[EN] The Ph.D. thesis "Weighted Banach Spaces of harmonic functions" presented here, treats several topics of functional analysis such as weights, composition operators, Fréchet and Gâteaux differentiability of the norm and isomorphism classes. The work is divided into four chapters that are preceded by one in which we introduce the notation and the well-known properties that we use in the proofs in the rest of the chapters. In the first chapter we study Banach spaces of harmonic functions on open sets of R^d endowed with weighted supremun norms. We define the harmonic associated weight, we explain its properties, we compare it with the holomorphic associated weight introduced by Bierstedt, Bonet and Taskinen, and we find differences and conditions under which they are exactly the same and conditions under which they are equivalent. The second chapter is devoted to the analysis of composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions. We characterize the continuity, the compactness and the essential norm of composition operators among these spaces in terms of their weights, thus extending the results of Bonet, Taskinen, Lindström, Wolf, Contreras, Montes and others for composition operators between spaces of holomorphic functions. We prove that for each value of the interval [0,1] there is a composition operator between weighted spaces of harmonic functions such that its essential norm attains this value. Most of the contents of Chapters 1 and 2 have been published by E. Jordá and the author in [48]. The third chapter is related with the study of Gâteaux and Fréchet differentiability of the norm. The \v{S}mulyan criterion states that the norm of a real Banach space X is Gâteaux differentiable at x\inX if and only if there exists x^* in the unit ball of the dual of X weak^* exposed by x and the norm is Fréchet differentiable at x if and only if x^* is weak^* strongly exposed in the unit ball of the dual of X by x. We show that in this criterion the unit ball of the dual of X can be replaced by a smaller convenient set, and we apply this extended criterion to characterize the points of Gâteaux and Fréchet differentiability of the norm of some spaces of harmonic functions and continuous functions with vector values. Starting from these results we get an easy proof of the theorem about the Gâteaux differentiability of the norm for spaces of compact linear operators announced by Heinrich and published without proof. Moreover, these results allow us to obtain applications to classical Banach spaces as the space H^\infty of bounded holomorphic functions in the disc and the algebra A(\overline{\D}) of continuous functions on \overline{\D} which are holomorphic on \D. The content of this chapter has been included by E. Jordá and the author in [47]. Finally, in the forth chapter we show that for any open set U of R^d and weight v on U, the space hv0(U) of harmonic functions such that multiplied by the weight vanishes at the boundary on U is almost isometric to a closed subspace of c0, extending a theorem due to Bonet and Wolf for the spaces of holomorphic functions Hv0(U) on open sets U of C^d. Likewise, we also study the geometry of these weighted spaces inspired by a work of Boyd and Rueda, examining topics such as the v-boundary and v-peak points and we give the conditions that provide examples where hv0(U) cannot be isometric to c0. For a balanced open set U of R^d, some geometrical conditions in U and convexity in the weight v ensure that hv0(U) is not rotund. These results have been published by E. Jordá and the author [46]. / [ES] La presente memoria, "Espacios de Banach ponderados de funciones armónicas ", trata diversos tópicos del análisis funcional, como son las funciones peso, los operadores de composición, la diferenciabilidad Fréchet y Gâteaux de la norma y las clases de isomorfismos. El trabajo está dividido en cuatro capítulos precedidos de uno inicial en el que introducimos la notación y las propiedades conocidas que usamos en las demostraciones del resto de capítulos. En el primer capítulo estudiamos espacios de Banach de funciones armónicas en conjuntos abiertos de R^d dotados de normas del supremo ponderadas. Definimos el peso asociado armónico, explicamos sus propiedades, lo comparamos con el peso asociado holomorfo introducido por Bierstedt, Bonet y Taskinen, y encontramos diferencias y condiciones para que sean exactamente iguales y condiciones para que sean equivalentes. El capítulo segundo está dedicado al análisis de los operadores de composición con símbolo holomorfo entre espacios de Banach ponderados de funciones pluriarmónicas. Caracterizamos la continuidad, la compacidad y la norma esencial de operadores de composición entre estos espacios en términos de los pesos, extendiendo los resultados de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes y otros para operadores de composición entre espacios de funciones holomorfas. Probamos que para todo valor del intervalo [0,1] existe un operador de composición sobre espacios ponderados de funciones armónicas tal que su norma esencial alcanza dicho valor. La mayoría de los contenidos de los capítulos 1 y 2 han sido publicados por E. Jordá y la autora en [48]. El capítulo tercero está relacionado con el estudio de la diferenciabilidad Gâteaux y Fréchet de la norma. El criterio de \v{S}mulyan establece que la norma de un espacio de Banach real X es Gâteaux diferenciable en x\in X si y sólo si existe x^* en la bola unidad del dual de X débil expuesto por x y la norma es Fréchet diferenciable en x si y sólo si x^*es débil fuertemente expuesto en la bola unidad del dual de X por x. Mostramos que en este criterio la bola del dual de X puede ser reemplazada por un conjunto conveniente más pequeño, y aplicamos este criterio extendido para caracterizar los puntos de diferenciabilidad Gâteaux y Fréchet de la norma de algunos espacios de funciones armónicas y continuas con valores vectoriales. A partir de estos resultados conseguimos una prueba sencilla del teorema sobre la diferenciabilidad Gâteaux de la norma de espacios de operadores lineales compactos enunciado por Heinrich y publicado sin la prueba. Además, éstos nos permiten obtener aplicaciones para espacios de Banach clásicos como H^\infty de funciones holomorfas acotadas en el disco y A(\overline{\D}) de funciones continuas en \overline{\D} que son holomorfas en \D. Los contenidos de este capítulo han sido incluidos por E. Jordá y la autora en [47]. Finalmente, en el capítulo cuarto mostramos que para cualquier abierto U contenido en R^d y cualquier peso v en U, el espacio hv0(U), de funciones armónicas tales que multiplicadas por el peso desaparecen en el infinito de U, es casi isométrico a un subespacio cerrado de c0, extendiendo un teorema debido a Bonet y Wolf para los espacios de funciones holomorfas Hv0(U) en abiertos U de C^d. Así mismo, inspirados por un trabajo de Boyd y Rueda también estudiamos la geometría de estos espacios ponderados examinando tópicos como la v-frontera y los puntos v-peak y damos las condiciones que proporcionan ejemplos donde hv0(U) no puede ser isométrico a c0. Para un conjunto abierto equilibrado U de R^d, algunas condiciones geométricas en U y sobre convexidad en el peso v aseguran que hv0(U) no es rotundo. Estos resultados han sido publicados por E. Jordá y la autora en [46]. / [CAT] La present memòria, "Espais de Banach ponderats de funcions harmòniques", tracta diversos tòpics de l'anàlisi funcional, com són les funcions pes, els operadors de composició, la diferenciabilitat Fréchet i Gâteaux de la norma i les clases d'isomorfismes. El treball està dividit en quatre capítols precedits d'un d'inicial en què introduïm la notació i les propietats conegudes que fem servir en les demostracions de la resta de capítols. En el primer capítol estudiem espais de Banach de funcions harmòniques en conjunts oberts de R^d dotats de normes del suprem ponderades. Definim el pes associat harmònic, expliquem les seues propietats, el comparem amb el pes associat holomorf introduït per Bierstedt, Bonet i Taskinen, i trobem diferències i condicions perquè siguen exactament iguals i condicions perquè siguen equivalents. El capítol segon està dedicat a l'anàlisi dels operadors de composició amb símbol holomorf entre espais de Banach ponderats de funcions pluriharmòniques. Caracteritzem la continuïtat, la compacitat i la norma essencial d'operadors de composició entre aquests espais en termes dels pesos, estenent els resultats de Bonet, Taskinen, Lindström, Wolf, Contreras, Montes i altres per a operadors de composició entre espais de funcions holomorfes. Provem que per a tot valor de l'interval [0,1] hi ha un operador de composició sobre espais ponderats de funcions harmòniques tal que la seua norma essencial arriba aquest valor. La majoria dels continguts dels capítols 1 i 2 han estat publicats per E. Jordá i l'autora en [48]. El capítol tercer està relacionat amb l'estudi de la diferenciabilitat Gâteaux y Fréchet de la norma. El criteri de \v{S}mulyan estableix que la norma d'un espai de Banach real X és Gâteaux diferenciable en x\inX si i només si existeix x^* a la bola unitat del dual de X feble exposat per x i la norma és Fréchet diferenciable en x si i només si x^* és feble fortament exposat a la bola unitat del dual de X per x. Mostrem que en aquest criteri la bola del dual de X pot ser substituïda per un conjunt convenient més petit, i apliquem aquest criteri estès per caracteritzar els punts de diferenciabilitat Gâteaux i Fréchet de la norma d'alguns espais de funcions harmòniques i contínues amb valors vectorials. A partir d'aquests resultats aconseguim una prova senzilla del teorema sobre la diferenciabilitat Gâteaux de la norma d'espais d'operadors lineals compactes enunciat per Heinrich i publicat sense la prova. A més, aquests ens permeten obtenir aplicacions per a espais de Banach clàssics com l'espai H^\infty de funcions holomorfes acotades en el disc i l'àlgebra A(\overline{\D}) de funcions contínues en \overline{\D} que són holomorfes en \D. Els continguts d'aquest capítol han estat inclosos per E. Jordá i l'autora en [47]. Finalment, en el capítol quart mostrem que per a qualsevol conjunt obert U de R^d i qualsevol pes v en U, l'espai hv0(U), de funcions harmòniques tals que multiplicades pel pes desapareixen en el infinit d'U, és gairebé isomètric a un subespai tancat de c0, estenent un teorema degut a Bonet y Wolf per als espais de funcions holomorfes Hv0(U) en oberts U de C^d. Així mateix, inspirats per un treball de Boyd i Rueda també estudiem la geometria d'aquests espais ponderats examinant tòpics com la v-frontera i els punts v-peak i donem les condicions que proporcionen exemples on hv0(U) no pot ser isomètric a c0. Per a un conjunt obert equilibrat U de R^d, algunes condicions geomètriques en U i sobre convexitat en el pes v asseguren que hv0(U) no és rotund. Aquests resultats han estat publicats per E. Jordá i l'autora en [46]. / Zarco García, AM. (2015). Weighted Banach spaces of harmonic functions [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/56461 / TESIS

