A classificação isomorfa dos espaços de Banach separáveis C(K) é devida a Milutin no caso em que K são não enumeráveis e a Bessaga e Pelczynski no caso em que K são enumeráveis. Neste trabalho apresentamos uma extensão vetorial dessa classificação e tiramos várias consequências, por exemplo, considerando o espaço métrico compacto infinito K e Y um espaço de Banach: &nbsp; &nbsp; 1. Sendo 1 < p < &infin; e &Gamma; um conjunto infinito, classificamos, a menos de isomorfismo, os espaços de Banach C(K, Y &oplus; lp(&Gamma;)), quando o dual de Y contém uma cópia de lq, onde 1/p+ 1/q =1. &nbsp; &nbsp; 2. Classificamos os espaços de Banach C(K, Y &oplus; l&infin;(&Gamma;)), quando a densidade de Y é estritamente menor que 2|&Gamma;|. &nbsp; &nbsp; 3. Classificamos os espaços de Banach C(K &times;(S&oplus; &beta;&Gamma;)) e C(S &oplus; (K&times; &beta;&Gamma;)), onde S é um compacto disperso de Hausdorff arbitrário e &beta;&Gamma; é a compactificação de Stone-Cech de &Gamma;. Obtemos, também, algumas leis de cancelamento para espaços de Banach da forma C(K1,X)&oplus; C(K2,Y), onde K1 e K2 são espaços compactos métricos infinitos de Hausdorff e X, Y espaços de Banach satisfazendo condições adequadas. Estabelecemos também um teorema de quase-dicotomia envolvendo os espaços C(K,X), onde X tem cotipo finito. Finalmente, apresentamos algumas majorações nas distorções de isomorfismos positivos de C([0,&omega;k]) em C([0,&omega;]) e também de C([0,&omega;]) em C([0,&omega;k]), k&isin; N, k &ge; 2. / The isomorphic classification of separable Banach spaces C(K) is due Milutin in the case when K are uncountable and to Bessaga and Pelczynski in the case when K are countable. In this work we prove a vectorial extention of this classification and provide several consequences, for example considering the infinite metric compact space K and Y a Banach space: &nbsp; &nbsp; 1. Let 1 < p < &infin; and &Gamma; a infinite set, we classify, up to an isomorphism, the Banach spaces C(K, Y &oplus; lp(&Gamma;)), in the case where the dual of Y contains no copy of lq, where 1/p+ 1/q =1. &nbsp; &nbsp; 2. We classify the Banach spaces C(K, Y &oplus; l&infin;(&Gamma;)), when the density character of Y is strictly less that 2|&Gamma;|. &nbsp; &nbsp; 3. We classify the Banach spaces C(K &times;(S&oplus; &beta;&Gamma;)) and C(S &oplus; (K&times; &beta;&Gamma;)) where S is an arbitrary dispersed compact and &beta;&Gamma; is the Stone-Cech compactification of &Gamma;. We obtain also some cancellation laws for Banach spaces in the form C(K1,X)&oplus; C(K2,Y), where K1 and K2 are metric compact Hausdorff spaces and X, Y Banach spaces satisfying appropriate conditions. We established also a quasi-dichotomy theorem envolving the C(K,X) spaces, where X is of finite cotype. Finally, we present some upper bounds of distortions of positive isomorphisms of C([0,&omega;k]) on C([0,&omega;]) and also of C([0,&omega;]) on C([0,&omega;k]), k&isin; N, k &ge; 2.

