Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.Comment: 36 pages, 9 figure

Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent trigonometry

A new method to obtain trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to trigonometry, and may well provide a path to systematically study trigonometry for any homogeneous symmetric space.Comment: 51 pages, LaTe

Observability and Controllability of Nonlinear Networks: The Role of Symmetry

Observability and controllability are essential concepts to the design of predictive observer models and feedback controllers of networked systems. For example, noncontrollable mathematical models of real systems have subspaces that influence model behavior, but cannot be controlled by an input. Such subspaces can be difficult to determine in complex nonlinear networks. Since almost all of the present theory was developed for linear networks without symmetries, here we present a numerical and group representational framework, to quantify the observability and controllability of nonlinear networks with explicit symmetries that shows the connection between symmetries and nonlinear measures of observability and controllability. We numerically observe and theoretically predict that not all symmetries have the same effect on network observation and control. Our analysis shows that the presence of symmetry in a network may decrease observability and controllability, although networks containing only rotational symmetries remain controllable and observable. These results alter our view of the nature of observability and controllability in complex networks, change our understanding of structural controllability, and affect the design of mathematical models to observe and control such networks.Comment: 19 pages, 9 figure

Англійська мова загальнотехнічного спрямування: Sensors and control

Практикум для студентів другого курсу факультету авіаційних і космічних систем укладено відповідно до чинної програми з дисципліни “Іноземна мова”, яка передбачає формування у студентів комунікативної компетенції, необхідної для ефективної участі в процесі навчання та в різноманітних ситуаціях професійного спілкування. Мета практикуму – розширення та закріплення студентами загальнотехнічної термінологічної лексики та опанування граматичними конструкціями, що дозволяють розуміти автентичні тексти, пов’язані з навчанням та спеціальністю. Розроблений практикум сприяє систематизації іншомовного навчального матеріалу, формуванню навичок, розвитку та удосконаленню вмінь у читанні, говорінні, письмі, аудіюванні та перекладі.The practicum for the 2nd year students of Faculty of Airspace Systems was designed according to the recommendations of the current curriculum that implies the students’ professional language competence formation. The objective of the practicum is to widen students’ technical thesaurus, to broaden students’ knowledge of specialty, to improve reading, writing, listening and speaking skills. The practicum helps to develop a broad understanding of international socio-cultural issues in order to operate appropriately in professional and cultural environment. The practicum highlights the requirements to students’ knowledge on different types of language activity.Практикум для студентов второго курса факультета авиационных и космических систем отвечает действующей программе дисциплины «Иностранный язык», которая предполагает формирование у студентов коммуникативных компетенций, необходимых для эффективного участия в процессе обучения. Цель практикума расширить и закрепить у студентов знание терминологической лексики, овладение грамматическими конструкциями, которые позволяют понимать аутентичные тексты. Данное учебное издание поможет систематизировать обучающий материал, сформировать навыки, развить и усовершенствовать умения в чтении, говорении, письме аудировании и переводе общетехнических текстов

EUSN 2021 Book of Abstracts, Fifth European Conference on Social Networks

Book of abstract of the fifth European conference on Social Networks EUSN 202