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In mathematics, the Legendre transformation (or Legendre transform), first introduced by Adrien-Marie Legendre in 1787 when studying the minimal surface problem, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions of one quantity (such as velocity, pressure, or temperature) into functions of the conjugate quantity (momentum, volume, and entropy, respectively). In this way, it is commonly used in classical mechanics to derive the Hamiltonian formalism out of the Lagrangian formalism (or vice versa) and in thermodynamics to derive the thermodynamic potentials, as well as in the solution of differential equations of several variables. For sufficiently smooth functions on the real line, the Legendre transform f ∗ {\displaystyle f^{*}} of a function f {\displaystyle f} can be specified, up to an additive constant, by the condition that the functions' first derivatives are inverse functions of each other. This can be expressed in Euler's derivative notation as where D {\displaystyle D} is an operator of differentiation, ⋅ {\displaystyle \cdot } represents an argument or input to the associated function, ( ϕ ) − 1 ( ⋅ ) {\displaystyle (\phi )^{-1}(\cdot )} is an inverse function such that ( ϕ ) − 1 ( ϕ ( x ) ) = x {\displaystyle (\phi )^{-1}(\phi (x))=x} , or equivalently, as f ′ ( f ∗ ′ ( x ∗ ) ) = x ∗ {\displaystyle f'(f^{*\prime }(x^{*}))=x^{*}} and f ∗ ′ ( f ′ ( x ) ) = x {\displaystyle f^{*\prime }(f'(x))=x} in Lagrange's notation. The generalization of the Legendre transformation to affine spaces and non-convex functions is known as the convex conjugate (also called the Legendre–Fenchel transformation), which can be used to construct a function's convex hull.
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https://en.wikipedia.org/wiki/Legendre_transformation
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In mathematics, the Lehmer mean of a tuple x {\displaystyle x} of positive real numbers, named after Derrick Henry Lehmer, is defined as: L p ( x ) = ∑ k = 1 n x k p ∑ k = 1 n x k p − 1 . {\displaystyle L_{p}(\mathbf {x} )={\frac {\sum _{k=1}^{n}x_{k}^{p}}{\sum _{k=1}^{n}x_{k}^{p-1}}}.} The weighted Lehmer mean with respect to a tuple w {\displaystyle w} of positive weights is defined as: L p , w ( x ) = ∑ k = 1 n w k ⋅ x k p ∑ k = 1 n w k ⋅ x k p − 1 . {\displaystyle L_{p,w}(\mathbf {x} )={\frac {\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p}}{\sum _{k=1}^{n}w_{k}\cdot x_{k}^{p-1}}}.} The Lehmer mean is an alternative to power means for interpolating between minimum and maximum via arithmetic mean and harmonic mean.
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https://en.wikipedia.org/wiki/Lehmer_mean
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In mathematics, the Lehmer–Schur algorithm (named after Derrick Henry Lehmer and Issai Schur) is a root-finding algorithm for complex polynomials, extending the idea of enclosing roots like in the one-dimensional bisection method to the complex plane. It uses the Schur-Cohn test to test increasingly smaller disks for the presence or absence of roots.
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https://en.wikipedia.org/wiki/Lehmer–Schur_algorithm
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In mathematics, the Leibniz formula for π, named after Gottfried Wilhelm Leibniz, states that an alternating series. It is sometimes called the Madhava–Leibniz series as it was first discovered by the Indian mathematician Madhava of Sangamagrama or his followers in the 14th–15th century (see Madhava series), and was later independently rediscovered by James Gregory in 1671 and Leibniz in 1673. The Taylor series for the inverse tangent function, often called Gregory's series, is: arctan x = x − x 3 3 + x 5 5 − x 7 7 + ⋯ {\displaystyle \arctan x=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-{\frac {x^{7}}{7}}+\cdots } The Leibniz formula is the special case arctan 1 = 1 4 π . {\textstyle \arctan 1={\tfrac {1}{4}}\pi .} It also is the Dirichlet L-series of the non-principal Dirichlet character of modulus 4 evaluated at s = 1 {\displaystyle s=1} , and, therefore, the value β(1) of the Dirichlet beta function.
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https://en.wikipedia.org/wiki/Leibniz_series
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In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence.
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https://en.wikipedia.org/wiki/Leray_spectral_sequence
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In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of the Künneth formula, which computes the cohomology of a product space as a tensor product of the cohomologies of the direct factors. It is a very special case of the Leray spectral sequence.
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https://en.wikipedia.org/wiki/Leray–Hirsch_theorem
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In mathematics, the Leray–Schauder degree is an extension of the degree of a base point preserving continuous map between spheres ( S n , ∗ ) → ( S n , ∗ ) {\displaystyle (S^{n},*)\to (S^{n},*)} or equivalently to a boundary sphere preserving continuous maps between balls ( B n , S n − 1 ) → ( B n , S n − 1 ) {\displaystyle (B^{n},S^{n-1})\to (B^{n},S^{n-1})} to boundary sphere preserving maps between balls in a Banach space f: ( B ( V ) , S ( V ) ) → ( B ( V ) , S ( V ) ) {\displaystyle f:(B(V),S(V))\to (B(V),S(V))} , assuming that the map is of the form f = i d − C {\displaystyle f=id-C} where i d {\displaystyle id} is the identity map and C {\displaystyle C} is some compact map (i.e. mapping bounded sets to sets whose closure is compact).The degree was invented by Jean Leray and Juliusz Schauder to prove existence results for partial differential equations. == References ==
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https://en.wikipedia.org/wiki/Leray–Schauder_degree
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In mathematics, the Lerch zeta function, sometimes called the Hurwitz–Lerch zeta function, is a special function that generalizes the Hurwitz zeta function and the polylogarithm. It is named after Czech mathematician Mathias Lerch, who published a paper about the function in 1887.
