Relation Between Dimension and Angular Momentum for Radially Symmetric Potential in N-Dimensional Space

It is proved that when solving Schr(o)dinger equations for radially symmetric potentials the effect of higher dimensions on the radial wave function is equivalent to the effect of higher angular momenta in lower-dimensional cases.This result is applied to giving solutions for several radially symmetric potentials in N dimensions.     

On the evolution of higher dimensional Heisenberg continuum spin systems

We consider the evolution of a classical Heisenberg ferromagnetic spin chain in its continuum limit in higher spatial dimensions. It is shown that the evolution of a radially symmetric chain could be identified with the motion of a helical space curve as in the linear case. The resulting invariant equations for the curvature (radial energy density) and torsion (related to current density) are shown to be equivalent to a generalized nonlinear Schrödinger equation, similar to the one derived by Ruijgrok and Jurkiewicz recently. Equivalent linear equations as well as special static solutions of point singular type are obtained. Similarity solutions, a class of which belong to Riccati type, are discussed in detail. For general higher dimensions, a potentially useful formulation is presented: Under stereographic projection of the unit sphere of spin, the equation of motion takes a neater form even with the inclusion of anisotropic interactions. Classes of explicit solutions are reported in higher dimensions. Propagating spin waves, static spin waves of point singular nature and of finite energy in some cases are also discussed.     

Analytical characterization of oscillon energy and lifetime

We develop an analytical procedure to compute all relevant physical properties of scalar field oscillons in models with quartic polynomial potentials: energy, radius, frequency, core amplitude, and lifetime. We compare our predictions to numerical simulations of models with symmetric and asymmetric double-well potentials in three spatial dimensions, obtaining excellent agreement. We also explain why oscillons have not been seen to decay in two spatial dimensions.     

Existence of Static BPS Monopoles and Dyons in Arbitrary (4p – 1)-Dimensional Spaces

We prove the existence and uniqueness of a static and radially symmetric BPS monopole of unit topological charge in an arbitrary (4p – 1)-dimensional space descended from the generalized Yang–Mills theory in 4p dimensions and formulated and presented in a recent study of Radu and Tchrakian. This monopole solution also gives rise to an electrically and magnetically charged soliton, called dyon, in the same spacetime setting through the well-known Julia–Zee correspondence. Our method is based on a dynamical shooting approach depending on two shooting parameters which provides an effective framework for constructing the BPS monopole and dyon solutions in general dimensions.     

An increase of liquid pressure in enclosure under thermal effect through walls

The problem of pressure increase at a thermal effect through a non-deformable wall on a liquid filling an enclosure is considered. The solutions are obtained for the cases of the plane, radially symmetric, and spherically symmetric problem statements, which describe the pressure variation due to thermal expansion and phase transitions. The influence of the initial liquid state and the enclosure shape on the pressure variation pace are analyzed.     

Emission tomography of radially symmetric objects and exponential Abelian transform

The properties of the exponential Abelian transform (EAT), which is defined as the exponential Radon transform (ERT) of a radially symmetric object just as the Abelian transform is the Radon transform of a radially symmetric object, are considered. A new approach to deriving the inverse EAT directly from the inverse ERT is suggested. The problems of numerical implementation of the EAT are discussed, including the problem of loss of information from deep-seated regions of the object, which is nonexistent in the case of the conventional Abel transform. The results obtained may be useful for reconstructing the spatial distribution of axisymmetric or spherically symmetric radiation sources.     

STED microscopy based on axially symmetric polarized vortex beams

A stimulated emission depletion(STED) microscopy scheme using axially symmetric polarized vortex beams is proposed based on unique focusing properties of such kinds of beams. The concept of axially symmetric polarized vortex beams is first introduced, and the basic principle about the scheme is described. Simulation results for several typical beams are then shown, including radially polarized vortex beams, azimuthally polarized vortex beams, and high-order axially symmetric polarized vortex beams. The results indicate that sharper doughnut spots and thus higher resolutions can be achieved, showing more flexibility than previous schemes based on flexible modulation of both phase and polarization for incident beams.     

