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This work presents a geometry-driven deep learning framework Geometry Orbital of Deep Learning (GOODLE) that accurately predicts all carbon properties in real space and momenta space. We focus on the hybrid carbon orbitals (sp2 and sp3) geometry to establish their deep learning “potential”. We demonstrate its excellent performance with the properties predictions in dual space, including energy, equation of states, phonon structure, electronic band structure, and optical absorption. Capable of obtaining the carbon orbital geometries from small lattice carbon structures, GOODLE can be used for both inorganic carbon allotropes and organic hydrocarbon molecules, including the magic-angle twisted bilayer graphene, with excellent transferability. GOODLE is also available for the nudged-elastic band calculation. Besides, we proposed an analytic geometry condition criterial for the carbon allotropes stability.

Empirical evidence in literature identified significant association between spatial ability and educational performance particularly in science, technology, engineering and mathematics (STEM). The purpose of this study was to explore pre-service teachers’ spatial skills in solving mathematics problems, in the context of coordinate geometry. It is envisaged that spatial skills allow for the perception of visual information and, therefore, spatial cognition has been considered as a key skill in teaching mathematics. However, literature asserts that teachers are ill prepared to teach mathematics, hence there is limited use and misuse of spatial skills in teaching the subject. This study, therefore, examines the spatial orientation of pre-service teachers in teaching coordinate geometry. This is a mixed methods study in which pre-service teachers answered a coordinate geometry test to explore their content knowledge and their ability to interpret, analyse and apply visual spatial models to solve mathematical problems in coordinate geometry. The study established that the spatial orientation skills of pre-service teachers determine their performance in mathematics, especially coordinate geometry.

International audience; Predicting analytically the coupling losses generated in a cable for fusion magnets is still a significant challenge. Difficulties are related to the complex geometry of the system: several multi-strand stages embedded in one another with different twist pitches length, difficulty to model multiplets of strands, including compaction to the final shape. A two-stage analytical geometry based model (COLISEUM) has previously been developed at CEA. We try to extend it to any n-stage cables We detail here an iterative enhancement method to an n-stage model. We validated it against experimental data and shown that it is robust enough to fit our measured coupling losses. Finally, this upgraded model can be used to assess coupling losses in fusion n-stage cables in a particularly precise way from only geometrical information and analytical tools.

This study aims to describe both fact and concept error of student s in solving space analytic geometry problems. The subjects of this research are two students of Mathematics Department of Universitas Negeri Makassar. Each of them represent s for each error (fact and concept errors). The collecting data were employed by using space analytic geometric tests and depth-interview. Interview guidelines and researcher were as research instruments. Data were qualitatively analyzed, using three stages of analysis: data reduction, data displa y and concluding. The main findings of this research are (1) the subject of fact error made mistakes in writing vector symbol s, (2) the subjects of concept error made mistakes in identifying an equation (plane equation or line equation), since his focus was only in the number of variables of the equation. Keywords: fact error, concept error, space analytic geometry

The purpose of this study was to analyze the needs of the GeoGebra application in mathematics learning, especially the learning of analytic geometry for prospective mathematics teacher students. This research is descriptive quantitative research with a survey method. Participants in this study were prospective teacher students who had taken a course in analytic geometry at a university in Bandung, Indonesia. The sample of this research is student-teacher candidates in the sixth, fourth and second semesters. The total sample was 72 students with 20 males and 52 females. The data collection instruments used questionnaires and interviews. Based on the results of the study, it is known that when learning analytical geometry, most students have difficulty visualizing geometric shapes so that students feel they need GeoGebra as a learning medium in lectures on analytic geometry. Keywords : analytical geometry ; geo g ebra, teacher candidate students DOI: http://dx.doi.org/10.23960/mtk/v9i2.pp223-232

Background and Aims: This study aims to develop and evaluate a practical approach for the complicated area and centroid cal culation of stable microemulsion regions. Pseudo ternary phase diagrams are used to determine the region of microemulsion existence. The effect of various surfactant/co-surfactant weight ratios on the extent of a stable microemulsion area can easily be observed with these diagrams. Furthermore, the optimum formulations are selected using the centroid of the microemulsion region. Methods: Microemulsion formulations were prepared by changing the weight ratios of the components. The titration method was used at a constant temperature. A pseudo ternary phase diagram was constructed using the spots of the stable micro emulsion formulations. Results: The area and centroid of the stable microemulsion region were calculated by using the formulas manually and the macro edited for Microsoft® Excel. Both results were the same. The macro was user-friendly, easy to use, and worked well in Microsoft® Excel. Conclusion: Definitions, formulas, algorithms, and calculations used in this research will be constructive for everyone inter ested in this field and can be modified very easily in every different case.

