Classical Geometry Research Articles

P -brane Galilean and Carrollian geometries and gravities

We study D-dimensional p-brane Galilean geometries via the intrinsic torsion of the adapted connections of their degenerate metric structure. These non-Lorentzian geometries are examples of G-structures whose characteristic tensors consist of two degenerate ‘metrics’ of ranks (p+1) and (D−p−1) . We carry out the analysis in two different ways. In one way, inspired by Cartan geometry, we analyse in detail the space of intrinsic torsions (technically, the cokernel of a Spencer differential) as a representation of G, exhibiting for generic (p, D) five classes of such geometries, which we then proceed to interpret geometrically. We show how to re-interpret this classification in terms of ( D−p−2 )-brane Carrollian geometries. The same result is recovered by methods inspired by similar results in the physics literature: namely by studying how far an adapted connection can be determined by the characteristic tensors and by studying which components of the torsion tensor do not depend on the connection. As an application, we derive a gravity theory with underlying p-brane Galilean geometry as a non-relativistic limit of Einstein–Hilbert gravity and discuss how it gives a gravitational realisation of some of the intrinsic torsion constraints found in this paper. Our results also have implications for gravity theories with an underlying ( D−p−2 )-brane Carrollian geometry.

The Potential of Spot Size Control in Shaping the Thickness Distribution in Ultrashort Laser Deposition.

The availability of new-generation femtosecond lasers capable of delivering pulses with energies in the hundreds of mJ, or even in the joules range, has called for a revision of the effect of scaling spot size on the material distribution within the plasma plume. Employing a state-of-the-art Szatmári-type hybrid dye-excimer laser system emitting 248 nm pulses with a maximum energy of 20 mJ and duration of 600 fs, copper films were grown in the classical pulsed laser deposition geometry. The exceptionally clean temporal profile of the laser pulses yielded a femtosecond component of 4.18 ± 0.19 mJ, accompanied by a 0.22 ± 0.01 mJ ASE pedestal on the target surface. While varying the spot sizes, the plasma plume consistently exhibited an extremely forward-peaked distribution. Deposition rates, defined as peak thickness per number of pulses, ranged from 0.030 to 0.114 nm/pulse, with a gradual narrowing of the thickness distribution as the spot area increased from 0.085 to 1.01 mm2 while keeping the pulse energy constant. The material distribution on the silicon substrates was characterized using the f(Θ) = AcoskΘ + (1 - A)cospΘ formalism, revealing exponents characterizing the forward-peaked component of the thickness profile of the film material along the axes, ranging from k = 15 up to exceptionally high values exceeding 50, as the spot area increased. Consequently, spot size control and outstanding beam quality ensured that majority of the ablated material was confined to the central region of the plume, indicating the potential of PLD (pulsed laser deposition) for highly efficient localized deposition of exotic materials.

On functional successive minima

AbstractIn the classical Geometry of Numbers, the calculation of successive minima may be quite difficult, even in using the norm coming from a distance function associated to a set. In the literature, there seem to be hardly any analogues when is replaced by the algebraic closure of a function field in one variable and one uses a norm arising from the absolute height. Here, we calculate a one‐parameter family of examples that naturally arose in our recent paper on bounded heights. We also comment on whether the minima are attained.

Visualisation of educational material in geometry using digital technologies on the Unity 3D platform

Relevance. Digital technologies are progressively being incorporated into the educational framework. Training with the use of digital technologies corresponds to the modern way of life, which is inextricably linked with access to, receipt, and use of information. This entails the modernization of not only the content of education, but also the forms, methods, and means of teaching. Purpose. This study aims to investigate the impact of augmented reality on the spatial abilities and academic performance of students. It involves the development and testing of an educational application for geometry classes to provide interactive material and improve learning efficiency. Methodology. The study used pedagogical experiments, observations, and assessments. The Geometria mobile application with augmented reality for visualizing spatial geometry problems was developed for a tenth-grade geometry textbook. Results. The study involved 40 tenth-graders studying the section "Parallelism in Space". Following the experiment, all participants except for four demonstrated improvements in their prior test outcomes, with ten out of nineteen participants achieving outstanding grades. According to the study results, teachers and students noted the originality of this product and the fact that its use increased the visibility and understanding of the geometric material. Conclusions. A positive experience of using augmented reality technology in the study of stereometry has been obtained. The study highlights the effectiveness of augmented reality in enhancing students' spatial abilities and academic performance in geometry. The results suggest that incorporating digital technologies like augmented reality can significantly improve learning outcomes and engagement in educational settings. Keywords: secondary education system; geometry; spatial thinking; stereometry

