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In mathematics, a Costas array can be regarded geometrically as a set of n points, each at the center of a square in an n×n square tiling such that each row or column contains only one point, and all of the n(n − 1)/2 displacement vectors between each pair of dots are distinct. This results in an ideal "thumbtack" auto-ambiguity function, making the arrays useful in applications such as sonar and radar. Costas arrays can be regarded as two-dimensional cousins of the one-dimensional Golomb ruler construction, and, as well as being of mathematical interest, have similar applications in experimental design and phased array radar engineering.
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https://en.wikipedia.org/wiki/Costas_array
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Costas arrays are named after John P. Costas, who first wrote about them in a 1965 technical report. Independently, Edgar Gilbert also wrote about them in the same year, publishing what is now known as the logarithmic Welch method of constructing Costas arrays. The general enumeration of Costas arrays is an open problem in computer science and finding an algorithm that can solve it in polynomial time is an open research question.
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https://en.wikipedia.org/wiki/Costas_array
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In mathematics, a Coulomb wave function is a solution of the Coulomb wave equation, named after Charles-Augustin de Coulomb. They are used to describe the behavior of charged particles in a Coulomb potential and can be written in terms of confluent hypergeometric functions or Whittaker functions of imaginary argument.
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https://en.wikipedia.org/wiki/Coulomb_functions
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In mathematics, a Countryman line (named after Roger Simmons Countryman Jr.) is an uncountable linear ordering whose square is the union of countably many chains. The existence of Countryman lines was first proven by Shelah. Shelah also conjectured that, assuming PFA, every Aronszajn line contains a Countryman line. This conjecture, which remained open for three decades, was proven by Justin Moore.
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https://en.wikipedia.org/wiki/Countryman_line
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups (Coxeter 1934), and finite Coxeter groups were classified in 1935 (Coxeter 1935).
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https://en.wikipedia.org/wiki/Coxeter_system
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Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras. Standard references include (Bourbaki 2002) (Humphreys 1992) and (Davis 2007).
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https://en.wikipedia.org/wiki/Coxeter_system
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In mathematics, a Cullen number is a member of the integer sequence C n = n ⋅ 2 n + 1 {\displaystyle C_{n}=n\cdot 2^{n}+1} (where n {\displaystyle n} is a natural number). Cullen numbers were first studied by James Cullen in 1905. The numbers are special cases of Proth numbers.
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https://en.wikipedia.org/wiki/Cullen_number
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In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes.
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https://en.wikipedia.org/wiki/Cunningham_chain
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In mathematics, a D-module is a module over a ring D of differential operators. The major interest of such D-modules is as an approach to the theory of linear partial differential equations. Since around 1970, D-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara.
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https://en.wikipedia.org/wiki/D-module
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The methods of D-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. The approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems (holonomic systems), and on the characteristic variety cut out by the symbols, which in the good case is a Lagrangian submanifold of the cotangent bundle of maximal dimension (involutive systems). The techniques were taken up from the side of the Grothendieck school by Zoghman Mebkhout, who obtained a general, derived category version of the Riemann–Hilbert correspondence in all dimensions.
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https://en.wikipedia.org/wiki/D-module
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In mathematics, a De Morgan algebra (named after Augustus De Morgan, a British mathematician and logician) is a structure A = (A, ∨, ∧, 0, 1, ¬) such that: (A, ∨, ∧, 0, 1) is a bounded distributive lattice, and ¬ is a De Morgan involution: ¬(x ∧ y) = ¬x ∨ ¬y and ¬¬x = x. (i.e. an involution that additionally satisfies De Morgan's laws)In a De Morgan algebra, the laws ¬x ∨ x = 1 (law of the excluded middle), and ¬x ∧ x = 0 (law of noncontradiction)do not always hold. In the presence of the De Morgan laws, either law implies the other, and an algebra which satisfies them becomes a Boolean algebra. Remark: It follows that ¬(x ∨ y) = ¬x ∧ ¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1 ∨ 0 = ¬1 ∨ ¬¬0 = ¬(1 ∧ ¬0) = ¬¬0 = 0).
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https://en.wikipedia.org/wiki/Kleene_algebra_(with_involution)
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Thus ¬ is a dual automorphism of (A, ∨, ∧, 0, 1). If the lattice is defined in terms of the order instead, i.e. (A, ≤) is a bounded partial order with a least upper bound and greatest lower bound for every pair of elements, and the meet and join operations so defined satisfy the distributive law, then the complementation can also be defined as an involutive anti-automorphism, that is, a structure A = (A, ≤, ¬) such that: (A, ≤) is a bounded distributive lattice, and ¬¬x = x, and x ≤ y → ¬y ≤ ¬x.De Morgan algebras were introduced by Grigore Moisil around 1935, although without the restriction of having a 0 and a 1. They were then variously called quasi-boolean algebras in the Polish school, e.g. by Rasiowa and also distributive i-lattices by J. A. Kalman.
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https://en.wikipedia.org/wiki/Kleene_algebra_(with_involution)
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(i-lattice being an abbreviation for lattice with involution.) They have been further studied in the Argentinian algebraic logic school of Antonio Monteiro.De Morgan algebras are important for the study of the mathematical aspects of fuzzy logic. The standard fuzzy algebra F = (, max(x, y), min(x, y), 0, 1, 1 − x) is an example of a De Morgan algebra where the laws of excluded middle and noncontradiction do not hold. Another example is Dunn's four-valued semantics for De Morgan algebra, which has the values T(rue), F(alse), B(oth), and N(either), where F < B < T, F < N < T, and B and N are not comparable.
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https://en.wikipedia.org/wiki/Kleene_algebra_(with_involution)
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In mathematics, a Delannoy number D {\displaystyle D} describes the number of paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (m, n), using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy.The Delannoy number D ( m , n ) {\displaystyle D(m,n)} also counts the number of global alignments of two sequences of lengths m {\displaystyle m} and n {\displaystyle n} , the number of points in an m-dimensional integer lattice or cross polytope which are at most n steps from the origin, and, in cellular automata, the number of cells in an m-dimensional von Neumann neighborhood of radius n while the number of cells on a surface of an m-dimensional von Neumann neighborhood of radius n is given with (sequence A266213 in the OEIS).