The SIR(D) epidemiological model is defined through a system of transcendental equations, not solvable by elementary functions. In the present paper those equations are successfully replaced by approximate ones, whose solutions are given explicitly in terms of elementary functions, originating, piece-wisely, from generalized logistic functions: they ensure {\em exact} (in the numerical sense) asymptotic values, besides to be quite practical to use, for example with fit to data algorithms; moreover they unveil a useful feature, that in fact, at least with very strict approximation, is also owned by the (numerical) solutions of the {\em exact} equations. The novelties in the work are: the way the approximate equations are obtained, using simple, analytic geometry considerations; the easy and practical formulation of the final approximate solutions; the mentioned useful feature, never disclosed before. The work's method and result prove to be robust over a range of values of the well known non-dimensional parameter called {\em basic reproduction ratio}, that covers at least all the known epidemic cases, from influenza to measles: this is a point which doesn't appear much discussed in analogous works.

This article presents a relative study to the circumference, with base to the program of Analytical Geometry of the first year of the course of Mathematics of the Pedagogic Superior School of Bie, following the logical-deductive beginnings introduced by Rene Descartes. The original vision that was in the base of this study, has to do with a diagnosis done to the first author through an abbreviation oral presentation of a mathematical theme of elementary level regarding the teaching of the circumference, in the extent of the plan curricular of the chair of Project of Teaching I of the Course of Master's degree in Mathematics for Teachers in the University of Interior Edge in Portugal. Whose result of the diagnosis reveals an immense list of inadequacies, relatively the teacher's incapacity in teaching the content of the circumference in the perspective of a teaching that leaves of the simplest for the more compound, demonstrating the results in the base of theorems and propositions adapted a priori, what promoted a less significant learning on the part of the students, in this extent. In this sense, it intends to invert that picture, with the accomplishment of this research whose objective is to contribute for the improvement of the teaching of the Analytical Geometry with matter it enhances to the theme of the circumference in the course of Mathematics of the Pedagogic Superior School of Bie. The same is directed with view to give answer the following subject of the research: how to improve the teaching and learning of the Analytical Geometry with emphasis to the theme of the circumference, in the students of the first year of the course of Mathematics of the Pedagogic Superior School of Bie? The methodological orientation is of the qualitative type and the descriptive approach, because it was observed, to analyze and to describe the resolution procedures used by the teachers and students in the study of the circumference, including the bibliographical revision of the support materials used as basic bibliography in the referred discipline.

In this article we propose to discuss a pedagogical proposal that integrates the History of Mathematics and ICTs for solving algebraic equations of the second degree. This study is the result of a thesis defended in 2018, in the program REAMEC - Amazonian Network of Education in Science and Mathematics. For this purpose, we adopted Descartes' book "Geometry", as an original source combined with the use of technology with the GeoGebra software. Through a bibliographic search in historical sources we found that Descartes' method for solving equations can be used in the classroom. The research presents a qualitative approach, for the production of data we use questionnaires, interviews, records in the field notebook, in addition to problem solving. The research collaborators are five undergraduate Mathematics students from PIBID, from the State University of Mato Grosso, who voluntarily participated in the research. The choice of this universe is due to the fact of being able to observe and analyze the impact of this proposal for the teaching of Analytical Geometry topics in High School. We believe that the development of this research will enable a broader discussion in relation to issues pertinent to the mathematics teaching proposal, such as: the relationship between theory and practice, as well as the implications of these practices in the initial training of undergraduate mathematics students. We emphasize the feasibility of using this pedagogical proposal that integrates the use of the History of Mathematics and Technologies for solving algebraic equations of the second degree.

This article is focused on the development of organizational and pedagogical conditions of college students’ vocational training in terms of social partnership and experimental study of their effectiveness. The leading methods in the study of this problem are the observations, conversations, monitoring, questioning, psychological testing and pedagogical experiment, allowing checking of the effectiveness of the proposed organizational and pedagogical conditions of students’ vocational training in terms of social partnership. The article deals with the role of social partnership of colleges and enterprises; identifies organizational and pedagogical conditions of students’ vocational training in terms of social partnership between the college and the company; justifies forms of cooperation between the educational institutions and its social partners; identifies criteria for production and pedagogical management of students’ vocational training in the conditions of Social Partnership between the college and the company; proves the effectiveness of college students’ vocational training in the conditions of social partnership. The paper submissions are of a definite value in the organization of social partnership in the system of secondary and higher vocational education.

