Splitting Of Space Research Articles

Nuclear Norm Spectrum Estimation From Uniformly Spaced Measurements

Subspace-based methods have been effectively used to estimate multi-input/multi-output, discrete-time, linear-time invariant systems from uniformly spaced spectrum samples. A critical step in these methods is the splitting of causal and noncausal invariant subspaces of a Hankel matrix built from spectrum measurements via singular-value decomposition in order to determine model order. Quite often, in particular when signal-to-noise ratio is low, unmodelled dynamics is present, and the number of measurements is small, this step is inconclusive since the assumed mirror image symmetry with respect to the unit circle between the eigenvalues of the invariant spaces is lost. In this paper, we propose a new model order selection and invariant space splitting scheme based on the regularized nuclear norm optimization in combination with a particular subspace method that is insensitive to noise and undermodelling. By a detailed numerical study, efficacy of the proposed scheme is shown for a broad range of signal-to-noise ratio and short data records.

Multivariate polynomial regression regularized via fit function uncertainty

This article describes a multivariate polynomial regression method where the uncertainty of the input parameters are approximated with Gaussian distributions, derived from the central limit theorem for large weighted sums, directly from the training sample. The estimated uncertainties can be propagated into the optimal fit function, as an alternative to the statistical bootstrap method. This uncertainty can be propagated further into a loss function like quantity, with which it is possible to calculate the expected loss function, and allows to select the optimal polynomial degree with statistical significance. Combined with simple phase space splitting methods, it is possible to model most features of the training data even with low degree polynomials or constants.

The geometry of the tangent bundle and the relativistic kinetic theory of gases

This paper discusses the relativistic kinetic theory for a simple collisionless gas from a geometric perspective. We start by reviewing the rich geometrical structure of the tangent bundle TM of a given spacetime manifold, including the splitting of the tangent spaces of TM into horizontal and vertical subspaces and the natural metric and symplectic structure it induces on TM. Based on these structures we introduce the Liouville vector field L and a suitable Hamiltonian function H on TM. The Liouville vector field turns out to be the Hamiltonian vector field associated to H. On the other hand, H also defines the mass shells as Lorentzian submanifolds of the tangent bundle. When restricted to these mass shells, the projection of the integral curves of L on the base manifold describes a family of future directed timelike geodesics. A simple collisionless gas is described by a distribution function on a particular mass shell, satisfying the Liouville equation. Together with the Liouville vector field the distribution function can be thought of as defining a fictitious incompressible fluid on the mass shells, with associated conserved current density. Flux integrals of this current density provide the averaged properties of the gas, while suitable fiber integrals of the distribution function define divergence-free tensor fields on the spacetime manifold such as the current density and stress–energy tensor. Finally, we discuss the relationship between symmetries of the spacetime manifold and symmetries of the distribution function. Taking advantage of the natural metric and symplectic structure on TM, we show that groups of isometries G of the spacetime manifold lift naturally to groups of isometries and symplectic flows on the tangent bundle. Motivated by these properties, we define a distribution function to be G-invariant whenever it is invariant under the lifted isometries. As a first application of our formalism we derive the most general spherically symmetric distribution function on any spherically symmetric spacetime and write the Einstein–Liouville equations as effective field equations on the two-dimensional radial manifold. As a second application we derive the most general collisionless distribution function on a Kerr black hole spacetime background.

Surgical anatomy of the axillary nerve branches to the deltoid muscle

Variations in the innervation of the posterior deltoid muscle by the anterior branch of the axillary nerve have been reported. The objective of this study is to clarify the anatomy of the axillary nerve branches to the deltoid muscle. One hundred and twenty-nine arms (68 right and 61 left) from 88 embalmed cadavers (83 male and 46 female) were included in the study. The anterior and posterior branches of the axillary nerve were identified and their lengths were measured from the point of emergence from the axillary nerve to their terminations in the deltoid muscle. In all cases, the axillary nerves split into two branches (anterior and posterior) within the quadrangular space and none split within the deltoid muscle. In all specimens, the anterior and middle parts of the deltoid muscle received their nerve supplies from the anterior branch of the axillary nerve. The posterior part of the deltoid muscle was supplied only by the anterior branch of the axillary nerve in 2.3% of the specimens, from the posterior branch in 8.5%, and from both branches in 89.1%. There were two sub-branches of the anterior branch in 4.7% of the specimens. The anterior branch of the axillary nerve supplied not only the anterior and middle parts of the deltoid muscle but also the posterior part in most cases (91.5%).

