Smooth Extension Research Articles

On the global Structure of Robinson–Trautman space-times

The global structure of Robinson–Trautman space-times is studied. When the space-time topology isR+xRxS2it is shown that all Robinson–Trautman space-time can beC117extended (in the vacuum Robinson–Trautman class of metrics) beyond ther= 2m'Schwarzschild-like' event horizon; evidence is given supporting the conjecture, that no smooth extensions beyond ther= 2mevent horizon exist unless the metric is the Schwarzschild one. When the space-time topology isR+xRx2M, with 2Ma higher genus surface, and the mass parametermis negative, Schwarzchild-like event horizons are shown to occur. The Proofs of these results are based on the derivation of a detailed asymptotic expansion describing the long-time behaviour of the solutions of a nonlinear parabolic equation.

Crystal growth and dendritic solidification

We present a numerical method which computes the motion of complex solid/liquid boundaries in crystal growth. The model we solve includes physical effects such as crystalline anisotropy, surface tension, molecular kinetics, and undercooling. The method is based on two ideas. First, the equations of motion are recast as a single history-dependent boundary integral equation on the solid/liquid boundary. A fast algorithm is used to solve the integral equation efficiently. Second, the boundary is moved by solving a “Hamilton-Jacobi” -type equation (on a fixed domain) formulated by Osher and Sethian for a function in which the boundary is a particular level set. This equation is solved by finite difference schemes borrowed from the technology of hyperbolic conservation laws. The two ideas are combined by constructing a smooth extension of the normal velocity off the moving boundary, in a way suggested by the physics of the problem. Our numerical experiments show the evolution of complex crystalline shapes, development of large spikes and corners, dendrite formation and side-branching, and pieces of solid merging and breaking off freely.

Anisotropic, time-dependent solutions in maximally Gauss-Bonnet extended gravity

In an arbitrary number of dimensions, we find the full exact anisotropic, time-dependent, diagonal-metric solutions to maximally Gauss-Bonnet extended gravity theory. This class of theories, for which the lagrangian is an arbitrary linear combination of dimensionally extended Euler forms, is the most general gravitational theory in which the field equations contain no more than second derivatives of the metric. We show that the space-time exponentially approaches an asymptotic state of constant, anisotropic curvature and prove three theorems concerning two generic types of singularities. The first theorem gives conditions for the existence of Kasner-like curvature singularities. For these the metric diverges as t Pi where ∑ p i = 2 k max- 1 and k max is the highest power of the curvature in the lagrangian. Other critical point singularities can arise from the polynomial nature of the theory. The remaining theorems demonstrate that the generic solution is extendible at all of these other critical points and that the generic critical points occur at moments of external volume density of space-time. We give an explicit coordinate transformation which produces a smooth extension through the critical point. The space-time may therefore alternately expand and contract for many cycles before expanding forever or contracting to a singularity. Many particular cases are treated in detail including several power series solutions, the generalized Kasner solution to general relativity with or without cosmological constant, the perturbative solution for quadratic string gravity, and five-dimensional extended gravity.

Computation of weighting functions for smoothing two-dimensional data by local polynomial approximation techniques

The problem of smoothing two-dimensional data by a local polynomial approximation technique (i.e., extension of Savitzky-Golay smoothing to two-dimensional applications) is considered. A simple analytical expression is proposed for computing the weighting functions used in the smoothing procedure with a polynomial of any degree and over any interval without loss of data at boundary points. The formulae obtained can be used for two dimensional smoothing by linear smoothing procedures, which considerably decreases the number of required operations.

Dynamic electromyography to assess elbow spasticity

Control of elbow motion was evaluated in 45 extremities of adults with spasticity resulting from traumatic brain injury with use of dynamic electromyography. Simultaneous recording of elbow motion was obtained using a double parallelogram goniometer. Thirty-four male and 9 female patients were studied. Mean elbow flexion was 85 degrees and mean extension was 20 degrees. The average time of elbow flexion was 1.8 seconds. Extension time was prolonged to a mean of 4.0 seconds. Dynamic electromyography revealed a consistent pattern of muscle activity. Severe spasticity was noted in the brachioradialis muscle. Moderate spasticity was present in the biceps and only mild spasticity was seen in the brachialis muscle. Normal phasic muscle activity was the rule in the triceps. All patients had active elbow flexion, but the flexor spasticity limited smooth extension. Elbow flexor spasticity, especially of the brachioradialis and biceps muscles, commonly interferes with hand placement. Lengthening of the biceps and brachialis tendons combined with release of the brachioradialis enhances elbow motion and improves hand placement.

Extensions of line-closed combinatorial geometries

In [4], line-closed combinatorial geometries were studied. Here, given a line-closed combinatorial geometry G( X), we determine all single point extensions of G( X) that are line-closed. Further, if H( X∪ Y) is a line-closed geometry that is a smooth extension of G( X) we give a natural necessary and sufficient condition for every geometry between G and H to be line-closed.

