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In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into R 3 {\displaystyle \mathbb {R} ^{3}} , which is tangent to the standard contact structure on R 3 {\displaystyle \mathbb {R} ^{3}} . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensio... | https://en.wikipedia.org/wiki/Legendrian_knot |
Many inequivalent Legendrian knots can be distinguished by considering their Thurston-Bennequin invariants and rotation number, which are together known as the "classical invariants" of Legendrian knots. More sophisticated invariants have been constructed, including one constructed combinatorially by Chekanov and using... | https://en.wikipedia.org/wiki/Legendrian_knot |
This Chekanov-Eliashberg invariant yields an invariant for loops of Legendrian knots by considering the monodromy of the loops. This has yielded noncontractible loops of Legendrian knots which are contractible in the space of all knots. Any Legendrian knot may be C 0 {\displaystyle C^{0}} perturbed to a transverse knot... | https://en.wikipedia.org/wiki/Legendrian_knot |
In mathematics, a Lehmer sequence is a generalization of a Lucas sequence. | https://en.wikipedia.org/wiki/Lehmer_sequence |
In mathematics, a Leray cover(ing) is a cover of a topological space which allows for easy calculation of its cohomology. Such covers are named after Jean Leray. Sheaf cohomology measures the extent to which a locally exact sequence on a fixed topological space, for instance the de Rham sequence, fails to be globally e... | https://en.wikipedia.org/wiki/Leray_cover |
Its definition, using derived functors, is reasonably natural, if technical. Moreover, important properties, such as the existence of a long exact sequence in cohomology corresponding to any short exact sequence of sheaves, follow directly from the definition. However, it is virtually impossible to calculate from the d... | https://en.wikipedia.org/wiki/Leray_cover |
On the other hand, Čech cohomology with respect to an open cover is well-suited to calculation, but of limited usefulness because it depends on the open cover chosen, not only on the sheaves and the space. By taking a direct limit of Čech cohomology over arbitrarily fine covers, we obtain a Čech cohomology theory that ... | https://en.wikipedia.org/wiki/Leray_cover |
However, like the derived functor cohomology, this cover-independent Čech cohomology is virtually impossible to calculate from the definition. The Leray condition on an open cover ensures that the cover in question is already "fine enough." The derived functor cohomology agrees with the Čech cohomology with respect to ... | https://en.wikipedia.org/wiki/Leray_cover |
Let U = { U i } {\displaystyle {\mathfrak {U}}=\{U_{i}\}} be an open cover of the topological space X {\displaystyle X} , and F {\displaystyle {\mathcal {F}}} a sheaf on X. We say that U {\displaystyle {\mathfrak {U}}} is a Leray cover with respect to F {\displaystyle {\mathcal {F}}} if, for every nonempty finite set {... | https://en.wikipedia.org/wiki/Leray_cover |
In mathematics, a Lidstone series, named after George James Lidstone, is a kind of polynomial expansion that can express certain types of entire functions. Let ƒ(z) be an entire function of exponential type less than (N + 1)π, as defined below. Then ƒ(z) can be expanded in terms of polynomials An as follows: f ( z ) = ... | https://en.wikipedia.org/wiki/Lidstone_series |
{\displaystyle f(z)=\sum _{n=0}^{\infty }\left+\sum _{k=1}^{N}C_{k}\sin(k\pi z).} Here An(z) is a polynomial in z of degree n, Ck a constant, and ƒ(n)(a) the nth derivative of ƒ at a. A function is said to be of exponential type of less than t if the function h ( θ ; f ) = lim sup r → ∞ 1 r log | f ( r e i θ ) | {\di... | https://en.wikipedia.org/wiki/Lidstone_series |
In mathematics, a Lie algebra (pronounced LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an alternating bilinear map g × g → g {\displaystyle {\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}} , that satisfies the Jacobi identity. Otherwise sai... | https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra |
Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining = x y − y x {\displaystyle =xy-yx} correctly defines a Lie bracket in addition to the already existing multiplication operation. Lie algebras are closely relat... | https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra |
This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. | https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra |
Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics. An elementary example (that is not derived from an associative algebra) is the space of three dimensional vectors g = R 3 {\displaystyle {\mathfrak {g}}=\mathbb {R} ^{3}} with the Lie bracket ... | https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra |
This is skew-symmetric since x × y = − y × x {\displaystyle x\times y=-y\times x} , and instead of associativity it satisfies the Jacobi identity: x × ( y × z ) = ( x × y ) × z + y × ( x × z ) . {\displaystyle x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).} This is the Lie algebra of the Lie group... | https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra |
In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebras g ≥ ≥ ] ≥ ] ] ≥ . . | https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra |
. {\displaystyle {\mathfrak {g}}\geq \geq ]\geq ]]\geq ...} We write g 0 = g {\displaystyle {\mathfrak {g}}_{0}={\mathfrak {g}}} , and g n = {\displaystyle {\mathfrak {g}}_{n}=} for all n > 0 {\displaystyle n>0} . If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nil... | https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra |
The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions. Note that the definition means ... | https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra |
In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie algebra g {\displaystyle {\mathfrak {g}}} is the subalgebra of g {\displaystyle {\mathfrak {g}}} , denoted {\displaystyle } that consists of all linear... | https://en.wikipedia.org/wiki/Derived_Lie_algebra |
. {\displaystyle {\mathfrak {g}}\geq \geq ,]\geq ,],,]]\geq ...} If the derived series eventually arrives at the zero subalgebra, then the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and solvable Lie algebras are analog... | https://en.wikipedia.org/wiki/Derived_Lie_algebra |
Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras and the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely those that can be obtained from semidirect products,... | https://en.wikipedia.org/wiki/Derived_Lie_algebra |
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra: g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} t... | https://en.wikipedia.org/wiki/Reductive_Lie_algebra |
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is a finite-dimensional Lie algebra over a field of characteristic 0. For such ... | https://en.wikipedia.org/wiki/Semi-simple_Lie_group |
In mathematics, a Lie algebroid is a vector bundle A → M {\displaystyle A\rightarrow M} together with a Lie bracket on its space of sections Γ ( A ) {\displaystyle \Gamma (A)} and a vector bundle morphism ρ: A → T M {\displaystyle \rho :A\rightarrow TM} , satisfying a Leibniz rule. A Lie algebroid can thus be thought o... | https://en.wikipedia.org/wiki/Lie_algebroid |
Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid. Lie algebroids were introduced in 1967 by Jean Pradines. | https://en.wikipedia.org/wiki/Lie_algebroid |
In mathematics, a Lie bialgebra is the Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication is skew-symmetric and satisfies a dual Jacobi identity, so that the dual vector space is a Lie algebra, whereas the comu... | https://en.wikipedia.org/wiki/Lie_bialgebra |
In mathematics, a Lie group (pronounced LEE) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the a... | https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group |
Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group SO ( 3 ) {\displaystyle {\text{SO}}(3)} ). Lie groups are widely used in many parts of modern mathematics and physics. Lie groups... | https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group |
These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model t... | https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group |
In mathematics, a Lie groupoid is a groupoid where the set Ob {\displaystyle \operatorname {Ob} } of objects and the set Mor {\displaystyle \operatorname {Mor} } of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source... | https://en.wikipedia.org/wiki/Lie_groupoid |
In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2‑grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements... | https://en.wikipedia.org/wiki/Lie_superalgebra |
In mathematics, a Lie-* algebra is a D-module with a Lie* bracket. They were introduced by Alexander Beilinson and Vladimir Drinfeld (Beilinson & Drinfeld (2004, section 2.5.3)), and are similar to the conformal algebras discussed by Kac (1998) and to vertex Lie algebras. | https://en.wikipedia.org/wiki/Lie-*_algebra |
In mathematics, a Lindelöf space is a topological space in which every open cover has a countable subcover. The Lindelöf property is a weakening of the more commonly used notion of compactness, which requires the existence of a finite subcover. A hereditarily Lindelöf space is a topological space such that every subspa... | https://en.wikipedia.org/wiki/Lindelöf_space |