Classifications of THA-surfaces in I^3

In classical differential geometry, the problem of obtaining Gaussian and mean curvatures of a surface is one of the most important problems. A surface M2 in I3 is a THA-surface of first type if it can be parameterized by r(s, t) = (s, t, Af(s + at)g(t) + B(f(s + at) + g(t))). A surface M2 in I3 is a THA- surface of second type if it can be parameterized by r(s, t) = (s, Af(s + at)g(t) + B(f(s + at) + g(t)), t), where A and B are non-zero real numbers [16, 17, 18]. In this paper, we classify two types THA-surfaces in the 3-dimensional isotropic space I3 and study THA-surfaces with zero curvature in I3.

An Exploration of Learners’ Understanding of Euclidean Geometric Concepts: A Case Study of Secondary Schools in the OR Tambo Inland District of the Eastern Cape

A major concern in South Africa is the poor performance of learners in mathematics, particularly in geometry. This paper therefore sought to explore the learners’ understanding of Euclidean geometric concepts. The Van Hiele model was a useful framework for understanding the reasoning and challenges that students encounter with geometry. This study focused on 15 participants from rural South African schools, including 5 grade 10 mathematics learners in each of the three secondary schools and three teachers who are teaching mathematics in grade 10, in three secondary schools. Through face-to-face interviews, learners’ comprehension of geometry was qualitatively assessed. Pedagogical and methodological difficulties, lack of learners’ interest and comprehension of numerous geometrical concepts, as well as the absence of technology use, were found to contribute to the challenges in learning and teaching Euclidean geometry. The recommendations suggest that teachers should plan and prepare their geometry classes with the students’ understanding in mind that Euclidean geometry has been a cornerstone of mathematical education for centuries, teaching students critical thinking, problem-solving, and logical reasoning skills. This study adds to the existing literature on introducing new concepts in mathematics into the educational system of South Africa. Keywords: Geometry, Students’ Interest, Self-efficacy, Understanding, Three-Dimensional Shape

Differentialgeometrie im Grossen

Over the past several decades, classical differential geometry has undergone a remarkable expansion, helped by the integration of tools and insights from neighboring fields like partial differential equations, complex analysis, and geometric topology. In keeping with the spirit of previous gatherings, this meeting aimed to bridge the gaps between researchers working in seemingly disparate subfields of differential geometry, illuminating the connections that unite them. Amongst other things, this meeting was centered around the theme of scalar curvature, which has recently emerged as a fundamental element across various fields, including differential geometry, metric geometry, topology, and complex geometry. This shared topic presented an ideal opportunity for scholars from these distinct areas to convene, discuss their individual progress, and foster a vibrant exchange of ideas.

Thermodynamics of the Einstein-Maxwell system

At first glance, thermodynamic properties of gravity with asymptotically AdS conditions and those with box boundary conditions, where the spatial section of the boundary is a sphere of finite radius, appear similar. Both exhibit a similar phase structure and Hawking-Page phase transition. However, when we introduce a U(1) gauge field to the system, discrepancies in thermodynamic properties between the two cases has been reported in [7] (JHEP 11 (2016) 041). In this paper, by accepting the assumption that all Euclidean saddles contribute to the partition function, I found that these discrepancies are resolved due to the contribution from the “bag of gold (BG),” which is the class of Euclidean geometries whose area of bolt is bigger than that of the boundary. As a result, the Hawking-Page phase structure is restored, with the unexpected properties that the upper bound of thermodynamic entropy is always larger than the boundary area divided by 4G when the chemical potential is non-zero, and that such high entropy states are realized at sufficiently high temperature.

