103 research outputs found
Transfinite thin plate spline interpolation
Duchon's method of thin plate splines defines a polyharmonic interpolant to
scattered data values as the minimizer of a certain integral functional. For
transfinite interpolation, i.e. interpolation of continuous data prescribed on
curves or hypersurfaces, Kounchev has developed the method of polysplines,
which are piecewise polyharmonic functions of fixed smoothness across the given
hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has
introduced boundary conditions of Beppo Levi type to construct a semi-cardinal
model for polyspline interpolation to data on an infinite set of parallel
hyperplanes. The present paper proves that, for periodic data on a finite set
of parallel hyperplanes, the polyspline interpolant satisfying Beppo Levi
boundary conditions is in fact a thin plate spline, i.e. it minimizes a Duchon
type functional
A Finslerian version of 't Hooft Deterministic Quantum Models
Using the Finsler structure living in the phase space associated to the
tangent bundle of the configuration manifold, deterministic models at the
Planck scale are obtained. The Hamiltonian function are constructed directly
from the geometric data and some assumptions concerning time inversion
symmetry. The existence of a maximal acceleration and speed is proved for
Finslerian deterministic models. We investigate the spontaneous symmetry
breaking of the orthogonal symmetry SO(6N) of the Hamiltonian of a
deterministic system. This symmetry break implies the non-validity of the
argument used to obtain Bell's inequalities for spin states. It is introduced
and motivated in the context of Randers spaces an example of simple 't Hooft
model with interactions.Comment: 25 pages; no figures. String discussion deleted. Some minor change
Nonholonomic Ricci Flows and Running Cosmological Constant: I. 4D Taub-NUT Metrics
In this work we construct and analyze exact solutions describing Ricci flows
and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It
is outlined a new geometric techniques of constructing Ricci flow solutions.
Some conceptual issues on spacetimes provided with generic off-diagonal metrics
and associated nonlinear connection structures are analyzed. The limit from
gravity/Ricci flow models with nontrivial torsion to configurations with the
Levi-Civita connection is allowed in some specific physical circumstances by
constraining the class of integral varieties for the Einstein and Ricci flow
equations.Comment: latex2e, final variant to be published in IJMP
A Geometric characterization of Finsler manifolds of constant curvature K
We prove that a Finsler manifold 𝔽m is of constant
curvature K=1 if and only if the unit horizontal Liouville vector
field is a Killing vector field on the indicatrix bundle IM of
𝔽m
Some constructions of almost para-hyperhermitian structures on manifolds and tangent bundles
In this paper we give some examples of almost para-hyperhermitian structures
on the tangent bundle of an almost product manifold, on the product manifold
, where is a manifold endowed with a mixed 3-structure
and on the circle bundle over a manifold with a mixed 3-structure.Comment: 10 pages; This paper has been presented in the "4th German-Romanian
Seminar on Geometry" Dortmund, Germany, 15-18 July 200
Finsler and Lagrange Geometries in Einstein and String Gravity
We review the current status of Finsler-Lagrange geometry and
generalizations. The goal is to aid non-experts on Finsler spaces, but
physicists and geometers skilled in general relativity and particle theories,
to understand the crucial importance of such geometric methods for applications
in modern physics. We also would like to orient mathematicians working in
generalized Finsler and Kahler geometry and geometric mechanics how they could
perform their results in order to be accepted by the community of ''orthodox''
physicists.
Although the bulk of former models of Finsler-Lagrange spaces where
elaborated on tangent bundles, the surprising result advocated in our works is
that such locally anisotropic structures can be modelled equivalently on
Riemann-Cartan spaces, even as exact solutions in Einstein and/or string
gravity, if nonholonomic distributions and moving frames of references are
introduced into consideration.
We also propose a canonical scheme when geometrical objects on a (pseudo)
Riemannian space are nonholonomically deformed into generalized Lagrange, or
Finsler, configurations on the same manifold. Such canonical transforms are
defined by the coefficients of a prime metric and generate target spaces as
Lagrange structures, their models of almost Hermitian/ Kahler, or nonholonomic
Riemann spaces.
Finally, we consider some classes of exact solutions in string and Einstein
gravity modelling Lagrange-Finsler structures with solitonic pp-waves and
speculate on their physical meaning.Comment: latex 2e, 11pt, 44 pages; accepted to IJGMMP (2008) as a short
variant of arXiv:0707.1524v3, on 86 page
Two-Connection Renormalization and Nonholonomic Gauge Models of Einstein Gravity
A new framework to perturbative quantum gravity is proposed following the
geometry of nonholonomic distributions on (pseudo) Riemannian manifolds. There
are considered such distributions and adapted connections, also completely
defined by a metric structure, when gravitational models with infinite many
couplings reduce to two--loop renormalizable effective actions. We use a key
result from our partner work arXiv:0902.0911 that the classical Einstein
gravity theory can be reformulated equivalently as a nonholonomic gauge model
in the bundle of affine/de Sitter frames on pseudo-Riemannian spacetime. It is
proven that (for a class of nonholonomic constraints and splitting of the
Levi-Civita connection into a "renormalizable" distinguished connection, on a
base background manifold, and a gauge like distortion tensor, in total space) a
nonholonomic differential renormalization procedure for quantum gravitational
fields can be elaborated. Calculation labor is reduced to one- and two-loop
levels and renormalization group equations for nonholonomic configurations.Comment: latex2e, 40 pages, v4, accepted for Int. J. Geom. Meth. Mod. Phys. 7
(2010
Fedosov Quantization of Lagrange-Finsler and Hamilton-Cartan Spaces and Einstein Gravity Lifts on (Co) Tangent Bundles
We provide a method of converting Lagrange and Finsler spaces and their
Legendre transforms to Hamilton and Cartan spaces into almost Kaehler
structures on tangent and cotangent bundles. In particular cases, the Hamilton
spaces contain nonholonomic lifts of (pseudo) Riemannian / Einstein metrics on
effective phase spaces. This allows us to define the corresponding Fedosov
operators and develop deformation quantization schemes for nonlinear mechanical
and gravity models on Lagrange- and Hamilton-Fedosov manifolds.Comment: latex2e, 11pt, 35 pages, v3, accepted to J. Math. Phys. (2009
Weak Gravitational Field in Finsler-Randers Space and Raychaudhuri Equation
The linearized form of the metric of a Finsler - Randers space is studied in
relation to the equations of motion, the deviation of geodesics and the
generalized Raychaudhuri equation are given for a weak gravitational field.
This equation is also derived in the framework of a tangent bundle. By using
Cartan or Berwald-like connections we get some types "gravito -
electromagnetic" curvature. In addition we investigate the conditions under
which a definite Lagrangian in a Randers space leads to Einstein field
equations under the presence of electromagnetic field. Finally, some
applications of the weak field in a generalized Finsler spacetime for
gravitational waves are given.Comment: 22 pages, matches version published in GER
Lagrange-Fedosov Nonholonomic Manifolds
We outline an unified approach to geometrization of Lagrange mechanics,
Finsler geometry and geometric methods of constructing exact solutions with
generic off-diagonal terms and nonholonomic variables in gravity theories. Such
geometries with induced almost symplectic structure are modelled on
nonholonomic manifolds provided with nonintegrable distributions defining
nonlinear connections. We introduce the concept of Lagrange-Fedosov spaces and
Fedosov nonholonomic manifolds provided with almost symplectic connection
adapted to the nonlinear connection structure.
We investigate the main properties of generalized Fedosov nonholonomic
manifolds and analyze exact solutions defining almost symplectic Einstein
spaces.Comment: latex2e, v3, published variant, with new S.V. affiliatio
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