Journal articles: 'Regular and singular curve'

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Zhao, Xin, and Donghe Pei. "Evolutoids of the Mixed-Type Curves." Advances in Mathematical Physics 2021 (December 23, 2021): 1–9. http://dx.doi.org/10.1155/2021/9330963.

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The evolutoid of a regular curve in the Lorentz-Minkowski plane ℝ 1 2 is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in ℝ 1 2 . As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in ℝ 1 2 and give the conception of the σ -transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.

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Honda, Shun'ichi, and Masatomo Takahashi. "Evolutes and focal surfaces of framed immersions in the Euclidean space." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 1 (January 26, 2019): 497–516. http://dx.doi.org/10.1017/prm.2018.84.

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AbstractWe consider a smooth curve with singular points in the Euclidean space. As a smooth curve with singular points, we have introduced a framed curve or a framed immersion. A framed immersion is a smooth curve with a moving frame and the pair is an immersion. We define an evolute and a focal surface of a framed immersion in the Euclidean space. The evolutes and focal surfaces of framed immersions are generalizations of each object of regular space curves. We give relationships between singularities of the evolutes and of the focal surfaces. Moreover, we consider properties of the evolutes, focal surfaces and repeated evolutes.

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Saad, M. Khalifa, H. S. Abdel-Aziz, and A. A. Abdel-Salam. "Evolutes of Fronts in de Sitter and Hyperbolic Spheres." International Journal of Analysis and Applications 20 (September 21, 2022): 47. http://dx.doi.org/10.28924/2291-8639-20-2022-47.

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The evolute of a regular curve is a classical object from the viewpoint of differential geometry. We study some types of curves such as framed curves, framed immersion curves, frontal curves and front curves in 2-dimensional de Sitter and hyperbolic spaces. Also, we investigate the evolutes and some of their properties of fronts at singular points under some conditions. Finally, some computational examples in support of our main results are given and plotted.

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Samoilenko, V. H., Yu I. Samoilenko, and V. S. Vovk. "Asymptotic analysis of the singularly perturbed Korteweg-de Vries equation." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 1 (2019): 194–97. http://dx.doi.org/10.17721/1812-5409.2019/1.45.

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The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

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Uvarov, Artem D. "Singular Points of Curves." Modeling and Analysis of Information Systems 25, no. 6 (December 19, 2018): 692–710. http://dx.doi.org/10.18255/1818-1015-2018-6-692-710.

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In this paper, we consider the key problem of geometric modeling, connected with the construction of the intersection curves of surfaces. Methods for constructing the intersection curves in complex cases are found: by touching and passing through singular points of surfaces. In the first part of the paper, the problem of determining the tangent line of two surfaces given in parametric form is considered. Several approaches to the solution of the problem are analyzed. The advantages and disadvantages of these approaches are revealed. The iterative algorithms for finding a point on the line of tangency are described. The second part of the paper is devoted to methods for overcoming the difficulties encountered in solving a problem for singular points of intersection curves, in which a regular iterative process is violated. Depending on the type of problem, the author dwells on two methods. The first of them suggests finding singular points of curves without using iterative methods, which reduces the running time of the algorithm of plotting the intersection curve. The second method, considered in the final part of the article, is a numerical method. In this part, the author introduces a function that achieves a global minimum only at singular points of the intersection curves and solves the problem of minimizing this function. The application of this method is very effective in some particular cases, which impose restrictions on the surfaces and their arrangement. In conclusion, this method is considered in the case when the function has such a relief, that in the neighborhood of the minimum point the level surfaces are strongly elongated ellipsoids. All the images given in this article are the result of the work of algorithms on methods proposed by the author. Images are built in the author’s software environment.

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Katz, Eric, and David Zureick-Brown. "The Chabauty–Coleman bound at a prime of bad reduction and Clifford bounds for geometric rank functions." Compositio Mathematica 149, no. 11 (October 9, 2013): 1818–38. http://dx.doi.org/10.1112/s0010437x13007410.

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AbstractLet $X$ be a curve over a number field $K$ with genus $g\geq 2$, $\mathfrak{p}$ a prime of ${ \mathcal{O} }_{K} $ over an unramified rational prime $p\gt 2r$, $J$ the Jacobian of $X$, $r= \mathrm{rank} \hspace{0.167em} J(K)$, and $\mathscr{X}$ a regular proper model of $X$ at $\mathfrak{p}$. Suppose $r\lt g$. We prove that $\# X(K)\leq \# \mathscr{X}({ \mathbb{F} }_{\mathfrak{p}} )+ 2r$, extending the refined version of the Chabauty–Coleman bound to the case of bad reduction. The new technical insight is to isolate variants of the classical rank of a divisor on a curve which are better suited for singular curves and which satisfy Clifford’s theorem.

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Streltsova, Irina. "Classification of curves on de Sitter plane." Proceedings of the International Geometry Center 13, no. 1 (April 1, 2020): 1–8. http://dx.doi.org/10.15673/tmgc.v13i1.1683.

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In 1917, de Sitter used the modified Einstein equation and proposed a model of the Universe without physical matter, but with a cosmological constant. De Sitter geometry, as well as Minkowski geometry, is maximally symmetrical. However, de Sitter geometry is better suited to describe gravitational fields. It is believed that the real Universe was described by the de Sitter model in the very early stages of expansion (inflationary model of the Universe). This article is devoted to the problem of classification of regular curves on the de Sitter space. As a model of the de Sitter plane, the upper half-plane on which the metric is given is chosen. For this purpose, an algebra of differential invariants of curves with respect to the motions of the de Sitter plane is constructed. As it turned out, this algebra is generated by one second-order differential invariant (we call it by de Sitter curvature) and two invariant differentiations. Thus, when passing to the next jets, the dimension of the algebra of differential invariants increases by one. The concept of regular curves is introduced. Namely, a curve is called regular if the restriction of de Sitter curvature to it can be considered as parameterization of the curve. A theorem on the equivalence of regular curves with respect to the motions of the de Sitter plane is proved. The singular orbits of the group of proper motions are described.

