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This study aims to examine the roles of tactile materials and origami models in the perception of concept definitions and representations in geometry applications of individuals with visual impairments. This case study draws on data from eight individuals with visual impairments selected according to the purposive sampling method in a mathematics village. Video recordings of focus group interviews conducted with tactile materials and origami models were analyzed by content analysis. Tactile materials made it possible for them to perceive concept definitions and visual representations, while the construction of origami models made every component designable, such as the relationship between concepts, visual prototypes and sub-concepts. Individuals with visual impairments have been quite successful in designing different representations and strategies for geometric concepts with perfect folding. Thus, it has been determined that origami is an accessible material for individuals with visual impairments to do geometry with creases and models.

This paper discusses the relevance of using physical situations to introduce the concept of angles at elementary school. We compare the effectiveness of two geometry teaching sequences. In the first one (physical sequence), the pupils learned the angle concept by experimenting on the playground (i.e. mesospace) and then modelling the situation. In the second one (paper-and-pencil sequence), the pupils worked solely in the space of a sheet of paper (i.e. microspace). In both sequences, pupils compared areas of space delineated by an angle between two directions. Pupils in two Grade 3 classes were exposed to one of the two teaching sequences. The unfolding of these sequences was videotaped and analyzed, and the pupils were tested individually, before and after teaching, to measure each sequence's effectiveness. Results showed that both sequences are effective to grasp the angle concept: Most pupils overcame the common erroneous conception of comparing angles' sides' lengths instead of angle openness. The comparison of areas of space delineated by an angle between two infinite directions, which is the two sequences' common core, seems to be the key factor underlying angle conceptualization. This paper ends with a discussion of these results' teaching implications and the merits of each sequence. [ABSTRACT FROM AUTHOR]

Today math teachers can best reach their students and show them how math surrounds us by using manipulatives, children's literature, and GeoGebra while teaching mathematics. These are some of the best pedagogical practices for teaching mathematics today. In our high-tech world, students need to be proficient in science, technology, engineering, and mathematics (STEM) fields. As endorsed by National Council of Teachers of Mathematics (2000) and stressed in common core state standards in mathematics, it is important that we teach with technology, address dispositions and math anxiety, and make the math that young people are learning pertinent and meaningful. Frequently, it may be best to start teaching young people geometry first as opposed to numbers, which are considered more abstract and difficult to learn. Geometry is one of the most tangible divisions of math and concentrating on this first can help students' whole view of mathematics and their insolences towards learning the subject. Nowadays teachers also need to be cognizant and checking for attitudes and dispositions toward learning mathematics, as math anxiety is an issue in today's classrooms. This paper will revisit the best pedagogical practices for teaching math, the review of the use of math manipulatives, children's books, and GeoGebra to help teachers create mathematically confident young people.

The research question studied in this paper is how teachers' task designing can be enhanced in professional development. We show that this can be done by scaffolding. Scaffolding is used in problem solving and we apply it by considering designing a task as a problem. In the paper, we describe a professional development of four teachers led by a mathematics educator. The professional development consisted of 10 meetings. The teachers were first exposed to a well-designed task in geometry as learners and then were asked to design new geometry tasks for their students. The mathematics educator helped them by supplying scaffolds such as modelling, reflection, articulation and worked examples. The data consisted of video-recordings of the workshops and a semi-structured interview with each of the teachers at the end of the project. The data analysis shows how the use of particular scaffolds was helpful while preserving the teachers' autonomy as task designers.

The paper describes the results of a study whose main aim was to find the interrelationship between pupils' school grades in Czech language (native), mathematics, and physics and pupils' cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils' performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables.

Geometry and spatial reasoning form the foundations of learning in mathematics. However, geometry is a subject often ignored by curriculum writers and teachers until high school, leading to students lacking in critical skills in geometric reasoning. As the United States moves into a new curriculum epoch, heralding the commencement of the national common core standards (CCS), one could question if CCS in geometry align with the essential understandings children need to be successful geometric thinkers. This paper begins with an examination of the essential understandings of geometric reasoning leading to an interpretation and critique of the elementary geometry CCS. Finally, the instructional implications are discussed, considering the common core progression through what we know about how children learn geometry.

Background/purpose: Awareness of the mathematical skills and knowledge children possess in their early years is widely accepted. This includes various common positive aspects, not only for educators but also for researchers and policymakers. This study presents a systematic review conducted to meticulously identify empirical studies published in the Scopus-Index Journal database about the mathematical skills children aged 3 to 8 years old have mastered. Materials/methods: This review followed the PRISMA guidelines and the research database comprised of Scopus-indexed journals. The technique followed used "keywords" and Boolean operators. The screening processes included reviewing abstracts, scanning complete texts of published articles, and rejecting those not meeting preset inclusion criteria. Moreover, systematic reviews, meta-analyses, and papers not written in English were also excluded. Of the 801 studies initially identified, a total of 15 empirical studies were included in the systematic review. Results: Children master various math skills from a very young age, mainly in "numbers and operations", but face difficulties in skills related to "algebra" as well as "geometry and measurement". Additionally, several preschoolers' characteristics help to explain the acquisition of these skills, with "age" being the primary factor. Researchers use various research instruments and mainly conduct individual semi-structured interviews. Children's geometry skills and knowledge appear to have been studied to a small extent. The areas of "measurement" and "data analysis and probability" were found to be under active investigation. Conclusion: It is worth noting that not only does a noticeable research gap exist for the math domains of "measurement," "geometry," and "data analysis and probability". Factors that seem to affect young children's math skills, such as "gender", "parents' educational level", and "attendance to a preparatory preschool" need further investigation. The implications of the current study's results extend beyond academia, providing valuable insights that educators and policymakers can leverage to enhance the quality of mathematics education during the early years period.

