Arbitrary Number Of Spatial Dimensions Research Articles

The Ricci decomposition of the inertia tensor for a rigid body in arbitrary spatial dimensions

The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric rank-2 tensors), and they are related by a rank-4 inertia tensor. Remarkably, this inertia tensor belongs to a well-studied class of algebraic curvature tensors that have the same index symmetries as the Riemann curvature tensor field used in general relativity. Any algebraic curvature tensor can be decomposed into irreducible representations of the orthogonal group via the Ricci decomposition. We calculate the Ricci decomposition of the inertia tensor for a rigid body in any number of dimensions, and we find that (unlike for the Riemann curvature tensor field) its traceless Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the traceless Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in n≥4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n \\ge 4$$\\end{document} dimensions.

The m-dimensional spatial Nyquist limit using the wave telescope for larger numbers of spacecraft

Abstract. Spacecraft constellations consisting of multiple satellites are becoming more and more interesting not only for commercial use but also for space science missions. The proposed and accepted scientific multi-satellite missions that will operate within Earth's magnetospheric environment, like HelioSwarm, require researchers to extend established methods for the analysis of multi-spacecraft data to more than four spacecraft. The wave telescope is one of those methods. It is used to detect waves and characterize turbulence from multi-point magnetic field data, by providing spectra in reciprocal position space. The wave telescope can be applied to an arbitrary number of spacecraft already. However, the exact limits of the detection for such cases are not known if the spacecraft, acting as sampling points, are irregularly spaced. We extend the wave telescope technique to an arbitrary number of spatial dimensions and show how the characteristic upper detection limit in k space imposed by aliasing, the spatial Nyquist limit, behaves for irregularly spaced sampling points. This is done by analyzing wave telescope k-space spectra obtained from synthetic plane wave data in 1D up to 3D. As known from discrete Fourier transform methods, the spatial Nyquist limit can be expressed as the greatest common divisor in 1D. We extend this to arbitrary numbers of spatial dimensions and spacecraft. We show that the spatial Nyquist limit can be found by determining the shortest possible basis of the spacecraft distance vectors. This may be done using linear combination in position space and transforming the obtained shortest basis to k space. Alternatively, the shortest basis can be determined mathematically by applying the modified Lenstra–Lenstra–Lovász (MLLL) algorithm combined with a lattice enumeration algorithm. Thus, we give a generalized solution to the determination of the spatial Nyquist limit for arbitrary numbers of spacecraft and dimensions without any need of a priori knowledge of the measured data. Additionally, we give first insights into the application to real-world data incorporating spacecraft position errors and minimizing k-space aliasing. As the wave telescope is an estimator for a multi-dimensional power spectrum substituting spatial Fourier transform, the results of this analysis can be applied to power spectral density estimation via Fourier transform or other methods making use of irregular sampling points. Therefore, our findings are also of interest to other fields of signal processing.

Dimensional reduction of the Dirac equation in arbitrary spatial dimensions

We investigate the general properties of the dimensional reduction of the Dirac theory, formulated in a Minkowski spacetime with an arbitrary number of spatial dimensions. This is done by applying Hadamard’s method of descent, which consists in conceiving low-dimensional theories as a specialization of high-dimensional ones that are uniform along the additional space coordinate. We show that the Dirac equation reduces to either a single Dirac equation or two decoupled Dirac equations, depending on whether the higher-dimensional manifold has even or odd spatial dimensions, respectively. Furthermore, we construct and discuss an explicit hierarchy of representations in which this procedure becomes manifest and can easily be iterated.

Theorem on extensive spectral degeneracy for systems with rigid higher symmetries in general dimensions

We establish, in the spirit of the Lieb-Schultz-Mattis theorem, lower bounds on the spectral degeneracy of quantum systems with higher (Gauge Like) symmetries with rather generic physical boundary conditions in an arbitrary number of spatial dimensions. Contrary to applying twists or equivalent adiabatic operations, we exploit the effects of modified boundary conditions. When a general choice of boundary geometry is immaterial in approaching the thermodynamic limit, systems that exhibit non-commuting Gauge Like symmetries, such as the orbital compass model, must have an exponential (in the size of the boundary) degeneracy of each of their spectral levels. We briefly discuss why, in spite of the proven large degeneracy associated with infrared-ultraviolet mixing, some systems may still exhibit conventional physical behaviors, i.e., of those of systems with non-extensive degeneracies, due to entropic "order by disorder" type effects.

