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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-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane. Two Legendrian knots are equivalent if they are isotopic through a family of Legendrian knots. There can be inequivalent Legendrian knots that are isotopic as topological knots.
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https://en.wikipedia.org/wiki/Legendrian_knot
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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 holomorphic discs by Eliashberg.
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https://en.wikipedia.org/wiki/Legendrian_knot
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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 (a knot transverse to a contact structure) by pushing off in a direction transverse to the contact planes. The set of isomorphism classes of Legendrian knots modulo negative Legendrian stabilizations is in bijection with the set of transverse knots.
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https://en.wikipedia.org/wiki/Legendrian_knot
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In mathematics, a Lehmer sequence is a generalization of a Lucas sequence.
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https://en.wikipedia.org/wiki/Lehmer_sequence
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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 exact.
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https://en.wikipedia.org/wiki/Leray_cover
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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 definition.
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https://en.wikipedia.org/wiki/Leray_cover
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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 does not depend on the open cover chosen. In reasonable circumstances (for instance, if the topological space is paracompact), the derived-functor cohomology agrees with this Čech cohomology obtained by direct limits.
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https://en.wikipedia.org/wiki/Leray_cover
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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 any Leray cover.
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https://en.wikipedia.org/wiki/Leray_cover
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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 { i 1 , … , i n } {\displaystyle \{i_{1},\ldots ,i_{n}\}} of indices, and for all k > 0 {\displaystyle k>0} , we have that H k ( U i 1 ∩ ⋯ ∩ U i n , F ) = 0 {\displaystyle H^{k}(U_{i_{1}}\cap \cdots \cap U_{i_{n}},{\mathcal {F}})=0} , in the derived functor cohomology. For example, if X {\displaystyle X} is a separated scheme, and F {\displaystyle {\mathcal {F}}} is quasicoherent, then any cover of X {\displaystyle X} by open affine subschemes is a Leray cover. == References ==
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https://en.wikipedia.org/wiki/Leray_cover
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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 ) = ∑ n = 0 ∞ + ∑ k = 1 N C k sin ( k π z ) .
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https://en.wikipedia.org/wiki/Lidstone_series
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{\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 ) | {\displaystyle h(\theta ;f)={\underset {r\to \infty }{\limsup }}\,{\frac {1}{r}}\log |f(re^{i\theta })|} is bounded above by t. Thus, the constant N used in the summation above is given by t = sup θ ∈ [ 0 , 2 π ) h ( θ ; f ) {\displaystyle t=\sup _{\theta \in [0,2\pi )}h(\theta ;f)} with N π ≤ t < ( N + 1 ) π . {\displaystyle N\pi \leq t<(N+1)\pi .}
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https://en.wikipedia.org/wiki/Lidstone_series
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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 said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x {\displaystyle x} and y {\displaystyle y} is denoted {\displaystyle } . The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative.
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https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra
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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 related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem).
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https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra
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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.
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https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra
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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 operation defined by the cross product = x × y . {\displaystyle =x\times y.}
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https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra
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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 of rotations of space, and each vector v ∈ R 3 {\displaystyle v\in \mathbb {R} ^{3}} may be pictured as an infinitesimal rotation around the axis v {\displaystyle v} , with velocity equal to the magnitude of v {\displaystyle v} . The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property = x × x = 0 {\displaystyle =x\times x=0} .
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https://en.wikipedia.org/wiki/Center_of_a_Lie_algebra
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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 ≥ ≥ ] ≥ ] ] ≥ . .
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https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra
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. {\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 nilpotent.
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https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra
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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 that, viewed as a non-associative non-unital algebra, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if it is nilpotent as an ideal.
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https://en.wikipedia.org/wiki/Nilpotent_Lie_algebra
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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 combinations of Lie brackets of pairs of elements of g {\displaystyle {\mathfrak {g}}} . The derived series is the sequence of subalgebras g ≥ ≥ , ] ≥ , ] , , ] ] ≥ . .
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https://en.wikipedia.org/wiki/Derived_Lie_algebra
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. {\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 analogs of solvable groups.
