Geometric Problems Research Articles

On non-parametric density estimation on linear and non-linear manifolds using generalized Radon transforms

Here we present a new non-parametric approach to density estimation and classification derived from theory in Radon transforms and image reconstruction. We start by constructing a “forward problem” in which the unknown density is mapped to a set of one dimensional empirical distribution functions computed from the raw input data. Interpreting this mapping in terms of Radon-type projections provides an analytical connection between the data and the density with many very useful properties including stable invertibility, fast computation, and significant theoretical grounding. Using results from the literature in geometric inverse problems we give uniqueness results and stability estimates for our methods. We subsequently extend the ideas to address problems in manifold learning and density estimation on manifolds. We introduce two new algorithms which can be readily applied to implement density estimation using Radon transforms in low dimensions or on low dimensional manifolds embedded in The code for our algorithms can be found here https://github.com/jameswebber1/On-nonparametric-density-estimation-on-linear-and-nonlinear-manifolds. We test our algorithms performance on a range of synthetic 2-D density estimation problems, designed with a mixture of sharp edges and smooth features. We show that our algorithm can offer a consistently competitive performance when compared to the state–of–the–art density estimation methods from the literature.

Development of competencies for teaching geometry through an analysis learning obstacle

Teaching and learning process that is carried out often results in learning obstacles. Based on the facts teachers rarely learn the thinking behind students’ learning obstacles. Through the analysis process of learning obstacles, is expected to develop the competency of a pre-service mathematics teacher to teach geometry. The design used in this study is a qualitative research design. This study uses the methodological framework of the Didactical Design Research (DDR). DDR or didactic design research that developed uses the philosophical foundation of hermeneutics, phenomenology, and ethnomethodology. The procedure of research through the examination of pre-service mathematics teachers in the implementation of Program Pengenalan Lapangan (PPL). Participants in the course are pre-service teachers who practice in their classrooms with various activities, such as geometric analysis and construction, presentation and resolution of geometrical problems, and the development of teaching units for geometry. Each of these activities allows learning obstacles to occur. The task of the pre-service teacher is to be able to analyze the learning obstacles faced by students. With the learning obstacle analysis process, it is the beginning of a pre-service mathematics teacher compiling in-depth knowledge related to the geometry concepts being studied.

New signal location method based on signal-range data for proximity tracing tools

Several technological solutions have emerged over the last several months to support proximity contact tracing to fight the COVID-19 pandemic. For this reason, today more than ever, accurate signal location is needed, even in indoor public areas (supermarkets, public transport, etc.). In a previous work, we proposed five methods to solve the problem of signal localization using elements of pole-polar geometry. The proposals were innovative, since they solved a geometric problem (locating a point in a coordinate system) only by applying concepts of geometry. Among these developed methods, the PPC (Pole-Polar Centroid model) was also presented. Although the PPC solves the problem of locating a device with better precision than conventional methods (based on numerical or optimization methods), its accuracy was found to be the worst among the five proposed geometric methods. In this context, this work proposes an extension to our PPC method, called the weighted Pole-Polar Centroid method (wPPC), which improves the accuracy of the previous PPC results. Such an extension does not change the complexity O(m2) or the minimum dimensionality (m = 2) of nodes, which integrate a location network to perform the triangulation of such signals. Moreover, this extension estimates a device's location coordinates by means of the interaction, via signals, of this device with the network nodes distributed in any coordinate system. An IEEE 802.11 network infrastructure is used to accomplish the experiments. Errors in signal data are common, and our new proposed method, the wPPC, can mitigate the influence of these errors, produce more accurate results than the PPC, and outperform some of the other four proposed geometric methods and current numeric methods. Despite the use of an IEEE 802.11 network infrastructure for testing here, this range-based method for signal triangulation can be applied to any signal type (such as Wi-Fi, Bluetooth, and light and sound propagation).