From kinetic to fluid models of liquid crystals by the moment method

<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>

Electromagnetic Casimir densities for planar boundaries in AdS spacetime

We investigate the vacuum expectation value of the energy-momentum tensor and the Casimir forces for the electromagnetic field in AdS spacetime for the geometry of two parallel plates. On the plates the field obeys the boundary condition that generalizes the perfect conductor boundary condition for an arbitrary number of spatial dimensions. The interaction forces between the plates are attractive. At separations larger than the curvature radius of the spacetime they decay exponentially.

AMFR-W-methods for parabolic problems with mixed derivates. Applications to the Heston model

The combination of Approximate Matrix Factorization (AMF), W-methods and iterative refinement in the solution of linear systems leads to the definition of AMFR-W-methods. This method class provides stable and accurate time integrators for parabolic PDEs with mixed derivatives discretized in space by means of Finite Differences (or Finite Volumes) in an arbitrary number of spatial dimensions. When the coefficients of the PDE actually depend on the spatial variables, the approximation of the pure diffusion coefficients by its respective maximum value produces simplified AMFR-W-methods requiring only a reduced number of LU decompositions of banded matrices with small bandwidth. The new class of methods is shown to be unconditionally stable regardless of the spatial dimension on a linear test problem relevant for homogeneous or periodic boundary conditions. Furthermore, high orders of convergence in PDE sense are observed when homogeneous boundary conditions are assumed. For general Robin boundary conditions, a simple algorithm is provided to convert a PDE problem into one where such conditions are homogeneous. Numerical experiments with the new simplified AMFR-W-methods on a linear parabolic problem with variable coefficients and the Heston problem from financial option pricing are presented.

A moving discontinuous Galerkin finite element method for flows with interfaces

SummaryA moving discontinuous Galerkin finite element method with interface condition enforcement is formulated for flows with discontinuous interfaces. The underlying weak formulation enforces the interface condition separately from the conservation law, so that the residual only vanishes upon satisfaction of both. In this formulation, the discrete grid geometry is treated as a variable, so that, in contrast to the standard discontinuous Galerkin method, this method has both the means to detect interfaces, via interface condition enforcement, and to satisfy, via grid movement, the conservation law and its associated interface condition. The method therefore directly fits interfaces, including shocks, preserving a high‐order representation up to the interface without requiring shock capturing or an upwind numerical flux to achieve stability. It can be generalized to flows with a priori unknown interfaces with nontrivial topology and curved interface geometry as well as to an arbitrary number of spatial dimensions. Unsteady flows are represented in a manner similar to steady flows using a space‐time formulation. In addition to computing flows with interfaces, the method can represent point singularities in a flow field by degenerating cuboid elements. In general, the method works in conjunction with standard local grid operations, including edge collapse, to ensure that degenerate cells are removed. Test cases are presented for up to three‐dimensional flows that provide an initial assessment of the stability and accuracy of the method.

Scalar Casimir effect in a linearly expanding universe

We investigate quantum vacuum effects for a massive scalar field, induced by two planar boundaries in background of a linearly expanding spatially flat Friedmann–Robertson–Walker spacetime for an arbitrary number of spatial dimensions. For the Robin boundary conditions and for general curvature coupling parameter, a complete set of mode functions is presented and the related Hadamard function is evaluated. The results are specified for the most important special cases of the adiabatic and conformal vacuum states. The vacuum expectation values of the field squared and of the energy–momentum tensor are investigated for a massive conformally coupled field. The vacuum energy–momentum tensor, in addition to the diagonal components, has nonzero off-diagonal component describing energy flux along the direction perpendicular to the plates. The influence of the gravitational field on the local characteristics of the vacuum state is essential at distances from the boundaries larger than the curvature radius of the background spacetime. In contrast to the Minkowskian bulk, at large distances the boundary-induced expectation values follow as power law for both massless and massive fields. Another difference is that the Casimir forces acting on the separate plates do not coincide if the corresponding Robin coefficients are different. At large separations between the plates the decay of the forces is power law. We show that during the cosmological expansion the forces may change the sign.

