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Toward identifying efficient sound absorbers, we have formulated and analyzed the micro-perforated panels (MPPs) comprising cylindrical pores of arbitrary axial profiles for out-of-plane wave propagation. First, the forward problem was solved wherein an exact analytical expression for the absorption coefficient α was developed for these MPPs. The dependence of α was analyzed for various profiles comprising convex and concave-shaped sections including divergent and convergent linear and sinusoidal profiles, conic sections, Kilroy, and chirp shapes. The MPPs with pores having a diverging shape at the entrance were found to have higher sound absorption. The non-symmetric fluctuations in the profile led to fluctuations in the spectrum of α. Aiming to address the long-standing challenge of low-frequency sound absorption, we solved the inverse problem to identify the axial profile of the cylindrical pore for maximum sound absorption for frequencies up to 2500 Hz. Given the advances in additive manufacturing, the results of this comprehensive work help in designing MPPs comprising complex-shaped pores suitable for a particular spectral regime. [ABSTRACT FROM AUTHOR]

In this article, Mehmet Vâsıf and his analytical geometry book, Hendese-i Halliyye-Musattaha and Kutû'-i Mahrûtiye were examined. Since this book is one of the first books written in its field in the Ottoman language, it has great importance for the Turkish history of mathematics and mathematics education. Mehmet Vâsıf taught mathematics at Mekteb-i Bahriyye between 1901-1908, and he taught geography at Tüccar Kaptan Mektebi Umûmî. He compiled many works in various fields and introduced them into the Ottoman language. In this article, information about Mehmet Vâsıf's life and works is given, and his book on analytical geometry and conic sections, Hendese-i Halliyye-i Musattaha and Kutû'-i Mahrûtiye is introduced. This study's scope includes the transliteration of the table of contents and the book's introduction. Mehmet Vâsıf didn't provide detailed information about his sources. Within the scope of this study, Mehmet Vâsıf's sources were also traced. [ABSTRACT FROM AUTHOR]

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

We show that a central oval in a Moufang Fano plane is a Moufang oval if and only if the oval is harmonic. This shows, for the first time, that harmonic ovals that are not conics exist. In addition, we show that, if a strongly harmonic oval in a Moufang Fano plane exists, then the plane is Pappian and the oval is a conic and that if a harmonic oval in a Moufang plane that is not a Fano plane exists, then the plane is Pappian and the oval is a conic. We also show local versions of these results, characterising conics by the property of: being harmonic at external points on a fixed secant line, being harmonic at external points on a fixed external line, being harmonic at points on a fixed tangent line and being harmonic at all secant lines on a fixed point of the oval. Finally, we give some related characterisation of conics in terms of degenerations of Pascal's theorem, including the theorem that an oval in a Moufang plane with the four point Pascal property is a conic in a Pappian plane. [ABSTRACT FROM AUTHOR]

The implementation of student projects in undergraduate mathematics and statistics courses is discussed with suggestions and recommendations. Advantages and pitfalls of this teaching tool are presented. A prototypical student project for an elementary mathematics course is described. The features of this sample project serve as models for components of other projects. This illustrative project concerns the locus of intersection points of pairs of tangent lines to a conic section, such that the products of the lines’ slopes are a constant. [ABSTRACT FROM AUTHOR]

This special issue of the Journal of Optimization Theory & Applications focuses on the work of Professors Cornelis Roos and Florian A. Potra in the field of optimization. The issue includes a biographical sketch of both professors, highlighting their achievements. The papers in the special issue cover various topics such as conic optimization, interior point methods, proximal methods, convex optimization, and stochastic optimal control. The authors of the papers have close connections to Professors Roos and Potra, either as students, visitors, or co-authors. The document is a compilation of research papers on interior-point methods (IPMs) and their applications in optimization and conic programming. The papers cover theoretical developments of IPMs, algorithms for linear complementarity problems (LCPs), weighted LCPs, and sufficient linear complementarity problems. There are also papers on new computational approaches for IPMs, applications of IPMs in different fields such as mechanics and support vector machines, and the theory and application of conic optimization. The document also includes papers on proximal methods for convex and non-convex problems, as well as papers on convexity and convex optimization. The volume concludes with acknowledgments to the authors, reviewers, and editor-in-chief. [Extracted from the article]

