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In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition.The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory.
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https://en.wikipedia.org/wiki/Elliptic_modular_form
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Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group S L 2 ( Z ) ⊂ S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\subset \mathrm {SL} _{2}(\mathbb {R} )} . The term "modular form", as a systematic description, is usually attributed to Hecke. Each modular form is attached to a Galois representation.
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https://en.wikipedia.org/wiki/Elliptic_modular_form
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In mathematics, a modular invariant of a group is an invariant of a finite group acting on a vector space of positive characteristic (usually dividing the order of the group). The study of modular invariants was originated in about 1914 by Dickson (2004).
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https://en.wikipedia.org/wiki/Modular_invariant_of_a_group
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In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operation of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology.
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https://en.wikipedia.org/wiki/Right_module
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In mathematics, a module over a (not necessarily commutative) ring is countably generated if it is generated as a module by a countable subset. The importance of the notion comes from Kaplansky's theorem (Kaplansky 1958), which states that a projective module is a direct sum of countably generated modules. More generally, a module over a possibly non-commutative ring is projective if and only if (i) it is flat, (ii) it is a direct sum of countably generated modules and (iii) it is a Mittag-Leffler module. (Bazzoni–Stovicek)
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https://en.wikipedia.org/wiki/Countably_generated_module
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In mathematics, a moduli scheme is a moduli space that exists in the category of schemes developed by Alexander Grothendieck. Some important moduli problems of algebraic geometry can be satisfactorily solved by means of scheme theory alone, while others require some extension of the 'geometric object' concept (algebraic spaces, algebraic stacks of Michael Artin).
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https://en.wikipedia.org/wiki/Fine_moduli_scheme
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In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.) Moment matrices play an important role in polynomial fitting, polynomial optimization (since positive semidefinite moment matrices correspond to polynomials which are sums of squares) and econometrics.
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https://en.wikipedia.org/wiki/Moment_matrix
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In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure μ to the sequence of moments m n = ∫ − ∞ ∞ x n d μ ( x ) . {\displaystyle m_{n}=\int _{-\infty }^{\infty }x^{n}\,d\mu (x)\,.} More generally, one may consider m n = ∫ − ∞ ∞ M n ( x ) d μ ( x ) . {\displaystyle m_{n}=\int _{-\infty }^{\infty }M_{n}(x)\,d\mu (x)\,.} for an arbitrary sequence of functions Mn.
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https://en.wikipedia.org/wiki/Moment_problem
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In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the subring Z of K generated by a. Then OK is a quotient of the polynomial ring Z and the powers of a constitute a power integral basis. In a monogenic field K, the field discriminant of K is equal to the discriminant of the minimal polynomial of α.
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https://en.wikipedia.org/wiki/Monogenic_field
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In mathematics, a monogenic semigroup is a semigroup generated by a single element. Monogenic semigroups are also called cyclic semigroups.
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https://en.wikipedia.org/wiki/Monogenic_semigroup
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In mathematics, a monoidal category (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗: C × C → C {\displaystyle \otimes :\mathbf {C} \times \mathbf {C} \to \mathbf {C} } that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, R-modules, or R-algebras into monoidal categories.
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https://en.wikipedia.org/wiki/Monoidal_category_of_endofunctors
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Monoidal categories can be seen as a generalization of these and other examples. Every (small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an abstraction, is that of a system of data types closed under a type constructor that takes two types and builds an aggregate type; the types are the objects and ⊗ {\displaystyle \otimes } is the aggregate constructor.
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https://en.wikipedia.org/wiki/Monoidal_category_of_endofunctors
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The associativity up to isomorphism is then a way of expressing that different ways of aggregating the same data—such as ( ( a , b ) , c ) {\displaystyle ((a,b),c)} and ( a , ( b , c ) ) {\displaystyle (a,(b,c))} —store the same information even though the aggregate values need not be the same. The aggregate type may be analogous to the operation of addition (type sum) or of multiplication (type product). For type product, the identity object is the unit ( ) {\displaystyle ()} , so there is only one inhabitant of the type, and that is why a product with it is always isomorphic to the other operand.
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https://en.wikipedia.org/wiki/Monoidal_category_of_endofunctors
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For type sum, the identity object is the void type, which stores no information and it is impossible to address an inhabitant. The concept of monoidal category does not presume that values of such aggregate types can be taken apart; on the contrary, it provides a framework that unifies classical and quantum information theory.In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category.
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https://en.wikipedia.org/wiki/Monoidal_category_of_endofunctors
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Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter physics. Braided monoidal categories have applications in quantum information, quantum field theory, and string theory.
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https://en.wikipedia.org/wiki/Monoidal_category_of_endofunctors
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In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, x 2 y z 3 = x x y z z z {\displaystyle x^{2}yz^{3}=xxyzzz} is a monomial. The constant 1 {\displaystyle 1} is a monomial, being equal to the empty product and to x 0 {\displaystyle x^{0}} for any variable x {\displaystyle x} .