This research work introduces a new concept of shape recognition where we obtain a geometric form from some given parameters related to architecture geometry. They include cell face slope, edge length initial value of the vertical alignment and boundary vertices of the horizontal alignment. Here we construct a conical mesh with horizontal planar latitudes from initial horizontal and vertical alignments. For this purpose, analytic geometry, interpolation, data structures using Visual C++ Programming, computational geometry and CAD features are used

A longitudinal failure criterion for UD composites based on a kinking model was re-constructed guided by the principles of analytic geometry. The proposed method for determining the fiber initial m...

A symbolic geometry system such as Geometry Expressions can generate symbolic measurements in terms of indeterminate inputs from a geometric figure. It has elements of dynamic geometry system and elements of automated theorem prover. Geometry Expressions is based on the analytical geometry method. We describe the method in the style used by expositions of semi-synthetic theorem provers such as the area method. The analytical geometry method differs in that it considers geometry from a traditional Euclidean/Cartesian perspective. To the extent that theorems are proved, they are only proved for figures sufficiently close to the given figure. This clearly has theoretical disadvantages, however they are balanced by the practical advantage that the geometrical model used is familiar to students and engineers. The method decouples constructions from geometrical measurements, and thus admits a broad variety of measurement types and construction types. An algorithm is presented for automatically deriving simple forms for angle expressions and is shown to be equivalent to a class of traditional proofs. A semi-automated proof system comprises the symbolic geometry system, a CAS and the user. The user’s inclusion in the hybrid system is a key pedagogic advantage. A number of examples are presented to illustrate the breadth of applicability of such a system and the user’s role in proof.

Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].

A aprendizagem de conceitos em matemática implica na percepção de relações entre diferentes abordagens, sejam elas por meio da aritmética, álgebra ou geometria. Nesse artigo, apresenta-se uma proposta alternativa para a resolução de um problema clássico da geometria analítica, que trata da divisão de um segmento em uma razão dada, percebido por uma acadêmica do curso de licenciatura em Matemática, ocorrido espontaneamente, no decorrer de uma disciplina denominada “Geometria Analítica”. A pesquisa bibliográfica indicou que existem diferentes modos de resolução, mas não foi encontrado um método semelhante ao proposto pela estudante. Concluiu-se que essa percepção, que implica no uso tanto de conceitos geométricos, como algébricos, pode favorecer o ensino e a aprendizagem significativa da matemática, pois permite estimular o uso de conceitos prévios já existentes, tais como os conceitos de ponto médio, pontos equidistantes e progressão aritmética, bem como possibilita estimular o desenvolvimento da competência em transitar entre diferentes registros semióticos.

Out-of-step protection has been widely studied, and it remains an important topic in the field. Given the crucial modernization of the power systems and the evolution of protective relaying, it is crucial to merge from traditionally fixed parameters to novel approaches that allow flexibility and possible time-domain protection. This article presents a new methodology to implement an out-of-step protection algorithm through the geometrical understanding of a power system oscillation using Clarke's transformation. The article is an extension of a virtual relay solution previously developed and improves the swing center voltage solution in the $\alpha$ $\beta$ plane with enhanced selectivity. The result brings a functional virtual relay capable of detecting and reacting under stable and unstable power swing events through playback techniques. The contribution of this article relies on the application of the time-domain approach to improving the function ANSI 78 as an extension of ongoing research on virtualization of power system protection and real-time prototyping.

In this essay, we sought to infer the method to design the most fitting geometrical shape for the plan of a ‘Salat’ space in the prayer hall in a mosque, from relevant scripture, and from textbooks of A’emah of Fiqh, making use of basic rules of analytic geometry. We established that any shape for the ‘Salat space’ plan besides the Orthogonal Polygon which takes the complete undivided ‘Musalliah unit’ for a strict module in its composition, would yield juristically unfavorable effects in the way rows of Musalleen are lined up behind the Imam, and would diminish the use that can be made of Waqf money in hosting the largest possible number of ‘Musalleen’ within the bounds of available space.

Scots-born Henry Parr Hamilton played a significant role in the revival of mathematical teaching in Cambridge. Following the promotion of the differential calculus by Herschel, Babbage and Peacock,...