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https://en.wikipedia.org/wiki/Derived_Lie_algebra
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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, starting from 0 and adding one dimension at a time.A maximal solvable subalgebra is called a Borel subalgebra. The largest solvable ideal of a Lie algebra is called the radical.
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https://en.wikipedia.org/wiki/Derived_Lie_algebra
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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}};} there are alternative characterizations, given below.
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https://en.wikipedia.org/wiki/Reductive_Lie_algebra
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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 a Lie algebra g {\displaystyle {\mathfrak {g}}} , if nonzero, the following conditions are equivalent: g {\displaystyle {\mathfrak {g}}} is semisimple; the Killing form, κ(x,y) = tr(ad(x)ad(y)), is non-degenerate; g {\displaystyle {\mathfrak {g}}} has no non-zero abelian ideals; g {\displaystyle {\mathfrak {g}}} has no non-zero solvable ideals; the radical (maximal solvable ideal) of g {\displaystyle {\mathfrak {g}}} is zero.
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https://en.wikipedia.org/wiki/Semi-simple_Lie_group
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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 of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones.
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https://en.wikipedia.org/wiki/Lie_algebroid
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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.
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https://en.wikipedia.org/wiki/Lie_algebroid
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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 comultiplication is a 1-cocycle, so that the multiplication and comultiplication are compatible. The cocycle condition implies that, in practice, one studies only classes of bialgebras that are cohomologous to a Lie bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in the study of the Yang–Baxter equations.
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https://en.wikipedia.org/wiki/Lie_bialgebra
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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 abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.
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https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group
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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 were first found by studying matrix subgroups G {\displaystyle G} contained in GL n ( R ) {\displaystyle {\text{GL}}_{n}(\mathbb {R} )} or GL n ( C ) {\displaystyle {\text{GL}}_{n}(\mathbb {C} )} , the groups of n × n {\displaystyle n\times n} invertible matrices over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } .
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https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group
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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 the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.
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https://en.wikipedia.org/wiki/Infinite_dimensional_Lie_group
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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 and target operations s , t: Mor → Ob {\displaystyle s,t:\operatorname {Mor} \to \operatorname {Ob} } are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name differentiable groupoids.
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https://en.wikipedia.org/wiki/Lie_groupoid
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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 to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
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https://en.wikipedia.org/wiki/Lie_superalgebra
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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.
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https://en.wikipedia.org/wiki/Lie-*_algebra
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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 subspace of it is Lindelöf.
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https://en.wikipedia.org/wiki/Lindelöf_space
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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.
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https://en.wikipedia.org/wiki/Lindelöf_space
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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.
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https://en.wikipedia.org/wiki/Lipschitz_domain
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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 of invariants: two different two-dimensional manifolds may have the same Listing numbers as each other.There are four Listing numbers associated with a space. The smallest Listing number counts the number of connected components of a space, and is thus equivalent to the zeroth Betti number. == References ==
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https://en.wikipedia.org/wiki/Listing_number
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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 for J. E. Littlewood who studied them in the 1950s.
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https://en.wikipedia.org/wiki/Littlewood_polynomial
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In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis.
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https://en.wikipedia.org/wiki/Loeb_space
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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.
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https://en.wikipedia.org/wiki/Loewy_ring
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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 ==
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https://en.wikipedia.org/wiki/Lorentz_surface
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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, ...) and the Fibonacci sequence (with a slight adjustment of the starting point 1, 2, 3, 5, 8, ...) are simple examples of Lucas chains. Lucas chains were introduced by Peter Montgomery in 1983. If L(n) is the length of the shortest Lucas chain for n, then Kutz has shown that most n do not have L < (1-ε) logφ n, where φ is the Golden ratio.
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https://en.wikipedia.org/wiki/Lucas_chain
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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 consideration some trivial cases like cubes of prime numbers, such as 8 or 27, which otherwise would be Lucas–Carmichael numbers (since n3 + 1 = (n + 1)(n2 − n + 1) is always divisible by n + 1). They are named after Édouard Lucas and Robert Carmichael.