In this paper we review selected significant developments in the use of digital technology in the teaching and learning of mathematics over the last five years. We focus on a number of important topics in this field, including the evolvement of STEAM and critical making as well as the process of redefining learning spaces in the transformation of the mathematics classroom. We also address the increasing use of computer algebra systems and dynamic geometry packages; and the issue of student collaboration online, especially using learning environments and social media. We briefly touch on artificial intelligence systems, including hyper-personalisation of learning, multimodality and videos. We include a brief discussion on the impact of COVID-19 on mathematics education, and lastly on the more theoretical perspective of the epistemology of digital technology and the construct of humans-with-media. We conclude the discussion with some possible concerns and mentioning some possible new topics for research in the field.

Performing hand gestures such as pointing and tracing while learning is an area of increasing focus in educational research. Previous studies have demonstrated that learners who performed these gestures while engaging with static paper- or computer-based learning materials performed better on posttests in multiple learning areas, such as mathematics, health sciences and language learning. In this paper, the effects of mimicking pointing and tracing gestures during a timed multimedia lesson on an iPad are investigated. Participants were asked to mimic or observe pointing and tracing gestures while engaging with timed multimedia worked-examples in geometry. Results did not replicate previous findings and instead showed that participants who performed pointing and tracing gestures achieved lower test scores than those who did not. Factors leading to these results are discussed, including redundancy of presented and performed hand gestures, the management of multiple modalities, the timed nature of worked example videos used and the studying of optimised learning materials. When considering these results, future research may consider the effects of pointing and tracing gestures when engaging in complex multimedia learning environments.

The combined use of origami and dynamic geometry software has recently appeared in mathematics education to enrich students' geometric thinking. The objective of this research is to study the roles played by the interaction of two artifacts, paper folding and GeoGebra, in a construction-proving problem as well as its generalization in the Euclidean geometry context. For this, we designed and implemented two mathematical tasks with 52 secondary education students (15-16 years old, 10th grade) during the COVID-19 emergency lockdown period in Italy. The tasks involved four phases: constructing, exploring, conjecturing, and proving. This article presents an epistemic analysis of the tasks and a cognitive analysis of the answers given by one of the students. The theoretical tools of the onto-semiotic approach supported these analyses. Cognitive analysis allows us to confront the intended meanings of the task and the meanings actually employed by a student, thus drawing specific conclusions about the roles of such artifacts in written arguments and give an interpretation of their combined use in mathematics education.

This paper reports some results of a study on how Colombian primary-school teachers, experienced in the use of technological tools, integrate digital resources into their practice. Specifically, we investigate the processes of selection and integration (including appropriation and orchestration) of these resources in the teaching of geometry. We aim to illustrate the potential paths (along with associated criteria) that teachers follow when selecting and integrating digital resources into their practice, using the Documentational Approach to Didactics. Accordingly, we present case studies of a 1st-grade teacher and a 5th-grade teacher, which describe their paths during their respective geometry lessons. The paths followed by the teachers were influenced by their knowledge and beliefs regarding the utilisation of digital technology resources and the potential student engagement with such resources. All these aspects come into play as teachers select and orchestrate digital resources. Our empirical data analysis enabled us to infer several criteria for resource selection and levels of orchestration and appropriation manifested by the teachers as they integrated digital resources into their classes. This work can provide valuable insights for further exploration of how teachers integrate digital resources, and can contribute to the development of professional development programmes.

Technological Pedagogical Content Knowledge (TPACK) is defined as the teacher knowledge needed for effective technology integration. This study aimed to investigate preservice elementary mathematics teachers' TPACK development. TPACK survey was administered to 33 preservice teachers (PSTs), and six of them were selected via maximum variation sampling to represent as different cases as possible based on their mathematical knowledge and technological self-assessment. These six PSTs implemented four technology-based lessons (two microteaching sessions in the mathematics teaching method course and two lessons in the student teaching). Participants' TPACK levels were examined based on the development model proposed by Niess, M. L., Sadri, P., & Lee, K. (2007, April). Dynamic spreadsheets as learning technology tools: Developing teachers' technology pedagogical content knowledge (TPCK). Paper presented at the meeting of the American Educational Research Association Annual Conference, Chicago, IL. Results showed that participants had not used technology effectively and efficiently in their microteaching sessions and the first lessons in their student teaching. After evaluating their first lessons in schools, PSTs improved their teaching considerably. We suggest giving more opportunities for PSTs to teach with technology in classrooms and to assess their teaching practices reflectively.

"Assumptions" play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in mathematical activity, with particular attention to the following two learning goals: recognize that (1) a conclusion depends on the assumption(s) underlying the argument that led to it; and (2) making the underlying assumption(s) explicit is crucial to reaching consensus on the conclusion. In the context of a 3-year design research study, we first used existing literature to construct an initial version of task design principles which we then empirically tested and refined by designing and implementing two tasks in Japanese school classrooms. One of the tasks was in the area of functions at the secondary level and the other in the area of geometry at the elementary level. We analyze two classroom episodes to discuss the promise and evolution of our proposed task design principles. In addition, our analysis sheds light on the role of the teacher's instructional moves and the students' mathematical knowledge during the implementation of the designed tasks.

