A Laplace-type representation for some generalized spherical functions of type $BC$

As we know the spherical functions are traditionally used in geodesy for modeling the gravitational field of the Earth. But the gravitational field is not stationary either in space or in time (but the latter is beyond the scope of this article) and can change quite strongly in various directions. By its nature, the spherical functions do not fully display the local features of the field. With this in mind it is advisable to use spatially localized basis functions. So it is convenient to divide the region under consideration into segments with a nearly stationary field. The complexity of the field in each segment can be characterized by means of an anisotropic matrix resulting from the covariance analysis of the field. If we approach the modeling in this way there can arise a problem of poor coherence of local models on segments’ borders. To solve the above mentioned problem it is proposed in this article to use new basis functions with Mahalanobis metric instead of the usual Euclidean distance. The Mahalanobis metric and the quadratic form generalizing this metric enables us to take into account the structure of the field when determining the distance between the points and to make the modeling process continuous.

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Exact Laplace-type asymptotic formulas for the Bogoliubov Gaussian measure: The set of minimum points of the action functional

Theoretical and Mathematical Physics ◽

10.1134/s0040577917060071 ◽

2017 ◽

Vol 191 (3) ◽

pp. 870-885 ◽

Cited By ~ 1

Author(s):

V. R. Fatalov

Keyword(s):

Gaussian Measure ◽

Asymptotic Formulas ◽

Action Functional ◽

Laplace Type ◽

Minimum Points

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Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽

10.3390/sym13061077 ◽

2021 ◽

Vol 13 (6) ◽

pp. 1077

Author(s):

Yarema A. Prykarpatskyy

Keyword(s):

Conservation Laws ◽

Hamiltonian Structure ◽

Necessary Condition ◽

Type Representation ◽

Infinite Hierarchy ◽

Polynomial Coefficients ◽

Kdv Type Equations ◽

Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

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Dirac Delta Functions in the Laplace‐Type Expansion of rnYlm(θ, φ)

The Journal of Chemical Physics ◽

10.1063/1.1672352 ◽

1969 ◽

Vol 51 (6) ◽

pp. 2359-2362 ◽

Cited By ~ 39

Author(s):

Kenneth G. Kay ◽

H. David Todd ◽

Harris J. Silverstone

Keyword(s):

Dirac Delta ◽

Delta Functions ◽

Laplace Type

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On Gromov's theorem andL 2-Hodge decomposition

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204210365 ◽

2004 ◽

Vol 2004 (1) ◽

pp. 25-44 ◽

Cited By ~ 6

Author(s):

Fu-Zhou Gong ◽

Feng-Yu Wang

Keyword(s):

Essential Spectrum ◽

Vector Bundles ◽

Functional Inequality ◽

Upper Bounds ◽

Hodge Decomposition ◽

Laplace Type

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the correspondingL 2-harmonic sections. In particular, some known results concerning Gromov's theorem and theL 2-Hodge decomposition are considerably improved.

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