46 research outputs found
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
Zero curvature representation for a new fifth-order integrable system
In this brief note we present a zero-curvature representation for one of the
new integrable system found by Mikhailov, Novikov and Wang in nlin.SI/0601046.Comment: 2 pages, LaTeX 2e, no figure
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
Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w
w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x +
(4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no
recursion operator or master symmetry was known so far, and prove that the
system (*) admits infinitely many local generalized symmetries that are
constructed using a nonlocal {\em two-term} recursion relation rather than from
a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and
definitions adde
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