On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

Coherent resonant interactions and slow light with molecules confined in photonic band-gap fibers

We investigate resonant nonlinear optical interactions and demonstrate induced transparency in acetylene molecules in a hollow-core photonic band-gap fiber at 1.5$\mu$m. The induced spectral transmission window is used to demonstrate slow-light effects, and we show that the observed broadening of the spectral features is due to collisions of the molecules with the inner walls of the fiber core. Our results illustrate that such fibers can be used to facilitate strong coherent light-matter interactions even when the optical response of the individual molecules is weak.Comment: 5 pages, 4 figure

Local and nonlocal solvable structures in ODEs reduction

Solvable structures, likewise solvable algebras of local symmetries, can be used to integrate scalar ODEs by quadratures. Solvable structures, however, are particularly suitable for the integration of ODEs with a lack of local symmetries. In fact, under regularity assumptions, any given ODE always admits solvable structures even though finding them in general could be a very difficult task. In practice a noteworthy simplification may come by computing solvable structures which are adapted to some admitted symmetry algebra. In this paper we consider solvable structures adapted to local and nonlocal symmetry algebras of any order (i.e., classical and higher). In particular we introduce the notion of nonlocal solvable structure

The Modern Irrationalities of American Criminal Codes: An Empirical Study of Offense Grading

The Model Penal Code made great advances in clarity and legality, moving most of the states from a mix of common law and ad hoc statutes to the modern American form of a comprehensive, succinct code that has served as a model around the world. Yet the decades since the wave of Model Code-based codifications have seen a steady degradation of American codes brought on by a relentless and accelerating rate of criminal law amendments that ignore the style, format, and content of the existing codes. The most damaging aspect of this trend is the exponentially increasing number of offense grading irrationalities found in most modern American codes. This Article documents the practical and prudential importance of getting offense grading right – that is, having the grade of each offense or suboffense reflect its relative seriousness in relation to all other offenses – then illustrates just how wrong things have gone, using a case study of offense grading in Pennsylvania, one of the better modern American codes. The critique of Pennsylvania, and its conclusions, does not rely upon the value judgments of the authors but rather upon an empirical study of the judgments of Pennsylvania residents regarding the relative seriousness of more than a hundred existing Pennsylvania offenses. The results suggest a startling conflict between the law\u27s grading judgments and those of the community it governs, as well as a variety of kinds of logical irrationalities and internal inconsistencies. The process by which these grading irrationalities have been and continue to be created is examined, and solutions for fixing and, perhaps, avoiding these problems in the future, are explored

On asymptotic nonlocal symmetry of nonlinear Schr\"odinger equations

A concept of asymptotic symmetry is introduced which is based on a definition of symmetry as a reducibility property relative to a corresponding invariant ansatz. It is shown that the nonlocal Lorentz invariance of the free-particle Schr\"odinger equation, discovered by Fushchych and Segeda in 1977, can be extended to Galilei-invariant equations for free particles with arbitrary spin and, with our definition of asymptotic symmetry, to many nonlinear Schr\"odinger equations. An important class of solutions of the free Schr\"odinger equation with improved smoothing properties is obtained

Few cycle pulse propagation

We present a comprehensive framework for treating the nonlinear interaction of few-cycle pulses using an envelope description that goes beyond the traditional SVEA method. This is applied to a range of simulations that demonstrate how the effect of a $\chi^{(2)}$ nonlinearity differs between the many-cycle and few-cycle cases. Our approach, which includes diffraction, dispersion, multiple fields, and a wide range of nonlinearities, builds upon the work of Brabec and Krausz[1] and Porras[2]. No approximations are made until the final stage when a particular problem is considered. The original version (v1) of this arXiv paper is close to the published Phys.Rev.A. version, and much smaller in size.Comment: 9 pages, 14 figure

Gravitational Waves: Just Plane Symmetry

We present some remarkable properties of the symmetry group for gravitational plane waves. Our main observation is that metrics with plane wave symmetry satisfy every system of generally covariant vacuum field equations except the Einstein equations. The proof uses the homothety admitted by metrics with plane wave symmetry and the scaling behavior of generally covariant field equations. We also discuss a mini-superspace description of spacetimes with plane wave symmetry.Comment: 10 pages, TeX, uses IOP style file

On the relation between standard and μ\mu-symmetries for PDEs

We give a geometrical interpretation of the notion of $\mu$-prolongations of vector fields and of the related concept of $\mu$-symmetry for partial differential equations (extending to PDEs the notion of $\lambda$-symmetry for ODEs). We give in particular a result concerning the relationship between $\mu$-symmetries and standard exact symmetries. The notion is also extended to the case of conditional and partial symmetries, and we analyze the relation between local $\mu$-symmetries and nonlocal standard symmetries.Comment: 25 pages, no figures, latex. to be published in J. Phys.