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https://en.wikipedia.org/wiki/Delannoy_number
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In mathematics, a Delzant polytope is a convex polytope in R n {\displaystyle \mathbb {R} ^{n}} such for each vertex v {\displaystyle v} , exactly n {\displaystyle n} edges meet at v {\displaystyle v} , and these edges form a collection of vectors that form a Z {\displaystyle \mathbb {Z} } -basis of Z n {\displaystyle \mathbb {Z} ^{n}} . Delzant's theorem, introduced by Thomas Delzant (1988), classifies effective Hamiltonian torus actions on compact connected symplectic manifolds by the image of the associated moment map, which is a Delzant polytope. The theorem states that there is a bijective correspondence between symplectic toric manifolds (up to torus-equivariant symplectomorphism) and Delzant polytopes -- more precisely, the moment polytope of a symplectic toric manifold is a Delzant polytope, every Delzant polytope is the moment polytope of such a manifold, and any two such manifolds with the equivalent moment polytopes (up to translations) admit a torus-equivariant symplectomorphism between them.
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https://en.wikipedia.org/wiki/Delzant's_theorem
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In mathematics, a Demazure module, introduced by Demazure (1974a, 1974b), is a submodule of a finite-dimensional representation generated by an extremal weight space under the action of a Borel subalgebra. The Demazure character formula, introduced by Demazure (1974b, theorem 2), gives the characters of Demazure modules, and is a generalization of the Weyl character formula. The dimension of a Demazure module is a polynomial in the highest weight, called a Demazure polynomial.
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https://en.wikipedia.org/wiki/Demazure_module
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In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns. A Diophantine set is a subset S of N j {\displaystyle \mathbb {N} ^{j}} , the set of all j-tuples of natural numbers, so that for some Diophantine equation P(x, y) = 0, x ¯ ∈ S ⟺ ( ∃ y ¯ ∈ N k ) ( P ( x ¯ , y ¯ ) = 0 ) . {\displaystyle {\bar {x}}\in S\iff (\exists {\bar {y}}\in \mathbb {N} ^{k})(P({\bar {x}},{\bar {y}})=0).} That is, a parameter value is in the Diophantine set S if and only if the associated Diophantine equation is satisfiable under that parameter value.
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https://en.wikipedia.org/wiki/Matiyasevich's_theorem
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The use of natural numbers both in S and the existential quantification merely reflects the usual applications in computability and model theory. It does not matter whether natural numbers refer to the set of nonnegative integers or positive integers since the two definitions for Diophantine set are equivalent. We can also equally well speak of Diophantine sets of integers and freely replace quantification over natural numbers with quantification over the integers.
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https://en.wikipedia.org/wiki/Matiyasevich's_theorem
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Also it is sufficient to assume P is a polynomial over Q {\displaystyle \mathbb {Q} } and multiply P by the appropriate denominators to yield integer coefficients. However, whether quantification over rationals can also be substituted for quantification over the integers is a notoriously hard open problem.The MRDP theorem (so named for the initials of the four principal contributors to its solution) states that a set of integers is Diophantine if and only if it is computably enumerable. A set of integers S is computably enumerable if and only if there is an algorithm that, when given an integer, halts if that integer is a member of S and runs forever otherwise.
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https://en.wikipedia.org/wiki/Matiyasevich's_theorem
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This means that the concept of general Diophantine set, apparently belonging to number theory, can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work. Matiyasevich's completion of the MRDP theorem settled Hilbert's tenth problem. Hilbert's tenth problem was to find a general algorithm which can decide whether a given Diophantine equation has a solution among the integers. While Hilbert's tenth problem is not a formal mathematical statement as such, the nearly universal acceptance of the (philosophical) identification of a decision algorithm with a total computable predicate allows us to use the MRDP theorem to conclude that the tenth problem is unsolvable.
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https://en.wikipedia.org/wiki/Matiyasevich's_theorem
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In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
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https://en.wikipedia.org/wiki/Diophantine_equation
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Diophantine problems have fewer equations than unknowns and involve finding integers that solve simultaneously all equations. As such systems of equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine geometry. The word Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra. The mathematical study of Diophantine problems that Diophantus initiated is now called Diophantine analysis. While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations (beyond the case of linear and quadratic equations) was an achievement of the twentieth century.
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https://en.wikipedia.org/wiki/Diophantine_equation
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In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function with the formula for some given period T {\displaystyle T} . Here t is a real variable and the sum extends over all integers k. The Dirac delta function δ {\displaystyle \delta } and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the δ {\displaystyle \delta } s as the comb's teeth), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol Ш ( t ) {\displaystyle \operatorname {\text{Ш}} \,\,(t)} , where the period is omitted, represents a Dirac comb of unit period.
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https://en.wikipedia.org/wiki/Dirac_comb
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This implies Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling and aliasing, in a single framework of continuous Fourier analysis on tempered distributions, without any reference to Fourier series. The Fourier transform of a Dirac comb is another Dirac comb. Owing to the Convolution Theorem on tempered distributions which turns out to be the Poisson summation formula, in signal processing, the Dirac comb allows modelling sampling by multiplication with it, but it also allows modelling periodization by convolution with it.
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https://en.wikipedia.org/wiki/Dirac_comb
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In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields.
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https://en.wikipedia.org/wiki/Dirac_measure
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In mathematics, a Dirac spectrum, named after Paul Dirac, is the spectrum of eigenvalues of a Dirac operator on a Riemannian manifold with a spin structure. The isospectral problem for the Dirac spectrum asks whether two Riemannian spin manifolds have identical spectra. The Dirac spectrum depends on the spin structure in the sense that there exists a Riemannian manifold with two different spin structures that have different Dirac spectra.
