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Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group $A$, we construct an operator solution of the Yang-Baxter equation generalizing the solution of the Faddeev-Volkov model. Based on a specific choice of a subgroup…
We have new solutions to the Yang-Baxter equation, from which we have constructed new link invariants containing more than two arbitrary parameters. This may be regarded as a generalization of the Jones' polynomial. We have also found…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators…
We study a nonlinear elliptic problem defined in a bounded domain involving fractional powers of the Laplacian operator together with a concave-convex term. We characterize completely the range of parameters for which solutions of the…
In this paper we discuss the existence and regularity of solutions of strongly indefinite systems involving fractional elliptic operators on a smooth bounded domain $\Omega$ in $\R^n$.
In this paper we propose and analyze a new Multiscale Method for solving semi-linear elliptic problems with heterogeneous and highly variable coefficient functions. For this purpose we construct a generalized finite element basis that spans…
We consider shape optimization problems for general integral functionals of the calculus of variations that may contain a boundary term. In particular, this class includes optimization problems governed by elliptic equations with a Robin…
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It…
We introduce an algebraic multiscale method for two--dimensional problems. The method uses the generalized multiscale finite element method based on the quadrilateral nonconforming finite element spaces. Differently from the…
The study of the limiting absorption principle for elliptic equations with periodic structures is very challenging when the dimension is greater than 1. The fundamental reason for the dimensional barrier is the mismatch between directional…
Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantum Yang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vector space and $\Omega$ is regarded as a space of parameters, a linear parametric YB map is a YB…
New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found…
We consider the numerical reconstruction of the spatially dependent conductivity coefficient and the source term in elliptic partial differential equations in a two-dimensional convex polygonal domain, with the homogeneous Dirichlet…
We investigate solutions to the functional equation $f(f(x)) = e^x$, which can be interpreted as the problem of finding a half iterate of the exponential map. While no elementary solution exists, we construct and analyze non-elementary…
The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the $R$ matrix…
Many nonlinear differential equations arising from practical problems may permit nontrivial multiple solutions relevant to applications, and these multiple solutions are helpful to deeply understand these practical problems and to improve…
Solutions of the classical Yang-Baxter equation provide a systematic method to construct integrable quantum systems in an algebraic manner. A Lie algebra can be associated with any solution of the classical Yang--Baxter equation, from which…
Functional digraphs are unlabelled finite digraphs where each vertex has exactly one out-neighbor. They are isomorphic classes of finite discrete-time dynamical systems. Endowed with the direct sum and product, functional digraphs form a…
We introduce a novel energy functional for ground-state electronic-structure calculations. Its fundamental variables are the natural spin-orbitals of the implied singlet many-body wave function and their joint occupation probabilities. The…
The method of fundamental solutions (MFS) is a numerical method for solving boundary value problems involving linear partial differential equations. It is well known that it can be very effective assuming regularity of the domain and…