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In mathematics, a doctrine is simply a 2-category which is heuristically regarded as a system of theories. For example, algebraic theories, as invented by William Lawvere, is an example of a doctrine, as are multi-sorted theories, operads, categories, and toposes. The objects of the 2-category are called theories, the 1-morphisms f: A → B {\displaystyle f\colon A\rightarrow B} are called models of the A in B, and the 2-morphisms are called morphisms between models. The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms.
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https://en.wikipedia.org/wiki/Strict_2-category
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It is this vocabulary that makes the theory of doctrines worth while. For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models.
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https://en.wikipedia.org/wiki/Strict_2-category
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As another example, one may take the subcategory of Cat consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms. This is the doctrine of multi-sorted algebraic theories. If one only wanted 1-sorted algebraic theories, one would restrict the objects to only those categories that are generated under products by a single object. Doctrines were discovered by Jonathan Mock Beck.
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https://en.wikipedia.org/wiki/Strict_2-category
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In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino. Since it has reflection symmetry, it is also the only one-sided domino (with reflections considered distinct). When rotations are also considered distinct, there are two fixed dominoes: The second one can be created by rotating the one above by 90°.In a wider sense, the term domino is sometimes understood to mean a tile of any shape.
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https://en.wikipedia.org/wiki/Domino_(mathematics)
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In mathematics, a double Mersenne number is a Mersenne number of the form M M p = 2 2 p − 1 − 1 {\displaystyle M_{M_{p}}=2^{2^{p}-1}-1} where p is prime.
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https://en.wikipedia.org/wiki/Double_Mersenne_number
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In mathematics, a double affine Hecke algebra, or Cherednik algebra, is an algebra containing the Hecke algebra of an affine Weyl group, given as the quotient of the group ring of a double affine braid group. They were introduced by Cherednik, who used them to prove Macdonald's constant term conjecture for Macdonald polynomials. Infinitesimal Cherednik algebras have significant implications in representation theory, and therefore have important applications in particle physics and in chemistry.
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https://en.wikipedia.org/wiki/Double_affine_Hecke_algebra
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In mathematics, a double affine braid group is a group containing the braid group of an affine Weyl group. Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras. For affine An groups, the double affine braid group is the fundamental group of the space of n distinct points on a 2-dimensional torus.
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https://en.wikipedia.org/wiki/Double_affine_braid_group
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In mathematics, a double vector bundle is the combination of two compatible vector bundle structures, which contains in particular the tangent T E {\displaystyle TE} of a vector bundle E {\displaystyle E} and the double tangent bundle T 2 M {\displaystyle T^{2}M} .
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https://en.wikipedia.org/wiki/Double_vector_bundle
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In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers u and v that are linearly independent as vectors over the field of real numbers. That u and v are periods of a function ƒ means that f ( z + u ) = f ( z + v ) = f ( z ) {\displaystyle f(z+u)=f(z+v)=f(z)\,} for all values of the complex number z.The doubly periodic function is thus a two-dimensional extension of the simpler singly periodic function, which repeats itself in a single dimension. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine, In the complex plane the exponential function ez is a singly periodic function, with period 2πi.
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https://en.wikipedia.org/wiki/Doubly_periodic_function
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In mathematics, a dual abelian variety can be defined from an abelian variety A, defined over a field K.
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https://en.wikipedia.org/wiki/Poincare_line_bundle
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In mathematics, a dual system, dual pair, or duality over a field K {\displaystyle \mathbb {K} } is a triple ( X , Y , b ) {\displaystyle (X,Y,b)} consisting of two vector spaces X {\displaystyle X} and Y {\displaystyle Y} over K {\displaystyle \mathbb {K} } and a non-degenerate bilinear map b: X × Y → K {\displaystyle b:X\times Y\to \mathbb {K} } . Duality theory, the study of dual systems, is part of functional analysis. It is separate and distinct to Dual-system Theory in psychology.
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https://en.wikipedia.org/wiki/Natural_pairing
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In mathematics, a dual wavelet is the dual to a wavelet. In general, the wavelet series generated by a square-integrable function will have a dual series, in the sense of the Riesz representation theorem. However, the dual series is not itself in general representable by a square-integrable function.
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https://en.wikipedia.org/wiki/Dual_wavelet
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In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings.
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https://en.wikipedia.org/wiki/Duality_(mathematics)
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It has been described as "a very pervasive and important concept in (modern) mathematics" and "an important general theme that has manifestations in almost every area of mathematics".Many mathematical dualities between objects of two types correspond to pairings, bilinear functions from an object of one type and another object of the second type to some family of scalars. For instance, linear algebra duality corresponds in this way to bilinear maps from pairs of vector spaces to scalars, the duality between distributions and the associated test functions corresponds to the pairing in which one integrates a distribution against a test function, and Poincaré duality corresponds similarly to intersection number, viewed as a pairing between submanifolds of a given manifold.From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces. This functor assigns to each space its dual space, and the pullback construction assigns to each arrow f: V → W its dual f∗: W∗ → V∗.
