THE REAL NUMBER MODULE IN PROBLEMS WITH GEOMETRIC CONTENT

В работе приводится решение задач, задаваемых уравнениями и неравенствами (или их системами), содержащими знак модуля. The paper presents the solution of problems defined by equations and inequalities (or their systems) containing the modulus sign.

Relative-perfectness of discrete gradient vector fields and multi-parameter persistent homology

The combination of persistent homology and discrete Morse theory has proven very effective in visualizing and analyzing big and heterogeneous data. Indeed, topology provides computable and coarse summaries of data independently from specific coordinate systems and does so robustly to noise. Moreover, the geometric content of a discrete gradient vector field is very useful for visualization purposes. The specific case of multivariate data still demands for further investigations, on the one hand, for computational reasons, it is important to reduce the necessary amount of data to be processed. On the other hand, for analysis reasons, the multivariate case requires the detection and interpretation of the possible interdepedance among data components. To this end, in this paper we introduce and study a notion of perfectness for discrete gradient vector fields with respect to multi-parameter persistent homology, called relative-perfectness. As a natural generalization of usual perfectness in Morse theory for homology, relative-perfectness entails having the least number of critical cells relevant for multi-parameter persistence. As a first contribution, we support our definition of relative-perfectness by generalizing Morse inequalities to the filtration structure where homology groups involved are relative with respect to subsequent sublevel sets. In order to allow for an interpretation of critical cells in 2-parameter persistence, our second contribution consists of two inequalities bounding Betti tables of persistence modules from above and below, via the number of critical cells. Our last result is the proof that existing algorithms based on local homotopy expansions allow for efficient computability over simplicial complexes up to dimension 2.

Models and scales for quality control: toward the definition of specifications (GOA-LOG) for the generation and re-use of HBIM object libraries in a Common Data Environment

The paper focuses on new opportunities of knowledge sharing, and comparison, thanks to the circulation and re-use of heritage HBIM models by means of Object Libraries within a Common Data Environment (CDE) and remotely-accessible Geospatial Virtual Hubs (GVH). HBIM requires a transparent controlled quality process in the model generation and its management to avoid misuses of such models once available in the cloud, freeing themselves from object libraries oriented to new buildings. The model concept in the BIM construction process is intended to be progressively enriched with details defined by the Level of Geometry (LOG) while crossing the different phases of development (LOD), from the pre-design to the scheduled maintenance during the long life cycle of buildings and management (LLCM). In this context, the digitization process—from the data acquisition until the informative models (scan-to-HBIM method)—requires adapting the definition of LOGs to the different phases characterizing the heritage preservation and management, reversing the new construction logic based on simple-to-complex informative models. Accordingly, a deeper understanding of the geometry and state of the art (as-found) should take into account the complexity and uniqueness of the elements composing the architectural heritage since the starting phases of the analysis, adopting coherent object modeling that can be simplified for different purposes as in the construction site and management over time. For those reasons, the study intends (i) to apply the well-known concept of scale to the object model generation, defining different Grades of Accuracy (GOA) related to the scales (ii) to start fixing sustainable roles to guarantee a free choice by the operators in the generation of object models, and (iii) to validate the model generative process with a transparent communication of indicators to describe the richness in terms of precision and accuracy of the geometric content here declined for masonry walls and vaults, and (iv) to identifies requirements for reliable Object Libraries.

Geometria w pierwszych latach szkolnej edukacji w świetle wybranych programów nauczania i poradników metodycznych dla nauczycieli z lat 1773 – 2020

Currently there is little geometry content in the early years of school. In my doctoral thesis Geometry in teaching children since the times of the Commission of National Education until today. Analysis of the successionof education concepts, supervised by Prof. Edyta Gruszczyk-Kolczynska, I examined how the teaching of geometry to younger children has changed since the time of KEN.In the article, I discuss curricula and methodological guide books for teachers in terms of the geometric content they cover in the first years of school education. I focus on three periods: the second half of the 19th century, the 1920s and the 1970s. These periods stand out from the others I studied n that there was a lot of geometric content in the first years of school. However, in the first one of these, the content was mainly included in the subject “drawings”, while in the others the main aim of teaching was to develop pupils’ computational skills. To a large extent, geometry has also served this purpose.

OBLIQUE IMAGES AND DIRECT PHOTOGRAMMETRY WITH A FIXED WING PLATFORM: FIRST TEST AND RESULTS IN HIERAPOLIS OF PHRYGIA (TK)

The complex archaeological site documentation benefits for a long time now from the aerial point of view and remote sensing methods. Moreover, the recent research on UAV photogrammetry platform equipment and flight planning actively contribute in this sense for a scaling improvement and cost-benefits balance. Frequently, the experiences on articulated topographic profiles in archaeological excavations require not only a multi-sensor approach but also and above all a multiscale one. According to this line, in a general time-cost ration framework, the geometric content of the generated DSMs should be complete of nadir and oblique point of view for the accurate 3D reconstruction of both upstanding buildings and excavations. In the same way, also the radiometric content closely depends on sensor payload quality and is strictly affected by excavation site condition, related to the site material and light. In this research, carried out in the impressive archaeological site of the ancient city of Hierapolis in Phrygia (Turkey) in the autumn 2019 campaign, the main goal was to evaluate and validate the overall performance of a novel UAV fix-wing ultralight platform with onboard GNSS receiver for RTK/PPK processing of cameras positioning and with the possibility of oblique images capturing. The expected contribute in terms of the acquisition, processing time, radiometric enhancement and geometry 3D reconstruction will be explored with preliminary test and outcomes, and with the results of the high-scale DSM and orthoimage generation of the complete Hierapolis site.