One of the major limitations of micro-milling applications in industries is its fast tool wear, which leads to low machining precision and efficiency. An accurate force model is fundamental for optimization micro-milling processes and minimize the tool wear. However, a generic model with tool runout and wear effect has not yet been established, which limits its practical application under varied working conditions. In this paper, a new idea is introduced by applying the spatial analytic geometry (SAG) method, under this framework the micro-milling force model is established based on the analysis of the geometrical relationship among the cutting edge positions, pre-processed workpiece morphology, and cutting force directions considering tool runout and wear effect. In this model, the tool runout is identified exclusively by only one parameter, namely the distance away from the center that perpendicular to the feed direction, so that it could be calibrated conveniently by calculating the ratio of resultant forces corresponding to different cutting edges. The tool wear–induced force is then modeled as increment of force coefficients to the original model. Therefore, the new force model with considering tool wear has the same form as the fresh tool. Finally, the accuracy and efficiency of the model are validated by experiments under varied working conditions.

After some studies on students with Down syndrome (DS) who learned algebra and problem solving and few single case studies on analytic geometry, here the experience on analytic geometry was extende...

In this work, intended for future teachers, we present a solution of a geometric locus problem using the GeoGebra software. This allows first to visualize the shape of the geometric locus, then to state a conjecture to validate a solution founded by analytical geometry methods. The search for a geometric locus using GeoGebra software makes the activity more attractive.

Learning programming in secondary school or on first university years may facilitate making mathematical or physical problem “alive”. With this in view, we are looking for such tasks for programming that are attractive to them, sufficiently easy but employed important and feasible mathematics and physics. A project has been suggested (several lessons and home works) how to create a MATLABprogram of analogue watch on the computer screen that pursues several educational aims. Initially, students are to split the whole task to several simpler ones that they are able to program. To solve them, students employ plotting graphs, trigonometry and analytical geometry, physical consideration and mental experiment, look for empirical equation to fit data, complete everything with rather easy programming and enjoy finally with pleasant widget on computer screen. Programs are given with comments. They serve, at the time, as an introduction to MATLAB. It may be seen that its environment do not hidden the sense of the work but rather facilitates it. That is why it is suggested for wide use in Ukrainian education.

In this work, we present the most recent numerical results in a nodal approach, which resulted in the development of a new numerical spectral nodal method, based on the spectral analysis of the multigroup, isotropic scattering neutron transport equations in the discrete ordinates ($S_N$) formulation for fixed-source calculations in non-multiplying media (shielding problems). The numerical results refer to simulations of typical problems from the reactor physics field, in rectangular two-dimensional Cartesian geometry, $X, Y$ geometry, and are compared with the traditional Diamond Difference ($DD$) fine-mesh method results, used as a reference, and the spectral coarse-mesh method Green's function ($SGF$) results.

In this work, we present a fully automated method for the construction of chemically meaningful sets of non-redundant internal coordinates (also commonly denoted as Z-matrices) from the cartesian coordinates of a molecular system. Particular focus is placed on avoiding ill-definitions of angles and dihedrals due to linear arrangements of atoms, to consistently guarantee a well-defined transformation to cartesian coordinates, even after structural changes. The representations thus obtained are particularly well suited for pathway construction in double-ended methods for transition state search and optimisations with non-linear constraints. Analytical gradients for the transformation between the coordinate systems were derived for the first time, which allows analytical geometry optimizations purely in Z-matrix coordinates. The geometry optimisation was coupled with a Symbolic Algebra package to support arbitrary non-linear constraints in Z-matrix coordinates, while retaining analytical energy gradient conversion. Sample applications are provided for a number of common chemical reactions and illustrative examples where these new algorithms can be used to automatically produce chemically reasonable structure interpolations, or to perform non-linearly constrained optimisations of molecules.

The computational heat conduction on a curvilinear grid, presented in the previous chapter for a complex geometry, is extended in this chapter for computational fluid dynamics. The curvilinear grid generation and the computation of geometrical parameters, presented in the previous chapter, are also applicable for this chapter. The mind map for this chapter is shown in Fig. 12.1. The figure shows that the present chapter starts with physical law-based FVM, and ends with the semi-explicit method-based solution methodology for a body-fitted curvilinear grid. The approximated algebraic formulation in the FVM is presented separately from the mass and momentum/energy conservation, and the solution methodology is presented as a two-step predictor-corrector method.

Knowledge about protein structure assignment enriches the structural and functional understanding of proteins. Accurate and reliable structure assignment data is crucial for secondary structure prediction systems. Since the 1980s, various methods based on hydrogen bond analysis and atomic coordinate geometry, followed by machine learning, have been employed in protein structure assignment. However, the assignment process becomes challenging when missing atoms are present in the protein files. Our method proposed a multi-class classifier program named DLFSA for assigning protein secondary structure elements (SSE) using convolutional neural networks (CNNs). A fast and efficient GPU-based parallel procedure extracts fragments from protein files. The model implemented in this work is trained with a subset of the protein fragments and achieves 88.1% and 82.5% train and test accuracy, respectively. The model uses only Cα coordinates for secondary structure assignments. The model has been successfully tested on a few full-length proteins also. Results from the fragment-based studies demonstrate the feasibility of applying deep learning solutions for structure assignment problems.