A stratospheric pseudolite (SP) is a pseudolite installed on a stratospheric airship. A stratospheric pseudolite network (SPN) is composed of multiple SPs, which shows promising potential in navigation applications because of its station-keeping capability, long service duration, and flexible deployment. Most traditional research about SPN geometry optimization has centered on geometric dilution of precision (GDOP). However, previous research rarely dealt with the topic of how SPN geometry configuration not only affects its GDOP, but also affects its energy balance. To obtain an optimal integrated performance, this paper employs the proportion of energy consumption in energy production as an indicator to assess SPN energy status and designs a composite indicator including GDOP and energy status to assess SPN geometry performance. Then, this paper proposes an SPN geometry optimization algorithm based on gray wolf optimization. Furthermore, this paper implements a series of simulations with an SPN composed of six SPs in a specific service area. Simulations show that the proposed algorithm can obtain SPN geometry solutions with good GDOP and energy balance performance. Also, simulations show that in the supposed scenarios and the specific area, a higher SP altitude can improve both GDOP and energy balance, while a lower SP latitude can improve SPN energy status. [ABSTRACT FROM AUTHOR]

IPCBs (Intelligent Pseudolite Constellations based on high-altitude balloons) are a novel type of air-based pseudolite application with many advantages. Compared with ground-based pseudolites and traditional air-based pseudolites, IPCBs have a wider coverage and a lower energy requirement. Compared with LEO satellite constellations, IPCBs have a stronger signal, a lower cost, and a shorter deployment period. These merits give promising potential to IPCBs. In IPCB applications, one of the key factors is geometry configuration, which is deeply influenced by the balloon's unique features. The basic idea of this paper is to pursue a strategy to improve IPCB geometry performance by using diverse winds at different altitudes and balloons' capability of altering flight altitude intelligently. Starting with a brief introduction to IPCBs, this paper defines an indicator to assess IPCB geometry performance, an approach to adjust IPCB geometry configuration and an IPCB geometry configuration planning algorithm. Next, a series of simulations are implemented with an IPCB composed of six pseudolites in winds with/without a quasi-zero wind layer. Some IPCB geometry configurations are analyzed, and their geometry performances are compared. Simulation results show the effectiveness of the proposed algorithm and the influence of the quasi-zero wind layer on IPCB performance. [ABSTRACT FROM AUTHOR]

A deep learning-based Visual Inertial SLAM technique is proposed in this paper to ensure accurate autonomous localization of mobile robots in environments with dynamic objects. Addressing the limitations of real-time performance in deep learning algorithms and the poor robustness of pure visual geometry algorithms, this paper presents a deep learning-based Visual Inertial SLAM technique. Firstly, a non-blocking model is designed to extract semantic information from images. Then, a motion probability hierarchy model is proposed to obtain prior motion probabilities of feature points. For image frames without semantic information, a motion probability propagation model is designed to determine the prior motion probabilities of feature points. Furthermore, considering that the output of inertial measurements is unaffected by dynamic objects, this paper integrates inertial measurement information to improve the estimation accuracy of feature point motion probabilities. An adaptive threshold-based motion probability estimation method is proposed, and finally, the positioning accuracy is enhanced by eliminating feature points with excessively high motion probabilities. Experimental results demonstrate that the proposed algorithm achieves accurate localization in dynamic environments while maintaining real-time performance. [ABSTRACT FROM AUTHOR]

In this paper, an optimisation procedure is introduced that uses a significantly cheaper, and CFD-free, objective function for aerodynamic optimisation than conventional CFD-driven approaches. Despite the reduced computational cost, we show that this approach can still drive the optimisation scheme towards a design with a similar reduction in drag coefficient for wave drag-dominated problems. The approach used is 'CFD-free', i.e., it does not require any computational aerodynamic analysis. It can be applied to geometries discretised using meshes more conventionally used for 'standard' CFD-based optimisation approaches. The approach outlined in this paper makes use of the transonic area rule and its supersonic extension, exploiting a mesh-based parameterisation and mesh morphing methodology. The paper addresses the following question: 'To what extent can an optimiser perform (wave) drag minimisation if using 'area ruling' alone as the objective (fitness) function measurement?'. A summary of the wave drag approximation in transonic and supersonic regimes is outlined along with the methodology for exploiting this theory on a typical CFD surface mesh to construct an objective function evaluation for a given geometry. The implementation is presented including notes on the considerations required to ensure stability, and error minimisation, of the numerical scheme. The paper concludes with the results from a number of (simple and complex geometry) examples of a drag-minimisation optimisation study and the results are compared with an approach using full-fidelity CFD simulation. The overall conclusions from this study suggest that the approach presented is capable of driving a geometry towards a similar shape to when using full-fidelity CFD at a significantly lower computational cost. However, it cannot account for any constraints, driven by other aerodynamic factors, that might be present within the problem. [ABSTRACT FROM AUTHOR]

This paper presents the numerical analysis of a convergent-divergent circular nozzle with the exit Mach number of 1.69 with and without passive control at the exit. The passive control method opted for this analysis was inward and outward ascending triangular protrusion. This paper explores the influence of the passive control geometry and its blockage area concerning the nozzle exit. The nozzle pressure ratio (NPR) used for carrying out the flow analysis were 3, 4.932, and 6. Two different inward and outward protrusions were used with a height of 1.5 mm and 3 mm. From the results, the potential core length of the protrusion 1.5 mm height was not much changed in the both outward and inward cases. But when the height of the protrusion was increased to 3 mm, there was a noticeable core length reduction at all NPR but with different cases. At the NPR of 6, the potential core length of the inward protrusions 3 mm was reduced by 44 % compared to the plain CD nozzle. [ABSTRACT FROM AUTHOR]

