On exact solutions for quantum particles with spin <Emphasis Type="Italic">S</Emphasis> = 0, 1/2, 1 and de Sitter event horizon

Exact wave solutions for particles with spin 0, 1/2 and 1 in the static coordinates of the de Sitter space–time model are examined in detail. Firstly, for scalar particle, two pairs of linearly independent solutions are specified explicitly: running and standing waves. A known algorithm for calculation of the reflection coefficient R_ej{R_{\epsilon j}} on the background of the de Sitter space–time model is analyzed. It is shown that the determination of R_ej{R_{\epsilon j}} requires an additional constrain on quantum numbers er/ (^h/_2p) c >> j{\epsilon \rho / \hbar c \gg j}, where ρ is a curvature radius. When taken into account of this condition, the R_ej{R_{\epsilon j}} vanishes identically. It is claimed that the calculation of the reflection coefficient R_ej{R_{\epsilon j}} is not required at all because there is no barrier in an effective potential curve on the background of the de Sitter space–time. The same conclusion holds for arbitrary particles with higher spins, it is demonstrated explicitly with the help of exact solutions for electromagnetic and Dirac fields.     

Spacelike helices in Minkowski 4-space $${E_1^4 }$$

In this paper, we give some characterizations for spacelike helices in Minkowski space–time E₁⁴{E_1^4}. We find the differential equations characterizing the spacelike helices and also give the integral characterizations for these curves in Minkowski space–time E₁⁴{E_1^4}.     

False vacuum decay in the de sitter space-time

In the example of the decay of a metastable scalar field state (the conformal vacuum of scalar particles over a false classical vacuum) in the background de Sitter metric, we propose a method to account for the initial quantum field state in the semiclassical calculation of the path integral in a curved space-time. Using this method, we justify the Coleman-De Luccia approach to calculating the decay probability. We interpret the Hawking-Moss instanton as a limit of constrained instantons. We find that the inverse process of the true vacuum going into a false one can occur in the de Sitter space and find the expression for the corresponding probability. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 3, pp. 451–472, September, 1999.     

Lightlike surfaces of spacelike curves in de Sitter 3-space

The Lorentzian space form with the positive curvature is called de Sitter space which is an important subject in the theory of relativity. In this paper we consider spacelike curves in de Sitter 3-space. We define the notion of lightlike surfaces of spacelike curves in de Sitter 3-space. We investigate the geometric meanings of the singularities of such surfaces. Work partially supported by Grant-in-Aid for formation of COE. ‘Mathematics of Nonlinear Structure via Singularities’     

On the maximal L p -regularity of parabolic mixed-order systems

We study maximal L _p -regularity for a class of pseudodifferential mixed-order systems on a space–time cylinder \mathbbRⁿ ×\mathbbR{\mathbb{R}^n \times \mathbb{R}} or X ×\mathbbR{X \times \mathbb{R}} , where X is a closed smooth manifold. To this end, we construct a calculus of Volterra pseudodifferential operators and characterize the parabolicity of a system by the invertibility of certain associated symbols. A parabolic system is shown to induce isomorphisms between suitable L _p -Sobolev spaces of Bessel potential or Besov type. If the cross section of the space–time cylinder is compact, the inverse of a parabolic system belongs to the calculus again. As applications, we discuss time-dependent Douglis–Nirenberg systems and a linear system arising in the study of the Stefan problem with Gibbs–Thomson correction.     

A Uniqueness Theorem for Degenerate Kerr–Newman Black Holes

We show that the domains of dependence of stationary, I ⁺-regular, analytic, electro-vacuum space–times with a connected, non-empty, rotating, degenerate event horizon arise from Kerr–Newman space–times.     

Submanifolds in de sitter space-time satisfying ΔH=λH

In [3] the author initiated the study of submanifolds whose mean curvature vectorH is an eigenvector of the Laplacian Δ and proved that such submanifolds are either biharmonic or of 1-type or of null 2-type. The classification of surfaces with ΔH=λH in a Euclidean 3-space was done by the author in 1988. Moreover, in [4] the author classified such submanifolds in hyperbolic spaces. In this article we study this problem for space-like submanifolds of the Minkowski space-timeE ₁ ^m when the submanifolds lie in a de Sitter space-time. As a result, we characterize and classify such submanifolds in de Sitter space-times.     

Singularities of Anti de Sitter torus Gauss maps

We study timelike surfaces in Anti de Sitter 3-space as an application of singularity theory. We define two mappings associated to a timelike surface which are called Anti de Sitter nullcone Gauss image and Anti de Sitter torus Gauss map. We also define a family of functions named Anti de Sitter null height function on the timelike surface. We use this family of functions as a basic tool to investigate the geometric meanings of singularities of the Anti de Sitter nullcone Gauss image and the Anti de Sitter torus Gauss map.     

Fermi Coordinates, Simultaneity, and Expanding Space in Robertson�CWalker Cosmologies

Explicit Fermi coordinates are given for geodesic observers comoving with the Hubble flow in expanding Robertson–Walker space–times, along with exact expressions for the metric tensors in Fermi coordinates. For the case of non-inflationary cosmologies, it is shown that Fermi coordinate charts are global, and space–time is foliated by space slices of constant Fermi (proper) time that have finite extent. A universal upper bound for the proper radius of any leaf of the foliation, i.e., for the proper radius of the spatial universe at any fixed time of the geodesic observer, is given. A general expression is derived for the geometrically defined Fermi relative velocity of a test particle (e.g., a galaxy cluster) comoving with the Hubble flow away from the observer. Least upper bounds of superluminal recessional Fermi velocities are given for space–times whose scale factors follow power laws, including matter-dominated and radiation-dominated cosmologies. Exact expressions for the proper radius of any leaf of the foliation for this same class of space–times are given. It is shown that the radii increase linearly with proper time of the observer moving with the Hubble flow. These results are applied to particular cosmological models.     

