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70 relations: Absolute convergence, Academic Press, Adjoint representation, Bernard de Wit, Bernoulli number, Cambridge University Press, Center (algebra), Center (group theory), Characteristic (algebra), Closed-form expression, Coalgebra, Commutative property, Commutator, Cosmas Zachos, Creation and annihilation operators, Derivative of the exponential map, Displacement operator, Emil Wolf, Eugene Dynkin, Exponential map (Lie theory), Felix Hausdorff, Formal power series, Free algebra, Free Lie algebra, Friedrich Schur, Gerard 't Hooft, Golden–Thompson inequality, Group (mathematics), H. F. Baker, Heisenberg group, Henri Poincaré, Hilbert space, Hilbert–Schmidt operator, Hopf algebra, If and only if, Jacobi identity, Jean-Pierre Serre, John Edward Campbell, Kurt Otto Friedrichs, Leonard Mandel, Lie algebra, Lie group, Lie group–Lie algebra correspondence, Lie product formula, Logarithm, Logarithm of a matrix, Magnus expansion, Martin Eichler, Martinus J. G. Veltman, Mathematics, ..., Mathematische Annalen, Matrix exponential, Matrix norm, Nathan Jacobson, Nilpotent group, Power series, Primitive element (co-algebra), Proceedings of the USSR Academy of Sciences, Quantum field theory, Quantum mechanics, Quantum optics, Ring homomorphism, Shlomo Sternberg, Springer Publishing, Stone–von Neumann theorem, Tensor product, Trace (linear algebra), Universal enveloping algebra, Wilfried Schmid, Zassenhaus. Expand index (20 more) »
Absolute convergence
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite.
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Academic Press
Academic Press is an academic book publisher.
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Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
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Bernard de Wit
Bernard Quirinus Petrus Joseph de Wit (born 1945 in Bergen op Zoom) is a Dutch theoretical physicist specialized in supergravity and particle physics.
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Bernoulli number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in number theory.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements.
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Center (group theory)
In abstract algebra, the center of a group,, is the set of elements that commute with every element of.
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Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char(R), is defined to be the smallest number of times one must use the ring's multiplicative identity (1) in a sum to get the additive identity (0) if the sum does indeed eventually attain 0.
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Closed-form expression
In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a finite number of operations.
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Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras.
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Commutative property
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.
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Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.
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Cosmas Zachos
Cosmas K. Zachos (Κοσμάς Ζάχος; born 1951, Athens) is a theoretical physicist.
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Creation and annihilation operators
Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems.
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Derivative of the exponential map
In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group into.
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Displacement operator
The displacement operator for one mode in quantum optics is the shift operator where \alpha is the amount of displacement in optical phase space, \alpha^* is the complex conjugate of that displacement, and \hat and \hat^\dagger are the lowering and raising operators, respectively.
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Emil Wolf
Emil Wolf (July 30, 1922 – June 2, 2018) was a Czech-born American physicist who made advancements in physical optics, including diffraction, coherence properties of optical fields, spectroscopy of partially coherent radiation, and the theory of direct scattering and inverse scattering.
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Eugene Dynkin
Eugene Borisovich Dynkin (Евге́ний Бори́сович Ды́нкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician.
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Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
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Felix Hausdorff
Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.
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Formal power series
In mathematics, a formal power series is a generalization of a polynomial, where the number of terms is allowed to be infinite; this implies giving up the possibility of replacing the variable in the polynomial with an arbitrary number.
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Free algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables.
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Free Lie algebra
In mathematics, a free Lie algebra, over a given field K, is a Lie algebra generated by a set X, without any imposed relations.
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Friedrich Schur
Friedrich Heinrich Schur (27 January 1856, Maciejewo, Krotoschin, Province of Posen – 18 March 1932, Breslau) was a German mathematician who studied geometry.
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Gerard 't Hooft
Gerardus (Gerard) 't Hooft (born July 5, 1946) is a Dutch theoretical physicist and professor at Utrecht University, the Netherlands.
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Golden–Thompson inequality
In physics and mathematics, the Golden–Thompson inequality is a trace inequality between exponentials of matrices proved independently by and.
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
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H. F. Baker
Henry Frederick Baker FRS FRSE (3 July 1866 – 17 March 1956) was a British mathematician, working mainly in algebraic geometry, but also remembered for contributions to partial differential equations (related to what would become known as solitons), and Lie groups.
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Heisenberg group
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form \end under the operation of matrix multiplication.
