On the Distance Between two Ellipses in \(\mathbb{R}^3\)

2020 ◽  
Vol 57 ◽  
pp. 111-122
Author(s):  
Ivaylo Tounchev ◽  
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Global Minimum ◽  
Saddle Points ◽  
Global Maximum

We prove that in the general case the number of critical points of the distance function between two ellipses in \(\mathbb{R}^3\) equals to 4, 6, 8, 10, 12, 14 or 16. As an example, the distance between the nowadays orbits of Neptune and Pluto has six critical points: one maximum two minima and three saddle points. The global minimum is 2.503 au (astronomical units), while the global maximum is 79.111 au. If we ignore the perturbations, then in year 21103 AD the distance between Neptune and Pluto would be 2.527 au.

A theoretical study of the structures and stabilities of C2H3S+ ions

1987 ◽  
Vol 65 (6) ◽  
pp. 1209-1213 ◽  
Author(s):  
Christopher F. Rodriquez ◽  
Alan C. Hopkinson
Keyword(s):  
Experimental Observation ◽  
Molecular Orbital ◽  
Global Minimum ◽  
Theoretical Study ◽  
Saddle Points ◽  
Intramolecular Rearrangement ◽  
Molecular Orbital Calculations ◽  
Dissociation Channel ◽  
Transition Structures

Abinitio molecular orbital calculations at the 6-31G* level have been used to locate eight minima and five saddle points on the C2H3S+ hypersurface. The thioacetyl ion, H3CCS+ (1), is the global minimum and the 1-thiovinyl cation (4), 106 kJ mol−1 higher, is the next lowest. Interconversion of 1 and 4 has the highest barrier calculated for this surface (250 kJ mol−1 above 4) but all transition structures to intramolecular rearrangement are lower than the energies of the dissociation products. These results are consistent with the experimental observation that there is only one dissociation channel for ions C2H3S+, regardless of the structure of organosulphur compound from which C2H3S+ was produced. Comparisons are made with the C2H3O+ hypersurface.

Topological vortex dynamics in axisymmetric viscous flows

1994 ◽  
Vol 260 ◽  
pp. 57-80 ◽  
Author(s):  
Mogens V. Melander ◽  
Fazle Hussain
Keyword(s):  
Phase Portrait ◽  
Critical Points ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Saddle Points ◽  
Time History ◽  
Vorticity Field ◽  
Vortex Lines ◽  
Short Period ◽  
Virtual Velocity

The topology of vortex lines and surfaces is examined in incompressible viscous axisymmetric flows with swirl. We argue that the evolving topology of the vorticity field must be examined in terms of axisymmetric vortex surfaces rather than lines, because only the surfaces enjoy structural stability. The meridional cross-sections of these surfaces are the orbits of a dynamical system with the azimuthal circulation being a Hamiltonian H and with time as a bifurcation parameter μ. The dependence of H on μ is governed by the Navier–Stokes equations; their numerical solutions provide H. The level curves of H establish a time history for the motion of vortex surfaces, so that the circulation they contain remains constant. Equivalently, there exists a virtual velocity field in which the motion of the vortex surfaces is frozen almost everywhere; the exceptions occur at critical points in the phase portrait where the virtual velocity is singular. The separatrices emerging from saddle points partition the phase portrait into islands; each island corresponds to a structurally stable vortex structure. By using the flux of the meridional vorticity field, we obtain a precise definition of reconnection: the transfer of flux between islands. Local analysis near critical points shows that the virtual velocity (because of its singular behaviour) performs ‘cut-and-connect’ of vortex surfaces with the correct rate of circulation transfer - thereby validating the long-standing viscous ‘cut-and-connect’ scenario which implicitly assumes that vortex surfaces (and vortex lines) can be followed over a short period of time in a viscous fluid. Bifurcations in the phase portrait represent (contrary to reconnection) changes in the topology of the vorticity field, where islands spontaneously appear or disappear. Often such topology changes are catastrophic, because islands emerge or perish with finite circulation. These and other phenomena are illustrated by direct numerical simulations of vortex rings at a Reynolds number of 800.

