Arbitrary Space Dimension Research Articles

Characterization of decoherence from an environmental perspective

For the case of phase damping (pure decoherence) we investigate the extent to which environmental traits are imprinted on an open quantum system. The dynamics is described using the quantum channel approach. We study what the knowledge of the channel may reveal about the nature of its underlying dynamics and, conversely, what the dynamics tells us about how to consistently model the environment. We find that for a Markov phase-damping channel, that is, a channel compatible with a time-continuous Markovian evolution, the environment may adequately be represented by a mixture of only a few coherent states. For arbitrary Hilbert space dimension $N\geq 4$ we refine the idea of {\it quantum phase damping}, of which we show a means of identification. Symmetry considerations are used to identify decoherence-free subspaces of the system.

Convolutions of singular measures and applications to the Zakharov system

Uniform L 2 -estimates for the convolution of singular measures with respect to transversal submanifolds are proved in arbitrary space dimension. The results of Bennett–Bez are used to extend previous work of Bejenaru–Herr–Tataru. As an application, it is shown that the 3D Zakharov system is locally well-posed in the full subcritical regime.

Nonrelativistic counterparts of twistors and the realizations of Galilean conformal algebra

Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of SO(2,1)⊕SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistors components. For arbitrary Hermitian quantum-mechanical Galilean twistor realizations we obtain the result that the representations of GCA with positive-definite Hamiltonian do not exist. For non-positive H we construct for N⩾2 the Hermitian Galilean N-twistor realizations of GCA; for N=2 such realization is provided explicitly.

Spectra of Ruelle transfer operators for Axiom A flows

For Axiom A flows on basic sets satisfying certain additional conditions we prove strong spectral estimates for Ruelle transfer operators similar to those of Dolgopyat (1998 Ann. Math. 147 357–90) for geodesic flows on compact surfaces (for general potentials) and transitive Anosov flows on compact manifolds with C1 jointly non-integrable horocycle foliations (for the Sinai–Bowen–Ruelle potential). Here we deal with general potentials and on spaces of arbitrary dimension, although under some geometric and regularity conditions. As is now well known, such results have deep implications in some related areas, e.g. in studying analytic properties of Ruelle zeta functions and partial differential operators, closed orbit counting functions, and in other areas.

Finite Element Approximation for Shape Optimization Problems with Neumann and Mixed Boundary Conditions

For optimal design problems, defined in domains of class $C$ and in arbitrary space dimension, governed by elliptic equations with boundary conditions of Neumann or mixed type, we introduce the corresponding discretized problems and we prove convergence results. The discretization method is of fixed domain type, in the sense that it is given in the domain that contains all the admissible open sets.

On Delaunay refinement for curved geometries

Purpose – The purpose of this paper is to investigate the extension of Delaunay refinement algorithms to work directly with a curved geometry in arbitrary dimensional spaces, which is also able to refine geometry pieces of different dimensions altogether.Design/methodology/approach – The extension of Delaunay refinement is based on ideas of the Bowyer‐Watson algorithm and Ruppert algorithm.Findings – The attempt to extend the fundamental ideas of Delaunay refinement to cope with curved geometries led to an algorithm whose performance in practice, regarding speed and mesh quality, is comparable to classical Delaunay refinement for flat geometries. Unfortunately, there are only theoretical guarantees that the refinement itself works under some conditions. No theoretical mesh quality bounds are provided.Research limitations/implications – A mesh refinement algorithm that deals with curved geometries is a key feature for adaptive mesh generators, so that points are inserted properly in the curved pieces inste...

Direct simulation of the infinitesimal dynamics of semi-discrete approximations for convection–diffusion–reaction problems

In this paper a scheme for approximating solutions of convection–diffusion–reaction equations by Markov jump processes is studied. The general principle of the method of lines reduces evolution partial differential equations to semi-discrete approximations consisting of systems of ordinary differential equations. Our approach is to use for this resulting system a stochastic scheme which is essentially a direct simulation of the corresponding infinitesimal dynamics. This implies automatically the time adaptivity and, in one space dimension, stable approximations of diffusion operators on non-uniform grids and the possibility of using moving cells for the transport part, all within the framework of an explicit method. We present several results in one space dimension including free boundary problems, but the general algorithm is simple, flexible and on uniform grids it can be formulated for general evolution partial differential equations in arbitrary space dimensions.

