Algorithms for minimal Picard-Fuchs operators of Feynman integrals

In even space-time dimensions the multi-loop Feynman integrals are integrals of rational function in projective space. By using an algorithm that extends the Griffiths--Dwork reduction for the case of projective hypersurfaces with singularities, we derive Fuchsian linear differential equations, the Picard--Fuchs equations, with respect to kinematic parameters for a large class of massive multi-loop Feynman integrals. With this approach we obtain the differential operator for Feynman integrals to high multiplicities and high loop orders. Using recent factorisation algorithms we give the minimal order differential operator in most of the cases studied in this paper. Amongst our results are that the order of Picard--Fuchs operator for the generic massive two-point $n-1$-loop sunset integral in two-dimensions is $2^{n}-\binom{n+1}{\left\lfloor \frac{n+1}{2}\right\rfloor }$ supporting the conjecture that the sunset Feynman integrals are relative periods of Calabi--Yau of dimensions $n-2$. We have checked this explicitly till six loops. As well, we obtain a particular Picard--Fuchs operator of order 11 for the massive five-point tardigrade non-planar two-loop integral in four dimensions for generic mass and kinematic configurations, suggesting that it arises from $K3$ surface with Picard number 11. We determine as well Picard--Fuchs operators of two-loop graphs with various multiplicities in four dimensions, finding Fuchsian differential operators with either Liouvillian or elliptic solutions.Comment: 57 pages. Results for differential operators are on the repository : https://github.com/pierrevanhove/PicardFuchs#readm

Novel procedures for graph edge-colouring

Orientador: Dr. Renato CarmoCoorientador: Dr. André Luiz Pires GuedesTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 05/12/2018Inclui referências e índiceÁrea de concentração: Ciência da ComputaçãoResumo: O índice cromático de um grafo G é o menor número de cores necessário para colorir as arestas de G de modo que não haja duas arestas adjacentes recebendo a mesma cor. Pelo célebre Teorema de Vizing, o índice cromático de qualquer grafo simples G ou é seu grau máximo , ou é ? + 1, em cujo caso G é dito Classe 1 ou Classe 2, respectivamente. Computar uma coloração de arestas ótima de um grafo ou simplesmente determinar seu índice cromático são problemas NP-difíceis importantes que aparecem em aplicações notáveis, como redes de sensores, redes ópticas, controle de produção, e jogos. Neste trabalho, nós apresentamos novos procedimentos de tempo polinomial para colorir otimamente as arestas de grafos pertences a alguns conjuntos grandes. Por exemplo, seja X a classe dos grafos cujos maiorais (vértices de grau ?) possuem soma local de graus no máximo ?2 ?? (entendemos por 'soma local de graus' de um vértice x a soma dos graus dos vizinhos de x). Nós mostramos que quase todo grafo está em X e, estendendo o procedimento de recoloração que Vizing usou na prova para seu teorema, mostramos que todo grafo em X é Classe 1. Nós também conseguimos resultados em outras classes de grafos, como os grafos-junção, os grafos arco-circulares, e os prismas complementares. Como um exemplo, nós mostramos que um prisma complementar só pode ser Classe 2 se for um grafo regular distinto do K2. No que diz respeito aos grafos-junção, nós mostramos que se G1 e G2 são grafos disjuntos tais que |V(G1)| _ |V(G2)| e ?(G1) _ ?(G2), e se os maiorais de G1 induzem um grafo acíclico, então o grafo-junção G1 ?G2 é Classe 1. Além desses resultados em coloração de arestas, apresentamos resultados parciais em coloração total de grafos-junção, de grafos arco-circulares, e de grafos cobipartidos, bem como discutimos um procedimento de recoloração para coloração total. Palavras-chave: Coloração de grafos e hipergrafos (MSC 05C15). Algoritmos de grafos (MSC 05C85). Teoria dos grafos em relação à Ciência da Computação (MSC 68R10). Graus de vértices (MSC 05C07). Operações de grafos (MSC 05C76).Abstract: The chromatic index of a graph G is the minimum number of colours needed to colour the edges of G in a manner that no two adjacent edges receive the same colour. By the celebrated Vizing's Theorem, the chromatic index of any simple graph G is either its maximum degree ? or it is ? + 1, in which case G is said to be Class 1 or Class 2, respectively. Computing an optimal edge-colouring of a graph or simply determining its chromatic index are important NP-hard problems which appear in noteworthy applications, like sensor networks, optical networks, production control, and games. In this work we present novel polynomial-time procedures for optimally edge-colouring graphs belonging to some large sets of graphs. For example, let X be the class of the graphs whose majors (vertices of degree ?) have local degree sum at most ?2 ? ? (by 'local degree sum' of a vertex x we mean the sum of the degrees of the neighbours of x). We show that almost every graph is in X and, by extending the recolouring procedure used by Vizing's in the proof for his theorem, we show that every graph in X is Class 1. We further achieve results in other graph classes, such as join graphs, circular-arc graphs, and complementary prisms. For instance, we show that a complementary prism can be Class 2 only if it is a regular graph distinct from the K2. Concerning join graphs, we show that if G1 and G2 are disjoint graphs such that |V(G1)| _ |V(G2)| and ?(G1) _ ?(G2), and if the majors of G1 induce an acyclic graph, then the join graph G1 ?G2 is Class 1. Besides these results on edge-colouring, we present partial results on total colouring join graphs, cobipartite graphs, and circular-arc graphs, as well as a discussion on a recolouring procedure for total colouring. Keywords: Colouring of graphs and hypergraphs (MSC 05C15). Graph algorithms (MSC 05C85). Graph theory in relation to Computer Science (MSC 68R10). Vertex degrees (MSC 05C07). Graph operations (MSC 05C76)

