Possibilities of Preparing Pupils for Proof Problems of Synthetic Plane Geometry Solvable by Deductive Methods

Proof problems, especially the ones of the synthetic plane geometry solvable by deductive methods, play a significant role in mathematical education and due to their demanding principle also in the above-standard education including mathematical competitions. Therefore, the issue of preparing pupils for solving the proof problems is very important. This study aimed to find out if the contemporary state of the system of pupils’ preparation for synthetic plane geometry proof problems is sufficient enough for the mentioned purpose. From the full set of schools of the Czech Republic, there were 14 schools identified as the successful ones based on the results of the national round of the Mathematical Olympiad. These schools were asked questions about literature used for pupils’ preparation and the publications named in the answers were then deeply inspected. The results showed a narrow range of the literature used by the schools and the didactic-methodical inspection of stated literature detected considerable space for improvements which led the author to the main theme of his dissertation.

The <i>M</i>_w 7.5 Tadine (Maré, Loyalty Islands) earthquake and related tsunami of 5 December 2018: seismotectonic context and numerical modeling

On 5 December 2018, a magnitude Mw 7.5 earthquake occurred southeast of Maré, an island of the Loyalty Islands archipelago, New Caledonia. This earthquake is located at the junction between the plunging Loyalty Ridge and the southern part of the Vanuatu Arc, in a tectonically complex and very active area regularly subjected to strong seismic crises and earthquakes higher than magnitude 7 and up to 8. Widely felt in New Caledonia, it was immediately followed by a tsunami warning, confirmed shortly after by a first wave arrival at the Loyalty Islands tide gauges (Maré and Lifou), and then along the east coast of Grande Terre of New Caledonia and in several islands of the Vanuatu Archipelago. Two solutions of the seafloor initial deformation are considered for tsunami generation modeling, one using a non-uniform finite-source model from USGS and the other being a uniform slip model built from the Global Centroid Moment Tensor (GCMT) solution, with the geological knowledge of the region and empirical laws establishing relationships between the moment magnitude and the fault plane geometry. Both tsunami generation and propagation are simulated using the Semi-implicit Cross-scale Hydroscience Integrated System Model (SCHISM), an open-source modeling code solving the shallow-water equations on an unstructured grid allowing refinement in many critical areas. The results of numerical simulations are compared to tide gauge records, field observations and testimonials from 2018. Careful inspection of wave amplitude and wave energy maps for the two simulated scenarios shows clearly that the heterogeneous deformation model is inappropriate, while it raises the importance of the fault plane geometry and azimuth for tsunami amplitude and directivity. The arrival times, wave amplitude and polarities obtained with the uniform slip model are globally coherent, especially in far-field locations (Hienghène, Poindimié and Port Vila). Due to interactions between the tsunami waves and the numerous bathymetric structures like the Loyalty and Norfolk ridges in the neighborhood of the source, the tsunami propagating toward the south of Grande Terre and the Isle of Pines is captured by these structures acting like waveguides, allowing it to propagate to the north-northwest, especially in the Loyalty Islands and along the east coast of Grande Terre. A similar observation results from the propagation in the Vanuatu islands, from Aneityum to Efate.

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

<p>This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p>

A Mechanical Verification of the Independence of Tarski's Euclidean Axiom

<p>This thesis describes the mechanization of Tarski's axioms of plane geometry in the proof verification program Isabelle. The real Cartesian plane is mechanically verified to be a model of Tarski's axioms, thus verifying the consistency of the axiom system. The Klein–Beltrami model of the hyperbolic plane is also defined in Isabelle; in order to achieve this, the projective plane is defined and several theorems about it are proven. The Klein–Beltrami model is then shown in Isabelle to be a model of all of Tarski's axioms except his Euclidean axiom, thus mechanically verifying the independence of the Euclidean axiom — the primary goal of this project. For some of Tarski's axioms, only an insufficient or an inconvenient published proof was found for the theorem that states that the Klein–Beltrami model satisfies the axiom; in these cases, alternative proofs were devised and mechanically verified. These proofs are described in this thesis — most notably, the proof that the model satisfies the axiom of segment construction, and the proof that it satisfies the five-segments axiom. The proof that the model satisfies the upper 2-dimensional axiom also uses some of the lemmas that were used to prove that the model satisfies the five-segments axiom.</p>