Smooth extensions for a finite 𝐶𝑊 complex

Smooth extensions for a finite 𝐶𝑊 complex

Smooth extension of norms and complementability of subspaces

Smooth extension of norms and complementability of subspaces

On smooth extensions of odd dimensional spheres and multidimensional Helton and Howe formula.

On smooth extensions of odd dimensional spheres and multidimensional Helton and Howe formula

Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings

A classical theorem of Carath6odory [8] states that every biholomorphic map f: D~ ~D2 between domains in the complex plane C bounded by simple closed Jordan curves extends to a homeomorphism of/31 onto/ )2 . There are some well-known generalizations of this result to domains in IE a. If D t and D2 are bounded pseudoconvex domains in 112" with if2 boundary and if the infinitezimal Kobayashi metric on D2 grows sufficiently fast near the boundary of D2(Ko2(g; X) >= [Xl/d(z, bDz) ~ for some e e (0, 1)), then every proper holomorphic map f : Dt--*D2 extends to a H61der-continuous map of/51 onto/32 [2, 3, 10, 25, 29-32]. This holds in particular if/92 is strictly pseudoconvex or if it is pseudoconvex with real-analytic boundary. The same result holds if D2 is only piecewise smooth strongly pseudoconvex [31]. Further results treat exceptional cases such as the balls [1 ], Reinhardt domains [5, 24], domains with many symmetries [4], and analytically bounded Hartogs domains in C 2 [I1]. Biholomorphic maps between certain types of nonpseudoconvex domains with real-analytic boundaries were treated in [13] and [27]. Besides the papers mentioned above there is a vast literature concerning smooth extension of proper holomorphic maps of smoothly bounded pseudoconvex domains ([6, 7, 12, 14], to mention just a few). In the present paper we obtain some results on the continuous extension of proper holomorphic maps f : D1--*/)2 under local assumptions on the boundaries bD1 and bD2. One of the main results is

Decay properties of highly excited nuclei produced in fast dissipative collisions

At bombarding energies of 10–20 MeV/u, fast dissipative collisions are found to represent a smooth extension of deep inelastic reactions, towards the intermediate energy regime.Although the main trends of the low energy systematics are confirmed new features emerge. In collisions among ( A ≈-100)-nuclei the large amount of kinetic energy deposited is used as a tool to produce nuclei of known excitation energy, up to values of ≈-=1 GeV. The evolution of the decay properties, with the emission of light charged particles as well as complex fragments and fission products, can be investigated continuosly as a function of excitation energy exploiting the full range of kinetic energy losses.The time scale of the emission can be investigated from proximity effects.

Rare gas clusters: Solids, liquids, slush, and magic numbers

Simulations by constant energy molecular dynamics have been performed for numerous clusters in the size range N=7–33. Detailed investigations have been conducted on the portions of the caloric curves in which the transition between rigid and nonrigid behavior occurs, to study the N dependence of the solid–liquid phase change. Clusters of several sizes display a coexistence of forms, each with a characteristic mean temperature, over a well-defined energy range in the transition region, as had been observed for the Ar13 cluster. Within the coexistence region, the high temperature form is solid-like and the low temperature form behaves in a liquid-like fashion. The caloric curves of state for these clusters take on two-valued forms when averages are calculated for each of the two ‘‘phases’’ separately; the two branches are smooth extensions of the curves from the single phase regions. Clusters of other sizes do not display this clear coexistence of phases, but appear to pass through a ‘‘slush-like’’ state during the melting transition. The coexistence behavior is not a smooth function of N. The clusters Ar13 and Ar19, and to a certain extent Ar7, display high stability properties indicative of magic number behavior.

Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples

This paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension. The concept of a lattice rule is introduced, and error bounds are developed in terms of the dual lattice. Examples of lattice rules discussed in the paper are the rectangle rule, the number-theoretic “good-lattice” methods of Korobov and others, the body-centred cubic rule, and a generalization of the latter to a family $\{ W_{nr} \} $. The $W_{nr} $ rules appear to have interesting extrapolation properties, both for fixed r and for $r = n$.

Forecasting irregularly spaced data: An extension of double exponential smoothing

Automated forecasts are often required, in practice, using data series from which certain points are missing and from data occurring at completely irregular time intervals. For instance, in computerised inventory control, fast methods of dealing with such data are required. There is an almost complete absence in the literature of computationally efficient methods for such a situation. This paper gives an extension of single and double exponential smoothing adapted to data occurring at irregular time intervals. These extensions are shown to have modest computational requirements and little sensitivity to initial conditions. Results of tests on sample data series are given showing only a minor decrease in accuracy with missing data, and indicating the appropriate method of choosing the smoothing parameter. Application of this method to published government time series is illustrated by two examples, firstly, to river water quality data originating from samples taken at irregular time intervals and, secondly, to divorce rate statistics from which certain points are missing due to summarizing the data. Successive summarizing of these series is found to have a negligible effect on forecast accuracy implying attractive cost saving possibilities in data collection and publication.

