The fundamental equations of electromagnetism, independent of metrical geometry

1934 ◽  
Vol 30 (4) ◽  
pp. 421-427 ◽  
Author(s):  
D. van Dantzig
Keyword(s):  
Differential Form ◽  
Orthogonal Group ◽  
Integral Form ◽  
Vector Calculus ◽  
Image Position ◽  
Fundamental Equations ◽  
Metrical Geometry

It is trivial that the fundamental equations of electromagnetism are invariant under orthogonal transformations of space. This invariance can be brought into evidence by using the calculus adapted to the orthogonal group, viz. the vector-calculus. The equations can be written either in their integral formor in their differential form

scholarly journals Extending the global-direction stencil with “face-area-weighted centroid” to unstructured finite volume discretization from integral form

2020 ◽  
Vol 2 (1) ◽  
Author(s):  
Lingfa Kong ◽  
Yidao Dong ◽  
Wei Liu ◽  
Huaibao Zhang
Keyword(s):  
Convergence Rate ◽  
Finite Volume ◽  
Differential Form ◽  
Integral Form ◽  
Finite Volume Methods ◽  
Reconstruction Method ◽  
Computational Accuracy ◽  
Face Area ◽  
Finite Volume Discretization ◽  
Skewness Reduction

AbstractAccuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work [Kong, et al. Chin Phys B 29(10):100203, 2020], we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Greatly inspired by the differential form, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has superiorities on both computational accuracy and convergence rate.

scholarly journals Central Extensions of Lie Groups Preserving a Differential Form

2020 ◽  
Author(s):  
Tobias Diez ◽  
Bas Janssens ◽  
Karl-Hermann Neeb ◽  
Cornelia Vizman
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Differential Form ◽  
Integral Form ◽  
Symplectic Manifolds ◽  
Central Extensions ◽  
Mapping Space ◽  
Canonical Class ◽  
Differential Characters

Abstract Let $M$ be a manifold with a closed, integral $(k+1)$-form $\omega $, and let $G$ be a Fréchet–Lie group acting on $(M,\omega )$. As a generalization of the Kostant–Souriau extension for symplectic manifolds, we consider a canonical class of central extensions of ${\mathfrak{g}}$ by ${\mathbb{R}}$, indexed by $H^{k-1}(M,{\mathbb{R}})^*$. We show that the image of $H_{k-1}(M,{\mathbb{Z}})$ in $H^{k-1}(M,{\mathbb{R}})^*$ corresponds to a lattice of Lie algebra extensions that integrate to smooth central extensions of $G$ by the circle group ${\mathbb{T}}$. The idea is to represent a class in $H_{k-1}(M,{\mathbb{Z}})$ by a weighted submanifold $(S,\beta )$, where $\beta $ is a closed, integral form on $S$. We use transgression of differential characters from $ S$ and $ M $ to the mapping space $ C^\infty (S, M) $ and apply the Kostant–Souriau construction on $ C^\infty (S, M) $.

Spinors in Hilbert space

1976 ◽  
Vol 80 (2) ◽  
pp. 337-347 ◽  
Author(s):  
R. J. Plymen
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Orthogonal Group ◽  
Double Cover ◽  
Projective Representation ◽  
Complex Vector Space ◽  
Complex Vector ◽  
True Representation ◽  
Dimension 2 ◽  
Image Position

In 1913, É. Cartan discovered that the special orthogonal groupSO(k) has a ‘two-valued’ representation (i.e. a projective representation) on a complex vector spaceSof dimension 2n, wherek= 2nor 2n+ 1. The projective representation in question lifts to a true representation of the double cover Spin (k) ofSO(k). We restrict attention to the casek= 2n. Under the action of Spin (2n),Sbreaks up into 2 irreducible subspaces:The vectors inSare calledspinors(relative toSO(2n)), those inS+orS−are calledhalf-spinors(4).

scholarly journals On Eddington's higher order equations of the gravitational field

1948 ◽  
Vol 8 (2) ◽  
pp. 89-94 ◽  
Author(s):  
H. A. Buchdahl
Keyword(s):  
Gravitational Field ◽  
Empty Space ◽  
Higher Order ◽  
Energy Tensor ◽  
Image Position ◽  
Fundamental Equations

Einstein's fundamental equations of the gravitational field arewhere Tμν are the components of the energy tensor and λ is the cosmical constant. In empty space these equations becomewhich may be reduced tosince G = 4λ, by contraction of (2).

