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Birkhoff's Theorem (electromagnetism)
Birkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff: * Birkhoff's theorem (relativity) * Birkhoff's theorem (electromagnetism) * Birkhoff's ergodic theorem It may also refer to theorems named for his son, Garrett Birkhoff: *Doubly_stochastic_matrix, Birkhoff–von Neumann theorem for doubly stochastic matrices * Birkhoff's HSP theorem, concerning the closure operations of homomorphism, subalgebra and product * Birkhoff's representation theorem for distributive lattices * Birkhoff's theorem (equational logic), stating that syntactic and semantic consequence coincide {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George David Birkhoff
George David Birkhoff (March21, 1884November12, 1944) was one of the top American mathematicians of his generation. He made valuable contributions to the theory of differential equations, dynamical systems, the four-color problem, the three-body problem, and general relativity. Today, Birkhoff is best remembered for the ergodic theorem. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark. Early life He was born in Overisel Township, Michigan, the son of two Dutch immigrants, David Birkhoff, who arrived in the United States in 1870, and Jane Gertrude Droppers. Birkhoff's father worked as a physician in Chicago while he was a child. From 1896 to 1902, he would attend the Lewis Institute as a teenager. Career Birkhoff was part of a generation of American mathematicians who were the first to study entirely within the United States and not participate in academics within Europe. Following his time at the Lewis Insti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Theorem (relativity)
In general relativity, Birkhoff's theorem states that any spherically symmetric solution of the vacuum field equations must be static and asymptotically flat. This means that the exterior solution (i.e. the spacetime outside of a spherical, nonrotating, gravitating body) must be given by the Schwarzschild metric. The converse of the theorem is true and is called Israel's theorem. The converse is not true in Newtonian gravity. The theorem was proven in 1923 by George David Birkhoff (author of another famous '' Birkhoff theorem'', the ''pointwise ergodic theorem'' which lies at the foundation of ergodic theory). Israel's theorem was proved by Werner Israel. Intuitive rationale The intuitive idea of Birkhoff's theorem is that a spherically symmetric gravitational field should be produced by some massive object at the origin; if there were another concentration of mass–energy somewhere else, this would disturb the spherical symmetry, so we can expect the solution to represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Theorem (electromagnetism)
Birkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff: * Birkhoff's theorem (relativity) * Birkhoff's theorem (electromagnetism) * Birkhoff's ergodic theorem It may also refer to theorems named for his son, Garrett Birkhoff: *Doubly_stochastic_matrix, Birkhoff–von Neumann theorem for doubly stochastic matrices * Birkhoff's HSP theorem, concerning the closure operations of homomorphism, subalgebra and product * Birkhoff's representation theorem for distributive lattices * Birkhoff's theorem (equational logic), stating that syntactic and semantic consequence coincide {{disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Ergodic Theorem
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Garrett Birkhoff
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory. The mathematician George Birkhoff (1884–1944) was his father. Life The son of the mathematician George David Birkhoff, Garrett was born in Princeton, New Jersey. He began the Harvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went to Cambridge University to study mathematical physics but switched to studying abstract algebra under Philip Hall. While visiting the University of Munich, he met Constantin Carathéodory who pointed him towards two important texts, Bartel Leendert van der Waerden, Van der Waerden on abstract algebra and Andreas Speiser, Speiser on group theory. Birkhoff held no Ph.D., a qualification British higher education did not emphasize at that time, and did not obtain an M.A. Nevertheless, after being a member of Harvard's Society of F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doubly Stochastic Matrix
In mathematics, especially in probability and combinatorics, a doubly stochastic matrix (also called bistochastic matrix) is a square matrix X=(x_) of nonnegative real numbers, each of whose rows and columns sums to 1, i.e., :\sum_i x_=\sum_j x_=1, Thus, a doubly stochastic matrix is both left stochastic and right stochastic. Indeed, any matrix that is both left and right stochastic must be square: if every row sums to 1 then the sum of all entries in the matrix must be equal to the number of rows, and since the same holds for columns, the number of rows and columns must be equal. Birkhoff polytope The class of n\times n doubly stochastic matrices is a convex polytope known as the Birkhoff polytope B_n. Using the matrix entries as Cartesian coordinates, it lies in an (n-1)^2-dimensional affine subspace of n^2-dimensional Euclidean space defined by 2n-1 independent linear constraints specifying that the row and column sums all equal 1. (There are 2n-1 constraints rather than 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's HSP Theorem
In universal algebra, a variety of algebras or equational class is the class of all algebraic structures of a given signature satisfying a given set of identities. For example, the groups form a variety of algebras, as do the abelian groups, the rings, the monoids etc. According to Birkhoff's theorem, a class of algebraic structures of the same signature is a variety if and only if it is closed under the taking of homomorphic images, subalgebras, and (direct) products. In the context of category theory, a variety of algebras, together with its homomorphisms, forms a category; these are usually called ''finitary algebraic categories''. A ''covariety'' is the class of all coalgebraic structures of a given signature. Terminology A variety of algebras should not be confused with an algebraic variety, which means a set of solutions to a system of polynomial equations. They are formally quite distinct and their theories have little in common. The term "variety of algebras" ref ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Representation Theorem
:''This is about lattice theory. For other similarly named results, see Birkhoff's theorem (other).'' In mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets. Here, a lattice is an abstract structure with two binary operations, the "meet" and "join" operations, which must obey certain axioms; it is distributive if these two operations obey the distributive law. The union and intersection operations, in a family of sets that is closed under these operations, automatically form a distributive lattice, and Birkhoff's representation theorem states that (up to isomorphism) every finite distributive lattice can be formed in this way. It is named after Garrett Birkhoff, who published a proof of it in 1937.. The theorem can be interpreted as providing a one-to-one correspondenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
