The widespread use of Dynamic Geometry Software (DGS) is raising many interesting questions and discussions as to the necessity, usefulness and meaning of proof in school mathematics. With these questions in mind, a didactical sequence on the topic “Conics” was developed in a teacher education course tailored for pre-service secondary math methods course. The idea of the didactical sequence is to introduce “Conics” using a concrete manipulative approach (paper folding) then an explorative DGS-based construction activity embedding the need for a proof. For that purpose, the DGS software serves as an intermediary tool, used to bridge the gap between the physical model and the formal symbolic system of proof. The paper will present an analysis of participants’ geometric thinking strategies, featuring proof as an embedded process in geometric construction situations.

Virtual manipulatives are employed by both preservice and inservice teachers to enhance the instructional effectiveness of physical manipulatives and related tools by addressing limitations of access, cost, and adaptability. While research into the use of emerging technologies continues, there are several variables to consider when measuring the effects of virtual manipulative use. Research design, sampling characteristics, and the type of manipulative used may influence achievement. Variables that may influence the effectiveness of virtual manipulatives include: previous experience with computers, grade level, mathematical topic, treatment length, student attitudes toward mathematics, and computer-to-student ratio.

Teaching mathematics means teaching learners to think – wrote Polya in How to Solve It? 1957. This paper intends to offer mathematics teachers suggestions for incorporating reading, writing, and speaking practices in the teaching of mathematics. Through explicit examples and explanations we intend to share ways of engaging students in deep learning of mathematics, especially using and producing written and oral texts. More specifically, we plan to broaden and deepen teachers’ understanding of strategies for guiding students’ thinking so that they grasp mathematical concepts and processes, and also bridge the divide between mathematical processes, and written and oral communication. This paper presents a core math lessons which provides numerous opportunities for the students to get actively engaged in the lesson and think about the new concepts, algorithms and ways of solving problems/ exercises. The lesson was designed for the 7th graders (13 year-olds). It was chosen to illustrate teaching by using reading and writing for understanding math processes. The teacher’s reflections after the lesson and some samples of the students’ work and feedback are included in the paper. The material in this paper is based on the author’s own extensive teaching experience; and her work in the Reading and Writing for Critical Thinking project in Romania.

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In dieser Dissertation untersuchen wir die Niedrigenergieentwicklung von Streuamplituden geschlossener Strings auf Einschleifenniveau (d.h. auf Genus eins) in einem zehndimensionalen Minkowski-Hintergrund mit Hilfe einer speziellen Klasse von Funktionen, den sogenannten modularen Graphenformen. Diese erlauben eine systematische Berechnung der Niedrigenergieentwicklung und erfüllen viele nicht-triviale algebraische- und Differentialgleichungen. Wir studieren diese Relationen detailliert und leiten Basiszerlegungen für eine große Zahl modularer Graphenformen her. Eines der Ergebnisse dieser Dissertation ist ein Mathematica-Paket, welches diese Vereinfachungen automatisiert. Wir benutzen diese Techniken, um die führenden Niedrigenergieordnungen der Streuamplitude von vier Gluonen im heterotischen String auf Einschleifenniveau zu berechnen. Für Stringamplituden auf Baumniveau bildet die Einwertigkeitsabbildung multipler Zetawerte offene Stringamplituden auf geschlossene Stringamplituden ab. Wir zeigen, dass ein bestimmter Vorschlag für die Definition einer geeigneten einschleifen-Verallgemeinerung, der sogenannten elliptische Einwertigkeitsabbildung, nicht alle Terme im heterotischen String reproduzieren kann. Ferner studieren wir eine Erzeugendenfunktion, die vermutlich die Torusintegrale aller perturbativen Theorien geschlossener Strings enthält. Wir bestimmen eine Differentialgleichung, die von dieser Erzeugendenfunktion erfüllt wird und lösen sie mit Hilfe von pfadgeordneten Exponentialen, was auf iterierte Integrale von holomorphen Eisensteinreihen führt. Da eine ähnliche Konstruktion im offenen String zur Verfügung steht, eröffnet dies außerdem eine neue Perspektive auf die elliptische Einwertigkeitsabbildung. / In this thesis, we investigate the low-energy expansion of scattering amplitudes of closed strings at one-loop level (i.e. at genus one) in a ten-dimensional Minkowski background using a special class of functions called modular graph forms. These allow for a systematic evaluation of the low-energy expansion and satisfy many non-trivial algebraic and differential relations. We study these relations in detail, leading to basis decompositions for a large number of modular graph forms which greatly reduce the complexity of the expansions of the integrals appearing in the amplitude. One of the results of this thesis is a Mathematica package which automatizes these simplifications. We use these techniques to compute the leading low-energy orders of the scattering amplitude of four gluons in the heterotic string at one-loop level. For tree-level string amplitudes, the single-valued map of multiple zeta values maps open-string amplitudes to closed-string amplitudes. The definition of a suitable one-loop generalization, a so-called elliptic single-valued map, is an active area of research and we show that a certain conjectural definition for this map, which was successfully applied to maximally supersymmetric amplitudes, cannot reproduce all terms in the heterotic string which has half-maximal supersymmetry. In order to arrive at a more systematic treatment of modular graph forms and at a different perspective on the elliptic single-valued map, we then study a generating function which conjecturally contains the torus integrals of all perturbative closed-string theories. We determine a differential equation satisfied by this generating function and solve it in terms of path-ordered exponentials, leading to iterated integrals of holomorphic Eisenstein series. Since a similar construction is available for the open string, this opens a new perspective on the elliptic single-valued map.