In the present paper, we describe a series of laboratory experiments on the friction between rigid indenters with different geometrical forms and an elastic sheet of elastomer as a function of the normal load. We show that the law of friction can be controlled by the shape of the surface profile. Since the formulation of the adhesive theory of friction by Bowden and Tabor, it is widely accepted and confirmed by experimental evidence that the friction force is roughly proportional to the real contact area. This means that producing surfaces with a desired dependence of the real contact area on the normal force will allow to "design the law of friction". However, the real contact area in question is that during sliding and differs from that at the pure normal contact. Our experimental studies show that for indenters having a power law profile f(r) = cnrn with an index n < 1, the system exhibits a constant friction coefficient, which, however, is different for different values of n. This opens possibilities for creating surfaces with a predefined coefficient of friction. [ABSTRACT FROM AUTHOR]

In this paper, we study the complete quadrangle. We started this investigation in a few of our previous papers. In those papers and here, the rectangular coordinates are used to enable us to prove the properties of the rich geometry of a quadrangle using the same method. Now, we are focused on the isoptic point of the complete quadrangle A B C D , which is the inverse point to A ′ , B ′ , C ′ , and D ′ with respect to circumscribed circles of the triangles B C D , A C D , A B D , and A B C , respectively, where A ′ , B ′ , C ′ , and D ′ are isogonal points to A , B , C , and D with respect to these triangles. In studying the properties of the quadrangle regarding its isoptic point, some new results are obtained as well. [ABSTRACT FROM AUTHOR]

As the variety of space targets expands, two-dimensional (2D) ISAR images prove insufficient for target recognition, necessitating the extraction of three-dimensional (3D) information. The 3D geometry reconstruction method utilizing energy accumulation of ISAR image sequence (ISEA) facilitates superior reconstruction while circumventing the laborious steps associated with factorization methods. Nevertheless, ISEA's neglect of valid information necessitates a high quantity of images and elongated operation times. This paper introduces a partitioned parallel 3D reconstruction method utilizing sorted-energy semi-accumulation with ISAR image sequences (PP-ISEA) to address these limitations. The PP-ISEA innovatively incorporates a two-step search pattern—coarse and fine—that enhances search efficiency and conserves computational resources. It introduces a novel objective function 'sorted-energy semi-accumulation' to discern genuine scatterers from spurious ones and establishes a redundant point exclusion module. Experiments on the scatterer model and simulated electromagnetic model demonstrate that the PP-ISEA reduces the minimum image requirement from ten to four for high-quality scatterer model reconstruction, thereby offering superior reconstruction quality in less time. [ABSTRACT FROM AUTHOR]

Landsat 9 (L9) was launched on 27 September 2021. This spacecraft contained two instruments, the Operational Land Imager-2 (OLI-2) and Thermal Infrared Sensor-2 (TIRS-2), that allow for a continuation of the Landsat program and the mission to acquire multi-spectral observations of the globe on a moderate scale. Following a period of commissioning, during which time the spacecraft and instruments were initialized and set up for operations, with the initial calibration performed, the mission moved to an operational mode This operational mode involved the same cadence and methods that were performed for the Landsat 8 (L8) spacecraft and the two instruments onboard, the Operational Land Imager-1 (OLI-1) and Thermal Infrared Sensor-1 (TIRS-1), with respect to calibration, characterization, and validation. This paper discusses the geometric operational aspects of the L9 instruments during the first year of the mission and post-commissioning, and compares these same geometric activities performed for L8 during the same time frame. During this time, optical axes of the two sensors, OLI-1 and OLI-2, were adjusted to stay aligned with the spacecraft's Attitude Control System (ACS), and the TIRS-1 and TIRS-2 instruments were adjusted to stay aligned with the OLI-1 and OLI-2 instruments, respectively. In this paper, the L9 operational adjustments are compared to the same operational aspects of L8 during this same time frame. The comparisons shown in this paper will demonstrate that both instruments aboard L8 and L9 performed very similar geometric qualities while fully meeting the expected requirements. This paper describes the geometric differences between the L9 imagery that was made available to the public prior to the reprocessing campaign that was performed using the new calibration updates to the sensor and to ACS and TIRS-to-OLI alignment parameters. This reprocessing campaign of L9 products involved data acquired from the launch of the spacecraft up to early 2023. [ABSTRACT FROM AUTHOR]

Conventional approaches to concrete three-dimensional (3-D) printing relies on printing concrete in a straight (linear) print path, with layers overlaid on top of each other. This results in interlayer and interfilament joints being potential weak spots that compromise the mechanical performance. This paper evaluates simple alterations to the print geometry to mitigate some of these effects. A printable mixture with 30% of limestone powder replacing cement (by mass), with a 28-day compressive strength of approximately 70 MPa in the strongest direction is used. S- and 3-shaped print paths are evaluated as alternatives to the linear print path. Staggering of the layers ensures that the interfilament joints do not lie on the same plane along the depth. Flexural strength enhancement is observed when print geometries are changed and/or layers are staggered. The study shows that print geometry modifications mitigate mechanical property reductions attributed to interfilament defects in 3-D concrete printing. [ABSTRACT FROM AUTHOR]