Asymptotically Almost Periodic Solutions for a Class of Stochastic Functional Differential Equations

This work is concerned with the quadratic-mean asymptotically almost periodic mild solutions for a class of stochastic functional differential equationsdxt=Atxt+Ft,xt,xtdt+H(t,xt,xt)∘dW(t). A new criterion ensuring the existence and uniqueness of the quadratic-mean asymptotically almost periodic mild solutions for the system is presented. The condition of being uniformly exponentially stable of the strongly continuous semigroup{Tt}t≥0is essentially removed, which is generated by the linear densely defined operatorA∶D(A)⊂L2(ℙ,ℍ)→L2(ℙ,ℍ), only using the exponential trichotomy of the system, which reflects a deeper analysis of the behavior of solutions of the system. In this case the asymptotic behavior is described through the splitting of the main space into stable, unstable, and central subspaces at each point from the flow’s domain. An example is also given to illustrate our results.

Solution of Mixed Combinatorial Optimization Problems on Arrangements by the Method of Construction of Lexicographic Equivalence

The paper considers the solution of mixed combinatorial optimization problems on arrangements by the method of construction of lexicographic equivalence. A generalization of the relation of equivalence, which is used for space splitting, is proposed and its properties are analyzed. The algorithms of the method known earlier are modified, an approximated algorithm is validated.

Multilevel preconditioners for discontinuous Galerkin approximations of elliptic problems with jump coefficients

We introduce and analyze two-level and multilevel preconditioners for a family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with large jumps in the diffusion coefficient. Our approach to IPDG-type methods is based on a splitting of the DG space into two components that are orthogonal in the energy inner product naturally induced by the methods. As a result, the methods and their analysis depend in a crucial way on the diffusion coefficient of the problem. The analysis of the proposed preconditioners is presented for both symmetric and non-symmetric IP schemes; dealing simultaneously with the jump in the diffusion coefficient and the non-nested character of the relevant discrete spaces presents additional difficulties in the analysis, which precludes a simple extension of existing results. However, we are able to establish robustness (with respect to the diffusion coefficient) and near-optimality (up to a logarithmic term depending on the mesh size) for both two-level and BPX-type preconditioners, by using a more refined Conjugate Gradient theory. Useful by-products of the analysis are the supporting results on the construction and analysis of simple, efficient and robust two-level and multilevel preconditioners for non-conforming Crouzeix-Raviart discretizations of elliptic problems with jump coefficients. Following the analysis, we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods.

Origin of giant Rashba spin splitting in Bi/Ag surface alloys

One-third of a monolayer of Bi alloyed into the Ag(111) surface yields a pair of Rashba spin-split free-electron-like surface states. The splitting in momentum space is the largest of all surface alloys investigated. Using first-principles calculations, we have determined the spin splitting in this system as a function of atomic corrugation of the top atomic layer and the strengths of the atomic spin-orbit coupling in Bi and Ag. The calculated splitting is proportional to the strengths of the atomic spin-orbit coupling, but it peaks at a certain surface corrugation parameter. These findings indicate that the observed giant spin splitting is caused by a near-optimal surface corrugation and a large atomic spin-orbit coupling in Bi. The results offer a useful guide for searching for two-dimensional systems with large surface spin effects.

Robust algebraic multilevel preconditioning in [formula omitted] and [formula omitted

An algebraic multilevel iteration method for solving a system of linear algebraic equations arising in H(curl) and H(div) spaces is presented. The algorithm is developed for the discrete problem obtained by using the space of lowest-order Nedelec and Raviart–Thomas–Nedelec elements. The theoretical analysis of the method is based only on some algebraic sequences and generalized eigenvalues of local (element-wise) problems. In the hierarchical basis framework, explicit recursion formulas are derived to compute the element matrices and the constant γ (which measures the quality of the space splitting) at any given level. It is proved that the proposed method is robust with respect to the problem parameters and is of optimal order complexity. Supporting numerical results, including the case when the parameters have jumps, are also presented.

