Gauss Equation Research Articles

Stability and eigenvalue estimates of linear Weingarten hypersurfaces in a sphere

Let M be an n-dimensional compact hypersurface without boundary in a unit sphere Sn+1(1). M is called a linear Weingarten hypersurface if cR+dH+e=0, where c,d and e are constants with c2+d2>0, R and H denote the scalar curvature and the mean curvature of M, respectively. By the Gauss equation, we can rewrite the condition cR+dH+e=0 as (n−1)ẽH2+aH=b, where H2 is the 2nd mean curvature, a, b and ẽ are constants such that a2+ẽ2>0, when ẽ=0, it reduces to the constant mean curvature case.In this paper, we obtain some stability results about linear Weingarten hypersurfaces, which generalize the stability results about the hypersurfaces with constant mean curvature or with constant scalar curvature. We show that linear Weingarten hypersurfaces satisfying (n−1)H2+aH=b, where a and b are constants, can be characterized as critical points of the functional ∫M(a+nH)dv for volume-preserving variations. We prove that such a linear Weingarten hypersurface is stable if and only if it is totally umbilical and non-totally geodesic. We also obtain optimal upper bounds for the first and second eigenvalues of the Jacobi operator of linear Weingarten hypersurfaces.

Linear Surface Reconstruction from Discrete Fundamental Forms on Triangle Meshes

AbstractWe present a linear algorithm to reconstruct the vertex coordinates for a surface mesh given its edge lengths and dihedral angles, unique up to rotation and translation. A local integrability condition for the existence of an immersion of the mesh in 3D Euclidean space is provided, mirroring the fundamental theorem of surfaces in the continuous setting (i.e. Gauss's equation and the Mainardi–Codazzi equations) if we regard edge lengths as the discrete first fundamental form and dihedral angles as the discrete second fundamental form. The resulting sparse linear system to solve for the immersion is derived from the convex optimization of a quadratic energy based on a lift from the immersion in the 3D Euclidean space to the 6D rigid motion space. This discrete representation and linear reconstruction can benefit a wide range of geometry processing tasks such as surface deformation and shape analysis. A rotation‐invariant surface deformation through point and orientation constraints is demonstrated as well.

A mathematical formulation of the random phase approximation for crystals

This works extends the recent study on the dielectric permittivity of crystals within the Hartree model [E. Cancès, M. Lewin, Arch. Ration. Mech. Anal. 197 (1) (2010) 139–177] to the time-dependent setting. In particular, we prove the existence and uniqueness of the nonlinear Hartree dynamics (also called the random phase approximation in the physics literature), in a suitable functional space allowing to describe a local defect embedded in a perfect crystal. We also give a rigorous mathematical definition of the microscopic frequency-dependent polarization matrix, and derive the macroscopic Maxwell–Gauss equation for insulating and semiconducting crystals, from a first order approximation of the nonlinear Hartree model, by means of homogenization arguments.

On conformally invariant equations on [formula omitted]-II. Exponential invariance

In this paper we provide a complete characterization of fully nonlinear differential operators of any integer order on Rn, which exhibit conformal invariance of exponential type. In this way we intend to complete the work that we undertook in Li et al. (2011) [2], where we introduced the family of elementary conformal tensors {Tm,αu} in order to describe all fully nonlinear differential operators of any integer order on Rn which are conformally invariant of degree α≠0. Examples of the differential operators that we study in this paper are those related to the Q–curvature equation on R4 and to the Gauss equation on R2.

Explicit simulation of electrified clouds: From idealized to real case studies

Original results obtained with the revised version of the explicit cloud electrification module developed in the French mesoscale model MesoNH are presented. The module includes three ingredients: (i) a prognostic budget equation of the electric charges carried by each type of hydrometeor and by the positive and negative ions, (ii) a computation of the electric field by inverting the Gauss equation and (iii) a statistical algorithm to produce intra-cloud and cloud-to-ground discharges. Electric charges are generated by non-inductive (NI) processes involving ice particles that collide in the presence of supercooled water. The geometry of the lightning flashes obeys a fractal law that characterizes the densely branched flash structures with a marked horizontal extent in electrically charged regions of the clouds.For the first time, the electrical module of MesoNH is used to simulate the electrical aspects of a heavy precipitation event over the Cevennes area in the South of France. MesoNH is initialized with “AROME” analyses of Meteo-France for the 6–8 September 2010 event that produced locally 250 to 300mm of precipitation over the Cevennes ridge. A 54hour simulation starting 6 September 2010 at 00 UTC and without electricity, is performed at 1km resolution over a 384×384km2 area in order to check the ability of the model to reproduce the very fine scale structure of the rainfall. MesoNH is then restarted on 7 September 2010 at noon for a 6hour simulation involving the full electrical module. Several aspects of the electrical state of the clouds could be simulated at this scale: the cloud charge structure, the electric field and the lightning flash characteristics. Finally a map of flash density is compared to observations taken from two conventional detection networks (LInet and ATDnet). Both networks suggest that the model produces ten times too many flashes but arguments are put forward to explain this discrepancy, e.g., filtering the less energetic flashes improves the comparison. In conclusion, this study calls for a reference evaluation of the electrical module of MesoNH against statistics of flash data at high resolution in a next planned field experiment in 2012.

