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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]

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]

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]

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]

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]

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]

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]

Symbolic 3-plithogenic sets as a generalization of classical concept of sets were applicable to algebraic structures. The symbolic 3-plithogenic rings and fields are good generalizations of classical corresponding systems. In this paper, we study the symbolic 3-plithogenic real functions with one variable by using a special algebraic function called AH-isometry. In addition, we discuss the symbolic 3-plithogenic simple differential equations and conic sections by using this isometry. Also, many examples will be presented to explain the novelty of this work. [ABSTRACT FROM AUTHOR]

Symbolic 2-plithogenic sets as a generalization of classical concept of sets were applicable to algebraic structures. The symbolic 2-plithogenic rings and fields are good generalizations of classical corresponding systems. In this paper, we study the symbolic 2-plithogenic real functions with one variable by using a special algebraic function called AH-isometry. In addition, we discuss the symbolic 2-plithogenic simple differential equations and conic sections by using this isometry. Also, many examples will be presented to explain the novelty of this work. [ABSTRACT FROM AUTHOR]

This paper addresses the equivalence problem of conic submanifolds in the tangent bundle of a smooth 2-dimensional manifold. Those are given by a quadratic relation between the velocities and are treated as nonholonomic constraints whose admissible curves are trajectories of the corresponding control systems, called quadratic systems. We deal with the problem of characterising and classifying conic submanifolds under the prism of feedback equivalence of control systems, both control-affine and fully nonlinear. The first main result of this work is a complete description of non-degenerate conic submanifolds via a characterisation under feedback transformations of the novel class of quadratic control-affine systems. This characterisation can explicitly be tested on structure functions defined for any control-affine system and gives a normal form of quadratisable systems and of conic submanifolds. Then, we consider the classification problem of regular conic submanifolds (ellipses, hyperbolas, and parabolas), which is treated via feedback classification of quadratic control-nonlinear systems. Our classification includes several normal forms of quadratic systems (in particular, normal forms not containing functional parameters as well as those containing neither functional nor real parameters) and, as a consequence, gives a classification of regular conic submanifolds. [ABSTRACT FROM AUTHOR]

The reverse k nearest neighbors (RkNN) query is a prominent yet time-consuming spatial query used in facility siting, influential domain analysis, potential customer analysis, etc. Its aim is to identify all points that consider the query point as one of their k closest points. However, when k is relatively large (e.g. k = 1000), existing RkNN techniques often struggle to provide acceptable response times (within a few seconds). To address this issue, we propose a verification approach called conic section discriminance (CSD). This method serves to determine whether points belong to the RkNN set. With CSD, only a small fraction of candidates require costly k nearest neighbors (kNN) queries for verification, while the rest can be rapidly verified with O(1) complexity. Furthermore, we propose a Voronoi-based candidate generation approach to curtail the candidate set size. By leveraging the VoR-tree structure, we integrate these two approaches to form a novel RkNN algorithm named CSD-RkNN. A comprehensive set of experiments is conducted to compare CSD-RkNN with Slice as the state-of-the-art RkNN algorithm, and VR-RkNN as the original RkNN algorithm on VoR-tree. The results indicate that CSD-RkNN consistently outperforms the other two algorithms, especially when k is relatively large. [ABSTRACT FROM AUTHOR]

This research addresses the problem regarding the efficient operation of photovoltaic (PV) plants in monopolar direct current (DC) distribution networks from a perspective of convex optimization. PV plant operation is formulated as a nonlinear programming (NLP) problem while considering two single-objective functions: the minimization of the expected daily energy losses and the reduction in the expected CO2 emissions at the terminals of conventional generation systems. The NLP model that represents the energy management system (EMS) design is transformed into a convex optimization problem via the second-order cone equivalent of the product between two positive variables. The main contribution of this research is that it considers the uncertain nature of solar generation and expected demand curves through robust convex optimization. Numerical results in the monopolar DC version of the IEEE 33-bus grid demonstrate the effectiveness and robustness of the proposed second-order cone programming model in defining an EMS for a monopolar DC distribution network. A comparative analysis with four different combinatorial optimizers is carried out, i.e., multiverse optimization (MVO), the salp swarm algorithm (SSA), the particle swarm optimizer (PSO), and the crow search algorithm (CSA). All this is achieved while including an iterative convex method (ICM). This analysis shows that the proposed robust model can find the global optimum for two single-objective functions. The daily energy losses are reduced by 44.0082% with respect to the benchmark case, while the CO2 emissions (kg) are reduced by 27.3771%. As for the inclusion of uncertainties, during daily operation, the energy losses increase by 22.8157%, 0.2023%, and 23.7893% with respect to the benchmark case when considering demand uncertainty, PV generation uncertainty, and both. Similarly, CO2 emissions increase by 11.1854%, 0.9102%, and 12.1198% with regard to the benchmark case. All simulations were carried out using the Mosek solver in the Yalmip tool of the MATLAB software. [ABSTRACT FROM AUTHOR]