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Fässler, Katrin, and Tuomas Orponen. "Singular integrals on regular curves in the Heisenberg group." Journal de Mathématiques Pures et Appliquées 153 (September 2021): 30–113. http://dx.doi.org/10.1016/j.matpur.2021.07.004.

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St�hr, Karl-Otto. "On the poles of regular differentials of singular curves." Boletim da Sociedade Brasileira de Matem�tica 24, no. 1 (March 1993): 105–36. http://dx.doi.org/10.1007/bf01231698.

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Overkamp, Otto. "Jumps and Motivic Invariants of Semiabelian Jacobians." International Mathematics Research Notices 2019, no. 20 (January 29, 2018): 6437–79. http://dx.doi.org/10.1093/imrn/rnx303.

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Abstract We investigate Néron models of Jacobians of singular curves over strictly Henselian discretely valued fields and their behavior under tame base change. For a semiabelian variety, this behavior is governed by a finite sequence of (a priori) real numbers between 0 and 1, called jumps. The jumps are conjectured to be rational, which is known in some cases. The purpose of this paper is to prove this conjecture in the case where the semiabelian variety is the Jacobian of a geometrically integral curve with a push-out singularity. Along the way, we prove the conjecture for algebraic tori which are induced along finite separable extensions and generalize Raynaud’s description of the identity component of the Néron model of the Jacobian of a smooth curve (in terms of the Picard functor of a proper, flat, and regular model) to our situation. The main technical result of this paper is that the exact sequence that decomposes the Jacobian of one of our singular curves into its toric and Abelian parts extends to an exact sequence of Néron models. Previously, only split semiabelian varieties were known to have this property.

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Benedini Riul, P., M. A. S. Ruas, and R. Oset Sinha. "The geometry of corank 1 surfaces in ℝ4." Quarterly Journal of Mathematics 70, no. 3 (December 21, 2018): 767–95. http://dx.doi.org/10.1093/qmath/hay064.

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Abstract We study the geometry of surfaces in ℝ4 with corank 1 singularities. For such surfaces, the singularities are isolated and, at each point, we define the curvature parabola in the normal space. This curve codifies all the second-order information of the surface. Also, using this curve, we define asymptotic and binormal directions, the umbilic curvature and study the flat geometry of the surface. It is shown that we can associate to this singular surface a regular one in ℝ4 and relate their geometry.

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Gradolato, Monique A., and Emilia Mezzetti. "Families of curves with ordinary singular points on regular surfaces." Annali di Matematica Pura ed Applicata 150, no. 1 (December 1988): 281–98. http://dx.doi.org/10.1007/bf01761471.

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Kindler, Lars. "Pullback of regular singular stratified bundles and restriction to curves." Mathematical Research Letters 22, no. 6 (2015): 1733–48. http://dx.doi.org/10.4310/mrl.2015.v22.n6.a10.

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Hristov, Jordan. "Response functions in linear viscoelastic constitutive equations and related fractional operators." Mathematical Modelling of Natural Phenomena 14, no. 3 (2019): 305. http://dx.doi.org/10.1051/mmnp/2018067.

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This study addresses the stress–strain relaxation functions of solid polymers in the framework of the linear viscoelasticity with aim to establish the adequate fractional operators emerging from the hereditary integrals. The analysis encompasses power-law and non-power-law materials, thus allowing to see the origins of application of the tools of the classical fractional calculus with singular memory kernels and the ideas leading towards fractional operators with non-singular (regular) kernels. A step ahead in modelling with hereditary integrals is the decomposition of non-power-law relaxation curves by Prony series, thus obtaining discrete relaxation kernels with a finite number of terms. This approach allows for seeing the physical background of the newly defined Caputo–Fabrizio time fractional derivative and demonstrates how other constitutive equations could be modified with non-singular fading memories. The non-power-law relaxation curves also allow for approximations by the Mittag–Leffler function of one parameter that leads reasonably into stress–strain hereditary integrals in terms of Atangana–Baleanu fractional derivative of Caputo sense. The main outcomes of the analysis done are the demonstrated distinguishes between the relaxation curve behaviours of different materials and are therefore the adequate modelling with suitable fractional operators.

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Zhou, Chun Bing, and Zhi Jia Zhang. "An Adaptive Image De-Noising Method Based on Singular Value Decomposition." Advanced Materials Research 532-533 (June 2012): 1041–45. http://dx.doi.org/10.4028/www.scientific.net/amr.532-533.1041.

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In this paper, we propose a new combined model for image de-noising. The singular value decomposition has been widely used in the image application such as de-noising till now. There is a problem, which is how to confirm the number of the singular values in the process of image reconstruction, need to solve in the process. Then the paper discusses a minimum energy model to confirm the number through the way of calculating the local minimum in the defined energy curve. The experiment results show that the established model is consistent to the fact under the situation that if the image be with regular structure.

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Takahashi, Masatomo, and Haiou Yu. "Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere." Mathematics 10, no. 8 (April 13, 2022): 1292. http://dx.doi.org/10.3390/math10081292.

