Το πρόβλημα Fermat-Torricelli για n μη συγγραμμικά σημεία με βαρύτητες στον R^3 (b.FT) διατυπώνεται ως εξής: Δοθέντος n μη συγγραμμικών σημείων στον R^3 να βρεθεί ένα σημείο το οποίο ελαχιστοποιεί το άθροισμα των αποστάσεων με θετικές βαρύτητες του σημείου αυτού από τα n δοσμένα σημεία. Το αντίστροφο πρόβλημα Fermat-Torricelli για n μη συγγραμμικά και μη συνεπίπεδα σημεία με βαρύτητες στον R^3 (αντ.FT) διατυπώνεται ως εξής: Δοθέντος ενός σημείου που ανήκει στο εσωτερικό ενός κλειστού πολυέδρου που σχηματίζεται από n δοσμένα μη συγγραμμικά και μη συνεπίπεδα σημεία στον R^3, υπάρχει μοναδικά προσδιορίσιμο σύνολο τιμών για τις βαρύτητες που αντιστοιχούν σε κάθε ένα από τα n δοσμένα σημεία, ώστε το σημείο αυτό να επιλύει για τις τιμές αυτές των βαρυτήτων το πρόβλημα b.FT στον R^3; Στην παρούσα διατριβή, αποδεικνύουμε μία γενίκευση της ισογώνιας ιδιότητας του σημείου b.FT για ένα γεωδαισιακό τρίγωνο σε ένα Κ-επίπεδο (Σφαίρα, Υπερβολικό επίπεδο, Ευκλείδειο επίπεδο). Στη συνέχεια, δίνουμε μία αναγκαία συνθήκη για να είναι το σημείο b.FT εσωτερικό σημείο ενός τετραέδρου και ενός πενταέδρου (πυραμίδες) στον R^3. Η δεύτερη ομάδα αποτελεσμάτων της διατριβής περιλαμβάνει τη θετική απάντηση στο αντ.FT πρόβλημα για τρία μη γεωδαισιακά σημεία στο Κ-επίπεδο και στο αντ.FT πρόβλημα για τέσσερα μη συγγραμμικά και μη συνεπίπεδα σημεία στον R^3. Η αρνητική απάντηση στο αντ.FT για τέσσερα μη συγγραμμικά σημεία στον R^2 θα μας οδηγήσει σε σχέσεις εξάρτησης των βαρυτήτων που ονομάζουμε εξισώσεις της δυναμικής πλαστικότητας των τετραπλεύρων. Ομοίως, δίνοντας αρνητική απάντηση στο αντ.FT πρόβλημα για πέντε μη συνεπίπεδα σημεία στον R^3, παίρνουμε τις εξισώσεις δυναμικής πλαστικότητας , διατυπώνουμε και αποδεικνύουμε την αρχή της πλαστικότητας των κλειστών εξαέδρων στον R^3, που αναφέρει ότι: Έστω ότι πέντε προδιαγεγραμμένα ευθύγραμμα τμήματα συναντώνται στο σημείο b.FT, των οποίων τα άκρα σχηματίζουν ένα κλειστό εξάεδρο. Επιλέγουμε ένα σημείο σε κάθε ημιευθεία που ορίζει το προδιαγεγραμμένο ευθύγραμμο τμήμα, τέτοιο ώστε το τέταρτο σημείο να βρίσκεται πάνω από το επίπεδο που σχηματίζεται από την πρώτη και δεύτερη προδιαγεγραμμένη ημιευθεία και το τρίτο και πέμπτο σημείο να βρίσκονται κάτω από το επίπεδο που σχηματίζεται από την πρώτη και δεύτερη προδιαγεγραμμένη ημιευθεία. Τότε η μείωση της τιμής της βαρύτητας που αντιστοιχεί στην πρώτη, τρίτη και τέταρτη προδιαγεγραμμένη ημιευθεία προκαλεί αύξηση στις βαρύτητες που αντιστοιχούν στη δεύτερη και πέμπτη προδιαγεγραμμένη ημιευθεία.Τέλος, ένα σημαντικό αποτέλεσμα της διατριβής αφορά την επίλυση του γενικευμένου προβλήματος του Gauss για κυρτά τετράπλευρα στο Κ-επίπεδο, θέτοντας δύο σημεία στο εσωτερικό του κυρτού τετραπλεύρου με ίσες βαρύτητες, τα οποία στη συνέχεια αποδεικνύουμε ότι είναι δύο σημεία b.FT με συγκεκριμμένες βαρύτητες, αποτέλεσμα το οποίο γενικεύει το πρόβλημα b.FT για τετράπλευρα στο Κ-επίπεδo. / The weighted Fermat-Torricelli for n non-collinear points in R^3 states the following: Given n non-collinear points in R^3 find a point (b.FT point) which minimizes the sum of the distances multiplied by a positive number which corresponds to a given point (weight). The inverse Fermat-Torricelli problem for n non-collinear points with weights in R^3 (inv.FT) states the following: Given a point that belongs to the interior of a closed polyhedron which is formed between n given non-collinear points in R^3, does there exist a unique set of weights which corresponds to each one of the n points such that this point solves the weighted Fermat-Torricelli problem