To meet the global hydrocarbon energy demand, it is imperative either to enhance the production from existing fields by applying innovative engineering solutions or discovering new field /resource areas. Both of these options are investigated by petroleum engineers intensively to tackle the challenges of meeting the ever-increasing demand. Meeting the energy demand as, like any other developing country, Jordan is facing a formidable challenge and requires exploration for conventional and unconventional hydrocarbon resources. As Jordan has a long exploration history for conventional reservoirs, Unconventional resource exploration and production seems to be the way to find new energy sources. Different exploration wells were drilled to evaluate the hydrocarbon potential. This research work is focusing on an experimental investigation to evaluate Jordanian hydrocarbon potential as well as to provide recommendations for future exploration activities in shale resources. The Evaluations were performed through comprehensive laboratory experiments that include measurements of Total Organic Content, Grain density, Pore Size Distribution, Specific Surface Area (BET), Mineralogy, Thermogravimetry Analysis, and Rock-Eval pyrolysis. The petrophysical properties (TOC, grain density, pore size distribution) of Jordanian shale (nine different wells) are investigated. The TOC and grain density are in an inversely proportional relationship. The TOC results show a gradual increment with the depth. All the samples have higher porosity dominated by macro pores. Fourteen (14) samples were selected primarily based on TOC (above 1.5%) for further analysis. The specific surface area results show a proportional relationship with the TOC content. Considering the petrophysical properties and mineralogy, these Jordanian shales broadly can be considered as high porosity clay and mudstone type of shale. Thermogravimetry analysis (TG/DTG) results indicate quantitative information related to organic and inorganic matter. Detection of thermos-reactive minerals, especially clay, carbonate, muscovite, pyrite is possible due to the combination of TG/DTG/DSC. The samples are examined under three different procedures which includes different heating programs. The oxidizing and inert atmospheric conditions (procedure i & ii) have the same heating program whereas procedure iii (inert atmospheric condition) has a heating program similar to the Rock-Eval pyrolysis program. The results of these samples show the complex nature of shale as well as organic matter by reacting in different stages (two or, three stages). Depending of the maturity of organic matter, the reaction occurring temperature range varies. Maximum oxidization reaction peaks happen between 479°C to 502°C. The maximum pyrolysis reaction peaks between 498°C to 521°C. Compared with complex heating (procedure iii) and rock Eval pyrolysis, S2 results indicate a high amount of inorganic compounds. Considering TGA reaction peaks and rock Eval pyrolysis results, these Jordanian shales indicate immature with low hydrocarbon generation potential. The Jordanian shale samples are analyzed by using Rock-Eval pyrolysis. Analysis results are used to interpret petroleum potential in rocks. The most important information includes organic matter types (also connected with the depositional settings), organic matter thermal maturity, and the remaining hydrocarbon generation potential in the current form. The organic geochemical analysis results indicate mostly poor to no source rock potential except JF2-760 samples. The hydrogen index (HI) and oxygen index (OI) result suggests that type iii kerogen and type iii/ iv kerogen are most likely from terrestrial and varied settings origin. The low hydrogen, as well as, low S2 value indicate very little hydrocarbon generation potential. Similarly, The Tmax and PI data indicate immature to early mature source rock status and low conversion scenario. Furthermore, the supercritical CO2 is injected into the samples, which is similar to gas flooding experiments to understand the recovery process. Hydrocarbon recovery or, CO2-shale interaction is determined by comparing three different properties (TOC, SSA, and TGA) pre-and-post supercritical CO2 injection. Supercritical CO2 injection in immature shale shows very limited property changes (TOC, SSA, and TGA) to the samples. However, in presence of hydrocarbon the pre-and post-injection property changes TOC, TGA, and SSA (BET) are noticeable enough to conclude HC recovery. Although in the case of immature shale with no hydrocarbon potential the kerogen or bitumen extraction has not been detected, which can be significant in the case of greenhouse gas storage, especially CCUS. This could reduce the risk of Organic Matter (OM) migration possibility in case immature shale formation is present in a suitable geological location.

Die Dissertation mit dem Titel 'Applications of Riesz Transforms and Monogenic Wavelet Frames in Imaging and Image Processing' beschäftigt sich mit modernen Verfahren der Signalverarbeitung in der Bildgebung sowie in der Bildverarbeitung. Hierzu werden Riesz-Transformationen und translationsinvariante Wavelet Frames zu monogenen Frames vereint und angewandt. Bekannte Techniken wie der Strukturtensor und der Energieoperator werden mit Hilfe der neuen Verfahren verbessert und für die Orientierungsbestimmung in Bildern genutzt. Eine weitere Anwendung stellt der Algorithmus 'Equalization of Brightness' dar. Er wird mit einigen Anpassungen verwendet, um eine Implementierung der monogenen Wavelet Frames mit Hilfe des NVIDIA CUDA Frameworks vorzustellen. Bei einem empirischen Vergleich der vorgestellten Techniken mit den ursprünglichen Verfahren konnten präzisere Ergebnisse mit niedrigerer Rauschanfälligkeit nachgewiesen werden. Ein weiterer Punkt der Arbeit beschäftigt sich mit den Möglichkeiten, monogene Wavelet Frames als Filter in optischen Systemen einzusetzen.:Preface Introduction 1. Time Frequency Analysis for Signal Processing 2. The Riesz Transform and Monogenic Wavelet Frames 3. Applications of Monogenic Wavelet Frames Conclusion A. Mathematical Appendix B. Source Code Listings Bibliography List of Figures