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https://en.wikipedia.org/wiki/Simple_expression
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If only a single variable x {\displaystyle x} is considered, this means that a monomial is either 1 {\displaystyle 1} or a power x n {\displaystyle x^{n}} of x {\displaystyle x} , with n {\displaystyle n} a positive integer. If several variables are considered, say, x , y , z , {\displaystyle x,y,z,} then each can be given an exponent, so that any monomial is of the form x a y b z c {\displaystyle x^{a}y^{b}z^{c}} with a , b , c {\displaystyle a,b,c} non-negative integers (taking note that any exponent 0 {\displaystyle 0} makes the corresponding factor equal to 1 {\displaystyle 1} ). A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial.
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https://en.wikipedia.org/wiki/Simple_expression
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A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is 1 {\displaystyle 1} . For example, in this interpretation − 7 x 5 {\displaystyle -7x^{5}} and ( 3 − 4 i ) x 4 y z 13 {\displaystyle (3-4i)x^{4}yz^{13}} are monomials (in the second example, the variables are x , y , z , {\displaystyle x,y,z,} and the coefficient is a complex number).In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers. Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial".
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https://en.wikipedia.org/wiki/Simple_expression
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In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., If u ≤ v {\displaystyle u\leq v} and w {\displaystyle w} is any other monomial, then u w ≤ v w {\displaystyle uw\leq vw} .Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order.
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https://en.wikipedia.org/wiki/Monomial_ordering
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In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle.
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https://en.wikipedia.org/wiki/Monopole_(mathematics)
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In mathematics, a monothetic group is a topological group with a dense cyclic subgroup. They were introduced by Van Dantzig (1933). An example is the additive group of p-adic integers, in which the integers are dense. A monothetic group is necessarily abelian.
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https://en.wikipedia.org/wiki/Monothetic_group
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In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
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https://en.wikipedia.org/wiki/Monotonically_non-decreasing
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In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
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https://en.wikipedia.org/wiki/Moving_frame
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In mathematics, a multiple is the product of any quantity and an integer. In other words, for the quantities a and b, it can be said that b is a multiple of a if b = na for some integer n, which is called the multiplier. If a is not zero, this is equivalent to saying that b / a {\displaystyle b/a} is an integer. When a and b are both integers, and b is a multiple of a, then a is called a divisor of b. One says also that a divides b. If a and b are not integers, mathematicians prefer generally to use integer multiple instead of multiple, for clarification. In fact, multiple is used for other kinds of product; for example, a polynomial p is a multiple of another polynomial q if there exists third polynomial r such that p = qr.
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https://en.wikipedia.org/wiki/Integer_multiple
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In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system. The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9.
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https://en.wikipedia.org/wiki/Multiplication_tables
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In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.
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https://en.wikipedia.org/wiki/Multiplicative_cascade
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In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field (Artin 1966), usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters.
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https://en.wikipedia.org/wiki/Multiplicative_character
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Dirichlet characters can be seen as a special case of this definition. Multiplicative characters are linearly independent, i.e. if χ 1 , χ 2 , … , χ n {\displaystyle \chi _{1},\chi _{2},\ldots ,\chi _{n}} are different characters on a group G then from a 1 χ 1 + a 2 χ 2 + ⋯ + a n χ n = 0 {\displaystyle a_{1}\chi _{1}+a_{2}\chi _{2}+\cdots +a_{n}\chi _{n}=0} it follows that a 1 = a 2 = ⋯ = a n = 0. {\displaystyle a_{1}=a_{2}=\cdots =a_{n}=0.}
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https://en.wikipedia.org/wiki/Multiplicative_character
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4.
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https://en.wikipedia.org/wiki/Reciprocal_value
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The reciprocal function, the function f(x) that maps x to 1/x, is one of the simplest examples of a function which is its own inverse (an involution). Multiplying by a number is the same as dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25).
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https://en.wikipedia.org/wiki/Reciprocal_value
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Therefore, multiplication by a number followed by multiplication by its reciprocal yields the original number (since the product of the number and its reciprocal is 1). The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797) to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocall in a 1570 translation of Euclid's Elements.In the phrase multiplicative inverse, the qualifier multiplicative is often omitted and then tacitly understood (in contrast to the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers.
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https://en.wikipedia.org/wiki/Reciprocal_value
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In these cases it can happen that ab ≠ ba; then "inverse" typically implies that an element is both a left and right inverse. The notation f −1 is sometimes also used for the inverse function of the function f, which is for most functions not equal to the multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is the cosecant of x, and not the inverse sine of x denoted by sin−1 x or arcsin x. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, probably for historical reasons (for example in French, the inverse function is preferably called the bijection réciproque).