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https://en.wikipedia.org/wiki/Dirac_spectrum
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In mathematics, a Dirichlet L-series is a function of the form L ( s , χ ) = ∑ n = 1 ∞ χ ( n ) n s . {\displaystyle L(s,\chi )=\sum _{n=1}^{\infty }{\frac {\chi (n)}{n^{s}}}.} where χ {\displaystyle \chi } is a Dirichlet character and s a complex variable with real part greater than 1.
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https://en.wikipedia.org/wiki/Dirichlet_L-function
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It is a special case of a Dirichlet series. By analytic continuation, it can be extended to a meromorphic function on the whole complex plane, and is then called a Dirichlet L-function and also denoted L(s, χ). These functions are named after Peter Gustav Lejeune Dirichlet who introduced them in (Dirichlet 1837) to prove the theorem on primes in arithmetic progressions that also bears his name.
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https://en.wikipedia.org/wiki/Dirichlet_L-function
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In the course of the proof, Dirichlet shows that L(s, χ) is non-zero at s = 1. Moreover, if χ is principal, then the corresponding Dirichlet L-function has a simple pole at s = 1. Otherwise, the L-function is entire.
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https://en.wikipedia.org/wiki/Dirichlet_L-function
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In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on X, whose real parts are dense in the algebra of bounded continuous real functions on X. The concept was introduced by Andrew Gleason (1957).
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https://en.wikipedia.org/wiki/Dirichlet_algebra
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In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region. The Dirichlet problem can be solved for many PDEs, although originally it was posed for Laplace's equation. In that case the problem can be stated as follows: Given a function f that has values everywhere on the boundary of a region in Rn, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary?This requirement is called the Dirichlet boundary condition. The main issue is to prove the existence of a solution; uniqueness can be proven using the maximum principle.
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https://en.wikipedia.org/wiki/Dirichlet's_problem
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In mathematics, a Dirichlet series is any series of the form where s is complex, and a n {\displaystyle a_{n}} is a complex sequence. It is a special case of general Dirichlet series. Dirichlet series play a variety of important roles in analytic number theory. The most usually seen definition of the Riemann zeta function is a Dirichlet series, as are the Dirichlet L-functions. It is conjectured that the Selberg class of series obeys the generalized Riemann hypothesis. The series is named in honor of Peter Gustav Lejeune Dirichlet.
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https://en.wikipedia.org/wiki/Formal_Dirichlet_series
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In mathematics, a Ditkin set, introduced by (Ditkin 1939), is a closed subset of the circle such that a function f vanishing on the set can be approximated by functions φnf with φ vanishing in a neighborhood of the set.
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https://en.wikipedia.org/wiki/Ditkin_set
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In mathematics, a Dold manifold is one of the manifolds P ( m , n ) = ( S m × C P n ) / τ {\displaystyle P(m,n)=(S^{m}\times \mathbb {CP} ^{n})/\tau } , where τ {\displaystyle \tau } is the involution that acts as −1 on the m-sphere S m {\displaystyle S^{m}} and as complex conjugation on the complex projective space C P n {\displaystyle \mathbb {CP} ^{n}} . These manifolds were constructed by Albrecht Dold (1956), who used them to give explicit generators for René Thom's unoriented cobordism ring. Note that P ( m , 0 ) = R P m {\displaystyle P(m,0)=\mathbb {RP} ^{m}} , the real projective space of dimension m, and P ( 0 , n ) = C P n {\displaystyle P(0,n)=\mathbb {CP} ^{n}} .
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https://en.wikipedia.org/wiki/Dold_manifold
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In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of complex multiplication theory. A shtuka (also called F-sheaf or chtouca) is a sort of generalization of a Drinfeld module, consisting roughly of a vector bundle over a curve, together with some extra structure identifying a "Frobenius twist" of the bundle with a "modification" of it. Drinfeld modules were introduced by Drinfeld (1974), who used them to prove the Langlands conjectures for GL2 of an algebraic function field in some special cases.
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https://en.wikipedia.org/wiki/Drinfeld_module
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He later invented shtukas and used shtukas of rank 2 to prove the remaining cases of the Langlands conjectures for GL2. Laurent Lafforgue proved the Langlands conjectures for GLn of a function field by studying the moduli stack of shtukas of rank n. "Shtuka" is a Russian word штука meaning "a single copy", which comes from the German noun “Stück”, meaning “piece, item, or unit". In Russian, the word "shtuka" is also used in slang for a thing with known properties, but having no name in a speaker's mind.
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https://en.wikipedia.org/wiki/Drinfeld_module
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In mathematics, a Dupin cyclide or cyclide of Dupin is any geometric inversion of a standard torus, cylinder or double cone. In particular, these latter are themselves examples of Dupin cyclides. They were discovered c.
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https://en.wikipedia.org/wiki/Dupin_cyclide
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1802 by (and named after) Charles Dupin, while he was still a student at the École polytechnique following Gaspard Monge's lectures. The key property of a Dupin cyclide is that it is a channel surface (envelope of a one-parameter family of spheres) in two different ways. This property means that Dupin cyclides are natural objects in Lie sphere geometry.
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https://en.wikipedia.org/wiki/Dupin_cyclide
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Dupin cyclides are often simply known as cyclides, but the latter term is also used to refer to a more general class of quartic surfaces which are important in the theory of separation of variables for the Laplace equation in three dimensions. Dupin cyclides were investigated not only by Dupin, but also by A. Cayley, J.C. Maxwell and Mabel M. Young. Dupin cyclides are used in computer-aided design because cyclide patches have rational representations and are suitable for blending canal surfaces (cylinder, cones, tori, and others).