XPand: an algorithm for perturbing homogeneous cosmologies

In this paper, we develop in detail a fully geometrical method for deriving perturbation equations about a spatially homogeneous background. This method relies on the 3 + 1 splitting of the background space–time and does not use any particular set of coordinates: it is implemented in terms of geometrical quantities only, using the tensor algebra package xTensor in the xAct distribution along with the extension for perturbations xPert. Our algorithm allows one to obtain the perturbation equations for all types of homogeneous cosmologies, up to any order and in all possible gauges. As applications, we recover the well-known perturbed Einstein equations for Friedmann–Lemaître–Robertson–Walker cosmologies up to second order and for Bianchi I cosmologies at first order. This work paves the way to the study of these models at higher order and to that of any other perturbed Bianchi cosmologies, by circumventing the usually too cumbersome derivation of the perturbed equations.

Orthogonal versus covariant Lyapunov vectors for rough hard disc systems

The Oseledec splitting of the tangent space into covariant subspaces for a hyperbolic dynamical system is numerically accessible by computing the full set of covariant Lyapunov vectors. In this paper, the covariant Lyapunov vectors, the orthogonal Gram–Schmidt vectors and the corresponding local (time-dependent) Lyapunov exponents are analyzed for a planar system of rough hard discs (RHDS). These results are compared to respective results for a smooth-hard-disc system (SHDS). We find that the rotation of the discs deeply affects the Oseledec splitting and the structure of the tangent space. For both the smooth and rough hard discs, the stable, unstable and central manifolds are transverse to each other, although the minimal angle between the unstable and stable manifolds of the RHDS typically is very small. Both systems are hyperbolic. However, the central manifold is precisely orthogonal to the rest of the tangent space only for the smooth-particle case and not for the rough discs. We also demonstrate that the rotations destroy the Hamiltonian (symplectic) character for the RHDS.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Lyapunov analysis: from dynamical systems theory to applications’.

A family of Multiscale Hybrid-Mixed finite element methods for the Darcy equation with rough coefficients

We aim at proposing novel stable finite element methods for the mixed Darcy equation with heterogeneous coefficients within a space splitting framework. We start from the primal hybrid formulation of the elliptic model for the pressure. Localization of this infinite-dimensional problem leads to element-level boundary value problems which embed multiscale and high-contrast features in a natural way, with Neumann boundary conditions driven by the Lagrange multipliers. Such a procedure leads to methods involving the space of piecewise constants for the pressure together with a discretization of the fluxes. Choosing (arbitrarily) polynomial interpolations, the lowest-order Raviart–Thomas element as well as some recent multiscale methods are recovered. In addition, the methods assure local mass conservation and can be interpreted as stabilized primal hybrid methods. Extensive numerical validation attests to the accuracy of the new methods on academic and more realistic problems with rough coefficients.

The tangent complex and Hochschild cohomology of -rings

AbstractIn this work, we study the deformation theory of${\mathcal {E}}_n$-rings and the${\mathcal {E}}_n$analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence$A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the${\mathcal {E}}_n$-tangent complex and${\mathcal {E}}_n$-Hochschild cohomology of an${\mathcal {E}}_n$-ring$A$. We give two proofs: the first is direct, reducing the problem to certain stable splittings of configuration spaces of punctured Euclidean spaces; the second is more conceptual, where we identify the sequence as the Lie algebras of a fiber sequence of derived algebraic groups,$B^{n-1}A^\times \rightarrow {\mathrm {Aut}}_A\rightarrow {\mathrm {Aut}}_{{\mathfrak B}^n\!A}$. Here${\mathfrak B}^n\!A$is an enriched$(\infty ,n)$-category constructed from$A$, and${\mathcal {E}}_n$-Hochschild cohomology is realized as the infinitesimal automorphisms of${\mathfrak B}^n\!A$. These groups are associated to moduli problems in${\mathcal {E}}_{n+1}$-geometry, a less commutative form of derived algebraic geometry, in the sense of the work of Toën and Vezzosi and the work of Lurie. Applying techniques of Koszul duality, this sequence consequently attains a nonunital${\mathcal {E}}_{n+1}$-algebra structure; in particular, the shifted tangent complex$T_A[-n]$is a nonunital${\mathcal {E}}_{n+1}$-algebra. The${\mathcal {E}}_{n+1}$-algebra structure of this sequence extends the previously known${\mathcal {E}}_{n+1}$-algebra structure on${\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)$, given in the higher Deligne conjecture. In order to establish this moduli-theoretic interpretation, we make extensive use of factorization homology, a homology theory for framed$n$-manifolds with coefficients given by${\mathcal {E}}_n$-algebras, constructed as a topological analogue of Beilinson and Drinfeld’s chiral homology. We give a separate exposition of this theory, developing the necessary results used in our proofs.