In this work, we used a modified Picard’s method to solve the Multigroup Neutron Space Kinetics Equations (MNSKE) in Cartesian geometry. The method consists in assuming an initial guess for the neutron flux and using it to calculate a fictitious source term in the MNSKE. A new source term is calculated applying its solution, and so on, iteratively, until a stop criterion is satisfied. For the solution of the fast and thermal neutron fluxes equations, the Laplace Transform technique is used in time variable resulting in a first order linear differential matrix equation, which are solved by classical methods in the literature. After each iteration, the scalar neutron flux and the delayed neutron precursors are reconstructed by polynomial interpolation. We obtain the fluxes and precursors through Numerical Inverse Laplace Transform by Stehfest method. We present numerical simulations and comparisons with available results in literature.

International audience; Tate introduced in [Ta71] the notion of Tate algebras to serve, in the context of analytic geometry over the-adics, as a counterpart of polynomial algebras in classical algebraic geometry. In [CVV19, CVV20] the formalism of Gröbner bases over Tate algebras has been introduced and advanced signature-based algorithms have been proposed. In the present article, we extend the FGLM algorithm of [FGLM93] to Tate algebras. Beyond allowing for fast change of ordering, this strategy has two other important benefits. First, it provides an efficient algorithm for changing the radii of convergence which, in particular, makes effective the bridge between the polynomial setting and the Tate setting and may help in speeding up the computation of Gröbner basis over Tate algebras. Second, it gives the foundations for designing a fast algorithm for interreduction, which could serve as basic primitive in our previous algorithms and accelerate them significantly.

The recently developed lpCMFD scheme has been shown to be unconditionally stable and more effective than CMFD scheme for 2D fixed source neutron transport problems by using Fourier analysis. Previous works have derived the Fourier analysis to study the convergence performance of the CMFD acceleration scheme for 1D k-eigenvalue neutron transport problem, but the study has not yet been performed for 2D k-eigenvalue neutron transport problem, which we believe is an important work that complements other studies of neutron transport acceleration methods. In this paper, we extend the Fourier analysis of CMFD and lpCMFD for k-eigenvalue neutron transport problems to 2D space, along with the detailed linearization and formulation derivation. The Fourier analysis results show that for both CMFD and lpCMFD, increasing the number of inner iterations per outer iteration can reduce the spectral radius up to an asymptotic value of the spectral radius. CMFD is found to be unstable at higher coarse mesh optical thickness while lpCMFD is unconditionally stable for scattering ratios c = 0.3, 0.6, 0.9, 0.99 and for a large range of coarse mesh optical thickness up to 100. The results presented here can support the application of lpCMFD for k-eigenvalue problems in 2D space for realistic neutronic simulations.

Neste artigo faremos um estudo bibliográfico com a finalidade de fazer uma reflexão a respeito do estudo da parábola a partir das noções de Quadros, Pontos de Vista e de Registros de Representação Semiótica. Não temos a intenção de esgotar o assunto, pois estamos interessados apenas no que se refere ao ensino básico. Assim, articulamos no quadro da Geometria os pontos de vista do corte entre plano e cone, e de lugar geométrico utilizando o registro material, o registro figural e o registro algébrico. A partir desse estudo mudamos o ponto de vista para o quadro da Geometria Analítica representando a parábola no plano cartesiano, que a caracteriza agora como um registro gráfico e o articulamos com o obtido como representação algébrica no quadro anterior para desenvolver outras representações algébricas, as equações.

This study examines the paper of Cardenas-Barron, entitled “The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra.” that was published in Applied Mathematical Modelling at 2011 to point out it contains questionable results. The algebraic geometry and algebra applied by Cardenas-Barron [2] violated the rule of the arithmetic-geometric mean (AGM) mentioned by Cardenas-Barron [74] . Moreover, we point out Sphicas [1] had already solved this kind of EOQ and EPQ models using algebraic methods.