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https://en.wikipedia.org/wiki/Duality_(mathematics)
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In mathematics, a dyadic compactum is a Hausdorff topological space that is the continuous image of a product of discrete two-point spaces, and a dyadic space is a topological space with a compactification which is a dyadic compactum. However, many authors use the term dyadic space with the same meaning as dyadic compactum above.Dyadic compacta and spaces satisfy the Suslin condition, and were introduced by Russian mathematician Pavel Alexandrov. Polyadic spaces are generalisation of dyadic spaces. == References ==
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https://en.wikipedia.org/wiki/Dyadic_space
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In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations.
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https://en.wikipedia.org/wiki/Dyadic_fraction
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Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number. The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula.
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https://en.wikipedia.org/wiki/Dyadic_fraction
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However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted Z {\displaystyle \mathbb {Z} } .
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https://en.wikipedia.org/wiki/Dyadic_fraction
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In advanced mathematics, the dyadic rational numbers are central to the constructions of the dyadic solenoid, Minkowski's question-mark function, Daubechies wavelets, Thompson's group, Prüfer 2-group, surreal numbers, and fusible numbers. These numbers are order-isomorphic to the rational numbers; they form a subsystem of the 2-adic numbers as well as of the reals, and can represent the fractional parts of 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalize mathematical analysis in reverse mathematics.
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https://en.wikipedia.org/wiki/Dyadic_fraction
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In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it.
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https://en.wikipedia.org/wiki/Global_dynamical_system
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At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifold. The evolution rule of the dynamical system is a function that describes what future states follow from the current state.
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https://en.wikipedia.org/wiki/Global_dynamical_system
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Often the function is deterministic, that is, for a given time interval only one future state follows from the current state. However, some systems are stochastic, in that random events also affect the evolution of the state variables.
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https://en.wikipedia.org/wiki/Global_dynamical_system
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In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives". In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized. The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics, biology, chemistry, engineering, economics, history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.
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https://en.wikipedia.org/wiki/Global_dynamical_system
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In mathematics, a dynamical system is chaotic if it is (highly) sensitive to initial conditions (the so-called "butterfly effect"), which requires the mathematical properties of topological mixing and dense periodic orbits.Alongside fractals, chaos theory ranks as an essentially universal influence on patterns in nature. There is a relationship between chaos and fractals—the strange attractors in chaotic systems have a fractal dimension. Some cellular automata, simple sets of mathematical rules that generate patterns, have chaotic behaviour, notably Stephen Wolfram's Rule 30.Vortex streets are zigzagging patterns of whirling vortices created by the unsteady separation of flow of a fluid, most often air or water, over obstructing objects. Smooth (laminar) flow starts to break up when the size of the obstruction or the velocity of the flow become large enough compared to the viscosity of the fluid.
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https://en.wikipedia.org/wiki/Geometry_of_natural_structure
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Meanders are sinuous bends in rivers or other channels, which form as a fluid, most often water, flows around bends. As soon as the path is slightly curved, the size and curvature of each loop increases as helical flow drags material like sand and gravel across the river to the inside of the bend. The outside of the loop is left clean and unprotected, so erosion accelerates, further increasing the meandering in a powerful positive feedback loop.
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https://en.wikipedia.org/wiki/Geometry_of_natural_structure
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In mathematics, a dynamical system is time-reversible if the forward evolution is one-to-one, so that for every state there exists a transformation (an involution) π which gives a one-to-one mapping between the time-reversed evolution of any one state and the forward-time evolution of another corresponding state, given by the operator equation: U − t = π U t π {\displaystyle U_{-t}=\pi \,U_{t}\,\pi } Any time-independent structures (e.g. critical points or attractors) which the dynamics give rise to must therefore either be self-symmetrical or have symmetrical images under the involution π.
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https://en.wikipedia.org/wiki/Reversed_process
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In mathematics, a eutactic lattice (or eutactic form) is a lattice in Euclidean space whose minimal vectors form a eutactic star. This means they have a set of positive eutactic coefficients ci such that (v, v) = Σci(v, mi)2 where the sum is over the minimal vectors mi. "Eutactic" is derived from the Greek language, and means "well-situated" or "well-arranged". Voronoi (1908) proved that a lattice is extreme if and only if it is both perfect and eutactic. Conway & Sloane (1988) summarize the properties of eutactic lattices of dimension up to 7.
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https://en.wikipedia.org/wiki/Eutactic_lattice
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In mathematics, a fact is a statement (called a theorem) that can be proven by logical argument from certain axioms and definitions.
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https://en.wikipedia.org/wiki/Scientific_fact
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In mathematics, a factor system (sometimes called factor set) is a fundamental tool of Otto Schreier’s classical theory for group extension problem. It consists of a set of automorphisms and a binary function on a group satisfying certain condition (so-called cocycle condition). In fact, a factor system constitutes a realisation of the cocycles in the second cohomology group in group cohomology.
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https://en.wikipedia.org/wiki/Factor_system
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In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorisation. The Schützenberger theorem relates the definition in terms of a multiplicative property to an additive property.Let A* be the free monoid on an alphabet A. Let Xi be a sequence of subsets of A* indexed by a totally ordered index set I. A factorisation of a word w in A* is an expression w = x i 1 x i 2 ⋯ x i n {\displaystyle w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}\ } with x i j ∈ X i j {\displaystyle x_{i_{j}}\in X_{i_{j}}} and i 1 ≥ i 2 ≥ … ≥ i n {\displaystyle i_{1}\geq i_{2}\geq \ldots \geq i_{n}} . Some authors reverse the order of the inequalities.