Wire arc additive manufacturing (WAAM) is a well-established additive manufacturing method that produces 3D profiles. A better deposition efficiency can be achieved by understanding the parameters that may influence the geometry of the bead. This paper provides a review that focuses on the factors that may influence the formation of the 3D profile. The included factors are the flow pattern of the molten pool after deposition, the built structure and orientation, the heat input and cooling conditions, the welding parameters, and other uncertainties. This review aims to facilitate a better understanding of these factors and achieve the optimum geometry of the 3D parts produced. According to the literature, the behavior of molten pools is identified as one of the major factors that can impact the deposition efficiency of a bead and govern its geometry. The review indicated that the flow behavior of the molten pool and the geometry of the deposited bead are significantly affected by most welding parameters, such as torch angle, wire travel speed, filler feed rate, and cooling conditions. Furthermore, this paper incorporates the technology utilized for comprehending the behaviors of the molten pool, as it constitutes an integral component of the control strategy. It has been concluded that automated planning and strategy are necessary to ensure efficient deposition by controlling those factors. The integration of artificial intelligence could bring benefits in planning to address the variation and complexity of shapes. [ABSTRACT FROM AUTHOR]

In this paper, we recover some traditional results in the geometry of probability distributions, and in particular the convergence of the alternating minimisation procedure, without actually referring to probability distributions. We will do this by discussing a new general concept of two types of points: admissible and agreeable, inspired by multi–agent uncertain reasoning and belief merging. On the one hand, this presents a unique opportunity to make traditional results accessible to a wider audience as no prior knowledge of the topic is required. On the other hand, it allows us to contemplate how a group of rational agents would seek an agreement given their beliefs without necessarily expressing it in terms of probability distributions, focusing instead on logical properties. Finally, we recover Euclidean and Hilbertian settings of discrete and continuous probability distributions. [ABSTRACT FROM AUTHOR]

In this paper, the influence of the construction process of large-span steel latticed arch on the geometric configuration and bearing capacity performance of the large-span steel latticed arch is calculated and analyzed by using the inverse analysis technique. By comparing the calculation results of single pre-deformation adjustment of the node co-ordinates in zero state of the whole structure and the dynamic adjustment of the node co-ordinates of the unconstructed part of the structure, it is found that the inverse analysis method of dynamic node co-ordinate adjustment is used to achieve the unification of the geometric configuration of the final structure and the bearing capacity performance of the structure. The results show that the inverse analysis method of dynamic node co-ordinate adjustment proposed in this paper can simulate and track the dynamic changes of structural geometry and stable bearing capacity performance in the actual construction process, which can not only consider the influence of the installed part of the structure on the co-ordinate adjustment value of the uninstalled part of the structure but also consider the deformation influence of the subsequent uninstalled structure on the installed structure and can realize the optimal approximation of the final state structure to the geometric configuration of the zero state structure and the accurate calculation of the bearing capacity performance with the progress of the construction process. [ABSTRACT FROM AUTHOR]

We analyze the well-posedness of certain field-only boundary integral equations (BIEs) for frequency domain electromagnetic scattering from perfectly conducting spheres. Starting from the observations that (1) the three components of the scattered electric field E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x) and (2) scalar quantity E\mathrm{s}\mathrm{c}\mathrm{a}\mathrm{t}(x)\cdot x are radiative solutions of the Helmholtz equation, we see that novel boundary integral equation formulations of electromagnetic scattering from perfectly conducting obstacles can be derived using Green's identities applied to the aforementioned quantities and the boundary conditions on the surface of the scatterer. The unknowns of these formulations are the normal derivatives of the three components of the scattered electric field and the normal component of the scattered electric field on the surface of the scatterer, and thus these formulations are referred to as field-only BIEs. In this paper we use the combined field methodology of Burton and Miller within the fieldonly BIE approach, and we derive new boundary integral formulations that feature only Helmholtz boundary integral operators, which we subsequently show to be well posed for all positive frequencies in the case of spherical scatterers. Relying on the spectral properties of Helmholtz boundary integral operators in spherical geometries, we show that the combined field-only boundary integral operators are diagonalizable in the case of spherical geometries and their eigenvalues are nonzero for all frequencies. Furthermore, we show that for spherical geometries one of the field-only integral formulations considered in this paper exhibits eigenvalues clustering at one--a property similar to second-kind integral equations. [ABSTRACT FROM AUTHOR]

Descriptive geometry has indispensable applications in many engineering activities. A summary of these is provided in the first chapter of this paper, preceded by a brief introduction into the methods of representation and mathematical recognition related to our research area, such as projection perpendicular to a single plane, projection images created by perpendicular projection onto two mutually perpendicular image planes, but placed on one plane, including the research of curves and movements, visual representation and perception relying on a mathematical approach, and studies on toothed driving pairs and tool geometry in order to place the development presented here among them. As a result of the continuous variability of the technological environment according to various optimization aspects, the engineering activities must also be continuously adapted to the changes, for which an appropriate approach and formulation are required from the practitioners of descriptive geometry, and can even lead to improvement in the field of descriptive geometry. The imaging procedures are always based on the methods and theorems of descriptive geometry. Our aim was to examine the spatial variation in the wear of the tool edge and the machining of the components of toothed drive pairs using two cameras. Resolving contradictions in spatial geometry reconstruction research is a constant challenge, to which a possible answer in many cases is the searching for the right projection direction, and positioning cameras appropriately. A special method of enumerating the possible infinite viewpoints for the reconstruction of tool surface edge curves is presented in the second part of this paper. In the case of the monitoring the shape geometry, taking into account the interchangeability of the projection directions, i.e., the property of symmetry, all images made from two perpendicular directions were taken into account. The procedure for determining the correct directions in a mathematically exact way is also presented through examples. A new criterion was formulated for the tested tooth edge of the hob to take into account the shading of the tooth next to it. The analysis and some of the results of the Monge mapping, suitable for the solution of a mechanical engineering task to be solved in a specific technical environment, namely defining the conditions for camera placements that ensure reconstructibility are also presented. Taking physical shadowing into account, conclusions can be drawn about the degree of distortion of the machined surface from the spatial deformation of the edge curve of the tool reconstructed with correctly positioned cameras. [ABSTRACT FROM AUTHOR]