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We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

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Shi, Tian Sheng, Jin Yu Lu, Liu Zhen Yao, and Yuan Lin Du. "On a Type of Single-Curved Cable-Strut Grids Generated by Semi-Regular Tensegrity." Applied Mechanics and Materials 66-68 (July 2011): 1781–85. http://dx.doi.org/10.4028/www.scientific.net/amm.66-68.1781.

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Tensegrity is a novel structure which attracts structure engineers’ interest because of its light weight and efficient structural behavior. Nowadays researches are mainly concentrated in the area of regular and irregular tensegrity, both of which are not suitable in many situations on account of their shapes or member length conditions. Thus, a new concept of semi-regular tensegrity was proposed in this paper. Based on the singular value decomposition of equilibrium matrix, an enumerative algorithm for the form-finding of semi-regular tensegrity was presented. According to the distribution of the minimum singular value of matrix, the configuration of semi-regular tensegrity was discovered. The obtained tensegrity was used as modulus for the generation of single curved cable-strut grid. A numerical example was illustrated to indicate that the proposed tensegrity modulus was feasible and advantageous in constructing single-curved tensegrity grid. Finally, the future research in the area of semi-regular tensegrity and its application was prospected.

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Poulakis, Dimitrios. "Integer points on algebraic curves with exceptional units." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 63, no. 2 (October 1997): 145–64. http://dx.doi.org/10.1017/s1446788700000628.

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AbstractLet F(X, Y) be an absolutely irreducible polynomial with coefficients in an algebraic number field K. Denote by C the algebraic curve defined by the equation F(X, Y) = 0 and by K[C] the ring of regular functions on Cover K. Assume that there is a unit ϕ in K[C] − K such that 1 − ϕ is also a unit. Then we establish an explicit upper bound for the size of integral solutions of the equation F(X, Y) = 0, defined over K. Using this result we establish improved explicit upper bounds on the size of integral solutions to the equations defining non-singular affine curves of genus zero, with at least three points at ‘infinity’, the elliptic equations and a class of equations containing the Thue curves.

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Lebedev, Pavel D., and Alexander A. Uspenskii. "COMBINED ALGORITHMS FOR CONSTRUCTING A SOLUTION TO THE TIME–OPTIMAL PROBLEM IN THREE-DIMENSIONAL SPACE BASED ON THE SELECTION OF EXTREME POINTS OF THE SCATTERING SURFACE." Ural Mathematical Journal 8, no. 2 (December 29, 2022): 115. http://dx.doi.org/10.15826/umj.2022.2.009.

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A class of time-optimal control problems in three-dimensional space with a spherical velocity vector is considered. A smooth regular curve $\Gamma$ is chosen as the target set. We distinguish pseudo-vertices that are characteristic points on $\Gamma$ and responsible for the appearance of a singularity in the function of the optimal result. We reveal analytical relationships between pseudo-vertices and extreme points of a singular set belonging to the family of bisectors. The found analytical representation for the extreme points of the bisector is taken as the basis for numerical algorithms for constructing a singular set. The effectiveness of the developed approach for solving non-smooth dynamic problems is illustrated by an example of numerical-analytical construction of resolving structures for the time-optimal control problem.

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Lemak, S. S., M. D. Belousova, and V. V. Alchikov. "To the Problem of Motion Cueing Simulation on a Robotic Stand for Aircraft Flight." Mekhatronika, Avtomatizatsiya, Upravlenie 23, no. 10 (October 9, 2022): 546–54. http://dx.doi.org/10.17587/mau.23.546-554.

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Various simulators with motion cueing simulation stands, which make it possible to create an acceleration environment for the pilot that is close to a real flight, are used for training aircraft pilots. The article considers the formulation of the motion cueing simulation on a stand based on an industrial manipulator. Motion cueing simulation algorithms include two phases: motion cueing simulation phase and phase of return to the working area center. During simulation phase the stand must implement such a movement that the angular accelerations acting on the person and the overload vector acting on the center of mass of the operator completely coincide with the real ones. If it is not possible then just the directions of these vectors should coincide. During the second phase the stand end point must return to the working area center with acceleration values below the threshold, but in the fastest way. This task can be presented as a generalization of the brachistochrone problem. The article considers the problem of the material point motion in a uniform gravity field along a curve located in a vertical plane, in the presence of restrictions on the trajectory curvature. It is necessary to choose the curve shape in such a way that the descent time is minimal. The problem solution is obtained by optimal control methods, the cases of regular and singular control realization are considered, the question of its conjugation. Also, the switching number between sections of regular and singular control is studied.

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Kim, Beom Jin, Chan Yeol Park, and Yong-Ki Ma. "Valuation of Credit Derivatives with Multiple Time Scales in the Intensity Model." Journal of Applied Mathematics 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/968065.

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We propose approximate solutions for pricing zero-coupon defaultable bonds, credit default swap rates, and bond options based on the averaging principle of stochastic differential equations. We consider the intensity-based defaultable bond, where the volatility of the default intensity is driven by multiple time scales. Small corrections are computed using regular and singular perturbations to the intensity of default. The effectiveness of these corrections is tested on the bond price and yield curve by investigating the behavior of the time scales with respect to the relevant parameters.

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Behrndt, Jussi, Pavel Exner, Markus Holzmann, and Vladimir Lotoreichik. "The Landau Hamiltonian with δ-potentials supported on curves." Reviews in Mathematical Physics 32, no. 04 (October 17, 2019): 2050010. http://dx.doi.org/10.1142/s0129055x20500105.