We analyze the sequence Xn+2 = imXn+1 + Xn, with X1 = X2 = 1+i, where i is the imaginary number and m is a real number. Plotting the sequence in the complex plane for different values of m, we see interesting figures from the conic sections. For values of m in the interval (-2,2) we show that the figures generated are ellipses. We also provide analysis which prove that for certain values of m, the sequence generated is periodic with even period. [ABSTRACT FROM AUTHOR]

Among the main defects present in optical systems are geometric aberrations, which are well characterized for spherical and aspherical surfaces. Special cases of aspherical surfaces are the Cartesian surfaces since they exhibit rigorous stigmatism and also, for a subfamily of these surfaces, aplanatism. However, for objects outside the stigmatic points, these surfaces also present geometric aberrations, so to evaluate their performance, it is necessary to characterize them. Therefore, in this work, the geometrical aberrations presented in Cartesian surfaces are studied, where the formulation of the coefficients for primary aberrations includes the corresponding expressions for the conic sections of revolution refractive surfaces as particular cases, and likewise the spherical refractive surface. [ABSTRACT FROM AUTHOR]

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of, including Poncelet's original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over C , or both. Here we show that Poncelet's theorem holds for any two conics C and D in the projective plane P 2 over an algebraically closed field k of any characteristic other than two. If C and D are osculating and char k > 2 , our result shows that there always exist infinitely many polygons of char k sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix. [ABSTRACT FROM AUTHOR]

This paper develops a new time difference of arrival (TDOA) emitter localization algorithm in the 3D space, employing conic approximations of hyperboloids associated with TDOA measurements. TDOA measurements are first converted to 1D angle of arrival (1D-AOA) measurements that define TDOA cones centred about axes connecting the corresponding TDOA sensor pairs. Then, the emitter location is calculated from the triangulation of 1D-AOAs, which is formulated as a system of nonlinear equations and solved by a low-complexity two-stage estimation algorithm composed of an iterative weighted least squares (IWLS) estimator and a Taylor series estimator aimed at refining the IWLS estimate. Important conclusions are reached about the optimality of sensor–emitter and sensor array geometries. The approximate efficiency of the IWLS estimator is also established under mild conditions. The new two-stage estimator is shown to be capable of outperforming the maximum likelihood estimator while performing very close to the Cramer Rao lower bound in poor sensor–emitter geometries and large noise by way of numerical simulations. [ABSTRACT FROM AUTHOR]

Nonnegative matrix factorization is the following problem: given a nonnegative input matrix V and a factorization rank K, compute two nonnegative matrices, W with K columns and H with K rows, such that WH approximates V as well as possible. In this paper, we propose two new approaches for computing high-quality NMF solutions using conic optimization. These approaches rely on the same two steps. First, we reformulate NMF as minimizing a concave function over a product of convex cones – one approach is based on the exponential cone and the other on the second-order cone. Then, we solve these reformulations iteratively: at each step, we minimize exactly, over the feasible set, a majorization of the objective functions obtained via linearization at the current iterate. Hence these subproblems are convex conic programs and can be solved efficiently using dedicated algorithms. We prove that our approaches reach a stationary point with an accuracy decreasing as O (1 i) , where i denotes the iteration number. To the best of our knowledge, our analysis is the first to provide a convergence rate to stationary points for NMF. Furthermore, in the particular cases of rank-1 factorizations (i.e. K = 1), we show that one of our formulations can be expressed as a convex optimization problem, implying that optimal rank-1 approximations can be computed efficiently. Finally, we show on several numerical examples that our approaches are able to frequently compute exact NMFs (i.e. with V = WH) and compete favourably with the state of the art. [ABSTRACT FROM AUTHOR]