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https://en.wikipedia.org/wiki/Reciprocal_value
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In mathematics, a multiplicative sequence or m-sequence is a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.
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https://en.wikipedia.org/wiki/Multiplicative_sequence
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In mathematics, a multiply perfect number (also called multiperfect number or pluperfect number) is a generalization of a perfect number. For a given natural number k, a number n is called k-perfect (or k-fold perfect) if the sum of all positive divisors of n (the divisor function, σ(n)) is equal to kn; a number is thus perfect if and only if it is 2-perfect. A number that is k-perfect for a certain k is called a multiply perfect number. As of 2014, k-perfect numbers are known for each value of k up to 11.It is unknown whether there are any odd multiply perfect numbers other than 1. The first few multiply perfect numbers are: 1, 6, 28, 120, 496, 672, 8128, 30240, 32760, 523776, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, ... (sequence A007691 in the OEIS).
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https://en.wikipedia.org/wiki/Multiperfect_number
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In mathematics, a multisection of a power series is a new power series composed of equally spaced terms extracted unaltered from the original series. Formally, if one is given a power series ∑ n = − ∞ ∞ a n ⋅ z n {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}\cdot z^{n}} then its multisection is a power series of the form ∑ m = − ∞ ∞ a q m + p ⋅ z q m + p {\displaystyle \sum _{m=-\infty }^{\infty }a_{qm+p}\cdot z^{qm+p}} where p, q are integers, with 0 ≤ p < q. Series multisection represents one of the common transformations of generating functions.
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https://en.wikipedia.org/wiki/Series_multisection
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In mathematics, a multiset (or bag, or mset) is a modification of the concept of a set that, unlike a set, allows for multiple instances for each of its elements. The number of instances given for each element is called the multiplicity of that element in the multiset. As a consequence, an infinite number of multisets exist which contain only elements a and b, but vary in the multiplicities of their elements: The set {a, b} contains only elements a and b, each having multiplicity 1 when {a, b} is seen as a multiset. In the multiset {a, a, b}, the element a has multiplicity 2, and b has multiplicity 1.
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https://en.wikipedia.org/wiki/Multiset
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In the multiset {a, a, a, b, b, b}, a and b both have multiplicity 3.These objects are all different when viewed as multisets, although they are the same set, since they all consist of the same elements. As with sets, and in contrast to tuples, the order in which elements are listed does not matter in discriminating multisets, so {a, a, b} and {a, b, a} denote the same multiset. To distinguish between sets and multisets, a notation that incorporates square brackets is sometimes used: the multiset {a, a, b} can be denoted by .The cardinality of a multiset is the sum of the multiplicities of all its elements.
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https://en.wikipedia.org/wiki/Multiset
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For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6. Nicolaas Govert de Bruijn coined the word multiset in the 1970s, according to Donald Knuth. : 694 However, the concept of multisets predates the coinage of the word multiset by many centuries.
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https://en.wikipedia.org/wiki/Multiset
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Knuth himself attributes the first study of multisets to the Indian mathematician Bhāskarāchārya, who described permutations of multisets around 1150. Other names have been proposed or used for this concept, including list, bunch, bag, heap, sample, weighted set, collection, and suite. : 694
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https://en.wikipedia.org/wiki/Multiset
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In mathematics, a multisymplectic integrator is a numerical method for the solution of a certain class of partial differential equations, that are said to be multisymplectic. Multisymplectic integrators are geometric integrators, meaning that they preserve the geometry of the problems; in particular, the numerical method preserves energy and momentum in some sense, similar to the partial differential equation itself. Examples of multisymplectic integrators include the Euler box scheme and the Preissman box scheme.
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https://en.wikipedia.org/wiki/Multisymplectic_integrator
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In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be viewed as asserting the existence of a multivalued function. In particular, the inverse function of a differentiable function is a multivalued function, and is single-valued only when the original function is monotonic. For example, the complex logarithm is a multivalued function, as the inverse of the exponential function.
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https://en.wikipedia.org/wiki/Many-valued_function
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It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy. Another common way for defining a multivalued function is analytic continuation, which generates commonly some monodromy: analytic continuation along a closed curve may generate a final value that differs from the starting value. Multivalued functions arise also as solutions of differential equations, where the different values are parametrized by initial conditions.