Global buckling analysis of composite honeycomb sandwich plate with negative Poisson’s ratio (CHSP-NPR) using variational asymptotic equivalent model

In present work, the variational asymptotic method is extended to study the bucking of composite honeycomb sandwich plate with negative Poisson’s ratio (CHSP-NPR), which is easily buckle under compressive loads or bending moments and may lead to catastrophic failure. For global bucking with small local rotations, the 3D geometrical nonlinear problem is decoupled into linear constitutive modeling using unit cell and geometrical nonlinear problem defined over the 2D equivalent plate model (2D-EPM). Results using 2D-EPM in global buckling analysis are compared with 3D-FEM results to evaluate the accuracy and efficiency of the present model, and parametric studies are carried out to reveal the relative influence of selective parameters on buckling behavior of CHSP-NPR. The results show the increase of the proportion of honeycomb core and the decrease of fiber volume fraction of face sheet may increase the negative Poisson’s ratio and weaken the buckling resistance of CHSP-NPR, but the buckling loads of CHSP-NPR are larger than those of conventional honeycomb sandwich plate under the same conditions.

Practical globally optimal consensus maximization by Branch-and-bound based on interval arithmetic

Consensus maximization is widely used in robust model fitting, and it is usually solved by RANSAC-type methods in practice. However, these methods cannot guarantee global optimality and sometimes return the wrong solutions. A series of Branch-and-bound (BnB) based globally optimal methods have been proposed, most of which involve deriving a complex bound. Interval arithmetic was utilized to derive simple bounds for BnB in solving geometric matching problems in 2003. However, this idea was somewhat forgotten in the community because it seems natural that the simple interval arithmetic based bounds might be worse than those elaborate bounds. Recently, some new globally optimal algorithms without using BnB were developed for consensus maximization, but they can only work with a small number of data points and low outlier ratios. In this work, we draw the idea of simple bounds by interval arithmetic back on the map and demonstrate its practicability by making substantial extensions. Concretely, we give detailed derivation of solving robust model fitting problems with both linear and quasi-convex residuals and propose practical methods to use them under Unit-Norm constraint and in a high-dimensional problem. Extensive experiments show that the proposed method can handle practical problems with large number of data points and high outlier ratios. It outperforms state-of-the-art global, RANSAC-type, and deterministic methods in terms of both accuracy and efficiency in low-dimensional problems. The source code is publicly available.22https://github.com/YiruWangYuri/Demo-for-GoIA.

ANALISIS KETERAMPILAN GEOMETRI SISWA DALAM MENYELESAIKAN MASALAH TRANSFORMASI DAN KESEBANGUNAN GEOMETRI RUMAH ADAT OSING

This research aims to determine the geometry ability of class XI students in solving transformation problems and geometric building problems of the Osingese traditional house. The subjects of this study were 6 students of class XI MIPA 4 SMAN 1 Tegaldlimo. Students are given questions of geometric transformation skills and geometric building problems tests, followed by interviews. This type of research is a qualitative descriptive study. The results obtained in the transformation of skills are students with high math scores first have all skills, students with low math scores first have all skills except applied skills, students with medium mathematics scores first only have drawing skills and applied skills, and students with moderate math scores second and second low math score students only have drawing skills. As for geometric building problem skills, almost all students have all skills. The second moderate math score students and first low math students had all skills except, applied skills.

The Pohozaev-Schoen identity on asymptotically Euclidean manifolds: Conservation laws and their applications

The aim of this paper is to present a version of the generalized Pohozaev-Schoen identity in the context of asymptotically Euclidean manifolds. Since these kind of geometric identities have proven to be a very powerful tool when analysing different geometric problems for compact manifolds, we will present a variety of applications within this new context. Among these applications, we will show some rigidity results for asymptotically Euclidean Ricci-solitons and Codazzi-solitons. Also, we will present an almost-Schur type inequality valid in this non-compact setting which does not need restrictions on the Ricci curvature. Finally, we will show how some rigidity results related with static potentials also follow from these type of conservation principles.