Einleitung Mit dem zunehmenden Anteil älterer Menschen an der Bevölkerung rücken altersassoziierte Erkrankungen wie die Demenz immer stärker in den Fokus. Die zerebrale Mikroangiopathie (cerebral small vessel disease, CSVD) aus der Gruppe der vaskulären Demenzen zählt nach der Alzheimerdemenz (AD) zu den häufigste Erkrankungen dieser Gruppe und beeinträchtigt in ihrer häufigsten Form (subkortikale arteriosklerotische Enzephalopathie, SAE; syn. M. Binswanger) insbesondere subkortikale Funktionen wie das Ausführen mehrteiliger zielgerichteter Handlungen (Exekutivfunktionen) und visuomotorische Geschwindigkeit. Zugrunde liegend sind zerebrale Durchblutungsstörungen, welche sich bilddiagnostisch (bevorzugt MRT) als sogenannte White Matter Lesions (Marklagerläsionen) zeigen. Nicht selten sind diese und liquorchemische Hinweise auf eine Alzheimerdemenz im Sinne einer Mischdemenz (MD, hier CSVD+) koexistent. Eine klinische Differentialdiagnose ist durch die große Variabilität der neuropsychologischen Defizite bei Patienten mit zerebraler Mikroangiopathie häufig erschwert. Zielsetzung / Methoden Ziel dieser Arbeit war es demzufolge, weitere Erkenntnisse über die neuropsychologischen Ausfallprofile zu gewinnen und neue differentialdiagnostische Optionen zu den genannten Demenzformen zu erproben. Zu diesem Zweck wurden 89 Patienten mit White Matter Lesions in der MRT rekrutiert und mit Hilfe des MMST (&lt;/≥ 28 Punkte) in neuropsychologisch unauffällige (Kontrollgruppe, n = 37 Patienten) und kognitive beeinträchtigte Patienten unterteilt. Letztere wurden durch Untersuchung auf alzheimertypische Liquorparameter (Aβ-Quotient (Aβ1-40/Aβ1-42) und/oder Aβ1-42) nochmals in die beiden Untersuchungsgruppen Gruppe 1 CSVD (n = 22 Patienten) und Gruppe 2 CSVD+ (n = 29 Patienten) unterteilt. Alle Patienten wurden einer ausführlichen neuropsychologischen Evaluation (CAMCOG, Clox-Test, Trailmaking-Test, Boston Naming Test) unterzogen und die MRT-Bilder wurden mit Hilfe der Scale for Age-Related White Matter Lesions (ARWMC) ausgewertet. Als mögliches neues differentialdiagnostisches Hilfsmittel wurde der CAMCOG-Quotient (Gedächtnis/Exekutivfunktionen) auf seine Aussagekraft untersucht. Um mögliche Einflussfaktoren auf die kognitive Leistungsfähigkeit zu identifizieren, wurden Alter, Ausbildungszeit, Ausmaß der White Matter Lesiosns (ARWMC) und Geschlecht diesbezüglich untersucht. Ergebnisse Die Untersuchungsgruppen (CSVD, CSVD+) und die Kontrollgruppe unterschieden sich signifikant im Bereich des Alters und der Ausbildungszeit. Die Gruppen CSVD und CSVD+ ließen sich lediglich im Mini-Mental-Status-Test (p = 0,044) von einander unterscheiden. In Bezug auf die Belastung mit White Matter Lesions (ARWMC) bestanden zwischen allen drei Gruppen keine signifikanten Unterschiede. Beide Untersuchungsgruppen zeigten deutliche kognitive Einschränkungen und ließen sich in jedem der durchgeführten neuropsychologischen Tests von der Kontrollgruppe unterscheiden. Jedoch zeigten sich vereinzelt auch in der Kontrollgruppe eindeutig pathologische Ergebnisse, die im MMST nicht erfasst worden waren. Zwischen den beiden Untersuchungsgruppen waren lediglich in Clox1 (p = 0,033) und dem neu entwickelten CAMCOG-Quotient (p < 0,01) Unterschiede feststellbar. Letzterer erzielte für die Detektion einer möglichen zusätzlichen Alzheimerdemenz bei einem Cutoff <1,18 eine Sensitivität von 61% und einer Spezifität von 81%. Bei der Untersuchung möglicher Einflussfaktoren auf die kognitive Leistungsfähigkeit konnte für ein höheres Alter ein negativer Effekt nachgewiesen werden. Bei ausreichender Stichprobengröße war eine längere Ausbildungszeit mit besseren Ergebnissen korrelierbar. Dahingegen waren für das Geschlecht und auch das Ausmaß der White Matter Lesions keine eindeutigen Korrelationen mit niedrigeren Testergebnissen festzustellen. Für den CAMCOG-Quotienten konnten für die Anwendung bei kognitiv eingeschränkten Patienten keine Einflussfaktoren identifiziert werden. Zusammenfassung Erwartungsgemäß waren in den beiden Untersuchungsgruppen deutliche kognitive Defizite feststellbar, wobei in fast allen durchgeführten Tests die Gruppe CSVD+ die schlechteren Ergebnisse erzielte. Statistische Signifikanz erreichte dies jedoch lediglich in einem durchgeführten Standardtest, was zeigt, dass Patienten mit reiner zerebraler Mikroangiopathie und einer Mischform bei gleichzeitiger Alzheimerdemenz anhand einer neuropsychologischen Untersuchung nur schwer von einander unterscheidbar sind. In dieser schwierigen differentialdiagnostischen Frage bietet der neu entwickelte CAMCOG-Quotient ein gutes Hilfsmittel. Von den hier untersuchten Einflussfaktoren auf die kognitiven Fähigkeiten der Patienten hatte lediglich das Alter einen eindeutig negativen Einfluss. Dieser war für den CAMCOG-Quotient nicht nachweisbar, womit dieser altersunabhängig eingesetzt werden kann. Als interessanten und klinisch relevanten Nebenaspekt zeigte sich bei dieser Arbeit, dass der MMST als Screeningtest bei Patienten mit White Matter Lesions kein ausreichendes Diagnostikum zum Ausschluss kognitiver Defizite darstellt.

