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Linear algebra tools are used to give a new approach to the open problem of the classification of Bernstein algebras. We prove that any Bernstein algebra (A,ω) is isomorphic to a semidirect product N ⋉(⋅,Ω)k associated to a commutative algebra (N,⋅) such that (x2)2 = 0, for all x ∈ N and an idempotent endomorphism Ω = Ω2 ∈End k(N) of N satisfying two compatibility conditions. The set of types of (1 + |I|)-dimensional Bernstein algebras is parametrized by an explicitly constructed classified object. The automorphisms group of any Bernstein algebra is described as a subgroup of the canonical semidirect product of groups (N, +) ⋉ GLk(N). [ABSTRACT FROM AUTHOR]

A new platform for event-based simulation is presented in this work based on a new proposed programming language. The platform supports modelling of any system as stages. The resources reside in what the language calls pools. The language supports topology description of the system as connections between stages and pools. It provides flexibility to choose the needed distribution for every stage, the type of data passed from a stage to another and the output statistics at the end of execution. One of the capabilities of this new platform is to optimize resources needed in the system through multiple simulation iterations. Optimization is based on a previously proposed reward system that balances between three satisfaction factors: patient, owner and resource. Accordingly, new resource allocations are suggested for system enhancement. The compiler of the language takes an English-like text and converts it into a Petri net. A case study is presented to improve the main performance measures of an emergency department located in Lebanon: patient length of stay, utilization rates and queues waiting times. [ABSTRACT FROM AUTHOR]

We prove that the celebrated Itô’s theorem for groups remains valid at the level of Leibniz algebras: ifis a Leibniz algebra such that, for two abelian subalgebrasand, thenis metabelian, i.e.. A structure-type theorem for metabelian Leibniz/Lie algebras is proved. All metabelian Leibniz algebras having the derived algebra of dimensionare described, classified and their automorphisms groups are explicitly determined as subgroups of a semidirect product of groupsassociated to any vector space. [ABSTRACT FROM PUBLISHER]

Letbe a Leibniz algebra,be a vector space andbe an epimorphism of vector spaces with. The global extension problem asks for the classification of all Leibniz algebra structures that can be defined onsuch thatis a morphism of Leibniz algebras: from a geometrical viewpoint this is to give the decomposition of the groupoid of all such structures in its connected components and to indicate a point in each component. All such Leibniz algebra structures onare classified by a global cohomological objectwhich is explicitly constructed. It is shown thatis the coproduct of all local cohomological objectswhich are classifying sets for all extensions ofby all Leibniz algebra structureson. The second cohomology groupof Loday and Pirashvili appears as the most elementary piece among all components of. Several examples are worked out in details for co-flag Leibniz algebras over, i.e. Leibniz algebrasthat have a finite chain of epimorphisms of Leibniz algebrassuch that, for all. Metabelian Leibniz algebras are introduced, described and classified using pure linear algebra tools. [ABSTRACT FROM AUTHOR]

For a given Hopf algebra A we classify all Hopf algebras E that are coalgebra split exten-sions of A by H4, where H4 is the Sweedler's four-dimensional Hopf algebra. Equiva-lently, we classify all crossed products of Hopf algebras A# H4 by computing explicitly two classifying objects: the cohomological "group" H²(H4,A) and CRP(H4, A) := the set of types of isomorphisms of all crossed products A# H4. All crossed products A#H4 are described by generators and relations and classified: they are parameterized by the set ZP(A) of all central primitive elements of A. Several examples are worked out in detail: in particular, over a field of characteristic p ≥ 3 an infinite family of non-isomorphic Hopf algebras of dimension 4p is constructed. The groups of automorphisms of these Hopf algebras are also described. [ABSTRACT FROM AUTHOR]

Let A be an algebra over a commutative ring k. We compute the center of the category of A-bimodules. There are six isomorphic descriptions: the center equals the weak center, and can be described as categories of noncommutative descent data, comodules over the Sweedler canonical A-coring, Yetter-Drinfeld type modules or modules with a flat connection from noncommutative differential geometry. All six isomorphic categories are braided monoidal categories: in particular, the category of comodules over the Sweedler canonical A-coring A ⊗ A is braided monoidal. We provide several applications: for instance, if A is finitely generated projective over k then the category of left k(A)-modules is braided monoidal and we give an explicit description of the braiding in terms of the finite dual basis of A. As another application, new families of solutions for the quantum Yang-Baxter equation are constructed: they are canonical maps Ω associated to any right comodule over the Sweedler canonical coring A ⊗ A and satisfy the condition Ω3 = Ω. Explicit examples are provided. [ABSTRACT FROM AUTHOR]

We address four problems regarding representable functors and give answers to them for functors connecting the category of comodules over a coring to the category of modules over a ring. A functor's property of being Frobenius is restated as a particular case of its representability by imposing the predefinition of the object of representability. Let R, S be two rings, C an R-coring and the category of left C-comodules. The category of all representable functors is shown to be equivalent to the opposite of the category . For U an (S, R)-bimodule we give necessary and sufficient conditions for the induction functor to be: a representable functor, an equivalence of categories, a separable or a Frobenius functor. The latter results generalize and unify the classical theorems of Morita for categories of modules over rings and the more recent theorems obtained by Brezinski, Caenepeel et al. for categories of comodules over corings. [ABSTRACT FROM PUBLISHER]

We survey the extensions of a group by a group using crossed products instead of exact sequences of groups. The approach has various advantages, one of them being that the crossed product is a universal object. Several new applications are given, a general Schreier type theorem is proved and a few open problems are posed. [ABSTRACT FROM AUTHOR]

A pair of adjoint functors (F, G) is called a Frobenius pair of the second type if G is a left adjoint of βFα for some category equivalences α and β. Frobenius ring extensions of the second kind provide examples of Frobenius pairs of the second kind. We study Frobenius pairs of the second kind between categories of modules, comodules, and comodules over a coring. We recover the result that a finitely generated projective Hopf algebra over a commutative ring is always a Frobenius extension of the second kind (cf.[27, 17]), and prove that the integral spaces of the Hopf algebra and its dual are isomorphic. [ABSTRACT FROM AUTHOR]