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)

Recent efforts to classify off-shell representations of supersymmetry without a central charge have focused upon directed, supermultiplet graphs of hypercubic topology known as Adinkras. These encodings of Super Poincare algebras, depict every generator of a chosen supersymmetry as a node-pair transformtion between fermionic bosonic component fields. This research thesis is a culmination of investigating novel diagrammatic sums of gauge-quotients by supersymmetric images of other Adinkras, and the correlated building of field theoretic worldline Lagrangians to accommodate both classical and quantum venues. We find Ref [40], that such gauge quotients do not yield other stand alone or "proper" Adinkras as afore sighted, nor can they be decomposed into supermultiplet sums, but are rather a connected "Adinkraic network". Their iteration, analogous to Weyl's construction for producing all finite-dimensional unitary representations in Lie algebras, sets off chains of algebraic paradigms in discrete-graph and continuous-field variables, the links of which feature distinct, supersymmetric Lagrangian templates. Collectively, these Adiankraic series air new symbolic genera for equation to phase moments in Feynman path integrals. Guided in this light, we proceed by constructing Lagrangians actions for the N = 3 supermultiplet YI /(iDI X) for I = 1, 2, 3, where YI and X are standard, Salam-Strathdee superfields: YI fermionic and X bosonic. The system, bilinear in the component fields exhibits a total of thirteen free parameters, seven of which specify Zeeman-like coupling to external background (magnetic) fluxes. All but special subsets of this parameter space describe aperiodic oscillatory responses, some of which are found to be surprisingly controlled by the golden ratio, [phi] = 1.61803, Ref [52]. It is further determined that these Lagrangians allow an N = 3 - > 4 supersymmetric extension to the Chiral-Chiral and Chiral-twistedChiral multiplet, while a subset admits two inequivalent such extensions. In a natural proiii gression, a continuum of observably and usefully inequivalent, finite-dimensional off-shell representations of worldline N = 4 extended supersymmetry are explored, that are variate from one another but in the value of a tuning parameter, Ref [53]. Their dynamics turns out to be nontrivial already when restricting to just bilinear Lagrangians. In particular, we find a 34-parameter family of bilinear Lagrangians that couple two differently tuned supermultiplets to each other and to external magnetic fluxes, where the explicit parameter dependence is unremovable by any field redefinition and is therefore observable. This offers the evaluation of X-phase sensitive, off-shell path integrals with promising correlations to group product decompositions and to deriving source emergences of higher-order background flux-forms on 2-dimensional manifolds, the stacks of which comprise space-time volumes. Application to nonlinear sigma models would naturally follow, having potential use in M- and F- string theories.

Text in Afrikaans. Abstracts in Afrikaans, English and Zulu / This study explored how nine adolescent learner-participants with cerebral palsy or other physical disabilities responded to an emotional support programme (EOP) of 20 intervention sessions, aimed at developing their emotional intelligence. The research study highlighted the utilisational value and importance of emotional intelligence, which is associated with problem-solving skills, career success, selfactualisation, stress management and the like. The rationale for compiling the EOP was in accordance with the rationale for the research study, being embedded in diverse learning needs, support for learners, and the design of