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https://en.wikipedia.org/wiki/Dupin_cyclide
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In mathematics, a Euclidean distance matrix is an n×n matrix representing the spacing of a set of n points in Euclidean space. For points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} in k-dimensional space ℝk, the elements of their Euclidean distance matrix A are given by squares of distances between them. That is A = ( a i j ) ; a i j = d i j 2 = ‖ x i − x j ‖ 2 {\displaystyle {\begin{aligned}A&=(a_{ij});\\a_{ij}&=d_{ij}^{2}\;=\;\lVert x_{i}-x_{j}\rVert ^{2}\end{aligned}}} where ‖ ⋅ ‖ {\displaystyle \|\cdot \|} denotes the Euclidean norm on ℝk. A = {\displaystyle A={\begin{bmatrix}0&d_{12}^{2}&d_{13}^{2}&\dots &d_{1n}^{2}\\d_{21}^{2}&0&d_{23}^{2}&\dots &d_{2n}^{2}\\d_{31}^{2}&d_{32}^{2}&0&\dots &d_{3n}^{2}\\\vdots &\vdots &\vdots &\ddots &\vdots &\\d_{n1}^{2}&d_{n2}^{2}&d_{n3}^{2}&\dots &0\\\end{bmatrix}}} In the context of (not necessarily Euclidean) distance matrices, the entries are usually defined directly as distances, not their squares.
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https://en.wikipedia.org/wiki/Euclidean_distance_matrix
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However, in the Euclidean case, squares of distances are used to avoid computing square roots and to simplify relevant theorems and algorithms. Euclidean distance matrices are closely related to Gram matrices (matrices of dot products, describing norms of vectors and angles between them). The latter are easily analyzed using methods of linear algebra.
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https://en.wikipedia.org/wiki/Euclidean_distance_matrix
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This allows to characterize Euclidean distance matrices and recover the points x 1 , x 2 , … , x n {\displaystyle x_{1},x_{2},\ldots ,x_{n}} that realize it. A realization, if it exists, is unique up to rigid transformations, i.e. distance-preserving transformations of Euclidean space (rotations, reflections, translations). In practical applications, distances are noisy measurements or come from arbitrary dissimilarity estimates (not necessarily metric).
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https://en.wikipedia.org/wiki/Euclidean_distance_matrix
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The goal may be to visualize such data by points in Euclidean space whose distance matrix approximates a given dissimilarity matrix as well as possible — this is known as multidimensional scaling. Alternatively, given two sets of data already represented by points in Euclidean space, one may ask how similar they are in shape, that is, how closely can they be related by a distance-preserving transformation — this is Procrustes analysis. Some of the distances may also be missing or come unlabelled (as an unordered set or multiset instead of a matrix), leading to more complex algorithmic tasks, such as the graph realization problem or the turnpike problem (for points on a line).
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https://en.wikipedia.org/wiki/Euclidean_distance_matrix
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In mathematics, a Euclidean field is an ordered field K for which every non-negative element is a square: that is, x ≥ 0 in K implies that x = y2 for some y in K. The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield. In other words, the constructible numbers form the Euclidean closure of the rational numbers.
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https://en.wikipedia.org/wiki/Euclidean_field
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In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space E n {\displaystyle \mathbb {E} ^{n}} ; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension n of the space, and is commonly denoted E(n) or ISO(n). The Euclidean group E(n) comprises all translations, rotations, and reflections of E n {\displaystyle \mathbb {E} ^{n}} ; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space.
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https://en.wikipedia.org/wiki/Indirect_isometry
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A Euclidean isometry can be direct or indirect, depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(n), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and rotations, but not reflections. These groups are among the oldest and most studied, at least in the cases of dimension 2 and 3 – implicitly, long before the concept of group was invented.
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https://en.wikipedia.org/wiki/Indirect_isometry
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In mathematics, a Euclidean plane is a Euclidean space of dimension two, denoted E2. It is a geometric space in which two real numbers are required to determine the position of each point. It is an affine space, which includes in particular the concept of parallel lines. It has also metrical properties induced by a distance, which allows to define circles, and angle measurement. A Euclidean plane with a chosen Cartesian coordinate system is called a Cartesian plane. The set R 2 {\displaystyle \mathbb {R} ^{2}} of the pairs of real numbers (the real coordinate plane), equipped with the dot product, is often called the Euclidean plane, since every Euclidean plane is isomorphic to it.
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https://en.wikipedia.org/wiki/Euclidean_plane
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In mathematics, a Fatou–Bieberbach domain is a proper subdomain of C n {\displaystyle \mathbb {C} ^{n}} , biholomorphically equivalent to C n {\displaystyle \mathbb {C} ^{n}} . That is, an open set Ω ⊊ C n {\displaystyle \Omega \subsetneq \mathbb {C} ^{n}} is called a Fatou–Bieberbach domain if there exists a bijective holomorphic function f: Ω → C n {\displaystyle f:\Omega \rightarrow \mathbb {C} ^{n}} whose inverse function f − 1: C n → Ω {\displaystyle f^{-1}:\mathbb {C} ^{n}\rightarrow \Omega } is holomorphic. It is well-known that the inverse f − 1 {\displaystyle f^{-1}} can not be polynomial.
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https://en.wikipedia.org/wiki/Fatou–Bieberbach_domain
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In mathematics, a Fedosov manifold is a symplectic manifold with a compatible torsion-free connection, that is, a triple (M, ω, ∇), where (M, ω) is a symplectic manifold (that is, ω {\displaystyle \omega } is a symplectic form, a non-degenerate closed exterior 2-form, on a C ∞ {\displaystyle C^{\infty }} -manifold M), and ∇ is a symplectic torsion-free connection on M . {\displaystyle M.} (A connection ∇ is called compatible or symplectic if X ⋅ ω(Y,Z) = ω(∇XY,Z) + ω(Y,∇XZ) for all vector fields X,Y,Z ∈ Γ(TM).
