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The notion of definition is a central building block of mathematics. In addition to specific definitions, it is desirable that students learn about meta-mathematical aspects of definitions such as their role of classification, and about characteristics of definitions such as arbitrariness. We investigated how such meta-mathematical aspects emerge when students are led to construct a definition such as point of inflection or tangent line, and how the constructing process interacts with meta-mathematical aspects. We used Abstraction in Context as theoretical framework because of its suitability to analyze processes of knowledge construction. We found that meta-mathematical aspects of definitions influence and are influenced by students' processes of constructing definitions. We argue that awareness of meta-mathematical aspects improves students' conception of mathematical definitions. [ABSTRACT FROM AUTHOR]

We propose an original machine that traces conics and some transcendental curves (oblique trajectories of confocal conics) by the solution of inverse tangent problems. For such a machine, we also provide the 3D-printable model to be used as an intriguing supplement for geometry, calculus, or ordinary differential equations classes. [ABSTRACT FROM AUTHOR]

A convex quadrilateral, Q, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of Q coincides with the midpoint of at least one of the diagonals of Q. A parallelogram, Þ, is a special case of a midpoint diagonal quadrilateral since the diagonals of Þ bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms. First, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has tangency chords which are parallel to one of the diagonals of Q. Second, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has a pair of conjugate diameters parallel to the diagonals of Q. Finally, we show that there is a unique ellipse, EI, of minimal eccentricity incribed in a midpoint diagonal quadrilateral, Q, and also that the unique pair of conjugate diameters parallel to the diagonals of Q are the equal conjugate diameters of EI. [ABSTRACT FROM AUTHOR]

Even before the beginnings of Calculus a variety of methods for construct- ing tangents to plane curves were known. But what about the converse problem— raised by Debaune in the seventeenth century: under what conditions will a given collection of straight lines be tangents to the same curve? Utilizing Hermite’s inter- polation theorem, we show in Section 2 that the members of any finite collection of lines are tangents to infinitely many differentiable plane curves. After first developing a prescient observation of Descartes in Section 3, we state and prove our main theo- rem in Section 4,. This describes sufficient conditions for a one-parameter collection of lines in complex form to be the family of tangents of a differentiable curve in the complex plane. As an application, we derive Jakob Steiner’s nineteenth century result that, all save three members of the collection of Wallace-Simson lines of a triangle, are tangents to a deltoid whose incircle is the nine-point circle of the triangle. [ABSTRACT FROM AUTHOR]

The ultrafiltration process as one of the membrane processes based on pressure can replace with the conventional method (liming-carbonation) as a new method in sugar industry. In this research, reduction of hardness, increasing the non-sugar rejection components and improving the purity of permeate flow were modeled during the treatment of raw beet syrup with ultrafiltration by artificial neural network. The ultrafiltration process was carried out at three temperatures of 30, 40 and 50 ° C, three pressures in the membrane 1, 2 and 3 times in eight intervals of equal time of 1 to 60 minutes. The best model for reduction of hardness was obtained with a hidden layer, the number of 13 neurons, the tangent transfer function, the momentum learning law, and the percentage of data 25, 35, and 40 for training, evaluation, and test, respectively. The variation of non-sugar rejection compounds with a hidden layer, 15 neurons, tangent transfer function, Levenberg learning law, and assigning 50, 5 and 45 percent of the data to training, evaluating and testing with the least error and the highest correlation coefficient during modeling. Improvement of the purity of permeate flow during ultrafiltration with a hidden layer, 18 neurons, sigmoid transfer function, Levenberg learning law and data percentages 35, 40 and 25 for training, evaluation, and testing created the best network. Also, the highest correlation coefficient between laboratory data and predicted values with the model was obtained for hardness variation, non-sugar rejection compounds and purity, 0.892, 0.985 and 0.985 respectively. [ABSTRACT FROM AUTHOR]

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers' thinking about a trigonometric relationship was impacted by a series of tasks that prompted uncertainty. Using dynamic geometry software, we asked preservice teachers to compare angle measures of lines on a coordinate grid to their slope values, beginning by investigating lines whose angle measures were in a near-linear relationship to their slopes. After encountering and resolving the uncertainty of the exact relationship between the values, preservice teachers connected what they learned to the tangent relationship and demonstrated new ways of thinking that entail quantitative and covariational reasoning about this trigonometric relationship. We argue that strategically using uncertainty can be an effective way of promoting preservice teachers' reasoning about the tangent relationship. [ABSTRACT FROM AUTHOR]

If a; b; c are non-zero integers, we considerer the following problem: for which values of n the line ax + by + cz = 0 may be tangent to the curve xn + yn = zn? We give a partial solution: if n = 5 or if n = 1 is a prime a number, then the answer is the line cannot be tangent to the curve. This problem is strongly related to Fermat's Last Theorem. [ABSTRACT FROM AUTHOR]

