published_or_final_version / Mathematics / Master / Master of Philosophy

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Ionospheric convection dynamics is an important window for understanding the coupling of the solar wind and interplanetary magnetic field to the Earth's ionosphere and upper atmosphere. In this study, we use measurements of ionospheric convection made by the SuperDARN radars to investigate the role of interhemispheric magnetic conjugacy in magnetosphere-ionosphere coupling and study the large-scale interactions between the magnetosphere and ionosphere. SuperDARN radars cover large geographic regions in both hemispheres and have a dataset spanning more than a decade, making them ideal for such studies. We begin in chapter 2 with an analysis of the degree of interhemispheric conjugacy exhibited in a Sub-Auroral Polarization Stream (SAPS). We present simultaneous observations of a SAPS event in both hemispheres made by mid-latitude SuperDARN radars with magnetically conjugate fields-of-view. An interhemispheric comparison of the characteristics of the SAPS channel reveals that the channel was conjugate in terms of potential variations across the channel even though substantial differences in latitudinal width and electric fields were observed in the channel. In chapter 3, we use interhemispheric SuperDARN observations of high latitude ionospheric convection in the noon-dusk sector to investigate the effects of IMF By penetrating into the closed magnetic field line region. The observations support the existence of an IMF By associated interhemispheric potential difference and field-aligned current system resulting in the generation of the interhemispheric asymmetries in ionospheric convection. Four events are analyzed in this study and the strength of interhemispheric currents associated with IMF By are estimated. Moreover, the strength of the interhemispheric currents is found to depend on the magnitude of IMF By, proximity of the currents to open-closed field line boundary, ionospheric conductivity and magnetic local time. In chapter 4, we use data from the mid-latitude SuperDARN radars between Jan-2011 and Aug-2012 to compile a database of SAPS events spanning about six hours in magnetic local time. The event database is used to analyze the average spatial variations in the occurrence rate and velocities of the SAPS channel under different geomagnetic conditions. An empirical model based on Dst-index is then developed to estimate the occurrence rate of SAPS at a given latitude and magnetic local time. / Ph. D.

International Telemetering Conference Proceedings / October 22-25, 2001 / Riviera Hotel and Convention Center, Las Vegas, Nevada / Based on the fact that two topological conjugacy chaotic maps have identical dynamical behaviors, this paper proposes a method generating spreadspectrum sequences by creating chaotic maps topological conjugacy to Kent- Map, and analyses the correlation properties of the chaotic spread-spectrum sequences. The results of simulation verified the correctness of the theoretical analysis.

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We proved that the topological conjugacy relation on $T_1$, a subclass of Toeplitz subshifts, is hyperfinite, extending Kaya's result that the topological conjugate relation of Toeplitz subshifts with growing blocks is hyperfinite. A close concept about the topological conjugacy is the flip conjugacy, which has been broadly studied in terms of the topological full groups. Particularly, we provided an equivalent characterization on Toeplitz subshifts with single hole structure to be flip invariant.