for this particular set of weights? In the present thesis, we prove a generalization of the isogonal property of the b.FT point for a geodesic triangle on the K-plane (Sphere, Hyperbolic plane, Euclidean plane). We proceed by giving a sufficient condition to locate the b.FT point at the interior of tetrahedra and pentahedra (pyramids) in R^3. The second group of results contains a positive answer on the inv.FT problem for three points that do not belong to a geodesic arc on the K-plane and on the inv.FT problem for four non collinear points and non coplanar in R^3. The negative answer with respect to the inv.FT problem for four non-collinear points in R^2 lead us to the relations of the dependence between the weights that we call the equations of dynamic plasticity for quadrilaterals. Similarly, by giving a negative answer with respect to the inv.FT problem for five points which do not belong in the same plane in R^3, we derive the equations of dynamic plasticity of closed hexahedra and we prove a plasticity principle of closed hexahedra in R^3, which states that: Considering five prescribed rays which meet at the weighted Fermat-Torricelli point, such that their endpoints form a closed hexahedron, a decrease on the weights that correspond to the first, third and fourth ray, causes an increase to the weights that correspond to the second and fifth ray, where the fourth endpoint is upper from the plane which is formed from the first ray and second ray and the third and fifth endpoint is under the plane which is formed from the first ray and second ray. Finally, a significant result of this thesis deals with the solution of the generalized Gauss problem for convex quadrilaterals on the K-plane in which by setting two points at the interior of the convex quadrilateral with equal weights we prove that these points are weighted Fermat-Torricelli points with specific weights, that generalizes the b.FT problem for quadrilaterals on the K-plane.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-22T13:57:08Z No. of bitstreams: 1 arquivototal.pdf: 1762696 bytes, checksum: 53bf47b7590ebc1271d2f0d81822f00c (MD5) / Made available in DSpace on 2017-08-22T13:57:08Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1762696 bytes, checksum: 53bf47b7590ebc1271d2f0d81822f00c (MD5) Previous issue date: 2017-02-24 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work aims to determine the maximum number of pairwise disjoint lines that a non-singular surface of degree d in P3 can contain. In the case of degrees d = 1 and d = 2 we found that these values are zero and in nite, respectively. Furthermore, in the case of degree d = 3 we did show that the maximum number of pairwise disjoint lines is 6, these con gurations were studied in 1863 by the Swiss Ludwig Schl a i (1814-1895) in [15]. For the case d = 4, in 1975 the Russian Viacheslav Nikulin in [10] showed that non-singular quartic surfaces contain at most 16 pairwise disjoint lines. In our work, we have been able to show that Schur's famous quartic achieves this bound and that Fermat's quartic has at most 12 pairwise disjoint lines. We also determined lower bounds for the maximum number of pairwise disjoint lines in the case of non-singular surfaces of degree d 5. For example, the Rams's family in [11] allows us to nd one of these lower bounds. / Este trabalho objetiva determinar a quantidade máxima de retas duas a duas disjuntas que uma superfície não singular de grau d em P3 pode conter. No caso dos graus d = 1 e d = 2 verificamos que estes