Simplices are the "simplest" examples of polytopes, and yet they exhibit much of the rich and subtle combinatorics and commutative algebra of their more general cousins. In this way they are sufficiently complicated --- insights gained from their study can inform broader research in Ehrhart theory and associated fields. In this dissertation we consider two previously unstudied properties of lattice simplices; one algebraic and one combinatorial. The first is the Poincar\'e series of the associated semigroup algebra, which is substantially more complicated than the Hilbert series of that same algebra. The second is the partial ordering of the elements of the fundamental parallelepiped associated to the simplex. We conclude with a proof-of-concept for using machine learning techniques in algebraic combinatorics. Specifically, we attempt to model the integer decomposition property of a family of lattice simplices using a neural network.

Made available in DSpace on 2014-12-17T14:36:10Z (GMT). No. of bitstreams: 1 IvanilkaLA_TESE_1_a_150.pdf: 4520585 bytes, checksum: 18c2643e49e2eac92f9fe5befa225c45 (MD5) Previous issue date: 2010-05-20 / This thesis represents a didactic research linked to the Post-graduation Programme in Education of the Universidade Federal do Rio Grande do Norte which aimed to approach the construction of the geometrical concepts of Volume of the Rectangular Parallelepiped, Area and Perimeter of the Rectangle adding a study of the Area of the Circle. The research was developed along with students from the 6th level of the Elementary School, in a public school in Natal/RN. The pedagogical intervention was made up of three moments: application of a diagnostic evaluation, instrument that enabled the creation of the teaching module by showing the level of the geometry knowledge of the students; introduction of a Teaching Module by Activities aiming to propose a reflexive didactic routing directed to the conceptual construction because we believed that such an approach would favor the consolidation of the learning process by becoming significant to the apprentice, and the accomplishment of a Final Evaluation through which we established a comparison of the results obtained before and after the teaching intervention. The data gathered were analyzed qualitatively by means of a study of understanding categories of mathematical concepts, in addition to using descriptive statistics under the quantitative aspect. Based on the theory of Richard Skemp, about categorization of mathematical knowledge, in the levels of Relational and Instrumental Understanding were achieved in contextual situations and varied proportions, thus enabling a contribution in the learning of the geometrical concepts studied along with the students who took part in the research. We believe that this work may contribute with reflections about the learning processes, a concern which remained during all the stages of the research, and also that the technical competence along with the knowledge about the constructivist theory will condition the implementation of a new dynamics to the teaching and learning processes. We hope that the present research work may add some contribution to the teaching practice in the context of the teaching of Mathematics for the intermediate levels of the Elementary School / Esta tese se constituiu em uma pesquisa did?tica vinculada ao Programa de P?s-gradua??o em Educa??o da Universidade Federal do Rio Grande do Norte que teve por objetivo abordar ? constru??o dos conceitos geom?tricos de volume do bloco retangular sua ?rea de superf?cie acrescentando um estudo sobre ?rea do c?rculo. Este foi desenvolvido junto a alunos do 6? ano do Ensino Fundamental, em uma escola da rede p?blica de ensino em Natal/RN. A interven??o pedag?gica constou de tr?s momentos: aplica??o de uma avalia??o diagn?stica, instrumento que subsidiou a elabora??o do m?dulo e ensino, ao indicar os conhecimentos pr?vios e fragilidades dos alunos; introdu??o do m?dulo de ensino por atividades e realiza??o de uma avalia??o final atrav?s da qual estabelecemos uma compara??o dos resultados obtidos antes e depois da interven??o de ensino. Os dados obtidos foram analisados qualitativamente mediante estudo de categorias de compreens?o de conceitos matem?ticos, al?m de utilizarmos a an?lise estat?stica validando-a sob o aspecto quantitativo. Com base nos resultados os n?veis de compreens?o relacional e instrumental foram atingidos em situa??es contextuais e propor??es variadas. Acreditamos que este possa contribuir ainda com reflex?es acerca dos processos do conhecer, preocupa??o que permeou todas as etapas da pesquisa. Nossa esperan?a ? que esta pesquisa possa acrescentar alguma contribui??o ao fazer docente no contexto do ensino matem?tico, haja vista concebermos que a compet?ncia t?cnica aliada ao conhecimento sobre a teoria construtivista implementaria uma nova din?mica aos processos de ensino e aprendizagem

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work, we will present vectors from its relation with a oriented segment, its main operations, linear combination, linear dependence and base, up to the alternating product part. We relate the alternating product of two vectors with the area of the parallelogram formed by them, as well as the alternating product of three vectors with the volume of the parallelepiped. We also show that the vector product and the mixed product are special cases of alternating product. / Neste trabalho, apresentaremos vetores desde a sua relação com um segmento orientado, suas principais operações, combinação linear, dependência linear e base, até a parte de produto alternado. Relacionamos o produto alternado de dois vetores com área do paralelogramo formado por eles, assim como, o produto alternado de três vetores com o volume do paralelepípedo. Mostramos ainda que o produto vetorial e o produto misto são casos especiais de produto alternado. / São Cristóvão, SE

