30,711 research outputs found
Computing Periods of Hypersurfaces
We give an algorithm to compute the periods of smooth projective
hypersurfaces of any dimension. This is an improvement over existing algorithms
which could only compute the periods of plane curves. Our algorithm reduces the
evaluation of period integrals to an initial value problem for ordinary
differential equations of Picard-Fuchs type. In this way, the periods can be
computed to extreme-precision in order to study their arithmetic properties.
The initial conditions are obtained by an exact determination of the cohomology
pairing on Fermat hypersurfaces with respect to a natural basis.Comment: 33 pages; Final version. Fixed typos, minor expository changes.
Changed code repository lin
Optimization in Differentiable Manifolds in Order to Determine the Method of Construction of Prehistoric Wall-Paintings
In this paper a general methodology is introduced for the determination of
potential prototype curves used for the drawing of prehistoric wall-paintings.
The approach includes a) preprocessing of the wall-paintings contours to
properly partition them, according to their curvature, b) choice of prototype
curves families, c) analysis and optimization in 4-manifold for a first
estimation of the form of these prototypes, d) clustering of the contour parts
and the prototypes, to determine a minimal number of potential guides, e)
further optimization in 4-manifold, applied to each cluster separately, in
order to determine the exact functional form of the potential guides, together
with the corresponding drawn contour parts. The introduced methodology
simultaneously deals with two problems: a) the arbitrariness in data-points
orientation and b) the determination of one proper form for a prototype curve
that optimally fits the corresponding contour data. Arbitrariness in
orientation has been dealt with a novel curvature based error, while the proper
forms of curve prototypes have been exhaustively determined by embedding
curvature deformations of the prototypes into 4-manifolds. Application of this
methodology to celebrated wall-paintings excavated at Tyrins, Greece and the
Greek island of Thera, manifests it is highly probable that these
wall-paintings had been drawn by means of geometric guides that correspond to
linear spirals and hyperbolae. These geometric forms fit the drawings' lines
with an exceptionally low average error, less than 0.39mm. Hence, the approach
suggests the existence of accurate realizations of complicated geometric
entities, more than 1000 years before their axiomatic formulation in Classical
Ages
Area-Constrained Planar Elastica
We determine the equilibria of a rigid loop in the plane, subject to the
constraints of fixed length and fixed enclosed area. Rigidity is characterized
by an energy functional quadratic in the curvature of the loop. We find that
the area constraint gives rise to equilibria with remarkable geometrical
properties: not only can the Euler-Lagrange equation be integrated to provide a
quadrature for the curvature but, in addition, the embedding itself can be
expressed as a local function of the curvature. The configuration space is
shown to be essentially one-dimensional, with surprisingly rich structure.
Distinct branches of integer-indexed equilibria exhibit self-intersections and
bifurcations -- a gallery of plots is provided to highlight these findings.
Perturbations connecting equilibria are shown to satisfy a first order ODE
which is readily solved. We also obtain analytical expressions for the energy
as a function of the area in some limiting regimes.Comment: 23 pages, several figures. Version 2: New title. Changes in the
introduction, addition of a new section with conclusions. Figure 14 corrected
and one reference added. Version to appear in PR
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