A complete characterization of irreducible cyclic orbit codes and their Plücker embedding

Constant dimension codes are subsets of the finite Grassmann variety. The study of these codes is a central topic in random linear network coding theory. Orbit codes represent a subclass of constant dimension codes. They are defined as orbits of a subgroup of the general linear group on the Grassmannian. This paper gives a complete characterization of orbit codes that are generated by an irreducible cyclic group, i.e. a group having one generator that has no non-trivial invariant subspace. We show how some of the basic properties of these codes, the cardinality and the minimum distance, can be derived using the isomorphism of the vector space and the extension field. Furthermore, we investigate the Plücker embedding of these codes and show how the orbit structure is preserved in the embeddin

On Generalized Galois Cyclic Orbit Flag Codes

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper, we present a new contribution to the study of such codes, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.This research was funded by Ministerio de Ciencia e Innovación (grant number PID2019-108668GB-I00) and Generalitat Valenciana y Fondo Social Europeo (Grant number ACIF/2018/196)

Geometric Analysis of Nonlinear Partial Differential Equations

This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects

On 4-Dimensional Point Groups and on Realization Spaces of Polytopes

This dissertation consists of two parts. We highlight the main results from each part. Part I. 4-Dimensional Point Groups. (based on a joint work with Günter Rote.) We propose the following classification for the finite groups of orthogonal transformations in 4-space, the so-called 4-dimensional point groups. Theorem A. The 4-dimensional point groups can be classified into * 25 polyhedral groups (Table 5.1), * 21 axial groups (7 pyramidal groups, 7 prismatic groups, and 7 hybrid groups, Table 6.3), * 22 one-parameter families of tubical groups (11 left tubical groups and 11 right tubical groups, Table 3.1), and * 25 infinite families of toroidal groups (2 three-parameter families, 19 two-parameter families, and 4 one-parameter families, Table 4.3.) In contrast to earlier classifications of these groups (notably by Du Val in 1962 and by Conway and Smith in 2003), see Section 1.7), we emphasize a geometric viewpoint, trying to visualize and understand actions of these groups. Besides, we correct some omissions, duplications, and mistakes in these classifications. The 25 polyhedral groups (Chapter 5) are related to the regular polytopes. The symmetries of the regular polytopes are well understood, because they are generated by reflections, and the classification of such groups as Coxeter groups is classic. We will deal with these groups only briefly, dwelling a little on just a few groups that come in enantiomorphic pairs (i.e., groups that are not equal to their own mirror.) The 21 axial groups (Chapter 6) are those that keep one axis fixed. Thus, they essentially operate in the three dimensions perpendicular to this axis (possibly combined with a flip of the axis), and they are easy to handle, based on the well-known classification of the three-dimensional point groups. The tubical groups (Chapter 3) are characterized as those that have (exactly) one Hopf bundle invariant. They come in left and right versions (which are mirrors of each other) depending on the Hopf bundle they keep invariant. They are so named because they arise with a decomposition of the 3-sphere into tube-like structures (discrete Hopf fibrations). The toroidal groups (Chapter 4) are characterized as having an invariant torus. This class of groups is where our main contribution in terms of the completeness of the classification lies. We propose a new, geometric, classification of these groups. Essentially, it boils down to classifying the isometry groups of the two-dimensional square flat torus. We emphasize that, regarding the completeness of the classification, in particular concerning the polyhedral and tubical groups, we rely on the classic approach (see Section 1.6). Only for the toroidal and axial groups, we supplant the classic approach by our geometric approach. We give a self-contained presentation of Hopf fibrations (Chapter 2). In many places in the literature, one particular Hopf map is introduced as “the Hopf map”, either in terms of four real coordinates or two complex