Static and dynamic response of sandwich beams with lattice and pantographic cores

Mechanical performance of 3d-printed polyamide sandwich beams with different type of the lattice cores is investigated. Four variants of the beams are considered, which differ in the type of connections between the elements in the lattice structure of the core. We consider the pantographic-type lattices formed by the two families of inclined beams placed with small offset and connected by stiff joints (variant 1), by hinges (variant 2) and made without joints (variant 3). The fourth type of the core has the standard plane geometry formed by the intersected beams lying in the same plane (variant 4). Experimental tests were performed for the localized indentation loading according to the three-point bending scheme with small span-to-thickness ratio. From the experiments we found that the plane geometry of variant 4 has the highest rigidity and the highest load bearing capacity in the static tests. However, other three variants of the pantographic-type cores (1–3) demonstrate the better performance under the impact loading. The impact strength of such structures are in 3.5–5 times higher than those one of variant 4 with almost the same mass per unit length. This result is validated by using numerical simulations and explained by the decrease of the stress concentration and the stress state triaxiality and also by the delocalization effects that arise in the pantographic-type cores.

PENGEMBANGAN ALAT PERAGA DARI LIMBAH PLASTIK UNTUK MATERI GEOMETRI BIDANG DATAR JENJANG SEKOLAH DASAR [CREATING TEACHING AID FROM PLASTIC WASTE ON PLANE GEOMETRY IN ELEMENTARY SCHOOL]

<p>Plastic waste that is difficult to decompose is a serious problem in environmental pollution. Geometry is a difficult subject for students, and plastic waste can be used as teaching aids to help students learn geometry. This research is a type of development research and aims to develop teaching aids from plastic waste in learning plane geometry in fifth grade elementary school. The results showed that the teaching aids developed were valid, practical and effective. Teaching aids are called practical if they meet the interesting, gradation, independent, auto-correction, and contextual aspects. The validity of the teaching aids is obtained through aspects of suitability, completeness, convenience, and clarity. Based on the assessment of the three validators, an average score of 3.50 was obtained so that the teaching aids were categorized as valid. The practicality of teaching aids by students showed that 95% of students stated that they were interesting, graded, and independent, 74% of students stated that it was auto-correction and 91% stated that it was contextual. The effectiveness of the teaching aids can be seen from the significant difference between the average pre-test score of 46.2 and the post-test average score of 77.3. The props from the plastic waste that were developed are stored and will be used later for the same lesson in the future.</p><p><strong>BAHASA INDONESIA ABSTRACT: </strong>Limbah plastik yang sulit terurai adalah masalah yang serius dalam pencemaran lingkungan. Geometri merupakan pelajaran yang sulit bagi siswa, dan limbah plastik dapat digunakan sebagai alat peraga untuk membantu siswa dalam belajar geometri. Penelitian ini adalah jenis pengembangan dan bertujuan mengembangkan alat peraga dari limbah plastik dalam pembelajaran geometri bidang datar di kelas V SD. Hasil penelitian menunjukkan bahwa alat peraga yang dikembangkan valid, praktis dan efektif. Alat peraga disebut praktis jika memenuhi aspek menarik, bergradasi, mandiri, <em>auto correction</em>, dan kontekstual. Kevalidan alat peraga diperoleh melalui aspek kesesuaian, kelengkapan, kemudahan, dan kejelasan. Berdasarkan penilaian tiga validator diperoleh skor rata-rata 3,50 sehingga alat peraga dikategorikan valid. Kepraktisan alat peraga oleh siswa menunjukkan bahwa 95% siswa menyatakan menarik, bergradasi, dan mandiri, 74% siswa menyatakan <em>auto correction </em>dan 91% menyatakan kontekstual. Keefektifan alat peraga terlihat dari perbedaan yang signifikan antara skor rata-rata pre tes 46,2 dan skor rata-rata pos tes 77,3. Alat peraga dari limbah plastik yang dikembangkan disimpan dan akan digunakan kemudian untuk pelajaran yang sama di kemudian hari.</p>