Applications of the Adversary Method in Quantum Query Algorithms

In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.Comment: PhD Thesis, 169 page

A comparative study of lower secondary mathematics textbooks from the Asia Pacific region

The rationale behind this study concerns the issues school administrators and teachers of expatriate students face over the progress and placement of the growing number of these students in mathematics classrooms in various countries brought about by the demographical changes occurring in this globalization era. This study aimed to present a method of examining lower secondary school mathematics textbooks with the purpose of evaluating students' expected past learning and comparing students' expected mathematics learning across the different curricula. It is anticipated that such an investigation will be of value to those responsible for the correct level of placement of these students.Six sets of textbooks from four countries on the Asia-Pacific rim, namely Australia, Brunei, China and Singapore, were selected for this study. The textbook content of each country was analyzed in terms of strand weighting and content details, and then coupled with information gained from interviews with teachers. This led to the findings which addressed the various issues raised.The findings facilitated a comparison of the learning paths offered by the various textbooks, fleshed out the differences and similarities of the various curricula and made available detailed comparisons of the textbooks' content in terms of topics covered. The analytical procedure of the examination of text content as presented in this study is itself a diagnostic technique for assessment of the students' past learning, which addressed the main objective of the study.The findings will be of interest to all who are interested in the mathematics taught in the countries involved.Outcomes will be particularly useful to curriculum planners and textbook writers as well as the administrators and teachers of International Schools and other schools enrolling expatriate students from these countries. The study offers a 'simplistic' way of evaluating textbooks to assess students' learning progress, and highlights the traits of the countries' curricula to provide a general idea of the mathematics ability expected from the expatriate students residing in these countries

Rooted structures in graphs: a project on Hadwiger's conjecture, rooted minors, and Tutte cycles

Hadwigers Vermutung ist eine der anspruchsvollsten Vermutungen für Graphentheoretiker und bietet eine weitreichende Verallgemeinerung des Vierfarbensatzes. Ausgehend von dieser offenen Frage der strukturellen Graphentheorie werden gewurzelte Strukturen in Graphen diskutiert. Eine Transversale einer Partition ist definiert als eine Menge, welche genau ein Element aus jeder Menge der Partition enthält und sonst nichts. Für einen Graphen G und eine Teilmenge T seiner Knotenmenge ist ein gewurzelter Minor von G ein Minor, der T als Transversale seiner Taschen enthält. Sei T eine Transversale einer Färbung eines Graphen, sodass es ein System von kanten-disjunkten Wegen zwischen allen Knoten aus T gibt; dann stellt sich die Frage, ob es möglich ist, die Existenz eines vollständigen, in T gewurzelten Minors zu gewährleisten. Diese Frage ist eng mit Hadwigers Vermutung verwoben: Eine positive Antwort würde Hadwigers Vermutung für eindeutig färbbare Graphen bestätigen. In dieser Arbeit wird ebendiese Fragestellung untersucht sowie weitere Konzepte vorgestellt, welche bekannte Ideen der strukturellen Graphentheorie um eine Verwurzelung erweitern. Beispielsweise wird diskutiert, inwiefern hoch zusammenhängende Teilmengen der Knotenmenge einen hoch zusammenhängenden, gewurzelten Minor erzwingen. Zudem werden verschiedene Ideen von Hamiltonizität in planaren und nicht-planaren Graphen behandelt.Hadwiger's Conjecture is one of the most tantalising conjectures for graph theorists and offers a far-reaching generalisation of the Four-Colour-Theorem. Based on this major issue in structural graph theory, this thesis explores rooted structures in graphs. A transversal of a partition is a set which contains exactly one element from each member of the partition and nothing else. Given a graph G and a subset T of its vertex set, a rooted minor of G is a minor such that T is a transversal of its branch set. Assume that a graph has a transversal T of one of its colourings such that there is a system of edge-disjoint paths between all vertices from T; it comes natural to ask whether such graphs contain a minor rooted at T. This question of containment is strongly related to Hadwiger's Conjecture; indeed, a positive answer would prove Hadwiger's Conjecture for uniquely colourable graphs. This thesis studies the aforementioned question and besides, presents several other concepts of attaching rooted relatedness to ideas in structural graph theory. For instance, whether a highly connected subset of the vertex set forces a highly connected rooted minor. Moreover, several ideas of Hamiltonicity in planar and non-planar graphs are discussed

