Barotropic Instability of the Bickley Jet at High Reynolds Numbers

We investigate the linear stability of the Bickley jet in the framework of the beta-plane approximation. Because singular inviscid neutral modes exist in the retrograde case     , it is necessary to add viscosity to interpret them. One of these modes was found in closed form by Howard and Drazin [1] . However, its critical point is at the center of the jet and it was therefore not possible for these authors to ascertain the relationship of this mode to the stability problem or to discuss how to continue the eigenfunction across the singularity.
The viscous critical layer problem associated with this singularity is considerably more difficult than the usual one (which leads to integrals of the Airy function) because     and, consequently, a second-order turning point is involved. Our analysis shows that the Howard–Drazin mode is degenerate in the domain where it is valid as a limit of the viscous problem (wavenumber  α²≤ 9/2  ), that is, it corresponds to both an odd and an even mode. This conclusion is confirmed by direct numerical solution of the Orr–Sommerfeld equation which shows, in addition, that viscosity is destabilizing along portions of the stability boundary. For a retrograde jet, instability is found to occur beyond the inviscid critical value of β, that is, in the region where the flow would be stable according to the Rayleigh–Kuo condition.     

Arithmetic on Groups of Positive Rationals

In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory of genera in algebraic number fields. We establish generalizations of both Dirichlet's theorem on primes in arithmetic progressions and the theory of genera for Abelian extensions of the rationals.     

Canonical Modules of Semigroup Rings and a Conjecture of Reiner

We prove that the homotopy types of two naturally defined simplicial complexes are related in the following way: one is homotopy equivalent to a multiple suspension of the canonical Alexander dual of the other. These simplicial complexes arise in free resolutions of semigroup rings and modules. The relation between their homotopy types was conjectured by Reiner, and was suggested by a homological consequence of a result due independently to Danilov and Stanley on canonical modules of normal semigroup rings. Our proof is purely topological, and gives an alternative proof of their result. We also prove a generalization of a result of Hochster saying that these rings are Cohen—Macaulay, and indicate a new proof of Ehrhart's reciprocity law for rational polytopes. Received October 3, 2000, and in revised form April 2, 2001, and May 10, 2001. Online publication November 7, 2001.     

Locally Periodic Versus Globally Periodic Infinite Words

We call a one-way infinite word w over a finite alphabet (ρ,l)-repetitive if all long enough prefixes of w contain as a suffix a ρth power (or more generally a repetition of order ρ) of a word of length at most l. We show that each (2,4)-repetitive word is ultimately periodic, as well as that there exist continuum many, and hence also nonultimately periodic, (2,5)-repetitive words. Further, we characterize nonultimately periodic (2,5)-repetitive words both structurally and algebraically.     

Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate

The nonlinear natural frequency of a rectangular box, which consists of one flexible plate and five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a simple piston. The behavior of the structural-acoustic coupling between the flexible plate and the air cavity is analyzed by using the proposed finite element modal method. The system finite element equation is reduced and expressed in terms of the modal coordinates with small degrees of freedom by using the proposed reduction method. The system nonlinear stiffness matrix representing the large amplitude vibration can be transformed to be a constant modal matrix. The natural frequencies are determined by using the harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth on the natural frequencies and convergence studies are discussed in detail.     

Possible Loss and Recovery of Gibbsianness¶During the Stochastic Evolution of Gibbs Measures

We consider Ising-spin systems starting from an initial Gibbs measure ν and evolving under a spin-flip dynamics towards a reversible Gibbs measure μ≠ν. Both ν and μ are assumed to have a translation-invariant finite-range interaction. We study the Gibbsian character of the measure νS(t) at time t and show the following: (1) For all ν and μ, νS(t) is Gibbs for small t. (2) If both ν and μ have a high or infinite temperature, then νS(t) is Gibbs for all t > 0. (3) If ν has a low non-zero temperature and a zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t and non-Gibbs for large t. (4) If ν has a low non-zero temperature and a non-zero magnetic field and μ has a high or infinite temperature, then νS(t) is Gibbs for small t, non-Gibbs for intermediate t, and Gibbs for large t. The regime where μ has a low or zero temperature and t is not small remains open. This regime presumably allows for many different scenarios. Received: 26 April 2001 / Accepted: 10 October 2001     

Stable multi-sets

In this paper we introduce a generalization of stable sets: stable multi-sets. A stable multi-set is an assignment of integers to the vertices of a graph, such that specified bounds on vertices and edges are not exceeded. In case all vertex and edge bounds equal one, stable multi-sets are equivalent to stable sets.  For the stable multi-set problem, we derive reduction rules and study the associated polytope. We state necessary and sufficient conditions for the extreme points of the linear relaxation to be integer. These conditions generalize the conditions for the stable set polytope. Moreover, the classes of odd cycle and clique inequalities for stable sets are generalized to stable multi-sets and conditions for them to be facet defining are determined.  The study of stable multi-sets is initiated by optimization problems in the field of telecommunication networks. Stable multi-sets emerge as an important substructure in the design of optical networks. Received: February 14, 2001/Revised version: September 7, 2001     

Every diassociative A-loop is Moufang

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

     

MODEL COMPARISON FOR ACOUSTIC SCATTERING BY A SPHERICAL AIR BUBBLE IN WATER

The oscillation of a bubble has many vibrational modes, contributing to different portions in acoustic scattering. Without approximations a priori, an hydrodynamic approach is applied to investigate acoustic scattering by a bubble in water, taking into account the heat exchange and viscosity. The scattering function is derived for a wide range of frequencies. Examples are shown to illustrate the thermal and viscous effects on sound scattering. Comparison is made with three existing models. While these models are known not to be applicable for high frequencies, it is shown here that even in the low-frequency region, there are also noticeable discrepancies between the exact solution and the three existing models with regard to the scattering properties such as the scattering cross-section, and the quality factor of resonant peaks. By numerical simulation, we claim that the discrepancies may be due to the incomplete consideration of the thermal exchange process in the previous models. The approach presented here is valid for any other fluid enclosure in liquids.