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Tesis por compendio / [ES] Un problema clásico en la teoría de grupos finitos es el estudio de cómo los tamaños de las clases de conjugación influyen sobre la estructura del grupo. En las últimas décadas, numerosos investigadores han obtenido nuevos avances en esta línea. Especialmente, se han probado resultados interesantes a partir de la información proporcionada por los tamaños de clase de algún subconjunto de elementos del grupo, como los elementos de orden potencia de primo, elementos p-regulares, etc. Además, ciertos subconjuntos de elementos definidos a través de la tabla de caracteres del grupo están siendo investigados recientemente, como los elementos anuladores y los elementos reales. Por otra parte, en los últimos años, el estudio de grupos factorizados como producto de subgrupos ha sido objeto de creciente interés. En particular, diversos autores han analizado la estructura de grupos factorizados en los que diferentes familias de subgrupos de los factores satisfacen ciertas condiciones de permutabilidad. En esta tesis pretendemos conjugar ambas perspectivas de actualidad en la teoría de grupos de manera novedosa. Así, en este contexto de literatura escasa, el objetivo es obtener nuevas contribuciones acerca de la estructura global de un grupo factorizado a partir de ciertas propiedades aritméticas de los tamaños de las clases de algunos elementos de sus factores. Estudiamos productos de dos subgrupos, eventualmente mutuamente permutables, donde los elementos (p-regulares) de orden potencia de primo de los factores tienen tamaños de clase libres de cuadrados. Analizamos el caso de tamaños de clase potencias de primos para grupos factorizados arbitrarios, evitando el uso de condiciones de permutabilidad entre los factores involucrados. El concepto de una core-factorización de un grupo, que extiende en particular a los productos mutuamente permutables, es introducido por primera vez en esta tesis y ha resultado crucial dentro de este contexto. Esta noción surge precisamente cuando consideramos las anteriores propiedades aritméticas para los tamaños de clase de elementos anuladores, interrelacionando novedosamente la teoría de caracteres con la investigación en grupos factorizados. Finalmente, estudiamos grupos que poseen una core-factorización cuyos tamaños de clase de pi-elementos (de orden potencia de primo) son pi-números o pi'-números. / [CA] Un problema clàssic dins de la teoria de grups finits és l'estudi de com els tamanys de les classes de conjugació influeixen sobre l'estructura del grup. En les últimes dècades, nombrosos investigadors han obtingut nous avanços en aquesta línia. Especialment, s'han provat resultats interessants a partir de la informació proporcionada pels tamanys de classe d'algun subconjunt d'elements del grup, com els elements d'ordre potència de primer, elements p-regulars, etc. A més, certs subconjunts d'elements definits a través de la taula de caràcters del grup estan sent investigats recentment, com els elements anul·ladors i els elements reals. D'altra banda, en els últims anys, l'estudi de grups factoritzats com a producte de subgrups ha sigut objecte de creixent interés. En particular, diversos autors han analitzat l'estructura de grups factoritzats en els quals diferents famílies de subgrups dels factors satisfan certes condicions de permutabilitat. En aquesta tesi pretenem conjugar ambdues perspectives d'actualitat en la teoria de grups de manera innovadora. En aquest context de literatura escassa, l'objectiu és obtenir noves contribucions sobre l'estructura global d'un grup factoritzat a partir de certes propietats aritmètiques dels tamanys de les classes d'alguns elements dels seus factors. Estudiem productes de dos subgrups, eventualment mútuament permutables, on els elements (p-regulars) d'ordre potència de primer dels factors tenen tamany de classe llibre de quadrats. Analitzem el cas de tamanys de classe potències de primers per a grups factoritzats arbitraris, evitant l'ús de condicions de permutabilitat entre els factors involucrats. El concepte d'una core-factorització d'un grup, que estén particularment als productes mútuament permutables, és introduït per primera vegada en aquesta tesi i ha resultat determinant dins d'aquest context. Aquesta noció sorgeix precisament quan considerem les propietats aritmètiques anteriors per als tamanys de classe d'elements anul·ladors, interrelacionant innovadorament la teoria de caràcters amb la investigació en grups factoritzats. Finalment, estudiem grups els quals posseeixen una core-factorització on els tamanys de classe dels pi-elements (d'ordre potència de primer) són pi-números o pi'-números. / [EN] The influence of the conjugacy class sizes on the structure of a group has been a widely investigated problem within finite group theory. In the last decades, several researchers have obtained new progress in this direction. Specially, some relevant information is provided by the class sizes of certain subsets of elements of the group, as prime power order elements, p-regular elements, etc. Other subsets of elements that have recently attracted interest are defined via the character table of the group, as vanishing elements and real elements. In parallel to this research on conjugacy classes, the study of groups which can be factorised as a product of two subgroups has gained increasing interest. In particular, the structure of factorised groups such that different families of subgroups of the factors satisfy certain permutability conditions has recently been analysed. In this thesis we aim to combine in a novel way both perspectives of group theory. In this framework of very scarce literature, our main purpose is to obtain new contributions about the global structure of a factorised group when the class lengths of some elements in its factors verify certain arithmetical properties. Square-free class length conditions on (p-regular) prime power order elements are considered for products of two subgroups, occasionally mutually permutable. Prime power class sizes are investigated for arbitrary products of two groups, avoiding the use of permutability conditions between the factors. The concept of a core-factorisation of a group, which particularly extends products of mutually permutable subgroups, is introduced for the first time in this dissertation, and it has been revealed determinant within this context. Precisely, this notion emerges when discussing the above arithmetical properties on the class sizes of vanishing elements, interplaying as a novelty character theory and the research on factorised groups. Core-factorisations are also exploited when analysing pi-number and pi'-number class lengths for (prime power order) pi-elements in the factors of a factorised group. / This dissertation has been elaborated at the Instituto Universitario de Matemática Pura y Aplicada de la Universitat Politècnica de València (IUMPA-UPV), thanks mainly to the financial support of the predoctoral grant ACIF/2016/170 from Generalitat Valenciana (Spain). The first academic year was supported by Proyecto Prometeo II/2013/013 from Generalitat Valenciana. The institute IUMPA has financed some travel expenses of the author’s attendances to research conferences. This research has been partially supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovación y Universidades. The mobility grant BEFPI/2018/025 from Generalitat Valenciana has allowed the author to perform a research stay of three months (March-May 2018) at the Dipartimento di Matematica e Informatica “U. Dini” (DIMAI) of Università di Firenze (Italy). The author has also been granted with a Borsa Ferran Sunyer i Balaguer for a research stay at Università di Firenze in April 2019. / Ortiz Sotomayor, VM. (2019). Conjugacy classes and factorised groups [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/125710 / Compendio