Optical properties of a class of generalized conics

The optical and geometric properties of a class of generalized conics are presented. The study makes it possible to find in an original way several results of classical geometry and to connect numerous curves by a common law. Applications in geometric optics are envisaged, with the production of diopters, lenses or mirrors.

The Effectiveness of Web-based Inductive Learning Assisted on GeoGebra

This research used web-based inductive learning that facilitated plane geometry learning, where the learning was carried out online. It was hypothesized that students could learn the concept of plane geometry better when students were involved in the induction process with the help of GeoGebra software. They did explore with GeoGebra software to form several examples of the concept, testing the patterns that existed in the examples, and generalized the findings into new knowledge. With the help of the web and GeoGebra software, students could carry out their own learning and make their own discoveries. To test the effectiveness of web-based inductive learning, two classes were involved in the research, they were students of the Mathematics education study program Faculty of Teacher Training and Education, University of Bengkulu who took plane geometry class in the even semester of the 2021/2022 academic year as many as 68 students. This research was a quasi-experimental research with a post-test-only control group design. The instrument used for measurement was a description test. Based on the data analysis, it was found that the learning outcomes using web-based inductive learning assisted on GeoGebra were better than those using the expository method.
 Keywords: web, inductive learning, geogebra

Quantum projections on conceptual subspaces: A deeper dive into methodological challenges and opportunities.

In alignment with the distributional hypothesis of language, the work "Quantum Projections on Conceptual Subspaces" (Martínez-Mingo A, Jorge-Botana G, Martinez-Huertas JÁ, et al. Quantum projections on conceptual subspaces. Cogn Syst Res 2023; 82: 101154) proposed a methodology for generating conceptual subspaces from textual information based on previous work (Martinez-Mingo A, Jorge-Botana G and Olmos R. Quantum approach for similarity evaluation in LSA vector space models. 2020). These subspaces enable the utilization of the quantum model of similarity put forth by Pothos and Busemeyer (Pothos E, Busemeyer J. A quantum probability explanation for violations of symmetry in similarity judgments. In Proceedings of the annual meeting of the cognitive science society, 2011, Vol. 33, No. 33), allowing for the empirical examination of the violations of assumptions concerning symmetry and triangular inequality (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327-352; Yearsley JM, Barque-Duran A, Scerrati E, et al. The triangle inequality constraint in similarity judgments. Prog Biophys Mol Biol 2017; 130: 26-32), as well as the diagnosticity effect (Tversky A. Features of similarity. Psychol Rev 1977; 84: 327-352; Yearsley JM, Pothos EM, Barque-Duran A, et al. Context effects in similarity judgments. J Exp Psychol Gen 2022; 151: 711-717), within a data-driven environment. These psychological biases, deeply studied by authors such as Tversky and Kahneman, inform us about the limitations of modeling psychological similarity measures using tools from classical geometry. This commentary aims to offer methodological clarifications, discuss theoretical and practical implications, and speculate on future directions in this field of research. Concretely, it aims to propose the use of different contours (conceptual or contextual) to generate the subspaces, which lead to subspaces of terms or contexts. Once these contours are defined, a differentiation is proposed between Aggregated Terms Subspaces (ATSs), Aggregated Contexts Subspaces (ACSs), and Aggregated Features Subspaces (AFSs) depending on whether we define the subspaces by grouping the terms or contexts within the contour, or from the latent dimensions of the semantic space obtained in the contour window. Finally, new data is provided on the violation of the triangular inequality assumption through the application of the quantum similarity model to ATSs.

LISR: Learning Linear 3D Implicit Surface Representation Using Compactly Supported Radial Basis Functions

Implicit 3D surface reconstruction of an object from its partial and noisy 3D point cloud scan is the classical geometry processing and 3D computer vision problem. In the literature, various 3D shape representations have been developed, differing in memory efficiency and shape retrieval effectiveness, such as volumetric, parametric, and implicit surfaces. Radial basis functions provide memory-efficient parameterization of the implicit surface. However, we show that training a neural network using the mean squared error between the ground-truth implicit surface and the linear basis-based implicit surfaces does not converge to the global solution. In this work, we propose locally supported compact radial basis functions for a linear representation of the implicit surface. This representation enables us to generate 3D shapes with arbitrary topologies at any resolution due to their continuous nature. We then propose a neural network architecture for learning the linear implicit shape representation of the 3D surface of an object. We learn linear implicit shapes within a supervised learning framework using ground truth Signed-Distance Field (SDF) data for guidance. The classical strategies face difficulties in finding linear implicit shapes from a given 3D point cloud due to numerical issues (requires solving inverse of a large matrix) in basis and query point selection. The proposed approach achieves better Chamfer distance and comparable F-score than the state-of-the-art approach on the benchmark dataset. We also show the effectiveness of the proposed approach by using it for the 3D shape completion task.

