Curvature invariants, volume functions and spectral expansions

1982 ◽  
Vol 13 (2) ◽  
Author(s):  
Zvi Har'el
Keyword(s):  
Spectral Expansions ◽  
Curvature Invariants ◽  
Volume Functions

scholarly journals Kinematic formula and tube formula in space of constant curvature

1990 ◽  
Vol 33 (1) ◽  
pp. 79-88
Author(s):  
Sungyun Lee
Keyword(s):  
Euclidean Space ◽  
Euler Characteristic ◽  
Constant Curvature ◽  
Kinematic Formula ◽  
Curvature Invariants ◽  
Space Of Constant Curvature

The Euler characteristic of an even dimensional submanifold in a space of constant curvature is given in terms of Weyl's curvature invariants. A derivation of Chern's kinematic formula in non-Euclidean space is completed. As an application of above results Weyl's tube formula about an odd-dimensional submanifold in a space of constant curvature is obtained.

Common volume functions and diffraction line profiles of polyhedral domains

2012 ◽  
Vol 45 (6) ◽  
pp. 1162-1172 ◽  
Author(s):  
Alberto Leonardi ◽  
Matteo Leoni ◽  
Stefano Siboni ◽  
Paolo Scardi
Keyword(s):  
Diffraction Pattern ◽  
Excellent Agreement ◽  
Numerical Algorithm ◽  
Theoretical Description ◽  
Fast Computation ◽  
Line Profiles ◽  
Cubic Materials ◽  
Equilibrium Shapes ◽  
The Common ◽  
Volume Functions

A general numerical algorithm is proposed for the fast computation of the common volume function (CVF) of any polyhedral object, from which the diffraction pattern of a corresponding powder can be obtained. The theoretical description of the algorithm is supported by examples ranging from simple equilibrium shapes in cubic materials (Wulff polyhedra) to more exotic non-convex shapes, such as tripods or hollow cubes. Excellent agreement is shown between patterns simulated using the CVF and the corresponding ones calculated from the atomic positionsviathe Debye scattering equation.

scholarly journals Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

2015 ◽  
Vol 24 (10) ◽  
pp. 1550079 ◽  
Author(s):  
Jens Boos
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
General Type ◽  
Coordinate Systems ◽  
Irreducible Decomposition ◽  
Asymptotically Flat ◽  
Type D ◽  
Curvature Invariants ◽  
Free Parameters ◽  
First Time

Analogies between gravitation and electromagnetism have been known since the 1950s. Here, we examine a fairly general type D solution — the exact seven parameter solution of Plebański–Demiański (PD) — to demonstrate these analogies for a physically meaningful spacetime. The two quadratic curvature invariants B2 - E2 and E⋅B are evaluated analytically. In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitoelectric mass and gravitoelectric current of the PD solution, respectively, if there are no gravitomagnetic monopoles present. Furthermore, the square of the Bel–Robinson tensor reads (B2 + E2)2 for the PD solution, reminiscent of the square of the energy density in electrodynamics. By analogy to the energy–momentum 3-form of the electromagnetic field, we provide an alternative way to derive the recently introduced Bel–Robinson 3-form, from which the Bel–Robinson tensor can be calculated. We also determine the Kummer tensor, a tensor cubic in curvature, for a general type D solution for the first time, and calculate the pieces of its irreducible decomposition. The calculations are carried out in two coordinate systems: In the original polynomial PD coordinates and in a modified Boyer–Lindquist-like version introduced by Griffiths and Podolský (GP) allowing for a more straightforward physical interpretation of the free parameters.

scholarly journals BORN–INFELD COSMOLOGIES

2000 ◽  
Vol 15 (27) ◽  
pp. 4341-4353 ◽  
Author(s):  
RICARDO GARCÍA-SALCEDO ◽  
NORA BRETÓN
Keyword(s):  
Electromagnetic Field ◽  
Early Universe ◽  
Big Bang ◽  
Linear Case ◽  
Conformal Metric ◽  
Curvature Invariants ◽  
Metric Function ◽  
Big Bang Singularity

