In this thesis we introduce and study the (i) Grassmannian, (ii) Lagrangian Grassmannian, and (iii) double Lagrangian Grassmannian of subspaces in ( A ⊗ B )n, where A and B are normed division algebras, i.e. R,C,H or O . / This gives a simple and uniform description of all symmetric spaces. This is analogous to Tits magic square description for simple Lie algebras. / We show that every irreducible compact Riemannian symmetric space X must be one of these Grassmannian spaces (up to a finite cover) or a compact simple Lie group. Furthermore, its noncompact dual symmetric space is the open sub-manifold of X consisting of spacelike linear subspaces, at least in the classical cases. / Huang, Yongdong. / "July 2007." / Adviser: Naichung Conan Leung. / Source: Dissertation Abstracts International, Volume: 69-01, Section: B, page: 0353. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2007. / Includes bibliographical references (p. 64-65). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

Thesis (Ph. D.)--Auburn University, 2008. / Abstract. Includes bibliographical references (p. 77-78).

Representations of SO(4,2;R) are constructed using 4 x 4 and 2 x 2 matrices with elements in H' ��� C . Using 2 x 2 matrix representations of C and H', the 4 x 4 representation is interpreted in terms of 16 x 16 real matrices. Finally, the known isomorphism between the conformal group and SO(4,2;R) is written explicitly in terms of the 4 x 4 representation. The 4 x 4 construction should generalize to matrices with elements in K' ��� K for K any normed division algebra over the reals and K' any split algebra over the reals. / Graduation date: 2013

Thesis (M.S.)--Rochester Institute of Technology, 2007. / Typescript. Includes bibliographical references (leaf 39).

This diploma thesis deals with basic features of magic squares and analyses these features with regard to usability during the teaching at elementary schools. Magic squares are known for hundreds years and since then they have changed due to various modifications, from which other kinds were derived. The first part of the thesis is therefore dedicated to the history. Next chapter deals with the construction of magic squares. The following chapters study similar games as Sudoku, Kakuro and Latin squares. The final part of the thesis is dedicated to the usability of magic squares in teaching mathematics. To practice the given topic, the worksheets which are divided according to their difficulty, were created.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Lehmer (1929) analisa matematicamente o método do passo uniforme para construção de quadrados mágicos de ordem impar. Ele divide sua análise em várias etapas. Na primeira delas, envolvendo a discussão de condições necessárias e suficientes para o preenchimento do quadrado pelo método, o autor afirma que se dois números guardarem entre si uma certa relação, eles serão designados a ocupar a mesma célula do quadrado causando seu não preenchimento. A análise do preenchimento pelo método do passo uniforme envolve a resolução de um sistema linear módulo n. Nesse trabalho, discutimos o comportamento das soluções desse sistema quando o método falha no preenchimento. Como consequência, concluímos que números que guardam a relação mencionada nunca ocupam a mesma célula. A análise das condições necessárias e suficientes para obter quadrados mágicos segundo a definição de Lehmer (1929) envolve a resolução de equações de congruências lineares a duas variáveis. Nesse trabalho, detalhamos os resultados de Lehmer (1929). A análise das condições necessárias e suficientes para obtenção de quadrados mágicos, como são reconhecidos usualmente, também envolve a resolução de equações de congruências lineares a duas variáveis. Discutimos o comportamento das soluções dessas equações para obter diagonais principais mágicas. Como consequência, mostramos que diagonais principais mágicas são obtidas se e somente se as coordenadas iniciais guardarem certas relações / Lehmer (1929) mathematically analyzes the uniform step method for constructing magic squares of odd order. He divides his analysis into several steps. In the first, involving a discussion of necessary and sufficient conditions for completing the square, the author states that if two numbers keep a certain relationship to each other, they will be designated to occupy the same cell of the square causing its non fulfillment. The analysis of the uniform step method involves solving a linear system module n. In this monograph, we discuss the behavior of solutions of this system when the method fails in fulfilling the square. Consequently, we conclude that numbers guarding the mentioned relationship never occupy the same cell. The analysis of necessary and sufficient conditions for obtaining magic square (as defined by Lehmer (1929)) involves solving linear congruences in two variables. In this work, we detail the results of Lehmer (1929). The analysis of the necessary and sufficient conditions for magic squares (as usually defined) also involves solving linear congruences in two variables. We discuss the behavior of solutions of these equations to obtain magic main diagonals. Then, we show that magic main diagonals are obtained if and only if the initial coordinates keep certain relationships