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Henri Poincaré
Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm where \|\ \| is the norm of H, \ an orthonormal basis of H, and Tr is the trace of a nonnegative self-adjoint operator.
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Hopf algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Jacobi identity
In mathematics the Jacobi identity is a property of a binary operation which describes how the order of evaluation (the placement of parentheses in a multiple product) affects the result of the operation.
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Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
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John Edward Campbell
John Edward Campbell (27 May 1862, Lisburn, Ireland – 1 October 1924, Oxford, Oxfordshire, England) was a mathematician, best known for his contribution to the Baker-Campbell-Hausdorff formula.
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Kurt Otto Friedrichs
Kurt Otto Friedrichs (September 28, 1901 – December 31, 1982) was a noted German American mathematician.
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Leonard Mandel
Leonard Mandel (May 9, 1927 – February 9, 2001) was the Lee DuBridge Professor Emeritus of Physics and Optics at the University of Rochester when he died at the age of 73 at his home in Pittsford, New York.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Lie group–Lie algebra correspondence
In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects.
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Lie product formula
In mathematics, the Lie product formula, named for Sophus Lie (1875), states that for arbitrary n × n real or complex matrices A and B, where eA denotes the matrix exponential of A. The Lie–Trotter product formula and the Trotter–Kato theorem extend this to certain unbounded linear operators A and B. This formula is an analogue of the classical exponential law which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to still hold.
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Logarithm
In mathematics, the logarithm is the inverse function to exponentiation.
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Logarithm of a matrix
In mathematics, a logarithm of a matrix is another matrix such that the matrix exponential of the latter matrix equals the original matrix.
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Magnus expansion
In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first order homogeneous linear differential equation for a linear operator.
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Martin Eichler
Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German number theorist.
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Martinus J. G. Veltman
Martinus Justinus Godefriedus "Tini" Veltman (born 27 June 1931) is a Dutch theoretical physicist.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Matrix exponential
In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function.
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Matrix norm
In mathematics, a matrix norm is a vector norm in a vector space whose elements (vectors) are matrices (of given dimensions).
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Nathan Jacobson
Nathan Jacobson (October 5, 1910 – December 5, 1999) was an American mathematician.
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Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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Primitive element (co-algebra)
In algebra, a primitive element of a co-algebra C (over an element g) is an element x that satisfies where \mu is the co-multiplication and g is an element of C that maps to the multiplicative identity 1 of the base field under the co-unit (g is called group-like).
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Proceedings of the USSR Academy of Sciences
The Proceedings of the USSR Academy of Sciences (Доклады Академии Наук СССР, Doklady Akademii Nauk SSSR (DAN SSSR), Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology.
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Quantum field theory
In theoretical physics, quantum field theory (QFT) is the theoretical framework for constructing quantum mechanical models of subatomic particles in particle physics and quasiparticles in condensed matter physics.
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
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Quantum optics
Quantum optics (QO) is a field of research that uses semi-classical and quantum-mechanical physics to investigate phenomena involving light and its interactions with matter at submicroscopic levels.
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Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the structure.
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Shlomo Sternberg
Shlomo Zvi Sternberg (born 1936), is an American mathematician known for his work in geometry, particularly symplectic geometry and Lie theory.
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Springer Publishing
Springer Publishing is an American publishing company of academic journals and books, focusing on the fields of nursing, gerontology, psychology, social work, counseling, public health, and rehabilitation (neuropsychology).
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Stone–von Neumann theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators.
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Tensor product
In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.
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Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.
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Universal enveloping algebra
In mathematics, a universal enveloping algebra is the most general (unital, associative) algebra that contains all representations of a Lie algebra.
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Wilfried Schmid
Wilfried Schmid (born May 28, 1943) is a German-American mathematician who works in Hodge theory, representation theory, and automorphic forms.
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Zassenhaus
Zassenhaus is a German surname.
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BCH formula, BCH identity, Baker-Campbell formula, Baker-Campbell-Hausdorff, Baker-Campbell-Hausdorff formula, Baker-Campbell-Hausdorff formulae, Baker-Campbell-Hausdorff formulas, Baker-Campbell-Hausdorff series, Baker-Hausdorff lemma, Bch formula, CBH identity, Campbell Hausdorff formula, Campbell-Baker-Hausdorff formula, Campbell-Hausdorff formula, Campbell-Hausdorff series, Campbell–Baker–Hausdorff formula, Campbell–Hausdorff series, Cbh formula, Zassenhaus formula.
References
[1] https://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formula