In this article we study the relations of some particular Möbius transformations related to vertices of polygons circumscribing a conic. We explore their geometric properties and apply them to prove the theorems of Siebeck-Marden, Siebeck-Linfield. We show also some necessary and sufficient conditions for three Möbius transformations which allow to define any conic inscribed in a triangle. [ABSTRACT FROM AUTHOR]

The advancement of technology and its integration into the educational field has had a transformative impact on the mathematics learning process. In this context, didactic tools have constantly evolved to address the needs and difficulties that students face when learning this subject. In this sense, it is essential to analyze the influence of GeoGebra software as a complementary tool in the teaching of mathematics. In order to evaluate the impact of GeoGebra, a quantitative study was carried out with high school students in which two groups were formed: an experimental and a control group. In the experimental group, students used GeoGebra as an additional tool in their learning process, while the control group continued to use traditional teaching. The results obtained revealed that the use of GeoGebra in teaching practice achieves in students a greater understanding of the analytical and graphical aspects of mathematical problems. In addition, it significantly increases motivation, commitment and interest in the subject; consequently, the development of autonomous learning. In conclusion, GeoGebra plays a transcendental role in the learning process of mathematics. [ABSTRACT FROM AUTHOR]

Let S be a 3 -dimensional quantum polynomial algebra, and f ∈ S 2 a central regular element. The quotient algebra A = S / (f) is called a noncommutative conic. For a noncommutative conic A , there is a finite-dimensional algebra C (A) which determines the singularity of A. In this paper, we mainly focus on a noncommutative conic such that its quadratic dual is commutative, which is equivalent to say, S is determined by a symmetric regular superpotential. We classify these noncommutative conics up to isomorphism of the pairs (S , f) , and calculate the algebras C (A). [ABSTRACT FROM AUTHOR]

The horopter's history may partly be responsible for its ambiguous psychophysical definitions and obscured physiological significance. However, the horopter is a useful clinical tool integrating physiological optics and binocular vision. This article aims to help understand how it could come to such different attitudes toward the horopter. After the basic concepts underlying binocular space perception and stereopsis are presented, the horopter's old ideas that influence today's research show their inconsistencies with the conceptualized binocular vision. Two recent geometric theories of the horopter with progressively higher eye model fidelity that resolve the inconsistencies are reviewed. The first theory corrects the 200-year-old Vieth-Müller circle still used as a geometric horopter. The second theory advances Ogle's classical work by modeling empirical horopters as conic sections in the binocular system with the asymmetric eye model that accounts for the observed misalignment of optical components in human eyes. Its extension to iso-disparity conics is discussed. [ABSTRACT FROM AUTHOR]

Polynomial nonnegativity constraints can often be handled using the sum of squares condition. This can be efficiently enforced using semidefinite programming formulations, or as more recently proposed by Papp and Yildiz (Papp D in SIAM J O 29: 822–851, 2019), using the sum of squares cone directly in an interior point algorithm. Beyond nonnegativity, more complicated polynomial constraints (in particular, generalizations of the positive semidefinite, second order and ℓ 1 -norm cones) can also be modeled through structured sum of squares programs. We take a different approach and propose using more specialized cones instead. This can result in lower dimensional formulations, more efficient oracles for interior point methods, or self-concordant barriers with smaller parameters. [ABSTRACT FROM AUTHOR]

In this paper, we examine stable exact relaxations for classes of parametric robust convex polynomial optimization problems under affinely parameterized data uncertainty in the constraints. We first show that a parametric robust convex polynomial problem with convex compact uncertainty sets enjoys stable exact conic relaxations under the validation of a characteristic cone constraint qualification. We then show that such stable exact conic relaxations become stable exact semidefinite programming relaxations for a parametric robust SOS-convex polynomial problem, where the uncertainty sets are assumed to be bounded spectrahedra. In addition, under the corresponding constraint qualification, we derive stable exact second-order cone programming relaxations for some classes of parametric robust convex quadratic programs under ellipsoidal uncertainty sets. [ABSTRACT FROM AUTHOR]

