Kinematic Quantities and Raychaudhuri Equations in a 5D5D Universe

Based on some ideas emerged from the classical Kaluza-Klein theory, we present a $5D$ universe as a product bundle over the $4D$ spacetime. This enables us to introduce and study two categories of kinematic quantities (expansions, shear, vorticity) in a $5D$ universe. One category is related to the fourth dimension (time), and the other one comes from the assumption of the existence of the fifth dimension. The Raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the $4D$ expansion and $5D$ expansion in a $5D$ universe.Comment: 27 page

Radially symmetric thin plate splines interpolating a circular contour map

Profiles of radially symmetric thin plate spline surfaces minimizing the Beppo Levi energy over a compact annulus $R_{1}\leq r\leq R_{2}$ have been studied by Rabut via reproducing kernel methods. Motivated by our recent construction of Beppo Levi polyspline surfaces, we focus here on minimizing the radial energy over the full semi-axis $0<r<\infty$. Using a $L$-spline approach, we find two types of minimizing profiles: one is the limit of Rabut's solution as $R_{1}\rightarrow0$ and $R_{2}\rightarrow\infty$ (identified as a `non-singular' $L$-spline), the other has a second-derivative singularity and matches an extra data value at $0$. For both profiles and $p\in\left[ 2,\infty\right]$, we establish the $L^{p}$-approximation order $3/2+1/p$ in the radial energy space. We also include numerical examples and obtain a novel representation of the minimizers in terms of dilates of a basis function.Comment: new figures and sub-sections; new Proposition 1 replacing old Corollary 1; shorter proof of Theorem 4; one new referenc

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

Paraquaternionic CR-submanifolds of paraquaternionic Kahler manifolds and semi-Riemannian submersions

In this paper we introduce paraquaternionic CR-submanifolds of almost paraquaternionic hermitian manifolds and state some basic results on their differential geometry. We also study a class of semi-Riemannian submersions from paraquaternionic CR-submanifolds of paraquaternionic Kaehler manifolds.Comment: 19 page

Ruled CR-submanifolds of locally conformal K\"{a}hler manifolds

The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal K\"{a}hler manifold.Comment: 10 pages, Journal of Geometry and Physics (to appear