The Notions of Center, Commutator and Inner Isomorphism for Groupoids

In this paper we introduce some algebraic properties of subgroupoids and normal subgroupoids. we define other things, we define the normalizer of a wide subgroupoid H of a groupoid G and show that, as in the case of groups, this normalizer is the greatest wide subgroupoid of G in which H is normal. Furthermore, we provide definitions of the center Z(G) and the commutator G' of the groupoid G and prove that both of them are normal subgroupoids. We give the notions of inner and partial isomorphism of G and show that the groupoid I(G) given by the set of all the inner isomorphisms of G is a normal subgroupoid of A(G), the set of all the partial isomorphisms of G. Moreover, we prove that I(G) is isomorphic to the quotient groupoid G/Z(G), which extends to groupoids the corresponding well-known result for groups.

COMPLEX SIMULATION-STATISTICAL METHOD FOR SYNTHESIZING CONDITIONALLY REAL DATA ARRAYS BASED ON A STRUCTURAL-PARAMETRIC MODEL OF INTERACTION BETWEEN USERS OF INFORMATION AND TELECOMMUNICATION SERVICES

The article presents a solution to the problem of synthesizing training tasks and data arrays when organizing a computer training platform for conducting practical exercises to investigate information security incidents. Two main stages of the synthesis of the background and situa-tional components of the conditionally real data arrays based on the structural-parametric model of interaction between users of information and telecommunication services are pro-posed: the formation of static and dynamic components. 14КОМПЛЕС НЫЙИ. ТОАИЦ-МЛИМПЧ К ЕЛЙИДЗ-ВЕИЛЛИУ МЙОДО Р 1Ь35Х / 2020Вступление в силу Федерального закона «О безопасности критической информацион-ной инфраструктуры Российской Федера-ции» свидетельствует об актуальности и зна-чимости решения задачи по обнаружению, предупреждению и ликвидации последствий компьютерных атак и реагирования на ком-пьютерные инциденты. Условием качествен-ного решения данных задач является практи-ко-ориентированная профессиональная под-готовка соответствующих специалистов. Для организации и проведения практи-ческих занятий по расследованию инциден-тов информационной безопасности в сетях документальной электросвязи и сети Интер-нет как на потоках магистратуры по направ-лению «Информационная безопасность», так и на потоках специалитета «Информацион-ная безопасность телекоммуникационных систем» и «Информационно-аналитические системы безопасности» необходимо созда-ние учебного компьютерного полигона, осна-щенного современными образцами инфор-мационно-аналитических систем безопасно-сти (далее – ИАСБ), такими как IBM I21, МФИ СОФТ «Январь»2, Lampyre3, Gephi4 и др. ИАСБ – это аппаратно-программные ком-плексы для проведения поисково-аналитиче-ской работы, имеющие возможность нака-пливать и анализировать данные о взаимо-действии пользователей информационно-те-1 http://www.ibm.com/software/ products/ru/analysts-notebook2 http://www.mfisoft.ru/direction/sorm/sorm-3/3 http://www.lampyre.io4 http://www.gephi.orgлекоммуникационных сервисов (далее – ИТ-сервисов). Однако подключение учебных ИАСБ к действующему оборудованию опера-торов связи, являющемуся источником ин-формации о взаимодействии пользователей ИТ-сервисов, невозможно в соответствии со ст. 64 Федерального закона «О связи». Также отсутствует возможность применения насто-ящих массивов биллинговой информации в силу того, что такие массивы содержат персо-нальные данные пользователей, а доступ к ним ограничен законодательно. Обзор литературы позволяет сделать вы-вод об отсутствии готовых методов и алгорит-мов генерации массивов данных, отражающих взаимодействие пользователей ИТ-сервисов.Для решения данной проблемы при соз-дании учебного компьютерного полигона по расследованию инцидентов информацион-ной безопасности в учебно-научном центре «Информационная безопасность» ИРИТ-РтФ УрФУ им. первого Президента России Б.Н. Ельцина разработано программное обеспе-чение (далее – ПО). Данное ПО работает в со-ответствии с созданной структурно-параме-трической моделью взаимодействия пользо-вателей в ИТ-сервисах и позволяет синтези-ровать массивы условно-реальных данных. Файлы, содержащие сгенерированные мас-сивы, в дальнейшем загружаются в базу дан-ных ИАСБ для проведения практических за-нятий по решению поисково-аналитических задач. На текущем этапе разработки ПО спо-собно генерировать массивы условно-реаль-ных данных о взаимодействии пользователей социальных сетей и сетей мобильной связи.Static components are synthesized based on the method of forming the structure of social graphs using a composition of models for constructing complex networks with various struc-tural parameters: the Watts-Strogatz model is used for a mobile communication service, and the Barabashi-Albert model is used for a social network service. To preserve the relationship between users in various services, a method is proposed for determining the largest common part of social graphs, based on the mutual differentiation of vertices and the allocation of a partial isomorphism of the compared graphs. When generating vertex attributes, the method of searching for social groups (families) is used, based on the Bron-Kerbosch algorithm for find-ing a click of a given size in a graph.For the synthesis of dynamic components of data arrays that describe the performance of communication events, the mathematical apparatus of color Petri nets is used. An interaction event in information and telecommunication services is represented in the form of a Petri net label, which contains the necessary set of parameters, depending on the type of service. It is proposed to use the structural, event, social and temporal statistical characterist