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https://en.wikipedia.org/wiki/Lerch_zeta_function
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In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a {\displaystyle a} can be constructed as a formal series of the form a = ∑ q ∈ Q a q ε q , {\displaystyle a=\sum _{q\in \mathbb {Q} }a_{q}\varepsilon ^{q},} where a q {\displaystyle a_{q}} are real numbers, Q {\displaystyle \mathbb {Q} } is the set of rational numbers, and ε {\displaystyle \varepsilon } is to be interpreted as a fixed positive infinitesimal. The support of a {\displaystyle a} , i.e., the set of indices of the nonvanishing coefficients { q ∈ Q: a q ≠ 0 } , {\displaystyle \{q\in \mathbb {Q} :a_{q}\neq 0\},} must be a left-finite set: for any member of Q {\displaystyle \mathbb {Q} } , there are only finitely many members of the set less than it; this restriction is necessary in order to make multiplication and division well defined and unique. The ordering is defined according to the dictionary ordering of the list of coefficients, which is equivalent to the assumption that ε {\displaystyle \varepsilon } is an infinitesimal. The real numbers are embedded in this field as series in which all of the coefficients vanish except a 0 {\displaystyle a_{0}} .
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https://en.wikipedia.org/wiki/Levi-Civita_field
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In mathematics, the Lichnerowicz conjecture is a generalization of a conjecture introduced by Lichnerowicz (1944). Lichnerowicz's original conjecture was that locally harmonic 4-manifolds are locally symmetric, and was proved by Walker (1949). The Lichnerowicz conjecture usually refers to the generalization that locally harmonic manifolds are flat or rank-1 locally symmetric. It has been proven true for compact manifolds with fundamental groups that are finite groups (Szabó 1990) but counterexamples exist in seven or more dimensions in the non-compact case (Damek & Ricci 1992)
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https://en.wikipedia.org/wiki/Lichnerowicz_conjecture
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In mathematics, the Lickorish–Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with ±1 surgery coefficients. Furthermore, each component of the link can be assumed to be unknotted. The theorem was proved in the early 1960s by W. B. R. Lickorish and Andrew H. Wallace, independently and by different methods.
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https://en.wikipedia.org/wiki/Lickorish–Wallace_theorem
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Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface. Wallace's proof was more general and involved adding handles to the boundary of a higher-dimensional ball. A corollary of the theorem is that every closed, orientable 3-manifold bounds a simply-connected compact 4-manifold. By using his work on automorphisms of non-orientable surfaces, Lickorish also showed that every closed, non-orientable, connected 3-manifold is obtained by Dehn surgery on a link in the non-orientable 2-sphere bundle over the circle. Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold.
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https://en.wikipedia.org/wiki/Lickorish–Wallace_theorem
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In mathematics, the Lie algebra E7½ is a subalgebra of E8 containing E7 defined by Landsberg and Manivel in order to fill the "hole" in a dimension formula for the exceptional series En of simple Lie algebras. This hole was observed by Cvitanovic, Deligne, Cohen and de Man. E7½ has dimension 190, and is not simple: as a representation of its subalgebra E7, it splits as E7 ⊕ (56) ⊕ R, where (56) is the 56-dimensional irreducible representation of E7. This representation has an invariant symplectic form, and this symplectic form equips (56) ⊕ R with the structure of a Heisenberg algebra; this Heisenberg algebra is the nilradical in E7½.
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https://en.wikipedia.org/wiki/E7½_(Lie_algebra)
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In mathematics, the Lie operad is an operad whose algebras are Lie algebras. The notion (at least one version) was introduced by Ginzburg & Kapranov (1994) in their formulation of Koszul duality.
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https://en.wikipedia.org/wiki/Lie_operad
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In mathematics, the Lie product formula, named for Sophus Lie (1875), but also widely called the Trotter product formula, named after Hale Trotter, states that for arbitrary m × m real or complex matrices A and B, e A + B = lim n → ∞ ( e A / n e B / n ) n , {\displaystyle e^{A+B}=\lim _{n\rightarrow \infty }(e^{A/n}e^{B/n})^{n},} where eA denotes the matrix exponential of A. The Lie–Trotter product formula (Trotter 1959) and the Trotter–Kato theorem (Kato 1978) extend this to certain unbounded linear operators A and B.This formula is an analogue of the classical exponential law e x + y = e x e y {\displaystyle e^{x+y}=e^{x}e^{y}\,} which holds for all real or complex numbers x and y. If x and y are replaced with matrices A and B, and the exponential replaced with a matrix exponential, it is usually necessary for A and B to commute for the law to still hold. However, the Lie product formula holds for all matrices A and B, even ones which do not commute. The Lie product formula is conceptually related to the Baker–Campbell–Hausdorff formula, in that both are replacements, in the context of noncommuting operators, for the classical exponential law.
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https://en.wikipedia.org/wiki/Lie_product_formula
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The formula has applications, for example, in the path integral formulation of quantum mechanics. It allows one to separate the Schrödinger evolution operator (propagator) into alternating increments of kinetic and potential operators (the Suzuki–Trotter decomposition, after Trotter and Masuo Suzuki). The same idea is used in the construction of splitting methods for the numerical solution of differential equations. Moreover, the Lie product theorem is sufficient to prove the Feynman–Kac formula.The Trotter–Kato theorem can be used for approximation of linear C0-semigroups.
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https://en.wikipedia.org/wiki/Lie_product_formula
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In mathematics, the Lie–Kolchin theorem is a theorem in the representation theory of linear algebraic groups; Lie's theorem is the analog for linear Lie algebras. It states that if G is a connected and solvable linear algebraic group defined over an algebraically closed field and ρ: G → G L ( V ) {\displaystyle \rho \colon G\to GL(V)} a representation on a nonzero finite-dimensional vector space V, then there is a one-dimensional linear subspace L of V such that ρ ( G ) ( L ) = L . {\displaystyle \rho (G)(L)=L.} That is, ρ(G) has an invariant line L, on which G therefore acts through a one-dimensional representation.
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https://en.wikipedia.org/wiki/Lie–Kolchin_theorem
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This is equivalent to the statement that V contains a nonzero vector v that is a common (simultaneous) eigenvector for all ρ ( g ) , g ∈ G {\displaystyle \rho (g),\,\,g\in G} . It follows directly that every irreducible finite-dimensional representation of a connected and solvable linear algebraic group G has dimension one. In fact, this is another way to state the Lie–Kolchin theorem. The result for Lie algebras was proved by Sophus Lie (1876) and for algebraic groups was proved by Ellis Kolchin (1948, p.19). The Borel fixed point theorem generalizes the Lie–Kolchin theorem.