Such a space is sometimes called strongly Lindelöf, but confusingly that terminology is sometimes used with an altogether different meaning. The term hereditarily Lindelöf is more common and unambiguous. Lindelöf spaces are named after the Finnish mathematician Ernst Leonard Lindelöf. | https://en.wikipedia.org/wiki/Lindelöf_space |
In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz. | https://en.wikipedia.org/wiki/Lipschitz_domain |
In mathematics, a Listing number of a topological space is one of several topological invariants introduced by the 19th-century mathematician Johann Benedict Listing and later given this name by Charles Sanders Peirce. Unlike the later invariants given by Bernhard Riemann, the Listing numbers do not form a complete set... | https://en.wikipedia.org/wiki/Listing_number |
In mathematics, a Littlewood polynomial is a polynomial all of whose coefficients are +1 or −1. Littlewood's problem asks how large the values of such a polynomial must be on the unit circle in the complex plane. The answer to this would yield information about the autocorrelation of binary sequences. They are named fo... | https://en.wikipedia.org/wiki/Littlewood_polynomial |
In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis. | https://en.wikipedia.org/wiki/Loeb_space |
In mathematics, a Loewy ring or semi-Artinian ring is a ring in which every non-zero module has a non-zero socle, or equivalently if the Loewy length of every module is defined. The concepts are named after Alfred Loewy. | https://en.wikipedia.org/wiki/Loewy_ring |
In mathematics, a Lorentz surface is a two-dimensional oriented smooth manifold with a conformal equivalence class of Lorentzian metrics. It is the analogue of a Riemann surface in indefinite signature. == Further reading == | https://en.wikipedia.org/wiki/Lorentz_surface |
In mathematics, a Lucas chain is a restricted type of addition chain, named for the French mathematician Édouard Lucas. It is a sequence a0, a1, a2, a3, ...that satisfies a0=1,and for each k > 0: ak = ai + aj, and either ai = aj or |ai − aj| = am, for some i, j, m < k.The sequence of powers of 2 (1, 2, 4, 8, 16, ...) a... | https://en.wikipedia.org/wiki/Lucas_chain |
In mathematics, a Lucas–Carmichael number is a positive composite integer n such that if p is a prime factor of n, then p + 1 is a factor of n + 1; n is odd and square-free.The first condition resembles the Korselt's criterion for Carmichael numbers, where -1 is replaced with +1. The second condition eliminates from co... | https://en.wikipedia.org/wiki/Lucas–Carmichael_number |
In mathematics, a Luzin space (or Lusin space), named for N. N. Luzin, is an uncountable topological T1 space without isolated points in which every nowhere-dense subset is countable. There are many minor variations of this definition in use: the T1 condition can be replaced by T2 or T3, and some authors allow a counta... | https://en.wikipedia.org/wiki/Luzin_set |
In mathematics, a Lüroth quartic is a nonsingular quartic plane curve containing the 10 vertices of a complete pentalateral. They were introduced by Jacob Lüroth (1869). Morley (1919) showed that the Lüroth quartics form an open subset of a degree 54 hypersurface, called the Lüroth hypersurface, in the space P14 of all... | https://en.wikipedia.org/wiki/Lüroth_quartic |
In mathematics, a Macbeath region is an explicitly defined region in convex analysis on a bounded convex subset of d-dimensional Euclidean space R d {\displaystyle \mathbb {R} ^{d}} . The idea was introduced by Alexander Macbeath (1952) and dubbed by G. Ewald, D. G. Larman and C. A. Rogers in 1970. Macbeath regions hav... | https://en.wikipedia.org/wiki/Macbeath_region |
In mathematics, a Madhava series is one of the three Taylor series expansions for the sine, cosine, and arctangent functions discovered in 14th or 15th century Kerala by the mathematician and astronomer Madhava of Sangamagrama (c. 1350 – c. 1425) or his followers in the Kerala school of astronomy and mathematics. Using... | https://en.wikipedia.org/wiki/Madhava_series |
+ θ 5 5 ! − θ 7 7 ! | https://en.wikipedia.org/wiki/Madhava_series |
+ ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos θ = 1 − θ 2 2 ! | https://en.wikipedia.org/wiki/Madhava_series |
+ θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! | https://en.wikipedia.org/wiki/Madhava_series |
θ 2 k , arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ = ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 x 2 k + 1 where | x | ≤ 1. {\displaystyle {\begin{alignedat}{3}\sin \theta &=\theta -{\frac {\theta ^{3}}{3! }}+{\frac {\theta ^{5}}{5! | https://en.wikipedia.org/wiki/Madhava_series |
}}-{\frac {\theta ^{7}}{7! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)! }}\theta ^{2k+1},\\\cos \theta &=1-{\frac {\theta ^{2}}{2! | https://en.wikipedia.org/wiki/Madhava_series |