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https://en.wikipedia.org/wiki/Many-valued_function
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In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example, x 2 + y 2 − 1 {\displaystyle x^{2}+y^{2}-1} is absolutely irreducible, but while x 2 + y 2 {\displaystyle x^{2}+y^{2}} is irreducible over the integers and the reals, it is reducible over the complex numbers as x 2 + y 2 = ( x + i y ) ( x − i y ) , {\displaystyle x^{2}+y^{2}=(x+iy)(x-iy),} and thus not absolutely irreducible. More generally, a polynomial defined over a field K is absolutely irreducible if it is irreducible over every algebraic extension of K, and an affine algebraic set defined by equations with coefficients in a field K is absolutely irreducible if it is not the union of two algebraic sets defined by equations in an algebraically closed extension of K. In other words, an absolutely irreducible algebraic set is a synonym of an algebraic variety, which emphasizes that the coefficients of the defining equations may not belong to an algebraically closed field. Absolutely irreducible is also applied, with the same meaning, to linear representations of algebraic groups. In all cases, being absolutely irreducible is the same as being irreducible over the algebraic closure of the ground field.
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https://en.wikipedia.org/wiki/Absolute_irreducibility
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In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first introduced by Max Koecher in his Jordan algebraic approach to Hermitian symmetric spaces and bounded symmetric domains of tube type.
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https://en.wikipedia.org/wiki/Isotope_(Jordan_algebra)
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Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compact subgroup. Both groups act transitively on the compactification. The theory has been extended to cover all Hermitian symmetric spaces using the theory of Jordan pairs or Jordan triple systems. Koecher obtained the results in the more general case directly from the Jordan algebra case using the fact that only Jordan pairs associated with period two automorphisms of Jordan algebras are required.
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https://en.wikipedia.org/wiki/Isotope_(Jordan_algebra)
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In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle F s ( M ) {\displaystyle F^{s}(M)} for some s ≥ 1 {\displaystyle s\geq 1} . It turns out that its transition functions depend functionally on local changes of coordinates in the base manifold M {\displaystyle M} together with their partial derivatives up to order at most s {\displaystyle s} .The concept of a natural bundle was introduced by Albert Nijenhuis as a modern reformulation of the classical concept of an arbitrary bundle of geometric objects.An example of natural bundle (of first order) is the tangent bundle T M {\displaystyle TM} of a manifold M {\displaystyle M} .
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https://en.wikipedia.org/wiki/Natural_bundle
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In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and b a {\displaystyle {\frac {b}{a}}} are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 60 5 = 12 {\displaystyle {\frac {60}{5}}=12} have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 60 6 = 10 {\displaystyle {\frac {60}{6}}=10} have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.
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https://en.wikipedia.org/wiki/Bi-unitary_divisor
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Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b. The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n): σ k ∗ ( n ) = ∑ d ∣ n gcd ( d , n / d ) = 1 d k . {\displaystyle \sigma _{k}^{*}(n)=\sum _{d\,\mid \,n \atop \gcd(d,\,n/d)=1}\!\!d^{k}.} If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number. The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931) who used the term block divisor.
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https://en.wikipedia.org/wiki/Bi-unitary_divisor
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In mathematics, a natural number in a given number base is a p {\displaystyle p} -Kaprekar number if the representation of its square in that base can be split into two parts, where the second part has p {\displaystyle p} digits, that add up to the original number. The numbers are named after D. R. Kaprekar.
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https://en.wikipedia.org/wiki/Kaprekar_number
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In mathematics, a natural number n is a Blum integer if n = p × q is a semiprime for which p and q are distinct prime numbers congruent to 3 mod 4. That is, p and q must be of the form 4t + 3, for some integer t. Integers of this form are referred to as Blum primes. This means that the factors of a Blum integer are Gaussian primes with no imaginary part. The first few Blum integers are 21, 33, 57, 69, 77, 93, 129, 133, 141, 161, 177, 201, 209, 213, 217, 237, 249, 253, 301, 309, 321, 329, 341, 381, 393, 413, 417, 437, 453, 469, 473, 489, 497, ... (sequence A016105 in the OEIS)The integers were named for computer scientist Manuel Blum.
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https://en.wikipedia.org/wiki/Blum_integer
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In mathematics, a near polygon is an incidence geometry introduced by Ernest E. Shult and Arthur Yanushka in 1980. Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. These structures generalise the notion of generalized polygon as every generalized 2n-gon is a near 2n-gon of a particular kind. Near polygons were extensively studied and connection between them and dual polar spaces was shown in 1980s and early 1990s. Some sporadic simple groups, for example the Hall-Janko group and the Mathieu groups, act as automorphism groups of near polygons.
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https://en.wikipedia.org/wiki/Near_polygon
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In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse.
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https://en.wikipedia.org/wiki/Near-field_(mathematics)
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In mathematics, a near-ring (also near ring or nearring) is an algebraic structure similar to a ring but satisfying fewer axioms. Near-rings arise naturally from functions on groups.
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https://en.wikipedia.org/wiki/Near-ring
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In mathematics, a near-semiring, also called a seminearring, is an algebraic structure more general than a near-ring or a semiring. Near-semirings arise naturally from functions on monoids.
