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>

Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions $(r,t,z)$ ( r , t , z ) due to the presence of the term $\frac{1}{r}$ 1 r , which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

Chess, visual memory and geometric transformations

This work shows how playing chess creates capacities in the student such as increasing visual memory. This helps to classify information in an orderly manner in the mind and contributes to a better understanding of geometric transformations such as displacements, turns and similarities. This was done with a mixed technique (Quantitative and Qualitative), starting with a structured questionnaire that was applied to 487 students. A case study was carried out with two students (one with and the other without notable chess skills) in two schools in Bogotá-Colombia, with the aim of understanding chess as a tool that can help the teacher to teach mathematics¡. In the quantitative part, data were collected by a structured questionnaire, and in the qualitative part, recordings and transcripts were made of what the two students reported in the case study. So, favorable results were achieved for students who usually play chess, because they show a great capacity for visual memory (in the long and short term) that contributes to a more optimal learning of displacements and similarities in the Cartesian plane. This research shows a powerful tool (chess) that can be used in the teaching of mathematics, thanks to the skills and concepts that are generated in the experience with the game.

Surgical Anatomy of the Medial Antebrachial Cutaneous Nerve: Clinical Application in Ulnar Nerve Decompression Surgery in the Elbow

Introduction Lesion to the posterior branch of the medial antebrachial cutaneous nerve (MACN) is one of the causes of revision of the ulnar nerve decompression surgery in the elbow.To avoid the morbidity associated with this injury, cadaver dissections were performed to identify this branch in its course through the ulnar tunnel. Methods We included 20 upper extremities of fresh cadaveric specimens. The posterior branch of the MACN was identified proximal to medial epicondyle and followed past the ulnar tunnel. The number of ramifications and their coordinates were recorded in a Cartesian plane, with the medial epicondyle as the central point. Results The posterior branch passed proximal and posterior to the medial epicondyle in all specimens, except one. The average of the adjusted x value is of 30 mm, and of the adjusted y value is -18 mm. Additionally, we determined that the posterior branch passes at an average angle of 30° with respect to the x axis. Conclusion The anatomical descriptions of this branch focused on surgical release of the ulnar nerve in the elbow are limited, and measures are only described in the horizontal plane (from proximal to distal). Schematizing the anatomy of this branch in its course throughout the ulnar tunnel will facilitate its identification during the procedures. However, variability and asymmetry in the branching pattern should be considered.

Estratégias didático-metodológicas com GeoGebra para o ensino e a aprendizagem de quadrantes no plano cartesianoDidactic-methodological strategies for teaching and learning Cartesian quadrants using GeoGebra

ResumoNa educação básica, as primeiras noções sobre o plano cartesiano iniciam com a identificação dos eixos coordenados e com a compreensão de que estes dividem o plano em quatro regiões chamadas de quadrantes. Esse entendimento se estende com a compreensão das noções relativas à posição, localização de figuras e deslocamentos no plano cartesiano. O objetivo do trabalho foi propor estratégias didático-metodológicas com o GeoGebra na abordagem matemática de quadrantes no plano cartesiano. Trata-se de uma pesquisa qualitativa, de caráter exploratório, desenvolvida com professores de matemática do ensino fundamental II (do 6º ao 9º ano) de escolas públicas. A escolha do GeoGebra foi pelas funcionalidades que podem contribuir para o entendimento da temática. As estratégias didático-metodológicas foram elaboradas em fichas padronizadas e detalhadas para que professores e alunos tenham um roteiro de orientação durante a atividade.Palavras-chave: Software gratuito, Ensino fundamental (6º ao 9º ano), Educação matemática.AbstractIn elementary education, initial notions of the Cartesian plane start with identification of coordinate axes and the recognition that these divide planes into four quadrants. This initial understanding extends to notions related to position, location, and displacement in the Cartesian plane. Our objective was to propose didactic-methodological strategies to introduce Cartesian quadrants using GeoGebra. This qualitative study had an exploratory nature and was developed with middle-school teachers (6th to 9th grades) from public schools. GeoGebra was chosen because of features that contribute to building an understanding of the topic. The didactic-methodological strategies were standardised and detailed so that teachers and students would be guided throughout the activity.Keywords: Free software, Middle school (6th to 9th grade), Mathematics education.ResumenEn la educación básica, las primeras nociones sobre el plano cartesiano comienzan con la identificación de los ejes coordinados y con la comprensión de que dividen el plano en cuatro regiones llamadas cuadrantes. Ese entendimiento se extiende con la comprensión de nociones relativas de posición, ubicación de figuras y desplazamientos en el plano cartesiano. El objetivo del trabajo fue proponer estrategias didáctico-metodológicas con el GeoGebra para el enfoque matemático de cuadrantes en el plano cartesiano. Trátase de una investigación cualitativa, de carácter exploratoria, desarrollada con maestros de matemáticas del liceo (del 6º al 9º grado) de la enseñanza pública. La elección del GeoGebra fue por sus funcionalidades poder contribuir para la comprensión del tema. Las estrategias didáctico-metodológicas fueron elaboradas en formularios estándares y detallados para que profesores y alumnos tengan una guía de orientación durante la actividad.Palabras clave: Software libre, Liceo (6º al 9º grado), Educación matemática.