}}+{\frac {\theta ^{4}}{4! }}-{\frac {\theta ^{6}}{6! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)! | https://en.wikipedia.org/wiki/Madhava_series |
}}\theta ^{2k},\\\arctan x&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}x^{2k+1}\quad {\text{where }}|x|\leq 1.\end{alignedat}}} All three series were later independently discovered in 17th century Europe. The series for sine and cosine were rediscov... | https://en.wikipedia.org/wiki/Madhava_series |
In recognition of Madhava's priority, in recent literature these series are sometimes called the Madhava–Newton series, Madhava–Gregory series, or Madhava–Leibniz series (among other combinations).No surviving works of Madhava contain explicit statements regarding the expressions which are now referred to as Madhava se... | https://en.wikipedia.org/wiki/Madhava_series |
In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947). | https://en.wikipedia.org/wiki/Maharam_algebra |
In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo (1911, 1912, 1913). As with all large cardinals, none of these varieties of Mahlo cardinals can be proven to exist by ZFC (assuming ZFC is consistent). A cardinal number κ {\displaystyle \kapp... | https://en.wikipedia.org/wiki/Hyper-Mahlo_cardinal |
In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal'cev 1949). | https://en.wikipedia.org/wiki/Malcev_Lie_algebra |
In mathematics, a Malcev algebra (or Maltsev algebra or Moufang–Lie algebra) over a field is a nonassociative algebra that is antisymmetric, so that x y = − y x {\displaystyle xy=-yx} and satisfies the Malcev identity ( x y ) ( x z ) = ( ( x y ) z ) x + ( ( y z ) x ) x + ( ( z x ) x ) y . {\displaystyle (xy)(xz)=((xy)z... | https://en.wikipedia.org/wiki/Malcev_algebra |
Malcev algebras play a role in the theory of Moufang loops that generalizes the role of Lie algebras in the theory of groups. Namely, just as the tangent space of the identity element of a Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as... | https://en.wikipedia.org/wiki/Malcev_algebra |
In mathematics, a Markov decision process (MDP) is a discrete-time stochastic control process. It provides a mathematical framework for modeling decision making in situations where outcomes are partly random and partly under the control of a decision maker. MDPs are useful for studying optimization problems solved via ... | https://en.wikipedia.org/wiki/Markov_Decision_Process |
They are used in many disciplines, including robotics, automatic control, economics and manufacturing. The name of MDPs comes from the Russian mathematician Andrey Markov as they are an extension of Markov chains. At each time step, the process is in some state s {\displaystyle s} , and the decision maker may choose an... | https://en.wikipedia.org/wiki/Markov_Decision_Process |
The process responds at the next time step by randomly moving into a new state s ′ {\displaystyle s'} , and giving the decision maker a corresponding reward R a ( s , s ′ ) {\displaystyle R_{a}(s,s')} . The probability that the process moves into its new state s ′ {\displaystyle s'} is influenced by the chosen action. ... | https://en.wikipedia.org/wiki/Markov_Decision_Process |
Thus, the next state s ′ {\displaystyle s'} depends on the current state s {\displaystyle s} and the decision maker's action a {\displaystyle a} . But given s {\displaystyle s} and a {\displaystyle a} , it is conditionally independent of all previous states and actions; in other words, the state transitions of an MDP s... | https://en.wikipedia.org/wiki/Markov_Decision_Process |
In mathematics, a Markov information source, or simply, a Markov source, is an information source whose underlying dynamics are given by a stationary finite Markov chain. | https://en.wikipedia.org/wiki/Markov_information_source |
In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer.The basic examp... | https://en.wikipedia.org/wiki/Markov_odometer |
In mathematics, a Marot ring, introduced by Marot (1969), is a commutative ring whose regular ideals are generated by regular elements. | https://en.wikipedia.org/wiki/Marot_ring |
In mathematics, a Menger space is a topological space that satisfies a certain basic selection principle that generalizes σ-compactness. A Menger space is a space in which for every sequence of open covers U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are finite sets F 1... | https://en.wikipedia.org/wiki/Menger_space |
In mathematics, a Mennicke symbol is a map from pairs of elements of a number field to an abelian group satisfying some identities found by Mennicke (1965). They were named by Bass, Milnor & Serre (1967), who used them in their solution of the congruence subgroup problem. | https://en.wikipedia.org/wiki/Mennicke_symbol |
