Second‐Order Transitions and Critical Points

1951 ◽  
Vol 19 (10) ◽  
pp. 1281-1283 ◽  
Author(s):  
P. J. Price
Keyword(s):  
Critical Points ◽  
Second Order

Second-Order Raman Spectra of Fluorites. I. Critical Point Analysis

1972 ◽  
Vol 50 (8) ◽  
pp. 849-857 ◽  
Author(s):  
N. Krishnamurthy ◽  
V. Soots
Keyword(s):  
Critical Point ◽  
Single Crystals ◽  
Raman Spectra ◽  
Brillouin Zone ◽  
Critical Points ◽  
Photon Counting ◽  
Second Order ◽  
Selection Rules ◽  
Point Analysis ◽  
Critical Point Analysis

With the use of a high-powered Ar+ laser and conventional photon counting techniques it has been possible to observe the second-order Raman spectra of single crystals of CaF2, SrF2, BaF2, and PbF2. The symmetries of the various parts of the spectra of the latter two were determined by using oriented single crystals of these two fluorides. The main features of the observed spectra have been analyzed, with the aid of group-theoretical selection rules, in terms of calculated phonon frequencies at the critical points of the Brillouin zone of these crystals.

Multiple solutions for a Dirichlet quasilinear system containing a parameter

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Shapour Heidarkhani ◽  
Johnny Henderson
Keyword(s):  
Differential Equations ◽  
Critical Points ◽  
Multiple Solutions ◽  
Second Order ◽  
Quasilinear System ◽  
Compact Interval ◽  
Three Solutions ◽  
Second Order Differential Equations

Abstract.In this paper, we apply two theorems from triple critical points theory to establish the existence of at least three solutions for the quasilinear second order differential equations on a compact interval

Contributions to the Theory of Mass Action Kinetics

1978 ◽  
Vol 33 (8) ◽  
pp. 989-992
Author(s):  
Klaus-Dieter Willamowski
Keyword(s):  
Critical Points ◽  
Dynamic Properties ◽  
Second Order ◽  
Mass Action ◽  
Mass Action Kinetics ◽  
Reaction Systems ◽  
Nonlinear Reaction ◽  
Stability Pattern

Second order mass action kinetics provide the simplest models of nonlinear reaction systems. Some of their dynamic properties, viz., location, number, and stability pattern of their critical points, will be analyzed. The occurrence of periodic and chaotic solutions of these systems is discussed.

scholarly journals Infinitely Many Periodic Solutions for Nonautonomous Sublinear Second-Order Hamiltonian Systems

2010 ◽  
Vol 2010 ◽  
pp. 1-10 ◽  
Author(s):  
Peng Zhang ◽  
Chun-Lei Tang
Keyword(s):  
Periodic Solutions ◽  
Hamiltonian Systems ◽  
Critical Points ◽  
Local Minimum ◽  
Second Order ◽  
The Other ◽  
Second Order Hamiltonian Systems ◽  
Minimax Methods ◽  
Infinitely Many Periodic Solutions ◽  
Minimum Points

Two sequences of distinct periodic solutions for second-order Hamiltonian systems with sublinear nonlinearity are obtained by using the minimax methods. One sequence of solutions is local minimum points of functional, and the other is minimax type critical points of functional. We do not assume any symmetry condition on nonlinearity.

scholarly journals Stability of viscous flow between rotating cylinders. III. Integration of a sixth order linear equation

1946 ◽  
Vol 187 (1011) ◽  
pp. 492-504 ◽  
Keyword(s):  
Differential Equation ◽  
Viscous Flow ◽  
Critical Points ◽  
Second Order ◽  
Sixth Order ◽  
Large Parameter ◽  
Rotating Cylinders ◽  
Linear Differential ◽  
High Reynolds ◽  
Second Order Equations