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The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian [Formula: see text] in [Formula: see text] with a [Formula: see text]-potential supported on a finite [Formula: see text]-smooth curve [Formula: see text] are studied. Here [Formula: see text] is the vector potential, [Formula: see text] is the strength of the homogeneous magnetic field, and [Formula: see text] is a position-dependent real coefficient modeling the strength of the singular interaction on the curve [Formula: see text]. After a general discussion of the qualitative spectral properties of [Formula: see text] and its resolvent, one of the main objectives in the present paper is a local spectral analysis of [Formula: see text] near the Landau levels [Formula: see text], [Formula: see text]. Under various conditions on [Formula: see text], it is shown that the perturbation smears the Landau levels into eigenvalue clusters, and the accumulation rate of the eigenvalues within these clusters is determined in terms of the capacity of the support of [Formula: see text]. Furthermore, the use of Landau Hamiltonians with [Formula: see text]-perturbations as model operators for more realistic quantum systems is justified by showing that [Formula: see text] can be approximated in the norm resolvent sense by a family of Landau Hamiltonians with suitably scaled regular potentials.

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Edmunds, D. E., V. Kokilashvili, and A. Meskhi. "Two-Weight Estimates For Singular Integrals Defined On Spaces Of Homogeneous Type." Canadian Journal of Mathematics 52, no. 3 (June 1, 2000): 468–502. http://dx.doi.org/10.4153/cjm-2000-022-5.

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AbstractTwo-weight inequalities of strong and weak type are obtained in the context of spaces of homogeneous type. Various applications are given, in particular to Cauchy singular integrals on regular curves.

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Guffroy, Sébastien. "Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps." Nagoya Mathematical Journal 171 (2003): 51–83. http://dx.doi.org/10.1017/s0027763000025514.

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AbstractSince J.Wahl ([27]), it is known that degree d plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say H, which can be singular. In Wahl’s example, H is singular along a subscheme F but the induced reduced scheme Hred is smooth along F. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of Hred.To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for d < 12 (B.2). Finally we construct our plane curves from the space curves represented by points of this hypersurface (4.7).

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Watanabe, Sumio. "Algebraic Analysis for Nonidentifiable Learning Machines." Neural Computation 13, no. 4 (April 1, 2001): 899–933. http://dx.doi.org/10.1162/089976601300014402.

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This article clarifies the relation between the learning curve and the algebraic geometrical structure of a nonidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points. By using a concept in algebraic analysis, we rigorously prove that the Bayesian stochastic complexity or the free energy is asymptotically equal to λ1 logn − (m1 − 1) loglogn + constant, where n is the number of training samples and λ1 and m1 are the rational number and the natural number, which are determined as the birational invariant values of the singularities in the parameter space. Also we show an algorithm to calculate λ1 and m1 based on the resolution of singularities in algebraic geometry. In regular statistical models, 2λ1 is equal to the number of parameters and m1 = 1, whereas in nonregular models, such as multilayer networks, 2λ1 is not larger than the number of parameters and m1 ≥ 1. Since the increase of the stochastic complexity is equal to the learning curve or the generalization error, the nonidentifiable learning machines are better models than the regular ones if Bayesian ensemble learning is applied.

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Burian, Sergei N. "Reaction forces of singular pendulum." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 9, no. 2 (2022): 278–93. http://dx.doi.org/10.21638/spbu01.2022.209.

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Various behavior types of reaction forces and Lagrange multipliers for the case of mechanical systems with configuration space singularities are studied in this paper. The motion of a one-dimensional double pendulum (or a singular pendulum) with a transversal singular point or a first order tangency singular point is considered. Properties of the configuration space of singular pendulum depends on the constraint line which the free vertex of the double pendulum moves along. Configuration space of singular pendulum could be represented by two smooth curves on a torus without common points, two transversely intersecting smooth curves or two curves with first-order tangency. In order to study the pendulum motion, the reaction forces on a two-dimensional torus are found. The expressions for the reaction forces are obtained analytically in angular coordinates. In the case of a transverse intersection it is proved that the reaction forces must be zero at the singular point. In the case of a first-order tangency singularity, the reaction forces are nonzero at the singular point. The Lagrange multiplier which depends on the motion along the ellipse becomes unlimited near to the singular point. Two mechanisms with a different type of singular points in the configuration space are described: a nonsmooth singular pendulum and a broken singular pendulum. There are no smooth regular curves passing through a singular point in the configuration spaces of these mechanical systems. In the case of nonsmooth singular pendulum, the Lagrange multiplier which depends on the motion along the ellipse becomes undefined when the singular point is passed. In the case of broken singular pendulum, the Lagrange multiplier makes a jump from a finite value to an infinite one.

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Bhaskar, Atul. "Elastic waves in Timoshenko beams: the ‘lost and found’ of an eigenmode." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2101 (October 8, 2008): 239–55. http://dx.doi.org/10.1098/rspa.2008.0276.

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This paper considers propagating waves in elastic bars in the spirit of asymptotic analysis and shows that the inclusion of shear deformation amounts to singular perturbation in the Euler–Bernoulli (EB) field equation. We show that Timoshenko, in his classic work of 1921, incorrectly treated the problem as one of regular perturbation and missed out one physically meaningful ‘branch’ of the dispersion curve (spectrum), which is mainly shear-wise polarized. Singular perturbation leads to: (i) Timoshenko's solution and (ii) a singular solution ; ϵ , ω * and k * are the non-dimensional slenderness, frequency and wavenumber, respectively. Asymptotic formulae for dispersion, standing waves and the density of modes are given in terms of ϵ . The second spectrum—in the light of the debate on its existence, or not—is discussed. A previously proposed Lagrangian is shown to be inadmissible in the context. We point out that Lagrangian densities that lead to the same equation(s) of motion may not be equivalent for field problems: careful consideration to the kinetic boundary conditions is important. A Hamiltonian formulation is presented—the conclusions regarding the validity (or not) of Lagrangian densities are confirmed via the constants of motion.