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https://en.wikipedia.org/wiki/Fedosov_manifold
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In other words, the symplectic form is parallel with respect to the connection, i.e., its covariant derivative vanishes.) Note that every symplectic manifold admits a symplectic torsion-free connection. Cover the manifold with Darboux charts and on each chart define a connection ∇ with Christoffel symbol Γ j k i = 0 {\displaystyle \Gamma _{jk}^{i}=0} . Then choose a partition of unity (subordinate to the cover) and glue the local connections together to a global connection which still preserves the symplectic form. The famous result of Boris Vasilievich Fedosov gives a canonical deformation quantization of a Fedosov manifold.
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https://en.wikipedia.org/wiki/Fedosov_manifold
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In mathematics, a Fekete polynomial is a polynomial f p ( t ) := ∑ a = 0 p − 1 ( a p ) t a {\displaystyle f_{p}(t):=\sum _{a=0}^{p-1}\left({\frac {a}{p}}\right)t^{a}\,} where ( ⋅ p ) {\displaystyle \left({\frac {\cdot }{p}}\right)\,} is the Legendre symbol modulo some integer p > 1. These polynomials were known in nineteenth-century studies of Dirichlet L-functions, and indeed to Dirichlet himself. They have acquired the name of Michael Fekete, who observed that the absence of real zeroes t of the Fekete polynomial with 0 < t < 1 implies an absence of the same kind for the L-function L ( s , x p ) . {\displaystyle L\left(s,{\dfrac {x}{p}}\right).\,} This is of considerable potential interest in number theory, in connection with the hypothetical Siegel zero near s = 1. While numerical results for small cases had indicated that there were few such real zeroes, further analysis reveals that this may indeed be a 'small number' effect.
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https://en.wikipedia.org/wiki/Fekete_polynomial
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In mathematics, a Feller-continuous process is a continuous-time stochastic process for which the expected value of suitable statistics of the process at a given time in the future depend continuously on the initial condition of the process. The concept is named after Croatian-American mathematician William Feller.
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https://en.wikipedia.org/wiki/Feller-continuous_process
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In mathematics, a Fermat number, named after Pierre de Fermat, the first known to have studied them, is a positive integer of the form F n = 2 2 n + 1 , {\displaystyle F_{n}=2^{2^{n}}+1,} where n is a non-negative integer. The first few Fermat numbers are: 3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... (sequence A000215 in the OEIS).If 2k + 1 is prime and k > 0, then k itself must be a power of 2, so 2k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023, the only known Fermat primes are F0 = 3, F1 = 5, F2 = 17, F3 = 257, and F4 = 65537 (sequence A019434 in the OEIS); heuristics suggest that there are no more.
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https://en.wikipedia.org/wiki/Fermat_numbers
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In mathematics, a Fermat quintic threefold is a special quintic threefold, in other words a degree 5, dimension 3 hypersurface in 4-dimensional complex projective space, given by the equation V 5 + W 5 + X 5 + Y 5 + Z 5 = 0 {\displaystyle V^{5}+W^{5}+X^{5}+Y^{5}+Z^{5}=0} .This threefold, so named after Pierre de Fermat, is a Calabi–Yau manifold. The Hodge diamond of a non-singular quintic 3-fold is
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https://en.wikipedia.org/wiki/Fermat_quintic_threefold
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In mathematics, a Ford circle is a circle in the Euclidean plane, in a family of circles that are all tangent to the x {\displaystyle x} -axis at rational points. For each rational number p / q {\displaystyle p/q} , expressed in lowest terms, there is a Ford circle whose center is at the point ( p / q , 1 / ( 2 q 2 ) ) {\displaystyle (p/q,1/(2q^{2}))} and whose radius is 1 / ( 2 q 2 ) {\displaystyle 1/(2q^{2})} . It is tangent to the x {\displaystyle x} -axis at its bottom point, ( p / q , 0 ) {\displaystyle (p/q,0)} . The two Ford circles for rational numbers p / q {\displaystyle p/q} and r / s {\displaystyle r/s} (both in lowest terms) are tangent circles when | p s − q r | = 1 {\displaystyle |ps-qr|=1} and otherwise these two circles are disjoint.
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https://en.wikipedia.org/wiki/Ford_circle
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In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are one of the objects of study in Fredholm theory. Fredholm kernels are named in honour of Erik Ivar Fredholm. Much of the abstract theory of Fredholm kernels was developed by Alexander Grothendieck and published in 1955.
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https://en.wikipedia.org/wiki/Fredholm_kernel
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In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve y 2 = x ( x − a ℓ ) ( x + b ℓ ) {\displaystyle y^{2}=x(x-a^{\ell })(x+b^{\ell })} associated with a (hypothetical) solution of Fermat's equation a ℓ + b ℓ = c ℓ . {\displaystyle a^{\ell }+b^{\ell }=c^{\ell }.} The curve is named after Gerhard Frey and (sometimes) Yves Hellegouarch.
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https://en.wikipedia.org/wiki/Frey_curve
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In mathematics, a Fricke involution is the involution of the modular curve X0(N) given by τ → –1/Nτ. It is named after Robert Fricke. The Fricke involution also acts on other objects associated with the modular curve, such as spaces of modular forms and the Jacobian J0(N) of the modular curve.
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https://en.wikipedia.org/wiki/Fricke_involution
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In mathematics, a Frink ideal, introduced by Orrin Frink, is a certain kind of subset of a partially ordered set.
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https://en.wikipedia.org/wiki/Frink_ideal
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In mathematics, a Frobenius group is a transitive permutation group on a finite set, such that no non-trivial element fixes more than one point and some non-trivial element fixes a point. They are named after F. G. Frobenius.
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https://en.wikipedia.org/wiki/Frobenius_kernel
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In mathematics, a Frobenius splitting, introduced by Mehta and Ramanathan (1985), is a splitting of the injective morphism OX→F*OX from a structure sheaf OX of a characteristic p > 0 variety X to its image F*OX under the Frobenius endomorphism F*. Brion & Kumar (2005) give a detailed discussion of Frobenius splittings. A fundamental property of Frobenius-split projective schemes X is that the higher cohomology Hi(X,L) (i > 0) of ample line bundles L vanishes.