Extremal Points, Critical Points, and Saddle Points of Analytic Functions

2007 ◽  
Vol 114 (6) ◽  
pp. 540-546 ◽  
Author(s):  
Joseph Bak ◽  
Pisheng Ding ◽  
Donald Newman
Keyword(s):  
Critical Points ◽  
Analytic Functions ◽  
Saddle Points ◽  
Extremal Points

Critical points in a turbulent near wake

1994 ◽  
Vol 275 ◽  
pp. 59-81 ◽  
Author(s):  
Y. Zhou ◽  
R. A. Antonia
Keyword(s):  
Critical Points ◽  
Phase Plane ◽  
Reynolds Shear Stress ◽  
Saddle Points ◽  
Normal Stresses ◽  
Flow Topology ◽  
Near Wake ◽  
Spanwise Vortex ◽  
Plane Technique ◽  
Insight Into

Velocity data were obtained in the turbulent wake of a circular cylinder with an orthogonal array of sixteen X-wires, eight in the (x, y)-plane and eight in the (x, z)-plane. By applying the phase-plane technique to these data, three types of critical points (where the velocity is zero and the streamline slope is indeterminate) were identified. Of these, foci and saddle points occurred most frequently, although a significant number of nodes was also found. Flow topology and properties associated with these points were obtained in each plane. Saddle-point regions associated with spanwise vortices provide the dominant contribution to the Reynolds shear stress and larger contributions to the normal stresses than focal regions. The topology was found to be in close agreement with that obtained from other methods of detecting features of the organized motion. The inter-relationship between critical points simultaneously identified in the two planes can provide some insight into the three-dimensionality of the organized motion. Foci in the (x, z)-plane correspond, with relatively high probability and almost negligible streamwise separation, to saddle points in the (x, y)-plane and are interpreted in terms of ribs aligned with the diverging separatrix between consecutive spanwise vortex rolls. Foci in the (x, z)-plane which correspond, with relatively weak probability, to foci in the (x, y)-plane seem consistent with a distortion of the vortex rolls in the (y, z)-plane.

scholarly journals Tracking the critical points of curves evolving under planar curvature flows

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Eszter Fehér ◽  
Gábor Domokos ◽  
Bernd Krauskopf
Keyword(s):  
Distance Function ◽  
Critical Points ◽  
Broad Class ◽  
Computational Cost ◽  
Radial Distance ◽  
Test Case ◽  
Case Examples ◽  
Planar Curve ◽  
Special Cases ◽  
Two Parameters

<p style='text-indent:20px;'>We are concerned with the evolution of planar, star-like curves and associated shapes under a broad class of curvature-driven geometric flows, which we refer to as the Andrews-Bloore flow. This family of flows has two parameters that control one constant and one curvature-dependent component for the velocity in the direction of the normal to the curve. The Andrews-Bloore flow includes as special cases the well known Eikonal, curve-shortening and affine shortening flows, and for positive parameter values its evolution shrinks the area enclosed by the curve to zero in finite time. A question of key interest has been how various shape descriptors of the evolving shape behave as this limit is approached. Star-like curves (which include convex curves) can be represented by a periodic scalar polar distance function <inline-formula><tex-math id="M1">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> measured from a reference point, which may or may not be fixed. An important question is how the numbers and the trajectories of critical points of the distance function <inline-formula><tex-math id="M2">\begin{document}$ r(\varphi) $\end{document}</tex-math></inline-formula> and of the curvature <inline-formula><tex-math id="M3">\begin{document}$ \kappa(\varphi) $\end{document}</tex-math></inline-formula> (characterized by <inline-formula><tex-math id="M4">\begin{document}$ dr/d\varphi = 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ d\kappa /d\varphi = 0 $\end{document}</tex-math></inline-formula>, respectively) evolve under the Andrews-Bloore flows for different choices of the parameters.</p><p style='text-indent:20px;'>We present a numerical method that is specifically designed to meet the challenge of computing accurate trajectories of the critical points of an evolving curve up to the vicinity of a limiting shape. Each curve is represented by a piecewise polynomial periodic radial distance function, as determined by a chosen mesh; different types of meshes and mesh adaptation can be chosen to ensure a good balance between accuracy and computational cost. As we demonstrate with test-case examples and two longer case studies, our method allows one to perform numerical investigations into subtle questions of planar curve evolution. More specifically — in the spirit of experimental mathematics — we provide illustrations of some known results, numerical evidence for two stated conjectures, as well as new insights and observations regarding the limits of shapes and their critical points.</p>

Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents

2020 ◽  
Vol 120 (3-4) ◽  
pp. 199-248
Author(s):  
Jianhua Chen ◽  
Xianjiu Huang ◽  
Dongdong Qin ◽  
Bitao Cheng
Keyword(s):  
Asymptotic Behavior ◽  
Ground State ◽  
Multiple Solutions ◽  
Global Minimum ◽  
Global Maximum ◽  
Quasilinear Schrödinger Equation ◽  
Critical Sobolev Exponents ◽  
Wave Solutions ◽  
Change Of Variable ◽  
Concentration Behavior

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3, ε > 0, 4 < p < 22 ∗ , g ∈ C 1 ( R , R + ), V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

CONSTRAINED HEBBIAN LEARNING: GRADIENT DESCENT TO GLOBAL MINIMA IN AN n-DIMENSIONAL LANDSCAPE

1991 ◽  
Vol 02 (01n02) ◽  
pp. 35-46
Author(s):  
Yves Chauvin
Keyword(s):  
Cost Function ◽  
Principal Components ◽  
Gradient Descent ◽  
Global Minimum ◽  
Hebbian Learning ◽  
Saddle Points ◽  
Local Maximum ◽  
Principal Component ◽  
Learning Trajectory ◽  
Computing Unit

This behavior of a constrained linear computing unit is analysed during “Hebbian” learning by gradient descent of a cost function corresponding to the sum of a variance maximization and a weight normalization term. The n-dimensional landscape of this cost function is shown to be composed of one local maximum and of n saddle points plus one global minimum aligned with the principal components of the input patterns. Furthermore, the landscape can be described in terms of hyperspheres, hypercrests, and hypervalleys associated with each of these principal components. Using this description, it is shown that the learning trajectory will converge to the global minimum of the landscape corresponding to the main principal component of the input patterns, provided some conditions on the starting weights and on the learning rate of the descent procedure. Extensions and implications of the algorithm are discussed.

CRITICAL POINTS OF DISTANCE TO AN ε-SAMPLING OF A SURFACE AND FLOW-COMPLEX-BASED SURFACE RECONSTRUCTION

2008 ◽  
Vol 18 (01n02) ◽  
pp. 29-61 ◽  
Author(s):  
TAMAL K. DEY ◽  
JOACHIM GIESEN ◽  
EDGAR A. RAMOS ◽  
BARDIA SADRI
Keyword(s):  
Surface Reconstruction ◽  
Distance Function ◽  
Critical Points ◽  
Geometric Modeling ◽  
Medial Axis ◽  
Reconstruction Algorithm ◽  
Three Dimensions ◽  
Distance Functions ◽  
Curve Reconstruction ◽  
Discrete Sample

The distance function to surfaces in three dimensions plays a key role in many geometric modeling applications such as medial axis approximations, surface reconstructions, offset computations and feature extractions among others. In many cases, the distance function induced by the surface can be approximated by the distance function induced by a discrete sample of the surface. The critical points of the distance functions are known to be closely related to the topology of the sets inducing them. However, no earlier theoretical result has found a link between topological properties of a geometric object and critical points of the distance to a discrete sample of it. We provide this link by showing that the critical points of the distance function induced by a discrete sample of a surface fall into two disjoint classes: those that lie very close to the surface and those that are near its medial axis. This closeness is precisely quantified and is shown to depend on the sampling density. It turns out that critical points near the medial axis can be used to extract topological information about the sampled surface. Based on this, we provide a new flow-complex-based surface reconstruction algorithm that, given a tight ε-sample of a surface, approximates the surface geometrically, both in distance and normals, and captures its topology. Furthermore, we show that the same algorithm can be used for curve reconstruction.

scholarly journals The proximities of asteroids and critical points of the distance function

2010 ◽  
pp. 91-102 ◽  
Author(s):  
Slavisa Milisavljevic
Keyword(s):  
Solar System ◽  
Distance Function ◽  
Critical Points ◽  
Efficient Method ◽  
Polynomial Solutions ◽  
Elliptical Orbits

The proximities are important for different purposes, for example to evaluate the risk of collisions of asteroids or comets with the Solar-System planets. We describe a simple and efficient method for finding the asteroid proximities in the case of elliptical orbits with a common focus. In several examples we have compared our method with the recent excellent algebraic and polynomial solutions of Gronchi (2002, 2005).

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