Exterior Geometry Research Articles

Synthesis of sterically encumbered organoselenium species and their selectivity towards Hg(II) ions

The synthesis and characterization data for an intriguing family of sterically encumbered tetrasubstituted organoselenium species are described. The open exterior geometry and flexibility of the multiple soft selenium-containing donor arms enhances conformational organization thus allowing the molecule to reveal a propensity for trapping Hg(II) ions. Preliminary studies on the ion-sensing properties of these molecules revealed that they can act as selective ionophores for Hg(II) ions.

Stable dark energy stars

The gravastar picture is an alternative model to the concept of a black hole, where there is an effective phase transition at or near where the event horizon is expected to form, and the interior is replaced by a de Sitter condensate. In this work a generalization of the gravastar picture is explored by considering matching of an interior solution governed by the dark energy equation of state, ω ≡ p/ρ < −1/3, to an exterior Schwarzschild vacuum solution at a junction interface. The motivation for implementing this generalization arises from the fact that recent observations have confirmed an accelerated cosmic expansion, for which dark energy is a possible candidate. Several relativistic dark energy stellar configurations are analysed by imposing specific choices for the mass function. The first case considered is that of a constant energy density, and the second choice that of a monotonic decreasing energy density in the star's interior. The dynamical stability of the transition layer of these dark energy stars to linearized spherically symmetric radial perturbations about static equilibrium solutions is also explored. It is found that large stability regions exist that are sufficiently close to where the event horizon is expected to form, so that it would be difficult to distinguish the exterior geometry of the dark energy stars, analysed in this work, from an astrophysical black hole.

Polymeric Membrane Ion-Selective Electrodes Based on Molecular Asterisk Ionophores

Ion-selective electrodes (ISEs) have been developed that incorporate a novel supramolecular class of ionophores called molecular asterisks. These ionophores are constructed with “arms” of repeating phenylene sulfide units that radiate outward from a core of either benzene or coronene. The flexibility of the arms, as well as the open exterior geometry and multiple soft Lewis base functionalities make these molecules potential candidates as ionophores for ISEs particularly for soft Lewis acids like transition metals. Studies with molecular asterisk-based ISEs show that these ionophores display a high selectivity relative to ion-exchanging ionophores toward Ag+ over a number of other cations. According to theoretical prediction, these ISEs demonstrate a super-Nernstian region of response toward silver from 10−6 to 10−5 M with a Nernstian response above 10−5 M, when primary ion is absent from the internal filling solution. Additionally, it was determined that both the nature of the core entity and the length (or generation) of the arms play a role in governing selectivity of these ionophores.

The enhançon and the consistency of excision

The enhancon mechanism removes a family of time-like singularities from certain supergravity spacetimes by forming a shell of branes on which the exterior geometry terminates. The problematic interior geometry is replaced by a new spacetime, which in the prototype extremal case is simply flat. We show that this excision process, made inevitable by stringy phenomena such as enhanced gauge symmetry and the vanishing of certain D-branes' tension at the shell, is also consistent at the purely gravitational level. The source introduced at the excision surface between the interior and exterior geometries behaves exactly as a shell of wrapped D6-branes, and in particular, the tension vanishes at precisely the enhancon radius. These observations can be generalised, and we present the case for non-extremal generalisations of the geometry, showing that the procedure allows for the possibility that the interior geometry contains an horizon. Further knowledge of the dynamics of the enhancon shell itself is needed to determine the precise position of the horizon, and to uncover a complete physical interpretation of the solutions.

Two questions on the exterior geometry of the plücker embeddings of the Grassmann manifolds

The Plücker model of the Grassmann manifold G +p,p+q is considered. The structure of intersections of G +p,p+q with tangent spaces of G +p,p+q regarded as subspaces of the ambient exterior algebra is described. An explicit formula for the second fundamental form of G +2,4 as of a hypersurface in the five-dimensional sphere is given. The level sets of the normal curvature functions for this hypersurface are studied. Bibliography: 3 titles.