Structurally damped plate and wave equations with random point force in arbitrary space dimensions

In this paper we consider structurally damped plate and wave equations with point and distributed random forces. In order to treat space dimensions more than one, we work in the setting of $L^q$--spaces with (possibly small) $q\in(1,2)$. We establish existence, uniqueness and regularity of mild and weak solutions to the stochastic equations employing recent theory for stochastic evolution equations in UMD Banach spaces.

Multi-class blue noise sampling

Sampling is a core process for a variety of graphics applications. Among existing sampling methods, blue noise sampling remains popular thanks to its spatial uniformity and absence of aliasing artifacts. However, research so far has been mainly focused on blue noise sampling with a single class of samples. This could be insufficient for common natural as well as man-made phenomena requiring multiple classes of samples, such as object placement, imaging sensors, and stippling patterns. We extend blue noise sampling to multiple classes where each individual class as well as their unions exhibit blue noise characteristics. We propose two flavors of algorithms to generate such multi-class blue noise samples, one extended from traditional Poisson hard disk sampling for explicit control of sample spacing, and another based on our soft disk sampling for explicit control of sample count. Our algorithms support uniform and adaptive sampling, and are applicable to both discrete and continuous sample space in arbitrary dimensions. We study characteristics of samples generated by our methods, and demonstrate applications in object placement, sensor layout, and color stippling.

Boundedness in the Higher-Dimensional Parabolic-Parabolic Chemotaxis System with Logistic Source

We consider nonnegative solutions of the Neumann boundary value problem for the chemotaxis system in a smooth bounded convex domain Ω ⊂ ℝ n , n ≥ 1, where τ > 0, χ ∈ ℝ and f is a smooth function generalizing the logistic source It is shown that if μ is sufficiently large then for all sufficiently smooth initial data the problem possesses a unique global-in-time classical solution that is bounded in Ω × (0, ∞). Known results, asserting boundedness under the additional restriction n ≤ 2, are thereby extended to arbitrary space dimensions.

FAST EMPIRICAL MODE DECOMPOSITIONS OF MULTIVARIATE DATA BASED ON ADAPTIVE SPLINE-WAVELETS AND A GENERALIZATION OF THE HILBERT–HUANG–TRANSFORMATION (HHT) TO ARBITRARY SPACE DIMENSIONS

The Hilbert–Huang-Transform (HHT) has proven to be an appropriate multiscale analysis technique specifically for nonlinear and nonstationary time series on non-equidistant grids. It is empirically adapted to the data: first, an additive decomposition of the data (empirical mode decomposition, EMD) into certain multiscale components is computed, denoted as intrinsic mode functions. Second, to each of these components, the Hilbert transform is applied. The resulting Hilbert spectrum of the modes provides a localized time-frequency spectrum and instantaneous (time-dependent) frequencies. For the first step, the empirical decomposition of the data, a different method based on local means has been developed by Chen et al. (2006). In this paper, we extend their method to multivariate data sets in arbitrary space dimensions. We place special emphasis on deriving a method which is numerically fast also in higher dimensions. Our method works in a coarse-to-fine fashion and is based on adaptive (tensor-product) spline-wavelets. We provide some numerical comparisons to a method based on linear finite elements and one based on thin-plate-splines to demonstrate the performance of our method, both with respect to the quality of the approximation as well as the numerical efficiency. Second, for a generalization of the Hilbert transform to the multivariate case, we consider the Riesz transformation and an embedding into Clifford-algebra valued functions, from which instantaneous amplitudes, phases and orientations can be derived. We conclude with some numerical examples.

A high-order, analytically divergence-free discretization method for Darcy’s problem

We develop and analyze a meshfree discretization method for Darcy's problem. Our approximation scheme is based upon optimal recovery which leads to a collocation scheme using divergence-free positive definite kernels. Besides producing analytically incompressible flow fields, our method can be of arbitrary order, works in arbitrary space dimension and for arbitrary geometries. After deriving the scheme, we investigate the approximation error for smooth target functions and derive optimal approximation orders. Finally, we illustrate the method by giving numerical examples.

Classification of Lorentzian surfaces with parallel mean curvature vector in pseudo-Euclidean spaces

Spatial surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension and index were classified by B.Y. Chen. In this work, we give a complete classification of Lorentzian surfaces with parallel mean curvature vector in pseudo-Euclidean spaces of arbitrary dimension and index. Consequently, the problem to classify all the surfaces with parallel mean curvature vector in pseudo-Euclidean spaces has been solved.

BLOW-UP OF SMOOTH SOLUTIONS TO THE NAVIER–STOKES EQUATIONS OF COMPRESSIBLE VISCOUS HEAT-CONDUCTING FLUIDS

AbstractWe give a simpler and refined proof of some blow-up results of smooth solutions to the Cauchy problem for the Navier–Stokes equations of compressible, viscous and heat-conducting fluids in arbitrary space dimensions. Our main results reveal that smooth solutions with compactly supported initial density will blow up in finite time, and that if the initial density decays at infinity in space, then there is no global solution for which the velocity decays as the reciprocal of the elapsed time.

Curvature operators and scalar curvature invariants

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterized by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of the alignment theory and the bivector form of the Weyl operator in higher dimensions and introduce the important notions of diagonalizability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.

Factorized ground state for a general class of ferrimagnets

We have found the exact (factorized) ground state of a general class of ferrimagnets in the presence of a magnetic field which includes the anisotropic and long-range interactions for arbitrary dimensional space. In particular cases our model represents many spin chains with bond alternation of antiferromagnetic-ferromagnetic coupling, ferrimagnetic spin ladders, and also homogeneous spin-$s$ models. The factorized ground state is a product of single-particle kets on a bipartite lattice composed of two different spins $(\ensuremath{\rho},\ensuremath{\sigma})$ which is characterized by two angles, a biangle state. The spin-wave analysis around the exact ground state shows two branch of excitations which are the origin of two dynamics of the model. The signature of these dynamics is addressed as a peak and a broaden bump in the specific heat.

Quadratic Morawetz inequalities and asymptotic completeness in the energy space for nonlinear Schrödinger and Hartree equations

Recently several authors have developed multilinear and in particular quadratic extensions of the classical Morawetz inequality. Those extensions provide (among other results) an easy proof of asymptotic completeness in the energy space for nonlinear Schrodinger equations in arbitrary space dimension and for Hartree equations in space dimension greater than two in the noncritical cases. We give a pedagogical review of the latter results.

The Continuous Shearlet Transform in Arbitrary Space Dimensions

This paper is concerned with the generalization of the continuous shearlet transform to higher dimensions. Similar to the two-dimensional case, our approach is based on translations, anisotropic dilations and specific shear matrices. We show that the associated integral transform again originates from a square-integrable representation of a specific group, the full n-variate shearlet group. Moreover, we verify that by applying the coorbit theory, canonical scales of smoothness spaces and associated Banach frames can be derived. We also indicate how our transform can be used to characterize singularities in signals.

ON RECTANGULAR COVERING PROBLEMS

Many applications like image processing, data compression or pattern recognition require a covering of a set of n points most often located in the (discrete) plane by rectangles due to specific cost constraints. In this paper we provide exact dynamic programming algorithms for covering point sets by regular rectangles, that have to obey certain boundary conditions, namely all rectangles must have lengths of at least k, which is a prescribed problem parameter. The concrete objective function underlying our optimization problem is the sum of area, circumference and a constant over all rectangles forming a covering. This objective function can be motivated by requirements of numerically solving PDE's by discretization over (adaptive multi-)grids. More precisely, we propose exact deterministic algorithms for such problems based on a (set theoretic) dynamic programming approach yielding a time bound of O(n23n). In a second step this bound is (asymptotically) decreased to O(n62n) by exploiting some structural features. Finally, a generalization of the problem and its solution methods is discussed for the case of arbitrary (finite) space dimension.

Phenomenologically symmetric local lie groups of transformations of the space R s

In this paper we define a phenomenologically symmetric local Lie group of transformations of an arbitrary-dimensional space. We take as a basis the axiom scheme of the theory of physical structures. Phenomenologically symmetric groups of transformations are nondegenerate both with respect to coordinates and to parameters. We obtain a multipoint invariant of this group of transformations and relate it with Ward quasigroups. We define a substructure of a physical structure as a certain phenomenologically symmetric subgroup of transformations. We establish a criterion for the phenomenological symmetry of the Lie group of transformations and prove the uniqueness of a structure with the minimal rank. We also introduce the notion of a phenomenologically symmetric product of physical structures.