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Robertson–Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the cosmological expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow. [ABSTRACT FROM AUTHOR]

This paper presents a taxonomy of divisibilism, a philosophical perspective advocating for the infinite divisibility of continua. The taxonomy is founded on various conceptualizations of indivisibles, enabling the identification of two types of divisibilism: ‘moderate’ and ‘strong’. The former denies indivisibles as constituent parts of magnitudes, whereas the latter rejects indivisibles as even intrinsic elements (such as limits or junctions) of magnitudes. The paper proceeds to demonstrate how Gregory of Rimini falls into the second category, utilizing geometry and non-entitism as primary tools to dismiss the very possibility of the existence of indivisibles. Within the framework of this ‘strong divisibilism’, the paper also examines Gregory’s interpretation of mathematical items (e.g. points, lines, surfaces) against the backdrop of one of his main sources, that is, Augustine. Lastly, the paper delves into Gregory’s interpretation of the fundamental Aristotelian divisibilist claim, namely, that a continuum is infinitely divisible. It highlights how Gregory’s stance ultimately conflicts with Aristotelian premises, particularly concerning his assertion that the division of all continua is made into infinitely many actual parts. [ABSTRACT FROM AUTHOR]

Research on topological interlocking (TI) assemblies indicates that the geometry of blocks plays a significant role in the performance of a configuration. The current TI generation methods can return assemblies of uniform antiprisms, tetrahedra, cubes, and octahedra. However, other shapes (both convex and concave) are well qualified for use in TI assemblies. This paper presents a framework to generate blocks for TI assembly. Starting from a seed polygon, evolution steps translate and reshape the polygon, contracting it eventually to a point, a line segment, or another polygon. Our framework generalizes and unifies previous-generation methods based on tilting angles and height parameters. We show how the proposed method systematically generates novel TI solids and previously reported others. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Möbius geometry of curves in R4. First, using moving frame methods we construct a complete system of Möbius invariants for regular curves in R4 by the isometric invariants. Second, we completely classify the Möbius homogeneous curves in R4 up to a Möbius transformation of R4. [ABSTRACT FROM AUTHOR]

For a harmonic diffeomorphism between the Poincaré disks, Wan [J. Differential Geom. 35 (1992), pp. 643–657] showed the equivalence between the boundedness of the Hopf differential and the quasi-conformality. In this paper, we will generalize this result from quadratic differentials to r-differentials. We study the relationship between bounded holomorphic r-differentials/(r-1)-differential and the induced curvature of the associated harmonic maps from the unit disk to the symmetric space SL(r,\mathbb R)/SO(r) arising from cyclic/subcyclic Higgs bundles. Also, we show the equivalence between the boundedness of holomorphic differentials and having a negative upper bound of the induced curvature on hyperbolic affine spheres in \mathbb {R}^3, maximal surfaces in \mathbb {H}^{2,n} and J-holomorphic curves in \mathbb {H}^{4,2}. Benoist-Hulin and Labourie-Toulisse have previously obtained some of these equivalences using different methods. [ABSTRACT FROM AUTHOR]

The geproci property is a recent development in the world of geometry. We call a set of points Z\subseteq \mathbb {P}_k^3 an (a,b)-geproci set (for GEneral PROjection is a Complete Intersection) if its projection from a general point P to a plane is a complete intersection of curves of degrees a\leq b. Nondegenerate examples known as grids have been known since 2011. Nondegenerate nongrids were first described in 2018, working in characteristic 0. Almost all of these new examples are of a special kind called half grids. In this paper, based partly on the author's thesis, we use a feature of geometry in positive characteristic to give new methods of producing geproci half grids and non-half grids. [ABSTRACT FROM AUTHOR]

AbstractTo dampen the loading sequences in the constructions, using the shear walls is recommended. Shear walls, which withstand the lateral forces (i.e., parallel to the plane of the wall) of wind and seismic activity, prevent buildings from collapsing while columns and load-bearing walls support the structure’s compression load all the way to the foundation. Besides, some new techniques are innovated for strengthening the structure like outrigger system. The steel shear walls are mostly used in the steel structures due to their high strength, ductility, and low weight. Recently, implementing Corrugated Steel Shear Walls (CSSWs) has gained more attention because they can absorb much amount of energy compared to flat steel shear walls. There are two important parameters in the CSSWs that play crucial roles in controlling their strength; corrugation geometry and orientation. Therefore, the current research paper aims to study numerically the effects of four corrugation geometries (trapezoidal, square, zigzag, and curved) with two orientations on the cyclic behavior of the CSSWs. Finite Element Analysis (FEA) is used to simulate the seismic loading conditions and finally, this procedure is validated through the comparison of the modeled flat shear wall with the experimentally published results. The difference between the peak loads of hysteresis curves obtained numerically and experimentally is about 8.5%. The stated FEA procedure in this paper can be regenerated by other researchers. The results confirmed the applicability of the CSSWs compared to the flat shear walls. Also, the vertical corrugation infill patterns result in higher strength in comparison to the horizontal ones. For instance, in the zigzag and square corrugation shapes, the plastic energy dissipation of the vertically oriented specimens (13.84 and 14.01 KJ, respectively) is higher around 7% and 29%, respectively than the horizontally oriented ones (12.9 and 10.84 KJ, respectively). Besides, the trapezoidal and curved corrugation geometries show the highest cumulated energy and plastic energy dissipation. [ABSTRACT FROM AUTHOR]

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve. [ABSTRACT FROM AUTHOR]