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Consistently, reports in the literature have identified that a sedentary lifestyle contributes to the progression of a range of chronic degenerative diseases. The measurement of energy expenditure and physical activity pattern in children is a challenge for all professionals interested in paediatric health and from a broader perspective, the public health fraternity charged with considering longer term health consequences of physical inactivity. The primary objective of this thesis was to identify a suitable indirect and objective measurement technique for the assessment of energy expenditure and physical activity pattern in children. The ideal characteristics of such a technique are that it should be reproducible and have been validated against a criterion reference method. To achieve this goal, a series of methodological studies were undertaken (Chapters II and III). This work was essential to increase accuracy during the individualised laboratory calibration process and further minimise prediction errors when analysing data from 7 days of monitoring under free-living conditions in the second part of the study (Chapters IV and V). In the first study to verify the combined effect of body position, apparatus and distraction on children's resting metabolic rate (RMR), experiments were carried out on 14 children aged 8-12 (mean age = 10.1 years ± 1.4). Each participant underwent 2 test sessions, one week apart under three different situations: a) using mouthpiece and nose-clip (MN) or facemask (FM); b) sitting (SEAT) or lying (LY) and c) TV viewing (TV) or no TV viewing. In the first session, following 20 min rest and watching TV, the following protocol was used: LY: 20 min - stabilisation; 10 min using MN and 10 min using FM. Body position was then changed to seated: 20 min stabilisation; 10 min using FM; 10 min using MN. In the second session, FM and MN order was changed and participants did not watch TV. Data were analysed according to the eight combinations among the three studied parameters. Repeated measures ANOVA indicated statistically significant differences for &VO2 (p=0.01) and RMR (p=0.02), with TVMNSEAT showing higher values than TVFMLY. Bland-Altman analysis showed a bias for &VO2, &VCO2, RQ and RMR between TVFMLY and TVMNSEAT of -17.8±14.5 ml.min-1, -8.8±14.5 ml. min-1, 0.03±0.05 and -115.2±101.9 kcal.d-1, respectively. There were no differences in RMR measurements due to body position and apparatus when each variable was isolated. Analyses of distraction in three of four combinations indicated no difference between TV and no TV. In summary, different parameter combinations can result in increased bias and variability and thereby reported differences among children's RMR measurement. The second study dealt with treadmill adaptation and determination of self-selected (SS) walking speed. Assessment of individual and group differences in metabolic energy expenditure using oxygen uptake requires that individuals are comfortable with, and can accommodate to, the equipment being utilised. In this study, a detailed proposal for an adaptation protocol based on the SS was developed. Experiments were carried out on 27 children aged 8-12 (mean age = 10.3±1.2 yr). Results from three treadmill tests following the adaptation protocol showed similar results for step length with no significant differences among tests and lower and no statistically significant variability within- and between-days. Additionally, no statistically significant differences between SS determined over-ground and on a treadmill were verified. These results suggest that SS speed determined over-ground is reproducible on a treadmill and the 10 min familiarisation protocol based on this speed provided sufficient exposure to achieve accommodation to the treadmill. The purpose of the third study was to verify within- and between-day repeatability and variability in children's oxygen uptake ( &VO2), gross economy (GE) [ &VO2 divided by speed] and heart rate (HR) during treadmill walking based on SS. 14 children (mean age = 10.2±1.4 yr) undertook 3 testing sessions over 2 days in which four walking speeds, including SS, were tested. Within- and between-day repeatability was assessed using the Bland and Altman method and coefficients of variability (CV) were determined for each child across exercise bouts and averaged to obtain a mean group CV value for &VO2, GE and HR per speed. Repeated measures ANOVA showed no statistically significant differences in within- or between-day CV for &VO2, GE or HR at any speed. Repeatability within and between-day for &VO2, GE and HR for all speeds was verified. These results suggest that submaximal &V O2 during treadmill walking is stable and reproducible at a range of speeds based on children's SS. In the fourth study, the objective was to establish the effect of walking speed on substrate oxidation during a treadmill protocol based on SS. Experiments were carried out on 12 girls aged 8-12 (mean age = 9.9±1.4 yr). Each participant underwent 2 test sessions, one week apart. Workloads on the treadmill included 2 speeds slower than SS (1.6 [V1] and 0.8 km.h-1 [V2] slower than SS), SS (V3), and a speed 0.8 km.h-1 faster than SS (V4). Indirect calorimetry from respired gas measurements enabled total fat (FO) and carbohydrate (CHO) oxidation rates to be calculated according to the non-protein respiratory quotient (Peronnet and Massicote, 1991) and percentage of CHO and FO calculations using equations from McGilvery and Goldstein (1983). Repeated measures ANOVA followed by a Tukey Post Hoc test (p< 0.05) was used to verify differences in CHO and FO rates among speeds. Paired T-test was used to verify differences in CHO and FO rates between tests per velocity. The reliability between-day was assessed using intraclass correlation coefficient (ICC). Results showed significant differences for CHO among all speeds, as well