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https://en.wikipedia.org/wiki/Near-semiring
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In mathematics, a nearly Kähler manifold is an almost Hermitian manifold M {\displaystyle M} , with almost complex structure J {\displaystyle J} , such that the (2,1)-tensor ∇ J {\displaystyle \nabla J} is skew-symmetric. So, ( ∇ X J ) X = 0 {\displaystyle (\nabla _{X}J)X=0} for every vector field X {\displaystyle X} on M {\displaystyle M} . In particular, a Kähler manifold is nearly Kähler. The converse is not true.
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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For example, the nearly Kähler six-sphere S 6 {\displaystyle S^{6}} is an example of a nearly Kähler manifold that is not Kähler. The familiar almost complex structure on the six-sphere is not induced by a complex atlas on S 6 {\displaystyle S^{6}} . Usually, non Kählerian nearly Kähler manifolds are called "strict nearly Kähler manifolds".
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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Nearly Kähler manifolds, also known as almost Tachibana manifolds, were studied by Shun-ichi Tachibana in 1959 and then by Alfred Gray from 1970 on. For example, it was proved that any 6-dimensional strict nearly Kähler manifold is an Einstein manifold and has vanishing first Chern class (in particular, this implies spin).
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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In the 1980s, strict nearly Kähler manifolds obtained a lot of consideration because of their relation to Killing spinors: Thomas Friedrich and Ralf Grunewald showed that a 6-dimensional Riemannian manifold admits a Riemannian Killing spinor if and only if it is nearly Kähler. This was later given a more fundamental explanation by Christian Bär, who pointed out that these are exactly the 6-manifolds for which the corresponding 7-dimensional Riemannian cone has holonomy G2. The only compact simply connected 6-manifolds known to admit strict nearly Kähler metrics are S 6 , C P 3 , P ( T C P 2 ) {\displaystyle S^{6},\mathbb {C} \mathbb {P} ^{3},\mathbb {P} (T\mathbb {CP} _{2})} , and S 3 × S 3 {\displaystyle S^{3}\times S^{3}} .
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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Each of these admits such a unique nearly Kähler metric that is also homogeneous, and these examples are in fact the only compact homogeneous strictly nearly Kähler 6-manifolds. However, Foscolo and Haskins recently showed that S 6 {\displaystyle S^{6}} and S 3 × S 3 {\displaystyle S^{3}\times S^{3}} also admit strict nearly Kähler metrics that are not homogeneous.Bär's observation about the holonomy of Riemannian cones might seem to indicate that the nearly-Kähler condition is most natural and interesting in dimension 6. This actually borne out by a theorem of Nagy, who proved that any strict, complete nearly Kähler manifold is locally a Riemannian product of homogeneous nearly Kähler spaces, twistor spaces over quaternion-Kähler manifolds, and 6-dimensional nearly Kähler manifolds.Nearly Kähler manifolds are also an interesting class of manifolds admitting a metric connection with parallel totally antisymmetric torsion.Nearly Kähler manifolds should not be confused with almost Kähler manifolds.
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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An almost Kähler manifold M {\displaystyle M} is an almost Hermitian manifold with a closed Kähler form: d ω = 0 {\displaystyle d\omega =0} . The Kähler form or fundamental 2-form ω {\displaystyle \omega } is defined by ω ( X , Y ) = g ( J X , Y ) , {\displaystyle \omega (X,Y)=g(JX,Y),} where g {\displaystyle g} is the metric on M {\displaystyle M} . The nearly Kähler condition and the almost Kähler condition are essentially exclusive: an almost Hermitian manifold is both nearly Kähler and almost Kahler if and only if it is Kähler. == References ==
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https://en.wikipedia.org/wiki/Nearly_Kähler_manifold
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In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset.
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https://en.wikipedia.org/wiki/Negative_number
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If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic.
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https://en.wikipedia.org/wiki/Negative_number
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For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced "minus three" or "negative three".
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https://en.wikipedia.org/wiki/Negative_number
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To help tell the difference between a subtraction operation and a negative number, occasionally the negative sign is placed slightly higher than the minus sign (as a superscript). Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3.
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https://en.wikipedia.org/wiki/Negative_number
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In general, the negativity or positivity of a number is referred to as its sign. Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers.
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https://en.wikipedia.org/wiki/Negative_number
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(Some definitions of the natural numbers exclude zero.) In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers.
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https://en.wikipedia.org/wiki/Negative_number
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It has been proposed that negative numbers were used on the Greek counting table at Salamis, known as the Salamis Tablet, dated to 300 BC. Negative numbers were also used in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220), but may well contain much older material.
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https://en.wikipedia.org/wiki/Negative_number
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Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers.
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https://en.wikipedia.org/wiki/Negative_number
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Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz (1646–1716) held that negative numbers were invalid, but still used them in calculations.
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https://en.wikipedia.org/wiki/Negative_number
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In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible.
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https://en.wikipedia.org/wiki/Negligible_set
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For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets. The opposite of a negligible set is a generic property, which has various forms.
