Geometry And Topology Research Articles

Legendre curves on 3-dimensional $C_{12}$-Manifolds

Legendre curves play a very important and special role in geometry and topology of almost contact manifolds.There are certain results known for Legendre curves in 3-dimensional normal almost contact manifolds. The aim of this paper is to study Legendre curves of three-dimensional $C_{12}$-manifolds which are non-normal almost contact manifolds and classifying all biharmonic Legendre curves in these manifolds.

Introduction

Introduction

Quest for the Gyroid Labyrinth: Geometry and Topology in Soft Matter

Quest for the Gyroid Labyrinth: Geometry and Topology in Soft Matter

The Effect of Image Resolution on the Geometry and Topological Characteristics of 3-D Reconstructed Images of Reservoir Rock Samples

The effect of image resolution on the measured geometry and topological characteristics of network models extracted from 3-D micro-computer tomography images has been investigated. The study was conducted by extracting geologically realistic networks from images of two rock samples, imaged at different resolutions. The rock samples involved were a Castlegate Sandstone and a Carbonate-28 reservoir rock. Two-dimensional images of these rocks were obtained at a magnification of ×50. The carbonate sample was studied at two different resolutions of 0.133 microns and 1.33 microns, while the sandstone was studied at 5.60 microns. Three-dimensional images of these 2-D images were obtained via image reconstruction, to generate the pore architecture models (PAMs) from which networks models of the imaged rocks were extracted with the aid of Pore Analysis software Tools (PATs). The measured geometry and topology (GT) properties included Coordination Number, Pore Shape Factor, Pore Size Distribution, and Pore Connectivity. The results showed that the measured geometry-topology (GT) characteristics of a network model depend greatly on the image resolution used for the model. Depending on the micro-structure of the reservoir rock, a minimum image resolution is necessary to properly define the geometrical and topological characteristics of the given porous medium.

Efficient extraction of networks from three‐dimensional porous media

Fluid flow through porous media, and the thermal, electrical, and acoustic properties of these materials, is largely controlled by the geometry and topology (GT) of the pore system, which can be considered as a network. Network extraction techniques have been applied in many research fields, including shape representation, pattern recognition, and artificial intelligence. However, the set of algorithms presented here significantly improves the efficiency of common thinning algorithms by introducing a sufficiency condition based on the idea of a simple set. This paper describes an efficient and accurate algorithm for extracting the geometrical/topological network that represents the pore structure of a porous medium, referred to as the GT network. The accurate medial axis and the specific GT description of the network are achieved by applying symmetrical and interval strategies during the erosion step in the image processing. The GT network extraction algorithm presented here involves a number of steps, including (1) calculation of the three‐dimensional Euclidean distance map; (2) clustering of voxels; (3) extraction of the network of the pore space; (4) partitioning of the pore space; and (5) computation of shape factors. The focus of this paper is mainly on the thinning method that underpins points 1–3. The paper is primarily a method description, but we illustrate the functionality of the technique by extracting a pore scale GT network from micro–computer tomography images of three sandstones.

Geometry and topology

Geometry and topology

Non-holonomy, critical manifolds and stability in constrained Hamiltonian systems

We approach the analysis of dynamical and geometrical properties of nonholonomic mechanical systems from the discussion of a more general class of auxiliary constrained Hamiltonian systems. The latter is constructed in a manner that it comprises the mechanical system as a dynamical subsystem, which is confined to an invariant manifold. In certain aspects, the embedding system can be more easily analyzed than the mechanical system. We discuss the geometry and topology of the critical set of either system in the generic case, and prove results closely related to the strong Morse-Bott, and Conley-Zehnder inequalities. Furthermore, we consider qualitative issues about the stability of motion in the vicinity of the critical set. Relations to sub-Riemannian geometry are pointed out, and possible implications of our results for engineering problems are sketched.

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press). - Volume 82 Issue 494

Geometry and topology of local minimal 2-trees

The aim of this paper is to get an effective restriction on the topologies of minimal 2-trees in terms of twisting numbers of the trees and the convexity levels number of the trees boundaries.

On a Neumann problem involving two critical Sobolev exponents

at one or more boundary points. By comparing it with the boundary conditionof the Yamabe Problem, ( ) appears as a natural condition in order to recovercompactness by breaking the conformal invariance.The proofs of the existence results of the above quoted papers rely on theminimization of a Sobolev–type quotient. By the geometrical assumption ( ) oneis able to overcome the lack of compactness due to the presence of the limitingexponent. Indeed, by means of energy estimates obtained by concentration ofsuitable test functions at some boundary points, one can prove that the infimumlies below the first energy level where the Palais–Smale condition fails. A similarperspective has been adopted by the authors in [14] where they solved problem(P) under the assumption that g is a linear function g(x;u)=l(x)uand ( ) holdsat one boundary point and (x) a given function. In [11], the same energy estimatewas obtained, using a more refined technique, when Ω is not conformally relatedto the unit ball (it has to be noticed that in the case of the ball one can exhibitthe explicit expression of the solution).However, this type of discussion can not be extended to the class of problems(P) when g has a superlinear rate of growth at infinity, even under assumption( ) on its linear part at zero, since, when one applies the blow–up technique, thecontribution of the superlinear term dominates that of the linear part. Further-more, when the opposite inequality holds in ( ) one can find examples where theminimization argument (or the application of the Mountain Pass lemma) fails.A natural question is then what is the real role of assumption ( ) (even in itsrelaxed form) in the solvability of (P).The purpose of the present work is to solve (P) under a condition weakerthan ( ) on the linearization l(x)ofgat zero; in addition, we allow a superlinear(but subcritical) rate of growth of g at infinity. As the reader will see, this studywill involve both topological and geometrical considerations. For this reason weshall restrict our analysis to the domain Ω having the simplest geometry andtopology: the unit ball B

Visual geometry and topology, by Anatolij Fomenko. Pp. 324. £55·50. 1994. ISBN 3-540-53361-3 (Springer-Verlag)

Visual geometry and topology, by Anatolij Fomenko. Pp. 324. £55·50. 1994. ISBN 3-540-53361-3 (Springer-Verlag)

Computers in geometry and topology, edited by M. C. Tangora. Pp 317. 1989. ISBN 0-8247-8031-0 (Marcel Dekker)

Computers in geometry and topology, edited by M. C. Tangora. Pp 317. 1989. ISBN 0-8247-8031-0 (Marcel Dekker)

Computers in Geometry and Topology.

Computers in Geometry and Topology.