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https://en.wikipedia.org/wiki/Lie–Kolchin_theorem
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In mathematics, the Lindelöf hypothesis is a conjecture by Finnish mathematician Ernst Leonard Lindelöf (see Lindelöf (1908)) about the rate of growth of the Riemann zeta function on the critical line. This hypothesis is implied by the Riemann hypothesis. It says that for any ε > 0, as t tends to infinity (see big O notation). Since ε can be replaced by a smaller value, the conjecture can be restated as follows: for any positive ε,
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https://en.wikipedia.org/wiki/Lindelöf_hypothesis
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In mathematics, the Lions–Lax–Milgram theorem (or simply Lions's theorem) is a result in functional analysis with applications in the study of partial differential equations. It is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear function can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Jacques-Louis Lions, Peter Lax and Arthur Milgram.
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https://en.wikipedia.org/wiki/Lions–Lax–Milgram_theorem
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In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself.
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https://en.wikipedia.org/wiki/Lions–Magenes_lemma
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In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory.
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https://en.wikipedia.org/wiki/Liouville–Neumann_series
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In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, a Liouvillian function is a function of one variable which is the composition of a finite number of arithmetic operations (+, −, ×, ÷), exponentials, constants, solutions of algebraic equations (a generalization of nth roots), and antiderivatives.
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https://en.wikipedia.org/wiki/Liouvillian_function
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The logarithm function does not need to be explicitly included since it is the integral of 1 / x {\displaystyle 1/x} . It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed under differentiation. It is not closed under limits and infinite sums.Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841.
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https://en.wikipedia.org/wiki/Liouvillian_function
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In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra. Developed as a bridge between the theory of crystal bases arising from the work of Kashiwara and Lusztig on quantum groups and the standard monomial theory of C. S. Seshadri and Lakshmibai, Littelmann's path model associates to each irreducible representation a rational vector space with basis given by paths from the origin to a weight as well as a pair of root operators acting on paths for each simple root. This gives a direct way of recovering the algebraic and combinatorial structures previously discovered by Kashiwara and Lusztig using quantum groups.
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https://en.wikipedia.org/wiki/Path_model
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In mathematics, the Littlewood conjecture is an open problem (as of May 2021) in Diophantine approximation, proposed by John Edensor Littlewood around 1930. It states that for any two real numbers α and β, lim inf n → ∞ n ‖ n α ‖ ‖ n β ‖ = 0 , {\displaystyle \liminf _{n\to \infty }\ n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0,} where ‖ x ‖ := min ( | x − ⌊ x ⌋ | , | x − ⌈ x ⌉ | ) {\displaystyle \Vert x\Vert :=\min(|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)} is the distance to the nearest integer.
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https://en.wikipedia.org/wiki/Littlewood_conjecture
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In mathematics, the Littlewood subordination theorem, proved by J. E. Littlewood in 1925, is a theorem in operator theory and complex analysis. It states that any holomorphic univalent self-mapping of the unit disk in the complex numbers that fixes 0 induces a contractive composition operator on various function spaces of holomorphic functions on the disk. These spaces include the Hardy spaces, the Bergman spaces and Dirichlet space.
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https://en.wikipedia.org/wiki/Littlewood_subordination_theorem
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In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say λ , μ , ν {\displaystyle \lambda ,\mu ,\nu } , of which λ {\displaystyle \lambda } and μ {\displaystyle \mu } describe the Schur functions being multiplied, and ν {\displaystyle \nu } gives the Schur function of which this is the coefficient in the linear combination; in other words they are the coefficients c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} such that s λ s μ = ∑ ν c λ , μ ν s ν . {\displaystyle s_{\lambda }s_{\mu }=\sum _{\nu }c_{\lambda ,\mu }^{\nu }s_{\nu }.} The Littlewood–Richardson rule states that c λ , μ ν {\displaystyle c_{\lambda ,\mu }^{\nu }} is equal to the number of Littlewood–Richardson tableaux of skew shape ν / λ {\displaystyle \nu /\lambda } and of weight μ {\displaystyle \mu } .
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https://en.wikipedia.org/wiki/Littlewood-Richardson_coefficient
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In mathematics, the Loewner differential equation, or Loewner equation, is an ordinary differential equation discovered by Charles Loewner in 1923 in complex analysis and geometric function theory. Originally introduced for studying slit mappings (conformal mappings of the open disk onto the complex plane with a curve joining 0 to ∞ removed), Loewner's method was later developed in 1943 by the Russian mathematician Pavel Parfenevich Kufarev (1909–1968). Any family of domains in the complex plane that expands continuously in the sense of Carathéodory to the whole plane leads to a one parameter family of conformal mappings, called a Loewner chain, as well as a two parameter family of holomorphic univalent self-mappings of the unit disk, called a Loewner semigroup. This semigroup corresponds to a time dependent holomorphic vector field on the disk given by a one parameter family of holomorphic functions on the disk with positive real part.
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https://en.wikipedia.org/wiki/Loewner_equation
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The Loewner semigroup generalizes the notion of a univalent semigroup. The Loewner differential equation has led to inequalities for univalent functions that played an important role in the solution of the Bieberbach conjecture by Louis de Branges in 1985. Loewner himself used his techniques in 1923 for proving the conjecture for the third coefficient. The Schramm–Loewner equation, a stochastic generalization of the Loewner differential equation discovered by Oded Schramm in the late 1990s, has been extensively developed in probability theory and conformal field theory.
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https://en.wikipedia.org/wiki/Loewner_equation
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In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d {\displaystyle d} -dimensional set by the sizes of its ( d − 1 ) {\displaystyle (d-1)} -dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas. The result is named after the American mathematicians Lynn Harold Loomis and Hassler Whitney, and was published in 1949.
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https://en.wikipedia.org/wiki/Loomis-Whitney_inequality
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In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue (Dunford & Schwartz 1958, III.3), although according to the Bourbaki group (Bourbaki 1987) they were first introduced by Frigyes Riesz (Riesz 1910). Lp spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines.
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https://en.wikipedia.org/wiki/L1_space
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In mathematics, the Lubin–Tate formal group law is a formal group law introduced by Lubin and Tate (1965) to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct the totally ramified abelian extensions of a local field. It does this by considering the (formal) endomorphisms of the formal group, emulating the way in which elliptic curves with extra endomorphisms are used to give abelian extensions of global fields.