scholarly journals Extending the global-direction stencil with "face-area-weighted centroid" to unstructured finite volume discretization from integral form

2020 ◽  
Author(s):  
Lingfa Kong ◽  
Yidao Dong ◽  
Wei Liu ◽  
Huaibao Zhang
Keyword(s):  
Finite Volume ◽  
Differential Form ◽  
Integral Form ◽  
Finite Volume Methods ◽  
Higher Order ◽  
Reconstruction Method ◽  
Computational Accuracy ◽  
Face Area ◽  
Finite Volume Discretization ◽  
Numerical Performance

Abstract Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction. For the commonly used k-exact reconstruction method, the cell centroid is always chosen as the reference point to formulate the reconstructed function. But in some practical problems, such as the boundary layer, cells in this area are always set with high aspect ratio to improve the local field resolution, and if geometric centroid is still utilized for the spatial discretization, the severe grid skewness cannot be avoided, which is adverse to the numerical performance of unstructured finite volume solver. In previous work, we explored a novel global-direction stencil and combine it with face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy. Note, however, that the differential form is hard to achieve higher-order accuracy, and in order to set stage for the method promotion on higher-order numerical simulation, in this research, we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization in integral form. Numerical examples governed by linear convective, Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension. Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid, the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid. As a result, on unstructured finite volume discretization from integral form, the method also has a better numerical performance.

Differential Form of Bivariate MMSE Estimator Based on Gaussian Noise

2016 ◽  
Vol 26 (01) ◽  
pp. 1750008
Author(s):  
P. Kittisuwan ◽  
C. Chinrungrueng
Keyword(s):  
Differential Form ◽  
Gaussian Noise ◽  
Additive White Gaussian Noise ◽  
Minimum Mean Square Error ◽  
Integral Form ◽  
Noise Signal ◽  
Original Form ◽  
Reduction Methods ◽  
Mmse Estimator ◽  
Heavy Tailed

In fact, the noise signal is an important problem in signal, circuits and systems. The minimum mean square error (MMSE) estimation technique is useful in several additive white Gaussian noise (AWGN) reduction methods. Original form of MMSE estimator is the integral form. Unfortunately, integral form of MMSE estimator cannot be obtained in simple form for any interesting peaked, heavy-tailed densities (also known as super-Gaussian densities). In this work, we proposed a differential form of bivariate MMSE estimator. The development depends on bivariate Taylor series. The proposed estimator requires no integration. In fact, the derivation is an extension of the existing results for differential form of univariate MMSE estimator.

scholarly journals Bipolar and Toroidal Harmonics

1915 ◽  
Vol 34 ◽  
pp. 102-108 ◽  
Author(s):  
G. B. Jeffery
Keyword(s):  
Integral Form ◽  
Mathematical Physics ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker, viz

scholarly journals THE VISUAL ACUITY AND INTENSITY DISCRIMINATION OF DROSOPHILA

1934 ◽  
Vol 17 (4) ◽  
pp. 517-547 ◽  
Author(s):  
Selig Hecht ◽  
George Wald
Keyword(s):  
Visual Acuity ◽  
Differential Form ◽  
Fiddler Crab ◽  
Integral Form ◽  
Visual Pattern ◽  
Resolving Power ◽  
Reflex Response ◽  
Intensity Discrimination ◽  
Functional Elements ◽  
General Statistical

Drosophila possesses an inherited reflex response to a moving visual pattern which can be used to measure its capacity for intensity discrimination and its visual acuity at different illuminations. It is found that these two properties of vision run approximately parallel courses as functions of the prevailing intensity. Visual acuity varies with the logarithm of the intensity in much the same sigmoid way as in man, the bee, and the fiddler crab. The resolving power is very poor at low illuminations and increases at high illuminations. The maximum visual acuity is 0.0018, which is 1/1000 of the maximum of the human eye and 1/10 that of the bee. The intensity discrimination of Drosophila is also extremely poor, even at its best. At low illuminations for two intensities to be recognized as different, the higher must be nearly 100 times the lower. This ratio decreases as the intensity increases, and reaches a minimum of 2.5 which is maintained at the highest intensities. The minimum value of ΔI/I for Drosophila is 1.5, which is to be compared with 0.25 for the bee and 0.006 for man. An explanation of the variation of visual acuity with illumination is given in terms of the variation in number of elements functional in the retinal mosaic at different intensities, this being dependent on the general statistical distribution of thresholds in the ommatidial population. Visual acuity is thus determined by the integral form of this distribution and corresponds to the total number of elements functional. The idea that intensity discrimination is determined by the differential form of this distribution—that is, that it depends on the rate of entrance of functional elements with intensity—is shown to be untenable in the light of the correspondence of the two visual functions. It is suggested that, like visual acuity, intensity discrimination may also have to be considered as a function of the total number of elements active at a given intensity.