This article analyzes Open City by Teju Cole as an example of a Theory Generation novel, positing that neither a strictly surface reading nor symptomatic reading approach alone is adequate for this sort of text. Expanding and complicating the novel's aside on the biology of bedbugs, the article argues that Julius, the book's protagonist, can best be understood not as a flâneur, fugueur, or model cosmopolitan, but, instead, as a parasite. His narratorial perspective is more analogous to the claustrophobic touch-driven irritations of the bedbugs that infest his mentor's Manhattan apartment than to the detached scopic flights of the birds that fill his free-associative reveries (and decorate the book's cover). Drawing on the work of J. Hillis Miller and Michel Serres, I posit that Julius's parasitic mode of interaction, which relies on brief moments of contact with others who are quickly mined for their stories and discarded, is deeply tied to the novel's reliance on tangential narration. The article argues for the unsettling ethical and affective implications of the similarities between the mechanisms of mathematical and narrative tangent-making vis-à-vis Julius's treatment of other characters in the novel. It finally asserts that Julius's rape of Moji is central to understanding both the affective discomfort of the novel as a whole and the limitations of similar protagonists who parasitically plot their way through Theory Generation novels. [ABSTRACT FROM AUTHOR]

Introduction. The aim of this study is determining the best configuration of artificial neural network, different networks with neuron number varying from 2 to 20, were designed. Their mean square errors, square normalized errors, absolute errors and correlation coefficients were investigated for different learning rules and transfer functions. Material and Methods. In this study, the potential of microfiltration process in reduction of hardness, improvement of purity and non-sugar rejection of raw beet juice was modeled with different parameters as temperature (30 and 60 °C) transmembrane pressure (1, 1.75 and 2.5 Bar) and time (regular time intervals from 1 to 60 min) by artificial neural network (ANN). ANN modeling was carried out by Neurosolution software v6 to determine the best type of transport function, learning rule, and determination of applied percentages for training, validation and testing stages. Results and discussion. The best neural network was the one hidden layer in Levenberg learning rules with tangent transfer function which included 8 neurons and resulted in maximum correlation coefficient for hardness according to temperature, pressure and time variation. The neural network with one hidden layer including 4 neurons with sigmoid transfer function under Levenberg learning rule had the least error and highest r for purity variation. Finally, the neural network with one hidden layer including 2 neurons, under Levenberg learning rule and tangent transfer function had the lowest error and highest correlation for non-sugar rejection percentage. Modeling was carried out with different percentages of data for training that the best prediction correlation for all parameters (turbidity, purity, non-sugar rejection) obtained when 60% of the data were used for training, 35% of them were employed for validation and 5% of the data were used for testing. The correlation of experimental data with the predicted values of the model obtained, too. According to the obtained models, ANN resulted in data with proper correlation with experimental data of hardness, purity and non-sugar rejection with respective correlation coefficients of 0.987, 0.980 and 0.981. This study also addressed the model sensitivity to input data. The best model sensitivity of the model for prediction of turbidity, purity and non-sugar rejection was related to time. Conclusion. The best rule for network training for prediction of hardness, purity and non-sugar rejection was Levenberg rule. The model was able to predict the hardness, purity and non-sugar rejection percentage under different operational models in a way that the modeled data showed high correlation with experimental data. [ABSTRACT FROM AUTHOR]

Several problems are considered in connection with the tangent points of circles inscribed in a triangle. A basic instrument in the problem solving is the application of Menelaus theorem. The work is awarded golden medal in the International competition “Mathematics and Projecting” in Moscow, Russia in May 2017. [ABSTRACT FROM AUTHOR]

The transition from the general Cartesian equation of the earth rotation ellipsoid to the parametric equation represents an exam subject for students of any faculties of geodesy. The present paper represents a method of obtaining this parametric equations which can be explained highschool seniors in applied mathematics courses. [ABSTRACT FROM AUTHOR]

ABSTRACT We present a modified interferometry method based on local tangent-phase analysis, which corrects the cross-correlated data before summation. The approach makes it possible to synthesize virtual signals usually vanishing in the conventional seismic interferometry summation. For a given pair of receivers and a set of different source positions, a plurality of virtual traces is obtained at new stationary projected points located along the signal wavefronts passing through the real reference receiver. The position of the projected points is estimated by minimizing travel times using wavefront constraint and correlation-signal tangent information. The method uses mixed processing, which is partially based on velocity-model knowledge and on data-based blind interferometry. The approach can be used for selected events, including reflections with different stationary conditions and projected points with respect to those of the direct arrivals, to extend the interferometry representation in seismic exploration data where conventional illumination coverage is not sufficient to obtain the stationary-phase condition. We discuss possible applications in crosswell geometry with a velocity anomaly and a time lapse. [ABSTRACT FROM AUTHOR]