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Aleksandrov, Alexander G. "The Poincaré Index on Singular Varieties." J 5, no. 3 (September 15, 2022): 380–401. http://dx.doi.org/10.3390/j5030026.

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In this paper, we discuss a few simple methods for computing the local topological index and its various analogs for vector fields and differential forms given on complex varieties with singularities of different types. They are based on properties of regular meromorphic and logarithmic differential forms, of the dualizing (canonical) module and related constructions. In particular, we show how to compute the index on Cohen–Macaulay, Gorenstein and monomial curves, on normal and non-normal surfaces and some others. In contrast with known traditional approaches, we use neither computers, nor integration, perturbations, deformations, resolution of singularities, spectral sequences or other related standard tools of pure mathematics.

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Wang, Yongqiao, Donghe Pei, and Ruimei Gao. "Generic Properties of Framed Rectifying Curves." Mathematics 7, no. 1 (January 3, 2019): 37. http://dx.doi.org/10.3390/math7010037.

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The position vectors of regular rectifying curves always lie in their rectifying planes. These curves were well investigated by B.Y.Chen. In this paper, the concept of framed rectifying curves is introduced, which may have singular points. We investigate the properties of framed rectifying curves and give a method for constructing framed rectifying curves. In addition, we reveal the relationships between framed rectifying curves and some special curves.

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SUZUKI, TOSHIHIRO, and SHUNJI MORI. "STRUCTURAL DESCRIPTION OF LINE IMAGES BY THE CROSS SECTION SEQUENCE GRAPH." International Journal of Pattern Recognition and Artificial Intelligence 07, no. 05 (October 1993): 1055–76. http://dx.doi.org/10.1142/s0218001493000534.

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In this paper, we propose the Cross Section Sequence Graph which describes line images in a simple and well structured form. It is composed of regular regions called cross section sequences and singular regions. A cross section sequence is a sequence of cross sections, each of which is constructed as a pair of boundary points almost perpendicular to the direction of the line. The sequence corresponds to a straight or curved line segment. The remaining regions are extracted as singular regions, each of which corresponds to an end point region, corner, branch, cross, and so on. The cross section sequence graph is useful for many kinds of feature extraction, especially for skeletonization since a singular region can be analyzed from adjacent regular regions. Experimental results show that the skeleton extracted from the cross section sequence graph is better than that of a pixel-wise skeletonization (thinning) in terms of both processing speed and the quality of the skeleton.

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INABA, Michi-aki, and Masa-Hiko SAITO. "Moduli of regular singular parabolic connections with given spectral type on smooth projective curves." Journal of the Mathematical Society of Japan 70, no. 3 (July 2018): 879–94. http://dx.doi.org/10.2969/jmsj/76597659.

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KUIKEN, K. F., and J. T. MASTERSON. "ON THE CLASSIFICATION OF SINGULAR AUTOMORPHIC DIFFERENTIAL EQUATIONS AND FLOW S ON THE RIEMANN SPHERE." Tamkang Journal of Mathematics 26, no. 1 (March 1, 1995): 13–19. http://dx.doi.org/10.5556/j.tkjm.26.1995.4373.

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In this paper, parameterizations are constructed for spaces of automor phic second order differential equations on certam subsets of $\hat C$. These equations have coefficients with a countable number of regular singular points on fundamen tal domains for bimeromorphic deformations of Kleiman groups. The equations considered are generalizations of classically-considered equations, including the hy pergeometric and Heun's equations, or have singular points on fam1hes of curves, including lines, conic sections, Joukowski airfoils or biconformal images of these curves. Global fluid flows associate with these equations are constructed and classified.

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33

Kossowski, Marek. "Fiber Completions, Contact Singularities and Single Valued Solutions for C∞-Second Order Ode." Canadian Journal of Mathematics 48, no. 4 (August 1, 1996): 849–70. http://dx.doi.org/10.4153/cjm-1996-043-9.

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AbstractAn implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

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34

Li, Zongguang, and Rui Liu. "Bifurcations and Exact Solutions in a Nonlinear Wave Equation." International Journal of Bifurcation and Chaos 29, no. 07 (June 30, 2019): 1950098. http://dx.doi.org/10.1142/s0218127419500986.

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The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the [Formula: see text]-family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

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35

Uspenskii, A. A., and P. D. Lebedev. "On the structure of the singular set of solutions in one class of 3D time-optimal control problems." Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki 31, no. 3 (September 2021): 471–86. http://dx.doi.org/10.35634/vm210309.

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A class of time-optimal control problems in terms of speed in three-dimensional space with a spherical velocity vector is considered. A smooth regular curve $\Gamma$ was chosen as the target set. Pseudo-vertices — characteristic points on $\Gamma,$ responsible for the appearance of a singularity in the optimal result function, are selected. The characteristic features of the structure of a singular set belonging to the family of bisectors are revealed. An analytical representation is found for the extreme points of the bisector corresponding to a fixed pseudo-vertex. As an illustration of the effectiveness of the developed methods for solving nonsmooth dynamic problems, an example of the numerical-analytical construction of resolving structures of a control problem in terms of speed is given.

