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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Myers's Theorem
Myers's theorem, also known as the Bonnet–Myers theorem, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following: In the special case of surfaces, this result was proved by Ossian Bonnet in 1855. For a surface, the Gauss, sectional, and Ricci curvatures are all the same, but Bonnet's proof easily generalizes to higher dimensions if one assumes a positive lower bound on the sectional curvature. Myers' key contribution was therefore to show that a Ricci lower bound is all that is needed to reach the same conclusion. Corollaries The conclusion of the theorem says, in particular, that the diameter of (M, g) is finite. The Hopf-Rinow theorem therefore implies that M must be compact, as a closed (and hence compact) ball of radius \pi/\sqrt in any tangent space is carried onto all of M by the exponential map. As a very particular case, this shows that any complete and nonc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard L
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include "Richie", "Dick", "Dickon", " Dickie", "Rich", "Rick", "Rico", "Ricky", and more. Richard is a common English, German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Catalan "Ricard" and the Italian "Riccardo", among others (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Andersen (other) * Richard Anderson (other) * Richard Cartwright (other) * Ri ... [...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] |
Bishop–Gromov Inequality
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem. Statement Let M be a complete ''n''-dimensional Riemannian manifold whose Ricci curvature satisfies the lower bound : \mathrm \geq (n-1) K for a constant K\in \R. Let M_K^n be the complete ''n''-dimensional simply connected space of constant sectional curvature K (and hence of constant Ricci curvature (n-1)K); thus M_K^n is the ''n''-sphere of radius 1/\sqrt if K>0, or ''n''-dimensional Euclidean space if K=0, or an appropriately rescaled version of ''n''-dimensional hyperbolic space if K<0. Denote by the ball of radius ''r'' around a point ''p'', defined with respect to the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length (vector Field)
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object. Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warner Comparison Theorem
Warner can refer to: People * Warner (writer) * Warner (given name) * Warner (surname) Fictional characters * Yakko, Wakko, and Dot Warner, stars of the animated television series ''Animaniacs'' * Aaron Warner, a character in '' Shatter Me series'' Education * Warner Pacific University, Portland, Oregon * Warner University, Lake Wales, Florida Places * Warner (crater), a lunar impact crater in the southern part of the Mare Smythii * Warner Theatre (other), several theatres ;Australia * Warner, Queensland ;In Canada * County of Warner No. 5, a municipal district in Alberta * Warner, Alberta, a village * Warner elevator row, Warner, Alberta ;In the United States * Warner, New Hampshire, a New England town ** Warner (CDP), New Hampshire, the main village in the town * Warner, Ohio, an unincorporated community * Warner, Oklahoma * Warner, South Dakota Organisations * Warner Aerocraft, an American aircraft manufacturer based in Seminole, Florida * Warner Ai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berger–Kazdan Comparison Theorem
In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the -dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan. Statement of the theorem Let be a closed -dimensional Riemannian manifold with injectivity radius . Let denote the Riemannian volume of and let denote the volume of the standard -dimensional sphere of radius one. Then :\mathrm (M) \geq \frac, with equality if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ... is isometric to the -sphere with its usual round met ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Berger Comparison Theorem
Berger is a surname in both German and French, although there is no etymological connection between the names in the two languages. The French surname is an occupational name for a shepherd, from Old French ''bergier'' (Late Latin ''berbicarius'', from ''berbex'' 'ram'). The German surname derives from the word ''Berg'', the word for "mountain" or "hill", and means "a resident on a mountain or hill", or someone from a toponym Berg, derived from the same. The pronunciation of the English name may sometimes be following the French phonetics (the German is ). Notable people with this surname include: Politics *Charles W. Berger (born 1936), American politician * James S. Berger (1903–1984), American politician * Jan Johannis Adriaan Berger, Dutch Labour Party politician. * Józef Berger (1901–1962), Polish theologian and politician. *Karine Berger (born 1973), French politician *Laurent Berger (born 1968), French trade unionist *Luciana Berger (born 1981), British Liberal Dem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
