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Cheng's Eigenvalue Comparison Theorem
In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains . Theorem Let ''M'' be a Riemannian manifold with dimension ''n'', and let ''B''''M''(''p'', ''r'') be a geodesic ball centered at ''p'' with radius ''r'' less than the injectivity radius of ''p'' ∈ ''M''. For each real number ''k'', let ''N''(''k'') denote the simply connected space form of dimension ''n'' and constant sectional curvature ''k''. Cheng's eigenvalue comparison theorem compares the first eigenvalue λ1(''B''''M''(''p'', ''r'')) of the Dirichlet problem in ''B''''M''(''p'', ''r'') with the first eigenvalue in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier. Academic Press publishes reference books, serials and online products in the subject areas of: * Communications engineering * Economics * Environmental science * Finance * Food science and nutrition * Geophysics * Life sciences * Mathematics and statistics * Neuroscience * Physical sciences * Psychology Well-known products include the ''Methods in Enzymology'' series and encyclopedias such as ''The International Encyclopedia of Public Health'' and the ''Encyclopedia of Neuroscience''. See also * Akademische Verlagsgesellschaft (AVG) — the German predecessor, founded in 1906 by Leo Jolowicz (1868–1940), the father of Walter Jolowicz Walter may refer to: People * Walter (name), both a surname and a given name * Little Walter, American blues harmonica player Marion Wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, formerly titled ''Proceedings of the Cambridge Philosophical Society'', has been published since 1843. See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics education in the United Kingdom Mathematics journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue Comparison Theorem
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by \lambda, is the factor by which the eigenvector is scaled. Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated. Formal definition If is a linear transformation from a vector space over a field into itself and is a nonzero vector in , then is an eigenvector of if is a scalar multiple of . This can be written as T(\mathbf) = \lambda \mathbf, where is a scalar in , known as the eigenvalue, characteristic value, or characteristic root ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comparison Theorem
In mathematics, comparison theorems are theorems whose statement involves comparisons between various mathematical objects of the same type, and often occur in fields such as calculus, differential equations and Riemannian geometry. Differential equations In the theory of differential equations, comparison theorems assert particular properties of solutions of a differential equation (or of a system thereof), provided that an auxiliary equation/inequality (or a system thereof) possesses a certain property. *Chaplygin inequality *Grönwall's inequality, and its various generalizations, provides a comparison principle for the solutions of first-order ordinary differential equations. *Sturm comparison theorem *Aronson and Weinberger used a comparison theorem to characterize solutions to Fisher's equation, a reaction--diffusion equation. * Hille-Wintner comparison theorem Riemannian geometry In Riemannian geometry, it is a traditional name for a number of theorems that compare vario ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sectional Curvature
In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R'' is the Riemann curvature tensor, defined here by the convention R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dirichlet Eigenvalue
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function ''u'' ≠ 0 and eigenvalue λ Here Δ is the Laplacian, which is given in ''xy''-coordinates by :\Delta u = \frac + \frac. The boundary value problem () is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in () is often known as the Dirichlet Laplacian when it is considered as accepting only fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