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https://en.wikipedia.org/wiki/Schützenberger_theorem
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In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the use of Casson handles and so does not work in the smooth category.
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https://en.wikipedia.org/wiki/Fake_4-ball
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In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type.
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https://en.wikipedia.org/wiki/Mumford_surface
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In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it. There are exactly 50 fake projective planes. Prasad & Yeung (2006) found four examples of fake projective 4-folds, and showed that no arithmetic examples exist in dimensions other than 2 and 4.
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https://en.wikipedia.org/wiki/Fake_projective_space
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In mathematics, a family F {\displaystyle {\mathcal {F}}} of sets is of finite character if for each A {\displaystyle A} , A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} if and only if every finite subset of A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} . That is, For each A ∈ F {\displaystyle A\in {\mathcal {F}}} , every finite subset of A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} . If every finite subset of a given set A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} , then A {\displaystyle A} belongs to F {\displaystyle {\mathcal {F}}} .
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https://en.wikipedia.org/wiki/Finite_character
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In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a family of real numbers, indexed by the set of integers, is a collection of real numbers, where a given function selects one real number for each integer (possibly the same) as indexing. More formally, an indexed family is a mathematical function together with its domain I {\displaystyle I} and image X {\displaystyle X} (that is, indexed families and mathematical functions are technically identical, just point of views are different). Often the elements of the set X {\displaystyle X} are referred to as making up the family.
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https://en.wikipedia.org/wiki/Indexed_set
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In this view, indexed families are interpreted as collections of indexed elements instead of functions. The set I {\displaystyle I} is called the index set of the family, and X {\displaystyle X} is the indexed set. Sequences are one type of families indexed by natural numbers. In general, the index set I {\displaystyle I} is not restricted to be countable. For example, one could consider an uncountable family of subsets of the natural numbers indexed by the real numbers.
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https://en.wikipedia.org/wiki/Indexed_set
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In mathematics, a fence, also called a zigzag poset, is a partially ordered set (poset) in which the order relations form a path with alternating orientations: a < b > c < d > e < f > h < i ⋯ {\displaystyle a c e h b < c > d < e > f < h > i ⋯ {\displaystyle a>b d f i\cdots } A fence may be finite, or it may be formed by an infinite alternating sequence extending in both directions. The incidence posets of path graphs form examples of fences. A linear extension of a fence is called an alternating permutation; André's problem of counting the number of different linear extensions has been studied since the 19th century. The solutions to this counting problem, the so-called Euler zigzag numbers or up/down numbers, are: 1 , 1 , 2 , 4 , 10 , 32 , 122 , 544 , 2770 , 15872 , 101042.
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https://en.wikipedia.org/wiki/Zigzag_poset
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{\displaystyle 1,1,2,4,10,32,122,544,2770,15872,101042.} (sequence A001250 in the OEIS).The number of antichains in a fence is a Fibonacci number; the distributive lattice with this many elements, generated from a fence via Birkhoff's representation theorem, has as its graph the Fibonacci cube.A partially ordered set is series-parallel if and only if it does not have four elements forming a fence.Several authors have also investigated the number of order-preserving maps from fences to themselves, or to fences of other sizes.An up-down poset Q(a,b) is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements.
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https://en.wikipedia.org/wiki/Zigzag_poset
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For instance, Q(2,9) has the elements and relations a > b > c < d > e > f < g > h > i . {\displaystyle a>b>c e>f h>i.} In this notation, a fence is a partially ordered set of the form Q(1,n).
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https://en.wikipedia.org/wiki/Zigzag_poset
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In mathematics, a fibrifold is (roughly) a fiber space whose fibers and base spaces are orbifolds. They were introduced by John Horton Conway, Olaf Delgado Friedrichs, and Daniel H. Huson et al. (2001), who introduced a system of notation for 3-dimensional fibrifolds and used this to assign names to the 219 affine space group types. 184 of these are considered reducible, and 35 irreducible.
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https://en.wikipedia.org/wiki/Fibrifold_notation
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In mathematics, a field F is algebraically closed if every non-constant polynomial in F (the univariate polynomial ring with coefficients in F) has a root in F.
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https://en.wikipedia.org/wiki/Algebraically_closed_field
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In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether, in a 1936 paper (Tsen 1936); and later by Serge Lang in his 1951 Princeton University dissertation and in his 1952 paper (Lang 1952). The idea itself is attributed to Lang's advisor Emil Artin. Formally, if P is a non-constant homogeneous polynomial in variables X1, ..., XN,and of degree d satisfying d < Nthen it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have P(x1, ..., xN) = 0.In geometric language, the hypersurface defined by P, in projective space of degree N − 2, then has a point over F.
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https://en.wikipedia.org/wiki/Quasi-algebraically_closed
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In mathematics, a field K is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation v and if its residue field k is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below.
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https://en.wikipedia.org/wiki/Normalized_valuation
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Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields.While Archimedean local fields have been quite well known in mathematics for at least 250 years, the first examples of non-Archimedean local fields, the fields of p-adic numbers for positive prime integer p, were introduced by Kurt Hensel at the end of the 19th century.
