A geometrical model for the double octahedral group based on a genus two Riemann surface

Geometrical models of double point groups are provided by double-sheeted Riemann surfaces of nonzero genus. This avoids the geometrically confusing picture of the doubling operation R as a rotation by 2π (i.e. 360°), which should instead be the identity operation E. In this manner a Riemann surface of genus 2 doubly covering a sphere and platonically tessellated b 16 equilateral triangles provides a geometrical model for the double octahedral group ²O_h. Stretching this Riemann surface along one axis to convert the 16 equilateral triangles to isosceles triangles and the underlying sphere to a prolate ellipsoid provides a model for the ²D_4h double group arising from the Jahn-Teller elongation of the regular octahedron into a prolate tetragonal bipyramid.     

Odd Laplacians: geometrical meaning of potential and modular class

A second-order self-adjoint operator \(\Delta =S\partial ^2+U\) is uniquely defined by its principal symbol S and potential U if it acts on half-densities. We analyse the potential U as a compensating field (gauge field) in the sense that it compensates the action of coordinate transformations on the second derivatives in the same way as an affine connection compensates the action of coordinate transformations on first derivatives in the first-order operator, a covariant derivative, \(\nabla =\partial +\Gamma \). Usually a potential U is derived from other geometrical constructions such as a volume form, an affine connection, or a Riemannian structure, etc. The story is different if \(\Delta \) is an odd operator on a supermanifold. In this case, the second-order potential becomes a primary object. For example, in the case of an odd symplectic supermanifold, the compensating field of the canonical odd Laplacian depends only on this symplectic structure and can be expressed by the formula obtained by K. Bering. We also study modular classes of odd Poisson manifolds via \(\Delta \)-operators, and consider an example of a non-trivial modular class which is related with the Nijenhuis bracket.     

The geometrical and vibrational properties of the Rh(111) surface

Low-energy electron diffraction (LEED) data have been used to characterize the clean Rh(111) surface. The surface geometry, the degree of surface relaxation, and the Debye temperature have been determined. In the Debye temperature measurement, specular LEED beam intensities were monitored as a function of temperature over a range of electron energies from approximately 30 to 1000 eV. It was found that the bulk Debye temperature is 380 ± 23 K, and the normal component of the Debye temperature at the lowest electron energy used is 197 ± 12 K. The Rh(111) surface relaxation has been determined both by a convolution-transform analysis and by dynamical calculations. Within experimental error, neither expansion nor contraction of the topmost layer has been detected. The results of the convolution-transform analysis of specular beams at two angles of incidence and of a nonspecular beam at normal incidence suggest an expansion of the topmost layer of 3 ± 5% of the bulk layer spacing. In agreement with this, comparisons between the results of the dynamical calculation and experimental data for five nonspecular beams at normal incidence suggest that the surface layer relaxes by 0 ± 5%. In addition, the dynamical calculations indicate that the topmost layer maintains an fcc structure.     

Fermi surface and band mapping of the cerium/palladium surface alloy

X-ray and UV excitation angle-resolved photoemission spectroscopy of ultra-thin films of cerium deposited on Pd(1 1 1) single-crystal surface has been carried out. Photoelectron diffraction pattern showed that deposition of 1 ML of Ce led to a formation of Ce-Pd substitutional alloy. Valence band spectra measured with high angular resolution permitted to plot valence band maps and Fermi surface scans and showed formation of surface alloy exhibiting d- and f-electron orbital hybridization. A shift of Pd 4d-derived states to higher binding energy in the Ce-Pd systems was observed.     

Calibration of geometrical systematic error in high-precision spherical surface measurement

Geometric aberrations in interferometric testing system can significantly influence the measurement results in the case of high-numerical-aperture test spherical surface, in which obvious high-order aberrations introduced by wavefront defocus could be observed and they cannot be removed with the traditional calibration method. A technique based on the rigorous model for the analysis of geometric aberrations introduced by wavefront tilt and defocus, is presented for the calibration of the corresponding geometrical systematic error. The calibration method can be carried out either with or without a prior knowledge of the spherical surface under test. The feasibility of the proposed method has been demonstrated by computer simulation, and the residual error less than 0.001λ is obtained. Experimental validation is carried out by testing a high-numerical-aperture spherical surface with the ZYGO interferometer, and an accuracy RMS about 0.003λ with the proposed calibration technique is achieved. The effect of geometric aberrations on the measurement is discussed in detail. The proposed calibration method provides a feasible way to lower the requirement on the adjusting precision of mechanical device, and is of great practicality for the high-precision measurement of high-numerical-aperture spherical surface.     

A geometrical setting for the Lax representation

A geometrical interpretation of the Lax representation is suggested; it is read as the condition on a section of a fiber bundle based on the phase manifold to be covariantly constant, with respect to a suitable connection, along the dynamics.     

Reduction of surface coverage of finite systems due to geometrical steps

The coverage of vicinal, stepped surfaces with molecules is simulated with the help of a two-dimensional Ising model including local distortions and an Ehrlich-Schwoebel barrier term at the steps. An effective two-spin model is capable to describe the main properties of this distorted Ising model. It is employed to analyze the behavior of the system close to the critical points. Within a well-defined regime of bonding strengths and Ehrlich-Schwoebel barriers we find a reduction of coverage (magnetization) at low temperatures due to the presence of the surface step. This results in a second, low-temperature transition besides the standard Ising order-disorder transition. The additional transition is characterized by a divergence of the susceptibility as a finite-size effect. Due to the surface step the mean-field specific heat diverges with a power law.     

