Regular Manifolds Research Articles

Relation Between Regge Calculus and BF Theory on Manifolds with Defects

In Regge calculus, the space–time manifold is approximated by certain abstract simplicial complex, called a pseudomanifold, and the metric is approximated by an assignment of a length to each 1-simplex. In this paper for each pseudomanifold, we construct a smooth manifold which we call a manifold with defects. This manifold emerges from the purely combinatorial simplicial complex as a result of gluing geometric realizations of its n-simplices followed by removing the simplices of dimension n-2. The Regge geometry is encoded in a boundary data of a BF theory on this manifold. We consider an action functional which coincides with the standard BF action for suitably regular manifolds with defects and fields. We show that the action evaluated at solutions of the field equations satisfying certain boundary conditions coincides with an evaluation of the Regge action at Regge geometries defined by the boundary data. As a result, the degrees of freedom of Regge calculus are traded for discrete degrees of freedom of topological BF theory.

On the flow of non-axisymmetric perturbations of cylinders via surface diffusion

We study the surface diffusion flow acting on a class of general (non-axisymmetric) perturbations of cylinders Cr in IR3. Using tools from parabolic theory on uniformly regular manifolds, and maximal regularity, we establish existence and uniqueness of solutions to surface diffusion flow starting from (spatially-unbounded) surfaces defined over Cr via scalar height functions which are uniformly bounded away from the central cylindrical axis. Additionally, we show that Cr is normally stable with respect to 2π-axially-periodic perturbations if the radius r>1, and unstable if 0<r<1. Stability is also shown to hold in settings with axial Neumann boundary conditions.

Computations of Homeomorphisms in the Contact Dynamical Systems

This paper studies the relations of the homeomorphisms in the contact dynamical systems on compact regular contact manifolds. Using some improved approximation lemmas and techniques of inequalities, this paper finally proves the coincidence of the topological strictly contact homeomorphisms and the modified continuous strictly contact homeomorphisms.

Embedded plateau problem

We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of D. The same result is also valid for homogenously regular manifolds with sufficiently convex boundary.

Thermodynamics of Photons on Fractals

A thermodynamical treatment of a massless scalar field (a photon) confined to a fractal spatial manifold leads to an equation of state relating pressure to internal energy, PV(s) = U/d(s), where d(s) is the spectral dimension and V(s) defines the "spectral volume." For regular manifolds, V(s) coincides with the usual geometric spatial volume, but on a fractal this is not necessarily the case. This is further evidence that on a fractal, momentum space can have a different dimension than position space. Our analysis also provides a natural definition of the vacuum (Casimir) energy of a fractal. We suggest ways that these unusual properties might be probed experimentally.

Separation of variables in the generalized 4th Appelrot class. II. Real solutions

We continue the analytical solution of the integrable system with two degrees of freedom arising as the generalization of the 4th Appelrot class of motions of the Kowalevski top for the case of two constant force fields [Kharlamov, RCD, vol. 10, no. 4]. The separated variables found in [Kharlamov, RCD, vol. 12, no. 3] are complex in the most part of the integral constants plane. Here we present the real separating variables and obtain the algebraic expressions for the initial Euler-Poisson variables. The finite algorithm of establishing the topology of regular integral manifolds is described. The article straightforwardly refers to some formulas from [Kharlamov, RCD, vol. 12, no. 3].

Basins of attraction of invariant regular manifolds

In this work, we present a new approach for the determination of the stability properties of regular invariant surfaces in the phase space of a system of ordinary differential equations. This approach also yields an estimation for basin of attraction of the invariant surface. The stability problem is reduced to the study of a system of non-autonomous linear differential equations and the use of the Wasewski’s inequalities. We illustrate our method for a system of equations describing the generalized mass action (GMA) model for biochemical reactions.

ON SECTIONALLY-ELLIPTICAL CONICAL SPACE–TIMES

We consider sectionally-elliptical conical space–times and compute the energy–momentum tensor of the source generating such configurations. It turns out that the problem can be reduced to the calculation of the Gaussian curvature of the bidimensional elliptical cone defined by the foliation t = const. , z = const. We then employ the method of smoothing the conical singularity by taking a sequence of regular bidimensional manifolds in order to compute the curvature. Finally, we conclude that the gravitational effects produced by sectionally-elliptical and sectionally-circular conical space–times are completely equivalent.

On traces for functional spaces related to Maxwell's equations Part II: Hodge decompositions on the boundary of Lipschitz polyhedra and applications

Hedge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L-2 tangential fields and then the attention is focused on some particular Sobolev spaces of order - 1/2. In order to reach this goal, it is required to properly define the first order differential operators and to investigate their properties. When the manifold Gamma is the boundary of a polyhedron Omega, these spaces are important in the analysis of tangential trace mappings for vector fields in H(curl, Omega) on the whole boundary or on a part of it. By means of,these Hedge decompositions, one can then provide a complete characterization of these trace mappings: general extension theorems, from the boundary, or from a part of it, to the inside; definition of suitable dualities and validity of integration by parts formulae. Copyright (C) 2001 John Wiley & Sons, Ltd.

Harmonic maps and $s$-regular manifolds

Harmonic maps and $s$-regular manifolds

Theory of Bessel potentials. III: Potentials on regular manifolds

Abstract : The present part of the 'Theory of Bessel Potentials' contains Chapter 4 dealing with potentials on regular Riemannian manifolds. (Author)

Focal sets of regular manifolds $M_{n-1}$ in $E_n$

Focal sets of regular manifolds $M_{n-1}$ in $E_n$

Focal sets of regular manifolds $M_{n-1}$ in $E_n$

Focal sets of regular manifolds $M_{n-1}$ in $E_n$