Educating and training mathematics teachers for secondary schools in Ireland: a new perspective on teacher education

This thesis is a record of experiments in the education of mathematics teachers for Irish Secondary schools conducted at Thomond College of Education, Limerick during the years 1975–77 inclusive. But it is more than a mere record of successes and failures. In its analyses and syntheses, based on experiments and programmes conducted under actual conditions, it endeavours in a true spirit of research in mathematical education to provide new insights. The research culminates in the redefinition of an old problem in mathematical education, and a first step towards a viable solution to the redefined problem is presented

Gender and other factors impacting on mathematics achievement at the secondary level in Mauritius

Mathematics has been seen to act as a ‘critical filter’ in the social, economic and professional development of individuals. The Island of Mauritius relies to a great extent on its human resource power to meet the challenges of recent technological developments, and a substantial core of mathematics is needed to prepare students for their involvements in these challenges. After an analysis of the School Certificate examination results for the past ten years in Mauritius, it was found that boys were out-performing girls in mathematics at that level. This study aimed to examine this gender difference in mathematics performance at the secondary level by exploring factors affecting mathematics teaching and learning, and by identifying and implementing strategies to enhance positive factors. The study was conducted using a mixed quantitative and qualitative methodology in three phases. A survey approach was used in the Phase One of the study to analyse the performance of selected students from seventeen schools across Mauritius in a specially designed mathematics test. The attitudes of these students were also analysed through administration of the Modified Fennema-Sherman Mathematics Attitude Scale questionnaire. In Phase Two a case study method was employed, involving selected students from four Mauritian secondary schools. After the administration of the two instruments used in Phase One to these selected students, qualitative techniques were introduced. These included classroom observations and interviews of students, teachers, parents and key informants. Data from these interviews assisted in analysing and interpreting the influence of these individuals on students, and the influence of the students’ own attitudes towards mathematics on their learning of mathematics.The results of Phases One and Two provided further evidence that boys were outperforming girls in mathematics at the secondary level in Mauritius. It was noted that students rated teachers highly in influencing their learning of mathematics. However, the teaching methods usually employed in the mathematics classrooms were found to be teacher-centered, and it was apparent that there existed a lack of opportunity for students to be involved in their own learning. It was also determined that parents and peers played a significant role in students’ learning of mathematics. After having analysed the difficulties students encountered in their mathematical studies, a package was designed with a view to enhance the teaching and learning of the subject at the secondary level. The package was designed to promote student-centred practices, where students would be actively involved in their own learning, and to foster appropriate use of collaborative learning. It was anticipated that the package would motivate students towards learning mathematics and would enhance their conceptual understanding of the subject. The efficacy of the package was examined in Phase Three of the study when students from a number of Mauritian secondary schools engaged with the package over a period of three months. Pre- and post-tests were used to measure students’ achievement gains. The What Is Happening in This Class (WIHIC) questionnaire also was used to analyse issues related to the affective domains of the students. An overall appreciation of the approaches used in the teaching and learning package was provided by students in the form of self-reports.The outcomes of the Third Phase demonstrated an improvement in the achievement of students in the areas of mathematics which were tested. The students’ perceptions of the classroom learning environment were also found to be positive. Through their self-reports, students demonstrated an appreciation for the package’s strategies used in motivating them to learn mathematics and in helping them gain a better understanding of the mathematical concepts introduced