valores s~ao zero e in nito, respectivamente. Al em disso, no caso de grau d = 3 mostramos que o n umero m aximo de retas duas a duas disjuntas e 6, ditas con gura c~oes foram estudadas em 1863 pelo sui co Ludwig Schl a i (1814-1895) em [15]. Para o caso d = 4, em 1975 o russo Viacheslav Nikulin em [10] mostrou que as superf cies qu articas n~ao singulares cont^em no m aximo 16 retas duas a duas disjuntas. No nosso trabalho, conseguimos mostrar que a famosa qu artica de Schur atinge essa cota e que qu artica de Fermat possui no m aximo 12 retas duas a duas disjuntas. Determinamos ainda cotas inferiores para o n umero m aximo de retas duas a duas disjuntas no caso de superf cies n~ao singulares de grau d 5. Por exemplo, a fam lia de Rams em [11] nos permite achar uma dessas cotas inferiores.

In recent years program transformation technology has matured into a practical solution for many software reengineering and migration tasks. FermaT, an industrial strength program transformation system, has demonstrated that legacy systems can be successfully transformed into efficient and maintainable structured C or COBOL code. Its core, a transformation engine, is based on mathematically proven program transformations and ensures that transformed programs are semantically equivalent to its original state. Its engine facilitates a Wide Spectrum Language (WSL), with low-level as well as high-level constructs, to capture as much information as possible during transformation steps. FermaT’s methodology and technique lack in provision of concurrent migration and analysis. This provision is crucial if the transformation process is to be further automated. As the constraint based program migration theory has demonstrated, it is inefficient and time consuming, trying to satisfy the enormous computation of the generated transformation sequence search-space and its constraints. With the objective to solve the above problems and to extend the operating range of the FermaT transformation system, this thesis proposes a Parallel Transformations Framework which makes parallel transformations processing within the FermaT environment not only possible but also beneficial for its migration process. During a migration process, many thousands of program transformations have to be applied. For example a 1 million line of assembler to C migration takes over 21 hours to be processed on a single PC. Various approaches of search, prediction techniques and a constraint-based approach to address the presented issues already exist but they solve them unsatisfactorily. To remedy this situation, this dissertation proposes a framework to extend transformation processing systems with parallel processing capabilities. The parallel system can analyse specified parallel transformation tasks and produce appropriate parallel transformations processing outlines. To underpin an automated objective, a formal language is introduced. This language can be utilised to describe and outline parallel transformation tasks whereas parallel processing constraints underpin the parallel objective. This thesis addresses and explains how transformation processing steps can be automatically parallelised within a reengineering domain. It presents search and prediction tactics within this field. The decomposition and parallelisation of transformation sequence search-spaces is outlined. At the end, the presented work is evaluated on practical case studies, to demonstrate different parallel transformations processing techniques and conclusions are drawn.

