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Birkhoff Decomposition (other)
{{surname, Birkhoff ...
Birkhoff is a surname. Notable people with the surname include: *George David Birkhoff (1884–1944), American mathematician *Garrett Birkhoff (1911–1996), American mathematician, son of George D. See also *Birkhoff (crater) * Birkhoff interpolation *Birkhoff's axioms *Birkhoff's theorem (other), multiple theorems * Birkhoff decomposition, two different decompositions *Birkhoff algorithm *Poincaré–Birkhoff–Witt theorem In mathematics, more specifically in the theory of Lie algebras, the Poincaré–Birkhoff–Witt theorem (or PBW theorem) is a result giving an explicit description of the universal enveloping algebra of a Lie algebra. It is named after Henri Poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George David Birkhoff
George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and during his time he was considered by many to be the preeminent American mathematician. The George D. Birkhoff House, his residence in Cambridge, Massachusetts, has been designated a National Historic Landmark. Personal life He was born in Overisel Township, Michigan, the son of David Birkhoff and Jane Gertrude Droppers. The mathematician Garrett Birkhoff (1911–1996) was his son. Career Birkhoff obtained his A.B. and A.M. from Harvard University. He completed his Ph.D. in 1907, on differential equations, at the University of Chicago. While E. H. Moore was his supervisor, he was most influenced by the writings of Henri Poincaré. After teaching at the University of Wisconsin–Madison and Princeton University, he taught at Harvard from 191 ... [...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 Carathéodory who pointed him towards two important texts, Van der Waerden on abstract algebra and Speiser on group theory. Birkhoff held no Ph.D., a qualification British higher education did not emphasize at that time, and did not even bother obtaining an M.A. Nevertheless, after being a member of Harvard's Society of Fellows, 1933–36, he spent the rest of h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff (crater)
Birkhoff is a giant lunar walled plain that is located on the far side of the Moon, in the northern hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes .... This formation is an ancient impact site that has been heavily eroded, and the surface reshaped by multiple craters in the interior and along the rim. The outer wall is bordered by the craters Carnot to the south, Rowland along the west rim, and Stebbins to the north. Just to the northeast is van't Hoff. What remains of the perimeter is now a rugged sloping rise along the inner wall, and the rim has been worn down until it is level with the irregular external terrain. The rim is pock-marked by small craters of various dimensions. Within the crater are several craters that are notable in their own right. Along the n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff Interpolation
In mathematics, Birkhoff interpolation is an extension of polynomial interpolation. It refers to the problem of finding a polynomial ''p'' of degree ''d'' such that certain derivatives have specified values at specified points: : p^(x_i) = y_i \qquad\mbox i=1,\ldots,d, where the data points (x_i,y_i) and the nonnegative integers n_i are given. It differs from Hermite interpolation in that it is possible to specify derivatives of ''p'' at some points without specifying the lower derivatives or the polynomial itself. The name refers to George David Birkhoff, who first studied the problem. Existence and uniqueness of solutions In contrast to Lagrange interpolation and Hermite interpolation, a Birkhoff interpolation problem does not always have a unique solution. For instance, there is no quadratic polynomial \textstyle p such that \textstyle p(-1)=p(1)=0 and \textstyle p'(0)=1. On the other hand, the Birkhoff interpolation problem where the values of \textstyle p'(-1), \textstyle p(0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Axioms
In 1932, G. D. Birkhoff created a set of four postulates of Euclidean geometry in the plane, sometimes referred to as Birkhoff's axioms. These postulates are all based on basic geometry that can be confirmed experimentally with a scale and protractor. Since the postulates build upon the real numbers, the approach is similar to a model-based introduction to Euclidean geometry. Birkhoff's axiom system was utilized in the secondary-school textbook by Birkhoff and Beatley. These axioms were also modified by the School Mathematics Study Group to provide a new standard for teaching high school geometry, known aSMSG axioms A few other textbooks in the foundations of geometry use variants of Birkhoff's axioms. Postulates The distance between two points and is denoted by , and the angle formed by three points is denoted by . Postulate I: Postulate of line measure. The set of points on any line can be put into a 1:1 correspondence with the real numbers so that for all point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff's Theorem (other)
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: * 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 :''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 ... 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 Algorithm
Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946. It has many applications. One such application is for the problem of fair random assignment: given a randomized allocation of items, Birkhoff's algorithm can decompose it into a lottery on deterministic allocations. Terminology A ''bistochastic matrix'' (also called: ''doubly-stochastic'') is a matrix in which all elements are greater than or equal to 0 and the sum of the elements in each row and column equals 1. An example is the following 3-by-3 matrix: \begin 0.2 & 0.3 & 0.5 \\ 0.6 & 0.2 & 0.2 \\ 0.2 & 0.5 & 0.3 \end A '' permutation matrix'' is a special case of a bistochastic matrix, in which each element is either 0 or 1 (so there is exactly one "1" in each row and each column). An example is the following 3-by-3 matrix: \begin 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
