The main purpose of this work is creation a helpful material for teachers at elementary school, where they could find the parting of the special tasks with an inspiration for math education in accordance with National school curriculum. The whole graduation theses is devided into three main parts. The first part is theoretical and contains the main ideas od National school curriculum, math textbook analyse and a part of mathematical competitions. The second part is special tasks classification with display of some pupils´solutions. The last part of the graduation theses is a textbook of special tasks.

This work deals with analytical representation of congruent transforma- tion in a plane at high schools. The work is a web page, it contains several interactive components which help student to understand the problem like hyperlinks, stepping of the construction, applets, etc. The work contains several solved problems. Main emphasis is placed on the connection of the synthetic and the analytic approach to congruent transformation. The work uses knowledge of analytic geometry students learn at high school. The web page is aimed at talented students of high school and for teaching at mathematical class. 1

The master thesis is about design and optimization of tractor front drive axle console. The first part, include the design of the new console geometry with respect of the surrounding area. After that is on the newly designed part performed a strain-stress analysis for three selected operating modes of the tractor, which have been defined based on critical operating situations. In thanks of the results of the individual states, the geometry of the component is adjusted according to meet the fatigue failure limit state. After inclusion of critical site modifications, the console design was subjected to design optimization in the ANSYS Workbench program to reduce the amount of material by 10%. The end of the work, is carried out the evaluation of these places and then the console control calculation.

The master’s thesis deals with computational fluid dynamics (CFD) of two orifices, that have different shapes of holes but similar cross-sectional flow areas. The first of them is orifice with circular-shaped hole, which is used for maintenance free measurement of flow. The second one is orifice with fractal-shaped hole, inspired by von Koch snow-flake. This thesis follows bachelor thesis, in which was experimentally examined, that fractal-shaped orifices have better hydraulic properties (hydraulic losses and lower pressure pulsations) than circle-shaped one. The main target is to confirm this conclusion based on experiment, this time using CFD with various types of turbulence modelling ap-proaches. Both single phase (cavitation free) and multiphase numerical simulations were realized. Each model was compared from perspective of hydraulic and dynamic charac-teristics.

This diploma thesis deals with the creation of computational model of arterial aneurysm, which respects its real geometry and constitutive behaviour. Subsequently there is explored the risk of rupture, comparing the values of the highest main stress in aneurysm´s and intact aorta´s wall. This thesis also offers the comprehensive summary and mutual comparision of current constitutive (hyperelastic) models, which are for simplicity isochoric considered. The main task of this thesis is to formulate the method for finding of „unloaded“ = „outstressed“ = „primary“ = reduced geometry, which is generally unavailable. Models of aneurysm´s geometry, used so far, present already deformed configurations from blood pressure, axial prestrech and rezidual strain.

This work deals with application problems regarding to geometry issues. A deep search has been performed throughout existing excercise books from various periods and of various difficulties. Then, 10 new application problems have been devised and solved, concerning miscellaneous mathematics education stages and geometry branches.

Navigation meshes are a widely used method for representing the world geometry in a format that can be used by pathfinding algorithms. Frequently used navigation mesh generation algorithms first discretize the input ge- ometry into a grid of voxels and then reconstruct the mesh out of them. This benefits the simplicity and performance of the algorithm, but comes with drawbacks. If the voxels are too large, the navigation mesh is not precise enough and may even have some pathways missing. If the voxels are too small, creation of the mesh takes too long. In this thesis we propose and implement an algorithm that creates a navigation mesh directly from the input geometry without using an intermediate voxel representation. This allows us to preserve original detail where required and results in a more precise navigation mesh. 1

