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Legendrian knot Summary Legendrian_knots 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 submanifo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Legendrian knot Summary Legendrian_knots 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 construc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Legendrian knot Summary Legendrian_knots 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 {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lehmer sequence Summary Lehmer_sequence In mathematics, a Lehmer sequence is a generalization of a Lucas sequence. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leray cover Summary Leray_cover 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leray cover Summary 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 im... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leray cover Summary 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 obtai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leray cover Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Leray cover Summary 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lidstone series Summary Lidstone_series 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lidstone series Summary 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 su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Center of a Lie algebra Summary Lie_algebra_homomorphism 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Center of a Lie algebra Summary Lie_algebra_homomorphism 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Center of a Lie algebra Summary Lie_algebra_homomorphism 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Center of a Lie algebra Summary Lie_algebra_homomorphism 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 {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Center of a Lie algebra Summary Lie_algebra_homomorphism 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\tim... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent Lie algebra Summary Nilpotent_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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent Lie algebra Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent Lie algebra Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived Lie algebra Summary Derived_algebra_of_a_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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived Lie algebra Summary Derived_algebra_of_a_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 su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Derived Lie algebra Summary Derived_algebra_of_a_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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reductive Lie algebra Summary Reductive_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 {\mat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Semi-simple Lie group Summary Root_system_of_a_semi-simple_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 fin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebroid Summary Lie_algebroid 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie algebroid Summary 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie bialgebra Summary Lie_bialgebra 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 spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite dimensional Lie group Summary Lie_Groups 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 ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite dimensional Lie group Summary Lie_Groups 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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite dimensional Lie group Summary Lie_Groups 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 mo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie groupoid Summary Lie_groupoid 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 inv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie superalgebra Summary Super_Jacobi_identity 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 sup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lie-* algebra Summary Lie-*_algebra 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lindelöf space Summary Lindelöf_space 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 to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lindelöf space Summary 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 Lindel... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lipschitz domain Summary Lipschitz_boundary 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 Ger... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Listing number Summary Listing_number 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 List... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Littlewood polynomial Summary Littlewood_polynomial 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 au... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Loeb space Summary Loeb_space In mathematics, a Loeb space is a type of measure space introduced by Loeb (1975) using nonstandard analysis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Loewy ring Summary Loewy_length 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lorentz surface Summary Lorentz_surface 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lucas chain Summary Lucas_chain 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 pow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lucas–Carmichael number Summary Lucas–Carmichael_number 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Luzin set Summary Luzin_space 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, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lüroth quartic Summary Lüroth_quartic 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Macbeath region Summary Macbeath_region 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary Madhava_series 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 scho... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary Madhava_series + θ 5 5 ! − θ 7 7 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary Madhava_series + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k + 1 ) ! θ 2 k + 1 , cos θ = 1 − θ 2 2 ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary Madhava_series + θ 4 4 ! − θ 6 6 ! + ⋯ = ∑ k = 0 ∞ ( − 1 ) k ( 2 k ) ! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary Madhava_series }}+{\frac {\theta ^{4}}{4! }}-{\frac {\theta ^{6}}{6! }}+\cdots &&=\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{(2k)! | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary 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 se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Madhava series Summary 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 w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Maharam algebra Summary Maharam_algebra In mathematics, a Maharam algebra is a complete Boolean algebra with a continuous submeasure (defined below). They were introduced by Dorothy Maharam (1947). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyper-Mahlo cardinal Summary Strongly_Mahlo_cardinal 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Malcev Lie algebra Summary Malcev_Lie_algebra 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Malcev algebra Summary Malcev_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 ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Malcev algebra Summary 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 for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov Decision Process Summary Markov_Decision_Processes 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov Decision Process Summary Markov_Decision_Processes 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov Decision Process Summary Markov_Decision_Processes 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov Decision Process Summary Markov_Decision_Processes 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov information source Summary Markov_source 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Markov odometer Summary Markov_odometer 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-equival... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Marot ring Summary Marot_ring In mathematics, a Marot ring, introduced by Marot (1969), is a commutative ring whose regular ideals are generated by regular elements. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Menger space Summary Menger_space 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mennicke symbol Summary Mennicke_symbol 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mersenne numbers Summary Mersenne_number 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 com... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mersenne numbers Summary Mersenne_number 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 =... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mersenne numbers Summary Mersenne_number 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 nu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mersenne numbers Summary Mersenne_number 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 afte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Metzler matrix Summary Quasipositive_matrix 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Meyer set Summary Meyer_set 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Minkowski plane Summary Minkowski_plane In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes (the others being Möbius plane and Laguerre plane). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Misiurewicz point Summary Misiurewicz_point 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moishezon manifold Summary Moishezon_manifold 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moishezon manifold Summary 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 varie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebraic hyperbolicity Summary Mordellic_variety 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 prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moufang loop Summary Moufang_loop 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Moufang set Summary Moufang_set In mathematics, a Moufang set is a particular kind of combinatorial system named after Ruth Moufang. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multibrot set Summary Multibrot_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. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multibrot set Summary 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mumford measure Summary Mumford_measure 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_strip 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 alrea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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 k... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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-in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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 we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mobius strip Summary Möbius_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 struct... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nekrasov matrix Summary Nekrasov_matrix 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 diag... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Table of Newtonian series Summary Table_of_Newtonian_series 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 \ch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary List_of_finite-dimensional_Nichols_algebras 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 certai... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary 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 summan... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary 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 al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
List of finite-dimensional Nichols algebras Summary 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 algebr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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