Related papers: R-Matrices and Generalized Inverses
Let $G$ be a connected graph with $V(G)=\{1,\dotsc,n\}$. Then the resistance distance between any two vertices $i$ and $j$ is given by $r_{ij}:=l_{ii}^{\dag} + l_{jj}^{\dag}-2 l_{ij}^{\dag}$, where $l_{ij}^\dag$ is the $(i,j)^{\rm th}$…
In this paper, we found the Moore-Penrose generalized inverse of adjacency matrix of an undirected graph, explicitly. We proved that the matrix $R_\lambda= [r_{ij}]$ is nonsingular where $r_{ii}=\frac{1}{\lambda}+ \deg v_i$ and $r_{ij}=\mid…
A family of integrable $GL(NM)$ models is described. On the one hand it generalizes the classical spin Ruijsenaars--Schneider systems (the case $N=1$), and on the other hand it generalizes the relativistic integrable tops on $GL(N)$ Lie…
In this paper, some more properties of the generalized principal pivot transform are derived. Necessary and sufficient conditions for the equality between Moore-Penrose inverse of a generalized principal pivot transform and its…
The aim of the paper is to build a universal R-matrix for the multiparameter deformation of any reductive Lie algebra. Such deformations, formulated in the recent past by Truini and Varadarajan, have the property of universality in a…
A new spectral parameter independent R-matrix (that depends on all of the dynamical variables) is proposed for the elliptic Calogero-Moser models. Necessary and sufficient conditions for this R-matrix to exist reduce to an equality between…
Generalized inverses of tensors play increasingly important roles in computational mathematics and numerical analysis. It is appropriate to develop the theory of generalized inverses of tensors within the algebraic structure of a ring. In…
Let $G$ be a strongly connected and balanced directed graph. The Laplacian matrix of $G$ is then the matrix (not necessarily symmetric) $L:=D-A$, where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal matrix such that the row sums…
New classical integrable systems of Camassa-Holm peakon type are proposed. They realize the maximal even piecewise-D_2 generalization of the Calogero-Francoise flows, yielding periodic and pseudoperiodic trigonometric/hyperbolic potentials.…
A reflexive generalized inverse and the Moore-Penrose inverse are often confused in statistical literature but in fact they have completely different behaviour in case the population covariance matrix is not a multiple of identity. In this…
The classical $R$-matrix structure for the $n$-particle Calogero-Moser models with (type IV) elliptic potentials is investigated. We show there is no momentum independent $R$-matrix (without spectral parameter) when $n\ge4$. The assumption…
The notion of classical $r$-matrix is re-examined, and a definition suitable to differential (-difference) Lie algebras, -- where the standard definitions are shown to be deficient, -- is proposed, the notion of an ${\mathcal O}$-operator.…
In this paper we analyse Cline's matrix equation, generalized Penrose's matrix system and a matrix system for k-commutative {1}-inverses. We determine reproductive and non-reproductive general solutions of analysed matrix equation and…
A large family of "standard" coboundary Hopf algebras is investigated. The existence of a universal R-matrix is demonstrated for the case when the parameters are in general position. Special values of the parameters are characterized by the…
The deformed algebra $\cal{A(R)}$, depending upon a Yang-Baxter R- matrix, is considered. The conditions under which the algebra is associative are discussed for a general number of oscillators. Four types of solutions satisfying these…
We develop a topological approach to prove the generalized Lax conjecture using the fact that determinants of sufficiently big symmetric linear pencils are able to express the rigidly convex sets of RZ polynomials of any degree $d$.…
We study the conditions for classical r-matrices to be compatible with the generalised Chern-Simons action for 3d gravity. Compatibility means solving the classical Yang-Baxter equations with a prescribed symmetric part for each of the real…
We consider a family of classical elliptic integrable systems including (relativistic) tops and their matrix extensions of different types. These models can be obtained from the "off-shell" Lax pairs, which do not satisfy the Lax equations…
It is well known that a square complex matrix is called EP if it commutes with its Moore-Penrose inverse. In this paper, new classes of matrices which extend this concept are characterized. For that, we consider commutative equalities given…
A class of transformations of $R_q$-matrices is introduced such that the $q\to 1$ limit gives explicit nonstandard $R_{h}$-matrices. The transformation matrix is singular itself at $q\to 1$ limit. For the transformed matrix, the…