Constraining graph layouts - that is, restricting the placement of vertices and the routing of edges to obey certain constraints - is common practice in graph drawing. In this book, we discuss algorithmic results on two different restriction types: placing vertices on the outer face and on the integer grid. For the first type, we look into the outer k-planar and outer k-quasi-planar graphs, as well as giving a linear-time algorithm to recognize full and closed outer k-planar graphs Monadic Second-order Logic. For the second type, we consider the problem of transferring a given planar drawing onto the integer grid while perserving the original drawings topology; we also generalize a variant of Cauchy's rigidity theorem for orthogonal polyhedra of genus 0 to those of arbitrary genus. / Das Einschränken von Zeichnungen von Graphen, sodass diese bestimmte Nebenbedingungen erfüllen - etwa solche, die das Platzieren von Knoten oder den Verlauf von Kanten beeinflussen - sind im Graphzeichnen allgegenwärtig. In dieser Arbeit befassen wir uns mit algorithmischen Resultaten zu zwei speziellen Einschränkungen, nämlich dem Platzieren von Knoten entweder auf der Außenfacette oder auf ganzzahligen Koordinaten. Für die erste Einschränkung untersuchen wir die außen k-planaren und außen k-quasi-planaren Graphen und geben einen auf monadische Prädikatenlogik zweiter Stufe basierenden Algorithmus an, der überprüft, ob ein Graph voll außen k-planar ist. Für die zweite Einschränkung untersuchen wir das Problem, eine gegebene planare Zeichnung eines Graphen auf das ganzzahlige Koordinatengitter zu transportieren, ohne dabei die Topologie der Zeichnung zu verändern; außerdem generalisieren wir eine Variante von Cauchys Starrheitssatz für orthogonale Polyeder von Geschlecht 0 auf solche von beliebigem Geschlecht.

TITLE: Experiments in DGS AUTHOR: Michal Sláma DEPARTMENT: Department of Mathematics and Mathematical Education SUPERVISOR: doc. RNDr. Antonín Jančařík, Ph. D. ABSTRACT The aim of this diploma thesis is to find out whether the pupils of the second stage of primary school, specifically the pupils of the 6th grade, are able to experiment independently in the environment of dynamic geometry with a suitably prepared material. The theoretical basis of the work includes the study of the valid intended curriculum of geometry at the second stage of primary school, the relationship of so-called inquiry-oriented teaching to the possibility of student experimentation in learning geometry, the study of pedagogical theories in connection with inquiry education and also in relation to mathematisation, which is in the teaching of mathematics and geometry necessarily required. The student experiment is conceived here as an independent didactic method. From the point of view of the feasibility of the experiment and subsequent research, the environment of dynamic geometry was chosen - the GeoGebra system. The theoretical part of the work ends with a description of parameters important for the feasibility of experimentation in the environment of dynamic geometry in the teaching of geometry. The research part deals with the...