Kelvin equationfor bridging transitions.

We study bridging transitions between a pair of nonplanar surfaces. We show that the transition can be described using a generalized Kelvin equationby mapping the system to a slit of finite length. The proposed equationis applied to analyze the asymptotic behavior of the growth of the bridging film, which occurs when the confining walls are gradually flattened. This phenomenon is characterized by a power-law divergence with geometry-dependent critical exponents that we determine for a wide class of walls' geometries. In particular, for a linear-wedge model, a covariance law revealing a relation between a geometric and Young's contact angle is presented. These predictions are shown to be fully in line with the numerical results obtained from a microscopic (classical) density functional theory.

The centaur-algebra of observables

This letter explores a transition in the type of von Neumann algebra for asymptotically AdS spacetimes from the implementations of the different gravitational constraints. We denote it as the centaur-algebra of observables. In the first part of the letter, we employ a class of flow geometries interpolating between AdS2 and dS2 spaces, the centaur geometries. We study the type II∞ crossed product algebra describing the semiclassical gravitational theory, and we explore the algebra of bounded sub-regions in the bulk theory following TT¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ T\\overline{T} $$\\end{document} deformations of the geometry and study the gravitational constraints with respect to the quasi-local Brown-York energy of the system at a finite cutoff. In the second part, we study arbitrary asymptotically AdS spacetimes, where we implement the boundary protocol of an infalling observer modeled as a probe black hole proposed by [1] to study modifications in the algebra. In both situations, we show how incorporating the constraints requires a type II1 description.

ABC‐Auxetics: An Implicit Design Approach for Negative Poisson's Ratio Materials

A novel methodology is introduced for designing auxetic (negative Poisson's ratio) structures based on topological principles and is demonstrated by investigating a new class of auxetics based on two‐dimensional (2D) textile weave patterns. Conventional methodology for designing auxetic materials typically involves determining a single deformable block (a unit cell) of material whose shape results in auxetic behavior. Consequently, patterning such a unit cell in a 2D (or 3D) domain results in a larger structure that exhibits overall auxetic behavior. Such an approach naturally relies on some prior intuition and experience regarding which unit cells may be auxetic. Second, tuning the properties of the resulting structures is typically limited to parametric variations of the geometry of a specific type of unit cell. Thus, most of the currently known auxetic structures belong to a selected few classes of unit cell geometries that are explicitly defined in accordance with a specified topological (i.e., grid structure). Herein, a new class of auxetic structures is demonstrated that, while periodic, can be generated implicitly, i.e., without reference to a specific unit cell design. The approach leverages weave‐based parameters (A–B–C), resulting in a rich design space for auxetics that is previously unexplored.

The points and localisations of the topos of $M$-sets

In previous papers, we were able to prove that, much like in classical algebraic geometry, it is possible to recover our monoid scheme X from the topos \mathsf{Qcoh}(X) . This was achieved using topos points and localisations of \mathsf{Qcoh}(X) . With this philosophy in mind, the aim of this paper is to study the topos of M -sets for non-commutative monoids, especially their points and localisations. We will classify the points and localisations of M -sets for finite monoids in terms of the idempotent elements of M and idempotent ideals of M , respectively. Some of the results obtained in this paper can already be found in previous works, in direct or indirect forms. See the last part of the introduction.