Relativistic Electromagnetic Interactions

2007 ◽  
Author(s):  
Kenneth G. Dyall ◽  
Knut Faegri
Keyword(s):  
Quantum Chemistry ◽  
Differential Operators ◽  
Relativistic Quantum ◽  
Time Derivative ◽  
Electromagnetic Interactions ◽  
Vector Calculus ◽  
Chemical Concepts ◽  
Relativistic Quantum Chemistry ◽  
Fundamental Equations

Chemical concepts are conveniently formulated in terms of molecules—aggregates of atoms linked by electromagnetic interactions. The proper relativistic description of these interactions is a prerequisite for the development of a theory of relativistic quantum chemistry. As a simple starting point we will consider classical systems made up of point charges, postponing the transition to a quantum mechanical description until later. From the previous chapter we know something about how an electron’s particle properties might be affected by relativity. In this chapter we describe the effects of relativity on the interaction with the electromagnetic field. Again, we adopt a minimalist approach. Electromagnetism and electrodynamics are subjects covered in numerous textbooks for a wide variety of target audiences. To develop the necessary theory from first principles is far beyond the scope of this book. We will only highlight those parts necessary for the later development and understanding of a theory of relativistic quantum chemistry. This means that some of the fundamental equations must be presented without derivation, requiring that the reader either knows these from before or that they must be taken on faith. In particular, in this chapter we make use of the Maxwell equations, the Lorentz force equation, and the generalized potential. The reader will be able to find descriptions or derivations of these in Jackson (1975), for example. We will also need to use a number of relations from vector calculus, and these will normally be introduced in the general form when required. In dealing with fields that vary over time and space, we will need various differential operators. In the nonrelativistic theory of electrodynamics the gradient operator, ∇, and the time derivative, d/dt , are used. From our experience in the previous chapter with mixing of space and time coordinates under Lorentz transformations, we might expect these to combine in a four-space differential operator also.

On the Nonexistence of a Feasible Solution in the Context of the Differential Form of Eringen’s Constitutive Model: A Proposed Iterative Model Based on a Residual Nonlocality Formulation

2017 ◽  
Vol 09 (07) ◽  
pp. 1750094 ◽  
Author(s):  
Fatin F. Mahmoud
Keyword(s):  
Constitutive Model ◽  
Differential Form ◽  
Feasible Solution ◽  
Differential Forms ◽  
Integral Form ◽  
Computational Method ◽  
Numerical Work ◽  
Iterative Model ◽  
Elasticity Problems ◽  
Ill Posed

The motivation of this paper is to highlight and discuss critically the details of two main aspects related to Eringen’s nonlocal constitutive model. The first aspect is to point out the inconsistency of the integral and differential forms of Eringen’s nonlocal constitutive model. In addition, to point out the ill-posed form and physical infeasibility of the results which may be obtained by using the differential form of the model. The critical analysis focuses on the lack of consistency between the set of the boundary constraints required by the differential form of Eringen’s model and the set of the prescribed boundary conditions of the nonlocal static equilibrium problem. Because of this lack of consistency between the two constitutive forms, it can be concluded that the formulation in context of the differential form is ill-posed and the existence of a feasible solution is questionable and might not be admitted. The second aspect deals with the intractability of the analytical solution of the nonlocal continuum problems based on the integral form of Eringen’s nonlocal constitutive model (IENCM). In the meantime, it mentions the cumbersome of numerical work required by the direct computational methods such as the nonlocal finite element method. The complexity of using the integral form of Eringen’s constitutive model and the lack of existence of a feasible solution by using the differential form of Eringen’s constitutive model lead to the mandatory need for developing an efficient iterative computational approach based on the integral form of Eringen’s constitutive model. In this paper, an iterative computational method, based on the nonlocality residual formulation for nonlocal continuums, capable to investigate different elasticity problems, is proposed. The traditional local continuum solution is taken as an initial solution. To point out the inconsistences between the integral and differential forms of Eringen’s nonlocal constitutive model, and to illustrate the efficiency and capability of the proposed iterative model, four static bending of beam problems with different nature are solved.

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