as significant differences for FO between V1 and V2 against V3 and V4 in both tests. Analyses between trials per velocity showed no significant substrate use differences as well as acceptable reliability. At the self-selected speed (V3) there was an accentuation in FO reduction as well as an increase in CHO oxidation. The purpose of the fifth study was to determine whether there were differences in substrate oxidation between girls (G) and women (W) during a treadmill protocol based on SS. Experiments were carried out on 12 G aged 8-12 (mean age = 9.9±1.4 yr) and 12 W aged 25-38 (mean age = 32.3±3.8 yr). The treadmill protocol included 6 min workloads followed by 5 min rest periods. Workloads included 2 speeds slower than SS (1.6 (V1) and 0.8 km.h-1 (V2) slower than SS), SS (V3), and a speed 0.8 km.h-1 faster than SS (V4). Total fat and carbohydrate (CHO) oxidation rates were calculated from indirect calorimetry according to the non-protein respiratory quotient. Repeated measures ANOVA followed by a Tukey Post Hoc test was used to verify intra-test differences in CHO and fat oxidation rates among speeds. Intergroup differences were analysed using paired T-test. Fat utilisation in W achieved a plateau at a relative velocity 0.8 km.h-1 slower than SS, but for G, fat utilisation increased until SS, and then stabilised upon reaching the higher velocity. CHO oxidation curves rose abruptly above V2 for W, while for G the acute increase occurred after SS (V3). Collectively, these results indicate that as walking intensity increases G are able to meet the energy demands of the work by increasing fat oxidation together with the increased CHO oxidation up to SS. In contrast for W, increasing CHO oxidation is associated with an early decrease in fat utilisation at a velocity slower than the self-selected speed. The sixth study dealt with validation of indirect techniques for the measurement of energy expenditure in free-living conditions against the DLW technique. Experiments were carried out on 19 children aged 8-12 (mean age = 10.3±1.0 yr). To indirectly predict energy expenditure 12 different procedures were used. Only one procedure, combining activity and heart rate (AHbranched), was based on a group equation, the others were based on individualised regression. Three of the individually-based techniques were able to accurately predict energy expenditure in free-living conditions. These tecniques were HRPAnetRMR using HRnet [HR exercise minus sleep HR (SHR)] against PAnet (measured PA exercise minus measured RMR) and upper and lower body equations corrected by RMR; HRPAnet4act using the same procedure but corrected by the mean resting &VO2 for 4 resting activities [(4act) = supine watching TV, sitting watching TV, sitting playing computer games and standing], and HRPALBnet4act using only lower body activities and corrected by 4act. HRPAnetRMR was only slightly more accurate than HRPAnet4act and HRPALBnet4act, but this technique is only adjusted by RMR whereas the other two are heavily dependent on more complex laboratory calibration. Bland and Altman (1986) analyses showed no significant differences between AHbranched predicted and measured TEE using the DLW technique. A SEE of 79 kcal.d-1 and a mean difference of 72 kcal.d-1, with a 95% CI ranging from -238 to 93.9 kcal.d-1 was found. In addition, no significant differences between predicted HRPAnetRMR and measured TEE using DLW were found, showing an SEE of 99 kcal.d-1 and a mean difference of -67 kcal.d-1, and a 95% CI ranging from -276.6 to 141.9 kcal.d-1. AHbranched and HRPAnetRMR were both valid and similarly suitable for the prediction of energy expenditure in children under free-living conditions. Significant associations between DLWAEE and the after-school time window indicated that this time window as an important discretionary period representative of children physical activity. However, the duration of the after-school time windows should be more carefully considered. Accelerometer data showed a better association between the largest after-school time window (3.5 hr) and measured TEE. The final study, completed with 19 children aged 8-12 (10.3±1.0 yr) highlighted, under laboratory conditions across a range of walking and running speeds, the inadequacy of the use of the standard MET in children. This traditional approach overestimates energy expenditure with an increased difference linearly related to speed increments. Minute-by-minute analyses of 7 days of free-living monitoring showed an average overestimation of 64 minutes per day for moderate-to-vigorousphysical- activity (MVPA) using the standard MET compared with the individually measured MET. For all intensities, these differences were statistically significant (p< 0.001). The second part of this study showed a variability of 20% in the average time spent at MVPA when comparing HR I 140 bpm and HR > 50%P &VO2 (P &VO2 = the highest &VO2 observed during an exercise test to exhaustion). Results of the current study compared to observations in the literature showed that HR I 140 bpm consistently estimates lower MVPA time than HR > 50%P &VO2. When these two PA indices were compared with individual and standard MET measured minute-byminute, statistically significant differences were verified among all of them at MPA, but no differences were verified at VPA, except between individual and standard METs. However, whether each one of the PA indices used are under- or overestimating time at MVPA is still debatable due to the lack of a gold standard. Finally, each index used in this study classified different numbers of participants as achieving the PA target of 60 min.d-1. The wide variability between indices when attempting to classify children who are achieving the recommended target is cause for great concern because habitually these indices are utilised as screening tools in paediatric and public health settings and used to guide behavioural interventions.