Spacelike surfaces in anti de Sitter four-space from a contact viewpoint

We define the notions of (S _t ¹ × S _s ²)-nullcone Legendrian Gauss maps and S ₊²-nullcone Lagrangian Gauss maps on spacelike surfaces in anti de Sitter 4-space. We investigate the relationships between singularities of these maps and geometric properties of surfaces as an application of the theory of Legendrian/Lagrangian singularities. By using S ₊²-nullcone Lagrangian Gauss maps, we define the notion of S ₊²-nullcone Gauss-Kronecker curvatures and show a Gauss-Bonnet type theorem as a global property. We also introduce the notion of horospherical Gauss maps which have geometric properties different from those of the above Gauss maps. As a consequence, we can say that anti de Sitter space has much richer geometric properties than the other space forms such as Euclidean space, hyperbolic space, Lorentz-Minkowski space and de Sitter space.     

Critical mass for a Patlak–Keller–Segel model with degenerate diffusion in higher dimensions

This paper is devoted to the analysis of non-negative solutions for a generalisation of the classical parabolic-elliptic Patlak–Keller–Segel system with d ≥ 3 and porous medium-like non-linear diffusion. Here, the non-linear diffusion is chosen in such a way that its scaling and the one of the Poisson term coincide. We exhibit that the qualitative behaviour of solutions is decided by the initial mass of the system. Actually, there is a sharp critical mass M _c such that if solutions exist globally in time, whereas there are blowing-up solutions otherwise. We also show the existence of self-similar solutions for . While characterising the possible infinite time blowing-up profile for M  =  M _c , we observe that the long time asymptotics are much more complicated than in the classical Patlak–Keller–Segel system in dimension two. This paper is under the Creative Commons licence Attribution-NonCommercial-ShareAlike 2.5.     

The lightlike flat geometry on spacelike submanifolds of codimension two in Minkowski space

We introduce the notion of the lightcone Gauss–Kronecker curvature for a spacelike submanifold of codimension two in Minkowski space, which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss–Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

The Problem of a Self-Gravitating Scalar Field with Positive Cosmological Constant

We study the Einstein-scalar field system with positive cosmological constant and spherically symmetric characteristic initial data given on a truncated null cone. We prove well-posedness, global existence and exponential decay in (Bondi) time, for small data. From this, it follows that initial data close enough to de Sitter data evolves to a causally geodesically complete spacetime (with boundary), which approaches a region of de Sitter asymptotically at an exponential rate; this is a non-linear stability result for de Sitter within the class under consideration, as well as a realization of the cosmic no-hair conjecture.     

Generalized coherent states for massive spinless fields in the de Sitter space

The de Sitter SO(4, 1) group conformal action is used to construct two different systems of generalized coherent states for the de Sitter group in the three-dimensional Euclidean space. The two systems of states have the physical sense of particles and antiparticles in the de Sitter space. The scalar product of two coherent states is a two-point function in the de Sitter space. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121. No. 2, pp. 258–263. November, 1999.     

Seasonal Space–Time Models for Climate Systems

A class of seasonal space–time models for general lattice systems is proposed. Covariance properties of spatial first-order models are studied. Estimation approaches in time series analysis are adopted and forecasting techniques using the seasonal space–time models are discussed. The models are applied to 516 consecutive fields of monthly averaged 500 mb geopotential heights over a 10 × 10 lattice in the extra-tropical northern hemisphere for the purpose of understanding the underlying statistical structure. It is found that space–time models with instantaneous spatial component give the best fit compared to other models in terms of maximizing the conditional likelihood function. The models are potentially useful for assessing the consistency of outputs from laboratory-based numerical models with field observations. Forecasting ability of the seasonal space–time models is also investigated. This revised version was published online in June 2006 with corrections to the Cover Date.     

Anti de Sitter horospherical flat timelike surfaces

We investigate a special timelike surfaces in Anti de Sitter 3-space.We call such a timelike surface an Anti de Sitter horospherical flat surface which belongs to a class of surfaces given by one parameter families of Anti de Sitter horocycle.We give a generic classification of singularities and study the geometric properties of such surfaces from the viewpoint of Legendrian singularity theory.     

On new geometrical concept of local quantum field

The key idea discussed in the paper is the hypothesis that the mass spectrum of elementary particles described by local quantum fields should be cut at some mass value M. The new universal parameter M called the “fundamental mass” is introduced in quantum field theory (QFT) in a pure geometric way; namely, in the framework of the Euclidean formulation of QFT we postulate that the 4-momentum space is the de Sitter space with radius M. It is of principal importance that the new version of QFT containing the fundamental mass M admits a local gauge invariant Lagrangian formulation and may serve as a basis for generalizing the Standard Model (SM) at high energies E ≥ M. Some correction terms to the SM Lagrangian, which may be compared in the future with LHC experimental data, are given.     

Applications of representation theory to wireless communications

In this paper, we demonstrate a new and original way to apply the representation theory of finite groups to the development of space–time codes for use in multiple antenna wireless communications. We include a summary of the relevant mathematical model for wireless communications on a multiple-antenna environment. We apply our construction in two examples and demonstrate that they outperform other designs in the literature.     

Ground states in non-relativistic quantum electrodynamics

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state – one that minimizes the energy and satisfies the Schr?dinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and the ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. Oblatum 21-IX-2000 & 8-IV-2001?Published online: 18 June 2001