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https://en.wikipedia.org/wiki/Lubin–Tate_formal_group
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In mathematics, the Lucas sequences U n ( P , Q ) {\displaystyle U_{n}(P,Q)} and V n ( P , Q ) {\displaystyle V_{n}(P,Q)} are certain constant-recursive integer sequences that satisfy the recurrence relation x n = P ⋅ x n − 1 − Q ⋅ x n − 2 {\displaystyle x_{n}=P\cdot x_{n-1}-Q\cdot x_{n-2}} where P {\displaystyle P} and Q {\displaystyle Q} are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U n ( P , Q ) {\displaystyle U_{n}(P,Q)} and V n ( P , Q ) . {\displaystyle V_{n}(P,Q).}
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https://en.wikipedia.org/wiki/Lucas_sequence
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More generally, Lucas sequences U n ( P , Q ) {\displaystyle U_{n}(P,Q)} and V n ( P , Q ) {\displaystyle V_{n}(P,Q)} represent sequences of polynomials in P {\displaystyle P} and Q {\displaystyle Q} with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas.
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https://en.wikipedia.org/wiki/Lucas_sequence
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In mathematics, the Lucas–Lehmer test (LLT) is a primality test for Mersenne numbers. The test was originally developed by Édouard Lucas in 1878 and subsequently proved by Derrick Henry Lehmer in 1930.
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https://en.wikipedia.org/wiki/Lucas–Lehmer_test
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In mathematics, the Lumer–Phillips theorem, named after Günter Lumer and Ralph Phillips, is a result in the theory of strongly continuous semigroups that gives a necessary and sufficient condition for a linear operator in a Banach space to generate a contraction semigroup.
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https://en.wikipedia.org/wiki/Lumer–Phillips_theorem
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In mathematics, the Lusternik–Schnirelmann theorem, aka Lusternik–Schnirelmann–Borsuk theorem or LSB theorem, says as follows. If the sphere Sn is covered by n + 1 closed sets, then one of these sets contains a pair (x, −x) of antipodal points. It is named after Lazar Lyusternik and Lev Schnirelmann, who published it in 1930.
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https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_theorem
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In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector δ Z 0 {\displaystyle \delta \mathbf {Z} _{0}} diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by | δ Z ( t ) | ≈ e λ t | δ Z 0 | {\displaystyle |\delta \mathbf {Z} (t)|\approx e^{\lambda t}|\delta \mathbf {Z} _{0}|} where λ {\displaystyle \lambda } is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector.
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https://en.wikipedia.org/wiki/Lyapunov_exponents
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Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time. The exponent is named after Aleksandr Lyapunov.
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https://en.wikipedia.org/wiki/Lyapunov_exponents
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In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent.
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https://en.wikipedia.org/wiki/Lyapunov_time
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In mathematics, the Lyapunov–Schmidt reduction or Lyapunov–Schmidt construction is used to study solutions to nonlinear equations in the case when the implicit function theorem does not work. It permits the reduction of infinite-dimensional equations in Banach spaces to finite-dimensional equations. It is named after Aleksandr Lyapunov and Erhard Schmidt.
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https://en.wikipedia.org/wiki/Lyapunov-Schmidt_reduction
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In mathematics, the Lyusternik–Fet theorem states that on every compact Riemannian manifold there exists a closed geodesic. It is named after Lazar Lyusternik and Abram Ilyich Fet.
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https://en.wikipedia.org/wiki/Lyusternik–Fet_theorem
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In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space X {\displaystyle X} is the homotopy invariant defined to be the smallest integer number k {\displaystyle k} such that there is an open covering { U i } 1 ≤ i ≤ k {\displaystyle \{U_{i}\}_{1\leq i\leq k}} of X {\displaystyle X} with the property that each inclusion map U i ↪ X {\displaystyle U_{i}\hookrightarrow X} is nullhomotopic. For example, if X {\displaystyle X} is a sphere, this takes the value two. Sometimes a different normalization of the invariant is adopted, which is one less than the definition above.
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https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category
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Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below). In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length.
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https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category
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In the modern normalization, the cup-length is a lower bound for the LS-category. It was, as originally defined for the case of X {\displaystyle X} a manifold, the lower bound for the number of critical points that a real-valued function on X {\displaystyle X} could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function). The invariant has been generalized in several different directions (group actions, foliations, simplicial complexes, etc.).
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https://en.wikipedia.org/wiki/Lusternik–Schnirelmann_category
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In mathematics, the Lévy C curve is a self-similar fractal curve that was first described and whose differentiability properties were analysed by Ernesto Cesàro in 1906 and Georg Faber in 1910, but now bears the name of French mathematician Paul Lévy, who was the first to describe its self-similarity properties as well as to provide a geometrical construction showing it as a representative curve in the same class as the Koch curve. It is a special case of a period-doubling curve, a de Rham curve.
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https://en.wikipedia.org/wiki/Lévy_C_curve
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In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.
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https://en.wikipedia.org/wiki/Lévy_metric
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In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e., a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Lévy and the Soviet mathematician Yuri Vasilyevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.
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https://en.wikipedia.org/wiki/Lévy–Prokhorov_metric
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In mathematics, the Lévy–Steinitz theorem identifies the set of values to which rearrangements of an infinite series of vectors in Rn can converge. It was proved by Paul Lévy in his first published paper when he was 19 years old. In 1913 Ernst Steinitz filled in a gap in Lévy's proof and also proved the result by a different method.In an expository article, Peter Rosenthal stated the theorem in the following way. The set of all sums of rearrangements of a given series of vectors in a finite-dimensional real Euclidean space is either the empty set or a translate of a subspace (i.e., a set of the form v + M, where v is a given vector and M is a linear subspace).
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https://en.wikipedia.org/wiki/Lévy–Steinitz_theorem
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In mathematics, the Maass–Selberg relations are some relations describing the inner products of truncated real analytic Eisenstein series, that in some sense say that distinct Eisenstein series are orthogonal. Hans Maass introduced the Maass–Selberg relations for the case of real analytic Eisenstein series on the upper half plane. Atle Selberg extended the relations to symmetric spaces of rank 1.