Fetal Alcohol Syndrome (FAS) is a preventable cause of mental retardation and is the severest category within Fetal Alcohol Spectrum Disorder (FASD). As gestational alcohol exposure affects fetal cognitive functioning, children with FAS present with intellectual deficits. Unfortunately FASD prevalence rates are increasing amongst infants and school-going children. The main goal of this study was to compare the neurodevelopmental subscales of infants diagnosed with FAS, Partial FAS and non- FAS. Seventy-four infants with confirmed FAS, Partial FAS or Non- FAS diagnoses were assessed using the Griffiths Mental Developmental Scale. Development assessed at 7-12 and 17-29 months of age showed that, regardless of a FAS, PFAS or Non-FAS diagnosis, all infants performed weaker at their assessment at 17-29 months. The Subscales significantly affected included Personal-Social, Eye- Hand Coordination and Performance. The infants with FAS and PFAS displayed the most marked developmental delays. From this study it can be concluded that there are definite neurodevelopmental profiles for infant's diagnosed with FAS, PFAS and/or Non-FAS, highlighting the significant impact of prenatal alcohol exposure on various aspects of infant development. / Social work / M.Diac.

A classificação isomorfa dos espaços de Banach separáveis C(K) é devida a Milutin no caso em que K são não enumeráveis e a Bessaga e Pelczynski no caso em que K são enumeráveis. Neste trabalho apresentamos uma extensão vetorial dessa classificação e tiramos várias consequências, por exemplo, considerando o espaço métrico compacto infinito K e Y um espaço de Banach: &nbsp; &nbsp; 1. Sendo 1 < p < &infin; e &Gamma; um conjunto infinito, classificamos, a menos de isomorfismo, os espaços de Banach C(K, Y &oplus; lp(&Gamma;)), quando o dual de Y contém uma cópia de lq, onde 1/p+ 1/q =1. &nbsp; &nbsp; 2. Classificamos os espaços de Banach C(K, Y &oplus; l&infin;(&Gamma;)), quando a densidade de Y é estritamente menor que 2|&Gamma;|. &nbsp; &nbsp; 3. Classificamos os espaços de Banach C(K &times;(S&oplus; &beta;&Gamma;)) e C(S &oplus; (K&times; &beta;&Gamma;)), onde S é um compacto disperso de Hausdorff arbitrário e &beta;&Gamma; é a compactificação de Stone-Cech de &Gamma;. Obtemos, também, algumas leis de cancelamento para espaços de Banach da forma C(K1,X)&oplus; C(K2,Y), onde K1 e K2 são espaços compactos métricos infinitos de Hausdorff e X, Y espaços de Banach satisfazendo condições adequadas. Estabelecemos também um teorema de quase-dicotomia envolvendo os espaços C(K,X), onde X tem cotipo finito. Finalmente, apresentamos algumas majorações nas distorções de isomorfismos positivos de C([0,&omega;k]) em C([0,&omega;]) e também de C([0,&omega;]) em C([0,&omega;k]), k&isin; N, k &ge; 2. / The isomorphic classification of separable Banach spaces C(K) is due Milutin in the case when K are uncountable and to Bessaga and Pelczynski in the case when K are countable. In this work we prove a vectorial extention of this classification and provide several consequences, for example considering the infinite metric compact space K and Y a Banach space: &nbsp; &nbsp; 1. Let 1 < p < &infin; and &Gamma; a infinite set, we classify, up to an isomorphism, the Banach spaces C(K, Y &oplus; lp(&Gamma;)), in the case where the dual of Y contains no copy of lq, where 1/p+ 1/q =1. &nbsp; &nbsp; 2. We classify the Banach spaces C(K, Y &oplus; l&infin;(&Gamma;)), when the density character of Y is strictly less that 2|&Gamma;|. &nbsp; &nbsp; 3. We classify the Banach spaces C(K &times;(S&oplus; &beta;&Gamma;)) and C(S &oplus; (K&times; &beta;&Gamma;)) where S is an arbitrary dispersed compact and &beta;&Gamma; is the Stone-Cech compactification of &Gamma;. We obtain also some cancellation laws for Banach spaces in the form C(K1,X)&oplus; C(K2,Y), where K1 and K2 are metric compact Hausdorff spaces and X, Y Banach spaces satisfying appropriate conditions. We established also a quasi-dichotomy theorem envolving the C(K,X) spaces, where X is of finite cotype. Finally, we present some upper bounds of distortions of positive isomorphisms of C([0,&omega;k]) on C([0,&omega;]) and also of C([0,&omega;]) on C([0,&omega;k]), k&isin; N, k &ge; 2.