Purpose This paper develops a vector variation score that quantifies the change in an array of data points from period-to-period. The array could be the amounts reported on an income tax return, the closing stock prices for a set of listed companies, the monthly sales amounts for retail locations or the monthly balances in general ledger accounts. Design/methodology/approach The score is grounded in analytic geometry. The angle θ measures whether the changes were uniformly spread across the line items. The item(s) with the largest contribution(s) to the score can be identified. Line items can be weighted such that they contribute less than fully to the score. Findings The method can identify tax returns with large year-on-year changes. The method can identify the fact that the price movements during earnings season are less dependent than is usually the case. The method can identify anomalies in reported sales amounts. The method should be able to identify ledger accounts’ large abnormal changes. Research limitations/implications Auditors will need to be trained to interpret the results and to reduce the number of false positives. Practical implications The score could be used in both external and internal audit applications where auditors want to quantify and rank period-on-period changes in a search for outliers. Originality/value The change score is normalized to the [0, 1] range. The results can be plotted as a polar plot for display on an auditing dashboard. The contribution of a single line item can be calculated and line items can be weighted to prevent them from having an undue influence on the results.

This study aims to describe the profile of students' critical thinking in solving analytical geometry problems that are viewed from gender. This research is a qualitative study with case studies. Data analysis technique used are Miles and Huberman analysis, namely data reduction, data presentation, and presented conclusions. The results of this study are male students and female students who have met the standard eight intellectual critical thinking standards of analytical geometry, namely clarity, precision, accuracy, relevance, consistency, logical correctness, completeness, and fairness. On the intellectual standard of precision, male students were more careful in giving reasons for taking problem solving strategies to female students. At the intellectual standard of fairness, male students are able to provide two alternative solutions to problems, while female students only provide one initial solution to problem solving.

The famous theorem of Van Aubel for quadrilaterals postulates that if squares are built externally on the sides of any quadrilateral, then the two segments that join the opposing centers of these squares are congruent and orthogonal. Inspired by this result and also by the results of Krishna, in this article we will prove the following result of plane geometry: each octagon is associated with a parallelogram, in some cases the parallelogram in question can be degenerate at a point or a segment. This is possible because of complex numbers and basics of analytical geometry.

We study a couple of geometrical elements of a monumental synagogue (Dohany Synagogue in Budapest). Then we describe how undergraduate students from the Jewish orthodox population have performed a ...

Methods are developed to calculate acoustic propagation paths through an atmospheric layer surrounding a spherical globe in order to more accurately model the propagation of infrasonic signals, which are often observed after propagating hundreds or thousands of kilometers. A generalized curvilinear coordinate system is used to define the ray tracing equations from the eikonal equation for a moving, inhomogeneous atmosphere, and the specific case of spherical coordinates is applied to obtain a system of coupled equations describing geometric propagation paths in an atmospheric layer surrounding a globe. Comparison with propagation predictions using a Cartesian geometry shows that even for relatively short infrasonic propagation distances of a few hundred kilometers, differences introduced by the change in geometry are significant. Characteristics of the stratospheric pair are considered, and it is found that differences in the upward and downward legs of the propagation paths corresponding to the fast and slow stratospheric arrivals are changed such that spherical coordinate geometry predicts a decrease in the relative arrival time between the two. Analysis of regional infrasonic signals produced by a series of surface explosions shows that predictions obtained using spherical geometry are more accurate in cases where the tropospheric and stratospheric waveguides are accurately characterized.

Charlotte Angas Scott (1858–1932) was an internationally renowned geometer, the first British woman to earn a doctorate in mathematics, and the chair of the Bryn Mawr mathematics department for forty years. There she helped shape the burgeoning mathematics community in the United States. Scott often motivated her research as providing a “geometric treatment” of results that had previously been derived algebraically. The adjective “geometric” likely entailed many things for Scott, from her careful illustration of diagrams to her choice of references and citations. This article will focus on Scott’s striking and consistent use of geometric to describe a reality of dynamic points, lines, planes, and spaces that could be manipulated analogously to physical objects. By providing geometric interpretations of algebraic derivations, Scott committed to an early-nineteenth-century aesthetic vision of a “whole” analytical geometry that she adapted to modern research areas.

Many procedures have been developed to design and optimize a heliostat field, however it is a rather challenging work partly because many influencing factors of a heliostat optical efficiency, especially the shading and blocking efficiency, are computationally intensive. And it’s difficult to judge whether a heliostat field layout is an ideal design intuitively and reliable. In this paper, two concise and accurate analytic geometry methods (AGM-I and AGM-II) are developed to identify the heliostats with the possibilities of shadowing or blocking a given heliostat firstly. The results compared with the Sassi method and bounding sphere & Sassi method show that the same accurate results are obtained while the computational time is significantly reduced by 32% and 23%, respectively. Then an intuitive approach is proposed to optimize and evaluate the rationality of a heliostat field layout by applying the maximum optical efficiency map (MOEM). Finally, a heliostat field based on the Gemasolar plant is studied, whose optical efficiency is optimized and evaluated by the proposed methods. Compared with the un-optimized result and the result optimized by using the reference method, the MOEM optimization result of the heliostat field with higher instantaneous optical efficiency and annual optical efficiency is more reasonable.