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 (P 3) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective. [ABSTRACT FROM AUTHOR]

Inspired by Schmidt's work on twisted cubics, we study wall crossings in Bridgeland stability, starting with the Hilbert scheme Hilb 2 m + 2 (P 3) parametrizing pairs of skew lines and plane conics union a point. We find two walls. Each wall crossing corresponds to a contraction of a divisor in the moduli space and the contracted space remains smooth. Building on work by Chen–Coskun–Nollet, we moreover prove that the contractions are K-negative extremal in the sense of Mori theory and so the moduli spaces are projective. [ABSTRACT FROM AUTHOR]

The numerical performance of algorithms can be studied using test sets or procedures that generate such problems. This paper proposes various methods for generating linear, semidefinite, and second-order cone optimization problems. Specifically, we are interested in problem instances requiring a known optimal solution, a known optimal partition, a specific interior solution, or all these together. In the proposed problem generators, different characteristics of optimization problems, including dimension, size, condition number, degeneracy, optimal partition, and sparsity, can be chosen to facilitate comprehensive computational experiments. We also develop procedures to generate instances with a maximally complementary optimal solution with a predetermined optimal partition to generate challenging semidefinite and second-order cone optimization problems. Generated instances enable us to evaluate efficient interior-point methods for conic optimization problems. [ABSTRACT FROM AUTHOR]

The article discusses the study of conic sections in mathematics and introduces a new class of curves called haricoids. It explains the definition of a conic in terms of its focus, directrix, and eccentricity. The article explores the concept of using a circle as a directrix for ellipses and hyperbolas, noting that the position of the focus relative to the circle determines the shape of the conic. The text then delves into the properties and equations of haricoids, including their construction and behavior as parameters vary. The author presents examples and graphs to illustrate these concepts and suggests further research on the applications and connections of haricoids. [Extracted from the article]

In this study, control and fault detection are examined for Takagi–Sugeno (T–S) fuzzy continuous-time systems with conic-type nonlinearity. In a conic-type dynamic system, Takagi–Sugeno fuzzy models are applied. A conic-type non-linearity must lie in an n-dimensional hyper-sphere with an uncertain centre to satisfy the restrictive condition. A combined error system of estimation and control error is obtained by combining the observations and the controller. Our main goal is to design an observer and controller for fault detection that minimizes the impact of unknown inputs and faults on the system in terms of the $ H_{\infty } $ H ∞ performance. The Lyapunov function is used to construct the residual signal using an observer and a controller. Using Linear Matrix Inequalities (LMIs), we can express the performance conditions for $ H_{\infty } $ H ∞ . The scheme's effectiveness is demonstrated with numerical examples. [ABSTRACT FROM AUTHOR]

One nappe of a right circular cone, cut by a transverse plane, splits the cone into an infinite frustum and a cone with an elliptical section of finite volume. There is a standard way of computing this finite volume, which involves finding the parameters of the so-called shadow ellipse, the characteristics of the oblique ellipse (the cut) and, finally, the projection of the vertex of the cone onto the oblique ellipse. This paper shows that it is possible to compute that volume just by using the information of the shadow ellipse and the height of the cone. Indeed, the finite slant cone has the same volume of an elliptic right cone, with the base being the shadow ellipse of the cut portion and with the height being the distance between the vertex of the cone and the intersection of the height of the original cone with the cutting plane. This is proved by introducing a volume-preserving shear transformation of the elliptical slant cone to a right cone, so that the standard volume formula for a cone can be straightforwardly applied. This implies a simplification in the procedure for computing the volume, since the oblique ellipse—i.e., the difficult part—can be neglected because only the shadow ellipse needs to be determined. [ABSTRACT FROM AUTHOR]