REFLECTIVE ABILITIES OF STUDENTS: WAYS AND MEANS OF DEVELOPMENT WHILE STUDYING GEOMETRY

A reflective ability as an ability to comprehend one’s experience, knowledge, evaluations is a psychological condition of thinking activity. But in researches, reflection is mainly considered as an indicator of a high level of thinking, creativity, the ability to analyze, types of reflection are not distinguished. For teaching mathematics, the development of intellectual reflection is especially important. In our study, the problem of the development of intellectual reflection is identified as an independent one. As a means of its development, we proposed geometric "many-valued" problems in which a situation of choice is organized. Three levels of development of reflective abilities in teaching geometry, in accordance with certain types of reflection are distinguished. The purpose of the study is to find out whether the level of development of students' reflection will change if "many-valued" problems are used in geometry. Is there a correlation of the manifestation of reflection on the geometric content and the content of another object? The experiment involved 375 students. The Pearson criterion was used in processing the results. The inclusion of "many-valued" problems in teaching geometry showed an increase in the level of formation of reflective skills of students and the transfer of these skills to another subject.

Tasks of the Final Stage of the 2019 Republican Olympiad in Mathematics, Methods and Criteria for Evaluating Their Solution

The example of real problems of the final stage of the 2019 Republican Mathematics Olympiad in the Kyrgyz Republic shows the methods of solution and criteria for a 10-point assessment of each problem. The final stage of the mathematics Olympiad was held in two rounds on March 30-31, 2019. The set of tasks for each round contained three tasks, one of them of geometric content. Thus, a total of 6 problems were proposed to the participants of the Olympiad, 2 of them in geometry and 1 in combinatorics. In the final stage, 318 winners of the previous stage took part, competing in 10 school subjects. In the individual event, 73 students became winners of the Olympiad, i.e. 3.57% of the number of all participants, starting from the regional stage. An increase in the number of winners from the regional regions of Kyrgyzstan was noted.

The Role and the Place of Geometry in the System of Mathematical School Olympiads

The purpose of this article is to determine the role and place of school geometry in the subject olympiad system. For this, the authors turn to the experience of Russia in organizing and conducting geometric olympiads for schoolchildren, exploring the specifics of the olympiads named after named I.F. Sharygin, named S.A. Anischenko, named A.P. Savina, Moscow and Iran olympiads. The objectives and themes of full-time, extramural, oral geometric olympiads are defined. It is revealed that the topics of topology, projective, affine, combinatorial sections of geometry constitute the content of olympiad geometry. The study showed that the tasks of the olympiad work on geometry checked mathematical skills to perform actions with geometric figures, coordinates and vectors; build and explore simple mathematical models; apply acquired knowledge and skills in practical activities. The conclusions are made about the need to include tasks of geometric content in the block of olympiad tasks for the development of spatial thinking of schoolchildren.

PELATIHAN PENGEMBANGAN SOAL GEOMETRI LEVEL HIGHER-ORDER THINKING SKILL (HOTS) BAGI GURU SEKOLAH DASAR DI KOTA KUPANG

Kebiasaan berpikir matematis khususnya pada level higher-order thinking skill (HOTS) merupakan sarana penting untuk mengembangkan gagasan secara terbuka dan divergen. Namun hal ini menjadi kendala karena para guru belum memiliki pemahaman yang komprehensif tentang HOTS serta bentuk instrument soal level HOTS. Permasalahan ini harus segera diatasi dengan memberikan pemahaman yang utuh tentang HOTS dan melatih mereka menyusun soal matematika level HOTS khususnya pada konten geometri.dalam bentuk kegiatan Pelatihan Pengembangan Soal Geometri Level HOTS. Sasaran kegiatan ini adalah guru SD Kota Kupang sebanyak 29 orang yang berlangsung di SDI Bertingkat Kelapa Lima 2 Kota Kupang. Metode kegiatan ini yakni ceramah, tanya jawab, diskusi dan presentasi. Setelah diberi pelatihan, guru dibimbing untuk membuat soal-soal level HOTS pada konten geometri yang akan digunakan dalam kegiatan pembelajaran maupun tes di kelas. Hasil yang diperoleh adalah 1) guru memiliki pemahaman yang sama tentang HOTS. Hasil pretest dan posttest menunjukkan adanya perubahan konsepsi tentang HOTS yang didefinisikan sebagai level berpikir analisis, kritis dan kreatif, 2) mampu mengembangkan keterampilan berpikir guru dalam menyusun instrumen soal level HOTS. 3) menumbuhkan komitmen mutu guru terhadap pengembangan kemampuan berpikir matematis siswa.Kata-kata kunci; geometri, higher-order thinking skillMathematical thinking habit, especially at the higher-order thinking skill (HOTS) level, is an important tool for developing ideas openly and diverging. But this is an problem because teachers do not have a comprehensive understanding of HOTS and the HOTS level questions yet. This problem must be solved immediately by providing a complete understanding of HOTS and training them to compile HOTS mathematics problems especially on geometry through the training of developing HOTS Level Geometry questions. The subjects of this training were 29 elementary school teachers which took place at SDI Bertingkat Kelapa Lima 2 Kota Kupang. The method of this activity is discourse, question and answer, discussion and presentation. After being given training, the teacher is guided to make HOTS level questions on geometric content that will be used in learning and test activities in the classroom. The results obtained are 1) the teacher has the same understanding of HOTS. The results of the pretest and posttest showed a change in conceptions about HOTS which was defined as the level of thinking analysis, critical and creative, 2) able to develop teacher thinking skills in preparing HOTS level question instruments. 3) growing the teacher's quality commitment to the development of students' mathematical thinking skills.Keywords; geometry, higher-order thinking skill