Scalar second order evolution equations possessing an irreducible sl2_2-valued zero curvature representation

We find all scalar second order evolution equations possessing an sl$_2$-valued zero curvature representation that is not reducible to a proper subalgebra of sl$_2$. None of these zero-curvature representations admits a parameter.Comment: 10 pages, requires nath.st

Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an sl(2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth century geometers. Finally, we characterize the associated normal congruences

The geometric sense of R. Sasaki connection

For the Riemannian manifold $M^{n}$ two special connections on the sum of the tangent bundle $TM^{n}$ and the trivial one-dimensional bundle are constructed. These connections are flat if and only if the space $M^{n}$ has a constant sectional curvature $\pm 1$. The geometric explanation of this property is given. This construction gives a coordinate free many-dimensional generalization of the connection from the paper: R. Sasaki 1979 Soliton equations and pseudospherical surfaces, Nuclear Phys., {\bf 154 B}, pp. 343-357. It is shown that these connections are in close relation with the imbedding of $M^{n}$ into Euclidean or pseudoeuclidean $(n+1)$-dimension spaces.Comment: 7 pages, the key reference to the paper of Min-Oo is included in the second versio

On the Invariant Theory of Weingarten Surfaces in Euclidean Space

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential equation. We apply these results to the special Weingarten surfaces: minimal surfaces, surfaces of constant mean curvature and surfaces of constant Gauss curvature.Comment: 16 page

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.

A unified approach to computation of integrable structures

We expose (without proofs) a unified computational approach to integrable structures (including recursion, Hamiltonian, and symplectic operators) based on geometrical theory of partial differential equations. We adopt a coordinate based approach and aim to provide a tutorial to the computations.Comment: 19 pages, based on a talk on the SPT 2011 conference, http://www.sptspt.it/spt2011/ ; v2, v3: minor correction

Sex-specific reproductive behaviours and paternity in free-ranging Barbary macaques (Macaca sylvanus)

In a wide variety of species, male reproductive success is determined by contest for access to females. Among multi-male primate groups, however, factors in addition to male competitive ability may also influence paternity outcome, although their exact nature and force is still largely unclear. Here, we have investigated in a group of free-ranging Barbary macaques whether paternity is determined on the pre- or postcopulatory level and how male competitive ability and female direct mate choice during the female fertile phase are related to male reproductive success. Behavioural observations were combined with faecal hormone analysis for timing of the fertile phase (13 cycles, 8 females) and genetic paternity analysis (n = 12). During the fertile phase, complete monopolisation of females did not occur. Females were consorted for only 49% of observation time, and all females had ejaculatory copulations with several males. Thus, in all cases, paternity was determined on the postcopulatory level. More than 80% of infants were sired by high-ranking males, and this reproductive skew was related to both, male competitive ability and female direct mate choice as high-ranking males spent more time in consort with females than low-ranking males, and females solicited copulations mainly from dominant males. As most ejaculatory copulations were female-initiated, female direct mate choice appeared to have the highest impact on male reproductive success. However, female preference was not directly translated into paternity, as fathers were not preferred over non-fathers in terms of solicitation, consortship and mating behaviour. Collectively, our data show that in the Barbary macaque, both sexes significantly influence male mating success, but that sperm of several males generally compete within the female reproductive tract and that therefore paternity is determined by mechanisms operating at the postcopulatory level