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Lehmer (1929) analisa matematicamente o método do passo uniforme para construção de quadrados mágicos de ordem impar. Ele divide sua análise em várias etapas. Na primeira delas, envolvendo a discussão de condições necessárias e suficientes para o preenchimento do quadrado pelo método, o autor afirma que se dois números guardarem entre si uma certa relação, eles serão designados a ocupar a mesma célula do quadrado causando seu não preenchimento. A análise do preenchimento pelo método do passo uniforme envolve a resolução de um sistema linear módulo n. Nesse trabalho, discutimos o comportamento das soluções desse sistema quando o método falha no preenchimento. Como consequência, concluímos que números que guardam a relação mencionada nunca ocupam a mesma célula. A análise das condições necessárias e suficientes para obter quadrados mágicos segundo a definição de Lehmer (1929) envolve a resolução de equações de congruências lineares a duas variáveis. Nesse trabalho, detalhamos os resultados de Lehmer (1929). A análise das condições necessárias e suficientes para obtenção de quadrados mágicos, como são reconhecidos usualmente, também envolve a resolução de equações de congruências lineares a duas variáveis. Discutimos o comportamento das soluções dessas equações para obter diagonais principais mágicas. Como consequência, mostramos que diagonais principais mágicas são obtidas se e somente se as coordenadas iniciais guardarem certas relações / Lehmer (1929) mathematically analyzes the uniform step method for constructing magic squares of odd order. He divides his analysis into several steps. In the first, involving a discussion of necessary and sufficient conditions for completing the square, the author states that if two numbers keep a certain relationship to each other, they will be designated to occupy the same cell of the square causing its non fulfillment. The analysis of the uniform step method involves solving a linear system module n. In this monograph, we discuss the behavior of solutions of this system when the method fails in fulfilling the square. Consequently, we conclude that numbers guarding the mentioned relationship never occupy the same cell. The analysis of necessary and sufficient conditions for obtaining magic square (as defined by Lehmer (1929)) involves solving linear congruences in two variables. In this work, we detail the results of Lehmer (1929). The analysis of the necessary and sufficient conditions for magic squares (as usually defined) also involves solving linear congruences in two variables. We discuss the behavior of solutions of these equations to obtain magic main diagonals. Then, we show that magic main diagonals are obtained if and only if the initial coordinates keep certain relationships

Thesis (PdD)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: A Latin square of order n is an n×n array containing an arrangement of n distinct symbols with the property that every row and every column of the array contains each symbol exactly once. It is well known that Latin squares may be used for the purpose of constructing designs which require a balanced arrangement of a set of elements subject to a number of strict constraints. An important application of Latin squares arises in the scheduling of various types of balanced sports tournaments, the simplest example of which is a so-called round-robin tournament — a tournament in which each team opposes each other team exactly once. Among the various applications of Latin squares to sports tournament scheduling, the problem of scheduling special types of mixed doubles tennis and table tennis tournaments using special sets of three mutually orthogonal Latin squares is of particular interest in this dissertation. A so-called mixed doubles table tennis (MDTT) tournament comprises two teams, both consisting of men and women, competing in a mixed doubles round-robin fashion, and it is known that any set of three mutually orthogonal Latin squares may be used to obtain a schedule for such a tournament. A more interesting sports tournament design, however, and one that has been sought by sports clubs in at least two reported cases, is known as a spouse-avoiding mixed doubles