M-Voronoi and other random open and closed-cell elasto-plastic cellular materials: Geometry generation and numerical study at small and large strains

The present study deals with a numerical design strategy of a novel class of three-dimensional random Voronoi-type geometries, called M-Voronoi. These materials comprise random, non-quadratic convex void shapes and non-uniform intervoid ligament thicknesses, and can span high-to-low relative densities. The starting point for their generation is a random adsorption algorithm (RSA) construction with spherical voids embedded in an incompressible, nonlinear elastic matrix phase. The initial RSA geometry is subjected to large elastic volume changes by prescribing Dirichlet boundary conditions. Due to the incompressibility of the matrix phase, the externally imposed volume changes lead to significant void growth. The numerical growth process may be stopped at any desired porosity. The proposed M-Voronoi process is general and allows the formation of isotropic (or anisotropic) designs. As a byproduct of the developed approach, we also present a novel remeshing technique allowing to read arbitrary geometries of one or multiple phases. The elasto-plastic properties of the M-Voronoi porous materials are numerically investigated at small strains as well as large compressive and shear loads. Their response is assessed by comparison with other well-known random and periodic porous geometries such as polydisperse porous materials with spherical voids (RSA), classical TPMS Gyroid geometries and random Spinodoid topologies. The results show that M-Voronoi and RSA (with spherical voids) geometries exhibit the stiffest elastic and highest flow stress response compared to the other two geometries. This study shows unambiguously that randomness may or may not lead to enhanced mechanical response such as higher stiffness or flow stress.

Curvature estimation for point cloud 2-manifolds based on the heat kernel

<p>The geometry processing of a point cloud 2-manifold (or point cloud surface) heavily depends on the discretization of differential geometry properties such as Gaussian curvature, mean curvature, principal curvature, and principal directions. Most of the existing algorithms indirectly compute these differential geometry properties by seeking a local approximation surface or fitting point clouds with certain polynomial functions and then applying the curvature formulas in classical differential geometry. This paper initially proposed a new discretized Laplace-Beltrami operator by applying an inherent distance parameter, which acts as the foundation for precisely estimating the mean curvature. Subsequently, the estimated mean curvature was taken as a strong constraint condition for estimating the Gaussian curvatures, principal curvatures, and principal directions by determining an optimal ellipse. The proposed methods are mainly based on the heat kernel function and do not require local surface reconstruction, thus belonging to truly mesh-free methods. We demonstrated the correctness of the estimated curvatures in both analytic and non-analytic models. Various experiments indicated that the proposed methods have high accuracy. As an exemplary application, we utilized the mean curvature for detecting features of point clouds.</p>

Didactic possibilities of using computer graphics programs in the educational process

This study investigated computer graphics programs and their several types and their capabilities in teaching drawing geometry classes in all technical and technological education areas of the Institute of Engineering Technology. - site created and supported with lesson notes and related learning resources on drawing techniques and theory. An exercise in drawing each line using the software and drawing the models was accompanied by video and audio recordings from local and international sources. Students had the opportunity to access the lessons in an interactive format. Throughout the program, hyperlinks to a variety of resource materials were provided to encourage non-linear access to knowledge, enhance repetition, and encourage exploration of the full potential of the and program. The learning model adopted simple and embedded technology, with ease of access being the focus. The study was successful in creating a suitable learning environment for skill development using technology. Overall, it can be concluded from the research that Solid Works is an effective tool for covering the topics of drawing geometry within several programs and for perfecting each detail using selected learning models and student needs.

Learning Difficulties of Ethnic Minority Students for Construction in Geometry

This paper reports the findings of a study that explored the learning difficulties of ethnic students for construction in geometry and analyzed the views and experiences of the students in practicing construction of geometrical figures like triangle and parallelogram with comparing the area of quadrilateral. This study was based on phenomenological research design on exploring the views and experiences of four ethnic minority students. Unstructured interview was used as the main tool for deriving the data from the participants. The collected data were analyzed and interpreted using thematic approach. The results indicated that the students felt conceptual complexity for the learning of geometry. It was found that students had such difficulties due to their irregularity in the geometry classes, teacher’s biasness to the below average students, shyness to ask questions to the teachers and emphasis on culturally privileged students over the minority group of students. Moreover, the ethnic minority students have their own problems as they miss their classes because of socio-economic status of their family and they are confined to their household works. Thus, the study implies that the conceptual complexity and socio-cultural reflection in classroom should be addressed in teaching geometry to the ethnic minority students.