While several studies have presented experimental research and simulations in tracking the focal point, a closed form mathematical model to understand the movement of the focal point of a convex lens when light is incident at an arbitrary angle is missing. Such closed form mathematical solutions allow significantly faster computation than numerical simulations. In addition, they provide significant physical insights that cannot be obtained from simulations or experiments. This work presents a closed form mathematical solution based on analytic geometry for calculating the position of the focal point of a convex lens at an arbitrary lens orientation and arbitrary angle of incidence. Results obtained from this method are verified using Lens Maker’s formula for thin lenses. The theoretical model presented here also agrees well with the experimental results. The verified method is then used to determine the criteria for the validity of thin lens approximation. As a consequence, the paper presents an explicit requirement on the radius of curvature of the lens in terms of its length, refractive index and four dimensionless constants for thin lens approximation to be applicable.

This paper addresses the issue of computing the shortest path in the plane connecting two assigned poses (positions and orientations) while crossing an assigned via-point without a prescribed orientation. The problem is a natural extension to the Dubins classical one, and it has been recently considered for the potential applications of its solution on path planning for teams of vehicles, in the presence of obstacles and for the Dubins’ traveling saleman’s problem. The solution that we propose is obtained using mathematical tools from analytic geometry and it provides easily computable algorithms well suited for on-line path planning. The effectiveness of the proposed planning method is validated by worked examples and simulations.

Any mathematical model is the interpretation of natural, technological, mental process in mathematical language. In scientific researches one faces interpretations method at every step. It is sufficient to mention the graph theory, analytic geometry, differential equations, Laplace transformation, Fourier transformation, encoding theory etc. As a rule in the interpretations method the problem of one branch of mathematics is interpreted in other branch, where it is either simplified or better responds to our intuition or allows usage of other approaches etc. We paid our attention to Gaussian quadratures not only because they are used in modern standard programs of integration. We made sure that there is certain didactic potential in Gaussian quadratures, which can be useful for those who study and teach mathematical modelling. We have selected problems in which nodes of Gaussian quadrature appear unexpectedly as a result of received solution. Traditionally the search of nodes and weighting factors of Gaussian quadrature involves making and solving the system of (non-linear!) algebraic equations, while simple mathematical folklore requires more ‘trivial’ proves which are good for simple understanding. Simple quadrature formula of Gauss (two nodes of integration) has been reviewed in the work. Examples of problems which contain latent connection to Gaussian quadrature are given. These problems are peculiar combination of simplicity and non-triviality in which a reader can find something interesting to his/her taste. It is natural that every problem is formulated on two ‘canonic’ intervals: [-1, 1] and [0, 1] , to cover two versions of quadrature: Gauss-Legendre and Gauss-Bernoulli ones. Reviewed examples give new subjects for reflections and observations. It is worth noting that approaches suggested in the work has been successfully tested for ‘clearness + briefness + convenience’ among students of higher education institutions. We agree with the point of view of Lithuanian mathematician R. Kashuba who thinks that interpolations method contributes to spread of democracy because it improves the ability to change point of view.

Na aprendizagem de conceitos de Álgebra Linear os alunos do ensino superior tendem a manifestar dificuldades devido à sua natureza abstrata, o que podem ser minimizadas caso se explore a conexão entre as diferentes representações de um mesmo conceito. Recorrendo a um ensino que explorou as representações algébrica e geométrica de conceitos de Álgebra Linear, procuramos averiguar como os alunos compreendem as transformações entre as diferentes representações, algébrica e geométrica, dos conceitos de coordenadas e mudança de base na resolução de uma tarefa após esse ensino. Adotamos uma abordagem qualitativa e interpretativa na procura de compreender como os alunos operacionalizam os conceitos de coordenadas e de mudança de base nessas representações. A análise das respostas dos alunos à tarefa proposta aponta para um melhor desempenho da transição da representação algébrica para a geométrica do que nas transformações realizadas dentro da própria representação algébrica. As maiores dificuldades relacionam-se com a interpretação equivocada dos conceitos envolvidos, a linguagem simbólica e com o uso inadequado de conceitos e procedimentos advindos da Geometria Analítica ou já abordados na própria Álgebra Linear.

