170 research outputs found
Spectral series of the Schrodinger operator with delta-potential on a three-dimensional spherically symmetric manifold
The spectral series of the Schrodinger operator with a delta-potential on a threedimensional compact spherically symmetric manifold in the semiclassical limit as h -> 0 are described
Noncompact Lagrangian manifolds corresponding to the spectral series of the Schrodinger operator with delta-potential on a surface of revolution
Melnikov theory to all orders and Puiseux series for subharmonic solutions
We study the problem of subharmonic bifurcations for analytic systems in the
plane with perturbations depending periodically on time, in the case in which
we only assume that the subharmonic Melnikov function has at least one zero. If
the order of zero is odd, then there is always at least one subharmonic
solution, whereas if the order is even in general other conditions have to be
assumed to guarantee the existence of subharmonic solutions. Even when such
solutions exist, in general they are not analytic in the perturbation
parameter. We show that they are analytic in a fractional power of the
perturbation parameter. To obtain a fully constructive algorithm which allows
us not only to prove existence but also to obtain bounds on the radius of
analyticity and to approximate the solutions within any fixed accuracy, we need
further assumptions. The method we use to construct the solution -- when this
is possible -- is based on a combination of the Newton-Puiseux algorithm and
the tree formalism. This leads to a graphical representation of the solution in
terms of diagrams. Finally, if the subharmonic Melnikov function is identically
zero, we show that it is possible to introduce higher order generalisations,
for which the same kind of analysis can be carried out.Comment: 30 pages, 6 figure
Punctual Hilbert Schemes and Certified Approximate Singularities
In this paper we provide a new method to certify that a nearby polynomial
system has a singular isolated root with a prescribed multiplicity structure.
More precisely, given a polynomial system f , we present a Newton iteration on an extended deflated system
that locally converges, under regularity conditions, to a small deformation of
such that this deformed system has an exact singular root. The iteration
simultaneously converges to the coordinates of the singular root and the
coefficients of the so called inverse system that describes the multiplicity
structure at the root. We use -theory test to certify the quadratic
convergence, and togive bounds on the size of the deformation and on the
approximation error. The approach relies on an analysis of the punctual Hilbert
scheme, for which we provide a new description. We show in particular that some
of its strata can be rationally parametrized and exploit these parametrizations
in the certification. We show in numerical experimentation how the approximate
inverse system can be computed as a starting point of the Newton iterations and
the fast numerical convergence to the singular root with its multiplicity
structure, certified by our criteria.Comment: International Symposium on Symbolic and Algebraic Computation, Jul
2020, Kalamata, Franc
The Topology of Parabolic Character Varieties of Free Groups
Let G be a complex affine algebraic reductive group, and let K be a maximal
compact subgroup of G. Fix elements h_1,...,h_m in K. For n greater than or
equal to 0, let X (respectively, Y) be the space of equivalence classes of
representations of the free group of m+n generators in G (respectively, K) such
that for each i between 1 and m, the image of the i-th free generator is
conjugate to h_i. These spaces are parabolic analogues of character varieties
of free groups. We prove that Y is a strong deformation retraction of X. In
particular, X and Y are homotopy equivalent. We also describe explicit examples
relating X to relative character varieties.Comment: 16 pages, version 2 includes minor revisions and some modified
proofs, accepted for publication in Geometriae Dedicat
On the Symmetries of Integrability
We show that the Yang-Baxter equations for two dimensional models admit as a
group of symmetry the infinite discrete group . The existence of
this symmetry explains the presence of a spectral parameter in the solutions of
the equations. We show that similarly, for three-dimensional vertex models and
the associated tetrahedron equations, there also exists an infinite discrete
group of symmetry. Although generalizing naturally the previous one, it is a
much bigger hyperbolic Coxeter group. We indicate how this symmetry can help to
resolve the Yang-Baxter equations and their higher-dimensional generalizations
and initiate the study of three-dimensional vertex models. These symmetries are
naturally represented as birational projective transformations. They may
preserve non trivial algebraic varieties, and lead to proper parametrizations
of the models, be they integrable or not. We mention the relation existing
between spin models and the Bose-Messner algebras of algebraic combinatorics.
Our results also yield the generalization of the condition so often
mentioned in the theory of quantum groups, when no parameter is available.Comment: 23 page
Non integrability of a self-gravitating Riemann liquid ellipsoid
We prove that the motion of a triaxial Riemann ellipsoid of homogeneous
liquid without angular momentum does not possess an additional first integral
which is meromorphic in position, impulsions, and the elliptic functions which
appear in the potential, and thus is not integrable. We prove moreover that
this system is not integrable even on a fixed energy level hypersurface.Comment: 14 pages, 8 reference
Total Degree Formula for the Generic Offset to a Parametric Surface
We provide a resultant-based formula for the total degree w.r.t. the spatial
variables of the generic offset to a parametric surface. The parametrization of
the surface is not assumed to be proper.Comment: Preprint of an article to be published at the International Journal
of Algebra and Computation, World Scientific Publishing,
DOI:10.1142/S021819671100680
Nonlinear analysis of spacecraft thermal models
We study the differential equations of lumped-parameter models of spacecraft
thermal control. Firstly, we consider a satellite model consisting of two
isothermal parts (nodes): an outer part that absorbs heat from the environment
as radiation of various types and radiates heat as a black-body, and an inner
part that just dissipates heat at a constant rate. The resulting system of two
nonlinear ordinary differential equations for the satellite's temperatures is
analyzed with various methods, which prove that the temperatures approach a
steady state if the heat input is constant, whereas they approach a limit cycle
if it varies periodically. Secondly, we generalize those methods to study a
many-node thermal model of a spacecraft: this model also has a stable steady
state under constant heat inputs that becomes a limit cycle if the inputs vary
periodically. Finally, we propose new numerical analyses of spacecraft thermal
models based on our results, to complement the analyses normally carried out
with commercial software packages.Comment: 29 pages, 4 figure
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