One of the geometric figures that has symmetry properties is the Cassini oval. The Cassini oval is a curve defined as the locus of points in the plane such that the product of the distances to two fixed points is constant. Cassini ovals are named after the astronomer Giovanni Domenico Cassini, who studied them in 1680. Today, the geometric properties of Cassini ovals are used in many fields: analytical geometry, nuclear physics, radiolocation, and industrial applications. The bistatic radar uses Cassini ovals to detect various targets in radiolocation. Until now, there have been no studies on the clustering capabilities of Cassini ovals. As a novelty, it is hypothesized that clustering possibilities could be used for Cassini ovals. In this article, a study of the capabilities of Cassini ovals in radiolocation was carried out, and their suitability for clustering purposes was shown. [ABSTRACT FROM AUTHOR]

We consider a system of two reaction-diffusion equations in a bounded domain of the -dimensional space with Neumann boundary conditions on the boundary for which the reaction terms and depend on two parameters and . Assume that the system has a spatially homogeneous solution , with and , where is the Jacobian of the corresponding linearized system in the diffusionless approximation and is a smooth monotonically increasing function. We propose some method for the analytical description of the domain of necessary and sufficient conditions of Turing instability on the plane of system parameters for a fixed diffusion coefficient . Also, we show that the domain of necessary conditions of Turing instability on the plane is bounded by the zero-trace curve, the discriminant curve, and the locus of points . Explicit expressions are found for the curves of sufficient conditions and we prove that the discriminant curve is the envelope of the family of these curves. It is shown that one of the boundaries of the Turing instability domain consists of the fragments of the curves of sufficient conditions and is expressed in terms of the function and the eigenvalues of the Laplace operator in the domain under consideration. We find the points of intersection of the curves of sufficient conditions and show that their abscissas do not depend on the form of and are expressed in terms of the diffusion coefficient and the eigenvalues of the Laplace operator. In the special case . For this case, the range of wave numbers at which Turing instability occurs is indicated. We obtain some partition of the semiaxis into half-intervals each of which corresponds to its own minimum critical wave number. The points of intersection of the curves of sufficient conditions lie on straight lines independent of the diffusion coefficient . By way of applications of the statements proven, we consider the Schnakenberg system and the Brusselator equations. [ABSTRACT FROM AUTHOR]

The author fully linearizes the stochastic volatility inspired (SVI) formula by rewriting it into the algebraic form of a conic section. This step reduces the complexity of the otherwise non-linear optimization problem drastically. Based on the conic representation, the author introduces the direct least-squares for SVI, which enables him to fit the model in a non-iterative and thus computationally efficient manner. The performance of the proposed method is evaluated upon empirical data of seven different asset classes. It significantly outperforms the existing benchmark algorithm in terms of computational speed while achieving a similar quality of fit. Concluding on that, the direct SVI fit qualifies as a robust method for calibrating implied volatilities in real-time. [ABSTRACT FROM AUTHOR]