In this paper, the design of a program algorithm for using path profile and coordinate geometry approach in the determination of the exact radius of curvature for rounded edge diffraction loss computation is presented. In addition to the radius of curvature, the program determined the values of other essential parameters required for the computation of rounded edge diffraction loss. The additional parameters include; occultation distance, the line of sight clearance height and the external angle between the tangent lines to the path profile. Importantly, the requisite mathematical expressions and detailed algorithm for the program are presented. Then a program was developed in Visual Basic for Application (VBA) based on the algorithm. Furthermore, sample elevation data was used to demonstrate the effectiveness of the program in the determination of the radius of curvature along with the other essential parameters required for the rounded edge diffraction loss computation. A sample path elevation profile was obtained using Geocontext path profile software for a 5233.692 m path that has a hilly obstruction with a maximum elevation of 593.0363 m that occurred at about a distance of 1861.649 m from the transmitter. The program result showed that for the sample path profile the rounded edge should be of a circle with a radius of 9,392.78 m, the occultation distance is 2,002.15 m, the line of sight clearance height which is 176.469 m and finally the angle between the tangent lines at their point of intersection is 0.212916 radians. The result is particularly useful because it is easy to generate the coordinates of elevation points which the program developed in this paper can use to automatically generate the essential parameters needed for the computation of rounded edge diffraction loss.

Monge’s mathematical work (descriptive geometry, analytic geometry, partial differential equations) belongs to the advanced science and teaching of all countries. Monge lived in Italy for almost 2 years as a commissioner for sciences and arts, at the behest of General Bonaparte, and giving the Constitution to the Jacobin Roman Republic of 1798. The small independent Republic of San Marino owes much of its existence to Monge, and even the well-known Parmesan cheese was studied scientifically for the first time by Monge. One section concerns the Monge–Ampere equation, obtained by Monge in 1784, and recently elucidated by Italian mathematicians.

To solve the problem of multi-module precise connection of support mechanism for modular space deployable antenna, a topology networking method is proposed. According to the general requirements of deployable antenna mesh surface, the composition and deployment principle of deployable antenna mechanism are analyzed. Based on the basic theories of differential geometry, analytical geometry and robotics, the geometric models of envelope circle from the first layer to the fifth layer are established respectively, and the modeling rules of geometric model for multi-layer envelope circle are analyzed, thus spatial geometry model of the deployable antenna meeting the requirements of large-scale development is constructed. On this basis, the influence relationships among spherical radius, envelope circle radius and module layers, and other parameters are analyzed respectively. Finally, the correctness of the model is validated with the numerical simulation software MATLAB, and the validation results show that the method can achieve accurate connection of multiple modules.

Five-coordinate geometry around ruthenium with highly exposed active sites has attracted intensive scientific interest due to its superior properties and extensive applications. Herein, we report a series of structurally controllable multi-Ru-bridged polyoxometalates, K5NaH10[{Ru4(H2O)n}(WO2)4(AsW9O33)4]·mH2O {1, 1-dehyd-373K, 1-dehyd-473K, 1-dehyd-573K; n = 4, m = 36; n = 4, m = 6; n = 4, m = 0; n = 0, m = 0} fabricated through a feasible assembly strategy using arsenotungstate {2, KNa12H17Cl2(As4W40O140)·29H2O} as a structure-directing unit. Systematic characterization methods identified that the six-coordinate geometry can successfully transform into five-coordinate geometry about active sites (Ru) by removing aqua ligands under high reaction temperatures. All the multi-Ru-bridged polyoxometalates demonstrated strong stability and catalytic effectiveness in the transformation of 1-(4-chlorophenyl)ethanol to 4′-chloroacetophenone under very mild conditions. 1-dehyd-573K, specifically, achieves the best catalytic effectiveness with a turnover frequency (TOF) = 25 100·h−1 owing to its unique five-coordinate geometry on the Ru sites. To our knowledge, 1-dehyd-573K outperforms other POM-based catalysts in the oxidative catalysis of 1-(4-chlorophenyl)ethanol. The heterogeneous polyoxometalates were also proven to be strongly reusable, with their structural integrities well maintained after multiple-cycle catalytic reactions.

Fundamental concepts in algebraic geometry such as algebraic cycles (with respect to intersection problems) or residues (with respect to division or interpolation ones) gain in flexibility once they are interpreted within the frame of currents in complex geometry ; such interpretation may even be transposed to the frame of arithmetics thanks to recent developments in analytic geometry over algebraically closed fields equipped with an ultrametric (instead of archimedean) absolute value. This survey intends to revisit from such currential interpretation (transposed if possible to the arithmetic setting, for example on Berkovich analytic spaces in place of complex spaces) improper intersection theory, together with the notions of local or global multiplicity, as well as multivariate residue theory with its articulations with Cauchy integral representation theory and Lagrange interpolation formula.