4,067 research outputs found
A Robust and Efficient Method for Solving Point Distance Problems by Homotopy
The goal of Point Distance Solving Problems is to find 2D or 3D placements of
points knowing distances between some pairs of points. The common guideline is
to solve them by a numerical iterative method (\emph{e.g.} Newton-Raphson
method). A sole solution is obtained whereas many exist. However the number of
solutions can be exponential and methods should provide solutions close to a
sketch drawn by the user.Geometric reasoning can help to simplify the
underlying system of equations by changing a few equations and triangularizing
it.This triangularization is a geometric construction of solutions, called
construction plan. We aim at finding several solutions close to the sketch on a
one-dimensional path defined by a global parameter-homotopy using a
construction plan. Some numerical instabilities may be encountered due to
specific geometric configurations. We address this problem by changing
on-the-fly the construction plan.Numerical results show that this hybrid method
is efficient and robust
Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories
The interplay rich between algebraic geometry and string and gauge theories
has recently been immensely aided by advances in computational algebra.
However, these symbolic (Gr\"{o}bner) methods are severely limited by
algorithmic issues such as exponential space complexity and being highly
sequential. In this paper, we introduce a novel paradigm of numerical algebraic
geometry which in a plethora of situations overcomes these short-comings. Its
so-called 'embarrassing parallelizability' allows us to solve many problems and
extract physical information which elude the symbolic methods. We describe the
method and then use it to solve various problems arising from physics which
could not be otherwise solved.Comment: 36 page
Relative pairing in cyclic cohomology and divisor flows
We construct invariants of relative K-theory classes of multiparameter
dependent pseudodifferential operators, which recover and generalize Melrose's
divisor flow and its higher odd-dimensional versions of Lesch and Pflaum. These
higher divisor flows are obtained by means of pairing the relative K-theory
modulo the symbols with the cyclic cohomological characters of relative cycles
constructed out of the regularized operator trace together with its symbolic
boundary. Besides giving a clear and conceptual explanation to all the
essential features of the divisor flows, this construction allows to uncover
the previously unknown even-dimensional counterparts. Furthermore, it confers
to the totality of these invariants a purely topological interpretation, that
of implementing the classical Bott periodicity isomorphisms in a manner
compatible with the suspension isomorphisms in both K-theory and in cyclic
cohomology. We also give a precise formulation, in terms of a natural Clifford
algebraic suspension, for the relationship between the higher divisor flows and
the spectral flow.Comment: 43 pages; revision 5.22; expanded by a factor of 1.5, in particular
even case adde
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