Extending a Euclidean Model of Ratio and Proportion

This paper is written for mathematics educators and researchers engaged at the elementary and middle school levels and interested in exploring ideas and representations for introducing students to ratio and proportion and for making a smooth transition from multiplication and division by whole numbers to their counterparts with fractions. Book V of Euclid’s Elements offers a scenario for deciding whether two ratios of magnitudes, embodied as a pair of line segments, are equal based on whether the ratios of magnitudes, when multiplied by the same whole numbers, n and m, each yield common products. This test of proportion can be performed using an educational software application where students are presented with a target ratio of commensurable magnitudes, A:B, and challenged to produce a selected ratio, C:D, that behaves like the target ratio under the critical conditions. The selected ratio is automatically constructed such that C:D = m:n, on the basis of a lattice point (n, m) chosen by the student. By adding partitive and Euclidean division to Euclid’s model, five new scenarios with similar goals are proposed. Representations in the Euclidean plane, on a number line, and in the Cartesian plane provide feedback that students may use to help identify a ratio of whole numbers corresponding to the targe ratio of magnitudes. The representations serve to highlight fractions as members of equivalence classes. The model remains to be investigated with teachers and students.

Chart for Flexible Curriculum in terms of Time and Similarity

Within the framework of the evaluation of curricular reforms made in the programs of the Faculty of Engineering of the Universidad Nacional de Colombia, this proposal is designed as an evaluation tool for a flexible curriculum, to characterize the enrollment behavior of students and their possible relationships with demographics and academic success. The principle consists of plotting the median time and similarity coefficients of each student in the program on the axes of a two-dimensional Cartesian plane. On the X-axis, the time coefficient was plotted, consisting in the relationship between the time proposed by the program curriculum for each course, and the time when the student takes it. On the Y-axis, the similarity coefficient was plotted, consisting in the number of courses that were taken at the time indicated in the curriculum grid. The conclusions suggest that, for the program analyzed, there are no demographic biases. However, the findings of this study suggest that even though students seek to take the proposed curriculum with the highest possible similarity, they spend more time than the estimated to achieve academic success

Exploring Marginality among Filipino Catholics in Japan: A Proposed Heuristic Device

The Church seeks to be inclusive; one that opens her doors to everyone. For many Filipino Catholics (FCs) in Japan, their ecclesial existence is marked by a history of negotiation as “guests” hosted by the Japanese Catholics (JCs). Within this field of host–guest interplay, this paper explores the dynamics of sociospatial seclusion by employing the ideation of marginality proffered by Loic Wacquant’s study on urban ghettos. The paper argues that the guest-identity of FCs must not be understood as a unilateral action imposed upon by the dominant hosts against the former’s subjugated narrative as powerless victims. Instead, its maintenance is perpetuated by FCs’ elective and chosen ethnic clustering. In attempt to obtain better analytical clarity of this dynamics, this paper employs the functional value of the Cartesian plane as a mapping device in plotting historical events of interplay within a spatial field. The techne inherent in the Cartesian plane is embedded with the episteme of Wacquant’s ideation. Fused together, its utility as a heuristic device is herewith proposed. It is hoped that this theoretical construct can also be useful to any analysis of marginality contained within a host–guest interplay.