Homology was among the concepts introduced in Jean Victor Poncelet's 1822 Traité des Propriétés Projectives des Figures. Homology is a projective transformation which has an axis, a line of fixed points. The Traité develops a straightedge construction of points under homology , essentially that found in work on perspective drawing and by Phillipe de la Hire, 1673. However, Poncelet's very distinct path to homology was through similitude , where the radical axis of a pair of circles became the axis of homology. We end with Poncelet's application of homology involving the focus of a conic section. [ABSTRACT FROM AUTHOR]

Given a conic \mathcal {C} over \mathbb {Q}, it is natural to ask what real points on \mathcal {C} are most difficult to approximate by rational points of low height. For the analogous problem on the real line (for which the least approximable number is the golden ratio, by Hurwitz's theorem), the approximabilities comprise the classically studied Lagrange and Markoff spectra, but work by Cha–Kim [ Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum , arXiv: 1903.02882, 2021] and Cha–Chapman–Gelb–Weiss [Monatsh. Math. 197 (2022), pp. 1–55] shows that the spectra of conics can vary. We provide notions of approximability, Lagrange spectrum, and Markoff spectrum valid for a general \mathcal {C} and prove that their behavior is exhausted by the special family of conics \mathcal {C}_n : XZ = nY^2, which has symmetry by the modular group \Gamma _0(n) and whose Markoff spectrum was studied in a different guise by A. Schmidt and Vulakh [Trans. Amer. Math. Soc. 352 (2000), pp. 4067–4094]. The proof proceeds by using the Gross-Lucianovic bijection to relate a conic to a quaternionic subring of \operatorname {Mat}^{2\times 2}(\mathbb {Z}) and classifying invariant lattices in its 2-dimensional representation. [ABSTRACT FROM AUTHOR]

In this paper we consider finding an approximate second-order stationary point (SOSP) of nonconvex conic optimization that minimizes a twice differentiable function over the intersection of an affine subspace and a convex cone. In particular, we propose a Newton-conjugate gradient based barrier method for finding an (ϵ,√ϵ)-SOSP of this problem. Our method not only is implementable but also achieves an iteration complexity of O(ϵ-3/2), which matches the best known iteration complexity of second-order methods for finding an (ϵ,√ϵ)-SOSP of unconstrained nonconvex optimization. The operation complexity, consisting of O(ϵ-3/2) Cholesky factorizations and Õ(ϵ-3/2min{n,ϵ-1/4}) other fundamental operations, is also established for our method, where n is the problem dimension and Õ(⋅) represents O(⋅) with logarithmic terms omitted. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of convex conic programming. In particular, we first propose an inexact augmented Lagrangian (I-AL) method that resembles the classical I-AL method for solving this problem, in which the augmented Lagrangian subproblems are solved approximately by a variant of Nesterov's optimal first-order method. We show that the total number of first-order iterations of the proposed I-AL method for finding an ϵ-KKT solution is at most O(ϵ-7/4). We then propose an adaptively regularized I-AL method and show that it achieves a first-order iteration complexity O(ϵ-1log ϵ-1), which significantly improves existing complexity bounds achieved by first-order I-AL methods for finding an ϵ-KKT solution. Our complexity analysis of the I-AL methods is based on a sharp analysis of the inexact proximal point algorithm (PPA) and the connection between the I-AL methods and inexact PPA. It is vastly different from existing complexity analyses of the first-order I-AL methods in the literature, which typically regard the I-AL methods as an inexact dual gradient method. [ABSTRACT FROM AUTHOR]

Primal and dual algorithms are developed for solving the n-dimensional convex optimization problem of finding the Euclidean ball of minimum radius that covers m given Euclidean balls, each with given center and radius. Each algorithm is based on a directional search method in which a search path may be a ray or a two-dimensional conic section in I R n . At each iteration, a search path is constructed by the intersection of bisectors of pairs of points, where the bisectors are either hyperplanes or n-dimensional hyperboloids. The optimal stopping point along each search path is determined explicitly. [ABSTRACT FROM AUTHOR]