Prior knowledge about the possible characteristics of demand and supply is vital in the planning and operation of economically sustainable isolated rural power systems. System modelling and simulation is one of the tools that can be used in planning and assessing the performance of these systems. This thesis is presenting a Monte Carlo simulation methodology for modelling, simulating and analysing the performance of isolated rural electricity markets applicable in developing countries. The definitions of possible power system operators managing these markets are introduced based on different economic objectives of operating the systems. The two system operators considered in the thesis are: altruistic and profit maximising operators. The concept used to define types of isolated rural electricity markets is combining the definitions of operators and the possible combinations of power supply options (purely thermal or hybrid system). It is anticipated that the rural electricity markets under consideration comprise of uncertainties in demand and supply (both demand and generation are modelled as random variables from assumed or estimated probability distributions). Demand is price sensitive and modelled as a product of two random variables, relative demand and peak demand. The price sensitivity of demand is shown by representing peak demand using an economic price-demand function. The parameters (price sensitivity and demand factor) of this function are modelled as random variables which reflect the randomness of consumers’ preferences. The simulation algorithm is based on the theory of correlated sampling, in order to compare the performance of systems under different operators. The thesis introduces the concept of nested Monte Carlo simulation to be able manage the simulation of different operators subjected to the same market conditions. The performance of electricity markets is assessed by analysing three parameters (tariffs, profit and reliability), which are random variables presented using probability distributions in form of duration curves. The methodology is tested on a theoretical case study system using load data obtained from a rural community in Africa.  The case study illustrates how to use the model, preparation of the input variables and how to use the output to estimate and assess the possible performance of isolated rural power systems under different power system operators. It is anticipated that the proposed methodology can be used by researchers, planners and academia as a tool for planning, estimating and assessing the performance of rural power systems in isolated areas of developing countries