A Proposal Of The FM-SN Graphs VisualisationApplied For A Crack Propagation Process

The SN characteristic which is applied in the fatigue investigations Wohler curve determines the number of cycles necessary for the entire destruction of an investigated element as a function of the load amplitude. The estimation of a crack propagation during a fatigue process can be described by the Paris equation and presented in a form of co-ordinates: the crack growth rate as a function of a stress intensity factor range depending on an exterior load and the exterior geometry of the investigated element, as well as on the geometry of a crack* In this paper the transformation method of the Paris curve into the Wohler curve, in the range of line fracture mechanics, FM, is presented. The obtained FM-SN curves make it possible to compare the fatigue crack propagation results with the SN curves. In the given example, concerning the fatigue crack calculations of the container hatch corners, the FM-SN curves were presented in comparison with the SN standard ones.

(2+1)-dimensional stars

We investigate, in the framework of (2+1)-dimensional gravity, stationary rotationally symmetric gravitational sources of the perfect fluid type, embedded in a space of an arbitrary cosmological constant. We show that the matching conditions between the interior and exterior geometries imply restrictions on the physical parameters of the solutions. In particular, imposing finite sources and the absence of closed timelike curves privileges negative values of the cosmological constant, yielding exterior vacuum geometries of rotating black hole type. In the special case of static sources, we prove the complete integrability of the field equations and show that the sources' masses are bounded from above and, for a vanishing cosmological constant, generally equal to 1. We also discuss and illustrate the stationary configurations by explicitly solving the field equations for constant mass-energy densities. If the pressure vanishes, we recover as interior geometries G\"odel-like metrics defined on causally well behaved domains, but with unphysical values of the mass to angular momentum ratio. The introduction of pressure in the sources cures the latter problem and leads to physically more relevant models.

EMBEDDING AS A SUBSTITUTE FOR THE KRUSKAL MAXIMAL EXTENSION

We consider the Minkowski embedding space of the Reissner–Nordstrom solution to show that the embedding space plays the same role as the Kruskal maximal extension. We point out that a static observer in Reissner–Nordstrom exterior geometry is a particular kind of uniformly accelerated observer in the six-dimensional Minkowski embedding space and therefore he experiences a thermal bath of Rindler particles. However, since the static observer is restricted to a 4-D manifold, his horizon, unlike accelerated observer's horizon, is finite and so is the entropy.

DEVEROPMENT OF MULTI-DEGREE-OE-FREEDOM AEROELASTIC MODEL OF A HIGH-RISE BUILDING

This paper describes new ideas and some improvements in the development of a multi-degree-of freedom aeroelastic model of a high-rise building. With this model, the mass, the eccentric distance, the frequency ratio of a torsional mode vibration to sway mode vibration, and the structural damping can be changed easily. A structural skin which reproduces the exterior geometry is discontinuous with slits separating the various zones of the model. The effect on the wind responses with these slits is in farther investigation.

A dynamical symmetry breaking model in Weyl space

The dynamical process following the breaking of Weyl geometry to Riemannian geometry is considered by studying the motion of de Sitter bubbles in a Weyl vacuum. The bubbles are given in terms of an exact, spherically symmetric thin shell solution to the Einstein equations in a Weyl–Dirac theory with a time-dependent scalar field of the form β=f(t)/r. The dynamical solutions obtained lead to a number of possible applications. An important feature of the thin shell model is the manner in which β provides a connection between the interior and exterior geometries since information about the exterior geometry is contained in the boundary conditions for β.