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https://en.wikipedia.org/wiki/Normalized_valuation
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Every local field is isomorphic (as a topological field) to one of the following: Archimedean local fields (characteristic zero): the real numbers R, and the complex numbers C. Non-Archimedean local fields of characteristic zero: finite extensions of the p-adic numbers Qp (where p is any prime number). Non-Archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series Fq((T)) over a finite field Fq, where q is a power of p.In particular, of importance in number theory, classes of local fields show up as the completions of algebraic number fields with respect to their discrete valuation corresponding to one of their maximal ideals. Research papers in modern number theory often consider a more general notion, requiring only that the residue field be perfect of positive characteristic, not necessarily finite. This article uses the former definition.
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https://en.wikipedia.org/wiki/Normalized_valuation
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In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty: B 1 ⊇ B 2 ⊇ ⋯ ⇒ ⋂ n ∈ N B n ≠ ∅ . {\displaystyle B_{1}\supseteq B_{2}\supseteq \cdots \Rightarrow \bigcap _{n\in {\mathbf {N} }}B_{n}\neq \emptyset .} The definition can be adapted also to a field K with a valuation v taking values in an arbitrary ordered abelian group: (K,v) is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection. Spherically complete fields are important in nonarchimedean functional analysis, since many results analogous to theorems of classical functional analysis require the base field to be spherically complete.
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https://en.wikipedia.org/wiki/Spherically_complete_field
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In mathematics, a field K {\displaystyle K} is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.
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https://en.wikipedia.org/wiki/Pseudo_algebraically_closed
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In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and p-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry.
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https://en.wikipedia.org/wiki/Field_(algebra)
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Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions.
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https://en.wikipedia.org/wiki/Field_(algebra)
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Among other results, this theory shows that angle trisection and squaring the circle cannot be done with a compass and straightedge. Moreover, it shows that quintic equations are, in general, algebraically unsolvable. Fields serve as foundational notions in several mathematical domains.
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https://en.wikipedia.org/wiki/Field_(algebra)
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This includes different branches of mathematical analysis, which are based on fields with additional structure. Basic theorems in analysis hinge on the structural properties of the field of real numbers. Most importantly for algebraic purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings of the field of rational numbers, are studied in depth in number theory. Function fields can help describe properties of geometric objects.
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https://en.wikipedia.org/wiki/Field_(algebra)
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In mathematics, a field of sets is a mathematical structure consisting of a pair ( X , F ) {\displaystyle (X,{\mathcal {F}})} consisting of a set X {\displaystyle X} and a family F {\displaystyle {\mathcal {F}}} of subsets of X {\displaystyle X} called an algebra over X {\displaystyle X} that contains the empty set as an element, and is closed under the operations of taking complements in X , {\displaystyle X,} finite unions, and finite intersections. Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over X {\displaystyle X} " is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory. Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
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https://en.wikipedia.org/wiki/Field_of_sets
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In mathematics, a figure is chiral (and said to have chirality) if it cannot be mapped to its mirror image by rotations and translations alone. For example, a right shoe is different from a left shoe, and clockwise is different from anticlockwise. See for a full mathematical definition. A chiral object and its mirror image are said to be enantiomorphs.
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https://en.wikipedia.org/wiki/Chirality
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The word enantiomorph stems from the Greek ἐναντίος (enantios) 'opposite' + μορφή (morphe) 'form'. A non-chiral figure is called achiral or amphichiral. The helix (and by extension a spun string, a screw, a propeller, etc.) and Möbius strip are chiral two-dimensional objects in three-dimensional ambient space.
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https://en.wikipedia.org/wiki/Chirality
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The J, L, S and Z-shaped tetrominoes of the popular video game Tetris also exhibit chirality, but only in a two-dimensional space. Many other familiar objects exhibit the same chiral symmetry of the human body, such as gloves, glasses (sometimes), and shoes. A similar notion of chirality is considered in knot theory, as explained below. Some chiral three-dimensional objects, such as the helix, can be assigned a right or left handedness, according to the right-hand rule.
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https://en.wikipedia.org/wiki/Chirality
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In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n-dimensional topological manifold X is the boundary of an (n + 1)-dimensional manifold W. Perhaps the most active area of current research is when n = 3, where one may consider certain types of fillings. There are many types of fillings, and a few examples of these types (within a probably limited perspective) follow. An oriented filling of any orientable manifold X is another manifold W such that the orientation of X is given by the boundary orientation of W, which is the one where the first basis vector of the tangent space at each point of the boundary is the one pointing directly out of W, with respect to a chosen Riemannian metric.
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https://en.wikipedia.org/wiki/Symplectic_filling
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Mathematicians call this orientation the outward normal first convention.All the following cobordisms are oriented, with the orientation on W given by a symplectic structure. Let ξ denote the kernel of the contact form α. A weak symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with ∂ W = X {\displaystyle \partial W=X} such that ω | ξ > 0 {\displaystyle \omega |_{\xi }>0} . A strong symplectic filling of a contact manifold (X,ξ) is a symplectic manifold (W,ω) with ∂ W = X {\displaystyle \partial W=X} such that ω is exact near the boundary (which is X) and α is a primitive for ω. That is, ω = dα in a neighborhood of the boundary ∂ W = X {\displaystyle \partial W=X} .