Correlation between mathematical creative thinking ability and mathematical creative thinking disposition in geometry

Mathematics educators commonly believe that an open-ended approach could facilitate students to build their creative thinking ability and its disposition, especially in geometry. This research belongs to a quasi-experiment that used a non-equivalent control group design. The purpose of this research was to analyze the correlation between creative thinking ability and creative thinking disposition through an open-ended learning approach toward area and volume of flat geometrical objects problems. Two classes are selected as the sample, they are the control group and experiment group. The open-ended approach was used in the experiment group then direct instruction in the control group. The population is all of the eighth-grade students at one of the junior high school in Bandung, West Java, Indonesia. This study used to test and non-test as the research instruments. The test instrument was a creative thinking ability test while the non-test instrument was a questionnaire. These instruments were given after the learning process then analyzed by using Bivariate Correlation through SPSS. According to the analysis results, the finding is: the correlation coefficient between creative thinking ability and creative thinking disposition is at the middle level. The conclusion is there is a significant correlation between students’ creative thinking ability and students’ creative thinking disposition.

A new algorithm for geometric optimization with a single-term exponent constrained by bipolar fuzzy relation equations

A geometric programming problem subject to bipolar max-product fuzzy relation equation constraints is studied in this paper. Some necessary and sufficient conditions are given for its solution existence. A lower and upper bound on the solution set of its feasible domain is obtained. Some sufficient conditions are proposed to determine some its optimal components without its resolution. A modified branch-and-bound method is extended to solve the problem. Moreover, an efficient algorithm is proposed to solve the problem based on the simplification operations and the modified branch-and-bound method. Its computational complexity is carefully analyzed. Some examples are given to show the importance of the problem and to illustrate the process of the algorithm. Finally, an analytic and comparative study is done to show the efficiency of the simplification procedures.

Pre-service elementary school teachers’ reasoning profile in solving geometry problems based on mathematics ability

A person’s reasoning can be seen in the solution of the problem he solved. Geometry is effective for developing the ability to think logically or reasoning students. Solve geometry problems that have an abstract object of study required a process of reasoning or reason. The reasoning of a student with a high mathematical ability may be different from the reasoning of a moderate or low-ability student in solving geometrical problems. The purpose of this study is to obtain information about pre-service elemantary school teachers’ reasoning profiles in solving geometry problems based on mathematical ability. The type of research that will be used is descriptive explorative research with qualitative approach. The subject of this research is 3 pre-service elementary school teachers’. Subject selection is based on a written math ability test. The instruments of this study include the main instruments and supporting instruments. The reasoning profile data was obtained through mathematical skills tests and problem-solving test-based interviews. The credibility of the data collected is tested through triangulation. Student reasoning profile of student of pre-service elementary school teacher is able to explain again the thinking process (reasoning or logical argument) clearly.

Students’ Critical Thinking Skill in Solving Geometry Problems Viewed from Their Emotional Intelligence

Critical thinking skills are important in solving geometric problems. The intelligence aspect also influences problem-solving. Therefore, this study aimed to describe students’ critical thinking skills in solving geometric problems in terms of emotional intelligence. This research was a qualitative descriptive study with 6 students as the research subject. Researchers used triangulation methods to collect data and determine the validity of the data, namely questionnaires, tests, and interviews. The results showed that as many as 32.26% of students who had low emotional intelligence could not do all the questions correctly, 32.26% of students who had medium emotional intelligence could not work on the questions with indicators of identify existing biases and compile and analyze arguments, and 35.48% of students who have high emotional intelligence cannot work on questions with indicators of make conclusions and analyze the veracity of statements. This was due to a lack of understanding of the definitions, formulas, concepts, and existing information to find information used in solving problems. So, it is necessary to provide more complex practice questions and emphasize understanding of the concept of geometry so that students can use ideas to solve non-routine questions.

