The centre of generic algebras of small PI algebras

Verbally prime algebras are important in PI theory. They are well known over a field $K$ of characteristic zero: 0 and $K$ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K$ is the free associative algebra with free generators $T$, $E$ is the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. Moreover $M_{ab}(E)$ are certain subalgebras of $M_{a+b}(E)$, defined below. The generic algebras of these algebras have been studied extensively. Procesi gave a very tight description of the generic algebra of $M_n(K)$. The situation is rather unclear for the remaining nontrivial verbally prime algebras. In this paper we study the centre of the generic algebra of $M_{11}(E)$ in two generators. We prove that this centre is a direct sum of the field and a nilpotent ideal (of the generic algebra). We describe the centre of this algebra. As a corollary we obtain that this centre contains nonscalar elements thus we answer a question posed by Berele.Comment: 15 pages. Misprints corrected. Provisionally accepted to publication in Journal of Algebr

On the polynomial identities of the algebra M11(E)M_{11}(E)

Verbally prime algebras are important in PI theory. They were described by Kemer over a field $K$ of characteristic zero: 0 and $K$ (the trivial ones), $M_n(K)$, $M_n(E)$, $M_{ab}(E)$. Here $K$ is the free associative algebra of infinite rank, with free generators $T$, $E$ denotes the infinite dimensional Grassmann algebra over $K$, $M_n(K)$ and $M_n(E)$ are the $n\times n$ matrices over $K$ and over $E$, respectively. The algebras $M_{ab}(E)$ are subalgebras of $M_{a+b}(E)$, see their definition below. The generic (also called relatively free) algebras of these algebras have been studied extensively. Procesi described the generic algebra of $M_n(K)$ and lots of its properties. Models for the generic algebras of $M_n(E)$ and $M_{ab}(E)$ are also known but their structure remains quite unclear. In this paper we study the generic algebra of $M_{11}(E)$ in two generators, over a field of characteristic 0. In an earlier paper we proved that its centre is a direct sum of the field and a nilpotent ideal (of the generic algebra), and we gave a detailed description of this centre. Those results were obtained assuming the base field infinite and of characteristic different from 2. In this paper we study the polynomial identities satisfied by this generic algebra. We exhibit a basis of its polynomial identities. It turns out that this algebra is PI equivalent to a 5-dimensional algebra of certain upper triangular matrices. The identities of the latter algebra have been studied; these were described by Gordienko. As an application of our results we describe the subvarieties of the variety of unitary algebras generated by the generic algebra in two generators of $M_{11}(E)$. Also we describe the polynomial identities in two variables of the algebra $M_{11}(E)$.Comment: 21 page

From algebra to analysis: new proofs of theorems by Ritt and Seidenberg

Ritt's theorem of zeroes and Seidenberg's embedding theorem are classical results in differential algebra allowing to connect algebraic and model-theoretic results on nonlinear PDEs to the realm of analysis. However, the existing proofs of these results use sophisticated tools from constructive algebra (characteristic set theory) and analysis (Riquier's existence theorem). In this paper, we give new short proofs for both theorems relying only on basic facts from differential algebra and the classical Cauchy-Kovalevskaya theorem for PDEs.Comment: 13 page

