As avaliações sensoriais tomam cada vez mais sua posição de importância dentro dos centros produtores e vendedores de alimentos e de outros produtos. Nestes, o objetivo final dos trabalhos realizados nas áreas de desenvolvimento, produção e `marketing\' e o consumidor, cuja avaliação se baseia, principalmente, na aceitabilidade e custos dos produtos. Nesses experimentos, uma serie de tratamentos e dada para cada provador, e um problema relevante e que a variável resposta dependa não só do tratamento aplicado atualmente, mas também do anterior seguido a ele, chamados de efeitos residuais. Visando uma melhor qualidade do produto, analises cada vez mais rigorosas são exigidas. Assim, um método frequentemente utilizado e o da analise descritiva, que tem por objetivo descrever e avaliar a intensidade dos atributos sensoriais dos produtos avaliados, orientando eventuais modificações das características das mesmas a m de atender as exigências do consumidor. Realizada por pessoas treinadas, com habilidade de discriminar, recebendo o nome de analise descritiva quantitativa (ADQ). Consequentemente, dadas as limitações quanto ao numero de provas sucessivas de degustação e presença frequente de efeitos residuais, o planejamento e analise dos experimentos para ADQ adquirem importância fundamental. Assim, de modo a resolver o problema apresentado, Williams (1949) apresentou os delineamentos quadrados latinos balanceados para vizinhança que, de forma geral, garantem que os efeitos residuais dos tratamentos não exerçam influência sobre a comparação dos efeitos dos tratamentos. Métodos adequados de construção, aleatorizacão e analise, utilizando o método ADQ de tais delineamentos são descritos e adaptados para o problema. São apresentados, analisados e discutidos, ainda, os resultados de um experimento de analise sensorial de diferentes cachaças, planejado e conduzido pela autora. Assim, com os resultados obtidos, concluiu-se que, para o planejamento de ensaios para a análise descritiva quantitativa (ADQ), os quadrados latinos balanceados para vizinhança, com a última coluna repetida, são uma alternativa importante / The sensory evaluations are increasingly taking its position of importance within the centers producers and sellers of food and other products. In these, the ultimate goal of the work in the areas of development, production and \'marketing\' is the consumer, whose evaluation is based mainly on the acceptability and cost of products. In these experiments, a series of treatments is given to each panelist, and a major problem is that the response depends not only on the treatment currently applied, but on the former followed by him. For a better quality product, analyzes increasingly stringent are required. Therefore, a method often used is descriptive analysis, which aims to describe and evaluate the intensity of sensory attributes of the products evaluated, guiding future modications of the same characteristics in order to meet consumer demands.Performed by trained people, with ability to discriminate, receiving the name of quantitative descriptive analysis (QDA). Therefore, given the limitations on the number of successive tasting trials and frequent presence of residual eects, planning and analysing the experiments for QDA are fundamentaly important. Thus, in order to solve the problem presented, Williams (1949) presented the Latin square design balanced for neighborhood that, in general, ensuring that the residual eects of the treatments do not in uence the comparison of treatment eects. Appropriate methods of construction, randomization and analysis, using the method of QDA such designs are described and adapted to the problem. Are presented, analyzed and discussed, yet, the results of an experiment of sensory analysis of dierent brandy, planned and conducted by the author. So with these results, we concluded that, for the planning of tests to quantitative descriptive analysis (QDA), the Latin squares balanced for neighborhoods, and repeated the last column, are an important alternative.

Let P be an n×n partial Latin square every non-empty cell of which lies in a fixed row r, a fixed column c or contains a fixed symbol s. Assume further that s is the symbol of cell (r,c) in P. We prove that P is completable to a Latin square if n≥8 and n is divisible by 4, or n≤7 and n∉{3,4,5}. Moreover, we present a polynomial algorithm for the completion of such a partial Latin square.

A net is a special configuration of lines and points in the projective plane. There are certain restrictions on the number of its lines and points. We proved that there cannot be any (4,4) nets in CP^2. In order to show this, we use tropical algebraic geometry. We tropicalize the hypothetical net and show that there cannot be such a configuration in CP^2.

