65,923 research outputs found
Pseudospherical surfaces on time scales: a geometric definition and the spectral approach
We define and discuss the notion of pseudospherical surfaces in asymptotic
coordinates on time scales. Thus we extend well known notions of discrete
pseudospherical surfaces and smooth pseudosperical surfaces on more exotic
domains (e.g, the Cantor set). In particular, we present a new expression for
the discrete Gaussian curvature which turns out to be valid for asymptotic nets
on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time
scale have constant negative Gaussian curvature. We present also the
quaternion-valued spectral problem (the Lax pair) and the Darboux-Backlund
transformation for pseudospherical surfaces (in asymptotic coordinates) on
arbitrary time scales.Comment: 20 page
Hidden symmetry of hyperbolic monopole motion
Hyperbolic monopole motion is studied for well separated monopoles. It is
shown that the motion of a hyperbolic monopole in the presence of one or more
fixed monopoles is equivalent to geodesic motion on a particular submanifold of
the full moduli space. The metric on this submanifold is found to be a
generalisation of the multi-centre Taub-NUT metric introduced by LeBrun. The
one centre case is analysed in detail as a special case of a class of systems
admitting a conserved Runge-Lenz vector. The two centre problem is also
considered. An integrable classical string motion is exhibited.Comment: 39 pages, 7 figures, references added, minor changes to section
Field reduction and linear sets in finite geometry
Based on the simple and well understood concept of subfields in a finite
field, the technique called `field reduction' has proved to be a very useful
and powerful tool in finite geometry. In this paper we elaborate on this
technique. Field reduction for projective and polar spaces is formalized and
the links with Desarguesian spreads and linear sets are explained in detail.
Recent results and some fundamental ques- tions about linear sets and scattered
spaces are studied. The relevance of field reduction is illustrated by
discussing applications to blocking sets and semifields
Distribution and density of the partition function zeros for the diamond-decorated Ising model
Exact renormalization map of temperature between two successive decorated
lattices is given, and the distribution of the partition function zeros in the
complex temperature plane is obtained for any decoration-level. The rule
governing the variation of the distribution pattern as the decoration-level
changes is given. The densities of the zeros for the first two
decoration-levels are calculated explicitly, and the qualitative features about
the densities of higher decoration-levels are given by conjecture. The Julia
set associated with the renormalization map is contained in the distribution of
the zeros in the limit of infinite decoration level, and the formation of the
Julia set in the course of increasing the decoration-level is given in terms of
the variations of the zero density.Comment: 8 pages,8figure
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