On statistical mechanics of systems with highly singular two-body potentials

A finite volume statistical mechanics for highly singular two-body potentials is studied. The case of “point” hard-core particles (Lennard-Jones type two-body potentials) is discussed and the mathematical background for this physical idealization is given. Firstly the definition of a self-adjoint Hamiltonian for such a system is considered. Then a natural cutoff procedure for two-body highly singular potentials is proposed. The main result is the proof of a convergence theorem for partition sums (or free energies) when the cutoff parameter is removed to infinity. The question of stability of the cutoff interactions is also discussed. These results are illustrated by a consideration of the Lennard-Jones potential (12-6). Our results are valid in all dimensions greater than two and for potentials being not inevitably spherically symmetric.     

Paraxial designing of planar waveguide variable-index focusing elements: Part 2—lenses of elliptical symmetry

A paraxial ray-optics formalism proposed for designing radially symmetric variable-index lenses is extended to the case of lenses of elliptical symmetry of the index profile.     

Implementation of the GRP scheme for computing radially symmetric compressible fluid flows

The study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view. Spherical explosion and implosion in air, water and other media are well-known problems in application. Typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center, in addition to the resolution of classical discontinuities (shocks and contact discontinuities). In the present paper we present the implementation of direct generalized Riemann problem (GRP) scheme to resolve this issue. The scheme is obtained directly by the time integration of the fluid flows. Our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center. Together with the vanishing of the momentum, we obtain new numerical boundary conditions at the center, which are then incorporated into the GRP scheme. The main ingredient is the passage from the Cartesian coordinates to the radially symmetric coordinates.     

Three-dimensional elastic compatibility and varieties of twins in martensites

We model a cubic-to-tetragonal martensitic transition by a Ginzburg-Landau free energy in the symmetric strain tensor. We show in three dimensions (3D) that solving the St. Venant compatibility relations for strain, treated as independent field equations, generates three anisotropic long-range potentials between the two order parameter components. These potentials encode 3D discrete symmetries, express the energetics of lattice integrity, and determine 3D textures. Simulation predictions include twins with temperature-varying orientation, helical twins, competing metastable states, and compatibility-induced elastic frustration. Our approach also applies to improper ferroelastics.     

Single higher-order transverse mode operation of a radially polarized Nd:YAG laser using an annularly reflectivity-modulated photonic crystal coupler

We demonstrate single higher-order transverse mode operation of a radially polarized Nd:YAG laser by using a polarization-selective and reflectivity-modulated output coupler. A narrow annular region with low reflectivity fabricated in a photonic crystal mirror behaves so as to select a higher-order transverse mode of a cylindrically symmetric laser beam. A double-ring-shaped radially polarized TM(02) mode beam is stably generated.     

A Uniqueness Criterion for Unbounded Solutions to the Vlasov–Poisson System

We prove uniqueness for the Vlasov–Poisson system in two and three dimensions under the condition that the L^p norms of the macroscopic density grow at most linearly with respect to p. This allows for solutions with logarithmic singularities. We provide explicit examples of initial data that fulfill the uniqueness condition and that exhibit a logarithmic blow-up. In the gravitational two-dimensional case, such states are intimately related to radially symmetric steady solutions of the system. Our method relies on the Lagrangian formulation for the solutions, exploiting the second-order structure of the corresponding ODE.     

Alternative method for measuring the phase retardation and fast axis of a wave plate

Optical Review - An alternative method is proposed for simultaneously measuring the phase retardation and fast axis angle of a wave plate by utilizing a radially polarized symmetric light beam. A...     

PLANE SYMMETRIC GENERAL SOLUTION TO EINSTEIN-MAXWELL EQUATIONS IN HIGHER DIMENSIONS

The plane symmetric general solution to the Einstein-Maxwell equations in D =n+2 dimensions is presented. In addition to the n(n+1)/2 spacelike Killing vector fields characterizing the higher dimensional plane symmetry, there is also an extra Killing vector field in the solution, suggesting that the generalized Birkhoff theorem proved for 4-dimensional spacetimes might also be valid in higher dimensions.     