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https://en.wikipedia.org/wiki/Maass–Selberg_relations
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Harish-Chandra generalized the Maass–Selberg relations to Eisenstein series of higher rank semisimple group (and named the relations after Maass and Selberg) and found some analogous relations between Eisenstein integrals, that he also called Maass–Selberg relations. Informally, the Maass–Selberg relations say that the inner product of two distinct Eisenstein series is zero. However the integral defining the inner product does not converge, so the Eisenstein series first have to be truncated. The Maass–Selberg relations then say that the inner product of two truncated Eisenstein series is given by a finite sum of elementary factors that depend on the truncation chosen, whose finite part tends to zero as the truncation is removed.
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https://en.wikipedia.org/wiki/Maass–Selberg_relations
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In mathematics, the Macdonald identities are some infinite product identities associated to affine root systems, introduced by Ian Macdonald (1972). They include as special cases the Jacobi triple product identity, Watson's quintuple product identity, several identities found by Dyson (1972), and a 10-fold product identity found by Winquist (1969). Kac (1974) and Moody (1975) pointed out that the Macdonald identities are the analogs of the Weyl denominator formula for affine Kac–Moody algebras and superalgebras.
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https://en.wikipedia.org/wiki/Macdonald_identities
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In mathematics, the Mahler measure M ( p ) {\displaystyle M(p)} of a polynomial p ( z ) {\displaystyle p(z)} with complex coefficients is defined as where p ( z ) {\displaystyle p(z)} factorizes over the complex numbers C {\displaystyle \mathbb {C} } as The Mahler measure can be viewed as a kind of height function. Using Jensen's formula, it can be proved that this measure is also equal to the geometric mean of | p ( z ) | {\displaystyle |p(z)|} for z {\displaystyle z} on the unit circle (i.e., | z | = 1 {\displaystyle |z|=1} ): By extension, the Mahler measure of an algebraic number α {\displaystyle \alpha } is defined as the Mahler measure of the minimal polynomial of α {\displaystyle \alpha } over Q {\displaystyle \mathbb {Q} } . In particular, if α {\displaystyle \alpha } is a Pisot number or a Salem number, then its Mahler measure is simply α {\displaystyle \alpha } . The Mahler measure is named after the German-born Australian mathematician Kurt Mahler.
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https://en.wikipedia.org/wiki/Mahler_measure
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In mathematics, the Mahler polynomials gn(x) are polynomials introduced by Mahler (1930) in his work on the zeros of the incomplete gamma function. Mahler polynomials are given by the generating function ∑ g n ( x ) t n / n ! = exp ( x ( 1 + t − e t ) ) {\displaystyle \displaystyle \sum g_{n}(x)t^{n}/n!=\exp(x(1+t-e^{t}))} Which is close to the generating function of the Touchard polynomials. The first few examples are (sequence A008299 in the OEIS) g 0 = 1 ; {\displaystyle g_{0}=1;} g 1 = 0 ; {\displaystyle g_{1}=0;} g 2 = − x ; {\displaystyle g_{2}=-x;} g 3 = − x ; {\displaystyle g_{3}=-x;} g 4 = − x + 3 x 2 ; {\displaystyle g_{4}=-x+3x^{2};} g 5 = − x + 10 x 2 ; {\displaystyle g_{5}=-x+10x^{2};} g 6 = − x + 25 x 2 − 15 x 3 ; {\displaystyle g_{6}=-x+25x^{2}-15x^{3};} g 7 = − x + 56 x 2 − 105 x 3 ; {\displaystyle g_{7}=-x+56x^{2}-105x^{3};} g 8 = − x + 119 x 2 − 490 x 3 + 105 x 4 ; {\displaystyle g_{8}=-x+119x^{2}-490x^{3}+105x^{4};}
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https://en.wikipedia.org/wiki/Mahler_polynomial
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In mathematics, the Malgrange preparation theorem is an analogue of the Weierstrass preparation theorem for smooth functions. It was conjectured by René Thom and proved by B. Malgrange (1962–1963, 1964, 1967).
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https://en.wikipedia.org/wiki/Malgrange_preparation_theorem
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In mathematics, the Malgrange–Ehrenpreis theorem states that every non-zero linear differential operator with constant coefficients has a Green's function. It was first proved independently by Leon Ehrenpreis (1954, 1955) and Bernard Malgrange (1955–1956). This means that the differential equation P ( ∂ ∂ x 1 , … , ∂ ∂ x ℓ ) u ( x ) = δ ( x ) , {\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=\delta (\mathbf {x} ),} where P is a polynomial in several variables and δ is the Dirac delta function, has a distributional solution u. It can be used to show that P ( ∂ ∂ x 1 , … , ∂ ∂ x ℓ ) u ( x ) = f ( x ) {\displaystyle P\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{\ell }}}\right)u(\mathbf {x} )=f(\mathbf {x} )} has a solution for any compactly supported distribution f. The solution is not unique in general. The analogue for differential operators whose coefficients are polynomials (rather than constants) is false: see Lewy's example.
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https://en.wikipedia.org/wiki/Malgrange–Ehrenpreis_theorem
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In mathematics, the Malliavin derivative is a notion of derivative in the Malliavin calculus. Intuitively, it is the notion of derivative appropriate to paths in classical Wiener space, which are "usually" not differentiable in the usual sense.
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https://en.wikipedia.org/wiki/Malliavin_derivative
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In mathematics, the Manin conjecture describes the conjectural distribution of rational points on an algebraic variety relative to a suitable height function. It was proposed by Yuri I. Manin and his collaborators in 1989 when they initiated a program with the aim of describing the distribution of rational points on suitable algebraic varieties.
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https://en.wikipedia.org/wiki/Manin_conjecture
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In mathematics, the Manin–Drinfeld theorem, proved by Manin (1972) and Drinfeld (1973), states that the difference of two cusps of a modular curve has finite order in the Jacobian variety.
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https://en.wikipedia.org/wiki/Manin–Drinfeld_theorem
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In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of non-linear operators acting on Lp spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators.
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https://en.wikipedia.org/wiki/Marcinkiewicz_interpolation_theorem
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In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order. It is a special case of the Burkholder-Davis-Gundy inequality in the case of discrete-time martingales.
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https://en.wikipedia.org/wiki/Marcinkiewicz–Zygmund_inequality
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In mathematics, the Markov brothers' inequality is an inequality proved in the 1890s by brothers Andrey Markov and Vladimir Markov, two Russian mathematicians. This inequality bounds the maximum of the derivatives of a polynomial on an interval in terms of the maximum of the polynomial. For k = 1 it was proved by Andrey Markov, and for k = 2,3,... by his brother Vladimir Markov.