Purpose. Investigation of the velocity fields of non-equilibrium fluid filtration in a layer under harmonic action on it and assessment of the influence of relaxation effects on the attenuation of the amplitude of initial disturbances within the framework of mathematical modeling of non-equilibrium plane-radial filtration. Methodology. A mathematical model of non-equilibrium plane-radial filtration with a generalized dynamic Darcy law in the form of a boundary value problem in a half-space with a harmonic excitation law at its boundary is considered. Based on the exact solutions of the boundary value problem, the attenuation of the amplitude of initial disturbances under the model’s parameters varying and influence of parameters on the size of the disturbed region are investigated. Findings. A differential equation modeling non-equilibrium filtration processes in the massif in the cylindrical reference frame was obtained. Using the method of separation of variables, a solution was constructed, bounded at infinity, to the model differential equation subjected to harmonic action at the layer boundary. The solution’s asymptotic approximation was constructed for large values of the argument. Using the asymptotic solution of the boundary value problem, the damping of velocity field during non-equilibrium filtration was analyzed depending on the frequency of the harmonic action, the ratio of the piezoconductivity coefficients of the layer, and the relaxation time. Profiles of the dependences of the size of the influence zone on the model parameters were plotted and the choice of parameters for optimal influence on the bottom-hole zone of the well was analyzed. Originality. On the basis of the non-equilibrium filtration model, it is shown that harmonic disturbances applied to the boundary of a semi-infinite layer can penetrate the reservoir over a greater distance under the conditions of manifestation of the relaxation mechanism of the fluid-skeleton interaction, compared to the equilibrium filtration process. Such an effect is observed at a finite interval of disturbance frequencies, while at high frequencies relaxation contributes to a more significant damping of disturbances. In the parametric space of excitation frequency – relaxation time, there is a locus of points that corresponds to the maximum size of influence zone of disturbances. Practical value. The obtained results are relevant for research on the impact of wave disturbances on the layer with the aim of intensifying filtration processes, as well as for creation of new wave technologies to increase the extraction of mineral resources from productive layers. [ABSTRACT FROM AUTHOR]

A conceptual understanding of mathematics will aid students in solving mathematical problems and is their starting point. The study's findings provide necessary information on the different sources such as teachers, the internet, modules, parents, and students' own metacognition in developing students' conceptual understanding of conic sections. As a result, this study is a quantitative study that aims to construct and validate an instrument that can be used to determine the extent to which the different sources help develop students' conceptual understanding. The study included 107 high school students and 2 mathematics content specialists. Exploratory factor analysis (EFA) was conducted for data reduction. The result of the exploratory factor analysis resulted in the removal of 14 items. Four items (19, 20, 24, and 48) failed to load on any dimension significantly, while 10 items (13, 16, 37, 38, 39, 40, 41, 42, 45, and 49) loaded onto factors other than their underlying factors. The final form of the instrument has 35 statements that are designed to determine the extent of different sources of conceptual understanding such as teacher, module, internet, parents, and own metacognition in learning conic sections. 10 statements under "teacher" as a source of conceptual understanding, 6 for the module, 9 under the internet, 6 under parents, and 4 statements under own metacognition. The scale has an overall Cronbach's Alpha of 0.93. [ABSTRACT FROM AUTHOR]

The 16-year old Blaise Pascal found an incidence relation that holds when six points lie on a conic. A century later, Braikenridge and Maclaurin extended Pascal's result to a straightedge construction that characterizes when six points lie on a conic. Nearly 400 years later, we develop a straightedge construction to check whether ten points lie on a cubic curve. [ABSTRACT FROM AUTHOR]

An algorithm for finding a control of a linear switched system by means of Geometric Algebra is designed. More precisely, we develop a switching path searching algorithm for a two‐dimensional linear dynamical switched system with a non‐singular matrix whose integral curves are formed by two sets of centralized ellipses. It is natural to represent them as elements of Geometric Algebra for Conics and construct the switching path by calculating switching points, i.e., intersections and contact points. For this, we use symbolic algebra operations or, more precisely, the wedge and inner products that are realizable by sums of products in the coordinate form. Therefore, no numerical solver to the system of equations is needed. Indeed, the only operation that may bring in an inaccuracy is vector normalization, i.e., square root calculation. The resulting switching path is formed by pieces of ellipses that are chosen, respectively, from the two sets of integral curves. The switching points are either intersections in the first or final step of our algorithm, or contact points. This choice guarantees the optimality of the switching path with respect to the number of switches. Two examples are provided to demonstrate the search for the intersections of the conics and, consequently, a description is presented of the construction of a switching path in both cases. [ABSTRACT FROM AUTHOR]

We study compactifications of the moduli space of unordered points in the plane via variation of GIT-quotients of their corresponding Hilbert scheme. Our VGIT considers linearizations outside the ample cone and within the movable cone. For that purpose, we use the description of the Hilbert scheme as a Mori dream space, and the moduli interpretation of its birational models via Bridgeland stability. We determine the GIT walls associated with curvilinear zero-dimensional schemes, collinear points, and schemes supported on a smooth conic. For seven points, we study a compactification associated with an extremal ray of the movable cone, where stability behaves very differently from the Chow quotient. Lastly, a complete description for five points is given. [ABSTRACT FROM AUTHOR]