The relation between spatial reasoning and creativity in solving geometric problems

Spatial reasoning has been used in daily life, such as in building designs, determining routes and using maps. Spatial reasoning is also used in mathematics, especially in geometry. Therefore, this study aims to find out how spatial reasoning is used by students in solving geometry problems, especially in their creativity to transform 2-dimensional images into 3-dimensional images. Creativity in this study is focused on fluency, flexibility, and originality. This study is an explorative descriptive study through a qualitative approach. The subjects of the study are 11 third grade junior high school students. Data were collected by geometry problem-solving tests with two types of questions in the form of true or false statements and open-ended problems. The results showed that: (1) in terms of fluency, students gave an average of three correct answers out of five, (2) in terms of flexibility, there were five different types of answers, and (3) in terms of originality, there are two students who gave different answers than other students. The results also showed that students applied their spatial reasoning in the form of spatial visualization and mental rotation in their creativity to solve the problems.

Shape optimization method for an inverse geometric source problem and stability at critical shape

<p style='text-indent:20px;'>This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.</p>

Connecting complex networks to nonadditive entropies

Boltzmann–Gibbs statistical mechanics applies satisfactorily to a plethora of systems. It fails however for complex systems generically involving nonlocal space–time entanglement. Its generalization based on nonadditive q-entropies adequately handles a wide class of such systems. We show here that scale-invariant networks belong to this class. We numerically study a d-dimensional geographically located network with weighted links and exhibit its ‘energy’ distribution per site at its quasi-stationary state. Our results strongly suggest a correspondence between the random geometric problem and a class of thermal problems within the generalised thermostatistics. The Boltzmann–Gibbs exponential factor is generically substituted by its q-generalisation, and is recovered in the q=1 limit when the nonlocal effects fade away. The present connection should cross-fertilise experiments in both research areas.

Aristotle on Attention

AbstractI argue that a study of theNicomachean Ethicsand of theParva Naturaliashows that Aristotle had a notion of attention. This notion captures the common aspects of apparently different phenomena like perceiving something vividly, being distracted by a loud sound or by a musical piece, focusing on a geometrical problem. For Aristotle, these phenomena involve a specific selectivity that is the outcome of the competition between different cognitive stimuli. This selectivity is attention. I argue that Aristotle studied the common aspects of the physiological processes at the basis of attention and its connection with pleasure. His notion can explain perceptual attention and intellectual attention as voluntary or involuntary phenomena. In addition, it sheds light on how attention and enjoyment can enhance our cognitive activities.

Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their existence. Then we show that if the fundamental group of a surface bundle over a surface is a ${\rm CAT}(0)$ group, the bundle must have injective monodromy (unless the monodromy has finite image). Finally, given a family of closed Riemann surfaces (of genus $\ge 2$) with injective monodromy $E\to B$ over a manifold $B$, we explain how to build a new family of Riemann surfaces with injective monodromy whose base is a finite cover of the total space $E$ and whose fibers have higher genus. We apply our construction to prove that the mapping class group of a once punctured surface virtually admits injective and irreducible morphisms into the mapping class group of a closed surface of higher genus.

Reinforcement-Learning-Based Asynchronous Formation Control Scheme for Multiple Unmanned Surface Vehicles

The high performance and efficiency of multiple unmanned surface vehicles (multi-USV) promote the further civilian and military applications of coordinated USV. As the basis of multiple USVs’ cooperative work, considerable attention has been spent on developing the decentralized formation control of the USV swarm. Formation control of multiple USV belongs to the geometric problems of a multi-robot system. The main challenge is the way to generate and maintain the formation of a multi-robot system. The rapid development of reinforcement learning provides us with a new solution to deal with these problems. In this paper, we introduce a decentralized structure of the multi-USV system and employ reinforcement learning to deal with the formation control of a multi-USV system in a leader–follower topology. Therefore, we propose an asynchronous decentralized formation control scheme based on reinforcement learning for multiple USVs. First, a simplified USV model is established. Simultaneously, the formation shape model is built to provide formation parameters and to describe the physical relationship between USVs. Second, the advantage deep deterministic policy gradient algorithm (ADDPG) is proposed. Third, formation generation policies and formation maintenance policies based on the ADDPG are proposed to form and maintain the given geometry structure of the team of USVs during movement. Moreover, three new reward functions are designed and utilized to promote policy learning. Finally, various experiments are conducted to validate the performance of the proposed formation control scheme. Simulation results and contrast experiments demonstrate the efficiency and stability of the formation control scheme.