Graphs provide a key means to model relationships between entities. They consist of vertices representing the entities, and edges representing relationships between pairs of entities. To make people conceive the structure of a graph, it is almost inevitable to visualize the graph. We call such a visualization a graph drawing. Moreover, we have a straight-line graph drawing if each vertex is represented as a point (or a small geometric object, e.g., a rectangle) and each edge is represented as a line segment between its two vertices. A polyline is a very simple straight-line graph drawing, where the vertices form a sequence according to which the vertices are connected by edges. An example of a polyline in practice is a GPS trajectory. The underlying road network, in turn, can be modeled as a graph. This book addresses problems that arise when working with straight-line graph drawings and polylines. In particular, we study algorithms for recognizing certain graphs representable with line segments, for generating straight-line graph drawings, and for abstracting polylines. In the first part, we first examine, how and in which time we can decide whether a given graph is a stick graph, that is, whether its vertices can be represented as vertical and horizontal line segments on a diagonal line, which intersect if and only if there is an edge between them. We then consider the visual complexity of graphs. Specifically, we investigate, for certain classes of graphs, how many line segments are necessary for any straight-line graph drawing, and whether three (or more) different slopes of the line segments are sufficient to draw all edges. Last, we study the question, how to assign (ordered) colors to the vertices of a graph with both directed and undirected edges such that no neighboring vertices get the same color and colors are ascending along directed edges. Here, the special property of the considered graph is that the vertices can be represented as intervals that overlap if and only if there is an edge between them. The latter problem is motivated by an application in automated drawing of cable plans with vertical and horizontal line segments, which we cover in the second part. We describe an algorithm that gets the abstract description of a cable plan as input, and generates a drawing that takes into account the special properties of these cable plans, like plugs and groups of wires. We then experimentally evaluate the quality of the resulting drawings. In the third part, we study the problem of abstracting (or simplifying) a single polyline and a bundle of polylines. In this problem, the objective is to remove as many vertices as possible from the given polyline(s) while keeping each resulting polyline sufficiently similar to its original course (according to a given similarity measure). / Graphen stellen ein wichtiges Mittel dar, um Beziehungen zwischen Objekten zu modellieren. Sie bestehen aus Knoten, die die Objekte repräsentieren, und Kanten, die Beziehungen zwischen Paaren von Objekten abbilden. Um Menschen die Struktur eines Graphen zu vermitteln, ist es nahezu unumgänglich den Graphen zu visualisieren. Eine solche Visualisierung nennen wir Graphzeichnung. Eine Graphzeichnung ist geradlinig, wenn jeder Knoten als ein Punkt (oder ein kleines geometrisches Objekt, z. B. ein Rechteck) und jede Kante als eine Strecke zwischen ihren beiden Knoten dargestellt ist. Eine sehr einfache geradlinige Graphzeichnung, bei der alle Knoten eine Folge bilden, entlang der die Knoten durch Kanten verbunden sind, nennen wir Polylinie. Ein Beispiel für eine Polylinie in der Praxis ist eine GPS-Trajektorie. Das zugrundeliegende Straßennetzwerk wiederum kann als Graph repräsentiert werden. In diesem Buch befassen wir uns mit Fragen, die sich bei der Arbeit mit geradlinigen Graphzeichnungen und Polylinien stellen. Insbesondere untersuchen wir Algorithmen zum Erkennen von bestimmten mit Strecken darstellbaren Graphen, zum Generieren von geradlinigen Graphzeichnungen und zum Abstrahieren von Polylinien. Im ersten Teil schauen wir uns zunächst an, wie und in welcher Zeit wir entscheiden können, ob ein gegebener Graph ein Stickgraph ist, das heißt, ob sich seine Knoten als vertikale und horizontale Strecken auf einer diagonalen Geraden darstellen lassen, die sich genau dann schneiden, wenn zwischen ihnen eine Kante liegt. Anschließend betrachten wir die visuelle Komplexität von Graphen. Konkret untersuchen wir für bestimmte Graphklassen, wie viele Strecken für jede geradlinige Graphzeichnung notwendig sind, und, ob drei (oder mehr) verschiedene Streckensteigungen ausreichend sind, um alle Kanten zu zeichnen. Zuletzt beschäftigen wir uns mit der Frage, wie wir den Knoten eines Graphen mit gerichteten und ungerichteten Kanten (geordnete) Farben zuweisen können, sodass keine benachbarten Knoten dieselbe Farbe haben und Farben entlang gerichteter Kanten aufsteigend sind. Hierbei ist die spezielle Eigenschaft der betrachteten Graphen, dass sich die Knoten als Intervalle darstellen lassen, die sich genau dann überschneiden, wenn eine Kanten zwischen ihnen verläuft. Das letztgenannte Problem ist motiviert von einer Anwendung beim automatisierten Zeichnen von Kabelplänen mit vertikalen und horizontalen Streckenverläufen, womit wir uns im zweiten Teil befassen. Wir beschreiben einen Algorithmus, welcher die abstrakte Beschreibung eines Kabelplans entgegennimmt und daraus eine Zeichnung generiert, welche die speziellen Eigenschaften dieser Kabelpläne, wie Stecker und Gruppen von zusammengehörigen Drähten, berücksichtigt. Anschließend evaluieren wir die Qualität der so erzeugten Zeichnungen experimentell. Im dritten Teil befassen wir uns mit dem Abstrahieren bzw. Vereinfachen einer einzelnen Polylinie und eines Bündels von Polylinien. Bei diesem Problem sollen aus einer oder mehreren gegebenen Polylinie(n) so viele Knoten wie möglich entfernt werden, wobei jede resultierende Polylinie ihrem ursprünglichen Verlauf (nach einem gegeben Maß) hinreichend ähnlich bleiben muss.