coordinates, leading to “the Hopf fibration”. In some sense, this is justified, as all Hopf bundles are (mirror-)congruent. However, for our characterization, we require the full generality of Hopf bundles. As a tool for working with Hopf fibrations, we introduce a parameterization for great circles in S^3 , which might be useful elsewhere. Our main tool to understand tubical groups are polar orbit polytopes. (Chapter 1). In particular, we study the symmetries of a cell of the polar orbit polytope for different starting points. Part II. Realization Spaces of Polytopes (based on a joint work with Rainer Sinn and Günter M. Ziegler.) Robertson in 1988 suggested a model for the realization space of a d-dimensional polytope P, and an approach via the implicit function theorem to prove that the realization space is a smooth manifold of dimension NG(P) := d(f_0 + f_{d−1} ) - f{0,d-1} . We call NG(P) the natural guess for (the dimension of the realization space of) P. We build on Robertson's model and approach to study the realization spaces of higher-dimensional polytopes. We conclude combinatorial criteria (Sections 9.3.3 and 9.4.1) to decide if the realization space of the polytope in consideration is a smooth manifold of dimension given by the natural guess. As another application, we study the realization spaces of the second hypersimplices (Section 9.4.2). We apply these criteria on 4-polytopes with small number of vertices, and along the way, we analyze examples where Robertson’s approach fails, identifying the three smallest examples of 4-polytopes, for which the realization space is still a smooth manifold, but its dimension is not given by the natural guess (Section 9.5). Finally, we investigate the realization space of the 24-cell (Section 9.5.2). We construct families of realizations of the 24-cell, and using them we show that the realization space of the 24-cell has points where it is not a smooth manifold. This provides the first known example of a polytope whose realization space is not a smooth manifold. We conclude by showing that the dimension of the realization space of the 24-cell is at least 48 and at most 52.Diese Dissertation befasst sich mit zwei verschiedenen Themen, von denen jedes seinen eigenen Teil hat. Der erste Teil befasst sich mit 4-dimensionalen Punktgruppen. Wir schlagen eine neue Klassifizierung für diese Gruppen vor (siehe Theorem A), die im Gegensatz zu früheren Klassifizierungen eine geometrische Sichtweise betont und versucht, die Aktionen dieser Gruppen zu visualisieren und zu verstehen. Im Folgenden werden diese Gruppen kurz beschrieben. Die polyedrischen Gruppen (Kapitel 5) sind mit den regelmäßigen Polytopen verwandt. Die axialen Gruppen (Kapitel 6) sind diejenigen, die eine Achse festhalten. Die schlauchartigen Gruppen (Kapitel 3) werden als solche charakterisiert, die genau eine invariantes Hopf-Bündel haben. Sie entstehen bei einer Zerlegung der 3-Sphäre in schlauchartige Strukturen (diskrete Hopf-Faserungen). Die toroidalen Gruppen (Kapitel 4) sind dadurch gekennzeichnet, dass sie einen invarianten Torus haben. Wir schlagen eine neue, geometrische Klassifizierung dieser Gruppen vor. Im Wesentlichen läuft sie darauf hinaus, die Isometriegruppen des zweidimensionalen quadratischen flachen Torus zu klassifizieren. Nebenbei geben wir eine in sich geschlossene Darstellung der Hopf-Faserungen (Kapitel 2). Als Hilfsmittel für die Arbeit mit ihnen führen wir eine Parametrisierung für Großkreise in S 3 ein, die an anderer Stelle nützlich sein könnte. Der zweite Teil befasst sich mit Realisierungsräumen von Polytopen. Wir bauen auf Robertsons Modell und Ansatz auf, um die Realisierungsräume von Polytopen zu untersuchen. Wir stellen kombinatorische Kriterien auf (Abschnitte 9.3.3 und 9.4.1), um zu entscheiden, ob der Realisierungsraum des betrachteten Polytops eine glatte Mannigfaltigkeit der durch die “natürliche Vermutung” gegebenen Dimension ist. Als weitere Anwendung, untersuchen wir die Realisierungsräume der zweiten Hypersimplices (Abschnitt 9.4.2). Nebenbei identifizieren wir die kleinsten Beispiele von 4-Polytopen, für die dieser Ansatz versagt (Abschnitt 9.5). Schließlich untersuchen wir den Realisierungsraum der 24-Zelle (Abschnitt 9.5.2). Wir zeigen, dass es Punkte gibt, an denen sie keine glatte Mannigfaltigkeit ist. Zuletzt zeigen wir, dass seine Dimension mindestens 48 und höchstens 52 beträgt

Geometrically uniform subspace codes and a proposal to construct quantum networks