In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n. They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. If n is a composite number then so is 2n − 1. Therefor... | https://en.wikipedia.org/wiki/Mersenne_numbers |
Numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, however, Mersenne numbers are defined to have the additional requirement that n be prime. The smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89. | https://en.wikipedia.org/wiki/Mersenne_numbers |
Mersenne primes were studied in antiquity because of their close connection to perfect numbers: the Euclid–Euler theorem asserts a one-to-one correspondence between even perfect numbers and Mersenne primes. Many of the largest known primes are Mersenne primes because Mersenne numbers are easier to check for primality. ... | https://en.wikipedia.org/wiki/Mersenne_numbers |
The largest known prime number, 282,589,933 − 1, is a Mersenne prime. Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project. In December 2020, a major milestone in the project was passed after all exponents below 100 million were ch... | https://en.wikipedia.org/wiki/Mersenne_numbers |
In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): ∀ i ≠ j x i j ≥ 0. {\displaystyle \forall _{i\neq j}\,x_{ij}\geq 0.} It is named after the American economist Lloyd Metzler. Metzler matrices appear in stability analysis of time delaye... | https://en.wikipedia.org/wiki/Metzler_matrix |
In mathematics, a Meyer set or almost lattice is a relatively dense set X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who introduced a... | https://en.wikipedia.org/wiki/Meyer_set |
In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). | https://en.wikipedia.org/wiki/Minkowski_plane |
In mathematics, a Misiurewicz point is a parameter value in the Mandelbrot set (the parameter space of complex quadratic maps) and also in real quadratic maps of the interval for which the critical point is strictly pre-periodic (i.e., it becomes periodic after finitely many iterations but is not periodic itself). By a... | https://en.wikipedia.org/wiki/Misiurewicz_point |
In mathematics, a Moishezon manifold M is a compact complex manifold such that the field of meromorphic functions on each component M has transcendence degree equal the complex dimension of the component: dim C M = a ( M ) = t r . d e g . C C ( M ) . | https://en.wikipedia.org/wiki/Moishezon_manifold |
{\displaystyle \dim _{\mathbf {C} }M=a(M)=\operatorname {tr.deg.} _{\mathbf {C} }\mathbf {C} (M).} Complex algebraic varieties have this property, but the converse is not true: Hironaka's example gives a smooth 3-dimensional Moishezon manifold that is not an algebraic variety or scheme. Moishezon (1967, Chapter I, Theo... | https://en.wikipedia.org/wiki/Moishezon_manifold |
In mathematics, a Mordellic variety is an algebraic variety which has only finitely many points in any finitely generated field. The terminology was introduced by Serge Lang to enunciate a range of conjectures linking the geometry of varieties to their Diophantine properties. | https://en.wikipedia.org/wiki/Algebraic_hyperbolicity |
In mathematics, a Moufang loop is a special kind of algebraic structure. It is similar to a group in many ways but need not be associative. Moufang loops were introduced by Ruth Moufang (1935). Smooth Moufang loops have an associated algebra, the Malcev algebra, similar in some ways to how a Lie group has an associated... | https://en.wikipedia.org/wiki/Moufang_loop |
In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang. | https://en.wikipedia.org/wiki/Moufang_set |
In mathematics, a Multibrot set is the set of values in the complex plane whose absolute value remains below some finite value throughout iterations by a member of the general monic univariate polynomial family of recursions. The name is a portmanteau of multiple and Mandelbrot set. The same can be applied to the Julia... | https://en.wikipedia.org/wiki/Multibrot_set |
z ↦ z d + c . {\displaystyle z\mapsto z^{d}+c.\,} where d ≥ 2. The exponent d may be further generalized to negative and fractional values. | https://en.wikipedia.org/wiki/Multibrot_set |
In mathematics, a Mumford measure is a measure on a supermanifold constructed from a bundle of relative dimension 1|1. It is named for David Mumford. | https://en.wikipedia.org/wiki/Mumford_measure |
In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and August Ferdinand Möbius in 1858, but it had already appeared in Roman mosaics from ... | https://en.wikipedia.org/wiki/Mobius_strip |
As an abstract topological space, the Möbius strip can be embedded into three-dimensional Euclidean space in many different ways: a clockwise half-twist is different from a counterclockwise half-twist, and it can also be embedded with odd numbers of twists greater than one, or with a knotted centerline. Any two embeddi... | https://en.wikipedia.org/wiki/Mobius_strip |