Deformation mapping and surface inspection of historical monuments

Electronic speckle pattern interferometry (EPSI) in various configurations (reference beam or shear) as well as double exposure speckle photography are applied for deformation monitoring in historical monuments. On-the-site measurements are made possible by a rigid construction of the optical head, common path configurations and portable image processing equipment. The delicate objects particularly call for non-intrusive measurement. Basic features of the techniques, their implementation and results from historical sites are presented. Inhomogeneities in plaster, paint or natural stones were detected and related to crack formation. The reversibility of deformations under mechanical load was checked. In frescos, debonds from the carrier wall could be discovered and the detrimental effects of crystalline efflorescences were studied. Correlation analysis in ESPI images showed the influence of moisture on a rough stone surface and exposed microbiological activity. Thus, valuable deformation data could be obtained that assist in the diagnostics of deterioration processes and provide information in the design of remedies.     

A geometrical relationship for the Einstein and Ricci tensors

Components of the Ricci and Einstein tensors are expressed in terms of the Gaussian curvatures of elementary two-spaces formed by the orthogonal coordinate planes, and the results are applied to some standard metrics.     

Economic mapping to the renormalization group scaling of stock markets

We make an attempt to map a simple economically motivated model for price evolution [J. Phys. A 33, 3637 (2000)] to the phenomenological renormalization group scaling of stock markets. This mapping gives insight into the critical exponents and the renormalization group predictions for the log-periodic oscillations preceding some stock market crashes from the perspective of non-linear changes in `the level of stock'. Received 7 August 2000     

A geometrical interpretation of the null sectional curvature

A geometrical interpretation is given for the null sectional curvature of degenerate planes in a Lorentzian manifold. This interpretation is based on a generalization to the indefinite case of the squaroids of Levi-Civita. Further, it is shown that a three-dimensional, conformally flat Lorentzian manifold has isotropic and spatially constant null sectional curvature if and only if it is locally a Robertson–Walker manifold.     

A new class of the surface shear magnetoacoustic waves in antiferromagnetic crystals

It is shown that the hybridization of magnetoelastic and dipolar interactions may give rise (even in a zero magnetic field) to a new type of surface shear magnetoacoustic waves near a mechanically free or an acoustically continuous interface between uniaxial ferromagnetic and nonmagnetic media. The effect of the conductivity of the nonmagnetic medium on the localization conditions of this surface mode is studied.     

非刚性表面声呐目标回波的修正几何亮点模型

基于修正的Kirchhoff近似公式,本文提出了包括幅度、时延、相位跳变和局部平面波反射系数的非刚性表面声呐目标的修正四参数几何亮点模型。用原来的和修正的几何亮点模型对刚硬、裸露和敷设消声瓦的充水椭圆锥台壳体的目标强度进行了计算。结果表明原来的三参数几何亮点模型是在高频情况下未敷瓦壳体的一级近似。对于实际壳体,无论敷瓦还是未敷瓦都应采用修正的四参数几何亮点模型计算目标强度。     

Quantitative mapping of diffusion characteristics under the cortical surface

Recent studies have demonstrated regional segregations on several peripheral white matter (WM) regions, which may imply different anatomical or functional characteristics [Cereb Cortex 17(4) 2007 816–25; Neuroimage 37(2) 2007 599–610; J Cogn Neurosci 16(7) 2004 1227–33]. Nonetheless, little is known about overall patterns of peripheral WM across the regions. In this study, diffusion tensor imaging with 2-mm isovoxel resolution and cortical surface mapping were combined to determine peripheral WM structure. Fractional anisotropy (FA) mapping showed consistent regional patterns across the young normal subjects while significant high or low FA values were shown in the motor-somatosensory cortex, prefrontal cortex, temporal, and medial occipital cortex. By adopting both region of interest and connectivity analysis, results were then discussed with structural network properties as well as WM maturation process.     

A geometrical description of local and global anomalies

The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected ‘gauge’ group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies.     

A geometrical theory of spin motion

A discussion of the fundamental interrelation of geometry and physical laws with Lie groups leads to a reformulation and heuristic modification of the principle of inertia and the principle of equivalence, which is based on the simple de Sitter group instead of the Poincaré group. The resulting law of motion allows a unified formulation for structureless and spinning test particles. A metrical theory of gravitation is constructed with the modified principle, which is structured after the geometry of the manifold of the de Sitter group. The theory is equivalent to a particular Kaluza-Klein theory in ten dimensions with the Lorentz group as gauge group. A restricted version of this theory excludes torsion. It is shown by a reformulation of the energy momentum complex that this version is equivalent to general relativity with a cosmologic term quadratic in the curvature tensor and in which the existence of spinning particle fields is inherent from first principles. The equations of the general theory with torsion are presented and it is shown in a special case how the boundary conditions for the torsion degree of freedom have to be chosen such as to treat orbital and spin angular momenta on an equal footing. The possibility of verification of the resulting anomalous spin-spin interaction is mentioned and a model imposed by the group topology ofSO(3,2) is outlined in which the unexplained discrepancy between the magnitude of the discrete valued coupling constants and the gravitational constant in Kaluza-Klein theories is resolved by the identification of identical fermions as one orbit. The mathematical structure can be adapted to larger groups to include other degrees of freedom.     

A geometrical approach to the integrability of soliton equations

A separability criterion in one-degree-of-freedom dynamics, suitable for soliton equations, is given in terms of a geometrical structure on the phase manifold. For solitonic degrees of freedom, i.e., those corresponding to the discrete spectrum of the associated Lax operator, integrability is a priori proved.     

Automatic computation of the geometrical characteristics of a surface modified by a scanning nanohardness tester

A technique for automatic determination of the contact area of a remnant impression obtained by nanoindentation with regard to a surrounding relief is suggested. An algorithm on which the suggested technique is based and an algorithm to automatically correct scanning defects are described. The hardness values determined by dynamic nanoindentation and by direct measurement of the impression are compared.