Considering the interaction among matter, vacuum, and radiation, this paper investigates the evolution of cosmic dynamics of the varying-vacuum model in a case of Finslerian geometry through dynamic analysis methods. Surprisingly, this model can alleviate the coincidence problem and allows for a stable later cosmological solution corresponding to the accelerating universe. [ABSTRACT FROM AUTHOR]

This paper presents a three-dimensional fluid-structure-coupled simulation of a flexible caudal fin with different trailing-edge shapes. The influences of caudal-fin shape on hydrodynamic performance are investigated by comparing the results of a simplified model of a square caudal fin with forked and deeply forked caudal fins under a wider range of non-dimensional flapping frequency, 0.6 < f* < 1.5, where f* is the ratio of flapping frequency to the natural frequency of each caudal fin, i.e., f* = f/fn. The leading edge of each caudal fin is forced to oscillate vertically in a water tank with zero free-stream conditions. The numerical results show that the amount of forking in the geometry of the caudal fin has significant effects on its hydrodynamic performance. A comparison of thrust coefficients shows that the square caudal fin has a greater thrust coefficient in the non-dimensional frequency range of 0.6 < f* < 1.2, while the deeply forked caudal fin generates higher thrust when 1.2 < f* < 1.5. In terms of propulsive efficiency, the square caudal fin is more efficient when 0.6 < f* < 0.9, while the propulsive efficiency of a deeply forked caudal fin is significantly enhanced when 0.9 < f* < 1.5. Based on our results, the deeply forked caudal fin has greater thrust coefficients and a higher propulsive efficiency in a higher frequency range than the natural frequency of each caudal fin. The thrust characteristics and flow fields around each caudal fin are investigated in detail. [ABSTRACT FROM AUTHOR]

Discovered as an apparent pattern, a universal relation between geometry and information called the holographic principle has yet to be explained. This relation is unfolded in the present paper. As it is demonstrated there, the origin of the holographic principle lies in the fact that a geometry of physical space has only a finite number of points. Furthermore, it is shown that the puzzlement of the holographic principle can be explained by a magnification of grid cells used to discretize geometrical magnitudes such as areas and volumes into sets of points. To wit, when grid cells of the Planck scale are projected from the surface of the observable universe into its interior, they become enlarged. For that reason, the space inside the observable universe is described by the set of points whose cardinality is equal to the number of points that constitute the universe's surface. [ABSTRACT FROM AUTHOR]

Track geometry measurements (TGMs) are a critical methodology for assessing the quality of track regularities and, thus, are essential for ensuring the safety and comfort of high-speed railway (HSR) operations. TGMs also serve as foundational datasets for engineering departments to devise daily maintenance and repair strategies. During routine maintenance, S-shaped long-wave irregularities (SLIs) were found to be present in the vertical direction from track geometry cars (TGCs) at the beginning and end of a vertical curve (VC). In this paper, we conduct a comprehensive analysis and comparison of the characteristics of these SLIs and design a long-wave filter for simulating inertial measurement systems (IMSs). This simulation experiment conclusively demonstrates that SLIs are not attributed to track geometric deformation from the design reference. Instead, imperfections in the longitudinal profile's design are what cause abrupt changes in the vehicle's acceleration, resulting in the measurement output of SLIs. Expanding upon this foundation, an additional investigation concerning the quantitative relationship between SLIs and longitudinal profiles is pursued. Finally, a method that involves the addition of a third-degree parabolic transition curve (TDPTC) or a full-wave sinusoidal transition curve (FSTC) is proposed for a smooth transition between the slope and the circular curve, designed to eliminate the abrupt changes in vertical acceleration and to mitigate SLIs. The correctness and effectiveness of this method are validated through filtering simulation experiments. These experiments indicate that the proposed method not only eliminates abrupt changes in vertical acceleration, but also significantly mitigates SLIs. [ABSTRACT FROM AUTHOR]

Array pattern synthesis with low sidelobe levels is widely used in practice. An effective way to incorporate sensor patterns in the design procedure is to use numerical optimization methods. However, the dimension of the optimization variables is very high for large-scale arrays, leading to high computational complexity. Fortunately, sensor arrays used in practice usually have symmetric structures that can be utilized to accelerate the optimization algorithms. This paper studies a fast pattern synthesis method by using the symmetry of array geometry. In this method, the problem of amplitude weighting is formulated as a second-order cone programming (SOCP) problem, in which the dynamic range of the weighting coefficients can also be taken into account. Then, by utilizing the symmetric property of array geometry, the dimension of the optimization problem as well as the number of constraints can be reduced significantly. As a consequence, the computational efficiency is greatly improved. Numerical experiments show that, for a uniform rectangular array (URA) with 1024 sensors, the computational efficiency is improved by a factor of 158, while for a uniform hexagonal array (UHA) with 1261 sensors, the improvement factor is 284. [ABSTRACT FROM AUTHOR]

R. Lyons, R. Pemantle and Y. Peres (Ann. Probab. 24 (4), 1996, 1993–2006) conjectured that for a Cayley graph G with a growth rate gr(G) > 1, the speed of a biased random walk exists and is positive for the biased parameter λ ∈ (1, gr(G)). And Gabor Pete (Probability and geometry ´ on groups, Chaper 9, 2024) sheds light on the intricate relationship between the spectral radius of the graph and the speed of the biased random walk. Here, we focus on an example of a Cayley graph, a free product of complete graphs. In this paper, we establish the continuity of the spectral radius of biased random walks with respect to the bias parameter in this class of Cayley graphs. Our method relies on the Kesten-Cheeger-Dodziuk-Mohar theorem and the analysis of generating functions. [ABSTRACT FROM AUTHOR]