[EN] The scope of this thesis is the abstract finite group theory. All the groups we will consider will be finite. hence, the word "group" will be understood as a synonimous of "finite group". We say that a subgroup H of a group G is solitary when no other subgroup of G is isomorphic to H. A normal subgroup H of a group G is said to be normal solitary when no other normal subgroup of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N. Solitary subgroups, normal solitary subgroups, and quotient solitary subgroups have been recently studied by authors like Thévenaz, who named the solitary subgroups as strongly characteristic subgroups, Kaplan and Levy, Tarnauceanu, and Atanasov and Foguel. The aim of this PhD thesis project is to deepen into the analysis of these subgroup embedding properties, by refining the knowledge of their lattice properties, by obtaining general properties related to classes of groups, and by analysing groups in which the members of some distinguished families of subgroups satisfy these embedding properties. The basic results of group theory that will be used in the memoir appear in Chapter 1. Among them, we comment on some results about soluble groups, supersoluble groups, nilpotent groups, classes of groups, and p-soluble and p-nilpotent groups for a prime p. In Chapter 2, we present the basic concepts about these embedding properties, as well as some basic results satisfied by them. Chapter 3 is devoted to the study of lattice properties of these types of subgroups. In this chapter we deepen into the study of the lattices of solitary subgroups and quotient solitary subgroups developed by Kaplan and Levy and by Tarnauceanu and we check that, even though these lattices consist of normal subgroups, they are not sublattices of the lattice of normal subgroups. We also check that the set of all normal solitary subgroups does not constitute a lattice, which motivates the introduction of the concept of subnormal solitary subgroup as a more suitable tool to deal with lattice properties. In Chapter 4, we study in depth the relations between these embedding properties and classes of groups. We observe that the subnormal solitary subgroups behave well with respect to radicals for Fitting classes and that the residuals for formations are quotient solitary subgroups. We also study conditions under which the radicals with respect to Fitting classes are quotient solitary subgroups and the residuals with respect to formations are solitary subgroups. To finish, we state the natural question of whether the solitary or subnormal solitary subgroups can be regarded as radicals for suitable Fitting classes or whether the quotient solitary subgroups are residuals for suitable Fitting classes. We give a negative answer to this question. Chapter 5 is devoted to the study of groups whose minimal subgroups are solitary, that is, groups with a unique subgroup of order p for each prime p dividing its order. We give a complete classification of these groups and we make some remarks about related problems. Our contributions to this research line have been accepted for their publication in two papers in Communications in Algebra and in Journal of Algebra and its Applications. / [ES] El ámbito de esta tesis es el de la teoría abstracta de grupos finitos. Todos los grupos que consideremos serán finitos. Por ello, la palabra «grupo» se entenderá como sinónima de «grupo finito». Decimos que un subgrupo H de un grupo G es solitario cuando ningún otro subgrupo de G es isomorfo a H. Un subgrupo normal H de un grupo G se dice normal solitario cuando ningún otro subgrupo normal de G es isomorfo a H. Un subgrupo normal N de un grupo G se dice que es solitario para cocientes cuando ningún otro subgrupo normal K de G da un cociente isomorfo a G/N. Los subgrupos solitarios, los subgrupos normales solitarios y los subgrupos solitarios para cocientes han sido recientemente estudiados por autores como Thévenaz, quien bautizó los subgrupos solitarios como subgrupos fuertemente característicos, Kaplan y Levy, Tarnauceanu y Atanasov y Foguel. El objeto de este proyecto de tesis doctoral es el de profundizar en el análisis de estas propiedades de inmersión de subgrupos, afinando en el conocimiento de sus propiedades reticulares, obteniendo propiedades generales en relación con clases de grupos y analizando grupos en los que los miembros de algunas familias destacadas de subgrupos satisfacen estas propiedades de inmersión. Los resultados básicos de teoría de grupos que se utilizan en la memoria aparecen en el capítulo 1. Entre ellos, comentamos algunos resultados sobre grupos resolubles, superresolubles, nilpotentes, clases de grupos y grupos p-resolubles y p-nilpotentes para un primo p. En el capítulo 2 presentamos los conceptos básicos sobre