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36

Manuylov, Gaik, Sergey Kosytsyn, and Irina Grudtsyna. "GEOMETRIC REPRESENTATIONS OF EQUILIBRIUM CURVES OF A COMPRESSED STIFFENED PLATE." International Journal for Computational Civil and Structural Engineering 17, no. 3 (September 29, 2021): 83–93. http://dx.doi.org/10.22337/2587-9618-2021-17-3-83-93.

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The work is aimed at studying the solutions of the stability problem (subcritical and postcritical equilibrium) of an infinitely wide regular compressed reinforced plate, using a selected T-shaped fragment that is equally stable with others. The authors have given a classification of possible analytical solutions for these plates. The results of the work are presented in the form of variants of spatial bifurcation diagrams, values of critical loads, as well as coordinates of singular points for different cases of solutions.

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37

Park, Hyeongki, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Nozomu Matsuura, and Yasuhiro Ohta. "Isoperimetric deformations of curves on the Minkowski plane." International Journal of Geometric Methods in Modern Physics 16, no. 07 (July 2019): 1950100. http://dx.doi.org/10.1142/s0219887819501007.

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We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing modified Korteweg-de Vries (mKdV) equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the [Formula: see text] functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class gives regular solutions to the defocusing mKdV equation. Some pictures illustrating the typical dynamics of the curves are presented.

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38

Meierovich, Boris E. "Macroscopic Theory of Dark Sector." Journal of Gravity 2014 (October 1, 2014): 1–23. http://dx.doi.org/10.1155/2014/586958.

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A simple Lagrangian with squared covariant divergence of a vector field as a kinetic term turned out to be an adequate tool for macroscopic description of the dark sector. The zero-mass field acts as the dark energy. Its energy-momentum tensor is a simple additive to the cosmological constant. Massive fields describe two different forms of dark matter. The space-like massive vector field is attractive. It is responsible for the observed plateau in galaxy rotation curves. The time-like massive field displays repulsive elasticity. In balance with dark energy and ordinary matter it provides a four-parametric diversity of regular solutions of the Einstein equations describing different possible cosmological and oscillating nonsingular scenarios of evolution of the Universe. In particular, the singular big bang turns into a regular inflation-like transition from contraction to expansion with the accelerated expansion at late times. The fine-tuned Friedman-Robertson-Walker singular solution is a particular limiting case at the lower boundary of existence of regular oscillating solutions in the absence of vector fields. The simplicity of the general covariant expression for the energy-momentum tensor allows displaying the main properties of the dark sector analytically. Although the physical nature of dark sector is still unknown, the macroscopic theory can help analyze the role of dark matter in astrophysical phenomena without resorting to artificial model assumptions.

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39

Wang, Yu-X., Y.-T. Li, and R.-Q. Guo. "Study on the method for 6-SPS Gough—Stewart platforms to pass through type-II singular points with its original configuration." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 222, no. 4 (April 1, 2008): 723–36. http://dx.doi.org/10.1243/09544062jmes863.

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It is well known that there exist many more singularities in parallel manipulators. At the singular point, the motion of the parallel manipulator is uncertain. In order to make parallel manipulators produce concrete output at singular points, and go away from singular points with the desired configuration, the method for the manipulator to pass through type-II singularities with its original configuration has been investigated in the paper. First, the semi-regular hexagon Gough—Stewart manipulator is taken as an example to analyse the configuration bifurcation characteristics at the vicinities of the type-II singularities going with the input parameters. The studies show that on different configuration branches, the singularity-free moving region is different, and there are two or four assembly configurations in the space above the base for the same group of input parameters. By researching transition behaviours of the configuration bifurcation curves under the disturbances applied to the other input parameters, it is found that under a suitable disturbance the perturbed persistent configuration will go away from the non-persistent configuration. Based on this kind of character, a novel method for the manipulator to pass through the turning point with its original configuration has been presented.

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40

LU, JIA-AN, and CHAO-GUANG HUANG. "WEAK FIELD APPROXIMATION IN A MODEL OF DE SITTER GRAVITY: SCHWARZSCHILD–DE SITTER SOLUTIONS." International Journal of Modern Physics D 22, no. 07 (June 2013): 1350048. http://dx.doi.org/10.1142/s021827181350048x.

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The weak field approximation in a model of de Sitter (dS) gravity is investigated in the static and spherically symmetric case, under the assumption that the vacuum spacetime without perturbations from matter fields is a torsion-free dS spacetime. It is shown on one the hand that any solution should be singular at the center of the matter field, if the exterior is described by a Schwarzschild–de Sitter (S–dS) spacetime and is smoothly connected to the interior. On the other, all the regular solutions are obtained, which might be used to explain the galactic rotation curves without involving dark matter.

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41

Hasegawa, Masaru, Atsufumi Honda, Kosuke Naokawa, Kentaro Saji, Masaaki Umehara, and Kotaro Yamada. "Intrinsic properties of surfaces with singularities." International Journal of Mathematics 26, no. 04 (April 2015): 1540008. http://dx.doi.org/10.1142/s0129167x1540008x.

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In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back metrics of wave fronts which admit only cuspidal edges and swallowtails in R3 are Kossowski metrics, and the pull-back metrics of surfaces consisting only of cross cap singularities are Whitney metrics. Since the singular sets of Kossowski metrics are the union of regular curves on the domains of definitions, and Whitney metrics admit only isolated singularities, these two classes of metrics are disjoint. In this paper, we give several characterizations of intrinsic invariants of cuspidal edges and cross caps in these classes of metrics. Moreover, we prove Gauss–Bonnet type formulas for Kossowski metrics and for Whitney metrics on compact 2-manifolds.