Τα ολοκληρωμένα ηλιακά συστήματα συλλέκτη αποθήκης αποτελούνται από μία δεξαμενή αποθήκευσης, της οποίας τμήμα της επιφάνειας της χρησιμοποιείται σαν ηλιακός συλλέκτης. Συνήθως το ρευστό της αποθήκης είναι το νερό χρήσης. Στο υπό εξέταση σύστημα το νερό χρήσης θερμαίνεται έμμεσα, διερχόμενο μέσα από σωληνώσεις εναλλάκτη θερμότητας που τοποθετείται στο εσωτερικό της παραλληλεπίπεδης δεξαμενής. Για την εντατικοποίηση της μετάδοσης θερμότητας προς το νερό χρήσης, δημιουργείται ανάδευση του ρευστού του δοχείου μέσω κυκλοφορητή, ο οποίος τίθεται σε λειτουργία μόνο όταν υπάρχει ζήτηση ζεστού νερού. Προς αποφυγή παραμορφώσεων τοποθετούνται πτερύγια συγκράτησης που ενώνουν τις δύο μεγάλες επιφάνειες της δεξαμενής. Στην παρούσα διδακτορική διατριβή διερευνήθηκαν τα ροϊκά φαινόμενα στο εσωτερικό του ICS που έχει περιγραφεί προηγουμένως, με στόχο τη μεγιστοποίηση της θερμανταλλαγής μεταξύ των δύο κυκλωμάτων νερού. Για την παρατήρηση του ροϊκού πεδίου καθώς και την λήψη μετρήσεων ταχυτήτων κατασκευάστηκε πειραματική συσκευή με διαφανή τοιχώματα από Plexiglas. Ελήφθησαν μετρήσεις ταχυτήτων και διακυμάνσεων με χρήση συστήματος Laser Doppler διπλής ακτίνας. Για την οπτικοποίηση του ροϊκού πεδίου τοποθετήθηκαν σωματίδια πολυστερίνης στο εσωτερικό της συσκευής. Ελήφθησαν ψηφιακές φωτογραφίες και βιντεοσκοπήσεις του ροϊκού πεδίου. Για την υπολογιστική προσομοίωση χρησιμοποιήθηκε το εμπορικό λογισμικό FLUENT. Αναπτύχθηκε υπολογιστικό μοντέλο και επιλύθηκε με όλα τα διαθέσιμα μοντέλα τύρβης. Στη συνέχεια πραγματοποιήθηκε σειρά υπολογιστικών προσομοιώσεων, στις οποίες διερευνήθηκε η βέλτιστη θέση και το μέγεθος των στομίων ανακυκλοφορίας, η βέλτιστη διάταξη των πτερυγίων συγκράτησης και η βέλτιστη θέση του εναλλάκτη. Επιπλέον προσδιορίστηκε υπολογιστικά και πειραματικά ο χρόνος αποκατάστασης του ροϊκού πεδίου. Τέλος, τα αποτελέσματα συγκρίθηκαν και με πειραματικά αποτελέσματα άλλων εργασιών. Τα συμπεράσματα που εξάγονται έχουν ως ακολούθως: Το μοντέλο τύρβης standard k-ω δίνει τα πιο αξιόπιστα αποτελέσματα. Το υπολογιστικό μοντέλο θεωρείται πιστοποιημένο μετά από πειραματική επιβεβαίωση ταχυτήτων και θερμοκρασιών. Το στόμιο ανακυκλοφορίας δεν πρέπει να τοποθετείται κάθετα στις μεγάλες επιφάνειες της δεξαμενής, ενώ η διάμετρος του πρέπει να είναι 1/2" ή και μικρότερη. Ο χρόνος αποκατάστασης του ροϊκού πεδίου είναι περίπου 35 s. Τα πτερύγια συγκράτησης πρέπει να ακολουθούν τις ροικές γραμμές. Ως βέλτιστη θέση του εναλλάκτη θεωρείται όταν τοποθετείται σε επαφή με το τοίχωμα. / Integrated Collector Storage (ICS) solar systems use part of the hot water storage as collector, i.e. half of the storage surface is used as absorber. Usually, the storage medium serves also as the energy transfer medium (service hot water). In the examined ICS, the service water is heated indirectly, passing through a serpentine heat exchanger placed inside the tank. The heat transfer from the stored water to the service water is intensified by the agitation of the stored water. A simple solution is the recirculation of the stored water by a small pump, which is functioning whenever a request for hot water exists. Fins in suitable positions, connect the front and back surface of the ICS, to withstand the deformation due to pressures by the tank water. In the present PhD thesis, the flow phenomena inside the ICS previous mentioned, are investigated. The aim is the maximization of the heat transfer between the two water circuits. An experimental device was constructed by transparent Plexiglas, for flow visualization and velocity measurements. A dual beam Laser Doppler Velocimetry (LDV) system was used to measure velocities. Polystyrene particles were added in the comprised water, for the visualization of the path lines. Photographs and video films were also taken. The commercial code FLUENT is used for the Computational Fluid Dynamics (CFD) simulations. A CFD model is developed and solutions are obtained using all the available turbulence models. Three main factors that influence the performance are optimized: the position and size of the recirculation ports, the arrangement and size of the interconnecting fins and the heat exchanger placement. The settling time, i.e., the time required for the flow field to be fully developed, is computed both numerically and experimentally. The previous analysis leads to the following conclusions: The standard k–ω model is selected as the most appropriate. The model is validated, with good agreement, against experimental measurements of velocities and temperatures. The placement of the inlet recirculation port perpendicular to the main surfaces of the ICS should be avoided, while its diameter should be 1/2" or less. The settling time is computed about 35s. The interconnecting fins, of the two main ICS surfaces should follow the flow filed path lines. The optimal placement of the tube heat exchanger is in contact with the two major surfaces of the storage tank.