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https://en.wikipedia.org/wiki/Lucas–Carmichael_number
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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 countable or even arbitrary number of isolated points. The existence of a Luzin space is independent of the axioms of ZFC. Luzin (1914) showed that the continuum hypothesis implies that a Luzin space exists. Kunen (1977) showed that assuming Martin's axiom and the negation of the continuum hypothesis, there are no Hausdorff Luzin spaces.
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https://en.wikipedia.org/wiki/Luzin_set
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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 quartics. Böhning & von Bothmer (2011) proved that the moduli space of Lüroth quartics is rational.
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https://en.wikipedia.org/wiki/Lüroth_quartic
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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 have been used to solve certain complex problems in the study of the boundaries of convex bodies. Recently they have been used in the study of convex approximations and other aspects of computational geometry.
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https://en.wikipedia.org/wiki/Macbeath_region
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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 modern notation, these series are: sin θ = θ − θ 3 3 !
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https://en.wikipedia.org/wiki/Madhava_series
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+ θ 5 5 ! − θ 7 7 !
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https://en.wikipedia.org/wiki/Madhava_series
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+ ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos θ = 1 − θ 2 2 !
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https://en.wikipedia.org/wiki/Madhava_series
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+ θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) !
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https://en.wikipedia.org/wiki/Madhava_series
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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!
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https://en.wikipedia.org/wiki/Madhava_series
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}}-{\frac {\theta ^{7}}{7! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k+1)! }}\theta ^{2k+1},\\\cos \theta &=1-{\frac {\theta ^{2}}{2!
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https://en.wikipedia.org/wiki/Madhava_series
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}}+{\frac {\theta ^{4}}{4! }}-{\frac {\theta ^{6}}{6! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)!
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https://en.wikipedia.org/wiki/Madhava_series
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}}\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 rediscovered by Isaac Newton in 1669, and the series for arctangent was rediscovered by James Gregory in 1671 and Gottfried Leibniz in 1673, and is conventionally called Gregory's series. The specific value arctan 1 = 1 4 π {\textstyle \arctan 1={\tfrac {1}{4}}\pi } can be used to calculate the circle constant π, and the arctangent series for 1 is conventionally called Leibniz's series.
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https://en.wikipedia.org/wiki/Madhava_series
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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 series. However, in the writing of later Kerala school mathematicians Nilakantha Somayaji and Jyeshthadeva one can find unambiguous attributions of these series to Madhava. These later works also include proofs and commentary which suggest how Madhava may have arrived at the series.
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https://en.wikipedia.org/wiki/Madhava_series
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In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947).
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https://en.wikipedia.org/wiki/Maharam_algebra
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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 \kappa } is called strongly Mahlo if κ {\displaystyle \kappa } is strongly inaccessible and the set U = { λ < κ ∣ λ is strongly inaccessible } {\displaystyle U=\{\lambda <\kappa \mid \lambda {\text{ is strongly inaccessible}}\}} is stationary in κ. A cardinal κ {\displaystyle \kappa } is called weakly Mahlo if κ {\displaystyle \kappa } is weakly inaccessible and the set of weakly inaccessible cardinals less than κ {\displaystyle \kappa } is stationary in κ {\displaystyle \kappa } . The term "Mahlo cardinal" now usually means "strongly Mahlo cardinal", though the cardinals originally considered by Mahlo were weakly Mahlo cardinals.
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https://en.wikipedia.org/wiki/Hyper-Mahlo_cardinal
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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).
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https://en.wikipedia.org/wiki/Malcev_Lie_algebra
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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)x+((yz)x)x+((zx)x)y.} They were first defined by Anatoly Maltsev (1955).
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https://en.wikipedia.org/wiki/Malcev_algebra
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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 a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop.
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https://en.wikipedia.org/wiki/Malcev_algebra
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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 dynamic programming. MDPs were known at least as early as the 1950s; a core body of research on Markov decision processes resulted from Ronald Howard's 1960 book, Dynamic Programming and Markov Processes.
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https://en.wikipedia.org/wiki/Markov_Decision_Process
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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 any action a {\displaystyle a} that is available in state s {\displaystyle s} .