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https://en.wikipedia.org/wiki/Frobenius_splitting
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In mathematics, a Fréchet surface is an equivalence class of parametrized surfaces in a metric space. In other words, a Fréchet surface is a way of thinking about surfaces independently of how they are "written down" (parametrized). The concept is named after the French mathematician Maurice Fréchet.
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https://en.wikipedia.org/wiki/Fréchet_surface
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In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R). Fuchsian groups are used to create Fuchsian models of Riemann surfaces.
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https://en.wikipedia.org/wiki/Fuchsian_group
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In this case, the group may be called the Fuchsian group of the surface. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry). General Fuchsian groups were first studied by Henri Poincaré (1882), who was motivated by the paper (Fuchs 1880), and therefore named them after Lazarus Fuchs.
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https://en.wikipedia.org/wiki/Fuchsian_group
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In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation. The concept is named after Lazarus Fuchs.
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https://en.wikipedia.org/wiki/Fuchsian_model
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In mathematics, a Følner sequence for a group is a sequence of sets satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is amenable. A more general notion of Følner nets can be defined analogously, and is suited for the study of uncountable groups. Følner sequences are named for Erling Følner.
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https://en.wikipedia.org/wiki/Følner_sequence
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In mathematics, a G-measure is a measure μ {\displaystyle \mu } that can be represented as the weak-∗ limit of a sequence of measurable functions G = ( G n ) n = 1 ∞ {\displaystyle G=\left(G_{n}\right)_{n=1}^{\infty }} . A classic example is the Riesz product G n ( t ) = ∏ k = 1 n ( 1 + r cos ( 2 π m k t ) ) {\displaystyle G_{n}(t)=\prod _{k=1}^{n}\left(1+r\cos(2\pi m^{k}t)\right)} where − 1 < r < 1 , m ∈ N {\displaystyle -1
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https://en.wikipedia.org/wiki/G-measure
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In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by two given elements. Equivalently, any two elements of R have a least common multiple (LCM).A GCD domain generalizes a unique factorization domain (UFD) to a non-Noetherian setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals (and in particular if it is Noetherian). GCD domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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https://en.wikipedia.org/wiki/GCD_domain
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In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.A result of Emil Artin allows one to construct Galois extensions as follows: If E is a given field, and G is a finite group of automorphisms of E with fixed field F, then E/F is a Galois extension.
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https://en.wikipedia.org/wiki/Galois_field_extension
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In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.
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https://en.wikipedia.org/wiki/Normal_integral_basis
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In mathematics, a Garside element is an element of an algebraic structure such as a monoid that has several desirable properties. Formally, if M is a monoid, then an element Δ of M is said to be a Garside element if the set of all right divisors of Δ, { r ∈ M ∣ for some x ∈ M , Δ = x r } , {\displaystyle \{r\in M\mid {\text{for some }}x\in M,\Delta =xr\},} is the same set as the set of all left divisors of Δ, { ℓ ∈ M ∣ for some x ∈ M , Δ = ℓ x } , {\displaystyle \{\ell \in M\mid {\text{for some }}x\in M,\Delta =\ell x\},} and this set generates M. A Garside element is in general not unique: any power of a Garside element is again a Garside element.
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https://en.wikipedia.org/wiki/Garside_element
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In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form and with parametric extension for arbitrary real constants a, b and non-zero c. It is named after the mathematician Carl Friedrich Gauss. The graph of a Gaussian is a characteristic symmetric "bell curve" shape. The parameter a is the height of the curve's peak, b is the position of the center of the peak, and c (the standard deviation, sometimes called the Gaussian RMS width) controls the width of the "bell". Gaussian functions are often used to represent the probability density function of a normally distributed random variable with expected value μ = b and variance σ2 = c2. In this case, the Gaussian is of the form Gaussian functions are widely used in statistics to describe the normal distributions, in signal processing to define Gaussian filters, in image processing where two-dimensional Gaussians are used for Gaussian blurs, and in mathematics to solve heat equations and diffusion equations and to define the Weierstrass transform.
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https://en.wikipedia.org/wiki/Error_curve
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In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted Q(i), obtained by adjoining the imaginary number i to the field of rationals Q.
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https://en.wikipedia.org/wiki/Gaussian_rational
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In mathematics, a Gelfand pair is a pair (G,K) consisting of a group G and a subgroup K (called an Euler subgroup of G) that satisfies a certain property on restricted representations. The theory of Gelfand pairs is closely related to the topic of spherical functions in the classical theory of special functions, and to the theory of Riemannian symmetric spaces in differential geometry. Broadly speaking, the theory exists to abstract from these theories their content in terms of harmonic analysis and representation theory. When G is a finite group the simplest definition is, roughly speaking, that the (K,K)-double cosets in G commute. More precisely, the Hecke algebra, the algebra of functions on G that are invariant under translation on either side by K, should be commutative for the convolution on G. In general, the definition of Gelfand pair is roughly that the restriction to K of any irreducible representation of G contains the trivial representation of K with multiplicity no more than 1. In each case one should specify the class of considered representations and the meaning of contains.
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https://en.wikipedia.org/wiki/Gelfand_pair
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In mathematics, a Gelfand ring is an associative ring R with identity such that if I and J are distinct right ideals then there are elements i and j such that iRj=0, i is not in I, and j is not in J. Mulvey (1979) introduced them as rings for which one could prove a generalization of Gelfand duality, and named them after Israel Gelfand.In the commutative case, Gelfand rings can also be characterized as the rings such that, for every a and b summing to 1, there exists r and s such that ( 1 + r a ) ( 1 + s b ) = 0 {\displaystyle (1+ra)(1+sb)=0} .Moreover, their prime spectrum deformation retracts onto the maximal spectrum. == References ==
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https://en.wikipedia.org/wiki/Gelfand_ring
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In mathematics, a Generalized Clifford algebra (GCA) is a unital associative algebra that generalizes the Clifford algebra, and goes back to the work of Hermann Weyl, who utilized and formalized these clock-and-shift operators introduced by J. J. Sylvester (1882), and organized by Cartan (1898) and Schwinger.Clock and shift matrices find routine applications in numerous areas of mathematical physics, providing the cornerstone of quantum mechanical dynamics in finite-dimensional vector spaces. The concept of a spinor can further be linked to these algebras.The term Generalized Clifford Algebras can also refer to associative algebras that are constructed using forms of higher degree instead of quadratic forms.