Electric field localizations in thin dielectric films with thickness non-uniformities

Thickness non-uniformities in electrostatic capacitors and in electroactive polymers – arising from manufacturing processes or electromechanical induced inhomogeneous deformations – may lead to drastic charge and electric field localizations and, ultimately, to an anticipated device failure. Based on a geometric interpretation of the Gauss equation enlightening the effect of the electrode curvature, we obtain an analytic expression of the electric field and of the surface charge density localization for non perfectly planar capacitors with symmetric thickness non-uniformities. The efficiency of the model is exploited by analyzing specific boundary value problems of technological interest.

CELLS v1.0: updated and parallelized version of an electrical scheme to simulate multiple electrified clouds and flashes over large domains

Abstract. The paper describes the fully parallelized electrical scheme CELLS which is suitable to simulate explicitly electrified storm systems on parallel computers. Our motivation here is to show that a cloud electricity scheme can be developed for use on large grids with complex terrain. Large computational domains are needed to perform real case meteorological simulations with many independent convective cells. The scheme computes the bulk electric charge attached to each cloud particle and hydrometeor. Positive and negative ions are also taken into account. Several parametrizations of the dominant non-inductive charging process are included and an inductive charging process as well. The electric field is obtained by inverting the Gauss equation with an extension to terrain-following coordinates. The new feature concerns the lightning flash scheme which is a simplified version of an older detailed sequential scheme. Flashes are composed of a bidirectional leader phase (vertical extension from the triggering point) and a phase obeying a fractal law (with horizontal extension on electrically charged zones). The originality of the scheme lies in the way the branching phase is treated to get a parallel code. The complete electrification scheme is tested for the 10 July 1996 STERAO case and for the 21 July 1998 EULINOX case. Flash characteristics are analysed in detail and additional sensitivity experiments are performed for the STERAO case. Although the simulations were run for flat terrain conditions, they show that the model behaves well on multiprocessor computers. This opens a wide area of application for this electrical scheme with the next objective of running real meterological case on large domains.

Numerical approximation of the Euler–Maxwell model in the quasineutral limit

We derive and analyze an Asymptotic-Preserving scheme for the Euler–Maxwell system in the quasi-neutral limit. We prove that the linear stability condition on the time-step is independent of the scaled Debye length λ when λ → 0. Numerical validation performed on Riemann initial data and for a model plasma opening switch device show that the AP-scheme is convergent to the Euler–Maxwell solution when Δ x/ λ → 0 where Δ x is the spatial discretization. But, when λ/Δ x → 0, the AP-scheme is consistent with the quasi-neutral Euler–Maxwell system. The scheme is also perfectly consistent with the Gauss equation. The possibility of using large time and space steps leads to several orders of magnitude reductions in computer time and storage.

Vacuum Spacetimes of Embedding Class Two

Doubt is cast on the much quoted results of Yakupov that the torsion vector in embedding class two vacuum space-times is necessarily a gradient vector and that class 2 vacua of Petrov type III do not exist. The first result is equivalent to the fact that the two second fundamental forms associated with the embedding necessarily commute and has been assumed in most later investigations of class 2 vacuum space-times. Yakupov stated the result without proof, but hinted that it followed purely algebraically from his identity: RijklCkl = 0 where Cij is the commutator of the two second fundamental forms of the embedding.From Yakupov's identity, it is shown that the only class two vacua with non-zero commutator Cij must necessarily be of Petrov type III or N. Several examples are presented of non-commuting second fundamental forms that satisfy Yakupovs identity and the vacuum condition following from the Gauss equation; both Petrov type N and type III examples occur. Thus it appears unlikely that his results could follow purely algebraically. The results obtained so far do not constitute definite counter-examples to Yakupov's results as the non-commuting examples could turn out to be incompatible with the Codazzi and Ricci embedding equations. This question is currently being investigated.