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42

COSCIA, V., and A. M. ROBERTSON. "EXISTENCE AND UNIQUENESS OF STEADY, FULLY DEVELOPED FLOWS OF SECOND ORDER FLUIDS IN CURVED PIPES." Mathematical Models and Methods in Applied Sciences 11, no. 06 (August 2001): 1055–71. http://dx.doi.org/10.1142/s0218202501001239.

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Steady, fully developed flows of second order fluids in curved pipes of circular cross-section have previously been studied using regular perturbation methods.2,3,12,20 These perturbation solutions are applicable for pipes with small curvature ratio: The cross sectional radius of the pipe divided by the radius of curvature of the pipe centerline. It was shown by Jitchote and Robertson12 that perturbation equations could be ill-posed when the second normal stress coefficient is nonzero. Motivated by the singular nature of the perturbation equations, here, we study the full governing equations without introducing assumptions inherent in perturbation methods. In particular, we examine the existence and uniqueness of solutions to the full governing equations for second order fluids. We show rigorously that a solution to the full problem exists and is locally unique for small non-dimensional pressure drop, in agreement with earlier results obtained using a formal expansion in the curvature ratio.12 The results obtained here are valid for arbitrarily shaped cross-section (sufficiently smooth) and for all curvature ratios. An operator splitting method has been employed which may be useful for numerical studies of steady and unsteady flows of second order fluids in curved pipes.

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MOIOLA, JORGE L., and GUANRONG CHEN. "FREQUENCY DOMAIN APPROACH TO COMPUTATION AND ANALYSIS OF BIFURCATIONS AND LIMIT CYCLES: A TUTORIAL." International Journal of Bifurcation and Chaos 03, no. 04 (August 1993): 843–67. http://dx.doi.org/10.1142/s0218127493000751.

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This paper introduces a frequency domain approach together with some techniques and methodologies for the computation and analysis of bifurcations and limit cycles arising in nonlinear dynamical systems. The frequency domain approach discussed in this paper originates from the classical feedback control systems theory, which has been proven to be successful and efficient for the computation and analysis of regular as well as singular bifurcations and stable as well as unstable limit cycles. While describing these techniques and methods, two representative yet distinct applications of the approach are studied in detail: The graphical analysis of multiple parametric bifurcation curves and the numerical computation of multiple limit cycles. Compared to the classical time domain methods, both the advantages and the limitations of the frequency domain approach are analyzed and discussed. It is believed that this frequency domain approach to the study of nonlinear dynamics has great potential and promising future in both theory and applications.

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44

Guiggiani, M., G. Krishnasamy, T. J. Rudolphi, and F. J. Rizzo. "A General Algorithm for the Numerical Solution of Hypersingular Boundary Integral Equations." Journal of Applied Mechanics 59, no. 3 (September 1, 1992): 604–14. http://dx.doi.org/10.1115/1.2893766.

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The limiting process that leads to the formulation of hypersingular boundary integral equations is first discussed in detail. It is shown that boundary integral equations with hypersingular kernels are perfectly meaningful even at non-smooth boundary points, and that special interpretations of the integrals involved are not necessary. Careful analysis of the limiting process has also strong relevance for the development of an appropriate numerical algorithm. In the second part, a new general method for the evaluation of hypersingular surface integrals in the boundary element method (BEM) is presented. The proposed method can be systematically applied in any BEM analysis, either with open or closed surfaces, and with curved boundary elements of any kind and order (of course, provided the density function meets necessary regularity requirements at each collocation point). The algorithm operates in the parameter plane of intrinsic coordinates and allows any hypersingular integral in the BEM to be directly transformed into a sum of a double and a one-dimensional regular integrals. Since all singular integrations are performed analytically, standard quadrature formulae can be used. For the first time, numerical results are presented for hypersingular integrals on curved (distorted) elements for three-dimensional problems.

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45

Sih, George. "Micro/macro fatigue crack growth rate model for 2024-T3 aluminum panel." International Journal of Structural Integrity 6, no. 4 (August 10, 2015): 522–40. http://dx.doi.org/10.1108/ijsi-05-2015-0014.

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Purpose – Fatigue crack growth rate data for 2024-T3 aluminum are found using three parameters d*, σ* and μ* for short and long cracks for Regions I-III in conventional fatigue. Asymptotic solution of a line crack with a micro-tip is found to yield a singular stress behavior of order 0.75 in contrast to the 0.50 order known for the macrocrack. The difference is due to the micro-macro interaction effects. The three parameters account for the combined effects of load, material and geometry via the tip region. Data for short and long cracks lie on a straight with a slope of about 3.9-4.8 for R values of 0.286-0.565. The results were based on an initial crack a1 mm where a is the half length for a central crack panel. The paper aims to discuss these issues. Design/methodology/approach – The belief that specimen fatigue data could assist the design of structural components was upended when FAA discovered that the NASGRO FCGD are not valid for short cracks that are tight and may even be closed. The regular ΔK vs da/dN model was limited to long cracks. The issue become critical for short cracks connecting the long ones of a few mm to cm or even m according to da/dN for the same crack history. The danger of short/long fatigue crack growth (SLFCG) prompted FAA to introduce an added test known as Limit of Validity (LOV), a way of setting empirical limits for structural components. The dual scale SLFCG data from ΔK micro/macro provide support for the LOV tests. Findings – Data for short and long cracks lie on a straight with a slope of about 3.9-4.8 for R values of 0.286-0.565. The single dual scale relation on ΔK micro/macro can switch from microscopic to macroscopic or vice and versa. The difference is fundamental. Order other than 0.75 can be obtained for simulating different microstructure effects as well as different materials and test conditions. Originality/value – Scale shifting from short to long fatigue cracks for 2024-T3 aluminum is new. The crack driving force is found to depend on the crack tightness. The sigmoidal curve based on the regular ΔK plot disappeared. The data from ΔK micro/macro for short cracks may supplement the FAA LOV tests for setting more reliable fatigue safe limits.