Electromagnetic quantum effects in anti-de Sitter spacetime

The two-point functions of the vector potential and of the field tensor for the electromagnetic field in the background of anti-de Sitter (AdS) spacetime are evaluated in an arbitrary number of spatial dimensions. First, we consider the two-point functions in the boundary-free geometry and then generalize the results in the presence of a reflecting boundary parallel to the AdS horizon. By using the expressions for the two-point functions of the field tensor, we investigate the vacuum expectation values of the electric field squared and of the energy–momentum tensor. Simple asymptotic expressions are provided near the AdS boundary and horizon.

ELECTROMAGNETIC CASIMIR DENSITIES FOR A PLATE IN ANTI-DE SITTER SPACETIME

We investigate the vacuum expectation values (VEV) of the electric field squared and of the energy-momentum tensor for the electromagnetic field in anti-de Sitter (AdS) spacetime induced by a plate parallel to the AdS boundary. On the plate the field obeys the boundary condition that generalizes the perfect conductor boundary condition for an arbitrary number of spatial dimensions. We show that the plate-induced contributions in the VEV of the electric field squared and the vacuum energy density are negative, whereas the normal stress is positive. The VEV vanish on the AdS boundary.

Electromagnetic Casimir effect for conducting plates in de Sitter spacetime

Two-point functions, the mean field squared and the vacuum expectation value (VEV) of the energy–momentum tensor are investigated for the electromagnetic field in the geometry of parallel plates on background of -dimensional dS spacetime. We assume that the field is prepared in the Bunch–Davies vacuum state and on the plates a boundary condition is imposed that is a generalization of the perfectly conducting boundary condition for an arbitrary number of spatial dimensions. It is shown that for the background gravitational field essentially changes the behavior of the VEVs at separations between the plates larger than the curvature radius of dS spacetime. At large separations, the Casimir forces are proportional to the inverse fourth power of the distance for all values of spatial dimension . For this behavior is in sharp contrast with the case of plates in Minkowski bulk where the force decays as the inverse power of the distance.

SPEEDUP Code for Calculation of Transition Amplitudes via the Effective Action Approach

AbstractWe present Path Integral Monte Carlo C code for calculation of quantum mechanical transition amplitudes for 1D models. The SPEEDUP C code is based on the use of higher-order short-time effective actions and implemented to the maximal order p=18 in the time of propagation (Monte Carlo time step), which substantially improves the convergence of discretized amplitudes to their exact continuum values. Symbolic derivation of higher-order effective actions is implemented in SPEEDUP Mathematica codes, using the recursive Schrödinger equation approach. In addition to the general 1D quantum theory, developed Mathematica codes are capable of calculating effective actions for specific models, for general 2D and 3D potentials, as well as for a general many-body theory in arbitrary number of spatial dimensions.

The bond-algebraic approach to dualities

An algebraic theory of dualities is developed based on the notion of bond algebras. It deals with classical and quantum dualities in a unified fashion explaining the precise connection between quantum dualities and the low temperature (strong-coupling)/high temperature (weak-coupling) dualities of classical statistical mechanics (or (Euclidean) path integrals). Its range of applications includes discrete lattice, continuum field, and gauge theories. Dualities are revealed to be local, structure-preserving mappings between model-specific bond algebras that can be implemented as unitary transformations, or partial isometries if gauge symmetries are involved. This characterization permits to search systematically for dualities and self-dualities in quantum models of arbitrary system size, dimensionality and complexity, and any classical model admitting a transfer matrix representation. Dualities like exact dimensional reduction, emergent, and gauge-reducing dualities that solve gauge constraints can be easily understood in terms of mappings of bond algebras. As a new example, we show that the (\mathbb{Z}_2) Higgs model is dual to the extended toric code model {\it in any number of dimensions}. Non-local dual variables and Jordan-Wigner dictionaries are derived from the local mappings of bond algebras. Our bond-algebraic approach goes beyond the standard approach to classical dualities, and could help resolve the long standing problem of obtaining duality transformations for lattice non-Abelian models. As an illustration, we present new dualities in any spatial dimension for the quantum Heisenberg model. Finally, we discuss various applications including location of phase boundaries, spectral behavior and, notably, we show how bond-algebraic dualities help constrain and realize fermionization in an arbitrary number of spatial dimensions.