Assume that C is a circle with center O and radius r. Fix a point H satisfying 0 < OH < 3r and set h = OH. By the (C;H) -locus TH, we mean the family of all the triangles inscribed in C with orthocenter H. The set AH of the symmedian points of all triangles in TH is called the symmedian locus of TH. We show that AH is the arc lying on the circle CH with center OH determined by 􀀀􀀀􀀀! OOH = 6r2 9r2􀀀h2 􀀀􀀀! OH and radius rH = 2rh2 9r2􀀀h2 which is contained in the open finite region enclosed by C. We also show that for each K 2 AH, we can construct a unique triangle in TH with symmedian point K. Hence there is a bijection from TH onto AH . [ABSTRACT FROM AUTHOR]

The article "Generalization of Conway's circle: a review" by Francisco Javier García Capitán, published in the Gazette of the Royal Spanish Mathematical Society, presents a generalization of Conway's circle. The properties of this generalization and the type of conic associated with each point in the plane of a triangle are studied. The article also mentions other mathematical concepts named after Conway, such as Conway's algebra and Conway's Game of Life. Additionally, a geometric construction is presented to trace the Tucker-Poncelet curve using the Cabri program, and related geometric concepts such as the radical circle and the cevian triangle are mentioned. [Extracted from the article]

• A DE-GDQ algorithm in free vibration analysis of shells of revolution with free-form meridian and their combined structures is proposed. • A complete solution procedure for free vibration analysis based on 3D inverse modeling, FSDT and DE-GDQ is established. • Validated numerically the reliability and accuracy of the DE-GDQ algorithm on multiple models, different boundary conditions and types of materials. In this paper, a data-driven enhanced generalized differential quadrature (DE-GDQ) algorithm in free vibration analysis is proposed, which can be applied to shells of revolution with free-form meridian and their combined structures. According to first order shear deformation theory (FSDT) and Hamilton's principle, the governing equation for doubly-curved shells is obtained. The general boundary conditions are considered through artificial springs. Then the GDQ is introduced to solve the governing equations and the DE-GDQ is proposed on the basis of GDQ. After a detailed solution procedure of DE-GDQ is introduced and the convergence analysis is finished, multiple models and boundary conditions are validated including shells of revolution with conic section meridian, Haack series meridian, rational Bézier curve meridian. Isotropic materials and anisotropic materials including functionally graded (FG) materials are both verified. The free vibration analysis of combined structures is also finished and compared with finite element method (FEM). The comparison among DE-GDQ, GDQ and FEM shows that the DE-GDQ has greater advantages in terms of calculation efficiency, accuracy, and scope of application. [Display omitted] [ABSTRACT FROM AUTHOR]

We study the minimum number of distinct distances between point sets on two curves in R 3. Assume that one curve contains m points and the other n points. Our main results: (a) When the curves are conic sections, we characterize all cases where the number of distances is O (m + n). This includes new constructions for points on two parabolas, two ellipses, and one ellipse and one hyperbola. In all other cases, the number of distances is Ω (min { m 2 / 3 n 2 / 3 , m 2 , n 2 }). (b) When the curves are not necessarily algebraic but smooth and contained in perpendicular planes, we characterize all cases where the number of distances is O (m + n). This includes a surprising new construction of non-algebraic curves that involve logarithms. In all other cases, the number of distances is Ω (min { m 2 / 3 n 2 / 3 , m 2 , n 2 }). [ABSTRACT FROM AUTHOR]