It has only a single boundary curve. Several geometric constructions of the Möbius strip provide it with additional structure. It can be swept as a ruled surface by a line segment rotating in a rotating plane, with or without self-crossings. | https://en.wikipedia.org/wiki/Mobius_strip |
A thin paper strip with its ends joined to form a Möbius strip can bend smoothly as a developable surface or be folded flat; the flattened Möbius strips include the trihexaflexagon. The Sudanese Möbius strip is a minimal surface in a hypersphere, and the Meeks Möbius strip is a self-intersecting minimal surface in ordi... | https://en.wikipedia.org/wiki/Mobius_strip |
A Möbius strip without its boundary, called an open Möbius strip, can form surfaces of constant curvature. Certain highly-symmetric spaces whose points represent lines in the plane have the shape of a Möbius strip. The many applications of Möbius strips include mechanical belts that wear evenly on both sides, dual-trac... | https://en.wikipedia.org/wiki/Mobius_strip |
Möbius strips appear in molecules and devices with novel electrical and electromechanical properties, and have been used to prove impossibility results in social choice theory. In popular culture, Möbius strips appear in artworks by M. C. Escher, Max Bill, and others, and in the design of the recycling symbol. Many arc... | https://en.wikipedia.org/wiki/Mobius_strip |
Performers including Harry Blackstone Sr. and Thomas Nelson Downs have based stage magic tricks on the properties of the Möbius strip. The canons of J. S. Bach have been analyzed using Möbius strips. Many works of speculative fiction feature Möbius strips; more generally, a plot structure based on the Möbius strip, of ... | https://en.wikipedia.org/wiki/Mobius_strip |
In mathematics, a Nekrasov matrix or generalised Nekrasov matrix is a type of diagonally dominant matrix (i.e. one in which the diagonal elements are in some way greater than some function of the non-diagonal elements). Specifically if A is a generalised Nekrasov matrix, its diagonal elements are non-zero and the diago... | https://en.wikipedia.org/wiki/Nekrasov_matrix |
In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence a n {\displaystyle a_{n}} written in the form f ( s ) = ∑ n = 0 ∞ ( − 1 ) n ( s n ) a n = ∑ n = 0 ∞ ( − s ) n n ! a n {\displaystyle f(s)=\sum _{n=0}^{\infty }(-1)^{n}{s \choose n}a_{n}=\sum _{n=0}^{\infty }{\frac {(-s)_{n}}{n! }}a_{... | https://en.wikipedia.org/wiki/Table_of_Newtonian_series |
In mathematics, a Nichols algebra is a Hopf algebra in a braided category assigned to an object V in this category (e.g. a braided vector space). The Nichols algebra is a quotient of the tensor algebra of V enjoying a certain universal property and is typically infinite-dimensional. Nichols algebras appear naturally in... | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
The following article lists all known finite-dimensional Nichols algebras B ( V ) {\displaystyle {\mathfrak {B}}(V)} where V {\displaystyle V} is a Yetter–Drinfel'd module over a finite group G {\displaystyle G} , where the group is generated by the support of V {\displaystyle V} . For more details on Nichols algebras ... | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
G {\displaystyle G} nonabelian. The rank is the number of irreducible summands V = ⨁ i ∈ I V i {\displaystyle V=\bigoplus _{i\in I}V_{i}} in the semisimple Yetter–Drinfel'd module V {\displaystyle V} . The irreducible summands V i = O χ {\displaystyle V_{i}={\mathcal {O}}_{}^{\chi }} are each associated to a conjugacy... | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
To any Nichols algebra there is by attached a generalized root system and a Weyl groupoid. These are classified in. | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
In particular several Dynkin diagrams (for inequivalent types of Weyl chambers). Each Dynkin diagram has one vertex per irreducible V i {\displaystyle V_{i}} and edges depending on their braided commutators in the Nichols algebra. The Hilbert series of the graded algebra B ( V ) {\displaystyle {\mathfrak {B}}(V)} is gi... | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
An observation is that it factorizes in each case into polynomials ( n ) t := 1 + t + t 2 + ⋯ + t n − 1 {\displaystyle (n)_{t}:=1+t+t^{2}+\cdots +t^{n-1}} . We only give the Hilbert series and dimension of the Nichols algebra in characteristic 0 {\displaystyle 0} .Note that a Nichols algebra only depends on the braided... | https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras |
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