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Noether's Theorem (other)
Noether's theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem may also refer to: Theorems by Emmy Noether * Noether's second theorem, on infinite-dimensional Lie algebras and differential equations * Noether normalization lemma, on finitely generated algebra over a field * Noether isomorphism theorems in abstract algebra Theorems by Max Noether * Max Noether's theorem, several theorems ** Noether's theorem on rationality for surfaces ** Noether inequality, a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold See also * Emmy Noether (1882–1935), German Jewish mathematician * Herglotz–Noether theorem, in special relativity * Lasker–Noether theorem, that states that every Noetherian ring is a Lasker ring * Skolem–Noether theorem, which characterizes the automorphisms of simple rings * Albert–Brauer–Hasse–N ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether's Theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space. Noether's theorem is used in theoretical physics and the calculus of variations. It reveals the fundamental relation between the symmetries of a physical system and the conservation laws. It also made modern theoretical physicists much more focused on symmetries of physical systems. A generalization of the formulations on constants of motion in Lagrangian and Hamiltonian mechanics (developed in 1788 and 1833, respectively), it does not apply to systems that cann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether's Second Theorem
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. :Translated in The action ''S'' of a physical system is an integral of a so-called Lagrangian function ''L'', from which the system's behavior can be determined by the principle of least action. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by ''k'' arbitrary functions and their derivatives up to order ''m'', then the functional derivatives of ''L'' satisfy a system of ''k'' differential equations. Noether's second theorem is sometimes used in gauge theory. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model. The theorem is named after Emmy Noether. See also * Noether's first theorem * Noether identities * Gauge symmetry (mathematics) In mathematics, any Lagrangian sys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether Normalization Lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field ''k'', and any finitely generated commutative ''k''-algebra ''A'', there exists a non-negative integer ''d'' and algebraically independent elements ''y''1, ''y''2, ..., ''y''''d'' in ''A'' such that ''A'' is a finitely generated module over the polynomial ring ''S'' = ''k'' 'y''1, ''y''2, ..., ''y''''d'' The integer ''d'' above is uniquely determined; it is the Krull dimension of the ring ''A''. When ''A'' is an integral domain, ''d'' is also the transcendence degree of the field of fractions of ''A'' over ''k''. The theorem has a geometric interpretation. Suppose ''A'' is integral. Let ''S'' be the coordinate ring of the ''d''-dimensional affine space \mathbb A^d_k, and let ''A'' be the coordinate ring of some other ''d''-dimensional affine variety ''X''. Then the inclusion map ''S'' → ''A'' induces a surjective f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism Theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences. History The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'', which was published in 1927 in Mathematische Annalen. Less general versions of these theorems can be found in work of Richard Dedekind and previous papers by Noether. Three years later, B.L. van der Waerden published his influential ''Moderne Algebra'' the first abstract algebra textbook that took the groups-rings-fields a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Max Noether's Theorem (other)
In algebraic geometry, Max Noether's theorem may refer to the results of Max Noether: * Several closely related results of Max Noether on canonical curves * AF+BG theorem, or Max Noether's fundamental theorem, a result on algebraic curves in the projective plane, on the residual sets of intersections * Max Noether's theorem on curves lying on algebraic surfaces, which are hypersurfaces in ''P''3, or more generally complete intersections * Noether's theorem on rationality for surfaces * Max Noether theorem on the generation of the Cremona group by quadratic transformations See also *Noether's theorem, usually referring to a result derived from work of Max's daughter Emmy Noether *Noether inequality *Special divisor *Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether's Theorem On Rationality For Surfaces
In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let ''S'' be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from ''S'' to the projective line, with ''general fibre'' also a projective line. Then the theorem states that ''S'' is rational. See also *Hirzebruch surface In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by . Definition The Hirzebruch surface \Sigma_n is the \mathbb^1-bundle, called a Projective bundle, over \mathbb^1 associated to the sheaf\mathca ... * List of complex and algebraic surfaces ReferencesCastelnuovo’s Theorem Notes Algebraic surfaces Theorems in algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noether Inequality