This paper proposes a recursive conic approximation methodology to deal with the optimal power flow (OPF) problem in unbalanced bipolar DC networks. The OPF problem is formulated through a nonlinear programming (NLP) representation, where the objective function corresponds to the minimization of the expected grid power losses for a particular load scenario. The NLP formulation has a non-convex structure due to the hyperbolic equality constraints that define the current injection/absorption in the constant power terminals as a function of the powers and voltages. To obtain an approximate convex model that represents the OPF problem in bipolar asymmetric distribution networks, the conic relation associated with the product of two positive variables is applied to all nodes with constant power loads. In the case of nodes with dispersed generation, a direct replacement of the voltage variables for their expected operating point is used. An iterative solution procedure is implemented in order to minimize the error introduced by the voltage linearization in the dispersed generation sources. The 21-bus grid is employed for all numerical validations. To validate the effectiveness of the proposed conic model, the power flow problem is solved, considering that the neutral wire is floating and grounded, and obtaining the same numerical results as the traditional power flow methods (successive approximations, triangular-based, and Taylor-based approaches): expected power losses of 95.4237 and 91.2701 kW, respectively. To validate the effectiveness of the proposed convex model for solving the OPF problem, three combinatorial optimization methods are implemented: the sine-cosine algorithm (SCA), the black-hole optimizer (BHO), and the vortex search algorithm (VSA). Numerical results show that the proposed convex model finds the global optimal solution with a value of 22.985 kW, followed by the VSA with a value of 22.986 kW. At the same time, the BHO and SCA are stuck in locally optimal solutions (23.066 and 23.054 kW, respectively). All simulations were carried out in a MATLAB programming environment. [ABSTRACT FROM AUTHOR]

We resolve Schinzel's Hypothesis (H) for 100 % of polynomials of arbitrary degrees. We deduce that a positive proportion of diagonal conic bundles over Q with any given number of degenerate fibres have a rational point, and obtain similar results for generalised Châtelet equations. [ABSTRACT FROM AUTHOR]

This paper documents a computational implementation of a projection and rescaling algorithm for solving one of the alternative feasibility problems find x ∈ L ∩ Ω or find x ^ ∈ L ⊥ ∩ Ω , where L is a linear subspace in R n , L ⊥ is its orthogonal complement, and Ω ⊆ R n is the interior of a direct product of second order cones. The gist of the projection and rescaling algorithm is to enhance a low-cost first-order method (a basic procedure) with an adaptive reconditioning transformation (a rescaling step). We give a full description of a Python implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our Python code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward. [ABSTRACT FROM AUTHOR]

Due to the high randomness and volatility of renewable energy sources such as wind energy, the traditional thermal unit commitment (UC) model is no longer applicable. In this paper, in order to reduce the possible negative effects of an inaccurate wind energy forecast, the chance-constrained programming (CCP) method is used to study the UC problem with uncertainty wind power generation, and chance constraints such as power balance and spinning reserve are satisfied with a predetermined probability. In order to effectively solve the CCP problem, first, we used the sample average approximation (SAA) method to transform the chance constraints into deterministic constraints and to obtain a mixed-integer quadratic programming (MIQP) model. Then, the quadratic terms were incorporated into the constraints by introducing some auxiliary variables, and some second-order cone constraints were formed by combining them with the output characteristics of thermal unit; therefore, a tighter mixed-integer second-order cone programming (MISOCP) formulation was obtained. Finally, we applied this method to some systems including 10 to 100 thermal units and 1 to 2 wind units, and we invoked MOSEK in MATLAB to solve the MISOCP formulation. The numerical results obtained within 24 h confirm that not only is the MISOCP formulation a successful reformulation that can achieve better suboptimal solutions, but it is also a suitable method for solving the large-scale uncertain UC problem. In addition, for systems of up to 40 units within 24 h that do not consider wind power and pollution emissions, the numerical results were compared with those of previously published methods, showing that the MISOCP formulation is very promising, given its excellent performance. [ABSTRACT FROM AUTHOR]