Orientador: Reginaldo Palazzo JuniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Códigos de subespaço se mostram muito úteis contra a propagação de erros em uma rede linear multicast. Em particular, a família dos códigos de órbita apresenta uma estrutura algébrica bem definida o que, possivelmente, resultará na construção de bons algoritmos de decodificação e uma forma sistemática para o cálculo dos parâmetros do código. Neste trabalho, apresentamos um estudo dos códigos de órbita vistos como códigos geometricamente uniformes. A caracterização destas duas classes segue direto da definição de códigos de órbita e, dado um particionamento geometricamente uniforme destes códigos a partir de subgrupos normais do grupo gerador, propomos uma redução sobre o número de cálculos necessários para a obtenção das distâncias mínimas de um código de órbita abeliano e de um código L-nível, além de um algoritmo de decodificação baseado nas regiões de Voronoi. No último capítulo deste trabalho, propomos uma ideia de como projetar, do ponto de vista teórico, uma possível rede capaz de transmitir e operar informações quânticas. Tais informações são representadas por estados quânticos emaranhados, onde cada ket destes estados está associado a um subespaço vetorialAbstract: Subspace codes have been very useful to solve the error propagation in a multicast linear network. In particular, the orbit codes family presents a well-defined algebraic structure, which it will probably provide constructions of good decoding algorithms and a systematic way to compute the parameters of the code. In this work, we present a study of orbit codes seen as geometrically uniform codes. The characterization of both classes is direct from the definition of orbit codes and, given a uniform geometrically partition of these orbit codes from their normal subgroups of the generator group, we propose a reduction of the computation necessary for obtaining the minimum distances of an abelian orbit code and an L-level code, in addition to a decoding algorithm based on Voronoi regions. In the last chapter, we propose a hypothetical quantum network coding for the transmission of quantum information. This network consists of maximum entangled pure quantum states such that each ket of these states is associated with a vector subspaceDoutoradoTelecomunicações e TelemáticaDoutor em Engenharia Elétrica142094/2013-7CAPESCNP

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Tropical Geometry in Singular

Das Ziel dieser Dissertation ist die Entwicklung und Implementation eines Algorithmus zur Berechnung von tropischen Varietäten über allgemeine bewertete Körper. Die Berechnung von tropischen Varietäten über Körper mit trivialer Bewertung ist ein hinreichend gelöstes Problem. Hierfür kombinieren die Autoren Bogart, Jensen, Speyer, Sturmfels und Thomas eindrucksvoll klassische Techniken der Computeralgebra mit konstruktiven Methoden der konvexer Geometrie. Haben wir allerdings einen Grundkörper mit nicht-trivialer Bewertung, wie zum Beispiel den Körper der $p$-adischen Zahlen $\mathbb{Q}_p$, dann stößt die konventionelle Gröbnerbasentheorie scheinbar an ihre Grenzen. Die zugrundeliegenden Monomordnungen sind nicht geeignet um Problemstellungen zu untersuchen, die von einer nicht-trivialen Bewertung auf den Koeffizienten abhängig sind. Dies führte zu einer Reihe von Arbeiten, welche die gängige Gröbnerbasentheorie modifizieren um die Bewertung des Grundkörpers einzubeziehen.$$\phantom{newline}$$ In dieser Arbeit präsentieren wir einen alternativen Ansatz und zeigen, wie sich die Bewertung mittels einer speziell eingeführten Variable emulieren lässt, so dass eine Modifikation der klassischen Werkzeuge nicht notwendig ist. Im Rahmen dessen wird Theorie der Standardbasen auf Potenzreihen über einen Koeffizientenring verallgemeinert. Hierbei wird besonders Wert darauf gelegt, dass alle Algorithmen bei polynomialen Eingabedaten mit ihren klassischen Pendants übereinstimmen, sodass für praktische Zwecke auf bereits etablierte Softwaresysteme zurückgegriffen werden kann. Darüber hinaus wird die Konstruktion des Gröbnerfächers sowie die Technik des Gröbnerwalks für leicht inhomogene Ideale eingeführt. Dies ist notwendig, da bei der Einführung der neuen Variable die Homogenität des Ausgangsideal gebrochen wird.$$\phantom{newline}$$ Alle Algorithmen wurden in Singular implementiert und sind als Teil der offiziellen Distribution erhältlich. Es ist die erste Implementation, welches in der Lage ist tropische Varietäten mit $p$-adischer Bewertung auszurechnen. Im Rahmen der Arbeit entstand ebenfalls ein Singular Paket für konvexe Geometrie, sowie eine Schnittstelle zu Polymake