In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces of general type over an algebraically closed field. Formulation of the inequality Let ''X'' be a smooth minimal projective surface of general type defined over an algebraically closed field (or a smooth minimal compact complex surface of general type) with canonical divisor ''K'' = −''c''1(''X''), and let ''p''g = ''h''0(''K'') be the dimension of the space of holomorphic two forms, then : p_g \le \frac c_1(X)^2 + 2. For complex surfaces, an alternative formulation expresses this inequality in terms of topological invariants of the underlying real oriented four manifold. Since a surface of general type is a Kähler surface, the dimension of the maximal positive subspace in intersection form on the second cohomology is given by ''b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Emmy Noether
Amalie Emmy NoetherEmmy is the ''Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noether'' (1907/08, NR. 2988); reproduced in: ''Emmy Noether, Gesammelte Abhandlungen – Collected Papers,'' ed. N. Jacobson 1983; online facsimile aphysikerinnen.de/noetherlebenslauf.html). Sometimes ''Emmy'' is mistakenly reported as a short form for ''Amalie'', or misreported as "Emily". e.g. (, ; ; 23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether's First and Second Theorem, which are fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Herglotz–Noether Theorem
Born rigidity is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics. The concept was introduced by Max Born (1909),Born (1909b) who gave a detailed description of the case of constant proper acceleration which he called hyperbolic motion. When subsequent authors such as Paul Ehrenfest (1909) tried to incorporate rotational motions as well, it became clear that Born rigidity is a very restrictive sense of rigidity, leading to the Herglotz–Noether theorem, according to which there are severe restrictions on rotational Born rigid motions. It was formulated by Gustav Herglotz (1909, who classified all forms of rotational motions)Herglotz (1909) and in a less general way by Fritz Noether (1909).Noether (1909) As a result, Born (1910)Born (1910) and others gave alternative, less restrictive definitions of rigidity. Definition Born rigidity is satisfied if the orthogonal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lasker–Noether Theorem
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finitely many ''primary ideals'' (which are related to, but not quite the same as, powers of prime ideals). The theorem was first proven by for the special case of polynomial rings and convergent power series rings, and was proven in its full generality by . The Lasker–Noether theorem is an extension of the fundamental theorem of arithmetic, and more generally the fundamental theorem of finitely generated abelian groups to all Noetherian rings. The theorem plays an important role in algebraic geometry, by asserting that every algebraic set may be uniquely decomposed into a finite union of irreducible components. It has a straightforward extension to modules stating that every submodule of a finitely generated module over a Noetherian ring is a finite intersection of primary submodules ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skolem–Noether Theorem
In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras. The theorem was first published by Thoralf Skolem in 1927 in his paper ''Zur Theorie der assoziativen Zahlensysteme'' (German: ''On the theory of associative number systems'') and later rediscovered by Emmy Noether. Statement In a general formulation, let ''A'' and ''B'' be simple unitary rings, and let ''k'' be the center of ''B''. The center ''k'' is a field since given ''x'' nonzero in ''k'', the simplicity of ''B'' implies that the nonzero two-sided ideal is the whole of ''B'', and hence that ''x'' is a unit. If the dimension of ''B'' over ''k'' is finite, i.e. if ''B'' is a central simple algebra of finite dimension, and ''A'' is also a ''k''-algebra, then given ''k''-algebra homomorphisms :''f'', ''g'' : ''A'' → ''B'', there exists a unit ''b'' in ''B'' such that for all ''a'' in ''A'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert–Brauer–Hasse–Noether Theorem
In algebraic number theory, the Albert–Brauer–Hasse–Noether theorem states that a central simple algebra over an algebraic number field ''K'' which splits over every completion ''K''''v'' is a matrix algebra over ''K''. The theorem is an example of a local-global principle in algebraic number theory and leads to a complete description of finite-dimensional division algebras over algebraic number fields in terms of their local invariants. It was proved independently by Richard Brauer, Helmut Hasse, and Emmy Noether and by Abraham Adrian Albert. Statement of the theorem Let ''A'' be a central simple algebra of rank ''d'' over an algebraic number field ''K''. Suppose that for any valuation ''v'', ''A'' splits over the corresponding local field ''K''''v'': : A\otimes_K K_v \simeq M_d(K_v). Then ''A'' is isomorphic to the matrix algebra ''M''''d''(''K''). Applications Using the theory of Brauer group, one shows that two central simple algebras ''A'' and ''B'' over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