round-robin (SAMDRR) tournament, and it is known that such a tournament may be scheduled using a self-orthogonal Latin square with a symmetric orthogonal mate (SOLSSOM). These applications have given rise to a number of important unsolved problems in the theory of Latin squares, the most celebrated of which is the question of whether or not a set of three mutually orthogonal Latin squares of order 10 exists. Another open question is whether or not SOLSSOMs of orders 10 and 14 exist. A further problem in the theory of Latin squares that has received considerable attention in the literature is the problem of counting the number of (essentially) different ways in which a set of elements may be arranged to form a Latin square, i.e. the problem of enumerating Latin squares and equivalence classes of Latin squares of a given order. This problem quickly becomes extremely difficult as the order of the Latin square grows, and considerable computational power is often required for this purpose. In the literature on Latin squares only a small number of equivalence classes of self-orthogonal Latin squares (SOLS) have been enumerated, namely the number of distinct SOLS, the number of idempotent SOLS and the number of isomorphism classes generated by idempotent SOLS of orders 4 n 9. Furthermore, only a small number of equivalence classes of ordered sets of k mutually orthogonal Latin squares (k-MOLS) of order n have been enumerated in the literature, namely main classes of 2-MOLS of order n for 3 n 8 and isotopy classes of 8-MOLS of order 9. No enumeration work on SOLSSOMs appears in the literature. In this dissertation a methodology is presented for enumerating equivalence classes of Latin squares using a recursive, backtracking tree-search approach which attempts to eliminate redundancy in the search by only considering structures which have the potential to be completed to well-defined class representatives. This approach ensures that the enumeration algorithm only generates one Latin square from each of the classes to be enumerated, thus also generating a repository of class representatives of these classes. These class representatives may be used in conjunction with various well-known enumeration results from the theory of groups and group actions in order to determine the number of Latin squares in each class as well as the numbers of various kinds of subclasses of each class. This methodology is applied in order to enumerate various equivalence classes of SOLS and SOLSSOMs of orders up to and including order 10 and various equivalence classes of k-MOLS of orders up to and including order 8. The known numbers of distinct SOLS, idempotent SOLS and isomorphism classes generated by idempotent SOLS are verified for orders 4 n 9, and in addition the number of isomorphism classes, transpose-isomorphism classes and RC-paratopism classes of SOLS of these orders are enumerated. The search is further extended to determine the numbers of these classes for SOLS of order 10 via a large parallelisation of the backtracking treesearch algorithm on a number of processors. The RC-paratopism class representatives of SOLS thus generated are then utilised for the purpose of enumerating SOLSSOMs, while existing repositories of symmetric Latin squares are also used for this purpose as a means of validating the enumeration results. In this way distinct SOLSSOMs, standard SOLSSOMs, transposeisomorphism classes of SOLSSOMs and RC-paratopism classes of SOLSSOMs are enumerated, and a repository of RC-paratopism class representatives of SOLSSOMs is also produced. The known number of main classes of 2-MOLS of orders 3 n 8 are verified in this dissertation, and in addition the number of main classes of k-MOLS of orders 3 n 8 are also determined for 3 k n−1. Other equivalence classes of k-MOLS of order n that are enumerated include distinct k-MOLS and reduced k-MOLS of orders 3 n 8 for 2 k n − 1. Finally, a filtering method is employed to verify whether any SOLS of order 10 satisfies two basic necessary conditions for admitting a common orthogonal mate with its transpose, and it is found via a computer search that only four of the 121 642 class representatives of RC-paratopism classes of SOLS satisfy these conditions. It is further verified that none of these four SOLS admits a common orthogonal mate with its transpose. By this method the spectrum of resolved orders in terms of the existence of SOLSSOMs is improved in that the non-existence of such designs of order 10 is established, thereby resolving a longstanding open existence question in the theory of Latin squares. Furthermore, this result establishes a new necessary condition for the existence of a set of three mutually orthogonal Latin squares of order 10, namely that such a set cannot