Tessellations are an important tool to model the microstructure of cellular and polycrystalline materials. Classical tessellation models include the Voronoi diagram and the Laguerre tessellation whose cells are polyhedra. Due to the convexity of their cells, those models may be too restrictive to describe data that includes possibly anisotropic grains with curved boundaries. Several generalizations exist. The cells of the generalized balanced power diagram are induced by elliptic distances leading to more diverse structures. So far, methods for computing the generalized balanced power diagram are restricted to discretized versions in the form of label images. In this work, we derive an analytic representation of the vertices and edges of the generalized balanced power diagram in 2d. Based on that, we propose a novel algorithm to compute the whole diagram. • Parametric representation of the edges of the generalized balanced power diagram. • Proposal of an algorithm to compute the generalized balanced power diagram. • Derivation of cell area and cell perimeter. [ABSTRACT FROM AUTHOR]

The use of subdivision schemes in applied and real-world contexts requires the development of conceptually simple algorithms that can be converted into fast and efficient implementation procedures. In the domain of interpolatory subdivision schemes, there is a demand for developing an algorithm capable of (i) reproducing all types of conic sections whenever the input data (in our case point-normal pairs) are arbitrarily sampled from them, (ii) generating a visually pleasing limit curve without creating unwanted oscillations, and (iii) having the potential to be naturally and easily extended to the bivariate case. In this paper we focus on the construction of an interpolatory subdivision scheme that meets all these conditions simultaneously. At the centre of our construction lies a conic fitting algorithm that requires as few as four point-normal pairs for finding new edge points (and associated normals) in a subdivision step. Several numerical results are included to showcase the validity of our algorithm. • We construct a univariate interpolatory subdivision scheme that reproduces conic sections. • It is a geometric subdivision scheme based on point-normal pairs. • We demonstrate our new scheme on several examples. [ABSTRACT FROM AUTHOR]

For any d ∈ { 1 , ... , 6 } , we prove that the web of conics on a del Pezzo surface of degree d carries a functional identity whose components are antisymmetric hyperlogarithms of weight 7 − d. Our approach is uniform with respect to d and relies on classical results about the action of the Weyl group on the set of lines on the del Pezzo surface. These hyperlogarithmic functional identities are natural generalizations of the classical 3-term and (Abel's) 5-term identities satisfied by the logarithm and the dilogarithm, which correspond to the cases when d = 6 and d = 5 respectively. [ABSTRACT FROM AUTHOR]

• A cylinder instead of sphere is used for calibration, which is easy to manufacture and can cover a larger robot workspace. • An unconstrained dimensionality reduction method is proposed to accurately fit the incomplete conic profiles as ellipses. • The calibration error of hand eye parameters is reduced by about 50% compared to feature points col-linearity based method. This paper proposed a hand-eye calibration method for a line structured-light (LSL) scanner mounted on industrial robot. Unlike widely used spherical targets, the target used in our work is a cylinder, which is inexpensive and easy to be manufactured in larger size to cover a wider calibration space. In order to address the profile self-occlusion and drastically shape changing of conic section during calibration, an unconstrained dimensionality reduction method is proposed to fit the scanned profiles. Based on fitted ellipse centers under various robot poses, the hand-eye calibration model is developed based on the difference between cylinder radius and the distance from the profile point to the cylinder axis. Simulation and actual experiments have shown that the proposed ellipse fitting method is more accurate and robust, and the calibration method achieves higher reconstruction accuracy under the same robot positioning error, need less robot operations to obtain more accurate hand-eye parameters. [ABSTRACT FROM AUTHOR]

We obtain the bifurcation of some special curves on generic 1-parameter families of surfaces in the Minkowski 3-space. The curves treated here are the locus of points where the induced pseudo metric is degenerate, the discriminant of the lines principal curvature, the parabolic curve and the locus of points where the mean curvature vanishes. [ABSTRACT FROM AUTHOR]