The aim of this paper is to derive the asymptotic integrals, and their transformations through the critical points, of a certain linear differential equation of the sixth order containing a large parameter. This particular equation is of importance in connexion with the question of stability of viscous flow between rotating cylinders. Since, however, similar equations occur in all questions of stability of viscous flow, a development of proper methods of solution of such equations is of very great importance for problems of viscous flow at high Reynolds numbers. The method of finding asymptotic integrals of linear differential equations containing a large parameter is well known; it was developed by Horn (1899), Sehlesinger (1907), Birkhoff (1908) and Fowler & Lock (1922). The main difficulty of the problem consists in the following. The coefficients of the differential equation are expressions like λΦ(x) , where λ is a large parameter, and Φ(x) is a slowly varying function of the independent variable; the function Φ(x) usually vanishes within the range of x under consideration, with the result that the asymptotic expansions become infinite at such critical points, lose their validity round these points and change their form in passing through such points. The main problem of integration consists, thus, in finding the transformations of the asymptotic integrals in passing through critical points. This problem was considered by Jeffreys (1924, 1942), Kramers (1926) and Goldstein (1928, 1932) for certain second-order equations. Langer (1931), using a different method, discussed several cases of second-order equations; a summary of methods used and results obtained was also given by Langer (1934). A case of a fourth-order equation was solved by Meksyn (in Press).

scholarly journals Erratum: On The Aubin Property of Critical Points to Perturbed Second-Order Cone Programs

2017 ◽  
Vol 27 (3) ◽  
pp. 2143-2151 ◽  
Author(s):  
Felipe Opazo ◽  
Jiří V. Outrata ◽  
Héctor Ramírez C.
Keyword(s):  
Critical Points ◽  
Second Order ◽  
Aubin Property ◽  
Second Order Cone

scholarly journals Toroidal Vortices of Energy in Tightly Focused Second-Order Cylindrical Vector Beams

Photonics ◽  
2021 ◽  
Vol 8 (8) ◽  
pp. 301
Author(s):  
Sergey S. Stafeev ◽  
Elena S. Kozlova ◽  
Victor V. Kotlyar
Keyword(s):  
Critical Points ◽  
Saddle Points ◽  
Numerical Aperture ◽  
Second Order ◽  
Optical Vortices ◽  
Annular Aperture ◽  
Toroidal Surface ◽  
Toroidal Vortices ◽  
532 Nm ◽  
Topological Charges

In this paper, we simulate the focusing of a cylindrical vector beam (CVB) of second order, using the Richards–Wolf formula. Many papers have been published on focusing CVB, but they did not report on forming of the toroidal vortices of energy (TVE) near the focus. TVE are fluxes of light energy in longitudinal planes along closed paths around some critical points at which the flux of energy is zero. In the 3D case, such longitudinal energy fluxes form a toroidal surface, and the critical points around which the energy rotates form a circle lying in the transverse plane. TVE are formed in pairs with different directions of rotation (similar to optical vortices with topological charges of different signs). We show that when light with a wavelength of 532 nm is focused by a lens with numerical aperture NA = 0.95, toroidal vortices periodically appear at a distance of about 0.45 μm (0.85 λ) from the axis (with a period along the z-axis of 0.8 μm (1.5 λ)). The vortices arise in pairs: the vortex nearest to the focal plane is twisted clockwise, and the next vortex is twisted counterclockwise. These vortices are accompanied by saddle points. At higher distances from the z-axis, this pattern of toroidal vortices is repeated, and at a distance of about 0.7 μm (1.3 λ), a region in which toroidal vortices are repeated along the z-axis is observed. When the beam is focused and limited by a narrow annular aperture, these toroidal vortices are not observed.

On the Aubin Property of Critical Points to Perturbed Second-Order Cone Programs

2011 ◽  
Vol 21 (3) ◽  
pp. 798-823 ◽  
Author(s):  
Jiří V. Outrata ◽  
Héctor Ramírez C.
Keyword(s):  
Critical Points ◽  
Second Order ◽  
Aubin Property ◽  
Second Order Cone
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