Absence of continuous symmetry breaking in a one-dimensional n−2 model

For a one-dimensional array ofS ^N–1 spins (N 2) with isotropic pair interactions (and more general systems) with J(j–i) obeying sup_n[n^–1 ₁ ⁿ j ²|J(j)|]<, we prove that every equilibrium state is invariant under the natural action ofSO(N). In particular, there is no long-range order of the conventional type. Included is the caseJ(n)=n ^–2.Research partially supported by U.S.N.S.F. Grant No. MCS-78-01885.S. Fairchild Scholar at Caltech. On leave from Departments of Mathematics and Physics, Princeton University, Princeton, New Jersey 08544.     

Noether symmetry and conserved quantity for a Hamilton system with time delay

In this paper, the Noether symmetries and the conserved quantities for a Hamilton system with time delay are discussed. Firstly, the variational principles with time delay for the Hamilton system are given, and the Hamilton canonical equations with time delay are established. Secondly, according to the invariance of the function under the infinitesimal transformations of the group, the basic formulas for the variational of the Hamilton action with time delay are discussed,the definitions and the criteria of the Noether symmetric transformations and quasi-symmetric transformations with time delay are obtained, and the relationship between the Noether symmetry and the conserved quantity with time delay is studied. In addition, examples are given to illustrate the application of the results.     

Modulated nematic structures and chiral symmetry breaking in 2D

ABSTRACT

We have studied the properties of biaxial particles interacting via an anisotropic pair potential, involving second-rank quadrupolar and third-rank octupolar coupling terms, using Monte Carlo simulation. The particles occupy the sites of a 2D square lattice and the interactions are restricted to nearest neighbours. The system exhibits spontaneous chiral symmetry breaking from an isotropic phase to a chiral modulated nematic phase, composed of ambidextrous chiral domains. When twofold axes of quadrupolar and octupolar tensors coincide this modulated phase appears to be the ambidextrous cholesteric phase with pitch comparable to a few lattice spacings. The associated phase transition is first order.     

Distributing points uniformly on the unit sphere under a mirror reflection symmetry constraint

Uniformly distributed point sets on the unit sphere with and without symmetry constraints have been found useful in many scientific and engineering applications. Here, a novel variant of the Thomson problem is proposed and formulated as an unconstrained optimization problem. While the goal of the Thomson problem is to find the minimum energy configuration of N electrons constrained on the surface of the unit sphere, this novel variant imposes a new symmetry constraint – mirror reflection symmetry with the x–y plane as the plane of symmetry. Qualitative features of the two-dimensional projection of the optimal configurations are briefly mentioned and compared to the ground-state configurations of the two dimensional system of charged particles laterally confined by a parabolic potential well.     

A Curtin–Hammett pentamethylene chain symmetrization process in the Bergman cyclization of an 11‐membered ring enediyne

Pentamethylene chain conformational effects for the Bergman cyclization of the 11‐membered ring enediyne, (3Z)‐3‐cycloundecene‐1,5‐diyne, 2, are examined theoretically with unrestricted Becke, three‐parameter, Lee–Yang–Parr/6‐31 G(d,p) calculations. A C₁ symmetric enediyne conformation was found to be the global minimum, where its nonsymmetric pentamethylene chain prevented π‐orbital alignment of the acetylene groups for C–C σ bond product formation. The Bergman cyclization of 2 was found to be conformationally dependent. In a Curtin–Hammett type process, the C₁ symmetric 2 inverts to one of the C_S or C₂ symmetric conformers required for the Bergman cyclization, which produced a C_S or C₂ symmetric 1,4‐diradical intermediate. The activation energy for the cyclization is slightly higher to reach the C₂ symmetry diradical compared with the C_S symmetry diradical. Copyright © 2012 John Wiley & Sons, Ltd.     

Implicit–explicit schemes for nonlinear nonlocal equations with a gradient flow structure in one space dimension

Nonlinear convection–diffusion equations with nonlocal flux and possibly degenerate diffusion arise in various contexts including interacting gases, porous media flows, and collective behavior in biology. Their numerical solution by an explicit finite difference method is costly due to the necessity of discretizing a local spatial convolution for each evaluation of the convective numerical flux, and due to the disadvantageous Courant–Friedrichs–Lewy (CFL) condition incurred by the diffusion term. Based on explicit schemes for such models devised in the study of Carrillo et al. a second‐order implicit–explicit Runge–Kutta (IMEX‐RK) method can be formulated. This method avoids the restrictive time step limitation of explicit schemes since the diffusion term is handled implicitly, but entails the necessity to solve nonlinear algebraic systems in every time step. It is proven that this method is well defined. Numerical experiments illustrate that for fine discretizations it is more efficient in terms of reduction of error versus central processing unit time than the original explicit method. One of the test cases is given by a strongly degenerate parabolic, nonlocal equation modeling aggregation in study of Betancourt et al. This model can be transformed to a local partial differential equation that can be solved numerically easily to generate a reference solution for the IMEX‐RK method, but is limited to one space dimension.