Inbound tour operators play a key role in sustainable tourism development, as they are centrally positioned in the distribution chain and provide the link between the supply and demand of tourism products and services. Embedded in this position, inbound tour operators can put pressure on their suppliers to operate more sustainably, while educating their customers on sustainable tourism practices, and influencing consumers’ decision-making before the purchasing of tourism products and services. Inbound tour operators can further implement sustainable tourism practices as part of their business operations. To date, little research has focussed on inbound tour operators’ contribution to sustainable tourism development, especially in a developing country context. Sustainable inbound tour operators can also become certified by a sustainable tourism certification programme to showcase their commitment to sustainability. Various studies have highlighted the history, benefits and issues related to certification programmes, but few studies have investigated the perspective that inbound tour operators have towards sustainable tourism certification programmes. Making use of a qualitative research approach, in-depth interviews were conducted with 22 South African inbound tour operators to investigate and identify the sustainable tourism practices currently being adopted within their organisations. Content analysis was used to analyse the data. The findings produced a list of sustainable tourism practices currently being adopted by inbound tour operators in South Africa. This study proposes that sustainable tourism organisations should become certified by a national or global sustainable tourism certification programme, to prove that they are truly operating sustainably, thus decreasing the effects of greenwashing. In addition, the certification of tourism organisations can assist inbound tour operators in identifying truly sustainable suppliers, fostering the development of a sustainable supply chain management strategy. / Dissertation (MCom)--University of Pretoria, 2020. / Tourism Management / MCom / Unrestricted