One of the most difficult tasks a programmer can be confronted with is the migration of a legacy system. Usually, these systems are unstructured, poorly documented and contain complex program logic. The reason for this, in most cases, is an emphasis on raw performance rather than on clean and structured code as well as a long period of applying quick fixes and enhancements rather than doing a proper software reengineering process including a full redesign during major enhancements. Nowadays, the old programming paradigms are becoming an increasingly serious problem. It has been identified that 90% of the costs of a typical software system arise in the maintenance phase. Many companies are simply too afraid of changing their software infrastructure and prefer to continue with principles like "never touch a running system". These companies experience growing pressure to migrate their legacy systems onto newer platforms because the maintenance of such systems is expensive and dangerous as the risk of losing vital parts of sources code or its documentation increases drastically over time. The FermaT transformation system has shown the ability to automatically or semi-automatically restructure and abstract legacy code within a special intermediate language called WSL (Wide Spectrum Language). Unfortunately, the current transformation process only supports the migration of assembler as WSL lacks the ability to handle data types properly. The data structures in assembler are currently directly translated into C data types which involves many assumptional “hard coded” conversions. The absence of an adequate type system for WSL caused several flaws for the whole transformation process and limits its abilities significantly. The main aim of the presented research is to tackle these problems by investigating and formulating how a type system can contribute to a safe and reliable migration of legacy systems. The described research includes the definition of key aspects of type related problems in the FermaT migration process and how to solve them with a suitable type system approach. Since software migration often includes a change in programming language the type system for WSL has to be able to support various type system approaches including the representation of all relevant details to avoid assumptions. This is especially difficult as most programming languages are designed for a special purpose which means that their possible programming constructs and data types differ significantly. This ranges from languages with simple type systems whose program sare prone to unintended side-effects, to languages with strict type systems which are constrained n their flexibility. It is important to include as many type related details as necessary to avoid making assumptions during language to language translation. The result of the investigation is a novel multi layered type system specifically designed to satisfy the needs of WSL for a sophisticated solution without imposing too many limitations on its abilities. The type system has an adjustable expressiveness, able to represent a wide spectrum of typing approaches ranging from weak typing which allows direct memory access and down casting, via very strict typing with a high diversity of data types to object oriented typing which supports encapsulation and data hiding. Looking at the majority of commercial relevant statically typed programming languages, two fundamental properties of type strictness and safety can be identified. A type system can be either weakly or strongly typed and may or may not allow unsafe features such as direct memory access. Each layer of the Wide Spectrum Type System has a different combination of these properties. The approach also includes special Type System Transformations which can be used to move a given WSL program among these layers. Other emphasised key features are explicit typing and scalability. The whole approach is based on a sound mathematical foundation which assures correctness and integrates seamlessly into the present mathematical definition of WSL. The type system is formally introduced to WSL by constructing an attribute grammar for the language. Type checking and type inference are used to annotate the Abstract Syntax Tree of a given WSL program with type derivations which can be used to reveal and indicate possible typing errors or to infer types if the program did not feature explicit type declarations in the first place. Notable in this approach is also the fact that object orientation is introduced to a procedural programming language without the introduction of new semantics. It is shown that object orientation can be introduced just by adjusting type checking rules and adding some syntactical notations. The approach was implemented and tested on two case studies. The thesis describes and discusses both cases in detail and shows how a migration which ignores type systems could accidentally introduce errors due to assumptions during translation. Both case studies use all important aspects of the approach, Including type transformations and object identification. The thesis finalises by summarising the whole work, identifying limitations, presenting future perspectives and drawing conclusions

Many properties have been found hidden in Pascal's triangle. In this paper, we will present several known properties in Pascal's triangle as well as the properties that lift to different extensions of the triangle, namely Pascal's pyramid and the trinomial triangle. We will tailor our interest towards Fermat numbers and the hockey stick property. We will also show the importance of the hockey stick properties by using them to prove a property in the trinomial triangle.

Šiame darbe matematine sistema MathCAD sukūrėme programą dideliems pirminiams skaičiams rasti. Išanalizavome pirminių, netikrų pirminių ir pseudopirminių skaičių pasiskirstymą. / In this thesis by mathematical system MathCAD we created the program for finding large prime numbers. Also, we analysed distribution of unreal prime, pseudoprime and prime numbers.