The intersection of an affine hyperplane in L4 with the light cone C is called a conic section. In this paper, it is proved that the conic sections in L4 are either Riemannian spheres, hyperbolic spaces or horospheres, depending on the causal character of the hyperplane. Analogous results for affine sections of de Sitter and hyperbolic spaces in L4 are also presented at the end. [ABSTRACT FROM AUTHOR]

Improving the detection accuracy of hyperbola in B-scan images has been a considerable challenge for ground penetrating radar (GPR) to detect tree roots. In this paper, a method for data enhancement and target detection, both based on deep learning was proposed to identify hyperbolas in GPR B-scan images. First, the authors used a cyclic consistent adversarial network (CycleGAN) to augment the original data. In this procedure, the hyperbolic features of the images were preserved and created a wider variety of training samples. Then, the authors could apply the enhanced dataset to the YOLOv5 detection model to evaluate the effectiveness of their method. Meanwhile, the detection effects of Yolov3, Yolov5, Faster R-CNN, and CenterNet detection models on the enhanced dataset were compared. The results showed that applying the enhanced dataset to the Yolov5 detection model exhibited better detection accuracy compared to other combinations of datasets and detection models. The authors demonstrate that the proposed method increases data diversity and the number of samples, improving the precision and recall of hyperbolic curves. These results provide a new method for tree root localization with important effects. [ABSTRACT FROM AUTHOR]

We introduce and study a new curvature function for plane curves inspired by the weighted mean curvature of M. Gromov. We call it radial being the difference between the usual curvature and the inner product of the normal vector field and the radial vector field. But, since the problem of vanishing of this curvature involves complicated expressions, we computed it for several examples. [ABSTRACT FROM AUTHOR]

For two given measure-preserving functions defined on the unit interval f , g : I → I , the function given by C f , g (u , v) : = λ (f − 1 ([ 0 , u ]) ∩ g − 1 ([ 0 , v ])) is a copula. Although the theoretical problem for constructing this copula is completely solved, in practice it is a rather difficult task. The principal problem is the reverse implication (that is, to prove that f and g are measure-preserving when C f , g is a copula). We provide new proof of this fact with a technique that is far from the previous ones already known in the literature. Indeed, finding two measure-preserving functions f and g , such that C f , g = C , for a given C , is equivalent to a suitable decomposition of such copula in the form C = C f , id ⁎ C id , g (the ⁎ -product), where id denotes the identity function. We also provide explicit algorithms which solve this problem in various contexts such as the measure preserving functions f and g are monotonic, as well as the copula C is a diagonal copula, an extreme copula, an extremal biconic copula, an Archimedean copula, a conic copula, a copula invariant under truncations, or an α -migrative copula. [ABSTRACT FROM AUTHOR]

We show that reflections on curves that can be obtained as the equidistant set to convex focal sets have exactly the same focal property as reflections on perpendicular bisectors when viewed on the level of first-order linear approximations. Hence, we obtain a suitable focal property for reflections on equidistant sets with convex focal sets. This focal property generalizes the classical focal properties of conic sections. All these results can be extended to higher-dimensional Euclidean spaces. [ABSTRACT FROM AUTHOR]

We investigate the hierarchy of conic inner approximations K n (r) (r ∈ N) for the copositive cone COP n , introduced by Parrilo (2000) [22]. It is known that COP 4 = K 4 (0) and that, while the union of the cones K n (r) covers the interior of COP n , it does not cover the full cone COP n if n ≥ 6. Here we investigate the remaining case n = 5 , where all extreme rays have been fully characterized by Hildebrand (2012) [12]. We show that the Horn matrix H and its positive diagonal scalings play an exceptional role among the extreme rays of COP 5. We show that equality COP 5 = ⋃ r ≥ 0 K 5 (r) holds if and only if every positive diagonal scaling of H belongs to K 5 (r) for some r ∈ N. As a main ingredient for the proof, we introduce new Lasserre-type conic inner approximations for COP n , based on sums of squares of polynomials. We show their links to the cones K n (r) , and we use an optimization approach that permits to exploit finite convergence results on Lasserre hierarchy to show membership in the new cones. [ABSTRACT FROM AUTHOR]