Analysis and adaptive modeling of highly heterogeneous elastic structures

In this paper, we develop a theory and methodology for obtaining approximate solutions to boundary value problems describing the deformation of highly heterogeneous linearly-elastic structures. The method, which represents a significant departure from traditional homogenization methods, provides a systematic and rigorous approach towards resolving the effects of microstructure of different scales on the macroscopic response of complex heterogeneous structures. An early variant of this method was first introduced in [10]. The method, referred to as HDPM (Homogenized Dirichlet Projection Method), proceeds by first solving an auxiliary homogenized boundary value problem, describing the deformation of a structure with the same exterior geometry, but with a selected set of uniform material properties. The adequacy of this homogenized solution is then determined using an a posteriori error estimate that provides a measure of the error of the homogenized solution in subdomains of the structure compared to the solution of the fine-scale heterogeneous problem, without specific knowledge of this fine-scale solution. In those subdomains where the homogenized solution is deemed acceptable, it is retained. However, in subdomains where the homogenized solution is inadequate, as is determined when the estimated error exceeds a preset tolerance, a local boundary value problem is constructed by projecting the homogenized displacements onto a partition designed to isolate these subdomains. These local boundary value problems are then solved in those subdomains where the homogenized solution is inaccurate using the exact microstructure with the approximate local boundary conditions. A posteriori error estimates are made to ascertain the quality of the resulting solution. If the quality of the solution remains inadequate, above a preset error tolerance, a two stage adaptive procedure is implemented. Stage-I (‘material adaptivity’) corresponds to modifying the homogenized structure's material properties. If, after Stage I, the solution quality is still inadequate, the subdomains of local solution are enlarged, thereby modeling in greater detail the actual microstructure, and the local solution process is repeated (Stage-II, ‘subdomain unrefinement’). The main feature of this method is that only in subdomains where the error in the usual homogenized solution is above a preset tolerance is the microstructure taken into account. The cost of this method is shown to be orders of magnitude cheaper than direct large-scale computational simulations of micromechanical events. The results of several numerical experiments are provided to demonstrate the method and verify theoretical estimates.

The energy localization problem and the renormalized vacuum energy in static Robertson-Walker universes

We calculate the renormalized quantum vacuum energy inside a spherical boundary for the massless conformal scalar field in curved background Robertson-Walker geometry. We use the mode sum method with an exponential cutoff. In our calculations we do not make assumptions about the exterior geometry or the global topology of the universe.

Breaking Weyl invariance in the interior of a bubble.

The basis on which Weyl's unified theory of gravitation and electromagnetism was rejected is reconsidered from a new perspective. It is argued that while Weyl's theory, as indeed any classical theory, is incapable of explaining atomic phenomena, this does not nullify the geometric interpretation of the exterior electromagnetic field; it simply reflects the fact that some form of quantization is needed to account for atomic standards of length. In support of this argument the Gauss-Mainardi-Codazzi formalism is employed to demonstrate that it is possible to construct a bubble in Weyl space where the exterior geometry is conformally invariant and the electromagnetic field can be given a geometric interpretation, while at the same time a standard of length can be introduced into the theory by breaking the conformal invariance in the interior of the bubble.

General-relativistic model of a spinning cosmic string.

We investigate the infinite, straight, rotating cosmic string within the framework of Einstein's general theory of relativity. A class of exact interior solutions is derived for which the source satisfies the weak and the dominant energy conditions. The interior metric is matched smoothly to the exterior vacuum. A subclass of these solutions has closed timelike curves both in the interior and the exterior geometries.

Surfaces with constant exterior geometry of negative curvature

Surfaces with constant exterior geometry of negative curvature

Discontinuity cylinder model of gravitating U(1) cosmic strings.

We introduce a model for an infinite-length, straight U(1) cosmic string as a cylindrical, singular shell enclosing a region of false vacuum. The properties of the geometry for the region exterior to the string are fully determined under the assumption that changes in the scalar and gauge field variables occur only at the cylindrical shell. This is consistent with a limiting form of the scalar potential V(\ensuremath{\varphi}) where a minimum at \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}=0 is separated by a large barrier from a global minimum at \ensuremath{\Vert}\ensuremath{\varphi}\ensuremath{\Vert}=\ensuremath{\eta}\ensuremath{\ne}0. The introduction of an approximately singular ``surface'' for the string allows the definition of a \ensuremath{\delta}-function stress-energy density that characterizes discontinuities in the fields. We show consistency of the model with the full coupled equations for the metric, and the scalar and gauge fields in curved space-time. It is found that for this model, in the absence of an ``external'' cosmological constant, the exterior geometry of the string approaches Minkowski space-time with a deficit angle, and it is shown that in the limit when the string becomes a line source, i.e., its radius vanishes, the deficit angle reduces to the well-known expression \ensuremath{\Delta}\ensuremath{\theta}=8\ensuremath{\pi}\ensuremath{\mu}, with \ensuremath{\mu} the proper mass per unit length of the string.