Continuous integration of AUTOSAR software

Die vorliegende Masterarbeit evaluiert und optimiert ein Konzept zu Continuous Integration (CI) von AUTOSAR Software der Firma Vector Informatik GmbH. Aus der Evaluierung des Konzepts folgte, dass die manuelle Konfiguration der Runtime Environment einer AUTOSAR Software, bestehend aus den Schritten Port-Mapping und Runnable-Task-Mapping, zu Problemen bei der Umsetzung der CI-Praktiken führte. Aus diesem Grund wurde als Ziel der Arbeit eine Optimierung des Konzepts durch die Automatisierung der Konfiguration festgelegt. Dazu wurde ein Ansatz zur Automatisierung des Port-Mappings und des Runnable-Task-Mappings entwickelt, welches sowohl auf Ideen aus wissenschaftlichen Publikation als auch Industriepraktiken basiert. Darauf folgte die Implementierung des Ansatzes als eine Groovy-Applikation, um anschließend eine Evaluierung basierend auf einem aktuellen AUTOSAR Projekt durchzuführen. Die Evaluierung lieferte ein positives Ergebnis in Bezug auf das Runnable-Task-Mapping, wohingegen die Verwendung des Port-Mappings für CI nur mit Einschränkungen möglich ist.This master thesis evaluates and optimizes a Continuous Integration (CI) concept for AUTOSAR software from Vector Informatik GmbH. The evaluation of the concept has shown, that the manual configuration of the runtime environment of a AUTOSAR software, consisting of the steps of port mapping and runnable task mapping, led to problems in the implementation of CI practices. For this reason, the goal of the work was to optimize the concept by automating the configuration. For this purpose, an approach for the automation of port mapping and runnable task mapping has been developed, which is based both on ideas from scientific publications as well as on industrial practices. This was followed by the implementation of the approach as a Groovy application in order to carry out an evaluation based on a current AUTOSAR project. The evaluation yielded a positive result with respect to the runnable task mapping, whereas the use of port mapping for CI is only possible with restrictions

СОВЕТСКО-ГВИНЕЙСКИЕ ОТНОШЕНИЯ В ПЕРИОД ПЕРВОЙ РЕСПУБЛИКИ (1958-1984 ГГ.)

The article is devoted to the history of the Guinean-Soviet relations in the period of Ahmed Secou Toure's presidency. Guinea and the USSR fought for Africa's liberation from colonialism and imperialism, and the Guinean leader was an initiator of the integration processes in Africa. The author proves that the character of the bilateral relations was under strong influence of the Guinean leader's controversial personality as well as the attempts to balance between the capitalist and socialist poles in the Cold War. As a result Guinea became estranged from the Soviet Union and restored the relationships with its former parent state, France, and other Western countries.Статья посвящена истории советско-гвинейских отношений в годы президентства Ахмеда Секу Туре. Гвинея и СССР боролись за освобождение Африки от колониализма и империализма, а гвинейский лидер был инициатором интеграционных процессов на Африканском континенте. Автор доказывает, что на характер двусторонних отношений сильное влияние оказывали противоречивая личность гвинейского лидера, а также его попытки балансировать между капиталистическим и социалистическим полюсами в холодной войне. В результате Гвинея отдалилась от Советского Союза во внешней политике, восстановив отношения с бывшей метрополией - Францией и остальными странами Запада

Simultaneous similarity and triangularization of sets of 2 by 2 matrices

Let $\mathcal{A}=(A_{1},...,A_{n},...)$ be a finite or infinite sequence of $2\times2$ matrices with entries in an integral domain. We show that, except for a very special case, $\mathcal{A}$ is (simultaneously) triangularizable if and only if all pairs $(A_{j},A_{k})$ are triangularizable, for $1\leq j,k\leq\infty$. We also provide a simple numerical criterion for triangularization. Using constructive methods in invariant theory, we define a map (with the minimal number of invariants) that distinguishes simultaneous similarity classes for non-commutative sequences over a field of characteristic $\neq2$. We also describe canonical forms for sequences of $2\times2$ matrices over algebraically closed fields, and give a method for finding sequences with a given set of invariants.Comment: 22 page

Algebraic Independence in SL(3,C) Character Varieties of Free Groups

Let X be the moduli space of SL(3,C) representations of a free group of rank r. In this paper we describe maximal algebraically independent subsets of certain minimal sets of coordinate functions on X. These subsets locally parametrize the moduli space.Comment: 13 pages, v2 contains minor change

Representability and Specht problem for G-graded algebras

Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A).Comment: 37 page

Commutative 2-cocycles on Lie algebras

On Lie algebras, we study commutative 2-cocycles, i.e., symmetric bilinear forms satisfying the usual cocycle equation. We note their relationship with antiderivations and compute them for some classes of Lie algebras, including finite-dimensional semisimple, current and Kac-Moody algebras.Comment: v7: minor changes; added ancillary file with GAP cod