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https://en.wikipedia.org/wiki/Negligible_set
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In mathematics, a nilcurve is a pointed stable curve over a finite field with an indigenous bundle whose p-curvature is square nilpotent. Nilcurves were introduced by Mochizuki (1996) as a central concept in his theory of p-adic Teichmüller theory. The nilcurves form a stack over the moduli stack of stable genus g curves with r marked points in characteristic p, of degree p3g–3+r.
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https://en.wikipedia.org/wiki/Nilcurve
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In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N / H {\displaystyle N/H} , the quotient of a nilpotent Lie group N modulo a closed subgroup H. This notion was introduced by Anatoly Mal'cev in 1951. In the Riemannian category, there is also a good notion of a nilmanifold.
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https://en.wikipedia.org/wiki/Nil_manifold
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A Riemannian manifold is called a homogeneous nilmanifold if there exist a nilpotent group of isometries acting transitively on it. The requirement that the transitive nilpotent group acts by isometries leads to the following rigid characterization: every homogeneous nilmanifold is isometric to a nilpotent Lie group with left-invariant metric (see Wilson). Nilmanifolds are important geometric objects and often arise as concrete examples with interesting properties; in Riemannian geometry these spaces always have mixed curvature, almost flat spaces arise as quotients of nilmanifolds, and compact nilmanifolds have been used to construct elementary examples of collapse of Riemannian metrics under the Ricci flow.In addition to their role in geometry, nilmanifolds are increasingly being seen as having a role in arithmetic combinatorics (see Green–Tao) and ergodic theory (see, e.g., Host–Kra).
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https://en.wikipedia.org/wiki/Nil_manifold
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In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable order.
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https://en.wikipedia.org/wiki/Non-Archimedean_ordered_field
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In mathematics, a non-Desarguesian plane is a projective plane that does not satisfy Desargues' theorem (named after Girard Desargues), or in other words a plane that is not a Desarguesian plane. The theorem of Desargues is true in all projective spaces of dimension not 2; in other words, the only projective spaces of dimension not equal to 2 are the classical projective geometries over a field (or division ring). However, David Hilbert found that some projective planes do not satisfy it. The current state of knowledge of these examples is not complete.
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https://en.wikipedia.org/wiki/Non-Desarguesian_plane
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In mathematics, a non-Euclidean crystallographic group, NEC group or N.E.C. group is a discrete group of isometries of the hyperbolic plane. These symmetry groups correspond to the wallpaper groups in euclidean geometry.
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https://en.wikipedia.org/wiki/Non-Euclidean_crystallographic_group
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A NEC group which contains only orientation-preserving elements is called a Fuchsian group, and any non-Fuchsian NEC group has an index 2 Fuchsian subgroup of orientation-preserving elements. The hyperbolic triangle groups are notable NEC groups. Others are listed in Orbifold notation.
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https://en.wikipedia.org/wiki/Non-Euclidean_crystallographic_group
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In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanics. To describe relativistic mechanics, one should consider a system of ordinary differential equations on a smooth manifold Q {\displaystyle Q} whose fibration over R {\displaystyle \mathbb {R} } is not fixed. Such a system admits transformations of a coordinate t {\displaystyle t} on R {\displaystyle \mathbb {R} } depending on other coordinates on Q {\displaystyle Q} .
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https://en.wikipedia.org/wiki/Relativistic_system_(mathematics)
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Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space Q = R 4 {\displaystyle Q=\mathbb {R} ^{4}} is of this type. Since a configuration space Q {\displaystyle Q} of a relativistic system has no preferable fibration over R {\displaystyle \mathbb {R} } , a velocity space of relativistic system is a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} of one-dimensional submanifolds of Q {\displaystyle Q} .
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https://en.wikipedia.org/wiki/Relativistic_system_(mathematics)
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The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle J 1 1 Q → Q {\displaystyle J_{1}^{1}Q\to Q} is projective and, following the terminology of Special Relativity, one can think of its fibers as being spaces of the absolute velocities of a relativistic system.