All the points, lines and curves encountered here are presumed to lie in a plane. A set of lines is called a pencil if any member of the set is a linear combination of two other members of the set. The lines are necessarily concurrent. Likewise, a set of conics (or cubics) form a pencil if each is expressible as a linear combination of two other members of the set. Each member of a pencil of conics will pass through four fixed points and for cubics there will be nine such points. These are referred to as the base points of the pencil. Here is presented a test for a general curve to be a pencil of lines and it is used to provide simplified proofs of standard results and also some new ones, particularly for the harmonic envelope of two conics and the associated problem for cubic curves. [ABSTRACT FROM AUTHOR]

Many convex optimization problems can be represented through conic extended formulations (EFs) using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. However, EFs are often significantly larger and more complex than equivalent conic natural formulations (NFs) represented using the much broader class of exotic cones. We define an exotic cone as a proper cone for which we can implement easily computable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. Our goal is to establish whether a generic conic interior point solver supporting NFs can outperform an advanced conic solver specialized for EFs across a variety of applied problems. We introduce Hypatia, a highly configurable open-source conic primal-dual interior point solver written in Julia and accessible through JuMP. Hypatia has a generic interface for exotic cones, some of which we define here. For seven applied problems, we introduce NFs using these cones and construct EFs that are necessarily larger and more complex. Our computational experiments demonstrate the advantages, especially in terms of solve time and memory usage, of solving the NFs with Hypatia compared with solving the EFs with either Hypatia or MOSEK 9. [ABSTRACT FROM AUTHOR]

Mixed-integer second-order cone programs (MISOCPs) form a novel class of mixed-integer convex programs, which can be solved very efficiently as a result of the recent advances in optimization solvers. This paper shows how various performance metrics of M/G/1 queues can be modeled by different MISOCPs. To motivate the reformulation method, it is first applied to a challenging stochastic location problem with congestion, which is broadly used to design socially optimal service systems. Three different MISOCPs are developed and compared on different sets of benchmark test problems. The new formulations efficiently solve very large-size test problems that cannot be solved by the two existing methods developed based on linear programming within reasonable time. The superiority of the conic reformulation method is next shown over a state-space decomposition method recently used to solve an assignment problem in queueing systems. Finally, the general applicability of the method is shown for similar optimization problems that use queue-theoretic performance measures to address customer satisfaction and service quality. [ABSTRACT FROM AUTHOR]

The object of research is the problem of torsion for a radially inhomogeneous transversally isotropic sphere and the study based on this three-dimensional stress-strain state. To establish the scope of applicability of existing applied theories and to create more refined applied theories of inhomogeneous shells, it is important to study the stress-strain state of inhomogeneous bodies based on threedimensional equations of elasticity theory. The problem of torsion of a radially inhomogeneous transversally isotropic non-closed sphere containing none of the poles 0 and π is considered. It is believed that the elastic moduli are linear functions of the radius of the sphere. It is assumed that the lateral surface of the sphere is free from stresses, and arbitrary stresses are given on the conic sections, leaving the sphere in equilibrium. The formulated boundary value problem is reduced to a spectral problem. After fulfilling the homogeneous boundary conditions specified on the side surfaces of the sphere, a characteristic equation is obtained with respect to the spectral parameter. The corresponding solutions are constructed depending on the roots of the characteristic equation. It is shown that the solution corresponding to the first group of roots is penetrating, and the stress state determined by this solution is equivalent to the torques of the stresses acting in an arbitrary section θ = const. The solutions corresponding to the countable set of the second group of roots have the character of a boundary layer localized in conic slices. In the case of significant anisotropy, some boundary layer solutions decay weakly and can cover the entire region occupied by the sphere. On the basis of the performed three-dimensional analysis, new classes of solutions (solutions having the character of a boundary layer) are obtained, which are absent in applied theories. In contrast to an isotropic radially inhomogeneous sphere, for a transversely isotropic radially inhomogeneous sphere, a weakly damped boundary layer solution appears, which can penetrate deep far from the conical sections and change the picture of the stress-strain state. [ABSTRACT FROM AUTHOR]