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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, South Africa, in ful lment of the requirements for the degree of Masters of Science. May 27, 2015. / Abstract The unsteady three-dimensional ow of a thin slender rivulet of incompressible Newtonian uid down an inclined porous plane is investigated. The leak-o velocity is not speci ed in the model but is determined in the process of deriving the invariant solution. A second order nonlinear partial di erential equation in two spatial variables and time and containing the leak-o velocity is derived for the height of the thin slender rivulet. Using Lie group analysis it is found that the partial di erential equation can be reduced in two steps to an ordinary di erential equation provided the leak-o velocity satis es a rst order linear partial di erential equation in three variables. An exact analytical solution with a dry patch in the central region is derived for a special leak-o velocity. Two models are considered, one with the leak-o velocity proportional to the height of the rivulet and the other with leak-o velocity proportional to the cube of the height. Numerical solutions are obtained for the height of the rivulet using a shooting method which also determines the two-dimensional boundary of the rivulet on the inclined plane. The e ect of uid leak-o on the height and width of the rivulet is investigated numerically and compared in the two models. The conservation laws for the partial di erential equation with no uid leak-o are investigated. Two conserved vectors are derived, the elementary conserved vector and a new conserved vector. The Lie point symmetry of the partial di erential equation associated with each conserved vector is obtained. Each associated Lie point symmetry is used to perform a double reduction of the partial di erential equation, but the solutions obtained are not physically signi cant.

Continued fractions have been extensively studied in number theoretic ways. In this text we will consider continued fraction expansions with partial quotients that are in Z = f x : x 2 Zg and where = 2 cos( q ); q 3 and with 1 < < 2. These continued fractions are expressed as the composition of M obius maps in PSL(2;R), that act as isometries on H2, taken at 1. In particular the subgroups of PSL(2;R) that are studied are the Hecke groups G . The Modular group is the case for q = 3 and = 1. In the text we show that the Hecke groups are triangle groups and in this way derive their fundamental domains. From these fundamental domains we produce the v-cell (P0) that is an ideal q-gon and also tessellate H2 under G . This tessellation is called the -Farey tessellation. We investigate various known -continued fractions of a real number. In particular, we consider a geodesic in H2 cutting across the -Farey tessellation that produces a \cutting sequence" or path on a -Farey graph. These paths in turn give a rise to a derived -continued fraction expansion for the real endpoint of the geodesic. We explore the relationship between the derived -continued fraction expansion and the nearest - integer continued fraction expansion (reduced -continued fraction expansion given by Rosen, [25]). The geometric aspect of the derived -continued fraction expansion brings clarity and illuminates the algebraic process of the reduced -continued fraction expansion.

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The Harris-Venkatesh conjecture posits a relationship between the action of derived Hecke operators on weight-one modular forms and Stark units. We prove the full Harris-Venkatesh conjecture for all CM dihedral weight-one modular forms. This reproves results of Darmon-Harris-Rotger-Venkatesh, extends their work to the adelic setting, and removes all assumptions on primality and ramification from the imaginary dihedral case of the Harris-Venkatesh conjecture. This is done by introducing the Harris-Venkatesh period on cuspidal one-forms on modular curves, introducing two-variable optimal modular forms, evaluating GL(2) × GL(2) Rankin-Selberg convolutions on optimal forms and newforms, and proving a modulo-ℓᵗ comparison theorem between the Harris-Venkatesh and Rankin-Selberg periods. Furthermore, these methods explicitly describe local factors appearing in the constant of proportionality prescribed by the Harris-Venkatesh conjecture. We also look at the application of our methods to non-dihedral forms.

In this paper, we examine the conjecture of Avner Ash, Darrin Doud, David Pollack, and Warren Sinnott relating Galois representations to the mod p cohomology of congruence subgroups of the general linear group of n dimensions over the integers. We present computational evidence for this conjecture (the ADPS Conjecture) for the case n = 3 by finding Galois representations which appear to correspond to cohomology eigenclasses predicted by the ADPS Conjecture for the prime p, level N, and quadratic nebentype. The examples include representations which appear to be attached to cohomology eigenclasses which arise from D8, S3, A5, and S5 extensions. Other examples include representations which are reducible as sums of characters, representations which are symmetric squares of two-dimensional representations, and representations which arise from modular forms, as predicted by Jean-Pierre Serre for n = 2.

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