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Matches in DBpedia 2016-04 for { <http://dbpedia.org/resource/Spectral_geometry> ?p ?o }

• Spectral_geometry abstract "Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems.Inverse problems seek to identify features of the geometry from information about the eigenvalues of the Laplacian. One of the earliest results of this kind was due to Hermann Weyl who used David Hilbert's theory of integral equation in 1911 to show that the volume of a bounded domain in Euclidean space can be determined from the asymptotic behavior of the eigenvalues for the Dirichlet boundary value problem of the Laplace operator. This question is usually expressed as ``Can one hear the shape of a drum?\", the popular phrase due to Mark Kac. A refinement of Weyl's asymptotic formula obtained by Pleijel and Minakshisundaram produces a series of local spectral invariants involving covariant differentiations of the curvature tensor, which can be used to establish spectral rigidity for a special class of manifolds. However as the example given by John Milnor tells us, the information of eigenvalues is not enough to determine the isometry class of a manifold (see isospectral). A general and systematic method due to Toshikazu Sunada gave rise to a veritable cottage industry of such examples which clarifies the phenomenon of isospectral manifolds.Direct problems attempt to infer the behavior of the eigenvalues of a Riemannian manifold from knowledge of the geometry. The solutions to direct problems are typified by the Cheeger inequality which gives a relation between the first positive eigenvalue and an isoperimetric constant (the Cheeger constant). Many versions of the inequality have been established since Cheeger's work (by R. Brooks and P. Buser for instance).".
• Spectral_geometry wikiPageID "20635289".
• Spectral_geometry wikiPageLength "3085".
• Spectral_geometry wikiPageOutDegree "34".
• Spectral_geometry wikiPageRevisionID "705885123".
• Spectral_geometry wikiPageWikiLink Asymptotic_analysis.
• Spectral_geometry wikiPageWikiLink Category:Differential_geometry.
• Spectral_geometry wikiPageWikiLink Category:Riemannian_geometry.
• Spectral_geometry wikiPageWikiLink Category:Spectral_theory.
• Spectral_geometry wikiPageWikiLink Cheeger_constant.
• Spectral_geometry wikiPageWikiLink Closed_manifold.
• Spectral_geometry wikiPageWikiLink Covariant_derivative.
• Spectral_geometry wikiPageWikiLink David_Hilbert.
• Spectral_geometry wikiPageWikiLink Differential_operator.
• Spectral_geometry wikiPageWikiLink Dirichlet_boundary_condition.
• Spectral_geometry wikiPageWikiLink Eigenvalues_and_eigenvectors.
• Spectral_geometry wikiPageWikiLink Euclidean_space.
• Spectral_geometry wikiPageWikiLink Hearing_the_shape_of_a_drum.
• Spectral_geometry wikiPageWikiLink Hermann_Weyl.
• Spectral_geometry wikiPageWikiLink Integral_equation.
• Spectral_geometry wikiPageWikiLink Isometry_(Riemannian_geometry).
• Spectral_geometry wikiPageWikiLink Isoperimetric_inequality.
• Spectral_geometry wikiPageWikiLink Isospectral.
• Spectral_geometry wikiPageWikiLink Jeff_Cheeger.
• Spectral_geometry wikiPageWikiLink John_Milnor.
• Spectral_geometry wikiPageWikiLink Laplace_operator.
• Spectral_geometry wikiPageWikiLink Laplace_operators_in_differential_geometry.
• Spectral_geometry wikiPageWikiLink Laplace–Beltrami_operator.
• Spectral_geometry wikiPageWikiLink Manifold.
• Spectral_geometry wikiPageWikiLink Mark_Kac.
• Spectral_geometry wikiPageWikiLink Mathematics.
• Spectral_geometry wikiPageWikiLink Riemann_curvature_tensor.
• Spectral_geometry wikiPageWikiLink Riemannian_manifold.
• Spectral_geometry wikiPageWikiLink Robert_W._Brooks.
• Spectral_geometry wikiPageWikiLink Spectral_invariants.
• Spectral_geometry wikiPageWikiLink Spectrum_(functional_analysis).
• Spectral_geometry wikiPageWikiLink Toshikazu_Sunada.
• Spectral_geometry wikiPageWikiLinkText "Spectral geometry".
• Spectral_geometry wikiPageWikiLinkText "spectral geometry".
• Spectral_geometry wikiPageWikiLinkText "spectral invariants".
• Spectral_geometry wikiPageUsesTemplate Template:Citation.
• Spectral_geometry subject Category:Differential_geometry.
• Spectral_geometry subject Category:Riemannian_geometry.
• Spectral_geometry subject Category:Spectral_theory.
• Spectral_geometry hypernym Field.
• Spectral_geometry type Algebra.
• Spectral_geometry type Physic.
• Spectral_geometry comment "Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators in differential geometry have also been examined.".
• Spectral_geometry label "Spectral geometry".
• Spectral_geometry sameAs Q3123529.
• Spectral_geometry sameAs Géométrie_spectrale.
• Spectral_geometry sameAs m.051xfr4.
• Spectral_geometry sameAs Q3123529.
• Spectral_geometry wasDerivedFrom Spectral_geometry?oldid=705885123.
• Spectral_geometry isPrimaryTopicOf Spectral_geometry.