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https://en.wikipedia.org/wiki/Markov_brothers'_inequality
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In mathematics, the Markov spectrum devised by Andrey Markov is a complicated set of real numbers arising in Markov Diophantine equation and also in the theory of Diophantine approximation.
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https://en.wikipedia.org/wiki/Markov_spectrum
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In mathematics, the Markov–Kakutani fixed-point theorem, named after Andrey Markov and Shizuo Kakutani, states that a commuting family of continuous affine self-mappings of a compact convex subset in a locally convex topological vector space has a common fixed point. This theorem is a key tool in one of the quickest proofs of amenability of abelian groups.
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https://en.wikipedia.org/wiki/Markov-Kakutani_fixed-point_theorem
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In mathematics, the Markus–Yamabe conjecture is a conjecture on global asymptotic stability. If the Jacobian matrix of a dynamical system at a fixed point is Hurwitz, then the fixed point is asymptotically stable. Markus-Yamabe conjecture asks if a similar result holds globally. Precisely, the conjecture states that if a continuously differentiable map on an n {\displaystyle n} -dimensional real vector space has a fixed point, and its Jacobian matrix is everywhere Hurwitz, then the fixed point is globally stable.
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https://en.wikipedia.org/wiki/Markus–Yamabe_conjecture
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The conjecture is true for the two-dimensional case. However, counterexamples have been constructed in higher dimensions. Hence, in the two-dimensional case only, it can also be referred to as the Markus–Yamabe theorem. Related mathematical results concerning global asymptotic stability, which are applicable in dimensions higher than two, include various autonomous convergence theorems. Analog of the conjecture for nonlinear control system with scalar nonlinearity is known as Kalman's conjecture.
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https://en.wikipedia.org/wiki/Markus–Yamabe_conjecture
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In mathematics, the Mathai–Quillen formalism is an approach to topological quantum field theory introduced by Atiyah and Jeffrey (1990), based on the Mathai–Quillen form constructed in Mathai and Quillen (1986). In more detail, using the superconnection formalism of Quillen, they obtained a refinement of the Riemann–Roch formula, which links together the Thom classes in K-theory and cohomology, as an equality on the level of differential forms. This has an interpretation in physics as the computation of the classical and quantum (super) partition functions for the fermionic analogue of a harmonic oscillator with source term. In particular, they obtained a nice Gaussian shape representative of the Thom class in cohomology, which has a peak along the zero section.
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https://en.wikipedia.org/wiki/Mathai–Quillen_formalism
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In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer. As a one-form, the Maurer–Cartan form is peculiar in that it takes its values in the Lie algebra associated to the Lie group G. The Lie algebra is identified with the tangent space of G at the identity, denoted TeG. The Maurer–Cartan form ω is thus a one-form defined globally on G which is a linear mapping of the tangent space TgG at each g ∈ G into TeG. It is given as the pushforward of a vector in TgG along the left-translation in the group: ω ( v ) = ( L g − 1 ) ∗ v , v ∈ T g G . {\displaystyle \omega (v)=(L_{g^{-1}})_{*}v,\quad v\in T_{g}G.}
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https://en.wikipedia.org/wiki/Maurer–Cartan_form
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In mathematics, the May spectral sequence is a spectral sequence, introduced by J. Peter May (1965, 1966). It is used for calculating the initial term of the Adams spectral sequence, which is in turn used for calculating the stable homotopy groups of spheres. The May spectral sequence is described in detail in (Ravenel 2003, pp. 67–74).
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https://en.wikipedia.org/wiki/May_spectral_sequence
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In mathematics, the Mazur–Ulam theorem states that if V {\displaystyle V} and W {\displaystyle W} are normed spaces over R and the mapping f: V → W {\displaystyle f\colon V\to W} is a surjective isometry, then f {\displaystyle f} is affine. It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach. For strictly convex spaces the result is true, and easy, even for isometries which are not necessarily surjective.
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https://en.wikipedia.org/wiki/Mazur–Ulam_theorem
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In this case, for any u {\displaystyle u} and v {\displaystyle v} in V {\displaystyle V} , and for any t {\displaystyle t} in {\displaystyle } , write and denote the closed ball of radius R around v by B ¯ ( v , R ) {\displaystyle {\bar {B}}(v,R)} . Then t u + ( 1 − t ) v {\displaystyle tu+(1-t)v} is the unique element of B ¯ ( v , t r ) ∩ B ¯ ( u , ( 1 − t ) r ) {\displaystyle {\bar {B}}(v,tr)\cap {\bar {B}}(u,(1-t)r)} , so, since f {\displaystyle f} is injective, f ( t u + ( 1 − t ) v ) {\displaystyle f(tu+(1-t)v)} is the unique element of and therefore is equal to t f ( u ) + ( 1 − t ) f ( v ) {\displaystyle tf(u)+(1-t)f(v)} . Therefore f {\displaystyle f} is an affine map. This argument fails in the general case, because in a normed space which is not strictly convex two tangent balls may meet in some flat convex region of their boundary, not just a single point.
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https://en.wikipedia.org/wiki/Mazur–Ulam_theorem
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In mathematics, the McKay graph of a finite-dimensional representation V of a finite group G is a weighted quiver encoding the structure of the representation theory of G. Each node represents an irreducible representation of G. If χ i, χ j are irreducible representations of G, then there is an arrow from χ i to χ j if and only if χ j is a constituent of the tensor product V ⊗ χ i . {\displaystyle V\otimes \chi _{i}.} Then the weight nij of the arrow is the number of times this constituent appears in V ⊗ χ i . {\displaystyle V\otimes \chi _{i}.} For finite subgroups H of GL ( 2 , C ) , {\displaystyle {\text{GL}}(2,\mathbb {C} ),} the McKay graph of H is the McKay graph of the canonical representation of H. If G has n irreducible characters, then the Cartan matrix cV of the representation V of dimension d is defined by c V = ( d δ i j − n i j ) i j , {\displaystyle c_{V}=(d\delta _{ij}-n_{ij})_{ij},} where δ is the Kronecker delta. A result by Steinberg states that if g is a representative of a conjugacy class of G, then the vectors ( ( χ i ( g ) ) i {\displaystyle ((\chi _{i}(g))_{i}} are the eigenvectors of cV to the eigenvalues d − χ V ( g ) , {\displaystyle d-\chi _{V}(g),} where χV is the character of the representation V. The McKay correspondence, named after John McKay, states that there is a one-to-one correspondence between the McKay graphs of the finite subgroups of SL ( 2 , C ) {\displaystyle {\text{SL}}(2,\mathbb {C} )} and the extended Dynkin diagrams, which appear in the ADE classification of the simple Lie algebras.