Gravitational collapse in free fall of a slowly rotating star

The Oppenheimer-Snyder model of a spherically symmetric collapse in free fall is generalized to the case in which the star possesses a small rotation. The exterior geometry is chosen to be the Kerr metric in synchronous coordinates, discarding terms of the order (a/r)2. The interior geometry is constructed by adding to the exact metric of the nonrotating case an off-diagonal first-order term in the parametera. This term is determined in part by requiring the validity of the junction conditions at the star's surface and, also, by demanding the conserved angular momentum of the source be equal toMa, in agreement with the value measured by a distant observer. The resulting stress-energy tensor describes a homogeneous, pressureless, ideal fluid (dust) nonuniformly rotating relative to the synchronous frame, which is no longer comoving with the stellar matter. The dynamics of collapse is qualitatively the same as in the spherically symmetric case. Again the star's surface crosses the event horizon when the mass density is finite everywhere, and space-time has not developed any singularity as viewed by freely falling observers at rest in the synchronous frame.

Confining the scalar field of the Kaluza–Klein monopole

A discussion is given of the Kaluza–Klein monopole and the inherent problem associated with its scalar field. A model that achieves a confinement of this scalar field is presented, whereby a thin boundary layer is imposed on the monopole geometry allowing an exterior geometry with a constant scalar field to be attached. The Gauss–Codazzi formalism then puts strong constraints on the effective stress energy of the boundary.

Three-dimensional cosmological gravity: Dynamics of constant curvature

In three space-time dimensions, Einstein's equations with cosmological constant Λ imply that the curvature is constant outside sources. When particles are present, they alter the global properties of the exterior geometry. In the De Sitter case, space is a two-sphere and static many-body solutions are quite different from their Λ = 0 counterparts; in particular particles lie at antipodal points, with the great-circle wedge between them excised. These configurations are analyzed in terms of the general static solution to the exterior field equations.

Geometric constraints on nonsingular, momentarily static, axisymmetric systems in general relativity

This paper attempts to examine the relationship between system size and gravitational collapse for the case of axial symmetry. The approach here is to construct noncollapsing systems, with momentarily static matter interiors and static vacuum exteriors, and to find limitations on the validity of the construction. Specifically, the exteriors are static, axisymmetric, asymptotically flat, vacuum geometries, described by Weyl solutions of the Einstein field equations. These solutions have singular sources (naked singularities, except for the Schwarzschild solution); here, regions of the Weyl solutions containing the singularities are replaced by momentarily static material bodies. These are described by axisymmetric solutions of Brill's time-symmetric initial-value equation with non-negative energy density, joining smoothly to the Weyl geometries at the bodies' boundaries. The consistency requirements of such a construction limit the choice of surfaces in the exterior geometry suitable as matter/vacuum boundaries; general constraints on the boundary location and geometry are derived here. For the explicit examples of the GAMMA metric and the Bach-Weyl ring metric as exteriors, these constraints forbid the boundary surface to be arbitrarily near the Weyl singularity. The ''hoop conjecture'' demands, roughly, that the largest circumference of the boundary surface of such a noncollapsing system always exceed a limit ofmore » the order of the system's mass. The specific examples studied here are all consistent with the hoop conjecture, but they show that the boundary constraints derived in this paper are not in general related to boundary surface size and thus that these constraints do not embody the hoop conjecture.« less