A computer generation of all pairs of mutually orthogonal Latin squares of order ten and dimension 35 or less is undertaken. All such pairs are successfully generated up to main class equivalence. No pairs of mutually orthogonal Latin squares of order ten exist for dimension 33. Six dimension 34 pairs, which are counterexamples to a conjecture by Moorehouse, are found. Eighty-five pairs can be formed with dimension 35. None of the pairs can be extended to a triple. If a triple of mutually orthogonal Latin squares exists for order ten, the pairs of Latin squares in the triple must be of dimension 36 or 37.

Submitted by Ricardo Pessoa Dos Santos null (ricopessoa@gmail.com) on 2017-12-14T17:35:33Z No. of bitstreams: 1 Dissertação.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) / Approved for entry into archive by Elza Mitiko Sato null (elzasato@ibilce.unesp.br) on 2017-12-14T18:53:30Z (GMT) No. of bitstreams: 1 santos_rp_me_sjrp.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) / Made available in DSpace on 2017-12-14T18:53:30Z (GMT). No. of bitstreams: 1 santos_rp_me_sjrp.pdf: 4489608 bytes, checksum: 2c9d751844c4b178546f2154b0718705 (MD5) Previous issue date: 2017-11-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Iremos apresentar a um grupo de alunos do Ensino Médio da rede pública de Ensino do Estado de São Paulo, o mundialmente conhecido quebra cabeças Sudoku, e realizar com eles várias atividades buscando apresentá-lo como subsídio didático na aprendizagem de conceitos matemáticos importantes, além de proporcionar oportunidades de aprimorar a concentração e o raciocínio lógico. Iremos explorar conceitos matemáticos ocultos por trás de suas linhas, colunas e blocos, partindo de uma das primeiras perguntas que podem ser feitas: Qual é a quantidade total de jogos válidos existentes? Para responde-la, será proposto a realização de diversas atividades, primeiramente com um Shidoku (matriz 4 × 4), em seguida iremos calcular o total desses jogos. O tamanho reduzido dessa grade, facilita os cálculos manuais, permitindo visualizar e compreender o processo utilizado, aproveitando para introduzir o princípio fundamental da contagem. A discussão principal desse trabalho, concentra-se na exploração de um método para se determinar a quantidade de jogos válidos existentes para um Sudoku, e para isso, utilizaremos as demonstrações de Bertrand Felgenhauer e Frazer Jarvis. Também apresentaremos um método capaz de gerar uma grade completa de Sudoku, partindo de uma matriz quadrada de ordem 3, que em seguida, será utilizada para gerar uma solução de Sudoku ortogonal. Finalizando, iremos apresentar e explorar algumas formas diferenciadas para os quebra cabeças Sudoku, mostrando variações no formato dos blocos, no tamanho das grades e uma variação que utiliza formas geométricas em suas pistas (Shapedoku). Como desafio de leitura, pesquisa e aprofundamento, será proposto o problema ainda em aberto do número mínimo de dados iniciais para se ter um jogo válido. Podemos afirmar que um dos objetivos esperados, é que tal atividade venha interferir na concentração e raciocínio, auxiliando nas atividades propostas nesse trabalho e que possam ser utilizadas em outros problemas do cotidiano. / We will present to a group of high school students of the public Education of Sao Paulo state, the world-known puzzle Sudoku, and perform with them several activities seeking to present it as a didactic subsidy in the learning important mathematical concepts, besides opportunities to enhance concentration and logical reasoning. We will explore hidden mathematical concepts behind their lines, columns and blocks, starting from one of the rst questions that can be asked: What is the total number of valid games in existence? To answer this question, it will be proposed to perform several activities, rst with a Shidoku (4 × 4 matrix), then we will calculate the total of these games. The reduced size of this grid facilitates manual calculations, allowing to visualize and understand the process used, taking advantage to introduce the fundamental principle of counting. The main discussion of this paper focuses on the exploration of a method to determine the amount of valid games existing for a Sudoku, and for that, we will use the demonstrations of Bertrand Felgenhauer and Frazer Jarvis. We will also present a method capable of generating a complete Sudoku grid, starting from a square matrix of order 3, which will then be used to generate an orthogonal Sudoku solution. Finally, we will introduce and explore some di erent shapes for the Sudoku puzzle, showing variations in the shape of the blocks, the size of the grids and a variation that uses geometric forms in their tracks (Shapedoku). As a challenge for reading, searching and deepening, the open problem of the minimum number of initial data to have a valid game will be proposed. We can say that one of the expected objectives is that such activity will interfere in concentration and reasoning, helping in the activities proposed in this paper and that can be used in other daily problems. / 3107510001F5

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.