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https://en.wikipedia.org/wiki/Symplectic_filling
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A Stein filling of a contact manifold (X,ξ) is a Stein manifold W which has X as its strictly pseudoconvex boundary and ξ is the set of complex tangencies to X – that is, those tangent planes to X that are complex with respect to the complex structure on W. The canonical example of this is the 3-sphere where the complex structure on C 2 {\displaystyle \mathbb {C} ^{2}} is multiplication by − 1 {\displaystyle {\sqrt {-1}}} in each coordinate and W is the ball {|x| < 1} bounded by that sphere.It is known that this list is strictly increasing in difficulty in the sense that there are examples of contact 3-manifolds with weak but no strong filling, and others that have strong but no Stein filling. Further, it can be shown that each type of filling is an example of the one preceding it, so that a Stein filling is a strong symplectic filling, for example. It used to be that one spoke of semi-fillings in this context, which means that X is one of possibly many boundary components of W, but it has been shown that any semi-filling can be modified to be a filling of the same type, of the same 3-manifold, in the symplectic world (Stein manifolds always have one boundary component).
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https://en.wikipedia.org/wiki/Symplectic_filling
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In mathematics, a filter on a set X {\displaystyle X} informally gives a notion of which subsets A ⊆ X {\displaystyle A\subseteq X} are "large". Filter quantifiers are a type of logical quantifier which, informally, say whether or not a statement is true for "most" elements of X . {\displaystyle X.} Such quantifiers are often used in combinatorics, model theory (such as when dealing with ultraproducts), and in other fields of mathematical logic where (ultra)filters are used.
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https://en.wikipedia.org/wiki/Filter_quantifier
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In mathematics, a filter on a set X {\displaystyle X} is a family B {\displaystyle {\mathcal {B}}} of subsets such that: X ∈ B {\displaystyle X\in {\mathcal {B}}} and ∅ ∉ B {\displaystyle \emptyset \notin {\mathcal {B}}} if A ∈ B {\displaystyle A\in {\mathcal {B}}} and B ∈ B {\displaystyle B\in {\mathcal {B}}} , then A ∩ B ∈ B {\displaystyle A\cap B\in {\mathcal {B}}} If A , B ⊂ X , A ∈ B {\displaystyle A,B\subset X,A\in {\mathcal {B}}} , and A ⊂ B {\displaystyle A\subset B} , then B ∈ B {\displaystyle B\in {\mathcal {B}}} A filter on a set may be thought of as representing a "collection of large subsets", one intuitive example being the neighborhood filter. Filters appear in order theory, model theory, and set theory, but can also be found in topology, from which they originate. The dual notion of a filter is an ideal. Filters were introduced by Henri Cartan in 1937 and as described in the article dedicated to filters in topology, they were subsequently used by Nicolas Bourbaki in their book Topologie Générale as an alternative to the related notion of a net developed in 1922 by E. H. Moore and Herman L. Smith. Order filters are generalizations of filters from sets to arbitrary partially ordered sets. Specifically, a filter on a set is just a proper order filter in the special case where the partially ordered set consists of the power set ordered by set inclusion.
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https://en.wikipedia.org/wiki/Elementary_filter
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In mathematics, a filter or order filter is a special subset of a partially ordered set (poset), describing "large" or "eventual" elements. Filters appear in order and lattice theory, but also topology, whence they originate. The notion dual to a filter is an order ideal. Special cases of filters include ultrafilters, which are filters that cannot be enlarged, and describe nonconstructive techniques in mathematical logic.
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https://en.wikipedia.org/wiki/Filter_(mathematics)
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Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book Topologie Générale, popularized filters as an alternative to E. H. Moore and Herman L. Smith's 1922 notion of a net; order filters generalize this notion from the specific case of a power set under inclusion to arbitrary partially ordered sets. Nevertheless, the theory of power-set filters retains interest in its own right, in part for substantial applications in topology.