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https://en.wikipedia.org/wiki/Markov_Decision_Process
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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. Specifically, it is given by the state transition function P a ( s , s ′ ) {\displaystyle P_{a}(s,s')} .
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https://en.wikipedia.org/wiki/Markov_Decision_Process
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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 satisfy the Markov property. Markov decision processes are an extension of Markov chains; the difference is the addition of actions (allowing choice) and rewards (giving motivation). Conversely, if only one action exists for each state (e.g. "wait") and all rewards are the same (e.g. "zero"), a Markov decision process reduces to a Markov chain.
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https://en.wikipedia.org/wiki/Markov_Decision_Process
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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.
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https://en.wikipedia.org/wiki/Markov_information_source
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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 example of such system is the "nonsingular odometer", which is an additive topological group defined on the product space of discrete spaces, induced by addition defined as x ↦ x + 1 _ {\displaystyle x\mapsto x+{\underline {1}}} , where 1 _ := ( 1 , 0 , 0 , … ) {\displaystyle {\underline {1}}:=(1,0,0,\dots )} . This group can be endowed with the structure of a dynamical system; the result is a conservative dynamical system. The general form, which is called "Markov odometer", can be constructed through Bratteli–Vershik diagram to define Bratteli–Vershik compactum space together with a corresponding transformation.
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https://en.wikipedia.org/wiki/Markov_odometer
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In mathematics, a Marot ring, introduced by Marot (1969), is a commutative ring whose regular ideals are generated by regular elements.
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https://en.wikipedia.org/wiki/Marot_ring
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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 ⊂ U 1 , F 2 ⊂ U 2 , … {\displaystyle {\mathcal {F}}_{1}\subset {\mathcal {U}}_{1},{\mathcal {F}}_{2}\subset {\mathcal {U}}_{2},\ldots } such that the family F 1 ∪ F 2 ∪ ⋯ {\displaystyle {\mathcal {F}}_{1}\cup {\mathcal {F}}_{2}\cup \cdots } covers the space.
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https://en.wikipedia.org/wiki/Menger_space
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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.
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https://en.wikipedia.org/wiki/Mennicke_symbol
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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. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2p − 1 for some prime p. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31, ... (sequence A000043 in the OEIS) and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ... (sequence A000668 in the OEIS).
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https://en.wikipedia.org/wiki/Mersenne_numbers
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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.
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https://en.wikipedia.org/wiki/Mersenne_numbers
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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. As of 2023, 51 Mersenne primes are known.
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https://en.wikipedia.org/wiki/Mersenne_numbers
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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 checked at least once.
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https://en.wikipedia.org/wiki/Mersenne_numbers
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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 delayed differential equations and positive linear dynamical systems. Their properties can be derived by applying the properties of nonnegative matrices to matrices of the form M + aI, where M is a Metzler matrix.
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https://en.wikipedia.org/wiki/Metzler_matrix
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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 and studied them in the context of diophantine approximation. Nowadays Meyer sets are best known as mathematical model for quasicrystals. However, Meyer's work precedes the discovery of quasicrystals by more than a decade and was entirely motivated by number theoretic questions.
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https://en.wikipedia.org/wiki/Meyer_set
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In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane).
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https://en.wikipedia.org/wiki/Minkowski_plane
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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 analogy, the term Misiurewicz point is also used for parameters in a multibrot set where the unique critical point is strictly pre-periodic (This term makes less sense for maps in greater generality that have more than one (free) critical point because some critical points might be periodic and others not). These points are named after mathematician Michał Misiurewicz who first studied them.
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https://en.wikipedia.org/wiki/Misiurewicz_point
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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 ) .
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https://en.wikipedia.org/wiki/Moishezon_manifold
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{\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, Theorem 11) showed that a Moishezon manifold is a projective algebraic variety if and only if it admits a Kähler metric. Artin (1970) showed that any Moishezon manifold carries an algebraic space structure; more precisely, the category of Moishezon spaces (similar to Moishezon manifolds, but are allowed to have singularities) is equivalent with the category of algebraic spaces that are proper over Spec(C).