Pseudo-Hermitian Geometry in 3-D

Pseudo-hermitian geometry is the study of contact form in conjunction with an almost complex structure on the contact planes. In this two part survey, we first discuss the theory of surfaces in 3-D pseudo-hermitian manifolds. The local geometry of the surface is governed by the p-mean curvature equation. In analogy with the Gauss and Codazzi equation of a surface in Euclidean three space, we develop their analogue of these two equations. Due to the degeneracy of the p-mean curvature equation, the regularity theory is quite interesting. In the second part of this article, we describe a CR-invariant condition for the embeddability question. In CR geometry in 3-D, there is no local integrability condition for the almost-complex structure, so the condition to impose is global in nature. The condition involves two geometrically defined linear operators which have their counterparts in conformal geometry. We also formulate a notion of CR mass and discuss a positive mass theorem to characterize the Heisenberg space. The solvability of the \({\bar\partial}\) Neumann problem is also a crucial ingredient in the positivity of the CR mass.

Acousto-Diffusive Waves in a Piezoelectric-Semiconductor-Piezoelectric Sandwich Structure

The propagation of acoustic waves in a homogeneous isotropic semiconducting layer sandwiched between two homogeneous transversely isotropic piezoelectric halfspaces has been investigated. The mathematical model of the problem is depicted by a set of partial differential equations of motion, Gauss equation in piezoelectric material and electron diffusion equation in semiconductor along with the boundary conditions to be satisfied at the piezoelectric-semiconductor interfaces. The secular equations describing the symmetric and asymmetric modes of wave propagation have been derived in compact form after obtaining the analytical expressions for various field quantities that govern the wave motion. The complex secular equation has been solved numerically using functional interaction method along with irreducible cardano method. The computer simulated results are obtained with the help of MATLAB software for 6 mm cadmium selenide (CdSe) piezoelectric material and n-type silicon (Si) semiconductor in respect of dispersion curve, attenuation and specific loss factor of energy dissipation for symmetric (sym) and asymmetric (asym) modes of wave propagation. The study may find applications in non-destructive testing, resonators, waveguides etc.

Surface Wave Characteristics at the Interface of Welded Elastic Halfspaces

The present article concentrates on the propagation of generalized surface acoustic waves in a composite struc- ture consisting of piezoelectric and non-piezoelectric semiconductor media. The mathematical model of the problem is depicted by a set of partial differential equations of motion, Gauss equation in piezoelectric and elec- tron diffusion equation in semiconductor along with boundary conditions to be satisfied at the interface. The secular equation that governs the propagation of surface waves has been derived in compact form after obtaining the formal solution. The analytic expressions for displacements, stresses, piezoelectric potential and electron concentration during the surface wave propagation at the interface have also been obtained. The numerical solu- tion of the secular equation is carried out for the cadmium selenide and silicon composite by employing fixed point functional iteration numerical method along with irreducible Cardano method. The computer simulated results with the help of MATLAB software in respect of dispersion curves, attenuation coefficient, displace- ments, stresses, carrier concentration and piezoelectric potential are presented graphically. This work may be useful in surface acoustic wave (SAW) devices and electronic industry.

Erratum to: The upperbound of the volume expansion rate in a Lorentzian manifold

where s(Ht ) and s M denote, respectively, the scalar curvature of level hypersurface Ht and M at the point γ (t) and θ(t) is the mean curvature of Ht along γ (t). To calculate the upperbound of the volume expansion rate along a timelike geodesic γ in a Lorentzian manifold, we used the above differential Eq. (1). But we neglected the sign when we take the trace of the Gauss equation (p. 408). Taking the sign of an unit speed timelike geodesic into account, Eq. (1) must be corrected to

Classification of integrable Weingarten surfaces possessing an (2)-valued zero curvature representation

In this paper we classify Weingarten surfaces integrable in the sense of soliton theory. The criterion is that the associated Gauss equation possesses an (2)-valued zero curvature representation with a nonremovable parameter. Under certain restrictions on the jet order, the answer is given by a third order ordinary differential equation to govern the functional dependence of the principal curvatures. Employing the scaling and translation (offsetting) symmetry, we give a general solution of the governing equation in terms of elliptic integrals. We show that the instances when the elliptic integrals degenerate to elementary functions were known to nineteenth-century geometers. Finally, we characterize the associated normal congruences.

Braneworld gravity in a symmetric space bulk

By considering the p-brane motion in a G/K symmetric space bulk we identify the G-invariant bulk metric in the solvable Lie algebra gauge of the brane action. After calculating the Levi-Civita connection of this bulk metric we use it in the Gauss equation to compute the braneworld curvature in terms of the bulk coordinates. Finally, by making use of the Gauss equation in the braneworld Einstein equation we present a geometrical method of implementing the first fundamental form in the gravitating brane dynamics for the specially chosen symmetric space bulk case leading to an Einstein equation expressed solely in terms of the bulk coordinates of the braneworld.