estas propiedades de inmersión, así como algunos resultados básicos que satisfacen. El capítulo 3 está dedicado al estudio de propiedades reticulares de estos tipos de subgrupos. En este capítulo se profundiza en el estudio de los retículos de subgrupos solitarios y solitarios para cocientes llevado a cabo por Kaplan y Levy y por Tarnauceanu y se comprueba que, a pesar de que estos retículos constan de subgrupos normales, no son subretículos del retículo de subgrupos normales. También comprobamos que el conjunto de subgrupos normales solitarios no constituye un retículo, lo que motiva la introducción del concepto de subgrupo subnormal solitario como herramienta más adecuada para tratar propiedades reticulares. En el capítulo 4 estudiamos con profundidad las relaciones entre estas propiedades de inmersión y clases de grupos. Observamos que los subgrupos subnormales solitarios se comportan bien respecto de radicales de clases de Fitting y que los residuales para formaciones son subgrupos solitarios para cocientes. Esto permite mejorar algunos resultados sobre subgrupos solitarios para cocientes. También estudiamos condiciones en que los radicales respecto de clases de Fitting son subgrupos solitarios para cocientes y los residuales respecto de formaciones son subgrupos solitarios. Por último, nos planteamos la cuestión natural de si los subgrupos solitarios o subnormales solitarios pueden verse como radicales para clases de Fitting adecuadas o si los subgrupos solitarios para cocientes son residuales para clases de Fitting adecuadas. Damos una respuesta negativa a esta cuestión. El capítulo 5 está dedicado al estudio de grupos cuyos subgrupos minimales son solitarios, es decir, grupos con un único subgrupo de orden p para cada primo p divisor de su orden. Damos una clasificación completa de estos grupos y hacemos algunas observaciones sobre problemas relacionados. Nuestras aportaciones a esta línea de investigación han sido aceptadas para su publicación en dos artículos en Communications in Algebra y en Journal of Algebra and its Applications. / [CAT] L'àmbit d'aquesta tesi és el de la teoria abstracta de grups finits. Tots els grups que hi considerem seran finits. Per això, la paraula «grup» s'entendrà com a sinònima de «grup finit». Direm que un subgrup H d'un grup G és solitari quan cap altre subgrup de G no és isomorf a H. Un subgrup normal H d'un grup G es diu normal solitari quan cap altre subgrup normal de G no és isomorf a H. Un subgrup normal N d'un grup G es diu que és solitari per a quocients quan cap altre subgrup normal K de G no dóna un quocient isomorf a G/N. Els subgrups solitaris, els subgrups solitaris normals i els subgrups solitaris per a quocients han sigut recentment estudiats per autors com Thévenaz, qui batejà els subgrups solitaris com a subgrups fortament característics, Kaplan i Levy, Tarnauceanu i Atanasov i Foguel. L'objecte d'aquest projecte de tesi doctoral és el d'aprofundir en l'anàlisi d'aquestes propietats d'immersió de subgrups, afinant en el coneixement de les seues propietats reticulars, obtenint propietats generals en relació amb classes de grups i analitzant grups en què els membres d'algunes famílies destacades de subgrups satisfan aquestes propietats d'immersió. Els resultats bàsics de teoria de grups que es fan servir en la memòria apareixen al capítol 1. Entre ells, comentem alguns resultats sobre grups resolubles, superresolubles, nilpotents, classes de grups i grups p-resolubles i p-nilpotents per a un primer p. Al capítol 2 presentem els conceptes bàsics sobre aquestes propietats d'immersió, així com alguns resultats bàsics que satisfan. El capítol 3 està dedicat a l'estudi de propietats reticulars d'aquests tipus de subgrups. En aquest capítol s'aprofundeix en l'estudi dels reticles de subgrups solitaris i solitaris per a quocients dut a terme per Kaplan i Levy i per Tarnauceanu i es comprova que, encara que aquests subgrups consten de subgrups normals, no són subreticles del reticle de subgrups normals. També comprovem que el conjunt de subgrups normals solitaris no constitueix un reticle, la qual cosa motiva la introducció del concepte de subgrup subnormal solitari com a eina més adient per tractar propietats reticulars. Al capítol 4 estudiem amb profunditat les relacions entre aquestes propietats d'immersió i classes de grups. Observem que els subgrups subnormals solitaris es comporten bé respecte de radicals de classes de Fitting i que els residuals per a formacions són subgrups solitaris per a quocients. Açò permet millorar alguns resultats sobre subgrups solitaris per a quocients. També estudien condicions en què els radicals respecte de classes de Fitting són subgrups solitaris per a quocients i els residuals respecte de formacions són subgrups solitaris. Per acabar, ens