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https://en.wikipedia.org/wiki/Filter_(mathematics)
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In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory. A filtered algebra over the field k {\displaystyle k} is an algebra ( A , ⋅ ) {\displaystyle (A,\cdot )} over k {\displaystyle k} that has an increasing sequence { 0 } ⊆ F 0 ⊆ F 1 ⊆ ⋯ ⊆ F i ⊆ ⋯ ⊆ A {\displaystyle \{0\}\subseteq F_{0}\subseteq F_{1}\subseteq \cdots \subseteq F_{i}\subseteq \cdots \subseteq A} of subspaces of A {\displaystyle A} such that A = ⋃ i ∈ N F i {\displaystyle A=\bigcup _{i\in \mathbb {N} }F_{i}} and that is compatible with the multiplication in the following sense: ∀ m , n ∈ N , F m ⋅ F n ⊆ F n + m . {\displaystyle \forall m,n\in \mathbb {N} ,\quad F_{m}\cdot F_{n}\subseteq F_{n+m}.}
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https://en.wikipedia.org/wiki/Filtered_ring
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In mathematics, a filtration F {\displaystyle {\mathcal {F}}} is an indexed family ( S i ) i ∈ I {\displaystyle (S_{i})_{i\in I}} of subobjects of a given algebraic structure S {\displaystyle S} , with the index i {\displaystyle i} running over some totally ordered index set I {\displaystyle I} , subject to the condition that if i ≤ j {\displaystyle i\leq j} in I {\displaystyle I} , then S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} .If the index i {\displaystyle i} is the time parameter of some stochastic process, then the filtration can be interpreted as representing all historical but not future information available about the stochastic process, with the algebraic structure S i {\displaystyle S_{i}} gaining in complexity with time. Hence, a process that is adapted to a filtration F {\displaystyle {\mathcal {F}}} is also called non-anticipating, because it cannot "see into the future".Sometimes, as in a filtered algebra, there is instead the requirement that the S i {\displaystyle S_{i}} be subalgebras with respect to some operations (say, vector addition), but not with respect to other operations (say, multiplication) that satisfy only S i ⋅ S j ⊆ S i + j {\displaystyle S_{i}\cdot S_{j}\subseteq S_{i+j}} , where the index set is the natural numbers; this is by analogy with a graded algebra. Sometimes, filtrations are supposed to satisfy the additional requirement that the union of the S i {\displaystyle S_{i}} be the whole S {\displaystyle S} , or (in more general cases, when the notion of union does not make sense) that the canonical homomorphism from the direct limit of the S i {\displaystyle S_{i}} to S {\displaystyle S} is an isomorphism. Whether this requirement is assumed or not usually depends on the author of the text and is often explicitly stated.
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https://en.wikipedia.org/wiki/Filtration_(mathematics)
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This article does not impose this requirement. There is also the notion of a descending filtration, which is required to satisfy S i ⊇ S j {\displaystyle S_{i}\supseteq S_{j}} in lieu of S i ⊆ S j {\displaystyle S_{i}\subseteq S_{j}} (and, occasionally, ⋂ i ∈ I S i = 0 {\displaystyle \bigcap _{i\in I}S_{i}=0} instead of ⋃ i ∈ I S i = S {\displaystyle \bigcup _{i\in I}S_{i}=S} ).
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https://en.wikipedia.org/wiki/Filtration_(mathematics)
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Again, it depends on the context how exactly the word "filtration" is to be understood. Descending filtrations are not to be confused with the dual notion of cofiltrations (which consist of quotient objects rather than subobjects). Filtrations are widely used in abstract algebra, homological algebra (where they are related in an important way to spectral sequences), and in measure theory and probability theory for nested sequences of σ-algebras. In functional analysis and numerical analysis, other terminology is usually used, such as scale of spaces or nested spaces.
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https://en.wikipedia.org/wiki/Filtration_(mathematics)
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In mathematics, a finitary relation over sets X1, ..., Xn is a subset of the Cartesian product X1 × ⋯ × Xn; that is, it is a set of n-tuples (x1, ..., xn) consisting of elements xi in Xi. Typically, the relation describes a possible connection between the elements of an n-tuple. For example, the relation "x is divisible by y and z" consists of the set of 3-tuples such that when substituted to x, y and z, respectively, make the sentence true. The non-negative integer n giving the number of "places" in the relation is called the arity, adicity or degree of the relation.
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https://en.wikipedia.org/wiki/Theory_of_relations
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A relation with n "places" is variously called an n-ary relation, an n-adic relation or a relation of degree n. Relations with a finite number of places are called finitary relations (or simply relations if the context is clear). It is also possible to generalize the concept to infinitary relations with infinite sequences.An n-ary relation over sets X1, ..., Xn is an element of the power set of X1 × ⋯ × Xn. 0-ary relations count only two members: the one that always holds, and the one that never holds.
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https://en.wikipedia.org/wiki/Theory_of_relations
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This is because there is only one 0-tuple, the empty tuple (). They are sometimes useful for constructing the base case of an induction argument.
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https://en.wikipedia.org/wiki/Theory_of_relations
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Unary relations can be viewed as a collection of members (such as the collection of Nobel laureates) having some property (such as that of having been awarded the Nobel prize). Binary relations are the most commonly studied form of finitary relations. When X1 = X2 it is called a homogeneous relation, for example: Equality and inequality, denoted by signs such as = and < in statements such as "5 < 12", or Divisibility, denoted by the sign | in statements such as "13|143".Otherwise it is a heterogeneous relation, for example: Set membership, denoted by the sign ∈ in statements such as "1 ∈ N".
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https://en.wikipedia.org/wiki/Theory_of_relations
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In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number.
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https://en.wikipedia.org/wiki/Galois_fields
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The order of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order pk, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory.
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https://en.wikipedia.org/wiki/Galois_fields
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In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals. Instead of repeating exactly the same design over and over, they have slight variations in each stage, allowing a richer structure while maintaining the elegant style of fractals. Subdivision rules have been used in architecture, biology, and computer science, as well as in the study of hyperbolic manifolds. Substitution tilings are a well-studied type of subdivision rule.
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https://en.wikipedia.org/wiki/Combinatorial_Riemann_mapping_theorem
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In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. Finite topological spaces are often used to provide examples of interesting phenomena or counterexamples to plausible sounding conjectures. William Thurston has called the study of finite topologies in this sense "an oddball topic that can lend good insight to a variety of questions".