On an invariant on isometric immersions into spaces of constant sectional curvature

In the present paper, we give an invariant on isometric immersions into spaces of constant sectional curvature. This invariant is a direct consequence of the Gauss equation and the Codazzi equation of isometric immersions. We apply this invariant on some examples. Further, we apply it to codimension 1 local isometric immersions of 2-step nilpotent Lie groups with arbitrary leftinvariant Riemannian metric into spaces of constant nonpositive sectional curvature. We also consider the more general class, namely, three-dimensional Lie groups G with nontrivial center and with arbitrary left-invariant metric. We show that if the metric of G is not symmetric, then there are no local isometric immersions of G into Q c 4.

A brief introduction to Cadabra: A tool for tensor computations in General Relativity

Cadabra is a powerful computer program for the manipulation of tensor equations. It was designed for use in high energy physics but its rich structure and ease of use lends itself well to the routine computations required in General Relativity. Here we will present a series of simple examples showing how Cadabra may be used, including verifying that the Levi-Civita connection is a metric connection and a derivation of the Gauss equation between induced and ambient curvatures.

A new method of characterizing equivalent strain for equal channel angular processing

In order to establish the quantitative relationship between equivalent strain and the performance index of the deformed material within the range of certain passes for equal channel angular processing (ECAP), a new approach to characterize the equivalent strain was proposed. The results show that there exists better accordance between mechanical property (such as hardness or strength) and equivalent strain after rolling and ECAP in a certain range of deformation amount, and Gauss equation can be satisfied among the equivalent strain and the mechanical properties for ECAP. Through regression analysis on the data of hardness and strength after the deformation, a more generalized expression of equivalent strain for ECAP is proposed as: e=k0exp[-(k1M-k2)^2], where M is the strength or hardness of the material, k1 is the modified coefficient (k1 ∈((0, 1)), k0 and k2 are two parameters dependent on the critical strain and mechanical property that reaches saturation state for the material, respectively. In this expression the equivalent strain for ECAP is characterized novelly through the mechanical parameter relating to material property rather than the classical geometry equation.

On the asymptotic behavior of holomorphic isometries of the Poincaré disk into bounded symmetric domains

In this article we study holomorphic isometries of the Poincare disk into bounded symmetric domains. Earlier we solved the problem of analytic continuation of germs of holomorphic maps between bounded domains which are isometries up to normalizing constants with respect to the Bergman metric, showing in particular that the graph V0 of any germ of holomorphic isometry of the Poincaré disk Δ into an irreducible bounded symmetric domain Ω ⋐ ℂN in its Harish-Chandra realization must extend to an affine-algebraic subvariety V C ⊂ = ℂ×ℂN = ℂN+1, and that the irreducible component of V∩(Δ×Ω) containing V0 is the graph of a proper holomorphic isometric embedding F: Δ → Ω. In this article we study holomorphic isometric embeddings which are asymptotically geodesic at a general boundary point b ∈ δΔ. Starting with the structural equation for holomorphic isometries arising from the Gauss equation, we obtain by covariant differentiation an identity relating certain holomorphic bisectional curvatures to the boundary behavior of the second fundamental form a of the holomorphic isometric embedding. Using the nonpositivity of holomorphic bisectional curvatures on a bounded symmetric domain, we prove that ∥ρ∥ must vanish at a general boundary point either to the order 1 or to the order ½ called a holomorphic isometry of the first resp. second kind. We deal with special cases of non-standard holomorphic isometric embeddings of such maps, showing that they must be asymptotically totally geodesic at a general boundary point and in fact of the first kind whenever the target domain is a Cartesian product of complex unit balls. We also study the boundary behavior of an example of holomorphic isometric embedding from the Poincare disk into a Siegel upper half-plane by an explicit determination of the boundary behavior of holomorphic sectional curvatures in the directions tangent to the embedded Poincare disk, showing that the map is indeed asymptotically totally geodesic at a general boundary point and of the first kind. For the metric computation we make use of formulas for symplectic geometry on Siegel upper half-planes.

IR-spectroscopic studies of hydrogen-bonding solutions: Lineshape analysis of ethanol + hexane system

IR (infrared) line shapes of OH stretching bands in hydrogen-bonding states of ethanol–hexane solutions have been examined using a theoretically rigorous Voigt function, instead of the commonly used empirical Lorentz–Gauss equation. An efficient and numerically-stable computer code with the Voigt function has been developed in order to analyze hydrogen-bonding effects in IR absorption spectra. Although the Voigt function is not an analytically-closed form, a numerical computer subroutine, with only three adjustable parameters for each IR band (or peak), provides a unique solution in a least-squares analysis of the line shapes. On the other hand, the Lorentz–Gauss equation is an analytical function with four adjustable parameters for each IR peak. However, this function makes non-linear least-squares analyses almost impossible due to the existence of strong correlations among the parameters. Example analyses, using the ethanol–hexane system, are given. The degree of hydrogen-bonding associations of ethanol and an effective IR-absorbance coefficient for hydrogen-bonded clusters have been estimated, and results are quite reasonable, compared with those in the literature.