Fraction‐Dimensional Accessible Solitons in a Parity‐Time Symmetric Potential

By using the modified Snyder‐Mitchell (MSM) model, which can describe the propagation of a paraxial beam in fractional dimensions (FDs), we find the exact "accessible soliton” solutions in the strongly nonlocal nonlinear media with a self‐consistent parity‐time (PT) symmetric complex potential. The exact solutions are constructed with the help of two special functions: the complex Gegenbauer and the generalized Laguerre polynomials in polar coordinates, parametrized by two nonnegative integer indices ‐ the radial and azimuthal mode numbers (n,m), and the beam modulation depth. By the choice of different soliton parameters, the intensity and angular profiles display symmetric and asymmetric structures. We believe that it is important to explore the MSM model in FDs and PT‐symmetric potentials, for a better understanding of nonlinear FD physical phenomena. Different physical systems in which the model might be of relevance are briefly discussed.     

Exotic Ground States of Directional Pair Potentials via Collective-Density Variables

Collective-density variables have proved to be a useful tool in the prediction and manipulation of how spatial patterns form in the classical many-body problem. Previous work has employed properties of collective-density variables along with a robust numerical optimization technique to find the classical ground states of many-particle systems subject to radial pair potentials in one, two and three dimensions. That work led to the identification of ordered and disordered classical ground states. In this paper, we extend these collective-coordinate studies by investigating the ground states of directional pair potentials in two dimensions. Our study focuses on directional potentials whose Fourier representations are non-zero on compact sets that are symmetric with respect to the origin and zero everywhere else. We choose to focus on one representative set that has exotic ground-state properties: two circles whose centers are separated by some fixed distance. We obtain ground states for this “two-circle” potential that display large void regions in the disordered regime. As more degrees of freedom are constrained the ground states exhibit a collapse of dimensionality characterized by the emergence of filamentary structures and linear chains. This collapse of dimensionality has not been observed before in related studies.     

Convolution Inequalities for the Boltzmann Collision Operator

We study integrability properties of a general version of the Boltzmann collision operator for hard and soft potentials in n-dimensions. A reformulation of the collisional integrals allows us to write the weak form of the collision operator as a weighted convolution, where the weight is given by an operator invariant under rotations. Using a symmetrization technique in L ^p we prove a Young’s inequality for hard potentials, which is sharp for Maxwell molecules in the L ² case. Further, we find a new Hardy-Littlewood-Sobolev type of inequality for Boltzmann collision integrals with soft potentials. The same method extends to radially symmetric, non-increasing potentials that lie in some L^s_weak{L^{s}_{weak}} or L ^s . The method we use resembles a Brascamp, Lieb and Luttinger approach for multilinear weighted convolution inequalities and follows a weak formulation setting. Consequently, it is closely connected to the classical analysis of Young and Hardy-Littlewood-Sobolev inequalities. In all cases, the inequality constants are explicitly given by formulas depending on integrability conditions of the angular cross section (in the spirit of Grad cut-off). As an additional application of the technique we also obtain estimates with exponential weights for hard potentials in both conservative and dissipative interactions.     

Optical Properties of a Polarization Converting Device Using a Nematic Liquid Crystal Cell

A radially and homogeneously aligned LC cell as a polarization controlling device is fabricated using a memory effect in an axially symmetric electrical field. Optical properties and the molecular orientation of the LC cell are investigated and the conversion property of the polarization states of the cell is discussed in detail from the molecular orientation in the cell.On leave from Advantest Laboratories, Ltd., 48-2 Matsubara, Kamiayashi, Aoba-ku, Sendai, 989-31 Japan.     

Multi-scale Analysis of Compressible Viscous and Rotating Fluids

We study a singular limit for the compressible Navier-Stokes system when the Mach and Rossby numbers are proportional to certain powers of a small parameter . If the Rossby number dominates the Mach number, the limit problem is represented by the 2-D incompressible Navier-Stokes system describing the horizontal motion of vertical averages of the velocity field. If they are of the same order then the limit problem turns out to be a linear, 2-D equation with a unique radially symmetric solution. The effect of the centrifugal force is taken into account.