contain a SOLS and its transpose / AFRIKAANSE OPSOMMING: ’n Latynse vierkant van orde n is ’n n × n skikking van n simbole met die eienskap dat elke ry en elke kolom van die skikking elke element presies een keer bevat. Dit is welbekend dat Latynse vierkante gebruik kan word in die konstruksie van ontwerpe wat vra na ’n gebalanseerde rangskikking van ’n versameling elemente onderhewig aan ’n aantal streng beperkings. ’n Belangrike toepassing van Latynse vierkante kom in die skedulering van verskeie spesiale tipes gebalanseerde sporttoernooie voor, waarvan die eenvoudigste voorbeeld ’n sogenaamde rondomtalietoernooi is — ’n toernooi waarin elke span elke ander span presies een keer teenstaan. Onder die verskeie toepassings van Latynse vierkante in sporttoernooi-skedulering, is die probleem van die skedulering van spesiale tipes gemengde dubbels tennis- en tafeltennistoernooie deur gebruikmaking van spesiale versamelings van drie paarsgewys-ortogonale Latynse vierkante in hierdie proefskrif van besondere belang. In sogenaamde gemengde dubbels tafeltennis (GDTT) toernooi ding twee spanne, elk bestaande uit mans en vrouens, op ’n gemengde-dubbels rondomtalie wyse mee, en dit is bekend dat enige versameling van drie paarsgewys-ortogonale Latynse vierkante gebruik kan word om ’n skedule vir s´o ’n toernooi op te stel. ’n Meer interessante sporttoernooi-ontwerp, en een wat al vantevore in minstens twee gerapporteerde gevalle deur sportklubs benodig is, is egter ’n gade-vermydende gemengde-dubbels rondomtalie (GVGDR) toernooi, en dit is bekend dat s´o ’n toernooi geskeduleer kan word deur gebruik te maak van ’n self-ortogonale Latynse vierkant met ’n simmetriese ortogonale maat (SOLVSOM). Hierdie toepassings het tot ’n aantal belangrike onopgeloste probleme in die teorie van Latynse vierkante gelei, waarvan die mees beroemde die vraag na die bestaan van ’n versameling van drie paarsgewys ortogonale Latynse vierkante van orde 10 is. Nog ’n onopgeloste probleem is die vraag na die bestaan van SOLVSOMs van ordes 10 en 14. ’n Verdere probleem in die teorie van Latynse vierkante wat aansienlik aandag in die literatuur geniet, is die bepaling van die getal (essensieel) verskillende maniere waarop ’n versameling elemente in ’n Latynse vierkant gerangskik kan word, m.a.w. die probleem van die enumerasie van Latynse vierkante en ekwivalensieklasse van Latynse vierkante van ’n gegewe orde. Hierdie probleem raak vinnig baie moeilik soos die orde van die Latynse vierkant groei, en aansienlike berekeningskrag word dikwels hiervoor benodig. Sover is slegs ’n klein aantal ekwivalensieklasse van self-ortogonale Latynse vierkante (SOLVe) in die literatuur getel, naamlik die getal verskillende SOLVe, die getal idempotente SOLVe en die getal isomorfismeklasse voortgebring deur idempotente SOLVe van ordes 4 n 9. Verder is slegs ’n klein aantal ekwivalensieklasse van geordende versamelings van k onderling ortogonale Latynse vierkante (k-OOLVs) in die literatuur getel, naamlik die getal hoofklasse voortgebring deur 2-OOLVs van orde n vir 3 n 8 en die getal isotoopklasse voortgebring deur 8-OOLVs van orde 9. Daar is geen enumerasieresultate oor SOLVSOMs in die literatuur beskikbaar nie. In hierdie proefskrif word ’n metodologie vir die enumerasie van ekwivalensieklasse van Latynse vierkante met behulp van ’n soekboomalgoritme met terugkering voorgestel. Hierdie algoritme poog om oorbodigheid in die soektog te minimeer deur net strukture te oorweeg wat die potensiaal het om tot goed-gedefinieerde klasleiers opgebou te word. Hierdie eienskap verseker dat die algoritme slegs een Latynse vierkant binne elk van die klasse wat getel word, genereer, en dus word ’n databasis van verteenwoordigers van hierdie klasse sodoende opgebou. Hierdie klasverteenwoordigers kan tesame met verskeie welbekende groepteoretiese telresultate gebruik word om die getal Latynse vierkante in elke klas te bepaal, asook die getal verskeie deelklasse van verskillende tipes binne elke klas. Die bogenoemde metodologie word toegepas om verskeie SOLV- en SOLVSOM-klasse van ordes kleiner of gelyk aan 10 te tel, asook om k-OOLV-klasse van ordes kleiner of gelyk aan 8 te tel. Die getal verskillende SOLVe, idempotente SOLVe en isomorfismeklasse voortgebring deur SOLVe word vir ordes 4 n 9 geverifieer, en daarbenewens word die getal isomorfismeklasse, transponent-isomorfismeklasse en RC-paratoopklasse voortgebring deur SOLVe van hierdie ordes ook bepaal. Die soektog word deur middel van ’n groot parallelisering van die soekboomalgoritme op ’n aantal rekenaars ook uitgebrei na die tel van hierdie klasse voortgebring deur SOLVe van orde 10. Die verteenwoordigers van RC-paratoopklasse voortgebring deur SOLVe wat deur middel van hierdie algoritme gegenereer word, word dan gebruik