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https://en.wikipedia.org/wiki/Finite_topological_space
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In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if V ∗ V = I {\displaystyle V^{*}V=I} , then V V ∗ = I {\displaystyle VV^{*}=I} . In terms of the comparison theory of projections, the identity operator is not (Murray-von Neumann) equivalent to any proper subprojection in the von Neumann algebra.
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https://en.wikipedia.org/wiki/Finite_von_Neumann_algebra
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In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,...,an of A such that every element of A can be expressed as a polynomial in a1,...,an, with coefficients in K. Equivalently, there exist elements a 1 , … , a n ∈ A {\displaystyle a_{1},\dots ,a_{n}\in A} s.t. the evaluation homomorphism at a = ( a 1 , … , a n ) {\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})} ϕ a: K ↠ A {\displaystyle \phi _{\bf {a}}\colon K\twoheadrightarrow A} is surjective; thus, by applying the first isomorphism theorem, A ≃ K / k e r ( ϕ a ) {\displaystyle A\simeq K/{\rm {ker}}(\phi _{\bf {a}})} . Conversely, A := K / I {\displaystyle A:=K/I} for any ideal I ⊂ K {\displaystyle I\subset K} is a K {\displaystyle K} -algebra of finite type, indeed any element of A {\displaystyle A} is a polynomial in the cosets a i := X i + I , i = 1 , … , n {\displaystyle a_{i}:=X_{i}+I,i=1,\dots ,n} with coefficients in K {\displaystyle K} . Therefore, we obtain the following characterisation of finitely generated K {\displaystyle K} -algebras A {\displaystyle A} is a finitely generated K {\displaystyle K} -algebra if and only if it is isomorphic to a quotient ring of the type K / I {\displaystyle K/I} by an ideal I ⊂ K {\displaystyle I\subset K} .If it is necessary to emphasize the field K then the algebra is said to be finitely generated over K . Algebras that are not finitely generated are called infinitely generated.
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https://en.wikipedia.org/wiki/Finitely-generated_algebra
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In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent modules all of which are defined below. Over a Noetherian ring the concepts of finitely generated, finitely presented and coherent modules coincide. A finitely generated module over a field is simply a finite-dimensional vector space, and a finitely generated module over the integers is simply a finitely generated abelian group.
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https://en.wikipedia.org/wiki/Module_of_finite_type
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In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form F ( x 1 , … , x n , u , u x 1 , … u x n ) = 0. {\displaystyle F(x_{1},\ldots ,x_{n},u,u_{x_{1}},\ldots u_{x_{n}})=0.\,} Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
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https://en.wikipedia.org/wiki/First-order_partial_differential_equation
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In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function.
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https://en.wikipedia.org/wiki/Stable_fixed_point
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In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms.
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https://en.wikipedia.org/wiki/Fixed_point_theory
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In mathematics, a flip graph is a graph whose vertices are combinatorial or geometric objects, and whose edges link two of these objects when they can be obtained from one another by an elementary operation called a flip. Flip graphs are special cases of geometric graphs. Among noticeable flip graphs, one finds the 1-skeleton of polytopes such as associahedra or cyclohedra.
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https://en.wikipedia.org/wiki/Flip_graph
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In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations.
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https://en.wikipedia.org/wiki/Flow_(mathematics)
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Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups.
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https://en.wikipedia.org/wiki/Flow_(mathematics)
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Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow.
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https://en.wikipedia.org/wiki/Flow_(mathematics)
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In mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there exist forms g 1 ( x ) , … , g k ( x ) {\displaystyle g_{1}(x),\ldots ,g_{k}(x)} of degree m such that Every form that is SOS is also a positive polynomial, and although the converse is not always true, Hilbert proved that for n = 2, 2m = 2, or n = 3 and 2m = 4 a form is SOS if and only if it is positive. The same is also valid for the analog problem on positive symmetric forms.Although not every form can be represented as SOS, explicit sufficient conditions for a form to be SOS have been found. Moreover, every real nonnegative form can be approximated as closely as desired (in the l 1 {\displaystyle l_{1}} -norm of its coefficient vector) by a sequence of forms { f ϵ } {\displaystyle \{f_{\epsilon }\}} that are SOS.
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https://en.wikipedia.org/wiki/Polynomial_SOS
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In mathematics, a formal distribution is an infinite sum of powers of a formal variable, usually denoted z {\displaystyle z} in the theory of formal distributions. The coefficients of these infinite sums can be many different mathematical structures, such as vector spaces or rings, but in applications most often take values in an algebra over a field. These infinite sums are allowed to have infinitely many positive and negative powers, and are not required to converge, and so do not define functions of the formal variable.
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https://en.wikipedia.org/wiki/Formal_distribution
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Rather, they are interpreted as distributions, that is, linear functionals on an appropriate space of test functions. They are closely related to formal Laurent series, but are not required to have finitely many negative powers. In particular, this means even if the coefficients are ring-valued, it is not necessarily possible to multiply two formal distributions. They are important in the study of vertex operator algebras, since the vertex operator playing a central role in the theory takes values in a space of endomorphism-valued formal distributions.