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https://en.wikipedia.org/wiki/McKay_graph
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In mathematics, the Mehler–Fock transform is an integral transform introduced by Mehler (1881) and rediscovered by Fock (1943). It is given by F ( x ) = ∫ 0 ∞ P i t − 1 / 2 ( x ) f ( t ) d t , ( 1 ≤ x ≤ ∞ ) , {\displaystyle F(x)=\int _{0}^{\infty }P_{it-1/2}(x)f(t)dt,\quad (1\leq x\leq \infty ),} where P is a Legendre function of the first kind. Under appropriate conditions, the following inversion formula holds: f ( t ) = t tanh ( π t ) ∫ 1 ∞ P i t − 1 / 2 ( x ) F ( x ) d x , ( 0 ≤ t ≤ ∞ ) . {\displaystyle f(t)=t\tanh(\pi t)\int _{1}^{\infty }P_{it-1/2}(x)F(x)dx,\quad (0\leq t\leq \infty ).}
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https://en.wikipedia.org/wiki/Mehler–Fock_transform
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In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(λ)n(x,φ) introduced by Meixner (1934), which up to elementary changes of variables are the same as the Pollaczek polynomials Pλn(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him. They are defined by P n ( λ ) ( x ; ϕ ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n , λ + i x 2 λ ; 1 − e − 2 i ϕ ) {\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n! }}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +ix\\2\lambda \end{array}};1-e^{-2i\phi }\right)} P n λ ( cos ϕ ; a , b ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n , λ + i ( a cos ϕ + b ) / sin ϕ 2 λ ; 1 − e − 2 i ϕ ) {\displaystyle P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n! }}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +i(a\cos \phi +b)/\sin \phi \\2\lambda \end{array}};1-e^{-2i\phi }\right)}
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https://en.wikipedia.org/wiki/Pollaczek_polynomials
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In mathematics, the Mellin inversion formula (named after Hjalmar Mellin) tells us conditions under which the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function.
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https://en.wikipedia.org/wiki/Mellin_inversion_theorem
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In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions. The Mellin transform of a function f is { M f } ( s ) = φ ( s ) = ∫ 0 ∞ x s − 1 f ( x ) d x . {\displaystyle \left\{{\mathcal {M}}f\right\}(s)=\varphi (s)=\int _{0}^{\infty }x^{s-1}f(x)\,dx.}
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https://en.wikipedia.org/wiki/Mellin_transform
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The inverse transform is { M − 1 φ } ( x ) = f ( x ) = 1 2 π i ∫ c − i ∞ c + i ∞ x − s φ ( s ) d s . {\displaystyle \left\{{\mathcal {M}}^{-1}\varphi \right\}(x)=f(x)={\frac {1}{2\pi i}}\int _{c-i\infty }^{c+i\infty }x^{-s}\varphi (s)\,ds.}
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https://en.wikipedia.org/wiki/Mellin_transform
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The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem. The transform is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.
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https://en.wikipedia.org/wiki/Mellin_transform
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In mathematics, the Melnikov method is a tool to identify the existence of chaos in a class of dynamical systems under periodic perturbation.
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https://en.wikipedia.org/wiki/Melnikov_distance
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In mathematics, the Menger curvature of a triple of points in n-dimensional Euclidean space Rn is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger.
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https://en.wikipedia.org/wiki/Menger_curvature
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In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.
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https://en.wikipedia.org/wiki/Sierpinski_sponge
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In mathematics, the Mercator series or Newton–Mercator series is the Taylor series for the natural logarithm: ln ( 1 + x ) = x − x 2 2 + x 3 3 − x 4 4 + ⋯ {\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-{\frac {x^{4}}{4}}+\cdots } In summation notation, ln ( 1 + x ) = ∑ n = 1 ∞ ( − 1 ) n + 1 n x n . {\displaystyle \ln(1+x)=\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}x^{n}.} The series converges to the natural logarithm (shifted by 1) whenever − 1 < x ≤ 1 {\displaystyle -1
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https://en.wikipedia.org/wiki/Mercator_series
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In mathematics, the Mersenne conjectures concern the characterization of a kind of prime numbers called Mersenne primes, meaning prime numbers that are a power of two minus one.
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https://en.wikipedia.org/wiki/Mersenne_conjectures
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In mathematics, the Mertens conjecture is the statement that the Mertens function M ( n ) {\displaystyle M(n)} is bounded by ± n {\displaystyle \pm {\sqrt {n}}} . Although now disproven, it had been shown to imply the Riemann hypothesis. It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Franz Mertens (1897), and disproved by Andrew Odlyzko and Herman te Riele (1985). It is a striking example of a mathematical conjecture proven false despite a large amount of computational evidence in its favor.
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https://en.wikipedia.org/wiki/Mertens_conjecture
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In mathematics, the Mestre bound is a bound on the analytic rank of an elliptic curve in terms of its conductor, introduced by Mestre (1986).
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https://en.wikipedia.org/wiki/Mestre_bound
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In mathematics, the Meyer signature cocycle, introduced by Meyer (1973). is an integer-valued 2-cocyle on a symplectic group that describes the signature of a fiber bundle whose base and fiber are both Riemann surfaces.
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https://en.wikipedia.org/wiki/Signature_cocycle
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In mathematics, the Mian–Chowla sequence is an integer sequence defined recursively in the following way. The sequence starts with a 1 = 1. {\displaystyle a_{1}=1.} Then for n > 1 {\displaystyle n>1} , a n {\displaystyle a_{n}} is the smallest integer such that every pairwise sum a i + a j {\displaystyle a_{i}+a_{j}} is distinct, for all i {\displaystyle i} and j {\displaystyle j} less than or equal to n {\displaystyle n} .