plantegem la qüestió natural de si els subgrups solitaris o subnormals solitaris poden veure's com a radicals per a classes de Fitting adients o si els subgrups solitaris per a quocients són residuals per a classes de Fitting adients. Donem una resposta negativa a aquesta qüestió. El capítol 5 està dedicat a l'estudi de grups els subgrups minimals dels quals són solitaris, és a dir, grups amb un únic subgrup d'ordre p per a cada primer p divisor del seu ordre. Donem una classificació completa d'aquests grups i fem algunes observacions sobre problemes relacionats. Les nostres aportacions a aquesta línia de recerca han sigut acceptades per a llur publicació a dos articles a Communications in Algebra i a Journal of Algebra and its Applications. / Liriano Castro, ODCDJ. (2015). Subgrupos solitarios de grupos finitos [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/59397 / TESIS

Treemaps are a popular space-filling visualization of hierarchical data that maps an attribute of a datum, or a data aggregate, to a proportional amount of area. Assuming a rectangular treemap consisting of nested rectangles (also called tiles), there are multiple possible valid tiling arrangements. A common criterion for optimization is aspect ratio. Nevertheless, treemaps usually consist of multiple rectangles, so the aspect ratios need be aggregated. The basic definition of aspect ratio (width divided by height) cannot be meaningfully aggregated. Given this, a definition of aspect ratio that does not differentiate height from width was suggested. This definition allows for meaningful aggregation, but only as long as there are no large differences in the data distribution, and the target aspect ratio is 1:1. Originally, a target aspect ratio of 1:1 was deemed to be axiomatically ideal. Currently, perceptual studies have found an aspect ratio of 1:1 to lead to the largest area estimation error. However, with any other target this definition of aspect ratio cannot be meaningfully aggregated. This thesis suggests a correction that can be applied to the current metric and would allow it to be meaningfully aggregated even when there are large value differences in the data. Furthermore, both the uncorrected and corrected metrics can be generalized for any target (i.e. targets other than 1:1). Another issue with current evaluation techniques is that algorithm fitness is evaluated through Monte Carlo trials. In this method, synthetic data is generated and then aggregated to generate a single final result. However, tiling algorithm performance is dependant on data distribution, so a single aggregateresult cannot generalize overall performance. The alternative suggested in this thesis is visual cluster analysis, which should hold more general predictive power.All of the above is put into practice with an experiment. In the experiment, a new family of tiling algorithms, based on criteria derived from the results of the perceptual tests in literature,is compared to the most popular tiling algorithm, Squarify. The results confirm that there are indeed vast but consistent value fluctuations for different normal distributions. At least for a target aspect ratio of 1.5, the new proposed algorithms are shown to perform better than Squarify for most use cases in terms of aspect ratio.

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)

This thesis deals with cohomologies on the symplectic quotient of a Frölicher space which is locally diffeomorphic to a Euclidean Frölicher subspace of Rn of constant dimension equal to n. The symplectic reduction under consideration in this thesis is an extension of the Marsden-Weinstein quotient (also called, the reduced space) well-known from the finite-dimensional smooth manifold case. That is, starting with a proper and free action of a Frölicher-Lie-group on a locally Euclidean Frölicher space of finite constant dimension, we study the smooth structure and the topology induced on a small subspace of the orbit space. It is on this topological space that we will construct selected cohomologies such as : sheaf cohomology, Alexander-Spanier cohomology, singular cohomology, ~Cech cohomology and de Rham cohomology. Some natural questions that will be investigated are for instance: the impact of the symplectic structure on these di erent cohomologies; the cohomology that will give a good description of the topology on the objects of category of Frölicher spaces; the extension of the de Rham cohomology theorem in order to establish an isomorphism between the five cohomologies. Beside the algebraic, topological and geometric study of these new objects, the thesis contains a modern formalism of Hamiltonian mechanics on the reduced space under symplectic and Poisson structures. / Mathematical Sciences / D. Phil. (Mathematics)