Related papers: Pairwise Comparisons Matrix Decomposition into App…
Our goal here is to see the space of matrices of a given size from a geometric and topological perspective, with emphasis on the families of various ranks and how they fit together. We pay special attention to the nearest orthogonal…
The central concept in the harmonic analysis of a compact group is the completeness of Peter-Weyl orthonormal basis as constructed from the matrix coefficients of a maximal set of irreducible unitary representations of the group, leading…
The KAK decomposition is a fundamental tool in Lie theory and quantum computing. Despite its widespread use, the mathematical foundations remain incomplete, particularly regarding the precise conditions for the decomposition and the…
Incomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely…
This is an expository article. We survey some fundamental trends in representation theory of symmetric groups and related objects which became apparent in the last fifteen years. The emphasis is on connections with Lie theory via…
We begin by reviewing the definition of 3-Lie algebras and the fundamental concepts of matched pairs. Subsequently, we introduce the representation theory of matched pairs and define the semidirect product. Building on this foundation, we…
In this paper we use some basic facts from the theory of (matrix) Lie groups and algebras to show that many of the classical matrix splittings used to construct stationary iterative methods and preconditioniers for Krylov subspace methods…
Let $A \subseteq E$ be a given extension of Hopf (respectively Lie) algebras. We answer the \emph{classifying complements problem} (CCP) which consists of describing and classifying all complements of $A$ in $E$. If $H$ is a given…
Matched pairs of Lie groupoids and Lie algebroids are studied. Discrete Euler-Lagrange equations are written for the matched pairs of Lie groupoids. As such, a geometric framework to analyse a discrete system by decomposing it into two…
A method for aggregation of expert estimates in small groups is proposed. The method is based on combinatorial approach to decomposition of pair-wise comparison matrices and to processing of expert data. It also uses the basic principles of…
For a perfect Lie algebra $\mathfrak{h}$ we classify all Lie algebras containing $\mathfrak{h}$ as a subalgebra of codimension $1$. The automorphism groups of such Lie algebras are fully determined as subgroups of the semidirect product…
Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\sH$ with a multivalued part $\mul A$. An operator $B$ in $\sH$ with $\ran B\perp\mul A^{**}$ is said to be an operator part of $A$ when $A=B \hplus (\{0\}\times \mul…
Convergence is a crucial issue in iterative algorithms. Damping is commonly employed to ensure the convergence of iterative algorithms. The conventional ways of damping are scalar-wise, and either heuristic or empirical. Recently, an…
In [7, Papadima and Suciu, When does the associated graded Lie algebra of an arrangement group decompose? Comment. Math. Helv. {\bf 81:4} (2006), 859--875] it is proved that the holonomy Lie algebra of an arrangement of hyperplanes through…
Pairwise comparisons between alternatives are a well-known method for measuring preferences of a decision-maker. Since these often do not exhibit consistency, a number of inconsistency indices has been introduced in order to measure the…
We aim to create the highest possible quality of treatment-control matches for categorical data in the potential outcomes framework. Matching methods are heavily used in the social sciences due to their interpretability, but most matching…
The relationship between associative composition algebras of dimensions 2 and 4 within the context of homogeneous spaces, with a particular focus on Hamiltonian quaternions, is explored. In the special case of Hamiltonian quaternions, the…
We will start from the beginning and define a matroid and its Orlik-Solomon algebra and holonomy Lie algebra, but first we give some background from topology and cohomology. A (central) hyperplane arrangement is a finite number of subspaces…
We generalize basic results relating the associated graded Lie algebra and the holonomy Lie algebra from finitely presented, commutator-relators groups to arbitrary finitely presented groups. In the process, we give an explicit formula for…
This book is mainly an exposition of the author's works and his joint works with his former students on explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic…