Mikhail Gromov’s work on hyperbolic groups in the late 1980s contributed to the formation of geometric group theory as a distinct branch of mathematics. The creation of hyperbolic metric spaces showed it was possible to define a large class of hyperbolic groups entirely geometrically yet still be able to derive significant algebraic properties. The objectives of this thesis are to provide an introduction to geometric group theory through the lens of quasi-isometry and show how hyperbolic groups have solvable word problem. Also included is the Stability Theorem as an intermediary result for quasi-isometry invariance of hyperbolicity.

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

TITLE: Word Problems in the First and Second Year of Primary School AUTHOR: Pavla Weinzettel DEPARTMENT: Departmement of Mathematics and Mathematical Education SUPERVISOR: PhDr. Michaela Kaslová ABSTRACT: This thesis focuses on word problems in maths books for Year One and Two of primary schools, published by Alter and Prodos. The objective is to analyse the word problems set out in these maths books and to classify them according to the following criteria: context, methods of solution, time management, and non- standard types of maths problems. A partial aim of the thesis is a comparison of the textbook word problem analysis and the analysis of those word problems which the students can create themselves. The conclusions of the work point out the importance of the teacher's role in maths teaching and learning. Teachers should use their judgement when using the published maths books and endeavour to expand on them, taking into consideration the above mentioned criteria in order to increase the students' eagerness to not only to solve the maths problems but also to understand as to why they solved them the way they did. The thesis recommends practical tools for teaching material evaluation and for assessing to what extent the students understand what the purpose of word problems is. KEYWORDS: word problem,...

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

Every knapsack problem may be relaxed to a cyclic group problem. In 1969, Gomory found the subadditive characterization of facets of the master cyclic group problem. We simplify the subadditive relations by the substitution of complementarities and discover a minimal representation of the subadditive polytope for the master cyclic group problem. By using the minimal representation, we characterize the vertices of cardinality length 3 and implement the shooting experiment from the natural interior point. The shooting from the natural interior point is a shooting from the inside of the plus level set of the subadditive polytope. It induces the shooting for the knapsack problem. From the shooting experiment for the knapsack problem we conclude that the most hit facet is the knapsack mixed integer cut which is the 2-fold lifting of a mixed integer cut. We develop a cutting plane algorithm augmenting cutting planes generated by shooting, and implement it on Wong-Coppersmith digraphs observing that only small number of cutting planes are enough to produce the optimal solution. We discuss a relaxation of shooting as a clue to quick shooting. A max flow model on covering space is shown to be equivalent to the dual of shooting linear programming problem.

The thesis deals with the connection between the way the mathematical word problems are formulated and their being understood correctly by the pupils. The main focus is on the semantic cues present in the assignment of the word problem. The thesis can be divided into two parts. The first part, which provides the theoretical framework for the whole thesis, contains the definition of a word problem. The phases of word problems solving and the possible classification of word problems are mentioned. More attention is paid to the problems with translation cues; new concepts of asymmetric context and symmetric context, called semantic cues, are introduced. An overview of several related research studies from abroad follows. The core of the thesis lies in its second part, containing a study that examines whether the pupils would be more successful in solving problems with asymmetric contexts than in solving problems with symmetric contexts. The influence had been examined on two types of problems and four age categories of pupils and students. The research was carried out through the short tests and it was attended by the total of 503 pupils of selected elementary schools and grammar schools in Prague and students of the Faculty of Education of Charles University. A higher success rate for the asymmetric...

With the advent of quantum computers, the security of number-theoretic cryptography has been compromised. Consequently, new cryptosystems have been suggested in the field of non-commutative group theory. In this thesis, we provide all the necessary background to understand and work with the Artin groups. We then show that Artin groups of finite type and Artin groups of large type possess an easily-computable normal form by explicitly writing the algorithms. This solution to the word problem makes these groups candidates to be cryptographic platforms. Finally, we present some combinatorial problems that can be used in group-based cryptography and we conjecture, through empirical evidence, that the conjugacy problem in Artin groups of large type is not a hard problem.

The diploma thesis is focused on the issue of word problems with different results, to which is not given systematic attention in the specialized literature. The focus of the work is to observe how pupils and teachers of elementary school work with a word problem with different results. The theoretical part focuses on the typology of word problems, the phases of their solution and solving strategies. Furthermore, the classification of word problems according to the number of results and the typology of word problems with different results is proposed. A significant part consists of the goals with which this type of task can be assigned in the teaching of mathematics and which change depending on the teacher and his approach to teaching. The work also deals with the basic characteristics of teaching styles. The selected qualitative research using the method of observation and questionnaire allows to describe the work of teachers with a word problem with different results and to identify elements of teaching styles, namely transmissive and constructivist approaches, which are reflected in the actions of teachers. An additional method is the analysis of selected textbooks in terms of the presence of word problems with different results, which reveals whether such tasks occur in textbooks and what...

Blockmodellen är en metod från Singapore som används inom problemlösning i matematiken. Det saknas svensk forskning på hur den fungerar i den svenska skolan trots att många skolor har läromedel och förutsättningar att använda metoden. Syftet med studien är att undersöka vad som bidrar till att elever i årskurs 6 förstår och kan använda blockmodellen och hur den kan bidra till deras problemlösningsförmåga utifrån planerade lektionstillfällen. Genom en designcykel skedde två lektionstillfällen, och med hjälp av inspelning av lektionerna kunde avgörande medierande handlingar identifieras utifrån Vygotskys medierande triangel. En deduktiv analys användes för att granska elevernas frågor och kommentarer i det inspelade materialet och utifrån det kunde elevernas mest avgörande handlingar presenteras. Eleverna visade sig förstå blockmodellen efter genomgångar om hur blocken ritas och med hjälp av en framtagen arbetsgång som fungerade som guide till en början. Eleverna visade även att algebraiska ekvationer var enklare att lösa med hjälp av blockmodellen, tack vare blockmodellens visuella framställning. När eleverna vet hur de ska rita och korrekt representera blocken i blockmodellen, är det en metod de kan tillämpa i sin problemlösning. / The bar model comes from Singapore and is used in word problems in mathematics. Swedish research on how the model works is missing even though many schools have the books and the conditions to use the method. The purpose of this study is to examine what it is that contributes to the understanding and usage of the method from pupils in year 6, and how it contributes to their problem-solving ability by planned lessons. Through a design cycle there were two lessons and with help from recording the lessons, crucial mediating actions could be found from the mediating triangle by Vygotsky. A deductive analysis was used to review the questions and comments from the recorded material and from that, the most crucial mediating actions could be presented. The pupils understood the bar model after briefings on how the bars should be drawn and with the help from a provided guide. The pupils also showed that algebraic equations were easier to solve with the bar model, thanks to its visual production. When the pupils know how to draw and correctly represent the bars in the bar model, they can apply the method in their problemsolving.

1 Solving strategies of word problems depending on the context topicality ABSTRACT The thesis deals with word problems solvable by linear equations. The aim of the work is to show if the context, on which the word problem depends, influences how pupils participating in the research solve it and verify if word problem phrasing and modern language usage help pupils solve the word problem. It is accomplished by assigning pupils a set of varied word problems and assessing their solutions. More results were gained by assessing questionnaires related to the context of the word problems being calculated. The second aim of the thesis is to explain various definitions of word problems, putting the term into the context of school mathematics and a brief summary of the historical development of mathematics focusing on word problems. The work consists of three parts. In the first part, problems and word problems are defined and it is shown how word problems are included in the Framework educational programme. Word problems are put into historical context. The second part shows the conception of motion word problems, word problems on joint work and word problems on dividing a whole into parts in the present-day mathematical textbooks in the second stage of elementary schools and in the lower classes of secondary grammar...