Weak-field limit off(R)gravity in three and more spatial dimensions

We investigate a pointlike massive source in nonlinear $f(R)$ theories in the case of an arbitrary number of spatial dimensions $D\ensuremath{\ge}3$. If $D>3$ then extra dimensions undergo toroidal compactification. We consider a weak-field approximation with Minkowski and de Sitter background solutions. In both these cases pointlike massive sources demonstrate good agreement with experimental data only in the case of ordinary three-dimensional ($D=3$) space. We generalize this result to the case of a perfect fluid with dustlike equations of state in the external and internal spaces. This perfect fluid is uniformly smeared over all extra dimensions and enclosed in a three-dimensional sphere. In ordinary three-dimensional ($D=3$) space, our formulas are useful for experimental constraints on parameters of $f(R)$ models.

Domain wall network evolution in (N+1)-dimensional FRW universes

We develop a velocity-dependent one-scale model for the evolution of domain wall networks in flat expanding or collapsing homogeneous and isotropic universes with an arbitrary number of spatial dimensions, finding the corresponding scaling laws in frictionless and friction dominated regimes. We also determine the allowed range of values of the curvature parameter and the expansion exponent for which a linear scaling solution is possible in the frictionless regime.

New features of a flat (4 + 1)-dimensional cosmological model with a perfect fluid in gauss-bonnet gravity

We investigated a flat multidimensional cosmological model in Gauss-Bonnet gravity in presence of a matter in form of perfect fluid. We found analytically new stationary regimes (these results are valid for arbitrary number of spatial dimensions) and studied their stability by means of numerical recipes in 4+1-dimensional case. In the vicinity of the stationary regime we discovered numerically another non-singular regime which appears to be periodical. Finally, we demonstrated that the presence of matter in form of a perfect fluid lifts some constraints on the dynamics of the 4+1-dimensional model which have been found earlier.

Some stationary properties of aQ-ball in arbitrary space dimensions

Introducing new physically motivated ansaetze, we explore both analytically and numerically the classical and absolute stabilities of a single Q-ball in an arbitrary number of spatial dimensions D, working in both the thin and thick wall limits.

New family of simple solutions of relativistic perfect fluid hydrodynamics

A new class of accelerating, exact and explicit solutions of relativistic hydrodynamics is found—more than 50 years after the previous similar result, the Landau–Khalatnikov solution. Surprisingly, the new solutions have a simple form, that generalizes the renowned, but accelerationless, Hwa–Bjorken solution. These new solutions take into account the work done by the fluid elements on each other, and work not only in one temporal and one spatial dimensions, but also in arbitrary number of spatial dimensions. They are applied here for an advanced estimation of initial energy density and life-time of the reaction in ultra-relativistic heavy ion collisions. New formulas are also conjectured, that yield further important increase of the initial energy density estimate and the measured life-time of the reaction if the value of the speed of sound is in the realistic range.

A fully adaptive reaction–diffusion integration scheme with applications to systems biology

Adaptive integration schemes for ODE systems typically function by adjusting the time step size so as to keep the truncation error below some desired value. For adaptive integration of PDE systems involving coupled kinetic reaction and diffusion operations, truncation error arises not only from the individual propagators but also from their method of coupling. A common second-order accurate method for coupling operators is Strang’s method of operator splitting. We derive an expression for the truncation error resulting from Strang splitting reaction and diffusion operators for an arbitrary number of spatial dimensions, and demonstrate its use in adaptive time step algorithms. In addition, we present explanations of the second order implicit reaction and diffusion operators, and their individual error calculations used in our implementation of the scheme. Finally, using example simulations we discuss the use of this calculation for problems in systems biology.