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https://en.wikipedia.org/wiki/Generalized_Clifford_algebra
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In mathematics, a Goldman domain or G-domain is an integral domain A whose field of fractions is a finitely generated algebra over A. They are named after Oscar Goldman. An overring (i.e., an intermediate ring lying between the ring and its field of fractions) of a Goldman domain is again a Goldman domain. There exists a Goldman domain where all nonzero prime ideals are maximal although there are infinitely many prime ideals.An ideal I in a commutative ring A is called a Goldman ideal if the quotient A/I is a Goldman domain.
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https://en.wikipedia.org/wiki/Goldman_domain
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A Goldman ideal is thus prime, but not necessarily maximal. In fact, a commutative ring is a Jacobson ring if and only if every Goldman ideal in it is maximal. The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal: the radical of an ideal I is the intersection of all Goldman ideals containing I.
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https://en.wikipedia.org/wiki/Goldman_domain
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In mathematics, a Golomb ruler is a set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its order, and the largest distance between two of its marks is its length. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays.
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https://en.wikipedia.org/wiki/Golomb_ruler
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The Golomb ruler was named for Solomon W. Golomb and discovered independently by Sidon (1932) and Babcock (1953). Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruent. This turned out to be false for six-point rulers, but true otherwise.There is no requirement that a Golomb ruler be able to measure all distances up to its length, but if it does, it is called a perfect Golomb ruler.
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https://en.wikipedia.org/wiki/Golomb_ruler
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It has been proved that no perfect Golomb ruler exists for five or more marks. A Golomb ruler is optimal if no shorter Golomb ruler of the same order exists. Creating Golomb rulers is easy, but proving the optimal Golomb ruler (or rulers) for a specified order is computationally very challenging.
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https://en.wikipedia.org/wiki/Golomb_ruler
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Distributed.net has completed distributed massively parallel searches for optimal order-24 through order-28 Golomb rulers, each time confirming the suspected candidate ruler.Currently, the complexity of finding optimal Golomb rulers (OGRs) of arbitrary order n (where n is given in unary) is unknown. In the past there was some speculation that it is an NP-hard problem. Problems related to the construction of Golomb rulers are provably shown to be NP-hard, where it is also noted that no known NP-complete problem has similar flavor to finding Golomb rulers.
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https://en.wikipedia.org/wiki/Golomb_ruler
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In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product. It is the most general structure in which projective properties are expressed in a coordinate-free way. The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear algebra. It is a form of modeling algebra for use in projective geometry.The technique uses subspaces as basic elements of computation, a formalism which allows the translation of synthetic geometric statements into invariant algebraic statements. This can create a useful framework for the modeling of conics and quadrics among other forms, and in tensor mathematics. It also has a number of applications in robotics, particularly for the kinematical analysis of manipulators.
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https://en.wikipedia.org/wiki/Grassmann–Cayley_algebra
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In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L {\displaystyle \operatorname {L} } is the linear differential operator, then the Green's function G {\displaystyle G} is the solution of the equation L G = δ {\displaystyle \operatorname {L} G=\delta } , where δ {\displaystyle \delta } is Dirac's delta function; the solution of the initial-value problem L y = f {\displaystyle \operatorname {L} y=f} is the convolution ( G ∗ f {\displaystyle G\ast f} ).Through the superposition principle, given a linear ordinary differential equation (ODE), L y = f {\displaystyle \operatorname {L} y=f} , one can first solve L G = δ s {\displaystyle \operatorname {L} G=\delta _{s}} , for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L. Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead. Under many-body theory, the term is also used in physics, specifically in quantum field theory, aerodynamics, aeroacoustics, electrodynamics, seismology and statistical field theory, to refer to various types of correlation functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.
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https://en.wikipedia.org/wiki/Green’s_function
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In mathematics, a Gregory number, named after James Gregory, is a real number of the form: G x = ∑ i = 0 ∞ ( − 1 ) i 1 ( 2 i + 1 ) x 2 i + 1 {\displaystyle G_{x}=\sum _{i=0}^{\infty }(-1)^{i}{\frac {1}{(2i+1)x^{2i+1}}}} where x is any rational number greater or equal to 1. Considering the power series expansion for arctangent, we have G x = arctan 1 x . {\displaystyle G_{x}=\arctan {\frac {1}{x}}.} Setting x = 1 gives the well-known Leibniz formula for pi. Thus, in particular, π 4 = arctan 1 {\displaystyle {\frac {\pi }{4}}=\arctan 1} is a Gregory number.
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https://en.wikipedia.org/wiki/Gregory_number
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In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.To every algebraic variety V {\displaystyle V} one can associate a Grothendieck category Qcoh ( V ) {\displaystyle \operatorname {Qcoh} (V)} , consisting of the quasi-coherent sheaves on V {\displaystyle V} . This category encodes all the relevant geometric information about V {\displaystyle V} , and V {\displaystyle V} can be recovered from Qcoh ( V ) {\displaystyle \operatorname {Qcoh} (V)} (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.
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https://en.wikipedia.org/wiki/Grothendieck_category
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In mathematics, a Grothendieck space, named after Alexander Grothendieck, is a Banach space X {\displaystyle X} in which every sequence in its continuous dual space X ′ {\displaystyle X^{\prime }} that converges in the weak-* topology σ ( X ′ , X ) {\displaystyle \sigma \left(X^{\prime },X\right)} (also known as the topology of pointwise convergence) will also converge when X ′ {\displaystyle X^{\prime }} is endowed with σ ( X ′ , X ′ ′ ) , {\displaystyle \sigma \left(X^{\prime },X^{\prime \prime }\right),} which is the weak topology induced on X ′ {\displaystyle X^{\prime }} by its bidual. Said differently, a Grothendieck space is a Banach space for which a sequence in its dual space converges weak-* if and only if it converges weakly.