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https://en.wikipedia.org/wiki/Moishezon_manifold
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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.
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https://en.wikipedia.org/wiki/Algebraic_hyperbolicity
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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 Lie algebra.
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https://en.wikipedia.org/wiki/Moufang_loop
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In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang.
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https://en.wikipedia.org/wiki/Moufang_set
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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 set, this being called Multijulia set.
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https://en.wikipedia.org/wiki/Multibrot_set
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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.
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https://en.wikipedia.org/wiki/Multibrot_set
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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.
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https://en.wikipedia.org/wiki/Mumford_measure
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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 the third century CE. The Möbius strip is a non-orientable surface, meaning that within it one cannot consistently distinguish clockwise from counterclockwise turns. Every non-orientable surface contains a Möbius strip.
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https://en.wikipedia.org/wiki/Mobius_strip
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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 embeddings with the same knot for the centerline and the same number and direction of twists are topologically equivalent. All of these embeddings have only one side, but when embedded in other spaces, the Möbius strip may have two sides.
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https://en.wikipedia.org/wiki/Mobius_strip
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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.
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https://en.wikipedia.org/wiki/Mobius_strip
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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 ordinary Euclidean space. Both the Sudanese Möbius strip and another self-intersecting Möbius strip, the cross-cap, have a circular boundary.
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https://en.wikipedia.org/wiki/Mobius_strip
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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-track roller coasters whose carriages alternate between the two tracks, and world maps printed so that antipodes appear opposite each other.
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https://en.wikipedia.org/wiki/Mobius_strip
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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 architectural concepts have been inspired by the Möbius strip, including the building design for the NASCAR Hall of Fame.
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https://en.wikipedia.org/wiki/Mobius_strip
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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 events that repeat with a twist, is common in fiction.
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https://en.wikipedia.org/wiki/Mobius_strip
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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 diagonal elements also satisfy, a i i > R i ( A ) {\displaystyle a_{ii}>R_{i}(A)} where, R i ( A ) = ∑ j = 1 i − 1 | a i j | R j ( A ) | a j j | + ∑ j = i + 1 n | a i j | {\displaystyle R_{i}(A)=\sum _{j=1}^{i-1}|a_{ij}|{\frac {R_{j}(A)}{|a_{jj}|}}+\sum _{j=i+1}^{n}|a_{ij}|} . == References ==
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https://en.wikipedia.org/wiki/Nekrasov_matrix
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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_{n}} where ( s n ) {\displaystyle {s \choose n}} is the binomial coefficient and ( s ) n {\displaystyle (s)_{n}} is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
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https://en.wikipedia.org/wiki/Table_of_Newtonian_series
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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 any pointed Hopf algebra and enabled their classification in important cases. The most well known examples for Nichols algebras are the Borel parts U q ( g ) + {\displaystyle U_{q}({\mathfrak {g}})^{+}} of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts u q ( g ) + {\displaystyle u_{q}({\mathfrak {g}})^{+}} of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity.
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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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 see Nichols algebra. There are two major cases: G {\displaystyle G} abelian, which implies V {\displaystyle V} is diagonally braided x i ⊗ x j ↦ q i j x j ⊗ x i {\displaystyle x_{i}\otimes x_{j}\mapsto q_{ij}x_{j}\otimes x_{i}} .
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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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 class ⊂ G {\displaystyle \subset G} and an irreducible representation χ {\displaystyle \chi } of the centralizer Cent ( g ) {\displaystyle \operatorname {Cent} (g)} .
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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To any Nichols algebra there is by attached a generalized root system and a Weyl groupoid. These are classified in.
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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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 given.
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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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 vector space V {\displaystyle V} and can therefore be realized over many different groups. Sometimes there are two or three Nichols algebras with different V {\displaystyle V} and non-isomorphic Nichols algebra, which are closely related (e.g. cocycle twists of each other). These are given by different conjugacy classes in the same column.
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https://en.wikipedia.org/wiki/List_of_finite-dimensional_Nichols_algebras
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