Fitting Tractable Convex Sets to Support Function Evaluations

The geometric problem of estimating an unknown compact convex set from evaluations of its support function arises in a range of scientific and engineering applications. Traditional approaches typically rely on estimators that minimize the error over all possible compact convex sets; in particular, these methods do not allow for the incorporation of prior structural information about the underlying set and the resulting estimates become increasingly more complicated to describe as the number of measurements available grows. We address both of these shortcomings by describing a framework for estimating tractably specified convex sets from support function evaluations. Building on the literature in convex optimization, our approach is based on estimators that minimize the error over structured families of convex sets that are specified as linear images of concisely described sets -- such as the simplex or the spectraplex -- in a higher-dimensional space that is not much larger than the ambient space. Convex sets parametrized in this manner are significant from a computational perspective as one can optimize linear functionals over such sets efficiently; they serve a different purpose in the inferential context of the present paper, namely, that of incorporating regularization in the reconstruction while still offering considerable expressive power. We provide a geometric characterization of the asymptotic behavior of our estimators, and our analysis relies on the property that certain sets which admit semialgebraic descriptions are Vapnik-Chervonenkis (VC) classes. Our numerical experiments highlight the utility of our framework over previous approaches in settings in which the measurements available are noisy or small in number as well as those in which the underlying set to be reconstructed is non-polyhedral.

АРБЕЛОС І ПОВ'ЯЗАНІ З НИМ КОЛА

Геометрія як наука зародилася в стародавній Греції, її аксіоматичні побудови описані в "Началах" Евкліда. Евклідова геометрія займалася вивченням найпростіших фігур на площині та в просторі. Грецькомовні математики, які жили в період між VI століттям до н.е. і V століттям н.е., поставили та розв’язали багато цікавих геометричних задач. Більшість цих задач розв’язувалася графічним шляхом, що вимагало виконання великої кількості різноманітних складних побудов. На той час вважалося, що "істинно геометричними" є ті задачі, які розв’язувалися тільки за допомогою таких "наукових інструментів" як циркуль та лінійка. Особливу увагу древньогрецькі математики приділяли одному із найважливіших геометричних образів – колу, яке навіть у ті часи знаходило широке практичне застосування. Суттєвий внесок у дослідження кола зробив Архімед Сіракузький, який вперше ввів поняття арбелос. Під арбелосом він розумів плоску геометричну фігуру, утворену деяким півколом, з якого вирізані два менших півкола з діаметрами, що лежать на діаметрі вихідного кола і розбивають його на дві частини. Таким чином, утворювався криволінійний трикутник, обмежений трьома півколами. У пропонованій роботі розглядається питання розв’язання відомих старовинних геометричних задач із застосуванням сучасних методів інженерної графіки, аналітичної геометрії та числових методів, без проведення додаткових побудов, які використовуються при графічному розв'язанні розглянутих задач. При числовій реалізації поставлена задача зводилася до розв’язання нелінійного рівняння з однією змінною. Нелінійні рівняння пов’язані із знаходженням радіусів вписаних або описаних кіл та координат їх центрів. У роботі, зокрема, побудовані коло, вписане в арбелос, спарені кола Архімеда, відомі як кола-близнюки, ланцюг Паппи Олександрійського. Спираючись на дослідження сучасних математиків, присвячених арбелосу, розв’язані задачі побудови кіл Банкова (Bankoff), Шоха (Schoch), Ву (Woo).

Inequalities on Projected Volumes

In this paper we study the following geometric problem: given $2^n-1$ real numbers $x_A$ indexed by the nonempty subsets $A\subset \{1,\dots,n\}$, is it possible to construct a body $T\subset \math...