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https://en.wikipedia.org/wiki/Grothendieck_space
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In mathematics, a Grothendieck universe is a set U with the following properties: If x is an element of U and if y is an element of x, then y is also an element of U. (U is a transitive set.) If x and y are both elements of U, then { x , y } {\displaystyle \{x,y\}} is an element of U. If x is an element of U, then P(x), the power set of x, is also an element of U. If { x α } α ∈ I {\displaystyle \{x_{\alpha }\}_{\alpha \in I}} is a family of elements of U, and if I is an element of U, then the union ⋃ α ∈ I x α {\displaystyle \bigcup _{\alpha \in I}x_{\alpha }} is an element of U.A Grothendieck universe is meant to provide a set in which all of mathematics can be performed.
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https://en.wikipedia.org/wiki/Grothendieck_universe
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(In fact, uncountable Grothendieck universes provide models of set theory with the natural ∈-relation, natural powerset operation etc.). Elements of a Grothendieck universe are sometimes called small sets. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.
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https://en.wikipedia.org/wiki/Grothendieck_universe
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The existence of a nontrivial Grothendieck universe goes beyond the usual axioms of Zermelo–Fraenkel set theory; in particular it would imply the existence of strongly inaccessible cardinals. Tarski–Grothendieck set theory is an axiomatic treatment of set theory, used in some automatic proof systems, in which every set belongs to a Grothendieck universe. The concept of a Grothendieck universe can also be defined in a topos.
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https://en.wikipedia.org/wiki/Grothendieck_universe
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In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem. Here, the idea was to map mathematical notation to a natural number (using a Gödel numbering).
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https://en.wikipedia.org/wiki/Data_encoding
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In mathematics, a Gödel numbering for sequences provides an effective way to represent each finite sequence of natural numbers as a single natural number. While a set theoretical embedding is surely possible, the emphasis is on the effectiveness of the functions manipulating such representations of sequences: the operations on sequences (accessing individual members, concatenation) can be "implemented" using total recursive functions, and in fact by primitive recursive functions. It is usually used to build sequential “data types” in arithmetic-based formalizations of some fundamental notions of mathematics. It is a specific case of the more general idea of Gödel numbering. For example, recursive function theory can be regarded as a formalization of the notion of an algorithm, and can be regarded as a programming language to mimic lists by encoding a sequence of natural numbers in a single natural number.
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https://en.wikipedia.org/wiki/Gödel_numbering_for_sequences
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In mathematics, a Hadamard manifold, named after Jacques Hadamard — more often called a Cartan–Hadamard manifold, after Élie Cartan — is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} that is complete and simply connected and has everywhere non-positive sectional curvature. By Cartan–Hadamard theorem all Cartan–Hadamard manifolds are diffeomorphic to the Euclidean space R n . {\displaystyle \mathbb {R} ^{n}.} Furthermore it follows from the Hopf–Rinow theorem that every pairs of points in a Cartan–Hadamard manifold may be connected by a unique geodesic segment. Thus Cartan–Hadamard manifolds are some of the closest relatives of R n . {\displaystyle \mathbb {R} ^{n}.}
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https://en.wikipedia.org/wiki/Hadamard_manifold
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In mathematics, a Hadamard matrix, named after the French mathematician Jacques Hadamard, is a square matrix whose entries are either +1 or −1 and whose rows are mutually orthogonal. In geometric terms, this means that each pair of rows in a Hadamard matrix represents two perpendicular vectors, while in combinatorial terms, it means that each pair of rows has matching entries in exactly half of their columns and mismatched entries in the remaining columns. It is a consequence of this definition that the corresponding properties hold for columns as well as rows.
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https://en.wikipedia.org/wiki/Hadamard_matrix
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The n-dimensional parallelotope spanned by the rows of an n×n Hadamard matrix has the maximum possible n-dimensional volume among parallelotopes spanned by vectors whose entries are bounded in absolute value by 1. Equivalently, a Hadamard matrix has maximal determinant among matrices with entries of absolute value less than or equal to 1 and so is an extremal solution of Hadamard's maximal determinant problem. Certain Hadamard matrices can almost directly be used as an error-correcting code using a Hadamard code (generalized in Reed–Muller codes), and are also used in balanced repeated replication (BRR), used by statisticians to estimate the variance of a parameter estimator.
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https://en.wikipedia.org/wiki/Hadamard_matrix
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In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind of Haefliger structure, which uniquely determines the foliation.
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https://en.wikipedia.org/wiki/Haefliger_structure
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In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken.
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https://en.wikipedia.org/wiki/Haken_hierarchy
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This conjecture was proven by Ian Agol.Haken manifolds were introduced by Wolfgang Haken (1961). Haken (1962) proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. William Jaco and Ulrich Oertel (1984) gave an algorithm to determine if a 3-manifold was Haken. Normal surfaces are ubiquitous in the theory of Haken manifolds and their simple and rigid structure leads quite naturally to algorithms.
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https://en.wikipedia.org/wiki/Haken_hierarchy
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In mathematics, a Hall plane is a non-Desarguesian projective plane constructed by Marshall Hall Jr. (1943). There are examples of order p2n for every prime p and every positive integer n provided p2n > 4.
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https://en.wikipedia.org/wiki/Hall_plane_of_order_9
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In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix J = {\displaystyle J={\begin{bmatrix}0_{n}&I_{n}\\-I_{n}&0_{n}\\\end{bmatrix}}} and In is the n-by-n identity matrix. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes the transpose.
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https://en.wikipedia.org/wiki/Hamiltonian_matrix
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