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https://en.wikipedia.org/wiki/Mian–Chowla_sequence
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In mathematics, the Milliken–Taylor theorem in combinatorics is a generalization of both Ramsey's theorem and Hindman's theorem. It is named after Keith Milliken and Alan D. Taylor. Let P f ( N ) {\displaystyle {\mathcal {P}}_{f}(\mathbb {N} )} denote the set of finite subsets of N {\displaystyle \mathbb {N} } , and define a partial order on P f ( N ) {\displaystyle {\mathcal {P}}_{f}(\mathbb {N} )} by α<β if and only if max α 0, let < k = { { ∑ t ∈ α 1 a t , … , ∑ t ∈ α k a t }: α 1 , ⋯ , α k ∈ P f ( N ) and α 1 < ⋯ < α k } . {\displaystyle _{<}^{k}=\left\{\left\{\sum _{t\in \alpha _{1}}a_{t},\ldots ,\sum _{t\in \alpha _{k}}a_{t}\right\}:\alpha _{1},\cdots ,\alpha _{k}\in {\mathcal {P}}_{f}(\mathbb {N} ){\text{ and }}\alpha _{1}<\cdots <\alpha _{k}\right\}.}
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https://en.wikipedia.org/wiki/Milliken–Taylor_theorem
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Let k {\displaystyle ^{k}} denote the k-element subsets of a set S. The Milliken–Taylor theorem says that for any finite partition k = C 1 ∪ C 2 ∪ ⋯ ∪ C r {\displaystyle ^{k}=C_{1}\cup C_{2}\cup \cdots \cup C_{r}} , there exist some i ≤ r and a sequence ⟨ a n ⟩ n = 0 ∞ ⊂ N {\displaystyle \langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N} } such that < k ⊂ C i {\displaystyle _{<}^{k}\subset C_{i}} . For each ⟨ a n ⟩ n = 0 ∞ ⊂ N {\displaystyle \langle a_{n}\rangle _{n=0}^{\infty }\subset \mathbb {N} } , call < k {\displaystyle _{<}^{k}} an MTk set. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MTk sets is partition regular for each k.
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https://en.wikipedia.org/wiki/Milliken–Taylor_theorem
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In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive. The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959. Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space.
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https://en.wikipedia.org/wiki/Milman–Pettis_theorem
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In mathematics, the Milne-Thomson method is a method for finding a holomorphic function whose real or imaginary part is given. It is named after Louis Melville Milne-Thomson.
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https://en.wikipedia.org/wiki/Milne-Thomson_method_for_finding_a_holomorphic_function
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In mathematics, the Milnor conjecture was a proposal by John Milnor (1970) of a description of the Milnor K-theory (mod 2) of a general field F with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of F with coefficients in Z/2Z. It was proved by Vladimir Voevodsky (1996, 2003a, 2003b).
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https://en.wikipedia.org/wiki/Milnor_conjecture
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In mathematics, the Milstein method is a technique for the approximate numerical solution of a stochastic differential equation. It is named after Grigori N. Milstein who first published it in 1974.
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https://en.wikipedia.org/wiki/Milstein_method
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In mathematics, the Minkowski–Hlawka theorem is a result on the lattice packing of hyperspheres in dimension n > 1. It states that there is a lattice in Euclidean space of dimension n, such that the corresponding best packing of hyperspheres with centres at the lattice points has density Δ satisfying Δ ≥ ζ ( n ) 2 n − 1 , {\displaystyle \Delta \geq {\frac {\zeta (n)}{2^{n-1}}},} with ζ the Riemann zeta function. Here as n → ∞, ζ(n) → 1.
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https://en.wikipedia.org/wiki/Minkowski–Hlawka_theorem
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The proof of this theorem is indirect and does not give an explicit example, however, and there is still no known simple and explicit way to construct lattices with packing densities exceeding this bound for arbitrary n. In principle one can find explicit examples: for example, even just picking a few "random" lattices will work with high probability. The problem is that testing these lattices to see if they are solutions requires finding their shortest vectors, and the number of cases to check grows very fast with the dimension, so this could take a very long time. This result was stated without proof by Hermann Minkowski (1911, pages 265–276) and proved by Edmund Hlawka (1943). The result is related to a linear lower bound for the Hermite constant.
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https://en.wikipedia.org/wiki/Minkowski–Hlawka_theorem
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In mathematics, the Minkowski–Steiner formula is a formula relating the surface area and volume of compact subsets of Euclidean space. More precisely, it defines the surface area as the "derivative" of enclosed volume in an appropriate sense. The Minkowski–Steiner formula is used, together with the Brunn–Minkowski theorem, to prove the isoperimetric inequality. It is named after Hermann Minkowski and Jakob Steiner.
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https://en.wikipedia.org/wiki/Minkowski–Steiner_formula
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In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for studying the Mathieu groups, binary Golay code and Leech lattice.
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https://en.wikipedia.org/wiki/Miracle_Octad_Generator
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In mathematics, the Mittag-Leffler function E α , β {\displaystyle E_{\alpha ,\beta }} is a special function, a complex function which depends on two complex parameters α {\displaystyle \alpha } and β {\displaystyle \beta } . It may be defined by the following series when the real part of α {\displaystyle \alpha } is strictly positive: E α , β ( z ) = ∑ k = 0 ∞ z k Γ ( α k + β ) , {\displaystyle E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (\alpha k+\beta )}},} where Γ ( x ) {\displaystyle \Gamma (x)} is the gamma function. When β = 1 {\displaystyle \beta =1} , it is abbreviated as E α ( z ) = E α , 1 ( z ) {\displaystyle E_{\alpha }(z)=E_{\alpha ,1}(z)} . For α = 0 {\displaystyle \alpha =0} , the series above equals the Taylor expansion of the geometric series and consequently E 0 , β ( z ) = 1 Γ ( β ) 1 1 − z {\displaystyle E_{0,\beta }(z)={\frac {1}{\Gamma (\beta )}}{\frac {1}{1-z}}} .
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https://en.wikipedia.org/wiki/Mittag-Leffler_function
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In mathematics, the Mittag-Leffler polynomials are the polynomials gn(x) or Mn(x) studied by Mittag-Leffler (1891). Mn(x) is a special case of the Meixner polynomial Mn(x;b,c) at b = 0, c = -1.
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https://en.wikipedia.org/wiki/Mittag-Leffler_polynomials
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