We present a model for an inhomogeneous and anisotropic early universe filled with a nonlinear electromagnetic field of Born–Infeld (BI) type. The effects of the BI field are compared with the linear case (Maxwell). Since the curvature invariants are well behaved then we conjecture that our model does not present an initial big bang singularity. The existence of the BI field modifies the curvature invariants at t=0 as well as sets bounds on the amplitude of the conformal metric function.

scholarly journals The approximation of the solution of wave problems by spectral expansions connected with elliptic differential operators

2021 ◽  
Vol 86 (2) ◽  
pp. 101-106
Author(s):  
Abdulkasim Akhmedov ◽  
Mohd Zuki Salleh ◽  
Abdumalik Rakhimov
Keyword(s):  
Wave Propagation ◽  
Differential Operators ◽  
Sufficient Conditions ◽  
Elastic Membrane ◽  
Conductive Material ◽  
Spectral Expansions ◽  
Vibration Process ◽  
Thermally Conductive ◽  
Elliptic Differential Operators ◽  
Wave Problems

In this research, we investigate the spectral expansions connected with elliptic differential operators in the space of singular distributions, which describes the vibration process made of thin elastic membrane stretched tightly over a circular frame. The sufficient conditions for summability of the spectral expansions connected with wave problems on the disk are obtained by taking into account that the deflection of the membrane during the motion remains small compared to the size of the membrane and for wave propagation problems, the disk is made of some thermally conductive material.

scholarly journals Impact of Boundary Condition and Kinetic Parameter Uncertainties on NOx Predictions in Methane-Air Stagnation Flame Experiments

2021 ◽  
Author(s):  
Antoine Durocher ◽  
Jiayi Wang ◽  
Gilles Bourque ◽  
Jeffrey M. Bergthorson
Keyword(s):  
Boundary Conditions ◽  
High Pressures ◽  
Laminar Flame ◽  
Combustion Process ◽  
Industrial Applications ◽  
Parameter Uncertainties ◽  
Nox Formation ◽  
Important Species ◽  
Spectral Expansions ◽  
Combustion Properties

Abstract A comprehensive understanding of uncertainty sources in experimental measurements is required to develop robust thermochemical models for use in industrial applications. Due to the complexity of the combustion process in gas turbine engines, simpler flames are generally used to study fundamental combustion properties and measure concentrations of important species to validate and improve modelling. Stable, laminar flames have increasingly been used to study nitrogen oxide (NOx) formation in lean-to-rich compositions in low-to-high pressures to assess model predictions and improve accuracy to help develop future low-emissions systems. They allow for non-intrusive diagnostics to measure sub-ppm concentrations of pollutant molecules, as well as important precursors, and provide well-defined boundary conditions to directly compare experiments with simulations. The uncertainties of experimentally-measured boundary conditions and the inherent kinetic uncertainties in the nitrogen chemistry are propagated through one-dimensional stagnation flame simulations to quantify the relative importance of the two sources and estimate their impact on predictions. Measurements in lean, stoichiometric, and rich methane-air flames are used to investigate the production pathways active in those conditions. Various spectral expansions are used to develop surrogate models with different levels of accuracy to perform the uncertainty analysis for 15 important reactions in the nitrogen chemistry and the 6 boundary conditions (ϕ, Tin, uin, du/dzin, Tsurf, P) simultaneously. After estimating the individual parametric contributions, the uncertainty of the boundary conditions are shown to have a relatively small impact on the prediction of NOx compared to kinetic uncertainties in these laboratory experiments. These results show that properly calibrated laminar flame experiments can, not only provide validation targets for modelling, but also accurate indirect measurements that can later be used to infer individual kinetic rates to improve thermochemical models.

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