Smooth extensions of a class of 2n soliton metrics

A particular class of 2n soliton solutions of Einstein's equations, obtained from a flat background by the inverse scattering method of Belinsky and Zakharov, (1978), is considered. Straightforward application of the method leads to metrics whose coefficients are singular along certain hypersurfaces. The geometrical nature of the singularities is analysed. It is found that in some cases they represent curvature singularities. In the other cases they originate only on the choice of coordinates and one can find explicitly smooth extensions through these hypersurfaces. Part of the discussion is restricted to the two soliton metrics, but the results are immediately applicable in the general case.

Study of the leaking channel modes of in‐seam exploration seismology by means of synthetic seismograms

If the dispersion characteristics of coal‐seam channel guides do not extend smoothly at frequencies below cut‐off, the number of parameters required to process inseam seismic data increases. Unwanted modes become more difficult to suppress, the arriving pulses from targets become more difficult both to identify and to pulse compress. The connection between leaking and normal channel mode dispersion is established here by first synthesizing and then analyzing theoretical in‐seam seismograms. Calculation of synthetic seismograms is based on numerical evaluation of the spatial Fourier integral for elastic displacements in the complex wavenumber plane. Theoretical seismograms are presented for three‐layered models. Phase velocity characteristics are recovered from these signals and compared with those obtained from the zero loci of the period equation in the complex wavenumber plane. Under cut‐off the former method yields smooth extensions of the normal mode dispersion characteristics, in contrast to the velocity curves obtained from the period equation only. It is found that the dispersion characteristics obtained from analyzing the seismograms can be used to recompress the dispersed arrivals.

On the physical interpretation of some simple soliton solutions of Einstein's equations

The simple soliton solutions of Einstein's equations obtained by Belinski and Zakharov using the inverse scattering method have been interpreted as gravitational (solitary) shock waves, partly on the basis of an analysis of certain (coordinate) singularities apparently inherent to the method. A closer study reveals, however, that such singularities can be removed by an appropriate extension of the solutions, which is given explicitly. A classification of inequivalent flat space-time metrics appropriate for the applications of the method is derived. The problem of matching the Belinski-Zakharov (B-Z) simple solitons to flat space-time is analyzed and found to have more than one solution depending on the type of singularity admitted in the Ricci tensor. This is further illustrated by considering a three-parameter solution, inequivalent to that of Belinski and Zakharov. For negative values of one of these parameters the ranges of the coordinates are limited only by curvature singularities. For positive values of the parameter, coordinate singularities, similar to those in the B-Z solution, are also present. In this case, however, a matching to flat space-time leads to a shock front whose intersection with any spacelike hypersurface is bounded, in contrast with the behavior displayed by the B-Z solutions. The limiting case when the parameter is zero is found to have some special properties. A smooth extension is also shown to exist.

Remarks on uniformity in hyperelastic materials

The basic notions of material uniformity, originally developed by Noll and Wang, are presented using more traditional mathematical tools. Attention is focused on hyperelastic materials and new results are obtained pertaining to the arbitrariness of the strain energy level. The role of the isotropy group in allowing for the smooth extension of a material connection over the whole body is brought out.

Potential-energy surfaces for chemical reactions. Dimerization of CH2 and SiH2, the S N2 reaction in gas-phase clusters and CH activation in transition-metal complexes

We present the results of three applications of the molecular-orbital method to problems of potential-energy surfaces that control chemical reactions. In the first application the dimerization of CH2 and SiH2 in both their singlet and triplet states is investigated in connection with the path of least motion as against the path of non-least motion. Two ground-state (3B1) methylenes in the path of non-least motion give ground-state ethylene with no barrier. The ground-state (1A1) silylenes give a ground-state disilene with a barrier in the path of least motion and no barrier in the path of non-least motion. In the second problem potential-energy surfaces are calculated for an SN2 reaction (H2O)nOH–+ CH3Cl → HOCH3+ Cl–+nH2O, where the reactants are complexed with one or two water molecules. When the hydroxide ion is solvated by two water molecules, the reaction takes place through the first step of reactant complex formation, followed by inversion of the methyl group. The transition state for methyl inversion has an energy comparable to that of the reactants. The migration of water molecules from the hydroxide side to the chloride side is not involved in the rate-determining process. The last problem is concerned with the activation of an inert CH bond in transition-metal complexes. In six-coordinate Ti d0 complexes the optimized geometry shows theoretical evidence that the distortion of the ethyl or methyl ligand, represented by a short M⋯H distance, a small M⋯C—C (or H) angle and a long CH bond, is of electronic origin. Electronegative axial ligands are essential for the existence of an agostic interaction, which is stabilized by CH σ→ Ti dxy charge transfer. A similarly distorted ethyl group has been found in the calculation of a three-coordinate Pd complex. The low-energy transition state for β-elimination lies along a smooth extension of the ethyl distortion.

Trapping and reaction rates on fractals: A random-walk study

In this Rapid Communication we study random walks on Sierpinski gaskets, which are fractal structures of simple geometry. We determine the probability of the walker to be captured by traps randomly distributed on the gaskets. The general reaction process is modeled through a kinetic approach. The decay laws which we find are smooth extensions to dimensionalities between one and two.