om SOLVSOMs te tel, terwyl bestaande databasisse van simmetriese Latynse vierkante as validasie van die resultate ook vir hierdie doel ingespan word. Op hierdie manier word die getal verskillende SOLVSOMs, standaardvorm SOLVSOMs, transponent-isomorfismeklasse voortgebring deur SOLVSOMs asook RC-paratoopklasse voortgebring deur SOLVSOMs bepaal, en word ’n databasis van verteenwoordigers van RC-paratoopklasse voortgebring deur SOLVSOMs ook opgebou. Die bekende getal hoofklasse voortgebring deur 2-OOLVs van ordes 3 n 8 word in hierdie proefskrif geverifieer, en so ook word die getal hoofklasse voortgebring deur k- OOLVs van ordes 3 n 8 bepaal, waar 3 k n−1. Ander ekwivalensieklasse voortgebring deur k-OOLVs van orde n wat ook getel word, sluit in verskillende k-OOLVs en gereduseerde k-OOLVs van ordes 3 n 8, waar 2 k n − 1. Laastens word daar van ’n filtreer-metode gebruik gemaak om te bepaal of enige SOLV van orde 10 twee basiese nodige voorwaardes om ’n ortogonale maat met sy transponent te deel kan bevredig, en daar word gevind dat slegs vier van die 121 642 klasverteenwoordigers van RC-paratoopklasse voortgebring deur SOLVe van orde 10 aan hierdie voorwaardes voldoen. Dit word verder vasgestel dat geeneen van hierdie vier SOLVe ortogonale maats in gemeen met hul transponente het nie. Die spektrum van afgehandelde ordes in terme van die bestaan van SOLVSOMs word dus vergroot deur aan te toon dat geen sulke ontwerpe van orde 10 bestaan nie, en sodoende word ’n jarelange oop bestaansvraag in die teorie van Latynse vierkante beantwoord. Verder bevestig hierdie metode ’n nuwe noodsaaklike bestaansvoorwaarde vir ’n versameling van drie paarsgewys-ortogonale Latynse vierkante van orde 10, naamlik dat s´o ’n versameling nie ’n SOLV en sy transponent kan bevat nie. / Harry Crossley Foundation / National Research Foundation

Traditionally, squaring the circle has been about bringing the incommensurable work of the gods within the realm of the commensurate by using infinite cosmic principles to regulate the finite world. The American architect Claude Bragdon (1866-1946) squared the circle using his Theosophic architectural theory that was based on a neo-Pythagorean emphasis on Number, which he believed to have contained the secret of the universe. America at the turn of the 20th century was interested in Eastern spirituality at the beginning of an age of scientific relativity when the world and universe were being questioned due to new scientific discoveries based on higher-dimensional mathematical speculations that challenged relationships between humankind and the cosmos. Paralleling this scientific search was the Western conquest of the world on earth, which brought back speculations about the Near and Far East, including translations of their ancient scriptures and encyclopedias of their architecture. The fourth dimension was an imaginary mathematical (re)creation of great interest to Bragdon and common to scientific relativity and Eastern spirituality; two cultural constructs that altered the perception of time and space to affect the American imagination and architectural production. Within this context, Squaring the Circle investigates the relationship of theory to practice by considering Bragdon's architecture as the material manifestation of his Theosophic architectural theory. / Ph. D.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-08T12:22:35Z No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) / Approved for entry into archive by Fernando Souza (fernandoafsou@gmail.com) on 2017-09-08T13:27:24Z (GMT) No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) / Made available in DSpace on 2017-09-08T13:27:24Z (GMT). No. of bitstreams: 1 QUADRADOS LATINOS E QUADRADOS MÁGICOS UMA PROPOSTA DIDÁTICA.pdf: 24072473 bytes, checksum: 1d47842f904bd89accec69224c2a3c26 (MD5) Previous issue date: 2017-03-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we study the Latin Squares and the Magic Squares. We explore the mathematical teory and, above all, we study the link between theses objects. We bring the necessary information to support the teacher in the usage of Latin Squares and Magic Squares as content. Our goal is to discuss the usage of games and challenges like didatic tools, and to find a proposal to applicate him in the classroom. / Neste trabalho fizemos uma pesquisa bibliográfica sobre os Quadrados Latinos e os Quadrados Magicos. Mostramos a teoria matematica envolvida e, sobretudo, estudamos a ligação entre esses objetos. Trouxemos as informações necessárias para subsidiar o professor a usar Quadrados Mágicos e Quadrados Latinos como con- teúdos. Nosso objetivo é discutir o uso de jogos e passatempos como ferramenta didática e chegar a uma proposta para utilização desses objetos em sala de aula.