A CHARACTERISATION FOR A GROUPOID GALOIS EXTENSION USING PARTIAL ISOMORPHISMS

Let $S|_{R}$ be a groupoid Galois extension with Galois groupoid $G$ such that $E_{g}^{G_{r(g)}}\subseteq C1_{g}$, for all $g\in G$, where $C$ is the centre of $S$, $G_{r(g)}$ is the principal group associated to $r(g)$ and $\{E_{g}\}_{g\in G}$ are the ideals of $S$. We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that $G=\dot{\bigcup }_{g\in G}\text{Isom}_{R}(E_{g^{-1}},E_{g})$ if and only if $E_{g}$ is a connected commutative algebra or $E_{g}=E_{g}^{G_{r(g)}}\oplus E_{g}^{G_{r(g)}}$, where $E_{g}^{G_{r(g)}}$ is connected, for all $g\in G$.

Representación y comprensión de la fotografia: uma epistemologia Del mundo de Cartier-Bresson

http://dx.doi.org/10.5007/1808-1711.2016v20n2p215 The article aims to conceptualize both representation and understanding in photography, an activity whose main goal consist in elucidating the process through which the photographic image is constructed on a partial isomorphism relationship, as well as in enabling to understand a meaningful message. We appeal to Nelson Goodman’s account, according to which such a construction is based on data provided by the image, on the one hand, and by viewer’s knowledge, on the other. Given that those sources give viewer a new knowledge about the world and that the inferential processes depend upon a general theory of symbols, we both show and account for the inferential procedure that raises from the photographic ‘information’ in several case-studies taken from Henry Cartier-Bresson’s work.

The affine structure of gravitational theories: Symplectic groups and geometry

We give a geometrical description of gravitational theories from the viewpoint of symmetries and affine structure. We show how gravity, considered as a gauge theory, can be consistently achieved by the nonlinear realization of the conformal-affine group in an indirect manner: due to the partial isomorphism between CA(3, 1) and the centrally extended Sp( 8), we perform a nonlinear realization of the centrally extended ( CE )Sp( 8) in its semi-simple version. In particular, starting from the bundle structure of gravity, we derive the conformal-affine Lie algebra and then, by the nonlinear realization, we define the coset field transformations, the Cartan forms and the inverse Higgs constraints. Finally, we discuss the geometrical Lagrangians where all the information on matter fields and their interactions can be contained.

Semilattices with a transitive automorphism group

If L is any semilattice, let TL denote the Munn semigroup of L, and Aut (L) the automorphism group of L.We show that every semilattice L can be isomorphically embedded as a convex subsemilattice in a semilattice L' which has a transitive automorphism group in such a way that (i) every partial isomorphism α of L can be extended to an automorphism of L', (ii) every partial isomorphism: α: eL → fL of L can be extended to a partial isomorphism αL′: eL′ → fL′ of L′ such that TL → TL′, α → αL′ embeds TL' isomorphically in TL′, (iii) every automorphism γ of L can be extended to an automorphism γL′ of L′ such that Aut (L) → Aut (L′), γ → γL embeds Aut (L) isomorphically in Aut (L′).

Categoricity regained

One of the earliest goals of modern logic was to characterize familiar mathematical structures up to isomorphism by means of properties expressed in a first order language. This hope was dashed by Skolem's discovery (cf. [6]) of a nonstandard model of first order arithmetic. A theory T such that any two of its models are isomorphic is called categorical. It is well known that if T has any infinite models then T is not categorical. We shall regain categoricity by(i) enlarging our language so as to allow expressions of infinite length, and(ii) enlarging our class of isomorphisms so as to allow isomorphisms existing in some Boolean valued extension of the universe of sets.Let and be mathematical structures of the same similarity type where say R is binary on A. We write if f is an isomorphism of onto , and if there is an f such that . We say that P is a partial isomorphism of onto and write if P is a nonempty set of functions such that(i) if f ∈ P then dom(f) is a substructure of , rng(/f) is a substructure of , and f is an isomorphism of its domain onto its range, and(ii) if f ∈ P, a ∈ A, and b ∈ B then there exist g,h ∈ P, both extending f such that a ∈ dom(g) and b ∈ rng(h). Write if there is a P such that .