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46

Brüning, Jochen, and Robert Seeley. "Regular singular asymptotics." Advances in Mathematics 58, no. 2 (November 1985): 133–48. http://dx.doi.org/10.1016/0001-8708(85)90114-8.

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47

Leibovich, S., and S. K. Lele. "The influence of the horizontal component of Earth's angular velocity on the instability of the Ekman layer." Journal of Fluid Mechanics 150 (January 1985): 41–87. http://dx.doi.org/10.1017/s0022112085000039.

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A systematic study is made of the effect of latitude on the linear, normal mode stability characteristics of the laminar barotropic Ekman layer. The outcome depends upon the direction of the geostrophic flow (in the case of flows modelling the atmospheric Ekman layer) or, alternatively, upon the direction of the applied stress (in the case of flows modelling the oceanic Ekman layer). The minimum critical Reynolds number Rc is a function of latitude. For the atmospheric Ekman layer Rc = 30.8 for all latitudes less than 26.2° and increases monotonically with latitude to 54.2. At a latitude of 45° N, Rc is 33.9 and arises for a geostrophic wind directed towards a compass heading 252° (clockwise from north), corresponding to rolls with axes pointing due west and having wavenumber k (with unit of length taken to be the Ekman layer depth) of 0.594. The minimum Rc for the oceanic boundary layer is 11.6 for latitudes less than 81.1°, and increases with latitude to 11.8. At 45° N latitude, the critical condition arises for a surface-current compass heading of 345.2°, roll axis of 351° and a wavenumber k = 0.33. The results for Rc are all symmetric about the equator, with roll axes and associated basic flow directions rotated by 180°. As the Reynolds number R increases, the effects of the perturbation Coriolis acceleration on the instability diminish, as has been previously shown, and the error caused by neglect of the horizontal component of angular velocity therefore decreases. The high Reynolds number limit is systematically explored. It is shown that the lower branch of the neutral curve is not inviscid as R → ∞; rather kR → constant. The upper branch is inviscid in the limit R → ∞, and corresponds to a regular or singular neutral mode depending on whether the angle ε between the outer geostrophic flow and the roll axis is greater or less than 15.93°. ‘Inflectional’ modes, thought to be relevant by some investigators, do not exist for ε < 15.93°. Lastly, the most unstable inviscid mode corresponding to zero phase speed, a condition to which certain well-known experiments are sensitive, occurs at ε = 11.8° with wavenumber k = 0.6. This is in good agreement with published experimental data.

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48

Reznik, Gregory, and Ziv Kizner. "Singular vortices in regular flows." Theoretical and Computational Fluid Dynamics 24, no. 1-4 (August 11, 2009): 65–75. http://dx.doi.org/10.1007/s00162-009-0150-5.

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49

Vainsencher, Israel. "Foliations singular along a curve." Transactions of the London Mathematical Society 2, no. 1 (2015): 80–92. http://dx.doi.org/10.1112/tlms/tlv004.

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50

Cisneros-Molina, José Luis, and Aurélio Menegon. "Equivalence of Milnor and Milnor-Lê fibrations for real analytic maps." International Journal of Mathematics 30, no. 14 (November 20, 2019): 1950078. http://dx.doi.org/10.1142/s0129167x19500782.

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In [J. Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61 (Princeton University Press, Princeton, NJ, 1968).] Milnor proved that a real analytic map [Formula: see text], where [Formula: see text], with an isolated critical point at the origin has a fibration on the tube [Formula: see text]. Constructing a vector field such that (1) it is transverse to the spheres, and (2) it is transverse to the tubes, he “inflates” the tube to the sphere, to get a fibration [Formula: see text], but the projection is not necessarily given by [Formula: see text] as in the complex case. In the case [Formula: see text] has isolated critical value, in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.] it was proved that if the fibers inside a small tube are transverse to the sphere [Formula: see text], then it has a fibration on the tube. Also in [J. L. Cisneros-Molina, J. Seade and J. Snoussi, Milnor fibrations and [Formula: see text]-regularity for real analytic singularities, Internat. J. Math. 21(4) (2010) 419–434.], the concept of [Formula: see text]-regularity was defined, it turns out that [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] is a fiber bundle equivalent to the one on the tube. In a more general setting, the corresponding facts are proved in [J. L. Cisneros-Molina, A. Menegon, J. Seade and J. Snoussi, Fibration theorems and [Formula: see text]-regularity for differentiable maps-germs with non-isolated critical value, Preprint (2017).], showing that if a locally surjective map [Formula: see text] has a linear discriminant [Formula: see text] with isolated singularity and a fibration on the tube [Formula: see text], then [Formula: see text] is [Formula: see text]-regular if and only if the map [Formula: see text] (with [Formula: see text] the radial projection of [Formula: see text] on [Formula: see text]) is a fiber bundle equivalent to the one on the tube. In this paper, we generalize this result for an arbitrary linear discriminant by constructing a vector field [Formula: see text] which inflates the tube to the sphere in a controlled way, it satisfies properties analogous to the vector field constructed by Milnor in the complex setting: besides satisfying (1) and (2) above, it also satisfies that [Formula: see text] is constant on the integral curves of [Formula: see text].

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Journal articles on the topic 'Regular and singular curve'

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