of difference equations of higher order. We discuss the periodic nature of the solutions and we investigate the stability character of the equilibrium points. We utilize Lie symmetry…
We consider quivers/skew-symmetric matrices under the action of mutation (in the cluster algebra sense). We classify those which are isomorphic to their own mutation via a cycle permuting all the vertices, and give families of quivers which…
In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…
We address combinatorial problems that can be formulated as minimization of a partially separable function of discrete variables (energy minimization in graphical models, weighted constraint satisfaction, pseudo-Boolean optimization, 0-1…
A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$. Using Voronov's derived bracket and a recent work of Lazarev et al., we construct an…
This paper studies topological definitions of chain recurrence and shadowing for continuous endomorphisms of topological groups generalizing the relevant concepts for metric spaces. It is proved that in this case the sets of chain recurrent…
We describe the notion of stability of coherent systems as a framework to deal with redundancy. We define stable coherent systems and show how this notion can help the design of reliable systems. We demonstrate that the reliability of…
An orthoset (also called an orthogonality space) is a set $X$ equipped with a symmetric and irreflexive binary relation $\perp$, called the orthogonality relation. In quantum physics, orthosets play a central role. In fact, a Hilbert space…
It was recently conjectured that every component of a discrete-time rational dynamical system is a solution to an algebraic difference equation that is linear in its highest-shift term (a quasi-linear equation). We prove that the conjecture…
The Lie-Hamilton approach for $t$-dependent Hamiltonians is extended to cover the so-called nonlinear Lie-Hamilton systems, which are no longer related to a linear $t$-dependent combination of a basis of a finite-dimensional Lie algebra of…
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its…
We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological…
In the same way decomposition spaces, also known as unital 2-Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they…
We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or…