Analysis of the application of the economic slope concept in determining the economically optimum gas injection rate for a gas lift process is presented in this work. The type of gas lift process used for the study is continuous gas lift. The study made use of PROSPER, a production modelling tool patented to Petroleum Experts Limited for the analysis. Well X was taken as a case for the gas lift design analysis. Reservoir, fluid and equipment data were obtained and used for the analysis. PROSPER was used to perform the gas lift equipment design and generate the IPR curve. PROSPER was also used to generate the gas lift design performance curve. Manual computations involving the economic slope concept were made based on the information read from the performance curve for the optimum gas injection rate. From the analysis, it is determined that the optimum gas injection rate for the gas lift process at Well X is 2.47MMscf/d. [ABSTRACT FROM AUTHOR]

The two main processes of calculus are integration and differentiation. These two processes are intimately related by being reciprocal to each other. This fact constitutes the fundamental theorem of calculus (FTC). In the present article, we explore this important theorem within a geometric context, based on Isaac Barrow's work from the seventeenth century. We argue that presenting and discussing the origins and development of the FTC is beneficial for first-year calculus courses. [ABSTRACT FROM AUTHOR]

Usually, the tangent line is considered to be a calculus notion. However, it is also a graphical and an algebraic notion. The graphical frame, where our primary conceptions are conceived, could give rise to algebraic methods to obtain the tangent line to a curve. In this pre-calculus perspective, two methods are described and discussed according to their potential for secondary students and teacher training. [ABSTRACT FROM AUTHOR]

Abstract: Differential operators are essential in many image processing applications. Previous work has shown how to compute derivatives more accurately by examining the image locally, and by applying a difference operator which is optimal for each pixel neighborhood. The proposed technique avoids the explicit computation of fitting functions, and replaces the function fitting process by a function classification process using a filter bank of feature detection templates. Both the feature detectors and the optimal difference operators have a specific shape and an associated cost, defined by a rigid mathematical structure, which can be described by Gröbner bases. This paper introduces a cost criterion to select the operator of the best approximating function class and the most appropriate template size so that the difference operator can be locally adapted to the digitized function. We describe how to obtain discrete approximates for commonly used differential operators, and illustrate how image processing applications can benefit from the adaptive selection procedure for the operators by means of two example applications: tangent computation for digitized object boundaries and the Laplacian of Gaussian edge detector. [Copyright &y& Elsevier]

We construct an example of a C 1-smooth real function of two variables whose gradient range is an arc with no tangent at any point. [ABSTRACT FROM AUTHOR]

Abstract: The purpose of this paper is to present an algorithm for computing the tangents of a real plane algebraic curve. By this algorithm, all the slopes of the tangents to a real plane algebraic curve at a particular point may be accurately represented via polynomial real root isolation. [Copyright &y& Elsevier]

Abstract: We present a family of unimodal maps, arising from a simple queueing model, which exhibits reverse bifurcations. We compare and contrast this with bifurcations occurring in the well-known logistic family of unimodal quadratic maps. Throughout this study, graphics generated via numerical simulations provide key insights. [Copyright &y& Elsevier]

This paper studies the asymptotic behaviour of steady supersonic flow past a piecewise smooth corner or bend. Under the hypothese that both vertex angle and the total variation of tangent along the boundary are small, it is shown that the solution can be obtained by a modified Glimm scheme, and that the asymptotic behaviour of the solution is determined by the velocity of incoming flow and the limit of the tangent of the boundary at infinity. [ABSTRACT FROM AUTHOR]

Introduction: In the XVII Century, Fermat d/eveloped the first general method to determine the minimums and maximums, in the memory Methodus ad disquirendam maximan et miniman et de tangentibus linearum curvarum, being a purely algorithmic procedure with no demonstrative foundation at all, and in which Fermat introduces the adequality technique, previously used by Diofantus in the Alexandria School. The vague and laconic way in which Fermat showed the Methodus has generated some interpretations under infinitesimal calculus terms, and one of them says that in the Methodus sub lies the calculation of a derivation that equals zero. Objective: To make a detailed study of the way Fermat explains his maximums and minimums method in the Methodus and discusses the anachronism interpretations given by some experts about his method. Methods: Based on the texts mentioned in the bibliography we make a historical scrutiny with the main objective of studying and analyzing the method showed by Fermat. Results: The detailed study is made, the method is applied to obtain a minimum in the problem of the segment division and the interpretations given by some experts are analyzed under the current standards. These interpretations must be seen carefully, because they force the method showed by Fermat. [ABSTRACT FROM AUTHOR]