Resistor network research is of great importance, yet many resistor networks and their large-scale fast computations have not received sufficient attention. This paper proposes a new resistor network with idiosyncratic shape, i.e., a α × β conic surface (CS) resistor network that resembles the upper part of a three-dimensional Dirac function. Utilizing the Recursion Transform (RT-V) method of Tan, a recursive matrix equation model is constructed based on Kirchhoff's law and nodal voltages, which contains the modified tridiagonal Toeplitz matrix. By using the orthogonal matrix transformation, the eigenvalues and eigenvectors of the modified tridiagonal Toeplitz are obtained. The discrete sine transform of the fourth type (DST − IV) is utilized to solve node voltages, while the explicit potential function is represented by the Chebyshev polynomials of the second kind. In addition, explicit potential functions for some special cases are provided, and the potential distribution is illustrated using dynamic three-dimensional graph. To achieve a rapid calculation of the potential, a fast algorithm based on the multiplication of DST-IV with a vector is proposed. In the end, analysis of computational efficiency for the explicit potential function and the fast algorithm are shown. • A new conic surface resistor network is established. • The explicit potential function of represented by Chebyshev polynomials. • The potential in special cases are displayed by a three-dimensional dynamic view. • A fast algorithm of computing potential is proposed by using famous DST-IV. • Experimental analysis of computational efficiency and computational scale. [ABSTRACT FROM AUTHOR]

We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval. • We present a systematic study on the rational cubic Bézier representation of conics. • The key ingredient is the conic associated with a given cubic Bézier polygon. • This conic results from applying a perspective to a semicircle embedded in a square. • A cubic is a conic if it coincides with this conic associated with the cubic polygon. • With positive weights, the cubic form duplicates the range of admissible rho-values. [ABSTRACT FROM AUTHOR]

In this article we explain that several integrable mechanical billiards in the plane are connected via conformal transformations. We first remark that the free billiards in the plane are conformal equivalent to infinitely many billiard systems defined in central force problems on a particular fixed energy level. We then explain that the classical Hooke-Kepler correspondence can be carried over to a correspondence between integrable Hooke-Kepler billiards. As part of the conclusion we show that any focused conic section gives rise to integrable Kepler billiards, which continues previous works of Panov [26] and Gallavotti-Jauslin [11]. We discuss several generalizations of integrable Stark billiards. We also show that any finite combinations of confocal conic sections give rise to integrable billiard systems of Euler's two-center problems. [ABSTRACT FROM AUTHOR]

This article presents an optimal power flow (OPF) formulation based on the branch flow model for bipolar DC grids with asymmetric loading. The proposal considers the power injection formulation of the OPF to obtain the branch flow model equations that allow solving the power flow problem in bipolar DC networks. Furthermore, a nonlinear programming (NLP) optimization model is deduced and transformed into a second-order conic programming one, which involves convex optimization and guarantees a global optimum for the relaxed model, the uniqueness of this solution, and fast convergence. Finally, an approximation based on balanced operation is proposed to simplify the OPF. Several case studies in two test systems composed of 21 and 85 nodes with a bipolar structure demonstrate that the proposal is accurate, fast, and close to the exact solution of the NLP model when compared with competitive literature reports. All optimization models were implemented in the Python language. • Bipolar DC networks can operate with balanced and unbalanced conditions as a function of the demands connected between poles. • A convex approximation can be applied to bipolar DC networks via second order cone programming. • A simplification based on expected balanced operation ca be reached to obtain a relaxed convex model with low estimation errors. • Dispersed generators and battery energy storage systems can be added to bipolar DC networks using a optimal power flow formulation. [ABSTRACT FROM AUTHOR]

The purpose of this note is to report, in narrative rather than rigorous style, about the nice geometry of 6-division points on the Fermat cubic F and various conics naturally attached to them. Most facts presented here were derived by symbolic algebra programs and the idea of the note is to propose a research direction in the search of conceptual proofs of facts stated here and their generalisations. Extensions in several directions seem possible (taking curves of higher degree and contact to F , studying higher degree curves passing through higher order division points on F , studying curves passing through intersection points of already constructed curves, taking the duals etc.) and we hope some younger colleagues might find pleasure in following proposed paths as well as finding their own. [ABSTRACT FROM AUTHOR]