As avaliações sensoriais tomam cada vez mais sua posição de importância dentro dos centros produtores e vendedores de alimentos e de outros produtos. Nestes, o objetivo final dos trabalhos realizados nas áreas de desenvolvimento, produção e `marketing\' e o consumidor, cuja avaliação se baseia, principalmente, na aceitabilidade e custos dos produtos. Nesses experimentos, uma serie de tratamentos e dada para cada provador, e um problema relevante e que a variável resposta dependa não só do tratamento aplicado atualmente, mas também do anterior seguido a ele, chamados de efeitos residuais. Visando uma melhor qualidade do produto, analises cada vez mais rigorosas são exigidas. Assim, um método frequentemente utilizado e o da analise descritiva, que tem por objetivo descrever e avaliar a intensidade dos atributos sensoriais dos produtos avaliados, orientando eventuais modificações das características das mesmas a m de atender as exigências do consumidor. Realizada por pessoas treinadas, com habilidade de discriminar, recebendo o nome de analise descritiva quantitativa (ADQ). Consequentemente, dadas as limitações quanto ao numero de provas sucessivas de degustação e presença frequente de efeitos residuais, o planejamento e analise dos experimentos para ADQ adquirem importância fundamental. Assim, de modo a resolver o problema apresentado, Williams (1949) apresentou os delineamentos quadrados latinos balanceados para vizinhança que, de forma geral, garantem que os efeitos residuais dos tratamentos não exerçam influência sobre a comparação dos efeitos dos tratamentos. Métodos adequados de construção, aleatorizacão e analise, utilizando o método ADQ de tais delineamentos são descritos e adaptados para o problema. São apresentados, analisados e discutidos, ainda, os resultados de um experimento de analise sensorial de diferentes cachaças, planejado e conduzido pela autora. Assim, com os resultados obtidos, concluiu-se que, para o planejamento de ensaios para a análise descritiva quantitativa (ADQ), os quadrados latinos balanceados para vizinhança, com a última coluna repetida, são uma alternativa importante / The sensory evaluations are increasingly taking its position of importance within the centers producers and sellers of food and other products. In these, the ultimate goal of the work in the areas of development, production and \'marketing\' is the consumer, whose evaluation is based mainly on the acceptability and cost of products. In these experiments, a series of treatments is given to each panelist, and a major problem is that the response depends not only on the treatment currently applied, but on the former followed by him. For a better quality product, analyzes increasingly stringent are required. Therefore, a method often used is descriptive analysis, which aims to describe and evaluate the intensity of sensory attributes of the products evaluated, guiding future modications of the same characteristics in order to meet consumer demands.Performed by trained people, with ability to discriminate, receiving the name of quantitative descriptive analysis (QDA). Therefore, given the limitations on the number of successive tasting trials and frequent presence of residual eects, planning and analysing the experiments for QDA are fundamentaly important. Thus, in order to solve the problem presented, Williams (1949) presented the Latin square design balanced for neighborhood that, in general, ensuring that the residual eects of the treatments do not in uence the comparison of treatment eects. Appropriate methods of construction, randomization and analysis, using the method of QDA such designs are described and adapted to the problem. Are presented, analyzed and discussed, yet, the results of an experiment of sensory analysis of dierent brandy, planned and conducted by the author. So with these results, we concluded that, for the planning of tests to quantitative descriptive analysis (QDA), the Latin squares balanced for neighborhoods, and repeated the last column, are an important alternative.