Vitamin E ameliorates histological and immunohistochemical changes in the cerebellar cortex of alloxan-induced diabetic rats

Introduction Diabetes mellitus causes a variety of functional and structural disorders in the central nervous system. Objectives To describe structural, ultrastructural, and immunohistochemical changes in the cerebellar cortex of alloxan-induced diabetic rats and to determine the possible protective effect of vitamin E. Materials and methods The rats were divided into three equal groups at random: group A (control), group B (alloxan-induced diabetic rats), and group C (alloxan-induced diabetic rats that received vitamin E). In rats of groups B and C, diabetes was induced by a single intraperitoneal injection of alloxan monohydrate. After 8 weeks, the rats of all groups were sacrificed and the cerebellum was dissected immediately and subjected to H&E stain, and immunostaining for glial fibrillary acidic protein (GFAP), and thin strips from the cortex were prepared and examined by transmission electron microscopy. Results Diabetic untreated rats showed shrunken Purkinje cells and apoptotic cells in the granular layer. Ultrastructurally, the nuclei of Purkinje cells were irregular and the cytoplasm contained numerous lysosomes, abnormal mitochondria, and dilated Golgi saccules. Myelinated nerve fibers showed splitting of myelin sheaths and wide axonal spaces. Immunohistochemically, a significant increase in the number of GFAP-positive astrocytes in diabetic untreated rats was found as compared with the control. These histological and ultrastructural alterations were ameliorated in vitamin E-treated diabetic rats. Moreover, a significant decrease in the number of GFAP-positive astrocytes was found in animals that received vitamin E as compared with diabetic untreated rats. Conclusion Vitamin E can be used as an adjuvant therapy for the prevention and treatment of the central nervous system complications of diabetes.

The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 1. The equations of motion in arbitrary Schwinger time gauges 1This paper is one of three companion papers published in the same issue of Can. J. Phys.

We study the coupling of N charged scalar particles plus the electromagnetic field to Arnowitt–Deser–Misner (ADM) tetrad gravity and its canonical formulation in asymptotically Minkowskian space–times without super-translations. To regularize the self-energies, both the electric charge and the sign of the energy of the particles are Grassmann-valued. The introduction of the noncovariant radiation gauge allows reformulation of the theory in terms of transverse electromagnetic fields and to extract the generalization of the Coulomb interaction among the particles in the riemannian instantaneous 3-spaces of global noninertial frames, the only ones allowed by the equivalence principle. Then we make the canonical transformation to the York canonical basis, where there is a separation between the inertial (gauge) variables and the tidal ones inside the gravitational field and a special role of the eulerian observers associated with the 3+1 splitting of space–time. The Dirac hamiltonian is weakly equal to the weak ADM energy. The Hamilton equations in Schwinger time gauges are given explicitly. In the York basis they are naturally divided into four sets: (i) the contracted Bianchi identities; (ii) the equations for the inertial gauge variables; (iii) the equations for the tidal ones; and (iv) the equations for matter. Finally, we give the restriction of the Hamilton equations and of the constraints to the family of nonharmonic 3-orthogonal gauges, in which the instantaneous riemannian 3-spaces have a nonfixed trace 3K of the extrinsic curvature but a diagonal 3-metric. The inertial gauge variable 3K (the general-relativistic remnant of the freedom in the clock synchronization convention) gives rise to a negative kinetic term in the weak ADM energy vanishing only in the gauges with 3K = 0: is it relevant for dark energy and back-reaction? In the second paper will appear the linearization of the theory in these nonharmonic 3-orthogonal gauges to obtain hamiltonian post-minkowskian gravity (without post-newtonian approximations) with asymptotic Minkowski background, nonflat instantaneous 3-spaces and no post-newtonian expansion. This will allow the exploration of the inertial effects induced by the York time 3K in nonflat 3-spaces (they do not exist in newtonian gravity) and to check how well dark matter can be explained as an inertial aspect of Einstein’s general relativity: this will be done in a third paper on the post-minkowskian 2-body problem in the absence of the electromagnetic field and on its 0.5 post-newtonian limit.