learner opportunities. The study focused among others on the development and implementation of an EOP based on Bar-On as conceptual model (2000), together with the five core competencies of social and emotional learning (SEL), namely: responsible decision-making, emotional self-awareness, self-management, social awareness, and relationship skills. The Lions Quest SEL-programme, Skills for Adolescence, formed the basis of the EOP, although the researcher had to adapt it for her learner-participants’ special educational needs. The EOP was further extended with the knowledge obtained through the literature study regarding emotional intelligence and adolescence, and was aligned with the aims of the South African school curriculum in respect of emotional intelligence skills, as stated in the aims of the subject Life Skills. The researcher’s embedded, mixed-method research design allowed the larger qualitative component to include the smaller quantitative component, so that the specific phenomenon could be understood from different participants’ perspectives. Qualitative data was collected by means of metaphor interviews and collage activities. The quantitative research component of the study refers to the standardised Bar-On Emotional Quotient Inventory: Youth Version (EQ-i:YV) together with the researcher’s own structured questionnaire. The researcher did not interpret the data obtained from the quantitative data collection instruments in isolation, but used it to expand and verify the results of her qualitative generated data. The integrated data results of the learner-participants brought the answer to the research problem: most of the adolescent learner-participants with special educational needs who had participated in the EOP, did indeed benefit through their involvement, and their emotional intelligence (the phenomenon under scrutiny) had improved. / Die studie ondersoek hoe nege adolessente leerder-deelnemers met serebrale en fisieke gestremdhede op ’n emosionele ondersteuningsprogram (EOP) van 20 intervensiesessies reageer met die doel om hulle emosionele intelligensie te ontwikkel. Die benuttingswaarde en belangrikheid van emosionele intelligensie word deur die navorsingstudie uitgelig. Hoë emosionele intelligensie word geassosieer met probleemoplossingsvaardighede, werksukses, selfverwesenliking en streshantering, onder andere. Die rasionaal vir die samestelling van die EOP stem ooreen met die rasionaal vir die navorsingstudie en is gevestig in uiteenlopende leerbehoeftes, ondersteuning aan leerders, en die skep van leergeleenthede. Die studie fokus op die samestelling en implementering van ’n emosionele ondersteuningsprogram (EOP) gebaseer op Bar-On (2000) se konstruk van emosionele intelligensie as konseptuele model van die studie saam met die vyf kernbevoegdhede van sosiale en emosionele leer (SEL), naamlik: verantwoordelike besluitneming, emosionele selfbewussyn, selfbestuur, sosiale bewussyn en verhoudingsvaardighede. Die Lions Quest SEL-program, Skills for Adolescence, vorm die grondslag van die EOP, alhoewel die navorser baie aanpassings vir haar leerder-deelnemers se spesiale onderwysbehoeftes moes maak. Die EOP is verder uitgebrei deur die kennis wat die navorser tydens die literatuurstudie ingewin het ten opsigte van emosionele intelligensie en adolessensie, en sluit aan by die Suid-Afrikaanse skoolkurrikulum-doelwitte wat verband hou met emosionele intelligensie-vaardighede soos uiteengesit in die doelwitte vir die vak Lewensvaardighede. Die navorser se ingebedde, gemengdemetode-navorsingsontwerp laat toe dat die groter kwalitatiewe komponent die kleiner kwantitatiewe komponent insluit, sodat die spesifieke fenomeen vanuit verskillende deelnemers se perspektiewe verstaan kon word. Kwalitatiewe data is deur middel van die metafooronderhoud