The Degasperis-Procesi equation as a non-metric Euler equation

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of $b$-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet space of all smooth functions on the circle.Comment: 17 page

Correspondence in Quasiperiodic and Chaotic Maps: Quantization via the von Neumann Equation

A generalized approach to the quantization of a large class of maps on a torus, i.e. quantization via the von Neumann Equation, is described and a number of issues related to the quantization of model systems are discussed. The approach yields well behaved mixed quantum states for tori for which the corresponding Schrodinger equation has no solutions, as well as an extended spectrum for tori where the Schrodinger equation can be solved. Quantum-classical correspondence is demonstrated for the class of mappings considered, with the Wigner-Weyl density $\rho(p,q,t)$ going to the correct classical limit. An application to the cat map yields, in a direct manner, nonchaotic quantum dynamics, plus the exact chaotic classical propagator in the correspondence limit.Comment: 36 pages, RevTex preprint forma

Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems

The measurement of high-dimensional entangled states of orbital angular momentum prepared by spontaneous parametric down-conversion can be considered in two separate stages: a generation stage and a detection stage. Given a certain number of generated modes, the number of measured modes is determined by the measurement apparatus. We derive a simple relationship between the generation and detection parameters and the number of measured entangled modes.Comment: 6 pages, 4 figure

GG-Strands

A $G$-strand is a map $g(t,{s}):\,\mathbb{R}\times\mathbb{R}\to G$ for a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. The SO(3)-strand is the $G$-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, $SO(3)_K$-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar\'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the $SO(3)_K$-strand is mapped into a completely integrable generalization of the classical chiral model for the SO(3)-strand. Analogous results are obtained for the $Sp(2)$-strand. The $Sp(2)$-strand is the $G$-strand version of the $Sp(2)$ Bloch-Iserles ordinary differential equation, whose solutions exhibit dynamical sorting. Numerical solutions show nonlinear interactions of coherent wave-like solutions in both cases. ${\rm Diff}(\mathbb{R})$-strand equations on the diffeomorphism group $G={\rm Diff}(\mathbb{R})$ are also introduced and shown to admit solutions with singular support (e.g., peakons).Comment: 35 pages, 5 figures, 3rd version. To appear in J Nonlin Sc

Predicting emotions and meta-emotions at the movies

Audiences are attracted to dramas and horror movies even though negative and ambivalent emotions are likely to be experienced. Research into the seemingly paradoxical enjoyment of this kind of media entertainment has typically focused on gender- and genre-specific needs and viewing motivations. Extending this line of research, the authors focus the role of the need for affect as a more general, gender- and genre-independent predictor of individual differences in the experience of emotions and meta-emotions (i.e., evaluative thoughts and feelings about one’s emotions). The article discusses a field study of moviegoers who attended the regular screening of a drama or a horror film. Results support the assumption that individuals high in need for affect experience higher levels of negative and ambivalent emotions and evaluate their emotions more positively on the level of meta-emotions. Controlling for the Big Five personality factors does not alter these effects. The results are discussed within an extended meta-emotion framework

Emotion, Meaning, and Appraisal Theory

According to psychological emotion theories referred to as appraisal theory, emotions are caused by appraisals (evaluative judgments). Borrowing a term from Jan Smedslund, it is the contention of this article that psychological appraisal theory is “pseudoempirical” (i.e., misleadingly or incorrectly empirical). In the article I outline what makes some scientific psychology “pseudoempirical,” distinguish my view on this from Jan Smedslund’s, and then go on to show why paying heed to the ordinary meanings of emotion terms is relevant to psychology, and how appraisal theory is methodologically off the mark by employing experiments, questionnaires, and the like, to investigate what follows from the ordinary meanings of words. The overarching argument of the article is that the scientific research program of appraisal theory is fundamentally misguided and that a more philosophical approach is needed to address the kinds of questions it seeks to answer

Local structure of the set of steady-state solutions to the 2D incompressible Euler equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions. Analogies with the finite-dimensional model of geodesics on a Lie group with left-invariant metric can be very instructive, but it is often difficult to prove analogues of finite-dimensional results in the infinite-dimensional setting of Euler's equations. In this paper we establish a result in this direction in the simple case of steady-state solutions in two dimensions, under some non-degeneracy assumptions. In particular, we establish, in a non-degenerate situation, a local one-to-one correspondence between steady-states and co-adjoint orbits.Comment: 81 page