Greedy and randomized versions of the multiplicative Schwarz method

We consider sequential, i.e., Gauss–Seidel type, subspace correction methods for the iterative solution of symmetric positive definite variational problems, where the order of subspace correction steps is not deterministically fixed as in standard multiplicative Schwarz methods. Here, we greedily choose the subspace with the largest (or at least a relatively large) residual norm for the next update step, which is also known as the Gauss–Southwell method. We prove exponential convergence in the energy norm, with a reduction factor per iteration step directly related to the spectral properties, e.g., the condition number, of the underlying space splitting. To avoid the additional computational cost associated with the greedy pick, we alternatively consider choosing the next subspace randomly, and show similar estimates for the expected error reduction. We give some numerical examples, in particular applications to a Toeplitz system and to multilevel discretizations of an elliptic boundary value problem, which illustrate the theoretical estimates.

NONCOMMUTATIVE SPACE CORRECTIONS ON THE KLEIN–GORDON AND DIRAC OSCILLATORS SPECTRA

We consider the influence of a noncommutative space on the Klein–Gordon and the Dirac oscillators. The nonrelativistic limit is taken and the θ-modified Hamiltonians are determined. The corrections of these Hamiltonians on the energy levels are evaluated in first-order perturbation theory. It is observed a total lifting of the degeneracy to the considered levels. Such effects are similar to the Zeeman splitting in a commutative space.

Macro-Cellular Silicon carbide Reactors for Nonstationary Combustion Under Piston Engine-Like Conditions

Strut lattice structures of reaction-bonded silicon infiltrated silicon carbide ceramics (RB-SiSiC) for air–fuel mixture formation and for nonstationary lean-burn under pressure applications were fabricated. The lattice design with a high porosity >80% was shaped by indirect three-dimensional printing. It was shown that pre-ignition processes in the porous reactor are much faster than in a free combustion, especially at lower temperatures. Interaction of high velocity diesel jets with cylindrical strut ligaments of the SiSiC lattice structure offers a new possibility for quick and efficient fuel distribution (multi-jet splitting) in space.

Sparse tensor spherical harmonics approximation in radiative transfer

The stationary monochromatic radiative transfer equation is a partial differential transport equation stated on a five-dimensional phase space. To obtain a well-posed problem, boundary conditions have to be prescribed on the inflow part of the domain boundary. We solve the equation with a multi-level Galerkin FEM in physical space and a spectral discretization with harmonics in solid angle and show that the benefits of the concept of sparse tensor products, known from the context of sparse grids, can also be leveraged in combination with a spectral discretization. Our method allows us to include high spectral orders without incurring the “curse of dimension” of a five-dimensional computational domain. Neglecting boundary conditions, we find analytically that for smooth solutions, the convergence rate of the full tensor product method is retained in our method up to a logarithmic factor, while the number of degrees of freedom grows essentially only as fast as for the purely spatial problem. For the case with boundary conditions, we propose a splitting of the physical function space and a conforming tensorization. Numerical experiments in two physical and one angular dimension show evidence for the theoretical convergence rates to hold in the latter case as well.

A tree search method for the container loading problem with shipment priority

This paper addresses a special kind of container loading problem with shipment priority. We present a tree search method, which is based on a greedy heuristic. In the greedy heuristic, blocks made up of identical items with the same orientation are selected for packing into a container. Five evaluation functions are proposed for block selection, and the different blocks selected by each evaluation function constitute the branches of the search tree. A method of space splitting and merging is also embedded in the algorithm to facilitate efficient use of the container space. In addition, the proposed algorithm covers an important constraint called shipment priority to solve practical problems. The validity of the proposed algorithm is examined by comparing the present results with those of published algorithms using the same data.