Le but de cette thèse est l'étude des Automates Cellulaires (AC) dans la perspective des fonctions booléennes et des dessins combinatoires. Au-delà de son intérêt théorique, cette recherche est motivée par ses applications à la cryptographie, puisque les fonctions booléennes et les dessins combinatoires sont utilisés pour construire des générateurs de nombres pseudo aléatoires (Pseudorandom Number Generators, PRNG) et des schémas de partage de secret (Secret Sharing Schemes, SSS). Les résultats présentés dans la thèse ont été développés sur trois lignes de recherche, organisées comme suit. La première ligne porte sur l'utilisation des algorithmes d'optimisation heuristique pour chercher des fonctions booléennes ayant des bonnes propriétés cryptographiques, à utiliser comme des règles locales dans des PRNG basés sur les AC. La motivation principale est l'amélioration du générateur de Wolfram basé sur la règle 30, qui a été montré être vulnérable vis à vis de deux attaques cryptanalytiques. La deuxième ligne s'occupe des fonctions booléennes vectorielles engendrées par les règles globales des AC. La première contribution considère la période des pré-images des configurations spatialement périodiques dans les AC surjectifs, et l'analyse des propriétés cryptographiques des règles globales des AC. La troisième ligne se concentre sur les dessins combinatoires engendrés par les AC, en considérant les Carrés Latins Orthogonaux (Orthogonal Latin Squares, OLS), qui sont équivalents aux SSS. En particulier, on donne une caractérisation algébrique des OLS engendrés par les AC linéaires, et on utilise des algorithmes heuristiques pour construire des OLS basés sur des AC non linéaires. / The goal of this thesis is the investigation of Cellular Automata (CA) from the perspective of Boolean functions and combinatorial designs. Beside its theoretical interest, this research finds its motivation in cryptography, since Boolean functions and combinatorial designs are used to construct Pseudorandom Number Generators (PRNG) and Secret Sharing Schemes (SSS). The results presented in the thesis are developed along three research lines, organized as follows. The first line considers the use of heuristic optimization algorithms to search for Boolean functions with good cryptographic properties, to be used as local rules in CA-based PRNG. The main motivation is to improve Wolfram's generator based on rule 30, which has been shown to be vulnerable against two cryptanalytic attacks. The second line deals with vectorial Boolean functions induced by CA global rules. The first contribution considers the period of preimages of spatially periodic configurations in surjective CA, and analyze the cryptographic properties of CA global rules. The third line focuses on the combinatorial designs generated by CA, specifically considering Orthogonal Latin Squares (OLS), which are equivalent to SSS. In particular, an algebraic characterization of OLS generated by linear CA is given, and heuristic algorithms are used to build OLS based on nonlinear CA.

A design is a set of points, V, together with a set of subsets of V called blocks. A classic type of design is a balanced incomplete block design, where every pair of points occurs together in a block the same number of times. This ‘balanced’ condition can be replaced with other properties. An adesign is a design where instead every pair of points occurs a different number of times together in a block. The number of times a specified pair of points occurs together is called the pair frequency. Here, a special type of adesign is explored, called a Sarvate-Beam design, named after its founders D.G. Sarvate and W. Beam. In such an adesign, the pair frequencies cover an interval of consecutive integers. Specifically the existence of Sarvate-Beam group divisible designs are investigated. A group divisible design, in the usual sense, is a set of points and blocks where the points are partitioned into subsets called groups. Any pair of points contained in a group have pair frequency zero and pairs of points from different groups have pair frequency one. A Sarvate-Beam group divisible design, or SBGDD, is a group divisible design where instead the frequencies of pairs from different groups form a set of distinct nonnegative consecutive integers. The SBGDD is said to be uniform when the groups are of equal size. The main result of this dissertation is to completely settle the existence question for uniform SBGDDs with blocks of size three where the smallest pair frequency, called the starting frequency, is zero. Higher starting frequencies are also considered and settled for all positive integers except when the SBGDD is partitioned into eight groups where a few possible exceptions remain. A relationship between these designs and graph decompositions is developed and leads to some generalizations. The use of matrices and linear programming is also explored and give rise to related results. / Graduate

A critical set in a Latin square of order n is a set of entries in an n x n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)<=n2-n. In Chapter 4, it is shown that lcs(n)<=n2-3n+3. Chapter 5 provides new bounds on the maximum number of intercalates in Latin squares of orders mX2^alpha (m odd, alpha>=2) and mX2^alpha+1 (m odd, alpha>=2 and alpha not equal to 3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n2 divided by 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares.