Abelian versus non-Abelian Baecklund Charts: some remarks

Connections via Baecklund transformations among different non-linear evolution equations are investigated aiming to compare corresponding Abelian and non Abelian results. Specifically, links, via Baecklund transformations, connecting Burgers and KdV-type hierarchies of nonlinear evolution equations are studied. Crucial differences as well as notable similarities between Baecklund charts in the case of the Burgers - heat equation, on one side and KdV -type equations are considered. The Baecklund charts constructed in [16] and [17], respectively, to connect Burgers and KdV-type hierarchies of operator nonlinear evolution equations show that the structures, in the non-commutative cases, are richer than the corresponding commutative ones.Comment: 18 page

Stokes-vector evolution in a weakly anisotropic inhomogeneous medium

Equation for evolution of the four-component Stokes vector in weakly anisotropic and smoothly inhomogeneous media is derived on the basis of quasi-isotropic approximation of the geometrical optics method, which provides consequent asymptotic solution of Maxwell equations. Our equation generalizes previous results, obtained for the normal propagation of electromagnetic waves in stratified media. It is valid for curvilinear rays with torsion and is capable to describe normal modes conversion in the inhomogeneous media. Remarkably, evolution of the Stokes vector is described by the Bargmann-Michel-Telegdi equation for relativistic spin precession, whereas the equation for the three-component Stokes vector resembles the Landau-Lifshitz equation for spin precession in ferromegnetic systems. General theory is applied for analysis of polarization evolution in a magnetized plasma. We also emphasize fundamental features of the non-Abelian polarization evolution in anisotropic inhomogeneous media and illustrate them by simple examples.Comment: 16 pages, 3 figures, to appear in J. Opt. Soc. Am.

The Depth and Breadth of John Bell's Physics

John Bell's investigations in foundations of quantum mechanics, particle physics, and quantum field theory are recalled.Comment: 46 pages, 1 figure, LaTeX; editorial corrections; email correspondence to R. Jackiw <[email protected]

On the derivation of complex linear models from simpler ones

Linear mixed models are useful in biology, genetics, medical research, agriculture, industry, and many other fields, providing a flexible approach in situations of correlated data. Based on the structure of the variance-covariance matrix, emerged a special class of linear mixed models, those of models with orthogonal block structure, which allows optimal estimation for variance components of blocks and contrasts of treatments. This approach triggered a more restrict class of mixed models, models with commutative orthogonal block structure, whose interest lies in the possibility of achieving least squares estimators giving best linear unbiased estimators for estimable vectors. Exploring the possibility of joint analysis of linear mixed models, obtained independently, and focusing on the approach based on the algebraic structure of the models, some authors have investigated the conditions in which the good properties of the estimators are preserved. In this work we intend to highlight the ideas underlying the techniques for the joint analysis of models, since these aspects were underexplored in the works where the theoretical formulation of the techniques were introduced. Given that these techniques were developed involving models with commutative orthogonal block structure, we provide a selective review of the literature focusing on the contributions addressing this special class of mixed linear models

An integrable 3D lattice model with positive Boltzmann weights

In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0<q<1 and three continuous field variables. The layer-to-layer transfer matrices of the model form a two-parameter commutative family. This is the first example of a solvable 3D lattice model with non-negative Boltzmann weights.Comment: HyperTex is disabled due to conflicts with some macro

The Role of Term Symmetry in E-Unification and E-Completion

A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied. The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings. The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory