Spinor field with induced nonlinearity in Bianchi VI cosmology: Exact and numerical solutions

A self-consistent system of interacting spinor and scalar fields in Bianchi type VI (B-VI) cosmology is studied in the presence of a cosmological constant. Exact solutions to the spinor, scalar and gravitational field equations are obtained for some special choice of the spinor field nonlinearity. It is shown that introduction of Λ > 0, which is often used to model dark energy, results in a rapid growth of the universe, while Λ < 0 gives rise to an oscillatory or non-periodic mode of expansion. If the metric functions a and b are taken to be inverse to each other (ab = 1), we have a singularity-free universe independently of the sign of Λ.     

On the div-curl lemma in a Galerkin setting

Given a sequence of Galerkin spaces X _h of square-integrable vector fields, we state necessary and sufficient conditions on X _h under which it is true that for any two sequences of vector fields u _h ,u _h ′∈X _h converging weakly in L² and such that u _h is discrete divergence free and curl u _h ′ is precompact in H⁻¹, the scalar product u _h ⋅u′ _h converges in the sense of distributions to the right limit. The conditions are related to super-approximation and discrete compactness results for mixed finite elements, and are satisfied for Nédélec’s edge elements. We also provide examples of sequences of discrete divergence free edge element vector fields converging weakly to 0 in L² but whose divergence is not precompact in H _loc ⁻¹.      

On black brane solutions and their fluid analogues

We review spherically symmetric solutions with a horizon in two models: (i) with scalar fields and fields of forms, and (ii) with a multi-component anisotropic fluid. The metrics of the solutions are defined on a manifold that contains a product of n − 1 Ricci-flat “internal” spaces. The solutions are governed by functions H _s obeying nonlinear differential equations with certain boundary conditions. Simulation of black-brane solutions is considered, and the Hawking temperature is calculated. For the fluid solution, the post-Newtonian parameters β and Γ corresponding to the 4-dimensional section of the metric are found.     

Interacting spinor and scalar fields in a bianchi type-I universe filled with viscous fluid: Exact and numerical solutions

We consider a self-consistent system of spinor and scalar fields in a Bianchi type I gravitational field filled with a viscous fluid in the presence of a cosmological constant. Exact solutions to the set of spinor, scalar and gravitational field equations are obtained in terms of v, the volume scale of Bianchi-I universe. A set of equations for v and ε, where ε is the energy of the viscous fluid, is deduced. Some special cases allowing exact solutions are thoroughly studied.     

On the billiard approach in multidimensional cosmological models

A short overview of the billiard approach for cosmological-type models with n Einstein factor spaces is presented. We start with the billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin. Then we consider cosmological models with a multicomponent perfect-fluid and with composite branes. The second case describes cosmological and spherically symmetric configurations in a theory with scalar fields and fields of forms. The conditions for appearance of asymptotic Kasnerlike and oscillating behaviors in the limits τ → +0 and τ → +∞ (where τ is the synchronous variable) are formulated (e.g., in terms of inequalities for Kasner parameters). Examples of billiards related to the hyperbolic Kac-Moody algebras E ₁₀, AE ₃ and A _1,II are given. Plenary talk given at the International Conference RUSGRAV-13, June 23–28, 2008, PFUR, Moscow.     

On particle creation and renormalization in a cosmological model with a Big Rip

An exact solution is obtained for a massive scalar field conformally coupled with the curvature in a cosmological model with the scale factor a(t) = a ₀/|t|, corresponding to background matter with the equation of state p = −5ε/3. An expression for the number density of created particles is obtained, and its behavior is studied as the model approached the instant of a Big Rip. Renormalization of the energy-momentum tensor is considered, and it is shown that back reaction of the quantum effects of a conformally coupled scalar field on the space-time metric can be neglected if the field mass is much smaller than the Planck mass and if the time left to the Big Rip is greater than the Planck time.     

On multidimensional cosmological solutions with scalar fields and 2-forms corresponding to rank-3 lie algebras: Acceleration and small variation of <Emphasis Type="Italic">G</Emphasis>

In a simple multidimensional model we study the possibility of accelerated expansion of a 3-dimensional subspace combined with variation of the effective 4-dimensional constant of gravity within experimental constraints. Multidimensional cosmological solutions with m 2-form fields and l scalar fields are presented. Solutions corresponding to rank-3 Lie algebras are singled out and discussed. Each of the solutions contains two factor spaces: the one-dimensional space M ₁ and the Ricci-flat space M ₂. A 3-dimensional subspace of M ₂ is interpreted as our space. We show that, if at least one of the scalar fields is of phantom nature, there exists a time interval where accelerated expansion of our 3D space is compatible with a small enough variation of the effective gravitational constant G(τ) (τ is the cosmological time). This interval contains τ ₀ at which G(gt) has a minimum. Special solutions with three phantom scalar fields are analyzed. It is shown that in the vicinity of τ ₀ the time variation of G(τ) decreases in the sequence of Lie algebras A ₃, C ₃ and B ₃ in the family of solutions with asymptotic power-law behavior of the scale-factors as τ → ∞. Exact solutions with asymptotically exponential accelerated expansion of 3D space are also considered.     