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https://en.wikipedia.org/wiki/Formal_distribution
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In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by S. Bochner (1946). The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology.
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https://en.wikipedia.org/wiki/Formal_group_law
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In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x n {\displaystyle ax^{n}} where x n {\displaystyle x^{n}} is the n {\displaystyle n} th power of a variable x {\displaystyle x} ( n {\displaystyle n} is a non-negative integer), and a {\displaystyle a} is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence.
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https://en.wikipedia.org/wiki/Formal_power_series_ring
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In a formal power series, the x n {\displaystyle x^{n}} are used only as position-holders for the coefficients, so that the coefficient of x 5 {\displaystyle x^{5}} is the fifth term in the sequence. In combinatorics, the method of generating functions uses formal power series to represent numerical sequences and multisets, for instance allowing concise expressions for recursively defined sequences regardless of whether the recursion can be explicitly solved.
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https://en.wikipedia.org/wiki/Formal_power_series_ring
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More generally, formal power series can include series with any finite (or countable) number of variables, and with coefficients in an arbitrary ring. Rings of formal power series are complete local rings, and this allows using calculus-like methods in the purely algebraic framework of algebraic geometry and commutative algebra. They are analogous in many ways to p-adic integers, which can be defined as formal series of the powers of p.
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https://en.wikipedia.org/wiki/Formal_power_series_ring
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In mathematics, a formal sum, formal series, or formal linear combination may be: In group theory, an element of a free abelian group, a sum of finitely many elements from a given basis set multiplied by integer coefficients. In linear algebra, an element of a vector space, a sum of finitely many elements from a given basis set multiplied by real, complex, or other numerical coefficients. In the study of series (mathematics), a sum of an infinite sequence of numbers or other quantities, considered as an abstract mathematical object regardless of whether the sum converges. In the study of power series, a sum of infinitely many monomials with distinct positive integer exponents, again considered as an abstract object regardless of convergence.
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https://en.wikipedia.org/wiki/Formal_sum
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In mathematics, a formula generally refers to an equation relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. However, having done this once in terms of some parameter (the radius for example), mathematicians have produced a formula to describe the volume of a sphere in terms of its radius: V = 4 3 π r 3 . {\displaystyle V={\frac {4}{3}}\pi r^{3}.}
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https://en.wikipedia.org/wiki/Formula
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Having obtained this result, the volume of any sphere can be computed as long as its radius is known. Here, notice that the volume V and the radius r are expressed as single letters instead of words or phrases. This convention, while less important in a relatively simple formula, means that mathematicians can more quickly manipulate formulas which are larger and more complex.
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https://en.wikipedia.org/wiki/Formula
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Mathematical formulas are often algebraic, analytical or in closed form.In a general context, formulas are often a manifestation of mathematical model to real world phenomena, and as such can be used to provide solution (or approximated solution) to real world problems, with some being more general than others. For example, the formula F = m a {\displaystyle F=ma} is an expression of Newton's second law, and is applicable to a wide range of physical situations. Other formulas, such as the use of the equation of a sine curve to model the movement of the tides in a bay, may be created to solve a particular problem.
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https://en.wikipedia.org/wiki/Formula
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In all cases, however, formulas form the basis for calculations. Expressions are distinct from formulas in that they cannot contain an equals sign (=). Expressions can be likened to phrases the same way formulas can be likened to grammatical sentences.
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https://en.wikipedia.org/wiki/Formula
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In mathematics, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the scale at which it is measured. It is also a measure of the space-filling capacity of a pattern, and it tells how a fractal scales differently, in a fractal (non-integer) dimension.The main idea of "fractured" dimensions has a long history in mathematics, but the term itself was brought to the fore by Benoit Mandelbrot based on his 1967 paper on self-similarity in which he discussed fractional dimensions. In that paper, Mandelbrot cited previous work by Lewis Fry Richardson describing the counter-intuitive notion that a coastline's measured length changes with the length of the measuring stick used (see Fig. 1).
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https://en.wikipedia.org/wiki/Fractal_dimension
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In terms of that notion, the fractal dimension of a coastline quantifies how the number of scaled measuring sticks required to measure the coastline changes with the scale applied to the stick. There are several formal mathematical definitions of fractal dimension that build on this basic concept of change in detail with change in scale: see the section Examples. Ultimately, the term fractal dimension became the phrase with which Mandelbrot himself became most comfortable with respect to encapsulating the meaning of the word fractal, a term he created.
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https://en.wikipedia.org/wiki/Fractal_dimension
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After several iterations over years, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants. "One non-trivial example is the fractal dimension of a Koch snowflake. It has a topological dimension of 1, but it is by no means rectifiable: the length of the curve between any two points on the Koch snowflake is infinite.
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https://en.wikipedia.org/wiki/Fractal_dimension
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No small piece of it is line-like, but rather it is composed of an infinite number of segments joined at different angles. The fractal dimension of a curve can be explained intuitively by thinking of a fractal line as an object too detailed to be one-dimensional, but too simple to be two-dimensional. Therefore its dimension might best be described not by its usual topological dimension of 1 but by its fractal dimension, which is often a number between one and two; in the case of the Koch snowflake, it is approximately 1.2619.
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https://en.wikipedia.org/wiki/Fractal_dimension
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