en collage-aktiwiteit ingesamel. Die kwantitatiewe navorsingskomponent van die studie verwys na die gestandaardiseerde Bar-On Emotional Quotient Inventory: Youth Version (EQ-i:YV) tesame met die navorser se eie gestruktureerde vraelys. Die navorser het die data wat sy uit die kwantitatiewe datainsamelingsinstrumente verkry het, nie net in isolasie geïnterpreteer nie, maar dit ook gebruik om die resultate van haar kwalitatief-gegenereerde data uit te brei en te verifieer. Die geïntegreerde data-resultate van die leerder-deelnemers bied die antwoord op die: die meerderheid van die adolessente leerderdeelnemers met spesiale onderwysbehoeftes wat aan die EOP deelgeneem het, het inderdaad gebaat by hulle deelname, en hulle emosionele intelligensie (die fenomeen wat ondersoek is) het verbeter. / Ucwaningo luhlola ukuthi ngabe ukuzibandakanya kwabafundi abangabadlalindima abayisishiyagalolunye abasesigabeni sobubhungu/sobutshitshi abakhubazeke ngokwengqondo noma ngayiphi indlela emzimbeni baphendule kanjani kuhlelo oluxhasa ngokommoya (emotional support programme) kumihlangano engama- 20 yokunceda, ehlose ekuqiniseni ubuhlakani bommoya. Isifundo socwaningo siveze ukubaluleka kokusebenzisa kanye nokubaluleka kobuhlakani bommoya, okuhambisana namakhono okuxazulula izinkinga, impumelelo yobizo lomsebenzi, ukuzibonakalisa, ukulawula ingcindezi yengqondo kanye nokunye. Isizathu sokuhlela i-ESP sasihambisana nenhloso yesifundo socwaningo, esitholakala kwizidingo zemfundo ezahlukahlukene, kanye nokuhleleka kwamathuba omfundi. Ucwaningo lubheke hlangana nokunye nokuthuthukiswa kanye nokusetshenziswa kohlelo lwe-ESP olususelwa ku-Bar-On njengemodeli yegama (2000), kanye namanye amacebo abalulekile amahlanu ohlelo lokufunda kwabantu kanye nokufunda ngokwemizwa (SEL), wona yilawa alandelayo: Ukumelana nezinqumo ozithethe, ukwazi imizwa yakho, ukuziphatha ngokwakho, ukuxwayiswa komphakathi kanye namakhono okwenza ubudlelwano. Uhlelo lwe- Lions Quest SEL, Skills for Adolescence, akha isisekelo se-ESP, yize umcwaningi kwakufanele aluguqule lolu hlelo ukwenzela ukuba luhambisane nezidingo ezikhethekile zemfundo yafundi abadlala indima. . Uhlelo lwe--ESP lwaqhubeka nokukhuliswa ngolwazi olwalutholakala ngocwaningo lombhalo wobuciko obumayelana nobuhlakani bokusebenzisa imizwa kanye nesigaba sobutshitshi/sobubhungu, kanti lolu hlelo lwaluhambisana nezinhloso zekharikhulami yezikole zaseNingizimu Afrika mayelana namakhono okusebenzisa obuhlakani bokusebenzisa imizwa, njengoba kushiwobkwizinhloso zesifundo samaKhono Empilo (Life Skills). Isakhiwo somcwaningi esequkethwe, sohlelo-oluvangene locwaningo siye savumela isigaba esikhulu socwaningo olugxile kukhwalithi (qualitative) ukuba sixube isigaba esincane esigxile kumanani (quantitative), ukuze kuzwisiseke uhlelo oluthize ngokwemiqondo yabadlalindima abehlukahlukene. Idatha yohlelo lwequalitative yaqoqwa ngokusebenzisa izinhlolombono zokungathekisa kanye nemisebenzi yokuhlanganisa imifanekiso eminingi. Isigaba socwaningo olugxile kumanani sichaza uhlelo olufanayo lwe--Bar-On Emotional Quotient Inventory: Youth Version (EQi:YV) kanye nemibhalo yemibuzo ehlelwe wumcwaningi. Umcwaningi akazange achaze idatha etholakele kumathuluzi okuqoqa idatha encike kumanani yodwa, kodwa le datha iye yasetshenziselwa ukukhulisa kanye nokuqinisekisa imiphumela yakhe eyakhiwe ngokwendlela yekhwalithi. Imiphumelo ehlangene yedatha yomfundi ongumdlalindima ilethe impendulo kwinkinga yocwaningo: Iningi labafundi abadlala indima abasesesigabeni sobubhungu/sobutshitshi abadinga imfundo ekhethekile abazibandakanye ohlelweni lwe-ESP, ngempela baye bazuza ngokuzibandakanya kwabo, kanti-ke izinga labo lokusebenzisa ikhono lobuhlakani bemizwa buye bathuthuka kakhulu (the phenomenon under scrutiny). / Inclusive Education / D. Ed. (Inklusiewe Onderwys)