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https://en.wikipedia.org/wiki/Relativistic_system_(mathematics)
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Given coordinates ( q 0 , q i ) {\displaystyle (q^{0},q^{i})} on Q {\displaystyle Q} , a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} is provided with the adapted coordinates ( q 0 , q i , q 0 i ) {\displaystyle (q^{0},q^{i},q_{0}^{i})} possessing transition functions q ′ 0 = q ′ 0 ( q 0 , q k ) , q ′ i = q ′ i ( q 0 , q k ) , q ′ 0 i = ( ∂ q ′ i ∂ q j q 0 j + ∂ q ′ i ∂ q 0 ) ( ∂ q ′ 0 ∂ q j q 0 j + ∂ q ′ 0 ∂ q 0 ) − 1 . {\displaystyle q'^{0}=q'^{0}(q^{0},q^{k}),\quad q'^{i}=q'^{i}(q^{0},q^{k}),\quad {q'}_{0}^{i}=\left({\frac {\partial q'^{i}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{i}}{\partial q^{0}}}\right)\left({\frac {\partial q'^{0}}{\partial q^{j}}}q_{0}^{j}+{\frac {\partial q'^{0}}{\partial q^{0}}}\right)^{-1}.} The relativistic velocities of a relativistic system are represented by elements of a fibre bundle R × T Q {\displaystyle \mathbb {R} \times TQ} , coordinated by ( τ , q λ , a τ λ ) {\displaystyle (\tau ,q^{\lambda },a_{\tau }^{\lambda })} , where T Q {\displaystyle TQ} is the tangent bundle of Q {\displaystyle Q} .
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https://en.wikipedia.org/wiki/Relativistic_system_(mathematics)
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Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads ( ∂ λ G μ α 2 … α 2 N 2 N − ∂ μ G λ α 2 … α 2 N ) q τ μ q τ α 2 ⋯ q τ α 2 N − ( 2 N − 1 ) G λ μ α 3 … α 2 N q τ τ μ q τ α 3 ⋯ q τ α 2 N + F λ μ q τ μ = 0 , {\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha _{2}\ldots \alpha _{2N}}}{2N}}-\partial _{\mu }G_{\lambda \alpha _{2}\ldots \alpha _{2N}}\right)q_{\tau }^{\mu }q_{\tau }^{\alpha _{2}}\cdots q_{\tau }^{\alpha _{2N}}-(2N-1)G_{\lambda \mu \alpha _{3}\ldots \alpha _{2N}}q_{\tau \tau }^{\mu }q_{\tau }^{\alpha _{3}}\cdots q_{\tau }^{\alpha _{2N}}+F_{\lambda \mu }q_{\tau }^{\mu }=0,} G α 1 … α 2 N q τ α 1 ⋯ q τ α 2 N = 1. {\displaystyle G_{\alpha _{1}\ldots \alpha _{2N}}q_{\tau }^{\alpha _{1}}\cdots q_{\tau }^{\alpha _{2N}}=1.} For instance, if Q {\displaystyle Q} is the Minkowski space with a Minkowski metric G μ ν {\displaystyle G_{\mu \nu }} , this is an equation of a relativistic charge in the presence of an electromagnetic field.
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https://en.wikipedia.org/wiki/Relativistic_system_(mathematics)
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In mathematics, a non-empty collection of sets R {\displaystyle {\mathcal {R}}} is called a δ-ring (pronounced "delta-ring") if it is closed under union, relative complementation, and countable intersection. The name "delta-ring" originates from the German word for intersection, "Durschnitt", which is meant to highlight the ring's closure under countable intersection, in contrast to a 𝜎-ring which is closed under countable unions.
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https://en.wikipedia.org/wiki/Delta-ring
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In mathematics, a non-measurable set is a set which cannot be assigned a meaningful "volume". The mathematical existence of such sets is construed to provide information about the notions of length, area and volume in formal set theory. In Zermelo–Fraenkel set theory, the axiom of choice entails that non-measurable subsets of R {\displaystyle \mathbb {R} } exist.
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https://en.wikipedia.org/wiki/Non-measurable_set
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The notion of a non-measurable set has been a source of great controversy since its introduction. Historically, this led Borel and Kolmogorov to formulate probability theory on sets which are constrained to be measurable. The measurable sets on the line are iterated countable unions and intersections of intervals (called Borel sets) plus-minus null sets.
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https://en.wikipedia.org/wiki/Non-measurable_set
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These sets are rich enough to include every conceivable definition of a set that arises in standard mathematics, but they require a lot of formalism to prove that sets are measurable. In 1970, Robert M. Solovay constructed the Solovay model, which shows that it is consistent with standard set theory without uncountable choice, that all subsets of the reals are measurable. However, Solovay's result depends on the existence of an inaccessible cardinal, whose existence and consistency cannot be proved within standard set theory.
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https://en.wikipedia.org/wiki/Non-measurable_set
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In mathematics, a nonabelian cohomology is any cohomology with coefficients in a nonabelian group, a sheaf of nonabelian groups or even in a topological space. If homology is thought of as the abelianization of homotopy (cf. Hurewicz theorem), then the nonabelian cohomology may be thought of as a dual of homotopy groups.
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https://en.wikipedia.org/wiki/Nonabelian_sheaf_cohomology
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In mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different. Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.
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https://en.wikipedia.org/wiki/Non-commutative_ring
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Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring. Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.
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https://en.wikipedia.org/wiki/Non-commutative_ring
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In mathematics, a noncommutative unique factorization domain is a noncommutative ring with the unique factorization property.