Made available in DSpace on 2014-08-13T14:50:58Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-10Bitstream added on 2014-08-13T17:59:49Z : No. of bitstreams: 1 000773210.pdf: 546618 bytes, checksum: fbdda0c883d37e49db72feaad1defd5d (MD5) / O foco deste trabalho está nos números primos, sobre os quais apresentamos algumas propriedades, primos especiais, avanços recentes e alguns testes de primalidade que detectam se um número é primo ou composto / Prime numbers have been studied for millennia and still hide many mysteries. The focus of this work is on prime numbers. We present some properties, special primes, recent advances and some primality tests that detect if a number is prime or composite

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T11:39:40Z No. of bitstreams: 2 arquivototal.pdf: 1037710 bytes, checksum: 4e3c7e69a8c60214c05fdcac3db1ec5e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T11:41:46Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 1037710 bytes, checksum: 4e3c7e69a8c60214c05fdcac3db1ec5e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-04T11:41:46Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 1037710 bytes, checksum: 4e3c7e69a8c60214c05fdcac3db1ec5e (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-08-09 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This paper is a survey on representation of integers as sums of squares for the cases where we have the sum of two, three and four squares. The idea is to study conditions so that we can ensure the representation of numbers that are written as the sum of two and four square. The central focus is the statement of the theorem of Lagrange four squares, although we have gone a little further studying Fermat' s technique of in nite descense and the case n = 3 of Fermat's last theorem. Finally, we work with the development of a didactic sequence that can be used in the nal grades of elementary school and middle school, addressing Chapter 2 of this dissertation. / Este trabalho tem como objetivo fazer uma pesquisa bibliográ fica sobre o tema da representação de inteiros como soma de quadrados, para os casos onde temos soma de dois, três e quatro quadrados. A ideia é estudar condições para que possamos garantir quando um número inteiro positivo poderá ser representado como uma soma de dois e quatro quadrados. O foco central está na demonstração do teorema dos quatro quadrados de Lagrange, apesar de termos ido um pouco adiante estudando a técnica do descenso in nito de Fernat e o caso n=3 do último teorema de Fermat. Por m, trabalhamos com a elaboração de uma sequência didática que pode ser utilizada nas séries nais do ensino fundamental e no ensino médio, cujo conteúdo abordado nesta sequência são os principais teoremas do capítulo 2 que remete a representação de inteiros como soma de quadrados.

Submitted by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-30T11:56:39Z No. of bitstreams: 1 arquivototal.pdf: 577523 bytes, checksum: 46a0fd78861d6616d08b4d54436a29ec (MD5) / Approved for entry into archive by Maria Suzana Diniz (msuzanad@hotmail.com) on 2015-11-30T12:38:08Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 577523 bytes, checksum: 46a0fd78861d6616d08b4d54436a29ec (MD5) / Made available in DSpace on 2015-11-30T12:38:08Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 577523 bytes, checksum: 46a0fd78861d6616d08b4d54436a29ec (MD5) Previous issue date: 2013-08-14 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work we present some resolution techniques of equations whit intergers coefficients. Such equations although they' re very similar, presents quite different techniques from each other. we show wich are the interger solutions of the equations like x²+y² = z² e x-² + y-² = z-², as the are know as the Pythagorean equations. we also present brief history about the Fermat' s last teorem and we show that the equation x + y4 = z4 has no interger solution. / Neste trabalho apresentamos algumas técnicas de resoluções de equações com coeficientes inteiros. Tais equações , apesar de serem bastantes,apresentam técnicas muito distintas umas das outras. Mostraremos quais são as soluções inteiras de equações do tipo x²+y² = z² e x-² + y-² = z-², que são conhecidas como equações de Pitágoras. Também apresentamos um breve histórico sobre o último teorema de Fermat e mostraremos que a equação x + y4 = z4 não possui solução inteira.

Orientadora: Carina Alves / Banca: Eliris Cristina Rizziolli / Banca: Grasiele Cristiane Jorge / Resumo: O foco deste trabalho está nos números primos, sobre os quais apresentamos algumas propriedades, primos especiais, avanços recentes e alguns testes de primalidade que detectam se um número é primo ou composto / Abstract: Prime numbers have been studied for millennia and still hide many mysteries. The focus of this work is on prime numbers. We present some properties, special primes, recent advances and some primality tests that detect if a number is prime or composite / Mestre