One major goal of spatial query processing is to mitigate I/O costs and minimize the search space. However, geometric computation can be heavy-duty for spatial queries, in particular for complex geometries such as polygons with many edges based on a vector-based representation. Many past techniques have been provided for spatial partitioning and indexing, which are mainly built on minimal bounding boxes or other approximation methods and are not optimized for reducing geometric computation. In this paper, we propose a novel vector-raster hybrid approach through rasterization, where rich pixel-centric information is preserved to help not only filter out more candidates but also reduce geometry computation load. Based on the hybrid model, we implement four typical spatial queries, which can be generalized for other types of spatial queries. We also propose cost models to estimate the latency for those query types. Our experiments demonstrate that the hybrid model can boost the performance of spatial queries on complex polygons by up to one order of magnitude. [ABSTRACT FROM AUTHOR]

Plagued by heterogeneous clutter, it is a serious challenge for airborne radars to detect low-altitude, weak targets. To overcome this problem, a novel matrix information geometry detector for airborne multi-channel radar is proposed in this paper. The proposed detector applies the given steering vector and array structure information to the matrix information geometry detection method so that it can be used for space–time adaptive detection. While improving the detection performance, the matrix information geometry detector's original anti-clutter advantage is enhanced as well. The simulation experiment results indicate that the proposed detector has advantages in several of the properties related to space–time adaptive detection, while its computational complexity does not increase significantly. Moreover, experiment results based on the measured data verify the superior performance of the proposed method. Sea-detecting data-sharing-program data, mountaintop data, and phased-array radar data are employed to verify the performance advantage of the proposed method in heterogeneous clutter and the ability for weak target detection. [ABSTRACT FROM AUTHOR]

Visualizing graphs and networks in non‐Euclidean space can have benefits such as natural focus+context in hyperbolic space and the familiarity of interactions in spherical space. Despite work on these topics going back to the mid 1990s, there is no survey, or a part of a survey for this area of research. In this paper we review and categorize over 60 relevant papers and analyze them by geometry, (e.g., spherical, hyperbolic, torus), by contribution (e.g., technique, evaluation, proof, application), and by graph class (e.g., tree, planar, complex). [ABSTRACT FROM AUTHOR]

In the territory of the former Yugoslavia, in the post-war period, numerous monumental sculptures, also referred to as "spomeniks" (monuments), were erected at various points, often in natural settings, to commemorate the People's Liberation War. Our focus in this paper is on one of them, the monument to the Kosmaj Partisan Detachment on mountain Kosmaj, near Belgrade. We examine the historical context in which the monument was created, directing the emphasis of the work towards the geometrical analysis of this object. We delve into the geometric elements of the monument, utilizing photogrammetry and descriptive geometry procedures. Through 'reverse engineering' based on the point cloud model and its orthogonal projections, we provide its idealized dimensions and plan. [ABSTRACT FROM AUTHOR]

The paper deals with the problematics of representation of the geometry of three-dimensional scenes in computer graphics using domain-specific hierarchical data structures based on octant trees or directed acyclic graphs as the means of space saving binary representation of this information. Several hierarchical data structures were introduced in this field in past decades. In this paper we are moving this problematic to the field of the representation of three-dimensional scenes which are rather dynamic in time, so their geometry is changing on the regular basis. We are investigating the influence of this time-variability on the size of the related hierarchical data structure on binary level. We are comparing the approach in which each time step is considered as the stand-alone static three-dimensional scene and therefore is represented by its own hierarchical data structure with the approach when this time-variable scene is considered as the four dimensional space and spatio-temporal sparse voxel octree hierarchical data structure is used for the representation of the geometry of this scene. We tested both approaches on three different scenes which were obtained by voxelization into two different resolutions of polygonal surface models represented originally as the mesh of triangles in Wavefront Object file format. Results of the tests show that spatio-temporal sparse voxel octree is better solution, when we obtained from 1.73 to 2.08 more compact binary representation, in comparison to the solution when time frames were encoded separately as the set of sparse voxel octrees. [ABSTRACT FROM AUTHOR]

This paper explores the possibility of using dual representation of geometry to speed up ray tracing and ensure robustness of light propagation simulation in complex optical systems with freeform surfaces defined by high-degree polynomials (up to order 34) or Jacobi polynomials. Traditional methods for representing this geometry both as a triangular mesh and as an analytical expression are analyzed. The analysis demonstrates the disadvantages of the traditional approaches due to the insufficient accuracy of calculating coordinates of the point at which the ray intersects the triangular mesh, as well as the non-robustness of the conventional procedures for finding the point of intersection between the tangent ray and the analytical surface. Thus, it is proposed to use a dual representation of geometry as a rough approximation of the surface by a triangular mesh, which is subsequently used as an initial approximation to find the point at which the ray intersects the surface defined by an analytical expression. This approach significantly speeds up convergence of the analytical methods and improves robustness of their solutions. The use of the Intel® Embree library to quickly find the point of intersection between the ray and the rough triangular mesh, as well as a vector model to refine the coordinates of the point of intersection between the ray and the analytically represented geometry, allows us to develop and implement a ray tracing algorithm for an optical system that has surfaces with dual representation of geometry. Experiments carried out using the developed algorithm show a significant speedup of ray tracing while preserving computational accuracy and high robustness of the results. The results are demonstrated by evaluating the point spread function and glare for two lenses with freeform surfaces defined by Jacobi polynomials. In addition, for these two lenses, an image formed by an RGB-D object that simulates a real scene is calculated. [ABSTRACT FROM AUTHOR]