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https://en.wikipedia.org/wiki/Noncommutative_unique_factorization_domain
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In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then p q − φ ( p q ) = p q − ( p − 1 ) ( q − 1 ) = p + q − 1 = n − 1.
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https://en.wikipedia.org/wiki/Noncototient
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{\displaystyle pq-\varphi (pq)=pq-(p-1)(q-1)=p+q-1=n-1.\,} It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1 = 2 − ϕ ( 2 ) , 3 = 9 − ϕ ( 9 ) {\displaystyle 1=2-\phi (2),3=9-\phi (9)} and 5 = 25 − ϕ ( 25 ) {\displaystyle 5=25-\phi (25)} . For even numbers, it can be shown 2 p q − φ ( 2 p q ) = 2 p q − ( p − 1 ) ( q − 1 ) = p q + p + q − 1 = ( p + 1 ) ( q + 1 ) − 2 {\displaystyle 2pq-\varphi (2pq)=2pq-(p-1)(q-1)=pq+p+q-1=(p+1)(q+1)-2} Thus, all even numbers n such that n+2 can be written as (p+1)*(q+1) with p, q primes are cototients.
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https://en.wikipedia.org/wiki/Noncototient
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In mathematics, a nonelementary antiderivative of a given elementary function is an antiderivative (or indefinite integral) that is, itself, not an elementary function (i.e. a function constructed from a finite number of quotients of constant, algebraic, exponential, trigonometric, and logarithmic functions using field operations). A theorem by Liouville in 1835 provided the first proof that nonelementary antiderivatives exist. This theorem also provides a basis for the Risch algorithm for determining (with difficulty) which elementary functions have elementary antiderivatives.
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https://en.wikipedia.org/wiki/Nonelementary_Integral
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In mathematics, a nonempty collection of sets is called a 𝜎-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.
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https://en.wikipedia.org/wiki/Sigma-ring
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In mathematics, a nonempty subset S of a group G is said to be symmetric if it contains the inverses of all of its elements.
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https://en.wikipedia.org/wiki/Symmetric_set
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In mathematics, a nonhypotenuse number is a natural number whose square cannot be written as the sum of two nonzero squares. The name stems from the fact that an edge of length equal to a nonhypotenuse number cannot form the hypotenuse of a right angle triangle with integer sides. The numbers 1, 2, 3 and 4 are all nonhypotenuse numbers. The number 5, however, is not a nonhypotenuse number as 52 equals 32 + 42.
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https://en.wikipedia.org/wiki/Nonhypotenuse_number
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The first fifty nonhypotenuse numbers are: 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 23, 24, 27, 28, 31, 32, 33, 36, 38, 42, 43, 44, 46, 47, 48, 49, 54, 56, 57, 59, 62, 63, 64, 66, 67, 69, 71, 72, 76, 77, 79, 81, 83, 84 (sequence A004144 in the OEIS)Although nonhypotenuse numbers are common among small integers, they become more-and-more sparse for larger numbers. Yet, there are infinitely many nonhypotenuse numbers, and the number of nonhypotenuse numbers not exceeding a value x scales asymptotically with x/√log x.The nonhypotenuse numbers are those numbers that have no prime factors of the form 4k+1. Equivalently, they are the number that cannot be expressed in the form K ( m 2 + n 2 ) {\displaystyle K(m^{2}+n^{2})} where K, m, and n are all positive integers. A number whose prime factors are not all of the form 4k+1 cannot be the hypotenuse of a primitive integer right triangle (one for which the sides do not have a nontrivial common divisor), but may still be the hypotenuse of a non-primitive triangle.The nonhypotenuse numbers have been applied to prove the existence of addition chains that compute the first n {\displaystyle n} square numbers using only n + o ( n ) {\displaystyle n+o(n)} additions.
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https://en.wikipedia.org/wiki/Nonhypotenuse_number
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In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M ( λ ) x = 0 , {\displaystyle M(\lambda )x=0,} where x ≠ 0 {\displaystyle x\neq 0} is a vector, and M {\displaystyle M} is a matrix-valued function of the number λ {\displaystyle \lambda } . The number λ {\displaystyle \lambda } is known as the (nonlinear) eigenvalue, the vector x {\displaystyle x} as the (nonlinear) eigenvector, and ( λ , x ) {\displaystyle (\lambda ,x)} as the eigenpair. The matrix M ( λ ) {\displaystyle M(\lambda )} is singular at an eigenvalue λ {\displaystyle \lambda } .
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https://en.wikipedia.org/wiki/Polynomial_eigenvalue_problem
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In mathematics, a nonlinear system is a system that is not linear—i.e., a system that does not satisfy the superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to solve for cannot be written as a linear sum of independent components. A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they can be transformed to a linear system as long as a particular solution is known.
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https://en.wikipedia.org/wiki/Dynamic_systems_theory
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