Third-order superintegrable systems separable in parabolic coordinates

In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and so admit two second-order integrals. The third-order integral is their Lie or Poisson commutator. We discuss how this situation is different from the Cartesian and polar cases where new potentials were discovered which are not multi-separable and which are expressed in terms of Painlev\'e transcendents or elliptic functions

Symmetries of Discrete Dynamical Systems Involving Two Species

The Lie point symmetries of a coupled system of two nonlinear differential-difference equations are investigated. It is shown that in special cases the symmetry group can be infinite dimensional, in other cases up to 10 dimensional. The equations can describe the interaction of two long molecular chains, each involving one type of atoms.Comment: 40 pages, no figures, typed in AMS-LaTe

Lie Symmetries and Exact Solutions of First Order Difference Schemes

We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted to the symmetries considered. The invariant difference schemes can be so chosen that their solutions coincide exactly with those of the original differential equation.Comment: Minor changes and journal-re

Third order superintegrable systems separating in polar coordinates

A complete classification is presented of quantum and classical superintegrable systems in $E_2$ that allow the separation of variables in polar coordinates and admit an additional integral of motion of order three in the momentum. New quantum superintegrable systems are discovered for which the potential is expressed in terms of the sixth Painlev\'e transcendent or in terms of the Weierstrass elliptic function

Major histocompatibility complex-associated odour preferences and human mate choice: near and far horizons

The major histocompatibility complex (MHC) is a core part of the adaptive immune system. As in other vertebrate taxa, it may also affect human chemical communication via odour-based mate preferences, with greater attraction towards MHC-dissimilar partners. However, despite some well-known findings, the available evidence is equivocal and made complicated by varied approaches to quantifying human mate choice. To address this, we here conduct comprehensive meta-analyses focusing on studies assessing i] genomic mate selection, ii] relationship satisfaction and iii] odour preference. Analysis of genomic studies reveals no association between MHC-dissimilarity and mate choice in actual couples; however, MHC effects appear to be independent of genomic background. The effect of MHC-dissimilarity on relationship satisfaction was not significant and we found evidence for publication bias in studies on this area. There was also no significant association between MHC-dissimilarity and odour preferences. Finally, combining effect sizes from all genomic, relationship satisfaction, odour preference and previous mate choice studies into an overall estimate showed no overall significant effect of MHC similarity on human mate selection. Based on these findings, we make a set of recommendations for future studies, focusing both on aspects that should be implemented immediately and those that lurk on the far horizon. We need larger samples with greater geographical and cultural diversity, that control for genome-wide similarity. We also need more focus on mechanisms of MHC-associated odour preferences and on MHC-associated pregnancy loss