Black brane solutions related to non-singular Kac-Moody algebras

A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M ₀ × M ₁ × … × M _n , where M _i are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g., solutions with harmonic functions and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g., hyperbolic ones) are considered. Some examples of black brane solutions are presented, e.g., those corresponding to hyperbolic KM algebras: H ₂(q, q) (q > 2), HA ₂⁽¹⁾ = A ₂⁺⁺ and to the Lorentzian KM algebra P ₁₀.     

Numerical solution of parabolic problems by the generalized Crank-Nicolson scheme

A Galerkin procedure is used to obtain a semi-discretization for parabolic equations such as the heat equationu _t =t _xx . The time variable being left continuous, the higher order approximation thus obtained for the space variable is then matched by a higher order discretization of the system of ordinary differential equations that results. Specifically we choose the Padé (2,2), and show how complex factorization it can be practically used. Moreover we prove that the operation count is 0 (h ⁻²) as compared to 0(h ⁻³) with the classical Crank-Nicolson. Numerical calculations are available. This work was supported by the Office of Naval Research, and the Lebanese Council for Scientific Research.     

Quantizing gravity using physical states of a superstring

A symmetric zero-mass tensor of rank two is constructed using the superstring modes of excitation, which satisfies the physical state constraints of a superstring. These states have a one to one correspondence with the quantized field operators and are shown to be the absorption and emission quanta of the Minkowski space Lorentz tensor, using the quantum field theory method of quantization. The principle of equivalence makes the tensor identical to the metric tensor at any arbitrary space-time point. The propagator for the quantized field is deduced. The gravitational interaction is switched on by going over from ordinary derivatives to co-derivatives. The Riemann-Christoffel affine connections are calculated, and the weak field Ricci tensor R _μν ⁰ is shown to vanish. The interaction part R _μν ^int is found, and the exact R _μν of the theory of gravity is expressed in terms of the quantized metric. The quantum-mechanical self-energy of the gravitational field in vacuum is shown to vanish. By the use of a projection operator, it is shown that gravitons are quanta of the general relativity field which gives the Einstein equation G _μν = 0. It is suggested that quantum gravity may be renormalizable by the use of the massless ground state of this superstring theory for general relativity, and a tachyonic vacuum creates and annihilates quanta of quantized gravitational field.     

On the Topology and Geometry of Spaces of Affine Shapes

We define the space of affine shapes of k points in R ⁿ to be the topological quotient of (R ⁿ ) ^k modulo the natural action of the affine group of R ⁿ . These spaces arise naturally in many image-processing applications, and despite having poor separation properties, have some topological and geometric properties reminiscent of the more familiar Procrustes shape spaces Σ _n ^k in which one identifies configurations related by an orientation-preserving Euclidean similarity transformation. We examine the topology of the connected, non-Hausdorff spaces Sh _n ^k in detail. Each Sh _n ^k is a disjoint union of naturally ordered strata, each of which is homeomorphic in the relative topology to a Grassmannian, and we show how the strata are attached to each other. The top stratum carries a natural Riemannian metric, which we compute explicitly for k>n, expressing the metric purely in terms of “pre-shape” data, i.e. configurations of k points in R ⁿ .     

Finite-difference method for computation of 3-D gas dynamics equations with artificial viscosity

A new numerical method for the solution of gas dynamics problems for three-dimensional (3D) systems in Eulerian variables is presented in the paper. The proposed method uses the approximation O(τ² + h _x ² + h _y ² + h _z ²) in the areas of the solution’s smoothness and beyond the compression waves; τ is the time step; and h _x , h _y , and h _z are space variable steps. In the proposed difference scheme, in addition to Lax-Wendroff corrections, artificial viscosity μ that monotonizes the scheme is introduced. The viscosity is obtained from the conditions of the maximum principle. The results of the computation of the 3D test problem for the Euler equation are presented.     

A desired state can not be found with certainty for Grover��s algorithm in a possible three-dimensional complex subspace

Using an accurate method, we prove that no matter what the initial superposition may be, neither a superposition of desired states nor a unique desired state can be found with certainty in a possible three-dimensional complex subspace, provided that the deflection angle Φ is not exactly equal to zero. By this method, we derive such a result that, if N is sufficiently large (where N denotes the total number of the desired and undesired states in an unsorted database), then corresponding to the case of identical rotation angles, the maximum success probability of finding a unique desired state is approximately equal to cos² Φ for any given F ? [0,p/2){\Phi\in\left[0,\pi/2\right)}.     

Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ² u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v _h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u _h , which is an approximation of u. A direct approximation v _h of v can be obtained from the approximation u _h of u. Using piecewise polynomials of degree p≥3, a priori error estimates of u−u _h in the broken H ¹ norm as well as in L ² norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v−v _h in L ² norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results. Supported by DST-DAAD (PPP-05) project.     

Exact Algorithms for the Bottleneck Steiner Tree Problem

Given n points, called terminals, in the plane ℝ² and a positive integer k, the bottleneck Steiner tree problem is to find k Steiner points from ℝ² and a spanning tree on the n+k points that minimizes its longest edge length. Edge length is measured by an underlying distance function on ℝ², usually, the Euclidean or the L ₁ metric. This problem is known to be NP-hard. In this paper, we study this problem in the L _p metric for any 1≤p≤∞, and aim to find an exact algorithm which is efficient for small fixed k. We present the first fixed-parameter tractable algorithm running in f(k)⋅nlog ² n time for the L ₁ and the L _∞ metrics, and the first exact algorithm for the L _p metric for any fixed rational p with 1<p<∞ whose time complexity is f(k)⋅(n ^k +nlog n), where f(k) is a function dependent only on k. Note that prior to this paper there was no known exact algorithm even for the L ₂ metric.     

Electric S-brane solutions corresponding to rank-2 Lie algebras: Acceleration and small variation of <Emphasis Type="Italic">G</Emphasis>

Electric S-brane solutions with two non-composite electric branes and a set of l scalar fields are considered. The intersection rules for branes correspond to Lie algebras A ₂, C ₂ and G ₂. The solutions contain five factor spaces. One of them, M ₀, is interpreted as our 3-dimensional space. It is shown that there exists a time interval where accelerated expansion of our 3-dimensional space is compatible with a small enough variation of the effective gravitational constant G(τ). This interval contains τ ₀, a point of minimum of the function G(τ). A special solution with two phantom scalar fields is analyzed, and it is shown that, in the vicinity of the point τ ₀, the time variation of G(τ) (calculated in the linear approximation) decreases in the sequence of Lie algebras A ₂, C ₂ and G ₂.     

Polarimetric scattering and transmitting of the Stokes vector from a layer of chiral small spheroids

To measure fully polarimetric scattering from random chiral small spheroids, the 2×2 dimensional (2×2-D) complex scattering amplitude matrix of randomly oriented, chiral small spheroid is derived. Polarimetric scattering from a bounded layer of non-uniformly oriented, chirally-active small spheroids in the Mueller matrix solution is obtained. Co-polarized and cross-polarized backscattering and polarization degree for any polarized incidence (χ, ψ) are numerically calculated. Transmitting of coherent Stokes parameters through the layer are also discussed. Either non-uniform orientation or chi-rality can yield non-diagonal extinction matrix ke and full eigenmatrix E due to the fact of forward depolarized-scattering functions , ≠ 0. Comparisons of fully polarimetric scattering from the chiral and achiral particulate media demonstrate the chirality effect on wave scattering and transmitting.     

Redundant modeling in permutation weighted constraint satisfaction problems

In classical constraint satisfaction, redundant modeling has been shown effective in increasing constraint propagation and reducing search space for many problem instances. In this paper, we investigate, for the first time, how to benefit the same from redundant modeling in weighted constraint satisfaction problems (WCSPs), a common soft constraint framework for modeling optimization and over-constrained problems. Our work focuses on a popular and special class of problems, namely, permutation problems. First, we show how to automatically generate a redundant permutation WCSP model from an existing permutation WCSP using generalized model induction. We then uncover why naively combining mutually redundant permutation WCSPs by posting channeling constraints as hard constraints and relying on the standard node consistency (NC*) and arc consistency (AC*) algorithms would miss pruning opportunities, which are available even in a single model. Based on these observations, we suggest two approaches to handle the combined WCSP models. In our first approach, we propose m\text -NC_{\text c}^*m\text {-NC}_{\text c}^* and m\text -AC_{\text c}^*m\text {-AC}_{\text c}^* and their associated algorithms for effectively enforcing node and arc consistencies in a combined model with m sub-models. The two notions are strictly stronger than NC* and AC* respectively. While the first approach specifically refines NC* and AC* so as to apply to combined models, in our second approach, we propose a parameterized local consistency LB(m,Φ). The consistency can be instantiated with any local consistency Φ for single models and applied to a combined model with m sub-models. We also provide a simple algorithm to enforce LB(m,Φ). With the two suggested approaches, we demonstrate their applicabilities on several permutation problems in the experiments. Prototype implementations of our proposed algorithms confirm that applying 2\text -NC_{\text c}^*,  2\text -AC_{\text c}^*2\text {-NC}_{\text c}^*,\;2\text {-AC}_{\text c}^*, and LB(2,Φ) on combined models allow far more constraint propagation than applying the state-of-the-art AC*, FDAC*, and EDAC* algorithms on single models of hard benchmark problems.